Hierarchical Optimization of DGs, BESSs, and D-STATCOMs for Reducing Energy Losses and CO2 Emissions in Unbalanced Off-Grid Networks

Contents lists available at ScienceDirect Results in Engineering journal homepage: www.sciencedirect.com/journal/results-in-engineering Research paper Hierarchical optimization of DGs, BESSs, and D-STATCOMs for reducing energy losses and CO2emissions in unbalanced off-grid networks Rubén Iván Bolañosa,e,, Jhony Andrés Guzmán-Henaoa, ,∗, Luis Fernando Grisales-Noreñab,e, Oscar Danilo Montoyac,, Jesús C. Hernándezd,e aFacultad de Ingenierías, Instituto Tecnológico Metropolitano, Campus Robledo, Medellín 050036, Colombia bGrupo de Investigación en Alta Tensión—GRALTA, Escuela de Ingeniería Eléctrica y Electrónica, Facultad de Ingeniería, Universidad del Valle, Cali 760015, Colombia cGrupo de Compatibilidad e Interferencia Electromagnética, Facultad de Ingeniería, Universidad Distrital Francisco José de Caldas, Bogotá 110231, Colombia dDepartment of Electrical Engineering, Universidad de Jaén, Campus Lagunillas s/n, Edificio A3, Jaén 23071, Spain ePrograma Doctorado: Computación Avanzada, Energía y Plasma, Dpto. Ingeniería Electrónica y de Computadores, Universidad de Córdoba, Edificio Leonardo Da Vinci, Campus de Rabanales, Ctra. N-IVa Km. 396, 14071 Córdoba, Spain A R T I C L E I N F O A B S T R A C T Keywords: Distributed energy resources CO2emission reduction Battery energy storage systems Distributed generation Distribution static compensators In developing countries, Non-Interconnected Zones often depend on costly, environmentally unsustainable, dieselbased power systems. These systems typically experience unbalanced load conditions, limited voltage regulation, and difficulty accommodating renewable energy sources. In this context, Distributed Energy Resources— including Distributed Generation (DG), Battery Energy Storage Systems (BESS), and Distribution Static Synchronous Compensators (D-STATCOMs)—represent a viable alternative, provided they are integrated and operated optimally under realistic technical and environmental constraints. This paper presents a methodology for the optimal integration and coordinated operation of DG, BESS, and D-STATCOMs in unbalanced three-phase distribution systems within NIZs. The approach is formulated as a bi-objective mixed-integer nonlinear programming problem to minimize annual energy losses and CO2emissions. To validate the proposed methodology, two benchmark distribution systems—one with 25 nodes and one with 37 nodes—were adapted to represent the operating conditions of Leticia and San Andrés Island (both in Colombia), using actual demand and solar irradiance data. Five metaheuristic algorithms—Whale Optimization Algorithm (WOA), Vortex Search Algorithm (VSA), Chu & Beasley Genetic Algorithm (CBGA), Multi-Verse Optimizer (MVO), and Black Widow Optimization Algorithm (BWOA)—were implemented and compared after 100 independent runs per case study. BWOA demonstrated the best performance in both scenarios: in Leticia, it reduced energy losses by 77.0164% and CO2 emissions by 61.1426%; in San Andrés, it reduced energy losses by 65.4489% and emissions by 59.9654%. All solutions met voltage and thermal constraints, confirming the technical feasibility and environmental benefits of the proposed strategy for DER integration in isolated and constrained distribution networks. Acronyms NIZ Non-Interconnected Zone DER Distributed Energy Resource DG Distributed Generation BESS Battery Energy Storage System D-STATCOM Distribution Static Synchronous Compensator MINLP Mixed-Integer Nonlinear Programming CO2Carbon Dioxide *Corresponding author. E-mail addresses: [email protected] (R.I. Bolaños), [email protected] (J.A. Guzmán-Henao), [email protected] (L.F. Grisales-Noreña), [email protected] (O.D. Montoya), [email protected] (J.C. Hernández). CBGA Chu & Beasley Genetic Algorithm VSA Vortex Search Algorithm BWOA Black Widow Optimization Algorithm EDS Electrical Distribution System ARO Artificial Rabbits Optimization QOBL Quasi-Opposition-Based Learning NSGA-II Non-Dominated Sorting Genetic Algorithm II SOC State of Charge DOD Depth of Discharge https://doi.org/10.1016/j.rineng.2025.107098 Received 14 July 2025; Received in revised form 24 August 2025; Accepted 2 September 2025 Results in Engineering 28 (2025) 107098 Available online 10 September 2025 2590-1230/© 2025 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY-NC license ( http://creativecommons.org/licenses/bync/4.0/ ). R.I. Bolaños, J.A. Guzmán-Henao, L.F. Grisales-Noreña et al. CO2Carbon Dioxide USD United States Dollar WOA Whale Optimization Algorithm MVO Multi-Verse Optimizer 1. Introduction 1.1. General context Access to reliable, sustainable, and environmentally responsible electricity remains a persistent challenge in many Non-Interconnected Zones (NIZs), especially in developing countries [1]. These regions usually have weak, isolated distribution systems and depend heavily on costly, polluting diesel-based power generation. Additionally, limited infrastructure and geographic barriers hinder the expansion of centralized grid systems [2]. In this context, Distributed Energy Resources (DERs)—including Distributed Generation (DG), Battery Energy Storage Systems (BESS), and Power Electronics Compensation Devices, such as Distribution Static Synchronous Compensators (D-STATCOMs)—have emerged as promising alternatives for improving the performance, reliability, and sustainability of local distribution networks [3]. However, effectively leveraging these technologies requires coordinated planning and operational strategies that address the technical and environmental constraints of isolated, unbalanced, three-phase distribution systems [4]. 1.2. Motivation The deployment of DERs in rural and isolated networks is a valuable opportunity to enhance technical performance and reduce environmental impact. This approach aligns with key Sustainable Development Goals, particularly Goal 7, which promotes access to affordable, reliable, sustainable, and modern energy for all, and Goal 13, which calls for urgent action to combat climate change and its impacts. Specifically, coordinating the operation of DG, BESS, and D-STATCOMs can lower energy losses, improve voltage regulation, and reduce CO2emissions [5,6]. However, most existing methodologies oversimplify the planning process by analyzing DER technologies in isolation or using single-phase equivalents, which fail to capture the operational characteristics of unbalanced networks. Moreover, the dynamic behavior of energy demand and renewable generation is often overlooked, limiting the practical applicability of these models in real-world scenarios. These limitations underscore the need for comprehensive, realistic methodologies that can integrate multiple DER technologies simultaneously, accounting for network asymmetries, solar resource variability, and time-dependent demand patterns. This study addresses this gap by proposing a robust strategy for jointly sizing, locating, and operationally dispatching DGs, BESSs, and D-STATCOMs in real distribution systems located in NIZs. 1.3. Literature review The integration of DERs—such as DG, BESS, and D-STATCOMs—has been widely studied due to their potential to improve the performance of Electrical Distribution Systems (EDSs) [7]. These technologies have demonstrated positive impacts on technical indicators, including reducing power loss, regulating voltage, and improving power quality, while supporting the energy transition by displacing fossil-fuel-based generation [8]. However, integrating DERs into EDSs presents challenges, especially in determining critical decision variables, such as nominal voltage, optimal placement, and the corresponding operational strategy for each device [9]. Various optimization techniques have been explored to address these challenges. Heuristic and metaheuristic algorithms have gained prominence due to their ability to efficiently solve nonlinear optimization problems with minimal computational effort [10–12]. Recent studies have investigated the optimal integration of one or more DER technologies under simplified conditions. For instance, Khalil et al. [13] proposed a hybrid algorithm that combines Artificial Rabbits Optimization (ARO) and Quasi-Opposition-Based Learning (QOBL) to determine the optimal sizing and location of photovoltaic DG and BESS units, with a focus on minimizing energy loss. They validated their methodology using the IEEE 33-bus test feeder modeled as a singlephase equivalent circuit. Similarly, in [14], an enhanced salp swarm algorithm was applied to optimize the placement of a DG unit. This approach aimed to reduce energy losses and operational costs while improving voltage profiles. Validation was performed using the IEEE 33and 69-bus radial distribution systems. Prasad et al. [15] investigated the coordinated allocation of DG units and reactive power compensation devices using the Non-Dominated Sorting Genetic Algorithm II (NSGA-II). They aimed to enhance the technical, economic, and environmental performance of electrical microgrids. Evaluation was conducted on single-phase equivalents of the IEEE 33and 69-bus systems. Another study employed a similar approach, using NSGA-II to optimize the capacity, location, and operational scheduling of BESS units in renewable-based distribution networks [16]. While these studies considered environmental and economic objectives, their system models were simplified by neglecting network unbalance and the dynamic variability of demand and generation. In addition, several recent studies have proposed efficient methods for the optimal planning of BESS in distribution systems. For example, in [17], a comparative analysis of metaheuristic algorithms was conducted to determine the optimal placement and capacity of BESS in networks with high penetration of photovoltaic generation and electric vehicles. In [18], the authors proposed a methodology for optimal siting and sizing of BESS from the perspective of Distribution System Operators (DSOs), incorporating both technical and regulatory criteria. Another recent study [19] introduced the Crayfish Optimization Algorithm to determine the optimal number, location, and capacity of multiple BESS units. These contributions offer valuable insights into the planning and operating specific DER combinations. However, they primarily rely on steady-state simulations and idealized test feeders, which do not capture the unbalanced, dynamic, and constrained conditions of real-world distribution systems. The simultaneous integration and coordinated operation of three distinct DER types—DG, BESS, and D-STATCOMs—under realistic technical constraints and environmental objectives has received limited attention in the literature. There is a clear need for methodologies that can jointly optimize the siting, sizing, and power injection or absorption levels of different DERs, while accurately representing the time-varying behavior of demand and renewable generation under the asymmetric and dynamic conditions of real, unbalanced, three-phase distribution systems. 1.4. Main contributions This study presents a technically robust and practically applicable methodology for the optimal integration and coordinated operation of multiple DERs—specifically, DG, BESS, and D-STATCOMs—in unbalanced three-phase distribution systems located in Non-Interconnected Zones (NIZs). Unlike previous works that often rely on simplified models using single-phase equivalents or address DER technologies in isolation, the proposed approach explicitly considers network asymmetries and the simultaneous interaction among different types of DERs under realistic operating conditions. A key novelty of this study lies in the joint resolution of sizing, siting, and operational dispatch problems for heterogeneous DERs using a unified bi-objective optimization model. This model minimizes (i) the active power losses in the distribution network and (ii) the total CO2emissions from diesel-based generation, incorporating detailed operational constraints and dynamic boundary conditions derived from real-world data. Results in Engineering 28 (2025) 107098 2 R.I. Bolaños, J.A. Guzmán-Henao, L.F. Grisales-Noreña et al. The main contributions of this study are summarized as follows: •Comprehensive methodology for the simultaneous integration and coordinated operation of DG, BESS, and D-STATCOMs, enabling the joint determination of their nominal capacities, network locations, and hourly operational profiles within realistic unbalanced threephase distribution systems. •Bi-objective mathematical formulation that incorporates both technical and environmental objectives, aiming to minimize active power losses and CO2emissions from diesel-based generation. The model includes operational constraints reflecting the physical and electrical limitations of both the network and DER technologies. •Development of two realistic case scenarios based on actual solar irradiance and load demand data from Leticia and San Andrés Island, capturing the operating conditions of non-interconnected distribution systems in Colombia. •Operational analysis to assess the impact of DER coordination on voltage regulation and power flow, quantifying the individual and combined technical contributions of each technology. •Statistical performance analysis based on multiple independent runs to evaluate the convergence, repeatability, and robustness of the optimization process under time-varying conditions. 