A Pareto Multiobjective Optimization Power Dispatch for Rural and Urban AC Microgrids with Photovoltaic Panels and Battery Energy Storage Systems

Abstract

This paper presents an economic–environmental power dispatch approach for a grid-connected microgrid (MG) with photovoltaic (PV) generation and battery energy storage systems (BESSs). The problem was formulated as a multiobjective optimization problem with functions such as minimizing fixed and variable generation costs, power losses, and CO₂ emissions. This study addresses the problem of intelligent energy management in microgrids with PV generation and BESSs to optimize their performance based on multiple criteria. This study focuses on optimizing the Energy Management System (EMS) with metaheuristic algorithms to achieve practical implementation with simpler algorithms to solve a complex optimization problem. This study employs four multiobjective optimization algorithms: Nondominated Sorting Genetic Algorithm II (NSGA-II), Harris Hawks Optimization (HHO), multiverse optimizer (MVO), and Salp Swarm Algorithm (SSA), which are classified as robust techniques for obtaining Pareto fronts. The computational resources employed to simulate the problem are presented. The optimal dispatch obtained from the Pareto front achieved reductions of 0.067% in fixed costs, 0.288% in variable costs, 3.930% in power losses, and 0.067% in CO₂ emissions, demonstrating the effectiveness of the proposed approach in optimizing both economic and environmental performance. The SSA stood out for its stability and computational efficiency, establishing itself as a promising method for energy management in urban and rural microgrids (MGs) and providing a solid framework for optimization in alternating current systems.

1. Introduction

Microgrids (MGs) have become an effective option for integrating renewable energy into power systems, thus improving their reliability, sustainability, and efficiency [1]. Due to their ability to operate in a coordinated way at the local level, MGs adjust the balance between generation and demand and increase autonomy in both urban and rural contexts [2,3]. A central challenge is the variability of renewable sources, particularly photovoltaic (PV) generators, which require the coordinated operation of generators, storage, power electronic converters, and loads under the network constraints.

In this setting, battery energy storage systems (BESSs) are critical because of their fast response, contribution to voltage and frequency support, and ability to maintain secure operation [4,5]. To balance supply and demand and integrate renewable energy under technical and economic criteria, energy management systems (EMSs) rely on optimization frameworks that include classical methods, stochastic approaches, predictive control of the model, artificial intelligence, and metaheuristics [1,6]. On a daily scale, PV-BESS coordination involves tracking irradiance and load variations while maintaining admissible power and voltage profiles.

Classical methods, such as mixed-integer linear programming (MILP) and Benders decompositions, have proven useful in well-structured formulations [7,8]; however, they face limitations when multiple objectives, uncertainty, and real-time scalability are required. Stochastic and predictive control approaches explicitly model variability and enable forecast-based decisions, at the cost of greater modeling complexity and data demands [9,10,11,12,13,14,15,16]. In parallel, metaheuristics and AI techniques provide flexibility and practical implementation for nonconvex multiobjective problems, while raising challenges in ensuring solution quality and transferability to larger scales [6]. In general, control and optimization strategies reduce losses and operating costs, and maintain service continuity under fluctuating generation and demand [17,18]. Against this background, the next section reviews the state of the art in energy management for MGs with PV generators and BESSs.

In [19], a real-time EMS was proposed for a low-voltage microgrid to optimize the operation of diesel, wind turbine (WT) and PV generators, and energy storage units. The study minimized costs, emissions, and power losses using NSGA-II enhanced with Dynamic Crowding Distance (DCD) and a local search algorithm (LSA), revealing a trade-off between costs and losses. The strategies included minimizing diesel use during the day and increasing it at night. However, they did not validate the system with a 24-h power flow, separate fixed and variable costs, incorporate CO₂ with power losses, or consider statistical analysis.

A study introduced an improved multiobjective brain storm optimization (BSO) algorithm to schedule hybrid AC/DC microgrids, improving economic efficiency and energy storage health [20]. It mitigated premature convergence through learning-based and adaptive operators, using fuzzy decision-making for optimal dispatch solutions. The algorithm outperformed MOPSO, MOABC, and NSGAII in terms of quality, stability, and speed, addressing renewable energy uncertainties in microgrids. However, it was limited to an isolated marine case, did not consider a 24 h power dispatch for AC networks, did not separate fixed and variable costs, did not include CO₂ emissions, and did not consider detailed Pareto front data.

Huang et al. (2024) introduced an economic optimization scheduling strategy for offshore fishing raft microgrid clusters to minimize operational, maintenance, and electricity exchange costs [21]. The study highlighted the effectiveness of the constraint multiobjective evolutionary algorithm based on decomposition (CMOEA/D) over NSGA-II in optimizing cost efficiency and power stability. However, the study was limited by testing only two algorithms, focusing solely on generation and maintenance costs, and lacking comprehensive cost analysis, CO₂ emissions, and statistical examination of the results.

Keshta et al. (2024) presented a bi-level energy management strategy for grid-tied microgrids to reduce operational costs while meeting technical constraints [22]. The strategy uses ANN-based day-ahead forecasting for improved PV, WT, and load predictions, and the coronavirus herd immunity optimizer (CHIO) for effective scheduling. Real-time scheduling adapts to changes in weather, demand, and tariffs, thus enhancing cost savings. However, they minimize total operational costs, but limitations include the lack of differentiation between fixed and variable costs and the absence of integrated losses and CO₂ penalties. The study is based on a single case of a grid-tied microgrid, without testing other systems.

Another study introduced a two-level optimization approach for energy management and sizing of a multi-energy microgrid. The purpose of the study was to minimize income and expenses [23]. This highlighted that hydrogen production can boost income through self-consumption and grid sales, despite the high electrolyzer costs. The optimal design depended on investment costs and hydrogen prices, evaluating only capacity configuration without considering 24 h operational dispatch and separating hourly variables from fixed costs. Multi-algorithm and statistical analyses were not performed in the study.

In [24], the research focused on optimizing the capacity configuration of MGs connected to the grid with photovoltaic and WT generators, and BESSs in China using an improved Beluga Whale Optimization (BWO) algorithm. The objective was to improve the utilization and economic efficiency of renewable energy by minimizing the levelized cost of energy (LCOE), maximizing the consumption of renewable energy, and minimizing the system costs. The improved BWO algorithm combined opposition-based learning and bee colony strategies to enhance population diversity and convergence accuracy. When applied to real-world local load data, it surpassed traditional methods in terms of solution quality, achieving an optimal LCOE of 0.192 yuan/kWh. This study highlighted the role of advanced optimization in the design of microgrids for low-carbon power and economic growth. However, the study did not validate the problem in a 24 h AC power dispatch, separate variables from fixed costs, or include multi-algorithm analyses.

In [25], the authors investigated the optimization of renewable energy microgrids using the Levy Flight Algorithm (LFA). They focused on hybrid hydrogen and battery storage to improve the energy management and reliability of the system with PV and WT generators. The LFA demonstrated superior cost-saving performance and improved levelized energy cost compared to other algorithms, highlighting the importance of hybrid storage in grid stability and reduction of carbon emissions. However, the study did not address 24-h AC power dispatch with network constraints, joint fixed and variable costs, power losses, CO₂ emissions in a multiobjective scheme with penalties, or provide Pareto fronts and variability indicators with statistical support.

Another study introduced the Golden Jackal Optimization (GJO) algorithm for managing microgrid energy, focusing on minimizing costs and addressing energy management challenges in systems with hybrid energy sources and battery storage [26]. It demonstrated superior efficiency and cost-effectiveness of GJO compared to other algorithms, such as Particle Swarm Optimization, Artificial Bee Colony, and Tabu Search. However, the GJO approach remained limited to economic objectives without incorporating multiobjective functions such as CO₂ emissions, and it lacked validation for 24-h power dispatch in AC networks and robustness analysis through multiple runs and nonparametric tests.

The study in [27] investigated multiobjective optimization for production scheduling in hybrid renewable power plants with battery energy storage systems (BESS) to enhance profits and reduce battery degradation. A particle swarm optimization (PSO) algorithm was used to balance profitability and battery life, and it was compared favorably with single-objective PSO and linear programming methods by extending battery life while maintaining similar profitability. It highlighted the importance of incorporating energy efficiency into cost functions and suggested alternative revenue strategies, such as ancillary services, owing to limited profits from energy arbitrage in markets such as the Iberian Electricity Market. The study also suggests improvements by using real production forecasts and considering the penalties for deviations. However, it did not perform 24 h power dispatch in an AC multinodal network, separate fixed and variable costs, integrate CO₂, or document Pareto front variability with confidence intervals.

Shaker et al. (2021) explored optimal strategies for charging and discharging energy storage in grid-connected microgrids using a multiobjective hunger game search optimizer (MOHGS) [28]. They focused on a hybrid microgrid with renewable energy and energy storage solutions to minimize costs, emissions, and loss of power supply while maximizing the use of renewable energy. The MOHGS outperformed the other optimizers in terms of cost efficiency and environmental benefits. The analysis incorporated the costs associated with battery degradation and employed supercapacitors to improve the system reliability. Despite its advantages, the study did not test 24 h operation under AC power constraints, articulate a comprehensive daily cost and emission function, or report Pareto fronts and statistical robustness.

In addition, a study explored the optimization of microgrid load dispatch with electric vehicles (EVs) using a modified gravitational search algorithm and particle swarm optimization (MGSA-PSO) [29]. To reduce costs, pollution, and load variance, unordered and ordered EV charging–discharging strategies were examined. The MGSA-PSO algorithm improved the convergence speed and accuracy, with ordered charging-discharging with DG cutting costs by 25.55% and load variance by 83.51%. This study emphasized the role of smart EV charging in grid efficiency and the trade-off between economic benefits and grid stability. However, the absence of multinodal PV–BESS co-management, fixed and variable cost separation, power loss consideration, and statistical Pareto front analysis were noted.

The study in [30] explored the optimization of P2P power trading in microgrids using coalition game theory. It focused on maximizing renewable energy self-consumption and monetary benefits for prosumers through a mixed-integer linear programming model. Simulations with 30 households considered load and market constraints, promoting local energy use and renewable investment. However, the model did not consider 24 h AC power dispatch and a comprehensive multiobjective scheme, and did not provide Pareto fronts or confidence intervals.

Parvin et al. (2023) explored the optimization of a renewable microgrid in Shiraz, Iran, using the MOPSO algorithm, and analyzed configurations combining wind turbines, PV, and CHP systems [31]. The focus is on minimizing the probability of power supply loss and energy cost, considering factors such as weather and demand. This study highlights the local impacts of climate change and suggests hydrogen-based energy storage for remote areas. The limitations included the lack of BESS integration, 24 h validation, cost differentiation, CO₂ considerations with losses, and the absence of a Pareto front or statistical support.

