A New Hybrid Framework for the MPPT of Solar PV Systems Under Partial Shaded Scenarios

Abstract

Nonlinear characteristics of solar photovoltaic (PV) and nonuniform surrounding conditions, including partial shading conditions (PSCs), are the major factors responsible for lower conversion efficiency in solar panels. One major condition is the cause of the multiple peaks and oscillation around the peak point leading to power losses. Therefore, this study proposes a novel hybrid framework based on an artificial neural network (ANN) and fractional order PID (FOPID) controller, where new algorithms are employed to train the ANN model and to tune the FOPID controller. The primary aim is to maintain the computed power close to its true peak power while mitigating persistent oscillations in the face of continuously varying surrounding conditions. Firstly, a modified shuffled frog leap algorithm (MSFLA) was employed to train the feed-forward ANN model using real-world solar PV data with the aim of generating a reference solar PV peak voltage. Subsequently, the parameters of the FOPID controller were tuned through the application of the Sanitized Teacher–Learning-Based Optimization (s-TLBO) algorithm, with a specific focus on achieving maximum power point tracking (MPPT). The robustness of the proposed hybrid framework was assessed using two different types (monocrystalline and polycrystalline) of solar panels exposed to varying levels of irradiance. Additionally, the framework’s performance was rigorously tested under cloudy conditions and in the presence of various partial shading scenarios. Furthermore, the adaptability of the proposed framework to different solar panel array configurations was evaluated. This work’s findings reveal that the proposed hybrid framework consistently achieves maximum power point with minimal oscillation, surpassing the performance of recently published works across various critical performance metrics, including the $MP{P}_{efficiency}$, relative error (RE), mean squared error (MSE), and tracking speed.

1. Introduction

India has experienced constant growth in its demand for electrical energy in different sectors. While nonrenewable energy sources used to dominate the electricity generation landscape, their usage poses serious challenges of economic dependency, geopolitical threats, and vulnerability of energy security related to the scanty and uneven distribution of reserves. It has resulted in a paradigm shift toward renewable energy sources and to exploring more sustainable options. Among the various renewable options, solar energy has emerged as the most promising choice in the last decade [1]. This is due to its numerous advantages, such as being pollution- and noise-free, having low maintenance costs, and the availability of an unlimited source of energy [2]. The solar PV system is an integration of solar panels in series with a DC–DC converter. The solar photovoltaic (PV) system presents several notable advantages, yet it is not without its inherent challenges. These challenges encompass issues such as sub-optimal power conversion efficiency and a reliance on environmental variables, such as irradiance and temperature [3]. Fluctuating environmental conditions and partial shading represent key contributors to overall power loss, thus lowering the power conversion efficiency [4].

To tackle the abovementioned challenges, several maximum power point tracking (MPPT) frameworks have been developed and employed for the optimal switching of a converter to achieve maximum power under changing environmental conditions [5]. Numerous conventional MPPT frameworks have been extensively explored in the literature, including perturb and observe (P and O) [6], incremental conductance (INC) [7], and fractional short-circuit current (FSCC) [8,9]. An adaptive P and O framework, which regulates the step size of the duty cycle of the converter, is proposed. The step size varies based on the distance of the calculated maximum power point (MPP) from the actual MPP [10]. This framework delivers output with sustained oscillation around the MPP. In [2], the authors introduced a modified P-and-O-based framework to track global maximum power point (GMPP) for a standalone PV system. The presented framework was simpler and easy to implement but tracked the global peak under partial shading conditions with a slow convergence rate. The authors in [11] introduced a modified incremental conductance (INC)-based framework to extract the maximum power from a solar PV system. This framework delivers the output with low efficiency within the range of 85 to 95%, along with oscillations, thus depicting the large power loss at the PV output. Conventional-based frameworks encounter challenges in reaching the global peak due to the presence of multiple peaks under partial shading conditions. Instead, they often become confined to local peaks, leading to a loss of power, and they also suffer with oscillations around the MPP, a low convergence rate, etc. In addition to the previously discussed frameworks, several soft computing-based frameworks for MPPT have emerged in the literature. These include the jellyfish search optimizer (JSO) [12], s-TLBO [13], particle swarm optimization (PSO) [14], the flower pollination algorithm (FPA) [15,16,17,18], etc. Furthermore, control-based methods have been deployed for MPPT, such as the utilization of the fuzzy logic controller (FLC) [19,20,21], other controller strategies [22,23,24,25], etc. The authors in [14] employed PSO in order to improve the artificial neural network (ANN) model accuracy, and they achieved this by utilizing it to determine the best topology. The presented framework was able to predict the maximum power, but the oscillation around the MPP still persisted, thus affecting the average output power efficiency. The work in [26] introduced a novel hybrid framework that combines gray wolf optimization (GWO) with perturb and observe (P and O) to achieve maximum power extraction. Their study investigated a system’s performance under rapidly varying solar irradiance conditions, and they also examined the outcomes in the presence of partial shading. To practically validate the proposed framework, the authors implemented it in MATLAB simulation and conducted experiments using an experimental setup. The results demonstrated that the proposed algorithm exhibits fast convergence in tracking the MPP, indicating its efficiency in real-world scenarios. The hybrid GWO and P and O framework showed promising potential for optimizing power generation in dynamic solar conditions. The study of [27] aimed to determine the optimal parameters for a single-diode PV model and the fractional-order proportional–integral (FOPI) controller, where its performance was compared with conventional PID controllers. The authors employed two meta-heuristic algorithms, the FPA and Water Cycle Algorithm (WCA), to identify the parameters of the solar PV mode, as well as the FOPI controller. The FPA was found to be better than WCA in both solar PV parameter identification and controller tuning. The presented framework is unable to accurately achieve maximum power and exhibits significant oscillations around the peak power point. In [28], the authors conducted an extensive experimental and AI-based analysis of solar floating photovoltaic (SFPV) and ground-mounted solar PV (GSPV) systems under harsh environmental conditions in Al-Khobar, Saudi Arabia. Their study included real-time testing and performance evaluation of power output, surface temperature, voltage, current, energy yield, and efficiency. To enhance prediction accuracy, they developed a novel hybrid AI framework that integrates LightGBM with GRU and LSTM models, optimized using the Brown Bear Optimization Algorithm (BBOA), for forecasting power generation and panel temperature. Additionally, in [29], the authors described the design and optimization of a BIPV/T system aimed at producing both electricity and thermal energy while reducing energy consumption in residential buildings. The system was dynamically modeled in MATLAB-Simulink 2024a, and its performance was evaluated through techno-economic and sensitivity analyses across various seasonal conditions. An artificial neural network (ANN) was employed to predict solar radiation and ambient temperature, thereby improving system accuracy. Furthermore, a multiobjective optimization using the NSGA-II algorithm was performed to maximize thermal efficiency and output while minimizing the system’s spatial footprint. The optimized system achieved a net thermal power output of 5320 W, an efficiency of 63%, and a levelized cost of electricity of 0.10/kWh, demonstrating the viability of BIPV/T systems as cost-effective and environmentally friendly energy solutions for residential applications.

