Electrical Power and Energy Systems 78 (2016) 96–107

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Electrical Power and Energy Systems

journal homepage: www.elsevier .com/locate / i jepes

Power distribution network reconfiguration for power loss minimizationusing novel dynamic fuzzy c-means (dFCM) clustering based ANNapproach

http://dx.doi.org/10.1016/j.ijepes.2015.11.0770142-0615/� 2015 Elsevier Ltd. All rights reserved.

⇑ Tel./fax: +30 210 7722018.E-mail address: h4477@hotmail.com

Hassan Fathabadi ⇑School of Electrical and Computer Engineering, National Technical University of Athens (NTUA), Greece

a r t i c l e i n f o

Article history:Received 18 August 2014Received in revised form 11 May 2015Accepted 25 November 2015Available online 17 December 2015

Keywords:Distribution network reconfigurationPower loss minimizationDynamic fuzzy c-means (dFCM)Clustering techniqueArtificial neural network (ANN)

a b s t r a c t

In this study, a three-layer artificial neural network (ANN) is proposed to reconfigure power distributionnetworks to obtain the optimal configuration in which the active power loss is minimal. Then, the pro-posed ANN is reduced in size by transforming the input space with kernels using a proposed modifieddynamic fuzzy c-means (dFCM) clustering algorithm to obtain a novel framework. The proposed frame-work and ANN both are implemented on the two IEEE 33-bus and IEEE 69-bus power distribution net-works. The ANN and framework both are trained using the training set consisting of only 64 trainingsamples. The simulated results are compared to the results obtained by performing a selected traditionalmethod which is the switching algorithm. The comparative results explicitly verify that using theproposed framework for distribution networks reconfiguration has some benefits such as a very shortprocess time that is far shorter than the others, a very simple structure including only a minimal numberof neurons and higher accuracy compared to the others. These features show that the proposed frame-work can be effectively used for real-time reconfiguration of power distribution networks.

� 2015 Elsevier Ltd. All rights reserved.

Introduction

A power distribution network and a transmission system arethe two important parts of an electric power generation and distri-bution system. The power loss in the distribution network is morethan that in the transmission system because the currents avail-able in the distribution part are generally much greater than thatin the transmission part. In electric power generation and distribu-tion systems, about 10% of the produced electric power is lost indistribution networks, so minimizing the electric power loss isone of the important problems related to electric power generationand distribution systems [1,2]. In practice, there are two methodsfor minimizing the power loss in a distribution network. The twomethods are the reconfiguration of the distribution network andcapacitors placement. The reconfiguration of distribution networkscan be also adopted to achieve the other goals such as better volt-age profile and better charge balance [3,4]. The limitations of apower distribution network such as radial structure, the capacityof the feeders and the acceptable voltage range of different busesshould be practically satisfied for the reconfigured network. In fact,power distribution networks reconfiguration is one of the impor-

tant problems related to the power systems, so that, there aremany recent researches addressing this issue [5–9]. For the distri-bution networks having a large number of the power switches, thereconfiguration is a multi-objective issue including a non-linearmapping between the input data and the desirable outputs[10,11]. The algorithms presented in the literature for reconfigur-ing the distribution networks can be divided into the several cate-gories including mathematical optimization methods, switchexchange methods, optimized flow pattern (OFP), and artificialintelligence algorithms [12,13]. A simple method which uses thebranches of the network graph and their limitations for networkreconfiguration was reported in [1]. A summarized version of thementioned method was presented in [2]. The summarized methoddetects the feeding path of each charge, and then, a simple sub-treeis used for each path reconfiguration. The defect of the mathemat-ical optimization techniques is to consume a long time for calcula-tion, so when these methods are implemented on a realdistribution network, increase in the size of the network leads toa serious problem. The switch exchange method (SEM) was intro-duced in 1988 [14]. The method estimates the power loss in eachstate of the positions of the power switches. The OFP is an innova-tive method which was introduced for the first time by Shirmo-hammadi in 1989. This method is also known as sequentialswitch opening method (SSOM). Application of genetic algorithm

H. Fathabadi / Electrical Power and Energy Systems 78 (2016) 96–107 97

(GA) for the reconfiguration of power distribution networks wasfirst reported in [15]. A research about providing load patterns,and then carrying out the feeders reconfiguration using patternsdetection was presented by Hoyong et al. [16]. A similar ANNapproach for power network reconfiguration was proposed in[17]. Hopfield network was used for the reconfiguration of distri-bution networks by Tang et al. [18]. The major defect of the meth-ods presented in [17,18] is that they can be implemented only forsmall size distribution networks. The process proposed by Hoyonget al. [16] together with classifying the loads into residential, com-mercial and industrial types was used for networks reconfigurationin [19].

The clustering techniques are used to classify different sets ofphysical parameters and events [20,21]. A number of clusteringtechniques such as local maxima search and search neighborhoodsare defined and reported in the literature [21]. Some clusteringtechniques such as deterministic annealing intensively dependon the data pattern while some other techniques such as genericclustering algorithm do not have this defect. Clustering techniquessuch as connected-cell search and k-means clustering are called‘‘hard” because they determine and assign a data point to a cluster.The assigned data point either lies in a cluster or not, so the clus-ters which have overlapping area cannot be effectively distin-guished [21]. To address this defect, fuzzy clustering techniqueswere presented. In a fuzzy clustering, data points are representedby a membership degree which indicates the dependence of a datapoint to a cluster. Thus, a data point may simultaneously lie inmore than one cluster, so an affective detection of overlappingclusters can be performed [21]. An important type of the fuzzyclustering techniques is called fuzzy c-means (FCM) [22–24]. Amodified version of the FCM algorithm in which the clusters aredynamically found was presented in [21]. The modified FCM whichhas high capability for specifying the non-uniformly distributedclusters is called dynamic FCM (dFCM).

