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Multiobjective evolutionary algorithm with a discrete differentialmutation operator developed for service restoration in distributionsystems

Danilo Sipoli Sanches a,⇑, João Bosco A. London Jr. b, Alexandre Cláudio B. Delbem c, Ricardo S. Prado d,Frederico G. Guimarães e, Oriane M. Neto e, Telma W. de Lima f

a Federal Technological University of Paraná, Cornélio Procópio, Brazilb São Carlos School of Engineering, University of São Paulo, São Carlos, SP, Brazilc Institute of Mathematical and Computing Sciences, University of São Paulo, São Carlos, SP, Brazil

d Federal Institute of Minas Gerais, Ouro Preto, Brazile Department of Electrical Engineering, Universidade Federal de Minas Gerais, UFMG, Brazilf Institute of Informatics, Universidade Federal de Goias, UFG, Brazil

a r t i c l e i n f o

Article history:

Received 22 March 2013

Received in revised form 24 April 2014

Accepted 10 May 2014

Available online 12 June 2014

Keywords:

Large-Scale Distribution Systems

Service restoration

Node-Depth EncodingMulti-Objective Evolutionary Algorithms

Differential Evolution

a b s t r a c t

Network reconfiguration for service restoration in distribution systems is a combinatorial complex opti-

mization problem that usually involves multiple non-linear constraints and objective functions. For large

scale distribution systems no exact algorithm has found adequate restoration plans in real-time. On the

other hand, the combination of Multi-Objective Evolutionary Algorithms (MOEAs) with the Node-Depth

Encoding (NDE) has been able to efficiently generate adequate restoration plans for relatively large dis-

tribution systems (with thousands of buses and switches). The approach called MEAN results from the

combination of NDE with a technique of MOEA based on subpopulation tables. In order to improve the

capacity of MEAN to explore both the search and objective spaces, this paper proposes a new approach

that results from the combination of MEAN with characteristics from the mutation operator of the Differ-

ential Evolution (DE) algorithm. Simulation results have shown that the proposed approach, called

MEAN-DE, is able to find adequate restoration plans for distribution systems from 3860 to 30,880

switches. Comparisons have been performed using the Hypervolume metric and the Wilcoxon rank-

sum test. In addition, a MOEA using subproblem Decomposition and NDE (MOEA/D-NDE) was investi-

gated. MEAN-DE has shown the best average results in relation to MEAN and MOEA/D-NDE.

2014 Elsevier Ltd. All rights reserved.

Introduction

Distribution system problems, such as service restoration

(SR) [1], power loss reduction [2], and expansion planning [3],

usually involve network reconfiguration procedures [4–7]. Asa consequence, they can be considered Distribution System

Reconfiguration (DSR) problems, which are usually formulated as

multiobjective and multiconstrained optimization problems

[8,9,5,10,1,11–13,6,14].

Several Evolutionary Algorithms (EAs) have been developed to

deal with DSR problems [8,4,9,5,10,14]. The results obtained by

such approaches have surpassed those obtained through both

Mathematical Programming and traditional Artificial Intelligence

[9]. However the majority of EAs still demands high running time

when applied to Large-Scale Distribution Systems (DSs) [9], that is,

DSs with thousands of buses and switches.The performance obtained by EAs for large-scale DSs is dramat-

ically affected by the data structure used to represent computa-

tionally the electrical topology of the DSs. Inadequate data

structure may reduce drastically the EA performance

[9,15,13,14,10]. Other critical aspects of EAs are the genetic opera-

tors that are used. Generally these operators do not generate radial

configurations [13].

In order to improve the EAs performance in DSR problems, the

approach proposed in [5] uses a vertex encoding based on the Pru-

fer number to encode the chromosomes. The Prufer number encod-

ing ensures system radiality avoiding the tedious ‘‘mesh check’’

algorithms, which are required to identify the existence of loops

http://dx.doi.org/10.1016/j.ijepes.2014.05.008

0142-0615/ 2014 Elsevier Ltd. All rights reserved.

⇑ Corresponding author. Tel.: +55 43 35204000; fax: +55 43 35204010.

E-mail addresses: [email protected] (D.S. Sanches), [email protected]

(J.B.A. London Jr.), [email protected] (A.C.B. Delbem), [email protected]

(R.S. Prado), [email protected] (F.G. Guimarães), [email protected]

(O.M. Neto), [email protected] (T.W. de Lima).

Electrical Power and Energy Systems 62 (2014) 700–711

Contents lists available at ScienceDirect

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(meshes) in the temporary solutions (DSs configurations). On the

other hand, in [16] it was proposed an innovative and effective

heuristic graph-based approach to SR problem. The idea is to min-

imize the switch operations in the de-energized areas. The

approach is based on the Prim algorithm [17].

In this context, the approaches proposed in [15,18,14] use a

new tree encoding, called Node-Depth Encoding (NDE), and its cor-

responding genetic operators [19]. As it was shown in those refer-ences, the NDE can improve the performance obtained by EAs in

DSR problems because of the following NDE properties: (i) The

NDE and its genetics operators produce exclusively feasible config-

urations, that is, radial configurations able to supply energy for the

whole re-connectable system1; (ii) The NDE can generate signifi-

cantly more feasible configurations (potential solutions) in relation

to other encoding in the same running time since its average time

complexity is Oð ffiffiffi

np Þ, where n is the number of graph nodes (each

graph node corresponds to a DS sector2); (iii) The NDE-based formu-

lation also enables a more efficient forward–backward Sweep Load

Flow Algorithm (SLFA) for DSs. Typically this kind of load flow

applied to radial networks requires a routine to sort network buses

into the Terminal-Substation Order (TSO) before calculating the

bus voltages [20–22]. Fortunately, each configuration generated by

the NDE has the buses naturally arranged in the TSO. Thus, the SLFA

can be significantly improved by the NDE-based formulation.

