A Novel Theoretical and Practical Methodology for Extracting the Parameters of the Single and Double Diode Photovoltaic Models

Solar Photovoltaic (PV) system is one of the most signiﬁcant forms of renewable energy resources, and it requires accuracy to assess, design, and extraction of its parameters. Several methods have been extensively applied to mimic the nonlinearity and multi-model behavior of the PV system. However, there is no method to date that can guarantee the extracted parameter of the PV model is the most accurate one. Therefore, this paper presents a unique approach known as Hybridized Arithmetic Operation Algorithm based on Efﬁcient Newton Raphson (HAOA ENR ) to experimentally extract the parameters of the single-diode and double-diode PV models at the variability of the climatic changes. Firstly, the objective function is efﬁciently designed to roughly predict the initial root values of the PV equation. Secondly, the Lévy ﬂight and Brownian strategies are integrated in the four operators of AOA to thoroughly analyze the feature space of this problem. Additionally, the four operators of the AOA is divided into two phases to equilibrium between the exploration and exploitation tendencies. Furthermore, the chaotic map and robust mutation techniques are systematically employed in the beginning and halves of generations to ensure the algorithm can reach globally at few numbers iterations. Finally, a nonlinearly adjustable damping parameter of the Levenberg-Marquardt technique is linked with the NR method to replicate the ﬂuctuation behaviours of the PV models. The experimental ﬁndings revealed that the proposed HAOA ENR outperformed all other methods found in the literature, with average RMSE values close to zero values for both PV models.


I. INTRODUCTION
The increasing depletion of traditional energy sources, price reductions, and permanently ubiquitous in nature have all accelerated the development of renewable energy applications in recent decades [1], [2].The Photovoltaic (PV) system is one of these renewable energy sources that can be The associate editor coordinating the review of this manuscript and approving it for publication was Padmanabh Thakur .
found almost everywhere and used in both cities and distant places.However, there is no perfection in industry and the PV panels that are manufactured do not accurately replicate the specifications in the datasheet.This opens the door for a slew of new difficulties and issues to arise as a result of the new PV technologies.Despite the solar PV system's exponential expansion, a number of problems such as solar irradiance fluctuations, temperature changes, technology, methods, high initial costs, aging, uncertainties of measurement, and poor energy conversion may restrict its separation [3].
To perform a realistic analysis, the I-V and P-V curves may be determined in a suitable manner either by utilizing the three known key points: short-circuit (SC), open-circuit (OC), and maximum power point (MPP), or by reducing the error between the proposed and experimental data points [4].The parameters extracted by three key points have various limitations, including the fact that the datasheet information is presented at standard test condition (STC), which reflects only ideal measurement and does not exist in actual operational situations.Such parameters are calculated using mathematical equations, which are solved and derived according to the authors' approximations and analyses [5], [6].The extraction of the parameters on the basis of the experimental data, on the other hand, is more exact and appropriate since the offered approaches can be tested and assessed utilizing many statistical criteria under different environmental circumstances [7].The main disadvantage is that the experimental data contains noises sounds that occurred during the measurement [8].
With a careful review and analysis of the literature, numerous approaches are extensively used to assess the performance of the PV cell and also to extract its parameters.Therefore, obtaining a precise and trustworthy PV model to represent the true behavior of the PV system in the light of the fluctuation of the weather conditions is crucial [9].The investigation rather than the vast gap between the theoretical research studies and employed actual applications.In the next part, we will go through the methodologies employed to extract the parameters of the PV model.
The numerical methods Newton Raphson [11], [12], Curve-fitting method [13], tabular [14], Maximum likelihood method [15], and Levenberg-Marquardt (LM) [16] have all been obtained to tackle the parameter extraction problem.The fundamental benefit of these methods is that they are assumed to be very precise especially when the step size of the initial condition is properly selected.They are also effective in dealing with nonlinearity and multidimensional optimization problems.Their biggest drawback is that they require a very computational time.Besides, the initial solutions must be accurately chosen.Sharp edge points, such as those seen at MPP of I-V and P-V curves, may also degrade accuracy.Finally, when tackling a large number of data points, they will evaluate the accumulation of error between each pair of points, which will diminish their accuracy.The second type of method is mathematical algebraic equations, which almost convert the equations into explicit and simple form considering the main three key points: OC, SC, and MPP at STC [17].The primary advantage of these methods are that they are fast and could be employed to determine the upper and lower boundary ranges of variables [12].These methods, however, face a number of challenges: To begin with, the manufacturer's datasheet is not always available, and if it is given, it is only represented at STC, which the nonlinearity and variability of the weather conditions are ignored [18].Furthermore, the derived mathematical equations for the same PV model may differ from one research study to the next, and any small deviation will result in a little decline in accuracy [19], [20].To end this, they have a high level of complexity in driving the DD and TD PV models.In fact, the analytical methods are not attractive and rarely adopted; so, it is recommended that they can be integrated with other approaches [21].
The limitations of the previous approaches have prompted researchers and scientists to design a new kind of techniques called stochastic, which they are inspired by the behavior of the evolutionary, swarm, physics, bio, and nature [22].These methods offer various benefits, including that there is no restriction of the initial conditions.The exploration and exploitation tactics can considerably improve the solution's quality, resulting in a lower error value based on obtained fitness function [23].However, one of the key disadvantages of these methods is that some of them may easily get trapped in local optima.In addition, the control settings, if exist, should be fine-tuned [24].Finally, the noise and nonlinearity may affect the performance of these methods marginally [25].The SD and DD PV models are the most frequently used in the literature, and their success is heavily reliant on the parameters that are required to be optimally extracted [23].Some of the well-known algorithms are Genetic Algorithm (GA) [26], [27], Differential Evolution (DE) [28], [29], Flower Pollination Algorithm (FPA) [30], Particle Swarm Algorithm [31], [32], Artificial Bee Colony [33], [34], Biogeography-based Optimization [35], Backtracking Search Algorithm [36], Teaching Learning-based Optimization (TLBO) [37], Whole Optimization Algorithm (WOA) [38], Jaya Algorithm (JA) [39], Firefly Algorithm (FA) [40], Cuckoo Search (CS) [41], and many more may be found in references [23], [24], [42]- [46].
According to the literature, the authors and scholars concentrate on the optimization methodologies without considering the wording of the formulation of the objective and how effectively the error between the simulated and experimental currents is minimized.Few studies employed Lambert W function [86]- [88], NR [60], [89], f-solve [90], Taylor series [91], Levenberg Marquardt [16], Bezier Curve [92], and least square nonlinear curve fitting method (lsqcurvefit function) [93].On the other hand, the nonlinear and multi-variable PV model equation is generally solved linearly [9], [28], [33], [36], [49], [82], [94], [95], showing a theoretical gap in this field.The downsides of these methods are that some of them need a considerable execution time and cannot properly imitate the experimental current especially when the number of the experimental data contains a large number of data points, hard edges, and noises.Additionally, minimizing the error is limited and cannot practically predict the real performance of the PV cell.
To the best of the authors' knowledge, no approach has been presented to date where the RMSE value is attained to zeros or near to it that can globally and realistically estimate the parameter of the PV model under variability of weather conditions for any type of PV model.Therefore, the purpose of this research is to provide practical and theoretical insights and clarification on the prediction of the output current of the PV cell using real experimental data under diverse weather conditions with zero or minor error values.We believe that our research may help to overcome the theoretical-practical divide.

