Application of an improved pelican optimization algorithm based on comprehensive strategy in PV parameter identification

Abstract

This paper proposes an improved Pelican optimization algorithm (IPOA) based on comprehansive strategy for the parameter identification of photovoltaic models. Firstly, the cubic chaotic mapping and the refraction reverse learning strategy are used to initialize the pelican population and enhance its diversity. Secondly, the position update formula of the Pelican optimization algorithm in the global detection phase is replaced by the position update formula of the red-tailed Eagle optimization algorithm in the soaring phase to obtain the adequacy of the Pelican optimization algorithm in solution space search. Further introducing the catchy variation strategy aims to improve the algorithm’s global search ability. Finally, the reverse solution generated by the lens imaging principle can provide a new search direction through the mirror reverse learning strategy when the Pelican optimization algorithm falls into the local optimal. The CEC2022 test function performed analysis and comparison with eight meta-heuristic algorithms. The Wilcoxon rank sum test verified the significance of the algorithm. In addition, the IPOA was used to optimize the critical parameters of the PV model to solve the problem of actual parameter identification of the single-diode and double-diode photovoltaic module models. The experimental results indicate that the IPOA outperforms other classical swarm intelligence algorithms in both convergence speed and solving accuracy. Furthermore, this optimization method yields the smallest mean square error across all types of solar cells, demonstrating the superiority of the proposed algorithm.

Introduction

Photovoltaic power generation systems¹ mainly include core components such as a controller, battery, inverter, and solar cell. Solar cells can convert solar energy into electricity by using the photovoltaic effect. The voltage generated by a single photovoltaic cell is limited and is insufficient to meet the electricity required for production and life. Therefore, multiple solar cells must be connected or combined into a module. Then, multiple components are connected to form a photovoltaic array, increasing the output voltage and current. The inverter converts the direct current generated by the photovoltaic array into alternating current, which is connected to the grid through the grid-connected system or directly supplied to the local load for home or industrial power supply. A photovoltaic power generation system comprises an off-grid and grid-connected system that meets different needs and application scenarios. However, there are always challenges with the photoelectric conversion efficiency and construction costs of photovoltaic power generation systems, making it crucial to improve their conversion efficiency. The photovoltaic cells, being susceptible to environmental factors, are the basis of the system. Therefore, accurately identifying the unknown parameters of photovoltaic cells in the photovoltaic system is of great significance. This task is crucial to stabilize the photovoltaic power generation system and improve photoelectric conversion efficiency.

The potential of meta-heuristic methods inspired by nature has become increasingly promising with the continuous advancement of computational intelligence technology. These methods have garnered significant attention and are being widely used to tackle nonlinear and multi-modal problems with their wide applicability. Their simplicity, efficiency, ease of implementation, and non-strict requirement of objective function make them particularly suitable for parameter identification of photovoltaic models. In recent years, the optimization algorithms and improvement strategies for PV model parameter identification have achieved many results. For example, a classified perturbation mutation-based particle swarm optimization (CPMPSO) algorithm was proposed in². It is used to optimize the model parameters of proton exchange membrane fuel cells and solar cells, and good experimental results are obtained under various light intensity and temperature conditions. Zhao Xiaohao et al.³ proposed a Fireworks algorithm with mixed differential mutation (DEFWA) to identify solar cell model parameters accurately. However, these methods may introduce additional adjustment parameters when solving other engineering problems. In their literature, Zhang Yiying et al.⁴ proposed a backtracking search algorithm by reusing differential vectors (BSARDVs). The algorithm can identify the model parameters of solar cells accurately and stably. Literature⁵ proposes an Improved symbiotic organism search algorithm (ImSOS) for PV module model parameter identification, which uses different strategies at different stages. The quasi-reflection learning mechanism is adopted in the initial stage, and the improved revenue factor strategy is adopted in the mutually symbiotic search stage, which realizes the balance of exploration and development ability and proves the superiority of ImSOS in the parameter identification of the solar cell model. Pankaj Sharma et al.⁶ predicted the parameters of solar photovoltaic cells/components, SDM, DDM, amorphous silicon aSi: H and PVM 752 GaAs thin films through the HFGD algorithm. Chappani Sankaran Sundar Ganesh et al.⁷ introduced a new method, namely the Golden Jackal Optimizer based on reinforcement learning (RL-GJO) and conducted strict tests on the photovoltaic parameter estimation benchmark dataset to highlight the superiority of RL-GJO. Ayşe Beşkirli et al.⁸ proposed a multi-strategy tree seed algorithm (MS-TSA) and applied it to the problem of estimating the parameters of solar photovoltaic models, achieving excellent results. Pankaj Sharma et al.⁹ proposed the hybrid flower gray difference (HFGD) algorithm, which has the additional advantages of FPA, GWO and DE algorithms, and solved the problem of parameter extraction for SDM, DDM, amorphous silicon aSi: H and pvm752 GaAs thin films. While improved algorithms can yield satisfactory results, no single technique is universally superior for all optimization problems. Challenges such as difficulty in selecting control parameters, slow convergence speeds, and extensive computational requirements can arise. The ongoing research is being conducted to enhance the efficiency of algorithms and tailor them to specific applications. A review of the literature indicates a pressing need for new optimization algorithms to address real-world problems effectively.

The Pelican Optimization Algorithm¹⁰ (POA) is an intelligent optimization algorithm inspired by pelican predation behavior. This algorithm was first proposed by Pavel Trojovsky and Mohammad Dehghani in 2022. The northern pelican is large, with a long beak and a large pouch in its throat for catching and swallowing prey. The birds enjoy groups and social life and live in groups of hundreds of pelicans. In the Pelican optimization algorithm, the behavior and strategy of pelicans during attack and hunting are simulated to update the candidate solution. Hunting is divided into two stages: approaching the prey (exploration phase) and surface flight (development phase). The Pelican optimization algorithm has shown application potential in many fields because of its unique search mechanism and efficient problem-solving ability, including engineering design optimization, machine learning, pattern recognition, photovoltaic models, and so on. The POA provides a novel optimization strategy to solve complex single and multi-objective optimization problems by simulating pelican hunting behavior. The original Pelican algorithm experiences issues due to the randomness in its initialization. This can lead to reduced population diversity and lower convergence efficiency, making it prone to getting trapped in local optimum solutions. During the iterative process, the Pelican Optimization Algorithm (POA) may struggle with insufficient population diversity, which further restricts its ability to search globally.

An integrated strategy improved Pelican optimization algorithm (IPOA) was proposed to improve the local optimization of swarm intelligent optimization algorithm and improve the global optimization ability. Firstly, Cubic chaotic initialization plus reverse refraction mechanism was used to initialize pelican population to enrich its diversity. Secondly, the position update formula of Pelican prey recognition stage is replaced by the position update formula of high-altitude flying stage of red-tailed eagle optimization algorithm for the adequacy of Pelican prey recognition stage in the solution space search and the performance of solving the optimization problem. Then, by using Cauchy variation strategy, the local exploration ability of the late iteration is enhanced, and the convergence speed of Pelican optimization algorithm is improved. Finally, the reverse solution generated by the lens imaging principle can provide a new search direction when the Pelican optimization algorithm falls into the local optimal, increase the probability of finding the global optimal solution, and improve the global optimization ability, so that it can jump out of the local optimal in the later iteration. The double diode model and the photovoltaic module model by testing the 12 test functions of CEC2022 and the actual parameter identification problems of the single diode model. The results show that the IPOA algorithm can break through the local optimal solution, obtain higher accuracy, and has stronger global search ability than other solutions.

The main contributions of this paper are as follows: (1) An Improved Pelican Optimization Algorithm (IPOA) is proposed to accurately estimate the parameters of the photovoltaic (PV) model. (2) To validate the effectiveness of the improved IPOA, eight algorithms were selected to conduct rigorous tests and evaluations of IPOA’s performance on the CEC2022 test functions. (3) In estimating the parameters of the PV model, the IPOA algorithm demonstrates superior performance compared to the original Pelican Optimization Algorithm and seven other optimization algorithms in terms of accuracy, convergence speed, and stability.

