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  • Electric Power Systems Research 49 (1999) 5561

    Power system network partitioning using tabu search

    C.S. Chang *, L.R. Lu, F.S. WenDepartment of Electrical Engineering, National Uni6ersity of Singapore, 10 Kent Ridge Crescent, Singapore 119260, Singapore

    Received 8 April 1998; accepted 27 July 1998

    Abstract

    This paper presents a new algorithm to power system network partitioning based on tabu search (TS). TS is a simple heuristicoptimization strategy that can achieve the optimal or near optimal solution within a reasonably short time. A 2-step TS-basedalgorithm is proposed to divide the system network into several subsystems to optimize the use of parallel computer systems forpower system analysis. Test results on two IEEE standard networks are presented and compared with those obtained by othercombinatorial optimization approaches, namely: simulated annealing parallel simulated annealing and genetic algorithm. It isshown that the proposed TS-based algorithm has higher efficiency and better convergence than the previously proposed methods. 1999 Elsevier Science S.A. All rights reserved.

    Keywords: Tabu search; Algorithm; Simulated annealing

    1. Introduction

    Recent developments in computer architecture havestimulated the interest of researchers towards applyingparallel processing to power system problems. With thegrowth in size and complexity of interconnected powersystems, conventional centralized scheme analysis meth-ods in power system operation and planning are be-coming too time-consuming and too expensive. Incontrast, parallel computing is attractive as it can over-come the above problems.

    Power system network partitioning is an importantproblem associated with the application of parallelprocessing before further analysis. The objective ofnetwork partitioning is to divide optimally the wholesystem into a number of subsystems in order to en-hance the efficiency of parallel processing. The divisionshould strike a balance among the size of each subsys-tem, the number of cut-set tie lines and the number ofcontrol variables.

    Mathematically, the network partitioning problembelongs to a class of NP-hard problems where nopolynomial-bounded algorithm for finding the globaloptimal solutions is likely to exist. This is a very

    complicated combinatorial optimization problem. Pre-vious research in this field included the application ofsuccessive approximation dynamic programming [1].Owing to the nonlinear property of the partitioningproblem, this method can only converge to local op-tima which depends on the initial specified solution. Aheuristic clustering approach using bottle optimal strat-egy [2] can obtain the local optima rapidly, however, itis not amenable to balance the objectives. Irving andSterling [3] proposed a stochastic optimization methodcalled simulated annealing (SA) based on the analogyto physical annealing. The results obtained indicate thatthe approach is sensitive to the pseudo-random numberand parameter setting, and that only local optima arefound. H. Mori et al. [4] extended the approach with adifferent objective function and a different method ofoptimization called parallel simulated annealing (PSA)which improves the speed and convergence of SA. Yetthe above-mentioned shortcoming still can not beavoided. H. Ding et al. [5] used genetic algorithms(GAs) to solve the problem as formulated in [3]. Theresults show that the global optimal solution can befound for large-scale power systems if a very largepopulation is used. This method is quite time-consum-ing and requires large computer memory.

    In this paper, a new technique based on tabu search(TS) is presented. TS [6,7] is a relatively new approach

    * Corresponding author. Tel.: 65-87-42109; fax: 65-77-91103;e-mail: elecs@leonis.nus.sg.

    0378-7796:99:$ - see front matter 1999 Elsevier Science S.A. All rights reserved.PII: S03 7 8 -7796 (98 )00119 -9

  • C.S. Chang et al. : Electric Power Systems Research 41 (1999) 556156

    to combinatorial optimization. It is characterized byaggressive local search during each iteration, and isable to avoid cycling in the solution space by keepinga short history of the attributes of the recent solu-tions. TS has been successfully applied to many com-plicated optimization problems such as schedulingproblem, traveling salesman problem, as well as alarmprocessing in power systems [8]. Reported results indi-cated TSs efficiency in solving optimization problems.

