Optimizing economic and operational performance in AC microgrids: An intelligent energy management strategy for BESS using the Generalized Normal Distribution Optimizer

Results in Engineering 27 (2025) 106005 Available online 14 July 2025 2590-1230/© 2025 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/bync-nd/4.0/). Contents lists available at ScienceDirect Results in Engineering journal homepage: www.sciencedirect.com/journal/results-in-engineering Research paper Optimizing economic and operational performance in AC microgrids: An intelligent energy management strategy for BESS using the Generalized Normal Distribution Optimizer Marco Alonso Llanos-Pinoa, Luis Fernando Grisales-Noreñab, ,∗, Daniel Sanin-Villac,, Oscar Danilo Montoyad,, Jesús C. Hernándeze, aUniversidad de Talca, Facultad de Ingeniería, Departamento de Ingeniería Eléctrica, Curicó, Chile bGrupo de Investigación en Alta Tensión—GRALTA, Escuela de Ingeniería Eléctrica y Electrónica, Universidad del Valle, Cali 760015, Colombia cÁrea Industria, Materiales y Energía, Universidad EAFIT, Medellín, 050022, Antioquia, Colombia dGrupo de Compatibilidad e Interferencia Electromagnética, Facultad de Ingeniería, Universidad Distrital Francisco José de Caldas, Bogotá 110231, Colombia eDepartment of Electrical Engineering, Universidad de Jaén, Campus Lagunillas s/n, Edificio A3, Jaén 23071, Spain A R T I C L E I N F O A B S T R A C T Keywords: Batery energy storage system (BESS) optimization PV generation variability AC microgrid energy management Generalized Normal Distribution Optimizer (GNDO) Operational cost minimization Energy loss reduction 𝐶𝑂2emissions mitigation This work addresses the problem of efficient management of battery energy storage systems (BESS) in alternating current microgrids (AC MGs), seeking to optimize technical (energy losses), economic (operating costs), and environmental (emissions) indicators in on-grid and off-grid operation scenarios. Previous studies often lack comprehensive comparisons with multiple optimization methodologies, thorough statistical validation, and adaptability to variable generation-demand scenarios in both grid-connected and isolated modes. To address this gap, the present study applies the Generalized Normal Distribution Optimization (GNDO) algorithm to the energy management of BESS in AC MGs. Its performance is validated against three established methods: a Continuous Genetic Algorithm (CGA), Particle Swarm Optimization (PSO), and the JAYA algorithm. The approach is evaluated on two test systems: 33-node on-grid MG and 27-node off-grid MG under photovoltaic generation and variable demand. Performance metrics include solution quality, standard deviation, and processing time, based on 100 independent runs per method. Results show that GNDO consistently outperforms the benchmark algorithms, with average improvements of 1.4349% in costs, 4.4426% in losses, and 0.1833% in emissions. The findings confirm GNDO’s robustness and efficiency in constrained energy management environments. 1. Introduction 1.1. Background Over the past two decades, the MG concept has become relevant worldwide [1]. This is mainly because they present an alternative to implementing a more renewable energy matrix, which reduces dependence on fossil fuels and thus reduces the environmental impact. MGs are characterized to a large extent by their versatility in operation, which can be reflected in their operation in off-grid or on-grid mode [2]. This last fact makes its implementation possible in densely populated urban areas and rural areas distant or isolated from the national interconnection system [3]. MGs are often considered as a “sup- *Corresponding author. E-mail addresses: [email protected] (M.A. Llanos-Pino), [email protected] (L.F. Grisales-Noreña), dsaninv2@eafit.edu.co (D. Sanin-Villa), [email protected] (O.D. Montoya), [email protected] (J.C. Hernández). port” to the primary grid; however, their operation goes far beyond that, mainly by offering a more robust, flexible and sustainable system [4]. Recent studies have increasingly explored the deployment of renewable energy-based microgrids in developing countries, particularly in remote and urbanizing areas of Bangladesh, as a strategy to alleviate power supply deficits and promote sustainable electrification. Ali et al. [5] proposed an off-grid MG incorporating PV, wind, and BESS to electrify an isolated riverside village, demonstrating that a 100% renewable configuration can deliver cost-effective, zero-emission energy with strong resilience to fluctuating demand and component costs. In residential contexts, grid-connected MGs combining solar PV and biogas generators have been shown to reduce CO2emissions by 46% and achieve competitive energy costs, as evidenced by techno-economic and https://doi.org/10.1016/j.rineng.2025.106005 Received 12 March 2025; Received in revised form 9 June 2025; Accepted 26 June 2025 Results in Engineering 27 (2025) 106005 2 M.A. Llanos-Pino, L.F. Grisales-Noreña, D. Sanin-Villa et al. predictive analyses in [6]. These hybrid systems integrate forecasting and demand response mechanisms to improve operational efficiency and ensure uninterrupted supply. Similar benefits are reported in institutional settings; for instance, a PV-biomass MG deployed at a university campus yielded significant environmental and financial savings, achieving a COE of $0.0347/kWh and reducing annual CO2emissions by over 1 million kg [7]. Furthermore, the integration of PV, wind, and BESS in educational facilities has proven effective in ensuring stable power with high renewable penetration and emission reductions exceeding 65% compared to conventional systems [8]. These outcomes are consistent with broader findings that hybrid MGs can offer scalable, low-emission solutions even in urbanized residential areas, where PV-biomass-wind systems have demonstrated both economic and ecological viability [9]. Collectively, these studies affirm the relevance of renewable-based MGs in diverse applications, emphasizing the need for high-performance energy management systems capable of adapting to technical constraints and varying load-generation conditions. However, most existing approaches focus on scenario-specific configurations and lack advanced optimization layers capable of optimizing BESS operation across multi-period dispatch problems, especially under voltageconstrained AC conditions. This underscores the necessity for flexible metaheuristic strategies, such as the one proposed in this study, that can handle grid-connected and isolated topologies while minimizing energy losses, emissions, and operational costs. At the Latin American level, MGs have developed rapidly because, due to its geographical location, the region has a high potential for renewable generation, which has benefited from the strengthening of the regulatory framework by the country, which has meant that their study has become increasingly relevant [10]. Globally, the characteristics of an electrical network in a given area to be considered MG are diffuse since, according to the legislation of each country, their requirements change. However, some characteristics could be considered universal [11]. MGs are low-voltage (LV) or medium-voltage (MV) electrical networks characterized by being composed of at least DG, ESS, and electrical loads associated with end users [12]. These operate as a single entity to supply local loads (specific to the MG) and non-local loads (demanded by the grid connection point) if necessary. Their operation depends to a large extent on the nature of the MG, since if it is mainly intended for urban applications, it will operate on-grid. However, if it is primarily intended for rural applications, it will commonly operate off-grid [13]. One of the fundamental characteristics of an MG, which offers a great number of advantages over conventional grids, is its high level of controllability and flexibility at the time of operation, which is mainly due to a great integration of power electronics and communications systems according to its needs [10]. However, the success of the MG is closely related to the efficient management of the energy resources available to it. Considering that the distributed generators operate in maximum power point tracking mode to make the best use of the renewable resources available in the region where the MG is located, the main responsibility for management falls on the BESS [14]. These systems, through the implementation of optimization strategies in their energy management system, guarantee the technical and operational conditions of the MG (such as power balance, nodal voltage profiles, and line currents, among others) while promoting economic, technical, and environmental improvements through intelligent management of the energy resource [15]. The positive or negative impact of the proposed energy management strategy depends directly on the intelligent algorithm employed for this task, so its selection and adjustment must be performed optimally to maximize the benefits that BESS can bring to the MG [16]. 