Multiobjective evolutionary algorithm with a ... - Semantic Scholar

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

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Electrical Power and Energy Systems journal homepage: www.elsevier.com/locate/ijepes

Multiobjective evolutionary algorithm with a discrete differential mutation operator developed for service restoration in distribution systems 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, Brazil São Carlos School of Engineering, University of São Paulo, São Carlos, SP, Brazil c Institute of Mathematical and Computing Sciences, University of São Paulo, São Carlos, SP, Brazil d Federal Institute of Minas Gerais, Ouro Preto, Brazil e Department of Electrical Engineering, Universidade Federal de Minas Gerais, UFMG, Brazil f Institute of Informatics, Universidade Federal de Goias, UFG, Brazil b

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 Encoding Multi-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 optimization 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 distribution 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 Differential 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 ranksum test. In addition, a MOEA using subproblem Decomposition and NDE (MOEA/D-NDE) was investigated. MEAN-DE has shown the best average results in relation to MEAN and MOEA/D-NDE. Ó 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]. As a 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 ⇑ 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). http://dx.doi.org/10.1016/j.ijepes.2014.05.008 0142-0615/Ó 2014 Elsevier Ltd. All rights reserved.

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 dramatically affected by the data structure used to represent computationally 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 operators 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 Prufer number to encode the chromosomes. The Prufer number encoding ensures system radiality avoiding the tedious ‘‘mesh check’’ algorithms, which are required to identify the existence of loops

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

(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 minimize 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 corresponding genetic operators [19]. As it was shown in those references, 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 configurations, that is, radial configurations able to supply energy for the whole re-connectable system1; (ii) The NDE can generate significantly more feasible configurations (potential solutions) in relation to other encoding in the same running time since its average time pffiffiffi complexity is Oð nÞ, where n is the number of graph nodes (each graph node corresponds to a DS sector2); (iii) The NDE-based formulation 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 objective function and the network operational constraints of a DSR problem can be analyzed solving a radial load-flow. The approach proposed in [15] uses NDE together with a conventional 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 problems 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 more easily leave the local toward global optima. Simulation results presented 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 literature. The Differential Evolution (DE) [26] has received increased interest from the Evolutionary Computation community, since DE has shown better performance over other well-known metaheuristics like Genetic Algorithm (GA) and Particle Swarm Optimization (PSO) [27]. The DE can identify promising regions of the search space by a relatively simple operator (called differential mutation) that generates new solutions (called individuals) from random calculation 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 performance has not been evaluated for a significant number of problems. Moreover, the available versions of DE for combinatorial optimization in the literature focus on permutation-based combinatorial problems, such as the job shop scheduling, flow shop

1 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.

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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. Moreover, the average time complexity of the proposed operator is pffiffiffi Oð nÞ, enabling efficient manipulation of large-scale networks. In addition, 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 MEANDE 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 largescale DSs’ addresses the SR problem for large-scale DSs; ‘Evolutionary 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 movements list’ presents the Discrete Differential Evolution algorithm with list of movements proposed in [46]; ‘Proposed approach’ proposed the MEAN-DE; ‘Experimental analyses’ evaluates and compares MEAN-DE to the MEAN and MOEA/D-NDE approaches; finally, ‘Conclusions’ summarizes the main contributions and concludes the paper. Service restoration problem Next sections present the nomenclature and the general formulation 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 represents sectors.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 identified 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 number 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.

