A Novel Solution Methodology Based on a Modiﬁed Gradient-Based Optimizer for Parameter Estimation of Photovoltaic Models

: In this paper, a modiﬁed version of a recent optimization algorithm called gradient-based optimizer (GBO) is proposed with the aim of improving its performance. Both the original gradient-based optimizer and the modiﬁed version, MGBO, are utilized for estimating the parameters of Photovoltaic models. The MGBO has the advantages of accelerated convergence rate as well as avoiding the local optima. These features make it compatible for investigating its performance in one of the nonlinear optimization problems like Photovoltaic model parameters estimation. The MGBO is used for the identiﬁcation of parameters of different Photovoltaic models; single-diode, double-diode, and PV module. To obtain a generic Photovoltaic model, it is required to ﬁt the experimentally obtained data. During the optimization process, the unknown parameters of the PV model are used as a decision variable whereas the root means squared error between the measured and estimated data is used as a cost function. The results veriﬁed the fast conversion rate and precision of the MGBO over other recently reported algorithms in solving the studied optimization problem.


Introduction
Fossil fuel depletion, greenhouse gas emission, and fluctuation of fuel prices in addition to the increased demand for electrical energy are the driving forces to exploit Renewable Energy Sources (RES). One of the most promising RES technologies is solar Photovoltaic (PV). There is a wide increase of installed capacities of PV where it is expected to reach 2.8 TW by 2030 and would reach 8.59 TW in 2050 according to an international renewable energy agency (IRENA) [1].
The modeling of the PV cell/module is not quite simple as it is based on variable operating conditions like temperature and solar irradiance. Moreover, the missed parameters and the data are not provided in the manufacturers' datasheets. In addition to the urgent need for accurate modeling especially with the wide increase of PV installed capacities. The PV model expresses the nonlinear relationship between the PV cell current, voltage, and power [2]. The ideal model of a PV cell comprises a current source that represents the photo-generated current, which is a function of the solar irradiance. However, there is a deviation from this model due to variant types of loss in PV cells. One of these losses is the recombination and diffusion loss in the quasi-neutral junction. The model that considers this type of loss and the simplest equivalent circuit model is the single diode model (SDM) [3]. However, at the low irradiance level and with temperature variations, ■ The results prove that the MGBO has the capability to improve the performance of the original GBO with better solutions and a fast convergence rate. This paper will be organized as follows; the mathematical formulation of PV models will be introduced in Section 2 while Section 3 will present an overview of the MGBO optimizer. In Section 4, the numerical simulation of MGBO for parameter extraction of single, double, and PV module models will take place. Finally, Section 5 outlines the main findings of this research work.

Mathematical Formulation
The mathematical formulation of the PV cell/module equivalent circuit parameters extraction and objected function formulation will be presented in this section.

Equivalent Circuit Model of PV Cell/Module
There are three popular equivalent circuit models of the PV cell which are SDM, DDM, and PV module model. In comparison with DDM, the PV module consumes more execution time as it is required to extract more parameters than DDM. The difference between the cost functions in the two models is relatively small [47]. In the following subsections, the equivalent circuit model of SDM, DDM, and PV module models will be shown.

Single Diode Model of Solar Cell
The equivalent circuit of the PV single diode model is shown in Figure 1. The output current I L can be computed as a function of the output voltage from the following equation [48,49]: Figure 1. Schematics of single diode model.

Double Diode Model of Solar Cell
The accuracy of the PV model can be enhanced by adding another diode that reflects the space charge loss in addition to diffusion and recombination loss considered in SDM. The DDM equivalent circuit PV model is depicted in Figure 2. The following equation can be used for output current calculation: { ℎ , ,1 , ,2 , , ℎ , 1 , 2 } = { ℎ , , , ℎ , } ( , , ) -' Figure 2. Schematics of the double diode model.

Objective Function Formulation
For a precise estimation of the different used PV model, the objective function is essentially defined. It will be used for the evaluation of the optimizer performance, in addition to guaranteeing the estimated parameter accuracy. In this research, the root means square error (RMSE) between the experimental and estimated current will be used as a cost function as given in Equation (4) [50].
where x is the vector of estimated parameters that are I ph , I sd,1 , I sd,2 , R S , R sh , n 1 , n 2 in case of DDM and x = I ph , I sd , R S , R sh , n in the case of SDM, and N is the number of measured values. The f (V m , I m , x) are used for the current calculation from Equations (1)-(3).