This paper is mainly divided into six chapters to explain. The main contents of each chapter are as follows: The first chapter briefly analyzes the research background, significance, and development trend of photovoltaic power generation systems—chapter 2 Overview of the photovoltaic model. In Chap. 3, the theoretical idea and mathematical model of the improved Pelican optimization algorithm are explained in detail. The CEC2022 test function is used to test and evaluate the improved algorithms, and the stability and accuracy of their performance are mainly observed. In addition, the simulation experiment is carried out in the corresponding simulation environment, and the comparison is made with the existing algorithm to prove the improved algorithm’s superiority further. In Chap. 5, the improved Pelican optimization algorithm is used to solve the problem of actual parameter identification of the single-diode model and double-diode model photovoltaic module model, and IPOA is used to optimize the critical parameters of the photovoltaic model, demonstrating the practical application of the research. The sixth chapter is the summary and prospect, expounds the main content of this research, and clarifies the relevance and importance of the research in the field of photovoltaic power generation and optimization algorithms.

Photovoltaic model

Photovoltaic cell model

Photovoltaic cells are devices that convert light energy into electrical energy, in which electrical loss and optical loss are the main factors affecting the efficiency of photovoltaic cells. Accurately constructing the equivalent circuit model of photovoltaic cells is the key to improving the photoelectric conversion efficiency. In recent years, a variety of photovoltaic cell models have been developed, including single-diode model, double-diode model and three-diode model.

Single diode model

As shown in Fig. 1, the single diode equivalent circuit¹¹ is composed of the element diode D₁, the photo-generated current I_ph, the series resistance R_s and the parallel resistance R_sh, where the series resistance R_s and the parallel resistance R_sh represent the loss part of the photovoltaic cell in the production and manufacture. According to the formula, the output current can be obtained as I:

1

Fig. 1.

Equivalent circuit diagram of single diode photovoltaic model.

In the formula, I_d is the diode current, and Ish is the parallel resistance current. Then, further following Kirchhoff ‘s Current Law (KCL), I_d and I_sh are expressed as follows:

2

3

Where I_sd represents the reverse saturation current of the diode, z represents the ideal coefficient of the diode, V represents the output voltage of the battery, and V_t represents the point voltage. The formula of V_t is as follows:

4

Here, k is the Boltzmann constant (1.380650310^− 23 J/K), q is the electron charge (1.6021764610^− 19 C), and T is the absolute temperature of the photovoltaic cell. It can be seen from Formulas (1)–(4) that the current output of the photovoltaic cell in the single diode model is as follows:

5

Obviously, there are five unknown parameters to be identified in the single diode model.

Dual diode model

In photovoltaic cells, the charge recombination effect leads to a large current loss. However, the single-diode model can only simplify the description of the case where there is no composite loss in the depletion region, so it is difficult to accurately describe such current losses. In order to further improve the simulation accuracy of the P-N junction, a second diode (D₂) needs to be added to obtain a more accurate double diode model¹², as shown in Fig. 2. In this model, I_d1 represents the current caused by the charge carrier flowing through the first diode, and Id₂ represents the current caused by the charge carrier flowing through the second diode. Two parameters are generated, and the model is also called a seven-parameter model due to the addition of a second diode. The current output I can be expressed by the following formula:

6

7

8

Fig. 2.

Equivalent circuit diagram of dual diode photovoltaic model.

Among them, I_sd1 and I_sd2 represent the diffusion current of the first diode and the saturation current of the second diode. z₁ and z₂ are the ideal coefficients of the first diode and the second diode, respectively.

Therefore, the current output I can be expressed as:

9

Three-diode model

The application of the three-diode model provides a solution to avoid the grain boundary and leakage current problems in the double-diode model and further improves the accuracy of photovoltaic cell modeling. The model of three diodes¹³ is shown in Fig. 3. The current flowing through the three diodes is I_d1, I_d2 and I_d3 in turn. I_d3 is used to describe the influence of grain boundary and leakage current. The output current of the three diodes is calculated according to formula (10).

10

11

12

13

Fig. 3.

Equivalent circuit diagram of three diode photovoltaic mode.

where I_sd3 is the saturation current of the third diode; z₃ the ideal coefficient of the third diode.

Thus, the current output I can be expressed as:

14

In the three-diode photovoltaic cell modeling process, nine parameters need to be determined, including (I_ph, I_sd1, I_sd2, I_sd3, z₁, z₂, z₃, R_s and R_sh) need to be determined.

Through the accurate establishment of the TDM photovoltaic cell model, its internal and external performance characteristics in extreme environment can be deeply analyzed, which has guiding significance for the evaluation and control of photovoltaic power generation system.

Photovoltaic module model

Single diode PV module models

Figure 4 shows the equivalent circuit diagram of the single-diode PV module model.

Fig. 4.

Single diode photovoltaic module model.

The PV module model (PVM) is based on a single diode¹⁴, which has N_s photovoltaic cells in series and N_p photovoltaic cells in parallel, the characteristic equation of the PVM model is:

15

Among them, N_s and N_p represent the number of photovoltaic cells in series and parallel in photovoltaic modules, respectively.

As shown in Formula (15), the unknown parameters in the single-diode PV module model are the same as those in the single-diode model.

Objective function

The solar cell model¹⁵ plays a key role in the simulation, control and optimization of solar photovoltaic (PV) power generation systems and is also influential in the evaluation of cell performance. However, the accuracy of the unknown parameters in the model is directly related to the accuracy of the model. Only if the unknown parameters are correctly obtained, can we obtain the current-voltage characteristic curve fitting results that match the actual cell behavior. This is the key to ensure that the solar cell model accurately simulates the solar PV power generation system. To obtain more accurate unknown parameters, the measured current samples are often compared with the simulated current data derived from the unknown parameters. Among them, the most used objective function is the root mean square error (RMSE), which can more accurately reflect the degree of fit between the simulated current data and the measured data. In this paper, RMSE is used as the objective function, and its expression is shown in Eq. (16).

16

Where N denotes the number of actual measured datasets, X denotes the unknown parameter to be identified in the PV model, and denotes the error function between the actual measured data and the predicted data, the error functions of the four models are as follows:

Single diode model error function:

17

Double diode model error function:

18

Three-diode model error function:

19

Single diode photovoltaic module model error function:

20

Improved pelican optimization algorithm

Pelican optimization algorithm

The Pelican Optimization Algorithm¹⁶ (POA) was proposed by Pavel et al.in 2022 under the influence of the behavior and strategic behavior of the hunting. The pelican has a large body, a long mouth, and a large bag in the throat to capture and devour the prey. When the prey is positioned, the pelican swoops toward the prey at 10 ~ 20 m. Then, it spreads its wings on the water surface and forces the fish to enter the shallow water area, so that it can easily capture the prey. When captured, a large amount of water enters the pelican ‘s mouth and swallows it in front of the fish. The algorithm model simulates the predatory behavior of pelicans, which is divided into a global exploration stage and local exploration stage.

Initialization

According to the boundary of the search area, the mathematical description of the initialization of the pelican population is as follows:

21

Where, is the j-dimensional position of the ith pelican; rand is a random number in the range of [0,1]; and are the upper and lower bounds of the jth dimension of the problem, respectively. In the pelican optimization algorithm, the pelican population can be represented by the following population matrix:

22

Where X is the population matrix of pelicans; is the location of the pelican; N is the population size of pelicans; m is the dimension of the solution problem. In the pelican optimization algorithm, the objective function of the solution problem can be used to compute the objective function value of the pelican; the objective function value of the pelican population can be expressed as a vector of objective function values:

23

Phase I: towards the prey (exploration phase)

In the first stage, the pelican recognizes the location of its prey and then moves towards this identified area. Modeling the pelican’s strategy of approaching the prey allows the POA algorithm to scan the search space, which in turn leverages the POA algorithm’s ability to explore different regions in the search space. One point to be emphasized in the POA algorithm is that the location of the prey is randomly generated in the search space, which increases the exploration ability of the POA algorithm in solving the exact search problem. Mathematical modeling of the above concepts and approximation of the prey strategy are as follows:

24

where is the position of the j dimension of the ith pelican after the phase 1 update; I is a random integer of 1 or 2; is the position of the j_th dimension of the prey; and is the value of the objective function of the prey.

In the POA, the new position of the pelican is accepted if the objective function value is improved at that position. In this type of update, the algorithm cannot move to a non-optimal region also known as effective update. This process can be described by the following equation:

25

Where: is the new position of the i pelican; is the objective function value of the new position of the i pelican after the first stage update.

Phase 2: wearing up on the water (mining phase)

In the second stage¹⁷, when pelicans reach the surface, they spread their wings above the water, move the fish upward, and then place the prey in their throat pouches. This strategy of pelicans’ surface flight allows them to catch more fish in the area they are attacked. This behavior of pelicans during hunting is mathematically modeled as:

26

where is the position of the j_th dimension of the ith pelican after the phase 2 update; R is a random integer of 0 or 2; t is the number of current iterations; and T is the maximum number of iterations.

