A multi-objective master–slave methodology for optimally integrating and operating photovoltaic generators in urban and rural electrical networks

Results in Engineering 24 (2024) 103059 Available online 10 October 2024 2590-1230/© 2024 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/bync-nd/4.0/). Contents lists available at ScienceDirect Results in Engineering journal homepage: www.sciencedirect.com/journal/results-in-engineering Research paper A multi-objective master–slave methodology for optimally integrating and operating photovoltaic generators in urban and rural electrical networks Jhony Andrés Guzmán-Henaoa, Rubén Iván Bolañosa, Brandon Cortés-Caicedo b, Luis Fernando Grisales-Noreña c,∗, Oscar Danilo Montoya d,∗, Jesús C. Hernández e,∗ aFacultad de Ingenierías, Instituto Tecnológico Metropolitano, Campus Robledo, Medellín 050036, Colombia bFacultad de Ingeniería, Institución Universitaria Pascual Bravo, Campus Robledo, Medellín 050036, Colombia cDepartamento de Ingeniería Eléctrica, Facultad de Ingeniería, Universidad de Talca, Curicó 3340000, Chile dGrupo de Compatibilidad e Interferencia Electromagnética, Facultad de Ingeniería, Universidad Distrital Francisco José de Caldas, Bogotá 110231, Colombia eDepartment of Electrical Engineering, Universidad de Jaén, Campus Lagunillas s/n, Edificio A3, Jaén 23071, Spain A R T I C L E I N F O A B S T R A C T Keywords: Distributed generation Multi-objective optimization Master–slave methodology Electrical distribution system Photovoltaic generation The integration of distributed generation (DG) sources, such as photovoltaic (PV) systems, into electrical power networks presents significant challenges and opportunities. With the increasing penetration of renewable energy sources, optimizing their placement and operation becomes crucial to ensure the reliability, efficiency, and economic viability of power systems. This study presents a master–slave methodology for optimally integrating and operating photovoltaic (PV) generators using multi-objective optimization. This methodology can simultaneously improve technical and economic aspects of the network by determining the best locations and power injection levels for distributed generation sources. Its master stage uses one out of three different algorithms—Multi-Objective Particle Swarm Optimization (MOPSO) algorithm, the Non-dominated Sorting Genetic Algorithm II (NSGA-II), or the Multi-Objective Ant Lion Optimizer (MOALO)—while the slave stage is always performed by a load flow analyzer. The three algorithms in the master stage were implemented considering variable generation and demand conditions in 33 and 27 bus feeders, representing urban and rural areas respectively. The results demonstrated the effectiveness of these algorithms. NSGA-II achieved the best performance, with reductions of 32.84% in energy losses and 42.41% in operating costs (with standard deviations of 0.21% and 0.39%, respectively) for the urban system; and reductions of 21.87% in energy losses and 43.36% in operating costs (with standard deviations of 0.07% and 0.24%, respectively) for the rural system. All of this was achieved within short solution processing times during a typical day in the proposed test scenarios. 1. Introduction The growing world population has brought about increasing electricity consumption [1]. Although traditional Electrical Power Systems (EPSs) have been able to meet most of the demand, they face different challenges in terms of efficiency and reliability [2]. As a result, the use of non-conventional power generation sources should be encouraged to satisfy the growing electricity demand. From the operational point of view, one of the main challenges of conventional generation is that electricity is usually produced in places far away from consumption centers; hence, it is necessary to transmit it over long distances [3]. This increases power losses and generation costs [4]. * Corresponding authors. E-mail addresses: [email protected] (J.A. Guzmán-Henao), [email protected] (R.I. Bolaños), [email protected] (B. Cortés-Caicedo), [email protected] (L.F. Grisales-Noreña), [email protected] (O.D. Montoya), [email protected] (J.C. Hernández). After it has been transmitted to big consumption centers, electricity should be delivered to different users connected to an Electrical Distribution System (EDS). Just like EPSs, EDSs face different challenges due to their direct interaction with users. These users can have multiple characteristics regarding power consumption and different types of connection to the EDS, which produces an imbalance in the operation that results in power losses, variations in voltage profiles, and increases in the current that flows through the lines in the EDS [5]. Recently, multiple power generation technologies have been developed to improve the operation of traditional EDSs [6]. These technologies are collectively known as Distributed Generation (DG) [7]. In addition to improving EDSs, DG seeks to enable a sustainable transition https://doi.org/10.1016/j.rineng.2024.103059 Received 22 July 2024; Received in revised form 24 September 2024; Accepted 3 October 2024 Results in Engineering 24 (2024) 103059 2 J.A. Guzmán-Henao, R.I. Bolaños, B. Cortés-Caicedo et al. Nomenclature and acronyms 𝐶𝑝𝑣 ℎRenewable energy generation curve during hour ℎin the area of interest. 𝐼𝑚𝑎𝑥 ℎMaximum allowable current during hour ℎin the system lines. 𝐼𝑙,ℎ Current in line 𝑙during hour ℎ. 𝑃𝑐𝑠,𝑚𝑎𝑥 𝑖Maximum active power assigned to the conventional generator at node 𝑖. 𝑃𝑐𝑠,𝑚𝑖𝑛 𝑖Minimum active power assigned to the conventional generator at node 𝑖. 𝑃𝑝𝑣,𝑚𝑎𝑥 𝑖Maximum active power assigned to the DG source at node 𝑖. 𝑃𝑝𝑣,𝑚𝑖𝑛 𝑖Minimum active power assigned to the DG source at node 𝑖. 𝑄𝑐𝑠 𝑖,ℎ Reactive power supplied by conventional generators at node 𝑖during hour ℎ 𝑄𝑐𝑠,𝑚𝑎𝑥 𝑖Maximum reactive power assigned to the conventional generator at node 𝑖. 𝑄𝑐𝑠,𝑚𝑖𝑛 𝑖Minimum reactive power assigned to the conventional generator at node 𝑖. 𝑉𝑚𝑎𝑥 𝑖Maximum allowable voltage at node 𝑖. 𝑉𝑚𝑖𝑛 𝑖Minimum allowable voltage at node 𝑖. ΔℎTime interval when electrical variables remain stable Set of all time intervals Set of all lines in the system Set of all the nodes in the network 𝜃𝑖,ℎ Voltage angle at node 𝑖during hour ℎ 𝜃𝑗,ℎ Voltage angle at node 𝑗during hour ℎ 𝜑𝑖𝑗 Angle of the admittance between nodes 𝑖and 𝑗 𝐶𝑘𝑊 ℎ Average cost of purchasing energy from the conventional grid 𝐶𝑂&𝑀Costs of operating and maintaining the DGs 𝑔1Cost of purchasing energy from conventional sources 𝑔2Maintenance and operation costs of DGs 𝐼𝑙Current flowing through line 𝑙 𝑃𝑐𝑠 𝑖,ℎ Active power supplied by conventional generators at node 𝑖during hour ℎ 𝑃𝑑 𝑖,ℎ Active power demanded by node 𝑖during hour ℎ 𝑃𝑝𝑣 𝑖,ℎ Active power injected by DGs at node 𝑖during hour ℎ 𝑄𝑐𝑠 𝑖,ℎ Reactive power acquired or injected during hour ℎby a conventional generator at node 𝑖. 