J. Parallel Distrib. Comput. 72 (2012) 936–943

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J. Parallel Distrib. Comput.

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Network partitioning using harmony search and equivalencing fordistributed computingG. Angeline Ezhilarasi ∗, K.S. SwarupDepartment of Electrical Engineering, Indian Institute of Technology Madras, Chennai - 600036, India

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 decompositionOptimal partitioningDistributed computingHarmony Search AlgorithmNetwork equivalencing

a b s t r a c t

Power system has a highly interconnected network that requires intense computational effort andresources for centralized control. Distributed computing needs the systems to be partitioned optimallyinto clusters. The network partitioning is an optimization problem whose objective is to minimize thenumber of nodes in a cluster and the tie lines between the clusters. Harmony Search(HS) Algorithm isone of the recently developed meta heuristic algorithms that can be applied to optimization problems.In this work, the HS algorithm is applied to the network partitioning problem and power flow basedequivalencing is done to represent the external system. Simulation is done on IEEE Standard Test Systems.The algorithm is found to be very effective in partitioning the system hierarchically and the equivalencingmethod gives accurate results in comparison to the centralized control.

© 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).

0743-7315/$ – see front matter© 2012 Elsevier Inc. All rights reserved.doi:10.1016/j.jpdc.2012.04.006

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 optimization

G. Angeline Ezhilarasi, K.S. Swarup / J. Parallel Distrib. Comput. 72 (2012) 936–943 937

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) = αN + βL + γM (1)where,N maximum number of nodes in a clusterL maximum number of lines in a clusterM maximum number of tie lines between the clustersα, β, γ 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 instrument’s pitch range correspondsto decision variable’s value range; musical harmony at a certaintime corresponds to the solution vector at a certain iteration;and audience’s 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 becomesunnecessary.The optimization procedure of the HS algorithm is as follows:

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

938 G. Angeline Ezhilarasi, K.S. Swarup / J. Parallel Distrib. Comput. 72 (2012) 936–943

(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 graphthat is not weighted, hence it is chosen for this work to check theconnectivity of the partitioned clusters.

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 arealso identified, which form the tie lines between the internaland the external system. For detailed analysis of the internal

G. Angeline Ezhilarasi, K.S. Swarup / J. Parallel Distrib. Comput. 72 (2012) 936–943 939

(a) Interconnected network. (b) Partitioned network.

Fig. 3. Schematic of power system network.

System, the external system is represented by its equivalentto reduce the computational burden in case of emergencies.The conventional techniques used for network equivalents arethe WARD Equivalents, Radial Equivalent Independent (REI)Equivalents [18] and Weighted least square approach in [24]. Theimpediments of these techniques lies in their inability tomodel anychange in the external network [23,24]. The networks are valid forincremental changes within the internal networks only.

Ward Equivalent is basically derived using the injection currentand voltage relationship [13]. In thismethod the internal, boundaryand external system are separated and the whole external systemis equivalenced. For an original unreduced system the networkequations can represented as,Yee Yeb 0

Ybe Y ebb + Y i

bb Ybi0 Yib Yii

EeEbEi

=

IeIbIi

(2)

where,

Yii admittance matrix of internal systemYbb admittance matrix of boundary systemYee admittance matrix of external system.

By Gaussian elimination the external system is reduced and theboundary power injections are found from the resulting voltagevector. However this involves a lot of computation in the case ofcontingency analysis of the entire system. It does not give accurateresults for large systems when the number of buses is more.

Another alternative is the Extended Ward Equivalent [16],where the external system is equivalenced by fictitious PV busesat all the boundary buses of the internal system. This is done tocorrect the reactive power response of the external system. Thismethod gives good accuracy but it is not very convenient to useunder emergencies.

REI Equivalent [18] is yet another method suggested for equiv-alencing 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 internalsystem [10]. Neural Network Techniques are useful when theinformation about the external system is inadequate.

Table 1Base case power flow in tie lines before partitioning.

Test case Tie lines Pflow (MW) Qflow (MVAR)

IEEE 57-Bus

25–30 7.5802 4.699922–38 −10.6410 −3.448311–41 8.7749 −2.530011–43 13.0250 −22.366015–45 35.6310 −44.268014–46 43.0590 −116.730013–49 28.9740 −28.154010–51 27.6010 −90.6380

