Hybrid Brown-Bear and Hippopotamus Algorithms with Fractional Order Chaos Maps for Precise Solar PV Model Parameter Estimation

Abstract

The rise in photovoltaic (PV) energy utilization has led to increased research on its functioning, as its accurate modeling is crucial for system simulations. However, capturing nonlinear current–voltage traits is challenging due to limited data from cells’ datasheets. This paper presents a novel enhanced version of the Brown-Bear Optimization Algorithm (EBOA) for determining the ideal parameters for the circuit model. The presented EBOA incorporates several modifications aimed at improving its searching capabilities. It combines Fractional-order Chaos maps (FC maps), which support the BOA settings to be adjusted in an adaptive manner. Additionally, it integrates key mechanisms from the Hippopotamus Optimization (HO) to strengthen the algorithm’s exploitation potential by leveraging surrounding knowledge for more effective position updates while also improving the balance between global and local search processes. The EBOA was subjected to extensive mathematical validation through the application of benchmark functions to rigorously assess its performance. Also, PV parameter estimation was achieved by combining the EBOA with a Newton–Raphson approach. Numerous module and cell varieties, including RTC France, STP6-120/36, and Photowatt-PWP201, were assessed using double-diode and single-diode PV models. The higher performance of the EBOA was shown by a statistical comparison with many well-known metaheuristic techniques. To illustrate this, the root mean-squared error values achieved by our scheme using (SDM, DDM) for RTC France, STP6-120/36, and PWP201 are as follows: (8.183847 × 10⁻⁴, 7.478488 × 10⁻⁴), (1.430320 × 10⁻², 1.427010 × 10⁻²), and (2.220075 × 10⁻³, 2.061273 × 10⁻³), respectively. The experimental results show that the EBOA works better than alternative techniques in terms of accuracy, consistency, and convergence.

1. Introduction

The growing requirement for energy in different nations throughout the world highlights the need for sustainable environmentally friendly energy solutions to fulfill the requirements of a fast-changing global economy and society [1]. The shift towards renewable energy sources has a crucial role in guaranteeing a sustainable future. By harnessing the power of natural resources, society can reduce its reliance on fossil fuels, addressing the challenges of climate change while promoting a cleaner world for the future [2,3]. In the end, these expenditures may save the environment for future generations and help create a more robust and stable economy. Ensuring the effectiveness of renewable energies requires modeling techniques, especially for forecasting photovoltaic energy production. PV systems offer significant potential as a renewable energy source because of their efficiency, eco-friendliness, simplicity of installation, and scalability; nevertheless, factors like solar radiation and temperature have a vital part in selecting the modeling approaches [4,5,6].

Exact determination of unknown values in the I-V curve is essential for simulating the I-V characteristics of PV systems. In order to accomplish a close likeness between measured and simulated curves, model attributes must be adjusted and refined. This allows for consistent performance evaluation and an ideal system scheme. Due to the lack of information provided by manufacturers, it is difficult to create precise mathematical models for PV cells and modules. Components like current sources, diodes, and resistors are found in standard electrical equivalent circuits for solar cells. There are many varieties of PV cell models, with varying levels of precision and complexity, such as single, double, and triple diodes. Compared to the simpler single-diode model, which has five parameters, the double-diode model, which has seven undefined parameters, is more accurate [7,8].

The following challenges have resulted in rising demands on the research axis despite significant attempts to treat PV parameter extraction difficulties through various approaches that have been detailed in the literature. Estimating PV parameters requires complex interactions between a number of variables, including temperature and sun irradiation, and the electrical characteristics of PV modules. Many times, these interactions are complicated, nonlinear, and unclear [9]. Researchers have published a number of approaches, including deterministic, metaheuristic, and analytical techniques, to ensure accurate parameter estimation in PV cell models [10]. When it comes to determining the parameters of the PV model, analytical techniques are seen to be more efficient and practical than other approaches. However, the precision and dependability of the answer are inadequate since assumptions must be established before analysis is permitted to begin. The parameter estimation was framed analytically using information from the PV datasheet (maximum power, voltage, and current, short circuit current, and open-circuit voltage) or its I-V characteristic profile [11]. Reduced Space Search (RSS) [12], Lambert-W-based approaches [13], and OSMP-based approaches [14] are a few of the analytical methodologies. When nonlinear equations are used to calculate the unknown parameters of the PV cell model, the computational complexity of the analytical method rises. One way to reduce the complexity of the model is to use simplifications, such setting the values of the series and shunt resistors to constants, which will reduce the number of unknown parameters. Deterministic approaches may also be iterative, obtaining parameters via trial and error or repetition. These methods include the Newton–Raphson Method (NRM) [15], the Gauss-Seidel method [16], and the least squares approach [17]. Iterative methods have limited applicability since they need system equations that are differentiable, continuous, and convex. To prevent local optima, beginning values must be chosen carefully. Furthermore, deterministic approaches are readily trapped in local maxima cycles because of the potential for deception arising from their basic assumptions. Similar to deterministic approaches, metaheuristic optimization techniques provide benefits including simplicity, ease of use, and dependable performance.

Though many attempts have been made in the literature to solve the problems using different approaches, photovoltaic parameter estimate research continues to confront growing hurdles. The process of estimating parameters in photovoltaic systems entails the complicated interplay between the electrical characteristics of the modules and environmental elements such as temperature and sun irradiation, which often result in convoluted and nonlinear correlations. To precisely identify these factors, a variety of methods have been put forward by researchers, including analytical, deterministic, and metaheuristic techniques. Analytical methods are often seen to be more efficient and user-friendly than other approaches, but since they need certain assumptions, they may not be as precise. Though dependable, deterministic approaches have the tendency to become trapped in local maxima because of underlying presumptions. Conversely, the advantages of metaheuristic optimization strategies are simplicity, ease of implementation, and dependable performance. The increasing significance of determining parameters from PV cells and modules in research has resulted in the creation of optimization techniques. The investigations conducted by scientists are committed to developing novel techniques for accurate and trustworthy PV parameter calculation. In order to improve accuracy, dependability, and efficiency, researchers create new methods and improve those that already exist, which helps to drive the green production of solar energy [18]. The primary objective of the optimization procedure is often to minimize the sum of squared variance between the measured output current and the forecasted data points of a solar PV module, as shown by the literature included in Table 1. Reducing the commonly used error criteria is necessary to solve this problem. The aforementioned tactics have proven effective in a quantity of undertakings, achieving goals including producing accurate findings, raising convergence efficiency, enhancing the harmony between exploitation and exploration, and mitigating the risk of local minima entrapment.

The parameter estimation of PV is one of the many technical difficulties that require the use of a thorough metaheuristic algorithm revolution to be addressed efficiently [19]. These algorithms include the following: Genetic Algorithm (GA) [20], Harmony-Search (HS) [21], Shuffled Frog Leaping Algorithm (SFLA) [22], Artificial Bee Colony [23], Multi-Verse Optimizer (MVO) [24], Time Varying Acceleration Coefficients Particle Swarm Optimizer (TVACPSO) [25], Generalized Oppositional Teaching Learning-Based Optimizer (GOTLBO) [26], Multiple Learning Backtracking Search Approach (MLBSA) [27], Enhanced JAYA (IJAYA) [28], Salp Swarm Algorithm (SSA) [29], Improved Adaptive Differential Evolution (EADE) technique [30], Enhanced Whale Optimization [31], Classified Perturbation Mutation Based PSO (CPMPSO) [32], Coyote Optimization Algorithm (COA) [33], Manta-Rays Foraging Optimizer (MRFO) [34], Harris Hawk’s Optimization (BHHO) [35], Modified Flower Pollination Algorithm (MFPA) [36], Improved Cuckoo Search (ICS) [37], Boosted Slime Mould optimization [38], Drone Squadron Optimizer (DSO) [39], Radial Movement Optimizer (RMO) [40], hybridized gradient-based optimizer with crisscross-based Nelder–Mead simplex technique [41], Kepler optimization technique [42,43], marine predators optimizer [44], Artificial Rabbits Algorithm (ARA) [45], Mantis Search Algorithm (MSA) [46], Henry Gas Solubility Optimizer (HGSO) [47], Artificial Hummingbird Optimizer (AHO) [48], Bonobo Optimizer (BO) [49], enhanced MRFO [50], Rime-Ice Growth Optimizer (RIGO) [51], hybrid White Shark and Artificial Rabbits Optimizer (hWSO-ARO) [52], Dwarf Mongoose Optimizer (DMO) [53], Elite Learning Adaptive DE (ELADE) [54], Material Generation Algorithm (MGA) [55].

This study utilized real-world case studies involving an RTC France, PWP201, and STP6-120/36 to estimate key parameters, which is a critical topic in the PV industry due to its significant impact on system-manufacturing costs. Accurate model parameter data are essential for several reasons: (1) PV system simulation relies on these parameters to estimate energy yield and optimize power converter selection; (2) they are key for managing and tracking the system’s Maximum Power Point Tracking (MPPT) to ensure its optimal performance; (3) they help maintain quality control throughout the production process; and (4) they are crucial for assessing the degradation of PV cells, as manufacturers’ data are typically based on Standard Test Conditions (STCs), while actual systems often operate under real-world, sub-optimal conditions.

Table 1. A range of optimization methods for photovoltaic parameter estimation.

Reference	PV Module	Method	Year	Comments and Annotations
[56]	RTC France, PWP201	WCA	2017	The main features are reduced user speedy convergence and adjustable parameters.
[57]	RTC France, PWP201	DE	2018	The efficiency of the scheme is evaluated by valuing the parameters of PV module of diverse manufactures
[58]	RTC France, PWP201	MLSHADE	2020	The MLSHADE algorithm can attain a greatly competitive performing contrasted with other state-of-the-art methods.
[59]	PWP201, STE4/100	PSOGWO	2021	The use of hybrid PSO and GWO based on investigational datasets of I-V characteristics.
[60]	PWP201, KC200GT	ABC-LS	2021	The RMSE rates for the parameter estimation of SD, DD and MM of the ABC-LS scheme were with satisfactory accuracy.
[61]	Adani ASB-7-355, Adani ASM-7-PERC-365, Tata TP280.	OBEO	2021	The present OBEO is an modernize process to generate the top solutions to discover superior search space.
[62]	RTC France, STM6-40/36, STP6-120/36, CTJ30	GSA	2022	The five parameters model offers the lowest RMSE value as contrasted to the procedures testified in the remaining literature.
[63]	RTC France, PWP201.	RKO	2022	Robust and enhanced convergence quickness.
[64]	RTC France, PWP201, ST40, SM55, STM6-40/36, STP6-120/36.	IQSODE	2023	Satisfactorily accurate PV parameters (SDM, DDM, and MM) are obtained using the suggested IQSODE approach.
[65]	RTC France, PWP201.	HHO	2023	Performance accuracy and dominance in identifying the PV parameters
[66]	STM6-40/36, KC200GT, PWP 201	AHT	2024	The AHT’s implementation is calculated employing the datasheet offered by the manufacturer.
[67]	RTC France, PWP201, STP6-120/36	ICOA	2024	A numerical analysis between numerous firm metaheuristic approaches was directed to display the excellent ICOA performing.
[68]	KC200GT, SM55	SAELDE	2024	The parameters are computed by computing the derivative matrix corresponding to the nonlinear parameters.
[69]	PWP201, STM6-40/36, STP6-120/36	ESCA	2024	Elevated convergence level, low and stable finest fitness values.
[70]	PWP201, STM6-40/36	µAFCSO	2024	The examination functions were reflected in order to establish that the process was used to solve usual problems managed applying population.
[71]	RTC France, PWP201	MGO	2024	The suggested optimizer performs best in identifying parameters for both SD, DD, and MM.

Table 1. A range of optimization methods for photovoltaic parameter estimation.

Reference	PV Module	Method	Year	Comments and Annotations
[56]	RTC France, PWP201	WCA	2017	The main features are reduced user speedy convergence and adjustable parameters.
[57]	RTC France, PWP201	DE	2018	The efficiency of the scheme is evaluated by valuing the parameters of PV module of diverse manufactures
[58]	RTC France, PWP201	MLSHADE	2020	The MLSHADE algorithm can attain a greatly competitive performing contrasted with other state-of-the-art methods.
[59]	PWP201, STE4/100	PSOGWO	2021	The use of hybrid PSO and GWO based on investigational datasets of I-V characteristics.
[60]	PWP201, KC200GT	ABC-LS	2021	The RMSE rates for the parameter estimation of SD, DD and MM of the ABC-LS scheme were with satisfactory accuracy.
[61]	Adani ASB-7-355, Adani ASM-7-PERC-365, Tata TP280.	OBEO	2021	The present OBEO is an modernize process to generate the top solutions to discover superior search space.
[62]	RTC France, STM6-40/36, STP6-120/36, CTJ30	GSA	2022	The five parameters model offers the lowest RMSE value as contrasted to the procedures testified in the remaining literature.
[63]	RTC France, PWP201.	RKO	2022	Robust and enhanced convergence quickness.
[64]	RTC France, PWP201, ST40, SM55, STM6-40/36, STP6-120/36.	IQSODE	2023	Satisfactorily accurate PV parameters (SDM, DDM, and MM) are obtained using the suggested IQSODE approach.
[65]	RTC France, PWP201.	HHO	2023	Performance accuracy and dominance in identifying the PV parameters
[66]	STM6-40/36, KC200GT, PWP 201	AHT	2024	The AHT’s implementation is calculated employing the datasheet offered by the manufacturer.
[67]	RTC France, PWP201, STP6-120/36	ICOA	2024	A numerical analysis between numerous firm metaheuristic approaches was directed to display the excellent ICOA performing.
[68]	KC200GT, SM55	SAELDE	2024	The parameters are computed by computing the derivative matrix corresponding to the nonlinear parameters.
[69]	PWP201, STM6-40/36, STP6-120/36	ESCA	2024	Elevated convergence level, low and stable finest fitness values.
[70]	PWP201, STM6-40/36	µAFCSO	2024	The examination functions were reflected in order to establish that the process was used to solve usual problems managed applying population.
[71]	RTC France, PWP201	MGO	2024	The suggested optimizer performs best in identifying parameters for both SD, DD, and MM.