The authors of [32] developed a multiobjective optimization strategy for home EMS that included photovoltaic and battery storage systems. This intelligent home EMS, featuring three adaptable strategies, aims to maximize economic benefits and consumer comfort by balancing costs, peak-valley balance, and user satisfaction through a multiobjective model with weighted coefficients. The study predicted PV power output using historical weather data and demonstrated significant improvements, such as a 39.81% reduction in electricity bills, a 50.37% decrease in peak load, and a 1.6 times increase in the user comfort index. However, the analysis was limited to residential users and did not account for the constraints of the multinode AC microgrid, cost separation, losses, CO₂ penalties, or provide Pareto fronts with confidence intervals and nonparametric tests.

Lu et al. presented a multiobjective optimal load dispatch model for microgrids that accounts for stochastic access to EVs. They aimed to minimize costs and load variance while considering uncertainties in EV charging [33]. The study included PV and WT generators, using Monte Carlo simulation and improved Particle Swarm Optimization. Three scenarios were examined: uncoordinated and coordinated charging states. The results showed that coordinated charging reduced costs by 3.09% and improved stability by shifting loads, with DGs adding a further 6.43% cost reduction. The study highlighted cost and stability trade-offs, but lacked an analysis of PV–BESS operation, 24 h AC dispatch with constraints, a unified cost formulation including CO₂, and robustness analysis.

The article by [34] explored the optimization of distribution networks with generators and BESS, with optimal network reconfiguration. The study introduced the Improved Bidirectional Coevolutionary (I-BiCo) algorithm to minimize energy costs, power losses, and voltage deviations. The algorithm addressed renewable energy variability through BESS and uses a multiobjective optimization approach to determine the best locations and sizes for RES and BESS. The I-BiCo algorithm outperformed existing methods in terms of convergence, diversity, and quality of the solution. Despite demonstrating the benefits of combining DG, BESS, and network reconfiguration, the study did not validate the 24 h AC dispatch within specific limits, differentiate between fixed and variable costs, integrate losses and CO₂ penalties simultaneously, or provide statistical evaluations of Pareto fronts.

Table 1. Literature review and gaps.

Ref.	Energy Cost	Power Losses	CO2 Emissions	Pareto	Multi DERs	Multinodal MG	Penalties	Statistics	Urban + Rural
[19]	✓	✓	✓	✓	✗	✗	✓	✗	✗
[20]	✓	✗	✗	✓	✓	✗	✗	✗	✗
[21]	✓	✗	✗	✓	✗	✓	✓	✗	✗
[22]	✓	✗	✗	✗	✗	✗	✗	✗	✗
[23]	✓	✗	✗	✓	✗	✗	✗	✗	✗
[24]	✓	✗	✗	✓	✗	✗	✗	✗	✗
[25]	✓	✗	✗	✗	✗	✗	✗	✗	✗
[26]	✓	✗	✗	✗	✗	✗	✗	✗	✗
[27]	✓	✗	✗	✗	✗	✗	✓	✗	✗
[28]	✓	✗	✓	✓	✗	✗	✗	✗	✗
[29]	✓	✗	✓	✗	✗	✗	✗	✗	✗
[30]	✓	✗	✗	✗	✗	✗	✗	✗	✗
[31]	✓	✗	✓	✓	✗	✗	✗	✗	✗
[32]	✓	✗	✗	✗	✗	✗	✓	✗	✗
[33]	✓	✗	✓	✗	✗	✗	✗	✗	✗
[34]	✓	✓	✗	✓	✓	✓	✗	✓	✗
Our approach	✓	✓	✓	✓	✓	✓	✓	✓	✓

presents a summary and comparison of the previous literature and the proposed method. The literature shows that most solutions have focused on multiobjective optimization based solely on energy costs. Applications have also concentrated on single-node tests with various energy resources, such as buildings, homes, maritime, and other small networks. However, more research is required to include additional objective functions, such as electrical network losses and CO₂ emissions. Moreover, these solutions should be applied to distribution networks based on microgrids with multiple nodes and energy resources at various locations. It is also desirable that the management system be applicable to both urban and rural networks. In addition, studies have been practically limited to the results of cost optimization, and no further statistical analyses have been presented to demonstrate the efficiency of the solutions obtained.

Therefore, this paper employs multiobjective optimization for an EMS in an AC microgrid using four objective functions: fixed energy costs, variable energy costs, power losses, and CO₂ emissions. The proposed algorithm is capable of managing energy for microgrids with PV generation, demand, and storage systems. Optimization can be performed for microgrids connected to the power grid or in island mode. Batteries previously located in the system are managed, considering charge and discharge states, which resulted in a total of 72 variables, considering the 24 SOC states of the batteries for one day of operation. This multiobjective function is solved by finding the Pareto frontiers for the objective functions and identifying the best solutions to the problem. The application of Pareto-based multiobjective optimization algorithms offers several advantages in solving complex problems with multiple conflicting objectives [35]. The mathematical formulation of this problem is explained in Section 2. The multiobjective formulation was solved using four algorithms: Harris Hawks Optimization (HHO), multiverse optimizer (MVO), Salp Swarm Algorithm (SSA), and nondominated sorting genetic algorithm II (NGSAII). First, the general algorithm is explained in Section 3. Subsequently, a general explanation of the advantages of each algorithm and the corresponding mathematical model is presented. Two power systems were used to test the methods: 27-node feeder and 33-node feeder test cases; the network parameters are included in Section 4. This section also presents the parameters of energy resources, power demand, and energy costs. The computational resources employed to simulate the problem are also presented. This study presents various results (Section 5) that focus on the details of decisions related to EMS. Finally, the conclusions and future work are presented (Section 6).

Based on the gaps found in previous studies, this study makes the following contributions:

• The literature shows that most previous research has focused on performing multiobjective optimization centered on energy costs. Few studies have used penalties on constraints to find valid solutions. In this study, multiobjective cost functions were addressed as separate functions: fixed costs, variable costs, power losses, and CO₂ emissions. In addition, penalties are considered in the constraints to obtain better solutions.
• Only a few works in the previous literature have used the Pareto front to present their solutions. In this study, the Pareto front was used to find solutions for each of the functions and to suggest the best results obtained from the problem.
• From the literature reviewed, only two works have proposed working with systems that have multiple DERs in microgrids, and these works use a single node and several energy resources. This study utilized multiple energy resources and nodes to optimize the operation.
• Most works only present conventional results and do not perform statistical analyses to select the best options. This study includes all statistical analyses that enable the characterization of the behaviors of the best solutions.
• The solutions proposed in the literature do not consider operational conditions for urban and rural networks within their mathematical formulations; they focus their solutions on one of these. This study proposes optimization solutions that can be applied to both types of networks located in urban and rural areas.

2. Optimization Problem

The optimal energy dispatch in AC microgrids, considering the integration of photovoltaic generation as a baseline case and its comparison with the additional incorporation of lithium-ion batteries, can be effectively formulated as a multiobjective optimization problem. To resolve this, it is essential to define a set of objective functions that structure the optimization methodology, aimed at the simultaneous minimization of costs, power losses, and CO₂ emissions. These functions were constructed according to the guidelines established in the specialized literature and sought to address the gaps identified in previous research. The general methodology is presented below, along with the respective mathematical formulations.

2.1. General Procedure

Figure 1 shows the general procedure of this study. In this procedure, the microgrid is modeled, including PV generators, power grid, and batteries. Additionally, the mathematical formulation of the multiobjective optimization problem based on Pareto was performed. In this case, the four objective functions defined in this study were considered (fixed energy cost, variable energy cost, power losses, and CO₂ emissions). Subsequently, the modeling of the multiobjective algorithms considered to solve the problem was carried out. It is important to adjust the algorithms to obtain a better solution. Finally, the results were obtained and statistical analyses were performed. Based on these results, the comparison and discussion are presented.

2.2. Objective Function

The main objective function serves as the core of the entire multiobjective optimization process, integrating the different targets to be minimized: fixed and variable costs, power losses, and CO₂ emissions. Its formulation depends on the operational context of the microgrid, distinguishing between urban and rural areas. In the urban scenario, four objective functions are considered: fixed costs, variable costs, power losses, and CO₂ emissions, as defined in Equation (1). In contrast, in the rural scenario, where the microgrid operates primarily in island mode with diesel generation, only three objective functions are considered: fixed costs, power losses, and CO₂ emissions, as expressed in Equation (2).

$$FO{B}_{(1,2,3,4)}=\left[\begin{array}{cccc}O{F}_1& O{F}_2& O{F}_3& O{F}_4\end{array}\right]+\beta ·PF$$

(1)

$$FO{B}_{(1,3,4)}=\left[\begin{array}{ccc}O{F}_1& O{F}_3& O{F}_4\end{array}\right]+\beta ·PF$$

(2)

In particular, each objective function incorporates a penalty factor $\left(PF\right)$ multiplied by a normalization factor $\beta =1000$, which aims to ensure a balance between the feasibility of the problem and the usefulness of the solutions generated by the optimization algorithms. These values were defined heuristically, as described in [36,37,38].

Each term represented in the $FOB$ vector is formulated to broadly structure the objective functions, constraints, and penalties that constitute the mathematical model of the proposed optimization problem. Although the specific meaning of each $O{F}_i$ component is not detailed at this stage, their individual formulations will be developed in the following sections, where their technical foundations and relevance to the microgrid energy management scheme will be explained in detail.

2.3. Operation Costs Minimization

The objective functions $O{F}_1$ (fixed costs) and $O{F}_2$ (variable costs) represent the minimization of the total operational costs associated with the energy supply in the AC microgrid during a typical 24 h operating day. In Equation (3), t denotes the hours in the analysis horizon.

$$\begin{array}{c}\hfill {\mathrm{OF}}_{1,2}=min\left(\sum _{t=1}^{24}\left(\underset{C_{EE}}{\underbrace{\sum _{i\in N}\left(C_{CGi}\left(t\right)·P_{CGi}\left(t\right)\right)}}+\underset{C_M}{\underbrace{\sum _{i\in N}\left(C_{MDGi}·P_{DGi}^{PV}\left(t\right)+C_{MBi}·P_{Bi}\left(t\right)\right)}}\right)\right)\end{array}$$

(3)

The term $C_{EE}$ represents the component of the energy cost, where $C_{CGi}\left(t\right)$ is the unit cost of the energy generated by the conventional generator at node i. This component distinguishes between fixed and variable costs, depending on the type of MG. In urban contexts, where energy prices fluctuate, both types of costs are considered in the analyses. In contrast, rural MGs operating in island mode assume a fixed energy cost derived from diesel consumption that remains constant throughout the operating period. The variable $P_{CGi}\left(t\right)$ denotes the power generated by the conventional generator at node i.

The term $C_M$ groups the maintenance costs associated with the DERs. Specifically, $C_{MDGi}$ represents the maintenance cost of PV generators, and $C_{MBi}$ corresponds to the maintenance costs of PV generators and batteries. The variables $P_{DGi}^{PV}\left(t\right)$ and $P_{Bi}\left(t\right)$ represent the power generated by the PV systems and the power managed (charged/discharged) by the batteries at node i.