Nevertheless, these frameworks exhibit certain limitations, particularly oscillations around the maximum power point (MPP) under varying environmental conditions, premature convergence during partial shading, and relatively slow tracking response. To overcome these challenges, this study proposes a novel hybrid framework based on an artificial neural network (ANN) and a fractional-order proportional–integral–derivative (FOPID) controller. The objective is to develop an enhanced ANN model capable of accurately predicting the PV voltage and an improved controller that efficiently regulates the converter’s duty cycle to reliably track the maximum power point.

This paper makes the following contributions:

• A new hybrid framework is proposed that leverages the strengths of both the ANN and FOPID controller for solar MPPT applications. A modified Shuffled Frog Leaping Algorithm (MSFLA) is employed to train the ANN model for various values of temperature (T) and irradiance (G). Additionally, the Sanitized Teacher–Learning-Based Optimization (s-TLBO) algorithm is introduced to tune the FOPID controller, aiming to efficiently regulate the duty cycle.
• The performance of the proposed hybrid framework is compared with [14,27], using different performance indices such as maximum power calculated, relative error, $MP{P}_{efficiency}$, oscillations, and MPP tracking time.
• Furthermore, tailored parameter modifications, such as variable step-size scaling, are introduced in the MSFLA. This enhancement helps avoid local minima during ANN training and accelerates convergence, particularly under nonlinear conditions like partial shading.
• The proposed hybrid framework improves MPP efficiency across the various scenarios considered in the study. It achieves the desired outcomes in minimal tracking time, with reduced oscillations around the MPP, compared to previously published works [14,27].
• The proposed framework accurately predicts the reference voltage under different environmental conditions and consistently delivers superior output across various scenarios and for different types of solar panels (monocrystalline and polycrystalline), demonstrating its robustness.
• It is finally validated by applying the proposed framework to different solar array configurations, namely, 5 × 1, 5 × 2, and 5 × 3 panel arrangements, under various partial shading conditions. The corresponding P–V and I–V characteristics curves are plotted, clearly illustrating the maximum power obtained for each case considered.

2. Solar PV Model

The fundamental building block of a solar photovoltaic (PV) panel is the solar cell, which is responsible for converting light energy into electrical power. A solar cell can be modeled as a parallel combination of a current source and a diode, along with series and shunt resistances. The schematic representation of an individual PV cell is shown in Figure 1. In an ideal PV cell, the series and shunt resistances are absent. However, in real-world applications, these resistances are significant. The shunt resistance ($r_{sh}$) accounts for losses due to leakage current, while the series resistance ($r_{se}$) represents losses caused by metal contacts. When PV cells are connected in series and parallel combinations, they form a PV panel. Further interconnection of multiple PV panels in series and parallel leads to the development of a solar array. The mathematical expression describing the behavior of a single-diode PV cell is given in Equation (1) [13].

$$i_l=i_p-i_r\left[exp\left({\displaystyle \frac{v_l+i_l{r}_{se}}{v_t}}\right)-1\right]-\left[{\displaystyle \frac{v_l+i_l{r}_{se}}{r_{sh}}}\right]$$

(1)

where

• $v_l$ = load voltage;
• $i_l$ = load current;
• $i_p$ = photon current;
• $i_r$ = saturation current;
• $r_{se}$ = series resistance;
• $r_{sh}$ = shunt resistance;
• $v_t$ = thermal voltage ($v_t$).

The generated current, known as photocurrent, primarily depends on two environmental factors: irradiance and temperature. This dependency is mathematically represented in Equations (2) and (3) [14], which illustrate how the solar PV current varies with changes in these two parameters.

$$i_p=i_p\left(G_o\right)\left[1+K_i\left(T-T_r\right)\right]$$

(2)

$$i_p\left(G\right)=i_p\left(G_o\right)\left[{\displaystyle \frac{G}{G_o}}\right]$$

(3)

here,

• $i_{ph}$ = photo generated current (A);
• $T_r$ = reference temperature;
• $G_o$ = reference irradiance;
• T = absolute temperature;
• $K_i$ = SC current temperature coefficient (A/K) under standard test conditions;
• G = irradiance (W/${\mathrm{m}}^2$).

The literature highlights that the one-diode PV model is a superior choice due to its simplicity, being characterized by a more streamlined set of equations compared to the more complex double-diode model [23,27]. As a result, for the scope of this research, the preference leans toward the one-diode model. In this study, two different types of solar panels are employed: the Kyocera KC200GT and the Sharp NU-S5E3E. The Kyocera panel is a high-efficiency polycrystalline solar module with a conversion efficiency of 16% [30]. Its cells are encapsulated between a tempered glass cover and a pottant backed by a black sheet, making it ideal for grid-connected applications. The Sharp NU series panel, on the other hand, is a monocrystalline solar module consisting of 48 cells and offers an efficiency of 14.1% [31]. It is designed for high-power applications and is well suited for both on-grid and off-grid PV system installations.

In this work, the study is conducted under two different scenarios. In the first scenario, the investigation focuses on a single solar panel, Kyocera KC200GT, referred to as SP-1. In the second scenario, the study examines the behavior of solar panel arrays configured in 5 × 1, 5 × 2, and 5 × 3 arrangements. Here, the solar panel used is Sharp NU-S5E3E, designated as SP-2. The parameters of both solar panels used in this study are provided in

Table 1. Solar PV panels datasheet.

Different Solar Panels/Parameters	Kyocera KC200GT	Sharp NU-S5E3E
Im (Amps)	7.61	7.71
Vm (Volts)	26.3	24
Pm (W)	200.1	185.04
Isc (Amps)	8.21	8.54
Voc (Volts)	32.9	30.2
Ns	54	48
No. of panels	1	5

.

In this study, the solar panel array is subjected to various partial shading conditions (PSCs). PSC occurs when certain panels in an array, or portions of a panel, receive different levels of irradiance compared to the rest. This condition results in multiple maximum power points on the P–V characteristics curve of the solar panel. These include several local maximum power points (LMPPs) and one global maximum power point (GMPP). The objective is to accurately track the GMPP without becoming trapped at an LMPP, as falling into an LMPP leads to significant power loss and reduces the overall efficiency of the solar PV system.

