Mathematical Formulation of Intelligent Management Algorithms for Isolated Microgrids: A Pareto-Based Critical Approach

Abstract

This study proposes a simplified mathematical formulation for optimizing isolated microgrids, enhancing computational efficiency while preserving solution quality. The research focuses on the influence of Operation and Maintenance (O&M) costs for Non-Dispatchable Generators (NDGs) and the relationship between costs and pollutant emissions. The proposed simplification reduces computational requirements, improves result interpretability, and increases the scalability of optimization techniques. The O&M costs of photovoltaic and wind systems were excluded from the initial optimization and calculated afterward. A Student’s t-test yielded a p-value of 87.3%, confirming no significant difference between the tested scenarios, ensuring that the simplification does not impact solution quality while reducing computational complexity. For emission-related costs, scenarios with single and multiple pollutant generators were analyzed. When only one generator type is present, modifications are needed to enable effective multi-objective optimization. To address this, two alternative mathematical formulations were tested, offering more suitable approaches for the problem. However, when multiple pollutant sources exist, cost and emission differences naturally define the problem as multi-objective without requiring adjustments. Future work will explore grid-connected microgrids and additional optimization objectives, such as loss minimization, voltage control, and device lifespan extension.

1. Introduction

Energy is a central pillar in the economic development of a nation, making the optimization of its energy resources an increasingly important priority. Recent studies have explored the potential of Microgrids (MGs) as decentralized and sustainable solutions, capable of integrating renewable energy sources, such as solar and wind, with storage systems to meet a broad range of applications, from individual households to entire communities [1,2]. These grids offer benefits such as reduced Operation and Maintenance (O&M) costs, improved energy quality, and decreased pollutant emissions ($C{O}_2$, $S{O}_2$, $N{O}_x$). These advantages position MGs as a key component in the transition to a more efficient and sustainable energy system [3]. This role is even more significant in isolated MGs, where the integration of smart technologies and renewable sources ensures greater energy resilience in remote regions or those disconnected from traditional grids.

Management is one of the central challenges in deploying an MG, as it aims to determine which energy generators should be activated at each time interval, minimizing the O&M costs of the MG, reducing the use of pollutant emissions, and ensuring the fulfillment of demand [4]. This process, essential for energy efficiency, is typically implemented through approaches such as Model Predictive Control (MPC), which integrates demand and generation forecasting algorithms with optimization techniques to develop efficient operational strategies [5]. MPC, by coordinating the power flow between different MG components, ensures a balance between cost and sustainability, optimizing the dispatch of energy in both renewable and conventional systems [6,7].

Optimization algorithms associated with MPC, such as Genetic Algorithms (GAs), NSGA-II (Non-Dominated Sorting Genetic Algorithm—II), and DE (Differential Evolution), play a pivotal role in determining the optimal strategy for distributing power among the devices that comprise the MG. These algorithms aim to identify solutions that balance multiple objectives, such as minimizing O&M costs and reducing pollutant emissions, while ensuring the system’s efficiency and reliability. However, as demonstrated by [8], the effectiveness of these algorithms is directly linked to the accurate formulation of the objective function, with several studies focusing on reformulating the problem to enhance solution quality.

An inadequate formulation of the objective function can significantly impair the performance of the optimization algorithm, leading to suboptimal solutions and inefficient utilization of computational resources, which are often constrained, particularly in embedded applications where MG management systems are typically implemented [7]. Therefore, a rigorous and precise definition of the objective function is critical for optimizing MG performance, ensuring that the solutions produced are not only technically feasible but also economically efficient and environmentally sustainable.

To achieve this, it is essential that the mathematical formulation accurately represents the complex interactions among the various components of isolated MGs, ensuring that the optimization process both achieves high-quality solutions and maximizes the efficiency of the limited computational resources, which are crucial for the feasibility of implementing energy management in real-world systems.

1.1. Literature Review

Many studies in the literature have been dedicated to the MG management problem, proposing optimization algorithms with different mathematical formulations, adapted to the specific characteristics and challenges of each application. The main objectives of these studies include reducing operational costs, minimizing pollutant gas emissions, and ensuring reliability and efficiency in energy supply.

Several reviews highlight the predominant role of multi-objective optimization in MG management. Reference [9] identifies PSO and GA as the most widely adopted techniques in this field. Expanding on this, ref. [10] notes that cost and emission minimization are the most common optimization objectives in MG studies. Additionally, ref. [11] suggests that future research should enhance optimization approaches by incorporating a stronger emphasis on economic, social, and environmental benefits, thereby increasing the applicability and impact of the proposed solutions.