At this stage, valid updates are also used to accept or reject new pelican locations, which are mathematically modeled as whether the location is accepted or not:

27

where is the new position of the i pelican; is the objective function value of the new position of the ith pelican after the first stage update.

According to the above algorithm design steps, the specific algorithm flow is represented by the following pseudo code.

Algorithm 1.

Pseudo-code of the POA.

Integrated strategy enhanced pelican optimization algorithm

The Pelican Optimization Algorithm (POA) is an intelligent algorithm that simulates the hunting behavior of pelicans. This algorithm has the advantages of strong global search ability and fast convergence speed, but also has shortcomings, such as: (1) Slow convergence speed: in some complex problems, the Pelican Optimization Algorithm may need more iterations to reach a satisfactory solution, which affects the efficiency of the algorithm. (2) Optimization accuracy problem: the algorithm may have a lack of accuracy in the search for the optimal solution, especially in the optimization problems with high dimensions and complexities. In high dimensionality and complexity optimization problems, this may result in finding a solution that is not globally optimal. (3) Easy to fall into local optimum: Although the Pelican optimization algorithm has strong global search ability, in some cases, the algorithm may still fall into the local optimum, especially in the search space where there are many local optimums. (4) Parameter tuning sensitivity: The performance of the algorithm relies on the initial parameter settings to a certain extent, and the improper choice of parameters may affect the search effect and the quality of the final solution of the algorithm. (5) Adaptability problem: the algorithm may need to be adapted and optimized for different optimization problems to adapt to the characteristics and needs of a specific problem, which may increase the complexity of the application of algorithms.

Improvements in population initialization

Elite individuals are selected in a larger range to realize the improvement of the diversity of the initial pelican population and the quality of the pelican individuals by combining the chaotic solution and the inverse solution through the elitism idea.

(1)Cubic chaos mapping.

Chaos theory reveals a principle of uncertain rule changes under the appearance of irregular changes. In the standard POA algorithm, the starting position of the pelican is randomly distributed and unevenly distributed. Therefore, we use chaotic sequences to initialize the population in the standard POA algorithm to improve the quality of solutions in the population.

In this study, the more homogeneous Cubic mapping (cubic chaotic mapping)¹⁸ is used to generate chaotic sequences for further homogenization of the population distribution. The Cubic mapping is defined as shown in Eq. (28):

28

where: x_n∈ ( 0,1);ρ is a control parameter, in this paper ρ = 2.595. Figure 5 shows the scatter plot and histogram of Cubic Chaos Mapping, respectively.

Fig. 5.

Cubic chaotic mapping.

(2)Refraction reverse learning mechanism.

According to the refraction principle of light, refraction reverse learning¹⁹ can be used to calculate the reverse solution, and thus expand the search space and improve the probability of selecting the optimal solution. By comparing the current solution with the refraction inverse solution, the best one is selected for iterative optimization, as shown in Fig. 6.

Fig. 6.

Refractive reverse learning mechanism.

In Fig. 6, the search interval for the solution on the x-axis is [a, b], the y-axis represents the normal line, the lengths of the incident and refracted rays are and , α and β are the angles of incidence and refraction respectively, and the point of intersection O is the mid-point of the interval [a, b]. From the geometric relationship of the line segments in the figure, we can obtain:

29

30

From the definition of refractive index, n=/, which is combined with the above two equations to get:

31

Let k =, which can be obtained by taking the above formula.

32

The calculation formula of refraction reverse learning solution is obtained by transforming Eq. (32)

33

Fusion red-tailed eagle optimization algorithm high-altitude flying position strategy

Red-tailed hawk algorithm (RTH)²⁰ is a new swarm intelligence optimization algorithm, which simulates the hunting behavior of red-tailed hawks, the actions taken and modeling in each hunting stage. The algorithm includes three stages: high-altitude glide, low-altitude glide, dive and dive. It has the characteristics of strong evolutionary ability, fast search speed and strong optimization ability.

The RTH high-altitude gliding phase of the red-tailed eagle algorithm is very similar to the search behavior of the evolutionary method. The RTH algorithm starts from the original position, passes to the best position and aggregates all search points. The pelican is accelerated to fish in the whole search space, the global search ability of the algorithm is enhanced by introducing the high-altitude flying position strategy of the Red-tailed Eagle optimization algorithm, the convergence speed is improved, and the exploration ability of the POA algorithm is increased. The improved global detection position formula is as follows:

34

35

36

Where X (t) represents the red-tailed eagle position at iteration t, X_best is the best position, X_mean is the average value of the position, and Levy represents the Levy flight distribution function that can be calculated according to the formula. And TF(t) represents the transfer factor function that can be calculated according to the formula. Where S is a constant (0.01), dim is the dimension of the problem, β is a constant (1.5), u and v are random numbers [0 to 1].

Cauchy’s mutation strategy

All the pelican individuals of the standard POA algorithm will gather to the optimal individual during the optimization search, and the diversity of the population will be reduced in the late stage of the algorithm, which leads to the algorithm easily falling into the local optimal condition. The Cauchy variation strategy²¹ is introduced to perturb the current optimal individuals to ensure that the algorithm can successfully jump out of the local extreme value region, thus reducing the impact of the local optimum on the algorithm’s ability to find the best. The Cauchy variation constant is derived from the Cauchy distribution, and its functional expression is as follows: the Cauchy distribution outliers:

37

The Cauchy variation operator is introduced into the POA algorithm to fully utilize its perturbation ability to adjust the value of the objective function of the optimal pelican individual with the following expression:

38

where is the standard Cauchy distribution function; denotes the objective function value of the optimal pelican individual.

Mirroring reverse learning strategies

Reverse learning is an optimization method that expands the search space by solving the inverse solution of the current position^22,23. Used in swarm intelligence optimization algorithm, it can moderately improve the performance of the algorithm in searching for the optimal solution; however, the inverse solution obtained by inverse learning is fixed, and if an individual falls into the local optimum and the inverse solution is inferior to the current solution, it is difficult to get rid of the local optimum by inverse learning. Lens imaging reverse learning can effectively solve this problem.

Lens imaging reverse learning strategy²⁴, i.e., the reverse learning measures implemented based on the lens imaging principle of light (law of refraction). The lens imaging reverse learning strategy dynamically adjusts the number of reverse solutions according to the scaling factor to achieve population size optimization and prevent the POA algorithm from falling into a local optimal situation. Lens imaging principle is shown in Fig. 7.

Fig. 7.

Lens imaging reverse learning.

As shown in Fig. 7, take the two-dimensional space as an example, [a, b] is the search range of the solution, and the y-axis denotes the convex lens. Suppose there is an object B with height H and projection A on the x-axis; the object is imaged through a convex lens to present an inverted solid image B’ on the other side of the convex lens with height h* and projection A’ on the x-axis. By the principle of convex lens imaging can be obtained:

39

Let k = A/A’, then Eq. (39) can be rewritten as:

40

Equation (40) is the solution formula for the inverse solution of the inverse learning strategy for convex lens imaging, and Eq. (40) reduces to when k = 1:

41

Through the above analysis, lens imaging inverse learning, i.e., specific inverse learning, is used to obtain a fixed inverse solution. And by adjusting the size of k, the inverse solution can be fixed using this method. With the help of adjusting the size of k value, lens reverse learning can obtain dynamic inverse solutions, which in turn improves the optimization effect of the algorithm. The k value used in this paper is calculated as:

42

The specific algorithm flow is represented by the following pseudo-code based on the above algorithm design steps.

Algorithm 2.

Pseudo-code of IPOA.

Algorithm performance testing

Contrast algorithm selection and parameterization

In this section, all the experiments in this paper are conducted in the same environment: a 64-bit Windows 10 system, 8G RAM, AMD Ryzen 76,800 H processor, and MATLAB R2017b simulation software. In order to verify the practicality of the IPOA algorithm, and its impact and performance in terms of algorithm performance, we chose the Harris Hawk Optimization algorithm²⁵ (HHO), the Goose Optimization Algorithm²⁶ (GOOSE), the Chernobyl Disaster Optimizer²⁷ (CDO), Osprey Optimization Algorithm²⁸ (OOA), Pelican Optimization Algorithm²⁹ (POA), Remora optimization algorithm³⁰ (ROA), Subtraction-Average-Based Optimizer (SABO)³¹, and Golf Optimization Algorithm³² (GOA). algorithms as a comparison object, and the relevant parameter settings of each algorithm are listed in Table 1.

