Enhanced Success History Adaptive DE for Parameter Optimization of Photovoltaic Models

School of Computer Science and Technology, Shandong Technology and Business University, Yantai 264005, China College of Economics & Management, Shanghai Ocean University, Shanghai 201306, China Operations Research Department, Institute of Statistical Studies and Research, Cairo University, Giza 12613, Egypt Wireless Intelligent Networks Center (WINC), School of Engineering and Applied Sciences, Nile University, Giza, Egypt School of Information and Engineering, Sichuan Tourism University, Chengdu 610100, China College of Electronic Information and Automation, Civil Aviation University of China, Tianjin 300300, China


Introduction
In recent years, in order to solve the problems of environmental pollution and burn out, the utilization of solar, wind, hydropower, nuclear, and so on has been increasing attention [1]. Among them, solar energy is considered as one of the most promising alternatives to inexhaustible and clean sources. At the present time, the PV systems play a very important role in power system, because solar energy can directly be converted into electric energy to supply power with humans. erefore, solar PV systems have been widely applied in the whole world and have continued growing. However, the PV systems are exposed to the external environment and their PV arrays are prone to aging, which seriously affects production efficiency of PV panels and are harmful to the work efficiency of solar energy [2,3]. Hence, in order to effectively design, simulate, estimate, control, and optimize PV systems, it is paramount to estimate the performance using exact model in operation. e most widely used mathematical models for describing the nonlinear behaviour and performance are the single and double diode models and the PV model. However, the accurateness of these models is dependent on the parameters' values of models. e parameters are the reflection of the intrinsic characteristics of the PV model, and the I-V equation can be determined by identifying the PV parameters in order to predict the output power of PV array. e parameter optimization problem of the PV model is to fastly and accurately identify the parameters of the PV model in order to obtain better output power prediction and maximum power point tracking. Due to the aging, failure, breakdown, and unstable working conditions, it is very difficult to determine these parameters. erefore, it is an extremely essential work to deeply study an effective method to optimize the parameters of PV and improve the solar energy utilizing efficiency.
Usually, the estimated problem of unknown parameters for the PV models is considered as an optimization objective function [4,5]. Due to the measured current data and voltage data involving noise, the constructed objective function is a nonlinear and multimodal function with multiple local optimums. Some researchers have studied and presented many methods to optimize the parameters of PV models in recent years, such as analytical method, deterministic method, and heuristic method. e heuristic method is a promising alternative to analytical method and deterministic method [6]. Due to no strict restrictions for the objective function, a lot of heuristic methods have obtained more and more attention for optimizing the parameters of PV models, such as simulated annealing (SA), genetic algorithm (GA), differential evolution (DE), particle swarm optimization (PSO), artificial bee colony (ABC), bacterial foraging algorithm (BFA), teaching-learning-based optimization (TLBO), whale optimization algorithm (COA), bird mating optimizer (BMO), month flame optimizer (MFO), and backtracking search algorithm (BSA) [7][8][9][10][11][12][13][14][15][16][17][18][19][20]. Recently, gaining-sharing knowledge-based algorithm has been proposed by Mohamed et al. [21]. Zagrouba et al. [22] used genetic algorithms to identify the parameters of PV solar cells and modules. Ishaque et al. [23] presented a penalty-based DE to extract the parameters of solar PV models. Merchaoui et al. [24] presented an improved PSO with adaptive mutation strategy to extract the parameters of PV solar cell/module. Yu et al. [25] presented a performance-guided JAYA algorithm to identify the parameters of PV cell and modules. Chen et al. [26] presented a teachinglearning-based ABC algorithm to estimate the parameters of the solar PV model. Li et al. [27] presented an improved teaching-learning-based optimization algorithm to extract the parameters of PV models. Oliva et al. [28] used ABC algorithm to identify the parameters of solar cells. Wu et al. [29] presented an improved ant lion optimizer to identify the parameters of the PV cell model. Ram et al. [30] presented a new hybrid bee pollinator flower pollination algorithm to estimate the parameters of solar PV. Oliva et al. [31] presented an improved chaotic WOA to estimate the parameters of PV cells. Xu and Wang [32] presented a hybrid flower pollination algorithm to estimate the parameters of PV modules. Chen et al. [33] presented a diversificationenhanced Harris Hawks optimizer to identify the parameters of PV modules. Long et al. [34] presented a new hybrid algorithm based on a grey wolf optimizer and cuckoo search to extract the parameters of solar PV models. Chen et al. [35] presented a novel opposition-based sine cosine approach with local search to estimate the parameters of PV models. Cai et al. [36] employed an orthogonal experiment method to optimize the parameters of the dust absorbing structure for PV panels. e results reported by these heuristic methods have achieved satisfied results, which indicate that these heuristic methods are promising alternative to identify the parameters of PV models. But because the parameter identification of PV models is a nonlinear and multimodal problem, their accuracy and reliability need to be further improved. In addition, many