Parameters Identification of Solar PV Using Hybrid Chaotic Northern Goshawk and Pattern Search

Abstract

This article proposes an effective evolutionary hybrid optimization method for identifying unknown parameters in photovoltaic (PV) models based on the northern goshawk optimization algorithm (NGO) and pattern search (PS). The chaotic sequence is used to improve the exploration capability of the NGO algorithm technique while evading premature convergence. The suggested hybrid algorithm, chaotic northern goshawk, and pattern search (CNGPS), takes advantage of the chaotic NGO algorithm’s effective global search capability as well as the pattern search method’s powerful local search capability. The effectiveness of the recommended CNGPS algorithm is verified through the use of mathematical test functions, and its results are contrasted with those of a conventional NGO and other effective optimization methods. The CNGPS is then used to extract the PV parameters, and the parameter identification is defined as an objective function to be minimized based on the difference between the estimated and experimental data. The usefulness of the CNGPS for extraction parameters is evaluated using three distinct PV models: SDM, DDM, and TDM. The numerical investigates illustrate that the new algorithm may produce better optimum solutions and outperform previous approaches in the literature. The simulation results display that the novel optimization method achieves the lowest root mean square error and obtains better optima than existing methods in various solar cells.

1. Introduction

Among many alternative energy sources, solar energy harvesting is a good contender, and the market share of solar-powered systems is rising quickly [1,2]. To build solar energy-generating structures, a few photovoltaic (PV) cells are linked together in serial or parallel patterns. Electricity supply systems and solar power plants are integrated and operate simultaneously [3,4,5]. The electricity generated by solar energy plants is influenced by operational and environmental factors. Power generating volatility has an impact on the commercial feasibility of solar energy projects. Therefore, it is recommended that PV cell or module modeling and parameters assessment be optimized in order to increase the effectiveness of solar energy production systems. A model based on observed current and voltage data is required for assessing and forecasting PV features in order to optimize energy conversion [6]. It is essential to evaluate PV panels with unknown properties and model PV systems appropriately. Parameter assessment is a difficult task since variables are nonlinear, multifactorial, and multimodal [5]. However, through mathematical modeling, researchers have made great progress in the previous few decades in understanding the characteristics of PV systems. PV cells can be successfully evaluated by using single (SDM), double (DDM), and triple-diode (TDM) models [7]. Unfortunately, no one is aware of these parameters’ values, which restricts the applicability of all of these models [8]. The inaccurate identification of these components can lead to errors in maximum power point tracking, quality control, and PV system act calculation. Numerous metaheuristic strategies have been successfully applied to parameterize PV systems and a variety of other applications [9]. To address challenging optimization problems, the computational intelligence paradigms known as metaheuristic algorithms are used [10]. Meta-heuristic algorithms have the benefit of not being constrained to continuum, differentiable, or convex situations [11,12,13,14,15,16,17,18,19]. Additionally, they search very effectively and can take a flexible approach to solving complex problems.

In order to overcome the shortcomings of numerical approaches, meta-heuristic optimization algorithms have recently become frequently employed to estimate the characteristics of PV cells. Meta-heuristic optimization techniques have a number of advantages, including improved conjunction, safety from initial guess, no singularity condition, and examination of all I-V data points rather than key locations on the I-V curve [6]. Numerous meta-heuristic optimization techniques have been utilized in the literature to obtain PV cell parameters. Some of these techniques include: Genetic Algorithms (GA) [20], Particle Swarm Optimization (PSO) [21,22], Simulated Annealing (SA) [23], Harmony Search (HS) [24], Bacterial Foraging Algorithm (BFA) [25,26], Cat Swarm Optimization (CSO) [27], Differential Evolution (DE) [28], Flower Pollination Algorithm (FPA) [29], Whale Optimization Algorithm (WOA) [30], Firefly algorithm (FA) [31,32], Gravitational Search Algorithm (GSA) [33], Grey Wolf optimization (GWO) [34], Moth–Flame Optimization (MFO) [35], Tunicate Swarm Algorithm (TSA) [36,37], Multi-Verse Optimizer (MVO) [38], Salp Swarm Algorithm (SSA) [39], and Golden Search Optimization Algorithm (GSO) [40]. Various heuristic techniques for extracting PV parameters have been successfully utilized in the literature [41,42]; some observations on these strategies are summarized in

Table 1. Literature in recent years.

Ref.	Objective Function	Algorithm	Type of Solar Cell	Remarks
[46]	RMSE	IMFOL	SDM DDM PV module	Appropriate levels of accuracy and robustness
[47]	RMSE	mBES	TDM	Low optimal fitness values
[48]	RMSE	TSA	PV module	Fast convergence, low fitness values
[49]	RMSE	FCHHHO	SDM DDM TDM	Superior performance, accuracy and robustness
[50]	RMSE	MSSA	SDM DDM PV module	Improved performance, avoiding local optimum
[51]	RMSE	DOLADE	SDM DDM PV Module	Great competition in terms of accuracy, reliability, and computational efficiency
[52]	RMSE	ITSA	SDM DDM PV Module	Higher convergence accuracy, better stability
[53]	RMSE	SDGBO	SDM DDM TDM PV Module	Enhance performance, capability to identify unknown parameters of PV models
[54]	RMSE	ARSO	SDM DDM PV Module	Accuracy, reliability, and convergence speed
[55]	RMSE	PSOCS	SDM DDM PV Module	High accuracy, enhanced exploration performance
[56]	RMSE SIAE	HSOA	SDM DDM PV module	Superior solution quality, convergence and reliability
[57]	RMSE NRMSD	IAOA	DDM	High level of reliability, accuracy, and effectiveness
[58]	RMSE	RN-ChOA	SDM DDM TDM PV Module	Better performance in estimating parameters of PV model
[59]	RMSE	GCAOAEmNR	SDM DDM TDM	Accuracy, stability, and convergence rate
[60]	RMSE	ADHHO	SDM DDM TDM PV Module	Proposed model can reasonably simulate output performance of PV and can be used as trustworthy method for extraction of unknown parameters of solar PV systems
[61]	RMSE NRMSD	IAOA	SDM	High level of reliability, accuracy, and effectiveness
[62]	RMSE	OLGBO	SDM DDM TDM PV Module	The OLGBO can be a fast, promising, reliable, and feasible optimization method for dealing with unknown parameter identification problems in PV models

. It can be observed that, in most situations, the goal of the optimization process is to minimize the sum of the squared difference between the measured solar PV cell output current and the corresponding estimated value from a certain number of data points. The objective of this problem is to reduce the root mean square error (RMSE), which is commonly employed. The abovementioned tactics achieved various goals, including providing accurate results, increasing convergence rate, improving exploration and exploitation, and preventing stuck-in local minima. A massive revolution of metaheuristic algorithms can be applied to solve different engineering problems efficiently.

Although metaheuristic algorithms can produce respectable results, no technique is superior to another for handling every optimization issue. In order to increase the efficiency of the initial metaheuristic algorithms and adapt them to a specific application, several studies have been carried out. According to the literature review, new optimization algorithms are desperately needed to solve real-world issues. The northern raptor optimization algorithm (NGO), a recently created bio-inspired meta-heuristic optimization technique, is informed by the searching and hunting habits of northern kestrels. Northern red tailed hawks that hunt for prey serve as the search agents in this strategy, which was first recommended by Dehghani et al. [43]. NGO outperforms other strategies in terms of identifying the best solutions and is perfectly suited to issues in real-world optimization. The updated version of the original NGO that is described in this study, called the chaos northern raptor optimization algorithm (CNGO), integrates chaotic sequences to improve the project’s skills for searching and exploring. A balance between utilization and investigation must be maintained throughout the search process in order to fully utilize any optimization strategy to its full potential. As a global search method, CNGO searches a vast area, hence, when used alone, it might not yield the best results. A search engine technology named Pattern Search (PS) leverages the local search but is capable of conducting a full search [44,45]. There is possibility for hybridization because of each approach’s various capacities. The anarchic north raptor optimization and correlation based method, or CNGPS, was created as a result of all of this and is currently being used in the present attempt to identify the RMSE that is lowest in order to extract the PV variables. By comparing the results of the suggested CNGPS technique with those of other methods in two benchmark problems taken from the literature, its efficacy is determined. The outcomes of the proposed technique were compared with six well-known methods for obtaining the characteristics of three different PV models, namely SDM, DDM, and TDM, in order to show the superior performance of CNGPS. The outcomes of the proposed method show that it outperforms earlier methods outlined in the literature and can produce better optimum solutions and attain a lower RMSE.

