Identiﬁcation of Parameters in Photovoltaic Models through a Runge Kutta Optimizer

: Recently, the resources of renewable energy have been in intensive use due to their environmental and technical merits. The identiﬁcation of unknown parameters in photovoltaic (PV) models is one of the main issues in simulation and modeling of renewable energy sources. Due to the random behavior of weather, the change in output current from a PV model is nonlinear. In this regard, a new optimization algorithm called Runge–Kutta optimizer (RUN) is applied for estimating the parameters of three PV models. The RUN algorithm is applied for the R.T.C France solar cell, as a case study. Moreover, the root mean square error (RMSE) between the calculated and measured current is used as the objective function for identifying solar cell parameters. The proposed RUN algorithm is superior compared with the Hunger Games Search (HGS) algorithm, the Chameleon Swarm Algorithm (CSA), the Tunicate Swarm Algorithm (TSA), Harris Hawk’s Optimization (HHO), the Sine–Cosine Algorithm (SCA) and the Grey Wolf Optimization (GWO) algorithm. Three solar cell models—single diode, double diode and triple diode solar cell models (SDSCM, DDSCM and TDSCM)—are applied to check the performance of the RUN algorithm to extract the parameters. the best RMSE from the RUN algorithm is 0.00098624, 0.00098717 and 0.000989133 for SDSCM, DDSCM and TDSCM, respectively.


Introduction
Researchers are making developments in sources of renewable energy to combat environmental pollution caused by use of fossil fuels. Solar, wind, nuclear and wave energies are the most renewable sources used in our lifetime; hence, researchers all over the world are focused on them [1][2][3][4][5]. Due to the cleanliness and availability of solar energy, it has been recognized as a capable renewable energy source [6]. Generation of electricity directly from solar energy is accomplished by photovoltaic (PV) systems [7] and their behaviour is successfully simulated based on several PV models [8,9]. They Algorithm (SCA) [66], the Chameleon Swarm Algorithm (CSA) [67] the and Grey Wolf Optimizer (GWO) [68]; the dataset from the R.T.C France cell is used. The experiments also include a statistical analysis to measure the performance evaluation of the proposed RUN algorithm and all competitor algorithms. This analysis contains several points such as the minimum, maximum, mean and standard deviation of the objective function over 30 independent runs. Finally a Friedman rank test is performed for the proposed RUN algorithm and all competitor algorithms in all cases study. The experiments conducted then validate the performance of the proposed approach based on the RUN algorithm to search for the optimal configuration of the parameters.
The goals of this article are listed as follows: • Introduce an alternative method to identify the parameters in solar cells using the RUN algorithm in combination with the diode models. • Test the RUN algorithm over a real multidimensional problem. • Accurately identify the best parameters in solar cells by using a modern metaheuristic.
This paper is organized as follows: The mathematical formulation of three photovoltaic diode models is described in Section 2. Section 3 discusses the objective function for the estimation of the photovoltaic parameters. Section 4 presents the Runge-Kutta optimizer (RUN). The experimental results are discussed in Section 5. The conclusions of the article are presented in Section 6.

Mathematical Photovoltaic Models for Solar Cells
This section analyses the mathematical formulation for the three models of photovoltaics: the Single diode solar cell model (SDSCM), the double diode solar cell model (DDSCM) and the triple diode solar cell model (TDSCM). Figure 1 shows the equivalent circuit of the single diode solar cell model. Based on this diagram the mathematical equation for SDSCM is defined as follows:

Single Diode Solar Cell Model
The output current of SDSCM is I, I pv is the generated photo-current, the current in shunt resistor is I sh , I D1 is the current in the first diode, R sh is the resistance shunted with the diode terminal, R s is the series resistance, n 1 is the diode ideality factor, the Boltzmann constant is K, the charge of electron is defined by q, I h is the diode reverse saturation current and the cell temperature is T c .  Figure 2 presents the equivalent circuit of DDSCM. Considering the circuit the DDSCM can be defined in the following equation:

Double Diode Solar Cell Model
where I D2 is the second diode current and n 2 is the second diode ideality factor.

