Constraint estimation in three‐diode solar photovoltaic model using Gaussian and Cauchy mutation‐based hunger games search optimizer and enhanced Newton–Raphson method

IET Renewable Power GenerationVolume 16, Issue 8 p. 1733-1772 ORIGINAL RESEARCHOpen Access Constraint estimation in three-diode solar photovoltaic model using Gaussian and Cauchy mutation-based hunger games search optimizer and enhanced Newton–Raphson method Manoharan Premkumar, Corresponding Author Manoharan Premkumar [email protected] orcid.org/0000-0003-1032-4634 Department of Electrical and Electronics Engineering, Dayananda Sagar College of Engineering, Bengaluru, Karnataka, 560078 India Correspondence Hassan Haes Alhelou, Department of Electrical Power Engineering, Tishreen University, 2230 Lattakia, Syria Arab Republic. Email: [email protected] M. Premkumar, Department of Electrical and Electronics Engineering, Dayananda Sagar College of Engineering, Bengaluru, Karnataka, India. Email: [email protected]Search for more papers by this authorPradeep Jangir, Pradeep Jangir Power and Energy Sector, Rajasthan Rajya Vidyut Prasaran Nigam Ltd., Sikar, Rajasthan, 332025 IndiaSearch for more papers by this authorChandrasekaran Kumar, Chandrasekaran Kumar orcid.org/0000-0002-1132-4794 Department of Electrical and Electronics Engineering, M. Kumarasamy College of Engineering, Karur, Tamil Nadu, 639113 IndiaSearch for more papers by this authorSomasundaram David Thanasingh Sundarsingh Jebaseelan, Somasundaram David Thanasingh Sundarsingh Jebaseelan Department of Electrical and Electronics Engineering, Sathyabama Institute of Science and Technology, Chennai, Tamil Nadu, 600119 IndiaSearch for more papers by this authorHassan Haes Alhelou, Corresponding Author Hassan Haes Alhelou [email protected] orcid.org/0000-0001-8176-1589 Department of Electrical Power Engineering, Tishreen University, Lattakia, 2230 Syrian Arab Republic Correspondence Hassan Haes Alhelou, Department of Electrical Power Engineering, Tishreen University, 2230 Lattakia, Syria Arab Republic. Email: [email protected] M. Premkumar, Department of Electrical and Electronics Engineering, Dayananda Sagar College of Engineering, Bengaluru, Karnataka, India. Email: [email protected]Search for more papers by this authorRajvikram Madurai Elavarasan, Rajvikram Madurai Elavarasan Department of Electrical and Electronics Engineering, Thiagarajar College of Engineering, Madurai, Tamil Nadu, 625015 IndiaSearch for more papers by this authorHuiling Chen, Huiling Chen College of Computer Science and Artificial Intelligence, Wenzhou University, Wenzhou, 325035 ChinaSearch for more papers by this author Manoharan Premkumar, Corresponding Author Manoharan Premkumar [email protected] orcid.org/0000-0003-1032-4634 Department of Electrical and Electronics Engineering, Dayananda Sagar College of Engineering, Bengaluru, Karnataka, 560078 India Correspondence Hassan Haes Alhelou, Department of Electrical Power Engineering, Tishreen University, 2230 Lattakia, Syria Arab Republic. Email: [email protected] M. Premkumar, Department of Electrical and Electronics Engineering, Dayananda Sagar College of Engineering, Bengaluru, Karnataka, India. Email: [email protected]Search for more papers by this authorPradeep Jangir, Pradeep Jangir Power and Energy Sector, Rajasthan Rajya Vidyut Prasaran Nigam Ltd., Sikar, Rajasthan, 332025 IndiaSearch for more papers by this authorChandrasekaran Kumar, Chandrasekaran Kumar orcid.org/0000-0002-1132-4794 Department of Electrical and Electronics Engineering, M. Kumarasamy College of Engineering, Karur, Tamil Nadu, 639113 IndiaSearch for more papers by this authorSomasundaram David Thanasingh Sundarsingh Jebaseelan, Somasundaram David Thanasingh Sundarsingh Jebaseelan Department of Electrical and Electronics Engineering, Sathyabama Institute of Science and Technology, Chennai, Tamil Nadu, 600119 IndiaSearch for more papers by this authorHassan Haes Alhelou, Corresponding Author Hassan Haes Alhelou [email protected] orcid.org/0000-0001-8176-1589 Department of Electrical Power Engineering, Tishreen University, Lattakia, 2230 