This research presents a novel approach for determining the mathematical model parameters of PV modules and cells using an upgraded version of the BOA. The methodology incorporates a robust search space design and adapts the parameters of the EBOA using FC maps to prevent premature convergence and stagnation, ensuring convergence to a local minimum. Initially, BOA’s random initialization is replaced by the FC maps. Better iterative beginning values are provided by EBOA. By modifying the static operations of traditional BOA with adaptive search boundary controls, the algorithm effectively balances exploration and exploitation. Additionally, the HO is first used to make use of historical events and direct the evolutionary process, which improves the algorithm’s capacity for exploration. In order to increase search space exploitation and speed up convergence during iterations, the BOA is then implemented. The proposed technique exhibits strong convergence performance, offering highly accurate solutions with reduced computational cost. Three real-world case studies were conducted, where the performance of the upgraded EBOA was rigorously compared to established numerical and optimization methods, demonstrating its superior accuracy and efficiency. Key contributions include the development of the adaptive EBOA with FC maps, the integration of HO operations, and its application in PV model parameter estimation. By merging the Newton–Raphson method with the EBOA, PV parameter estimation was accomplished. Using double-diode and single-diode PV models, many module and cell types, such as RTC France, STP6-120/36, and Photowatt-PWP201, were evaluated.

Figure 1 displays the basic paper outline, while the primary contribution of this research is enumerated in the following list of contributions:

• The integration of HO algorithm operations and FC maps into BOA enhances global and local search balance, maintains variety, and enables adaptive parameter fine-tuning.
• The EBOA’s potency was thoroughly analyzed using benchmark functions.
• A comprehensive study demonstrated that the EBOA-with-NR technique outperformed other established metaheuristic methods in accurately predicting optimal PV parameters.
• Various PV module/cell scenarios were evaluated using various PV models, including SDM and DDM, to estimate parameters.
• Experimental results confirmed the effectiveness of the EBOA in PV parameter extraction through its accuracy, consistency, and convergence.

Figure 1. Basic paper outline.

Figure 1. Basic paper outline.

2. PV Models and Their Objective Functions

The current–voltage characteristics of PV cells and modules are often shown in the literature by means of models such as the PV module model (MM), DDM, and SDM. Section 2.1 contains further information about these models and the corresponding objective functions.

2.1. Single-Diode Model (SDM)

As seen in Figure 2, the SDM consists of a current source that describes the radiation of incoming flux. This element is connected in series resistance $R_s$, which represents resistance at the contacts and parallel resistive elements $R_{sh}$, which models the leakage current in parallel with a diode symbolizing the P-N junction [44]. The computations performed to obtain the output current $I_L$ are as outlined below:

$$I_L=I_{ph}-I_D-I_{sh}$$

(1)

where $I_{ph}$ is the photo-generated current, $I_D$ represents the diode current defined using the Shockley diode equation, and $I_{sh}$ is the current flowing through the shunt resistor.

$$I_D=I_{sd}\left[exp \left({\displaystyle \frac{q(V_L+R_s{I}_L)}{akT}}\right)-1\right]$$

(2)

$$I_{sh}={\displaystyle \frac{V_L+R_s{I}_L}{R_{sh}}}$$

(3)

where $a$ denotes the diode ideality variable, and $q$ stands for the electron charge of 1.60217646 $\times$ 10⁻¹⁹ C. The Boltzmann constant, or 1.3806503 $\times$ 10⁻²³ J/K, is also represented by the letters $k$, where $T$ represents the cell temperature expressed in Kelvin (K).

The formula for the output current that results from applying Equations (1)–(3) is as follows:

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

(4)

The model has five unknown parameters: $I_{ph}$, $I_{sd}$, $R_s$, $R_{sh}$, and $a$.

2.2. Double-Diode Model (DDM)

Recombination current dissipation in the space charge section is not taken into account by the SDM, as seen in Figure 2. Contrarily, the DDM, shown in Figure 3 [39], accounts for this dissipation by adding a second diode, producing two diodes that are designated I_D1 and I_D2. As a result, the following is the equation for output current using DDM:

$$I_L=I_{ph}-I_{D1}-I_{D2}-I_{sh}$$

(5)

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

(6)

Seven unknown parameters are included in the model; they are designated as $I_{ph}$, $I_{sd1}$, $a_1$, $R_s$, $R_{sh}$, $a_2$, and $I_{sd2}$.

2.3. PV Module Model (MM)

Connecting several identical solar cells in parallel and series arrangements results in the PV (MM) [40]. By using the SDM, the equation for the PV module model’s output current $I_L$ may be expressed in the subsequent manner:

$$I_L=I_{ph}{N}_p-I_{sd}{N}_p\left[exp\left({\displaystyle \frac{q(V_L{N}_p+R_s{I}_L{N}_s)}{akT{N}_s{N}_p}}\right)-1\right]-{\displaystyle \frac{V_L{N}_P+R_s{I}_L{N}_s}{R_{sh}{N}_s}}$$

(7)

Five unknown parameters (I_ph, I_sd, a, R_s, and R_sh) in the SDM need to be estimated.

2.4. Objective Function

The main goal of modeling PV systems mathematically is to properly identify the unknowns that differentiate the different models. This involves specifying the SDM’s five parameters and the DDM’s seven parameters. The final objective is to leverage experimental data from the current–voltage characteristics of a PV cell and module to guarantee that the predicted current closely matches the actual current. Consequently, the estimate process turns into an optimization formulation where the goal is to reduce the RMSE within the fitness function. The variance between the estimated and measured current values divided by the complete set of data points (n) is the chosen fitness function.

$$RMSE=\sqrt{{\displaystyle \frac{\sum _{i=0}^n{(I_{est i}-I_i)}^2}{n}}}$$

(8)

After considering the output current from the earlier models, the following expressions are used to characterize the identified current value for used model (Equation (9) for SDM and Equation (10) for DDM):

$$\left\{\begin{array}{c}{\mathrm{I}}_{\mathrm{s}\mathrm{i}\mathrm{m}}=\mathrm{f}\left({\mathrm{I}}_{\mathrm{L}},{\mathrm{V}}_{\mathrm{L}},\mathrm{X}\right)={\mathrm{I}}_{\mathrm{p}\mathrm{h}}-{\mathrm{I}}_{\mathrm{s}\mathrm{d}}\left[\mathrm{e}\mathrm{x}\mathrm{p}\left({\displaystyle \frac{\mathrm{q}({\mathrm{V}}_{\mathrm{L}}+{\mathrm{R}}_{\mathrm{s}}{\mathrm{I}}_{\mathrm{L}})}{\mathrm{a}\mathrm{k}\mathrm{T}}}\right)-1\right]-{\displaystyle \frac{{\mathrm{V}}_{\mathrm{L}}+{\mathrm{R}}_{\mathrm{S}}{\mathrm{I}}_{\mathrm{L}}}{{\mathrm{R}}_{\mathrm{s}\mathrm{h}}}}\\ \mathrm{X}=\left\{{\mathrm{I}}_{\mathrm{p}\mathrm{h}},{\mathrm{I}}_{\mathrm{s}\mathrm{d}}{,\mathrm{R}}_{\mathrm{s}},{\mathrm{R}}_{\mathrm{s}\mathrm{h}},\left.\mathrm{a}\right\}\right.\end{array}\right.$$

(9)

$$\left\{\begin{array}{c}{\mathrm{I}}_{\mathrm{s}\mathrm{i}\mathrm{m}}=\mathrm{f}\left({\mathrm{I}}_{\mathrm{L}},{\mathrm{V}}_{\mathrm{L}},\mathrm{X}\right)={\mathrm{I}}_{\mathrm{p}\mathrm{h}}-{\mathrm{I}}_{\mathrm{s}\mathrm{d}}\left[\mathrm{e}\mathrm{x}\mathrm{p}\left({\displaystyle \frac{\mathrm{q}\left({\mathrm{V}}_{\mathrm{L}}+{\mathrm{R}}_{\mathrm{s}}{\mathrm{I}}_{\mathrm{L}}\right)}{\mathrm{a}\mathrm{k}\mathrm{T}}}\right)-1\right]-{\mathrm{I}}_{\mathrm{s}\mathrm{d}1}\left[\mathrm{e}\mathrm{x}\mathrm{p}\left({\displaystyle \frac{\mathrm{q}\left({\mathrm{V}}_{\mathrm{L}}+{\mathrm{R}}_{\mathrm{s}}{\mathrm{I}}_{\mathrm{L}}\right)}{{\mathrm{a}}_2\mathrm{k}\mathrm{T}}}\right)-1\right]-{\displaystyle \frac{{\mathrm{V}}_{\mathrm{L}}+{\mathrm{R}}_{\mathrm{s}}{\mathrm{I}}_{\mathrm{L}}}{{\mathrm{R}}_{\mathrm{s}\mathrm{h}}}}\\ \\ \mathrm{X}=\left\{{\mathrm{I}}_{\mathrm{p}\mathrm{h}},{\mathrm{I}}_{\mathrm{s}\mathrm{d}1}{,\mathrm{R}}_{\mathrm{s}},{\mathrm{R}}_{\mathrm{s}\mathrm{h}}{,\mathrm{a}}_1,{\mathrm{I}}_{\mathrm{s}\mathrm{d}2},\left.{\mathrm{a}}_2\right\}\right.\end{array}\right.$$

(10)

There is an intrinsic nonlinearity in the calculated current equation. Because of this, in order to calculate the simulated current, a numerical method that can solve nonlinear algebraic problems must be used. Resolving a nonlinear problem using the NRM is considered the most effective approach. Calculating the estimated current for the purpose of determining the fitness function is its main duty. However, as was already said, the existence of certain unknown factors is an important factor. To tackle the above-described problems, the NRM is combined with the EBOA to ascertain the properties of PV cells and modules under diverse point conditions, thereby permitting the extraction of every variable in the two investigated models.

A range of performance metrics are used to assess the acquired outcomes. The absolute individual error (AIE) is then computed to compare the simulated values with the observed data. The latter is computed as the absolute difference in the power (AIE_P) and current (AIE_I) experimental and calculated values. Apart from AIE, additional performance indices like mean absolute error (MAE) and relative error (RE) are also applied to provide a comprehensive evaluation of the contract between the experimental and simulated-current data. The complete set of data measurement is denoted by n and the deviation between the estimated and measured current values by $I_i$ and $I_{est i}$, respectively.

$$AIE\_I=\left|I_{est i}-I\right|$$

(11)

$$AIE\_P=\left|P_{est i}-P\right|$$

(12)

$$RE={\displaystyle \frac{\left|I_{est i}-I\right|}{I}}$$

(13)

3. Proposed EBOA

Any optimization algorithm’s exploitation and exploration skills determine how well it performs. While the global search represents the exploration phase, the local search represents the exploitation phase. First, the FC maps are used instead of BOA’s random initialization. The iterative starting values that EBOA provides are better. The location phase of the HO in the suggested hybrid algorithm handles the exploring phase. Two randomly selected vectors from the scenarios in the populations are used to update the hippopotamuses’ location vector. During the pedal scent-marking phase in the BOA, the local search is carried out. In order to update its location until the first one-third of the total iterations, which are characterized by sub-phases of the distinctive gait while walking, the algorithm makes use of the surrounding information. The locations are updated with the best and worst pedal marks in the sub-phase of attentive stepping until the second-third part of the iterations when the surroundings are discovered. By shifting the position vector of the pedal marks away from the least favorable pedal marks and towards the most favorable pedal marks, the last third of the generations explains the exploitation phase. The merging of the fractional sine map with the HO exploration phase results in an improvement in BOA.

Every population member is updated depending on the exploration and exploitation process after the completion of each EBOA iteration. Until the last iteration, the population is updated continuously. As the algorithm runs, the optimal possible solution is continuously monitored and saved. The best candidate, also known as the dominant solution, is defined as the final solution to the issue when the whole method is finished. The other BOA operators are described in the subsequent sections. In Algorithm 1, the EBOA pseudo-code is shown.

