An Improved Cheetah Optimizer for Accurate and Reliable Estimation of Unknown Parameters in Photovoltaic Cell and Module Models

: Solar photovoltaic systems are becoming increasingly popular due to their outstanding environmental, economic, and technical characteristics. To simulate, manage, and control photovoltaic (PV) systems, the primary challenge is identifying unknown parameters accurately and reliably as early as possible using a robust optimization algorithm. This paper proposes a newly developed cheetah optimizer (CO) and improved CO (ICO) to extract parameters from various PV models. This algorithm, inspired by cheetah hunting behavior, includes several basic strategies: searching, sitting, waiting, and attacking. Although this algorithm has shown remarkable capabilities in solving large-scale problems, it needs improvement concerning its convergence speed and computing time. Here, an improved CO (ICO) is presented to identify solar power model parameters for this purpose. The ICO algorithm’s search phase is controlled based on the leader’s position. The step length is adjusted following the sorted population. As a result of this updated operator, the algorithm can perform global and local searches. Furthermore, the interaction factor during the attack phase is adjusted based on the position of the prey, and a random value controls the turning factor. Single-, double-, and PV module models are investigated to test the ICO’s parameter estimation performance. Statistical analysis uses the minimum, mean, maximum, and standard deviation. Furthermore, to improve conﬁdence in the test results, Wilcoxon and Freidman rank nonparametric tests are also performed. Compared with other state-of-the-art optimization algorithms, the CO and ICO algorithms are proven to be highly reliable and accurate when identifying PV parameters. According to the results, the ICO and CO obtained the ﬁrst-and second-best sum ranking results for the studied PV models among 12 applied algorithms. Despite this, the ICO algorithm reduces the CO’s computation time by 40% on average. Additionally, ICO’s convergence speed is high, reaching an optimal solution in less than 25,000 function evaluations in most cases.


Introduction 1.Motivation and Incitement
A growing number of solar photovoltaic systems are being integrated with electric utilities due to their outstanding environmental, economic, and technical characteristics [1].The availability of solar radiation in most regions of the world makes solar energy generation and storage systems an attractive option for customers looking for a quick and efficient method for upgrading their electrical systems.In PV systems, solar energy is converted into electricity.In addition to solar radiation and temperature, several other factors affect the capacity of solar energy to generate electricity.As a result, it is essential to analyze how Appl.Sci.2023, 13, 9997 2 of 31 PV systems perform in real time so that they are capable of being optimized, managed, and modeled [2].The single-diode model (SDM), double-diode model (DDM), and PV module model (PVMM) are typically used despite the existence of many mathematical models for PV nonlinearity.These models must include parameters that can change with environmental changes, faults, and aging [3].Thus, regardless of the model used, it is essential to accurately determine unknown parameters as early as possible by using a robust optimization algorithm.Therefore, developing an optimization algorithm capable of accurately estimating the properties of PV models using the current-voltage measurements of the PV cell and module is imperative [4].
An optimization problem can be established to extract PV cell and module parameters, which involves the formulation of an objective function and the establishment of a set of constraints.There is noise in the measured current-voltage data.There are, therefore, several local optima in the search space, resulting in a nonlinear and multimodal search space [5,6].Deterministic and metaheuristic algorithms are commonly used to solve this challenging optimization problem.The former method makes use of gradient information as well as the initial points.As a result, classical techniques are ineffective in identifying the parameters of photovoltaic models due to their nonlinear and non-convex nature [7][8][9].There is a consensus that metaheuristic algorithms are more modern and easier to use than deterministic algorithms.Since then, there has been an increase in interest in metaheuristic algorithms for optimizing PV systems more efficiently and flexibly.

Contribution and Paper Organization
Although researchers are developing and modifying meta-heuristic algorithms in light of the "No Free Lunch" theorem [31] to determine the parameters of PV models, according to the authors' knowledge, past algorithms have not provided a satisfactory balance between accuracy and reliability while maintaining a reasonable computing time.To improve the performance of metaheuristic algorithms, new ideas must be developed to produce simple and efficient methods for dealing with practical optimization problems.
To optimize, manage, and model PV systems, it is necessary to analyze their real-time performance.However, the SDM, DDM, and PVMM are the most employed mathematical models for PV nonlinearity in the literature.Thus, no matter what model is used, it is essential to implement a robust optimization algorithm that can determine unknown parameters as early as possible and with as much accuracy as possible.Considering this, it is imperative to develop an optimization algorithm capable of accurately estimating PV models' properties based on the current-voltage measurements of the PV cell and module.Many methods are currently available in the literature for identifying the unknown parameters of PV cell and module models.However, few of them combine low complexity and computational cost with high accuracy and reliability.Innovative approaches must be taken to develop methods that optimize accuracy, computational cost, and reliability to achieve this.
Recently, Akbari et al. [32] introduced a new and powerful algorithm, namely the CO algorithm, which is inspired by the behavior of cheetahs during the hunting process.The CO has demonstrated excellent performance when used to solve standard test functions at various scales.However, it is necessary to test this algorithm on different optimization problems.This is so that its strengths are recognized and weaknesses are identified and resolved.This article uses this algorithm for the first time to identify PV parameters.Based on the authors' experiences, this algorithm requires some modifications and simplifications to be easily applied to various real-world optimization problems.
The purpose of this article is to introduce a simplified and improved version of the CO, namely the ICO, that can improve the features of the CO while requiring less computational effort.As part of the ICO algorithm, the search phase is controlled according to the leader's position, and its step length is also adjusted following the sorted population.This updated operator also aids the algorithm's global and local search.In addition, the interaction factor in the attack phase is adjusted based on the prey's position, and a random value controls the turning factor.It is believed that the proposed attack operator will improve the behavior of the algorithm in the global search as well as its convergence speed.When it comes to estimating the optimal parameters for PV cells and models, the CO and proposed ICO are compared to two recent well-established algorithms for parameter extraction of PV models (i.e., PGJAYA [6] and SEDE [14]) and eight well-known original algorithms: DE [33], PSO [34], GA [35], TLBO [36], JAYA [37], SSA [38], WSO [13], and GWO [39].
This paper contributes the following: • A simplified and improved version of the CO is introduced for parameter estimation of PV models.

