PV Cells and Modules Parameter Estimation Using Coati Optimization Algorithm

Citation: Elshara, R.; Hançerlio˘gullari, A.; Rahebi, J.; Lopez-Guede, J.M. PV Cells and Modules Parameter Estimation Using Coati Optimization Algorithm. Energies 2024,17, 1716. https://doi.org/10.3390/en17071716 Academic Editor: Abdul-Ghani Olabi Received: 15 November 2023 Revised: 3 March 2024 Accepted: 1 April 2024 Published: 3 April 2024 Copyright: © 2024 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). energies Article PV Cells and Modules Parameter Estimation Using Coati Optimization Algorithm Rafa Elshara 1, Aybaba Hançerlio˘gullari 2, Javad Rahebi 3and Jose Manuel Lopez-Guede 4,* 1Department of Material Science and Engineering, University of Kastamonua, Kastamonu 37150, Turkey; [email protected] 2Department of Physics, University of Kastamonu, Kastamonu 37150, Turkey; [email protected] 3Department of Software Engineering, Istanbul Topkapi University, Istanbul 34087, Turkey; [email protected] 4Department of Systems and Automatic Control, Faculty of Engineering of Vitoria-Gasteiz, University of the Basque Country (UPV/EHU), C/Nieves Cano 12, 01006 Vitoria-Gasteiz, Spain *Correspondence: [email protected] Abstract: In recent times, there have been notable advancements in solar energy and other renewable sources, underscoring their vital contribution to environmental conservation. Solar cells play a crucial role in converting sunlight into electricity, providing a sustainable energy alternative. Despite their significance, effectively optimizing photovoltaic system parameters remains a challenge. To tackle this issue, this study introduces a new optimization approach based on the coati optimization algorithm (COA), which integrates opposition-based learning and chaos theory. Unlike existing methods, the COA aims to maximize power output by integrating solar system parameters efficiently. This strategy represents a significant improvement over traditional algorithms, as evidenced by experimental findings demonstrating improved parameter setting accuracy and a substantial increase in the Friedman rating. As global energy demand continues to rise due to industrial expansion and population growth, the importance of sustainable energy sources becomes increasingly evident. Solar energy, characterized by its renewable nature, presents a promising solution to combat environmental pollution and lessen dependence on fossil fuels. This research emphasizes the critical role of COAbased optimization in advancing solar energy utilization and underscores the necessity for ongoing development in this field. Keywords: coati optimization algorithm (COA); chaos theory; opposition-based learning; solar systems; optimization of PV parameters 1. Introduction In recent years, the increase in energy demand has caused environmental concerns to increase. Although fossil fuels are relatively cheap, they are a continuous source of global warming and air pollution. Oil and gas extraction provide energy resources, but they have become the main factor for environmental pollution in the past decades. Furthermore, estimates show that energy sources such as oil and gas will run out in the coming years, so finding new and renewable energy sources is vital [1,2]. In recent years, clean and renewable energies have developed and progressed, so now, they are a suitable alternative to fossil fuels (oil and gas). Renewable energy, such as the power obtained from the sun, wind, tides, water, hot springs, etc., has been attracting the attention of scientists. Among renewable energies, solar energy is more accessible than other energies and used in most areas of the earth. Solar energy is a renewable energy source. Unlike electricity generation by fossil fuels such as gas and oil, solar energy does not cause any environmental pollution [ 3 ]. Estimates show that the share of solar energy in electricity production will reach about USD 194 billion by 2027, which shows the importance of this energy [ 4 ]. Photovoltaic systems help transform solar energy Energies 2024,17, 1716. https://doi.org/10.3390/en17071716 https://www.mdpi.com/journal/energies Energies 2024,17, 1716 2 of 26 into valuable electricity. Photovoltaic (PV) systems convert solar energy into electricity using semiconductor technology [ 5 ]. Progress in renewable energy has influenced many countries and governments to use this clean energy to produce electricity. Solar energy is an economical and abundant clean energy source. Due to the easy access to sunlight, solar systems are used in many countries in Africa, the Middle East, and America [6]. Solar energy conversion into electricity in solar cells needs devices based on electronic semiconductors with a thin layer. A solar system based on semiconductor crystals is usually made of two types: multi-crystalline [ 7 ] and monocrystalline [ 8 ] solar modules. Singlecrystal semiconductors are more efficient for generating power and have better electrical properties. However, despite having a large production capacity, monocrystalline PV modules are unprofitable because of the high cost of the crystalline wafer-based technology. An alternate process to single-crystal modules for solar power production is thin film technology. This method typically uses amorphous silicon [ 8 ] or other semiconductor materials such as gallium arsenide [ 9 ], copper indium gallium selenide [ 10 ], or cadmium telluride [11]. Solar energy is a clean, renewable source because it directly converts solar energy into electricity using semiconductor technology in photovoltaic devices. The amount of energy production from PV systems depends on weather conditions, solar radiation, ambient temperature, type of modules, etc. [11]. An accurate understanding of PV’s electrical structure and modules may help researchers increase power production efficiency. PV systems comprise a power converter, a solar generator, and other parts connected to the power grid by control circuits, power converters, or inverters. The production efficiency of PV devices must be increased through effective design and implementation. PV power generation is non-linear by nature and is impacted by environmental variables, including temperature, light intensity, and load characteristics. Therefore, PV devices need control circuits for high power-generation performance. Control circuits in PV devices optimize current and output voltage to increase power production and system efficiency [10]. The two main models for modeling solar PV cells include the double-diode model (DDM) [ 12 ] and the single-diode model (SDM) [ 13 ]. DDM can effectively represent a PV cell since it is more accurate than SDM. Planning and optimizing solar PV cells is complicated because there are about ten unknown variables in this problem [ 9 ]. Investigations show that current curves in terms of voltage or I-V in PV cells have a non-linear and complex nature. A controller is an essential component in any PV circuit. In photovoltaic systems, controllers are usually used to manage the charging of batteries or the power supplies to the grid [ 14 ]. Maximum power point tracking (MPPT) is one of the capabilities that some controllers use to increase solar cells’ power efficiency. MPPT algorithms, especially in large systems, are essential in increasing power production and management. For a solar module, the MPPT tracks the current and voltage to reach the maximum power and then absorbs it [ 14 ]. There are three ways to find MPPT when solving non-linear equations in a controller. Numerical approaches [ 10 ], evolutionary algorithms [ 15 ], and analytical methods [ 16 ] are a few of these techniques. In order to solve non-linear calculations with numerous unknowns, the first method, which mostly uses linear calculations, must be initiated. Predicting the initial value for solving the equation, which determines the convergence rate of the solution, is the most difficult aspect of such approaches. The second approach relies on local search techniques. Although numerical approaches have several limitations, they can generate properties of PV cells that are more precise than analytical methods. First, the convergence of this technique heavily depends on the initial explanation. Second, they are prone to errors, and third, obtaining the appropriate model parameters takes time and effort. In recent years, optimization and meta-heuristic methods have played an essential role in solving non-linear calculations and providing solutions very close to the optimal solution. Using these techniques, searching for the ideal solution takes little time. Various meta-heuristic techniques have recently been employed to pinpoint solar cell model parameters in PV systems; for example, GA [ 13 ], PSO, MCDM [ 12 ], WOA [ 17 ], Energies 2024,17, 1716 3 of 26 GWO [ 18 ], and IHHO-VMD [ 16 ] algorithms are used for this purpose. Table 1presents the application of meta-heuristic, bio-inspired, and hybrid optimization algorithms for parameter extraction of PV cell models. Table 1. Utilizing meta-heuristic optimization algorithms for parameter extraction of PV cell models. Ref Aim Algorithm Limitations Advantages Results [10,13] The study seeks to create a method for designing zero-energy residential structures