Algorithm 1. The pseudo code of the EBOA

Start EBOA

Involve: $N_{pop}$, $N_{iter}$, $D$ and variables limits Confirm: objective function $f\left(P\right)$ is described 1: Initialize random bears group including FC maps based on Equation (32) 2: Form group of bears represented by population P in Equation (15) 3: Initialize best objective function $k$ = 1 and $f^{best}=\mathrm{i}\mathrm{n}\mathrm{f}$ 4: while ($k\le N_{iter}$) do 5: Hippopotamuses phase exploration starts 6:   for ($j$ = 1: size ($P$;1)) do 7:        Compute the new location of the ith hippopotamus based on Equations (24) and (27) 8:        Update the location of the ith hippopotamus using Equations (29) and (30) 9:        Assess objective function $f\left(P_j\right)$. 10:     if ($f^{best}$ > $f\left(P_j\right)$)) then 11:      $f^{best}$ = $f\left(P_j\right)$; 12:      $P_k^{best}$ = $P (j; :);$ 13:     end if 14:   end for 15: Hippopotamuses phase exploration ends 16: Pedal scent marking activity commences 17: Choose the best and worst bears group. 18:   ${\theta }_k={\displaystyle \frac{k}{N_{iter}}}$; 19:   for ($i$ = 1: size ($P$;1)) do 20:     % Starting of Characteristic gait while walking 21:     if (${\theta }_k$ > 0 && ${\theta }_k\le$ ${\displaystyle \frac{2{ N}_{iter} }{3}}$) then 22:       Compute the updated value of $P_{i,k}$ via Equation (16) 23:       % Starting of Careful stepping characteristic 24:     else if (${\theta }_k>$ ${\displaystyle \frac{{ N}_{iter} }{3}}$ && ${\theta }_k\le$ ${\displaystyle \frac{2{ N}_{iter} }{3}}$) then 25:           Compute the updated value of $P_{i,k}$ based on Equation (18) 26:           % Starting of Twisting feet characteristic 27:       else if (${\theta }_k\le$ ${\displaystyle \frac{2{ N}_{iter} }{3}}$ && ${\theta }_k\le$ 1) then 28:           Compute the updated value of $P_{i,k}$ via (22) 29:       end if 30:   end for 31:   Choose most favorable bears group. 32:   Behavior of pedal scent marking terminates 33:   Behavior of sniffing begins 34:   for ($m$ = 1: size($P$;1)) do 35:     Choose one random of bears group $P_{n,k}$ for $m\ne n$ 36:     Compute the updated value of $P_{m,k}$ via (23) 37:   end for 38:   Choose better bears group 39:   behavior of sniffing terminates 40:   $k=k+1$; 41: end while 42: print $f^{best}$ and $P^{best}$.

End EBOA

3.1. BOA: Mathematical Model

The suggested optimization technique focuses on brown bears’ pedal-scent-marking and sniffing behaviors, dividing them into three subcategories with equal chances of occurring [72].

3.1.1. Group Formation

This algorithm uses brown-bear groups to represent solution sets in a population, with pedal scent marks representing solution parameters. The territory is a trouble search area, and the initialization of populations is performed. Different groups are randomly produced with a specified number of marks, with their limits defined by problem decision variables. The following is a mathematical equation for randomly initializing the groups inside a region:

$$P_{i,j}=P_{i,j}^{min}+\lambda .(P_{i,j}^{max}-P_{i,j}^{min})$$

(14)

$P_{i,j}$ represents the jth pedal mark of a brown-bears group, where $N_{pop}$ and $D$ represent the total amount of groups and decision variables. The set P, representing the solutions, is described as follows:

$$P=\left[\begin{array}{cccc}{P}_{\mathrm{1,1}}& P_{\mathrm{1,2}}& \cdots & P_{1,D}\\ P_{\mathrm{2,1}}& P_{2,j}& \cdots & P_{2,D}\\ ⋮& ⋮& \ddots & ⋮\\ P_{N_{pop},1}& P_{N_{pop},2}& \cdots & P_{N_{pop},D}\end{array}\right]$$

(15)

3.1.2. Pedal-Scent-Marking Behavior

As was previously established, brown bears exhibit a particular kind of pedal-scent-marking behavior. It is distinguished by a unique walking style, cautious stepping, and foot twisting onto the ground’s depressions. The behavior of pedal scent marking is represented mathematically by these three features. There is an equal possibility of each of these traits occurring. The incidence of each of the aforementioned characteristics is equal to ${\displaystyle \frac{N_{iter}}{3}}$, assuming that the algorithm’s total number of generations equals $N_{iter}$. Below is a definition of the attributes’ mathematical model.

Characteristic Gait While Walking

Only male members of the majority exhibit the behavior of marking pedal scent. For the sake of simplicity, each group is assumed to have one male member. Each group’s male members have a unique walking gait. Because of this, the pedal scent markings that male bears from each group make are distinctly different. Up to the first third of the total number of generations, $N_{iter}$, it is expected that pedal smell traces produced based on typical gait while walking are maintained. This trait can be stated mathematically as follows:

$$P_{i,j,k}^{new}=P_{i,j,k}^{old}-({\theta }_k·{\alpha }_{i,j,k}·P_{i,j,k}^{old})$$

(16)

In the kth iteration, the updated pedal mark of the ith brown-bears group is represented through $P_{i,j,k}^{new}$, while the prior pedal mark is represented by $P_{i,j,k}^{old}$. A random integer, ${\alpha }_{i,j,k}$, is used to represent the jth pedal mark. The occurrence factor, denoted as ${\theta }_k$, rises linearly with the iterations number, expressed as the ratio of completed to in progress iterations and represented as follows:

$${\theta }_k={\displaystyle \frac{C_{iter}}{N_{iter}}}$$

(17)

where the number of iterations is represented by $C_{iter}$.

Brown bears update pedal marks in the first to second third of iterations, using a careful stepping characteristic to make the marks more recognizable to other group members. The following is the expression of this characteristic’s mathematical model:

$$P_{i,j,k}^{new}=P_{i,j,k}^{old}+F_k (P_{j,k}^{best}-L_k·P_{j,k}^{worst})$$

(18)

The occurrence factor ${\theta }_k$ determines the step factor for the kth iteration, $F_k$, based on the pedal smell marks $P_{j,k}^{worst}$ and $P_{j,k}^{best}$ across all brown-bear groups.

$$F_k={\beta }_{1,k}·{\theta }_k$$

(19)

The algorithm uses a random number ${\beta }_{1,k}$ and step length $L_k$ to modify pedal marks derived from the highest and lowest pedal marks within the population. A male brown bear in a group steps carefully in either forward or backward track, forming new pedal marks with step lengths of 1 or 2. The step length $L_k$ may be stated mathematically as follows:

$$L_k=round(1+{\beta }_{2,k})$$

(20)

where ${\beta }_{2,k}$ represents any random number in the interval [0, 1] that is evenly distributed.

Twisting-Feet Characteristic

The updating of pedal marks in a group is determined by the typical twisting of the feet seen in male brown bears, which contributes to the formation of firmer pedal scent marks. Additional group members use this capability to make scent maps. Prior pedal marks are selected based on the best and worst pedal marks. The following equation represents the angular velocity at which each male brown bear in a group rotates his feet:

$${\omega }_{i,k}=2\pi ·{\theta }_k·{\gamma }_{i,k}$$

(21)

The angular velocity of twist in a brown bear is determined by ${\omega }_{i,k}$ and ${\gamma }_{i,k}$, where ${\omega }_{i,k}$ is the ith twist velocity, and ${\gamma }_{i,k}$ is a random number between 0 and 1. The following phrase serves to define this property.

$$P_{i,j,k}^{new}=P_{i,j,k}^{old}+{\omega }_{i,k}· (P_{j,k}^{best}-\left|P_{i,j,k}^{old}\right|)-{\omega }_{i,k}· (P_{j,k}^{worst}-\left|P_{i,j,k}^{old}\right|)$$

(22)

It is imperative to say that the best bears group who were chosen for this round go on to the next one.

3.1.3. Sniffing Behavior

Brown bears use sniffing behavior to communicate and control their movement within their territory. They randomly select pedal marks to move towards their group’s marks, leaving other marks behind. This behavior is mathematically modeled as follows:

$$P_{m,j,k}^{new}=\left\{\begin{array}{c}{P}_{m,j,k}^{old}+{\lambda }_{i,k}·(P_{m,j,k}^{old}-P_{n,j,k}^{old}) \mathrm{if} f\left(P_{m,k}^{old}\right)<f\left(P_{n,k}^{old}\right)\\ P_{m,j,k}^{old}+{\lambda }_{i,k}·(P_{n,j,k}^{old}-P_{m,j,k}^{old}) \mathrm{if} f\left(P_{n,k}^{old}\right)<f\left(P_{m,k}^{old}\right)\end{array}\right.$$

(23)

In the kth iteration, the jth updated pedal markings of the mth and nth bear groups are represented by $P_{i,j,k}^{new}$ and $P_{i,j,k}^{old}$, respectively, with m and n being different. The fitness function values for these groups are denoted by $f\left(P_{m,k}^{old}\right)$ and $f\left(P_{n,k}^{old}\right)$, respectively. For the jth pedal marks during the kth iteration, ${\lambda }_{i,k}$ corresponds to a consistently distributed random value in the interval [0, 1].

3.2. HO Exploration Phase

A novel approach to the metaheuristic technique is shown by the HO [73], which was designed with inspiration from the innate behaviors of hippopotamuses. A trinary-phase model is used to conceptually characterize the HO. This model includes the evasive techniques, defensive measures against predators, and location updates in rivers or ponds, all of which are mathematically expressed. The following is the mathematical modeling of HO exploration phase:

Numerous adult female hippopotamuses, calves, numerous adult male hippopotamuses, and dominant male hippopotamuses (the herd leader) make up a hippopotamus herd. The maximum value for the maximization problem and the minimum for the minimization trouble are the iterations of the objective function that decide which hippopotamus is dominant. Hippopotamus species often congregate next to one another. Dominant male hippopotamuses defend the area and herd against intruders. Once they reach maturity, the dominant male hippopotamuses kick the other males out of the herd. Afterwards, to assert their own superiority, these expelled males must either attract females or compete with other established males in the herd. The location of the male hippopotamus in the herd inside the lake or pond is represented mathematically in Equation (24):

$$\begin{gathered} x_i^{Mhippo}{:}_{x_{i,j}}^{Mhippo}=x_{i,j}+y_1·\left(D_{hippo}-I_1{x}_{i,j}\right) \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}For i=\mathrm{1,2},\dots , \left[{\displaystyle \frac{N}{2}}\right] and j=\mathrm{1,2},...,m \end{gathered}$$

(24)

The male hippopotamus position is represented by $x_i^{Mhippo}$, $y_1$ is a random integer between 0 and 1, while the dominant hippopotamus position is indicated by $D_{hippo}$ (the hippopotamus with the optimal solution for this iteration). $I_1$ is an integer between 1 and 2.

$$h=\left\{\begin{array}{c}{I}_2\times \overrightarrow{r_1}+\left(~{\varrho }_1\right)\\ 2\times \overrightarrow{r_2 }-1\\ \overrightarrow{r_3}\\ I_1\times \overrightarrow{r_4}+\left(~{\varrho }_2\right)\\ \overrightarrow{r_5}\end{array}\right.$$

(25)

$$T=exp(-{\displaystyle \frac{l}{\Gamma }})$$

(26)

$$x_i^{\mathcal{F}\mathcal{B}hippo}:x_{i,j}^{\mathcal{F}\mathcal{B}hippo}=\left\{\begin{array}{c}{x}_{i,j}+h_1.(Dhippo-.I_{2}M{\mathcal{G}}_i) T>0.6\\ ⊑ else\end{array}\right.$$

(27)

$$\begin{gathered} \mathsf{\Xi }=\left\{\begin{array}{c}{x}_{i,j}+h_2.(M{\mathcal{G}}_i-Dhippo) r_6>0.5\\ {\mathcal{l}\mathcal{b}}_j+r_7.({\mathcal{u}\mathcal{b}}_j-{\mathcal{l}\mathcal{b}}_j) else\end{array}\right. \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}For i=\mathrm{1,2},\dots , \left[{\displaystyle \frac{N}{2}}\right] and j=\mathrm{1,2},...,m \end{gathered}$$

(28)

In Equation (25), $\overrightarrow{r_{1\dots 5}}$ is a randomly generated value between 0 and 1, ${\varrho }_1$ and ${\varrho }_2$, are random integers which may have values of one or zero. In Equation (27), ${m\mathcal{g}}_i$ denotes the mean values of few randomly chosen hippos with an equal likelihood of containing the present measured hippos ($x_i$), $I_2$ is an integer between 1 and 2. This study describes the location of female or immature hippopotamus in a herd using Equations (27) and (28). Young hippopotamuses may wander apart from their mothers out of curiosity. An immature hippopotamus separates from its mother if $T$ is more than 0.6. If $r_6$, a number in the range [0, 1], is larger than 0.5, the young hippopotamus has parted from its mother but is still in or close to the herd. If not, it has split apart entirely. The variables $h_1$ and $h_2$ are chosen randomly from five possible outcomes. The objective function is $\mathcal{F}$. The position update of immature or male and female hippos within the herd is provided by Equations (29) and (30).

$$x_i=\left\{\begin{array}{c}{x}_i^{\mathcal{M}hippo}{\mathcal{F}}_i^{\mathcal{M}hippo} <{\mathcal{F}}_i\\ x_i else\end{array}\right.$$

(29)

The suggested algorithm’s exploration and global search are both improved by the $I_1$ and $I_2$ scenarios’ use of $h$ vectors. It improves the suggested algorithm’s exploration process and results in a better global search.

$$x_i=\left\{\begin{array}{c}{x}_i^{\mathcal{F}\mathcal{B}hippo}{\mathcal{F}}_i^{\mathcal{M}hippo} <{\mathcal{F}}_i\\ x_i else\end{array}\right.$$

(30)

3.3. FC Maps

Randomness is a key component of meta-heuristic optimization techniques, with Gaussian and uniform distributions being the most popular options. However, because of their dynamic features and pseudo-randomness, chaos maps have drawn a lot of interest [74,75,76]. These maps allow meta-heuristic algorithms to successfully escape local optima by modifying their behavior to boost inclinations towards exploration and exploitation. FC maps, which provide new dynamic distributions, have been presented by researchers. These maps are intended to enhance certain characteristics, especially in high-dimensional troubleshooting, such accuracy, speed of convergence, and consistency. The selected chaotic map in the article is the fractional sine map, which takes the place of conventional distributions. New versions of FC maps have been developed because of this advancement. This map’s mathematical formulation is given in [77] and is shown below:

$$x_{k+1}=x_0+{\displaystyle \frac{\mu }{\Gamma \left(\alpha \right)}}{\sum }_{j=1}^k{\displaystyle \frac{\Gamma (k-j+\alpha )}{\Gamma (k-j+1)}}sin\left(x_{j-1}\right)$$

(31)

where $\mu =3.8$ and $\alpha =0.8$. We assume $0.3$ for the starting value $x\left(0\right)$.