•
The search phase is controlled according to the leader's position, and its step length is adjusted following the sorted population.Hence, the proposed search strategy facilitates both global and local search capabilities.

•
In the attack phase, the interaction factor is adjusted based on the prey's position, while the turning factor is determined based on a random value.Using the proposed attack operator, the algorithm is expected to perform better during global searches and achieve faster convergence.

•
An extensive study is conducted and compared with other well-established metaheuristic algorithms to evaluate the ICO's effectiveness.
The following is a summary of an overview of the remainder of the paper.In Section 2, we describe in detail the SDM, DDM, and PVMM.The proposed ICO algorithm is presented in Section 3. A simulation and evaluation of the results of the experiment are presented in Section 4. Finally, Section 5 makes some closing remarks.

PV Modeling and Problem Formulation
In the literature, many PV models have been presented to describe the characteristics of solar cells and PV module models.Among these models are the SDM, DDM, and PVMM.This section describes the mathematical model used to formulate the optimization problem of determining the optimal parameters for these models.
2.1.The Model of a Solar Cell 2.1.1.SDM For demonstrating the real-time characteristics of PV systems, their mathematical modeling is required under practical considerations.A PV array can be modeled by using the cell as its basic unit.SDMs are widely used due to their simplicity and ease of implementation.According to Figure 1a, the equivalent circuit for the SDM consists of a parallel resistor, a series resistor, a diode, and a current source.Calculating the output current can be accomplished using the following formula [40]: where I p , I sh , and I D are the photogenerated, shunt resistor, and diode currents, respectively.Calculating I sh and I D can be accomplished using Kirchhoff's voltage law (KVL) and Shockley's equation as follows: where u represents the non-physical diode ideality factor, whereas I represents the diode reverse saturation current, V o represents the cell output voltage, R sh represents the shunt resistance, and R s represents the series resistance.

DDM
Although it is widely employed to simulate PV cells, the SDM ignores the recombination current in the depletion region.As shown in Figure 1b, by combining the photogenerated current source, the shunt resistance, two rectifying diodes, and the series resistance, the DDM can solve the problem.
Using KCL, one can calculate the output current in a DDM as follows: The junction thermal voltage can be calculated using the electron charge q (1.60217646 × 10 −19 C) the junction temperature T, and Boltzmann's constant k (1.8865033 × 10 −23 J/K) as follows: Combining Equations ( 1) and (4) will result in the cell output current (I o ) for the SDM as follows:

DDM
Although it is widely employed to simulate PV cells, the SDM ignores the recombination current in the depletion region.As shown in Figure 1b, by combining the photogenerated current source, the shunt resistance, two rectifying diodes, and the series resistance, the DDM can solve the problem.Using KCL, one can calculate the output current in a DDM as follows: Current flows through the first and second diodes (i.e., I D1 and I D2 , respectively) as described by the Shockley diode equations in Equations ( 7) and (8).Diodes also have two ideality factors known as u 1 and u 2 .The diffusion and saturation currents are I 1 and I 2 , respectively.Thus, by substituting Equations ( 3), (7), and ( 8), Equation ( 6) can be rewritten as follows:

PVMM
A photovoltaic module may be designed to increase the voltage and current by arranging several PV cells in parallel or series (see Figure 1c).Using the PVMM, the output current can be calculated as follows: Here, a parallel arrangement consists of M solar cells, and a series arrangement consists of N solar cells.

Problem Formulation
The goal of the proposed optimization problem is to determine the unknown parameters of PV cells and modules accurately.An optimization algorithm is commonly employed to minimize the differences between the estimated and experimental I-V data obtained from the PV systems.Hence, as a rule, it is common to consider that minimization of the root mean square error (RMSE) is an objective function that should be considered when determining an estimate of the current: subject to x i,min ≤ x i ≤ x i,max ; ∀i = 1, 2, . . . 5 (SDM and PVMM); ∀i = 1, 2, . . ., 7 (DDM) (12)    14) Here, S is the amount of experimental paired sample data, while Îo,s and I o,s are the sth measured sample and the determined value of the PV output current, respectively.The constraints in Equation ( 12) indicate the upper (x i,max ) and lower (x i,min ) bounds on the PV parameters (decision variables).For the SDM and PVMM, the five unknown parameters are x = I p , I, R sh , R s , u , and the seven decision variables (i.e., x = I p , I 1 , I 2 , R sh , R s , u 1 , u 2 ) should be defined for the DDM using an optimization technique.Finally, the calculated PV output current in each sample s, I o,s , can be expressed using Equations ( 13)- (15) for the SDM, DDM, and PVMM, respectively.

Overview of the CO Algorithm
Akbari et al. [32] recently developed the CO algorithm as a powerful optimization algorithm for mimicking specific cheetahs' hunting strategies.This algorithm utilizes three important strategies: searching for prey, sitting and waiting, and attacking.The algorithm introduces leaving the prey and returning home to avoid getting stuck in local optimal points.In this section, the mathematical model of the CO algorithm is explained, and then the ICO algorithm is presented.
Based on these strategies, as shown in Figure 2, cheetah populations are formed in different arrangements.The probable hunting arrangements of each cheetah are considered equivalent to the solution to the problem.It is assumed that the best position among the population determines the prey (best solution).Cheetahs adjust their possible arrangements to optimize their performance during the hunting period.the PV parameters (decision variables).For the SDM and PVMM, the five unknown parameters are  = [ , ,  ,  , ] , and the seven decision variables (i.e.,  = [ ,  ,  ,  ,  ,  ,  ]) should be defined for the DDM using an optimization technique.
Finally, the calculated PV output current in each sample s,  , , can be expressed using Equations ( 13)- (15) for the SDM, DDM, and PVMM, respectively.