II. PHOTOVOLTAIC MODELS AND OBJECTIVE FUNCITON DESIGN
In this study, real experimental data under seven climatic conditions is applied to verify and assess the performance of the SD and DD PV models.Tables 1 and 2 shows the specification of the utilized PV module under Standard Test Condition (STC) and the number of the I-V data points under climatic conditions [53].While the objective function is constructed based on NR and LM parameters using RMSE statistical criteria.

A. SINGLE DIODE PV MODEL
The electrical equivalent circuit of the SD PV model is shown in Fig. 2. The single diode is simple and widely used in the literature as it has only five parameter that required to be optimally extracted.The five physical parameters are current source I Ph in (A) connected in parallel, diode ideality factor d linked in the opposite way in order to mirror the PV cells' output voltage, the diode saturation current I o in (A), a small series resistor R s in ( ), and a large shunt resistance R p in ( ).Kirchhoff's law may be utilized for computing the output current in (A) of the PV cell, which is defined as [50], where I d represents the forward diode current in (A) and I p refers to the shunt resistor current in (A).As a result, I d may be computed by using the Shockley diode law as follows: where V denotes to the output voltage (V) and V t represents the thermal voltage of the diode (V) and may be represented as [50]: where KB is the constant of Boltzmann (1.38 × 10 −23 J /K ), T c denotes to the cell temperature (K ), and q represents the electron charge 1.60 × 10 −19 C. The I p may be written as: As a result of solving Eqs. ( 1)-( 4), the I is written as follows [60]: It is worth noting that the five parameters are subject to operating weather conditions and must be properly extracted in order to depict I-V and P-V curves [96].

B. DOUBLE DIODE PV MODEL
The electrical equivalent circuit of the two diode PV model is illustrated in Fig. 2. The main benefit of utilizing the DD PV model is that it provides improved accuracy when the solar irradiance levels are high.The second diode d 2 is included to reflect the recombination losses that occur in the depletion zone.Thus, the DD PV model's output current is given by [53]: where I o1 and I o2 are the reverse saturation currents (A) in diodes, and V t1 and V t2 are the diode thermal voltages (V) written as [53]: where d 1 and d 2 are the ideality factors of the first and second diodes, respectively.Despite the excellent precision of the DD PV module, determining the seven DD PV model parameter (I Ph , I o1 , I o1 , R s , R p , d 1 and d 2 ) requires substantial calculation.