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The nodes are arranged in the TSO for each produced configuration 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-connectable system. Eq. (1) can be rewritten as follows:

wðG; G0 Þ; cðGÞ and

Min:

xx XðGÞ þ xb BðGÞ þ xv VðGÞ subject to

ð2Þ

G is a forest generated by the NDE; Load flow calculated using the NDE 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.

of switching operations is important since the time required by the restoration process basically depends on the number of switching operations. The SR problem can be formalized as follows:

Min:

/ðGÞ; wðG; G0 Þ and cðGÞ

s:t:

Ax ¼ b XðGÞ 6 1 BðGÞ 6 1

where xx ; xb and xv are weights balancing among the network operational constraints. In this paper, these weights are set as follows:

xx ¼ xb ¼ xv ¼

ð1Þ

VðGÞ 6 1 G is a forest; where G is a spanning forest of the graph representing a system configuration [31] (each tree of the forest [32] corresponds to a feeder or to an out-of-service area, nodes correspond to sectors and edges to switches); /ðGÞ is the number of consumers that are outof-service in a configuration G (considering only the re-connectable system); wðG; G0 Þ 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 configuration 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); XðGÞ is called network loading of configuration G, that is, XðGÞ is the highest ratio xj =xj , where xj is the upper bound of current magnitude for each line current magnitude xj 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 a bus in a substation); VðGÞ is called the maximal relative voltage drop of configuration G, that is, VðGÞ is the highest value of jv s v k j=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 acceptable voltage drop (in this paper d ¼ 0:1, i.e. the voltage drop is limited to 10%). The formulation of Eq. (1) can be synthesized by considering: Penalties for violated constraints XðGÞ; BðGÞ and VðGÞ. 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 (a feasible configuration).

1; if; XðGÞ > 1

0; otherwise; 1; if; BðGÞ > 1

0; otherwise; 1; if; VðGÞ > 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 not require a specific routine to verify and to correct unfeasible configurations. Those aspects enable the construction of new configurations in a fast way for large-scale DSs (average-time complexity pffiffiffi Oð nÞ, 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 pffiffiffiffiffi complexity Oð nb 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 relevant 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 few approaches were developed for them. Fortunately, MOEA combined with the NDE proposed in [14] has properly solved DSR problems formulated with more than two objectives. Node-Depth Encoding A graph G is a pair ðNðGÞ; EðGÞÞ, where NðGÞ is a finite set of elements called nodes and EðGÞ is a finite set of elements called edges. A DS can be represented by graphs, where nodes represent the sectors 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 a linear list containing the tree nodes and their depths. It can be


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Multi-Objective EA with subpopulation tables (MEAN)

Fig. 2. (a) A graph, a spanning tree (thick edges) and (b) its NDR assuming node 1 as root.

implemented by an array of pairs ðnx ; dx Þ, where nx is the node label and dx 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 (nx ; dx ) in the list each time a node nx 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 pffiffiffi operators are computationally efficient, requiring Oð nÞ 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 multimodal, 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 multiobjective problems [23].

MEAN was proposed in [14] and uses a simple and computationally efficient strategy to deal with several objectives and constraints. The basic idea is to subdivide a population into subpopulation tables related to different objectives and constraints. The MEAN is different from VEGA (Vector Evaluated Genetic Algorithm [37]), since it adds a fundamental subpopulation table that stores individuals assessed by at least one aggregation function (see Eq. (3)), moreover, any individual can be simultaneously evaluated using weighted (by table(s) of aggregation 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 simultaneously searching for the extreme points of the Pareto-front4 and the best values of the aggregation function makes MEAN more similar 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 VðGÞ; (c) T 3 – solutions with low XðGÞ; (d) T 4 – solutions with low BðGÞ; (e) T 5 – solutions with low values of an aggregation function, defined as follows:

fagg ðGÞ ¼ wðG; G0 Þ þ cðGÞ þ xx XðGÞ þ xb BðGÞ þ xv VðGÞ;

