Parameter Estimation of Three Diode Photovoltaic Model Using Grasshopper Optimization Algorithm

While addressing the issue of improving the performance of Photovoltaic (PV) systems, the simulation results are highly influenced by the PV model accuracy. Building the PV module mathematical model is based on its I-V characteristic, which is a highly nonlinear relationship. All the PV cells’ data sheets do not provide full information about their parameters. This leads to a nonlinear mathematical model with several unknown parameters. This paper proposes a new application of the Grasshopper Optimization Algorithm (GOA) for parameter extraction of the three-diode PV model of a PV module. Two commercial PV modules, Kyocera KC200GT and Solarex MSX-60 PV cells are utilized in examining the GOA-based PV model. The simulation results are executed under various temperatures and irradiations. The proposed PV model is evaluated by comparing its results with the experimental results of these commercial PV modules. The efficiency of the GOA-based PV model is tested by making a fair comparison among its numerical results and other optimization method-based PV models. With the GOA, a precise three-diode PV model shall be established.


Introduction
Solar energy is considered as a highly promising renewable energy resource [1]. For the 8th year now, solar power gained the largest share of innovative investments in renewable energies [2]. Driven by the International Governmental Support and the competition between the Photovoltaic (PV) manufacturers, PV modules with higher efficiencies and lower prices are released to the market. The global solar annual installed capacity exceeded 98 GW by the end of 2017 with expectations to reach 162 GW by 2021 [3,4]. An accurate simulation of a solar photovoltaic (PV) system is a prerequisite for actual implementation. Precise modeling of PV modules is the prime step for improving the performance of PV systems during simulation and design procedures. PV module is simulated by the I-V characteristic mathematical model. However, PV modeling is a complicated problem due to the nature of the I-V characteristic, which is a nonlinear relation that is highly affected with the variation of temperatures and solar irradiations [5][6][7][8][9][10][11].
The I-V mathematical model contains a large number of unknowns. The single and double diode model are the leading mathematical models for getting precise system modeling [12]. Single-Diode module and Solarex MSX-60 PV modules are obtained. Using MATLAB program, simulations at various environmental conditions are carried out to validate the robustness of the models. The GOA-based PV models simulation results are compared with that determined by other metaheuristic optimization methods. The proposed PV model is evaluated by implementing a fair comparison between its results and the experimental results of these commercial PV modules. A comparison between the current error of the GOA model and other PV models is made in this study. GOA succeeds to build an accurate, reliable PV model that can be used in simulating and studying the PV system. This work is arranged in its current form as follows: Section 2 explains PV module modeling. Section 3 discusses the problem formulation. In Section 4, the GOA is presented. Section 5 discusses in detail the simulation outcomes and its fair comparison with that achieved through the experimental tests. In Section 6, conclusions are inferred.

Mathematical Modeling of a PV Module
In this approach, the three-diode model is chosen to model the PV cell. The three diode model is shown in Figure 1. The three currents in the three diodes are I d1 , I d2 , I d3 . I d1 represents the current due to diffusion and recombination in the emitter and bulk regions of the P-N junction. I d2 is the recombination current in the depletion region. I d3 represents the effect of grain boundaries and leakage current. The series resistance (R s ) signifies the semiconductor material resistance at the neutral regions of the solar cell. The parallel resistance (R p ) is the leakage current at the surface of the solar cell [21].
Energies 2020, 13, x FOR PEER REVIEW 3 of 15 method is implemented as an objective function [36]. GOA-based PV models for both Kyocera KC200GT PV module and Solarex MSX-60 PV modules are obtained. Using MATLAB program, simulations at various environmental conditions are carried out to validate the robustness of the models. The GOA-based PV models simulation results are compared with that determined by other metaheuristic optimization methods. The proposed PV model is evaluated by implementing a fair comparison between its results and the experimental results of these commercial PV modules. A comparison between the current error of the GOA model and other PV models is made in this study. GOA succeeds to build an accurate, reliable PV model that can be used in simulating and studying the PV system. This work is arranged in its current form as follows: Section 2 explains PV module modeling. Section 3 discusses the problem formulation. In Section 4, the GOA is presented. Section 5 discusses in detail the simulation outcomes and its fair comparison with that achieved through the experimental tests. In Section 6, conclusions are inferred.