by employing building performance simulation technology and multi-objective optimization, aiming to attain the most efficient energy-saving solutions adaptable to various climate zones in China. NSGA-II Challenges entail the necessity for enhancing the building energy calculation model to encompass regional nuances and the simplification of some parameters for research practicality. The strengths of the research include the creation of a parametric design platform to determine energy consumption thresholds and photovoltaic replacement rates specific to diverse climate zones, providing valuable guidance for policymakers and standard-setting bodies. Furthermore, the study underscores the significance of holistic considerations beyond mere energy usage in residential architectural planning. The findings indicate that near-zero energy consumption is viable in select climate zones in China, with defined photovoltaic replacement rates. Additionally, the research offers crucial guidance on fine-tuning design parameters to harmonize energy efficiency, economic viability, and residential comfort, highlighting the need to avoid overly zealous pursuit of zero-energy targets to maintain a balanced approach. [12] The study aims to compare decision tree and particle swarm optimization algorithms for identifying optimal solar power plant locations, providing insights for renewable energy planning in Iran. PSO, MCDM The research recognizes constraints such as the unavailability of specific data such as transmission line information, potentially impacting the precision of the findings. The article presents the application of the decision tree method in identifying prime sites for solar power plants, offering a fresh approach to environmental studies. Furthermore, it assesses the efficacy of the decision tree against the PSO method, emphasizing the decision tree’s advantage in this particular domain. The decision tree method outperforms PSO in predicting high-potential solar energy areas, emphasizing its effectiveness in identifying optimal sites for solar projects. Employing spatial data mining techniques is advised to improve site suitability assessment for power plants, with eastern and southeastern Iran highlighted as especially favorable regions. [17] The study investigates the impact of PVs and EVs on economic emission dispatch, proposes a modified WOA for optimization, and verifies its performance with simulations. WOA Constraints of the study encompass the intricate nature of the optimization issue, non-linear and non-convex constraints, and the necessity for meticulous consideration of diverse factors like valve point loading effects, restricted areas, and transmission losses. The strengths of the study are found in its capacity to efficiently handle the intricate and contradictory aims of economic emission dispatching through the integration of PVs and EVs, along with proposing a customized WOA algorithm that surpasses other optimization methods in delivering superior quality results. The findings of the study indicate that incorporating PVs and EVs into economic dispatching leads to lower emissions and energy generation costs, while the proposed modified WOA algorithm consistently yields superior quality results when compared to other optimization algorithms employed for economic load dispatch. Energies 2024,17, 1716 4 of 26 Table 1. Cont. Ref Aim Algorithm Limitations Advantages Results [18] The research aims to improve the performance of MPPT controllers by optimizing them with the GWO algorithm and assessing their efficiency compared to traditional methods in various conditions, focusing on response time, efficiency, and power generation. GWO Challenges in the study encompass the need for fine-tuning the metaheuristic algorithm when applied within a PLC mandating historical irradiation data across diverse weather conditions. The strengths of the study are evident in the enhanced efficacy of the MPPT controller, optimized using the GWO algorithm, resulting in superior efficiency and power generation when contrasted with conventional approaches. Furthermore, the research offers valuable observations regarding the response time of various algorithms amidst changing conditions. The research demonstrates that optimizing the MPPT controller with GWO enhances efficiency and power generation while minimizing overshooting, with GWO exhibiting faster response times than traditional algorithms. [16] The research aims to mitigate the variability in photovoltaic output by suggesting a hybrid energy storage setup strategy. IHHO-VMD Challenges involve the possibility of constraining photovoltaic output and diminishing power generation, alongside the restricted energy storage capacity of the HESS, indicating the need for exploring improved decomposition techniques and integrating electric hydrogen into HESS, warranting further investigation. Benefits encompass a 6.15% decrease in the hybrid energy storage system cost relative to the original algorithm as well as mitigated power fluctuations, leading to enhanced system economy and stability. The IHHO-VMD algorithm effectively reduces energy storage system costs, enhances power allocation, and stabilizes photovoltaic grid-connected power. While MA helps mitigate power fluctuations, challenges remain with photovoltaic output and HESS capacity, warranting further exploration of improved decomposition methods and HESS expansion. Finding appropriate model parameters is a challenging optimization problem for evaluating PV parameters. In general, classical optimization algorithms cannot find the optimal solution. New meta-heuristic algorithms have demonstrated better ability in global and local search and search the problem space more optimally. Optimization methods such as genetics and particles use basic mechanisms for searching, while optimizing the parameters of a PV is a complex problem with several variables. In this research, we undertook parameter optimization for the SDM, DDM, and PV solar cell models to enhance solar cell efficiency. We thus propose a tailored COA [ 19 ], specifically crafted to improve the efficiency of PV devices. Moreover, we introduce an enhanced version of the COA that fosters mutual learning. Additionally, we conducted parameter estimation for PV models, comparing them with recent meta-heuristic methods. Finally, we herein present an upgraded iteration of the coati optimization algorithm, which integrates chaos theory, to provide a comprehensive approach to optimizing solar cell parameters and improving efficiency. This study introduces a novel optimization strategy aimed at enhancing the efficiency of solar cells and optimizing their parameters. The main contributions of this manuscript are as follows: ■Parameter optimization for the SDM, DDM, and PV solar cell models; Energies 2024,17, 1716 5 of 26 ■Improving the efficiency of solar cells; ■ Introducing a coati optimization algorithm [ 15 ] designed to enhance the efficiency of PV devices; ■ Introducing an upgraded version of the coati optimization algorithm that fosters mutual learning; ■ Estimating parameters for PV models and comparing them with recent metaheuristic methods; ■ Introducing an enhanced version of the coati optimization algorithm integrated with chaos theory. This research work was compiled and is herein presented in five sections. Section 2 introduces solar cells and their components and related works for optimizing PV device parameters review. Section 3shows the proposed method for improving the COA or the PV parameters’ optimal estimation. Section 4explains tests and implementation, and the results are analyzed and evaluated. Finally, Section 5includes the conclusions and future work. 2. Related Works A strategy for optimizing the photovoltaic model’s parameters was presented in research [ 20 ] that applied northern goshawk optimization (NGO). In order to determine the triple-diode model’s parameters (PV module), this research applied an optimization algorithm known as NGO. Three commercial PV modules were applied in the current research. The simulation results demonstrate that NGO outperformed the other optimization algorithms in terms of speed and accuracy. Furthermore, with this technique, the cost function for the Canadian Solar CS6K-M module may be reduced to 0.000195. A strategy for optimizing the solar photovoltaic models’ parameters by applying differential evolution and queue search optimization was discussed in another paper [21]. The PV model has a multi-model and non-linear specification, making it challenging to determine its ideal values. The algorithms employed to address this problem are prone to stick in local optima because of the non-linear nature of the problem. Due to their large impact on the PV system’s current and power generation performance, the parameters’ appropriate estimation is crucial. In order to extract the ideal PV parameter values, this study provided an enhanced queue search optimization (QSO) based on the differential evolution (DE) method. Their method outperformed other methods like genetic algorithms and particles for obtaining the best parameters, e.g., single-diode, double-diode, and PV module models. A previous study introduced a fuzzy solar PV and wind turbine system