3. Preliminaries

3.1. Artificial Neural Network (ANN)

This framework is based on a distributed processing architecture, which offers the advantage of not requiring an in-depth understanding of the system during the modeling phase. Instead, it depends on the availability of accurate data to make predictions that closely align with real-world outcomes. The algorithm is provided with training data as input, which it uses to establish a nonlinear mapping between the input and output nodes. In artificial neural networks (ANNs), two primary topologies are commonly used: feed-forward and feedback networks. Among these, the feed-forward topology is preferred due to its efficient memory usage during implementation and its strong performance in managing nonlinear systems, such as solar PV systems. A feed-forward network consists of three layers: the input layer, hidden layer, and output layer, with nodes in each layer interconnected through weighted connections. The mathematical representation of the ANN system is provided in Equation (4) [14].

$$y=\sum _{i=1}^n{w}_{ij}{x}_j+b_j$$

(4)

here,

• $x_j$ = input;
• $w_{ij}$ = weights;
• $b_j$ = bias.

Different algorithms are employed to improve the accuracy of the ANN model by iteratively updating the weights and biases until the desired outcome is achieved. The mean square error (MSE) is used as the cost function for this process and is defined in Equation (5) [14].

$$MSE={\displaystyle \frac{1}{N}}\sum _{i=1}^N\sum _{j=1}^M{[y_j\left(i\right)-t_j\left(i\right)]}^2$$

(5)

here,

• N = no. of input data;
• M = no. of output data;
• $y_j\left(i\right)$ = calculated output;
• $t_j\left(i\right)$ = desired output.

Real-time temperature (T) and irradiance (G) data were generated using hourly variations sourced from [32]. Subsequently, a dataset of PV voltage values corresponding to each combination of T and G was created using MATLAB-2024a code. The dataset is structured as an array containing 10,000 entries, comprising temperature, irradiance, and the resulting solar PV voltage values. This dataset is further divided into three subsets: training data (70%), testing data (15%), and validation data (15%). Finally, the compiled dataset is used to train the ANN model.

3.2. Sanitized Teacher–Learning-Based Optimization

This framework emulates a learning mechanism similar to the teacher–student paradigm [13]. It represents an enhanced version of the basic Teaching–Learning-Based Optimization (TLBO) technique. The method operates in two distinct phases: the teacher phase and the learner phase, as described below.

3.2.1. Teacher Phase

In this phase, the learning process is modeled as students acquiring knowledge from an experienced teacher. The teacher is assumed to possess significant expertise and a deep understanding of the subject matter, which is utilized to enhance the learning outcomes of the students. The teacher’s objective is to elevate the average knowledge level of the class, guiding it progressively closer to their own level of expertise.

3.2.2. Learner Phase

In this phase, knowledge is acquired from a classmate with the aim of enhancing the knowledge of an individual student.

3.3. FOPID Controller

The FOPID controller is a distinct controller utilized in the field of control systems engineering. It expands upon the conventional proportional–integral–derivative (PID) controller by integrating fractional calculus concepts. Unlike the PID controller, which relies on integer-order calculus for its proportional, integral, and derivative components, the FOPID controller leverages fractional-order calculus. This unique framework offers increased adaptability in comprehending intricate system behaviors and dynamics. The transfer function of the controller is expressed by Equation (6) [27].

$$G_c\left(s\right)={\displaystyle \frac{Y\left(s\right)}{R\left(s\right)}}=K_p+K_i{s}^{-\lambda }+K_d{s}^{\mu }$$

(6)

here, ($\lambda$, $\mu$) > 0. In this form of controller, two additional parameters are included in order to enhance the degree of freedom, thus improving the system’s controllability and stability.

3.4. DC–DC Boost Converter

The primary objective of the converter is to align the input resistance with the load resistance, effectively regulating its switching cycle to attain MPP. In this investigation, a boost converter is employed for this purpose. To ensure equilibrium in power levels at both ends of the converter, the output current of the converter is intentionally reduced in comparison to the supply current. This relationship between output and input voltages is mathematically described by Equation (8) [33].

$${\displaystyle \frac{v_o}{v_i}}={\displaystyle \frac{1}{(1-d)}}$$

(7)

here, d is the duty cycle.

In this study, the comprehensive transfer function of the system, encompassing both the converter and the FOPID controller, is established. This transfer function considers the duty cycle (d) as the input and the resulting voltage ($V_o$) as the output, and it is precisely defined in Equation (8) [33].

$${\displaystyle \frac{V_o}{d}}={\displaystyle \frac{s{V}_{c}RC+V_c-I_{l}R(D-1)}{s^{2}RLC+sL+{(D-1)}^{2}R}}$$

(8)

The parameters of the DC–DC boost converters are as follows:

• Inductance (L) = 10 mH;
• Capacitance (C) = 500 μF;
• Resistance (R) = 25 ohms.

4. Proposed Hybrid-Framework

To extract maximum power from a solar PV system, the controller requires a reference value, which it compares with the current system output to minimize the error and achieve the desired performance. In real-world scenarios, environmental conditions change continuously, causing the PV voltage to fluctuate accordingly. Due to its inherent ability to handle nonlinearity and the advantage of not requiring explicit knowledge of PV parameters, the artificial neural network (ANN) is selected for this study. The ANN is trained to provide the reference peak voltage corresponding to various combinations of temperature and irradiance values. The FOPID controller, with its greater degrees of freedom compared to conventional controllers, is well suited for managing nonlinear systems such as solar PV systems. This makes it an ideal choice for enhancing control accuracy and adaptability under dynamically changing operating conditions.

Therefore, this work proposes a novel hybrid framework that integrates two complementary modules:

• An artificial neural network (ANN) trained using a Modified Shuffled Frog Leaping Algorithm (MSFLA), and
• A fractional-order PID (FOPID) controller dynamically tuned using a sanitized Teacher–Learning-Based Optimization (s-TLBO) algorithm.

This dual-layer architecture is designed to enhance prediction accuracy and adaptive control responsiveness under nonlinear and dynamically changing environmental conditions, such as partial shading and low-irradiance scenarios. Furthermore, the novelty in the implementation of the MSFLA lies in the tailored parameter modifications, including variable step-size scaling, which are discussed in detail in Section 4.1. The study involves two distinct solar PV panels on which the proposed hybrid framework is implemented. The training dataset comprises varying irradiance and temperature inputs, along with the corresponding PV voltage outputs. This experimentally recorded data is used to train the ANN in offline mode, enabling it to generate a reference voltage for comparison with the actual PV voltage. The resulting error is then fed into the controller, which regulates the converter’s duty cycle to maximize power extraction. A block diagram of the proposed hybrid framework is shown in Figure 2, while the detailed workflow is illustrated in Figure 3. Initially, the ANN model is trained using temperature and irradiance as input features and the recorded PV voltage as the output, with the training process facilitated by the MSFLA. Once trained, the ANN provides a reference voltage, which is continuously compared with the real-time PV voltage. The resulting error signal is passed to the proposed tuned FOPID controller, which adjusts the duty cycle in a manner that ensures maximum power point (MPP) tracking, thereby maximizing energy generation. The proposed framework is evaluated for peak power extraction under various operating scenarios, as elaborated in the following sections.

