Meta-Heuristic Optimization for Hybrid Renewable Energy System in Durgapur: Performance Comparison of GWO, TLBO, and MOPSO

Abstract

This paper aims to find an efficient optimization algorithm to bring down the cost function without compromising the stability of the system and respect the operational constraints of the Hybrid Renewable Energy System. To accomplish this, MATLAB simulations were carried out using three optimization techniques: Grey Wolf Optimization (GWO), Teaching–Learning-Based Optimization (TLBO), and Multi-Objective Particle Swarm Optimization (MOPSO). The study compared their outcomes to identify which method yielded the most effective performance. The research included a statistical analysis to evaluate how consistently and stably each optimization method performed. The analysis revealed optimal values for the output power of photovoltaic systems (PVs), wind turbines (WTs), diesel generator capacity (DGs), and battery storage (BS). A one-year period was used to confirm the optimized configuration through the analysis of capital investment and fuel consumption. Among the three methods, GWO achieved the best fitness value of 0.24593 with an LPSP of 0.12528, indicating high system reliability. MOPSO exhibited the fastest convergence behaviour. TLBO yielded the lowest Net Present Cost (NPC) of 213,440 and a Cost of Energy (COE) of 1.91446/kW, though with a comparatively higher fitness value of 0.26628. The analysis suggests that GWO is suitable for applications requiring high reliability, TLBO is preferable for cost-sensitive solutions, and MOPSO is advantageous for obtaining quick, approximate results.

1. Introduction

The merging of HRES with conventional electricity networks serves as a promising approach to a functional and current trend in energy engineering. The hybrid system configuration delivers substantial improvements in power supply stability where renewable energy sources experience fluctuations. Off-grid systems that combine PV panels with WT stand out as powerful solutions among renewable technologies by effectively meeting local power needs [1]. Recent studies increasingly focus on how fossil fuel extraction and combustion contribute to environmental and climate issues, including global warming and air pollution. The Earth contains abundant renewable energy resources that can satisfy the growing global energy demand for future generations despite current limitations [2]. Certain renewable energy (RE) sources produce power that lacks consistent output quality because of their inherent variability [3]. The complete substitution of fossil fuels with renewable resources presents a substantial challenge because it requires merging different energy sources. Regional energy requirements need multiple sources, ranging from renewables like wind, solar, and hydropower to sources like biomass, geothermal, nuclear, hydrogen, and fossil fuels, to be used together in complementary ways instead of operating separately [4]. HRE systems achieve higher reliability through the combination of different renewable elements, including photovoltaic panels and wind turbines [5,6,7]. RE sources deliver multiple benefits, but they also come with distinct drawbacks. At times, energy expenses become substantial, while power generation stability and reliability problems often arise. The ability to predict long-term energy production remains difficult because atmospheric conditions that impact solar and wind energy sources are unpredictable [8,9]. The integration of battery banks into hybrid systems helps maintain efficiency and addresses energy deficits during periods of low solar radiation or wind speed, as well as high energy demand. Battery banks must be used to maintain a continuous power flow [10,11]. Hybrid systems combining multiple energy sources for cogeneration provide a potential answer to the problem of inconsistent power production. The variability of renewable energy in hybrid systems can be managed through energy storage technologies spanning electrochemical and mechanical storage solutions, such as batteries, FC, flywheels, compressed air, and hydro storage [12]. Researchers have investigated multiple system configurations to enhance HRE system performance. For a location in Ghardaia, Algeria, researchers created a hybrid off-grid power system with PVs, WTs, diesel generators, and battery storage (BS), which then underwent cost effective and reliable techno economic optimization. An iterative approach to optimization was adopted for predicting the optimal configuration to achieve the lowest possible cost. To avoid energy deficits, a technical and economic assessment analysis was performed through a reliability model measuring the total energy shortfall, combined with an economic evaluation structure that assesses the net present cost and cost of electricity. The best system configuration identified in this study for the combination of PV/wind/diesel/battery hybrid generator includes an 8.5 kW solar array, a 1 kW WT unit, a DG with 4.2 kVA output, and 86.4 kWh battery storage capacity [13]. Diab et al. published a 2020 study [14] about modelling grid-connected hybrid systems bringing together wind and solar energy alongside pumped storage solutions. The model demonstrates a hybrid grid-connected system that integrates wind energy generation with photovoltaic technology and pumped storage solutions. The optimal system configuration was identified through the application of multiple optimization algorithms within MATLAB R2017b that were customized for the case study’s unique parameters. The results from the simulation show that the Whale Optimization Algorithm (WOA) achieved a superior performance by minimizing both the COE and LPSP among all tested algorithms. The authors, in their paper [15], discussed an HRE setup designed for household use, combining solar panels, wind turbines, batteries, heat pumps, solar collectors, and insulation. GA was used to find an efficient balance between cost and environmental impact. Unlike many earlier studies, this work includes a full life cycle analysis using the Product Environmental Footprint (PEF) method to capture a wide range of environmental effects. The findings highlight that the cheapest system design often differs from the most eco-friendly one. Among the components, solar PV was found to be the most effective in cutting emissions, while the benefits of batteries and solar collectors were limited when a grid connection was available. The Energy Filter Algorithm (EFA) was implemented to achieve both cost efficiency and system reliability optimization [16]. In their study [17], the authors present a hybrid photovoltaic fuel cell (PV-FC) system integrated with an electrolyzer and hydrogen storage for green hydrogen and power generation. A techno-enviro-economic analysis was performed using MATLAB/Simulink (R2021b), and key system parameters were optimized using Gaussian Process Regression (GPR) in combination with NSGA-II and TOPSIS. The objective was to maximize fuel cell output and emission reduction while minimizing cost and stack area. The optimal configuration achieved a stack power of 1589 kW, reduced CO₂ emissions by 1268 tons, and maintained the levelized cost of energy (LCOE) below 2 USD/kWh, highlighting the system’s potential for reliable and sustainable energy production. GA-PSO was used as a single-objective method in the optimization process and a multi-objective PSO-based approach. Both algorithms aimed to improve system reliability while reducing total costs [18,19]. The HOMER(Pro v3.6) tool performed a simulation and techno-economic analysis to determine the best configuration for an HRES. PSO provides a superior configuration alternative for HRES planning despite HOMER’s popularity as a planning tool [20,21]. A techno-economic analysis of a green hydrogen production system powered by onshore wind farms in Al-Bida, Saudi Arabia, was carried out using HOMER Pro. The design incorporates 26 wind turbines (160 MW total), a 120 MW alkaline electrolyzer, and 300 tons of hydrogen storage. The system achieved a levelized cost of hydrogen (LCOH) of 5.26 USD/kg and maintained high reliability with a minimal unmet load. Sensitivity analysis indicated that wind speed, demand level, and hydrogen technology costs are key factors affecting the overall project cost. The study concluded that with proper system sizing and supportive policies, wind-powered hydrogen production at a large scale is both technically viable and economically attractive [22]. PSO, GA, and HOMER methods were compared in the research to determine the most efficient design for the PV/biomass/hydro/DG HRE system. The PSO optimization method produced the minimum COE among all the investigated techniques [23]. The ideal setup for hybrid energy systems has been identified by researchers through the use of multiple optimization approaches utilizing evolutionary and swarm intelligence methods. The success of optimization techniques, including HBMO [24], Cuckoo Search [25], MOSaDE [26], Pattern Search, and SMCS [27], has been shown to vary based on the complexity of the system and its objectives. Recent research applied cutting-edge meta-heuristic techniques like FFA, MFFA [28], and MSOA [29] to develop optimal hybrid energy systems. An optimized off-grid hybrid system with WT and PV panels and hydrogen storage units has been suggested to provide reliable, sustainable water production for desalination purposes. A stepwise iterative approach is employed to fine-tune the dimensions of all system components, such as solar PV arrays, WT systems, hydrogen storage tanks, FC, electrolyzers, and the desalination unit. One of the primary objectives of designing the hybrid system is to minimize energy storage requirements as well as total expenses while assuring a constant power supply. The rich solar and wind energy sources in the Kerkennah Islands have decreased the necessary hydrogen tank capacity, which improved system reliability [30]. Optimal sizing of a stand-alone hybrid renewable energy system (PV, wind, biomass, and battery) using PSO, Harmony Search, and Jaya algorithms was studied in [31]. Among these, the Harmony Search algorithm achieved the best performance, with the lowest energy cost (0.254 USD/kWh), total system cost (USD 581,218), and a 91% reduction in GHG emissions. It also demonstrated faster convergence and more consistent results, making it the most effective optimization method in the study. The GWO algorithm mimics grey wolves’ hunting patterns and social behaviour to achieve both easy implementation and strong exploitation capabilities [32]. GWO has become increasingly popular since 2015 because it produces superior results compared to GA and PSO. However, GWO has difficulty achieving optimal performance in proximity to local optima and experiences slow convergence rates. Khobaragade et al., in their paper [33], explored an enhanced method for solving the Economic Load Dispatch (ELD) problem by applying the TLBO algorithm to a system with ten thermal power units. It studied how Plug-in Electric Vehicles (PEVs) affect load dispatch under different charging patterns, including overall, off-peak, peak, and random scenarios. By including these charging models in the dispatch plan, the work treated PEVs as flexible resources that help manage power demand more effectively. The results showed that the TLBO-based method can achieve cost savings and stable system operation when PEV integration was considered. In another study [34], the authors looked at ways to improve electricity access in rural areas using isolated microgrids that relied on local renewable energy sources. To manage energy use more effectively, they introduced a local system where people could buy or sell extra electricity within a group of microgrids. They also developed a new method called Modified JAYA Learning-Based Optimization (MJLBO), which combined ideas from two existing algorithms to avoid common problems like becoming stuck in poor solutions. This approach helps households share energy better, lower their power bills, and reduce inconvenience during peak times. The results showed that MJLBO performs better than other methods, cutting the peak-to-average load ratio by 65.38%. Researchers increasingly point out that using optimization models guided by economic indicators can lead to energy systems that are not only more cost-effective but also more sustainable over the long term [35,36,37].

