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Article

Application of the Generalized Normal Distribution Optimization Algorithm to the Optimal Selection of Conductors in Three-Phase Asymmetric Distribution Networks

1
Grupo de Compatibilidad e Interferencia Electromagnética, Facultad de Ingeniería, Universidad Distrital Francisco José de Caldas, Bogotá 110231, Colombia
2
Laboratorio Inteligente de Energía, Facultad de Ingeniería, Universidad Tecnológica de Bolívar, Cartagena 131001, Colombia
*
Author to whom correspondence should be addressed.
Energies 2023, 16(3), 1311; https://doi.org/10.3390/en16031311
Submission received: 23 December 2022 / Revised: 18 January 2023 / Accepted: 24 January 2023 / Published: 26 January 2023
(This article belongs to the Special Issue Emerging Topics in Power Electronic Converters of Microgrids)

Abstract

:
This article addresses the problem of the optimal selection of conductors in asymmetric three-phase distribution networks from a combinatorial optimization perspective, where the problem is represented by a mixed-integer nonlinear programming (MINLP) model that is solved using a master-slave (MS) optimization strategy. In the master stage, an optimization model known as the generalized normal distribution optimization (GNDO) algorithm is proposed with an improvement stage based on the vortex search algorithm (VSA). Both algorithms work with discrete-continuous coding that allows us to represent the locations and gauges of the different conductors in the electrical distribution system. For the slave stage, the backward/forward sweep (BFS) algorithm is adopted. The numerical results obtained in the IEEE 8- and 27-bus systems demonstrate the applicability, efficiency, and robustness of this optimization methodology, which, in comparison with current methodologies such as the Newton metaheuristic algorithm, shows significant improvements in the values of the objective function regarding the balanced demand scenario for the 8- and 27-bus test systems (i.e., 10.30% and 1.40% respectively). On the other hand, for the unbalanced demand scenario, a reduction of 1.43% was obtained in the 27-bus system, whereas no improvement was obtained in the 8-bus grid. An additional simulation scenario associated with the three-phase version of the IEEE33-bus grid under unbalanced operating conditions is analyzed considering three possible load profiles. The first load profile corresponds to the yearly operation under the peak load conduction, the second case is associated with a daily demand profile, and the third operation case discretizes the demand profile in three periods with lengths of 1000 h, 6760 h, and 1000 h with demands of 100%, 60% and 30% of the peak load case. Numerical results show the strong influence of the expected demand behavior on the plan’s total costs, with variations upper than USD/year 260,000.00 between different cases of analysis. All implementations were developed in the MATLAB® programming environment.

1. Introduction

The network operator of the Local Distribution System depends fundamentally on the physical assets of the National Interconnected System to be able to transport electrical energy from the generation plants to the final consumer, with the distribution networks being the greatest asset that the power system represents [1]. The constant growth of societies brings with it the need to expand and repower networks. In addition, the high energy demand, the use of new technologies and the transition to e-mobility systems are factors that accelerate the thermal limits being reached in conductors [2]. All of the above entails new challenges for the network operator, who seeks to ensure the quality, reliability and safety in the provision of public service, developing an adequate and efficient planning of its emerging electric power systems [1]. Solving problems associated with the loadability in transmission lines has marked the trend in smart electrical systems, and factors such as environmental awareness have led to the promotion of new plans for the effective and efficient expansion of distribution networks [2,3].
However, there is a risk implied in the improper selection of conductors for the electrical distribution networks, which can compromise the correct operation of the network if the conductors selected do not meet the electricity demand of all the users, making the system unreliable and inefficient, or by oversizing them which would be detrimental to the investment in addition to being an unnecessary expense of material [4]. Properly planning the electrical system better secures its good health in economic and technical terms; since energy loss is minimized, voltage profiles are optimized, and reliability and quality in service provision are improved [3]. In this sense, applying new methods that will guide the appropriate planning and expansion of electrical distribution networks will not only ensure that the system is operating correctly but will also reduce the costs associated with its implementation and development [5]. An efficient network distribution expansion plan must ensure the economic viability of the distribution company in compliance with the quality standards required by the regulatory entities [3]. The optimal conductor selection problem is a classic subproblem in the efficient expansion of the electrical distribution system [6] and can be approached through different optimization methodologies, among which heuristic algorithms and exact methods stand out. In this way, carrying out arbitrary expansions in the electrical system can lead to the detriment of the general economy of the system, which is why an adequate management and design is necessary. Thus, based on mixed-integer programming and heuristic algorithms, an efficient tool can be obtained which allows solving these types of problems [7,8,9], reducing costs thanks to their ability to analyze the solution space and correctly assign the elements to be installed in the electrical distribution system [10]. Linear and nonlinear approximations also allow for an adequate formulation of this type of problem since these models are solved through exact techniques available in commercial optimization packages. Implementing heuristic algorithms allows evaluating the solution space with the objective of maximizing the savings associated with investment and operating costs [11,12]. This does not involve enormous computational efforts and contributes greatly to the development of electrical networks. By improving efficiency in this type of project, investment is encouraged, bringing with it more job opportunities, greater development and technology in this sector, in addition to being able to make this type of business a market opportunity.
The following in this document is organized as follows: Section 2 presents the literature review and contributions of this. Section 3 presents the general mathematical formulation for the problem of optimal selection of conductors in asymmetric three-phase distribution networks. Section 4 presents the methodology proposed in the master stage based on the GNDO metaheuristic algorithm with an improvement stage based on hybridizing the discrete version of the vortex search algorithm (VSA) that is integrated with the slave stage based on the three-phase matricial iterative sweep power flow method. Section 5 presents information on all the test systems used, as well as the different simulation cases. Section 6 presents the information related to the computational validation of the proposed method. Section 7 presents all the numerical results of the proposed GNDO method in the test systems presented. Section 8 presents the conclusions obtained from the work presented and proposes future studies.