Overview of GBO
The gradient-based optimizer (GBO) is a proposed metaheuristic optimization algorithm by (Iman Ahmadianfar et al., 2020) [51]. It was inspired by Newton's gradient-based method. This optimization algorithm has a unique feature as it results from the combination of gradient-based methods and population methods concepts. This feature makes the GBO an efficient and effective optimization algorithm as it will be capable of escaping from the local optimum problem besides the fast convergence rate. To explore the search space, the GBO uses two operators namely Gradient Search Rule (GSR) and Local Escaping Operator (LEO) in addition to a set of vectors.

GBO Initialization
The GBO comprises an N vector (members of populations) in the D-dimensional search space as Equation (5) where members of the population are randomly generated by Equation (6).
where X min , X max the border are limits of the decision variables and rand(0, 1) is a randomly generated number in the range of (0, 1).

Gradient Search Rule (GSR)
GSR is based on the concept of the gradient-based method where the extreme point at which the gradient is equal to zero must be identified to determine the optimal solution. Exploration tendency enhancement and convergence rate acceleration are the aims of using GSR. Based on the numerical gradient approach and with the aids of the Taylor series, the new position X n+1 can be obtained by: Equation (7) will be changed to accommodate the population-based search concept which is given by Equation (8).
where x worst , x best are the worst and best candidate solutions through the process of optimization, randn is a normally distributed random number, ε is a small number arbitrarily choose in the range of [0, 0.1], and ∆x is the change in position at each iteration. To achieve the balance between the exploration and exploitation process and seeking for search capability improvement, the GSR will be modified accordingly to be: where the randomly generated parameter ρ 1 is given by: where β min and β max are 0.2 and 1.2, respectively, m is the number of iterations, and M is the total number of iterations. The symbol α is a sine function for the transition from exploration to exploitation. In addition, randn is a normally distributed random number, and ε is a small number within the range of [0, 0.1]. The change ∆x between the best candidate solution x best and a randomly selected position x m r1 is given by: where rand(1 : N) is a random number with N dimensions, r1, r2, r3, and r4 (r1 = r2 = r3 = r4 = n) are different integers randomly chosen from [1, N], step is a step size, which is determined by x best and x m r1 .The updated position X n+1 in Equation (7) can be updated based on the GSR as given in Equation (16): For better exploitation of the nearby area of X n , the direction of movement (DM) is added, which is calculated as below: Equation (19) is used to obtain the updated position taking into consideration the GSR and DM.
By replacing the position of the best vector (x best ) with the current vector (x m n ) in Equation (21), the new vector (X2 m n ) can be generated as follows: in which Based on the positions X1 m n ,X2 m n , and the current position (X m n ), the new solution at the next iteration (x m+1 n ) can be defined as

Local Escaping Operator (LEO)
The LEO is introduced to promote the efficiency of the GBO algorithm for solving complex problems. The LEO generates a solution with a superior performance (X m LEO ) by using several solutions, which include the best position (x best ), the solutions X1 m n and X2 m n , two random solutions x m r1 and x m r2 , and a new randomly generated solution (x m k ). The solution X m LEO is generated by the following scheme: where f 1 is a uniform random number in the range of (−1, 1), f 2 is a random number from a normal distribution with a mean of 0 and a standard deviation of 1, pr is the probability, and u 1 , u 2 , and u 3 are three random numbers, which are defined as: where rand is a random number in the range of (0, 1), and µ 1 is a number in the range of (0, 1). The above equations can be simplified: where L 1 is a binary parameter with a value of 0 or 1. If parameter µ 1 is less than 0.5, the value of L 1 is 1, otherwise, it is 0. To determine the solution x m k in Equation (6), the following scheme is suggested.
where x rand is a new solution, x m p is a randomly selected solution of the population (p ∈ [1, 2, . . . , N]), and µ 2 is a random number in the range of (0, 1). Equations (6)-(7) can be simplified as: where L 2 is a binary parameter with a value of 0 or 1. If µ 2 is less than 0.5, the value of L 2 is 1, otherwise, it is 0.