References [12,13] investigated MG management using optimization algorithms aimed at minimizing operational costs, including the O&M costs of Non-Dispatchable Generators (NDGs), while also considering environmental variables. To address multiple objectives, these studies adopted a weighted-sum approach, wherein specific weights are assigned to each objective within the mathematical formulation. Although this strategy simplifies the optimization process, it presents significant limitations, such as the loss of diversity in the solution set, as potentially promising solutions that do not align with the predefined weights are automatically excluded. This limitation is also highlighted in [14], which examined the operation of an isolated MG with the goal of minimizing total operational cost and carbon emissions, facing similar challenges in balancing the optimization criteria.

Certain studies focus exclusively on minimizing the operational costs of isolated MGs. Reference [15] explores various optimization techniques to determine the optimal number of energy-generating modules for a hybrid renewable energy system in Pulau Perhentian, Malaysia. The objective function focuses on minimizing the Net Present Cost (NPC) while ensuring system stability and efficiency. Reference [16] presents an approach utilizing the Transient Search Optimization algorithm for energy management in islanded hybrid microgrids. The objective function is formulated to minimize operational costs, incorporating various components such as diesel generator units, photovoltaic systems, and battery energy storage. In these approaches, operational costs include the O&M costs of NDGs.

Some papers also minimize the operational cost in connected MGs. Reference [17] presents an optimization framework utilizing various metaheuristic algorithms, including GA and Particle Swarm Optimization (PSO) and Mixed-Integer Distributed Ant Colony Optimization (MIDACO), for effective energy management in a connected microgrid system. The primary objectives are to minimize operational costs, enhance resource allocation, and ensure a reliable power supply while integrating renewable energy sources. In [18], an improved Gazelle Optimization Algorithm (IGOA) is presented for the operational optimization of a microgrid system, incorporating both Power-to-Gas equipment and hybrid energy storage systems. The microgrid operates in a connected mode, aiming to minimize operational costs and carbon emissions while maintaining a balance between electricity and hydrogen power.

Additionally, reference [19] presents an advanced optimization algorithm for microgrid energy management, integrating a hybrid neural-fuzzy network with an improved Modified Particle Swarm Optimization (MPSO) algorithm. The microgrid aims to minimize generation costs and maximize power consumption efficiency. The system utilizes multiple renewable energy sources and an energy storage system, along with a microturbine, to ensure reliability and balance during peak demand periods. The proposed approach demonstrates superior performance in cost-effectiveness and energy savings compared to traditional methods.

The work present in [20] introduces an energy management system tailored for grid-tied MGs, employing an Improved Adaptive Genetic Algorithm (IAGA) for optimization. The MG integrates various renewable sources, specifically wind and photovoltaic systems, alongside a standalone diesel generator as the sole pollutant-emitting backup. Key objectives include optimizing resource allocation, reducing operational costs, and enhancing energy efficiency amid fluctuating demand. The IAGA enhances adaptability and efficiency compared to traditional methods, demonstrating practical applicability for sustainable energy management. These works consider the costs and emissions of polluting generating sources as conflicting objectives in multiple pollutant-based generators, emphasizing the importance of balancing economic feasibility and environmental sustainability.

Other studies have addressed MG optimization with the objective of minimizing both costs and emissions with multiple pollutant-based generator sources. Reference [21] presents the Bi-Objective Ant Colony Algorithm (BOACA) for optimizing hybrid isolated microgrid systems in Egypt, focusing on minimizing costs and GHG emissions using multiple fuel options. In [22], the authors propose a PSO-based methodology with a fuzzy system for managing microgrids, aiming to reduce operational costs and $C{O}_2$ emissions, with tests conducted on a six-bus AC/DC grid. Comparisons were made with a multi-objective genetic algorithm with controlled elitism.

Some studies expand the optimization scope by incorporating additional objectives beyond costs and emissions. For instance, ref. [23] addresses the minimization of operational costs, voltage levels, and system stability, while [24] includes not only minimizing costs and emissions but also reducing electrical losses and integrating the O&M costs of NDGs into the mathematical formulation. Reference [25] presents an adaptive AI-based Home Energy Management System (HEMS), utilizing the African Vultures Optimization Algorithm (AVOA). The study focuses on isolated systems, aiming to minimize costs, energy surplus, and the Loss of Power Supply Probability (LPSP). It also incorporates the O&M costs of NDGs. The microgrid integrates renewable energy sources along with backup sources, and the study formulates a multi-objective optimization problem to coordinate the operation of these backup sources.

Some studies focus on minimizing costs and emissions in isolated systems with multiple pollutant-generating sources. Reference [26] presents an Improved Multi-Objective Crow Search Algorithm (IMOCSA) designed to optimize the performance of rural microgrids, which can operate in both isolated and interconnected configurations. The optimization targets three main objectives: minimizing operational costs, reducing voltage deviation to ensure power quality, and decreasing pollutant emissions from multiple generation sources, including microturbines and fuel cells. IMOCSA improves upon the standard Crow Search Algorithm by incorporating an adaptive chaotic awareness probability, a mutation mechanism to prevent premature convergence, and a K-means clustering method to enhance efficiency.