2. Literature Review and Contributions

Some studies, reported in the scientific literature, consider the objective function economically in the problem of optimal selection of conductors in asymmetric three-phase distribution systems. For example, in 1982 the authors of [13] developed a procedure for grading the cross section of conductors in radial distribution feeders and minimizing costs. They represented the problem using multistage dynamic programming, formulating models to represent the costs associated with the feeder, energy losses and voltage regulation as a function of conductor cross section. For the planning of optimal expansion of medium voltage power grids in [5] they proposed the improved Tabu Search/Particle Swarm algorithm (TS/IPSO), being a hybrid algorithm that proved to have better performance compared to other algorithms (such as Simulated Annealing (SA), Tabu Search (TS), Improved Genetic Algorithm (IGA), (SA/TS) Y (TS/IGA)) in high dimensional grids. The authors of [14] proposed a heuristic method where they incorporate biologically inspired structures and operators, such as recombination, mutation and fitness-based selection, the Evolutionary Strategy (ES) proves to be successful in the 8-bus system, composed of a substation and 7 radial feeders.The authors of [15] implemented an algorithm with evolutionary approaches (GA) and iterative sweep as a method to calculate power flows, voltage magnitudes and power losses, the results demonstrated the effectiveness of the proposed approach on a 69-bus radial network. For optimal driver selection, the authors of [16] proposed to solve the problem using the Harmony Search with a Differential Operator algorithm (HSDE), whose objective function was modeled to minimize the costs associated to capital investment and energy losses. The model was implemented on 16-bus and 85-bus systems, the results show better performances compared to the metaheuristic algorithm of Population Search (HSA) and Evolutionary Programming (EP).The authors of [17] implemented the Particle Swarm optimization technique for optimal single-wire ground return (SWER) conductor selection in rural distribution networks, the method was implemented on the 13-bus IEEE system describing the real 13-bus network extracted from the Namibian SWER system, the results show the effectiveness of the algorithm in terms of cost effectiveness compared to the base case, reducing the active power losses by 90.12% and improving the network voltage profiles. In 2016, the authors of [18] used the discrete particle swarm optimization algorithm (DPSO), minimizing the energy benefit cost, reducing energy losses and minimizing depreciation. This algorithm was implemented on a 26-bus and 32-bus system, with five different conductors types, the results showed that the algorithm is feasible and efficient as it improves the performance compared to the GA. The authors of [19] proposed a modified differential evolution (MDE) algorithm for conductor selection and implemented a direct approximation load flow. The objective function takes into account capital investment, active power loss and that classical technical constraints are satisfied, they considered variables such as load growth. The method was implemented on a 32-bus system, the results of which reduce the active power losses by 60% with respect to the base case by improving the voltage profiles.The authors of [20] proposed the metaheuristic crow search algorithm (CSA), taking into account the objective function and classical technical constraints. The above is implemented in a 16-bus and 85-bus system, and the results obtained show significant reductions in energy losses compared to the original network of 33.32% and 19.563%, respectively. The authors of [21] implemented the sine-cosine optimization algorithm (SCA) in the networks of the Egyptian distribution system for the optimal selection of conductors, considering the annual growth rate of the load over a period of ten years and maintaining the voltage and ampacity constraints, using a catalog of twenty conductors. Their results are satisfactory in terms of the result and the computational processing time.The authors of [22] implemented the metaheuristic salp swarm optimization algorithm to solve the optimal conductor selection in a real radial distribution system in Egypt, the results showed the effectiveness of the algorithm in satisfying the objective function and constraints, however it highlights the importance of considering feeder reinforcement to improve system hosting capacity (HC) levels, giving the possibility to connect distributed generator (DG) units.The authors of [23] implemented the Bifurcation Minimization Technique (BWMT) for optimal conductors selection, it is highlighted that the method takes into account the environmental effects on the branches resistance in addition to the load sensitivity to weathering in a 16-bus system, thus allowing a more accurate calculation of power losses in the system, the results are shown to be satisfactory when compared with the original network.The authors of [24] implemented the Whale Optimization Algorithm (WOA) to solve the optimal radial distribution network conductor selection problem, where techno-economic aspects such as high annual load growth and payback period were considered. The objective function minimizes the cost associated with the investment and energy losses, taking into account the basic constraints, the model is implemented in a system of 16 and 85 buses obtaining favorable results compared to literature results, maximizing the overall savings and maintaining the constraints for a period of five years.The authors of [25] did simulations where customer losses were evaluated before and after implementing the model, whose objective is to achieve the best selection of conductors for the distribution system, implementing the optimization technique based on the Teaching-Learning algorithm in a 16-bus system, simulating heavy load during the Indian agricultural season; the results were satisfactory, reducing losses and improving voltage profiles.
Given the nature of the electrical power system, new methods for calculating the three-phase flow for both balanced and unbalanced loads are sought, continually improving the accuracy of proposed models, as in [26], where the authors propose the optimal selection of conductors in three-phase distribution networks through a discrete version of the metaheuristic vortex search method using two systems structured with 8- and 27-buses under three different maximum load scenarios for 8670 hours, three percentages of consumption and a real daily load curve.
As described in the previous review of state of the art, regardless of the technique or methodology of solution used, the main objective is to reduce, as much as possible, the costs associated with planning electrical distribution networks, keeping the technical parameters within the allowable values according to the current territorial regulation, seeking to improve the quality of service received by end users and to generate a profitable environment for the companies providing the service.
Considering the literature review presented in the previous section, this paper makes the following contribution: An improvement stage is added to the master stage of the Generalized Gaussian Normal Distribution (GNDO) based metaheuristic algorithm, which combines its evolution rules with the discrete version of the Vortex Search Algorithm (VSA) to obtain a new hybrid optimizer. The main advantage of this proposed method is that it guarantees to reach a optimal local solution with the minimum standard deviation, obtaining the convergence of the objective function in a single evaluation, which allows to have a higher performance in the computational processing times.
Table 1 summarizes the studies consulted in developing state of the art, presenting the numerical method applied, the objective function under analysis, the year of publication and the corresponding citation.
The main advantages of the proposed GNDO to deal with the problem in electrical distribution network are as follows:
  • The proposed solution methodology can be implemented for any three-phase balanced and unbalanced distribution system with radial topology regardless of the load connections or the number of buses.
  • It’s exploration and exploitation stages make the proposed GNDO a robust and efficient solution methodology, in which with the help vortex search algorithm could obtain an optimal solution in a single evaluation besides using shorter processing times.
  • Implementation and adaptation of complementary software such as Microsoft Power BI® allowed us to rigorously examine both technically and economically, the analysis of the algorithm proposed.
Table 1. Summary of the methodologies used in the literature for the problem of optimal conductor selection in distribution networks.
Table 1. Summary of the methodologies used in the literature for the problem of optimal conductor selection in distribution networks.
Sol. MethodologyObjetive FunctionOperation ScenarioRefs.Year
Evolutionary strategyMinimize network investment costsMaximum load on 7-bus system[14]2006
Genetic AlgorithmsMinimizing active power losses and maximizing conductor material savingsMaximum load on 13-bus system[27]2011
Harmony searchMinimize the sum of capital investment and energy lossesHeavy load, simulating the agricultural season in a 16 and 85-bus system.[16]2011
Particle swarm optimizationMinimize capital investment depreciation, energy loss costs and maximum current capacityMaximum load for 26- and 32-bus system[17,18]2012
2016
Modified Differential EvolutionMinimize fixed costs associated with network investment and variable costs associated with operation.Two scenarios projecting annual load growth[19]2014
Differential evolutionSavings in conductive material costs and energy losses.Maximum load, implemented in an unbalanced three-phase system of 19 and 34 buses.[28]2016
Crow search algorithmMinimize capital cost and energy lossesscenario with peak demand in 16- and 85-buses system[20]2017
Sine cosine algorithmMinimizing the annual cost of energy losses and capital investmentPeak demand, projecting 10 consecutive years of load growth in a 22-buses system.[21]2017
Salp swarm optimizerMinimize energy losses and investment cost of conductorsThree scenarios at 50, 100 and 150% of nominal load and different levels of distributed generation[22]2018
Improved tabu search algorithmMinimize investment cost and energy lossMaximum load on 23-, 48- and 71-buses system[5]2019
Heuristic search methodDriver depression cost and annual energy losses.Maximum load on 69- and 85-buses system[23,29]2016
2019
Whale optimization algorithmMinimize the cost of energy loss and investment of conductorsPeak power in 16- and 85-buses system[24]2019
Teaching—learningMinimize investment and the annual cost of lossesPeak demand scenario in 16-buses system[25]2020
Vortex search algorithmMinimization of conductor investment costs and network technical lossesBalanced and unbalanced load under three scenarios, at 100% of the maximum load, 50%, and 44% of the nominal load.[26]2021
Newton-based metaheuristic optimizerminimize annual energy losses and investment in conductorsBalanced and unbalanced load scenarios[30]2022
It’s important to mention that in this study’s scope only covers applying a new efficient optimization technique to select conductors in three-phase asymmetric networks with radial topologies, uncertainties regarding demand behavior are not considered. However, to present the effectiveness of the proposed GNDO, a simulation scenario with three different anual demand cases was exposed. In future works derived from this study, uncertainty in demand curves can be studied since, as evidenced in Table 1, the studied problem continues to be relevant and essential for academics and distribution companies, implying that more research is required. Additionally, we present the effectiveness of the proposed GNDO in the 8-bus grid with five different combinatorial optimization methods. The best three approaches are select for comparison in the 27-bus grid. However, for the 33-bus grid, based on the results of the 27-bus grid, we only present numerical results with the proposed GNDO because no literature reports exist for the IEEE 33-bus grid in its three-phase version.

3. Mathematical Formulation

The problem of the optimal selection of conductors in balanced (single-phase representation) and unbalanced (three-phase representation) local distribution systems (LDS) can be mathematically modeled through mixed integer nonlinear programming (MINLP) [1] where the binary (or integer) variables are related to the selection of the conductor gauge in each network section. In contrast, the continuous variables are associated with the voltages, currents and powers. The product between the trigonometric functions and the voltage causes the power balance equations to have a non-convex and non-linear behavior, making this the most significant problem in the model when it comes to determining the optimal set of conductors [10]. The mathematical model that describes the optimization problem studied in this research is presented below:

3.1. Objective Function

The objective function considers an operational evaluation horizon of one year, which seeks to minimize the costs associated with the investment in the conductor sizes and the annual energy losses. In (1)–(4), the general structure of the objective function is represented.
C l o s s = C p h Ω h p Ω p q Ω p i Ω b j Ω b V h , i p V h , j q Y i j p q λ i j c cos ϕ h , i p ϕ h , j q ϕ h , i j p q λ i j c Δ h ,
C i n v = c Ω C k m Ω L C k m c L k m λ k m c ,
C p e n = c Ω c k m Ω L P λ k m c ,
Z = min C l o s s + C i n v + C p e n ,
In (1) the variable C l o s s represents the operating costs associated with energy losses for the evaluation period, C p refers to the average cost of energy; Δ h is the magnitude in hours of the evaluated period, Y i j p q and Φ h , i j p q represent the magnitude and angle of the admittance formed between the buses i and j of the phases p and q respectively, these are nonlinear functions of the binary variable λ i j c , this variable defines the “ c ” gauges for the conductor that joins the buses i and j ; for a period of time h , V h , i p and V h , j p are the variables that define the voltage magnitudes, while Φ h , i p and Φ h , j p define the angles of the voltages at buses i and j of phase p respectively.
In (2), C i n v represents the cost of the investment, according to the selected gauge of the conductors belonging to the three-phase distribution system networks, where C k m c associates the cost of the “ c ” type conductor along a kilometer in length, the variable L k m refers to the network section in kilometers formed by the buses k and m ; λ k m c is the binary decision variable that defines the installation of the type “ c ” conductor in the network fragment formed by the buses k and m .
In (3), C p e n represents the penalty cost, where P is a fixed-integer monetary value established by the violation of the I m a x c according to the conductor used for each network section, as described in (8). This penalty cost P was arbitrarily determined to be one million dollars.
In (4), Z is the value of the objective function, which corresponds to the sum of energy loss costs in the period of one year ( C l o s s ), costs associated with the investment of conductors ( C i n v ) and penalties ( C p e n ) derived from violating the constraints of the distribution system, such as the thermal limit in conductors.