4.1. MSFLA-Based Trained ANN

In this study, recorded data from two distinct solar panels is used as input for training an ANN. The input features consist of varying levels of irradiance and temperature, while the desired output is the corresponding PV voltage. To optimize the performance of the ANN model, the MSFLA is employed. This algorithm assists in selecting the most appropriate feed-forward ANN topology and facilitates the training process to generate the reference voltage required for analysis. The MSFLA is a nature-inspired algorithm that emulates the local exploitation and global exploration behavior observed in frog populations [34], making it well suited for solving complex optimization problems such as ANN training for nonlinear systems [35].

The modified parameters of the MSFL algorithm are set to obtain the trained ANN model for both solar panels as follows:

• Number of weights = 30;
• Number of bias = 11;
• Learning rate ($\alpha$) = 0.05;
• ANN training function = ‘trainlm’;
• ANN activation function = ‘Levenberg–Marquardt’;
• Number of unknown variables = 41;
• Number of memeplexes = 3;
• Number of frogs in each memeplex = 3;
• Maximum iteration = 50;
• Constant values, $c_1$ = $c_2$ = 1.5, w = 1.2.

Further, to optimize the ANN weights, MSFLA is employed with the following addition:

• Variable step size scaling (s) for each frog is carried out and it is mathematically expressed in Equation (9),

$$s=\alpha \times rand\left(\right)\times \left[F_b-F_w\right]$$

(9)

where $F_b$ and $F_w$ are the best and worst fitness function, and $\alpha$ is the learning rate.

These modifications enhance the exploration–exploitation trade-off, prevent premature convergence, and lead to faster and more stable learning in the ANN, particularly under the dynamic input–output characteristics commonly observed in PV systems.

The flowchart illustrating the process of training the ANN model using the MSFL algorithm is presented in Figure 4. According to this process, the recorded data is divided into three subsets: training, validation, and testing. The training function and activation function are then defined, followed by the initialization of additional parameters as previously described. Next, the algorithm initializes the population. The individuals in the initial population are grouped into equally sized subgroups known as memeplexes. Within each memeplex, individuals evolve by moving toward either the local best or the global best solution within that subgroup. After a predetermined number of local evolutionary iterations, the entire population is reshuffled to disperse the memes, enhancing diversity. This process is repeated iteratively until the predefined stopping criteria are met.

The proposed algorithm identifies the optimal ANN topology, which consists of a single hidden layer with 10 neurons, two input nodes, and one output node. The Levenberg–Marquardt algorithm is employed as the training function in this study. The results demonstrate that the ANN topology generated using the proposed algorithm achieves higher prediction accuracy compared to the model presented in [14], thereby highlighting the effectiveness and robustness of the proposed framework.

4.2. FOPID Tuned Using s-TLBO

In this section, the s-TLBO algorithm is applied to fine-tune the parameters of the FOPID controller. A total of five controller parameters are adjusted using this algorithm. The overall transfer function of the system comprising the controller and the converter is evaluated. This resultant transfer function is employed within the s-TLBO algorithm, where root mean square error (RMSE) serves as the objective function. This framework is used to determine the optimal values for the FOPID controller’s parameters. These parameters yield the transfer function of the controller that contributes to delivering optimum values of the duty cycle for achieving the peak power of the PV system. For solar panel-1, the converter’s transfer function is mathematically defined in Equation (10) [33].

$${\displaystyle \frac{V_{o1}}{d_1}}={\displaystyle \frac{230.75s+219.19}{0.000355s^2+0.0001s+16.378}}$$

(10)

where $V_{o1}$ = output voltage of the converter, and $d_1$ = duty cycle.

As the objective is to control the solar PV voltage by regulating the converter’s duty cycle, the aforementioned transfer function can be modified as given in Equation (11) [33],

$${\displaystyle \frac{V_{pv1}}{d_1}}={\displaystyle \frac{93.38s+88.71}{0.000355s^2+0.0001s+16.378}}$$

(11)

where $V_{pv1}$ = solar panel-1 voltage.

The system’s (converter and FOPID-controller) closed-loop transfer function for solar panel-1 is expressed in Equation (14). The s-TLBO algorithm is employed for the estimation of the controller’s parameters using Equation (14), having RMSE as an objective function.

Equation (12) expresses the transfer function of the converter in the case of solar panel-2:

$${\displaystyle \frac{V_{o2}}{d_2}}={\displaystyle \frac{443.75s+536.93}{0.00008875s^2+0.0001s+4.095}}$$

(12)

where $V_{o2}$ is output voltage of SP-2, and $d_2$ is duty cycle.

Similarly, the transfer function of the converter in terms of solar PV voltage as an output is expressed as given below:

$${\displaystyle \frac{V_{pv2}}{d_2}}={\displaystyle \frac{179.58s+217.29}{0.00008875s^2+0.0001s+4.095}}$$

(13)

where $V_{pv2}$ = solar panel-2 voltage.

The system’s closed-loop transfer function for solar panel-2 is expressed in Equation (15). The s-TLBO algorithm is employed for the estimation of the controller’s parameters using Equation (15), having RMSE as an objective function. In the system’s transfer function, the parameters of the FOPID controller are expressed as follows: $x_1$ is $K_p$, $x_2$ is $K_i$, $x_3$ is $K_d$, $x_4$ is $\mu$, and $x_5$ is $\lambda$.

$${\displaystyle \frac{V_{pv1}}{e_1}}={\displaystyle \frac{\begin{array}{c}\hfill 93.38(x_1{s}^{x_5+1}+x_{2}s+x_3{s}^{x_4+x_5+1})+88.71(x_1{s}^{x_5}+x_2+x_3{s}^{x_4+x_5})\end{array}}{\begin{array}{c}\hfill 0.000355s^{2+x_5}+(0.0001+93.38x_1)s^{x_5+1}+88.71(x_1{s}^{x_5}\\ \hfill +x_3{s}^{x_4+x_5}+x_2)+93.38(x_{2}s+x_3{s}^{x_4+x_5+1})+16.38x_5\end{array}}}$$