Observe that, once a feasible configuration is guaranteed, the objec-

tive function and the network operational constraints of a DSR prob-

lem can be analyzed solving a radial load-flow.

The approach proposed in [15] uses NDE together with a con-

ventional EA, that is, an EA based on just one objective function

that weights the multiple objectives and penalizes the violation

of constraints. According to [11,23] this strategy for treating prob-

lems with multiple objectives and constraints suffers from various

disadvantages. In this sense, in [14] NDE was combined with a

technique of Multi-Objective Evolutionary Algorithm (MOEA)

based on subpopulation tables, where each subpopulation stores

the found solutions that better satisfy an objective or a constraint

of a DSR problem. The MOEA with subpopulation tables can moreeasily leave the local toward global optima. Simulation results pre-

sented in [14] demonstrate that the Multi-Objective EA with NDE

(MEAN) is an efficient alternative to deal with DSR problems in

large-scale DSs. Also simulations results obtained by MEAN to treat

the SR problem in large-scale DSs have surpassed those obtained

by the NSGA-II, the SPEA-2 and the integration of MEAN with both

NSGA-II and SPEA-2 [24,25].

Beyond the scopes of multiobjective and data structures for DSR

problems, new relevant EAs have been investigated in the litera-

ture. The Differential Evolution (DE) [26] has received increased

interest from the Evolutionary Computation community, since DE

has shown better performance over other well-known metaheuris-

tics like Genetic Algorithm (GA) and Particle Swarm Optimization

(PSO) [27]. The DE can identify promising regions of the searchspace by a relatively simple operator (called differential mutation)

that generates new solutions (called individuals) from random cal-

culation of pairs of differences between individuals.

Nevertheless, there are few proposals of implementation of this

operator in the space of discrete variables [28–30] and their perfor-

mance has not been evaluated for a significant number of prob-

lems. Moreover, the available versions of DE for combinatorial

optimization in the literature focus on permutation-based combi-

natorial problems, such as the job shop scheduling, flow shop

scheduling and the travelling salesman problem, precluding their

use for SRs in DSs.

The main contribution of this paper is to propose a differential

mutation operator based on the NDE. The new operator can extract

the essential difference between two DS feasible configurations

and use it in order to compose new feasible configurations. More-

over, the average time complexity of the proposed operator is

Oð ffiffiffi np

Þ, enabling efficient manipulation of large-scale networks. Inaddition, the differential mutation operator based on the NDE is

combined with MEAN, producing a new powerful MOEA (called

MEAN-DE) to solve SR problems for large-scale DSs. Experimental

results have indicated that MEAN-DE can find restoration plans

with one-fourth less switching operations than MEAN plans for

the largest DS used in the tests (30,880 buses and 5166 switches).

Although the majority of MOEA has successfully worked with

combinatorial Multi-Objective Problems (MOPs) with at most

two objectives, the MEAN have solved the SR problem formulated

with more than two objectives [14]. Other MOEA that has obtained

interesting results for MOPs with more than two objectives is the

MOEA based on Decomposition (MOEA/D) [38]. As a consequence,

we also proposed an extension of MOEA/D using NDE, called

MOEA/D-NDE, adapted for the SR problem. However, the MEAN-

DE also presented better results in relation to MOEA/D-NDE for

the SR problem.

This paper is organized as follows: ‘Service restoration problem’

formalizes the SR problem; ‘Addressing the SR problem for large-

scale DSs’ addresses the SR problem for large-scale DSs; ‘Evolution-

ary Algorithms with NDE’ summarizes the MEAN, proposes the

extension of MOEA/D using NDE (MOEA/D-NDE) and describes

the recombination operator for the NDE; ‘Discrete DE with move-

ments list’ presents the Discrete Differential Evolution algorithm

with list of movements proposed in [46]; ‘Proposed approach’ pro-

posed the MEAN-DE; ‘Experimental analyses’ evaluates and com-

pares MEAN-DE to the MEAN and MOEA/D-NDE approaches;

finally, ‘Conclusions’ summarizes the main contributions and con-

cludes the paper.

Service restoration problem

Next sections present the nomenclature and the general formu-

lation for the SR problem.

Nomenclature

After the faulted areas have been identified and isolated, the

out-of-service areas must be connected to other feeders by closing

and/or opening switches. Fig. 1 shows an example of SR in a DS

with three feeders. Nodes 1, 2 and 3 represent power sources in

a feeder, solid lines are Normally Closed (NC) switches, dashed

lines are Normally Open (NO) switches, and each circle representssectors.3 Suppose sector 4 is in fault (Fig. 1). Then, sector 4 must be

isolated from the system by opening switches A and B. Sectors 7 and

8 are in an out-of-service area (gray box in Fig. 1). One way to restore

energy for those sectors is by closing switch C.

Mathematical formulation

The SR problem emerges after the faulted areas have been iden-

tified and isolated. The desired solution is the minimal number of

switching operations that results in a configuration with minimal

number of out-of-service loads, without violating the operational

and radial constraints of the DS. The minimization of the number1 The term ‘‘re-connectable system’’ means all areas having at least one switch

linking them to energized areas. Some out-of-service areas may not have any switch

to re-connect them to the remaining energized areas.2 A sector is a set of buses connected by lines without switches.