2.4. Power Loss Minimization

The objective function $O{F}_3$ represents the minimization of power losses ($P_{Loss}$) in the electrical system under evaluation during a typical operating day. These losses are a key parameter in MGs, as their reduction directly contributes to enhancing the efficiency, reliability, and overall sustainability of the system, particularly when energy storage systems, such as batteries, are integrated.

$$\begin{array}{c}O{F}_3=min\left(\sum _{t=1}^{24}\underset{P_{Loss}}{\underbrace{{\displaystyle \sum _{i\in N}\sum _{j\in N}}\left(Y_{ij}·V_i\left(t\right)·V_j\left(t\right)·cos({\theta }_i\left(t\right)-{\theta }_j\left(t\right)-{\phi }_{ij}\left(t\right))\right)}}\right)\end{array}$$

(4)

In this expression, $V_i\left(t\right)$ and $V_j\left(t\right)$ denote the voltage magnitudes at the nodes i and j at time t, while ${\theta }_i\left(t\right)$ and ${\theta }_j\left(t\right)$ are their corresponding phase angles. The parameter ${\phi }_{ij}\left(t\right)$ represents the angle of the complex admittance of the line connecting nodes i and j, and $Y_{ij}$ is the magnitude of the admittance derived from the admittance matrix of the system. This formulation enables the accurate quantification of the losses of the technical lines within the MG by considering the voltage profile and both active and reactive power flows, thus providing a comprehensive measure of the technical performance of the system.

2.5. CO₂ Emission Minimization

The objective function $O{F}_4$ represents the minimization of total carbon dioxide (CO₂) emissions generated by the power sources within the microgrid during a typical operating day. This component is essential for integrating environmental sustainability criteria into the optimization process, particularly in scenarios with a high share of fossil-based or nonrenewable generation.

$$\begin{array}{c}O{F}_4=min\left(\sum _{t=1}^{24}\underset{\mathrm{E}}{\underbrace{\left({\displaystyle \sum _{i\in N}}{P}_{CGi}\left(t\right)·C{E}^{CG}i+\sum j\in N{P}_{GDj}\left(t\right)·C{E}_j^{GD}\right)}}\right)\end{array}$$

(5)

In this expression, E denotes the total CO₂ emissions per hour generated by the microgrid. The term $P_{CGi}\left(t\right)$ refers to the power generated by the conventional generators at node i, and $C{E}_i^{CG}$ is the associated emission factor of CO₂. Similarly, $P_{GDj}\left(t\right)$ represents the power generated by the generators in the MG at node j, with $C{E}_j^{GD}$ being the corresponding emission coefficient. This formulation enables a quantitative assessment of the environmental impact of energy dispatch, incorporating the ecological dimension into the optimal energy management model of the MG.

2.6. Power Balance Constraints

Once the objective functions that form the multiobjective optimization framework have been defined, it is essential to incorporate a set of constraints that ensure the technical and operational feasibility of the electric power system under analysis. In particular, maintaining the balance of active and reactive power is crucial for accurately modeling the operation of AC microgrids with renewable energy, conventional power sources, and energy storage systems.

The active power balance constraint, shown in Equation (9), ensures that at each node i and every time step t, the sum of the active power injected by conventional generators $P_{CGi}\left(t\right)$, renewable generators $P_{DGi}{\left(t\right)}^{PV}$, and batteries $P_{Bi}\left(t\right)$—which may either inject or absorb power, depending on their operational mode—must equal the active demand $P_{Di}\left(t\right)$ plus the power losses associated with the network transfers. These losses are modeled using the admittance parameters $Y_{ij}$ and phase angles ${\theta }_i\left(t\right)$ and ${\theta }_j\left(t\right)$.

$$\begin{array}{c}{P}_{CGi}\left(t\right)+P_{DGi}^{PV}\left(t\right)\pm P_{Bi}\left(t\right)-P_{Di}\left(t\right)\\ =V_i\left(t\right)·\sum _{j\in N}{Y}_{ij}·V_j\left(t\right)·cos\left({\theta }_i\left(t\right)-{\theta }_j\left(t\right)-{\phi }_{ij}\left(t\right)\right)\\ \forall i,j\in N,\forall t\in H\end{array}$$

(6)

Additionally, the reactive power balance constraint, shown in Equation (10), aims to ensure voltage stability across the electrical network by requiring that the reactive power generated adequately meets the demand of the users at each node. This formulation considers the reactive power supplied by conventional generators $Q_{CGi}\left(t\right)$ and photovoltaic generators to the MG $Q_{DGi}{\left(t\right)}^{PV}$, along with the reactive power demand of the loads $Q_{Di}\left(t\right)$ at each node of the system. As with the active power balance, this equation incorporates electrical system parameters, such as the admittances $Y_{ij}$ and phase angles ${\theta }_i\left(t\right)$ and ${\theta }_j\left(t\right)$, which are now applied to calculate the reactive power flows.

$$\begin{array}{c}{Q}_{CGi}\left(t\right)+Q_{DGi}^{PV}\left(t\right)-Q_{Di}\left(t\right)\\ =V_i\left(t\right)·\sum _{j\in N}{Y}_{ij}·V_j\left(t\right)·sin\left({\theta }_i\left(t\right)-{\theta }_j\left(t\right)-{\phi }_{ij}\left(t\right)\right)\\ \forall i,j\in N,\forall t\in H\end{array}$$

(7)

In this context, it is essential to adhere to the operational generation limits defined for each energy source within the active and reactive power balances of the MG. This ensures the technical feasibility of the system, particularly for configurations that integrate intermittent renewable sources with battery-based energy storage. These generation constraints play a critical role in maintaining the operational stability and reliability of the power system, ensuring that all generating units operate within their technical capabilities during the analysis period.

Consequently, conventional generation is constrained by the upper and lower bounds for both active and reactive power, as expressed in Equations (8) and (9), respectively.

$$\begin{array}{c}{P}_{CGi}^{min}\le P_{CGi}\left(t\right)\le P_{CGi}^{max}\\ \forall i\in N,\forall t\in H\end{array}$$

(8)

$$\begin{array}{c}{Q}_{CGi}^{min}\le Q_{CGi}\left(t\right)\le Q_{CGi}^{max}\\ \forall i\in N,\forall t\in H\end{array}$$

(9)

Additionally, the power generated by photovoltaic systems must remain within their defined operational limits, as stated in Equation (10). These constraints ensure the safe and efficient operation of renewable resources, considering their intermittency.

$$\begin{array}{c}{P}_{DGi}^{min}\le P_{DGi}{\left(t\right)}^{PV}\le P_{DGi}^{max}\\ \forall i\in N,\forall t\in H\end{array}$$

(10)

Finally, battery operation is constrained by the maximum allowable power during the charging and discharging processes ($P_{Bi}$), as specified in Equation (11). These constraints aim to improve energy management efficiency and preserve and extend the useful life of storage systems.

$$\begin{array}{c}{P}_{Bi}^{{\mathrm{Char}}_{max}}\le P_{Bi}\left(t\right)\le P_{Bi}^{{\mathrm{Disch}}_{max}}\\ \forall i\in N,\forall t\in H\end{array}$$

(11)

2.7. Operational Constraints

Additionally, within the set of constraints imposed on the power balance and the formulation of objective functions, it is essential to include those that ensure the correct technical operation of the MG. First, the voltage levels at each node must be maintained within the allowable operational margins, as defined in Equation (12).

$$V_i^{min}\le V_i\left(t\right)\le V_i^{max}\forall i\in N,\forall t\in H$$

(12)

Similarly, the current flowing through the network lines must be limited to prevent overloads that could compromise the stability and integrity of electrical systems. This condition is expressed in Equation (13).

$$I_{ij}\left(t\right)\le I_{ij}^{max}\forall i,j\in N,\forall t\in H$$

(13)

Regarding batteries, constraints play a crucial role in ensuring the efficiency and reliability of energy management in the MG. One of the key variables to control is the state of charge (SoC), which must remain within safe limits to preserve the lifespan of the storage system and ensure an adequate response to fluctuations in demand and renewable generation. The SoC is calculated as a function of the analysis period and the charge/discharge factor $\left({\varphi }_{Bi}\right)$, as shown in Equation (14).

$${\mathrm{SoC}}_{Bi}\left(t\right)={\mathrm{SoC}}_{Bi}(t-1)-\left({\varphi }_{Bi}\times P_{Bi}\left(t\right)\right)\times \Delta t$$

(14)

The factor ${\varphi }_{Bi}$ is determined by the technical characteristics of the battery—in particular, its charging $\left(t{c}_{Bi}\right)$ and discharging $\left(t{d}_{Bi}\right)$ times—and is defined according to Equation (15).

$${\varphi }_{Bi}={\displaystyle \frac{1}{t{c}_{Bi}\times P_{Bi}^{{\mathrm{char}}_{-}min}}}={\displaystyle \frac{1}{t{d}_{Bi}\times P_{Bi}^{{\mathrm{disch}}_{-}max}}}$$

(15)

The state of charge (SoC) must remain within a safe operating range, typically between 10% and 90% of the battery’s total capacity. This constraint is formulated as shown in Equation (16).

$$\begin{array}{c}{\mathrm{SoC}}_i^{min}\le {\mathrm{SoC}}_{Bi}\left(t\right)\le {\mathrm{SoC}}_i^{max}\\ \forall i\in N,\forall t\in H\end{array}$$

(16)

Finally, Equation (17) presents the formulation of the penalty function ($PF$). This function is designed to ensure that all technical constraints imposed on the system are satisfied during the iterative optimization process. In practical terms, $PF$ serves as a corrective mechanism that increases the value of the objective function whenever any critical operational conditions, such as voltage levels, line currents, or battery SoC, are outside the allowable limits.

$$PF=\sum _{t=1}^{24}\left\{\begin{array}{c}max\left\{0,\sum _{i\in \mathcal{N}}\left(P_{Bi}\left(t\right)-P_{Bi}^{{\mathrm{char}}_{max}}\right)\right\}\\ +min\left\{0,\sum _{i\in \mathcal{N}}\left(P_{Bi}\left(t\right)-P_{Bi}^{{\mathrm{disch}}_{max}}\right)\right\}\\ +max\left\{0,\sum _{i\in \mathcal{N}}\left({\mathrm{SoC}}_{Bi}\left(t\right)-{\mathrm{SoC}}_i^{max}\right)\right\}\\ +min\left\{0,\sum _{i\in \mathcal{N}}\left({\mathrm{SoC}}_{Bi}\left(t\right)-{\mathrm{SoC}}_i^{min}\right)\right\}\\ +max\left\{0,\sum _{i\in \mathcal{N}}\left(V_i\left(t\right)-V_i^{max}\right)\right\}\\ +min\left\{0,\sum _{i\in \mathcal{N}}\left(V_i\left(t\right)-V_i^{min}\right)\right\}\\ +max\left\{0,\sum _{i\in \mathcal{N}}\sum _{j\in \mathcal{N}}\left(I_{ij}\left(t\right)-I_{ij}^{max}\right)\right\}\end{array}\right\}$$

(17)

2.8. Hourly Power Flow

The optimization model considers a daily analysis horizon divided into 24 h intervals. For each individual in the candidate population, an hourly power flow simulation was performed on the microgrid, integrating the operation of conventional generators, photovoltaic generators, and battery storage systems.