1.5. Document structure The remainder of this paper is organized as follows. Section 2 presents the mathematical formulation for modeling the integration and operation of DERs in unbalanced three-phase distribution networks. Section 3details the proposed methodology, including the optimization algorithms and implementation procedure. Section 4discusses the case studies based on real data from Leticia and San Andrés Island and analyzes the technical and environmental outcomes. Finally, Section 5 summarizes the main conclusions and outlines future research directions. 2. Mathematical model for DER integration in NIZ distribution systems This study develops a mathematical model to determine the optimal integration and coordinated operation of DERs in unbalanced threephase distribution systems within NIZs. The model includes two objective functions—technical and environmental—and a set of constraints that capture the physical limits and operational behavior of the network and integrated DERs. 2.1. Objective functions The objective functions address two critical aspects of system performance: operational efficiency and environmental sustainability. These objectives are optimized independently and provide complementary perspectives on the impact of DERs in isolated systems. 2.1.1. Technical objective: minimization of energy losses The first objective function aims to minimize active power losses, a key indicator of efficiency in distribution systems with high DER penetration [20]. This objective is defined in Equation (1): min (Losses)=min⎛⎜⎜⎜⎝ real⎧ ⎪ ⎨ ⎪ ⎩∑ 𝑓∈⎛⎜⎜⎝∑ 𝑖∈ 𝑉3𝜑𝑖,𝑓 ⎛⎜⎜⎝∑ 𝑗∈ 𝑌3𝜑𝑖𝑗,𝑓 𝑉3𝜑𝑗,𝑓 ⎞⎟⎟⎠⎞⎟⎟⎠ ∗⎫ ⎪ ⎬ ⎪ ⎭ Δℎ⎞⎟⎟⎟⎠(1) In Equation (1), 𝑉3𝜑𝑖,𝑓 and 𝑉3𝜑𝑗,𝑓 represent the complex three-phase voltages at nodes 𝑖and 𝑗, respectively, for phase 𝑓. The term 𝑌3𝜑𝑖𝑗,𝑓 denotes the complex admittance of the line connecting these nodes in phase 𝑓. The set includes all network nodes, while represents the set of phases (typically 𝑎, 𝑏, and 𝑐in three-phase systems). The time step Δℎdefines the temporal resolution of the analysis and is assumed constant throughout each evaluation step. 2.1.2. Environmental objective: minimization of CO2emissions The second objective function seeks to reduce CO2emissions by minimizing the energy supplied by conventional diesel-based generators, thereby promoting the use of zero-emission renewable resources. This objective is defined in Equation (2): min (CO2Emissions)=min(𝐸𝐹𝐶𝐺 ∑ ℎ∈∑ 𝑖∈ 𝑝𝐶𝐺 3𝜑𝑖,ℎ Δℎ)(2) In Equation (2), 𝑝𝐶𝐺 3𝜑𝑖,ℎ represents the active power injected by conventional generation units at node 𝑖during time step ℎ, while 𝐸𝐹𝐶𝐺 denotes the emission factor of conventional technologies, expressed in kilograms of CO2per kilowatt-hour generated. The set defines the time horizon of the optimization problem, which contains all discrete time steps considered in the analysis, and Δℎis the duration of each step. 2.2. Set of constraints The proposed model incorporates a set of operational constraints that capture the nominal characteristics of the DERs and the specific conditions of unbalanced three-phase distribution systems in NIZs. These constraints ensure the technical feasibility of the solution and are organized according to the system elements they govern. 2.2.1. Active and reactive power balance at each node These constraints ensure that, at each node and for every time step, the active and reactive power supplied by conventional generators and DERs equals the power demanded by the loads. The formulation explicitly accounts for voltage magnitudes and phase angles at each node, as well as the magnitude and phase angle of the line admittances. The active and reactive power balance equations are provided in Equations (3) and (4), respectively: 𝑝𝐶𝐺 3𝜑𝑖,ℎ +𝑝𝐷𝐺 3𝜑𝑖,ℎ +∑ 𝐵𝐴𝑇 ∈ 𝑝𝐵𝐴𝑇 3𝜑𝑖,ℎ −𝑃𝑑 3𝜑𝑖,ℎ = 𝑉3𝜑𝑖,ℎ ∑ 𝑗∈ 𝑌3𝜑𝑖𝑗 𝑉3𝜑𝑗,ℎ cos (𝜃3𝜑𝑖,ℎ −𝜃3𝜑𝑗,ℎ −𝜑3𝜑𝑖𝑗 ),{∀𝑖∈ ∀ℎ∈}(3) 𝑞𝐶𝐺 3𝜑𝑖,ℎ +𝑞𝐷𝑆𝑇 3𝜑𝑖,ℎ −𝑄𝑑 3𝜑𝑖,ℎ = 𝑉3𝜑𝑖,ℎ ∑ 𝑗∈ 𝑌3𝜑𝑖𝑗 𝑉3𝜑𝑗,ℎ sin (𝜃3𝜑𝑖,ℎ −𝜃3𝜑𝑗,ℎ −𝜑3𝜑𝑖𝑗 ),{∀𝑖∈ ∀ℎ∈}(4) In Equation (3) and Equation (4), 𝑃𝑑 3𝜑𝑖,ℎ and 𝑄𝑑 3𝜑𝑖,ℎ denote the active and reactive power demanded at node 𝑖during time step ℎ, while 𝑞𝐶𝐺 3𝜑𝑖,ℎ represents the reactive power supplied by conventional generators. The term 𝑝𝐷𝐺 3𝜑𝑖,ℎ refers to the active power delivered by DG units, and 𝑝𝐵𝐴𝑇 3𝜑𝑖,ℎ corresponds to the active power injected or absorbed by BESS. Additionally, 𝑞𝐷𝑆𝑇 3𝜑𝑖,ℎ indicates the reactive power contribution from D-STATCOMs. The voltages 𝑉3𝜑𝑖,ℎ and 𝑉3𝜑𝑗,ℎ represent the three-phase voltage magnitudes at nodes 𝑖and 𝑗, with phase angles 𝜃3𝜑𝑖,ℎ and 𝜃3𝜑𝑗,ℎ , respectively. The admittance between these nodes is denoted by 𝑌3𝜑𝑖𝑗 with a phase angle of 𝜑3𝜑𝑖𝑗 . Finally, represents the set of BESS integrated into the distribution network. 2.2.2. Limits on active and reactive power from conventional generators These constraints ensure that the active and reactive power outputs of conventional generators remain within their nominal capacities. Additionally, to reflect the operational characteristics of isolated systems, conventional generators are prevented from absorbing power by setting Results in Engineering 28 (2025) 107098 3 R.I. Bolaños, J.A. Guzmán-Henao, L.F. Grisales-Noreña et al. the lower limits to zero. The corresponding constraints are shown in Equations (5) and (6): 𝑃𝐶𝐺,min 3𝜑𝑖 ≤𝑝𝐶𝐺 3𝜑𝑖,ℎ ≤𝑃𝐶𝐺,max 3𝜑𝑖,(5) 𝑄𝐶𝐺,min 3𝜑𝑖 ≤𝑞𝐶𝐺 3𝜑𝑖,ℎ ≤𝑄𝐶𝐺,max 3𝜑𝑖.(6) In Equation (5) and Equation (6), 𝑃𝐶𝐺,min 3𝜑𝑖and 𝑃𝐶𝐺,max 3𝜑𝑖represent the minimum and maximum allowable active power outputs of the conventional generator at node 𝑖, while 𝑄𝐶𝐺,min 3𝜑𝑖and 𝑄𝐶𝐺,max 3𝜑𝑖indicate the corresponding reactive power limits. 2.2.3. Active power limits for DG units and batteries This model considers the contributions of DG units and batteries exclusively in terms of active power. The allowable active power ranges for these devices are forced through the constraints defined in Equations (7) and (8): 𝑃𝐷𝐺,min 3𝜑𝑖 ≤𝑝𝐷𝐺 3𝜑𝑖,ℎ ≤𝑃𝐷𝐺,max 3𝜑𝑖𝐶𝐷𝐺 ℎ,(7) 𝑃𝐵𝐴𝑇 ,min 3𝜑𝑖 ≤𝑝𝐵𝐴𝑇 3𝜑𝑖,ℎ ≤𝑃𝐵𝐴𝑇 ,max 3𝜑𝑖.(8) In these expressions, 𝑃𝐷𝐺,min 3𝜑𝑖and 𝑃𝐷𝐺,max 3𝜑𝑖specify the minimum and maximum active power limits of the DG unit at node 𝑖, while 𝐶𝐷𝐺 ℎ represents the renewable generation profile at time step ℎ. Similarly, 𝑃𝐵𝐴𝑇 ,min 3𝜑𝑖and 𝑃𝐵𝐴𝑇,max 3𝜑𝑖define the active power bounds for each battery unit. 2.2.4. Reactive power limits for D-STATCOMs The reactive power output of each D-STATCOM is limited by its nominal capacity. The upper and lower bounds are enforced by the inequality presented in Equation (9): 𝑄𝐷𝑆𝑇 ,min 3𝜑𝑖 ≤𝑞𝐷𝑆𝑇 3𝜑𝑖,ℎ ≤𝑄𝐷𝑆𝑇 ,max 3𝜑𝑖.(9) Here, 𝑄𝐷𝑆𝑇 ,min 3𝜑𝑖and 𝑄𝐷𝑆𝑇 ,max 3𝜑𝑖represent the minimum and maximum allowable reactive power outputs for the D-STATCOM installed at node 𝑖. 2.2.5. Voltage limits at network nodes To ensure safe and reliable operation, voltage magnitudes at all network nodes must remain within specified operational limits. These constraints are enforced using the inequality shown in Equation (10): 𝑉min 3𝜑𝑖 ≤𝑣3𝜑𝑖,ℎ ≤𝑉max 3𝜑𝑖.(10) In Equation (10), 𝑉min 3𝜑𝑖and 𝑉max 3𝜑𝑖denote the minimum and maximum allowable three-phase voltage magnitudes at node 𝑖, while 𝑣3𝜑𝑖,ℎ represents the actual voltage magnitude at node 𝑖during time step ℎ. 2.2.6. Current limits on distribution lines The integration of DERs modifies current flows within the network. To maintain system security and ensure that conductors operate within thermal limits, the constraint in Equation (11) imposes an upper bound on the line current: 𝐼3𝜑𝑙,ℎ ≤𝐼max 3𝜑𝑙,(11) where 𝐼3𝜑𝑙,ℎ is the current magnitude through line 𝑙during time step ℎ, and 𝐼max 3𝜑𝑙is the corresponding thermal current limit of the line. The set includes all lines in the distribution network. 2.2.7. State of charge and power exchange management in batteries The active power injected or absorbed by a battery affects its State of Charge (SOC), which evolves according to the linear relationship described in Equation (12): 𝑆𝑂𝐶𝐵𝐴𝑇 𝑖,ℎ =𝑆𝑂𝐶𝐵𝐴𝑇 𝑖,ℎ−1 −𝜑𝐵𝐴𝑇 𝑖𝑝𝐵𝐴𝑇 3𝜑𝑖,ℎ Δℎ. (12) In this Equation, 𝑆𝑂𝐶𝐵𝐴𝑇 𝑖,ℎ is the SOC of the battery at node 𝑖during time step ℎ, 𝜑𝐵𝐴𝑇 𝑖is a scaling factor that relates active power exchange to SOC variation, and Δℎis the duration of the time step. The coefficient 𝜑𝐵𝐴𝑇 𝑖is calculated based on the rated power of the battery, as defined in Equation (13): 𝜑𝐵𝐴𝑇 𝑖=1 𝑃𝐵𝐴𝑇 3𝜑𝑖 ,(13) where 𝑃𝐵𝐴𝑇 3𝜑𝑖is the nominal three-phase power rating of the battery at node 𝑖. 2.2.8. Initial and final SOC conditions Following best practices reported in the literature [21–23], the initial and final SOC of each battery is set to 50% of its nominal capacity. These conditions are specified in Equations (14) and (15): 𝑆𝑂𝐶𝐵𝐴𝑇 𝑖,ℎ𝑖=𝑆𝑂𝐶𝐵𝐴𝑇,initial 𝑖,(14) 𝑆𝑂𝐶𝐵𝐴𝑇 𝑖,ℎ𝑓=𝑆𝑂𝐶𝐵𝐴𝑇,final 𝑖.(15) Here, 𝑆𝑂𝐶𝐵𝐴𝑇 𝑖,ℎ𝑖and 𝑆𝑂𝐶𝐵𝐴𝑇 𝑖,ℎ𝑓denote the SOC at the beginning and end of the operating horizon, while 𝑆𝑂𝐶𝐵𝐴𝑇 ,initial 𝑖and 𝑆𝑂𝐶𝐵𝐴𝑇,final 𝑖are the reference values, which are usually set to 50% of capacity. The indices ℎ𝑖and ℎ𝑓represent the initial and final time steps of the simulation horizon, respectively. 2.2.9. Maximum charge and discharge power per battery (16) and (17) define the maximum charging and discharging power for each battery based on its nominal capacity and characteristic charge/discharge durations: 𝑃𝐵𝐴𝑇 ,charge 3𝜑𝑖=− 𝑃𝐵𝐴𝑇 3𝜑𝑖 𝑡𝑐𝐵𝐴𝑇 𝑖 ,(16) 𝑃𝐵𝐴𝑇 ,discharge 3𝜑𝑖= 𝑃𝐵𝐴𝑇 3𝜑𝑖 𝑡𝑑𝐵𝐴𝑇 𝑖 .(17) In these expressions, 𝑡𝑐𝐵𝐴𝑇 𝑖and 𝑡𝑑𝐵𝐴𝑇 𝑖represent the characteristic charge and discharge times for the battery at node 𝑖. During charging, the power exchange is negative and limited by 𝑃𝐵𝐴𝑇,charge 3𝜑𝑖, while during discharging, the power is positive and limited by 𝑃𝐵𝐴𝑇,discharge 3𝜑𝑖. These constraints ensure that the charging and discharging rates align with the nominal capacity and operational time constraints of each battery unit. 2.2.10. SOC operating limits To prevent overcharging or deep discharging, the SOC of each battery must remain within specified operational limits, enforced through the inequality in Equation (18): 𝑆𝑂𝐶𝐵𝐴𝑇,min 𝑖≤𝑆𝑂𝐶𝐵𝐴𝑇 𝑖,ℎ ≤𝑆𝑂𝐶𝐵𝐴𝑇,max 𝑖,(18) where 𝑆𝑂𝐶𝐵𝐴𝑇,min 𝑖and 𝑆𝑂𝐶𝐵𝐴𝑇,max 𝑖are typically set to 10% and 90% of the nominal capacity of the battery, respectively. 3. Proposed methodology for the optimal integration and operation of DERs in EDSs of NIZs This study proposes a leader–follower optimization methodology for the optimal integration and operation of multiple DER types in EDSs located in NIZs. Specifically, the approach considers the coordinated deployment of DG sources, BESS, and D-STATCOMs. The addressed problem is highly complex due to the nonlinear behavior of electrical variables in unbalanced three-phase EDSs and the stochastic nature of several DER operational parameters. Furthermore, Results in Engineering 28 (2025) 107098 4 R.I. Bolaños, J.A. Guzmán-Henao, L.F. Grisales-Noreña et al. Fig. 1. Operational flow of the leader–follower optimization methodology. the decision variables exhibit mixed mathematical characteristics: the nominal capacities and power injection or absorption levels of the DERs are modeled as continuous variables, while their location decisions are represented as discrete variables. The proposed methodology simultaneously optimizes the sizing, siting, and operational scheduling of each DER technology to minimize energy losses and reduce CO2emissions, thereby improving the technical and environmental performance of the system. The leader–follower structure enables an iterative optimization process in which the leader generates candidate solutions for the decision variables, while the follower evaluates these solutions by computing the objective functions and verifying compliance with operational constraints. Based on this feedback, the leader updates the candidate solutions using a metaheuristic optimization algorithm. This iterative cycle continues until a predefined convergence criterion is met. The operational flow of the proposed methodology is shown in Fig. 1. 3.1. Leader stage At this stage, a metaheuristic optimization technique is employed to explore the solution space and identify optimal configurations for DER integration and operation. Metaheuristics are particularly well-suited for addressing nonlinear, non-convex, and computationally intensive problems involving mixed discrete and continuous variables [24]. In this study, five metaheuristic algorithms are considered: the Chu & Beasley Genetic Algorithm (CBGA), the Vortex Search Algorithm (VSA), the Black Widow Optimization Algorithm (BWOA), the Multi-Verse Optimizer (MVO), and the Whale Optimization Algorithm (WOA). Each algorithm is adapted to generate decision variable values and interact with the follower stage in search of optimal solutions. 3.1.1. Chu & Beasley genetic algorithm The CBGA is a variant of the classical genetic algorithm specifically designed for constrained combinatorial optimization problems [25]. Its application in electrical engineering has been widely documented, which demonstrates its robustness, fast convergence, and ability to produce high-quality solutions [26–28]. The algorithm operates through an iterative evolutionary process that begins with the generation of a random initial population of candidate solutions within the defined variable bounds. Each individual is evaluated using the objective function to determine its fitness, and the best initial solution is identified. A tournament selection mechanism is applied to select parents with superior fitness, increasing the likelihood of passing desirable traits to the next generation. Offspring are generated by recombining the genetic information of selected parents using a crossover operator, while a mutation operator introduces random variations to maintain genetic diversity and prevent premature convergence. An elitism-based selection strategy is then employed to form the next generation by retaining the best individuals from the previous generation and complementing the population with the most promising offspring. Throughout the evolutionary process, the algorithm continuously updates the best global solution whenever an improvement is found. The search process terminates after completing a predefined number of generations, returning the best solution identified along with its corresponding fitness value. The pseudocode for the CBGA is presented in Algorithm 1. Algorithm 1: Pseudocode of the Chu & Beasley genetic algorithm. Input: Number of variables (dimension), population size (population), number of generations (generations), lower and upper variable bounds, crossover probability (crossover), mutation probability (mutation). Output: Best solution found (best_solution) and its fitness value (best_fitness). 