Parameter Estimation of Three-Diode Photovoltaic Model Using Reinforced Learning-Based Parrot Optimizer with an Adaptive Secant Method

Abstract

In the developing landscape of photovoltaic (PV) technology, accuracy in simulating PV cell behaviour is dominant for enhancing energy conversion efficiency. This study introduces a new approach for parameter estimation in the three-diode PV model, a basis in the representation of PV cell characteristics. The methodology combines a reinforced learning-based parrot optimizer (RLPO) with an adaptive secant method (ASM) to fine-tune the parameters governing the PV model. The RLPO algorithm is inspired by the mimetic ability of parrots, i.e., foraging, staying, communicating, and fear noticed in trained Pyrrhura Molinae parrots, as it influences reinforced learning mechanisms to adaptively explore and exploit the search space for optimal parameter sets. Simultaneously, the ASM enhances the convergence rate through an iterative adjustment mechanism, responding to the curvature of the objective function, thereby ensuring accuracy in parameter estimation. The combination of the RLPO and ASM addresses the complexities and non-linearities inherent in the PV model, offering a robust framework for parameter estimation. Through extensive simulations, the proposed method demonstrated superior performance in terms of accuracy, convergence speed, and reliability when compared to existing algorithms. The empirical results emphasize the effectiveness of integrating a reinforced learning strategy with an adaptive method in handling the details of PV model parameterization. These outcomes show that the algorithm can handle issues related to optimization in PV systems, opening the door to progress in sustainable energy technologies.

1. Introduction

The global shift towards renewable energy sources highlights the importance of photovoltaic (PV) technologies in meeting the world’s increasing energy demands. The transition to renewable energy is essential to achieving global sustainability goals, and PV systems are at the forefront of this transformation. At the heart of these technologies lies the need for accurate PV models that can predict the performance of PV cells under various environmental conditions. Such models are crucial for designing and optimizing PV systems for maximum energy conversion efficiency [1,2,3]. Among the array of PV models developed, the single-diode model (SDM), double-diode model (DDM), and three-diode model (TDM) are key in understanding and simulating PV cell behaviour [4,5]. The SDM provides a basic representation, focusing on essential semiconductor physics to simulate PV cell operation. Its simplicity makes it widely used, though it may not capture all aspects of cell behaviour accurately [6,7]. The DDM adds complexity by incorporating a second diode to better account for recombination losses, offering improved accuracy over the SDM [8,9]. The most comprehensive of these is the TDM, which further elaborates on PV cell dynamics by including mechanisms for recombination losses in the depletion region, diffusion losses, and shunt resistance effects. This model offers the highest level of detail and accuracy in simulating PV cell behaviour [10,11,12]. However, the complexity of the TDM introduces significant challenges in parameter estimation. Accurate parameter estimation is critical across all PV models to ensure they accurately reflect the behaviour of actual PV cells. This process becomes more complex and crucial with more detailed models. Traditional methods for parameter estimation often struggle with the non-linear nature of the three-diode model, leading to potential inaccuracies and convergence issues [13,14,15].

In the domain of PV model parameter estimation, analytical and numerical methods have placed the foundational framework that supports the development of more complex and efficient optimization techniques. The exploration of these methods, their advantages, and their limitations sets the stage for understanding the shift towards metaheuristic methods and the motivation behind their hybridization with numerical methods [16,17]. Analytical methods for PV parameter estimation involve solving sets of equations derived from the electrical characteristics of PV cells and modules. These methods often employ simplified assumptions to make the problem tractable analytically. One common approach is the explicit extraction of diode parameters using equations that relate the open-circuit voltage, short-circuit current, and maximum power point to the diode’s ideality factors and series resistance. However, the simplified assumptions may not capture the full complexity of PV cell behaviour, especially under varying environmental conditions [18,19,20]. Numerical methods, including the Newton–Raphson method, secant method, Taylor series expansions, Lambert W function, and Levenberg–Marquardt algorithms, handle parameter estimation by iteratively refining parameter values to minimize the difference between the model data and experimental data. These methods are rooted in the calculus of optimization, employing derivatives to navigate the parameter space towards the optimum set of values [21,22]. The secant method is a numerical technique used for finding the roots of equations, an essential process in parameter estimation for PV models [23,24,25]. Unlike the Newton–Raphson method [11,26], which requires the derivative of the function, the secant method uses two initial guesses to approximate the derivative, making it particularly useful in scenarios where calculating the derivative is computationally expensive. However, the method does not guarantee convergence to a root, especially if the initial guesses are not close to the actual root. Taylor series expansions offer a method to approximate non-linear functions as a sum of polynomial terms based on derivatives of the function at a single point [27]. Taylor series can be used to linearize the equations governing the PV model, simplifying the estimation of parameters by breaking down complex, non-linear relationships into manageable linear components. However, the accuracy of the approximation depends on the number of terms included and the point around which the series is expanded. The Lambert W function, which solves equations of the form $x=y{e}^y$, is particularly relevant in PV parameter estimation for solving the transcendental equations that arise in modelling diode characteristics [28,29]. This function enables explicit solutions in certain PV model parameters, facilitating more direct and sometimes analytically controllable parameter estimation. However, its utility is limited to specific forms of equations, and not all parameter estimation problems can be formulated to fit these constraints.

The limitations of analytical and numerical methods, particularly concerning computational intensity and convergence problems, have motivated the exploration of metaheuristic methods. Metaheuristics, such as the Genetic Algorithm (GA) [30], Particle Swarm Optimization (PSO) [31], and others, offer a probabilistic approach to exploring the parameter space. The need for derivative information does not constrain these methods and can efficiently search large and complex spaces, potentially avoiding the pitfalls of local minima. Hybridizing metaheuristic methods with numerical methods leverages the strengths of both approaches [32,33,34]. Metaheuristic methods excel in globally searching the parameter space and identifying regions where the optimum parameters are likely to be found. Once these promising regions are identified, numerical methods can be employed to fine-tune the parameters with high precision. This hybrid approach aims to combine the global search capability and robustness of metaheuristics with the local search efficiency and accuracy of numerical methods. The hybrid approach can significantly reduce the computational time by using metaheuristics for the initial global search and then switching to faster, more precise numerical methods for local optimization. By combining these methods, the approach aims to achieve higher accuracy in parameter estimation, effectively addressing the complex behaviour of PV cells and systems. This methodology enhances the robustness of the optimization process, improving the reliability of parameter estimates across different PV models and conditions. While analytical and numerical methods have provided valuable insights into PV model parameter estimation, the evolution towards metaheuristic methods and their hybridization with numerical approaches represents a significant advancement in the field. This paradigm shift promises to address the inherent challenges of parameter estimation, paving the way for more accurate, efficient, and reliable PV system modelling and optimization [35,36,37].

To overcome these challenges, this paper introduces a methodology that combines the reinforced learning-based parrot optimizer (RLPO) with the adaptive secant method (ASM). This study proposes a new optimization strategy that not only improves modelling precision but also directly supports the development of efficient and sustainable energy solutions. This approach aims to improve the precision of parameter estimation for the three-diode model, thereby enhancing the predictive accuracy and efficiency of PV systems. By integrating RLPO’s adaptive exploration capabilities with the precision adjustments of the ASM, this method aims to address the limitations of conventional parameter estimation techniques, presenting a significant advancement in the simulation and optimization of PV cell performance. The following are the major contributions of this study:

• Development of a new variant of the parrot optimizer (PO) using a reinforcement learning-based method for solving real-time engineering problems.
• Development of a new adaptive secant method to improve the solution diversity and reduce the computational time.
• Validation of the proposed algorithm through simulations of different operating conditions.
• Performance comparison among other reputed algorithms.

The paper is organized as follows. Section 2 introduces the literature study on parameter estimation of various PV models using various algorithms. Section 3 discusses the mathematical modelling and the problem formulations. Section 4 presents the basic concepts of the parrot optimizer and also discusses in detail the proposed reinforced learning-based parrot optimizer. Section 5 comprehensively discusses the results obtained by various algorithms. Section 6 discusses the conclusion and future scope of this research.

2. Literature Review

Numerous researchers have identified limitations in traditional analytical and numerical methods when it comes to parameter estimation for PV models. Specifically, these conventional approaches often fall short due to their reliance on precise initial conditions, assumptions, and the necessity for explicit mathematical formulations. In contrast, metaheuristic algorithms, inspired by natural processes and phenomena, have emerged as superior in extracting and analysing PV parameters. These methods avoid the rigid prerequisites of analytical and numerical techniques, offering a more flexible and often more accurate approach to parameter estimation [38,39,40]. Metaheuristic algorithms, leveraging principles observed in natural systems, have gained widespread application in identifying optimal PV parameters. Their superiority over previous methodologies is notable in terms of both precision and computational efficiency. These algorithms excel by directly minimizing the difference between measured experimental data and model simulations, thereby refining the parameter extraction process [41,42,43]. Among the array of metaheuristic techniques employed are the GA [44], differential evolution [45], PSO [46], the artificial bee colony [47], cuckoo search [40], teaching–learning-based optimization [48], etc. Each of these methods contributes uniquely to the field, offering diverse strategies for directing the complex parameter space of PV models. The primary challenge associated with metaheuristic algorithms is their computational intensity. The search for optimal parameters across a broad and complex landscape can be time-consuming, often requiring significant computational resources. Additionally, the stochastic nature of metaheuristic algorithms means that achieving a global optimum solution is not guaranteed [49,50]. While these methods statistically improve the likelihood of identifying optimal parameters, they cannot assure absolute precision in every instance. The extraction and analysis of PV parameters through metaheuristic methods underscore a significant advancement in the field. These approaches demonstrate a marked improvement in accuracy and efficiency compared to traditional methods. However, the ongoing challenge of reducing computational time without compromising the quality of the solution remains. Future enhancements in metaheuristic algorithms will likely focus on optimizing computational efficiency, thereby broadening their applicability and effectiveness in PV parameter estimation [16,51].

The GA has gained widespread recognition for its effectiveness in solving various optimization challenges. This algorithm has been specifically utilized to deduce the electrical parameters of various PV cells. For instance, the authors of [30] developed a GA program aimed at extracting parameters of PV cells, showcasing the adaptability of GAs to PV parameter estimation. Contrastingly, the authors of [52] employed a numerical approach, the Newton–Raphson method, for determining the photo-generated current and diode reverse saturation current while utilizing the GA for extracting series resistance, shunt resistance, and the diode ideality factor. The integration of the GA with the interior point method further underscores efforts to refine the precision of PV model parameters [53]. An adaptive GA approach, guided by a multi-objective function focusing on a least mean square error and residual error reduction, has been proposed to locate optimal PV parameters, demonstrating the algorithm’s adaptability [54]. Differential evolution (DE) has been extensively applied to optimization tasks, including PV model parameter extraction [55]. Various adaptations of the DE algorithm have been crafted to enhance solution feasibility and quality. For example, a penalty-based DE algorithm aimed at generating viable solutions with improved quality was proposed, illustrating DE’s efficacy in accuracy and computational efficiency [56]. An adaptive DE algorithm, adjusting its control parameters based on fitness value, along with an improved adaptive DE method featuring a crossover rate sort mechanism and a dynamic population reduction scheme, highlights the continuous evolution of DE algorithms for better PV parameter extraction. Hybrid algorithms have emerged as a new approach in the extraction of PV model parameters, combining different optimization strategies to improve the strengths. The authors of merged DE with reinforcement learning (RL), employing a unique feedback loop during the duplication process to fine-tune parameters based on environmental model assessments. The authors of [57] introduced a gaining–sharing knowledge-based algorithm to handle PV model optimization problems. This algorithm operates through two distinct phases, transitioning from beginner–intermediate to intermediate–expert levels.