1.2. Literature review A review of the state of the art on energy management strategies applied to BESS in AC MGs for the improvement of economic, technical, and environmental indicators highlights the work reported in [17], where an EMS was developed for urban ultralight electric vehicles based on the Jaya algorithm, designed to optimize the distribution of energy between batteries and ultracapacitors in hybrid architectures. The objective function minimizes the system’s total energy loss, considering the dynamic characteristics of the storage sources and the demand requirements. The proposed method stands out for its simplicity, as it does not require parameter adjustment and its accuracy, approaching the optimum of Dynamic Programming (DP) with a difference of 3.1% and achieving 1.9% less losses than PSO. This work offers an efficient and practical solution for energy management in electric vehicles, with potential for future optimizations. This methodology can be used for battery management in multinode MGs to improve economic, technical, and environmental conditions, taking advantage of the simplicity and high efficiency of JAYA. In [18], the battery energy management problem is addressed, using as an objective function the reduction of energy purchase costs, the reduction of energy losses associated with transportation, and 𝐶𝑂2 emissions; considering the constraints associated to the grid and batteries: active and reactive power balance, power limits in equipment and lines, voltage limit, battery SoC, among others. As solution strategies, they proposed the optimization method based on a vortex search algorithm (VSA) using the hourly FC based on successive approximations (SA) to evaluate the impact of power configurations on the charging and discharging actions. They employed CGA and the PSO method as comparison methods. A connected network of 33 nodes and an isolated network of 27 nodes with variable power generation and demand were used as test systems. The simulation results demonstrated the effectiveness of the proposed solution methodology in terms of the best solution, the average solution, and the required processing times. This work identifies the need to propose high-performance optimization strategies to improve the impact of battery management within the network, proposing test scenarios for such methodologies. In [19], the problem of optimal operation of PV generators operating in conjunction with BESS is addressed. The authors used the Siler-Evans adaptation of Hawke’s method, which aims to arbitrage emissions for the system by charging the batteries when emissions are low and discharging when emissions increase. The proposed objective function seeks to reduce 𝐶𝑂2emissions from the algorithm mentioned above, considering a system composed of the PV-battery set with real consumption measurements between February 2012 and January 2013 in a UK household with an annual consumption of 3845 kWh. The results presented by the authors show that it is possible to offset the 𝐶𝑂2emissions generated in manufacturing using the intelligent management of a battery system in a household by operating with the emissions arbitrage algorithm; however, the authors did not compare the algorithm proposed with other methods, did not perform statistical analysis, nor did they analyze the computation times. On the other hand, in [20], the integration and operation of ESS in power grids were studied. The authors proposed a methodology based on the gravitational search algorithm and a hybrid methodology based on PSO and GA. The objective functions defined in the study were the reduction of operating costs, the reduction of voltage deviation, and the reduction of greenhouse gas emissions in a 30-node system. Different scenarios were considered to test the algorithms, thus allowing comparison of the proposed methodologies, the associated results, and their respective computation times; however, statistical analysis of the results was not performed. In [21], the integration and operation of ESS in AC distribution systems were analyzed. The authors proposed a master-slave stage, based on the Chu & Beasley optimization algorithm for the optimization stage (master stage) and the SA method to solve the load-flow problem (slave stage). The objective function was the reduction of energy losses in a 69node system. Although the method demonstrated efficiency, it was not compared with other metaheuristic methods, and statistical variables or processing times were not analyzed. Results in Engineering 27 (2025) 106005 3 M.A. Llanos-Pino, L.F. Grisales-Noreña, D. Sanin-Villa et al. In [22], an investigation on battery management in an urban network is presented. The methodology uses a parallel version of the PSO algorithm. The objective function focuses on reducing conventional generator power purchase costs, thus reducing greenhouse gas emissions. The study was conducted in an urban system of 22 nodes with distributed PV generation. The results were compared with three methods from the literature (black hole, continuous GA, and traditional GA Chu % Beasley), corroborating solution quality, standard deviation, and processing times. In [23], the problem associated with off-grid operation of a power system is addressed. The authors developed a strategy focusing on distributed PV generators, orienting the rest of the system elements to optimize their operation. The proposed methodology demonstrated effectiveness; however, it was not compared with studies of similar characteristics. On the other hand, [24] presents an applied case of ESS operation for an average operating day in Romania. A GA-based methodology considered three different operation scenarios. The objective was the reduction of energy losses in an urban AC network in northern Romania. Results highlighted industrial applicability, yet comparisons with other methodologies, statistical analysis, and processing times were not included. In [25], they study the intelligent operation of ESS. The authors describe a nonlinear programming model, which operates from the GAMS software to solve the associated problem. The objective function of the study is oriented to the minimization of greenhouse gas emissions and energy losses for an average operating day, considering systems of 33 and 69 nodes as test scenarios, which correspond to urban networks in both cases. The results demonstrate the proposed solution’s effectiveness; however, the authors did not compare this result with those presented by other methods in the specialized literature. Additionally, the processing times associated with resolving the problem were not analyzed. Another study, [26], investigated the optimal dispatch of BESS operating in coordination with wind generators and capacitors. For this purpose, the authors presented an optimization methodology based on an unsorted GA, complemented by ordering techniques that selected the solution closest to the ideal one. The proposed objective function focused on reducing energy losses and operating costs in 33and 108node systems. Although the results demonstrated the efficiency of the proposed method, the authors did not perform statistical analyses or evaluate the processing times associated with the methodology. Reference [27] examined MGs with multiple generation technologies using a moving-horizon strategy to reduce operational costs, validated in an experimental MG in Huatacondo, Chile. Statistical analysis and processing-time evaluation were not included. On the other hand, [28] presents a comprehensive review of the controls used in MG, highlighting the relevance of power and EMS, and exploring their potential market share. The results highlight the contributions in terms of controllability of MGs, underlining the main differences concerning traditional power systems. In addition, the growth of this technology is evidenced by the increasing interest in the multiple benefits it offers to operators and customers who participate in it. Finally, in [29], BESS’s real-time configuration and operation in power grids are analyzed. The authors used a convex optimization-based approach to solve both problems. The objective function focused on minimizing BESS configuration and operation costs, considering key parameters such as capacity, losses, maximum charge, and discharge rates. The results were compared with literature references, and an analysis of processing times was also incorporated. These results evidenced the efficiency in both the solutions’ quality and the computational times required. Despite the growing body of literature on energy management in microgrids, several critical gaps remain. Many previous studies lack comprehensive algorithmic comparisons, do not provide statistical robustness over multiple simulation runs, and often rely on