Swarm and Evolutionary Computation 7 (2012) 1–6

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Swarm and Evolutionary Computation

2210-65

http://d

n Corr

E-m

journal homepage: www.elsevier.com/locate/swevo

Regular Paper

Network decomposition using Kernighan–Lin strategy aided harmonysearch algorithm

G.A. Ezhilarasi, K.S. Swarup n

Department 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 1 October 2011

Received in revised form

6 July 2012

Accepted 9 July 2012Available online 28 September 2012

Keywords:

Network decomposition

Distributed computing

Harmony search algorithm

Kernighan–Lin algorithm

02/$ - see front matter & 2012 Elsevier B.V. A

x.doi.org/10.1016/j.swevo.2012.07.002

esponding author. Tel.: þ91 44 2257 4440; fa

ail address: swarup@ee.iitm.ac.in (K.S. Swaru

a b s t r a c t

Power system is a large-scale network with a number of components and interconnections for which

centralized control becomes cumbersome. For multi-area computations, decentralization is necessary.

For implementation of this approach network decomposition becomes an essential task. In this paper

the network decomposition problem is solved as an optimization problem using the harmony search

(HS) algorithm. To improve the performance of the HS algorithm, a widely used graph bi-partitioning

method called Kernighan–Lin (KL) strategy is used in the improvisation process. KL strategy is used in

the partitioning of digital and VLSI circuits and is suitable for bi-partitioning networks. The connectivity

of the partitioned clusters are checked by means of graph traversal techniques. Simulation are carried

out on IEEE Standard systems and found to be very effective in decomposition of the system

hierarchically.

& 2012 Elsevier B.V. All rights reserved.

1. Introduction

Power system is a complicated grid interconnected betweengeographically distributed multi areas. The centralized controland monitoring of the system is complex as the size of the systemincreases rapidly. On the other hand the advancements incomputing technology had given an insight into distributedcomputing and parallel processing in power systems. For theimplementation of distributed computing the problem has to bebroken down into subproblems and the physical network must betorn into subnetworks. Hence network decomposition becomesan essential part of distributed computing of power systemproblems.

In the literature several approaches have been proposed fornetwork decomposition like matrix decomposition, successiveapproximation and heuristic clustering techniques. The conven-tional 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 simulatedannealing 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 fornetwork decomposition in [5]. These methods formulate thenetwork partitioning as an optimization problem and tries to

ll rights reserved.

x: þ91 44 2257 4401.

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-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.

In this paper one of the recently evolving heuristic algorithmcalled harmony search algorithm (HSA) is used to solve thepartitioning problem of large-scale power networks. The harmonysearch algorithm has been applied to many optimization pro-blems in engineering and design. To mention a few it is used forsolving structural optimization problems in [11]. In [12,13] animproved harmony search is applied to optimal economic powerdispatch, dynamic economic dispatch with wind energy is solvedin [14,15] a hybrid swarm intelligence based harmony search isused 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 hasbeen applied to combined heat and power economic dispatch

G.A. Ezhilarasi, K.S. Swarup / Swarm and Evolutionary Computation 7 (2012) 1–62

problem and in [20] it is used for optimal scheduling of dieselgenerators. In [21], multi-objective HS algorithm is used to solvethe optimal power flow 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.

This paper proposes the application of harmony search algo-rithm aided by Kernighan–Lin 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 Kernighan–Lin 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.

Generate Random HM

Check connectivity of clustersusing BFS

Calculate Fitness Value

Improvise new HM usingHMCR

Start

Input system data and initializethe HS parameters

Interchange between Paritionsusing KL Strategy

Stopping CriteriaReached ?

No

Yes

Obtain the best variablein the HM

Stop

Fig. 1. Decomposition using Kernighan–Lin aided harmony search algorithm.

2. Network decomposition

Power system is a interconnected network that can be repre-sented by a graph GðV ,EÞ, 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 nodeis also handled by balancing the number of nodes in each of thedecomposed networks. The number of edges in each subnetworkwill also increase the computations involved in each subnetwork.

