# Network decomposition using Kernighan–Lin strategy aided harmony search algorithm

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<ul><li><p>n6000</p><p>cale</p><p>es c</p><p>s ap</p><p>n p</p><p>e th</p><p>Lin</p><p>nd</p><p>are</p><p>stem</p><p>grid ineas. Tplex ahandan inin po</p><p>putingthe phork d</p><p>network decomposition in [5]. These methods formulate the</p><p>theonypro-d for</p><p>in [14,15] a hybrid swarm intelligence based harmony search is</p><p>Contents lists available at SciVerse ScienceDirect</p><p>.els</p><p>Swarm and Evolutio</p><p>Swarm and Evolutionary Computation 7 (2012) 16been applied to combined heat and power economic dispatchE-mail address: swarup@ee.iitm.ac.in (K.S. Swarup).network partitioning as an optimization problem and tries to used to solve economic dispatch. HS algorithm has been used fortransmission network planning [16]. The exploratory power ofthe HS algorithm is analyzed in [17] and the work in [18] givesparameter setting free harmony search algorithm. In [19] HS has</p><p>2210-6502/$ - see front matter & 2012 Elsevier B.V. All rights reserved.</p><p>http://dx.doi.org/10.1016/j.swevo.2012.07.002</p><p>n Corresponding author. Tel.: 91 44 2257 4440; fax: 91 44 2257 4401.annealing was done by Irving and Sterling in [3]. Later Chang et al.proposed the application of tabu search for network partitioningin [4]. A new ant colony optimization has also been used for</p><p>solving structural optimization problems in [11]. In [12,13] animproved harmony search is applied to optimal economic powerdispatch, dynamic economic dispatch with wind energy is solvedtional methods generate connected clusters with less number ofinterconnections but fails to balance the size of the clusters [1].Ding et al. proposed clustering of power networks using geneticalgorithm in [2]. Optimal tearing of network using simulated</p><p>called harmony search algorithm (HSA) is used to solvepartitioning problem of large-scale power networks. The harmsearch algorithm has been applied to many optimizationblems in engineering and design. To mention a few it is usean essential part of distributed computing of power systemproblems.</p><p>In the literature several approaches have been proposed fornetwork decomposition like matrix decomposition, successiveapproximation and heuristic clustering techniques. The conven-</p><p>posed so far in the literature are computation intensive andinvolves procedures based on natural selection crossover andmutation. It also requires a large population size and occupiesmore memory.</p><p>In this paper one of the recently evolving heuristic algorithm1. Introduction</p><p>Power system is a complicatedgeographically distributed multi arand monitoring of the system is comincreases rapidly. On the othercomputing technology had givencomputing and parallel processingimplementation of distributed combroken down into subproblems andtorn into subnetworks. Hence netwterconnected betweenhe centralized controls the size of the systemthe advancements insight into distributedwer systems. For thethe problem has to beysical network must beecomposition becomes</p><p>minimize a common objective function [6]. The objective functionis formed in such a way that it represents the computational loadand the communication between the clusters. The networkpartitions are done for a wide range of applications in powersystems control and monitoring. In [7], multi-partitioning ofpower network is done using simulated annealing for stateestimation. Zhongxu et al. proposed network partitioning fordistributed reactive power optimization in [8]. A new methodfor partitioning is proposed in [9] for voltage collapse margincalculations. Ref. [10] gives a metaheuristic technique for cluster-ing web documents. However, the evolutionary methods pro-Regular Paper</p><p>Network decomposition using Kernighasearch algorithm</p><p>G.A. Ezhilarasi, K.S. Swarup n</p><p>Department of Electrical Engineering, Indian Institute of Technology Madras, Chennai</p><p>a r t i c l e i n f o</p><p>Article history:</p><p>Received 1 October 2011</p><p>Received in revised form</p><p>6 July 2012</p><p>Accepted 9 July 2012Available online 28 September 2012</p><p>Keywords:</p><p>Network decomposition</p><p>Distributed computing</p><p>Harmony search algorithm</p><p>KernighanLin algorithm</p><p>a b s t r a c t</p><p>Power system is a large-s</p><p>centralized control becom</p><p>For implementation of thi</p><p>the network decompositio</p><p>(HS) algorithm. To improv</p><p>method called Kernighan</p><p>the partitioning of digital a</p><p>of the partitioned clusters</p><p>out on IEEE Standard sy</p><p>hierarchically.