𝑄𝑑 𝑖,ℎ Reactive power demanded by node 𝑖during hour ℎ 𝑄𝑝𝑣 𝑖,ℎ Reactive power injected by DGs at node 𝑖during hour ℎ 𝑅𝑙Electrical resistance in line 𝑙 𝑉𝑖,ℎ Voltage at node 𝑖during hour ℎ 𝑉𝑗,ℎ Voltage at node 𝑗during hour ℎ 𝑌𝑖𝑗 Admittance between nodes 𝑖and 𝑗 DG Distributed Generation EDS Electrical Distribution System GOMPSO Grid-Oriented Multi-Particle Swarm Optimization GWO Grey Wolf Optimization IGJO Improved Golden Jackal Optimization MOALO Multi-Objective Ant Lion Optimizer MOFFA Multi-objective Fuzzy Firefly Algorithm MOPSO Multi-Objective Particle Swarm Optimization MOWOA Multi-Objective Whale Optimization Algorithm NSGA-II Non-dominated Sorting Genetic Algorithm II PSO Particle Swarm Optimization USD United States Dollars ZOA Zebra Optimization Algorithm toward renewables, e.g., wind and solar power [8]. One of the most common types of DG are photovoltaic (PV) technologies. This is because solar energy is abundant in different regions and PV technologies are easy to implement. If electricity is generated close to final consumption centers, DG devices can reduce the need to transmit power from big power plants and, therefore, power losses associated with transmission. In addition, installing DG devices contributes to improving the voltage profiles of the nodes in the network and reduces current flows through the lines [9]. Implementing DG can enhance the operation of EDSs. However, decisive aspects—in particular, the location and size of the Distributed Generators (DGs)—should be analyzed to enjoy all the benefits and avoid negative impacts on the network [10]. Therefore, network operators need strategies that help them to determine the most adequate locations to install DG devices in an EDS [11][12]. These strategies should also enable an efficient management and operation of these devices by determining the most adequate level of power injection to meet the needs of the system—thus influencing economic, environmental, and technical aspects of its operation [13]. 1.1. Literature review Where to install DG sources and how big they should be in an EDS is a multivariate problem due to the size and characteristics of this kind of systems [14]. Regarding location, each one of the demand nodes is a candidate for the installation of a generator. Therefore, a strategy to determine their best possible location should evaluate the operational impact of DG sources at the candidate nodes. Then, in order to establish the size of the DGs, it is essential to guarantee that their nominal power and the amount of power they inject at a given time respond to the power demand in the system [15]. Operating an EDS is a complex task. As a result, the problem of the optimal location and size of DGs involves binary as well as continuous variables. For that reason, novel efficient strategies should be formulated to adequately integrate and manage DG sources in EDSs [16]. Multiple authors have proposed strategies that use optimization techniques to improve operational aspects of EDSs in scenarios that include DG sources. These optimization techniques can be classified into analytical [17], numerical [18], exact [19], heuristic [20], and metaheuristic [21]. Some of the optimization objectives most widely studied in this field include, operating cost minimization, power loss reduction, and CO2emission reduction [22]. These strategies can have a single objective, i.e., mono-objective optimization. In this kind of optimization, it is possible to solve the problem of integrating DGs by quantifying the improvements obtained in EDSs in one regard in particular, whether technical, economic, or environmental [23]. An example of this is the paper by Serrano and Leite [24]. They combined the Greedy Randomized Adaptive Search Procedure (GRASP) and tabu search in a single strategy to minimize power losses by correctly locating and sizing DGs in a real system that provides electricity to 834 loads. In another study, Elattar and Elsayed [25] proposed a strategy to minimize the operating costs of an EDS in a scenario in which renewable DG sources had been installed. Their strategy utilized the Modified Moth-Flame Optimization (MMFO) algorithm and a 69-node test system to validate their results. In addition, they took into account some operational constraints for the EDS, such as voltage profile and current flowing through the distribution lines. Purlu et al. [26] proposed a strategy to minimize annual power losses by finding the optimal size and location of DGs. For that purpose, they used Particle Swarm Optimization (PSO) and the 33-node test system as a validation scenario. One of the main contributions of their study is that they were able to reduce an- Results in Engineering 24 (2024) 103059 3 J.A. Guzmán-Henao, R.I. Bolaños, B. Cortés-Caicedo et al. nual power losses by 39.8% implementing PV generators and by 64.3% using wind turbines. The research conducted by Kandel et al. [27] proposes a methodology based on the Zebra Optimization Algorithm (ZOA) to determine the optimal locations and the amount of power injected by DG sources in a 33-node radial distribution system. The study separately analyzes, under a mono-objective optimization model, the energy losses and the operational stability of the system. However, it does not account for the system’s behavior under dynamic conditions of generation and energy demand, which would offer a more realistic scenario for system operation. Aeggegn et al. [28] proposed an optimization strategy for electrical microgrids in an environment integrating various distributed energy resources, including DG sources and energy storage systems. Using the Grey Wolf Optimization (GWO) algorithm, the strategy determines the optimal sizing of the different devices, with the objective of minimizing operating costs in a 14-node test electrical system. However, the study does not provide an analysis of the network’s operational conditions, which would allow for an evaluation of the technical feasibility of the proposed solutions. Although mono-objective optimization can be useful, the real problems network operators have to face usually involve a simultaneous optimization of different aspects. This is known as multi-objective optimization [29]. When a problem is addressed using multi-objective optimization, the objective functions to be optimized should be in conflict with each other, that is, when one of them is improved another one tends to get worse. As a consequence, finding the optimal solution to a given problem will depend on the criterion applied by the network operator [29]. Multi-objective optimization uses sets of solutions called “Pareto fronts.” These fronts represent a group of non-dominated solutions, that is, those solutions in which an objective function cannot be improved without worsening another [30]. Optimization techniques can employ learning, genetic, and population-based algorithms to perform an adequate exploration of the solution space, which facilitates the calculation of Pareto fronts. This is the reason why these algorithms are commonly used in multi-objective optimization problems [31]. An example of this approach is described in the paper by Agarwal et al. [32], who proposed a strategy based on a Grid-Oriented Multi-Particle Swarm Optimization (GOMPSO) algorithm. They implemented said strategy to determine the location and size of DG sources in a multi-objective optimization environment, focusing on several technical aspects of the EDS (e.g., reducing power losses and improving the voltage profiles of the system). The performance of their strategy was validated in a 33-bus system, and they used indices for (a) reductions in reactive and active power losses and (b) voltage profile deviations. However, in their strategy, they did not analyze different time intervals during which the system conditions could change. In addition, they did not consider economic aspects that are essential to make this kind of projects viable. The authors in [33] addressed the problem of locating DG sources in EDSs using a multi-objective optimization approach. For this, they employed the Improved Golden Jackal Optimization (IGJO) algorithm, with technical