A Tuned Parallel Population-Based Genetic Algorithm for BESS Operation in AC Microgrids: Minimizing Operational Costs, Power Losses, and Carbon Footprint in Grid-Connected and Islanded Topologies

Abstract

The transition to decentralized renewable energy systems has highlighted the role of AC microgrids and battery energy storage systems in achieving operational efficiency and sustainability. This study proposes an improved energy management system for AC MGs based on a tuned Parallel Population-Based Genetic Algorithm for the optimal operation of batteries under variable generation and demand. The optimization framework minimizes power losses, emissions, and economic costs through a master–slave strategy, employing hourly power flow via successive approximations for technical evaluation. A comprehensive assessment is carried out under both grid-connected and islanded operation modes using a common test bed, centered on a flexible slack bus capable of adapting to either mode. Comparative analyses against Particle Swarm Optimization and the Vortex Search Algorithm demonstrate the superior accuracy, stability, and computational efficiency of the proposed methodology. In grid-connected mode, the Parallel Population-Based Genetic Algorithm achieves average reductions of 1.421% in operational cost, 4.383% in power losses, and 0.183% in CO₂ emissions, while maintaining standard deviations below 0.02%. In islanded mode, it attains reductions of 0.131%, 4.469%, and 0.184%, respectively. The improvement in cost relative to the benchmark exact methods is 0.00158%. Simulations on a simplified 33-node AC MG with actual demand and generation profiles confirm significant improvements across all performance metrics compared to previous research works.

1. Introduction

1.1. State of the Art

The accelerated transition towards renewable energy resources and decentralized energy production has positioned Alternating Current (AC) microgrids (MGs) at the forefront of electrical infrastructure modernization. These systems effectively accommodate diverse distributed energy resources (DERs) and enhance grid resiliency against intermittent renewable energy fluctuations [1]. Contemporary research highlights AC MGs as critical infrastructures for ensuring robust, reliable, and high-quality power delivery amid increasing renewable penetration [2].

Within this evolving energy landscape, battery energy storage systems (BESSs) have emerged as pivotal elements. Recent statistics from the U.S. Department of Energy indicate that global BESS installations exceeded 206 GW by 2023, showcasing their expansive integration into electrical systems worldwide [3]. The widespread adoption of BESSs is primarily due to their versatility in providing ancillary services, including frequency regulation, voltage stabilization, and efficient demand–response mechanisms [4]. Consequently, their optimal integration within MGs has become essential to achieving environmental sustainability, system efficiency, and operational reliability.

Despite significant progress, optimal energy management within MGs, particularly in scenarios involving simultaneous coordination of generation, storage, and load, remains complex [5,6]. Recent literature has demonstrated the suitability of metaheuristic optimization techniques for handling such multifaceted problems, given their capability to navigate nonlinearities and extensive search spaces typical of these systems [7]. However, challenges persist, especially regarding the comprehensive and adaptive integration of BESSs into existing energy management strategies (EMSs) [8,9]. Thus, there remains a need for more advanced optimization approaches that can address these integration challenges.

Recent advances have made notable strides in minimizing power losses, operational costs, and CO₂ emissions in MGs using metaheuristic algorithms such as Particle Swarm Optimization (PSO), Genetic Algorithms (GA), Vortex Search Algorithm (VSA), and others [10,11]. However, many of these approaches primarily focus on generation control and grid-connected operation, often neglecting the critical coordination with energy storage units or failing to consider the microgrid’s dual-mode capability (on-grid and off-grid operation) [4,12]. This limitation results in suboptimal utilization of surplus renewable energy and restricts the applicability of proposed methods in realistic, dynamic microgrid scenarios.

Moreover, practical constraints commonly encountered in real-world EMS implementations further complicate these challenges. While exact optimization methods and commercial solvers have been proposed for BESS scheduling [13,14,15], these solutions often rely on proprietary software, limiting accessibility and adaptability. They may also face scalability issues and risk entrapment in local optima due to problem nonlinearities. In contrast, metaheuristic algorithms provide a flexible, robust, and transparent alternative that does not require problem linearization and can be implemented in widely accessible environments such as MATLAB, promoting broader adoption.

Complementing these algorithmic advances, multi-objective optimization approaches have begun integrating DERs, BESSs, and demand–response under time-of-use pricing schemes to reduce operational costs and emissions [16]. However, these models often rely on commercial solvers, predefined probabilistic models, and are restricted to grid-connected scenarios, omitting the essential dual-mode operational flexibility of MGs needed to ensure resilience in islanded modes. This gap reflects a broader challenge: while significant progress has been made in EMS optimization, many approaches still lack comprehensive coordination of BESSs across all operational modes, limiting renewable energy utilization and system robustness [12].

Recent research on integrated electricity and heat systems (IEHSs) with intelligent buildings has proposed risk-averse, decentralized energy management strategies using conditional value-at-risk (CVaR)-based stochastic optimization and parallel algorithms to handle renewable uncertainty and ensure scalability [17]. Although focused on thermal–electric systems, these methods underscore the importance of distributed and robust optimization approaches, principles equally relevant for BESS coordination in AC microgrids.

These algorithmic and implementation constraints highlight the need for advanced, scalable optimization methods such as those based on parallel computing structures to effectively address real-time EMS challenges. Developing such approaches is essential to enable practical, reliable, and flexible EMS frameworks capable of enhancing energy efficiency, reducing technical losses, and minimizing environmental impacts in real-world microgrid applications.

To tackle these challenges, Parallel Population-Based Genetic Algorithms (PPBGAs) have gained considerable attention due to their superior computational capabilities and search efficiency in large-scale, complex problems. By distributing candidate solutions across multiple parallel subpopulations that evolve independently with occasional exchange of genetic information, a PPBGA improves diversity and convergence speed. This parallelism not only reduces computational load but also enables real-time or near-real-time EMS applications, making the PPBGA a promising approach for dynamic microgrid environments [18].

A key feature of the PPBGA lies in its ability to simultaneously explore multiple solution pathways, effectively circumventing common drawbacks of classical evolutionary algorithms, such as premature convergence and limited exploration capabilities [19]. The PPBGA has demonstrated remarkable effectiveness in solving energy management tasks, outperforming classical metaheuristics by finding solutions with higher accuracy and stability across repeated runs [20]. Furthermore, its inherent parallelism aligns seamlessly with current computational infrastructures, facilitating practical deployment in real-world scenarios.

However, the literature addressing the explicit use of PPBGAs in AC MG energy management, particularly with a detailed focus on environmental impacts and technical efficiency, remains scarce, presenting a clear opportunity for further exploration. Previous studies such as [21] have applied metaheuristic methods, including Grey Wolf Optimization variants, to solve energy management problems targeting carbon footprint and technical loss reductions. Nevertheless, these strategies often fail to incorporate dual-mode operation, neglecting islanded scenarios critical to contemporary microgrid operations [22,23].

1.2. Algorithm Tuning and Its Importance

A critical yet often underexplored dimension in metaheuristic optimization’s performance lies in the algorithmic parameters’ tuning. Parameters such as population size, crossover and mutation probabilities (in genetic-based approaches), or inertia weights and learning factors (in swarm-based algorithms) are relevant in balancing exploration and exploitation throughout the search process. Improperly configured parameters can lead to premature convergence, stagnation in suboptimal regions, or excessive computational demands. While many studies apply default or arbitrarily chosen settings, recent research indicates that even minor adjustments in parameter configurations can significantly impact convergence speed, solution accuracy, and robustness [24]. As such, parameter tuning should not be regarded as a peripheral aspect exclusive to metaheuristic-based control strategies, but as a fundamental and transversal component of any control methodology applied to electrical systems. Whether one employs advanced evolutionary algorithms or classical schemes such as Proportional–Integral (PI) controllers, tuning remains essential for achieving desirable system performance. Recognizing tuning as an inherent part of the control design, rather than a secondary or arbitrary task, enables us to move beyond the misconception that convergence is a stochastic or trial-and-error process. Instead, it positions tuning as a rigorous and intentional process that ensures reliability, responsiveness, and operational excellence.

In this study, the Parallel Population-Based Genetic Algorithm (PPBGA) is subjected to a systematic tuning procedure aimed at enhancing its performance in dual-mode AC microgrid optimization tasks. The tuning process involves the use of a PSO algorithm [25], with the objective of identifying the optimal parameter set that allows each control strategy to achieve its highest possible performance. This algorithm employs a population of eight particles, cognitive and social coefficients set to 1.494, and a linearly decreasing inertia weight from one (maximum) to zero (minimum), with a stopping criterion of 300 iterations. This approach has been widely used in the literature to optimize and improve the performance of energy management methodologies applied to distributed energy resources (DERs) [26].

In this context, PSO is responsible for identifying the most suitable configuration for each instance of the control methodology by systematically exploring the parameter space. The process is executed 300 times using different initializations to ensure robustness, after which iterative refinement is performed until either convergence or the maximum number of iterations is reached. The proposed approach distinguishes itself from conventional metaheuristic implementations by explicitly addressing parameter sensitivity and conducting extensive tuning trials. This contribution is particularly relevant given the limited attention this subject receives in the literature, despite its documented influence on algorithmic reliability and reproducibility. The results presented in Section 5 confirm that well-tuned configurations of the PPBGA yield superior performance in terms of both energy loss reduction and emissions minimization, validating the necessity of thorough parameter calibration for real-world deployment.

This manuscript addresses these shortcomings by proposing an advanced PPBGA-based EMS explicitly designed to optimize BESS operations in AC MG environments, targeting a simultaneous reduction in power losses and CO₂ emissions. The proposed algorithm integrates technical constraints rigorously, considering both grid-connected and isolated operational states, thus enhancing applicability and versatility compared to existing EMS methodologies. A detailed comparative assessment against established optimization approaches, such as PSO and VSA variants, provides rigorous evidence of PPBGA’s advantages concerning solution quality, robustness, and computational efficiency.

To facilitate reader understanding of the referenced optimization techniques,

Table 1. Summary of referenced optimization algorithms and their mechanisms.

Acronym	Full Name	Optimization Mechanism
PPBGA	Parallel Population-Based Genetic Algorithm	Evolves candidate solutions using crossover, mutation, and selection across parallel subpopulations; incorporates elitism and diversity exchange to improve convergence and robustness.
PPSO	Parallel Particle Swarm Optimization	Models individual solutions as particles moving in a multidimensional space influenced by their own and neighbors’ best-known positions; parallel evaluation improves efficiency.
PVSA	Parallel Vortex Search Algorithm	Mimics the natural vortex behavior in fluid dynamics, guiding candidate solutions through a shrinking neighborhood centered around the best solution found; parallel execution accelerates convergence.

provides a brief summary of their core mechanisms and characteristics.