At each hour t, the power flow is solved based on the alternating current (AC) model, using the system’s active and reactive demands as input. The estimated active generation for that hour was derived from conventional generators, photovoltaic generators, and the power exchanged by batteries, which was calculated based on the changes in the state of charge (SoC). These values are used to compute the net load power, which serves as the input for solving the nodal system of complex voltages, as expressed in Equation (18).

$$\begin{array}{c}\hfill \mathbb{V}{d}^{t+1}=-\mathbb{Z}dd\left[\mathbb{Z}ds\mathbb{V}s+{\mathrm{diag}}^{-1}\left(\mathbb{V}{d}^{t,\ast }\right)\mathbb{S}{d}^{\ast }\right]\end{array}$$

(18)

In this model, the voltages at the load nodes are represented by the vector $\mathbb{V}d$, and the voltage at the main supply node of the electrical grid is denoted as $\mathbb{V}s$. The admittance matrices $\mathbb{Z}dd$ and $\mathbb{Z}ds$ describe the electrical connections among the load nodes and between the load and supply nodes, respectively. This procedure is repeated iteratively until the convergence criterion based on voltage variations is satisfied. Once the nodal voltage vector ${\mathbb{V}}_d$ is determined, all intrinsic variables associated with the objective functions are computed.

To conclude this section and introduce the multiobjective optimization algorithms, it is essential to emphasize that the energy management problem—integrating photovoltaic generation, battery storage, and multiobjective functions in AC microgrids—requires following the step-by-step sequence outlined in Algorithm 1, as it provides a clear structure for implementation and adaptation to the selected optimization techniques.

Algorithm 1 General pseudocode for multiobjective evaluation in alternating current (AC) microgrids
1:	Load system parameters: nodal topology, demand curves $P_{Di}\left(t\right)$ and $Q_{Di}\left(t\right)$, generation capacities $P_{CGi}^{min,max}$, and battery operation limits.
2:	Initialize the population of individuals ($N_i$) (Optimization algorithm), number of iterations.
3:	Compute the evolution of the state of charge ${\mathrm{SoC}}_{Bi}\left(t\right)$ using Equation (14), with charge/discharge factors ${\varphi }_{Bi}$ (Equation (15)).
4:	Validate the trajectories of ${\mathrm{SoC}}_{Bi}\left(t\right)$ according to the limits in Equation (16).
5:	for each individual $i=1$:$N_i$ do
6:	Assign $P_{Bi}\left(t\right)$ profiles based on SoC.
7:	Set $P_{DGi}^{PV}\left(t\right)$ and $P_{CGi}\left(t\right)$ according to the defined scenarios.
8:	Initialize electrical state matrices: $V_i\left(t\right)$, $P_{CGi}\left(t\right)$, $P_{Bi}\left(t\right)$, $P_{DGi}^{PV}\left(t\right)$.
9:	for each hour $t=1$ to 24 do
10:	Update net demand ($S_{dx}\left(t\right)$) considering battery operation.
11:	Solve the AC load flow using the successive approximations model (Equation (18)).
12:	Update nodal voltages $V_i\left(t\right)$ and angles ${\theta }_i\left(t\right)$.
13:	Compute injections of $P_{CGi}\left(t\right)$, $P_{DGi}\left(t\right)$ and losses in the network $P_{Loss}\left(t\right)$ (Equation (4)).
14:	Check current constraints on lines $I_{ij}\left(t\right)$ (Equation (13)) and voltage limits (Equation (11)).
15:	Apply active and reactive power balance (Equations (6) and (7)).
16:	end for
17:	Evaluate objective functions (Equation (1) or (2)):
18:	Apply penalty $PF$ if violations exist (Equation (17)):
19:	Build multiobjective vector (Equation (1) or (2)):
20:	end for
21:	Return all $FOB$ vectors for use in the multiobjective optimization algorithm

2.9. Pareto Multiobjective Optimization

Multiobjective optimization algorithms are advantageous because they allow the evaluation of multiple functions; therefore, it is essential to highlight the role of the Pareto front. These fronts are used in problems with multiple objectives because, in these scenarios, there is generally no single solution capable of minimizing or maximizing all the objective functions simultaneously. Therefore, the focus is on identifying a set of solutions known as the Pareto optimal set. In this context, a solution ${\mathbf{X}}_1$ is said to dominate another solution ${\mathbf{X}}_2$ if and only if the criterion in Equation (19) is satisfied.

$$\forall i:F_i\left({\mathbf{X}}_1\right)\le Fi\left({\mathbf{X}}_2\right)\mathrm{y}\exists j:Fj\left({\mathbf{X}}_1\right)<Fj\left({\mathbf{X}}_2\right)$$

(19)

The set of solutions that are not dominated by any other solution is called a Pareto front. This set represents the optimal trade-offs among the various objective functions under consideration in the vector-solution space. Each point on the front corresponds to an efficient equilibrium, where improving one objective necessarily worsens at least another objective. In contrast, solutions that lie outside this frontier are considered dominated, as there exists at least one other solution that improves one or more objectives without degrading the others (See Figure 2).

3. Optimization Algorithms

Once the multiobjective behavior with Pareto frontiers was established, a general description was provided for each of the optimization algorithms implemented to offer a solution to the problem of intelligent energy management applied to microgrids (MG) with photovoltaic solar systems and batteries. For each algorithm, special emphasis is placed on its method of progression or evolution. Thus, this study employs four multiobjective optimization algorithms, NSGA-II, HHO, MVO, and SSA, which are classified as robust techniques for obtaining Pareto fronts. Their evaluation functions inherently involve nonlinear and nonconvex problems within their models, allowing for a homogeneous assessment of their objective functions. This portfolio ensures broad and comparable Pareto fronts for technical-economic and environmental dispatch in MG with PV–BESS. NSGA-II was adopted as the baseline because of its nondominated sorting and effective preservation of diversity. The HHO contributes to adaptive exploitation–exploration with jumps that help to escape local optima. The MVO algorithm provides early global coverage of space with controlled jumps, which is useful for high-dimensional hourly vector data. Finally, the SSA smooths the trajectories (leader–chain), favoring a stable convergence with a low variance.

3.1. Harris Hawks Optimization (HHO)

The HHO is a bioinspired optimization technique that emulates the cooperative hunting behavior of Harris’s hawks. This bird species is known for its intelligence and sociability, as it can execute coordinated harassment and ambush tactics on agile prey, such as rabbits, simulating a progressive chase through rapid jumps in different directions. This strategy, known as the pounce surprise or the seven attacks, forms the basis of a mathematical model that guides the HHO optimization process [39].

This algorithm operates in two fundamental phases: exploration and exploitation, which alternate dynamically based on the remaining energy of the prey species. This energy, represented by the parameter E, decreases over the iterations as the process progresses and is modeled by Equation (20).

$$E=2E_0\left(1-{\displaystyle \frac{t}{T}}\right)$$

(20)

where $E_0$ is in the range (−1, 1) and is taken randomly to simulate the initial state of the prey, t represents the current iteration, and T is the total number of iterations. It is important to note that for the algorithm to enter the exploration or exploitation phase, it is compared with the value of $\left|E\right|$; if this is greater than or equal to 1, the exploration phase is activated; if, on the other hand, it is less than 1, the algorithm transitions to the exploitation phase of the search space. This allows for smooth adaptation between the two phases of the search space. It is worth mentioning that in the exploratory phase, hawks explore the search environment from random or strategic positions close to other members of the group, as denoted in Equation (21).

$$X(t+1)=\left\{\begin{array}{cc}{X}_{rand}\left(t\right)-r_1\left|X_{rand}\left(t\right)-2r_{2}X\left(t\right)\right|\hfill & q\ge 0.5\hfill \\ X_{rabbit}\left(t\right)-X_m-r_3\left(Lb+r_4\left(Ub-Lb\right)\right)\hfill & q<0.5\hfill \end{array}\right.$$

(21)

where $X\left(t\right)$ represents the current position of the hawk, $X_{rand}$ is a random hawk, $X_{rabbit}$ corresponds to the best current solution, and $X_m$ is the average of all positions in the population. $r_1$, $r_2$, $r_3$, $r_4$, and q are uniformly distributed random variables in the range of $(0, 1)$, and $Lb$ and $Ub$ are the lower and upper bounds of the search space, respectively. However, during the exploitation phase, a single strategy is not applied; instead, it adapts its behavior according to the energy level E of the prey and the probability r that it will be able to escape. This adaptability gives rise to four attack modes that are dynamically activated as the search space evolves.

For these possible phases, the process begins with the possibility of a soft besiege, which is triggered when $\left|E\right|\ge 0.5$ and $r\ge 0.5$. In this scenario, the prey retains sufficient energy but fails to escape. Therefore, the hawks strategically surround its position, which is modeled by Equation (22).

$$X(t+1)=\Delta X\left(t\right)-E·\left|J·X_{rabbit}\left(t\right)-X\left(t\right)\right|$$

(22)

where $\Delta X=X_{rabbit}\left(t\right)-X\left(t\right)$ and $J=2(1-r_5)$ represent the random force of the jump of the prey. The possible second phase is referred to as hard besiege, which occurs when $\left(\right|E|<0.5)$ and $(r\ge 0.5)$, indicating that the prey is exhausted. In this case, the hawks intensify their approach and perform aggressive and direct movements, as indicated by Equation (23).

$$X(t+1)=X_{rabbit}\left(t\right)-E·\left|\Delta X\left(t\right)\right|$$

(23)

Using this mechanism, convergence towards the best solution found up to that point can be accelerated. Additionally, the third siege is characterized as gentle with rapid progressive dives and occurs when $\left(\right|E|\ge 0.5)$ and $(r<0.5)$; that is, it occurs when the prey has sufficient energy to carry out evasive maneuvers. Therefore, hawks apply more sophisticated and adaptive movements that combine prediction and random jumps modeled by Lévy flights, as detailed in Equations (24)–(26).

$$\begin{array}{cc}\hfill Y& =X_{rabbit}\left(t\right)-E·\left|J·X_{rabbit}\left(t\right)-X\left(t\right)\right|\hfill \end{array}$$

(24)

$$\begin{array}{cc}\hfill Z& =Y+S·\mathrm{Levy}\left(D\right)\hfill \end{array}$$

(25)

where the positions must be compared to select the best candidate solution obtained by increasing the population size. Here, Y is the tentative position based on the direct movement toward the prey, S is a random vector, and $\mathrm{Levy}\left(D\right)$ represents a jump generated with a Lévy distribution in a D dimension.

$$X(t+1)=\left\{\begin{array}{cc}Y,\hfill & f\left(Y\right)<f\left(X\right(t\left)\right)\hfill \\ Z,\hfill & f\left(Z\right)<f\left(X\right(t\left)\right)\hfill \end{array}\right.$$

(26)

The fourth and final strategy is a hard siege with rapid progressive dives, which occurs when $\left(\right|E|<0.5)$ and $(r<0.5)$, implying that the prey is exhausted and making erratic movements. Therefore, hawks adjust their trajectory based on the population mean and apply a Lévy flight to search for more refined solutions, as expressed in Equation (27).

$$\begin{array}{cc}\hfill Y& =X_{rabbit}\left(t\right)-E·\left|J·X_{rabbit}\left(t\right)-X_m\left(t\right)\right|\hfill \end{array}$$

(27)

Finally, the decision for the next move is made by comparing the fitness of vectors Y and Z in the same manner as in Equations (25) and (26). The intention is to ensure that the algorithm can approach a global solution without losing population diversity.

