Parameter Identification of Solar Cell Mathematical Models Using Metaheuristic Algorithms

Abstract

Lately, there has been a notable interest among researchers in constructing a precise mathematical representation using experimentally gathered data from solar cells and photovoltaic (PV) modules. This representation serves as a means to simulate and assess the performance of PV systems. In this study, the Walrus Optimization Algorithm (WaOA) and Cheetah optimizer (CO) were employed to deduce the unknown parameters inherent in various modes of solar cells and PV modules, specifically the single-diode model (SDM) and double-diode model (DDM). Furthermore, the evaluation criterion for this work involved measuring the route mean square error (RMSE) between the simulated outcomes generated using identified parameters for each mathematical model and the actual voltage derived from measurements of solar cells and PV modules. Notably, a comprehensive statistical analysis was carried out to validate the efficacy and stability of the WaOA and CO algorithms. These algorithms were compared against other optimization techniques for their effectiveness in solving the optimization challenge of accurately estimating the design parameters of PV systems. The outcomes of simulations and the extensive statistical assessment substantiate the superior performance and reliability of the Walrus Optimization Algorithm in effectively extracting parameter values from diverse PV modules under various operational scenarios.

1. Introduction

The finite nature of fossil fuel reserves and their excessive utilization have not only put human health at risk but have also had detrimental effects on the ecological balance. Hence, the imperative to develop sustainable energy alternatives has become exceedingly urgent [1, 2]. Among the array of renewable energy options available such as solar, hydro, wind, geothermal, and biomass energy, these sources are either practically inexhaustible or can be replenished in the short term [1, 3].

Solar energy, in particular, stands as a formidable candidate, possessing an immense reservoir of energy capable of satisfying the entire spectrum of contemporary human energy needs. This potential has propelled the integration of solar energy into diverse applications, including desalination processes, heating installations, and the generation of photovoltaic (PV) power [4, 5, 6]. Given its dual attributes of cleanliness and widespread accessibility, PV power generation has evolved into a pivotal initiative within the larger framework of advancing renewable energy sources [3].

Notably, the pursuit of PV power generation aligns seamlessly with the growing demand for clean electrical energy across various sectors. As a result, it has assumed a critical role in the ongoing efforts to cultivate and expand renewable energy alternatives. The trajectory of shifting toward PV power generation underscores a proactive stance in mitigating the environmental repercussions of fossil fuel reliance while fostering a sustainable energy landscape for the future [7, 8].

Doping, the intentional introduction of impurities into semiconductor materials, plays a pivotal role in modulating electronic properties, and understanding its effects is essential for precise solar cell parameter extraction. The study presented in [9] explores the correlation between secondary electron doping contrast and Fermi level pinned surfaces in silicon samples, prepared through HF-based wet-chemical treatment or focused ion beam micromachining. This investigation employs energy-resolved SE imaging techniques and finite element analyses, revealing surface band-bending effects and challenging the conventional belief exclusively associating doping contrast with patch fields or adventitious metal-semiconductor contacts. In a complementary approach, Chee et al. [10] introduce a Monte Carlo model incorporating a finite element method and a ray tracing algorithm to detail the computation of electrostatic fields inside and outside a semiconductor for doping contrast in a scanning electron microscope (SEM). This numerical method effectively distinguishes effects on doping contrast arising from surface band bending, external patch fields, and macroscopic external fields in the SEM detection system. The presented theory aligns well with experimental observations, advancing our understanding of doping contrast mechanisms and facilitating quantitative dopant profiling using the SEM. Building on this foundation, Chee [11] discusses doping contrast utilization with the secondary electron (SE) signal in the SEM for quantitative dopant profiling in alignment with International Roadmap for Semiconductors (ITRS) requisites. This research specifically focuses on site-specific dopant profiling of silicon p–n junction specimens, employing a 30-kV Ga + focused ion beam (FIB) for trench side-wall cutting and successive milling at lower voltages in a dual-beam FIB/SEM system. Despite the protective platinum strap’s effective control of “curtaining” ‘effects, reduced doping contrast from the side wall, compared to a cleaved surface with the same ion beam energy, is attributed to material effects from prior milling steps and differences in geometries between milling and imaging. The study introduces novel principles underlying the doping contrast mechanism, considering ion implantation depth and concentration, and amorphization damage as linear functions of the final milling voltage. Although reaching only half the contrast achievable from a freshly-cleaved specimen, the research demonstrates the feasibility of site-specific dopant profiling in situ in the SEM with doping contrast increasing as milling voltage decreases. The continual advancement of semiconductor fabrication processes stands as a cornerstone in the relentless pursuit of highly efficient inorganic photovoltaic (PV) technologies. Through dedicated research and development efforts, these processes have undergone refinement, unlocking new frontiers in the realm of solar energy harnessing. This evolution is pivotal in enhancing the overall performance, durability, and cost-effectiveness of inorganic PV systems.

The effective design, forecasting, sizing, diagnostics, and maintenance of photovoltaic (PV) system installations rely heavily on a dependable and precise model encompassing the cell, module, and PV array components [12, 13]. In the realm of literature, three distinct models have been established, each featuring one, two, or three diodes. These models are characterized by parameters that demand accurate extraction. The process of obtaining these diverse parameters remains both critical and complex. To tackle this challenge, numerous techniques have emerged in scholarly works to deduce the optimal PV parameters. These techniques span numerical, analytical, evolutionary, and hybrid methodologies [14, 15].

Among the various methods available for parameter extraction in PV systems, the equation governing the current-voltage characteristic holds particular promise. This equation encapsulates the entirety of PV properties and parameters. However, its transcendental nature introduces complexities in solving it, transforming it into an optimization predicament. In navigating this complexity, metaheuristic approaches have surfaced as particularly effective tools. Their demonstrated success in addressing a spectrum of challenges across diverse domains attests to their potential in resolving the intricacies of PV parameter optimization. As such, these metaheuristics offer a promising avenue to surmount the complexities inherent in solving the multifaceted equation governing PV behavior, advancing the domain of PV system analysis and application [16, 17].

The precision of modeling holds significant importance in the evaluation, enhancement of efficiency, fault analysis, and simulation of photovoltaic (PV) systems. These systems, composed of aggregated PV cells, are typically subject to modeling through equivalent circuits, encompassing single-diode (SDM), double-diode (DDM), and triple-diode (TDM) models [18]. These circuit representations effectively emulate the electrical attributes of PV cells. In these models, parameters are present in quantities of five, seven, and nine, respectively, demanding precise extraction [19].

Intricacies arise with the expansion of diode numbers, which introduces a greater number of parameters needing extraction, thus intensifying computational complexities. The challenges posed encompass not only the heightened intricacy of solutions due to multiple unknown variables but also the interdependence between electrical values, which results in a highly implicit function. Furthermore, the incorporation of exponential functions within the characteristic equations exacerbates the difficulty in solving nonlinear attributes [20, 21].

These compounded challenges coalesce to create a perplexing puzzle when it comes to establishing accurate PV models. This enigma requires innovative approaches that can surmount the intertwined complexities of parameter extraction and nonlinear behavior to arrive at models that truly reflect the intricacies of PV systems [21].

In the research conducted by Senturk and Eke [22], a novel empirical relationship was utilized to extract parameters within the framework of the one-diode model. This empirical approach involves determining the initial value of the series resistance using the slope of the I-V curve as given by the producer. However, acquiring numerical data for the current-voltage characteristic is often not readily available upon purchasing a photovoltaic (PV) module. To address this limitation, image processing techniques are necessitated to extract the data from technical documentation, yet the accuracy of this extracted data may vary.

Analytical methods exhibit effectiveness under standard weather conditions; however, their reliability diminishes in the face of fluctuating atmospheric conditions [23]. Furthermore, approximating equations in these methods substantially compromises result accuracy. On a different note, Tossa et al. [24] introduced an innovative approach for accurately modeling the SD model of a PV module. This method, implemented in the MATLAB/Simulink environment, relies on the Levenberg-Marquardt algorithm. Another avenue, proposed by Ghani et al. [25], involves scrutinizing the current-voltage characteristic to derive parameter values. Here, a model of five formulas is solved utilizing the Newton-Raphson method. Yet, this algorithm not only requires solving a system of equations but also entails handling a Jacobian matrix composed of twenty-five first and second derivative terms, adding a layer of complexity to the process.

