A multi-objective evolutionary approach for planning and optimal condition restoration of secondary distribution networks

Applied Soft Computing Journal 90 (2020) 106182

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Applied Soft Computing Journal journal homepage: www.elsevier.com/locate/asoc

A multi-objective evolutionary approach for planning and optimal condition restoration of secondary distribution networks ∗

J.P. Avilés , J.C. Mayo-Maldonado, O. Micheloud Tecnológico de Monterrey, Monterrey, Mexico

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Article history: Received 5 June 2018 Received in revised form 13 January 2020 Accepted 13 February 2020 Available online 17 February 2020 Keywords: Distribution network planning Distribution network reconfiguration Multi-objective optimization Nondominated sorting genetic algorithm Particle swarm optimization

a b s t r a c t A secondary distribution network (SDN), corresponding to the final user low voltage distribution circuit, is continuously growing due to a persistent increase in load demand. Consequently, the performance of any optimized design will inevitably degrade over time. To avoid the associated repercussions such as faults, congestion, voltage drops, and other major quality issues, we are eventually prompted to redesign this part of the grid. To do so, we propose a Two-Stage Multi-Objective Evolutionary Approach (TS-MOEAP), which is able to find a new optimal network configuration, circumventing the associated quality issues. The proposed approach is oriented to improve the performance of SDNs by combining the concepts of network reconfiguration (NR) and optimal placement of distribution transformers (DTs). Due to the large and complex topology of SDNs, we deal with a hard combinatorial, non-convex, and nonlinear optimization problem. Consequently, to facilitate the resolution of the problem, the proposal is divided into two stages: (1) optimal placement and sizing of distribution transformers, as well as conductor sizing and branch routing, and (2) optimal network reconfiguration. For the first stage, an improved particle swarm optimization technique (IPSO) combined with a greedy algorithm is used, and for the second stage, an improved nondominated sorting genetic algorithm with a heuristic mutation operator (NSGA-HO) is implemented. The approach redesigns SDNs by minimizing total power loss and investment costs while satisfying quality issues and technical constraints. The proposed approach is validated by improving a real-life SDN with critical quality and technical issues. We also compare the results with respect to other state-of-the-art algorithms. © 2020 Elsevier B.V. All rights reserved.

1. Introduction One of the biggest challenges faced by utilities is the constant increase in load demand, which prompts to continuously expand and/or reconfigure the network. Furthermore, utilities must mitigate potential issues such as voltage deviation in consumer nodes with respect to the nominal voltage, power quality issues and any problem that affects the system reliability. To overcome these challenges, in practice the distribution network (DN) components can be over-sized; however, though effective, this is not an economical and efficient solution. For this reason, optimization methods to improve DN performance have been proposed, which adopt the following approaches: Network Reconfiguration (NR) [1]: this approach consists in changing the topology of the network by opening or closing existing switches. It is focused on finding a new radial configuration where one or more objectives can be minimized, e.g. power losses and/or nominal voltage deviations. ∗ Corresponding author. E-mail address: [email protected] (J.P. Avilés). https://doi.org/10.1016/j.asoc.2020.106182 1568-4946/© 2020 Elsevier B.V. All rights reserved.

Optimal Placement and Sizing of Distributed Generation (DG): this is another well-known method to optimize DNs (e.g. see [2, 3]). It consists in the optimal location of new or existing distributed generators in radial distribution systems. It is focused on minimizing the total power loss, reactive power flow, and voltage deviation. Though highly effective, these methods are tailored to improve only primary distribution networks (PDNs) (with nominal voltages greater than 4 kV). Unfortunately, there is no contribution or insight about the optimization of secondary distribution networks (SDNs) (110 V–240 V), which must not be belittled, since they correspond to the distribution circuits that feed the final users. Moreover, optimization of SDNs involves completely different considerations, as well as an increasingly challenging scenario, e.g. SDNs have no switches that permit an easy reconfiguration as in PDNs. On the other hand, a key element of SDNs is the fact that distribution transformers (DTs) can be easily changed in quantity and location, due to their small size and low cost. This is a particularity that cannot be exploited in PDNs; consequently, the well-established optimization algorithms do not consider this fact. Furthermore, in the design of SDNs, there is more freedom to change loads from one transformer to another. Also SDNs involve

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J.P. Avilés, J.C. Mayo-Maldonado and O. Micheloud / Applied Soft Computing Journal 90 (2020) 106182

low voltages and short distances hence their conductors have a high R/X ratio (ratio of the line reactance with respect to the line resistance), consequently, SDNs have more I 2 R losses (Joule effect) than PDNs as well as more voltage drops (∆V , less voltage along the conductor). These situations prompt the designer to include the conductor size in the optimization problem, something that is not relevant in the optimization of PDNs. As a summary, the optimization of SDNs is more complex due to a large number of possible configurations, contrary to PDNs where the topology remains almost fixed and fewer parameters are considered. For the reasons mentioned above, in this paper we propose a novel Two-Stage Multi-Objective Evolutionary Approach for the planning or improvement of SDNs, based on the concepts of NR and optimal placement of DG. The design of the proposed approach is completely application-driven, i.e. the optimization algorithm is tailored to deal with the complexity of SDNs. We divide the solution into two stages: (1) optimal placement and sizing of distribution transformers, as well as conductor sizing and branch routing, and (2) optimal network reconfiguration. For the first stage, an improved particle swarm optimization technique along with a greedy algorithm is used, while for the second stage an improved nondominated sorting genetic algorithm with a heuristic mutation operator (NSGA-HO) is implemented. The algorithm can find the best topological configuration of SDNs to solve existing quality issues and to improve its design, trying to minimize power losses and investment costs. The proposal is tested using a real-life SDN with critical issues, which must be solved economically. This paper is organized as follows. Section 2 presents the bibliographic review. Section 3 describes the contributions of this paper. Section 4 explains the multi-objective problem. Section 5 presents the TS-MOEAP method. Section 6 provides numerical results and comparisons with other optimization algorithms, and Section 7 summarizes the main contribution and conclusions of this paper. 2. Related works In this section, we examine contributions that are part of the state-of-the-art. For the closest possible comparison with the present paper, we consider those contributions with similar aims, regarding NR and optimal placement of electric components. As argued in this paper, these algorithms are oriented to PDNs while optimization of SDNs requires a significant reformulation. A comprehensive survey on traditional PDN reconfiguration techniques considering different objectives and constraints is presented in [4] and [5]. Common issues treated by NR are the minimization of power losses or voltage deviations of PDNs, for which different meta-heuristic methods such as BiogeographyBased Optimization (BBO) [6], Selective Particle Swarm (SPSO) [7], Genetic Algorithms (GA) [8,9], Binary Particle Swarm Optimization (BPSO) [10], Cuckoo Search Algorithm (CSA) [11], and Ant Colony Algorithms (ACA) [12], have been applied. Lately, NR has been implemented to improve quality parameters of distribution networks [13], using as objective functions reliability indices, stability, or harmonic distortion [14,15]. However, due to the complexity of such combinatorial optimization problems, these metaheuristic methods require considerable time to converge to a solution, therefore compensation techniques such as switch exchange [16], power flow simplifications [17], branch exchange methods [13], or improved mutation operators [18] can be implemented. Other authors have tried to simplify the problem of NR using Lagrange relaxation [19] and heuristic rules with a Minimum Spanning Tree (MST) algorithm [20]. For the case of optimal DG placement [2,21], and [3] present a review of different techniques and models with different objectives. Among the most relevant, [22] presents a Krill Herd