The main contributions of this work can be summarized as follows:

(1) An efficient hybrid optimization algorithm, CNGPS, is proposed for numerical function optimization and parameter estimation of solar cell SDM, DDM, and TDM models;

(2) The proposed hybrid algorithm utilizes the exploration ability of CNGO and the exploitation ability of PS, which can significantly improve the accuracy of results;

(3) Compared with the original NGO and five state-of-the-art algorithms for the parameter estimation problem of PV models, CNGPS demonstrated significant advantages in terms of accuracy, convergence speed, and stability.

The remaining sections of the current study are arranged as follows: The proposed hybrid optimization approach is described in Section 2. The PV models used in this investigation are introduced in Section 3. Additionally included is the derivation of the mathematical equation used to assess the PV model’s parameters. Section 4 presents and discusses the experimental results. Section 5 describes the verdict and subsequent actions.

2. Proposed Procedure

The suggested hybrid technique CNGPS will be defined in this subdivision after a brief indication of the northern goshawk optimization algorithm and pattern search methods.

2.1. Northern Goshawk Optimization

A hunter of average size, the northern goshawk is a member of the Slip-up genus. This northeastern bird hunts a variety of prey, including small and big birds, perhaps even other raptors; smaller animals like rodents, rabbits, and squirrels; as well as foxes and raccoons. The two stages of the northern goshawk’s hunting process entail quickly approaching its prey in the first stage after spotting it, then briefly pursuing the prey in the second [63].

The NGO algorithm, an inhabitant’s technique, involves northern goshawks as players. Based on their position in the search space, each member of the population suggests solutions to the problem [26]. In the first step of the NGO, population members are randomly modified according to the problem’s lower and upper bounds based on Equation (1).

$$x_{i,j}=l_j+\left(u_j-l_j\right), i=1,2,\cdots ,N j=1,2,\cdots ,m$$

(1)

where x_i,j is the rate of the jth variable definite by the ith applicant resolution and N is the number of population members.

A matrix known as the demographic matrix (X) in Equation (2) is used to identify the northern goshawk individuals from the population in the NGO. The columns of this matrix display recommended readings for the issue variables, while the rows represent potential solutions.

$$X=\left[\begin{array}{c}{X}_1\\ ⋮\\ {\begin{array}{c}{X}_i\\ ⋮\\ X\end{array}}_N\end{array}\right]=\left[\begin{array}{ccccc}{x}_{1,1}& \cdots & x_{1,j}& \cdots & x_{1,m}\\ ⋮& \cdots & ⋮& \cdots & ⋮\\ x_{i,1}& \cdots & x_{i,j}& \cdots & x_{i,m}\\ ⋮& \cdots & ⋮& \cdots & ⋮\\ x_{N,1}& \cdots & x_{N.i}& \cdots & x_{N,m}\end{array}\right]$$

(2)

The NGO can evaluate the optimal solution of the issue at hand based on each prospective solution. Equation (3) determines the values achieved for the objective using a vector called the optimization problem vector. Any of the alternative solutions can be used to assess the situation’s optimal value. In Equation (3), the values for the optimal solution are resolute using the optimal solution vector.

$$F=\left[\begin{array}{c}F(X_1)\\ ⋮\\ \begin{array}{c}F(X_i)\\ ⋮\\ F\left(X_N\right)\end{array}\end{array}\right]$$

(3)

The standard by which the best explanation is chosen is the value of the objective function. The value of the goal function decreases in minimization problems and increases in solving complex optimization problems in direct proportion to how good the proposed solution is. The best appropriate solution should be updated after each iteration because fresh values for the goal function were acquired with each iteration [25].

The NGO imitates the behavior of attacking and hunting northern goshawks in order to update potential solutions.

Identification of the prey and attack, or Phase of Exploration: In the first phase, the northern goshawks locate their prey, which they then immediately attack. By mimicking the activity of this northern goshawk, the NGO is able to explore and identify several search spaces. This increases the NGO’s ability for exploration inside the particular investigation of the problem-solving domain. In Equations (4)–(6) [25], the aforementioned concepts as well as the northern goshawk’s first-phase approach are mathematically modeled, [25].

$$P_i=X_k, i=1,2,\dots ,N, k=1,2,i-1,i+1,\dots ,N$$

(4)

$$x_{i,j}^{new,p_1}=\left\{\begin{array}{c}{x}_{i,j}+rand.\left(p_{i,j}-I.x_{i,j}\right), F_{pi}<F_i\\ x_{i,j}+rand.\left(x_{i,j}-p_{i,j}\right), else\end{array}\right.$$

(5)

$$X_i=\left\{\begin{array}{c}{X}_i^{new,p_1} F_i^{new,p_1}<F_i\\ X_i else\end{array}\right.$$

(6)

where p_i is the location of prey in the ith northern goshawk, F_pi is its objective function value, k is a random number in interval [1, N], $X_i^{new,p_1}$ is the new status of the ith solution, $x_{i,j}^{new,p_1}$ is the new position of the ith solution in the jth dimension based on phase 1, and $F_i^{p_1}$ is its objective function value based on phase.

The exploitation phase, also known as the chase and escape operation, is when the victim tries to run after being assaulted by the northern goshawk. As a result, the northern goshawk continues to chase after its prey. The extraordinary speed of the northern goshawks allows them to pursue their prey in almost any situation and eventually engage in hunting. With the aid of this tactic, the attacked region’s northern goshawks are able to catch more fish, and the hunting region’s NGO gathers in more advantageous locations. The local search and exploitation capabilities of NGO are improved by this technique. To arrive at a more accurate result, the computer needs to numerically investigate the points close to the location of the northern goshawk. Analytical simulations of the northern goshawk’s activity during a hunt are provided by Equations (7)–(9) [25].

$$x_{i,j}^{new, p_2}=x_{i,j}+R.\left(2.rand-1\right).x_{i,j}$$

(7)

$$R=0.02(1-\frac{t}{T})$$

(8)

$$\begin{array}{c}\hfill X_i=\left\{\begin{array}{cc}{X}_i^{new,p_2}& \hspace{1em}{F}_i^{new,p_2}<F_i\\ X_i& \mathit{else}\end{array}\right.\end{array}$$

(9)

where $x_{i,j}^{p_2}$ is the new position of the ith northern goshawk in the jth dimension based on phase 2, t is the iteration counter, and T is the maximum number of iterations. $X_i^{new,p_2}$ is the new status of the ith northern goshawk and $F_i^{new,p_2}$ is its objective function value based on phase 2.

The coefficient R describes the area of the region that each person in the population can locally explore to find a more accurate response. This coefficient affects the NGO’s ability to exploit resources in order to get closer to the ideal global solution. Since this coefficient has a high value in the initial iterations, a greater area around each person is looked at. The R coefficient falls as the process progresses, which results in a reduction in the radius of each person’s area. This procedure enables the NGO to obtain findings that are closer to the global optimum and to undertake more extensive searches at more regular intervals in the area surrounding each member of the population [25].

Once everyone has been modified in following the previous first and second stages, the strongest candidate answer will be changed with the new demographic status and the parameters of the objective function. The NGO continues the procedure based on Equations (4)–(9) until the entire calculation is complete before moving on to the following iteration. The perfect reaction to the current situation is finally proposed as the best candidate solution identified during the algorithm rounds. Algorithm 1 presents the several NGO steps as pseudo-code.

Algorithm 1. Pseudo-code of northern goshawk optimization algorithm.

Determine the NGO population size (N) and the number of iterations (T)

Initialization of the position of northern goshawk randomly based on Equation (1)

Calculate the objective function of the population

For t = 1:T

For I = 1:N

Phase 1: prey identification (exploration phase)

Select a random prey using Equation (4)

For j = 1:m

Calculate new status of the jth dimension using Equation (5)

End

Update the ith population member using Equation (6)

Phase 2: tail and chase operation (exploitation phase)

Update R using Equation (7)

For j = 1:m.