    This paper is organized into five sections. Section 2defines the power system partitioning problem andestablishes the required mathematical models. Section3 presents the essential elements of TS, and itsapplication to the problem and the formulation. Sec-tion 4 presents the test results on the IEEE 30- andIEEE 118- node systems. It also compares the TSscomputing time and convergence with SA, PSA asproposed in previous work. Section 5 concludes thepaper.

    2. Mathematical model

    In the context of the problem, a power networkcomprises nodes (representing busbars, including gen-erators, shunt capacitors and loads) which are con-nected in pairs by branches (representing transmissionlines and transformers). System partitioning is carriedout to assign these nodes and branches into subsys-tems with appropriate size and few cut-set tie lines,and to allocate the subsystems to individual proces-sors for parallel power system analysis. However, it isdifficult to describe this problem in a strictly mathe-matical manner for the following reasons:1. It is hard to describe strictly the relation between

    individual processor assignment and the size andstructure of the respective subsystems. Above allthe relation depends on the problem to be solvedafterwards. For example, the solution time for par-allel loadflow is related to each subsystems num-ber of nodes, structure and branch and loadparameters, while in parallel stability simulation itmainly depends on the number and parameters ofgenerators and large dynamic loads. Moreover,even in a specific problem such as parallel loadflow, it is hard to give an exact formula to de-scribe the solution time.

    2. It is hard to give appropriate weighting factors tocompromise the different requirements of the solu-tion time of individual processors and the amountof communications among them. This depends onnot only the problem to be solved but also theparticular parallel processor system to be adopted.

    The optimal assignment of nodes and branches canbe formulated into an optimization problem by mini-mizing some cost functions [35]. Reference [3] for-mulated its cost function to represent the solution

    time in a typical parallel processor solution algorithm.SA was then applied to minimize:

    Min CaM2b L3 (1)

    where: M is the maximum number of nodes in anysubsystem; L is the number of cut-set tie lines be-tween subsystems; a, b are weighting factors.

    The first term on the right hand side of Eq. (1)reflects the solution time for individual subsystems,while the second term reflects the communication timebetween subsystems. Eq. (1) was also minimized byGA in [5]. Eq. (1) does not however consider thefeasibility of candidate solutions.

    An alternative cost function has a multiobjectivestructure[4]. It was chosen to equally assign the num-ber of nodes to each subsystem and minimize thenumber of cut-set tie lines and constraint violationsamong subsystems. PSA was employed for the mini-mization.

    Min C %T

    i1

    ni2b2b %T

    i1

    (FiFgiFciFti) (2)

    where: T is the total busbar number of subsystems; niis the total busbar number in the i-th subsystem; b isthe number of cut-set tie lines; Fi is the index for thefeasibility of the solution. If the i-th subsystem is aconnected network, it is feasible; otherwise unfeasible.For feasible solution Fi0, and for unfeasible solu-tion Fi1.

    Fgi, Fci, Fti are indices for indicating violations ofthe number of generators, capacitors and transformersrespectively. If the number of generators in subsystemi is larger than the specified lower bound of the num-ber of generators, then Fgi0, otherwise Fgi1. Sim-ilar explanation is applicable to Fci and Fti.

    b is a penalty factor.The first term on the right hand side of Eq. (2)

    reflects the requirement that the number of busbarsshould be evenly assigned to each subsystem. The sec-ond term considers minimizing the number of cut-settie lines in partitioning the total system. The thirdterm indicates the penalty for constraint violationsand solution unfeasibility.

    By modifying Eq. (1) by including the feasibility ofeach subsystem as a constraint, an improved costfunction is obtained:

    `

    Min Ca M2b L3

    s.t.Fi0; i1, 2,, T.

    (3)

    By dealing with Fi in Eq. (2) as a constraint ratherthan a penalty term in the cost function, improvedconvergence is obtained:

  • C.S. Chang et al. : Electric Power Systems Research 41 (1999) 5561 57

    `

    Min C %T

    i1

    ni2b2b %T

    i1

    (FgiFciFti)

    s.t.Fi0; i1, 2, , T.

    (4)

    Eq. (3) is more suited for preparing parallel loadflow,while Eq. (4) can be used for preparing parallel voltagecontrol. In this paper, both these cost functions areminimized using TS (Section 4).