Overview of the CO Algorithm
Akbari et al. [32] recently developed the CO algorithm as a powerful optimization algorithm for mimicking specific cheetahs' hunting strategies.This algorithm utilizes three important strategies: searching for prey, sitting and waiting, and attacking.The algorithm introduces leaving the prey and returning home to avoid getting stuck in local optimal points.In this section, the mathematical model of the CO algorithm is explained, and then the ICO algorithm is presented.
Based on these strategies, as shown in Figure 2, cheetah populations are formed in different arrangements.The probable hunting arrangements of each cheetah are considered equivalent to the solution to the problem.It is assumed that the best position among the population determines the prey (best solution).Cheetahs adjust their possible arrangements to optimize their performance during the hunting period.

Searching Strategy
A cheetah scans its surroundings or searches for suitable prey based on environmental conditions and hunting behavior.A mathematical model's searching phase looks like this [32]: where  , represents the current arrangement and  , represents the new arrangement of cheetah  at hunting time .The inverse of a normally distributed random number ̂ , represents the randomization parameter.Aside from that, the random step length is defined by  , , which is expressed for the leader as follows [32]:

Searching Strategy
A cheetah scans its surroundings or searches for suitable prey based on environmental conditions and hunting behavior.A mathematical model's searching phase looks like this [32]: where X t i,j represents the current arrangement and X t+1 i,j represents the new arrangement of cheetah i at hunting time t.The inverse of a normally distributed random number ri,j represents the randomization parameter.Aside from that, the random step length is defined by α t i,j , which is expressed for the leader as follows [32]: where U j and L j are the upper and lower limits of the variable j, respectively.The length of a hunting time is represented by T. For other members of a group of cheetahs, the random step length is expressed based on the distance of the cheetah i and an arbitrarily selected cheetah k in a group as follows [32]: 3.1.2.Sitting-and-Waiting Strategy Cheetahs are swift hunters.During the chase, speed and flexibility require much energy.Therefore, the duration of the attack and chase cannot be long.As a result, one of the important strategies of cheetahs during the hunting process is to wait until the prey is close enough to them.Then, they start the attack.Hunting success can be increased by this behavior, which is modeled as follows [32]:

Attacking Strategy
At the appropriate time, cheetahs attack their prey.Speed and flexibility are two critical factors that the cheetah exploits during its attack.Cheetahs attack with maximum speed since the cheetah must reach a close distance from their prey in the shortest possible time.In this case, the prey notices the cheetah's attack and starts to run away.Because of the cheetah's high speed and short distance from the prey, the prey prefers to escape by changing directions suddenly.Therefore, the cheetah uses its high flexibility to place the prey in unstable conditions and catch it.Attacks may take place individually or in groups.In solo attack mode, the cheetah's position change is adjusted based on the position of the prey.This can be carried out interactively in a group attack based on the status of other members of the group and the prey.This strategy can be expressed using the following mathematical model [32]: ři,j = |r i,j | exp(r i,j /2) sin 2πr i,j where X t B,j is the prey position; ři,j is the turning factor which reflects the sudden changes of the prey while fleeing; and r i,j is a randomly chosen value from a normal distribution.The interaction factor is defined by β t i,j in Equation (20), which is expressed as follows [32]:

Strategy Selection Mechanism
Choosing the right strategy in the CO algorithm is performed randomly [32].Let r 2 and r 3 be two random numbers from a uniform distribution.If r 2 is greater than r 3 , then the sit-and-wait strategy is selected; otherwise, one of the search or attack strategies takes place.There is a condition between the two strategies of search and attack, which is controlled based on the H factor (see Figure 3).This factor decreases over time, which is expressed as follows [32]: where r 1 is a random value in the range [0, 1].A condition has been set between these two strategies so that searching is the most likely choice at the start of hunting season.An attack will likely occur as the time of hunting progresses.
= e 2(1−/) (2 1 − 1) where  1 is a random value in the range [0, 1].A condition has been set between these two strategies so that searching is the most likely choice at the start of hunting season.An attack will likely occur as the time of hunting progresses.The pseudo-code of the CO is summarized in Algorithm 1 [32].
Algorithm  itial population of cheetahs   ( = 1, 2, … , ), and evaluate the fitness of each cheetah opulation's home, leader, and prey solutions  do ≤  ≤ ) members of cheetahs randomly mber  ∈  do eighbor agent of member  rbitrary arrangement  ∈ {1, 2, … , } do te , , , , and  the strategy selection mechanism to generate a new arrangement according to Equation ( 16), (20) e solutions of member  and the leader ave-the-prey-and-go-back-home strategy The pseudo-code of the CO is summarized in Algorithm 1 [32].The pseudo-code of the CO is summarized in Algorithm 1 [32].For each member  ∈  do 8: Define the neighbor agent of member  9: For each arbitrary arrangement  ∈ {1, 2, … , } do 10: Calculate , , , , and

11:
Apply the strategy selection mechanism to generate a new arrangement according to Equation ( 16), (19), or (20) 12: End 13: Update the solutions of member  and the leader 14: End 15: Apply the leave-the-prey-and-go-back-home strategy 16: ←  + 1

Improved Cheetah Optimizer (ICO) Algorithm
The CO algorithm has shown good capabilities in solving large-scale problems.However, as we will show in the experimental results, it needs improvement in terms of convergence speed and computing time in identifying the parameters of photovoltaic models.For this purpose, a modified version of the CO algorithm is presented to cover these shortcomings.