Reference [27] presents the Fractional-Order Fish Migration Optimization (FOFMO) algorithm, which is applied to minimize costs and emissions in power systems integrating both renewable energy sources and natural gas units. The algorithm is tested on a modified IEEE 30-bus system. Three test scenarios were considered: the first includes three thermal units, while the second and third incorporate a mix of thermal and natural gas units. The three thermal and three natural gas units have different emission factors, enabling the application of multi-objective optimization algorithms. Even in the first scenario, where all generators are of the same type, variations in their emission factors create a natural trade-off between cost and emissions, maintaining the conflict between these objectives.

Reference [28] presents an innovative optimization methodology based on the JAYA algorithm for the energy management of AC MGs operating in both isolated and grid-connected modes. The primary objectives include minimizing operational costs, energy losses, and $C{O}_2$ emissions while effectively integrating photovoltaic generation. The study demonstrates the algorithm’s superior performance compared to traditional methods such as PSO and GA. Notably, this work is unique in the literature as it specifically addresses the optimization of isolated microgrids with a single pollutant-based generator source and a single pollution coefficient, highlighting its contribution to advancing sustainable energy management solutions.

1.2. Motivation and Research Gap

The analyzed studies proposed the optimization of microgrid management through optimization algorithms. These studies focused on minimizing operational costs and, in some cases, also reducing GHG emissions. However, the presented mathematical modeling contains inaccuracies that compromise the coherence of the formulation.

A direct inclusion of O&M costs for NDGs in the optimization model is unwarranted, as these costs remain largely invariant over time. Instead, they can be calculated separately after the optimization process without compromising solution quality. This simplification reduces computational complexity while maintaining the accuracy and reliability of the results.

Independent of this aspect, most studies on microgrid optimization incorporate multiple pollutant-generating sources, as this approach more effectively captures the trade-offs between economic and environmental objectives [12,14,19,21,22,23,24,26]. Additionally, some of studies focus on grid-connected systems [17,18,20] or do not explicitly aim to minimize pollutant emissions [15,16,25], further limiting research on isolated microgrids with a single pollutant-emitting generator. As a result, only the studies by [27,28] have addressed similar scenarios.

Despite the prevalence of remote microgrids operating with a single fossil-fuel-based generator—such as diesel-based systems in isolated communities—there is a lack of dedicated research exploring the implications of single-source pollutant generation on optimization strategies. Existing formulations generally assume inherent conflicts between cost and emissions, but when only one pollutant-emitting source is present, this assumption may not hold.

Addressing this gap is essential, as the formulation of multi-objective optimization problems may require adjustments under such conditions. If cost and emissions are directly proportional due to a single fuel source, traditional multi-objective approaches may be unnecessary or ineffective. This study aims to investigate whether conventional optimization methods remain valid in these cases or if alternative formulations are required, contributing to a more accurate and efficient approach to microgrid management.

1.3. Contributions and Scope

This study introduces novel modifications to the mathematical formulation of optimization algorithms for isolated MGs, aiming to simplify computations while maintaining solution quality. It critically evaluates the inclusion of O&M costs for NDGs, demonstrating that their incorporation within the optimization process is unnecessary. Additionally, it addresses the underexplored scenario of MGs relying on a single pollutant-based generator, revealing inaccuracies in existing mathematical models. The study also reassesses the cost–emission trade-off, ensuring conflicting objectives when only one pollutant-emitting source is considered. By refining the mathematical framework, the proposed approach enhances optimization efficiency, reduces unnecessary calculations, and improves the robustness of cost–emission strategies.

The main contributions of this paper are as follows:

• Demonstrate that few studies in the literature focus on testing MGs with a single pollutant-based generator;
• Identify inaccuracies in the existing mathematical modeling of isolated MGs in this specific case;
• Propose a simplified cost formulation that eliminates unnecessary calculations during the optimization process;
• Develop a novel cost–emission modeling approach for systems with a single pollutant-based generator, ensuring a true conflict between objectives;
• Introduce a single-objective formulation for the cost–emission optimization problem, reducing the number of objectives and enhancing the model’s robustness and effectiveness.

2. Materials and Methods

This study proposes modifications to the mathematical formulation of optimization algorithms for isolated single pollutant-based source MGs, with the goal of simplifying the formulation while reducing the demand for computational resources, without compromising performance or solution quality.

The proposal focuses on evaluating the inclusion of O&M costs for NDGs in the objective function formulation based on the assumption that these costs can be considered invariant over time and can be incorporated at the end of the search process without negatively impacting the results.