Algorithm (KHA) and [23] uses a Symbiotic Organisms Search (SOS) to minimize the total power loss of the network. In [24], instead of DG optimal placement, the power quality of DNs is improved by optimal sitting, sizing, and harmonic tuning orders of LC filters. In [25], a combination of NR and DG optimal placement is presented to maximize the DG owner’s profit, minimizing the distribution company’s costs. Here the problem is simplified using a single-objective function and applying a ε -constraint method. However, the use of a single-objective function does not ensure the resolution of all the quality issues in the network, therefore multi-objective approaches are preferred. For example in [26] a multi-objective differential evolution algorithm (DE) is presented for optimal NR, minimizing power losses and voltage deviations. Despite the satisfactory results, the drawback is the use of scalarization, which requires a new weighting when the user’s preferences change. In [27], the optimal placement of DGs and capacitors is used to minimize the real power loss and the net reactive power flow of 12.66 kV systems. Although satisfactory results are achieved, the number of DGs to install must be prespecified. Moreover, other approaches such as [28,29] and [30] solve these multi-objective optimization problems using fuzzy logic. However, the drawback of these methods is the proper tuning of membership function parameters (MF) when the algorithm is intended to be applied to other systems. To avoid these disadvantages, nondominated sorting based algorithms can be implemented. For example, in [31] a Nondominated Sorting Particle Swarm Optimizer (NSPSO) is presented for NR, minimizing power losses and voltage deviation. In [32] an Improved Nondominated Sorting Genetic Algorithm (INSGA) is proposed to minimize power losses, voltage deviations, and to improve the voltage stability, and in [33] a Nondominated Sorting Genetic Algorithm II (NSGA-II) is used for optimal placement of storage systems considering reliability and investment costs as main objectives. Considering the complexity to optimize large scale PDNs, some authors have proposed hybrid or two-stage algorithms. For example, in [34], a two-stage evolutionary optimization method is implemented for multi-year expansion planning of primary distribution systems (e.g. 20 kV); the first stage calculates the operational cost and the second stage solves the optimal power flow problem. In [35] a hybrid evolutionary algorithm is proposed combining Shuffled Frog Leaping Algorithm (SFLA) and Particle Swarm Optimization (PSO). This study considers the power loss, Voltage Stability Index (VSI), and the number of switching as objective functions. A survey of these related works is presented in Table 1. Please note that all these works are useful only for primary networks. Other important contributions that are not completely related to these topics, but that deserve further exploration can be found e.g. in [36–39]. A comparison with an algorithm of very similar nature to NSGA-HO is included in this paper in Section 6. 3. Contributions In sharp contrast with the works presented in Section 2 (which focus on the optimization of primary networks), our proposal is intended for the optimization of secondary distribution networks (SDNs), also known as low voltage distribution networks. Due to the constant increase of users (loads), the capacity and optimal operation of an SDN degrades over time. Among the most common problems we have: voltage drops, over-currents (thermal limits), and overloaded transformers. These problems are represented in Fig. 1. To solve these problems and return the network to an optimal state, it is necessary to change its original design. One way to do so is by increasing the capacity of the elements that compose

J.P. Avilés, J.C. Mayo-Maldonado and O. Micheloud / Applied Soft Computing Journal 90 (2020) 106182

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Table 1 Survey of related works applied in primary networks. Work

NR

[6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [22] [23] [24] [25] [26] [27] [28] [29] [30]

x x x x x x x x x x x x x x x

[31] [32]

x

[33] [34] [35]

x x

Optimal placement

x x x x x

x x x

x x x x

Algorithm/Method

Objectives

BBO SPSO GA DE BPSO CSA ACA Branch exchange Heuristic approach EA Switch exchange MIP GA MIP MST KHA SOS BFO ε-Constraint DE MOEA/D Fuzzy/RRA Fuzzy-Firefly Fuzzy-PSO

PLoss PLoss Voltage sag costs PLoss PLoss PLoss, Voltage magnitude PLoss, Load balance PLoss, Quality issues Reliability Stability PLoss Minimizing energy cost PLoss PLoss PLoss PLoss PLoss PLoss Profit PLoss, Voltage deviation PLoss, Reactive power PLoss, Load balance PLoss, Voltage magnitude PLoss, Voltage deviation, Load balance PLoss, Voltage deviation PLoss, Voltage deviation, Stability Reliability, Investment costs Total operational cost PLoss, Stability, Number of switching

NSPSO INSGA NSGA-II Two-stage, BMICA/ISSO Hybrid, SFLA/PSO

Fig. 1. Common problems of SDNs.

it, e.g. we can increase the thickness of the conductors and the capacity of the transformers. Another way to improve the SDN is changing its topology/configuration, e.g. users could be redistributed between the transformers; we could change the location of the transformers, or we can add more transformers and assign them a load. For example, for the network shown in Fig. 1, we could increase the size of the u4-u6 conductor to solve the overcurrent; we could move the transformer T1 to u4 to improve the voltage in u2; we could increase the capacity of T2 to solve the overload, or we could divide its network into two and locate another transformer to feed the users of u5 and u7. As we can see, the possibilities are endless, but each of them carries a monetary cost. So, the question is, what is the optimal configuration that we must implement to fix all these problems in the most economical way? The works presented in Table 1 do not change the entire network topology, and the optimization problem is simplified by opening or closing pre-established switches at specific points of the network, i.e. the number of possible configurations is substantially reduced. On the other hand, to improve an SDN we must

Fig. 2. TS-MOEAP structure.

necessarily modify the entire topology, and given a large number of possible configurations, it becomes an extremely complex, non-convex, and non-linear combinatorial problem. To overcome these difficulties, we have proposed a Two-Stage Evolutionary Algorithm (TS-MOEAP) as illustrated in Fig. 2. Stage-2 will be responsible for optimizing the network configuration (grouping the loads in different ways), and Stage-1 will be responsible for designing the networks, locating the transformers, and selecting the conductors. 4. Problem formulation In this section, we show the fundamental formulation of the SDN optimization problem addressed in this paper.

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J.P. Avilés, J.C. Mayo-Maldonado and O. Micheloud / Applied Soft Computing Journal 90 (2020) 106182

4.1.2. Minimization of investment cost The second objective is the minimization of the total investment cost expressed as min f2 (c) = min (

λ ∑

qn ρn +

n=1

ς ∑

ln ϱn + µ ζ )

(3)

n=1

where λ is the total number of available transformer capacities, qn is the number of transformers, and ρn is the cost per unit. ς is the total number of available conductor sizes, ln is the total length, and ϱn is the cost per meter of the conductor. µ is the total number of utility poles, and ζ is the cost of ceramic insulators per utility pole. Power losses and quality issues of an SDN can be minimized by improving the network infrastructure and equipment capacity. This implies high investment costs, therefore, (3) is crucial to balance the optimization problem. 4.2. Constraints

Fig. 3. TS-MOEAP operation. (a) Stage-1. Route of conductors. Optimal size and location of transformers. Optimal selection of conductor’s size. (b) Stage-2. Network reconfiguration is applied to change the grouping of loads.

The equality constraints of the problem are represented by the power-flow equations (4), where (Pgi , Qgi ) are the active and reactive generation outputs, and (Pli , Qli ) are the active and reactive loads at node i. Gij and Bij are the conductance and susceptance of the admittance matrix, respectively. Pgi − Pli = Vi

µ ∑

Vj (Gij cos θij + Bij sin θij )

j=1

4.1. Objective functions

Qgi − Qli = Vi

µ ∑

(4) Vj (Gij sin θij + Bij cos θij )

j=1

The intention of our proposal (in the real world) is to restore or improve the optimal operation of SDNs, which eventually will present quality issues as shown in Fig. 1. In particular, to solve these problems we need to reinforce the network structure, i.e. increasing the capacity and number of their components (transformers, conductors) or changing their topology. A value that is directly related to this reinforcement is the power loss (greater reinforcement causes fewer power losses), therefore, this can be applied in the optimization model as one of the objective functions. On the other hand, while the network is reinforced, the investment costs are also increased. Consequently, such a cost-oriented metric is considered as the second objective function. In summary, the optimization approach must identify an optimal network configuration (c) to minimize power losses (f1 ) and investment costs (f2 ), expressed as min [f1 (c), f2 (c)], c ∈ Π ,

(1)

where Π is a feasible solution space. In the following, we describe each objective function and the constraints of the optimization problem. 4.1.1. Minimization of power losses The first objective is the minimization of power losses of the entire system (see Sec. 9.1 of [40]), expressed as min f1 (c) = min

η ∑

gij (Vi2

+

Vj2

− 2Vi Vj cos θij )

(2)

n=1

where η is the total number of branches in the system, (i, j) are the nodes of the branch (also named as utility poles ui,j ), and gij is the conductance between the respective nodes. Vi and Vj are the voltage magnitudes at each node and θij is the difference between phase angles of the respective voltages.