    3. Tabu search based algorithm

    3.1. A brief re6iew of tabu search

    TS is an iterative procedure for solving discrete com-binatorial optimization problems. It was first suggestedby Glover [6] and has since become increasingly popu-lar. It has been successfully applied to obtain the opti-mal or sub-optimal solutions to problems such as thetraveling salesman problem (TSP), timetable schedulingand network layout design [9].

    TS is a restricted neighborhood search technique,and is an iterative algorithm. The fundamental idea ofTS is the use of flexible memory of search history whichthus guides the search process to surmount local opti-mal solutions. The basic steps of TS are described inAppendix A.

    3.2. Tabu search-based power system networkpartitioning algorithm

    The following is a description of the proposed 2-stepTS-based algorithm for power system network parti-tioning.

    Step 1: The original network is first simplified togenerate a reduced system, in which at least two pathsare used to connect a pair of nodes.

    The network partitioning problem in this work dealsmainly with transmission networks, which have gener-ally a loop structure. This means that the majority ofload busbars are linked with at least two transmissionlines to ensure electric supply reliability. There arehowever the exception of a few tree-structure partscomprising: a few unimportant load busbars, which arelinked with a single transmission line as well as genera-tor busbars, which are linked to the system throughstep-up transformers. For example, in the IEEE 14-node system shown in Fig. 1 (a), only busbars 7 and 8form a tree-structure part [shown in Fig. 1 (b)]. Wemust assign all busbars in a tree-structure part to thesame subsystem. Otherwise isolated busbars may beassigned and an unfeasible solution may be obtained.To prevent such cases, it will be beneficial to simplify

    Fig. 1. (a) Original IEEE 14-node system (b) tree-structure betweenbuses 7 and 8 (c) simplification of (b).

    the original system beforehand in order to reduce com-plexity of power system partitioning. After all the tree-structure parts of the original system are identified (Fig.1b), all busbars within each tree-structure part aremerged in a manner as in Fig. 1c. Fig. 2 shows thereduced system after simplifying the original IEEE 14-node system as in Fig. 1a.

    The simplification of an original system with a moregeneral tree-structure part is illustrated in Fig. 3. Asshown in Fig. 3, busbars 1, 2, 3, and 4 consist of atree-structure part, and the root node is 1. After sim-plification, node 1 replaces the four original busbars(busbars 1, 2, 3 and 4), with the total number of allthese four busbars generators, transformers and capac-itors included [note: 1 (g, c) identifies generator andcapacitor on busbar 1; t identifies transformer, so itrelates two conjoint nodes].

    Step 2: A random initial partitioning of the abovereduced system is created and TS is adopted to improvethe partitioning solution iteratively.

    Fig. 2. Modified IEEE 14-node system.

  • C.S. Chang et al. : Electric Power Systems Research 41 (1999) 556158

    Fig. 3. Simplification example: (a) original graph; (b) specific graph.(Note: 1 (g, c) identifies generator and capacitor on busbar 1; tidentifies transformer, so it relates two conjoint nodes).

    subsystem P1 to P2 together with node i. In the case thatthe moving of node i does not result in the non-connec-tion in the current system, it is not necessary to moveother nodes.

    In the TS design, we use a circular list to record movesexecuted in the last t iterations. Each element of the tabulist describes the attributes of one above-mentionedmove, i.e. all moving nodes from the current subsystemto a destination subsystem in one executed move. Thenall the moves listed in this tabu list and their reversemoves are tabooed. Tabu list length Tmax can be set toa number such as seven. We use the cost of the best visitedsolution as the aspiration level.

    4. Test results

    The TS-based algorithm is applied to the IEEE 30-node test network using Eq. (3) as the cost function. Theimportant parameters are specified as: ab1, T3.The optimal partitioning results are shown in Fig. 5.

