1.1 Background and motivation

The demand for energy has increased worldwide as modern societies’ population and energy use have dramatically increased. Total energy consumption is estimated to increase by 53% by 2035. Since the 2000s, there has been a growing awareness of greenhouse gas (GHG) emissions and environmental degradation caused by the substantial reliance on fossil fuels. At present, there are many international calls and strategies to face these issues [1–3], such as using renewable energy sources (RESs) as an alternative energy source. Unfortunately, the transportation and energy production sectors play a role in carbon dioxide (CO₂) emissions, contributing to 64% of total emissions [4]. These emissions have resulted in consequences leading to widespread public concern [5]. Thus, integrating RESs with electric vehicles (EVs) is crucial to meet the necessary reduction goals for CO₂ emissions in both the transportation and power sectors [6]. Additionally, the transportation industry has become a significant energy consumer, significantly impacting the need for electrical power sources [7].

Extensive research has been conducted to study the distinctive load characteristics of electrical power systems. EVs, a new type of electrical load/supply, are gaining recognition for their potential expansion. Plug-in electric vehicles (PEVs) are becoming more popular due to their reduced carbon emissions and the presence of government incentives [8]. Renewable energy sources like photovoltaics (PVs) and wind turbines (WTs) are often combined with EVs and battery storage systems (BSSs) in hybrid power plants, energy hubs, and microgrids (μGs) [9]. BSS plays a crucial role in the efficient operation and administration of μGs [10]. BSS enables load shifting by storing excess energy during periods of low demand and supplying it during peak demand periods. By optimizing the use of stored energy, BSS helps reduce peak demand from the grid, which can be costly [11]. This load management capability reduces the need for additional generation capacity and grid infrastructure upgrades, resulting in cost savings. BSS facilitates the integration of intermittent RESs, such as solar and wind, into microgrids. It stores surplus energy generated during periods of high renewable energy production and releases it when there is insufficient generation. This smoothes out the variability of renewable energy, enhances grid stability, and reduces reliance on fossil fuel-based backup generation, leading to lower operational costs and reduced emissions. BSS can provide ancillary services to the microgrid and the main grid, enhancing system reliability and stability [12]. BSS can respond rapidly to frequency fluctuations, voltage regulation, and grid imbalances, improving power quality and reducing the need for costly grid infrastructure upgrades. By providing these grid support services, BSS contributes to cost savings and facilitates the integration of renewable energy, ultimately reducing emissions. BSS allows microgrid operators to optimize energy usage based on time-of-use pricing. During periods of low electricity prices, BSS can store energy, which can be discharged during high-price periods, reducing the overall cost of electricity consumption. This optimization helps reduce electricity expenses and can incentivize the use of renewable energy sources during peak demand periods, further reducing emissions [13].

The system’s efficiency and power generation costs are greatly impacted by the strategic use of batteries and resource management. Energy management systems (EMSs) are essential for regulating energy in a μG to match the load need [14]. With the rising adoption of RESs and the emergence of new technologies such as energy storage systems and μGs [15], the optimization of energy resource management has become more crucial. EMSs provide real-time monitoring and control of energy distribution, enabling the seamless integration of μGs into power grids [16]. An EMS is responsible for overseeing a μG’s component known as a μG central control (μGCC) unit to ensure safe, dependable, and cost-effective operation [17]. Fig 1 illustrates the necessity of employing a hierarchical control approach with multiple layers to represent the μG environment as a unified entity.