In addition, the authors of [58] proposed a comprehensive learning Rao-1 optimization technique, emphasizing the algorithm’s simplicity, ease of use, and cost-effectiveness while incorporating quantum and Levy flight strategies to enhance exploration and exploitation. Another advanced approach involved modifying the honey badger algorithm for PV parameter extraction, initially incorporating a Gauss map-based chaotic strategy for handling randomness in exploration and exploitation, complemented by opposition-based learning to improve search space exploration through contrasting candidate solutions [59]. The authors of [60] developed an adaptive Harris hawk optimizer augmented with sine–cosine differences, aiming to boost the algorithm’s global search capability. The modification of energy components sought to balance exploration and exploitation phases, demonstrating improved efficiency in CPU time reduction and parameter extraction across various PV models under different environmental conditions. These advancements reflect the dynamic and large landscape of optimization algorithms in the realm of PV parameter extraction, highlighting the shift towards more sophisticated, hybrid, and adaptive strategies to address the complexities of accurately modelling PV systems.

The development of the chaotic tunicate swarm algorithm represents a significant advancement in the predictive analysis of PV cell parameters [61,62]. This approach employs a unique mechanism characterized by an adaptive weight mathematical model designed to enhance the algorithm’s capability in efficiently navigating the search space. This model proficiently balances the exploratory and exploitative potential of the algorithm by dynamically responding to both negative and positive feedback, akin to the natural behaviour observed in tunicate swarms. In another notable development, the Gradient-Based Optimizer (GBO) was improved for detailed parameter analysis of PV modules, encompassing SDMs, DDMs, and TDMs [63,64,65,66]. This approach underscores the flexibility and adaptability of the GBO in addressing the complexities associated with various PV models, thereby enhancing the accuracy and reliability of parameter estimation across a broad range of PV systems. Further, an enhanced butterfly optimization algorithm was introduced specifically for the extraction of unknown parameters within the three PV models [67,68]. A significant feature of this algorithm is the incorporation of a dynamic control parameter meticulously designed to refine the position-updating equation adaptively. This adjustment mechanism, coupled with the introduction of a good-point set method for initial population generation, significantly augments the diversity and efficacy of the algorithm, ensuring a more comprehensive exploration of the parameter space. An improved arithmetic optimization algorithm was also proposed as a method for estimating PV cell parameters, highlighting the continuous origination within the field [69]. This approach not only emphasizes the refinement of optimization strategies but also includes the development of a specialized test bench designed to facilitate the accurate interpretation of measured I-V characteristics. This integration of experimental validation tools represents a holistic approach to enhancing the precision of parameter estimation. Moreover, the war strategy optimization algorithm was conceptualized for the extraction of unknown parameters in solar PV systems [70]. The incorporation of the algorithm with the Newton–Raphson technique further exemplifies the potential of hybrid approaches in increasing the accuracy and reliability of solution outputs. These advancements underscore the dynamic evolution of optimization algorithms in the context of PV parameter estimation, showcasing a trend towards more specialized, adaptive, and integrative strategies. Each of these methodologies contributes uniquely to the main goal of enhancing the precision and efficiency of PV system modelling and analysis, paving the way for more effective and reliable solar energy solutions.

The literature reveals a wide range of heuristic approaches successfully applied for extracting parameters from PV systems, highlighting the significant advancements in this area. These approaches include simulated annealing by [71,72], which simulates the physical process of heating and cooling to minimize system defects. Further notable methods include the artificial bee colony algorithm by [47], based on honey bees’ food foraging behaviour, and the harmony search-based algorithm (HSA) by [73], mimicking musicians’ improvisation process to find a harmonious state. Additionally, techniques like sunflower optimization by [74] and PSO by [31], along with several others, have been applied to PV parameter estimation, demonstrating the diversity of heuristic algorithms in this field. These heuristic and metaheuristic methods, despite showing improved performance and accuracy over deterministic approaches, often require many iterations to achieve varied results upon repeated applications. This has led to the development of enhanced methods aimed at reducing the number of iterations while ensuring consistent outcomes. These advancements indicate a continued effort to optimize the efficiency and reliability of extracting PV system parameters, addressing the challenge of a large number of iterations with more refined and effective optimization techniques.

Despite many metaheuristic algorithms proposed in the literature for obtaining PV model parameters, accurately and reliably finding these parameter values remains a challenge. This leads to an important question: Can we solve this problem with the current algorithms while still keeping accuracy and robustness? The “No-Free-Lunch (NFL)” theorem indicates that no single algorithm can solve every optimization problem perfectly [75]. An algorithm might work exceptionally well for one specific problem but may not achieve the same level of success for others. Therefore, the search for an all-encompassing metaheuristic algorithm that can effectively handle all types of optimization problems is still an open area for research. The NFL theorem is a key consideration in this field, encouraging the adaptation of existing algorithms for new types of problems. This motivated the authors to improve the exploration and exploitation of one of the very recent algorithms called the parrot optimizer by incorporating RL.

3. Mathematical Modelling and Problem Formulation

The PV panel system fundamentally operates in a manner similar to the diode activated by sunlight. This foundational similarity underpins the widespread application of Shockley diode-based equivalent circuits for the theoretical description of PV cell operations. Typically, a PV system is conceptualized as an ideal source of photocurrent paired with an ideal diode. However, realizing the ideal characteristics of the cells in practical applications is challenging due to the presence of various loss mechanisms and electrical conduction phenomena that significantly influence the operational characteristics of PV cells. Accurate modelling and analysis must account for these factors to ensure a realistic representation of PV system behaviour. In the context of PV system modelling, three primary models have garnered attention for their applicability and depth of detail: the SDM, DDM, and TDM [76,77]. While more simplistic models exist, they do not provide the same level of detail or accuracy. The SDM is widely utilized for its balance between simplicity and analytical precision, making it an optimal choice for applications demanding a manageable level of complexity without sacrificing significant accuracy. Conversely, the TDM, despite its increased complexity and the large number of variables it incorporates, is preferred for scenarios where a higher degree of modelling precision is required [78,79]. Central to the analysis of PV panel systems is the determination of the output current $I_{pv}$ based on the output voltage $V_{pv}$, which facilitates the derivation of the complete current–voltage (I-V) characteristics. These characteristics are crucial for assessing the performance efficiency of the PV system, highlighting the importance of selecting an appropriate modelling approach to capture the nuances of PV cell behaviour accurately.

3.1. Three-Diode Photovoltaic Model

The TDM equivalent circuit for the PV cell, as depicted in Figure 1, presents a sophisticated representation that integrates a photocurrent source with three parallel-connected diodes. Each diode within this configuration plays a distinct role in mimicking the various physical phenomena that occur within the PV cell. Diode D1 is tasked with replicating the diffusion process of minority carriers across the depletion layer, a critical mechanism for the generation of photocurrent in response to solar irradiation. This diode effectively models the movement of these carriers from regions of higher concentration to lower concentration within the cell, contributing to the overall current flow. The role of Diode D2 is to represent the recombination of carriers within the space-charge region of the junction. This area, crucial for the cell’s operation, is where the electric field assists in separating the photo-generated electron–hole pairs. However, some of these carriers recombine before contributing to the current loss mechanism that D2 accounts for within the model. Lastly, Diode D3 explores the effects of significant leakage currents and recombination losses, specifically in the defect regions of the cell. These areas, characterized by imperfections within the crystal lattice, can significantly impact the cell’s efficiency. D3 models the influence of these defects, which can lead to additional recombination and leakage paths for carriers, detracting from the cell’s overall performance. Through the incorporation of these three diodes, the TDM offers a comprehensive framework for understanding the intricate internal processes of the SPV cell, enabling more accurate predictions of its behaviour under various operating conditions.

Within the TDM of a PV cell, as illustrated in Figure 1, the incorporation of two lumped resistors is essential for capturing the inherent electrical losses. These resistors are denoted as the series resistance, $R_{se}$, and the shunt resistance, $R_{sh}$, and serve distinct functions in the model. The series resistance accounts for the resistive losses that occur as the current travels through the conductive paths of the cell, including the metal contacts and semiconductor material. This resistance directly influences the fill factor and overall efficiency of the PV cell by reducing the voltage output. Conversely, the shunt resistance models the leakage current paths within the cell, where current bypasses the intended path due to defects or imperfections, leading to a decrease in the cell’s open-circuit voltage. The photocurrent, $I_{ph}$, represents the current generated by the PV cell in response to solar irradiation at a specific ambient temperature. This current is a direct measure of the cell’s ability to convert solar energy into electrical energy and is influenced by the intensity of the incident light and the cell’s characteristics.

To describe the behaviour of the TDM under a given irradiance and operating temperature, Kirchhoff’s current law is applied to the equivalent circuit. This approach enables the formulation of the following equations that account for the currents flowing through each diode, the generated photocurrent, and the current contributions due to the series and shunt resistances. By integrating these elements, the model provides a comprehensive understanding of the electrical behaviour of the PV cell, facilitating the analysis of how changes in irradiance, temperature, and cell properties affect its performance.

$$I_{pv}=I_{ph}-I_{d1}-I_{d2}-I_{d3}-I_{sh}$$

(1)

$$I_{d1}=I_{sd1}\left[\mathrm{e}\mathrm{x}\mathrm{p}\left({\displaystyle \frac{V_{pv}+{I_{pv}.R}_{se}}{n_1{V}_{th}}}\right)-1\right]$$

(2)

$$I_{d2}=I_{sd2}\left[\mathrm{e}\mathrm{x}\mathrm{p}\left({\displaystyle \frac{V_{pv}+{I_{pv}.R}_{se}}{n_2{V}_{th}}}\right)-1\right]$$

(3)

$$I_{d3}=I_{sd3}\left[\mathrm{e}\mathrm{x}\mathrm{p}\left({\displaystyle \frac{V_{pv}+{I_{pv}.R}_{se}}{n_3{V}_{th}}}\right)-1\right]$$

(4)

$$V_{th}={\displaystyle \frac{K.T}{q}}$$

(5)

$$I_{sh}={\displaystyle \frac{V_{pv}+{I_{pv}.R}_{se}}{R_{sh}}}$$

(6)

In the context of the TDM for a PV cell, $I_{pv}$ and $I_{ph}$ denote the output current of the cell and the photocurrent, respectively. The photocurrent is the current generated due to the absorption of solar radiation. $I_{d1}$, $I_{d2}$, and $I_{d3}$ represent the currents flowing through each of the three diodes within the model. Correspondingly, $I_{sd1}$, $I_{sd2}$, and $I_{sd3}$ are the reverse saturation currents for each of these diodes, which characterize the current through a diode when it is reverse-biased. The variables $V_{pv}$, $R_{se}$, and $R_{sh}$ signify the terminal voltage of the PV cell, the series resistance, and the shunt resistance, respectively. The series resistance accounts for the resistive losses within the cell’s pathways, while the shunt resistance represents the leakage current paths across the cell. $V_{th}$, the thermal voltage, is a critical parameter that depends on the cell’s operating temperature and embodies the energy provided by thermal agitation to the charge carriers. The Boltzmann’s constant is denoted as $K$, playing a pivotal role in defining the thermal voltage alongside the absolute temperature $T$ and the charge of an electron $q$. The ideality factors of the diodes, represented by $n_1$, $n_2$, and $n_3$, indicate the deviation of each diode’s behaviour from that of an ideal diode. These factors are crucial for accurately modelling the diode currents under various operating conditions. To fully characterize the TDM of a PV cell, nine parameters must be identified: $I_{ph}$, $I_{sd1}$, $I_{sd2}$, $I_{sd3}$, $n_1$, $n_2$, $n_3$, $R_{se}$, and $R_{sh}$. These parameters are essential for accurately predicting the cell’s performance and understanding how it responds to changes in environmental conditions and operating states.