Furthermore, this study investigates how the objective function behaves in the presence of a single pollutant-based generator compared to multiple pollutant-emitting sources. It analyzes the interactions between multiple objectives, assessing whether the conventional mathematical formulation adequately represents the trade-offs in both cases. In this context, scenarios with one or multiple pollutant-emitting sources may have significant implications for the optimization process, influencing the feasibility and effectiveness of cost–emission strategies.

2.1. MG Management

The evaluation of the impacts of the mathematical formulation involves understanding the interactions between different system parameters, such as the power curves of DGs and operational constraints, and how these interactions affect optimization efficiency and the quality of the obtained solutions. An example of the operation of the management system for an isolated MG based on MPC is presented in Figure 1.

The process of managing isolated MGs begins with the collection of load and climatic data. Subsequently, a simulation of climatic data is conducted to determine the generation potential of NDGs. Next, load and energy generation forecasts are made for a future time horizon, typically 24 h. At this stage, advanced forecasting algorithms, such as Long Short-Term Memory (LSTM) and Seasonal Autoregressive Integrated Moving Average (SARIMA), are widely employed.

Based on the forecasts, the management system employs optimization algorithms to determine the optimal power curves for each dispatchable component of the MG, such as generators and energy storage systems. These algorithms perform iterative searches, evaluating various parameter combinations and running simulations until identifying a set of solutions that meet the predefined objectives—namely, minimizing O&M costs and pollutant emissions while adhering to all operational constraints of the grid. The final solutions are represented on a Pareto frontier, which illustrates the most efficient trade-offs among the optimization criteria, ensuring compliance with operational constraints.

Finally, decision-making algorithms select the most suitable solution from the Pareto frontier, aligning with the operational and strategic objectives of MG management. These algorithms enable the prioritization of different criteria based on specific operational conditions and objectives. Once the optimal solution is selected, the plan is translated into operational commands and implemented in the MG, ensuring efficient system operation within the established constraints and objectives.

It is worth emphasizing that the generated plan serves as a strategic guide, providing decision-making guidelines for real-time device operations. This planning approach ensures that system actions are grounded in optimized, pre-calculated analyses, promoting greater operational efficiency and alignment with MG management objectives.

2.2. Mathematical Formulation

The existing literature presents various mathematical formulations for modeling energy management in isolated MGs, often framed as multi-objective optimization problems. The primary goal is to simultaneously minimize operational costs—encompassing fuel expenses and O&M costs—while also reducing pollutant gas emissions.

To consolidate these approaches, Equation (1) provides a summary of the most commonly used formulas in the literature, used in optimization algorithms tailored for this context, along with the constraints inherent to such problems.

$$\begin{array}{cc}\hfill & min\left\{\begin{array}{cc}\hfill & f1=C_{total}=\sum _t^W[\sum _i^D(C_{fuel}\left(P_i^t\right)+C_{O\&M}\left(P_i^t\right))+\sum _j^N{C}_{O\&M}\left(P_j^t\right)+\sum _k^M{C}_{O\&M}\left(P_k^t\right)]\hfill \\ \hfill & f2=E_{total}=\sum _t^W\left[\sum _i^D{E}_{fuel}\left(P_i^t\right)\right]\hfill \end{array}\right.\hfill \\ \hfill & s.t.\left\{\begin{array}{c}\sum _i^D{P}_i+\sum _j^N{P}_j-P_{load}=0\\ 0.95<V_{MG}<1.05\\ P_D^{min}\le P_D\le P_D^{max}\\ P_N^{min}\le P_N\le P_N^{max}\\ So{C}_{stor}^{min}\le So{C}_{stor}\le So{C}_{stor}^{max}\end{array}\right.\hfill \end{array}$$

(1)

where $C_{total}$ and $E_{total}$ represent the total cost and emissions of the MG, respectively; t represents the time instant considered in the analysis horizon; W is the size of the analyzed time window, corresponding to the total planning period; i, j, and k are the indices of DGs, NDGs, and other MG devices (such as batteries), respectively; D, N, and M correspond to the total number of DGs, NDGs, and other MG devices, respectively; P is the power of a given device; $C_{fuel}$ represents the cost of the fuel consumed by a device; $C_{O\&M}$ is the $O\&M$ cost of a device; and $E_{fuel}$ denotes the $C{O}_2$ emissions generated by a device’s fuel consumption. $P_{load}$ is the power demanded by the MG; $V_{MG}$ is the voltage level of the MG (in p.u.); and $So{C}_{stor}$ is the state of charge of the MG storage devices.