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Fig 1. Energy management hierarchical system.

https://doi.org/10.1371/journal.pone.0307810.g001

Also, Fig 1 shows that initially, the data for power demand, power generation, and market price is collected. EM is done to determine the output of each unit considering all operation constraints of each power generation and μG, and then this is implemented in reality [18, 19]. The integration of EV charging with RESs and storage systems is a concept that aims to maximize the benefits of clean energy generation while efficiently managing EV charging and grid interactions. By integrating EV charging with RESs like PV or WT, we can significantly reduce our reliance on fossil fuels for transportation and electricity generation. This integration allows us to tap into clean, RES, which in turn leads to a reduction in greenhouse gas emissions [20]. EVs produce zero tailpipe emissions, meaning they don’t release any harmful pollutants while driving. However, it’s important to consider the source of electricity used to charge these vehicles. If the electricity comes from fossil fuel-based power plants, the overall emissions reduction potential is limited. That’s where the integration with RESs becomes crucial. The widespread adoption of EVs can present challenges and negative impacts on the power network [21].

According to the global EV outlook 2024 report published by the International Energy Agency (IEA), the global new registrations of EVs reached nearly 14 million units in 2023. This substantial increase in EV adoption has led to the total number of EVs on the world’s roads surpassing 40 million vehicles. Therefore, EVs have become an essential load in the electrical network that must be considered. The simultaneous charging of multiple EVs, particularly during peak hours, can strain the power grid and require costly infrastructure upgrades. It can lead to grid congestion, voltage stability issues, and increased peak demand. Distribution network planning becomes crucial to handle the increased load and ensure reliable power distribution. Additionally, EV charging can introduce power quality issues and require careful management to address harmonics and power factor concerns [22]. A BSS is one of the promising facilities that can mitigate the challenges that result from the adoption of EVs in an MG. BSS may improve the hosting capacity of the EV charging station to increase the number of EVs that can charge from it simultaneously and boost MG’s dependability to raise its resilience to recover swiftly from disruptions caused by the station [12]. This motivated the authors to conduct research on the optimization of μG performance through the integration of EV charging with renewable energy and storage systems.

Thus, connecting BSS with EV charging stations in microgrids offers several benefits in terms of operational efficiency, cost reduction, and environmental impact [23]. BSS can help balance the load by absorbing excess energy during periods of low demand and supplying it to EV charging stations during peak demand. This load balancing optimizes the utilization of available energy resources, reduces strain on the grid, and improves the overall operational efficiency of the microgrid [10]. By using BSS to manage the charging of EVs, microgrids can mitigate grid congestion issues caused by multiple EVs charging simultaneously. BSS can distribute the charging load intelligently, considering grid constraints and available capacity, to prevent overloading and ensure a reliable power supply to both EVs and other critical loads [23]. BSS can help shave peak demand during charging periods, reducing demand charges imposed by utilities. By avoiding peak demand spikes, microgrids can significantly lower electricity costs associated with high-demand tariffs, thus reducing operational expenses [13]. BSS can store excess energy during low-cost periods and discharge it during high-cost periods. By leveraging time-of-use pricing, microgrids can optimize the charging of EVs to align with cheaper electricity rates, resulting in cost savings. BSS coupled with EV charging stations enables better integration of renewable energy sources into microgrids. Excess renewable energy generated during periods of high production can be stored in the BSS and subsequently used to charge EVs [20]. This reduces the reliance on fossil fuel-based energy sources and promotes the use of clean energy, thereby reducing greenhouse gas emissions and environmental impact. By promoting the adoption of electric vehicles and utilizing renewable energy sources, the overall emissions associated with transportation and energy consumption can be reduced [22]. As EVs replace conventional combustion engine vehicles, the direct emissions from transportation are eliminated, while using renewable energy for charging further reduces indirect emissions from the electricity generation process [20].

1.2 Literature overview

Nodehi et al. [24] investigated the effect of the optimal EM of distributed generations (DGs) and EVs on the total operation cost (T_cost) and total emission (T_em) of the μG, considering the demand response. Rezaei et al. [25] presented robust optimization to scheduling the energy of μG with the integration of RESs and EVs, considering the uncertainty of RESs to increase μG profits and reduce its emissions. Tan and Chen [26] presented an optimization method to obtain the economic scheduling of μG to minimize operation costs, power losses, and total emissions, considering the operation constraints of transmission lines. Mohamed et al. [27] suggested a stochastic fuzzy technique to get optimal energy management for an μG, assessing EVs and different types of RESs to minimize the total operation cost of the μG. Mohamed et al. [28] presented an optimal approach based on scenario generation to schedule the energy, considering the uncertainties of RESs on the power losses, cost function, and voltage of the system. The model maintains the proper precision for modeling uncertainty while requiring less computing work than Monte Carlo.