Table 1 provides an illustration of the methods for optimization employed in HRES. RE systems experience a higher power output during sunny or windy days because their efficiency is closely influenced by meteorological conditions. The optimization of RE sources requires an extensive analysis of the technological and economic feasibility of the hybrid system. Designers of hybrid energy systems are increasingly implementing optimal sizing methods. The unique strengths and weaknesses of each artificial intelligence optimization algorithm result in an ongoing search for the perfect solution that might never reach a conclusion. The effectiveness of an algorithm is closely linked to the features of the problem it tries to solve. To find the most effective solution for optimization problems, various algorithms are typically applied. Many studies test optimization techniques by analyzing data from just one day without conducting a substantial statistical analysis.

Table 1. Comparison of some optimization methods used.

Ref No	Optimization Method	System Components	Methods Compared	Objective Function	Findings
[38]	ALO, GWA	PV/WT/SB/GT	CS, FPA	Total cost annually and system emission	GWO and ALO perform better than CS and FPA.
[39]	GWA	PV/WT/SB	PSO	Minimize TNPC	GWO is more cost-efficient and reliable than PSO.
[40]	GWA	PV/WT/BM	GA/SA	Minimize TNPC/LCOE	GWA delivers superior results compared to GA and SA.
[41]	FPA	PV/WT/FC/HS	TLBO/PSO	Minimize TNPC	Simplifies implementation, decision-making, and accelerates convergence.
[42]	TLBO	PV/SB	GA, PSO	Minimize NPC, COE	TLBO excels over GA and PSO in solving efficiency.
[43]	GA-PSO, MOPSO	PV/WT/SB	HOMER	Minimize TPC, maximize reliability	MOPSO delivers a more extensive set of optimal points while optimizing two objectives together.
[44]	TLBO	PV/WT/BM/SB/VDG	BFSO, GA, PSO	Minimize COE, LPSP, and PMI, maximizing RF, HDI, and JCI	TLBO outperforms in all areas.

Table 1. Comparison of some optimization methods used.

Ref No	Optimization Method	System Components	Methods Compared	Objective Function	Findings
[38]	ALO, GWA	PV/WT/SB/GT	CS, FPA	Total cost annually and system emission	GWO and ALO perform better than CS and FPA.
[39]	GWA	PV/WT/SB	PSO	Minimize TNPC	GWO is more cost-efficient and reliable than PSO.
[40]	GWA	PV/WT/BM	GA/SA	Minimize TNPC/LCOE	GWA delivers superior results compared to GA and SA.
[41]	FPA	PV/WT/FC/HS	TLBO/PSO	Minimize TNPC	Simplifies implementation, decision-making, and accelerates convergence.
[42]	TLBO	PV/SB	GA, PSO	Minimize NPC, COE	TLBO excels over GA and PSO in solving efficiency.
[43]	GA-PSO, MOPSO	PV/WT/SB	HOMER	Minimize TPC, maximize reliability	MOPSO delivers a more extensive set of optimal points while optimizing two objectives together.
[44]	TLBO	PV/WT/BM/SB/VDG	BFSO, GA, PSO	Minimize COE, LPSP, and PMI, maximizing RF, HDI, and JCI	TLBO outperforms in all areas.

To address this research gap, Grey Wolf Optimizer (GWO), Multi-Objective PSO (MOPSO), and Teaching–Learning-Based Optimization (TLBO) were used to develop a hybrid system capable of utilizing real-world data collected over a full year for a case study conducted in Durgapur, India. Extensive statistical investigations were carried out to analyze and assess the performance of the optimization techniques and their stability. Optimization of the system size involved examining supply shortages, while simulation outcomes validated the practicality and performance of the proposed technique. In addition, a thorough comparison of three optimization techniques highlights their effectiveness in enhancing system reliability while minimizing costs. This research aims to reduce total expenses and maintain energy supply reliability while managing long-term storage charge levels. The study uses local weather information to analyze how climate conditions affect outcomes.

This paper is divided into the following sections: Section 2 presents the mathematical formulation of the HRE system. Section 3 explains the energy management strategy along with the corresponding objective functions and boundary conditions. Section 4 describes the optimization techniques employed in this study. The results of the algorithms and their performance evaluations are discussed in Section 5, followed by a statistical analysis in Section 6. The conclusion part is provided in Section 7.

2. HRE System Modelling

Each component’s performance is fundamental to designing an effective HRES. Figure 1 presents the block diagram of the proposed hybrid system. Accurate performance predictions depend on the individual modelling of each system component.

A dependable power production system relies on a thorough evaluation of how all components work together. Accurate power output predictions from each component help identify the most efficient combination for delivering a cost-effective energy supply. The complete specifications for all selected components are provided in

Table 2. Input component parameters.