ð3Þ

0

where wðG; G Þ; cðGÞ; XðGÞ; BðGÞ; VðGÞ; xx ; xb and xv were defined in ‘Service restoration problem’5; 2. Tables denoted T 5þp that are related to the required pair of switching operations after fault isolation: (a) Each Table T 5þp , with p = 1,. . . , 5, stores the best solutions found with more than p 1 and at most p pairs of switching operations. In these tables the solutions are ranked (in increasing order) according to the value of VðGÞ þ XðGÞ. Solutions with similar value, considering precision 102 , are randomly ranked. A new generated individual (Inew ) is included in subpopulation table T i if this table is not full or if Inew is better than the worst solution 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; STi 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 i is 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, Inew . Subpopulation table T i receives Inew if T i is not full (since T i has size bounded by ST i ) or if Inew is better (according to the cri4 Pareto front contains the non-dominated solutions of the whole set of found solutions. A solution Gi dominates another Gj if Gi is better than Gj according to at least one objective and Gi is not worse than Gj in all other objectives. 5 Note that all configurations generated by MEAN are feasible, that is, they are radial 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 algorithm simultaneously 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 neighborhood of ki [38]. The coefficient vectors should spread uniformly in the objective space. The number of vectors is V ¼ C o1 Hþo1 , where o is the number of problem objectives and H þ 1 is the size of weight set 0 1 ; ; . . . ; HH used to construct coefficient vectors. As a conseH H quence, a tradeoff between o and H should be found in order to bound V, generating a number of subproblems that is computationally 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 algorithm 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 parameters 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 Px , which stores the positions of node x in each individual, 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 successive 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 vertices 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 modification 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, consisting 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

705

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. Combining these two modifications, it is possible to obtain a new individual 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 general, 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 decision-making process [41].

Fig. 7. Individual 2 generated by CAO.

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

Fig. 8. Common ancestor of the individuals in Figs. 6 and 7. Fig. 5. NDE for the feeders in Fig. 4.

Fig. 6. Individual 1 generated by PAO.

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

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To quantify these three characteristics in a set of nondominated 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 an approximated 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 vector v eci 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 scattering of solutions and a better convergence to the true Pareto front. The HV of PF is given by Eq. (4) [41].

HV ¼

X

v oli ; v eci 2 PF:

ð4Þ

i

Discrete DE with movements list In [46] the authors propose an optimization approaches for the Differential Evolution, called Discrete Differential Evolution algorithm 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 sj 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 sj ;

ð5Þ

where is a special binary minus operator that returns a list of movements Mij that represents a path from si towards sj . 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 Mij , where M ij is the size of the list.

v i ¼ x0 F ðx1 x2 Þ v i ¼ x0 F M12 v i ¼ x0 M012 ; which is the proposed discrete version of the typical differential mutation equation. We emphasize that in this paper we extend the ideas in [46] by proposing 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 x2 is a list of movements M 12 composed by a sequence of triples ðp; ; aÞ and ðp; 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 common ancestor. Thus, the EHR can be used to implement a discrete differential pffiffiffi mutation operator that is computationally efficient (Oð nÞ in average). 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 ðx1 x2 Þ v i ¼ x0 F M12 ;

where the list of movements M12 is obtained from the history of applications of PAO, CAO and EHR, which are stored into the modified array pm . To illustrate the proposed differential mutation operator based on 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:

Mc1 ¼ ½ð11; ; 17Þ ð7; ; 6Þ ð24; ; 23Þ Mc2 ¼ ½ð21; 20; 14Þ ð11; 12; 13Þ

ð6Þ

Finally, the application of a list of movements to a given solution 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 s0k :

s0k ¼ sk M 0ij :

ð7Þ

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

ð9Þ

The list of movements M 12 between individuals x1 and x2 is the junction of this two lists. M12 is built by choosing alternately one

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 :

ð8Þ

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

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Fig. 11. Individual x1 generated by three applications of the PAO operator into the common ancestor of Fig. 10.

Fig. 14. The result mutant vector

vi.

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 differential mutation operator as the search engine within the MEAN framework.

Experimental analyses

Fig. 12. Individual x2 generated by three applications of the CAO operator into the common ancestor of Fig. 10.