Sedighizadeh et al. [29] investigated the effects of responsive loads and the stochastic behavior of electric cars as demand-side management tools on the effective functioning of a grid-connected μG that combines power, heat, and cooling systems. Li et al. [30] presented an optimization algorithm to obtain optimal operation of the μG, including EVs and RESs, to minimize the total operation cost of the μG. Gholami et al. [31] introduced a new management strategy to schedule the output power of EVs and DG units to minimize the operation cost of the distribution system, considering the uncertainty of electrical parameters. Doosti et al. [32] presented a management strategy to obtain optimal scheduling of EVs, energy storage, and RESs, considering the uncertainty of RESs to save energy. Zhang and Chen [33] introduced an optimization approach to optimize the energy of EVs and investigate the effect of EV load on the main grid. Kurukuru et al. [34] presented a fuzzy interference system-based EM of a single μG; μG profit is maximized using a genetic algorithm without any consideration of how the suggested approach would be affected by renewable generation. Pirouzi and Aghaei [35] introduced a model to manage the energy of the smart distribution network based on cooperation between the distribution system operator (DSO) and EV parking lots to improve network voltage security. Momen et al. [36] introduced a two-stage stochastic approach to obtain the EM of the distribution system based on the load control and the master-slave control techniques considering the diesel generators and PV units with EVs.

Within the context of integrating traffic networks and power grids, Lixun et al. [37] developed a comprehensive evaluation system and methodology for EV charging networks. Initially, an EV travel model is constructed, utilizing a trip probability matrix to analyze the geographical and temporal characteristics of EV usage. Subsequently, the interconnectedness among users, the road network, the power grid, and the charging infrastructure is examined. The study proposes four critical criteria: user feedback, the operational impact on the road network, the functionality of the charging network, and the influence on the power grid’s operation. For each criterion, specific evaluation indices are established, culminating in a holistic evaluation index system.

Subramaniam and Singh [38] detailed a strategic optimization approach for the charging of EVs and the selection of optimal installation sites. The main objective is to develop a charging network that is cost-effective while maintaining the operational integrity of the distribution network. The methodology addresses these challenges by employing renewable energy sources and meta-heuristic algorithms for optimal planning, taking into consideration the impacts of various factors. Consequently, this study introduces a novel perspective on managing the distribution of RES and charging station challenges, advocating for a multi-objective approach that incorporates the characteristics of charging stations.

Wenchao et al. [39] proposed a methodology to ascertain the optimal number of charging stations alongside a pricing strategy for EVs, considering various configurations of component commonality. The study explored four distinct scenarios characterized by differing common components and levels of quality. For each scenario, the optimal quantity of charging stations and the corresponding EV pricing were determined. Subsequently, through numerical simulations, the researchers evaluated the most favorable solutions and manufacturer profits across these scenarios, yielding insightful managerial implications.

To meet the growing demand for EVs, Hao et al. [40] developed a two-stage optimization framework aimed at enhancing the capacity and efficiency of charging stations. This framework systematically identifies the optimal capacity for generating renewable energy and the most effective timing for electricity distribution. By analyzing EV arrival patterns and demand profiles with real-world data, they established a viable model for EV service requests. Numerical results illustrate the optimal configuration for an EV charging station powered by renewable sources, showcasing the best mix of wind and solar power generation alongside the optimal electricity distribution schedule. Ahmad et al. [41] investigated the economic advantages of employing EVs as a temporary BSS within an μG that incorporates PV units. The formulation of the problem considered various factors, including the feed-in tariff, incentives for EV owners, and the capital costs associated with charging stations. To optimize the profitability of the proposed system, a planning algorithm was developed to determine the requisite number of EV charging stations (EVCSs).