(14)

$${\displaystyle \frac{V_{pv2}}{e_2}}={\displaystyle \frac{\begin{array}{c}\hfill 179.58(x_1{s}^{x_5+1}+x_{2}s+x_3{s}^{x_4+x_5+1})+217.29(x_1{s}^{x_5}+x_2+x_3{s}^{x_4+x_5})\end{array}}{\begin{array}{c}\hfill 0.00008875s^{2+x_5}+0.0001s^{x_5+1}+4.095s^{x_5}+179.58(x_1{s}^{x_5+1}\\ \hfill +x_{2}s+x_3{s}^{x_4+x_5+1})+217.29(x_1{s}^{x_5}+x_2+x_3{s}^{x_4+x_5})\end{array}}}$$

(15)

where $V_{pv1}$ is the PV voltage of solar panel-1, $e_1$ is the error fed to the controller in solar panel-1, $V_{pv2}$ is the PV voltage of solar panel-2, and $e_2$ is the error fed to the controller in solar panel-2. Now, the s-TLBO algorithm is employed for the FOPID’s parameters’ estimation.

The following are the variables of s-TLBO considered in this study:

• $N_{pop}$ = 100;
• Unknown variables (D) = 5;
• Upper limit (UL) = [4 4 4 1 1];
• Lower limit (LL) = [0 0 0 0 0];
• Max Iteration (i) = 100.

To estimate the suitable upper and lower bound for this particular application, ten trials were carried out.

The population initialization is the first step, as expressed in Equation (16) [13]:

$$pop=\left[\begin{array}{ccccc}{x}_{1_1}& x_{2_1}& x_{3_1}& x_{4_1}& x_{5_1}\\ x_{1_2}& x_{2_2}& x_{3_2}& x_{4_2}& x_{5_2}\\ ⋮& ⋮& ⋮& ⋮& ⋮\\ x_{1_i}& x_{2_i}& x_{3_i}& x_{4_i}& x_{5_i}\end{array}\right]$$

(16)

Now, the parameters in this algorithm are updated using Equations (17) and (18) [13]:

$$NewSolution=OldSolution+rand(1,D)\ast \left(BestSolution-TF\ast Mean\right)$$

(17)

$$\left[\begin{array}{c}{x}_{1_n}\\ x_{2_n}\\ x_{3_n}\\ x_{4_n}\\ x_{5_n}\end{array}\right]=\left[\begin{array}{c}{x}_{1_o}\\ x_{2_o}\\ x_{3_o}\\ x_{4_o}\\ x_{5_o}\end{array}\right]+rand(1,D)\ast \left(\left[\begin{array}{c}{x}_{1_b}\\ x_{1_b}\\ x_{1_b}\\ x_{1_b}\\ x_{1_b}\end{array}\right]-TF\ast \left[\begin{array}{c}{x}_{1_m}\\ x_{2_m}\\ x_{3_m}\\ x_{4_m}\\ x_{5_m}\end{array}\right]\right)$$

(18)

where TF defines teaching factor ranges between 1 and 2.

If the NewSolution is better than the OldSolution, then a partner is selected. Further, the fitness value is checked to see if the fitness of $i^{th}$ iteration is less than the fitness of the partner, and the solution is updated using the Equation (19) [13]; otherwise, it is updated using the Equation (20) [13]. Integral squared error is considered to be the fitness function for the presented problem, as shown in Equation (21) [13].

$$\left[\begin{array}{c}{x}_{1_n}\\ x_{2_n}\\ x_{3_n}\\ x_{4_n}\\ x_{5_n}\end{array}\right]=\left[\begin{array}{c}{x}_{1_o}\\ x_{2_o}\\ x_{3_o}\\ x_{4_o}\\ x_{5_o}\end{array}\right]+rand(1,D)\times \left(\left[\begin{array}{c}{x}_{1_o}\\ x_{1_o}\\ x_{1_o}\\ x_{1_o}\\ x_{1_o}\end{array}\right]-TF\times \left[\begin{array}{c}{x}_{1_p}\\ x_{2_p}\\ x_{3_p}\\ x_{4_p}\\ x_{5_p}\end{array}\right]\right)$$

(19)

$$\left[\begin{array}{c}{x}_{1_n}\\ x_{2_n}\\ x_{3_n}\\ x_{4_n}\\ x_{5_n}\end{array}\right]=\left[\begin{array}{c}{x}_{1_o}\\ x_{2_o}\\ x_{3_o}\\ x_{4_o}\\ x_{5_o}\end{array}\right]+rand(1,D)\times \left(\left[\begin{array}{c}{x}_{1_p}\\ x_{1_p}\\ x_{1_p}\\ x_{1_p}\\ x_{1_p}\end{array}\right]-TF\times \left[\begin{array}{c}{x}_{1_o}\\ x_{2_o}\\ x_{3_o}\\ x_{4_o}\\ x_{5_o}\end{array}\right]\right)$$

(20)

$$Fit\left(i\right)=\int e^{2}dt$$

(21)

here, e represents the error, defined as the difference between the target value and the calculated value. The flowchart illustrating the tuning process of the FOPID controller using the s-TLBO algorithm is shown in Figure 5.

The MSFLA-trained ANN model generates the appropriate reference voltage corresponding to various environmental conditions for both solar panels. This reference voltage is then compared with the actual PV voltage, producing an error signal. The error signal is subsequently fed into the tuned FOPID controller, which dynamically adjusts the converter’s duty cycle. This adaptive regulation mechanism ensures the effective tracking of MPP for the solar panels under diverse operating conditions.

5. Computational Result Studies

The proposed hybrid framework is employed on two different solar panels. Table 1 presents the solar panel’s datasheet. For this study, solar panel-1 (SP-1) is Kyocera 200GT, and Sharp NU-S5E3E is solar panel-2 (SP-2). The study involves different scenarios for each solar panel for which the proposed framework is tested and validated. The performance indices discussed below help in measuring the superiority of the proposed hybrid framework when compared with previously published work.

• Relative error (RE) is a measure used to quantify the accuracy by comparing the calculated value with the true value. In the proposed work, relative error is measured by comparing the calculated power with the actual peak power.

  $$RE={\displaystyle \frac{\left|P_{calculated}-P_{actual}\right|}{P_{actual}}}$$

  (22)

  where $P_{calculated}$ is the peak power calculated and $P_{actual}$ is the actual peak power of the panel.
• MPP efficiency is the measure of the precision obtained in measuring the actual peak power for the particular operating condition. It is expressed as

  $$MP{P}_{efficiency}={\displaystyle \frac{P_{calculated}}{P_{actual}}}\times 100\%$$

  (23)
• Tracking speed is the measure of the speed by which the curve reaches its global peak and it is measured in seconds.
• Root mean squared error (RMSE) measures the average magnitude of the errors between the calculated values and the actual values. The smaller the RMSE, the better the model is at calculating the values.

  $$RMSE=\sqrt{{\displaystyle \frac{\sum {[Calculate{d}_{value}-Actua{l}_{value}]}^2}{n}}}$$

  (24)

5.1. Solar Panel-1

Initially, SP-1 is considered for the study. The trained ANN model produces reference PV voltage as its output. Subsequently, the FOPID controller is precisely tuned using the s-TLBO algorithm. This tuning process utilizes the system’s transfer function given in Equation (14).