Table 1.

Parameters settings.

Arithmetic	Parameterization
ROA	C = 0.1
OOA	ri, j are random numbers in the interval [0, 1], Ii, j are random numbers from the set {1, 2}
SABO	P = 0.2
GOA	a decrease from 2 to 0
IPOA	I = round (1 + rand (1,1))
POA	I = round (1 + rand (1,1))
CDO	Sαare random numbers in the interval (1, 300,000), Sβare random numbers in the interval (1, 270,000) r are random numbers in the interval(0, 1)

Standard test functions

In this thesis, we critically test and evaluate the performance of different algorithms based on the test function CEC2022²⁹shown in Table 2. To increase the fairness of the comparison experiments, we wrote the algorithms and simulated them using MATLAB R2017b. The algorithms were run on a Windows 10 64-bit system with 8 GB of RAM. The standard test functions were categorized into four groups: single-peak functions (F1), basis functions (F2-F5), hybrid functions (F6-F8), and combinatorial functions (F9-F12), as shown in Table 2. Figure 8 is the three-dimensional function diagram of CEC2022.In the field of swarm intelligent optimization algorithms, they are often used to evaluate their feasibility, efficiency and robustness.。.

Table 2.

CEC2022 test function.

	NO.	Functions	Fi*
Unimodal Function	1	Shifted and Full Rotated Zakharov Function	300
Basic Functions	2	Shifted and Full Rotated Rosenbrock’s Function	400
	3	Shifted and Full Rotated Expanded Schaffer’s F6 Function	600
	4	Shifted and Full Rotated Non-Continuous Rastrigin’s Function	800
	5	Shifted and full Rotated Levy Function	900
Hybrid Functions	6	Hybrid Function 1(N = 3)	1800
	7	Hybrid Function 2(N = 6)	2000
	8	Hybrid Function 3(N = 5)	2200
Composition Functions	9	Composition Function 1 (N = 5)	2300
	10	Composition Function 2 (N = 4)	2400
	11	Composition Function 3 (N = 5)	2600
	12	Composition Function 4 (N = 6)	2700

Search range: [-100,100]^D.

Fig. 8.

CEC2022 test function.

Each test function runs 30 times to ensure the reliability of the results. All algorithms use the same population size and number of iterations, i.e., in the test experiments, the population size is set to N = 30, the maximum number of iterations is T = 1000, and the number of dimensions is D = 20 to ensure the reliability of the results.

Comparative analysis of algorithm performance

Testing the performance of algorithms using test functions with known global optima is a common approach in the field. In this thesis, we critically test and evaluate the performance of different algorithms based on the test functions shown in Table 2. We write the algorithms and simulate the implementation using Matlab2017b. The following test method is used: the optimal value, the worst value, the mean, the median and the standard variance in Table 3 show the eight algorithms running independently. In order to ensure the accuracy of the test results and reduce the possibility of errors, we ran each test 30 times to ensure that credible results were obtained. By analyzing the results obtained, we can draw conclusions as shown in Table 3.

Table 3.

Results of the different algorithms on the CEC2022 functions.

		IPOA	GOOSE	POA	OOA	CDO	ROA	HHO	ABSO	GAO
F1	min	300	300.008	2816.779	22266.26	24238.9	22928.98	3674.763	15677.17	27809.25
	std	2.642	2576.21	2998.261	17396.42	730.3642	18502.63	4277.645	6552.127	22848.16
	avg	300.0001	2147.489	8364.096	48735.63	25776.66	52014.34	9230.986	27596.11	53420.47
	median	300	1084.957	7843.358	50845.66	25848.06	46110.58	8972.424	27442.39	49882.73
	worse	301.580	11299.65	15484.21	84104.3	27091.42	103,188	20243.61	39001.74	126008.8
F2	min	400.2511	400.052	475.5131	1902.493	1995.894	830.6314	450.7786	470.7327	2522.795
	std	15.02062	19.79253	126.6039	592.6621	31.11843	635.3913	39.38651	78.05335	607.3406
	avg	439.1145	450.2086	615.2864	2945.028	2039.335	1920.927	507.7808	620.2165	3634.042
	median	444.9776	449.0914	596.4697	2900.497	2035.277	1845.42	500.575	621.2347	3534.499
	worse	467.8211	474.8659	1139.731	4281.683	2117.142	3346.253	590.7842	811.0503	4806.837
F3	min	600	657.5102	633.3946	660.8505	650.0681	640.3556	635.3535	619.4072	664.2975
	std	0	9.008905	8.866235	9.24341	6.478677	13.82049	9.016577	15.11427	8.069702
	avg	600	672.5237	652.2583	675.5964	664.186	672.2247	661.9385	647.8099	684.5407
	median	600	670.904	650.6307	674.5173	663.7918	671.3465	663.4253	649.7109	685.66
	worse	600	692.0388	667.7938	693.154	683.7772	696.2729	675.8365	681.4281	698.6935
F4	min	825.1706	888.5512	862.7884	932.8829	926.0931	937.5953	848.8081	909.6007	951.3939
	std	12.83737	38.16286	12.46685	15.20388	13.3928	17.9401	16.21861	20.62088	12.15672
	avg	858.1597	931.176	884.8665	971.396	948.1966	967.5795	886.135	950.2769	979.9129
	median	857.0845	928.349	882.8598	975.3606	947.012	966.6638	885.0674	951.3984	980.883
	worse	894.9415	1007.944	910.1836	995.8777	985.0664	1010.859	912.4607	993.6885	1005.68
F5	min	900	2456.168	1530.395	2567.599	2855.387	2153.379	2047.107	1217.147	2943.666
	std	0.579224	1266.718	260.8783	444.0286	307.3461	483.7589	299.9458	677.9486	330.3222
	avg	900.4804	3843.986	2249.802	3389.234	3408.443	3377.066	2779.464	2336.497	3607.03
	median	900.2721	3724.482	2330.756	3370.652	3416.627	3309.788	2819.74	2295.013	3615.155
	worse	901.8319	7434.541	2621.804	4263.985	4006.909	4312.886	3269.509	3848.727	4528.509
F6	min	1977.673	1959.637	2619.743	4.42E + 08	8.38E + 08	19,503,339	29653.31	389465.8	1.55E + 09
	std	3365.769	4207.315	557093.2	1.12E + 09	1.81E + 09	9.53E + 08	67068.82	11,238,143	9.51E + 08
	avg	3696.161	5026.364	269709.8	2.01E + 09	4.71E + 09	7.48E + 08	135836.8	10,062,298	3.39E + 09
	median	2111.172	2900.477	25147.78	1.76E + 09	5.95E + 09	2.99E + 08	149583.3	4,744,462	3.47E + 09
	worse	15719.17	16329.19	2,407,047	4.31E + 09	5.95E + 09	3.72E + 09	272722.7	43,567,024	5.23E + 09
F7	min	2022.402	2096.29	2063.296	2110.853	2251.985	2086.397	2105.513	2099.703	2155.354
	std	21.15624	135.9814	28.265	40.98153	36.63176	65.79644	72.95057	60.85222	47.68962
	avg	2047.356	2338.151	2107.001	2201.688	2334.143	2201.775	2193.161	2201.333	2236.893
	median	2034.075	2324.505	2108.035	2198.816	2336.988	2195.567	2174.877	2189.445	2225.061
	worse	2166.659	2582.587	2166.863	2280.03	2464.493	2378.726	2368.524	2357.512	2333.607
F8	min	2217.431	2229.773	2225.015	2242.514	2239.668	2236.13	2229.171	2246.883	2302.775
	std	1.845523	203.916	50.16526	108.8513	8.83663	154.6729	56.82941	77.71238	280.4284
	avg	2224.538	2589.234	2261.936	2357.164	2250.524	2342.053	2266.144	2369.357	2670.903
	median	2224.495	2584.256	2233.455	2319.193	2248.399	2278.206	2241.698	2371.763	2576.788
	worse	2227.419	3094.227	2357.252	2612.378	2270.066	2977.893	2466.97	2522.951	3332.303
F9	min	2480.781	2480.951	2485.999	2866.516	3310.721	2663.042	2481.507	2611.19	3000.89
	std	1.46E-13	42.85361	36.501	365.7496	32.10963	385.1058	32.90409	55.06773	429.791
	avg	2480.781	2509.531	2543.452	3431.358	3478.108	3103.795	2504.441	2675.275	3750.213
	median	2480.781	2494.532	2536.499	3399.973	3484.351	2961.707	2494.458	2665.718	3852.624
	worse	2480.781	2643.735	2648.049	4142.42	3492.973	4179.853	2656.434	2909.124	4412.42
F10	min	2500.298	2501.22	2501.067	2721.811	4468.623	2584.62	2501.594	2729.444	4704.426
	std	612.0069	706.9913	962.5547	834.4684	467.5293	1543.971	679.6715	1002.984	715.4717
	avg	2778.038	5329.514	3377.092	6535.828	6228.425	5703.191	4015.274	6558.345	6730.85
	median	2500.443	5327.774	2690.643	6624.299	6287.543	6353.223	3779.871	6923.237	6962.676
	worse	4378.078	6621.384	5144.007	7432.041	7051.766	7288.347	5741.869	7350.14	7438.231
F11	min	2600	2900.38	3352.438	6969.073	8461.471	6976.687	2958.094	3559.233	7113.266
	std	58.3292	53910.52	724.5744	622.5514	26.51302	755.1161	277.6492	1018.164	711.5441
	avg	2893.333	60783.16	4477.982	9097.982	8507.39	8747.933	3169.654	4976.709	8997.201
	median	2900	72334.84	4499.204	9170.054	8501.273	8756.37	3052.98	4846.591	9110.676
	worse	3000	176106.8	6154.771	9981.271	8597.988	9902.995	4294.647	7357.762	10140.62
F12	min	2935.655	3174.61	2958.804	3512.097	3465.355	3048.484	3012.604	2977.493	3629.685
	std	13.35905	302.4612	54.13511	266.679	43.84836	187.057	198.8371	37.29596	252.8275
	avg	2953.18	3742.417	3029.975	4017.458	3536.423	3312.476	3235.709	3056.348	4211.89
	median	2948.918	3756.419	3015.557	3990.89	3534.716	3317.674	3168.069	3055.565	4228.168
	worse	2986.95	4444.184	3165.689	4588.857	3674.357	3684.186	3800.236	3138.432	4700.205