A survey in the literature shows that there are other types offuzzy clustering dynamic algorithms that inside evolving systemssuch as dynamically evolving clustering (DEC) [25], hyper-ellipsoidal clustering for evolving data stream (HECES) [26], onlineevolving fuzzy clustering algorithm based on maximum likelihoodestimator [27], density-based clustering for evolving uncertaindata stream [28], evolving soft subspace clustering [29], evolvingclustering method (ECM) [30] and adaptive learning evolving clus-tering method (ALECM) [30]. DEC uses cluster weight and distancebefore generating new clusters that is unlike other approaches thatconsider either the data density or distance from existing clustercenters [25]. In HECES, sliding window model is used to handleincoming stream of data to minimize the impact of the obsoleteinformation on recent clustering results, and shrinkage techniqueis used to avoid the singularity issue in finding the covariance ofcorrelated data [26]. In the algorithm proposed in [27], the distancefrom a point to center of the cluster is computed by maximum like-lihood similarity of data. The density-based algorithm presented in[28] gives a method for discovering clusters in evolving uncertaindata stream, and probability distance was introduced as a similar-ity measure. The evolving soft subspace clustering proposed in [29]leverages on the effectiveness of online learning scheme and scal-able clustering methods for streaming data by revealing the impor-tant local subspace characteristics of high dimensional data. ECM isa kind of efficient online clustering method, which evolved theclusters automatically from data streams. It is a distance- andprototype-based clustering method. The distance of a new incom-ing sample to the closest cluster center cannot be larger than athreshold value; otherwise a new cluster is evolved [30]. Firstdefect of ECM is that when performing incremental learning fromscratch, it is quite not appropriate to set the predefined thresholdfor a good performing adaptation. As second defect, ECM is quite

sensitive to different data orders. To overcome the two mentioneddefects, ALECM was proposed in [30] that uses the on-line learningcapability by adjusting and evolving the clusters automaticallywith new incoming samples.

There are also some researches on ANN pruning reported in theliterature. Self-adaptive evolutionary constructive and pruningalgorithm (SAECPA) that is a structural algorithm was reported in[31]. SAECPA considers an ANN in which one hidden neuron islinked towards single input node, then using cluster pruning (CP)and survival selection (SS) the ANN is pruned. Another method thatuses equation synthesis and correlated activation pruning (CAPing)was introduced in [32]. Equation synthesis involves the incremen-tal increase in the number of connections of the trained ANN untilsatisfactory prediction is achieved. CAPing involves the identifica-tion of nodes that have similar effects on the desired output. Com-parison of the inputs to these nodes can lead to useful dependencyrelationships. A method for designing ANNs for prediction prob-lems based on an evolutionary constructive and pruning algorithm(ECPA) was also proposed in [33]. The proposed ECPA begins with aset of ANNs with the simplest possible structure, one hidden neu-ron connected to an input node, and employs crossover and muta-tion operators to increase the complexity of an ANN population.Additionally, cluster-based pruning (CBP) and age-based survivalselection (ABSS) were proposed as two new operators for ANNpruning. The CBP operator retains significant neurons and prunesinsignificant neurons on a probability basis and therefore preventsthe exponential growth of an ANN [33].

In this study, an ANN is proposed for distribution networksreconfiguration to obtain the optimal configuration in which theactive power loss is minimal. Then, the proposed ANN is reducedin size by transforming the input space with kernels using a pro-posed modified dFCM clustering algorithm to obtain a novel frame-work. The proposed framework and ANN both are implemented ontwo power distribution networks. The simulated results are com-pared to the results obtained by performing the switching algorithm[34]. The comparative results explicitly show that the proposedframework has higher performance compared to the others.

This paper is organized as follows. Fuzzy clustering and the FCMalgorithm are discussed in Section ‘‘Fuzzy clustering and FCM algo-rithm”. Section ‘‘Dynamic fuzzy c-means algorithm and clustervalidity” deals with the proposed dFCM algorithm and the conceptsof cluster validity. Distribution network reconfiguration is for-mulized in Section ‘‘Distribution network reconfiguration and theproposed ANN and framework” and the proposed ANN and frame-work are presented. Simulated results of implementing the ANNand framework on two distribution networks are presented in Section ‘‘Simulated results”. Finally, Section ‘‘Conclusion” concludesthe paper.