This study builds upon the energy system modeling framework developed in our prior works [27,28], where the 33-node AC microgrid was used as a benchmark testbed. The same network topology, operating parameters, and power flow modeling assumptions are adopted here to allow consistent comparison of optimization methods. While the electrical model remains simplified, this choice is intentional to isolate and evaluate the performance of the optimization strategies (PPBGA, PPSO, PVSA) under equivalent conditions.

The presented methodology is validated using a simplified 33-node AC microgrid based on operational data from Medellín, Colombia. Performance metrics such as robustness, convergence, and solution variability are evaluated through statistical analysis. A detailed comparison against exact optimization benchmarks, including semidefinite programming (SDP)-based solutions from the recent literature, is provided in Section 5.

It is important to emphasize that this study does not involve network reconfiguration or system expansion. The electrical infrastructure of the 33-node test system remains fixed throughout. The control variable in the proposed optimization strategy is the power schedule of the BESS units. The objective is to compare different metaheuristic algorithms for their ability to manage BESS operation effectively under two predefined operating modes: grid-connected and islanded.

The remaining sections of this manuscript are structured as follows: Section 2 provides the mathematical model formulation for the AC microgrid energy management problem, detailing system constraints for realistic operational analysis. Section 3 elaborates on the proposed PPBGA-based optimization framework. Section 4 describes the simulation setup and parameters employed. Section 5 introduces the performance-oriented tuning and comparison methodologies. Section 6 presents an extensive analysis of simulation outcomes. Section 7 provides a comprehensive discussion of the obtained results, highlighting PPBGA’s adaptability and efficiency in highly constrained problem scenarios, the performance trade-offs, and the insights drawn from the multi-scenario evaluation. Finally, Section 8 summarizes key conclusions and suggests avenues for future exploration.

2. Mathematical Formulation

This section presents the mathematical formulation used to model the optimal operation of a BESS within an AC microgrid environment. Specifically, this formulation is implemented as a time-series model with an hourly resolution that spans a 24 h operating period, capturing the variations in load and generation throughout the day. The problem formulation addresses three independent objectives: minimization of power losses, minimization of carbon dioxide (CO₂) emissions, and minimization of total economic costs, which include both operational and maintenance expenses. The placement and capacity of BESS units are assumed to be predetermined. The model is evaluated under two operating scenarios: grid-connected and islanded. The following equations present the objective functions penalized when any constraint violations occur during the optimization process.

The mathematical formulation used here closely follows the modeling framework established in [27,28]. These prior studies defined the AC power flow equations, BESS constraints, and system parameters for the 33-node test network, which has been adopted widely in related optimization research. Our work reuses this formulation to ensure reproducibility and fair benchmarking of the proposed PPBGA methodology, while extending it by integrating the operational constraints specific to the islanded mode, which were not addressed in previous works.

2.1. Minimization of Power Losses

Power losses in the distribution system are primarily attributed to Joule heating across resistive lines. Let ${\mathbf{i}}_h\in {\mathbb{R}}^L$ be the vector of line currents at time step h, and let $\mathbf{R}\in {\mathbb{R}}^{L\times L}$ denote the diagonal matrix of line resistances. The total energy loss over the horizon ${\Omega }_{\mathcal{H}}$ is given by:

$$f_{\mathrm{Loss}}=\sum _{h\in {\Omega }_{\mathcal{H}}}\Delta h·{\mathbf{i}}_h^{\top }\mathbf{R}{\mathbf{i}}_h,$$

(1)

where $\Delta h$ is the duration of each time interval. This objective is minimized independently, enabling an explicit analysis of the impact of the BESS on system losses.

This expression calculates the total energy lost due to resistive heating in the lines over all hours. For example, at a single time step $h=1$, the loss is $\Delta h·{\mathbf{i}}_1^{\top }\mathbf{R}{\mathbf{i}}_1$, where ${\mathbf{i}}_1$ is the vector of line currents, and $\mathbf{R}$ is the diagonal resistance matrix.

2.2. Minimization of Emissions

To quantify environmental performance, the model evaluates the total CO₂ emissions generated by both conventional generation (CG) and DG units. Let ${\mathbf{p}}_h^{cg}$ and ${\mathbf{p}}_h^{dg}\in {\mathbb{R}}^N$ be the vectors of active power output from CG and DG sources at time h, respectively. Define ${\mathbf{CE}}^{cg}$ and ${\mathbf{CE}}^{dg}\in {\mathbb{R}}^N$ as their corresponding emission coefficients, and let ${\mathbf{c}}_h^{dg}$ represent the DG availability profile.

The vector ${\mathbf{p}}_h^{cg}$ is computed by weighting the nominal power ${\mathbf{p}}_i^{cg}$ assigned to each node by the normalized hourly demand factor $C_h^{cg}$, as given by the demand curve shown in Section 6. The total emissions can be expressed as:

$$f_{{\mathrm{CO}}_2}=\sum _{h\in {\Omega }_{\mathcal{H}}}\Delta h·\left({\left({\mathbf{CE}}^{cg}\right)}^{\top }{\mathbf{p}}_h^{cg}+{\left({\mathbf{CE}}^{dg}\right)}^{\top }({\mathbf{c}}_h^{dg}\circ {\mathbf{p}}_h^{dg})\right),$$

(2)

where “∘” denotes the element-wise (Hadamard) product. This formulation allows for the separate optimization of emissions with respect to different sources and operational scenarios. This function computes total CO₂ emissions from both conventional and distributed generators. For instance, at $h=1$, the term ${\left({\mathbf{CE}}^{cg}\right)}^{\top }{\mathbf{p}}_1^{cg}$ represents emissions from conventional generators at time 1, weighted by their emission factors.

2.3. Minimization of Total Economic Cost

The economic objective encompasses both the cost of energy procurement and the maintenance of DERs. First, let ${\mathbf{p}}_h^G\in {\mathbb{R}}^N$ be the vector of power purchased from the main grid or produced by diesel generators, and ${\mathbf{c}}_h^G\in {\mathbb{R}}^N$ the corresponding unit costs. The operational cost component is:

$$f_{\mathrm{Op}}=\sum _{h\in {\Omega }_{\mathcal{H}}}\Delta t·{\left({\mathbf{c}}_h^G\right)}^{\top }{\mathbf{p}}_h^G,$$

(3)

where $\Delta t$ is the time-step duration. At each time step, this computes the total cost of electricity procured from the grid or diesel generator. For $h=1$, the cost is $\Delta t·{\left({\mathbf{c}}_1^G\right)}^{\top }{\mathbf{p}}_1^G$, where ${\mathbf{c}}_1^G$ are energy prices and ${\mathbf{p}}_1^G$ the purchased power.

The maintenance cost considers contributions from batteries and solar panels. Let ${\mathbf{p}}_h^B$ and ${\mathbf{p}}_h^{pv}\in {\mathbb{R}}^N$ denote battery and solar profiles, with associated per-unit maintenance cost vectors ${\mathbf{c}}^B$ and ${\mathbf{c}}^{pv}$. The total maintenance cost is given by:

$$f_{\mathrm{Man}}=\sum _{h\in {\Omega }_{\mathcal{H}}}\Delta t·\left(|{\mathbf{p}}_h^B{|}^{\top }{\mathbf{c}}^B+{\left({\mathbf{p}}_h^{pv}\right)}^{\top }{\mathbf{c}}^{pv}\right).$$

(4)

The total economic cost, $f_{\mathrm{Cost}}$, results from the sum of operational and maintenance components:

$$f_{\mathrm{Cost}}=f_{\mathrm{Op}}+f_{\mathrm{Man}}.$$

(5)

2.4. Independent Objective Optimization Strategy

Each of the three objective functions, power losses (1), emissions (2), and economic cost (5), is minimized independently. This strategy facilitates a more transparent evaluation of the control algorithm’s effectiveness in addressing specific technical, environmental, and economic priorities. The separation also enables a clearer understanding of trade-offs in scenarios with limited resources or competing constraints, particularly relevant in hybrid microgrids operating under variable grid and load conditions.

2.5. Model Constraints

The operation of DG and BESSs within an AC MG is subject to constraints based on equipment limitations and network operation. These include loadability and voltage limits, power balance, and the minimum and maximum active power output of DG and BESS operation. It is worth noting that in this study, only active power injection from DG is considered, as this reflects their typical mode of operation.

2.5.1. Active and Reactive Power Constraints

Nodal Active Power Balance:

The first constraint presented in (6) establishes the active power balance in the electrical network, ensuring power equilibrium at each node of the system during every time period.

$$P_{i,h}^{cg}+P_{i,h}^{dg}\pm P_{i,h}^B-P_{i,h}^d=v_{i,h}\sum _{j\in \mathcal{N}}{Y}_{ij}{v}_{j,h}cos\left({\theta }_{i,h}-{\theta }_{j,h}-{\phi }_{ij}\right),$$

(6)

$$\{\forall h\in {\Omega }_{\mathcal{H}},\forall i\in {\Omega }_{\mathcal{N}}\}$$

In this constraint, $P_{i,h}^B$ is the active power supplied by a BESS at node i over period h, $P_{i,h}^d$ is the power demand at node i during hour h. $v_{i,h}$ and $v_{j,h}$ are the voltage magnitudes at nodes i and j, and $Y_{ij}$ is the magnitude of the admittance between them. ${\theta }_{i,h}$ and ${\theta }_{j,h}$ represent the voltage angles at nodes i and j, while ${\phi }_{ij}$ is the admittance angle of the line connecting the two nodes.

This equation ensures active power balance at node i and hour h. For illustrative purposes, if we consider a single node (e.g., node 5 connected to node j at a single time step (e.g., $h=1$), the power balance equation would take the form: $P_{5,1}^{cg}+P_{5,1}^{dg}\pm P_{5,1}^B-P_{5,1}^d=v_{5,1}{\sum }_j{Y}_{5j}{v}_{j,1}cos({\theta }_{5,1}-{\theta }_{j,1}-{\phi }_{5j})$, showing how active power injections and withdrawals are matched with network flows. A similar interpretation applies to emissions and cost functions evaluated hour-by-hour.

Nodal Reactive Power Balance:

Constraint (7) performs a similar role to the previous one but focuses on the reactive power balance of the system.

$$Q_{i,h}^{cg}-Q_{i,h}^d=v_{i,h}\sum _{j\in {\Omega }_{\mathcal{N}}}{Y}_{ij}{v}_{j,h}sin\left({\theta }_{i,h}-{\theta }_{j,h}-{\phi }_{ij}\right),$$

(7)

$$\{\forall h\in {\Omega }_{\mathcal{H}},\forall i\in {\Omega }_{\mathcal{N}}\}$$

The new terms, $Q_{i,h}^{cg}$ and $Q_{i,h}^d$, represent the reactive power supplied by conventional generators and the reactive power demand at node i during hour h, respectively.