By ensuring that the initial brown-bear distribution is homogeneously dispersed over the search space, the conventional random initialization generates uncertain solution quality. This outcome is the result of certain groups of brown bears being far from the global optimum. To solve the issue, the initial search space is chosen at random and may be modified in conventional BOA using FC maps. Grounded in the ergodic nature and non-repetitiveness of the fractional chaotic employment, the EBOA can perform higher-degree overall searches than stochastic searches. The method process is that the FC maps completely replace brown-bear positions within the population, therefore skillfully augmenting the search space. The fundamental benefit of the FC maps lies in the algorithm it drives, which encompasses a seamless transition between exploration and exploitation. Similarly, in contrast to the standard optimizer exploration with self-constructed distributions, the authors employed FC maps in this work to restore randomly chosen starting values that carry out the systematic redistribution of the search space. To avoid becoming stuck in local optima, BOA and FC maps are mixed during calculation. The parameter of Equation (32) in the range of [0, 1] in the typical BOA is a random integer. It has been chosen to represent fractional chaotic numbers symbolized by (FCM) between 0 and 1 in the improved BOA, as shown by Equation (32):

$$P_{i,j}=P_{i,j}^{min}+FCM.(P_{i,j}^{max}-P_{i,j}^{min})$$

(32)

4. Results and Discussion on Benchmark Functions Evaluation

This part involves a thorough performance analysis of our recent EBOA using the multi-modal and unimodal benchmark functions. This includes a wide range of multiple unique benchmark tasks, as indicated in

Table 2. Unimodal benchmark functions and their characteristics.

Function	Formula	Dimensions	Interval
F1	$f(x)={\displaystyle {\sum }_{i=1}^n{x}_i^2}$	30	[100, 100]
F2	$f(x)={\displaystyle {\sum }_{i=1}^n\left|x_i\right|}+{\displaystyle {\prod }_{i=1}^n\left|x_i\right|}$	30	[10, 10]
F3	$f(x)={\displaystyle {\sum }_{i=1}^n{\left({\displaystyle {\sum }_{j-1}^i{x}_j}\right)}^2}$	30	[100, 100]
F4	$f(x)={\max }_i\left\{\left|x_i\right|,\hspace{0.17em}1\le i\le n\right\}$	30	[100, 100]
F5	$f(x)={\displaystyle {\sum }_{i=1}^{n-1}\left[100{\left(x_{i+1}-x_i^2\right)}^2+{(x_i-1)}^2\right]}$	30	[30, 30]
F6	$f(x)={\displaystyle {\sum }_{i=1}^n{\left(\left[x_i+0.5\right]\right)}^2}$	30	[100, 100]
F7	$f(x)={\displaystyle {\sum }_{i=1}^{n}i{x}_i^4+random(0,1)}$	30	[1.28, 1.28]

and

Table 3. Multimodal benchmark functions and their characteristics.

Function	Formula	Dimensions	Interval
F8	$f(x)={\displaystyle \sum _{i=1}^n}-x_i\mathit{sin}\left(\sqrt{\left|x_i\right|}\right)$	30	[−500, 500]
F9	$f(x)={\displaystyle \sum _{i=1}^n}\left[x_i^2-10\mathit{cos}\left(2\pi x_i\right)+10\right]$	30	[−5.12, 5.12]
F10	$f(x)=-20exp(-0.2\sqrt{{\displaystyle \frac{1}{n}}{\displaystyle \sum _{i=0}^n}{x}_i^2)}-exp({\displaystyle \frac{1}{n}}{\displaystyle \sum _{i=1}^n}\mathit{cos}(2\pi x_i))+20+e$	30	[−30, 30]
F11	$f(x)={\displaystyle \frac{1}{4000}}{\displaystyle \sum _{i=1}^n}{x}_i^2-{\prod }_{i=1}^n\mathit{cos}({\displaystyle \frac{x_i}{\sqrt{i}}})+1$	30	[−600, 600]
F12	$f(x)={\displaystyle \frac{\pi }{n}}\left\{10\mathit{sin}(\pi y_1)+{\displaystyle \sum _{i=1}^{n-1}}{(y_i-1)}^2[1+10{sin}^2(\pi y_{i+1})]{(y_n-1)}^2\right\}+{\displaystyle \sum _{i=1}^n}u\left(x_i,10,100,4\right), where y_i=1+{\displaystyle \frac{x_i+1}{4}}$$\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} u(x_i,a,k,m)=\left\{\begin{array}{c}k{(x_i-a)}^m{x}_i>a\\ 0 -a<x_i<a\\ k{({-x}_i-a)}^m{x}_i<-a\end{array}\right.$	30	[−50, 50]
F13	$f(x)=0.1{sin}^2(3\pi x_1)+{\displaystyle \sum _{i=1}^n}{(x_i-1)}^2[1+{sin}^2(3\pi x_i+1\left)\right]+{(x_n-1)}^2[1+{sin}^2(2\pi x_n)]+{\displaystyle \sum _{i=1}^n}u(x_i,5,100,4)$	30	[−50, 50]

. They exhibit a wide range of features and complexities, making it possible to perform a thorough evaluation of the efficacy obtained from the incorporation of DHL search method and fractional chaos maps into the initial BOA. Between this wide range of benchmark functions, two main classes have been isolated: functions F1 through F7, which exhibit unimodal comportment, and functions F8 through F13, which are known for the complexity arising from their multimodal features. Every experiment in this assessment is designed using 30 individuals over 500 generations in a 30-dimensional space. In our evaluation procedure, we conduct 30 separate runs to further improve the robustness and dependability of our results. We used the analytical power of the fitness error’s mean and standard deviation to precisely assess the algorithm’s performance.

Unimodal functions evaluate an optimization algorithm’s convergence speed and precision. Benchmark functions are used to assess the algorithm’s basic search capability, as they have a single global optimum. Whereas multi-modal functions are complex and have multiple local minima, posing challenges for optimization algorithms. These functions, like the highly non-linear functions, are ideal for evaluating the robustness and exploration capabilities of the EBOA.

To provide a fair and comprehensive validation of the EBOA’s performance, we decided to use a combination of unimodal and multi-modal benchmark functions. Multi-modal functions examine the algorithm’s capacity to strike a balance between exploration (finding a variety of solutions) and exploitation (fine-tuning solutions), while unimodal functions aid in assessing the algorithm’s convergence efficiency. This combination guarantees that the advantages and disadvantages of the EBOA are sufficiently represented for solar cells and photovoltaic module parameter estimation.

Figure 4 shows the distribution behavior of the considered chaotic maps against generations. The dynamic distribution of chaotic maps successfully escapes and climbs from local optima solutions, replacing the uniform and Gaussian distribution of traditional optimization techniques. By showing distinct distributions from their integer-order equivalents, fractional calculus improves the dynamical behavior of maps.

The results of this examination demonstrate a number of significant benefits that fractional-order chaotic maps have over integer-order chaotic maps. Initially, the incorporation of the fractional order allows for the formation of larger chaotic areas. Second, it makes sure that there are more chaotic, unpredictable sequences, which helps to improve stability and security. Lastly, fractional-order chaotic maps exhibit improved ergodicity and distribution properties, further distinguishing them from their integer-order counterparts.

We conducted a thorough comparison study, comparing the suggested method against a wide range of heuristics, to evaluate its efficacy. They comprise both traditional algorithms like PSO and newer, well-known ones like BOA, SCA, WOA, BA, and STOA. For each method that was examined, performance indicators such as worst (worst), best (best), standard deviation (std), and average (average) were calculated.

Table 4. Optimization analysis of proposed optimizers.

		Alternative Optimizers
Fun No.	Measure	EBOA	BOA	PSO	BA	SCA	WOA	STOA
F1	Worst	34,346 × 10−273	10,052 × 10−266	19,834 × 10−2	17,959 × 104	15,341 × 102	24,018 × 10−70	13,549 × 10−6
	Best	52,067 × 10−293	12,440 × 10−282	63,112 × 10−4	59,745 × 103	54,241 × 10−2	28,197 × 10−84	36,303 × 10−9
	Std	00,000 × 100	00,000 × 100	51,431 × 10−3	30,936 × 103	41,459 × 101	47,996 × 10−71	33,224 × 10−7
	Average	11,729 × 10−274	41,086 × 10−268	45,987 × 10−3	10,425 × 104	20,479 × 101	98,326 × 10−72	21,639 × 10−7
F2	Worst	24,748 × 10−138	19,809 × 10−134	20,208 × 101	40,200 × 103	66,422 × 10−2	72,149 × 10−50	32,774 × 10−5
	Best	65,314 × 10−153	97,044 × 10−145	18,009 × 10−3	27,217 × 101	18,852 × 10−4	26,103 × 10−58	58,972 × 10−7
	Std	63,245 × 10−139	44,975 × 10−135	44,282 × 100	82,025 × 102	16,957 × 10−2	15,461 × 10−50	89,905 × 10−6
	Average	21,896 × 10−139	15,969 × 10−135	12,307 × 100	25,682 × 102	13,443 × 10−2	59,194 × 10−51	93,523 × 10−6
F3	Worst	18,583 × 10−23	31,483 × 10−197	64,410 × 103	57,757 × 104	21,516 × 104	79,270 × 104	44,111 × 10−1
	Best	31,202 × 10−244	35,588 × 10−228	63,474 × 102	11,693 × 104	13,525 × 103	14,808 × 104	45,720 × 10−4
	Std	00,000 × 100	00,000 × 100	16,760 × 103	12,635 × 104	51,042 × 103	15,495 × 104	10,940 × 10−1
	Average	10,829 × 10−24	10,972 × 10−198	19,388 × 103	33,990 × 104	78,818 × 103	46,307 × 104	89,518 × 10−2
F4	Worst	27,954 × 10−127	22,129 × 10−126	91,165 × 100	52,999 × 101	70,165 × 101	84,214 × 101	63,421 × 10−2
	Best	89,258 × 10−138	40,058 × 10−137	35,114 × 100	31,939 × 101	60,155 × 100	43,429 × 10−1	83,971 × 10−3
	Std	52,243 × 10−128	47,362 × 10−127	13,663 × 100	65,284 × 100	12,567 × 101	29,925 × 101	16,015 × 10−2
	Average	14,384 × 10−128	12,243 × 10−127	71,187 × 100	40,989 × 101	35,716 × 101	49,933 × 101	31,625 × 10−2
F5	Worst	25,700 × 101	26,054 × 101	90,082 × 104	22,728 × 107	82,106 × 104	28,773 × 101	28,806 × 101
	Best	00,000 × 100	00,000 × 100	28,195 × 101	17,384 × 106	48,723 × 101	27,127 × 101	27,228 × 101
	Std	46,921 × 100	47,516 × 100	33,590 × 104	57,591 × 106	27,232 × 104	44,863 × 10−1	48,681 × 10−1
	Average	85,666 × 10−1	94,793 × 10−1	14,665 × 104	87,099 × 106	20,712 × 104	27,932 × 101	28,142 × 101
F6	Worst	00,000 × 100	00,000 × 100	38,476 × 10−2	20,137 × 104	43,454 × 101	86,177 × 10−1	39,225 × 100
	Best	00,000 × 100	00,000 × 100	57,243 × 10−4	62,942 × 103	48,363 × 100	95,794 × 10−2	14,988 × 100
	Std	00,000 × 100	00,000 × 100	10,007 × 10−2	34,385 × 103	12,446 × 101	22,783 × 10−1	61,282 × 10−1
	Average	00,000 × 100	00,000 × 100	73,990 × 10−3	11,425 × 104	14,673 × 101	47,106 × 10−1	26,549 × 100
F7	Worst	60,190 × 10−4	60,391 × 10−4	66,283 × 10−2	87,852 × 100	15,955 × 100	13,495 × 10−2	15,884 × 10−2
	Best	34,587 × 10−6	43,499 × 10−5	32,056 × 10−2	12,009 × 100	31,976 × 10−3	13,618 × 10−4	91,766 × 10−4
	Std	16,747 × 10−4	17,083 × 10−4	10,008 × 10−2	18,354 × 100	31,816 × 10−1	36,706 × 10−3	44,573 × 10−3
	Average	20,755 × 10−4	24,228 × 10−4	45,341 × 10−2	38,237 × 100	14,348 × 10−1	35,971 × 10−3	71,961 × 10−3
F8	Worst	−12,569 × 104	−12,569 × 104	−72,577 × 103	−22,880 × 103	−31,586 × 103	−71,131 × 103	−46,269 × 103
	Best	−12,569 × 104	−12,569 × 104	−93,896 × 103	−56,689 × 103	−46,273 × 103	−12,566 × 104	−64,860 × 103
	Std	18,501 × 10−12	18,501 × 10−12	62,398 × 102	97,894 × 102	33,048 × 102	18,249 × 103	40,391 × 102
	Average	−12,569 × 104	−12,569 × 104	−84,657 × 103	−34,206 × 103	−38,027 × 103	−10,092 × 104	−52,231 × 103
F9	Worst	00,000 × 100	00,000 × 100	81,588 × 101	12,537 × 102	11,313 × 102	11,369 × 10−13	47,643 × 101
	Best	00,000 × 100	00,000 × 100	33,846 × 101	27,863 × 101	17,147 × 10−1	00,000 × 100	22,843 × 10−7
	Std	00,000 × 100	00,000 × 100	13,276 × 101	23,732 × 101	34,314 × 101	26,863 × 10−14	13,240 × 101
	Average	00,000 × 100	00,000 × 100	54,966 × 101	68,775 × 101	38,762 × 101	90,949 × 10−15	11,408 × 101
F10	Worst	88,818 × 10−16	88,818 × 10−16	23,166 × 100	16,111 × 101	20,334 × 101	79,936 × 10−15	19,963 × 101
	Best	88,818 × 10−16	88,818 × 10−16	30,331 × 10−3	11,996 × 101	68,122 × 10−2	88,818 × 10−16	19,958 × 101
	Std	00,000 × 100	00,000 × 100	81,994 × 10−1	99,839 × 10−1	70,099 × 100	21,707 × 10−15	12,397 × 10−3
	Average	88,818 × 10−16	88,818 × 10−16	54,677 × 10−1	14,538 × 101	16,572 × 101	42,988 × 10−15	19,960 × 101
F11	Worst	00,000 × 100	00,000 × 100	88,791 × 10−2	17,538 × 102	32,011 × 100	19,550 × 10−1	98,008 × 10−2
	Best	00,000 × 100	00,000 × 100	10,016 × 10−2	60,727 × 101	20,689 × 10−1	00,000 × 100	10,233 × 10−8
	Std	00,000 × 100	00,000 × 100	21,190 × 10−2	27,626 × 101	54,354 × 10−1	39,101 × 10−2	29,995 × 10−2
	Average	00,000 × 100	00,000 × 100	34,690 × 10−2	11,058 × 102	97,475 × 10−1	78,201 × 10−3	23,729 × 10−2
F12	Worst	15,705 × 10−32	97,443 × 10−31	92,698 × 10−1	14,889 × 107	48,595 × 103	83,802 × 10−2	78,377 × 10−1
	Best	15,705 × 10−32	15,705 × 10−32	60,281 × 10−6	68,350 × 103	67,866 × 10−1	46,345 × 10−3	48,884 × 10−2
	Std	55,674 × 10−48	18,165 × 10−31	21,713 × 10−1	39,597 × 106	97,714 × 102	24,888 × 10−2	13,895 × 10−1
	Average	15,705 × 10−32	73,317 × 10−32	12,499 × 10−1	31,634 × 106	24,296 × 102	26,900 × 10−2	25,525 × 10−1
F13	Worst	13,498 × 10−32	34,844 × 10−30	34,537 × 10−1	53,121 × 107	15,841 × 106	13,944 × 100	24,467 × 100
	Best	13,498 × 10−32	13,498 × 10−32	12,213 × 10−3	10,410 × 106	26,987 × 100	20,232 × 10−1	14,266 × 100
	Std	55,674 × 10−48	64,213 × 10−31	95,813 × 10−2	11,521 × 107	33,066 × 105	29,866 × 10−1	24,581 × 10−1
	Average	13,498 × 10−32	16,819 × 10−31	82,392 × 10−2	15,490 × 107	10,981 × 105	65,070 × 10−1	19,136 × 100

