Extremal Nelder–Mead colony predation algorithm for parameter estimation of solar photovoltaic models

Abstract

Measurement data based on current and voltage of photovoltaic (PV) systems and the establishment of more accurate and stable solar system models are of typical significance for the design, control, evaluation and optimization of PV systems. Accurate and stable parameter evaluation for PV systems needs to be based on more efficient optimization techniques to achieve efficient energy conversion from solar energy. Therefore, this paper proposes a novel and efficient optimization technique enhanced colony predation algorithm to solve the complex PV parameter identification problem named ECPA. By fusing extremal optimization strategy and Nelder–Mead simplex method enables ECPA to further develop in the neighborhood of potential optimal solutions while improving the position of inferior agent candidates, and finally has the ability to search globally beyond the local optimum. To verify the optimization efficiency of ECPA, the first part verifies the efficiency of ECPA in solving complex high‐dimensional and multimodal problems by conducting competitive comparison experiments at the IEEE CEC 2020 benchmark case. In the second part, ECPA is compared with nine similar published state‐of‐the‐art algorithms, and competitive tests for PV parameter identification under single diode model, double diode model, triple diode model and PV module model (PV) are conducted. Finally, we focused on three different commercial PV models (thin film ST40, monocrystalline SM55, and multicrystalline KC200GT) to test the accuracy of ECPA in evaluating PV parameters. The test results show that ECPA is able to maintain a high level of accuracy and stability when dealing with commercial PV models in complex environments. The experimental results demonstrate that ECPA outperforms other algorithms in terms of data fitting, stability, convergence speed and convergence accuracy. All the competitive experimental results show that ECPA can be a novel technique with the best performance for identifying the parameters to be determined in solar PV systems.

INTRODUCTION

As the supply of large areas of nonrenewable energy sources is decreasing, the use of renewable energy sources is becoming more and more important. And nowadays, solar energy has become one of the most valuable and efficient energy sources.¹ Solar photovoltaic (PV) systems have become the main energy technology in the world after wind energy, which is a near-pollution-free distributed power generation system. Large-scale investments and funding of PV systems have been recorded in detail more recently,² but PV systems also have disadvantages such as low efficiency and instability, so the accuracy and efficiency of PV system model design is even more crucial for researchers. Designing a stable and accurate solar PV model is even more of a challenging project. Modeling of PV systems often requires accurate mathematical expressions and parameter extraction, however, in many solar industries the parameters given in the manufacturer's data sheets are not accurate enough, and we need to more accurately evaluate the performance of PV arrays, power–voltage (P–V) and current–voltage (I–V),^3,4 as a way to optimize the efficiency of light-generated electricity. The single diode model (SDM) and the double diode model (DDM) are frequently used by researchers today.⁵ Both models contain electronic components capable of analyzing the nonlinear effects of real PV model circuits.

Classical methods for identifying PV models include some gradient-based deterministic methods, such as exact, Newtonian, and curve fitting,^6,7 but these methods often have some limitations, such as being too sensitive to the generated initial values requiring user-defined parameters,^8,9 easily falling into local optima, and so on. So, the use of metaheuristic techniques in the field of solar energy helps us to solve some problems effectively. It does not have any requirements for constructing the objective function and has a wide range of search behaviors. Thus, it can solve the parameter evaluation problem of complex solar PV systems more accurately. So the application of some metaheuristic techniques can better solve such problems.^10,11 Awadallah et al.¹² used the bacterial foraging algorithm to study the relevant variations of PV system parameters extracted from manufacturer's data sheets. Zhang et al.¹³ studied the distribution optimization problem of generalized normal and its application direction in solar model parameter evaluation. Yang et al.¹⁴ applied grouped gray wolf optimizer to the problem related to maximum power point tracking of doubly fed wind turbine. Bastidas-Rodriguez et al.¹⁵ applied genetic algorithm to identify the parameters of SDM of PV panel. Askarzadeh et al.¹⁶ applied artificial swarm optimization algorithm to the problem of solar PV cell parameter identification and obtained highly accurate results. Nayak et al.¹⁷ applied grey wolf optimization algorithm (GWO) to single diode PV module parameter estimation and obtained accurate results. Abbassi et al.¹⁸ applied salp swarm-inspired algorithm to solar cell model parameter identification problem. Askarzadeh et al.¹⁹ designed a novel bird mating optimization method for the extraction of PV system parameters under different operating conditions. Compared with the traditional analytical methods, the metaheuristic algorithm does not need to be designed into an exact mathematical model,^20,21 but only needs to define the search range of parameters and a designed objective function to achieve the solution for complex problems, which largely reduces the computational burden.

Compared with the traditional gradient-based methods, metaheuristic algorithms do show better results in solving tedious derivation arithmetic problems. These metaheuristics has shown great effectiveness in solving the practical problems such as multiattribute decision making,^22–25 object tracking,^26,27 design of power electronic circuit,^28,29 neural network training,³⁰ gate resource allocation,^31,32 fault diagnosis of rolling bearings,^33,34 traveling salesman problem,³⁵ fractional-order controller,³⁶ feature selection,^37,38 and medical diagnosis.^39,40 However, when faced with these multimodal nonlinear objective functions of PV models containing multiple local optima, convergence is often slow and enters into deceptive convergence. According to the No Free Lunch (NFL) theorem,⁴¹ there does not exist an optimizer that can solve all the optimization parameters extraction in an all-purpose way. Therefore, some variants of metaheuristic algorithms have emerged with relatively good results. Jordehi et al.⁴² proposed a time-varying acceleration coefficient particle swarm optimization algorithm for efficient evaluation of pending parameters of PV system components. Hasanien et al.⁴³ improved the whale optimization algorithm WOA for accurate modeling of dual diodes to further improve the utilization of PV systems for light energy. Qais et al.⁴⁴ completed a parsing-based and sunflower optimization algorithm for electrical parameter identification of three-diode PV models. Ye et al.⁴⁵ designed a particle swarm algorithm-based solar cell parameter extraction scheme to demonstrate that the accuracy of parameter extraction under different PV cell models is better than that of the genetic algorithm. Chen et al.⁴⁶ designed a hybrid cuckoo search technique with biogeography-based optimization techniques to estimate PV model parameters. Xu et al.⁴⁷ further improved political optimization by combining the quantum rotation gate strategy and NMs local search scheme to finally and accurately evaluate the unknown parameters of the PV system. Song et al.⁴⁸ proposed an adaptive harris hawk optimization (ADHHO) with continuous triangular (sine cosine) difference for parameter identification of PV systems. Yu et al.⁴⁹ proposed gradient-based optimization (GBO) used in combination with orthogonal learning to solve the PV parameter identification problem. Weng et al.⁵⁰ proposed an improved slime mold algorithm (ISMA) to solve the complex parameter extraction problem. Fan et al.⁵¹ solved the PV parameter identification problem with the PSOCS algorithm based on particle swarm optimization (PSO) of the core components and a random reselection strategy of the parasitic nests that appear in the cuckoo search. Zhou et al.⁵² proposed an enhanced spherical evolutionary algorithm (SE) based on a novel dynamic sine cosine mechanism (DSCSE) solved the parameter extraction problem of solar cells. Yu et al.⁵³ proposed an efficient sine cosine differential gradient based optimization method for identifying unknown parameters of PV models. Wang et al.⁵⁴ used an improved ensemble multistrategy driven shuffle frog leadership algorithm (EMSFLA) to stably extract the parameters to be determined for PV systems. Iman Ahmadianfar et al.⁵⁵ proposed an enhanced metaphor-free gradient optimizer (EGBO) for fast, accurate and reliable extraction of PV parameters. Liu et al.⁵⁶ proposed an advanced clay algorithm (SMA) integrated with Nelder–Mead strategy and chaotic map, called CNMSMA, which is able to extract the unknown parameters of PV solar cells accurately. Weng et al.⁵⁷ proposed an enhanced spherical evolutionary algorithm without metaphor for identifying unknown parameters in PV models. Ridha et al.⁵⁸ proposed an enhanced version of the LSHADE (ELSHADE) method to determine the parameters of three diode PV models by integrating various contributions in the algorithm itself and the objective function. Fan et al.⁵⁹ proposed a delayed dynamic step mechanism to overcome this drawback based on the characteristics of the shuffling frog-leaping algorithm, called the DDSFLA algorithm, in which the problem of PV parameter extraction is solved. A comprehensive review of recent developments in SAPV system design based on multiobjective optimization and multicriteria decision making (MCDM) approaches, including mathematical models for estimating the output power of PV modules and batteries, is also presented by Ridha et al.⁶⁰ Wang et al.⁶¹ designed an enhanced antlion optimizer to efficiently and accurately estimate the PV model parameters. Zhang et al.⁶² proposed an orthogonal moth flame optimization (MFO) with local search for identifying the parameters of the PV cell model, which was named NMSOLMFO. Abbassi et al.⁶³ proposed an opposition-based learning modified sarp swarm algorithm (OLMSSA) for the accurate identification of the two diode model parameters of the electrical equivalent circuit of the PV cell. Chen et al.⁶⁴ proposed an improved sine cosine algorithm to efficiently approach the unknown parameters of solar cells and PV modules.

However, most of the aforementioned hybrid metaheuristic algorithms can only obtain satisfactory root mean square error values and do not use relatively few computational resources. Some algorithms still suffer from degradation of the fitness function, excessive interactivity and slow transformation speed. Moreover, for algorithms based on a single search, they are more dependent on the starting point where the algorithm starts performing the search. In multimodal optimization problems, experimental data sets that contain noise caused by the process of collecting imperfect data may lead to deterioration of the performance of the multipeak multimodal characteristics of the solar PV system objective function using metaheuristic algorithms, further leading to problems such as low solution accuracy and falling into deceptive optimality. Therefore, the existing metaheuristic method still needs continuous improvement to further enhance its search capability. Developing a more competitive technique to solve this multimodal function optimization problem more comprehensively in any environment where the parameters of the PV model need to be accurately evaluated to achieve the actual energy efficient utilization of the PV system. This is still a complex and challenging optimization problem.

Colony predation algorithm (CPA)⁶⁵ is a state-of-the-art stochastic continuous optimization algorithm, which is constructed with the idea of collective predation behavior of animals in nature, using the relevant mathematical mapping to perform operations such as scattering prey, encircling prey, and supporting the closest companion of hunters while using the difference in success rates to simulate the animal. The selective neglect of prey in hunting can reflect the coexistence of hunters. Due to its high search efficiency and the absence of complex concepts related to the introduction of gradients, CPA requires fewer parameters to be tuned, and is being followed in various fields such as local sequence comparison, maximum power tracking for PV models, multiobjective problems, and training of neural networks. However, due to the simplicity of the structure, CPA still suffers from some shortcomings. For example, when dealing with highly complex nonlinear multiple deceptively optimal optimization problems, it leads to less than perfect search space and poorer diversity of populations, then the convergence trend and speed at later stages will be greatly reduced. The performance of traditional CPA algorithms in the search space still has much room for improvement. So far, no CPA-related techniques have emerged to deal with the complex multimodal nonlinear optimization problems of PV models and PV parameter extraction problems. CPA also suffers from incompleteness and immaturity of convergence for the population search space. Therefore, in this study, an enhanced colony predation algorithm (ECPA) based on the extremal optimization (EO) scheme and Nelder–Mead simplex strategies (NMs) is proposed for the first time. The scheme makes full use of the differential vectors and the target information generated by the search population, enhances the intensification of the algorithm search, extends the dimensionality of the optimization problem, and possesses a stronger search and exploitation capability to extract the parameters of the solar PV system. To validate the performance of the algorithm, this study compares with some advanced state-of-the-art algorithms in solar PV such as generalized oppositional teaching learning based optimization (GOTLBO),⁶⁶ GOFPANM,⁶⁷ improved JAYA (IJAYA) optimization algorithm,⁶⁸ multiple learning backtracking search algorithm (MLBSA),⁶⁹ random learning gradient based optimization (RLGBO),⁷⁰ enhanced Harris hawks optimization (EHHO),⁷¹ slime mould algorithm (SMA),⁷² gradient-based optimizer (GBO),⁷³ weighted mean of vectors (INFO)⁷⁴ on different PV models (including single diode, double diode, triple diode, and PV modules) for efficient extraction of PV parameters, ECPA compared with other ECPA shows the fastest extraction capability and the most accurate results compared to other strong competing algorithms. In addition, statistical and competitive experiments confirm the highly stable performance of ECPA, highlighting higher efficacy results, by performing stability analysis of EPCA for varying temperature and irradiation under two real datasets obtained from the manufacturer's data sheet.

In short, the main contributions of this study are as follows:

• (1)

  Based on the idea of introducing EO scheme and NMs local search strategy, an enhanced evolutionary algorithm ECPA is proposed in this study. The proposed ECPA algorithm can identify the unknown parameters of multiple PV models more accurately, rapidly and stably.

• (2)

  The EO strategy incorporated into CPA is firstly able to generate variants that successively mutate the inferior genes in the original CPA, enhancing the diversity of the population while ensuring the breadth of the algorithm search. At the same time, the introduction of NMs local search strategy in the iterative process can enhance the algorithm's trend of mining the optimal solution, accurately scanning the optimal solution neighborhood, and improving the accuracy of the final convergence to the global optimal solution. Therefore, ECPA can ensure a more mature balance between exploration capability and exploitation capability.

• (3)

  Based on experimental results on the IEEE CEC 2020 function test suite, the efficiency of the ECPA technique in solving single-mode functions, multimode functions, hybrid functions, and composite functions is verified.

