Front. Energy Res. Frontiers in Energy Research Front. Energy Res. 2296-598X Frontiers Media S.A. 1739604 10.3389/fenrg.2025.1739604 Original Research A stochastic multi-objective enhanced energy management system in smart microgrid based on an improved grey wolf optimizer and reduced unscented transformation Heydari et al. 10.3389/fenrg.2025.1739604 Heydari Mohammad 1 Investigation Writing - original draft Software Visualization Resources Writing - review and editing Methodology Validation Formal Analysis Conceptualization Data curation Niknam Taher 1 Methodology Validation Writing - review and editing Visualization Project administration Supervision Dehghani Moslem 2 * Writing - review and editing Methodology Visualization Validation Conceptualization Data curation Formal Analysis Funding acquisition Investigation Resources Software Writing - original draft Garces Alvaro Hoffer 2 * Validation Methodology Writing - review and editing Funding acquisition Visualization Aly Mokhtar 2 Visualization Methodology Writing - review and editing Validation Funding acquisition Rodriguez Jose 2 Methodology Visualization Funding acquisition Validation Project administration Writing - review and editing Supervision Department of Electrical Engineering, Shiraz University of Technology, Shiraz, Iran Centro de Transición Energética (CTE), Facultad de Ingeniería, Universidad San Sebastián, Santiago, Chile Correspondence: Moslem Dehghani, ext.moslem.dehghani@uss.cl; Alvaro Hoffer Garces, alvaro.hoffer@uss.cl 02 02 2026 2025 13 1739604 04 11 2025 28 11 2025 05 12 2025 Copyright © 2026 Heydari, Niknam, Dehghani, Garces, Aly and Rodriguez. 2026 Heydari, Niknam, Dehghani, Garces, Aly and Rodriguez https://creativecommons.org/licenses/by/4.0/ This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

The aim of this article is to examine how energy management systems (EMSs) in smart microgrids (MGs) can be achieved with the increased use of renewable energy, energy storage systems (ESSs), and time-of-use tariffs, which introduce variability and uncertainty in the market, supply, and demand. As a result, operators will be able to decrease operation costs and pollutant emissions while improving the flexibility of energy in various conditions. In this regard, the mathematical model for an EMS is first developed, which considers the real-time energy pricing, wind turbines (WTs), photovoltaics (PVs), fuel cells, microturbines, and ESSs, as well as the energy sold to the grid by the smart MG. Second, a stochastic optimization framework was examined that accounts for system uncertainty using the reduced unscented transformation layout. In addition, by applying the penalty function approach, the balance between generation and consumption, as well as grid limitations, is taken into account. To address the system’s uncertainties, the stochastic optimization was used to model load demand, PVs, WTs, and market price uncertainties. To demonstrate the performance of the proposed framework, the suggested energy management is considered under different case studies, such as with and without ESS, and limitations on exchanged power with the grid. Different optimization algorithms, such as grey wolf optimizer (GWO), improved GWO (IGWO), whale optimization algorithm, particle swarm optimization (PSO), and improved PSO (IPSO), are applied to solve the suggested stochastic multi-objective optimization problem by using the weighting factor ratios. To solve a stochastic multi-objective problem with grid limitations and an ESS, IGWO achieves the best rank in finding the best solution of 2,366.88 with a low standard deviation of 72.43. In contrast, the best solutions of GWO, WOA, PSO, and IPSO are 2,676.76, 2,818.56, 2,640.87, and 2,439.42, respectively, with standard deviations of 15,083.78, 146,046.86, 4,352.86, and 403.57. The total cost and emission of the best solution of the stochastic optimization problem for IGWO are 893.93 cents/kWh and 736.48 kg/MWh, respectively. This shows that the IGWO clearly outperforms GWO, WOA, PSO, and IPSO. In addition, as shown in simulation results, this model reduces energy prices and environmental pollution, optimizes the MG operations, and demonstrates the effectiveness of IGWO. In addition, different weighting factor ratios are considered to assess the sensitivity of the results to the weighting factor ratios.

 energy management system improved grey wolf optimizer reduced unscented transform smart microgrid stochastic multi-objective problem Agencia Nacional de Investigación y Desarrollo 10.13039/501100020884 11251914 11230430 3250347 AFB240002 The author(s) declared that financial support was received for this work and/or its publication. This work was supported by La Agencia Nacional de Investigación y Desarrollo (ANID), Chile Fondo Nacional de Desarrollo Científico y Tecnológico (FONDECYT) de Postdoctorado 2025 under Grant 3250347, and in part by ANID, Chile FONDECYT Iniciacion under Grants 11251914 and 11230430. J. Rodriguez acknowledges the support of ANID through project CIA250006. section-at-acceptance Sustainable Energy Systems Introduction

Power systems are currently promoting a variety of renewable energy resources (RERs), energy storage systems (ESSs), and different eco-friendly energy management systems (EMSs) to provide pollution-free energy and decrease fossil fuel usage (Varasteh et al., 2019). RERs are producing more energy than they were previously (Varasteh et al., 2019; Ma et al., 2025; Le et al., 2024a), and ESSs in a smart power grid with EMSs and demand response programs can adjust the timing of load consumption by scheduling the charging/discharging of the ESSs (Scrocca et al., 2025). Hence, the power systems with ESSs and RERs can reduce pollutant emissions, thereby improving their technical, environmental, and financial aspects. Nevertheless, this is only applicable to EMSs utilized in the grid (Varasteh et al., 2019; Scrocca et al., 2025). In general, this field begins with a coordinated effort with a central organization. Due to the difficulty in coordinating the large number of elements with the power system operator, centralized frameworks for RERs, ESSs, and load demand are needed to enhance energy performance effectively. A microgrid (MG) comprises various types of RERs that enable the smart grid to efficiently provide power and meet customer demands. It is crucial to provide reliable energy to customers and loads in the smart grids (Shabanpour-Haghighi and Seifi, 2015; SHI et al., 2025; Li T. et al., 2025). Power is exchanged between the MG and the main grid to ensure provision of the required power. By redesigning the system, the costs and pollutant emissions can be reduced and optimized (Mewafy et al., 2024). Due to the uncertainties in RERs and their stochastic behavior, the energy management (EM) in an MG is more complex with a high computational burden.

