Artificial bee colony algorithm based on a new local search approach for parameter estimation of photovoltaic systems

In this study, an ABC-Local Search (ABC-Ls) method was proposed by including a new local search procedure into the standard artificial bee colony (ABC) algorithm to perform the parameter estimation of photovoltaic systems (PV). The aim of the proposed ABC-Ls method was to improve the exploration capability of the standard ABC with a new local search procedure in addition to the exploitation and exploration balance of the standard ABC algorithm. The proposed ABC-Ls method was first tested on 15 well-known benchmark functions in the literature. In the results of the Friedman Mean Rank test used for statistical analysis, ABC-Ls method successfully ranked first with a value of 1.300 in benchmark functions. After obtaining successful results on the benchmark tests, the proposed ABC-Ls method was applied to the single diode, double diode and Photowatt-PWP-201 PV modules of PV systems for parameter estimations. In addition, the proposed ABC-Ls method has been applied to the KC200GT PV module for parameter estimation under different temperature and irradiance conditions of the PV modules. The success of ABC-Ls method was compared with genetic algorithm (GA), particle swarm optimization (PSO) algorithm, ABC algorithm, tree seed algorithm (TSA), Jaya, Atom search optimization (ASO). The comparison results were presented in tables and graphics in detail. The RMSE values for the parameter estimation of single diode, double diode and Photowatt-PWP-201 PV module of the proposed ABC-LS method were found as 9.8602E−04, 9.8257E−04 and 2.4251E−03, respectively. In this context, the proposed ABC-LS method has been compared with the literature for parameter estimation of single diode, double diode and Photowatt-PWP-201 PV module and it has been found that it provides a parameter estimation similar or better than other studies. The proposed ABC-Ls method for parameter estimation of the KC200GT PV module under different conditions is shown in convergence graphs and box plots, where it achieves more successful, effective and stable results than GA, PSO, ABC, TSA, Jaya and ASO algorithms.