Fuzzy clustering and FCM algorithm

Clustering is the process of grouping or dividing a series of datafrom unlabeled patterns into a number of groups which are calledclusters, so that, the similar patterns are allocated to one cluster.Each pattern can be shown with a vector which has differentparameters and properties. Clustering technique includes twobasic criteria which are adjacency measurement and grouping.Adjacency measurement shows the similarity between two pointsand grouping is used to find an appropriate target function and therelated algorithm. Each clustering method determines the similar-ity between the patterns by calculating the distance between therelated patterns. For patterns with metric properties, differenttypes of distance measurement such as Euclid distance or Maha-lanobis can be used [21]. Fuzzy clustering is a technical methodto allocate data points to different clusters using fuzzy logic which

98 H. Fathabadi / Electrical Power and Energy Systems 78 (2016) 96–107

provides effective means for separating overlapping clusters. Fuzzyclustering is more appropriate for the applications which have con-tinuous or overlapping profiles [21]. The most common fuzzy clus-tering algorithm is fuzzy c-means (FCM) which is a k-meansalgorithm that uses fuzzy logic to determine the association of adata point to a cluster [22–24,35]. The association to a cluster isdetermined by calculating the inverse distance to the cluster cen-ter. The cluster centers determined by FCM directly depend onthe geometric locations of the data points on the plane or space.Some applications of the FCM algorithm for tracking gamma raysand detecting ions have been reported in [36,37], respectively.

In the FCM algorithm, an objective function which should beminimized is considered as:

FðY; Z;a;XÞ ¼Xn

k¼1

Xmi¼1

yikð Þa xk � zik k2 ð1Þ

where a is the fuzzy factor, m is the number of clusters,Z ¼ z1; z2; . . . ; zmð ÞT is cluster center vector consisting of the centersof the m clusters, n is the number of the data points, X ¼x1; x2; . . . ; xnð ÞT is the data points vector, Y ¼ yik½ �m�n is the member-shipmatrix consisting of themembership yik which shows themem-bership of xk in the ith cluster, and k � k shows the Euclidean distance

norm (kVk ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiVT � V

p). The fuzzy factor a is used to normalize and

fuzzify the memberships the sum of which should be equal to 1.Minimization of FðY; Z;a;XÞ is carried out through an iterative tech-niques such as alternating optimization (AO) [23]. When a > 1, anoptimal solution which minimizes FðY; Z;a;XÞ is found as [23]:

yik ¼Xmj¼1

kxk � zikkxk � zjk

� �2=ða�1Þ" #�1

ð2Þ

where 1 6 i 6 m, 1 6 k 6 n, and the center of the ith cluster isobtained as:

zi ¼Pn

k¼1 yikð ÞaxkPnk¼1 yikð Þa ð3Þ

After clustering the data, a validity index is used to show how wellthe data have been clustered. There are different validity indexessuch as Xie–Beni index and modified partition coefficient (PCC)index [24,36,38]. In fact, all indexes present a numerical aspect todetermine how well the data have been clustered.

Dynamic fuzzy c-means algorithm and cluster validity

As shown in Eq. (1), a drawback of the FCM algorithm is thatclustering significantly depends on the fuzzy factor a whichexplicitly varies from one data set to other [21]. Another draw-back of the FCM clustering algorithm is that it deals with outlierssame as data points to put them in the data bulk, so some mod-ifications have been made to improve the FCM algorithm over theyears. Using the suppressed FCM algorithm which holds the big-gest memberships in high regard and suppresses the other mem-berships was a solution to decrease the two drawbacks [39]. TheFCM clustering algorithm uses the Euclidean distance betweendata samples, so there is equal importance for each data pointand each dimension which refers to a feature. To address thisconcern and to improve the FCM algorithm, using feature-weight learning in the FCM algorithm was proposed [21]. A mod-ified version of the FCM algorithm which is called dynamic fuzzyc-means (dFCM) was presented in [21]. The dFCM clustering tech-nique is more suitable for the applications including online anal-

ysis of incoming data in which the process needs adaptiveinformation or the incoming data are not uniform. An applicationof the dFCM clustering technique for calorimetric data recon-struction in high-energy physics was reported in [21]. The dFCMclustering is a general technique which can be applied to a largenumber of different applications [40,41]. The dFCM clusteringtechnique is more suitable for the applications including onlineanalysis of incoming data in which the process needs adaptiveinformation or the incoming data are not uniform. On the otherhand, in a power distribution network, incoming data should beanalyzed online. Furthermore, the input data (load flows) arenot uniform and adaptive information is needed to adapt the loadflows to the practical load patterns to obtain the optimal config-uration based on the least mismatching between the instant inputload flow and one of the practical pattern. Thus, in this study, thedFCM clustering technique is first modified to make it appropriateto use in a power distribution network, and then it is used for dis-tribution networks reconfiguration to reduce the active powerlosses.

Dynamic fuzzy c-means algorithm

The dFCM algorithm dynamically finds clusters, and further-more, it deletes and regenerates clusters if it is necessary whenthe incoming data flow for clustering. It fits the data pattern con-tinuously, and the clusters are selected using a validity index. Amodified version of the dFCM clustering technique presented in[21] is proposed in this study. The flow chart of the proposed dFCMclustering algorithm is shown in Fig. 1. The proposed dFCM cluster-ing algorithm can be summarized as follows:

(1) Membership threshold (ythr:) is defined as the maximumacceptable level for the memberships and FCM error (EFCM)is also defined as the maximum acceptable differencebetween the two clusters centers obtained in the twosequential steps using the FCM algorithm. At first, thereare a few of the incoming data points, so the incoming datarange, the membership threshold, the FCM error and theboundary of the clusters number (m) are estimated.