As can be seen from Table 3, in 20 dimensions, the IPOA function is better than the other eight algorithms in terms of accuracy and stability on the single-peak function F1, reaching the optimal value; on the basis function (F2-F5), the IPOA achieves the optimal value of accuracy and stability on F3, and the standard deviation is slightly lower than that of the POA and the GOA on F4; on the hybrid function (F6-F8), the solution accuracy of IPOA is greatly improved by optimizing and improving the IPOA algorithm on the basis of the original POA algorithm. On the mixed functions (F6-F8), the optimization improvement based on the original POA algorithm, the solution accuracy of the IPOA algorithm has been greatly improved. Although the optimal solution is not reached in the hybrid function, the IPOA algorithm has obvious advantages and is significantly more competitive than the other eight algorithms; in the combined function (F9-F12), for function F11, the optimization effect of IPOA is slightly worse than that of the CDO algorithm, but compared with the other seven algorithms, its standard deviation is much smaller than that of the other algorithms. For each of the tested functions, the average of the solutions of the IPOA is close to the global optimum with relatively small standard deviations. This situation indicates that the fluctuation of the solution data is small, and the optimization process of the algorithm shows more reliable stability. In summary, IPOA performs well in both single-peak, basis function, hybrid function, and combined function problems, proving the good optimization-seeking performance and stability of IPOA in global exploration and local mining.

Figure 9 shows the convergence curves for the single-peak function (F1), the basic function (F2-F5), the hybrid function (F6-F8) and the combined function (F9-F12) to visually compare the convergence speeds of these nine algorithms.The IPOA has an absolute advantage in convergence speed on the 12 test functions, which is significantly better than HHO, OOA, GOOSE, POA, ROA, ABSO, GOA and CDO. On the test functions, IPOA converges to the optimal value faster than the other eight algorithms. According to the convergence curve, IPOA obtains a better fitness value in the initial stage, and it can be considered that IPOA is competitive compared with the other eight algorithms. The smoothness and stability of the convergence curve is an important index for evaluating the performance of the algorithm. The convergence curve of IPOA shows a good convergence trend, which indicates that the IPOA algorithm is stable in gradually approaching the target value. Compared with the other eight algorithms, the convergence curve of IPOA always stays at the bottom, which indicates that its convergence speed is faster, and the smaller the correlation value is, the higher the accuracy of the solution is.

Fig. 9.

The convergence curves of different algorithms on CEC2022.

Figure 10 shows the boxplots of different algorithms³⁰ for measuring the stability and robustness of the algorithms. The box lengths of the IPOA algorithm are smaller than those of the original POA algorithm and the other algorithms for the single-peak function (F1), the basis function (F2-F5), the hybrid function (F6-F8), and the combinatorial function (F9-F12), indicating that the IPOA algorithm has a more centralized distribution of the results after running it for 30 times for these functions. For the remaining functions, the median of the data also needs to be considered. On the CEC2022 function, the median of IPOA is smaller than the other functions. Therefore, considering the length of the box and the median, the improved algorithms have an advantage in terms of stability and robustness, which is due to the introduction of these four improved strategies, which maintains a better balance between mining and exploration capabilities.

Fig. 10.

Comparison of 12 test function box plots.

Wilcoxon’s rank sum test

The optimization performance of the IPOA algorithm was verified by using the Wilcoxon rank sum test³¹ at a significance level of 5% on the mean results of the test function when the dimension is 30. The test results are expressed in terms of P. When P < 5%, “+” is used to indicate that there is a significant difference between the two compared algorithms; when P > 5%, “-” is used to indicate that the gap between the two algorithms is not significant. When NaN is displayed, it indicates that the two algorithms have comparable performance. Under the same experimental conditions, IPOA is tested for differences with HHO, OOA, GOOSE, POA, ROA, ABSO, GOA and CDO, respectively. The results are shown in Table 4, in comparing the eight algorithms, most of the rank sum test P-values are less than the significant level of 5%, which indicates that the IPOA algorithm is significantly different from the other algorithms, i.e., the IPOA algorithm has a better performance of optimization search.

Table 4.

Rank sum test for p-values.

	GOOSE	POA	OOA	CDO	ROA	HHO	SABO	GOA
F1	1.16E-07	3.02E-11	3.02E-11	3.02E-11	3.02E-11	3.02E-11	3.02E-11	3.02E-11
F2	0.141278	3.02E-11	3.02E-11	3.02E-11	3.02E-11	2.61E-10	3.02E-11	3.02E-11
F3	1.21E-12	1.21E-12	1.21E-12	1.21E-12	1.21E-12	1.21E-12	1.21E-12	1.21E-12
F4	8.84E-07	9.76E-10	3.69E-11	4.5E-11	3.34E-11	0.002312	8.15E-11	3.02E-11
F5	3.02E-11	3.02E-11	3.02E-11	3.02E-11	3.02E-11	3.02E-11	3.02E-11	3.02E-11
F6	0.001501	7.39E-05	3.02E-11	3.02E-11	3.02E-11	3.02E-11	3.02E-11	3.02E-11
F7	4.5E-11	2.39E-08	4.98E-11	3.02E-11	8.99E-11	1.61E-10	9.92E-11	4.08E-11
F8	4.98E-11	5.19E-07	3.02E-11	3.02E-11	3.02E-11	5.49E-11	3.02E-11	3.02E-11
F9	3.15E-12	3.15E-12	3.15E-12	3.15E-12	3.15E-12	3.15E-12	3.15E-12	3.15E-12
F10	2.15E-10	1.27E-05	2.15E-10	1.29E-09	1.29E-06	2E-06	1.85E-08	8.15E-11
F11	9.13E-11	2.37E-12	2.37E-12	2.37E-12	2.37E-12	5.18E-09	2.37E-12	2.37E-12
F12	3.02E-11	9.92E-11	3.02E-11	3.02E-11	3.02E-11	3.02E-11	4.08E-11	3.02E-11

Experimental results and analysis of improved algorithms for photovoltaic model parameter identification

In this paper, two diode models (i.e., single diode model and dual diode model) are investigated and analyzed based on three experimental datasets of three different models of PV cells, which have been widely used for testing parameter identification algorithms for PV cells³². First, we used the RTC France silicon model PV cell dataset, with the experimental conditions set to a sunlight irradiance of 1000 W/m² and an ambient temperature of 33 °C. This dataset contains 26 pairs of PV cell discharge data, which will be used as the basis for our analysis and comparison of the performance of the different methods. Secondly, we used the dataset of PV cell model STM6-40/36, which is manufactured by Schutten Solar and consists of 36 monocrystalline cells connected in series. The experimental conditions were an ambient temperature of 51 °C and a light intensity of 1000 W/m²³³. Finally, we used a PV cell dataset of model PVM752GaAs with standard experimental conditions: ambient temperature 25 °C and light intensity 1000 W/m²³⁴. In addition, we also selected a photovoltaic module dataset of model Photowatt-PW201, and 36 polycrystalline silicon cells in series were used in the experiments under the experimental conditions of external ambient irradiance of 1000 W/m² and ambient temperature of 45 °C.