3.2. Problem Formulation

The primary aim is to develop a mathematical model for a PV cell or module by accurately determining the various unknown parameters that define this model. To achieve this, curve-fitting techniques are applied to the current–voltage (I-V) curve of a three-diode PV module at several points along the curve. The objective function, which measures the difference between the measured and modelled currents, is typically evaluated using the root mean square error (RMSE) method [80,81,82]. The RMSE is defined as

$$\mathrm{RMSE}=\sqrt{{\displaystyle \frac{1}{N}}{\sum }_{i=1}^N{\left(I_{e,i}-I_{m,i}\right)}^2}$$

(7)

$$\begin{array}{c}{I}_m=I_{ph}-I_{sd1}\left[\mathrm{e}\mathrm{x}\mathrm{p}\left({\displaystyle \frac{V_{pv}+{I_{pv}.R}_{se}}{n_1{V}_{th}}}\right)-1\right]-I_{sd2}\left[\mathrm{e}\mathrm{x}\mathrm{p}\left({\displaystyle \frac{V_{pv}+{I_{pv}.R}_{se}}{n_2{V}_{th}}}\right)-1\right]-\\ I_{sd3}\left[\mathrm{e}\mathrm{x}\mathrm{p}\left({\displaystyle \frac{V_{pv}+{I_{pv}.R}_{se}}{n_3{V}_{th}}}\right)-1\right]-{\displaystyle \frac{V_{pv}+{I_{pv}.R}_{se}}{R_{sh}}}\end{array}$$

(8)

where $I_{e,i}$ and $I_{m,i}$ represent the experimental and modelled current values at the $i^{th}$ point on the I-V curve, respectively, and $N$ is the total number of data points. The modelled current $I_m$ for a PV cell, incorporating the effects of three diodes, can be expressed through a complex relationship involving the output voltage, series resistance, shunt resistance, and the parameters encapsulated within vector $\mathsf{\theta }$, which includes the photocurrent $\left(I_{ph}\right)$, diode saturation currents, and ideality factors, among others. Equation (8) takes into account the exponential behaviour of diode currents and the linear behaviour associated with resistive losses. Due to the non-linear nature of Equation (8), which prevents a direct solution, Equation (8) has been reformulated as Equation (9).

$$\begin{array}{c}f\left(I_m\right)=I_{ph}-I_{sd1}\left[\mathrm{e}\mathrm{x}\mathrm{p}\left({\displaystyle \frac{V_{pv}+{I_{pv}.R}_{se}}{n_1{V}_{th}}}\right)-1\right]-I_{sd2}\left[\mathrm{e}\mathrm{x}\mathrm{p}\left({\displaystyle \frac{V_{pv}+{I_{pv}.R}_{se}}{n_2{V}_{th}}}\right)-1\right]-\\ I_{sd3}\left[\mathrm{e}\mathrm{x}\mathrm{p}\left({\displaystyle \frac{V_{pv}+{I_{pv}.R}_{se}}{n_3{V}_{th}}}\right)-1\right]-{\displaystyle \frac{V_{pv}+{I_{pv}.R}_{se}}{R_{sh}}}-I_m\end{array}$$

(9)

The ultimate goal is to minimize the RMSE by adjusting the model parameters $\mathsf{\theta }$, thereby aligning the modelled I-V characteristics as closely as possible with the experimental data, which is indicative of a successful parameter estimation process and, by extension, a reliable mathematical model of the PV cell or module.

The objective function in Equation (6) is characterized as a transcendental equation. To avoid the complexities associated with solving transcendental equations, some researchers have approximated the estimated current to match the measured current. However, to enhance the precision of the results, this study has addressed this issue by incorporating an adaptive secant method during the evaluation of the objective function. In numerical analysis, the secant method is a technique employed to find the root of a function $f$ [23,25]. This method serves as a viable alternative to Newton’s method in situations where the function $f$ is non-differentiable or lacks a clear derivative expression, $f^{\prime }\left(x\right)$. Therefore, instead of using the tangent line as in Newton’s method, the secant line is used to approximate the curve of the function. Consider that the function $f$ has a root within the interval $\left[x_{-1},x_0\right]$. The initial step towards estimating this root involves replacing $f$ with its linear interpolation between $x_{-1}$ and $x_0$, expressed as

$$f\left(x\right)=f\left(x_0\right)+{\displaystyle \frac{\left(x-x_0\right)\left(f\left(x_0\right)-f\left(x_{-1}\right)\right)}{x_0-x_{-1}}}$$

(10)

This expression defines a unique linear function that matches the values of $f$ at $x_{-1}$ and $x_0$. The next approximation, $x_1$, is derived by setting $f\left(x_1\right)=0$, leading to

$$x_1=x_0-{\displaystyle \frac{f\left(x_0\right)\left(x_0-x_{-1}\right)}{f\left(x_0\right)-f\left(x_{-1}\right)}}$$

(11)

Geometrically, as illustrated in Figure 2, this approximation is similar to replacing the curve $y=f\left(x\right)$ with the secant line AB, where $x_{n+1}$ is the intersection point of AB with the x-axis.

As depicted in Figure 2, $x_{n+1}$ is closer to the actual root than either $x_n$ or $x_{n-1}$. By iterating this process with the new points $x_n$ and $x_{n+1}$, it is possible to refine the approximation. This iterative method generates a sequence as follows:

$$x_{n+1}=x_n-{\displaystyle \frac{f\left(x_n\right)\left(x_n-x_{n-1}\right)}{f\left(x_n\right)-f\left(x_{n-1}\right)}}$$

(12)

where $n=0, 1, 2, \dots , \mathrm{\infty }$. The secant method, while efficient and straightforward for finding the roots of equations, has some drawbacks that can affect its performance in certain situations. One primary limitation is its dependency on the choice of initial points. If these points are not close to the actual root, the method can converge slowly or even diverge. Additionally, the secant method assumes the function to be approximately linear in the interval of interest, which might not hold true for functions with high curvature, leading to inaccurate approximations. Another challenge with the secant method is that it does not guarantee convergence to a root if the function exhibits steep gradients or multiple roots in the vicinity of the initial estimates. This can result in oscillations or a failure to converge, especially in complex functions derived from physical models like those in PV cell analysis.

To address these limitations and enhance the performance of the secant method, an adaptive approach can be introduced, as illustrated by the use of an adaptive parameter, $\mathsf{\lambda }$, in the given algorithm. The adaptive parameter $\mathsf{\lambda }$ is dynamically adjusted based on the iteration count $t$, modulating the step size as the iterations proceed [83,84]. This adaptation aims to refine the approximation of the root with increased precision over iterations, especially as the solution nears the true root. The $\mathsf{\lambda }$ is defined as follows.

$$\mathsf{\lambda }=\exp \left(-{\left(5\cdot t/T\right)}^{2.5}\right)$$

(13)

where $T$ denotes the maximum number of iterations, and the term $\mathsf{\lambda }$ allows the method to adjust the impact of each iteration’s correction based on the progress towards convergence. Early in the iteration process, when far from the root, $\mathsf{\lambda }$ allows for larger adjustments, facilitating faster convergence. As the solution approaches the actual root, $\mathsf{\lambda }$ decreases, making better adjustments that help in stabilizing the convergence and reducing the likelihood of overshooting the root or oscillating around it. In the context of PV cell modelling, where the objective function $f\left(x\right)$ and its derivative $f^{\prime }\left(x\right)$ involve complex exponential terms that represent the cell’s I-V characteristics, the adaptive secant method enhances the ability to estimate the PV parameters accurately. By adjusting the update step for the estimated current $I_m\left(i\right)$ using $\mathsf{\lambda }$, the method improves robustness and convergence speed, especially in the non-linear and complex landscape of PV cell analysis. This adaptive modification is particularly beneficial for dealing with the non-linearities and the multiple parameters involved in PV models, ensuring a more reliable and efficient root-finding process.

The sequence converges to the solution $x_{\mathrm{\infty }}$, and to ensure the convergence meets the desired precision, a stopping criterion is established such that

$$\left|x_{n+1}-x_n\right|\le \mathsf{\beta }$$

(14)

This approach systematically narrows down the approximation to the root, enhancing the accuracy of the solution with each iteration until the specified precision threshold is met.

4. Proposed Reinforced Learning-Based Parrot Optimizer

This section briefly introduces the concepts of the parrot optimizer and reinforced learning. Also, this section discusses the formulation of the proposed algorithm.

4.1. Parrot Optimizer

The parrot optimizer (PO) is a new and effective optimization technique that draws inspiration from distinctive behaviours, such as foraging, staying, communicating, and fear of strangers, exhibited by trained Pyrrhura Molinae parrots [85].

The initial setup for the PO algorithm, given a population size of $N$, a maximum number of iterations $T$, and the search domain defined by the lower bound $\left(lb\right)$ and upper bound $\left(ub\right)$, is articulated as follows:

$$X_i^0=lb+\mathrm{rand}\left(\mathrm{0,1}\right)\times \left(ub-lb\right)$$

(15)

Here, $\mathrm{rand}\left(\mathrm{0,1}\right)$ generates a uniformly distributed random value within the interval [$\mathrm{0,1}$], and $X_i^0$ represents the initial position of the $i^{\mathrm{th}}$ Pyrrhura molinae within the defined search space. Within the PO framework, the foraging behaviour is modelled by estimating the food’s location through direct observation or by gauging the position of the parrot’s owner, subsequently propelling the parrot towards the identified location. The algorithm considers this behaviour via the following positional update equation.

$$X_i^{t+1}=\left(X_i^t-X_{best}\right)\cdot \mathit{Levy}\left(\mathit{dim}\right)+\mathit{rand}\left(\mathrm{0,1}\right)\cdot {\left(1-{\displaystyle \frac{t}{T}}\right)}^{{\displaystyle \frac{2t}{T}}}\cdot X_{mean}^t$$

(16)

In this equation, $X_i^t$ signifies the current position of the $i$^th parrot, whereas $X_i^{t+1}$ indicates the parrot’s position at the subsequent time step. The term $X_{mean}^t$ denotes the average position within the current population, reflecting a collective orientation towards the food source. The Levy distribution, represented as $\mathit{Levy}\left(\mathit{dim}\right)$, characterizes the flight patterns of parrots, capturing the stochastic and long-range movements observed in their search for food. The best position encountered from the beginning of the search up to the current iteration is denoted by $X_{best}$, which also symbolizes the position of the parrot’s owner. The component $\left(X_i^t-X_{best}\right)\cdot \mathrm{Levy}\left(\dim \right)$ models the parrot’s movement relative to the owner or the best-known location, incorporating a random component influenced by the Levy distribution to simulate the parrot’s exploratory flights. Additionally, $(\mathrm{rand}\left(\mathrm{0,1}\right)\cdot {\left(1-{\displaystyle \frac{t}{T}}\right)}^{{\displaystyle \frac{2t}{T}}}\cdot X_{mean}^t)$ captures the tendency to consider the population’s overall direction towards the food source, progressively refining the search orientation as the optimization process advances. This dual approach enables a balanced exploration and exploitation strategy, guiding the algorithm towards optimal solutions. The Levy random distribution function is represented in Equation (15), where the value of $\gamma$ is selected as 1.5.