presents a complete overview of the experimental outcomes, including the best solutions, with strong accents highlighting the best results

The algorithm ability to effectively discovery and utilize the best solution may be assessed with the use of unimodal test functions. However, multimodal functions, which include several local minima, provide a reliable measure of the algorithm’s exploration and navigational skills away from local optima. The EBOA method regularly produces very competitive results, as seen by a careful analysis of Table 4’s findings, which show especially notable gains on functions F1–F4, F12, and F13. Due in large part to its strong DHL approach, this evaluation indicates that the EBOA performs very well when examining the optimal solution zone. The other algorithms, namely PSO and BA, on the other hand, are less successful in carrying out an exhaustive and complete search. Further evidence of EBOA’s potent exploratory ability comes from the table, which displays its excellent performance on multimodal functions. The previous research highlights that, in comparison to the normal BOA technique, The BOA framework’s use of fractional-chaos maps enhances exploration and effectively lowers the chance of local minima.

As was already noted, multi-modal and composite test functions are suitable for assessing the exploratory capabilities of an algorithm. Thus, applied to these issues, the EBOA’s ability to avoid local optima becomes apparent. Moreover, we may assess the algorithm’s capacity to accomplish a balance between exploitation and exploration using these functions. The results clearly show the responses that EBOA and other algorithms have obtained for composite functions, with EBOA routinely producing better results than its competitors. A careful look at Table 4’s data reveals that the EBOA often produces very competitive results, with function F8, F9, F10, F12, and F13 showing very notable enhancements. These findings highlight EBOA’s skill in maintaining a balanced state between exploration and exploitation, which facilitates the successful avoidance of local optima. Based on the previous discussion, it is shown that fractional chaos maps are essential for enhancing BOA’s exploratory and exploitative features. This is also mostly because the EBOA utilizes a strong movement that is based on places selected from a predetermined neighborhood radius using DHL technique, which effectively increases the exploitation capacity.

We evaluate and compare EBOA’s convergence trends with those of its rivals in the studies that follow. We provide the convergence profiles for a selection of multimodal and unimodal functions in Figure 5. Combined from 30 separate runs, these curves show the most effective solutions at each iteration. The provided graphs clearly show the progression of EBOA convergence, with significant fluctuations in the early iterations and decreasing variances in the latter phases. With an increase in the generations number, the additional algorithm components, such as the fractional chaotic maps and efficacy movement scheme, cooperate to improve outcomes by persistently adjusting their locations, which appears in best results. The decline in these curves signifies a cooperative endeavor within the population. Furthermore, it can be easily said that the research highlights the ability of fractional-chaos maps to significantly improve the overall effectiveness of the BOA, especially with regard to its exploitation potential. Moreover, improved convergence patterns are consistently shown by the EBOA across all curves. This means that, over the course of an iteration, EBOA outperforms its rival algorithms by successfully striking a harmonic balance between exploration and exploitation.

4.1. Results and Discussion on PV Cell/Modules

Using experimental data from the situation, the EBOA approach is validated. Three separate case studies using one PV cell and two PV modules were obtained under certain temperature and irradiation conditions as shown in

Table 5. Optimized parameters with boundaries limits.

Parameter	RTC France		PWP201		STP6-120/36
	SDM/DDM		SDM/DDM		SDM/DDM
	LB	UB	LB	UB	LB	UB
Iph (A)	0	1	0	2	0	8
I01, I02 (μA)	0	1	0	50	0	50
Rs (Ω)	0	0.5	0	2	0	0.36
Rsh (Ω)	0	100	0	2000	0	1500
a1, a2	1	2	1	50	1	2

. The experimental data used in the proposed cases came from earlier research investigations. In case study 1, The RTC France commercial silicon PV cell comes with a 57 mm diameter that operates at $33 °\mathrm{C}$ temperature and $1000 \mathrm{W}/{\mathrm{m}}^2$ of irradiance [78]. We examine the STP6-120/36, which has 36 series-connected cells, in case study 2 [79]. In this case study, 24 sets of current–voltage data are collected under certain circumstances, such as a temperature of $55 °\mathrm{C}$ and a sun irradiation of $1000 \mathrm{W}/{\mathrm{m}}^2$ is examined. The Photowatt-PWP201 PV module, which consists of 36 polycrystalline silicon cells connected in series and $45 °\mathrm{C}$ temperature and $1000 \mathrm{W}/{\mathrm{m}}^2$ of sun radiation as the testing conditions [80], is the subject of case study 3. This specific cell is used as a standard for comparing and evaluating algorithms since it has been extensively described in the literature.

It is noteworthy to highlight that the parameters dimension D, which is equivalent to 5 for SDM and 7 for DDM, represents the total number of identified parameters. For every algorithm under study, the maximum number of iterations and population size counts are fixed at 500 and 30, respectively. The PV module and cell limit boundaries are taken from previous research sources to guarantee fair comparisons. By defining their lower and upper boundaries, Table 5 offers a comprehensive view of the search limit for PV-cell- and -modules-identified parameters. The EBOA is used to calculate the difference between estimated and measured voltage and current values, using the objective function given in Equation (8), up to the maximum number of iterations. A variety of commercial PV modules are shown in numerical simulations below.

4.1.1. Simulation Results for RTC France PV Cell

Using experimental data of the RTC France PV cell, five and seven unknown variables for the SDM and DDM, respectively, were estimated at $1000 \mathrm{W}/{\mathrm{m}}^2$ of irradiance and $33 °\mathrm{C}$. The measured profile was captured using data from 26 sets of I-V measurements. Twenty separate runs of each model are used to generate this procedure. However, the focus of this research is on the objective function and optimized parameter values results that originate from the best favorable trial.

The RMSE fitness values and obtained parameter outcomes for SDM and DDM, separately, in the current study are shown in

Table 6. Optimized parameters and RMSE value results in recent literature for SDM.

PV Module/Cell	Optimizer	${\mathit{I}}_{\mathit{p}\mathit{h}}\left(\mathit{A}\right)$	$\mathit{a}$	${\mathit{I}}_0\left(\mathit{A}\right)$	${\mathit{R}}_{\mathit{s}}\left(\mathit{\Omega }\right)$	${\mathit{R}}_{\mathit{s}\mathit{h}}\left(\mathit{\Omega }\right)$	$\mathit{R}\mathit{M}\mathit{S}\mathit{E}\left(\mathit{A}\right)$
RTC France	EBOA	0.76074	1.46584	2.76774 × 10−7	0.03656	48.75049	8.18384 × 10−4
	HBA [81]	0.76077	1.48118	3.23020 × 10−7	3.63770 × 10−2	53.71851	9.86021 × 10−4
	FPA [82]	0.76077	1.48105	3.22617 × 10−7	3.63804 × 10−2	53.65319	9.86021 × 10−4
	MLBSA [27]	0.76077	1.48118	3.23020 × 10−7	3.63770 × 10−2	53.71852	9.86021 × 10−4
	ITLBO [83]	0.7608	1.4812	3.2310 × 10−7	3.6340 × 10−2	53.7187	9.86020 × 10−4
	DMTLBO [84]	0.7608	1.4812	3.2302 × 10−6	3.6340 × 10−2	53.7183	9.86020 × 10−4
	SEDE [85]	0.76077	1.48118	3.23020 × 10−7	3.63770 × 10−2	53.71852	9.86021 × 10−4
	DPDE [86]	0.76077	1.48118	3.23020 × 10−7	3.63770 × 10−2	53.71852	9.86021 × 10−4
	MIGTO [87]	0.76077	1.481183	3.23020 × 10−7	3.63770 × 10−2	53.71852	9.86021 × 10−4
	IJAYA [27]	0.76076	1.48137	3.23622 × 10−7	3.63708 × 10−2	53.83082	9.86021 × 10−4
	HDE [88]	0.76078	1.48118	3.23020 × 10−7	3.63770 × 10−2	53.71852	9.86021 × 10−4
STP6-120/36	EBOA	7.48156	1.2418	1.8694 × 10−6	0.16923	432.1108	1.430320 × 10−2
	HBA [81]	7.4603	1.2699	2.6278 × 10−6	0.1638	1500	1.67843 × 10−2
	FPA [82]	7.4724	1.2606	2.3491 × 10−6	0.1652	814.0852	1.66007 × 10−2
	MLBSA [27]	7.4715	1.2803	2.4673 × 10−6	0.1643	902.6820	1.66006 × 10−2
	ITLBO [83]	7.4725	1.2601	2.3350 × 10−6	0.1656	799.9164	1.66010 × 10−2
	DMTLBO [84]	7.4725	1.2601	2.3350 × 10−6	0.1656	799.9308	1.66010 × 10−2
	SEDE [85]	7.4725	1.2756	2.3349 × 10−6	0.1654	799.9160	1.66006 × 10−2
	DPDE [86]	7.4725	1.2601	2.3349 × 10−6	0.1654	799.9166	1.66006 × 10−2
	MIGTO [87]	7.4718	1.3004	3.7256 × 10−6	0.1573	297.3660	1.66006 × 10−2
	IJAYA [27]	7.4672	1.2753	2.2536 × 10−6	0.1654	771.8252	1.66013 × 10−2
	HDE [88]	7.4725	1.2601	2.3349 × 10−6	0.1654	799.9160	1.66006 × 10−2
PWP201	EBOA	1.03322	46.15385	1.7588 × 10−6	1.27924	634.95259	2.22008 × 10−3
	HBA [81]	1.03051	48.64283	3.48226 × 10−6	1.20127	981.98275	2.42507 × 10−3
	FPA [82]	1.03048	48.63156	3.47234 × 10−6	1.20163	985.04957	2.42520 × 10−3
	MLBSA [27]	1.03051	48.64283	3.48226 × 10−6	1.20127	981.98222	2.42507 × 10−3
	ITLBO [83]	1.03050	48.64330	3.48270 × 10−6	1.20130	982.40380	2.42510 × 10−3
	DMTLBO [84]	1.03050	48.64280	3.48230 × 10−6	1.20130	981.98220	2.42510 × 10−3
	SEDE [85]	1.03051	48.64283	3.48226 × 10−6	1.20127	981.98246	2.42507 × 10−3
	DPDE [86]	1.03051	48.64283	3.48220 × 10−6	1.20271	981.98227	2.42507 × 10−3
	MIGTO [87]	1.02462	56.55348	20.3362 × 10−6	0.02676	1323.48643	2.42507 × 10−3
	IJAYA [27]	1.03050	48.64631	3.48544 × 10−6	1.20120	983.27900	2.42507 × 10−5
	HDE [88]	1.03051	48.64280	3.48226 × 10−6	1.20127	981.98224	2.42507 × 10−3

and

Table 7. Optimized parameters and RMSE value results in recent literature for DDM.