2.1. Problem formulation

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:

Min CðM,N,LÞ ¼ aM3þbN2

þgL ð1Þ

where 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.

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.

2.2. Solution methodology

Network decomposition has been in the literature in variousfields of engineering for the past years. It has been solved using

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 fixed number of improvisations. Improvisation is done based onthe HS parameter harmony memory considering rate (HMCR). Inthis work the pitch adjustment is done based on Kernighan–Linstrategy in order to improve the search strategy. This method istermed as the Kernighan–Lin strategy aided harmony searchalgorithm (KL-HS).

Table 1Partitioning details of IEEE test cases.

Test Nodes in Cluster 1 Nodes in Cluster 2 Tie

case lines

IEEE 14 Bus 1 2 3 4 5 7 8 6 9 10 11 12 13 14 3

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

IEEE 57 Bus 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

19 20 21 22 23 24 25 26 27 28 29 52 53 54 55

30 31 32 33 34 35 36 37 38 39 40 41 42 43 44

45 46 47 48 49 50 51 56 57

8

IEEE 118 Bus 1:59 113 114 115 117 60:110 111 112 116 118 9

0 10 20 30 40 50 60 70 80 90 1000.5

1

1.5

2

2.5

3

3.5x 104

Number of Improvisations

Fitn

ess

Val

ue

0 10 20 30 40 50 60 70 80 90 1000

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5x 105

Number of Improvisations

Fitn

ess

Val

ue

Fig. 2. Convergence characteristics of HS algorithm for IEEE 57 and 118 Bus

systems. (a) Convergence of 57 Bus system. (b) Convergence of 118 Bus system.

050

100150

020

40600

20

40

60

80

100

120

Number of Variables

Size of Harmony Memory

Ran

ge o

f Var

iabl

es

050

100150

020

40600

20

40

60

80

100

120

Number of VariablesSize of Harmony Memory

Ran

ge o

f Var

iabl

es

Fig. 3. Initial and final harmony Memories showing the improvisations for IEEE 118

Bus system. (a) 118 Bus initial harmony memory. (b) 118 Bus final harmony memory.

G.A. Ezhilarasi, K.S. Swarup / Swarm and Evolutionary Computation 7 (2012) 1–6 3

3. KL strategy aided HS algorithm

Kernighan–Lin algorithm is used in solving the graph parti-tioning problem, especially in the field 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.

Step1: Initialization of the problem and HS algorithm parameters.

The optimization problem is well defined initially as follows:

Minimize f ðxÞ subject to xi AXi, i¼ 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 ðxÞ ¼ aM3

þbN2þgL. The size of the harmony memory (HMS)

representing the number of solution vectors and the harmonymemory considering rate (HMCR) is initialized.

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

G.A. Ezhilarasi, K.S. Swarup / Swarm and Evolutionary Computation 7 (2012) 1–64

follows:

HM¼

N11 N1

2 � � � N1n�1 N1

n

N21 N2

2 � � � N2n�1 N2

n

^ ^ ^ ^ ^

NHMS�11 NHMS�1

2 � � � NHMS�1n�1 NHMS�1

n

NHMS1 NHMS

2 � � � NHMSn�1 NHMS

n

0BBBBBBB@

1CCCCCCCA

ð2Þ

Step3: Improvise the new harmony from the HM. Improvisationis the process of creating a new harmony from the existingmemory. A new harmony vector X0 ¼ ðN1

0,N10, . . . ,Nn

0Þ is generated

based on three rules: (a) memory consideration, (b) randomselection and (c) KL strategy. The value of the first decisionvariable is chosen from one of the harmonies betweenN1

1 and NHMS1 . The other decision variables in the harmony mem-

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

Ni0¼

Ni0AfN1

i ,N2i , . . . ,NHMS

i g with probability HMCR,

Ni0AXi with probability ð1 - HMCRÞ:

(ð3Þ

Now the new harmony vector is pitch adjusted using KL strategy.According to the KL strategy a pair of nodes between partitions are

1

2

3

4

5

6

7

8

9

10

1112

13

14

15

16 17

18

19

20

21

29

52

53

54

55

Fig. 4. Partitioned clusters of IEEE 57 Bus system.