• (4)

  Based on several PV models, we compare ECPA with some state-of-the-art algorithms through detailed and comprehensive experiments and analyses as a way to verify the highest efficiency of ECPA and its strong competitiveness in dealing with complex parameter extraction problems of PV systems.

• (5)

  Based on three solar PV models, ECPA was tested at different temperatures and irradiances to verify its efficiency and high stability.

The structure of this paper is as follows. Section 2 introduces the definition related to the PV problem. Section 3 presents the algorithm proposed in this paper in detail. Section 4 validates the performance of ECPA by conducting competitive comparison experiments of the proposed algorithm on the IEEE CEC 2020 function test suite and four PV models. Simulation experiments are also introduced for stability experimental analysis. Finally, Section 5 provides a summary of this study and an outlook and plan for the future.

PROBLEM DEFINITION

With the highest potential to solve the system of converting light energy into electrical energy, the stable establishment of PV models and the evaluation of unknown parameters can directly affect the efficiency of converting solar energy into electrical energy. Therefore, researchers have set up some accurate PV models that can stably demonstrate the basic current-voltage characteristics of solar modules. The models commonly used by researchers are single diode model (SDM), double diode model (DDM), triple diode model (TDM) and PV module. The above-mentioned equivalent models and the corresponding objective functions are described in detail below.

Solar cell model Single diode model

SDM are often used in the characterization of solar PV cells and in the field of PVs because of their simpler structural model and high simulation accuracy. Among others, SDM can be exploited in low cost and fast response PV modules. Figure 1 shows the basic structure of the SDM, which consists of three parts. The first part is a current source, which is connected in parallel with the diode. The second part is a current divider capable of indicating the leakage current. The third part is a series-connected resistive cell, whose action corresponds to the energy consumption associated with the load current. In this case, the output current of the SDM is expressed as the equation below. [Image Omitted. See PDF]where I_L is the output current, I_ph stands for photo-generated current, I_d denotes diode current, which can be computed by Equation (2) according to Shockley equation, and I_sh is shunt resistor current calculated by Equation (3). [Image Omitted. See PDF] [Image Omitted. See PDF]where R_s and R_sh are on behalf of the series and shunt resistances, respectively. V_L denotes the output voltage and I_sd is the reverse saturated current of SDM. And n denotes the diode ideality factor, k means Boltzmann constant (1.3806503 × 10⁻²³ J/K), q stands for the charge of the electron (1.60217646 × 10⁻¹⁹ C), and T stands for the Kelvin temperature of the battery. Based on this, Equation (1) can be reset to the following equation: [Image Omitted. See PDF]

Figure 1. Equivalent circuit image of single diode model

From Equation (4) we can see that the single diode has five unknown parameters (I_ph, I_sd, R_s, R_sh, n). Optimization techniques to more consistently and accurately evaluate these parameters have led to a significant increase in the energy conversion rate of solar PV cells.

Double diode model

The structural simplicity of SDM makes it widely used in PV systems, but it has the limitation that it does not calculate the effects related to the compound current losses. Therefore, the DDM was introduced to improve the accuracy of the model. In the DDM, an additional diode is added than the SDM. The role of the first diode is responsible for modeling the compound currents and uncertainties, and the second is used as a rectifier. A shunt resistor is also present to divert the current generated by the solar energy. Figure 2 shows the structural circuit diagram of the DDM model. The simple expression for the output current is represented by the following equation: [Image Omitted. See PDF]where I_d1 and I_d2 denote the currents of the first and second diodes. A more detailed expression for the output current is shown below. [Image Omitted. See PDF]where I_sd1 and I_sd2 mean the diffusion and saturation currents, n₁ and n₂ represent the ideal factors for diffusion and conforming diodes, respectively. The seven different parameters (I_ph, I_sd1, I_sd2, n₁, n₂, R_s, R_sh) that need to be extracted under the DDM model can be seen in Equation (6).

Figure 2. Equivalent circuit image of double diode model

Three diode model

The TDM is used to solve industrial optimization problems for large-scale solar industry, to better determine the current components of solar PV cells, and to fit the current curves well but the hardware implementation is cumbersome. Figure 3 shows the equivalent circuit of the TDM model. The current generated by the solar light source through the triple diode and two equivalent resistors is recorded by $${I}_{{\rm{d}}1}$$, $${I}_{{\rm{d}}2}$$, $${I}_{{\rm{d}}3}$$, $${I}_{\mathrm{sh}}$$, and $${I}_{{\rm{L}}}$$. Based on Kirchhoffss current law, the simple output current expression of the TDM is shown in equation as follows: [Image Omitted. See PDF]

Figure 3. Equivalent circuit image of three diode model

The following equation shows a more detailed expression for the output current under the DDM model. [Image Omitted. See PDF]where $${I}_{\mathrm{sd}3}$$ denotes the current of third diode and $${n}_{3}$$ stands for the ideal factor. Under the TDM model, the same parameters represent the same meaning as under the SDM and DDM models. Therefore, TDM contains nine control parameters, namely I_ph, I_sd1, I_sd2, I_sd3, n₁, n₂, n₃, R_s, and R_sh.

PV module model

PV module model is an important device for converting light energy into electrical energy, which is mainly obtained from some solar cells connected in series or in parallel. The equivalent circuit of the PV module is shown in Figure 4. The output current expression of the PV module can be systematically expressed as the following equations: [Image Omitted. See PDF] [Image Omitted. See PDF]

Figure 4. Equivalent circuit image of PV module

In Equations (9) and (10), N_p denotes the number of parallel solar cells while N_s is the number of series ones. The PV model also has five parameters that need to be evaluated precisely: I_ph, I_sd, R_s, R_sh, and n.

Objective function

In solar PV systems, to solve the most accurate model and the minimum value of the simulated objective function in each PV model, the primary goal is to simulate an objective function that accurately reflects the real performance of the PV system.^75–77 Also to make the error between the experimentally derived current data and the measured current data as small as possible.^78,79 In this paper, for the four PV models under SDM, DDM, TDM, and PV components, the objective functions are shown in Equations (11), (12), (13), and (14). Their overall differences are indicated by the root mean square error (RMSE) values exhibited by Equation (15). A smaller value of the RMSE indicator means that the data simulated by experiment is closer to the measured value, that is, the optimization has better accuracy. The goal of this study is to obtain the optimal value of $$\mathrm{RMSE}({\rm{X}})$$ by adjusting the target vector X. [Image Omitted. See PDF] [Image Omitted. See PDF] [Image Omitted. See PDF] [Image Omitted. See PDF] [Image Omitted. See PDF]

In Equation (15), N stands for the number of current data in the experiment and $$X$$ means the set of unknown parameters. $${X}_{s}$$ denotes the parameter set of $$[{I}_{\mathrm{ph}},{I}_{\mathrm{sd}},{R}_{{\rm{S}}},{R}_{\mathrm{sh}},n]$$. $${X}_{{\rm{D}}}$$ is on behalf of the parameter set of $$[{I}_{\mathrm{ph}},{I}_{\mathrm{sd}1},{{I}_{\mathrm{sd}2},R}_{{\rm{S}}},{R}_{\mathrm{sh}},{n}_{1},{n}_{2}]$$. $${X}_{{\rm{T}}}$$ represents the parameter set of $$[{I}_{ph},{I}_{\mathrm{sd}1},{{I}_{\mathrm{sd}2},{I}_{\mathrm{sd}2},R}_{{\rm{S}}},{R}_{\mathrm{sh}},{n}_{1},{n}_{2},{n}_{3}]$$. $${X}_{\mathrm{PV}}$$ is the parameter set of $$[{I}_{\mathrm{ph}},{I}_{\mathrm{sd}},{R}_{S},{R}_{\mathrm{sh}},n]$$.

THE PROPOSED ECPA METHOD An overview of the CPA

Many optimizers have been proposed to tackle the practicle problems, such as Runge Kutta optimizer (RUN)*https://aliasgharheidari.com/RUN.html,⁸⁰ slime mould algorithm (SMA)†

https://aliasgharheidari.com/SMA.html

,⁷² weighted mean of vectors (INFO)‡

http://aliasgharheidari.com/INFO.html

,⁷⁴ Harris hawks optimization (HHO)§

https://aliasgharheidari.com/HHO.html

,⁸¹ and hunger games search (HGS).⁸² They can be the great potential tools to solve many problems in the field of engineering,^83,84 medicine,^85–87 education,^88,89 finance,^90,91 and energy.^62,71 The CPA⁶⁵ is an advanced algorithm developed based on the group predation behavior of animals. Group predation is usually carried out by a number of herd animals, and this group predation behavior usually improves the survival rate of each individual animal through communication and cooperation to capture higher quality prey. This group behavior is usually more efficient than individual animal predation.⁹² Group predation increases predation success by dividing and surrounding the target. The principle of survival of the fittest⁹³ is also used in the algorithm to select individual animal leaders by this rule.

Mathematical model

This section describes the mathematical model of the CPA algorithm and its establishment. This model is based on the behavior of group predation and is easy to understand and highly feasible. Figure 5 shows the search directions of groups and individuals in two-dimensional space and in three-dimensional space. The scattered predators can determine the update target by ($${X}_{\mathrm{best}},{Y}_{\mathrm{best}}$$).

Figure 5. Search area in (A) two-dimensional space and in (B) three-dimensional space

Figure 6 illustrates the process by which the search agent updates its position by the position of the animal leader and other hunters in the two-dimensional search region. The orange individuals on the top right form a circle of encircled individuals represent a single predator encountering difficulties, through the left side of the green individual mutual assistance to find a partner, mutual communication, so as to complete the process of common hunting. The left side of the figure represents the stage where the blue leader leads the orange predator below to surround the prey, tan(l*π/4) represents the curve where the hunter surrounds the prey, S represents the search radius formed by the search force of the prey, and D represents the distance between the prey and the predator. The right side represents the potential location of the predator, which searches for the next prey potential location after no search target is found in the neighborhood, and D₁ represents the distance between them. From this process, it can be seen that the optimal search position is based on some random position inside the circle defined by the dispersed predators and the leader during the search. The gray circle represents the desired direction of the search agent to update its position.

Figure 6. Colony predation algorithm's location update strategy

Communication and cooperation

Individual animals that hunt in groups improve their hunting success by communicating and cooperating. The following Equation (16) represents the pattern of search behavior of group-hunting animals that hunt through communication and cooperation. [Image Omitted. See PDF]where $${\vec{{\boldsymbol{X}}}}_{j}^{i}(t+1)$$ represents the updated position of the search agent. $${\vec{{\boldsymbol{X}}}}_{j}^{{i}_{1}}(t)$$ represents that the animal is pursuing food and the range of $$r$$ belongs to [0,1]. $${\vec{{\boldsymbol{X}}}}_{1}(t)$$ and $${\vec{{\boldsymbol{X}}}}_{2}(t)$$ represent the two positions in the jth dimension that are located closest to the prey.

Disperse food

Individual animals are driven to different directions to separate prey from the population during the dispersal food phase. The predatory behavior of an individual animal during hunting is modeled by below equation [Image Omitted. See PDF]where $$\vec{{\boldsymbol{X}}}(t+1)$$ represents the location of the population and $${\vec{{\boldsymbol{X}}}}_{\text{best}}$$ represents the location of the optimal solution prey. $${{\boldsymbol{r}}}_{1}$$ represents the random number in [0,1]. $$\mathrm{ub}$$ and $$\mathrm{lb}$$ represent the upper and lower boundaries, respectively. $$S$$ represents the power of the target individual, which can be expressed by Equation (19). [Image Omitted. See PDF] [Image Omitted. See PDF]where $$N$$ is the number of individual animals and $$t$$ represents the current number of iterations. $${r}_{2}$$ is a random number in [0,1]. As the number of iterations increases, $${S}_{0}$$ decreases from $$a$$ to 0. In the case where ω equals 9, $$a$$ is defined as follows. [Image Omitted. See PDF]

Surrounding the prey

Animal hunters will approach and surround their targets with a second strategy, a search that can be represented by the following equation. [Image Omitted. See PDF]where $$l$$ represents a random number between [0,1] and $$\tan \unicode{x0200A}\left(l\frac{\pi }{4}\right)$$ is the curve surrounded by the hunter. $${\boldsymbol{D}}$$ represents the distance between the prey and the predator expressed in Equation (22). [Image Omitted. See PDF]where $$\vec{{\boldsymbol{X}}}(t)$$ represents the number of hunters.

The probability of execution of the two predatory behaviors is expressed in Equation (23). [Image Omitted. See PDF]

Assisting the closest individual

When a single hunter is in trouble when hunting, this is when an assistance strategy is introduced, represented by the below equation. [Image Omitted. See PDF]where $${\vec{P}}_{\text{nearest}}$$ represents the position of the closest hunter in the mutual aid group. $$\vec{{{\boldsymbol{P}}}_{J}}$$ represents the closest hunter to the target, as indicated below. $${r}_{3}$$ represents a random number in [0,1]. [Image Omitted. See PDF]

Finding prey

If no search target is found in the neighborhood, individuals identify a substitute, and this pattern is shown below. [Image Omitted. See PDF] [Image Omitted. See PDF]where $${D}_{1}$$ is the distance of the animal population transition. $${r}_{4}$$ is a randomly generated number in [0,1]. $${\vec{{\boldsymbol{X}}}}_{\text{rand}}$$ is a randomly generated new individual, expressed by Equation (28), and $${r}_{5}$$ is a randomly generated number in [0,1]. [Image Omitted. See PDF]

The animal group and the prey target determine the probability that an individual animal assists the nearest individual and finds the target, expressed by the below equation [Image Omitted. See PDF]where $${r}_{6}$$ is a randomly generated number in [−2,2].