The inequality constraints include: (a) transformer capacity (5), where S max is the maximum available capacity and τ is the total number of transformers/groups in the system; (b) voltage magnitude (6), where V min and V max are established limits; and (c) thermal limits (7), where Iij is the magnitude of the current between the nodes (i, j), I max is the conductor ampacity, (V i , V j ) are voltages in phasor representation, and zij is the branch impedance. 0 < STi ≤ S max ; i = 1, 2, . . . , τ . V

min

≤ Vi ≤ V

max

(5)

; i = 1, 2, . . . , µ.

(6)

Iij = |(V i − V j )/zij | ≤ I max

(7)

Unlike primary distribution lines, SDN planning also needs geographical constraints to avoid using: (d) restricted branches (8), where Γ is the set of branches with restriction (Uijrst ) and Uij is the branch between the nodes (i, j); and (e) restricted utility poles (9), where the transformers cannot be located, for this Λ is the set of utility poles with restriction (urst n ) and uTi is the utility pole where a transformer is installed. Uij ̸ = ∀ Uijrst ∈ Γ ; Uij := [ui uj ]; uT i ̸ = ∀

urst n

(8)

∈ Λ; i = 1, 2, . . . , τ ;

(9)

4.3. Reformulation of objective functions in terms of constraints The equality constraints (4) can be satisfied during the powerflow calculation and the inequality constraint (8) can be satisfied during the network construction. The inequality constraints (6), (7) and (9) can be satisfied through penalizing the objective function f1 (c), and (5) can be satisfied through penalizing the objective function f2 (c). Finally, the constrained optimization problem can be reformulated as min [f1′ (c), f2′ (c)]; c ∈Π

f1′ (c) = f1 + fp1 + fp2 + fp3 f2′ (c) = f2 + fp4

(10)

J.P. Avilés, J.C. Mayo-Maldonado and O. Micheloud / Applied Soft Computing Journal 90 (2020) 106182 µ

fp1 = w1

∑

|min(Vi − V min , 0, V max − Vi )|

(11)

max(Iij − I max , 0)

(12)

max([uTi = urst n ] ⇒ [1], 0)

(13)

max(STi − S max , 0, [STi = 0] ⇒ [w5 ]).

(14)

i=1

fp2 = w2

η ∑ n=1

fp3 = w3

τ ∑ i=1

fp4 = w4

τ ∑ i=1

Eqs. (11)–(14) are penalty functions applied when voltage magnitudes are out of limits, there are overcurrents in the branches, transformers are located at restricted utility poles, and when transformers are not supplying power or cannot feed the required demand, respectively. w1 , w2 , w3 , w4 , and w5 are penalty factors to delimit in a lesser or greater way the search space, and to ensure the desired operation regarding standard concerns. 5. Two-stage multi-objective evolutionary approach (TS-MOEAP) TS-MOEAP can be defined as an algorithm composed of two stages of optimization, in order to improve an SDN which may have several problems as shown in Fig. 1. The operation of each stage can be summarized as follows:

• Stage-1. Optimal placement and sizing of DTs, as well as optimal branch routing and conductor sizing, see Fig. 3(a).

• Stage-2. Optimal network reconfiguration, see Fig. 3(b). This division was made to facilitate the resolution of the problem, because in each stage we have different types of variables, number of objectives, and procedures to carry out. For example, in Stage-1 we first need to find the optimal route for conductors using the GPS coordinates of the nodes (float variables); find the optimal placement of DTs (non-negative integer variable); and finally, evaluate each network through a power flow analysis (set of non-linear equations). At this stage, the DT location is selected based on the lowest power loss (one objective). On the other hand, Stage-2 needs to change the topological configuration of the SDN to minimize both power losses and investment costs. For the Stage-2, we propose an integer vector representation, based on groups of loads instead of preset switches, a novel solution that allows us to change the entire network topology. We should note that if the topology of the SDN changes in Stage-2, we must repeat the whole process of Stage-1 (easily thousands of iterations), i.e. Stage-1 should be nested in Stage2, that is why we require a fast and reliable sub-optimization algorithm for Stage-1. The optimal placement of DTs is based solely on power loss and an integer representation (for each location), therefore, we propose an IPSO due to its simplicity and outstanding performance (there is no need to implement a more complex algorithm for that task). The IPSO will be combined with a greedy algorithm for the optimal route of conductors, i.e. for the placement of each particle an SDN must be built (please see Fig. 4). For the second stage, we deal with a combinatorial problem, two objectives, and an integer vector representation (in this case the whole vector is a single variable). For these reasons, an NSGA-HO is proposed to optimize the topology of the SDN. This algorithm is able to solve complex combinatorial multiobjective problems, without the need to calibrate several parameters [41]. Additionally, it implements a heuristic mutation operator to take advantage of some particularities of these SDNs. This improved mutation operator can perform specific changes in the

5

genotypes (chromosomes) to find better network topologies. We must note that meta-heuristic algorithms have been shown to be an effective way to solve the reconfiguration of distribution networks [5]. In order to develop TS-MOEAP, the following assumptions are made: (1) a projected unit demand (PUD ) is used for the network design and transformer sizing; (2) primary lines can be extended to any utility pole to feed the transformers; (3) the quantity and location of utility poles do not change, something common in practice; (4) the utility poles connected to one transformer will form a group Tk , and (5) each utility pole or node has a (xi , yi ) GPS coordinate. In the following sections, we explain in detail the implementation of each stage and their respective algorithms. 5.1. STAGE-1 In this section, we describe how the IPSO and a greedy algorithm (PRIM) are applied to construct and optimize a radial distribution network (please see Fig. 4), by selecting the optimal placement and capacity of the DT, as well as the optimal conductor size for each branch. 5.1.1. IPSO algorithm For the optimal placement of DTs, an IPSO algorithm is proposed, based on a regular PSO [42,43] which is a meta-heuristic method inspired by the social behavior of bird flocking or fish schooling. The algorithm works with a population where each particle is a candidate solution, i.e. a possible transformer location. Unlike the PSO, the IPSO does not have a fixed population, and the particles do not start with random positions. Due to the radial configuration of the SDNs, the particles can be located uniformly excluding the most extreme nodes (utility poles). This ensures that the transformer is located in an optimal node, excluding unnecessary locations from the analysis. Furthermore, the IPSO has an additional memory to store all the results found so far by all the particles. In this way, we avoid evaluating the particles that fall into positions already explored. This helps us to reduce computational time since the evaluation of each particle requires a power-flow analysis (e.g. Gauss–Seidel method, see section 9 of [40]). The equation that governs the movement of each particle is represented by

vi(t +1) = wI vi(t) + wC r1 [γi − χi(t) ] + wS r2 [σ − χi(t) ]; (t) i

(15)

(t) i

where v is the velocity and χ is the current position of the particle; γi is the best position found by each particle, and σ is the best position found by the entire group. wI , wC and wS are constant weights for inertia, cognitive behavior and social behavior, respectively; and r1,2 are random values in [0, 1]. The position of each particle is then updated by

χi(t +1) = χi(t) + vi(t +1) .