Mathematical Modeling of a PV Module
In this approach, the three-diode model is chosen to model the PV cell. The three diode model is shown in Figure 1. The three currents in the three diodes are Id1, Id2, Id3. Id1 represents the current due to diffusion and recombination in the emitter and bulk regions of the P-N junction. Id2 is the recombination current in the depletion region. Id3 represents the effect of grain boundaries and leakage current. The series resistance (Rs) signifies the semiconductor material resistance at the neutral regions of the solar cell. The parallel resistance (Rp) is the leakage current at the surface of the solar cell [21]. The current of such PV module is mathematically formulated as follows [20]: where IPV is the cell-generated photocurrent; Io1, Io2, and Io3 are the saturation currents of the three diodes; α1, α2, α3 are the diode ideality factor for the three diodes; Vth is the PV module thermal voltage; I is the current of PV module. V is the PV module output voltage. The values of some parameters change with the variation in temperature and irradiance. Thus, these variations are formulated in the following equations [37,38]: The current of such PV module is mathematically formulated as follows [20]: where I PV is the cell-generated photocurrent; I o1 , I o2 , and I o3 are the saturation currents of the three diodes; α 1 , α 2 , α 3 are the diode ideality factor for the three diodes; V th is the PV module thermal voltage; I is the current of PV module. V is the PV module output voltage. The values of some parameters change with the variation in temperature and irradiance. Thus, these variations are formulated in the following equations [37,38]: where I PV , I on , E gn , G n , R pn , and T n are considered photocurrent, the saturation current, material band gap, solar irradiation, circuit shunt resistance, and cell temperature at the standard test condition (STC), respectively. E gn has a value of 1.121 eV for silicon cells [37], ∆T is the temperature difference between T and T n . K i is the coefficient of short-circuit current.

Formulation of The Optimization Problem
Estimation of PV cell model parameters problem is formulated as an objective function. The proper designation of the fitness function is very significant for accurate identification of unknown parameters. The extracted parameter values must guarantee that the model behaves exactly as the PV panel.
In this study, the target of the fitness function is to minimize the root mean square error between estimated model current and its practical values [36]. This fitness function is written by the following equation: where N of the last equation represents the number of experimental samples, ∅ represents the design variables vector such that ∅ = {I PV , I o1 , I o2 , I o3 , R S , R P , α 1 , α 2 , α 3 }; f k (V, I, ∅) is represented by the following equation: This study aims at optimizing Equation (7) with respect to ∅. The GOA is used to solve this objective function to obtain the unknown parameters of the three diode PV models.

GOA
The GOA is a recent swarm intelligence algorithm developed by Mirjalili and other researchers. The GOA is imitating the social behavior of grasshopper swarms. This algorithm is a population-based method. Grasshoppers pass through two phases of development in their life cycle beginning as nymphs then adults. The nymph grasshoppers are wingless, slowly herbivores. When they mature, adults grow wings and fly fast on a large-scale area. Grasshoppers gather to form one of the hugest known swarms. The distinctive feature in grasshoppers is that they exhibit the swarming behavior in the larval and adulthood phase. The larval phase is characterized by the slow motion and little steps of grasshoppers. While long-range and quick movement are the fundamental features of the swarm in adulthood. Another vital feature in the grasshoppers' swarming is the food source seeking. Grasshoppers naturally move abruptly and locally in small areas to search for food. This divides the search process into two main stages that represent exploration and exploitation.
Inspecting the swarm motion, there are three different forces that specifies the location of the grasshopper in the swarm. The position of each grasshopper specifies a possible solution in the population. The three forces on each grasshopper are the social interaction between it and the other grasshoppers (S i ), the gravitational force applied on it (G i ) and the wind advection (A i ). The total forces on each grasshopper is represented as follows: where X i is the position of ith grasshopper. r 1 , r 2 , r 3 are random variables. The social interaction force between each grasshopper and the other grasshopper can be defined as follows: (9) where d ij dictates the distance between the grasshopper i and j. d ij is distance unit vector from the ith grasshopper to the jth grasshopper. s is a function that represents the strength of two social forces, attraction and repulsion, between grasshoppers, and it can be formulated as follows: where f, l are the attraction intensity and the attractive length scale, respectively. The function s divides the search space into comfort, repulsion, and attraction zones; however, its ability decreases to zero when the distance between two grasshoppers is greater than 10. To avoid this problem, the distance between grasshoppers is mapped between 1 and 4. If the distance between two grasshoppers is between 0 and 2.079 then a repulsion force exhibits, but if it is greater than 2.079, attraction force appears but it diminishes gradually after it reaches 4. When the distance between two grasshoppers is 2.079, there will be a comfortable zone, where there is neither attraction nor repulsion. The gravity force applied on each grasshopper is calculated as follows: where g indicates the gravitational constant and e g is a center of earth unity vector. The nymph grasshoppers' motion depends to a great extent on the wind direction as they lack wings. The direction of wind can be estimated as indicated: where u refers to a constant drift and e w is a wind direction unity vector. The grasshopper position is calculated as follows: This equation is reformulated as shown to avoid any fast arrival to the comfort zone that could lead to a local optimum solution: where ub d , lb d are the higher and lower boundaries in the dth dimension. T d is the target value in the dth dimension which is the best solution found so far. T d simulates the tendency to move towards the source of food. The parameter c is a decreasing coefficient that achieves balance between both the exploration and the exploitation procedures in the GOA and it can be estimated as shown: where c max , c min are the maximum and minimum values respectively, l is the current iteration and L is its maximum value. The pseudo code of the GOA is illustrated in Figure 2. The algorithm begins by setting the initial values of c max , c min , f, l and L. The first population is generated randomly. The value of each solution in the population is calculated by the fitness function, then the best solution is assigned. At the beginning of each new iteration, the coefficient parameter c is updated to shrink the three interaction zones as shown in Equation (15). Each solution in the population is updated as illustrated in Equation (14). If any of the updated solutions violate its lower and upper boundaries, it is returned to its position. The updated solutions are evaluated, and the best global solution is allocated. The overall operations are repeated until reaching to L which is the termination criterion in this algorithm and the best global solution T is returned.
Energies 2020, 13, x FOR PEER REVIEW 6 of 15 assigned. At the beginning of each new iteration, the coefficient parameter c is updated to shrink the three interaction zones as shown in Equation (15). Each solution in the population is updated as illustrated in Equation (14). If any of the updated solutions violate its lower and upper boundaries, it is returned to its position. The updated solutions are evaluated, and the best global solution is allocated. The overall operations are repeated until reaching to L which is the termination criterion in this algorithm and the best global solution T is returned.

