# Network partitioning using harmony search and equivalencing for distributed computing

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Optimal partitioningDistributed computingHarmony Search AlgorithmNetwork equivalencing

2012 Elsevier Inc. All rights reserved.

1. Introduction

A power system is a complex network with a large numbercomponents and interconnections. The ever growing size andcomplexity of the system and the advances in computingtechnology have given an insight into parallel processing anddistributed computing for power system computations. Networkdecomposition is an essential for parallel processing and adistributed computing approach to any power system problem.The large interconnected network may be optimally divided intoclusters. In order to reduce the overall execution time thereshould be a balance between the size of the cluster and theinterconnection between the clusters. An objective function isformed such that it reflects these factors and is solved using theevolutionary algorithm.

Over the past decades a number of algorithms have been pro-posed in literature for optimal network tearing [1]. The techniquesincludematrix decomposition, successive approximation, dynamicprogramming and heuristic clustering approaches. These methodstend to form clusters with fewer interconnections but fail to bal-ance with the size of the clusters. Some of the evolutionary op-timization techniques such as simulated annealing [14], geneticalgorithm [7], ant colony optimization [20] and tabu search [2]have also been applied to optimal partitioning of the network.These methods try to minimize an objective function that reflects

Corresponding author.E-mail address: angel.ezhil@gmail.com (G. Angeline Ezhilarasi).

a balance between the two, so the objective function is formedsuch that it reflects the features of parallel and distributed process-ing. Network partitioning based on voltage variation at the loadbuses relative to other buses is proposed for voltage margin cal-culations in [32]. A decomposition algorithm is used for partition-ing the network for distributed reactive power optimization in apower system in [33]. Meta heuristic algorithms have been usedfor clustering web documents. However these methods are com-putation intensive and involve procedures based on natural selec-tion crossover andmutation. It also requires a large population sizeand occupies more memory.

In this paper one of the recently evolving heuristic algorithmscalled the Harmony Search Algorithm (HSA) is used to solve thepartitioning problem of large scale power networks. The harmonysearch algorithm has been applied tomany optimization problemsin engineering and design. To mention a few it is used for solvingstructural optimization problems in [15]. An improved harmonysearch is applied to optimal economic power dispatch [4,17],dynamic economic dispatch with wind energy is solved in [22]and a hybrid swarm intelligence based harmony search is usedto solve economic dispatch in [21]. The HS algorithm has beenused for transmission network planning [30]. In [26], a multiobjective HS algorithm is used to solve the optimal power flowproblem and in [27] environmental economic dispatch is solvedusing the same. In software engineering the HS algorithm hasbeen used for the task assignment problem [34] and a novelglobal harmony search algorithm is described in [35]. Self adaptiveharmony search is proposed in [31] for expert system applications.A novel derivative of harmony search for discrete optimizationJ. Parallel Distrib. Comp

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Network partitioning using harmony seadistributed computingG. Angeline Ezhilarasi , K.S. SwarupDepartment of Electrical Engineering, Indian Institute of Technology Madras, Chennai - 60

a r t i c l e i n f o

Article history:Received 19 January 2011Received in revised form21 December 2011Accepted 20 April 2012Available online 8 May 2012

Keywords:Network decomposition

a b s t r a c t

Power system has a highlyresources for centralized coninto clusters. The network pnumber of nodes in a clusteone of the recently developeIn this work, the HS algorithequivalencing is done to reprThe algorithm is found to bemethod gives accurate result0743-7315/$ see front matter 2012 Elsevier Inc. All rights reserved.doi:10.1016/j.jpdc.2012.04.006ut. 72 (2012) 936943

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interconnected network that requires intense computational effort andtrol. Distributed computing needs the systems to be partitioned optimallyartitioning is an optimization problem whose objective is to minimize theand the tie lines between the clusters. Harmony Search(HS) Algorithm isd meta heuristic algorithms that can be applied to optimization problems.m is applied to the network partitioning problem and power flow basedesent the external system. Simulation is done on IEEE Standard Test Systems.ery effective in partitioning the system hierarchically and the equivalencings in comparison to the centralized control.

aG. Angeline Ezhilarasi, K.S. Swarup / J. Par

problems has been proposed in [11]. In [9] a hybrid method hasbeen proposed combining the harmony search method with thesequential quadratic programming method and a global harmonysearch algorithm is proposed for unconstrained optimizationproblems as well. The exploratory power of the HS algorithm infinding the optimumsolution is given in [5] and [12] discuses aboutthe parameter setting free harmony search algorithm. Hence theliterature shows the applicability of the HS algorithm to a widerange of optimization problems.

