Growth Optimizer for Parameter Identification of Solar Photovoltaic Cells and Modules

Abstract

One of the most significant barriers to broadening the use of solar energy is low conversion efficiency, which necessitates the development of novel techniques to enhance solar energy conversion equipment design. The correct modeling and estimation of solar cell parameters are critical for the control, design, and simulation of PV panels to achieve optimal performance. Conventional optimization approaches have several limitations when solving this complicated issue, including a proclivity to become caught in some local optima. In this study, a Growth Optimization (GO) algorithm is developed and simulated from humans’ learning and reflection capacities in social growing activities. It is based on mimicking two stages. First, learning is a procedure through which people mature by absorbing information from others. Second, reflection is examining one’s weaknesses and altering one’s learning techniques to aid in one’s improvement. It is developed for estimating PV parameters for two different solar PV modules, RTC France and Kyocera KC200GT PV modules, based on manufacturing technology and solar cell modeling. Three present-day techniques are contrasted to GO’s performance which is the energy valley optimizer (EVO), Five Phases Algorithm (FPA), and Hazelnut tree search (HTS) algorithm. The simulation results enhance the electrical properties of PV systems due to the implemented GO technique. Additionally, the developed GO technique can determine unexplained PV parameters by considering diverse operating settings of varying temperatures and irradiances. For the RTC France PV module, GO achieves improvements of 19.51%, 1.6%, and 0.74% compared to the EVO, FPA, and HTS considering the PVSD and 51.92%, 4.06%, and 8.33% considering the PVDD, respectively. For the Kyocera KC200GT PV module, the proposed GO achieves improvements of 94.71%, 12.36%, and 58.02% considering the PVSD and 96.97%, 5.66%, and 61.20% considering the PVDD, respectively.

1. Introduction

Solar power is one of the most widely available clean forms of energy because it can be produced almost anywhere, including open clearings, building roofs, mountains, prairies, and other places where sunlight is visible. Renewable energy sources for electricity primarily consist of green energies like hydropower, geothermal, solar, and wind. Solar energy also means no exploitation and is non-polluting [1,2,3]. Even so, the internal components of photovoltaic (PV) cells and models are particularly susceptible to changes, such as higher resistance caused by a rise in metal conductor temperature and a weaker photovoltaic current caused by a fall in panel light intensity, among other things. For PV system design and operation management, accurate identification of the PV cell/module parameters is essential [4,5].

Over the last several decades, significant advancements have been made in comprehending the way the attributes of PV systems function by using mathematical representations of the PV system. The majority of models tend to be the ones that match observed current-voltage measurements taken from PV cells in all operating conditions and mimic the behavior of real PV cells [6]. Typically, comparable circuit models made of diodes are used. Considering light of that, the PV single-diode (PVSD) and PV double-diode (PVDD) types are indeed particularly commonly used [7]. The three-diode design also describes the current-voltage (I–V) characteristics. Such identical designs each necessitate the identify five, seven, and nine parameter values [8].

Identifying the PV model’s unknown parameters is a challenging subject that many scholars are studying due to its nonlinearity, nonconvexity, and multiparameter characteristics. Metaheuristic optimization algorithms have been frequently employed in recent years to solve the parameter estimation of PV systems and have revealed superior performance than standard techniques. This technique is ideal for handling complex problems with optimization since it is gradient-free, is simple to construct, and performs well. These solving methods involve a genetic algorithm (GA) [9], a particle swarm optimizer (PSO) [10], differential evolution (DE) [11], supply–demand optimization [12,13], a seagull optimization algorithm (SOA) [14], a biogeographic-based optimizer (BBO) [15], a sunflower algorithm (SFA) [16], a bonobo optimizer [17], social networking search technique (SNST) [18], a butterfly optimization algorithm (BOA) [19], teaching-learning-based optimization (TLBO) [20], a marine predator algorithm (MPA) [21], etc. Nonetheless, most optimization techniques in the scientific literature have certain limitations.

For instance, the PVDD framework and the memetic adaptive differential evolution algorithm (MADEA) interacted poorly in [22]. Similar to this, artificial bee colonies (ABC) were underused [23]. Despite the BBO’s quick convergence, its elitist strategy effectively confines it to the best local values. The cuckoo search method (CSM) also has a sluggish convergence rate [15]. A generalized normal distribution algorithm (GNDA) that has been improved has been introduced in [24] to simulate the behaviors of PV systems. To enhance the GNDA’s performance, two different ranking-based update and premature convergence methods were included in the work of [24], although this added computing complexity doubled that of the original GNDA. The use of a GA technique with evolving non-uniform mutation (NUM) [25] for obtaining the characteristics from PV representations in a dependable, precise, and time-efficient manner has been demonstrated. This study uses the GA with NUM technique on PVSD and PVDD modules. It has not, however, been deployed on the three-diode PV design. GA [26] is highly dependent on the beginning PV parameter choice. Once the initial parameters are incorrectly chosen, the parameters derived from the following improvements will likely descend to a locally optimal solution. Such a problem leads to the inaccurate operating performance of the PV system due to inaccurate PV model extracted parameters. In [27], a solar three-diode PV prototype utilizing a mSi cell, KC200GT module, and STM6-40/36 module was used to mimic an artificial hummingbird optimizer. The PV parameter optimization procedure was stalled in [14] because of SOA’s extensive search of the parameterized space throughout the migration cycle. Even while modifying the global and local searches by adding a non-linear regulating element alleviated the deterioration, it was only able to improve the stability and precision of the generated variables. In [19], the inadequate exploration of BOA frequently ignores potential areas in the parameter’s area, rendering more precise PV system information hard to obtain. According to the study mentioned above, currently, the available extraction techniques for addressing the parameter finding of the PV framework have various flaws, such as unsuitable exploration, unsatisfactory convergence, and straightforward falling into the local optimum [14,19,26], which significantly affect the reliability and accuracy of the calculated parameter values. The properties of the real I-V curves cannot be precisely and efficiently recreated, and the PV system’s energy conversion efficiency has not increased considerably. This implies that developing a new effective technique to precisely represent the PV equivalent model and determine the unknown parameters is complex and time-consuming.