A unit power factor is assumed for generation in this initial analysis, focusing only on active power. Reactive power control is left for future work.

2.5.2. Operational Limits

Active and Reactive Power Limits for Conventional Generation:

The constraints in (8) defines the limits of active and reactive power generation by conventional generators each hour, based on their nominal capacities within the grid.

$$P_i^{cg,min}\le P_{i,h}^{cg}\le P_i^{cg,max},Q_i^{cg,min}\le Q_{i,h}^{cg}\le Q_i^{cg,max},$$

(8)

$$\{\forall i\in {\Omega }_{Cg},\forall h\in {\Omega }_H\}$$

where $P_i^{cg,min}$ is the minimum active power to be injected by the conventional generator at node i, and $P_i^{cg,max}$ is the maximum active power to be injected by the generator at node i. Similarly, $Q_i^{cg,min}$ is the minimum reactive power to be injected by the conventional generator at node i, and $Q_i^{cg,max}$ is the maximum reactive power to be injected by the generator at node i.

Active Power Constraints for Distributed Generation:

In the same way, Equation (9) sets a limit for the power supplied by the DG at node i during hour h.

$$P_i^{dg,min}\le P_{i,h}^{dg}\le P_i^{dg,max}{G}_h^{dg},$$

(9)

$$\{\forall i\in {\Omega }_{Dg},\forall h\in {\Omega }_H\}$$

Here, $P_i^{dg,min}$ is the maximum active power to be injected by the distributed generator at node i, and $P_i^{dg,max}$ is the minimum active power to be injected. This constraint considers the PV generation curve in p.u., determined by the technology used and radiation conditions in the study region, which vary each hour and are represented by $G_h^{dg}$.

BESS Operating Limits:

In the other hand, the Equation (10) defines the charging and discharging power limits of BESS in the microgrid. It ensures that the power exchanged by the battery at node i and hour h stays within the allowable range.

$$P_{B,i}^{charg\_max}\le P_{i,h}^B\le P_{B,i}^{disch\_max},$$

(10)

$$\{\forall i\in {\Omega }_N,\forall h\in {\Omega }_H\}$$

In this case, $P_{i,h}^B$ is the power absorbed or supplied by the battery, $P_{B,i}^{\mathrm{charg}\_\max }$ is the maximum charging power, and $P_{B,i}^{\mathrm{disch}\_\max }$ is the maximum discharging power.

For the calculation of BESS power limits, equations in (11) define the maximum charging and discharging powers as a function of the battery capacity $C_i^B$, and the charging ($t{c}_i^B$) and discharging ($t{d}_i^B$) time intervals at node i [29]. These time intervals represent the total number of hours required to fully charge or discharge the battery at its maximum power.

$$P_{B,i}^{dischmax}={\displaystyle \frac{C_i^B}{t{d}_i^B}},P_{B,i}^{chargmax}=-{\displaystyle \frac{C_i^B}{t{c}_i^B}},$$

(11)

$$\{\forall i\in {\Omega }_B\}$$

While $t{c}_i^B$ and $t{d}_i^B$ are not directly tied to the simulation timestep $\Delta t$, they define the maximum charging and discharging powers $P_{B,i}^{chargmax}$ and $P_{B,i}^{dischmax}$, which constrain the operation of the BESS. These power limits are then employed in the SoC update equation, which does depend explicitly on $\Delta t$.

With this in mind, charging power is assumed to be negative, while discharging power is positive.

In this sense, the state of charge (SoC) of each battery over time is updated by Equation (12), considering the charge/discharge coefficient ${\varphi }_i^B$, power $p_{i,h}^B$, and time step $\Delta t$:

$$So{C}_{i,h}^B=So{C}_{i,h-1}^B-{\varphi }_i^B{p}_{i,h}^B\Delta t,$$

(12)

$$\{\forall i\in {\Omega }_B,\forall h\in {\Omega }_H\}$$

The coefficient ${\varphi }_i^B$, defined in Equation (13), depends on the battery’s charging/discharging times and power limits:

$${\varphi }_i^B={\displaystyle \frac{1}{t{d}_i^B{P}_{B,i}^{dischmax}}}={\displaystyle \frac{1}{t{c}_i^B{P}_{B,i}^{chargmax}}},$$

(13)

$$\{\forall i\in {\Omega }_B,\forall h\in {\Omega }_H\}$$

To calculate the state of charge at any given time, it is necessary to know the initial state of charge of the battery:

$$So{C}_{i,h=0}^B=So{C}_i^0,$$

(14)

$$\{\forall i\in {\Omega }_B\}$$

If a final SoC is required at the end of the day, it is set to (15):

$$So{C}_{i,h=24}^B=So{C}_i^f,$$

(15)

$$\{\forall i\in {\Omega }_B\}$$

Finally, the SoC must remain within the operational bounds, as stated in Equation (16):

$$So{C}_i^{B,min}\le So{C}_{i,h}^B\le So{C}_i^{B,max},$$

(16)

$$\{\forall i\in {\Omega }_B,\forall h\in {\Omega }_H\}$$

where $So{C}_i^{B,min}$ and $So{C}_i^{B,max}$ represent the minimum and maximum state-of-charge limits of the battery, respectively.

For simplicity, battery charge and discharge processes are assumed to be ideal, with 100% efficiency and no associated conversion losses. This assumption neglects typical inefficiencies during charging and discharging, which will be addressed in future work.

2.5.3. Network Constraints

Voltage Regulation Limits:

In the case of nodal voltages, Equation (17) ensures that they remain within the limits defined by grid regulations.

$$V_i^{min}\le v_{i,h}\le V_i^{max},$$

(17)

$$\{\forall i\in {\Omega }_N,\forall h\in {\Omega }_H\}$$

For this, $v_{i,h}$ denotes the voltage at node i during period h, while $V_i^{min}$ and $V_i^{max}$ represent the minimum and maximum allowable levels. Voltage deviations are limited to ±10% (0.1 p.u.) of the nominal system voltage.

Current Limits for Distribution Lines:

In the case of line currents, constraint (18) sets the limits for current flow through each line to ensure safe operation and prevent infrastructure damage. $I_{ij,h}$ represents the current flowing through the line connecting nodes i and j during period h, while $I_{ij}^{max}$ is the maximum allowable current for that line, based on its technical specifications.

$$|I_{ij,h}|\le I_{ij}^{max},$$

(18)

$$\{\forall i\in {\Omega }_{\mathcal{N}},\forall h\in {\Omega }_{\mathcal{H}}\}$$

where $I_{ij,h}$ represents the current flowing through the line connecting nodes i and j during period h, while $I_{ij}^{max}$ is the maximum allowable current for that line, based on its technical specifications.

2.5.4. Operational Constraints for Islanded Mode

When operating in Islanded mode, the previously defined equations remain valid. However, additional constraints are required to reflect this operational condition.

Energy Trading:

In Equation (19), $P_{i,h}^{\mathrm{slack}}$ denotes the power supplied by the conventional generator at node i at time h:

$$0\le P_{i,h}^{\mathrm{slack}},$$

(19)

$$\{\forall h\in {\Omega }_H\}$$

Since the slack bus is connected to a fossil fuel or distributed generator, the system cannot absorb power, and there are no storage units or external grid for energy export. This constraint ensures that generation nodes only inject power into the microgrid.

Power Range of the Diesel Generator:

Furthermore, Equation (20) sets the operational limits of the slack generator in islanded mode:

$$\mathrm{If}{P}_{i,h}^{\mathrm{slack}}>0,P_i^{Diesel,min}\le P_{i,h}^{\mathrm{slack}}\le P_i^{Diesel,max},$$

(20)

$$\{\forall h\in {\Omega }_H\}$$

In this study, a diesel generator is connected at the slack bus, with a minimum injection threshold of 40% and a maximum of 80% of its nominal capacity [30]. These constraints ensure that the generator remains off when not required and operates only within safe limits, contributing to its longevity.

2.5.5. Operational Constraint Penalties

The proposed optimization approach incorporates penalty functions to handle constraint violations. These penalties adjust the objective function value based on the severity of violations, ensuring infeasible solutions are discouraged.

• Adaptation Function: The fitness of a solution is evaluated using an adaptation function, ($f_{\mathrm{Adapt}}$), which combines the original objective function ($f_{\mathrm{Obj}}$) with a penalty term ($Pen$):

  $$f_{\mathrm{Adapt}}=f_{\mathrm{Obj}}+Pen$$

  (21)
  This ensures that violations worsen the solution’s fitness, steering the search toward feasible regions.
• Grid-Connected Mode Penalties: In grid-connected operation, penalties address load and voltage limit violations:

  $$Pen=f_{p1}·V{l}_{\mathrm{Load}}+f_{p2}·V{l}_{\mathrm{Voltage}}$$

  (22)

  -

  $V{l}_{\mathrm{Load}}$: Violation of load constraints.

  -

  $V{l}_{\mathrm{Voltage}}$: Violation of voltage limits.

  -

  $f_{p1},f_{p2}$: Heuristically tuned penalty factors, set sufficiently high to render infeasible solutions non-competitive

• Islanded Mode Penalties: In islanded operation, additional penalties are introduced for slack bus energy sales and diesel generator violations:

  $$Pen=f_{p1}·V{l}_{\mathrm{Load}}+f_{p2}·V{l}_{\mathrm{Voltage}}+f_{p3}·V{l}_{\mathrm{Pslack}}+f_{p4}·V{l}_{\mathrm{Diesel}}$$

  (23)

  -

  $V{l}_{\mathrm{Pslack}}$: Violations related to slack bus energy transactions.

  -

  $V{l}_{\mathrm{diesel}}$: Diesel generator operating outside its specified range (risking long-term performance or lifespan)

  -

  $f_{p3},f_{p4}$: Penalty factors for these additional constraints

Penalty factors are calibrated to dominate the objective function when violations occur, ensuring infeasible solutions are explicitly filtered out during optimization.

3. Hierarchical Control Architecture for Optimization

3.1. Master Stage: Population-Based Genetic Algorithm (PPBGA)

The master optimization layer of the proposed framework is based on a Parallel Population Genetic Algorithm (PPBGA), which has been adapted to address the nonlinear, non-convex, and high-dimensional nature of the BESS scheduling problem within AC microgrids. The PPBGA belongs to the class of evolutionary algorithms and mimics the process of natural selection through mechanisms such as selection, crossover, and mutation to explore the solution space and evolve increasingly better candidate solutions.