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 M12 is given by:

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

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 compose other three DSs with size varying from two to eight times the original DS. System 2 is composed of two Systems 1 interconnected 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 available in [47]). Those DSs have the following general characteristics:

ð10Þ Assuming that F ¼ 0:6 [46], the number of movements used from M12 is dF jM12 je ¼ d0:6 5e ¼ 3. Thus M012 results in:

M 012 ¼ ½ð11; ; 17Þ; ð21; 20; 14Þ; ð7; ; 6Þ:

ð11Þ

The mutant vector v i is obtained by applying each movement of M012 into the base vector x0 :

v i ¼ x0 M012 :

ð12Þ M012

Thus, using x0 from Fig. 13 and 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].

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

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 NC and 541 NO switches), 12 substations, and 92 feeders. System 4 (S4): 30,880 buses, 4256 sectors, 5166 switches (4072 NC and 1094 NO switches), 24 substations, and 184 feeders. The approaches MEAN, MEAN-DE and MOEA/D-NDE were run using a Core 2 Quad 2.4 GHz, 8G RAM, with Linux Operating System 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 ST 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 number 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 experiments with these DSs evaluated the approaches according to: (a) the performance of them for the SR problem; and (b) the relative performance of those MOEAs concerning HV. MEAN, MEAN-DE

708


and MOEA/D-NDE were run 50 times (with different seeds for the used random number generator) for each test case. Each run evaluated 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, substation and network loading). Table 1 enables a comparison of MEAN, MEAN-DE and MOEA/DNDE 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 ranksum test (‘Experimental statistical analyses’) confirmed the significance 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 switching 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 feeder in S1, S2, S3 and S4. Tables 4–6 synthesize other electrical aspects of the best solutions 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 according to those aspects, thus, the critical aspect is the number of switching operations, as shown by Table 3.

Table 2 Simulations with single fault in System 1.

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.


MEAN-DE

Power losses (kW) Voltage ratio (%) Network load (%) Substation load (%) Running time (s)

S1 Minimum Average Maximum Standard deviation

7 9 11 1.46

MOEA/D-NDE

Av.

Std

Av.

Std.

Av.

Std.

377.2 4.1 77.7 53.9 13.6

29.8 0.8 7.7 2.1 0.2

353.8 3.8 74.4 53.3 12.8

36.1 0.7 8.4 1.5 1.2

370.19 3.3 86.3 53.1 5.7

28.6 0.07 6.9 2.9 0.2

Table 3 Simulations with single fault in Systems 2, 3 and 4. Switching operations MEAN-DE

MEAN


7 16 69 10.73

9 17 47 7.51


7 20 77 14.07

11 25 107 20.11


7 24 79 20.08

11 36 181 36.81

Table 4 Simulations with single in System 2. MEAN-DE


MEAN

Av.

Std.

Av.

Std.

639.9 3.9 79.4 55.1 14.7

40.8 0.7 8.3 1.9 0.2

636.3 3.7 75.2 54.4 13.7

45.7 0.7 10.4 1.7 1.6

MEAN-DE


MEAN

Av.

Std.

Av.

Std.

1195.6 3.7 80.7 55.2 16.3

50.8 0.6 10.1 1.9 0.6

1191.6 3.9 83.1 57.1 14.9

69.5 0.8 9.5 4.8 0.4

Table 6 Simulations with Single Fault in System 4.

Switching operations MEAN-DE

MEAN

MEAN 7 13 29 5.48

MEAN-DE MOEA/D-NDE 9 19 73 13.41


MEAN

Av.

Std.

Av.

Std.

2321.4 3.4 85.2 55.8 17.1

54.3 0.3 8.6 3.1 0.9

2301.2 3.7 84.6 56.8 15.5

90.91 0.7 10.5 5.2 0.5


709

MEAN MEAN− DE

Fig. 15. Pareto front obtained from System 1.

Fig. 18. Pareto front obtained from System 4.

Fig. 19. Box plots of HV values obtained from System 1. Fig. 16. Pareto front obtained from System 2.

Fig. 20. Box plots of HV values obtained from System 2. Fig. 17. Pareto front obtained from System 3.