Parameters	Values	Parameter	Values
PV Components		WT Components
Cost Initial	3400 USD/kW	Model	ZEYUFD-2 KW
Power Rated	60 kWh	Cost Initial	2000 USD/kW
Regulator Efficiency	95%	P_Rated	2 kW
Regulator Cost	USD 1500	V_Rated	9.5 m/s
Lifespan	24 years	Cut-out Limit	40 m/s
		Cut-in Limit	2.5 m/s
Battery Bank		Regulator Cost	USD 1000
Cost Initial	280 USD/kWh	Regulator Efficiency	95%
Power Rated	40 kWh	Lifespan	24 years
Efficiency	85%	Height	62 m
Lifespan	12 years
Diesel Generator		Inverter
Cost Initial	1000 USD/kW	Cost Initial	USD 2500
Power Rated	4 kW	Efficiency	92%
Lifespan	24,000 h	Lifespan	24 years
Economic Parameters
Rate of Discount	8%	Project Lifespan	24 years
Interest Rate	12%	O&M + running cost	20%
Inflation Rate (Fuel)	5%

.

2.1. Solar PV System Modelling

The number of PV modules requires adjustment and optimization to minimize energy costs while meeting load requirements. A PV system’s performance depends on solar radiation levels combined with surrounding temperature conditions and its efficiency [45]. Solar cells are normally connected in a series to produce a PV module that delivers the required voltage. Arrays are formed by combining multiple modules where voltage increases through series connections and current output is enhanced by parallel connections. The output power of an individual PV system at any given time t, referred to as P_PV(t), can be determined through the application of a specific equation [46].

$${\mathrm{P}}_{\mathrm{PV}}\left(\mathrm{t}\right)=\mathrm{I}\left(\mathrm{t}\right) \times \mathrm{A} \times { \mathsf{\eta }}_{\mathrm{PV}},$$

(1)

The symbol I refers to solar irradiance measured in watts per square metre, while A represents the PV panel surface area in square metres, and η_pv indicates photovoltaic module efficiency. The total electrical power from multiple PV systems is determined by adding together the outputs from each system.

$${\mathrm{P}}_{\mathrm{n}-\mathrm{PV}\_\mathrm{out}}{=\mathrm{N}}_{\mathrm{PV}\_\mathrm{no}}\times { \mathrm{P}}_{\mathrm{PV}}\left(\mathrm{t}\right),$$

(2)

Here, N_{PV_no} is the Count of PV units.

The PV output, P_PV-outp, is determined by a temperature-adjusted model based on incident solar energy and environmental temperature.

$${\mathrm{P}}_{\mathrm{PV}-\mathrm{outp}}{=\mathrm{P}}_{\mathrm{n}-\mathrm{PV}\_\mathrm{out}}\times {\displaystyle \frac{\mathrm{G}}{{\mathrm{G}}_{\mathrm{refr}}}}\times { [1+\mathrm{K}}_{\mathrm{t}}{(\mathrm{T}}_{\mathrm{c}}-{\mathrm{T}}_{\mathrm{ref}}\left)\right],$$

(3)

G represents solar irradiance in W/m², G_refr is 1000 W/m², T_ref is set at 25 °C, and K_t is −3.7 × 10³ (1/°C).

T_c = T_ambi + (0.0256 × G); T_ambi denotes the ambient temperature.

2.2. WT System Modelling

Power generation in a wind turbine starts when wind speeds surpass the minimum level called the ‘cut-in’ limit of wind speed, V_cut-in. When wind speed hits the turbine’s maximum specified level, called V_r or ‘rated’ speed, power output from the turbine stabilizes at a steady rate. The system disables the turbine when wind velocity reaches the ‘cut-out’ threshold V_cut−out to protect the generator. The research examines a WT with a power rating of 2 kW, designed for small-scale applications.

A wind turbine generates electrical energy from the kinetic energy of wind. Figure 2 demonstrates that the power output by wind turbines depends on both wind speed and hub height. The equation in [47] allows for wind speed calculations at any desired height.

$${\mathrm{V}=\mathrm{V}}_1{\left({\displaystyle \frac{{\mathrm{h}}_2}{{\mathrm{h}}_1}}\right)}^{\mathsf{\alpha }},$$

(4)

Here, V stands for wind speed at turbine hub height h₂, while V₁ denotes wind speed at standard height h₁. The parameter α stands for the friction factor. The wind turbine produces power at time t based on its characteristic parameters.

$${\mathrm{P}}_{\mathrm{W}}=\left(\begin{array}{cc}0,\hfill & {\mathrm{V}<\mathrm{V}}_{\mathrm{cut-in},\hspace{1em}}{\mathrm{V}>\mathrm{V}}_{\mathrm{cut-out}}\hfill \\ {\mathrm{V}}^3({\displaystyle \frac{{\mathrm{P}}_{\mathrm{r}}}{{\mathrm{V}}_{\mathrm{r}}^3 {-\mathrm{V}}_{\mathrm{cut-in}}^3}}{)-\mathrm{p}}_{\mathrm{r}}({\displaystyle \frac{{\mathrm{V}}_{\mathrm{cut-in}}^3}{{\mathrm{V}}_{\mathrm{r}}^3{-\mathrm{V}}_{\mathrm{cut-in}}^3}}),\hfill & \hspace{1em}{\mathrm{V}}_{\mathrm{cut-in}}^3{< \mathrm{V} < \mathrm{V}}_{\mathrm{rated}}\hfill \\ {\mathrm{P}}_{\mathrm{r}} ,\hfill & {\mathrm{V}}_{\mathrm{rated}}{< \mathrm{V} < \mathrm{V}}_{\mathrm{cut-out}}\hfill \end{array}\right),$$

(5)

The cumulative energy generated by N_Wind turbines is calculated by combining the output of each turbine.

$${\mathrm{P}}_{\mathrm{WT}}{\left(\mathrm{t}\right)=\mathrm{N}}_{\mathrm{Wind}}{\times \mathrm{P}}_{\mathrm{W}}\left(\mathrm{t}\right),$$

(6)

2.3. Battery System Modelling

The kilowatt capacity of battery systems is established by the predicted energy use and the duration they must run autonomously. This estimation is derived using the following [48]:

$${\mathrm{C}}_{\mathrm{B}}={\displaystyle \frac{{\mathrm{E}}_{\mathrm{L}}.\mathrm{AD}}{\mathrm{DOD}{.\mathsf{\eta }}_{\mathrm{Inv}}{.\mathsf{\eta }}_{\mathrm{bat}}}},$$

(7)

The energy load is indicated by E_L while AD represents the system’s autonomy duration, which typically ranges between 3 and 6 days. DOD usually reaches around 80%, η_Inv represents the efficiency (95%), and η_bat shows the efficiency of the battery (85%).