simplified or static operating conditions. Moreover, the application of newer probabilistic optimization methods, such as GNDO, remains largely unexplored. This study addresses these gaps by applying GNDO to the energy management of AC microgrids in both on-grid and off-grid scenarios, incorporating dynamic pricing and demand profiles, and benchmarking its performance against CGA, PSO, and JAYA using a rigorous statistical framework. Several recent studies have focused on advanced metaheuristic strategies for the optimal management of distributed energy resources in AC microgrids, demonstrating the growing relevance of combining renewable generation, power electronics, and storage devices within robust energy management frameworks. This work [30] proposed a master–slave optimization strategy using a hybrid discrete-continuous multiverse optimizer combined with matrix-based power flow analysis to minimize both investment and operational costs associated with PV and D-STATCOM integration. Their results, validated across 33and 69node networks with varying demand and solar generation, highlighted MVO’s superiority over traditional algorithms. Extending this line of research, the authors also investigated optimal energy dispatch from distributed wind turbines using four optimization algorithms in both connected and isolated AC microgrid scenarios [31]. Their study emphasized the importance of rigorous statistical validation (via ANOVA and Tukey’s post hoc tests). Furthermore, Lobos-Cornejo et al. [32] proposed a comprehensive energy management strategy for hybrid AC MGs incorporating BESS, wind turbines, and D-STATCOMs. By applying the Gray Wolf Optimizer in conjunction with successive approximations, their approach achieved notable cost reductions and voltage profile improvements. These studies reinforce the importance of metaheuristic-based master–slave frameworks for multi-device coordination in MGs. 1.3. Research objectives and contributions The need to develop BESS energy management strategies within MGs is identified based on the above works. These strategies aim to improve operational conditions, technical (energy losses), economic (energy cost), and environmental (𝐶𝑂2emissions), through intelligent management of the available energy resources. The proposed strategies must adapt to multiple network topologies and variable generation and demand behaviors in both connected and isolated modes. Additionally, optimization methodologies should yield high-quality solutions with adequate repeatability and reduced processing times. Based on this motivation, the main research objectives of this study are as follows: •Apply the GNDO to the BESS scheduling problem in AC MGs, demonstrating its suitability through comparison with established optimization methods. •Formulate a unified energy management framework to optimize operating costs, energy losses, and 𝐶𝑂2emissions under both on-grid and off-grid conditions. •Compare the GNDO algorithm with three widely used optimization methods (CGA, PSO, and JAYA) under identical conditions to assess solution quality, robustness, and efficiency. •Validate the feasibility of the proposed EMS strategies with respect to technical-operational constraints such as nodal voltages, line currents, and battery SoC limits. The specific quantitative performance improvements resulting from this methodology (such as reductions in costs, losses, and emissions) are discussed in the results and conclusion sections. The remainder of this article is organized as follows: Section 2details the mathematical formulation; Section 3presents the proposed methodology and comparison algorithms; Section 4describes the test scenarios; Section 5discusses simulation results; and Section 6outlines conclusions and future work. Results in Engineering 27 (2025) 106005 4 M.A. Llanos-Pino, L.F. Grisales-Noreña, D. Sanin-Villa et al. 2. Mathematical formulation This section develops a mathematical formulation representing the energy management problem of the batteries installed in the [18] MG. It describes the objective functions that represent the operating conditions of the grid in terms of cost, energy losses, and 𝐶𝑂2emissions. As well as the technical and operational constraints of the MG. The mathematical representation described in this section allows modeling the operation of energy storage systems in both on-grid and off-grid MGs, considering the variations between both operating modes. In the case of the off-grid system, it is impossible to sell energy, so the modeling must avoid a power flow to the diesel generator, which acts as the main generation source of MG. On the other hand, in the on-grid mode, there can be power transfer between the MG and the grid. In this context, the term objective function refers to the mathematical expression that evaluates system performance according to one or more predefined criteria, such as economic cost, energy losses, or 𝐶𝑂2emissions, which are to be minimized during optimization. The adaptation function, by contrast, is an internal formulation used by the optimization algorithm. It extends the objective function by adding penalty terms for constraint violations, thereby guiding the search process toward technically feasible and optimal solutions. 2.1. Objective functions For this problem, three objective functions are defined, which seek to improve the economic, technical, and environmental indicators of an alternating current microgrid. These functions focus on minimizing the operating costs of the MG (including the purchase of energy and maintenance of distributed energy devices), the reduction of losses associated with energy transportation, and 𝐶𝑂2emissions from conventional generation sources, respectively. 𝐹𝑂1=𝑚𝑖𝑛(𝑓𝑐 ⏟ ⏟ ⏟ 𝑜𝑝𝑒𝑟𝑎𝑡𝑖𝑜𝑛 + 𝑀𝑎𝑖𝑛𝑡𝑒𝑛𝑎𝑛𝑐𝑒 ⏞ ⏞ ⏞ 𝑓𝑀)(1) The first objective function analyzed (𝐹𝑂1) is associated with the economic indicator and is described in Equation (1). It seeks to minimize the energy costs associated with purchasing conventional generators (grid or diesel) and the cost of DER maintenance. This function is constituted by two elements: 𝑓𝑐and 𝑓𝑀. The element 𝑓𝑐is associated with the MG’s purchase/sale of energy to conventional generators. Meanwhile, 𝑓𝑀is associated with the maintenance costs of the MG equipment, i.e., the BESS and DG elements. 𝑓𝑐=⎛⎜⎜⎝∑ ℎ∈Ω∑ 𝑖∈Ω𝑐𝑔 𝐶𝑜𝑠𝑡𝑠𝑐𝑔 𝑖,ℎ ⋅𝑃𝑐𝑔 𝑖,ℎ ⋅Δ𝑡⎞⎟⎟⎠ , {∀𝑖∈Ω 𝑐𝑔,∀ℎ∈Ω }(2) Equation (2)represents the costs of purchasing energy from the grid (on-grid mode) or the production of energy by generation sources that operate based on polluting sources (off-grid mode). In this equation, 𝐶𝑜𝑠𝑡𝑠𝑐𝑔 𝑖,ℎ corresponds to the costs of buying or selling energy from or to the conventional generator located at node 𝑖in hour ℎ, 𝑃𝑐𝑔 𝑖,ℎ corresponds to the power generated by the conventional generator located at node 𝑖 in hour ℎ. During the operation of the system under study and depending on each operator (remember that this is a requirement of the MG), a period Δ𝑡is chosen to analyze the data, i.e., the duration of the periods of duration of the generation, demand and battery charge and discharge actions is selected. Within this equation, the variable Ω𝑐𝑔 represents the set of conventional generators in the grid, Ωrepresents the periods in daily operation and Ω𝑔𝑑 the set of PV generators installed in the system. 𝑓𝑀=⎛⎜⎜⎝∑ ℎ∈∑ 𝑖∈Ω𝐵|𝑝𝐵 𝑖,ℎ|⋅𝐶𝐵 𝑖,𝑂&𝑀 ⋅Δ𝑡+∑ ℎ∈∑ 𝑖∈Ω𝑔𝑑 𝑝𝑔𝑑 𝑖,ℎ ⋅𝐶𝑔𝑑 𝑖,𝑂&𝑀 ⋅Δ𝑡⎞⎟⎟⎠ , {∀𝑖∈Ω 𝐵,∀𝑖∈Ω 𝑔𝑑,∀ℎ∈Ω } (3) The maintenance costs of DERs installed within the grid are represented in the Equation (3). In this equation, 𝑝𝐵 𝑖,ℎ denotes the power stored or delivered by the battery located at node 𝑖in hour ℎ, 𝐶𝐵 𝑖,𝑂&𝑀corresponds to the maintenance cost per kW of the battery located at node 𝑖, 𝑝𝑔𝑑 𝑖,ℎ is associated to the active power injected by the DG located at node 𝑖in hour ℎ, finally, 𝐶𝑔𝑑 𝑖,𝑂&𝑀corresponds to the maintenance cost per kW of the PV generator located at node 𝑖;. At the same time, Ω𝐵represents the set of batteries installed in the system. 𝐹𝑂2=𝑚𝑖𝑛⎛⎜⎜⎝∑ ℎ∈Ω∑ 𝑖∈Ω∑ 𝑗∈Ω 𝑌𝑖𝑗 ⋅𝑣𝑖,ℎ ⋅𝑣𝑗,ℎ ⋅𝑐𝑜𝑠(𝜃𝑖,ℎ −𝜃𝑗,ℎ −𝜑𝑖𝑗 )⋅Δ𝑡⎞⎟⎟⎠ , {∀𝑖∈Ω ,∀ℎ∈Ω } (4) The second objective function (𝐹𝑂2) seeks to minimize the energy losses within the network, which correspond to the active power lost when transporting energy through the lines, which is dissipated by the Joule effect in the lines. This objective function is described by the Equation (4), where 𝑌𝑖𝑗 corresponds to the magnitude of the admittance associated to the line connecting nodes 𝑖with node 𝑗, 𝑣𝑖,ℎ and 𝑣𝑗,ℎ are associated to the voltage magnitudes at node 𝑖and node 𝑗in hour ℎrespectively, 𝜃𝑖,ℎ and 𝜃𝑗,ℎ correspond to the voltage angles at node 𝑖and node 𝑗at time ℎrespectively, finally, 𝜑defines the angle of the admittance of the line interconnecting nodes 𝑖with node 𝑗and Ωrepresents the set of nodes that conform the system. 