1Generate a random initial population within the defined boundaries. 2Evaluate the objective function of each individual in the population. 3Identify the best solution in the initial population. 4for 𝑡=1to generations do 5Select pairs of individuals using tournament selection. 6Apply crossover to generate offspring with the specified crossover probability. 7Apply mutation to offspring with the specified mutation probability. 8Form the next generation using elitism. 9Update the best solution if an improved candidate is found. 10 Return: the best solution identified and its corresponding fitness value. 3.1.2. Vortex search algorithm The VSA is inspired by the fluid dynamics of vortex formation and dissipation and operates by updating candidate solutions through a rotational movements around the current best solution [29]. The algorithm begins by generating a random initial population within the defined variable bounds, evaluating their objective function values, and identifying the best-performing candidate. A search radius, based on the variable range, is defined to set the scope of local exploration. In each iteration, new candidate solutions are generated by perturbing the current best solution within the search radius, and these candidates are evaluated. If a new candidate outperforms the current solution, it replaces it in the population. The global best solution is updated whenever a superior candidate is found. To ensure convergence, the search radius is progressively reduced using a predefined reduction factor, gradually narrowing the search around the best-known region. This process continues until the maximum number of iterations is reached or a convergence criterion is met. The VSA effectively balances diversification and intensification, which makes it well-suited for solving highly nonlinear, non-convex problems such as the optimal integration and operation of DERs in EDSs of NIZs. Its application in related studies has demonstrated robustness Results in Engineering 28 (2025) 107098 5 R.I. Bolaños, J.A. Guzmán-Henao, L.F. Grisales-Noreña et al. and computational efficiency [30–32]. The pseudocode of the VSA is presented in Algorithm 2. Algorithm 2: Pseudocode of the vortex search optimization algorithm. Input: Problem dimension (dimension), number of iterations (iterations), population size (population), lower bounds (lower_bound) and upper bounds (upper_bound). Output: Best solution found (best_solution) and its fitness value (best_fitness). 1Generate an random initial population within the variable bounds. 2Evaluate the objective function for each solution. 3Identify the best initial solution and its fitness value. 4Set the initial search radius as the maximum variable range. 5Define the radius reduction factor (reduction_factor). 6for 𝑡=1to iterations do 7for 𝑖=1to population do 8Generate a new solution by randomly perturbing the current best solution within the search radius. 9Ensure the solution remains within the variable bounds. 10 Evaluate its fitness value. 11 if the new fitness is better than the current solution then 12 Replace the current solution with the new candidate. 13 Identify the best solution in the population. 14 if its fitness is better than the global best then 15 Update the global best solution and its fitness value. 16 Multiply the search radius with the reduction_factor. 17 if the convergence criterion is met then 18 Terminate the algorithm. 19 Return: the best solution found and its fitness value. 3.1.3. Black widow optimization algorithm The BWOA is a nature-inspired metaheuristic based on the reproductive and survival behaviors of black widow spiders [33]. It emulates biological processes such as mating, cannibalism, and competitive selection to enhance solution quality across iterations. The algorithm begins by generating a random initial population within the defined variable bounds, followed by evaluating and ranking individuals based on their fitness to identify the best initial solution. During the reproduction phase, new candidate solutions are created by crossing pairs of the fittest individuals. To maintain population quality and prevent the propagation of weak solutions, a portion of the least fit offspring is removed through a process analogous to cannibalism. A mutation mechanism is then applied to a subset of the offspring, introducing random variations that enhance search space exploration. The next generation is formed by combining the strongest individuals from the current population with the surviving offspring, retaining the best solutions after sorting. Whenever a superior global solution is identified, it is updated to guide the search of the algorithm. This iterative process continues until a predefined termination criterion is met. Due to its aggressive selection strategy and rapid convergence towards high-quality solutions, the BWOA has been successfully applied to complex optimization problems with large and constrained search spaces, including the optimal planning and operation of EDSs with integrated DERs [34–36]. The pseudocode of the BWOA is presented in Algorithm 3. 3.1.4. Multi-verse optimizer The Multi-Verse Optimizer (MVO) was originally introduced by Mirjalili et al. [37] as a nature-inspired metaheuristic based on three concepts from cosmology: white holes, black holes, and wormholes. These mechanisms are mathematically modeled to guide the exploration and exploitation of the search space. For this study, the improved version of the algorithm proposed in [38] was implemented, as it provides a Algorithm 3: Pseudocode of the black widow optimization algorithm. Input: Population size (population), number of variables (dimension), maximum number of iterations (iterations), lower and upper variable bounds, cannibalism rate (cannibalism), mutation rate (mutation). Output: Best solution found (best_solution) and its fitness value (best_fitness). 1Generate a random initial population within the variable bounds. 2Evaluate the objective function of each individual. 3Sort the population by fitness and select the best initial solution. 4for 𝑡=1to iterations do 5for 𝑖=1to population - cannibalism do 6Select two parents from the top-ranked individuals. 7Generate a new offspring by crossing the parents. 8Remove a percentage of the weakest offspring. 9for each offspring individual do 10 if mutation probability is met then 11 Apply a random mutation to the solution. 12 Combine the offspring with the top individuals from the current population. 13 Sort the combined population and select the best individuals for the next generation. 14 if a better global solution is found then 15 Update the best solution. 16 Return: the best solution found and its fitness value. more effective balance between global and local search. The MVO has been successfully applied in electrical engineering and energy-related optimization tasks, as reported by several studies [39,40]. The algorithm begins with a set of universes (candidate solutions) randomly generated within the feasible region. The exchange of information between universes is controlled by the white hole and black hole operators, allowing better solutions to share their features with others according to their fitness values. The wormhole mechanism perturbs solutions around the best universe found, enhancing local search and intensification. Two adaptive parameters guide the search process: the exploration probability, which increases linearly from a minimum to a maximum value across iterations, and the traveling distance rate, which decreases dynamically to refine the search in later iterations. This mechanism ensures a progressive transition from exploration to exploitation. The process continues until a predefined number of iterations is reached, returning the best solution and its corresponding fitness value. The pseudocode for the MVO is presented in Algorithm 4. 3.1.5. Whale optimization algorithm The Whale Optimization Algorithm (WOA) was first reported by Mirjalili and Lewis [41], inspired by the bubble-net hunting strategy of humpback whales. Its mathematical model captures both the encircling behavior around prey and the spiral updating position strategy. In this study, the enhanced version of the WOA presented in [42] was adopted, which improves the convergence speed and global search capability of the algorithm. The effectiveness of WOA in electrical engineering optimization has been confirmed by several recent applications [43,44]. The algorithm starts by initializing a population of whales (candidate solutions) randomly distributed within the feasible region. During each iteration, the whales update their positions either by shrinking encircling movements toward the current best solution or by performing spiral trajectories to simulate bubble-net hunting. A control parameter governs the probability of switching between these two mechanisms, thus balancing exploration and exploitation. Additionally, whales may perform a random search with respect to other individuals in the population, which helps to avoid premature convergence. This iterative process continues until the termination criterion is reached, yielding the best soResults in Engineering 28 (2025) 107098 6 R.I. Bolaños, J.A. Guzmán-Henao, L.F. Grisales-Noreña et al. Algorithm 4: Pseudocode of the Multi-Verse Optimizer. Input: Number of variables (dimension), number of universes (population), maximum iterations (iterations), lower and upper variable bounds, minimum exploration probability (WEP_min), maximum exploration probability (WEP_max), traveling distance rate parameter (p). Output: Best solution found (best_solution) and its fitness value (best_fitness). 1Generate an initial population of universes randomly within the defined boundaries. 2Evaluate the objective function of each universe. 3Identify the best initial universe. 4for 𝑡=1to iterations do 5Calculate the wormhole existence probability (WEP) as a linear function of iteration between WEP_min and WEP_max. 6Calculate the traveling distance rate (TDR) as a nonlinear decreasing function governed by parameter p. 7Apply white hole and black hole operators to exchange information between universes according to fitness. 8Apply wormhole search around the best universe with probability WEP and distance controlled by TDR. 9Update the best universe if an improved candidate is found. 10 Return: the best solution identified and its corresponding fitness value. lution and its corresponding fitness value. The pseudocode for the WOA is presented in Algorithm 5. Algorithm 5: Pseudocode of the Whale Optimization Algorithm. Input: Number of variables (dimension), population size (population), maximum iterations (iterations), lower and upper variable bounds. Output: Best solution found (best_solution) and its fitness value (best_fitness). 1Generate an initial population of whales randomly within the defined boundaries. 2Evaluate the objective function of each whale. 3Identify the best initial whale. 4for 𝑡=1to iterations do 5Update whales’ positions using encircling or spiral updating mechanisms. 6Switch between exploration and exploitation according to the adaptive parameter. 7Perform random search with respect to other whales when necessary. 8Update the best solution if an improved whale is found. 9Return: the best solution identified and its corresponding fitness value. 3.1.6. Tuning of algorithms used in the leader stage The leader-stage optimization process employs three metaheuristic algorithms: CBGA, VSA, BWOA, MVO and WOA. Each algorithm requires the configuration of control parameters that directly impact convergence behavior and solution quality. Given the sensitivity of the optimization results to these parameters, a dedicated tuning procedure was conducted to ensure robust performance under the proposed formulation. For this purpose, Particle Swarm Optimization (PSO) was implemented as a meta-optimizer to calibrate the control parameters of each algorithm. The PSO was configured with a swarm size of 30 particles, a linearly decreasing inertia weight from 0.7 to 0.001, and cognitive and social acceleration coefficients set to 1.494. The tuning process was executed over a maximum of 100 iterations. This configuration was selected based on prior studies that have effectively applied PSO for parameter tuning in power system optimization problems [45–48]. The parameter search intervals were defined based on literature recommendations and preliminary testing. Table 1summarizes the explored parameter ranges and the resulting tuned values for each algorithm. These tuned parameters were subsequently applied during the execution of the proposed methodology. 3.2. Follower stage As established in the leader–follower scheme, the leader stage generates candidate solutions, which the follower stage then evaluates. This evaluation involves verifying compliance with the operational constraints of the system and calculating the impact of the proposed solution on the objective functions. To perform this assessment, the three-phase Successive Approximations Method (SAM) was used to solve the power flow in unbalanced EDSs. This method iteratively estimates the threephase voltages at the demand nodes until a predefined convergence criterion is met. A key advantage of AST is its ability to operate directly with complex variables, avoiding the need for additional transformations that would otherwise increase mathematical complexity. The recursive voltage update equation is provided in Equation (19). 