Dielektrik

1. Power distribution network reconfiguration for power loss minimization
using novel dynamic fuzzy c-means (dFCM) clustering based ANN
approach
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 2014
Received in revised form 11 May 2015
Accepted 25 November 2015
Available online 17 December 2015
Keywords:
Distribution network reconfiguration
Power loss minimization
Dynamic fuzzy c-means (dFCM)
Clustering technique
Artificial 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 distribution
networks 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 modified
dynamic 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 training
samples. The simulated results are compared to the results obtained by performing a selected traditional
method which is the switching algorithm. The comparative results explicitly verify that using the
proposed framework for distribution networks reconfiguration has some benefits such as a very short
process time that is far shorter than the others, a very simple structure including only a minimal number
of 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.
Ó 2015 Elsevier Ltd. All rights reserved.
Introduction
A power distribution network and a transmission system are
the two important parts of an electric power generation and distri-
bution system. The power loss in the distribution network is more
than that in the transmission system because the currents avail-
able in the distribution part are generally much greater than that
in the transmission part. In electric power generation and distribu-
tion systems, about 10% of the produced electric power is lost in
distribution networks, so minimizing the electric power loss is
one of the important problems related to electric power generation
and distribution systems [1,2]. In practice, there are two methods
for minimizing the power loss in a distribution network. The two
methods are the reconfiguration of the distribution network and
capacitors placement. The reconfiguration of distribution networks
can be also adopted to achieve the other goals such as better volt-
age profile and better charge balance [3,4]. The limitations of a
power distribution network such as radial structure, the capacity
of the feeders and the acceptable voltage range of different buses
should 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 are
many recent researches addressing this issue [5–9]. For the distri-
bution networks having a large number of the power switches, the
reconfiguration is a multi-objective issue including a non-linear
mapping 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, switch
exchange methods, optimized flow pattern (OFP), and artificial
intelligence algorithms [12,13]. A simple method which uses the
branches of the network graph and their limitations for network
reconfiguration was reported in [1]. A summarized version of the
mentioned method was presented in [2]. The summarized method
detects the feeding path of each charge, and then, a simple sub-tree
is 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 real
distribution network, increase in the size of the network leads to
a serious problem. The switch exchange method (SEM) was intro-
duced in 1988 [14]. The method estimates the power loss in each
state 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 sequential
switch opening method (SSOM). Application of genetic algorithm
http://dx.doi.org/10.1016/j.ijepes.2015.11.077
0142-0615/Ó 2015 Elsevier Ltd. All rights reserved.
⇑ Tel./fax: +30 210 7722018.
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Electrical Power and Energy Systems 78 (2016) 96–107
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journal homepage: www.elsevier.com/locate/ijepes

2. (GA) for the reconfiguration of power distribution networks was
first reported in [15]. A research about providing load patterns,
and then carrying out the feeders reconfiguration using patterns
detection was presented by Hoyong et al. [16]. A similar ANN
approach 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 for
small size distribution networks. The process proposed by Hoyong
et al. [16] together with classifying the loads into residential, com-
mercial and industrial types was used for networks reconfiguration
in [19].
The clustering techniques are used to classify different sets of
physical parameters and events [20,21]. A number of clustering
techniques such as local maxima search and search neighborhoods
are defined and reported in the literature [21]. Some clustering
techniques such as deterministic annealing intensively depend
on the data pattern while some other techniques such as generic
clustering algorithm do not have this defect. Clustering techniques
such 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 techniques
were presented. In a fuzzy clustering, data points are represented
by a membership degree which indicates the dependence of a data
point to a cluster. Thus, a data point may simultaneously lie in
more than one cluster, so an affective detection of overlapping
clusters can be performed [21]. An important type of the fuzzy
clustering techniques is called fuzzy c-means (FCM) [22–24]. A
modified version of the FCM algorithm in which the clusters are
dynamically found was presented in [21]. The modified FCM which
has high capability for specifying the non-uniformly distributed
clusters is called dynamic FCM (dFCM).