The growth optimizer (GO) approach, a metaheuristic algorithm that uses a population-based approach, was very recently presented by Q. Zhang et al. in [28]. The human capacity for reflection and learning in socially developing activities serves as the design’s main source of inspiration. People develop via learning and reflection, where they take in knowledge from others and assess their flaws to help them get better. In [28], the GO approach was used for many mathematical benchmarks, and it has shown significant advantages over other comparative strategies. It outperformed alternative metaheuristic strategies about the effectiveness of the solutions and avoiding local optimism. Based on those advantages, the GO technique is assessed and developed to handle one of the important optimization targets in electrical power engineering. Due to the distinctive features of the GO method, it focuses on improving the electrical characteristics of different models of equivalent circuits of PV systems, which considers the composition of PVSD and PVDD.

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

• According to the author, the GO technique is developed for the first time in this study for optimally extracting the PV parameters.
• The suggested GO approach is utilized for numerous applications in PV technologies, using two commercial PV panels of RTC France and Kyocera KC200GT PV modules.
• Its effectiveness is demonstrated considering the PVSD and PVDD compared to previous optimization strategies such as energy valley optimizer (EVO) [29], Five Phases Algorithm (FPA) [30], and Hazelnut tree search (HTS) algorithm [31].
• Furthermore, the suggested GO technique’s capacity to determine unexplained PV parameters is proved by considering diverse operating settings of varying temperatures and irradiances.
• When identifying the relevant PV parameters at variable irradiance and temperature, the proposed GO technique outperforms commercial PV systems, indicating the efficacy of the proposed methodology and its capacity to offer highly consistent fit-ted I-V data with manufacturers’ data sheets.

The five sections of this study are as follows: Section 2 describes the mathematical description of the PVSD and PVDD models, while Section 3 shows the developed GO procedure. Furthermore, a complete description of the acquired simulation results by GO, EVO, FPA, HTS, and several reported techniques is illustrated in Section 4. Section 5 also includes the paper’s final observations.

2. Problem Formulation of Solar PV Parameters Extraction

Solar PV Parameters extraction is a very important topic that helps effectively dealing with PV integration in power systems [32,33]. Many electrical equivalent circuits were developed to demonstrate the I-V characteristics of solar PV panels. In practice, the PVSD and PVDD models as equivalent circuits are used the most. These PV models are described in the paragraphs that follow [34]. In recent years, the Shockley diode equivalent circuits are the most well-known depiction of PV cells.

2.1. PVSD Model

Mathematical modeling has become crucial in analyzing the constantly changing relationships between different PV system components. The PVSD is frequently used to depict solar cell properties. Figure 1 shows the corresponding circuit of PVSD.

A diode, two resistors, and a current source are the essential elements of a PVSD model comprising PV cells. The PVSD model’s load current equation is theoretically displayed using Kirchoff’s Current Law. Series resistance (R_S) and shunt resistance (R_Sh) are the two lumped resistors’ losses. Equation (1) represents the output current (I), which is estimated using the Shockley diode as follows [35]

$$I=I_{ph}-I_{S1}\left[\exp \left(\frac{I{R}_S}{{\eta }_1 V_{th}}+\frac{V}{{\eta }_1 V_{th}}\right)-1\right]-\frac{I{R}_S}{R_{sh}}+\frac{V}{R_{sh}}$$

(1)

where η₁ and I_S1 represent the ideality factor and the reverse saturation current related to the diode (D1), V is the output voltage at the terminal. R_S and R_Sh are, respectively, the series and shunt resistances. I_Ph and I indicate the cell photocurrent and the output current. V_th represents the PV cell’s thermal voltage and may be computationally described as shown in Equation (2).

$$V_{th}=\frac{K_{B}T}{q_c}$$

(2)

where (q_c) and (T) represent the charge on an electron and the temperatures in absolute terms, accordingly, and K_B denotes Boltzmann’s constant.

Using the PV panels’ I–V measurements, this model’s five unnamed parameters—I_Ph, I_S1, R_Sh, R_S, and η₁—must be calculated.

2.2. PVDD Model

The PVSD model is thought of as being changed by this model. This variant expands on Figure 2’s basic PVSD model by adding an extra diode that illustrates parallel space charge recombination. PVDD model’s load current equation is theoretically expressed in Equation (3):

$$I=I_{ph}-I_{S1}\left[\exp \left(\frac{I{R}_S}{{\eta }_1 V_{th}}+\frac{V}{{\eta }_1 V_{th}}\right)-1\right]-I_{S2}\left[\exp \left(\frac{I{R}_S}{{\eta }_2 V_{th}}+\frac{V}{{\eta }_2 V_{th}}\right)-1\right]-\frac{I{R}_S}{R_{sh}}+\frac{V}{R_{sh}}$$

(3)

where: η₂ and I_S2 symbols are the ideality factor and the reverse saturation current related to the diode (D2), respectively.

Using the PV panels’ I–V measurements, this model’s seven unnamed parameters—I_Ph, I_S1, I_S2, R_Sh, R_S, η₁, and η₂—must be calculated.

2.3. PV Modules Handling

An example of a PV module composed of N_s cells connected in series and N_p cells linked in parallel can be used to demonstrate the equations of the PVSD and PVDD models. As a result, for the PVSD and PVDD models, Equations (1) and (3) are improved and changed to ((4) and (5)) as follows:

$$I=N_p\left(I_{ph}-I_{S1}\left[\exp \left(\frac{1}{{\eta }_1 N_s{V}_{th}}\times \left(\frac{V}{N_p}+\frac{I{N}_s{R}_S}{N_p}\right)\right)-1\right]\right)-\frac{1}{N_s{N}_p{R}_{sh}}\times \left(\frac{V}{N_p}+\frac{I{N}_s{R}_S}{N_p}\right)$$

(4)

$$I=N_p\left(\begin{array}{l}{I}_{ph}-I_{S1}\left[\exp \left(\frac{1}{{\eta }_1 N_s{V}_{th}}\times \left(\frac{V}{N_p}+\frac{I{N}_s{R}_S}{N_p}\right)\right)-1\right]\\ -I_{S2}\left[\exp \left(\frac{1}{{\eta }_2 N_s{V}_{th}}\times \left(\frac{V}{N_p}+\frac{I{N}_s{R}_S}{N_p}\right)\right)-1\right]\end{array}\right)-\frac{1}{N_s{N}_p{R}_{sh}}\times \left(\frac{V}{N_p}+\frac{I{N}_s{R}_S}{N_p}\right)$$

(5)

Variables that are not considered in the PV cell/module for the PVSD and PVDD models are computed using optimization methods.