1.3 Contribution and organization

In this study, the optimal EM of an μG, encompassing PVs, FCs, WTs, ESSs, and EVCSs, is optimized using the innovative krill herd algorithm (KHA), an algorithm that has garnered significant attention. This paper delivers an optimization strategy for managing the energy of an μG to fulfill multi-objective functions, including the minimization of total operational costs, maximization of BSS profits, and reduction of total emissions. The novelty of this work can be summarized as follows:

1. Multi-objective Optimization: The manuscript presents a multi-objective optimization model that simultaneously considers the microgrid’s total operation cost and emissions. This approach allows for a comprehensive analysis and decision-making process that aims to achieve the best balance between reducing expenses and minimizing the environmental impact of greenhouse gas emissions.
2. Integration of EV Charging and BSS: The manuscript highlights the benefits of connecting BSS with EV charging stations in microgrids. By intelligently managing the charging load and utilizing stored energy during peak demand, the integration of EVs and BSSs optimizes the utilization of available energy resources, reduces strain on the grid, and improves the overall operational efficiency of the microgrid.
3. Comparison with Other Optimization Methods: The study utilizes the krill herd algorithm (KHA) as an optimization technique and compares the results obtained with those other optimization methods. This comparison demonstrates the validity and effectiveness of the proposed solution. It shows that integrating EV charging infrastructure with renewable energy significantly enhances the microgrid’s operational efficiency, reduces operating costs, and minimizes environmental impact.
4. Sustainability and Energy Demand: The manuscript emphasizes the importance of integrating RESs with EVs to sustainably meet the increasing energy demand. By tapping into clean energy sources and reducing reliance on fossil fuels, adopting EVs and utilizing renewable energy help reduce greenhouse gas emissions and address environmental concerns.

The main contributions of this research are outlined as follows:

• A day-ahead scheduling model for optimal EM is introduced for interconnected μGs, incorporating PVs, FCs, WTs, ESSs, and EVCSs.
• A day-ahead scheduling model is introduced to determine the optimal operation schedule for each distributed generator within the interconnected microgrids depending on the objective function, considering factors such as energy generation, energy demand, grid constraints, and economic considerations.
• The study examines three case studies that focus on different objective functions related to implementing Energy Management (EM) within the microgrid (μG).
• The first case study aims to lower the total operational costs of the μG and increase the profits of the BSS. The EM strategies and scheduling decisions are optimized to minimize costs associated with energy generation, storage, and consumption while maximizing the revenue generated by the BSS.
• The second case study focuses on reducing the total emissions from the μG. The EM strategies and scheduling decisions are optimized to minimize the carbon footprint of the μG by maximizing the utilization of renewable energy sources, optimizing energy dispatch, and minimizing reliance on fossil fuel-based generation.
• The third case study combines the objectives of the first two studies. It aims to implement EM strategies that simultaneously decrease the total operational costs, enhance the profits of the BSS, and reduce the total emissions from the μG. The optimization model considers the trade-offs and synergies between these objectives to find a balanced solution that provides economic benefits, environmental sustainability, and efficient energy management.
• The optimization strategy employed in this study aims to achieve an optimal balance between cost reduction and the mitigation of environmental impacts associated with greenhouse gas (GHG) emissions. By considering both economic and environmental objectives, the strategy seeks to identify solutions that minimize operational costs while simultaneously reducing the carbon footprint of the microgrid (μG).
• To demonstrate the validity and effectiveness of the proposed solution, the study utilizes the Krill Herd Algorithm (KHA) and compares the obtained results with those achieved by other optimization methods, namely the Firefly (FF), Cuckoo (CK), and Golden Section Search (GSS) algorithms. This comparative analysis aims to evaluate the performance and efficiency of the KHA in solving the optimization problem at hand.