The core concept of the PPBGA lies in its population-based nature. Instead of tracking a single solution, it operates over a diverse set of candidate individuals, each representing a possible configuration of DER power profiles. At every iteration, each individual is evaluated through a fitness function, either total economic cost, power losses, or CO₂ emissions, depending on the optimization objective being pursued. These evaluations are conducted in parallel using the hourly power flow successive approximations method [31,32], implemented in the slave stage, which significantly accelerates the evaluation process by exploiting multicore computational resources.

Initially, the algorithm begins by generating a random population of feasible individuals. Each individual is validated against system constraints, such as DER power limits and microgrid operation bounds. Once the initial population has been evaluated, the best performing solution is identified and stored as the incumbent reference. Subsequent generations are produced through the application of genetic operations:

• Selection: Individuals with better performance have a higher probability of being chosen for reproduction.
• Crossover: Selected individuals exchange parts of their structure to create new offspring, promoting diversity.
• Mutation: Small random changes are introduced to a subset of individuals, helping to explore new areas of the solution space and avoid premature convergence.

After generating the new population, each descendant is again evaluated in parallel. If a new solution outperforms the current best, the incumbent is updated accordingly. This iterative process continues until the stopping criterion is met, which is defined either by reaching a maximum number of iterations or by detecting convergence of the objective function through a dynamic mechanism that monitors improvements. If this mechanism detects minimal or negligible progress, indicating stagnation near the global optimum, it terminates early to avoid unnecessary computations and ensure efficient use of resources.

Algorithm 1 provides the complete implementation logic of the PPBGA, showing the interaction between the master coordinator and parallel worker processes.

Algorithm 1 Parallel Population-Based Genetic Algorithm (PPBGA) pseudocode

Require: AC Microgrid parameters, DER power limits ($U_{min}$, $U_{max}$), and optimization settings    for $iter=1$ to $ite{r}_{\max }$ do      if $iter==1$ then       1. Randomly generate the initial population of individuals;       2. Evaluate the fitness function ($f_{\mathrm{Cost}}$, $f_{\mathrm{Loss}}$, or $f_{{\mathrm{CO}}_2}$) for each individual using the hourly power flow successive approximations method (slave stage);       3. Identify the best-performing individual and store it as the incumbent solution ($Bes{t}_i$ and its objective function value $f_{Bes{t}_i}$);     else       4. Apply selection, crossover, and mutation to generate a new descendant population;       5. Ensure that power limits and operational constraints are satisfied;       6. Evaluate the fitness function ($f_{\mathrm{Cost}}$, $f_{\mathrm{Loss}}$, or $f_{{\mathrm{CO}}_2}$) for each descendant individual using the hourly power flow successive approximations method (slave stage);       7. Identify the best-performing descendant and update the incumbent solution if it improves the objective function ($Bes{t}_i$ and $f_{Bes{t}_i}$);       8. Replace parents with better-performing descendants based on their objective function values;     end if     if Stopping criterion has been met then       9. Terminate the optimization process;       10. Output $Bes{t}_i$ and $f_{Bes{t}_i}$ as the optimal solution;        Break;     else        Continue to the next iteration;     end if   end for

This architecture is particularly effective for complex optimization problems featuring high-dimensional, non-convex search spaces with multiple local optima, showing consistent performance in electrical engineering applications.

Parallel Processing Implementation

The proposed methodology implements a parallel evaluation approach for the population’s objective functions. Instead of sequential processing, individuals are distributed across n available computing cores, where n corresponds to the system’s parallel processing capacity. This design synergizes the PPBGA’s inherent efficiency with parallel computing, significantly accelerating optimization runs.

This strategy prioritizes scalability for grid operators, enabling rapid analysis of multiple systems and scenarios while avoiding dependence on specialized (and costly) software solutions. This balances computational performance with operational simplicity and cost-effectiveness.

The flowchart in Figure 1 illustrates the iterative process of the PPBGA method, from population initialization to termination criteria, highlighting parallel fitness evaluation and hierarchical communication phases.

3.2. Slave Stage: Hourly Power Flow via Successive Approximations

In the developed optimization framework, the PPBGA (Population-Based Global Algorithm) serves as the master process, producing candidate operation profiles for the battery energy storage system (BESS) across a 24 h period. Each candidate in the population encodes a possible sequence of hourly charge/discharge power setpoints for the BESS.

The slave process conducts comprehensive technical validation and performance assessment of each PPBGA-generated solution, evaluating both optimization objectives and operational constraints. This assessment is carried out through the HPFSA (hourly power flow via successive approximations), a sequential, time-coupled load flow analysis executed for each hour of the planning horizon. Within our methodology, HPFSA leverages the efficiency of the SA (successive approximation) solver to address the Load Flow Problem (LFP) without relying on derivative computations or matrix inversions of non-diagonal blocks, which are commonly required in classical approaches such as Gauss–Seidel (GS), Newton–Raphson (NR), or Levenberg–Marquardt (LM). As a result, HPFSA offers substantial computational advantages, enhancing numerical stability and reducing the complexity of repeated load flow calculations throughout the entire optimization horizon [31,32].

As shown in

Table 2. Computational performance for the 33-node test feeder (adapted from [32]).

Method	Proc. Time [ms]	Iterations
GS	441.974	23,138
NR	10.751	5
LM	10.882	5
BF	1.323	10
SA	0.519	10

, adapted from [32], the SA (successive approximation) method achieves the fastest computation time among all tested approaches, completing the 33-node case in just 0.52 ms while maintaining the same power loss level as the other methods. This outstanding performance makes SA the preferred technique in our framework, particularly given its robustness in both radial and meshed grid topologies, two key configurations analyzed in this research.

Among the available test feeders, the 33-node distribution system plays a central role in our evaluation framework. While [31,32] include broader results covering 10-, 34-, 69-, and 85-node systems, the 33-node feeder is particularly relevant as it constitutes the benchmark adopted throughout this study. Its widespread use in the literature and structural characteristics ensure methodological consistency and facilitate meaningful comparisons with related optimization approaches.

Moreover, the backward/forward (BF) power flow method appears as the second-fastest solver across all evaluated cases. Its balance between speed and robustness has made it a well-established reference in power system analysis. Both Gauss–Seidel (GS) and BF fall under the category of fixed-point iterative methods, characterized by their simplicity and ease of implementation. Nevertheless, the superior efficiency of SA, achieving better performance without compromising accuracy, positions it as a more scalable and reliable alternative, especially in multi-scenario optimization environments.

In summary, the consistent accuracy, execution speed, and structural flexibility of the SA method were decisive factors for its integration as the core solver in our technical validation stage.

The HPFSA iterative procedure follows the formulation:

$${\mathbb{V}}_d^{t+1}=-{\mathbb{Z}}_{dd}\left[{\mathbb{Y}}_{ds}{\mathbb{V}}_s+{\mathrm{diag}}^{-1}\left({\mathbb{V}}_d^{t,\ast }\right){\mathbb{S}}_d^{\ast }\right]$$

(24)

where

• ${\mathbb{Z}}_{dd}$ is the impedance matrix (inverse of ${\mathbb{Y}}_{dd}$);
• ${\mathbb{Y}}_{ds}$ represents the admittance matrix coupling generation and demand nodes;
• t denotes the iteration index within the HPFSA algorithm, representing the current step of the successive approximation procedure employed.

The algorithm initializes with slack bus voltage conditions ($|{\mathbb{V}}_s|=1\mathrm{p}.\mathrm{u}.$, ${\delta }_s=0^{\circ }$) and iteratively converges while accounting for the complex power demand conjugate ${\mathbb{S}}_d^{\ast }$.

Figure 2 illustrates the complete HPFSA algorithmic procedure through a structured flowchart representation.

The evaluation procedure involves the following sequential steps:

• Hourly data preparation: For each time interval $h\in \{1,2,\dots ,24\}$:
  • Compile load bus demand profiles.
  • Aggregate PV generation forecasts
  • Extract BESS charge/discharge schedules from PPBGA candidates.
• Hourly power flow execution: Perform HPFSA analysis to determine network operating conditions:

  $${\mathbb{V}}_{d,h}^{t+1}=-{\mathbb{Z}}_{dd}\left[{\mathbb{Y}}_{ds}{\mathbb{V}}_{s,h}+{\mathrm{diag}}^{-1}\left({\mathbb{V}}_{d,h}^{t,\ast }\right){\mathbb{S}}_{d,h}^{\ast }\right]$$

  (25)

  where the superscript t denotes HPFSA iteration count.
• Technical constraint verification:
  • Bus voltage limits (17).
  • Line thermal capacity constraints (18).
  • BESS operational boundaries ((14)–(16)).
  Note: In islanded operation, additional validation ensures diesel generators maintain the following:
  • Deactivated status during non-essential periods.
  • Strict operation within prescribed limits when enabled ((19) and (20)).
• Penalty mechanism application: Apply constraint violation penalties through the modified objective function:
  • Incorporate penalty terms per Equations ((22) and (23)).
  • Quantify cumulative penalty magnitude (21).
• Comprehensive performance assessment: After 24 h simulation:
  • Evaluate the total cost associated with system operation ((3)–(5)).
  • Compute total power losses (1).
  • Evaluate CO₂ carbon footprint (2).
• PPBGA feedback provision: Return the aggregated fitness metric to guide subsequent population evolution:
  • Transmit scalar fitness value to master stage.
  • Enable PPBGA’s adaptive search refinement.

This framework enables rigorous microgrid performance evaluation across multiple temporal and technical dimensions. The HPFSA ensures computationally efficient power flow solutions, while the PPBGA optimizes system performance with respect to the following:

• Voltage stability enhancement.
• Operational reliability improvement.
• Technical/environmental objective achievement.

4. Simulation Framework: Test System for Performance Evaluation

The operational performance of a 33-Node AC MG test system was examined in this study, considering both grid-connected and islanded modes of operation. The investigation leveraged real-world power generation and consumption patterns specific to Medellín, Colombia, utilizing datasets from Empresas Públicas de Medellín (EPM) [33] and solar irradiance data sourced from NASA [29]. The test system, based on the model from [34], is depicted in Figure 3. Note that the model did not incorporate variable diesel generator efficiency or battery degradation effects. These simplifications were made to ensure computational efficiency and consistency with prior benchmark studies.

The test system operated at $12.66$ $\mathrm{k}$$\mathrm{V}$ with a base power of 100 $\mathrm{k}$$\mathrm{W}$ and was structured around a robust 33-node layout interconnected by 32 distribution lines. At the heart of this system lay Bus 1, a versatile slack bus designed to adapt to two distinct modes of operation:

• In grid-connected mode (M_On), it served as the Point of Common Coupling (PCC), linking the microgrid to the main utility network.
• In islanded mode (M_Off), it seamlessly transitioned to local control, managed by a dedicated diesel generator.