Triple Diode Solar Cell Model
In Figure 3 is presented the circuit that defines the TDSCM. In Equation (5) the TDSCM is mathematically defined where the current passes in the third diode are I D3 and n 3 is the third diode ideality factor.

Problem Definition
The two effective matters in any optimization problem are the objective function and boundary limits of the search space. In this paper, the objective function of this problem concerns the minimization of the root mean square error (RMSE) between the experimental recorded current data and the current data extracted from the simulation. The mathematical definition of the RMSE is presented in the following equation: where I exp , the experimental recorded current data, N is the number of reading data, X is variable with the extracted parameters and I sim corresponds to the estimated current obtained by one of the diode models. For the SDSCM the variables to be identified are X = (R s , I h1 , n 1 , R sh and I pv ). In the DDSCM the variables to be extracted are X = (R s , I h1 , n 1 , R sh , I pv , I h2 and n 2 ). Finally, for the TDSCM the elements to be identified are X = (R s , I h1 , n 1 , R sh , I pv , I h2 , n 2 , I h3 and n 3 ). The lower and upper boundaries of the variables for all the models are reported in Table 1.

The Runge-Kutta Optimizer
This section introduces a brief description of the Runge-Kutta optimizer (RUN) optimization algorithm [69]. The RUN is based on the theory of the Runge-Kutta method that is employed for solving ordinary differential equations in numerical methods. The RUN has two stages; the first is the search procedure that uses the Runge-Kutta theory and the second is called enhanced solution quality (ESQ). The following subsection will explain the basics of the RUN.

Updating Solutions
The RUN algorithm uses a search mechanism (SM), see Appendix A, based on the Runge-Kutta method to update the position of the current solution at each iteration, which is defined Algorithm 1.
From the previous explanation r is an integer number that takes the values of 1 or −1 and helps to improve the diversity. g is a random number in the range [0, 2]; also µ is a random number too. The formula to compute SM is defined in Appendix A. Finally SF is an adaptive factor that is computed as follows: where Maxi stands for the largest number of iterations. Besides, the values of x c and x m are as follows: From Equations (11) and (12), φ is a random number in the range of (0,1). x best is the best-so-far solution. x lbest is the best position obtained at each iteration.

Enhanced Solution Quality
In the RUN algorithm, the enhanced solution quality (ESQ) is a step employed to increase the quality of the solutions and avoid local optima in each iteration. The scheme shown in Algorithm 2 is executed to create the solution (x new2 ) by using the ESQ: From the previous steps, the values of w, x avg and x new are computed by using the following equations: where β is a random number defined in the range [0, 1]. c = 5 × rand, where rand is a random value. r is an integer number that could take the values 1, 0 or −1. x best is the best solution explored so far. The solution x new2 does not always have a better fitness than the current solutions. In this case, the RUN provides another opportunity to enhance the fitness by using x new3 . This procedure is performed as shown in as Algorithm 3.

Algorithm 3 Enhancing the new solution x new3
if rand < w then where v is a random number with a value of 2rand. The pseudo-code of the standard RUN is presented in Algorithm 4.

Stage 1. Initialization
Initialize a, b Generate the RUN population X n (n = 1,2,. . . ,N) Calculate the objective function of each member of population Determine the solutions x w , x b , and x best

Experimental Results
This section explains the identified parameters for the SDSCM, DDSCM and TDSCM models using the RUN algorithm. The proposed implementation for the RUN algorithm is compared with other meta-heuristic approaches such as the Hunger Games Search (HGS) algorithm [63], the Tunicate Swarm Algorithm (TSA) [64], Harris Hawks Optimization (HHO) [65], the Sine-Cosine Algorithm (SCA) [66], the Chameleon Swarm Algorithm (CSA) [67] and the Grey Wolf Optimizer (GWO) algorithm [68]. The recorded R.T.C France solar cell data at 1000 w/m 2 solar radiation and 33°C temperature is applied to evaluate the algorithms' accuracy and reliability and accuracy.
Each algorithm's settings are described in Table 2 for clear comparison. The next subsections will present and discuss the experimental results obtained by the SDSCM, the DDSCM and the TDSCM. A statistical analysis is also conducted; the outcomes are also discussed in a respective subsection.