This paper is organized as follows. Section 2 deals with theproblem formulation of network partitioning. Section 3 outlinesthe Harmony Search Algorithm and Section 4 explains theimplementation of the HS algorithm to the network partitioningproblem. Section 5 presents the simulation results to assess theeffectiveness of the proposed method on its application to thenetwork partitioning problem.

2. Network partitioning problem

A power system is a highly interconnected network, geograph-ically distributed over a wide area. In order to enhance the com-putational performance with the available resources distributedcomputing is adopted, for which the network should be optimallypartitioned into clusters. The Partitioning problem has the mainobjectives namely (1) To minimize the number of nodes in a clus-ter; (2) To minimize the number of lines in a cluster; and (3) Tominimize the number of tie lines between the clusters. The firsttwo objectives represent the computational load on each clusterand the third represents the communication between the clusters.Hence network partitioning can be viewed as a combinatorial opti-mizationproblem. It can bemathematically represented as follows.Min C(N, L,M) = N + L+ M (1)where,N maximum number of nodes in a clusterL maximum number of lines in a clusterM maximum number of tie lines between the clusters, , weighting factors for each term.

This optimization problem is subject to the constraint thatthe nodes in each cluster must form a connected graph. Thisconstraint checks the observability of the network at the instantof decomposition of the network.

3. Harmony search algorithm

The Harmony Search (HS) algorithm is a meta heuristicalgorithm developed recently based on the improvisation ofharmony in music composition. This method is based on theimprovisations done by the musician to obtain a better harmony.Musical improvisation is analogous to the optimization processseeking an optimal solution. Each musician corresponds to eachdecision variable; a musical instruments pitch range correspondsto decision variables value range; musical harmony at a certaintime corresponds to the solution vector at a certain iteration;and audiences aesthetics corresponds to objective function. Justlike musical harmony is improved time after time, the solutionvector is improved iteration by iteration. The harmony in themusicrepresents the solution vector of any optimization problem andthe improvisations made by the musician represents the local andglobal searches towards the optimum solution. There is no needfor initial values of decision variables in the HS algorithm. Unlikeother heuristic optimization algorithms that use a gradient search,HS uses a stochastic random search. The search is based on theharmonymemory considering rate and the pitch adjusting rate andso the derivative information of the previous iteration becomes

unnecessary.The optimization procedure of the HS algorithm is as follows:llel Distrib. Comput. 72 (2012) 936943 937

Fig. 1. Harmony search methodology for network partitioning.

1. Initialize the optimization problem and the algorithm parame-ters.

2. Initialize the Harmony Memory (HM).3. Improvise a new harmony from the Harmony Memory.4. Update the Harmony Memory.5. Repeat Steps 3 and 4 until the termination criterion is reached.

4. Implementation of HS to network partitioning

The HS algorithm is proposed to optimally partition the powernetworks into individual clusters, such that the clusters areconnected networks. The implementation of the HS algorithm tothe partitioning problem is explained by means of a flowchart inFig. 1.

In this clustering problem, the variables in the harmonymemory are the nodes or the buses in the network. The initialsystem data such as nodes and lines and their connectivity is takenfrom the original interconnected network. A set of random vectorsare generated varying between 1 to n without any redundancyto create the initial harmony memory, n being the number ofnodes in the network. The random memory is partitioned intothe desired number of clusters. The connectivity of the derivedclusters is determined using the graph traversing methods. Thenumber of tie lines linking the clusters, the number of nodes in acluster and the number of lines in a cluster is obtained from thepartitionedmemory. If the cluster has no isolated nodes then based

on the fitness value the improvisations are done as explained in theprevious section.

a(a) Breadth first search. (b) Depth first search.