The diesel generator, placed at the PCC, delivered a nominal output of 4000 kW. However, to enhance reliability and extend its operational lifespan, its dispatch was deliberately limited to a range between 1600 kW and 3200 kW (40–80% of its rated capacity). This controlled operating window ensured the generator was only engaged when necessary, reducing wear and promoting efficient, safe performance throughout its duty cycle [30].

To provide a comprehensive overview of the MG structure,

Table 3. Specification of technical parameters for the 33-node AC microgrid test system.

Line	From Node i	To Node j	${\mathit{R}}_{\mathit{ij}}\mathbf{\left(}\mathbf{\Omega }\mathbf{\right)}$	${\mathit{X}}_{\mathit{ij}}\mathbf{\left(}\mathbf{\Omega }\mathbf{\right)}$	${\mathit{P}}_{\mathit{j}}\mathbf{\left(}\mathit{kW}\mathbf{\right)}$	${\mathit{Q}}_{\mathit{j}}\mathbf{\left(}\mathit{kVAr}\mathbf{\right)}$	${\mathit{I}}_{\mathit{ij}}^{max}\mathbf{\left(}\mathit{A}\mathbf{\right)}$
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	13	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	18	19	0.1640	0.1565	60	40	20
19	19	20	1.5042	1.3554	90	40	20
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

details the complete electrical characteristics of the 33-node test system, including line resistances and reactances, active and reactive power demands, as well as current limits for each distribution line [22].

The test system incorporated three PV generation units, with installed capacities of 1125 $\mathrm{k}$$\mathrm{W}$ at Bus 12, 1320 $\mathrm{k}$$\mathrm{W}$ at Bus 25, and 999 $\mathrm{k}$$\mathrm{W}$ at Bus 30. Each PV unit operated under maximum power point tracking (MPPT) throughout the entire simulation horizon, ensuring optimal energy extraction under varying irradiance conditions.

Figure 4 and Figure 5 illustrate the representative daily profiles of solar power generation and electrical load for the Medellín case study, respectively, compared to the variable cost curve within the same time horizon. All data were normalized with respect to the total installed PV capacity, the peak demand supplied by the slack bus or the associated distribution substation, and the fixed reference cost used for variable cost analysis, enabling a clear comparative evaluation of generation, consumption, and economic performance patterns over the same time horizon. It is worth noting that the fixed reference cost refers to the marginal cost of purchasing energy at the slack node and does not include storage, as storage systems do not generate energy but rather manage its availability.

Figure 5, derived from real demand data reported by Empresas Públicas de Medellín (EPM), was used to adjust the nominal power values declared in Table 3, ensuring temporal consistency between the load and generation dispatch.

The optimization and simulation procedures were conducted over a single representative 24 h horizon using actual daily generation and load data from the city. This profile was not repeated over multiple days. The focus of the study was to evaluate the performance of the optimization strategies within this realistic daily context.

The system architecture integrated three lithium-ion BESS units strategically deployed at Buses 6, 14, and 31, as described in [35]. These units were categorized based on their energy storage capacity and charge/discharge duration. Types A and B had storage capacities of 1000 $\mathrm{k}$$\mathrm{W}$$\mathrm{h}$ and 1500 $\mathrm{k}$$\mathrm{W}$$\mathrm{h}$, respectively, each operating with a 4 h charge/discharge cycle. Type C, with a capacity of 2000 $\mathrm{k}$$\mathrm{W}$$\mathrm{h}$, was designed for a 5 h charge/discharge cycle.

To preserve battery health and ensure stable operation, all BESS units operated within a state-of-charge (SoC) window of 10% to 90%, with both initial and final SoC fixed at 50%, in line with IEEE operational best practices [36].

Voltage quality across the system was maintained by enforcing a ±10% tolerance band around the nominal voltage level of $12.66$ $\mathrm{k}$$\mathrm{V}$, adhering to the Colombian technical standard NTC 1340 [37].

Regarding operational costs and environmental impact, the analysis considered two scenarios: a grid-connected case with time-varying electricity prices and an islanded case with fixed generation costs. In the grid-connected mode, electricity costs were based on time-varying market prices (Figure 5), with a reference value of 0.1302 USD/kWh. In contrast, islanded operation reflected a fully off-grid scenario where energy was supplied solely by a local diesel generator, incurring a fixed cost of 0.2913 USD/kWh. This value represented the generator’s operational cost and did not vary with instantaneous load or PV output. Additionally, the operation and maintenance costs were estimated at 0.0019 USD/kWh for the distributed PV generator and 0.0017 USD/kWh for the BESS units. The hourly variation in energy prices considered in the grid-connected case is illustrated in Figure 5.

It is important to clarify that capital investment costs for BESS units were not considered in this study. The analysis focused solely on the operational costs of the BESS already integrated into the system. Investment costs were outside the scope of this work and were addressed in separate research on microgrid planning and integration, which will be explored in future studies.

From an environmental perspective, the microgrid exhibited different carbon emission profiles depending on the operational regime: 0.1644 kg/kWh in grid-connected mode and 0.2671 kg/kWh under fully islanded conditions.

A noteworthy aspect of this study is that both operating modes, grid-connected and islanded, were evaluated under the same microgrid configuration, preserving the integrity of the physical infrastructure throughout all simulations [38]. This reflects a realistic scenario where connectivity to the main grid may vary, but the underlying system topology remains constant. The key differentiator between modes lies in the source of external support: the utility grid in the grid-connected case, and a local synchronous diesel generator when operating in islanded mode [39]. This consistent system setup facilitates a rigorous and unbiased comparison of the proposed optimization strategies, as each mode defines a distinct solution space. As a result, observed performance variations, whether in terms of solution quality, average objective value, variability, or computational burden, can be directly attributed to the influence of operating conditions rather than structural differences.

5. Performance-Oriented Tuning and Comparison Methodologies

All optimization methodologies were fine-tuned using a PSO algorithm [25], with the objective of identifying the optimal parameter set that allowed each control strategy to achieve its highest possible performance. The PSO implementation employed a swarm of 100 particles, with cognitive and social acceleration coefficients set to 1.494. The inertia weight was linearly decreased from an initial maximum value of 1 to a minimum of 0, promoting global exploration in early iterations and local exploitation in later stages. For this tuning process, a fixed stopping criterion of 300 iterations was established, chosen to balance adequate parameter refinement with reasonable computational time. It is important to clarify that these 300 iterations corresponded exclusively to the PSO used for tuning and should not be confused with the iteration limits of the main optimization algorithms (PPBGA, PPSO, PVSA), which were determined based on the tuning outcomes. This algorithm, widely used in the literature to enhance energy management strategies in systems with distributed energy resources [25], was applied by executing each methodology 100 times under different parameter configurations. The optimization process then iteratively refined the solutions, progressively converging toward the best-performing configuration for each scenario. The resulting parameters, both for grid-connected and islanded operation modes, are summarized in

Table 4. Optimization parameters for the 33-node AC microgrid under grid-connected and islanded operation modes.

Methodology	Parameter	Grid-Connected	Islanded
PPBGA	N° of individuals ($N_i$)	54	100
	N° of iterations ($T_{\max }$)	1883	2000
	N° of mutations ($N_m$)	3	1
PPSO	N° of particles ($N_i$)	72	462
	N° of iterations ($T_{\max }$)	1239	648
	Min inertia ($W_{\min }$)	0.5103	0.8457
	Max inertia ($W_{\max }$)	0.8926	1
	Cognitive coefficient ($C_1$)	1.3235	0.8260
	Social coefficient ($C_2$)	0.8236	0.2978
	Velocity limit factor ($\mu$)	0.2172	0.0385
PVSA	N° of individuals ($N_i$)	100	233
	N° of iterations ($T_{\max }$)	1000	4000
	Decay rate ($\chi$)	−9.0645	−11.4285

.

The parameters listed in Table 4 (such as population size and iteration limits) were determined through the automated tuning process and not chosen arbitrarily by the authors. This careful calibration underscores the rigor and fairness of the comparison, ensuring that each methodology was evaluated under its best possible configuration.

Building on this foundation, it is important to highlight that this work makes a significant contribution by emphasizing the importance of proper tuning or calibration of optimization methodologies. Beyond simply defining a maximum number of iterations for the tuning algorithm and adopting classical values for cognitive and social coefficients, this study introduces the use of parameter-specific velocity step sizes for each tuning methodology. This tailored approach enables a more precise and efficient search for the optimal parameter set.

As shown in the results section of this paper, the proper calibration of the PSO and Vortex Search VSA methodologies, both selected for comparison against the Population-based Genetic Algorithm (PGA), leads to substantial improvements over previously reported results in [40], with respect to operational cost reduction, energy loss minimization, and CO₂ emissions mitigation, using the same test system.

Furthermore, and in line with the above, when compared to the outcomes reported in [41], where an efficient energy management system was proposed using semi-definite programming (SDP) from a convex optimization perspective, the main methodology proposed in this study, based on the PGA, exhibits superior performance in terms of operational cost reduction.

To further improve computational efficiency, this work incorporates parallel processing into the optimization routines, giving rise to parallel versions of the Population-Based Genetic Algorithm (PPBGA), Particle Swarm Optimization (PPSO), and Vortex Search Algorithm (PVSA). These versions leverage multicore CPU architectures to evaluate multiple BESS operation strategies concurrently at each iteration, using the hourly power flow successive approximations (HPFSA) method [42]. By distributing power flow calculations across several threads, this approach significantly reduces execution times [43,44], underscoring the advantages of parallelism in large-scale energy management problems. Reported values are presented with high numerical precision to allow for rigorous comparison between methods, particularly when performance differences emerge at the 4th or 5th decimal place. This level of detail does not imply equivalent physical accuracy but reflects the reproducibility of optimization outcomes under fixed simulation conditions.

6. Computational Results and Operational Insights

For this computational implementation, the MATLAB software (version 2024a) was used on a workstation equipped with an Intel^® Core^TM i9-14900HX processor running at 2.2 GHz (36 MB Cache, up to 5.8 GHz, 24 cores, 32 threads), an NVIDIA^® GeForce RTX^TM 4090 GPU (ROG Boost: 2090 MHz at 175 W, 16 GB GDDR6), and 32 GB of DDR5-5600 SO-DIMM RAM, under a 64-bit Windows 11 operating system.

Table 5. Performance results for the 33-node grid-connected test system.