On-grid MGs are widely examined in various studies. Mohammadi et al. (2012) examine MGs, including photovoltaic (PV) arrays, fuel cells (FCs), and batteries, alongside other distributed generators (DGs) under a pooled and hybrid electricity market layout. The genetic algorithm (GA) is used to solve the optimization problem, but the uncertainties of the RERs and load demand are not considered, and only the economic aspect is optimized. The uncertainties of RERs and load demand were forecasted using support vector machines, and then an enhanced teaching-learning-based optimization algorithm was applied to solve the optimization problem and minimize costs efficiently (Krishna and Hemamalini, 2024), but only the operating cost is considered as a single-objective problem. To reduce pollutant emissions, consider uncertainties, and also increase resiliency, the technologies, issues, and key aspects are reviewed for the MGs (Hirsch et al., 2018). An improved particle swarm optimization (PSO) algorithm is applied for solving an optimization problem in MGs (Guan et al., 2024), which considers operating prices and the environmental issues for an on-grid MG, but the uncertainties of RERs, load demand, and market pricing are not considered. Masoudi et al. (2025) considered a deterministic multi-objective optimization problem for the scheduling problem in power systems to optimize the objective functions, including the operator and environmental costs, by using an improved PSO algorithm. However, they did not investigate the stochastic behaviors of the RERs, demand, and market pricing. Khaloie et al. (2020) examined a nonlinear programming scheme to optimize the energy efficiency of cogeneration plants and minimize environmental impacts, but only the wind turbines (WTs) and ESS are considered. Al-Bahran and Abdulrasool (2021) used the modified ant colony algorithm to solve the constrained optimal power flow in a single-objective problem with respect to minimizing the total fuel cost, emission, power losses, and improving the voltage profile. Simulated annealing was used by Ekren and Ekren (2010) to optimize the size of a hybrid energy conversion system, which includes only a PV array, a WT, and an ESS, but the stochastic behaviors of the RERs, demand, and market pricing are not considered. An improved bat algorithm was examined by Bahmani-Firouzi and Azizipanah-Abarghooee (2014) and produced corrective methods and dispatches at the lowest cost, but the uncertainties are not included. A grid-connected MG evaluates the effectiveness of the suggested method and determines the ideal ESS size before implementation. In multi-MG clusters, multi-objective EMSs employing the Multi-Verse Optimizer are being developed, which improve distributed energy efficiencies while balancing costs and reliability (Aeggegn et al., 2025), and only the PVs, WTs, FCs, and ESSs are considered in the MG. Rajagopalan et al. (2022) applied a grey wolf optimizer (GWO) based on the oppositional gradient to solve a multi-objective problem for scheduling an on-grid MG. In addition, the Gaussian walk and Levy flight techniques are used to improve the exploration and exploitation abilities of the proposed optimization technique. Both operational costs and emissions are considered as objective functions, and the proposed method is shown to outperform other optimization algorithms. Ghiasi et al. (2021) examined a nonlinear equation of a multi-objective optimization problem, using different inequality and equality constraints to minimize the overall operating prices while taking into account pollution impacts on the environment by using an improved differential evolution algorithm, but the uncertainties of RERs, load demand, and market pricing are not considered. Zandrazavi et al. (2022) examined a stochastic multi-objective EMS for minimizing voltage deviations and total operational costs in a grid-connected MG. The epsilon-constraint and fuzzy methods are used to solve the optimization problem, but the environmental aspects, such as emission pollution, are not considered. Kumar and Karthikeyan (2024) used a golden jackal optimization algorithm to optimize energy dispatch in MGs in a deterministic multi-objective problem to minimize the cost and emissions. An additive relative decision-making method is examined by Chen et al. (2022) for optimizing the operational costs in grid-tied multi-energy MGs by taking into account the uncertainty, but the emission is not considered. Lu et al. (2017) optimized on-grid MGs through a two-phase optimization process that considers both operational and environmental features, and an improved PSO is used to solve the deterministic optimization problem, but the stochastic behaviors of the RERs, demand, and market pricing are not considered. Li et al. (2021) used a hybrid gravitational search PSO algorithm to solve a deterministic single-objective problem to optimize the operation of electric vehicles, loads, and RERs in an on-grid MG to reduce the MG’s total operational costs. Sun et al. (2022) used a bat algorithm with a fuzzy scheme to solve a multi-objective optimization approach to reduce the operational costs and emission pollutants of the MG and also overcome the uncertainties of forecasted wind power and schedule the generation power of distributed generation units. A grey wolf optimizer (GWO) was used by Miao and Hossain (2020) to optimize the sizing of ESSs in the MGs for determining the least operational costs. Le et al. (2024b) investigated the combustion performance of a diesel engine fueled with biomass gasification gas, and extensive experimental data were collected under various engine conditions (load, equivalence ratio, and compression ratio). Then, the response surface methodology and the GWO were used to simultaneously predict and optimize engine efficiency and emissions. In addition, the ideal engine setting with high accuracy is specified by combining experimental data and optimization schemes. Paramasivam et al. (2025) developed and investigated the performance of a biogas-fueled dual-fuel engine, which includes practical experiments under different engine conditions. The response surface methodology is considered for modeling and optimization. Predictive and analytical models have been created using Shapley additive explanations and extreme gradient boosting. In this study, by combining experimental data and advanced modeling methods, the optimal engine conditions and the role of key variables are accurately identified.

A grid-connected MG was examined to minimize the operating prices and pollution by a stochastic model with beta and Weibull distributions, and the one-to-one-based optimizer is used to solve the optimization problem (Fathy, 2025). The chaos theory is used by Zhao (2024) to overcome the uncertainties of RERs in the MGs, and the salp swarm optimization algorithm is used to solve the optimization problem in order to reduce the costs and pollutant emissions. The MG-EMS was considered using multi-objective artificial hummingbird algorithms to reduce operating cost and pollution and extend the lifespan of the batteries (Li LL. et al., 2025). The two-layer methods of cooperative MG operations were investigated to minimize costs and pollution and improve reliability (Moazzen and Hossain, 2025). A three-level stochastic EMS was also developed for interlinked MGs using hydrogen storage, which efficiently addresses the uncertainty of load demand and RERs. Particularly in cases that incorporated a high number of RERs, the method reduced costs and pollution significantly (Zhang et al., 2024).

EMSs are unable to efficiently manage the uncertainties in traditional power systems. Hence, it is essential to utilize stochastic optimization schemes to manage the uncertainties of the RERs and load demands (Alruwaili et al., 2025). By using the stochastic methods, the reliability and sustainability of the system are increased. As energy demand increases, it is essential to enhance the integration of RERs and ESSs. The high interconnectedness among systems by modern technology makes it challenging to find the best solution in an EMS for the smart MG that exchanges power with the main grid at time-of-tariff costs and with uncertainty about RERs, load demands, and market prices. Recent reports and studies show that renewable energy sources are highly variable. For example, PV generation in distribution networks can fluctuate by 18%–25%, and wind generation can vary by up to 20%–30% depending on weather conditions (Graabak and Korpås, 2016; Ringkjøb et al., 2020). Time-of-use (ToU) tariffs have significant differences of approximately 2–3 times between peak and off-peak periods (Liu et al., 2025). Uncertainties arising from electric vehicle charging and residential loads can cause total load deviations (Farhadi et al., 2024; Dehghani et al., 2025a). Traditional EMSs face difficulties in maintaining efficient performance under such variable conditions and uncertainties. Therefore, robust stochastic optimization methods are essential to overcome these uncertainties for EMSs of microgrids.

This article examines a stochastic optimization structure for optimizing an MG. A grid-connected MG has been proposed to exchange power according to market prices and ToU tariffs. An optimization approach is proposed to schedule microturbine (MT), FC, ESS, and the purchasing and selling power with the grid that uses the reduced unscented transformation (RUT) method for capturing the uncertainties of the loads, generation power of WTs and PVs, and the energy price based on ToU. It also uses an improved GWO (IGWO) algorithm for solving the proposed multi-objective stochastic optimization problem. Due to the growing difficulty of energy flow management in smart MGs with uncertainties of RERs and market price, the main contributions are summarized in the following:

PVs, WTs, FCs, MTs, ESSs, and the upstream grid are all integrated into the proposed EMS. The ability of the ESS to charge/discharge causes the load demand to shift into low tariff times, thereby reducing costs.

The energy cost based on the different costs of each generation unit, market price based on ToU tariffs, and also operating costs of distributed generation units are considered.

This article uses a multi-objective stochastic optimization problem using two different objective functions for optimizing the costs and pollutant emissions.

It uses the RUT approach for capturing the uncertainty of loads, PVs, WTs, and ToU tariffs of the market price.

Different optimization algorithms such as GWO, IGWO, whale optimization algorithm (WOA), PSO, and improved PSO (IPSO) are used to solve the suggested stochastic multi-objective optimization problem with and without grid limitation in exchanging power.

A multi-objective issue is analyzed to determine the impact of various weight factors on two objective functions: costs and pollutant emissions for economic and environmental aspects.