Fig. 2. Graph traversal of IEEE 14 bus system for connectivity check.

4.1. Cluster connectivity

The graph traversing is mainly used in two major applicationsin the power system namely, topological problems and networkflow problems. The topological or structural problems deal withfinding the parts of the graph connected and defines one specificor all spanning trees. It also determines how strongly the graphcomponents are connected and how to color different parts ofthe network. The network flow problems include solving theshortest path problem, finding feasible optimal flow pattern andrecognizing the loop flows and wheeling problems. Breadth FirstSearch (BFS) andDepth First Search (DFS) are the two techniques ingraph theory having fundamentally different traversal techniques.

4.1.1. Breadth first searchBFS is a uninformed search algorithm that does not have any

heuristics. It starts from the root node of a graph and explores allthe child nodes systematically before any other node begins. It isused extensively for a range of applications like finding the shortestpath, testing for bipartite graphs, computing themaximum flow ina network and many others. In this method all the nodes that havebeen generated so far are stored increasing the space complexity.If there a is complete traversal possible, BFS guarantees to find it,by the fact that longer paths are never explored until all shorterones have been examined. BFS is complete and optimal for a graph

4.1.2. Depth first searchDFS is a uninformed search technique, which starts with a root

a node and traverses a child node till it reaches a node that hasno child nodes. Then it backtracks and follows the next child nodethat was left unexplored. The leaves of a spanning tree are reachedin the fastest possible way in this traversal. This is mostly usedfor topological sorting, planarity testing, solving puzzles such asmazes and many others. Depth first search requires less memorysince only the nodes on the current path are stored. In contrast toBFS, DFS finds a long path to a solution from one part of the tree,when a shorter path exists in some other unexplored part of thetree. The graph traversal of IEEE 14 Bus system is shown in Fig. 2which clearly distinguishes the methods of BFS and DFS. An initialopen list is created with all the nodes of the cluster. After completegraph traversal, the visited nodes are removed from the initialopen list. If the open list is empty at the end of the traversal, thecluster is connected. The nodes in the open links indicate that thereare isolated nodes in the cluster. This ensures that the partitionedcluster is connected completely.

5. Network equivalents

Network Equivalencing methods aims to identify the internalarea of the system to be fully preserved and the external areathat is to reduced and represented by its equivalent [6,25]. Theboundary buses at the internal and the external system are938 G. Angeline Ezhilarasi, K.S. Swarup / J. Parthat is not weighted, hence it is chosen for this work to check theconnectivity of the partitioned clusters.llel Distrib. Comput. 72 (2012) 936943also identified, which form the tie lines between the internaland the external system. For detailed analysis of the internal

aoalencing the external network for system planning applications. Inthis case the external system is transformed into a passive networkby grouping all the generations into one bus and all the loads of theexternal system in another bus. It creates a lossless network as afunction of the injected power of the buses. However it is difficultto simulate the precise behavior of the generators once they aregrouped together [19].

The network topology of the system changes only duringdisturbances, or it may be a scheduled change. But the load onthe system changes every instant and hence the power injectionsat the boundary buses will also change from time to time. Thereare pattern recognition techniques to match the boundary powerinjections, such that it reflects the topology changes of the internal

stability analysis.The equivalent power injections at the boundary buses can be

obtained from the base case power flow in the tie lines as given inTable 1. The sum of the real power (Pflow) and the reactive power(Qflow) flows at the sending end of the bus are summed up toget the load representation at the boundary buses for the internalsystem. Similarly the line flows at the receiving end are added up toget the generations at the boundary buses in the external system.Depending on the direction of the flows the bus type is changed togenerator bus or load bus.

6. Simulation results

To study the effectiveness of the proposed method of clus-G. Angeline Ezhilarasi, K.S. Swarup / J. Par

(a) Interconnected network.

Fig. 3. Schematic of p

System, the external system is represented by its equivalentto reduce the computational burden in case of emergencies.The conventional techniques used for network equivalen...

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