heuristic methods have their own parameters to be experimentally tuned to improve their efficacy, accuracy, reliability, and scalability. Differential evolution (DE), a random evolution algorithm based on population evolution, was proposed by Storn and Price in 1997 [37], which has been regarded a simple and efficient optimization algorithm. Due to the fast convergence, robustness, and search ability, various advanced DE variants have been presented and widely applied in various fields of engineering design, operations research, biomedicine, and so on. Most of the various advanced DE variants are used to optimize the parameters of different PV models. Chang [38] presented an ant direction hybrid differential evolution algorithm. Gong and Cai [39] presented an improved adaptive differential evolution with crossover rate repairing technique and ranking-based mutation, as R-cr-IJADE. Muhsen et al. [40] presented an improved differential evolution with adaptive mutation per iteration algorithm (DEAM). Chellaswamy and Ramesh [41] presented adaptive differential evolution algorithm. Mohamed and Abdulaziz [42] proposed a differential evolution with novel mutation and adaptive crossover strategies to solve largescale global optimization problem. Zhang et al. [43] presented a gradient decent-based multiobjective cultural differential evolution (GD-MOCDE) to improve the optimal efficiency. Mohamed [44] proposed an enhanced adaptive differential evolution algorithm to solve large-scale global optimization problems. Rashidi and Khorshidi [45] presented a multiobjective differential evolution algorithm. Mohamed [44] presented a multiobjective self-adaptive differential evolution (MOSaDE) algorithm. Mohamed [46] proposed a new approach to differential evolution algorithm for solving stochastic programming problem. Ramli et al. [47] presented a multiobjective self-adaptive differential evolution algorithm. Mohamed and Suganthan [48] proposed enhanced fitness-adaptive differential evolution algorithm with novel mutation to solve real-parameter unconstrained optimization problem. Mohamed and Mohamed [49] proposed an adaptive guided directed differential evolution algorithm to solve unconstrained optimization problems. Tey et al. [50] presented an improved global search space differential evolution algorithm. Xiong et al. [51] presented an effective hybrid method based on the exploration of DE with the exploitation of WOA as DE/ WOA. Hadi et al. [52] proposed a LSHADE-SPA memetic framework to solve large-scale optimization problem. Mohamed and Mohamed [53] proposed an enhanced AGDE algorithm for real-parameter unconstrained optimization problem. Li et al. [54] presented a memetic adaptive differential evolution as MADE algorithm. Mohamed et al. [55] proposed an enhanced DE algorithm (EDDE) that utilizes the information given by good individuals and bad individuals in the population.
e new mutation scheme maintains effectively the exploration/exploitation balance. Essiet et al. [56] presented an improved enhanced 2 Complexity differential evolution algorithm for implementing demand response between the aggregator and consumer. In the team of professor Mohamed, a lot of work has been done to improve the DE algorithm. Mohamed et al. [53] proposed an enhanced directed differential evolution algorithm for solving constrained nonlinear integer and mixed-integer global optimization problems. Mohamed et al. [57] proposed an EBLSHADE algorithm based on novel mutation strategy. In this work, two mutation operators are introduced, ord_best and ord_pbest, which are versions of the classical DE/current-to-best/1 scheme. e proposed mutations were incorporated into SHADE and LSHADE algorithms in order to enhance their performances.
Due to the better robustness, stability, and quality of the solution of LSHADE, an enhanced SHADE (EBLSHADE) algorithm is applied to propose a parameter optimization method to optimize the parameters of PV models quickly, accurately, and reliably. In the EBLSHADE, a less and more greedy mutation strategy is used to enhance the exploitation capability and the exploration capability. A linear population size reduction strategy is used to gradually reduce population for improving the convergence speed, balancing the exploration and exploitation capabilities in the process of evolution. e EBLSHADE is employed to propose a parameter optimization method to optimize the unknown parameter estimation of the single diode model, double diode model, and PV model. e experimental results demonstrate that the parameter optimization method can exactly and reliably optimize the unknown parameters of different PV models and provide highly competitive results compared with other algorithms. e main contributions of this paper are described as follows: (i) An EBLSHADE is applied to effectively optimize the parameters of PV models. Based on quantified performance, the proper update strategy can adaptively be selected for individuals to immeasurably elevate the related searching performance. (ii) e less and more greedy mutation strategy in the EBLSHADE is used to enhance the exploitation capability and the exploration capability, respectively. (iii) e linear population size reduction strategy is employed to gradually reduce population to further improve the convergence speed, balance the exploration and exploitation capabilities, and avoid to falling into local optimum in the process of evolution. (iv) e performance of EBLSHADE has been extensively investigated by the parameter estimation of different PV models. e rest of this paper is arranged as follows. Section 2 describes different PV models and their objective functions. e differential evolution algorithm is briefly described in Section 3. Section 4 introduces an enhanced SHADE with multistrategies in detail. e experimental results on different PV models are shown and analyzed in Section 5. Finally, the conclusion is given in Section 6.

