Operating Cost Reduction in Distribution Networks Based on the Optimal Phase-Swapping Including the Costs of the Working Groups and Energy Losses

: The problem of optimal phase-balancing in three-phase asymmetric distribution networks is addressed in this research from the point of view of combinatorial optimization using a master– slave optimization approach. The master stage employs an improved sine cosine algorithm (ISCA), which is entrusted with determining the load reconﬁguration at each node. The slave stage evaluates the energy losses for each set of load connections provided by the master stage by implementing the triangular-based power ﬂow method. The mathematical model that was solved using the ISCA is designed to minimize the annual operating costs of the three-phase network. These costs include the annual costs of the energy losses, considering daily active and reactive power curves, as well as the costs of the working groups tasked with the implementation of the phase-balancing plan at each node. The peak load scenario was evaluated for a 15-bus test system to demonstrate the effectiveness of the proposed ISCA in reducing the power loss (18.66%) compared with optimization methods such as genetic algorithm (18.64%), the classical sine cosine algorithm (18.42%), black-hole optimizer (18.38%), and vortex search algorithm (18.59%). The IEEE 37-bus system was employed to determine the annual total costs of the network before and after implementing the phase-balancing plan provided by the proposed ISCA. The annual operative costs were reduced by about 13% with respect to the benchmark case, with investments between USD 2100 and USD 2200 in phase-balancing activities developed by the working groups. In addition, the positive effects of implementing the phase-balancing plan were evidenced in the voltage performance of the IEEE 37-bus system by improving the voltage regulation with a maximum of 4% in the whole network from an initial regulation of 6.30%. All numerical validations were performed in the MATLAB programming environment.


Introduction
Three-phase distribution networks are responsible for interfacing transmission and sub-transmission networks at high-to-medium-voltage substations with end users at medium-and low-voltage levels [1,2]. These grids are typically built with a radial configuration to minimize the cost of investment in conductors and to simplify the process of protective device coordination [3]. The main challenge in three-phase distribution networks is to minimize the energy losses in the grid at minimal investment costs [4]. The energy losses in three-phase networks are mainly caused by the asymmetric nature of the matrix of impedances in all distribution lines, as well as the presence of multiple single-, two-, and three-phase loads [5]. Energy losses in electrical distribution networks can vary from 6 to 18% in the Colombian context [6]. These variations in the distribution networks depend used a linear version of the three-phase power flow problem, which introduces estimation errors in the final solution. Authors of [23] presented a mixed-integer convex model to solve the load redistribution problem in three-phase asymmetric networks. The convex reformulation makes it possible find the global optimum of the approximated proposed model without considering the effect of the power balance equations in the optimization model; however, when the solution is evaluated in the exact phase-balancing formulation, the solutions provided by [23] correspond to locally optimal solutions. The authors of [24] proposed a linear programming formulation for the problem of phase balancing in threephase networks. They did not consider the effect of the voltage magnitudes, since they were close to the substation voltage (i.e., 1.0 p.u.). Numerical results validated the proposed linear integer programming model in a small grid with six nodes; however, the authors did not provide comparisons with metaheuristics to confirm the effectiveness of their proposal. Table 1 summarizes the main approaches reported in the literature to solve the problem of phase balancing in three-phase asymmetric distribution networks, which are based on metaheuristics and approximated convex models. Power losses minimization 2020 [5] Mixed-integer conic reformulation Expected energy losses 2021 [22] Mixed-integer convex approximation Average unbalance level 2021 [23] The main characteristic of the previous metaheuristic optimization methods listed in Table 1 is that they follow the leader-follower strategy. Here, the leader stage is entrusted by defining the load connections, and the follower stage solves the three-phase power flow problem to evaluate the objective function value, thereby helping guide the solution methodology through the solution space [25,26]. In addition, most of these studies only evaluated the phase-balancing methodology in the peak load condition and did not consider daily active and reactive power curves or the costs of the working groups that implement the final phase balancing along the test feeder.
The present research study makes the following contributions to the field: • The use of an improved sine cosine algorithm (ISCA) to solve the phase-balancing problem; the algorithm was modified by changing the number of points in the codification that can change in each iterative cycle. This was achieved by using an adaptive rule as a function of the maximum number of nodes of the network. • Hybridization of the ISCA (master stage) with the triangular-based three-phase power flow method (slave stage), which can solve power flow problems in asymmetric networks with loads with Yand ∆-connections.