Calculate new status of the jth dimension using Equation (8)

End

Update the ith population member using Equation (9)

End

Update best candidate solution

End

Output best solution obtained by NGO

2.2. Chaotic Northern Goshawk Optimization

Utilizing the NGO’s global search functions is the study’s goal. The chaotic series is employed in both the discovery phase and the location, updating expressions defined in Equation (5) to do this and boost the algorithm’s exploring capability. Chaotic systems, which are deterministic structures affected by the beginning conditions, exhibit erratic behavior, irregularity, and stochastic properties. Chaotic parameters can go outside of defined constraints without repeating due to their inherent irregularity. A chaotic map is one with a chaotic pattern and the ability to induce chaotic movement. This study employs the well-known logistic map, which is founded on Equation (10):

µ (t + 1) = a × µ (t) × (1 − µ (t))

(10)

where µ (t) represents the chaotic map and t denotes the iteration number. a denotes a constant equal to four.

The chaotic NGO (CNGO) improves the stochastic behavior of the algorithm while preventing premature convergence by using the chaotic map rather than a simple random number in the northern goshawk’s approach to the prey equation (i.e., Equation (5)). Therefore, for guiding northern goshawks toward the prey during the exploration phase, the following Equation (11) is given.

$$x_{i,j}^{new,p_1}=\left\{\begin{array}{c}{x}_{i,j}+{\mu }_t.\left(p_{i,j}-I.x_{i,j}\right), F_{pi}<F_i\\ x_{i,j}+{\mu }_t.\left(x_{i,j}-p_{i,j}\right), else\end{array}\right.$$

(11)

Additionally, in the recommended CNGO, to improve the algorithm’s exploration and search ability, at every iteration the lowliest northern goshawk contribution of the maximum fitness value will be different by a new random northern goshawk as offered in Equation (12):

$$X_{worst}=X_{i min}+rand\times \left(X_{i max}-X_{i min}\right)$$

(12)

where $X_{worst}$ is a northern goshawk with the highest fitness value. The process of the proposed CNGO is depicted in Figure 1.

2.3. Pattern Search (PS)

PS is an easy gradient-free approach for enhancing local search. The PS scheme creates a list of locations that may or may not be near the optimal position [25]. An element is encircled by a mesh in the first round. The old element is replaced if a new element in the grid has a higher efficiency in the subsequent round.

The PS starts the exploration with a user-defined initial position P₀. The mesh level is reserved as 1 in the first round, and the pattern elements are produced as [0 1] + P₀, [1 0] + P₀, [−1 0] + P₀, and [0 −1] + P₀, and original mesh elements are added as described in Figure 2. The fitness function is then considered for each created sample element until a value less than P₀ is exposed. The survey is effective if there is such an element (f (P₁) < f (P₀)), and the PS technique accepts this element as the basis point. The process doubles the present mesh size by 2 after a positive election and changes on to the second round with the following novel elements: 2 × [0 1] + P₁, 2 × [1 0] + P₁, 2 × [−1 0] + P₁, and 2 × [0 −1] + P₁. Then, P₂ is recognized if a value less than P₁ is exposed, the mesh size is extended by two, and iterations continue. If the poll fails at any round, the current element is left alone, and the mesh size is reduced by a contraction factor. Until the lowest value was attained, or the termination criteria were satisfied, these processes were repeated.

2.4. Chaotic Northern Goshawk Optimization—Pattern Search (CNGPS)

A hybrid strategy combines two or more approaches to the same issue. Hybridization aims to incorporate the benefits of each technique in order to boost the precision of the output [36]. The CNGPS approach, which integrates the CNGO and PS methods, has been developed in the existing research. The chaotic northern goshawk algorithm, a based optimization method, successfully searches the complex problem and, therefore, is likely to yield an ideal or nearly optimum. Therefore, it can be utilized in conjunction with local optimization strategies like pattern search. Pattern search makes it simpler to exploit a limited area, but it rarely makes it simpler to investigate a larger area.

The proposed hybrid technique may make advantage of both the robust global and local finding characteristics of the CNGO and the PS algorithm. The CNGO algorithm easily avoids the global minimum and achieves excellent global optimum performance. The CNGO can increase the output accuracy by boosting the number of repeats. However, CNGO is unable to increase the outcomes’ accuracy if there are enough generations. This results in CNGO’s local search functionality still being below standard. The starting point of the pattern search algorithm has a significant impact on the algorithm’s output as a local optimization criterion; different starting points provide outcomes that are notably different. However, if a great beginning point is chosen, pattern search will be a rapid and effective strategy. In this work, we successfully combine the benefits of CNGO as a global optimization with opportunity to discover as an optimal solution to identify the best response. The CNGO is where the suggested hybrid strategy starts because the PS relies on the first answer. The CNGO is hired to continue taking care of a specific number of iterations. The PS begins its local search with the best choice provided by CNGO as its starting point. Figure 3 shows the process flow of the suggested hybrid algorithm.

3. PV Models and Problem Formulation

This section provides a brief overview of the SDM, DDM, and TDM models in order to illustrate the I-V properties of PV. The parameter estimation for each PV model is abstracted as mathematical expressions with related optimization issues [24,64].

3.1. SDM

SDM is simple, and Figure 4 explains its corresponding circuit diagram. In this figure, there are many important parameters, such as I_ph, I_sd, I_L, R_s and R_sh. Because of its simplicity and accuracy, the SDM is the most extensively used PV model [64,65]. The current output I_L may be written using Equation (13), and five unknown parameters, I_ph, I_sd, n, R_s, and R_sh, need to be correctly estimated in the SDM.

$$I_L=I_{ph}-I_{sd}\left(exp\left(\frac{q(V_L+R_s{I}_L)}{nkT}\right)-1\right)-\frac{V_L+R_s{I}_L}{R_{sh}}$$

(13)

where I_ph is the photo-current; I_L is the output-current; and I_d and I_sd denote the diode current and reverse saturation current of D₁, respectively. R_s and R_sh are series and shunt resistance, respectively. n is the D₁ ideality factor; T is the temperature in Kelvin; k = 1.3806503 × 10⁻²³ (J/K); q = 1.60217646 × 10⁻¹⁹ C.

3.2. DDM

The compound power loss inside the depletion zone is better taken into account by the DDM than the SDM, as seen in Figure 5. The DDM is frequently used in PV cell modeling in this fashion [28]. Two diodes are linked in parallel in the analogous DDM circuit. The output’s current IL can be approximated as follows:

$$I_L=I_{ph}-I_{sd1}\left(exp\left(\frac{q(V_L+R_s{I}_L)}{n_{1}kT}\right)-1\right)-I_{sd2}\left(exp\left(\frac{q(V_L+R_s{I}_L)}{n_{2}kT}\right)-1\right)-\frac{V_L+R_s{I}_L}{R_{sh}}$$

(14)

where I_sd1 and I_sd2 denotes the saturation current of D₁ and D₂, respectively. n₁ and n₂ are the ideality factors of D₁ and D₂, respectively. For DDM, seven parameters, including I_ph, I_sd1, I_sd2, R_s, R_sh, n₁, and n₂, require accurate identification.

3.3. TDM

The TDM, as shown in Figure 6, employs a third diode to replicate leakage current in the intergranular of commercial solar cells. This model can be used to simulate the output I-V properties of big manufacturing silicon solar cells [27]. Equation (15) may be used to compute the output current I_L. Nine parameters, I_ph, I_sd1, I_sd2, I_sd3, R_s, R_sh, n₁, n₂, and n₃, need appropriate fitting for TDM.

$$I_L=I_{ph}-I_{sd1}\left(exp\left(\frac{q(V_L+R_s{I}_L)}{n_{1}kT}\right)-1\right)-I_{sd2}\left(exp\left(\frac{q(V_L+R_s{I}_L)}{n_{2}kT}\right)-1\right)-I_{sd3}\left(exp\left(\frac{q(V_L+R_s{I}_L)}{n_{3}kT}\right)-1\right)-\frac{V_L+R_s{I}_L}{R_{sh}}$$

(15)

where n₃ is the ideality factor of the D₃ and I_sd3 represents the saturation current of D₃.

3.4. Objective Function

The equations are changed into linked optimization issues that can be resolved by an optimization algorithm to determine precise parameter values. When the optimal parameter values are employed, the objective function exhibits the least difference between the recorded and experimental data. The objective function is to extract the best parameter settings from the PV models by reducing the variation between simulated results and the measured data. The target function of SDM, DDM, and TDM is provided by Equations (16)–(18).