Differential Evolution
e main operations of DE contain initialization population, mutation operation, crossover operation, selection operation, and so on. Its main thoughts are to differentiate and scale between two different individual vectors in the same population and add a third individual vector in this population to obtain a mutation individual vector, which is crossed with the parent individual vector with a certain probability to generate an attempted individual vector. Finally, the attempted individual vector and the parent individual vector are executed greedy selection, and the better individual vector is saved to the next generation population.
e basic evolution processes of e DE are described.

Abbreviations and Acronyms.
e DE uses NP D-dimension vectors as the initial solution. Setting population number N, each individual can be expressed as where G represents the G th generation, x max represents the maximum search space value, x min represents the minimum search space value, and rand(0, 1) represents a random number.

Mutation Operation.
It generates a mutation vector V i,G for x i,G , namely, target vector. For each generated target vector, the mutation strategy is used to obtain a corresponding mutation vector. e mutation operation is the most important operation of DE. According to the different generation methods of the mutation individuals, several different mutation strategies of DE are formed. e most common mutation strategies are described as follows: (1) DE/best/2/bin (2) DE/rand/2/bin (3) DE/current-to-best/1/bin (4) DE/current-to-rand/1/bin

Crossover Operation.
Each pair of target vectors x i,G and the corresponding mutation vectors V i,G are crossed to obtain a test vector U i,G � (u 1,G , u 2,G , . . . , u i,G ). In the basic DE, it uses a binomial crossover to define.
where the CR is a constant between 0 and 1, which is used to control the duplicated proportion from the mutation vector. j rand is a selected integer within [1, D], j � 1, 2, 3, . . . , D.

Selection Operation.
Comparing the objective function value f(U i,G ) of each test vector, the objective function value of the test vector is less than the corresponding target vector, and then the target vector is replaced by the experimental vector. e selection operation is given as follows:

Implementation of DE.
e DE algorithm evolves generation by generation until the result or ending condition has been met. e flow chart of DE algorithm is shown in Figure 1.