objective functions such as improving voltage profiles and system stability. To validate the proposed methodology, the study used the IEEE 69-bus and 118-bus test systems and compared its effectiveness with other algorithms reported in the literature that solve this problem. However, the research did not include an economic analysis to assess the feasibility of installing DG sources, nor did it consider the system’s behavior under dynamic conditions of generation and energy demand. The authors in [34] addressed the integration of DG in EDSs using the multi-objective fuzzy firefly algorithm (MOFFA). They applied the method to a 33-node IEEE test system and a real 62-node system in Tamil Nadu, India, evaluating three scenarios with different objectives, such as minimizing energy losses, costs, and emissions, as well as improving voltage stability. However, the analysis did not include scenarios where energy generation and demand varied over time, nor did it include a statistical study of the solutions found. Prasad et al. [35] proposed a methodology that employs the MultiObjective Whale Optimization Algorithm (MOWOA) to determine the best locations and sizes for DG devices installed in an EDS. Their objective was to simultaneously minimize power losses and reduce annual operating costs, and their methodology was evaluated in 33and 69bus test feeders. Nevertheless, they did not take into consideration real scenarios in which generation and demand are variable, which are the main characteristics of a real EDS. Akbar et al. [36] proposed a strategy to optimally assign DG units in radial EDSs. In particular, they implemented a multi-objective optimization approach that included reductions in voltage deviation and power losses, as well as a stability index. Their strategy combined Improved Grey Wolf Optimization with Particle Swarm Optimization (I-GWOPSO). The efficiency of their I-GWOPSO was validated in 33and 69-bus test systems applying different power factors and obtaining reductions of up to 98% in power losses. However, they did not analyze scenarios in which the demand and generation conditions changed over time or incorporated, into the mathematical model, the analysis of economic aspects that can be used to determine the viability of this kind of projects. Table 1provides a summary of the main studies found in the literature. Although multiple studies have focused on the optimization of DG integration into EDSs in terms of size and location, many of them lack a mathematical model that includes all the operational constraints inherent in systems of this kind. Furthermore, they did not implement test scenarios where electricity generation and demand were variable, which caused the proposed methodologies to be further from reality. They have also used specialized tools that limit the adaptability of the proposed strategies and increase implementation costs. As a result, the strategies they have been implemented in the literature to optimally integrate DG resources into EDSs—i.e., the problem examined here—do not offer adequate repeatability. In addition, some papers lack a statistical analysis that establishes the repeatability and quality of the solutions and their processing times in multi-objective optimization environments. For all the reasons stated above, it is necessary to propose multi-objective optimization methodologies with mathematical models that consider the operational constraints of the system and variable conditions (i.e., power generation and demand). These methodologies should implement simple, low-cost tools that are flexible and adaptable to different problems. They should also include statistical and operational validations of the solutions obtained for an EDS in particular. 1.2. Contributions and scope Previous studies have extensively examined the optimal location and sizing of DGs in distribution systems, focusing on various economic, technical, and environmental objectives over both shortand long-term horizons. However, a notable research gap remains regarding the integration of PV generators, specifically within urban and rural distribution grids, under a unified multi-objective framework. This study addresses this gap. The main contribution is a methodology for the optimization of the integration and operation of photovoltaic generators in urban and rural electricity distribution networks under a multi-objective approach. This methodology simultaneously optimizes both technical and economic aspects while ensuring compliance with the operational constraints of the network. Unlike previous studies, which often treat technical and economic optimization separately or within a limited scope, our approach offers a comprehensive solution that enhances both the technical performance and economic feasibility of distribution networks.. The following aspects are highlighted in this work: • Development of a mathematical model that incorporates specific operational constraints reflective of DG source scenarios. The model Results in Engineering 24 (2024) 103059 4 J.A. Guzmán-Henao, R.I. Bolaños, B. Cortés-Caicedo et al. Table 1 Related studies found in the specialized literature and their characteristics. Ref. Strategy Solution Technique Test System (Number of Nodes) Optimization Variables Objective Function Analysis of Dynamic Conditions Statistical analysis [24] Single-objective GRASP Real system of 834 nodes Locating and sizing DGs Technical No No [25] Single-objective MMFO 69-node test system Locating and sizing DGs Technical and economic No No [26] Single-objective PSO 33-node test system Locating and sizing DGs Technical No Yes [27] Single-objective ZOA 33-node test system Locating and sizing DGs Technical No No [28] Single-objective GWO 14-node test system Sizing of DGs Economic Yes No [32] Multi-objective GOMPSO 33-node test system Locating and sizing DGs Technical No No [33] Multi-objective IGJO 69-and 118-node test systems Sizing of DGs Technical No Yes [34] Multi-objective MOFFA 33-node test system and a real 62-node system Locating DGs Technical and economic No Yes [35] Multi-objective MOWOA 33and 69-node test systems Locating and sizing DGs Technical and economic No No [36] Multi-objective I-GWOPSO 33and 69-node test systems Locating DGs Technical No Yes features a multi-objective function that concurrently addresses technical and economic facets of PV generator integration. • Implementation of a two-stage methodology wherein the master stage determines the optimal installation points and power output for DG sources, while the slave stage assesses operational conditions and quantifies improvements post-integration. An iterative hourly load flow process is employed to enhance economic and operational aspects, utilizing three multi-objective algorithms—NSGA-II, MOPSO, and MOALO—to validate the methodology. • Creation of two test scenarios designed to represent variable conditions in urban and rural environments, including power generation and demand variations. • Introduction of a flexible, straightforward, and rapid methodology that supports the integration of various multi-objective algorithms in the master stage. This adaptability facilitates tailored solutions for different systems and reduces reliance on specialized tools for efficient DG installation and operation. • Development of a tool that generates non-dominated solution fronts, with extreme points representing the lowest values in each objective function. This tool allows network operators to apply diverse criteria to select the most suitable solution, providing flexibility and customization in decision-making. • Comprehensive evaluation of the solutions based on operational constraints, repeatability, processing times, and overall quality. This analysis ensures the effectiveness and robustness of the proposed methodology. 