</p><p>journal homepage: wwwLin strategy aided harmony</p><p>36, India</p><p>network with a number of components and interconnections for which</p><p>umbersome. For multi-area computations, decentralization is necessary.</p><p>proach network decomposition becomes an essential task. In this paper</p><p>roblem is solved as an optimization problem using the harmony search</p><p>e performance of the HS algorithm, a widely used graph bi-partitioning</p><p>(KL) strategy is used in the improvisation process. KL strategy is used in</p><p>VLSI circuits and is suitable for bi-partitioning networks. The connectivity</p><p>checked by means of graph traversal techniques. Simulation are carried</p><p>s and found to be very effective in decomposition of the system</p><p>& 2012 Elsevier B.V. All rights reserved.</p><p>evier.com/locate/swevo</p><p>nary Computation</p></li><li><p>is also handled by balancing the number of nodes in each of the</p><p>the conventional graph theory methods and many evolutionarymethods like genetic algorithm, simulated annealing and tabusearch. In this work we propose to solve the network decomposi-tion using a recently evolved metaheuristic algorithm calledharmony search (HS) algorithm. HS evolved from the process ofmusic composition based on the improvisation of harmony. Theharmony in the music composition is analogous to the solutionvector of the optimization problem and the improvisations madeby the musicians represents the search towards the optimum.This algorithm is independent of the initial values and theprevious iteration values. The search is based on the harmonymemory considering rate and pitch adjustment rate. The optimi-zation procedure of the HS algorithm involves two main stepsnamely initialization of the harmony and HS parameters andimprovisation of the existing harmony. This is done iteratively fora xed number of improvisations. Improvisation is done based onthe HS parameter harmony memory considering rate (HMCR). Inthis work the pitch adjustment is done based on KernighanLinstrategy in order to improve the search strategy. This method istermed as the KernighanLin strategy aided harmony searchalgorithm (KL-HS).</p><p>Generate Random HM</p><p>Check connectivity of clustersusing BFS</p><p>Calculate Fitness Value</p><p>Start</p><p>Input system data and initializethe HS parameters</p><p>G.A. Ezhilarasi, K.S. Swarup / Swarm and Evolutionary Computation 7 (2012) 162decomposed networks. The number of edges in each subnetworkwill also increase the computations involved in each subnetwork.</p><p>2.1. Problem formulation</p><p>Network decomposition can be viewed as a combinatorialoptimization problem with main objective being to minimizethe number of tie lines between the subnetworks. It also needs tobalance the number of nodes and lines within the subnetworksfor load balancing during computation. Hence the problem can bemathematically formulated as follows:</p><p>Min CM,N,L aM3bN2gL 1where M is the maximum number of tie lines between theclusters, N is the maximum number of nodes in a cluster, L isthe maximum number of lines in a cluster, a,b,g are weightingfactors for each term.</p><p>This optimization is subject to the constraint that the nodes ineach cluster must form a connected graph. This constraint checksthe observability of the network at the instant of decompositionof the network.