A New Application of Chaos Game Optimization Algorithm for Parameters Extraction of Three Diode Photovoltaic Model

The precision of the PV model greatly influences the simulation results to enhance the effectiveness of photovoltaic (PV) energy systems. The PV mathematical model is based on a remarkably nonlinear relationship of its I-V characteristic. The data sheets of overall PV cells do not supply complete information of its parameters. This leads to a nonlinear mathematical model of PV with numerous unknown parameters. Consequently, in this paper, a new application of an appropriate optimization algorithm called Chaos Game Optimization algorithm (CGO) is proposed for estimating the unknown parameters of the three-diode (TD) PV model. The simulation results are carried out for PV real cells and PV module which with varying the temperature and irradiation. The proposed model of the PV module is evaluated by matching its results with the actual PV modules experimental results. To confirm the performance of the CGO algorithm in extracting the parameters of the PV model, its results are compared with the most present and robust techniques results in the literature. The results show that the CGO algorithm attains the least Root Mean Square Error (RMSE), the mean and standard deviation as the best solution. In addition, CGO provides the smallest implementation time compared with the other investigated algorithms.


I. INTRODUCTION
Solar energy is considered a significant type from the numerous types of renewable energy resources as a result of providing the appropriate environment for its spread around the world and its cleanliness. It is deemed a worthy solution to cope with fossil fuels shortage [1]. Nowadays, PV modules are being introduced into the markets which have high efficiency and little prices motivated by international government subsidies as well as competition among PV manufacturers [2]. For actual implementation of solar PV systems, an accurate simulation must be prepared in advance. Thus, to enhance the performance of PV systems during the procedures of simulation and design, accurate modeling of the PV modules must be prepared. Nevertheless, the modeling of the PV module is an intricate problem as a result of the I-V characteristic The associate editor coordinating the review of this manuscript and approving it for publication was Lorenzo Ciani .
behavior, which is known as a nonlinear relationship that is extremely affected by the changing in temperatures and solar radiations [3]- [8]. There are enormous unknown parameters in the I-V mathematical model. Commonly, there are three types of the PV solar cells mathematical model; single diode (SD) model as the simplest one, double diode (DD) model as an elaborated model, and TD model as a more detailed model. SD model is considered a simple and very acceptable degree of accuracy [9], [10]. The I-V characteristic of the SD model has five unknown parameters, and they are the Photo-generated current (Iph); the diode saturation current (I0); the factor of ideality (a); the series resistance (R s ); and shunt resistance (R sh ) [11]. Nevertheless, the SD model has a deficiency of accuracy in the case of the opencircuit voltage with low radiation because of the neglecting of the losses of carrier recombination in the depletion region [12], [13]. To vanquish this problem, the DD model is presented. Another diode is added to show the losses of recombination. The DD model presents more precision but this model is more complicated due to the increase of unknown parameters; saturation current and ideality factor parameters of the other diode are added to the unknown parameters in the SD case [13]- [15]. The TD model as a more detailed model is a significant aspect to cope with low radiation conditions in massive PV installations [16]. This model deals with the special effects of leakage current and the boundaries [17]. Although this model can fulfill most of the PV solar cell physical requirements, it includes the calculation of nine parameters and this makes the TD model more complex than SD and DD models [18].