Syrian Arab Republic Correspondence Hassan Haes Alhelou, Department of Electrical Power Engineering, Tishreen University, 2230 Lattakia, Syria Arab Republic. Email: [email protected] M. Premkumar, Department of Electrical and Electronics Engineering, Dayananda Sagar College of Engineering, Bengaluru, Karnataka, India. Email: [email protected]Search for more papers by this authorRajvikram Madurai Elavarasan, Rajvikram Madurai Elavarasan Department of Electrical and Electronics Engineering, Thiagarajar College of Engineering, Madurai, Tamil Nadu, 625015 IndiaSearch for more papers by this authorHuiling Chen, Huiling Chen College of Computer Science and Artificial Intelligence, Wenzhou University, Wenzhou, 325035 ChinaSearch for more papers by this author First published: 23 April 2022 https://doi.org/10.1049/rpg2.12475Citations: 1AboutSectionsPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat Abstract The reliability of the photovoltaic models is strongly reliant on their parameters, which are primarily determined by the optimization algorithm and the objective function. As a result, obtaining the parameters under different environmental conditions is critical for increasing their performance, reliability and significantly lowering cost. Many optimization techniques are reported to address this problem based on the complexity. As a result, an enhanced version of the recently reported Hunger Games Search Optimizer (HGSO) method called Gaussian and Cauchy Mutation-based HGSO (GCMHGSO) algorithm for defining the requirements of the Three-Diode equivalent Model (TDeM) by utilizing multiple representations in the algorithm along with an efficient objective function. The Cauchy mutation increases the exploration ability, and Gaussian mutation increases the exploitation ability of the basic HGSO. Furthermore, an Enhanced Newton–Raphson Method (ENRM) is presented to effectively solve the behaviour of the current–voltage relation of the TDeM. The robust optimization is also considered to demonstrate the impact of the measurement error. Comparing the GCMHGSO-ENRM to other competitors reveals that the proposed GCMHGSO-ENRM can accurately find the best solution, and its effectiveness is verified in many statistical parameters. It is found that the GCMHGSO-ENRM algorithm is stable and robust compared to other competitors. 1 INTRODUCTION After wind energy reforming structures, PhotoVoltaic (PV) generation has become a major global energy technology. Recently, substantial investments have been made in the PV market, and several funding has been recorded for research studies in the PV systems sector [1]. The prime motive for such a path is the severe cost reduction in PV system parts, indicating the PV sector's incredible improvement [2, 3]. Over the last decade, the price of solar components has plummeted, including the PV system itself and its connectors. This PV cost reduction increases its competitiveness against other renewable energy sources [4, 5]. As per estimates from the global market prediction for PV systems, 50% of PV developments were completed in 2018, with a total of 512 GW. Alternative energy benefits are increasingly emphasized to address various issues [6, 7]. The energy crisis, coal depletion, pollution of the environment, and changes in global climate patterns are the main issues in this problem. Solar electricity, also known as photovoltaic (PV), is a renewable source that supports the atmosphere and promotes long-term development [8, 9]. The photovoltaic systems are inactive structures that turn solar cell energy directly into electric energy. The prediction of PV system output is based on a specific model system that identifies the analogous circuit direct current variables. The reliability in which the variables are calculated is essential for various factors, depending on the quantity of direct current variables displayed in the framework. It is unquestionably helpful for extracting PV cell variables, describing the impact on model efficiency, and optimizing development practices [10-12]. Nevertheless, it is unfortunate that most