Introduction
The demand for alternative, renewable and non-polluting energy sources has increased day by day due to the decreased fossil fuel sources and increased environmental concerns around the world [1,2]. Similarly, renewable energy sources play an important role in electricity generation [3]. Among the renewable energy sources, solar energy is widely used to produce electrical energy as it can directly convert the potential energy obtained from the sun into electrical energy via photovoltaic (PV) cell modules [1][2][3]. Because PV systems are installed in sun-exposed environments, they are highly affected by environmental conditions. An accurate and proper solar PV cell modeling is critical in solar PV energy systems [4,5]. Therefore, it is important to obtain the energy using the PV systems with maximum efficiency [6,7]. Simulation and design calculations of PV systems require that the parameters defining the nonlinear electricity model of solar cells be accurately estimated [8]. In the literature, the parameters of PV systems are estimated using deterministic methods and the methods based on artificial intelligence optimization algorithms. In deterministic methods, the parameter estimation of PV systems has mostly been performed through classical mathematical modeling [8][9][10][11][12][13]. For the PV parameter estimation using deterministic methods, the study performed by Easwarakhanthan et al. (1986) can be considered as a reference. This study compared the results obtained in experimental studies with a nonlinear leastsquares algorithm based on the Newton model modified with the Levenberg parameter. It also determined the current-voltage (I-V) parameters in experimental studies [8].
Because the parameter estimation of PV systems is a nonlinear and multivariate problem, there is a high probability of being stuck to local optimum during the use of deterministic techniques [1,14]. Therefore, recently, researchers have started to use various solutions based on artificial intelligence optimization algorithms, as an alternative to deterministic solutions, for the parameter estimation of PV systems to obtain better solutions [1,4,[14][15][16][17][18]. As seen in the examination of some of the studies based on artificial intelligence optimization algorithms, Jervase et al. (2001) proposed a technique based on genetic algorithms (GA) to increase the accuracy of solar cell parameters extracted using traditional methods. Their approach is based on formulating the parameter extraction as a search and optimization problem [19]. Ye et al. (2009) extracted the solar cell parameters of single and double diode models using the particle swarm optimization (PSO) algorithm [20]. AlHajri et al. (2012) applied the pattern search (PS) method for optimum extraction of solar cell parameters [14]. Askarzadeh et al. (2013) used artificial bee swarm optimization (ABSO) algorithm to define the parameters of solar cell models [15]. Yuan et al. (2014) designed the chaotic asexual reproduction optimization (CARO) algorithm by improving the global search capability of the asexual reproduction optimization algorithm with the chaos-based optimization and then used this algorithm for the parameter extraction of solar cell models [16]. Yu et al. (2017) applied a self-adaptive weight parameter and the Improved Jaya (IJaya) method to the population update procedure of the Jaya algorithm for the parameter estimation of single diode, double diode and PV models [1]. Jordehi (2017) proposed a gravity search algorithm with a linearly decreasing gravity constant (GSA with linear G) to analyze the PV cell parameter extraction problem [21]. Oliva et al. (2017) developed the chaotic improved artificial bee colony algorithm (CIABC) for parameter estimation of photovoltaic cells. CIABC combines the search capabilities of the ABC approach with the use of chaotic maps instead of random variables [22]. Xiong et al. (2018) stated that the whale optimization algorithm (WOA) stagnates and suffers from early convergence while dealing with multimodal problems. Therefore, they designed the improved whale optimization algorithm (IWOA) to effectively balance local exploitation and global exploration. They extracted the parameters of three different single diode, double diode and PV models using the IWOA [17]. Yu et al. (2019) proposed a performance-guided JAYA (PGJAYA) algorithm to extract the parameters of different PV models.  achieved the parameter estimation of the PV models with Rao II and Rao III heuristic algorithms which do not have any algorithm-specific parameter value [4]. In the literature, the recent optimization algorithms obtained by developing standard optimization algorithms are applied for the parameter estimation or extraction of PV systems [17,18,[23][24][25][26][27][28][29]. Some of the recently used optimization algorithms and studies for PV module parameter estimation are as follows: Estimation of PV model parameters using improved chaotic gradient-based optimization algorithm [24], a new stochastic slime mold optimization algorithm for the estimation of PV module parameters [25], Enhanced chaotic JAYA algorithm for parameter estimation of single diode and double diode [26], estimation of single diode photovoltaic module and KC200GT and SQ85 PV module parameters using hybrid particle swarm optimization and gray wolf optimization algorithm [26], determination of maximum power output with PV power generation systems with a new salp swarm assisted hybrid maximum power point tracking algorithm [27], wind-driven optimization algorithm as new method for estimation of PV parameters [29], using the political optimizer algorithm to estimate the parameters of single diode, double diode PV cells [30].
In the present study, an ABC Local search (ABC-Ls) was designed by including a new local search procedure into the standard artificial bee colony (ABC) algorithm for the parameter estimation of single-diode module, double-diode module and PV modules. The ABC algorithm is a wellknown and well-established optimization algorithm that balances local search and global search. However, depending on the characteristics of the optimization problems, there are many studies in the literature on improving the local search and global search capacity of the ABC algorithm [31][32][33][34][35][36]. Therefore, ABC-Ls method is proposed in this study for parameter estimation of PV modules, which is a specific and real-world optimization problem. The ABC-Ls method, which was developed by including the standard ABC into the balance of exploitation in local search and exploration in global search, repositions some unnoticed solutions. Accordingly, with the ABS-Ls method, it was aimed to improve the exploitation capability of the standard ABC in local search and obtain the optimum solution or solutions close to the optimum solution. To determine the success of the proposed ABC-Ls method, it was first tested on 15 benchmark functions well known in the literature. The commonly used standard ABC, GA, PSO optimization algorithms and state-of-the-art tree seed algorithm (TSA) [37], Jaya [38] and atom search optimization (ASO) [39] algorithms were used to compare test results. The parameter estimation performance of the PV systems with the proposed ABC-Ls method was evaluated using four standard data sets: single diode, double diode, Photowatt-PWP-201 and KC200GT PV modules. The benchmark test results of the ABC-Ls and the parameter estimation results of the PV systems were presented comparatively and statistically in tables and graphs. Moreover, the comparisons made with the other studies in the literature were also provided in tables. After determining the success of the proposed method based on the results of benchmark functions, parameter estimation of the PV systems was performed and the results were found to be better than those in the literature. In this regard, the proposed ABC-Ls method was seen to obtain accurate and reliable parameters in the parameter estimation of PV systems.