3.2. Multiverse Optimizer (MVO)

The MVO algorithm is inspired by three fundamental concepts of cosmology: the expansion of the universe, black holes, and wormholes. These phenomena are used as metaphors to represent the transfer of information between solutions (universes) and the adaptive evolution of a population in the search space. Therefore, in MVO, each candidate solution is represented as a universe that adapts according to an inflation rate that is related to the quality of the solution. In this context, universes with better fitness tend to attract information from worse universes through processes similar to black holes, which is referred to as the exploitation phase of the search space. All universes can exchange random information through wormholes, which are categorized as exploration phases [40]. Therefore, the movement of a solution is guided by a random process regulated by the normalized inflation rate of each universe, as denoted in Equation (28).

$$NI\left(U_j\right)={\displaystyle \frac{f\left(U_j\right)-f_{worst}}{f_{best}-f_{worst}}}$$

(28)

where $NI\left(U_j\right)$ is the normalized inflation rate of the universe j, $f\left(U_j\right)$ is the fitness value, and $f_{best}$ and $f_{worst}$ are the best and worst fitness values in the population, respectively. This allows the value to regulate the probability that a universe gives or receives information during its evolution. As mentioned earlier, for the exploitation strategy of the algorithm, a process must be carried out through the transfer of matter through black holes, as expressed in Equation (29).

$$X_{i,j}^{t+1}=\left\{\begin{array}{cc}{X}_{k,j}^t,\hfill & r_1<NI\left(U_k\right)\hfill \\ X_{i,j}^t,\hfill & r_1\ge NI\left(U_k\right)\hfill \end{array}\right.$$

(29)

where $X_{i,j}^{t+1}$ is the value of the j-th parameter of the i-th universe in iteration $t+1$, $X_{k,j}^t$ is the corresponding value of a random universe k, and $r_1$ is a random number ranging from $(0,1)$. Therefore, this mechanism ensures that universes with better inflation rates influence the others, with the intention of promoting the intensification of solutions. However, the exploration phase is managed through wormholes, which allow any universe to make random jumps towards regions of space near the best universe found so far. This dynamic is regulated by the Wormhole Existence Probability ($WEP$) and Travel Distance Rate ($TDR$), defined by Equations (30) and (31), respectively.

$$WEP=WE{P}_{min}+{\displaystyle \frac{(WE{P}_{max}-WE{P}_{min})·t}{T}}$$

(30)

$$TDR=1-{\left({\displaystyle \frac{t}{T}}\right)}^{1/p}$$

(31)

where t is the current iteration, T is the total number of iterations, and p is a parameter that regulates the convergence rate of the algorithm. The complete process of the wormholes is modeled using the following update rule defined in Equation (32).

$$X_{i,j}^{t+1}=\left\{\begin{array}{cc}\begin{array}{cc}\hfill X_{\mathrm{best},j}^t+TDR\times \left((U{b}_j-L{b}_j)\times r_4+L{b}_j\right),& r_3<0.5\hfill \\ \hfill X_{\mathrm{best},j}^t-TDR\times \left((U{b}_j-L{b}_j)\times r_4+L{b}_j\right),& r_3\ge 0.5\hfill \end{array}\hfill & r_2<WEP\hfill \\ X_{i,j}^t,\hfill & r_2\ge WEP\hfill \end{array}\right.$$

(32)

where $X_{best,j}^t$ represents the best solution found in dimension j, and $U{b}_j$ and $L{b}_j$ are the limits of the search space. Thus, this phase allows for controlled jumps around the best solutions, enhancing the ability to escape local optima and improving the exploration of the search space.

3.3. Salp Swarm Algorithm (SSA)

SSA is a bioinspired optimization technique based on the collective movement of salps, marine organisms that travel in linear chains while searching for food. Within this chain, the first salp is called the leader, and the remaining salps act as followers who adjust their positions according to the individuals ahead of them. This natural behavior is mathematically modeled to achieve an adequate balance between the exploration and exploitation of the search space [41].

Here, the movement of the leader salp is defined by its position in dimension j, which is updated based on the best known solution $F_j$ and search boundaries $L{b}_j$ and $U{b}_j$, as shown in Equation (33).

$$X_{1,j}=\left\{\begin{array}{cc}{F}_j+c_1\times \left[(U{b}_j-L{b}_j)\times c_2+L{b}_j\right],\hfill & c_3\ge 0.5\hfill \\ F_j-c_1\times \left[(U{b}_j-L{b}_j)\times c_2+L{b}_j\right],\hfill & c_3<0.5\hfill \end{array}\right.$$

(33)

Considering that $c_1$ is an adaptive decay coefficient that controls the transition between the exploration and exploitation phases, it is defined as in Equation (34).

$$c_1=2e^{-{\left({\displaystyle \frac{4t}{T}}\right)}^2}$$

(34)

The parameters $c_2$ and $c_3$ are random numbers within the interval $[0,1]$, which introduce randomness in the direction and sense of movement of the leading salp. In contrast, the follower salps update their position by taking as a reference the salp immediately preceding them in the chain, applying a smoothed numerical integration to simulate a progressive and coordinated movement, as denoted in Equation (35).

$$X_{i,j}={\displaystyle \frac{X_{i,j}+X_{i-1,j}}{2}},i\ge 2$$

(35)

where $X_{i,j}$ is the position of the i-th salp and $x_{i-1,j}$ is the position of the preceding salp in the dimension j. This allows the equation to adjust the trajectories of the follower salps, reducing the distance with respect to those ahead of them. By contrast, parameter $c_1$ is essential for balancing exploration and exploitation. At the beginning of the optimization, its value is high, causing the leader to make large movements in the search space for a broad exploration. As the iterative process progresses, $c_1$ decreases exponentially, concentrating the movement of the salp chain around the best solutions found, thus allowing for the intense and precise exploitation of the search space.

3.4. Nondominated Sorting Genetic Algorithm II (NSGA-II)

The foundation of NSGA-II originates from classical genetic algorithms that use evolutionary operators, such as crossover and mutation, to generate new solutions from the initial population. However, unlike traditional single-objective optimization algorithms, NSGA-II introduces a mechanism for partial comparison between solutions based on Pareto dominance [42].

Given a set of solutions p, the solution p is considered to dominate another solution q if it meets the conditions described in Equation (36).

$$p\prec q\iff \left\{\begin{array}{c}{f}_m\left(p\right)\le f_m\left(q\right),\forall m\in \{1, 2, \dots , M\}\hfill \\ \exists m\in \{1, 2, \dots , M\}\mathrm{such}\mathrm{that}{f}_m\left(p\right)<f_m\left(q\right)\hfill \end{array}\right.$$

(36)

where M represents the number of objectives, and $f_m$ is the value of objective function m. This criterion allows the identification of all nondominated solutions that constitute the first Pareto front. During the iterative process of the algorithm, dominated solutions are eliminated and replaced by new nondominated solutions, thus forming successive Pareto fronts. In other words, solution p dominates another solution q if it simultaneously meets the following conditions: p is not worse than q in any of the objectives, and p is strictly better than q in at least one objective.

To preserve the diversity within each Pareto front, NSGA-II incorporates the crowding distance ($C{D}_i$), which prevents solutions from concentrating solely in one region of the search space. The calculation of this distance is presented in Equation (37).

$$C{D}_i=\sum _{m=1}^M{\displaystyle \frac{f_{i+1}^m-f_{i-1}^m}{f_{max}^m-f_{min}^m}}$$

(37)

where $f_{i+1}^m$ and $f_{i-1}^m$ are the values of the objective function m for neighboring solutions within the same front, and $f_{max}^m$ and $f_{min}^m$ are the extreme values of the function. This mechanism favors less crowded solutions and promotes balanced distribution in the solution space.

In general, NSGA-II begins by generating an initial population and evaluating its objective functions to sort them according to the Pareto criterion. Subsequently, the crowding distance is calculated to diversify the population, and genetic operators of crossover, mutation, and recombination are applied, along with an elitist selection process that allows the combination of parents and offspring into a new population. In this population, nondominated sorting and the calculation of the crowding distance were performed again, repeating the process iteratively until an efficient and well-distributed approximation of the Pareto front was achieved.

4. Test Case and Materials

Simulations of MGs with photovoltaic generation and battery energy storage systems were conducted to identify the multiobjective optimization method that best suits the resolution of the intelligent energy management problem. This approach seeks to minimize the technical, economic, and environmental impacts associated with the optimal energy dispatch, ensuring that each test scenario is validated under the same computational conditions.

4.1. Network Parameters

Figure 3 shows the case of the test system powered by 27 nodes and 26 lines, where the system operates at a base voltage of 23 kV and a base power of 100 kVA. It is important to note that this is considered a rural scenario connected to a single diesel generator operating in island mode. The parameters of the rural network are listed in

Table 2. Electrical parameters for the AC MG 27-node feeder test case system.

Line l	Node i	Node j	${\mathit{R}}_{\mathit{ij}}\mathbf{(}\Omega \mathbf{)}$	${\mathit{X}}_{\mathit{ij}}\mathbf{(}\Omega \mathbf{)}$	${\mathit{P}}_{\mathit{j}}$ (kW)	${\mathit{Q}}_{\mathit{j}}$ (kVAr)	${\mathit{I}}_{\mathit{ij}}^{max}$ (A)
1	1	2	0.0140	0.6051	0	0	240
2	2	3	0.7463	1.0783	0	0	165
3	3	4	0.4052	0.5855	297.50	184.37	95
4	4	5	1.1524	1.6650	0	0	85
5	5	6	0.5261	0.7601	255.00	158.03	70
6	6	7	0.7127	1.0296	0	0	55
7	7	8	1.6628	2.4024	212.50	131.70	55
8	8	9	5.3434	3.1320	0	0	20
9	9	10	2.1522	1.2615	266.05	164.88	20
10	2	11	0.4052	0.5855	85.00	52.68	70
11	11	12	1.1524	1.6650	340	210.71	70
12	12	13	0.5261	0.7601	297.50	184.37	55
13	13	14	1.2358	1.1332	191.25	118.53	30
14	14	15	2.8835	2.6440	106.25	65.85	20
15	15	16	5.3434	3.1320	255.00	158.03	20
16	3	17	1.2942	1.1867	255.00	158.03	70
17	17	18	0.7027	0.6443	127.50	79.02	55
18	18	19	3.3234	1.9480	297.50	184.37	40
19	19	20	1.5172	0.8893	340	210.71	25
20	20	21	0.7127	1.0296	85.00	52.68	20
21	4	22	8.2528	2.9911	106.25	65.85	20
22	5	23	9.1961	3.3330	55.25	34.24	20
23	6	24	0.7463	1.0783	69.70	43.20	20
24	8	25	2.0112	0.7289	255.00	158.03	20
25	8	26	3.3234	1.9480	63.75	39.51	20
26	26	27	0.5261	0.7601	170	105.36	20

, which provides the electrical parameters of an MG. In this table, the first column shows the line l, the second column the sending node of the line i, the third column the receiving node of the line j, the fourth column presents the resistance parameters of the line given in ohms $(\Omega )$, the fifth column also represents the reactance of the line $X_{ij}$ in $(\Omega )$, and the sixth, seventh, and eighth columns represent the active power at the receiving node $\left(P_j\right)$ in kilowatts (kW), the reactive power at the receiving node $\left(Q_j\right)$ in kilovolt-amperes (kVAr), and the maximum admissible current in the lines $\left(I_{ij}^{max}\right)$ in amperes (A).