• For SDM:

$$\left\{\begin{array}{c}f\left(V_L.I_L.\mathit{X}\right)=I_{ph}-I_{sd}\left(exp\left(\frac{q(V_L+R_s{I}_L)}{nkT}\right)-1\right)-\frac{V_L+R_s{I}_L}{R_{sh}}-I_L \\ \mathit{X}=\left\{I_{ph}.I_{sd}. R_s. R_{sh}. n \right\} \end{array}\right.$$

(16)

• For DDM:

$$\left\{\begin{array}{c}f\left(V_L·I_L·\mathit{X}\right)=I_{ph}-I_{sd1}\left(\mathit{exp}\left(\frac{q\left(V_L+R_S{I}_L\right)}{n_{1}kT}\right)-1\right)\\ -I_{sd2}\left(\mathit{exp}\left(\frac{q\left(V_L+R_s{I}_L\right)}{n_{2}kT}\right)-1\right)-\frac{V_L+R_S{I}_L}{R_{sh}}-I_L\\ \mathit{X}=\left\{I_{ph}·I_{sd1}·I_{sd2}·R_s·R_{sh}·n_1·n_2\right\}\end{array}\right.$$

(17)

• For TDM:

$$\left\{\begin{array}{c}f\left(V_L·I_L·\mathit{X}\right)=I_{ph}-I_{sd1}\left(\mathit{exp}\left(\frac{q\left(V_L+R_s{I}_L\right)}{n_{1}kT}\right)-1\right)\\ -I_{sd2}\left(\mathit{exp}\left(\frac{q\left(V_L+R_s{I}_L\right)}{n_{2}kT}\right)-1\right)\\ -I_{sd3}\left(\mathit{exp}\left(\frac{q\left(V_L+R_s{I}_L\right)}{n_{3}kT}\right)-1\right)-\frac{V_L+R_s{I}_L}{R_{sh}}-I_L\\ \mathit{X}=\left\{I_{ph}·I_{sd1}·I_{sd2}·R_s·R_{sh}·n_1·n_2\right\}\end{array}\right.$$

(18)

According to [37], the vast majority of algorithms use the total root mean square error (RMSE) as their goal function. It is also possible to compare the outcomes of simulations with measurements using the RMSE. Therefore, RMSE is selected as the objective function:

$$RMSE\left(\mathit{X}\right)=\sqrt{\frac{1}{M}{\displaystyle {{\displaystyle \sum }}_{d=1}^M}f{\left(V_L.I_L.\mathit{X}\right)}^2}$$

(19)

where X denotes the vector containing the unknown parameters to be estimated.

4. Results and Analysis

The acquired results of the CNGPS method for numerical optimizer and parameter extraction of PV models will be assessed and compared with certain other optimization techniques to confirm the viability of the proposed strategy.

4.1. Proposed Method for Benchmark Functions

This section compares and validates the effectiveness of the proposed northern goshawk optimization algorithm-pattern search using a set of numerical reference test functions (CNGPS). In the literature on empirical evidence, these functions are widely employed to evaluate the efficacy of optimizers [20,27].

The mathematical typical and characteristics of these test functions are presented in

Table 2. Description of unimodal benchmark functions.

Function	Name	Range	${\mathit{f}}_{\mathit{m}\mathit{i}\mathit{n}}$	n (Dim)
$F_1\left(X\right)={{\displaystyle \sum }}_{i=1}^n{x}_i^2$	Sphere function	${\left[-100, 100\right]}^n$	0	30
$F_2\left(X\right)={{\displaystyle \sum }}_{i=1}^n\left|x_i\right|+{{\displaystyle \prod }}_{i=1}^n\left|x_i\right|$	Schwefel’s problem 2.22	${\left[-10, 10\right]}^n$	0	30
$F_3\left(X\right)={{\displaystyle \sum }}_{i=1}^n{\left({{\displaystyle \sum }}_{j=1}^i{x}_j\right)}^2$	Schwefel’s problem 1.2	${\left[-100, 100\right]}^n$	0	30
$F_4\left(X\right)=\underset{i}{\max } \left\{\left|x_i\right|, 1\le i\le n \right\}$	Schwefel’s problem 2.22	${\left[-100, 100\right]}^n$	0	30
$F_5\left(X\right)={{\displaystyle \sum }}_{i=1}^{n-1}\left[100{\left(x_{i+1}-x_i^2\right)}^2+{\left(x_i-1\right)}^2\right]$	Generalized Rosenbrock’s function	${\left[-30, 30\right]}^n$	0	30
$F_6\left(X\right)={{\displaystyle \sum }}_{i=1}^n{\left(\left[x_i+0.5\right]\right)}^2$	Step function	${\left[-100, 100\right]}^n$	0	30
$F_7\left(X\right)={{\displaystyle \sum }}_{i=1}^{n}i{x}_i^4+random\left[0,1\right)$	Quartic function with noise	${\left[-1.28, 1.28\right]}^n$	0	30

and

Table 3. Description of multimodal benchmark functions.

Function	Name	Range	${\mathit{f}}_{\mathit{m}\mathit{i}\mathit{n}}$	n (Dim)
$F_8\left(X\right)={{\displaystyle \sum }}_{i=1}^n-x_i\sin \left(\sqrt{\left|x_i\right|}\right)$	Generalized Schwefel’s problem 2.26	${\left[-500, 500\right]}^n$	428.9829 × n	30
$F_9\left(X\right)={{\displaystyle \sum }}_{i=1}^n\left[x_i^2-10\cos \left(2\pi x_i\right)+10\right]$	Generalized Rastrigin’s function	${\left[-5.12, 5.12\right]}^n$	0	30
$F_{10}\left(X\right)=-20 \exp \left(-0.2\sqrt{\frac{1}{n}{{\displaystyle \sum }}_{i=1}^n{x}_i^2}\right)-exp\left(\frac{1}{n}{{\displaystyle \sum }}_{i=1}^n\cos \left(2\pi x_i\right)\right)+20+e$		${\left[-32, 32\right]}^n$	0	30
$F_{11}\left(X\right)=\frac{1}{4000}{{\displaystyle \sum }}_{i=1}^n{x}_i^2-{{\displaystyle \prod }}_{i=1}^n\cos \left(\frac{x_i}{\sqrt{i}}\right)+1$	Ackley’s function	${\left[-600, 600\right]}^n$	0	30
$\begin{array}{c}\begin{array}{c}\hfill F_{12}\left(X\right)=\hfill \\ \frac{\pi }{n}\left\{10sin\left(\pi y_1\right)+\sum _{i=1}^{n-1}{\left(y_i-1\right)}^2\left[1+10{sin}^2\left(\pi y_{i+1}\right)\right]+{\left(y_n-1\right)}^2\right\}\hfill \\ \hfill +\sum _{i=1}^{n}u\left(x_i,10,100,4\right)\hfill \\ y_i=1+\frac{x_{i+4}}{4}u\left(x_i,a,k,m\right)=\left\{\begin{array}{cc}k{\left(x_i-a\right)}^m\hfill & x_i>a\\ 0\hfill & a<x_i<a\\ k{\left(-x_i-a\right)}^m\hfill & x_i<-a\end{array}\right.\hfill \end{array}\end{array}$	Generalized penalized function	${\left[-50, 50\right]}^n$	0	30
$\begin{array}{c}\begin{array}{c}{F}_{13}\left(X\right)=0.1\left\{{sin}^2\left(3\pi x_1\right)\right.\hfill \\ \hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+\sum _{i=1}^n{\left(x_i-1\right)}^2\left[1+{sin}^2\left(3\pi x_i+1\right)\right]\\ \left.\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+{\left(x_n-1\right)}^2\left[1+{sin}^2\left(2\pi x_n\right)\right]\right\}\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+\sum _{i=1}^{n}u\left(x_i,5,100,4\right)\hfill \end{array}\end{array}$	Generalized penalized function	${\left[-50, 50\right]}^n$	0	30

. Three-dimensional representations of these benchmark functions are shown in Figure 7 and Figure 8. This standard set is divided into two categories: number of co-processes with numerous different minimum standards and a worldwide ideal for testing algorithms’ ability to avoid global optimum and perform exploratory analysis, and unimodal features with a single global best for testing algorithms’ speed and ability to enslave. In MATLAB R2020b, the recommended algorithms were created. It is better to keep each of these duties to a minimum. Each function also has a dimension of 30.