Knowing the values of the PV cell parameter is considered a problem of optimization. There are numerous optimization methods have been proposed for the identification of the PV solar cell unknown parameters. These techniques can be divided into two classifications; conventional algorithms and meta-heuristic algorithms. Several types of the conventional methods have been presented in [19]- [21] for DD model and for TD model in [17]. Nevertheless, these algorithms consume more time and fail their capability to provide precise solutions particularly with an increasing number of unknown estimated parameters [22]. Thus, meta-heuristic algorithms are considered a better solution to avoid the disadvantages of conventional algorithms. The most recent literature for the meta-heuristic algorithms for estimating the unknown PV model parameters is as the following: Bacterial Foraging (BF) algorithm has been presented in [23] that is executed under normal and shading conditions. In [24] Generalized Oppositional Teaching Learning Based Optimization (GOTLBO) has been developed to achieve the optimum solution in little time consuming. In [25], [26], A Differential Evolution with Integrated Mutation per iteration (DEAM) and Hybrid Evolutionary algorithm (DEIM) have been suggested, respectively, that have presented a worthy execution time of CPU and get a good precision. In [1], the Flower pollination algorithm has exhibited unobserved divergence between the estimated and the experimental (I-V) curves, particularly with low solar radiation levels. Additionally, Pattern Search technique [27], Simulated Annealing algorithm (SA) [4], Harmony Search based algorithm (HS) [28], Artificial Bee Swarm Optimization algorithm (ABSO) [29], Artificial Bee Colony algorithm (ABC) [30], Modified Gradient-Based Optimizer (MGBO) [41], an enhanced teaching-learning-based optimization (ETLBO) [42], and Mutative-scale Parallel Chaos Optimization algorithm (MPCOA) [31] are presented to estimate the unknown parameters of SD and DD depending on an experimental data that presented in [32]. In [33], Genetic algorithm is executed to estimate the global optimum unknown parameters for SD and DD. Furthermore, in [34], Bird Mating (BM) algorithm is used to extract the SD unknown parameters for a PV solar array.
From the above literature, it is noticed that most researches are utilizing the common types of PV cell models such as SD and DD. The complex model such as TD is rarely used in modern investigations to evade extracting a higher number of unknown parameters. Nevertheless, it has been confirmed in this research that this model is more efficacious for coping with the more intricate physical attitude of the PV module. Furthermore, the results were less accurate and time consuming for the estimation process when the researchers used the TD model and their research depend on the traditional algorithms [17]. Consequently, a more effective algorithm should be proposed to extract the unknown parameters of this complicated model with more precision and with minimum time consuming.
In this paper, The TD model which is considered the more complicated model is investigated based on two sources of data. The application is the experimental data for 57 mm diameter commercial silicon R.T.C France solar cell. The other one is experimental data measured at the laboratory for PV module Photowatt-PWP201. The main contributions of this work can be shortened as follows: • The CGO optimization algorithm is applied for accurately and reliably identification the parameters of PV models with the TD using the measured I-V curves.
• The performance and results of the CGO are compared to the recent and efficient algorithms proposed in the literature.
• For more validation of the results, an evaluation analysis which includes RMSE, absolute output current error, and absolute power error are performed.
• The results demonstrate that the CGO technique has more accurate results with the best execution time when compared to the other compared algorithms.
The rest of the paper is arranged as follow: Section 2 presents the mathematical Model of the PV module. The problem formulation is discussed in section 3. Section 4 discusses the theory of the CGO technique. The simulation results are presented in section 5. The conclusion is added in section 6.

II. MATHEMATICAL MODEL
There are numerous mathematical models of the PV module that depict its process and the actual performance. It is becoming essential to achieve this actual behavior of solar cell. Therefore, a detailed model such as TD model is highly recommended to represent the effect of leakage current and boundaries [15], [16] and [36]. Fig. 1 shows the TD model that is represented by the I ph shunted with three parallel diodes (D 1 , D 2 and D 3 ) and R sh and series with R s . The current I D1 pass through the diode D 1 indicates the current as a result of recombination and diffusion in the quasi-neutral regions of the bulk regions and the emitter of the P-N junction. The current I D2 pass through the diode D 2 indicates the current as a result of recombination in the space charge regions [36]- [38]. The current I D3 pass through the diode D 3 designates the impact of the large leakage current and the grain boundaries. The R sh is used to calculate the leakage current. Furthermore, the R S represents the material resistance of the PV cell [1]. So in this model, the PV output current for SD and DD are presented in equation 1 and 2.

1) THE THREE DIODE (TD) MODEL
The detailed model of TD is counted as follows where I sd3 is the saturation current of D 3 N 3 is and ideality factor of D 3 . Some parameters modify as temperature and radiation change according to the following equations: where I Phs , G s , I sds , E gs , T s , and R shs are represented the generated photocurrent, solar radiation, the saturation current, band gap of the PV material, cell absolute temperature, and the shunt resistance of the circuit at the standard test condition (STC), respectively. The value of E gs is 1.121 eV for the  0  100  0  1000  Iph  0  1  0  2  Id1  0  1  0  1  Id2  0  1  0  1  Id3  0  1  0  1  n1  1  2  1  50  n2  1  2  1  50  n3  1  2  1  50 silicon cells [39], T is the difference temperature between T s and T . K i is the short circuit current coefficient.