3 Lines and buses without sectionalizing or tie-switches are inside a sector, thus,they are not shown in DS representations, similar to the one in Fig. 1.

D.S. Sanches et al. / Electrical Power and Energy Systems 62 (2014) 700–711 701



of switching operations is important since the time required by therestoration process basically depends on the number of switching

operations.

The SR problem can be formalized as follows:

Min: /ðGÞ; wðG;G0Þ and cðGÞ

s:t: Ax ¼ b

X ðGÞ 6 1

BðGÞ 6 1

V ðGÞ 6 1

G is a forest;

ð1Þ

where G is a spanning forest of the graph representing a system

configuration [31] (each tree of the forest [32] corresponds to a fee-

der or to an out-of-service area, nodes correspond to sectors andedges to switches); /ðGÞ is the number of consumers that are out-

of-service in a configuration G (considering only the re-connectable

system); wðG; G0Þ is the number of switching operations to reach a

given configuration G from the configuration just after the isolation

of the faulted areas G0; cðGÞ are the power losses, in p.u., of config-

uration G; A is the incidence matrix of G [32]; x is a vector of line

current flow; b is a vector containing the load complex currents

(constant) at buses with bi 6 0 or the injected complex currents

at the buses with bi > 0 (substation); X ðGÞ is called network loading

of configuration G, that is, X ðGÞ is the highest ratio x j= x j, where x j is

the upper bound of current magnitude for each line current magni-

tude x j on line j; BðGÞ is called substation loading of configuration G,

that is, BðGÞ is the highest ratio bs=bs, where bs is the upper bound of

current injection magnitude provided by a substation (s means abus in a substation); V ðGÞ is called the maximal relative voltage

drop of configuration G, that is, V ðGÞ is the highest value of

jv s v kj=d, where v s is the node voltage magnitude at a substation

bus s in p.u. and v k the node voltage magnitude at network bus k in

p.u. (obtained from a SLFA for DSs) and d is the maximum accept-

able voltage drop (in this paper d ¼ 0:1, i.e. the voltage drop is lim-

ited to 10%). The formulation of Eq. (1) can be synthesized by

considering:

Penalties for violated constraints X ðGÞ; BðGÞ and V ðGÞ.

The use of the NDE [14], i.e. an abstract data type for graphs that

can efficiently manipulate a network configuration (spanning

forest) and guarantee that the performed modifications always

produce a new configuration G that is also a spanning forest (afeasible configuration).

The nodes are arranged in the TSO for each produced configura-

tion G in order to solve Ax ¼ b using an efficient SLFA for DSs

[33]. The NDE stores nodes in the TSO. Through x obtained from

a backward sweep, the complex node voltages are calculated

from a forward sweep.

/ðGÞ ¼ 0. The NDE always generates forests that correspond to

networks without out-of-service consumers in the re-connect-

able system.

Eq. (1) can be rewritten as follows:

Min: wðG; G0Þ; cðGÞ and

x x X ðGÞ þxbBðGÞ þ xv V ðGÞ

subject to

G is a forest generated by the NDE ;

Load flow calculated using the NDE

ð2Þ

where x x; xb and xv are weights balancing among the network

operational constraints. In this paper, these weights are set as

follows:

x x ¼

1; if ; X ðGÞ > 1

0; otherwise;

xb ¼ 1; if ; BðGÞ > 1

0; otherwise;

xv ¼ 1; if ; V ðGÞ > 1

0; otherwise:

Addressing the SR problem for large-scale DSs

The SR problem, as formulated in the previous section, is based

on NDE, thus, the efficiency in solving it depends on such encoding.

The NDE operators generate only feasible configurations (radial

configurations able to supply energy for the whole re-connectable

system). As a consequence, such abstract data type does notrequire a specific routine to verify and to correct unfeasible config-

urations. Those aspects enable the construction of new configura-

tions in a fast way for large-scale DSs (average-time complexity

Oð ffiffiffi n

p Þ, where n is the number of sectors in DS). In addition, a SLFA

[20,21] based on the TSO provided by the NDE fast evaluates each

new produced configuration for large-scale DSs [33] (average-time

complexity Oð ffiffiffiffiffi nb

p Þ, where nb is the number of load buses of the

DS).

Moreover, the formulations of Eqs. (1) and (2) correspond to a

Multi-Objective Problem (MOP). MOEAs are among the most rele-

vant methods to deal with MOPs [23,34]. However, these and other

methods have shown success to work with combinatorial MOPs

with at most two objectives. In fact, problems with more objectives

have been called many-objective problems and relatively fewapproaches were developed for them. Fortunately, MOEA com-

bined with the NDE proposed in [14] has properly solved DSR prob-

lems formulated with more than two objectives.

Node-Depth Encoding

A graph G is a pair ðN ðGÞ; E ðGÞÞ, where N ðGÞ is a finite set of ele-

ments called nodes and E ðGÞ is a finite set of elements called edges.

A DS can be represented by graphs, where nodes represent the sec-

tors and edges represent the sectionalizing- and tie-switches. A

tree is a connected and acyclic subgraph of a graph. The depth of

a node is the length of the unique path from the root of its tree

to the node.

The NDE is basically a representation of a graph tree in alinear list containing the tree nodes and their depths. It can be

Fig. 1. Ilustration of DS modeled by a graph and the restoration process: (a) an

original configuration in fault; and (b) a configuration with service restorated.