The basic NGO and other well-known optimizations, including Particle Swarm Optimization (PSO) [2], Firefly Algorithm (FA) [15], Multi-Verse Optimizer (MVO) [22], Tunicate Swarm Algorithm (TSA) [20], and Salp Swarm Algorithm (SSA) [23], are compared to the proposed CNGPS. The size of solutions (N) and the maximum number of iterations (tmax) are set to 30 and 1000, respectively, to allow for a fair comparison of all methods. One metaheuristic method run’s results are uncertain and might not be precise. In order to provide a valid assessment and evaluate the efficiency of the algorithms statistically, analysis should be conducted.

Table 4. Comparison of different methods in solving unimodal test functions.

F	Index	CNGPS	NGO	PSO	FA	MVO	SSA	TSA
F1	Mean	0.00	2.42 × 10−97	4.98 × 10−9	7.11 × 10−3	2.81 × 10−1	3.29 × 10−7	8.31 × 10−56
	Std.	0.00	7.22 × 10−97	1.40 × 10−8	3.21 × 10−3	1.11 × 10−1	5.92 × 10−7	1.02 × 10−58
F2	Mean	0.00	1.16 × 10−52	7.29 × 10−4	4.34 × 10−1	3.96 × 10−1	1.9111	8.36 × 10−35
	Std.	0.00	2.55 × 10−52	1.84 × 10−3	1.84 × 10−1	1.41 × 10−1	1.6142	9.86 × 10−35
F3	Mean	0.00	7.84 × 10−81	1.40 × 10	1.66 × 103	4.31 × 10	1.50 × 103	1.51 × 10−14
	Std.	0.00	3.49 × 10−80	7.13	6.72 × 102	8.97	707.05	6.55 × 10−14
F4	Mean	0.00	4.57 × 10−46	6.00 × 10−1	1.11 × 10−1	8.80 × 10−1	2.44 × 10−5	1.95 × 10−5
	Std.	0.00	9.98 × 10−46	1.72 × 10−1	4.75 × 10−2	2.50 × 10−1	1.89 × 10−5	4.49 × 10−4
F5	Mean	7.22 × 10−8	2.80 × 10	4.93 × 10	7.97 × 10	1.18 × 102	136.56	28.4
	Std.	0.00	2.42 × 10−97	4.98 × 10−9	7.11 × 10−3	2.81 × 10−1	3.29 × 10−7	8.31 × 10−56
F6	Mean	0.00	7.22 × 10−97	1.40 × 10−8	3.21 × 10−3	1.11 × 10−1	5.92 × 10−7	1.02 × 10−58
	Std.	0.00	1.16 × 10−52	7.29 × 10−4	4.34 × 10−1	3.96 × 10−1	1.9111	8.36 × 10−35
F7	Mean	0.00	2.55 × 10−52	1.84 × 10−3	1.84 × 10−1	1.41 × 10−1	1.6142	9.86 × 10−35
	Std.	0.00	7.84 × 10−81	1.40 × 10	1.66 × 103	4.31 × 10	1.50 × 103	1.51 × 10−14

and

Table 5. Comparison of different methods in solving multimodal test functions.

	Index	CNGPS	NGO	PSO	FA	MVO	SSA	TSA
F8	Mean	–1.25× 104	–1.01 × 104	–6.01 × 103	–5.85 × 103	–6.92 × 103	–7.46 × 103	–7.89 × 103
	Std.	0.00	1.70 × 103	1.30 × 103	1.61 × 103	9.19 × 102	634.67	599.26
F9	Mean	0.00	0.00	4.72 × 10	1.51 × 10	1.01 × 102	55.45	151.45
	Std.	0.00	0.00	1.03 × 10	1.25 × 10	1.89 × 10	18.27	35.87
F10	Mean	8.88 × 10−16	8.77 × 10−16	3.86 × 10−2	4.58 × 10−2	1.15	2.84	2.409
	Std.	0.00	0.00	2.11 × 10−1	1.20 × 10−2	7.87 × 10−1	6.58 × 10−1	1.392
F11	Mean	0.00	0.00	5.50 × 10−3	4.23 × 10−3	5.74 × 10−1	2.29 × 10−1	0.0077
	Std.	0.00	0.00	7.39 × 10−3	1.29 × 10−3	1.12 × 10−1	1.29 × 10−1	0.0057
F12	Mean	1.57 × 10−32	1.25 × 10−1	1.05 × 10−2	3.13 × 10−4	1.27	6.82	6.373
	Std.	2.88 × 10−48	5.41 × 10−2	2.06 × 10−2	1.76 × 10−4	1.02	2.72	3.458
F13	Mean	1.35 × 10−32	1.99	4.03 × 10−1	2.08 × 10−3	6.60 × 10−2	21.31	2.897
	Std.	2.95 × 10−48	2.51 × 10−1	5.39 × 10−1	9.62 × 10−4	4.33 × 10−2	16.99	0.643

present the results of 30 runs using the a strategic fix for this issue. Table 4 and Table 5 show that, regarding the mean values of the objective functions, CNGPS may provide solutions that are better than both conventional NGO and other optimization techniques for all functions.

The results also confirm that the mean and standard deviation of the CNGPS algorithm are meaningfully lower than those of the other approaches, proving the algorithm’s stability. The findings demonstrate that CNGPS outperforms both the traditional approach and additional optimization strategies.

4.2. Simulation of CNGPS on Three Models

In this part, a series of experiments on several PV models (SDM, DDM, TDM) are carried out to assess the performance of the proposed CNGPS for parameter estimation. The three types in question are the SDM, DDM, TDM, and poly-crystalline Photowatt-PWP201 modules. SDM, DDM, and TDM I–V values are collected as measured on a 57 mm diameter commercial silicon R.T.C France solar cell at 33 °C under 1000 W/m² [10]. In this study, the Photowatt-PWP201 PVM is presented, which contains 36 linked cells in series at 45 C [66,67].

The variable search ranges for three PV models are again displayed in

Table 6. Lower and upper bounds of each parameter.

Parameter	Lower Limit	Upper Limit
Iph(A)	0	1
Isd1, Isd2, Isd3, Isd (μA)	0	1
Rs(Ω)	0	0.5
Rsh(Ω)	0	100
n1, n2, n3,n	1	2

[29,30,31]. Six algorithms were compared to CNGPS, including NGO, TSA, SSA, PSO, MVO, and FA. All algorithms were developed using the same experimental parameters. In this experiment, each algorithm was run 30 times independently, resulting in a total of 30 solutions for all the approaches that were being compared. The maximum number of function evaluations permitted was 20,000. In Table 6, the parameter structure is displayed. In the hypothetical experiment, the error RMSE of the algorithms was calculated using the largest number (Max), the smallest number (Min), the average amount (Mean), and the variance (std). The checking signals in the following tables are all “+,” suggesting that CNGPS and the comparison methods are significantly different in each test situation.

The suggested algorithm was used to predict the values of the SDM, DDM, and TDM parameters of a 57 mm diameter commercial (RTC France) silicon solar cell with a power density of less than 1000 W/m² at 33 °C. SDM’s five unknown variables, DDM’s seven unknown variables, and TDM’s nine parameters have all been calculated.

4.2.1. Results on the SDM

Table 7. The standards of RSME obtained by related methods on SDM.

Algorithm	Max	Min	Mean	Std
CNGPS	9.86027 × 10−4	9.86021 × 10−4	9.86022 × 10−4	1.12317 × 10−9
NGO	1.56519 × 10−3	9.88767 × 10−4	1.19734 × 10−3	1.59523 × 10−4
TSA	1.45946 × 10−3	9.86126 × 10−4	1.08655 × 10−3	1.25814 × 10−4
SSA	2.45685 × 10−3	1.02799 × 10−3	2.37701 × 10−3	1.70656 × 10−4
PSO	1.45946 × 10−3	9.90726 × 10−4	1.08655 × 10−3	1.25814 × 10−4
MVO	1.30477 × 10−2	1.63882 × 10−3	3.66904 × 10−3	2.71439 × 10−3
FA	1.80225 × 10−3	1.02787 × 10−3	1.34978 × 10−3	1.91189 × 10−4

in this paragraph shows the Max, Min, Mean, and Std of RMSE for CNGPS, NGO, TSA, SSA, PSO, MVO, and FA on SDM, with the minimum values of the four indicators bolded. As seen in the table, CNGPS achieves the highest values: 9.86027 × 10⁻⁴, 9.86021 × 10⁻⁴, 9.86022 × 10⁻⁴ and 1.12317 × 10⁻⁹. As a result, it demonstrates that CNGPS produces a lower RMSE than the other algorithms tested.

