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A Chaotic Second Order Oscillation JAYA Algorithm for Parameter Extraction of Photovoltaic Models

A Chaotic Second Order Oscillation JAYA Algorithm for Parameter Extraction of Photovoltaic Models hv photonics Article A Chaotic Second Order Oscillation JAYA Algorithm for Parameter Extraction of Photovoltaic Models Xianzhong Jian * and Yougang Cao School of Optical Electrical and Computer Engineering, University of Shanghai for Science and Technology, Shanghai 200093, China; 203590775@st.usst.edu.cn * Correspondence: jianxz@usst.edu.cn Abstract: In order to identify the parameters of photovoltaic (PV) cells and modules more accurately, reliably and efficiently, a chaotic second order oscillation JAYA algorithm (CSOOJAYA) is proposed. Firstly, both logical chaotic map and the mutation strategy are brought in enhancing the population di- versity and improving exploitation. Secondly, in order to better balance exploration and exploitation, the second order oscillation factor is added, which not only improves the diversity of the population, but also has strong exploration at the beginning of the iteration and strong exploitation at the end of the iteration. In order to balance the two abilities, the self-adaptive weight is introduced into the CSOOJAYA algorithm to regulate individuals the tendency of moving toward the optimal solution and escaping from the worst solution, so as to enhance the search efficiency and exploitation. In order to validate the behavior of CSOOJAYA, it is employed to the parameter identification problem of PV models. Finally, the experimental results show that CSOOJAYA delivers excellent behavior in the aspects of convergence, reliability and accuracy. Keywords: CSOOJAYA algorithm; parameter identification; PV models; optimization algorithm 1. Introduction Citation: Jian, X.; Cao, Y. A Chaotic In order to alleviate the greenhouse effect and solve problems such as global warming, Second Order Oscillation JAYA the application of renewable energy has attracted much attention [1]. Among the renewable Algorithm for Parameter Extraction energy sources, solar energy is considered as one of the most promising renewable energy of Photovoltaic Models. Photonics source because of its green, clean, widely distributed and free energy [2]. Solar PV systems 2022, 9, 131. https://doi.org/ can directly convert solar energy into electrical energy, so it has become one of the most 10.3390/photonics9030131 widely used power generation technologies [3]. However, PV systems are susceptible to Received: 20 January 2022 external factors, especially irradiance and temperature [4]. PV arrays tend to degrade Accepted: 22 February 2022 over time under harsh environmental conditions, significantly affecting the behavior and Published: 25 February 2022 utilization efficiency of PV systems [5]. Therefore, in order to accurately control PV systems, it is significant to adopt the experimentally measured current-voltage (I-V) data to estimate Publisher’s Note: MDPI stays neutral the actual behavior of PV systems at work [6]. The single diode model (SDM) and the with regard to jurisdictional claims in double diode model (DDM) are widely employed and can accurately reflect the nonlinear published maps and institutional affil- behavior of PV panels [7]. The parameter accuracy of these models is essential for modeling, iations. control and optimization of PV panels [8]. However, these parameters are susceptible to various environmental factors and become unavailable [9]. PV models are implicitly nonlinear transcendental equations [10]. It is very significant and challenging to employ an Copyright: © 2022 by the authors. efficient and reliable methods to identify the PV parameters. Licensee MDPI, Basel, Switzerland. Many methods have been proposed to solve PV parameter identification. In principle, This article is an open access article these methods can be divided into analytical methods and optimization methods [11]. The distributed under the terms and analytical methods mainly employ several key data (short-circuit current, open-circuit conditions of the Creative Commons voltage, and maximum power) provided by the manufacturer, and use mathematical Attribution (CC BY) license (https:// equations to deduce the parameters [12]. The analytical methods have the advantages of creativecommons.org/licenses/by/ simple, fast and direct calculation. However, these data are obtained under standard test 4.0/). Photonics 2022, 9, 131. https://doi.org/10.3390/photonics9030131 https://www.mdpi.com/journal/photonics Photonics 2022, 9, 131 2 of 21 conditions [13]. Therefore, the extracted parameters above cannot precisely predict the I-V curves at different temperatures and solar irradiance [14]. Furthermore, this approach depends heavily on the key points of the selected I-V characteristics. If the crucial points are wrongly selected, the computational accuracy will be significantly reduced [11]. Instead of choosing a few key points, the optimization method takes into account all the actual measured current and voltage data. From the algorithm point of view, optimization methods can be divided into meta-heuristic methods and deterministic methods [15]. The optimization methods transform the PV parameter identification problem into an optimization problem, and then the parameters are identified according to all the reference points on the I-V characteristic curve. Deterministic methods, such as Newton–Raphson method [16,17], iterative curve fitting [18,19] and Lambert W-function [20,21], require the objective function to be continuous, convex, and differentiable [22]. The I-V curve of the PV cell model is non-linear, multi-peak and multi-mode. And the deterministic method is easily affected by the initial conditions and gradient information, so it is free to become trapped in a local optimum when resolving such complicated problems [23]. The heuristic methods have no strict requirements on the form of the optimization problem and are not affected by initial conditions and gradient information [24]. There- fore, many heuristic methods have been used to identify the parameters of PV models in the past ten years. These include the classified perturbation mutation based particle swarm optimization algorithm (CPMPSO) [1], efficient teaching-learning-based optimiza- tion algorithm (MTLBO) [16], multiswarm spiral leader particle swarm optimization algo- rithm (MSLPSO) [18], enhanced adaptive differential evolution algorithm (EJADE) [22], memetic adaptive differential evolution (MADE) [23], modified Rao-1 optimization algo- rithm (MRAO-1) [25], improved gaining-sharing knowledge algorithm (IGSK) [26] and chaos induced coyote algorithm (CICA) [27]. The characteristics of the three existing methods are summarized in Table 1. Table 1. The characteristics of the three methods. Method Characteristic Instance - Advantages such as simple, fast and direct calculation. - If the initial value is not selected well, it is necessary to repeatedly assign a new initial value for a new round Analytical method of solution, and the error in the solution process will also increase with the increase of the number of identification parameters, which requires a large amount of calculation and a long time. - Convert the PV parameter identification problem into an optimization problem - The objective function has characteristics such as Newton–Raphson method [16,17], Deterministic method continuity, convexity and differentiability. Easily iterative curve fitting [18,19] and affected by initial conditions and gradient information. Lambert W-function [20,21] - It is easy to get stuck in local optima when dealing with complex multimodal problems. - Overcome the shortcomings of the above two CPMPSO [1], MTLBO [16], methods, the solution speed is significantly improved, Heuristic method MSLPSO [18], EJADE [22], MADE and the accuracy is improved [23], MRAO-1 [25] - contains some control parameters Compared with deterministic methods, heuristic methods can acquire more precise and robust PV parameter identification results. However, PV models are nonlinear and multi-mode, so some heuristic methods may become trapped in local optima during the optimization process [8]. In addition to the basic parameters of population size and Photonics 2022, 9, 131 3 of 21 termination condition, most heuristic algorithms have specific parameters related to their own mechanisms. The behavior of the heuristic algorithm depends to some extent on these specific parameters. Choosing inappropriate parameters will not only increase the computational burden of the algorithm, but it may also converge prematurely or fall into a local optimum [28]. Rao proposed a new and efficient heuristic algorithm in 2016, namely the JAYA algorithm [29]. The structure of the JAYA algorithm is relatively simple, with only two parameters: population size and termination conditions [28]. Therefore, the advantage of the JAYA algorithm is that it can reduce the time of the optimization process and avoid the difficulties caused by parameter adjustment. However, the individual position is only affected by the current optimal individual and the worst individual in the JAYA algorithm, which cannot effectively maintain the diversity of the population [8]. Therefore, JAYA algorithm is free to trap in local optimum when solving complicated problems such as multi-peak and multi-mode. Although the performance of some improved JAYA algorithms such as the performance- guided JAYA algorithm or PGJAYA [8], comprehensive learning JAYA algorithm (CL- JAYA) [10], improved JAYA algorithm (IJAYA) [28], logistic chaotic JAYA algorithm (LC- JAYA) [29], and elite opposition-based JAYA algorithm (EO-JAYA) [30] have been improved to some extent, the imbalance between exploration and exploitation is not fully considered. The corresponding summary of the extraction parameters of existing heuristic methods is shown in Table 2. Table 2. Summary of some existing heuristic methods. Algorithm Summary - Introducing the classification perturbation mutation strategy to improve the local search CPMPSO [1] ability and population diversity of the algorithm. - The performance of each individual is quantified by probability. PGJAYA [8] An adaptive chaotic perturbation mechanism is introduced to improve the quality of the population. - Introducing a comprehensive learning mechanism to improve the global search ability of CLJAYA [10] the algorithm. According to the score level of each learner, each teaching stage is divided into three levels to MTLBO [16] update the population. - Combine multiple populations with different search mechanisms to maintain a balance between exploration and exploitation. MSLPSO [18] - Guided by several different spiral trajectories, the algorithm maintains the diversity of the population while exploring different regions of the multidimensional space. - Introduce crossover rate sorting mechanism to retain excellent individuals. EJADE [22] - Introduce a dynamic population reduction strategy to improve convergence speed and balance exploration and exploitation. - Adaptive differential evolution strategy based on success-history improves global search ability. MADE [23] - A ranking-based elimination strategy is proposed to retain promising individuals in the population. Photonics 2022, 9, 131 4 of 21 Table 2. Cont. Algorithm Summary MRAO-1 [25] - A bidirectional update strategy is proposed to improve population diversity. - Introduce an adaptive mechanism to automatically adjust the knowledge rate parameter. - A constraint handling method is proposed to improve the convergence speed and maintain IGSK [26] the balance between exploration and exploitation. - Automatically select a binding processing strategy from three algorithm pools to improve the performance of the algorithm. - Chaos map can solve problems such as easy to fall into local optimum and slow algorithm CICA [27] convergence speed. - Experience-based learning strategies maintain the diversity of the population and improve the exploration ability of the population. IJAYA [28] - A chaotic elite learning method is proposed to improve the quality of the best solution in each generation. LCJAYA [29] - Introduces a chaotic mutation strategy to improve and balance exploration and exploitation. EO-JAYA [30] - Introducing an elite opposition learning strategy to increase population diversity. In order to improve the performance of JAYA algorithm, this paper proposes a chaotic second order oscillation JAYA algorithm (CSOOJAYA), which can identify parameters more accurately and stably. Firstly, the introduction of logical chaotic mapping mechanism improves the population diversity and exploration. Secondly, the introduction of the second order oscillation mechanism not only improves the diversity of the population, but also has strong exploration ability due to the oscillation convergence of the solution vector in the early iteration. The solution vector converges asymptotically in the later iteration and has strong exploitation ability. The self-adaptive weight mechanism is introduced to adjust the tendency of individuals to approach the optimal solution and escape the worst solution, improve the search efficiency of the population and the exploitation. The algorithm improves and balances the global search ability and local optimization ability as a whole. The introduction of mutation mechanisms ensures that individuals avoid getting stuck in local optima. The structure of CSOOJAYA is similar to the JAYA algorithm, and it only has two parameters: population size and termination condition. In order to verify the effectiveness of the proposed CSOOJAYA algorithm, CSOOJAYA and other novel algorithms are used to extract the parameters of three PV models. The experimental results show that CSOOJAYA has the best performance in terms of identification accuracy, stability and convergence speed. 