C. OBJECTIVE FUNCTION DESIGN
The main principle of model formulation and parameter determination is to decrease the difference between the estimated current and experimental current in a variety of weather situations.Therefore, the estimate of the nine parameters may be stated by treating RMSE as an fitness function that must to be reduced.The currents of the SD and DD PV equations are nonlinearity implicit transcendental and cannot be solved directly.Therefore, several research proposed methods for optimally extracting these parameters, including f-solve [90], Levenberg Marquardt [16], Bezier Curve [92], NR [53], [97], [98], Taylor series (TS) [91], Lambert W function [99], and curve fitting method least square error (lsqcurvefit function) [93].According to the literature, the NR and the Lambert W function methods are often employed for determining the PV models' parameters as follows: Estimating the five and seven parameters of the SD and DD PV models at a variety of climatic conditions, the NR method is utilized to solve I-V curve equation, which is provided by [53], [100] (8) and ( 9), as shown at the bottom of the page.

2) LAMBERT W FUNCTION METHOD
To compensate for the voltage-current implicit coupling, the lambert W function is obtained.Because of its considerable level of accuracy, the Lambert W function is obtained to optimally determine the parameters of the SD and DD PV models, particularly at heterogeneous operating temperatures [25], [87], [88], [101].Therefore, the output currents of the SD and DD PV models utilizing the Lambert W function may be given by the following [87]: For the SD PV model: where For the DD PV model: where 3) THE PROPOSED ENHANCED NEWTON-RAPHSON (ENR) METHOD The NR and LW function methods have limited optimality to the approximate root solutions, resulting in loss of their accuracy even if the number of the derivative equation iterations is increased.Despite the fact that the NR has rapid coverage and LW function has been assumed to be more accurate at certain level of weather conditions [8], [50], there are several barriers that cannot be released in any way, which are: • Effects of noise associated with laboratory measurements.
• Oscillatory activity in PV cells and fluctuating of solar radiation in nature may result in poor estimating performance of the PV cell.
• Their accuracy suffers when there are data points with sharp edges at I-V and P-V curves.We have adopted a line from our earlier research method [102], constant step size, to lessen the oscillation effects and nonlinearity behavior of the I-V curve.When compared with traditional NR and LW function methods, the improvement approximately is around 50%.In terms of global viewpoint, the global minimum solutions can be obtained by neglecting the oscillation, nonlinearity, and fluctuation behaviours combined with the PV model's equation.This can be accomplished by using a new adaptive Levenberg parameter (λ) as given below [11]: where λ is computed by, For SD PV model: For DD PV model: In this work, two more power points are included: the new efficient NR method reduces the number of iterations up to two, and the epsilon value is extended to 1e-50 to reach the best global and minimize the value of RMSE [103], [104].The premature convergence is therefore avoided at all weather conditions and even at acute data points of the I-V and P-V curves [105].The LW function, NR, and INR methods, in theory, cannot simulate the distance between each data point along the I-V curve.Hence, the NR has a direct curvature, the Lambert W function has an exponential convergence and the INR has a static behavior convergence to the data of the I-V curve.Thus, the formulation of the objective function is represented as follows [53], where N is the length of the I-V data curve, V e and I e are the PV model's experimental voltage and current, and θ is the vector containing the parameters that are need to be optimally extracted.

III. ARITHMETIC OPTIMIZATION ALGORITHM
The Arithmetic optimization algorithm (AOA) is similar to other population-based optimization methods introduced lately [106] by Abualigah et al. in 2021.The core operation of all stochastic methods is based on exploration and exploitation strategies.To prevent slipping into local optima, it is essential to explore the search space as much as possible.Small changes in any variable value will cause divergence during the exploitative phase.Thus, the best algorithm is the one that can achieve a balance between both phases, particularly when the optimization problem involves parameters extraction of the PV model, which has several local optima, multi-variable, and non-linearity.The arithmetic operators in math, Multiplication (M ''×''), Division (D ''÷''), Subtraction (S ''−''), and Addition (A ''+''), implemented the diversification and exploitation mechanisms in AOA.

A. INSPIRATION
Arithmetic is fundamental to number theory, modern mathematics, geometry, analysis, and algebra.The four arithmetic operators (Multiplication, Division, Subtraction, and Addition) are basically used in various computational sciences [107].AOA is successfully applied to find a proper model identification of the Proton Exchange Membrane Fuel Cell (PEMFC) [108].Therefore, these four simple operators can be utilized to find the most optimum solutions while balancing between the exploration and exploitation phases [22], [109].

B. INITIALIZATION PHASE
At the beginning, the initialization step is implemented to randomly generate a set of candidate solutions (X ).
The best solution in all iterations is stored as the best optimal solution [106].
where x is the vector of a candidate solution, N refers to the number of search agents, and n represents the number of dimensions.To determine the exploration and exploitation phases, the AOA employs a function named as Math Optimizer Accelerated (MOA) and it is calculated as follows [106]: where MOA(C iter ) is the function value at the tth iteration, while C iter is the current iteration and its range between 1 and the maximum number of iteration (M iter ).Max and Min are the maximum and minimum values of the accelerated function.The next steps will discuss the exploration and exploitation phases in detail.