5.1. Proposed equivalencing

In this work a simple method of representing the externalsystem by means of the power injections at the boundary busesis proposed. The overview of this approach is shown in Fig. 3(a)and (b). For verification of the proposed method of equivalencingthe load flow is done on the internal system under study and theexternal system. The simulation is done on IEEE standard test caseswhich are discussed in detail in the later sections. As depicted inFig. 3 the boundary buses at the internal system and the externalsystem are separated. The boundary buses in the internal systemare treated a load buses with the load being the power flow in thetie lines from that bus. Similarly the boundary buses at the externalsystem are assumed to be generator buses with the power flowin the tie lines as the generation at that bus. The power flow inthe tie lines are represented as loads at the boundary buses of theinternal system. The base case load flow results can be obtainedfrom the centralized control center or from the historical dataduring contingencies. A network based distributed slack busmodelis presented in [28] and selection of slack bus for transmissionnetwork cost allocation based on network utilization is givenin [29]. In this work the internal system slack bus of the internalsystem is unchanged but for the external system, it is chosen tobe the generator bus with maximum generation and its voltageand angle being fixed using the base case power flow results. Thecreated partitions along with the external equivalent can be usedfor detailed and decentralized study as done in [8] for transientstability analysis.

The equivalent power injections at the boundary buses can beobtained 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-tering, simulation was done using the IEEE standard test sys-tems. Parameters of the HS algorithm were set as follows:

940 G. Angeline Ezhilarasi, K.S. Swarup / J. Parallel Distrib. Comput. 72 (2012) 936–943

Table 2Partitioning details of IEEE test cases.

Test case Nodes in cluster1 Nodes in cluster2 Tie lines Fitness value

IEEE 30 bus 1 2 3 4 5 6 7 8 9 10 11 12 13 16 17 14 15 18 19 20 21 22 23 24 25 26 27 28 29 30 7 119IEEE 57 bus 1–29 52 53 54 55 30–51 56 57 8 228IEEE 118 bus 1–59 113 114 115 117 60–110 111 112 116 118 9 490

244

242

240

238

236

234

232

230

228

226

Fitn

ess

Val

ue

Number of Improvisations0 20 40 60 80 100

(a) Convergence characteristics of 57 bus system.

020

4060

0

50

1000

20

40

60

Number of VariablesSize of HM

Ran

ge o

f V

aria

bles

020

4060

0

50

1000

20

40

60

Number of VariablesSize of HM

Ran

ge o

f V

aria

bles

(b) Initial harmony memory. (c) Final harmony memory.

Fig. 4. Convergence characteristics of HS algorithm with the initial and final harmony memory showing the improvisations for IEEE 57 test system.

Harmony Memory Size = 100,Number of Improvisations = 100,Number of trials = 50,Harmony Memory Considering Rate = 0.8,Pitch Adjustment Rate = 0.6 and Bandwidth bw = 1.The partitioning details of test systems are shown in Table 2.The performance of the HS algorithm is verified by means ofthe convergence characteristics. The initial and final harmonymemory is observed for the various test cases. This indicates therandomness in the initial harmony and the hierarchical clusteringin the final harmonymemory. Theperformance of theHS algorithmis shown for 57 bus system in Fig. 4.

The partitioned clusters of IEEE 57 bus system are shown inFig. 5. The connectivity of the clusters is verified by the artificialintelligence techniques such as breadth first search and depth firstsearch and the clusters are found to contain no isolated nodes.

7. Case study

Simulations of the proposed method of partitioning andequivalencing the power network for distributed load flowanalysis

is carried out on IEEE 30 Bus, IEEE 57 Bus and IEEE 118 BusSystem [3]. The analysis was done on an Intel Pentium IV, 3.40 GHzProcessor with 1 GB RAM using Matlab. The partitions are createdby sequential testing of all possible combinations and singularitychecks. The system is separated into Internal and External Systemand the Boundary Buses are identified. Conventional load flowis done using fast decoupled load flow method and the same isapplied on the partitioned system. The results are justified in termsof the error in the voltage magnitude and the phase angle. It isfound that for the system under study, the error in voltage isminimum. The line flows in the internal and external system arealso compared with that of the original unreduced system.

7.1. IEEE 57 bus system

This test case is a system with 57 nodes and 80 linesinterconnection them. The optimal partition for the system isfound to be 2. The number of buses in the system are not equalbecausemost of the almost equal partitions leads to isolated noted

G. Angeline Ezhilarasi, K.S. Swarup / J. Parallel Distrib. Comput. 72 (2012) 936–943 941

(a) 57 bus—cluster 1. (b) 57 bus—cluster 2.

Fig. 5. Partitioned clusters of IEEE 57 bus system.

in the partitioned network. This may lead to load balancing inthe case of simultaneous runs of the internal and the externalsystem. But an isolated node in the sub networks would lead to thesingularity of the Jacobian in load flow computations. The Systemhas 7 generators and the rest are the load buses. After partitioningthe internal system has 6 generators and the external system hasone generator andmost of the loads. Equivalent power injection ofthe external system is added to the boundary buses. Load flow isdone on the internal and the external system independently. Erroranalysis between the actual and distributed load flow of the IEEE57 bus system is shown in Fig. 6.