(16)

5.1.2. Representation and initialization An SDN consists of several DTs, which feed a group of loads connected to different utility poles (un ). In this case, an integer vector (chromosome, ci ) could represent an entire system by assigning to each utility pole a transformer/group number, e.g. the system configuration of Fig. 3(a) and Fig. 3(b) can be represented by c1 and c2 , respectively, as follows: c1 c2

[

u1

u2

u3

u4

u5

u6

u7

u8

u9

2 2

1 2

2 2

1 2

2 3

1 1

2 3

1 1

1 1

] .

where 1, 2, and 3 are the possible transformers to connect the loads. We must note that the utility poles are sorted in ascending order with respect to the x-GPS coordinate, and this order is

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J.P. Avilés, J.C. Mayo-Maldonado and O. Micheloud / Applied Soft Computing Journal 90 (2020) 106182

Fig. 4. Stage-1 procedure. Construction of a radial SDN using PRIM and optimal placement of DTs using the IPSO.

maintained throughout the process, i.e. {u1 u2 . . . , uµ }, where ui (xi , yi ). From ci , we can get the Tk sets, selecting the ui elements with the same transformer number, e.g. c1 = [2 1 2 1 2 1 2 1 1] →

T1 = [u2 u4 u6 u8 u9 ] T2 = [u1 u3 u5 u7 ].

The maximum number of sets is defined by τ = max(ci ), and each set will become a search space for the Stage-1. Unlike the PSO, the proposed IPSO algorithm assigns a εk number of particles to each Tk set according to its length ψk , i.e.

εk = max [round(ψk · ϑ ), 2]; k = 1, 2, . . . , τ

(17)

where ϑ is a constant value assigned by the user. Considering that the Tk set represents a real network, the particles can be linearly distributed, excluding the extremes for practical reasons, i.e. T1 = [u2 u4 u6 u8 u9 ]. ↑ χ1

↑ χ2

5.1.3. Network construction For each Tk set, the DN must be built in a radial topology. In this case, the network can be considered as a weighted undirected graph from which we can obtain a minimum spanning tree using a greedy algorithm. A well-known greedy algorithm is PRIM, which takes a graph as input and finds the subset of edges/branches (Uij ) which will form a tree (the network) with the minimum amount of weight (wire length). The algorithm starts by calculating the Euclidean distance dij between all the ui nodes contained in Tk , using the (xi , yi ) GPS coordinates, i.e. u1 u1 u2

u2

···

0 ⎢ d21

d12 0

.. .

··· ··· .. .

dψ 1

dψ 2

···

⎡

.. ⎢ ⎣ .

uψ

.. .

dij i̸=j =

uψ

d1ψ d2ψ ⎥

⎤

.. .

⎥; ⎦

(18)

0

5.1.4. Particle evaluation In Stage-1, each particle is a possible location for the DT, and its fitness will be f1′ (χj ). With the set of branches, found in the j previous section, it is possible to calculate the admittance matrix Y since each branch has a zij impedance (see Sec. 7.3 of [40]). The particle location can be considered as the slack bus of the system, and the remaining nodes as PQ type. With this data, the set of power-flow equations (4) can be solved to find: voltage nodes, branch currents, and therefore the total power loss f1′ (χj ) (see j Sec. 9.2 of [40]). This power-flow analysis is repeated for each particle. The position with the lowest fitness (i.e. power loss) found so far by the particle is saved as γj , and the best position found among all particles is saved as σ . After this, the velocity and position of each particle are updated through (15) and (16), respectively. The power-flow calculation is a process that consumes valuable computational time, therefore the IPSO records all the solutions found so far by the particles. This improvement avoids repeating the power-flow computation if another particle falls in the same place. At the end of the process, the best value f1′ (χj ) is j saved as the fitness of the group Tk , i.e. f1′k (Tk ) = f1′ (σ ).

(xi − xj )2 + (yi − yj )2 .

√

If within Tk there are two utility poles (ui , uj ) that can form a restricted branch Uijrst ∈ Γ , a penalty w6 is added to dij to avoid selecting that specific branch, i.e.

[Uij = Uijrst ] ⇒ [dij = dij + w6 ],

Fig. 5. Network construction procedure. PRIM adds at each step the shortest Uij branch until all ui nodes are in the network.

(19)

this satisfies the inequality constraint (8). After that, PRIM operates by building the network one node at a time, from an arbitrary starting node ui , adding at each step the shortest branch Uij , until all nodes are included in the network. This process is illustrated in Fig. 5.

(20)

5.1.5. Conductor optimization For the DT optimal placement, the algorithm builds the network with the thickest conductor available, this way we are sure to apply a penalty to the fitness value f1′ (χj ) if a voltage node or a j branch current is out of limits. When the DT placement is found, and the Tk network does not have any technical problem, the algorithm reduces the wire size of the branches and recalculates the power-flow. If the network still does not present any problem,

J.P. Avilés, J.C. Mayo-Maldonado and O. Micheloud / Applied Soft Computing Journal 90 (2020) 106182

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Fig. 6. Nondominated sorting and crowding distance procedure.

the wire size is reduced again to minimize costs; otherwise, the algorithm stops and keeps the last wire. i.e.

∀ Uij ∈ Tk |3/0, 2/0, 1/0, 2AWG .

(21)

where 3/0, 2/0, 1/0 and 2 AWG are available ACSR wire sizes. 5.1.6. Transformer sizing and cost Calculation The transformer capacity for each group Tk is calculated with TCk =

(Hk0.91

· PUD + Lk · PSL ) · ξ

(22)

where Hk is the total number of houses connected to the transformer, and PUD is the projected unit demand according to the type of users. Lk is the number of street lights, and PSL is their power. ξ is an overload factor usually used by utility companies for DT sizing. Once calculated TCk , this value is compared with a list of available transformers, and the closest one is selected. If TCk exceeds the biggest available transformer, the penalty function (14) is applied. Finally, when the conductor size, transformer capacity, and network topology are established, the algorithm is allowed to calculate f2′ (Tk ).

found, and so on. This process is repeated until |A(t +1) ∪ Fr | ≥ N, in this case, the nondominated sorting is stopped, and the remaining members of the population A(t +1) must be chosen from the last front Fr by crowding distance (Di ), as illustrated in Fig. 6. The operator Di estimates the density of solutions surrounding a particular ith solution and it is calculated as the sum of individual distance values corresponding to each objective, i.e. Di = Di + [fn (ci+1 ) − fn (ci−1 )]/[fnmax − fnmin ]; n = 1, 2, . . . , m (24)

where fnmax and fnmin represent the maximum and minimum fitness values of the nth objective function. A solution with a smaller D value is more crowded by other solutions, therefore to maintain diversity, the members with greater D are selected. In our case, the fitness values used for nondominated sorting and crowding distance calculation can be obtained by adding the individual fitness values of each Tk group found at the previous optimization stage, i.e. f1′i (ci ) = ′

f2i (ci ) =

τ ∑

i = 1, 2, . . . , 2N

(25)

′

f2k (Tk )

k=1

In the following, we describe how the proposed NSGA-HO algorithm is implemented to optimize the topology of the network by minimizing power loses and investment costs. 5.2.1. NSGA-HO algorithm The proposed NSGA-HO algorithm is based on the multiobjective evolutionary algorithm NSGA-II [41] that uses nondominated sorting and sharing. For a multi-objective problem (1), where m is the number of objective functions, a solution ci (for us a network configuration) is said to dominate cj (represented as ci ≺ cj ) when

∀n ∈ {1, 2, . . . , m} : ci ≺ cj ⇔ fn (ci ) ≤ fn (cj ).

f1′k (Tk )

k=1

k

5.2. STAGE-2

τ ∑

(23)

Taking as reference Fig. 6, each solution is assigned a rank equal to its nondominated level. ci solutions not dominated by any other have rank 1 (they belong to the Pareto front, F1 ), the next best-solutions have rank 2 (front F2 ) and so on. The NSGA-HO algorithm does not find all nondominated fronts at once, first, it finds F1 ; if its size is smaller than N, their members are chosen to form A(t +1) = A(t +1) ∪ F1 , and the next front can be found. If the size of |A(t +1) ∪ F2 | is smaller than N, the members of F2 are chosen to form A(t +1) = A(t +1) ∪ F2 , and the next front F3 is