The study is organized as follows: Section 2 illustrates the problem formulation, which involves two objective functions: energy price and emission, along with their respective limitations. Section 3 discusses the RUT approach. Section 4 presents the IGWO to solve the suggested stochastic optimization problem. Section 5 discusses the test system and presents the simulation outcomes. Section 6 discusses the major conclusions.

 Problem formulation

The proposed MG shown in Figure 1 includes diverse DG units such as MTs, FCs, PVs, WTs, and ESS and is also connected to the main grid. In the proposed EMS, two objective functions are considered for minimization: operational cost and pollutant emission. An optimal stochastic multi-objective EMS is developed to minimize the above two objective functions while also satisfying all constraints.

Schematic of a smart MG.

Diagram of a microgrid system connected to a main grid. It includes a wind turbine (WT), photovoltaic system (PV), fuel cell (FC), microturbine (MT), and energy storage system (ESS). The bus numbers in the system are labeled from 1 to 16. Objective functions

Here, the two key objective functions, including costs and pollutant emissions, are illustrated in a stochastic optimization problem that should be minimized with respect to satisfying the constraints.

First objective function: costs

The total operational costs of the on-grid MG are considered as the first objective function, which should be minimized as a stochastic optimization problem for a proper energy management (EM) to schedule the operation of the generation units, ESS, and power exchanged with the main grid. This objective is mathematically expressed by Equation 1: O F 1 = min t o t a l   O p e r a t i o n a l   C o s t = ∑ i = 1 N p ∑ t = 1 T C t o t a l i t . P r o b a b i l i t y i ,

Where the total operation cost for each scenario can be computed by Equation 2: C t o t a l t = C W T t + C P V t + C F C t + C M T t + C E S S D i s c h − l o a d t + C F C S T / S H t + C M T S T / S H t + C G r i d B u y t − C E S S D i s c h − G r i d t .

Here, N p is the number of generated scenarios; Probability i defines the occurrence probability of each scenario; C total t is the total cost; C WT t , C PV t , C FC t , and C MT t define the costs of buying power from PVs, WTs, FCs, and MTs, respectively. C ESS Disch − load t presents the cost of buying power from an ESS to feed the loads. C FC ST / SH t , and C MT ST / SH t defines the startup and shutdown cost of the FCs and MTs, respectively. C ESS Disch − Grid t defines the cost of selling energy from ESS to the main grid. In this study, it is assumed that only an ESS can sell energy to the grid.

Based on the bid of each generation unit and ToU tariffs, the cost of each unit can be determined by Equations 3–11: C P V t = B i d P V * P P V t , C W T t = B i d W T * P W T t , C F C t = B i d F C * P F C t , C M T t = B i d M T * P M T t , C E S S D i s c h − l o a d t = B i d E S S * P E S S D i s c h − l o a d t , C F C S T / S H t = B i d F C S T / S H * m a x ⁡ 0 , U F C t − U F C t − 1 , C M T S T / S H t = B i d M T S T / S H * ⁡ max ⁡ 0 , U M T t − U M T t − 1 , C G r i d B u y t = T o U t * P G r i d B u y t , C E S S D i s c h − G r i d t = 1 − t a x T o U t * P E S S D i s c h − G r i d t ,

where Bi d PV , Bi d WT , Bi d FC , Bi d MT , and Bi d ESS represent the electricity bids of PVs, WTs, FCs, MTs, and ESSs, respectively. Bid FC ST / SH and Bid MT ST / SH define the startup/shutdown cost of the FC and MT, respectively. ToU t defines the ToU tariffs for power purchased from the grid. tax shows the amount of tax that the MG should pay for selling power to the grid. U FC and U MT are the on/off status of the FC and MT, respectively, and are 0 or 1. P PV t , P WT t , P MT t , and P FC t are the generated power of the PVs, WTs, MTs, and FCs, respectively. P ESS Disch − load t is the discharging power of the ESS that is used to feed the loads. P ESS Disch − Grid t shows the discharging power of the ESS that is sold to the grid. P Grid Buy t is power purchased from the grid.

 Second objective function: pollutant emissions

The goal of the MG-EMS is to minimize emissions, especially for systems that aim to be sustainable and environmentally friendly. The study focuses on four main sources of emissions: MT, FC, ESS, and the main grid. According to their operating features and emission factors, they all contribute to total emissions. Therefore, keeping pollution low is the second objective function, which is expressed by Equation 12; so key pollutants like C O 2 , S O 2 , and N O x are taken into consideration, resulting in O F 2 = min p o l l u t i o n = ∑ i = 1 N p ∑ t = 1 T E m i s s i o n t o t a l i t . P r o b a b i l i t y i ,

Where the total emission for each scenario can be achieved by Equations 13–17: E m i s s i o n t o t a l t = E F C t + E M T t + E E S S t + E G r i d t , E F C t = P F C t . E F C c o 2 + E F C N O x + E F C s o 2 , E M T t = P M T t . E M T c o 2 + E M T N O x + E M T s o 2 , E E S S t = P E S S D i s c h t . E E S S c o 2 + E E S S N O x + E E S S s o 2 , E G r i d t = P G r i d B u y t . E G r i d c o 2 + E G r i d N O x + E G r i d s o 2 , where Emission total t shows the entire emission that should be minimized; E FC t , E MT t , E ESS t , and E Grid t represent the pollutant emissions from the FC, MT, ESS, and grid, respectively; P ESS Disch t is the entire discharging power of the ESS. E FC c o 2 , E MT c o 2 , E ESS c o 2 , and E Grid c o 2 are the C O 2 emission of the FC, MT, ESS, and grid, respectively; E FC N O x , E MT N O x , E ESS N O x , and E Grid N O x show the N O x emission of the FC, MT, ESS, and grid, respectively; E FC s o 2 , E MT s o 2 , E ESS s o 2 , and E Grid s o 2 are the SO 2 emission of the FC, MT, ESS, and grid, respectively.

 Weighting factor-based multi-objective problem

The weighting factor approach has been applied to solve the proposed stochastic multi-objective problem to minimize the costs and pollutant emissions. The weighting factors should be determined in such a way that none of the objectives sacrifices another objective, and a balance between them is determined based on the policy of each operator. Equation 18 shows the proposed objective function: O F t o t a l = w 1 . O F 1 + w 2 . O F 2 + K B a l a n c e . ∑ t = 1 T P l o s s t + K L i m i t a t i o n . ∑ t = 1 T Δ P lim ⁡ i t a t i o n t .

Here, O F total shows the total objective function that should be minimized; w 1 , and w 2 defines the weighting factors of the first and second objective functions ( O F 1 , and O F 2 ), respectively; K Balance and K Limitation are the penalty ratios for the power balance constraint and power limitation of the grid, respectively; P loss is the lost power in the MG due to the excess generation of units over the load demand that can be achieved by Equation 19; Δ P limitation is the difference between the exchange power with the grid and the grid limitation, which can be calculated by Equation 20: P loss t = P WT t + P PV t + P FC t + P MT t + P Grid Buy t + P ESS Disch − load t − P load t − P ESS Charge t , Δ P limitation t = P Grid Buy t − P Grid max , if P Grid max < P Grid Buy t , Δ P limitation t = 0 ,   else ,

where P ESS Charge t is the charging ratio of ESS at time t . P Grid max is the maximum power that can be transferred between the grid and the MG. As we assumed that only the ESS can sell energy to the grid, the power injected into the grid is limited by the ESS’s discharging rate.

 Constraints

An overview of the main limitations necessary for operating MGs reliably and efficiently is provided in this section. Ultimately, the following constraints guarantee a balanced power between generation and consumption, a safe operation, and a commitment to operating limitations.