industries provide limited knowledge in the datasheets, which is required to accurately model the characteristics of a PV system, especially under different conditions [13, 14]. In order to obtain the excellent function of PV cells and panels under various operating conditions, reliable modelling strategies must be established in response to some of these uncertainties. The solar cell's design method is primarily based on the analogous mathematical model of the solar cell. The solar cell model's significance is evident in its potential to enhance all parameters that control the actual solar cell's complex characteristics [15-17]. Furthermore, the current–voltage (I–V) curves can be easily accessed by analysing the behaviour using the mathematical model, which aids in efficiently predicting the characteristics of solar cells and PV modules under various weather conditions. The Single-Diode equivalent Model (SDeM) and Double-Diode equivalent Model (DDeM) are the two primary models used to model the PV cell [18-20]. Both models comprise electrical components that constitute a circuit capable of analysing and incorporating the nonlinear effects of an actual PV cell. The diode-saturation current Isd, the shunt resistance Rsh, the photo-generated current Iph, the ideality factor a, and the series resistance Rse relevant to the cell are included in both SDeM and DDeM versions. The solar cell function is represented by five variables (Isd, Rsh, Iph, a, and Rse) in the SDeM. In the meantime, the DDeM has about seven parameters (Isd1, Isd2, Rsh, Iph, a1, a2, and Rse). The key point about deriving an efficient mathematical model is that it is entirely dependent on the computation of such variables. Therefore, the variables must be derived appropriately and accurately to achieve an optimal balance between voltage and current for PV cells and modules [8, 21–23]. Although their potential to provide a deeper explanation of the cell or module, such models (SDeM and DDeM) have been restricted to domestic application, implying that the complex output has only been evaluated for a small operating temperature range. To address this problem, the Three-Diode equivalent Model (TDeM) was introduced [16, 22, 24–28]. The TDeM includes nine parameters (Isd1, Isd2, Isd3, Rsh, Iph, a1, a2, a3, and Rse) used to stimulate and enhance the behaviour of the cells and modules. In particular, the ideality factor is determined for each diode (three diodes in TDeM) to achieve the best I–V curve, ranging from 1 to 2 depending on the material quality and fabrication of the cell [29, 30]. Furthermore, the actual recombination process due to the boron-oxygen deficiency or interlayer iron is dependent. As a result of using such variables, detailed modelling of PVs can be achieved [16, 28]. To obtain a robust and accurate PV model that plays a crucial role in the nonlinear response of grid-tied PV systems via optimization studies, it is necessary to obtain the nine parameters accurately. Using the nine parameters, it is possible to model real-world industrial applications of the PV cells and modules with relatively high accuracy. In particular, using diode models necessitates the configuration of variables to produce the desired output, which is regarded as a challenge. On the other hand, this challenge can be approached and developed as an optimization problem. Various computational methodologies are used to examine the best parameter values for PV modules [31]. According to the studies, several different approaches for solving the parameter identification problem have been reported, including analytical [32], Newton-Raphson Method (NRM) [33], numerical [34], curve-fitting process [35, 36], and tabular [37]. A nonlinear least-squares method was implemented using the Newton model [33, 36] to determine the unknown solar cell design variables. As described in [38], analytical processes and approaches were used to introduce I-V features using a co-content framework. Three analytical techniques for obtaining the parameters based on the SDeM were analysed and discussed in [39], with the finding that