Results of SDSCM
The results of the decision parameters for SDSCM estimated by implementing the RUN algorithm and the comparison with the other method for finding the best RMSE are illustrated in Table 3. The optimization method that achieves the best RMSE of value 0.000986242 is the RUN algorithm, this value is in a bold number in Table 3. The order of algorithms based on the minimum value of the RMSE is RUN, GWO, HHO, TSA, CSA, SCA and HGS. The I-V and P-V characteristics of the tested SDSCM are performed based on the extracted results from the RUN algorithm in Table 3. Figure 4 shows the I-V curve and the absolute error of current (AEI) for SDSCM based on the extracted data from the RUN algorithm. Figure 5

Results of DDSCM
The results of the variables of DDSCM estimated using the RUN optimizer and its comparison with the rest of the algorithms to obtain the optimal RMSE are illustrated in Table 4. The algorithm that obtains the optimal RMSE is the RUN and the value achieved is 0.000987168, this value is in a bold number in Table 4. The compared approaches can be ranked according to the minimum value of RMSE obtained. The rank for the DDSCM from the lower to the upper value is RUN, GWO, HHO, TSA, CSA, SCA and HGS. The I-V and P-V characteristics of the tested DDSCM are performed based on the extracted results from the RUN algorithm in Table 4. In Figure 6 are plotted the I-V curve and the absolute error of current (AEI) for DDSCM based on the extracted data from the RUN algorithm. Figure 6 explains the P-V curve and the absolute error of power (AEP) for DDSCM based on the extracted data from the RUN algorithm. Based on Figures 6 and 7 the lower value of absolute current error is 0.0000146823091732307 and the lower value of absolute power error is 0.00000171970722786467. The error values presented before can be interpreted with a high degree of accuracy due to the optimal configuration of parameters obtained by using the RUN algorithm. Moreover, these figures explain the high closeness between the simulated and experimental recorded data so that the proposed RUN algorithm achieves high performance and more accuracy in extracting decision variables of DDSCM. Notice that for Figures 6 and 7 Nu of samples refers to the number of samples used to compute the absolute error of the variables.

Results of TDSCM
For the TDSCM the values estimated by all the optimizers in a comparative way including the RMSE are presented in Table 5. The best RMSE value is 0.000989133 and it is reached by the RUN algorithm, this value is in a bold number in Table 5. The rank of the optimization methods based on its performance using the RMSE as objective function is RUN, GWO, HHO, TSA, SCA, HGS and CSA, where they are sorted from the minimum to the maximum value. The I-V and P-V characteristics of the tested TDSCM are computed by using the parameters obtained by the RUN method and they are shown in Table 5. Figure 8 explains the I-V curve and the absolute error of current (AEI) for TDSCM based on the extracted data from the RUN algorithm. Figure 9 explains the P-V curve and the absolute error of power (AEP) for TDSCM based on the extracted data from the RUN algorithm. Based on Figures 8 and 9 the minimum value of the absolute current error is 0.000144871336086205 and the minimum value of the absolute power error is 0.00000162546142774399. The error values presented before can be interpreted with a high degree of accuracy due to the optimal configuration of parameters obtained by using the RUN algorithm. Moreover, these figures explain the high closeness between the simulated and experimental recorded data so that the proposed RUN algorithm achieves high performance and more accuracy in extracting decision variables of TDSCM. Notice that for Figures 6 and 7 Nu of samples refers to the number of samples used to compute the absolute error of the variables.