III. THE PROBLEM FORMULATION
The essential purpose of presenting a model of PV solar cell is to reduce the variance between the estimated data and the experimental behavior with different ecological conditions by estimating the optimum values for the anonymous parameters of the PV cell model. To fulfill the main requirements for the application of any optimization algorithm by defining the solution vector (X), the range of the search, and the objective function [1].
The vector of solutions for TD PV model is realized as the following; X = (R S , R sh , I ph , I sd1 , I sd2 , I sd3 , N 1 , N 2 , and N3). The search range (lower and upper boundaries) of the unknown parameters In line with previous literature is speechified, in Table1.
The purpose of the objective function is to diminish the (RMSE) between estimated PV cell model and its experimental values [22]. The current error I err is the variance between the estimated and the experimental currents. The power error P err is the variance between the estimated and the experimental power. I err and P err can be calculated as the following: P err = P out − P out−exp (10) where I out is the estimated value of the output current that can be calculated from (4) for TD model of the PV cell model. This objective function is addressed by the following equation: where N represents the number of the measured data. The suggested optimization techniques (CGO), improved Grey wolf optimizer (IGWO), manta ray foraging optimization (MRFO), Generalized Predictive Control (GPC), Honey Bee Optimization (HBO) and Arithmetic Optimization Algorithm (AOA)) are mostly utilized to optimal estimation of the unknown parameters for the TD model with minimum error of objective function (RMSE).

A. CHAOS GAME OPTIMIZATION (CGO)
The essential idea of the CGO algorithm depends on certain rules of chaos theory where the arrangement of fractals by chaos game idea and the fractals self-similarity concerns are in perspective [35]. Chaos theory focuses on some properties in dynamic systems, and these properties are very sensitive to their initial conditions. Despite the randomness present in these dynamic systems, chaos theory indicates that there are some elementary patterns, for example, similar loops, repeating shapes, fractals, and multiple subsystems in the performance of these dynamic systems, so they can be described as self-similar and self-organizing dynamic systems. The generality of the chaotic procedures has graphical forms of fractals. A fractal, in math, is a subgroup of Euclidean space in which a particular geometric form is repetitive in various ranges. The well-known fractals similarity of the Mandelbrot group is drawn in diverse ranges as shown in Fig. 2.
Chaos theory poses that slight variations in the initial conditions for a dynamic system will lead to some severe differences in the coming conditions of these systems because of their reliance on its initial conditions. According to this theory, the current condition of a system can determine the subsequent state of that system while the approximate current condition of the system does not roughly define the subsequent condition of that system. The chaos game, in math, is the approach of producing fractals exploiting a primary polygon form and a randomly nominated primary point. The essential objective is to construct a concatenation of points in a repeated attitude to attain a shape that has a similar form in diverse ranges.
To understand the approach of the chaos game, a simple example can be taken to construct a Sierpinski triangle as a straightforward fractal. Initially, three vertices are nominated to construct the main fractal shape that results in a triangle in this situation. Every vertex from nominated vertices is highlighted in one of the colors red, green and blue. In this case, a die used must have two sides in red color, two sides in blue color and two sides in green color. A first random point  is nominated as the initial point of the fractal that is deemed in this case as a seed. As the die rolls and depending on the color that appears, the seed is moved from the first point towards the relevant vertex midway the space between the vertex and the seed. When the die is rolled again, the new location of the seed is exploited as the beginning point in the next reiteration and consequently, the seed is shifted into the intentional vertex. The Sierpinski triangle is accomplished as the final form when the die is rolled several times. Fig. 3 indicates the schematic view of the CGO approach while Fig. 4 presents the self-similarity of Sierpinski triangle in its final form with various ranges.