    Further investigation of the method for this IEEE30-node network is performed with different tabu listlength Tmax and different initial solutions. In general, theresults show that TS with short term memory tabu list(Tmax is set to about 10) works better than that with longterm memory tabu list. On the other hand, the requirednumber of iterations for obtaining the optimal solutionsis related to the initial specified solution. If the initialsolution is far from the optimal solution, then the searchprocess should climb a long distance hill to reach theoptimal solution. However, whether the initial solutionis good or bad, TS can all converge to the optimalsolution. Obviously, TS guides the heuristic to continueexploration without becoming confounded by an absenceof improving moves, and can overclimb a local optimum.

    We also apply the TS approach to the IEEE 118-nodetest system on Pentium-166 and compare the results withthose obtained by using GA, SA and PSA.

    Test case 1: Eq. (3) is used as the cost function, andthe parameters are specified as same in [5]: T3;ab1. Test results using the TS-based method arepresented in Table 1, and the comparison with theresults as given in [5] by GA is listed in Table 2.Test case 2: Eq. (4) is used as the cost function, andthe parameters are specified as same as in [4]: T4,the lower bound of generators8, the lower bound ofcapacitors2, the lower bound of transformers1,and b1100. In this IEEE test system, there are 33generators, 21 capacitors and 10 transformers. Testresults using TS-based method are presented in Table3, and the comparison with the results as given in [3]by SA and [4] in PSA is listed in Table 4.Some general observations can be made from the

    results shown in Tables 14. The computational effi-ciency of the TS-based algorithm is much higher than

    To optimize the partitioning performance, TS is ag-gressively oriented to find the best available move at eachstep. To make a trade-off between greedy search and timeconsumption, TS favors node move, i.e. vertex move,rather than vertex exchange. It is because vertex movehas a smaller neighbor size n*(T1) (n is the totalnumber of the nodes in the specific graph, and T is thenumber of subsystems). Let S be the set of all definedmoves for moving one node from the current partitioningto another, and m is move (mS).

    However, under some circumstances, the above nodemove must be modified because the execution of a nodemove may result in a partition without respecting theconstraint of connectivity of each subsystem. For exam-ple, in Fig. 4, let P1, P2 and P3 be the set of all nodesin the subsystems 1, 2, and 3 respectively, and P1 can bedivided into three parts G1, G2 and G3. G1, G2 and G3 arethe set of nodes in the subsystem P1. They are connectedinside P1, respectively and are connected with each otherby node i. When node i is moved from subsystem P1 toP2, P1 will be decomposed into three isolated parts, andthis means the solution is unfeasible.

    In order to make a move feasible, we modify the nodemove as follows: first, identify the largest part among allthose isolated parts of the current subsystem, say G1, thenretain all the nodes of this largest part in the originalsubsystem. Afterwards, move all nodes in the otherisolated parts (here, nodes (G2@G3)) from the current

    Fig. 4. Move example (T3).

  • C.S. Chang et al. : Electric Power Systems Research 41 (1999) 5561 59

    Fig. 5. Partitioning result of IEEE 30-bus system (M14, L5, cost321).

    that of the SA- and GA-based algorithms. The qualityof the partitioning result obtained by the TS method isbetter than those obtained by SA and PSA.

    5. Conclusion

    This paper has presented a TS-based algorithm forpower system network partitioning problem to enhancethe efficiency of parallel processing in some powersystem analysis problems. Test results reveal that thisalgorithm gives a good compromise between the qualityof partitions and time required for the solution. Com-pared with SA, GA and PSA, the TS-based algorithm isshown to have converged faster for a better solution.This is achieved by using the tabu list and aspirationlevel to override the local optima, while in GA largeamount of calculation is required to keep a sufficientlarge number of individuals in mating pool.

    The paper also investigates the effectiveness of the TSbased approach with regards to variance of the initialsolution and tabu list length. It is demonstrated by testresults that the TS algorithm with short-term memoryworks better, and an appropriate initial solution is

    important in enhancing the computational efficiency ofthe proposed algorithm.