•
Evaluation of the phase-balancing plan by modifying the objective function to account for the annual costs of the energy losses (including daily active and reactive power curves) and the costs of the working groups that travel along the feeder to implement the optimization plan.
Importantly, the use of the ISCA to solve the phase-swapping problem in three-phase networks using master-slave optimization methodology has not previously been reported in the specialized literature. This is a gap in the current research that this work aims to fill. In addition, the results reported in this paper take into account the annual operating costs of the network by additionally considering the costs of deploying the working groups. This could be considered a reference point for the proposal of future optimization methodologies designed to address the problem of phase balancing in asymmetric three-phase networks.
The remainder of this document is structured as follows: Section 2 presents the general mixed-integer nonlinear programming model to represent the optimal phasebalancing problem in three-phase networks with a multi-period formulation; Section 3 describes the proposed optimization methodology based on the improved version of the sine cosine algorithm in the master stage and the triangular-based three-phase power flow method in the slave stage; Section 4 presents the main characteristics of the threephase networks under study, which are composed of 15 and 37 nodes, respectively. Then, Section 5 presents the main numerical results obtained by the proposed ISCA in the 15and IEEE 37-bus systems, considering the peak load scenario and daily active and reactive power curves. Furthermore, comparisons with four metaheuristic optimizers are provided to validate the effectiveness and robustness of our master-slave optimization proposal. Finally, Section 6 presents the concluding remarks derived from our research and outlines possible future work.

General Optimization Model
Electrical distribution networks are typically constructed with a three-phase structure such that they are as balanced as possible in terms of the impedances; however, the nature of the loads makes it impossible to ensure that the network is completely symmetrical, owing to the presence of single-, two-, and three-phase loads. This implies that the currents through all the lines are generally asymmetric [27,28], which significantly increases the energy losses of the network with respect to the ideal balanced case. Ultimately, this has a negative impact on the net profit of the distribution company in the electricity commercialization activity [29]. In order to address the problem of load balancing in three-phase networks, we propose an optimization model from the family of mixedinteger nonlinear programming (MINLP) methods, which are suitable for the analysis of multiperiod operating environments. The proposed optimization model attempts to minimize the annual operating cost of the network. These costs include the total costs of the energy losses summed with the costs of the phase-balancing task associated with the costs of displacing the crew working along the feeder. Note that the nonlinearities in the phase-balancing problem are related to the products of the trigonometric function and voltage magnitudes in the power balance constraints [30]. The integer part of the optimization model is related to the six possible load connections that are allowed at each node of the network [22]. The complete MINLP model is given in the following.

Objective Function
The structure of the objective function considered in this study corresponds to the summation of the costs incurred as a result of the annual energy loss and the cost of the phase balancing, which is associated with the displacement of the working group along the distribution feeder to implement the phase-balancing plan. Equation (1) defines the objective function used in this study.
where A cost represents the total annual cost of the network operation, f 1 is the annual operating costs related with the total energy losses in all the branches of the network, and f 2 is the cost of the phase-balancing activity. C kWh represents the average cost of the energy losses, T represents the number of hours in an ordinary year (i.e., 8760 h), Y k f mg represents the magnitude of the admittance that relates node k at phase f with node m at phase g, V k f t (V m f t ) corresponds to the voltage magnitude at node k (m) in phase f (g) at time period t, δ k f t (δ mgt ) is the angle of the voltage at node k (m) in phase f (g) at time period t, θ k f mg represents the angle of the admittance that relates node k at phase f with node m at phase g, ∆ t is the time period during which the power demands remain constant, and C k,bal is the cost of interchanging load phases at node k. Observe that F , N , and T are the sets that contain all the phases, nodes, and time periods, respectively.

