A Deep Learning-Based Ensemble Method for Parameter Estimation of Solar Cells Using a Three-Diode Model

Abstract

Accurate parameter estimation of solar cells is critical for early-stage fault diagnosis in photovoltaic (PV) power systems. A physical model based on three-diode configuration has been recently introduced to improve model accuracy. However, nonlinear and recursive relationships between internal parameters and PV output, along with parameter drift and PV degradation due to long-term operation, pose significant challenges. To address these issues, this study proposes a deep learning-based ensemble framework that integrates outputs from multiple optimization algorithms to improve estimation precision and robustness. The proposed method consists of three stages. First, the collected data were preprocessed using some data processing techniques. Second, a PV power generation system is modeled using the three-diode structure. Third, several optimization algorithms with distinct search behaviors are employed to produce diverse estimations. Finally, a hybrid deep learning model combining convolutional neural networks (CNNs) and long short-term memory (LSTM) networks is used to learn from these results. Experimental validation on a 733 kW PV power generation system demonstrates that the proposed method outperforms individual optimization approaches in terms of prediction accuracy and stability.

1. Introduction

Accurate estimation of solar cell parameters is essential for predicting power output and facilitating early-stage fault detection in photovoltaic (PV) power generation systems [1]. Manufacturers typically provide key specifications, such as short-circuit current, open-circuit voltage, and the maximum power point (MPP) parameters of voltage and current under standard test conditions (STCs). These specifications serve as the basis for developing physical models that estimate real-world power generation under various weather conditions.

Several physical models have been proposed for PV modules, including the single-diode, two-diode, and three-diode models. The single-diode model incorporates parameters such as photocurrent, dark saturation current, ideality factor, series resistance, and shunt resistance [2,3,4,5], offering a balance between simplicity and accuracy. The two-diode model [6,7,8,9] extends this by adding a second diode, which improves performance under low irradiance conditions.

In the three-diode model [10,11,12,13,14,15,16,17,18,19], the dark saturation current is represented by three distinct diodes, accounting for (i) diffusion and recombination currents associated with the emitter and the P–N junction, (ii) recombination current in the depletion region, and (iii) currents arising from leakage and grain boundary effects. This model provides the most comprehensive representation, improving accuracy by capturing a wider range of internal physical mechanisms. Nevertheless, the parameters count increases to nine, thereby introducing greater complexity to the model.

Despite improvements in modeling, estimating parameters remains challenging due to the exponential nonlinearities and recursive dependencies between internal parameters and output power. Conventional iterative methods converge rapidly but are susceptible to local optima if initialized poorly [20,21,22,23]. In contrast, multi-agent optimization approaches, such as evolutionary computing (EC), swarm intelligence (SI), and physics-based (PB) methods, leverage multi-point search and parallelism to achieve more reliable solutions.

EC algorithms simulate natural evolution through mutation and crossover, with examples including the flower pollination algorithm (FPA) [24], differential evolution (DE) [25], and genetic algorithm (GA) [26]. SI methods are inspired by collective behaviors, such as particle swarm optimization (PSO) [27], whale optimization algorithm (WOA) [28], bonobo optimizer (BO) [29], and salp swarm algorithm (SSA) [30]. PB algorithms rely on physical phenomena, including charged system search (CSS) [31] and lightning search algorithm (LSA) [32]. References [33,34] provide a detailed introduction and review of the parameter estimation methods for single-diode and two-diode power generation models.

Table 1. Comparison of the three-diode models reported in the literature.

Reference	No. of Parameter	Optimization Algorithm
		EC	SI	PB	Newton
[1] *	16	√ **	√
[10]	9		√
[11]	9				√
[12] *	16		√
[13]	9		√
[14]	9		√	√
[15]	9		√		√
[16]	9			√
[17]	9		√
[18]	9		√	√
[19]	9		√

*: Expand nine parameters to 16 parameters for more precise estimates. In this study, the nine parameters are transformed into 14 parameters. **: A check mark indicates that the reference belongs to that category of optimization method.

summarizes the three-diode models reported in the literature. Most studies employed SI-based optimization algorithms to obtain accurate parameter estimates. As presented in [1,12], the three-diode model achieves improved estimation accuracy compared with the single-diode model, yielding an MRE reduction of approximately 0.017–0.040%. Relative to the two-diode model, the accuracy improvement ranges from 0.001% to 0.008%, depending on the optimization technique applied. Furthermore, References [1,12] expanded the parameter set from nine to sixteen, enabling more refined estimates. A similar transformation is adopted in this study.