3.2. Set of Constraints

The optimal selection of conductors in three-phase distribution networks is subject to a set of constraints associated with the balance of active and reactive power, operating limits of the conductors, voltage regulation at buses and the binary decision variables. From (5)–(12), the set of constraints mentioned above is presented.
P g i , h p P d i , h p = p Ω h q Ω p i Ω b j Ω b V h , i p V h , j q Y i j p q λ i j c c o s ϕ h , i p ϕ h , j q ϕ h , i j p q λ i j c , i Ω b h Ω h p Ω p ,
Q g i , h p Q d i , h p = p Ω h q Ω p i Ω b j Ω b V h , i p V h , j q Y i j p q λ i j c s e n ϕ h , i p ϕ h , j q ϕ h , i j p q λ i j c , i Ω b h Ω h p Ω p ,
I k m , h p = f V h , k p , V h , m p , ϕ h , k p , ϕ h , m p , λ k m c , R k m c , X k m c , k m Ω L h Ω h p Ω p ,
I k m , h p C Ω λ k m c I c m a x , k m Ω L h Ω h p Ω p ,
V i m i n V i , h p V i m a x , i Ω b h Ω h p Ω p ,
c Ω c λ k m c = 1 k m Ω L ,
k m Ω L c Ω c λ k m c = n 1 ,
λ k m c 0 , 1 k m Ω L , c Ω c ,
From the previous equations, Ω L contains the set of network sections of the distribution system, Ω b establishes all the buses belonging to the system, Ω h defines the periods of load duration and the set Ω p contains the phases of the system.
In (5) and (6), the active and reactive power balance is defined for each bus, phase and time period, where P g i , h p and P d i , h p represent active power generation and demand; similarly, Q g i , h p and Q d i , h p represent reactive power generation and demand, respectively. In (7), the current is calculated that flows in the network section formed by the buses k and m , of phase p , in a period of time h , where I k m , h p is a function of the magnitude of the voltages, angles, gauge of the conductor and its resistance ( R k m c ) and reactance ( X k m c ) parameters. The inequality constraint presented in (8) guarantees that the current flow in the network section formed by the buses k and m of the phase p , in a period of time h , does not exceed the thermal limit of the gauge selected for the network section, i.e., I c m a x . Constraint (9) ensures that for each bus i of phase p in a period of time h , the system voltage is within the regulatory range defined by the limit greater than or equal to the minimum voltage V i m i n and the limit less than or equal to the maximum voltage V i m a x . Equation (10) guarantees that for the network section formed by the buses k and m , only one “ c ” type conductor is selected; Additionally, in (11), it is guaranteed that the total number of conductors selected is equal to the number of buses in the system n minus 1. Finally, in (12), the discrete binary nature of the decision variable is defined with respect to the “ c ” type gauge for the section of network formed between the buses k and m .
The MINLP mathematical model defined in (1)–(12) represents the general formulation of the problem of optimal selection of conductors in three-phase systems with both balanced and unbalanced structures [30]. According to the analyzed case study, the dimension of the solution space can be as large as desired, due to its dependence on the number of network sections “ l ” and the number of candidate gauges “ c ”, with c l as solution space. This represents one of the biggest complications for this type of problem [31].
As an example of the above, considering a radial configuration of the distribution system, which is made up of 20 network sections ( l = 20), and additionally, there are 5 candidate gauges ( c = 5), the solution space will be 5 20 , which is a size equivalent to 95, 367, 431, 640, 625, with approximately 95 billion possible solutions, so evaluating each of these possible solutions becomes a cumbersome and inefficient task. To face this difficulty and to obtain an optimal solution in reasonable computation times, a master-slave optimization approach is proposed to solve the MINLP mathematical model based on the generalized normal distribution optimization algorithm GNDO and the three-phase matricial iterative sweep algorithm as a power flow method.

4. Solution Methodology

This document proposes the implementation of the metaheuristic algorithm GNDO [32] in the master stage and three-phase matricial formulation of the iterative sweep method for calculating power flow at the slave stage. The master stage will be responsible for the optimal selection of gauges for all branches using integer coding [26] and in the slave stage, each set of conductors will be evaluated to determine the value of the objective function (total costs of energy losses) for a given planning period. Similar to most metaheuristic optimization algorithms, the GNDO works with an initial population x i t that has the following structure (13):
x i t = [ 3 , 5 , , g , , 4 ] ,
where t is the iteration counter of the GNDO and “ g ” refers to a gauge available in the set of conductors. It is important to note that the dimension of the matrix associated with the initial population x i t is of the form N i × d , where N i is the number of individuals in the initial population and d is the dimension of the problem corresponding to the number of lines in the distribution system. In the following subsections, the most important aspects of the master and slave stages are presented.

4.1. Slave Stage: Three-Phase Power Flow

Distribution networks have a radial or tree-like structure. It is common in the analysis of the calculation of the power flow in this type of electrical network to find methods based on graph theory, where with the help of the incidence matrix, it is possible to represent the topological structure of the distribution network relating the buses with the branches of the system; in this way, if there is a branch between two buses of a graph, both buses will be connected [32]. The matricial formulation of the iterative sweep method is an approach for calculating power flow based on graph theory, in which Kirchhoff’s laws are used to calculate the currents of the system buses, starting from the terminal buses toward the slack bus (backward sweep), as well as the voltage drops in the different segments of the network, starting from the slack bus toward the terminal buses (forward sweep) [33]. This solution method is commonly known as the backward/forward sweep method.
To illustrate the structure of the incidence matrix, consider the distribution system shown in Figure 1. This radial network is made up of 4 branches and 5 buses.
To represent the topology of the electrical network, the relationship between the buses and the phases of the branches of the system is established by the three-phase incidence matrix Λ , where the positive direction of the branch current is defined so that it always flows out of the bus, as shown in (14).
Λ i x , j z = + 1 i f l i n e i i s c o n n e c t e d t o b u s j a n d t h e c u r r e n t o f p h a s e x l e a v e s t h e b u s . 1 i f l i n e i i s c o n n e c t e d t o b u s j a n d t h e c u r r e n t o f p h a s e x e n t e r s t h e b u s . 0 i f p h a s e x o f b r a n c h i i s n o t c o n n e c t e d t o p h a s e z o f b u s j
where i corresponds to the branches of the system, x to the phases of the branches, j to the buses of the system and z to the phases of the buses; in this way, the bus connections and the three-phase incidence matrix Λ are shown in Table 2 and in (15), respectively.
Λ i x , j z = 1 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 1 | 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1
This three-phase incidence matrix is divided into two submatrices, which are related with respect to generation (this is possible because the slack bus is located on bus 1) and demand, as shown in (16).
Λ i x , j z Λ 3 ϕ = Λ s 3 ϕ Λ d 3 ϕ ,
where Λ s 3 Φ corresponds to the first three columns of Λ 3 Φ (slack bus) and Λ d 3 Φ corresponds to the rest of the columns of Λ 3 Φ (demand buses).
Now, we proceed to define the voltage drops per phase for each of the branches of the system. These voltage drops are defined as the difference between the node voltages per phase that connect each of the branches; in this way, we get:
E r 3 ϕ = Λ s 3 ϕ V s 3 ϕ + Λ d 3 ϕ V d 3 ϕ ,
where E r 3 Φ is a matrix that includes all voltage drops, in which V s 3 Φ = V { 1 3 } 3 Φ and V d 3 Φ is a matrix that includes all the voltages in the demand buses. On the other hand, it is possible to define the bus currents ( I n 3 Φ ) as
I n 3 ϕ = I s 3 ϕ I d 3 ϕ T ,
where I s 3 Φ and I d 3 Φ are the generation and demand currents per phase, respectively. In the same way, the generation and demand currents can be deduced as shown below:
I s 3 ϕ = Λ s 3 ϕ T J r 3 ϕ ,
I d 3 ϕ = Λ d 3 ϕ T J r 3 ϕ ,
In contrast, the voltage drops per phase in each of the branches can be defined through the three-phase impedance matrix for distribution lines, which contains information of each three-phase line impedance and the currents flowing around each one of the branches, as expressed in matrix form in (21).
E a A E a B E a C E b A E b B E b C E c A E c B E c C E d A E d B E d C = Z a A A Z a A B Z a A C 0 0 0 0 0 0 0 0 0 Z a B A Z a B B Z a B C 0 0 0 0 0 0 0 0 0 Z a C A Z a C B Z a C C 0 0 0 0 0 0 0 0 0 0 0 0 Z b A A Z b A B Z b A C 0 0 0 0 0 0 0 0 0 Z b B A Z b B B Z b B C 0 0 0 0 0 0 0 0 0 Z b C A Z b C B Z b C C 0 0 0 0 0 0 0 0 0 0 0 0 Z c A A Z c A B Z c A C 0 0 0 0 0 0 0 0 0 Z c B A Z c B B Z c B C 0 0 0 0 0 0 0 0 0 Z c C A Z c C B Z c C C 0 0 0 0 0 0 0 0 0 0 0 0 Z d A A Z d A B Z d A C 0 0 0 0 0 0 0 0 0 Z d B A Z d B B Z d B C 0 0 0 0 0 0 0 0 0 Z d C A Z d C B Z d C C J a A J a B J a C J b A J b B J b C J c A J c B J c C J d A J d B J d C
From (21) the following can be obtained:
E r 3 ϕ = Z r 3 ϕ J r 3 ϕ ,
Thus, from (22) J r 3 Φ can be found and replaced in (20) as follows:
I d 3 ϕ = Λ d 3 ϕ T Z r 3 ϕ 1 E r 3 ϕ ,
Replacing (16) in (23) obtains:
I d 3 ϕ = Λ d 3 ϕ T Z r 3 ϕ 1 ( Λ s 3 ϕ V s 3 ϕ + Λ d 3 ϕ V d 3 ϕ ) ,
I d 3 ϕ = Λ d 3 ϕ T Z r 3 ϕ 1 Λ s 3 ϕ V s 3 ϕ + Λ d 3 ϕ T Z r 3 ϕ 1 Λ d 3 ϕ V d 3 ϕ ,
Taking into account that:
Y d g 3 ϕ = Λ d 3 ϕ T Z r 3 ϕ 1 Λ s 3 ϕ ,
Y d d 3 ϕ = Λ d 3 ϕ T Z r 3 ϕ 1 Λ d 3 ϕ ,
Equations (26) and (27) can be replaced in (25)
I d 3 ϕ = Y d g 3 ϕ V s 3 ϕ + Y d d 3 ϕ V d 3 ϕ ,
The main objective is to find a function of the form V d 3 Φ = f ( I d 3 Φ ) in which the three-phase demand voltages are related to the three-phase demand currents, clearing V d 3 Φ from (28), the expression (29) is obtained
V d 3 ϕ = Y d d 3 ϕ 1 ( I d 3 ϕ + Y d g 3 ϕ V s 3 ϕ )
However, in (29) I d 3 Φ is still an unknown variable. This three-phase demand current will depend on the type of load that will be connected, i.e., Star connection ( Y ) or in Delta ( Δ ). Therefore, in the case in which the loads that are connected to the electrical system have a star connection, we get (30)–(32).
I k A = S k A V k A * ,
I k B = S k B V k B * ,
I k c = S k C V k C * ,
On the other hand, in the case that the connection of the loads is given by Delta connections, we obtain (33)–(35).
I k A = S k A V k A V k B * S k C V k C V k A * ,
I k B = S k B V k B V k C * S k A V k A V k B * ,
I k C = S k C V k C V k A * S k B V k B V k C * ,
for k Ω b where k refers to the bus number. Thus, rewriting Equation (29) in terms of the following iteration, we get (36).
V d 3 ϕ t + 1 = Y d d 3 ϕ 1 ( I d 3 ϕ + Y d g 3 ϕ V s 3 ϕ ) ,
where t is the iteration counter. It is common for the power flow analysis to handle a representation per unit of all the electrical variables, which implies that the voltage of the substation can be defined with a positive sequence, as shown below: V s 3 Φ = 1 0 , 1 120 , 1 120 T . In the same way, for expression (36) to be valid, it must meet the expression defined in (37).
V d 3 ϕ t + 1 V d 3 ϕ t ϵ ,
where ϵ is the maximum acceptable error value, with a recommended value of 1 × 10 10 . To determine the three-phase power losses, expression (38) is used, which describes the Joule effect.
P l o s s 3 ϕ = R e ( s u m ( E r 3 ϕ t + 1 ) T ( J 3 ϕ t + 1 ) * ) ,
Finally, to represent the matricial reformulation of the three-phase iterative sweep, the flow diagram is presented in Figure 2.
Observation 1. Once the power losses are found for the proposed time period h, with the help of Equation (38), the value of the objective function associated with the costs of energy losses expressed in (1) is obtained, which can be found according to the following expression:
C l o s s = C p P l o s s 3 ϕ Δ h ,