𝐹𝑂3=𝑚𝑖𝑛⎛⎜⎜⎝∑ ℎ∈Ω∑ 𝑖∈Ω𝑐𝑔 𝑃𝑐𝑔 𝑖,ℎ ⋅𝐶𝐸𝑐𝑔 𝑖 ⋅Δ𝑡+ ∑ ℎ∈Ω∑ 𝑖∈Ω𝑔𝑑 𝑃𝑔𝑑 𝑖 ⋅𝐶𝑔𝑑 ℎ ⋅𝐶𝐸𝑔𝑑 𝑖 ⋅Δ𝑡⎞⎟⎟⎠ , {∀𝑖∈Ω 𝑐𝑔,∀𝑖∈Ω 𝑔𝑑,∀ℎ∈Ω } (5) The third objective function, 𝐹𝑂3, is presented in Equation (5). This seeks to minimize 𝐶𝑂2emissions to the environment by reducing the use of conventional generators and prioritizing the consumption supplied by PV generators or energy stored in the BESS. Within this equation, 𝐶𝐸𝑐𝑔 𝑖corresponds to the emission coefficient of the conventional generator located at node 𝑖, 𝑃𝑔𝑑 𝑖is associated with the nominal power of the PV generator located at node 𝑖, 𝐶𝑔𝑑 ℎdefines the parameter that describes the behavior of the expected production curve for the PV generator in hour ℎ, finally, 𝐶𝐸𝑔𝑑 𝑖corresponds to the emission coefficient of the PV generator located at node 𝑖. It is important to note that the PV distributed generators integrated within this model consider the operation at the maximum power point, as their operation is conventionally within the MGs [33]. 2.2. Technical and operational constraints of the AC MG The intelligent operation of BESS is subject to multiple constraints associated with the equipment and operation of the MG. First, the constraints associated with the network under study and the generation elements are analyzed, and then a similar analysis is performed for the constraints associated with the BESS. 𝑝𝑐𝑔 𝑖,ℎ +𝑝𝑔𝑑 𝑖,ℎ ±𝑝𝐵 𝑖,ℎ −𝑃𝑑 𝑖,ℎ =𝑣𝑖,ℎ ∑ 𝑗∈Ω 𝑌𝑖𝑗 ⋅𝑣𝑗,ℎ ⋅𝑐𝑜𝑠(𝜃𝑖,ℎ −𝜃𝑗,ℎ −𝜑𝑖𝑗 ), {∀𝑖∈Ω ,∀ℎ∈Ω } (6) Results in Engineering 27 (2025) 106005 5 M.A. Llanos-Pino, L.F. Grisales-Noreña, D. Sanin-Villa et al. 𝑞𝑐𝑔 𝑖,ℎ −𝑄𝑑 𝑖,ℎ =𝑣𝑖,ℎ ∑ 𝑗∈Ω 𝑌𝑖𝑗 ⋅𝑣𝑖,ℎ ⋅𝑠𝑖𝑛(𝜃𝑖,ℎ −𝜃𝑗,ℎ −𝜑𝑖𝑗), {∀𝑖∈Ω ,∀ℎ∈Ω } (7) Equations (6)and (7)represent the global power balance of active and reactive power, respectively. In Equation (6), 𝑃𝑑 𝑖,ℎ corresponds to the active power demanded at node 𝑖in hour ℎ. In the case of Equation (7), 𝑞𝑐𝑔 𝑖,ℎ corresponds to the reactive power injected by the conventional generator located at node 𝑖in hour ℎ, 𝑄𝑑 𝑖,ℎ is associated to the reactive power demanded at node 𝑖in hour ℎ. 𝑃𝑐𝑔,𝑚𝑖𝑛 𝑖≤𝑝𝑐𝑔 𝑖,ℎ ≤𝑃𝑐𝑔,𝑚𝑎𝑥 𝑖, {∀𝑖∈Ω 𝑐𝑔,∀ℎ∈Ω }(8) 𝑄𝑐𝑔,𝑚𝑖𝑛 𝑖≤𝑞𝑐𝑔 𝑖,ℎ ≤𝑄𝑐𝑔,𝑚𝑎𝑥 𝑖, {∀𝑖∈Ω 𝑐𝑔,∀ℎ∈Ω }(9) The equations (8)and (9)establish the operating limits for conventional generators for both their active and reactive power, respectively. In these equations, 𝑃𝑐𝑔,min 𝑖and 𝑃𝑐𝑔,max 𝑖correspond to the minimum and maximum active power injected by the conventional generator located at node 𝑖. Similarly, 𝑄𝑐𝑔,min 𝑖and 𝑄𝑐𝑔,max 𝑖represent the minimum and maximum reactive power injected by the conventional generator located at the same node and time period. 𝑃𝑔𝑑,𝑚𝑖𝑛 𝑖≤𝑝𝑔𝑑 𝑖,ℎ ≤𝑃𝑔𝑑,𝑚𝑎𝑥 𝑖, {∀𝑖∈Ω 𝑔𝑑,∀ℎ∈Ω }(10) Equation (10)establishes the operating limits for DGs, in this case PV, concerning their active power. In the present study, their operation with the unity power factor is considered. In this equation, 𝑃𝑔𝑑,min 𝑖 and 𝑃𝑔𝑑,max 𝑖correspond to the minimum and maximum active power injected by the DG located at node 𝑖. 𝑉𝑚𝑖𝑛 𝑖≤𝑣𝑖,ℎ ≤𝑉𝑚𝑎𝑥 𝑖, {∀𝑖∈Ω ,∀ℎ∈Ω }(11) |𝐼𝑖𝑗,ℎ|≤𝐼𝑚𝑎𝑥 𝑖𝑗 , {∀𝑖∈Ω ,∀ℎ∈Ω }(12) Equations (11)and (12)establish the operating limits of the system for line voltages and currents, respectively. In Equation (11), 𝑉𝑚𝑖𝑛 𝑖and 𝑉𝑚𝑎𝑥 𝑖correspond to the minimum and maximum voltages, respectively, at node 𝑖. For the case of equation (12), 𝐼𝑖𝑗,ℎ corresponds to the current of the line connecting node 𝑖with node 𝑗at time ℎ, while 𝐼𝑚𝑎𝑥 𝑖𝑗 defines the maximum current allowed by the line connecting the nodes mentioned above. 𝑃𝑐ℎ𝑎𝑟𝑔𝑒𝑚𝑎𝑥 𝐵,𝑖 ≤𝑝𝐵 𝑖,ℎ ≤𝑃𝑑𝑖𝑠𝑐ℎ𝑎𝑟𝑔𝑒𝑚𝑎𝑥 𝐵,𝑖 , {∀𝑖∈Ω 𝐵,∀ℎ∈Ω }(13) Equation (13)represents the operating limits for charging and discharging batteries, which is essential to preserve their integrity throughout their lifetime. In this equation, 𝑃chargemax 𝐵,𝑖 and 𝑃dischargemax 𝐵,𝑖 correspond to the maximum allowable charge and discharge power for the battery located at node 𝑖. 𝑃𝑑𝑖𝑠𝑐ℎ𝑎𝑟𝑔𝑒𝑚𝑎𝑥 𝐵,𝑖 =𝐶𝐵 𝑖 𝑡𝑑𝐵 𝑖 , {∀𝑖∈Ω 𝐵}(14) 𝑃𝑐ℎ𝑎𝑟𝑔𝑒𝑚𝑎𝑥 𝐵,𝑖 =−𝐶𝐵 𝑖 𝑡𝑐𝐵 𝑖 , {∀𝑖∈Ω 𝐵}(15) Equations (14)and (15) present the way to calculate the maximum active power discharge and charge, respectively, which allows defining the limits previously established in Equation (13). In these equations, 𝐶𝐵 𝑖corresponds to the nominal capacity in kWh of the battery located at node 𝑖, while 𝑡𝑑𝐵 𝑖is associated with the battery discharge time interval at node 𝑖. On the other hand, in Equation (15), 𝑡𝑐𝐵 𝑖represents the charging time interval of the battery located at node 𝑖. 𝑆𝑂𝐶𝐵 𝑖,ℎ =𝑆𝑂𝐶𝐵 𝑖,ℎ−1 −𝜙𝐵 𝑖 ⋅𝑝𝐵 𝑖,ℎ ⋅Δ𝑡, {∀𝑖∈Ω 𝐵,∀ℎ∈Ω }(16) Equation (16)describes how to calculate the SoC of the battery located at node 𝑖for period ℎ. In this equation, 𝑆𝑜𝐶𝐵 𝑖,ℎ and 𝑆𝑜𝐶𝐵 𝑖,ℎ−1 represent the current and previous SoC of the battery, respectively. On the other hand, 𝜙𝐵 𝑖denotes the charge/discharge coefficient of the battery located at node 𝑖. 𝜙𝐵 𝑖=1 𝑡𝑑𝐵 𝑖 ⋅𝑃𝑑𝑖𝑠𝑐ℎ𝑎𝑟𝑔𝑒𝑚𝑎𝑥 𝐵,𝑖 =1 𝑡𝑐𝐵 𝑖 ⋅𝑃𝑐ℎ𝑎𝑟𝑔𝑒𝑚𝑎𝑥 𝐵,𝑖 , {∀𝑖∈Ω 𝐵,∀ℎ∈Ω } (17) Equation (17)presents the way to calculate 𝜙𝐵 𝑖, which is inversely proportional to the maximum active charging and discharging powers, multiplied by the charging (𝑡𝑐𝐵 𝑖) and discharging (𝑡𝑑𝐵 𝑖) times of the batteries located at node 𝑖. 𝑆𝑂𝐶𝐵 ℎ=0 =𝑆𝑂𝐶0 𝑖, {∀𝑖∈Ω 𝐵}(18) 𝑆𝑂𝐶𝐵 ℎ=24 =𝑆𝑂𝐶𝑓 𝑖, {∀𝑖∈Ω 𝐵}(19) Equations (18)and (19)establish the initial and final states of the SoC. It is worth mentioning that both will have a value of 0.5for each battery in the system, as shown in Fig. 1. This is associated with an IEEE recommendation that allows the potentiation of the effect of the batteries on the economic, technical, and environmental indicators of the MG. In these equations, 𝑆𝑜𝐶0 𝑖and 𝑆𝑜𝐶𝐹 𝑖correspond to the initial and final SoC of the battery located at node 𝑖. 𝑆𝑂𝐶𝐵,𝑚𝑖𝑛 𝑖≤𝑆𝑂𝐶𝐵 𝑖,ℎ ≤𝑆𝑂𝐶𝐵,𝑚𝑎𝑥 𝑖, {∀𝑖∈Ω 𝐵,∀ℎ∈Ω }(20) Equation (20)presents the operating limits associated with the SoC of the batteries in the system. For the problem to be addressed, variations between 10% and 90% for the SoC correspond to lithium-ion batteries [34]. In this equation, 𝑆𝑂𝐶𝐵,min 𝑖and 𝑆𝑂𝐶𝐵,max 𝑖represent the minimum and maximum SoC. 3. Proposed methodology This section describes the proposed solution methodology to optimize the operation of the BESS in AC MGs, with the objective of improving the economic, technical and environmental indicators of the network. The strategy used corresponds to a “master-slave” type approach, widely employed in nonlinear optimization problems such as the one addressed in this paper [18]. This methodology divides the problem into two stages: the master stage, responsible for the intelligent operation of the batteries, and the slave stage, which evaluates the hourly load flow to analyze the impact of the proposed operation on the objective function and the problem constraints. This division allows the problem to be addressed efficiently, ensuring high-quality solutions and reduced processing times. 3.1. Coding The coding proposes possible solutions to the operation problem by assigning values to the power dispatch in each hour of operation within the time horizon considered in the analysis, which comprises a total of 24 hours. Fig. 1shows the coding to model the operating scheme of power injected or absorbed by the batteries within the electrical grid in each period contained in an average day of operation, in this particular case 24 hours. The power levels assigned within each hour of operation for the BESS integrated within the MG are established in a controlled manner within the maximum charge and discharge power ranges set by the parameters of each battery used. In the case of this example, three batteries installed within the grid are considered, which have 24 power levels within the coding for each battery in the analysis hours, generating a coding of size 1𝑥72, where the 72 is obtained by multiplying the 24-hour operation horizon in 1-hour study periods by the number of batteries installed within the MG. It is relevant to mention that the operation of the BESS requires constant supervision by the algorithm, since, as previously discussed in this study, the SoC must move between 10% and 90%, so that if this condition is not met in a certain time window, Results in Engineering 27 (2025) 106005 6 M.A. Llanos-Pino, L.F. Grisales-Noreña, D. Sanin-Villa et al. Fig. 1. Proposed coding for the problem of battery operation in MGs [18]. the algorithm would be running incorrectly. Additionally, the batteries must have an SoC equivalent to 50% at the beginning and end of their daily operation, following the IEEE recommendation described in [18]. The generation of individuals using the proposed coding satisfies all of the above, so the SoC violation is not integrated within the adaptation function. 