Figure 4 shows the test case powered by 33 nodes and 32 lines, where the system operates at a base voltage of 12.66 kV and a base power of 100 kVA. Notably, this is considered an urban scenario connected to the electrical grid with a single generator. This information was obtained from [43,44] and its electrical configurations, which are detailed in

Table 3. Electrical parameters for the AC MG 33-node feeder test case system.

Line l	Node i	Node j	${\mathit{R}}_{\mathit{ij}}(\Omega )$	${\mathit{X}}_{\mathit{ij}}(\Omega )$	${\mathit{P}}_{\mathit{j}}$ (kW)	${\mathit{Q}}_{\mathit{j}}$ (kVAr)	${\mathit{I}}_{\mathbf{ij}}^{max}$ (A)
1	1	2	0.0922	0.0477	100	60	385
2	2	3	0.4930	0.2511	90	40	355
3	3	4	0.3660	0.1864	120	80	240
4	4	5	0.3811	0.1941	60	30	240
5	5	6	0.8190	0.7070	60	20	240
6	6	7	0.1872	0.6188	200	100	110
7	7	8	1.7114	1.2351	200	100	85
8	8	9	1.0300	0.7400	60	20	70
9	9	10	1.0400	0.7400	60	20	70
10	10	11	0.1966	0.0650	45	30	55
11	11	12	0.3744	0.1238	60	35	55
12	12	13	1.4680	1.1550	60	35	55
13	13	14	0.5416	0.7129	120	80	40
14	14	15	0.5910	0.5260	60	10	25
15	15	16	0.7463	0.5450	60	20	20
16	16	17	1.2890	1.7210	60	20	20
17	17	18	0.7320	0.5740	90	40	20
18	2	19	0.1640	0.1565	90	40	40
19	19	20	1.5042	1.3554	90	40	25
20	20	21	0.4095	0.4784	90	40	20
21	21	22	0.7089	0.9373	90	40	20
22	3	23	0.4512	0.3083	90	50	85
23	23	24	0.8980	0.7091	420	200	85
24	24	25	0.8960	0.7011	420	200	40
25	6	26	0.2030	0.1034	60	25	125
26	26	27	0.2842	0.1447	60	25	110
27	27	28	1.0590	0.9337	60	20	110
28	28	29	0.8042	0.7006	120	70	110
29	29	30	0.5075	0.2585	200	600	95
30	30	31	0.9744	0.9630	150	70	55
31	31	32	0.3105	0.3619	210	100	30
32	32	33	0.3410	0.5302	60	40	20

, have the same structure as those described in Table 2. It is important to highlight that both the urban and rural cases were assigned line current limits for the MG under analysis in each of the study cases, as it is crucial to have the information from the base case without photovoltaic DGs or batteries and using a nodal voltage limit of +/−10% of the nominal voltage of the systems.

4.2. Energy Resources

The MG integrates PV systems and lithium-ion batteries, which are strategically connected to different nodes of the network. The main objective is to ensure that these energy resources operate under strict safety and efficiency criteria, avoid system overload, and ensure optimal energy management that allows for a balanced and reliable distribution of energy.

Table 4. Locations and power ratings of the PV DGs in each test system.

27-Node Feeder Test Case System (Rural)		33-Node Feeder Test Case System (Urban)
Node	${\mathit{P}}_{\mathit{DG}}$ (kW)	Node	${\mathit{P}}_{\mathit{DG}}$ (kW)
5	1012.5	12	1125
9	1188	25	1320
19	899.1	30	999

lists the technical specifications of the elements installed in the different MGs. The first column represents the number of generators located in the MG, the second column indicates the location of the DGs at the node where they are installed, and the third column details the power of each generator.

Table 5. Location and characteristics of the batteries in the test systems.

Battery 27-Node Feeder Test Case System (Rural)
Node	Type	${\mathit{P}}_{\mathit{B}}$ (kW)	${\mathit{t}}_{\mathit{c}}$ (h)	${\mathit{t}}_{\mathit{d}}$ (h)
3	C1	2000	5	5
8	A1	1000	4	4
19	B1	1500	4	4
Battery 33-Node Feeder Test Case System (Urban)
Node	Type	${\mathit{P}}_{\mathit{B}}$ (kW)	${\mathit{t}}_{\mathit{c}}$ (h)	${\mathit{t}}_{\mathit{d}}$ (h)
6	C1	2000	5	5
14	A1	1000	4	4
31	B1	1500	4	4

lists the specifications of the batteries in the test system. Here, the first column denotes the node where the battery is installed, the second column represents the type of battery, and columns three, four, and five detail the battery power and the charging and discharging times of the batteries, respectively. Each of these characteristics is specified for the different test systems used [36,45].

4.3. Power Demand and Solar Resources

Once the technical parameters of the test systems were defined, the use of generation and demand profiles to simulate an average day of MG operation was highlighted (Figure 5). It is worth mentioning that these average daily data were obtained by collecting a year’s worth of data based on solar irradiation and ambient temperature data provided by NASA POWER [44]. Therefore, they can be used for other types of generation and demand conditions. Additionally, the data are provided per unit (pu), which makes it easier to modify them for data extrapolation according to the technical and operational requirements of MGs [44,46].

4.4. Energy Cost

It is important to highlight that, for the analysis of variable energy costs in urban areas, the energy costs of the supply grid (the grid connected to the MG) must be included, because nonregulated users can generate significant variations, as shown in Figure 6. These costs are adjusted according to market fluctuations and are represented by existing nonregulated users, which facilitates their integration into the models applied for optimization and in the test systems. Data related to the price of the costs of energy production are taken from [37,47].

The elements used in the MGs, as well as the purchase and maintenance of their components, are described below. Here, the economic and environmental parameters associated with MGs and their configurations are represented by MGs connected to the power grid and autonomous MGs. In the first case, the cost of purchasing energy from the grid was 0.1302 USD/kWh, the operating cost of the diesel generator was 0.0019 USD/kWh, and the operating cost of the batteries was 0.2913 USD/kWh. The emissions associated with the grid electricity were 0.1644 kg/kWh. In the autonomous mode, both the operating cost of the battery and the cost equivalent to the energy that would come from the grid were 0.2913 USD/kWh, whereas the diesel generator maintained a cost of 0.0019 USD/kWh. The specific emissions in this case were 0.2671 kg/kWh, suggesting greater reliance on fossil fuel sources.

4.5. Computational Resources

All simulations were implemented and executed in MATLAB^® R2024a on a workstation equipped with an Intel^® Xeon^® Silver4110 (up to 2.10 GHz), 16 GB of DDR5 RAM, a 1 TB SATA solid-state drive, and Windows 11. The multiobjective optimizers (NSGA-II, HHO, MVO, and SSA) were coded within a unified framework with vectorized evaluation of the four objectives (fixed costs, variable costs, power losses, and CO₂ emissions). The AC power flow was solved at each discrete time step under quasi–steady-state assumptions, and operational constraints (voltage and current limits, battery SoC bounds, and charge/discharge power) were enforced through an additive penalty function integrated into the objective vector. The demand, PV generation, and tariff profiles were handled in the same environment to ensure consistent data management across the experiments.

This setup is intended for dispatch and short-term planning studies and does not emulate the fast switching dynamics of power electronic converters, inner control loops, or electromagnetic transients. When detailed dynamic analyses are required, the proposed formulation is directly portable to specialized tools (e.g., DIgSILENT PowerFactory, OpenDSS, and GridLAB-D) by retaining the multiobjective layer and feeding it with network states and flows computed on those platforms. The random seeds, population sizes, iterations, and stopping criteria used in the comparative study are reported in Section 5.1 to support replicability and fair comparison across methods.

5. Results and Analysis

Based on the methodology defined in Section 2, the results were obtained by validating the different test scenarios, considering the generation and demand profiles established for an average day of operation. These tests were evaluated using four multiobjective optimization algorithms. In the results analysis, the first step corresponds to the tuning or adjustment of the algorithms, with the aim of ensuring the highest possible quality solutions for the problem under study. Subsequently, the optimization methodologies for intelligent energy management with the PV DERs and the BESS are executed, with the purpose of comparing the results obtained against the baseline case. Likewise, both the performance of the algorithms and the behavior of the systems under the operating conditions of the MG were analyzed in detail to validate the proper functioning of the proposed approaches.

5.1. Algorithm Tuning

The optimization algorithms described in Section 3 were tuned to identify the parameters that best fit the problem under analysis. The adjusted parameters correspond to the maximum number of iterations, maximum number of iterations without convergence, maximum number of particles or individuals, and maximum and minimum ranges defined in the progression or evolution methodologies specific to each optimization algorithm. Tuning is performed using another algorithm responsible for optimizing and determining these parameters. In this case, a continuous genetic algorithm was used, in which the search space for the considered parameters was defined as follows: [1–100] particles or individuals, [100–20,000] maximum iterations, and [100–20,000] iterations without convergence. Meanwhile, the intrinsic parameters of each optimization algorithm were set within a working range of [0–1], with the sole exception of the parameter p of the MVO, whose range is [0–10] and which is responsible for defining the precision of the exploitation of the search space throughout the iterative process. The tuning results are presented in

Table 6. Optimal parameters obtained after the tuning of the HHO, NGSAII, SSA, and MVO algorithms.

Method	Optimization Parameters	Value
HHO	Number of particles	65
	Maximum number of iterations	8890
	Maximum number of repetitions without improvement	6065
NGSAII	Number of particles	30
	Maximum number of iterations	5215
	Maximum number of repetitions without improvement	1890
	Crossover probability $\left(pc\right)$	0.983
	Mutation probability $\left(pm\right)$	0.104
	Mutation intensity $\left(ms\right)$	0.0177
SSA	Number of particles	96
	Maximum number of iterations	9175
	Maximum number of repetitions without improvement	7142
MVO	Number of particles	94
	Maximum number of iterations	9070
	Maximum number of repetitions without improvement	7950
	Accuracy of exploitation (p)	6

.