4.2. Master Stage: GNDO Metaheuristic Algorithm

The GNDO is a metaheuristic method also known as generalized normal distribution optimization, which is based on the Gaussian distribution model, where each individual makes use of this curve to improve their position. The most important characteristic of the GNDO is that it does not need any type of special control parameter, but on the contrary, it only requires the initial population and assigning the terminal condition before running the algorithm. The general model of Gaussian distribution is represented in (40), where x is a random variable that corresponds to the distribution probability, which has a location parameter μ and scale parameter δ [32].
f ( x ) = 1 2 π δ e x p x μ 2 2 δ 2 ,
According to (39), the most important variables in the normal distribution are the location parameter ( μ ) and the scale parameter ( δ ), both used to express the mean value and standard variance of random variables, respectively [32].

4.2.1. Local Exploitation

In this stage of exploitation, a broad search is carried out to find the best solution found thus far to verify whether there are better solutions in an attempt to accelerate the speed of convergence [34]. The process of local exploitation involves finding better optimal solutions around the current positions of all the individuals in the solution [32]. Therefore, taking as a reference the relationship between all the individuals of the solution in the current population and the normal distribution, in addition to seeking to accelerate the speed of convergence, we seek to maintain the quality of the solutions found [34], thus obtaining the following model:
v i t = μ i + δ i η , i = 1 , 2 , . . . , N i . .
In (41), i represents the ith individual solution, n corresponds to a penalty factor, δ i represents the generalized standard variation, μ i represents the generalized mean position, and v i t corresponds to a vector of the current solution, it is important to note that N i corresponds to the total number of individuals in the population. The terms μ i , δ i and η can be obtained as follows:
μ i = 1 3 x i t + x b e s t t + M ,
δ i = 1 3 ( ( x i t μ i ) 2 + ( x b e s t t μ i ) 2 + ( M μ i ) 2 ) 1 2 ,
η = log λ 1 1 2 cos 2 π λ 2 , i f a b log λ 1 1 2 cos 2 π λ 2 + π , i f a > b ,
where λ 1 , λ 2 , a and b correspond to random numbers between 0 and 1, with a uniform distribution; x b e s t t corresponds to a vector containing the best solution found so far; and M represents a vector containing the average position of the current individuals of the population t , as defined in (45).
M = 1 N i i = 1 N i x i t ,

4.2.2. Global Exploration

The purpose of this exploration stage is to determine and locate promising new regions of the large solution space [32]. This is because μ i may end up being a local minimum, as the case may be, so performing the search around this parameter may not be sufficient to improve the quality of the solutions. In this way, exploring the search space as much as possible will avoid being trapped in local minima [34]. The global exploration in the GNDO can be defined as follows:
v i t = x i t + β λ 3 . v 1 + 1 β λ 4 . v 2 ,
where λ 3 and λ 4 are two random numbers subject to the standard normal distribution, β represents the adjustment parameter, which is randomly selected between 0 and 1, and both v 1 and v 2 are two vectors defined as observed in (47) and (48).
v 1 = x i t x j t , i f A f ( x i t ) < A f ( x j t ) x j t x i t , O t h e r w i s e ,
v 2 = x k t x m t , i f A f ( x k t ) < A f ( x m t ) x m t x k t , O t h e r w i s e ,
In (47) and (48), j , k , and m are three random integers, whose values range between 1 and the total number of individuals in the population. It is important to verify that these values are different from each other, as well as from the current individual i for iteration t . On the other hand, A f ( x i , j , k , m t ) represents the value of the objective function for the vectors i , j , k s and m of the current population t . To ensure the viability of each generation in the solution space, (49) is used.
v i , l t = v i , l t , i f x l m i n v i , l t x l m a x x b e s t , l t , O t h e r w i s e ,
where x l m i n and x l m a x are the integer values corresponding to the minimum and maximum values of the set of conductors available for installation in the distribution grid under study, which are evaluated from the first position to position d of the vector v i , l t with the help of the decision variable l . It is important to clarify that d represents the dimension of the problem studied. v i , l t represents the gauge value of the ith individual solution for a given position in the vector according to the decision variable l . On the other hand, x b e s t , l t refers to the value of the decision variable l for the best current solution found so far.
To maintain the best solution within the next generation of the population, the following expression is used:
x i t + 1 = v i t , i f A f ( v i t ) A f ( x i t ) x i t , O t h e r w i s e ,

4.3. Improvement of the Exploration and Exploitation of the Solution Space

In order to obtain better results in the exploration and exploitation of the solution space, a modified version of the GNDO base algorithm with the vortex search algorithm is proposed [26]. The vortex search algorithm (VSA) is inspired by the vortex pattern created by the vortex flow of agitated fluids, making it a fairly simple algorithm in addition to being computationally efficient, so it is a good candidate for the solution of real-life optimization problems [35]. The VSA works with continuous variables [33]. Therefore, a discrete version will be used to determine the correct assignment of gauges to be installed in the branches of the different distribution systems. Thus, the initial center of the hyper-ellipse is defined in (51).
μ 0 = x m i n + x m a x 2 ,
where x m i n N d x 1 and x m a x N d x 1 are vectors that contain integers and have a size determined by the dimension of the solution space d ( d = n 1), where n is the number of buses. It is important to clarify that μ 0 must be rounded up to the nearest integer value. On the other hand, the generation of candidate solutions is obtained with the help of the Gaussian normal distribution as seen in (52).
v i t = p ς i t , μ t , Σ = 1 2 π d Σ e x p 1 2 ς i t μ t τ Σ 1 ς i t μ t ,
where ζ i t R d x 1 is a vector with dimension d full of random values, μ t R d x 1 which represents the center of the hyper-ellipse for each iteration t , and Σ ∈ R d x 1 is the covariance matrix. This covariance matrix is worked as shown in [26], where it is described as a matrix that has identical values in its diagonal and null values outside it, as (53) shows.
Σ = σ 0 I ,
where σ 0 is the variance of the Gaussian distribution and I is the identity matrix with the appropriate dimensions. In the same way, the value σ 0 is found as follows:
σ 0 = max x max min x min 2 ,
Here, σ 0 can be considered the initial radius (i.e., r 0 ). Because an adequate exploration of the solution space is required in the initial phases. r 0 is chosen to be a large value [36] (note that max x max obtains the maximum value of the vector x max and max x min obtains the minimum value x max , respectively). The use the incomplete gamma function (see [33]) allows finding the optimal solution during the search process the optimal solution can be found. This, because the radius progressively decreases until it reaches zero, which can be seen in (55).
Y x , a = 0 x e t t a 1 d t ,
In (55), a is a shape parameter greater than zero, and x is a random parameter greater than or equal to zero. Thus, in order to determine the adaptive size of the radius for each iteration, the incomplete gamma function is used, as presented in [33] and (56)
r t = σ 0 Y 1 x , a t ,
where a t is a parameter obtained as shown in (57)
a t = t m a x t t m a x ,
where t m a x corresponds to the total number of iterations ( t m a x = 1000 ); in the same way as in (55), the parameter x is selected as 0.1, as reported in [33]. To advance the process of the exploration and exploitation search, it is necessary to update the new center of the hyper-ellipse ( μ t + 1 ), which will be selected as the best individual solution contained in v i t , where the minimum value of the objective function will be presented, which in turn is known as x b e s t t .
It is important to mention that there is the possibility of having a candidate solution outside the limits established by the solution space, so it is necessary and essential to review the values of the minimum and maximum limits, as shown in (49).
Finally, to represent the general improved implementation of the GNDO based on ellipses of variable radius, the flow diagrama is presented in Figure 3.