1.3. Paper structure This paper is organized as follows. Section 2presents the mathematical model employed in this study to describe the multi-objective optimization of the integration of Distributed Generators (DGs) into an EDS. Section 3describes the test scenarios and their adaptation. These scenarios represent real systems and were used to validate the approach proposed here for integrating DGs. Section 4details the methodology, including the steps that are necessary in a possible replication of this study. Then, the results are presented and analyzed in Section 5. The last part, Section 6, draws the conclusions and suggests future research in this area. 2. Mathematical model The integration of distributed generators into EDSs is represented as a multivariate problem in mathematical models that are used to quantify operational aspects of the system. In addition, to optimize the installation of DG sources, the problem should be formulated including two elements that refer to the operation of the EDS: objective functions and a set of constraints. 2.1. Formulating the objective functions The aim of this study is to develop a tool to optimize several aspects of the operation of EDSs that include DGs. The literature in this field has widely covered this optimization in technical and economic terms. Consequently, this study employed two objective functions: a technical one and an economic one. 2.1.1. Technical objective function In the literature in this area, one of the most common objective functions has been the reduction of power losses, which have a considerable impact on operational aspects of EDSs. In this paper, Equation (1)is the technical objective function: 𝑚𝑖𝑛 𝑓1=∑ ℎ∈∑ 𝑙∈ 𝑅𝑙𝐼2 𝑙Δℎ(1) where 𝑅𝑙is the electrical resistance found in line 𝑙; 𝐼𝑙, the current that flows through line 𝐼𝑙; and Δℎ, a time interval when the electrical variables remain stable. In addition, all the time intervals are denoted as ; and the total number of lines in the system, as . 2.1.2. Economic objective function The literature in this area has also thoroughly explored two economic criteria: (i) minimizing power purchase costs and (ii) reducing the maintenance and operating costs of DGs [21][7]. Hence, to model the economic objective function (𝑓2), this paper integrates—in a single expression—two types of costs: (a) that of purchasing power from conventional sources (𝑔1) and (b) the maintenance and operating costs of the DGs (𝑔2), as shown in Equation (2). 𝑚𝑖𝑛 𝑓2=𝑔1+𝑔2(2) Equations (3)and (4) detail each one of the aspects used here to calculate generation and maintenance costs. 𝑔1=C 𝑘𝑊 ℎ (∑ ℎ∈∑ 𝑖∈ 𝑃𝑐𝑠 𝑖,ℎΔℎ)(3) 𝑔2=C 𝑂&𝑀(∑ ℎ∈∑ 𝑖∈ 𝑃𝑝𝑣 𝑖,ℎ Δℎ)(4) In the equations above, C𝑘𝑊 ℎ represents the average cost of buying power from the conventional grid; and C𝑂&𝑀, the costs of operating and maintaining the DGs. Variable 𝑃𝑐𝑠 𝑖,ℎ denotes the active power that a conventional generator connected to node 𝑖should inject during hour ℎ, and 𝑃𝑝𝑣 𝑖,ℎ indicates the amount of active power that a DG installed at node 𝑖should inject during the same hour ℎ. Furthermore, Δℎrefers to the hours when the electrical variables remain constant. denotes all the nodes in the network; and , all the hours in the operation. Results in Engineering 24 (2024) 103059 5 J.A. Guzmán-Henao, R.I. Bolaños, B. Cortés-Caicedo et al. Fig. 1. Optimal Pareto front and dominated solution. 2.1.3. Multi-objective technical-economic function Although mono-objective optimization can be useful to operate EDSs, the real problems network operators have to face usually require a simultaneous optimization of different aspects. Therefore, this study aims to provide a tool that can simultaneously improve the technical and economic objective functions. The functions that are simultaneously optimized in this methodology are defined in expression 𝑓𝑖(𝑢), where 𝑢is a vector that contains the decision variables in the problem, and 𝑖is the specific optimized objective function; in this case, 𝑓1and 𝑓2. Likewise, 𝑈represents the solution space of the problem. Equation (5)is the multi-objective function employed in this study (𝑚𝑖𝑛 𝐹(𝑢)). 𝑚𝑖𝑛 𝐹𝑖(𝑢)=[𝑓1(𝑢),𝑓 2(𝑢).....𝑓𝑛(𝑢)],𝑢∈𝑈(5) A non-dominated classification approach was adopted to determine the set of best solutions obtained after the optimization. Said approach focuses on two aspects: •Domination: A decision vector, 𝑢1, dominates 𝑢2if and only if 𝑢1 is at least as good as 𝑢2regarding all the objective functions and 𝑢1 is strictly better than 𝑢2in at least one objective function. •Optimal Pareto front: It is composed of a set of non-dominated solutions that represent points in the solution space, as shown in Fig. 1. After the optimal Pareto front has been obtained, a selection criterion should be employed in accordance with the purpose of the optimization. 2.2. Constraints The set of constraints included here represent the maximum and minimum operating levels of the EDS in an environment that includes DG sources. These constraints are formulated as a set of inequalities and equations that should be respected during the multi-objective optimization process. 2.2.1. Balance between active and reactive power at the nodes This balance relates the active and reactive power supplied by the conventional and distributed generators to the active and reactive power demanded by the loads at individual nodes. It also considers the amplitude and voltage phase recorded at individual nodes during a preset interval, in addition to the amplitude and phase of the admittance in the lines. Equations (6)and (7) describe the mathematical models employed in this paper to ensure the balance between active and reactive power in the system. 𝑃𝑐𝑠 𝑖,ℎ +𝑃𝑝𝑣 𝑖,ℎ −𝑃𝑑 𝑖,ℎ =𝑉𝑖,ℎ ∑ 𝑗∈ 𝑌𝑖𝑗 𝑉𝑗,ℎ Cos (𝜃𝑖,ℎ −𝜃𝑗,ℎ −𝜑𝑖𝑗 ),{∀𝑖∈ ∀ℎ∈}, (6) 𝑄𝑐𝑠 𝑖,ℎ +𝑄𝑝𝑣 𝑖,ℎ −𝑄𝑑 𝑖,ℎ =𝑉𝑖,ℎ ∑ 𝑗∈ 𝑌𝑖𝑗 𝑉𝑗,ℎ Cos (𝜃𝑖,ℎ −𝜃𝑗,ℎ −𝜑𝑖𝑗 ),{∀𝑖∈ ∀ℎ∈}, (7) In the expressions above, 𝑃𝑐𝑠 𝑖,ℎ and 𝑄𝑐𝑠 𝑖,ℎ represent the active and reactive power, respectively, supplied by conventional generators at node 𝑖during hour ℎ. In turn, 𝑃𝑑 𝑖,ℎ and 𝑄𝑑 𝑖,ℎ indicate the active and reactive power required by node 𝑖during the same hour. Likewise, 𝑃𝑝𝑣 𝑖,ℎ and 𝑄𝑝𝑣 𝑖,ℎ denote the active and reactive power, respectively, injected at node 𝑖by DGs during hour ℎ. Variables 𝑉𝑖,ℎ and 𝑉𝑗,ℎ denote the voltage at nodes 𝑖 and 𝑗, respectively, during hour ℎ, while 𝜃𝑖,ℎ and 𝜃𝑗,ℎ represent the voltage angles at nodes 𝑖and 𝑗, respectively, during said hour. Additionally, 𝑌𝑖𝑗 is the admittance of the line that connects nodes 𝑖and 𝑗, and 𝜑𝑖𝑗 is the admittance angle of said line. 2.2.2. Constraints for conventional generators: active and reactive power limits The constraints described in this subsection are derived from the physical characteristics of conventional generators and established in accordance with this type of system. Inequalities (8)and (9)are used to represent these limits. 𝑃𝑐𝑠,𝑚𝑖𝑛 𝑖≤𝑃𝑐𝑠 𝑖,ℎ ≤𝑃𝑐𝑠,𝑚𝑎𝑥 𝑖,{∀𝑖∈ ∀ℎ∈}(8) 𝑄𝑐𝑠,𝑚𝑖𝑛 𝑖≤𝑄𝑐𝑠 𝑖,ℎ ≤𝑄𝑐𝑠,𝑚𝑎𝑥 𝑖,{∀𝑖∈ ∀ℎ∈}(9) In the inequalities above, 𝑃𝑐𝑠 𝑖,ℎ and 𝑄𝑐𝑠 𝑖,ℎ represent the active and reactive power, respectively, acquired or injected during hour ℎby a conventional generator at node 𝑖. In addition, 𝑃𝑐𝑠,𝑚𝑖𝑛 𝑖and 𝑃𝑐𝑠,𝑚𝑎𝑥 𝑖are the maximum and minimum active power, respectively, assigned to the conventional generator at node 𝑖. In turn, 𝑄𝑐𝑠,𝑚𝑖𝑛 𝑖and 𝑄𝑐𝑠,𝑚𝑎𝑥 𝑖are the minimum and maximum reactive power, respectively, assigned to said generator. 