2.4. Objective Model

The following equation was used to carry out the statistical evaluation in this article, which centered on the root mean square error (RMSE) [36]:

$$RMSE=\sqrt{\frac{1}{PN}\left({\displaystyle \sum _{K=1}^{PN}{(I_{cal}^K(V_{\mathit{exp}}^K,x)-I_{\mathit{exp}}^K)}^2}\right)}$$

(6)

where x designates the searching individual, which contains the candidate PV parameters; PN denotes the number of measured data points, and I_exp^K and V_exp^K describe the observed current and voltage.

3. Developing Growth Optimizer for Optimal Extraction of PV Parameters

For the most effective extraction of PV parameters, this study developed the growth optimizer (GO), a distinctive and reliable metaheuristic method. The basic design of social development activities is influenced by participants’ capacities for learning and self-reflection. The process is divided into two parts. The learning phase, the initial part of the process, is when the person puts what they’ve learned about other people’s differences into practice. Second, the reflection phase is used when a person uses a variety of techniques to identify and correct their shortcomings.

An organized set of design variables makes up a social population with a certain number of members (N), which is often represented as a matrix. The ith individual is characterized for i = 1, 2, 3, …, N in the searching space ${\overrightarrow{x}}_i=(x_{i,1}^{},x_{i,2},\dots ,x_{i,D})$, manifesting the ith individual’s Dth element.

3.1. Phase 1: Learning

An individual can develop by addressing interpersonal distance, looking into its causes, and taking lessons from it. Four frequent distances are mathematically modeled throughout the GO learning phase: the distance between the leader and the elite (Dis₁), the distance between the leader and the bottom (Dis₂), the distance between the elite and the bottom (Dis₃), and the distance among two arbitrary individuals (Dis₄). Equation (7) describes the mathematical model for each set of distances.

$$\left\{\begin{array}{l}D\overrightarrow{i}{s}_1={\overrightarrow{x}}_{best}-{\overrightarrow{x}}_{better}\\ D\overrightarrow{i}{s}_2={\overrightarrow{x}}_{best}-{\overrightarrow{x}}_{worse}\\ D\overrightarrow{i}{s}_3={\overrightarrow{x}}_{better}-{\overrightarrow{x}}_{worse}\\ D\overrightarrow{i}{s}_4={\overrightarrow{x}}_{L1}-{\overrightarrow{x}}_{L2}\end{array}\right.$$

(7)

where x_best denotes the society’s leader, while x_better indicates the elite, extracted, and ranked inside the bound of 2 to P₁, representing the global sub-optimal solutions. x_worse is located at the bottom of the social pyramid and represents a solution vector from the P₁ lowest-ranked searching agents. The symbols (x_L1 and x_L2) manifest random individuals distinct from the ith individual. Dis_k (k = 1, 2, 3, 4) is the distance between two individuals, which permits learners to completely appreciate the distinctions between two individuals to gain knowledge from them.

A learning factor (LF) is added to each of the four disparity measurements to account for this variation. The term (LF_k) is going to impact how well the kth group distance learns for the ith individual, and it can be modeled as follows:

$$L{F}_k=\frac{‖D\overrightarrow{i}{s}_k‖}{{\displaystyle {\sum }_{k=1}^4‖D\overrightarrow{i}{s}_k‖}}, (k=1,2,3,4)$$

(8)

where LF_k implies the kth group’s standardized proportion to the Euclidean distance Dis_k and has a range of [0, 1].

People view themselves distinctly depending on where they are in the growing cycle. Each person (i) utilizes SF_i to determine his or her suitable level of knowledge. SF_i can be mathematically modeled as follows:

$$S{F}_i=\frac{G{R}_i}{G{R}_{\max }}$$

(9)

where GR_max is the highest growing resistance, and GR_i is the ith individual’s growing resistance. A lower GR_i generally indicates that a person will gather and assimilate knowledge in greater detail the higher his level. As a result, the person ought to develop a lesser SF_i that is more inclined to engage in localized exploitation trends.

For the kth group of distances Dis_k, the ith person learns something from them, representing the kth knowledge accumulation collective KA_k.

Following the LF_k and SF_i actions on the group distance (k), it is defined by KA_k which is produced for the ith individual via Equation (10).

For the ith individual, KA_k is obtained after the operations of LF_k and SF_i on the kth group distance, a process described by Equation (10).

$$K{\overrightarrow{A}}_k=S{F}_i·L{F}_k·D\overrightarrow{i}{s}_k, (k=1:4)$$

(10)

where KA_k represents the knowledge that the ith person gathered from group (k) of the distance has learned. SF_i evaluates its circumstances. With the guidance of the two evaluations, the person (i) completes the learning process by determining himself to possess the necessary expertise and retrieving KA_k from Dis_k. Therefore, Equation (11) provides the ith people’s unique learning process updating strategy.

$${\overrightarrow{x}}_i^{Iter+1}={\overrightarrow{x}}_i^{Iter}+K{\overrightarrow{A}}_1+K{\overrightarrow{A}}_2+K{\overrightarrow{A}}_3+K{\overrightarrow{A}}_4$$

(11)

where Iter denotes the present iteration and x_i signifies the ith person who grows by absorbing the information learned throughout the learning stage.

