SCSO: snake optimization with sine-cosine algorithm for parameter extraction of solar photovoltaic models

Abstract

Solar power generation is a clean and sustainable energy source. To ensure the efficient operation of photovoltaic (PV) systems, it is essential to develop accurate equivalent models of PV cells and precisely determine their unknown parameters. However, due to the nonlinear and multimodal characteristics of PV systems, accurately extracting PV parameters remains a significant challenge. This paper proposes a hybrid Snake Optimization combined with a Sine–Cosine Algorithm (SCSO) to address the PV parameter extraction problem. The proposed algorithm incorporates three key improvements: (1) integration of the Sine–Cosine Algorithm to enhance the bio-inspired Snake Optimization, balancing exploration and exploitation; (2)The parameters C₁ and C₂ are adaptively adjusted, and the Newton–Raphson method is introduced to accelerate the algorithm’s convergence speed which accelerates convergence; and (3) application of a lens imaging reverse learning strategy to improve exploration capabilities and population diversity, preventing the algorithm from becoming trapped in local optima. First, the performance of the SCSO algorithm is qualitatively analyzed using the CEC2022 test functions. Then, the algorithm is applied to extract parameters for three different PV modules. Finally, two commercial models (TFST 40 and MCSM 55) are tested under varying environmental conditions to validate the algorithm’s accuracy. Experimental results demonstrate that SCSO outperforms several state-of-the-art metaheuristic algorithms, achieving higher precision and faster convergence.

Article Highlights

This paper proposes a new hybrid Snake Optimization combined with the Sine–Cosine Algorithm (SCSO) and conducts a qualitative analysis of the improved algorithm using CEC2022 test functions, demonstrating its superior performance.

The SCSO is applied to the extraction of unknown parameters in six solar photovoltaic module models, including the Single Diode Model (SDM), Double Diode Model (DDM), and PV module model.

Compared to other metaheuristic algorithms, the SCSO achieves faster and more precise parameter extraction, as demonstrated on two commercial PV models, TFST 40 and MCSM 55.

Introduction

With the progress of society, fossil fuels are no longer capable of meeting the demands of people. Moreover, the utilization of fossil fuels contributes to environmental pollution and the greenhouse effect. Therefore, renewable clean energy is gradually becoming a substitute choice for individuals. PV energy represents a key form of renewable, clean energy. As an inexhaustible resource, solar energy offers distinct advantages. In comparison to other renewable energy sources, solar PV systems generally incur lower operational and maintenance costs [1]. However, due to constantly changing weather conditions, coupled with the multivariate and nonlinear characteristics of solar photovoltaic modules components, the extraction of parameters from solar photovoltaic models remains a important and challenging task [2]. Hence, accurate PV modeling is imperative for managing, monitoring, simulating, and diagnosing faults in PV installations. PV modeling involves designing parameter models for PV systems based on fundamental physical principles and simplified assumptions, followed by estimating parameters using actual measurement data and specialized optimization method [2]. The optimization method utilizes actual measured current–voltage (I-V) data to generate estimates. Unfortunately, the characteristics of the actual measured I-V data include its nonlinear and sensitivity to conditions such as solar radiation and battery temperature changes [3]. Due to these characteristics, establishing an accurate photovoltaic model is crucial. PV models are represented by equivalent circuits, and there are three widely used PV models in the literature: the single-diode model (SDM), the double-diode model (DDM), and the PV module model. These models are popular due to their simplicity and high accuracy [4]. The current and voltage information obtained through optimization methods must match the measured current and voltage information of the solar photovoltaic generator perfectly. The SDM has five unknown parameters, namely, the photo-current (I_ph), reverse saturation current of the diode (I_sd), series resistance (R_s), shunt resistance (R_sh), and ideal factor (n₁). The DDM has seven unknown parameters. Compared with SDM, DDM has an additional diode reverse saturation current I_sd2 and diode ideal factor n₂ to be extracted. The reason for the addition of these two parameters is that DDM includes an extra diode to simulate some non-ideal characteristics of photovoltaic cells, such as leakage current and carrier recombination effects.

Over the past few decades, numerous metaheuristic algorithms have been proposed to tackle nonlinear, multimodal optimization problems. Although they do not guarantee an optimal solution, they can provide feasible solutions within an acceptable time, making them a leading approach in optimization problem-solving. For instance, Khurma, R. et al. utilized the Snake Optimization Algorithm for disease diagnosis [5]. Paul, K. et al. proposed an Improved Manta Ray Foraging Optimization (IMRFO) algorithm to address congestion cost issues in power systems [6]. Al-Shourbaji, I. et al. applied the Snake Optimization Algorithm for feature selection [7]. Paul, K. also introduced the Gravitational Search Algorithm in congestion management techniques [8]. Additionally, Paul, K. proposed a cuckoo search algorithm-based strategy for wind farm grid connection congestion mitigation [9]. Sharma, P. et al. used the Bonobo Optimizer (BO) algorithm for solar photovoltaic parameter estimation [10]. Paul, K. further proposed an Improved Crow Search Algorithm (ICSA) to solve congestion management issues in power systems [11].

The problem of solar PV parameter extraction is also a multimodal, nonlinear challenge. Typically, solving such problems can be divided into two approaches: deterministic methods or metaheuristic algorithms. In the literature, deterministic methods such as the Lambert W function and Newton–Raphson method have been used for parameter extraction in PV models. In recent years, various swarm intelligence optimization algorithms have been applied to extract these parameters. For instance, Particle Swarm Optimization (PSO) [12], Simulated Annealing (SA) [13], Differential Evolution (DE) [14], Harmony Search (HS) [15], Pattern Search (PS) [16], Bacterial Foraging Algorithm (BFA) [17], Artificial Bee Colony (ABC) [18], JAYA algorithm [19], Grey Wolf Optimizer [20], Firefly Algorithm(FA) [21], Teaching–Learning-Based Optimization (TLBO) [22], Moth Flame Optimization (MFO) [23], and Tree Seed Algorithm [24]. These methods have shown effectiveness through experimental validation, yet they still possess certain limitations. Consequently, ongoing research efforts focus on developing more robust methodologies to overcome the limitations identified in many current approaches.

In pursuit of achieving more precise outcomes, hybrid optimization algorithms have been proposed that combine the strengths of multiple algorithms. For example, Paul, K. proposed a hybrid improved Grey Wolf Optimization-Sine Cosine Algorithm for multi-objective optimization scheduling in hybrid power systems, Ram et al. introduced the hybrid Bee Pollinator Flower Pollination Algorithm (BPFPA) [25], Li et al. proposed Memetic Adaptive Differential Evolution (MADE) [26], Abd Elaziz et al. presented an Improved Opposition-Based Whale Optimization Algorithm (OBWOA) [27], Beigi et al. introduced Hybrid Firefly and Pattern Search Algorithm (HFAPS) [28], Chen et al. proposed Teaching–Learning-Based Artificial Bee Colony (TLABC) [29], Yu et al. suggested Improved JAYA Algorithm (IJAYA) [30], Zhou et al. introduced an Adaptive Differential Evolution algorithm with Dynamic Opposition Learning Strategy (DOLADE) [31], Aoufi et al. developed Nested Loop Biogeography based Optimization-Differential Evolution (NLBBODE) [2], Yu et al. proposed a hybrid adaptive JAYA algorithm [32], and another work by Yu et al. introduced a Improved Grey Wolf Optimizer with Gaussian Mutation and Levy Flight [33]. Furthermore, Li et al. introduced a Landscape-Aware Particle Swarm Optimization (LAPSO) for estimating PV system parameters [34]. Paul, K. et al. utilized a hybrid Harris Hawks Optimization and Sine–Cosine Algorithm to solve home energy management problems, optimizing the scheduling of smart appliances in buildings to minimize power consumption [35]. Yang, Q. et al. proposed an adaptive cuckoo search algorithm with a self-adaptive selection operator [36]. Sundar Ganesh et al. combined reinforcement learning with the Golden Jackal Optimizer to enhance algorithm performance [37]. Charu, K. M. et al. applied an improved Particle Swarm Optimization (PSO) and Harmony Search Algorithm for photovoltaic (PV) parameter extraction [38]. Mohamed, R. et al. introduced a parameter estimation technique based on an improved Kepler Optimization Algorithm [39].

Above mentioned methods have made a significant contribution to the extraction of photovoltaic parameters, but the photovoltaic parameter extraction problem is a multi-parameter, multi peak optimization problem, and many heuristic algorithms find it difficult to find a more accurate global optimal solution. The Snake Optimization (SO) Algorithm is a recently proposed population-based Bio-inspired algorithm inspired by the foraging and mating behaviors of snakes. It has been applied to a variety of optimization problems, such as disease diagnosis [5], feature selection [7], and other complex optimization problems. The SO Algorithm has achieved promising results in these optimization problems, demonstrating its effectiveness in handling complex optimization challenges. Due to the No Free Lunch Theorem, no algorithm can perfectly solve every problem. Therefore, to overcome the limitations of a single algorithm and improve the accuracy of the results, this paper introduces a hybrid snake optimizer that combines the Sine Cosine Algorithm. By replacing the position update formula of the snake optimization algorithm exploration phase with that of the sine–cosine algorithm, the algorithm ensures exploration capability while speeding up convergence. Additionally, converting fixed-value parameters into variable values enhances the algorithm’s ability to perform random walks between exploration and exploitation and accelerates convergence. Furthermore, the use of the lens imaging reverse learning strategy makes it easier for the algorithm to escape local optima. Through these improvements, the method addresses limitations observed in other optimization algorithms, allowing for more accurate and rapid identification of position parameters.

The primary contributions of this paper can be summarized as follows:

1. This paper proposes a new hybrid Snake Optimization combined with the Sine–Cosine Algorithm (SCSO) and conducts a qualitative analysis of the improved algorithm using CEC2022 test functions, demonstrating its superior performance.

2. SCSO is applied to the extraction of unknown parameters in six solar photovoltaic module models, including the Single Diode Model (SDM), Double Diode Model (DDM), and PV module model.

3. Compared to other metaheuristic algorithms, the SCSO achieves faster and more precise parameter extraction, as demonstrated on two commercial PV models, TFST 40 and MCSM 55.

The structure of this paper is organized as follows: Sect. 2 presents a concise overview of the foundational knowledge relevant to this study. Section 3 focuses on the modeling of the photovoltaic power generation system. Section 4 includes the analysis and discussion of the experimental results. Section 5 concludes the paper and offers prospects for future work. The symbols and abbreviations used in this article are listed in Table 1.