Improved Implementation of the GNDO Based on Ellipses of Variable Radius

The improved implementation of the GNDO based on ellipses of variable radius is presented in Algorithm 1.
Algorithm 1. Improved implementation of the GNDO based on the VSA.
Data : Select the AC electrical network under study;
Find the equivalent electrical network in values per unit;
Assign the population size ( N i ), the maximum iterations ( t m a x ) and t = 0 ;
Find the initial center ( μ 0 ) and the standard desviation ( σ 0 ) as shown in (51) and (54), respectively;
Generate the initial population according to (13);
Establish the best adaptive initial function as A f ( x b e s t t ) = ;
for t t m a x do
    Obtain the value of the objective function for each of the individuals belonging to the initial
    population x i t with the help of the slave stage;
    Select the individual x b e s t t as the one that obtains the minimum value in the objective function
    according to the results found in A f ( x i t ) ;
     for   i = 1 : N i do
       Generate two random numbers ϑ and ϱ between 0 and 1;
        Enter the value of the vector M as described in (45);
        if   ϑ < 1 2 then
           Generate three random numbers j , k and m with each of them different from each other
            and different from i ;
            Find the value of v 1 and v 2 as described in (47) and (48), respectively;
            if   ρ 1 2 then
              Calculate the value of μ i as described in (42);
               Obtain the value of δ i as described in (43);
               Find the penalty factor η according to (44);
               Obtain v i t as described in (41);
            Else
              Perform the global exploration process as explained in (46);
            End
            Verify and adjust the values of v i t according to the limits established in (49);
            Evaluate v i t in the slave stage in order to obtain the value of the objective function
            A f ( v i t ) ;
            Select next generation x i t + 1 as explained in (50);
        Else
            Obtain the radius r t as expressed in (56);
            Generate the candidate solutions using the Gaussian distribution with center at μ t
           based on (52);
            Verify and adjust the values of v i t according to the limits established in (49);
            Evaluate v i t in the slave stage in order to obtain the value of the objective function
            A f ( v i t ) ;
            Select next generation x i t + 1 as explained in (50);
        End
     End
End

5. Characteristics of the Test Systems

In this section, two different radial distribution systems composed of 8 and 27 buses are presented. The test systems will be analyzed under two study scenarios. The first case corresponds to the analysis of the test systems operating under balanced loads, while the second case corresponds to the analysis of the test systems operating under unbalanced loads [26]. For both radial distribution systems, Star connections are used for all loads. The general operating parameters for both test systems are listed in Table 3.

5.1. First Test System

This first test system is a three-phase electrical distribution network consisting of 7 lines and 8 buses operating at an average nominal voltage of 13.8 kV between phase and neutral with a unity power factor. Due to its radial topology, the slack bus corresponds to bus 1 [26]. The electrical network is presented in Figure 4.
Table 4 shows the corresponding information for the case of operating under a balanced load scenario for the 8-bus system. It also presents the length for each network section, the load levels per phase present for each load bus (PQ) bus and the topology of the electrical network.
Table 5 shows the corresponding information for the case of operating under an unbalanced load scenario for the 8-bus system, in which the different load levels per phase present for each PQ bus are shown.

5.2. Second Test System

This second test system is a three-phase electrical distribution network made up of 26 lines and 27 buses operating at an average nominal voltage of 13.8 kV between phase and neutral with a unitary power factor. Due to its radial topology, the slack bus corresponds to bus 1 [26]. The electrical network is presented in Figure 5.
Table 6 shows the corresponding information for the case of operating under a balanced load scenario for the 27-bus system, in which the length for each network section and the load levels per phase present for each PQ bus and the topology of the electrical network are shown.
Table 7 shows the corresponding information for the case of operating under an unbalanced load scenario for the 27-bus system, in which the different load levels per phase present for each PQ bus are shown.

5.3. Set of Available Conductors

Eight different types of conductors are considered to be assigned within the different test systems exposed. Table 8 shows the different values of impedances, maximum thermal currents and cost per kilometer for each of them.

6. Computational Validation

The proposed optimization methodology is implemented on a personal computer using MATLAB® R2021a software. The computer has an Intel (R) Core (TM) i7-8565U CPU @1.80 GHz 1.99 GHz, 8 Gb RAM (Intel, Santa Clara, CA, USA), and Windows 11 Home operating system of ×64 bits. The results are analyzed with the help of Microsoft Power BI Desktop® Version 2.109.642.0 ×64 bits (Redmond, WA, USA).

7. Results and Discussion

This section presents the numerical results obtained for both test systems considering an operating scenario in peak demand, i.e., T = 8760 h and with demand of 100% at every instant of time.

7.1. Results in a Balanced Operating Scenario

The results obtained for IEEE 8 and 27-bus test systems of are presented considering the optimization method presented with the master-slave structure for balanced operating scenarios.

7.1.1. Results in the IEEE 8-Bus Test System

Table 9 presented in [30] shows the solutions obtained by different comparative methods, such as the tabu search algorithm (TBA), traditional genetic algorithm (TGA), Chu and the Beasley genetic algorithm (CBGA), the exact MINLP solver model in General Algebraic Modeling System (GAMS), Newton’s metaheuristic algorithm (NMA) and the vortex search algorithm (VSA). In the same way, the solution reached by the proposed GNDO is presented.
The results presented in Table 9 show that (i) the proposed methodology based on the GNDO improves the numerical results obtained by the NMA and discrete version of the VSA (DVSA) algorithms, with a solution equal to US $ 455,969,791, of which 50.03% corresponds to the costs associated with energy losses stipulated for the established planning horizon and 49.97% corresponds to the costs associated with the investment in conductors. (ii) The second-best solution methodology is the NMA and DVSA algorithms, both with an objective function value equivalent to US$ 508,357,959, followed in third place by the CBGA solution methodology with an objective function value of US$ 516,231,965. (iii) It is noted how the solution obtained by the GNDO presents the highest level of investment in conductors compared to the other methods. However, this increase in the investment cost is compensated by a lower level of cost in energy losses, so in that sense the best combination of gauges is achieved for this test system. (iv) The proposed methodology based on the GNDO reduces annual costs by 10.30% compared to the second-best solution presented in Table 9.
In contrast, Figure 6 shows the behavior of the voltage profiles at the eight buses of the system, where it can be observed that the minimum voltage value is present in bus six with a value of 0.9904 pu, yet it should be clarified that because the system is in a balanced load scenario, the behavior of the different voltage profiles for the different phases A, B and C will be the same.
On the other hand, Figure 7 shows the total apparent power losses (S) for the phases A, B and C in the different branches belonging to the eight-bus system, it can be observed that the greatest value of losses is present in branch four with a value of 58.53 kVA. It is worth clarifying that because the system is in a balanced load scenario, the corresponding contribution of each of the phases A, B and C to the total power losses in each branch will be the same.
Finally, Figure 8 represents the loadability of each of the branches in the 8-bus system, which shows the comparison between the maximum current value obtained from the phases A, B and C for a given branch X against the maximum-established current limits according to gauge-type obtained for said network section shown in Table 8. In this way, it is important to highlight that the maximum flowing currents for each of the branches are quite far from the limits of the established maximum current.