Improved Cheetah Optimizer (ICO) Algorithm
The CO algorithm has shown good capabilities in solving large-scale problems.However, as we will show in the experimental results, it needs improvement in terms of convergence speed and computing time in identifying the parameters of photovoltaic models.For this purpose, a modified version of the CO algorithm is presented to cover these shortcomings.

Searching Strategy
In the search mode of the CO algorithm, each cheetah updates its position based on its previous position.This is when cheetahs usually follow the leader of the group.On this basis, the searching strategy in Equation ( 16) is modified based on the leader's position (second-best cheetah's position in the population) X t L,j as follows: where the randomization parameter (r t ) and random step length (α t i,j ) are modified as follows: Here, r and r are random values of the normal distribution function, and X t k ,j and X t i ,j are the positions of the kth and ith cheetahs in the sorted population, respectively.It is worth noting that updating the position of each cheetah around the position of the group leader can help the local search phase.In addition, the second term on the right side of the relationship in Equation ( 24) causes diversity in the solutions and thus contributes to the global search phase (exploitation phase).Also, by creating long steps during the hunting period, the random parameter will cause the solution to extend out of the range of variables and thus be replaced with the new random solution in the population.Consequently, in addition to diversifying the solutions, it can prevent the algorithm from getting stuck in local optimal points.

Attacking Strategy
Moreover, the attacking strategy in the ICO algorithm is reformulated as follows: where řt is a random value in the range [0, 1].
In the CO algorithm, the interaction factor is expressed using the position of the adjacent cheetah (see Equation ( 22)).Cheetahs usually attack their prey singly.Therefore, their positions should be adjusted based on the position of the prey.Hence, in this proposed attack strategy, each cheetah updates his or her position relative to the prey during the attack mode and moves toward it, which is defined as follows: This proposed attack strategy helps the CO algorithm to find the near-optimal solution faster.Therefore, the local search capability (exploitation phase) of the CO algorithm is enhanced, and its convergence speed will be increased.

Experimental Results
The CO and ICO algorithms are evaluated in this section to show their performance for the parameter estimation of three types of PV models: SDM, DDM, and PVMM.The SDM and DDM tests were conducted on silicon solar cells with 57 mm diameters (RTC France) to collect current-voltage data [41].Moreover, under 1000 W/m2 irradiance, a PV module (Photo Watt-PWP 201) with 36 polycrystalline PV cells was used [41].Experimental data were used to estimate the parameters of the PV models using a variety of algorithms.The maximum and minimum limits for each parameter of the PV model are given in Table 2 [16].
Moreover, two recently developed algorithms, PGJAYA [6] and SEDE [14], as well as eight well-known original algorithms (i.e., DE [33], PSO [34], GA [35], TLBO [36], JAYA [37], SSA [38], WSO [13], and GWO [39]) were chosen to validate and verify the effectiveness of the CO and ICO for identifying the PV parameters.A maximum number of 50,000 function evaluations was assumed for all case studies.As in the original literature, the other parameters of the applied algorithms were maintained.The statistical analysis was performed by running each algorithm 30 times independently in MATLAB 2021b.

Experimental Results
The CO and ICO algorithms are evaluated in this section to show their performance for the parameter estimation of three types of PV models: SDM, DDM, and PVMM.The SDM and DDM tests were conducted on silicon solar cells with 57 mm diameters (RTC France) to collect current-voltage data [41].Moreover, under 1000 W/m2 irradiance, a PV module (Photo Watt-PWP 201) with 36 polycrystalline PV cells was used [41].Experimental data were used to estimate the parameters of the PV models using a variety of algorithms.The maximum and minimum limits for each parameter of the PV model are given in Table 2 [16].Moreover, two recently developed algorithms, PGJAYA [6] and SEDE [14], as well as eight well-known original algorithms (i.e., DE [33], PSO [34], GA [35], TLBO [36], JAYA [37], SSA [38], WSO [13], and GWO [39]) were chosen to validate and verify the effectiveness of the CO and ICO for identifying the PV parameters.A maximum number of 50,000 function evaluations was assumed for all case studies.As in the original literature, the other parameters of the applied algorithms were maintained.The statistical analysis was performed by running each algorithm 30 times independently in MATLAB 2021b.

Population Size Analysis
One of the parameters influencing the performance of any evolutionary algorithm is the size of the initial population.Therefore, the behavior of the proposed algorithm for optimal extraction of the parameters of the three PV models was investigated with population sizes (n) of 10, 20, 40, 50, 80, and 100.For each of these population sizes, the ICO was run 30 times, and the statistical results are summarized in Table 3.As can be seen, for the SD and PVM models, the algorithm could achieve the best solution (Min value) for all population sizes.For the DD model, the algorithm reached the best solution in 9.824849 × 10 −4 with n = 80.In addition, the proposed algorithm showed significant robustness with all initial populations, except for the population of 10.The CPU times and Friedman test results through 30 runs are represented in the last three columns of Table 3, and their average values for the three models are shown in Figure 4. Based on these results, it can be seen that the proposed algorithm with n = 80 had the best relative performance in the three models, with an average sum rank of 80 and a CPU time of 39.5 s.The population sizes of 40 and 50 ranked second and third among all examined population sizes, respectively.In addition, the convergence characteristics of the algorithm with different population numbers are shown in Figure 5 for three models.When the population number was set above 10, almost the same convergence behavior was seen.However, for the first model, when the population was 40, the speed of convergence was almost better.When the populations of 80 and 100 were considered, the speed of convergence in the second model was the best.For the third model, all populations except 100 almost converged on the same point.For the population of 10, the speed of convergence in the SDM and DDM showed the worst situation among the tested populations, while for the third model, it showed a significant convergence behavior.
set above 10, almost the same convergence behavior was seen.However, for the first model, when the population was 40, the speed of convergence was almost better.When the populations of 80 and 100 were considered, the speed of convergence in the second model was the best.For the third model, all populations except 100 almost converged on the same point.For the population of 10, the speed of convergence in the SDM and DDM showed the worst situation among the tested populations, while for the third model, it showed a significant convergence behavior.