employing particle swarm optimization (PSO) [ 22 ] to boost efficiency. However, accurately creating a power forecasting model allows a researcher to regulate the randomized behavior of solar and wind energy sources. Solar PV and wind forecasting algorithms based on fuzzy logic may better handle this unpredictable and random aspect. Furthermore, the performance of the forecasting model was improved by using hybrid fuzzy–PSO intelligent forecasting, which also enhanced the system’s restrictions. Their tests revealed that the proposed fuzzy model was more effective in boosting the power of solar and wind systems than the fuzzy model used in conjunction with a genetic algorithm (GA). In another study [ 23 ], the optimization of electricity production by solar energy was presented using improved MPPT techniques. In this research, they optimized the parameters of the photovoltaic module with optimization methods. In a research paper [ 24 ], a hybrid approach based on the bat algorithms (BA) and grasshopper optimization algorithms (GOA) was presented to maximize power generation through solar photovoltaics. This research used a combined meta-heuristic algorithm to extract the maximum power from PV using the XSG controller. The proposed algorithm performed well for power extraction, according to experiments, and is more capable and effective at boosting power than the BA and GOA algorithms. Energies 2024,17, 1716 6 of 26 One paper [ 25 ] validated the firefly algorithm’s efficacy in optimizing solar cell and photovoltaic module parameters, especially compared to experimental data and the existing literature, highlighting its effectiveness in minimizing error metrics and accurately reproducing current-voltage characteristics under varying irradiance and temperature conditions. The research from [ 26 ] endeavored to create a dependable approach employing the Lambert W function to precisely estimate single-diode PV parameters, tackling issues present in current methodologies, such as inaccuracies in RMSE computation and excessive dependence on optimization methods. The suggested analytical solution enhanced parameter estimation accuracy, notably observable in single-diode PV equivalent circuits, by rectifying RMSE calculation inaccuracies identified in the existing literature. The study endeavored to develop a dependable method for accurately estimating PV parameters, utilizing an innovative hybrid strategy that integrates diversification and intensification mechanisms from different metaheuristics (MHs). It tackled issues such as computational complexity and parameter sensitivity, highlighting its ability to adapt to various optimization challenges, explore multiple search spaces simultaneously, and enhance accuracy and reliability as evidenced by comparisons with alternative MHs and benchmark functions [27]. The reference [ 19 ] presented the arithmetic optimization algorithm (IAOA) as a solution to improve the estimation of PV model parameters, tackling issues such as parameter sensitivity and computational complexity. Despite its potential drawbacks, IAOA showcased notable precision and dependability in estimating solar cell parameters, efficiently optimizing PV models across various scenarios and surpassing other algorithms in terms of accuracy and performance. The study of [ 28 ] presented and assessed the mountain gazelle optimizer (MGO) algorithm’s effectiveness in pinpointing PV model parameters, specifically targeting the SDM and DDM of photovoltaic systems. MGO showcased benefits like rapid processing, consistent convergence, and precise results, outperforming other algorithms with the lowest RMSE across 30 separate iterations. The use of the new optimization technique to predict the ideal parameters in solar modules was observed in one study [ 29 ]. This study considered the Harris hawks optimizer (HHO) to acquire the PV systems’ model parameters. The modified HHO offers a worldwide search capacity, high efficiency, and high convergence speed compared to the conventional method. In addition, the research revealed that the HHO method has a lower error value for voltage power (P-V) and current-voltage (I-V) features. Another research [ 30 ] presented the evaluation and improvement of the photovoltaic grid-connected system using the VPFOTADF controller with the improved version of Wall’s algorithm. This paper aimed to design solar cells to reduce harmonic distortion and improve the solar system’s performance connected to the photovoltaic grid using group intelligence. The solar PV system has components such as a booster converter, photovoltaic array, multi-level inverter, and controller. This study optimized the amplifier converter using the improved WOA algorithm. Evaluation and tests showed that their method increases the production capacity more than the WOA algorithm. In another study, an improved arithmetic optimization algorithm (AOA) was proposed for extracting parameters of a single-diode photovoltaic solar cell model [ 31 ]. The experimental findings demonstrated that IAOA outcomes are more effective and accurate than those obtained using AOA. A method to determine the solar PV model’s optimal parameters using the chimp optimization algorithm (ChOA) was mentioned in one research paper [ 21 ]. To produce precise and trustworthy PV models, including single-diode, dual-diode, triple-diode, and PV module models, this research suggested a novel technique called ChOA that is inspired by nature. The fundamental difficulty in predicting the PV models’ parameters using optimization methods is convergence to the local optimum. Therefore, this study integrated the best distinctive aspects of PSO and a local search technique. The tests revealed that their suggested algorithm outperforms the EHHOA, BMO, FPSO, CBBO, and GOTLA Energies 2024,17, 1716 7 of 26 algorithms in terms of optimizing the three commercial modules’ parameters that are often used: KC200GT, SW255, and SM55 multi-crystal. 3. Methodology In the present study, the COA algorithm was selected due to its pioneering methodology, which amalgamates elements of biological inspiration, population-based optimization, modeling of natural behaviors, mutual learning, opposition-based learning, and integration of chaos theory. This comprehensive approach empowers the algorithm to effectively navigate intricate search spaces and discover high-quality solutions, rendering it a compelling option for optimizing PV system parameters. The benefits of utilizing the COA for optimizing PV systems are outlined in Table 2. Table 2. Advantages of the COA for PV System Optimization. Ref Advantage Description Benefit for PV System Optimization [32] Biological inspiration Mimics coatis’ intelligent hunting and evasion behaviors. Novel optimization perspective, potentially leading to adaptive and resourceful solutions. [15]Population-based approach Explores multiple solutions simultaneously. Efficiently finds global optimums in complex PV system problems. [33]Integration of natural behaviors Models coatis’ hunting and evasion behaviors for optimization. Achieves faster convergence and more robust solutions. [34]Opposition-based Learning Diversifies exploration by generating opposite solutions. Prevents premature convergence and encourages exploration of diverse regions. [35] Chaos theory integration Introduces randomness to escape local optima. Enhances exploration capabilities and avoids stagnation. The P-V and I-V curve properties of PV modules and solar cells are created using two common mathematical models described in this section. First, the SDM model and formulation are explained, then the DDM model is explained, and finally, the circuit’s optimum parameters are discovered using the coati optimization algorithm. 3.1. SMD Circuit Figure 1shows the electrical circuit equivalent to SDM. In this circuit, I ph represents the current produced by the photogenerated current. The SDM framework is relatively simple. First, the generated diode current I d , output current I L , shunt current I sh , and hot Iph are defined. Energies 2024, 17, x FOR PEER REVIEW 7 of 27 algorithms in terms of optimizing the three commercial modules’ parameters that are often used: KC200GT, SW255, and SM55 multi-crystal. 3. Methodology In the present study, the COA algorithm was selected due to its pioneering methodology, which amalgamates elements of biological inspiration, population-based optimization, modeling of natural behaviors, mutual learning, opposition-based learning, and integration of chaos theory. This comprehensive approach empowers the algorithm to effectively navigate intricate search spaces and discover high-quality solutions, rendering it a compelling option for optimizing PV system parameters. The benefits of utilizing the COA for optimizing PV systems are outlined in Table 2. Table 2. Advantages of the COA for PV System Optimization. Ref Advantage Description Benefit for PV System Optimization [32] Biological inspiration Mimics coatis’ intelligent hunting and evasion behaviors. Novel optimization perspective, potentially leading to adaptive and resourceful solutions. [15] Population-based approach Explores multiple solutions simultaneously. Efficiently finds global optimums in complex PV system problems. [33] Integration of natural behaviors Models coatis’ hunting and evasion behaviors for optimization. Achieves faster convergence and more robust solutions. [34] Opposition-based Learning Diversifies exploration by generating opposite solutions. Prevents premature convergence and encourages exploration of diverse regions. [35] Chaos theory integration Introduces randomness to escape local optima. Enhances exploration capabilities and avoids stagnation. The P-V and I-V curve properties of PV modules and solar cells are created using two common mathematical models described in this section. First, the SDM model and formulation are explained, then the DDM model is explained, and finally, the circuit’s optimum parameters are discovered using the coati optimization algorithm. 