Power balance

The power demand for loads and charging the ESS should be supplied and be equal to the generated power by PVs, WTs, FCs, MTs, power purchased from the grid, and the discharging power of the ESS for the loads (Dehghani et al., 2025b; Chak et al., 2024), so the power balance constraint is achieved by Equation 21: P load t + P ESS Charge t = P WT t + P PV t + P FC t + P MT t + P ESS Disch − load t + P Grid Buy t .

Power constraint

The operation limitations for the generation units, such as WTs, PVs, FCs, and MTs, are presented by Equations 22–25: 0 ≤ P WT t ≤ P WT max , 0 ≤ P PV t ≤ P PV max , P FC min . U FC t ≤ P FC t ≤ P FC max . U FC t , P MT min . U MT t ≤ P MT t ≤ P MT max . U MT t ,

where P WT max , P PV max , P FC max , and P MT max are the maximum generation power of the WTs, PVs, FCs, and MTs, respectively. P FC min and P MT min show the minimum generated power of FCs and MTs, respectively.

 ESS

It is important to note that ESSs change their modes according to whether they are in a charge or discharge state; so, if they are charged, they are regarded as loads, and if they are discharged, they are regarded as energy resources. The charging and discharging rated power of the ESS can be written by Equations 26, 27, respectively: 0 ≤ P ESS Charge t ≤ U ESS Charge t . P ESS Charge − max , 0 ≤ P ESS Disch t = P ESS Disch − Grid t + P ESS Disch − load t ≤ U ESS Disch t . P ESS Disch − max .

Here, P ESS Charge t and P ESS Disch t show the charging and discharging amount of power in the ESS over the period t , respectively; U ESS Charge t and U ESS Disch t are the binary variables (0 or 1) that define the charging mode and discharging mode at period t , respectively; P ESS Charge − max and P ESS Disch − max define the maximum allowable charging and discharging amount at each period t in the ESS, respectively.

The ESS can be charged or discharged in each period; hence, the constraint that is shown in Equation 28 should be applied, which shows the status of the ESS in charging or discharging mode: 0 ≤ U ESS Charge t + U ESS Disch t ≤ 1 .

Equation 29 shows the model of the accumulated energy level of the ESS at time t (Dehghani et al., 2025b; Chak et al., 2024); and the SOC limitations of ESS is expressed by Equation 30: SoC ESS t = SoC ESS t − 1 + η P ESS Charge t − 1 η P ESS Disch S o C E s s min ≤ So C ESS t ≤ SO C ESS max where S o C E S S t shows the energy level of the ESS (state of charge) that is stored in it at the time t ; η shows the efficiency of charging and discharging of the ESS. S o C E s s min and S O C E S S max represent the minimum and maximum capacities of the ESS.

 Grid constraints

The power exchanged with the grid is limited by the allowable minimum and maximum amount at a time t (Dehghani et al., 2025b; Chak et al., 2024; Sun et al., 2024), which are expressed by Equations 31, 32: 0 ≤ P Grid Buy t ≤ U Grid t . P Grid max , 0 ≤ P Grid Sell t = P ESS Disch − Grid ≤ 1 − U Grid t . P Grid max , where P Grid Buy t is the power purchased from the grid at time t ; P Grid Sell t shows the power sold to the grid at the time t ; P Grid max shows the maximum transferred power between the grid and the MG. Power can only be sold to or purchased from the grid, so U Grid t is the grid buying or selling energy status at time t , which is a binary variable (0 or 1).

 Stochastic optimization approach for optimal scheduling of the smart MG

An optimization problem with uncertainty is analyzed in this part using a stochastic optimization method. As the smart MG generates a large number of calculations, it is necessary to develop powerful and fast approaches for addressing uncertainties and problem-solving. Uncertainty problems can be solved more quickly with the RUT approach (the speed has doubled). Therefore, the RUT method was applied for modeling market price uncertainty. The IGWO algorithm was used in addition to look for the optimal solution to the stochastic optimization problem.

RUT

It is possible to model uncertainty using three major techniques: Monte Carlo simulation (MCS), analytical methods, and approximate techniques (Shokri et al., 2024; Aien et al., 2012). Random variables in a problem are investigated using the MCS method to create various cases and solve the problem based on each of them. After that, statistical characteristics, such as mean values, are computed for decision variables and objective functions. Although the MCS is accurate, it is highly computationally intensive. This makes it unsuitable for short-term planning. Comparatively, analytical methods perform better in terms of computation but remain simplified due to some mathematical presuppositions (Shokri et al., 2024; Gupta et al., 2022). The unscented transformation (UT) is a method based on approximate methods that overcomes these limitations. Table 1 compares MSC, UT, and RUT in terms of computational burden, sample size requirements, and accuracy.

Comparison between the RUT and traditional MCS and UT methods.

Method Sample size Computational burden Accuracy
RUT Low Low High
UT High High High
MSC Very high Very high Very high

y = f x is a nonlinear problem in which X is a random input with an unknown number of parameters. A nonlinear function is defined by f , and an output vector is defined by y . The mean value is shown by μ , and P xx shows the covariance. UT uncertainties are modeled by solving 2 m + 1 times, similarly to the two-point estimation method. RUT, on the other hand, solves the problem m + 2 times. Due to this, RUT is quicker than UT and can solve complex stochastic optimization problems more quickly and easily. A step-by-step explanation of the RUT method is provided below (Shokri et al., 2024; Julier and Uhlmann, 2002).

Step 1. The free parameter W 0 is chosen in the range of [0,1].

Step 2. The weight sequence is chosen by: W i = 1 − W 0 m + 1 .

Step 3. The vector sequence is initialized by Equation 33:

 x 0 1 = 0 , x 1 1 = − 1 2 W 1 , x 2 1 = 1 2 W 1 .

Step 4. The vector sequence on k set is expanded by Equation 34:

 x i k = x 0 k − 1 0 , x i k − 1 − 1 k k + 1 W 1 , i = 1 , . . . , k , 0 k − 1 k k k + 1 W 1 , i = k + 1 .

Step 5. m + 2 input sigma points are given to the nonlinear function, and the output samples are obtained by: Y i = f x k .

Step 6. The mean and covariance matrix of the output Y are calculated as follows: μ k = ∑ i = 0 P W i y i and P yy = ∑ i = 0 P W i y i − μ i y i − μ i T .

Improved GWO GWO

Mirjalili et al. (2014) introduced the GWO algorithm in 2014 as a meta-heuristic optimization approach. Grey wolves have a hierarchical leadership structure and a similar hunting method is proposed in the GWO. There are four level of wolves in the leadership hierarchy: the alpha (the best candidate), the beta (the second-best candidate), the delta (the third-best candidate), and the omega (the other candidates). The algorithm consists of three main steps: searching, encircling, and attacking.

A grey wolf encircles its prey and moves toward it as part of its hunt. The Equations 35, 36 are used to model this behavior: D → = C →   X p → t − X → t , X → t + 1 = X p → t − A → . D . →

Here, t shows the current iteration. D → is the movement vector. X p → is the position vector of the target. A → and C → represent coefficient vectors. X → defines the position vector of the wolf. Equations 37, 38 are used to calculate the coefficient vectors ( A → and C → ): A → = 2 a →   . r 1 → − a → , C → = 2   . r 2 . →

Here, r 1 , r 2 have been selected at random within [0,1]. With each iteration, a → components decrease linearly from 2 to 0. Grey wolves are capable of approaching prey at random, repositioning themselves around it according to Equation 36.

The subsequent step involves updating the position of other search agents (like the omegas), based on information from the alpha (optimal solution), beta, and delta by Equations 39–41: D α → = C 1 → . X α → − X → , D β → = C 2 → . X β → − X → , D δ → = C 3 → . X δ → − X → , X 1 → = X α → − A 1 → . D α , → X 2 → = X β → − A 2 → . D β → X 3 → = X δ → − A 3 → . D δ , → , X H → t + 1 = X 1 → + X 2 → + X 3 → 3 .