Table 1. List of symbols and abbreviations

Symbol	Meaning	Abbreviations	Meaning
I	Output current	PV	Photovoltaic
Isd,Isd1,Isd2	The diode reverse saturation current	SDM	The single-diode model
n, n1, n2	The ideal factor	DDM	The double-diode model
Iph	Photo-current	LIRL	Lens Imaging Reverse Learning Strategy
Ish	Parallel resistance current	HBO	Heap-Based Optimizer
Id,Id1,Id2	Diode current	WOA	Whale Optimization Algorithm
V	The output voltage	SCA	Sine Cosine Algorithm
Rs	The series resistance	GWO	Grey Wolf Optimization Algorithm
Rsh	The parallel resistance	HHO	Harris Hawks Optimization Algorithm
q	The charge of an electron (1.60217646 × 10−19C)	EHHO	Enhanced Harris Hawks Optimization Algorithm
T	The temperature in Kelvin	SCSO	Hybrid Snake Optimization with Sine Cosine Algorithm
k	The Boltzmann constant (1.380653 × 10−23 J/K)	RLWOA	Refraction-learning-based Whale Optimization Algorithm

Preliminary

Snake optimization algorithm

Snake Optimization (SO) Algorithm is inspired by the mating and foraging behavior of snakes. The search process is divided into two stages: exploration and exploitation. The quantity of food serves as the boundary between exploration and exploitation. The content of this section is derived from the rewriting of reference [40]. First, the population is initialized, and then it is divided into males and females. We assume an equal number of males and females, using Eq. (1) to divide the population.

1

$$N_m=N_f=N\left./\phantom{N2}\right)2$$where N represents the total population, N_m and N_f represent the number of males and females, respectively. Then, Eq. (2) and Eq. (3) are used to determine the temperature T and the quantity of food Q_food.

2

$$T={\text{e}}^{\left(\frac{-t}{T_{max}}\right)}$$

3

$$Q_{\mathit{food}}=C_1\ast {\text{e}}^{\left(\frac{t-T_{max}}{T_{max}}\right)}$$where t represents the current iteration count, T_max represents the maximum iteration count, C₁ is a constant equal to 0.5. We now enter the exploration phase (without food). If Q_food is less than the threshold (threshold = 0.25), the snake searches for food by randomly selecting positions and updates their positions relative to food according to Eq. (4).

4

$$S_{i,m(\text{f)}}\left(t+1\right)=S_{rand,m(f)}\left(t\right)\pm C_2\times A_{m(f)}\times \left(\left(S_{max}-S_{min}\right)\times r+S_{min}\right)$$where S_i,m(f) represents the position of the i-th male or female, S_rand,m(f) represents the position of a random male or female, r is a random number between 0 and 1, C₂ is a constant equal to 0.05, and A_m(f) is the ability of males or females to search for food, calculated using Eq. (5).

5

$$A_{m(f)}={\text{e}}^{\left(\frac{-f_{rand,m(f)}}{f_{i,m(f)}}\right)}$$where f_rand,m(f) is the fitness of S_rand,m(f), f_i,m(f) is the fitness of the i-th individual in the male or female group. If Q is greater than the threshold (threshold = 0.25), the algorithm enters the development phase. Take the temperature threshold is 0.6, if T is greater than 0.6, the temperature is classified as hot; otherwise, it is classified as cold. If the external environment is hot, the snake will only move toward the food. The position movement formula is as shown in Eq. (6).

6

$$S_i\left(t+1\right)=P_{\mathit{food}}\pm C_3\times T\times \text{r}\times \left(P_{\mathit{food}}-S_i\left(t\right)\right)$$where P_food represents the location of the food, i.e., the position of the fittest individual in terms of fitness, C₃ is a constant and equal to 2. If the external environment is cold, the snake is in either combat mode or mating mode. A random number is chosen, and if it is greater than 0.6, the snake enters combat mode. If the random number is less than 0.6, the snake will enter mating mode.

7

$$S_{i,m(f)}\left(t+1\right)=\left\{\begin{array}{c}{S}_{i,m(f)}\left(t\right)+C_3\times F_m(F_f)\times r\times \left(Q_{\mathit{food}}\times S_{best,f(m)}-S_{i,m(f)}\left(t\right)\right),if\text{r}>0.6\\ S_{i,m(f)}\left(t\right)+C_3\times M_m(M_f)\times r\times \left(Q_{\mathit{food}}\times S_{i,f(m)}\left(t\right)-S_{i,m(f)}\left(t\right)\right),others\\ \end{array}\right)$$where S_best,m(f) represents the position of the best individual in the male or female group, F_m and F_f represents the combat power of males and females, respectively. M_m and M_f represent the mating abilities of males and females, respectively, calculated using Eq. (8).

8

$$\left\{\begin{array}{c}{F}_m={\text{e}}^{-\left(\frac{f_{best,f}}{f_i}\right)}\\ F_f={\text{e}}^{-\left(\frac{f_{best,m}}{f_i}\right)}\\ M_m={\text{e}}^{-\left(\frac{f_{i,f}}{f_{i,m}}\right)}\\ M_f={\text{e}}^{-\left(\frac{f_{i,m}}{f_{i,f}}\right)}\\ \end{array}\right)$$where the f_best,f and f_best,m represents the fitness of the best individual in the female group and the male group, respectively, while f_i denotes the fitness of the i-th individual. f_i,f and f_i,m represent the fitness of the i-th individual in the female group and the male group, respectively.

If the mating process takes place within the defined search space, there is a likelihood that the female will deposit eggs, which subsequently hatch into new offspring. The hatched individuals replace the worst-performing males and females.

9

$$S_{worst,m(f)}=S_{min}+r\times \left(S_{max}-S_{min}\right)$$where S_worst,m(f) is the lowest fitness value in the male or female group.

Proposed hybrid snake optimization with sine cosine (SCSO) algorithm

Sine cosine algorithm

Sine Cosine Algorithm (SCA) [41] utilizes the periodic oscillatory nature of sine and cosine functions to construct an iterative equation that serves two essential purposes: global exploration and local exploitation. This concise iterative equation introduces perturbations and updates the solution set. There are two specific types of iterative equations within SCA: the sine iteration equation and the cosine iteration equation.

10

$$X_{\text{i}}\left(t+1\right)=\left\{\begin{array}{c}{X}_{\text{i}}\left(t\right)+r_1\ast sin(r_2)\ast \left|r_3\ast Bes{t}_{\text{i}}^t-X_{\text{i}}\left(t\right)\right|(r_4<0.5)\\ X_{\text{i}}\left(t\right)+r_1\ast cos(r_2)\ast \left|r_3\ast Bes{t}_{\text{i}}^t-X_{\text{i}}\left(t\right)\right|(r_4\ge 0.5)\\ \end{array}\right)$$where t represents the current iteration count, Best_i^t represents the best position of the i-th individual, r₁, r₂, r₃, and r₄ are random parameters. Specifically, r₁ = a—a * t/T, where a = 2, T represents the maximum iteration count, r₂ ∈ [0, 2π], r₃ ∈ [0, 2], and r₄ ∈ [0, 1].

Lens imaging reverse learning strategy (LIRL)

The principle of the LIRL is derived from the imaging of convex lenses in optics. When an object passes through a convex lens, an image is formed. According to the relationship between object distance and focal length, the resulting image can be a combination of magnification, reduction, upright, inverted, real, and virtual images. Based on this principle, we assume that individuals in the search space are objects in front of a convex lens. Through the convex lens, a new individual can be obtained. Due to the relationship between object distance and focal length, the new individuals obtained each iteration are not fixed. This enhances the diversity of the population and helps the algorithm escape local optima. The calculation formula for the new individual using the LIRL strategy is shown in Eq. (11).

11

$$x_j^{_{}^{\prime }\ast }=\frac{a_j+b_j}{2}+\frac{a_j+b_j}{2k}-\frac{x_j^{\ast }}{k}$$where $k={\left(1+{\left(\frac{t}{T}\right)}^{0.5}\right)}^{10}$$x_j^{_{}^{\prime }\ast }$ and $x_j^{\ast }$ are the j-th dimensional components of $x_{}^{_{}^{\prime }\ast }$ and $x_{}^{\ast }$, and a_j and b_j are the j-th dimensional upper and lower bounds of the decision variables.

It can be seen that when k equals 1 (object distance equals twice the focal length), it represents the general reverse learning (RL) strategy, and Eq. (11) can be simplified to Eq. (12). Therefore, the general RL strategy is just a special case of the LIRL strategy [42].

12

$$x_{}^{_{}^{\prime }\ast }=a+b-x_{}^{\ast }$$

Parameter settings

The key to determine whether the algorithm is in the exploration stage or the exploitation stage is the quantity of food [43]. In the original algorithm, the parameter C₁, which controls the amount of food, is set as a constant value of 0.5. This setup results in a relatively rigid transition between exploration and exploitation. Therefore, to enable the algorithm to have the capability of random walking between exploration and exploitation, this article introduces a nonlinear change to C₁, as shown in Eq. (13). When C₁ is not changed, the food quantity change, and threshold curve are shown in Fig. 1a. After improvement, the food quantity change, and threshold curve are shown in Fig. 1b.

13

$$C_1=C_1+\frac{1}{10}\times cos(r^4\times \frac{\pi }{2})$$where r ∈ [0,1].

Fig. 1 [Images not available. See PDF.]

Curve of food quantity change before a and after bC₁ change

To expedite the convergence speed during the exploitation phase of the algorithm, the parameter C₃, which controls the position updates during the exploitation phase, is changed from a fixed value of 2 to a variable value. The change rule is to reduce C₃ from 2 nonlinear to 0. The variation formula is illustrated in Eq. (14). The comparison between changing C3 and not changing the value of C3 is shown in Fig. 2.

14

$$C_3=C_3-2\times sin\left({\left(\frac{t}{T}\right)}^4\times \frac{\pi }{2}\right)$$

Fig. 2 [Images not available. See PDF.]

Value changes before and after C₃ change

Newton–Raphson method

The Newton–Raphson method is a numerical technique used to solve nonlinear equations. It is an iterative method that finds the roots of the equation f(x) = 0 through successive approximations. This method is renowned for its rapid convergence, particularly when the initial guess is close to the actual solution. In this paper, we introduce a Newton–Raphson optimizer [44] that employs the Newton–Raphson search rule to perform local fine-tuning on the individuals in the algorithm after each iteration. By leveraging the fast convergence advantage of the Newton–Raphson method, we aim to achieve quicker convergence and more accurate solutions for complex nonlinear optimization problems.

Hybrid snake optimization with sine–cosine algorithm

This paper hybrid the Snake Optimization Algorithm in the following ways: Firstly, by integrating the Sine Cosine Algorithm, the exploration is conducted using the Sine Cosine Algorithm, ensuring an expanded search range while expediting convergence. Secondly, the parameter C₁ controlling the control between exploration and exploitation is changed from a fixed value to a variable value, enabling the algorithm to perform random walks between exploration and exploitation. Furthermore, to accelerate the convergence speed of the algorithm, the parameter C₃, which governs the snake’s position update during the development phase, is changed from a fixed value of 2 to a variable value. Additionally, a Newton–Raphson optimizer was introduced. Thirdly, the LIRL strategy is employed to facilitate the algorithm’s escape from local optima. The pseudo code for the SCSO is shown in Table 2, and the algorithm flowchart is presented in Fig. 3.

Table 2. The pseudo code for the Hybrid SCSO

Fig. 3 [Images not available. See PDF.]