PV Module/Cell	Optimizer	${\mathit{I}}_{\mathit{p}\mathit{h}}\left(\mathit{A}\right)$	${\mathit{a}}_1$	${\mathit{I}}_{01}\left(\mathit{A}\right)$	${\mathit{a}}_2$	${\mathit{I}}_{02}\left(\mathit{A}\right)$	${\mathit{R}}_{\mathit{s}}\left(\mathit{\Omega }\right)$	${\mathit{R}}_{\mathit{s}\mathit{h}}\left(\mathit{\Omega }\right)$	$\mathit{R}\mathit{M}\mathit{S}\mathit{E}\left(\mathit{A}\right)$
RTC France	EBOA	0.76080	1.40680	1.25660 × 10−7	1.81563	7.51505 × 10−7	0.03730	55.05508	7.478488 × 10−4
	HBA [81]	0.76078	1.4500	2.2345 × 10−7	1.9999	7.7121 × 10−7	3.6751 × 10−2	55.5443	9.82488 × 10−4
	MLSBA [27]	0.76080	1.4703	2.8411 × 10−7	1.9824	2.7277 × 10−7	3.6486 × 10−2	54.2657	9.84184 × 10−4
	FPA [82]	0.76073	1.9917	1.0000 × 10−6	1.4442	2.0632 × 10−7	3.6710 × 10−2	58.2248	9.90582 × 10−4
	SEDE [85]	0.76077	1.9999	2.7410 × 10−7	1.4703	2.8473 × 10−7	3.6510 × 10−2	54.2881	9.83996 × 10−4
	MIGTO [87]	0.76078	1.4510	2.2597 × 10−7	1.9999	7.4934 × 10−7	3.6740 × 10−2	55.4854	9.82484 × 10−4
	DPDE [86]	0.76078	1.4510	2.2596 × 10−7	2.0000	7.4949 × 10−7	3.6740 × 10−2	55.4869	9.82484 × 10−4
	HDE [88]	0.76078	2.000	7.4934 × 10−7	1.451	2.2597 × 10−7	3.6740 × 10−2	55.4854	9.82484 × 10−4
	IJAYA [27]	0.76010	1.2185	5.0445 × 10−9	1.6247	7.5094 × 10−7	3.7600 × 10−2	77.8519	9.82930 × 10−3
STP6-120/36	EBOA	7.47353	44.55638	1.58944 × 10−6	47.20850	4.42635 × 10−7	0.16883	635.78665	1.427010 × 10−2
	SEDE [85]	7.4765	1.2754	2.4125 × 10−6	1.0607	1.9140 × 10−8	0.1688	582.4523	1.68465 × 10−2
	MLSBA [27]	7.4768	1.2584	2.2461 × 10−6	1.3008	6.8850 × 10−8	0.1654	633.7942	1.66258 × 10−2
	DPDE [86]	7.4725	1.8827	2.1127 × 10−18	1.2601	2.3349 × 10−6	0.1654	799.9163	1.66006 × 10−2
	IJAYA [27]	7.4759	1.2650	2.4601 × 10−6	1.9990	3.6638 × 10−6	0.1635	837.2170	1.69522 × 10−2
PWP201	EBOA	1.03143	46.21139	5.60179 × 10−8	47.71761	2.63904 × 10−6	1.23296	822.95825	2.061273 × 10−3
	HBA [81]	1.0305	48.6388	3.1176 × 10−7	48.6293	3.1590 × 10−6	1.20164	981.0549	2.42510 × 10−3
	MLSBA [27]	1.0304	48.5828	7.6463 × 10−8	48.6496	3.4107 × 10−6	1.20118	987.2186	2.425099 × 10−3
	FPA [82]	1.0292	50	9.6636 × 10−7	48.4082	2.6346 × 10−6	1.20165	1161.7881	2.45263 × 10−3
	SEDE [85]	1.0305	48.6436	1.8570 × 10−6	48.6351	1.6223 × 10−6	1.20134	980.8042	2.425076 × 10−3
	DPDE [86]	1.0305	48.6428	1.7389 × 10−6	48.6428	1.7433 × 10−6	1.20127	981.9822	2.425074 × 10−3
	IJAYA [27]	1.0306	47.5070	1.7940 × 10−8	48.4311	3.2690 × 10−6	1.20676	925.9600	2.44562 × 10−3

. The statistical results for RMSE are shown in

Table 8. Results of designed optimization algorithms for SDM.

		Employed Algorithms
PV Module/Cell	Measure	EBOA	BOA	PSO	BA	SCA	STOA	WOA
RTC France	Std	0.000466	0.006503	0.035204	0.018983	0.013285	0.010446	0.014241
	Average	0.001209	0.005391	0.051791	0.026258	0.033528	0.005799	0.013858
	Worst	0.002090	0.024013	0.119509	0.050426	0.050795	0.037994	0.048406
	Best	0.000818	0.000986	0.001409	0.003521	0.015852	0.000988	0.001209
STP6-120/36	Std	0.008785	0.048210	0.589954	0.540366	0.077886	0.048703	0.007593
	Average	0.018352	0.037107	0.459092	0.383753	0.107564	0.035143	0.025806
	Worst	0.052952	0.214284	1.819860	1.413122	0.319543	0.236849	0.044955
	Best	0.014303	0.014427	0.016869	0.027154	0.031466	0.014559	0.015399
PWP201	Std	0.003124	0.006152	0.123922	0.114793	0.061612	0.068141	0.028330
	Average	0.004067	0.008293	0.065658	0.087338	0.045335	0.033881	0.014573
	Worst	0.016220	0.027346	0.497088	0.275789	0.275792	0.225691	0.131506
	Best	0.002220	0.002477	0.002328	0.003095	0.006938	0.002472	0.002512

and

Table 9. Results of suggested algorithms for DDM.

PV Module/Cell	Measure	Comparative Algorithms
		EBOA	BOA	PSO	BA	SCA	STOA	WOA
RTC France	Std	0.000260	0.009401	0.282503	0.036805	0.030387	0.269096	0.011963
	Average	0.000942	0.004622	0.383483	0.032667	0.035264	0.189322	0.011971
	Worst	0.001623	0.040751	1.068761	0.098770	0.097183	0.734283	0.039495
	Best	0.000748	0.000854	0.008265	0.002921	0.008545	0.002207	0.002505
STP6-120/36	Std	0.000918	0.116976	1.854783	0.104066	0.079437	0.998330	0.083037
	Average	0.014652	0.089204	3.155799	0.101195	0.113995	0.390756	0.057722
	Worst	0.018421	0.383094	7.608302	0.394694	0.338034	3.403038	0.394565
	Best	0.014270	0.014427	0.045460	0.029041	0.041901	0.016217	0.014514
PWP201	Std	0.001520	0.001662	0.282503	0.036805	0.030387	0.269096	0.011963
	Average	0.002985	0.004620	0.383483	0.032667	0.035264	0.189322	0.011971
	Worst	0.007236	0.008145	1.068761	0.098770	0.097183	0.734283	0.039495
	Best	0.002061	0.002133	0.008265	0.002921	0.008545	0.002207	0.002505

. They include the mean, maximum (worst), and lowest (best) values. Figure 6 shows the performance-ranking chart of the RTC France PV cell for DDM. These findings are the result of 20 different runs of the EBOA and other methods. Remarkably, compared to the other approaches, the suggested approach reliably yields reduced best, mean, and worst RMSE values, as seen in the table. Additionally,

Table 10. Optimized parameters and RMSE value results obtained for SDM.

		Identified Parameters
PV Module/Cell	Optimizer	${\mathit{I}}_{\mathit{p}\mathit{h}}\left(\mathit{A}\right)$	$\mathit{a}$	${\mathit{I}}_0\left(\mathit{A}\right)$	${\mathit{R}}_{\mathit{s}}(\mathit{\Omega })$	${\mathit{R}}_{\mathit{s}\mathit{h}}(\mathit{\Omega })$	$\mathit{R}\mathit{M}\mathit{S}\mathit{E}\left(\mathit{A}\right)$
RTC France	EBOA	0.76074	1.46584	0.27677	0.03656	48.75049	8.18384 × 10−4
	BOA	0.76122	1.51146	0.43339	0.03511	56.23840	9.855673 × 10−4
	PSO	0.76108	1.51794	0.45818	0.03457	49.93541	1.408906 × 10−3
	BA	0.76021	1.50952	0.42635	0.03534	71.14412	9.881127 × 10−4
	SCA	0.76727	1.00000	0.00030	0.06783	29.84489	1.585196 × 10−2
	STOA	0.75797	1.47589	0.31153	0.03700	89.32690	3.520927 × 10−3
	WOA	0.75994	1.52588	0.49769	0.03482	85.76491	1.208675 × 10−3
STP6-120/36	EBOA	7.48156	44.70655	1.86937	0.16923	432.11083	1.430320 × 10−2
	BOA	7.46384	45.00911	2.07425	0.16809	1476.52501	1.442695 × 10−2
	PSO	7.45750	43.13704	1.07257	0.17991	859.03909	1.686895 × 10−2
	BA	7.47148	45.58258	2.51101	0.16422	991.97891	1.455934 × 10−2
	SCA	7.52296	50.00000	9.49796	0.13764	1156.23808	3.146588 × 10−2
	STOA	7.49039	50.00000	9.40477	0.13613	1451.76317	2.715378 × 10−2
	WOA	7.47311	46.28204	3.14966	0.15984	1264.51024	1.539900 × 10−2
PWP201	EBOA	1.03322	46.15385	1.75875	1.27924	634.95259	2.220075 × 10−3
	BOA	1.03521	45.24490	1.34505	1.30553	535.69807	2.476718 × 10−3
	PSO	1.03438	48.06259	2.97706	1.21093	646.53044	2.328492 × 10−3
	BA	1.02907	46.43618	1.91615	1.28118	970.19848	2.471905 × 10−3
	SCA	1.03250	49.90009	4.75089	1.23856	790.00686	6.937771 × 10−3
	STOA	1.03302	50.00000	4.90493	1.18077	994.67292	3.094647 × 10−3
	WOA	1.03243	49.54180	4.36880	1.16691	856.39004	2.511538 × 10−3

and

Table 11. Optimized parameters and RMSE value results obtained for DDM.

		Identified Parameters
PV Module/Cell	Optimizer	${\mathit{I}}_{\mathit{p}\mathit{h}}\left(\mathit{A}\right)$	${\mathit{a}}_1$	${\mathit{I}}_{01}\left(\mathit{A}\right)$	${\mathit{a}}_2$	${\mathit{I}}_{02}\left(\mathit{A}\right)$	${\mathit{R}}_{\mathit{s}}(\mathit{\Omega })$	${\mathit{R}}_{\mathit{s}\mathit{h}}(\mathit{\Omega })$	$\mathit{R}\mathit{M}\mathit{S}\mathit{E}\left(\mathit{A}\right)$
RTC France	EBOA	0.76080	1.40680	0.12566	1.81563	0.75151	0.03730	55.05508	7.478488 × 10−4
	BOA	0.76132	1.93167	0.99188	1.40295	0.12812	0.03793	46.86802	8.544780 × 10−4
	PSO	0.76040	1.45675	0.17265	1.60314	0.31346	0.03562	66.27633	9.073320 × 10−4
	BA	0.76066	1.89816	0.09667	1.49682	0.37219	0.03548	59.71745	8.648217 × 10−4
	SCA	0.74824	1.56751	0.60294	1.71896	0.37346	0.02377	68.52579	1.219726 × 10−2
	STOA	0.76630	1.71386	0.98691	1.37948	0.06311	0.03628	29.89405	4.074129 × 10−3
	WOA	0.75960	1.77440	0.31112	1.47839	0.28799	0.03556	76.61059	1.073366 × 10−3
STP6-120/36	EBOA	7.47353	44.55638	1.58944	47.20850	0.44263	0.16883	635.78665	1.427010 × 10−2
	BOA	7.46619	2.19263	0.00244	1116.41491	0.16899	45.31182	36.30835	1.442730 × 10−2
	PSO	7.50724	47.10182	3.61314	46.19926	0.23003	0.17416	1025.82920	4.545994 × 10−2
	BA	7.47602	38.00916	0.04711	50.00000	5.96205	0.16636	1499.96852	1.621662 × 10−2
	SCA	7.47319	47.96793	0.10131	43.01288	1.00572	0.17905	415.98699	4.190054 × 10−2
	STOA	7.49433	50.00000	7.10648	46.54970	0.78533	0.14783	1188.25439	2.904129 × 10−2
	WOA	7.47328	49.24992	0.46384	44.46723	1.62315	0.16832	627.30669	1.451427 × 10−2
PWP201	EBOA	1.03143	46.21139	0.05602	47.71761	2.63904	1.23296	822.95825	2.061273 × 10−3
	BOA	1.03284	46.68684	2.04592	49.41975	3.31756	1.26116	673.99531	2.132806 × 10−3
	PSO	1.03164	45.46348	1.41689	43.46383	0.00102	1.17539	415.76347	8.264875 × 10−3
	BA	1.03089	50.00000	0.00693	48.93449	3.74726	1.19242	959.83935	2.206877 × 10−3
	SCA	1.05162	50.00000	0.00545	50.00000	4.83721	1.13918	294.41053	8.545024 × 10−3
	STOA	1.03196	50.00000	4.87471	36.68460	0.00002	1.16122	895.55094	2.921392 × 10−3
	WOA	1.03524	42.81119	0.25117	49.93793	2.86465	1.23859	574.93359	2.504610 × 10−3

compare the RMSE values and estimated parameters from the optimal test for DDM and SDM, respectively. The values of the parameters are shown with several decimal places of precision due to the influence of parameter sensitivity on model correctness. Notably, when compared to other algorithms, the designed EBOA approach consistently yields the lowest RMSE. The planned EBOA routinely achieves the best RMSE values, as the data illustrate. For SDM and DDM, the magnitude order is in the range of 8.183847 × 10⁻⁴ A and 7.478488 × 10⁻⁴ A, respectively. Other algorithms’ fitness values are very similar to one another.