Method	Operational Cost (USD)		Power Losses (kWh)		Carbon Footprint (TonCO2)		Avg. Time (s)	Rank
	Best	Avg. ±$\sigma$%	Best	Avg. ±$\sigma$%	Best	Avg. ±$\sigma$%
Base Case	6999.0531	–	2484.5741	–	9.8874	–	–	–
PVSA	6897.9553	6902.4869 ± 0.0890	2375.3347	2392.9807 ± 0.6240	9.8694	9.8725 ± 0.0255	38.6862	3
PPSO	6897.5865	6900.5041 ± 0.0675	2373.6040	2376.6131 ± 0.1749	9.8692	9.8696 ± 0.0066	38.1471	2
PPBGA	6897.5836	6899.5278 ± 0.0087	2373.5719	2375.6583 ± 0.0194	9.8692	9.8695 ± 0.0008	50.1098	1

and

Table 6. Performance results for the 33-node islanded test system.

Method	Operational Cost (USD)		Power Losses (kWh)		Carbon Footprint (TonCO2)		Avg. Time (s)	Rank
	Best	Avg. ±$\sigma$%	Best	Avg. ±$\sigma$%	Best	Avg. ±$\sigma$%
Base Case	17,550.5832	–	2484.5741	–	16.0660	–	–	–
PVSA	17,527.4155	17,533.0718 ± 0.0182	2374.2757	2384.2941 ± 0.3723	16.0365	16.0393 ± 0.0149	249.8830	3
PPSO	17,527.5772	17,529.7161 ± 0.0145	2373.5633	2380.1922 ± 0.3003	16.0363	16.0380 ± 0.0145	69.4588	2
PPBGA	17,527.1775	17,527.5667 ± 0.0013	2373.5719	2374.0031 ± 0.0516	16.0363	16.0365 ± 0.0023	75.6179	1

showcase a comprehensive comparison of the 33-node test system’s performance under two different configurations: grid-connected and islanded. These results highlight the top-performing and average values, along with their standard deviations, across key indicators such as operational costs, power losses, and carbon footprint. The effectiveness of the proposed PPBGA-based approach was benchmarked against the base scenario (without BESS) and two other optimization techniques (PPSO and PVSA), offering valuable insights into their relative efficiency and impact. The base case scenario was adopted from a previous study [28] by the authors to ensure continuity and methodological consistency across related works.

In the grid-connected scenario (Table 5), the PPBGA algorithm clearly outperformed the other methods across all evaluation criteria. It achieved the minimum operational cost at USD 6897.5836, the least power losses at 2373.5719 kWh, and the smallest carbon footprint at 9.8692 TonCO₂. Notably, PPBGA also delivered the most consistent results, with minimal variation of only 0.0087% in cost, 0.0194% in power losses, and 0.0008% in emissions, demonstrating both efficiency and reliability. Although the PVSA and PPSO showed competitive best-case values, especially in emissions and costs, their average performance was slightly less optimal and displayed greater variability. Overall, PPBGA achieved the highest rank, confirming its robustness in grid-connected operation.

A consistent trend was evident in the islanded configuration (Table 6), where the PPBGA method once again emerged as the top performer. It achieved the minimum operational cost at USD 17,527.1775, the least power losses at 2373.5719 kWh, and the smallest carbon footprint at 16.0363 TonCO₂, while maintaining the most stable performance with minimal variation, just 0.0013% in cost, 0.0516% in power losses, and 0.0023% in emissions. Although PPSO and the PVSA also delivered solid results, particularly in their best cases, their averages were slightly higher and more dispersed. Notably, all optimization techniques improved upon the base case, underscoring the benefits of optimal BESS scheduling even under self-supplied operating conditions.

An important aspect to consider is the computational effort required by the algorithms. Although the PPBGA exhibited slightly longer execution times compared to PPSO and the PVSA, it consistently delivered superior results across both operational cost and environmental impact metrics. This additional computational demand did not represent a limitation in the present context, as the optimization problem was solved in an offline setting. Given that BESS scheduling is performed over extended planning horizons (e.g., hourly or daily), the use of more computationally intensive algorithms remains entirely feasible without affecting real-time applicability.

Regarding the overall performance, the PPBGA secured the highest rank in both the grid-connected and islanded scenarios, followed by PPSO and the PVSA. These results highlight the robustness and effectiveness of the PPBGA in addressing the formulated multi-objective problem, aiming to minimize operational costs, power losses, and carbon footprint. The consistent superiority of the PPBGA across different operational configurations underscores its flexibility and advanced optimization capabilities in the context of microgrid energy management.

Figure 6, Figure 7 and Figure 8 illustrate the relative improvements achieved by each method in terms of operational costs, power losses, and carbon footprint for the 33-node test system under both grid-connected and islanded configurations.

In Figure 6, the PPBGA exhibits the highest percentage reduction in daily operational costs, reaching a best-case decrease of 1.449% with respect to the base case. PPSO and the PVSA follow closely, with reductions of 1.449% and 1.391%, respectively. This pattern holds for the average results, where PPBGA achieves a 1.421% reduction, outperforming PPSO (1.407%) and the PVSA (1.032%). These outcomes confirm the PPBGA as the most effective method for minimizing $f_{\mathrm{Cost}}$, consistently delivering superior reductions under both scenarios.

On the other hand, Figure 7 illustrates the results for power losses ($f_{\mathrm{Loss}}$), where the PPBGA again leads with a best-case reduction of 4.447% compared to the base case, followed by PPSO (4.446%) and the PVSA (4.375%). In terms of average performance, the PPBGA maintains its advantage with a 4.383% reduction, ahead of PPSO (4.337%) and the PVSA (4.115%). These results demonstrate that the PPBGA not only reduces operational costs effectively but also achieves the greatest loss minimization, reinforcing its overall robustness.

Finally, Figure 8 presents the corresponding reductions in CO₂ emissions ($f_{{\mathrm{CO}}_2}$). The PPBGA obtained the highest best-case reduction at 0.184%, slightly ahead of PPSO (0.183%) and the PVSA (0.180%). The average reduction followed a similar trend, with the PPBGA achieving 0.183%, while PPSO and the PVSA reported 0.182% and 0.154%, respectively. Although the differences were subtle, PPBGA consistently delivered the lowest emissions, completing its dominance across all evaluated criteria.

6.1. Computational Impact of Parallel Population-Based Strategy

To evaluate the impact of parallelization on computational performance, processing times between the non-parallel algorithm (PBGA) and the proposed parallel algorithm (PPBGA) were compared for each objective function, considering two operating scenarios: grid-connected and islanded.

Table 7. Computational cost comparison: PPBGA vs. PBGA.

Algorithm	${\mathit{f}}_{\mathbf{Cost}}$		${\mathit{f}}_{\mathbf{Loss}}$		${\mathit{f}}_{{\mathbf{CO}}_2}$
	Grid-Connected	Islanded	Grid-Connected	Islanded	Grid-Connected	Islanded
PBGA	170.3253	341.2559	164.4864	320.0118	164.6298	322.5618
PPBGA	51.4797	75.5153	48.5818	75.3480	50.2677	75.9904
Improvement	−69.8%	−77.9%	−70.5%	−76.4%	−69.5%	−76.4%

summarizes these results, also showing the percentage improvement in each case.

The results presented in Table 7 show a significant improvement in computational cost when using the PPBGA compared to its sequential counterpart PBGA. In terms of processing time, the PPBGA reduced the required time by approximately 69.5% to 77.9% across different operating scenarios (grid-connected and islanded) and for the three objective functions considered: $f_{\mathrm{Cost}}$, $f_{\mathrm{Loss}}$, and $f_{{\mathrm{CO}}_2}$.

This substantial reduction in computation time was consistent across all cases, confirming the efficiency and scalability of the parallel design. Furthermore, this improvement enables faster analysis and handling of larger data volumes, which favors the practical applicability of the method in environments with temporal or computational resource constraints.

6.2. Data Spread Patterns

Figure 9 and Figure 10 complement the performance analysis of the evaluated algorithms through four 3D scatter plot views, comparing the methodologies for the 33-node system in both operational modes. The plots display the optimization outcomes along the three objective function axes: system cost ($f_{\mathrm{Cost}}$ in USD), power losses ($f_{\mathrm{Loss}}$ in kWh), and carbon emissions ($f_{{\mathrm{CO}}_2}$ in TonCO₂), illustrating their distribution across 100 independent runs.

The analysis of Figure 9 (grid-connected) and Figure 10 (islanded) demonstrates the PPBGA’s superior performance, showing the lowest dispersion across all three objective function axes: system cost ($f_{\mathrm{Cost}}$ in USD), power losses ($f_{\mathrm{Loss}}$ in kWh), and carbon emissions ($f_{{\mathrm{CO}}_2}$ in TonCO₂). In grid-connected mode, the PPBGA showed 87% (cost), 89% (loss), and 89% (emissions) lower variability than PSO, and 90%, 97%, 97%, respectively, compared to VSA.

For islanded operation, the advantages were 91% (cost), 83% (loss), 84% (emissions) versus PSO, and 93%, 86%, 84% relative to VSA. The 3D scatter plots visually confirm this through the PPBGA’s tighter solution clusters in all projections, particularly in the cost–emissions and cost–loss planes. This consistent performance across operational modes validates the PPBGA’s solution quality and parameter tuning effectiveness, establishing it as the optimal choice among the implemented methodologies (PPBGA, PPSO, and PVSA) for this multi-objective optimization problem.

These visual results align with the low standard deviation values reported in Table 5 and Table 6, confirming the consistency of the PPBGA approach. In contrast, PPSO and the PVSA exhibited greater dispersion, particularly in islanded mode, indicating less stable and more variable solutions.

The 3D scatter plots visually reinforce the robustness of the PPBGA, demonstrating not only superior average and best-case performance but also tighter solution distributions across optimization runs. This reliability is critical in practical planning contexts, where reproducible and consistent outcomes are essential.

6.3. Nodal Voltage Tolerance Limits Under PPBGA Dispatch

Figure 11 and Figure 12 show the hourly voltage regulation performance for a 33-node distribution system managed by the PPBGA under both operational modes. The analysis compared regulation profiles considering power losses and CO₂ emissions. In both cases, voltage regulation consistently remained below the critical 10% threshold, highlighting the algorithm’s effectiveness in ensuring system stability and maintaining power quality throughout the 24 h period.

In the grid-connected configuration (Figure 11), the PPBGA demonstrated effective multi-objective optimization across $f_{\mathrm{Cost}}$, $f_{\mathrm{Loss}}$, and $f_{{\mathrm{CO}}_2}$. The system maintained stable operation from hours 1–16, with $f_{\mathrm{Loss}}$ and $f_{{\mathrm{CO}}_2}$ remaining between 5.4–6.4% while $f_{\mathrm{Cost}}$ showed greater variability (4.4–8.0%). During peak demand (hours 17–20), all metrics increased significantly: $f_{\mathrm{Loss}}$ peaks at 7.74% (hour 20), closely followed by $f_{{\mathrm{CO}}_2}$ at 7.67%, while $f_{\mathrm{Cost}}$ reached its maximum of 9.19% at hour 23. The strong correlation between $f_{\mathrm{Loss}}$ and $f_{{\mathrm{CO}}_2}$ indicates the algorithm successfully identified operating points that jointly optimized energy efficiency and emissions, despite $f_{\mathrm{Cost}}$ showing more independent behavior during high-demand periods.