𝕍𝑡+1 𝑑3𝜑,ℎ =−𝕐−1 𝑑𝑑3𝜑(𝕐𝑑𝑔3𝜑𝕍𝑔3𝜑,ℎ −𝕀𝑡 𝑑3𝜑,ℎ)(19) In this equation, 𝕍𝑡+1 𝑑3𝜑,ℎ denotes the vector of three-phase voltages at the demand nodes during time step ℎat iteration 𝑡+1. The matrix 𝕐𝑑𝑑3𝜑is the submatrix of the three-phase admittance matrix that relates demand nodes to each other, while 𝕐𝑑𝑔3𝜑connects demand nodes with generation nodes. The term 𝕍𝑔3𝜑,ℎ represents the three-phase voltages at generation nodes for the same time step, and 𝕀𝑡 𝑑3𝜑,ℎ is the vector of complex three-phase currents injected at the demand nodes at iteration 𝑡. Once the voltage values are calculated, the operating conditions of the system are evaluated to verify compliance with the optimization model constraints, including voltage and current limits. If any constraint is violated, the proposed solution is penalized using a fitness function that includes a penalty term 𝛽. This penalization discourages the selection of infeasible solutions while still allowing their consideration in subsequent iterations, which may lead to improved feasible alternatives. The penalized fitness function is defined in Equation (20). 𝐹𝑓𝑖𝑡𝑛𝑒𝑠𝑠 =Objective Function +𝛽 ⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝ +max{0,𝕍𝑑3𝜑,ℎ −𝑉max 3𝜑𝑖} −min{0,𝕍𝑑3𝜑,ℎ −𝑉min 3𝜑𝑖} +max{0,𝕀𝑖𝑗3𝜑,ℎ −𝐼max 3𝜑𝑖𝑗 } −min{0,𝑝 𝐶𝐺 3𝜑𝑖,ℎ −𝑃𝐶𝐺,min 3𝜑𝑖} +max{0,SOC𝐵𝐴𝑇 𝑖,ℎ −SOC 𝐵𝐴𝑇 ,max 𝑖} −min{0,SOC𝐵𝐴𝑇 𝑖,ℎ −SOC 𝐵𝐴𝑇 ,min 𝑖} ⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ (20) In this expression, 𝕍𝑑3𝜑,ℎ represents the vector of three-phase voltages at the demand nodes, while 𝑉min 3𝜑𝑖and 𝑉max 3𝜑𝑖indicate the lower and upper voltage limits at node 𝑖, respectively. The term 𝕀𝑖𝑗3𝜑,ℎ denotes the three-phase current through the line connecting nodes 𝑖and 𝑗at time ℎ, with 𝐼max 3𝜑𝑖𝑗 representing its upper limit. The variable 𝑝𝐶𝐺 3𝜑𝑖,ℎ corresponds to the active power injected by conventional generation at node 𝑖and time ℎ, while 𝑃𝐶𝐺,min 3𝜑𝑖defines its lower operating limit. Similarly, SOC𝐵𝐴𝑇 𝑖,ℎ denotes the state of charge of the BESS at node 𝑖and time ℎ, constrained within the permissible range of operation defined by SOC𝐵𝐴𝑇,min 𝑖and SOC𝐵𝐴𝑇 ,max 𝑖. The penalty coefficient 𝛽is set to 1×10 6, a value selected heuristically based on sensitivity analyses and supported by previous studies. Voltage limits adhere to regional regulatory standards, while Results in Engineering 28 (2025) 107098 7 R.I. Bolaños, J.A. Guzmán-Henao, L.F. Grisales-Noreña et al. Table 1 Tuned parameters for CBGA, VSA, BWOA, MVO, and WOA. Parameter CBGA VSA BWOA MVO WOA Range Tuned value Range Tuned value Range Tuned value Range Tuned value Range Tuned value Number of individuals [1–1000] 752 [1–1000] 751 [0–2000] 1429 [0–1000] 345 [0–2000] 1083 Maximum iterations [1–200000] 179981 [1–1000] 213 [0–2000] 1508 [0–1000] 822 [0–500] 122 Search radius reduction factor – – [0–1] 0.9638 – – – – – – Cannibalism rate – – – – [0–1] 0.2498 – – – – Mutation probability – – – – [0–1] 0.4882 – – – – Minimum exploration probability – – – – – – [0–1] 0.1467 – – Maximum exploration probability – – – – – – [0–1] 0.9638 – – Traveling distance rate (p) – – – – – – [0–30] 27.8558 – – Spiral constant – – – – – – – – [0–3] 1.5236 current limits are determined by the physical and thermal characteristics of the distribution conductors. Finally, Algorithm 6presents the implementation steps for the successive approximations method used to solve the unbalanced threephase power flow within the proposed leader–follower optimization framework. Algorithm 6: Implementation of the successive approximations method for unbalanced three-phase power flow. Input: Configuration of the unbalanced three-phase distribution system, three-phase admittance matrix, per-phase load powers, initial voltages and angles at each node, algorithm parameters (maximum iterations, convergence tolerance, and the set of time periods for analysis). Output: Three-phase voltage values at each node and the computed value of the fitness function. 1Collect system data and calculate the three-phase admittance matrix. 2Register initial node voltages and angles. 3Set algorithm parameters, including the number of iterations, convergence threshold, and time horizon. 4for each hour ℎin a 24-hour period do 5Initialize the voltage profile for hour ℎ. 6Execute the iterative process until convergence or the iteration limit is reached. 7Calculate load currents based on connection type (star or delta). 8Update voltage values using Equation (19). 9Evaluate the error relative to the previous iteration. 10 Store the voltage values for hour ℎ. 11 Calculate the fitness function, applying penalties as needed. 12 return the voltage profiles and the fitness value. 3.3. Selection of DERs and coding scheme for the optimization process This study addresses the integration and optimal operation of three types of DERs in the EDSs of NIZs: photovoltaic-based DG units, lithiumion BESS, and D-STATCOMs. These technologies were chosen due to their increasing relevance in the literature and their complementary roles in enhancing technical, economic, and environmental system performance [49–55]. While various reactive power compensation devices are available in power systems, D-STATCOMs were selected for this study due to their proven effectiveness in improving voltage profiles in distribution networks, particularly under unbalanced conditions [56]. Compared to other FACT devices such as Static Var Compensators (SVCs) and Unified Power Flow Controllers (UPFCs), D-STATCOMs offer faster response times, finer reactive power control, and phase-specific compensation, making them especially suitable for radial, low-voltage feeders as commonly found in NIZs [57,58]. Although UPFCs allow broader control over power flow, their high cost and implementation complexity often limit their use [59]. SVCs, on the other hand, present slower dynamic behavior and are less effective under fast-changing load conditions [60]. DSTATCOMs, despite being limited to reactive power support, contribute significantly to voltage stability when integrated with complementary technologies such as DGs and BESSs. This combination enables a coordinated control strategy that addresses both active and reactive power challenges in distribution systems with high penetration of DERs. To maintain methodological consistency, three units of each DER type were considered in the analysis. The selection of three DG units was based on the cost-benefit analysis by Montoya et al. [61], which demonstrated that this configuration significantly reduces power losses while avoiding the diminishing returns associated with adding more units. For BESS, previous studies recommend installing an equal number of storage units as DG units to enable coordinated energy management and enhance grid flexibility [22,62]. Similarly, the choice of three DSTATCOMs aligns with prior research showing that this configuration effectively improves voltage profiles and system stability [63,64]. Nominal power limits for each device were defined according to regulatory and technical guidelines. For DG units, the maximum capacity was set at 5 MW in line with Colombian regulation CREG 030 of 2018 [65]. The same limit was adopted for BESS units to ensure consistency in power management strategies [66]. For D-STATCOMs, a nominal capacity of 500 kvar was selected, consistent with values reported in similar studies [63,67]. To encode candidate solutions within the optimization process, a discrete–continuous hybrid coding scheme was implemented using a 1 × 204 dimensional vector. This structure reflects the variables associated with DER integration and operation, including device locations, nominal power ratings, and temporal operation profiles across the planning horizon. Specifically, the first 45 positions of the vector correspond to the DG units: positions 1–3 define the nominal power of each unit, positions 4–6 specify their locations within the EDS, and positions 7–45 capture hourly active power injections over the 13 time periods with solar generation potential. The next segment (positions 46–126) represents the BESS: positions 46–48 store their nominal power ratings, positions 49–51 indicate their locations, and positions 52–126 capture the hourly SOC evolution over a 24-hour cycle, including the initial SOC. Finally, positions 127–204 correspond to the D-STATCOMs: positions 127–129 define their nominal reactive power ratings, positions 130–132 specify their locations, and positions 133–204 record their hourly reactive power injection or absorption over the 24-hour period. Fig. 2illustrates the structure of the solution vector, detailing the segments assigned to each DER type and specifying the nature of the encoded variables. 3.4. Software tool used for algorithm implementation The proposed optimization algorithms were implemented and executed using original code developed in MATLAB®, Academic Version R2024b. All simulations were conducted on a Dell Precision 3450 workResults in Engineering 28 (2025) 107098 8 R.I. Bolaños, J.A. Guzmán-Henao, L.F. Grisales-Noreña et al. Fig. 2. Structure of the solution vector, showing segment allocation by DERs type and variable classification. station equipped with an 11th-generation Intel®CoreTM i9-11900 processor (8 cores at 2.5 GHz) and 64 GB of RAM. This configuration provides sufficient computational capacity to handle the complexity and dimensionality of the leader–follower optimization methodology. 3.5. Selection of implementation scenarios To validate the proposed methodology for the integration and optimal operation of DERs, two NIZs in Colombia were selected as case studies: Leticia, located in the Amazon region, and San Andrés Island, situated in the Caribbean Sea. Both regions face critical challenges due to their reliance on diesel-based generation, which results in high operational costs, vulnerability to fuel logistics, and significant environmental impacts. However, these zones also exhibit considerable potential for the deployment of distributed energy technologies, which makes them suitable for evaluating the applicability and benefits of the proposed approach. The selection of these scenarios was guided by the need to analyze diverse operating conditions. On one hand, Leticia is characterized by lower energy demand and significant logistical and infrastructure limitations. On the other hand, San Andrés Island presents a more complex load profile with higher peak demand and pronounced daily variations. These differences allow the robustness of the methodology to be tested under varying technical and environmental conditions while reflecting the real challenges faced by NIZs in Colombia. Due to the limited availability of detailed electrical data in these regions—a common issue in NIZs—representative radial test systems were adopted to emulate the behavior of their distribution networks. Specifically, a 25-node system was assigned to Leticia, while a 37-node system was used for San Andrés Island, which reflects the relative size and estimated energy demand of each region. This approach ensures proportional representation of their distribution infrastructure, providing a relevant and realistic basis for assessing the performance of the proposed optimization strategy under conditions consistent with actual field constraints. 3.5.1. Leticia, Amazonas Leticia, the capital of Amazonas department, is located in the southernmost region of Colombia at the tri-border area with Brazil and Peru. According to the 2018 National Population and Housing Census by DANE, Leticia has an estimated population of 48,144 and serves as the principal administrative, commercial, and urban center of the Colombian Amazon. The local economy is driven primarily by ecotourism and cross-border trade, leveraging its strategic location and rich biodiversity. Fig. 3. Graphical representation of the electrical layout of the 25-node unbalanced three-phase test system. However, the region’s power system faces critical limitations. Electricity supply in Leticia relies entirely on diesel-based generation, with fuel transported via the Amazon River, which results in high operational costs, limited supply reliability, and significant environmental impacts on an ecologically sensitive area [68,69]. In this context, the integration of DERs—specifically photovoltaicbased DG units, BESS, and D-STATCOMs—presents an opportunity to improve energy reliability, reduce generation costs, and mitigate environmental impacts. To represent the electrical characteristics of the distribution network in Leticia, the three-phase 25-node radial test system was employed. Originally proposed by Ramana et al. [70], this system has been widely used in the literature for validating DER planning and operational methodologies [71,72]. The 25-node system operates at a nominal voltage of 4.16 kV and comprises 24 distribution lines. It features unbalanced loads with a total demand of 946 kW and 648 kvar on phase A, 573.6 kW and 430.6 kvar on phase B, and 771.8 kW and 554 kvar on phase C. The detailed electrical parameters of lines and loads—including line impedances, load profiles, and node configurations—were obtained from [71], ensuring consistency with previous studies and enabling realistic simulation of network behavior under DERs integration. The electrical layout of the adopted test system is shown in Fig. 3. 