Table 8. The outcome of parameter approximation by CNGPS and others on SDM.

Method	Iph (A)	Isd (A)	Rs (Ω)	Rsh (Ω)	n	RMSE	Sig
CNGPS	0.760775	3.23021 × 10−7	0.0363771	53.7185	1.48118	9.86021× 10−4	∼
NGO	0.760776	3.22978 × 10−7	0.0363775	53.7101	1.48117	9.88767 × 10−4	+
TSA	0.760775	3.23021 × 10−7	0.0363771	53.7186	1.48118	9.86126 × 10−4	+
SSA	0.760980	3.21318 × 10−7	0.0362752	49.1335	1.48078	1.02799 × 10−3	+
PSO	0.770684	7.78089 × 10−7	0.0358563	30.4120	1.57304	9.90726 × 10−4	+
MVO	0.760833	3.16023 × 10−7	0.0364796	53.0828	1.47898	1.63882 × 10−3	+
FA	0.760776	3.19527 × 10−7	0.0364149	53.4137	1.48009	1.02787 × 10−3	+

shows the SDM parameter values discovered by various linked methodologies. When the predicted five parameters (Iph, Isd, Rs, Rsh, and n) are 0.760775, 3.23021 × 10⁻⁷, 0.0363771, 53.7185 and 1.48118 respectively, the RMSE obtained by CNGPS is 9.86021 × 10⁻⁴. A total of 26 sets of estimated data and experimental data are listed in

Table 9. Statistics of measured and simulated data in SDM.

Item	Measured Data		Simulated Current Data		Simulated Power Data
	V (V)	I (A)	Isim (A)	IAE_I (A)	P (W)	Psim (W)	IAE_P (W)
1	−0.26	0.7640	0.764088	8.7704 × 10−5	−0.15716	−0.1571728	1.8041 × 10−5
2	−0.13	0.7620	0.762663	6.6309 × 10−4	−0.09838	−0.0984598	8.5604 × 10−5
3	−0.06	0.7605	0.761355	8.5531 × 10−4	−0.04472	−0.0447677	5.0292 × 10−5
4	0.0057	0.7605	0.760154	3.4601 × 10−4	0.004335	0.0043329	1.9723 × 10−6
5	0.0646	0.7600	0.759055	9.4479 × 10−4	0.049096	0.0490350	6.1034 × 10−5
6	0.1185	0.7590	0.758042	9.5765 × 10−4	0.089942	0.0898280	1.1348 × 10−4
7	0.1678	0.7570	0.757092	9.1654 × 10−5	0.127025	0.1270400	1.5380 × 10−5
8	0.2132	0.7570	0.756141	8.5864 × 10−4	0.161392	0.1612093	1.8306 × 10−4
9	0.2545	0.7555	0.755087	4.1313 × 10−4	0.192275	0.1921696	1.0514 × 10−4
10	0.2924	0.7540	0.753664	3.3612 × 10−4	0.220470	0.2203713	9.8282 × 10−5
11	0.3269	0.7505	0.751391	8.9097 × 10−4	0.245338	0.2456297	2.9126 × 10−4
12	0.3585	0.7465	0.747354	8.5385 × 10−4	0.267620	0.2679264	3.0611 × 10−4
13	0.3873	0.7385	0.740117	1.6172 × 10−3	0.286021	0.2866474	6.2635 × 10−4
14	0.4137	0.7280	0.727382	6.1777 × 10−4	0.301174	0.3009180	2.5557 × 10−4
15	0.4373	0.7065	0.706973	4.7265 × 10−4	0.308952	0.3091591	2.0669 × 10−4
16	0.4590	0.6755	0.675280	2.1985 × 10−4	0.310055	0.3099536	1.0091 × 10−4
17	0.4784	0.6320	0.630758	1.2417 × 10−3	0.302349	0.3017548	5.9404 × 10−4
18	0.4960	0.5730	0.571928	1.0716 × 10−3	0.284208	0.2836765	5.3153 × 10−4
19	0.5119	0.4990	0.499607	6.0702 × 10−4	0.255438	0.2557488	3.1073 × 10−4
20	0.5265	0.4130	0.413649	6.4879 × 10−4	0.217445	0.2177861	3.4159 × 10−4
21	0.5398	0.3165	0.317510	1.0101 × 10−3	0.170847	0.1713920	5.4526 × 10−4
22	0.5521	0.2120	0.212155	1.5494 × 10−4	0.117045	0.1171307	8.5542 × 10−5
23	0.5633	0.1035	0.102251	1.2487 × 10−3	0.058302	0.0575982	7.0339 × 10−4
24	0.5736	−0.0100	−0.008718	1.2825 × 10−3	−0.005736	−0.0050004	7.3562 × 10−4
25	0.5833	−0.1230	−0.125507	2.5074 × 10−3	−0.071746	−0.0732085	1.4626 × 10−3
26	0.5900	−0.2100	−0.208472	1.5277 × 10−3	−0.123900	−0.1229987	9.0133 × 10−4

. The dependability of the CNGPS is seen in Figure 9. Figure 9a displays the estimated data as well as the experimental voltage and current data as dots. Figure 9b displays the trend of the power data, both actual and estimated, as it varies with voltage. Figure 9c displays the absolute error (IAE) of the flow rate as the voltage increases. Figure 9d depicts the current variance (RE) trend. Figure 9e illustrates the absolute inaccuracy of power as voltage increases. Figure 9f depicts the connection between voltage and relative power error. The results of the test suggest that the better the outcome, the less the IAE inaccuracy. In conclusion, CNGPS performs better than other methods for identifying unidentified SDM parameters.

4.2.2. Results on the DDM

This section thoroughly examines the outcomes of CNGPS and other comparative DDM algorithms.

Table 10. The values of RSME obtained by related methods on DDM.

Algorithm	Max	Min	Mean	Std
CNGPS	9.87761 × 10−4	9.82508 × 10−4	9.85251 × 10−4	1.27917 × 10−6
NGO	2.43158 × 10−3	9.85467 × 10−4	1.53577 × 10−3	4.15711 × 10−4
TSA	9.31141 × 10−3	1.73608 × 10−3	4.25210 × 10−3	2.31125 × 10−3
SSA	1.43838 × 10−3	9.83532 × 10−4	1.13956 × 10−3	1.78691 × 10−4
PSO	2.48306 × 10−3	1.06254 × 10−3	1.47935 × 10−3	3.01999 × 10−4
MVO	1.68606 × 10−3	9.89404 × 10−4	1.15083 × 10−3	1.88651 × 10−4
FA	1.99227 × 10−3	9.83832 × 10−4	1.05121 × 10−3	1.90533 × 10−4

displays the findings of four RMSE indicators: Max, Min, Mean, and Std. The smaller the RMSE number, the smaller the disparity between the simulated and measured data. CNGPS optimizes the values of Max, Min, Mean, and Std to be the least, which are 9.87761 × 10⁻⁴, 9.82508 × 10⁻⁴, 9.85251 × 10⁻⁴ and 1.27917 × 10⁻⁶ respectively. We might obtain the conclusion that CNGPS can forecast unknown DDM parameters reasonably. The parameters produced by various approaches are also shown in

Table 11. The result of parameter estimation by CNGPS and others on DDM.

Algorithm	Iph(A)	Isd1(A)	Rs(Ω)	Rsh(Ω)	n1	Isd2(A)	n2	RSME	Sig
CNGPS	0.76078	6.9844 × 10−7	0.036715	55.370	1.9999	2.3182 × 10−7	1.4535	9.82508 × 10−4	∼
NGO	0.76047	3.4174 × 10−7	0.037050	58.052	1.9101	2.3812 × 10−7	1.4546	9.85467 × 10−4	+
TSA	0.71684	1.0000 × 10−6	0.036840	84.676	2.0000	5.8794 × 10−7	1.5439	1.73608 × 103	+
SSA	0.76083	2.1636 × 10−7	0.036828	54.557	1.4475	6.4258 × 10−7	1.9525	9.83532 × 10−4	+
PSO	0.76084	6.5779 × 10−7	0.033321	84.745	1.5563	1.8922 × 10−10	1.8273	1.06254 × 10−3	+
MVO	0.76083	5.7427 × 10−7	0.036739	55.185	1.9097	2.1647 × 10−7	1.4435	9.89404 × 10−4	+
FA	0.76035	2.5315 × 10−7	0.036842	53.800	1.4654	6.9214 × 10−8	1.6940	9.83832 × 10−4	+

. Clearly, CNGPS has the lowest RMSE, demonstrating that the anticipated value and experimental value are most closely related. The results of the Wilcoxon rank test and the RSME are also included. The results show that CNGPS performs better than other similar algorithms.