Here, the subscripts of α , β , and δ show the alpha, beta, and delta wolves. a   → values are reduced from 2 to 0, and A → varies between − 2 a → and 2 a → . This means that decreasing a → would also result in a reduction in A → because wolves were forced toward prey with A → < 1 . A grey wolf follows the leader wolf, diverges from the leader wolf, and converges to attack and search for prey. When A → exceeds unity, wolves can diverge to search for food.

 Improvement scheme

Dehghani et al. (2025b) introduced the improvement in 2025. This improvement is inspired by the secondary competition between predators for prey, in which a piece of prey is stolen by other animals, like hyenas. This causes some variables to change again when finding a solution, increasing the diversity of the population and preventing the algorithm from getting stuck in a local minimum. In this improvement, it is assumed that three wild animals are trying to steal all or a piece of the prey; thus, Equation 42 is presented: R 1 → = r 3 X p − X R 1 → t , R 2 → = r 4 X p − X R 2 → t , R 3 → = r 5 X p − X R 5 → t .

Here, r 3 , r 4 , and r 5 are selected randomly in 0 , 1 ; X R 1 → t , X R 2 → t , and X R 3 → t show the wild animals that are attacking to steal a part or the entire prey, and these are chosen from the population members at random. Equations 43, 44 are presented to model this modification: X R → t + 1 = X R → t − b R . R 1 → + R 2 → + R 3 → , b R = 1 − t ite r max .

Here, b R shows a coefficient in the range of [0, 1], which, by increasing the iteration number, is reduced linearly from 1 to 0. It is assumed that the robber animals and the grey wolves have an equal chance to rob or save the prey, so a 50% chance is considered for each group; which is modeled by Equation 45: X → t + 1 = X H → t + 1 , if ch > 0.5 , X R → t + 1 , if ch ≤ 0.5 , where ch shows the chance of each group to have the prey that is selected randomly in the range of [0, 1].

 Simulation results

A low-voltage MG is examined to consider the proposed stochastic EMS. Figure 1 depicts a single-line diagram of the proposed MG test system, including various energy resources, such as PVs, WTs, FCs, MTs, and ESSs, that are connected to the grid (Guan et al., 2024; Chak et al., 2024). The normalized ToU tariffs, output power of PVs and WTs, and load power demand are shown in Table 2. The bid of each unit, pollution emission, and power limitation of each generation unit for the proposed MG components are presented in Table 3. The RUT method is considered to model the uncertainties of PVs, WTs, load demands, and ToU energy prices, and the input-generated scenarios are shown in Figure 2. Figures 2a,b show the generated scenarios of PVs and WTs, respectively. The load demand and ToU energy price scenarios are depicted in Figures 2c,d, respectively.

Normalized ToU tariffs, output power of PVs and WTs, and load power demand.

Hour ToU tariffs (cents/kWh) Output power of PVs (kW) Output power of WTs (kW) Load power demand (kW)
1 0.23 0 9.3397 50
2 0.19 0 8.7736 48
3 0.14 0 10.1887 48
4 0.12 0 11.3208 49
5 0.12 0 11.3208 53
6 0.2 0 10.7548 62
7 0.2 0 10.1887 67
8 0.2 5.3125 10.0472 73
9 1.5 15 10.3302 74
10 4 21.2500 10.8963 78
11 4 25 11.4623 75
12 4 22.815 11.8868 72
13 1.5 20.3125 13.8680 70
14 4 18.7500 15.0001 70
15 2 20.9375 14.8586 74
16 1.95 15.6250 14.4340 78
17 0.6 8.1250 13.7265 83
18 0.41 0.9375 13.3019 86
19 0.35 0 12.4529 87
20 0.63 0 11.8868 85
21 1.17 0 11.4623 76
22 0.64 0 11.0378 70
23 0.3 0 10.6133 62
24 0.26 0 10.1887 53

Test system details.

Unit Power (kW) Bid (cents/kWh) Startup/Shutdown cost (cents/kWh) Emission coefficient (kg/MWh)
Minimum Maximum C O 2 N O x S O 2
PV According to Table 2 2.584 0 0 0 0
WT 1.073 0 0 0 0
ESS Charging/discharging rate (kW) −30 30 0.38 0 10 0.001 0.0002
State of energy (kW) 60 300
FC 3 30 0.294 1.65 460 0.0075 0.003
MT 6 30 0.457 0.96 720 0.1 0.0036
Grid −30 30 According to Table 2 0 950 2.1 0.5

Input-generated scenarios: (a) PV generation scenario, (b) WT generation scenario, (c) load demand scenario, and (d) ToU energy price scenarios.

Four line graphs show scenarios over 24 hours. (a) PV Generation peaks around 11:00, (b) WT Generation fluctuates around 10-20 kW, (c) Load Demand peaks at 80-100 kW midday, (d) ToU Tariff spikes around 11:00 and 14:00. Each graph uses different colored lines to denote various scenarios.

The suggested stochastic multi-objective problem is considered under four case studies to present the performance of suggested method to solve the stochastic optimization problem: 1) suggested test system without grid limitations or an ESS, 2) suggested test system with an ESS and without grid limitations, 3) suggested test system with both an ESS and grid limitations, 4) Sensity analysis of suggested test system with an ESS and grid limitations under different weighting factors. All case studies are solved by IGWO, GWO, WOA, and PSO (cognitive and social constants are: C 1 = C 2 = 2 ; local constant ( W ) is 0.7), and IPSO ( C 1 = C 2 = 2 ; and W is linearly decreased from 0.9 to 0.1) algorithms over 30 independent runs, and they are compared with each other. For the first three case studies, the weighting factors, w 1 , and w 2 , are selected as 1 and 2, respectively. The penalty ratios: K Balance and K Limitation are selected as 1000. For the entire run, the population number is 150, and the maximum number of iterations is 2000. It is assumed that the initial energy level of the ESS at t = 1 is 180 KW.

The simulations are performed using MATLAB Software Package 2024a on an Intel Core i9-14900KF CPU, GeForce RTX 5090, and RAM 128 GB of 5600 MHz DDR5 memory, running Windows 11 Pro 64-bit.

Case Study 1: stochastic multi-objective problem: without grid limitations or ESS

In Case Study 1, the MG operation is considered without the ESS and grid limitation in exchanging power to feed the load demand. Minimizing the costs and emissions is the main goal of the stochastic optimization problem to schedule the MG without an ESS. Table 4 presents the results of the suggested stochastic multi-objective problem without an ESS or grid limitations over 30 independent runs and also illustrates the performance of IGWO compared to the GWO, WOA, PSO, and IPSO algorithms. As can be seen, IGWO achieves the best rank in finding the best solution with a low standard deviation of 24.83. The total cost and emission of the best solution of the stochastic optimization problem for IGWO are 1,238.41 cents and 715.82 kg, respectively. Table 5 shows the optimal stochastic scheduling of the FC, MT, and grid for the best solution of the IGWO algorithm. As can be seen in Table 5, the FC is active and on at all hours due to the low cost of energy production and pollution compared to the grid and the MT. However, the MT is off due to its higher price and emissions compared to the FC in 1–4, 11–15, and 23 h, and during these hours, due to the low cost of purchasing from the grid, the excess required energy is purchased from the upstream grid. However, due to the high pollution of the upstream grid, during the hours when the cost of purchasing power from the upstream grid increases, the EMS switches to purchasing from the MT, and it is turned on from 5:00 to 10:00, 16:00 to 22:00, and for 24 h. By combining purchasing from the MT and the upstream grid, the EMS tries to create a balance between costs and pollution.