Modified GBO
One of the methods for enhancing the performance of optimization algorithms seeking to obtain the best solution and decreasing the search space is to find the stability between the capability of exploitation and exploration [52]. The convergence of a technique depends on how the solutions are moved in the search space. In the GBO algorithm, the direction of movement (DM) is used to converge around the area of the solution. Therefore, we suggest changing the DM-value gradually in the MGBO according to: where D value increases gradually from 1 to 2 as follows [53]: The same modification has been used to improve the balancing between exploration and exploitation phases in the search process of water cycle algorithm in [53]. The flow chart of the MGBO algorithm is summarized in

Results and Evaluation
The numerical simulation of the MGBO for estimating parameters of single-diode, double-diode, and PV-Module models is illustrated in this section. As mentioned previously, the root means squared value of the error (RMSE) between measures and correspondingly estimated current is used as a cost function in this research. The MGBO-based parameters estimation is accomplished using MATLAB 2016a platform using an Intel ® core TM i5-4210U CPU, 1.70 GHz, 8 GB RAM Laptop. Table 1 lists the boundary limits of the estimated parameters of all used PV models in this research.

Scenario #1: Single Diode Model
Parameters identification of the single diode model of PV cells will be investigated in this subsection. In comparison with recently proposed metaheuristic optimizers, the MGBO attains the lowest RMSE with a comparatively fast convergence rate as can be depicted in Figures 5 and 6. The statistical results reflect this superiority as indicated in Table 2 below. The best-attained results of the PV cell single diode model after 20 runs using MGBO and some of the different used optimization algorithms are given in Table 3. In addition, Table 4 lists the individual absolute error between measured and simulated data of the PV cell output current, output voltage, and output power. A graphical plot of the Integral absolute error (IAE) of the simulated current and output power of the PV cell single diode model using MGBO is displayed in Figure 7. The coincidence between the measured and estimated data points for the I-V and P-V curves is depicted in Figure 8a

Scenario #2: Double Diode Model
As mentioned previously, the DDM is more accurate than the SDM for parameters estimation of the PV cell. This subsection will be used to present this feature of the DDM in addition to the application of the MGBO and other comparative algorithms for parameters estimation of the PV cell model. The best-obtained parameters of the double diode model equivalent circuit model using MGBO and the other comparative optimization algorithms are presented in Table 5. The MGBO reaches the lowest RMSE in comparison with the other used optimizers as can be visualized in Figure 9. Furthermore, the statistical results in Table 6 stress these superiorities over the comparative algorithms besides the provided boxplot in Figure 10. Additionally, in Figure 11, the IAE of the measured and simulated current and power using the DDM of PV is tabulated in Table 7. The accurate estimation of the PV DDM parameters can be expressed by the accurate matching of the experimental and estimated I-V and P-V curves as shown in Figure 12.

Scenario #3: PV Module Model
For the sake of generality of the MGBO, this subsection will introduce its usage in the PV module model parameters estimation. The estimated parameters of the PV module model using MGBO are given in Table 8. As given in SDM and DDM, the MGBO succeeded to reach the lowest RMSE in a comparatively short time with a comparison with the other comparative algorithms, which can be concluded from Figures 13 and 14, and Table 9. Table 10 lists the IAE of the measured and estimated data points based on the estimated parameters of the PV module model supported by the plot in Figure 15. Besides, the matching between the experimental and simulated quantities is visualized in Figure 16.   Figure 13. Convergence graphs of different algorithms for the PV module model.

Conclusions
In this paper, a novel solution methodology based on a modified version of gradientbased optimizer for extracting the optimal parameters of different photovoltaic models has been presented. A modification to the GBO has been proposed with the aim of enhancing its performance through the integration with the local escaping operator. The MGBO has been used in this research due to its ability to find a global solution, in addition to its fast convergence. A comprehensive comparison has been conducted to prove the superiority of the modified algorithm over the previously used algorithms. Three different equivalent circuit models of PV, single-diode, double-diode, and PV module have been tested with the modified algorithm. The numerical simulation results reflect the capability of the MGBO for parameter extraction of all models with comparatively high precision.
The results obtained by the MGBO have been compared with those obtained by several optimization techniques, GBO, BO, MRFO, TLBO, and AEO. The main finding confirmed the effectiveness of the proposed strategy using MGBO in solving the PV parameter extraction problem compared with the other optimizers. Finally, the enhancement of the MGBO algorithm using chaotic maps will be presented in future work to be used with energy storage systems and fuel cell power generation systems.