The flowchart of the SCSO

Qualitative evaluation

Jump out of local optimal analysis

To verify that the lens imaging reverse learning strategy can help the algorithm easily escape local optima and achieve the global optimum, the CEC2022 test function was utilized for testing with a dimension of 20. In this case, SO represents the original algorithm without any improvement strategy, while SCSO represents the algorithm incorporating the lens imaging reverse learning strategy. Their convergence curves are illustrated in Fig. 4. Through the observation of the convergence curves, it was found that the convergence curve of SCSO, which integrates the lens imaging reverse learning strategy, is smoother. It obtains more precise values compared to the original algorithm for functions 1, 3, 4, 5, 6, 7, 8, 9, 10, 11, and 12. Therefore, the LIRL strategy can assist the algorithm in escaping local optima.

Fig. 4 [Images not available. See PDF.]

Compared SO and SCSO jump out of local optimal convergence curves

Accelerated convergence analysis

Due to the fixed value of 2 for the parameter C₃ controlling the snake’s movement during the development phase in the SO, the snake moves slowly in each iteration, leading to a slow convergence speed of the algorithm. The improvement of this article is to transform this fixed value into a variable value and integrate the Newton–Raphson method to accelerate the convergence speed of the algorithm. To verify the effectiveness of the algorithm improvement, the CEC2022 test function was employed for testing. Each function was independently tested 30 times, with a selected dimension of 20, and the best result was chosen as the final results. Figure 5 depicts the test results for the 12 test functions, where SO represents the original algorithm, and SCSO represents the improved algorithm with the variable C₃ value. Based on Fig. 5, it can be concluded that the improvement has a more pronounced effect on functions 1, 2, 3, 5, 6, 8, 9, 10, 11, and 12, as the algorithms with variable C₃ values converge earlier than those without this modification.

Fig. 5 [Images not available. See PDF.]

SO and SCSO Accelerated Convergence Analysis convergence curve

Exploration and exploitation analysis

Exploration and exploitation are a pair of contradictions. Exploration focuses on the scope of search, while exploitation focuses on the accuracy of search. A wide search range will inevitably lead to poor accuracy, while high accuracy leads to a small search range. Therefore, it is necessary to ensure a balance between the exploration and exploitation stages of the algorithm, so that the algorithm will not be unable to find the optimal solution or converge to the local optimum early. In the algorithm exploration stage, the search range should be as large as possible; In the algorithm exploitation stage, higher precision solutions are pursued. The proportion of exploration and exploitation is calculated using Eqs. (15) and (16). Where D (t) is the distance between individuals, calculated using Eq. (17). D_max is the maximum diversity in the entire iteration, and parameter x_id is the position of the i-th individual in dimension [45]. The CEC2022 test was used to evaluate the exploration and exploitation capabilities of the original SO algorithm, as well as the SO algorithm integrated with the sine–cosine algorithm and the modified C₁ parameter value. The testing was conducted with a dimension of 20. The results are depicted in Fig. 6. According to the exploration–exploitation ratio graph, it can be inferred that the improved algorithm can explore at a higher ratio in the early stages, thereby bypassing local optima. In the later stages, it can focus on exploitation at a higher ratio to discover more accurate solutions.

15

$$T\left(\%\right)=\frac{D(t)}{D_{max}}\times 100$$

16

$$K(\%)=\frac{\left|D(t)-D_{max}\right|}{D_{max}}\times 100$$

17

$$D(\text{t) =}\frac{1}{D}\sum _{d=1}^D\frac{1}{N}\sum _{i=1}^N\left|median(x_d(t))-x_{\mathit{id}}(t)\right|$$

Fig. 6 [Images not available. See PDF.]

Exploration and exploitation percentage

Research workflow

This section provides a flowchart of the entire research workflow in Fig. 7, summarizing the entire process of our study. The flowchart highlights the key stages of the work, including the proposed algorithm and model validation.

Fig. 7 [Images not available. See PDF.]

Research Workflow Flowchart

Photovoltaic power generation problem modeling

Single diode model (SDM)

The SDM is often used to describe the I-V characteristics of solar photovoltaic generators due to its simplicity and high accuracy [29]. In ideal conditions, solar cells are equivalent to the parallel circuit of SDM as shown in Fig. 8. The output current I of the entire circuit can be expressed as:

18

$$I=I_{\mathit{ph}}-I_{\mathit{sh}}-I_d$$where I_ph represents photocurrent, I_sh represents parallel resistance current, and I_d represents diode current. The parallel resistance current I_sh is calculated using the following formula:

19

$$I_{\mathit{sh}}=(V+I·R_s)\left./\phantom{V+I·R_s)R_{\mathit{sh}}}\right)R_{\mathit{sh}}$$where V represents the output voltage, R_s represents the series resistance, and R_sh represents the parallel resistance. The I_d is calculated using Eq. (20).

20

$$I_d=I_{\mathit{sd}}·\left[{\text{e}}^{\left(\frac{V+I·R_s}{n·V_t}\right)}-1\right]$$where n represents the ideal factor of the diode, I_sd represents the reverse saturation current of the diode and V_t represents the thermal voltage, calculated by Eq. (21).

21

$$V_t=\frac{k·T}{q}$$

Fig. 8 [Images not available. See PDF.]

Equivalent Circuit of the SDM

Summing up the above, we obtain the formula for calculating the I, as shown in Eq. (22).

22

$$I=I_{\mathit{ph}}-I_{\mathit{sd}}·\left[{\text{e}}^{\left(\frac{q·\left(V+I·R_s\right)}{n·k·T}\right)}-1\right]-\frac{V+I·R_s}{R_{\mathit{sh}}}$$

According to Eq. (22), it can be concluded that the SDM involves the extraction of five variables (I_ph, I_sd, R_s, R_sh, and n).

Double diode model

In ideal conditions, solar cells are equivalent to the parallel circuit of DDM as shown in Fig. 9. In this model, the I is calculated using the Eq. (23):

23

$$I=I_{\mathit{ph}}-I_{\mathit{sh}}-I_{d1}-I_{d2}$$

Fig. 9 [Images not available. See PDF.]

Equivalent Circuit of the DDM

Just like in the SDM, the formula for calculating the I in the DDM is as shown in Eq. (24).

24

$$I=I_{\mathit{ph}}-I_{sd1}·\left[{\text{e}}^{\left(\frac{q·\left(V+I·R_s\right)}{n_1·k·T}\right)}-1\right]-I_{sd2}·\left[{\text{e}}^{\left(\frac{q·\left(V+I·R_s\right)}{n_2·k·T}\right)}-1\right]-\frac{V+I·R_s}{R_{\mathit{sh}}}$$

According to Eq. (24), it can be inferred that there are seven variables (I_ph, I_sd1, I_sd2, R_s, R_sh, n₁, and n₂) that need to be extracted in the DDM.

Photovoltaic modules

As shown in Fig. 10, solar photovoltaic modules are obtained by connecting photovoltaic cells in series and parallel. Each module consists of N_p parallel-connected cell strings, and each cell string is composed of N_s photovoltaic cells connected in series. For comprehensive consideration, this article uses the equivalent circuit of a SDM and a DDM in photovoltaic cells to simulate the nonlinear I-V characteristics of photovoltaic modules.

Fig. 10 [Images not available. See PDF.]

Equivalent Circuit of the Photovoltaic Modules

Under uniform illumination and non-mismatch conditions, the I of the photovoltaic module, composed of N_p × N_s photovoltaic cells using the SDM, is calculated using Eq. (25).

25

$$I=N_p·I_{\mathit{ph}}-N_p·I_{\mathit{sd}}·\left[{\text{e}}^{\left(\frac{q·\left(V+I·N_s·R_s\left./\phantom{R_s{N}_p}\right)N_p\right)}{n·k·N_s·T}\right)}-1\right]-\frac{V+(I·N_s·R_s)\left./\phantom{V+(I·N_s·R_s)N_p}\right)N_p}{N_s·R_{\mathit{sh}}\left./\phantom{R_{\mathit{sh}}{N}_p}\right)N_p}$$

This article only considers the series connection case, hence setting the value of Np to 1. From this, we obtain the formula for calculating the I, as shown in Eq. (26):

26

$$I=I_{\mathit{ph}}-I_{\mathit{sd}}·\left[{\text{e}}^{\left(\frac{q·\left(V+I·N_s·R_s\right)}{n·k·N_s·T}\right)}-1\right]-\frac{V+I·N_s·R_s}{N_s·R_{\mathit{sh}}}$$

Objective function formulations

Determining whether the extracted parameters are optimal essentially involves assessing whether the error between the calculated current values using the extracted parameters and the actual measured value from the manufacturer is minimized. Ideally, we aim for this error to be 0. Hence, the fitness function chosen for this study is the Root Mean Square Error (RMSE). The computation of RMSE is presented in Eq. (27):

27

$$RMSE\left(X\right)=\sqrt{\frac{1}{N}\sum _{i=1}^{N}F{\left(V,I,X\right)}^2}$$where N represents the total number of measurements, x represents the extracted unknown parameters, and V and I represent the measured voltage and current.

The formula for calculating the detailed error function $F\left(V,I,X\right)$ for different photovoltaic models is as follows:

SDM:

28

$$\left\{\begin{array}{c}F\left(V,I,X\right)=I_{\mathit{ph}}-I_{\mathit{sd}}·\left[{\text{e}}^{\left(\frac{q·\left(V+I·R_s\right)}{n·k·T}\right)}-1\right]-\frac{V+I·R_s}{R_{\mathit{sh}}}-I\\ X=\left\{I_{\mathit{ph}},I_{\mathit{sd}},R_s,R_{\mathit{sh}},n\right\}\\ \end{array}\right)$$

DDM:

29

$$\left\{\begin{array}{c}F\left(V,I,X\right)=I_{\mathit{ph}}-I_{sd1}·\left[{\text{e}}^{\left(\frac{q·\left(V+I·R_s\right)}{n_1·k·T}\right)}-1\right]-I_{sd2}·\left[{\text{e}}^{\left(\frac{q·\left(V+I·R_s\right)}{n_2·k·T}\right)}-1\right]-\frac{V+I·R_s}{R_{\mathit{sh}}}-I\\ X=\left\{I_{\mathit{ph}},I_{sd1},I_{sd2},R_s,R_{\mathit{sh}},n_1,n_2\right\}\\ \end{array}\right)$$

The photovoltaic module:

SDM:

30

$$\left\{\begin{array}{c}F\left(V,I,X\right)=I_{\mathit{ph}}-I_{\mathit{sd}}·\left[{\text{e}}^{\left(\frac{q·\left(V+I·N_s·R_s\right)}{n·k·N_s·T}\right)}-1\right]-\frac{V+I·N_s·R_s}{N_s·R_{\mathit{sh}}}-I\\ X=\left\{I_{\mathit{ph}},I_{\mathit{sd}},R_s,R_{\mathit{sh}},n\right\}\\ \end{array}\right)$$

DDM:

31

$$\left\{\begin{array}{c}F\left(V,I,X\right)=I_{\mathit{ph}}-I_{sd1}·\left[{\text{e}}^{\left(\frac{q·\left(V+I·N_s·R_s\right)}{n_1·k·N_s·T}\right)}-1\right]-I_{sd2}·\left[{\text{e}}^{\left(\frac{q·\left(V+I·N_s·R_s\right)}{n_2·k·N_s·T}\right)}-1\right]-\frac{V+I·N_s·R_s}{R_{\mathit{sh}}·N_s}-I\\ X=\left\{I_{\mathit{ph}},I_{sd1},I_{sd2},R_s,R_{\mathit{sh}},n_1,n_2\right\}\\ \end{array}\right)$$

Results analysis

The simulation environment utilized for this experiment is MATLAB 2018a, executed on a system equipped with an Intel Core i3 - 6100 CPU operating at a clock speed of 3.70 GHz and 8 GB of RAM.