    Appendix A

    To describe the basic steps of TS, we consider acombinatorial optimization problem in the followingform:

    Min F(X) (5)

    where X is a vector of dimension N, and its elementsare integers. F (X) is the objective function (cost orpenalty function), which can be linear or nonlinear. Thefirst step of TS is to produce an initial (current) solu-tion X current either randomly or using an existing(heuristic) method to the given problem. The secondstep is to define a set of moves that may be applied tothe current solution to produce a set of trial solutions.As an example, the move can take the form of X trialX currentDX. Here, DX is a vector with the samedimension as X. Among all the trial solutions thus

    Table 2Comparisons of results on IEEE 118-node system using TS and GA(objective function given in Eq. (3))

    Algorithm RemarkComputationalCost valuetime

    2329 Iteration25TS 6.8 s8.33 min Popsize2000GA 2976

    generation1002329 Popsize800018.4 minGA

    generation100

    Table 1Results on IEEE 118-node system using TS (objective function givenin Eq. (3))

    Subsystem Number of nodes Result

    M40391392 L9403 C2329

  • C.S. Chang et al. : Electric Power Systems Research 41 (1999) 556160

    Table 3Results on IEEE 118-node system using TS (objective function given in Eq. (4))

    Number of generators Number of capacitorsSubsystem Number of nodes Number of transformers

    6831 2134 9 5 42

    2482538 64 128

    Tie Lineb15 C3751

    produced, TS seeks the one that most improves theobjective function. In certain situations, if there is noimproving move, a fact which means some local opti-mum exists. TS chooses the one that least degradesthe objective function. The basic components of theTS are the moves, tabu list and aspiration level (crite-rion), which are introduced below.1. Moves

    The search process of TS is implemented by themoves. A trial solution can be created by a move.During each iteration, a specified number of moves(the neighborhood sampling number), NSmax, are exe-cuted. Many kinds of moves are currently available[6,7]. Vertex move and vertex exchange are two popu-lar classes of moves [10] for graph partitioning prob-lem. Vertex move means moving one vertex awayfrom its current partition to any other possible parti-tions. Vertex exchange means exchanging two verticesbelonging to two different partitions.2. Tabu list

    In order to prevent from returning to the localoptimum just visited, the reverse move that is detri-mental to achieving the optimum solution must beforbid [6,7]. This is implemented by storing this movein a tabu list, which stores the attributes of somemoves made. The elements of the tabu list are calledtabu moves. The reverse moves are restricted fromregions the search already explored.

    The condition to determine whether a move to be atabu move or not can be problem specific. For exam-ple, a move may be tabu if it leads to a solutionwhich has already been considered in the last itera-tions (this is called as recency or short term condi-tion). Recency based tabu list is a short-term memorythat restricts the moves recently made. A tabu status

    for a move is in the memory for a number of itera-tions, which is known as the tabu tenure. The tabutenure can be chosen either using static rules or dy-namic rules. Static rules allow a value for the tabutenure that remains fixed throughout the search. Dy-namic rules allow the value of tabu tenure to vary.The examples of these two kinds of rules are as fol-low [9]:

    Static rules choose tabu tenure (Tmax) to be aconstant such that Tmax7 or TmaxN, where Nis a measure of problem dimension.Dynamic rules choose Tmax to vary (randomlyor by systematic pattern) between bounds T1 and T2such as T15 and T211, or T10.9N andT21.1N.The indicated values such as 7 and N are only

    suggestive. They should be obtained by experiments fora particular class of problems.

    Frequency based tabu list [9] provides a type ofinformation that complements the information pro-vided by recency-based tabu list, broadening the foun-dation for selecting preferred moves. Frequency basedtabu list is a long-term memory structure. It records thenumber of moves adopted in the search. This type ofmemory is important to diversify the search into movesthat are less frequently used if the search appears to betrapped in a local neighborhood.

    The dimension of the tabu list is called the tabu listsize. The tabu list size should grow with the size of thegiven problem, but how to specify the optimal tabu listsize is still an open problem. In addition, how tomanage the tabu list such as how long (how manyiterations) a move can be retained in the tabu list is alsoan important problem. Many methods to implementand manage the tabu list are available, and the methodsused here are described below.