When the search agent exceeds the upper or lower bound, Equation (30) is introduced to replace the position beyond the bound using the current optimal position. [Image Omitted. See PDF]where $${\vec{X}}_{\mathrm{ofb}}(j)$$ denotes the position beyond the boundary and $${\vec{X}}_{\text{best}}(j)$$ denotes the current optimal position.

An overview of the EO

It is wellknown that metaheauristics can effectively solve many real-world problems sucha as image segmentation,^94–96 plant disease recognition,⁹⁷ big data optimization problems,⁹⁸ scheduling problems,^99,100 green supplier selection,¹⁰¹ economic emission dispatch problem,¹⁰² engineering design,^103,104 and combination optimization problems.¹⁰⁵ EO¹⁰⁶ is also one of these kind of metaheauristics. EO¹⁰⁶ emerged to validate the concept of self-organized criticality (SOC), a concept used to represent burstiness. EO reserves only a single chromosome for defining local fitness functions of multiple genes during optimization, which are characterized by the ability to linearly combine into more high-quality solutions for solving the target. And the optimization of high quality solutions can be achieved by co-evolution of genes to reach the goal.

Inspired by the Bak–Sneppen model, EO is able to pick a single solution from multiple solutions of the problem population to adjust the mutation and finally find the optimal target solution. EO is characterized by focusing on the adjustment of elements in a single solution strategy, being able to generate variants in the optimization process, and getting new fitness values for ranking in each evaluation process, thus randomly replacing the solutions about the variables. This motivates a large-scale dynamic movement of the population. Thus, it brings us a new way of thinking that allows the worse solutions or suboptimal solutions to be adjusted by mutations, for successive mutations of inferior genes, continuously improving their constituents and thus not being limited to optimal solutions in a narrow space. The pseudocode of EO can be represented by Algorithm 1.

1 Algorithm

The pseudocode of EO

1. Generate a random solution X, X = ($${x}_{1},{x}_{2},\text{\unicode{x02026}},{x}_{i},$$…,$${x}_{i,D}$$), $$i\in (\mathrm{1,2},\text{\unicode{x02026}},D$$). Put the current solution as the best solution, that is, $${X}_{\text{best}}=X$$. At this point the value of the objective function is assigned, that is $$F({X}_{\text{best}})=F(X)$$. 2. Facing solution X: 1) While ensuring that other variables remain unchanged, the variables $${x}_{i}$$ that make up $$X$$ are introduced one by one. This generates D new solutions $${NX}=[{X}_{n1},{X}_{n2},\text{\unicode{x02026}},{X}_{{nj}},\text{\unicode{x02026}},{X}_{{nD}}]$$. 2) For each variable $${x}_{i}$$ the fitness evaluation $${l}_{j}={OBJ}({X}_{{nj}})-{OBJ}({X}_{\text{best}})$$ was performed. 3) After comparing the fitness values and finding the optimal fitness value, assuming it is the mth variable, set $${l}_{m}\le {l}_{j}$$ for all j. 4) The fitness value of $$f({X}_{\mathrm{new}})$$ is updated after adding the mutation of $${X}_{i}$$, where $${X}_{\mathrm{new}}$$ = ($${x}_{1},\text{\unicode{x02026}},n{x}_{m}{,\text{\unicode{x02026}},x}_{D}$$), where $$n{x}_{m}$$ is the element of mutation. 5) If $${OBJ}({X}_{\text{new}})\lt {OBJ}(X)$$ after the operation by OBJ, then update $$X$$ and $$F(X)$$ as follows, that is, $$X={X}_{\mathrm{new}}$$, $$F(X)=F({X}_{\mathrm{new}})$$. Otherwise, keep $$X$$ the same as the previous time. 6) If a better solution is obtained, then the global optimal solution is set to $${X}_{\text{new}}$$, that is, $${X}_{\text{best}}={X}_{\text{new}}$$. 3. If the termination condition is not met, then it returns to the second part. 4. Output the best solution $${X}_{\text{best}}$$ and $$F({X}_{\text{best}})$$.

NMs method

The use of the NMs method technique¹⁰⁷ is designed to solve minimization problems without gradient constraints. The NMs iterative technique is used to search for the optimal solution by generating a simplex polyhedron with D + 1 vertices in the face of a D-dimensional minimization function. While iterating, the fitness value of each vertex of the simplex polyhedron is calculated, and some poorly performing vertices emerge. The poorly behaved vertices are then reflected, expanded, continuously compressed and contracted as follows to change the vertex positions, thus generating a new simplex polyhedron. This allows the vertex positions to be continuously updated during the iterative process, so that the simplex is always transformed towards the optimal solution target.

The details of the implementation of NMs technology can be achieved in six steps as follows.

Step 1: The reflection coefficient $$\alpha $$, expansion coefficient $$\beta $$, compression coefficient $$\gamma $$ and shrinkage coefficient $$\delta $$ are set in advance, and then the fitness values computed for each vertex of the simplex are recorded and sorted using the equation [Image Omitted. See PDF]

Step 2: Use Equation (32) to calculate the position of the reflection point and its fitness value $$f({x}_{r})$$ calculated by the function. After replacing $${x}_{D+1}$$ with $${x}_{r}$$ to make a judgment: if the fitness function value of $${x}_{r}$$ is between $${x}_{D}$$ and $${x}_{1}$$, that is, $$f({x}_{1})\le f({x}_{r})\le f({x}_{D})$$, then perform the sixth step. If the fitness function value of $${x}_{r}$$ is smaller than the fitness function value of $${x}_{1}$$, step three is executed at this point. If the value of the fitness function of $${x}_{r}$$ is greater than the value of the fitness function of $${x}_{D}$$, then the fourth step is executed. [Image Omitted. See PDF]where $$\alpha $$ denotes the reflection factor, $$\mathop{x}\limits^{̅}$$ is the center position of the vertex, and $$\mathop{x}\limits^{̅}$$ can be calculated using the equation below [Image Omitted. See PDF]

Step 3: Use Equation (34) to calculate the position of the unfolding point $${x}_{e}$$ and the value of the fitness function $$f({x}_{e})$$ calculated by the function. At this point, if $$f({x}_{e})\le f({x}_{r})$$, then the position of the unfolding point $${x}_{e}$$ is used to replace $${x}_{D+1}$$, otherwise $${x}_{r}$$ is used to replace $${x}_{D+1}$$. Then the circular judgment condition in step 6 is executed. [Image Omitted. See PDF]where $$\beta $$ is the expansion coefficient.

Step 4: Compare the values of the fitness functions of $${x}_{r}$$ and $${x}_{D+1}$$. If $$f({x}_{r})\le f({x}_{D+1})$$, then use Equation (35) to calculate the external compression point $${x}_{{oc}}$$, otherwise use Equation (36) to calculate the internal compression point $${x}_{{ic}}$$. [Image Omitted. See PDF] [Image Omitted. See PDF]where $$\gamma $$ represents the compression coefficient.

When the external compression point $${x}_{\mathrm{oc}}$$ is computed the judgment of the fitness value is performed. If $$f({x}_{\mathrm{oc}})\lt f({x}_{r})$$, then replace $${x}_{D+1}$$ with the external compression point $${x}_{\mathrm{oc}}$$ before executing the sixth step, otherwise skip to the fifth step. Correspondingly, when the internal compression point $${x}_{\mathrm{ic}}$$ is computed, if $$f({x}_{\mathrm{ic}})\lt f({x}_{D+1})$$, then replace $${x}_{D+1}$$ with the internal compression point $${x}_{\mathrm{ic}}$$ and skip to step 6, otherwise go to step 5.

Step 5: Use Equation (37) to shrink all the points with $${x}_{1}$$ removed, with the aim of being able to generate a new simplex polyhedron, and then perform step 6. [Image Omitted. See PDF]where $$\delta $$ is the shrinkage coefficient.

Step 6: Determine whether the termination condition of the iteration is satisfied, if so, stop the step of updating the simplex polyhedra, otherwise continue the iterative process.

The proposed ECPA

The basic CPA algorithm suffers from the disadvantages of low solution efficiency and immature convergence, poor original population diversity, and immature exploration and exploitation balance when solving optimization complex optimization problems, and an enhanced algorithm with respect to CPA has not yet appeared for stable and highly accurate identification of unknown parameters of PV systems. Therefore, this study proposes an evolutionary version of CPA, ECPA, which incorporates the idea of EO in CPA, improves the location of candidate solutions for inferior agents, and enhances the variational scalability of the algorithm. After improving the diversity of the population, the fusion NMs strategy further enhances the algorithm's accurate development capability, allowing the dimensionality of the optimization problem to be extended and converge to the optimal solution faster. PV experiments further verified that ECPA's precise development capabilities have further improved the accuracy of extracting PV parameters. The performance of the ECPA algorithm is substantially improved by the improved ECPA algorithm, which has better metrics in both exploration and exploitation stages.

In this study first starts initializing the agent population by cyclically acquiring individuals and calculating the fitness value of each individual, then updating $${S}_{0}$$ with $${S}_{0}=a-t\left(\frac{a}{N}\right)$$ to prepare for the next update of the prey intensity. Immediately afterward, we enter the communication and cooperation phase, choosing a population of one dimension and using the two best individuals to change the position of the population. The loop then updates the prey intensity using $$S=2{S}_{0}{r}_{2}-{S}_{0}$$, which decreases with the number of iterations. The loop chooses to disperse the prey or surround it depending on the size relationship between $${r}_{2}$$ and 0.5, choosing to execute the following fundamental strategy of population evolution. [Image Omitted. See PDF]

Then, after entering the phase of searching for prey based on the relationship between animal hunters and prey and their probability of finding the target, mutations are adjusted using EO to generate variants, and a selected scenario can generate D mutation scenarios. In each evaluation process replace the solution about the variant. In this case, the mutation step is adjusted by the following form. [Image Omitted. See PDF] [Image Omitted. See PDF]where $${x}_{m}$$ and is $${x}_{{Nm}}$$ are the mth decision variable before the mutation occurs and the decision variable after the mutation, respectively, and $${N}_{m}(\mathrm{0,1})$$ and $${C}_{m}(\mathrm{0,1})$$ are the random numbers satisfying Gaussian and Cauchy distributions, respectively.

Figure 7 represents the process of new solutions generated by ECPA after the introduction of EO, and it is clear that the mutation operation has a crucial role in the change of the composition of the inferior genes and the solution out of the narrow space.

Figure 7. Enhanced colony predation algorithm process for generating new solutions (A) without using EO (B) with EO. EO, extremal optimization

While EO motivates the population to move dynamically to continuously adjust the suboptimal solutions, the strategy of NMs is introduced to search for better solutions in the continuous space. In this case, after constructing a simplex with the current optimal solution, the simplex is updated, and the position of the optimal solution in the population is selected as the starting position for the search, and the improved simplex trajectory is searched in the tolerance domain of the current solution. If a better improved solution can be found, then it is used as a trial run trajectory for the next simplex search cycle. Otherwise, the iterative process is repeated while the search steps are gradually reduced until the termination condition is satisfied. In this way, the undesirable cost of repeated redundant calculations is avoided, and the potential optimal solution neighborhood of the algorithm is further developed to improve the accuracy of convergence to the optimal solution.

We found that there is an important parameter μ in the ECPA algorithm, and the number of iterations of ECPA and the rate of convergence of the algorithm to the global optimal solution depend to some extent on the value of μ. After testing, if μ will be set too large, our compression and expansion operations in NMs local search strategy dominate the weight and the variational strategy of EO is drastically scaled down. It also greatly wastes the average CPU time occupied by the algorithm. If the parameter μ is set too small, the local search capability of NMs is not fully exploited, and the convergence accuracy of ECPA becomes poor. After testing, we set the value of μ to be more suitable as dim + 1, and further iterate the NMs local search method μ times to search the global optimal solution.

Therefore, the pseudocode of the ECPA method proposed in this study is shown in Algorithm 2 and the flowchart is illustrated in Figure 8.