Once the learning stage is adjusted, the level of excellence of the possible solution exhibited by every person could be enhanced or could underperform. Therefore, it is essential to confirm if it has advanced. If advancement happens, the ith person’s GR_i will decline, and the ranking will rise. Since learning requires the time and effort of the ith individual, there is a tiny chance that the ith person would continue to reveal the information that has been learned. However, if the ith person slows down, there is a significant likelihood of losing some gained understanding. In this case, P₂ takes charge of managing the retention probability. Equation (12) describes this procedure.

$${\overrightarrow{x}}_i^{Iter+1}=\left\{\begin{array}{l}{\overrightarrow{x}}_i^{Iter+1} if f({\overrightarrow{x}}_i^{Iter+1})<f({\overrightarrow{x}}_i^{Iter})\\ \left\{\begin{array}{l}{\overrightarrow{x}}_i^{Iter+1} if r_1<P_2\\ {\overrightarrow{x}}_i^{Iter} else\end{array}\right. else\end{array}\right.$$

(12)

where r₁ indicates a randomized number within range [0, 1] following uniform distribution, ind(i) refers to the rank of solution individual (i), and P₂ decides how much the recently gained information is preserved when the ith member fails to upgrade. P₂ is 0.001 in this instance.

3.2. Phase 2: Reflection

Both strategies of the GO techniques, reflection and learning, go hand in hand. The preliminary information ought to be disregarded, and systematic learning would be resumed whenever the understanding of a particular issue cannot be repaired. Equations (13) and (14) provide a mathematical model for GO’s reflecting process.

$$x_{i,j}^{Iter+1}=\left\{\begin{array}{l}\left\{\begin{array}{l}L+r_4\times (U-L) if r_3<AF\\ x_{i,j}^{Iter}+r_5\times (R_j-x_{i,j}^{Iter}) else\end{array}\right. if r_2<P_3\\ x_{i,j}^{Iter} else\end{array}\right.$$

(13)

$$AF=0.01+0.99\times (1-\frac{FEs}{MaxFEs})$$

(14)

where r₂, r₃, r₄, and r₅ indicate randomized values inside bound [0, 1]. U and L represent the maximum and bottom boundaries of the searching control variables.

The chance of reflecting is controlled by the coefficient of P₃, typically set at 0.3. The present number of functions evaluation (FEs) and their total amount (MaxFEs) make up the attenuation factor (AF). The regard of AF would eventually settle to 0.01 as the process continues, indicating that they avoid spending time on repeated startup as a person advances. The jth component of the ith person will receive guidance from an individual at a higher level (R) during the reflection stage. R acts as a reflective learning tool for the present searching member i. R_j refers to the knowledge of the jth issue of R. When the jth feature of the ith human needs to learn from others, there will be an upper-level individual as R to guide it since the class of R is specified as the top P1 + 1 human. The created GO technique’s flowchart for solving the extraction optimization problem of PV parameters is shown in Figure 3.

4. Simulation Results

Using the suggested GO approach, this section investigates R.T.C France and the Kyocera KC200GT PV module. The first case study involves the commercial silicon solar R.T.C France module, which operates at 33 degrees Celsius with a sun radiance of 1000 W/m². It possesses a (0.5727 V) open circuit voltage and a (0.7605 A) short-circuit current. Furthermore, R.T.C France’s maximum point voltage and current are 0.4590 V and 0.6755 A. The second case study relates to the Kyocera KC200GT PV Module, composed of 54 multi-crystalline cells with series connection, which has (32.90 V) open circuit voltage and (8.21 A) short circuit current. This module’s maximum current, voltage, and power points are 7.61 A, 26.30 V, and 200 W, respectively. This module’s maximum voltage, current, and power are 26.30 V, 7.61 A, and 200 W, respectively.

Table 1. The margins range for the cell parameters.

Parameter	RTC France PV Cell		Kyocera KC200GT PV Module
	Lower Bound	Upper Bound	Lower Bound	Upper Bound
IS1, IS2 (μA)	0	1	0	10
Iph (A)	0	1	0	10
Rsh (Ω)	0	100	0	100
RS (Ω)	0	0.5	0	2
η1, η2 per cell	1	2	1	2
No series cells	1		54

shows the upper and lower bounds for the retrieved parameters of the RTC France and KC200GT PV modules. In this section, the GO technique is investigated and applied to parameter extraction issues for various solar cells/modules of the SD and DD models for comparison with relatively recent optimization techniques, energy valley optimizer (EVO) [29], Five Phases Algorithm (FPA) [30], and Hazelnut tree search (HTS) algorithm [31]. All the compared algorithms, GO, EVO, FPA, and HTS, are applied with the same iterations’ number of 1000 and individuals’ number of 200.

4.1. Applications of RTC France Silicon Cell

First, the suggested GO, EVO, FPA, and HTS approaches are used to minimize the RMSE function objective for the RTC France silicon PV cell using the PVSD model.

Table 2. Obtained parameters by the proposed GO, EVO, FPA, and HTS techniques for the PVSD of RTC France.

Applied Technique	EVO	FPA	HTS	GO
IPh (A)	0.761388	0.760689	0.760810	0.760776
Rsh (Ω)	54.164970	53.818103	52.002386	53.718745
RS (Ω)	0.035133	0.036117	0.036627	0.036377
IS1 (A)	4.347 × 10−7	3.369 × 10−7	3.032 × 10−7	3.23 × 10−7
η1	1.511825	1.485471	1.474822	1.481183
RMSE	1.2250 × 10−3	1.0021 × 10−3	9.9339 × 10−4	9.8602 × 10−4
Difference compared to GO	2.39 × 10−4	1.61 × 10−5	7.37 × 10−6	-
Improvement	19.51%	1.60%	0.74%	-

tabulates the appropriate values, whereas Figure 4 depicts the convergence trends of the suggested GO, EVO, FPA, and HTS approach for this scenario. It is illustrated from the table that the proposed GO successfully achieves the minimum RMSE value of 9.8602 × 10⁻⁴, where the HTS a comparable RMSE outcome of 9.9339 × 10⁻⁴. Conversely, the FPA and EVO record RMSE values of 1.0021 × 10⁻³ and 1.2250 × 10⁻³, respectively. Based on these outcomes, the presented GO technique achieves improvements of 19.51%, 1.6%, and 0.74% compared to the EVO, FPA, and HTS techniques.

