Adaptive slime mould algorithm for optimal design of photovoltaic models

Solar energy is becoming more popular as it is a clean source of electricity. The design of photovoltaic (PV) cells has therefore captivated experts worldwide. The two key issues are the lack of an excellent model to define solar cells and the lack of data regarding PV cells. This scenario even impacts solar module performance (panels). The behavior of solar cells is described by the current versus voltage. Considering these values, the design challenge entails solving complicated nonlinear multimodal objectives. Different methods to figure out the parameters of the PV cells and panels have been suggested. They do not come up with the best solutions most of the time. Hence, a powerful and reliable optimizer is needed to derive the optimal parameters of these models. To this end, this study has developed an adaptive slime mould algorithm (ASMA) as a robust and precise optimization method. To implement the ASMA, four improvements are proposed: (1) a trigonometric‐based mutation and a double‐based best mutation are introduced to promote the global and local search; (2) a suitable mechanism to adaptively select the control parameters of the SMA; (3) a local escaping strategy; (4) an opposition‐based learning operator to improve the best solution. The ASMA is employed to derive optimal parameters of PV models and assessed utilizing a total number of eight well‐known optimization algorithms. The findings show that the ASMA is very competitive in terms of accuracy and convergence speed and that this is supported by a wealth of evidence. As a result, when assessing the parameters of the PV model, ASMA is a very efficient and robust optimizer. The source codes of the proposed ASMA will be uploaded for the public at http://imanahmadianfar.com and http://aliasgharheidari.com.