The evolution of the RMSE averages is shown in Figure 7, contrasting the suggested EBOA with other methods, to highlight the improved performance of the EBOA. The convergence curves demonstrate that the EBOA performs very well when it comes to convergence, particularly when evaluated against optimizers such as SCA, BA, and WOA. The findings collected offer compelling evidence that the EBOA can offer simulated data with great precision and accuracy, leading to the best alignment with experimental data. The current–voltage and power–voltage profiles of SDM and DDM shown, respectively, in Figure 8 and Figure 9, which illustrate a strong agreement between the simulated and experimental data, further supporting this alignment. The extra algorithmic components working together harmoniously is the main reason for this accomplishment. These elements, which include the fractional chaotic maps and the efficient search space, work together to provide better outcomes by iteratively shifting their places in quest of better solutions.

To calculate the simulated current and power data, the obtained values using the designed EBOA are integrated into the objective function (Regarding data are tabulated in

Table A1. Current and power data results of the RTC France solar cell (DDM).

	Experimental Data			Estimated Current Data			Estimated Power Data
	$\mathit{V}\left(\mathit{V}\right)$	$\mathit{I}\left(\mathit{A}\right)$	$\mathit{P}\left(\mathit{W}\right)$	${\mathit{I}}_{\mathit{e}}\left(\mathit{A}\right)$	$\mathit{A}\mathit{I}\mathit{E}\_\mathit{I}\left(\mathit{A}\right)$	$\mathit{R}\mathit{E}\_\mathit{I}\left(\mathit{A}\right)$	${\mathit{P}}_{\mathit{e}}\left(\mathit{W}\right)$	$\mathit{A}\mathit{I}\mathit{E}\_\mathit{P}\left(\mathit{W}\right)$	$\mathit{R}\mathit{E}\_\mathit{P}\left(\mathit{W}\right)$
1	−0.2057	0.7640	−0.1572	0.7640	1.9572 × 10−5	2.5618 × 10−5	−0.1572	4.0260 × 10−6	−2.5618 × 10−5
2	−0.1291	0.7620	−0.0984	0.7626	6.2910 × 10−4	8.2559 × 10−4	−0.0985	8.1217 × 10−5	−8.2559 × 10−4
3	−0.0588	0.7605	−0.0447	0.7614	8.5271 × 10−4	1.1212 × 10−3	−0.0448	5.0139 × 10−5	−1.1212 × 10−3
4	0.0057	0.7605	0.0043	0.7602	3.1944 × 10−4	4.2004 × 10−4	0.0043	1.8208 × 10−6	4.2004 × 10−4
5	0.0646	0.7600	0.0491	0.7591	8.9347 × 10−4	1.1756 × 10−3	0.0490	5.7718 × 10−5	1.1756 × 10−3
6	0.1185	0.7590	0.0899	0.7581	8.8771 × 10−4	1.1696 × 10−3	0.0898	1.0519 × 10−4	1.1696 × 10−3
7	0.1678	0.7570	0.1270	0.7572	1.7037 × 10−4	2.2505 × 10−4	0.1271	2.8587 × 10−5	2.2505 × 10−4
8	0.2132	0.7570	0.1614	0.7562	7.8389 × 10−4	1.0355 × 10−3	0.1612	1.6713 × 10−4	1.0355 × 10−3
9	0.2545	0.7555	0.1923	0.7551	3.6315 × 10−4	4.8068 × 10−4	0.1922	9.2423 × 10−5	4.8068 × 10−4
10	0.2924	0.7540	0.2205	0.7537	3.3344 × 10−4	4.4223 × 10−4	0.2204	9.7498 × 10−5	4.4223 × 10−4
11	0.3269	0.7505	0.2453	0.7513	8.2460 × 10−4	1.0987 × 10−3	0.2456	2.6956 × 10−4	1.0987 × 10−3
12	0.3585	0.7465	0.2676	0.7472	7.1706 × 10−4	9.6056 × 10−4	0.2679	2.5707 × 10−4	9.6056 × 10−4
13	0.3873	0.7385	0.2860	0.7399	1.4235 × 10−3	1.9275 × 10−3	0.2866	5.5132 × 10−4	1.9275 × 10−3
14	0.4137	0.7280	0.3012	0.7272	7.6294 × 10−4	1.0480 × 10−3	0.3009	3.1563 × 10−4	1.0480 × 10−3
15	0.4373	0.7065	0.3090	0.7069	3.7472 × 10−4	5.3040 × 10−4	0.3091	1.6387 × 10−4	5.3040 × 10−4
16	0.4590	0.6755	0.3101	0.6753	1.5431 × 10−4	2.2844 × 10−4	0.3100	7.0829 × 10−5	2.2844 × 10−4
17	0.4784	0.6320	0.3023	0.6311	9.4396 × 10−4	1.4936 × 10−3	0.3019	4.5159 × 10−4	1.4936 × 10−3
18	0.4960	0.5730	0.2842	0.5723	6.8964 × 10−4	1.2036 × 10−3	0.2839	3.4206 × 10−4	1.2036 × 10−3
19	0.5119	0.4990	0.2554	0.4997	6.8432 × 10−4	1.3714 × 10−3	0.2558	3.5030 × 10−4	1.3714 × 10−3
20	0.5265	0.4130	0.2174	0.4136	5.7303 × 10−4	1.3875 × 10−3	0.2177	3.0170 × 10−4	1.3875 × 10−3
21	0.5398	0.3165	0.1708	0.3172	6.5643 × 10−4	2.0740 × 10−3	0.1712	3.5434 × 10−4	2.0740 × 10−3
22	0.5521	0.2120	0.1170	0.2119	7.6362 × 10−5	3.6020 × 10−4	0.1170	4.2159 × 10−5	3.6020 × 10−4
23	0.5633	0.1035	0.0583	0.1025	9.9889 × 10−4	9.6512 × 10−3	0.0577	5.6268 × 10−4	9.6512 × 10−3
24	0.5736	−0.0100	−0.0057	−0.0094	5.9296 × 10−4	−5.9296 × 10−2	−0.0054	3.4012 × 10−4	−5.9296 × 10−2
25	0.5833	−0.1230	−0.0717	−0.1244	1.3577 × 10−3	−1.1038 × 10−2	−0.0725	7.9195 × 10−4	−1.1038 × 10−2
26	0.5900	−0.2100	−0.1239	−0.2090	1.0409 × 10−3	−4.9566 × 10−3	−0.1233	6.1412 × 10−4	−4.9566 × 10−3
$\sum AIE \left(A\right)$					1.7124 × 10−2			6.4650 × 10−3
$MAE \left(A\right)$					6.8215 × 10−4

in the Appendix A). In the RTC France cell, for example, the AIE metrics range from 1.9572 × 10⁻⁵ A to 1.4235 × 10⁻³ A, yet the corresponding $MAE$ is still quite low at 6.8215 × 10⁻⁴ A.

4.1.2. Simulation Results for STP6-120/36

Experimental data from the STP6-120/36, consisting of 36 cells connected in series and 24 I-V data points measured at 1000 $\mathrm{W}/{\mathrm{m}}^2$ at $55 °\mathrm{C}$, were used to determine seven unknown DDM parameters and five undefined SDM parameters. This particular case is comparable to the one we previously discussed. The aggregate results of 20 separate runs for both models form the basis of this process. Nonetheless, the majority of the inquiry is concentrated on the fitness function outcomes and parameter values derived from the most successful test.

Table 10 and Table 11 present a comprehensive collection of numerical outcomes pertaining to the objective function, including maximum, mean, and minimum values. These findings are an aggregate of twenty independent trials conducted using various algorithms and the recommended EBOA technique. Surprisingly, the data demonstrate that the designed procedure reliably yields minimum RMSE values in the best, worst, and medium classes in contrast to earlier approaches. A comprehensive comparison of the optimized parameters and RMSE values from the optimal test is presented in Table 10 and Table 11, which correspond to the SDM and DDM, respectively. It is important to note that the EBOA approach developed in this work often excels over competing algorithms by obtaining the reduced RMSE values. The particularly developed EBOA consistently assures the best RMSE values, which, for the STP6-120/36, are usually in the region of 1.430320 × 10⁻² A for SDM and 1.427010 × 10⁻² A for DDM, as the Tables demonstrate. Other methods’ fitness function values, however, exhibit comparable convergence.

As shown in Figure 10, the EBOA strategy outperforms other methods like BOA, WOA, and PSO in convergence efficiency. Also, it produces precise simulated data, as shown in Figure 11 and Figure 12, which aligns perfectly with the experimental data. Figure 13 shows the performance-ranking chart of the STP6-120/36 for DDM. The strategy minimizes local minimums and facilitates exploration, thanks to the incorporation of fractional chaotic maps and an effective search space structure in BOA procedures. The fitness function that was previously mentioned is combined with the parameters that were obtained using the recommended EBOA approach to calculate the simulated current and power values (the data are tabulated in

Table A2. Current and power data results of the STP6-120/36 PV module (DDM).

	Experimental Data			Estimated Current Data			Estimated Power Data
	$\mathit{V}\left(\mathit{V}\right)$	$\mathit{I}\left(\mathit{A}\right)$	$\mathit{P}\left(\mathit{W}\right)$	${\mathit{I}}_{\mathit{e}}\left(\mathit{A}\right)$	$\mathit{A}\mathit{I}\mathit{E}\_\mathit{I}\left(\mathit{A}\right)$	$\mathit{R}\mathit{E}\_\mathit{I}\left(\mathit{A}\right)$	${\mathit{P}}_{\mathit{e}}\left(\mathit{W}\right)$	$\mathit{A}\mathit{I}\mathit{E}\_\mathit{P}\left(\mathit{W}\right)$	$\mathit{R}\mathit{E}\_\mathit{P}\left(\mathit{W}\right)$
1	19.2100	0.0000	0.0000	0.0041	4.1211 × 10−3	1.0000	0.0792	7.9166 × 10−2	1.0000
2	17.6500	3.8300	67.5995	3.8285	1.5383 × 10−3	4.0164 × 10−4	67.5723	2.7151 × 10−2	4.0164 × 10−4
3	17.4100	4.2900	74.6889	4.2708	1.9165 × 10−2	4.4673 × 10−3	74.3552	3.3366 × 10−1	4.4673 × 10−3
4	17.2500	4.5600	78.6600	4.5438	1.6216 × 10−2	3.5561 × 10−3	78.3803	2.7972 × 10−1	3.5561 × 10−3
5	17.1000	4.7900	81.9090	4.7839	6.0722 × 10−3	1.2677 × 10−3	81.8052	1.0383 × 10−1	1.2677 × 10−3
6	16.9000	5.0700	85.6830	5.0809	1.0870 × 10−2	2.1440 × 10−3	85.8667	1.8370 × 10−1	2.1440 × 10−3
7	16.7600	5.2700	88.3252	5.2733	3.2870 × 10−3	6.2372 × 10−4	88.3803	5.5090 × 10−2	6.2372 × 10−4
8	16.3400	5.7500	93.9550	5.7780	2.7954 × 10−2	4.8615 × 10−3	94.4118	4.5677 × 10−1	4.8615 × 10−3
9	16.0800	6.0000	96.4800	6.0394	3.9432 × 10−2	6.5719 × 10−3	97.1141	6.3406 × 10−1	6.5719 × 10−3
10	15.7100	6.3600	99.9156	6.3514	8.5707 × 10−3	1.3476 × 10−3	99.7810	1.3465 × 10−1	1.3476 × 10−3
11	15.3900	6.5800	101.2662	6.5710	9.0357 × 10−3	1.3732 × 10−3	101.1271	1.3906 × 10−1	1.3732 × 10−3
12	14.9300	6.8300	101.9719	6.8179	1.2111 × 10−2	1.7733 × 10−3	101.7911	1.8082 × 10−1	1.7733 × 10−3
13	14.5800	6.9700	101.6226	6.9612	8.8201 × 10−3	1.2654 × 10−3	101.4940	1.2860 × 10−1	1.2654 × 10−3
14	14.1700	7.1000	100.6070	7.0903	9.6816 × 10−3	1.3636 × 10−3	100.4698	1.3719 × 10−1	1.3636 × 10−3
15	13.5900	7.2300	98.2557	7.2190	1.0997 × 10−2	1.5211 × 10−3	98.1062	1.4945 × 10−1	1.5211 × 10−3
16	13.1600	7.2900	95.9364	7.2847	5.3234 × 10−3	7.3023 × 10−4	95.8663	7.0055 × 10−2	7.3023 × 10−4
17	12.7400	7.3400	93.5116	7.3314	8.5863 × 10−3	1.1698 × 10−3	93.4022	1.0939 × 10−1	1.1698 × 10−3
18	12.3600	7.3700	91.0932	7.3627	7.2863 × 10−3	9.8865 × 10−4	91.0031	9.0059 × 10−2	9.8865 × 10−4
19	11.8100	7.3800	87.1578	7.3948	1.4765 × 10−2	2.0007 × 10−3	87.3322	1.7438 × 10−1	2.0007 × 10−3
20	11.1700	7.4100	82.7697	7.4187	8.7090 × 10−3	1.1753 × 10−3	82.8670	9.7279 × 10−2	1.1753 × 10−3
21	10.3200	7.4400	76.7808	7.4372	2.7852 × 10−3	3.7435 × 10−4	76.7521	2.8743 × 10−2	3.7435 × 10−4
22	9.7400	7.4200	72.2708	7.4448	2.4754 × 10−2	3.3361 × 10−3	72.5119	2.4110 × 10−1	3.3361 × 10−3
23	9.0600	7.4500	67.4970	7.4506	5.7674 × 10−4	7.7414 × 10−5	67.5022	5.2252 × 10−3	7.7414 × 10−5
24	0.0000	7.4800	0.0000	7.4715	8.4539 × 10−3	1.1302 × 10−3	0.0000	0.0000 × 10	0.0000 × 10
$\sum AIE \left(A\right)$					2.6911 × 10−1			3.8391 × 10
$MAE \left(A\right)$					1.1557 × 10−2