Table 2. FOPID-controller parameters for the SP-1.

Parameter	FOPI-FPA [27]	FOPI-WCA [27]	Proposed FOPID-s-TLBO
RMSE	0.050992	0.06892	0.01414
Kp	0.0121	0.015	0.8029
Ki	1.12	0.9	1.7407
Kd	0	0	0.3931
Lambda	1.52	1.62	0.9629
Mu	0	0	0.1350

presents the derived parameters of the FOPID and FOPI controllers obtained through the proposed method and alternative algorithms. Additionally, the root mean square error (RMSE) associated with the estimation of controller parameters using the s-TLBO, WCA, and FPA algorithms is detailed in Table 2. To evaluate the effectiveness of our proposed framework, two distinct scenarios are considered for the study, as elaborated below.

5.1.1. Scenario-1

In the first scenario, the irradiance is changed in a step-wise fashion, and the temperature remains constant at 25 deg C. Four irradiance levels are considered for this study: 0 ${\mathrm{W}/\mathrm{m}}^2$, 300 ${\mathrm{W}/\mathrm{m}}^2$, 500 ${\mathrm{W}/\mathrm{m}}^2$, and 1000 ${\mathrm{W}/\mathrm{m}}^2$. The power, current, and voltage response obtained using the proposed and other frameworks for the first part are compared and depicted in Figure 6, Figure 7 and Figure 8. The graph shows that the calculated maximum power reaches the actual maximum power faster using the proposed framework as compared to power calculated using other frameworks. Thereafter, the P–V and I–V characteristics curves are drawn for G = 1000 ${\mathrm{W}/\mathrm{m}}^2$, 500 ${\mathrm{W}/\mathrm{m}}^2$, and 300 ${\mathrm{W}/\mathrm{m}}^2$ as depicted by Figure 9, Figure 10 and Figure 11. The presented curves reveal that the power extracted using the proposed framework is more than that of other frameworks for different values of irradiances. Further, numerical data obtained using proposed and other frameworks are compared, and the comparison is shown in

Table 3. Comparison based on various parameters for the SP-1 at varying irradiances.

Irradiance	Frameworks	Pactual (W)	Pcalculated (W)	Vcalculated (V)	Relative Error	MPPEfficiency (%)
1000	FOPI-FPA [27]	200.14	199.98	26.28	0.000799	99.92%
	FOPI-WCA [27]	200.14	199.96	26.28	0.000899	99.91%
	Proposed framework	200.14	200.08	26.29	0.00029	99.97%
500	FOPI-FPA [27]	104.8	103.95	27.37	0.00811	99.18%
	FOPI-WCA [27]	104.8	103.75	27.39	0.01001	98.99%
	Proposed framework	104.8	104.28	27.39	0.00496	99.50%
300	FOPI-FPA [27]	63.53	62.98	27.60	0.00865	99.13%
	FOPI-WCA [27]	63.53	62.52	27.70	0.01589	98.41%
	Proposed framework	63.53	63.17	27.54	0.00566	99.43%

. The comparative analysis is carried out based on calculated voltage, relative error, and MPP efficiency obtained for the part first. It is observed that the proposed framework is able to track the actual peak and delivers superior outcomes in terms of the following PIs: relative error of 0.00029, and MPP efficiency close to 99.97%, for G = 1000 ${\mathrm{W}/\mathrm{m}}^2$; relative error of 0.00496, and MPP efficiency close to 99.50%, for G = 500 ${\mathrm{W}/\mathrm{m}}^2$; and relative error of 0.00566, and MPP efficiency close to 99.43%, for G = 300 ${\mathrm{W}/\mathrm{m}}^2$.

Further,

Table 4. Oscillation measurements for scenario-1 for SP-1.

Irradiance (G)	Frameworks	ΔP (W)	ΔT (s)	1/ΔT (Hz)	ΔP/ΔT (W/s)
300	Proposed	0.00580	0.00704	141.90400	0.83070
	FOPI-WCA [27]	0.02020	0.01104	90.56900	1.82900
	FOPI-FPA [27]	0.02970	0.01725	57.94900	1.72500
500	Proposed	0.00170	0.01297	77.07000	0.13120
	FOPI-WCA [27]	0.01404	0.01499	66.67900	0.93590
	FOPI-FPA [27]	0.01380	0.01501	66.60450	0.91910
1000	Proposed	0.00375	0.00498	200.80500	0.75390
	FOPI-WCA [27]	0.00959	0.00600	166.66800	1.59900
	FOPI-FPA [27]	0.01486	0.01097	91.09900	1.35450

shows the measurements obtained from the power vs. time graph, showing the rate of change of power (W) with respect to time (s). Here, ΔT represents the time period of one complete cycle of the waveform in seconds, while ΔP represents the difference between peak to peak power value observed in one cycle of the waveform. Thus, ΔP/ΔT gives the idea of an oscillation around the maximum power point. From Table 4, it is deduced that the proposed framework delivers power with minimum oscillations close to 0.83070 W/s at G = 300 ${\mathrm{W}/\mathrm{m}}^2$, 0.13120 W/s at G = 500 ${\mathrm{W}/\mathrm{m}}^2$, and 0.75390 W/s at G = 1000 ${\mathrm{W}/\mathrm{m}}^2$. Thus, it is perceived from the above analysis that the proposed framework delivers better performance by delivering maximum power with minimum oscillations exhibiting the robustness.

5.1.2. Scenario-2

In the second scenario, the continuously varied irradiance is fed as an input while keeping the temperature constant at 25 deg C. For this case, the power, current, and voltage response obtained using the proposed and other frameworks are compared and depicted in Figure 12, Figure 13 and Figure 14. It is observed that the proposed framework tracks the MPP more efficiently, and the extracted power is more than the power obtained using other frameworks.

5.2. Solar Panel-2

This part of the study consists of an array of five solar panels. The proposed hybrid framework is employed for the maximum power extraction, and the outcomes for different scenarios are compiled, analyzed, and compared with [27]. The tuned parameters of the FOPID controller are as follows: $K_p$ is 0.8029, $K_i$ is 1.7407, $K_d$ is 0.3931, $\lambda$ is 0.9629, $\mu$ is 0.1350, and RMSE is 0.01414.