| INTRODUCTION
With the increasing global population and demand for energy on the one hand and the problems associated with the use of fossil fuels, such as carbon-induced environmental pollution and resource depletion on the other, clean energies such as solar, wind, geothermal, biomass, and tidal energy are receiving a lot of attention. 1,2 Among all of the aforementioned types of energy, solar energy has been designated as one of the cleanest accessible and usable energy systems due to its abundance across the globe and lack of greenhouse gas emissions. [3][4][5] In many developed countries, such as the United States and China, about 2%-3% of the total power produced is from solar energy, and this amount has been increasing in recent years. 6 Renewable energy systems such as solar photovoltaic (PV) systems, which convert solar energy into electricity directly, have recently attracted a lot of attention due to their evaluable capabilities. PV systems are one of the most widely used renewable energy technologies for directly converting solar energy into electricity. 6,7 This equipment, on account of being noise-free, ease of installation, and various environmental benefits, is considered the most challenging topic. 8 Generally, the solar PV models based on the gathered voltage and current data are developed through two significant stages: the mathematical model selection and identifying the best model parameters. [8][9][10][11] So far, various mathematical models have been proposed to describe the behavioral complexity in PV systems. 12 The standard mathematical PV models include the single diode model (SDM), double diode model (DDM), and PV module model. The SDM and DDM are the most widespread models for PV systems. 13,14 Although the use of PV systems has been a successful experience in the world, the result of field investigations points out that these systems, due to harsh environmental conditions on the one hand and the lack of accurate design of its components, on the other hand, maybe exposed to severe damage, fail, or degrade. Therefore, a valid and reliable model design of PV systems can accurately predict both PV panels and PV plant performance before installing them in real-time. 15 There are various approaches for estimating the PV model parameters and extracting parameters of the current-voltage characteristic. According to recent literature, the PV parameter extraction approaches can be divided into three categories, including deterministic, 16 numerical, and metaheuristic algorithms (MAs). 17,18 Deterministic methodologies include analytical and iterative approaches as the most conventional technique to extract the PV parameters. 19 Heretofore, deterministic methods have been widely used to derive the parameters of PV models on account of their simplicity, high speed, and straightforward. Lambert function based method (i.e., "Cocontent function"), 20 iterative curve fitting, 21 analytical five-point method, Newton-Raphson methods, and iterative fivepoint method 21,22 are the most popular techniques which have been mostly employed to extract the SDM and DDM PV models. 23,24 However, due to simplifications in mathematical models on the one hand and different shortcomings such as convexity and sensitivity to initial conditions, on the other hand, may quickly fall into the trap of local optima and divergence to inaccurate results.
Recently, to conquer the drawbacks of deterministic approaches in PV parameters modeling, meta-heuristic techniques have been proposed due to global search and multimodal optimization problems capabilities. Bioinspired computing has found an increasing load of applications in recent years due to the limitation of the computational sources and time in complex feature spaces. [25][26][27][28][29][30] In the last decade, many researchers have made great efforts to optimize the parameters of PV systems. For example, for different aspects of determining PV system parameters, the genetic algorithm (GA), 31,32 particle swarm optimization (PSO), [33][34][35][36] differential evolution (DE), 37 simulated annealing (SA), [38][39][40] and ant lion optimizer (ALO) 41,42 were utilized.
Recently, various attempts have been made to improve the parameters of PV models using modified evolutionary algorithms (EAs) and hybrid meta-heuristic approaches. For instance, a chaotic whale optimization algorithm (CWOA) 43 ; a hybrid of firefly and pattern search algorithm 44 ; a logistic chaotic JAYA algorithm (LCJAYA) 7 ; a novel hybrid algorithm, namely, fractional chaotic ensemble particle swarm optimizer (FC-EPSO) 45 ; random reselection particle swarm optimization (PSOCS) 46 ; shuffled frog leading algorithms (SFLA) [47][48][49] ; artificial electric field algorithm (AEF) 50 ; marine predators algorithm (MPA) 51 ; slime mould algorithm (SMA) 52 ; spherical evolution algorithm (SE) 53,54 ; Harris hawks optimization (HHO) 4,55-57 ; coyote optimization algorithm (COA) 58 ; and a novel hybrid biogeography-based optimization (BBO) and cuckoo search (CS) 59 was provided to identify the parameters of PV cells and panels. In recent years, Chen et al. 60 proposed a hybridized teaching-learning-based optimization (TLBO) with the artificial bee colony (ABC) for accurately estimating the parameters of different PV systems. Abbassi et al. 11 presented a new salp swarm algorithm (SSA) to determine seven parameters of the DDM PV cell models. The obtained results were more accurate with the sine cosine algorithm (SCA) and virus colony search algorithm (VCS). Moreover, Li et al. 61 developed efficient parameter extraction techniques using an adaptive differential evolution algorithm, namely EJADE. In this study, the SDM, DDM, and PV modules were examined to validate the accuracy and reliability of the proposed algorithm. Zhang et al. 62 presented a local search-based orthogonal moth flame optimization (MFO) combined with an orthogonal learning (OL) strategy and Nelder-Mead simplex (NMS) method, namely, NMSOLM-FO for optimization of the parameters in PV cell models. Liang et al. 63 proposed a multitask optimization algorithm to extract the parameters of the solar PV model, including SDM, DDM, and PV module models, and finally achieved excellent performance.
Ridha et al. 4 provided a boosted HHO (BHHO) scheme to obtain the parameters of the single-diode PV module. In this study, two adaptive strategies enriched by the flower pollination algorithm (FPA) and differential evolution (DE) were implemented to enhance convergence rate acceleration and better exploration performance. Liang et al. 15 developed a self-adaptive ensemble-based DE (SEDE) based on three different mutation strategies to assess the PV model parameters. Also, novel meta-heuristic techniques comprised of multiple learning backtracking search algorithm (MLBSA) 64 ; BSA algorithm hybrid with Lévy flight (LFBSA) 6 ; improved adaptive NMS hybridized with the artificial bee colony (ABC) 65 ; self-adaptive teachinglearning-based optimization (SATLBO) 66 and sine cosine algorithm (SCA) enriched by NMS and opposition-based learning (OBL) approaches 18 have been proposed for accurate estimation of the parameters of various PV systems, which have high efficiency and flexibility.
Generally, improved optimization algorithms are created by expanding the algorithm's local and global search capabilities to produce a new algorithm intended to exceed the algorithm's original form. [67][68][69][70] Even though numerous metaheuristic algorithms have been utilized to extract PV parameters, establishing metaheuristic algorithms based on strong exploration and exploitation terms may further increase the reliability and accuracy of an optimization technique. Thus, in this study, an improved version of the SMA 71 known as the adaptive SMA (ASMA) is presented for identifying the parameters of PV models more effectively and accurately than the previous approach described. Indeed, an improved SMA method is proposed to address SMA shortcomings such as sluggish convergence rate, poor exploration mechanism, and easy premature convergence. [72][73][74] It seems that by enhancing the MAs, the algorithm's performance will become more efficient and appealing for solving complex optimization issues. In this context, four enhancements are being proposed for the ASMA to optimize various kinds of PV models that the original SMA may be unable to effectively address. First, in the contraction mode of the SMA, two mutation vectors, namely (1) a trigonometric-based mutation (TBM) operator and (2) a double-based best mutation operator, are used to improve the global and local search. Second, a novel mechanism is implemented to adapt the control parameters of the proposed algorithm. Third, a local escaping strategy (LES) introduced by Ahmadianfar et al. 68 is provided to increase the prevention of trapping in the local solutions. Finally, an opposition-based learning (OBL) operator is developed to improve the position of the best solution at each iteration. To assess the efficiency of ASMA, the proposed algorithm was applied to derive the different PV model parameters. The calculated results prove that the proposed approach can accurately and reliably optimize various PV models also present encouraging results compared with other optimizers.
The present study is organized as follows. The mathematical formulations of the PV models are described in Section 2. The SMA algorithm is described in Section 3. Section 4 shows the proposed ASMA method carefully. Section 5 studies the results of the tests to check the efficiency of the ASMA. Finally, Section 6 describes the conclusion and the future perspectives of this study.