3.1. SMD Circuit Figure 1 shows the electrical circuit equivalent to SDM. In this circuit, Iph represents the current produced by the photogenerated current. The SDM framework is relatively simple. First, the generated diode current Id, output current IL, shunt current Ish, and hot Iph are defined. Figure 1. SDM circuit structure in PV systems [36]. Then, Kirchhoff’s current law is used to obtain IL using Equation (1). The equations of the diode Shockley and Ohm laws are, respectively, used to obtain Id using Equation (2) and 𝐼𝑠ℎ applying Equation (3). Figure 1. SDM circuit structure in PV systems [36]. Then, Kirchhoff’s current law is used to obtain I L using Equation (1). The equations of the diode Shockley and Ohm laws are, respectively, used to obtain I d using Equation (2) and Ish applying Equation (3). IL=Iph −Id−Ish (1) Energies 2024,17, 1716 8 of 26 Id=Isd ·expq·(VL+RS·IL) n·k·T−1(2) Ish =VL+RS·IL Rsh (3) Equation (1) is expanded using Equations (2)–(4): IL=Iph −Id·expq·(VL+RS·IL) n·k·T−1−VL+RS·IL Rsh (4) R S is a series resistor, R sh is a parallel resistor, V L stands for output voltage k,q stands for initial charge (1.60217646 × 10 −19 ), and nis the ideality factor of the diode and refers to Boltzmann’s constant and equals (1.3806503 × 10 −23 J/K q). Here, SDM identifies five different unknown parameters ( Iph , Id , RS , Rsh , and n), and Trepresents the absolute temperature. 3.2. DDM Circuit The corresponding electrical circuit for DDM is shown in Figure 2. DDM can more precisely depict the voltage–current relationship given the inherent limitations of SDM. A diagram of the DDM circuit is shown in Figure 2. Because DDM has an additional diode (in parallel) compared to SDM, Figure 2demonstrates how DDM differs from SDM. I L is calculated using Equation (5): IL=Iph −Id1·expq·(VL+RS·IL) n1·k·T−1−Id2·expq·(VL+RS·IL) n2·k·T−1−VL+RS·IL Rsh (5) Here, Id2 stands for saturation current, n 1 and n 2 represent the ideal saturation coefficient of two diodes, and Id1 is the diode emission current. DDM needs to extract seven different parameters (Iph,Id1,Id2,RS,Rsh,n1, and n2). Energies 2024, 17, x FOR PEER REVIEW 8 of 27 𝐼𝐿=𝐼𝑝ℎ−𝐼𝑑−𝐼𝑠ℎ (1) 𝐼𝑑=𝐼𝑠𝑑⋅[exp⁡(𝑞⋅(𝑉𝐿+𝑅𝑆⋅𝐼𝐿) 𝑛⋅𝑘⋅𝑇 )−1] (2) 𝐼𝑠ℎ=𝑉𝐿+𝑅𝑆⋅𝐼𝐿 𝑅𝑠ℎ (3) Equation (1) is expanded using Equations (2)–(4): 𝐼𝐿=𝐼𝑝ℎ−𝐼𝑑⋅[exp⁡(𝑞⋅(𝑉𝐿+𝑅𝑆⋅𝐼𝐿) 𝑛⋅𝑘⋅𝑇 )−1]−𝑉𝐿+𝑅𝑆⋅𝐼𝐿 𝑅𝑠ℎ (4) RS is a series resistor, Rsh is a parallel resistor, VL stands for output voltage k, q stands for initial charge (1.60217646 × 10−19), and n is the ideality factor of the diode and refers to Boltzmann’s constant and equals (1.3806503 × 10−23 J/K q). Here, SDM identifies five different unknown parameters (𝐼𝑝ℎ, 𝐼𝑑, 𝑅𝑆, 𝑅𝑠ℎ, and n), and T represents the absolute temperature. 3.2. DDM Circuit The corresponding electrical circuit for DDM is shown in Figure 2. DDM can more precisely depict the voltage–current relationship given the inherent limitations of SDM. A diagram of the DDM circuit is shown in Figure 2. Because DDM has an additional diode (in parallel) compared to SDM, Figure 2 demonstrates how DDM differs from SDM. IL is calculated using Equation (5): 𝐼𝐿=𝐼𝑝ℎ−𝐼𝑑1⋅[exp⁡(𝑞⋅(𝑉𝐿+𝑅𝑆⋅𝐼𝐿) 𝑛1⋅𝑘⋅𝑇 )−1]−𝐼𝑑2⋅[exp⁡(𝑞⋅(𝑉𝐿+𝑅𝑆⋅𝐼𝐿) 𝑛2⋅𝑘⋅𝑇 )−1]−𝑉𝐿+𝑅𝑆⋅𝐼𝐿 𝑅𝑠ℎ (5) Here, 𝐼𝑑2 stands for saturation current, n1 and n2 represent the ideal saturation coefficient of two diodes, and 𝐼𝑑1 is the diode emission current. DDM needs to extract seven different parameters (𝐼𝑝ℎ, 𝐼𝑑1, 𝐼𝑑2, 𝑅𝑆, 𝑅𝑠ℎ, 𝑛1, and 𝑛2). Figure 2. DDM circuit structure in PV systems [36]. 3.3. PV Module Modeling Figure 3 shows the PV module’s circuit diagram with numerous PV cells connected in parallel or series. IL is calculated using Equation (6): 𝐼𝐿=𝑁𝑃⋅𝐼𝑝ℎ−𝑁𝑃⋅𝐼𝑑⋅[exp⁡(𝑞⋅(𝑉𝐿/𝑁𝑆+𝑅𝑆⋅𝐼𝐿/𝑁𝑃) 𝑛⋅𝑘⋅𝑇 )−1]−𝑁𝑃⋅𝑉𝐿/𝑁𝑆+𝑅𝑆⋅𝐼𝐿 𝑅𝑠ℎ (6) The terms NS and NP represent the quantity of parallel and serial connections between PV cells. The SDM of PV modules is used in this research. The PV module should determine five unknowns (Iph, Id, RS, Rsh, and n). Figure 2. DDM circuit structure in PV systems [36]. 3.3. PV Module Modeling Figure 3shows the PV module’s circuit diagram with numerous PV cells connected in parallel or series. ILis calculated using Equation (6): IL=NP·Iph −NP·Id·expq·(VL/NS+RS·IL/NP) n·k·T−1−NP·VL/NS+RS·IL Rsh (6) The terms N S and N P represent the quantity of parallel and serial connections between PV cells. The SDM of PV modules is used in this research. The PV module should determine five unknowns (Iph,Id,RS,Rsh, and n). Energies 2024,17, 1716 9 of 26 Energies 2024, 17, x FOR PEER REVIEW 9 of 27 Figure 3. Equivalent circuit of a PV module [36]. 3.4. Objective Function The discrepancy between the determined calculated value and the actual measured value is objectively assessed using an objective function. The calculated data points of the SDM, DDM, and PV modules and error functions for the experiments are represented by Equations (7)–(9). {𝑓𝑆𝐷(𝑉𝐿,𝐼𝐿,𝑿)=𝐼𝐿−𝐼𝑝ℎ+𝐼𝑑∗[exp⁡(𝑞(𝑉𝐿+𝑅𝑔−𝐼𝐿) 𝑛−𝑘𝑇 )−1]+𝑉𝐿+𝑅𝑆⋅𝐼𝐿 𝑅Lh 𝑿={𝐼ph,𝐼𝑑,𝑅𝑆,𝑅sh ,𝑛} (7) { 𝑓𝐷𝐷(𝑉𝐿,𝐼𝐿,𝑿)=𝐼𝐿−𝐼𝑝ℎ+𝐼𝑑1⋅[exp⁡(𝑞(𝑉𝐿+𝑅𝑆⋅𝐼𝐿) 𝑛1⋅𝑘𝑇 )−1] +𝐼𝑑2⋅[exp⁡(𝑞−(𝑉𝐿+𝑅𝑠−𝐼𝐿) 𝑛2⋅𝑘−𝑇 )−1]+𝑉𝐿+𝑅𝑆⋅𝐼𝐿 𝑅𝑠ℎ 𝑿={𝐼𝑝ℎ,𝐼𝑑𝑑,𝐼𝑑2,𝑅𝑆,𝑅𝑠ℎ,𝑛1,𝑛2} (8) { 𝑓𝑀𝐷(𝑉𝐿,𝐼𝐿,𝑿)=𝐼𝐿− 𝑁𝑃⋅𝐼𝑀ℎ+𝑁𝑃⋅𝐼𝑑⋅[exp⁡(𝑞(𝑉𝐿/𝑁𝑠+𝑅𝑆⋅𝐼𝐿/𝑁𝑃) 𝑛⋅𝑘⋅𝑇 )−1] ⁡+𝑁𝑃⋅𝑉𝐿/𝑁𝑆+𝑅𝑦⋅𝐼𝐿 𝑅𝑠ℎ 𝑿= {𝐼𝑝ℎ,𝐼𝑑,𝑅𝑆,𝑅𝑠ℎ,𝑛} (9) Equation (10) employs RMSE as an objective function to objectively assess the total disparity between experimental and calculated data. RMSE⁡(𝑿)=√1 𝑁∑𝑘=1 𝑁 𝑓𝑘(𝑉𝐿,𝐼𝐿,𝑿)2 (10) In this formula, X represents the solution consisting of different unknown parameters, and N shows the number of actual measured data. 3.5. Error Metrics The paper investigates PV parameter extraction using the firefly algorithm, emphasizing its accuracy in estimating parameters for solar cells and PV modules. A three-step methodology assesses the algorithm’s efficacy, comparing its performance with other methods and evaluating error metrics like RE and IAE. The results indicate the firefly algorithm’s superior performance and accuracy, confirming its effectiveness in PV parameter estimation compared to alternative algorithms [37]. Figure 3. Equivalent circuit of a PV module [36]. 3.4. Objective Function The discrepancy between the determined calculated value and the actual measured value is objectively assessed using an objective function. The calculated data points of the SDM, DDM, and PV modules and error functions for the experiments are represented by Equations (7)–(9).      fSD(VL,IL,X)=IL−Iph +Id*expq(VL+Rg−IL) n−kT −1+VL+RS·IL RLh X=nIph,Id,RS,Rsh ,no(7)          fDD(VL,IL,X)=IL−Iph +Id1·hexpq(VL+RS·IL) n1·kT −1i +Id2·hexpq−(VL+Rs−IL) n2·k−T−1i+VL+RS·IL Rsh X=nIph,Idd,Id2,RS,Rsh,n1,n2o (8)        fMD(VL,IL,X)=IL−NP·IMh +NP·Id·hexpq(VL/Ns+RS·IL/NP) n·k·T−1i +NP·VL/NS+Ry·IL Rsh X=nIph,Id,RS,Rsh,no (9) Equation (10) employs RMSE as an objective function to objectively assess the total disparity between experimental and calculated data. RMSE(X) = r1 N∑N k=1fk(VL,IL,X)2(10) In this formula, Xrepresents the solution consisting of different unknown parameters, and Nshows the number of actual measured data. 3.5. Error Metrics The paper investigates PV parameter extraction using the firefly algorithm, emphasizing its accuracy in estimating parameters for solar cells and PV modules. A three-step methodology assesses the algorithm’s efficacy, comparing its performance with other methods and evaluating error metrics like RE and IAE. The results indicate the firefly algorithm’s superior performance and accuracy, confirming its effectiveness in PV parameter estimation compared to alternative algorithms [37]. In the reference [ 20 ], analysis of error metrics such as RMSE reveals TSA’s superior accuracy in parameter extraction compared to other algorithms. TSA exhibits faster con- Energies 2024,17, 1716 16 of 26 The fundamental necessities outlined in this article are presented in Table 3. Table 3. The advantages of solar PV systems. Requirements Description Optimization of solar PV parameters Maximize the efficiency of solar PV systems by accurately optimizing their parameters. Reduction in environmental impact Mitigate environmental pollution and reduce reliance on non-renewable energy sources like fossil fuels. Advancement of optimization techniques Represents a significant advancement in optimization techniques for solar PV systems. Enhanced stability and accuracy Improve the stability and accuracy of parameter optimization in PV modules and solar cells, thereby increasing the reliability and performance of solar energy systems. 