the curve-fitting methodology produced excellent outcomes. In [40], an appropriate model for the Root-Mean-Square-Error (RMSE) of five-parameter SDeM was suggested using an RMSE based on the Lambert-W function. Nevertheless, because of the high nonlinearity of the current representations of such models, no accurate numerical solution based on the Lambert-W function has yet been found for DDeM or TDeM. The Lambert W-function was defined in [41], and it was used to obtain the DDeM model's variables. Tabular strategies for PV modules were introduced in [37] with accurate performance, but the computation complexity was large. For tuning the parameters of PV panels, the authors [42] present a thorough comparison of the Levenberg–Marquardt algorithm and the Newton–Raphson technique. These optimization methods are categorized as deterministic strategies, and they can take into account a variety of boundary limits, such as differentiability and convexity, to ensure effective implementation. Regrettably, since the outputs are dependent on primary solutions, this behaviour led to local optima stagnation. The authors of [33, 43] utilized the Newton–Raphson method to obtain the DDeM's unknown parameters. The results revealed a significant difference between the actual and expected current and voltage values. This is in contrast to the methodologies' strong computational demands. Another consideration is the calculated I–V curve relying on datasheet knowledge, usually done in ideal depictions with linear definitions for measured values [44, 45]. Besides, extracting the variables of the three-diode PV model involves intensive computation. To combat this, the parameters' values can be modified throughout the day. They are location-dependent, and then using shifted equations, such approaches cannot test all of the parametric uncertainties or balance the actual output. The evolutionary computation techniques for obtaining the parameters of PV cells and panels were already introduced and implemented in the last decade. Such strategies do have several advantages, including the capacity to execute multidimensional discovery in search spaces utilizing various arrangements before the optimal result is obtained [45-47]. Numerous methods have been suggested based on the quality of evolutionary computation techniques. Genetic Algorithms (GA) increase the precision of model parameters for the DDeM of PV cells and modules, as stated in [48]. Particle Swarm Optimization (PSO) was used to test the characteristics of solar cells by using SDeM and DDeMs [49-52]. In [51-53], an effective method called Simulated Annealing (SA) was used to measure the variables of the SDeM and DDeMs of the PV cells and modules. Furthermore, metaheuristic optimization algorithms have been shown to construct an active PV device based on various factors such as accuracy, reliability, fast convergence, measurement efficiency, and fewer control variables [54]. The metaheuristic algorithms used for parameter estimation problem is classified as follows: Evolutionary algorithms (GA and its variants [55-58], Differential Evolutionary (DE) algorithm, and its variants [59-63] etc.), physics-based algorithms (Gravitational Search Algorithm (GSA) [9], Flower Pollination Algorithm (FPA) [64, 65], Wind Driven Optimization (WDO) [66, 67], Heap-Based Optimizer (HBO) [68], Henry–Gas Solubility Algorithm (HGSA) [69, 70], Equilibrium Optimizer (EO) [71, 72] etc.), swarm-based algorithms (Artificial Bee Colony [73, 74], Cat Swarm Optimization (CSO) [75], Grey Wolf Optimizer (GWO) [76], Salp–Swarm Algorithm (SSA) [8, 77], Moth-Flame Optimizer (MFO) [78], Whale Optimization Algorithm (WOA) [79], Marine-Predator Algorithm (MPA) [80, 81] etc.), and population-based algorithms (Teaching-Learning-based Optimizer (TLBO) and its variants [82, 83], Jaya algorithm and its variants [84-86], Political Optimizer (PO) [87], Rao algorithm (RAO) [88-90], Coyote Optimization Algorithm (COA) [91], Gradient-Based Optimizer (GBO) [92, 93] etc.,). Some hybrid variants, such as PSO-GWO [94], PSO-WOA [95], GSA-PSO [96] etc., and some improved