Moreover, a statistical comparison is conducted between the proposed GO, EVO, FPA, and HTS techniques for the PVSD of RTC France, where the box plot of the proposed GO, EVO, FPA, and HTS techniques is exemplified in Figure 5 for the PVSD of RTC France. It can be noticed that the proposed GO technique displays superior performance compared to the EVO, FPA, and HTS techniques. According to the average RMSE objective score, the proposed GO technique achieves improvements of 44.45%, 6.04%, and 22.32% compared to the EVO, FPA, and HTS techniques. Based on the standard deviation of the RMSE objective score, the proposed GO technique achieves the smallest index of 0.00099 compared to the EVO, FPA, and HTS techniques which obtain counterparts of 0.00248, 0.00114, and 0.00188, respectively.

Table 3. Comparative assessment of the presented techniques for RTC France with the PVSD model.

Algorithms	RMSE
Proposed GO	9.8602 × 10−4
EVO	1.2250 × 10−3
FPA	1.0021 × 10−3
HTS	9.9339 × 10−4
GA with NUM [25]	9.8618 × 10−4
BBO with mutation [37]	9.8634 × 10−4
TLBO [38]	9.8733 × 10−4
JAYA optimizer [42]	9.8946 × 10−4
Improved DE [38]	9.89 × 10−4
CSA [38]	9.91184 × 10−4
HSBA [40]	9.95146 × 10−4
Comprehensive Learning PSO [43]	9.9633 × 10−4
ABC [39]	10 × 10−4
Chaotic PSO [38]	13.8607 × 10−4
GWO [41]	75.011 × 10−4

shows a comparison between the suggested GO approach and various optimizers that are reported in the literature for the PVSD model. It compares the proposed HPO technique’s RMSE value to other optimizers in the literature, including GA with NUM [25], BBO with mutation [37], TLBO [38], ABC [39], harmony search-based algorithm (HSBA) [40], grey wolf optimizer (GWO) [41], JAYA optimizer [42], and comprehensive learning PSO [43]. As demonstrated, the suggested GO outperforms its competitors in finding the lowest RMSE.

In addition, Figure 6a,b show the real and simulated P-V and I-V characteristics for the PVSD model. It can be shown that the data computed by the suggested GO approach roughly corresponded with the experimental data, demonstrating that the suggested GO methodology effectively extracts the required PV parameters.

Second, the suggested GO, EVO, FPA, and HTS approaches are used to minimize the RMSE function objective for the RTC France silicon PV cell using the PVDD model.

Table 4. Obtained parameters by the proposed GO, EVO, FPA, and HTS techniques for the PVDD of RTC France.

Applied Technique	EVO	FPA	HTS	GO
IPh (A)	0.760002	0.760655	0.760191	0.760791
Rsh (Ω)	99.666910	58.872740	56.835980	54.977810
RS (Ω)	0.032481	0.035891	0.036797	0.036658
IS1 (A)	5.57 × 10−7	1.58 × 10−8	2.96 × 10−7	2.38 × 10−7
η1	1.599326	1.993556	1.472788	1.455712
IS2 (A)	2.56 × 10−7	3.68 × 10−7	7.18 × 10−9	5.74 × 10−7
η2	1.548337	1.494564	1.673216	1.968428
RMSE	2.045 × 10−3	1.025 × 10−3	1.072 × 10−3	9.830 × 10−4
Difference compared to GO	1.06 × 10−3	4.20 × 10−5	8.90 × 10−5	-
Improvement	51.93%	4.10%	8.30%	-

tabulates the appropriate values, whereas Figure 7 depicts the convergence trends of the suggested GO, EVO, FPA, and HTS approaches for this scenario. It is illustrated from the table that the proposed GO successfully achieves the minimum RMSE value of 9.83 × 10⁻⁴ where the FPA, HTS, and EVO record RMSE values of 1.025 × 10⁻³, 1.072 × 10⁻³, and 2.045 × 10⁻³, respectively. The difference between the optimization values of different optimizers is very small. However, the presented GO technique achieves improvements of 51.92%, 4.06%, and 8.33% compared to the EVO, FPA, and HTS techniques. In addition, the proposed GO method displays excellent capability in rapid convergence through the first one hundred iterations compared to the others, as demonstrated in Figure 7.

Moreover, a statistical comparison is conducted between the proposed GO, EVO, FPA, and HTS techniques for the PVSD of RTC France, where Figure 8 depicts the box plot of the proposed GO, EVO, FPA, and HTS techniques for the PVSD of RTC France. It can be noticed that the proposed GO technique displays superior performance compared to the EVO, FPA, and HTS techniques. According to the average RMSE objective score, the proposed GO technique achieves improvements of 73.38%, 14.08%, and 31.19% compared to the EVO, FPA, and HTS techniques. Based on the standard deviation of the RMSE objective score, the proposed GO technique achieves the smallest index of 4.78091 × 10⁻⁶ compared to the EVO, FPA, and HTS techniques which obtain counterparts of 0.001628, 8.941 × 10⁻⁵, and 0.00026, respectively.

Table 5. Comparative assessment of the presented techniques for RTC France with the PVDD model.

Algorithms	RMSE
Proposed GO	9.8602 × 10−4
EVO	1.2250 × 10−3
FPA	1.0021 × 10−3
HTS	9.9339 × 10−4
TLBO [44]	1.52057 × 10−3
Sine cosine approach [45]	9.86863 × 10−4
ABC [23]	1.28482 × 10−3
Teaching–learning–based ABC [46]	1.50482 × 10−3
Generalized oppositional TLBO [47]	4.43212 × 10−3
Comprehensive learning PSO [48]	1.3991 × 10−3
Flower pollination algorithm [49]	1.934336 × 10−3
Cat swarm algorithm [50]	1.22 × 10−3

shows a comparison between the suggested GO approach and various optimizers that are reported in the literature for the PVSD model. It compares the proposed HPO technique’s RMSE value to other optimizers in the literature, including TLBO [44], sine cosine approach [45], ABC [23], teaching–learning–based ABC [46], generalized oppositional TLBO [47], comprehensive learning PSO [48], flower pollination algorithm [49] and cat swarm algorithm [50]. As demonstrated, the suggested GO outperforms its competitors in finding the lowest RMSE.

4.2. Applications for Multi-Crystalline KC200GT PV Module

First, the suggested GO, EVO, FPA, and HTS approaches are used to minimize the RMSE function objective for the KC200GT PV module using the PVSD model considering the operating conditions at an irradiance of (1000 W/m²) and temperature of 25 °C, respectively.