3.2. Master-slave strategy The master-slave strategy allows the problem to be divided into two stages. The master stage proposes the power schemes for the charging and discharging actions of the BESSs inside the MG. On the other hand, the slave stage is in charge of evaluating the quality of the solutions executing an hourly load flow, which, in this case, makes use of the SA method to validate the effect of the operation proposed by the master stage on the objective function and set of constraints that represent the problem, which are grouped in an adaptation function that penalizes the objective function when any constraint is violated, adding a value that is directly proportional to the violation by the proposed operating scheme to the batteries. The adaptation function comprises the objective function and the sum of the penalties since this paper addresses a minimization problem. The penalties included in this paper are associated with violating nodal voltage limits, line current limits, and power sales to the grid in the case of the off-grid system. When the penalty associated with the three elements described above is zero, the adaptation function is equivalent to the objective function. The adaptation function used by this methodology is presented in the following equation: 𝐹𝐴=𝐹𝑂+𝐹𝑝1⋅𝑉+𝐹𝑝2⋅𝐼+𝐹𝑝3⋅𝑉𝐸 (21) Where 𝐹𝐴 represents the adaptation function, 𝐹𝑂𝑖defines the objective function used, 𝐹𝑝1, 𝐹𝑝2, and 𝐹𝑝3represent penalty factors for each electrical parameter that could violate the established limits, which take a value of 1000 in this case, this value is set heuristically. Finally, 𝑉, 𝐼, and 𝑉𝐸correspond to the nodal voltage, current, and energy sale violations in the slack node, respectively. These variables are penalized if they exceed their respective limits as established in the corresponding regulations. Equation (21)operates based on the violations that occur in the operation of the system, subsequently obtaining the product with its respective penalty factor and generating that the adaptation function greatly increases its value, implying that the solution is less feasible since it is penalized. 3.3. Master stage The master stage executes the optimization algorithms that allow generating the potential solutions to the proposed optimization problem, i.e., identifying the loading and unloading schemes of the BESS installed within the MG to improve the economic, technical, and environmental conditions. This is done by generating daily operation schemes for the BESS. This stage is closely linked to the slave stage (adaptation function) since, depending on this adaptation function, the master stage will allow the optimization algorithms to seek to maximize or minimize a certain system variable. Within the master stage, metaheuristic optimization techniques were selected as an optimization method to solve the problem of intelligent operation of BESS in AC MG. These are the most widely implemented in the operation of distributed energy resources in electrical grids due to their simplicity and high efficiency. This type of method is divided into different families: evolutionary algorithms where the Population Genetic Algorithm (PGA) [35] belongs, algorithms based on probabilities as the GNDO [36], bio-inspired algorithms based on populations as the PSO algorithm, hybrid algorithms as the JAYA optimization algorithm [37]. Being proposed as a solution methodology in the particular case of this research the GNDO, while the JAYA, CGA and PSO were reported in the literature to solve the problem addressed here, being employed as comparison methods. The respective citations detail all the information related to the description of the comparison methods and their application to the problem. The GNDO optimization algorithm applied to solve the master stage is described below. 3.3.1. Generalized Normal Distribution Optimization algorithm The GNDO is based on the Gaussian-type probability distribution, also known as a normal distribution. In this context, the characteristic parameters of this distribution are essential in the exploration of the solution space by the algorithm, mainly the mean (𝜇) and the standard deviation (𝜎) [38]. The algorithm uses two types of searches, a local one (exploitation) and a global one (exploration), allowing it to explore the solution space widely. It is important to note that for each individual in the population, the GNDO randomly selects the individual update method, generating a random number 𝛿, which, if it is less than or equal to 0.5, the possible replacement individual is generated using exploitation, on the contrary, if it is greater than or equal to this value, exploration is used. When it is decided to generate the individual by applying exploitation, it advances according to the quality of the solutions contained in the current population. One of the first relevant expressions for the algorithm, when using this type of advancement, is the overall mean of the population, which is calculated as follows: 𝑦𝑡=1 𝑛𝑠 𝑛𝑠 ∑ 𝑗=1 𝑥𝑡 𝑗(22) In Equation (22), 𝑦𝑡corresponds to the new individual associated with iteration 𝑡, 𝑛𝑠represents the number of individuals in the population and 𝑥𝑡 𝑗defines the j-th individual of iteration 𝑡. Considering that the best solution within a given population is available, along with its associated standard deviation, this study calculates the generalized variables for each case (mean and standard deviation): 𝜇𝑡 𝑗=1 3(𝑥𝑡 𝑗+𝑥𝑡 𝑏𝑒𝑠𝑡 +𝑦𝑡)(23) 𝜎𝑡 𝑗=√((𝑥𝑡 𝑗−𝜇𝑡 𝑗)2 +(𝑥𝑡 𝑏𝑒𝑠𝑡 −𝜇𝑡 𝑗)2 +(𝑦𝑡−𝜇𝑡 𝑗)2)(24) In Equation (23), 𝜇𝑡 𝑗corresponds to the generalized mean of the j-th individual at iteration 𝑡, 𝑥𝑡 𝑏𝑒𝑠𝑡 represents the best individual contained in the current population. While in the equation (24), 𝜎𝑡 𝑗corresponds to the generalized standard deviation of the j-th individual at iteration 𝑡. For the local search, the potential individual that is part of the solution to the problem posed is constructed from the following expression: 𝑣𝑡 𝑗=𝜇𝑡 𝑗+𝜎𝑡 𝑗 ⋅𝜂(25) In Equation (25), 𝑣𝑡 𝑗corresponds to the j-th potential individual in the iteration 𝑡that gives a solution to the problem, 𝜂defines a factor that allows to give more or less relevance to the standard deviation in the generation of the potential individual 𝑣𝑡 𝑗. It is worth mentioning that Results in Engineering 27 (2025) 106005 7 M.A. Llanos-Pino, L.F. Grisales-Noreña, D. Sanin-Villa et al. the product shown in the equation (25)corresponds to the term-by-term product between both vectors. Constructing 𝜂as follows: 𝜂={√−𝑙𝑜𝑔(𝜆1)𝑐𝑜𝑠(2𝜋𝜆2)𝑟1≤𝑟2 √−𝑙𝑜𝑔(𝜆1)𝑐𝑜𝑠(2𝜋𝜆2+𝜋)𝑟1>𝑟 2 (26) Where 𝜆1and 𝜆2are 72x1 vectors with elements whose values are uniformly distributed within the range 0 and 1, while 𝑟1and 𝑟2are randomly generated scalars between 0 and 1. The global search, exploration based on random values and current population information of the GNDO algorithm, allows the algorithm to escape from local optima by analyzing areas of the solution space that have not yet been explored. This global search is performed from three points on the population, 𝑝1, 𝑝2and 𝑝3, which must be different from each other, different from the j-th individual under analysis and must belong to the range [0 𝑁𝐼], where 𝑁𝐼 represents the number of individuals. After originating the points associated with the global search, two tracking vectors, named 𝑣1and 𝑣2, are defined: 𝑣1={𝑥𝑡 𝑗−𝑥𝑡 𝑝1𝑓(𝑥𝑡 𝑗)≤𝑓(𝑥𝑡 𝑝1) 𝑥𝑡 𝑝1−𝑥𝑡 𝑗𝑓(𝑥𝑡 𝑗)>𝑓(𝑥𝑡 𝑝1)(27) 𝑣2={𝑥𝑡 𝑝2−𝑥𝑡 𝑝3𝑓(𝑥𝑡 𝑝2)≤𝑓(𝑥𝑡 𝑝3) 𝑥𝑡 𝑝3−𝑥𝑡 𝑝2𝑓(𝑥𝑡 𝑝3)>𝑓(𝑥𝑡 𝑝2)(28) In Equation (27), 𝑥𝑡 𝑝1corresponds to the individual associated with row 𝑝1of the population at iteration 𝑡, while 𝑓(𝑥𝑡 𝑗)and 𝑓(𝑥𝑡 𝑝1)are the adaptive functions of the j-th and 𝑝1individual respectively for iteration 𝑡. In equation (28), 𝑥𝑡 𝑝2and 𝑥𝑡 𝑝3correspond to the individuals in rows 𝑝2and 𝑝3respectively for iteration 𝑡, while 𝑓(𝑥𝑡 𝑝2)and 𝑓(𝑥𝑡 𝑝3)represent the adaptive functions of 𝑝2and 𝑝3respectively for iteration 𝑡. Once the tracking vectors are defined, the potential solution 𝑣𝑡 𝑗is obtained: 𝑣𝑡 𝑗=𝑥𝑡 𝑗+𝛽⋅(|𝜆3|⋅𝑣1)+(1−𝛽)⋅(|𝜆4|⋅𝑣2)(29) Where 𝜆3, 𝜆4and 𝛽are random scalar values between 0 and 1 that allow defining the importance of each tracking vector. Once the potential solution of the respective search is obtained for the j-th individual of the population under analysis using exploration or exploitation, it is compared with the maximum and minimum limits of the decision variable of the problem, in this case, the power limits for each battery. In case the potential solution exceeds the upper or lower limit, it is replaced by the upper or lower limit, respectively. Finally, the adaptation function of the potential replacement solution is compared with that of the current individual. Replacing it within the population 𝑥𝑡 𝑗in the case that it improves the adaptation function. The GNDO applies this process to each individual in the population, updating the best solution for each iteration. It converges when the maximum number of iterations is reached, presenting only the population size and the maximum number of iterations as optimization parameters. Algorithm 1describes in detail the iterative process of GNDO. This algorithm generates in the first iteration a population of individuals at random 𝑥𝑡. Subsequently, it evaluates the adaptation function of each individual using the slave stage from parallel processing [39], proceeding with these values to identify the individuals that generate the best solution within the current population. Subsequently, it creates a random variable called 𝛿, which allows to select the type of search performed, in