    In our work, the tabu list is updated iteration byiteration until the maximum permitted iteration numberTmax has been reached. At the end of each iteration, thenew move is added to the tabu list, and an old movemay be removed if it has been in the tabu list for Tmaxiterations.3. Aspiration criterion (level)

    The aspiration criterion is introduced in TS to deter-mine when a tabu move can be overridden. The main

    Table 4Comparisons of results on IEEE 118-node system using TS, SA andGA (objective function given in Eq. (4))

    Algorithm Cost value Computational time (s)

    TS 3751 9.818.1507306SA

    3776PSA 27.417

  • C.S. Chang et al. : Electric Power Systems Research 41 (1999) 5561 61

    purpose is to enable tabu moves that could possiblylead to an optimal solution. This criterion provides anadded flexibility to choose good moves by allowing atabu move to be overridden if its aspiration level isattained. Many implementation strategies for the aspi-ration level are available [6,7]. Choice of the aspirationcriterion depends on the specific applications, and inthis work the aspiration level is defined as: if a tabumove from the current solution X current can reach asolution which is better than the best solution found sofar, then the aspiration level for this tabu move isattained and can be overridden.4. A general tabu search algorithm

    A general TS algorithm for solving the minimizationproblem of F(X) can be described as follows:1. Initially, X0 is produced randomly and chosen as the

    initial solution. And let X* be X0.2. Assume that there are p moves, where moves are

    M{m1, m2,, mp}, then the next possible solutionis M(X*){m1(X*), m2(X*),, mp(X*)}.

    3. The neighborhood of the current solution is the setof feasible solutions that can be reached by applyingthe moves. The neighborhood set N(X*)M(X*).

    4. In the neighborhood set, there could be some solu-tions that are reached by applying tabu moves.These solutions (tabu set) are denoted as T(X*),and T(X*)N(X*).

    5. Within the tabu set, some solutions might havesurpassed the aspiration criteria. This set of solu-tions is known as the aspirant set, denoted byA(X*), and A(X*)T(X*).

    6. The next solution is chosen from the neighbor which

    is either as aspirant or not tabus and for which F(X)is minimal.

    The search process then iterates from Steps 26 untilthe terminating condition is met. The terminating con-dition should be determined based on the characteris-tics of the specific problems, and a commonly used oneis that the maximum permitted iteration number, Kmax,has been reached.

    References

    [1] J.M. Undrill, H.H. Happ, Automatic sectionalization of powersystem networks for network solution, IEEE Trans. PowerApp. Syst. 90 (1) (1971) 4652.

    [2] A. Sangiovanni-Vincentelli, L. Chen, L. Chua, An efficientheuristic cluster algorithm for tearing large-scale networks,IEEE Trans. Circuits Syst. 24 (12) (1977) 709717.

    [3] M.R. Irving, M.J.H. Sterling, Optimal network tearing usingsimulated annealing, IEE-Proc. Part C 137 (1) (1990) 6972.

    [4] H. Mori, K. Takeda, Parallel simulated annealing for powersystem decomposition, in: Proceedings of IEEE ConferencePICA93, 1993, pp. 366372.

    [5] H. Ding, A.A. El-Keib, R. Smith, Clustering of power net-works using genetic algorithms, Electr. Power Syst. Res. 30(1994) 209214.

    [6] F. Glover, Tabu searchpart l, ORSA J. Comput. 1 (1989)190206.

    [7] F. Glover, Tabu searchpart 2, ORSA J. Comput. 2 (1990)132.

    [8] F.S. Wen, C.S. Chang, Tabu search approach to alarm pro-cessing in power system, Proc. IKE Gener. Transm. Distrib.144 (1) (1997) 3138.

    [9] F. Glover, M. Laguna, Tabu Search, Kluwer Academic Pub-lishers, Boston, USA, 1997.

    [10] L. Tao, Y. Zhao, Multi-way graph partition by stochasticprobe, Comput. Ops. Res. 20 (3) (1993) 321347.

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