The battery stores energy when generation surpasses consumption but supplies stored energy when consumption exceeds generation. The iterative optimization of hybrid systems frequently requires state of charge data under specific time points or defined load conditions. When solar panels and wind turbines together produce an energy surplus over consumption needs, the excess energy is stored through battery charging. When the energy generated by solar and wind power does not adequately satisfy the demand, the system switches to battery power. The mathematical expression provides the calculation for the accumulated energy in the battery bank at time t [49].

$${\mathrm{E}}_{\mathrm{Batt}}\left(\mathrm{t}\right)= {\mathrm{E}}_{\mathrm{Batt}}(\mathrm{t}-1).(1-\mathsf{\sigma })+[{\mathrm{E}}_{\mathrm{Gen}}\left(\mathrm{t}\right)-{\displaystyle \frac{{\mathrm{E}}_{\mathrm{L}}\left(\mathrm{t}\right)}{{\mathsf{\eta }}_{\mathrm{Inv}}}} ].{\mathsf{\eta }}_{\mathrm{bat}} ,$$

(8)

The battery energy levels at the current and previous hour are represented by E_Batt(t) and E_Batt(t − 1), while σ indicates the battery’s hourly rate of self-discharge, with E_Gen(t) showing the net renewable energy post-controller losses and E_L(t) being the energy needed by the load at time t.

The battery bank’s charging capacity is restricted by specific guidelines.

E_Bmin ≤ E_B(t) ≤ E_Bmax

E_Bmax and E_Bmin define the maximum and minimum charge limits of the battery bank.

2.4. DG System Modelling

During high demand periods when the battery runs out, diesel functions as a backup power. The DG supplies additional power through the hybrid system when PV and WT outputs, together with battery storage, cannot satisfy the load demand. The calculation method for both the DG fuel consumption and its associated cost is presented as follows [50].

$${\mathrm{C}}_{\mathrm{fl}\_\mathrm{DG} }{=\mathsf{\alpha }}_{\mathrm{DG}}{\times \mathrm{P}}_{\mathrm{DG}}\left(\mathrm{t}\right){+\mathsf{\beta }}_{\mathrm{DG}}{\times \mathrm{P}}_{\mathrm{rated}\_\mathrm{DG}},$$

(9)

C_{fl_DG} is the fuel usage, P_DG(t) is the average power, and P_{rated_DG} is the rated power of DG. The consumption coefficients, α_DG and β_DG, are 0.246 and 0.08145 l/kWh, respectively.

2.5. Modelling of DC/AC Converter

The energy transfer between the AC and DC parts of the system is managed by electronic converters. The converters and inverters transform electrical power between AC and DC to match the load’s needed frequency. The converter efficiency is determined by the following [44]:

$${\mathsf{\eta }}_{\mathrm{inv}}={\displaystyle \frac{\mathrm{P}}{{\mathrm{P}+\mathrm{P}}_0{+\mathrm{k} \mathrm{P}}^2}}$$

(10)

where the values of P, P₀, and k are obtained using the equations below:

$${\mathrm{P}}_0=1-99{\left({\displaystyle \frac{10}{{\mathsf{\eta }}_{10}}}-{\displaystyle \frac{1}{{\mathsf{\eta }}_{100}}}-9\right)}^2; \mathrm{k}={\displaystyle \frac{1}{{\mathsf{\eta }}_{100}}}{-\mathrm{P}}_0-1; \mathrm{P}={\displaystyle \frac{{\mathrm{P}}_{\mathrm{out}}}{{\mathrm{P}}_{\mathrm{n}}}},$$

(11)

The parameters η₁₀ and η₁₀₀, as provided by the manufacturer, indicate the inverter’s efficiency at 10% and 100% of its nominal power, respectively. P, P_n, and P₀ refer to the inverter’s output power, its rated nominal capacity, and the load-independent standby (vacuum) losses, respectively.

2.6. Optimal Sizing Parameters

This study considers the NPC, COE, and LPSP as critical parameters for determining the optimal configuration.

2.6.1. Mathematical Representation of COE and NPC

The overall system cost per year (C_{ann_tot}) is calculated by adding up four components: yearly capital expenditure (C_{ann_cap}), annualized replacement expenditure (C_{ann_rep}), annual operational cost (C_{ann_op}), and annual maintenance cost (C_{ann_mant}) [51].

C_{ann_tot} = C_{ann_cap} + C_{ann_rep} + C_{ann_op} + C_{ann_mant},

(12)

The following formula is used to compute the NPC of the hybrid system:

$$\mathrm{NPC}={\displaystyle \frac{{\mathrm{C}}_{\mathrm{ann}\_\mathrm{tot}}}{\mathrm{CRF}}},$$

(13)

CRF represents the capital recovery factor.

The COE generated by the hybrid system, expressed in USD/kWh, can be determined using the following equation:

$$\mathrm{COE}={\displaystyle \frac{{\mathrm{C}}_{\mathrm{ann}\_\mathrm{tot}}}{{\displaystyle {\sum }_{\mathrm{t}=1}^{8760}{\mathrm{P}}_{\mathrm{L}}\left(\mathrm{t}\right)}}}$$

(14)

CRF calculates the yearly capital expenditure for the hybrid system by applying it to the starting investment amounts of key components like PV arrays and wind turbines, as well as battery storage units and diesel generators, according to the provided equation:

$$\mathrm{CRF}={\displaystyle \frac{{ \mathrm{r}(1+\mathrm{r})}^{{\mathrm{P}}_{\mathrm{lyf}}}}{{(1+\mathrm{r})}^{{\mathrm{P}}_{\mathrm{lyf}}}-1}} ,$$

(15)

C_{ann_cap} = C_{i_cap} × CRF,

(16)

The interest rate (%) is represented by r while C_{ann_cap} and C_{i_cap} stand for the annual and initial capital costs, respectively, alongside P_lyf, which indicates the total system lifespan, and P_lyfi, which specifies the lifetime span of each part.

The project assesses expenses for replacing parts that require renewal during the project timeline because of their limited operational lifespan.

$${\mathrm{C}}_{\mathrm{rep}}{=\mathrm{C}}_{\mathrm{i}\_\mathrm{cap}}{\displaystyle \frac{\left({\mathrm{P}}_{\mathrm{lyf}}-{\mathrm{P}}_{\mathrm{lyfi}}\right)}{{\mathrm{P}}_{\mathrm{lyfi}}}},$$

(17)

Operational and maintenance expenses (O&M) represent ongoing expenses required to ensure the system’s functionality and are expressed as follows:

$${\mathrm{C}}_{\mathrm{O}\&\mathrm{M}\_\mathrm{tot}}{=\mathrm{C}}_{\mathrm{O}\&\mathrm{M}\_\mathrm{PV}}{\times \mathrm{N}}_{\mathrm{PV}}{+\mathrm{C}}_{\mathrm{O}\&\mathrm{M}\_\mathrm{WT}}{\times \mathrm{N}}_{\mathrm{WT}} {+\mathrm{C}}_{\mathrm{O}\&\mathrm{M}\_\mathrm{bat}}{\times \mathrm{N}}_{\mathrm{bat}}{+\mathrm{C}}_{\mathrm{O}\&\mathrm{M}\_\mathrm{DG}}{\times \mathrm{N}}_{\mathrm{DG}}{+\mathrm{C}}_{\mathrm{O}\&\mathrm{M}\_\mathrm{inv}}{\times \mathrm{N}}_{\mathrm{inv}}$$

(18)

2.6.2. Mathematical Representation of LPSP

Over a full year (8760 h), LPSP is calculated as the proportion of the total power shortages (LPS) to the total load, serving as a key reliability indicator [52].

$$\mathrm{LPSP}={\displaystyle \frac{{\displaystyle {\sum }_{\mathrm{t}=1}^{8760}\mathrm{LPS}\left(\mathrm{t}\right)}}{{\displaystyle {\sum }_{\mathrm{t}=1}^{8760}{\mathrm{P}}_{\mathrm{L}}\left(\mathrm{t}\right)\times \mathsf{\Delta }\mathrm{t}}}},$$

(19)

LPSP quantifies the mismatch between the power supply and demand in the hybrid system and is expressed by the equation below [14]:

$$\mathrm{LPSP}={\displaystyle \frac{{\displaystyle {\sum }_{\mathrm{t}=1}^{8760}\left({\mathrm{P}}_{\mathrm{L}}\left(\mathrm{t}\right)-{\mathrm{P}}_{\mathrm{tot}\_\mathrm{sys}}\left(\mathrm{t}\right)\right)}}{{\displaystyle {\sum }_{\mathrm{t}=1}^{8760}{\mathrm{P}}_{\mathrm{L}}\left(\mathrm{t}\right)}}},$$

(20)

where P_L signifies the load power, and P_{tot_sys} is the total generated power by the system.