Main contribution and motivation
In this study, ABC-Ls method, in which a new local search procedure is added to the standard ABC algorithm, is proposed for parameter estimation of PV modules. The main purpose of the proposed ABC-Ls method is to improve exploitation success in local search by relocating the overlooked candidate solutions of the standard ABC in the search space. It is an advantage to reposition overlooked candidate solutions in the search space and obtain new solutions. The new local search procedure added in the proposed ABC-Ls method is called every ten iterations. Therefore, it is also an advantage to add a new local search procedure without disturbing the balance of standard ABC's local search and global search. Adjusting parameters or adding new parameters is a disadvantage in metaheuristic optimization algorithms. Adjusting the search range coefficient (SRC), which is the algorithm parameter in the proposed ABC-Ls method, can be considered as a disadvantage. However, in this study, the disadvantageous situation was eliminated by performing sensitivity analysis for the SRC value in the proposed ABC-Ls method. It is seen in the experimental study results that the proposed ABC-Ls method scans the search space more effectively and avoids local minimums. The originality and novelty of the proposed ABC-Ls method are given below as highlights: • A new improved method named ABC-LS is proposed. • The ABC-LS method is supported with a new local search added to the standard ABC. • In ABC-LS, the local search capability of the standard ABC has been increased. • The proposed ABC-LS method has been applied to benchmark functions and PV models.
• ABC-LS has good performance in both benchmark function solutions and setting parameters of PV models.
For the rest of the article, the standard ABC is explained in Sect. 2, the proposed ABC-Ls method is detailed in Sect. 3, the problem definition of PV system is given in Sect. 4, experimental study is given in Sect. 5, Benchmark and PV module results and discussion are presented in Sect. 6, sensitivity analysis is given in Sect. 7 and the study is concluded in Sect. 8.

Standard Artificial Bee Colony (ABC) Algorithm
Artificial Bee Colony (ABC) is an optimization algorithm inspired by the unique methods used by swarms when searching for food [40][41][42]. The ABC algorithm is used to solve the optimization problems based on the behavior of swarms while they are trying to find food. In the ABC, there are three bee groups in a colony: employed bees, onlooker bees, and scout bees. The ABC consists of three phases: Fig. 1 General flow chart of ABC [44] employed and onlooker bees are sent to sources, the nectar amount of the sources are calculated, and the scout bee is identified and randomly located to a new source [42]. Food sources correspond to the possible solutions of the optimized problem. The nectar amount of a source is the quality value of the solution expressed with that source [43]. The ABC algorithm finds the point (solution) providing the minimum or maximum of the problem among the solutions in the search space by trying to find the location of the source with the highest nectar amount [40][41][42][43][44]. The ABC algorithm can be divided into four phases: initialization, employed bee, onlooker bee and scout bee [40][41][42][43][44]. The general flow chart of ABC is shown in Fig. 1.
In the initialization phase, the algorithm should be started with various basic parameters such as the number of food sources (SN), the end condition (i.e., the maximum number of iterations (Maxiter)), the bound parameter value and the counter recording the number of consecutive failed updates for each. At this phase, an initial population can be generated as follows: where fit i is the fitness value of the i th food source, X i , and f(X i ) is the target function value of X i for the optimization problem.
In the employed bee phase, each employed bee is associated with a food source (solution). Each employed bee tries to find a better food source, i.e., a better solution, by searching near its own food source. The employed bees' search is given in Eq. (3).
where V i,j is the new solution of the ith candidate food source at the jth dimension; X i,j and X k,j are the jth dimension of the ith and kth food source, respectively; k is a random number selected from (1, 2, …, SN) and k ≠ i; j is randomly selected from (1,2, …, D); and ϕ i,j is a random real number in the [− 1,1] interval. According to the fitness values, if the candidate solution is better than the previous one, the previous solution is replaced with the new solution. If the new fitness value is worse than the previous one, the previous solution is used.
In the onlooker bee phase, each onlooker bee selects a food source for further searches with the probability Pi calculated by Eq. (4). The more the nectar amount of the source is, the higher the probability of being selected by an onlooker bee will be for this source.
In the scout bee phase, the food source with the highest bound value in the population is selected. If the bound value is greater than the predefined threshold bound, this food source is abandoned by the bee and the process goes back to a scout bee to randomly create a new food source according to Eq. (1).