| Single diode model
A single diode model is widely utilized because of its easy implementation and simplicity. As it is illustrated in Figure 1A, it can be seen that the model has a photogenerated power supply in parallel with the diode, a shunt resistor to characterize the loss owing to surface leakage end to end the edge of the cell or because of crystal defects. A resistor in series to reflect the losses related with the load current. The formula of output current I is given below 75 : where I signifies the output current of the SDM, I ph represents the photogenerated current, I d represents the where V is the cell output voltage; I sd means the diode reverse saturation current; R s is the series resistance and R sh is the shunt resistance; β denotes the nonphysical diode ideality factor. indicates the junction thermal voltage, which is given below: where k indicates the Boltzmann constant (1.3806503 × 10 J/K), −23 T is the temperature of junction in Kelvin, and q denotes the electron charge (1.60217646 × 10 C) −19 .
Combining Equations (1)-(4), the output current I will be obtained as: where five parameters, namely: I ph , I sd , R s , R sh , and β, are required to be recognized and identified in an SDM.

| Double diode model
In view of the influence of current recombination loss in the depletion region, the DDM is more accurate and precise. Figure 2 illustrates DDM's equivalent circuit. As shown in the figure, two diodes are connected in parallel with the photogenerated power supply. One of the diodes is utilized as a rectifier, and the other is employed to model the composite current. The output current's mathematical rule is as 75 : where I d1 is the first diode current and I d2 means the second diode currents, which are defined as below, where I sd1 is the diffusion current and I sd2 denotes the saturation current. β 1 is the first diode ideality factor and β 2 is the second diode ideality factors. Similarly, DDM's output current I is formulated as Equation (9), where seven unknown parameters, namely:

| PV module model
In a scientific view, the magnitude of the voltage provided by a PV cell is limited, and usually, the actual needs cannot be met. For solving this drawback, the PV panel module model is introduced. The PV panel module model's current-voltage relationship is given as follows based on the SDM 43 : where M s is the number of solar cells connected in series and M p indicated the number of solar cells connected in parallel. As the SMM implemented in this study are all in series, M p is considered to be 1. Thus, Equation (10) is given below, Also, for the SMM, there are five unknown parameters as (I ph , I sd , R s , R sh , and β) that which are required to be derived. Figure 3 depicts SMM's equivalent circuit.

| Optimization objective function
In the PV model parameter identification issue, it is vital to ensure that the derived and extracted parameters minimize the difference between the current produced from the model simulation and the actual measured current error. To determine the optimal value for an unknown parameter. To verify the models accurately, the root mean square error (RMSE) has been utilized as the optimization objective, 18,43,63,76 and it is described as: where K is the number of measured I-V data; x represents the parameter to be identified; f(V, I, x) denotes the error function, which is specified by Equations (13)-(15) for the SDM and DDM and PV panel models, correspondingly.
Objective function of SDM: Objective function of DDM: Objective function of PV panel module model:

| SLIME MOULD ALGORITHM
SMA is a new optimizer that has been developed by Li et al., 71 and the oscillation mode of slime mould inspires it. Accurate mathematical modeling regarding constraints and requirements of problems is a crucial step for any real-world problem in optimization and machine learning. [77][78][79][80][81][82] Hence, SMA is not an exception as well. More details about this algorithm are delineated below:

| Approach food
The mathematical equation is utilized in this section to represent the slime mould's approach and behavior. To simulate the contraction mode, the following formulae are provided.
where μ b denotes to the parameter in the range of a a [− , ], μ c reduces linearly from one to zero. it denotes the current iteration, x best is the individual location with the highest odor concentration currently found, x k it denotes the location of slime mould, x a and x b denote two individuals who have been selected F I G U R E 3 Equivalent circuit diagram of photovoltaic module model randomly from the swarm, w is the weight of slime mould. The formula of p is described as below equation: where  k N p 1,2, …, , S k ( ) is the fitness of x k , DF denotes the best fitness which is acquired in all iterations.
The equation of μ b is formulated as below: The equation of w is formulated as: where condition represents that S k ( ) ranks first half of the population, r is the random value in the interval of [0,1] , bF represents the optimal fitness acquired in the current iterative process, wF represents the worst fitness value which is obtained in the iterative process, Smell Index represents the sorted fitness values' sequence.

| Wrap food
To update the location of slime mould, we utilize a mathematical equation as below: where X min is the lower boundaries of the search range, and X max represents the upper boundaries of the search range, rand and r represent the random value in [0,1].

| Grabble food
The μ b the value fluctuates randomly in the range of a a [− , ], and it gradually leads to zero as the iterations increase. The μ c the value fluctuates between [−1,1], and finally, it approaches zero.

| Improved approach to food
The fundamental approach to food in the original SMA can keep the capacity to look for food in the local solution area. In contrast, it causes the SMA algorithm's convergence rate to slow down. Due to this, this study presented a mutation technique based on weighted vectors to increase local search ability, global exploration capacity and accelerate the convergence of SMA. As a result, two mutation vectors are employed to enhance the contraction mode of the SMA algorithm: (1) a trigonometric-based mutation (TBM) operator 83

(Equation 23) and (2) a double-based best mutation operator (Equation 24
). This phase of the algorithm is formulated as follows: The algorithm uses the TBM and its three weight a c ) to move from a solution with a larger objective function value toward a solution with a smaller objective function value. 83 Besides this, these weights automatically scale the largeness of the vector differentials to make a suitable perturbation in the solution space. μ b and μ c are two random parameters.

| Proposed wrap food
This research's suggested wrap food strategy is comparable to the basic wrap food approach in the original SMA.
The key distinction is that the better approach to food mentioned in the preceding section is employed instead of the basic formulation used in the original SMA. In this section, the main formulation of the proposed algorithm to update the position of slime mould is defined as follows:

| Crossover operator
To increase the diversity of the population in this study, a binomial crossover operator is used. This type of crossover is broadly used in previous studies. 61,84,85 The proposed crossover combines the components of two vectors and creates a new vector (x new ). The scheme of this operator is expressed as follows: where j rnd is a random integer number in the range of [1, D]. In this study, the DE/current-to-best/1 mutation operator of the DE algorithm can improve the global search ability of the proposed algorithm (Equation 27). pc k is the probability crossover rate is determined by using Equation (28). 86 In addition, based on this scheme, the chance of selecting the current solution (x k ) is high in the initial iterations, whereases in the last iterations the probability of choosing the best solution is very high.
Calculate the positions x sma using Equation (25) Applying the crossover operator and creating the x new using Equation (26) if rand < LM Update the parameters μ b , μ c , and pc using Equation (29)

| Adaptive parameters
In general, the proper selection of the control parameters of an optimization algorithm has a significant role in its optimization efficiency. In this regard, an adaptive mechanism is provided in this study to improve the performance of the proposed algorithm. The formulation of this mechanism is defined in Equation (29).
Based on Equation (29), when the objective function is larger than the objective function current solution f x ( ) k , this means that three parameters (i.e., μ b k , , μ c k , , pc k ) generated using Equation (29) are not suitable for this iteration, and their old values maintained for these parameters (μold b , μold c , and pcold). This mechanism assists the algorithm in selecting the best values for these parameters. In fact, this process can conduct the algorithm to explore the most promising regions in the search space. It can facilitate the procedure of finding the global solution during the optimization process.