C. EXPLORATION PHASE
In AOA, the two operators Division (D) or Multiplication (M) are employed for the exploratory phase due to their highdistributed values in design space.The D and M mechanisms, on the other hand, have a great dispersion ability, making it harder to approach the target.The exploration phase is conditioned by using the MOA function, where r 1 is picked at random, and if the r 1 > MOA is detected, the D and M operators will be executed; otherwise, the A and S will be used.The equation of the exploration part is described by the following [106] (24), as shown at the bottom of the page, where r 2 is a random number and conditioned between the D and M operators.UB j and LB j are the lower and upper bounds values of the jth position, is a small integer value, and µ is a control parameter and set to be 0.5 to adjust the search process.
where Math Optimizer Probability is a coefficient named as (MOP) and α is a sensitive control parameter and chosen to be 5 which defines the exploitation accuracy over the iterations [106].

D. EXPLOITATION PHASE
The exploitation strategy is introduced by utilizing A and S operators.The A and S have high-dense performance, but they can easily approach due to the low dispersion.The S and A can be modeled as follows [106] (26), as shown at the bottom of the page, where r 3 is a randomly number chosen to distinguish between the A and S operators.

E. HYBRIDIZED AOA PROCEDURE
In the original AOA, one of four arithmetic mathematical operations is performed during all iterations of the program based on random mechanism.This implies that the balance between exploration and exploitation cannot be achieved.Furthermore, neither the Multiplication (M) nor the Addition (A) operators can be acquired for the exploration or exploitation phases.The justifications behind this can be explained by the following: • In Central Processing Unit (CPU), the M operator is represented as an A operator.Therefore, the M operator should be used in the exploitation strategy.
• The D is represented as the S operator in the CPU.Since the CPU deals with these four operations logically.
• The best optimal solution (best x j ) is chosen to lead all other candidate solutions by including the LB j of variables, resulting in easy trapping into local optima.
• Finally, the MOP control parameter has negligible effect on the perturbation of optimal solutions which has numerous unfavorable repercussions for diversification and/or intensification strategies.In view of the many previous constraints in the basic AOA, we have not only presented new improvements in the discovering new regions, but also enriching the quality of optimal solutions.

1) IMPROVED EXPLORATION PHASE
We have separated the population into two parties using four arithmetic mathematical operators: the following presents the exploration part (27), as shown at the bottom of the previous page, where Mu is given by, Therefore, Mu plays an important role to intensify the diversification in the early stages and to ensure the intensification in later stages of optimization, RL i,j is Levy flight strategy which has randomly step sizes based on probability function and is given by [110], (29) where L j denotes to the flight's length, which the exponent of the power-law is ranged between 1 and 2 [110].The probability density of the Levy distribution in integral form can be represented as [111]: where α is the distribution index that can control the scale properties of the process, γ is used to select the scale unit.
The integral is used when the α = 2 representing Gaussian distribution and when α = 1 representing a Cauchy distribution [112].A series expansion method is required when x has a huge value as shown by the following: where represents gamma function in which (1 + α) is equal to α!.According to [111], the value of α ranged within 0.3 and 1.99.In this study, the Mantegna method is used to generate a random number using Levy distribution as shown by the following: where y and x are two normal distribution variables which can be written as follows: x ), and y, where the σ x is computed as follows [110]: where, σ y = 1 and α = 1.5 The Levy strategy has movements with small steps size combined with long jumps, while the Brownian has the ability to cover most of the regions within the domain associated with uniform and controlled step sizes [112].Bst j and Wrst j are the best and worst solutions in the current iteration, and rand is a number chosen randomly.
The Levy flight assists the D operator in reaching further areas in the design space and simulating its behavior by moving small steps in long jumps.Meanwhile, the M is retained in the first part of the population to ensure the import of highquality solutions by considering the best and worst solutions in each iteration.