2.3. Analyzing the Influence of Including O&M Costs

Equation (2) highlights the component of Equation (1) that is exclusively related to the O&M costs of the MG.

$$C_{total}=\sum _t^W[\sum _i^D(C_{fuel}\left(P_i^t\right)+C_{O\&M}\left(P_i^t\right))+\sum _j^N{C}_{O\&M}\left(P_j^t\right)+\sum _k^M{C}_{O\&M}\left(P_k^t\right)]$$

(2)

In this formulation, both the costs associated with DGs and NDGs are considered. However, as discussed in Section 2.1, the optimization algorithms operate directly on the variables associated with DGs, iteratively adjusting their power outputs to minimize the cost function. Consequently, the output of the objective function is exclusively influenced by the variables of these components, as the optimization process does not alter parameters of the NDGs. Therefore, the O&M costs of the latter are considered fixed in the optimization process, since they remain constant and do not participate in the evolutionary dynamics of the proposed solutions.

The main issue with this modeling lies in the fact that the optimization algorithm, in its search for optimal solutions, performs a high number of iterative simulations, which can easily reach hundreds or even thousands of executions. This process requires the constant recalculation of all parameters defined in the objective function, including the O&M costs of the NDGs. However, since the costs of these devices do not directly affect the energy dispatch decisions, their inclusion in the calculation of each iteration is unnecessary, leading to an inefficient use of computational resources. This redundancy undermines the algorithm’s efficiency, especially in embedded systems with limited capacities, where optimizing calculation cycles is crucial for operational feasibility. Therefore, it is methodologically appropriate to consider these costs as fixed only after the completion of the optimization process.

By avoiding the initial inclusion of these costs, the optimization algorithm can focus its computational resources on dynamic operational variables that directly impact the operational efficiency of the MG. Given the importance of these cost values, they could be added to the total O&M cost after the optimization algorithm has been executed, ensuring the accuracy of the optimization process without compromising available computational resources. This strategy not only enhances the efficiency of the algorithm but also ensures the practical feasibility of its application in isolated MGs.

Thus, the simplification of the mathematical formulation for the optimization algorithm is expressed in Equation (3), with the removal of the calculations for the O&M costs of the NDGs.

$$C_{total}=\sum _t^W[\sum _i^D(C_{fuel}\left(P_i^t\right)+C_{O\&M}\left(P_i^t\right))+\sum _k^M{C}_{O\&M}\left(P_k^t\right)]$$

(3)

2.4. Analyzing the Relationship Between Costs and Emissions

Multi-objective optimization algorithms are widely regarded as the most efficient approach for addressing problems that involve multiple criteria. These algorithms typically seek solutions that form the Pareto frontier, representing the set of non-dominated solutions where no objective can be improved without causing a deterioration in another. A key feature of these algorithms is the inherent trade-off relationship between conflicting objectives, such as minimizing O&M costs and pollutant emissions, where improvement in one objective often results in a compromise of another. Therefore, the mathematical formulation must reflect this relationship, ensuring that one or more objectives are in conflict, which enables Pareto-based algorithms to effectively solve the optimization problem.

Upon analyzing Equation (4) (a reformulation of Equation (1), incorporating Equation (3)), it becomes evident that the defined objectives, at first glance, appear to exhibit characteristics of direct conflict, a typical feature in multi-objective optimization problems.

$$min\left\{\begin{array}{c}f1=C_{total}=\sum _t^W[\sum _i^D(C_{fuel}\left(P_i^t\right)+C_{O\&M}\left(P_i^t\right))+\sum _k^M{C}_{O\&M}\left(P_k^t\right)]\\ f2=E_{total}=\sum _t^W\left[\sum _i^D{E}_{fuel}\left(P_i^t\right)\right]\end{array}\right.$$

(4)

In particular, it is observed that an increase in pollutant emissions is often associated with a reduction in O&M costs, while minimizing emissions tends to increase costs. However, a more detailed analysis reveals that this conflict may not occur in the conventional way, as the operating curves associated with dispatchable pollutant energy sources play the same role in both objective functions. When these devices are triggered, both the cost function and emissions are simultaneously altered, presenting a direct correlation rather than a characteristic trade-off. Thus, changes in the use of these sources impact both metrics in the same direction, highlighting a differentiated dynamic in the formulation of the problem.

This study presents two viable solutions to address this issue. The first approach consists of removing the term related to the costs of pollutant generators from the cost objective function. In this way, the costs associated with these generators would be minimized indirectly along with the emissions function, reducing computational costs and improving the quality of the solutions obtained. The resulting mathematical formulation is presented in Equation (5). After the optimization process is completed, the excluded costs can be added to the final calculation, ensuring the integrity of the result.

$$min\left\{\begin{array}{c}f1=C_{total}=\sum _t^W\sum _k^M{C}_{O\&M}\left(P_k^t\right)\\ f2=E_{total}=\sum _t^W\sum _i^D{E}_{fuel}\left(P_i^t\right)\end{array}\right.$$

(5)

The second solution considers transforming the multi-objective problem into a single-objective problem by eliminating the emissions objective. In this case, only the cost function would be minimized directly, and the emissions produced by the polluting generators would be reduced as a consequence of the cost optimization. The emissions, in this context, would be calculated later for analysis, generating the mathematical formulation presented in Equation (6).