Results of Parameter Extraction Based on the SDM
For the SDM, the best solutions found by competitive algorithms with n = 40 and n = 80, including the PV parameters and objective function (RMSE) values, are summarized in Table 4. From the results, for the two tested population sizes, the best RMSE value of 9.860218778914 × 10 −4 was obtained from the CO and ICO.For n = 40, SEDE and WSO (and for n = 80, WSO) gave the second-best solutions.It should be noted that the lower values of the RMSE indicate a higher accuracy of the estimation of the model parameters.The curves of the current and power in terms of voltage are illustrated in Figure 6a,b to verify the accuracy of the algorithm.Additionally, the values of IAEI and IAEP are drawn in Figure 6c,d over the voltage ranges.In all cases, the individual absolute error of current (IAEI) was less than 2.52 × 10 −3 , and the individual absolute error of power (AIEP) was less than 1.375 × 10 −3 , indicating that the CO and ICO were highly accurate in estimating the SDM parameters.

Results of Parameter Extraction Based on the DDM
The best solutions found by competitive algorithms for the DDMs with n = 40 and n = 80, including the PV parameters and optimal RMSE values, are represented in Table 5.From the table, it can be seen that the CO obtained the best RMSE value of 9.824848822723 × 10 −4 for n = 40, followed by the ICO with an RMSE of 9.824860991382 × 10 −4 .Additionally, for n = 80, these optimizers obtained the best results out of the 12 algorithms.Conversely, SSA and GWO produced the worst results.Figure 7a,b illustrates the I-V and P-V curves, respectively, using the measured and estimated data for the DDM model.The corresponding IAEI and IAEP are illustrated in Figure 7c,d

Results of Parameter Extraction Based on the DDM
The best solutions found by competitive algorithms for the DDMs with n = 40 and n = 80, including the PV parameters and optimal RMSE values, are represented in Table 5.
From the table, it can be seen that the CO obtained the best RMSE value of 9.824848822723 × 10 -4 for n = 40, followed by the ICO with an RMSE of 9.824860991382 × 10 -4 .Additionally, for n = 80, these optimizers obtained the best results out of the 12 algorithms.Conversely, SSA and GWO produced the worst results.Figure 7a,b illustrates the I-V and P-V curves, respectively, using the measured and estimated data for the DDM model.The

PVMM-Based Photo Watt-PWP 201
For the PVMM, it can be observed from Table 6 that the best solutions were obtained from CO, ICO, SEDE, and WSO.However, in terms of the RMSE, the best result was related to the CO, and then the ICO, WSO, and SEDE were placed in the following ranks.Among the competitive algorithms, for n = 40 and n = 80, the CO, ICO, and WSO showed stable behavior in finding the best optimal solution.In Figure 8a,b, the I-V and P-V characteristics of the measured data were very similar to those obtained by the ICO and CO.It can be seen that the IAEI and IAEP in this example were less than 0.0048 and 0.0798, respectively (see Figure 8c,d).The results of this study demonstrate the high accuracy of the estimated parameters under the CO and ICO for the PVMM.

Comparison of Statistical Results
For a clearer understanding of the comparison, the statistical results were also saved.We recorded the maximum (Max), the minimum (Min), the mean (Mean), and the standard deviation (SD) of the RMSE over 30 independent runs for each algorithm.By comparing the Min, Mean, and SD values of the RMSE, one can measure the accuracy, the average accuracy, and the robustness of the applied algorithms.Tables 7-9 present the statistical results for the 12 algorithms with n = 40 and n = 80 through 30 runs to identify the unknown parameters of the three PV models.A bold value indicates the algorithm that produced the best results.A Freidman test was used to determine the performance ranking of the comparative algorithms.In the Freidman test, the smallest value of the mean or sum rank indicates that the applied algorithm was superior to the other 12 algorithms.To measure the significance between the ICO and its competitors, the Wilcoxon signed rank test was used with a significance level of 0.05.The symbols of "+" and "≈" in Tables 7-9 0 2 4 6 8 10 12 14