2) IMPROVED EXPLOITATION PHASE
In the improved exploitation phase, the S and A operators are responsible to escape from falling into the locality and allowing the decision-making to choose the most near-optimum solutions by obtaining the stochastic step size of the Brownian motion (RB) and a very small step size of Mu mechanism.To increase the diversity of the best global solution so far and local optimal avoidance, the best current optimal solution (Bst j ) is obtained rather than the best optimal global solution (best(x j )) (34), as shown at the bottom of the page.
Brownian motion: It is a stochastic step size process inspired by Gaussian distribution with unit variance (σ 2 = 1) and zero mean (µ = 0).For this motion, the Probably Density Function (PDF) at point x is given by following [113]: After the completion of the exploration and exploitation phases, the chaotic distribution is utilized at the first half of generations to efficiently find the further areas while avoiding the local solutions in the search space [114].Also, this randomized search mechanism aids in the speed of convergence in few numbers of generations as given by the following [49]: where Chaos offers advantage of pseudo-random patterns inside cycles [115].The chaotic distribution is utilized to explore new search regions, which leads to improvement in search direction from locality to globally.From a theoretical standpoint and after multiple attempts, we discovered that the chaotic strategy becomes ineffective after around half of the population.According to the previous finding, we provided an extended methodology for obtaining an optimum solution while accounting for any slight changes in each variable's value.Therefore, a well-known and promising (DE/best/2/bin) mutation strategy is used to assure completion of such a challenging task, as described in [116]: ) This procedure will eventually be repeated until the stopping criterion is met, the operation and detailed process of the HAOA is illustrated in Fig. 4.
In the HAOA −ENR , the balancing achieves by splitting the population into two groups, especially with the influence of two randomized step size of the Lévy flight and Brownian strategies, as well as nonlinear formula of Eq.( 28).The inclusion of the chaotic map and robust mutations techniques increases the complexity of the proposed hybrid algorithm, resulting in a very high-level-of-accuracy.In practice, addressing the parameter extraction of PV model optimization issue needs a proactive approach to avoiding local minima and premature convergence affectively.Hence, even little changes in variables have a large impact on accuracy.

IV. RESULTS AND DISCUSSIONS
In this work, a novel methodology based on Hybridized Arithmetic optimization Algorithm (HAOA) and the Efficient Newton Raphson (ENR) method is proposed to precisely extract the parameters of the single and double-diode models using real experimental data obtained under various weather conditions.The proposed HAOA ENR is verified based on several statistical data and compared with three variants of the AOA algorithm and with recently well-published papers such as: HAOA LW , HAOA NR , AOA L , Farmland Fertility Optimization (FFA NR ) [117], Chaotic Heterogeneous Comprehensive Learning Particle Swarm Optimization (CHCLPSO NR ) [118], Improved Slime Mould optimizer (ImSMA LW ) [119], Marine Predators Algorithm (MPA LW ) [50], Self-adaptive Ensemble-based Differential Evolution algorithm (SEDE L ) [94], and Time-Varying Acceleration Coefficients PSO (TVACPSO NR ) [60].As per the literature, the output current equation based on RMSE statistical criteria is solved using the Linear (L), Lambert W function (LW), and Newton Raphson (NR) methods.As a consequence, the articles have been carefully selected to cover the majority of the most modern, advanced, and high-levelof-accuracy methods used to tackle the parameter extraction optimization issue.To provide a fair comparison, the environmental setting for all methods are made to be same.This is due the need to scientifically understand the performance of each method and what degree of accuracy is optimum for the defined objective function.The number of generations (M iter ) is set to 500, and the population (X ) is set to 30, with each method being executed 30 times.The search space of the SD and DD PV models is given in Table 3 [53], [102].