702 D.S. Sanches et al./ Electrical Power and Energy Systems 62 (2014) 700–711



implemented by an array of pairs ðn x; d xÞ, where n x is the node labeland d x is the node depth in the tree. The order the pairs are

disposed on the linear list is important. A depth search [17] in the

graph spanning tree can produce the proper ordering by inserting

a pair (n x; d x) in the list each time a node n x is visited by the search.

This processing can be executed off-line. Fig. 2 presents a graph,

where thick edges highlight a spanning tree of it and the NDE

corresponding to such spanning tree, assuming node 1 as root

node. The proposed forest representation is composed of the union

of the encodings of all trees that compose the forest. Therefore, the

forest data structure can be easily implemented using an array of

pointers, where each pointer indexes an NDE of a tree.

Two operators were developed to efficiently manipulate a

forest stored in NDEs producing a new one: the Preserve

Ancestor Operator (PAO) and the Change Ancestor Operator(CAO). Each operator modifies the forest encoded by NDE arrays,

which is equivalent to pruning and grafting a sub-tree of a forest

generating a new forest. The CAO produces more complex

modifications than PAO in a forest, as described in [14]. Both

operators are computationally efficient, requiring Oð ffiffiffi n

p Þ average

time for the construction of a new forest. Additional information

about the NDE and its operators applied to DSR problems are

described in [14].

Evolutionary Algorithms with NDE

EAs are stochastic search algorithms based on principles of

natural selection and recombination. They have been used to

solve difficult problems with objective functions that are multi-modal, discontinuous or non-smooth functions. These algorithms

attempt to find the optimal solution to the problem in hand by

manipulating a set (called population) of candidate solutions

(called individuals) [35]. The individuals are evaluated according

to a fitness function (based on a criterion that evaluates solutions

of the problem in hand). Better solutions have a higher probability

of being selected to reproduce, generating a new population.

The process of constructing a population from another is called

generation. After several generations, very fit individuals dominate

the current population, increasing the average quality of the

generated solutions. An EA does not guarantee a global optimal

solution [36]. However, as the literature shows, this technique

often finds useful solutions to relatively complex problems. EAs

have also shown superior performance to work with multiobjectiveproblems [23].

Multi-Objective EA with subpopulation tables (MEAN)

MEAN was proposed in [14] and uses a simple and computa-

tionally efficient strategy to deal with several objectives and con-

straints. The basic idea is to subdivide a population into

subpopulation tables related to different objectives and con-

straints. The MEAN is different from VEGA (Vector Evaluated

Genetic Algorithm [37]), since it adds a fundamental subpopula-tion table that stores individuals assessed by at least one aggrega-

tion function (see Eq. (3)), moreover, any individual can be

simultaneously evaluated using weighted (by table(s) of aggrega-

tion function(s)) and non-weighted scores (through the remaining

tables) from objectives and no additional heuristic is required to

induce middling values as proposed in [37]. The ability of simulta-

neously searching for the extreme points of the Pareto-front4 and

the best values of the aggregation function makes MEAN more sim-

ilar to the MOEA/D [38].

The whole set of tables is organized as follows:

1. Tables associated with each objective and constraint:

(a) T 1 – solutions with low cðGÞ;

(b) T 2 – solutions with low V ðGÞ;

(c) T 3 – solutions with low X ðGÞ;

(d) T 4 – solutions with low BðGÞ;

(e) T 5 – solutions with low values of an aggregation function,

defined as follows:

f agg ðGÞ ¼ wðG;G0Þ þ cðGÞ þ x x X ðGÞ þ xbBðGÞ

þ xv V ðGÞ; ð3Þ

where wðG; G0Þ; cðGÞ; X ðGÞ; BðGÞ; V ðGÞ; x x; xb and xv were

defined in ‘Service restoration problem’5;2. Tables denoted T 5þ p that are related to the required pair of

switching operations after fault isolation:

(a) Each Table T 5þ p, with p = 1,. . . , 5, stores the best solutions

found with more than p 1 and at most p pairs of switchingoperations. In these tables the solutions are ranked (in

increasing order) according to the value of V ðGÞ þ X ðGÞ. Solu-

tions with similar value, considering precision 102, are ran-

domly ranked.

A new generated individual (I new) is included in subpopulation

table T i if this table is not full or if I new is better than the worst solu-

tion in T i, then replacing it. Note that the same individual may be

included in more than one subpopulation table. The MEAN requires

the following input parameters:

Gmax is the maximum number of individuals generated by the

MEAN. It is also used as the stopping criterion;

S Ti is the size of the subpopulation table T i indicating how many

individuals can be stored in T i, with i ¼ 1; . . . ; 10.

The reproduction operators used to generate new individuals

are the NDE operators PAO and CAO. First a solution is selected

from the subpopulation tables as follows: a subpopulation T iis randomly chosen, then, an individual from it is randomly

picked up. Next, PAO or CAO (according to a dynamic probability

[14]) is applied to such individual, generating a new one, I new.

Subpopulation table T i receives I new if T i is not full (since T ihas size bounded by S T i ) or if I new is better (according to the cri-

Fig. 2. (a) A graph, a spanningtree(thickedges) and (b) its NDR assuming node1 as

root.

4 Pareto front contains the non-dominated solutions of the whole set of found

solutions. A solution G i dominates another G j if Gi is better than G j according to at

least one objective and Gi is not worse than G j in all other objectives.

5 Note that all configurations generated by MEAN are feasible, that is, they areradial networks able to supply energy for the whole re-connectable system.


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terion associated with T i) than the worst solution in T i, then

replacing it.