SHADE.
e SHADE is one of the most successful variants of DE [57]. In the SHADE, a historical memory with H entries is used. It is made up of MCR and MF, which can adaptively control parameters CR and F. In each iteration, each individual (x i ) has its own F i and CR i to generate a new test vector u i . e two parameters are expressed as follows: where r i is a random integer on [1, H], randn is the Gaussian normal distribution, and the scale parameter is 0.1, and randc is the Cauchy distribution and the variance is 0.1. e CR and F of the generated test vectors are recorded as S CR and S F . eir average values are stored in M CR and M F . e SHADE keeps H parameters to guide the control parameters in order to achieve self-adaptive search. Even if the S CR and S F of some offsprings contain a set of poor values, the stored parameters of the previous generation will not be affected. e control parameters are shown in Table 1.
en, the M CR and M F are updated according to the following expressions: where t is an index to determine the saved position and k is the set 1. When a pair new S CR and S F are added in the history, k is incremented by 1. Mean WA (S CR ) and Mean WL (S F ) are weight mean values of S CR and weight Lehmer mean value of S F .

LSHADE.
An improved SHADE based on the reduction strategy of the linear population size, namely, LSHADE is developed. In the LSHADE, the population size is gradually lessened. erefore, the linear function is described as follows: where round returns the nearest integer number. NFE is the fitness optimization, Max NFE is the optimization with the maximum iterations, NP 0 is the initial population size, and NP min is the possible minimum population size, which is the minimum number of individuals (NP min � 4).

A Mutation Strategy.
e DE/current-to-best/1 strategy can find the best solution in the evolution. But it may deteriorate or lose the population diversity and exploration capability. To overcome these shortcomings, a variant of DE/current-to-best/1, namely, DE/current-to-or_best/1 is used to balance the local exploitation and the global exploration abilities and enhance the convergence speed. In the strategy, all individuals are sorted to divide into three vectors. e best vector is referred as x or_best , G, the median vector is referred as x or_median , G, and the worst vector is referred as x or_worst , G. erefore, the trial vector is described as follows: As can be seen from the new mutation strategy equation (12), the added objective function value has two advantages. e difference vector of the best vector and the target vector is the first perturbation part of the new mutation strategy, which can substantially avoid prematurity and accelerate convergence.
e second perturbation part of the new mutation strategy is the difference vector of the median vector and the worst vector. erefore, the DE/current-to-or_best/1 can get the global optimal solution.
When the population size reduction strategy can improve optimization performance of the algorithm, the initial population size will be reduced to 18 dimensions in the LSHADE. e increased population size will affect DE/ current-to-or_best/1; the probability of x or_best , G, will be decreased to be the global best solution. erefore, the behaviour of the DE/current-to-or_best/1 will approximate to DE/rand/1. To solve this problem, an enhanced version of DE/current-to-or_pbest/1 is used. In this mutation strategy, one vector from top p best vectors is included. e other two vectors are chosen randomly. en, three vectors are sorted.
e best vector is referred as x or_pbest , G, the median vector is referred as x or_pmedian , G, and the worst vector is referred as x or_pworst , G. erefore, the trial vector is obtained as follows:

Model of EBLSHADE.
e flow of the EBLSHADE is shown in Figure 2.

SDM.
is model can describe the behaviour of solar cell effectively [17]. Equivalent circuit is given in Figure 3.
In this model, it is made up of a leakage current shunt resistance, a parallel current source with a diode, and several resistors for the related load current loss.
erefore, the output current is described as follows: From the abovementioned equations, the parameters of I ph , I d , R s , R sh , and α need to be optimized. erefore, the objective functions can be formulated:

DDM.
It is used to take the recombination current loss effect. Equivalent circuit is given in Figure 4. e output current of this model is described as follows: From the abovementioned equations, the parameters of I ph , I sd1 , I sd2 , R s , R sh , α 1 , and α 2 need to be estimated. erefore, the objective functions can be formulated: Generate randomly the initial population Meet the end condition?
Calculate fitness value of each individual Do difference evolution operation Output the result