The upper and lower limits of the parameters of the photovoltaic cells in the photovoltaic system, that is, the parameter range is shown in Table 5. By testing the data obtained from the experiment, the photovoltaic cell model under different light intensity or temperature environment is verified, and the performance of different algorithms is evaluated, including the comparison of IPOA algorithm with other algorithms³⁵. The parameters of these algorithms should comply with the provisions of Sect. Contrast algorithm selection and parameterization to achieve a fair comparison. The maximum number of iterations is set to 1000 times, and the operation is repeated 5 times. Each algorithm performs 30 independent runs. The simulation experiment uses 64-bit Windows 10 system, 8G memory, AMD Ryzen 76,800 H processor, and MATLAB R2017 b simulation software.

Table 5.

The range of values for different parameters of photovoltaic cell models in photovoltaic systems.

Parameters	SDM/DDM		PV module model
	The lower limit	The upper limit	The lower limit	The upper limit
Iph(A)	0	1	0	2
Isd (uA)	0	1	0	50
n, n1, n2, n3	1	2	1	50
Isd1, Isd2, Isd3(uA)	0	1	0	50
Rs (Ω)	0	0.5	0	2
Rsh (Ω)	0	100	0	2000

Diode model parameter identification

For the single diode model, multiple algorithms were run independently for 30 times and the minimum RMSE values obtained are included in Table 6. Based on this result, we derive the circuit model parameters obtained by each optimization algorithm during the single diode model identification process, including I_ph, I_SD, R_s, R_sh, and n. It is worth noting that these parameters are the optimal RMSE values obtained by all the algorithms after 30 independent runs. RMSE values.。.

Table 6.

Circuit model parameter values identified through different optimization algorithms in the single diode model.

	Iph(A)	ISD(UA)	Rs(Ω)	Rsh(Ω)	n	RMSE
IPOA	0.76081	3.22E-07	0.030382	53.71351	1.480953	9.8602E-04
GOOSE	0.761821	0.000001	0.031271	68.80999	1.60483	2.6591E-03
POA	0.76152	4.75E-07	0.034551	51.51235	1.521378	1.4163E-03
OOA	0.781897	3.95E-07	0.069296	68.21633	1.541471	1.4623E-01
CDO	0.758721	4.5E-07	0.030731	73.93854	1.516851	8.6907E-03
ROA	0.758458	2.84E-07	0.038106	34.00905	1.469747	6.1324E-03
HHO	0.760223	7.19E-07	0.032989	99.77717	1.566243	1.8443E-03
SABO	0.760009	0.000001	0.034019	100	1.603384	6.8460E-03
GAO	0.810029	3.68E-07	0.066776	46.45392	1.488834	6.3774E-02

It is known from Table 6 that although the difference of each parameter data is small, there are still some differences, which strongly proves that each algorithm achieves the optimal value within a limited number of runs³⁶. The values listed in the table intercept a finite number of digits that can reflect the differences in the results of different methods. Among them, the RMSE value obtained by IPOA algorithm reaches 9.8602E-04, which is the best in the table, and the corresponding parameters are the best. At the same time, the results show that the optimal parameters of the solar cell SDM model extracted by the proposed IPOA algorithm are very close to the actual ones, so the parameters can be accurately identified.

The result data of 30 independent experiments are counted in Table 7, where Best, Mean, Median, Worst and Std represent the best, mean, median, worst and standard deviation of the 30 experiments, respectively. From Table 7, the IPOA algorithm outperforms the other algorithms in terms of best, mean, median, and worst values in the SDM model parameter identification results. Among them, the best value, mean and median of the IPOA algorithm are 0.000986, and its standard deviation is only 3.61E-07, which is much lower than the calculation results of the POA algorithm, which clearly reflects that the proposed IPOA algorithm improves the optimization accuracy and stability relative to other algorithms, indicating that the algorithmic improvement has a certain degree of effectiveness.

Table 7.

Results of five evaluation indicators for single diode model.

	IPOA	GOOSE	POA	OOA	CDO	ROA	HHO	SABO	GAO
worst	0.000989	0.006045	0.002636	0.218774	0.021722	0.07077	0.006514	0.035722	0.159747
best	0.000986	0.002659	0.001416	0.146233	0.008691	0.006132	0.001844	0.006846	0.063774
std	3.61E-07	0.001455	0.000544	0.029776	0.005254	0.025092	0.00205	0.012381	0.039455
mean	0.000986	0.003457	0.002074	0.179906	0.014335	0.028089	0.004018	0.018651	0.123541
median	0.000986	0.002871	0.002353	0.167273	0.013193	0.020819	0.003	0.01484	0.124121

Table 8 shows the 26 predicted currents of the SDM model optimized by the IPOA algorithm, and the predicted power is obtained by the power calculation formula. I_cal and P_cal indicate that the photovoltaic cell current and power output values calculated by the identified parameters will predict the current and 26 experimental current data to calculate the absolute current error IAE_I and absolute power error IAE_P. IPOA can maintain a small level in 26 current error data estimates, even up to 1.77E-06. Obviously, the predicted data using IPOA are highly consistent with the experimental data, which effectively reflects that the extracted parameters are accurate enough. Therefore, it can be considered that the IPOA algorithm is a potentially effective algorithm for extracting SDM model parameters.

Table 8.

Error statistics of IPOA algorithm in SDM model.

	V(V)	I(A)	P(W)	Ical(A)	Pcal(W)	IAEI(A)	IAEP(W)
1	-0.2057	0.764	-0.15715	0.764123	-0.15718	0.000123	2.53E-05
2	-0.1291	0.762	-0.09837	0.762698	-0.09846	0.000698	9.01E-05
3	-0.0588	0.7605	-0.04472	0.76139	-0.04477	0.00089	5.23E-05
4	0.0057	0.7605	0.004335	0.760189	0.004333	0.000311	1.77E-06
5	0.0646	0.76	0.049096	0.75909	0.049037	0.00091	5.88E-05
6	0.1185	0.759	0.089942	0.758077	0.089832	0.000923	0.000109
7	0.1678	0.757	0.127025	0.757126	0.127046	0.000126	2.12E-05
8	0.2132	0.757	0.161392	0.756176	0.161217	0.000824	0.000176
9	0.2545	0.7555	0.192275	0.755122	0.192178	0.000378	9.63E-05
10	0.2924	0.754	0.22047	0.753699	0.220382	0.000301	8.8E-05
11	0.3269	0.7505	0.245338	0.751427	0.245641	0.000927	0.000303
12	0.3585	0.7465	0.26762	0.747391	0.26794	0.000891	0.00032
13	0.3873	0.7385	0.286021	0.740157	0.286663	0.001657	0.000642
14	0.4137	0.728	0.301174	0.727425	0.300936	0.000575	0.000238
15	0.4373	0.7065	0.308952	0.707018	0.309179	0.000518	0.000227
16	0.459	0.6755	0.310055	0.675328	0.309975	0.000172	7.91E-05
17	0.4784	0.632	0.302349	0.630806	0.301777	0.001194	0.000571
18	0.496	0.573	0.284208	0.571973	0.283698	0.001027	0.00051
19	0.5119	0.499	0.255438	0.499645	0.255768	0.000645	0.00033
20	0.5265	0.413	0.217445	0.413676	0.217801	0.000676	0.000356
21	0.5398	0.3165	0.170847	0.317525	0.1714	0.001025	0.000553
22	0.5521	0.212	0.117045	0.212154	0.11713	0.000154	8.5E-05
23	0.5633	0.1035	0.058302	0.102234	0.057588	0.001266	0.000713
24	0.5736	-0.01	-0.00574	-0.00875	-0.00502	0.001249	0.000716
25	0.5833	-0.123	-0.07175	-0.12556	-0.07324	0.002557	0.001492
26	0.59	-0.21	-0.1239	-0.20853	-0.12303	0.001467	0.000866

Figure 11 shows the convergence curve of IPOA and other improved algorithms on the SDM model, which shows the average RMSE value of each algorithm after 30 independent runs. Compared with other methods, IPOA converges very fast. In addition, it can be seen more intuitively from Fig. 11 that as the number of evaluations increases, the convergence curve of IPOA is always below other algorithms, which means that the average convergence accuracy of IPOA is high and better than other comparison algorithms. Based on this, it can be asserted that IPOA is useful in evaluating the unknown parameters of the SDM model. Figure 12 shows the I-V and P-V characteristics identified by IPOA. It can be seen from Fig. 14 that the experimental data of all voltage measurement points are highly consistent with the estimated data, which highlights the consistency of the algorithm and strongly reflects the excellent performance of IPOA on the model.