The proposed ABC-Ls method
In this study, an ABC local search (ABC-Ls) was proposed by adding a new local search phase after the employed bee, onlooker bee and scout bee phases of the standard ABC algorithm. With the proposed ABC-Ls method, it was aimed to improve the exploitation capability of the standard ABC in local search and reach the optimum solution (or a solution as close to the optimum solution as possible) more efficiently.
In the proposed ABC-Ls method, the local search procedure was created being inspired by the local search function of the multiple trajectory search (MTS) algorithm developed by Tseng and Chen (2008) [45]. The proposed ABC-Ls method runs every ten iterations ( mod(iter, 10) = 0 ) of the standard ABC algorithm. The pseudo-codes of the proposed ABC-Ls and the local search (Ls) method are shown in Algorithm 1 and Algorithm 2, respectively. In the ABC-Ls method, the search range coefficient (SRC) is determined between 0 and 1 (SRC ∈(0, 1]). The SRC value is entered at the initialization of Algorithm 1. In Algorithm 2, the SRC value decreases by half until achieving success in the fit i local search and is redetermined within the maximum and minimum bounds of the function when it reaches 1E − 15 . According to the value calculated after the employed bee, onlooker bee and scout bee phases of Algorithm 1, the local search procedure in Algorithm 2 is run. Individuals as many as the number of food are repositioned under the maximum and minimum bounds based on the SRC value in Algorithm 2 and then applied to the objective function. If the obtained solution is good, it is used and the position of the individuals achieving that solution is stored. The SRC value is repeatedly calculated and the good new solutions are given to the output of Algorithm 2. The new solutions obtained in Algorithm 2 are reevaluated in the employed bee, onlooker bee and scout bee phases in Algorithm 1 and the bees are repositioned. Thus, in Algorithm 1, a balance is achieved between exploration in global search and exploitation in local search in the employed bee, onlooker bee and scout bee phases. The ABC-Ls repeats the local searches until the maximum number of iterations is reached. By doing so, the proposed ABC-Ls method tries to obtain the best solution by recreating different combinations with the local search procedure for the unnoticed values in the calculation. Thus, the exploitation capability of the standard ABC in local search was improved with the ABC-Ls method, which performs a new local search without distorting the balance of exploration and exploitation of the standard ABC. Figure 2 shows the flow chart of the ABC-Ls method.
In order to explain the operation of the proposed ABC-Ls method in Algorithm 1 and Algorithm 2, a demonstration was made by applying it to the F1 function (Sphere) in Table 2. The proposed ABC-Ls method was run to the F1 function in 100 iterations for 5 design variables and 5 population numbers. In the ABC-LS method, Algorithm 2 is called every ten iterations, as in Algorithm 1. In the local search procedure in Algorithm 2, the function value and  design variable values obtained in Algorithm 1 are processed repeatedly and overlooked values are re-evaluated. The local search procedure in Algorithm 2 is run as much as the population number (SN). Depending on the minimum and maximum limits in Algorithm 2, the SRC value, which determines the design variables, is continuously halved until it is less than 1E−15. The newly determined design variables are applied to the objective function. Algorithm 1 continues to work with the new design variables obtained at the end of Algorithm 2. In this way, it is aimed to improve the intensification and exploitation capability of the standard ABC in local search. Redetermination of design variables every ten iterations is as in Table 1. The resulting new design variables are applied to the objective function. To illustrate with an example in Table 1, between the 1st and 9th iterations, the function value drops from 7.47E 03 to 1.24E 03 with standard ABC. However, in the 10th iteration, the proposed ABC-Ls method is called and the new function value depending on the new design variables becomes 2.19E−01 with a significant decrease. In Fig. 3, the proposed ABC-Ls method was demonstrated based on the values in Table 1. In Fig. 3, it is seen that the proposed ABC-Ls method converges better to a solution every ten iterations.

Benchmark functions and problem definition of PV system
The proposed ABC-Ls method, standard ABC, GA, PSO, TSA, Jaya and ASO algorithms have been applied to 15 benchmark functions commonly used in the literature for experimental study. The commonly used benchmark functions are shown in Table 2. Although there are different definitions for PV systems in the literature, PV is a system that generally examines the (I-V) relationship between current and voltage. Although many equivalent circuit models have been developed and proposed to describe the (I-V) characteristics of PV systems, three models are widely used in practice: single diode module, double diode module, Photowatt-PWP-201 and KC200GT PV modules [4,[15][16][17][18][46][47][48]. This study presents the mathematical models of three different PV systems: single diode module, double diode module and PV module models. The more accurate the parameters of PV systems are, the better the performance of the solar cell will be [8,49].