2.2.3. Constraints for the DGs: active power limits In this study, the contribution by DG sources is considered in terms of active power. Therefore, (10)is proposed as a representation of the active power limits for the DG sources. 𝑃𝑝𝑣,𝑚𝑖𝑛 𝑖≤𝑃𝑝𝑣 𝑖,ℎ ≤𝑃𝑝𝑣,𝑚𝑎𝑥 𝑖𝐶𝑝𝑣 ℎ,{∀𝑖∈ ∀ℎ∈},(10) In this equation, 𝑃𝑝𝑣,𝑚𝑖𝑛 𝑖and 𝑃𝑝𝑣,𝑚𝑎𝑥 𝑖are the minimum and maximum power, respectively, assigned to the DGs in accordance with the requirements of the system. Also, 𝐶𝑝𝑣 ℎis the curve of renewable energy generation in the area of interest. 2.2.4. Constraints for the nodes: voltage limits Voltage limits are subject to the technical standards applicable in the location where the EDS is operating. This paper proposes (11)to establish these limits: 𝑉𝑚𝑖𝑛 𝑖≤𝑉𝑖,ℎ ≤𝑉𝑚𝑎𝑥 𝑖,{∀𝑖∈ ∀ℎ∈},(11) where 𝑉𝑖 𝑚𝑖𝑛 and 𝑉𝑖 𝑚𝑎𝑥 represent the minimum and maximum allowable voltages, respectively. 2.2.5. Constraints for the lines: current limits These limits are some the most critical elements in the model because they reflect construction-related aspects of the EDS. These limits should be strictly respected, especially when installing DG sources can alter the current flow through the system lines. Said limits are represented in (12): Results in Engineering 24 (2024) 103059 6 J.A. Guzmán-Henao, R.I. Bolaños, B. Cortés-Caicedo et al. Fig. 2. Graphical representation of the topology of the 33-node test system. 𝐼𝑙,ℎ ≤𝐼𝑚𝑎𝑥 ℎ,{∀𝑙∈ ∀ℎ∈}.(12) 3. Test scenarios The impact of the proposed strategy was evaluated in test scenarios simulating the behavior of urban and rural EDS. In this context, the most crucial information includes the curves representing power demand and DG. These data are essential for developing a model that accurately reflects the actual behavior of the network. 3.1. Urban scenario Medellín (Colombia) stands out as one of the most densely populated cities around the world. As a result, in the EDS in Medellín, the loads are noticeably concentrated in small areas, which poses a series of operational challenges that can be addressed by integrating DG sources [37]. Among the most significant issues are power losses, voltage drops, and the high percentage of loading on system lines [38]. These factors directly impact the stability and reliability of the electrical supply, compromising service quality and increasing the risk of operational failures [39]. This city has a warm climate and plenty of solar irradiance, characteristics that make it an attractive location for solar energy projects. In order to model a typical EDS in Medellín, this study used the radial 33-bus test feeder, which has been extensively investigated in the literature. 3.1.1. Radial 33-node test system Introduced by Baran and Wu in 1989 [40], this test system has been widely adopted by multiple authors to validate the strategies they have proposed [41–43]. In this system, there are 32 lines and 33 nodes (i.e., one generation and 32 demand nodes). These nodes are located in a radial configuration, as shown in Fig. 2. The base voltage of this system is 12.66 kV; and its base power, 100 kVA. Those two characteristics define its nominal capacity. Furthermore, its total demand for active and reactive power is 3,715 kW and 2,300 kVAR, respectively. All parametric information regarding the lines and nodes was extracted from the study presented by Guzmán et al. [44]. This scenario was configured with data about the radial 33-node system and the characteristic power demand and generation curves in Medellín over a typical day of operation divided into 24 one-hour intervals. 3.2. Rural scenario Capurganá, located in the department of Chocó, Colombia, was selected as a rural setting due to its warm climate and abundant solar radiation—conditions that are ideal for the use of photovoltaic generators. This region is home to approximately 2,500 inhabitants dispersed over a vast area, resulting in distribution lines operating at low load levels. Additionally, energy generation in the area relies heavily on fossil fuel sources, which not only increase energy costs but also have significant negative environmental impacts [45]. We selected Capurganá Fig. 3. Graphical representation of the topology of the 27-node test system. Fig. 4. Photovoltaic generation curve in the regions under analysis. due to the potential positive impact that the integration of DG sources could have on energy purchase costs and on the operational aspects of the system, such as reducing power losses and improving voltage profiles. Furthermore, the incorporation of DG has the potential to enhance the reliability and resilience of the electrical system by diversifying the energy matrix [46]. From an environmental perspective, the use of renewable resources helps to reduce the carbon footprint, promoting sustainability and the preservation of the natural environment. To model the typical EDS of Capurganá, this study utilized the 27-bus radial test feeder, which has been extensively studied in the literature. 3.2.1. Radial 27-node test system Introduced by Falaghi et al. in 2005 [47], this test system has been widely adopted by multiple authors to validate the strategies they have proposed [14,44,48]. In this system, there are 26 lines and 27 nodes (i.e., one generation and 26 demand nodes). These nodes are located in a radial configuration, as shown in Fig. 3. The base voltage of this system is 23 kV; and its base power, 100 kVA. Those two characteristics define its nominal capacity. Furthermore, its total demand for active and reactive power is 4,131 kW and 2,560 kVAR, respectively. All parametric information regarding the lines and nodes was extracted from the study presented by Guzmán et al. [44]. This scenario was configured with data about the radial 27-node system and the characteristic power demand and generation curves in Capurganá over a typical day of operation divided into 24 one-hour intervals. 3.3. Generation and demand curves The solar generation curve and energy demand provide crucial information on the amount of electricity that can be generated and the amount needed to meet energy demand in each test scenario. This study employed data on both urban and rural scenarios obtained by Cortés et al. [14]. Fig. 4shows the electricity generation potential in the regions under analysis, and Fig. 5details the demand for electricity there. 4. Methodology 4.1. Selecting the number of generators and their nominal power In this paper, the number of generators to be implemented is based on the article by Montoya et al. [19]. They conducted a cost–benefit analysis of multiple numbers of generators installed in different test systems. In their analysis, the decrease in power losses was the central Results in Engineering 24 (2024) 103059 7 J.A. Guzmán-Henao, R.I. Bolaños, B. Cortés-Caicedo et al. Fig. 5. Characteristic power demand curve in the regions under analysis. Fig. 6. Master–slave methodology for optimal integration of DG. parameter. Their results indicate that incorporating three generators reduces power losses by 17%. In contrast, adding 4, 5, or 6 generators produces extra reductions of only 1%, 2%, and 3%, respectively. Therefore, considering the cost of each DG source, it can be concluded that the best cost–benefit ratio is achieved with three units. For that reason, this study used three PV generators. The nominal power of each generator is, without a doubt, a decisive factor in the economic optimization of DG integration. This is because an oversized DG could entail additional unnecessary costs in terms of acquisition and operation. For the test system discussed here, several authors have established that the most adequate nominal power is around 2400 kW [49][50][51]— that is the value used in this study. 