| Local escaping strategy
In this study, the LES introduced by Ahmadianfar et al. 68 is employed to promote the capability of escaping from the local optimal solutions in the proposed algorithm. This strategy is only applied when the condition rand < LM is met. The main formulation of this strategy is expressed in the next rule.
where LM is the logistic map (LM = 4 × LM × (1 − LM)) with the initial value of 0.7. In mathematics, chaos is known as randomness and nonlinear phenomena. The chaos is broadly applied in previous studies [55][56][57] to boost the efficiency of optimizers for solving nonlinear systems. In this regard, the logistic map is implemented to better use the LES. x cur-best is the best solution achieved in each iteration.

| Proposed opposition-based learning operator
OBL was first proposed by Tizhoosh, 87 and its ability to improve the performance of optimization algorithms has been proven in many previous studies. The main idea of OBL is to investigate the opposite position of an original candidate solution to explore more promising solutions. In this way, to calculate the opposite positions of the solution x k , the following formula is defined, This study uses the enhanced opposition-based learning (EOBL) to promote the position of the best solution x best . The mathematical scheme of OBL is expressed as follows: where m denotes the number of times that the proposed OBL is performed at each iteration. n Max and n Min are the maximum and minimum number of times that the OBL is executed. x d denotes a randomly selected member of the population in the range of [1, Np]. The proposed OBL is performed when the condition rand < LM is satisfied. The proposed operator is repeated m times at each iteration. The parameter m can be varied based on Equation (33). In this study, two formulas are considered to implement the OBL. The first formula employs the x best to calculate the opposite solution (Xobl), and then uses it to calculate the final solution (X OB ), according to Equation (34). In the case of the second formula, a random solution selected from the population (x d ) is utilized to calculate the X obl, and the final solution is calculated based on the X obl and x d (Equation 35). The condition rand < 0.5 is employed to specify when these formulas can be applied. Eventually, the best solution is chosen by comparing the objective function of the X OB and x best . The pseudocode of the proposed algorithm is shown in Table 1. Figure 4 displays the flowchart of the ASMA algorithm.

| RESULTS AND DISCUSSION
In this part, the PV models' parameter identification problem, that is, the single diode (SDM (M1)), the double diode (DDM (M2)), and PV module (M3 and M4) model are addressed simultaneously for verifying the proposed algorithm's efficiency as compared to other state-of-the-art  66 hybrid of differential evolution (DE) and PSO with multistrategy (MS-DEPSO), 94 and hybrid teaching-learning-based artificial bee colony (TLABC) 60 have been utilized to compare with the proposed ASMA approach. For a fair comparison, as per optimization and machine learning rules, [95][96][97][98][99] the MaxNFE for all of these algorithms is set to be same, which is equal to 20,000 in our tests. Also, the parameter of each algorithm is given in detail in Table 3.
To further analyze the superiority of the developed ASMA method, we compare the data obtained from the simulations with the existing data containing the relevant NFE, RMSE values, and the identified parameter values. In addition, the optimal values obtained in the table are marked in bold. It should be mentioned that the suggested ASMA algorithm and other comparative techniques were implemented in this study using MATLAB R2017b software on a Windows Server 2008 R2 operating system with an Intel (R) Xeon (R) CPU E5-2650 v3 (2.30 GHz) and 16 GB of RAM.