3.5.2. San Andrés Island San Andrés is an insular territory located in the Caribbean Sea, approximately 700 km from mainland Colombia. According to the 2018 National Population and Housing Census by DANE, the island is home to roughly 61,280 inhabitants. Its economy relies heavily on tourism and commerce, industries that demand a reliable and cost-effective electricity supply. Currently, the region depends entirely on diesel-based generation, with fuel transported by sea. This dependency leads to high electricity production costs, vulnerability to supply disruptions, and significant environmental impacts in a resource-constrained and ecologically sensitive setting [73,74]. In this context, the integration of DERs into the local power system offers a strategic opportunity to diversify the energy mix, reduce operational costs, and lower greenhouse gas emissions. Broadly, the deployment of DERs can support the sustainable development of the island by improving the performance of critical industries, such as tourism, which is particularly sensitive to energy availability and environmental quality. To emulate the electrical characteristics of San Andrés’ distribution network, this study employs the IEEE 37-node unbalanced three-phase test system. Originally based on a real distribution feeder documented in [75] and later adapted by Granada-Echeverri et al. [76], the system retains its radial configuration while incorporating the detailed load and conductor specifications for planning and operational analysis. The test system consists of 37 nodes and 35 lines, supplying 25 loads with unbalanced demand profiles. It operates at a nominal voltage of Results in Engineering 28 (2025) 107098 9 R.I. Bolaños, J.A. Guzmán-Henao, L.F. Grisales-Noreña et al. Fig. 13. Location node (𝑁) and nominal power values (𝑃, 𝑄) assigned to each device by BWOA for the environmental objective in Leticia. Table 11 Hourly operation profile of DG units (G1, G2, and G3) and D-STATCOMs (D1, D2, and D3) delivered by the best environmental solution in Leticia. Hour DG Power Output (kW) D-STATCOM Reactive Power Output (kvar) G1 G2 G3 D1 D2 D3 1 0.0000 0.0000 0.0000 205.3025 237.6199 220.3251 2 0.0000 0.0000 0.0000 191.6371 179.9452 205.3706 3 0.0000 0.0000 0.0000 221.0254 232.5826 192.5992 4 0.0000 0.0000 0.0000 156.8819 168.9566 191.9782 5 0.0000 0.0000 0.0000 179.3562 195.0592 252.0538 6 4.1214 4.0256 6.0330 162.5460 185.3768 253.2922 7 258.3040 241.8158 376.6510 165.1657 224.7703 200.8034 8 458.8170 563.0324 764.2132 215.7098 184.0150 254.4090 9 676.3378 615.0737 1019.0123 211.8286 215.6884 241.8470 10 693.4731 844.9977 849.9080 221.2023 237.7312 211.5041 11 828.1845 735.9220 879.9965 191.9289 174.2531 201.4248 12 650.5899 752.1053 904.7419 190.5683 191.7523 247.3184 13 738.0794 753.3628 932.0212 175.7645 192.4161 208.3170 14 734.3246 789.1203 860.4414 203.2381 197.5552 198.1329 15 639.5252 779.1365 896.7905 220.0247 185.4388 247.6957 16 554.6347 659.7750 717.3551 217.9544 231.7571 249.3921 17 440.3084 329.4480 568.2313 180.2627 209.3190 233.9215 18 32.8593 29.3245 44.5243 183.4356 223.8040 238.9695 19 0.0000 0.0000 0.0000 168.8416 186.8566 186.2738 20 0.0000 0.0000 0.0000 189.1511 212.0131 189.6712 21 0.0000 0.0000 0.0000 175.6752 231.3489 228.8488 22 0.0000 0.0000 0.0000 167.9108 183.1012 188.6271 23 0.0000 0.0000 0.0000 216.1560 181.9164 235.3643 24 0.0000 0.0000 0.0000 164.4879 191.1104 252.4095 Fig. 14. Maximum generation potential and active power injection of DG units G1, G2, and G3 proposed by BWOA for the environmental objective in Leticia. Meanwhile, the WOA reached the lowest reductions (63.6339% for energy losses and 39.4913% for emissions) but proved to be the fastest algorithm, with runtimes close to 200 seconds per solution, and the most stable performance in the environmental objective. Overall, all five algorithms produced high-quality, repeatable solutions, confirming the robustness of the proposed methodology. For long-term planning applications where maximizing technical and environmental performance is critical, BWOA stands out as the most effective approach, while WOA and CBGA offer attractive trade-offs when computational efficiency is prioritized. Fig. 15. Reactive power injection by D-STATCOM units D1, D2, and D3 proposed by BWOA for the environmental objective in Leticia. 4.2. Implementation of the methodology in the San Andrés test scenario The application of the proposed leader–follower optimization framework to the San Andrés test scenario offers a complementary perspective on the effectiveness of DER integration in non-interconnected power systems. Compared to Leticia, the San Andrés network exhibits a distinct load profile and topological configuration, which directly affect DER siting, sizing, and operational strategies. This section analyzes the results Results in Engineering 28 (2025) 107098 16 R.I. Bolaños, J.A. Guzmán-Henao, L.F. Grisales-Noreña et al. Table 12 SOC and power output of BESS units for the environmental objective in Leticia. Interval SOC [p.u.] Power Output [kW] B1 B2 B3 B1 B2 B3 1 0.5000 0.5000 0.5000 2490.5000 2500.0000 1625.0000 2 0.3096 0.6083 0.3111 1540.5091 3041.4422 1011.1890 3 0.3278 0.4100 0.3143 1633.2227 2050.0219 1021.9133 4 0.1840 0.4607 0.2131 915.5842 2303.6444 692.6389 5 0.2044 0.2705 0.2201 1017.9892 1352.3269 715.1832 6 0.1520 0.1587 0.2537 757.3837 793.5286 824.5193 7 0.1000 0.1000 0.2027 498.1000 500.0000 659.8159 8 0.1753 0.1033 0.1000 872.7807 516.4461 325.0000 9 0.1611 0.2774 0.1123 802.3526 1387.2479 365.1032 10 0.2400 0.4016 0.1604 1195.9984 2007.7653 521.4100 11 0.2755 0.5800 0.2122 1372.9706 2899.8605 690.1169 12 0.4592 0.6217 0.2283 2286.3216 3108.3546 742.5752 13 0.5310 0.7058 0.3328 2645.6480 3528.9122 1081.7330 14 0.4945 0.8187 0.5686 2461.7294 4093.7214 1843.5181 15 0.5628 0.9000 0.7060 2801.3360 4500.0000 2294.5867 16 0.7336 0.8862 0.8066 3654.1028 4431.2039 2621.4607 17 0.8095 0.9000 0.9000 4029.5426 4500.0000 2925.0000 18 0.9000 0.9000 0.8153 4482.9000 4500.0000 2659.1347 19 0.8702 0.7934 0.7111 4333.9672 3967.9584 2311.9938 20 0.9000 0.6342 0.6479 4482.9000 3171.0669 2104.9194 21 0.8090 0.4536 0.7406 4027.7710 2267.8927 2406.8434 22 0.7719 0.4353 0.8278 3844.6930 2176.4842 2690.3117 23 0.5751 0.5433 0.9000 2863.2281 2716.4976 2925.0000 24 0.4398 0.6445 0.6500 2189.7590 3222.4799 2112.5000 25 0.5000 0.5000 0.5000 2490.5000 2500.0000 1625.0000 Table 13 Performance summary of WOA, VSA, CBGA, MVO, and BWOA for technical and environmental objectives in Leticia. Technical objective function Algorithm Baseline energy losses (kW) Optimized energy losses (kW) Reduction (%) Avg. time per solution (s) Relative Std. Dev. WOA 251197.9100 91350.8332 63.6339 198.1670 0.0993 VSA 251197.9100 83783.5805 66.6463 285.8830 0.1422 CBGA 251197.9100 76808.1910 69.4232 266.8781 0.0411 MVO 251197.9100 75095.0521 70.1052 592.2412 0.0884 BWOA 251197.9100 57734.2719 77.0164 2672.0248 0.0657 Environmental objective function Algorithm Baseline emissions (kg CO2) Optimized emissions (kg CO2) Reduction (%) Avg. time per solution (s) Relative Std. Dev. WOA 3625229.3312 2193579.1511 39.4913 199.5651 0.0412 VSA 3625229.3312 2006822.9784 44.6428 291.6325 0.0544 CBGA 3625229.3312 1869197.8382 48.4391 262.3496 0.0567 MVO 3625229.3312 1963160.4916 45.8473 929.3892 0.0788 BWOA 3625229.3312 1408667.4079 61.1426 2804.4987 0.0573 Fig. 16. Hourly SOC profile proposed by BWOA for the environmental objective in Leticia. for both technical and environmental objectives, emphasizing key differences in system behavior and optimization outcomes. 4.2.1. Performance of optimization algorithms in San Andrés Tables 14 and 15 summarize the performance of the five metaheuristic algorithms in minimizing annual energy losses and CO2emissions in the San Andrés test system. As in the Leticia case, the BWOA achieved the most significant improvements, reducing energy losses by 65.4489% and CO2emissions by 59.9654%. These outcomes highlight its strong exploratory capability, although they required the highest computational effort, with average runtimes above 3300 s per solution. While all five algorithms achieved significant improvements in both objective functions, each exhibited distinct advantages. The WOA consistently delivered the fastest execution times, whereas the CBGA achieved the lowest variability across runs, ensuring stable convergence. The VSA provided balanced performance between quality of solutions and runtime, while the MVO produced competitive reductions at the cost of longer runtimes. Finally, the BWOA consistently achieved the greatest reductions in both energy losses and CO2emissions, albeit with the highest computational demand. These performance trends are visually supported by the boxplots in Figs. 18 and 19, which illustrate the statistical distribution of outcomes for energy losses and emissions, respectively. 4.2.2. Operational behavior of DERs under the BWOA solutions For both the technical and environmental objectives, the topperforming solutions obtained via BWOA were selected for detailed Results in Engineering 28 (2025) 107098 17 R.I. Bolaños, J.A. Guzmán-Henao, L.F. Grisales-Noreña et al. Fig. 17. Active power dispatch of conventional generators before and after applying the BWOA solution for the environmental objective in Leticia. Table 14 Comparison of optimization techniques in terms of energy loss reduction, runtime, and consistency in San Andrés. Algorithm Baseline Losses (kW) Best Solution (kW) Reduction (%) Avg. Time (s) Relative Std. Dev. WOA 387870.0243 176839.3879 54.4076 215.2985 0.0925 VSA 387870.0243 175642.8623 54.7160 328.6940 0.0858 CBGA 387870.0243 155384.6824 59.9389 371.5789 0.0347 MVO 387870.0243 164437.1347 57.6051 698.5774 0.0652 BWOA 387870.0243 134013.1437 65.4489 3342.9084 0.0446 Table 15 Comparison of optimization techniques in terms of CO2emissions, runtime, and consistency in San Andrés. Algorithm Baseline Emissions (kg CO2) Best Solution Reduction (%) Avg. Time (s) Relative Std. Dev. WOA 4731862.2151 2752451.8001 41.8315 205.7002 0.0553 VSA 4731862.2151 2359021.1604 50.1460 333.6773 0.0833 CBGA 4731862.2151 2562139.5057 45.8534 374.4212 0.0342 MVO 4731862.2151 2559110.3957 45.9175 1005.0494 0.0404 BWOA 4731862.2151 1894382.0221 59.9654 3360.2052 0.0467 Fig. 18. Statistical distribution of energy losses (kW) for WOA, VSA, CBGA, MVO, and BWOA in San Andrés. Fig. 19. Statistical distribution of CO2emissions (kg) for WOA, VSA, CBGA, MVO, and BWOA in San Andrés. operational analysis. The decision variables, including DER location and nominal power ratings, are illustrated in Figs. 20 and 21. In the technical configuration, nodes 18 and 34 were identifies as critical deployment points, hosting both DG and BESS units. This localized reinforcement strategy effectively reduced long-distance power transfers, thereby minimizing line losses. Only one BESS unit reached its maximum rated capacity, while the D-STATCOMs operated at average reactive power levels exceeding 300 kvar. In contrast, the environmental configuration concentrated all three DER types at node 10, indicating a localized support strategy designed to maximize renewable energy utilization and minimize emissions. In this layout, BESS units were consistently sized at their maximum allowable power, and D-STATCOMs operated at an average nominal rating of 138 kvar. The hourly operation profiles under the BWOA solutions are illustrated in Figs. 22, 23, and 24 for the technical objective, and in Figs. 25, 26, and 27 for the environmental objective. These figures depict DG active power injection, D-STATCOM reactive power dispatch, and BESS SOC profiles over a representative 24-hour period. For the technical objective, DG units (Fig. 22) tracked the regional solar irradiance profile, injecting power during peak solar hours without exceeding nominal capacities. This approach reflects a moderate dispatch strategy to minimize energy losses while avoiding voltage violations or overloading the network. D-STATCOMs reactive power dispatch (Fig. 23) varied significantly throughout the day, with highest demands at nodes 2, 18, and 28 during Results in Engineering 28 (2025) 107098 18 R.I. Bolaños, J.A. Guzmán-Henao, L.F. Grisales-Noreña et al. Fig. 20. Node locations (𝑁) and nominal power values (𝑃, 𝑄) assigned to DERs by BWOA for the technical objective in San Andrés. Fig. 21. Node locations (𝑁) and nominal power values (𝑃, 𝑄) assigned to DERs by BWOA for the environmental objective in San Andrés. low-generation intervals, reflecting localized voltage regulation requirements. BESS SOC profiles (Fig. 24) exhibited charging from 6 to hour 17, when photovoltaic generation was available, and discharging overnight, thereby reducing the load on conventional generators. All BESS units complied with the specified constraints and returned to 50% SOC at the end of the cycle, as mandated by the optimization model. For the environmental objective, the DG units (Fig. 25) showed slightly higher dispatch levels than in the technical case, especially units G2 and G3. Although their outputs remained below nominal capacity, they sustained levels for longer periods, which reflects a stronger emphasis on displacing fossil-fuel-based generation with renewable sources. As shown in Fig. 26, the D-STATCOMs exhibited smoother and less variable power injection than in the technical case. Devices at nodes 10 and 19 maintained moderate, stable reactive power injections, sugFig. 