2. PV Models and Objective Function 2.1. Formula of SDM Figure 1 presents the circuit diagram of the SDM. Obviously, the framework of the SDM is relatively simple. There are photogenerated current I , output current I , diode ph L current I , and shunt current I in the circuit. According to Kirchhoff current law, I can d sh L be get by Equation (1). According to the diode Shockley equation and Ohm law, I can be get by Equation (2), and I can be obtained by Equation (3): sh I = I I I (1) L ph d sh q(V + R I ) L S L I = I  exp 1 (2) d sd nkT Photonics 2022, 9, 131 5 of 21 V + R I L S L I = (3) sh sh q(V + R I ) V + R I L S L L S L I = I I  exp 1 (4) L ph d nkT R sh Figure 1. The equivalent circuit of SDM. R refers to the series resistance, R stands for the parallel resistance, V stands for S sh L the output voltage, n is the ideality factor of the diode. k refers to the Boltzmann constant 23 19 (1.3806503  10 J/K), q stands for the elementary charge (1.60217646  10 C), and T refers to the absolute temperature. Therefore, we can substitute Equations (2) and (3) into Equation (1) to get Equation (4). Obviously, SDM needs to identify five unknown different parameters (I , I , R , R , n). ph d S sh 2.2. Formula of DDM Taking into account the inherent drawbacks of SDM, DDM can more accurately illustrate the relationship between voltage and current. Figure 2 presents the circuit diagram of the DDM. It can be seen from the Figure 2, DDM differs from SDM in that it has one more diode in parallel with the current source than SDM. I can be obtained by Equation (5): q(V + R I ) q(V + R I ) V + R I L S L L S L L S L I = I I  exp 1 I  exp 1 (5) ph d1 d2 n kT n kT R 1 2 sh Figure 2. The equivalent circuit of DDM. I refers to the diode diffusion current, and I stands for the saturation current. n d1 d2 1 and n represent the ideal saturation factor of two diodes, respectively. Obviously, the DDM needs to extract seven unknown different parameters (I , I , I , R , R , n , n ). ph d1 d2 S sh 1 2 Photonics 2022, 9, 131 6 of 21 2.3. Formula of PV Module Figure 3 presents the circuit diagram of a PV module, which consists of several PV cells in parallel or in series. I can be obtained by Equation (6): q V /N + R I /N N V /N + R I ( ) L S S L P P L S S L I = N I N I  exp 1 (6) L P ph P d nkT R sh Figure 3. The equivalent circuit of a PV module. N and N refer to the number of serial and parallel connections of PV cells, respec- S P tively. This paper adopts the SDM of PV modules. Therefore, the PV module needs to identify five unknown different parameters (I , I , R , R , n). ph d S sh 2.4. Objective Function An objective function is determined to quantitatively evaluate the difference between the actual measured value and the calculated value. Equations (7)–(9) represent the error functions for the experimental and calculated data points of the SDM, DDM, and PV module, respectively: 8 h   i q(V +R I ) V +R I L S L L S L f (V , I , X) = I I + I  exp 1 + SD L L L ph d nkT R sh (7) n o X = I , I , R , R , n ph d S sh 8 h   i q(V +R I ) L S L > f (V , I , X) = I I + I  exp 1 DD L L L ph d1 > n kT < h   i q(V +R I ) V +R I L S L L S L +I  exp 1 + (8) d2 n kT R 2 sh > n o X = I , I , I , R , R , n , n ph d1 d2 S sh 1 2 h   i q(V /N +R I /N ) L S S L P f (V , I , X) = I N I + N I  exp 1 MD L L L P P > ph d nkT N V /N +R I P L S S L + (9) sh n o X = I , I , R , R , n ph d S sh RMSE X = f (V , I , X) (10) ( ) L L å k k=1 Photonics 2022, 9, 131 7 of 21 However, the error function cannot reflect all the differences between the calculated and experimental data. The root mean square error (RMSE) defined by Equation (10) is used as the objective function to quantitatively evaluate the overall difference between the experimental and calculated data. References like [10,11,14] generally use the objective function. X represents the solution composed of different unknown parameters, and N represents the number of actual measured data. 3. JAYA Algorithm JAYA is a new population-based intelligent optimization algorithm [29]. In each generation of the JAYA algorithm, the solution vector is optimized by approaching the optimal solution while staying away from the worst solution. JAYA does not need to tune algorithm-specific parameters. The only two parameters are population size and maximum number of evaluation functions (max ). FES For the objective function with d unknown variables (j = 1,2,3...d), x represent the i,j value of the j-th variable of the i-th unknown solution, then X = (x , x ...... x ) represents i i,1 i,2 i,d the i-th unknown solution. If an individual can get the worst/best value of f (X) among all individuals, then this is the worst/best individual, expressed as x = (x , x , . . . best best,1 best,2 x )/x = (x , x ,... x ). Each solution x of the JAYA algorithm can be best,d worst worst,1 worst,2 worst,d i,j updated by Equation (11): x = x + rand  x x rand  x x (11) newi,j i,j 1 best,j i,j 2 worst,j i,j x represents the absolute value of x , and x represents the updated value of i,j i,j newi,j x . rand and rand represent random numbers in [0,1]. The second and third terms in i,j 1 2 the above Equation (11) represent the tendency of the solution vector x to move towards the optimal solution and escape from the worst solution, respectively. In order to retain a better solution vector, the updated solution x = (x , x , . . . , x ) is received if newi newi,1 newi,2 newi,d it can provide a better function value. 4. Chaotic Second Order Oscillation JAYA Algorithm 4.1. Motivation At present, the identification of PV parameters is considered to be a multi-peak and multi-modal problem, that is, there are multiple local optimal solutions. When the JAYA algorithm deals with such problems, it easily falls into a local optimum because it cannot effectively explore different regions of the space. How to effectively enhance the behavior of the algorithm while balancing the exploitation and exploration? Therefore, a chaotic second order oscillation JAYA (CSOOJAYA) algorithm is proposed in this research. 4.2. Logistic Chaotic Map Strategy In the optimization process of the JAYA algorithm, it is hard to use two uniformly distributed random numbers to maintain the population diversity. However, the num- ber of chaotic sequences generated by the logical chaotic mapping strategy is aperiodic and does not converge to a specific value. It enhances the population diversity of the heuristic algorithm, improves the global search ability, and avoids falling into the local optimum. Therefore, two random numbers are replaced by chaotic sequence numbers, and Equation (12) represents the new update solution: x = x + C (x x ) C (x x ) (12) newi,j i,j 1i,j best,j i,j 2i,j worst,j i,j where C and C represent two chaotic sequence numbers, which are generated by 1i,j 2i,j Equation (13): C = 4 C (1 C ) (13) n+1 n n Photonics 2022, 9, 131 8 of 21 C represents the chaotic value of the n-th iteration. Its value range is [0,1], and C is n 1 usually set to 0.8. 4.3. Second Order Oscillation Strategy According to the update equation of the JAYA algorithm, When the i-th solution is the best/worst solution, one term on the right side of Equation (11) is 0, which cannot maintain the diversity of the population. There are only two factors that affect the population position of the JAYA algorithm, namely the current optimal individual and the worst individual. Although the algorithm can speed up the convergence speed and improve the exploitation, it cannot effectively enhance the diversity of the population and weaken the exploration ability in the optimization process. Considering the above two situations, the population individuals may fall into a local optimum in the JAYA algorithm. It can be seen from Equations (11) and (12) that x and best x have a significant impact on the performance of JAYA. Therefore, taking advantages worst of population information and enhancing exploration capabilities are the available meth- ods to improve the behavior of the JAYA. How to balance exploitation and exploration while improving population diversity. Therefore, the second order oscillation factor is introduced into Equation (12), that is, two unequal second order oscillation factors are added respectively to the two rightmost terms of Equation (12), as shown in Equation (14). The difference between the current optimal individual and the optimal individual of the previous generation and the difference between the current worst individual and the worst individual of the previous generation can guide the individual to update a better range, maintain population diversity and improve exploration ability. After adopting the second order oscillation strategy, because the solution vector has oscillation convergence in the early iteration, the algorithm has strong global search ability; The solution vector converges asymptotically in the later stage of iteration, and the algorithm has strong local optimization ability. This strategy improves the convergence accuracy of the algorithm as a whole: x = x + C  (1 + k1)x k1x x C ((1 newi,j i,j 1i,j best,j bestp,j i,j 2i,j (14) +k2) x k2x x worst,j worstp,j i,j where x and x represent the optimal solution and the worst solution of the jth bestp,j worstp,j variable of the previous generation of population individuals, respectively. k1 and k2 are two randomly generated numbers in [0,1], and k1 is not equal to k2. 4.4. Self-Adaptive Weight Equation (12) enhances the population diversity and exploration, and Equation (14) not only improves exploitation and exploration, but also effectively balances the two abilities of the algorithm. However, how to balance the two abilities as a whole is the core problem of the algorithm. Considering that the strategy is dedicated to improving exploitation ability, the self-adaptive weight strategy is introduced into Equation (14) to obtain Equation (15). w is obtained from Equation (16). It can be seen from Equation (16) that the value of w increases gradually with the increase of the number of function evaluations. Because the difference between the optimal solution and the worst solution becomes smaller and smaller as the search process progresses. Therefore, in the optimization process of the JAYA algorithm, the population can approach the promising solution at the beginning of the iteration and conduct local optimization in the area of the promising solution at the end of the iteration. Thus, it enhances the search efficiency of the population and the exploitation of the algorithm: Photonics 2022, 9, 131 9 of 21 x = x + C  (1 + k1)x k1x x wC ((1 newi,j i,j 1i,j best,j bestp,j i,j 2i,j (15) +k2) x k2x x worst,j worstp,j i,j f (x ) best ( ) , if f(x ) 6= 0 worst f (x ) w = worst (16) 1 , otherwise f (x ) and f (x ) represent the values of the optimal and worst individual of the worst best objective function, respectively. Furthermore, the self-adaptive weight is automatically determined, so there is no need to adjust parameters. 