4.2. Proposed strategy The proposed strategy uses multi-objective optimization to establish the optimal location and size of DGs in an EDS. This approach seeks to have a simultaneous positive impact on both the economic and technical aspects of the system. Considering the complexity of the variables included in this mathematical model, a master–slave methodology was implemented. In this methodology, different values can be proposed for the variables of interest to evaluate their impact on different aspects of the EDS. In the master stage, specific demand nodes are selected for the installation of DG sources, as well as the power injection for each generator during the evaluation period. Subsequently, the slave stage utilizes the information provided by the master stage to determine the impact on the system’s operating conditions resulting from the installation and operation of the DG sources, as established in the master stage, as illustrated in Fig. 6. The interaction between these two stages, where the slave stage provides feedback on the system’s response, helps refine and validate the solution proposed in the master stage, thereby forming a two-stage optimization framework. 4.2.1. Workflow of the master–slave methodology First, the master stage employs a multi-objective optimization technique to establish the optimal locations for the DG sources and the amount of power that each one should inject during each hour in a typical day of network operation. Second, these data are shared with the slave stage, which evaluates the power flow and analyzes the operational impact of installing the generators, calculating the values associated with the objective functions. This optimization model is configured with a set of operational constraints that should be satisfied for a solution to be considered feasible. Third, if these constraints are violated, a penalty is applied to the value obtained during the optimization process. As the case examined in this study is a minimization problem, the lowest values of each objective function are selected. These values are later fed back to the master stage. Finally, this step is repeated in an iterative process to determine the best installation points and power injection levels for the DG sources. Fig. 7 details this process. 4.2.2. Optimization techniques implemented in the methodology The master stage employed three optimization techniques that have been widely explored in the literature to address problems in electrical distribution systems. •Non-dominated Sorting Genetic Algorithm II (NSGA-II): Amultiobjective optimization technique, the NSGA-II was proposed in 2002 [52]as an improvement to the NSGA. It was inspired by Darwin’s concepts of evolution and natural selection. In optimization processes, this algorithm can represent candidate solutions as individuals that evolve and improve themselves over generations using genetic operators such as selection, crossover, and mutation. In each generation, solutions reproduce and give birth to new solutions, among which only the best survive. Thanks to this multiobjective approach, the NSGA-II can be used to address problems with different aims—which are frequently in conflict—and propose non-dominated solutions. These solutions are complemented with an expert selection criterion that is adjusted to the particular characteristics of the problem being addressed. •Multi-Objective Particle Swarm Optimization (MOPSO): Proposed by Coello and Lechuga in 2002 [53], this optimization technique is based on Particle Swarm Optimization (PSO) [54], originally developed by Eberhart et al. in 1995. As the PSO technique, MOPSO is inspired by the behavior observed in certain animal species (e.g., birds or fish), which use collective intelligence to locate the best food sources. Using an iterative process, the MOPSO moves particles over the solution space of the problem. This adjustment in positions depends on their individual as well as collective learning, thanks to which it possible to obtain better solutions at each iteration and classify them into non-dominated solution fronts. This strategy has been investigated in several studies and highlighted due to its outstanding results [55][56][57]. •Multi-Objective Ant Lion Optimizer (MOALO): Proposed in 2017 [58], this metaheuristic algorithm is based on the Ant Lion Optimizer (ALO) [59]. It is inspired by the hunting strategy of an insect called ant lion, which builds traps to catch its prey by making coneshaped holes in the sand. Then, it hides at the base of the cone, waiting for the prey to fall into its trap. This algorithm performs simultaneously two important parts in the optimization process: exploration and exploitation of the solution space. After the possible solutions have been generated, they are evaluated to select the most promising ones. This algorithm is employed to address multiobjective optimization problems where several objective functions should be optimized simultaneously. The Pareto fronts are used to classify the best non-dominated solutions that are produced by the iterative process. These fronts provide useful information for decision-making based on expert criteria according to the problem that should be optimized. The MOALO has been employed in multiple studies that have reported its effectiveness in the optimization of engineering problems [60][61][14]. 4.2.3. Evaluating the operating status using the hourly power flow The operating status of the EDS is evaluated using power flow analysis based on measurements of the power demanded and generated in Results in Engineering 24 (2024) 103059 8 J.A. Guzmán-Henao, R.I. Bolaños, B. Cortés-Caicedo et al. Fig. 7. Workflow of the proposed master–slave methodology. the system. This process uses essential parameters of the electrical power system, such as data about the lines and constant impedance values associated with the nodes. The main purpose of this analysis is to establish the voltage magnitudes, phase angles, and power flows for each node using equations that relate these variables [62]. In the literature, different mathematical models have been implemented to solve the power flow in an EDS. Some of the most remarkable are the Gauss–Seidel [63], Newton–Raphson [64], iterative sweeping [65], triangular [66], and other methods [67]. This study used the Successive Approximation (SA) method, which is based on the Gauss–Seidel technique, but it is different from it mainly in its capacity to handle the variables directly in their complex representation. Thanks to this characteristic, it is not necessary to perform different transformations, which, ultimately, would create more mathematical complexity and, therefore, pose more significant computational challenges [62]. 