Comparison of Statistical Results
For a clearer understanding of the comparison, the statistical results were also saved.We recorded the maximum (Max), the minimum (Min), the mean (Mean), and the standard deviation (SD) of the RMSE over 30 independent runs for each algorithm.By comparing the Min, Mean, and SD values of the RMSE, one can measure the accuracy, the average accuracy, and the robustness of the applied algorithms.Tables 7-9 present the statistical results for the 12 algorithms with n = 40 and n = 80 through 30 runs to identify the unknown parameters of the three PV models.A bold value indicates the algorithm that produced the best results.A Freidman test was used to determine the performance ranking of the comparative algorithms.In the Freidman test, the smallest value of the mean or sum rank indicates that the applied algorithm was superior to the other 12 algorithms.To measure the significance between the ICO and its competitors, the Wilcoxon signed rank test was used with a significance level of 0.05.The symbols of "+" and "≈" in Tables 7-9 indicate that the ICO's performance was significantly superior or almost similar to its competitor, respectively.For the SDM, as shown in Table 7, when n = 40, the ICO, CO, and SEDE gave the best and average accuracy results in terms of the Min and Mean values.However, in terms of robustness, the ICO with an SD value of 3.091 × 10 −17 showed the best performance among the competitive algorithms.PGJAYA and WSO showed the second-and thirdbest accuracies, respectively.According to the Friedman test, the CO showed the best performance, and the ICO had the second-best performance among the 12 algorithms.Aside from that, when n = 80, the ICO and CO showed the best accuracy, and WSO showed the second-best accuracy.However, in terms of reliability, the ICO and CO with SD values of 5.21 × 10 −17 and 1.02 × 10 −16 were the best and second-best among the competitive algorithms, respectively.Based on the Wilcoxon signed rank test, there was no significant difference between the ICO and CO, while they obtained significantly superior results compared with the other competitive algorithms.
When it comes to the DDM, as represented in Table 8, the CO resulted in the best accuracy, and the ICO obtained the best average accuracy among the tested algorithms.The ICO's accuracy in terms of the minimum value of the RMSE was very similar to the result of the CO, and other algorithms were unable to approach it.The ICO and CO rank first and second in terms of robustness, respectively.In addition, when the population size was set to 40, SEDE provided the best performance, and the ICO provided the second-best performance based on Friedman's test.When n = 80, however, the ICO was determined to be the best-performing algorithm, while the CO was determined to be the second-best-performing algorithm among the 12 competing algorithms.Furthermore, when the population size was 40, the Wilcoxon signed rank test did not indicate a significant difference between the ICO, CO, SEDE, and PGJAYA.Moreover, based on the Wilcoxon tests with n = 80, the ICO, CO, and PGJAYA performed similarly.
The best results were from utilizing the ICO, CO, and SEDE for the PVMM when looking at the Min and Mean RMSE of Table 9.Despite WSO's ability to achieve the highest accuracy, its average accuracy and robustness could not compete with the ICO, CO, and SEDE.The lowest SD value was achieved by SEDE (2.319 × 10 −17 ), and the second-and third-best values were obtained by the ICO and CO (4.998 × 10 −17 and 1.105 × 10 −16 , respectively).According to Friedman's test, the ICO provided the best performance, and the CO provides the second-best performance when n = 40.For n = 80, this ranking was shifted.The final ranking of the comparative algorithms for identifying the unknown parameters of the SDM, DDM, and PVMM is shown in Figure 9.For these models, the best sum rank result among the 12 algorithms with n = 40 was obtained by the ICO, followed by the CO and SEDE.While n = 80, the CO, ICO, and PGJAYA exhibited the first, second, and third sum rank results in the three models, respectively.Additionally, Figure 10 shows a box plot diagram of all competitive algorithms for a visual representation of the distribution of optimal RMSEs obtained for the three investigated models over 30 runs.Based on the distribution of answers, it is clear that the ICO and CO performed the best in terms of robustness in finding the optimal solution.SEDE and PGJAYA also provided acceptable robustness.
Additionally, Figure 10 shows a box plot diagram of all competitive algorithms for a visual representation of the distribution of optimal RMSEs obtained for the three investigated models over 30 runs.Based on the distribution of answers, it is clear that the ICO and CO performed the best in terms of robustness in finding the optimal solution.SEDE and PGJAYA also provided acceptable robustness.

Computational Time
To further evaluate the performance of the competing algorithms, we recorded the computing times for 30 runs of each algorithm with the three models and presented them in Figure 11.As can be seen from this figure, different times were spent to identify the parameters of each model of the algorithm.Among the 12 algorithms, the GAs took the

Computational Time
To further evaluate the performance of the competing algorithms, we recorded the computing times for 30 runs of each algorithm with the three models and presented them in Figure 11.As can be seen from this figure, different times were spent to identify the parameters of each model of the algorithm.Among the 12 algorithms, the GAs took the longest time to solve the three models, while JAYA took the least amount of time for the SDM and DDM.Aside from that, SSA required the least computational time to solve the PVMM, followed by GWO, JAYA, and the ICO.For the SDM, after JAYA, the ICO required the least computing time.The DDM, PSO, GWO, WSO, and SSA had almost the same computing times, and the ICO needed a little more time than them.Compared with the original algorithms such as JAYA, GWO, WSO, and SSA, the time spent by the ICO was comparable.Its superior performance over these algorithms, however, is significant from a statistical perspective.In addition, although the CO, SEDE, and PGJAYA showed significant performance in terms of statistical results, they required more computational time than the ICO.Compared with the CO, the main advantage of the ICO is its ability to reduce the computing time while maintaining or even improving its performance.
in Figure 11.As can be seen from this figure, different times were sp parameters of each model of the algorithm.Among the 12 algorithms longest time to solve the three models, while JAYA took the least amo SDM and DDM.Aside from that, SSA required the least computation PVMM, followed by GWO, JAYA, and the ICO.For the SDM, after JAYA the least computing time.The DDM, PSO, GWO, WSO, and SSA ha computing times, and the ICO needed a little more time than them.C original algorithms such as JAYA, GWO, WSO, and SSA, the time spe comparable.Its superior performance over these algorithms, however a statistical perspective.In addition, although the CO, SEDE, and PGJA icant performance in terms of statistical results, they required more c than the ICO.Compared with the CO, the main advantage of the ICO duce the computing time while maintaining or even improving its per

Convergence Characteristics
In Figure 12, each algorithm's convergence curve is depicted for each model and indicates the average RMSE performance across 30 independent runs.As can be seen from Figure 12, it is evident that the ICO achieved a competitive or faster convergence rate than the other algorithms for the three PV models, demonstrating its capability to maintain a good balance between exploration and exploitation.It is worth noting that the convergence behavior of the CO seems better than that of the other conventional algorithms, such as DE, GA, PSO, WGO, SSA, TLBO, JAYA, and WSO.
dicates the average RMSE performance across 30 independent runs.As can be seen from Figure 12, it is evident that the ICO achieved a competitive or faster convergence rate than the other algorithms for the three PV models, demonstrating its capability to maintain a good balance between exploration and exploitation.It is worth noting that the convergence behavior of the CO seems better than that of the other conventional algorithms, such as DE, GA, PSO, WGO, SSA, TLBO, JAYA, and WSO.