Experimental statistical analyses The HV values obtained for MEAN and MEAN-DE (‘Results using HV metric’) are not normally distributed, moreover, the variance is noticeably 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 test assumes the sample of differences is randomly selected and the

710


Conclusions

Fig. 21. Box plots of HV values obtained from System 3.

Fig. 22. Box plots of HV values obtained from System 4.

Table 7 Analyses with Wilcoxon rank-sum test. System test

p-value

System System System System

0.001 0.002 0.001 0.001

1 2 3 4

probability distributions (from which the sample of paired differences 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-DE and MEAN for HV metric.

This paper presented a new MOEA using NDE with a powerful differential mutation operator to solve the SR problem in largescale 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 based on 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 satisfied 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 largescale 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 Wilcoxon 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 memoriam prof. Oriane Magela Neto. References [1] Manjunath K, Mohan M. A new hybrid multi-objective quick service restoration technique for electric power distribution systems. Int J Electr Power Energy Syst 2007;29(1):51–64. http://dx.doi.org/10.1016/j.ijepes.2005. 12.012. [2] Kumar KS, Jayabarathi T. Power system reconfiguration and loss minimization for an distribution systems using bacterial foraging optimization algorithm. Int J Electr Power Energy Syst 2012;36(1):13–7. http://dx.doi.org/10.1016/ j.ijepes.2011.10.016. [3] González A, Echavarren F, Rouco L, Gómez T, Cabetas J. Reconfiguration of large-scale distribution networks for planning studies. Int J Electr Power Energy Syst 2012;37(1):86–94. http://dx.doi.org/10.1016/j.ijepes.2011.12.009.