Figure 8. Flowchart of enhanced colony predation algorithm

2 Algorithm

The pseudocode of ECPA

Initialize the parameters of the ECPA algorithm, such as N, dim, Max_FEs, values;

Initialize the location of all individuals $${X}_{i}(i=\mathrm{1,2},\text{\unicode{x02026}},N){\rm{;}}$$

While ($${fes}$$ ≤ Max_FEs)

For i = 1: N do

Check the solution if the search space is exceeded and replace the exceeded position with Equation (30);

Calculate the fitness value for each target individual;

Update the best location $${\vec{X}}_{\text{best}}$$;

End For

Use Equation (18) and Equation (19) to update the target individual's strength $$S$$;

Update a by Equation (5);

Update $${r}_{1}$$;

For j = 1:dim do

Update $${X}_{1},{X}_{2},{i}_{1},r$$;

If rand() < fes/Max_FEs

$${\vec{X}}_{i}={\vec{X}}_{\text{best}}$$;

End If

Calculate the $${X}_{j}^{i}$$ by Equation (16);

End For

Perform the operation of Extremal optimization to update the best location $${\vec{X}}_{\text{best}}$$;

For i =1:N do

Update the target individual's strength $$S$$ again;

If abs(S) < 2*a/3 do

Calculate $${X}_{i}$$ using Equation (23);

Else

Calculate $${X}_{i}$$ using Equation (29)

End If

End For

Perform NMs operation with the current best position as the initial value;

End While

Output $${\vec{X}}_{\text{best}}$$

Computational complexity analysis

The computational complexity of our proposed ECPA algorithm is related to the maximum number of evaluations (Max_FEs), the population size (N), and the dimensionality of the problem (D). So the overall computational complexity is [Image Omitted. See PDF] In this case, our location update process includes the location update of all agents in the original CPA, which updates all agents by EO and NMs local search strategy (which also includes multiple evaluations of fitness values). The complexity of computing the fitness values of all agents is $$O(n)$$. The computational complexity of updating the agent's position in the original CPA is $$O\left(N\times \left(1+\frac{D+{N}^{2}}{4}\right)\right)$$. The computational complexity of updating the agent's position by EO is $$O(D+N\mathrm{log}N)$$. The computational complexity of updating the agent's location by NMs local search strategy is $$O(D)$$. Therefore, the final computational complexity of ECPA is $$O(\mathrm{EPCA})=\,O(\mathrm{Max\_FEs}* (n+N\times \left(1+\frac{D+{N}^{2}}{4}\right)+D)).$$

EXPERIMENTAL INVESTIGATIONS AND ANALYSIS

Fair experimental setup is very important for computational science issues, such as location-based services^108,109 and information retrieval services.^110–112 In the experimental part, we validate the performance performance and results analysis of the ECPA in a more complete way. First, the efficiency and accuracy of the algorithm is verified by a well-known series of benchmark function tests (IEEE CEC 2020 function set), where it is evaluated against a variety of state-of-the-art algorithms to test the potential of ECPA to solve complex problems. Then, we apply ECPA to several classical parameter extraction problems of solar PV cell systems and compare it with some well-known algorithms to verify the effectiveness and accuracy of the ECPA method on solar PV cells. All the experimental results verify that the ECPA method can accurately and efficiently complete problems such as nonlinear multimodality under PV models, and can extend the dimensionality of optimization problems and find the optimal solution more precisely.

The Wilcoxon singed-rank test¹¹³ with a significance level of 0.05 was used in our experiments to evaluate the performance tests of ECPA and related competing algorithms. The symbols “+,” “−,” and “=” were used to determine the performance differences between ECPA and the competitors. For example, “+” is used when the performance of ECPA is better than the competitor, “−” is used when the performance result is not enough to catch up with the competitor, and “=” is used when the performance result is not significantly different from the competing algorithms. The performance comparison between ECPA and some advanced algorithms can be seen more intuitively and simply by using the Wilcoxon singed-rank test. In this study, the experiments were done on a desktop computer with AMD 3700X CPU, 3.60 GHz, Win10 system and 16 GB of RAM using MATLAB 2015b.

Detailed results on the IEEE CEC 2020 benchmark functions

In this section, to verify the effectiveness of the proposed ECPA in solving complex function optimization problems, we test and compare the ECPA with some advanced algorithms including OBSCA,¹¹⁴ SCADE,¹¹⁵ HGWO,¹¹⁶ CESCA, CBA,¹¹⁷ AMFOA,¹¹⁸ RCBA,¹¹⁹ BMWOA,¹²⁰ MSFOA,¹²¹ EM on the IEEE CEC 2020 function set comparisons. These advanced competing algorithms are often used by researchers to solve complex nonlinear multimodal high-dimensional problems.^122–124 The IEEE CEC 2020 function set includes functions that provide a complete and multifaceted assessment of the algorithms' strengths and weaknesses. It consists of a total of 10 functions: including basic function F11, hybrid functions F21 on CEC 2014 and unimodal function F1, basic function F7, F19, hybrid functions F17, F16, composition functions F22, F24, F25 on CEC 2017. More detailed evaluation criteria and problem definitions for these functions can be found in the IEEE CEC 2020 related literature. A complete analytical test of the functions allows a better evaluation of the optimal solutions to different optimization problems.^{85,86,123–130} The comparison of ECPA with advanced competing algorithms on a CEC 2020 function set with dimensionality set to 30 is set according to the requirements of the competition. The maximum number of evaluations (Max_FEs) was set to 300,000 and the population size was set to 40. To eliminate the effect of algorithm randomness, all competing algorithms were run 30 times and the final results were averaged. Table 1 shows that all competing algorithms maintained the same parameters as the original literature when tested on CEC 2020.

Table 1 Parameter records of the competition algorithms

Algorithm	Parameters
SCADE	$${beta\_}{\min }\,=0.2$$; $${beta\_}{\max }\,=0.8$$; $${pCR}=0.8$$; $${r}_{1}=a-{fes}* (a/{Max\_FE}s)$$
RCBA	$${f}_{\min }=0$$; $${f}_{\max }=2$$; $$r={ones}(1,N)* 0.5$$
BMWOA	$$a=2-{FEs}* ((2)/{Max\_FE}s)$$; $$a2=-1+{FEs}* ((-1)/{Max\_FE}s)$$
MSFOA	$${W}_{0}=1$$; α $$=0.95$$; $$M=5$$
HGWO	$$a=2-{FEs}* (2/{Max\_FE}s)$$; $$A=2* {rand}(\mathrm{0,1})* a-a$$; $$C=2* {rand}(\mathrm{0,1})$$
OBSCA	$${r}_{1}=a-{fes}* (a/{Max\_FE}s)$$; $$a=2$$; $${r}_{2}=(2* {pi})* {rand}()$$
CBA	$${C}_{w}=3$$; $${f}_{\max }=2.5$$; $${r}_{1}={r}_{2}=0.5+{rand}(\mathrm{0,1})$$; $$\omega =\left(\frac{{FEs}}{{Max\_FE}s}\right)+0.1$$; $$u=\left(\frac{{FEs}}{{Max\_FE}s}\right)* 0.4+0.3$$
CESCA	$${r}_{1}=a-{fes}* ((a)/{Max\_FE}s)$$; $$a=2$$; $${r}_{2}=(2* {pi})* {rand}()$$
AMFOA	$${W}_{0}=1$$; α $$=0.95$$; $$M=5$$
EM	$${IndexLB}=0$$; $${LSITER}=5$$; $${delta}=0.1$$
ECPA	$$a={\exp }(9-18* {fes}/{Max\_FE}s)$$; $$S0=a* (1-{fes}/{Max\_FE}s)$$; $$S=2* S0* {rand}-S0$$

The optimization statistics generated by running ECPA with all competing algorithms on the CEC 2020 function set are documented in Table 2. The table details the statistics for all competing algorithms, including the mean (Mean) and standard deviation (SD). Bold represents the performance of the algorithm in finding the optimal value. Among them, ECPA is able to find the theoretical optimum by showing the best results on F1, F2, F4, F5, F7, F9, and F10. ECPA has the most experimental results in bold for all functions, indicating that it finds the theoretical optimum on the most functions compared to other competing algorithms. This also demonstrates that ECPA is able to perform the most efficient development when solving complex optimization function problems.

Table 2 Comparison data between ECPA and advanced competitive algorithms on IEEE CEC 2020

[Table omitted. See PDF]

The results of the evaluation of ECPA against other advanced algorithms and based on the Wilcoxon singed-rank test are documented in Table 3. The table includes both the overall comparative performance of the different algorithms on all functions and a more detailed segmental comparative performance by function classification. Based on the results of the full function comparison, it can be seen that ECPA performs much better than its competitors on all 10 functions of the CEC 2020 test suite.

Table 3 Comparison of ECPA with other advanced competing algorithms on CEC 2020 based on Wilcoxon singed-rank test

Comparison groups	+ (better)/− (worse)/= (no sig.)
	Unimodal function (F1)	Basic functions (F2–F4)	Hybrid functions (F5–F7)	Composition functions (F8–F10)	All functions
ECPA vs. CBA	1/0/0	3/0/0	3/0/0	3/0/0	10/0/0
ECPA vs. RCBA	1/0/0	3/0/0	2/0/1	3/0/0	9/0/1
ECPA vs. AMFOA	1/0/0	3/0/0	3/0/0	3/0/0	10/0/0
ECPA vs. EM	1/0/0	1/0/2	2/1/0	2/1/0	6/2/2
ECPA vs. OBSCA	1/0/0	2/0/1	3/0/0	3/0/0	9/0/1
ECPA vs. SCADE	1/0/0	2/0/1	3/0/0	1/0/2	7/0/3
ECPA vs. HGWO	1/0/0	2/0/1	2/0/1	2/0/1	7/0/3
ECPA vs. CESCA	1/0/0	3/0/0	3/0/0	3/0/0	10/0/0
ECPA vs. BMWOA	1/0/0	3/0/0	2/0/1	3/0/0	9/0/1
ECPA vs. MSFOA	1/0/0	3/0/0	3/0/0	3/0/0	10/0/0

A detailed comparison of the results after classifying the functions by extreme points shows that ECPA works better than any of the algorithms on the simple single-peaked function of F1, which has a better solution to the problem of evaluating a single extreme point. Second, ECPA still shows a competitive performance on the three simple multipeak functions F2–F4. Its ability to solve multipeak functions is no less than any other advanced competing algorithms. In particular, when facing the algorithms CBA, AMFOA, RCBA, CESCA, BMWOA, and MSFOA, ECPA outperforms these comparative algorithms for all three functions. When facing the better performing EM algorithm, ECPA outperforms EM in one function and is evenly matched in two functions. Again, ECPA still outperforms all the remaining competing algorithms on the three hybrid functions, F5–F7, to the best level. Finally, when comparing ECPA with the advanced competing algorithms on the very complex composition functions, ECPA does not decrease its optimization power as the complexity of the functions increases. ECPA still shows the best performance and achieves the best evaluation results for all composition functions. In summary, ECPA is able to find exact optimal solutions not only for simple unimodal functions and basic functios, but also for very complex hybrid functions and composition functions by extending the dimensionality of the optimization problem to avoid local optimal solutions and thus produce high-quality optimization results.

Table 4 records the ranking results of ECPA after the Friedman test compared with the above competing algorithms. Based on the Friedman test ranking results, it can be seen that ECPA achieves the minimum value of 1.90333 for all algorithms and ranks first compared to other advanced algorithms. The next algorithms are EM and BMWOA, and so on. The results of the evaluation with the advanced algorithms show that ECPA has a better ability to find the best in solving core search problems and complex multimode evaluation problems. Although the candidate solution positions of the inferior agents are changed by extremal optimization strategies to perform extended variants, the NMs local search strategy is used to avoid redundant computations in the potentially continuous search space and further develop better solutions. Therefore, the convergence curves of competing algorithms on the CEC 2020 test suite shown in Figure 9 similarly demonstrate that ECPA maintains the best convergence accuracy and the fastest convergence speed compared to other competing algorithms in the process of finding the optimal solution. Based on this, ECPA can be a potential optimization technique for accurate mining of optimal solutions.

Table 4 Ranking of the algorithms generated by the Friedman test results

Algorithms	Friedman ranking
CBA	5.33667
RCBA	5.27333
AMFOA	8.62000
EM	3.82333
OBSCA	5.36833
SCADE	5.52667
HGWO	5.43500
CESCA	9.11333
BMWOA	5.23333
MSFOA	10.36667
ECPA	1.90333

Figure 9. Convergence curves based on F1, F2, F3, F5, F7 functions on the CEC 2020 test suite

Evaluation based on PV models

In this section, we want to verify the performance performance of ECPA for complex parameter extraction of solar PV systems, using ECPA for four classical parameter extraction of SDM, DDM, TDM, and PV modules that convert light energy into electricity.¹³¹ In this case, each algorithm was tested for 30 independent runs on different problems, the maximum number of evaluations was set to 20,000 and the population size was taken to 30. When modeling these modules, the input variables were set to some meteorological data, such as temperature and irradiation, and the output variables were power, current, or voltage. The following two sets of data were used in this set of experiments, which correlate with the Photowatt-PWP 201 PV module and the RTC France PV cell, respectively. The first set of data was obtained from the RTC France PV cell, where the temperature was set to 33°C and the light intensity was set to 1000 W/m². Meanwhile, the second data set is from the Photowatt-PWP 201 PV module, a solar system containing 36 polycrystalline cells, where the temperature is set to 45°C and the light intensity is also set to 1000 W/m². These data sets were used extensively in the performance demonstration process to validate a new method of its kind for solar PV systems.^16,66 Second, to fairly compare the performance differences between the different algorithms, the upper and lower bounds of the parameters over the different models were kept consistent with the original paper, as shown in Table 5. To further verify the performance merits of the proposed ECPA method, it is compared here with some state-of-the-art algorithms, such as GOFPANM, GOTLBO, IJAYA, MLBSA, RLGBO, EHHO, SMA, GBO, and INFO. These similar well-known metaheuristics possess a strong capability to solve nonlinear multimodal PV parameter extraction problems. The parameter list of the contrast algorithm we adopted in the PV experiments is recorded in Table 6, and they are consistent with the original literature.

Table 5 The range of unknown parameters for different PV models⁴⁷

Parameters	Single diode/double diode/three diode		PV module
	Lower bound	Upper bound	Lower bound	Upper bound
Iph (A)	0	1	0	2
Isd (μA)	0	1	0	50
Rs (Ω)	0	0.5	0	2
Rsh (Ω)	0	100	0	2000
n, n1, n2, n3	1	2	1	50
Isd1 (μA)	0	1	0	50
Isd2 (μA)	0	1	0	50
Isd3 (μA)	0	1	0	50

Table 6 Parameters involved in the comparison algorithm in photovoltaic experiments

Algorithm	Parameters
IJAYA	$$w=1$$; $$z={rand}()$$
GOTLBO	$$l=0$$; $${Jr}=0.3$$
MLBSA	$${DIM\_RATE}=1$$; $$F=3* {randn}$$
EHHO	$${beta}=1.5$$; $$Q=5$$; $$F=6$$
RLGBO	$${pr}=0.5$$;
SMA	$$z=0.03$$
GBO	$${pr}=0.5$$
INFO	$$I={rand}i([25])$$
GOFPANM	$$p=0.8$$; $${beta}=3/2$$; $$d=3$$
CPA	$$a={\exp }(9-18* {fes}/{Max\_iter})$$; $$S0=a* (1-{fes}/{Max\_iter})$$
ECPA	$$a={\exp }(9-18* {fes}/{Max\_iter})$$; $$S0=a* (1-{fes}/{Max\_iter})$$; $$\mu =0.1* {\dim }$$

The performance of the proposed ECPA algorithm can be discerned from several performance metrics as follows.