22. Active power injection by DG units under the technical objective in San Andrés. Fig. 23. Reactive power injection by D-STATCOMs under the technical objective in San Andrés. Fig. 24. Hourly SOC profiles of BESS units under the technical objective in San Andrés. Fig. 25. Active power output of DG units under the environmental objective in San Andrés. gesting a preventive voltage regulation strategy that supports emission reduction by stabilizing the network under higher renewable conditions. The BESS SOC profiles (Fig. 27) indicate more intensive cycling than in the technical case. Batteries charged intensively during daylight hours and discharged more deeply overnight. This enhanced utilization supports the reduction of CO2emissions by integrating renewable energy in the total energy mix during peak demand periods. 4.2.3. Operational feasibility and system impact The feasibility of the BWOA-based solutions for both objectives was assessed by examining key operational indicators, including voltage proResults in Engineering 28 (2025) 107098 19 R.I. Bolaños, J.A. Guzmán-Henao, L.F. Grisales-Noreña et al. Fig. 26. Reactive power injection by D-STATCOMs under the environmental objective in San Andrés. Fig. 27. Hourly SOC profiles of BESS units under the environmental objective in San Andrés. files, line loading, and the active power output of the conventional generator. Under the technical objective, the minimum voltage was 0.9736 p.u. at node 21, phase C, during hour 24, while the maximum voltage reached 1.0260 p.u. at node 22, phase B, during hour 15. The highest line loading was observed on line 17, phase B, at hour 16, reaching 80.4733%. These results confirm full compliance with voltage and current constraints. Fig. 28 shows the active power injection of the conventional generator before and after DER integration. The minimum generation constraint (strictly positive values) was maintained at all times. A significant decrease in generator dispatch was observed, particularly during peak solar hours, contributing directly to the reduction in energy losses. For the environmental objective, voltage levels ranged from a minimum of 0.9554 p.u. at node 19, phase A, at hour 24, to a maximum of 1.0467 p.u. at node 22, phase B, at hour 13. The highest line loading— 98.9834%—was also observed on line 17, phase B, at hour 13. Although this configuration imposed greater operational stress, all parameters remained within the allowable operating limits defined by the model. Fig. 29 illustrates the active power dispatch of the conventional generator under the environmental objective. A significant reduction in power output is evident, particularly between hours 9 and 17, driven by the coordinated operation of DG units, BESS, and D-STATCOMs. This synergistic behavior was instrumental in achieving the 59.9654% reduction in CO2emissions with the BWOA-based solution. 4.2.4. Complementary economic assessment in San Andrés Although the main objective was to optimize technical and environmental criteria, the sizing and operation of DG, BESS, and D-STATCOMs also entailed economic effects. A complementary assessment was therefore carried out for the San Andrés system using (i) the best solution vector obtained when minimizing losses and (ii) the best solution vector obtained when minimizing emissions, to verify that total annualized cost decreases with respect to the base case even when economics were not the optimization target. As shown in Equation (21), the total annualized cost was computed as the sum of three components: 𝑓1(purchase of electricity from the conventional grid), 𝑓2(operation and maintenance of the distributed energy resources), and 𝑓3(initial investment): Costs =𝑓1+𝑓2+𝑓3.(21) Energy purchases. To compute 𝑓1, Equation (22)was used: 𝑓1=𝐶𝑘𝑊 ℎ 𝑇 𝐴𝑎𝐴𝑐(∑ ℎ∈∑ 𝑖∈∑ 𝑝∈ 𝑝𝐶𝐺 3𝜑𝑖,ℎ Δℎ).(22) Here, 𝐶𝑘𝑊 ℎ denotes the average grid purchase price, 𝑇the number of days per year, 𝑝𝐶𝐺 3𝜑𝑖,ℎ the active power from conventional generation at node 𝑖and hour ℎ, and Δℎthe time step; , , and are the sets of hours, nodes, and phases. The factors 𝐴𝑎(annuity) and 𝐴𝑐(tariff escalation) were computed with Equations (23)–(24), where 𝑡𝑎is the internal rate of return, 𝑁𝑡the planning horizon, 𝑡𝑒the annual price growth, and the set of years: 𝐴𝑎=𝑡𝑎 1−(1+𝑡𝑎)−𝑁𝑡 ,(23) 𝐴𝑐=∑ 𝑡∈(1+𝑡𝑒 1+𝑡𝑎)𝑡 .(24) Operation and maintenance. To compute 𝑓2, Equation (25)was used: 𝑓2= 𝑇∑ ℎ∈∑ 𝑖∈∑ 𝑝∈(𝑝𝐷𝐺 3𝜑𝑖,ℎ 𝐶𝐷𝐺 𝑂&𝑀+|||𝑝𝐵𝐴𝑇 3𝜑𝑖,ℎ |||𝐶𝐵𝐴𝑇 𝑂&𝑀+𝑞𝐷𝑆𝑇 3𝜑𝑖,ℎ 𝐶𝐷𝑆𝑇 𝑂&𝑀)Δℎ. (25) In this expression, 𝐶𝐷𝐺 𝑂&𝑀, 𝐶𝐵𝐴𝑇 𝑂&𝑀, and 𝐶𝐷𝑆𝑇 𝑂&𝑀are the unit O&M costs of DG, BESS, and D-STATCOMs; 𝑝𝐷𝐺 3𝜑𝑖,ℎ , 𝑝𝐵𝐴𝑇 3𝜑𝑖,ℎ , and 𝑞𝐷𝑆𝑇 3𝜑𝑖,ℎ are their hourly active/reactive injections. Annualized investments. To compute 𝑓3, Equation (26)was used: 𝑓3= 𝐴𝑎∑ 𝑖∈∑ 𝑝∈(𝐶DG 𝑃𝐷𝐺 3𝜑𝑖+𝐶BAT 𝑃𝐵𝐴𝑇 3𝜑𝑖) +𝛾∑ 𝑖∈∑ 𝑝∈ 𝑄DST 3𝜑𝑖(𝜔1(𝑄DST 3𝜑𝑖)2+𝜔2𝑄DST 3𝜑𝑖+𝜔3).(26) Here, 𝑃𝐷𝐺, 𝑃𝐵𝐴𝑇 , and 𝑄𝐷𝑆𝑇 are the installed capacities of DG, BESS, and D-STATCOMs; 𝐶DG and 𝐶BAT are their unit investment costs; and 𝛾,𝜔1,𝜔 2,𝜔 3parameterize the D-STATCOM cost curve. Parameterization and sources. To compute the total system cost, the following aspects were considered: •Average energy price 𝐶𝑘𝑊 ℎ =0.2912 USD/kWh for diesel supply (SUI [83]), consistent with [26,46,62]. Annualization used 𝑇= 365 days/year and Δℎ=24h; hourly tariff variability followed the per–unit profile in [84]. •Financial factors in (23)–(24): 𝑁𝑡=20years and 𝑡𝑎= 10% yielded 𝐴𝑎=0.11746; 𝑡𝑒=2%yielded 𝐴𝑐=9.9338 [21]. •O&M unit costs: 𝐶𝐷𝐺 𝑂&𝑀=0.0019 USD/kWh and 𝐶𝐵𝐴𝑇 𝑂&𝑀= 0.0017 USD/kWh [21]; 𝐶𝐷𝑆𝑇 𝑂&𝑀=0.0068 USD/kvar [85]. •Investment parameters: 𝐶DG = 1036.4900 USD/kW⋅h [32]; 𝐶BAT = 47.9351 USD/kW⋅h and 𝛾=0.0500 [64]; D-STATCOM cost curve with 𝜔1=0.0003, 𝜔2=0.3051, 𝜔3= 127.3800 (USD/kvar terms) from [86]. Findings. In San Andrés, the base–case annual cost was 5,402,367.9561 USD. Evaluating the best loss–oriented solution yielded 3,664,289.5228 USD (a reduction of 1,738,078.4333 USD; 32.1725%). Evaluating the best emission–oriented solution yielded 3,258,259.0682 USD (a reduction of 2,144,108.8879 USD; 39.6883%). These results indicate that total annualized cost decreased even when the optimization objective was non–economic, because energy purchases, O&M, and investment were explicitly accounted for. Results in Engineering 28 (2025) 107098 20 R.I. Bolaños, J.A. Guzmán-Henao, L.F. Grisales-Noreña et al. Fig. 28. Active power output of the conventional generator before and after implementing the BWOA-based solution for the technical objective in San Andrés. Fig. 29. Active power output of the conventional generator before and after implementing the BWOA-based solution for the environmental objective in San Andrés. Table 16 Performance summary of WOA, VSA, CBGA, MVO, and BWOA for technical and environmental objectives in San Andrés. Technical objective Algorithm Baseline losses (kW) Best losses (kW) Reduction (%) Avg. time (s) Rel. Std. Dev. WOA 387870.0243 176839.3879 54.4076 215.2985 0.0925 VSA 387870.0243 175642.8623 54.7160 328.6940 0.0858 CBGA 387870.0243 155384.6824 59.9389 371.5789 0.0347 MVO 387870.0243 164437.1347 57.6051 698.5774 0.0652 BWOA 387870.0243 134013.1437 65.4489 3342.9084 0.0446 Environmental objective Algorithm Baseline emissions (kg CO2) Best emissions (kg CO2) Reduction (%) Avg. time (s) Rel. Std. Dev. WOA 4731862.2151 2752451.8001 41.8315 205.7002 0.0553 VSA 4731862.2151 2359021.1604 50.1460 333.6773 0.0833 CBGA 4731862.2151 2562139.5057 45.8534 374.4212 0.0342 MVO 4731862.2151 2559110.3957 45.9175 1005.0494 0.0404 BWOA 4731862.2151 1894382.0221 59.9654 3360.2052 0.0467 4.2.5. Synthesis of results in San Andrés The comparative performance of the five algorithms—WOA, VSA, CBGA, MVO, and BWOA—in the San Andrés test system is summarized in Table 16. Consistent with the Leticia case, the BWOA achieved the largest reductions in both annual energy losses and CO2emissions, albeit with the longest average runtime. In contrast, the WOA delivered the shortest execution times, the CBGA exhibited the lowest variability across runs, the VSA offered a balanced trade-off between solution quality and runtime, and the MVO produced competitive reductions at a higher computational cost. Under the technical objective, the BWOA delivered the largest improvement, reducing annual energy losses from 387 870.0243 kW to 134 013.1437 kW (65.4489%). The next best results were obtained by the CBGA and MVO, with reductions of 59.9389% and 57.6051%, respectively, while the VSA and WOA achieved 54.7160% and 54.4076%. This ranking is mirrored by the computational effort: the WOA was the fastest method (average 215.2985 s), followed by the VSA (328.6940 s) and CBGA (371.5789 s), whereas the MVO (698.5774 s) and especially the BWOA (3342.9084 s) required longer runtimes. In terms of repeatability, the CBGA showed the lowest relative standard deviation (0.0347), indicating stable convergence across runs. For the environmental objective, the BWOA again provided the best outcome, decreasing annual CO2emissions from 4 731 862.2151 kg to 1 894 382.0221 kg (59.9654%). The VSA attained the second-highest reduction (50.1460%), followed by the MVO (45.9175%) and CBGA (45.8534%), whereas the WOA yielded 41.8315%. The runtime pattern remained consistent with the technical objective: the WOA was the fastest (205.7002 s) and the BWOA the slowest (3360.2052 s). Variability was low for all methods (relative standard deviations below 0.10), with the CBGA again exhibiting the smallest dispersion (0.0342). Overall, the five algorithms produced high-quality and repeatable solutions under San Andrés conditions. When maximizing technical and Results in Engineering 28 (2025) 107098 21 R.I. Bolaños, J.A. Guzmán-Henao, L.F. Grisales-Noreña et al. environmental performance is the priority, the BWOA stands out despite its computational cost. When computational efficiency or repeatability is critical, the WOA and CBGA become attractive alternatives, while the VSA offers a balanced compromise—particularly for the environmental objective—and the MVO provides competitive reductions at the expense of longer runtimes. Additionally, a complementary economic assessment was conducted for San Andrés to quantify the cost implications of the proposed deployments. Using the annualized total cost definition in Equation (21)—which aggregates energy purchases, operation and maintenance, and initial investments—the base-case annual cost of 5,402,367.9561 USD decreased to 3,664,289.5228 USD when evaluating the best loss-oriented design (a saving of 1,738,078.4333 USD; 32.1725%) and to 3,258,259.0682 USD when evaluating the best emission-oriented design (a saving of 2,144,108.8879 USD; 39.6883%). These results show that, even when the optimization target was noneconomic, the optimized siting, sizing, and operation of DG, BESS, and D-STATCOMs produced meaningful reductions in total annualized cost, reinforcing the practical viability of the methodology for isolated systems such as San Andrés. 5. Conclusions and future work This study presented a comprehensive and technically robust methodology for the optimal integration and coordinated operation of DG, BESS, and D-STATCOMs in unbalanced three-phase distribution networks located in NIZs. The proposed leader–follower optimization framework minimizes annual energy losses and CO2emissions by optimizing the siting, sizing, and hourly dispatch of DERs under realistic operating conditions. Two real-world test cases—Leticia and San Andrés Island—were evaluated using actual hourly demand and solar irradiance profiles. Results confirmed the effectiveness of the methodology in improving system performance while reducing environmental impact. In Leticia, the BWOA-based solution reduced annual energy losses from 251,197.9100 kW to 57,734.2719 kW (a 77.0164% reduction) and CO2emissions from 3,625,229.3312 kg to 1,408,667.4079 kg (a 61.1426% reduction). In San Andrés, losses decreased from 387,870.0243 kW to 134,013.1437 kW (a 65.4489% reduction), and emissions were reduced from 4,731,862.2151 kg to 1,894,382.0221 kg (a 59.9654% reduction). In both scenarios, the DER configurations respected all voltage and thermal constraints, with voltage magnitudes ranging from 0.9554 to 1.0467 p.u. and line loadings remaining below 98.9834%. Coordinated DG, BESS, and D-STATCOM operation produced clear synergistic effects, particularly under dynamic and unbalanced conditions. DG units injected active power during solar peak hours, BESS units performed charge–discharge cycles to balance temporal mismatches, and D-STATCOMs provided fast, phase-specific reactive support to maintain voltage stability. This coordination was especially evident in nodes with high unbalance and reactive demand. Although both DG and BESS units contribute to reducing energy losses and CO2emissions by injecting active power and managing demand over time, their capability to regulate voltage is limited. D-STATCOMs provide fast and precise reactive power compensation, which plays a key role in maintaining voltage stability and improving power quality, particularly under unbalanced or dynamic operating conditions. While DGs and BESSs enhance system performance from an energy perspective, D-STATCOMs address voltage-related challenges that are critical for ensuring reliable operation and protecting end-use equipment. This complementary functionality reinforces the need to coordinate all three technologies when integrating DERs in distribution systems with high levels of asymmetry and variability. The methodology also demonstrated strong robustness across multiple runs with five leader-stage algorithms (WOA, VSA, CBGA, MVO, and BWOA). BWOA delivered the largest reductions in both objectives but incurred the highest computational effort, with average runtimes of approximately 2,672 s in Leticia and 3,360 s in San Andrés. At the other end of the spectrum, WOA consistently achieved the shortest execution times (about 198–215 s) with the lowest reductions. CBGA exhibited the most stable behavior—showing the smallest relative standard deviations—while MVO produced competitive reductions at higher runtimes and VSA offered a balanced compromise between solution quality and runtime. These trade-offs remained consistent across both test systems. The consistent and technically sound results obtained in both case studies support the practical applicability of the proposed methodology. By incorporating real demand and irradiance profiles, detailed DER models, and strict operational constraints, the optimization framework captures the dynamics and limitations of actual distribution systems in NIZs. The resulting DER configurations comply with voltage and thermal operating limits, facilitating implementation without substantial modifications to existing infrastructure. Furthermore, the modular structure of the methodology enables integration into utility-level planning tools or decentralized energy management systems, providing a scalable and adaptable solution for rural electrification projects. Future work will incorporate economic cost models, uncertainty analysis for renewable generation and load variability, and additional DER types such as electric vehicles and demand response mechanisms. In parallel, surrogate models or reduced-order approximations will be investigated to accelerate the optimization process. Finally, research will focus on real-time deployment of the strategy within decentralized control architectures for rural electrification initiatives. CRediT authorship contribution statement 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. 