Table 6. Obtained parameters by the proposed GO, EVO, FPA, and HTS for the PVSD of the KC200GT PV module.

Applied Technique	EVO	FPA	HTS	GO
IPh (A)	8.214036	8.189381	8.190356	8.192967
Rsh (Ω)	13.696883	25.536521	53.111481	15.103921
RS (Ω)	0.004290	0.004690	0.004418	0.004710
IS1 (A)	1.42038 × 10−7	4.64227 × 10−8	1.12138 × 10−7	4.31808 × 10−8
η1	1.326768	1.248929	1.309576	1.244346
RMSE	0.023070	0.011226	0.017998	0.008515
Difference compared to GO	1.46 × 10−2	2.71 × 10−3	9.48 × 10−3	-
Improvement	63.09%	24.15%	52.69%	-

tabulates the appropriate values, whereas the convergence trends of the suggested GO, EVO, FPA, and HTS approaches are exemplified in Figure 9 for this scenario. It is illustrated from the table that the proposed GO successfully achieves the minimum RMSE value of 0.008515, where the EVO, FPA, and HTS record RMSE values of 0.023070, 0.011226, and 0.017998, respectively. Based on these outcomes, the presented GO technique achieves improvements of 63.09%, 24.14%, and 52.69% compared to the EVO, FPA, and HTS techniques. Furthermore, the suggested GO technique outperforms the others in fast convergence within the first three hundred iterations, as shown in Figure 9.

Furthermore, a statistical evaluation is made between the suggested GO, EVO, FPA, and HTS approaches, as manifested in Figure 10, for the PVSD of the KC200GT PV module, which illustrates the box plot of the proposed GO, EVO, FPA, and HTS techniques for the PVSD of the KC200GT PV module. As demonstrated, the suggested GO approach outperforms the EVO, FPA, and HTS optimizers. Based on the average RMSE objective score, the suggested GO approach outperforms the EVO, FPA, and HTS procedures by 94.71%, 12.36%, and 58.02%, respectively. Based on the worst RMSE objective score, the suggested GO approach outperforms the EVO, FPA, and HTS procedures by 96.97%, 5.66%, and 61.20%, respectively.

Secondly, using the PVDD model, the suggested GO, EVO, FPA, and HTS approaches are used to minimize the RMSE function objective for the KC200GT PV module. The relevant parameters are organized in

Table 7. Obtained parameters by the proposed GO, EVO, FPA, and HTS for the PVDD of KC200GT PV module.

Applied Technique	EVO	FPA	HTS	GO
IPh (A)	8.204245	8.184637	8.203824	8.193643
Rsh (Ω)	66.836979	100	92.463951	16.378287
RS (Ω)	0.004219	0.004592	0.004562	0.004689
IS1 (A)	1.82124 × 10−7	1.4257 × 10−7	5.66339 × 10−8	6.01244 × 10−8
η1	1.346599	1.777218	1.749083	1.832451
IS2 (A)	5.75716 × 10−7	6.293 × 10−8	8.0133 × 10−8	4.57891 × 10−8
η2	1.933055	1.269921	1.286121	1.248347
RMSE	0.027177	0.014006	0.020515	0.009049
Difference compared to GO	1.81 × 10−2	4.96 × 10−3	1.15 × 10−2	-
Improvement	66.70%	35.39%	55.89%	-

, while the convergence trends of the recommended GO, EVO, FPA, and HTS techniques for this situation are depicted in Figure 11. As demonstrated, the suggested GO approach achieves a minimal RMSE value of 0.009049, whereas the EVO, FPA, and HTS achieve RMSE values of 0.027177, 0.014006, and 0.020515, respectively. Based on these results, the provided GO approach outperforms the EVO, FPA, and HTS procedures by 66.7%, 35.39%, and 55.89%, respectively. Furthermore, the suggested GO technique outperforms other metaheuristics regarding fast convergence during the first one hundred iterations, as demonstrated in Figure 11.

Moreover, a statistical comparison is conducted between the proposed GO, EVO, FPA, and HTS techniques for the PVSD of the KC200GT PV module, where the box plot of the proposed GO, EVO, FPA, and HTS techniques are exemplified in Figure 12 for the PVSD of KC200GT PV module. The proposed GO technique performs better than the EVO, FPA, and HTS techniques. Based on the average RMSE objective score, the suggested GO approach achieves a minimal RMSE value of 0.011929, whereas the EVO, FPA, and HTS achieve RMSE values of 0.221039, 0.017136, and 0.033682, respectively. Therefore, the proposed GO technique achieves improvements of 94.6%, 30.38%, and 64.58% compared to the EVO, FPA, and HTS techniques. Based on the worst RMSE objective score, the suggested GO approach achieves a minimal RMSE value of 0.017417, whereas the EVO, FPA, and HTS achieve RMSE values of 0.26969, 0.025162, and 0.04527, respectively. Therefore, the proposed GO technique achieves improvements of 93.54%, 30.78%, and 61.53% compared to the EVO, FPA, and HTS techniques. According to the standard deviation of the RMSE objective scores, the suggested GO approach achieves a minimal RMSE value of 0.002021, whereas the EVO, FPA, and HTS achieve RMSE values of 0.658340, 0.003030, and 0.006198, respectively. Therefore, the proposed GO technique achieves improvements of 99.69%, 33.3%, and 67.38% compared to the EVO, FPA, and HTS techniques.

Moreover, the proposed GO technique’s efficiency is validated compared to EVO, FPA, and HTS under various weather conditions. The results were collected at a constant temperature of 25 °C using five distinct irradiances, which are 200 W/m², 400 W/m², 600 W/m², 800 W/m², and 1000 W/m². In addition, the KC200GT is available in three temperatures (25 °C, 50 °C, and 75 °C) with a fixed irradiance of 1000 W/m².

Table 8. Obtained parameters by the proposed GO, EVO, FPA, and HTS for the PVSD of KC200GT PV module at different weather conditions.