4.5. Mutation Mechanism The three position update equations based on the JAYA algorithm not only improve the overall exploration and exploitation, but also effectively balance the two abilities. However, the optimal individual may still have a very small probability of being in a local optimal state to solve the identification of PV parameters. In order to make the individual jump out of the local optimum, the mutation mechanism is proposed. If from the first function evaluation to a quarter of the maximum function evaluation time, and so on, if the RMSE values of the two are equal, the population individual is reinitialized. And if the best RMSE of the previous generation is better than the current optimal RMSE, the RMSE of the previous generation is accepted. 4.6. Framework of CSOOJAYA The overall update equation of the improved algorithm based on the above strategy is shown in Equation (17). p is the probability, which represents random numbers in [0,1]: x + C  (1 + k1)x k1x x > i,j 1i,j best,j bestp,j i,j C  (1 + k1)x k2x x , > 2i,j worst,j worstp,j i,j i f 0  p < 1/2 x + C  (1 + k1)x k1x x i,j 1i,j best,j bestp,j i,j x = (17) newi,j C w (1 + k2)x k2x x , 2i,j worst,j worstp,j i,j > i f  p < 3/4 x + C  x x C  x x , > i,j 1i,j best,j i,j 2i,j worst,j i,j > 3 i f  p  1 The pseudocode of the CSOOJAYA algorithm can be summarized by the Algorithm 1. In addition, Figure 4 shows the flow chart of the specific implementation of CSOO- JAYA. Similar to the JAYA algorithm, the CSOOJAYA does not need to adjust the specific algorithm parameters. Photonics 2022, 9, 131 10 of 21 Algorithm 1: CSOOJAYA algorithm. 1. Set the population size (NP) and max ; FES 2. Randomly initialize the population and RMSE values of all individuals are calculated; 3. FES = NP; 4. While FES < max do FES 5. Identify the best x and worst x of all individuals; i i 6. For i = 1 to NP do 7. if (0  p < 1/2) then 8. Update the x using Equation (14); 9. else if (1/2  p < 3/4) then 10. Calculate the w using Equation (15); 11. Update the x using Equation (16) 12. else 13. Update the x using Equation (12) 14. End if 15. Obtain the new population and Calculate the RMSE for the updated individual x ; newi 16. Receive the updated solution if it is more excellent than the pre-update one 17. End For 18. If the conditions of the mutation strategy are met 19. Randomly initialize the population and Save the current optimal solution and RMSE 20. End if 21. End While Figure 4. The flowchart of CSOOJAYA. 5. Experimental Results and Discussion In order to estimate the effective behavior of CSOOJAYA, it was applied to the identifi- cation of PV parameters. The benchmark data are extracted from reference [11]. where the Photonics 2022, 9, 131 11 of 21 data for SDM and DDM were obtained from a commercial RTC silicon PV cell (1000 w/m at 33 C, Photowatt, Bourgoin-Jallieu, France) with a diameter of 57 mm, and the data for the PV module was obtained from a Photowatt-PWP201 (1000 w/m at 45 C) solar module consisting of 36 polycrystalline silicon cells connected in series (Photowatt, Bourgoin-Jallieu, France). In order to ensure fairness, the search space of the solution vector is the same, and the value range of the identification parameter is the same as that adopted in the previous literature. Table 3 shows the value ranges of the parameters corresponding to different PV models. Table 3. Parameters ranges of PV models. SDM/DDM PV Module Parameter Lower Upper Lower Upper 0 1 0 2 0 1 0 50 0 0.5 0 2 0 100 0 2000 1 2 1 50 CSOOJAYA is compared with other novel algorithms to verify its superior perfor- mance. These algorithms are DE/WOA, MLBSA, EJADE, MTLBO, MPPCEDE, MADE, ITLBO, CPMPSO, DERAO, JAYA, IJAYA, LCJAYA and PGJAYA. Because these optimization algorithms have better performance in PV model parameter identification, they are used for comparison with CSOOJAYA. Table 4 presents the corresponding parameter settings for the selected algorithms, which are extracted from their respective references. For fair comparison, set max to 50,000 for all algorithms and run 30 times independently. The FES results of DE/WOA [11], MLBSA [5], EJADE [22], MTLBO [16], MPPCEDE [15], MADE [23], ITLBO [2], CPMPSO [1], DERAO [25], JAYA [31], IJAYA [28], LCJAYA [29] and PGJAYA [8] were obtained directly from the corresponding references. The selected algorithms are all executed in MATLAB R2016b on a PC configured with an Intel(I) Core (TM) i5-7200u CPU operating at 3.1 GHz and equipped with 4 Gb of RAM. Table 4. Parameter settings of the compared algorithms. Algorithm Parameters DE/WOA Np = 40, F = rand (0.1,1), Cr = rand (0,1) MLBSA Np = 50 EJADE Np = 50, Np = 4 max min MTLBO Np = 50 MPPCEDE Np = 40; F = rand (0.1,1); Cr = rand (0,1) MADE NP = 20; 0.05 for the SDM and PV module; 0.01 for the DDM IJAYA NP = 20 ITLBO Np = 50 CPMPSO NP = 50, w = 0.729, c1 = 1.49455, c2 = 1.49455 DERAO NP = 20 Photonics 2022, 9, 131 12 of 21 Table 4. Cont. Algorithm Parameters JAYA NP = 20 PGJAYA NP = 20 LCJAYA NP = 20 CSOOJAYA NP = 20 5.1. Results on the SDM For SDM, Table 5 illustrates the best parameters and RMSE obtained for all selected algorithms. The overall best RMSE values of all compared algorithms are indicated in bold. It is obvious from Table 5 that CSOOJAYA, DE/WOA, MLBSA, EJADE, MTLBO, MPPCEDE, MADE, ITLBO, CPMPSO, DERAO, LCJAYA and PGJAYA obtained the smallest RMSE (9.8602  10 ), followed by IJAYA. Although accurate parameter value informa- tion cannot be obtained, it can be seen from the literature [1,2] that the RMSE can be employed to represent the accuracy. Furthermore, the smaller the RMSE, the more precise the identified parameters. Table 5. Comparison of the extracted optimal parameters of the SDM. Algorithm I (A) I (A) R (W) R (W) n RMSE ph d S sh DE/WOA 0.760776 0.323021 0.036377 53.718524 1.481184 9.860219  10 MLBSA 0.7608 0.32302 0.0364 53.7185 1.4812 9.8602  10 EJADE 0.7608 0.3230 0.0364 53.7185 1.4812 9 8602  10 MTLBO 0.76077553 0.32300 0.03637709 53.7185251 1.48118359 9 860219  10 MPPCEDE 0.7608 0.32302 0.0364 53.7185 1.4812 9.8602  10 MADE 0.7608 0.3230 0.0364 53.7185 1.4812 9.8602  10 IJAYA 0.7608 0.3228 0.0364 53.7595 1.4811 9.8603  10 ITLBO 0.7608 0.3230 0.0364 53.7185 1.4812 9.8602  10 CPMPSO 0.760776 0.323021 0.036377 53.718522 1.481184 9.860219  10 DERAO 0.760776 0.323021 0.036377 53.718522 1.481135 9.860219  10 JAYA 0.7608 0.3281 0.0364 54.9298 1.4828 9.8946  10 PGJAYA 0.7608 0.3230 0.0364 53.7185 1.4812 9.8602  10 LCJAYA 0.7608 0.3230 0.0364 53.7185 1.4819 9.8602  10 CSOOJAYA 0.760776 0.323021 0.036377 53.718525 1.481184 9.860219  10 To further confirm the performance of CSOOJAYA, the IAE of power and current between the measured data and the calculated data was used. The IAE can be current calculated by Equation (18), and the IAE can be obtained by Equation (19): power IAE = jI I j (18) current measured estimated IAE = jV  I V  I j (19) power measured estimated where I represents the output current estimated by different models through estimated Equations (4), (5) or (6), and I represents the actual measured current. V repre- measured sents the actual measured voltage. IAE represents the individual absolute error of current current, and IAE represents the individual error of power. power The IAE of power and current are shown in Table 6. All values of IAE are less current 3 3 than 2.50  10 and all values of IAE are less than 1.46  10 , proving the accuracy power of extracted parameters. To further verify the accuracy of the results, Figure 5 illustrates the best results obtained using CSOOJAYA to plot the I-V and P-V curves. Obviously, the data calculated by CSOOJAYA are in good agreement with the measured data across the entire voltage range. Photonics 2022, 9, 131 13 of 21 Table 6. IAE of CSOOJAYA for the SDM. Item V /V I /A I /A IAE P /W IAE measured measured estimated current estimated power 1 0.2057 0.7640 0.7640881747 0.0000881747 0.1571729375 0.0000181375 2 0.1291 0.7620 0.7626635570 0.0006635570 0.0984598652 0.0000856652 3 0.0588 0.7605 0.7613557780 0.0008557780 0.0447677197 0.0000503197 4 0.0057 0.7605 0.7601544618 0.0003455381 0.0043328804 0.0000019695 5 0.0646 0.7600 0.7590556797 0.0009443202 0.0490349969 0.0000610030 6 0.1185 0.7590 0.7580428161 0.0009571838 0.0898280737 0.0001134262 7 0.1678 0.7570 0.7570921248 0.0000921248 0.1270400585 0.0000154585 8 0.2132 0.7570 0.7561418358 0.0008581641 0.1612094393 0.0001829606 9 0.2545 0.7555 0.7550873439 0.0004126560 0.1921697290 0.0001050209 10 0.2924 0.7540 0.7536643499 0.0003356500 0.2203714559 0.0000981440 11 0.3269 0.7505 0.7513914392 0.0008914392 0.2456298615 0.0002914115 12 0.3585 0.7465 0.7473543260 0.0008543260 0.2679265258 0.0003062758 13 0.3873 0.7385 0.7401176997 0.0016176997 0.2866475851 0.0006265351 14 0.4137 0.7280 0.7273827075 0.0006172924 0.3009182261 0.0002553738 15 0.4373 0.7065 0.7069731390 0.0004731390 0.3091593537 0.0002069037 16 0.4590 0.6755 0.6752806425 0.0002193574 0.3099538149 0.0001006850 17 0.4784 0.6320 0.6307587591 0.0012412408 0.3017549903 0.0005938096 18 0.4960 0.5730 0.5719288231 0.0010711768 0.2836766963 0.0005313036 19 0.5119 0.4990 0.4996074317 0.0006074317 0.2557490442 0.0003109442 20 0.5265 0.4130 0.4136491106 0.0006491106 0.2177862567 0.0003417567 21 0.5398 0.3165 0.3175102788 0.0010102788 0.1713920485 0.0005453485 22 0.5521 0.2120 0.2121548945 0.0001548945 0.1171307172 0.0000855172 23 0.5633 0.1035 0.1022509879 0.0012490120 0.0575979815 0.0007035684 24 0.5736 0.0100 0.0087182129 0.0012817870 0.0050007669 0.0007352330 25 0.5833 0.1230 0.1255085017 0.0025085017 0.0732091090 0.0014632090 26 0.5900 0.2100 0.2084737660 0.0015262339 0.1229995219 0.0009004780 Figure 5. Experimental and calculated data of CSOOJAYA for SDM. 5.2. Results on the DDM For DDM, the extraction of seven parameters increases the difficulty of parameter identification. Table 7 illustrates the best parameters and RMSE obtained for all selected algorithms. The overall best RMSE value of all compared algorithms are indicated in bold. Photonics 2022, 9, 131 14 of 21 It is obvious from Table 7 that CSOOJAYA, DE/WOA, EJADE, MTLBO, ITLBO, CPMPSO, DERAO have the smallest RMSE (9.8248  10 ), followed by MLBSA and MPPCEDE. To further verify the accuracy of the results, Figure 6 illustrates the best results obtained using CSOOJAYA to plot the I-V and P-V curves. Obviously, the data calculated by CSOOJAYA are in good agreement with the measured data across the entire voltage range. Table 7. Comparison of the extracted optimal parameters of the DDM. Algorithm I (A) I (A) I (A) R (W) R (W) n1 n2 RMSE ph d1 d2 S sh DE/WOA 0.760781 0.225974 0.749346 0.036740 55.485437 1.451017 2 9.824849  10 MLBSA 0.7608 0.22728 0.73835 0.0367 55.4612 1.4515 2 9.8249  10 EJADE 0.7608 0.2260 0.7493 0.0367 55.4854 1.4510 2 9.8248  10 MTLBO 0.7607810 0.22597 0.7493 0.03674043 55.485447 1.451016 2 9.824849  10 MPPCEDE 0.7608 0.22728 0.73835 0.0367 55.4612 1.4515 2 9.82485  10 MADE 0.7608 0.2246 0.7394 0.0368 55.4329 1.4505 1.9963 9.8261  10 IJAYA 0.7601 0.0050445 0.75094 0.0376 77.8519 1.2186 1.6247 9.8293  10 ITLBO 0.7608 0.2260 0.7493 0.0367 55.4854 1.4510 2 9.8248  10 CPMPSO 0.76078 0.22597 0.74935 0.03674 55.48544 1.45102 2 9.824849  10 DERAO 0.760781 0.225959 0.749667 0.036740 55.486045 1.450963 2 9.824849  10 JAYA 0.7607 0.0060763 0.31507 0.0364 52.6575 1.4788 1.8436 9.8934  10 PGJAYA 0.7608 0.21031 0.88534 0.0368 55.8135 1.4450 2 9.8263  10 LCJAYA 0.7608 0.22596 0.74640 0.0367 55.4815 1.4518 2.0000 9.8250  10 CSOOJAYA 0.760781 0.225974 0.749345 0.036740 55.485437 1.451017 2.000000 9.824849  10 Figure 6. Experimental and calculated data of CSOOJAYA for DDM. The IAE of power and current are shown in Table 8. All values of IAE are less current 3 3 than 2.54  10 and all values of IAE are less than 1.48  10 , which demonstrates power the accuracy of CSOOJAYA parameter identification. 