The article organizes the study into key sections, following the introduction that sets the context by discussing the global increase in energy demand, the significance of RESs and EVs for reducing carbon emissions, and the challenges in managing μGs, and explores the literature on EM, and the models used for efficient μGs operation. Further, Section 2 presents a schematic representation of the system under investigation with the formulation and discussion of its various components, PVs, FCs, WTs, ESSs, and EVCSs. Section 3 presents the problem formulation, objectives and constraints, and the optimization algorithm used. The main contributions include a proposed day-ahead scheduling model for interconnected microgrids, the benefits of incorporating BSS with EV charging stations, and the analysis of case studies to evaluate the model’s effectiveness in optimizing operation costs, maximizing BSS profits, and minimizing emissions. In this regard, Section 4 presents the results obtained and the discussion. The paper concludes with a summary of findings and potential areas for future research in Section 5.

2.1 Photovoltaic (PV) units

The method of generating PV power involves converting solar radiation energy into electrical power through physical means. The solar cell is similar to a pn junction diode [44]. The output voltage is used to express the output current as follows: (1) where P_pv,t represents the output power of the PV units, and denotes the units’ rated power. N^PV indicates the total number of PV modules. G and G₀ refer to the actual global solar irradiance and the standard test condition irradiance, measured in watts per square meter (W/m²), respectively. T_A stands for the ambient temperature, while T_C is the coefficient that describes how the PV’s maximum power output changes with temperature. η^inv and η^rel are the inverter’s efficiency and the PV modules’ relative efficiency, respectively [45].

2.2 Wind turbine (WT) units

Wind serves as the fundamental energy source for these turbines, as they convert wind energy into electrical power. The power generation capacity of these turbines is directly proportional to the speed of the wind [46]. (2) where, P_wt,t and denote the output power and the rated power of the WTs respectively. The terms and correspond to the wind speed at the current time step and the WT’s rated wind speed, respectively. Moreover, and refer to the minimum and maximum wind speeds at which the WTs start and stop operating, respectively [47].

2.3 Fuel cells (FCs)

A fuel cell (FC) is an electrochemical apparatus that generates electrical power through the reaction of specific substances. One prevalent model, known as the polymer electrolyte membrane (PEM) fuel cell, PEMFC, harnesses a chemical process involving hydrogen and oxygen to produce electricity, with water as a byproduct [48]. This PEMFC is characterized by its semipermeable membrane, which facilitates the movement of protons but prevents electrons from passing through, thus distinguishing it as a proton-exchange membrane fuel cell due to its unique operation [49].

In the operation of the FC, hydrogen atoms are separated into protons and electrons upon arrival at the anode side. The electrons then take a path through an external circuit, generating electrical current, whereas the protons move directly through the membrane to the other side [50]. At the cathode end, the reunion of protons and electrons with incoming oxygen results in the production of heat and water, completing the chemical process. FCs, by bypassing the need for the combustion of traditional fuels, avoid the pollution associated with conventional fuel burning, thereby representing a sustainable and clean energy source. Fig 3 illustrates the fundamental principles of a PEMFC’s operation [51].

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Fig 3. Configuration of PEMFC.

https://doi.org/10.1371/journal.pone.0307810.g003

The Nernst equation relates the cell voltage (E) of the FC to the standard cell potential (E⁰), the gas constant (R), the temperature (T) in Kelvin, the Faraday constant (F), number of electrons transferred in the reaction (n) and the concentrations of reactants and products [52], where Q represents the reaction quotient (ratio of product concentrations to reactant concentrations).