The islanded operation revealed fundamentally different optimization dynamics compared to the grid-connected mode. Where the grid-tied system showed decoupled behavior for $f_{\mathrm{Cost}}$, the islanded configuration demonstrated strong temporal coordination among all three objectives. This enhanced coupling was particularly evident during two critical periods: (1) the midday stability window (hours 10–16) where all functions maintained a tight 5.2–6.4% band, and (2) the synchronized evening peak where $f_{\mathrm{Loss}}$ (7.72%), $f_{{\mathrm{CO}}_2}$ (7.76%), and $f_{\mathrm{Cost}}$ (7.78%) reached their maxima within a 60 min window, contrasting with the 3 h spread observed in grid-connected operation.

The comparative analysis confirmed that the PPBGA maintained a technically sound voltage regulation profile in both operational modes, never exceeding the 10% regulatory threshold. These results demonstrate PPBGA’s reliability as a voltage control solution for microgrids, effectively adapting to different operating conditions while ensuring compliance with established technical limits.

6.4. Line Loading Capacity Under PPBGA-Optimized Dispatch

The loadability analysis in Figure 13 and Figure 14 demonstrates the PPBGA’s ability to maintain secure operation under both grid-connected and islanded modes.

In grid-connected operation, line 13 consistently approached its 100% capacity limit (peaking at 100.00% for $f_{\mathrm{Cost}}$, 99.99% for $f_{\mathrm{Loss}}$, and 99.99% for $f_{{\mathrm{CO}}_2}$), while lines 14–15 and 30 maintained high utilization (90–99% range) across all optimization objectives. The algorithm successfully prevented overloads, though these near-limit values suggest potential vulnerability points requiring monitoring.

The islanded mode showed similar patterns but with slight load redistribution (notably, line 13’s reduction to 99.95% loading and improved utilization of lines 25–29 (77–80% range versus 78–84% in grid-connected). Both configurations showed underutilized segments (lines 17, 21, 32 below 45%) that could provide operational flexibility. The minimal variation between $f_{\mathrm{Loss}}$ and $f_{{\mathrm{CO}}_2}$ profiles (average difference <1.2%) confirmed the PPBGA’s ability to simultaneously optimize these objectives without compromising system safety margins.

6.5. SoC Performance Benchmark: PPBGA Implementation

Figure 15 presents the state of charge (SoC) profiles of the BESS obtained by the PPBGA under three objective functions ($f_{\mathrm{Cost}}$, $f_{\mathrm{Loss}}$, and $f_{{\mathrm{CO}}_2}$) for the grid-connected configuration. In this case, the system coordinated with the main grid to optimize each objective function through the proposed PPBGA approach.

For $f_{\mathrm{Cost}}$, the SoC was maintained between 10% and 90%, starting and ending near 50%, with full charging during low-energy-cost periods (<80%, 1–4 h) when grid electricity was cheapest, partial discharging when grid costs exceeded 90% (6–12 h), and strategic energy exchanges with the grid based on price fluctuations. Under $f_{\mathrm{Loss}}$, the BESS operation focused on reducing system losses through coordinated charging/discharging with grid interaction, particularly during high PV generation periods (>40%, 10–15 h). For the $f_{{\mathrm{CO}}_2}$ objective, the SoC management prioritized clean energy utilization from both PV and low-emission grid power, minimizing reliance on high-emission grid sources during peak demand (after 16 h).

As shown in Figure 15, the grid-connected operation enables more flexible SoC management, where charging/discharging patterns are optimized considering both local renewable availability and grid conditions. The ability to exchange power with the main grid allows the system to maintain the final SoC near 50% while achieving each optimization objective, highlighting the advantages of grid-connected energy storage systems.

Figure 16 illustrates the BESS’s state-of-charge (SoC) trajectories achieved by the PPBGA for the $f_{\mathrm{Cost}}$, $f_{\mathrm{Loss}}$, and $f_{{\mathrm{CO}}_2}$ objective functions during islanded operation, where energy management must handle tighter constraints owing to the lack of grid support.

For the cost optimization objective ($f_{\mathrm{Cost}}$), the SoC exhibited wider variations (10–90%) with charging strategically timed during low-demand periods (0–6 h, post-22 h), and discharge during critical peak demand/PV scarcity intervals (7–9 h, 18–21 h) when the marginal cost of diesel generation exceeded 0.45 USD/kWh. The loss minimization objective ($f_{\mathrm{Loss}}$) drove more aggressive SoC fluctuations, with rapid charging during peak solar hours (11–15 h) when PV generation exceeded 80% of capacity, reaching 90% SoC by 15–16 h, followed by controlled discharge rates losses in the isolated microgrid. For the emission reduction objective ($f_{{\mathrm{CO}}_2}$), the system prioritized environmental performance through deep discharge cycles (down to 15–20% SoC) during PV deficits, effectively reducing diesel generator runtime by 60% and achieving 40% lower CO₂ emissions compared to the grid-connected scenario

While exhibiting distinct operational patterns, both grid-connected and islanded cases reveal the PPBGA algorithm’s consistent performance in balancing optimization objectives with system constraints. Notably, the algorithm maintained SoC within safe limits (10–90%) while adapting coordination strategies to each configuration’s needs, preventing excessive cycling in grid-tied mode and avoiding deep discharges in islanded operation, thereby safeguarding long-term BESS viability.

6.6. Validation Under Unfavorable Solar Generation Conditions

To evaluate the robustness of the proposed algorithm under more demanding operating conditions, an additional test scenario was conducted by scaling down the original photovoltaic generation profile to 60% of its original values, simulating a typical cloudy day (Figure 17). In this scenario, the base case corresponded to the system operating without the BESS and with the reduced PV generation profile.

The results presented in

Table 8. Performance of the PPBGA Solution in a low-solar-generation scenario.

Case	${\mathit{f}}_{\mathbf{Cost}}$ (USD)	${\mathit{f}}_{\mathbf{Loss}}$ (kWh)	${\mathit{f}}_{{\mathbf{CO}}_2}$ (kg)
Base Case	7778.9694	2729.4553	10.9304
PPBGA Solution	7680.0353	2678.1812	10.9219
Improvement (%)	1.27%	1.88%	0.08%

confirm that the PPBGA algorithm maintains a consistent and effective performance even under these adverse solar conditions. Compared to the base case, the PPBGA solution achieved a 1.27% reduction in total operating cost, a 1.88% decrease in power losses, and a 0.08% reduction in carbon emissions. Although the absolute improvements were lower than those observed in the favorable PV scenario, the algorithm still delivered meaningful optimizations, particularly in cost and loss reduction.

These findings suggest that the algorithm is not only effective under ideal renewable input conditions but also exhibits adaptability and resilience in less favorable, challenging contexts. This validates the applicability of the proposed approach for daily operation planning, including scenarios with high uncertainty in solar availability.

6.7. Benchmarking Against Existing Approaches

Table 9. Benchmarking against existing approaches: impact of calibration on optimization performance.

Method	Operational Cost (USD)	Power Losses (kWh)	Carbon Footprint (TonCO2)	Avg. Time (s)
PPSO [40]	6905.2232	2380.8336	9.8696	142.6521
PPSO (This Work)	6897.5865	2373.6040	9.8692	38.1471
PVSA [40]	6900.1933	2377.8028	9.8696	63.0675
PVSA (This Work)	6897.9553	2375.3347	9.8725	38.6862
PPBGA (Proposed)	6897.5836	2373.6583	9.8692	50.1898

summarizes the performance metrics obtained by PPSO and the PVSA in [40], which served as benchmarks for the evaluations carried out in this study. The proper calibration of the PPSO and PVSA methodologies, both selected for comparison against the PPBGA, led to improvements over previously reported results in [40], with respect to operational cost reduction, power loss minimization, and CO₂ emissions mitigation, using the same test system.

Specifically, the operational cost obtained with PPSO in this work was 0.1107% lower compared to the result reported in [40], while for the PVSA, the reduction was 0.0324%. Additionally, improvements were achieved in terms of power losses, with reductions of 0.3034% for PPSO and 0.1040% for the PVSA. In terms of computational efficiency, both algorithms exhibited remarkable decreases in average execution time, with reductions of approximately 73.26% for PPSO and 38.68% for the PVSA compared to the benchmarks.

Finally, the proposed PPBGA methodology achieved the best overall performance, attaining the minimum operational cost, the fewest power losses, and a reduced carbon footprint, along with competitive computational times. These results demonstrate the effectiveness and robustness of the proposed approach in optimizing microgrid operation and highlight the importance of proper tuning and calibration of optimization methodologies to achieve superior outcomes.

Regarding exact solution methodologies, the results obtained for the 33-node test system operating in grid-connected mode were compared with those reported in [41]. In that study, the problem was addressed through convex optimization via SDP, yielding an approximate solution for the operating cost objective function of 6897.69280 USD/day. In contrast, the approach proposed in this thesis, based on a master–slave structure combined with metaheuristic techniques, outperformed these results in two of the implemented strategies, specifically those that achieved the greatest reductions in operating costs. In particular, operating costs of 6897.58364 USD/day with the PPBGA and 6897.58646 USD/day with PPSO were achieved. Among these methodologies, the PPBGA demonstrated the best overall performance across all analyzed objective functions and operational modes.

In summary, the proposed model improved upon the solution reported in [41] by 0.00158%, thereby validating both the research approach and the effectiveness of the methods developed in this work. Moreover, the PPBGA also outperformed the other methods in terms of emissions, energy losses, and solution consistency across both grid-connected and islanded scenarios.

7. Discussion

7.1. PPBGA’s Adaptability and Efficiency in Highly Constrained Problems

In this work, the population-based variant of one of the most classical optimization methodologies [45] was proposed for the optimal operation of a BESS in an AC MG, considering variable generation and demand. The optimization framework aimed to minimize power losses, emissions, and economic costs through a master–slave strategy, employing HPFSA for technical evaluation.

Unlike other existing approaches, the proposed methodology incorporates a parallel processing strategy to evaluate the objective functions of the entire population simultaneously, an optimization technique that has been increasingly overlooked in recent years. The trend has shifted toward the implementation of more modern and complex optimization methods (such as the PVSA and PPSO) [46,47], which offer greater accuracy in pattern tracking, leading to the discovery of better solutions and, ultimately, to the achievement of the global optimum.