(2) The m clusters centers are uniformly located in the inputspace, and the memberships of the initial data points are cal-culated using Eq. (2).

(3) For a new incoming data point, its memberships in the exist-ing clusters are calculated using Eq. (2). If the maximummembership is greater than or equal to the membershipthreshold, it means that the data point belongs to at leastone of the clusters, and consequently, an update is necessaryto modify the membership threshold and the clusterscenters.

(4) It is assumed that m is the clusters number obtained by theiterative process, so that, the maximum membership of thedata point is less than ythr:. To check whether a better selec-tion is available or not?, the validity of the m obtained clus-ter centers is compared with the validities of the cases ofm � 1 and m + 1 available clusters, and then the cluster cen-ters that provide a better validity index are selected. Ifm � 1or m + 1 clusters provide better validity index, then m � 1 orm + 1 clusters centers are created using the new data points.

(5) The dFCM process ends when there are not any new incom-ing data.

Evaluating cluster validity

It is assumed that Znew and Zold are the new and old cluster cen-ter vectors, respectively. Now, a condition is defined as:

Fig. 1. The flow chart of the dynamic FCM (dFCM) algorithm.

H. Fathabadi / Electrical Power and Energy Systems 78 (2016) 96–107 99

kZnew � Zoldk ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiXm

i¼1zi�new � zi�oldð Þ2

q> ythr: ð4Þ

If the above condition is satisfied, then evaluating the clustervalidity is necessary. In other words, if the distance between thenew and old cluster center vectors is greater than ythr:, then thecluster validity is checked. In this case, when a new data pointarrives, regardless of its membership, the cluster centers are

updated using the alternating optimization (AO). After updatingthe cluster centers, ythr: is used to evaluate whether the new clustercenters are significantly different from the old cluster centers ornot (inequality (4) is satisfied or not)? Thus, the dFCM processdecides whether a new cluster validity check is necessary or not?It is obvious that using the condition expressed by inequality (4)effectively reduces the calculations of the proposed dFCM

100 H. Fathabadi / Electrical Power and Energy Systems 78 (2016) 96–107

clustering process. The selected value for the membership thresholddepends on the application type which uses the dFCM clustering. Itis clear that the inequality (4) prevents the check of the clustervalidity when a new a data point arrives. In fact, if the new datapoint belongs to a specified acceptable neighbor of a cluster, thenit is not checked that whether the other clusters are better or not?

Validity index

After clustering, a validity index is used to determine how wellthe data have been represented by the obtained clusters. Differentvalidity indices have been defined and reported in the literature[24]. The Xie–Beni validity index is one of the best validity indexeswhich is widely used because it depends on not only membershipsbut also geometric distances. In this study, the Xie–Beni index ischosen to check how well the incoming data have been clustered.The Xie–Beni index is defined as:

I:VXBðY; Z;XÞ ¼Pn

k¼1

Pmi¼1 yikð Þ2kxk � zik2

n:ðMini–jfkzi � zjkg

¼ FðY ; Z;2;XÞn � ðMini–jfkzi � zjkg ð5Þ

where I:VXBðY; Z;XÞ is the Xie–Beni index. This index indicates aratio of the total variation of the cluster centers together with thememberships in the obtained clusters to the distance between thecluster centers, so minimizing the Xie–Beni index results a betterclustering. In fact, a minimum amount of the Xie–Beni index showsa larger separation between the clusters together with the moreconcentration of the data points around the related cluster centers,and thus the obtained clusters have been perfectly selected. Eq. (5)shows that there is not any upper bound for the Xie–Beni index.

Distribution network reconfiguration and the proposed ANNand framework

The most important goal of the reconfiguration of power distri-bution systems is to determine the topology in which the activepower loss is the least possible amount [19]. The active power lossof a power distribution system consisting of N buses is expressedas:

PLoss ¼XNi¼1i–j

XNj¼1

rijjIijj2 ð6Þ

where rij is the total ohmic resistance of the branch i � j, and Iij isthe electrical current flowing between ith and jth buses. The totalactive power (PTot) distributed on the power distribution system isobtained as:

PTot ¼XNi¼1

pi þ PLoss ð7Þ

where pi is the active power absorbed by ith bus. Similarly, the totalreactive power (QTot) distributed on the power distribution systemcan be also found as:

QTot ¼XNi¼1

qi þXNi¼1i–j

XNj¼1

xijjIijj2 ð8Þ

where xij is the total reactance of the branch i � j, qi is the reactivepower absorbed by ith bus. The goal is to find an ANN which deter-mines the optimal configuration of the power distribution system inwhich the active power loss is minimal. The limitations are that thereconfigured distribution network should be a radial network with

an acceptable range of ½0:95 p:u:;1:05 p:u:� for the voltage profile.In a power distribution network, the instant active and reactivepowers absorbed by each bus of the network continuously and per-petually changes according to the load demand, so it is inevitable tosimultaneously apply the instant active and reactive powers of allthe buseswhich is called ‘‘load flow” to the ANN or framework. Thus,the input vector of the proposed ANN is ðp1; q1Þ; ðp2; q2Þ; . . . ; ðpi; qiÞ;½. . . ; ðpN; qNÞ�T and the output of the ANN is the determined optimalconfiguration. It is clear that there is an intensive non-linear relationbetween the inputs and outputs of the ANN. In this study, the three-layer perceptron neural network shown in Fig. 2 has been used toperform the mentioned nonlinear mapping.