A survey in the literature shows that there are other types of
fuzzy clustering dynamic algorithms that inside evolving systems
such as dynamically evolving clustering (DEC) [25], hyper-
ellipsoidal clustering for evolving data stream (HECES) [26], online
evolving fuzzy clustering algorithm based on maximum likelihood
estimator [27], density-based clustering for evolving uncertain
data stream [28], evolving soft subspace clustering [29], evolving
clustering method (ECM) [30] and adaptive learning evolving clus-
tering method (ALECM) [30]. DEC uses cluster weight and distance
before generating new clusters that is unlike other approaches that
consider either the data density or distance from existing cluster
centers [25]. In HECES, sliding window model is used to handle
incoming stream of data to minimize the impact of the obsolete
information on recent clustering results, and shrinkage technique
is used to avoid the singularity issue in finding the covariance of
correlated data [26]. In the algorithm proposed in [27], the distance
from 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 uncertain
data 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 is
a kind of efficient online clustering method, which evolved the
clusters automatically from data streams. It is a distance- and
prototype-based clustering method. The distance of a new incom-
ing sample to the closest cluster center cannot be larger than a
threshold value; otherwise a new cluster is evolved [30]. First
defect of ECM is that when performing incremental learning from
scratch, it is quite not appropriate to set the predefined threshold
for a good performing adaptation. As second defect, ECM is quite
sensitive to different data orders. To overcome the two mentioned
defects, ALECM was proposed in [30] that uses the on-line learning
capability by adjusting and evolving the clusters automatically
with new incoming samples.
There are also some researches on ANN pruning reported in the
literature. Self-adaptive evolutionary constructive and pruning
algorithm (SAECPA) that is a structural algorithm was reported in
[31]. SAECPA considers an ANN in which one hidden neuron is
linked towards single input node, then using cluster pruning (CP)
and survival selection (SS) the ANN is pruned. Another method that
uses 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 until
satisfactory 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 dependency
relationships. 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 a
set 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 survival
selection (ABSS) were proposed as two new operators for ANN
pruning. The CBP operator retains significant neurons and prunes
insignificant neurons on a probability basis and therefore prevents
the exponential growth of an ANN [33].
In this study, an ANN is proposed for distribution networks
reconfiguration to obtain the optimal configuration in which the
active power loss is minimal. Then, the proposed ANN is reduced
in 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 on
two 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 proposed
framework has higher performance compared to the others.
This paper is organized as follows. Fuzzy clustering and the FCM
algorithm are discussed in Section ‘‘Fuzzy clustering and FCM algo-
rithm”. Section ‘‘Dynamic fuzzy c-means algorithm and cluster
validity” deals with the proposed dFCM algorithm and the concepts
of cluster validity. Distribution network reconfiguration is for-
mulized in Section ‘‘Distribution network reconfiguration and the
proposed ANN and framework” and the proposed ANN and frame-
work are presented. Simulated results of implementing the ANN
and framework on two distribution networks are presented in Sect
ion ‘‘Simulated results”. Finally, Section ‘‘Conclusion” concludes
the paper.
Fuzzy clustering and FCM algorithm
Clustering is the process of grouping or dividing a series of data
from unlabeled patterns into a number of groups which are called
clusters, so that, the similar patterns are allocated to one cluster.
Each pattern can be shown with a vector which has different
parameters and properties. Clustering technique includes two
basic criteria which are adjacency measurement and grouping.
Adjacency measurement shows the similarity between two points
and grouping is used to find an appropriate target function and the
related algorithm. Each clustering method determines the similar-
ity between the patterns by calculating the distance between the
related patterns. For patterns with metric properties, different
types of distance measurement such as Euclid distance or Maha-
lanobis can be used [21]. Fuzzy clustering is a technical method
to allocate data points to different clusters using fuzzy logic which
H. Fathabadi / Electrical Power and Energy Systems 78 (2016) 96–107 97

3. provides effective means for separating overlapping clusters. Fuzzy
clustering 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-means
algorithm that uses fuzzy logic to determine the association of a
data point to a cluster [22–24,35]. The association to a cluster is
determined by calculating the inverse distance to the cluster cen-
ter. The cluster centers determined by FCM directly depend on
the geometric locations of the data points on the plane or space.
Some applications of the FCM algorithm for tracking gamma rays
and detecting ions have been reported in [36,37], respectively.