(a) Operating Irradiance of 200 W/m2 at Temperature of 25 °C
Applied Technique	EVO	FPA	HTS	GO
IPh (A)	1.614391	1.641477	1.635220	1.641655
Rsh (Ω)	55.950700	6.679449	8.863031	6.661734
RS (Ω)	0.000533	0.004758	0.001718	0.004559
IS1 (A)	4.13 × 10−7	1.96 × 10−8	1.33 × 10−7	2.22 × 10−8
η1	1.456835	1.215165	1.357627	1.223433
RMSE	0.013567	0.001513	0.004668	0.000955
Difference compared to GO	1.26 × 10−2	5.58 × 10−4	3.71 × 10−3	-
Improvement	92.96%	36.88%	79.54%	-
(b) Operating Irradiance of 400 W/m2 at Temperature of 25 °C
Applied Technique	EVO	FPA	HTS	GO
IPh (A)	3.258167	3.280930	3.276746	3.277192
Rsh (Ω)	28.182600	8.180287	10.695920	8.306541
RS (Ω)	0.004266	0.004235	0.003628	0.004447
IS1 (A)	6.4 × 10−8	4.63 × 10−8	1.14 × 10−7	3.7 × 10−8
η1	1.285174	1.262536	1.328550	1.247147
RMSE	0.012554	0.004800	0.007983	0.003933
Difference compared to GO	8.62 × 10−3	8.67 × 10−4	4.05 × 10−3	-
Improvement	68.67%	18.06%	50.73%	-
(c) Operating Irradiance of 600 W/m2 at Temperature of 25 °C
Applied Technique	EVO	FPA	HTS	GO
IPh (A)	4.911306	4.908892	4.906687	4.918437
Rsh (Ω)	12.053640	14.518550	22.410870	8.889982
RS (Ω)	0.004805	0.004507	0.003882	0.004639
IS1 (A)	2.32 × 10−8	5.43 × 10−8	1.6 × 10−7	3.51 × 10−8
η1	1.210601	1.266975	1.346080	1.237893
RMSE	0.012870	0.009391	0.014001	0.005880
Difference compared to GO	6.99 × 10−3	3.51 × 10−3	8.12 × 10−3	-
Improvement	54.31%	37.39%	58.00%	-
(d) Operating Irradiance of 800 W/m2 at Temperature of 25 °C
Applied Technique	EVO	FPA	HTS	GO
IPh (A)	8.320163	8.441114	8.410788	8.441114
Rsh (Ω)	11.251770	1.602796	2.046745	1.602797
RS (Ω)	0.005796	0.005628	0.005674	0.005628
IS1 (A)	1 × 10−6	1 × 10−6	1 × 10−6	1 × 10−6
η1	1.211071	1.212172	1.211876	1.212172
RMSE	0.058525	0.039212	0.040739	0.039212
Difference compared to GO	1.93 × 10−2	0.00 × 100	1.53 × 10−3	-
Improvement	33.00%	0.00%	3.75%	-
(e) Operating Irradiance of 1000 W/m2 at Temperature of 50 °C
Applied Technique	EVO	FPA	HTS	GO
IPh (A)	8.266066	8.298574	8.296297	8.301133
Rsh (Ω)	33.152470	6.134123	6.661412	5.624210
RS (Ω)	0.004752	0.004803	0.004790	0.004840
IS1 (A)	8.91 × 10−7	6.83 × 10−7	7.23 × 10−7	6.15 × 10−7
η1	1.339864	1.318481	1.323004	1.310146
RMSE	0.011639	0.002195	0.002984	0.001562
Difference compared to GO	1.01 × 10−2	6.33 × 10−4	1.42 × 10−3	-
Improvement	86.58%	28.84%	47.65%	-
(f) Operating Irradiance of 1000 W/m2 at Temperature of 75 °C
Applied Technique	EVO	FPA	HTS	GO
IPh (A)	8.320163	8.441114	8.410788	8.441114
Rsh (Ω)	11.251770	1.602796	2.046745	1.602797
RS (Ω)	0.005796	0.005628	0.005674	0.005628
IS1 (A)	1 × 10−6	1 × 10−6	1 × 10−6	1 × 10−6
η1	1.211071	1.212172	1.211876	1.212172
RMSE	0.058525	0.039212	0.040739	0.039212
Difference compared to GO	1.93 × 10−2	0.00 × 100	1.53 × 10−3	-
Improvement	33.00%	0.00%	3.75%	-

and

Table 9. Obtained parameters by the proposed GO, EVO, FPA, and HTS for the PVDD of KC200GT PV module at different weather conditions.