Without using dFCM clustering technique, the proposed neuralnetwork should have 2N neurons in the input layer where N is thenumber of buses because the input vector ( ðp1; q1Þ; ðp2; q2Þ; . . . ;½ðpi; qiÞ; . . . ; ðpN; qNÞ�T) consists of 2N elements. There are also K neu-rons in the hidden layer and C neurons in the output layer asshown in Fig. 2, where C is the number of the optimal distributionnetwork configurations obtained by applying the switching algo-rithm, and K is selected, so that, the desired minimum amount ofthe sum of the squares of errors and minimum process time canbe achieved after training the proposed ANN using the ‘‘BatchLearning-LMS algorithm” [42,43]. After training, the jth neuron ofthe output layer can only produce 0 or 1, where ‘‘1” means thatthe jth optimal configuration obtained by performing the switch-ing algorithm [34] has been chosen as the optimal structure bythe proposed ANN. All the biases of the hidden and output layersand also all the weights of the input layer have been chosen oneto reduce the training time. In fact, the weight coefficients matrixesof the hidden and output layers are determined by training theproposed three-layer ANN. It is clear that the neurons number ofthe proposed ANN significantly increases when the buses numberof the distribution network increases. For example, for the IEEE 33-bus network, the neurons number of the entrance layer is 66, andfor the IEEE 69-bus network, it is 138. Increasing in the neuronsnumber of the ANN results a longer training time, a considerablereduction in the convergence speed of the ANN, and more difficultyin implementation of the proposed ANN [19].

The neurons number of the proposed three-layer feed-forwardANN can be effectively reduced using the dFCM clustering tech-nique. In fact, the neurons number of the input layer decreasesto 2m, where m is the clusters number explained in Eqs. (1) and(2). Sincem << N, the proposed ANN is simplified to a new versionincluding much less neurons. The final framework including thethree-layer feed-forward ANN reduced in size using the proposeddFCM clustering technique is shown in Fig. 3.

In a power distribution network, the number of load levels isdivided into Q levels based on the maximum demand in the net-work [16]. The network buses are also divided into l types such

as residential and commercial, so there are totally Ql combinations(load patterns) for the available load levels. It is clear that the com-binations number is equal to the elements number of the trainingset. For instance, if two load levels are only considered as 100% and60% of the full load (Q ¼ 2), and the network buses are also dividedinto three types (l ¼ 3) consisting of residential, commercial andindustrial types, then the training set consisting of 23 ¼ 8 elementsis expressed as:

Training Set ¼ fðComb:1; Conf :1Þ; ðComb:2; Conf :2Þ; . . . ;ðComb:8; Conf :8Þg ð9Þ

where Comb:j and Conf :j are the jth load pattern and the relatedoptimal distribution network configuration in which the activepower loss is minimal, respectively.

Fig. 2. Proposed three-layered perceptron neural network without using dFCM clustering technique.

Fig. 3. The proposed framework including a three-layer feed-forward ANN reduced in size using the proposed dFCM clustering technique.

H. Fathabadi / Electrical Power and Energy Systems 78 (2016) 96–107 101

Simulated results

In this section, to validate the theoretical results and to checkthe performance of the proposed ANN and framework, two IEEE33-bus and IEEE 69-bus distribution networks have been consid-ered. For the each network, the reconfiguration has been carriedout by applying both the proposed ANN shown in Fig. 2 and theproposed framework shown in Fig. 3. The results have been com-pared to each other to verify the benefits of using the proposeddFCM clustering technique.

IEEE 33-bus distribution network

The IEEE 33-bus distribution network consisting of 33 buses isshown in Fig. 4. Other detailed specifications of the IEEE 33-busdistribution network have been reported in previously publishedarticles such as [44]. Based on subscribers’ demand the load typesare divided into three types consisting of residential, commercialand industrial loads (l ¼ 3). According to the statistical datareported for the practical distribution networks [19,35], four prac-tical load levels have been also considered as 100%, 82%, 61% and39% of the full load (Q ¼ 4). Thus, there are 43 ¼ 64 load patterns,

and the training set consists of 64 elements. In this study, theswitching algorithm presented in [34] has been used for specifyingthe optimal network configuration for the each combination of the64 available load levels. For the each load pattern, the switchingalgorithm determines the related optimal network configuration.The limitation is that the determined optimal configuration shouldbe a radial network with an acceptable range of ½0:95 p:u:;1:05 p:u:�for the voltage profile. In practice, a number of the optimal net-work configurations obtained for different load patterns are same,so the number of all the obtained optimal network configurationsis less than 64. As mentioned before, the number of the distinctoptimal network configurations is equal to the neurons numberin the output layers of the proposed ANN shown in Fig. 2 and theproposed framework shown in Fig. 3. The switching algorithm isa conventional method that is widely used in power dispatchingcenters yet. It works based on the following sequential steps:

(1) For a present load flow available in a distribution network, itfinds all possible configurations (sets of ‘‘closed” and ‘‘open”switches) of the distribution network by setting each switchto ‘‘closed” or ‘‘open” status.

(2) Each obtained configuration that is not a radial network isrejected.

Fig. 4. IEEE 33-bus distribution network.

Table 1Optimal configurations of the IEEE 33-bus distribution network related to thedifferent load patterns.