• (1)

  RMSE: The value of RMSE, the objective function, indicates how good the quality results of different algorithms produce optimization. The smaller the value of RMSE, the closer the simulation value and the real value are to the experimental optimization.

• (2)

  The Wilcoxon singed-rank test metric was used to visually express the variability of the different algorithms, where the significance level was taken as 0.05.

• (3)

  Convergence trend curve: The optimal solution obtained by 30 independent averaging experiments is plotted into a curve, which can clearly visualize the speed, direction and trend of convergence of different algorithms toward the optimal solution.

• (4)

  Absolute error (IAE) and relative error (RE): two precise error indicators enable a clearer survey of the gap between the simulated and real data at different voltages, thus verifying whether the proposed method is able to achieve an accurate state for identifying unknown parameters of the PV system. The expressions are shown as follows:

[Image Omitted. See PDF] [Image Omitted. See PDF]

Experimental results of SDM model

In this section, we examine the capability of ECPA in handling the unknown parameters under the SDM model. Figure 10 vividly illustrates the I–V and P–V characteristic curves plotted by the ECPA technique proposed in this study when simulating the actual performance of the SDM model. It can be clearly seen that the performance curves simulated by ECPA in different voltage ranges match well with the actual data set used in this experiment, demonstrating the high degree of fit of ECPA. Figure 11 shows the error characteristics of IAE and RE between the estimated current data and the measured current data under the SDM model. From the error characteristics plot, it can be seen that the value of IAE is always below 2.50741E−03 and the value of RE is always between [−1.99782E−02, 1.47108E−01], indicating that ECPA has the ability to extract the parameters accurately. Next, Table 7 shows more precisely the statistics of IAE between the simulated and measured values of current and power generated by the ECPA under the simulated SDM model. The sum of the current IAE is 2.15269E−02 and the sum of the power IAE is 8.73077E−03. The maximum value of all IAEs is only 2.50741E−03. In summary, all the low error values clearly demonstrate that the ECPA is able to accurately simulate the actual performance of the SDM model.

Figure 10. (A) I–V and (B) P–V characteristics of SDM estimated with ECPA

Figure 11. Error values between simulated and experimental current data under the SDM model. (A) IAE values and (B) RE values.

Table 7 IAE of ECPA on SDM

Item	Measured data		Simulated current data		Simulated power data
	V (V)	I (A)	Isim (A)	IAEI (A)	Psim (W)	IAEp (W)
1	−0.2057	0.7640	0.764087754	0.000087754	−0.157172851	0.000018051
2	−0.1291	0.7620	0.762663121	0.000663121	−0.098459809	0.000085609
3	−0.0588	0.7605	0.761355328	0.000855328	−0.044767693	0.000050293
4	0.0057	0.7605	0.760153999	0.000346001	0.004332878	0.000001972
5	0.0646	0.7600	0.759055205	0.000944795	0.049034966	0.000061034
6	0.1185	0.7590	0.758042331	0.000957669	0.089828016	0.000113484
7	0.1678	0.7570	0.757091631	0.000091631	0.127039976	0.000015376
8	0.2132	0.7570	0.756141333	0.000858667	0.161209332	0.000183068
9	0.2545	0.7555	0.755086835	0.000413165	0.192169600	0.000105150
10	0.2924	0.7540	0.753663837	0.000336163	0.220371306	0.000098294
11	0.3269	0.7505	0.751390926	0.000890926	0.245629694	0.000291244
12	0.3585	0.7465	0.747353818	0.000853818	0.267926344	0.000306094
13	0.3873	0.7385	0.740117201	0.001617201	0.286647392	0.000626342
14	0.4137	0.7280	0.727382224	0.000617776	0.300918026	0.000255574
15	0.4373	0.7065	0.706972673	0.000472673	0.309159150	0.000206700
16	0.4590	0.6755	0.675280193	0.000219807	0.309953609	0.000100891
17	0.4784	0.6320	0.630758326	0.001241674	0.301754783	0.000594017
18	0.4960	0.5730	0.571928411	0.001071589	0.283676492	0.000531508
19	0.5119	0.4990	0.499607058	0.000607058	0.255748853	0.000310753
20	0.5265	0.4130	0.413648807	0.000648807	0.217786097	0.000341597
21	0.5398	0.3165	0.317510098	0.001010098	0.171391951	0.000545251
22	0.5521	0.2120	0.212154904	0.000154904	0.117130723	0.000085523
23	0.5633	0.1035	0.102251267	0.001248733	0.057598139	0.000703411
24	0.5736	−0.0100	−0.008717572	0.001282428	−0.005000399	0.000735601
25	0.5833	−0.1230	−0.125507412	0.002507412	−0.073208474	0.001462574
26	0.5900	−0.2100	−0.208472277	0.001527723	−0.122998643	0.000901357
Sum of IAE	NA	NA	NA	2.15269E−02	NA	8.73077E−03

Table 8 precisely records the RMSE values and optimal parameters obtained by ECPA with the above published advanced techniques such as GOTLBO, GOFPANM, IJAYA, MLBSA, RLGBO, EHHO, SMA, GBO, INFO evaluated on the SDM model. The table shows that ECPA achieves the optimal RMSE value of 9.8602E−04 together with some published advanced algorithms GOFPANM, MLBSA, GBO, INFO and RLGBO, which is smaller than the remaining competing algorithms. Based on the Wilcoxon singed-rank test it can be seen that ECPA has the potential to solve the PV parameter extraction problem in the SDM case.

Table 8 Comparison of different algorithms on SDM case

Algorithm	Iph (A)	Isd (A)	Rs (Ω)	Rsh (Ω)	n	RSME	sig
GOTLBO	0.760796	3.18387E−07	0.0364	52.9922	1.4797	9.8690E−04	+
GOFPANM	0.760776	3.23021E−07	0.0364	53.7185	1.4812	9.8602E−04	=
IJAYA	0.760725	3.24017E−07	0.0364	54.2724	1.4815	9.8674E−04	+
MLBSA	0.760776	3.23021E−07	0.0364	53.7185	1.4812	9.8602E−04	=
RLGBO	0.760776	3.23021E−07	0.0364	53.7185	1.4812	9.8602E−04	=
EHHO	0.760253	3.71316E−07	0.0359	64.1091	1.4953	1.0569E−03	+
SMA	0.729659	7.48383E−07	0.0232	52.6564	1.6638	1.3022E−01	+
GBO	0.760775	3.23022E−07	0.0364	53.7189	1.4812	9.8602E−04	=
INFO	0.760776	3.23021E−07	0.0364	53.7185	1.4812	9.8602E−04	=
CPA	0.760787	3.75498E−07	0.0357	56.7208	1.4965	1.0295E−03	+
ECPA	0.760776	3.23015E−07	0.0364	53.7180	1.4812	9.8602E−04

In Table 9 the results of the RMSE statistics and Wilcoxon singed-rank test demonstrated by the ECPA method under the processing of the SDM model are recorded in detail, and the performance of the advanced algorithms mentioned above is compared under 30 independent experiments. In this table it is possible to see the minimum, maximum, standard deviation, and mean values of the RMSE metrics demonstrated by the different competing algorithms in the evaluation of the simulated SDM model. The results from the Wilcoxon singed-rank test metric show that ECPA successfully achieves the lowest standard deviation of the RMSE metric of all competing algorithms and outperforms them all, showing a more accurate and stable performance. Figure 12 records the convergence curves of ECPA and the above competitors under the SDM model, from which it is obvious that the ECPA algorithm converges to the optimal solution much faster than the original CPA algorithm, and faster than all the remaining competitors. Moreover, when all competing algorithms of the same kind converge to the optimal solution, it can be seen from the local zoomed-in plot that the ECPA algorithm is able to achieve the best accuracy. We can see that the three slower convergence algorithms, CPA, SMA, and EHHO, have fallen into the local optimum prematurely in the middle of the iteration, resulting in poor convergence to the final result. The EPCA algorithm has made a qualitative change to the convergence speed and accuracy of the original CPA algorithm, effectively improving the shortcomings of the slow and immature convergence of CPA. The well-known and quite effective GOFPANM lags behind the ECPA proposed in this study in terms of both convergence speed and convergence accuracy, reaching the second place, while the remaining techniques such as GOTLBO and RLGBO are far from the performance of ECPA. Based on the above discussion about ECPA, it can be seen that ECPA is much better than its similar competitors in evaluating the SDM model parameters.

Table 9 Competitive statistics of RMSE metrics under SDM model

Algorithm	Max	Min	Mean	SD	sig
GOTLBO	1.48663E−03	9.86899E−04	1.11084E−03	1.37145E−04	+
GOFPANM	9.86022E−04	9.86022E−04	9.86022E−04	8.13789E−12	+
IJAYA	1.43908E−03	9.86745E−04	1.11186E−03	1.38578E−04	+
MLBSA	2.21804E−03	9.86022E−04	1.03108E−03	2.24430E−04	+
RLGBO	1.43848E−03	9.86022E−04	1.05002E−03	1.27865E−04	+
EHHO	3.84355E−02	1.05687E−03	6.32020E−03	8.04555E−03	+
SMA	3.01565E−01	1.30219E−01	2.22567E−01	5.13776E−02	+
GBO	1.43848E−03	9.86022E−04	1.06132E−03	1.38014E−04	+
INFO	9.98063E−04	9.86022E−04	9.86428E−04	2.19760E−06	+
CPA	1.00000E+00	1.02954E−03	1.33046E−01	2.98367E−01	+
ECPA	9.86022E−04	9.86022E−04	9.86022E−04	1.69365E−12

Figure 12. Comparison of the convergence of ECPA with competing algorithms in the SDM model

Experimental results of DDM model

This section describes the experimental results of the parameter evaluation using the ECPA technique under the DDM model. First, the P–V and I–V curves are shown in Figure 13 for the simulated and experimental data using the ECPA technique in the DDM model. As with the SDM model, the simulated current data matches the measured current data, demonstrating the efficient and accurate performance of the ECPA simulated DDM model. Next, Figure 14 shows the IAE and RE metrics between the estimated and measured values of the ECPA currents under the DDM model. Where the value of IAE is not greater than 2.54343E−03 and the value of RE is in the interval of [−2.02594E−02, 1.37430E−01], proving that ECPA is able to simulate the performance of DDM accurately. Table 10 records the sum of IAE between the experimental and measured values of current and power under the DDM model, which are 2.12752E−02 and 8.77665E−03, respectively. The low level of error ensures the accuracy of the simulation.

Figure 13. (A) I–V and (B) P–V characteristics of double diode model estimated with ECPA

Figure 14. Error values between simulated and experimental current data under the double diode model. (A) IAE values and (B) RE values.

Table 10 IAE of ECPA on DDM

Item	Measured data		Simulated current data		Simulated power data
	V (V)	I (A)	Isim (A)	IAEI (A)	Psim (W)	IAEp (W)
1	-0.2057	0.7640	0.763983416	0.000016584	-0.157151389	0.000003411
2	-0.1291	0.7620	0.762604101	0.000604101	-0.098452189	0.000077989
3	-0.0588	0.7605	0.761337704	0.000837704	-0.044766657	0.000049257
4	0.0057	0.7605	0.760173794	0.000326206	0.004332991	0.000001859
5	0.0646	0.7600	0.759107687	0.000892313	0.049038357	0.000057643
6	0.1185	0.7590	0.758121427	0.000878573	0.089837389	0.000104111
7	0.1678	0.7570	0.757188621	0.000188621	0.127056251	0.000031651
8	0.2132	0.7570	0.756243614	0.000756386	0.161231138	0.000161262
9	0.2545	0.7555	0.755177307	0.000322693	0.192192625	0.000082125
10	0.2924	0.7540	0.753722357	0.000277643	0.220388417	0.000081183
11	0.3269	0.7505	0.751399135	0.000899135	0.245632377	0.000293927
12	0.3585	0.7465	0.747301441	0.000801441	0.267907567	0.000287317
13	0.3873	0.7385	0.740010655	0.001510655	0.286606127	0.000585077
14	0.4137	0.7280	0.727246946	0.000753054	0.300862061	0.000311539
15	0.4373	0.7065	0.706850292	0.000350292	0.309105633	0.000153183
16	0.4590	0.6755	0.675210539	0.000289461	0.309921637	0.000132863
17	0.4784	0.6320	0.630760758	0.001239242	0.301755946	0.000592854
18	0.4960	0.5730	0.571994736	0.001005264	0.283709389	0.000498611
19	0.5119	0.4990	0.499706140	0.000706140	0.255799573	0.000361473
20	0.5265	0.4130	0.413733678	0.000733678	0.217830781	0.000386281
21	0.5398	0.3165	0.317546209	0.001046209	0.171411443	0.000564743
22	0.5521	0.2120	0.212122996	0.000122996	0.117113106	0.000067906
23	0.5633	0.1035	0.102163275	0.001336725	0.057548573	0.000752977
24	0.5736	-0.0100	-0.008791751	0.001208249	-0.005042948	0.000693052
25	0.5833	-0.1230	-0.125543432	0.002543432	-0.073229484	0.001483584
26	0.5900	-0.2100	-0.208371578	0.001628422	-0.122939231	0.000960769
Sum of IAE	NA	NA	NA	2.12752E−02	NA	8.77665E−03

For the DDM model, Table 11 records the results of ECPA's comparison with some similar competitors on DDM. Among them, the RMSE metric of ECPA reaches 9.8248E−04, which is in the highest standard among all competing algorithms. From the Wilcoxon singed-rank test, we can see that the ECPA algorithm has outstanding performance and possesses a strong ability to simulate the actual performance of the DDM model.