| Results of the SDM model
The statistical results for the SDM (M1) model based on the RMSE values are given in Table 3, where it is evident that the best (BS), worst (WS), average (AVG), standard deviation (SD), and CPU time have been considered. The Wilcoxon singed-rank test evaluated the significance between ASMA and other competing algorithms. 100 R + denotes the sum of ranks for various runs in which ASMA exceeded other similar methods in performance. R − is the sum of ranks for different runs in which other similar methods exceeded ASMA in performance. p value indicates the importance of results in the statistical hypothesis test (α = 0.05). The symbol "+" indicates that  ASMA is performing promisingly as compared to its competitor while "=" denotes that ASMA is similar to its competitor. Based on Table 4, ASMA, MS-DEPSO, and SHADE achieved the best BS, WS, and AVG, indicating that these algorithms can achieve the problem's optimal solution, followed by SATLBO JADE GOTLBO, PGJAYA, TLABC, and IJAYA. Regarding the SD reflecting the algorithm's robustness, ASMA, MS-DEPSO, SHADE, and SATLBO demonstrated promising performance analysis, while IJAYA yielded the most unsatisfactory results. Also, the ASMA can achieve the smallest SD (4.0740E−17) compared with the other methods. This indicates the proposed method has high reliability and precision to solve the SDM model. The ASMA is ranked fifth based on the CPU time (s). Moreover, based on the Wilcoxon singed-rank test results (see Table 5), ASMA is very similar to MS-DEPSO. However, it performs better than other algorithms, especially the IJAYA, PGIAYA, TLABC, and GOTLBO. This superior efficiency of the proposed method is due to its LES operator and adaptive parameters (i.e., μ b , μ c , and pc), which can assist the algorithm in searching in the solution domain better locally. Table 6, which consists of the RMSE results of the algorithms, shows that most of the algorithms can finally attain the optimal value of RSME (9.8602E0−4). The corresponding parameters also have the same value. Additionally, Figure 5 describes the derived parameter values' accuracy, which has been obtained using ASMA. The I-V curve can intuitively reflect the degree of fit between the data obtained from the ASMA evaluation and the actual measured experimental data set. Also, the current's absolute individual error (IAE) is relatively small. It can be found that the IAE values of all the measurement points are below 1.6E−03 for the whole measured voltage (see Figure 5B). Table 7 displays the relevant IAE values of simulated current and simulated power attained from the trials. The results of the top five optimization approaches have also been listed in this table. Figure 6 shows the convergence curves of all these algorithms. As it is evident from this figure, the fastest convergence speed is for ASMA, followed by the GOTLBO, SATLVO, MS-DEPSO, SHADE, and JADE. Besides this, the convergence curves of the top six optimization approaches have been depicted in this figure. The Friedman rank test has been utilized for calculating the ranks of all optimizers. All these optimizers' rank on the M1 model is shown in Figure 7. The ASMA (2.9) and MS-DEPSO (2.9) have the best rank, followed by the SHADE, SATLBO, JADE, GOTLBO, PGJAYA, TLABC, and IJAYA.
In Figure 8, the box plot patterns clearly demonstrate how amounts in data are enlarged, resulting in less required space, and it is obvious that it could be beneficial for distribution analyses of datasets for

| Results of the DDM model
In Section 2, it was mentioned that seven unknown parameters are required to be derived in the DDM. and SD (1.6642E−06) as compared to other optimization algorithms. The proposed ASMA is more stable to achieve a suitable solution in 30 various runs based on the SD values. In fact, this superior performance is due to the efficient wrap food operator provided in ASMA and its proposed LES and OBL operators. These operators can assist the algorithm to an appropriate search globally in the feasible domain and then an effective local search to reach a promising solution.
In the case of CPU time (s), the rank of the ASMA algorithm is 5. The developed method performs significantly better than all other methods (Table 5). Table 9 shows the results of the RMSE values of ASMA compared to the related competing algorithms which ATLDE has acquired and the compared algorithms, where ASMA achieved the best BS (9.8248E−04), followed closely by MS-DEPSO (9.8261E−04), GOTLBO (9.8270E−04), SATLBO (9.8316E−04) and other methods. Although the RMSE value acquired by ASMA is slightly smaller than that of MS-DEPSO and GOTLBO, in reality, a slight difference can also cause huge benefits. The simulated data obtained by the ASMA is depicted in Figure 9, where the I-V curve between the measured data and simulated data indicates the accuracy of the derived parameters.
As shown in Figure 9B, all IAE values fall within the range of [0, 1.49E−03]. According to the comparison results among the measured and simulated data, which have been obtained by various methods available in Table 10, the ASMA reaches the smallest total IAE (1.732E−02). The ASMA achieves a better RMSE and IAE as compared to SDM. Moreover, DDM has better accuracy than SDM. The convergence curve for various algorithms is depicted in Figure 10. At the early stage, it is clear that the PGJAYA and MS-DEPSO have a fast convergence speed, while the ASMA converges rapidly right after 3500 NFE. The convergence of JADE, GOTLBO, and SATLBO is not fast enough. Figure 11 depicts the ranks of all these optimization algorithms on the M1 model. As it is clear, the best rank is for the ASMA (1.6), followed by the MS-DEPSO (3.97), SATLBO (4.00), PGJAYA (4.57), GOTLBO (5.20), TLABC (5.37), IJAYA (6.00), JADE (7.07), and SHADE (7.23). Figure 12 illustrates the boxplots of RMSE over 30 different independent runs for nine optimizers. According to the figure, the top and lower values are close when the ASMA box-plot is compared to all other approaches. The ASMA approach for generating optimum parameters for the M2 model, as demonstrated by the findings of this study, is thus the most exact method for obtaining optimal parameters for the M2 model among all the optimization strategies examined in this study. As is clear, the developed ASMA algorithm has the lower distribution RMSE over 30 independent runs, which is followed by the PGJAYA, MS-DEPSO, SATLBO, GOTLBO, IJAYA, JADE, TLABC, and SHADE.