B. MATHEMATICAL MODEL
The CGO algorithm has been discussed which depends on the principles of chaos theory. The main conceptions of the chaos game theory and fractals are exploited to formulate the CGO algorithm mathematical model. Due to the numerous algorithms inspired from the natural evolution, a set of solutions are developed in random modifications and selection, the proposed CGO algorithm deems a number of candidate solutions (X) in this target which signifies some of the eligible points within the triangle of Sierpinski. Each solution candidate (X i ), involves some decision variables (x i,j ) that represents the location of these eligible points inner the triangle of Sierpinski. In the CGO optimization algorithm, the triangle of Sierpinski is deemed as a search area for solution candidates. The following equation indicates the mathematical model of these concepts: where: n refers to the eligible points number (solution candidates) within the triangle of Sierpinski (search area).
d refers to the points dimension. The first locations of the eligible seeds are randomly located within the search range as the following: where: As described above, the essentials of chaos theory relate to the presence of certain elementary patterns in the attitude of dynamic systems that assort these systems in selforganized and self-similar systems. The initial points that randomly formed are considered as eligible points; it represents the basic forms of dynamic systems depending on the chaos theory. In order to deem these eligible points as basic forms of dynamic systems appearing the self-similarity, it is mathematically modeled by deeming the superiority of the candidates of the solution (X). The candidates of solution with the uppermost and lowermost levels of eligibility are equipollent to the greatest and poorest values of fitness for overall solution candidates, respectively. The core idea of this model is to produce various eligible points within the search area to complete the entire form of the Sierpinski Triangle. Regarding to this, the approach for constructing new points within the triangle of Sierpinski as shown in Fig. 4 is employed for this target. For every initial eligible point in the search area (X i ), an interim triangle is sketched and considered with three points as the following: -GB: Global Best position of the so far, -MG i : Mean Group position, -X i : i th is the solution candidate position as the chosen initial eligible point.
The GB denotes the away far found best solution candidate that has the highest level of eligibility and MG i denotes the average values of various randomly chosen initial eligible points with a similar probability of involving the present deemed initial eligible point (X i ). GB and MG i combined with the nominated initial eligible point (X i ) are deemed as the triangle of Sierpinski three vertices. As declared earlier, for every initial eligible point within the search range, an interim triangle is formulated with the aim of constructing various new points within the search area that could be  deemed as the new eligible points for finishing the Sierpinski triangle points. Fig. 5 shows a schematic view of constructing interim triangles while Fig. 6 provides a supplementary detailed schematic depiction of this purpose.
The basic target from constructing the temporary triangles is to produce new points with the search range that could have the opportunity to be the new eligible points. Four methods are described to accomplish this objective. This procedure is characterized for the i th interim triangle containing the i th initial eligible point while the identical procedure is iterated for overall the eligible points and interim triangles within the search range.
About the i th interim triangle, the Sierpinski triangle three vertices within the search range including three i th starting eligible point; the green point (GB), blue point (X i ) and red point (MG i ). A die and three seeds are exploited for forming new points according to the chaos game approach in this interim triangle. The first, the second and the third seeds are positioned in X i , GB and MG i , respectively. With regard to the first seed that is located in the X i , a die that has three red sides and three green sides are exploited. When the die is rolled and based on the color that appears (green or red), the seed is moved in Xi in the direction of GB (green face) or MGi (red face). Whenever rolling the die and depending on the color that appears (red or green), the seed in X i is shifted to the GB (green side) or MG i (red side). This feature can be modeled through a certain random integer producing function that constructs just two integers [0 and 1] for the opportunity of choosing red or green sides. When the green side appears, the seed located in the X i is shifting on the way to the GB but when the red side appears, the seed located in the X i is shifting on the way to the MG i . Irrespective of the reality that each of the red or green sides have an equivalent probability of appearing in this game, the possibility of constructing two random integers for both MG i and GB are also deemed that the seed located in the X i is shifted along the way of the linked lines between the MG i and the GB. Several randomly produced factorials are also exploited depending on the reality of the motion of the seeds within the search area must be restricted due to the chaos game procedure that controls this feature. Fig. 7 describes a schematic presentation of this procedure for the first seed whilst the mathematical presentation for it is as the following: where: X i refers to the solution candidate (i th ). GB refers to the so far global solution. MG i refers to the average values of certain starting eligible points that are deemed as three vertices in the i th interim triangle.