In order to create an offspring population E (t) , a binary tournament selection, recombination, and mutation operators are used. In the tournament selection, two random individuals are compared by rank and crowding distance. The individual with the best rank is selected, but if both individuals belong to the same front, the individual with greater crowding distance is selected. 5.2.2. Heuristic mutation operation Considering the combinatorial complexity of SDNs, the mutation operator has been modified to obtain better network topologies and to turn an unfeasible solution into a feasible one. Something well-known about distribution networks is that utility poles usually are connected to the nearest transformer group. Using this characteristic, the heuristic mutation operator (H) finds better solutions than a regular operator and can help to satisfy some constraints (fp1 , fp2 , and fp3 ). For example, as shown in Fig. 7, the heuristic operator first calculates the average center of each group (e.g. a, b, c) and then takes the farthest utility pole to each center (e.g. u1 , u7 , u8 ). Later, the distances to the center of other groups are compared, e.g. u7 − b is compared with u7 − a and u7 − c. If a shorter distance is found the utility pole is switched, e.g. u7 is switched to T3 , u8 is switched to T1 and u1 is maintained in T1 . A ci

8

J.P. Avilés, J.C. Mayo-Maldonado and O. Micheloud / Applied Soft Computing Journal 90 (2020) 106182

6.1. Case study In order to prove the effectiveness of TS-MOEAP, a real SDN, with critical quality issues and technical problems, is considered. The information, which can be found in [44], is shared by the utility company E.E.R.C.S in collaboration with the Ecuadorian government. As background, in the last few years, this country has built several hydroelectric plants causing an excess of generation capacity. Due to this particularity, the government is motivating residential users to change gas stoves by induction stoves. The new electric loads, with an average power of 4 kW, have caused an increment of demand of almost twice, as shown in Fig. 8. The current SDNs are not prepared to support this increased demand; therefore, optimal planning to improve them is needed. The SDN considered for the application of TS-MOEAP comes from a real urban area, as shown in Fig. 9, and presents the following characteristics: 3-phase system, a primary voltage of 22 kV, a secondary voltage of 220/127 V, 400 residential users, 88 utility poles, and 6 DTs with an installed capacity of 335 kVA. Active and reactive loads, at each utility pole, are shown in Appendix.

Fig. 7. Heuristic mutation operation.

Algorithm 1: TS-MOEAP

Fig. 8. Demand profile of a residential user. The curve presents three defined peaks due to the implementation of an induction stove.

1

2

configuration mutates if a random value between [1,0] is superior to a mutation rate ϖ , i.e. H(ci ),

if rand[0, 1] ≥ ϖ

ci ,

if rand[0, 1] < ϖ .

{ cmi =

(26)

Applying the heuristic operator H to the configuration c1 = [1 1 2 2 1 2 2 3 3 3 3], of Fig. 7, the result is:

3 4 5 6

7 8 9 10

H(c1 ) = [1 1 2 2 1 2 3 1 3 3 3].

11

Data: System parameters, GPS coordinates, users and loads, geographical and technical constraints, available equipment, and costs. STAGE-2: NSGA-HO Create a random population R(t) of size 2N ci ∈ R(t) , i = 1, 2, ..., 2N for t = 1 : t limit do for i = 1 : 2N do Identify Tk groups inside ci , Tk ∈ ci . f1′ (ci ) ←− 0; f2′ (ci ) ←− 0; i

i

for k = 1 : τ do STAGE-1: IPSO Initialize χj particles within Tk Network construction using PRIM while Population diversity > Tol. do Power-flow computation Calculate objective function f1′ (χj ) j

Save the global best result, f1′ (Tk )

12

5.2.3. Recombination operation Due to the integer encoding, and the random order of Tk groups, a uniform crossover is used (see Sec. 4.2 of [42]). This operator takes two parents from the population A(t +1) and creates two new individuals by selecting each gene from the first or second parent at random. For each position of the new chromosome, if a random value between [0,1] is below a crossover rate δ , the gene is inherited from the first parent, otherwise from the second. For example, assuming that parents are ci = [ai1 ai2 . . . , aiµ ] and cj = [bj1 bj2 . . . , bjµ ], the offspring is created by ain ,

if rand[0, 1] ≤ δ

bjn ,

if rand[0, 1] > δ

{ rijn =

16

17

f1′ (ci ) ←− f1′ (ci ) + f1′ (Tk )

18

f2′ (ci ) ←− f2′ (ci ) + f2′ (Tk )

14 15

19

n = 1, 2, . . . , µ

(27)

cij = [rij1 rij2 ... rijµ ]

20 21

22

6. Implementation and results The proposed algorithm is implemented in MATLAB, and its general pseudo-code is shown in Algorithm 1.

k

Update vj velocity and χj position end Optimize conductor size Calculate transformer capacity Calculate objective function f2′ (Tk )

13

k

i

i

k

i

i

k

end end Sort R(t) by nondominated fronts and crowding distance Truncate R(t) to form A(t) (size N) Make selection, recombination, and mutation operations on A(t) to form an offspring E (t) Combine current population and offspring, R(t +1) ←− A(t) ∪ E (t) end Result: The most economical ci configuration from the Pareto set.

J.P. Avilés, J.C. Mayo-Maldonado and O. Micheloud / Applied Soft Computing Journal 90 (2020) 106182

9

Fig. 11. The best result found by TS-MOEAP. This configuration is the most economical that solves all the quality issues of the original system. Fig. 9. Test system. The SDN presents several problems due to the increased demand.

Fig. 10. Random configuration from the first generation.

The problems that affect the system are also shown in Fig. 9 and summarized in Table 2, accounting for a total of 33 quality issues, including nodes with low voltage magnitude, branches with overcurrent, and overloaded transformers. Furthermore, the system has a total power loss of 19.6 kW with a total investment cost of $23 837, reference values that must be minimized. 6.2. TS-MOEAP results The main input data for the algorithm are: 1. GPS coordinates, and loads per utility pole, see Fig. 9 and Appendix, 2. Geographical and technical constraints, see Table 3 3. Available main equipment and their average costs, see Table 4

4. Projected unit demand PUD = 1.36 kVA and overload factor ξ = 0.8 5. Iteration limits for the IPSO and NSGA-HO, 100 and 200 respectively 6. Penalty factors w1,2,3,6 = 1000, w4 = 300, and w5 = 10 000 [42] 7. NSGA-HO mutation rate ϖ = 0.8, crossover rate δ = 0.5, and population size 100 [41] 8. IPSO factors wI = 0.4, wC = 1, wS = 3 and ϑ = 0.3. With these values, the variance is almost zero with an average convergence time of 0.25 s [43]. Applying TS-MOEAP to solve the test system, the following results are the best obtained after 100 simulations. Fig. 10 shows a random SDN when the optimization process starts, and Fig. 11 shows an optimal SDN when the optimization process ends. In the first generations, the configurations have poor quality, because the initial population is generated at random. However, as this population evolves, better results are found. As we can see in Fig. 11, TS-MOEAP converges to a solution where all the quality issues are solved, minimizing power losses and investment costs. Therefore we can say that the optimal condition of the test system is restored. Fig. 12(a) shows a comparison between voltage magnitudes (in descending order), before and after the optimization. Voltage drop is the most common problem in SDNs; however, TS-MOEAP found a new network configuration where voltage magnitudes are above the limit. Fig. 12(b) shows branch currents for both the original and the optimized system. As we can see, the original system exceeds the ampacity limit, even when the thickest conductor (3/0 AWG) has been used. In contrast, the optimized system corrected this problem and still has room to add more loads. Similarly, Fig. 12(c) shows a comparison between branch power losses. The optimized system has less power loss per branch, even when it has fewer transformers. This is possible thanks to the optimal network reconfiguration. TS-MOEAP can find several nondominated configurations that will form a Pareto set, from which the most economical is chosen, as shown in Fig. 12(d). A summary of results, before and after the optimization, is shown in Table 5, where voltage deviation (VD)

10

J.P. Avilés, J.C. Mayo-Maldonado and O. Micheloud / Applied Soft Computing Journal 90 (2020) 106182 Table 2 Quality issues of the test system. Overloaded DTs Nodes with a low voltage magnitude Branches with overcurrent

Qty.