4.2.4. Proposed coding The problem addressed in this article—i.e., the optimal integration of DG sources into an EDS—requires a type of coding that includes discrete as well as continuous variables to represent the location of the generators and the amount of power that they should inject at each time interval, respectively. This study uses a discrete-continuous coding that is represented using a vector of size 1*(NG + (NI * NG)), where NG is Fig. 8. Proposed coding. the number of DG sources installed and NI is the number of time intervals (during which the active power injected by each generator should be established). An example of this coding is presented in Fig. 8, which shows a system with three DG sources (i.e., G1, G2, and G3). The power each one of them should inject is established in 13 one-hour intervals (i.e., the number of sunshine hours in the region under analysis). Therefore, the first three columns of the vector contain the locations of the generators. Columns 4 to 16 contain the power (in kW) that Generator 1 (G1) should inject during the 13 one-hour intervals. Then, columns 17 to 29 detail the power that G2 should inject. Finally, columns 30 to 42 contain the power that G3 should inject at those hours. The resulting vector is 1 by 42. Some aspects of the example presented in Fig. 8should be highlighted: Results in Engineering 24 (2024) 103059 9 J.A. Guzmán-Henao, R.I. Bolaños, B. Cortés-Caicedo et al. Table 2 Tuned parameters for the MOPSO algorithm. Parameter Range Value obtained Number of individuals [0–100] 83 Maximum number of iterations [0–1000] 728 Minimum inertia [0.1–0.5] 0.468 Maximum inertia [0.5–0.9] 0.762 Cognitive coefficient [0–2] 1.388 Social coefficient [0–2] 1.171 •G1 is located at Node 12, G2 at Node 8, and G3 at Node 19. •During the first hour, G1 injects 100.0 kW of active power; at Hour 13, it injects 0.3 kW. 4.2.5. Selecting the best solution from the Pareto front Each one of the optimization techniques used here provides a set of solutions grouped into an optimal Pareto front, as proposed in Section 2.1.3. This front offers minimum values in each one of the objective functions that compose the multi-objective optimization model. However, what is considered the best solution depends on the specific criterion applied in a specific optimization and the particular interests behind it. This study adopted the criterion explained and used in [68], that is, selecting a solution that is equidistant to the limits of each objective function as a solution to the problem. This method for selecting equidistant points was implemented here to identify an adequate solution that would enable a statistical analysis and an evaluation of the consistency of the solutions obtained. 4.2.6. Software to implement the algorithms The algorithms were developed and solved using code in MATLAB (R2022a release). This code was run on a Dell Precisión 3450 desktop computer with the following characteristics: an 8-core Intel®Core™ i9-11900 processor @ 2.5 GHz and 64 GB of RAM. Note that the programming of the three algorithms in MATLAB was optimized because the methodology utilized the eight cores available in the processor and MATLAB’s capacity for parallel processing of “for” loops using the “parfor” command. Parallel processing significantly reduced the processing times required to find a solution. 5. Results 5.1. Tuning the algorithms The master stage in the proposed methodology incorporated three algorithms: NSGA-II, MOPSO, and MOALO. Each one of them has specific parameters that play a crucial role in the results obtained during the optimization process. To correctly adjust these parameters in each one of the three techniques, this study implemented PSO, which significantly improved the results in the scenarios previously described. The parameters for the PSO algorithm were a population of 30 individuals, minimum inertia of 0.001, maximum inertia of 0.7, cognitive/social coefficients of 1.494, and maximum 100 iterations. These specific values were selected here because other authors that have used the PSO algorithm to tune optimization algorithms for electrical engineering problems have reported good results with them [69] [14][48]. 5.1.1. Tuning the MOPSO algorithm The iterative process of this optimization technique uses several parameters: number of individuals, maximum number of iterations, maximum inertia, minimum inertia, and cognitive and social coefficients. Table 2reports the parameters established after the tuning procedure, along with the respective ranges that were evaluated. Table 3 Tuned parameters for the NSGA-II. Parameter Range Value obtained Number of individuals [0–100] 63 Maximum number of iterations [0–1000] 888 Crossover probability [0.1–0.5] 0.451 Mutation probability [0.5–0.9] 0.042 Mutation intensity [0–2] 0.408 Table 4 Tuned parameters for the MOALO algorithm. Parameter Range Value obtained Number of individuals [0–100] 85 Maximum number of iterations [0–1000] 985 5.1.2. Tuning the NSGA-II The iterative process of the NSGA-II uses several parameters: number of individuals, maximum number of iterations, crossover probability, mutation probability, and mutation intensity. Table 3details the parameters tuned for this algorithm, as well as the respective ranges that were evaluated. 5.1.3. Tuning the MOALO algorithm Only two parameters can be tuned in the iterative process of the MOALO algorithm: number of individuals and maximum number of iterations. Table 4reports the values obtained after the tuning, along with the corresponding ranges of the parameters. 5.2. Operating limits of the test system before the installation of DG sources The mathematical model proposed here included an important operational constraint: a limit to the current flowing through the lines in the EDS. First, the 33-node and 27-node test systems were evaluated prior to the installation of DG sources, thereby obtaining current values for each line of the systems. Second—to implement a scenario that reflects typical generation and demand conditions in Medellín and Capurganá—the Colombian standards currently in force for selecting electrical conductors were applied in accordance with Table 310-16 in NTC 2050. Then, the highest allowable current in each line in the EDS was established as reported in Table 5. The maximum current in each line of both systems was evaluated for every operational hour. Since this study incorporates variable conditions throughout a typical day of operation, the next step was to integrate typical generation and demand curves in Medellín and Capurganá, as detailed in Section 3.3. These curves were used to adjust the test systems to variable demand and generation, which could change hourly over a typical 24-hour period. Another essential constraint in the proposed model is the allowable voltage levels for the nodes. As in the case of the current, the Colombian standard currently in force was applied to establish these levels. In accordance with this standard, the actual voltage of this kind of systems must be between -10% and +5% of their nominal voltage. Finally, this information was used to determine the initial operating limits for each system. That includes identifying the network segment with the highest current flow and the nodes that exhibited the highest and lowest voltages during the day of operation. These conditions were applied to evaluate the objective functions, each one individually, before the DG sources were installed. As a result, a base case was established to quantify the improvements obtained after DG sources were integrated and the proposed methodology was applied. In the base case of the urban scenario (33-node system), power losses amounted to 3,379.06 kWh and the operational cost of the system was 9,931.66 USD. For the rural scenario (27-node system), power losses were 691.14 kWh and the operational cost of the system was 18,544.01 USD. Results in Engineering 24 (2024) 103059 16 J.A. Guzmán-Henao, R.I. Bolaños, B. Cortés-Caicedo et al. power losses and operating costs, respectively. The mathematical