The notable drawback of numerical methods, such as the Newton-Raphson, is their demand for extensive calculations to ensure convergence. This method’s effectiveness diminishes as the number of parameters that have to be identified rise. Despite their efficiency, the sluggish convergence of numerical techniques does not always guarantee optimal outcomes as they might converge to a local minima. Additionally, selecting appropriate initial conditions for these methods can be challenging [23]. In light of these complexities, the field continues to seek innovative solutions that can offer improved accuracy and efficiency in parameter extraction while addressing the limitations of current techniques.

In the study by Oliva et al. [25], a novel approach utilizing a chaotic artificial bee colony (ABC) algorithm was introduced to estimate parameters for photovoltaic (PV) panels. The innovation in this proposed chaotic ABC lies in the replacement of the arbitrary number in the onlooker bee phase with a number produced through chaos theory. This chaotic number is obtained from a carefully selected chaotic map, with the tent map being identified as the optimal choice and subsequently incorporated into the proposed chaotic ABC algorithm. A comprehensive exploration of various chaotic maps was conducted, culminating in the selection of the tent map. Yang introduced a novel, versatile population-based optimization technique called hunger games search (HGS) in 2021 [26]. This optimizer demonstrated scalability and adaptability, making it suitable for a wide range of optimization challenges spanning application and structural domains. A notable feature of HGS was its Laplacian-based crossover search mechanism, which effectively diversified solution exploration. Additionally, the integration of the Nelder-Mead (NM) local search technique heightened the precision of convergence toward the optimal solution. Building upon the original HGS framework, the study extended its capabilities by incorporating the Laplacian mechanism and the Nelder-Mead simplex strategy. This extension resulted in an enhanced parameter optimization performance, successfully applied to the realm of static photovoltaic models. The proposed method emulated the hunger-induced behavior and attributes of individuals within the optimization process. Despite HGS finding utility in various engineering problems, its potential in parameter estimation for solar PV models had remained untapped. To address this gap, a new algorithm emerged, combining elements from the simplified swarm optimizer (SSO) with the Nelder-Mead simplex approach. This innovative fusion led to a dual achievement of precise and swift parameter identification for both static deflection model (SDM) and dynamic deflection model (DDM). The novel approach exhibited the capability to finely balance accuracy and speed in parameter estimation, demonstrating its potential for practical applications in the solar PV domain [26].

The simulation outcomes, utilizing both SD and DD models, clearly underscore the advantages of the developed chaotic ABC over a range of competing techniques. These include the conventional ABC, a chaotic variant of the particle swarm optimization (PSO), the artificial bee swarm optimization algorithm, simulated annealing, the cat swarm optimization algorithm, teaching learning-based optimization algorithm, and the harmony search method. To provide concrete context, the study employed real-world case studies involving an RTC-France PV cell, a polycrystalline panel labeled STM6-120/36, and a monocrystalline PV panel denoted as STM6-40/36.

The utilization of the chaotic ABC technique presents a significant step forward in enhancing the accuracy and efficiency of parameter estimation for PV panels. By harnessing the dynamics of chaos theory, this approach demonstrates its capacity to outperform several established optimization methods. As the realm of PV technology continues to evolve, these innovative methodologies hold the potential to substantially contribute to the improvement of PV system performance, thereby promoting the broader adoption of renewable energy sources.

The work by Oliva et al. [27] presents a notable advancement by introducing a chaotic variation of the whale optimization algorithm (WOA) for the precise estimation of parameters associated with photovoltaic (PV) cells and panels. The foundation of the WOA draws inspiration from the behavioral patterns of whales. In the WOA framework, during each iteration, the repositioning of whales involves the generation of a random number within the range of [0,1]. This number then dictates the likelihood of selecting either a spiral model or a shrinking encircling mechanism for adjustment. The innovation in the proposed chaotic WOA lies in the replacement of this randomly generated number with a value generated through chaos theory.

In the pursuit of the optimal chaotic map for this endeavor, four distinct chaotic maps—namely, Singer, Sinusoidal, Logistic, and Tent—underwent testing. Through rigorous evaluation, the singer map emerged as the most fitting choice for integration into the chaos-based WOA, proving its mettle in tackling the parameter estimation challenge effectively. Simulation outcomes, encompassing both single- and double-diode models, serve as compelling evidence of the superiority of the proposed chaos-infused WOA over a spectrum of cutting-edge optimization algorithms. Among these algorithms are the ABC, PSO, artificial bee swarm optimization, simulated annealing, bird mating optimization, differential evolution, and harmony search. These results underscore the potential and advantages of this chaotic WOA variant, showcasing its ability to surpass established optimization techniques in addressing the complexities of parameter estimation in PV systems. By harnessing the power of chaos theory within optimization frameworks, this research contributes to the refinement of parameter estimation processes within the domain of photovoltaics. Such innovative methodologies are integral to advancing the efficiency and effectiveness of PV systems, aligning with the broader global push for sustainable and renewable energy sources.

The study conducted by Mughal et al. [28] introduces an innovative approach that merges the strengths of PSO and SA to estimate parameters related to PV cells. In the proposed PSO-SA hybrid framework, the advantages of both algorithms are combined by applying the operators of PSO and SA to the search agents during each iteration. The optimization algorithm employs the RMSE as its primary objective. The simulation outcomes, which involve an RTC-France PV cell encompassing both single- and double-diode models, unequivocally demonstrate the superior performance of the proposed hybrid algorithm when compared to PSO, SA, harmony search (HS), and pattern search techniques.

Furthermore, in Gong et al. [29], a distinct application of PSO is presented. This version utilizes varying acceleration coefficients and inertia weights for parameter extraction of PV modules. In the research by Fathy and Rezk [30], an imperialistic competitive algorithm (ICA) is leveraged to estimate circuit model parameters associated with PV modules. The investigation spans both single- and double-diode models, considering various PV technologies such as mono-crystalline, polycrystalline, thin-film, and amorphous materials. The incorporation of amorphous on crystalline materials into the analysis signifies recognition of the evolving landscape of PV technologies. This nuanced exploration becomes particularly pertinent in instances where focused ion beam (FIB) processing of crystalline materials is employed, resulting in hybrid structures that merge the characteristics of amorphous and crystalline semiconductors. The interplay between these materials introduces a unique set of challenges and opportunities, making them a crucial consideration in the development and application of single- and double-diode models [31]. The attainments underscore the superiority of ICA over several alternative optimization techniques, including pattern search, chaos-based PSO, bird mating optimization, adaptive differential evolution (DE), ABSO, SA, and HS [30].

These studies collectively highlight the ongoing quest for refining parameter estimation methodologies in the realm of photovoltaics. The fusion of optimization algorithms and the exploration of their variations underscore the significance of devising novel strategies to enhance the accuracy and efficiency of PV system performance analysis. As the renewable energy landscape continues to expand, the advancements in parameter estimation techniques contribute to the broader mission of harnessing sustainable energy sources and minimizing environmental impact. In addition to the research mentioned, other studies have also employed distinct algorithms for the parameter extraction of photovoltaic cells. For instance, Ram et al. [32] introduced an enhanced version of the flower pollination algorithm, reference [33] utilized the fireworks optimization technique, and Abdelghany et al. [34] employed the water cycle algorithm for this purpose. While metaheuristics offer several advantages over alternative parameter estimation methods, it is important to acknowledge a potential pitfall known as premature convergence, especially when dealing with complex PV cell parameter estimation issues, which often exhibit multimodal characteristics.

Premature convergence occurs when metaheuristic algorithms become trapped in local optima instead of reaching the global optimum. This concern is particularly relevant in the context of PV cell parameter estimation, where the problem’s multi-peaked nature can hinder the ability of algorithms to find the most accurate parameter values. This challenge is further evidenced by the relatively high root mean square error (RMSE) values observed in the application of various metaheuristic algorithms to PV cell parameter estimation [35].

To address this issue of premature convergence, extensive efforts are being dedicated to developing strategies that enhance the performance of metaheuristic techniques in the context of PV cell parameter estimation. Researchers are actively exploring techniques to ensure that these algorithms have a higher likelihood of escaping local optima and converging toward more accurate parameter values. By mitigating the premature convergence problem, the potential of MAs can be maximized in accurately estimating the parameters of PV cells and modules.