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.

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 find the fitness 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 ¼ ðx1

0,x10, . . . ,xn

0Þ

is 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 thefitness value of the harmony.

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 flowchart in Fig. 1.

3.1. Cluster connectivity

The graph traversing is mainly used in two major applicationsin power system namely, topological problems and network flow

22

23

24

2526

27

28

30

31

32

33

34

35

36

37

38 39

40

41

4243

44

45

46

47

48

49

50

51

56

57

(a) 57 Bus—Cluster 1. (b) 57 Bus—Cluster 2.

Fig. 5. Partitioned clusters of IEEE 118 Bus system. (a) 118 Bus—Cluster 1. (b) 118 Bus—Cluster 2.

G.A. Ezhilarasi, K.S. Swarup / Swarm and Evolutionary Computation 7 (2012) 1–6 5

G.A. Ezhilarasi, K.S. Swarup / Swarm and Evolutionary Computation 7 (2012) 1–66

problems. The topological or structural problems deals 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 of thenetwork.The network flow problems include solving the shortestpath problem, finding feasible optimal flow pattern and recogniz-ing the loop flows and wheeling problems.

Breadth first search (BFS) and depth first search (DFS) are thetwo techniques in graph theory having fundamentally differenttraversal techniques. While in BFS a node is fully explored beforethe exploration of any other node begins, in DFS the leaves of thespanning trees are reached in the fastest possible way. Depth firstsearch requires less memory since only the nodes on the currentpath are stored, whereas in breadth first search all the nodes thathave been generated so far has to be stored. If there a is completetraversal possible, BFS guarantees to find it, by the fact that longerpaths are never explored until all shorter ones have beenexamined. This contradicts with DFS, which finds a long path toa solution from one part of the tree, when a shorter path exists insome other unexplored part of the tree. In this paper BFS is usedfor the graph traversal and the visited nodes are removed fromthe initial open list. If the open list is empty at the end the clusteris connected, other wise there are isolated nodes in the cluster.This ensures that the partitioned cluster is completely observable.

4. Simulation results

To study the effectiveness of the proposed method of cluster-ing, simulation was done using the IEEE standard test systems[28]. Parameters of the HS algorithm were set as follows:harmony memory size¼100, number of improvisations¼100,number of trials¼50, harmony memory considering rate¼0.8.

The decomposition details of test systems are shown inTable 1.

The performance of the HS algorithm is verified by means ofthe convergence characteristics as is shown in Fig. 2 for the IEEE57 bus system and IEEE 118 bus system. The initial and finalharmony memories are shown for the various test cases. Thisindicates the randomness in the initial harmony and the hier-archical clustering in the final harmony memory as shown inFig. 3 for 118 bus system.

The partitioned clusters of IEEE 57 bus and IEEE 118 bussystem is shown in Figs. 4 and 5 respectively. The connectivity ofthe clusters is verified by the graph traversing techniques such asbreadth first search and depth first search and the clusters arefound to contain no isolated nodes.

5. Conclusion

A novel Kernighan–Lin aided harmony search algorithm isproposed to solve the network decomposition problem of large-scale power systems. Network decomposition is formulated as anoptimization problem whose objective is to create well balancedclusters with the minimum number of interconnections betweenthem. This is done for the implementation of distributed comput-ing of power systems applications like load flow, optimal powerflow and state estimation. The partitioned clusters together withthe decomposition and coordination methods will aid the appli-cation of distributed computing and parallel processing in powersystems. The communication of data between the clusters will bethe only overhead of such applications in real time. This approachcan be easily implemented in a decentralized energy manage-ment systems of very large-scale distributed power networks.

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