A. RESULTS ON SINGLE DIODE MODEL
The estimated parameters of the single diode model for all methods are tabulated in Table 4 based on genuine experimental data obtained under various arbitrary weather conditions.The ideality factor (d) exhibits an inverse relationship with weather conditions due to the effects of Joule heating [120], where its value decreases from 1.4215 to 1.0629, indicating that this value decreases when the ambient temperature increases.While the photocurrent (I Ph ) behaves linearly due as solar irradiance levels increases.Moreover, its values are somewhat larger than those of other proposed models, implying a very high degree of agreement with experimental data.R s and R P decease dramatically when the solar irradiance rises except for R P at level S 6 .This is may be attributed to the fluctuation in the nature or the influence of other parameters, as is the case with most optimization methods that are strongly dependent on the formulation of the objective function.The above arguments confirm that the proposed HAOA ENR has distinct extraction parameter values when compared to the same method with various forms of objective function (LW, NR, and L).It is important to note that this optimization problem is non-linear, multi-variable has numerous local minima.Finally, the saturation current I o lies between (1.00E-5 and 5.17E-7), with I o values being greatly impacted by changes in ambient temperature owing to the expansion of photon absorption.
The convergence behavior of the HAOA ENR model for the single diode PV model is analyzed and shown utilizing seven genuine experimental data in Fig. 5.In all figures, the first column refers to the initialized candidate solutions.Following that, the rest of the other figures are plotted subsequently after every 120 iterations.The graphic depicts the behavior of the five parameters of the SD PV model using HAOA ENR from initial solutions to global solutions.The data points at the right of each row corroborate the final values of each parameter during the optimization, as tabulated in Table 4. Except for some of the solutions in the matrix, whose values are changeable during the optimization, the parameters values may potentially take the optimum direction after roughly 240 iterations as seen in the third column.After 480 generations, the five parameter values are globally optimized and the final matrix is constructed.We have observed that the convergence speed relies on the optimization method implemented in terms of exploration and exploitation tendencies, as well as how well the formulation of the objective function is handled.The possibility of premature convergence, if it happens, can be seen clearly in these figures due to the structure adopted by the candidate solutions during the optimization.As a result, the uniform solution in the figure can be illustrated in the last column, which refers that all candidates are convergent globally.
The extracted parameters of the single diode PV model are obtained for all methods based on its objective function formulation and are used to individually plot the I-V and P-V data characteristics curves in Fig. 6 (a) and (b), avoiding overlapping and to demonstrating how precisely each model matches the experimental data.As can be observed, the real experimental data points are not straightforward and there are multiple zigzag zones that reflect the meteorological data intermittently.The level of accuracy decreases as the quantity of data points increases.This is due to the fact that the distance between each data point is considered, and each pair of data points may have a level of error.Speaking from a practical perspective, the proposed innovative HAOA ENR can exactly match all the experimental data points at all weather conditions.The zoomed-in figures can slightly affirm that the proposed HAOA ENR is superior to other models, even at hard edges.
In terms of statistical criteria analysis of the SD PV model, the new and promising HAOA ENR the error values against other peers, resulting in zero errors at S 4 and S 7 trivial errors at others as shown in Table 5.Whereas the average RMSE value is 4.96E-35, the least RMSE value is reported in this field, indicating its excellent levels of capability in terms  it has failed to obtain the optimal solutions, especially when the length of the data points becomes large.This is because the other solar irradiance levels necessitate heavy and strong mechanisms to escape from the locality.In meanwhile, the HAOA L and SEDE L solved the output current equation linearly, which makes the optimal solutions unattainable.For the rest of the statistical criteria, It is clear that HAOA dominates other methods by a wide margin for the MBE, R 2 , d i , and TS, where their average values are 7.03E-69, 1, 5.32E-69, and 4.04E-34, respectively.When compared to versions of HAOA-LW/NR , the proposed HAOA method achieves a significant improvement in terms of the accuracy, stability, and reliability.This is due to the fact that it can effectively identify more solutions in search space and precisely return global parameters of the PV module.Also, up to date, the current version of the HAOA ENR reveals the most practical model is used to extract the parameters of the PV model and can cope with a transcendent equation with no or close to near-zero errors.
Another important statistical criteria is the Absolute Error (AE), which is employed to evaluate the performance of the proposed HAOA ENR and other methods for the SD PV model.High stability and reliability indicate a small value of AE especially when the experimental data contains noise and a large number of data points.Therefore, minimizing the error between each distance of data point remains a challenging task that requires not only a powerful approach but also an excellent and practical design of the objective function.From Table 6 shows that the proposed HAOA ENR has the capacity to deficit absolute error in all weather conditions when compared to other models.Which demonstrates a positive correlation with the experimental currents.Following HAOA ENR in the first place, the HAOA LW and MPA LW share the second rank with a comparable average AE of 0.0436, followed by HAOA NR at third rank.CHCLPSO NR and FFA NR have been ranked fourth and fifth, respectively.The worst average AE values are recorded for the SEDE L and HAOA L , where improper formulation of objective function leads to a considerable error between the planned and experimental currents, followed by ImSMA LW , TVACPSO NR , respectively, as shown in Fig 7 .For additional results analysis based on methodology and dealing with the objective function formulation itself, the main principle to reflect the real behavior of the PV model at the variability of the climatic changes is to provide a resilient and dependable algorithm based on a highly accurate objective function.Thus, Table 7 demonstrates the lowest (Min), Maximum (Max), and average (Avg) values of the fitness function (RMSE) for the SD PV model using various methods.There is no doubt that HAOA ENR outperforms other approaches at the minimum and average fitness function values for all environmental conditions.For more emphases on the superiority of the HAOA ENR , the convergence of the instantaneous RMSE at several real experimental conditions is shown in Fig. 8.As shown in Fig. 8, the proposed HOAO ENR may lower the RMSE values before reaching 200 iterations.Despite the high solar irradiance levels at S 4 to S 7 , which contain several noises and many errors between each distance of experimental data points, the proposed HAOA ENR exhibits fast convergence, very precise results, and can theoretically and practically deal with multi-model optimization in terms of optimization methods, as well as solve    DD PV model.Whilst, the d is decreased when the solar radiation increases for the SD PV model as shown in Table 4.The previous analysis state that there are no specific and unique solutions for the extracted parameter of the SD and DD PV models.In addition, we observed that the extracted optimal parameters have different ranges depending on the obtained method and the objective function design.Finally, these parameters are affected by the initial conditions and variations in the weather conditions, as well as the set of the interdependence of the parameters.convergence might be stuck in local optimal considering the number of parameters of the DD PV model that should be globally extracted.Therefore, the HAOA ENR can practically cope with the premature convergence.