In this chapter, the investigation revolves around the utilization of both the Walrus Optimization Algorithm (WaOA) [36, 37] and the Cheetah optimizer (CO) [38] as potent tools for addressing the intricate challenge of PV cell parameter estimation. By employing these advanced optimization techniques, the aim is to enhance the accuracy and efficiency of estimating parameters crucial to the performance of photovoltaic cells. The Walrus Optimization Algorithm, characterized by its unique nature-inspired strategies, and the Cheetah optimizer, drawing inspiration from the swift and agile cheetah’s hunting behaviors, are harnessed as innovative approaches to tackle the complexities inherent in this parameter estimation problem.

The remainder of this chapter follows a structured organization. In Section II, the formulation of the PV cell parameter estimation problem is presented. Section III introduces the methodologies that have been proposed. Moving forward, the fourth section presents the outcomes and analysis. Ultimately, the fifth section contains the concluding remarks to wrap up the discussion.

2. Models of solar cells

A mathematical model must be utilized to determine the solar cell properties analytically in order to construct the solar cells and PV modules. Diode-based electronic circuits are employed to model solar cells based on this. Along with the TD circuit, which was very recently developed, the SD and DD models are the most often used tools for determining the parameters of solar cells and PV modules [38].

2.1 Single-diode model

The model, which can be seen in Figure 1, simply has one diode that is utilized to parallelize the current source I_ph, which displays the photo-generated electrical current. The diode functions as a rectifier of half-waves. The nonphysical accessibility factor of the ideality of the diode is taken into account by the mathematical framework [39]. The model has a fairly straightforward structure, making it simple to put into practice. The biggest problem with this straightforward model is that there are only five parameters that need to be carefully specified.

Figure 1.

Equivalent circuit for SD model.

According to Figure 1, I_t can be expressed as;

$${\mathit{I}}_{\mathit{t}}={\mathit{I}}_{\mathit{ph}}-{\mathit{I}}_{\mathit{sd}}-{\mathit{I}}_{\mathit{sh}}$$E1

where ${\mathit{I}}_{\mathit{t}}$, $I_{\mathit{ph}}$, $I_{\mathit{sd}}$, and $I_{\mathit{sh}}$ imply the total current, photo current, diode current, and current through shunt resistance, respectively.

In order to create a more accurate model for the inside functioning of the diode, Shockley’s equivalent diode formula might be used. Eq. (1) is capable of being rewritten as follows [39, 40]:

$$I_t=I_{\mathit{ph}}-I_{\mathit{sd}}\left[\mathit{exp}\left(\frac{q\left(V_t+R_s{I}_t\right)}{\mathit{nkt}}\right)-1\right]-\frac{V_t+R_s{I}_t}{R_{\mathit{sh}}}$$E2

Where $V_t$ implies the model terminal voltage, $I_{\mathit{sd}}$ implies reverse saturation current of the diode, $R_s$ implies the resistance in series connection, $R_{\mathit{sh}}$ implies the resistance in the parallel combination, n denotes the ideality factor, q implies the electron charge ($\mathit{q}=1.602\ast {10}^{-23}\mathrm{J}/\mathrm{K}$), k implies the Boltzmann constant ($\mathit{k}=1.380\ast {10}^{-23}\mathrm{J}/\mathrm{K}$), and T implies cell temperature i. Therefore, precise estimation of these variables, which will be carried out using various optimization strategies in the subsequent sections, is necessary to ensure the model operates properly.

2.2 Double-diode model

The double-diode (DD) model, shown in Figure 2, is suggested as a replacement for the single-diode model, which is typically not a good option for use with a variety of applications [41]. The solar spectrum containing sufficient energy irradiating the p-n junction can induce a photocurrent that changes the junction potential. The excess electron-hole pairs generated can forward bias the p-n junction [42, 43]. The surface or interface traps/charges also play a crucial role in the nonideal behavior of the solar cell [44, 45], as well as the carrier distributions following the electric potentials influenced by the solar irradiation [46].

Figure 2.

Equivalent circuit for DD model.

As seen in Figure 2, there are two diodes: one serves as a rectifier, and the other is utilized to account for the effects of the solar cell’s non-idealities plus the current generated from the combination. The following equation could be used to depict the current balance in the corresponding circuit of Figure 2 [40]:

$${\mathit{I}}_{\mathit{t}}={\mathit{I}}_{\mathit{ph}}-{\mathit{I}}_{\mathit{d}1}-{\mathit{I}}_{\mathit{d}2}-{\mathit{I}}_{\mathit{sh}}$$E3

where $I_{d1}$, and $I_{d2}$ signify current of the first and second diodes. The two diodes’ internal configuration is updated using the Shockley equivalency. Therefore, Eq. (3) can be rewritten as given [40],

$$I_t=I_{\mathit{ph}}-I_{d1}\left[\mathit{exp}\left(\frac{q\left(V_t+R_s{I}_t\right)}{n_1\mathit{kt}}\right)-1\right]-I_{d2}\left[\mathit{exp}\left(\frac{q\left(V_t+R_s{I}_t\right)}{n_2\mathit{kt}}\right)-1\right]-\frac{V_t+R_s{I}_t}{R_{\mathit{sh}}}$$E4

where $I_{d1}$ and $I_{d2}$ symbolize saturation currents for the two diodes in the circuit. n₁, n₂ signify ideality factors. The DD model contains seven unidentified parameters that have to be precisely determined, that is, $R_{s,}{R}_{\mathit{sh}},I_{\mathit{ph}},I_{d1},I_{d2,}{n}_1,\mathrm{\text{and}}{n}_{2.}$

2.3 Problem formulation

It is feasible to approach each of the mathematical models for the SD and DD as an optimization issue, with the best values for the unidentified model parameters serving as the answer. To determine if the optimized parameters meet their real values or not, a fitness function needs to be initially applied. Examining the level of agreement between the I-V curves listed in the datasheet of an actual solar cell and those anticipated regarding the identified parameters of the theoretical empirical model is another way to determine the quality of the methodology used for estimation. Consequently, the fitness function for the SD can be illustrated as [37, 41]:

$$f_{\mathit{SD}}\left(V_t,I_t,x\right)=I_t-x_3+x_4\left[\mathit{exp}\left(\frac{q\left(V_t+x_1{I}_t\right)}{x_5\mathit{kt}}\right)-1\right]-\frac{V_t+x_1{I}_t}{x_2}$$E5

meanwhile, for the DD model, the fitness function is written as:

$$f_{\mathit{DD}}\left(V_t,I_t,x\right)=I_t-x_3+x_4\left[\mathit{exp}\left(\frac{q\left(V_t+x_1{I}_t\right)}{x_6\mathit{kt}}\right)-1\right]+x_5\left[\mathit{exp}\left(\frac{q\left(V_t+x_1{I}_t\right)}{x_7\mathit{kt}}\right)-1\right]-\frac{V_t+x_1{I}_t}{x_2}$$E6

Where $V_t,\mathit{\text{and}}{I}_t$ imply the experimental values of voltage and current of the solar cell. For SD model, $\mathit{x}=\left[{\mathit{R}}_{\mathit{s}},{\mathit{R}}_{\mathit{sh}},{\mathit{I}}_{\mathit{ph}},{\mathit{I}}_{\mathit{sd}},\mathit{n}\right]$ symbolizes the solution vector, wherein the case of the DD model the solution vector is $\mathit{x}=\left[{\mathit{R}}_{\mathit{s}},{\mathit{R}}_{\mathit{sh}},{\mathit{I}}_{\mathit{ph}},{\mathit{I}}_{\mathit{d}1},{\mathit{I}}_{\mathit{d}2},{\mathit{n}}_1,{\mathit{n}}_2\right]$. The functions $f_{\mathit{SD}},f_{\mathit{DD}}$ assess and investigate how closely each circuit’s outcomes resemble those obtained through experimental measurement.