Table 11 Comparison of different algorithms on DDM case

Algorithm	Iph (A)	Isd1 (A)	Isd2 (A)	Rs (Ω)	Rsh (Ω)	n1	n2	RSME	sig
GOTLBO	0.76063	2.09918E−07	1.63097E−07	0.0367	54.5531	1.5901	1.4407	1.0016E−03	+
GOFPANM	0.76078	2.25974E−07	7.49344E−07	0.0367	55.4854	1.4510	2.0000	9.8248E−04	=
IJAYA	0.76078	4.02922E−07	2.63879E−07	0.0366	54.5093	1.9805	1.4641	9.8341E−04	+
MLBSA	0.76077	5.69547E−07	2.09888E−07	0.0368	55.2238	1.8929	1.4460	9.8384E−04	+
RLGBO	0.76079	9.29128E−07	2.05144E−07	0.0368	55.8528	2.0000	1.4429	9.8275E−04	+
EHHO	0.76072	4.31401E−07	3.72599E−07	0.0346	67.2448	1.5141	1.9726	1.3258E−03	+
SMA	0.77286	3.04286E−07	8.01849E−07	0.0062	39.4151	1.8941	1.5988	5.8204E−02	+
GBO	0.76073	2.61384E−07	4.78711E−07	0.0366	55.5441	1.4634	1.9963	9.8375E−04	+
INFO	0.76078	7.37901E−07	2.25355E−07	0.0367	55.4715	1.9941	1.4508	9.8255E−04	+
CPA	0.76057	5.17633E−07	6.08020E−07	0.0329	92.9175	1.9475	1.5530	1.9398E−03	+
ECPA	0.76078	2.25970E−07	7.49382E−07	0.0367	55.4855	1.4510	2.0000	9.8248E−04	+

The statistical data of the RMSE metrics obtained by the ECPA algorithm proposed in this study with competing algorithms under the DDM model are recorded in Table 12. Among them, the maximum value, minimum value, mean and standard deviation of the RMSE statistical metrics of ECPA achieve the most advantageous results compared to the above mentioned competitors. Referring to the Wilcoxon rank sum test result metrics it can be seen that ECPA outperforms all competitors in terms of the effectiveness of the evaluation parameters on the DDM model. Second, the convergence curves of ECPA and the above competitors converge to the optimal solution under the DDM model can be seen in Figure 15, where it can be intuitively seen that ECPA maintains the fastest convergence speed and optimal convergence accuracy both at the beginning, middle, and end of the iteration. Based on the above discussion on the various aspects under the DDM case, ECPA is able to maturely smooth the exploration phase and exploitation phase, outperforms all competitors, and can produce higher quality optimization solutions.

Table 12 Competitive statistics of RMSE metrics under DDM model

Algorithm	Max	Min	Mean	SD	sig
GOTLBO	2.44346E−03	1.00163E−03	1.34767E−03	3.43664E−04	+
GOFPANM	9.89172E−04	9.82485E−04	9.84562E−04	2.03779E−06	+
IJAYA	1.50228E−03	9.83411E−04	1.13922E−03	1.63881E−04	+
MLBSA	1.16372E−03	9.83838E−04	1.01723E−03	5.11670E−05	+
RLGBO	1.43846E−03	9.82747E−04	1.06636E−03	1.56082E−04	+
EHHO	1.80916E−02	1.32584E−03	5.98293E−03	5.32951E−03	+
SMA	3.07698E−01	5.82039E−02	2.04135E−01	7.52167E−02	+
GBO	1.89676E−03	9.83754E−04	1.18610E−03	2.47451E−04	+
INFO	1.47219E−03	9.82553E−04	1.08369E−03	1.78968E−04	+
CPA	1.00000E+00	1.93977E−03	4.16465E−01	4.65841E−01	+
ECPA	9.88641E−04	9.82485E−04	9.84353E−04	1.93792E−06

Figure 15. Comparison of the convergence of ECPA with competing algorithms in the DDM model

Experimental results of TDM model

Similarly, we modeled the ECPA on the TDM for the experiments. For the experiments on TDM, the I–V and P–V characteristics that can describe the simulated and experimental data can be represented by Figure 16. The high degree of fit shown in the figure again demonstrates that the simulated data over the specified voltage range gives a good indication of the actual performance of the TDM. Figure 17 also shows the IAE and RE values of the ECPA for the estimated and measured currents under the TDM model, with the IAE values always less than 2.54345E−03 and the RE values fluctuating in the range [−2.02595E−02, 1.37425E−01]. The low error level also ensures the high accuracy optimization results of ECPA.

Figure 16. (A) I–V and (B) P–V characteristics of TDM estimated with ECPA

Figure 17. Error values between simulated and experimental current data under the TDM model. (A) IAE values and (B) RE values.

The IAE metrics for current and power under the TDM model are recorded in detail in Table 13. 2.12752E−02 is the sum of IAEs belonging to current and 8.77667E−03 is the sum of IAEs belonging to power. The low level of absolute errors again proves that ECPA is more stable and accurate in evaluating parameters on TDM.

Table 13 IAE of ECPA on TDM

Item	Measured data		Simulated current data		Simulated power data
	V (V)	I (A)	Isim (A)	IAEI (A)	Psim (W)	IAEp (W)
1	−0.2057	0.7640	0.763983369	0.000016631	−0.157151379	0.000003421
2	−0.1291	0.7620	0.762604072	0.000604072	−0.098452186	0.000077986
3	−0.0588	0.7605	0.761337691	0.000837691	−0.044766656	0.000049256
4	0.0057	0.7605	0.760173797	0.000326203	0.004332991	0.000001859
5	0.0646	0.7600	0.759107702	0.000892298	0.049038358	0.000057642
6	0.1185	0.7590	0.758121453	0.000878547	0.089837392	0.000104108
7	0.1678	0.7570	0.757188654	0.000188654	0.127056256	0.000031656
8	0.2132	0.7570	0.756243649	0.000756351	0.161231146	0.000161254
9	0.2545	0.7555	0.755177338	0.000322662	0.192192632	0.000082118
10	0.2924	0.7540	0.753722375	0.000277625	0.220388423	0.000081177
11	0.3269	0.7505	0.751399134	0.000899134	0.245632377	0.000293927
12	0.3585	0.7465	0.747301416	0.000801416	0.267907558	0.000287308
13	0.3873	0.7385	0.740010611	0.001510611	0.286606110	0.000585060
14	0.4137	0.7280	0.727246892	0.000753108	0.300862039	0.000311561
15	0.4373	0.7065	0.706850247	0.000350247	0.309105613	0.000153163
16	0.4590	0.6755	0.675210518	0.000289482	0.309921628	0.000132872
17	0.4784	0.6320	0.630760768	0.001239232	0.301755951	0.000592849
18	0.4960	0.5730	0.571994772	0.001005228	0.283709407	0.000498593
19	0.5119	0.4990	0.499706187	0.000706187	0.255799597	0.000361497
20	0.5265	0.4130	0.413733713	0.000733713	0.217830800	0.000386300
21	0.5398	0.3165	0.317546219	0.001046219	0.171411449	0.000564749
22	0.5521	0.2120	0.212122974	0.000122974	0.117113094	0.000067894
23	0.5633	0.1035	0.102163228	0.001336772	0.057548546	0.000753004
24	0.5736	−0.0100	−0.008791790	0.001208210	−0.005042971	0.000693029
25	0.5833	−0.1230	−0.125543450	0.002543450	−0.073229494	0.001483594
26	0.5900	−0.2100	−0.208371531	0.001628469	−0.122939203	0.000960797
Sum of IAE	NA	NA	NA	2.12752E−02	NA	8.77667E−03

In Table 14, the parameters and RMSE metrics of ECPA are recorded in comparison with the above competing algorithms under the TDM model. Among all the solving algorithms, ECPA achieves the best RMSE value of 9.8248E−04 together with GOFPANM, which is also superior to the remaining competitors. This is followed by INFO, RLGBO, GBO, MLBSA, IJAYA, and EHHO. Also, the metrics based on the Wilcoxon rank sum test show that ECPA achieves the best performance under the TDM model and is highly competitive in evaluating the unknown PV parameters.

Table 14 Comparison among different algorithms on TDM

Algorithm	Iph (A)	Isd1 (A)	Isd2 (A)	Isd3 (A)	Rs (Ω)	Rsh (Ω)	n1	n2	n3	RSME	sig
GOTLBO	0.76081	1.180E−07	1.218E−07	2.273E−07	0.03647	55.17204	1.61704	1.88967	1.45916	9.8976E−04	+
GOFPANM	0.76078	2.260E−07	4.112E−13	7.493E−07	0.03674	55.48544	1.45102	1.45079	2.00000	9.8248E−04	=
IJAYA	0.76085	2.242E−07	2.917E−07	1.812E−08	0.03645	54.50943	1.98944	1.47274	1.93387	9.8604E−04	+
MLBSA	0.76078	1.941E−07	6.856E−08	4.308E−07	0.03675	55.18837	1.44135	1.98862	1.80788	9.8459E−04	+
RLGBO	0.76078	5.702E−07	2.465E−07	7.642E−10	0.03665	55.02026	1.99999	1.45856	1.44959	9.8271E−04	+
EHHO	0.76052	9.995E−07	1.305E−07	9.995E−07	0.03690	68.38694	1.99640	1.40826	1.99949	1.0441E−03	+
SMA	0.78826	1.000E−12	1.000E−06	6.909E−07	0.03662	61.47905	1.00000	2.00000	1.57917	6.9149E−02	+
GBO	0.76078	4.925E−07	5.121E−09	2.579E−07	0.03657	55.56671	1.98986	1.99902	1.46236	9.8378E−04	+
INFO	0.76081	7.822E−07	2.361E−23	2.224E−07	0.03675	55.32192	2.00000	1.00000	1.44970	9.8265E−04	+
CPA	0.75869	1.670E−07	8.493E−08	3.096E−07	0.03810	92.48047	1.54948	1.39006	1.86727	1.8180E−03	++
ECPA	0.76078	2.259E−07	2.719E−08	7.225E−07	0.03674	55.48621	1.45100	2.00000	2.00000	9.8248E−04

Then, in Table 15, the four statistical metrics on the RMSE values for ECPA and similar competing algorithms for the more complex TDM model are represented. The statistical metrics based on the Wilcoxon rank sum test show that the ECPA approach is able to outperform all competitors and achieve better results. Finally, the optimization convergence graph of ECPA compared with its competitors on TDM is shown in Figure 18. It is clear from the graph that ECPA again outperforms all competing algorithms, gets the optimal convergence accuracy and the fastest convergence speed, and is able to make better use of individual information to jump out of the immature optimal solution. This shows that ECPA can be a very competitive technique for the study and extraction of PV parameters.

Table 15 Statistics of RMSE obtained by competing algorithms on TDM

Algorithm	Max	Min	Mean	SD	sig
GOTLBO	2.91773E−03	9.89758E−04	1.48219E−03	3.96541E−04	+
GOFPANM	9.94427E−04	9.82488E−04	9.85451E−04	2.49695E−06	+
IJAYA	1.96859E−03	9.86037E−04	1.22805E−03	2.36893E−04	+
MLBSA	1.46441E−03	9.84585E−04	1.11274E−03	1.61879E−04	+
RLGBO	1.42333E−03	9.82712E−04	1.07685E−03	1.50788E−04	+
EHHO	3.15813E−02	1.04411E−03	6.54243E−03	6.84803E−03	+
SMA	3.66903E−01	6.91491E−02	2.13797E−01	7.97426E−02	+
GBO	1.86484E−03	9.83779E−04	1.25733E−03	2.39768E−04	+
INFO	1.43351E−03	9.82647E−04	1.11810E−03	1.78402E−04	+
CPA	1.00000E+00	1.81800E−03	6.09188E−01	4.88079E−01	+
ECPA	9.87808E−04	9.82485E−04	9.84104E−04	1.76360E−06

Figure 18. Comparison of the convergence of ECPA with competing algorithms in the TDM model

Experimental results of PV model

This section records the analytical experiments performed by ECPA on the simulated PV assembly. Similarly, the I–V and P–V curves of current and power obtained from the simulations on the PV model are recorded in Figure 19. As with the previous three PV models, the same conclusion is reached, and the perfect fit between the estimated and experimental data proves that ECPA can also accurately simulate the performance of TDM with high stability. In addition, Figure 20 shows a record plot of the ECPA technique by simulating the current data compared to the error values IAE and RE produced by the experimental data. Among them, the recorded values of IAE are less than 4.83284E−03 and the values of RE are in the range [−3.90848E−02, 5.02545E−02]. Among the more detailed values recorded in Table 16, the sum of IAEs for current and power reached 4.89237E−02 and 5.16890E−01. The results demonstrate that the proposed ECPA technique is capable of estimating the unknown parameters of PV components with low error and stability and accuracy.