8. Conclusion

In this paper the harmony search algorithm is used to solvethe network partitioning problem for large scale power systems.

The proposed method is based on the concept of improvisation inmusic to determine the optimal setting of the control variables.The network partitioning is formulated as an optimization problemwhere the objective is to minimize the number of tie linesbetween the clusters and to attain a balance of nodes betweenthe clusters. The connectivity of the clusters obtained is checkby means of the AI search algorithms namely BFS and DFS. Thepower flow based equivalencing proposed is simple and accurateand can be used to study a part of the large scale systems indetail. The proposed method has been tested on IEEE standardtest cases. The simulation results clearly show the efficiency ofthe proposed method has been validated using load flow studyon the partitioned and equivalenced network. This partitioningcan be easily implemented in a decentralized control centersfor distributed computing applications of a large interconnectedpower system network.

942 G. Angeline Ezhilarasi, K.S. Swarup / J. Parallel Distrib. Comput. 72 (2012) 936–943

(a) Voltage magnitude. (b) Phase angle.

(c) Error in voltage magnitude. (d) Error in phase angle.

Fig. 6. Simulation results of IEEE 57 bus system load flow.

References

[1] G. Cafaro, E. Margarita, Method for optimal tearing of electrical networks,International Journal of Electrical Power & Energy Systems 4 (3) (1982)185–191.

[2] C. Chang, L. Lu, F. Wen, Power system network partitioning using tabu search,Electric Power Systems Research 49 (1) (1999) 55–61.

[3] R.D. Christie, Power system test case archive,http://www.ee.washington.edu/research/pstca/.

[4] L. Coelho, V. Mariani, An improved harmony search algorithm for powereconomic load dispatch, Energy Conversion and Management 50 (10) (2009)2522–2526.

[5] S. Das, A. Mukhopadhyay, A. Roy, A. Abraham, B. Panigrahi, Exploratory powerof the harmony search algorithm: analysis and improvements for globalnumerical optimization, IEEE Transactions on Systems, Man, and Cybernetics,Part B: Cybernetics 41 (1) (2011) 89–106.

[6] S. Deckmann, A. Pizzolante, A. Monticelli, B. Stott, O. Alsac, Studies on powersystem load flow equivalencing, IEEE Transactions on Power Apparatus andSystems (6) (1980) 2301–2310.

[7] H. Ding, A. El-Keib, R. Smith, Optimal clustering of power networks usinggenetic algorithms, Electric Power Systems Research 30 (3) (1994) 209–214.

[8] S. Esmaeili, S. Kouhsari, A distributed simulation based approach for detailedand decentralized power system transient stability analysis, Electric PowerSystems Research 77 (5–6) (2007) 673–684.

[9] M. Fesanghary, M. Mahdavi, M. Minary-Jolandan, Y. Alizadeh, Hybridizingharmony search algorithm with sequential quadratic programming for

engineering optimization problems, ComputerMethods in AppliedMechanicsand Engineering 197 (33–40) (2008) 3080–3091.

[10] Y. Fu, T. Chung, A hybrid artificial neural network (ann) and ward equivalentapproach for on-line power system voltage security assessment, ElectricPower Systems Research 53 (3) (2000) 165–171.

[11] Z. Geem, Novel derivative of harmony search algorithm for discrete designvariables, Applied Mathematics and Computation 199 (1) (2008) 223–230.

[12] Z. Geem, K. Sim, Parameter-setting-free harmony search algorithm, AppliedMathematics and Computation 217 (8) (2010) 3881–3889.

[13] E. Housos, G. Irisarri, R. Porter, A. Sasson, Steady state network equivalents forpower system planning applications, IEEE Transactions on Power Apparatusand Systems (1980) 2113–2120.

[14] M. Irving, M. Sterling, Optimal network tearing using simulated annealing,IEE Proceedings C Generation, Transmission and Distribution 137 (1) (1990)69–72.

[15] K. Lee, Z. Geem, A new structural optimization method based on the harmonysearch algorithm, Computers & Structures 82 (9–10) (2004) 781–798.

[16] K. Lo, L. Peng, J. Macqueen, A. Ekwue, D. Cheng, An extended ward equivalentapproach for power system security assessment, Electric Power SystemsResearch 42 (3) (1997) 181–188.

[17] M. Mahdavi, M. Fesanghary, E. Damangir, An improved harmony searchalgorithm for solving optimization problems, Applied Mathematics andComputation 188 (2) (2007) 1567–1579.

[18] L. Min, A. Abur, Rei-equivalent based decomposition method for multi-areattc computation, in: Transmission andDistribution Conference and Exhibition,2005/2006 IEEE PES, IEEE, 2006, pp. 506–510.