(3)

The relationship between power (P_fc,t) in watts, cell voltage in the FC (voltage across the FC) and I represents the electric current flowing through the FC can be described by the following equation [53] and: (4)

2.4 Battery storage system (BSS)

Rechargeable batteries, also known as energy storage devices, play a vital role in storing energy derived from both renewable and non-renewable sources for subsequent use [54]. These devices, functioning on direct current (DC), effectively bridge the gap between energy supply and demand by releasing stored electrochemical energy as required to meet electrical demands. BSS emerge as a pivotal technology for μGs in light of the increasing reliance on RESs and the global ambition to achieve net-zero carbon emissions. The deployment of BSS is instrumental in advancing toward net-zero energy goals, offering a critical pathway towards the adoption of green energy solutions [55].

2.5 Electric vehicles (EVs)

The charging and discharging patterns of EVs are influenced by several key factors, including the number of EVs connected for charging, the type and speed of the charging stations, the state of charge (SoC), battery capacity, duration of charging, and the start time of charging [56]. EV charging stations, also known as EV charging points, provide the essential infrastructure for recharging EV batteries [57]. These stations vary in their power outputs and charging speeds, encompassing different types such as Level 1 charging, which offers basic charging through a standard household outlet with power typically ranging from 1.4 to 2.3 kW; Level 2 charging, which provides faster charging speeds requiring specific charging equipment and delivers 3.3 kW to 22 kW of power; and DC fast charging (Level 3), the fastest charging option, delivering power from 50 kW to 350 kW or more, allowing for rapid recharging of EVs [58–63]. EV charging stations play a crucial role in facilitating the widespread adoption of electric vehicles by providing the necessary infrastructure for battery recharging during travel. Investments by companies, governments, and other entities in the expansion of charging infrastructure are critical to meet the increasing demand for EVs [61].

The integration of EVs into μGs presents both opportunities and challenges. As the adoption of EV grows, there may be an increase in energy demand on the μG. However, EVs can also serve as a flexible load, capable of being managed to match electricity generation and demand profiles within the μG. By charging during periods of high generation and discharging stored energy back to the μG during peak demand or low renewable generation times, EVs can help optimize the use of RES such as solar or wind. Nevertheless, accommodating the rising power demand from EV charging infrastructure, especially from fast-charging stations, may require upgrades to the μG’s distribution system [63]. Thus, the impact of EVs on μGs encompasses both potential benefits and challenges. With appropriate design, smart charging strategies, and integration technologies, EVs can enhance the flexibility, resilience, and sustainability of μGs by managing energy demand, leveraging RESs, and providing distributed energy storage capabilities. A microgrid is defined as a small distribution network equipped with distributed energy resources, BSS, active loads, and EVs, as depicted in Fig 4 [64]. Finally, the μG considered in this study can be described as a compact distribution network that incorporates distributed energy resources, BSS, active loads, and EVs, as depicted in Fig 4 [64].

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Fig 4. Microgrid configuration.

https://doi.org/10.1371/journal.pone.0307810.g004

3.1 Cost function formulation

This work is dedicated to minimizing the operating costs of an μG, which incorporates PVs, WTs, FCs, BSSs, and EVCSs. The first objective function under consideration (OF₁) is formulated as follows: (5)

To clarify the provided variables clearly, MP_t represents the market price at time t for the utility grid, P_grid,t denotes the output power from the main utility at t, and P_pv,t refer to the bidding cost and the output power of the PVs, respectively. and P_wt,t indicate the bidding cost and the output power of the WTs. and P_fc,t are the cost of the natural gas and the output power of the FC. CB_t and P_B,t indicate the bidding cost and the charge or discharge power of the BSS at time t.

(6)

(7)

For the FC, the terms , μ_SU, and μ_SD denote the start-up cost, shut-down cost, and the coefficients for start-up and shut-down costs, respectively. The variable indicates the operational status of the FC at time t, where a value of 1 implies that the FC is active (on), and a value of 0 signifies that the FC is inactive (off).

3.2 Emission function formulation

The second objective function (OF₂) is represented as follows: (8) , and represent the carbon dioxide emitted from the grid and FC. , and represent the sulfur dioxide emitted from the grid and FC. , and represent the nitrogen dioxide that is harmful to human health, affecting the respiratory system and is emitted from the grid and FC.