Set of Constraints
The general set of constraints related to the phase-balancing problem in three-phase asymmetric networks includes active and reactive power balance equations, voltage regulation bounds, and conditions over the decision variables to maintain the feasibility of the solution space [5,22]. The set of constraints is as follows: where P s k f t and Q s k f t represent the active and reactive power generated by source s connected at node k in phase f in time period t, P d kgt and Q d kgt correspond to the active and reactive power demands connected at bus k in phase g at time period t, x k f g is a binary variable that determines the connection of the constant power consumption at bus k at f in phase g, and V min and V max are the minimum and maximum voltage regulation limits permitted for all buses of the electrical grid in each time period, respectively. Remark 1. The equality constraints (2) and (3), regarding the active and reactive power balance equations at each node, phase, and period of time, respectively, show the complexity of the three-phase unbalanced power flow problem in distribution grids, even when the connection of the loads at these nodes is perfectly known. Note that the main complication in these equations arises because of the need to calculate the products of the voltages and trigonometric functions, which necessitates the use of numerical methods to reach the solution within the desired convergence error [28].

General Model Interpretation
To understand the general formulation of the phase-balancing problem in asymmetric distribution networks, the following interpretations of Equations (1) and (6) are provided. Equation (1) is the objective function of the optimization problem. This function is intended to minimize the annual operating costs of the network, combining the costs of the energy losses with the cost of the phase-balancing implementation using working groups that travel along the feeder to implement the optimization plan. Equations (2) and (3) correspond to the active and reactive power balance equations for each node, phase, and time period in the three-phase distribution network, respectively. Note that in metaheuristic optimization methods with master-slave strategies these equations are solved numerically using methods such as the backward/forward power flow method [31], triangular-based power flow method [28], Newton-Raphson method, and graph-based approaches [32].
Equations (4) and (5) ensure that the loads are connected to the phases in a unique form by employing a matrix connection at each bus (i.e., node k), formed by the variables x k f g , with the values in each column and each row of the matrix required to be equal to 1. Finally, the box-type Constraint (6) determines the upper and lower voltage regulation limits applicable to the electric distribution grid, which are typically ±10% for medium-voltage levels [5].

Solution Methodology
In this research, the problem of optimal phase balancing in three-phase networks is addressed from the point of view of master-slave optimization [15]. For the master stage, an improved sine cosine algorithm (ISCA) is proposed, whereas the slave stage is governed by solving the power flow problem for three-phase networks by using the recently developed triangular-based power flow method [28]. Note that the master stage is entrusted with determining the load configuration at each node of the network, whereas the slave stage is entrusted with the calculation of the costs incurred because of the energy loss during the period of analysis. In the following subsections, the master and slave stages are described in detail.

Master Stage: Improved Sine Cosine Algorithm
To determine the nodal connections of the three-phase loads, we propose an improved version of the sine cosine algorithm that explores and exploits the solution space by evolving through the solution space with trigonometric sine and cosine functions [33,34]. The main idea of the ISCA is to sweep the solution space using the information of the best current solution at iteration t (i.e., x best ) and each individual x t i in the population with weighted sine/cosine factors and a reduced number of modifications in the structure of the individual x t i , based on the adaptive rule. The codification adopted to solve the phase-balancing problem is presented in Table 2. Note that the binary variable x k f g can be easily represented with integer numbers between 1 and 6 [11]. Table 2. Possible load connections in a three-phase node [5].

Connection Type Phases Sequence
Binary Variable The proposed classification for each individual x t i in the ISCA is presented in Equation (7): which produces an initial population with the form I t where n i is the number of individuals in the population.

Remark 2.
Importantly, the codification presented in Equation (7) is feasible because each component of the individual x t i , i.e., x t ij , is generated with the following rule thereof: where x min = 1, x max = 6, round(y) is a function that obtains the integer part of a real number, and rand is a random number between 0 and 1 generated with a normal distribution.
To generate the descending individuals, we propose a variation of the sine cosine algorithm that does not modify all the information of an individual in the same iteration. The general evolution of the ISCA is presented in Algorithm 1.