case 𝛿is less than or equal to 1∕2, the search will be local, otherwise, the search will be global. In the case of the local search, the characteristic parameters of the normal distribution, i.e., mean and standard deviation, are calculated first, then the variable 𝜂is calculated as described in the equation (26), and finally obtain the potential solution of the problem. When the search to be performed is of global type, the tracking vectors are first generated as described in the equations (27) and (28); after this, the potential solution of the problem is calculated. Data: Read system electrical data, generation power, demand curves and algorithm parameters. Define the characteristics of the equipment; Define the algorithm parameters; for iter =1to 𝑁𝑖𝑡𝑒𝑟 do if iter =1then Create the initial population; Evaluate the adaptive function for each individual in the population using the slave stage ←→ Parallel processing; Determine the best solution for the population; else Generate a random value for 𝛿; if 𝛿≤1 2then Calculate the relevant parameters of the normal distribution (mean and standard deviation); Compute 𝜂according to equation (26); Calculate the potential solution according to the equation (25); elseCalculate the tracking vectors according to the following equations (27)and (28); Calculate the potential solution according to the equation (29); end Verify the limits of the potential solution; Evaluate the adaptive functions of the current individual and the potential solution using the slave stage ←→ Parallel processing; Replace individual if applicable; end end Identify the best solution from all iterations; Report the best solution found; Algorithm 1: Algorithm for the application of GNDO. Independently of the search to be performed, the feasibility of the potential solution is subsequently checked in terms of the power limits. If any of them is exceeded, the maximum or minimum power is assigned to this value, as appropriate. With the corrected values (if necessary), the potential solution is evaluated in the slave stage (load flow), allowing it to obtain its adaptation function. After obtaining the adaptation function of the potential solution, it is compared with the best solution so far, updating the population if appropriate. Finally, this process is repeated a number of times, determined by the variable 𝑁𝑖𝑡𝑒𝑟; if the maximum number of iterations is met, the algorithm ends. Otherwise, it returns to generate potential individuals according to a random search. After tuning the GNDO parameters for this paper, a number of 500 individuals and a maximum number of iterations of 2000 were obtained. 3.4. Comparison algorithms This research aims to analyze the results obtained using the proposed methodology based on the GNDO algorithm. These results will be compared with the CGA and PSO optimization methods, selected due to the excellent performance reported by the authors in solving the energy management problem in AC MGs operating in connected and isolated mode [18]. Additionally, considering the outstanding results reported in [17] for battery power management using the JAYA algorithm, this methodology is used as a benchmark for comparison in this study. The optimization parameters reported in the literature for CGA and PSO ensure proper validation and accurate comparison since the tests are performed under the same scenarios. On the other hand, for the case of JAYA, a tuning of the algorithm employing PSO was carried out, as described in [18], obtaining a number of individuals of 3000 and 2000 iterations, respectively. This same methodology was employed for the GNDO tuning. Results in Engineering 27 (2025) 106005 8 M.A. Llanos-Pino, L.F. Grisales-Noreña, D. Sanin-Villa et al. 3.5. Slave stage This stage is responsible for evaluating the effect on the adaptation function of the potential solutions proposed by the master stage within its iterative process. In practical terms, the slave stage executes an hourly load flow that calculates the impact on the objective function of the power configurations assigned to the BESSs for each individual generated by the master stage. In addition, it penalizes the objective function in case any system constraint is exceeded, such as voltage limits, line current or power violations at the slack node, especially in the case of isolated networks. The hourly load flow used seeks to find the magnitude and angle of voltage in the different bars of the system, as well as the active and reactive power flowing through each line of the system. The load flow studies are relevant for the planning of the operation and design of the expansion of power systems, as well as for the determination of the best operating conditions for a given indicator [40]. In the context of this study, the load flow analysis enables the validation of the impact of the power references assigned to the batteries during hourly operation, for which in each hour of operation, a load flow should be executed to validate the impact on the objective function and constraints, in such a way that the impact on the objective function and constraints of a full day of operation can be condensed in an adaptive function. This analysis is known in the literature as hourly load flow. The main load flow methods used to solve the hourly load flow problem are described below. Regarding the resolution of the hourly load flow problem in AC electrical networks, the method of SA presents excellent results, greatly decreasing the computation times exposed by the classical methods and other modern methods of the literature [41]; therefore, it is the method selected to validate the hourly load flow of the charging and discharging actions of the batteries proposed by the energy management strategies developed in this study. Data: Read system electrical data and algorithm parameters. for h=1:24 do Charging the power demand of the loads during the h-hour; Charging the power generated by the PV generators during the h-hour; Charge the power supplied or demanded by the batteries in the h-hour; Solve the load flow AC problem for the h-hour; Calculate the objective function associated with the h-hour; Evaluate the adaptation function at the h-hour; end Sum of the adaptation function values obtained for the contained period within the analyzed time horizon; Return to the master stage the adaptation function value corresponding to the battery power dispatch proposed by the individual under analysis; Algorithm 2: Hourly load flow based on Successive Approximations. The Algorithm 2, describes the iterative process of the SA method. It starts its execution by defining the system electrical data and algorithm parameters, which in this case corresponds to 1000 iterations and a convergence error of 1𝑥10−10. After reading the data, the algorithm begins to iterate by loading the hourly demand and PV generation data, as well as the power levels assigned by the individual under analysis for charging and discharging the batteries. Subsequently, the load flow problem is processed to be solved with the SA-based load flow method in each operating period contained in the analyzed time horizon. Then, for each operation period, the effect on the objective function and grid constraints are calculated, which will be condensed into an adaptive function that stores such data for the operation of an average operating day. In this way, the effect of the loading and unloading actions of the BESS on the network is evaluated under a generation and demand scenario projected for the following day. This value finally returns to the Table 1 Parameters according to battery type. Type Capacity [kWh] Charging time [h] Discharge time [h] A 1000 4 4 B 1500 4 4 C 2000 5 5 master stage to identify the quality of the operation of the condensed batteries in each individual generated by the master stage, allowing this stage to make decisions based on this value within its iterative process. 