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. 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 All we affirm that have participated in (a) conception and design, or analysis and interpretation of the data; (b) drafting the article or revising it critically for important intellectual content; and (c) approval of the final version. In addition, we affirm that our manuscript has not been submitted to, nor is under review at, another journal or other publishing venue. Lastly, we affirm that we do not have any conflicts of interest. 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. The authors acknowledge the support provided by Thematic Network 723RT0150 Results in Engineering 28 (2025) 107098 22 R.I. Bolaños, J.A. Guzmán-Henao, L.F. Grisales-Noreña et al. “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 thank ITM Translation Agency ([email protected]) for editing the manuscript. Data availability No data was used for the research described in the article. References [1] A.-F. Papathanasiou, E. Baltas, Achieving water and energy independence, economic sustainability, and co2 reduction through hybrid renewable systems: a case study of skyros island, Water 17 (9) (2025), https://doi.org/10.3390/w17091267, https:// www.mdpi.com/2073-4441/17/9/1267. [2] J. Valencia, I. Dyner, F. Mesa, A. Aristizábal, Evaluating demand response through the time of use model in off-grid regions, Renew. Sustain. Energy Trans. 7 (2025) 100115, https://doi.org/10.1016/j.rset.2025.100115, https://www.sciencedirect. com/science/article/pii/S2667095X25000145. [3] O. Bystrom, Next-generation distribution planning: how do we capture the value of distributed energy resources?, IEEE Power Energy Mag. 20 (2) (2022) 32–38, https://doi.org/10.1109/MPE.2021.3134146. [4] S. Lobos-Cornejo, L.F. Grisales-Noreña, F. Andrade, O.D. Montoya, D. Sanin-Villa, Smart energy strategy for ac microgrids to enhance economic performance in gridconnected and standalone operations: a gray wolf optimizer approach, Sci. 7 (2) (2025), https://doi.org/10.3390/sci7020073, https://www.mdpi.com/2413-4155/ 7/2/73. [5] Z.U. Islam, M. Hossain Lipu, S.T. Meraj, A.M. Fuad, T. Rahman, M.A. Islam, M.R. Sarker, Hybrid renewable energy systems towards sustainable development in Bangladesh: configurations, optimizations, applications, challenges and future pathways, Results Eng. 27 (2025) 105728, https://doi.org/10.1016/j.rineng.2025. 105728, https://www.sciencedirect.com/science/article/pii/S2590123025017992. [6] M. Bilal, P.N. Bokoro, G. Sharma, Hybrid optimization for sustainable design and sizing of standalone microgrids integrating renewable energy, diesel generators, and battery storage with environmental considerations, Results Eng. 25 (2025) 103764, https://doi.org/10.1016/j.rineng.2024.103764, https://www. sciencedirect.com/science/article/pii/S2590123024020073. [7] V. Saxena, S. Manna, S.K. Rajput, P. Kumar, B. Sharma, M.H. Alsharif, M.-K. Kim, Navigating the complexities of distributed generation: integration, challenges, and solutions, Energy Rep. 12 (2024) 3302–3322, https://doi.org/10.1016/j.egyr.2024. 09.017, https://www.sciencedirect.com/science/article/pii/S2352484724005948. [8] M. Cavus, Advancing power systems with renewable energy and intelligent technologies: a comprehensive review on grid transformation and integration, Electronics 14 (6) (2025), https://doi.org/10.3390/electronics14061159, https://www.mdpi. com/2079-9292/14/6/1159. [9] A. Paaso, N. Burica, R. Burg, Realizing the value of ders: a utility perspective, IEEE Power Energy Mag. 20 (2) (2022) 39–46, https://doi.org/10.1109/MPE.2021. 3134147. [10] V. Tomar, M. Bansal, P. Singh, Metaheuristic algorithms for optimization: a brief review, Eng. Proc. 59 (1) (2023), https://doi.org/10.3390/engproc2023059238, https://www.mdpi.com/2673-4591/59/1/238. [11] E.H. Houssein, M.K. Saeed, G. Hu, et al., Metaheuristics for solving global and engineering optimization problems: review, applications, open issues and challenges, Arch. Comput. Methods Eng. 31 (2024) 4485–4519, https://doi.org/10. 1007/s11831-024-10168-6. [12] S. Selvarajan, A comprehensive study on modern optimization techniques for engineering applications, Artif. Intell. Rev. 57 (8) (2024) 194, https://doi.org/10.1007/ s10462-024-10829-9. [13] M.A. Khalil, T.M. Elkhodragy, W.A.A. Salem, A novel hybrid algorithm based on optimal size and location of photovoltaic with battery energy storage systems for voltage stability enhancement, Electr. Eng. 107 (1) (2025) 1009–1034, https://doi. org/10.1007/s00202-024-02508-3. [14] O.M. Neda, Optimal amalgamation of dg units in radial distribution system for techno-economic study by improved ssa: practical case study, Electr. Power Syst. Res. 241 (2025) 111365, https://doi.org/10.1016/j.epsr.2024.111365, https://www. sciencedirect.com/science/article/pii/S0378779624012513. [15] K. Prasad, R. Kollu, A. Ramkumar, A. Ramesh, A multi-objective strategy for optimal dg and capacitors placement to improve technical, economic, and environmental benefits, Int. J. Electr. Power Energy Syst. 165 (2025) 110491, https://doi.org/10.1016/j.ijepes.2025.110491, https://www.sciencedirect.com/ science/article/pii/S0142061525000420. [16] S. Jeon, S. Bae, Integrated optimization for sizing, placement, and energy management of hybrid energy storage systems in renewable power systems, J. Energy Storage 106 (2025) 114793, https://doi.org/10.1016/j.est.2024.114793, https:// www.sciencedirect.com/science/article/pii/S2352152X24043792. [17] N. Pompern, S. Premrudeepreechacharn, A. Siritaratiwat, S. Khunkitti, Optimal placement and capacity of battery energy storage system in distribution networks integrated with pv and evs using metaheuristic algorithms, IEEE Access 11 (2023) 68379–68394, https://doi.org/10.1109/ACCESS.2023.3291590. [18] P. Boonluk, A. Siritaratiwat, P. Fuangfoo, S. Khunkitti, Optimal siting and sizing of battery energy storage systems for distribution network of distribution system operators, Batteries 6 (4) (2020), https://doi.org/10.3390/batteries6040056, https:// www.mdpi.com/2313-0105/6/4/56. [19] K. Wichitkrailat, S. Premrudeepreechacharn, A. Siritaratiwat, S. Khunkitti, Optimal sizing and locations of multiple besss in distribution systems using crayfish optimization algorithm, IEEE Access 12 (2024) 94733–94752, https://doi.org/10.1109/ ACCESS.2024.3425963. [20] J. Guzman-Henao, L.F. Grisales-Noreña, B.J. Restrepo-Cuestas, O.D. Montoya, Optimal integration of photovoltaic systems in distribution networks from a technical, financial, and environmental perspective, Energies 16 (2023), https://doi.org/10. 3390/en16010562, https://www.mdpi.com/1996-1073/16/1/562. [21] B. Cortés-Caicedo, L.F. Grisales-Noreña, O.D. Montoya, R.I. Bolaños, Minimizing the annual costs in ac distribution microgrids through optimal bess location, selection, and operation using a hybrid approach, J. Energy Storage 84 (2024) 110894, https://doi.org/10.1016/j.est.2024.110894, https://www. sciencedirect.com/science/article/pii/S2352152X2400478X. [22] B. Cortés-Caicedo, L.F. Grisales-Noreña, O.D. Montoya, R.I. Bolaños, J. Muñoz, A multi-objective optimization approach based on the non-dominated sorting genetic algorithm ii for power coordination in battery energy storage systems for dc distribution network applications, J. Energy Storage 113 (2025) 115430, https://doi.org/10.1016/j.est.2025.115430, https://www.sciencedirect. com/science/article/pii/S2352152X25001434. [23] V.M. Garrido-Arévalo, O.D. Montoya, W. Gil-González, L.F. Grisales-Noreña, J.C. Hernández, An sdp relaxation in the complex domain for the efficient coordination of bess and dgs in single-phase distribution grids while considering reactive power capabilities, J. Energy Storage 90 (2024) 111913, https://doi.org/10.1016/j.est.2024. 111913, https://www.sciencedirect.com/science/article/pii/S2352152X24014981. [24] H. Rezk, A.G. Olabi, E.T. Sayed, T. Wilberforce, Role of metaheuristics in optimizing microgrids operating and management issues: a comprehensive review, Sustainability 15 (6) (2023), https://doi.org/10.3390/su15064982, https://www.mdpi.com/ 2071-1050/15/6/4982. [25] P.C. Chu, J.E. Beasley, A genetic algorithm for the multidimensional knapsack problem, J. Heuristics 4 (1998) 63–86. [26] J.A. Guzmán-Henao, B. Cortés-Caicedo, B.J. Restrepo-Cuestas, R.I. Bolaños, L.F. Grisales-Noreña, Optimal integration of photovoltaic generators into urban and rural power distribution systems, Sol. Energy 270 (2024) 112400, https:// doi.org/10.1016/j.solener.2024.112400, https://www.sciencedirect.com/science/ article/pii/S0038092X2400094X. [27] D. Sanin-Villa, H.A. Figueroa-Saavedra, L.F. Grisales-Noreña, Efficient bess scheduling in ac microgrids via multiverse optimizer: a grid-dependent and self-powered strategy to minimize power losses and co2 footprint, Appl. Syst. Innov. 8 (3) (2025), https://doi.org/10.3390/asi8030085, https://www.mdpi.com/2571-5577/8/3/85. [28] V.P. Kolanchinathan, K.R.N. Aswini, N. Rajendran, B. Chinthamani, S. Bhuvana, S.N. Deepa, S.B. Mohan, N.R. Shanker, J. Gayathri, Optimal location identification of solar pv systems in distributed generators based on prediction of load flow and factor using rule based deep learning algorithm, J. Electr. Eng. Technol. 20 (3) (2025) 1269–1281, https://doi.org/10.1007/s42835-024-02063-8. [29] B. Doğan, T. Ölmez, A new metaheuristic for numerical function optimization: vortex search algorithm, Inf. Sci. 293 (2015) 125–145, https://doi.org/10.1016/j.ins.2014. 08.053, https://www.sciencedirect.com/science/article/pii/S0020025514008585. [30] A. Kumar, A. Maulik, K. Chinmaya, A decentralized energy management scheme for a dc microgrid with correlated uncertainties and integrated demand response, Electr. Power Syst. Res. 238 (2025) 111093, https://doi.org/10.1016/j.epsr.2024.111093, https://www.sciencedirect.com/science/article/pii/S0378779624009787. [31] A. Paz-Rodríguez, J.F. Castro-Ordoñez, O.D. Montoya, D.A. Giral-Ramírez, Optimal integration of photovoltaic sources in distribution networks for daily energy losses minimization using the vortex search algorithm, Appl. Sci. 11 (10) (2021), https:// doi.org/10.3390/app11104418, https://www.mdpi.com/2076-3417/11/10/4418. [32] B. Cortés-Caicedo, S. Bustamante-Mesa, D.L. Rodríguez-Salazar, O.D. Montoya, M. Rico-García, Integration and operation of energy storage systems in active distribution networks: economic optimization via salp swarm optimization, Electricity 6 (1) (2025), https://doi.org/10.3390/electricity6010011, https://www.mdpi.com/ 2673-4826/6/1/11. [33] V. Hayyolalam, A.A. Pourhaji Kazem, Black widow optimization algorithm: a novel meta-heuristic approach for solving engineering optimization problems, Eng. Appl. Artif. Intell. 87 (2020) 103249, https://doi.org/10.1016/j.engappai.2019.103249, https://www.sciencedirect.com/science/article/pii/S0952197619302283. [34] R. Shaikh, A. Stojcevski, M. Seyedmahmoudian, J. Chandran, A multi-objective approach for optimal sizing and placement of distributed generators and distribution static compensators in a distribution network using the black widow optimization algorithm, Sustainability 16 (11) (2024), https://doi.org/10.3390/su16114577, https://www.mdpi.com/2071-1050/16/11/4577. [35] S.A. Salimon, I.G. Adebayo, G.A. Adepoju, O.B. Adewuyi, Optimal allocation of distribution static synchronous compensators in distribution networks considering various load models using the black widow optimization algorithm, Sustainability 15 (21) (2023), https://doi.org/10.3390/su152115623, https://www.mdpi.com/ 2071-1050/15/21/15623. Results in Engineering 28 (2025) 107098 23 R.I. Bolaños, J.A. Guzmán-Henao, L.F. Grisales-Noreña et al. [36] D. Xu, J. Yin, An improved black widow optimization algorithm for engineering constrained optimization problems, IEEE Access 11 (2023) 32476–32495, https:// doi.org/10.1109/ACCESS.2023.3262600. [37] S. Mirjalili, S.M. Mirjalili, A. Hatamlou, Multi-verse optimizer: a nature-inspired algorithm for global optimization, Neural Comput. Appl. 27 (2) (2016) 495–513, https://doi.org/10.1007/s00521-015-1870-7. [38] E. Hosseini, K.Z. Ghafoor, A. Emrouznejad, A.S. Sadiq, D.B. Rawat, Novel metaheuristic based on multiverse theory for optimization problems in emerging systems, Appl. Intell. 