The cost function is then developed based on identifying the parameters that provide the least amount of error between the actual, as determined by observations, and the predicted, as determined by diode models. The cost function can then be defined using a set of NE samples to broaden the search until global optima are found and expressed as:

$$\mathit{\text{RMSE}}\left(x\right)=\sqrt{\frac{1}{N}{\sum }_{c=1}^{\mathit{NE}}{\left(f_M^c\left(V_t^c,I_t^c,x\right)\right)}^2}$$E7

where $\mathit{M}$ aids in determining, which diode model should be used and $\mathrm{\text{RMSE}}$ stands for root mean square error. The Walrus Optimization Algorithm (WaOA) and Cheetah optimizer (CO) are employed to determine the best values for the design parameters of various PV modules and solar cells that produce the objective function’s least value.

3. Optimization algorithms

3.1 Walrus optimization algorithm (WaOA)

A large marine mammal with flippers, the walrus has a patchy distribution in the Arctic Ocean and the waters surrounding the North Pole in the Northern Hemisphere [35, 36]. Large tusks and whiskers on adults make them easy to recognize. Walruses are sociable animals who spend the most of their time searching for benthic bivalve mollusks to feed on the sea ice. The tusks on walruses, which are this animal’s most noticeable trait, are very long. These are elongated canines that are present in both male and female animals and can grow to be up to 1 m long and 5.4 kg in weight. In the late summer, walruses prefer to move to outcrops or rocky beaches when the weather warms and the ice begins to melt. These migrations are quite dramatic and include large gatherings. Due to their size and tusks, the polar bear and the killer whale (orca) are the walrus’ only two natural predators.

Walruses exhibit intellectual behavior in their social interactions and everyday actions. Three of these clever actions stand out as being the most inescapable:

1. Directing the individuals to eat under the direction of the tribe member with the longest tusks: During the search process, the algorithm is guided toward areas of greatest potential by keeping track of the best population participant. The dominant walrus, which is distinguished by having a longer tusk, is in charge of leading the other walruses in the social life of the species. This method involves moving walruses, which significantly alters their posture. The algorithm’s capacity for global search and exploration is improved by simulating these huge displacements [35, 36].

2. Migration toward rocky beaches: The migration of walruses in response to summer’s warming temperatures is one of their normal behaviors. Walruses shift dramatically in this process, going toward rocky outcrops or beaches. The position of other walruses is taken into account as a walrus’s migration destination in the WaOA simulation for a walrus. One of these locations is arbitrary pointed out, and the walrus migrates in its direction [35, 36]. WaOA’s architecture, which follows this tactic, enhances the possibilities of global search and discovery. The foraging approach used by the strongest walrus differs from the migratory strategy in that the population update process is not allowed to rely just on one individual, such as the population’s best member. This updating procedure stops the algorithm from stalling in local optima as well as premature convergence.

3. Fight or run away from predators: When battling their predators, such as polar bears and killer whales, walruses adopt a protracted chasing technique. The walrus position is only little altered by this chasing activity, which occurs in a tiny area nearby. WaOA’s capacity to seek locally and exploit to converge to better solutions is therefore improved by imitating the minor movements of the walrus by looking for good locations throughout the struggle.

3.1.1 Initialization

Walruses serve as the search population in WaOA, and a potential answer to the optimization problem is represented by each walrus in WaOA. The potential values for the problem variables are thus determined by each walrus’ location inside the search space. Walrus populations are arbitrarily initialized at the start of WaOA deployment. Utilizing (8), we arrive at this WaOA population matrix [35, 36].

$$\mathit{X}=\left[\begin{array}{c}{\mathit{X}}_1\\ ⋮\\ \begin{array}{c}{\mathit{X}}_{\mathit{i}}\\ \begin{array}{c}⋮\\ {\mathit{X}}_{\mathit{N}}\end{array}\end{array}\end{array}\right]=\left[\begin{array}{ccc}{\mathit{x}}_{1,1}& \cdots & {\mathit{x}}_{1,\mathit{j}}\cdots {\mathit{x}}_{1,\mathit{m}}\\ ⋮& \ddots & \begin{array}{ccc}⋮& ⋰& ⋮\end{array}\\ \begin{array}{c}{\mathit{x}}_{\mathit{i},1}\\ ⋮\\ {\mathit{x}}_{\mathit{N},1}\end{array}& \begin{array}{c}\cdots \\ \ddots \\ \cdots \end{array}& \begin{array}{c}\begin{array}{ccc}{\mathit{x}}_{\mathit{i},\mathit{j}}& \cdots & {\mathit{x}}_{\mathit{i},\mathit{m}}\end{array}\\ \begin{array}{ccc}⋮& ⋰& \cdots \end{array}\\ \begin{array}{ccc}{\mathit{x}}_{\mathit{N},\mathit{j}}& \cdots & {\mathit{x}}_{\mathit{N},\mathit{m}}\end{array}\end{array}\end{array}\right]$$E8

Where $\mathit{N}$ denotes the number of walruses, m denotes the number of choice variables, $\mathit{X}$ signifies the population of walruses, ${\mathit{X}}_{\mathit{i}}$ signifies the ith walrus (candidate solution), and ${\mathit{x}}_{\mathit{i},\mathit{j}}$ signifies the value of the jth decision variable recommended by the ith walrus. For the fitness function derived from walruses, the generated values are described in (9).

$$\mathit{F}=\left[\begin{array}{c}{\mathit{F}}_1\\ ⋮\\ \begin{array}{c}{\mathit{F}}_{\mathit{i}}\\ \begin{array}{c}⋮\\ {\mathit{F}}_{\mathit{N}}\end{array}\end{array}\end{array}\right]=\left[\begin{array}{c}\mathit{F}\left({\mathit{X}}_1\right)\\ ⋮\\ \begin{array}{c}\mathit{F}\left({\mathit{X}}_{\mathit{i}}\right)\\ \begin{array}{c}⋮\\ \mathit{F}\left({\mathit{X}}_{\mathit{N}}\right)\end{array}\end{array}\end{array}\right]$$E9

where $\mathit{F}$ signifies the vector of the fitness function and ${\mathit{F}}_{\mathit{i}}$ signifies the value of the fitness function calculated depending on the ith walrus.

3.1.2 Mathematical modeling of WaOA

3.1.2.1 Phase 1: feeding strategy (exploration)

Walruses eat a wide variety of marine creatures, including more than 60 different species of sea cucumbers, tunicates, soft corals, tube worms, prawns, and different mollusks. The walrus prefers benthic bivalve mollusks, particularly clams, and forages by grazing on the ocean floor while using its active flipper movements and sensitive vibrissae to search out and find food. The more powerful walrus in the group, the one with the largest tusks, leads the other walruses in their search for nourishment [35, 36]. The degree of accuracy of the fitness values of the potential solutions is comparable to the size of the tusks in walruses. The most powerful walrus in the team is pointed out as the best candidate solution having the best fitness. The WaOA’s exploration effectiveness in the global search is increased as a result of the walruses’ searching behavior, which results in varied scanning regions of the search field. Following the direction of the most important member of the group, (10) and (11) are used to mimic the process of changing the walruses’ position depending on the grazing strategy [35, 36]. A new position for the walrus is first created in this process using (10). If this novel location increases the fitness value, it replaces the earlier location; this idea is modeled in Eq. (11).

$$x_{i,j}^{P_1}=x_{i,j}+{\mathit{rand}}_{i,j}·\left({\mathit{SW}}_j-I_{i,j}·x_{i,j}\right)$$E10

$$X_i=\left\{\begin{array}{cc}{X}_i^{P_1},& F_i^{P_1}<F_i\\ X_i,& \mathit{\text{else}}\end{array}\right.$$E11

where $X_i^{P_1}$ denotes the updated position for the ith walrus according to the feeding stage, $x_{i,j}^{P_1}$ denotes its jth dimension, $F_i^{P_1}$ denotes the fitness value, ${\mathit{rand}}_{i,j}$ signifies arbitrary numbers ranged from 0 to 1, $\mathit{SW}$ denotes the finest potential solution related to the most powerful walrus, and $I_{i,j}$ is an integer number selected randomly between 1 and 2. $I_{i,j}$ is utilized to enhance the algorithm’s exploration capability so that if it is selected as 2, it creates more significant and broader changes in the location of candidates compared to the case when the value is selected as 1, which presents the normal condition of this displacement. These conditions help enhance the global searching of the WaOA in escaping from the local optima and finding the global optimal spot in the problem-solving area.