$$min\left\{\begin{array}{c}f1=C_{total}=\sum _t^W[\sum _i^D(C_{fuel}\left(P_i^t\right)+C_{O\&M}\left(P_i^t\right))+\sum _k^M{C}_{O\&M}\left(P_k^t\right)]\end{array}\right.$$

(6)

2.5. Testing and Validation Procedures

To validate the points discussed in this study, tests were conducted on MG 03, as previously described in [29]. This study presents four microgrids (MGs) to be used in benchmarks, all of which can operate in isolated mode. MG 03 is based on the IEEE-69 bus network [30] and features a structure composed of 18 buses. Bus 301 includes three DGs, which, depending on the test scenario, were simulated as diesel, biomass, or natural gas generators. Additionally, the MG contains 6 photovoltaic systems installed on buses 303, 304, 305, 306, 307, and 315, as well as 2 wind generators located on buses 314 and 317.

The MG 03 configuration operates in an 11 kV three-phase system and also includes energy storage systems located on the same busbars as the photovoltaic systems. These storage systems have the capacity to sustain the operation of the MG independently for up to two hours, providing flexibility and additional support to intermittent distributed generation. Figure 2 shows the single-line diagram of the analyzed MG (Figure 2a), along with the generation (Figure 2b,c) and load curves (Figure 2d) used in the tests.

It provides a comprehensive benchmark for testing, including a set of MGs with detailed load and generation data for a one-year period. In the present work, data referring to a single day of operation from the study in [29] were used, with adjustments made to the loads to increase the variety and complexity of the analyzed system (Figure 2d). The O&M values and the emission factor of the MG devices are presented in

Table 1. O&M costs and emission factors.

Device	O&M $(\mathit{\$}/\mathbf{MW})$	Emission $({\mathbf{tCO}}_2/\mathbf{MW})$
Batteries	5	N/A
Photovoltaic	5	N/A
Wind Generator	12	N/A
Diesel Generator	20	0.8
Biomass	20	0.072
Natural Gas	4	0.5

.

The optimization algorithms used in the tests were the GA for the single-objective optimization and the NSGA-II for the multi-objective approach. The main focus of this study is to compare the mathematical formulation of isolated microgrid energy management; therefore, no comparison was made between the optimization algorithms. However, the number of individuals for both algorithms was adjusted to approximate the results of the comparison presented in Section 3.2. The parameters of the optimization algorithms are presented in

Table 2. Parameters of the optimization algorithms.

Parameters	GA/NSGA-II
Number of Generations	200
Population Size	100
Crossover Rate (%)	0.1
Mutation Rate (%)	0.7
Number of Parents for Crossover	2
Number of Offspring after Crossover	2

.

The implementation was carried out in Python 3.10.11, using the Pymoo library (version 0.6.0.1) [31] to execute the optimization algorithms. In addition, the PandaPower library (version 2.13.1) [32] was employed to perform the load flow calculations necessary to verify the operational constraints of the MG, ensuring the consistency of the obtained results.

The tests were conducted on an Ubuntu 22.04.5 LTS operating system, using a computer with an AMD Ryzen 9 7950X processor (Advanced Micro Devices, Santa Clara, CA, USA), featuring 32 threads running at 5.881 GHz and 32 GB of RAM.

3. Results

3.1. Comparing the Mathematical Formulation Regarding the O&M Costs

To assess the influence of O&M costs of NDGs on MG management, as outlined in Section 2.3, we compare the computational complexity of two mathematical formulations for calculating the total cost $\left(C_{total}\right)$ in microgrid management. Equation (2) includes three summations: one for DGs, one for NDGs, and one for other MG devices. Equation (3) simplifies the expression by removing the summation over NDGs, thereby reducing the number of operations required.

In Equation (2), for each time period t, the first summation over dispatchable generators $\left(i\right)$ involves two operations per generator (one for fuel costs and one for O&M costs). The second summation over non-dispatchable generators $\left(j\right)$ involves only one operation per generator (O&M costs), while the third summation over other MGs devices k also involves one operation per device.

The total number of operations for this formulation is proportional to the following:

$$W\ast (2D+N+M)$$

(7)

where W is the number of time periods, D is the number of dispatchable generators, N is the number of non-dispatchable generators, and M is the number of other MG devices.

For this simplified formulation (Equation (3)), for each time period j, the summation over dispatchable generators $\left(i\right)$ involves two operations per generator (fuel costs and O&M costs of DGs) and one operation for the other MG devices.

The total number of operations for this simplified formulation is as follows:

$$W\ast (2D+M)$$

(8)

Equation (2) requires more computational operations than Equation (3) due to the additional summation over N, which introduces extra operations. Equation (3), by excluding the NDGs costs, results in fewer operations and is computationally more efficient. This simplification in Equation (3) leads to a reduction in computational complexity, especially as the number of non-dispatchable generators increases.