Fig. 11.

Convergence curve of single diode model results.

Fig. 12.

IPOA evaluation of SDM’s alpha and alpha characteristics.

Fig. 14.

IPOA evaluation of DDM’s alpha and alpha characteristics.

Parameter identification of dual-diode model

The dual diode model has one more diode compared to the single diode model, so the number of parameters to be identified increases from five to seven, which are I_ph, I_SD1, R_s, R_sh, n₁, I_SD2 and n₂. Based on the number of runs set for each algorithm, the values of the internal parameters of the two-diode model corresponding to the minimum RMSE values were identified and recorded in Table 9. Among all the algorithms listed, the best RMSE value is highlighted. During the mathematical identification of the dual diode model, the values of the internal parameters extracted by different algorithms vary significantly and their respective RMSE values also show significant differences which are easy to observe³⁷. The RMSE value obtained by the IPOA algorithm is the best. When the result is 9.8423E-04, it shows that IPOA algorithm recognizes the parameters with higher accuracy than other algorithms, and it is especially excellent among the seven recognized parameters. The IPOA algorithm shows its effectiveness and accuracy when compared to other algorithms.。.

Table 9.

Circuit model parameter values identified through different optimization algorithms in the dual diode model.

	Iph(A)	Rs(Ω)	Rsh(Ω)	ISD1(UA)	n 1	ISD2(UA)	n 2	RMSE
IPOA	0.760785	0.036627	54.63868	2.4E-07	1.457167	3.9E-07	1.883529	9.8423E-04
GOOSE	0.760917	0.031166	90.82189	0.000001	1.604641	0	1.077096	2.4928E-03
POA	0.760381	0.035945	65.29723	4.79E-09	1.343233	3.88E-07	1.506815	1.0662E-03
OOA	0.765497	0.04001	46.87734	5.17E-07	1.919586	8.54E-07	1.602623	2.9665E-02
CDO	0.764128	0.040237	38.46964	9.2E-08	1.368157	7.63E-07	2	3.8261E-03
ROA	0.767837	0.0338	81.684	9.59E-07	1.663144	7.12E-07	1.670861	9.6058E-03
HHO	0.761923	0.035309	46.30588	1.99E-07	1.453581	5.63E-07	1.750491	1.4842E-03
SABO	0.770099	0.028266	83.92565	8.59E-07	1.635792	6.23E-07	1.678339	8.5154E-03
GAO	0.804596	0.079077	66.28935	9.15E-07	1.588636	3.23E-07	1.715756	1.2140E-01

Table 10 details the results of 30 independent experiments, which contain key metrics such as Best, Mean, Median, Worst, and Standard Deviation. From these data, we can see that the IPOA algorithm performs particularly well in the experiments for DDM model parameter identification, and its Best, Mean, Median, Worst, and Std are significantly better than the other algorithms. Specifically, the optimal value of the IPOA algorithm reaches an astonishing 0.000984 and the median is as high as 0.000989, while its standard deviation is only 8.33E-06, which is much lower than that of the POA algorithm, and the optimal value of GOA is 0.121398, which is the worst performance. This fully proves the superiority of the proposed IPOA algorithm in terms of optimization seeking accuracy and stability and further verifies the effectiveness of the algorithm improvement.

Table 10.

Five evaluation indicators of the dual diode model.

	IPOA	GOOSE	POA	OOA	CDO	ROA	HHO	SABO	GAO
worst	0.00105	0.036913	0.002903	0.28033	0.02067	0.061209	0.009727	0.020247	0.358485
best	0.000984	0.002493	0.001066	0.029665	0.003826	0.009606	0.001484	0.008515	0.121398
std	2.74E-05	0.015227	0.000765	0.100384	0.006056	0.022513	0.003487	0.004764	0.098792
mean	0.001004	0.009683	0.001765	0.106676	0.013106	0.028144	0.004809	0.014725	0.195139
median	0.000989	0.002876	0.001339	0.084518	0.013722	0.015958	0.003478	0.014836	0.155101

The current output and output power of the dual-diode model are calculated based on the internal parameters of the dual-diode model identified in Table 11. The calculation results are shown in Table 11. The current, power calculation value and error value, the double diode model obtained by the IPOA algorithm. The estimated current and power values are not very different from the experimental values, and the results are convincing, which also proves the accuracy of the parameters identified by the IPOA algorithm.

Table 11.

Error statistics of IPOA algorithm in DDM model.

	V(V)	I(A)	P(W)	Ical(A)	Pcal(W)	IAEI(A)	IAEP(W)
1	-0.2057	0.764	-0.15715	0.764039	-0.15716	3.87E-05	7.96E-06
2	-0.1291	0.762	-0.09837	0.762638	-0.09846	0.000638	8.24E-05
3	-0.0588	0.7605	-0.04472	0.761352	-0.04477	0.000852	5.01E-05
4	0.0057	0.7605	0.004335	0.760171	0.004333	0.000329	1.88E-06
5	0.0646	0.76	0.049096	0.759089	0.049037	0.000911	5.88E-05
6	0.1185	0.759	0.089942	0.75809	0.089834	0.00091	0.000108
7	0.1678	0.757	0.127025	0.757149	0.12705	0.000149	2.5E-05
8	0.2132	0.757	0.161392	0.7562	0.161222	0.0008	0.00017
9	0.2545	0.7555	0.192275	0.755137	0.192182	0.000363	9.23E-05
10	0.2924	0.754	0.22047	0.753694	0.22038	0.000306	8.95E-05
11	0.3269	0.7505	0.245338	0.751389	0.245629	0.000889	0.000291
12	0.3585	0.7465	0.26762	0.747313	0.267912	0.000813	0.000291
13	0.3873	0.7385	0.286021	0.740041	0.286618	0.001541	0.000597
14	0.4137	0.728	0.301174	0.727288	0.300879	0.000712	0.000295
15	0.4373	0.7065	0.308952	0.706887	0.309122	0.000387	0.000169
16	0.459	0.6755	0.310055	0.675229	0.30993	0.000271	0.000124
17	0.4784	0.632	0.302349	0.630754	0.301753	0.001246	0.000596
18	0.496	0.573	0.284208	0.571967	0.283695	0.001033	0.000513
19	0.5119	0.499	0.255438	0.499667	0.25578	0.000667	0.000341
20	0.5265	0.413	0.217445	0.4137	0.217813	0.0007	0.000368
21	0.5398	0.3165	0.170847	0.317529	0.171402	0.001029	0.000555
22	0.5521	0.212	0.117045	0.212129	0.117116	0.000129	7.13E-05
23	0.5633	0.1035	0.058302	0.102189	0.057563	0.001311	0.000738
24	0.5736	-0.01	-0.00574	-0.00877	-0.00503	0.001232	0.000706
25	0.5833	-0.123	-0.07175	-0.12553	-0.07322	0.002531	0.001476
26	0.59	-0.21	-0.1239	-0.2084	-0.12296	0.001598	0.000943

The convergence curves in Fig. 13 show that the IPOA algorithm has the highest convergence speed and convergence accuracy. This is followed by the HHO and POA algorithms. The GOA algorithm has the worst optimization performance. At the beginning of iterations, the IPOA algorithm converges faster than the other eight algorithms. However, as the number of iterations increases, the convergence accuracy of the IPOA algorithm is higher than the other algorithms. The GOA, ROA, and OOA algorithms all have poor convergence ability. It is obvious from Fig. 14 that the I-V and P-V curves obtained from the simulation model are basically consistent with the actual measured data. Therefore, the IPOA proposed in this paper can accurately characterize the actual characteristics of solar cells.

Fig. 13.

Convergence curve of the results of the dual diode model.

PV module model parameter identification

The PV module model consists of either series or parallel PV cells. As a result of the previous analysis, five necessary parameters are derived: I_ph, I_SD, R_s, R_sh and n. Table 12 provides an exhaustive list of the internal parameters of the PV module model as well as the values of the circuit model parameters identified by various optimization algorithms. Among them, the internal parameters of the PV cell extracted by the IPOA algorithm have the lowest calculated RMSE values, which is also reflected in the corresponding data in the table³⁸. Therefore, the IPOA algorithm can accurately obtain the internal parameters of the PV model, which can provide a reference and basis for the analysis of the PV cell, and provide a strong support for the analysis of the PV cell and the construction of the system.。.