5.2. Simulation with the Test Case System

5.2.1. Rural Analysis Results 27-Node MG Feeder Test System

Different results were obtained based on the test scenarios and generation and demand profiles. The analysis began with the first scenario, consisting of 27 nodes, which was considered representative of a rural case. In this scenario, the MG operates in island mode, meaning it is isolated from the public utility grid, and its main generation system is a diesel generator. In this context,

Table 7. Results of the optimization with the four algorithms application of rural MG with PV DERs and BESSs.

Minimum Objective Function
Method	Fixed Cost (USD)	Emissions (kgCO2/kWh)	Power Losses (kWh)
Base case + PV	15,424.030	582.377	14,125.660
HHO	15,414.215	548.676	14,116.662
NGSAII	15,412.460	542.651	14,115.053
SSA	15,412.675	543.388	14,115.250
MVO	15,423.260	554.064	14,124.127
Average Objective Function
Method	Fixed Cost (USD)	Emissions (kgCO2/kWh)	Power Losses (kWh)
HHO	15,415.660	553.636	14,117.987
NGSAII	15,413.930	547.697	14,116.401
SSA	15,413.740	547.047	14,116.227
MVO	15,596.104	742.729	14,296.431
Standard Deviation (%)
Method	Fixed Cost (USD)	Emissions (kgCO2/kWh)	Power Losses (kWh)
HHO	5.267$\times {10}^{-5}$	5.035$\times {10}^{-3}$	5.274$\times {10}^{-5}$
NGSAII	4.031$\times {10}^{-5}$	3.895$\times {10}^{-3}$	4.037$\times {10}^{-5}$
SSA	2.899$\times {10}^{-5}$	2.805$\times {10}^{-3}$	2.903$\times {10}^{-5}$
MVO	1.946$\times {10}^{-2}$	4.126$\times {10}^{-1}$	2.123$\times {10}^{-2}$

and Figure 7 present the results obtained from the simulation performed on the test case. The information contained in Table 7 is organized as follows: the first column indicates the algorithm used; the second reports the values obtained for energy management considering a fixed energy cost, expressed in U.S. dollars (USD); the third shows the CO₂ emissions calculated for each algorithm, expressed in kg of CO₂/kWh; finally, the last column presents the total energy losses recorded in kWh.

It is important to note that the table presents the global minimum values reached by the objective functions. Additionally, the average values and standard deviations corresponding to each optimization algorithm are included. This analysis allows for the identification of the methods’ performance in terms of the quality of the solutions offered for the optimal energy management problem, aimed at reducing costs, CO₂ emissions, and power losses in AC microgrids. Figure 7 shows the results in terms of the reduction percentages obtained with respect to the base case for the rural area. This radar chart facilitates interpretation by clearly indicating that the algorithm with the best performance, that is, the greatest reductions, is located farther from the origin, whereas the one with the least benefit, or with the worst performance compared to the base case, is positioned closer to the center of the chart. The purpose of this representation is to provide a clearer view of the results obtained through the implementation of the optimization methodologies (HHO, NGSA-II, SSA, and MVO). Additionally, it is important to note that this figure makes it possible to jointly analyze different indicators, showing the reduction percentages in terms of minimum costs, CO₂ emissions, power losses, and averages of these objective functions.

It is worth noting that the table presents the global minimum values achieved by the objective functions. Additionally, it includes the average values and standard deviations corresponding to each optimization algorithm. This analysis allows for the identification of the performance of the methods in terms of the quality of the solutions offered for the optimal energy management problem, aimed at reducing costs, CO₂ emissions, and power losses in AC microgrids. Figure 7 shows the results in terms of the percentage reductions obtained compared with the baseline case. This radar-type graph facilitates interpretation by illustrating that the algorithm with the best results—that is, the greatest reductions—is located farther from the origin, whereas the one with the smallest benefits or the worst performance compared to the baseline case is positioned closer to the center of the graph. The purpose of this representation is to provide a clearer view of the results obtained through the implementation of the optimization methodologies (HHO, NGSA-II, SSA, and MVO). Additionally, it is important to note that this figure allows for a joint analysis of different indicators, showing the percentage reductions in terms of minimum costs, CO₂ emissions, power losses, and averages of these objective functions.

Figure 8 presents another radar analysis, but unlike the previous one, this focuses on comparing the technique that achieved the best results with the others. In this case, SSA was selected as the most stable technique because, in the previously obtained results regarding average percentages, it offered the best performance. It is important to highlight that, unlike the previous radar figure, in this representation, the technique that is closest to the origin is the one that matches or approaches the best technique, while the one that is furthest away corresponds to the one that showed the worst results.

Note that, with respect to NGSAII, in minimal terms, it surpasses SSA with improvement values of 0.001% for costs, 0.136% for CO₂ emissions, and 0.001% for power losses. However, in a global comparison with the other techniques, SSA achieved better average results, with values of 0.026%, 0.918%, and 0.024%, respectively. However, regarding the average values and standard deviations of SSA compared to the other optimization techniques, it was superior in all cases. Thus, in the average results, SSA outperformed HHO by 0.012% in fixed energy costs, by 1.190% in terms of CO₂ emissions, and by 0.012% in power losses. Compared to NGSAII, the improvements were 0.001%, 0.119%, and 0.001%, respectively. As for MVO, the SSA achieved advantages of 1.169% in fixed costs, 26.346% in CO₂ emissions, and 1.260% in power losses. Consequently, considering the overall average fixed costs for reductions, SSA was better than all other techniques by 0.394%, by 9.218% in CO₂ emissions, and by 0.425% in power losses. Finally, overall, the SSA was also superior in the standard deviations of the results compared to all techniques, with average values of 57.626% for fixed costs, 57.202% for CO₂ emissions, and 57.630% for power losses.

As part of the results analysis, it is possible to observe that this information can be reinforced by Figure 9, which represents the global iterations of each optimization algorithm with respect to the objective functions analyzed, relating to fixed energy costs, CO₂ emissions, and power losses. This figure shows why the MVO was the technique with the greatest dispersion of results, due to the spread of some of its solutions in terms of minimization.

Similarly, these results can be further strengthened through the nonparametric Friedman analysis, detailed in

Table 8. Classification of the three objective functions based on multiobjective algorithms using Friedman’s nonparametric test in the rural AC grid with PV and BESS DERs.

Method	Fixed Cost	Rank	CO2 Emissions	Rank	Power Loss	Rank
SSA	15,413.740	1	547.047	1	14,116.227	1
NGSAII	15,413.930	2	547.697	2	14,116.401	2
HHO	15,415.660	3	553.636	3	14,117.987	3
MVO	15,596.104	4	742.729	4	14,296.431	4

, whose approach consists of analyzing the information provided by the global solutions. This test allows for the evaluation of disparities among the values obtained in the reductions of the objective functions during the implementation of different optimization techniques. Table 8 consolidates the information derived from the Friedman test, which demonstrates significant differences in the results. For the rural tests with 27 nodes, the SSA stands out as the best optimization technique, and the MVO emerges as the least effective technique for the intelligent energy management problem.

In addition to the statistical and results analyses, Figure 10 details the analysis of confidence intervals, where the lines corresponding to the optimization algorithms and objective functions are represented. The light black shaded area indicates the range in which the mean of the population of solutions obtained by the optimization techniques is found, with a 95% confidence level. This analysis was performed to provide a deeper understanding of the variability of the results and the uncertainty associated with the optimization algorithms.

It is important to highlight the behavior of the charging and discharging states in batteries, as these are a fundamental part of intelligent energy management for reducing fixed costs, CO₂ emissions, and power losses in the MG. Therefore, Figure 10 shows the best SoC behaviors obtained by the optimization methodologies; furthermore, the figure shows that at no point do any of the optimization algorithms violate the technical-operational constraints, and they remain within the SoC operating ranges of 10–90%. Figure 11 shows the behavior of the hours with the highest battery power contribution is reflected in the periods when there is no solar energy; namely, those in which there is a greater dependence on diesel generation for the rural case: hours 1–6 and 18–24 h.

Similarly, Figure 12 shows that the algorithms in each of the iterative processes do not exceed the voltage and current limits or restrictions of the lines and nodes of the MG during the 24-h operation period. This demonstrates that the objective functions and mathematical models applied within the intelligent energy management methodologies ensure operations that respect the technical and operational constraints and conditions of the MG.

5.2.2. Annual Cost Minimization for Rural AC MG

For the rural case of the 27-node test system, the optimization algorithms and energy management made it possible to achieve results that highlighted SSA and NGSAII as the best optimization techniques for intelligent energy management applied to the optimal technical-environmental dispatch of the MG that integrates PV DERs and BESSs. Since SSA was classified as the algorithm that achieved the optimal benefits for the problem, it obtained values of 10.29 USD in fixed costs for an average day of operation, 35.33 kg of CO₂/kWh in CO₂ emissions, and 9.433 kWh in power losses. These results allow for an average annual analysis projecting benefits of 3755.676 USD in fixed energy costs for the rural MG, 12.896 tons in CO₂ emissions, and 34.430 MW in energy loss reductions.

5.2.3. Urban Analysis Results 33-Node MG Feeder Test System

Once the rural system has been analyzed, the analysis proceeds with the 33-node test system, which is considered a rural scenario in an MG connected to the main service grid, such as the EPM in the case of Medellín, Colombia. In this context,

Table 9. Results of the optimization with the four algorithms application of urban MG with PV DERs and BESSs.

Minimum Objective Function
Method	Fixed Cost (USD)	Variable Cost (USD)	Emissions (kgCO2/kWh)	Power Losses (kWh)
Base case + PV	7859.027	6999.053	2484.575	9887.048
HHO	7854.3777	6998.6600	2492.1575	9885.9473
NGSAII	7846.0257	6969.7423	2384.7094	9870.9924
SSA	7848.7657	6949.6422	2405.7577	9874.4523
MVO	7854.1309	7000.1955	2483.0672	9895.8229
Mean Objective Function
Method	Fixed Cost (USD)	Variable Cost (USD)	Emissions (kgCO2/kWh)	Power Losses (kWh)
HHO	7863.6552	7004.6647	2489.2424	9891.7415
NGSAII	7847.4987	6954.5174	2396.0252	9872.8525
SSA	7850.3563	6936.1009	2417.9767	9876.4609
MVO	7863.9351	7003.3036	2489.6415	9893.0226
Standard Deviation [%]
Method	Fixed Cost (USD)	Variable Cost (USD)	Emissions (kgCO2/kWh)	Power Losses (kWh)
HHO	7.035$\times {10}^{-4}$	8.482$\times {10}^{-4}$	2.356$\times {10}^{-3}$	5.819$\times {10}^{-4}$
NGSAII	1.013$\times {10}^{-4}$	1.334$\times {10}^{-3}$	2.548$\times {10}^{-3}$	1.016$\times {10}^{-4}$
SSA	9.410$\times {10}^{-5}$	1.021$\times {10}^{-3}$	2.347$\times {10}^{-3}$	9.445$\times {10}^{-5}$
MVO	7.667$\times {10}^{-4}$	8.596$\times {10}^{-4}$	2.196$\times {10}^{-3}$	5.795$\times {10}^{-4}$

and Figure 13 present the results derived from the simulation of the 33-node test case for the urban area. These results are organized in the same way as in the 27-node system (Table 7); however, in Table 9, a column is added that contains the energy costs, expressed in United States dollars (USD).