As shown in Table 1 and References [24,25,26,27,28,29,30,31,32], most prior studies employ either single or hybrid optimization methods for parameter estimation. Although multi-agent optimization algorithms have shown promising results, various optimization algorithms with distinct search behaviors often yield diverse estimates under varying weather conditions, parameter drift, and PV modules degradation after long-term operation. To address these limitations, this study proposes a deep learning-based ensemble framework that combines convolutional neural networks (CNNs) and long short-term memory (LSTM) networks [35] to integrate the estimated outputs from distinct optimization algorithms. The key contributions are as follows:

• Estimation results using the three-diode model are compared with those from other diode models to verify its accuracy.
• Multiple optimization algorithms, including EC, SI, and PB methods, are employed to generate complementary estimates, thereby enhancing ensemble effectiveness and mitigating overfitting.
• A hybrid CNN–LSTM architecture is developed to improve predictive accuracy and stability in three-diode parameter estimation.
• As the output power of PV systems decreases due to degradation and parameter drift, the proposed ensemble method integrates multiple optimization algorithms to enhance prediction accuracy, thereby improving the reliability of practical deployments.

The remainder of this paper is organized as follows. Section 2 introduces the three-diode model. Section 3 presents the proposed parameter estimation methodology. Section 4 reports the simulation results for a 733 kW PV power generation system. Section 5 provides discussions, and Section 6 concludes the paper.

2. The Three-Diode Model

Figure 1 illustrates the configuration of a three-diode model. The output current ($I_{pv}$) can be expressed as:

$$\begin{array}{cc}{I}_{pv}\hfill & =I_{ph}-I_{sat1}\left[\exp \left({\displaystyle \frac{V_{pv}+I_{pv}{R}_s}{n_{idl1}{V}_T}}\right)-1\right]-I_{sat2}\left[\exp \left({\displaystyle \frac{V_{pv}+I_{pv}{R}_s}{n_{idl2}{V}_T}}\right)-1\right]\hfill \\ & -I_{sat3}\left[\exp \left({\displaystyle \frac{V_{pv}+I_{pv}{R}_s}{n_{idl3}{V}_T}}\right)-1\right]-{\displaystyle \frac{V_{pv}+I_{pv}{R}_s}{R_h}},\hfill \end{array}$$

(1)

where $I_{ph}$ is the photocurrent, $R_s$ is the series resistance, $R_{sh}$ is the parallel resistance, $I_{sat1},I_{sat2}\mathrm{a}\mathrm{n}\mathrm{d}{I}_{sat3}$ are the dark saturation currents, ${ n}_{idl1},$ ${ n}_{idl2},{ n}_{idl3}$ are the ideality factors, $V_{pv}$ is the output voltage, and $V_T$ is the thermal voltage which is varied with the absolute temperature.

Since the open-circuit voltage of a solar cell is approximately 0.6 V, multiple solar cells are connected in series to form a solar module. These modules are further arranged in series and parallel to construct a PV array, thereby achieving the desired voltage, current, and power output, as shown in Figure 2. The output current for a PV array can be rewritten from (1) as:

$$\begin{array}{cc}{I}_{pv}\hfill & ={N_{p}I}_{ph}-N_p{I}_{sat1}\left[\mathit{exp}\left({\displaystyle \frac{V_{pv}+I_{pv}{R}_s{N}_{s/}{N}_p}{N_s{n}_{idl1}{V}_T}}\right)-1\right]-N_p{I}_{sat2}\left[\mathit{exp}\left({\displaystyle \frac{V_{pv}+I_{pv}{R}_s{N}_{s/}{N}_p}{N_s{n}_{idl2}{V}_T}}\right)-1\right]\hfill \\ & -N_p{I}_{sat3}\left[\mathit{exp}\left({\displaystyle \frac{V_{pv}+I_{pv}{R}_s{N}_{s/}{N}_p}{N_s{n}_{idl3}{V}_T}}\right)-1\right]-{\displaystyle \frac{V_{pv}+I_{pv}{R}_s{N}_{s/}{N}_p}{R_h}},\hfill \end{array}$$

(2)

where $N_s$ and $N_p$ denote the number of modules connected in series and in parallel, respectively.

The aforementioned nine parameters ($I_{ph}$, $I_{sat1}$, $I_{sat2}, I_{sat3},{ n}_{idl1}$,${ n}_{idl2},{ n}_{idl3}$ $R_s$, and $R_{sh}$) directly influence the output characteristics of the PV power generation system. However, these key parameters ($I_{ph}$, $I_{sat1}$,${ n}_{idl1}$, $R_s$, and $R_{sh}$) are in turn affected by the short-circuit current ($I_{sc}$), open-circuit voltage ($V_{oc}$), and the voltage ($V_m$) and current ($I_m$) at the MPP as follows [2,3,4,5]:

$$I_{ph}=\left(1+{\displaystyle \frac{R_s}{R_{sh}}}\right)I_{sc},$$

(3)

$$I_{sat1}=\left(I_{sc}-{\displaystyle \frac{V_{oc}}{R_{sh}}}\right)\left[\mathit{exp}\left(-{\displaystyle \frac{V_{oc}}{n_{idl1}{V}_T}}\right)\right],$$