Multi-Objective EA based on Decomposition (MOEA/D)

MOEA/D is a Multi-Objective EA that uses a technique of

decomposition [38] of a problem into subproblems. This algorithmsimultaneously optimizes V single objective subproblems, each of

them corresponds to an aggregation function. MOEA/D usually

employs the Tchebycheff approach [39] for the decomposition of

a Multi-Objective Problem into subproblems. A coefficient vector

ki defines each aggregation function and a set with the U coefficient

vectors that are the closest to ki in fk0; . . . ;kV g composes the neigh-

borhood of k i [38].

The coefficient vectors should spread uniformly in the objective

space. The number of vectors is V ¼ C o1H þo1, where o is the number

of problem objectives and H þ1 is the size of weight set0H

; 1H

; . . . ; H H

used to construct coefficient vectors. As a conse-

quence, a tradeoff between o and H should be found in order to

bound V , generating a number of subproblems that is computa-

tionally tractable.

At the first generation, MOEA/D generates a random population.

In the next generations, genetic operators applied on solutions of

the same neighborhood produce new solutions. Then, the algo-

rithm updates solutions of each neighborhood. The non-dominated

solutions from the population compose what is called External

Population (EP), completing a generation of MOEA/D or finishing

it when a stop criterion is achieved.

In relation to MEAN, MOEA/D requires the additional parame-ters U and H . To work with the SR problem, we adapted MOEA/D

to use NDE, which was called MOEA/D-NDE.

Evolutionary history recombination

In this section the recombination operator for the NDE is

described: Evolutionary History Recombination (EHR) proposed

in [40]. As the name of the operator suggests, EHR is based on

the evolutionary history of the operators PAO and CAO, that is,

on the sequence of vertices ( p; a), for PAO, and ( p; r ; a),X for CAO,

applied in the generation of new individuals. The history of each

individual can be retrieved by using the auxiliary structures from

NDE: matrix P x, which stores the positions of node x in each indi-

vidual, and array p, which stores the ancestor of each individual.

Fig. 3. History of applications of PAO and CAO operators.




Fig. 3 shows an example of the evolutionary history of succes-

sive applications of PAO and CAO operators. In this figure, starting

from forest F0 and applying the operator PAO using vertices p = F

and a = B, we obtain forest F1 (left). Forests F2 and F3 (right) are

generated from the application of the operator CAO to F0 (with ver-

tices p = E , r = D and a = H ) and to F2 (with vertices p = G, r = J and

a = A) respectively.

In order to simplify the utilization of EHR, we propose a modi-fication in array p , called pm, such that it can store not only the

index of the ancestor but also a triple of nodes (a; r and p) that were

used in the application of the operator PAO or CAO (in the case of

PAO, the value in r is null). In this way, a sequence of movements to

generate individual F i from any ancestor can be accessed from pm.

EHR – illustrative example

To better illustrate EHR, let us consider the DS in Fig. 4, consist-

ing of 3 feeders. The NDE of each feeder is shown in Fig. 5.

Figs. 6 and 7 show two individuals randomly selected in the

population. They were generated by operators PAO and CAO,

respectively. These two individuals have a common ancestor (see

Fig. 8), which is the forest shown in Fig. 4. PAO with p

¼ 11 and

a ¼ 17 generates individual 1 and CAO with p ¼ 21; r ¼ 20 and

a ¼ 14 produces individual 2 from the common ancestor. Combin-

ing these two modifications, it is possible to obtain a new individ-

ual such as the one shown in Fig. 9.

Performance assessment

The performance of MOEAs is usually assessed by the quality of

the approximated Pareto fronts found by the algorithms. In gen-

eral, three characteristics are taken into account to evaluate an

approximated Pareto front: (1) proximity to the Pareto-optimal

front, (2) diversity of solutions along the front and (3) uniformity

of solutions along the front. These three criteria guide the search

to a high-quality and diversified set of solutions which enable

the choice of the most appropriate solution in a posterior deci-

sion-making process [41].

Fig. 4. DS with 3 feeders modeled by a graph with three trees.

Fig. 5. NDE for the feeders in Fig. 4.

Fig. 6. Individual 1 generated by PAO.

Fig. 7. Individual 2 generated by CAO.

Fig. 8. Common ancestor of the individuals in Figs. 6 and 7.

Fig. 9. New individual obtained from the combination of modifications thatgenerated other two individuals from a common ancestor.


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To quantify these three characteristics in a set of non-

dominated solutions, various metrics have been developed, for

instance, Error Ratio [42], Generational Distance [42], the R2

and R3 [43], Hypervolume (HV) [44] and -indicator [44]. In this

paper, HV is used to assess the performance of the proposed

approach (denominated MEAN-DE, that is, MEAN with DE) and

MEAN.

The HV metric uses the covered volume, dominated by anapproximated Pareto front PF , as a measure of quality of such front.

The calculus of the covered volume requires a reference point,

which usually consists of an anti-utopian point or ‘‘worst values’’

point in the objective space [45]. For each generated decision vec-

tor v ec i a hypercube v oli is constructed in relation to the reference

point, after this, the hypercubes of all decision vectors are joined.

Higher values of Hypervolume are expected to mean a larger scat-

tering of solutions and a better convergence to the true Pareto

front. The HV of PF is given by Eq. (4) [41].

HV ¼Xi

v oli; v ec i 2 PF : ð4Þ

Discrete DE with movements list

In [46] the authors propose an optimization approaches for the

Differential Evolution, called Discrete Differential Evolution algo-

rithm with List of Movements (DDELM). DDELM was applied to

combinatorial problems where the operators difference, addition,

and product by scalar in the differential mutation equation are

redefined in the space of discrete variables.