$$\left\{\begin{array}{c}Levy\left(\dim \right)={\displaystyle \frac{\mu \cdot \sigma }{{\left|v\right|}^{\frac{1}{\gamma }}}}\hfill \\ \mu \sim N(0,\dim )\hfill \\ v\sim N(0,\dim )\hfill \\ \sigma =({\displaystyle \frac{\Gamma (1+\gamma )\cdot \mathit{sin}(\frac{\pi \gamma }{2})}{\Gamma \left(\frac{1+\gamma }{2}\right)\cdot \gamma \cdot 2^{\frac{1+\gamma }{2}}}}{)}^{\gamma +1}\hfill \end{array}\right.$$

(17)

The Pyrrhura Molinae exhibits highly social behaviour, particularly demonstrated through its tendency to fly abruptly towards and then remain perched on a specific part of its owner’s body for an extended duration. This behavioural pattern is mathematically modelled as follows:

$$X_i^{t+1}=X_i^t+X_{best}\cdot \mathrm{Levy}\left(\dim \right)+\mathrm{rand}\left(\mathrm{0,1}\right)\cdot \mathrm{ones}\left(1,\dim \right)$$

(18)

where $\mathrm{ones}\left(1,\dim \right)$ represents a vector of dimension $\dim$ filled with ones. The term $X_{best}\cdot \mathrm{Levy}\left(\dim \right)$ models the parrot’s flight towards its owner, reflecting the influence of the owner’s position on the parrot’s movement, with the Levy distribution capturing the variability in flight patterns. Additionally, $\mathrm{rand}\left(\mathrm{0,1}\right)\cdot \mathrm{ones}\left(1,\dim \right)$ symbolizes the parrot’s action of selecting a random stopping point on the owner’s body. This equation encapsulates the combination of targeted movement towards the owner under the guidance of $X_{best}$ and the stochastic, exploratory behaviour influenced by both the Levy distribution and random positioning, thereby reflecting the parrot’s natural staying behaviour within the optimization process.

The parrots are known for their social nature and exhibit distinct communication behaviours within their groups, including converging towards the flock for interaction and engaging in communication without physically joining the flock. In the PO, these two modes of behaviour are modelled to occur with an equal likelihood, with the average position of the current population representing the flock’s centroid. This concept is mathematically expressed as

$$X_i^{t+1}=\left\{\begin{array}{c}0.2\cdot rand\left(\mathrm{0,1}\right)\cdot (1-{\displaystyle \frac{t}{T}})\cdot (X_i^t-X_{mean}^t), P\le 0.5\hfill \\ 0.2\cdot rand\left(\mathrm{0,1}\right)\cdot \mathit{exp}(-{\displaystyle \frac{t}{rand\left(\mathrm{0,1}\right)\cdot T}}), P>0.5\hfill \end{array}\right.$$

(19)

where the expression $0.2\cdot rand\left(\mathrm{0,1}\right)\cdot (1-{\displaystyle \frac{t}{T}})\cdot (X_i^t-X_{mean}^t)$ symbolizes the action of a parrot moving towards the group to partake in communication, while $0.2\cdot rand\left(\mathrm{0,1}\right)\cdot \mathit{exp}(-{\displaystyle \frac{t}{rand\left(\mathrm{0,1}\right)\cdot T}})$ represents a parrot departing immediately post-interaction. The decision to engage in either behaviour is governed by a probability $P$, generated randomly within the interval $[0, 1]$. This approach facilitates the emulation of both congregative and solitary communication behaviours observed in parrots, thereby enriching the PO’s mimicry of natural flock dynamics. By nature, avian species commonly display an instinctual caution towards unfamiliar entities, a trait shared by Pyrrhura Molinae parrots. This cautious behaviour is characterized by a tendency to distance themselves from unknown individuals and gravitate towards their owners for a sense of security. The mathematical model capturing this behaviour is formulated as follows:

$$\begin{array}{c}{X}_i^{t+1}=X_i^t+\mathrm{rand}\left(\mathrm{0,1}\right)\cdot \cos \left(0.5\mathsf{\pi }\cdot {\displaystyle \frac{t}{T}}\right)\cdot \left(X_{\mathrm{best}}-X_i^t\right)-\cos \left(\mathrm{rand}\left(\mathrm{0,1}\right)\cdot \mathsf{\pi }\right)\cdot {\left({\displaystyle \frac{t}{T}}\right)}^{{\displaystyle \frac{2}{T}}}\cdot \\ \left(X_i^t-X_{\mathrm{best}}\right)\end{array}$$

(20)

Here, the term $\mathrm{rand}\left(\mathrm{0,1}\right)\cdot \cos \left(0.5\pi \cdot {\displaystyle \frac{t}{T}}\right)\cdot \left(X_{\mathrm{best}}-X_i^t\right)$ mathematically represents the parrot’s action of realigning its flight towards the owner, signifying a move towards safety and familiarity. Conversely, $-\cos \left(\mathrm{rand}\left(\mathrm{0,1}\right)\cdot \pi \right)\cdot {\left({\displaystyle \frac{t}{T}}\right)}^{{\displaystyle \frac{2}{T}}}\cdot \left(X_i^t-X_{\mathrm{best}}\right)$ depicts the motion away from unfamiliar entities or strangers. This dual-component model encapsulates the instinctual behaviours of avoidance and seeking refuge, reflecting the Pyrrhura Molinae’s navigational choices when faced with the unfamiliar. Algorithm 1 presents the pseudo-code of the PO algorithm.

Algorithm 1: Pseudo-code for the PO algorithm

Initialize PO parameters Randomly initialize the positions of solutions For $i = 1 \mathrm{t}\mathrm{o} T$ do      Calculate the fitness function for each solution      Determine the best and worst positions among the solutions      For $j = 1 \mathrm{t}\mathrm{o} N$ do      Randomly select a strategy $S$ from 1 to 4 (Uniform random distribution, i.e., $S=randi\left(\right[\mathrm{1,4}]$).      If $S$ equals 1 (Foraging Behavior)       Update the position according to Equation (16).      Else If $S$ equals 2 (Staying Behavior)       Update the position according to Equation (18).      Else If $S$ equals 3 (Communicating Behavior)       Update the position according to Equation (19).      Else If $S$ equals 4 (Fear of Strangers Behavior)       Update the position according to Equation (20).      End If      End For Return the best solution found End For

4.2. Reinforcement Learning-Based Parrot Optimizer

The integration of reinforcement learning (RL) principles into the PO algorithm represents a new approach to enhancing its exploration and exploitation capabilities [86,87,88]. This fusion aims to dynamically adjust the behaviour of solutions based on their performance history, fostering a more efficient search process that is adaptive to the problem landscape. RL is a type of machine learning where an agent learns to make decisions by taking actions in an environment to achieve some goals. The agent receives rewards or penalties based on the outcomes of its actions and uses this feedback to learn the best strategy or policy for maximizing cumulative rewards over time [89,90,91]. The core idea behind integrating RL into optimization algorithms like the PO is to treat the selection of solution behaviours as a decision-making process, where the goal is to minimize the objective function in optimization problems.

In the context of the PO, each solution (or “parrot”) can exhibit different behaviours (e.g., foraging, staying, communicating, fear of strangers) as it navigates the search space. The RL-enhanced PO introduces a Q-learning framework, where each behaviour is associated with a Q-value representing its expected utility or effectiveness in improving the solution’s fitness. The Q-values are updated based on the rewards received, which are determined by the improvements in the fitness values following the application of the behaviours. Q-learning is a model-free reinforcement learning algorithm employed to update the Q-values without requiring a model of the environment [86]. It uses the following update rule:

$$Q\left(s,a\right)\leftarrow Q\left(s,a\right)+\mathsf{\alpha }\left[R+\mathsf{\gamma }\underset{a^{\prime }}{\max }Q\left(s^{\prime },a^{\prime }\right)-Q\left(s,a\right)\right]$$

(21)

where $Q\left(s,a\right)$ is the Q-value of taking action $a$ in state $s$, $\mathsf{\alpha }$ is the learning rate (=0.25), $R$ is the reward received after taking action, $\mathsf{\gamma }$ is the discount factor (=0.65), and $\underset{a^{\prime }}{\max }Q\left(s^{\prime },a^{\prime }\right)$ represents the maximum Q-value achievable in the next state $s^{\prime }$. This update rule allows the algorithm to improve its estimates of the Q-values iteratively, guiding the solutions towards behaviours that are more likely to improve their fitness. During the reinforcement learning update, the step size (represented here by the learning rate α) is the responsibility of the learning rate α. After testing several of the candidates (such as 0.1, 0.2, 0.25, and 0.3), this value was chosen. The best compromise between fast convergence and stability during updates was 0.25. The learning process was slowed down by lower values of α, and the divergence and instability were caused by higher values of α. By choosing 0.25 as a prefixed value, it was possible to achieve stable and efficient learning of the parameters. The reinforcement learning framework emphasizes the future rewards, and this is set by the discount factor γ. The study used a grid search from 0.5 to 0.8 and picked a value of 0.65. This value helped the algorithm strike a balance between how immediate rewards and long-term optimizer goals could be optimized. The algorithm was short-sighted with smaller values of γ (i.e., ≤0.5) and only cared about immediate gains; larger values of γ (i.e., ≥0.5) made the algorithm converge slowly, giving excessive value to future rewards. It selected a value of 0.65 to ensure an optimal trade-off.

Behaviour selection is conducted through an ε-greedy strategy, balancing exploration and exploitation. With a probability of $\mathsf{ϵ}$, a random behaviour is selected to encourage the exploration of new strategies. Conversely, with a probability of $(1-\mathsf{ϵ})$, the behaviour with the highest Q-value is chosen, favouring the exploitation of known effective strategies. This approach ensures that the PO algorithm does not prematurely converge to suboptimal solutions and remains capable of discovering better solutions throughout the optimization process. The integration of RL into the PO enhances the algorithm’s adaptability, allowing it to dynamically adjust its search strategy based on the evolving landscape of the optimization problem. This can lead to improved convergence rates and the ability to escape local optima. However, the challenge lies in appropriately balancing the learning rate and the exploration–exploitation trade-off to prevent the algorithm from becoming too greedy or too exploratory, which could reduce the performance. The pseudo-code of the proposed RLPO is shown in Algorithm 2.

Algorithm 2: Pseudo-code of the RL-based parrot optimizer (RLPO)

Initialize the parameters of the proposed algorithm ($N$, $T$, $lb$, $ub$, $\gamma$, and $\alpha$) Initialize population positions within [$lb$, $ub$] and evaluate the fitness of each parrot Initialize Q-values for each of the 4 behaviours For $t=1 to T$ do     For $i = 1 to N$ do      Select behaviour for parrot $i$ using an ε-greedy strategy based on Q-values      Apply behaviour b to update position considering boundaries $[lb, ub]$      Evaluate the new fitness of parrot $i$      If new fitness is better than previous:        Update the parrot’s position and fitness        Calculate reward based on improvement      Else:        Calculate penalty      Update Q-value based on the received reward or penalty      If new fitness is better than best fitness:        Update global best with new position and fitness     Update $\epsilon$ to decrease exploration over time Return: Global best as the best solution found and its fitness as the best score.

4.3. Time and Space Complexity

Several factors, including the size of the population, the number of dimensions in the search space, the maximum number of iterations, and the complexity of the objective function, influence the time and space complexity of the RL-based PO. Additionally, the inclusion of RL concepts introduces complexity related to managing and updating the Q-values for each behaviour strategy.