To evaluate this, comparative tests were conducted under two distinct scenarios. In the first scenario, the O&M costs of NDGs were included (Equation (2)), while in the second scenario, these costs were excluded (Equation (3)). Fifteen simulations were run for each scenario, as detailed in

Table 3. Comparative results between Equations (2) and (3).

	Current Methodology (Equation (2))	Average	Proposed Methodology (Equation (3))	Average
1	$33,719.08	$32,311.52	$31,791.52	$32,229.64
2	$32,052.54		$32,374.59
3	$32,311.52		$32,158.35
4	$34,926.78		$33,006.85
5	$32,692.05		$31,843.69
6	$31,978.37		$35,063.20
7	$36,772.56		$32,198.60
8	$31,997.99		$32,556.67
9	$30,776.29		$32,229.64
10	$36,274.52		$35,596.39
11	$34,191.52		$32,076.50
12	$31,423.62		$35,233.22
13	$32,687.50		$34,070.44
14	$31,615.55		$31,126.25
15	$31,397.64		$32,156.72

. A Student’s t-test was conducted to assess the statistical significance of the differences between the results. This statistical test helps determine whether the observed differences in the means are significant, with p-values greater than 5% indicating no significant difference between the algorithms.

The results of the Studen’s t-test showed a p-value of 87.3%, indicating that there was no statistically significant difference between the values of the two scenarios. Thus, the calculation of O&M costs for NDGs during the optimization process proved to be unnecessary. Figure 3 illustrates the average evolution of both cases, highlighting the standard deviations of each group. Figure 3a shows the operational cost including O&M and Figure 3b excluding O&M cost. This visual representation reinforces the absence of discrepancies between the samples, showing that both approaches produce equivalent results.

The execution time comparison between the two approaches is presented in Figure 4, which depicts the average execution time across all 15 simulations for both methods. Although the inclusion of O&M costs introduces some additional computational overhead, the results indicate that the increase in execution time remains minimal across all runs (approximately 24.5 s). In environments with limited computational resources, this difference becomes significantly more pronounced, emphasizing the potential benefits of the proposed simplification in enhancing computational efficiency under constrained conditions.

3.2. Comparing the Cost vs. Emissions Relation

As presented in Section 2.4, two possible solutions to the specified problem are formulated in Equations (5) and (6). To demonstrate the effectiveness and implications of the proposed approaches, tests were conducted in two distinct scenarios: the first simulating a MG with only one DG, and the second scenario with three different polluting generators, each characterized by distinct cost and emission values. For each scenario, 10 simulations were performed for Equations (4)–(6), resulting in 30 executions for the case with a single polluting generator (three diesel generators). Additionally, 10 simulations were conducted using Equation (4) in a scenario with multiple polluting generators, consisting of a diesel generator, a biomass generator, and a natural gas generator.

Figure 5 displays the populations of all multi-objective executions applied to the mathematical formulations of Equations (4) and (5) in scenarios with a single generator and multiple polluting generators. A comparison with Equation (6) is not shown, as it specifically addresses the dispersion of the population in the context of Pareto front generation.

The results indicate that, in all cases involving Equation (4) with a single type of pollutant source (Figure 5a), a strong correlation between the objective functions was observed, as evidenced by the diagonal linear characteristic of the generated population. This relationship limits the applicability of this formulation in scenarios with a single source of pollutant generation. However, this limitation does not arise in the presence of multiple types of pollutant generation sources (Figure 5b), where the varying emission rates and costs ensure the multi-objective nature of the problem.

It is important to note that, even in cases with multiple types of pollutant generation sources, a linear trend can still be observed when compared to the results obtained with Equation (5). Nevertheless, this trend does not undermine the use of multi-objective optimization algorithms, thus making Equation (4), with the results presented in Figure 5c, suitable for these contexts.

With Equation (4) discarded in cases involving a single type of pollutant generation, we now analyze the comparative results between Equations (5) and (6) for this scenario. The results are presented in Figure 6, which shows the Pareto fronts obtained with Equation (5) (represented in blue) and the best results from the mono-objective executions of Equation (6) (indicated in red). For the latter, emission values were calculated later.

The results indicated that, in some cases, the mono-objective executions outperformed the multi-objective executions. This can be seen in Figure 7, where several multi-objective solutions are located within the dominance region of a mono-objective solution. However, this solution would not be fully dominant, as some solutions from the multi-objective executions would not be dominated by it. In summary, the findings suggest that both approaches, represented by Equations (5) and (6), are viable alternatives to Equation (4). These approaches offer more robust and consistent solutions in terms of performance and alignment with the established objectives.