4. Test scenarios and operating conditions Two test systems are used in this paper, associated with an MG in on-grid operation and an off-grid MG operating with a diesel generator. The mentioned systems have the participation of conventional generators with active and reactive power injection, PV generators with active power injection operating with maximum power point tracking, and with BESS operating considering a unity power factor using three different types of lithium-ion batteries, being both test systems reported in [18]. The dynamics of the two systems considered for the study come from Colombia, more specifically from Medellín (on-grid) and Capurganá (off-grid); this considers elements such as demand curves and PV generation, among others. Table 1presents the type, capacity in kWh, charge, and discharge time in hours of the three batteries used, with which it is possible to model the operation of the batteries within the electric MG. The simulation scenarios developed in this study are based on realworld conditions to ensure the practical applicability and replicability of the proposed energy management methodology. Specifically, both PV generation and demand profiles were derived from monitored or modeled data associated with actual microgrid deployments in Colombia [42]. For the urban (on-grid) microgrid, representing conditions in Medellín, solar radiation and ambient temperature data were retrieved from the NASA database [43], using hourly resolution values. These data were averaged to construct a representative PV generation profile for a typical day. The power consumption profile for Medellín was based on historical demand data provided by the local utility Empresas Públicas de Medellín (EPM) [44], also processed on an hourly basis and averaged to obtain a daily consumption pattern. For the rural (off-grid) microgrid, modeled after the locality of Capurganá, Colombia, demand data were obtained from historical monitoring reports provided by the Instituto de Planificación y Promoción de Soluciones Energéticas para las Zonas No Interconectadas (IPSE) [42]. Similar to the urban case, hourly demand data were averaged to construct a representative daily consumption profile. Additionally, the photovoltaic (PV) generation profile for Capurganá was developed using the same methodology applied in the on-grid scenario, combining irradiance and temperature data with standard PV performance models. This modeling approach ensures that both test systems—on-grid and off-grid—are based on realistic operating conditions observed in Colombian microgrids, thereby supporting the validity and practical relevance of the proposed energy management strategy. 4.1. On-grid microgrid Fig. 2presents the system associated with the 33-node on-grid MG. This system consists of 33 nodes and 32 lines. It operates with a base voltage of 12.66 kV and a base power of 100 kW. The system has three PV generators located at nodes 12, 25, and 30 with nominal capacities of 1125, 1320, and 999 kW, respectively, which operate with maximum power point tracking. In the case of the BESS, these are lithium-ion technology and are located at nodes 6 (Type C), 14 (Type A) and 31 (Type B). Results in Engineering 27 (2025) 106005 9 M.A. Llanos-Pino, L.F. Grisales-Noreña, D. Sanin-Villa et al. Fig. 2. Electrical diagram of 33-node on-grid test system. Table 2 Technical parameters of the 33-node on-grid microgrid. Line Node 𝑖Node 𝑗 𝑅𝑖𝑗 (Ω) 𝑋𝑖𝑗 (Ω) 𝑃𝑗(kW) 𝑄𝑗(kVAr) 𝐼𝑚𝑎𝑥 𝑗(A) 1 1 2 0,0922 0,0477 100 60 385 2 2 3 0,4930 0,2511 90 40 355 3 3 4 0,3660 0,1864 120 80 240 4 4 5 0,3811 0,1941 60 30 240 5 5 6 0,8190 0,7070 60 20 240 6 6 7 0,1872 0,6188 200 100 110 7 7 8 1,7114 1,2351 200 100 85 8 8 9 1,0300 0,7400 60 20 70 9 9 10 1,0400 0,7400 60 20 70 10 10 11 0,1966 0,0650 45 30 55 11 11 12 0,3744 0,1238 60 35 55 12 12 13 1,4680 1,1550 60 35 55 13 13 14 0,5416 0,7129 120 80 40 14 14 15 0,5910 0,5260 60 10 25 15 15 16 0,7463 0,5450 60 20 20 16 16 17 1,2890 1,7210 60 20 20 17 17 18 0,7320 0,5740 90 40 20 18 2 19 0,1640 0,1565 90 40 40 19 19 20 1,5042 1,3554 90 40 25 20 20 21 0,4095 0,4784 90 40 20 21 21 22 0,7089 0,9373 90 40 20 22 3 23 0,4512 0,3083 90 50 85 23 23 24 0,8980 0,7091 420 200 85 24 24 25 0,8960 0,7011 420 200 40 25 6 26 0,2030 0,1034 60 25 125 26 26 27 0,2842 0,1447 60 25 110 27 27 28 1,0590 0,9337 60 20 110 28 28 29 0,8042 0,7006 120 70 110 29 29 30 0,5075 0,2585 200 600 95 30 30 31 0,9744 0,9630 150 70 55 31 31 32 0,3105 0,3619 210 100 30 32 32 33 0,3410 0,5302 60 40 20 Table 2presents the parameters of the lines, the power demand associated with the loads, as well as the current limits of the lines. From left to right, this table shows the line number, the sending node, the receiving node, the line resistance and reactance, the active and reactive power demanded at the receiving node, and the maximum current limit that can flow through each line of the system. The above table shows the parameters associated with the 33-node system under study, where the characteristic parameters of the lines, such as resistance, reactance, and maximum equivalent current, are shown. In contrast, for the nodes, the consumption associated with each one is presented. In addition, and due to its connection to the main grid, the costs associated with generating the MG are variable, which allows incentives from the economic point of view for the algorithms. A maximum variation of ±10% from the nominal MG voltage is considered for system operational voltage limits. Finally, Fig. 3presents the demand and PV generation curves for an average day in Medellín, Colombia. Additionally, for this scenario, the connection to the grid considers a variable cost of energy sale, whose behavior in percentage is described in Fig. 4. Furthermore, for this system, an energy cost of 0.1302 USD/kWh is considered, as well as a 𝐶𝑂2emissions factor associated to the power grid of 0.1644 kg/kWh, reported by the grid operator of Medellín, Colombia. Finally, the maintenance costs of the PV and BESS generators correspond to 0.0019 USD/kWh and 0.0017 USD/kWh, respectively. 4.2. Off-grid microgrid Fig. 5presents the system associated with the 27-node off-grid MG. This system consists of 27 nodes and 26 lines. It operates with a base voltage of 23 kV and a base power of 100 kW. The system has three Results in Engineering 27 (2025) 106005 16 M.A. Llanos-Pino, L.F. Grisales-Noreña, D. Sanin-Villa et al. Fig. 13. Worst voltages for each time period contained in an average operating day for the 27-node off-grid microgrid. In all the curves, voltage drops can be observed between nodes 7 and 21, mainly attributed to the high consumption in that region of the MG. It is important to highlight that for all the executions applied to the algorithms described above, the voltage limits established for the MG are respected. This also occurs for the CGA and PSO methodologies, which are not integrated to not contaminate the analysis and because their analysis does not contribute significantly to the state-ofthe-art. Fig. 14 shows the worst voltages obtained by the best solutions according to each objective function, considering the GNDO and JAYA algorithms in the on-grid MG. It can be observed that when contrasting the curves with those present in Fig. 13, the voltage drops are relevant in the on-grid scenario, which is essentially attributed to the higher consumption per node in the on-grid MG. Additionally, the on-grid MG has fewer branches, which causes higher current levels through the lines and, consequently, higher voltage drops in the lines. Fig. 15 shows the worst loadings obtained in each hour of system operation for the GNDO and JAYA algorithms considering the off-grid MG. The curves show that the lowest loadability coincides with the lowest system demand, which can be evidenced by contrasting Fig. 15 with Fig. 6. On the other hand, the highest loadability occurs when the PV generation of the system is maximized, which allows higher current levels through the lines and may pose a risk to the system’s operation if this constraint is not considered. However, the system always remains under the current limit (100%), allowing for proper operation. As for the voltage limits, the loadability limits are respected for all runs of the optimization algorithms. Fig. 16 shows the worst loadings in each hour of system operation for the GNDO and JAYA algorithms considering the on-grid MG. In Fig. 16(a), a significant difference is observed between the curves associated with GNDO and JAYA between hours 1 and 3, which is due to the feed-forward and convergence nature of the optimization algorithms, showing that the problem under analysis contains multiple good quality solutions. The other curves present in Fig. 16 (b and c) present similar behaviors between them, with a practically constant value between hours 8 and 16, which coincide with the points of highest demand and generation, which generates high levels of current through the system lines. As in Fig. 15, the lowest loadings correspond to the hours of lower demand since the current levels required are lower in those hours, so the system operates more loosely. As mentioned above, the system remains within operating limits at all times. The consistently superior performance of the GNDO algorithm across all objective functions can be attributed to several inherent characteristics that align well with the specific requirements of the BESS energy management problem. First, GNDO’s search mechanism, based on a combination of the population mean, the global best solution, and a generalized normal perturbation, helps maintain solution diversity throughout the iterative process. This prevents premature convergence, which is particularly important given the high dimensionality of the decision space. Second, GNDO requires minimal parameter tuning, relying only on the population size and iteration count, unlike CGA and PSO, which depend on several hyperparameters such as crossover rate, mutation rate, inertia, and acceleration coefficients. This makes GNDO more robust to sub-optimal configurations under identical experimental conditions. Third, the problem under study involves strict operational constraints (voltage, line current, and SoC), and GNDO’s adaptive step sizes facilitate effective navigation of narrow feasible regions defined by these bounds. Finally, the scenarios evaluated include variable pricing and highly dynamic demand profiles in both on-grid and off-grid configurations, conditions under which GNDO’s balance between exploration and exploitation offers a distinct advantage over the other methods tested. Therefore, the observed performance is not an artifact of implementation, but a reflection of GNDO’s intrinsic suitability for this class of optimization problems. 