Table 12. Statistics of measured and simulated data in DDM.

Item	Measured Data		Simulated Current Data		Simulated Power Data
	V (V)	I (A)	Isim (A)	IAE_I (A)	P (W)	Psim (W)	IAE_P (W)
1	−0.2057	0.7640	0.763983	1.6588 × 10−5	−0.157155	−0.1571514	3.4121 × 10−6
2	−0.1291	0.7620	0.762604	6.0410 × 10−4	−0.098374	−0.0984522	7.7989 × 10−5
3	−0.0588	0.7605	0.761338	8.3770 × 10−4	−0.044717	−0.0447667	4.9257 × 10−5
4	0.0057	0.7605	0.760174	3.2621 × 10−4	0.004335	0.0043330	1.8594 × 10−6
5	0.0646	0.7600	0.759108	8.9232 × 10−4	0.049096	0.0490384	5.7644 × 10−5
6	0.1185	0.7590	0.758121	8.7858 × 10−4	0.089942	0.0898374	1.0411 × 10−4
7	0.1678	0.7570	0.757189	1.8861 × 10−4	0.127025	0.1270562	3.1649 × 10−5
8	0.2132	0.7570	0.756244	7.5639 × 10−4	0.161392	0.1612311	1.6126 × 10−4
9	0.2545	0.7555	0.755177	3.2270 × 10−4	0.192275	0.1921926	8.2127 × 10−5
10	0.2924	0.7540	0.753722	2.7765 × 10−4	0.220470	0.2203884	8.1184 × 10−5
11	0.3269	0.7505	0.751399	8.9913 × 10−4	0.245338	0.2456324	2.9393 × 10−4
12	0.3585	0.7465	0.747301	8.0144 × 10−4	0.267620	0.2679076	2.8732 × 10−4
13	0.3873	0.7385	0.740011	1.5107 × 10−3	0.286021	0.2866061	5.8508 × 10−4
14	0.4137	0.7280	0.727247	7.5305 × 10−4	0.301174	0.3008621	3.1154 × 10−4
15	0.4373	0.7065	0.706853	3.5030 × 10−4	0.308952	0.3091056	1.5319 × 10−4
16	0.4590	0.6755	0.675211	2.8946 × 10−4	0.310055	0.3099216	1.3286 × 10−4
17	0.4784	0.6320	0.630761	1.2392 × 10−3	0.302349	0.3017559	5.9285 × 10−4
18	0.4960	0.5730	0.571995	1.0053 × 10−3	0.284208	0.2837094	4.9861 × 10−4
19	0.5119	0.4990	0.499706	7.0614 × 10−4	0.255438	0.2557996	3.6147 × 10−4
20	0.5265	0.4130	0.413734	7.3367 × 10−4	0.217445	0.2178308	3.8628 × 10−4
21	0.5398	0.3165	0.317546	1.0462 × 10−3	0.170847	0.1714114	5.6474 × 10−4
22	0.5521	0.2120	0.212123	1.2300 × 10−4	0.117045	0.1171131	6.7906 × 10−5
23	0.5633	0.1035	0.102163	1.3367 × 10−3	0.058302	0.0575486	7.5298 × 10−4
24	0.5736	−0.0100	−0.008792	1.2082 × 10−3	−0.005736	−0.0050429	6.9305 × 10−4
25	0.5833	−0.1230	−0.125543	2.5434 × 10−3	−0.071746	−0.0732295	1.4836 × 10−3
26	0.5900	−0.2100	−0.208371	1.6284 × 10−3	−0.123900	−0.1229392	9.6076 × 10−4

additionally contains 26 sets of scientific results as well as estimated data. The outcomes of the simulation are depicted in Figure 10. Figure 10a also displays the measuring trend lines for both voltage and current. The experimental findings of power varying with voltage measurement are shown in Figure 10b. Figure 10c illustrates the absolute error of measured current as it varies with measured voltage. The relative accuracy of the current changing with voltage measurement is shown in Figure 10d. Figure 10e displays the IAE of power as a function of voltage. Figure 10f depicts the relationship between RE, power, and voltage. The figures show how well CNGPS simulates real data and how factor identification on DDM may be possible.

4.2.3. Experiment on TDM

Table 13. The value of RSME obtained by related methods on TDM.

Method	Max	Min	Mean	std
CNGPS	1.4385 × 10−3	9.8249× 10−4	1.08848× 10−3	1.52726× 10−4
NGO	4.8644 × 10−3	9.8256 × 10−4	1.27573 × 10−3	1.97125 × 10−4
TSA	5.5563 × 10−2	1.4814 × 10−3	9.2738 × 10−3	1.1422 × 10−2
SSA	6.3369 × 10−1	3.0098 × 10−2	5.3167 × 10−2	1.41972 × 10−2
PSO	1.9304 × 10−3	1.0301 × 10−3	1.1310 × 10−3	2.2646 × 10−4
MVO	2.4031 × 10−3	1.1167 × 10−3	1.5250 × 10−3	3.1907 × 10−4
FA	7.4451 × 102	2.2916 × 10−2	4.9223 × 10−2	1.1877 × 10−2

shows the result of the TDM simulation. Max, Min, Mean, and Std are the four RMSE values that are displayed on TDM. The superior performance of CNGPS over NGO, TSA, SSA, PSO, MVO, and FA indicates that CNGPS is capable of accurately and successfully identifying unidentified TDM parameters. According to the various approaches employed,

Table 14. The result of parameter estimation by CNGPS and others on TDM.

Method	CNGPS	NGO	TSA	SSA	PSO	MVO	FA
Iph (A)	0.760782	0.760781	0.765642	0.792467	0.760815	0.760763	0.760650
Isd1 (A)	2.28420 × 10−7	3.07069 × 10−7	1.74707 × 10−7	1.95578 × 10−7	8.51528 × 10−10	4.02028 × 10−7	3.80608 × 10−7
Isd2 (A)	7.26329 × 10−7	5.42394 × 10−7	9.57003 × 10−9	7.67869 × 10−7	2.71697 × 10−7	4.28194 × 10−7	1.64373 × 10−7
Isd3 (A)	4.91844 × 1015	2.14569 × 10−7	3.68999 × 10−7	1.74886 × 10−7	3.31786 × 10−7	5.46206 × 10−7	6.04885 × 10−7
Rs (Ω)	0.0368917	0.0368200	0.0365547	0.0533244	0.0366112	0.0359268	0.036813
Rsh (Ω)	55.3988	55.7599	36.6203	12.0785	57.6676	75.5573	62.4033
n1	1.45190	1.99999	1.42543	1.85770	1.73659	1.51261	1.85961
n2	1.99999	1.99999	1.45788	1.63053	1.99189	1.98739	1.42802
n3	1.99926	1.44670	2.00000	1.55133	1.48587	1.94060	1.91769
RMSE	9.82490 × 10−4	9.82560 × 10−4	1.4814 × 10−3	0.0300982	1.03013 × 10−3	1.1167 × 10−3	0.022916
Sig	∼	+	+	+	+	+	+

presents the TDM parameters. There are 26 sets of results for current and power, as shown in

Table 15. Statistics of measured and simulated data in TDM.