In the FCM algorithm, an objective function which should be
minimized is considered as:
FðY; Z; a; XÞ ¼
Xn
k¼1
Xm
i¼1
yikð Þa
xk À 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 centers
of the m clusters, n is the number of the data points, X ¼
x1; x2; . . . ; xnð ÞT
is the data points vector, Y ¼ yik½ ŠmÂn is the member-
ship matrix consisting of the membership yik which shows the mem-
bership of xk in the ith cluster, and k Á k shows the Euclidean distance
norm (kVk ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffi
VT
Á 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, an
optimal solution which minimizes FðY; Z; a; XÞ is found as [23]:
yik ¼
Xm
j¼1
kxk À zik
kxk À zjk
2=ðaÀ1Þ
#À1
ð2Þ
where 1 6 i 6 m, 1 6 k 6 n, and the center of the ith cluster is
obtained as:
zi ¼
Pn
k¼1 yikð Þa
xk
Pn
k¼1 yikð Þa ð3Þ
After clustering the data, a validity index is used to show how well
the data have been clustered. There are different validity indexes
such as Xie–Beni index and modified partition coefficient (PCC)
index [24,36,38]. In fact, all indexes present a numerical aspect to
determine 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 that
clustering significantly depends on the fuzzy factor a which
explicitly varies from one data set to other [21]. Another draw-
back of the FCM clustering algorithm is that it deals with outliers
same as data points to put them in the data bulk, so some mod-
ifications have been made to improve the FCM algorithm over the
years. 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]. The
FCM clustering algorithm uses the Euclidean distance between
data samples, so there is equal importance for each data point
and each dimension which refers to a feature. To address this
concern 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 fuzzy
c-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 adaptive
information or the incoming data are not uniform. An application
of the dFCM clustering technique for calorimetric data recon-
struction in high-energy physics was reported in [21]. The dFCM
clustering is a general technique which can be applied to a large
number of different applications [40,41]. The dFCM clustering
technique is more suitable for the applications including online
analysis of incoming data in which the process needs adaptive
information or the incoming data are not uniform. On the other
hand, in a power distribution network, incoming data should be
analyzed online. Furthermore, the input data (load flows) are
not uniform and adaptive information is needed to adapt the load
flows to the practical load patterns to obtain the optimal config-
uration based on the least mismatching between the instant input
load flow and one of the practical pattern. Thus, in this study, the
dFCM clustering technique is first modified to make it appropriate
to use in a power distribution network, and then it is used for dis-
tribution networks reconfiguration to reduce the active power
losses.
Dynamic fuzzy c-means algorithm
The dFCM algorithm dynamically finds clusters, and further-
more, it deletes and regenerates clusters if it is necessary when
the incoming data flow for clustering. It fits the data pattern con-
tinuously, and the clusters are selected using a validity index. A
modified version of the dFCM clustering technique presented in
[21] is proposed in this study. The flow chart of the proposed dFCM
clustering 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 maximum
acceptable level for the memberships and FCM error (EFCM)
is also defined as the maximum acceptable difference
between the two clusters centers obtained in the two
sequential steps using the FCM algorithm. At first, there
are a few of the incoming data points, so the incoming data
range, the membership threshold, the FCM error and the
boundary of the clusters number (m) are estimated.
(2) The m clusters centers are uniformly located in the input
space, 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 maximum
membership is greater than or equal to the membership
threshold, it means that the data point belongs to at least
one of the clusters, and consequently, an update is necessary
to modify the membership threshold and the clusters
centers.
(4) It is assumed that m is the clusters number obtained by the
iterative process, so that, the maximum membership of the
data 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 of
m À 1 and m + 1 available clusters, and then the cluster cen-
ters that provide a better validity index are selected. If m À 1
or m + 1 clusters provide better validity index, then m À 1 or
m + 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:
98 H. Fathabadi / Electrical Power and Energy Systems 78 (2016) 96–107

4. kZnew À Zoldk ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiXm
i¼1
ziÀnew À ziÀoldð Þ2
q
ythr: ð4Þ
If the above condition is satisfied, then evaluating the cluster
validity is necessary. In other words, if the distance between the
new and old cluster center vectors is greater than ythr:, then the
cluster validity is checked. In this case, when a new data point
arrives, regardless of its membership, the cluster centers are
updated using the alternating optimization (AO). After updating
the cluster centers, ythr: is used to evaluate whether the new cluster
centers are significantly different from the old cluster centers or
not (inequality (4) is satisfied or not)? Thus, the dFCM process
decides 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
Fig. 1. The flow chart of the dynamic FCM (dFCM) algorithm.
H. Fathabadi / Electrical Power and Energy Systems 78 (2016) 96–107 99

5. clustering process. The selected value for the membership threshold
depends on the application type which uses the dFCM clustering. It
is clear that the inequality (4) prevents the check of the cluster
validity when a new a data point arrives. In fact, if the new data
point belongs to a specified acceptable neighbor of a cluster, then
it is not checked that whether the other clusters are better or not?