5.3. Results on the PV Module The effectiveness of CSOOJAYA is further verified using the PhotoWatt-PWP201 PV module. Table 9 illustrates the best parameters and RMSE obtained for all selected algorithms. The overall best RMSE value of all the compared algorithms are marked in bold. Among all the comparison algorithms, CSOOJAYA, DE/WOA, MLBSA, EJADE, MTLBO, MPPCEDE, MADE, ITLBO, CPMPSO, PGJAYA, IJAYA, LCJAYA and DERAO obtained the best RMSE value, followed by JAYA. To further verify the accuracy of the results, Figure 7 illustrates the best results obtained using CSOOJAYA to plot the I-V and P-V curves. Obviously, the data calculated by CSOOJAYA are in good agreement with the measured data across the entire voltage range. The IAE of power and current are shown in Table 10. All values of IAE are less than 4.83  10 and all values of current IAE are less than 7.99  10 , which verifies the accuracy of CSOOJAYA parameter power identification again. Photonics 2022, 9, 131 15 of 21 Table 8. IAE of CSOOJAYA for the DDM. Item V /V I /A I /A IAE P /W IAE measured measured estimated current estimated power 1 0.2057 0.7640 0.7639834118 0.0000165881 0.1571513878 0.0000034121 2 0.1291 0.7620 0.7626040957 0.0006040957 0.0984521887 0.0000779887 3 0.0588 0.7605 0.7613376980 0.0008376980 0.0447666566 0.0000492566 4 0.0057 0.7605 0.7601737877 0.0003262122 0.0043329905 0.0000018594 5 0.0646 0.7600 0.7591076801 0.0008923198 0.0490383561 0.0000576438 6 0.1185 0.7590 0.7581214198 0.0008785801 0.0898373882 0.0001041117 7 0.1678 0.7570 0.7571886136 0.0001886136 0.1270562493 0.0000316493 8 0.2132 0.7570 0.7562436068 0.0007563931 0.1612311369 0.0001612630 9 0.2545 0.7555 0.7551773015 0.0003226984 0.1921926232 0.0000821267 10 0.2924 0.7540 0.7537223535 0.0002776464 0.2203884161 0.0000811838 11 0.3269 0.7505 0.7513991342 0.0008991342 0.2456323769 0.0002939269 12 0.3585 0.7465 0.7473014432 0.0008014432 0.2679075674 0.0002873174 13 0.3873 0.7385 0.7400106607 0.0015106607 0.2866061288 0.0005850788 14 0.4137 0.7280 0.7272469525 0.0007530474 0.3008620642 0.0003115357 15 0.4373 0.7065 0.7068502977 0.0003502977 0.3091056352 0.0001531852 16 0.4590 0.6755 0.6752105425 0.0002894574 0.3099216390 0.0001328609 17 0.4784 0.6320 0.6307607576 0.0012392423 0.3017559464 0.0005928535 18 0.4960 0.5730 0.5719947329 0.0010052670 0.2837093875 0.0004986124 19 0.5119 0.4990 0.4997061352 0.0007061352 0.2557995706 0.0003614706 20 0.5265 0.4130 0.4137336728 0.0007336728 0.2178307787 0.0003862787 21 0.5398 0.3165 0.3175462057 0.0010462057 0.1714114418 0.0005647418 22 0.5521 0.2120 0.2121229960 0.0001229960 0.1171131061 0.0000679061 23 0.5633 0.1035 0.1021632765 0.0013367234 0.0575485736 0.0007529763 24 0.5736 0.0100 0.0087917509 0.0012082490 0.0050429483 0.0006930516 25 0.5833 0.1230 0.1255434350 0.0025434350 0.0732294856 0.0014835856 26 0.5900 0.2100 0.2083715891 0.0016284108 0.1229392376 0.0009607623 Table 9. Comparison of the extracted optimal parameters of the PV module. Algorithm I (A) I (A) R (W) R (W) n RMSE ph d S sh DE/WOA 1.030514 3.482263 1.201271 981.982143 48.642835 2.425075  10 MLBSA 1.0305 3.4823 1.2013 981.9823 48.6428 2.4251  10 EJADE 1.0305 3.4823 1.2013 981.9824 48.6428 2.4251  10 MTLBO 1.0305143 3.4823 1.2012710 981.9823732 48.6428349 2425075  10 MPPCEDE 1.030514 3.482263 1.201271 981.982143 48.642835 2.42507  10 MADE 1.0305 3.4823 1.2013 981.9823 48.6428 2.4250  10 IJAYA 1.0305 3.4703 1.2016 977.3752 48.6298 2.4251  10 ITLBO 1.0305 3.4823 1.2013 981.9823 48.6428 2.4251  10 CPMPSO 1.030514 3.4823 1.201271 981.9823 48.64284 2.425075  10 DERAO 1.030514 3.4823 1.201271 981.9821 48.64131 2.425075  10 JAYA 1.0302 3.4931 1.2014 1022.5 48.6531 2.427785  10 PGJAYA 1.0305 3.4818 1.2013 981.8545 48.6424 2.425075  10 LCJAYA 1.0305 3.4823 1.2024 981.9828 48.6684 2.425075  10 CSOOJAYA 1.030514 3.482263 1.201271 981.982243 48.642834 2.425075  10 Photonics 2022, 9, 131 16 of 21 Figure 7. Experimental and calculated data of CSOOJAYA for PV module. Table 10. IAE of CSOOJAYA for PV module. Item V /V I /A I /A IAE P /W IAE measured measured estimated current estimated power 1 0.1248 1.0315 1.0291188622 0.0023811377 0.1284340340 0.0002971659 2 1.8093 1.0300 1.0273807740 0.0026192259 1.8588400345 0.0047389654 3 3.3511 1.0260 1.0257414978 0.0002585021 3.4373623333 0.0008662666 4 4.7622 1.0220 1.0241068556 0.0021068556 4.8770016680 0.0100332680 5 6.0538 1.0180 1.0222915054 0.0042915054 6.1887483154 0.0259799154 6 7.2364 1.0155 1.0199303816 0.0044303816 7.3806242138 0.0320600138 7 8.3189 1.0140 1.0163628063 0.0023628063 8.4550205496 0.0196559496 8 9.3097 1.0100 1.0104958517 0.0004958517 9.4074132312 0.0046162312 9 10.2163 1.0035 1.0006286698 0.0028713301 10.2227226794 0.0293343705 10 11.0449 0.9880 0.9845480779 0.0034519220 10.8742350664 0.0381261335 11 11.8018 0.9630 0.9595213746 0.0034786253 11.3240793592 0.0410540407 12 12.4929 0.9255 0.9228385152 0.0026614847 11.5289292866 0.0332496633 13 13.1231 0.8725 0.8725993581 0.0000993581 11.4512086365 0.0013038865 14 13.6983 0.8075 0.8072739566 0.0002260433 11.0582808396 0.0030964103 15 14.2221 0.7265 0.7283361680 0.0018361680 10.3584698161 0.0261141661 16 14.6995 0.6345 0.6371376868 0.0026376868 9.3656054279 0.0387726779 17 15.1346 0.5345 0.5362127464 0.0017127464 8.1153654320 0.0259217320 18 15.5311 0.4275 0.4295110044 0.0020110044 6.6707783610 0.0312331110 19 15.8929 0.3185 0.3187741584 0.0002741584 5.0662458227 0.0043571727 20 16.2229 0.2085 0.2073891784 0.0011108215 3.3644539035 0.0180207464 21 16.5241 0.1010 0.0961668397 0.0048331602 1.5890704768 0.0798636231 22 16.7987 0.0080 0.0083257217 0.0003257217 0.1398613011 0.0054717011 23 17.0499 0.1110 0.1109368217 0.0000631782 1.8914617173 0.0010771826 24 17.2793 0.2090 0.2092476082 0.0002476082 3.6156521970 0.0042784970 25 17.4885 0.3030 0.3008639322 0.0021360677 5.2616588799 0.0373566200 5.4. Statistical Results and Convergence Curve The above three sections demonstrate the superior accuracy of CSOOJAYA in solving parameter identification on different PV models. However, convergence and robustness should also be considered. Therefore, this section compares the convergence curves and statistical results of the three PV models. Photonics 2022, 9, 131 17 of 21 The maximum RMSE (Max), minimum RMSE(Min), average RMSE (Mean) and stan- dard deviation (SD) of RMSE obtained by running all the comparison algorithms 30 times independently on three different PV models are shown in Table 11. The overall best RMSE values for all compared algorithms are marked in bold. where Max represents the worst accuracy, Min is the best accuracy, and Mean is the average accuracy, and SD represents the stability of algorithm. It can be seen from the Table 11: (1) For the best RMES value, only CSOOJAYA, DE/WOA, EJADE, MTLBO, ITLBO, CPMPSO and DERAO get the best RMSE values for all PV models. (2) For the worst RMSE value, only CSOOJAYA and DE/WOA can obtain the optimal worst value of all PV models (3) For the average RMSE value, only CSOOJAYA can obtain the optimal average value for all PV models. (4) Taking into account the SD, it can reflect the reliability of the algorithm. Although the SD of MTLBO, MPPCEDE, ITLBO, CPMPSO and DERAO are slightly better than those of the CSOOJAYA algorithm in the SDM and the PV module, the SD of CSOOJAYA is far superior to other algorithms in the DDM. On the whole, the CSOOJAYA algorithm has stronger reliability than other algorithms. Based on the superior stability and accuracy of three PV models, CSOOJAYA can obtain the best parameter identification results. Table 11. Comparison of statistical results for PV models. Model Algorithm RMSE Min Max Mean SD 4 4 4 17 SDM DE/WOA 9.860219  10 9.860219  10 9.860219  10 3.545178  10 4 4 4 12 MLBSA 9.8602  10 9.8602  10 9.8602  10 9.1461  10 4 4 4 17 EJADE 9.8602  10 9.8602  10 9.8602  10 5.13  10 4 4 4 17 MTLBO 9.860219  10 9.860219  10 9.860219  10 1.9092749  10 4 4 4 17 MPPCEDE 9.86022  10 9.86022  10 9.86022  10 2.44932  10 4 4 4 15 MADE 9.8602  10 9.8602  10 9.8602  10 2.74  10 4 3 4 5 IJAYA 9.8603  10 1.0622  10 9.9204  10 1.4033  10 4 4 4 17 ITLBO 9.8602  10 9.8602  10 9.8602  10 2.19  10 4 4 4 17 CPMPSO 2.17556  10 9.86022  10 9.86022  10 9.86022  10 4 4 4 17 DERAO 9.860219  10 9.860219  10 9.860219  10 3.642031  10 4 3 3 4 JAYA 9.8946  10 1.4783  10 1.1617  10 1.8796  10 4 4 4 9 PGJAYA 9.8602  10 9.8603  10 9.8602  10 1.4485  10 4 4 4 16 LCJAYA 9.8602  10 9.8602  10 9.8602  10 5.6997  10 4 4 4 17 CSOOJAYA 9.860219  10 9.860219  10 9.860219  10 4.717305  10 4 4 4 7 DDM DE/WOA 9.824849  10 9.860377  10 9.829703  10 9.152178  10 4 4 4 6 MLBSA 9.8249  10 9.8518  10 9.8798  10 1.3482  10 4 4 4 6 EJADE 9.8248  10 9.8602  10 9.8363  10 1.36  10 4 4 4 9 MTLBO 9.824849  10 9.825026  10 9.824855  10 3.30000  10 4 4 4 8 MPPCEDE 9.82485  10 9.82908  10 9.82504  10 8.02951  10 4 4 4 5 MADE 9.8261  10 9.8786  10 9.8608  10 8.02  10 4 3 3 5 IJAYA 9.8293  10 1.4055  10 1.0269  10 9.8325  10 Photonics 2022, 9, 131 18 of 21 Table 11. Cont. Model Algorithm RMSE Min Max Mean SD 4 4 4 6 ITLBO 9.8248  10 9.8812  10 9.8497  10 1.54  10 4 4 4 6 CPMPSO 9.82485  10 9.83137  10 9.86022  10 1.3398  10 4 4 4 9 DERAO 9.824841  10 9.824897  10 9.824845  10 1.01560  10 4 3 3 4 JAYA 9.8934  10 1.4793  10 1.1767  10 1.9356  10 4 4 4 6 PGJAYA 9.8263  10 9.9499  10 9.8582  10 2.5375  10 4 4 4 6 LCJAYA 9.8250  10 9.8602  10 9.8308  10 1.3118  10 4 4 4 17 CSOOJAYA 5.576332  10 9.824849  10 9.824849  10 9.824849  10 3 3 3 8 PV module DE/WOA 2.425075  10 2.425442  10 2.425092  10 6.270718  10 3 3 3 8 MLBSA 2.425075  10 2.425084  10 2.425312  10 4.336794  10 3 3 3 17 EJADE 2.4251  10 2.4251  10 2.4251  10 3.27  10 3 3 3 17 MTLBO 2.425075  10 2.425075  10 2.425075  10 1.3070107  10 3 3 3 17 MPPCEDE 2.42507  10 2.42507  10 2.42507  10 2.23156  10 3 3 3 17 MADE 3.07  10 2.4250  10 2.4251  10 2.4251  10 3 3 3 6 IJAYA 2.4251  10 2.4393  10 2.4289  10 3.7755  10 3 3 3 17 ITLBO 2.4251  10 2.4251  10 2.4251  10 1.27  10 3 3 3 17 CPMPSO 2.425075  10 2.425075  10 2.425075  10 1.515959  10 3 3 3 17 DERAO 2.425075  10 2.425075  10 2.425075  10 1.716113  10 3 3 3 5 JAYA 2.427785  10 2.595873  10 2.453710  10 3.456290  10 3 3 3 7 PGJAYA 2.425075  10 2.426764  10 2.425144  10 3.071420  10 3 3 3 16 LCJAYA 2.425075  10 2.425075  10 2.425075  10 2.415229  10 3 3 3 17 CSOOJAYA 2.425075  10 2.425075  10 2.425075  10 2.699858  10 In order to validate the convergence speed of CSOOJAYA, Figures 8–10 plot the iterative curves of several algorithms on the PV models. Obviously, CSOOJAYA not only get the best results among these three models, but also converges faster than other algorithms. Figure 8. Convergence curves of the seven algorithms on SDM. Photonics 2022, 9, 131 19 of 21 Figure 9. Convergence curves of the seven algorithms on DDM. Figure 10. Convergence curves of the seven algorithms on PV. In summary, the above comparison shows that CSOOJAYA has faster convergence, better accuracy and robustness in extracting the parameters of PV models. Furthermore, the behavior of CSOOJAYA is competitive when compared to other algorithms. 6. Conclusions This paper proposes a chaotic second order oscillation JAYA algorithm to provide an accurate and stable way to identify the PV parameters. The algorithm introduces the logical chaotic map strategy, and the generated random sequence numbers are non- periodic and non-convergent, which can enhance the diversity of the population and improve the exploration. In order to improve and balance the above two abilities, the Photonics 2022, 9, 131 20 of 21 second order oscillation strategy is introduced into the algorithm. The strategy can not only maintain the diversity of the population, but also has strong exploration ability due to the oscillation convergence of the solution vector in the early iteration; The solution vector converges asymptotically in the later iteration and has strong exploitation ability. In order to balance the exploitation and exploration as a whole, the self-adaptive weight mechanism is introduced to adjust the tendency of individuals to approach the optimal solution and escape the worst solution and improve the search efficiency of the population and the exploitation. The introduction of mutation mechanisms ensures that individuals avoid getting stuck in local optima. In order to validate the behavior of CSOOJAYA, which is employed to the parameter identification problem of PV models. Finally, the experimental results present that CSOOJAYA delivers excellent behavior in the aspects of convergence, reliability and accuracy. The data of the PV cells used in this article are measured under certain temperature and light irradiance conditions. In order to further verify the applicability of CSOOJAYA algorithm to parameter identification under different PV cell or module operating scenarios. The next step of the research group will be to study the parameter identification of PV cells or components in multiple scenarios. Author Contributions: Conceptualization, X.J.; methodology, X.J.; software, Y.C.; validation, Y.C.; writing—original draft preparation, X.J.; writing—review and editing, Y.C.; supervision, X.J. All authors have read and agreed to the published version of the manuscript. Funding: This research was funded by the National Natural Science Foundation of China (NSFC), grant number 11774017. Institutional Review Board Statement: Not applicable. 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A Chaotic Second Order Oscillation JAYA Algorithm for Parameter Extraction of Photovoltaic Models