The randomized steps strategies of the Levy flight and Brownian strategies are very useful in obtaining a systematic explorer and exploiter phases when dealing with nonlinearity and multi-variable optimization problem.The logistic map strategy is used in the first half of optimization guarantee escape from falling in the locality, while the robust mutation is employed in the second part of optimization to produce the most exploited solutions.Finally, the efficient formulation of objective function allows the ability the to reduce the RMSE to zero or close to zero and beat all the published approach in this topic of parameter optimization problem based on experimental data under various weather conditions.
There are no doubt that the most accurate method is one that can match all points of the experimental data, especially when the I-V curve has a hard edge.This makes solving the optimization problem a challenging task, since it is much hard to cover all linear, and nonlinearity points under all weather conditions, especially for the DD PV model.The proposed HAOA ENR demonstrates a very high level of satisfying performance in meeting the majority of data points, and if there is a divergence, it can be trivial as the error value is relatively tiny in terms of the practical engineering applications, as illustrated in Fig. 10 (a-b).Among all methods, the HAOA ENR can undoubtedly match the real experimental data points at all environmental conditions, as seen in the magnified sections of Fig. 10.
The statistical criteria are also implemented to establish the degree of stability, dependability, accuracy of the proposed HAOA ENR and other peers as demonstrated in Table 9.In terms of accuracy, the HAOA ENR has best RMSE value, with an average value of 9.43E-36.It can confirm that the proposed HAOA ENR is the most practical model for estimating the parameters of the PV model.Furthermore, the DD PV model outperforms the SD PV model based on HAOA ENR in terms of accuracy.The promising performance is shown not only for the HAOA ENR but for the HAOA LW also, which obtained the second rank and has an average RMSE value of 0.0576, followed by MPA LW , HAOA NR , CHCLPSO NR , FFA NR , and TVACPSO NR , respectively.The poorest average RMSE values, on the other hand, are produced by SEDE L , HAOA L , and ImSMA LW , with values of 0.7342, 0.7342, and 0.6032, respectively.The aim of this research study is not only to demonstrate the superiority of the proposed HAOA ENR , but also to instruct other researchers on how to cope with the nonlinearity and multi-modality of the PV model's equation in practice and theory.Based on statistical criterion comparison, the consistent performance of HOA ENR considering the MBE, R 2 , d i , and TS, with average values of 2.70E-70, 1, 2.11E-70, and 7.95E-35, respectively.The previous statistical criteria findings indicate that the proposed HAOA ENR is the most efficient, stable, and accurate model found to date.
The AE also is also measured at each level of experimental data for better visual definition of performance of the proposed HAOA ENR relative to other models, as presented in Table 11, and Fig. 11.Observations show that, the proposed HAOA ENR has a lower AE value than other models at all climatic conditions.The proposed HAOA ENR has an average AE value of 1.19E-36, followed by another variant HAOA LW , which has an AE value of 0.0408.Similar to the AE assessment ranking, the MPA LW , HAOA NR , CHCLPSO NR , FFA NR , and TVACPSO NR are ranked as third, fourth, fifth, sixth, and seventh, respectively, as demonstrated in Table 10.Contrarily, the worst average AE values are for SEDE L , HAOA L , and ImSMA LW , in that order.As a conclusion, advancements in dealing with the I-V curve equation, as well as methodology is necessary.
Finally, Table 11 exhibits the minimum, maximum, and average RMSE values of the DD PV model for the proposed HAOA ENR and other peers.Among all of these approaches, the HAOA ENR has the lowest RMSE under all environmental conditions.Besides, for average RMSE values, the proposed HAOA ENR dominates other models, reflecting the method's excellent stability and reliability.However, the maximum RMSE value relates to the initialization solutions before the algorithm begins to optimize the problem, which varies from one algorithm to another.Fig. 12 illustrates the performance of the proposed HAOA ENR to minimize the fitness function under the variability of climatic changes for the DD PV model.The average number of iterations required to get the lowest RMSE value under all weather conditions is 180, indicating a good approach in solving the parameter extraction optimization problem.
The computational time of each method is considered in this research study for the both SD and DD PV models under seven climatic conditions as tabulated in Table 12.It can clearly observed that the CPU execution time depends not only on the methodology but also on the basis of the employed fitness function.Therefore, the TVACPSO NR has minimum average CPU time among other methods, followed by CHCLPSO NR , SEDE L , HAOA L , HAOA NR , FFA NR , HAOA ENR as shown in Fig. 13.While the longest CPU execution time are taken by MPA LW , HAOA LW , IMSMA LW , respectively.This is because the objective function based on lambert W function is slower by 2.8-4.1 times than the that of the linear objective function [121].Despite the delayed in CPU execution time of the proposed HAOA ENR compared with 1 st method is 13-14 seconds for both models.However, the novelty of the HAOA ENR registers the minimized RMSE values for both models.Additionally, the measurement of this research study is off-line which reduces the error has more priority in such optimization task.To end this, the complexity and CPU execution time of the proposed HAOA ENR is increased due to the significantly improvement in terms of the methodology and objective function.
In terms of balancing exploration and exploitation inclinations, previous performance results and statistical analysis prove the superiority of the proposed HAOA ENR .In terms of objective function design, the derivative equations of the ENR method, with the helping of the Levenberg parameter, can confidently predict and choose the initial root values of the I-V data curve equation in fewer number of iterations.Also, each pairs (current and voltage) of the experimental data are included individually in the designing the objective function.Finally, the epsilon is assumed to very small to check the error between the proposed and experimental currents for ensuring a high-level-of-accuracy.Therefore, the extracted parameters of the SD and DD PV models based on HAOA ENR , on the other hand, are somewhat different from the other formulations of the objective function techniques such as LW and the original NR.For most levels of meteorological conditions in the SD and DD PV models, the R p value is almost large.Also, during certain of the climatic changes, the d 1 for the DD PV model is being high.Therefore, a new mechanism to handle the out variables from the search space is strongly recommended, one that is centered on attaining optimum solutions rather than the typical use of lower and upper bounds.Last but not the least, there is a need for further investigation into the lower and upper parameter ranges.Previous experimental findings demonstrate that the DD PV model is assumed to be more accurate than the SD PV model one, especially at high levels of environmental conditions.However, the DD PV model needs additional computing time.