Validity index
After clustering, a validity index is used to determine how well
the data have been represented by the obtained clusters. Different
validity indices have been defined and reported in the literature
[24]. The Xie–Beni validity index is one of the best validity indexes
which is widely used because it depends on not only memberships
but also geometric distances. In this study, the Xie–Beni index is
chosen to check how well the incoming data have been clustered.
The Xie–Beni index is defined as:
I:VXBðY; Z; XÞ ¼
Pn
k¼1
Pm
i¼1 yikð Þ2
kxk À zik2
n:ðMini–jfkzi À zjkg
¼
FðY; Z; 2; XÞ
n Á ðMini–jfkzi À zjkg
ð5Þ
where I:VXBðY; Z; XÞ is the Xie–Beni index. This index indicates a
ratio of the total variation of the cluster centers together with the
memberships in the obtained clusters to the distance between the
cluster centers, so minimizing the Xie–Beni index results a better
clustering. In fact, a minimum amount of the Xie–Beni index shows
a larger separation between the clusters together with the more
concentration 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 ANN
and framework
The most important goal of the reconfiguration of power distri-
bution systems is to determine the topology in which the active
power loss is the least possible amount [19]. The active power loss
of a power distribution system consisting of N buses is expressed
as:
PLoss ¼
XN
i¼1
i–j
XN
j¼1
rijjIijj2
ð6Þ
where rij is the total ohmic resistance of the branch i À j, and Iij is
the electrical current flowing between ith and jth buses. The total
active power (PTot) distributed on the power distribution system is
obtained as:
PTot ¼
XN
i¼1
pi þ PLoss ð7Þ
where pi is the active power absorbed by ith bus. Similarly, the total
reactive power (QTot) distributed on the power distribution system
can be also found as:
QTot ¼
XN
i¼1
qi þ
XN
i¼1
i–j
XN
j¼1
xijjIijj2
ð8Þ
where xij is the total reactance of the branch i À j, qi is the reactive
power absorbed by ith bus. The goal is to find an ANN which deter-
mines the optimal configuration of the power distribution system in
which the active power loss is minimal. The limitations are that the
reconfigured distribution network should be a radial network with
an acceptable range of ½0:95 p:u:;1:05 p:u:Š for the voltage profile.
In a power distribution network, the instant active and reactive
powers absorbed by each bus of the network continuously and per-
petually changes according to the load demand, so it is inevitable to
simultaneously apply the instant active and reactive powers of all
the buses which 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; qNÞŠT
and the output of the ANN is the determined optimal
configuration. It is clear that there is an intensive non-linear relation
between the inputs and outputs of the ANN. In this study, the three-
layer perceptron neural network shown in Fig. 2 has been used to
perform the mentioned nonlinear mapping.
Without using dFCM clustering technique, the proposed neural
network should have 2N neurons in the input layer where N is the
number of buses because the input vector ( ðp1; q1Þ; ðp2; q2Þ; . . . ;½
ðpi; qiÞ; . . . ; ðpN; qNÞŠT
) consists of 2N elements. There are also K neu-
rons in the hidden layer and C neurons in the output layer as
shown in Fig. 2, where C is the number of the optimal distribution
network configurations obtained by applying the switching algo-
rithm, and K is selected, so that, the desired minimum amount of
the sum of the squares of errors and minimum process time can
be achieved after training the proposed ANN using the ‘‘Batch
Learning-LMS algorithm” [42,43]. After training, the jth neuron of
the output layer can only produce 0 or 1, where ‘‘1” means that
the jth optimal configuration obtained by performing the switch-
ing algorithm [34] has been chosen as the optimal structure by
the proposed ANN. All the biases of the hidden and output layers
and also all the weights of the input layer have been chosen one
to reduce the training time. In fact, the weight coefficients matrixes
of the hidden and output layers are determined by training the
proposed three-layer ANN. It is clear that the neurons number of
the proposed ANN significantly increases when the buses number
of the distribution network increases. For example, for the IEEE 33-
bus network, the neurons number of the entrance layer is 66, and
for the IEEE 69-bus network, it is 138. Increasing in the neurons
number of the ANN results a longer training time, a considerable
reduction in the convergence speed of the ANN, and more difficulty
in implementation of the proposed ANN [19].