5.2.1. Scenario-1

In this scenario, the irradiance is varied in step- and ramp-wise fashion, and the temperature remains constant at 25 deg C. Two irradiance levels considered for this study are 1000 ${\mathrm{W}/\mathrm{m}}^2$ and 200 ${\mathrm{W}/\mathrm{m}}^2$. The simulation model is run for five seconds. The irradiance at the input is kept at 1000 ${\mathrm{W}/\mathrm{m}}^2$ for one second, and for the next one second, the irradiance is set to gradually decreasing order until the value of 200 ${\mathrm{W}/\mathrm{m}}^2$. The irradiance at 200 ${\mathrm{W}/\mathrm{m}}^2$ is kept constant for the next one secone, and then it is set to gradually increasing order until 1000 ${\mathrm{W}/\mathrm{m}}^2$. The power obtained using the proposed and other frameworks is compared and depicted in Figure 15. The graph shows that the calculated maximum power reaches the actual maximum power faster using the proposed framework as compared to power calculated using other frameworks. Thereafter, the P–V and I–V characteristic curves are drawn for G = 1000 ${\mathrm{W}/\mathrm{m}}^2$, and 200 ${\mathrm{W}/\mathrm{m}}^2$, as depicted by Figure 16 and Figure 17. The presented curves reveal that the power extracted using the proposed framework is more than that of other frameworks for different values of irradiances. Further, numerical data obtained using proposed and other frameworks are compared, and the comparison is shown in

Table 5. Comparison based on various parameters for the SP-2 at varying irradiances.

Irradiance	Frameworks	Pactual (W)	Pcalculated (W)	Vcalculated (V)	Relative Error	MPPEfficiency (%)
1000	Using Conventional ANN [14]	925.2	924	119.9	0.00129	99.87
	Using ANN-PSO [14]	925.2	924.60	119.8	0.00064	99.93
	Proposed framework	925.2	925.09	120.02	0.00011	99.98
200	Using Conventional ANN [14]	198.9	193.12	126.8	0.02905	97.09
	Using ANN-PSO [14]	198.9	192.98	126.9	0.02976	97.02
	Proposed framework	198.9	195.70	127.2	0.01608	98.39

. The comparative analysis is carried out based on calculated voltage, relative error, and MPP efficiency obtained for the part first. It is observed that the proposed framework is able to track the actual peak and delivers superior outcomes in terms of following PIs: For G = 1000 ${\mathrm{W}/\mathrm{m}}^2$, relative error is found to be 0.00011, and MPP efficiency is close to 99.98%; and for G = 200 ${\mathrm{W}/\mathrm{m}}^2$, relative error is found to be 0.01608, and MPP efficiency is close to 98.39%.

Table 6. Comparison based on various parameters for the SP-2.

Frameworks/Parameters	MSE	Epoch	MPP Tracking Time (s)	Oscillations
Proposed	0.00000409	04	0.049	Low
ANN-PSO [14]	0.00068	17	0.06	Low
Conventional ANN [14]	0.0079	68	0.08	High

shows the comparison of the proposed framework with other considered frameworks based on several parameters such as mean square error (MSE), number of epochs, MPP tracking time, and occurrence of oscillation. Further,

Table 7. Oscillation measurements for scenario-1 for SP-2.

Irradiance (G)	Frameworks	ΔP (W)	ΔT (s)	1/ΔT (Hz)	ΔP/ΔT (W/s)
200	Proposed	0.02076	0.02111	47.38210	0.98402
	ANN-PSO [14]	0.05051	0.02585	38.68170	1.95400
	Conv-ANN [14]	0.06510	0.02588	38.62790	2.51500
1000	Proposed	0.01930	0.02011	49.82800	0.96402
	ANN-PSO [14]	0.05239	0.03098	32.27700	1.69100
	Conv-ANN [14]	0.13830	0.03205	31.19500	4.31500

shows the measurements obtained from the power vs. time graph, showing the rate of change of power (W) with respect to time (s). From Table 7, it is deduced that the proposed framework delivers power with minimum oscillations close to 0.98402 W/s at G = 200 ${\mathrm{W}/\mathrm{m}}^2$, and 0.96402 W/s at G = 1000 ${\mathrm{W}/\mathrm{m}}^2$. Thus, it is perceived from the above analysis that the proposed framework delivers better performance by delivering maximum power with minimum oscillations exhibiting the robustness.

5.2.2. Scenario-2

In the second scenario, partial shading condition is considered in the form of cloudy weather, and the power obtained using the proposed and other frameworks is compared and depicted in Figure 18. It is deduced that the proposed framework successfully tracks the maximum power along with minimum oscillation in the case of continuously changing irradiances, thus projecting the robustness of the proposed framework.

5.2.3. Scenario-3

In this scenario, the effect of partial shading conditions is studied on an array of a solar panel-2. Five different cases are considered in order to test the competence of the proposed framework. The cases considered for the study under partial shading conditions are depicted in Figure 19. In Case-A, only two panels are under shade, having an irradiance of 200 W/${\mathrm{m}}^2$, and the three panels have an irradiance of 1000 W/${\mathrm{m}}^2$. In Case-B, three panels are under shade, having an irradiance of 200 W/${\mathrm{m}}^2$, and the two panels have an irradiance of 1000 W/${\mathrm{m}}^2$. In Case-C, the irradiance is different for all the panels. In Case-D, one string consist of five panels connected in series, and three such strings are connected in parallel to make an array of 5 × 3. In this case, two strings have an irradiance of 1000 W/${\mathrm{m}}^2$, and one string is under shade, having an irradiance of 200 W/${\mathrm{m}}^2$. In Case-E, the same array of 5 × 3 is analyzed for different partial shading conditions. In this case, one string has an irradiance of 1000 W/${\mathrm{m}}^2$ while the other two strings are under shaded conditions, having an irradiance of 500 W/${\mathrm{m}}^2$ and 200 W/${\mathrm{m}}^2$. The responses for all the cases are recorded in the shading condition in terms of power, PV voltage, and boost voltage that is fed to the grid-connected inverter. The temperature for all the cases is considered to be 25 °C. The power responses using the proposed hybrid framework for all five cases are depicted in Figure 20 and tabulated in

Table 8. Various performance indices obtained for the SP-2 array under PSC using the proposed hybrid framework.