3.1.2.2 Phase 2: migration

Due to the rising temperature of the air in the final days of summer, one of the normal behaviors of walruses is their movement to outcrops or rocky beaches. The WaOA uses this migratory process to direct the walruses in the search space to find appropriate regions in the search space [32, 33]. Employing (12) and (13), this behavioral process is quantitatively modeled. This modeling makes the assumption that each walrus moves to a different walrus location (chosen at random) in a different region of the search field. As a result, the suggested alternative location is initially derived using (12). Then, in accordance with (13), the most recent position supersedes the earlier one of the walrus if it increases the fitness value [35, 36].

$$x_{i,j}^{P_2}=\left\{\begin{array}{cc}{x}_{i,j}+{\mathit{rand}}_{i,j}·\left(x_{k,j}-I_{i,j}·x_{i,j}\right),& F_k<F_i\\ x_{i,j}+{\mathit{rand}}_{i,j}·\left(x_{i,j}-x_{k,j}\right),& \mathit{\text{else}}\end{array}\right.$$E12

$$X_i=\left\{\begin{array}{cc}{X}_i^{P_2},& F_i^{P_2}<F_i\\ X_i,& \mathit{\text{else}}\end{array}\right.$$E13

where $X_i^{P_2}$ denotes the updated location of the ith walrus according to the migration stage, $x_{i,j}^{P_2}$ denotes the jth dimension, $F_i^{P_2}$ denotes the fitness value, $X_{\mathit{k}}$, $\mathit{k}\in \left\{1,2,\dots ,\mathit{N}\right\}$ and k ≠ i, signifies the position of the pointed walrus toward the ith walrus will move, $x_{k,j}$ denotes the jth dimension, and $F_k$ denotes its fitness value.

3.1.2.3 Phase 3: running out and fighting against predators (exploitation)

Polar bear and killer whale assaults are constant threats to walruses. The walruses move around in the area around where they are located as a result of their technique of evading and combating these predators. The WaOA’s ability to utilize this aspect of walrus behavior in the small search area surrounding potential solutions is enhanced. Since this process takes place close to each walrus’s location, in the WaOA algorithm, it is considered that this range of walrus location change takes place in a neighborhood centered on walruses that has a specific radius. The radius of this neighborhood is thought of as a variable because it initially begins at its greatest value and then decreases throughout the algorithm repetitions because, in the early iterations, priority is given to global search to attempt to find the ideal area in the search field. Because of this, a variable radius with algorithm iterations has been created in this stage of WaOA using localized lower/upper boundaries [34, 35]. For the purpose of simulating the scenario in WaOA, a neighborhood is assumed to be present around each individual walrus, which is then given an arbitrary new location inside that neighborhood using steps (14) and (15). Then, in accordance with (16), this new location substitutes the prior one if the fitness value is enhanced [35, 36].

$$x_{i,j}^{P_3}=x_{i,j}+({\mathit{lb}}_{\mathit{\text{local}},j}^t+\left({\mathit{ub}}_{\mathit{\text{local}},j}^t-\mathit{rand}·{\mathit{lb}}_{\mathit{\text{local}},j}^t\right),$$E14

$$\mathit{\text{Local Bounds}}:\left\{\begin{array}{c}{\mathit{lb}}_{\mathit{\text{local}},j}^t=\frac{{\mathit{lb}}_j}{t},\\ {\mathit{ub}}_{\mathit{\text{local}},j}^t=\frac{{\mathit{ub}}_j}{t},\end{array}\right.$$E15

$$X_i=\left\{\begin{array}{cc}{X}_i^{P_3},& F_i^{P_3}<F_i\\ X_i,& \mathit{\text{else}}\end{array}\right.$$E16

where ${\mathit{lb}}_j$ and ${\mathit{ub}}_j$ denote the lower and upper boundaries of the jth variable, ${\mathit{lb}}_{\mathit{\text{local}},j}^t$and ${\mathit{ub}}_{\mathit{\text{local}},j}^t$ denote local lower and local upper boundaries allowable for the jth variable. $X_i^{P_3}$ denotes its jth dimension. $F_i^{P_3}$ denotes its fitness value. Figure 3 displays the flowchart for implementing the WaOA.

Figure 3.

Flowchart of WaOA technique.

3.2 Cheetah optimizer (CO)

3.2.1 Inspiration

The cheetah (Acinonyx jubatus) stands out as the primary feline species and is recognized as the fastest land animal, inhabiting the central regions of Iran and Africa. These remarkable creatures are capable of reaching speeds exceeding 120 kilometers per hour. Their exceptional speed and agility are attributed to their physical characteristics, including a long tail, slender legs, lightweight build, and a flexible spine. Cheetahs are known for their swiftness, adept stealth, rapid pursuit during hunting, and distinctive spotted coats. However, it is important to note that these visually-oriented predators cannot sustain their high-speed actions for extended periods, limiting their chases to less than half a minute [37]. After capturing their prey, the cheetah’s speed dramatically drops from 93 kilometers per hour (58 mph) to 23 kilometers per hour (14 mph) in just three strides. To overcome this limitation, cheetahs employ a keen sense of observation, often perching on small branches or hills to scan their surroundings for potential prey. Their ability to blend seamlessly into high, dry grass due to their unique coat pattern further aids in successful hunting. Cheetahs typically target animals such as Thomson’s gazelles, impalas, antelopes, hares, birds, rodents, and young calves of larger herbivores [37]. The hunting strategy of cheetahs involves slow, stealthy approaches to minimize the distance between themselves and their prey. They maintain a crouched posture and patiently wait for the prey to draw near as they tend to abandon the hunt if detected. The preferred minimum distance for a successful chase ranges from 60 to 70 m (200 to 230 feet), but it increases to 200 meters (660 feet) if they fail to remain concealed adequately. The pursuit itself typically lasts about 60 seconds, covering an average distance ranging from 173 m (568 feet) to 559 m (1834 feet).

To bring down their prey, cheetahs employ a tactic involving a swift forepaw strike to the prey’s rump, causing the prey to lose its balance. Subsequently, the cheetah utilizes its strength to overpower the prey and affect a swift kill. The cheetah’s muscular tail plays a pivotal role in achieving sharp turns during the pursuit. Generally, it is easier for them to hunt animals that have strayed from their herds or display less vigilance. Predation outcomes are influenced by various factors such as the age and gender of the cheetah, the number of predators involved, and the level of alertness displayed by the prey. Additionally, coalitions of cheetahs or mothers with cubs tend to be more successful in hunting larger prey.

Biological studies have revealed that cheetahs possess extraordinary spinal flexibility and long tails that contribute to their physical balance. Their shoulder blades, which are not connected by collarbones, enable a wider range of shoulder movement, enhancing their hunting prowess. Despite these exceptional attributes, it is important to acknowledge that not all cheetah predations result in a successful capture.

3.2.2 Mathematical model and algorithm

It is probable to spot prey when a cheetah patrols or examines its immediate surroundings. The cheetah may sit in one spot after spotting its victim, watch until it approaches, and then launch an assault. There are phases of rushing and capturing in the attack phase. The cheetah might cease hunting for a number of reasons, including power limitations, the ability to catch prey quickly, etc. The CO algorithm’s overall foundation is the intelligent application of different hunting techniques during hunting sessions [37].

• Searching: Cheetahs must seek, either actively or passively, throughout their territory (search space) or the surrounding region in order to find their prey.

• Sitting and waiting: If the prey is discovered but the circumstances are not ideal, cheetahs may wait for the prey to approach or for the situation to improve;

• Attacking: This tactic entails two crucial steps.

  • Rushing: The cheetah will move as quickly as possible toward its victim when it intends to attack.

  • Capturing: The cheetah approached its prey while moving quickly and maneuverably.

• Leave the prey and return home: Two scenarios are taken into consideration for this tactic. (1) If the cheetah is unable to catch its prey, it should move or go back to its home zone. (2) It may move to the location of the most recent prey found and search the area around it if there has been no successful hunting activity for a period of time (Figure 4).

Figure 4.

Hunting behaviors of cheetah: (a) Scanning, (b) sitting and waiting, (c) rushing, and (d) capturing.