3. Power Management Strategy

Energy management in HRE systems faces complexity due to renewable resource variability, which necessitates strategic planning to match the energy supply with demand. The configuration of hybrid systems requires meticulous power management to function properly. Renewable sources primarily supply energy to the load while excess power is stored in the batteries until both the load’s needs and storage capacity are fulfilled, after which the surplus energy flows to a dump load. The diesel generator provides power and charges the battery bank when renewable sources and battery levels fail to satisfy the demand. The operation of the hybrid system is divided into four cases based on how well renewable sources meet the energy demand. In the first case, when solar and wind power are enough to meet the load, any extra energy is directed to charge the battery. The second case takes place when there is still leftover power after meeting both the load and battery needs—this extra energy is sent to a dump load to avoid waste. In the third case, if the renewables cannot fully supply the demand, the battery steps in to cover the shortfall before the diesel generator is used. The final case occurs when both the battery and renewable sources cannot keep up with the load; here, the diesel generator takes over to supply power and also recharge the battery. This approach helps balance supply and demand efficiently while limiting generator use. Figure 3 demonstrates the power management strategies employed within the HRES. The corresponding flowchart is depicted in Figure 4.

Objective Function

When working with HRE systems, the optimization process often involves balancing multiple goals, keeping annual costs low while ensuring the system remains reliable. LPSP functions as a standard reliability index for determining the size of HRES. The objective requires optimizing a linear function while fulfilling all inequality conditions to locate the optimal solution, which manifests as a unique point on the Pareto front. The fitness function is computed as the following [48]:

$$\mathrm{fitness}=\min \left\{{\displaystyle \sum _{\mathrm{i}=1}^{\mathrm{K}}{\mathrm{w}}_{\mathrm{i}}\frac{{\mathrm{f}}_{\mathrm{i}}\left(\mathrm{x}\right)}{{\mathrm{f}}_{\mathrm{i}}^{\max }}}\right\} \mathrm{with} {\mathrm{w}}_{\mathrm{i}}\ge \mathrm{o} \mathrm{and} {\displaystyle \sum _{\mathrm{i}=1}^{\mathrm{k}}{\mathrm{w}}_{\mathrm{i}}=1} ,$$

(21)

The set of decision variables is represented by x, while w_i shows the weights that represent the significance of each objective, k denotes the total number of objectives, f represents the optimization function, and f_i^max serves as the maximum limit for the ith objective function.

4. Optimization Technique

In recent years, meta-heuristic optimization techniques have gained significant popularity because of their straightforward application, adaptable nature, derivation independence, and their capability to bypass local optima. Meta-heuristics maintain simplicity at their foundation because they derive inspiration from basic concepts related to natural phenomena and animal instincts as well as evolutionary mechanisms. In the No Free Lunch (NFL) theorem [53], it has been established that there exists no single meta-heuristic that is optimal for solving all optimization problems. Any single algorithm could be highly effective for specific problem sets while failing to produce good results for different problems. Despite the differences across meta-heuristics, they commonly involve two key phases throughout the search process [54]: during the exploration phase of the algorithm’s process, it conducts a broad search across the most optimal areas of the search space. Stochastic operators need to execute random global searches throughout the entire space during this phase. In exploitation, the search process hones in on the promising areas identified during exploration. The stochastic features of meta-heuristics make it difficult to balance exploration with exploitation during optimization. Our study implemented three optimization techniques in a real case scenario.

4.1. Grey Wolf Optimization

Grey wolves’ leadership structure and hunting behaviour in nature serves as the inspiration behind the GWO algorithm. To simulate the optimization method, the GWO algorithm utilizes four categories of grey wolves: alpha, beta, delta, and omega. Figure 5a illustrates the social hierarchy; alpha (α) stands for the best solution, while beta (β) and delta (δ) indicate the second and third most optimal solutions. The algorithm labels all leftover candidate solutions as omega (ω). There are three main steps to reach the optimal solution, which encompass encircling the prey, followed by the hunting and attacking phases.

4.1.1. Encircling Prey

To describe the encircling behaviour mathematically, the equations below are proposed [32]:

$${\mathrm{D}}_{\mathrm{gw}}=\left| {\overrightarrow{\mathrm{C}}}_{\mathrm{gw}}. {\overrightarrow{\mathrm{X}}}_{\mathrm{p}}\left(\mathrm{t}\right)-\overrightarrow{\mathrm{X}}\left(\mathrm{t}\right) \right|,$$

(22)

$$\overrightarrow{\mathrm{X}}(\mathrm{t}+1)={\overrightarrow{\mathrm{X}}}_{\mathrm{p}}\left(\mathrm{t}\right)-{\overrightarrow{\mathrm{A}}}_{\mathrm{gw}}.{\overrightarrow{\mathrm{D}}}_{\mathrm{gw}},$$

(23)

Here, t stands for the ongoing iteration, ${\overrightarrow{\mathrm{A}}}_{gw}$ and ${\overrightarrow{\mathrm{C}}}_{gw}$ represent the coefficient matrices, ${\overrightarrow{\mathrm{X}}}_{\mathrm{P}}$ is the location vector of the prey, while $\overrightarrow{\mathrm{X}}$ indicates the location vector of a grey wolf. The calculation for vectors $\overrightarrow{\mathrm{A}}$ and $\overrightarrow{\mathrm{C}}$ is as given below:

$${\overrightarrow{\mathrm{A}}}_{\mathrm{gw}}=2{\overrightarrow{\mathrm{a}}}_{\mathrm{gw}}{. \mathrm{r}}_1-{\overrightarrow{\mathrm{a}}}_{\mathrm{gw}}; {\overrightarrow{\mathrm{C}}}_{\mathrm{gw}}=2{.\mathrm{r}}_2,$$

(24)

The parameter ${\overrightarrow{a}}_{gw}$ is progressively reduced from two to zero across the iterations, while r₁ and r₂ are random values ranging from zero to one. Figure 5b offers a graphical representation of the encircling concept.