Exploration and Exploitation Analysis
Keeping exploration and exploitation in balance can be achieved by ensuring sufficient diversity among individuals.In this way, an algorithm can avoid being trapped in a local solution and ultimately produce a better solution to a particular optimization problem.However, exploration-exploitation and diversity measurements alone cannot prove that one algorithm is better than another for solving optimization problems.Some experiments are presented in this section to evaluate the exploration-exploitation and diversity of solutions in the comparative algorithms for the SDM problem. Figure 13 shows variations in exploration, exploitation, and population diversity among the individuals of competitive algorithms during the iterations.Calculations were made according to the procedure detailed in [42].

Exploration and Exploitation Analysis
Keeping exploration and exploitation in balance can be achieved by ensuring suffi cient diversity among individuals.In this way, an algorithm can avoid being trapped in local solution and ultimately produce a better solution to a particular optimization pro lem.However, exploration-exploitation and diversity measurements alone cannot pro that one algorithm is better than another for solving optimization problems.Some exp iments are presented in this section to evaluate the exploration-exploitation and divers of solutions in the comparative algorithms for the SDM problem. Figure 13 shows var tions in exploration, exploitation, and population diversity among the individuals of co petitive algorithms during the iterations.Calculations were made according to the proc dure detailed in [42].Figure 13a,b shows that, in contrast to the other algorithms, SSA and GWO exhibited a greater percentage of exploration during the iterations than exploitation.It is shown in Figure 14 that these two algorithms had average exploration-exploitation ratios of 80%:20% and 76%:24%, respectively.This was due to the high diversity in the population in these two algorithms, as shown in Figure 13c.Comparatively, the GAs and WSO provided the greatest level of exploitation capabilities, with an average value of 99%.Further evidence of this can be found in Figure 13c, demonstrating that these two algorithms were unable to provide sufficient diversity throughout the iteration process.Thus, premature convergence is one of the main weaknesses of these algorithms.
80%:20% and 76%:24%, respectively.This was due to the high diversity in the population in these two algorithms, as shown in Figure 13c.Comparatively, the GAs and WSO provided the greatest level of exploitation capabilities, with an average value of 99%.Further evidence of this can be found in Figure 13c, demonstrating that these two algorithms were unable to provide sufficient diversity throughout the iteration process.Thus, premature convergence is one of the main weaknesses of these algorithms.
Figure 13a,b also shows that the ICO, CO, DE, PSO, GA, TLBO, SEDE, JAYA, PGJAYA, and WSO were all explorative at first, but after a few iterations, they were deemed exploitative algorithms.Similar results can be observed for the diversity measure in these algorithms which, after a few iterations, dropped (see Figure 13c).It must be noted, however, that spikes in the population diversity characteristic were observed in the CO due to the leave-the-prey-and-go-back-home strategy.

Results for STM6-40/36 PV Module
Further investigation of the proposed algorithm for determining the parameters of solar module STM6-40/36 was undertaken.The upper and lower limits of the variables are given in [43].Irradiation of 1000 W/m 2 and a temperature of 51 °C were used to measure 20 pairs of current and voltage values [44].Table 10 presents the optimal parameters and statistical results of the proposed algorithm and recent well-established methods for the SDM and DDM of STM6-40/36.Aside from the algorithms used in the previous sections, the African vultures optimizer (AVO) [45], tuna swarm optimizer (TSO) [46], and artificial hummingbird technique (AHT) [43] were also compared based on the results reported in [43].The results indicate that the CO and ICO performed better and were more reliable than the other algorithms for determining the optimal solutions.Also, SEDE and PGJAYA Figure 13a,b also shows that the ICO, CO, DE, PSO, GA, TLBO, SEDE, JAYA, PGJAYA, and WSO were all explorative at first, but after a few iterations, they were deemed exploitative algorithms.Similar results can be observed for the diversity measure in these algorithms which, after a few iterations, dropped (see Figure 13c).It must be noted, however, that spikes in the population diversity characteristic were observed in the CO due to the leave-the-prey-and-go-back-home strategy.

Results for STM6-40/36 PV Module
Further investigation of the proposed algorithm for determining the parameters of solar module STM6-40/36 was undertaken.The upper and lower limits of the variables are given in [43].Irradiation of 1000 W/m 2 and a temperature of 51 • C were used to measure 20 pairs of current and voltage values [44].Table 10 presents the optimal parameters and statistical results of the proposed algorithm and recent well-established methods for the SDM and DDM of STM6-40/36.Aside from the algorithms used in the previous sections, the African vultures optimizer (AVO) [45], tuna swarm optimizer (TSO) [46], and artificial hummingbird technique (AHT) [43] were also compared based on the results reported in [43].The results indicate that the CO and ICO performed better and were more reliable than the other algorithms for determining the optimal solutions.Also, SEDE and PGJAYA provided good performance.In conclusion, this algorithm offers the most reliable and efficient method for obtaining the most efficient results for various solar modules.The performance of the proposed algorithm was compared with the results reported in other articles that included various developed, hybrid, and improved algorithms.These algorithms included SATLBO [5], IJAYA [16], hARS-PS [47], APLO [48], CMM-DE/BBO [49], DE/BBO [50], BLPSO [51], CLPSO [52], MSSA [53], TLABC [54], GOTLBO [55], STLBO [56], MVO [57], QMVO [58], Rcr-JADE [59], and IWOA [60].The statistical results from the point of view of the SD, Mean, Max, and Min values of the RMSEs for solving three types of PV models of the RTC France solar cell and Photo Watt-PWP 201 PV module are summarized in Table 11.The results show that most algorithms obtained the optimal solution.However, from the perspective of robustness, it can be seen that for the SDM, the highest performance was related to the ICO with the SD value of 3.09 × 10 −17 .After that, the CO and APLO were placed in the following ranks.For the DDM, hARS-PS showed the best performance in terms of standard deviation with a value of 1.45 × 10 −7 , followed by MSSA and the ICO, which were second and third best, respectively.For the PVMM, Rcr-JADE provided the best stability in comparison with the other algorithms.The ICO could also achieve the second-best SD value of 3.19 × 10 −17 , followed by the CO with the SD of 4.37 × 10 −17 .The results indicate that the ICO provided the most accurate and reliable algorithm for identifying solar PV model parameters.Furthermore, the original CO algorithm also exhibited reasonable performance compared with other hybrid and improved algorithms.