Algorithm 1: Generation of the descending individual.
Data: Take the information of the individual x t i for i ≤ n i do Set y t i = x t i ; Generate the probability ρ = rand; Select the positions of the individual y t i that will be modified, that is, In Algorithm 1, the parameter ρ is determined for each individual if the evolution is made to occur using the sine or cosine function, r 2 is a random number between 0 and 2π generated with a normal distribution, and r 3 is a random number between 0 and 1 that determines the level of importance of the best current solution in the generation of the new individual. Note that r 1 is a variable factor that controls the balance between the exploration and exploitation of the solution space. This parameter is calculated as where t max is the maximum number of iterations specified for the ISCA.
Note that an important step prior to the evaluation of the descending individual y t i is the verification of its upper and lower bounds, that is, x min ≤ y t ij ≤ x max . If either the upper or lower limit is violated, then the individual y t ij is corrected by applying Equation (8). To determine whether the descending individual y t i will occupy the position x t+1 i , the objective function value associated with the annual operating costs must be evaluated in the slave stage, which implies that if A cost y t i ≤ A cost x t i , then x t+1 i = y t i . The evolution process of the ISCA ends if one of the following criteria is met: • If the maximum number of iterations t max is reached, then x t max best is reported as the optimal solution. • If the objective function of the best solution does not improve during k max iterations, x t t best is reported as the optimal solution.
Remark 3. The main advantage of using part of an individual to implement changes (i.e., the p m positions of the vector) is that the algorithm does not undertake large jumps through the solution space. This is necessary to control the exploration and exploitation properties of the algorithm, especially because of the discrete nature of the solution variables. The number of positions p m that will change varies between 1 and 10% of the number of variables of the problem, that is, n nodes.

Slave Stage: Triangular-Based Power Flow Method
To evaluate each configuration provided by the master stage, we used the triangularbased three-phase power method that was recently proposed in [28]. The main advantages of this approach are its shorter processing times and its guarantee of convergence via the Banach fixed-point theorem [31]. The general formula for the triangular-based three-phase power flow method is presented below (taken from [28]): where V 3ϕ ∈ C 3(n−1)×1 corresponds to a vector that contains all the phase voltages ordered per node, 1 3ϕ ∈ R 3(n−1)×3 corresponds to a rectangular matrix filled by 3 × 3 identity matrices, V 13ϕ ∈ C 3×1 is a vector that contains the voltages of the substation node (voltage-controlled node), T 3ϕ ∈ R 3b×3b represents the three-phase equivalent of the upper-triangular matrix, Z 3ϕ ∈ C 3b×3b represents the primitive three-phase impedance matrix, which has a three-diagonal form, and I 3ϕ ∈ C 3(n−1)×1 is a vector that contains all the phase currents ordered per node. Note that Equation (9) must be solved iteratively because the voltages in all the demand nodes are indeed a function of the demanded currents (I 3ϕ ), which are functions of these voltages, that is, I 3ϕ = f V 3ϕ . Equation (10) presents the recursive formula to solve the three-phase power-flow problem considering symmetric and asymmetric loads.
where m is the iterative counter. In addition, the convergence criterion adopted in this research corresponds to the maximum error between the voltage variables in two consecu- where ε is the maximum tolerance and it is assigned as 1 × 10 −10 , as recommended in [35].
It is important to emphasize that the triangular-based power flow method defined by the recursive Formula (10) has the ability to work with loads with ∆and Y-connections, which implies that the current I m 3ϕ must be calculated as a function of the type of load. If we consider that at node k there exists a load with a Y-connection, then the method for calculating the demand current is defined by Equation (11).
where I ka , I kb , and I kc are the current demands in phases a, b, and c, respectively; V ka , V kb , and V kc are the voltages per phase, considering that the neutral point of the load is solidly grounded; and S ka , S kb , and S kc correspond to the loads connected between each phase and the neutral point. Observe that X presents the conjugate operator of the variable or parameter X.
In the case of loads with a ∆-connection at node k, the demanded current is calculated as defined in Equation (13).
where S kab , S kbc , and S kca are the apparent power consumption between the connections between phases a and b, b and c, and c and a, respectively.
To generalize the implementation of the triangular-based power flow method, Algorithm 2 presents the pseudo-code to solve the recursive Formula (10) for a general three-phase network with symmetric and asymmetric loads. Set m = 0; Determine the initial voltage as V m 3ϕ = 1 3ϕ V 13ϕ ; k = 1; for m ≤ m max do for k ≥ n − 1 do if Load in node k connected to Y then Compute the demanded current I k3ϕ using Equation (11) ; else Compute the demanded current I k3ϕ using Equation (13); Calculate the new voltages V m+1 3 f using Equation (10); Report the nodal voltages as V = V 13ϕ ; V m+1 3ϕ ; Report the final three-phase current, i.e., I 3ϕ ; break; else Once the power flow problem is solved using the triangular-based power flow methodology presented in Algorithm 1, the evaluation of the annual energy losses of the network necessitates the calculation of the total power losses in each period of time. This is accomplished using the following expressions: where J 3ϕ represents the vector that contains all the currents in the branches of the network, and E 3ϕ represents the voltage drops. Note that if we combine these vectors, then the grid power losses are reached as defined in Equation (15).
where S loss corresponds to the apparent power losses in the network.