Cases/Parameters	PMPP (W)	PCalculated (W)	VCalculated (V)	VBoost (V)	MPPEfficiency (%)	MPP Tracking Time (s)
Case-A	630.21	630.20	69.15	204.5	99.99	0.08
Case-B	486.30	486.20	130.10	222	99.97	0.11
Case-C	565.32	565.30	103.20	224	99.99	0.1
Case-D	2035.2	2035	120.20	225	99.99	0.14
Case-E	1594.8	1594	121.60	221	99.94	0.2

. The results depicted in Table 8 show the robustness of the proposed framework in achieving the maximum power point with high efficiency and rapid tracking time under PSC. Further, P–V and I–V curves for different configurations under partial shading conditions are drawn. Figure 21 depicts the P–V and I–V curves for Case-A, Case-B, and Case-C, showing the global maximum power point. From Figure 21, it is deduced that the proposed framework is able to track down the global maximum power point for different conditions of partial shading. Further, an array of 5 × 2 and 5 × 3 of solar panel-2 are placed under normal surrounding conditions, i.e., at G = 1000 W/${\mathrm{m}}^2$ and T = 25 °C, and the proposed framework is implemented. The configurations of an array for this study are shown in Figure 22. The power responses for the mentioned configurations are recorded and depicted in Figure 23 and tabulated in

Table 9. Various performance indices obtained for the SP-2 array under normal conditions using the proposed hybrid framework.

Cases/Parameters	PMPP (W)	PCalculated (W)	VCalculated (V)	VBoost (V)	MPPEfficiency (%)	MPP Tracking Time (s)
Case-A	1850.2	1849.8	119.9	220	99.97	0.15
Case-B	2775.6	2774.8	120.2	225	99.97	0.12

. Further, Figure 24 shows the P–V and I–V curves of Case-D and Case-E, depicting that the maximum power is successfully extracted for both the cases.

Based on the evaluation, it is said that the proposed framework exhibits better performance and tracks the maximum power faster than other frameworks with minimum oscillation under vast varying conditions. It is also analyzed that the proposed framework works well and is able to track the global maximum power point when an array of 5 × 3 is subjected to various partial shading conditions.

The results presented in

Table 10. Power improvements in % for both solar panels.

From Table 3-SP-1
S.No	Irradiances (G)	Proposed vs. FPA	Proposed vs. WCA
1	1000	+0.05%	+0.06%
2	500	+0.32%	+0.51%
3	300	+0.30%	+1.04%
From Table 5-SP-2
S.No	Irradiances (G)	Proposed vs. ANN-PSO	Proposed vs. Conv. ANN
1	1000	+0.053%	+0.12%
2	200	+1.41%	+1.34%

consolidate the percentage power improvements of the proposed hybrid ANN–FOPID MPPT framework over several benchmark algorithms across varying irradiance levels, using data from Table 3 (SP-1 system) and Table 5 (SP-2 system) in the manuscript. The proposed framework consistently shows positive improvement over conventional methods (FPA, WCA, ANN-PSO, and conventional ANN) under both high and low irradiance levels. For example, under standard test conditions (1000 W/m²), improvements range between +0.05% to +0.12%, confirming precise MPPT behavior when solar input is stable. Under lower irradiance levels (e.g., 200–300 W/m²), which are typically more challenging for MPPT due to increased nonlinearity and reduced voltage gradients, the proposed framework delivers significant improvements:

• +1.04% over WCA at 300 W/m².
• +1.41% over ANN-PSO and +1.34% over conventional ANN at 200 W/m².

The power improvement comparison for different values of irradiances is diagrammatically presented in Figure 25. This clearly reflects the robustness of the hybrid control strategy in handling nonideal and dynamic environmental conditions, such as partial shading or low solar insulation. These improvements are attributed to the accurate voltage reference prediction by the ANN (trained using MSFLA), and the dynamic duty cycle control achieved by the self-tuned TLBO-based FOPID controller.

The intelligent combination of AI prediction and adaptive control enables the system to track the maximum power point (MPP) more effectively than traditional algorithms. Even a 0.5% to 1% improvement in power output can translate to substantial annual energy gains and economic benefits, especially in large-scale PV systems or in locations with frequent irradiance fluctuations.

6. Conclusions

This paper presents an innovative hybrid framework that integrates an artificial neural network (ANN) with a fractional-order proportional–integral–derivative (FOPID) controller, specifically designed to enhance maximum power point (MPP) extraction in solar photovoltaic (PV) systems. The study considered two different types of solar panels. Within this framework, the ANN was trained using the Modified Shuffled Frog Leaping Algorithm (MSFLA), while the FOPID controller was optimized via the Sanitized Teacher–Learning-Based Optimization (s-TLBO) algorithm. The resulting hybrid model was implemented in a PV system and tested through comprehensive simulations conducted under various environmental conditions. The obtained results were rigorously compared with previously published works.

The comparative analysis demonstrates that the proposed hybrid framework consistently outperforms existing methods reported in [14,27]. The framework reliably delivers power values close to the actual peak power across a wide range of operating conditions. Validation was performed using multiple performance indices, including mean square error (MSE), root mean square error (RMSE), MPP tracking time, relative error (RE), and efficiency. Key findings include the following:

• The performance of solar panel-1 was systematically evaluated and benchmarked against prior studies in two phases. Across a range of conditions, the proposed hybrid framework demonstrated superior performance, with consistently high efficiency, as evidenced in Table 2, Table 3 and Table 4.
• To further assess its robustness, the framework was applied to an array configuration of solar panel-2. This evaluation included testing under various environmental conditions, such as cloudy weather and different partial shading scenarios. The framework was also tested on array configurations of 5 × 1, 5 × 2, and 5 × 3, yielding improved performance relative to prior studies, as shown in Table 5, Table 6, Table 7, Table 8 and Table 9. Notably, the proposed approach achieved fast and stable MPP tracking within 0.049 s, with minimal oscillations.
• The effectiveness of the proposed framework was further validated through rigorous simulations conducted using MATLAB software.

The outcomes achieved using the proposed hybrid framework indicated notable improvements in overall system performance. These enhancements were systematically presented using various performance metrics across a spectrum of environmental scenarios. The results establish the framework as a robust and efficient solution for optimizing solar PV system performance, validated by extensive testing and favorable comparisons with existing techniques. While the current model demonstrates strong predictive capability using irradiance (G) and temperature (T) as input features, its performance could be further enhanced. Future work will explore the inclusion of additional dynamic parameters, such as load variations and wind-induced cooling, which significantly influence PV system behavior. Incorporating these factors would enable the ANN model to capture a more comprehensive representation of real-world operating conditions, thereby improving its accuracy, adaptability, and robustness under complex and variable scenarios. Furthermore, exploring the practical implementation and real-time deployment of the proposed framework remains a promising direction for future research.