in the Appendix A). The extraordinary efficiency was achieved by combining the recommended processes with the original BOA at the same time, which also allowed for the successful discovery of the optimal parameter values. These important findings demonstrate the efficacy of the established global approach and its capacity to raise the accuracy and potency of optimization techniques, particularly in intricate models featuring nonlinear functions.

4.1.3. Simulation Results for Photowatt-PWP201 PV Module

In a manner comparable to the case study mentioned above, experimental data from the Photowatt-PWP201 PV module were used to retrieve the five undetermined parameters for the SDM and seven for the DDM (operating at $1000 \mathrm{W}/{\mathrm{m}}^2$ and $33 °\mathrm{C}$). The dataset contains twenty-six pairs of data points for I-V values. Although data from twenty different runs were considered for both models, this study only examines the parameter and objective function’s RMSE values from the best-performing trial.

The objective function’s mean, maximum, and lowest values are all included in the combined set of statistical data seen in Table 10 and Table 11. Figure 14 shows the Photowatt-PWP201 PV module’s performance-ranking chart for DDM. Notably, the evaluation continually yields reduced RMSE values in the worst, medium, and best categories in contrast to other approaches, indicating the persistent superiority of the suggested method. As shown in Table 10 and Table 11, the EBOA approach suggested in this research consistently performs better than other methods by attaining the lowest RMSE values. The designed EBOA consistently ensures the lowest RMSE values, which are usually in the region of 2.220075 × 10⁻³ A for SDM and 2.061273 × 10⁻³ A for DDM, as the data show. On the other hand, the values of the fitness functions of the different methods show close outcome to each other. The performance of the EBOA is demonstrated through a comparison of its convergence speed and accuracy with other algorithms, such as PSO, SCA, and STOA, as shown in Figure 15. The EBOA exhibits superior convergence speed and precision, generating highly accurate simulated results that closely match the experimental data. This strong consistency is evident in the I-V and P-V profiles illustrated in Figure 16 and Figure 17, where the estimated current and power values align closely with experimental results. The enhanced performance is attributed to the synergistic integration of FC maps and an effective search space scheme, which dynamically adjust positions during iterations to improve fitness. The results, supported by statistical data (Regarding data are tabulated in

Table A3. Current and power data results of the Photowatt-PWP201 PV module (DDM).

	Experimental Data			Estimated Current Data			Estimated Power Data
	$\mathit{V}\left(\mathit{V}\right)$	$\mathit{I}\left(\mathit{A}\right)$	$\mathit{P}\left(\mathit{W}\right)$	${\mathit{I}}_{\mathit{e}}\left(\mathit{A}\right)$	$\mathit{A}\mathit{I}\mathit{E}\_\mathit{I}\left(\mathit{A}\right)$	$\mathit{R}\mathit{E}\_\mathit{I}\left(\mathit{A}\right)$	${\mathit{P}}_{\mathit{e}}\left(\mathit{W}\right)$	$\mathit{A}\mathit{I}\mathit{E}\_\mathit{P}\left(\mathit{W}\right)$	$\mathit{R}\mathit{E}\_\mathit{P}\left(\mathit{W}\right)$
1	0.1248	1.0315	0.1287	1.0297	1.7717 × 10−3	1.7176 × 10−3	0.1285	2.2111 × 10−4	1.7176 × 10−3
2	1.8093	1.0300	1.8636	1.0277	2.3360 × 10−3	2.2680 × 10−3	1.8594	4.2265 × 10−3	2.2680 × 10−3
3	3.3511	1.0260	3.4382	1.0257	2.7028 × 10−4	2.6343 × 10−4	3.4373	9.0573 × 10−4	2.6343 × 10−4
4	4.7622	1.0220	4.8670	1.0238	1.8395 × 10−3	1.7999 × 10−3	4.8757	8.7602 × 10−3	1.7999 × 10−3
5	6.0538	1.0180	6.1628	1.0218	3.8186 × 10−3	3.7510 × 10−3	6.1859	2.3117 × 10−2	3.7510 × 10−3
6	7.2364	1.0155	7.3486	1.0193	3.8230 × 10−3	3.7646 × 10−3	7.3762	2.7665 × 10−2	3.7646 × 10−3
7	8.3189	1.0140	8.4354	1.0157	1.7229 × 10−3	1.6991 × 10−3	8.4497	1.4333 × 10−2	1.6991 × 10−3
8	9.3097	1.0100	9.4028	1.0100	4.8964 × 10−5	4.8479 × 10−5	9.4023	4.5584 × 10−4	4.8479 × 10−5
9	10.2163	1.0035	10.2521	1.0004	3.1421 × 10−3	3.1312 × 10−3	10.2200	3.2101 × 10−2	3.1312 × 10−3
10	11.0449	0.9880	10.9124	0.9847	3.3371 × 10−3	3.3777 × 10−3	10.8755	3.6858 × 10−2	3.3777 × 10−3
11	11.8018	0.9630	11.3651	0.9601	2.9176 × 10−3	3.0297 × 10−3	11.3307	3.4433 × 10−2	3.0297 × 10−3
12	12.4929	0.9255	11.5622	0.9238	1.7489 × 10−3	1.8897 × 10−3	11.5403	2.1849 × 10−2	1.8897 × 10−3
13	12.6490	0.9120	11.5359	0.9130	1.0498 × 10−3	1.1511 × 10−3	11.5492	1.3279 × 10−2	1.1511 × 10−3
14	13.1231	0.8725	11.4499	0.8734	9.4543 × 10−4	1.0836 × 10−3	11.4623	1.2407 × 10−2	1.0836 × 10−3
15	14.2221	0.7265	10.3324	0.7285	1.9887 × 10−3	2.7373 × 10−3	10.3606	2.8283 × 10−2	2.7373 × 10−3
16	14.6995	0.6345	9.3268	0.6366	2.1171 × 10−3	3.3366 × 10−3	9.3580	3.1120 × 10−2	3.3366 × 10−3
17	15.1346	0.5345	8.0894	0.5355	9.5355 × 10−4	1.7840 × 10−3	8.1039	1.4432 × 10−2	1.7840 × 10−3
18	15.5311	0.4275	6.6395	0.4283	7.5683 × 10−4	1.7704 × 10−3	6.6513	1.1754 × 10−2	1.7704 × 10−3
19	15.8929	0.3185	5.0619	0.3179	5.7513 × 10−4	1.8057 × 10−3	5.0527	9.1405 × 10−3	1.8057 × 10−3
20	16.2229	0.2085	3.3825	0.2071	1.4148 × 10−3	6.7858 × 10−3	3.3595	2.2953 × 10−2	6.7858 × 10−3
21	16.5241	0.1010	1.6689	0.0977	3.2897 × 10−3	3.2571 × 10−2	1.6146	5.4359 × 10−2	3.2571 × 10−2
22	16.7987	−0.0080	−0.1344	−0.0086	5.5144 × 10−4	−6.8930 × 10−2	−0.1437	9.2634 × 10−3	−6.8930 × 10−2
23	17.0499	−0.1110	−1.8925	−0.1110	2.9072 × 10−5	−2.6191 × 10−4	−1.8920	4.9568 × 10−4	−2.6191 × 10−4
24	17.2793	−0.2090	−3.6114	−0.2087	3.4971 × 10−4	−1.6732 × 10−3	−3.6053	6.0427 × 10−3	−1.6732 × 10−3
25	17.4885	−0.3030	−5.2990	−0.3010	1.9809 × 10−3	−6.5375 × 10−3	−5.2644	3.4642 × 10−2	−6.5375 × 10−3
$\sum AIE \left(A\right)$					4.2779 × 10−2			4.5310 × 10−1
$MAE \left(A\right)$					1.7623 × 10−3

in the Appendix A), validate the effectiveness of the EBOA combined with the NRM in reducing optimization errors and identifying optimal parameters. This research highlights the ability of the EBOA to enhance the accuracy and efficiency of parameter estimation, particularly in complex models with nonlinear fitness functions.

4.1.4. Sensitivity Analysis

Through sensitivity analysis, the effect of parameter changes on results is evaluated. In this procedure, regions that produce extreme output values are identified and prioritized, which helps direct the ongoing progress of more dependable and operative PV models.

Table A4. Sensitivity analysis of the optimized parameters.

		RTC France	STP6−12/36	PWP201
Parameter	% Variation	RMSE (A)	RMSE (A)	RMSE (A)
Ipℎ	+5	3.386141 × 10−2	3.307724 × 10−1	4.269409 × 10−2
	−5	3.390541 × 10−2	3.318923 × 10−1	4.281554 × 10−2
$I_{01}$	+5	8.858318 × 10−3	6.173882 × 10−2	2.113757 × 10−3
	−5	9.118680 × 10−3	6.314808 × 10−2	2.113776 × 10−3
$I_{02}$	+5	1.887529 × 10−3	1.612278 × 10−2	1.449201 × 10−2
	−5	1.895500 × 10−3	1.612954 × 10−2	1.498657 × 10−2
$R_{sh}$	+5	8.003946 × 10−4	1.430694 × 10−2	2.134671 × 10−3
	−5	8.116866 × 10−4	1.430688 × 10−2	2.151207 × 10−3
$R_s$	+5	2.566258 × 10−3	3.945228 × 10−2	4.867641 × 10−3
	−5	2.607148 × 10−3	3.954989 × 10−2	4.955742 × 10−3
$a_1$	+5	1.111639 × 10−1	7.087283 × 10−1	4.839321 × 10−3
	−5	1.658334 × 10−1	1.087732 × 10	9.417220 × 10−3
$a_2$	+5	1.516574 × 10−2	7.412821 × 10−2	1.619020 × 10−1
	−5	2.876266 × 10−2	1.536272 × 10−1	2.090213 × 10−1
Best RMSE		7.478488 × 10−4	1.427010 × 10−2	2.061273 × 10−3

shows what happens when the mentioned PV models are changed by ±5%. The experiment demonstrates that the $I_{ph}\left(A\right)$ is a sensitive parameter since the output changes significantly with even a little variation in its value. Additionally sensitive, the diode ideality parameters show noticeable variations in both directions. However, the $I_0$ and $R_{sh}$ seem to have little impact on the model. However, it was also observed that a critical component was the $R_s$. The significance of emphasizing the most crucial elements is shown by the employed sensitivity analysis. The PV cell output is significantly impacted by the best values of these parameters, which directly affects the PV system’s efficiency.

5. Conclusions

This paper develops a novel EBOA for estimating the PV parameters of DDM and SDM using the NRM by optimizing the cost function. Initially, the fundamental BOA is merged with FC maps to adaptively adjust its parameters. This combined version is then enhanced by the addition of operators from HO algorithm to maintain algorithmic variety and strike a better balance between local and global search efforts. Three distinct case studies were conducted using the Two PV modules and one PV cell comprising the Photowatt-PWP201, STP6-120/36, and RTC France solar cell. The data were collected under certain temperature and irradiation conditions to validate the EBOA approach. The findings indicated that extracting PV parameters in DDM was more challenging than expected, although it was found to be more precise than SDM. Considering the wide range of comparative analysis, we conclude that the suggested approach reliably and exactly identifies the mathematical model parameters that specify the features of the solar modules and cells. Compared to more recent optimization approaches, the EBOA obtained a fast convergence rate and effectively minimized the fitness function. In addition, robust performance indices, including hybrid numerical and optimization methods previously published in the literature, proved EBOA’s superiority over rival optimizers.