General Algorithm for the Proposed Master-Slave Optimizer
Algorithm 3 presents the pseudo-code with the optimization methodology designed to simplify the implementation of the proposed master-slave optimization approach. This method optimizes phase balancing in three-phase asymmetric networks with the ultimate aim of lowering the annual operating costs of the grid.  (10); Evaluate the initial population using Algorithm 1 (i.e., slave stage); for t ≤ t max do Select the best current individual x t best ; for i ≤ n i do Generate each of the descending individuals y t i using algorithm 1; Verify that each y t i is in its upper and lower bounds using Equation (10); if is one of the stopping criteria met? then Result: Report the optimal solution contained in x t best .

Test Feeders
The proposed master-slave optimization approach was validated using two test feeders in medium-voltage distribution networks composed of 15 and 37 nodes with radial topology, respectively. A description of each of these test feeders is presented below.

15-Bus Test Feeder
This test feeder is composed of 15 nodes and 14 branches, with a voltage-controlled source connected at node 1, which is operated with a voltage magnitude of 13.2 kV. A single-phase diagram of this test feeder is presented in Figure 1. In addition, all the data for the 15-bus system are presented in Tables 3 and 4. Note that this information was adapted from [5].

Behavior of the Demand in a Typical Working Day in Colombia
The effectiveness of the ISCA, in terms of performing the phase-balancing task for three-phase distribution networks, is demonstrated by considering the demand curves presented in Figure 3. The data used to plot these curves are also reported in Table 7 to enable our results to be compared with those in future research (note that the scaling factor of the data reported in Table 7 for active and reactive power demands is 2). Additional details regarding the active and reactive power daily behaviors presented in Table 7 can be consulted in [36].  Note that in the objective function we assume that the average energy cost is US-D/kWh 0.1390, which corresponds to the average cost of the energy in Bogotá, Colombia in May 2019 [36]. The number of days is considered to be 365 for an ordinary year, and the length of the power flow period, ∆ t , is 0.5 h. In addition, the cost of the phase balancing of the working group is specified as USD 100 per node that requires intervention.

Numerical Analysis
In this section, we present the numerical results of the proposed master-slave optimization for solving the phase-balancing problem in three-phase asymmetric networks to minimize the annual grid operational costs. The following simulation scenarios are considered: (i) the evaluation of the numerical performance of the proposed ISCA on the 15-bus system considering the peak load condition, and (ii) the analysis of the annual operating cost of the network for the IEEE 37-bus system by employing the proposed ISCA.