Specification

2 25 6

T 3, T 4 u6 , u8 , u9 , u13 , u16 , u17 , u23 , u24 , u30 , u34 , u36 –u38 , u65 , u73 , u78 –u88 U7,12 , U11,12 , U51,53 , U57,62 , U66,68 , U68,74

Fig. 12. Different results before and after the optimization. (a) Voltage magnitude. (b) Magnitude of currents. (c) Branch power losses. (d) Pareto front obtained by TS-MOEAP.

Fig. 13. Algorithm comparisons for total investment cost, power loss, and quality issues. ANMT presents the highest investment costs. The GA and NSGA-II do not solve all the quality issues. The GA-HO, with a single-objective function, gets better results than NSGA-II. NSGA-HO/IPSO got the cheapest configuration.

and installed capacity (IC ) are defined as follows

VD =

µ ∑ i=1

IC =

τ ∑

TCk .

(29)

k=1

|V ref − Vi |/V ref

(28)

V ref is the reference voltage of the SDN. We should note that the optimized system obtained a lower investment cost than the

J.P. Avilés, J.C. Mayo-Maldonado and O. Micheloud / Applied Soft Computing Journal 90 (2020) 106182 Table 3 Geographical and technical constraints. Constraints Restricted branches Restricted utility poles for transformers Conductor ampacity for ACSR 3/0 AWG Voltage magnitude limits Maximum available transformer capacity

U1,4 , U18,20 , U37,38 , U59,71 , U80,85 u8 , u30 , u39 , u52 , u60 , u79 315 A ±4.5% 100 kVA

Table 4 Available equipment and costs. Specification

Price/U.

Transformers

30 kVA 50 kVA 75 kVA 100 kVA

$ $ $ $

ACSR conductors

2 AWG 1/0 AWG 2/0 AWG 3/0 AWG

0.67 0.75 0.87 1.10

Ceramic insulators

Type 53–2

$ 15

2050 2300 2830 3500 $/m $/m $/m $/m

Table 5 Results for the test system, before and after the optimization. Total investment cost [$] Total power loss [kW] Voltage deviation [p.u.] Installed capacity [kVA] Max. ∆V per node [%] Total transformers Quality issues

Before

After

Difference

23 837 19.6 1.45 335 9.5 6 33

23 062 12.8 0.99 350 4.4 5 0

−3.3% −34.7% −31.7% +4.3% – – –

original one. This is quite impressive considering that to solve all quality issues the network usually has to be oversized, therefore investment costs also increase. 6.3. Comparison of TS-MOEAP using other algorithms Since there is no similar proposal to optimize SDNs, TS-MOEAP will be tested with different algorithms at Stage-2, to verify the effectiveness of NSGA-HO. The algorithms selected for the comparison are: NSGA-II [41], a multiobjective evolutionary algorithm based on decomposition (MOEA/D) [45], a conventional GA [8], and an analytic method (ANMT). The analytical method will represent a manual procedure used by the utility company to make corrective maintenance of SDNs as follows: 1. For each Tk network with voltage drop problems, try to relocate the transformer, if the problems persist, increase the conductor size. 2. For each Tk network with branch overcurrents, try to relocate the transformer, if the problems persist, increase the conductor size. 3. If 1. and 2. do not fix up voltage drops and overcurrents, split the network on the branch with the highest current. 4. If a transformer is overloaded, select a transformer with greater capacity. If the maximum available transformer is installed and the problem persists, split the network on the branch with the highest current. The NSGA-II, MOEA/D, and the GA were selected primarily for their ability to solve combinatorial problems with an integer vector representation [42]; therefore, they can be implemented in the Stage-2 of TS-MOEAP without major modifications. NSGA-II and MOEA/D are recent algorithms focused on solving multiobjective problems; however, the NSGA-HO can find better results

11

thanks to the implementation of the new heuristic mutation operator. To prove this, the new mutation operator (see Section 5.2.2) will also be implemented in the GA and MOEA/D, to compare results against their basic models. For the GA, the multiobjective problem is transformed into a single-objective problem using scalarization, i.e min f = α · f1′ (c) + β · f2′ (c)

(30)

where α and β are constant weights, used to balance the objective functions (10 and 0.08 respectively). The GA parameter setting can be found in [42], and the parameter setting for the NSGA-II will be the same as those of the NSGA-HO (please see Section 6.2). For the MOEA/D, the main parameter settings are Tchebycheff decomposition, maximum iterations 100, population size 200, and the number of weight vectors 15 [45]. For the Stage1, the IPSO will be implemented only when the heuristic mutation operator is used in Stage-2; otherwise, a conventional PSO will be used [43]. 6.3.1. Comparison results As shown in Fig. 13, the combination of the NSGA-HO and the IPSO (both proposed for TS-MOEAP) found the most economical configuration, amending all the problems despite an acceptable increase in power losses. On the other hand, ANMT which does not implement optimal NR, nor optimal DTs placement, has the highest investment cost. Furthermore, we must note that the GAHO and MOEA/D-HO found better configurations than their basic models, thanks to the implementation of the heuristic mutation operator. Additionally, the GA-HO obtained better results than the NSGA-II even though the first one is using scalarization for the multiobjective problem. With respect to MOEA/D, we found that this algorithm tends to install a greater number of transformers than other approaches. This causes an increase in the total investment cost, as shown in Fig. 13. After an analysis, we conclude that due to the discrete nature of the problem, the solutions to neighboring subproblems are not very close in the decision space, causing a lack of ability to explore new areas, and therefore a lack of dominant solutions. This can be verified by observing the Pareto front obtained by MOEA/D and NSGA-HO in Fig. 14. The authors in [45] reported a similar drawback. However, if the limits for the number of transformers to install are narrowed (e.g. τ = 5:6), and the heuristic mutation along with the IPSO are implemented, results as good as those of NSGA-HO can be found, in less computational time. Finally, Table 6 shows a comparison of different variables. Among the most important, the installed capacity and the oversize index are used to prove how well the loads are distributed among transformers. The proposed approach found a configuration that only needs an installed capacity of 350 kVA, this means an oversizing of 10.4% with respect to an average load of 317 kVA. As before, ANMT got the worst results, with an oversizing of 29.3%. The algorithms that implemented the novel heuristic mutation operator got better results in general. In summary, the aforementioned results provide evidence that TS-MOEAP can obtain better configurations using the proposed NSGA-HO and IPSO algorithms. 7. Conclusions TS-MOEAP is proposed for the planning and optimal condition restoration of secondary distribution networks, which can present several quality issues. Due to the complexity of the entire optimization problem, this was divided into two stages: • Stage-1, for the optimal placement of transformers, conductor sizing, and branch routing using an improved particle swarm optimization technique and a greedy algorithm;

12

J.P. Avilés, J.C. Mayo-Maldonado and O. Micheloud / Applied Soft Computing Journal 90 (2020) 106182 Table 6 Algorithm comparison.

Investment cost [$] Power loss [kW] Voltage deviation [p.u.] Total transformers Installed capacity [kVA] Cost/Avg. load [$/kVA] Oversizing [%]

ANMT

GA

GA HO

NSGAII

MOEA /D

MOEA /D-HO

NSGA HO

27 185 10.5 0.85 7 410 85.8 29.3

26 670 9.9 0.83 6 385 84.1 21.5

24 421 11.1 0.81 6 360 77.0 13.6

25 351 11.9 0.91 6 365 80.0 15.1

26 844 10.5 0.79 8 385 84.7 21.4

23 145 11.6 0.90 6 325 73.0 2.5

23 062 12.8 0.99 5 350 72.8 10.4

Table 7 Active and reactive loads per node of the test system.