D.S. Sanches et al. / Electrical Power and Energy Systems 62 (2014) 700–711 [4] Zhu JZ. Optimal reconfiguration of electrical distribution network using the refined genetic algorithm. Electric Power Syst Res 2002;62(1):37–42. http:// dx.doi.org/10.1016/S0378-7796(02)00041-X. [5] Hong Y-Y, Ho S-Y. Determination of network configuration considering multiobjective in distribution systems using genetic algorithms. IEEE Trans Power Syst 2005;20(2):1062–9. [6] Taher, Niknam. An efficient hybrid evolutionary algorithm based on pso and hbmo algorithms for multi-objective distribution feeder reconfiguration. Energy Convers Manage 2009;50(8):2074–82. http://dx.doi.org/10.1016/ j.enconman.2009.03.029. [7] Singh S, Raju G, Rao G, Afsari M. A heuristic method for feeder reconfiguration and service restoration in distribution networks. Int J Electr Power Energy Syst 2009;31(7–8):309–14. http://dx.doi.org/10.1016/j.ijepes.2009.03.013. [8] Augugliaro A, Dusonchet L, Sanseverino ER. Multiobjective service restoration in distribution networks using an evolutionary approach and fuzzy sets. Int J Electr Power Energy Syst 2000;22:103–10. [9] Delbem A, de Carvalho A, Bretas N. Main chain representation for evolutionary algorithms applied to distribution system reconfiguration. IEEE Trans Power Syst 2005;20(1):425–36. http://dx.doi.org/10.1109/TPWRS.2004.840442. [10] Carrano E, Soares L, Takahashi R, Saldanha R, Neto O. Electric distribution network multiobjective design using a problem-specific genetic algorithm. IEEE Trans Power Deliv 2006;21(2):995–1005. http://dx.doi.org/10.1109/ TPWRD.2005.858779. [11] Kumar Y, Das B, Sharma J. Multiobjective, multiconstraint service restoration of electric power distribution system with priority customers. IEEE Trans Power Deliv 2008;23(1):261–70. http://dx.doi.org/10.1109/TPWRD.2007. 905412. [12] Garcia VJ, França PM. Multiobjective service restoration in electric distribution networks using a local search based heuristic. Euro J Operat Res 2008;189(3):694–705. http://dx.doi.org/10.1016/j.ejor.2006.07.048. [13] Carreno E, Romero R, Padilha-Feltrin A. An efficient codification to solve distribution network reconfiguration for loss reduction problem. IEEE Trans Power Syst 2008;23(4):1542–51. [14] Santos A, Delbem A, London J, Bretas N. Node-depth encoding and multiobjective evolutionary algorithm applied to large-scale distribution system reconfiguration. IEEE Trans Power Syst 2010;25(3):1254–65. http:// dx.doi.org/10.1109/TPWRS.2010.2041475. [15] Santos A, Delbem A, Bretas N. Energy restoration for large-scale distribution system using ea and a new data structure, in: 2008 IEEE power and energy society general meeting – conversion and delivery of electrical energy in the 21st century; 2008. p. 1–8. http://dx.doi.org/10.1109/PES.2008.4596388. [16] Dimitrijevic S, Rajakovic N. An innovative approach for solving the restoration problem in distribution networks. Electric Power Syst Res 2011;81(10):1961–72. http://dx.doi.org/10.1016/j.epsr.2011.06.005. [17] Cormen TH, Leiserson CE, Rivest RL, Stein C. Introduction to algorithms. 2nd ed. The MIT Press; 2001. [18] Mansour M, Santos A, London J, Delbem A, Bretas N. Energy restoration in distribution systems using multi-objective evolutionary algorithm and an efficient data structure. In: PowerTech, 2009 IEEE Bucharest; 2009. p. 1 –7. http://dx.doi.org/10.1109/PTC.2009.5282205. [19] Delbem ACB, de Carvalho ACPLF, Policastro CA, Pinto AKO, Honda K, Garcia AC. Node-depth encoding for evolutionary algorithms applied to network design, in: GECCO (1); 2004. p. 678–87. [20] Srinivas M. Distribution load flows: a brief review. Power engineering society winter meeting, 2000, vol. 2. IEEE; 2000. p. 942–5. http://dx.doi.org/10.1109/ PESW.2000.850058. [21] Shirmohammadi D, Hong H, Semlyen A, Luo G. A compensation-based power flow method for weakly meshed distribution and transmission networks. IEEE Trans Power Syst 1988;3(2):753–62. http://dx.doi.org/10.1109/59.192932. [22] Das D, Nagi HS, Kothari DP. Novel method for solving radial distribution networks. IEE Proc – Gener Trans Distrib 1994;141(4):291–8. [23] Deb K. Multi-objective optimization using evolutionary algorithms. New York: Wiley; 2001. [24] Sanches D, Gois M, London J, Delbem A. Integrating several subpopulation tables with node-depth encoding and strength pareto for service restoration in large-scale distribution systems. In: Power and energy society general meeting; 2013. p. 1–5. [25] Gois MM, Sanches DS, Martins J, London Jr JBA, Delbem A. Multi-objective evolutionary algorithm with node-depth encoding and strength pareto for service restoration in large-scale distribution systems. In: 7th international conference on evolutionary multi-criterion optimization, 2013, sheffield. Lecture notes in computer science. Berlin Heidelberg: Springer; 2013. p. 1–5.