Optimalconfigurationnumber

Load pattern numbers Power switchesthat should beopened

1 1, 5, 10, 17, 26, 29, 32, 36, 37, 39, 42,49, 52, 53, 56, 59, 60, 63

S5, S7, S12, S29,S34

2 2, 8, 9, 12, 13, 35, 38, 41, 48, 49, 51,57, 60

S5, S7, S12, S25,S29

3 3, 11, 15, 16, 47, 48, 50, 57 S5, S7, S12, S25,S33

4 4, 14, 44, 58 S7, S12, S25, S30,S33

5 7, 18, 19, 20, 21, 22, 23, 24, 25, 27, 28,30, 31, 33, 34, 40, 43, 54, 55, 62

S5, S7, S12, S28,S34

6 15, 16 S5, S7, S12, S15,S25

7 36, 49 S5, S8, S12, S29,S34

8 45, 46, 61, 64 S7, S12, S25, S29,S30

102 H. Fathabadi / Electrical Power and Energy Systems 78 (2016) 96–107

(3) Each obtained configuration that has not an acceptablerange of voltage profile (in this study, in the range of½0:95 p:u:;1:05 p:u:�) among all the buses of the distributionnetwork is rejected.

(4) Now, between the remaining configurations, the configura-tion(s) that results the least active power loss is (are)selected as optimal configuration(s).

The main defect of the switching algorithm is that it needs along time to find the optimal configuration(s). For instant and aswill be shown in this study, using a 2.2 GHz processor the switch-ing algorithm needs 44.24 s and 89.33 s to find an optimal config-uration of the very simple IEEE 33-bus and IEEE 69-busdistribution networks based on a given load flow. It can be summa-rized that, the switching algorithm finds the optimal configurationwith a considerable time delay based on a load flow occurring inpast time, so the optimal configuration determined by the switch-ing algorithm is not reliable and cannot be chosen as an optimalconfiguration for the distribution network at any time because ofprobable variation in the load flow during the mentioned delaytime. For instant, the load flow in the IEEE 33-bus distribution net-work may vary during 44.24 s, so the optimal configurationobtained by the switching algorithm is just valid for 44.24 s ago,not now.

For the 64 available load patterns of the IEEE 33-bus distributionnetwork, 8 distinct optimal network configurations have beenobtained by performing the switching algorithm, and thus the neu-rons numbers in the output layers of the proposed ANN and frame-work are eight (C = 8). The obtained optimal networkconfigurations are presented in Table 1. Since N = 33 and C = 8,the size of the proposed ANN can be determined as follows. Theneurons number of the input layer is 2N = 66 and the neurons num-ber of the output layer is C = 8, so the weights matrix of the hiddenlayer consisting of K neurons is defined as:

W2 ¼

w1;1 w1;2 . . . w1;66

w2;1 w2;2 . . . w2;66

. . . . . . . . . . . .

wK;1 wK;2 . . . wK;66

26664

37775 ð10Þ

Similarly, the weights matrix of the output layer consisting of 8neurons is expressed as:

W3 ¼

w01;1 w0

1;2 . . . w01;K

w02;1 w0

2;2 . . . w02;K

. . . . . . . . . . . .

w08;1 w0

8;2 . . . w08;K

26664

37775 ð11Þ

The weights number that should be determined by training theANN is 66K þ 8K ¼ 74K. Since the training set has 64 elements, theleast possible amount of K which sets the sum of the squares of theerrors equal to zero is one (K = 1). Thus, the weights matrixesexpressed by Eqs. (10) and (11) can be simplified as:

W2 ¼ w1;1 w1;2 . . . w1;66½ � ð12Þand

W3 ¼

w01;1

w02;1

. . .

w08;1

26664

37775 ð13Þ

Based on the above explanation, it can be summarized that theneurons number of the input, hidden and output layers of the pro-posed ANN are 66, 1, and 8, respectively. The ANN proposed foranalyzing the IEEE 33-bus distribution network is shown with allthe specifics in Fig. 5. It can be seen that the number of the optimalnetwork configurations listed in Table 1 is equal to the neuronsnumber of the output layer. After training, each neuron of outputlayer can only produce 0 or 1. For a specific load pattern, if thejth neuron becomes 1, this means that the jth optimal distributionnetwork configuration listed in Table 1 has been selected as theoptimal structure by the proposed ANN. If more than one neuronbecome 1, for instances, the ith and jth neurons both become 1,it means that the ith and jth optimal distribution network config-urations listed in Table 1 have been selected as the optimalstructures.

Using the dFCM clustering technique, the proposed frameworkshown in Fig. 3 can be obtained by reducing the size of the ANNshown in Fig. 5. As mentioned, there are 64 available load patterns.The first load pattern (load pattern #1) in which all the 33 buseshave the full load (100%) is shown in Fig. 6. The buses belong tothe three load types consisting of residential, commercial andindustrial loads. The load of the each bus is shown with a red point.Now, by applying the proposed dFCM clustering algorithm, the

Fig. 5. Proposed ANN for analyzing the IEEE 33-bus distribution network.

Fig. 6. 33 buses (red points) and the clusters centers (green points). (For interpretation of the references to color in this figure legend, the reader is referred to the web versionof this article.)

Fig. 7. Calculated Xie–Beni index for the IEEE 33-bus distribution network.