G. Angeline Ezhilarasi, K.S. Swarup / J. Parallel Distrib. Comput. 72 (2012) 936–943 943

[19] A. Monticelli, S. Deckmann, A. Garcia, B. Stott, Real-time external equivalentsfor static security analysis, IEEE Transactions on Power Apparatus and Systems(2) (1979) 498–508.

[20] H. Mori, Y. Komatsu, Power network decomposition with new ant colonyoptimization, in: International Conference on Probabilistic Methods Appliedto Power Systems, PMAPS 06, 2006, pp. 1–6.

[21] V. Pandi, B. Panigrahi, R. Bansal, S. Das, A. Mohapatra, Economic load dispatchusing hybrid swarm intelligence based harmony search algorithm, ElectricPower Components and Systems 39 (8) (2011) 751–767.

[22] V. Pandi, B. Panigrahi, S. Das, Z. Cui, Dynamic economic load dispatch withwind energy using modified harmony search, International Journal of Bio-Inspired Computation 2 (3) (2010) 282–289.

[23] Y. Phulpin, M. Begovic, M. Petit, J. Heyberger, D. Ernst, Evaluation of networkequivalents for voltage optimization in multi-area power systems, IEEETransactions on Power Systems 24 (2) (2009) 729–743.

[24] I. Rahim, et al., A weighted least squares method for determination of powersystem equivalents, Electric Power Systems Research 5 (3) (1982) 207–218.

[25] A. Rahimi, K. Kato, S. Ansari, V. Brandwajn, G. Cauley, D. Sobajic, On externalnetwork model development, IEEE Transactions on Power Systems 11 (2)(1996) 905–910.

[26] S. Sivasubramani, K. Swarup, Multi-objective harmony search algorithm foroptimal power flow problem, International Journal of Electrical Power &Energy Systems 33 (3) (2011) 745–752.

[27] S. Sivasubramani, K. Swarup, Environmental/economic dispatch using multi-objective harmony search algorithm, Electric Power Systems Research 81 (9)(2011) 1778–1785.

[28] S. Tong, M. Kleinberg, K. Miu, A distributed slack bus model and its impact ondistribution system application techniques, IEEE International Symposium onCircuits and Systems, vol. 5, IEEE, 1999, 2005, p. 4743.

[29] C. Vazquez, L. Olmos, I. Pérez-Arriaga, On the selection of the slack busin mechanisms for transmission network cost allocation that are based onnetwork utilization, in: Proceedings of the 15th Power Systems ComputingConference, PSCC, Seville, 2002.

[30] A. Verma, B. Panigrahi, P. Bijwe, Harmony search algorithm for transmissionnetwork expansion planning, Generation, Transmission & Distribution, IET 4(6) (2010) 663–673.

[31] C. Wang, Y. Huang, Self-adaptive harmony search algorithm for optimization,Expert Systems with Applications 37 (4) (2010) 2826–2837.

[32] D. Zambroni, et al., New technique of network partitioning for voltagecollapse margin calculations, IEE Proceedings, Generation, Transmission andDistribution 141 (6) (1994) 630–636.

[33] R.L. Zhongxu Li, Yutian Liu, X. Niu, Network partition for distributed reactivepower optimization in power systems, in: IEEE International Conference onNetworking, Sensing and Control, 2008, pp. 6–8.

[34] D. Zou, L. Gao, S. Li, J. Wu, X. Wang, A novel global harmony search algorithmfor task assignment problem, Journal of Systems and Software 83 (10) (2010)1678–1688.

[35] D. Zou, L. Gao, J. Wu, S. Li, Novel global harmony search algorithm forunconstrained problems, Neurocomputing 73 (16–18) (2010) 3308–3318.

G. Angeline Ezhilarasi received her Bachelor’s Degreein Electrical and Electronics Engineering from MadrasUniversity and Master’s in Power Systems Engineeringfrom Anna University. She is currently doing her PhD in IITMadras in the area of Power System Analysis. Earlier, sheworked as a project associate at IIT Madras in the NationalProject on Technology Enhanced Learning. Her researchinterests are power system operation, high performancecomputing techniques such as distributed computing andparallel processing in power system applications.

Email: angel.ezhil@gmail.com

K.S. Swarup is with the Department of Electrical Engineer-ing, Indian Institute of Technology, Madras, India. Prior tojoining the department as a visiting faculty member, heheld positions at the Mitsubishi Electric Corporation, Os-aka, Japan, and Kitami Institute of Technology, Hokkaido,Japan, serving as a visiting research scientist and visitingprofessor, respectively, from 1992 to 1999. His areas ofresearch include AI, knowledge-based systems, computa-tional intelligence, soft computing, and object modelingand design of electric power systems.

Email: swarup@ee.iitm.ac.in