Parametrization of the Optimization Algorithms
To determine the optimal population size for the proposed ISCA and the comparative optimization methodologies, we employed a grid generation with population sizes from 10 to 100 in steps of 5 in the 15-bus system for the peak load simulation case. The results were compared with those obtained using the following comparative methods: the classical Chu and Beasley genetic algorithm (CBGA) [11], classical sine cosine algorithm (SCA) [37], black-hole optimizer (BHO) [38], and vortex search algorithm (VSA) [5]. Table 8 presents the general behavior of each of the optimization algorithms considering as performance indicators the minimum, maximum, mean and standard deviation. The numerical results in Table 8 allow us to observe that: All the optimization methods required at least 50 or more individuals in the population to achieve an adequate objective function performance; the lowest minimum population size was 50, for the proposed ISCA, and the largest minimum population size was 95, for the BHO.
The minimum power losses were obtained by the proposed ISCA, with a value of 109.1980 kW, followed by the CBGA with a value of 109.2218 kW; however, the reliability of the ISCA was better since it had the lowest standard deviation of all the methods compared (0.1144 kW), followed by the CBGA as the second-best method with a standard deviation of about 0.2075 kW. The ISCA presented a small variation between the extreme solutions, with a difference between the minimum and maximum values of 0.4916 kW; the largest difference was obtained with the BHO, with a value of 3.5207 kW. Note that the small difference between the minimum and maximum values in conjunction with the low standard deviation confirms that the ISCA obtained all the solutions inside of a small hypersphere, with the main advantage that most of the solutions obtained near to the optimal are better than the best results of the comparative methods. As an example, see the mean value of the ISCA in comparison with the minimum values of the BHO and the classical SCA.
On the other hand, to demonstrate the effectiveness of the proposed ISCA to solve the phase-balancing problem on the 15-bus system for the peak load scenario, Table 9 presents all the numerical results for all the optimization algorithms after 100 consecutive iterations, with the optimal number of individuals in the population reported in Table 8, for a total of 1000 iterations. Table 9. Reduction of the power losses after phase balancing considering all the loads with Y-connections in the 15-bus system.

Method Solution Losses (kW) Reduction (%)
Benchmark case {1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 The results in Table 9 indicate the following: (i) all the comparative methods reduced the power losses by more than 18%; (ii) the ISCA achieved the best solution with a reduction of 18.66% regarding power losses, followed by the CBGA, with a difference of 0.02%; (iii) the worst-performing method was the BHO with a reduction of 18.38%, which implies a difference of 0.3735 kW with respect to the solution reached by the ISCA. Figure 4 presents the 10 best objective function values determined by the ISCA after 100 consecutive evaluations. The results in this figure indicate that the difference between the first and the tenth values is only 0.04 kW, which implies that these 10 solutions are closer to one another. In addition, a comparison of the tenth solution (i.e., 109.2427 kW) with the results in Table 9 reveals that this result is superior to those obtained by all the comparative methods, except CBGA, which confirms the high efficiency of our proposed ISCA regarding the reduction of power losses.
On the other hand, Figure 5 shows the general evolution of the objective function value as a function of the number of iterations for each of the comparative methods and the one proposed reported in Table 9.
The behavior of the objective function value with respect to the number of iterations depicted in Figure 5 for the different metaheuristic optimizers indicates the following: (i) all of them were stabilized by about 500 iterations, since after this number none of the optimizers presented variations in the final objective function value; (ii) after the first 200 iterations, all the optimizers reached an objective function value lower than 112 kW (i.e., a reduction of about 16.57%), which implies that these iterations were used to explore the solution space while the remaining iterations were used to exploit the promissory regions of the solutions space identified in the exploration stage; and (iii) the classical SCA showed a premature convergence to the local optimum when compared with the proposed ISCA, since the classical SCA did not present objective function reduction after the first 150 iterations while the ISCA reached improvements during the first 450 iterations. These demonstrate the positive effect of the hybridization with the vortex search algorithm to balance the exploration and exploitation of the solution space. Power losses (kW)