51 (6) (2021) 3275–3292, https://doi.org/10.1007/s10489-020-01920z. [39] L.F. Grisales-Noreña, D. Sanin-Villa, O.D. Montoya, Optimal integration of pv generators and d-statcoms into the electrical distribution system to reduce the annual investment and operational cost: a multiverse optimization algorithm and matrix power flow approach, e-Prime, Adv. Electr. Eng. Electron. Energy 9 (2024) 100747, https://doi.org/10.1016/j.prime.2024.100747, https://www. sciencedirect.com/science/article/pii/S2772671124003279. [40] Y. Han, W. Chen, A.A. Heidari, H. Chen, X. Zhang, Balancing exploration– exploitation of multi-verse optimizer for parameter extraction on photovoltaic models, J. Bionics Eng. 21 (2) (2024) 1022–1054, https://doi.org/10.1007/s42235-02400479-6. [41] S. Mirjalili, A. Lewis, The whale optimization algorithm, Adv. Eng. Softw. 95 (2016) 51–67, https://doi.org/10.1016/j.advengsoft.2016.01.008, https://www. sciencedirect.com/science/article/pii/S0965997816300163. [42] G. Sun, Y. Shang, R. Zhang, An efficient and robust improved whale optimization algorithm for large scale global optimization problems, Electronics 11 (9) (2022), https://doi.org/10.3390/electronics11091475, https://www.mdpi. com/2079-9292/11/9/1475. [43] A.M. Alshannaq, M.A. Haj-ahmed, M. Aldwaik, D. Abualnadi, Interline power flow controller allocation for active power losses enhancement using whale optimization algorithm, Energies 17 (24) (2024), https://doi.org/10.3390/en17246318, https:// www.mdpi.com/1996-1073/17/24/6318. [44] T. Boonraksa, W. Pinthurat, P. Wongdet, P. Boonraksa, B. Marungsri, B. Hredzak, Optimal capacity and cost analysis of hybrid energy storage system in standalone dc microgrid, IEEE Access 11 (2023) 65496–65506, https://doi.org/10.1109/ACCESS. 2023.3289821. [45] R. Eberhart, J. Kennedy, A new optimizer using particle swarm theory, in: MHS’95. Proceedings of the Sixth International Symposium on Micro Machine and Human Science, 1995, pp. 39–43. [46] L.F. Grisales-Noreña, B. Cortés-Caicedo, O.D. Montoya, J.C. Hernandéz, G. Alcalá, A battery energy management system to improve the financial, technical, and environmental indicators of colombian urban and rural networks, J. Energy Storage 65 (2023) 107199, https://doi.org/10.1016/j.est.2023.107199, https://www. sciencedirect.com/science/article/pii/S2352152X23005960. [47] B. Cortés-Caicedo, L.F. Grisales-Noreña, O.D. Montoya, M.A. Rodriguez-Cabal, J.A. Rosero, Energy management system for the optimal operation of pv generators in distribution systems using the antlion optimizer: a colombian urban and rural case study, Sustainability (Switzerland) 14 (12 2022), https://doi.org/10.3390/ su142316083. [48] L.F. Grisales-Noreña, O.D. Montoya, J.C. Hernández, C.A. Ramos-Paja, A.-J. PereaMoreno, A discrete-continuous pso for the optimal integration of d-statcoms into electrical distribution systems by considering annual power loss and investment costs, Mathematics 10 (14) (2022), https://doi.org/10.3390/math10142453, https://www.mdpi.com/2227-7390/10/14/2453. [49] L.O. Shobayo, C.D. Dao, Smart integration of renewable energy sources employing setpoint frequency control—an analysis on the grid cost of balancing, Sustainability 16 (22) (2024), https://doi.org/10.3390/su16229906, https://www.mdpi.com/ 2071-1050/16/22/9906. [50] M. Ikram, M. Aslam, K. Aurangzeb, S. Ahmed, S.N.K. Marwat, S.I. Haider, M. Alhussein, Integrating renewable energy sources for optimal demand-side management using decentralized multi-agent control, Sustain. Energy Grids Netw. 36 (2023) 101193, https://doi.org/10.1016/j.segan.2023.101193, https://www. sciencedirect.com/science/article/pii/S2352467723002011. [51] A.K. ALAhmad, R. Verayiah, H. Shareef, A. Ramasamy, S. Ba-swaimi, Optimizing renewable energy and green technologies in distribution systems through stochastic planning of distributed energy resources, Energy Convers. Manag. X 25 (2025) 100834, https://doi.org/10.1016/j.ecmx.2024.100834, https://www.sciencedirect. com/science/article/pii/S259017452400312X. [52] G. Carpinelli, A. Di Fazio, S. Perna, A. Russo, M. Russo, A-priori multi-objective optimization for the short-term dispatch of distributed energy resources, Int. J. Electr. Power Energy Syst. 164 (2025) 110410, https://doi.org/10.1016/j.ijepes.2024. 110410, https://www.sciencedirect.com/science/article/pii/S0142061524006331. [53] A.P.A. Amorim, K.V. Pontes, B. Dorneanu, H. Arellano-Garcia, Optimizing microgrid design and operation: a decision-making framework for residential distributed energy systems in Brazil, Chem. Eng. Res. Des. 214 (2025) 251–268, https://doi.org/ 10.1016/j.cherd.2024.12.033, https://www.sciencedirect.com/science/article/pii/ S0263876224007123. [54] M. Kiasari, M. Ghaffari, H.H. Aly, A comprehensive review of the current status of smart grid technologies for renewable energies integration and future trends: the role of machine learning and energy storage systems, Energies 17 (16) (2024), https:// doi.org/10.3390/en17164128, https://www.mdpi.com/1996-1073/17/16/4128. [55] C.P. Ohanu, S.A. Rufai, U.C. Oluchi, A comprehensive review of recent developments in smart grid through renewable energy resources integration, Heliyon 10 (3) (2024) e25705, https://doi.org/10.1016/j.heliyon.2024.e25705. [56] M.F. Alwaeli, S. Galvani, V. Talavat, Addressing power quality challenges in hybrid renewable energy systems through statcom devices and advanced gray wolf optimization technique, Results Eng. 25 (2025) 104405, https:// doi.org/10.1016/j.rineng.2025.104405, https://www.sciencedirect.com/science/ article/pii/S2590123025004852. [57] K.M. Berdugo Sarmiento, J.I. Silva-Ortega, V. Sousa Santos, J.E. Candelo-Becerra, F.E. Hoyos, Improving the operation of transmission systems based on static var compensator, Electricity 6 (3) (2025), https://doi.org/10.3390/electricity6030040, https://www.mdpi.com/2673-4826/6/3/40. [58] A. Mandal, K.M. Muttaqi, M. Islam, D. Sutanto, A novel architecture of the solidstate unified power flow controller and its integration into the power grid, Electr. Power Syst. Res. 241 (2025) 111296, https://doi.org/10.1016/j.epsr.2024.111296, https://www.sciencedirect.com/science/article/pii/S0378779624011829. [59] M. Dehghani, M.R. Yousefi, A. Mosavi, A. Fathollahi, Unified power flow controller: operation, modelling and applications, in: 2023 IEEE 17th International Symposium on Applied Computational Intelligence and Informatics (SACI), 2023, pp. 000699–000704. [60] Z. Wang, L. Fan, Z. Miao, Stability analysis of oscillations in svcs, in: 2022 North American Power Symposium (NAPS), 2022, pp. 1–5. [61] O.D. Montoya, W. Gil-González, L. Grisales-Noreña, An exact minlp model for optimal location and sizing of dgs in distribution networks: a general algebraic modeling system approach, Ain Shams Eng. J. 11 (2) (2020) 409–418, https://doi.org/10.1016/j.asej.2019.08.011, https://www.sciencedirect. com/science/article/pii/S2090447919301200. [62] L.F. Grisales-Noreña, B. Cortés-Caicedo, O.D. Montoya, R.I. Bolaños, C.A.M. Moreno, Nonlinear programming for bess operation for the improvement of economic, technical and environmental indices by considering grid-connected and standalone networks: an application to the territory of Colombia, J. Energy Storage 98 (2024) 112856, https://doi.org/10.1016/j.est.2024.112856, https://www. sciencedirect.com/science/article/pii/S2352152X24024423. [63] L.F. Grisales-Noreña, D. Sanin-Villa, O.D. Montoya, Optimal integration of pv generators and d-statcoms into the electrical distribution system to reduce the annual investment and operational cost: a multiverse optimization algorithm and matrix power flow approach, e-Prime, Adv. Electr. Eng. Electron. Energy 9 (2024) 100747, https://doi.org/10.1016/j.prime.2024.100747, https://www. sciencedirect.com/science/article/pii/S2772671124003279. [64] O.D. Montoya, L.F. Grisales-Noreña, W. Gil-González, Simultaneous siting and sizing of pvs and d-statcoms in medium-voltage grids using the cauchy-based distribution optimizer, Results Eng. 25 (2025) 104407, https://doi.org/10.1016/j.rineng.2025. 104407, https://www.sciencedirect.com/science/article/pii/S2590123025004876. [65] Comisión de Regulación de Energía y Gas (CREG), Gestor normativo de la creg, https://gestornormativo.creg.gov.co/gestor/entorno/docs/resolucion_creg_ 0030_2018.htm, 2025. [66] Y. Wang, C. Deng, D. Liu, Y. Xu, J. Dai, Unified real power sharing of generator and storage in islanded microgrid via distributed dynamic event-triggered control, IEEE Trans. Power Syst. 36 (3) (2021) 1713–1724, https://doi.org/10.1109/TPWRS. 2020.3039530. [67] L.S. Avellaneda-Gómez, B. Cortés-Caicedo, O.D. Montoya, Minimizing energy losses in unbalanced distribution networks via the optimal integration of pv sources and d-statcoms, in: 2024 IEEE Colombian Conference on Applications of Computational Intelligence (ColCACI), 2024, pp. 1–6. [68] Departamento Administrativo Nacional de Estadística (DANE), Informe estadístico Leticia, https://www.dane.gov.co/files/investigaciones/planes-departamentosciudades/220502-InfoDane-Leticia-Amazonas-fin.pdf, 2022. (Accessed 7 February 2025). [69] Ministerio de Comercio, Industria y Turismo de Colombia (MinCIT), Informe sobre Amazonas, https://www.mincit.gov.co/getattachment/d590efc5-9b03-4943-9255929554b8f45b/Amazonas, 2024. (Accessed 7 February 2025). [70] T. Ramana, V. Ganesh, S. Sivanagaraju, Distributed generator placement and sizing in unbalanced radial distribution system, Cogen. Distrib. Gen. J. 25 (1) (2010) 52–71, https://doi.org/10.1080/15453661009709862. [71] B. Cortés-Caicedo, L.S. Avellaneda-Gómez, O.D. Montoya, L. Alvarado-Barrios, H.R. Chamorro, Application of the vortex search algorithm to the phase-balancing problem in distribution systems, Energies 14 (5) (2021), https://www.mdpi.com/19961073/14/5/1282. [72] J.A. Guzmán-Henao, B. Cortés-Caicedo, R.I. Bolaños, L.F. Grisales-Noreña, O.D. Montoya, Optimal conductor selection and phase balancing in three-phase distribution systems: an integrative approach, Results Eng. 24 (2024) 103416, https://doi.org/10.1016/j.rineng.2024.103416, https://www.sciencedirect.com/ science/article/pii/S2590123024016682. [73] Departamento Nacional de Planeación de Colombia (DNP), Documento CONPES 3855 -Crédito San Andrés, https://colaboracion.dnp.gov.co/CDT/Conpes/ Econ%C3%B3micos/3855_Credito_San_Andres_VPublicaci%C3%B3n.pdf, 2016. (Accessed 7 February 2025). [74] Comisión Económica para América Latina y el Caribe (CEPAL), Título oficial del documento, https://repositorio.cepal.org/server/api/core/bitstreams/e00955b2-08d84a5a-8a17-50cf8eb82af6/content, 2017. (Accessed 7 February 2025). Results in Engineering 28 (2025) 107098 24 R.I. Bolaños, J.A. Guzmán-Henao, L.F. Grisales-Noreña et al. [75] W. Kersting, Radial distribution test feeders, IEEE Trans. Power Syst. 6 (3) (1991) 975–985, https://doi.org/10.1109/59.119237. [76] M. Granada Echeverri, R.A. Gallego Rendón, J.M. López Lezama, Optimal phase balancing planning for loss reduction in distribution systems using a specialized genetic algorithm, Ing. Cienc. 8 (15) (2012) 121–140, https://doi.org/10.17230/ingciencia. 8.15.6. [77] Q. Hassan, M. Jaszczur, E. Przenzak, J. Abdulateef, The pv cell temperature effect on the energy production and module efficiency, Contemp. Probl. Power Eng. Env. Protect. 33 (2016) 1. [78] NASA Prediction of Worldwide Energy Resources (POWER), NASA POWER Data Access Viewer, https://power.larc.nasa.gov/, 2025. (Accessed 7 February 2025). [79] Instituto de Planificación y Promoción de Soluciones Energéticas para las Zonas No Interconectadas, Ipse - instituto de planificación y promoción de soluciones energéticas, https://ipse.gov.co/, 2025, Consultado el 10 de febrero de 2025. [80] Instituto de Planificación y Promoción de Soluciones Energéticas para las Zonas No Interconectadas, Informe mensual de telemetría - diciembre 2023, https:// ipse.gov.co/documentos_cmn/documentos/informes_mensuales_de_telemetria/ 2023/diciembre/INFORME_MENSUAL_TELEMETRIA_DICIEMBRE_DE_2023.pdf, 2023, Consultado el 10 de febrero de 2025. [81] A.A. Eras-Almeida, T. Vásquez-Hernández, M.J. Hurtado-Moncada, M.A. EgidoAguilera, A comprehensive evaluation of off-grid photovoltaic experiences in noninterconnected zones of Colombia: integrating a sustainable perspective, Energies 16 (5) (2023), https://www.mdpi.com/1996-1073/16/5/2292. [82] Academia Colombiana de Ciencias Exactas, Físicas y Naturales, Factores de emisión de los combustibles colombianos, https://bdigital.upme.gov.co/bitstream/handle/ 001/1285/17%20Factores%20de%20emision%20de%20combustibles.pdf, 2016. (Accessed 9 May 2023). [83] Sistema de Servicios Públicos, Consolidado de información técnica operativa zni, https://sui.superservicios.gov.co/Reportes-delsector/Energia/Reportescomerciales/Consolidado-de-informacion-tecnicaoperativa-ZNI, 2023. (Accessed 9 March 2024). [84] W. Gil-González, O.D. Montoya, E. Holguín, A. Garces, L.F. Grisales-Noreña, Economic dispatch of energy storage systems in dc microgrids employing a semidefinite programming model, J. Energy Storage 21 (2019) 1–8, https://doi. org/10.1016/j.est.2018.10.025, https://www.sciencedirect.com/science/article/ pii/S2352152X18302962. [85] S.K. Dash, S. Mishra, A.Y. Abdelaziz, A critical analysis of modeling aspects of dstatcoms for optimal reactive power compensation in power distribution networks, Energies 15 (19) (2022), https://doi.org/10.3390/en15196908, https://www.mdpi. com/1996-1073/15/19/6908. [86] L. Cai, I. Erlich, G. Stamtsis, Optimal choice and allocation of facts devices in deregulated electricity market using genetic algorithms, in: IEEE PES Power Systems Conference and Exposition, vol. 1, 2004, pp. 201–207. Results in Engineering 28 (2025) 107098 25