| Results of the PV model
The developed ASMA algorithm has been utilized for identifying the parameters of the SDM and DDM. In this session, ASMA has been employed to determine the parameters of two practical PV panel models: monocrystalline STM6-40/36 (M3) and polycrystalline STP6-120/36 (M4). It is worth noting that, for the sake of fairness, both models have the same evaluation criteria.  smaller than other optimization algorithms indicating that the ASMA is more robust than the others. The ASMA algorithm is ranked fifth, similar to the previous PV models (M1 and M2). According to the Wilcoxon singed-rank test results (see Table 5), it is evident that the ASMA performs promisingly as compared to other algorithms, including GOTLBO, IJAYA, JADE, SHADE, PGJAYA, SATLBO, MS-DEPSO, and TLABC. Table 12 shows the results of the RMSE values of ASMA compared to the related competing algorithms. Figure 13 illustrates the I-V curve, which the ASMA has acquired. As it is clear from Table 12, Figure 13A, the I-V curves in the tests have a high grade of fitting with the collected data inside the normal range. Also, it is evident that the IAE has more fluctuations at the 11th data point, but it does not cross the 10−3 value (see Figure 13B). Table 13 gives the comparisons' results among the measured and simulated data. The ASMA and MS-DEPSO provide the smallest total IAE value for this model. The algorithms' Friedman ranks are depicted in Figure 14. Based on the results of this figure, ASMA (1.12) has the best rank, and it is followed by the MS-DEPSO (2.85), SHADE (2.87), PGJAYA (4.37), JADE (5.57), TLABC (5.93), SATLBO (6.83), IJAYA (6.93), and GOTLBO (8.53).

| Results on the M3 model
Besides this, the convergence curves for various algorithms are shown in Figure 15. The results indicate that the fastest convergence rate belongs to the developed ASMA. When NFE is 6500, ASMA achieves the optimal value. In contrast, the GOTLBO and PGJAYA fell into the local optimum. The boxplots of RMSE over 30 various runs for these optimizers are displayed in Figure 16. It is clear that the lower distribution RMSE over 30 runs belongs to the proposed ASMA, followed by the SHADE, MS-DEPSO, PGJAYA, JADE, IJAYA, TLABC, SATLBO, and GOTLBO.