The initialization of the population and Q-values is $O\left(N\cdot dim\right)$, as it requires setting up initial positions for each individual in the population across all dimensions and initializing the Q-values for the behaviour strategies. The main loop of the algorithm, which includes updating positions based on selected behaviours, evaluating the objective function, and updating Q-values, runs for $T$ iterations. The complexity within each iteration involves the following: (i) $O\left(N\right)$ for selecting behaviours for each individual and applying the behaviours, which might include additional operations depending on the behaviour logic. However, the complexity of applying each behaviour can generally be considered $O\left(dim\right)$, making this step $O\left(N\cdot dim\right)$. (ii) $O\left(N\cdot C_{fobj}\right)$, where $C_{fobj}$ is the complexity of evaluating the objective function once; and (iii) $O\left(N\right)$ for updating the Q-values based on the rewards received after applying the behaviours. Given these considerations, the overall time complexity per iteration is $O\left(N\cdot dim+N\cdot C_{fobj}+N\right)$, which simplifies to $O\left(N\cdot \left(dim+C_{fobj}+1\right)\right)$. Over iterations, the time complexity is $O\left(T\cdot N\cdot \left(dim+C_{fobj}+1\right)\right)$.

$O\left(N\cdot dim\right)$ is used for storing the positions of all individuals in the population. $O\left(M\right),$ where $M$ is the number of behaviours, and $O\left(N\right)$ for storing the fitness value of each individual in the population apply. Additional space is required for intermediate calculations, including new positions and fitness values, but this does not significantly exceed $O\left(N\cdot dim\right).$ Thus, the space complexity is dominated by the storage of the population’s positions and the fitness values, resulting in $O\left(N\cdot dim+N\right)=O\left(N\cdot \left(dim+1\right)\right)$.

5. Results and Discussions

In an extensive investigation, the RLPO method, when integrated with an adaptive secant method, was accurately analysed to verify its dependability, stability, accuracy, and convergence efficiency. This evaluation utilized experimental datasets sourced from the ST40 PV module under varied operational conditions. Moreover, a comparative study was performed against other algorithms employing several performance indicators to benchmark its efficacy. The ST40 PV module was selected for its relevance and extensive use in photovoltaic system research. The specific characteristics and performance metrics of this PV module were carefully logged under standardized test conditions, as outlined in Table 1. For each testing scenario, a comprehensive dataset was collected, with the total count of data samples for these scenarios methodically listed in Table 2. This organization offers a clear view of the experimental scope undertaken in this study. The evaluation of the RLPO approach, alongside a comparison with other algorithms, was anchored on the following metrics:

• RMSE: Quantifies the discrepancies between the model’s predictions and the actual observed values, calculated as $\mathrm{RMSE}=\sqrt{{\displaystyle \frac{1}{N}}{\sum }_{i=1}^N{\left(I_{pv,m}-I_{pv,e}\right)}^2}$.
• Relative Error (RE): Provides an error measurement relative to the true value, calculated as $RE={\displaystyle \frac{\left|I_{pv,m}-I_{pv,e}\right|}{I_{pv,m}}}$.
• Absolute Error (AE): Directly measures the magnitude of error, computed as $AE=\left|I_{pv,m}-I_{pv,e}\right|$.
• Determination Coefficient (R²): Illustrates the variance percentage in the dependent variable that can be predicted from the independent variable(s), with $R^2=1-{\displaystyle \frac{{\sum }_{i=1}^N{\left(I_{pv,m}-I_{pv,e}\right)}^2}{{\sum }_{i=1}^N{\left(I_{pv,m}-I_{pv,m,avg}\right)}^2}},\mathrm{w}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{e} I_{pv,m,avg}={\displaystyle \frac{1}{N}}{\sum }_{i=1}^N{I}_{pv,m}^2$.
• Mean Bias Error (MBE): Signifies the model’s average bias in predictions, determined by $MBE={\displaystyle \frac{1}{N}}{\sum }_{i=1}^N\left(I_{pv,m}-I_{pv,e}\right)$.
• Standard Deviation (STD): Assesses the variability or dispersion among a set of values.

Table 1. Specifications of the ST40 PV module.

Parameters	Specifications
Maximum power voltage	16.6 V
Maximum power current	2.41 A
Maximum power	40 W
Open-circuit voltage	23.3 V
Short-circuit current	2.68 A
Temperature coefficient of open-circuit voltage	−100 × 10−3 V/°C
Temperature coefficient of short-circuit current	0.35 × 10−3 A/°C

Table 1. Specifications of the ST40 PV module.

Parameters	Specifications
Maximum power voltage	16.6 V
Maximum power current	2.41 A
Maximum power	40 W
Open-circuit voltage	23.3 V
Short-circuit current	2.68 A
Temperature coefficient of open-circuit voltage	−100 × 10−3 V/°C
Temperature coefficient of short-circuit current	0.35 × 10−3 A/°C

Table 2. Experimental scenarios of the ST40 module under operating conditions.

	1000 W/m2				25 °C
Operating conditions	25 °C	40 °C	50 °C	70 °C	800 W/m2	600 W/m2	400 W/m2	200 W/m2
	C1	C2	C3	C4	C5	C6	C7	C8
Data samples	26	25	23	22	26	25	25	23

Table 2. Experimental scenarios of the ST40 module under operating conditions.

	1000 W/m2				25 °C
Operating conditions	25 °C	40 °C	50 °C	70 °C	800 W/m2	600 W/m2	400 W/m2	200 W/m2
	C1	C2	C3	C4	C5	C6	C7	C8
Data samples	26	25	23	22	26	25	25	23

The proposed RLPO algorithm was contrasted with various algorithms, all augmented with an adaptive secant method. These encompassed the RLPO, the Opposition-Based GBO (OBGBO) [66], the Opposition-Based Exponential Distribution Optimizer (OBEDO) [92], the Reinforced Learning-Based Grey Wolf Optimizer (RLGWO) [89,91], the RL-based Particle Swarm Optimizer (RLPSO) [89], the Leader Slime Mould Algorithm (LSMA) [93], Heterogeneous Differential Evolution (HDE) [14], and the original PO [85]. The selection of these algorithms for the comparative study aimed to ensure a comprehensive and diverse analysis across a broad range of methodologies within the domain. The strategic hybridization with an adaptive secant method facilitated a balanced comparison, enabling a reasonable and practical evaluation of the performance across varied algorithms.

To maintain a fair and unbiased comparison across various optimization algorithms, the environmental conditions for their evaluation were carefully standardized. This step is critical in optimization algorithm research to ensure that comparisons are valid and reliable, as uniform testing conditions eliminate potential biases stemming from different starting points. This principle, widely recognized in previous studies on optimization algorithms, highlights the necessity of consistent experimental settings [94,95]. A key component of ensuring a level playing field in this comparative analysis was the standardization of the number of function evaluations for all algorithms under consideration. This uniformity guarantees that each algorithm is afforded an equal chance to adjust and improve its proposed solutions. The count of search agents, pivotal in determining an algorithm’s search capability, was set at 40 for every algorithm, facilitating a direct comparison of their efficiency and effectiveness under identical conditions. The experiment also defined a maximum iteration limit of 1000 for each algorithm, a parameter that significantly impacts the algorithm’s convergence behaviour and its success in identifying optimal solutions. Additionally, the problem’s dimensionality was established at nine, a choice that directly influences the search space’s complexity and the algorithm’s proficiency in exploring and exploiting this space. Recognizing the inherent variability and stochastic characteristics of these optimization algorithms, each was executed 30 times [96,97]. Furthermore, to enhance the study’s transparency and facilitate its reproducibility, the control parameters for all selected algorithms were precisely determined and are provided in

Table 3. Various parameters of the algorithms under consideration.

Algorithm	Parameters	Values
PO	$\beta$	1.5
RLPO	β	1.5
	α	0.25
	γ	0.65
RLGWO	a	Linearly decreases from 2 to 0
	α	0.25
	γ	0.65
RLPSO	ω	0.9
	$c_1$ and $c_2$	2
	$\alpha$	0.25
	$\mathsf{\gamma }$	0.65
OBEDO	$\alpha$	0.5
OBGBO	$\epsilon$	0.01
	$pr$	0.5
LSMA	$z$	0.03
	$b$	Linearly decreases from 1 to 0
HDE	p	0.1
	H	5

, allowing for the accurate imitation of the experiment by other researchers or the evaluation of new algorithms under equivalent conditions, promoting further research and validation in the field of optimization algorithms.

In the optimization of various parameters, specific search bounds were established for each parameter to delineate the optimization algorithm’s search space for optimal solutions. For ideality factor $n_1$, the bounds were defined from 1 to 2, reflecting the typical behaviour observed in PV panels. The search range for the ideality factors $n_2$ and $n_3$ was broadened to 1 to 5 to accommodate the variability seen in certain PV panel types, especially in multi-crystalline modules. This adjustment acknowledges that while ideality factors commonly range from 1 to 2 across PV panels, they may escalate, particularly in thin film modules, due to an increase in defect density. Such a rise in the ideality factor increases the electron–hole recombination rates, often resulting from an increased presence of Donor–Acceptor pairs. Additionally, the ideality factor is susceptible to the effects of joule heating, which can reduce the factor over time, thereby impacting the PV module’s efficiency. The reverse saturation current for the diodes, denoted as $I_{sd1}$, $I_{sd2}$, and $I_{sd3}$, was set within a range of 0 to 100 µA, mirroring the typical currents measured in PV diodes under reverse bias conditions. The series resistance parameters were limited to a range of 0 to 2 Ω, encapsulating the common resistance values found in the internal wiring and connections of PV modules. The shunt resistance was given a wider range of 0 to 5000 Ω to account for the variability in shunt resistance across different PV modules, which can markedly influence their performance. Lastly, the range for photocurrent was established between 0 and twice the short-circuit current, with the upper limit set to reflect the maximal current generation under optimal conditions.

Each algorithm was configured with a consistent population size of 40 and executed for a maximum span of 1000 iterations. The findings from these simulations, especially across different case studies, are displayed in

Table 4. TDM parameters of KC200GT PV module estimated by all algorithms.