α i refers to the randomly produced factorial for modeling the seeds motion limitations.
β i and γ i refer trandom integer of 0 or 1 for modeling the rolling a die probability.
With regard to the second seed that is located in the GB, a die has in this case three sides in blue color and three sides in red color are exploited. With rolling the die and depending on the color that appears (red or blue), the seed in the GB is shifted to the X i (blue side) or the MG i (red side). This feature can be modeled in a random certain integer production function that produces just two integers as 0 and 1 for the opportunity of choosing red or blue sides. The location of the seed in the GB is shifted to the X i when the blue side appears, whilst the location of the seed in the GB is shifted to the MG i when the red side appears. Irrespective of the reality that each blue side or red side has an equivalent probability of appearing in this game, the possibility of constructing two random integers of 1 for X i and MG i are also deemed that the seed located in GB is shifted along the way of the linked lines between MG i and X i . Depending on the reality of the seed movement within the search area ought to be restricted according to the chaos game procedure; certain randomly produced factorials are exploited to control this feature. Fig. 8 describes a schematic presentation of this procedure for the second seed whilst the mathematical presentation for it is as the following: Also, for the third seed that is located in MG i , a die has in this case three sides in blue and three sides in green are exploited. With rolling the die and depending on the color that appears (green or blue), the seed in MG i is shifted to the GB (green side) or the X i (blue side). This feature can be modeled through a certain random integer production function that produces just two integers as 0 and 1 for the opportunity of choosing green or blue sides. The location of the seed in the MG i is shifted to the X i when the blue side appears, whilst the location of the seed in the MG i is shifted to the GB when the green side appears. Irrespective of the reality that each of the green or blue sides have an equivalent probability of appearing in this game, the possibility of constructing two random integers of 1 for both X i and GB are also deemed that the seed located in the MG i is shifted along the way of the linked lines between the GB and the X i . Depending on the reality of the seed movement within the search area ought to be restricted according to the chaos game procedure; certain randomly produced factorials are exploited in this purpose to control this feature. Fig. 9 describes a schematic presentation of this procedure for the third seed whilst the mathematical VOLUME 9, 2021 FIGURE 9. The presentation of schematic the updated location for the third seed within the search area [35]. presentation for it is as the following: To carry out the phase of mutation in the location updates of the starting eligible points within the search range, the extra seed is also exploited as a fourth seed that is located in the X i . The location updating procedure for the fourth seed is determined depending on certain random variations in the randomly selected decision variables. Fig. 10 describes a schematic presentation of this procedure for the fourth seed whilst the mathematical presentation for it is as the following: where: k refers to an integer in random range of [1, d] R refers to an uniformly random number that distributed in range of [0,1].
In the suggested CGO algorithm, for controlling and adjusting the rate of exploration and exploitation, four equations are exploited to determine the α i , as indicated in Eq. (18), that is used for modeling the seeds motion restrictions. To determine the location of the first to third seeds, these four equations are used randomly. where: Rand refers to a random number distributed uniformly within range [0,1].
δ and ε refer to integers random values within range [0,1]. Depending on the matter of the self-similarity in the fractals, the initial eligibility points together with the newly generated seeds using the conception of the chaos game must be deemed to select whether the newly produced seeds ought to   be involved or not with the overall eligible points within the search range. The new candidate solutions are compared to the initial points, so that the initial points (seeds) are replaced by the new points, in the event that the new points reach the best levels of self-similarity or reservation of the initial points in the event that the worst level of self-similarity is reached with using the new points This must be taken into account that the replacement procedure is executed in the mathematical methodology to get model in less complexity. Actually,    on the total number of iterations that the optimization procedure ends after a specified number of iterations. The steps CGO algorithm is as the following whilst the pseudo-code of this optimization is indicated in Fig. 11.