Figure 19. (A) I–V and (B) P–V characteristics of PV model estimated with ECPA

Figure 20. Error values between simulated and experimental current data under the PV model. (A) IAE values and (B) RE values.

Table 16 IAE of ECPA on the PV model

Item	Measured data		Simulated current data		Simulated power data
	V (V)	I (A)	Isim (A)	IAEI (A)	Psim (W)	IAEp (W)
1	0.1248	1.0315	1.029119192	0.002380808	0.128434075	0.000297125
2	1.8093	1.0300	1.027381086	0.002618914	1.858840598	0.004738402
3	3.3511	1.0260	1.025741793	0.000258207	3.437363322	0.000865278
4	4.7622	1.0220	1.024107136	0.002107136	4.877003004	0.010034604
5	6.0538	1.0180	1.022291773	0.004291773	6.188749938	0.025981538
6	7.2364	1.0155	1.019930640	0.004430640	7.380626084	0.032061884
7	8.3189	1.0140	1.016363059	0.002363059	8.455022649	0.019658049
8	9.3097	1.0100	1.010496103	0.000496103	9.407415567	0.004618567
9	10.2163	1.0035	1.000628925	0.002871075	10.222725285	0.029331765
10	11.0449	0.9880	0.984548343	0.003451657	10.874237992	0.038123208
11	11.8018	0.9630	0.959521654	0.003478346	11.324082661	0.041050739
12	12.4929	0.9255	0.922838813	0.002661187	11.528933005	0.033245945
13	13.1231	0.8725	0.872599674	0.000099674	11.451212778	0.001308028
14	13.6983	0.8075	0.807274287	0.000225713	11.058285369	0.003091881
15	14.2221	0.7265	0.728336509	0.001836509	10.358474664	0.026119014
16	14.6995	0.6345	0.637138032	0.002638032	9.365610507	0.038777757
17	15.1346	0.5345	0.536213091	0.001713091	8.115370653	0.025926953
18	15.5311	0.4275	0.429511346	0.002011346	6.670783663	0.031238413
19	15.8929	0.3185	0.318774494	0.000274494	5.066251158	0.004362508
20	16.2229	0.2085	0.207389509	0.001110491	3.364459259	0.018015391
21	16.5241	0.1010	0.096167165	0.004832835	1.589075848	0.079858252
22	16.7987	−0.0080	−0.008325396	0.000325396	−0.139855838	0.005466238
23	17.0499	−0.1110	−0.110936495	0.000063505	−1.891456148	0.001082752
24	17.2793	−0.2090	−0.209247278	0.000247278	−3.615646485	0.004272785
25	17.4885	−0.3030	−0.300863594	0.002136406	−5.261652962	0.037362538
Sum of IAE	NA	NA	NA	4.89237E−02	NA	5.16890E−01

The optimal RMSE values and performance parameters determined by ECPA and some published competing algorithms of the same type for the PV model under the PV component are documented by Table 17. As can be seen, the ECPA method competitively achieves again the RMSE value of 2.42507E−03, the optimal metric, with a high potential for evaluating the PV model parameters.

Table 17 Comparison results for PV model

Item	Iph (A)	Isd (A)	Rs (Ω)	Rsh (Ω)	n	RMSE	sig
GOTLBO	1.03052	3.44316E−06	1.20200	968.52219	48.60038	2.42637E−03	+
GOFPANM	1.03051	3.48226E−06	1.20127	981.98229	48.64283	2.42507E−03	=
IJAYA	1.03046	3.51682E−06	1.20022	990.77992	48.68077	2.42527E−03	+
MLBSA	1.03051	3.48226E−06	1.20127	981.98384	48.64284	2.42507E−03	=
RLGBO	1.03051	3.48226E−06	1.20127	981.98084	48.64283	2.42507E−03	=
EHHO	1.02574	2.87896E−06	1.23398	1865.23123	47.90173	3.08151E−03	+
SMA	0.85920	8.10115E−07	0.68625	774.99571	44.21801	1.19818E−01	+
GBO	1.03051	3.48251E−06	1.20127	982.26559	48.64310	2.42507E−03	=
INFO	1.03051	3.48226E−06	1.20127	981.98236	48.64284	2.42507E−03	=
CPA	1.02267	3.32035E−06	1.12921	1112.50479	48.48872	8.66247E−03	+
ECPA	1.03051	3.48224E−06	1.20127	981.97170	48.64281	2.42507E−03

Table 18 further documents the statistical results of the RMSE values for the ECPA and competing algorithms under the PV model. From the table, it can be seen that the ECPA method not only achieves the optimal 2.42507E−03 for the maximum, minimum, and mean values of the RMSE metrics, but also achieves the optimal 5.07771E−13 for the standard deviation. The Wilcoxon rank sum test metric also proves that the ECPA solution has better accuracy and stability. Finally, Figure 21 visualizes the convergence curve plotted by ECPA for solving the PV model parameter estimation. Again, at each stage of the algorithm execution, ECPA maintains the fastest search speed and the best convergence accuracy compared to its strong competitors, so the ECPA method can be a more efficient and accurate technique to solve the PV system parameter evaluation problem.

Table 18 Statistics of RMSE obtained by competing algorithms on PV model

Algorithm	Max	Min	Mean	SD	sig
GOTLBO	2.74662E−03	2.42637E−03	2.48742E−03	9.30119E−05	+
GOFPANM	2.42507E−03	2.42507E−03	2.42507E−03	1.70014E−12	+
IJAYA	2.99555E−03	2.42527E−03	2.48707E−03	1.11521E−04	+
MLBSA	2.89530E−03	2.42507E−03	2.45467E−03	8.89495E−05	+
RLGBO	2.74251E−01	2.42507E−03	2.12270E−02	6.88630E−02	+
EHHO	4.38170E−01	3.08151E−03	6.55448E−02	1.05279E−01	+
SMA	1.00000E+00	1.19818E−01	5.09072E−01	1.89754E−01	+
GBO	2.55077E−03	2.42507E−03	2.43802E−03	2.59910E−05	+
INFO	6.56520E+00	2.42507E−03	7.64909E−01	1.50543E+00	+
CPA	1.00000E+00	8.66247E−03	8.04056E−01	3.78762E−01	+
ECPA	2.42507E−03	2.42507E−03	2.42507E−03	5.07771E−13

Figure 21. Comparison of the convergence of ECPA with competing algorithms in the PV model

Average CPU time

To more fully evaluate the computational efficiency of the ECPA algorithm, the key metric of CPU time consumed by the algorithm needs to be further documented as well. In this section, the average CPU time in milliseconds consumed by ECPA is recorded for 30 experiments run independently under four different models. Detailed results of the times are recorded in Table 19, and more visual results are also plotted by Figure 22. From the figure, it can be visualized that the algorithm time spent on evaluating the optimal parameters on different PV models varies for different techniques. Combining the results generated from the experiments performed on the four models shows that the original CPA algorithm, although spending the least average CPU time compared to some complex algorithms, performs so poorly for PV parameter extraction and performance simulation of the actual PV model that it often falls into immature convergence. However, the ECPA algorithm, after a mature balance between exploration and exploitation, is able to better solve the PV parameters and simulate the actual PV model performance with qualitative improvements in both solution accuracy and convergence speed, while only adding a few milliseconds and not significantly increasing the CPU time consumed. On the contrary, the GOFPANM algorithm, which is second only to ECPA in terms of solution performance, consumes a substantial increase in CPU time, reaching a 100 times or more than the ECPA algorithm. In particular, on the complex model of TDM, the average CPU time spent by ECPA is very close to that on the SDM, DDM, and PV models, while the time spent by GOFPANM on the TDM model is very disparate compared to the other three models. Furthermore, although the CPU time spent by GOTLBO, IJAYA, MLBSA, EHHO, SMA, GBO, INFO, and ECPA are very close, the results obtained from the experiments show that the actual performance demonstrated in simulating the actual parameters of the PV model is far behind that of ECPA. Therefore, the consistency of the ECPA method in evaluating the performance and efficiency of PV systems further demonstrates the efficiency of ECPA as a potential method that can evaluate the most accurate parameters in a short period of time while achieving the highest performance.

Table 19 Average CPU time of competitive algorithms on four PV models (ms)

Item	ECPA	GOTLBO	GOFPANM	IJAYA	MLBSA	RLGBO	EHHO	SMA	GBO	INFO	CPA
SDM	22.4260	21.8161	4785.4516	19.3063	19.0807	20.5234	23.0620	19.1964	19.1724	18.9359	17.4313
DDM	23.2104	22.3682	7119.3313	18.4964	18.3396	19.8813	22.6219	19.0193	19.0490	19.1656	17.8000
TDM	22.9052	21.7599	9291.8932	17.8786	17.6344	19.1927	21.8844	18.5146	18.6708	18.7089	17.4625
PV	21.6578	21.0667	4685.6031	19.4948	19.1422	20.5745	23.2365	19.0995	19.0974	18.9661	17.2964

Figure 22. Average CPU time of ECPA compared with other competing algorithms on four PV models

ECPA results through manufacturer's datasheet

This part of the experiment identifies valid parameters for thin-film (ST40), monocrystalline (SM55) and polycrystalline (KC200GT) PV modules to verify the stability and effectiveness of the proposed ECPA method under two PV models. The datasets used in the experiments were obtained from reliable information in the manufacturer's data sheets. Among them, these data set suites were always used by the researchers to detect the efficiency of the evaluation of different unknown parameters for the PV models with more precise test implications. The above three PV modules were used by Alam et al.⁵ in the flower pollination algorithm (FPA) to test the evaluation capability on PV models, and Yu et al. also used the data sets from these PV modules to test the detailed performance of the JAYA (PGJAYA) algorithm. In summary, these three extracted accurate data sets play a crucial role in the evaluation of the efficiency and stability of the algorithm under PV systems.

Two sets of experiments under SDM and DDM models were conducted by controlling different temperatures and different irradiations and extracting data precisely from the above data set. First, the first set of experiments was conducted when the temperature was constant at 20°C, and by varying the irradiation in the light conditions to 200, 400, 600, 800, and 1000 W/m², the exact parameter tables mined by the ECPA method are shown in Tables 20, 21, and 22, and ECPA demonstrates very low RMSE values. At this point, to verify the accuracy of the parameters extracted by the evaluation, the I–V characteristic diagrams of the approximate currents simulated by ECPA under the three PV modules ST40, SM55 and KC200GT are plotted by Figures 23, 24, and 25. It can be visually seen that the tapped PV parameters make the estimated and measured curves match highly and the ECPA demonstrates a high degree of stability performance. Next, the second set of experiments was conducted, when the irradiation of the control light condition was a constant 1000 W/m², and the ambient temperature was set at 25°C, 40°C, 50°C, and 70°C for comparison experiments. At this point the results evaluated by the ECPA method for the real parameters of the SDM and DDM models are recorded in Tables 23, 24, and 25. Similarly, the I–V characteristic diagrams of the simulated currents were used to verify the match between the simulated data and the measured data extracted from the data set by the ECPA method for different temperatures, as documented in Figures 26, 27, and 28. The two sets of experiments allow us to conclude that the ECPA technique proposed in this study is able to simulate the actual performance of the PV model stably and extract the important unknown parameters of the PV model accurately when the ambient temperature and the irradiation of the light undergo turbulent changes. The unity of efficiency and stability determines that ECPA can be a new competitive method to deal with the parameter evaluation of PV systems.

Table 20 Estimation of ST40 parameters using ECPA at different irradiations and the same temperature of 25°C

Parameters	Irradiance
	200 W/m2	400 W/m2	600 W/m2	800 W/m2	1000 W/m2
SDM
Iph (A)	0.533137328	1.067544952	1.604810356	2.138014261	2.675799818
Isd (A)	1.42971E−06	1.84852E−06	1.44187E−06	1.15821E−06	1.52898E−06
Rs (Ω)	1.185706678	1.080603387	1.112612769	1.125282297	1.113215120
Rsh (Ω)	344.9840184	362.5015039	347.6870904	332.8996835	357.5964697
n	1.747111231	1.778514047	1.745124233	1.718684331	1.750340344
RMSE	4.77201E−04	6.30725E−04	6.74036E−04	7.73905E−04	7.34099E−04
DDM
Iph (A)	0.533248145	1.067529700	1.604774442	2.137827099	2.675799817
Isd1 (A)	4.33406E−07	1.83805E−06	1.33899E−06	2.94244E−11	1.52880E−06
Isd2 (A)	2.58026E−05	3.79261E−08	1.96085E−06	1.21562E−06	3.23820E−13
Rs (Ω)	1.442600810	1.080547183	1.116743193	1.124795675	1.113225921
Rsh (Ω)	356.6061365	362.8945703	349.3612557	336.4094110	357.5983749
n1	1.606718347	1.784578974	1.736635002	1.208495394	1.750326676
n2	3.061556909	1.667657351	2.860894360	1.725751899	1.729278210
RMSE	4.55474E−04	6.31744E−04	6.77171E−04	7.86618E−04	7.34099E−04

Table 21 Estimation of SM55 parameters using ECPA at different irradiations and the same temperature of 25°C