3.8. The Challenges Tackled by This Paper The challenges tacked by this paper comprise the following: • Complexity of Parameter Optimization: The process of optimizing parameters within PV modules and solar cells entails grappling with intricate mathematical models and inherent uncertainties stemming from fluctuations in solar radiation and temperature; • Requirement for Improved Optimization Algorithms: Traditional optimization algorithms may encounter difficulties in adequately managing the intricacies of parameter optimization within solar PV systems, underscoring the necessity for novel approaches such as the COA algorithm proposed herein; • Attaining Global Optima: The task of identifying the global optimum solution for parameter optimization in solar PV systems is arduous due to the existence of numerous local optima and the expansive search space with high dimensions. 3.9. The Contributions Made by This Paper to the Field The contributions made by this paper to the corresponding field are as follows: (1) Innovative Optimization Strategy: Introducing a fresh optimization strategy utilizing the COA algorithm and chaotic functions, which exhibits superior effectiveness compared to existing meta-heuristic algorithms in terms of minimizing RMSE and standard deviation; (2) Enhanced Parameter Optimization: The proposed approach ensures more precise and consistent optimization of parameters within SDM, DDM, and PV modules, consequently improving power generation efficiency and the overall reliability of solar PV systems; (3) Potential for Future Research: The paper outlines future research prospects, including the exploration of LSTM neural networks for optimizing solar cell parameters and forecasting solar radiation and panel temperature. This indicates promising avenues for advancing techniques in solar energy optimization. 4. Results and Discussion The suggested approach is based on four popular PV models that were applied and proven through SDM assault in the DDM attack in the RTC France cell, RTC France cell, STP6-120/36 module, and Photowatt-PWP201 module. Current-voltage data for DDM and SDM were measured on a 57 mm diameter commercial silicon RTC. At 33 ◦ C, French solar cells were measured below 1000 W/m 2 . A total of 36 polycrystalline silicon cells comprising a Photowatt-PWP201 module were linked in series, and at 45 ◦ C, the currentvoltage data were measured. Moreover, a total of 36 polycrystalline silicon cells, measured at 55 ◦ C, made up the STP6-120/36 modules. A few earlier investigations were studied to gather the STP6-120/36 module’s current-voltage measurement data [ 29 , 38 ]. The proposed Energies 2024,17, 1716 17 of 26 algorithm was implemented on the MATLAB 2021 platform using an Intel ® i7-HQ CPU with 16 GB memory. Through parameter optimization for distinct PV modules employing the COA, the approach not only boosts output power but also reduces errors and standard deviation, surpassing traditional algorithms such as ITLBO, JSO, CPMPSO, WOA, SCA, GNDO, and MJSO. Furthermore, it outperformed competing algorithms in terms of RMSE and standard deviation, indicating its superior performance and suitability across various circuit configurations. Additionally, the method’s efficient execution time further bolsters its practical applicability. In summary, these results underscore the effectiveness and versatility of the proposed method in tackling diverse optimization challenges within solar energy systems. 4.1. The Range of Parameters Table 4demonstrates the SM55 and ST40 parameters, specifically their upper and lower ranges. In addition to the parameters used to implement the proposed algorithm, the number of repetitions is the considered variable, and the population size is 50. In the COA algorithm, r has a random value between zero and one, and I equals a random number that equals 1 or 2. The values of the chaotic function in the proposed method were set at z_0 = 0.125 and β= 2.59. Table 4. Lower and upper range in the modules [31]. Parameters Low Range Upper Range Iph(A)0 2∗Isc Isd(A)0100 ×10−6 Rs(Ω)0 2 Rsh(Ω)0 5000 n,n1,n21 4 4.2. Results Based on SDM Table 5displays the optimal parameters’ values extracted for the SDM circuit in the proposed method, namely the improved COA (ICOA), and compares ITLBO, JSO, CPMPSO, WOA, SCA, GNDO, MJSO, and COA. The experiments showed that all meta-heuristic algorithms except WOA and SCA have minimized RMSE values. Table 5. Comparison among optimal values of parameters in SDM. Algorithms Iph(A)Isd(A)Rs(Ω)Rsh(Ω)nRMSE ITLBO [21] 0.76078 3.11 ×10−70.03654 52.8897 1.47726 0.000773006 JSO [22] 0.76079 3.11 ×10−70.03654 52.8882 1.47727 0.000773006 CPMPSO [39] 0.76078 3.11 ×10−70.03654 52.8897 1.47726 0.000773006 WOA [38] 0.76162 3.86 ×10−70.03530 45.9308 1.49953 0.001085820 SCA [29] 0.7582 4.09 ×10−70.03595 68.8388 1.50500 0.002483415 GNDO [30] 0.76078 3.11 ×10−70.03654 52.8897 1.47726 0.000773006 MJSO [31] 0.76078 3.11 ×10−70.03654 52.8897 1.47726 0.000773006 COA [15] 0.76078 3.11 ×10−70.03654 52.8897 1.47726 0.000773006 ICOA (proposed method) 0.76078 3.11 ×10−70.03654 52.8897 1.47726 0.000773006 Figure 9displays the I-V and P-V characteristic curves based on the ICOA’s extracted optimal parameters for SDM. It is clear from Figure 9that there is a good agreement between the measured and simulated ICOA data. Energies 2024,17, 1716 18 of 26 Energies 2024, 17, x FOR PEER REVIEW 18 of 27 Table 4. Lower and upper range in the modules [31]. Parameters Low Range Upper Range 𝐼𝑝ℎ(A) 0 2∗Isc 𝐼𝑠𝑑(A) 0 100 × 10−6 𝑅𝑠(Ω) 0 2 𝑅𝑠ℎ(Ω) 0 5000 n, 𝑛1,⁡𝑛2 1 4 4.2. Results Based on SDM Table 5 displays the optimal parameters’ values extracted for the SDM circuit in the proposed method, namely the improved COA (ICOA), and compares ITLBO, JSO, CPMPSO, WOA, SCA, GNDO, MJSO, and COA. The experiments showed that all metaheuristic algorithms except WOA and SCA have minimized RMSE values. Table 5. Comparison among optimal values of parameters in SDM. Algorithms 𝑰𝒑𝒉(𝑨) 𝑰𝒔𝒅(𝑨) 𝑹𝒔(𝛀) 𝑹𝒔𝒉(𝛀) n RMSE ITLBO [21] 0.76078 3.11 × 10−7 0.03654 52.8897 1.47726 0.000773006 JSO [22] 0.76079 3.11 × 10−7 0.03654 52.8882 1.47727 0.000773006 CPMPSO [39] 0.76078 3.11 × 10−7 0.03654 52.8897 1.47726 0.000773006 WOA [38] 0.76162 3.86 × 10−7 0.03530 45.9308 1.49953 0.001085820 SCA [29] 0.7582 4.09 × 10−7 0.03595 68.8388 1.50500 0.002483415 GNDO [30] 0.76078 3.11 × 10−7 0.03654 52.8897 1.47726 0.000773006 MJSO [31] 0.76078 3.11 × 10−7 0.03654 52.8897 1.47726 0.000773006 COA [15] 0.76078 3.11 × 10−7 0.03654 52.8897 1.47726 0.000773006 ICOA (proposed method) 0.76078 3.11 × 10−7 0.03654 52.8897 1.47726 0.000773006 Figure 9 displays the I-V and P-V characteristic curves based on the ICOA’s extracted optimal parameters for SDM. It is clear from Figure 9 that there is a good agreement between the measured and simulated ICOA data. (a) (b) Energies 2024, 17, x FOR PEER REVIEW 19 of 27 (c) Figure 9. Comparison of the measured and estimated data obtained by ICOA for SDM: (a) I-V characteristic, (b) P-V characteristic, and (c) IAE curves. 4.3. Results Based on DDM Table 6 compares the value of optimal parameters extracted for the DDM circuit using the proposed method, ITLBO, JSO, CPMPSO, WOA, SCA, GNDO, MJSO, and COA. Table 6. Comparison among optimal values of parameters of DDM. Algorithms 𝑰𝒑𝒉(𝑨) 𝑰𝒔𝒅𝟏(𝑨) Rs(Ω) Rsh(Ω) n1 𝑰𝒔𝒅𝟐(𝑨) n2 RMSE ITLBO [21] 0.7608 2.47 × 10−7 0.0368 53.9599 1.4579 4.78 × 10−7 1.9949 0.000742264 JSO [22] 0.7608 5.38 × 10−7 0.0371 54.4640 1.7980 1.61 × 10−7 1.4262 0.000754167 CPMPSO [39] 0.7608 7.03 × 10−8 0.0378 56.2715 1.3642 1.00 × 10−6 1.7963 0.000741937 WOA [38] 0.7608 2.67 × 10−7 0.0368 51.8538 1.4662 4.10 × 10−8 1.6133 0.000776464 SCA [29] 0.7684 0.00 × 10+0 0.0324 38.3064 1.1740 3.84 × 10−7 1.4970 0.007351184 GNDO [30] 0.7608 1.00 × 10−6 0.0373 55.6033 1.9051 1.40 × 10−7 1.4130 0.000742327 MJSO [31] 0.7608 7.03 × 10−8 0.0378 56.2715 1.3642 1.00 × 10−6 1.7963 0.000741937 COA [15] 0.7607 7.01 × 10−8 0.0377 56.2716 1.3641 1.00 × 10−6 1.7964 0.000741936 ICOA (proposed method) 0.7608 7.02 × 10−8 0.0377 56.2714 1.3642 1.00 × 10−6 1.7962 0.000741936 The comparison and tests results show that in the DDM circuit, the proposed method provided more optimal values for the DDM circuit than the JSO, CPMPSO, WOA, SCA, GNDO, MJSO, and COA methods. The RMSE error index in the proposed method shows a lower value. The COA algorithm ranks second after the proposed method, and the CPMPSO and MJSO algorithms rank third. Figure 10 displays the I-V and P-V characteristic curves based on the ICOA’s extracted optimal parameters for DDM. It is clear from Figure 10 that there is a good agreement between the measured and simulated ICOA data. Figure 9. Comparison of the measured and estimated data obtained by ICOA for SDM: (a) I-V characteristic, (b) P-V characteristic, and (c) IAE curves. 