The difference between two candidate solutions is a list of

movements in the search space defined as:

Definition 1. A list of movements M ij is a list containing a

sequence of valid movements mk such that the application of

these movements to a solution si 2 S leads to the solution s j 2 S ,

where S is the search space, that is, the set of all possible

combinations of values for the variables.In this way, the ‘‘difference’’ between two solutions is defined

as being the corresponding list of movements:

M ij ¼ si s j; ð5Þ

where is a special binary minus operator that returns a list of

movements M ij that represents a path from si towards s j. This list,

in some sense, captures the differences between these two

solutions.

The multiplication of the list of movements by a constant is

defined as:

Definition 2. The multiplication of the list of movements, M ij, by a

constant F

2 ½0; 1

, returns a list M 0ij with the F

M ij

movements

of M ij, where M ij is the size of the list.

Thus, the multiplication of the list of movements by a constant

can be denoted, using the special binary multiplication operator ,

by:

M 0ij ¼ F M ij: ð6Þ

Finally, the application of a list of movements to a given solu-

tion is defined as follows:

Definition 3. The application of the sequence of movements in the

list M 0ij into a solution sk, returns a new solution s 0k

:

s0k ¼ sk M 0ij: ð7Þ

With the definition above, one can generate a mutant vectordefined as:

v i ¼ x0 F ð x1 x2Þ

v i ¼ x0 F M 12

v i ¼ x0 M 012;

which is the proposed discrete version of the typical differential

mutation equation.

We emphasize that in this paper we extend the ideas in [46] byproposing a list of movements based on the NDE, which is suitable

for representing candidate solutions in DSR problems.

Proposed approach

The proposed approach is called MEAN-DE, which consists

basically of MEAN and the mutation operator of DE re-designed

from the EHR operator. In other words, the list of movements

[46] is obtained from the application of EHR operator. In this sense,

the difference between any two individuals x1 and x 2 is a list of

movements M 12 composed by a sequence of triples ð p; ; aÞ and

ð p; r ; aÞ obtained from pm (‘Evolutionary history recombination’).

In fact, the list is the concatenation of two sequences: one from

x1 to xc and another from xc to x2, where xc is their commonancestor.

Thus, the EHR can be used to implement a discrete differential

mutation operator that is computationally efficient (Oð ffiffiffi n

p Þ in aver-

age). The implementation is straightforward as follows:

Be x0; x1, and x2 three individuals randomly selected from the

current population to participate in the differential mutation

equation:

v i ¼ x0 F ð x1 x2Þ

v i ¼ x0 F M 12;ð8Þ

where the list of movements M 12 is obtained from the history of

applications of PAO, CAO and EHR, which are stored into the mod-

ified array pm.

To illustrate the proposed differential mutation operator basedon EHR, consider the tree representation of the common ancestor

xc (Fig. 10) of individuals x1 and x2 (shown in Figs. 11 and 12,

respectively) generated through the application of CAO and PAO.

Individual 1 comes from ancestor xc by the following sequence of

PAO applications (11,_,17), (7,_,6) and (24,_,23). Individual 2

derives from xc by applying CAO as follows: ð21; 20; 14Þ, and

ð11; 12; 13Þ.

Thus, with the aid of the EHR, the list of movements for each

individual is written as:

M c 1 ¼ ð11; ; 17Þ ð7; ; 6Þ ð24; ; 23Þ½

M c 2 ¼ ð21; 20; 14Þ ð11; 12; 13Þ½ ð9Þ

The list of movements M 12 between individuals x1 and x2 is the

junction of this two lists. M 12 is built by choosing alternately one

Fig. 10. Common ancestor xc of individuals x1 and x2.


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movement from each list of the individuals in order to avoid a bias

of the movements list from one only individual. So, the movement

list M 12 is given by:

M 12 ¼ ½ð11; ; 17Þ; ð21; 20; 14Þ; ð7; ; 6Þ; ð11; 12; 13Þ; ð24; ; 23Þ

ð10Þ

Assuming that F ¼ 0:6 [46], the number of movements used from

M 12 is dF jM 12je ¼ d0:6 5e ¼ 3. Thus M 012 results in:

M 012 ¼ ð11; ; 17Þ; ð21; 20; 14Þ; ð7; ; 6Þ½ : ð11Þ

The mutant vector v i is obtained by applying each movement of M 012

into the base vector x0:

v i ¼ x0 M 012: ð12Þ

Thus, using x0 from Fig. 13 and M 012 obtained above, the resultant

mutant vector is the one shown in Fig. 14.

In summary, the proposed approach allows the implementation

of the differential mutation operator for DSR problems by using the

EHR operator to build the list of movements as proposed in [46].

The list of movements captures the differences between any two

individuals and can be scaled and applied to the base individual

in order to generate a mutant solution. MEAN-DE employs the dif-

ferential mutation operator as the search engine within the MEAN

framework.

Experimental analyses

In order to analyze how the MEAN and MEAN-DE approaches

behave with the increase in the network size for the SR problem,

the real Sao Carlos city DS (System 1 hereafter) was used to com-

pose other three DSs with size varying from two to eight times

the original DS. System 2 is composed of two Systems 1 intercon-

nected by 13 NO new additional switches. System 3 is composed of

four Systems 1 interconnected by 49 NO new additional switches.

Finally, System 4 is composed of eight Systems 1 interconnected by

110 NO new additional switches (the data of the four DSs are avail-

able in [47]).Those DSs have the following general characteristics:

System 1 (S1): 3860 buses, 532 sectors, 632 switches (509 NC

and 123 NO switches), 3 substations, and 23 feeders.