The neurons number of the proposed three-layer feed-forward
ANN can be effectively reduced using the dFCM clustering tech-
nique. In fact, the neurons number of the input layer decreases
to 2m, where m is the clusters number explained in Eqs. (1) and
(2). Since m N, the proposed ANN is simplified to a new version
including much less neurons. The final framework including the
three-layer feed-forward ANN reduced in size using the proposed
dFCM clustering technique is shown in Fig. 3.
In a power distribution network, the number of load levels is
divided 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 training
set. For instance, if two load levels are only considered as 100% and
60% of the full load (Q ¼ 2), and the network buses are also divided
into three types (l ¼ 3) consisting of residential, commercial and
industrial types, then the training set consisting of 23
¼ 8 elements
is 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 related
optimal distribution network configuration in which the active
power loss is minimal, respectively.
100 H. Fathabadi / Electrical Power and Energy Systems 78 (2016) 96–107

6. Simulated results
In this section, to validate the theoretical results and to check
the performance of the proposed ANN and framework, two IEEE
33-bus and IEEE 69-bus distribution networks have been consid-
ered. For the each network, the reconfiguration has been carried
out by applying both the proposed ANN shown in Fig. 2 and the
proposed framework shown in Fig. 3. The results have been com-
pared to each other to verify the benefits of using the proposed
dFCM clustering technique.
IEEE 33-bus distribution network
The IEEE 33-bus distribution network consisting of 33 buses is
shown in Fig. 4. Other detailed specifications of the IEEE 33-bus
distribution network have been reported in previously published
articles such as [44]. Based on subscribers’ demand the load types
are divided into three types consisting of residential, commercial
and industrial loads (l ¼ 3). According to the statistical data
reported for the practical distribution networks [19,35], four prac-
tical load levels have been also considered as 100%, 82%, 61% and
39% 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, the
switching algorithm presented in [34] has been used for specifying
the optimal network configuration for the each combination of the
64 available load levels. For the each load pattern, the switching
algorithm determines the related optimal network configuration.
The limitation is that the determined optimal configuration should
be a radial network with an acceptable range of ½0:95 p:u:;1:05 p:u:Š
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 configurations
is less than 64. As mentioned before, the number of the distinct
optimal network configurations is equal to the neurons number
in the output layers of the proposed ANN shown in Fig. 2 and the
proposed framework shown in Fig. 3. The switching algorithm is
a conventional method that is widely used in power dispatching
centers yet. It works based on the following sequential steps:
(1) For a present load flow available in a distribution network, it
finds all possible configurations (sets of ‘‘closed” and ‘‘open”
switches) of the distribution network by setting each switch
to ‘‘closed” or ‘‘open” status.
(2) Each obtained configuration that is not a radial network is
rejected.
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

7. (3) Each obtained configuration that has not an acceptable
range of voltage profile (in this study, in the range of
½0:95 p:u:;1:05 p:u:Š) among all the buses of the distribution
network 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 a
long time to find the optimal configuration(s). For instant and as
will 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-bus
distribution networks based on a given load flow. It can be summa-
rized that, the switching algorithm finds the optimal configuration
with a considerable time delay based on a load flow occurring in
past time, so the optimal configuration determined by the switch-
ing algorithm is not reliable and cannot be chosen as an optimal
configuration for the distribution network at any time because of
probable variation in the load flow during the mentioned delay
time. For instant, the load flow in the IEEE 33-bus distribution net-
work may vary during 44.24 s, so the optimal configuration
obtained 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 distribution
network, 8 distinct optimal network configurations have been
obtained 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 network
configurations are presented in Table 1. Since N = 33 and C = 8,
the size of the proposed ANN can be determined as follows. The
neurons 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 hidden
layer 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
2
6
6
6
4
3
7
7
7
5
ð10Þ
Similarly, the weights matrix of the output layer consisting of 8
neurons is expressed as:
W3 ¼
w0
1;1 w0
1;2 . . . w0
1;K
w0
2;1 w0
2;2 . . . w0
2;K
. . . . . . . . . . . .
w0
8;1 w0
8;2 . . . w0
8;K
2
6
6
6
4
3
7
7
7
5
ð11Þ
The weights number that should be determined by training the
ANN is 66K þ 8K ¼ 74K. Since the training set has 64 elements, the
least possible amount of K which sets the sum of the squares of the
errors equal to zero is one (K = 1). Thus, the weights matrixes
expressed by Eqs. (10) and (11) can be simplified as:
W2 ¼ w1;1 w1;2 . . . w1;66½ Š ð12Þ
and
W3 ¼
w0
1;1
w0
2;1
. . .