Analysis of Annual Operating Costs
The results of the evaluation of the annual operating cost of the network (considering the costs of the working groups) for the IEEE 37-bus system are reported in Table 10, where the five best solutions for this system are presented. The results in Table 10 indicate the following: (i) the optimal solution obtained by the ISCA (i.e., Solution 1) allows an annual operating cost reduction of USD 5774.3627 with respect to the benchmark case; (ii) the difference in cost between the first and the final solutions reached by the ISCA was only USD 100.6840, which is the same amount a phase change by the working group would incur; (iii) the percentage reduction with respect to the benchmark case for all the solutions was contained between 13.36% and 13.13% when Solutions 1 and 5 were respectively analyzed. These amounts imply that the ISCA generally allows the annual costs of operating the network to be lowered by approximately 13%. This would translate into a net profit for the utility company that could increase in subsequent years because the working groups would not incur additional cost in future years; (iv) the cost of the working groups oscillated between USD 2100 and USD 2200, implying that between 21 and 22, changes in the phase configuration are needed to improve the general grid performance; and (v) the average processing time to solve the phase-balancing problem with the ISCA was approximately 220 s for all 100 consecutive evaluations of the methodology. This could be considered the minimum processing time owing to the complexity of the phase-balancing optimization problem, which for the IEEE 37-bus system has a solution space with 6 n−1 dimensions (with n = 37)-tfhat is, 1.031 × 10 28 solutions are possible.
The positive impact of the phase-balancing strategy in three-phase asymmetric networks is illustrated in Figure 6, which depicts the percentage reduction in the cost of the energy losses of the network, i.e., the reduction of component f 1 of the objective function defined in Equation (1) Energy losses reduction (%) Figure 6. Level of reduction in the energy losses for the five best solutions reached by the ISCA for the IEEE 37-bus system.
The general conclusion based on the results in Figure 6 is that the average reduction in the costs associated with the energy loss from the network was approximately 18%, with a maximum of 18.45% for solution 1 and 17.98% for solution 5.
It is worth mentioning that the cost of the energy losses, i.e., the objective function f 1 , had a participation about 94.1259% in the total annual operative costs for Solution 1 and 94.4079% in the case of Solution 5 (see Table 10). This implies that, with inversions lower than 6% of the A cost , it is possible to reach important reductions in the amount of grid power losses and their costs, as presented in Figure 6. This is indeed the most important component of the objective function in the case of the IEEE 37-bus system.
The numerical values in Figure 7 indicate that phases a and c involved reductions of 126.98 kWh/day and 408.17 kWh/day regarding the daily energy losses, respectively, whereas phase c experienced an increase of 220.80 kWh/day. This behavior is expected in the phases of the network because the objective of the phase-balancing problem is to reduce the level of asymmetry of the network, which is then achieved by redistributing all the loads of the network as uniformly as possible. Note that even if the daily energy losses of phase b were to increase, the general effect of phase balancing is a global daily reduction of 314.35 kWh/day or 18.45%, as presented in Figure 4 for Solution 1.
In addition, to observe the benefit derived from the reduction in the energy losses in terms of the general electrical performance of the system, we present the total daily energy losses per phase for the benchmark case and for the best solution produced by the proposed ISCA in Figure 7. Daily energy losses (kWh/day) Figure 7. Daily energy losses per phase before and after using the proposed ISCA for optimal phase-balancing in the IEEE 37-bus system.