System 2 (S2): 7720 buses, 1064 sectors, 1277 switches (1018

NC and 259 NO switches), 6 substations, and 46 feeders.

System 3 (S3): 15,440 buses, 2128 sectors, 2577 switches (2036


System 4 (S4): 30,880 buses, 4256 sectors, 5166 switches (4072


The approaches MEAN, MEAN-DE and MOEA/D-NDE were run

using a Core 2 Quad 2.4 GHz, 8G RAM, with Linux Operating Sys-

tem Ubuntu 10.04 version, and the language compiler C gcc-4.4.Parameters of MEAN and MEAN-DE are the subpopulation table

sizes, which were all setup to S T i ¼ 5. MEAN, MEAN-DE and MOEA/

D-NDE used dynamic probability of PAO and CAO applications. We

evaluated different values of parameters U and H of MOEA/D-NDE

in order to keep the total number of evaluations closest to the

number used in MEAN and MEAN-DE. Some sets of values did

not found feasible solutions in all runs, thus, they were discarded.

Among the sets of values U and H that found feasible solutions in

all runs, we chose the set that corresponded to the smallest num-

ber of switching operations, returning U ¼ 30 and H ¼ 10.

All the tests refers to a fault at the largest feeder in Systems 1, 2,

3 and 4, interrupting the service for the whole feeder. The experi-

ments with these DSs evaluated the approaches according to: (a)

the performance of them for the SR problem; and (b) the relativeperformance of those MOEAs concerning HV. MEAN, MEAN-DE

Fig. 11. Individual x1 generated by three applications of the PAO operator into the

common ancestor of Fig. 10.

Fig. 12. Individual x2 generated by three applications of the CAO operator into the

common ancestor of Fig. 10.

Fig. 13. Individual x0, base vector, randomly choosing in a current population.

Fig. 14. The result mutant vector v i.


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and MOEA/D-NDE were run 50 times (with different seeds for the

used random number generator) for each test case. Each run eval-

uated 100,000 solutions. In this paper the MEAN, MEAN-DE and

MOEA/D-NDE approaches will be search for SR plans which restore

the entire out-of-service area (full restoration cases) respecting

radiality and all the operational constraints (voltage drop, substa-

tion and network loading).

Table 1 enables a comparison of MEAN, MEAN-DE and MOEA/D-NDE for S1 according to the number of switching operations for SR

plans (the most critical aspect for the SR problem). Those results

concern only the feasible solutions with the smallest number of

switching operations found by each approach in each run. Clearly,

MEAN-DE and MEAN outperform MOEA/D-NDE according to the

number of switching operations for SR plans. The Wilcoxon rank-

sum test (‘Experimental statistical analyses’) confirmed the signif-

icance of such difference ( p-value equal to 0.001).

Table 2 presents results concerning other electrical aspects.

Some aspects in the solutions from MOEA/D-NDE are improved

in relation to solutions from MEAN and MEAN-DE, nevertheless,

such benefits costed higher number of switching operations.

The next experiments focus on comparing MEAN and MEAN-DE,

since they outperform MOEA/D-NDE according to the number of

switching operations for SR plans. First it was evaluated how the

MEAN and MEAN-DE approaches behave with the increase in the

DS size for the SR problem. Results for MEAN from Table 3 show

that it cannot find the same solutions with the increase in the DS

size (note that the system SN is composed of N systems S1). Such

behavior indicates that finding adequate configurations for very

large networks is hard, since the space of feasible configurations

is huge.

On the other hand, Table 3 also shows that MEAN-DE can deal

with relatively complex networks finding lower number or switch-

ing operations and reaching the smallest number (seven) found for

S1. It is important to highlight that initially were required three

switching operations to isolated the faulted areas at the largest fee-

der in S1, S2, S3 and S4.

Tables 4–6 synthesize other electrical aspects of the best solu-tions found by both MEAN and MEAN-DE for each objective and

constraint. Basically, they emphasize that the found solutions are

all feasible and do not significantly differ from each other accord-

ing to those aspects, thus, the critical aspect is the number of

switching operations, as shown by Table 3.

Results using HV metric

The analysis of the results, according to the HV metric used to

compare MOEAs, shows that MEAN-DE outperforms MEAN in the

four test problems (Systems 1, 2, 3 and 4) preserving a diverse

set of nondominated solutions (see Figs. 15–18).The distribution of HV values for Systems 1, 2, 3 and 4 are

shown in Figs. 19–22. MEAN-DE found in average larger HV for

all systems, indicating that in general it obtains fronts more diverse

and uniformly distributed when compared with MEAN.

Table 1

Simulations with single fault in System 1.

Switching operations

MEAN-DE MEAN MOEA/D-NDE

S1

Minimum 7 7 9

Average 9 13 19

Maximum 11 29 73

Standard deviation 1.46 5.48 13.41

Table 2


MEAN-DE MEAN MOEA/D-NDE

Av. Std Av. Std. Av. Std.

Power losses (kW) 377.2 29.8 353.8 36.1 370.19 28.6

Voltage ratio (%) 4.1 0.8 3.8 0.7 3.3 0.07

Network load (%) 77.7 7.7 74.4 8.4 86.3 6.9

Substation load (%) 53.9 2.1 53.3 1.5 53.1 2.9

Running time (s) 13.6 0.2 12.8 1.2 5.7 0.2

Table 3

Simulations with single fault in Systems 2, 3 and 4.