711

[26] Storn R, Price K. Differential evolution: a simple and efficient heuristic for global optimization over continuous spaces. J Global Optim 1997;11(4):341–59. http://dx.doi.org/10.1023/A:1008202821328. [27] Clerc M, Kennedy J. The particle swarm-explosion, stability, and convergence in a multidimensional complex space. IEEE Trans Evol Comput 2002;6(1):58–73. [28] Das D, Suganthan PN. Differential evolution: a survey of the state-of-the-art. Evol IEEE Trans Comput 2011;15(1):4–31. [29] Guimarães FG, Silva RCP, Prado RS, Neto OM, Davendra DD. Flow shop scheduling using a general approach for differential evolution. In: Handbook of optimization. Springer; 2012. p. 597–614. [30] Onwubolu GC, Davendra D. Differential evolution: a handbook for global permutation-based combinatorial optimization. Springer Publishing Company; 2009. [31] Diestel R. Graph theory. Graduate texts in mathematics, 3rd ed., vol. 173. Heidelberg: Springer-Verlag; 2005. . [32] Ahuja RK, Magnanti TL, Orlin JB. Network flows: theory, algorithms,and applications. Englewood Cliffs: Prentice Hall; 1993. [33] Santos AC, Nanni M, Mansour MR, Delbem ACB, London JBA, Bretas NG. A power flow method computationally efficient for large-scale distribution systems. In: Power and energy society latin america transmission and distribution conference and exposition, 2012 IEEE PES T&D; 2008. p. 1–6. [34] Coello CAC, Lamont GB, Veldhuizen DAV. Evolutionary algorithms for solving multi-objective problems (genetic and evolutionary computation). Secaucus (NJ, USA): Springer-Verlag New York Inc; 2006. [35] De Jong K. Evolutionary computation: a unified approach. New York (NY, USA): ACM; 2008. [36] Willis H, Tram H, Engel M, Finley L. Selecting and applying distribution optimization methods. Comput Appl Power IEEE 1996;9(1):12–7. http:// dx.doi.org/10.1109/67.475954. [37] Schaffer JD. Multiple objective optimization with vector evaluated genetic algorithms. In: Proceedings of the 1st international conference on genetic algorithms. Hillsdale (NJ, USA): L. Erlbaum Associates Inc; 1985. p. 93–100. [38] Zhang Q, Li H. Moea/d: a multiobjective evolutionary algorithm based on decomposition. IEEE Trans Evol Comput 2007;11(6):712–31. http://dx.doi.org/ 10.1109/TEVC.2007.892759. [39] Miettinen K. Nonlinear multiobjective optimization. International series in operations research & management science. US: Springer; 1999. . [40] Sanches D, Lima T, Santos A, Delbem A, London J. Node-depth encoding with recombination for multi-objective evolutionary algorithm to solve loss reduction problem in large-scale distribution systems. In: 2012 IEEE power and energy society general meeting; 2012. p. 1–8. http://dx.doi.org/10.1109/ PESGM.2012.63450432012. [41] Coelho G, Von Zuben F, da Silva A. A multiobjective approach to phylogenetic trees: selecting the most promising solutions from the pareto front. In: ISDA 2007, Seventh international conference on intelligent systems design and applications, 2007; 2007. p. 837–42. http://dx.doi.org/10.1109/ISDA.2007.87. [42] Van Veldhuizen DA. Multiobjective evolutionary algorithms: classifications, analyses, and new innovations. PhD thesis. Wright-Patterson AFB, OH; 1999. [43] Hansen M, Jaszkiewicz A. Evaluating the quality of approximations to the nondominated set. Tech rep. Poznan University of Technology; March 1998. [44] Zitzler E, Thiele L, Laumanns M, Fonseca C, da Fonseca V. Performance assessment of multiobjective optimizers: an analysis and review. IEEE Trans Evol Comput 2003;7(2):117–32. http://dx.doi.org/10.1109/TEVC.2003.810758. [45] While L, Bradstreet L, Barone L, Hingston P. Heuristics for optimizing the calculation of hypervolume for multi-objective optimization problems. In: The 2005 IEEE congress on evolutionary computation, 2005, vol. 3; 2005. p. 2225– 32. http://dx.doi.org/10.1109/CEC.2005.1554971. [46] Prado RS, Silva RCP, Guimarães FG, Neto OM. A new differential evolution based metaheuristic for discrete optimization. Int J Nat Comput Res 2010;1(2):15–32. [47] .Source project. . [48] Van Veldhuizen D, Lamont G. On measuring multiobjective evolutionary algorithm performance. In: Proceedings of the 2000 congress on evolutionary computation, 2000, vol. 1; 2000. p. 204–11. http://dx.doi.org/10.1109/ CEC.2000.870296. [49] McClave J, Benson P, Sincich T. Statistics for business and economics. MyStatLab series. Pearson Prentice Hall; 2008.