V. CONCLUSION AND FUTUTRE DIRECTIONS
A unique HAOA ENR theoretical and practical approach is presented to experimentally extract the parameters of the SD and DD PV models validated by using several statistical criteria.The essential improvements are divided into two sections: the first is applied for estimating the parameters of the PV model quickly and initially using a nonlinear mathematical model for the objective function to simulate the fluctuations behavior of the PV system.The second key of improvement has been acquired to enrich the exploration and exploitation tendencies by including randomly step size movements of Levy flight and Brownian strategy for the four arithmetic operations based on the nonlinear formula.The chaotic map and resilient mutation strategies are then used in the first and second halves of generation to systematically equilibrium between the exploration and exploitation phases.The objective function is designed using nonlinear equation based on adaptive damping parameter of the LM method, and it is then integrated with NR method.The experimental findings of numerous statistical criteria reveal that the proposed HAOA ENR is the most efficient and scrupulous model to date for extracting the parameters of the SD and DD PV models with zero or negligible RMSE value for both models.Also, even at hard and challenging edges of the I-V and P-V curves, there is a highly agreement between the projected and experimental currents.
The PV model optimization problem needs to be handled using multi-objective methods for the future work direction.

FIGURE 2 .
FIGURE 2. The electrical circuit of the SD PV model.

FIGURE 3 .
FIGURE 3. The electrical circuit of the DD PV model.
The proposed HAOA ENR and other models' accuracy, reliability, stability, and convergence speed are evaluated using RMSE, Mean Bias Error (MBE), Coefficient of Determination ( R 2 ), Standard Test Deviation (STD), Test Statistical (TS), RMSE deviation considering all solar radiations' level d i criteria, and Absolute Error (AE) The best model exhibits extremely good congruence between the experimental data and the proposed model.

FIGURE 5 .
FIGURE 5. Convergence behavior of the proposed HAOA ENR objective function for the SD PV model at seven operator conditions.

FIGURE 6 .TABLE 5 .
FIGURE 6.Comparison between the real experimental data at seven climatic conditions and proposed HAOA ENR model for the SD PV model (a) I-V curve (b) P-V curve.

FIGURE 7 .
FIGURE 7. Average AE values of the HAOA ENR and other models for the SD PV model under seven weather conditions.

FIGURE 8 .
FIGURE 8. Development of the fitness function of the HAOA ENR for the SD PV model under variability of experimental data.

FIGURE 9 .
FIGURE 9. Convergence behavior of the proposed HAOA ENR objective function for the DD PV model at seven operator conditions.

FIGURE 10 .
FIGURE 10.Comparison between the real experimental data at seven climatic conditions and proposed HAOA ENR model for the DD PV model (a) I-V curve (b) P-V curve.

Fig. 9
depicts the convergence of optimum solutions from initializations to 480 iterations for the DD PV model under seven environmental conditions.According to Fig.9, the proposed HAOA ENR has the entire capability to fast convergence and extensive search around global optimum solutions under even the rough conditions.It is vital to highlight that, a fastVOLUME 10, 2022

FIGURE 11 .
FIGURE 11.Average AE values of the HAOA ENR and other models for the DD PV model under seven weather conditions.

FIGURE 13 .
FIGURE 13.Average CPU execution time for the proposed HAOA ENR and other methods.

TABLE 3 .
The search space ranges for the SD/DD PV models.

TABLE 4 .
Results of several methods for extracting the parameters of the SD PV model.

TABLE 6 .
Average AE values of different methods for the SD PV model at various weather conditions.

TABLE 7 .
Minimum, maximum, and average values of the RMSE for the SD PV model.

TABLE 8 .
(Continued.) Results of several methods for extracting the parameters of the DD PV model.

TABLE 9 .
Statistical results of different methods for the DD PV model.

TABLE 10 .
Average AE values of different methods for the DD PV model at various weather conditions.

TABLE 11 .
Minimum, maximum, and average values of the RMSE for the DD PV model.

TABLE 12 .
CPU execution time for the SD and DD PV models.Development of the fitness function of the HAOA ENR for the DD PV model under variability of experimental data.