Simulation Results
In this research, KC200GT and MSX-60 polycrystalline PV modules are used to verify the validity of the GOA-based PV model. The specifications of the electrical performance of these commercial PV modules under the STC condition are illustrated in Table 1 [39,40]. Figure 3a,b points out the objective function convergence for both modules. The number of iterations set for every module is 1000 and the number of search agents is 60. The convergence speed is very high for each

Simulation Results
In this research, KC200GT and MSX-60 polycrystalline PV modules are used to verify the validity of the GOA-based PV model. The specifications of the electrical performance of these commercial PV modules under the STC condition are illustrated in Table 1 [39,40]. Figure 3a,b points out the objective function convergence for both modules. The number of iterations set for every module is 1000 and the number of search agents is 60. The convergence speed is very high for each module. The best optimal value of the objective function found by GOA is 9.9775 × 10 −11 and 7.592 × 10 −10 for KC200GT and MSX-60 modules, respectively. The obtained results of the proposed GOA are compared with that obtained using the whale optimization algorithm (WOA), genetic algorithm (GA), Simulated Annealing technique (SA) for KC200GT PV modules in Table 2. To inspect the results of MSX-60 module, the fitness function is optimized with GA technique and SA technique using Optimization Toolbox embedded in MATLAB, the fitness function is also optimized using WOA and the results are compared with the GOA in Table 3. The values of the undetermined parameters of these PV models using the GOA are around the values obtained from other methods and are within an acceptable range. Figure 4a illustrates the I-V curves of the proposed GOA model and those for the practical data of the KC200GT PV module under various temperature conditions. Figure 4b shows the power versus voltage (P-V) curves of the GOA PV model and those for the practical data of the KC200GT PV module at various temperature conditions. It can be seen clearly that the I-V and P-V curves of the GOA PV model and those of practical data are matching. This demonstrates the great precision in the proposed PV model regardless of temperature variations. Figure 5a illustrates the I-V curves of the proposed GOA model and those for the practical data of the KC200GT PV module at various irradiations conditions. Figure 5b shows the P-V curves of the GOA model and the practical data of the KC200GT PV module at various irradiation conditions.      No deviations can be noticed between the simulation and experimental results. These graphical comparisons judge and verify the validity of the novel GOA PV model. Figure 7a,b shows another graphical comparison between the current errors for KC200GT and MSX-60 modules and the error of other techniques. In case of KC200GT PV module, the current error of the GOA PV model is very small, and it shows better results than most of the introduced models except at a small portion on the curve. In case of MSX-60 PV module, the current error of the GOA PV model is less than that achieved by using other PV models. This reflects the proper design of the proposed GOA and its high performance to achieve good results.
Energies 2020, 13, x FOR PEER REVIEW 11 of 15 small, and it shows better results than most of the introduced models except at a small portion on the curve. In case of MSX-60 PV module, the current error of the GOA PV model is less than that achieved by using other PV models. This reflects the proper design of the proposed GOA and its high performance to achieve good results.

Conclusions
In this paper, the GOA as a novel application to extract the undetermined parameters of the PV model of a PV module is exhibited. The problem formulation is based on minimizing the root mean-square error between calculated current and measured data by adjusting parameters of the proposed model. Simulation is carried out under different environmental conditions of temperature and irradiance level. The PV model results nearly coincide with its experimental results. The efficiency of the GOA PV model is tested by comparing the simulation results with the results of other optimization methods-based PV models. The results are within an acceptable range. The effectiveness of such model is evaluated by making a fair comparison among its current error and the current error of other PV models. The GOA has proved its superiority and high performance in case of KC200GT PV modules and MSX-60 PV modules. The GOA has succeeded in optimizing the parameters of the three-diode model. The proposed GOA may be extended to solve other optimization problems in several research fields such as wind energy systems, other renewable energy systems and smart grids.