To evaluate the effectiveness of the proposed SCSO algorithm, it is applied to the parameter extraction problem in photovoltaic models and components. The SCSO optimizer was utilized to extract parameters for three PV models: RTC France solar cell (RTC), Poly-crystalline Photowatt-PWP- 201 module (PWP- 201), and Mono-crystalline STM6 - 40/36 module (STM6 - 40/36). The voltage and current data for these three PV models were sourced from reference [2]. SDM and DDM were used to construct the above three models for experimental verification. In this study, we compared the performance of several algorithms: Multi-Strategy-Based Tree Seed Algorithm (MS-TSA) [24], Adaptive Operator Selection Cuckoo Search Algorithm (AOSCS) [36], Reinforcement Learning-based Golden Jackal Optimizer (RL-GJO) [37], Hybrid Kepler Optimization Algorithm [39], Whale Optimization Algorithm (WOA) [46], Refraction-learning-based Whale Optimization Algorithm (RLWOA) [47], Grey Wolf Optimization Algorithm (GWO) [20], Harris Hawks Optimization Algorithm (HHO) [48], Enhanced Harris’ Hawks Optimization Algorithm (EHHO) [48], Heap-Based Optimizer (HBO) [49], Sine Cosine Algorithm (SCA) [49], and Snake Optimization Algorithm (SO) [40]. All algorithms were independently run 30 times on different PV models, and the best result was chosen as the final experimental result. (Note: The experimental data and results for MS-TSA, AOSCS, RL-GJO, and HKOA are sourced from their respective references.) The search range limits for the unknown parameters of the three photovoltaic models are presented in Table 3, and the operational conditions for each PV model are outlined in Table 4. Table 5 illustrates the parameter settings for each algorithm, with all parameter values consistent with those in the literature. Finally, in this section, we selected two commercial models, TFST 40 and MCSM 55, for unknown parameter extraction to validate the accuracy and effectiveness of SCSO. Each algorithm mentioned in the text was run 30 times, with 1,000 iterations per run. The result obtained on the 1,000 th iteration is considered the final result for each run. The optimal result is the smallest value among the 30 runs.

Table 3. bounds of unknown parameters for different PV models [2]

Unknown parameter	RTC		PWP- 201		STM6 - 40/36
	L	U	L	U	L	U
Iph (A)	0	1	0	2	0	2
Isd (μA)	0	1	0	50	0	50
Rs (Ω)	0	0.5	0	2	0	0.36
Rsh (Ω)	0	100	0	2000	0	1000
n	1	2	1	50	1	60

Table 4. PV generators working conditions [2]

PV	PV temperature (K)	Incident solar irradiance (W/m2)	Number of series cell
RTC	306.15	1000	1
PWP- 201	318.15	1000	36
STM6 - 40/36	328.15	1000	36

Table 5. Algorithm Parameter setting table

Algorithms	Parameters
WOA	NP = 50, Max_iter = 1000, a = 2 ~ 0
RLWOA	NP = 50, Max_iter = 1000
GWO	NP = 50, Max_iter = 1000, a = 2 ~ 0
HHO	NP = 50, Max_iter = 1000, Beta = 1.5
EHHO	NP = 50, Max_iter = 1000, Beta = 1.5,r1 = 2
HBO	NP = 50, Max_iter = 1000
SCA	NP = 50, Max_iter = 1000,r1 = 2 ~ 0, r2 ∈ [0,2π], r3 ∈ [0, 2], r4 ∈ [0, 1]
SO	NP = 50, Max_iter = 1000, Qfood = 0.25, T = 0.6, C1 = 0.5, C2 = 0.05, C3 = 2
SCSO	NP = 50, Max_iter = 1000, Qfood = 0.2, T = 0.45, C1 = 0.5 ~ 0.6, C2 = 0.05, C3 = 2 ~ 0

Analysis of the best parameters and RMSE results

For RTC SDM and DDM, Table 6 presented the best values of unknown parameters and the RMSE values obtained by SCSO and several comparative algorithms, with the best experimental results highlighted in bold. According to Table 6, SCSO achieved the best RMSE value for SDM is 7.7300e- 04, and the best RMSE value for DDM is 6.8721e- 04.

Table 6. The optimal results of SCSO and various comparison algorithms for SDM and DDM

Model	Algorithm	Iph (A)	Isd1 (A)	Isd2 (A)	Rs (Ω)	Rsh (Ω)	n1	n2	RMSE
SDM	WOA	0.760031	3.6583e- 07	–	3.6186e- 02	69.80778	1.493672	–	9.7235e- 04
	RLWOA	0.761062	3.4962e- 07	–	3.5932e- 02	51.08432	1.489289	–	8.2974e- 04
	GWO	0.760648	5.1315e- 07	–	0.034779	80.02836	1.529093	–	0.001259
	HHO	0.7597e- 01	3.2587e- 07	–	3.6709e- 02	75.53326	1.481778	–	0.001108
	EHHO	0.760377	4.6518e- 07	–	3.4830e- 02	69.67121	1.518788	–	0.001022
	HBO	0.760869	3.19e- 07	–	0.036398	52.156285	1.480028	–	0.000989
	SCA	0.768091	2.70e- 10	–	0.053797	8.865369	1	–	0.015554
	SO	0.760788	3.1144e- 07	–	0.036536	52.93122	1.477508	–	7.7302e- 04
	MS-TSA [24]	0.760784	3.2040E- 07	–	0.036410	53.46859	1.480365	–	9.8642e- 04
	HKOA [39]	0.7608	3.107e- 07	–	0.0365	52.8898	1.4773	–	7.7300e- 04
	RL-GJO [37]	0.7608	3.23e- 07	–	0.0364	53.72	1.4812	–	9.8602e- 04
	AOSCS [36]	0.760776	0.323021	–	0.036377	53.718526	1.481184	–	9.8602e- 04
	SCSO	0.760788	3.1058e- 07	–	0.036549	52.88328	1.477230	–	7.7293e- 04
DDM	WOA	0.759851	1.0389e- 07	9.9007e- 08	0.05710521	76.50837	1.623544	1.410669	0.001312
	RLWOA	0.762071	4.8239e- 08	4.6843e- 08	7.6748e- 02	42.01894	1.437308	1.433985	0.001062
	GWO	0.761585	1.3569e- 07	4.6505e- 07	0.04378568	42.83866	1.411843	1.952340	9.0958e- 04
	HHO	0.761231	9.4589e- 08	1.4872e- 07	6.0184e- 02	56.99807	1.417661	1.641890	0.001314
	EHHO	0.759977	1.0155e- 07	1.8882e- 08	6.9191e- 02	75.72138	1.443669	1.415365	0.001245
	HBO	0.760714	1.9535e- 07	2.8364e- 07	3.9142e- 02	58.63873	1.903733	1.475414	8.4080e- 04
	SCA	0.760097	0	4.8465e- 07	3.2936e- 02	56.88643	1.546042	1.524735	0.003548
	SO	0.761597	2.5774e- 07	4.2401e- 08	0.065644	48.39592	1.673745	1.367628	7.9197e- 04
	MS-TSA [24]	0.760802	2.6259e- 07	4.6281e- 07	0.036557	54.69441	1.463723	1.999760	9.8356e- 04
	HKOA [39]	0.761	8.66e- 08	2.16e- 06	0.038	58.356	1.373	2.000	7.326e- 04
	RL-GJO [37]	0.7608	2.16e- 7	8.39e- 07	0.0368	55.76	1.4472	2.000	9.8255e- 04
	AOSCS [36]	0.760781	0.225947	0.749349	0.036740	55.485442	1.451017	2.000	9.8248e- 04
	SCSO	0.761239	7.9571e- 07	2.1963e- 08	0.065992	55.37284	1.807436	1.321050	6.8720e- 04

The convergence curves for SCSO and several comparative algorithms for RTC SDM and DDM are shown in Figs. 11, and 12, respectively. Clearly, SCSO produced the best results and the fastest convergence speed. The I-V and P–V fitting curves for the actual measurement data and the calculated estimated data for SDM and DDM are depicted in Figs. 13 and 14, respectively. Figures 13 and 14 demonstrate that the estimated data obtained through SCSO align closely with the actual measurement data, confirming the consistency between the two.

Fig. 11 [Images not available. See PDF.]

Convergence curves of SCSO and various comparison algorithms of the SDM

Fig. 12 [Images not available. See PDF.]

I-V a and P–V b measured curves and calculated curves of a SDM

Fig. 13 [Images not available. See PDF.]

Convergence curves of SCSO and various comparison algorithms of the DDM

Fig. 14 [Images not available. See PDF.]

I-V a and P–V b measured curves and calculated curves of the DDM

For the PWP- 201 SDM and DDM, Table 7 provides the best values of unknown parameters and the RMSE values obtained by SCSO and several comparative algorithms, with the best experimental results highlighted in bold. According to Table 7, SCSO achieved the best RMSE value for SDM (0.002061) and the best RMSE value for DDM (0.002053).