| Results on the M4 model
The statistical results are reported in Table 14. The results show that the smallest BS (1.6601E−02) belongs to ASMA, GOTLBO, SATLBO, and MS-DEPSO. In contrast, only the ASMA achieved the best BS, AVG, WS, and SD of RMSE, particularly the SD (2.4929E− 16) value. ASMA has a minimal SD value which indicates good robustness. Due to the efficient operators provided in the ASMA (i.e., the wrap food, LES, and OBL) and its adaptive parameters (i. e., μ b , μ c , and pc) can explore the promising area in the search space in the initial generations of the optimization process and then it can efficiently move toward the global solution. Also, concerning the CPU time (s), the ASMA rank is 5. According to the Wilcoxon singed-rank test result, ASMA has a considerable R+ value higher than other algorithms (see Table 5).
All algorithms' corresponding derived parameter values are reported in Table 15, where four algorithms achieve the best RMSE value (1.6601E−02). In contrast, the other algorithm cannot reach this value (i.e., IJAYA (2.0459E−02), JADE (1.6607E−02), SHADE (1.6743E−02), PGJAYA (1.6762E−02), and TLABC (1.6647E−02). As can be seen from Figure 17A, the simulated and measured data are highly coherent with each other, and also, the   Figure 17B). Comparison results among the measured and simulated data for various approaches are given in Table 16. It is also apparent that the ASMA and MS-DEPSO give the minimum total value of IAE (0.2447). Moreover, the convergence curves for various algorithms are displayed in Figure 18. It is clear that the fastest convergence speed belongs to the developed method, followed by MS-DEPSO, PGJAYA, JADE, GOTLBO, and SATLBO. The ranks of nine optimizers for the M4 model are shown in Figure 19. It is clear proof enough t hat the ASMA gets the first ranking which is followed by the SHADE (4.   algorithm. We experienced that the proposed ASMA-based idea is not limited to a specific class of problem in parameter identification and its performance encourage us to apply it to several open new areas in future works, including hybridization with power systems, 101 hybridization with machine learning ideas, 102 hybridization with other algorithms, 103 decision-making models, 104 computational models, 105 and dealing with other energy optimization problems. 106

| Comparison of the basic SMA with ASMA
For indicating and evaluating the efficiency of the developed ASMA algorithm, four models are solved using the ASMA. The maximum number of NFE (MaxNFE) for the two algorithms equals 20000. Table 17 reports the statistical results of two optimizers for four models. Compared with SMA, it is evident that the ASMA has an entirely noticeable improvement in terms of the BS, WS, AVG, and SD. Particularly, ASMA has a smaller SD value than the SMA algorithm, indicating perfect accuracy and reliability. Additionally, Figure 21 illustrates the convergence curves of two optimizers (i.e., ASMA and SMA). Based on this figure, the developed ASMA has a faster convergence rate than the SMA. Based on Table 17 and Figure 21, it can be concluded that the original SMA cannot explore the promising region in the initial generations of the optimization process. Therefore, the original SMA suffers In addition, the time complexity is an important consideration (i.e., the number of NFEs consumed for the adaptive evaluation), and it is clear that ASMA consumes the least amount of computer resources and performs well on each model, indicating the efficiency and reliability of our proposed approach.  In this study, an ASMA has been developed to optimize PV models' parameters. In the ASMA, a trigonometricbased mutation (TBM) operator and a double-based best mutation operator are first utilized to improve the original SMA algorithm's local and global search proficiency. Next, a crossover operator has been employed to increase the diversity of the population. After that, a LES is implemented to avoid the local optimal solutions and improve the convergence speed. Finally, an enhanced opposition-based learning operator is performed to promote the best solution at each iteration. It is noteworthy that an adaptive mechanism is carried out to select the best values for the control parameters at each iteration. The performance analysis of ASMA has been evaluated by evaluating the parameters of various solar cell models, that is, the SDM, the DDM, and the PV model. The experimental results acquired by ASMA have been compared with the GOTLBO, IJAYA, JADE, SHADE, PGJAYA, SATLBO, MS-DEPSO, and TLABC and also with other state-of-the-art algorithms available in the literature. The results have shown the superior performance of ASMA in terms of efficiency, robustness, and convergence speed. Significantly, the mean RMSE values achieved by ASMA are 9.8602E−04, 9.8449E−04, 1.7298E−03, and 1.6601E−02 for The M1, M2, M3, and M4 models, respectively. In addition, the proposed ASMA indicated that it could achieve the best solution with less NFE (20,000) compared with the other methods such as ATLDE (NFE = 30,000), SGDE (NFE = 50,000), and SEDE (NFE = 40,000). This confirms the superior performance of the proposed method to solve the different PV models.
We will focus on employing the ASMA for the realistic solar cell model at different temperatures for future work perspectives. Moreover, we need to focus more on the DDM since the results obtained by many methods still are not satisfactory, and maybe machine learning technology is a feasible method. In addition, several dynamic parameters will be used and evaluated for ASMA. To lessen ASMA's computing burden, the surrogate model may be used. Other optimization problems such as scheduling and extension of the ASMA to be used for multiobjective problems may be our next direction.