Step 1: Using random nomination method to define the initial locations of the solution candidates (X) or the starting eligible points within the search area.  Step 2: Calculate the fitness values of the staring solution candidates depending on the self-similarity of the starting eligible points.
Step 3: Determined the Global Best (GB) that relevant to the point that has high eligibility grade.
Step 4: For every eligible point (X i ) within the search range, determine a Mean Group (MG i ) using a random selection procedure.
Step 5: For every eligible point (X i ) within the search area, determine an interim triangle with specified three vertices of X i , MG i , and GB.
Step 6: For every interim triangles, Calculate the values of the α i , β i , and γ i .
Step 8: For the new produced seeds with the x j i external the variables scope, conduct a boundary condition check.
Step 9: Calculate the objective function of the new seeds (new solution candidates) depending on the matter of self-similarity.
Step 10: Substitute the initial eligible points by the new seeds when the objective function of the new sees give high self-similarity levels.  Step 11: Check the terminating criterion according to the specified maximum iteration.

V. SIMULATION RESULTS AND DISCUSSIONS
In this section, CGP is applied to optimize the parameters of the TD model. Part 1 discusses a comparative study between optimized parameters of the TD model obtained by CGO and other recent algorithms for commercial silicon R.T.C France solar cell. Part 2 discusses the application of CGO to estimate the parameters of the TD model for a more complicated real system a polycrystalline PV panel STM6-120/36. The detailed results are presented as follow:

A. PART 1 RESULTS
This part discusses the application of CGO for estimating the parameters of the TD PV model. The real data of 57 mm diameter commercial silicon R.T.C France solar cell (under 1000 W = m2 at 33 • C) is considered in this application [40]. Table 2 presents the nine estimated parameters of the TD model using CGO and other algorithms. The RMSE values obtained using the CGO and other compared   algorithms are also presented in Table 2. From Table 2, the results of the CGO algorithm for the TD model estimated parameter are more accurate in comparison with other algorithms. It is highly clear that the CGO technique gives the least RMSE value. There are many references in the   literature that discuss the application of different algorithms on the same example (silicon R.T.C France solar cell). From the authors' point of view, this value is more accurate as to the most general best RMSE value than Single Diode model 9.8600E-04 and 9.8300 E-04 for the double diode model [40]. The speed of CGO is cleared in Fig. 12 which presents the  Table 3. The graphical presentation of the statistical results is presented in Fig. 13 through the boxplot figure. Table 4 gives the value of experimental measurements of voltage, current and power. Also, it presents the current absolute error (IAE) and power Absolute error (PAE) declared by Eq. (19). In this part, the CGO is applied to estimate the parameters of the TD model for the real PV module Photowatt-PWP201 [40]. This module contains 36 polycrystalline silicon cells connected in series and operating at an irradiance of 1000 W/m 2 and temperature of 45C.In the same manner of part 1, the results are arranged as follow: Table 5 presents the nine estimated parameters of the TD model and the RMSE for CGO and other algorithms. CGO gives the best RMSE and this proves that the proposed technique gives the most accurate TD model parameters. The convergence curve of the CGO when applied to estimate the PV module parameters in comparison with other compared algorithms is presented in Fig.18 that presents the fast response of the CGO algorithm. The statistical results for 15 independent runs for all algorithms are presented in Table 6. The slight change in the RMSE value of CGO during these independent runs is an explicit indication of the robust of CGO in comparison with other algorithms. The boxplot for the statistical results is presented in Fig. 19. Table 7

VI. CONCLUSION
In this paper, a new application of the Chaos Game Optimization algorithm has been proposed for estimating the parameters of the TD PV model. The TD model has been selected as it is the most detailed model for PV cell. The obtained results of CGO have been compared with other recent optimization algorithms. The results were compared via different evaluation factors such as RMSE values and the statistical analysis also calculating the IAE and PAE values for all the real measured values. For deep check, the CGO has been applied for estimating the parameters of PV module Photowatt-PWP201 contained 36 polycrystalline silicon cells. In all cases, the results obtained by the CGO algorithm were more accurate than those obtained by other optimization algorithms. Moreover, it has a faster response.