Results of the proposed stochastic multi-objective problem without an ESS or grid limitations over 30 independent runs.

Algorithm IGWO GWO WOA IPSO PSO
Best O F T 2,730.06 2,868.75 3,176.96 2,907.84 2,979.80
O F 1 1,298.41 1,295.96 1,442.68 1,360.35 1,319.94
O F 2 715.82 786.40 867.14 773.74 829.923
Mean O F T 2,779.65 2,967.48 3,390.07 3,028.62 3,100.43
O F 1 1,308.31 1,331.23 1,539.02 1,375.26 1,385.28
O F 2 735.67 818.13 925.53 826.68 857.58
Worst O F T 2,837.84 3,086.19 3,534.48 3,189.35 3,286.13
O F 1 1,334.57 1,328.49 1,584.61 1,436.76 1,424.52
O F 2 751.63 878.85 974.94 876.29 930.81
Standard deviation O F T 24.83 58.73 80.17 75.01 73.23
O F 1 11.96 27.82 46.19 40.28 31.70
O F 2 9.33 23.02 29.20 24.863 27.66
CPU time 57.6907 56.9503 55.5323 49.5369 45.0644
Rank-based best results 1 2 5 3 4
Rank-based average results 1 2 5 3 4

Optimal stochastic scheduling of the smart MG in Case Study 1 for the best solution of the IGWO algorithm.

Hour MT (kW) FC (kW) Grid (kW)
1 0 29.80 10.86
2 0 29.94 9.29
3 0 29.99 7.83
4 0 29.98 7.70
5 7.16 29.95 4.56
6 16.86 29.81 4.57
7 10.85 29.89 16.07
8 23.64 30 4.01
9 13.45 29.96 5.25
10 9.22 29.88 6.76
11 0 30 8.54
12 0 30 7.30
13 0 30 5.82
14 0 30 6.25
15 0 29.98 8.22
16 12.35 30 5.59
17 25.41 30 5.74
18 29.97 29.93 11.86
19 29.72 29.98 14.84
20 30 30 13.12
21 29.99 29.99 4.56
22 24.82 30 4.14
23 0 29.97 21.42
24 8.56 29.92 4.33
Case Study 2: stochastic multi-objective problem: with an ESS and without grid limitations

In Case Study 2, the MG operation is considered with the ESS and without grid limitation in exchanging power to feed the load demand. Minimizing the costs and emissions is the main goal of the stochastic optimization problem to schedule the MG with ESS. Table 6 illustrates the results of the suggested stochastic multi-objective problem with an ESS and without grid limitations over 30 independent runs and also illustrates the performance of IGWO compared to the other algorithms. IGWO achieves the best rank in finding the best solution with a low standard deviation of 31.14. The total cost and emission of the best solution of the stochastic optimization problem for IGWO are 829.29 cents and 697.34 kg, respectively. By participating in the ESS, the costs and emissions are reduced to 409.12 cents and 18.48 kg, respectively, compared to Case Study 1. Table 7 shows the optimal stochastic scheduling of the FC, MT, ESS, power purchased from the grid, and power sold to the grid for the best solution of the IGWO algorithm. Figure 3 shows the load demand with and without the ESS, and the power sold to the grid. Figure 3 and Table 7 show that during the hours of 9:00 to 16:00, when the purchase and sale price with the upstream grid is high, the ESS is discharged to supply part of the load and also to sell electricity to the upstream grid to increase the income and reduce costs. During other hours when the sale price is low, the ESSis charged to supply part of the load during high price times to reduce costs and also to sell to the grid. Therefore, the ESS is charged during the low tariff periods and discharged to feed loads and sell power during the high tariff periods. The FC also produces close to the maximum power at all hours due to its low energy price and lower pollutant emissions than other units. On the other hand, because the energy purchase price and pollutant emissions of the MT relative to the grid are different at different hours, during hours when the upstream grid price is much lower, the MT is turned off, and the excess energy required is purchased from the grid. At times when the upstream grid price increases, and considering that the MT pollution is lower than the grid, the energy management system moves toward simultaneous purchase from the MT and the grid to maintain a balance between price and pollutant emissions.

Results of the proposed stochastic multi-objective problem with an ESS and without grid limitations over 30 independent runs.

Algorithm IGWO GWO WOA IPSO PSO
Best O F T 2,223.96 2,410.34 2,873.53 2,405.26 2,434.13
O F 1 829.29 894.61 1,153.31 872.04 872.82
O F 2 697.34 757.86 860.11 766.61 780.65
Mean O F T 2,284.26 2,506.55 3,163.97 2,623.38 2,729.48
O F 1 851.52 900.60 1,403.37 949.72 1,019.63
O F 2 716.37 802.97 880.30 836.83 854.93
Worst O F T 2,358.46 2,599.46 3,375.29 2,809.08 2,986.12
O F 1 901.95 876.74 1,601.53 993.35 1,262.50
O F 2 728.26 861.36 886.88 907.86 861.81
Standard deviation O F T 31.14 47.23 131.42 97.05 112.81
O F 1 19.44 31.99 116.83 62.42 97.66
O F 2 14.31 24.35 32.45 38.44 32.66
CPU time 75.5642 67.5248 64.7497 57.9698 55.6961
Rank-based best results 1 3 5 2 4
Rank-based average results 1 2 5 3 4

Optimal stochastic scheduling of the smart MG in Case Study 2 for the best solution of the IGWO algorithm.

Hour MT (kW) FC (kW) ESS (kW) Grid (kW)
Charging Discharging
Feed loads Sell to grid
1 0 29.82 20.38 0 0 31.21
2 0 29.97 2.08 0 0 11.34
3 0 30 0 2.88 0 4.93
4 0 30 15.801 0 0 23.49
5 16.67 29.84 13.28 0 0 8.45
6 16.98 30 0.11 0 0 4.38
7 22.49 29.89 0.95 0 0 5.38
8 28.08 29.88 4.15 0 0 3.84
9 8.34 29.95 0 10.32 0.87 0.06
10 8.87 29.96 0 7.02 22.95 0
11 0 30 0 8.54 21.41 0
12 0 30 0 7.30 22.67 0
13 0 29.99 0 5.77 1.14 0.06
14 0 29.99 0 6.26 23.74 0
15 0 30 0 8.20 9.67 0
16 0 29.85 0 17.94 0.09 0.16
17 25.81 29.99 0 0 0 5.34
18 29.95 29.98 0 0 0 11.83
19 29.26 29.98 0.04 0 0 15.35
20 29.85 29.81 0 0 0 13.45
21 29.92 29.97 0.020 0 0 4.66
22 24.61 29.97 0 0.05 0 4.34
23 17.17 29.97 0 0 0 4.24
24 0 29.99 0.04 0 0 12.86

Load demand of the proposed stochastic multi-objective problem with an ESS and without grid limitations.

Load demand power and sold power in kilowatts are on the y-axis and time in hours is on the x-axis. A blue line represents load demand without energy storage system (ESS), a red line shows load demand with ESS, and yellow bars indicate sold energy to the grid. Case Study 3: stochastic multi-objective problem: with ESS and grid limitations

In Case Study 3, the MG operation is considered with the ESS and grid limitation in exchanging power to feed the load demand. Minimizing the costs and emissions is the main goal of the stochastic optimization problem to schedule the MG with an ESS and grid limitations. Table 8 shows the results of the suggested stochastic multi-objective problem with an ESS and grid limitations over 30 independent runs, and also compares the performance of the IGWO with the other mentioned algorithms. As can be seen, IGWO achieves the best rank in finding the best solution with a low standard deviation of 72.43. The total cost and emission of the best solution of the stochastic optimization problem for IGWO are 893.93 cents and 736.48 kg, respectively. By considering the grid limitations, the costs and emissions are increased to 64.64 cents and 39.14 kg, respectively, compared to Case Study 2. Table 9 illustrates the optimal stochastic scheduling of the FC, MT, ESS, power purchased from the grid, and power sold to the grid for the best solution of the IGWO algorithm. Figure 4 shows the load demand with and without ESS, and the power sold to the grid. As shown in Figure 4 and Table 9, the ESS sells energy to the upstream grid during the hours of 9:00 to 16:00 when energy sales are high, to increase revenue, and also provides part of the load. By imposing restrictions on energy purchases from the upstream grid and also uncertainties in resources and load, the participation of microturbines in the energy supply increases.