Case Study	Algorithm	Iph (A)	Isd1 (A)	Rse (Ω)	Rsh (Ω)	n1	Isd2 (A)	n2	Isd3 (A)	n3
C1	RLPO	2.6760	8.63 × 10−9	1.1193	354.31	4.1887	1.82 × 10−7	4.9995	1.45 × 10−6	1.7439
	PO	2.6606	1.00 × 10−4	1.1600	1941.43	4.8415	1.53 × 10−7	1.5238	1.00 × 10−4	3.0125
	OBGBO	2.6703	1.51 × 10−5	1.0480	483.09	4.0874	2.60 × 10−6	1.8145	5.82 × 10−5	4.7785
	RLPSO	2.6573	3.74 × 10−6	0.7502	2597.55	2.3719	1.71 × 10−7	2.3369	2.15 × 10−5	2.1317
	LSMA	−0.2566	2.67 × 10−1	264.71	835.83	1.0788	5.41 × 10−1	6.2599	−1.00 × 102	2.5847
	RLGWO	2.6589	3.74 × 10−5	0.9445	1515.05	4.1075	6.57 × 10−6	1.9426	2.21 × 10−5	3.2635
	HDE	2.6760	4.67 × 10−6	1.1202	355.87	4.9851	1.43 × 10−6	1.7422	4.31 × 10−6	4.9909
	OBEDO	2.6736	1.00 × 10−4	1.1009	422.17	5.0000	1.00 × 10−4	5.0000	1.47 × 10−6	1.7454
C2	RLPO	2.6812	5.41 × 10−6	1.1334	358.21	1.7165	2.67 × 10−11	4.6837	0.00 × 100	4.9991
	PO	2.6778	9.94 × 10−5	1.0731	455.31	5.0000	8.22 × 10−6	1.7706	9.96 × 10−5	5.0000
	OBGBO	2.6734	7.61 × 10−6	1.0039	560.96	4.9695	1.36 × 10−5	3.0469	1.36 × 10−5	1.8390
	RLPSO	2.6674	9.73 × 10−5	0.9409	4314.89	2.9302	3.47 × 10−5	2.8206	2.22 × 10−5	1.9398
	LSMA	2.3675	2.08 × 10−1	158.60	4330.35	0.4133	−6.81 × 10−1	0.6192	−8.02 × 10−1	3.1605
	RLGWO	2.6660	4.27 × 10−5	1.0564	1733.81	3.1788	3.32 × 10−6	1.6777	5.09 × 10−5	2.4430
	HDE	2.6812	1.62 × 10−10	1.1334	358.21	4.8665	9.62 × 10−10	4.9997	5.41 × 10−6	1.7165
	OBEDO	2.6808	1.00 × 10−4	1.1339	374.92	5.0000	0.00 × 100	5.0000	5.19 × 10−6	1.7113
C3	RLPO	2.6904	1.69 × 10−13	1.1229	317.75	4.5116	2.33 × 10−5	1.7492	3.40 × 10−14	4.9967
	PO	2.6797	1.00 × 10−4	0.9782	728.09	4.9310	3.72 × 10−5	3.7480	6.75 × 10−5	1.9169
	OBGBO	2.6807	1.49 × 10−5	1.1237	549.94	3.2956	4.29 × 10−6	5.0000	2.95 × 10−5	1.7860
	RLPSO	2.6735	6.86 × 10−5	0.9441	2937.45	3.5476	6.68 × 10−5	2.5268	6.70 × 10−5	1.9276
	LSMA	3.0902	−1.18 × 101	179.40	2461.61	0.6797	5.61 × 100	2.2959	1.07 × 101	1.1642
	RLGWO	2.6797	2.47 × 10−7	0.9555	787.59	3.3146	3.68 × 10−5	1.8488	9.15 × 10−5	2.3270
	HDE	2.6904	9.61 × 10−8	1.1229	317.76	5.0000	6.86 × 10−12	4.9992	2.33 × 10−5	1.7492
	OBEDO	2.6891	1.00 × 10−4	1.1039	344.31	5.0000	2.63 × 10−5	1.7662	0.00 × 100	1.0000
C4	RLPO	2.6930	6.97 × 10−5	1.1472	365.82	1.6924	6.14 × 10−5	3.5672	9.89 × 10−5	3.4249
	PO	2.6856	2.59 × 10−5	1.1997	591.43	4.9982	1.00 × 10−4	2.3705	4.59 × 10−5	1.6406
	OBGBO	2.6913	5.44 × 10−5	1.1271	375.29	1.6973	4.02 × 10−5	1.8372	3.12 × 10−5	5.0000
	RLPSO	2.6795	9.99 × 10−5	0.9054	2834.31	1.8913	9.99 × 10−5	1.9207	9.99 × 10−5	2.0497
	LSMA	2.2897	−9.95 × 10−1	170.01	4844.27	0.6088	7.19 × 10−1	1.7404	−5.39 × 10−1	4.2969
	RLGWO	2.6850	2.34 × 10−5	1.1040	657.98	1.9749	9.84 × 10−5	1.7591	3.36 × 10−7	3.0636
	HDE	2.6926	1.00 × 10−4	1.1537	383.14	3.2120	9.77 × 10−5	2.8335	5.90 × 10−5	1.6692
	OBEDO	2.6930	7.92 × 10−5	1.1374	355.15	1.7113	0.00 × 100	5.0000	9.66 × 10−5	5.0000
C5	RLPO	2.6760	1.45 × 10−6	1.1194	354.37	1.7438	1.53 × 10−7	5.0000	5.37 × 10−7	4.9092
	PO	2.6616	1.00 × 10−4	1.2826	1719.81	2.9201	1.21 × 10−7	4.9998	1.02 × 10−8	1.3157
	OBGBO	2.6707	1.23 × 10−5	1.0473	540.24	4.9386	3.12 × 10−5	2.7935	1.65 × 10−6	1.7687
	RLPSO	2.6661	3.54 × 10−6	0.6633	967.37	1.9782	3.88 × 10−5	2.3644	3.14 × 10−5	2.8067
	LSMA	2.4251	4.82 × 10−1	212.85	1362.59	1.2111	−7.69 × 100	0.8734	1.64 × 100	3.1965
	RLGWO	2.6655	2.09 × 10−5	0.8595	757.83	3.0206	1.81 × 10−6	1.8837	9.80 × 10−6	2.0864
	HDE	2.6758	2.82 × 10−5	1.1227	361.16	4.8604	1.36 × 10−6	1.7370	5.55 × 10−6	4.9872
	OBEDO	2.6736	0.00 × 100	1.0539	393.97	5.0000	0.00 × 100	5.0000	2.51 × 10−6	1.8096
C6	RLPO	2.1382	1.09 × 10−5	1.1390	331.24	4.8464	1.04 × 10−6	1.7062	0.00 × 100	1.0000
	PO	2.1241	6.31 × 10−5	1.0332	815.92	4.2965	3.35 × 10−6	1.8547	9.93 × 10−6	4.2683
	OBGBO	2.1316	1.96 × 10−5	1.0146	422.86	3.2827	1.68 × 10−6	1.7590	1.12 × 10−5	5.0000
	RLPSO	2.1152	7.21 × 10−6	0.8159	4097.27	3.6336	1.08 × 10−5	2.0172	8.75 × 10−10	4.0451
	LSMA	−0.4905	−1.32 × 10−1	939.53	2581.04	3.0508	1.60 × 10−1	5.1556	3.84 × 10−1	5.5430
	RLGWO	2.1298	7.75 × 10−6	0.8232	549.53	1.9696	2.73 × 10−5	4.3172	4.39 × 10−5	3.7817
	HDE	2.1382	1.06 × 10−6	1.1369	330.15	1.7092	1.76 × 10−6	4.9983	1.19 × 10−8	4.9960
	OBEDO	2.1264	0.00 × 100	0.6718	686.15	1.0000	1.88 × 10−5	2.1095	0.00 × 100	5.0000
C7	RLPO	1.6049	6.05 × 10−7	1.1232	346.03	5.0000	1.36 × 10−6	1.7386	1.03 × 10−9	4.9697
	PO	1.5977	4.68 × 10−6	1.0509	537.65	1.9933	3.46 × 10−5	3.5136	9.06 × 10−8	1.5340
	OBGBO	1.6011	1.40 × 10−6	1.1086	450.23	1.7461	3.92 × 10−5	4.5916	1.80 × 10−5	3.0373
	RLPSO	1.6021	1.19 × 10−5	0.6476	462.08	2.0546	1.01 × 10−5	3.5326	8.38 × 10−5	3.6943
	LSMA	−8.1819	5.71 × 102	201.58	1302.96	1.7744	−9.8 × 10−1	5.6157	3.63 × 100	0.4987
	RLGWO	1.5960	4.15 × 10−5	0.8056	683.76	3.0592	8.54 × 10−6	2.4768	3.89 × 10−6	1.8977
	HDE	1.6049	1.29 × 10−6	1.1286	349.73	1.7319	2.22 × 10−7	4.9161	2.55 × 10−5	4.9737
	OBEDO	1.6011	0.00 × 100	0.9074	429.32	5.0000	4.76 × 10−6	1.9049	0.00 × 100	5.0000
C8	RLPO	1.0679	1.00 × 10−4	1.2094	369.35	4.9767	1.04 × 10−6	1.7080	5.62 × 10−8	4.7167
	PO	1.0626	1.34 × 10−5	0.9009	504.81	3.7927	1.00 × 10−4	4.9972	4.12 × 10−6	1.8934
	OBGBO	1.0650	4.26 × 10−5	0.6733	411.73	4.9926	7.44 × 10−5	5.0000	4.45 × 10−6	1.8951
	RLPSO	1.0586	1.89 × 10−5	0.0814	823.60	2.4394	4.11 × 10−5	2.3835	1.81 × 10−6	2.8115
	LSMA	2.0078	8.84 × 100	378.32	1937.33	0.6172	2.36 × 10−1	6.2047	1.27 × 100	3.3094
	RLGWO	1.0631	3.95 × 10−6	0.2145	437.95	3.0625	5.29 × 10−5	3.4349	9.20 × 10−6	2.0084
	HDE	1.0679	9.41 × 10−5	1.2148	369.14	4.9431	1.02 × 10−6	1.7050	5.77 × 10−6	4.8998
	OBEDO	1.0651	3.27 × 10−6	0.8402	432.65	1.8559	1.00 × 10−4	5.0000	1.00 × 10−4	5.0000

. This table elaborates on the comparative analysis of nine critical parameters of the TDM of the photovoltaic module as evaluated by each algorithm. Table 4 reveals a predictable decrease in photocurrent corresponding with falling irradiance levels, a result that is in agreement with established physical principles indicating that the photocurrent in PV modules is directly proportional to irradiance intensity. The ideality factors for the diodes were adjustable up to a maximum value of 5, a range specifically chosen to cover potential variances in defect density and other intrinsic properties of PV modules. The observed values for the reverse saturation currents of the diodes, labelled $I_{sd1}$, $I_{sd2}$, and $I_{sd3}$, largely spanned from 2.33 × 10⁻⁵ to 2.67 × 10¹¹. These findings are linked to various factors, such as shifts in ambient temperature impacting photon absorption and an increase in defect density that influences the diode’s ideality factor, with a noted positive relationship between these currents and temperature. It was observed that an adaptive secant technique, while highly responsive to fluctuations in ideality factor values, could prompt divergences in the RLPO algorithm’s optimization process if ideality factors are inaccurately set. A significant decrease was observed in the series and shunt resistance values, moving from roughly 4844.27 and 1.1334 to 317.75 and 1.1193, respectively. Based on the insights from Table 4, the RLPO algorithm exhibited superior parameter dispersion capabilities, underscoring its proficiency in accurately identifying the root values of the parameters and maintaining stability throughout its process. The most favourable results are highlighted in bold, making it straightforward to pinpoint the optimum outcomes yielded by each algorithm.

The detailed findings illustrated in Table 4 effectively demonstrate the ability of various algorithms to simulate the parameters of the PV module, with particular importance placed on the RLPO algorithm’s superior performance. This algorithm distinguishes itself through its precise parameter estimation and consistent operational stability throughout the optimization process, as exposed through an in-depth analysis of the TDM parameters of the PV module. The careful recording of these results, notably with optimal results emphasized, provides critical insights into the effectiveness of these algorithms in optimizing PV modules. Figure 3 offers a comprehensive graphical depiction of the current–voltage (I-V) and power–voltage (P-V) behaviours predicted for the TDM of a PV module employing the RLPO method and all other algorithms in combination with an adaptive secant technique. This visualization is crucial for demonstrating the proposed and other methods’ accuracy in capturing PV module performance. The curves portrayed showcase the dynamic between I-V and P-V, essential for defining the electrical characteristics of the PV module under diverse operational scenarios. A significant takeaway from Figure 3 is the notable alignment of the RLPO algorithm’s predictions with empirical data. The reliability of I-V and P-V characteristic predictions highlights the practical relevance of the RLPO algorithm. It suggests the algorithm’s reliability in forecasting PV module performance.