7.1.2. Results in the IEEE 27-Bus Test Scenario

In Table 10 presented in [30] the results obtained by different comparative methods are presented, such as the vortex search algorithm (VSA) and Newton’s metaheuristic algorithm (NMA). In the same way, the solution reached by the proposed GNDO is presented.
The results presented in Table 10 show that (i) the proposed methodology based on the GNDO improves the numerical results obtained by the NMA and DVSA algorithms, with a solution equal to US $ 549,883.572, of which 41.85% corresponds to the costs associated with the energy losses stipulated for the established planning horizon and 58.15% corresponds to the costs associated with the investment in conductors; (ii) It is noted how the solution obtained by the GNDO presents the lowest level of investment in conductors compared to the other methods. However, this lower investment cost promotes an increase in the cost of energy losses, so, although there is a slight increase in the latter, it is not high enough, so the lowest value of the objective function is obtained when compared with the objective functions of the different methods shown in Table 10. In this sense, the best combination of gauges is achieved for this test system. (iii) The GNDO methodology reduces annual costs by 1.40% with respect to the second-best solution presented in Table 10.
In contrast, Figure 9 shows the behavior of the voltage profiles in the 27 buses of the system, where it can be observed that the minimum voltage value is present in bus 10 with a value of 0.9771 pu, and should be clarified that because the system is in a balanced load scenario, the behavior of the different voltage profiles for the different phases A, B and C will be the same.
On the other hand, Figure 10 shows the total apparent power losses (S) for the phases A, B and C in the different branches belonging to the 27-bus system, it can be observed that the highest losses occur in branch two with a value of 42.42 kVA. Because the system is in a balanced load scenario, the corresponding contribution of each of the phases A, B and C to the total power losses in each branch will be the same.
Finally, Figure 11 represents the loadability of each of the branches in the 27-bus system, which shows the comparison between the maximum current value obtained from the phases A, B and C for a given branch X against the maximum-established current limits according to gauge-type obtained for said network section shown in Table 8. In this way, it is important to highlight that the maximum flowing currents for each of the branches are quite far from the limits of the established maximum current.

7.2. Results in an Unbalanced Operating Scenario

The results obtained for IEEE 8- and 27-bus test systems are presented considering the optimization method used with master-slave structure for the unbalanced operating scenarios.

7.2.1. Results in the IEEE 8-Bus Test System

Table 11 presented in [30] shows the results obtained by different comparative methods, such as the vortex search algorithm (VSA) and Newton’s metaheuristic algorithm (NMA). In the same way, the solution reached by the proposed GNDO is presented.
The results presented in Table 11 show that (i) the proposed methodology based on the GNDO does not improve but matches the numerical results obtained by the NMA and VSA algorithms because a solution equal to US $ 558,758.394 is obtained, of which 48.15% corresponds to the costs associated with the energy losses stipulated for the established planning horizon and 51.85% corresponds to the costs associated with the investment in conductors.
In contrast, Figure 12 shows the behavior of the voltage profiles in the eight buses of the system, where it can be observed that the minimum voltage value is present in bus six for the phase B with a value of 0.9869 pu, it should be noted that because the system is in an unbalanced load scenario, the behavior of the different voltage profiles for the different phases A, B and C will no longer be the same.
On the other hand, Figure 13 shows the total apparent power losses (S) for the phases A, B and C in the different branches belonging to the eight-bus system. It can be observed that the highest losses value is present in branch three in phase C with a value of 50.89 kVA. It should be clarified that because the system is in an unbalanced load scenario, the corresponding contribution of each of the phases A, B and C to the total power losses in each branch will no longer be the same.
Finally, Figure 14 represents the loadability of each of the branches in the 8-bus system, which shows the comparison between the maximum current value obtained of phases A, B and C for a given branch X against the maximum-established current limits according to gauge-type obtained for said network section shown in Table 8. In this way, it is important to highlight that the maximum flowing currents for each of the branches are quite far from the limits of the established maximum current.

7.2.2. Results in the IEEE 27-Bus Test Scenario

Table 12 presented in [30] shows results obtained by different comparative methods, such as the vortex search algorithm (VSA) and Newton’s metaheuristic algorithm (NMA). In the same way, the solution reached by the proposed GNDO is presented.
The results presented in Table 12 show that (i) the proposed methodology based on the GNDO improves the numerical results obtained by the NMA and DVSA algorithms, with a solution equal to US $ 589,018,800, of which 43.66% corresponds to the costs associated with the energy losses stipulated for the established planning horizon and 56.34% corresponds to the costs associated with the investment in conductors; (ii) It is indicated how the solution obtained by the GNDO presents the lowest investment in conductors compared to the other methods. However, this lower investment cost promotes an increase in the cost of energy losses, so, although there is a slight increase in the latter, it is not high enough, so the lowest value of the objective function is obtained when compared with the objective functions of the different methods presented in Table 12. In this sense, the best combination of gauges is achieved for this test system. (iii) The proposed methodology based on the GNDO reduces annual costs by 1.43% compared to the second-best solution presented in Table 12.
In contrast, Figure 15 shows the behavior of the voltage profiles in the 27 buses of the system, where it can be observed that the minimum voltage value is present in bus 10 for phase C with a value of 0.9596 pu. This is due to the active and reactive power demand value present in bus 10 for phase C, with values of 798.3 kW and 497.7 kvar, respectively, which constitute the highest demand values for this phase in all the buses of the distribution system in the unbalanced scenario. In this sense, by comparing phases A, B and C, it can be seen that, since the first two lack demand, their behavior with respect to phase C remains more stable. In the same way, when performing a topological analysis of the network, it is possible to observe that bus 10 is one of the furthest nodes from the slack node, which entails more significant voltage variations. It should be clarified that because the system is in an unbalanced load scenario, the behavior of the different voltage profiles for the different phases A, B and C is not the same.
On the other hand, Figure 16 shows the total apparent power losses (S) for the phases A, B and C in the different branches belonging to the 27-bus system; it can be observed that the greatest losses are present in phase B for branch two with a value of 17.06 kVA and in phase C for branch seven with a value of 17.15 kVA. It is valid to clarify that because the system is in an unbalanced load scenario, the contribution corresponding to phases A, B, and C to the total power losses in each branch will no longer be the same.
Finally, Figure 17 represents the loadability of each of the branches in the 27-bus system, which shows the comparison between the maximum current value obtained of phases A, B and C for a given branch X against the maximum-established current limits according to gauge-type obtained for said network section shown in Table 8. In this way, it is important to highlight that the maximum flowing currents for each of the branches are pretty far from the limits of the established maximum current.

7.3. Complementary Analysis

To analyze the behavior of the proposed algorithm, Figure 18 and Figure 19 are shown. These results show how the proposed GNDO algorithm behaved in the search to minimize the value of the objective function throughout all iterations ( t m a x ). It is observed how the IEEE 8-bus system for both balanced and unbalanced loads manages to converge and stabilize in a few iterations; on the other hand, the IEEE 27-bus system for both balanced and unbalanced loads takes many more iterations ( t ) to converge.
In the same way, according to the results obtained from the IEEE 8- and 27-bus systems for the test scenarios under balanced and unbalanced loads, the following is obtained:
  • The proposed GNDO only needs a single evaluation to obtain the numerical results presented in this research, which guarantees very reduced computation times, making this algorithm the most efficient one reported in the specialized literature.
  • The proposed GNDO improves the results in most of the case studies presented, with the exception of the 8-bus test system under an unbalanced operating scenario, in which it matches the results obtained by algorithms such as the NMA or the VSA. In this way, it can be inferred that for this particular case study, the overall optimal solution has been obtained.
  • The levels of loadability shown for each of the lines in the different cases of study guarantee good functioning in the electrical distribution systems tested, additionally ensuring the possibility of future expansions and new load connections.
    Figure 18. Behavior of the value of the objective function for: ( a ) 8-bus system with balanced load; ( b ) 27-bus system with balanced load.
    Figure 18. Behavior of the value of the objective function for: ( a ) 8-bus system with balanced load; ( b ) 27-bus system with balanced load.
    Energies 16 01311 g018
    Figure 19. Behavior of the value of the objective function for: ( a ) 8-bus system with unbalanced load; ( b ) 27-bus system with unbalanced load.
    Figure 19. Behavior of the value of the objective function for: ( a ) 8-bus system with unbalanced load; ( b ) 27-bus system with unbalanced load.
    Energies 16 01311 g019