Jian, Xianzhong; Cao, Yougang
Photonics , Volume 9 (3) – Feb 25, 2022
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hv photonics Article A Chaotic Second Order Oscillation JAYA Algorithm for Parameter Extraction of Photovoltaic Models Xianzhong Jian * and Yougang Cao School of Optical Electrical and Computer Engineering, University of Shanghai for Science and Technology, Shanghai 200093, China; 203590775@st.usst.edu.cn * Correspondence: jianxz@usst.edu.cn Abstract: In order to identify the parameters of photovoltaic (PV) cells and modules more accurately, reliably and efficiently, a chaotic second order oscillation JAYA algorithm (CSOOJAYA) is proposed. Firstly, both logical chaotic map and the mutation strategy are brought in enhancing the population di- versity and improving exploitation. Secondly, in order to better balance exploration and exploitation, the second order oscillation factor is added, which not only improves the diversity of the population, but also has strong exploration at the beginning of the iteration and strong exploitation at the end of the iteration. In order to balance the two abilities, the self-adaptive weight is introduced into the CSOOJAYA algorithm to regulate individuals the tendency of moving toward the optimal solution and escaping from the worst solution, so as to enhance the search efficiency and exploitation. In order to validate the behavior of CSOOJAYA, it is employed to the parameter identification problem of PV models. Finally, the experimental results show that CSOOJAYA delivers excellent behavior in the aspects of convergence, reliability and accuracy. Keywords: CSOOJAYA algorithm; parameter identification; PV models; optimization algorithm 1. Introduction Citation: Jian, X.; Cao, Y. A Chaotic In order to alleviate the greenhouse effect and solve problems such as global warming, Second Order Oscillation JAYA the application of renewable energy has attracted much attention [1]. Among the renewable Algorithm for Parameter Extraction energy sources, solar energy is considered as one of the most promising renewable energy of Photovoltaic Models. Photonics source because of its green, clean, widely distributed and free energy [2]. Solar PV systems 2022, 9, 131. https://doi.org/ can directly convert solar energy into electrical energy, so it has become one of the most 10.3390/photonics9030131 widely used power generation technologies [3]. However, PV systems are susceptible to Received: 20 January 2022 external factors, especially irradiance and temperature [4]. PV arrays tend to degrade Accepted: 22 February 2022 over time under harsh environmental conditions, significantly affecting the behavior and Published: 25 February 2022 utilization efficiency of PV systems [5]. Therefore, in order to accurately control PV systems, it is significant to adopt the experimentally measured current-voltage (I-V) data to estimate Publisher’s Note: MDPI stays neutral the actual behavior of PV systems at work [6]. The single diode model (SDM) and the with regard to jurisdictional claims in double diode model (DDM) are widely employed and can accurately reflect the nonlinear published maps and institutional affil- behavior of PV panels [7]. The parameter accuracy of these models is essential for modeling, iations. control and optimization of PV panels [8]. However, these parameters are susceptible to various environmental factors and become unavailable [9]. PV models are implicitly nonlinear transcendental equations [10]. It is very significant and challenging to employ an Copyright: © 2022 by the authors. efficient and reliable methods to identify the PV parameters. Licensee MDPI, Basel, Switzerland. Many methods have been proposed to solve PV parameter identification. In principle, This article is an open access article these methods can be divided into analytical methods and optimization methods [11]. The distributed under the terms and analytical methods mainly employ several key data (short-circuit current, open-circuit conditions of the Creative Commons voltage, and maximum power) provided by the manufacturer, and use mathematical Attribution (CC BY) license (https:// equations to deduce the parameters [12]. The analytical methods have the advantages of creativecommons.org/licenses/by/ simple, fast and direct calculation. However, these data are obtained under standard test 4.0/). Photonics 2022, 9, 131. https://doi.org/10.3390/photonics9030131 https://www.mdpi.com/journal/photonics Photonics 2022, 9, 131 2 of 21 conditions [13]. Therefore, the extracted parameters above cannot precisely predict the I-V curves at different temperatures and solar irradiance [14]. Furthermore, this approach depends heavily on the key points of the selected I-V characteristics. If the crucial points are wrongly selected, the computational accuracy will be significantly reduced [11]. Instead of choosing a few key points, the optimization method takes into account all the actual measured current and voltage data. From the algorithm point of view, optimization methods can be divided into meta-heuristic methods and deterministic methods [15]. The optimization methods transform the PV parameter identification problem into an optimization problem, and then the parameters are identified according to all the reference points on the I-V characteristic curve. Deterministic methods, such as Newton–Raphson method [16,17], iterative curve fitting [18,19] and Lambert W-function [20,21], require the objective function to be continuous, convex, and differentiable [22]. The I-V curve of the PV cell model is non-linear, multi-peak and multi-mode. And the deterministic method is easily affected by the initial conditions and gradient information, so it is free to become trapped in a local optimum when resolving such complicated problems [23]. The heuristic methods have no strict requirements on the form of the optimization problem and are not affected by initial conditions and gradient information [24]. There- fore, many heuristic methods have been used to identify the parameters of PV models in the past ten years. These include the classified perturbation mutation based particle swarm optimization algorithm (CPMPSO) [1], efficient teaching-learning-based optimiza- tion algorithm (MTLBO) [16], multiswarm spiral leader particle swarm optimization algo- rithm (MSLPSO) [18], enhanced adaptive differential evolution algorithm (EJADE) [22], memetic adaptive differential evolution (MADE) [23], modified Rao-1 optimization algo- rithm (MRAO-1) [25], improved gaining-sharing knowledge algorithm (IGSK) [26] and chaos induced coyote algorithm (CICA) [27]. The characteristics of the three existing methods are summarized in Table 1. Table 1. The characteristics of the three methods. Method Characteristic Instance - Advantages such as simple, fast and direct calculation. - If the initial value is not selected well, it is necessary to repeatedly assign a new initial value for a new round Analytical method of solution, and the error in the solution process will also increase with the increase of the number of identification parameters, which requires a large amount of calculation and a long time. - Convert the PV parameter identification problem into an optimization problem - The objective function has characteristics such as Newton–Raphson method [16,17], Deterministic method continuity, convexity and differentiability. Easily iterative curve fitting [18,19] and affected by initial conditions and gradient information. Lambert W-function [20,21] - It is easy to get stuck in local optima when dealing with complex multimodal problems. - Overcome the shortcomings of the above two CPMPSO [1], MTLBO [16], methods, the solution speed is significantly improved, Heuristic method MSLPSO [18], EJADE [22], MADE and the accuracy is improved [23], MRAO-1 [25] - contains some control parameters Compared with deterministic methods, heuristic methods can acquire more precise and robust PV parameter identification results. However, PV models are nonlinear and multi-mode, so some heuristic methods may become trapped in local optima during the optimization process [8]. In addition to the basic parameters of population size and Photonics 2022, 9, 131 3 of 21 termination condition, most heuristic algorithms have specific parameters related to their own mechanisms. The behavior of the heuristic algorithm depends to some extent on these specific parameters. Choosing inappropriate parameters will not only increase the computational burden of the algorithm, but it may also converge prematurely or fall into a local optimum [28]. Rao proposed a new and efficient heuristic algorithm in 2016, namely the JAYA algorithm [29]. The structure of the JAYA algorithm is relatively simple, with only two parameters: population size and termination conditions [28]. Therefore, the advantage of the JAYA algorithm is that it can reduce the time of the optimization process and avoid the difficulties caused by parameter adjustment. However, the individual position is only affected by the current optimal individual and the worst individual in the JAYA algorithm, which cannot effectively maintain the diversity of the population [8]. Therefore, JAYA algorithm is free to trap in local optimum when solving complicated problems such as multi-peak and multi-mode. Although the performance of some improved JAYA algorithms such as the performance- guided JAYA algorithm or PGJAYA [8], comprehensive learning JAYA algorithm (CL- JAYA) [10], improved JAYA algorithm (IJAYA) [28], logistic chaotic JAYA algorithm (LC- JAYA) [29], and elite opposition-based JAYA algorithm (EO-JAYA) [30] have been improved to some extent, the imbalance between exploration and exploitation is not fully considered. The corresponding summary of the extraction parameters of existing heuristic methods is shown in Table 2. Table 2. Summary of some existing heuristic methods. Algorithm Summary - Introducing the classification perturbation mutation strategy to improve the local search CPMPSO [1] ability and population diversity of the algorithm. - The performance of each individual is quantified by probability. PGJAYA [8] An adaptive chaotic perturbation mechanism is introduced to improve the quality of the population. - Introducing a comprehensive learning mechanism to improve the global search ability of CLJAYA [10] the algorithm. According to the score level of each learner, each teaching stage is divided into three levels to MTLBO [16] update the population. - Combine multiple populations with different search mechanisms to maintain a balance between exploration and exploitation. MSLPSO [18] - Guided by several different spiral trajectories, the algorithm maintains the diversity of the population while exploring different regions of the multidimensional space. - Introduce crossover rate sorting mechanism to retain excellent individuals. EJADE [22] - Introduce a dynamic population reduction strategy to improve convergence speed and balance exploration and exploitation. - Adaptive differential evolution strategy based on success-history improves global search ability. MADE [23] - A ranking-based elimination strategy is proposed to retain promising individuals in the population. Photonics 2022, 9, 131 4 of 21 Table 2. Cont. Algorithm Summary MRAO-1 [25] - A bidirectional update strategy is proposed to improve population diversity. - Introduce an adaptive mechanism to automatically adjust the knowledge rate parameter. - A constraint handling method is proposed to improve the convergence speed and maintain IGSK [26] the balance between exploration and exploitation. - Automatically select a binding processing strategy from three algorithm pools to improve the performance of the algorithm. - Chaos map can solve problems such as easy to fall into local optimum and slow algorithm CICA [27] convergence speed. - Experience-based learning strategies maintain the diversity of the population and improve the exploration ability of the population. IJAYA [28] - A chaotic elite learning method is proposed to improve the quality of the best solution in each generation. LCJAYA [29] - Introduces a chaotic mutation strategy to improve and balance exploration and exploitation. EO-JAYA [30] - Introducing an elite opposition learning strategy to increase population diversity. In order to improve the performance of JAYA algorithm, this paper proposes a chaotic second order oscillation JAYA algorithm (CSOOJAYA), which can identify parameters more accurately and stably. Firstly, the introduction of logical chaotic mapping mechanism improves the population diversity and exploration. Secondly, the introduction of the second order oscillation mechanism not only improves the diversity of the population, but also has strong exploration ability due to the oscillation convergence of the solution vector in the early iteration. The solution vector converges asymptotically in the later iteration and has strong exploitation ability. The self-adaptive weight mechanism is introduced to adjust the tendency of individuals to approach the optimal solution and escape the worst solution, improve the search efficiency of the population and the exploitation. The algorithm improves and balances the global search ability and local optimization ability as a whole. The introduction of mutation mechanisms ensures that individuals avoid getting stuck in local optima. The structure of CSOOJAYA is similar to the JAYA algorithm, and it only has two parameters: population size and termination condition. In order to verify the effectiveness of the proposed CSOOJAYA algorithm, CSOOJAYA and other novel algorithms are used to extract the parameters of three PV models. The experimental results show that CSOOJAYA has the best performance in terms of identification accuracy, stability and convergence speed. 