PVM.
It is usually based on series or parallel solar cells. Its equivalent circuit is described in Figure 5. is circuit can be extended to the N-diode PV model. e output current of this model is formulated.
Due to the used PV model with series in the experiment, there is N P � 1.
e abovementioned equation can be reformulated: From the abovementioned equations, the parameters of I ph , I d , R s , R sh , and α will be optimized. e objective functions of the PV model can be formulated:

Parameter Setting.
In order to guarantee the comparative fairness, the lower bound (LB) and upper bound (UP) are described in Table 2.
To prove the superior ability of the EBLSHADE, some existing algorithms are selected in Table 3

Root Mean Square
Error. Root mean square error (RMSE) is the ratio square root of the square of the deviation between the predicted value and the actual value and the number of observations n. In the actual measurement, the observation number n is always limited, and the true value can only be replaced by the most reliable value. It is used to measure the deviation between the observed value and the true value and illustrates the dispersion degree of the data. e root mean square error (RMSE) is defined as follows: where N is the experiment number and x is a vector.

Experimental Results of SDM.
Here, the RMSE is the obtained computation value according to the equation of root mean square error (equation 20) by each independent run of the algorithm. MRMSE is the mean value of RMSE of different set of 30 runs; BRMSE is the best value of RMSE among different set of 30 runs. erefore, the compared results of five parameters, statistical results of the best RMSE (BRMSE), the mean RMSE (MRMSE), the standard deviation (SD), and the least computing resources (NFE) are given in Table 3, where the obtained best results are bold. e boxplot of RMSE is presented in Figure 6.
From Table 3 and Figure 6, the TLABC, MLBSA, JADE, SHADE, MADE, and EBLSHADE obtain the BRMSE value (9.8602E − 04). Especially for MLBSA, SHADE, MADE, and EBLSHADE, the best, worst, and mean RMSE values are the same value (9.8602E − 04). Due to the unavailable information, the RMSE is usually used to express the more accurate parameters. Although the second BRMSE value (9.8603E − 04) of IJAYA is infinitely close to the BRMSE value (9.8602E − 04), it is significant to reduce the order of objective function and improve the true value of parameters. For standard deviation, the EBLSHADE obtains the third best standard deviation value, which is close to the best To further prove the effectiveness of the EBLSHADE, the detailed results of the individual absolute error of current (IAEI) and power (IAEP) between the experimental data and the measured data are shown in Table 4. See references for measurement data. e I-V and P-V characteristics curve obtained by EBLSHADE are shown in Figures 7-10. Note that Ic is calculated by the optimized parameters using EBLSHADE.
From Table 4 and Figures 7-10, all IAEI values are lesser than 2.5075E − 03 and all IAEP values are lesser than 1.4626E − 03. It can be seen that the EBLSHADE can accurately optimize these parameters. It is also evident that the obtained experimental data by EBLSHADE are highly consistent with the measured data, which effectively reflects the optimized parameters to be accurate enough. erefore,  Complexity the EBLSHADE can be considered as a significant method for the parameter optimization of SDM.

Experimental Results of DDM.
Seven optimized parameters will increase the optimization difficulty. e EBLSHADE is compared with the selected methods in Table 5. e results are shown in Table 6. e boxplot of RMSE is shown in Figure 11.
From Table 5 and Figure 11, the SHADE only achieved the BRMSE value ( Figure 5: e equivalent circuit.    are shown in Table 6. e obtained I-V and P-V curves are shown in Figures 12-15. See references for measurement data.
From     10 Complexity the optimized parameters with higher accuracy.
e experimental results show that the experimental data by EBLSHADE are highly in agreement with the measured data. So, the EBLSHADE is regarded as a significant algorithm to optimize the parameters of DDM.