7.4. Selection of Conductors Considering Different Load Periods

To validate the effectiveness of the proposed GNDO in dealing with medium-scale distribution networks considering demand variations, this subsection presents some numerical validations for the three-phase asymmetric version of the IEEE 33-bus grid. It´s single line diagram is presented in Figure 20. The nodal connections and the line lengths are listed in Table 13.
The information regarding power consumption per node in the three-phase version of the 33-bus grid is listed in Table 14.
To evaluate the effectiveness of the proposed GNDO in solving the problem of the optimal selection of conductors in three-phase unbalanced distribution networks against different demand variations, three numerical validations are carried out, as defined below.
  • Case 1: The evaluation of the optimization methodology during the peak load condition, condition, i.e., under the same simulation conditions used for the 8- and 27-bus grids.
  • Case 2: The evaluation of the optimization methodology considering daily load variations, according to Table 15 data.
  • Case 3: The evaluation of the optimization methodology considering three different demand periods over the year as follows: (i) 1000 h at 100 percent demand, (ii) 6760 h at 60 percent demand, (iii) 1000 h at 30 percent demand.
    Table 15. Daily demand variation for the 33-bus grid.
    Table 15. Daily demand variation for the 33-bus grid.
    Time (h)Demand (pu)Time (h)Demand (pu)Time (h)Demand (pu)
    10.68451133549247590.706039245570585170.874071251666984
    20.644122690036197100.787007048961707181
    30.613069156029720110.839016955610593190.983615926843208
    40.599733282530006120.852733854067441200.936368832158506
    50.588874071251667130.870642027052772210.887597637645266
    60.598018670222900140.834254143646409220.809297008954087
    70.626786054486569150.816536483139646230.745856353591160
    80.651743189178891160.819394170318156240.733473042484283
Note that no comparison with the previous literature reports is made in this simulation scenario since this is the first time that the IEEE 33-bus grid is adapted to a three-phase unbalanced version of studying the problem of the optimal selection of conductors. This implies that the results reported here will become a reference for future developments in this research area.
Table 16 presents the list of calibers assigned for each operative scenario. The main characteristic of the conductors selected for each operative scenario is that these solutions fulfill the telescopic operating condition typically implemented by distribution companies to reduce investment costs in conducting material.
Table 17 presents the investment, operating, and total costs for the IEEE 33-bus grid. Note that C p e n is zero for all the solutions listed in Table 16 since each final solution is 100% feasible.
The numerical results presented in Table 17 reveal the following:
i.
The investment and operating cost in Case 1, where the peak load condition is analyzed during the year, shows that to minimize the costs of energy losses, the size of the conductors is increased considerably, being this investment costs about 90.1518 % of the total costs of the plan. This is an expected result since, as observed in Table 16, the solution of Case 1 has the largest conductor sizes compared with the remainder of the simulation cases.
ii.
The operation scenario considering the daily load profile shows the more realistic simulation scenario (see Case 2) where the energy losses considerably reduce their costs when compared to the peak load operation scenario (about 91.8576 % ), which has a direct effect on the conductors selected to supply the energy load within this daily operation case. Note that the reduction in the investment costs when compared to Case 1 and Case 2 is about 76.0402 % . This is an expected reduction in the investment costs since these are nonlinearly related to the energy loss costs, which implies that smaller conductor sizes can be found as an adequate balance between investment and operating costs.
iii.
Case 3 shows an intermediate solution scenario between Cases 1 and 2, with a total investment of USD/year 642 , 128.413 distributed about 92.4702 % on investment in conductors and 7.5298 % in costs of the energy losses. Note that what happened with Case 1, the expected energy losses costs of the network, have an essential influence on the size of the conductors selected (see Table 16); in addition, note that for the distribution company, the solution in Case 3 can be the most attractive alternative since it has moderate investment costs with the main advantage that the selected conductors can support future load increments without any changes in the grid structure.
iiii.
Analyzing the processing times, we obtained that for Case 1 it takes 215.5422075 seconds to converge in a single evaluation, while Case 2 and 3 take 4514.526068 and 488.239934 seconds, respectively. Based on the results obtained we can observe that Cases 1 and 2 don’t represent big computational processing times, in other hand for third case its time increases among other things for daily demand variations.

8. Conclusions and Future Work

The problem of the optimal selection of conductors in asymmetric three-phase distribution networks was analyzed in this work with an improved version of the generalized normal distribution algorithm called GNDO. This algorithm included the exploration and exploitation stages based on hyper-ellipses of variable radius, which were uniformly distributed using a Gaussian distribution compared to the best current solution X b e s t t if the parameter ϑ was greater than 50%. The numerical results show that for the 8-bus system, the proposed GNDO improves the results reported for the balanced case with a value of US $455,969.791, on the other hand, for the unbalanced case it is the same as the reported results with a value of USD 558,758.394. In the case of the 27-bus system, the proposed GNDO improves the results reported for both the balanced case and the unbalanced case with US $ values 549,883.572 and US $589,018.800, respectively. This makes the proposed method the reference methodology to compare new optimization methods that are applicable to the problem of optimal selection of conductors in asymmetric three-phase distribution networks.
The processing times of the GNDO show that it takes on average 31.24 s to obtain a solution in the 8-bus system under the scenarios presented, and it takes 168.43 s on average to obtain a solution in the 27-bus system under the scenarios presented, which corresponds to extremely shorter computation times taking into account that the dimensions of the solution space for these systems are in the order of 8 7 and 8 26 representing 2097.152 million and 3.02231454903657 × 10 23 three hundred thousand trillion possibilities.
It’s notable that when the IEEE system buses increase, the processing time will increase in the same way. When comparing the processing time in the IEEE 27-bus system under peak demand and the processing time in the IEEE 33-bus system under peak demand (Case 1), it’s possible to observe that it wasn’t a representative incremental time. The delta value was only 47.1122 seconds. For this reason, the proposed GNDO is suitable for large-scale power grids, and its implementation doesn’t have an upper limit of application.
In future works, the following works can be made: (i) apply the proposed metaheuristic methodology can be applied to solve problems associated with the local distribution system, such as the location of possible charging stations for electric vehicles, possible distributed generation, electrical compensation systems, and protection and load-balancing devices; (ii) to apply new combinatorial optimization methods using efficient initialization algorithms for selecting conductors in large-scale distribution networks by including intelligent evolution strategies that consider the telescopic nature of the studied problem and the possibility of integrating distributed energy resources in the distribution network; and (iii) to develop a convex approximation model for the three-phase networks with a mixed-integer structure that ensure the repeatability of the solution methodology and high-quality solutions with low computational effort.

Author Contributions

Conceptualization, methodology, software, and writing (review and editing): O.D.M., J.A.V.-F. and J.S.R.-C. All authors have read and agreed to the published version of the manuscript.

Funding

This research received no external funding.

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Data Availability Statement

No new data were created or analyzed in this study. Data sharing does not apply to this article.

Acknowledgments

To God, to our families and to the teachers who were part of our professional training process. Dedicated especially to Zuleny Castellanos and Juan de Jesús Vega. This work is derived from the undergraduate project Selección óptima de conductores en sistemas de distribución trifásicos asimétricos empleando el algoritmo de optimización de distribución normalizada, submitted by Julián Alejandro Vega Forero and Jairo Stiven Ramos Castellanos to the Electrical Engineering Program of the Department of Engineering of Universidad Distrital Francisco José de Caldas as a partial requirement for obtaining a Bachelor’s degree in Electrical Engineering.

Conflicts of Interest

The authors declare no conflict of interest.