</p><p>2.2. Solution methodology</p><p>Network decomposition has been in the literature in variousproblem and in [20] it is used for optimal scheduling of dieselgenerators. In [21], multi-objective HS algorithm is used to solvethe optimal power ow problem and in [22] environmentaleconomic dispatch is solved using the same. In software engi-neering the HS algorithm has been used for the task assignmentproblem [23] and a novel global harmony search algorithm isdescribed in [24]. Self-adaptive harmony search is proposed in[25] for expert system applications. A novel derivative of har-mony search for discrete optimization problems has been pro-posed in [26]. In [27] a hybrid method has been proposedcombining the harmony search method with the sequentialquadratic programming method and a global harmony searchalgorithm is proposed for unconstrained optimization problemsas well. Hence the literature shows the applicability of the HSalgorithm to a wide range of optimization problems in powersystems and other engineering applications.</p><p>This paper proposes the application of harmony search algo-rithm aided by KernighanLin strategy to solve the networkdecomposition problem and it is organized as follows. Section 2deals with the problem formulation of network partitioning.Section 3 describes the KernighanLin strategy aided harmonysearch algorithm (KL-HS) and the implementation methodologyin detail to the network partitioning problem. Section 4 presentsthe simulation results done on IEEE standard test cases to assessthe effectiveness of the proposed method.</p><p>2. Network decomposition</p><p>Power system is a interconnected network that can be repre-sented by a graph GV ,E, with V vertices and E edges. Theobjective of network decomposition is to group the closelycoupled parts of the network and localize it, in order to make itidentical to a network of computers connected through commu-nication. This will facilitate the distributed computing of powersystem applications over a computing network. The major phy-sical coupling factor of a interconnected network is the number oflines between the decomposed networks. In concern to distrib-uted computing, the computational load on each computing nodeelds of engineering for the past years. It has been solved usingImprovise new HM usingHMCR</p><p>Interchange between Paritionsusing KL Strategy</p><p>Stopping CriteriaReached ?</p><p>No</p><p>Yes</p><p>Obtain the best variablein the HM</p><p>StopFig. 1. Decomposition using KernighanLin aided harmony search algorithm.</p></li><li><p>G.A. Ezhilarasi, K.S. Swarup / Swarm and Evolutionary Computation 7 (2012) 16 3Table 1Partitioning details of IEEE test cases.</p><p>Test Nodes in Cluster 1</p><p>case</p><p>IEEE 14 Bus 1 2 3 4 5 7 8</p><p>IEEE 30 Bus 1 2 3 4 5 6 7 8 9 10 11 12 13 16 17</p><p>IEEE 57 Bus 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18</p><p>19 20 21 22 23 24 25 26 27 28 29 52 53 54 55</p><p>IEEE 118 Bus 1:59 113 114 115 1173. KL strategy aided HS algorithm</p><p>KernighanLin algorithm is used in solving the graph parti-tioning problem, especially in the eld of VLSI circuits. Thisinvolves interchange of vertex pairs between partitions whichwill have the largest decrease in cost of partitioning in terms ofcut size. This strategy is made use of the HS algorithm for thepitch adjustment. The implementation methodology is explainedstep by step in this section.</p><p>Step1: Initialization of the problem and HS algorithm parameters.The optimization problem is well dened initially as follows:</p><p>0 10 20 30 40 50 60 70 80 90 1000.5</p><p>1</p><p>1.5</p><p>2</p><p>2.5</p><p>3</p><p>3.5x 104</p><p>Number of Improvisations</p><p>Fitn</p><p>ess </p><p>Val</p><p>ue</p><p>0 10 20 30 40 50 60 70 80 90 1000</p><p>0.5</p><p>1</p><p>1.5</p><p>2</p><p>2.5</p><p>3</p><p>3.5</p><p>4</p><p>4.5</p><p>5x 105</p><p>Number of Improvisations</p><p>Fitn</p><p>ess </p><p>Val</p><p>ue</p><p>Fig. 2. Convergence characteristics of HS algorithm for IEEE 57 and 118 Bussystems. (a) Convergence of 57 Bus system. (b) Convergence of 118 Bus system.Nodes in Cluster 2 Tie</p><p>lines</p><p>6 9 10 11 12 13 14 3</p><p>14 15 18 19 20 21 22 23 24 25 26 27 28 29 30 7</p><p>30 31 32 33 34 35 36 37 38 39 40 41 42 43 44</p><p>45 46 47 48 49 50 51 56 57</p><p>8</p><p>60:110 111 112 116 118 9Minimize f x subject to xi AXi, i 1,2,3, . . .n where, f(x) is theobjective function, xi is the set of decision variables, Xi is the set ofpossible range of the design variables and n is the number ifdecision variables. In the network decomposition problemf x aM3bN2gL. The size of the harmony memory (HMS)representing the number of solution vectors and the harmonymemory considering rate (HMCR) is initialized.