Results of the suggested stochastic multi-objective problem with grid limitations and an ESS over 30 independent runs.

Algorithm IGWO GWO WOA IPSO PSO
Best O F T 2,366.88 2,676.76 2,818.56 2,439.42 2,640.87
O F 1 893.93 1,211.70 1,357.12 1,009.27 1,169.53
O F 2 736.48 732.53 730.72 715.07 735.67
Mean O F T 2,545.10 7,563.82 73,820.66 2,745.09 5,515.03
O F 1 1,055.14 1,427.67 2,760.88 1,163.52 1,498.63
O F 2 744.98 3,068.08 35,529.89 790.78 2008.20
Worst O F T 2,736.01 66,121.84 552,567.35 4,778.83 22,873.21
O F 1 1,269.00 2,674.26 12,293.81 1,028.33 1892.00
O F 2 733.50 31,723.79 270,136.77 1875.25 10,490.61
Standard deviation O F T 72.43 15,083.78 146,046.86 403.57 4,352.86
O F 1 85.47 333.73 2,939.85 130.68 152.51
O F 2 18.40 7,379.28 71,554.33 203.62 2,116.10
CPU time 52.2087 48.5298 47.7768 48.2651 40.4613
Rank-based best results 1 3 5 2 4
Rank-based average results 1 4 5 2 3

Optimal stochastic scheduling of the smart MG in Case Study 3 for the best solution of the IGWO algorithm.

Hour MT (kW) FC (kW) ESS (kW) Grid (kW)
Charging Discharging
Feed loads Sell to grid
1 7.52 29.95 7.41 0 0 10.60
2 0 29.91 0 3.19 0 6.12
3 18.99 29.97 25.76 0 0 14.60
4 0 30 8.83 0 0 16.51
5 0 30 7.19 0 0 18.87
6 27.03 0 0 8.44 0 15.78
7 28.94 29.99 12.95 0 0 10.82
8 16.42 29.70 0 6.49 0 5.04
9 30 0 0 18.61 1.28 0.06
10 29.26 9.66 0 6.93 23.07 0
11 29.53 0 0 9.00 20.99 0
12 0 29.67 0 7.63 22.37 0
13 0 30 0 5.09 0.06 0.73
14 29.90 0 0 6.35 23.53 0
15 0 30 0 7.09 0.07 1.11
16 17.73 23.79 0 6.37 2.54 0.06
17 21.65 29.95 0 1.58 0 7.97
18 29.98 29.78 2.09 0 0 14.09
19 29.67 28.70 0 1.17 0 15.01
20 28.82 29.97 0 0.38 0 13.94
21 29.97 30 0.27 0 0 4.84
22 19.74 29.90 0 2.19 0 7.13
23 29.95 17.17 0 0.01 0 4.26
24 0 30 0 0.02 0 12.79

Load demand of the suggested stochastic multi-objective problem with grid limitations and an ESS.

Load demand power and sold power in kilowatts are on the y-axis, and time in hours is on the x-axis over 24 hours. A blue line represents load demand without an energy storage system (ESS), a red line shows load demand with ESS, and yellow bars indicate sold energy to the grid. Case Study 4: sensitivity analysis under different weighting factors

In Case Study 4, the proposed stochastic optimization problem has been solved to schedule the operation of the MT, FC, ESS, and exchanged power with the grid for diverse weighting factors , w 1 and w 2 , to consider the effects of the proposed objective functions. Table 10 presents the efficiency of the IGWO, GWO, WOA, IPSO, and PSO algorithms in solving the proposed stochastic optimization problem for scheduling the MG with diverse weighting factors. As can be seen in Table 10, the performance of the IGWO in finding the best solution for the suggested stochastic optimization problem is better than the other mentioned optimization algorithms with a lower standard deviation.

Results of the proposed stochastic multi-objective problem with grid limitations and an ESS over 30 independent runs for different weighting factors.

w 1 w 2 Algorithm Best result Worst result Average Standard deviation
O F T O F 1 O F 2 O F T O F 1 O F 2 O F T O F 1 O F 2 O F T O F 1 O F 2
1 0 IGWO 777.56 777.56 855.03 1,105.76 1,105.76 836.76 907.33 907.33 823.40 71.14 71.14 24.52
GWO 1,093.33 1,093.33 870.15 1,501.13 1,501.13 2,614.41 1,235.18 1,235.18 1,470.52 106.7045 106.70 866.52
WOA 1,178.97 1,178.97 1,041.50 6,668.812 6,668.81 138,909.86 2,226.97 2,226.97 24,418.11 1,572.07 1,572.07 37,899.21
IPSO 896.92 896.92 813.34 8,216.73 8,216.730 171,506.03 1,296.21 1,296.21 6,564.66 1,293.54 1,293.54 30,629.62
PSO 1,104.17 1,104.17 791.02 13,131.42 13,131.42 285,483.31 1,779.25 1,779.25 11,686.49 2,114.46 2,114.46 50,885.13
1 1 IGWO 1,647.99 863.936 784.05 1974.48 1,239.76 734.71 1,782.41 1,013.34 769.06 81.80 84.29 15.49
GWO 1817.79 1,073.81 743.98 3,754.26 1,421.68 2,332.58 2,444.57 1,294.40 1,150.17 632.54 78.09 611.77
WOA 1941.71 1,199.18 742.53 364,748.37 15,825.43 348,922.94 48,740.70 48,740.70 45,601.84 96,835.46 3,894.15 92,942.46
IPSO 1,660.56 886.11 774.45 329,984.39 14,466.93 315,517.46 12,927.03 1,575.99 11,351.04 58,877.53 2,397.57 56,483.31
PSO 1969.62 1,201.76 767.85 181,675.58 8,745.62 172,929.95 9,662.45 1,730.61 7,931.84 32,061.20 1,315.43 30,749.68
1 2 IGWO 2,366.88 893.93 736.48 2,736.01 1,269.00 733.50 2,545.10 1,055.14 744.98 72.43 85.47 18.40
GWO 2,676.76 1,211.70 732.53 66,121.84 2,674.26 31,723.79 7,563.82 1,427.67 3,068.08 15,083.78 333.73 7,379.28
WOA 2,818.56 1,357.12 730.72 552,567.35 12,293.81 270,136.77 73,820.66 2,760.88 35,529.89 146,046.86 2,939.85 71,554.33
IPSO 2,439.42 1,009.27 715.07 4,778.83 1,028.33 1,875.25 2,745.09 1,163.52 790.78 403.57 130.68 203.62
PSO 2,640.87 1,169.53 735.67 22,873.21 1,892.00 10,490.61 5,515.03 1,498.63 2008.20 4,352.86 152.51 2,116.10
2 1 IGWO 2,501.46 859.71 782.03 3,137.57 1,179.32 778.94 2,778.68 998.50 781.68 179.35 91.27 16.03
GWO 2,825.91 1,017.74 790.43 6,352.35 1,424.80 3,502.74 4,065.39 1,319.37 1,426.64 911.99 116.99 781.77
WOA 3,168.02 1,209.16 749.70 645,502.09 25,886.64 593,728.82 64,232.78 3,660.58 56,911.63 148,714.74 5,716.485 137,283.30
IPSO 2,594.65 896.01 802.63 109,868.25 5,164.15 99,539.95 6,614.48 1,235.87 4,142.73 19,179.01 742.81 17,716.12
PSO 2,982.50 1,108.52 765.45 21,450.87 2,378.33 16,694.21 5,384.51 1,504.53 2,375.46 3,658.97 224.22 3,259.03
0 1 IGWO 647.15 1,293.64 647.15 1,490.27 1,396.28 1,490.27 704.87 1,354.65 704.87 146.82 34.60 146.82
GWO 668.70 1,367.40 668.70 3,181.19 1,560.22 3,181.19 1,054.72 1,419.93 1,054.72 667.43 60.41 667.43
WOA 736.89 1,345.39 736.89 177,282.70 8,681.34 177,282.70 18,605.39 2,054.74 18,605.39 35,541.98 1,467.52 35,541.98
IPSO 663.79 1,278.24 663.79 171,441.04 8,633.36 171,441.04 6,423.91 1,627.22 6,423.91 30,643.55 1,302.13 30,643.55
PSO 703.79 1,337.62 703.79 9,069.65 1,773.54 9,069.65 1,674.42 1,533.94 1,674.42 1,890.77 117.80 1,890.77