4.3. Results Based on DDM Table 6compares the value of optimal parameters extracted for the DDM circuit using the proposed method, ITLBO, JSO, CPMPSO, WOA, SCA, GNDO, MJSO, and COA. Table 6. Comparison among optimal values of parameters of DDM. Algorithms Iph(A)Isd1(A)Rs(Ω)Rsh(Ω)n1Isd2(A)n2RMSE ITLBO [21] 0.7608 2.47 × 10 −70.0368 53.9599 1.4579 4.78 × 10 −71.9949 0.000742264 JSO [22] 0.7608 5.38 × 10 −70.0371 54.4640 1.7980 1.61 × 10 −71.4262 0.000754167 CPMPSO [39] 0.7608 7.03 × 10 −80.0378 56.2715 1.3642 1.00 × 10 −61.7963 0.000741937 WOA [38] 0.7608 2.67 × 10 −70.0368 51.8538 1.4662 4.10 × 10 −81.6133 0.000776464 SCA [29] 0.7684 0.00 ×10+0 0.0324 38.3064 1.1740 3.84 × 10 −71.4970 0.007351184 GNDO [30] 0.7608 1.00 × 10 −60.0373 55.6033 1.9051 1.40 × 10 −71.4130 0.000742327 MJSO [31] 0.7608 7.03 × 10 −80.0378 56.2715 1.3642 1.00 × 10 −61.7963 0.000741937 COA [15] 0.7607 7.01 × 10 −80.0377 56.2716 1.3641 1.00 × 10 −61.7964 0.000741936 ICOA (proposed method) 0.7608 7.02 × 10 −80.0377 56.2714 1.3642 1.00 × 10 −61.7962 0.000741936 The comparison and tests results show that in the DDM circuit, the proposed method provided more optimal values for the DDM circuit than the JSO, CPMPSO, WOA, SCA, GNDO, MJSO, and COA methods. The RMSE error index in the proposed method shows Energies 2024,17, 1716 19 of 26 a lower value. The COA algorithm ranks second after the proposed method, and the CPMPSO and MJSO algorithms rank third. Figure 10 displays the I-V and P-V characteristic curves based on the ICOA’s extracted optimal parameters for DDM. It is clear from Figure 10 that there is a good agreement between the measured and simulated ICOA data. Energies 2024, 17, x FOR PEER REVIEW 20 of 27 Figure 10. Comparison of the measured and estimated data obtained by ICOA for DDM model: (a) I-V characteristic, (b) P-V characteristic, and (c) IAE curves. 4.4. Results Based on STP6-120/36 Table 7 compares the value of optimal parameters of the STP6-120/36 circuit in the proposed method with similar meta-heuristic methods. The conducted tests show that in the STP6-120/36 circuit, the proposed method’s RMSE value was lower than the metaheuristic algorithms JSO, CPMPSO, WOA, SCA, GNDO, MJSO, and COA. In these experiments, the COA method’s error is the second lowest in terms of minimum. On the other hand, the experiments show that the CPMPSO, ITLBO, MJSO, and GNDO methods rank third for RMSE error. Table 7. Comparison among optimal parameter values in STP6-120/36. Algorithms 𝑰𝒑𝒉(𝑨) 𝑰𝒔𝒅(𝑨) 𝑹𝒔(𝛀) 𝑹𝒔𝒉(𝛀) N RMSE ITLBO [21] 7.47528 1.93 × 10−6 0.16891 570.1972 44.80042 0.014251063 JSO [22] 7.47525 1.93 × 10−6 0.16890 571.5660 44.80254 0.014251066 CPMPSO [39] 7.47528 1.93 × 10−6 0.16891 570.1975 44.80042 0.014251063 WOA [38] 7.50318 3.27 × 10−6 0.15781 307.7831 46.40846 0.017581962 SCA [29] 7.56027 1.70 × 10−6 0.17318 323.9495 44.38346 0.052443825 GNDO [30] 7.47528 1.93 × 10−6 0.16891 570.1972 44.80042 0.014251063 MJSO [31] 7.47528 1.93 × 10−6 0.16891 570.1975 44.80042 0.014251063 COA [15] 7.47528 1.92 × 10−6 0.16891 570.1975 44.80041 0.014251063 ICOA (proposed method) 7.47528 1.93 × 10−6 0.16891 570.1974 44.80041 0.014251063 Figure 10. Comparison of the measured and estimated data obtained by ICOA for DDM model: (a) I-V characteristic, (b) P-V characteristic, and (c) IAE curves. 4.4. Results Based on STP6-120/36 Table 7compares the value of optimal parameters of the STP6-120/36 circuit in the proposed method with similar meta-heuristic methods. The conducted tests show that in the STP6-120/36 circuit, the proposed method’s RMSE value was lower than the meta-heuristic algorithms JSO, CPMPSO, WOA, SCA, GNDO, MJSO, and COA. In these experiments, the COA method’s error is the second lowest in terms of minimum. On the other hand, the experiments show that the CPMPSO, ITLBO, MJSO, and GNDO methods rank third for RMSE error. According to the experimentation, the worst algorithm for this situation is the SCA algorithm, which has the highest error among the compared algorithms. Figure 11 displays the I-V and P-V characteristic curves based on the ICOA’s extracted optimal parameters for STP6-120/36. It is clear from Figure 11 that there is a good agreement between the measured and simulated ICOA data. Energies 2024,17, 1716 20 of 26 Table 7. Comparison among optimal parameter values in STP6-120/36. Algorithms Iph(A)Isd(A)Rs(Ω)Rsh(Ω)NRMSE ITLBO [21] 7.47528 1.93 ×10−60.16891 570.1972 44.80042 0.014251063 JSO [22] 7.47525 1.93 ×10−60.16890 571.5660 44.80254 0.014251066 CPMPSO [39] 7.47528 1.93 ×10−60.16891 570.1975 44.80042 0.014251063 WOA [38] 7.50318 3.27 ×10−60.15781 307.7831 46.40846 0.017581962 SCA [29] 7.56027 1.70 ×10−60.17318 323.9495 44.38346 0.052443825 GNDO [30] 7.47528 1.93 ×10−60.16891 570.1972 44.80042 0.014251063 MJSO [31] 7.47528 1.93 ×10−60.16891 570.1975 44.80042 0.014251063 COA [15] 7.47528 1.92 ×10−60.16891 570.1975 44.80041 0.014251063 ICOA (proposed method) 7.47528 1.93 ×10−60.16891 570.1974 44.80041 0.014251063 Energies 2024, 17, x FOR PEER REVIEW 21 of 27 According to the experimentation, the worst algorithm for this situation is the SCA algorithm, which has the highest error among the compared algorithms. Figure 11 displays the I-V and P-V characteristic curves based on the ICOA’s extracted optimal parameters for STP6-120/36. It is clear from Figure 11 that there is a good agreement between the measured and simulated ICOA data. Figure 11. Comparison of the measured and estimated data obtained by ICOA for STP6-120/36 model, (a) I-V characteristic, (b) P-V characteristic, and (c) IAE curves. 4.5. Ranking This section discusses the ranking of the proposed algorithm and competing algorithms such as WOA, GWO, HHO, AVOA, and COA in two error indicators, namely RMSE and standard deviation, for SMD, DDM, and PV modules, respectively, which are shown in Figures 12 and 13. Figure 11. Comparison of the measured and estimated data obtained by ICOA for STP6-120/36 model, (a) I-V characteristic, (b) P-V characteristic, and (c) IAE curves. 4.5. Ranking This section discusses the ranking of the proposed algorithm and competing algorithms such as WOA, GWO, HHO, AVOA, and COA in two error indicators, namely RMSE and standard deviation, for SMD, DDM, and PV modules, respectively, which are shown in Figures 12 and 13. Energies 2024,17, 1716 21 of 26 Energies 2024, 17, x FOR PEER REVIEW 22 of 27 Figure 12. Ranking of algorithms in terms of the minimum RMSE error. Figure 13. Algorithm ranking based on standard deviation (SD) index. In the experiments, Friedman’s test was applied to rank different algorithms. In Friedman’s test, any algorithm that shows a lower value has a better ranking to find the optimal solution. A lower number means the algorithm has obtained a better ranking in finding the optimal solution. According to the experiments, if the circuit is SDM-type, the ranking of WOA, GWO, HHO, AVOA, JSO, COA algorithm, and the proposed method is 1.79, 1.87, 1.98, 1.73, 1.56, 1.48, and 1.36, respectively. In this case, the proposed method obtained the best ranking, and HHO showed the worst performance. Experiments pertaining to the DDM circuit show that the ranks of algorithms, including WOA, GWO, HHO, AVOA, JSO, COA, and the proposed method for calculating the minimum RMSE are 2.54, 2.73, 2.36, 1.73, 1.52, 1.39, and 1.24, respectively, and the proposed method ranks best in minimizing RMSE. The PV circuit’s rating values of WOA, GWO, HHO, AVOA, JSO, COA, and the proposed method are 1.63, 1.88, 1.79, 1.82, 1.64, 1.25, and 1.13, respectively, so the proposed method again showed the top performance. Figure 12. Ranking of algorithms in terms of the minimum RMSE error. Energies 2024, 17, x FOR PEER REVIEW 22 of 27 Figure 12. Ranking of algorithms in terms of the minimum RMSE error. Figure 13. Algorithm ranking based on standard deviation (SD) index. In the experiments, Friedman’s test was applied to rank different algorithms. In Friedman’s test, any algorithm that shows a lower value has a better ranking to find the optimal solution. A lower number means the algorithm has obtained a better ranking in finding the optimal solution. According to the experiments, if the circuit is SDM-type, the ranking of WOA, GWO, HHO, AVOA, JSO, COA algorithm, and the proposed method is 1.79, 1.87, 1.98, 1.73, 1.56, 1.48, and 1.36, respectively. In this case, the proposed method obtained the best ranking, and HHO showed the worst performance. Experiments pertaining to the DDM circuit show that the ranks of algorithms, including WOA, GWO, HHO, AVOA, JSO, COA, and the proposed method for calculating the minimum RMSE are 2.54, 2.73, 2.36, 1.73, 1.52, 1.39, and 1.24, respectively, and the proposed method ranks best in minimizing RMSE. The PV circuit’s rating values of WOA, GWO, HHO, AVOA, JSO, COA, and the proposed method are 1.63, 1.88, 1.79, 1.82, 1.64, 1.25, and 1.13, respectively, so the proposed method again showed the top performance. Figure 13. Algorithm ranking based on standard deviation (SD) index. In the experiments, Friedman’s test was applied to rank different algorithms. In Friedman’s test, any algorithm that shows a lower value has a better ranking to find the optimal solution. A lower number means the algorithm has obtained a better ranking in finding the optimal solution. According to the experiments, if the circuit is SDM-type, the ranking of WOA, GWO, HHO, AVOA, JSO, COA algorithm, and the proposed method is 1.79, 1.87, 1.98, 1.73, 1.56, 1.48, and 1.36, respectively. In this case, the proposed method obtained the best ranking, and HHO showed the worst performance. Experiments pertaining to the DDM circuit show that the ranks of algorithms, including WOA, GWO, HHO, AVOA, JSO, COA, and the proposed method for calculating the minimum RMSE are 2.54, 2.73, 2.36, 1.73, 1.52, 1.39, and 1.24, respectively, and the proposed