3.2.3 Searching strategy

Cheetahs hunt for prey using one of two methods: either vigorously patrolling the territory while sitting or standing to scan the surroundings. When the prey is numerous and grazing while traversing the plains, scanning manner is more appropriate. However, if the prey is dispersed and active, it is preferable to choose an active style that consumes much power than the scan method. A series of such searching strategies may, therefore, be chosen by the cheetah throughout the time of hunting, subject to the state of the prey, the area’s coverage, and the cheetahs’ own health [37].

Mathematically, the cheetah’s states (other configurations) create a population, and each victim is a spot of a decision variable matching to the best option (see Figure 5a). Then, using an arbitrary step size and the present position of each cheetah in the arrangement as a starting point, this equation is suggested:

Figure 5.

(a–d) Graphical representation of cheetah’s hunting strategies.

$$X_{i,j}^{t+1}=X_{i,j}^t+r_{i,j}^{-1}·{\alpha }_{i,j}^t$$E17

Where $X_{i,j}^t$ signifies the present location of cheetah i$\left(\mathit{i}=1,2,\dots ,\mathit{n}\right)$ in group j (j = 1, 2, …, D),$\mathit{n}$ implies the population size, and $D$ denotes the size of the problem. $X_{i,j}^{t+1}$i signifies the following locations of cheetah $\mathit{i}$ in arrangement $\mathit{j}$, respectively. $\mathit{t}$ implies the present hunting time, and $\mathit{T}$ is the maximum duration of hunting period. $r_{i,j}^{-1}$ and ${\alpha }_{i,j}^t$ denote the random number and step length for cheetah $\mathit{i}$ in group $\mathit{j}$.

3.2.4 Sitting and waiting strategy

The prey might become visible to a cheetah’s area of vision, while it is in the searching mode. Each move the cheetah makes in this scenario has the potential to alert the victim to his or her existence and cause the prey to flee. The cheetah may decide to ambush to get sufficiently nearby to the prey in order to allay this worry (by reclining on the ground or lurking amid the bushes) [37]. As a result, the cheetah waits until the prey gets closer while maintaining his or her posture (see Figure 5b). The following can be used to simulate this behavior:

$$X_{i,j}^{t+1}=X_{i,j}^t$$E18

where $X_{i,j}^{t+1}$and $X_{i,j}^t$ denote the modified and present locations of cheetah $i$ in arrangement $j$. In order to boost hunting effectiveness (find a better solution), this method needs the CO approach to refrain from changing all cheetahs concurrently in each group. This can help the algorithm avoid early convergence.

3.2.5 Attacking strategy

A cheetah runs to the prey at maximum speed when it intends to attack. The prey eventually becomes aware of the cheetah’s onslaught and starts to flinch. As seen in Figure 5c, the cheetah swiftly chases the prey in the line of interception. The cheetah tracks the prey’s location and modifies its course so that it blocks the prey’s path at a certain point. The prey has to flee and change its location quickly in order stay alive because the cheetah has only traveled a short interval from it at full speed, as illustrated in Figure 5d, that is. the cheetah’s upcoming location is close to the prey’s last position [37]. During this phase, the cheetah captures the prey by moving quickly and maneuvering about. Each cheetah in a group hunt has the ability to change positions depending on the location of the leader or nearby cheetah and the location of the prey. Simply put, all of cheetahs’ attacking strategies can be defined numerically as follows:

$$X_{i,j}^{t+1}=X_{B,j}^t+r_{i,j}^{^}·{\beta }_{i,j}^t$$E19

where $X_{B,j}^t$ denotes the present location of the animal in group j. It is the present optimal location of the population. $r_{i,j}^{^}$ and ${\beta }_{i,j}^t$ signify the turning coefficient and interaction coefficient related to the cheetah$\mathit{i}$ in group $\mathit{j}$.

Based on the hunting behaviors of cheetahs, the proposed CO algorithm incorporates the following assumptions and strategies:

3.2.5.1 Individual representation

In the CO algorithm, each row in the population represents a cheetah in different states. Each column corresponds to a specific arrangement of cheetahs concerning the prey, representing the best solution for each decision variable. Cheetahs emulate the behavior of tracking their prey (the best value for a variable). To identify the optimal solution, cheetahs must successfully capture the prey in each arrangement. A cheetah’s performance is assessed through its fitness in all arrangements with higher performance indicating a greater likelihood of successful hunting.

3.2.5.2 Diverse reactions

Just as real cheetahs exhibit different reactions during group hunting, the CO algorithm allows each cheetah to be in various states in each arrangement. Some may be in attack mode while others are in searching, sitting-and-waiting, or attacking modes. Energy levels of cheetahs are considered independent of the prey, and the algorithm introduces random parameters to prevent premature convergence during extensive evolution processes. These random variables act as an energy source for the cheetahs during the hunting process.

3.2.5.3 Random behavior

The behaviors of cheetahs during searching and attacking strategies are assumed to be entirely random, ensuring diversity in the search. In contrast, during the rushing and capturing phases, the prey changes direction abruptly. Randomization parameters model these movements, and varying step lengths and interaction factors with random variables contribute to an effective optimization process.

3.2.5.4 Adaptive strategy

The choice between searching and attacking strategies is random, but searching becomes more likely as a cheetah’s energy decreases. Initial steps may be dedicated to searching, while attacking is preferred for larger values of time (t) to achieve better solutions. The selection of strategies is influenced by random factors and energy considerations, much like the cheetah’s behavior in the wild.

3.2.5.5 Scanning and sitting-and-waiting

In the CO algorithm, scanning and sitting-and-waiting strategies are considered equivalent, indicating that a cheetah (search agent) remains stationary during the hunting period.

3.2.5.6 Leader adaptation

If the lead cheetah consistently fails in hunting, a randomly selected cheetah’s position is changed to the last successful hunting position (i.e., the prey’s location). This approach maintains the prey’s position among a small population and strengthening the exploration phase.

3.2.5.7 Energy limitations and home range

Each group of cheetahs in the CO algorithm has a time limit for hunting due to energy constraints. If a group fails in a hunting period, they abandon the current prey and return to their home range (initial position). The leader’s position is also updated. This strategy helps prevent getting stuck in local optimum solutions.

3.2.5.8 Iterative evolution

In each iteration of the CO algorithm, a subset of the population actively participates in the evolutionary process.

These assumptions and strategies in the CO algorithm draw inspiration from the behavior of cheetahs during hunting, aiming to create an effective optimization technique that mimics their adaptability, randomness, and energy considerations in the quest for optimal solutions.

The fundamental stages of the CO algorithm can be depicted through the pseudo-code outlined in Algorithm 1, drawing inspiration from cheetah hunting tactics and underlying assumptions.

Algorithm 1: The CO methodology

Specify the problem data, dimension (D), and the initial population size (n).

Create the initial population of cheetahs $X_i\left(i=1,2,\dots ,n\right)$ and assess the fitness of each cheetah.

Set the starting solutions for the population’s home, leader, and prey positions.

$t\to 0$

$\mathit{it}\to 1$

$\mathit{\text{MaxIt}}\to \mathit{\text{maximum nubmer of iterations}}$

$T\to 60\times \frac{D}{10}$

$\mathit{\text{while}}\mathit{it}\le \mathit{\text{MaxIt}}$

 Randomly select m individuals ($2\le m\le n$)

 for each individual i

 Define its neighbor agent

  for each random arrangement $j\in \left\{1,2,\dots ,D\right\}$

  Calculate $r_{i,j}^{^},\alpha ,\beta ,H$

  $r_2,r_3\to \mathit{\text{random numbers between}}\left[0,1\right]$

  $\mathit{\text{if}}{r}_2\le r_3$

   $r_4\to 1\mathit{\text{random numbers between}}\left[0,3\right]$

   $\mathit{\text{if}}{r}_4\le H$

    Determine the updated position of a member using (19)

   Else

    Determine the updated position of a member using (17)

   End

  Else

    Determine the updated position of a member using (18)

   End

  End

 Revise the solutions of the member and the leader.

End

$t\to t+1$

$\mathit{\text{if}}t>\mathit{rand}\times T$ the leader’s position remains unchanged for a certain time, then execute the strategy of abandoning the current prey and returning to the home location, followed by adjusting the leader’s position.