4.1.2. Hunting

Without precise knowledge of the optimum (prey) location within abstract space, it is considered that the alpha (top solution), together with beta and delta, has a greater insight of the prey’s location. The following expressions represent their behaviour in mathematical terms:

$${\overrightarrow{\mathrm{D}}}_{\mathrm{gw}\mathsf{\alpha }}=\left|{\overrightarrow{\mathrm{C}}}_{\mathrm{gw}1}. {\overrightarrow{\mathrm{X}}}_{\mathsf{\alpha }}-\overrightarrow{\mathrm{X}}\right|; {\overrightarrow{\mathrm{D}}}_{gw\beta }=\left|{\overrightarrow{\mathrm{C}}}_{gw2}. {\overrightarrow{\mathrm{X}}}_{\beta }-\overrightarrow{\mathrm{X}}\right|; {\overrightarrow{\mathrm{D}}}_{gw\delta }=\left|{\overrightarrow{\mathrm{C}}}_{gw3}. {\overrightarrow{\mathrm{X}}}_{\delta }-\overrightarrow{\mathrm{X}}\right|,$$

(25)

$${\overrightarrow{\mathrm{X}}}_1={\overrightarrow{\mathrm{X}}}_{\mathsf{\alpha }}-{\overrightarrow{\mathrm{A}}}_{\mathrm{gw}1} . ({\overrightarrow{\mathrm{D}}}_{\mathrm{gw}\mathsf{\alpha }}); {\overrightarrow{\mathrm{X}}}_2={\overrightarrow{\mathrm{X}}}_{\beta }-{\overrightarrow{\mathrm{A}}}_{gw2} . ({\overrightarrow{\mathrm{D}}}_{gw\beta }); {\overrightarrow{\mathrm{X}}}_3={\overrightarrow{\mathrm{X}}}_{\delta }-{\overrightarrow{\mathrm{A}}}_{gw3} . ({\overrightarrow{\mathrm{D}}}_{gw\delta }),$$

(26)

$$\overrightarrow{\mathrm{X}}(\mathrm{t}-1)={\displaystyle \frac{{\overrightarrow{\mathrm{X}}}_1+{\overrightarrow{\mathrm{X}}}_2+{\overrightarrow{\mathrm{X}}}_3}{3}},$$

(27)

The process by which the search agent refines its position is depicted in Figure 6a.

4.1.3. Attacking Prey

The fluctuation range of A becomes smaller when parameter ‘a’ decreases. The wolves initiate their assault when the absolute value of A falls below one. Once |A| goes beyond one, the wolves disperse, looking for prey with higher potential. The C vector creates randomness in how prey influences wolf movement during the GWO algorithm’s exploration stage. The GWO process becomes less predictable through this mechanism, which simultaneously strengthens its exploration potential and lowers the chances of encountering local optima. The flow diagram for the GWO algorithm appears in Figure 6b, and the pseudo-code is as follows.

Setup the initial population of grey wolves (X_i) randomly within the search space.

Initialize parameters:

• MaxIterations: Maximum iterations number.
• a: Reduces linearly from 2 down to 0.
• A, C: Coefficient vectors.

Determine the fitness of each wolf using the objective function (e.g., cost minimization or reliability maximization).

Identify the alpha, beta, and delta wolves depending on fitness values.

While (t < Max_Iterations):

• For each wolf:
  • Update position using:
    • D_gwα = |C_gw1 × X_α − X_i|
    • D_gwβ = |C_gw2 × X_β − X_i|
    • D_gwδ = |C_gw3 × X_δ − X_i|
    • X₁ = X_α − A_gw1 × D_gwα
    • X₂ = X_β − A_gw2 × D_gwβ
    • X₃ = X_δ − A_gw3 × D_gwδ
    • X_i(t + 1) = (X₁ + X₂ + X₃)/3
• Update a, A, and C:
  • a_gw = 2 − t × (2/Max_Iterations)
  • A_gw = 2a × r₁ − a
  • C_gw = 2 × r₂
• Find the fitness for each wolf.
• Update α, β, and δ wolves.
• t = t + 1

Output the best solution (alpha wolf).

4.2. Multi-Objective Particle Swarm Optimization

MOPSO extends traditional PSO algorithms to solve problems with multiple conflicting objectives. MOPSO functions as a population-based optimization method that organizes solutions into a swarm where every solution functions as a particle. The position of each particle is updated according to two values: each particle in MOPSO utilizes its personal best, which represents its top performance, and the global best, which indicates the superior solution discovered within the entire swarm [55]. The MOPSO algorithm enables each particle to change its position by evaluating its present location while taking into account its velocity and the distances to both its personal best and the global best positions. The fitness function evaluates multiple objectives to detect the best possible optimal solution from all feasible options. The MOPSO flowchart is illustrated in Figure 7.

Pseudo-code of MOPSO is as follows:

Initialize:

• Population size (N)
• MaxItr
• Inertia weight (w)
• Cognitive and social coefficients (c₁, c₂)
• Empty Pareto archive

For each particle i in population:

• Initialize random position (x_i) and velocity (v_i)
• Evaluate fitness

Detect the non-dominated solutions

While (t < Max_Itr):

• For each particle i:
  • velocity Update:
  • position Update:
  • Apply constraints
  • Evaluate fitness
  • Update p_best
• Update Pareto archive:
  • Add newly non-dominated solutions.
  • Remove dominated solutions.
  • Maintain diversity using adaptive grid.
• Select g_Best from archive
• t = t + 1

Output the Pareto-optimal solutions.

4.3. Teaching–Learning-Based Optimization

The optimization method TLBO, developed by Rao et al. [56,57], originates from the teaching and learning process philosophy. This optimization technique functions by representing the collective learners as the population. TLBO captures the interaction between a teacher and students and mirrors the environment found in classroom learning settings. The design variables function in the same way as subjects taught to learners because they directly relate to the outcomes that define the objectives. In TLBO, the idea is simple: just as the most experienced person in a group often helps others learn, the algorithm picks the best solution to act as the teacher for the rest. TLBO operates in two phases: in the teacher stage, learners gain knowledge from the teacher before moving into the learner stage, where they participate in discussions and exchanges to deepen their understanding.

4.3.1. Teacher Phase

At the beginning of the TLBO algorithm, learners focus on enhancing their knowledge through learning from the teacher. In the teaching phase of the TLBO algorithm, the teacher tries to move the group’s average solution from X₁ to a better solution X2, which is confirmed to outperform X₁ based on the teacher’s skill. The newly generated solution requires a fitness function recalculation before selecting the best result through a greedy selection method.

X_new = X + r (X_best − T_f X_mean),

(28)

X_new represents the new solution, while X refers to the ongoing solution. X_best is the previous best solution, which is considered the teacher. The random number, denoted by r, lies between zero and one, and T_f is the teaching factor.

T_f = round (1 + rand)

(29)

4.3.2. Learner Phase

This phase involves learners improving their understanding through mutual interaction. A new solution is generated through collaboration with a partner solution (X_P).

X_new = X + r (X − X_P);    if  fitness_x < fitness_P

(30)

X_new = X + r (X_P − X)     if  fitness_x > fitness_P

(31)

The procedure for implementing TLBO in hybrid system sizing consists of several steps:

• Set TLBO variables, including population size and maximum iteration count.
• Population generation occurs randomly, subject to constraints related to population size and design variables, which must fit within their upper and lower limits.
• Calculate the fitness function with each solution.
• Teacher phase: The solution with the lowest fitness score functions as the teacher labelled X_best. Start by determining the learners’ average performance and then find X_new to evaluate its fitness before implementing greedy selection.
• Learner phase: Compare fitness values with partners to find X_new before calculating the updated fitness function, followed by the greedy selection.
• Terminate the process and store the optimal values when the maximum generation limit has been reached.

Figure 8 shows the flowchart of the TLBO system.

Pseudo-code for TLBO is as follows:

Initialize:

• Population size (N)
• Max_Iterations
• Randomly generate initial population (X)

Evaluate fitness for all solutions (e.g., Cost, Reliability).