variants of algorithms based on opposition-based learning [15, 86], Gaussian Mutation (GM) [97], Cauchy Mutation (CM) [57, 98], Chaotic GBO [99], Opposition-based GBO [100], Chaos random number generation [22, 49, 101], Nelder–Mead Simplex [78, 102] etc. are also applied to parameter estimation problem of solar cell and modules. The following conclusions are drawn from the preceding analysis: Disturbance, as well as changes in temperature and irradiation, may affect the efficiency of such techniques. To find the best solution, consider the effect of different objective functions on the efficiency of metaheuristic methods, such as measurement and energy estimation errors, RMSE, Relative Error (RE), Integral Absolute Error (IAE), Mean Bias Error (MBE), and so on. Despite the enormous amount of studies that have been published in the literature, there is also a conceptual deficit in identifying global solutions under all climatic conditions due to the fast development of the PV module and improvement in its performance, especially with the use of bifacial PV technology. This is due to the PV model equation's difficulty, lack of empirical observations under varying weather conditions, and irrational behaviour. Regrettably, several experiments often measure the PV cell's current without considering its non-linearity or the diversity of each variable, resulting in inaccurate results. Many methods, such as the f-solve command in MATLAB [63], the Lambert W function [103-105], Newton–Raphson Method (NRM) [30, 43], and the Taylor series [106], have been introduced to deal with electrical equations to address this problem adequately. On the other hand, the Taylor series method has the disadvantage of taking longer and achieving lower convergence than the NRM [107]. Since the NRM converges quickly, it generates fewer AE. Concurrently, the Lambert-W function consumes more CPU time while providing greater accuracy, particularly at high output voltages. As a result, difficulties in balancing the exploration and exploitation stages and preserving higher reliability and accuracy levels. The authors of [54] contrasted several optimization techniques to obtain parameters of the SDeM, DDeM, and TDeM PV models via the NRM and direct linear PV model equation method using RMSE objective function. The lambert-W function technique is used to estimate which approach is more reliable. The test results demonstrated that one of the most modern techniques based on the NRM outperforms other research techniques in terms of accuracy and convergence speed. The parameter extraction for the TD PV model based on the Lambert-W function, on the other hand, received no attention. Therefore, a strong and robust optimization tool is required to estimate the variables of TDeM of the PV cells and modules. The researchers have proposed a few variants of the basic algorithms based on CM and GM strategies. By integrating cultural learning and GM, the authors of [108] suggested an improved MFO. The numerical simulations revealed that the two operators could considerably enhance MFO's efficiency to demonstrate effectiveness and robustness considerably. The algorithms were also enhanced by combining opposition-based learning, CM, and adaptive boundary constraint handling [101, 109]. When introduced to complicated tasks with minimal iterations, all of the variants mentioned above improve searching capabilities or convergence speeds, but they have trouble escaping from local optimal. As a result, the following assumptions are probable. The basic algorithm's restricted searching capacity allows it to collapse into local optima, leading to premature convergence quickly. The individual mutation equation struggles to strike a balance between exploitation and exploration capability. Finally, the basic versions have trouble obtaining the global optima for complicated situations with minimal iterations. Based on the above-all discussions and as per the no-free lunch principle, no optimization algorithm can achieve the global solution for all challenges [110]. According to the researchers' understanding, no