Figure 3 provides a graphical representation illustrating the proposed algorithm’s robustness and accuracy in modelling the behaviour of PV modules across varied operational scenarios. The close correlation of its outcomes with empirical data not only confirms the algorithm’s efficacy but also highlights its significant applicability in photovoltaic research. The precise simulation of I-V and P-V curves is crucial for enhancing the design and functionality of PV systems, positioning the RLPO as a crucial asset in advancing renewable energy technologies. To showcase the superior capabilities of the proposed algorithm, improved by an adaptive secant method, extensive comparative analyses were conducted against a range of algorithms applied for PV panel optimization under diverse conditions. The outcomes of these examinations are concisely summarized in

Table 5. Evaluation metrics of the selected algorithms of the KC200GT PV module.

Metrics	Algorithm	RLPO	PO	OBGBO	RLPSO	LSMA	RLGWO	HDE	OBEDO
RMSE	C1	6.473 × 10−6	6.992 × 10−6	5.984 × 10−6	4.460 × 10−6	5.110 × 10−6	4.060 × 10−6	2.136 × 10−6	1.052 × 10−6
	C2	4.561 × 10−5	1.677 × 10−5	3.619 × 10−5	2.823 × 10−5	4.579 × 10−5	1.840 × 10−5	9.881 × 10−6	1.592 × 10−6
	C3	2.291 × 10−5	3.137 × 10−5	3.438 × 10−5	1.419 × 10−5	2.622 × 10−5	1.311 × 10−5	7.876 × 10−6	2.535 × 10−6
	C4	9.088 × 10−5	8.860 × 10−5	5.287 × 10−5	4.960 × 10−5	7.688 × 10−5	3.726 × 10−5	2.336 × 10−6	1.328 × 10−5
	C5	3.119 × 10−5	4.632 × 10−5	2.785 × 10−5	2.591 × 10−5	2.855 × 10−5	1.367 × 10−5	4.821 × 10−6	1.976 × 10−6
	C6	6.355 × 10−5	4.707 × 10−5	4.115 × 10−5	2.282 × 10−5	4.107 × 10−5	2.862 × 10−5	1.558 × 10−5	1.932 × 10−6
	C7	6.471 × 10−6	6.992 × 10−6	5.984 × 10−6	4.654 × 10−6	5.113 × 10−6	4.064 × 10−6	2.137 × 10−6	1.063 × 10−6
	C8	1.093 × 10−5	7.258 × 10−6	7.531 × 10−6	4.950 × 10−6	5.638 × 10−5	1.520 × 10−5	6.316 × 10−6	2.041 × 10−6
	Mean	4.533 × 10−6	2.531 × 10−5	1.907 × 10−5	5.409 × 10−5	2.254 × 10−5	3.272 × 10−5	4.560 × 10−6	1.383 × 10−5
R2	C1	1.00000	0.98751	0.98987	0.99000	0.96588	0.98756	0.98985	0.98756
	C2	1.00000	0.98746	0.98746	0.98000	0.97459	1.00000	1.00000	0.99875
	C3	1.00000	0.99000	0.99874	0.96700	0.96651	0.97530	1.00000	0.98875
	C4	1.00000	0.99995	0.96589	0.98970	0.96524	0.98786	1.00000	0.98369
	C5	1.00000	1.00000	0.97854	0.98966	0.97846	0.94759	1.00000	0.98746
	C6	1.00000	0.97854	0.99357	0.99000	0.97419	0.99875	0.98573	0.97336
	C7	1.00000	1.00000	1.00000	0.99000	0.96423	0.99854	1.00000	0.98756
	C8	1.00000	0.96589	0.98556	1.00000	0.97856	1.00000	0.98578	1.00000
	Mean	1.00000	0.98867	0.98745	0.98704	0.97096	0.98695	0.99517	0.98839
MBE	C1	1.001 × 10−9	3.014 × 10−8	3.934 × 10−8	1.174 × 10−6	2.260 × 10−8	1.621 × 10−7	1.031 × 10−9	4.266 × 10−9
	C2	1.603 × 10−9	5.183 × 10−8	3.046 × 10−7	2.041 × 10−7	4.400 × 10−8	1.025 × 10−7	1.603 × 10−9	1.662 × 10−9
	C3	4.003 × 10−9	1.464 × 10−7	2.213 × 10−8	2.364 × 10−7	1.877 × 10−8	2.253 × 10−7	4.001 × 10−9	1.138 × 10−8
	C4	2.791 × 10−9	1.134 × 10−7	0.237 × 10−8	1.375 × 10−6	1.683 × 10−8	7.794 × 10−9	3.120 × 10−9	1.129 × 10−9
	C5	1.679 × 10−9	5.167 × 10−8	2.412 × 10−7	7.605 × 10−7	2.387 × 10−8	4.554 × 10−7	1.523 × 10−9	1.258 × 10−6
	C6	9.322 × 10−10	1.465 × 10−8	8.857 × 10−9	9.142 × 10−8	1.007 × 10−8	4.616 × 10−8	9.579 × 10−10	1.672 × 10−8
	C7	7.300 × 10−9	1.056 × 10−8	2.768 × 10−7	1.841 × 10−7	3.434 × 10−9	1.660 × 10−6	7.881 × 10−9	4.303 × 10−8
	C8	5.221 × 10−10	3.134 × 10−9	9.258 × 10−9	2.308 × 10−7	1.771 × 10−9	6.966 × 10−9	4.918 × 10−10	6.066 × 10−9
	Mean	2.479 × 10−9	5.273 × 10−8	1.143 × 10−7	5.321 × 10−7	1.767 × 10−8	3.333 × 10−7	2.576 × 10−9	1.677 × 10−7
AE	C1	5.444 × 10−6	3.436 × 10−5	1.955 × 10−5	7.776 × 10−5	1.992 × 10−5	5.479 × 10−5	5.439 × 10−6	8.474 × 10−6
	C2	5.635 × 10−6	1.447 × 10−5	2.647 × 10−5	7.476 × 10−5	4.340 × 10−5	3.870 × 10−5	5.635 × 10−6	6.027 × 10−6
	C3	4.357 × 10−6	3.194 × 10−5	2.911 × 10−5	4.482 × 10−5	1.875 × 10−5	3.695 × 10−5	4.357 × 10−6	5.734 × 10−6
	C4	3.684 × 10−6	2.445 × 10−5	1.271 × 10−5	3.867 × 10−5	1.813 × 10−5	1.706 × 10−5	3.725 × 10−6	3.695 × 10−6
	C5	4.358 × 10−6	3.652 × 10−5	1.956 × 10−5	6.285 × 10−5	2.396 × 10−5	3.544 × 10−5	4.368 × 10−6	4.794 × 10−5
	C6	3.588 × 10−6	1.532 × 10−5	1.159 × 10−5	2.997 × 10−5	9.836 × 10−6	2.512 × 10−5	3.572 × 10−6	1.284 × 10−5
	C7	1.596 × 10−6	8.380 × 10−6	5.890 × 10−6	2.062 × 10−5	2.970 × 10−6	1.065 × 10−5	1.782 × 10−6	5.244 × 10−6
	C8	9.037 × 10−7	1.348 × 10−6	2.126 × 10−6	1.176 × 10−5	1.850 × 10−6	1.536 × 10−6	8.806 × 10−7	1.564 × 10−6
	Mean	3.696 × 10−6	2.085 × 10−5	1.588 × 10−5	4.515 × 10−5	1.735 × 10−5	2.753 × 10−5	3.720 × 10−6	1.144 × 10−5
RE	C1	2.362 × 10−7	2.081 × 10−8	4.155 × 10−6	1.932 × 10−5	1.021 × 10−5	5.263 × 10−6	2.573 × 10−7	8.966 × 10−7
	C2	4.125 × 10−8	5.309 × 10−6	1.550 × 10−5	5.027 × 10−6	2.018 × 10−5	5.993 × 10−6	4.264 × 10−8	1.104 × 10−7
	C3	7.627 × 10−7	7.183 × 10−6	1.804 × 10−8	9.076 × 10−6	9.896 × 10−6	7.989 × 10−6	7.616 × 10−7	1.869 × 10−6
	C4	1.185 × 10−6	9.482 × 10−6	1.375 × 10−6	3.380 × 10−5	9.972 × 10−6	4.169 × 10−7	1.196 × 10−6	5.150 × 10−7
	C5	5.813 × 10−7	1.539 × 10−6	1.397 × 10−5	1.682 × 10−5	1.355 × 10−5	1.363 × 10−5	5.124 × 10−7	1.921 × 10−5
	C6	1.029 × 10−7	2.205 × 10−8	1.272 × 10−6	4.723 × 10−7	7.510 × 10−6	3.522 × 10−7	1.248 × 10−7	3.604 × 10−6
	C7	1.628 × 10−6	8.521 × 10−8	1.179 × 10−5	4.387 × 10−6	3.887 × 10−6	2.740 × 10−5	1.706 × 10−6	4.271 × 10−6
	C8	2.950 × 10−8	4.883 × 10−7	3.764 × 10−7	7.890 × 10−6	4.021 × 10−6	7.469 × 10−7	5.021 × 10−9	8.069 × 10−7
	Mean	5.709 × 10−7	3.016 × 10−6	6.056 × 10−6	1.210 × 10−5	9.904 × 10−6	7.723 × 10−6	5.757 × 10−7	3.911 × 10−6
STD	C1	1.946 × 10−5	2.976 × 10−5	2.398 × 10−5	1.493 × 10−4	2.150 × 10−5	1.227 × 10−4	2.649 × 10−5	1.417 × 10−4
	C2	1.189 × 10−5	2.076 × 10−5	2.561 × 10−5	1.111 × 10−4	2.127 × 10−5	1.431 × 10−4	2.010 × 10−5	2.304 × 10−5
	C3	5.029 × 10−6	1.820 × 10−5	1.116 × 10−5	1.424 × 10−4	2.412 × 10−5	5.056 × 10−5	1.418 × 10−5	1.571 × 10−5
	C4	1.539 × 10−7	2.818 × 10−5	3.140 × 10−5	7.480 × 10−5	1.292 × 10−7	3.129 × 10−5	1.466 × 10−5	7.206 × 10−5
	C5	4.032 × 10−6	1.220 × 10−5	2.078 × 10−5	8.684 × 10−5	1.231 × 10−5	1.976 × 10−5	2.451 × 10−5	7.062 × 10−5
	C6	7.109 × 10−8	1.004 × 10−5	1.204 × 10−5	4.370 × 10−5	2.821 × 10−5	7.439 × 10−6	1.141 × 10−5	1.498 × 10−5
	C7	4.026 × 10−6	7.703 × 10−6	5.345 × 10−6	1.964 × 10−5	6.327 × 10−6	5.998 × 10−6	5.236 × 10−6	9.279 × 10−6
	C8	2.115 × 10−7	4.039 × 10−6	1.854 × 10−6	1.428 × 10−6	3.147 × 10−7	4.688 × 10−6	5.606 × 10−7	5.439 × 10−6
	Mean	5.609 × 10−6	1.636 × 10−5	1.652 × 10−5	7.865 × 10−5	3.040 × 10−5	4.819 × 10−5	1.464 × 10−5	4.410 × 10−5
RT	C1	22.436	22.413	47.296	23.272	70.407	22.876	23.965	59.296
	C2	22.007	21.996	46.713	24.231	70.220	23.159	23.924	58.494
	C3	20.539	20.294	43.163	21.677	65.233	21.953	22.178	54.849
	C4	22.354	22.328	46.771	24.573	70.618	23.586	24.388	59.429
	C5	21.214	21.004	44.198	21.772	67.547	22.080	22.639	55.728
	C6	21.793	21.491	44.792	22.523	67.770	23.542	23.219	56.557
	C7	16.443	16.234	34.593	17.579	53.683	18.657	18.300	44.398
	C8	21.856	21.688	46.009	22.786	69.325	22.891	23.395	58.203
	Mean	21.080	20.931	44.192	22.302	66.850	22.343	22.751	55.869

, which outlines the comparative performance of each algorithm. Table 5 acts as a detailed collection, sorting the efficacy of various optimization algorithms across a range of operational conditions, thereby offering insights into their relative performance.

