Power system network partitioning using tabu search

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<ul><li><p>Electric Power Systems Research 49 (1999) 5561</p><p>Power system network partitioning using tabu search</p><p>C.S. Chang *, L.R. Lu, F.S. WenDepartment of Electrical Engineering, National Uni6ersity of Singapore, 10 Kent Ridge Crescent, Singapore 119260, Singapore</p><p>Received 8 April 1998; accepted 27 July 1998</p><p>Abstract</p><p>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.</p><p>Keywords: Tabu search; Algorithm; Simulated annealing</p><p>1. Introduction</p><p>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.</p><p>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.</p><p>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</p><p>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.</p><p>In this paper, a new technique based on tabu search(TS) is presented. TS [6,7] is a relatively new approach</p><p>* Corresponding author. Tel.: 65-87-42109; fax: 65-77-91103;e-mail: elecs@leonis.nus.sg.</p><p>0378-7796:99:$ - see front matter 1999 Elsevier Science S.A. All rights reserved.PII: S03 7 8 -7796 (98 )00119 -9</p></li><li><p>C.S. Chang et al. : Electric Power Systems Research 41 (1999) 556156</p><p>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.</p><p>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.</p><p>2. Mathematical model</p><p>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</p><p>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.</p><p>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.</p><p>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</p><p>time in a typical parallel processor solution algorithm.SA was then applied to minimize:</p><p>Min CaM2b L3 (1)</p><p>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.</p><p>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.</p><p>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.</p><p>Min C %T</p><p>i1</p><p>ni2b2b %T</p><p>i1</p><p>(FiFgiFciFti) (2)</p><p>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.</p><p>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.</p><p>b is a penalty factor.The first term on the right hand side of Eq. (2)</p><p>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.</p><p>By modifying Eq. (1) by including the feasibility ofeach subsystem as a constraint, an improved costfunction is obtained:</p><p>`</p><p>Min Ca M2b L3</p><p>s.t.Fi0; i1, 2,, T.</p><p>(3)</p><p>By dealing with Fi in Eq. (2) as a constraint ratherthan a penalty term in the cost function, improvedconvergence is obtained:</p></li><li><p>C.S. Chang et al. : Electric Power Systems Research 41 (1999) 5561 57</p><p>`</p><p>Min C %T</p><p>i1</p><p>ni2b2b %T</p><p>i1</p><p>(FgiFciFti)</p><p>s.t.Fi0; i1, 2, , T.</p><p>(4)</p><p>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).</p><p>3. Tabu search based algorithm</p><p>3.1. A brief re6iew of tabu search</p><p>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].</p><p>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.</p><p>3.2. Tabu search-based power system networkpartitioning algorithm</p><p>The following is a description of the proposed 2-stepTS-based algorithm for power system network parti-tioning.</p><p>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.</p><p>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</p><p>Fig. 1. (a) Original IEEE 14-node system (b) tree-structure betweenbuses 7 and 8 (c) simplification of (b).</p><p>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.</p><p>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].</p><p>Step 2: A random initial partitioning of the abovereduced system is created and TS is adopted to improvethe partitioning solution iteratively.</p><p>Fig. 2. Modified IEEE 14-node system.</p></li><li><p>C.S. Chang et al. : Electric Power Systems Research 41 (1999) 556158</p><p>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).</p><p>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.</p><p>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.</p><p>4. Test results</p><p>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.</p><p>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.</p><p>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.</p><p>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</p><p>results shown in Tables 14. The computational effi-ciency of the TS-based alg...</p></li></ul>