w0
8;1
2
6
6
6
4
3
7
7
7
5
ð13Þ
Based on the above explanation, it can be summarized that the
neurons number of the input, hidden and output layers of the pro-
posed ANN are 66, 1, and 8, respectively. The ANN proposed for
analyzing the IEEE 33-bus distribution network is shown with all
the specifics in Fig. 5. It can be seen that the number of the optimal
network configurations listed in Table 1 is equal to the neurons
number of the output layer. After training, each neuron of output
layer can only produce 0 or 1. For a specific load pattern, if the
jth neuron becomes 1, this means that the jth optimal distribution
network configuration listed in Table 1 has been selected as the
optimal structure by the proposed ANN. If more than one neuron
become 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 optimal
structures.
Using the dFCM clustering technique, the proposed framework
shown in Fig. 3 can be obtained by reducing the size of the ANN
shown in Fig. 5. As mentioned, there are 64 available load patterns.
The first load pattern (load pattern #1) in which all the 33 buses
have the full load (100%) is shown in Fig. 6. The buses belong to
the three load types consisting of residential, commercial and
industrial loads. The load of the each bus is shown with a red point.
Now, by applying the proposed dFCM clustering algorithm, the
Fig. 4. IEEE 33-bus distribution network.
Table 1
Optimal configurations of the IEEE 33-bus distribution network related to the
different load patterns.
Optimal
configuration
number
Load pattern numbers Power switches
that should be
opened
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

8. amount of the Xie–Beni index, the clusters number and the clusters
centers are obtained. The calculated Xie–Beni index is shown in
Fig. 7. Based on the amount of the Xie–Beni index, the clusters
numbers 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 with
green points in Fig. 6. The number of the clusters is 7 (m = 7), so
the 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
2
6
6
6
4
3
7
7
7
5
ð14Þ
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 version
of 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

9. Similarly, the weights matrix of the output layer consisting of 8
neurons is expressed as:
W3 ¼
w00
1;1 w00
1;2 . . . w00
1;L
w00
2;1 w00
2;2 . . . w00
2;L
. . . . . . . . . . . .
w00
8;1 w00
8;2 . . . w00
8;L
2
6
6
6
4
3
7
7
7
5
ð15Þ
The weights number that should be determined by training the
framework is 14L þ 8L ¼ 22L. Since the training set has 64 ele-
ments, the least possible amount of L which sets the sum of the
squares 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. The
framework presented for analyzing the IEEE 33-bus distribution
network is shown in Fig. 8.
After training the ANN shown in Fig. 5 and the framework
shown 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 test
samples. 80 load patterns are out of the training set and 20 pat-
terns have been chosen from the training set. The optimal network
configurations obtained using the proposed ANN (for K = 1 and
K = 2) and framework have been compared to the results obtained
by applying the switching algorithm. The comparative results
including the average process time using a 2.2 GHz processor for
each input load pattern are reported in Table 2. The results listed
in Table 2 shows that there are 6 and 5 distinct answers between
the proposed ANN and the switch algorithm while there is only
one distinct answer between the optimal configurations specified
by 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 configurations
which 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 s
and 44.24 s for the proposed ANN and the switch algorithm,
respectively. The comparative results shown in Table 2 explicitly
verify the excellent performance of the proposed framework.
IEEE 69-bus distribution network
IEEE 69-bus distribution network has been considered as
another distribution network for implementing the proposed
ANN and framework. The IEEE 69-bus distribution network is a
12.66 kV radial distribution system with 69 buses which is shown
in Fig. 9. Similar to IEEE 33-bus distribution network, the load
types are divided into three types consisting of residential, com-
mercial and industrial loads (l ¼ 3). Four practical load levels have
been 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 set
consists of 64 elements. For the 64 available load patterns of the
IEEE 69-bus distribution network, 9 distinct optimal network con-
figurations presented in Table 3 have been obtained using the
switching algorithm. Since N = 69 and C = 9, the size of the
Fig. 8. Proposed framework for analyzing the IEEE 33-bus distribution network.
Table 2
Simulation results of the test samples for the IEEE 33-bus distribution network.