Table 7. The optimal results of SCSO and various comparison algorithms for the PWP- 201

Model	Algorithm	Iph (A)	Isd1 (A)	Isd2 (A)	Rs (Ω)	Rsh (Ω)	n1	n2	RMSE
SDM	WOA	1.029959	4.3179e- 06	–	1.196574	1273.129	1.374455	–	0.002616
	RLWOA	1.035091	1.5766e- 05		0.965645	963.4884	1.535733		0.005996
	GWO	1.033859	1.6909e- 06	–	1.284168	608.1432	1.278250	–	0.002252
	HHO	1.031359	9.7958e- 06	–	1.069274	1341.231	1.472105	–	0.004199
	EHHO	1.029716	6.2273e- 06	–	1.154700	1677.167	1.416222	–	0.003286
	HBO	1.030964	2.5745e- 06	–	1.240715	851.5889	1.319770	–	0.002099
	SCA	1.019864	1.8674e- 07	–	1.527171	926.1990	1.095905	–	0.008165
	SO	1.028900	3.7934e- 06	–	1.195296	1243.440	1.360228	–	0.002208
	MS-TSA [24]	1.030485	3.5002e- 06	–	1.200725	987.5209	48.662452	–	0.002425
	RL-GJO [37]	1.0305	3.48e- 06	–	1.2013	981.98	48.64	–	0.00243
	AOSCS [36]	1.030514	3.482263	–	0.033369	27.2773	1.35119	–	0.002425
	SCSO	1.031478	2.4491e- 06	–	1.245529	804.2290	1.314576	–	0.002060
DDM	WOA	1.046352	1.5528e- 05	6.2671e- 06	1.084296	361.4087	18.91299	1.419483	0.006438
	RLWOA	1.025960	3.7674e- 05	6.9250e- 07	1.459406	1122.003	38.58226	1.197494	0.005897
	GWO	1.030392	5.2841e- 06	4.3839e- 06	1.179817	1191.079	33.46125	1.376156	0.002365
	HHO	1.036083	1.6136e- 05	2.5346e- 06	0.9535839	870.9306	1.539068	21.91145	0.006270
	EHHO	1.026516	2.0041e- 05	6.6393e- 07	1.439533	1094.674	43.23313	1.193721	0.005098
	HBO	1.030746	4.7029e- 05	1.0798e- 05	1.048509	2000.000	50.00000	1.484102	0.004075
	SCA	1.036055	0	3.7170e- 05	0.832222	1143.565	40.47775	1.666321	0.009961
	SO	1.032990	1.8324e- 05	2.9968e- 06	1.212371	723.6495	50.00000	1.335611	0.002182
	RL-GJO [37]	1.0305	6.89e- 11	3.48e- 06	1.2013	981.98	30.86	48.64	0.00243
	SCSO	1.031388	2.6591e- 06	0	1.234679	826.8299	1.322981	1.000000	0.002051

The convergence curves for SCSO and several comparative algorithms for the PWP- 201 SDM and DDM are shown in Figs. 15, and 16, respectively. Clearly, SCSO produced the best results and the fastest convergence speed. The I-V and P–V fitting curves for the actual measurement data and the calculated estimated data for SDM and DDM are depicted in Figs. 17 and 18, respectively. Figures 17 and 18 demonstrate that the estimated data obtained through SCSO closely match the actual measurement data, confirming their consistency.

Fig. 15 [Images not available. See PDF.]

Convergence curves of SCSO and various comparison algorithms of the PWP- 201SDM

Fig. 16 [Images not available. See PDF.]

I-V a and P–V b measured curves and calculated curves of PWP- 201 SDM

Fig. 17 [Images not available. See PDF.]

Convergence curves of SCSO and various comparison algorithms of the PWP- 201 DDM

Fig. 18 [Images not available. See PDF.]

I-V a and P–V b measured curves and calculated curves of PWP- 201 DDM

For the STM6 - 40/36 SDM and DDM, Table 8 provides the best values of unknown parameters and the RMSE values obtained by SCSO and several comparative algorithms, with the best experimental results highlighted in bold. According to Table 8, SCSO and SO achieved the best RMSE value for SDM is 0.001722 and SCSO achieved the best RMSE value for DDM is 0.001703.

Table 8. The optimal results of SCSO and various comparison algorithms for the for STM6 - 40/36

Model	Algorithm	Iph (A)	Isd1 (A)	Isd2 (A)	Rs (Ω)	Rsh (Ω)	n1	n2	RMSE
SDM	WOA	1.669224	8.6999e- 06	–	0.023062	849.3170	1.721395	–	0.009397
	RLWOA	1.677290	1.0615e- 06	–	0.256915	356.4031	1.469056	–	0.007968
	GWO	1.667143	1.4780e- 06	–	0.145627	444.8986	1.502835	–	0.002796
	HHO	1.672932	1.6031e- 05	–	0.073701	919.6308	1.816032	–	0.024168
	EHHO	1.713568	3.6457e- 07	–	0.101597	134.8103	1.368394	–	0.022979
	HBO	1.660381	5.6297e- 06	–	9.0699e- 03	1000	1.661322	–	0.003311
	SCA	1.675260	2.0369e- 06	–	0.176185	375.6714	1.537416	–	0.010506
	SO	1.663870	1.7687e- 06	–	0.151794	576.1062	1.522185	–	0.001722
	MS-TSA [24]	1.663548	1.9447e- 06	–	0.003909	16.55890	1.532741	–	0.001753
	AOSCS [36]	1.663905	1.738657	–	0.004274	15.928294	1.520303	–	0.001730
	SCSO	1.663608	2.0130e- 06	–	1.3635e- 01	592.8653	1.526586	–	0.001722
DDM	WOA	1.693873	4.1889e- 06	1.3152e- 05	7.4838e- 02	390.7884	7.078880	1.784932	0.025835
	RLWOA	1.674336	4.3944e- 05	3.5236e- 07	0.311506	318.3247	57.25203	1.363348	0.011571
	GWO	1.665994	1.5328e- 06	1.3227e- 06	0.189014	499.8356	2.016841	1.490843	0.002072
	HHO	1.666141	5.1434e- 06	1.7560e- 06	0.288962	821.4767	8.725672	1.522213	0.011284
	EHHO	1.668986	0	1.6401e- 06	0.242945	545.8632	43.98573	1.514839	0.007962
	HBO	1.660755	5.2678e- 06	4.8253e- 05	1.4511e- 02	935.2319	1.652654	56.61047	0.003174
	SCA	1.649304	0	4.1706e- 07	0.360000	750.3256	5.611486	1.380569	0.011857
	SO	1.663528	1.7800e- 06	2.6605e- 05	0.152971	593.5367	1.522824	12.85328	0.001737
	SCSO	1.663587	5.0000e- 05	8.9587e- 07	0.195551	651.1012	2.922849	1.455536	0.001703

The convergence curves for SCSO and several comparative algorithms for the STM6 - 40/36 SDM and DDM are shown in Figs. 19, and 20, respectively. Clearly, SCSO produced the best results and the fastest convergence speed.

Fig. 19 [Images not available. See PDF.]

Convergence curves of SCSO and various comparison algorithm of STM6 - 40/36 SDM

Fig. 20 [Images not available. See PDF.]

I-V a and P–V b measured curves and calculated curves of the STM6 - 40/36 SDM

The I-V and P–V fitting curves for the actual measurement data and the calculated estimated data for SDM and DDM are depicted in Figs. 21 and 22, respectively. Figures 21 and 22 demonstrate that the estimated data obtained through SCSO closely match the actual measurement data, confirming their consistency.

Fig. 21 [Images not available. See PDF.]

Convergence curves of SCSO and various comparison algorithm of STM6 - 40/36 DDM

Fig. 22 [Images not available. See PDF.]

I-V a and P–V b measured curves and calculated curves of the STM6 - 40/36 DDM

Individual absolute error (IAE)

IAE is one of the important evaluation indicators for the quality of solar photovoltaic parameter extraction. It can evaluate the difference between actual measured values and experimentally calculated values. Among them, I_m and V_m are the actual measured current and voltage, I_cal is the current calculated through experiments, P_cal is the power calculated through experiments, IAEI is the absolute error of current, and IAEP is the absolute error of power.

Table 9 provides 26 sets of data for RTC SDM, each comprising actual measured voltage and current data, experimentally estimated current and power data, as well as the current error and power error between the actual measured data and experimental estimated data. Figure 23 displays a bar graph of IAEI and IAEP for SDM. According to Table 9, the maximum IAEI value is 0.001585, and the maximum IAEP value is 0.000794. Both of these error values are very small, indicating the accuracy of the photovoltaic parameter extraction by SCSO.

Table 9. IAE of SCSO for the SDM

	Measured data		Calculated current data		Calculated power data
	Im	Vm	Ical	IAEI	Pcal	IAEP
1	0.7640	− 0.2057	0.764150	0.000150	− 0.157186	0.000031
2	0.7620	− 0.1291	0.762703	0.000703	− 0.098465	0.000091
3	0.7605	− 0.0588	0.761374	0.000874	− 0.044769	0.000051
4	0.7605	0.0057	0.760155	0.000345	0.004333	0.000002
5	0.7600	0.0646	0.759039	0.000961	0.049034	0.000062
6	0.7590	0.1185	0.758011	0.000989	0.089824	0.000117
7	0.7570	0.1678	0.757046	0.000046	0.127032	0.000008
8	0.7570	0.2132	0.756085	0.000915	0.161197	0.000195
9	0.7555	0.2545	0.755022	0.000478	0.192153	0.000122
10	0.7540	0.2924	0.753597	0.000403	0.220352	0.000118
11	0.7505	0.3269	0.751327	0.000827	0.245609	0.000270
12	0.7465	0.3585	0.747305	0.000805	0.267909	0.000289
13	0.7385	0.3873	0.740085	0.001585	0.286635	0.000614
14	0.7280	0.4137	0.727427	0.000573	0.300936	0.000237
15	0.70650	0.4373	0.707027	0.000527	0.309183	0.000230
16	0.6755	0.4590	0.675401	0.000099	0.310009	0.000045
17	0.6320	0.4784	0.630999	0.001001	0.301870	0.000479
18	0.5730	0.4960	0.572175	0.000825	0.283799	0.000409
19	0.4990	0.5119	0.499539	0.000539	0.255714	0.000276
20	0.4130	0.5265	0.413484	0.000484	0.217699	0.000255
21	0.3165	0.5398	0.317160	0.000660	0.171203	0.000356
22	0.2120	0.5521	0.212015	0.000015	0.117054	0.000009
23	0.1035	0.5633	0.102636	0.000864	0.057815	0.000487
24	− 0.0100	0.5736	− 0.009299	0.000701	− 0.005334	0.000402
25	− 0.1230	0.5833	− 0.124361	0.001361	− 0.072540	0.000794
26	− 0.2100	0.5900	− 0.209100	0.000900	− 0.123369	0.000531
Sum of IAE				0.017631		0.006480

Bold indicates the maximum IAEP value

Fig. 23 [Images not available. See PDF.]

RTC SDM IAEI a and IAEP b bar graphs

Table 10 provides 26 sets of data for RTC DDM, each comprising actual measured voltage and current data, experimentally estimated current and power data, as well as the current error and power error between the actual measured data and experimental estimated data. Figure 24 displays a bar graph of IAEI and IAEP for DDM. According to Table 10, the maximum IAEI value is 0.001257, and the maximum IAEP value is 0.000733. All these error values are very small and are smaller than the errors in SDM, indicating that DDM is more accurate than SDM.