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Figure 1. Single-line diagram of the proposed example.
Figure 1. Single-line diagram of the proposed example.
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Figure 2. Flowchart used for the BFS method.
Figure 2. Flowchart used for the BFS method.
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Figure 3. Flowchart used for the improved GNDO.
Figure 3. Flowchart used for the improved GNDO.
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Figure 4. Single-line diagram for the 8-bus test system.
Figure 4. Single-line diagram for the 8-bus test system.
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Figure 5. Single-line diagram for the 27-bus test system.
Figure 5. Single-line diagram for the 27-bus test system.
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Figure 6. Behavior of the voltage profiles in the 8-bus three-phase distribution system under a balanced load scenario.
Figure 6. Behavior of the voltage profiles in the 8-bus three-phase distribution system under a balanced load scenario.
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Figure 7. Power losses present in the branches of the three-phase 8-bus distribution system under a balanced load scenario.
Figure 7. Power losses present in the branches of the three-phase 8-bus distribution system under a balanced load scenario.
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Figure 8. Loadability levels present in the branches of the three-phase 8-bus distribution system under a balanced load scenario.
Figure 8. Loadability levels present in the branches of the three-phase 8-bus distribution system under a balanced load scenario.
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Figure 9. Behavior of the voltage profiles in the three-phase 27-bus distribution system under a balanced load scenario.
Figure 9. Behavior of the voltage profiles in the three-phase 27-bus distribution system under a balanced load scenario.
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Figure 10. Power losses present in the branches of the three-phase 27-bus distribution system under a balanced load scenario.
Figure 10. Power losses present in the branches of the three-phase 27-bus distribution system under a balanced load scenario.
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Figure 11. Country house with poor lighting.
Figure 11. Country house with poor lighting.
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Figure 12. Behavior of the voltage profiles in the three-phase 8-bus distribution system under an unbalanced load scenario.
Figure 12. Behavior of the voltage profiles in the three-phase 8-bus distribution system under an unbalanced load scenario.
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Figure 13. Power losses present in the branches of the three-phase 8-bus distribution system under an unbalanced load scenario.
Figure 13. Power losses present in the branches of the three-phase 8-bus distribution system under an unbalanced load scenario.
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Figure 14. Loadability levels present in the branches of the three-phase 8-bus distribution system under an unbalanced load scenario.
Figure 14. Loadability levels present in the branches of the three-phase 8-bus distribution system under an unbalanced load scenario.
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Figure 15. Behavior of the voltage profiles in the three-phase 27-bus distribution system under an unbal-anced load scenario.
Figure 15. Behavior of the voltage profiles in the three-phase 27-bus distribution system under an unbal-anced load scenario.
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Figure 16. Power losses present in the branches of the three-phase 27-bus distribution system under an unbalanced load scenario.
Figure 16. Power losses present in the branches of the three-phase 27-bus distribution system under an unbalanced load scenario.
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Figure 17. Loadability levels present in the branches of the three-phase 27-bus distribution system under an unbalanced load scenario.
Figure 17. Loadability levels present in the branches of the three-phase 27-bus distribution system under an unbalanced load scenario.
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Figure 20. IEEE 33-bus single line diagram.
Figure 20. IEEE 33-bus single line diagram.
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Table 2. Bus connections for the example shown in Figure 1.
Table 2. Bus connections for the example shown in Figure 1.
Nomenclature of Branch i From Bus j To Bus k
a12
b23
c24
d15
Table 3. General simulation parameters.
Table 3. General simulation parameters.
ParameterValueUnit
Energy cost0.1390(US$/kWh)
Iterations1000-
Population size30-
Tolerance 1 × 10 10 -
Table 4. Information corresponding to the 8-bus test system for the balanced case.
Table 4. Information corresponding to the 8-bus test system for the balanced case.
LineBus i Bus j L ij (km) P j , h D (kW) Q j , h D (kvar)
1121.001054.20
2231.00806.50
3141.002632.50
4151.006090
5561.002034.50
6371.00932.80
7381.001731.40
Table 5. Information corresponding to the 8-bus test system for the unbalanced case.
Table 5. Information corresponding to the 8-bus test system for the unbalanced case.
Bus j P j , a D (kW) Q j , a D (kvar) P j , b D (kW) Q j , b D (kvar) P j , c D (kW) Q j , c D (kvar)
23162.600000
3002419.5000
400007897.50
5913.50913.5000
6003051.603051.60
72798.400000
81298.5502597.101298.550
Table 6. Information corresponding to the 27-bus test system for the balanced case.
Table 6. Information corresponding to the 27-bus test system for the balanced case.
LineBus i Bus j L i , j (km) P j , h D (kW) Q j , h D (kvar)
1120.5500
2231.5000
3340.45297.5184.4
4450.6300
5560.70255158
6670.5500
7781.00212.5131.7
8891.2500
99101.00266.1164.9
102111.008552.7
1111121.23340210.7
1212130.75297.5184.4
1313140.56191.3118.5
1414151.00106.365.8
1515161.00255158
163171.00255158
1717180.60127.579
1818190.90297.5184.4
1919200.95340210.7
2020211.008552.7
214221.00106.365.8
225231.0055.334.2
236240.4069.743.2
248250.60255158
258260.6063.839.5
2626270.80170105.4
Table 7. Information corresponding to the 27-bus test system for the unbalanced case.
Table 7. Information corresponding to the 27-bus test system for the unbalanced case.
Bus j P j , a D (kW) Q j , a D (kvar) P j , b D (kW) Q j , b D (kvar) P j , c D (kW) Q j , c D (kvar)
2000000
3000000
4892.5553.20000
5000000
60076547400
7000000
80000637.5395.1
9000000
100000798.3494.7
1100255158.100
121020632.10000
13446.25276.6446.25276.600
1400286.95177.75286.95177.75
15159.4598.700159.4598.7
1600382.5237382.5237
171076547400
18382.52370000
19446.25276.6446.25276.600
2000510316.05510316.05
21127.579.0500127.579.05
2200159.7598.7159.7598.7
23165.9102.60000
240000209.1129.6
25255158255158255158
2663.839.563.839.563.839.5
27170105.4170105.4170105.4
Table 8. Set of conductors considered for the different test systems.
Table 8. Set of conductors considered for the different test systems.
Gauge (c) r ( Ω / km ) x ( Ω / km ) I c , max (A) C c (US$/km)
10.87630.41331801986
20.69600.41332002790
30.55180.40772303815
40.43870.39832705090
50.34800.38993008067
60.27650.361034012,673
70.09660.120160023,419
80.08530.095072030,070
Table 9. Solutions obtained by different optimization methods.
Table 9. Solutions obtained by different optimization methods.
MethodGaugesInvestment in Conductors (US$)Losses (US$)Annual Costs (US$)
TGA{6,5,3,4,4,1,4}125,433406,222.461531,655.461
CBGA{6,6,4,4,4,1,4}143,076373,155.965516,231.965
GAMS{6,4,4,5,4,1,2}122,358416,681.580539,039.580
TBA{6,5,4,4,4,1,3}125,433397,754.442523,187.442
VSA{6,6,5,5,4,2,4}163,350345,007.959508,357.959
NMA{6,6,5,5,4,2,4}163,350345,007.959508,357.959
GNDO{7,7,5,5,4,2,4}227.826228,143.791455,969.791
Table 10. Solutions obtained by different optimization methods.
Table 10. Solutions obtained by different optimization methods.
MethodGaugesInvestment in Conductors (US$)Losses (US$)Annual Costs (US$)
VSA{7,7,5,4,4,3,3,1,1,4,4,2,3 2,1,4,4,2,2,2,1,1,2,2,1,1}344,352.150217,672.327562,024.477
NMA{7,7,4,4,4,4,3,1,1,4,4,3,3 1,2,4,3,2,1,1,1,1,2,2,1,1}337,744.800219,950.455557,695.255
GNDO{7,7,4,4,4,3,3,1,1,4,4,2,1 1,1,3,2,2,1,1,1,1,1,1,1,1}319,768.08230,115.492549,883.572
Table 11. Solutions obtained by different optimization methods.
Table 11. Solutions obtained by different optimization methods.
MethodGaugesInvestment in Conductors (US$)Losses (US$)Annual Costs (US$)
VSA{7,7,7,5,5,4,4}289,713269,045.394558,758.394
NMA{7,7,7,5,5,4,4}289,713269,045.394558,758.394
GNDO{7,7,7,5,5,4,4}289,713269,045.394558,758.394
Table 12. Solutions obtained by different optimization methods.
Table 12. Solutions obtained by different optimization methods.
MethodGaugesInvestment in Conductors (US$)Losses (US$)Annual Costs (US$)
VSA{7,7,5,4,4,4,4,2,2,4,4,3,2 1,1,2,3,2,1,2,2,1,2,2,4,1}350,392.95257,999.185608,392.135
NMA{7,7,4,4,4,3,4,2,1,4,4,4,2 1,1,4,3,2,2,1,1,1,2,2,2,1}344,954.40252,624.608597,579.008
GNDO{7,7,4,4,4,4,4,1,1,4,4,3,1 1,1,4,2,2,1,1,1,1,1,1,1,1}331,828.08257,190.720589,018.800
Table 13. Information corresponding to the 33-bus test system.
Table 13. Information corresponding to the 33-bus test system.
LineBus i Bus j L ij (km)
1120.0699
2230.3720
3340.2762
4450.2876
5560.7630
6670.4030
7781.4733
8890.8850
99100.8900
1010110.1308
1111120.2491
1212131.3115
1313140.6272
1414150.5585
1515160.6457
1616171.5050
1717180.6530
182190.1603
1919201.4298
2020210.4439
2121220.8231
223230.3798
2323240.8035
2424250.7985
256260.1532
2626270.2145
2727280.9963
2828290.7524
2929300.3830
3030310.9687
3131320.3362
3232330.4356
Table 14. Information corresponding to the 33-bus test system.
Table 14. Information corresponding to the 33-bus test system.
Bus j P j , a D (kW) Q j , a D (kvar) P j , b D (kW) Q j , b D (kvar) P j , c D (kW) Q j , c D (kvar)
2100501006000
3900904000
4120751208015090
5602060306030
6601860206020
720015000100100
8200020010000
960600000
106060602000
11453045304530
12006035155100
136011060356035
14120801908012080
15601060506010
166020110806020
1760201509500
18904010009040
1900009040
202105085407075
2190401104011020
22300400009040
23905070000
24420200420200420200
251207500150100
266025802500
2700802500
28602048246020
29120701857512070
30200600400400500600
31150701209015070
3221010012035210100
33604010075000
Table 16. Conductors selected for the IEEE 33-bus grid at each operative scenario.
Table 16. Conductors selected for the IEEE 33-bus grid at each operative scenario.
LineCase 1Case 2Case 3
1747
2747
3747
4747
5747
6747
7747
8747
9747
10747
11736
12736
13734
14624
15511
16511
17111
18415
19412
20411
21111
22534
23534
24111
25747
26745
27615
28613
29613
30311
31211
32211
Table 17. Investment and operating costs found with the application of the GNDO to the three-phase version of the IEEE 33-bus grid for each operative scenario.
Table 17. Investment and operating costs found with the application of the GNDO to the three-phase version of the IEEE 33-bus grid for each operative scenario.
Case C inv (USD) C loss (USD)Z (USD)
1814,647.15188,991.787903,638.938
2195,187.4567246.061202,433.517
3593,777.67248,350.741642,128.413
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MDPI and ACS Style

Vega-Forero, J.A.; Ramos-Castellanos, J.S.; Montoya, O.D. Application of the Generalized Normal Distribution Optimization Algorithm to the Optimal Selection of Conductors in Three-Phase Asymmetric Distribution Networks. Energies 2023, 16, 1311. https://doi.org/10.3390/en16031311

AMA Style

Vega-Forero JA, Ramos-Castellanos JS, Montoya OD. Application of the Generalized Normal Distribution Optimization Algorithm to the Optimal Selection of Conductors in Three-Phase Asymmetric Distribution Networks. Energies. 2023; 16(3):1311. https://doi.org/10.3390/en16031311

Chicago/Turabian Style

Vega-Forero, Julián Alejandro, Jairo Stiven Ramos-Castellanos, and Oscar Danilo Montoya. 2023. "Application of the Generalized Normal Distribution Optimization Algorithm to the Optimal Selection of Conductors in Three-Phase Asymmetric Distribution Networks" Energies 16, no. 3: 1311. https://doi.org/10.3390/en16031311

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