2. PV Models and Objective Function 2.1. Formula of SDM Figure 1 presents the circuit diagram of the SDM. Obviously, the framework of the SDM is relatively simple. There are photogenerated current I , output current I , diode ph L current I , and shunt current I in the circuit. According to Kirchhoff current law, I can d sh L be get by Equation (1). According to the diode Shockley equation and Ohm law, I can be get by Equation (2), and I can be obtained by Equation (3): sh I = I I I (1) L ph d sh q(V + R I ) L S L I = I  exp 1 (2) d sd nkT Photonics 2022, 9, 131 5 of 21 V + R I L S L I = (3) sh sh q(V + R I ) V + R I L S L L S L I = I I  exp 1 (4) L ph d nkT R sh Figure 1. The equivalent circuit of SDM. R refers to the series resistance, R stands for the parallel resistance, V stands for S sh L the output voltage, n is the ideality factor of the diode. k refers to the Boltzmann constant 23 19 (1.3806503  10 J/K), q stands for the elementary charge (1.60217646  10 C), and T refers to the absolute temperature. Therefore, we can substitute Equations (2) and (3) into Equation (1) to get Equation (4). Obviously, SDM needs to identify five unknown different parameters (I , I , R , R , n). ph d S sh 2.2. Formula of DDM Taking into account the inherent drawbacks of SDM, DDM can more accurately illustrate the relationship between voltage and current. Figure 2 presents the circuit diagram of the DDM. It can be seen from the Figure 2, DDM differs from SDM in that it has one more diode in parallel with the current source than SDM. I can be obtained by Equation (5): q(V + R I ) q(V + R I ) V + R I L S L L S L L S L I = I I  exp 1 I  exp 1 (5) ph d1 d2 n kT n kT R 1 2 sh Figure 2. The equivalent circuit of DDM. I refers to the diode diffusion current, and I stands for the saturation current. n d1 d2 1 and n represent the ideal saturation factor of two diodes, respectively. Obviously, the DDM needs to extract seven unknown different parameters (I , I , I , R , R , n , n ). ph d1 d2 S sh 1 2 Photonics 2022, 9, 131 6 of 21 2.3. Formula of PV Module Figure 3 presents the circuit diagram of a PV module, which consists of several PV cells in parallel or in series. I can be obtained by Equation (6): q V /N + R I /N N V /N + R I ( ) L S S L P P L S S L I = N I N I  exp 1 (6) L P ph P d nkT R sh Figure 3. The equivalent circuit of a PV module. N and N refer to the number of serial and parallel connections of PV cells, respec- S P tively. This paper adopts the SDM of PV modules. Therefore, the PV module needs to identify five unknown different parameters (I , I , R , R , n). ph d S sh 2.4. Objective Function An objective function is determined to quantitatively evaluate the difference between the actual measured value and the calculated value. Equations (7)–(9) represent the error functions for the experimental and calculated data points of the SDM, DDM, and PV module, respectively: 8 h   i q(V +R I ) V +R I L S L L S L f (V , I , X) = I I + I  exp 1 + SD L L L ph d nkT R sh (7) n o X = I , I , R , R , n ph d S sh 8 h   i q(V +R I ) L S L > f (V , I , X) = I I + I  exp 1 DD L L L ph d1 > n kT < h   i q(V +R I ) V +R I L S L L S L +I  exp 1 + (8) d2 n kT R 2 sh > n o X = I , I , I , R , R , n , n ph d1 d2 S sh 1 2 h   i q(V /N +R I /N ) L S S L P f (V , I , X) = I N I + N I  exp 1 MD L L L P P > ph d nkT N V /N +R I P L S S L + (9) sh n o X = I , I , R , R , n ph d S sh RMSE X = f (V , I , X) (10) ( ) L L å k k=1 Photonics 2022, 9, 131 7 of 21 However, the error function cannot reflect all the differences between the calculated and experimental data. The root mean square error (RMSE) defined by Equation (10) is used as the objective function to quantitatively evaluate the overall difference between the experimental and calculated data. References like [10,11,14] generally use the objective function. X represents the solution composed of different unknown parameters, and N represents the number of actual measured data. 3. JAYA Algorithm JAYA is a new population-based intelligent optimization algorithm [29]. In each generation of the JAYA algorithm, the solution vector is optimized by approaching the optimal solution while staying away from the worst solution. JAYA does not need to tune algorithm-specific parameters. The only two parameters are population size and maximum number of evaluation functions (max ). FES For the objective function with d unknown variables (j = 1,2,3...d), x represent the i,j value of the j-th variable of the i-th unknown solution, then X = (x , x ...... x ) represents i i,1 i,2 i,d the i-th unknown solution. If an individual can get the worst/best value of f (X) among all individuals, then this is the worst/best individual, expressed as x = (x , x , . . . best best,1 best,2 x )/x = (x , x ,... x ). Each solution x of the JAYA algorithm can be best,d worst worst,1 worst,2 worst,d i,j updated by Equation (11): x = x + rand  x x rand  x x (11) newi,j i,j 1 best,j i,j 2 worst,j i,j x represents the absolute value of x , and x represents the updated value of i,j i,j newi,j x . rand and rand represent random numbers in [0,1]. The second and third terms in i,j 1 2 the above Equation (11) represent the tendency of the solution vector x to move towards the optimal solution and escape from the worst solution, respectively. In order to retain a better solution vector, the updated solution x = (x , x , . . . , x ) is received if newi newi,1 newi,2 newi,d it can provide a better function value. 4. Chaotic Second Order Oscillation JAYA Algorithm 4.1. Motivation At present, the identification of PV parameters is considered to be a multi-peak and multi-modal problem, that is, there are multiple local optimal solutions. When the JAYA algorithm deals with such problems, it easily falls into a local optimum because it cannot effectively explore different regions of the space. How to effectively enhance the behavior of the algorithm while balancing the exploitation and exploration? Therefore, a chaotic second order oscillation JAYA (CSOOJAYA) algorithm is proposed in this research. 4.2. Logistic Chaotic Map Strategy In the optimization process of the JAYA algorithm, it is hard to use two uniformly distributed random numbers to maintain the population diversity. However, the num- ber of chaotic sequences generated by the logical chaotic mapping strategy is aperiodic and does not converge to a specific value. It enhances the population diversity of the heuristic algorithm, improves the global search ability, and avoids falling into the local optimum. Therefore, two random numbers are replaced by chaotic sequence numbers, and Equation (12) represents the new update solution: x = x + C (x x ) C (x x ) (12) newi,j i,j 1i,j best,j i,j 2i,j worst,j i,j where C and C represent two chaotic sequence numbers, which are generated by 1i,j 2i,j Equation (13): C = 4 C (1 C ) (13) n+1 n n Photonics 2022, 9, 131 8 of 21 C represents the chaotic value of the n-th iteration. Its value range is [0,1], and C is n 1 usually set to 0.8. 4.3. Second Order Oscillation Strategy According to the update equation of the JAYA algorithm, When the i-th solution is the best/worst solution, one term on the right side of Equation (11) is 0, which cannot maintain the diversity of the population. There are only two factors that affect the population position of the JAYA algorithm, namely the current optimal individual and the worst individual. Although the algorithm can speed up the convergence speed and improve the exploitation, it cannot effectively enhance the diversity of the population and weaken the exploration ability in the optimization process. Considering the above two situations, the population individuals may fall into a local optimum in the JAYA algorithm. It can be seen from Equations (11) and (12) that x and best x have a significant impact on the performance of JAYA. Therefore, taking advantages worst of population information and enhancing exploration capabilities are the available meth- ods to improve the behavior of the JAYA. How to balance exploitation and exploration while improving population diversity. Therefore, the second order oscillation factor is introduced into Equation (12), that is, two unequal second order oscillation factors are added respectively to the two rightmost terms of Equation (12), as shown in Equation (14). The difference between the current optimal individual and the optimal individual of the previous generation and the difference between the current worst individual and the worst individual of the previous generation can guide the individual to update a better range, maintain population diversity and improve exploration ability. After adopting the second order oscillation strategy, because the solution vector has oscillation convergence in the early iteration, the algorithm has strong global search ability; The solution vector converges asymptotically in the later stage of iteration, and the algorithm has strong local optimization ability. This strategy improves the convergence accuracy of the algorithm as a whole: x = x + C  (1 + k1)x k1x x C ((1 newi,j i,j 1i,j best,j bestp,j i,j 2i,j (14) +k2) x k2x x worst,j worstp,j i,j where x and x represent the optimal solution and the worst solution of the jth bestp,j worstp,j variable of the previous generation of population individuals, respectively. k1 and k2 are two randomly generated numbers in [0,1], and k1 is not equal to k2. 4.4. Self-Adaptive Weight Equation (12) enhances the population diversity and exploration, and Equation (14) not only improves exploitation and exploration, but also effectively balances the two abilities of the algorithm. However, how to balance the two abilities as a whole is the core problem of the algorithm. Considering that the strategy is dedicated to improving exploitation ability, the self-adaptive weight strategy is introduced into Equation (14) to obtain Equation (15). w is obtained from Equation (16). It can be seen from Equation (16) that the value of w increases gradually with the increase of the number of function evaluations. Because the difference between the optimal solution and the worst solution becomes smaller and smaller as the search process progresses. Therefore, in the optimization process of the JAYA algorithm, the population can approach the promising solution at the beginning of the iteration and conduct local optimization in the area of the promising solution at the end of the iteration. Thus, it enhances the search efficiency of the population and the exploitation of the algorithm: Photonics 2022, 9, 131 9 of 21 x = x + C  (1 + k1)x k1x x wC ((1 newi,j i,j 1i,j best,j bestp,j i,j 2i,j (15) +k2) x k2x x worst,j worstp,j i,j f (x ) best ( ) , if f(x ) 6= 0 worst f (x ) w = worst (16) 1 , otherwise f (x ) and f (x ) represent the values of the optimal and worst individual of the worst best objective function, respectively. Furthermore, the self-adaptive weight is automatically determined, so there is no need to adjust parameters. 