Parameters	Irradiance
	200 W/m2	400 W/m2	600 W/m2	800 W/m2	1000 W/m2
SDM
Iph (A)	0.692013715	1.382843964	2.070898191	2.760380985	3.450104156
Isd (A)	1.30912E−07	1.00439E−07	1.55494E−07	1.43951E−07	1.71121E−07
Rs (Ω)	0.312431233	0.396636759	0.330507142	0.337583871	0.329155854
Rsh (Ω)	438.0397623	427.0584105	450.0317518	459.8843916	483.8760151
n	1.370891394	1.352004261	1.387523390	1.381144407	1.395736829
RMSE	5.20542E−04	7.07608E−04	8.23950E−04	6.68581E−04	1.14622E−03
DDM
Iph (A)	0.692013614	1.382729384	2.070896545	2.760377006	3.450150505
Isd1 (A)	9.54792E−19	8.51172E−09	1.55514E−07	8.22861E−08	1.67052E−07
Isd2 (A)	1.30925E−07	9.62699E−07	7.86038E−18	6.30748E−08	2.98040E−08
Rs (Ω)	0.312405751	0.481738323	0.330502520	0.337721257	0.329675493
Rsh (Ω)	438.0433675	445.0229737	450.0685123	460.0307621	482.2060735
n1	3.140284438	1.190345502	1.387534220	1.371254531	1.393855846
n2	1.370900438	1.765605764	3.977983180	1.398732991	1.966742394
RMSE	5.20542E−04	6.58271E−04	8.23949E−04	6.69216E−04	1.15022E−03

Table 22 Estimation of KC200GT parameters using ECPA at different irradiations and the same temperature of 25°C

Parameters	Irradiance
	200 W/m2	400 W/m2	600 W/m2	800 W/m2	1000 W/m2
SDM
Iph (A)	1.646133747	3.287851232	4.934308104	6.571125421	8.216889308
Isd (μA)	5.32604E−10	1.48942E−09	3.86161E−09	9.91520E−10	2.24202E−09
Rs (Ω)	0.379319097	0.353587980	0.337337800	0.356650142	0.343813883
Rsh (Ω)	690.9887710	752.0305726	742.9984321	755.2939167	763.6023949
n	1.004244581	1.055024114	1.104022895	1.037115878	1.076411792
RMSE	1.41875E−03	1.42620E−03	1.29767E−03	1.64751E−03	1.53903E−03
DDM
Iph (A)	1.646107285	3.287850399	4.934307782	6.570691140	8.216842190
Isd1 (A)	4.83112E−10	4.50486E−10	3.86152E−09	5.42511E−10	1.39405E−06
Isd2 (A)	9.72285E−09	1.04577E−09	6.83653E−13	2.96896E−08	2.23488E−09
Rs (Ω)	0.382464864	0.353612099	0.337337745	0.360386735	0.343839393
Rsh (Ω)	693.3349292	752.0144040	743.0167469	787.3127145	767.1529583
n1	1.000017009	1.063741319	1.104021720	1.012018978	3.850109410
n2	1.607272657	1.052032596	2.636693560	1.464403353	1.076258987
RMSE	1.41454E−03	1.42645E−03	1.29767E−03	1.51068E−03	1.53893E−03

Figure 23. Estimated and measured ST40 properties at the same temperature and under different irradiation. (A) Single diode model (SDM) and (B) double diode model (DDM).

Figure 24. Estimated and measured SM55 properties at the same temperature and under different irradiation. (A) Single diode model (SDM) and (B) double diode model (DDM).

Figure 25. Estimated and measured KC200GT properties at the same temperature and under different irradiation. (A) Single diode model (SDM) and (B) double diode model (DDM).

Table 23 Estimation of ST40 parameters using ECPA at different temperatures and the same irradiation of 1000 W/m²

Parameters	Temperatures
	25°C	40°C	50°C	70°C
SDM
Iph (A)	2.675799332	2.680912203	2.691968060	2.692329482
Isd (A)	1.52887E−06	5.66629E−06	1.86808E−05	8.75214E−05
Rs (Ω)	1.113220715	1.129294243	1.149590463	1.125889237
Rsh (Ω)	357.5969515	364.1031140	295.0300872	367.7528496
n	1.750331493	1.722562130	1.717576917	1.727319018
RMSE	7.34099E−04	1.32141E−03	1.82326E−03	7.77718E−04
DDM
Iph (A)	2.675686219	2.680798650	2.688901054	2.692329477
Isd1 (A)	9.58702E−07	1.01114E−07	1.86572E−09	8.75213E−05
Isd2 (A)	9.51564E−07	5.79537E−06	3.81945E−05	5.61665E−10
Rs (Ω)	1.116203874	1.129926168	1.271904170	1.125888711
Rsh (Ω)	360.5769305	366.7055432	381.6014442	367.7532252
n1	1.713414625	1.554284243	1.016591054	1.727319869
n2	1.947343552	1.735663456	1.902407336	1.727298804
RMSE	7.45421E−04	1.32669E−03	1.58958E−03	7.77718E−04

Table 24 Estimation of SM55 parameters using ECPA at different temperatures and the same irradiation of 1000 W/m²

Parameters	Temperatures
	25°C	40°C	60°C
SDM
Iph (A)	3.450103669	3.469137175	3.494609802
Isd (A)	1.71176E−07	1.14510E−06	6.90967E−06
Rs (Ω)	0.329144438	0.313096115	0.318704964
Rsh (Ω)	483.9146774	533.0796934	484.8695861
n	1.395763810	1.417838765	1.405144373
RMSE	1.14622E−03	3.78881E−03	3.78039E−03
DDM
Iph (A)	3.450103564	3.469065374	3.494608469
Isd1 (A)	1.71154E−07	3.58867E−07	6.90950E−06
Isd2 (A)	1.42068E−08	1.21908E−06	1.34448E−17
Rs (Ω)	0.329147705	0.316904117	0.318705726
Rsh (Ω)	483.9004829	541.7888911	484.8839654
n1	1.395752859	1.349550930	1.405141753
n2	3.911259362	1.537469533	3.934011168
RMSE	1.14621E−03	3.89753E−03	3.78039E−03

Table 25 Estimation of KC200GT parameters using ECPA at different temperatures and the same irradiation of 1000 W/m²

Parameters	Temperatures
	25°C	50°C	75°C
SDM
Iph (A)	8.216892762	8.295305561	8.377663878
Isd (A)	2.24181E−09	1.25954E−07	1.63080E−06
Rs (Ω)	0.343814913	0.335654348	0.342498214
Rsh (Ω)	763.4733387	953.8752252	790.5069951
n	1.076407256	1.117292974	1.101478803
RMSE	1.53903E−03	2.74651E−03	4.47293E−03
DDM
Iph (A)	8.216858068	8.295308004	8.377662978
Isd1 (A)	2.20345E−09	6.95879E−12	4.64020E−14
Isd2 (A)	5.15054E−09	1.25959E−07	1.63082E−06
Rs (Ω)	0.343908527	0.335656497	0.342497964
Rsh (Ω)	765.8512056	953.7567005	790.5575257
n1	1.075616661	1.030106987	3.919760669
n2	1.652379076	1.117311128	1.101479928
RMSE	1.53941E−03	2.74653E−03	4.47293E−03

Figure 26. Estimated and measured ST40 properties under the same irradiation and at different temperatures. (A) Single diode model (SDM) and (B) double diode model (DDM).

Figure 27. Estimated and measured SM55 properties under the same irradiation and at different temperatures. (A) Single diode model (SDM) and (B) double diode model (DDM).

Figure 28. Estimated and measured KC200GT properties under the same irradiation and at different temperatures. (A) Single diode model (SDM) and (B) double diode model (DDM).

Owing to the strong optimization capability, the ECPA can also be a great potential tool to deal with some complex problems such as drug discovery,^132,133 disease module identification,^134,135 and pharmacoinformatic data mining.^136,137 It also can be combined with other tools for tackling the practicle problems such as fault diagnosis,^138,139 active surveillance,¹⁴⁰ urban road planning,¹⁴¹ human motion capture,¹⁴² and kayak cycle phase segmentation.¹⁴³

CONCLUSIONS AND FUTURE DIRECTIONS

In this study, an enhanced ECPA algorithm is proposed to better handle the optimization problem of high-dimensional multimodal functions to accurately and stably evaluate the parameters to be evaluated in solar PV systems, and ultimately to achieve more efficient utilization of solar energy as a renewable energy source. The proposed ECPA incorporates extremal optimization and local search strategy based on NMs, which allows more frequent communication between individuals in the population, thus enabling a more comprehensive exploration and more accurate exploitation of the algorithm. ECPA adjusts the mutation scheme of individuals through extremal optimization strategy to promote the migration of the population toward the direction close to the optimal solution, and the incorporation of NMs local search strategy can continuously improve the genes of inferior individuals so that they can escape from the immature convergence solution, thus overcoming the defects such as stagnation and easy to fall into immature convergence in the development stage of the original CPA.

Based on the multiple competitive experiments explored in this study and their results the following conclusions can be drawn.

• I.

  ECPA is tested on the IEEE CEC 2020 function set with 10 advanced competitive algorithms such as OBSCA, SCADE, and HGWO, and the competitive performance and result comparison of ECPA are well analyzed. The mean, standard deviation, Wilcoxon singed-rank test metrics and Friedman test ranking results of ECPA are further reliable and accurate compared to other competitive algorithms, both on simple unimodal and multimodal functions and on complex composite and hybrid functions.

• II.

  Based on the convergence curves, it can be seen that ECPA is able to achieve better convergence accuracy and faster convergence in the process of CEC 2020 function test suite and PV model unknown parameter extraction.

• III.

  The competitive experiments for PV parameter evaluation demonstrate that ECPA has the lowest RMSE value, maximum value, minimum value, mean value and standard deviation when comparing nine already existing competitive algorithms such as GOTLBO, GOFPANM and IJAVA, showing a more accurate and efficient level.

• IV.

  The stability simulation experiments under the manufacturer's datasheet showed that the ECPA maintained the same highly stable performance under different irradiation and temperature.

• V.

  All evaluation experiments and results demonstrate that ECPA can be a novel and promising technique to solve optimization problems of multimodal functions and parametric evaluation of solar PV parameters. Thus, it can become a novel and high standard technique for identifying parameters to be determined in complex PV systems.

It is worth noting that the values of the parameters in ECPA need to be carefully tuned because the values of the parameters affect the quality of the solution and the parameter settings depend on the different problems. Second, the same limitation as many advanced hybrid metaheuristic algorithms exists, that is, the ECPA consumes additional computational resources compared to the original CPA due to the revision of the new structure, and it is still worthwhile to further investigate how to balance the computational resources consumed and the accuracy exhibited by the algorithm.

Although the above experiments have verified the efficiency of ECPA in solving PV parameter problems, there are still more aspects that deserve further exploration to explore a wider range of application scenarios. For example, ECPA can be used to extend to other solar cell datasets to expand its performance more. Second, to further improve the stability of ECPA in actual PV system applications, more environmental variables can be added for stability extension testing, such as wind speed, humidity, air pressure, and other variables. In the future, ECPA can also be extended to solve discrete optimization problems, multiobjective problems,^144,145 or parametric evaluation of renewable energy sources such as geothermal energy, biomass energy, and so on.

NOMENCLATURE

• CPA
• colony predation algorithm
• DDM
• double diode model
• ECPA
• enhanced colony predation algorithm
• IAE
• absolute error
• KC200GT
• multicrystal photovoltaic module
• N
• the number of individual
• NMs
• Nelder-Mead simplex algorithm
• Photowatt-PWP 201
• photovoltaic module
• PV
• photovoltaic
• RE
• relative error
• RMSE
• root mean square error
• RTC France
• solar cell
• SDM
• single diode model
• SM55
• mono-crystalline PV module
• ST40
• thin-film PV module
• STC
• standard test conditions
• TDM
• three diode model
• $${I}_{{\rm{L}}}$$
• output current
• $${I}_{\mathrm{ph}}$$
• photo-generated current
• $${I}_{{\rm{d}}}$$, $${I}_{{\rm{d}}1}$$, $${I}_{{\rm{d}}2}$$, $${I}_{{\rm{d}}3}$$
• diode current
• $${I}_{\mathrm{Sh}}$$
• shunt resistor current
• $${I}_{{sd}}$$, $${I}_{{sd}1}$$, $${I}_{{sd}2}$$, $${I}_{{sd}3}$$
• reverse saturated current
• $${R}_{S}$$
• series resistor
• $$n$$, $${n}_{1}$$, $${n}_{2}$$, $${n}_{3}$$
• diode ideality factor
• $$k$$
• Boltzmann constant
• $$T$$
• Kelvin temperature
• $${N}_{P}$$
• number of parallel solar cells
• $${N}_{S}$$
• number of series solar cells
• $${I}_{{SC}}$$
• short circuit current
• $$G$$
• irradiance
• $$\alpha $$
• temperature coefficient
• $${I}_{{SC\_STC}}$$
• short circuit current at STC
• $${G}_{{STC}}$$
• irradiance at STC
• $${T}_{{STC}}$$
• Kelvin temperature at STC
• $$S$$
• the power of the target individual
• $${D}_{1}$$
• the distance of the animal population transition
• $${r}_{i}$$
• random number
• $$\alpha $$
• reflection coefficient
• $$\beta $$
• expansion coefficient
• $$\gamma $$
• compression coefficient
• $$\delta $$
• Shrinkage coefficient
• $${ub}$$
• upper boundary
• $${lb}$$
• lower boundary
• $${Max\_FE}s$$
• maximum number of function evaluations
• $$D$$
• the dimension of the problem
• $${R}_{\mathrm{sh}}$$
• parallel resistor

ACKNOWLEDGMENTS

This study was supported in part by the National Natural Science Foundation of China (62076185 and U1809209), Wenzhou Scienceand Technology Bureau (ZG2020030), Humanities and Social Science Research Planning Fund Project of the Ministry of Education (20YJA790090). We acknowledge Fangjun Kuang for her contribution to the analysis of the data, and the comments of the reviewers.