Within the competitive landscape, the RLPO algorithm distinguished itself by delivering the most accurate, consistent, and robust results. It consistently surpassed other algorithms, achieving the lowest RMSE under all conditions examined, with an exceptional average RMSE of 4.533 × 10⁻⁶. This level of accuracy exemplifies the algorithm’s matchless efficiency in closely approximating true parameter values with minimal deviation. Subsequent to the RLPO, HDE, the OBEDO, and the OBGBO exhibited commendable performance, recording average RMSEs of 4.560 × 10⁻⁶, 1.383 × 10⁻⁵, and 1.907 × 10⁻⁵, respectively, with the PO, RLGWO, RLPSO, and LSMA following in order of effectiveness. Notably, the PO and LSMA recorded the highest average RMSE values, reflecting their relatively lower performance metrics. The RLPO algorithm’s proficiency lies in its proficient exploration and exploitation within the search space, enabling the precise determination of nine critical parameters integral to PV panel optimization. It proficiently directs the complexities of transcendental non-linear equations characteristic of the TDM of PV modules, ensuring the accurate estimation of optimal solutions. In evaluating performance using MBE, R², and STD, the RLPO again led the field. It reported average metrics of 2.479 × 10⁻⁹ for MBE, 1.0000 for R², and 5.609 × 10⁻⁶ for STD, further confirming its dominant performance. Following closely was HDE, with the OBGBO, OBEDO, RLGWO, PO, RLPSO, and LSMA arranged behind in descending order of performance effectiveness. In order to analyse the computational time of each algorithm, the runtime (RT) metric is also recorded in Table 5. Based on the mean RT metric value, it is confirmed that the original PO is the fastest algorithm, with a 20.931 s mean value, followed by the RLPO, the RLPSO, the RLGWO, HDE, the OBGBO, the OBEDO, and the LSMA. Though the mean RT of the RLPO is slightly higher than that of the PO, i.e., 0.149 s, other metrics are very much better than the original PO algorithm. It is obvious that the RT is slightly higher for the RLPO compared to the PO due to the additional learning strategy. The comparative analyses presented in Table 5 conclusively position the RLPO algorithm as an outstanding solution for optimizing PV panels under a variety of operational scenarios. Its accuracy, as evidenced by the lowest RMSE and strong performance across other statistical measures like MBE, R², and STD, not only shows its effectiveness but also highlights its proficiency in accurately estimating and optimizing essential parameters of PV modules.

Figure 4 introduces a box plot that visually contrasts the performance of several models, particularly highlighting the RLPO algorithm’s exemplary performance. This graphical illustration is required to validate the algorithm’s superiority by showing the distribution of RMSE values across different models and accentuating the RLPO’s ability to attain minimal RMSE. Such visual depiction aids in the easier understanding of the relative performance among various algorithms. In comparison, the PO and RLPSO demonstrate variable effectiveness, with the box plot revealing their potential limitations in overcoming local optima and handling complex, multimodal problems under noisy conditions. Further, Figure 5a–d illustrates a detailed performance evaluation, presenting statistical metrics for each model. Throughout these statistical measures, the RLPO consistently surpassed its counterparts, illustrating its superior capability in yielding enhanced results across RMSE, MBE, R², and STD. This consistent performance is attributed to the synergetic integration of the PO algorithm, RL, and an adaptive secant method in formulating the fitness function, which promotes exceptional exploitation and exploration dynamics, adept at navigating complicated optimization landscapes.

The TDM involves addressing multimodal optimization, which poses significant challenges, particularly in the presence of noisy experimental data. In such contexts, any reduction in RMSE values is remarkable. The observed decrease in RMSE values with the RLPO algorithm signifies its exceptional precision, an achievement particularly acceptable given the complexity of the optimization challenges met. The combined visual and statistical analysis provided by Figure 4 and Figure 5 clearly validates the RLPO model’s dominant performance. Its consistent attainment of lower RMSE values and robust performance across various statistical metrics, even among multimodal optimization challenges and noise, accentuates its precision and effectiveness as a good optimization strategy for PV systems.

To better understand how well the proposed method works, a detailed study was done using real data, shown in Table 5 and Figure 6. This study focused on figuring out the AE and RE at different irradiation levels and temperatures to see how the RLPO compares to other algorithms. Using real data makes this study reliable and gives a strong basis for comparing the algorithms. The RLPO was found to have lower AE and RE values than the others, showing it can more accurately match the real currents measured in PV panels. Getting the parameters right in the TDM is important because it helps improve the function we are trying to optimize, which makes the whole optimization process better. The RLPO’s ability to do this well shows that it is a strong approach. When looking at average AE and RE values, the RLPO did the best, followed by HDE, the OBGBO, the RLGWO, the OBEDO, the RLPSO, the PO, and the LSMA.

Figure 6 visually compares the estimated data to the real data, showing where the AE and RE values did not match. This makes it easier to see how closely the RLPO and other algorithms match what was actually observed. The combination of the PO, RL, and an adaptive secant method in the RLPO algorithm helps it find high-quality solutions, especially for modelling PV modules in the TDM. Overall, the study presented in Table 5 and Figure 6, and the comparison of AE and RE values, clearly show that the RLPO performs better than the other algorithms.

The analysis shown in Figure 7 offers a key visual comparison that is essential for verifying the convergence efficiency across different optimization strategies. This analysis plays a crucial role in evaluating the performance of each algorithm with respect to finding the optimal solution efficiently. The figure is deliberately designed to illustrate the convergence behaviour of the algorithms, acting as a comparative visual aid to assess the speed and efficiency with which each method achieves convergence. Among the methods evaluated, the RLPO algorithm is highlighted as the most efficient. It distinguishes itself through its rapid and consistent convergence, surpassing other approaches in both speed and uniformity. An important highlight from the figure is the RLPO’s proficiency in attaining convergence with fewer iterations across a broad range of operational settings. This capability underscores the algorithm’s versatility and resilience, showcasing its effective performance in diverse conditions. However, despite its generally superior efficacy, the RLPO algorithm encounters situations necessitating a higher iteration count to reach convergence. These scenarios are typically defined by their high non-linearity, the presence of multimodal environments, and extensive datasets. Such conditions present a rigorous framework for testing the performance of optimization algorithms, especially in terms of attaining the minimum RMSE value.

The performance of the RLPO algorithm in challenging scenarios emphasizes its proficiency in navigating complex and diverse optimization problems. Thus, Figure 7 serves as an essential demonstration of the comparative convergence abilities across various optimization methods, with the RLPO algorithm emerging as the top peer in terms of convergence speed and efficiency. Its proficiency at handling most conditions, combined with its flexibility in tackling complex scenarios, emphasizes the RLPO’s effectiveness and adaptability as an optimization tool. Within the domain of parameter estimation for PV systems, the runtime (RT) of algorithms is an aspect that often receives insufficient attention. Nonetheless, RT is crucial for evaluating the efficiency of different algorithms. The study presented in

Table 6. RT of all algorithms tested under different test conditions.

Algorithms	C1	C2	C3	C4	C5	C6	C7	C8	Average
RLPO	23.4365	23.0073	21.5385	22.1042	22.2135	22.7927	16.4427	22.8563	21.7990
PO	23.4125	22.9958	21.7938	22.1167	21.8042	22.0906	16.7344	22.8875	21.7294
OBGBO	70.4073	70.2198	65.2333	67.0750	67.5469	67.7698	53.6833	69.3250	66.4076
RLPSO	59.2958	58.4938	54.8490	56.0344	55.7281	56.5573	44.3979	58.2031	55.4449
LSMA	47.2958	46.7125	43.1625	44.6479	44.1979	44.7917	34.5927	46.0094	43.9263
RLGWO	22.8760	23.1594	21.9531	22.2281	22.0802	22.5417	17.6573	22.8906	21.9233
HDE	23.9646	23.9240	22.1781	22.8760	22.6385	23.2188	18.3000	23.3948	22.5618
OBEDO	23.2719	23.2313	21.6771	22.2615	21.7719	22.5229	17.5792	22.7865	21.8878

fills this research gap by presenting the RTs for various algorithms, providing a detailed perspective on their time efficiency. The RLPO algorithm showcases an optimal balance between efficiency and effectiveness, with an average RT of 21.7294 s, ranking it as the second fastest among the evaluated algorithms. Interestingly, the algorithm’s basic variant achieves a lower average RT, placing it at the forefront in terms of speed. However, this variant falls short of the enhanced version regarding accuracy, robustness, and overall performance.

Conversely, the OBGBO and RLPSO algorithms and the LSMA exhibit the longest RTs, highlighting a relative time inefficiency compared to their counterparts. As detailed in Table 6, the proposed algorithm not only presents a competitive RT but also stands out for its reliability, precision, robustness, and convergence ability. This comprehensive performance profile positions it as a formidable contender in PV system optimization. A key factor contributing to the RLPO’s success is the integration of an adaptive secant technique, which efficiently selects initial root values for the I-V curve equation, thereby reducing the number of iterations required to achieve optimal convergence and, consequently, lowering the overall RT. This thorough evaluation emphasizes the significance of including RT in the assessment of algorithms, particularly within the context of optimizing PV systems, highlighting the need to evaluate time efficiency alongside other performance metrics.

6. Conclusions and Recommendations

This study successfully demonstrates the application of a reinforced learning-based parrot optimizer integrated with an adaptive secant method for parameter estimation of a three-diode photovoltaic model. This new approach has proven to be highly effective in accurately determining the parameters crucial for modelling PV systems. The adaptive nature of the secant method and reinforced learning principles, combined with the exploration capabilities of the PO, allow for an efficient search of the parameter space, leading to highly accurate estimations. The performance of the proposed algorithm, benchmarked against state-of-the-art algorithms, showcases its superiority in terms of convergence speed, accuracy, and robustness under varying operational conditions. By employing the RL strategy, the proposed approach proficiently directs the complexities of PV model parameter estimation, effectively tackling problems such as non-linearity and the challenge of multiple local optima. The findings of this study not only contribute to the optimization literature but also offer practical insights for enhancing the design and performance analysis of PV systems. The proposed RLPO algorithm, with its adaptive learning mechanism, demonstrates its potential to enhance PV system efficiency, reducing dependency on conventional energy sources and promoting environmental sustainability. The proposed method’s adaptability and efficiency highlight its potential as a valuable tool for researchers in the renewable energy domain, paving the way for more accurate and reliable PV system models.

The proposed strategy opens several paths for future research and application enhancements. Firstly, the integration of additional machine learning techniques could further improve the optimization process, potentially improving the accuracy and efficiency of parameter estimation even in more complex and dynamic environmental conditions. Secondly, the application of this strategy can be extended to other types of PV models, such as those incorporating newer materials or hybrid systems. Additionally, adapting the algorithm for real-time parameter estimation could significantly benefit the operation and maintenance of PV installations, allowing for dynamic adjustments to optimize performance in response to environmental changes. Another promising direction is the applicability in other domains of renewable energy, such as wind or hydroelectric power generation, where similar optimization challenges exist. The development of a more user-friendly software implementation of the proposed strategy could enhance its accessibility to researchers, developing wider adoption and iterative improvements based on user feedback. Lastly, long-term studies can be conducted to assess the performance of PV systems optimized using this strategy under various operational conditions.