method ranks best in minimizing RMSE. The PV circuit’s rating values of WOA, GWO, HHO, AVOA, JSO, COA, and the proposed method are 1.63, 1.88, 1.79, 1.82, 1.64, 1.25, and 1.13, respectively, so the proposed method again showed the top performance. Energies 2024,17, 1716 22 of 26 Based on the simulations conducted in MATLAB, the proposed method ranks first in the optimization of SDM, DDM, and PV parameters. The proposed method also increases the optimal calculation rank in these three circuits by 8.1%, 10.79%, and 9.6%, respectively, compared to the COA algorithm. Figure 13 displays the average rank of meta-heuristic algorithms and the proposed algorithm in the standard deviation index. Figure 13 compares the standard deviation (SD) of three modes, namely SDM, DDM, and PV, to the SD values of the proposed method and meta-heuristic algorithms. The standard deviation index is an essential and critical index for measuring the optimization stability for optimizing the parameters of DDM, PV, and SDM circuits. The rank of the proposed algorithm in the standard deviation index is equal to 1.22, and it has the lowest standard deviation among the competing algorithms. This means that the proposed algorithm has more stability in terms of optimizing the parameters of SDM, DDM, and PV circuits than the WOA, GWO, HHO, AVOA, JSO, and COA algorithms. The worst algorithm in terms of stability in finding optimal solutions is the HHO algorithm. The second meta-heuristic algorithm in terms of stability is the WOA algorithm. 4.6. Time Complexity To ensure equitable comparison, we incorporated outcomes from alternative methods rather than solely relying on external references. Ensuring uniformity, we maintained identical objective functions and parameters across all experiments as outlined in the original papers. Furthermore, each method received an equal number of attempts to address the problem, enabling a fair comparison within comparable computational constraints. By concentrating on methods addressing identical optimization challenges, we assessed the effectiveness of our proposed model against existing approaches within a controlled framework. This method facilitated an impartial and straightforward evaluation of the strengths and limitations of each approach. The improved COA algorithm has more equations and complexity than the COA algorithm, so it was expected to be more time-consuming than the COA algorithm. However, contrary to the expectation of the ICOA algorithm, it needs fewer iterations to reach an error level because it has faster convergence than the COA algorithm. Figure 14 compares the optimization time of the solar system parameters in the proposed method and other meta-heuristic methods. Energies 2024, 17, x FOR PEER REVIEW 23 of 27 Based on the simulations conducted in MATLAB, the proposed method ranks first in the optimization of SDM, DDM, and PV parameters. The proposed method also increases the optimal calculation rank in these three circuits by 8.1%, 10.79%, and 9.6%, respectively, compared to the COA algorithm. Figure 13 displays the average rank of meta-heuristic algorithms and the proposed algorithm in the standard deviation index . Figure 13 compares the standard deviation (SD) of three modes, namely SDM, DDM, and PV, to the SD values of the proposed method and meta-heuristic algorithms. The standard deviation index is an essential and critical index for measuring the optimization stability for optimizing the parameters of DDM, PV, and SDM circuits. The rank of the proposed algorithm in the standard deviation index is equal to 1.22, and it has the lowest standard deviation among the competing algorithms. This means that the proposed algorithm has more stability in terms of optimizing the parameters of SDM, DDM, and PV circuits than the WOA, GWO, HHO, AVOA, JSO, and COA algorithms. The worst algorithm in terms of stability in finding optimal solutions is the HHO algorithm. The second meta-heuristic algorithm in terms of stability is the WOA algorithm. 4.6. Time Complexity To ensure equitable comparison, we incorporated outcomes from alternative methods rather than solely relying on external references. Ensuring uniformity, we maintained identical objective functions and parameters across all experiments as outlined in the original papers. Furthermore, each method received an equal number of attempts to address the problem, enabling a fair comparison within comparable computational constraints. By concentrating on methods addressing identical optimization challenges, we assessed the effectiveness of our proposed model against existing approaches within a controlled framework. This method facilitated an impartial and straightforward evaluation of the strengths and limitations of each approach. The improved COA algorithm has more equations and complexity than the COA algorithm, so it was expected to be more time-consuming than the COA algorithm. However, contrary to the expectation of the ICOA algorithm, it needs fewer iterations to reach an error level because it has faster convergence than the COA algorithm. Figure 14 compares the optimization time of the solar system parameters in the proposed method and other meta-heuristic methods. Figure 14. Comparison of the calculation time of optimal PV parameters. Figure 14. Comparison of the calculation time of optimal PV parameters. The experiments conducted in MATLAB show that the proposed algorithm’s execution time is about 6.34 s, and it needs less time to reach a certain error level than other algorithms. Energies 2024,17, 1716 23 of 26 The performance of the COA algorithm is slightly worse than the proposed method in terms of time index, but it is in second place for execution time. Among the compared algorithms, the worst algorithm in terms of time index is the HHO algorithm. The reason behind the significant execution time of the HHO algorithm is the large number of equations and the high complexity of this algorithm. 5. Conclusions Global power demand is escalating due to industrial expansion and population growth, with electricity being the primary energy source for industries. However, conventional electricity production methods such as fossil fuels pose environmental challenges. Transitioning to solar energy through PV modules offers a renewable solution to mitigate environmental pollution. Solar cells efficiently convert solar energy into electrical energy, yet their output fluctuates due to varying radiation intensity and angles, posing a challenge for maximizing power generation. To address this, the proposed method utilized the COA to optimize parameters for different PV modules, enhancing output power by minimizing errors and reducing standard deviation compared to conventional algorithms. However, challenges persist, including uncertainty in finding optimal solutions and reliance on precise input data, which may hinder optimization efficacy. Acknowledging practical obstacles like hardware limitations and maintenance complexities is crucial for real-world implementation, underscoring the need for further validation of scalability and applicability in large-scale solar energy systems. Future research could focus on mitigating uncertainty, enhancing algorithm robustness, and integrating predictive modeling techniques like LSTM neural networks to improve parameter optimization and solar energy forecasting, aiming to address the inherent limitations and guide future advancements in solar energy optimization. Author Contributions: Conceptualization, R.E. and A.H.; methodology, A.H.; software, R.E.; validation, A.H., J.R. and J.M.L.-G.; formal analysis, R.E. and A.H.; investigation, R.E.; resources, A.H.; data curation, R.E.; writing—original draft preparation, R.E. and A.H.; writing—review and editing, J.R. and J.M.L.-G.; visualization, R.E.; supervision, J.R. and J.M.L.-G.; project administration, J.R.; funding acquisition, J.M.L.