4. Discussion

The analyses conducted were based on various mathematical formulations and optimization approaches, using computational tools to calculate energy dispatches and validate operational constraints. The objective of the tests was to evaluate the impact of the proposals on cost reduction, pollutant emissions, and computational efficiency, providing a critical view of the performance and applicability of the proposed solutions.

Regarding the inclusion of O&M costs for NDGs, the results indicated that adding these costs to the objective function during the optimization process did not significantly impact the obtained results, as evidenced by statistical analysis and visualizations in Figure 3. This finding suggests that it is feasible to exclude these fixed costs from the initial formulation, simplifying the mathematical model and reducing computational complexity without compromising the quality of the generated solutions. Therefore, the proposed approach proves to be an efficient alternative for managing isolated MGs.

Concerning the interaction between costs and emissions, the use of Equation (4) proved inefficient in the multi-objective optimization process with a single type of pollutant-based generator source, requiring modifications in the mathematical formulation to improve results. However, in scenarios with multiple types of polluting generator sources, the model remains applicable.

The modifications in the objective function expressed in Equations (5) and (6) proved to be viable alternatives to Equation (4), especially in the context of only one type of polluting generator source. The mono-objective approach demonstrated greater efficiency in terms of cost–benefit, due to its lower computational complexity and faster solution acquisition. On the other hand, the multi-objective approach offered a more diverse set of solutions, making it more suitable for scenarios that require greater control and flexibility in decision-making. Thus, the choice between approaches should be guided by the project’s priorities, such as simplicity and efficiency or solution diversity.

In summary, while the mono-objective approach is more suitable for fast and low computational cost solutions, the multi-objective approach remains essential for dealing with multiple criteria and heterogeneous generating sources. The final choice of the most appropriate approach should be based on the specific demands of each project, considering the previously defined technical and operational criteria.

Therefore, the proposed modifications provide a practical balance between computational efficiency and optimization flexibility. While the mono-objective formulation significantly reduces computational costs and accelerates solution acquisition, its applicability is primarily suited to scenarios that prioritize rapid decision-making over solution diversity. Conversely, the multi-objective approach, despite its higher computational demand, remains essential for cases requiring more refined trade-offs between costs and emissions, particularly in systems with heterogeneous generating sources.

Another advantage of the mono-objective approach is the reduction in the number of objectives, which decreases the complexity of the algorithms. If additional objectives are introduced, this method allows for a lower overall problem complexity, facilitating the optimization of MG management. Thus, these findings underscore the importance of selecting the appropriate optimization strategy based on the specific constraints and objectives of each microgrid implementation.

It is important to note that this article focuses exclusively on analyzing the relationship between cost and emissions in a single pollutant-based source MG, while other objectives—such as maximizing component lifespan, improving energy quality, and minimizing technical losses—are not addressed. The selection of these criteria is justified by their relevance to the economic feasibility and environmental impact mitigation of MG operation, with additional objectives potentially being incorporated in future studies.

As a continuation, the plan is to investigate the impact of these modifications in grid-connected MGs, where the interaction between the local generation and the main grid introduces additional operational constraints and optimization challenges. Additionally, future work will explore the modeling of other objectives, such as loss minimization, voltage control, and extending the lifespan of MG devices. By incorporating these aspects, the proposed methodology can be further refined to enhance system reliability, improve power quality, and optimize long-term operational efficiency. These advancements will contribute to expanding the applicability and robustness of the developed solutions, making them more suitable for a wider range of real-world MG configurations.

5. Conclusions

Efficient MG management relies on optimization techniques that balance economic and environmental objectives while overcoming challenges such as high computational costs, limitations in existing mathematical models, and the complexity of handling multiple objectives. This study proposed a simplified mathematical formulation for optimizing isolated MG management, improving computational efficiency without compromising solution quality. The results demonstrate that incorporating O&M costs for NDGs, such as photovoltaic and wind systems, does not interfere with the optimization process. This allows these costs to be calculated separately without affecting the results.

Additionally, the study analyzed the relationship between emissions and costs in pollutant-based generation. When the MG relies on a single pollutant-emitting source, such as diesel generators, modifications to the mathematical model are necessary to enable effective optimization. However, in scenarios with multiple pollutant-generating sources, the differences in cost and emission coefficients naturally create a multi-objective optimization problem, eliminating the need for additional modifications.

Overall, the findings of this study reinforce the importance of adapting optimization models to the specific characteristics of MG configurations. The proposed simplifications enhance computational efficiency, making optimization more accessible in resource-constrained environments, while still ensuring effective decision-making. Furthermore, the analysis of cost–emission interactions highlights the need for tailored mathematical formulations depending on the diversity of generation sources. Future work will focus on extending these models to grid-connected MGs and incorporating additional objectives, such as voltage regulation and component lifespan maximization, to further improve the robustness and applicability of MG optimization strategies.