SDSCM, DDSCM and TDSCM Statistical Analysis
The statistical analysis of all algorithms is performed in this section based on 30 independent runs for all the methods. This analysis is based on calculating the minimum, mean, maximum and standard deviation of the objective function (in this case the RMSE). The algorithm with high accuracy is determined according to the best value of RMSE. The algorithm with more reliability is specified according to the standard deviation of the RMSE value. Tables 6-8 present the statistical analysis of the RUN algorithm and other comparative algorithms for the three models, SDSCM, DDSCM and TDSCM, respectively. Based on these recorded results the proposed RUN algorithm achieves the minimum RMSE and the best value of standard deviation for all SDSCM, SDSCM and TDSCM. Considering this fact, the proposed RUN algorithm is the superior algorithm on all competitor algorithms due to its better reliability and higher accuracy. The robustness and convergence curves of each algorithm are very important items in the evaluation of the performance of algorithms. The robustness curves are explained in Figures 10-12 for SDSCM, DDSCM and TDSCM, respectively. These figures explain that the RUN algorithm achieves the best fitness at each run compared with all used algorithms. Considering the above, the proposed RUN algorithm realizes high robustness reliability for the best solution concerning all algorithms. The convergence curves are explained in Figures 13-15 for SDSCM, DDSCM and TDSCM, respectively. The convergence curves show how the algorithm behaves along the iterations until reaching the global optimal value of the objective functions. Based on these figures, the proposed RUN algorithm achieves faster convergence to the optimal solution than all algorithms. The Friedman test is a non-parametric statistical test that is used to detect differences in treatments across multiple test attempts. The Friedman rank test is performed for the SDSCM, DDSCM and TDSCM results reported in Tables 6-8, respectively. The extension of the Wilcoxon test is the Friedman test. The analysis of the simulated data can be done by using this test. The extracted data of 30 runs for all algorithms is used in measuring the Friedman test for all case studies. Figure 16 presents the Friedman rank for SDSCM. It is observed that RUN obtained the best rank, i.e., rank 1 followed by TSA, GWO, HHO, SCA, HGS and CSA. Figure 17 shows the Friedman rank for DDSCM. It is observed that RUN obtained rank 1 followed by GWO, TSA, HHO, SCA, HGS and CSA. Figure 18 presents the Friedman rank for TDSCM. Here it is observed that RUN obtained the best rank followed by GWO, TSA, SCA, HHO, HGS and CSA . Thus, it can be concluded that RUN obtained the best rank for all the three cases.          Based on the results analysis previously described, the RUN is an optimization mechanism that has been tested over a huge range of real-world problems. However, since it has been recently proposed its performance needs to be proven in different areas of application as energy systems. In this paper the RUN is used to estimate the parameters of solar cells by using three different diode models. The accuracy of the results provided by the RUN helps to verify its performance regarding the optimization. Moreover, the analysis of the results from the application point of view permits to understand that the solar cells designed by the RUN are more efficient in terms of voltage and power. This means that they can be used in real applications due to their capabilities to follow the input design parameters. Besides, the robustness of the RUN is confirmed since the values are stable along different experiments.
The following merits revealed the efficiency of RUN for solving various complex optimization problems:

•
In the RUN algorithm, the enhanced solution quality (ESQ) is employed to increase the quality of the solutions and to avoid local optima at each iteration. • The Scale factor (SF) has a randomized adaptation nature, which assists RUN in further improving the exploration and exploitation steps. • Using the average position of solutions can promote RUN's exploration tendency in the early iterations. • RUN is based on the Runge-Kutta (RK) method; this permits a proper balance between exploration and exploitation. • The ESQ also helps to promote the quality of solutions and improve the convergence speed.

Conclusions and Future Work
In this work, a new optimization technique called Runge-Kutta optimizer (RUN) is applied for the parameter estimation of PV diode models. Comparisons between the RUN algorithm and the Hunger Games Search (HGS) algorithm, the Chameleon Swarm Algorithm (CSA), the Tunicate Swarm Algorithm (TSA), Harris hawk's optimization (HHO), the Sine-Cosine Algorithm (SCA) and the Grey Wolf Optimization (GWO) algorithm are conducted for the same dataset of R.T.C France solar cells. The performance of the RUN algorithm is compared with all algorithms according to statistical analyses concerning minimum, average, maximum, standard deviation and Friedman rank test of 30 independent runs. The findings show that the closeness between the extracted I-V and P-V curves achieved by the RUN algorithm compared with the measured data is very high. The best RMSE for DDSCM is better than for SDSCM. The TDSCM achieves better RMSE than SDSCM and DDSCM. The performance of robustness and convergence rates is superior for the RUN algorithm for all tested models compared with all competitor algorithms. In future studies, the RUN can be applied for several optimization problems in different fields such as economic load dispatch problem, optimal power flow in power system and estimation of fuel cell parameters.