-G. All authors have read and agreed to the published version of the manuscript. Funding: The authors were supported by the Vitoria-Gasteiz Mobility Lab Foundation, an organization of the government of the Provincial Council of Araba and the City Council of Vitoria-Gasteiz through the following project grant (“Utilización de drones en la movilidad de mercancías”). Institutional Review Board Statement: Not applicable. Informed Consent Statement: Not applicable. Data Availability Statement: The data presented in this study are available on request from the corresponding author. The data are not publicly available due to privacy. Conflicts of Interest: The authors declare no conflicts of interest. Abbreviations Definitions of abbreviations used throughout the article. COA Coati optimization algorithm PV Photovoltaic SDM Single-diode model DDM Double-diode model MPPT Maximum power point tracking IEA Energy agency MPP Maximum power point PWM Pulse width modulation GWO Gray wolf optimization NGO Northern goshawk optimization EVs Electric vehicles Energies 2024,17, 1716 24 of 26 HESS Hybrid energy storage system ICOA Improved coati optimization algorithm SD Standard deviation MHs Meta-heuristics RE Relative error MJSO Modified artificial jellyfish search optimizer GNDO Generalized normal distribution optimization SCA Sine cosine algorithm JSO Jellyfish search optimizer QSO Queue search optimization DE Differential evolution PSO Particle swarm optimization GA Genetic algorithm GOA Grasshopper optimization algorithms BA Bat algorithms HHO Harris hawks optimizer AOA Arithmetic optimization algorithm ChOA Chimp optimization algorithm WOA Whale optimization algorithm IHHO-VMD Improved Harris hawk optimization algorithm–variational mode decomposition MCDM Multiple-criteria decision making NSGA-II Non-dominated sorting genetic algorithm II PLC Programmable logic controller MGO Mountain gazelle optimizer IAE Individual absolute error RMSE Root mean square error TSA Tree seed algorithm CPMPSO Classified perturbation mutation-based particle swarm optimization ITLBO Improved teaching-learning-based optimization AVOA African vultures optimization algorithm IAOA Arithmetic optimization algorithm References 1. Rauf, A.; Nureen, N.; Irfan, M.; Ali, M. The current developments and future prospects of solar photovoltaic industry in an emerging economy of India. Environ. Sci. Pollut. Res. 2023,30, 46270–46281. [CrossRef] [PubMed] 2. Yusupov, Z.; Almagrahi, N.; Yaghoubi, E.; Yaghoubi, E.; Habbal, A.; Kodirov, D. Modeling and Control of Decentralized Microgrid Based on Renewable Energy and Electric Vehicle Charging Station. In Proceedings of the 12th World Conference “Intelligent System for Industrial Automation” (WCIS-2022), Tashkent, Uzbekistan, 25–26 November 2022; pp. 96–102. 3. Toufik, T.; Dekkiche, M.; Denai, M. Techno-Economic Comparative Study of Grid-Connected Pv/Reformer/Fc Hybrid Systems with Distinct Solar Tracking Systems. Energy Convers. Manag. X 2023,18, 100360. [CrossRef] 4. Sharma, A.; Sharma, A.; Dasgotra, A.; Jately, V.; Ram, M.; Rajput, S.; Averbukh, M.; Azzopardi, B. Opposition-based tunicate swarm algorithm for parameter optimization of solar cells. IEEE Access 2021,9, 125590–125602. [CrossRef] 5. Van Gompel, J.; Spina, D.; Develder, C. Cost-effective fault diagnosis of nearby photovoltaic systems using graph neural networks. Energy 2023,266, 126444. [CrossRef] 6. Wu, X.; Liao, B.; Su, Y.; Li, S. Multi-objective and multi-algorithm operation optimization of integrated energy system considering ground source energy and solar energy. Int. J. Electr. Power Energy Syst. 2023,144, 108529. [CrossRef] 7. Balamurugan, N.; Karuppasamy, P.; Ramasamy, P. Investigation on Different Crystal Grains from the Multi-crystalline Silicon (mc-Si) Wafer. Silicon 2023,15, 1465–1474. [CrossRef] 8. Irshad, A.S. Design and comparative analysis of grid-connected BIPV system with monocrystalline silicon and polycrystalline silicon in Kandahar climate. MATEC Web Conf. 2023,374, 3002. [CrossRef] 9. Hamed, S.B.; Abid, A.; Ben Hamed, M.; Sbita, L.; Bajaj, M.; Ghoneim, S.S.; Zawbaa, H.M.; Kamel, S. A robust MPPT approach based on first-order sliding mode for triple-junction photovoltaic power system supplying electric vehicle. Energy Rep. 2023,9, 4275–4297. [CrossRef] 10. Kumar, C.M.S.; Singh, S.; Gupta, M.K.; Nimdeo, Y.M.; Raushan, R.; Deorankar, A.V.; Kumar, T.A.; Rout, P.K.; Chanotiya, C.S.; Pakhale, V.D.; et al. Solar energy: A promising renewable source for meeting energy demand in Indian agriculture applications. Sustain. Energy Technol. Assess. 2023,55, 102905. [CrossRef] Energies 2024,17, 1716 25 of 26 11. Afzal, A.; Buradi, A.; Jilte, R.; Shaik, S.; Kaladgi, A.R.; Arıcı, M.; Lee, C.T.; Nižeti´c, S. Optimizing the thermal performance of solar energy devices using meta-heuristic algorithms: A critical review. Renew. Sustain. Energy Rev. 2023,173, 112903. [CrossRef] 12. Shorabeh, S.N.; Samany, N.N.; Minaei, F.; Firozjaei, H.K.; Homaee, M.; Boloorani, A.D. A decision model based on decision tree and particle swarm optimization algorithms to identify optimal locations for solar power plants construction in Iran. Renew. Energy 2022,187, 56–67. [CrossRef] 13. Wu, H.; Deng, F.; Tan, H. Research on parametric design method of solar photovoltaic utilization potential of nearly zero-energy high-rise residential building based on genetic algorithm. J. Clean. Prod. 2022,368, 133169. [CrossRef] 14. Bonthagorla, P.K.; Mikkili, S. A Novel Hybrid Slime Mould MPPT Technique for BL-HC Configured Solar PV System Under PSCs. J. Control. Autom. Electr. Syst. 2023,34, 782–795. [CrossRef] 15. Dehghani, M.; Montazeri, Z.; Trojovská, E.; Trojovský, P. Coati Optimization Algorithm: A new bio-inspired metaheuristic algorithm for solving optimization problems. Knowl. Based Syst. 2023,259, 110011. [CrossRef] 16. Zhang, Y.; Wu, Y.; Li, L.; Liu, Z. A Hybrid Energy Storage System Strategy for Smoothing Photovoltaic Power Fluctuation Based on Improved HHO-VMD. Int. J. Photoenergy 2023,2023, 9633843. [CrossRef] 17. Zadehbagheri, M.; Abbasi, A.R. Energy cost optimization in distribution network considering hybrid electric vehicle and photovoltaic using modified whale optimization algorithm. J. Supercomput. 2023,79, 14427–14456. [CrossRef] 18. Aguila-Leon, J.; Vargas-Salgado, C.; Chiñas-Palacios, C.; Díaz-Bello, D. Solar photovoltaic Maximum Power Point Tracking controller optimization using Grey Wolf Optimizer: A performance comparison between bio-inspired and traditional algorithms. Expert Syst. Appl. 2023,211, 118700. [CrossRef] 19. Abbassi, A.; Ben Mehrez, R.; Bensalem, Y.; Abbassi, R.; Kchaou, M.; Jemli, M.; Abualigah, L.; Altalhi, M. Improved Arithmetic Optimization Algorithm for Parameters Extraction of Photovoltaic Solar Cell Single-Diode Model. Arab. J. Sci. Eng. 2022,47, 10435–10451. [CrossRef] 20. Be¸skirli, A.; Da˘g, ˙ I. Parameter extraction for photovoltaic models with tree seed algorithm. Energy Rep. 2023,9, 174–185. [CrossRef] 21. Li, S.; Gong, W.; Yan, X.; Hu, C.; Bai, D.; Wang, L.; Gao, L. Parameter extraction of photovoltaic models using an improved teaching-learning-based optimization. Energy Convers. Manag. 2019,186, 293–305. [CrossRef] 22. Chou, J.-S.; Truong, D.-N. A novel metaheuristic optimizer inspired by behavior of jellyfish in ocean. Appl. Math. Comput. 2021, 389, 125535. [CrossRef] 23. Patra, B.; Nema, P.; Khan, M.Z.; Khan, O. Optimization of solar energy using MPPT techniques and industry 4.0 modelling. Sustain. Oper. Comput. 2023,4, 22–28. [CrossRef] 24. Rayaguru, N.K.; Lindsay, N.M.; Crespo, R.G.; Raja, S.P. Hybrid bat–grasshopper and bat–modified multiverse optimization for solar photovoltaics maximum power generation. Comput. Electr. Eng. 2023,106, 108596. [CrossRef] 25. Louzazni, M.; Khouya, A.; Amechnoue, K.; Gandelli, A.; Mussetta, M.; Crăciunescu, A. Metaheuristic algorithm for photovoltaic parameters: Comparative study and prediction with a firefly algorithm. Appl. Sci. 2018,8, 339. [CrossRef] 26. ´ Calasan, M.; Aleem, S.H.A.; Zobaa, A.F. On the root mean square error (RMSE) calculation for parameter estimation of photovoltaic models: A novel exact analytical solution based on Lambert W function. Energy Convers. Manag. 2020,210, 112716. [CrossRef] 27. Nunes, H.; Pombo, J.; Bento, P.; Mariano, S.; Calado, M. Collaborative swarm intelligence to estimate PV parameters. Energy Convers. Manag. 2019,185, 866–890. [CrossRef] 28. Abbassi, R.; Saidi, S.; Urooj, S.; Alhasnawi, B.N.; Alawad, M.A.; Premkumar, M. An Accurate Metaheuristic Mountain Gazelle Optimizer for Parameter Estimation of Singleand Double-Diode Photovoltaic Cell Models. Mathematics 2023,11, 4565. [CrossRef] 29. Mirjalili, S. SCA: A Sine Cosine Algorithm for solving optimization problems. Knowl. Based Syst. 2016,96, 120–133. [CrossRef] 30. Zhang, Y.; Jin, Z.; Mirjalili, S. Generalized normal distribution optimization and its applications in parameter extraction of photovoltaic models. Energy Convers. Manag. 2020,224, 113301. [CrossRef] 31. Abdel-Basset, M.; Mohamed, R.; Chakrabortty, R.K.; Ryan, M.J.; El-Fergany, A. An Improved Artificial Jellyfish Search Optimizer for Parameter Identification of Photovoltaic Models. Energies 2021,14, 1867. [CrossRef] 32. Perles, L.; Martins, T.F.; Barreto, W.T.G.; de Macedo, G.C.; Herrera, H.M.; Mathias, L.A.; Labruna, M.B.; Barros-Battesti, D.M.; Machado, R.Z.; André, M.R. Diversity and Seasonal Dynamics of Ticks on Ring-Tailed Coatis Nasua nasua (Carnivora: Procyonidae) in Two Urban Areas from Midwestern Brazil. Animals 2022,12, 293. [CrossRef] [PubMed] 33. Hasanien, H.M.; Alsaleh, I.; Alassaf, A.; Alateeq, A. Enhanced coati optimization algorithm-based optimal power flow including renewable energy uncertainties and electric vehicles. Energy 2023,283, 129069. [CrossRef] 34. Sun, X.; Zhu, L.; Liu, D. Quantification of early bruises on blueberries using hyperspectral reflectance imaging coupled with band ratio and improved multi-threshold coati optimization algorithm method. Microchem. J. 2024,199, 110078. [CrossRef] 35. Yin, X.; Tian, H.; Zhang, F.; Li, A. Quantitative analysis of millet mixtures based on terahertz time-domain spectroscopy and improved coati optimization algorithm. Spectrosc. Lett. 2024,57, 31–44. [CrossRef] 36. Jian, X.; Cao, Y. A Chaotic Second Order Oscillation JAYA Algorithm for Parameter Extraction of Photovoltaic Models. Photonics 2022,9, 131. [CrossRef] 37. Louzazni, M.; Khouya, A.; Amechnoue, K.; Mussetta, M.; Crăciunescu, A. Comparison and evaluation of statistical criteria in the solar cell and photovoltaic module parameters’ extraction. Int. J. Ambient. Energy 2020,41, 1482–1494. [CrossRef]