Figure 7 shows the results in terms of the percentage reductions obtained with respect to the base case, as analyzed in the 27-node system. The radar chart allows for easier interpretation of the information; in this figure, it can be seen that SSA and NGSAII are the best optimization techniques, while HHO and MVO yielded the worst results compared to the base case, considering that the analysis of the radar chart is the same as previously explained. Analyzing the results from the table and the figure, it can be deduced that in terms of minimum cost reductions, NGSAII ranked first with reductions of 0.165%, SSA was in second place with 0.131%, MVO took third place with 0.062%, and HHO was fourth with 0.059%. Regarding the variable energy costs and their maximum reductions obtained, SSA achieved first place with 0.706%, followed by NGSAII with 0.419%, while HHO and MVO obtained values of 0.006% and −0.016%, respectively. It is worth noting that the negative value of MVO is due to the fact that, in terms of variable energy costs, it did not achieve significant reductions in the system under consideration. As for CO₂ emissions and power losses, NGSAII achieved first place with reductions of 4.019% and 0.162%, respectively. Second, SSA achieved reductions of 3.172% and 0.127%. Meanwhile, HHO and MVO, which ranked third and fourth, respectively, presented results of −0.305% and 0.011%, and 0.061% and −0.089% in terms of CO₂ emissions and power losses, respectively.

On the other hand, Figure 14 shows the average percentages obtained by the optimization algorithms. Once again, SSA and NGSAII delivered the best results within the framework of the 100 global runs. In this regard, NGSAII achieved values of 0.0021%, 0.0060%, 0.1618%, and 0.0016% for fixed energy costs, variable energy costs, CO₂ emissions, and power losses, respectively. In turn, the SSA obtained values of 0.0017%, 0.0101%, 0.1277%, and 0.0013% for the same objective functions.

This can be seen more clearly in

Table 10. Classification of the four objective functions based on multiobjective algorithms using Friedman’s nonparametric test in the rural AC grid with PV and BESS DERs.

Method	Fixed Cost	Rank	Variable Cost	Rank	Power Loss	Rank	CO2 Emissions	Rank
NGSAII	7847.499	1	6954.517	2	2396.025	1	9872.853	1
SSA	7850.356	2	6936.101	1	2417.977	2	9876.461	2
HHO	7863.655	3	7003.304	3	2489.242	3	9891.742	3
MVO	7863.935	4	7004.665	4	2489.642	4	9893.023	4

, in which, upon applying the nonparametric Friedman test again, the order and hierarchy of the optimization algorithms stand out in terms of the minimum and average values obtained using the methodology applied for intelligent energy management in rural AC microgrids with PV DERs and BESSs.

Once the behavior of the different optimization algorithms was analyzed in terms of the minimum and average reduction percentages, it was necessary to rank the best optimization technique based on the quality of the solutions. Although NSGAII outperformed various aspects of the objective functions, SSA was selected as the best optimization technique because, despite ranking second in some metrics, it showed a higher level of accuracy in terms of standard deviations compared to the other techniques. Consequently, on average, compared to the other optimization techniques, the SSA achieved average percentage minimizations of 0.035% for fixed energy costs, 0.570% for variable energy costs, 1.899% for CO₂ emissions, and 0.099% for power losses; whereas, for the overall mean values, the SSA reached 0.102%, 0.734%, 1.608%, and 0.095%, respectively, for the same objective functions. Moreover, in terms of standard deviations, SSA outperformed NSGAII with percentages of 7.082% for fixed energy costs over the one hundred global iterations, 23.476% for variable costs, 7.892% for CO₂ emissions, and 7.082% for power losses. This highlights that the accuracy of SSA surpasses that of NSGAII within the global iterative processes concerning the execution and validation of the optimization algorithms.

This can be seen in Figure 15, which shows the global iterations of each optimization algorithm with respect to the objective functions analyzed, relating the fixed energy costs, CO₂ emissions, and power losses. Note that, although NGSAII presents good results in terms of variation, SSA exhibits boxplots with greater reductions than the other optimization techniques, which classifies it as the most suitable alternative in terms of the quality of solutions offered by the stochastic nature of the optimization algorithms.

However, for this case study, confidence intervals are also used in the scenario with 33 nodes, as shown in Figure 16. As in the 27-node system, this figure details the analysis of the confidence intervals with respect to the objective functions and optimization algorithms. Similar to the previous case, the light black shaded edges represent the mean of the solutions and fall within a 95%.

Additionally, it is important to highlight the behavior of the algorithms regarding the SoC of the batteries located in the AC MG of the 33-node system within these optimization methodologies. In this case, Figure 17 shows the performance of the three battery banks. Note that technical and operational constraints of the battery SoC are not violated.

Finally, Figure 18 shows that the system never exceeds the voltage and current limits in the lines and nodes of the 33-node MG, displaying the hour-by-hour behavior for voltage at all 33 nodes and current in the 32 lines. This demonstrates and guarantees that the applied mathematical model and implemented methodologies ensure that the technical and operational constraints of the MG are satisfied at every demand level of the system.

5.2.4. Annual Cost Minimization for Urban AC MG

The results obtained so far allow us to analyze that, for the urban case of the 33-node test system in intelligent energy management applied to the technical-environmental optimal dispatch of the MG containing PV DERs and BESS, the SSA achieved the optimal benefits. Among these, the highlights are 8671 USD in fixed energy costs, 62,952 USD in variable energy costs for an average day of operation, 66,598 kg of CO₂/kWh in CO₂ emissions, and 10,587 kWh in power losses. When these results are projected to an average year of MG operation, they amount to 3,164,959 USD in fixed costs, 22,977,510 USD in variable energy costs, 24.308 tons of CO₂ emissions, and 38.643 MW in energy loss reductions for the 33-node urban MG.

Finally, after the analysis, it is important to demonstrate that optimization algorithms exhibit certain behaviors and constraints through Pareto algorithms, as shown in Figure 19, which illustrates how different optimization algorithms behave when determining optimized solutions in terms of the quality of the solutions for each objective function. The trends and limits of the algorithms can be observed, as this represents the point of incidence where the MG functions can be further reduced.

Figure 19 summarizes the behavior of the four algorithms in the objective space (fixed cost, CO₂ emissions and power losses). In each panel, the semitransparent cloud represents the population evaluated during the iterations, and the red markers correspond to the final nondominated set. The contraction of the cloud towards the red edge and the continuity of that edge are two visual indicators of the convergence and quality/diversity of the Pareto front. The figure shows that the SSA and NSGA-II form compact and well-extended fronts in the desired region (low cost, low emissions, and low losses). It is also worth noting that all the red points remain within the operational limits (voltage, current, and SoC), which explains the visible barrier that no feasible solution crosses; thus, any further improvement in one objective implies deterioration in at least one other.

6. Conclusions and Future Work

This paper presented an intelligent EMS for AC microgrids with PV–BESS through a multiobjective formulation that minimizes fixed and variable costs, power losses, and CO₂ emissions. The problem was solved using Pareto fronts and hourly AC power flow with penalties. A methodology was proposed to solve scenarios related to rural MG operating in island mode and urban MG connected to the grid while maintaining operation within technical and operational limits. The following conclusions were obtained:

In the rural system with 27 nodes, coordinated PV–BESS dispatch shifted the demand during non-irradiance periods (1–6 and 18–24 h), thus reducing diesel usage and decreasing losses. The state of charge was maintained within 10–90% without violating constraints, and voltage and current profiles remained within range throughout the 24 h. Projecting to a typical year, with the best configuration found, estimated benefits are 3755.676 USD in fixed costs, 12.896 t of avoided CO₂, and 34.430 MW of reduced losses.

In the urban system with 33 nodes, the inclusion of variable costs allowed the exploitation of hourly price signals through energy arbitrage without compromising the voltage quality. The scheme found a cost–technical–environmental compromise with batteries operating within the safe range of 10 to 90% and without overloading lines. The annual projection yields 3164.959 USD in fixed costs, 22,977.510 USD in variable costs, 24.308 t of CO₂, and 38.643 MW of avoided losses.

Regarding the algorithms, SSA showed the most robust behavior on average (narrow confidence bands and low dispersion), making it preferable when repeatability and statistical stability are prioritized. NGSAII achieved the best minima for several objectives, serving as a reference when exploring the extremes of the Pareto front. HHO was consistent and competitive, whereas MVO showed greater sensitivity to dimensionality and tuning, with higher dispersion in some runs. In general, the portfolio {SSA, NGSAII, HHO} offers a solid balance between exploration, exploitation, and front quality.

Methodologically, combining the multiobjective formulation with AC power flow, explicit SoC management, and penalties, along with an ex-post statistical analysis (distributions and 95% confidence intervals), provides a reproducible pathway to compare techniques and support operational decisions in MGs with high renewable penetration.

From a practical standpoint, the Pareto fronts obtained in this study translate into hourly dispatch policies that can be implemented within an EMS as follows: in urban or industrial microgrids with hourly tariff signals, the solutions exploit the price signal through energy arbitrage while preserving AC feasibility (voltages, currents) and safe SoC windows; in rural or islanded settings, they systematically reduce diesel consumption and technical losses without violating operational limits. Because feasibility is enforced within the AC power flow and through explicit constraints and penalties, the schedules require no post-hoc repairs and can be ported to EMS/SCADA or specialized tools (e.g., DIgSILENT, OpenDSS, GridLAB-D) for deployment and operator training. This strengthens the direct applicability of the results to real microgrids seeking economic, technical, and environmental improvements under the integration conditions of PV-BESS.

The following extensions are proposed for future work: (i) incorporating uncertainty with robust/stochastic approaches and probabilistic forecasts of demand, irradiance, and prices, incorporating day-ahead/real-time levels; (ii) co-optimizing PV–BESS location and sizing along with hourly dispatch, including per-cycle costs and degradation models; (iii) coupling Volt/VAR control of inverters and network reconfiguration to reduce losses and improve voltage profiles while retaining the AC model; (iv) integrating demand response and flexible loads (e.g., EVs) under dynamic tariffs; (v) exploring hybrid and adaptive metaheuristics (e.g., NGSAII + SSA) with online tuning and post-Pareto decision criteria; (vi) assessing real-time operation with digital twins and hardware-in-the-loop validation; (vii) analyzing resilience to failures, unintentional islanding and extreme events (N − 1 criterion), as well as ancillary services (synthetic inertia, black start); and (viii) studying P2P and multi-energy (electric–thermal) models for community MGs. These lines strengthen the field transfer and scalability in urban and rural contexts with high PV–BESS penetration. Additionally, an experimental validation is planned in two stages: hardware-in-the-loop (HIL) using OPAL-RT coupled with MATLAB/Simulink or DIgSILENT PowerFactory to reproduce measured profiles and verify the executability of the dispatch under AC power flow, SoC management, and voltage/current limits; and a field test on a city feeder or pilot microgrid, contrasting the dispatch schedules with measurements under urban and rural scenarios.