End

$\mathit{it}\to \mathit{it}+1$

Modify the prey (global best) location

End

4. Results and discussion

The validation process of the introduced WaOA and CO optimization algorithms involved the precise determination of optimal parameters for various PV models. To achieve this, the estimating models, specifically the single-diode model (SDM) and the double-diode model (DDM), were leveraged to compute the PV characteristics, creating power-voltage curves and current-voltage profiles. A comprehensive comparative analysis was undertaken, juxtaposing the estimated performance of each module, both inter-module and against the reference datasheet values of the assessed cells and modules.

The application of the proposed optimization algorithms extended to a range of commercial solar cells and PV modules. This encompassed a standard silicon solar cell from RTC France and the Photo Watt-PWP 201 PV module. Precise measured data from multiple manufacturers’ datasheets and references (cited as Refs. [42, 43]) served as the foundational input for these evaluations.

In configuring the optimization process, each of the proposed techniques adhered to a standardized setup. The maximum iteration count was capped at 200 iterations, while each population was composed of 20 search agents. The upper and lower limits of the parameters have to be identified are tabulated in Table 1. To anchor the entire validation procedure, the MATLAB R2018a platform was utilized, providing a robust and reliable foundation for executing and assessing the optimization algorithms.

	SDM and DDM		PV module
	Min	Max	Min	Max
Iph (A)	0	1.00	0	2.00
Isd (μA)	0	1.00	0	50.00
Rs (Ω)	0	0.50	0	2.00
Rsh (Ω)	0	1000	0	10,000
n1, n2, n3	1.00	2.00	1.00	50.00

Table 1.

Boundary limits for the optimized parameters in solar cell models.

In this comprehensive validation procedure, the proposed WaOA and CO optimization algorithms have been rigorously evaluated and benchmarked against established models. The utilization of diverse PV modules, combined with standardized optimization parameters and a trusted computational platform, underscores the meticulous nature of this research effort. The results obtained from this meticulous validation endeavor contribute to the veracity and robustness of the optimization techniques introduced in this study.

4.1 R.T.C. France solar cell

In this study, the researchers employed a novel WaOA and CO optimization technique to identify the parameters of two distinct models of R.T.C. France solar cells. The I–V characteristic curves of the R.T.C. France solar cell were obtained from existing literature sources [41, 42]. To refine the parameters of the SDM (Single-Diode Model) and DDM (Double-Diode Model), the WaOA and CO optimization techniques were employed. The results, including those from the CO-based SDM and WaOA-based SDM models, are tabulated in Table 2. This table also encompasses parameter estimates from alternative optimization methods such as An.5-Pt. [44], LW [45], ABSO [46], Newton [50], CM [51], HS [47], PSO [48], GA [49], and PS [52]. Notably, the application of the proposed WaOA algorithm led to the lowest root mean square error (RMSE) value of 7.730062E-04 for the SDM model.

Technique	Iph (A)	Isd (μA)	Rs (Ω)	Rsh (Ω)	n	RMSE
WaOA	0.7607879	0.310682709	0.03654698	52.889880	1.47726717	7.730062E-04
CO	0.760772	0.32384593	0.036366463	53.8160016	1.48143923	7.7912E-04
TGA [16]	0.7606	0.21656	0.0383	50.3996	1.4419	9.90895E-04
COA [15]	0.7607692	0.3083945	0.0365546	52.826661	1.4765477659	7.75467E-04
ABSO [46]	0.76080	0.30623	0.03659	52.2903	1.47583	9.9124E-04
HS [47]	0.7607	0.30495	0.03663	53.5946	1.47538	9.9510E-04
PSO [48]	0.7607	0.400	0.0354	59.012	1.5033	1.3900E-03
GA [49]	0.7619	0.8087	0.0299	42.3729	1.5751	1.8704E-02

Table 2.

Optimized parameters for R.T.C. France solar cell – SDM.

Table 3 further presents outcomes from the utilization of the WaOA and CO techniques to extract parameters from the DDM of the R.T.C. FRANCE solar cell. In order to validate the effectiveness of the employed techniques, this table includes results from other methods such as ABSO [46], HS [47], PSO [48], GA [49], ABC [53], SBMO [54], SSO [55], and MSSO [55]. The results underscore the superior performance of the CO optimization technique, displaying a minimal RMSE objective function value of 7.631566E-04.

Technique	Iph (A)	Isd1 (μA)	Isd2 (μA)	Rs (Ω)	Rsh (Ω)	n1	n2	RMSE
WaOA	0.760814	0.280943	0.25799796	0.03672	53.40962	1.915624	1.4619251	7.631566E-04
CO	0.7607855	0.237803	0.28038851	0.036603	53.75604	1.98260	1.4688138	7.757378E-04
TGA[16]	0.7607	0.16740	0.22083	0.0356	58.2574	1.4999	1.4999	8.48824E-04
COA [15]	0.7607194	0.244676	0.380190	0.03692	53.51296	1.45635	1.989923	7.696186E-04
ABSO[46]	0.76078	0.26713	0.38191	0.03657	54.6219	1.46512	1.98152	9.8344E-04
ABC [53]	0.760813	0.192684	0.999587	0.036861	55.933515	1.438003	1.983721	9.8387E-04
SBMO [54]	0.760786	0.200798	0.74373	0.036917	55.104367	1.441256	1.947888	9.8485E-04
SSO [55]	0.760651	0.287201	0.065979	0.036255	55.853271	1.510345	1.433838	9.9129E-04
MSSO [55]	0.760748	0.234925	0.671593	0.036688	55.714662	1.454255	1.995305	9.8281E-04

Table 3.

Optimized parameters for R.T.C. France solar cell – DDM.

Furthermore, the study conducted a comprehensive comparison of the SDM and DDM models utilizing both the WaOA and CO optimization strategies. Figure 6 graphically illustrates the convergence patterns of the RMSE objective function for both SDM and DDM models of the R.T.C. France solar cell. Notably, the DDM model exhibited faster and more efficient convergence compared to the SDM model. To validate the accuracy of the identified parameters and the efficacy of the proposed optimization methodologies. Moreover, the study depicted the characteristics of the studied solar cell by plotting the estimated parameters against the measured ones for both SDM and DDM models. These plots are presented in Figures 7 and 8, respectively.

Figure 6.

Convergence curves for WaOA and CO for RTC France solar cell: (a) SDM and (b) DDM.

Figure 7.

Characteristics of RTC France solar cell SDM using WaOA: (a) I/V curve and (b) P/V curve.

Figure 8.

Characteristics of RTC France solar cell DDM using WaOA: (a) I/V curve and (b) P/V curve.

To establish the robustness and reliability of the optimization techniques, the researchers executed the optimization algorithms 20 times and recorded the best minimum objective function from each run. Statistical analysis was performed, encompassing metrics such as mean, standard deviation, relative error, as well as best and worst values over the 20 implements. The results of this statistical study are summarized in Table 4. Additionally, the final values of the fitness function through the 20 times are graphically presented in Figure 9. This analysis convincingly demonstrates that the WaOA algorithm is a highly effective approach for addressing the parameter identification optimization challenges across various mathematical models of the R.T.C. France solar cell.

	SDM		DDM
Index	WaOA	CO	WaOA	CO
Best	0.0007730062	0.0007791220	0.00076315	0.00077573785
Worst	0.000850113	0.0010605183	0.00113209	0.0014436476
Average	0.000785616	0.0008886448	0.00082953	0.0009957970
Median	0.000778313	0.000877633	0.00079522	0.0009532422
STD	0.002131738	0.008997239	0.00928636	0.0200651278
RE	0.326251792	2.81144352	1.73949234	5.67354655
RMSE	2.4304620e-05	0.0001403051	0.000112241	0.000294404

Table 4.

Statistical analysis of R.T.C. France solar cell for both SDM and DDM.

Figure 9.

Values of RMSE over the 20 runs for WaOA and CO methods for RTC France solar cell: (a) SDM and (b) DDM.

4.2 Photowatt-PWP201 module

To further validate the effectiveness of the employed WaOA and CO techniques, an assessment was carried out to estimate the parameters for various mathematical models of the Photowatt-PWP201 module. This module comprises 36 silicon in a series combination, operating under conditions of 1000 W/m² solar radiation and a cell temperature of 45°C [42, 43]. This chapter aimed to not only assess the proposed methodologies accuracy but also to compare its outcomes with alternative techniques in literature.