Table 10. IAE of SCSO for the DDM

	Measured data		Calculated current data		Calculated power data
	Im	Vm	Ical	IAEI	Pcal	IAEP
1	0.7640	− 0.2057	0.764161	0.000161	− 0.157188	0.000033
2	0.7620	− 0.1291	0.762685	0.000685	− 0.098463	0.000088
3	0.7605	− 0.0588	0.761330	0.000830	− 0.044766	0.000049
4	0.7605	0.0057	0.760086	0.000414	0.004332	0.000002
5	0.7600	0.0646	0.758949	0.001051	0.049028	0.000068
6	0.7590	0.1185	0.757901	0.001099	0.089811	0.000130
7	0.7570	0.1678	0.756919	0.000081	0.127011	0.000014
8	0.7570	0.2132	0.755946	0.001054	0.161168	0.000225
9	0.7555	0.2545	0.754878	0.000622	0.192116	0.000158
10	0.7540	0.2924	0.753459	0.000541	0.220312	0.000158
11	0.7505	0.3269	0.751214	0.000714	0.245572	0.000234
12	0.7465	0.3585	0.747243	0.000743	0.267887	0.000266
13	0.7385	0.3873	0.740101	0.001601	0.286641	0.000620
14	0.7280	0.4137	0.727544	0.000456	0.300985	0.000189
15	0.70650	0.4373	0.707243	0.000743	0.309277	0.000325
16	0.6755	0.4590	0.675684	0.000184	0.310139	0.000085
17	0.6320	0.4784	0.631285	0.000715	0.302007	0.000342
18	0.5730	0.4960	0.572389	0.000611	0.283905	0.000303
19	0.4990	0.5119	0.499626	0.000626	0.255758	0.000320
20	0.4130	0.5265	0.413424	0.000424	0.217668	0.000223
21	0.3165	0.5398	0.316979	0.000479	0.171106	0.000259
22	0.2120	0.5521	0.211775	0.000225	0.116921	0.000124
23	0.1035	0.5633	0.102420	0.001080	0.057693	0.000608
24	− 0.0100	0.5736	− 0.009400	0.000600	− 0.005392	0.000344
25	− 0.1230	0.5833	− 0.124257	0.001257	− 0.072479	0.000733
26	− 0.2100	0.5900	− 0.208792	0.001208	− 0.123187	0.000713
Sum of IAE				0.018202		0.006613

Bold indicates the maximum IAEP value

Fig. 24 [Images not available. See PDF.]

RTC DDM IAEI a and IAEP b bar graphs

Table 11 provides 25 sets of data for the PWP201 SDM, each comprising actual measured voltage and current data, experimentally estimated current and power data, as well as the current error and power error between the actual measured data and experimentally estimated data. Figure 25 displays a bar graph of IAEI and IAEP for the PWP201 SDM. According to Table 11, the maximum IAEI value is 0.003682, and the maximum IAEP value is 0.058895. Both of these error values are very small, indicating the accuracy of photovoltaic parameter extraction by SCSO.

Table 11. IAE of SCSO for the Photowatt-PWP201 SDM

	Measured data		Calculated current data		Calculated power data
	Im	Vm	Ical	IAEI	Pcal	IAEP
1	1.0315	0.1248	1.029723	0.001777	0.128509	0.000222
2	1.03	1.8093	1.027613	0.002387	1.859260	0.004319
3	1.026	3.3511	1.025638	0.000362	3.437017	0.001212
4	1.022	4.7622	1.023717	0.001717	4.875143	0.008157
5	1.018	6.0538	1.021677	0.003677	6.185027	0.022258
6	1.0155	7.2364	1.019182	0.003682	7.375207	0.026642
7	1.014	8.3189	1.015610	0.001610	8.448759	0.013394
8	1.01	9.3097	1.009902	0.000098	9.401887	0.000910
9	1.0035	10.2163	1.000411	0.003089	10.220496	0.031561
10	0.988	11.0449	0.984844	0.003156	10.877509	0.034853
11	0.963	11.8018	0.960393	0.002607	11.334368	0.030766
12	0.9255	12.4929	0.924152	0.001348	11.545344	0.016835
13	0.8725	13.1231	0.873862	0.001362	11.467781	0.017876
14	0.8075	13.6983	0.808446	0.000946	11.074336	0.012958
15	0.7265	14.2221	0.728683	0.002183	10.363404	0.031048
16	0.6345	14.6995	0.636629	0.002129	9.358133	0.031300
17	0.5345	15.1346	0.535294	0.000794	8.101457	0.012013
18	0.4275	15.5311	0.427969	0.000469	6.646831	0.007285
19	0.3185	15.8929	0.317573	0.000297	5.047160	0.014728
20	0.2085	16.2229	0.206740	0.001760	3.353915	0.028559
21	0.101	16.5241	0.097436	0.003564	1.610040	0.058895
22	− 0.008	16.7987	− 0.008693	0.000693	− 0.146030	0.011640
23	− 0.111	17.0499	− 0.110935	0.000065	− 1.891424	0.001115
24	− 0.209	17.2793	− 0.208401	0.000599	− 3.601018	0.010356
25	− 0.303	17.4885	− 0.300532	0.002468	− 5.255858	0.043157
Sum of IAE				0.043468		0.472079

Bold indicates the maximum IAEP value

Fig. 25 [Images not available. See PDF.]

PWP201 SDM IAEI a and IAEP b bar graphs

Table 12 provides 25 sets of data for the PWP201 DDM, each comprising actual measured voltage and current data, experimentally estimated current and power data, as well as the current error and power error between the actual measured data and experimentally estimated data. Figure 26 displays a bar graph of IAEI and IAEP for the PWP201 DDM. According to Table 12, the maximum IAEI value is 0.004226, and the maximum IAEP value is 0.048226. Both of these error values are very small, and the IAEP value is smaller than the error in SDM, indicating that DDM is more accurate than SDM.

Table 12. IAE of SCSO for thePhotowatt-PWP201 DDM

	Measured data		Calculated current data		Calculated power data
	Im	Vm	Ical	IAEI	Pcal	IAEP
1	1.0315	0.1248	1.029424	0.002076	0.128472	0.000259
2	1.03	1.8093	1.027551	0.002449	1.859149	0.004430
3	1.026	3.3511	1.025789	0.000211	3.437522	0.000707
4	1.022	4.7622	1.024047	0.002047	4.876719	0.009751
5	1.018	6.0538	1.022145	0.004145	6.187860	0.025092
6	1.0155	7.2364	1.019726	0.004226	7.379144	0.030579
7	1.014	8.3189	1.016146	0.002146	8.453213	0.017849
8	1.01	9.3097	1.010322	0.000322	9.405791	0.002994
9	1.0035	10.2163	1.000597	0.002903	10.222400	0.029657
10	0.988	11.0449	0.984703	0.003297	10.875948	0.036413
11	0.963	11.8018	0.959895	0.003105	11.328484	0.036649
12	0.9255	12.4929	0.923367	0.002133	11.535534	0.026645
13	0.8725	13.1231	0.872960	0.000460	11.455940	0.006035
14	0.8075	13.6983	0.807645	0.000145	11.063370	0.001993
15	0.7265	14.2221	0.728176	0.001676	10.356197	0.023842
16	0.6345	14.6995	0.636522	0.002022	9.356554	0.029721
17	0.5345	15.1346	0.535586	0.001086	8.105878	0.016435
18	0.4275	15.5311	0.428574	0.001074	6.656220	0.016675
19	0.3185	15.8929	0.318350	0.000150	5.059510	0.002378
20	0.2085	16.2229	0.207529	0.000971	3.366722	0.015752
21	0.101	16.5241	0.098081	0.002919	1.620708	0.048226
22	− 0.008	16.7987	− 0.008329	0.000329	− 0.139909	0.005520
23	− 0.111	17.0499	− 0.110966	0.000034	− 1.891952	0.000587
24	− 0.209	17.2793	− 0.208915	0.000085	− 3.609901	0.001473
25	− 0.303	17.4885	− 0.301592	0.001408	− 5.274396	0.024620
Sum of IAE				0.041417		0.414278

Bold indicates the maximum IAEP value

Fig. 26 [Images not available. See PDF.]

PWP201 DDM IAEI a and IAEP b bar graphs

Table 13 provides 20 sets of data for the STM6 - 40/36 SDM, each including actual measured voltage and current data, experimentally estimated current and power data, as well as the current error and power error between the actual measured data and experimentally estimated data. Figure 27 displays a bar graph of IAEI and IAEP for the STM6 - 40/36 SDM. According to Table 13, the maximum IAEI value is 0.003949, and the maximum IAEP value is 0.098237. All these error values are very small, indicating the accuracy of photovoltaic parameter extraction by SCSO.

Table 13. IAE of SCSO for the STM6 - 40/36 SDM

	Measured data		Calculated current data		Calculated power data
	Im	Vm	Ical	IAEI	Pcal	IAEP
1	1.6630	0	1.661049	0.001951	0.000000	0.000000
2	1.6630	0.1180	1.660895	0.002105	0.195986	0.000248
3	1.6610	2.2370	1.658123	0.002877	3.709221	0.006436
4	1.6530	5.4340	1.653867	0.000867	8.987114	0.004712
5	1.650	7.2600	1.651267	0.001267	11.988202	0.009202
6	1.6450	9.6800	1.646968	0.001968	15.942648	0.019048
7	1.640	11.5900	1.641091	0.001091	19.020246	0.012646
8	1.6360	12.6000	1.635492	0.000508	20.607196	0.006404
9	1.6290	13.3700	1.628829	0.000171	21.777439	0.002991
10	1.6190	14.0900	1.619468	0.000468	22.818302	0.006592
11	1.5970	14.8800	1.603602	0.006602	23.861597	0.098237
12	1.5810	15.5900	1.581430	0.000430	24.654490	0.006700
13	1.5420	16.4000	1.541386	0.000614	25.278727	0.010073
14	1.5240	16.7100	1.520051	0.003949	25.400052	0.065988
15	1.5000	16.9800	1.497896	0.002104	25.434279	0.035721
16	1.4850	17.1300	1.483923	0.001077	25.419601	0.018449
17	1.4650	17.3200	1.464293	0.000707	25.361554	0.012246
18	1.3880	17.9100	1.386688	0.001312	24.835581	0.023499
19	1.1180	19.0800	1.121199	0.003199	21.392475	0.061035
20	0	21.0200	− 0.000319	0.000319	− 0.006713	0.006713
Sum of IAE				0.033586		0.406239

Bold indicates the maximum IAEP value

Fig. 27 [Images not available. See PDF.]

STM6 - 40/36 SDM IAEI a and IAEP b bar graphs

Table 14 provides 20 sets of data for the STM6 - 40/36 DDM, each comprising actual measured voltage and current data, experimentally estimated current and power data, as well as the current error and power error between the actual measured data and experimentally estimated data. Figure 28 displays a bar graph of IAEI and IAEP for the STM6 - 40/36 DDM. According to Table 14, the maximum IAEI value is 0.006048, and the maximum IAEP value is 0.089997. Both of these error values are very small, and the IAEP value is smaller than the error in SDM, indicating that DDM is more accurate than SDM.