Fig. 14. Pareto fronts of MOEA/D and NSGA-HO from the test system.

• Stage-2, for the optimal topological reconfiguration of the network using an improved nondominated sorting genetic algorithm. For the multi-objective problem in Stage-2, two objective functions are considered: minimization of power losses and minimization of investment costs. Knowing the special characteristics of SDNs, a heuristic mutation operator was developed to find better topological configurations. TS-MOEAP was successfully applied to improve an urban network that presented several problems. The result was compared with those of other algorithms, concluding that the combination of the proposed NSGA-HO and IPSO can converge towards better configurations, satisfying all the imposed constraints. As future research, it is proposed to combine the optimization of primary and secondary distribution networks. Declaration of competing interest No author associated with this paper has disclosed any potential or pertinent conflicts which may be perceived to have impending conflict with this work. For full disclosure statements refer to https://doi.org/10.1016/j.asoc.2020.106182. CRediT authorship contribution statement J.P. Avilés: Conceptualization, Methodology, Software, Investigation, Project administration, Writing - original draft. J.C. MayoMaldonado: Validation, Formal analysis, Writing - review & editing, Visualization. O. Micheloud: Supervision, Writing - review & editing. Appendix See Table 7. The information can be found in [44].

ui

P kW

Q kVAr

ui

P kW

Q kVAr

ui

P kW

Q kVAr

ui

P kW

Q kVAr

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22

6.5 5.4 6.5 4.4 4.4 0.0 5.4 3.3 9.8 1.1 9.8 0.0 1.1 6.5 4.4 5.4 8.7 3.3 4.4 3.3 1.1 0.0

4.9 4.1 4.9 3.3 3.3 0.0 4.1 2.4 7.3 0.8 7.3 0.0 0.8 4.9 3.3 4.1 6.5 2.4 3.3 2.4 0.8 0.0

23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44

5.4 16.3 0.0 5.4 3.3 4.4 5.4 14.1 3.3 2.2 0.0 12.0 5.4 3.3 8.7 0.0 5.4 0.0 3.3 0.0 5.4 6.5

4.1 12.2 0.0 4.1 2.4 3.3 4.1 10.6 2.4 1.6 0.0 9.0 4.1 2.4 6.5 0.0 4.1 0.0 2.4 0.0 4.1 4.9

45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66

0.0 4.4 10.9 3.3 6.5 19.6 0.0 0.0 7.6 0.0 7.6 16.3 6.5 6.5 0.0 6.5 10.9 5.4 2.2 5.4 10.9 0.0

0.0 3.3 8.2 2.4 4.9 14.7 0.0 0.0 5.7 0.0 5.7 12.2 4.9 4.9 0.0 4.9 8.2 4.1 1.6 4.1 8.2 0.0

67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88

5.4 1.1 3.3 5.4 1.1 1.1 0.0 3.3 9.8 7.6 4.4 7.6 19.6 3.3 2.2 0.0 2.2 6.5 2.2 10.9 0.0 3.3

4.1 0.8 2.4 4.1 0.8 0.8 0.0 2.4 7.3 5.7 3.3 5.7 14.7 2.4 1.6 0.0 1.6 4.9 1.6 8.2 0.0 2.4

References [1] M.E. Baran, F.F. Wu, Network reconfiguration in distribution systems for loss reduction and load balancing, IEEE Trans. Power Deliv. 4 (2) (1989) 1401–1407. [2] P.S. Georgilakis, N.D. Hatziargyriou, Optimal distributed generation placement in power distribution networks: Models, methods, and future research, IEEE Trans. Power Syst. 28 (3) (2013) 3420–3428. [3] C. Wang, M.H. Nehrir, Analytical approaches for optimal placement of distributed generation sources in power systems, IEEE Trans. Power Syst. 19 (4) (2004) 2068–2076. [4] L. Tang, F. Yang, J. Ma, A survey on distribution system feeder reconfiguration: Objectives and solutions, in: 2014 IEEE Innovative Smart Grid Technologies - Asia, ISGT ASIA, 2014, pp. 62–67. [5] A.E. Abu-Elanien, M. Salama, K.B. Shaban, Modern network reconfiguration techniques for service restoration in distribution systems: A step to a smarter grid, Alexandria Eng. J. 57 (4) (2018) 3959– 3967, [Online]. Available: http://www.sciencedirect.com/science/article/pii/ S1110016818301893. [6] B.Y. Bagde, B.S. Umre, R.D. Bele, H. Gomase, Optimal network reconfiguration of a distribution system using biogeography based optimization, in: 2016 IEEE 6th International Conference on Power Systems, ICPS, 2016, pp. 1–6. [7] A. Tandon, D. Saxena, Optimal reconfiguration of electrical distribution network using selective particle swarm optimization algorithm, in: 2014 International Conference on Power, Control and Embedded Systems, ICPCES, 2014, pp. 1–6. [8] S. Bahadoorsingh, J.V. Milanovic, Y. Zhang, C.P. Gupta, J. Dragovic, Minimization of voltage sag costs by optimal reconfiguration of distribution network using genetic algorithms, IEEE Trans. Power Deliv. 22 (4) (2007) 2271–2278. [9] A. Kouzou, R.D. Mohammedi, Optimal reconfiguration of a radial power distribution network based on meta-heuristic optimization algorithms, in: 2015 4th International Conference on Electric Power and Energy Conversion Systems, EPECS, 2015, pp. 1–6. [10] R. Pegado, Z.. Naupari, Y. Molina, C. Castillo, Radial distribution network reconfiguration for power losses reduction based on improved selective

J.P. Avilés, J.C. Mayo-Maldonado and O. Micheloud / Applied Soft Computing Journal 90 (2020) 106182

[11]

[12]

[13]

[14]

[15]

[16] [17]

[18]

[19]

[20]

[21]

[22]

[23]

[24]

[25]

[26]