model in this methodology is composed of a multi-objective function that covers both aspects and a set of constraints to accurately represent the operation of this kind of systems in an environment of variable power generation and demand. The multi-objective optimization procedure returns a set of non-dominated solutions that, in all the cases, improve the economic and technical characteristics of the EDS under examination. This front of optimal solutions—where the extreme points contain the lowest value of each objective function—enables network operators to employ different criteria to select the most adequate solution according to their particular needs. This methodology used a master–slave approach to integrate different algorithms, which can be useful because each technique has particular characteristics that have a considerable influence on the optimization process. Furthermore, this model facilitates the analysis of the operational impact of integrating DG sources in different contexts. MATLAB, which was used to implement the algorithms, was found to be flexible and easily adaptable to the proposed strategy in different scenarios. In addition, it reduces the dependency on specialized software. A statistical analysis was conducted to establish the repeatability and processing times of the solutions obtained. 6.2. Main results The proposed strategy generated the following results: •In general terms, the evaluation indicated that the implementation of DG resources has a better effect when the generators are located a considerable distance from the generation node and distributed among the multiple branches that compose the system. •Full power injection by the generators does not always produce the best results in the objective function. At different times, it was more convenient to inject less than that. •NSGA-II achieved the optimal values for the two objective functions on a non-dominated solution front in the 33-node test system. Within this front, the two extreme points corresponded to reductions of 32.84% in power losses and 42.41% in operating costs, respectively, compared to the base case. •NSGA-II achieved the optimal values for the two objective functions on a non-dominated solution front in the 27-node test system. Within this front, the two extreme points corresponded to reductions of 21.87% in power losses and 43.36% in operating costs, respectively, compared to the base case. •Although the MOPSO algorithm did not achieve the highest reductions in objective functions on the 33-node test system, it required only 27.70 seconds on average to perform the optimization, marking the shortest processing time in this study. Similarly, NSGA-II achieved an average time of 29.76 seconds among the solutions in the 27-node test system, representing the best time among the algorithms used in this system. •The three techniques employed here exhibited adequate standard deviations over the 500 evaluations. Among them, the NSGA-II stands out with only 0.21% and 0.39% standard deviations in power losses and operating costs, respectively. •The three techniques employed exhibited appropriate standard deviations throughout the 500 evaluations. Notably, NSGA-II demonstrated standard deviations of 0.21% and 0.39% in power losses and operating costs, respectively, in the 33-node test system. For the 27-node test system, the standard deviations were 0.07% and 0.24% for losses and costs, respectively. 6.3. Limitations and future work Although the optimization model of this study was validated in test scenarios, these strategies should be applied in real environments subject to varying generation and demand over longer periods. Future studies in this field should integrate aspects such as seasonality and economic models that determine electricity prices. This paper analyses the operating conditions of two test systems using a single-phase representation or an equivalent model, which facilitates some calculations. However, the actual operation of this type of electrical network includes unbalanced three-phase systems, which adds a certain degree of complexity to the analysis. Therefore, future research in this field could focus on the development of a multi-objective optimization model that includes these features. CRediT authorship contribution statement Jhony Andrés Guzmán-Henao: Writing – review & editing, Writing – original draft, Visualization, Validation, Supervision, Software, Resources, Project administration, Methodology, Investigation, Funding acquisition, Formal analysis, Data curation, Conceptualization. Rubén Iván Bolaños: Writing – review & editing, Writing – original draft, Visualization, Validation, Supervision, Software, Resources, Project administration, Methodology, Investigation, Funding acquisition, Formal analysis, Data curation, Conceptualization. Brandon Cortés-Caicedo: Writing – review & editing, Writing – original draft, Visualization, Validation, Supervision, Software, Resources, Project administration, Methodology, Investigation, Funding acquisition, Formal analysis, Data curation, Conceptualization. Luis Fernando GrisalesNoreña: Writing – review & editing, Writing – original draft, Visualization, Validation, Supervision, Software, Resources, Project administration, Methodology, Investigation, Funding acquisition, Formal analysis, Data curation, Conceptualization. Oscar Danilo Montoya: Writing – review & editing, Writing – original draft, Visualization, Validation, Supervision, Software, Resources, Project administration, Methodology, Investigation, Funding acquisition, Formal analysis, Data curation, Conceptualization. Jesús C. Hernández: Writing – review & editing, Writing – original draft, Visualization, Validation, Supervision, Software, Resources, Project administration, Methodology, Investigation, Funding acquisition, Formal analysis, Data curation, Conceptualization. Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgements This work was supported by the Council of Andalucía (Junta de Andalucía, Consejería de Transformación Económica, Industria, Conocimiento y Universidades, Secretaría General de Universidades, Investigación y Tecnología) under Project ProyExcel_00381. This research has been funded by the Ministry of Science, Technology and Innovation of Colombia (MINCIENCIAS) and the National Fund for Science, Technology, and Innovation (Fondo Francisco José de Caldas) through the support received for the realization of the research internship under Valorization Contingent Funding Contract No. 112721-358-2023, Call. 934 of 2023. Additionally, part of this research has been founded by the ANID/FONDECyT Iniciación 2024 Folio 11240006 under the project: “Smart energy management methods for improving the economic, technical, and environmental indexes of the alternating current microgrids including variable generation and demand profiles”, at the Engineering Faculty of Talca University, Campus Curicó, Chile. The authors acknowledge the support provided by Thematic Network 723RT0150 “Red para la integración a gran escala de energías renovables en sistemas eléctricos (RIBIERSE-CYTED)” funded by the 2022 Call for Thematic Networks by the Ibero-American Program of Science and Technology for Development (CYTED in Spanish). Finally, we would like to thank ITM Translation Agency (traducciones @itm .edu .co) for the translation of the manuscript into English. Results in Engineering 24 (2024) 103059 17 J.A. Guzmán-Henao, R.I. Bolaños, B. Cortés-Caicedo et al. 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