4.5. Mutation Mechanism The three position update equations based on the JAYA algorithm not only improve the overall exploration and exploitation, but also effectively balance the two abilities. However, the optimal individual may still have a very small probability of being in a local optimal state to solve the identification of PV parameters. In order to make the individual jump out of the local optimum, the mutation mechanism is proposed. If from the first function evaluation to a quarter of the maximum function evaluation time, and so on, if the RMSE values of the two are equal, the population individual is reinitialized. And if the best RMSE of the previous generation is better than the current optimal RMSE, the RMSE of the previous generation is accepted. 4.6. Framework of CSOOJAYA The overall update equation of the improved algorithm based on the above strategy is shown in Equation (17). p is the probability, which represents random numbers in [0,1]: x + C  (1 + k1)x k1x x > i,j 1i,j best,j bestp,j i,j C  (1 + k1)x k2x x , > 2i,j worst,j worstp,j i,j i f 0  p < 1/2 x + C  (1 + k1)x k1x x i,j 1i,j best,j bestp,j i,j x = (17) newi,j C w (1 + k2)x k2x x , 2i,j worst,j worstp,j i,j > i f  p < 3/4 x + C  x x C  x x , > i,j 1i,j best,j i,j 2i,j worst,j i,j > 3 i f  p  1 The pseudocode of the CSOOJAYA algorithm can be summarized by the Algorithm 1. In addition, Figure 4 shows the flow chart of the specific implementation of CSOO- JAYA. Similar to the JAYA algorithm, the CSOOJAYA does not need to adjust the specific algorithm parameters. Photonics 2022, 9, 131 10 of 21 Algorithm 1: CSOOJAYA algorithm. 1. Set the population size (NP) and max ; FES 2. Randomly initialize the population and RMSE values of all individuals are calculated; 3. FES = NP; 4. While FES < max do FES 5. Identify the best x and worst x of all individuals; i i 6. For i = 1 to NP do 7. if (0  p < 1/2) then 8. Update the x using Equation (14); 9. else if (1/2  p < 3/4) then 10. Calculate the w using Equation (15); 11. Update the x using Equation (16) 12. else 13. Update the x using Equation (12) 14. End if 15. Obtain the new population and Calculate the RMSE for the updated individual x ; newi 16. Receive the updated solution if it is more excellent than the pre-update one 17. End For 18. If the conditions of the mutation strategy are met 19. Randomly initialize the population and Save the current optimal solution and RMSE 20. End if 21. End While Figure 4. The flowchart of CSOOJAYA. 5. Experimental Results and Discussion In order to estimate the effective behavior of CSOOJAYA, it was applied to the identifi- cation of PV parameters. The benchmark data are extracted from reference [11]. where the Photonics 2022, 9, 131 11 of 21 data for SDM and DDM were obtained from a commercial RTC silicon PV cell (1000 w/m at 33 C, Photowatt, Bourgoin-Jallieu, France) with a diameter of 57 mm, and the data for the PV module was obtained from a Photowatt-PWP201 (1000 w/m at 45 C) solar module consisting of 36 polycrystalline silicon cells connected in series (Photowatt, Bourgoin-Jallieu, France). In order to ensure fairness, the search space of the solution vector is the same, and the value range of the identification parameter is the same as that adopted in the previous literature. Table 3 shows the value ranges of the parameters corresponding to different PV models. Table 3. Parameters ranges of PV models. SDM/DDM PV Module Parameter Lower Upper Lower Upper 0 1 0 2 0 1 0 50 0 0.5 0 2 0 100 0 2000 1 2 1 50 CSOOJAYA is compared with other novel algorithms to verify its superior perfor- mance. These algorithms are DE/WOA, MLBSA, EJADE, MTLBO, MPPCEDE, MADE, ITLBO, CPMPSO, DERAO, JAYA, IJAYA, LCJAYA and PGJAYA. Because these optimization algorithms have better performance in PV model parameter identification, they are used for comparison with CSOOJAYA. Table 4 presents the corresponding parameter settings for the selected algorithms, which are extracted from their respective references. For fair comparison, set max to 50,000 for all algorithms and run 30 times independently. The FES results of DE/WOA [11], MLBSA [5], EJADE [22], MTLBO [16], MPPCEDE [15], MADE [23], ITLBO [2], CPMPSO [1], DERAO [25], JAYA [31], IJAYA [28], LCJAYA [29] and PGJAYA [8] were obtained directly from the corresponding references. The selected algorithms are all executed in MATLAB R2016b on a PC configured with an Intel(I) Core (TM) i5-7200u CPU operating at 3.1 GHz and equipped with 4 Gb of RAM. Table 4. Parameter settings of the compared algorithms. Algorithm Parameters DE/WOA Np = 40, F = rand (0.1,1), Cr = rand (0,1) MLBSA Np = 50 EJADE Np = 50, Np = 4 max min MTLBO Np = 50 MPPCEDE Np = 40; F = rand (0.1,1); Cr = rand (0,1) MADE NP = 20; 0.05 for the SDM and PV module; 0.01 for the DDM IJAYA NP = 20 ITLBO Np = 50 CPMPSO NP = 50, w = 0.729, c1 = 1.49455, c2 = 1.49455 DERAO NP = 20 Photonics 2022, 9, 131 12 of 21 Table 4. Cont. Algorithm Parameters JAYA NP = 20 PGJAYA NP = 20 LCJAYA NP = 20 CSOOJAYA NP = 20 5.1. Results on the SDM For SDM, Table 5 illustrates the best parameters and RMSE obtained for all selected algorithms. The overall best RMSE values of all compared algorithms are indicated in bold. It is obvious from Table 5 that CSOOJAYA, DE/WOA, MLBSA, EJADE, MTLBO, MPPCEDE, MADE, ITLBO, CPMPSO, DERAO, LCJAYA and PGJAYA obtained the smallest RMSE (9.8602  10 ), followed by IJAYA. Although accurate parameter value informa- tion cannot be obtained, it can be seen from the literature [1,2] that the RMSE can be employed to represent the accuracy. Furthermore, the smaller the RMSE, the more precise the identified parameters. Table 5. Comparison of the extracted optimal parameters of the SDM. Algorithm I (A) I (A) R (W) R (W) n RMSE ph d S sh DE/WOA 0.760776 0.323021 0.036377 53.718524 1.481184 9.860219  10 MLBSA 0.7608 0.32302 0.0364 53.7185 1.4812 9.8602  10 EJADE 0.7608 0.3230 0.0364 53.7185 1.4812 9 8602  10 MTLBO 0.76077553 0.32300 0.03637709 53.7185251 1.48118359 9 860219  10 MPPCEDE 0.7608 0.32302 0.0364 53.7185 1.4812 9.8602  10 MADE 0.7608 0.3230 0.0364 53.7185 1.4812 9.8602  10 IJAYA 0.7608 0.3228 0.0364 53.7595 1.4811 9.8603  10 ITLBO 0.7608 0.3230 0.0364 53.7185 1.4812 9.8602  10 CPMPSO 0.760776 0.323021 0.036377 53.718522 1.481184 9.860219  10 DERAO 0.760776 0.323021 0.036377 53.718522 1.481135 9.860219  10 JAYA 0.7608 0.3281 0.0364 54.9298 1.4828 9.8946  10 PGJAYA 0.7608 0.3230 0.0364 53.7185 1.4812 9.8602  10 LCJAYA 0.7608 0.3230 0.0364 53.7185 1.4819 9.8602  10 CSOOJAYA 0.760776 0.323021 0.036377 53.718525 1.481184 9.860219  10 To further confirm the performance of CSOOJAYA, the IAE of power and current between the measured data and the calculated data was used. The IAE can be current calculated by Equation (18), and the IAE can be obtained by Equation (19): power IAE = jI I j (18) current measured estimated IAE = jV  I V  I j (19) power measured estimated where I represents the output current estimated by different models through estimated Equations (4), (5) or (6), and I represents the actual measured current. V repre- measured sents the actual measured voltage. IAE represents the individual absolute error of current current, and IAE represents the individual error of power. power The IAE of power and current are shown in Table 6. All values of IAE are less current 3 3 than 2.50  10 and all values of IAE are less than 1.46  10 , proving the accuracy power of extracted parameters. To further verify the accuracy of the results, Figure 5 illustrates the best results obtained using CSOOJAYA to plot the I-V and P-V curves. Obviously, the data calculated by CSOOJAYA are in good agreement with the measured data across the entire voltage range. Photonics 2022, 9, 131 13 of 21 Table 6. IAE of CSOOJAYA for the SDM. Item V /V I /A I /A IAE P /W IAE measured measured estimated current estimated power 1 0.2057 0.7640 0.7640881747 0.0000881747 0.1571729375 0.0000181375 2 0.1291 0.7620 0.7626635570 0.0006635570 0.0984598652 0.0000856652 3 0.0588 0.7605 0.7613557780 0.0008557780 0.0447677197 0.0000503197 4 0.0057 0.7605 0.7601544618 0.0003455381 0.0043328804 0.0000019695 5 0.0646 0.7600 0.7590556797 0.0009443202 0.0490349969 0.0000610030 6 0.1185 0.7590 0.7580428161 0.0009571838 0.0898280737 0.0001134262 7 0.1678 0.7570 0.7570921248 0.0000921248 0.1270400585 0.0000154585 8 0.2132 0.7570 0.7561418358 0.0008581641 0.1612094393 0.0001829606 9 0.2545 0.7555 0.7550873439 0.0004126560 0.1921697290 0.0001050209 10 0.2924 0.7540 0.7536643499 0.0003356500 0.2203714559 0.0000981440 11 0.3269 0.7505 0.7513914392 0.0008914392 0.2456298615 0.0002914115 12 0.3585 0.7465 0.7473543260 0.0008543260 0.2679265258 0.0003062758 13 0.3873 0.7385 0.7401176997 0.0016176997 0.2866475851 0.0006265351 14 0.4137 0.7280 0.7273827075 0.0006172924 0.3009182261 0.0002553738 15 0.4373 0.7065 0.7069731390 0.0004731390 0.3091593537 0.0002069037 16 0.4590 0.6755 0.6752806425 0.0002193574 0.3099538149 0.0001006850 17 0.4784 0.6320 0.6307587591 0.0012412408 0.3017549903 0.0005938096 18 0.4960 0.5730 0.5719288231 0.0010711768 0.2836766963 0.0005313036 19 0.5119 0.4990 0.4996074317 0.0006074317 0.2557490442 0.0003109442 20 0.5265 0.4130 0.4136491106 0.0006491106 0.2177862567 0.0003417567 21 0.5398 0.3165 0.3175102788 0.0010102788 0.1713920485 0.0005453485 22 0.5521 0.2120 0.2121548945 0.0001548945 0.1171307172 0.0000855172 23 0.5633 0.1035 0.1022509879 0.0012490120 0.0575979815 0.0007035684 24 0.5736 0.0100 0.0087182129 0.0012817870 0.0050007669 0.0007352330 25 0.5833 0.1230 0.1255085017 0.0025085017 0.0732091090 0.0014632090 26 0.5900 0.2100 0.2084737660 0.0015262339 0.1229995219 0.0009004780 Figure 5. Experimental and calculated data of CSOOJAYA for SDM. 5.2. Results on the DDM For DDM, the extraction of seven parameters increases the difficulty of parameter identification. Table 7 illustrates the best parameters and RMSE obtained for all selected algorithms. The overall best RMSE value of all compared algorithms are indicated in bold. Photonics 2022, 9, 131 14 of 21 It is obvious from Table 7 that CSOOJAYA, DE/WOA, EJADE, MTLBO, ITLBO, CPMPSO, DERAO have the smallest RMSE (9.8248  10 ), followed by MLBSA and MPPCEDE. To further verify the accuracy of the results, Figure 6 illustrates the best results obtained using CSOOJAYA to plot the I-V and P-V curves. Obviously, the data calculated by CSOOJAYA are in good agreement with the measured data across the entire voltage range. Table 7. Comparison of the extracted optimal parameters of the DDM. Algorithm I (A) I (A) I (A) R (W) R (W) n1 n2 RMSE ph d1 d2 S sh DE/WOA 0.760781 0.225974 0.749346 0.036740 55.485437 1.451017 2 9.824849  10 MLBSA 0.7608 0.22728 0.73835 0.0367 55.4612 1.4515 2 9.8249  10 EJADE 0.7608 0.2260 0.7493 0.0367 55.4854 1.4510 2 9.8248  10 MTLBO 0.7607810 0.22597 0.7493 0.03674043 55.485447 1.451016 2 9.824849  10 MPPCEDE 0.7608 0.22728 0.73835 0.0367 55.4612 1.4515 2 9.82485  10 MADE 0.7608 0.2246 0.7394 0.0368 55.4329 1.4505 1.9963 9.8261  10 IJAYA 0.7601 0.0050445 0.75094 0.0376 77.8519 1.2186 1.6247 9.8293  10 ITLBO 0.7608 0.2260 0.7493 0.0367 55.4854 1.4510 2 9.8248  10 CPMPSO 0.76078 0.22597 0.74935 0.03674 55.48544 1.45102 2 9.824849  10 DERAO 0.760781 0.225959 0.749667 0.036740 55.486045 1.450963 2 9.824849  10 JAYA 0.7607 0.0060763 0.31507 0.0364 52.6575 1.4788 1.8436 9.8934  10 PGJAYA 0.7608 0.21031 0.88534 0.0368 55.8135 1.4450 2 9.8263  10 LCJAYA 0.7608 0.22596 0.74640 0.0367 55.4815 1.4518 2.0000 9.8250  10 CSOOJAYA 0.760781 0.225974 0.749345 0.036740 55.485437 1.451017 2.000000 9.824849  10 Figure 6. Experimental and calculated data of CSOOJAYA for DDM. The IAE of power and current are shown in Table 8. All values of IAE are less current 3 3 than 2.54  10 and all values of IAE are less than 1.48  10 , which demonstrates power the accuracy of CSOOJAYA parameter identification. 