Table 12.

Circuit model parameter values identified through different optimization algorithms in photovoltaic module models.

	Iph(A)	ISD(UA)	Rs(Ω)	Rsh(Ω)	n	RMSE
IPOA	0.206066	7.07E-07	1.999855	1684.127	16.23284	2.4255E-03
GOOSE	0.209797	0.00005	0.640858	1974.901	24.5428	1.9233E-02
POA	0.208411	7.01E-06	1.407773	1342.631	19.86193	1.0607E-02
OOA	0.21166	2.56E-05	1.032852	911.6185	22.71789	2.8363E-03
CDO	0.213403	3.53E-05	0.908217	669.3331	23.58527	2.2162E-02
ROA	0.210921	2.29E-05	1.030202	1030.202	22.44107	1.7139E-02
HHO	0.208547	2.15E-05	0.90207	1441.974	22.30122	1.6363E-02
SABO	0.214284	0.00005	0.890886	1519.742	24.57735	3.1064E-02
GAO	0.212041	1.5E-05	0.566402	489.4194	21.51024	3.1491E-02

Table 13 exhaustively shows the simulation results of the nine algorithms in the PVM model, from which it can be observed that IPOA consistently maintains excellent optimization accuracy and algorithmic stability. In the experiments, it is found that there are individual algorithms that can obtain higher accuracy and good results in algorithm stability though. From the analysis of the experimental results in Tables 7 and 10, and 13, the experimental metrics are set appropriately. Evaluating the performance of an algorithm needs to be carried out from multiple perspectives, and the stability index is especially critical. Obviously, based on the simulation optimization accuracy data of the three models, it can be concluded that the IPOA algorithm has strong competitiveness in both optimization accuracy and stability.

Table 13.

Five evaluation indicators for photovoltaic module models.

	IPOA	GOOSE	POA	OOA	CDO	ROA	HHO	SABO	GAO
worst	0.002433	0.43137	0.015935	0.066875	0.029134	0.034899	0.032139	0.05959	0.056771
best	0.002425	0.019233	0.010607	0.018695	0.022162	0.017139	0.016363	0.031064	0.031491
std	6.76E-05	0.20605	0.002135	0.019109	0.002718	0.007307	0.006213	0.010413	0.011095
mean	0.002467	0.241677	0.01349	0.03578	0.024745	0.022227	0.021531	0.043787	0.044339
median	0.00244	0.331473	0.013319	0.033048	0.02357	0.020698	0.01953	0.042717	0.046984

Table 14 shows the data about the 25 errors of IPOA on the PVM model, from which the 25 predicted parameters maintain very small errors. The predicted data using IPOA is highly consistent with the experimental data, which effectively reflects that the extracted parameters are accurate enough. In summary, the IPOA algorithm extracts the unknown parameters of the three PV models with high accuracy and is an effective parameter prediction algorithm.

Table 14.

Error statistics of IPOA algorithm in PVM model.

	V(V)	I(A)	P(W)	Ical(A)	Pcal(W)	IAEI(A)	IAEP(W)
1	0.1248	1.0315	0.128731	1.028976	0.128416	0.002524	0.000315
2	1.8093	1.03	1.863579	1.027286	1.858668	0.002714	0.004911
3	3.3511	1.026	3.438229	1.02569	3.437189	0.00031	0.00104
4	4.7622	1.022	4.866968	1.024094	4.87694	0.002094	0.009971
5	6.0538	1.018	6.162768	1.022312	6.188873	0.004312	0.026104
6	7.2364	1.0155	7.348564	1.019978	7.380971	0.004478	0.032406
7	8.3189	1.014	8.435365	1.01643	8.455581	0.00243	0.020216
8	9.3097	1.01	9.402797	1.010573	9.40813	0.000573	0.005333
9	10.2163	1.0035	10.25206	1.000704	10.2235	0.002796	0.028562
10	11.0449	0.988	10.91236	0.984611	10.87493	0.003389	0.037428
11	11.8018	0.963	11.36513	0.959564	11.32458	0.003436	0.040556
12	12.4929	0.9255	11.56218	0.922856	11.52914	0.002644	0.033035
13	13.1231	0.8725	11.4499	0.872594	11.45114	9.38E-05	0.00123
14	13.6983	0.8075	11.06138	0.807254	11.058	0.000246	0.003373
15	14.2221	0.7265	10.33236	0.728312	10.35812	0.001812	0.025769
16	14.6995	0.6345	9.326833	0.63712	9.365349	0.00262	0.038516
17	15.1346	0.5345	8.089444	0.536211	8.115332	0.001711	0.025888
18	15.5311	0.4275	6.639545	0.429526	6.671019	0.002026	0.031474
19	15.8929	0.3185	5.061889	0.318807	5.06677	0.000307	0.004881
20	16.2229	0.2085	3.382475	0.207434	3.365183	0.001066	0.017292
21	16.5241	0.101	1.668934	0.096218	1.589914	0.004782	0.07902
22	16.7987	-0.008	-0.13439	-0.00829	-0.13919	0.000286	0.004801
23	17.0499	-0.111	-1.89254	-0.11091	-1.89108	8.58E-05	0.001462
24	17.2793	-0.209	-3.61137	-0.20925	-3.61571	0.000251	0.00434
25	17.4885	-0.303	-5.29902	-0.30091	-5.26238	0.002095	0.036636

The convergence curve shown in Fig. 15 shows that the IPOA algorithm performs best in terms of convergence performance. It has superiority in convergence accuracy. In contrast, the convergence effect of GOA algorithm is the most unsatisfactory. The HHO, POA and CDO algorithms show convergence. In Fig. 16, we construct the output characteristics of photovoltaic cells, including the relationship between output current and output voltage, and the relationship between output power and output voltage, and show the experimental data points. The output curve of the battery fitted according to the internal parameters of the extracted battery model has a high degree of coincidence with the experimental measurements, which accurately reflects the output characteristics of the photovoltaic cell in the actual environment. The experimental results verify the effectiveness of the IPOA algorithm and the feasibility of modeling the photovoltaic system by identifying the internal parameters of the photovoltaic cell.

Fig. 15.

Convergence curve of photovoltaic model results.

Fig. 16.

IPOA evaluation of PVM’s alpha and alpha characteristics.

Conclusion

This paper presents an improved pelican optimization algorithm with comprehensive strategy to enhance the performance of the PV system parameters. The Cubic chaotic initialization plus the reverse refraction mechanism to initialize the pelican population first. Secondly, the position update formula of the pelican prey identification phase is replaced by the position update formula of the red-tailed eagle optimization algorithm in the high-altitude soaring phase for the adequacy of the pelican prey identification phase in the solution space search and the solution performance in the optimization problem. Then the convergence speed of the pelican optimization algorithm is enhanced by using the Cauchy variation strategy, which enhances the local exploration ability at the late iteration stage. Finally, the inverse solution generated by the lens imaging principle can provide a new search direction when the pelican optimization algorithm falls into the local optimum and increases the probability of finding the global optimal solution to improve the global optimality search capability, so that it can jump out of the local optimum in the late iteration. The results show that the IPOA algorithm can break out of the local optimum solution, obtain higher accuracy and have stronger global search ability than other solutions by testing the 12 test functions of CEC2022 and the real parameter identification problems of single diode model, double diode model and PV module model,.In this paper, the performance improvement based on the Pelican optimization algorithm is aimed at solving the function optimization and PV model parameter problems. However, there are still the following deficiencies that need to be further investigated: (1) The study of the mathematical principle and parameter adjustment of the pelican optimization algorithm needs to be deepened. (2) The proposed comprehensive strategy-enhanced Pelican optimization algorithm can effectively solve the PV model parameter problem, however, its application to more complex real-world optimization problems are yet to be explored. Therefore, the application scope of the Pelican optimization algorithm needs to be expanded.

Author contributions

Conceptualization, Zhang yi.; methodology, Sang beicong; software, Sang beicong; investigation, Sang beicong; resources, Xu Yong.; data curation, Xu Yong; writing—original draft preparation, Xu Yong; writing—review and editing, Zhang yi.

Data availability

The data that support the findings of this study are available from the corresponding author upon reasonable request.

Declarations

Competing interests

The authors declare no competing interests.