3.3 Combined, multi-objective, function formulation

To minimize both the operational costs and pollutant emissions over a 24-hour period, this study introduces multi-objective optimization functions. Below is the formulation of the multi-objective function: (9)

For optimizing both the total operational cost and pollution emissions of the μG over a 24-hour period, m₁ and m₂ represent the weighting factors for the total operational cost and pollution emissions of the μG, respectively. These factors are used to balance the importance of reducing costs versus minimizing environmental impact in the optimization process.

The optimization problem in this paper involves two conflicting objective functions—minimizing operational costs (OF₁) and minimizing environmental impact/pollutant emissions (OF₂). To balance these two objectives, a multi-objective optimization approach can be employed, utilizing weight factors to prioritize the relative importance of each objective. The weight factors, denoted as m₁ and m₂, represent the importance placed on OF₁ and OF₂ respectively. Typically, the range for these weights would be: (10) (11) (12)

By adjusting the values of m₁ and m₂, the optimizer can explore the trade-off between the two competing objectives. For example: If m₁ = 1 and m₂ = 0, the optimization would solely focus on minimizing operational costs (OF₁), disregarding environmental impact. If m₁ = 0 and m₂ = 1, the optimization would solely focus on minimizing emissions/environmental impact (OF₂), disregarding operational costs. By using intermediate values, such as m₁ = 0.6 and m₂ = 0.4, the optimization would attempt to find a balanced solution that accounts for both operational costs and environmental impact to some degree. The specific choice of weight factors would depend on the priorities and constraints of the particular μG system being analyzed. For instance, if there are strict emissions regulations or environmental targets to meet, a higher weight (m₂) may be placed on the emissions objective (OF₂). Conversely, if operational costs are the primary concern, a higher weight (m₁) may be placed on the cost objective (OF₁). In this study, the weight factors utilized in the multi-objective optimization were set to m₁ = 0.5 and m₂ = 0.5. This configuration reflected a balanced prioritization between the two competing objective functions: minimizing operational costs (OF₁) and minimizing environmental impact/emissions (OF₂). By assigning equal weights of 0.5 to each objective, the optimization sought to find solutions that provide an equitable compromise between the objectives of reducing operational expenditures and lowering the environmental footprint of the microgrid system.

3.4 Constraints considered in the optimization process

The three objective functions are subject to the following constraints:

3.4.1 Energy storage limits.

Several constraints related to energy storage should be taken into account in formulating the problem. The first constraint, as presented in Eq (10), relates the charging power to the maximum charging capacity of the BSS. In a similar look, the second constraint, outlined in Eq (11), ensures that the discharged power does not exceed the BSS’s maximum discharging capacity . Eq (15) sets the boundaries for the energy stored within the BSS, adhering to its minimum and maximum SoC storage limits.

(13)(14)(15)

Moreover, the power charged into the battery, accounting for its efficiency (η_bat), should match the power discharged, as depicted in Eq (16). The battery’s state of charge at any given time (SoC_t) is determined by its previous state (SOC_t-1), alongside the quantities of power charged and discharged at time t, as specified in Eq (17). It’s important to highlight that for the initial period (t = 1), the initial state of charge (SoC₀) must be taken into account. Additionally, the state of charge at the conclusion of the period should revert to SoC₀ to ensure consistency in the battery’s state of charge, as articulated in Eq (18). Eq (19) introduces a constraint related to battery aging, further refining the model’s accuracy.

(16)(17)(18)(19)

3.4.2 Loads balance.

At any given time, t, the sum of power generated from FCs, PVs, WTs, power imported from (or exported to) the grid, and power discharged from (or charged to) the battery, must balance the total load demand (P_L−t) and the EVs’ load (P_EVL). This relationship is given in Eq (20).

(20)

3.4.3 Power limit of the grid.

The power imported from or exported to the grid (P_g−t) must adhere to its predefined limits at each time interval, as described in Eq (21).