Table 14. IAE of SCSO for the STM6 - 40/36 DDM

	Measured data		Calculated current data		Calculated power data
	Im	Vm	Ical	IAEI	Pcal	IAEP
1	1.6630	0	1.663475	0.000475	0.000000	0.000000
2	1.6630	0.1180	1.663270	0.000270	0.196266	0.000032
3	1.6610	2.2370	1.659573	0.001427	3.712464	0.003193
4	1.6530	5.4340	1.653941	0.000941	8.987517	0.005115
5	1.650	7.2600	1.650595	0.000595	11.983319	0.004319
6	1.6450	9.6800	1.645460	0.000460	15.928052	0.004452
7	1.640	11.5900	1.639257	0.000743	18.998992	0.008608
8	1.6360	12.6000	1.633730	0.002270	20.585004	0.028596
9	1.6290	13.3700	1.627295	0.001705	21.756932	0.022798
10	1.6190	14.0900	1.618310	0.000690	22.801995	0.009715
11	1.5970	14.8800	1.603048	0.006048	23.853357	0.089997
12	1.5810	15.5900	1.581553	0.000553	24.656404	0.008614
13	1.5420	16.4000	1.542285	0.000285	25.293470	0.004670
14	1.5240	16.7100	1.521182	0.002818	25.418948	0.047092
15	1.5000	16.9800	1.499165	0.000835	25.455816	0.014184
16	1.4850	17.1300	1.485233	0.000233	25.422034	0.003984
17	1.4650	17.3200	1.465609	0.000609	25.384354	0.010554
18	1.3880	17.9100	1.387594	0.000406	24.851808	0.007272
19	1.1180	19.0800	1.118500	0.000500	21.340985	0.009545
20	0	21.0200	− 0.000065	0.000065	− 0.001373	0.001373
Sum of IAE				0.021928		0.284116

Bold indicates the maximum IAEP value

Fig. 28 [Images not available. See PDF.]

STM6 - 40/36 DDM IAEI a and IAEP b bar graphs

Statistical analysis of results

In this article, eleven other metaheuristic techniques, WOA, RLWOA, GWO, HHO, EHHO, HBO, SCA, MS-TSA [24], HKOA [39], RL-GJO [37], AOSCS [36], along with the non-improved SO, are selected as comparative algorithms from the literature. Each algorithm was independently run thirty times on three different PV models. The selected evaluation criteria for the algorithm are as follows: best RMSE value (Min), average RMSE value (Mean), worst RMSE value (Max), standard deviation (STD), and Average ranking. STD reflects the dispersion of a data set and can indicate the stability of a set of data. A smaller standard deviation signifies better stability. In this study, we conducted a Friedman test on the experimental data and obtained the average ranks, where smaller values indicate better results. (Note: The experimental data and results for MS-TSA [24], AOSCS [36], RL-GJO [37], and HKOA [39] are sourced from their respective references. Since the literature for MS-TSA and RL-GJO does not provide mean and worst values, these two algorithms are excluded from the statistical analysis in this section.)

Table 15 presents the statistical results for the RTC of the SDM and DDM, PWP- 201 of the SDM and DDM,STM6 - 40/36 of the SDM and DDM. The results highlighted in bold indicate the best performance.

Table 15. Statistical results of SCSO compared to other algorithms on Different Models

Cell& Modules	Model	Algorithm	Min	Mean	Max	STD	CPU time (s)	Average ranking	Rank
RTC (Data Set [50])	SDM	WOA	9.724e- 04	0.0048	0.0213	0.0047	3.2081	7.967	7
		RLWOA	8.297e- 04	0.0060	0.0179	0.0058	8.6520	8.200	9
		GWO	0.00126	0.0027	0.0086	0.0016	2.8802	6.800	6
		HHO	0.00111	0.0078	0.0261	0.0075	7.4989	7.977	8
		EHHO	0.00102	0.0051	0.0217	0.0061	7.7889	8.867	10
		HBO	7.739e- 04	9.536e- 04	0.0017	2.015e- 04	3.0041	4.167	4
		SCA	0.00605	0.0128	0.0231	0.0040	3.0291	10.333	11
		SO	7.730e- 04	9.529e- 04	0.0014	6.928e- 04	2.9077	3.500	3
		HKOA [39]	7.730e- 04	7.730e- 04	7.730e- 04	4.040e- 17	–	1	1
		AOSCS [36]	9.860e- 04	9.860e- 04	9.860e- 04	2.090e- 17	–	4.400	5
		SCSO	7.7293e- 04	9.496e- 04	0.0013	5.050e- 04	5.8084	2.800	2
	DDM	WOA	0.00131	0.0049	0.0148	0.0036	4.8409	7.567	7
		RLWOA	0.00106	0.0050	0.0110	0.0028	15.826	7.967	8
		GWO	9.096e- 04	0.0021	0.0063	0.0011	5.0049	6.500	6
		HHO	0.00131	0.0071	0.0223	0.0056	13.052	8.633	10
		EHHO	0.00125	0.0064	0.0218	0.0049	13.975	8.600	9
		HBO	8.408e- 04	0.0014	0.0021	3.013e- 04	5.0963	4.533	4
		SCA	0.00355	0.0121	0.0378	0.0063	5.4250	10.333	11
		SO	7.920e- 04	0.0017	0.0021	3.082e- 04	4.9403	4.733	5
		HKOA [39]	7.326e- 04	7.473e- 04	7.730e- 04	1.232e- 05	–	1	1
		AOSCS [36]	9.825e- 04	1.021e- 03	1.245e- 03	6.340e- 05	–	2.633	2
		SCSO	6.872e- 04	0.0016	0.0130	0.0020	9.9863	3.500	3
PWP- 201 (Data set [19])	SDM	WOA	0.00262	0.0079	0.0124	0.0027	2.9602	6.633	7
		RLWOA	0.00560	0.0077	0.0099	0.0010	8.3834	6.367	6
		GWO	0.00225	0.0070	0.0104	0.0025	2.8669	5.900	5
		HHO	0.00420	0.0113	0.0266	0.0045	8.8459	8.167	8
		EHHO	0.00329	0.0127	0.0335	0.0072	8.0320	8.233	9
		HBO	0.00367	0.0149	0.0475	0.0114	3.1034	3.867	4
		SCA	0.00994	0.0213	0.0641	0.0107	2.9844	9.600	10
		SO	0.00221	0.0026	0.0033	2.727e- 04	2.8573	2.233	2
		AOSCS [36]	0.00245	0.0025	0.0025	1.290e- 17	–	2.400	3
		SCSO	0.00206	0.0023	0.0031	1.850e- 04	6.3305	1.600	1
	DDM	WOA	0.00644	0.0671	0.2743	0.0796	4.4854	7.100	8
		RLWOA	0.00590	0.0361	0.2747	0.0656	13.353	6.033	6
		GWO	0.00231	0.0105	0.0473	0.0100	4.5684	3.900	5
		HHO	0.00627	0.0858	0.2743	0.0917	11.863	6.267	7
		EHHO	0.00510	0.0387	0.2742	0.0571	12.292	7.367	9
		HBO	0.00408	0.0061	0.0083	0.0012	4.7003	2.967	4
		SCA	0.00996	0.0834	0.2743	0.1081	4.5157	7.400	10
		SO	0.00218	0.0038	0.0047	6.630e- 04	4.7991	1.700	1
		SCSO	0.00205	0.0039	0.0135	0.0017	9.1535	2.233	2
STM6 - 40/36 (Data set [19])	SDM	WOA	0.00940	0.1278	0.3108	0.1242	2.4179	8.600	8
		RLWOA	0.00797	0.1192	0.3108	0.1196	6.6825	9.233	11
		GWO	0.00280	0.0083	0.0163	0.0036	2.4735	5.333	6
		HHO	0.02417	0.1596	0.3108	0.1275	6.6441	9.033	9
		EHHO	0.02298	0.1643	0.3108	0.1166	6.1825	9.067	10
		HBO	0.00331	0.0046	0.0369	0.0061	2.4057	4.733	5
		SCA	0.01051	0.1110	0.3108	0.1333	2.2588	8.200	7
		SO	0.00172	0.0047	0.0355	0.0080	2.3180	3.567	3
		AOSCS [36]	0.00173	0.00173	0.00173	4.054e- 17	–	1.083	1
		SCSO	0.00172	0.0045	0.0346	0.00852	5.1917	3.065	2
	DDM	WOA	0.02584	0.0748	0.3108	0.0667	4.4335	6.767	6
		RLWOA	0.01157	0.1816	0.3108	0.1329	12.173	7.833	9
		GWO	0.00207	0.0088	0.0191	0.0054	4.0241	3.600	4
		HHO	0.01128	0.0810	0.3108	0.0828	10.525	7	8
		EHHO	0.00796	0.1267	0.3108	0.1239	10.226	6.967	7
		HBO	0.00317	0.0039	0.0069	0.0009	4.3214	2.867	3
		SCA	0.01186	0.0835	0.3108	0.1160	4.0830	6.233	5
		SO	0.00174	0.0036	0.0126	0.0020	3.8811	2.267	2
		SCSO	0.00170	0.0032	0.0292	0.0057	7.5787	1.667	1

Bold indicates the maximum IAEP value

Real world manufacturer data experiment

We conducted experiments using data provided by real-world manufacturers to validate the accuracy of SCSO parameter extraction. The data used in this paper is sourced from reference [51]. Two commercial models utilized are TFST 40 and MCSM 55. The data were collected under two different conditions: one with the same temperature of 25 °Cand varying solar radiation intensities of 1000 W/m², 800 W/m², 600 W/m², 400 W/m², and 200 W/m². The second condition had a constant solar radiation intensity of 1000 W/m² but different temperatures; TFST 40 temperatures were 25 °C, 40 °C, 55 °C, and 70 °C, while MCSM 55 temperatures were 25 °C, 40 °C, and 60 °C.

We used SCSO for parameter extraction, running the program independently 30 times. The optimal parameter values obtained for TFST 40 and MCSM 55 under a temperature of 25 °C with varying solar radiation intensities are shown in Table 16. The optimal parameters obtained for TFST 40 and MCSM 55 under a solar radiation intensity of 1000 W/m² with different temperatures are shown in Table 17. Table 18 presents the statistical results of RMSE (Root Mean Square Error) for TFST 40 and MCSM 55 using SCSO to extract unknown parameters under the same temperature but varying solar radiation intensities. Table 19 shows the statistical results of RMSE for TFST 40 and MCSM 55 using SCSO to extract unknown parameters under the same solar radiation intensity but different temperatures. The statistical results include the Max, the Min, the Mean, and the STD, providing comprehensive metrics for evaluating the performance of SCSO from multiple perspectives. Based on the information presented in Tables 17 and 18, it can be observed that even in the worst-case scenarios, the values of RMSE are not significantly high, and the standard deviation (STD) is low. This indicates that SCSO provides stable parameter extraction even under challenging conditions.