BPSO, Electr. Power Syst. Res. 169 (2019) 206–213, [Online]. Available: http://www.sciencedirect.com/science/article/pii/S0378779618304279. T.T. Nguyen, A.V. Truong, Distribution network reconfiguration for power loss minimization and voltage profile improvement using cuckoo search algorithm, Int. J. Electr. Power Energy Syst. 68 (2015) 233– 242, [Online]. Available: http://www.sciencedirect.com/science/article/pii/ S0142061514008023. Y.K. Wu, C.Y. Lee, L.C. Liu, S.H. Tsai, Study of reconfiguration for the distribution system with distributed generators, IEEE Trans. Power Deliv. 25 (3) (2010) 1678–1685. Y. Ch, S. Goswami, D. Chatterjee, Effect of network reconfiguration on power quality of distribution system, Int. J. Electr. Power Energy Syst. 83 (2016) 87–95, [Online]. Available: http://www.sciencedirect.com/science/ article/pii/S0142061516305713. S. Ghasemi, Balanced and unbalanced distribution networks reconfiguration considering reliability indices, Ain Shams Eng. J. 9 (4) (2018) 1567– 1579, [Online]. Available: http://www.sciencedirect.com/science/article/pii/ S2090447916301617. J. Shukla, B. Das, V. Pant, Stability constrained optimal distribution system reconfiguration considering uncertainties in correlated loads and distributed generations, Int. J. Electr. Power Energy Syst. 99 (2018) 121– 133, [Online]. Available: http://www.sciencedirect.com/science/article/pii/ S0142061517318343. R.A. Jabr, I. Džafić, I. Huseinagić, Real time optimal reconfiguration of multiphase active distribution networks, IEEE Trans. Smart Grid (2017). S. Huang, Q. Wu, L. Cheng, Z. Liu, Optimal reconfiguration-based dynamic tariff for congestion management and line loss reduction in distribution networks, IEEE Trans. Smart Grid 7 (3) (2016) 1295–1303. M. Abdelaziz, Distribution network reconfiguration using a genetic algorithm with varying population size, Electr. Power Syst. Res. 142 (2017) 9–11, [Online]. Available: http://www.sciencedirect.com/science/article/pii/ S0378779616303273. N.V. Kovački, P.M. Vidović, A.T. Sarić, Scalable algorithm for the dynamic reconfiguration of the distribution network using the Lagrange relaxation approach, Int. J. Electr. Power Energy Syst. 94 (2018) 188– 202, [Online]. Available: http://www.sciencedirect.com/science/article/pii/ S0142061516321019. H. Li, W. Mao, A. Zhang, C. Li, An improved distribution network reconfiguration method based on minimum spanning tree algorithm and heuristic rules, Int. J. Electr. Power Energy Syst. 82 (2016) 466– 473, [Online]. Available: http://www.sciencedirect.com/science/article/pii/ S0142061516306494. S. Jain, S. Kalambe, G. Agnihotri, A. Mishra, Distributed generation deployment: State-of-the-art of distribution system planning in sustainable era, Renew. Sustain. Energy Rev. 77 (2017) 363–385, [Online]. Available: http://www.sciencedirect.com/science/article/pii/S1364032117305245. S. Sultana, P.K. Roy, Krill herd algorithm for optimal location of distributed generator in radial distribution system, Appl. Soft Comput. 40 (2016) 391– 404, [Online]. Available: http://www.sciencedirect.com/science/article/pii/ S1568494615007541. B. Das, V. Mukherjee, D. Das, DG placement in radial distribution network by symbiotic organisms search algorithm for real power loss minimization, Appl. Soft Comput. 49 (2016) 920–936, [Online]. Available: http://www. sciencedirect.com/science/article/pii/S1568494616304744. M. Mohammadi, Bacterial foraging optimization and adaptive version for economically optimum sitting, sizing and harmonic tuning orders setting of LC harmonic passive power filters in radial distribution systems with linear and nonlinear loads, Appl. Soft Comput. 29 (2015) 345– 356, [Online]. Available: http://www.sciencedirect.com/science/article/pii/ S1568494615000253. E. kianmehr, S. Nikkhah, A. Rabiee, Multi-objective stochastic model for joint optimal allocation of DG units and network reconfiguration from DG owner’s and DisCo’s perspectives, Renew. Energy 132 (2019) 471– 485, [Online]. Available: http://www.sciencedirect.com/science/article/pii/ S0960148118309820. A. Tiguercha, A.A. Ladjici, M. Boudour, Optimal radial distribution network reconfiguration based on multi objective differential evolution algorithm, in: 2017 IEEE Manchester PowerTech, 2017, pp. 1–6.

13

[27] P.P. Biswas, R. Mallipeddi, P. Suganthan, G.A. Amaratunga, A multiobjective approach for optimal placement and sizing of distributed generators and capacitors in distribution network, Appl. Soft Comput. 60 (2017) 268– 280, [Online]. Available: http://www.sciencedirect.com/science/article/pii/ S1568494617304052. [28] T.T. Nguyen, T.T. Nguyen, A.V. Truong, Q.T. Nguyen, T.A. Phung, Multi-objective electric distribution network reconfiguration solution using runner-root algorithm, Appl. Soft Comput. 52 (2017) 93– 108, [Online]. Available: http://www.sciencedirect.com/science/article/pii/ S1568494616306366. [29] M. Kaur, S. Ghosh, Network reconfiguration of unbalanced distribution networks using fuzzy-firefly algorithm, Appl. Soft Comput. 49 (2016) 868– 886, [Online]. Available: http://www.sciencedirect.com/science/article/pii/ S1568494616304781. [30] T. Kaboodi, J. Olamaei, H. Siahkali, R. Bita, Optimal distribution network reconfiguration using fuzzy interaction and MPSO algorithm, in: 2014 Smart Grid Conference, SGC, 2014, pp. 1–5. [31] S.R. Tuladhar, J.G. Singh, W. Ongsakul, Multi-objective approach for distribution network reconfiguration with optimal DG power factor using NSPSO, IET Gener. Transm. Distrib. 10 (12) (2016) 2842–2851. [32] W. Sheng, K.Y. Liu, Y. Liu, X. Meng, Y. Li, Optimal placement and sizing of distributed generation via an improved nondominated sorting genetic algorithm II, IEEE Trans. Power Deliv. 30 (2) (2015) 569–578. [33] A.V. Pombo, J. Murta-Pina, V.F. Pires, Multiobjective formulation of the integration of storage systems within distribution networks for improving reliability, Electr. Power Syst. Res. 148 (2017) 87–96, [Online]. Available: http://www.sciencedirect.com/science/article/pii/S0378779617301128. [34] M. Ahmadigorji, N. Amjady, S. Dehghan, A novel two-stage evolutionary optimization method for multiyear expansion planning of distribution systems in presence of distributed generation, Appl. Soft Comput. 52 (2017) 1098–1115, [Online]. Available: http://www.sciencedirect.com/ science/article/pii/S1568494616304793. [35] A. Azizivahed, H. Narimani, E. Naderi, M. Fathi, M.R. Narimani, A hybrid evolutionary algorithm for secure multi-objective distribution feeder reconfiguration, Energy 138 (2017) 355–373, [Online]. Available: http: //www.sciencedirect.com/science/article/pii/S036054421731280X. [36] S. Datta, A. Ghosh, K. Sanyal, S. Das, A radial boundary intersection aided interior point method for multi-objective optimization, Inform. Sci. 377 (2017) 1–16, [Online]. Available: http://www.sciencedirect.com/science/ article/pii/S0020025516311276. [37] A. Peimankar, S.J. Weddell, T. Jalal, A.C. Lapthorn, Evolutionary multiobjective fault diagnosis of power transformers, Swarm Evol. Comput. 36 (2017) 62–75, [Online]. Available: http://www.sciencedirect.com/science/ article/pii/S2210650216301699. [38] H.M. Dubey, M. Pandit, B. Panigrahi, An overview and comparative analysis of recent bio-inspired optimization techniques for wind integrated multi-objective power dispatch, Swarm Evol. Comput. 38 (2018) 12– 34, [Online]. Available: http://www.sciencedirect.com/science/article/pii/ S2210650217306387. [39] S.N. Naserabad, A. Mehrpanahi, G. Ahmadi, Multi-objective optimization of HRSG configurations on the steam power plant repowering specifications, Energy 159 (2018) 277–293, [Online]. Available: http://www.sciencedirect. com/science/article/pii/S0360544218311952. [40] J.J. Grainger, W.D. Stevenson, Power System Analysis, Vol. 621, McGraw-Hill New York, 1994. [41] K. Deb, A. Pratap, S. Agarwal, T. Meyarivan, A fast and elitist multiobjective genetic algorithm: NSGA-II, IEEE Trans. Evol. Comput. 6 (2) (2002) 182–197. [42] A.E. Eiben, J.E. Smith, et al., Introduction to Evolutionary Computing, Vol. 53, Springer, 2003. [43] Y. del Valle, G.K. Venayagamoorthy, S. Mohagheghi, J.C. Hernandez, R.G. Harley, Particle swarm optimization: Basic concepts, variants and applications in power systems, IEEE Trans. Evol. Comput. 12 (2) (2008) 171–195. [44] EERCS, Geoportal, 2018, [Online]. Available: http://geoportal.centrosur.com. ec/geoportal/Default.aspx. (Accessed 5 January 2018). [45] Q. Zhang, H. Li, MOEA/D: A multiobjective evolutionary algorithm based on decomposition, IEEE Trans. Evol. Comput. 11 (6) (2007) 712–731.