While (t < Max_Iterations):

**Teacher Phase**:

• Identify the best solution (Teacher).
• Calculate mean (M) of all solutions.
• For each student (X_i):
  • Compute T_f = round(1 + rand(0,1)) //T_f = 1 or 2
  • Update:
    X_{new_sol} = X_i + rand() x (Teacher − T_f x M)
  • Evaluate X_new.
  • Replace X_i if X_{new_sol} is better.
  **Learner Phase**:
• For each student (X_i):
  • Randomly select another student (X_j, where j ≠ i).
  • If X_i is better than X_j:
    X_{new_sol} = X_i + rand() x (X_i − X_j)
  • Else:
    X_{new_sol} = X_i + rand() x (X_j − X_i)
  • Evaluate X_new.
  • Replace X_i if X_{new_sol} is better.
• t = t + 1

Output the optimum solution (Teacher).

5. Results and Discussion

Our study examines PV and WT for renewable energy generation alongside a battery storage system and diesel generator backup. The research was conducted in Durgapur, which is situated in West Bengal, India, at 23.55 latitude and 87.29 longitude. Our research included five houses, which collectively had an average daily power demand of 12.72 kW. Load variations produce substantial effects on the system’s LPSP. A significant reduction in LPSP occurs when the load decreases while all other system constraints remain unchanged. During our analysis, we aimed to maintain LPSP levels under 25%, which led us to treat the PV panel’s rated power as a changing factor instead of settling on a fixed quantity of panels. The desired number of PV panels can be determined by dividing the total required rated power by the manufacturer-provided output rating of each panel, which differs across PV models. The PV’s maximum rated power capacity was established at 60 kW. The PV needs 113 panels when each one produces 400 W of power, and we also installed WT rated at 2 kW each at a height of 62 m and limited to a total of 10 turbines. Figure 9a displays the hourly average power outputs throughout an entire year. The calculation of the battery bank capacity required treating the autonomy day as a flexible variable because storage capacity design mostly depends on this parameter. Our HRE system maintains an autonomy day range of 0 to 3 days. The generator runs at its rated power continuously to achieve optimal economic efficiency during operations. Solar radiation averages 4.38 kWh/m²/day each year, and wind velocity averages 4.89 m/s. The system requires 12.72 kW on average and reaches a maximum power draw of 25 kW. Figure 9b shows the daily power demand in kW. The system is projected to function for 24 years under a 12% interest rate. MATLAB performed the optimization analysis.

Analysis (

Table 3. Results of optimization methods.

	After 10 Iterations			Final Optimization
	GWO	TLBO	MOPSO	GWO	TLBO	MOPSO
Best Fitness	0.25651	0.269396	0.25296	0.24593	0.26628	0.25296
Pnpv (kW)	60	56	60	60	60	60
NWT (No.)	10	10	10	10	10	10
AD (day)	3	2	3	3	3	3
NDG (No.)	4	3	4	3	3	4
Iteration no. for optimal solution	-	-	-	11	12	8
COE	2.78865	1.96358	2.788658	1.91446	1.91446	2.78865
LPSP	0.07516	0.13067	0.073471	0.12528	0.12528	0.07516
Dump load (kW)	32,537	32,548	32,537	31,846	31,846	32,537
NPC	310,910	218,920	310,910	213,440	213,440	310,910

) comparing GWO, TLBO, and MOPSO shows specific trade-offs exist among the convergence speed, solution quality, and economic efficiency. GWO achieves the highest solution accuracy by reaching the lowest fitness value of 0.24593 in 11 iterations and maintains superior system reliability with an LPSP of 0.12528. The Teaching–earning-Based Optimization (TLBO) method achieves optimal cost savings by reaching the lowest Net Present Cost (NPC = 213,440) and Cost of Energy (COE = 1.91446/kW) but registers a marginally higher fitness value (0.26628) and slightly lower system reliability (LPSP = 0.12528). MOPSO achieves rapid convergence in just eight iterations but ends with a lesser fitness of 0.25296 and a higher Net Present Cost of USD 310,910, indicating a compromise between quick convergence and optimal precision.

The objective space plot (Figure 10 and Figure 11) shows GWO’s steady performance in fitness reduction compared to TLBO and MOPSO, which display more variable outcomes. The LPSP comparison demonstrates GWO’s superiority in reliability, while TLBO achieves economic benefits according to Figure 12’s cost minimization analysis. All algorithms reach comparable configurations for wind turbines (10 units) and diesel generators (3–4 units), but TLBO stands out by minimizing the dump load to 31,846 kW. The research results indicate that GWO should be chosen for applications demanding high reliability, while TLBO should be used when cost efficiency is essential, and MOPSO can provide quick approximate answers. A subsequent investigation should examine hybrid techniques to achieve equilibrium between these conflicting goals.

The energy contribution breakdown (Figure 13) further highlights these trade-offs: the TLBO configuration (56 kW, 10 WT, 2 AD, and 3 DG) achieves a 16% diesel reduction compared to GWO’s 26% while enhancing PV integration by 33%, which explains its cost benefits. GWO uses a greater diesel percentage (26%) to boost its reliability performance at the expense of efficiency, while MOPSO’s system setup of (60 kW, 10 WT, 3 AD, and 4 DG) displays higher waste rates (9%) and fuel expenditures, which reveal resource allocation inefficiencies. TLBO emerges as the balanced choice for cost-sensitive applications, while GWO remains the preferred option for scenarios demanding high reliability. System design decisions and component utilization patterns emerge directly from the optimization focus on cost versus reliability. GWO uses a greater diesel percentage (26%) to boost its reliability performance at the expense of efficiency, while MOPSO’s system setup of (60 kW, 10 WT, 3 AD, and 4 DG) displays higher waste rates (9%) and fuel expenditures, which reveal resource allocation inefficiencies. TLBO emerges as the balanced choice for cost-sensitive applications while GWO remains the preferred option for scenarios demanding high reliability. System design decisions and component utilization patterns emerge directly from the optimization focus on cost versus reliability.

The hourly performance data for different components of the optimally sized HRES throughout an entire year (8760 h) has been presented in Figure 14 based on results generated through the Grey Wolf Optimizer (GWO). The PV output displays a consistent daily pattern with a maximum generation of approximately 55 kW during daylight hours and heightened activity during hours 1000–2500 and 6000–7500, which matches the high-irradiance summer months. The photovoltaic system produces no electricity during nighttime hours between 0–500, 3000–3500, and 8500–8760. The WT output shows intermittent spikes that reach approximately 20 kW, which illustrates the unpredictable and variable nature of wind energy. The intervals between hours 4000–5500 and 7500–8500 show the most frequent occurrences of higher wind outputs. The diesel generator (DG) displays high usage because the system operates on a small scale and sets limited upper bounds for each individual component. The system design dictates that the DG operates at a constant 4 kW power level and demonstrates high activity throughout the fourth subplot. The generator creates surplus energy, which remains unadjustable because of its constant power output capacity. The fifth subplot demonstrates significant energy dumping in the intervals between 0–2500 and 6000–8000 h, where dump loads regularly surpass 10 kW. The energy waste represents a necessary sacrifice to maintain system reliability and keep the LPSP under 25%, which was specified in the study. The battery system reaches peak energy usage of up to 20 kWh during times when both PV and wind power generation cannot meet demand during hours 1500–2500 and 7000–8000. The battery works to fill power supply gaps but fails to provide full system reliability because its autonomy lasts only between zero to three days, and the system size is limited. When generation remains low for extended periods, the system relies primarily on the diesel generator to satisfy demand. The patterns identified (Figure 15) demonstrate system reliability issues in small scale HRES and explain why diesel generator usage remains high with associated power wastage. The PV output reaches its maximum of 50 kW at hour 1594, while the wind output stays below 10 kW throughout. At 1590 h, a major power deficit of approximately −35 kW causes the battery to discharge and activates the diesel generator (DG) at 1596 h, which leads to a sharp power waste increase since the DG can only operate at rated power (4 kW) and causes the dump load to peak at nearly 50 kW. The limited size of the system and the reduced maximum capacity of components require trade-offs between system dependability (low LPSP) and optimal energy usage.