Item	Measured Data		Simulated Current Data		Simulated Power Data
	V (V)	I (A)	Isim (A)	IAE_I (A)	P (W)	Psim (W)	IAE_P (W)
1	−0.2057	0.7640	0.7639834	1.6588 × 10−5	−0.15716	−0.157151	3.4121 × 10−6
2	−0.1291	0.7620	0.7626041	6.0410 × 10−4	−0.09837	−0.098452	7.7989 × 10−5
3	−0.0588	0.7605	0.7613377	8.3770 × 10−4	−0.04472	−0.044767	4.9257 × 10−5
4	0.0057	0.7605	0.7601738	3.2621 × 10−4	0.004335	0.0043330	1.8594 × 10−6
5	0.0646	0.7600	0.7591077	8.9232 × 10−4	0.049096	0.0490384	5.7644 × 10−5
6	0.1185	0.7590	0.7581214	8.7858 × 10−4	0.089942	0.0898374	1.0411 × 10−4
7	0.1678	0.7570	0.7571886	1.8861 × 10−4	0.127025	0.1270562	3.1649 × 10−5
8	0.2132	0.7570	0.7562436	7.5639 × 10−4	0.161392	0.1612311	1.6126 × 10−4
9	0.2545	0.7555	0.7551773	3.2270 × 10−4	0.192275	0.1921926	8.2127 × 10−5
10	0.2924	0.7540	0.7537224	2.7765 × 10−4	0.220470	0.2203884	8.1184 × 10−5
11	0.3269	0.7505	0.7513991	8.9913 × 10−4	0.245338	0.2456324	2.9393 × 10−4
12	0.3585	0.7465	0.7473014	8.0144 × 10−4	0.267620	0.2679076	2.8732 × 10−4
13	0.3873	0.7385	0.7400107	1.5107 × 10−3	0.286021	0.2866061	5.8508 × 10−4
14	0.4137	0.7280	0.7272470	7.5305 × 10−4	0.301174	0.3008621	3.1154 × 10−4
15	0.4373	0.7065	0.7068503	3.5030 × 10−4	0.308952	0.3091056	1.5319 × 10−4
16	0.4590	0.6755	0.6752105	2.8946 × 10−4	0.310055	0.3099216	1.3286 × 10−4
17	0.4784	0.6320	0.6307608	1.2392 × 10−3	0.302349	0.3017559	5.9285 × 10−4
18	0.4960	0.5730	0.5719947	1.0053 × 10−3	0.284208	0.2837094	4.9861 × 10−4
19	0.5119	0.4990	0.4997061	7.0614 × 10−4	0.255438	0.2557996	3.6147 × 10−4
20	0.5265	0.4130	0.4137337	7.3367 × 10−4	0.217445	0.2178308	3.8628 × 10−4
21	0.5398	0.3165	0.3175462	1.0462 × 10−3	0.170847	0.1714114	5.6474 × 10−4
22	0.5521	0.2120	0.2121230	1.2300 × 10−4	0.117045	0.1171131	6.7906 × 10−5
23	0.5633	0.1035	0.1021633	1.3367 × 10−3	0.058302	0.0575486	7.5298 × 10−4
24	0.5736	−0.0100	−0.008792	1.2082 × 10−3	−0.00574	−0.0050429	6.9305 × 10−4
25	0.5833	−0.1230	−0.125543	2.5434 × 10−3	−0.07175	−0.0732295	1.4836 × 10−3
26	0.5900	−0.2100	−0.208372	1.6284 × 10−3	−0.12390	−0.1229392	9.6076 × 10−4

. Figure 11 displays a number of factors that alter when voltage is measured. Figure 11a displays the current long-term trend because of voltage measurement. Figure 11b displays the experiments for the power variation with voltage. Figure 11c illustrates the total inaccuracy of the voltage-dependent current. Figure 11d depicts the relative accuracy of power and voltage. Figure 11e displays the total measurement error for power in proportion to voltage. The relative imprecision of power varies with voltage measurement, as seen in Figure 11f. Therefore, employing CNGPS, variables on TDM may be reliably and properly determined.

4.2.4. Discussion

In this section, the prior algorithms are extensively compared, and some conclusions can be made based on the statistical findings, which are shown in

Table 16. Statistical comparison of RMSE values realized by the algorithms for three models.

	Algorithm	Max	Min	Mean	Std
SDM	CNGPS	9.8602× 10−4	9.8602× 10−4	9.8602× 10−4	1.1231× 10−9
	NGO	1.56529 × 10−3	9.8877 × 10−4	1.19734 × 10−3	1.5952 × 10−4
	TSA	1.4595 × 10−3	9.8613 × 10−4	1.08655 × 10−3	1.2581 × 10−4
	SSA	2.4569 × 10−3	1.0278 × 10−3	2.37701 × 10−3	1.7066 × 10−4
	PSO	1.4595 × 10−3	9.9073 × 10−4	1.08655 × 10−3	1.2584 × 10−4
	MVO	1.3048 × 10−2	1.6389 × 10−3	3.66904 × 10−3	2.7140 × 10−3
	FA	1.8023 × 10−3	1.0279 × 10−3	1.34978 × 10−3	1.9119 × 10−4
DDM	CNGPS	9.8777× 10−4	9.8251× 10−4	9.8525× 10−4	1.2792× 10−6
	NGO	2.4316 × 10−3	9.8547 × 10−4	1.5358 × 10−3	4.1571 × 10−4
	TSA	9.3214 × 10−3	1.7361 × 10−3	4.2531 × 10−3	2.3113 × 10−3
	SSA	1.4384 × 10−3	9.8133 × 10−4	1.1386 × 10−3	1.7869 × 10−4
	PSO	2.4831 × 10−3	1.0635 × 10−3	1.4774 × 10−3	3.0199 × 10−4
	MVO	1.6961 × 10−3	9.8941 × 10−4	1.1518 × 10−3	1.8861 × 10−4
	FA	1.9823 × 10−3	9.8383 × 10−4	1.0512 × 10−3	1.9053 × 10−4
TDM	CNGPS	1.4385× 10−3	9.8249× 10−4	1.0885× 10−3	1.5272× 10−4
	NGO	4.8645 × 10−3	9.8256 × 10−4	1.2757 × 10−3	1.9712 × 10−4
	TSA	5.5563 × 10−2	1.4814 × 10−3	9.2738 × 10−3	1.1422 × 10−2
	SSA	6.3369 × 10−1	3.0098 × 10−2	5.3167 × 10−2	1.41972 × 10−2
	PSO	1.9304 × 10−3	1.0301 × 10−3	1.1310 × 10−3	2.2646 × 10−4
	MVO	2.4031 × 10−3	1.1167 × 10−3	1.5250 × 10−3	3.1907 × 10−4
	FA	7.4451 × 10−2	2.2916 × 10−2	4.9223 × 10−2	1.1877 × 10−2

. As shown in Table 16, the CNGPS technique performs better than the other six strategies in terms of model reliability and average accuracy. According to Table 16, the suggested approach performs better than alternative strategies in terms of average and standard deviation.

Based on the simulation results under three different models, it can be observed that CNGPS can obtain the most satisfactory performance compared to its competitors. The analysis results show that the proposed algorithm is very effective, converges faster than other methods, and could find a smaller RMSE value for all models. As per the results and findings, the proposed CNGPS can jump out of the local optimum and search in a more feasible area. In addition, the local search ability of the pattern search method increases the accuracy of the results obtained by CNGPS. As far as the overall effect is concerned, these improvements are very promising for the parameter extraction of other complex PV models.

5. Conclusions

The chaotic northern goshawk optimization method and pattern search (CNGPS) is used to locate unknown parameters in various PV models. The proposed methodology exploits the robust exploring capability of the chaotic north goshawk optimization algorithm and the effective local search capability of the pattern search technique. A number of benchmark unimodal and mixed functions are utilized to assess the effectiveness of the suggested technique. The findings show that for the majority of test functions, the CNGPS outperforms conventional NGO and other approaches in terms of global solution determination. To obtain the PV model parameters with the lowest RMSE, the proposed CNGPS is then used. The experiments demonstrate that the solutions obtained by the proposed CNGPS are better than the comparative algorithms for optimizing and operating real-world solar systems. CNGPS may, therefore, be thought of as a good candidate approach for deriving parameters from more intricate PV models. The mean RMSE values generated by CNGPS are 9.8602 × 10⁻⁴, 9.8525 × 10⁻⁴, and 1.0885 × 10⁻³ for SDM, DDM, and TDM, respectively. According to the experimental results, the solutions produced by the suggested CNGPS are superior options for optimizing and managing real-world solar systems than the comparison algorithms. The outcomes of the competition simulation imply that CNGPS can efficiently and accurately estimate parameters. Therefore, it is believed that it is worth considering the effects of the developed algorithms on constrained multi-objective optimization problems as well as dynamic multi-objective optimization problems in future works.