Experimental Results of PVM.
For the PV models, to further prove the performance of the EBLSHADE, photowatt-PWP201, STM6-40/36, and STP6-120/36 are used in here. e experimental results of the PVMs are shown in Tables 7-9. e boxplot of RMSE are presented in Figures 16-18.
From Table 7 and Figure 16 on the photowatt-PWP201 model, all compared algorithms can obtain the same BRMSE (2.4251E − 03), while the EBLSHADE only consumed the least 5,000 NFE. Especially for the SHADE, MADE, PGJAYA, and EBLSHADE, the best, worst, and mean RMSE values are the same value (2.4251E − 03), which indicate that the SHADE, MADE, PGJAYA, and EBLSHADE take on better stability. For standard deviation, the EBLSHADE obtained the second best standard deviation value (2.8821E − 17), which is close to the best standard deviation value (2.0700E − 17). Although the EBLSHADE did no't obtain the best standard deviation value, it only consumed 5,000 NFE to obtain the better optimization results.
From Table 7 and Figure 17 7298E − 03), which indicate that the MADE and EBLSHADE take on better stability. For standard deviation, the EBLSHADE obtained the best standard deviation value (6.40591E − 14). e max NFE of EBLSHADE is 10,000 NFE, which utilizes less computational resources compared to other compared algorithms except for MADE. Although the EBLSHADE is not less NFE for the STM6-40/36 model, it consumed 10,000 NFE to obtain the better optimization results.
From Table 8 and Figure 18 on the STP6-120/36 model, except for IJAYA and JADE, the other compared algorithms can get the same BRMSE value (1.6601E − 02). Especially for SATLBO, MADE, and EBLSHADE, the best, worst, and mean RMSE values are the same value (1.6601E − 02), which  Complexity indicate that the SATLBO, MADE, and EBLSHADE take on better stability. For standard deviation, the EBLSHADE obtained the best standard deviation value (8.0544E − 16). e max NFE of EBLSHADE is 15,000 NFE, which are much less computational resources than the other compared algorithms except for MADE. Although the EBLSHADE is not less NFE for the STP6-120/36 model, it consumed 15,000 NFE to obtain the better optimization results. erefore, it is clear that the EBLSHADE can effectively and consistently provide better results of integrating the RMSE and NFE considerations by comparing with other methods.         It is also clear that the experimental data by EBLSHADE are highly in agreement with the measured data. erefore, the EBLSHADE is considered as a significant method to optimize the parameters of the PV models.

Discussion of the Results.
As the experiment results are given in Section 5.3, the EBLSHADE is applied in the parameter optimization of PV models, respectively. e superiority of the EBLSHADE has been proved by comparing with some other algorithms. It can be seen from the I-V curves and P-V curves that the experimental results of EBLSHADE are highly in agreement with the measured data for all datasets, which effectively reflects the optimized Photowatt-PWP201 model          parameters to be accurate enough by using EBLSHADE. All the compared results containing the optimal parameters, the BRMSE, WRMSE, MRMSE, SD, and NFE indicate that the EBLSHADE has better capacities of the exploration and exploitation. e reason is that the novel mutation strategies of DE/current-to-or_best/1 and DE/current-to-or_pbest/1 for EBLSHADE can enhance and balance the exploitation capability and exploration capability and enhance the convergence. Rregarding the whole result, the EBLSHADE is considered as a significant method for the parameter optimization of SDM, DDM, and PVM. It is also expected to be applied for optimizing parameters of other PV models.

Conclusion
In this paper, an enhanced success history adaptive DE(EBLSHADE) has been applied to propose a parameter optimization method for deferent PV models. e effectiveness of the EBLSHADE and parameter optimization method has been verified on estimating parameters of different PV models. e experimental and statistical results with the reliability, accuracy, and computing efficiency show that the EBLSHADE is superior to the other compared algorithms, and the parameter optimization method is an effective method and can design, control, and optimize the PV systems. erefore, the EBLSHADE is considered as a significant method for optimizing parameters of other PV models.
Data Availability e data are available in this paper.

Conflicts of Interest
e authors declare that they have no conflicts of interest.  Complexity