</p><p>Step2: Initialize the harmony memory (HM). The harmonymemory is a matrix in which the sets of decision variables arestored. In this step the initial harmony memory will consist of arandom combination of nodes of the network to be decomposed.Partitions are created in this random initial harmony. N isany node of the interconnected network and is represented as</p><p>050</p><p>100150</p><p>020</p><p>40600</p><p>20</p><p>40</p><p>60</p><p>80</p><p>100</p><p>120</p><p>Number of V</p><p>ariables</p><p>Size of Harmony Memory</p><p>Ran</p><p>ge o</p><p>f Var</p><p>iabl</p><p>es</p><p>050</p><p>100150</p><p>020</p><p>40600</p><p>20</p><p>40</p><p>60</p><p>80</p><p>100</p><p>120</p><p>Number of </p><p>Variables</p><p>Size of Harmony Memory</p><p>Ran</p><p>ge o</p><p>f Var</p><p>iabl</p><p>es</p><p>Fig. 3. Initial and nal harmony Memories showing the improvisations for IEEE 118Bus system. (a) 118 Bus initial harmony memory. (b) 118 Bus nal harmony memory.</p></li><li><p>follows:</p><p>HM</p><p>N11 N12 N1n1 N1n</p><p>N21 N22 N2n1 N2n</p><p>^ ^ ^ ^ ^</p><p>NHMS11 NHMS12 NHMS1n1 NHMS1n</p><p>NHMS1 NHMS2 NHMSn1 NHMSn</p><p>0BBBBBBB@</p><p>1CCCCCCCA</p><p>2</p><p>Step3: Improvise the new harmony from the HM. Improvisationis the process of creating a new harmony from the existingmemory. A new harmony vector X0 N1 0,N1 0, . . . ,Nn 0 is generatedbased on three rules: (a) memory consideration, (b) randomselection and (c) KL strategy. The value of the rst decisionvariable is chosen from one of the harmonies betweenN11 and N</p><p>HMS1 . The other decision variables in the harmony mem-</p><p>ory are also chosen in the same manner with the probability ofthe Harmony Memory Considering Rate (HMCR) that variesbetween 0 and 1. HMCR is the rate of choosing a variable fromthe historical data in the HM and (1-HMCR) is the rate ofrandomly selecting one value from the possible range of values</p><p>Ni0 Ni</p><p>0AfN1i ,N2i , . . . ,NHMSi g with probability HMCR,Ni</p><p>0AXi with probability 1 - HMCR:</p><p>(3</p><p>Now the new harmony vector is pitch adjusted using KL strategy.According to the KL strategy a pair of nodes between partitions are</p><p>interchanged only if the gain of interchange is more than the gain ofall other interchanges of node pairs. The number of lines connectedbetween the nodes and also the lines connecting the pair of nodes totheir own sub networks sum to give the gain of the interchange.</p><p>Step4: Update the harmony memory. The number of tie lineslinking the clusters, the number of nodes in a cluster and thenumber of lines in a cluster are obtained from the partitionedmemory after pitch adjustment to nd the tness value of thenew harmony vectors. The connectivity of the clusters aredetermined using the graph traversal methods to be discussedin the next section. If the new harmony vector X0 x1 0,x1 0, . . . ,xn 0is better than the worst harmony in the HM, then the newharmony memory is included in the HM and the existing worstharmony is excluded from the HM. This is decided based on thetness value of the harmony.</p><p>Step5: Check the stopping criteria. The HS algorithm is stoppedtill the number of improvisations is met and there is no furtherimprovement in the harmony. Steps 3 and 4 are continued untilthe stopping criteria is met. This implementation methodology iswell described by means of a owchart in Fig. 1.</p><p>3.1. Cluster connectivity</p><p>The graph traversing is mainly used in two major applicationsin power system namely, topological problems and network ow</p><p>1</p><p>2</p><p>3</p><p>4</p><p>5 18</p><p>21</p><p>22</p><p>23</p><p>24</p><p>2526</p><p>28</p><p>30</p><p>31</p><p>32</p><p>34...</p></li></ul>

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