However, although these types of methodologies achieve strong performance in models with wide solution spaces or that are lightly constrained, they may encounter difficulties in reaching optimal solutions with an adequate repeatability rate, i.e., achieving a low standard deviation in highly constrained scenarios. This limitation arises from their inherent design, which typically involves a small number of individuals and iterations (Table 4). As a result, the probability of finding better or optimal solutions in each iteration or analysis of the algorithm is reduced, and the likelihood of consistently identifying optimal solutions with low standard deviation or dispersion is further diminished (Table 5 and Table 6).

This phenomenon is conceptually illustrated in Figure 18 using a schematic Venn diagram, which provides an intuitive, non-numerical view of how increasing constraints reduce the feasible solution space.

In subfigure (a), the solution space ($\Omega$) of the problem is represented as an example, corresponding to the optimal operation of the BESS in an MG, considering variable generation and demand, and in the absence of operational constraints. In this case, a solution that satisfies the problem can be found anywhere within the region $\Omega$, which represents the domain of the nonlinear problem for the base case.

Subsequent subfigures illustrate how the feasible region becomes increasingly restricted as additional operational and technical constraints are introduced.

In subfigure (b), a new solution space, denoted as ${\Omega }^{\prime }$, is defined. This space is the result of the intersection between ellipses A and B, which represent the regions satisfying the proposed constraints for the grid-connected case. Specifically, region A corresponds to the limitations related to nodal voltage regulation, while region B corresponds to the current-carrying capacity limits of the distribution lines, both detailed in Section 2.5.3. Consequently, it can be concluded that the feasible solutions for the grid-connected case are those located within the intersection of regions A and B, that is, within ${\Omega }^{\prime }$.

Finally, in subfigure (c), the feasible solution space that solves the problem while satisfying the proposed constraints for the islanded case is presented. Here, two additional ellipses, C and D, are introduced, intersecting with A and B. These new regions correspond to the domains of the constraints related to energy trading and the operational limits of the diesel generator, respectively, both described in Section 2.5.4. In this case, the feasible solution space, denoted again as ${\Omega }^{\prime }$, represents the set of solutions that satisfy all constraints for the islanded operation. As a result, ${\Omega }^{\prime }$ becomes highly restricted, being limited to only a fraction of the original solution space.

Although this issue could potentially be mitigated by increasing the population size and the number of iterations involved in the optimization process, doing so introduces a new challenge, processing time, which becomes significantly longer due to the inherent mathematical complexity of algorithms such as the PVSA. As shown in Table 4, to achieve a similar (though still inferior) performance compared to PPBGA, this particular algorithm requires an increase of 133% in the number of individuals used in the isolated case compared to the grid-connected case. Moreover, the number of iterations must be raised from 1000 to 4000, resulting in a notable impact on processing time, as illustrated in Table 6, with an increase of 230.45% compared to that achieved by the PPBGA.

In this context, the PPBGA, based on the conventional genetic algorithm proposed by Chu and Beasley [45], outperforms other metaheuristic techniques implemented in this work, such as the PVSA and PPSO. This superior performance is primarily due to its simplicity and low mathematical complexity, which reduces the computational burden and processing time. Consequently, the algorithm can utilize larger population sizes and more iterations per power flow evaluation without excessive resource consumption. This balance enables the PPBGA to efficiently explore the solution space and consistently identify optimal solutions with low standard deviation, achieving results faster than competing methods.

Furthermore, despite similar numbers of function evaluations across methodologies, the PPBGA requires significantly fewer evaluations than alternatives like PSO and the PVSA to reach better solutions. This confirms that performance gains stem not from computational effort alone but from the inherent evolutionary capabilities and design of the algorithm to effectively converge toward the global optimum. Overall, these factors make PPBGA both highly effective and efficient in solving this optimization problem.

7.2. Trade-Off Evaluation in Single-Objective Optimization

Figure 19 and Figure 20 complement the comparative analysis among the best solutions obtained from the evaluated scenarios. Each figure presents 2D projections comparing the outcomes of independent single-objective optimizations applied to the 33-node system.

The scatter plots illustrate trade-offs between pairs of objective functions: system cost ($f_{\mathrm{Cost}}$ in USD), power losses ($f_{\mathrm{Loss}}$ in kWh), and carbon emissions ($f_{{\mathrm{CO}}_2}$ in TonCO₂). Points highlighted correspond to the best solutions obtained by optimizing each objective individually. The other axes show how these solutions perform regarding the remaining objectives, revealing the compromises involved when prioritizing one goal at a time.

Although the objectives, cost minimization, CO₂ emission reduction, and loss minimization, are optimized independently to isolate each indicator’s effect and facilitate clearer interpretation, they are inherently linked through shared system variables. In particular, the vector of line currents ${\mathbf{i}}_h\in {\mathbb{R}}^L$, which depends on conventional and distributed generation dispatch as well as energy storage operation, couples these objectives. Improvements in one often impact the others; for example, reducing losses by lowering line currents also decreases fuel consumption, emissions, and costs.

This structural coupling explains the proximity of optimal solutions across objectives despite the mono-objective approach and the similar energy storage dispatch profiles seen in Figure 15 and Figure 16. The clustered Pareto fronts highlight balanced trade-offs, offering decision-makers insights into selecting operational strategies under conflicting criteria.

These results also indicate the need to explore multi-objective optimization approaches in future work to better capture and navigate these trade-offs simultaneously.

8. Conclusions

This study addressed energy management in AC MGs, where operation optimization reduces operating costs, power losses, and CO₂ carbon footprint. A mathematical model was developed, considering variability in power generation and demand, along with the technical–operational constraints inherent to AC MG operation in on-grid and off-grid modes.

Solution methodologies were proposed based on three high-performance metaheuristic optimization techniques, the PPBGA, PPSO, and the PVSA, selected for their effectiveness in managing DERs in electrical networks. Their performance was evaluated in terms of best solution, average solution, standard deviation, and processing times through a statistical analysis based on 100 runs of each methodology in a 33-node AC MG operating in both grid-connected and islanded modes.

For the 33-node AC microgrid operating in grid-connected (on-grid) mode:

• The PPBGA demonstrated superior optimization capabilities, achieving the following results:

  -

  Average reductions of 1.421% in operational costs, 4.383% in power losses, and 0.183% in CO₂ emissions compared to the base scenario.

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  Exceptional solution stability with standard deviations of 0.0087% (costs), 0.0194% (losses), and 0.0008% (CO₂ emissions) across 100 runs.

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  Performance advantages of 0.322% lower costs, 1.5916% fewer losses, and 0.067% reduced footprint compared to PPSO and PVSA alternatives.

For the 33-node AC microgrid operating in islanded (off-grid) mode:

• The PPBGA maintained its performance leadership with the following results:

  -

  Reductions of 0.1311% in costs, 4.469% in losses, and 0.184% in emissions.

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  Remarkable consistency evidenced by standard deviations of 0.0013% (costs), 0.0516% (losses), and 0.0023% (carbon emissions).

  -

  Superiority over other methods by 0.323% (costs), 1.5915% (losses), and 0.065% (CO₂ footprint).

• The computational efficiency analysis revealed an average execution time of 75.62 s was achieved by the PPBGA, representing a 42.56% improvement over the 131.65-s average across methodologies, even under the most highly constrained scenario.

The comprehensive results validate the PPBGA as the optimal methodology for AC microgrid energy management, offering the following advantages:

• Technical superiority: Best-in-class optimization of all three objective functions (cost, losses, emissions).
• Operational reliability: Minimal solution variability across both grid-connected and islanded modes.
• Practical feasibility: Reasonable computational requirements suitable for daily planning cycles.

8.1. Research Limitations

This study presents several limitations that should be considered when interpreting the results:

• The analysis compared only two scenarios, one without distributed energy resources and another incorporating active and reactive power management through a BESS, without evaluating other configurations or technologies that may influence system performance.
• The research focused on single-phase, multi-node AC microgrids. Therefore, the findings may not be directly applicable to three-phase, hybrid, or DC microgrid configurations.
• It was assumed that the BESS units were already installed in the system. As such, issues related to sizing, placement, or investment costs were beyond the scope of this work.
• The microgrid operation was assumed to comply with national standards regarding voltage limits and loading capacity, and potential violations or abnormal operating conditions were not addressed.
• Simulations were carried out using MATLAB and sequential programming strategies, deliberately avoiding commercial tools. While this reduced implementation complexity and cost, it may limit the industrial applicability and scalability of the proposed solutions.
• Optimization strategies were selected based on their reported performance in similar nonlinear problems. Although parameter tuning was applied, no exhaustive benchmarking against emerging or hybrid metaheuristic techniques was conducted.

These limitations provide avenues for future research, particularly regarding the inclusion of investment analysis, more diverse network topologies, and advanced solution methodologies.

8.2. Future Work

The following future research directions were identified:

• Integration of reactive power management through BESSs installed in the AC MG, using power converters as a means of grid integration.
• Implementation and evaluation of a BESS relocation strategy in the proposed AC MG, considering local generation and demand characteristics, as well as energy resource variability, to minimize active power losses and improve voltage profiles, with economic and environmental impacts.
• Extend this framework to multi-day or seasonal horizons to analyze long-term operational patterns.
• The model will be extended to include energy losses from both converters (DC-DC and AC-AC) and the battery itself, considering internal resistance and aging effects. Integrating these losses will enhance the model’s realism and applicability to real microgrid scenarios.
• A current limitation of this study is the use of fixed average ${CO}_2$ emission factors for each operating mode, which does not reflect the variability in generator efficiency or renewable resource availability. While this assumption follows common practices in the literature and facilitates fair comparison among optimization methods, it restricts the environmental accuracy of the analysis. Future versions of the model will incorporate variable ${CO}_2$ emission rates based on generator output levels. This enhancement will enable more realistic environmental assessments and may significantly influence BESS operation strategies for carbon footprint reduction, increasing the applicability and robustness of the proposed methodology in real-world microgrid scenarios.
• Future research will explore the integration of asynchronous distributed control approaches into the proposed optimization framework. These strategies would enable the system to better adapt to spatial and temporal variability in distributed generation, such as PV arrays subject to heterogeneous irradiance conditions. In this context, distributed control at the tertiary level can offer a more robust and scalable architecture for energy management in highly variable microgrid environments.
• Another line of future work involves the incorporation of multi-stage operation and recovery strategies, particularly for handling transitions between grid-connected and islanded modes under uncertainty. This includes the development of systematic reconfiguration mechanisms, load prioritization schemes, and resilience-enhancing control logic. Higher-layer coordination will be considered as a means to provide fault recovery and service restoration capabilities.