Single diode module
This module has one diode and one parallel-connected current source. It includes a shunt resistor for residual current and a series resistor showing the load current loss. Due to its simplicity, this model is often used to determine the static property of the solar cell. The equivalent circuit of this model is shown in Fig. 4. The output current of the circuit is given in Eqs. (5)-(7). (5) where I L is the output current of the solar cell, I ph is the total current generated by the solar cell, I d is the diode current calculated by Shockley in Eq. (5), and I sh is the shunt current calculated in Eq. (6). R S is the series resistance while R sh is the shunt resistance. V L is the output cell voltage and I sd is the reverse saturation current of the diode. n is the ideal factor of the diode. k is the Boltzmann constant and its value is 1.3806503 × 10 −23 J/K, while q is the electron constant and its value is 1.60217646 × 10 −19 C. Finally, T is the absolute temperature of the cell and indicated in Kelvin. Equation (8) is obtained by combining Eqs. (5), (6) and (7).
Equation (8) shows that there are five different parameters for the single diode model. When these parameters are predicted correctly, the actual behavior of the solar cell is seen. Step

Double diode module
This module has two diodes and one parallel-connected current source. It includes a shunt resistor for residual current and a series resistor showing the load current loss. This circuit was created considering the recombination current loss effect. The equivalent circuit of this module is shown in Fig. 5. The output current of the circuit is shown in Eqs. (9) and (10) where I sd1 is the diffusion current, I sd2 is the saturation current, n 1 is the diffusion diode ideal factor, and n 2 is the recombination diode ideal factor.
Equation (10) shows that there are seven different parameters for the double diode model. When these parameters are predicted correctly, the actual behavior of the solar cell is seen.

PV module model
This module has many series and parallel diodes and one parallel-connected current source. It includes a shunt resistor for residual current and a series resistor showing the load current loss. The equivalent circuit of this module is shown in Fig. 6. The output current of the circuit is presented in Eq. (11).
where N p indicates the number of parallel solar cells and N S is the number of series solar cells. As in the single diode module, this module has five different parameters shown in Eq. (11). When these parameters are predicted correctly, the actual behavior of the PV module is seen [49].

Objective function of the problem
The optimization techniques frequently used in parameter estimations are also used in the parameter estimation of the (11) (12) The objective function, which is used to measure the difference between experimental and simulated current data, is given in Eq. (15). In optimization problems, the objective function minimizes the RMSE obtained by the vector x, which depends on the standard data V L and I L in the search space.
where x indicates that the solution vector consists of unknown parameters. N is the number of experimental data.

Experimental study
In this section, first of all, experimental studies of benchmark functions are carried out to determine the effectiveness of the proposed ABC-L method. Then, the performance of the proposed method is applied to the parameter values estimation of single diode module, double diode module and PV module problems. Well-known literature algorithms such as GA, PSO, ABC and state-of-the-art algorithms such as TSA, Jaya and ASO were used to compare the performance of the proposed ABC-Ls algorithm. All studies were carried out in Matlab R2016a environment on a Windows 10 operating system laptop with i7-6700Hq CPU 2.60 Ghz, 16 GB RAM hardware configuration. All algorithms were run 30 times under the same conditions and the maximum number of function evaluations (MaxFEs) was taken as 50,000. Control parameter settings specific to the algorithms were designed according to the literature and are presented in Table 3.
The performance of the proposed ABC-L method for PV parameter estimation was first applied to the single and double diode models of the RTC France PV Cell for experimental study. Secondly, the proposed method was applied to the parameter estimation of the Photowatt-PWP-201 PV module and finally to the KC200GT PV module for parameter estimation in different conditions. The processes are performed using solar cells and solar module current voltage data. The data used for the single and double diode models of the RTC France PV Cells were measured at 33 °C from a PV cell with a diameter of 57 mm under 1000 W/m 2 irradiance [8,24,26]. The data used for the Photowatt-PWP-201 PV was measured at 45 °C from 36 series polycrystalline silicon PV cells under 1000 W/m 2 irradiance [8,24]. The KC200GT PV module is a 54-cell multi-crystalline panel. A total of 223 samples were collected in the datasheet of the KC200GT PV module under different irradiance and temperature conditions [24,25,46]. The parameter values required to solve the RTC France PV Cells, Photowatt-PWP-201 PV module and KC200GT PV modules are taken based on the literature and shown in Table 4 [8,24,25,46].
The I sc (short-circuit current) value in Table 4 is calculated as in Eq. (16) [24].
where I sc−STC represents I sc at STC. GSTC and TSTC represent irradiance and temperature under different conditions at STC. G and T indicate actual irradiance and temperature.