current approach can claim that the optimized parameters are global solutions. An accurate, reliable, effective, and consistent method is needed to reach the PV system's actual output under any environmental conditions. Therefore, investigators must use various optimization techniques to extract the uncertain parameters of the TDeM. This tends to be the authors' primary reason for selecting the improved version of the recently introduced Hunger Games Search Optimization (HGSO) algorithm [111] called Gaussian and Cauchy Mutation-based Hunger Games Search Optimization (GCMHGSO) algorithm to minimize the formulated objective function. The proposed GCMHGSO is combined with the enhanced NRM (GCMHGSO-ENRM) and conventional NRM (GCMHGSO-NRM) and obtains the optimal nine parameters of the PV cells and modules. The main contributions of the paper are listed as follows. A new variant of the HGSO algorithm combined with an enhanced Newton–Raphson (ENRM) and conventional NRM called GCMHGSO-ENRM and GCNMHGSO-NRM, respectively, is introduced to find the optimal solutions for the TDeM of the PV cells and modules. The exploration and exploitation ability of the basic HGSO algorithm is improved by the GM and CM strategies. Using damping to replicate the nonlinearity behaviour of the PV model and prevent large iterations is recommended as an ENRM method to provide highly accurate initial solutions. The performance of both the GCMHGSO-ENRM and GCMHGSO-NRM algorithms is validated using experimental values of the PV cells and PV modules under different operating conditions and compared with other state-of-the-art algorithms. Robust optimization is also proposed to demonstrate the impact of the measurement error. The structure of the formulated is as follows. Section 2 explains the mathematical modelling of the TDeM of the PV cells and PV modules and discusses the formulation of the objective functions. Section 3 presents the basic version of the HGSO algorithm and the formulation of the improved variant called the GCMHGSO algorithm. Section 4 demonstrates the experimental validations on the cells and modules under different operating conditions. Section 5 concludes the paper. 2 MATHEMATICAL MODELLING AND COST FUNCTION Mathematical modelling of TDeM The electrical equivalent of the three-diode photovoltaic model is illustrated in Figure 1, and from Figure 1, it is observed that there are nine unknown parameters for the TDeM of the PV cell or module. The unknown parameters are as follows: The photo-generated current (Iph) in Amps., the ideality factor of three diodes (a1,a2, and a3), the diode's saturation currents (Isd1, Isd2, and Isd3) in Amps., a large value of shunt resistance (Rsh) in Ω connected in parallel to explain the saturation current of diodes, and small value of series resistance (Rse) in Ω [54, 112–114]. By applying Kirchhoff's law, the output current of the PV cell is derived as follows. I p v = I p h − I d − I s h \begin{equation}{I_{pv}} = {I_{ph}} - {I_d} - {I_{sh}}\end{equation} (1)where Ish denotes the current through the shunt resistor, Id denotes the diode current, and Ipv denotes the total output current of the PV cell. As per the Shockley equation, the expression for the diode current is written: I d = I s d 1 exp V p v + I p v R s e V T 1 − 1 + I s d 2 exp V p v + I p v R s e V T 2 − 1 + I s d 3 exp V p v + I p v R s e V T 3 − 1 \begin{eqnarray} {I_d} &=& {I_{sd1}}\left[ {\exp \left( {\frac{{{{\rm{V}}_{pv}} + {I_{pv}}{R_{se}}}}{{{V_{{T_1}}}}}} \right) - 1} \right] \nonumber\\ &&+\ {I_{sd2}}\left[ {\exp \left( {\frac{{{{\rm{V}}_{pv}} + {I_{pv}}{R_{se}}}}{{{V_{{T_2}}}}}} \right) - 1} \right] \nonumber\\ &&+\, {I_{sd3}}\left[ {\exp \left( {\frac{{{{\rm{V}}_{pv}} + {I_{pv}}{R_{se}}}}{{{V_{{T_3}}}}}} \right) - 1} \right] \end{eqnarray} (2)where Vpv denotes the total output voltage of the PV cell and V T 1 , V T 2 ${V_{{T_1}}},\ {V_{{T_2}}}$ , and V T 3 ${V_{{T_3}}}$ are the thermal voltage of each diode in the TDeM. The expressions for the same are written as follows. V T 1 = a 1 k T q V T 2 = a 2 k T q V T 3 = a 3 k T q \begin{equation} \def\eqcellsep{&}\begin{array}{@{}*{1}{c}@{}} {{V_{{T_1}}} = \dfrac{{{a_1}kT}}{q}}\\[10pt] { \def\eqcellsep{&}\begin{array}{@{}*{1}{c}@{}} {{V_{{T_2}}} = \dfrac{{{a_2}kT}}{q}}\\[10pt] {{V_{{T_3}}} = \dfrac{{{a_3}kT}}{q}} \end{array} } \end{array} \end{equation} (3)where q denotes the electron charge (= 1.60 × 10–19 C), k denotes the Boltzmann constant (= 1.38 × 10–23 J/K), and T denotes the cell temperature. The shunt current of the PV cell is written as follows. I s h = V p v + I p v R s e R s h \begin{equation}{I_{sh}} = \frac{{{{\rm{V}}_{pv}} + {I_{pv}}{R_{se}}}}{{{R_{sh}}}}\end{equation} (4)From the above-all equations, the total output current of the PV cell is written as follows. I p v = I p h − I s d 1 exp V p v + I p v R s e V T 1 − 1 − I s d 2 exp V p v + I p v R s e V T 2 − 1 − I s d 3 exp V p v + I p v R s e V T 3 − 1 − V p v + I p v R s e R s h \begin{eqnarray} {I_{pv}} &=& {I_{ph}} - {I_{sd1}}\left[ {\exp \left( {\frac{{{{\rm{V}}_{pv}} + {I_{pv}}{R_{se}}}}{{{V_{{T_1}}}}}} \right) - 1} \right] \nonumber\\ &&-\ {I_{sd2}}\left[ {\exp \left( {\frac{{{{\rm{V}}_{pv}} + {I_{pv}}{R_{se}}}}{{{V_{{T_2}}}}}} \right) - 1} \right] \nonumber\\ &&-\, {I_{sd3}}\left[ {\exp \left( {\frac{{{{\rm{V}}_{pv}} + {I_{pv}}{R_{se}}}}{{{V_{{T_3}}}}}} \right) - 1} \right] - \frac{{{{\rm{V}}_{pv}} + {I_{pv}}{R_{se}}}}{{{R_{sh}}}} \end{eqnarray} (5)From Equation (5), the non-linear behaviour of the PV cell can be observed, and note that all the parameters are sensitive to temperature and solar irradiance. FIGURE 1Open in figure viewerPowerPoint The circuit structure of the TDeM of the PV cell The PV module consists of the series-connected cells (Nse) to increase the voltage rating and parallel-connected cells (Nsh) to increase the current rating based on the overall PV module rating. Therefore, the output current expression for the PV module is written as follows. I p v = N s h I p h − N s h I s d 1 exp V p v + N s e N s h I p v R s e N s e V T 1 − 1 − N s h I s d 2 exp V p v + N s e N s h I p v R s e N s e V T 2 − 1 − N s h I s d 3 exp V p v + N s e N s h I p v R s e N s e V T 3 − 1 − V p v + N s e N s h I p v R s e N s e N s h R s h \begin{eqnarray} {I_{pv}} &=& {N_{sh}}{I_{ph}} - {N_{sh}}{I_{sd1}}\left[ {\exp \left( {\frac{{{{\rm{V}}_{pv}} + \left( {\frac{{{N_{se}}}}{{{N_{sh}}}}} \right){I_{pv}}{R_{se}}}}{{{N_{se}}{V_{{T_1}}}}}} \right) - 1} \right] \nonumber\\ &&-\, {N_{sh}}{I_{sd2}}\left[ {\exp \left( {\frac{{{{\rm{V}}_{pv}} + \left( {\frac{{{N_{se}}}}{{{N_{sh}}}}} \right){I_{pv}}{R_{se}}}}{{{N_{se}}{V_{{T_2}}}}}} \right) - 1} \right]\nonumber\\ &&-\,{N_{sh}}{I_{sd3}}\left[ {\exp \left( {\frac{{{{\rm{V}}_{pv}} + \left( {\frac{{{N_{se}}}}{{{N_{sh}}}}} \right){I_{pv}}{R_{se}}}}{{{N_{se}}{V_{{T_3}}}}}} \right) - 1} \right] \nonumber\\ &&-\, \frac{{{{\rm{V}}_{pv}} + \left( {\frac{{{N_{se}}}}{{{N_{sh}}}}} \right){I_{pv}}{R_{se}}}}{{\left( {\frac{{{N_{se}}}}{{{N_{sh}}}}} \right){R_{sh}}}} \end{eqnarray} (6)Therefore, from the above-all discussions, it is observed that there are nine unknown variables (Isd1, Isd2, Isd3, Rsh, Iph, a1, a2, a3, and Rse) to be extracted to establish P–V and I–V characteristics of the PV cells and modules. Problem formulation The aim of model development and parameter identification is to reduce the error between simulated and experimental values under various climatic conditions. As a consequence, the RMSE can be used as a fitness function that aims to minimize the error by optimizing the nine parameters. As per the observation, Equation (6) is a non-linear implicit transcendental expression with nine uncertain variables. Numerous researches, such as NRM, Lambert-W function, and Taylor series, are suggested for efficiently identifying such unknown variables. As per the literature, the NRM and the Lambert-W function were frequently used to identify any PV models' parameters. However, in this paper, classical NRM and an enhanced NRM are utilized for identifying the parameters of the PV cells and modules. 2.2.1 Classical Newton-Raphson method The Newton-Raphson method is used to solve I–V characteristic equations, which can estimate the nine variable