Switching operations

MEAN-DE MEAN

S2

Minimum 7 9

Average 16 17

Maximum 69 47

Standard deviation 10.73 7.51

S3

Minimum 7 11Average 20 25

Maximum 77 107


S4

Minimum 7 11

Average 24 36

Maximum 79 181


Table 4

Simulations with single in System 2.

MEAN-DE MEAN

Av. Std. Av. Std.Power losses (kW) 639.9 40.8 636.3 45.7

Voltage ratio (%) 3.9 0.7 3.7 0.7

Network load (%) 79.4 8.3 75.2 10.4

Substation load (%) 55.1 1.9 54.4 1.7

Running time (s) 14.7 0.2 13.7 1.6

Table 5


MEAN-DE MEAN

Av. Std. Av. Std.

Power losses (kW) 1195.6 50.8 1191.6 69.5

Voltage ratio (%) 3.7 0.6 3.9 0.8Network load (%) 80.7 10.1 83.1 9.5

Substation load (%) 55.2 1.9 57.1 4.8

Running time (s) 16.3 0.6 14.9 0.4

Table 6

Simulations with Single Fault in System 4.

MEAN-DE MEAN

Av. Std. Av. Std.

Power losses (kW) 2321.4 54.3 2301.2 90.91

Voltage ratio (%) 3.4 0.3 3.7 0.7

Network load (%) 85.2 8.6 84.6 10.5

Substation load (%) 55.8 3.1 56.8 5.2

Running time (s) 17.1 0.9 15.5 0.5




Experimental statistical analyses

The HV values obtained for MEAN and MEAN-DE (‘Results using

HV metric’) are not normally distributed, moreover, the variance isnoticeably different for different MOEAs [42]. This data may not

satisfy necessary assumptions for parametric mean comparisons

[48]. Thus, the use of a non-parametric statistical technique for

comparing results (HV) from both MOEAs is interesting.

In this sense, Wilcoxon rank-sum test can compare those results

to determine the MOEA that achieved better performance. This testassumes the sample of differences is randomly selected and the

MEAN

MEAN−DE

Fig. 15. Pareto front obtained from System 1.




Fig. 19. Box plots of HV values obtained from System 1.



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probability distributions (from which the sample of paired differ-

ences is drawn) is continuous [49]. The found HV values meet

these criteria, then the following hypotheses were tested:

HV 0: The probability distributions of HV values obtained by

MEAN-DE and MEAN from the tests with the four DSs are

identical.

HV 1: Distributions of the HV values differ between MEAN-DE

and MEAN.

The Wilcoxon rank-sum test results are termed p-values, also

called observed significance levels. We reject the null hypothesis

whenever p-values 6 a. Using a significance level a ¼ 0:05 applied

to HV results from the four systems, all p-values obtained are equal

or less than 0.02 (see Table 7). Thus, there is enough evidence to

support the alternative hypothesis and conclude that these tests

show a significant statistical difference between the MEAN-DEand MEAN for HV metric.

Conclusions

This paper presented a new MOEA using NDE with a powerful

differential mutation operator to solve the SR problem in large-

scale DSs (i.e., DSs with thousands of buses and switches).

The proposed approach, called MEAN-DE, combines the main

characteristics of MEAN, EHR operator and list of movements of

the DDELM proposed in [46]. MEAN and MEAN-DE are both basedon the strategy of subpopulation tables to deal with multiobjective

optimization involving more than two objectives. The MEAN-DE

uses a new differential mutation operator structured from the

EHR operator, providing computational efficiency in order to deal

with relatively large DSs.

In addition, a MOEA using subproblem Decomposition and NDE

(MOEA/D-NDE) was investigated.

In the experiments, the approaches MEAN, MOEA/D-NDE and

MEAN-DE were applied to DS called S1, with 3860 buses. Results

indicated that MOEA/D-NDE requires in general more switching

operations for SR plans than MEAN and MEAN-DE, what is critical

since the number of switching operations must be reduced for SR.

MEAN and MEAN-DE presented similar results for S1.

Performances of MEAN and MEAN-DE were also evaluated for

other three larger Systems: S2 (7720 buses), S3 (15,440 buses)

and S4 (30,880 buses). The results show that they enabled SR in

large-scale DSs and solutions were found where: energy was

restored to the entire out-of-service area after a feeder-fault in

the largest feeder of each DS, the operational constraints were sat-

isfied and a reduced number of switching operations was obtained.

Moreover, armed with the relatively low running time required by

restoration plans for all the tested systems, we can conclude that

those approaches can generate appropriately SR plans for large-

scale DSs.

A statistical analysis performed using 50 experiments for each

approach shows MEAN-DE performs better than MEAN for the

tested DSs. MEAN-DE has obtained the best average results for

lower switching operations while preserving the diversity of

solutions.To measure the quality of obtained solutions of MEAN and

MEAN-DE, the HV metric was used. According to the simulation

results, MEAN-DE showed a better performance in terms of HV in

relation to MEAN. The nonparametric statistical analysis, the Wil-

coxon rank-sum test, showed MEAN-DE is better than MEAN when

applied to the four DSs tested.

Finally, this work provided interesting basis for combining

promising aspects of different EA approaches into a new approach

that shows relevant performance on all tested DSs.

Acknowledgments

The authors would like to acknowledge CAPES, CNPq, FAPEMIG

and FAPESP for the financial support given to this research. The

authors would like to acknowledge especially Professor Oriane

M. Neto by numerous contributions given to the paper. In memo-

riam prof. Oriane Magela Neto.

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