Benchmark results
The benchmark functions used for experimental study were applied to GA, PSO, TSA, Jaya, ABC and the proposed ABC-Ls algorithms in 50 dimensions (D = 50). Algorithms  Table 5. The best values calculated by GA, PSO, TSA, Jaya, ABC and the proposed ABC-Ls algorithms are indicated in bold font. In this section, the proposed ABC-Ls algorithm for the best results in Table 5 is compared with the GA, PSO, ABC, TSA, Jaya and ASO algorithms. In the F7 function, the proposed ABC-Ls method, GA, PSO, ABC, TSA and ASO algorithms have found the optimum result, except for the Jaya algorithm. Similarly, in the F13 function, the proposed ABC-Ls, PSO and ASO found optimum results. In the F9 function, the best result was obtained by PSO. The proposed ABC-L method yielded successful results in a total of 14 functions, including F1-F8 and F10-F15 functions. In this context, it can be said that the ABC-Ls algorithm is much more successful than the standard ABC algorithm, as it is developed with local search capability. In addition, Fig. 7a shows that the convergence performance of the proposed ABC-Ls algorithm was surprisingly better than other algorithms in the graphs of all functions except F9 and F14. The box plot in Fig. 7b shows that the ABC-Ls algorithm achieves a more stable result and the proposed local search strategy is a significant improvement for ABC.
Friedman Mean Rank Test was used for statistical analysis of experimental study results of benchmark functions. The Friedman Mean Rank test is a nonparametric statistical analysis technique for repeated measures developed by Milton Friedman [53]. The Friedman mean rank test is a multiple comparison test that aims to detect significant differences between the behavior of two or more algorithms [54]. Friedman Rank Test results of the proposed ABC-Ls, GA, PSO, TSA, Jaya and ASO algorithms are listed in Table 6. The P Value being less than 0.00001 confirms the importance and usability of the Friedman mean ranks test. [54,55].
In Fig. 8, Friedman mean rank bar graph of all algorithms is given according to benchmark test results. It is seen in Fig. 8 that the rank value of the proposed ABC-Ls method is 1.3000 compared to other algorithms. Similarly, it is seen in Table 6 that ABC-Ls is in the 1st rank.

Single diode module results
The min RMSE values of the GA, PSO, TSA, Jaya, ASO, ABC and ABC-Ls algorithms for parameter estimation of the single diode model are listed in Table 7. Figure 9 shows the convergence graphs and box-plots of the minimum RMSE values according to the parameter values estimated by algorithms. Table 8 shows the standard and estimated current and voltage values used for the single diode as well as the absolute error (AE). The graphical representations of the standard and simulation data in Table 8 are shown in Fig. 10, and Table 9 compares the ABC-Ls algorithm with the results of some studies in the literature. In Table 7 and Table 9, the best RMSE values calculated by algorithms are given in bold. Table 7 indicates that the ABC-Ls was the algorithm obtaining the best value, indicated in bold. GA and ASO had the worst value, followed by the ABC algorithm. Although PSO and TSA algorithm obtained a value close to the ABC-Ls algorithm, the success of the ABC-Ls algorithm can be seen in Table 7, which shows the values obtained from the parameter estimation according to the best results of GA, PSO, ABC, TSA, Jaya, ASO and proposed ABC-Ls. The convergence graphs and box-plots in Fig. 9 indicate that the ABC-Ls algorithm converged faster than the standard ABC while the ABC-Ls and ABC obtained more balanced and stable results in the box-plot. Table 8 shows the voltage, current and power values in the standard data. Simulation data, on the other hand, includes the current and power values obtained by the ABC-Ls. It also shows the AE values of the current and power values. Table 9 compares the ABC-Ls algorithm with some previous studies that achieved values close to those of the ABC-Ls, the same values, or worse results. Generally, it is seen from the results that the ABC-Ls algorithm achieved a very high performance for the single diode model compared to many studies in the literature, and its success was the same only with PGJaya.