H. Fathabadi / Electrical Power and Energy Systems 78 (2016) 96–107 103

amount of the Xie–Beni index, the clusters number and the clusterscenters are obtained. The calculated Xie–Beni index is shown inFig. 7. Based on the amount of the Xie–Beni index, the clustersnumbers have been obtained as 2, 3 and 2 for residential, commer-cial and industrial loads, respectively, and then, the clusters cen-ters have been calculated. The clusters centers are shown withgreen points in Fig. 6. The number of the clusters is 7 (m = 7), sothe neurons number of the input layer of the proposed framework

is 2m = 14, and the neurons number of the output layer is C = 8.Thus, the weights matrix of the hidden layer consisting of L neu-rons is defined as:

W2 ¼

�w1;1 �w1;2 . . . �w1;14

�w2;1 �w2;2 . . . �w2;14

. . . . . . . . . . . .

�wL;1 �wL;2 . . . �wL;14

26664

37775 ð14Þ

Fig. 8. Proposed framework for analyzing the IEEE 33-bus distribution network.

Table 2Simulation results of the test samples for the IEEE 33-bus distribution network.

Method Number ofthe testsamples(loadpatterns)

Number of theobtained optimalconfigurationswhich are same asthe results of theswitching algorithm

Averageprocessingtime using a2.2 GHzprocessor (s)

Proposed ANN (K = 1) 100 94 1.29Proposed ANN (K = 2) 100 95 1.43Proposed framework 100 99 0.37Switching algorithm 100 100 44.24

Fig. 9. IEEE 69-bus distribution network.

104 H. Fathabadi / Electrical Power and Energy Systems 78 (2016) 96–107

Similarly, the weights matrix of the output layer consisting of 8neurons is expressed as:

W3 ¼

w001;1 w00

1;2 . . . w001;L

w002;1 w00

2;2 . . . w002;L

. . . . . . . . . . . .

w008;1 w00

8;2 . . . w008;L

26664

37775 ð15Þ

The weights number that should be determined by training theframework is 14Lþ 8L ¼ 22L. Since the training set has 64 ele-ments, the least possible amount of L which sets the sum of thesquares of the errors equal to zero is three (L = 3). It can be summa-rized that the neurons number of the input, hidden and output lay-ers of the proposed framework are 14, 3, and 8, respectively. Theframework presented for analyzing the IEEE 33-bus distributionnetwork is shown in Fig. 8.

After training the ANN shown in Fig. 5 and the frameworkshown in Fig. 8, to check whether the proposed schemes are suc-cessful to determine the optimal configurations of the IEEE 33-bus distribution network, 100 load patterns have been used as testsamples. 80 load patterns are out of the training set and 20 pat-terns have been chosen from the training set. The optimal networkconfigurations obtained using the proposed ANN (for K = 1 andK = 2) and framework have been compared to the results obtainedby applying the switching algorithm. The comparative resultsincluding the average process time using a 2.2 GHz processor foreach input load pattern are reported in Table 2. The results listedin Table 2 shows that there are 6 and 5 distinct answers betweenthe proposed ANN and the switch algorithm while there is onlyone distinct answer between the optimal configurations specifiedby the proposed framework and the switch algorithm. It is worth-while to note that the only distinct structure specified by the pro-posed framework also belongs to the set of the three configurationswhich have the least active power loss. For the proposed frame-work, the average process time of the optimal configuration deter-mination for each input load pattern is only 0.37 s while it is 1.29 sand 44.24 s for the proposed ANN and the switch algorithm,respectively. The comparative results shown in Table 2 explicitlyverify the excellent performance of the proposed framework.

IEEE 69-bus distribution network

IEEE 69-bus distribution network has been considered asanother distribution network for implementing the proposedANN and framework. The IEEE 69-bus distribution network is a12.66 kV radial distribution system with 69 buses which is shownin Fig. 9. Similar to IEEE 33-bus distribution network, the loadtypes are divided into three types consisting of residential, com-mercial and industrial loads (l ¼ 3). Four practical load levels havebeen also considered as 100%, 82%, 61% and 39% of the full load(Q ¼ 4). Again, there are 43 ¼ 64 load patterns, so the training setconsists of 64 elements. For the 64 available load patterns of theIEEE 69-bus distribution network, 9 distinct optimal network con-figurations presented in Table 3 have been obtained using theswitching algorithm. Since N = 69 and C = 9, the size of the

Table 3Optimal configurations of the IEEE 69-bus distribution network related to thedifferent load patterns obtained by the switching algorithm.