Method Number of
the test
samples
(load
patterns)
Number of the
obtained optimal
configurations
which are same as
the results of the
switching algorithm
Average
processing
time using a
2.2 GHz
processor (s)
Proposed ANN (K = 1) 100 94 1.29
Proposed ANN (K = 2) 100 95 1.43
Proposed framework 100 99 0.37
Switching algorithm 100 100 44.24
Fig. 9. IEEE 69-bus distribution network.
104 H. Fathabadi / Electrical Power and Energy Systems 78 (2016) 96–107

10. proposed ANN can be determined as follows. The neurons number
of 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 should
be determined by training the ANN is 138K þ 9K ¼ 147K. The
training set has 64 elements, so the least possible amount of K
which sets the sum of the squares of the errors equal to zero is
one (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 framework
shown in Fig. 3 can be obtained by reducing the size of the ANN
shown in Fig. 10. The first load pattern (load pattern #1) in which
all the 69 buses have full load (100%) is shown in Fig. 11. The load
of the each bus is shown with a red point. By applying the dFCM
clustering algorithm, the amount of the Xie–Beni index, the clus-
ters number and the clusters centers are obtained. The calculated
Xie–Beni index is shown in Fig. 12. Based on the amount of the
Xie–Beni index, the clusters numbers have been obtained as 3, 3
and 2 for residential, commercial and industrial loads, respectively,
and then, the clusters centers have been calculated. The clusters
centers are shown with green points in Fig. 11. The number of
the clusters is 8 (m = 8), so the neurons number of the input layer
of the proposed framework is 2m = 16 and the neurons number of
the output layer is C = 9. The number of the weight coefficients that
should be determined by training the framework is 16L þ 9L ¼ 25L.
Since the training set has 64 elements, the least possible amount of
L which sets the sum of the squares of the errors equal to zero is
three (L = 3). It can be summarized that the neurons number of
the input, hidden and output layers of the proposed framework
are 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, 100
load patterns have been used as test samples to check whether the
proposed 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
Table 3
Optimal configurations of the IEEE 69-bus distribution network related to the
different load patterns obtained by the switching algorithm.
Optimal
configuration
number
Load pattern numbers Power
switches that
should be
opened
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
Fig. 10. Proposed ANN for analyzing the IEEE 69-bus distribution network.
Fig. 11. 69 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
version of this article.)
H. Fathabadi / Electrical Power and Energy Systems 78 (2016) 96–107 105

11. network configurations obtained using the proposed ANN (for K = 1
and K = 2) and framework have been compared to the results
obtained by applying the switching algorithm. The comparative
results for each input load pattern are presented in Table 4. The
results listed in Table 4 shows that there are 8 and 6 distinct
answers between the proposed ANN and the switch algorithm
while there are only two distinct answers between the optimal
configurations specified by the proposed framework and the
switch algorithm. Again, it is worthwhile to note that the two dis-
tinct structures specified by the proposed framework also belong
to the set of the three configurations which have the least active
power 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 proposed
ANN and the switch algorithm, respectively. Again, the results
shown in Table 4 explicitly verify that the proposed framework
has 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 of
distribution networks.
The simulation results reported in Tables 2 and 4 also verify
that by choosing K P 2 for the two ANN, the sum of the squares
of the errors is again zero but the process time increases while
the number of the distinct answers between the ANN and the
switch algorithm does not significantly decrease.
Conclusion
This study proposed a three-layer framework to obtain the
optimal configuration of a power distribution network. The
proposed framework was obtained by reducing the size of the pro-
posed three-layer ANN. Reduction in the size was done using the
proposed dFCM clustering algorithm. The proposed framework
and ANN both were implemented on the two IEEE 33-bus and IEEE
69-bus power distribution networks. The ANN and framework were
trained using the training set obtained by performing the switching
algorithm. The simulated results were compared to the results
obtained by performing the switching algorithm. The comparative
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 4
Simulation results of the test samples for the IEEE 69-bus distribution network.
Method Number of
the test
samples
(load
patterns)
Number of the
obtained optimal
configurations
which are same as
the results of the
switching algorithm
Average
processing
time using a
2.2 GHz
processor (s)
Proposed ANN (K = 1) 100 92 2.86
Proposed ANN (K = 2) 100 94 3.17
Proposed framework 100 98 0.51
Switching algorithm 100 100 89.33
106 H. Fathabadi / Electrical Power and Energy Systems 78 (2016) 96–107

12. 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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