Double diode module results
For the parameter estimation of the double diode module, the results obtained by GA, PSO, standard ABC, TSA, Jaya, ASO and the proposed ABC-Ls method are given in Table 10. In Table 10, the proposed ABC-Ls method calculated the lowest RMSE value as 9.8257E−04. TSA and Jaya calculated the closest result to the proposed ABC-Ls method. Table 10 shows that the performance of GA and ASO is not very good, ABC and PSO algorithms reach similar results. It is deduced from Table 10 that the proposed ABC-Ls method achieved the best min RMSE result and outperformed other algorithms. Figure 11 shows the convergence graphs and box-plots of the minimum RMSE values according to the parameter values estimated by the algorithms. According to the results of the algorithms in Fig. 11, the convergence graph of the ABC-Ls was the highest, and (a) (b) Fig. 9 The algorithms' a convergence graphs and b box-plots for single diode module the box plots show that it performed well in terms of stability. Table 11 shows the standard and estimated current and voltage values used for the double diode module as well as the absolute error (AE). The graphical representations of the standard and simulation data in Table 11 are shown in Fig. 12. In Table 10 and Table 12, the best RMSE values calculated by algorithms are given in bold. Table 12 shows that the proposed ABC-Ls method performed a better parameter estimation than the standard ABC in the double diode module [4]. The proposed ABC-Ls method achieved better results than all other studies except the Rao-II algorithm [4].

Photowatt-PWP-201 PV module results
It is seen in Table 13 that the proposed ABC-Ls method calculates the min RMSE value of 2.4251E−03 for parameter estimation of the Photowatt-PWP-201 PV module. It is listed in Table 13 that the result of the proposed ABC-Ls method is better than other algorithms. The closest to the min RMSE results of the proposed ABC-Ls method were Jaya, TSA and GA algorithms, while the worst result was given by ASO. It is seen in Fig. 13a that the proposed ABC-Ls method and PSO converge closely and similarly. The box plot in Fig. 13b shows that the RMSE values calculated by the proposed ABC-Ls method are more stable. It is also seen in the Fig. 10 Comparison graphs between the ACB-Ls' standard and simulation data for single diode module: a I-V characteristics, b P-V characteristics, c current, and d power box plot that the proposed ABC-Ls method is better than the standard ABC. Although the GA algorithm achieved good success in min RMSE values, it is seen in the box plot in Fig. 13b that it did not perform well in terms of stability. In Table 13 and Table 15, the best RMSE values calculated by algorithms are given in bold. Table 14 shows the standard and estimated current and voltage values used for the double diode module as well as the absolute error. The graphical representations of the standard and simulation data in Table 14 are shown in Fig. 14.
For the Photowatt-PWP-201 PV module parameter estimation, the proposed ABC-Ls method achieved a better estimation result compared to other studies in the literature, as seen in Table 15. It is understood from Table 15 that the proposed ABC-Ls method in local search was more usable than other standard and developed methods in the PV module parameter estimation.  Fig. 11 The algorithms' (a) convergence graphs and (b) box-plots for the double diode module