(a) Operating Irradiance of 200 W/m2 at Temperature of 25 °C
Applied Technique	EVO	FPA	HTS	GO
IPh (A)	1.623979	1.645062	1.642334	1.641698
Rsh (Ω)	20.390730	6.167409	6.904484	6.745461
RS (Ω)	1.01 × 10−19	0.004099	0.001720	0.003971
IS1 (A)	2.16 × 10−7	0	1.02 × 10−7	8.37 × 10−8
η1	2.000000	1.835084	1.336849	1.953584
IS2 (A)	5.38 × 10−7	2.75 × 10−8	9.8 × 10−8	3.1 × 10−8
η2	1.484407	1.238244	1.890871	1.246916
RMSE	0.011151	0.001825	0.003912	0.000917
Difference compared to GO	1.02 × 10−2	9.08 × 10−4	3.00 × 10−3	-
Improvement	91.78%	49.75%	76.56%	-
(b) Operating Irradiance of 400 W/m2 at Temperature of 25 °C
Applied Technique	EVO	FPA	HTS	GO
IPh (A)	3.269870	3.277503	3.276352	3.281489
Rsh (Ω)	16.742480	8.571691	10.398770	7.399506
RS (Ω)	0.002429	0.004064	0.003023	0.004712
IS1 (A)	1.21 × 10−7	0	4.02 × 10−7	1.85 × 10−8
η1	1.844696	2.000000	2.000000	1.204604
IS2 (A)	3.65 × 10−7	6.3 × 10−8	1.77 × 10−7	5.72 × 10−7
η2	1.425606	1.284554	1.364486	1.838341
RMSE	0.013536	0.005031	0.010082	0.001997
Difference compared to GO	1.15 × 10−2	3.03 × 10−3	8.09 × 10−3	-
Improvement	85.25%	60.31%	80.19%	-
(c) Operating Irradiance of 600 W/m2 at Temperature of 25 °C
Applied Technique	EVO	FPA	HTS	GO
IPh (A)	4.904594	4.910920	4.924728	4.919192
Rsh (Ω)	27.028030	12.765290	12.860540	9.116196
RS (Ω)	0.004365	0.004341	0.004266	0.004490
IS1 (A)	6.61 × 10−8	6.99 × 10−8	8.48 × 10−8	4.81 × 10−8
η1	1.282090	1.284889	1.300237	1.259016
IS2 (A)	5.46 × 10−7	4.58 × 10−8	1.67 × 10−7	9.19 × 10−9
η2	1.903013	1.967746	1.728064	1.698114
RMSE	0.012289	0.009196	0.013487	0.005683
Difference compared to GO	6.61 × 10−3	3.51 × 10−3	7.80 × 10−3	-
Improvement	53.76%	38.20%	57.86%	-
(d) Operating Irradiance of 800 W/m2 at Temperature of 25 °C
Applied Technique	EVO	FPA	HTS	GO
IPh (A)	8.311589	8.423242	8.358966	8.423041
Rsh (Ω)	7.206907	1.992924	4.847170	1.993537
RS (Ω)	0.005719	0.005411	0.005527	0.005411
IS1 (A)	6.71 × 10−7	1 × 10−6	1 × 10−6	1 × 10−6
η1	1.223431	1.265644	1.265694	1.266804
IS2 (A)	4.26 × 10−7	1 × 10−6	1 × 10−6	1 × 10−6
η2	1.210971	1.268029	1.266909	1.266877
RMSE	0.055535	0.029195	0.037941	0.029191
Difference compared to GO	2.63 × 10−2	4.00 × 10−6	8.75 × 10−3	-
Improvement	47.44%	0.01%	23.06%	-
(e) Operating Irradiance of 1000 W/m2 at Temperature of 50 °C
Applied Technique	EVO	FPA	HTS	GO
IPh (A)	8.273769	8.298369	8.285531	8.300973
Rsh (Ω)	20.073080	6.191240	10.458260	5.643489
RS (Ω)	0.004803	0.004800	0.004766	0.004841
IS1 (A)	9.9 × 10−7	6.72 × 10−7	7.66 × 10−7	5.03 × 10−9
η1	1.754878	1.317407	1.757351	1.700812
IS2 (A)	6.76 × 10−7	2.27 × 10−7	7.61 × 10−7	6.14 × 10−7
η2	1.319326	1.822645	1.328574	1.310004
RMSE	0.009872	0.002403	0.006670	0.001564
Difference compared to GO	8.31 × 10−3	8.39 × 10−4	5.11 × 10−3	-
Improvement	84.16%	34.91%	76.55%	-
(f) Operating Irradiance of 1000 W/m2 at Temperature of 75 °C
Applied Technique	EVO	FPA	HTS	GO
IPh (A)	8.311589	8.423242	8.358966	8.423041
Rsh (Ω)	7.206907	1.992924	4.847170	1.993537
RS (Ω)	0.005719	0.005411	0.005527	0.005411
IS1 (A)	6.71 × 10−7	1 × 10−6	1 × 10−6	1 × 10−6
η1	1.223431	1.265644	1.265694	1.266804
IS2 (A)	4.26 × 10−7	1 × 10−6	1 × 10−6	1 × 10−6
η2	1.210971	1.268029	1.266909	1.266877
RMSE	0.055535	0.029195	0.037941	0.029191
Difference compared to GO	2.63 × 10−2	4.00 × 10−6	8.75 × 10−3	-
Improvement	47.44%	0.01%	23.06%	-

illustrate the findings of the optimum parameters derived by the suggested GO, EVO, FPA, and HTS approaches for PVSD and PVDD models. Additionally, Figure 13 and Figure 14 depict the box plot of the proposed GO, EVO, FPA, and HTS techniques for the KC200GT PV module for PVSD and PVDD models. The proposed GO technique’s superior performance is demonstrated for all the weather conditions investigated for PVSD and PVDD models.

Figure 13 and Figure 14 show minimal RMSE values obtained by the proposed GO technique at different irradiance and temperatures. For a better understanding of the ability of the proposed GO technique for the determination of unknown PV parameters, the I-V data modeled by the proposed GO technique at various temperatures regarding 1000 W/m² irradiance is shown in Figure 15a and at 25 °C with various irradiance is manifested in Figure 15b. Figure 16a,b shows the P-V data simulated by the proposed GO approach at different temperatures with 1000 W/m² irradiance and 25 °C for varying irradiance, respectively. Those simulated curves were fitted to the manufacturer’s I-V and P-V data sheets. Consequently, the suggested GO approach outperforms commercial PV systems when identifying the PV parameters at varying temperatures and irradiance, demonstrating the suggested methodology’s effectiveness and ability to deliver highly consistent fitted I-V data with manufacturers’ data sheets.

5. Conclusions

According to the manufacturing technology and solar cell modeling of PV, a unique GO approach has been proposed in this paper for extracting the parameters of PVSD and PVDD solar PV cell/panel models of RTC France and Kyocera KC200GT PV modules. The suggested GO approach is designed to improve the parameters for optimum RMSE value performance. The suggested GO technique’s performance is compared to that of three recent algorithms of the EVO, FPA, and HTS algorithms. Compared to them, the proposed GO technique derives great superiority and statistical robustness by acquiring the least four RMSE indicators with higher improvement percentages. In addition, the proposed GO technique derives a higher convergence speed than EVO, FPA, and HTS algorithms. The proposed GO approach outperforms commercial PV systems when determining the pertinent PV parameters under fluctuating irradiance and temperature, demonstrating the suggested methodology’s effectiveness and ability to provide fitted I-V data that are highly consistent with the manufacturer’s data sheets. Further evidence that the suggested GO achieves superior results compared to recently published optimizers in the literature comes from the precise solution of PV cell/panel models. Additionally, the prescribed GO technique’s ability to ascertain unknown PV characteristics is demonstrated by taking into account various operational conditions with varied temperatures and irradiances.