Optimalconfigurationnumber

Load pattern numbers Powerswitches thatshould beopened

1 1, 5, 6, 7, 17, 18, 19, 20, 21, 22, 23, 24, 25,53, 54, 55

S12, S19, S43,S53, S54

2 2, 8, 9, 10, 26, 27, 29, 30, 32, 33, 34, 56 S12, S20, S43,S53, S54

3 2, 3, 11, 12, 13, 26, 27, 28, 29, 30, 31, 34,35, 36, 37, 38, 39, 40, 41, 42, 43, 56, 57,59, 60, 62, 63

S11, S18, S20,S43, S53

4 4, 15 S12, S21, S43,S53, S54

5 14, 58 S12, S21, S44,S53, S54

6 16, 45, 47, 48, 50, 51 S11, S18, S22,S43, S53

7 40, 41 S11, S18, S21,S43, S53

8 46, 48, 49, 52, 61, 64 S11, S18, S21,S44, S53

9 44 S12, S22, S43,S53, S54

H. Fathabadi / Electrical Power and Energy Systems 78 (2016) 96–107 105

proposed ANN can be determined as follows. The neurons numberof the input layer is 2N = 138 and the neurons number of the out-put layer is C = 9. The number of the weight coefficients that shouldbe determined by training the ANN is 138K þ 9K ¼ 147K. Thetraining set has 64 elements, so the least possible amount of K

Fig. 10. Proposed ANN for analyzing the

Fig. 11. 69 buses (red points) and the clusters centers (green points). (For interpretatioversion of this article.)

which sets the sum of the squares of the errors equal to zero isone (K = 1). The ANN proposed for analyzing the IEEE 69-bus distri-bution network is shown with all the specifics in Fig. 10.

Using the dFCM clustering technique, the proposed frameworkshown in Fig. 3 can be obtained by reducing the size of the ANNshown in Fig. 10. The first load pattern (load pattern #1) in whichall the 69 buses have full load (100%) is shown in Fig. 11. The loadof the each bus is shown with a red point. By applying the dFCMclustering algorithm, the amount of the Xie–Beni index, the clus-ters number and the clusters centers are obtained. The calculatedXie–Beni index is shown in Fig. 12. Based on the amount of theXie–Beni index, the clusters numbers have been obtained as 3, 3and 2 for residential, commercial and industrial loads, respectively,and then, the clusters centers have been calculated. The clusterscenters are shown with green points in Fig. 11. The number ofthe clusters is 8 (m = 8), so the neurons number of the input layerof the proposed framework is 2m = 16 and the neurons number ofthe output layer is C = 9. The number of the weight coefficients thatshould be determined by training the framework is 16Lþ 9L ¼ 25L.Since the training set has 64 elements, the least possible amount ofL which sets the sum of the squares of the errors equal to zero isthree (L = 3). It can be summarized that the neurons number ofthe input, hidden and output layers of the proposed frameworkare 16, 3, and 9, respectively. The framework presented for analyz-ing the IEEE 69-bus distribution network is shown in Fig. 13. Sim-ilar to previous section, after training the ANN and framework, 100load patterns have been used as test samples to check whether theproposed schemes are successful to determine the optimal config-urations? 80 load patterns are out of the training set and 20 pat-terns have been chosen from the training set. The optimal

IEEE 69-bus distribution network.

n of the references to color in this figure legend, the reader is referred to the web

Fig. 12. Calculated Xie–Beni index for the IEEE 69-bus distribution network.

Fig. 13. Proposed framework for analyzing the IEEE 69-bus distribution network.

Table 4Simulation results of the test samples for the IEEE 69-bus distribution network.

Method Number ofthe testsamples(loadpatterns)

Number of theobtained optimalconfigurationswhich are same asthe results of theswitching algorithm

Averageprocessingtime using a2.2 GHzprocessor (s)

Proposed ANN (K = 1) 100 92 2.86Proposed ANN (K = 2) 100 94 3.17Proposed framework 100 98 0.51Switching algorithm 100 100 89.33

106 H. Fathabadi / Electrical Power and Energy Systems 78 (2016) 96–107

network configurations obtained using the proposed ANN (for K = 1and K = 2) and framework have been compared to the resultsobtained by applying the switching algorithm. The comparativeresults for each input load pattern are presented in Table 4. Theresults listed in Table 4 shows that there are 8 and 6 distinctanswers between the proposed ANN and the switch algorithmwhile there are only two distinct answers between the optimalconfigurations specified by the proposed framework and theswitch algorithm. Again, it is worthwhile to note that the two dis-tinct structures specified by the proposed framework also belongto the set of the three configurations which have the least activepower loss. For the proposed framework, the average process time

of the optimal configuration determination for each input load pat-tern is only 0.51 s while it is 2.86 s and 89.33 s for the proposedANN and the switch algorithm, respectively. Again, the resultsshown in Table 4 explicitly verify that the proposed frameworkhas excellent capability for determining the optimal configura-tions. Because of a very short process time, the proposed frame-work can be effectively used for real-time reconfiguration ofdistribution networks.

The simulation results reported in Tables 2 and 4 also verifythat by choosing K P 2 for the two ANN, the sum of the squaresof the errors is again zero but the process time increases whilethe number of the distinct answers between the ANN and theswitch algorithm does not significantly decrease.

Conclusion

This study proposed a three-layer framework to obtain theoptimal configuration of a power distribution network. Theproposed framework was obtained by reducing the size of the pro-posed three-layer ANN. Reduction in the size was done using theproposed dFCM clustering algorithm. The proposed frameworkand ANN both were implemented on the two IEEE 33-bus and IEEE69-bus power distribution networks. The ANN and frameworkweretrained using the training set obtained by performing the switchingalgorithm. The simulated results were compared to the resultsobtained by performing the switching algorithm. The comparative

H. Fathabadi / Electrical Power and Energy Systems 78 (2016) 96–107 107

results explicitly verified that a very short process time, a very sim-ple structure and high accuracy are some benefits of using the pro-posed framework to reconfigure the power distribution networks.

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