5.3. Results on the PV Module The effectiveness of CSOOJAYA is further verified using the PhotoWatt-PWP201 PV module. Table 9 illustrates the best parameters and RMSE obtained for all selected algorithms. The overall best RMSE value of all the compared algorithms are marked in bold. Among all the comparison algorithms, CSOOJAYA, DE/WOA, MLBSA, EJADE, MTLBO, MPPCEDE, MADE, ITLBO, CPMPSO, PGJAYA, IJAYA, LCJAYA and DERAO obtained the best RMSE value, followed by JAYA. To further verify the accuracy of the results, Figure 7 illustrates the best results obtained using CSOOJAYA to plot the I-V and P-V curves. Obviously, the data calculated by CSOOJAYA are in good agreement with the measured data across the entire voltage range. The IAE of power and current are shown in Table 10. All values of IAE are less than 4.83  10 and all values of current IAE are less than 7.99  10 , which verifies the accuracy of CSOOJAYA parameter power identification again. Photonics 2022, 9, 131 15 of 21 Table 8. IAE of CSOOJAYA for the DDM. Item V /V I /A I /A IAE P /W IAE measured measured estimated current estimated power 1 0.2057 0.7640 0.7639834118 0.0000165881 0.1571513878 0.0000034121 2 0.1291 0.7620 0.7626040957 0.0006040957 0.0984521887 0.0000779887 3 0.0588 0.7605 0.7613376980 0.0008376980 0.0447666566 0.0000492566 4 0.0057 0.7605 0.7601737877 0.0003262122 0.0043329905 0.0000018594 5 0.0646 0.7600 0.7591076801 0.0008923198 0.0490383561 0.0000576438 6 0.1185 0.7590 0.7581214198 0.0008785801 0.0898373882 0.0001041117 7 0.1678 0.7570 0.7571886136 0.0001886136 0.1270562493 0.0000316493 8 0.2132 0.7570 0.7562436068 0.0007563931 0.1612311369 0.0001612630 9 0.2545 0.7555 0.7551773015 0.0003226984 0.1921926232 0.0000821267 10 0.2924 0.7540 0.7537223535 0.0002776464 0.2203884161 0.0000811838 11 0.3269 0.7505 0.7513991342 0.0008991342 0.2456323769 0.0002939269 12 0.3585 0.7465 0.7473014432 0.0008014432 0.2679075674 0.0002873174 13 0.3873 0.7385 0.7400106607 0.0015106607 0.2866061288 0.0005850788 14 0.4137 0.7280 0.7272469525 0.0007530474 0.3008620642 0.0003115357 15 0.4373 0.7065 0.7068502977 0.0003502977 0.3091056352 0.0001531852 16 0.4590 0.6755 0.6752105425 0.0002894574 0.3099216390 0.0001328609 17 0.4784 0.6320 0.6307607576 0.0012392423 0.3017559464 0.0005928535 18 0.4960 0.5730 0.5719947329 0.0010052670 0.2837093875 0.0004986124 19 0.5119 0.4990 0.4997061352 0.0007061352 0.2557995706 0.0003614706 20 0.5265 0.4130 0.4137336728 0.0007336728 0.2178307787 0.0003862787 21 0.5398 0.3165 0.3175462057 0.0010462057 0.1714114418 0.0005647418 22 0.5521 0.2120 0.2121229960 0.0001229960 0.1171131061 0.0000679061 23 0.5633 0.1035 0.1021632765 0.0013367234 0.0575485736 0.0007529763 24 0.5736 0.0100 0.0087917509 0.0012082490 0.0050429483 0.0006930516 25 0.5833 0.1230 0.1255434350 0.0025434350 0.0732294856 0.0014835856 26 0.5900 0.2100 0.2083715891 0.0016284108 0.1229392376 0.0009607623 Table 9. Comparison of the extracted optimal parameters of the PV module. Algorithm I (A) I (A) R (W) R (W) n RMSE ph d S sh DE/WOA 1.030514 3.482263 1.201271 981.982143 48.642835 2.425075  10 MLBSA 1.0305 3.4823 1.2013 981.9823 48.6428 2.4251  10 EJADE 1.0305 3.4823 1.2013 981.9824 48.6428 2.4251  10 MTLBO 1.0305143 3.4823 1.2012710 981.9823732 48.6428349 2425075  10 MPPCEDE 1.030514 3.482263 1.201271 981.982143 48.642835 2.42507  10 MADE 1.0305 3.4823 1.2013 981.9823 48.6428 2.4250  10 IJAYA 1.0305 3.4703 1.2016 977.3752 48.6298 2.4251  10 ITLBO 1.0305 3.4823 1.2013 981.9823 48.6428 2.4251  10 CPMPSO 1.030514 3.4823 1.201271 981.9823 48.64284 2.425075  10 DERAO 1.030514 3.4823 1.201271 981.9821 48.64131 2.425075  10 JAYA 1.0302 3.4931 1.2014 1022.5 48.6531 2.427785  10 PGJAYA 1.0305 3.4818 1.2013 981.8545 48.6424 2.425075  10 LCJAYA 1.0305 3.4823 1.2024 981.9828 48.6684 2.425075  10 CSOOJAYA 1.030514 3.482263 1.201271 981.982243 48.642834 2.425075  10 Photonics 2022, 9, 131 16 of 21 Figure 7. Experimental and calculated data of CSOOJAYA for PV module. Table 10. IAE of CSOOJAYA for PV module. Item V /V I /A I /A IAE P /W IAE measured measured estimated current estimated power 1 0.1248 1.0315 1.0291188622 0.0023811377 0.1284340340 0.0002971659 2 1.8093 1.0300 1.0273807740 0.0026192259 1.8588400345 0.0047389654 3 3.3511 1.0260 1.0257414978 0.0002585021 3.4373623333 0.0008662666 4 4.7622 1.0220 1.0241068556 0.0021068556 4.8770016680 0.0100332680 5 6.0538 1.0180 1.0222915054 0.0042915054 6.1887483154 0.0259799154 6 7.2364 1.0155 1.0199303816 0.0044303816 7.3806242138 0.0320600138 7 8.3189 1.0140 1.0163628063 0.0023628063 8.4550205496 0.0196559496 8 9.3097 1.0100 1.0104958517 0.0004958517 9.4074132312 0.0046162312 9 10.2163 1.0035 1.0006286698 0.0028713301 10.2227226794 0.0293343705 10 11.0449 0.9880 0.9845480779 0.0034519220 10.8742350664 0.0381261335 11 11.8018 0.9630 0.9595213746 0.0034786253 11.3240793592 0.0410540407 12 12.4929 0.9255 0.9228385152 0.0026614847 11.5289292866 0.0332496633 13 13.1231 0.8725 0.8725993581 0.0000993581 11.4512086365 0.0013038865 14 13.6983 0.8075 0.8072739566 0.0002260433 11.0582808396 0.0030964103 15 14.2221 0.7265 0.7283361680 0.0018361680 10.3584698161 0.0261141661 16 14.6995 0.6345 0.6371376868 0.0026376868 9.3656054279 0.0387726779 17 15.1346 0.5345 0.5362127464 0.0017127464 8.1153654320 0.0259217320 18 15.5311 0.4275 0.4295110044 0.0020110044 6.6707783610 0.0312331110 19 15.8929 0.3185 0.3187741584 0.0002741584 5.0662458227 0.0043571727 20 16.2229 0.2085 0.2073891784 0.0011108215 3.3644539035 0.0180207464 21 16.5241 0.1010 0.0961668397 0.0048331602 1.5890704768 0.0798636231 22 16.7987 0.0080 0.0083257217 0.0003257217 0.1398613011 0.0054717011 23 17.0499 0.1110 0.1109368217 0.0000631782 1.8914617173 0.0010771826 24 17.2793 0.2090 0.2092476082 0.0002476082 3.6156521970 0.0042784970 25 17.4885 0.3030 0.3008639322 0.0021360677 5.2616588799 0.0373566200 5.4. Statistical Results and Convergence Curve The above three sections demonstrate the superior accuracy of CSOOJAYA in solving parameter identification on different PV models. However, convergence and robustness should also be considered. Therefore, this section compares the convergence curves and statistical results of the three PV models. Photonics 2022, 9, 131 17 of 21 The maximum RMSE (Max), minimum RMSE(Min), average RMSE (Mean) and stan- dard deviation (SD) of RMSE obtained by running all the comparison algorithms 30 times independently on three different PV models are shown in Table 11. The overall best RMSE values for all compared algorithms are marked in bold. where Max represents the worst accuracy, Min is the best accuracy, and Mean is the average accuracy, and SD represents the stability of algorithm. It can be seen from the Table 11: (1) For the best RMES value, only CSOOJAYA, DE/WOA, EJADE, MTLBO, ITLBO, CPMPSO and DERAO get the best RMSE values for all PV models. (2) For the worst RMSE value, only CSOOJAYA and DE/WOA can obtain the optimal worst value of all PV models (3) For the average RMSE value, only CSOOJAYA can obtain the optimal average value for all PV models. (4) Taking into account the SD, it can reflect the reliability of the algorithm. Although the SD of MTLBO, MPPCEDE, ITLBO, CPMPSO and DERAO are slightly better than those of the CSOOJAYA algorithm in the SDM and the PV module, the SD of CSOOJAYA is far superior to other algorithms in the DDM. On the whole, the CSOOJAYA algorithm has stronger reliability than other algorithms. Based on the superior stability and accuracy of three PV models, CSOOJAYA can obtain the best parameter identification results. Table 11. Comparison of statistical results for PV models. Model Algorithm RMSE Min Max Mean SD 4 4 4 17 SDM DE/WOA 9.860219  10 9.860219  10 9.860219  10 3.545178  10 4 4 4 12 MLBSA 9.8602  10 9.8602  10 9.8602  10 9.1461  10 4 4 4 17 EJADE 9.8602  10 9.8602  10 9.8602  10 5.13  10 4 4 4 17 MTLBO 9.860219  10 9.860219  10 9.860219  10 1.9092749  10 4 4 4 17 MPPCEDE 9.86022  10 9.86022  10 9.86022  10 2.44932  10 4 4 4 15 MADE 9.8602  10 9.8602  10 9.8602  10 2.74  10 4 3 4 5 IJAYA 9.8603  10 1.0622  10 9.9204  10 1.4033  10 4 4 4 17 ITLBO 9.8602  10 9.8602  10 9.8602  10 2.19  10 4 4 4 17 CPMPSO 2.17556  10 9.86022  10 9.86022  10 9.86022  10 4 4 4 17 DERAO 9.860219  10 9.860219  10 9.860219  10 3.642031  10 4 3 3 4 JAYA 9.8946  10 1.4783  10 1.1617  10 1.8796  10 4 4 4 9 PGJAYA 9.8602  10 9.8603  10 9.8602  10 1.4485  10 4 4 4 16 LCJAYA 9.8602  10 9.8602  10 9.8602  10 5.6997  10 4 4 4 17 CSOOJAYA 9.860219  10 9.860219  10 9.860219  10 4.717305  10 4 4 4 7 DDM DE/WOA 9.824849  10 9.860377  10 9.829703  10 9.152178  10 4 4 4 6 MLBSA 9.8249  10 9.8518  10 9.8798  10 1.3482  10 4 4 4 6 EJADE 9.8248  10 9.8602  10 9.8363  10 1.36  10 4 4 4 9 MTLBO 9.824849  10 9.825026  10 9.824855  10 3.30000  10 4 4 4 8 MPPCEDE 9.82485  10 9.82908  10 9.82504  10 8.02951  10 4 4 4 5 MADE 9.8261  10 9.8786  10 9.8608  10 8.02  10 4 3 3 5 IJAYA 9.8293  10 1.4055  10 1.0269  10 9.8325  10 Photonics 2022, 9, 131 18 of 21 Table 11. Cont. Model Algorithm RMSE Min Max Mean SD 4 4 4 6 ITLBO 9.8248  10 9.8812  10 9.8497  10 1.54  10 4 4 4 6 CPMPSO 9.82485  10 9.83137  10 9.86022  10 1.3398  10 4 4 4 9 DERAO 9.824841  10 9.824897  10 9.824845  10 1.01560  10 4 3 3 4 JAYA 9.8934  10 1.4793  10 1.1767  10 1.9356  10 4 4 4 6 PGJAYA 9.8263  10 9.9499  10 9.8582  10 2.5375  10 4 4 4 6 LCJAYA 9.8250  10 9.8602  10 9.8308  10 1.3118  10 4 4 4 17 CSOOJAYA 5.576332  10 9.824849  10 9.824849  10 9.824849  10 3 3 3 8 PV module DE/WOA 2.425075  10 2.425442  10 2.425092  10 6.270718  10 3 3 3 8 MLBSA 2.425075  10 2.425084  10 2.425312  10 4.336794  10 3 3 3 17 EJADE 2.4251  10 2.4251  10 2.4251  10 3.27  10 3 3 3 17 MTLBO 2.425075  10 2.425075  10 2.425075  10 1.3070107  10 3 3 3 17 MPPCEDE 2.42507  10 2.42507  10 2.42507  10 2.23156  10 3 3 3 17 MADE 3.07  10 2.4250  10 2.4251  10 2.4251  10 3 3 3 6 IJAYA 2.4251  10 2.4393  10 2.4289  10 3.7755  10 3 3 3 17 ITLBO 2.4251  10 2.4251  10 2.4251  10 1.27  10 3 3 3 17 CPMPSO 2.425075  10 2.425075  10 2.425075  10 1.515959  10 3 3 3 17 DERAO 2.425075  10 2.425075  10 2.425075  10 1.716113  10 3 3 3 5 JAYA 2.427785  10 2.595873  10 2.453710  10 3.456290  10 3 3 3 7 PGJAYA 2.425075  10 2.426764  10 2.425144  10 3.071420  10 3 3 3 16 LCJAYA 2.425075  10 2.425075  10 2.425075  10 2.415229  10 3 3 3 17 CSOOJAYA 2.425075  10 2.425075  10 2.425075  10 2.699858  10 In order to validate the convergence speed of CSOOJAYA, Figures 8–10 plot the iterative curves of several algorithms on the PV models. Obviously, CSOOJAYA not only get the best results among these three models, but also converges faster than other algorithms. Figure 8. Convergence curves of the seven algorithms on SDM. Photonics 2022, 9, 131 19 of 21 Figure 9. Convergence curves of the seven algorithms on DDM. Figure 10. Convergence curves of the seven algorithms on PV. In summary, the above comparison shows that CSOOJAYA has faster convergence, better accuracy and robustness in extracting the parameters of PV models. Furthermore, the behavior of CSOOJAYA is competitive when compared to other algorithms. 6. Conclusions This paper proposes a chaotic second order oscillation JAYA algorithm to provide an accurate and stable way to identify the PV parameters. The algorithm introduces the logical chaotic map strategy, and the generated random sequence numbers are non- periodic and non-convergent, which can enhance the diversity of the population and improve the exploration. In order to improve and balance the above two abilities, the Photonics 2022, 9, 131 20 of 21 second order oscillation strategy is introduced into the algorithm. The strategy can not only maintain the diversity of the population, but also has strong exploration ability due to the oscillation convergence of the solution vector in the early iteration; The solution vector converges asymptotically in the later iteration and has strong exploitation ability. In order to balance the exploitation and exploration as a whole, the self-adaptive weight mechanism is introduced to adjust the tendency of individuals to approach the optimal solution and escape the worst solution and improve the search efficiency of the population and the exploitation. The introduction of mutation mechanisms ensures that individuals avoid getting stuck in local optima. In order to validate the behavior of CSOOJAYA, which is employed to the parameter identification problem of PV models. Finally, the experimental results present that CSOOJAYA delivers excellent behavior in the aspects of convergence, reliability and accuracy. The data of the PV cells used in this article are measured under certain temperature and light irradiance conditions. In order to further verify the applicability of CSOOJAYA algorithm to parameter identification under different PV cell or module operating scenarios. The next step of the research group will be to study the parameter identification of PV cells or components in multiple scenarios. Author Contributions: Conceptualization, X.J.; methodology, X.J.; software, Y.C.; validation, Y.C.; writing—original draft preparation, X.J.; writing—review and editing, Y.C.; supervision, X.J. All authors have read and agreed to the published version of the manuscript. Funding: This research was funded by the National Natural Science Foundation of China (NSFC), grant number 11774017. Institutional Review Board Statement: Not applicable. 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Journal

Photonics – Multidisciplinary Digital Publishing Institute

Published: Feb 25, 2022

Keywords: CSOOJAYA algorithm; parameter identification; PV models; optimization algorithm

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