When the pollution emission weighting factor ( w 2 ) is 0, it means that the proposed stochastic optimization problem is a single-objective problem, with minimizing costs as the primary goal. Consequently, the costs decrease, but the pollution emissions increase. When the weighting factor of costs ( w 1 ) is 0, it means that the proposed stochastic optimization problem is a single-objective problem to minimize the emissions; thus, without the cost objective function, the emission is the primary goal, and it is decreased, and the costs are increased compared to the multi-objective problem.

 Statistical analysis

The standard deviation and mean indices can be used to evaluate the results of the objective functions obtained from optimization algorithms; however, due to the random behavior of evolutionary algorithms, an algorithm’s solutions may be better than those of other methods after several runs. Therefore, the use of statistical analysis methods such as the Wilcoxon rank sum test is necessary to show the efficiency and superiority of one algorithm over other evolutionary algorithms (de Barros et al., 2018). In the Wilcoxon rank sum test, the p-value is used as an indicator of the statistical significance of the differences between two algorithms, and a p-value less than 0.05 indicates the superiority of the algorithm over the competing algorithms. The p-value of the proposed IGWO algorithm over other mentioned algorithms is shown in Table 11 for all case studies and is less than 0.05, which indicates the superiority of the proposed algorithm over the other mentioned methods.

p-values obtained based on the Wilcoxon rank sum test.

Case study IGWO vs. GWO IGWO vs. WOA IGWO vs. IPSO IGWO vs. PSO
Case Study 1 3.0199 × 10−11 3.0199 × 10−11 3.0199 × 10−11 3.0199 × 10−11
Case Study 2 3.0199 × 10−11 3.0199 × 10−11 3.0199 × 10−11 3.0199 × 10−11
Case Study 3 5.4941 × 10−11 3.0199 × 10−11 1.0407 × 10−04 3.6897 × 10−11
Case Study 4 w 1 = 1 ;   w 2 = 0 4.0772 × 10−11 3.0199 × 10−11 7.2208 × 10−06 3.3384 × 10−11
w 1 = 0 ;   w 2 = 1 4.4440 × 10−07 4.5043 × 10−11 3.6709 × 10−03 4.6159 × 10−10
w 1 = 1 ;   w 2 = 1 9.9186 × 10−11 3.0199 × 10−11 3.8307 × 10−05 3.0199 × 10−11
w 1 = 2 ;   w 2 = 1 5.5727 × 10−10 4.5043 × 10−11 1.6798 × 10−03 6.0658 × 10−11
Conclusion

An optimal stochastic EMS is developed in this study for scheduling the operation of diverse energy resources in an MG, including PVs, WTs, FCs, MTs, and ESSs, that can exchange power with the grid in order to feed the loads. The primary goals of the proposed stochastic optimization problem are to minimize the costs and emissions. For this purpose, four important case studies, including 1) an MG without an ESS or grid limitation, 2) an MG with an ESS and without grid limitation, 3) an MG with an ESS and grid limitation, and 4) various weighting factors, are considered by using the IGWO algorithm to solve the optimization problem. In addition, the performance of IGWO is compared to the GWO, WOA, PSO, and IPSO algorithms in finding the best solutions. The bid cost of PVs, WTs, ESSs, FCs, MTs, the startup/shutdown cost of the FC and the MT, and ToU tariffs for the exchanged power with the grid are taken into account in the suggested stochastic EM model. In addition, the power sold by the MG to the larger grid has been evaluated, and the RUT scheme is used to consider the uncertainties of the WTs, PVs, load demands, and ToU tariff.

The results of four case studies demonstrate that the performance of IGWO in solving the proposed stochastic optimization problem surpasses that of the GWO, WOA, PSO, and IPSO algorithms. The average and standard deviation of best solutions in 30 independent runs of IGWO are lower than those of the other methods, which shows the effectiveness and robustness of the IGWO algorithm in solving the proposed stochastic optimization problem. Statistical analysis is done by the Wilcoxon rank sum test, and the obtained p-value for the IGWO versus the GWO, WOA, IPSO, and PSO is less than 0.05, which shows the superiority of the proposed IGWO algorithm over the competing algorithms. The suggested stochastic scheme ensures the adaptability of the MG in the face of the uncertainties in the RERs, load demand, and ToU tariffs. Comparing Case Study 1 and 2 shows that by using the ESS, the costs and emissions are reduced by 409.12 cents (33.03%), and 18.48 kg (2.58%), respectively. Comparing Case Study 2 and 3 shows that by applying the grid limitations, the costs and emissions are increased by 64.64 cents (7.79%), and 39.14 kg (5.61%), respectively.

In future research, the model can be extended to the standard distribution systems that include several types of renewable energy resources, a hydrogen storage system, transportation systems, and smart buildings. It can consider different objectives, such as grid stability, reliability, security, and power losses. In addition, the system can include multiple microgrids that can communicate with each other in a secure cloud-fog computing scheme. In addition, deep machine learning methods can be used to estimate the RERs generation and forecast the loads.

 Data availability statement

The original contributions presented in the study are included in the article/supplementary material, further inquiries can be directed to the corresponding authors.

 Author contributions

MH: Investigation, Writing – original draft, Software, Visualization, Resources, Writing – review and editing, Methodology, Validation, Formal analysis, Conceptualization, Data curation. TN: Methodology, Validation, Writing – review and editing, Visualization, Project administration, Supervision. MD: Writing – review and editing, Methodology, Visualization, Validation, Conceptualization, Data curation, Formal analysis, Funding acquisition, Investigation, Resources, Software, Writing – original draft. AG: Validation, Methodology, Writing – review and editing, Funding acquisition, Visualization. MA: Visualization, Methodology, Writing – review and editing, Validation, Funding acquisition. JR: Methodology, Visualization, Funding acquisition, Validation, Project administration, Writing – review and editing, Supervision.

 Conflict of interest

The author(s) declared that this work was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

 Generative AI statement

The author(s) declared that generative AI was not used in the creation of this manuscript.

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 Publisher’s note

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Edited by: Abdullahi Mas’Ud, Jubail Industrial College, Saudi Arabia

Reviewed by: Prabhu Paramasivam, Mattu University, Ethiopia

Songyu Jiang, Rajamangala University of Technology Rattanakosin, Thailand

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