The prowess of the proposed algorithms was put to the test in parameter estimation for the SDM concerning the Photowatt-PWP201 module. The findings are detailed in Table 5, which additionally featured a comparative analysis against outcomes from other methods, including Newton [56], PS [52], OIS [57], and 1DAB [45]. This comparison effectively demonstrated the COA’s superiority in contrast to the other techniques. Notably, the application of WaOA yielded the lowest RMSE value of 0.00212629, indicating its exceptional performance.

Technique	Iph (A)	Isd (μA)	Rs (Ω)	Rsh (Ω)	n	RMSE
WaOA	1.0316505	3.287862	1.2076800	872.238281	1.3978671	0.00212629
CO	1.0315416	3.394903	1.20431447	891.945365	1.4013906	0.00225002
TGA[16]	1.0263	9.5710	0.0298	6842.2	1.5255	0.00381949
COA [15]	1.0296281	4.8155440	1.1728971	2000	1.4395994	0.00362202
Newton [56]	1.0318	3.2875	1.2057	555.5556	1.3474016	0.7805
PS [52]	1.0324	3.1859	1.304	843.5233	48.2467	0.0127
OIS [57]	1.03674	3.1946	1.32897	1184.58	49.0435	0.004783
1DAB[45]	1.04276	3.4265	1.73762	948.845	49.2843	0.00536

Table 5.

Optimized parameters for Photowatt-PWP201 module – SDM.

To further solidify the validation process, both WaOA and CO algorithms were employed to estimate parameters for the DDM of the Photowatt-PWP201 PV module. The optimized DDM parameters achieved through WaOA and CO are presented in Table 6. This table also facilitated a comparison between DDM-based WaOA, DDM-based CO, and other techniques such as WDOWOAPSO [50], GCPSO [58], TVACPSO [59], and ABC-DE [60]. The results highlighted the clear supremacy of the suggested WaOA technique, reflected in the minimized RMSE value of 0.00257992. Importantly, the cumulative results in Tables 5 and 6 substantiated the WaOA algorithm’s efficacy and high precision in parameter extraction for diverse models of the Photowatt-PWP201 module, showcasing a reduction in RMSE value.

Technique	Iph (A)	Isd1 (μA)	Isd2 (μA)	Rs (Ω)	Rsh (Ω)	n1	n2	RMSE
WaOA	1.034168	18.27992	1.13725	1.3265	561.174	3.44090	1.24156	2.5799E-03
CO	1.029743	6.457305	7.33164	1.1241	1587.47	1.42048	47.0336	4.9436E-03
TGA	1.0265	9.2998	2.2586	0.0301	6719.0	1.5225	1.4164	3.7559E-03
COA [15]	1.0265	9.2998	2.2586	1.2163	1019.78	1.34098	50	2.2090E-03
WDOWOAPSO [50]	1.03062	3.171702	5.00	1.2382	744.714	1.31730	1.31730	2.0465E-03
GCPSO [58]	1.032382	2.512916	1.00005	1.2392	744.715	1.31730	1.31693	2.0465E-03
TVACPSO [59]	1.031434	2.638124	1.00	1.2356	821.652	1.32099	2.77777	2.0530E-03
ABC-DE [60]	1.0318	0.32774	2.4305	1.2062	845.249	1.3443	1.3443	2.400E-03

Table 6.

Optimized parameters for Photowatt-PWP201 module – DDM.

The convergence trends of the RMSE based on both WaOA and CO optimization methods across the two proposed models are depicted in Figure 10. To visually demonstrate the application of the optimized parameters in the SDM and DDM, estimated I-V (Current-Voltage) and P-V (Power-Voltage) curves of the Photowatt-PWP201 module were generated. These curves, derived from the CO method, were compared with the experimental data, and the results are presented in Figures 11 and 12 for SDM and DDM, respectively. These figures clearly depicted the alignment between the estimated curves using the CO method and the empirical data.

Figure 10.

Convergence curves for WaOA and CO for Photowatt-PWP201 module: (a) SDM and (b) DDM.

Figure 11.

Characteristics of Photowatt-PWP201 module SDM using WaOA: (a) V/I curve and (b) V/P curve.

Figure 12.

Characteristics of Photowatt-PWP201 module DDM using WaOA: (a) V/I curve and (b) V/P curve.

In order to establish the robustness of the proposed optimization techniques, a statistical analysis was carried out. The final values of the fitness function through the 20 times are graphically presented in Figure 13. The outcomes of this analysis were summarized in Table 7, demonstrating the CO algorithm’s favorable performance with respect to statistical indices such as SD and RE for both estimated models. This comprehensive validation process solidified the CO algorithm’s reliability and effectiveness in parameter extraction across different models of the Photowatt-PWP201 PV module.

Figure 13.

Values of RMSE over the 20 runs for WaOA and CO methods for Photowatt-PWP201 module: (a) SDM and (b) DDM.

	SDM		DDM
Index	WaOA	CO	WaOA	CO
Best	0.002126291	0.002250025	0.002579920	0.0049436296
Worst	0.04809182	0.009377722	0.047042367	0.009555128
Average	0.01895375	0.006957418	0.015732435	0.007914192
Median	0.015838395	0.007241249	0.009331317	0.008001565
STD	1.4961108	0.216848518	1.48913560	0.14630791
RE	158.27992	41.84303421	101.9606167	12.01774000
RMSE	0.0222667117	0.005160113	0.019587075	0.003295119

Table 7.

Statistical analysis of Photowatt-PWP201 module for both SDM and DDM.

5. Conclusion

In this chapter, we have harnessed the Walrus Optimization Algorithm (WaOA) and the Cheetah optimizer (CO) to tackle the intricate optimization challenge of determining parameters for solar cells and a variety of PV modules. To thoroughly evaluate the effectiveness of our proposed optimization approach, we have leveraged data from manufacturer datasheets and real-world measurements gathered from literature sources. This comprehensive dataset encompasses diverse solar cells and PV modules, accounting for varying solar radiation intensities and temperatures. Our investigation has encompassed two distinct models—the single-diode model and the double-diode model—for both solar cells and PV modules.

The outcomes resulting from the application of the WaOA and CO have been systematically compared with findings previously documented in the literature pertaining to alternative optimization methodologies. Notably, our proposed algorithms have consistently outperformed others, consistently yielding optimal values for the objective function, often quantified through root mean square error (RMSE). This underscores the robustness and effectiveness of the Walrus Optimization Algorithm and Cheetah optimizer in this context.

Furthermore, we have not limited our assessment to comparative analysis alone; our exploration has delved deeper into the results. By subjecting the outcomes of solar cell parameter optimization to both parametric and nonparametric statistical scrutiny, we have fortified the confirmation of the efficacy of WaOA and CO in solving this optimization puzzle. A significant achievement has been the remarkable alignment between the V-I curves obtained through the optimized parameters and the corresponding data from manufacturer datasheets. This alignment serves as a clear indication of the validity of our approach.

For instance, when applied to the RTC France solar cell, the WaOA algorithm produced the lowest root mean square error (RMSE) values, measuring at 7.730062E-04 for the SDM model and 7.631566E-04 for the DDM model. Similarly, for the Photowatt-PWP201 module, the WaOA algorithm achieved minimal values for the objective function, measuring at 0.00212629 for the SDM and 0.00257992 for the DDM. These results are a testament to the effectiveness and reliability of our proposed optimization methods.

In summary, the Walrus Optimization Algorithm has proven its mettle by standing alongside established optimization algorithms, firmly establishing itself as a formidable contender for parameter extraction across a diverse range of solar cells and PV modules. Through rigorous testing and analysis, we have substantiated the potential of our methodology to make a significant contribution to the ongoing advancement of solar technology optimization.

For future directions, it is worth considering the application of these optimization algorithms to more complex solar systems, exploring potential enhancements in convergence speed, and adapting the approach to real time, dynamic operational scenarios for solar cells and PV modules. Additionally, the incorporation of machine learning and artificial intelligence techniques for predictive modeling and optimization could further advance the state of the art in solar technology.