6. Conclusions and future work This work addressed the problem of energy management in BESS in AC MGs to improve the economic, technical, and environmental conditions of the grid. This implies applying a second and third-level control to the batteries to find the power dispatch configurations that allow op- Results in Engineering 27 (2025) 106005 17 M.A. Llanos-Pino, L.F. Grisales-Noreña, D. Sanin-Villa et al. Fig. 14. Worst voltages for each time period contained in an average operating day for the 33-node on-grid microgrid. Fig. 15. Worst loadings for each period contained in an average operating day for the 27-node off-grid microgrid. eration at minimum cost, with lower energy losses, or with lower 𝐶𝑂2 emissions rate according to the grid operator’s and MG users’ needs. In this work, the GNDO algorithm was identified as the most efficient algorithm for optimal solution, repeatability, and low processing times to solve the energy management problem in AC MGs operating in on-grid and off-grid mode. This algorithm demonstrated a significant improvement in the economic and operational conditions of the MG when compared to three high-performance optimization methodologies widely used in the literature: CGA, PSO, and JAYA. In addition to outperforming the baseline and benchmark methods, GNDO achieved average improvements of 1.4349% in operating costs, 4.4426% in energy losses, and 0.1833% in 𝐶𝑂2emissions, demonstrating Results in Engineering 27 (2025) 106005 18 M.A. Llanos-Pino, L.F. Grisales-Noreña, D. Sanin-Villa et al. Fig. 16. Worst loadings for each time period contained in an average operating day for the 33-node on-grid microgrid. its robustness and consistency across both microgrid configurations. The algorithm’s effectiveness is further supported by a statistical evaluation based on 100 independent runs, where metrics such as the best solution, average solution, standard deviation, and processing times confirmed its competitive performance. The results also validate the importance of including variable energy cost scenarios in the EMS formulation, particularly for off-grid microgrids, as this variation introduces incentives that enhance the economic impact of battery-based strategies. In the case of energy losses and 𝐶𝑂2 emissions, the impact of batteries tends to adapt to a greater extent to the behavior of energy demand and power injection from renewable generation sources. In the on-grid MG, relevant improvements are observed in the objective function of operating costs, dynamics that are not present in the off-grid MG; this is because the electric energy in the latter scenario is taken and injected to the MG at a fixed price during the entire window under study (24 hours), which does not allow the batteries to take advantage of variability in energy costs. This implies the absence of an incentive that would allow the algorithm to improve the objective function for the isolated scenario. This is evident from Tables 6and 9. Thus highlighting the importance of variable energy cost scenarios in enhancing the effect of energy management using BESS in electrical MGs. While in the case of energy losses and 𝐶𝑂2emissions, the impact of batteries tends to adapt to a greater extent to the behavior of energy demand and power injection by renewable generation sources. In terms of processing times, the proposed methodology presents reduced times, which will allow AC MG operators or owners to evaluate the greatest number of scenarios in the pre-dispatch of energy to obtain the greatest benefits from the existing energy resource within the MG, as well as from the BESS. In addition, it was verified that the proposed scenarios respected the technical-operational limitations of the MG, both in connected and isolated operation, ensuring compliance with the voltage limits, the load capacity of the lines, and the maximum load states allowed for the batteries. Thus, recognizing the solutions defined by the GNDO algorithms as feasible concerning the technical standard also applies to the other optimization methodologies. From a practical standpoint, the GNDO-based energy management strategy can be integrated into existing industrial microgrid infrastructures without requiring hardware modifications. As a software-level optimization module, it can operate within the supervisory EMS layer, using standard forecast data and grid parameters. Its output (battery dispatch schedules) can be interfaced with lower-level control devices such as inverters and PLCs, ensuring seamless implementation under existing communication protocols. The minimal tuning requirements of GNDO further enhance its appeal for real-world deployment. Based on the work developed within this study, the following lines of research and/or development in the area of energy management have been identified as potential future work: •Study GNDO for operating scenarios with different characteristics (presence of wind turbines, D-STATCOMS, etc.) •Study other algorithms that have demonstrated high performance in operating distributed energy resources in power grids. •Integrate the effects of the operation on the SoH of the batteries within the mathematical formulation, defining this variable as an objective function. •To develop and implement multi-objective optimization algorithms as solution methodologies for the proposed mathematical formulation, enabling the simultaneous improvement of economic, technical, and environmental performance indicators through the intelligent operation of integrated battery systems within the microgrid. •To study the operation of the BESS, taking advantage of the inverters to perform the injection of reagents to the network in a controlled manner to improve the impact on the indicators mentioned above. •To study the impact of variations in infrastructure-level parameters —such as battery sizing and technology, PV capacity, and the spatial allocation of distributed elements—on the performance of the microgrid, to integrate long-term design considerations into the energy management strategy. Results in Engineering 27 (2025) 106005 19 M.A. Llanos-Pino, L.F. Grisales-Noreña, D. Sanin-Villa et al. CRediT authorship contribution statement Marco Alonso Llanos-Pino: Writing – original draft, Resources, Investigation, Formal analysis, Data curation. Luis Fernando GrisalesNoreña: Writing – review & editing, Writing – original draft, Resources, Project administration, Funding acquisition, Formal analysis, Conceptualization. Daniel Sanin-Villa: Writing – review & editing, Writing – original draft, Software, Methodology, Investigation, Formal analysis, Conceptualization. Oscar Danilo Montoya: Writing – review & editing, Supervision, Methodology, Formal analysis. Jesús C. Hernández: Writing – review & editing, Supervision, Resources, Project administration, Conceptualization. Funding This research has been funded by the initiation project 2024 ANID/FONDECyT with number 11240006 and name: “Smart energy management methods for improving the economic, technical, and environmental indexes of the alternating current microgrids including variable generation and demand profiles”. Developed at the Faculty of Engineering of Universidad de Talca, within the Department of Electrical Engineering. Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgement This research has been founded by the ANID FONDECYT Iniciación 2024 Folio (11240006) under the project: “Smart energy management methods for improving the economic, technical, and environmental indexes of the alternating current microgrids including variable generation and demand profiles”. Furthermore, 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 “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). Data availability Data will be made available on request. References [1] H. Xie, S. Zheng, M. Ni, Microgrid development in China: a method for renewable energy and energy storage capacity configuration in a megawatt-level isolated microgrid, IEEE Electrif. Mag. 5 (2) (2017) 28–35, https://doi.org/10.1109/MELE.2017. 2685818. [2] R. Lasseter, Microgrids, in: Conference Proceedings (Cat. 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