Table 16. The best parameter values obtained for TFST 40 and MCSM 55 at a temperature of 25 °C under varying solar radiation intensities are as follows

Condition	Model	Iph (A)	Isd (A)	Rs (Ω)	Rsh(Ω)	n	RMSE
1000 W/m2	TFST40	2.671001	2.117718e- 06	1.085347	428.3788	1.789866	0.001537
	MCSM55	3.437906	5.039160e- 07	0.290629	1893.701	1.490597	0.005660
800 W/m2	TFST40	2.136650	1.695792e- 06	1.085170	355.2165	1.764527	0.001196
	MCSM55	2.754886	3.190134e- 07	0.298282	701.1435	1.448930	0.002734
600 W/m2	TFST40	1.603643	1.758847e- 06	1.086168	365.8219	1.769911	0.000723
	MCSM55	2.068529	2.149669e- 07	0.309982	524.3558	1.415063	0.001334
400 W/m2	TFST40	1.066402	2.492601e- 06	1.026623	381.9867	1.819039	0.000854
	MCSM55	1.382227	1.386829e- 07	0.349015	441.3141	1.378140	0.000812
200 W/m2	TFST40	0.5329070	2.003003e- 06	0.999178	348.1467	1.793492	0.000513
	MCSM55	0.6914812	1.499986e- 07	0.279213	448.8824	1.382770	0.000320

Table 17. The best parameters obtained for TFST 40 and MCSM 55 under a solar radiation intensity of 1000 W/m² and varying temperatures are as follows

Condition	Model	Iph (A)	Isd (A)	Rs (Ω)	Rsh(Ω)	n	RMSE
25 °C	TFST40	2.671001	2.117718e- 06	1.085347	428.3788	1.789866	0.001537
	MCSM55	3.437906	5.039160e- 07	0.290629	1893.701	1.490597	0.005660
40 °C	TFST40	2.677531	7.185857e- 06	1.111444	431.9276	1.753893	0.001310
	MCSM55	3.468614	1.092962e- 06	0.314547	542.1227	1.413376	0.002776
55 °C	TFST40	2.690594	2.038594e- 05	1.142850	309.2309	1.703632	0.001141
	MCSM55	–	–	–	–	–	–
60 °C	TFST40	–	–	–	–	–	–
	MCSM55	3.492422	6.953358e- 06	0.321941	593.3906	1.405718	0.002712
70 °C	TFST40	2.692798	8.407851e- 05	1.129893	356.3277	1.720746	0.000539
	MCSM55	–	–	–	–	–	–

Table 18. The statistical results for TFST 40 and MCSM 55 using SCSO under varying solar radiation intensities at a temperature of 25 °C are as follows

Condition	Model	Min	Mean	Max	STD	CPU time
1000 W/m2	TFST40	0.001537	0.006349	0.015767	0.003039	5.809883
	MCSM55	0.005660	0.007501	0.014304	0.001692	5.427686
800 W/m2	TFST40	0.001196	0.006020	0.014329	0.003171	5.969990
	MCSM55	0.002734	0.005909	0.006846	0.001250	5.497350
600 W/m2	TFST40	0.000723	0.004604	0.008943	0.003044	5.882341
	MCSM55	0.001334	0.005335	0.006856	0.001676	5.270297
400 W/m2	TFST40	0.000854	0.004141	0.007551	0.002745	5.626533
	MCSM55	0.000812	0.002502	0.005810	0.000927	5.380637
200 W/m2	TFST40	0.000513	0.000946	0.001394	0.000243	4.955716
	MCSM55	0.000320	0.000750	0.002182	0.000360	4.847575

Table 19. The statistical results for TFST 40 and MCSM 55 using SCSO under varying temperatures at a solar radiation intensity of 1000 W/m² are as follows

Condition	Model	Min	Mean	Max	STD	CPU time
25 °C	TFST40	0.001537	0.006349	0.015767	0.003039	5.809883
	MCSM55	0.005660	0.007501	0.014304	0.001692	5.427686
40 °C	TFST40	0.001310	0.005736	0.027187	0.005208	5.447734
	MCSM55	0.002776	0.011788	0.034220	0.011469	5.246707
55 °C	TFST40	0.001141	0.004471	0.011375	0.002367	5.097529
	MCSM55	–	–	–	–	–
60 °C	TFST40	–	–	–	–	–
	MCSM55	0.002712	0.009425	0.020085	0.007137	4.952606
70 °C	TFST40	0.000539	0.004032	0.018378	0.004487	5.084638
	MCSM55	–	–	–	–	–

Based on the extracted unknown parameters, it is possible to calculate the current and power for the experiments. Figure 29 displays the actual I-V and P–V curves alongside the experimental curves for TFST40 under constant temperature at 25 °C and varying solar radiation intensities of 1000 W/m², 800 W/m², 600 W/m², 400 W/m², and 200 W/m². As seen in Fig. 29, the actual data matches the data calculated through experimentation, indicating that SCSO has successfully extracted the unknown parameters for TFST40. Figure 30 shows the I-V and P–V curves for TFST40 under constant solar radiation intensity of 1000 W/m² and varying temperatures at 25 °C, 40 °C, 55 °C, and 70 °C. Similar to Fig. 29, the actual data closely aligns with the experimentally calculated data, confirming the successful parameter extraction by SCSO for TFST40. Figure 31 illustrates the I-V and P–V curves for MCSM55 under constant temperature at 25 °C and varying solar radiation intensities of 1000 W/m², 800 W/m², 600 W/m², 400 W/m², and 200 W/m². Once again, the actual data closely matches the experimentally calculated data, indicating a successful extraction of unknown parameters by SCSO for MCSM55. Lastly, Fig. 32 presents the I-V and P–V curves for MCSM55 under constant solar radiation intensity of 1000 W/m² and varying temperatures at 25 °C, 40 °C, and 60 °C. As with the previous figures, the actual data aligns well with the experimentally calculated data, demonstrating that SCSO has effectively extracted the unknown parameters for MCSM55.These results collectively highlight the reliability of SCSO in parameter extraction for both TFST40 and MCSM55 across various experimental conditions.

Fig. 29 [Images not available. See PDF.]

The I-V a and P–V b measurement data and calculated data fitting curves for TFST40 under varying solar radiation intensities at 25 °C

Fig. 30 [Images not available. See PDF.]

The I-V a and P–V b measurement data and calculated data fitting curves for TFST40 under varying temperatures at a solar radiation intensity of 1000 W/m²

Fig. 31 [Images not available. See PDF.]

The I-V a and P–V b measurement data and calculated data fitting curves for MCSM55 under varying solar radiation intensities at 25 °C

Fig. 32 [Images not available. See PDF.]

The I-V a and P–V b measurement data and calculated data fitting curves for MCSM55 under varying temperatures at a solar radiation intensity of 1000 W/m²

In summary, the proposed SCSO method accurately extracts unknown parameters of various photovoltaic models under different conditions, indicating that SCSO can serve as an effective alternative approach to the problem of extracting unknown parameters in photovoltaic models.

Results and discussion

By analyzing the convergence curves of the six photovoltaic models mentioned above, it can be concluded that the proposed SCSO algorithm demonstrates the fastest convergence speed and the highest accuracy. It generally converges to the optimal solution within 200 generations, and the smoothness of the convergence curve indicates that it is less likely to fall into local optima. This shows that the improvements introduced in this paper effectively enhance the algorithm’s performance. Through the analysis of the experimental data provided, it is easy to find that although the algorithms are not the same, the RMSE error of DDM is always smaller than that of SDM for different models of SDM and DDM, indicating that DDM is more accurate than SDM. This more accurate result is not obtained out of thin air, and the cost paid is that DDM has two unknown parameters more than SDM. Through the analysis of IAE, it was found that for the different models in the text, the sum of IAEI and IAEP of all data did not exceed 0.05 and 0.6, respectively. This indicates that the experimental data simulated through SCSO parameter extraction is very close to the actual data, proving the accuracy of SCSO. Through analyzing the statistical results, it was found that although SCSO extracts more accurate parameters, the STD value is relatively large compared to other algorithms, indicating that the algorithm has poor stability compared to other algorithms. This can enhance the stability of the algorithm in future work. Through the analysis of P-values, it was found that the P-values of SCSO and other algorithms are generally less than 0.05, indicating that SCSO is significantly different from other algorithms. Finally, by analyzing the I-V and P–V curves of two commercial models at the same temperature, different solar radiation intensities, and the same solar radiation intensity at different temperatures, it was found that when the temperature was the same, the photo-generated current would increase with the increase of solar radiation intensity. When the solar radiation intensity is the same, the reverse saturation current of the diode will increase with the increase of temperature. Except for these two parameter values, there is little change in other values. This situation indicates that from a physical perspective, the parameters extracted by SCSO under different conditions are reasonable. In summary, SCSO extraction of photovoltaic parameters is accurate, effective, and reasonable.

Conclusion and future work

This paper introduces a hybrid Snake Optimization algorithm combined with the Sine–Cosine Algorithm (SCSO) for extracting the unknown parameters of PV models. The enhanced version of the Snake Optimization integrates the Sine–Cosine Algorithm to make the bio-inspired Snake Optimization more closely resemble the movement of real snakes, balancing exploration and exploitation to enhance search capabilities. Additionally, the parameter C₁, which controls the balance between exploration and exploitation, is changed from a fixed value to a variable one, enabling the algorithm to perform random walks between these two phases. To further accelerate the algorithm’s convergence, the parameter C₃, which controls the snake’s position update during the exploitation phase, was modified from a fixed value of 2 to a variable value that nonlinearly decreases from 2 to 0 and incorporates the Newton–Raphson method. The use of the LIRL strategy allows the algorithm to escape local optima. The SCSO is applied to different PV models for parameter extraction, and the experimental results show that for both the Single Diode Model (SDM) and Double Diode Model (DDM), the RMSE error of DDM is consistently smaller than that of SDM, indicating that DDM is more accurate. Compared to other metaheuristic algorithms, SCSO achieved the most precise results with the fastest convergence speed. Finally, SCSO was tested on two commercial PV modules, TFST 40 and MCSM 55, confirming the algorithm’s accuracy and effectiveness. By analyzing the I-V and P–V curves of the two commercial modules under varying solar irradiance and temperature conditions, it was observed that when the temperature remains constant, the photocurrent increases as the solar irradiance increases. When solar irradiance is constant, the diode’s reverse saturation current increases with rising temperature, which, from a physics perspective, validates the rationale of SCSO in extracting unknown parameters. Therefore, SCSO stands as an alternative solution for addressing PV parameter extraction challenges.

However, SCSO has some limitations, such as slightly lower performance on certain problems and longer runtime. Additionally, experiments showed that SCSO’s stability is weaker than the original algorithm, likely due to the rigid integration of the Sine–Cosine Algorithm. Future research could focus on using reinforcement learning to flexibly combine different algorithms. Moreover, the data provided by the manufacturers for the experiments were obtained under standard conditions, which are difficult to replicate in real-world scenarios. Therefore, future studies should explore experiments under varying conditions. The development of dynamic PV models could also be considered for future work. Combining the algorithm with the Newton–Raphson method or other numerical methods to improve its accuracy is also a potential direction to consider.

Author contributions

Q. L.: Investigation, experiment, writing—draft; Y.Z.: Supervision, writing—review and editing; Qi. L.: Algorithm design, writing—review.

Funding

This work was supported by the National Natural Science Foundation of China under Grant No. U21 A20464, 62066005.

Data availability

The data is provided within the manuscript or in the supplementary information files.

Code availability

Data is provided within the manuscript or supplementary information files.

Declarations

Competing interests

The authors declare no conflict of interests.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

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