Complementary Analysis
Here, we present some additional results and comments for the IEEE 37-bus system that allow us to confirm the positive effects that allow the implementation of the phasebalancing plan through the working groups along the grid. One of the most significant effects of the phase-balancing in three-phase asymmetric networks was the general improvement of the voltage profile in all nodes of the network. Figure 8 presents the general performance of the voltage profile before and after implementing the phase-balancing plan in Solution 1 (see Table 10).  The behavior of the voltage profiles in Figure 8 indicates the following: (i) Some magnitudes of the voltages in phases a and c had a regulation voltage higher than 5%, namely, nodes 13 to 22 with voltages lower than 0.95 p.u. (Figure 8a). (ii) Phase b showed a better voltage performance in the benchmark case (see Figure 8a) when compared to phases a and c, which is an expected behavior since this phase had lower active and reactive power demand consumption when compared with the other phases. The active and reactive power consumption in all the phases were 727 kW and 357 kvar for phase a, 639 kW and 314 kvar for phase b, and 1091 kW and 530 kvar for phase c. (iii) After the implementation of the phase-balancing plan (i.e., Solution 1 in Table 10), all the phases had a maximum voltage regulation of 4%, which implies that all the voltage profiles were above or equal to 0.96 p.u. In addition, all the phases presented a closer voltage behavior, which is the product of the load redistribution in all the phases, which had the following final consumptions: 763 kW and 371 kvar for phase a; 941 kW and 461 kvar for phase b; and 753 kW and 369 kvar for phase c.
An additional important result after implementing the phase-balancing plan is the improvement of the general grade of unbalance of the network. The grade of unbalance was measured for each phase with respect to the ideal consumption of the phase (perfectly balanced case), which was obtained through the average of the active and reactive power consumption in all of the phases. The ideal active and reactive power consumptions for the IEEE 37-bus system would be 819 kW and 400.33 kvar. With these values, the grades of unbalance before and after the implementation of the phase-balancing plan are reported in Table 11. The results in Table 11 show the following: (i) Phase c had the highest level of unbalance with respect to the ideal case in the benchmark scenario (greater than 30% for active and reactive power demands); this was because this phase had the highest demand in this scenario, with 1091 kW and 530 kvar (i.e., with an additional 272 kW and 129.67 kvar with respect to the ideal case). (ii) After implementing the phase-balancing plan, the highest level of unbalance corresponding to phase b was about 14.90% for the active power and 15.15% for the reactive power. (iii) Phases a and c reduced their active and reactive power unbalances from two digits to one digit, beginning their variation in phase c with more than 25% in both power demands.
Regarding the implementation of the phase-balancing plan by part of the working groups, it is important to mention that only a few trained staff with the ability to disconnect and reconnect transformers at the point of common coupling between the loads and the distribution feeder are required. However, before making any physical interventions in the network that will affect the electricity supply, the utility staff must follow the norms of each country regarding the notification of the end-users.

Conclusions
The problem of optimal phase balancing in three-phase asymmetric distribution networks was addressed in this research. The problem was solved from the point of view of metaheuristic optimization using a newly proposed master-slave optimization approach. The master stage employs an improved version of the sine cosine algorithm to determine the load connections among phases by using integer codification. In the slave stage, the triangular-based power flow method is used for three-phase networks that operate with loads with Yand ∆-connections. Two test feeders composed of 15 and 37 nodes were employed for the numerical validation of the proposed ISCA to solve the phase-balancing problem.
The numerical results for the 15-bus system demonstrated that the ISCA allowed a total power loss reduction of 18.66% with respect to the benchmark case. These results were superior to those of the genetic algorithm (18.64%), black hole optimizer (18.04%), sine cosine algorithm (18.51%), and vortex search algorithm (18.57%). In addition, the difference in power losses among the 10 best solutions reached by the proposed ISCA was approximately 0.04 kW, demonstrating its efficiency with respect to solving the phasebalancing problem in three-phase asymmetric networks.
The computational validations of the IEEE 37-bus system demonstrated that the proposed ISCA allowed reductions of approximately 13% in the annual operating costs of the network, with investments between USD 2100 and USD 2200 in working groups to implement the optimal phase-balancing plan along the test feeder in nodes where changes are required. In addition, the voltage regulation in the IEEE 37-bus system was improved from 6.3% to 4% when compared to the benchmark case with the implementation of the phase-balancing plan. In addition, the average processing time to solve the optimization problem was approximately 220 s, which can be considered a relatively small computational requirement, given that the dimension of the solution space for the IEEE 37-bus system is higher than 1 × 10 28 .
In the future, the following research aims should be addressed: (i) to use the proposed ISCA to solve the problem of the optimal selection of conductors for three-phase asymmetric networks, including the phase-balancing strategy in a unique codification; and (ii) to propose a convex programming model that allows the global optimal solution of the phasebalancing problem to be reached using conic programming, and to compare these results with those of the proposed ISCA to determine the discrepancy between the global optimal solution and the solutions reached using metaheuristics.

Data Availability Statement:
No new data were created or analyzed in this study. Data sharing is not applicable to this article.