KC200GT PV module results (for dynamic conditions)
The KC200GT PV module is used for parameter estimation when the PV module is under different temperature and irradiance conditions. The proposed ABC-Ls method and other standard algorithms have been applied to the parameter estimation of the KC200GT PV module. First of all, the proposed ABC-Ls method and other algorithms were applied at a constant temperature of 25 °C and at different irradiances of 200, 400, 600, 800 and 1000 W∕m 2 , respectively. Then, the proposed ABC-Ls algorithm and other algorithms were applied to parameter estimation at different temperatures of 25, 50 and 75 °C at constant 1000 W∕m 2 irradiance. In Table 16, the parameter estimates of the proposed ABC-Ls method and other algorithms at different irradiance conditions of the KC200GT PV module are given. Likewise, parameter estimates of the KC200GT PV module under different temperature conditions are given in Table 17. In Table 16, the proposed ABC-Ls method found the best min RMSE value among all algorithms under different irradiance conditions. In Table 17  other algorithms under different irradiance and temperature conditions. In Fig. 15, the I-V curve of the proposed ABC-Ls method under different irradiance conditions is demonstrated. Figure 16 shows the I-V curve of ABC-Ls under different temperature conditions. In Table 16 and Table 17, the best RMSE values calculated by algorithms are given in bold.
The proposed ABC-Ls method, the convergence curve of GA, PSO, TSA, Jaya and ABC algorithms under different irradiance conditions are given in Fig. 17 and the box plot graph is given in Fig. 18. In the convergence curve in Fig. 17, it is seen that the convergence performance of the proposed ABC-Ls is better than other algorithms. In the box plot curve in Fig. 18, it is seen that the results of the proposed ABC-Ls method are more stable compared to the min RMSE values in 30 different experimental studies. It can be said in Fig. 18 that TSA is second in terms of stability after the proposed ABC-Ls method in box plot chart.
The proposed ABC-Ls method, the convergence curve of GA, PSO, TSA, Jaya and ABC algorithms under different temperature conditions are given in Fig. 19 and the box plot graph is given in Fig. 20. In the convergence curve in Fig. 19, it is seen that the convergence performance of the proposed ABC-Ls at 25 °C and 75 °C is better than other algorithms. At 50 °C, it is seen that the min RMSE values of the ABC-Ls method are similar to the Jaya algorithm. According to the min RMSE values in 30 different experimental studies in the box plot curve in Fig. 20, it was   Fig. 13 The algorithms' a convergence graphs and b box-plots for the Photowatt-PWP-201 PV module model

Sensitivity analysis
The control parameter Search Range Coefficient (SRC) (SRC ∈(0,1]) in the input parameters of Algorithm 1 affects the performance of the proposed ABC-Ls method. In this study, the SRC coefficient of ABC-Ls is used to measure the sensitivity of performance to parameter changes. While the SRC parameter of the proposed ABC-Ls was changed for each experiment, the values of the other parameters (CS,n o ,n e , n s ) in Table 3 were kept constant. The mean RMSE values and Standard deviation (Std.) values obtained according to the results of 30 experimental studies for parameter estimation of single, double and PV modules are given in Table 18. The best results obtained in the experimental studies are shown in bold in Table 18. According to Table 18, when SRC = 0.2, it was determined that the proposed ABC-Ls method performed better. In this study, the SRC value was taken as 0.2. Therefore, the SRC parameter value in Table 3 was determined accordingly. As a result, it was determined that the probability of finding the global optimum result of the proposed ABC-Ls method increases when the SRC value approaches from 1 to 0.

Conclusion
In this study, an ABC-Local Search (ABC-Ls) method was proposed to estimate the parameters of the PV system. The proposed ABC-Ls method runs a new local search procedure for every 10 iterations in addition to the employed bee, onlooker bee and scout bee phases of the standard ABC algorithm. It was tested on 15 different benchmark functions to show the effectiveness of the ABC-Ls method. The experimental results indicated that the success of the proposed method was very high compared to its standard version. The proposed method was applied to four real-world problems of the PV systems: single diode, double diode, Photowatt-PWP-201 and KC200GT PV modules. The convergence graph and box-plots of the RMSE values indicated that the ABC-Ls method provided better results in single diode module, double diode, Photowatt-PWP-201 and KC200GT PV module problems compared to the GA, PSO, TSA, Jaya, ASO and standard ABC methods. According to the convergence graphs and box-plots, ABC-Ls had very good convergence rate and stability. Accordingly, it can be said that the ABC-Ls algorithm showed a good performance. As a result, it can be suggested that ABC-Ls is a suitable method for different PV systems and it should be used in the studies to be conducted with other PV systems. In addition, the sensitivity analysis of the search range coefficient (SCR) in the local search procedure in the proposed ABC-Ls method     was also performed. In the proposed ABC-Ls method, it was also found that the PV module parameter estimation is better when the SCR value approaches from 1 to 0.
Authors contribution MFT was responsible for methodology, writing-original draft, data curation, validation, writing-review & editing.

Declaration
Conflict of interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.