6. Quantitative Performance Analysis

Here, we deliver a complete analysis of the outcomes generated by optimization methods using multiple statistical evaluation tools. The study performs a sensitivity analysis to evaluate the strength and reliability of each algorithm when exposed to different conditions. The evaluation metrics comprise maximum and minimum cost function values along with Standard Deviation (SD), Root Mean Square Error (RMSE), Mean Absolute Error (MAE), Mean, Median, Relative Error (RE), and efficiency values. The expressions that correspond to these parameters can be found in the section below.

$$\mathrm{Std} \mathrm{Dv}=\sqrt{{\displaystyle \frac{{\displaystyle \sum _{\mathrm{i}=1}^{\mathrm{n}}{{( \mathrm{F}}_{\mathrm{i}}-\overline{\mathrm{F}})}^2}}{\mathrm{n}-1}}}; \mathrm{RMSE}=\sqrt{{\displaystyle \frac{{\displaystyle \sum _{\mathrm{i}=1}^{\mathrm{n}}{{(\mathrm{F}}_{\mathrm{i}}{-\mathrm{F}}_{\min })}^2}}{\mathrm{n}}}},$$

(32)

$$RE=\sqrt{{\displaystyle \frac{{\displaystyle \sum _{i=1}^n(F_i-F_{\min })}}{F_{\min }}} } \times 100; \mathrm{MAE}={\displaystyle \frac{{\displaystyle \sum _{\mathrm{i}=1}^{\mathrm{n}}{ \mathrm{F}}_{\mathrm{i}}-{\mathrm{F}}_{\min }}}{\mathrm{n}}}; \mathrm{Eff}={\displaystyle \frac{{\mathrm{F}}_{\min }}{{\mathrm{F}}_{\mathrm{i}}}} \times 100$$

(33)

The variable F_i stands for the optimization technique’s fitness function value during the ith execution. The variable Fmin stands for the lowest fitness function value achieved through all experimental runs, and n represents the total number of independent MATLAB simulations. The convergence behaviour for 20 independent runs across the three algorithms GWO, TLBO, and MOPSO is represented in Figure 16.

Table 4. Statistical analysis of optimization methods.

Optima Technique	Max	Min	Mean	Median	SD	RMSE	MAE	RE (%)	Min Efficiency (%)	Mean Efficiency (%)
GWO	0.5717	0.2459	0.3466	0.3378	0.0714	0.1189	0.1007	40.96	100.00%	70.96%
TLBO	0.5183	0.2612	0.3245	0.3039	0.0692	0.0929	0.0786	30.10%	94.14%	75.78%
MOPSO	0.3672	0.2594	0.2968	0.2913	0.0286	0.0583	0.0509	20.70%	94.80%	82.85%

contains statistical data that outlines the performance characteristics for each method.

Analysis of optimization techniques shows that MOPSO and TLBO excel in particular areas, while GWO consistently achieves a balanced performance throughout different metrics, which confirms its overall superiority. GWO presents slightly elevated Mean (0.3466) and RMSE (0.1189) values but achieves perfect minimum efficiency at 100%, which neither TLBO nor MOPSO manage. Despite MOPSO achieving the lowest RMSE (0.0583) and MAE (0.0509), its high maximum value (0.3672) and narrow efficiency range indicate potential adaptability issues under diverse conditions. GWO offers strong performance stability against diverse system changes due to its top maximum value of 0.5717 and strong efficiency metrics, even though its deviation rate stands at SD = 0.0714. GWO emerges as the superior technique for achieving optimal balance between statistical accuracy and operational efficiency.

The outcomes of 30 independent runs for the objective function, COE, and LPSP across all optimization techniques are shown in Figure 17, which reflects the system’s reliability.

Figure 18 presents the highest performance results from the runs, which shows that the GWO algorithm achieves the best results while maintaining exceptional consistency and stability compared to other methods. The efficient GWO technique used in this study helps achieve a better fitness value compared to the other two techniques considered. This ensures that all major components, such as PV panels, WTs, batteries, and DGs, are utilized more efficiently and maintained at a high utility throughout their operational lifetime while maintaining the optimized system cost and reliability. This reflects the core philosophy of the Circular Economy, which focuses on maximizing asset value throughout their entire life cycle while minimizing premature waste and reducing dependence on virgin materials. Improved optimization helps lower both initial and operational costs while delaying the need for resource-intensive replacements, ultimately reducing the embedded carbon emissions associated with manufacturing and transportation. A longer battery life significantly reduces the ecological footprint of the HRES. In addition, better scheduling of the diesel generator minimizes material use and environmental harm. As a result, the system encourages the responsible use of materials, saves fuel, and helps reduce environmental degradation. At the same time, it improves the overall financial sustainability of the project.

7. Conclusions

The research successfully applied and assessed three meta-heuristic optimization methods, including GWO, TLBO, and MOPSO, to develop the best HRES system design for Durgapur, India. The study evaluated each algorithm’s performance by examining its COE, LPSP, convergence behaviour, and statistical metrics.

GWO demonstrated superior performance by attaining the minimum value of 0.2459, which surpassed MOPSO’s 0.2594 and TLBO’s 0.2612, thereby proving its exceptional exploration ability and efficiency in overcoming local optima. The greater variability seen in GWO (SD = 0.0714) demonstrates its strong searching capabilities essential for effective space exploration, which matches the research findings of Mirjalili et al. [32]. Grey Wolf Optimizer (GWO) maintained superior reliability and system stability while demonstrating better overall optimization capability compared to TLBO and MOPSO despite their strengths in early convergence and moderate COE.

The qualitative analysis of optimization techniques presented in

Table 5. Qualitative analysis.

Criterion	MPSO	TLBO	GWO
Exploration	Low	High	Moderate
Exploitation	High	Moderate	High
Convergence Speed	Fast	Moderate	Moderate
Robustness	Good	Good	Excellent
Parameter Sensitivity	Moderate	Low	Low
Computational Cost	Moderate	Low	Low

demonstrates GWO’s superiority through its lowest objective function values and consistent robustness over multiple runs according to convergence curves and performance metrics. The study showed that the small system configuration and constant rated operation of the diesel generator led to significant energy wastage, which emphasizes the need for accurate sizing and system reliability to achieve low LPSP. GWO stands as the best solution for complex optimization problems that need optimal results, even though it needs more computational power. The study recommends that researchers focus on using GWO for critical optimization problems and investigate hybrid solutions to manage its exploration capability alongside its convergence stability. The next steps in research should concentrate on fine-tuning GWO’s parameters and exploring its effectiveness in higher-dimensional search spaces to maximize its capabilities.