Conclusions
This paper introduced a simplified and improved version of the CO algorithm and investigated its performance when identifying unknown parameters of PV cells and modules.An extensive set of experiments was conducted to assess the performance of the CO and ICO when identifying the parameters of different PV models, including the SDM, DDM, and PVMM.We examined how the size of the initial population affected the performance of the ICO.It was found that the algorithm performed well for populations with a number greater than 10.The results obtained from the ICO and CO were also compared to those obtained from other well-known algorithms in terms of accuracy, robustness, computing time, and convergence characteristics.Based on the Wilcoxon signed rank test and Friedman test, the performance of the algorithms was compared and determined.The results of these tests indicated the superiority of the ICO compared with other competitive algorithms in terms of accuracy, reliability, good convergence speed, and computation.Moreover, the further improvement made in the CO revealed that the ICO was able to significantly reduce the computing time by maintaining or improving its features, and it also demonstrated enhanced performance.The ICO and CO achieved the best sum rank results among the 12 applied algorithms for the studied PV models.However, the ICO algorithm reduced the computation time of the CO by approximately 40%.It is also worth noting that the ICO's convergence speed was high, reaching an optimal solution within a few thousand function evaluations in most cases.
Accordingly, the ICO can be considered a promising candidate method for extracting the model parameters of PV cells and modules.Hence, the ICO will be applied in our future studies to solve a variety of power systems' operation and planning optimization problems, including placement of renewable distributed generations, economic load dispatch, and feeder reconfiguration.Furthermore, we intend to develop binary and multiobjective versions of the ICO algorithm.

Figure 2 .
Figure 2. Representation of the CO algorithm.

Figure 2 .
Figure 2. Representation of the CO algorithm.

Figure 3 .
Figure 3.An overview of the strategy selection mechanism in the CO algorithm.

Figure 3 .
Figure 3.An overview of the strategy selection mechanism in the CO algorithm.

Algorithm 1 . 33 Figure 3 .
Figure 3.An overview of the strategy selection mechanism in the CO algorithm.

Figure 4 .Figure 4 .
Figure 4. Average ranks of the utilized population sizes in three models.In addition, the convergence characteristics of the algorithm with different population numbers are shown in Figure5for three models.When the population number was

Figure 5 .Figure 5 .
Figure 5. Convergence curves of ICO with different population sizes in solving (a) SDM, (b) DDM, and (c) PVMM.4.2.Results of Parameter Extraction Based on the SDMFor the SDM, the best solutions found by competitive algorithms with n = 40 and n = 80, including the PV parameters and objective function (RMSE) values, are summarized in Table4.From the results, for the two tested population sizes, the best RMSE value of 9.860218778914 × 10 -4 was obtained from the CO and ICO.For n = 40, SEDE and WSO (and

Figure 6 .
Figure 6.Estimated and measured data of the RTC France silicon solar cell based on the SDM with the ICO: (a) I-V, (b) P-V, (c) IAEI, and (d) IAEP.

Figure 6 .
Figure 6.Estimated and measured data of the RTC France silicon solar cell based on the SDM with the ICO: (a) I-V, (b) P-V, (c) IAEI, and (d) IAEP.

Figure 8 .
Figure 8.Estimated and measured data yielded by the ICO for the PV module model based on Photo Watt-PWP 201: (a) I-V, (b) P-V, (c) IAEI, and (d) IAEP.

Figure 8 .
Figure 8.Estimated and measured data yielded by the ICO for the PV module model based on Photo Watt-PWP 201: (a) I-V, (b) P-V, (c) IAEI, and (d) IAEP.

Figure 9 .
Figure 9. Final ranking of applied algorithms for three models based on the Friedm 40 and (b) n = 80.

Figure 9 .
Figure 9. Final ranking of applied algorithms for three models based on the Friedman test: (a) n = 40 and (b) n = 80.

Figure 11 .Figure 11 .
Figure 11.CPU time over 30 runs by different algorithms with n = 40 for SDM

Figure 14 .
Figure 14.Mean exploration-exploitation of comparative algorithms on SDM.

Figure 14 .
Figure 14.Mean exploration-exploitation of comparative algorithms on SDM.

Table 1 .
A review of the solar PV parameter extraction methods.

Table 2 .
Parameters' bounds for three models.

Table 2 .
Parameters' bounds for three models.

Table 3 .
An analysis of the effect of population size on ICO performance for different PV models.

Table 3 .
An analysis of the effect of population size on ICO performance for different PV models.

Table 4 .
Optimal parameters of SDM obtained by different algorithms with n = 40.
, respectively, indicating that the CO and ICO were incredibly accurate in estimating the DDM parameters.

Table 5 .
Optimal parameters for DDM obtained by different algorithms with n = 40.
nAlgorithm I

Table 6 .
Optimal parameters for PVMM by different algorithms with n = 40 and n = 80.

Table 7 .
Statistical results of different algorithms with n = 40 and n = 80 for SDM.

Table 8 .
Statistical results of different algorithms with n = 40 and n = 80 for DDM.

Table 9 .
Statistical results of different algorithms with n = 40 and n = 80 for PVMM.