(21)

3.4.4 Power limits of DGs.

The output powers of FC, PV, and WT must be within their required limits as expressed in (22)–(24).

(22)(23)(24)

3.5 Krill Herd Optimization (KHO) employed in this work

Krill Herd Optimization (KHO) is a bio-inspired metaheuristic algorithm modeled after the swarming behavior of krill in the natural world. It has become increasingly popular for its proficiency in addressing complex optimization challenges [65]. In KHO, the objective is often defined by the distance between a krill individual’s position and the target location, representing the search for optimal solutions. The optimization process in KHO comprises three distinct phases: movement influenced by the presence of other krill, foraging behavior, and physical random diffusion. These phases are mathematically formulated to simulate the natural dynamics of krill behavior in seeking optimal positions [66], as follows: (25) where P_j denotes the diffusion of the jth krill individuals, V_j denotes the foraging motion, and M_j represents the motion induced by the other krill.

3.5.1 Motion induced by other krill.

The initial motion’s (γ_j) three main components are the target effect, the local effect, and the repulsive effect. With respect to the krill j, the following numerical expression is possible [67]. (26) (27) M^max represents the maximum induced speed, σ_n is the inertia weight within [0,1], denotes the old motion induced, denotes the local effect, denotes the target direction effect.

(28)(29)(30)

The fitness of the jth and qth krill, respectively, are B_j and B_q. On the krill scale, the best fitness is B_best, and the lowest is B_worst. MM is the total number of neighbors, B is the related positions, where j = 1,2, …, MM.

(31)

The optimal fitness coefficient for the jth krill individual is denoted as W_best.

3.5.3 Physical diffusion.

In its most basic form, physical diffusion is an unpredictable process [69] expressed as follows: (34) where T_max represents the maximum diffusion speed, and μ is a random vector within the range [−1,1]. Drawing inspiration from evolutionary computation, two genetic reproduction mechanisms—namely the crossover and mutation operators—are integrated into the basic KHO algorithm [62]. Detailed descriptions of these genetic reproduction operators and the KHO algorithm are available in [69]. The process is illustrated in the flowchart presented in Fig 5. The steps of how the KHO algorithm is implemented and how nonlinear constraints are considered in the model [70, 71]:

• Initialization: The algorithm begins by initializing a population of potential solutions, referred to as "krill." Each krill represents a candidate solution to the optimization problem.
• Objective Function Evaluation: The objective function, which represents the quantity to be optimized, is evaluated for each krill in the population. The objective function could be defined based on the specific optimization problem, such as minimizing costs or maximizing efficiency.
• Movement and Interaction: The krill in the population move and interact with each other using predefined rules inspired by the behavior of real krill. These rules include attraction to food (good solutions) and repulsion from predators (poor solutions). The movement and interaction aim to explore the search space and improve the solutions iteratively.
• Krill Update: The krill update step involves modifying the positions and characteristics of the krill based on their movement and interaction. The update can be performed using mathematical equations that determine the new positions and characteristics of the krill based on the previous values and the defined rules.
• Constraints Handling: Nonlinear constraints, if present in the optimization problem, need to be considered during the update step. Various approaches can be employed to handle constraints, such as penalty functions, repair mechanisms, or constraint handling techniques like feasibility-based rules. These techniques ensure that the updated solutions satisfy the constraints imposed by the problem.
• Iteration and Termination: The movement, interaction, and update steps are repeated iteratively until a termination condition is met. This condition can be a maximum number of iterations, convergence criteria, or reaching a satisfactory solution based on predefined metrics.

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Fig 5. Flowchart of KHO.

https://doi.org/10.1371/journal.pone.0307810.g005

By iteratively updating the positions and characteristics of the krill based on their movement and interaction, the KHO algorithm explores the search space and converges towards optimal solutions. The consideration of nonlinear constraints ensures that the updated solutions satisfy the constraints imposed by the problem, leading to feasible solutions.