(4)

$$n_{idl1}=\left[I_{sat1}\left({\displaystyle \frac{V_m+I_m{R}_s}{I_m}}\right)exp\left({\displaystyle \frac{V_m+I_m{R}_s}{n_{idl1}{V}_T}}\right)\right]/V_T,$$

(5)

$$R_s={\displaystyle \frac{\left(I_{sat1}\left(\frac{V_m+I_m{R}_s}{I_m}\right)exp\left(\frac{V_m+I_m{R}_s}{n_{idl1}{V}_T}\right)\right)ln\left(\frac{I_{sc}-I_m}{I_{sat1}}\right)-V_m}{I_m}},$$

(6)

$$R_{sh}=\left({\displaystyle \frac{I_{sc}}{I_{ph}-I_{sc}}}\right)R_s.$$

(7)

To simplify the parameter extraction process, Ishaque et al. [9] assumed $I_{sat1}$=${ I}_{sat2}$= $I_{sat3}$, and based on the Shockley diode diffusion theory [36], set $n_{idl1}\le 1.2$, and selected $n_{idli2}>1.2$ to achieve the best estimation results.

Moreover, the manufacturer-provided specifications ($I_{sc}$, $V_{oc}$, $V_m$, and $I_m$) vary with solar irradiance and module temperature as shown below:

$$I_{sc}={\displaystyle \frac{G}{G_r}}{I}_{scr}+{\beta }_{I_{sc}}\left(T-T_r\right),$$

(8)

$$V_{oc}=V_{ocr}+{\beta }_{V_{oc}}\left(T-T_r\right),$$

(9)

$$V_m=V_{mr}+{\beta }_{V_{oc}}\left(T-T_r\right),$$

(10)

$$I_m={\displaystyle \frac{G}{G_r}}{I}_{mr}+{\beta }_{I_{sc}}\left(T-T_r\right),$$

(11)

where $I_{scr}$, $V_{ocr}$, $V_{mr}$, and $I_{mr}$ represent the values under STC, with irradiance of 1000 W/m², temperature of 298 K, and air mass of 1.5. ${\beta }_{I_{sc}}$ and ${\beta }_{V_{oc}}$denote temperature coefficients for short-circuit current and open-circuit voltage, respectively.

To improve the estimation accuracy of the three-diode model under varying irradiance and temperature conditions, the original nine parameters are extended to 14, as conducted from Equation (3) to Equation (11). The parameter control framework is illustrated in Figure 3, where these 14 parameters act as control variables for determining the nine key parameters. Optimization algorithms are then employed to identify the optimal set of control variables. Once obtained, the output current and power are computed using Equation (2).

3. The Proposed Method

Figure 4 illustrates the overall framework of the proposed ensemble learning approach, which comprises four stages: (i) data preprocessing (Stage 1), (ii) model establishment (Stage 2), (iii) preliminary estimation using individual optimization algorithms (Stage 3), and (iv) final estimation with the proposed ensemble method (Stage 4). The following subsections provide detailed descriptions of these stages.

3.1. Data Preprocessing

3.1.1. Missing Data Compensation

Missing data are compensated using interpolation or regression techniques. If more than 50% of the data for a given day are missing, the data from that day are excluded from further analysis.

3.1.2. Outlier Removal

Outlier detection and removal are essential for improving data quality. Several approaches exist, including the interquartile range (IQR), median absolute deviation (MAD), classification models, clustering-based methods, and the Z-score. In this study, the Z-score method is employed, as it provides a normalized measure of deviation from the mean, defined as:

$$Z={\displaystyle \frac{x-\mu }{\sigma }} ,$$

(12)

where x is a data point, $\mu$ is the mean, and $\sigma$ is the standard deviation. A higher $\left|Z\right|$ value indicates a greater deviation from the mean.

3.1.3. Data Smoothing

Data smoothing reduces noise and emphasizes long-term trends in time series analysis. The moving average method is applied, defined as:

$${MA}_t(n)={\displaystyle \frac{1}{n_w}}{\sum }_{i=t-n+1}^t{x}_i,$$

(13)

where $n_w$ is the moving window size and ${MA}_t\left(n\right)$ is the moving average at time t.

3.1.4. Feature Selection

To select the most relevant input variables for PV power generation, the Pearson correlation coefficient (PCC) is calculated as:

$$r={\displaystyle \frac{\sum xy-\frac{\sum x\sum y}{n_d}}{\sqrt{\left(\sum x^2-\frac{{\left(\sum x\right)}^2}{n_d}\right)\left(\sum y^2-\frac{{\left(\sum y\right)}^2}{n_d}\right)}}},$$

(14)

where x is the weather variable, y is the PV power output, $n_d$ is the number of samples, and r is the correlation coefficient.

3.1.5. Data Classification

K-means clustering [37] is employed to partition n data point (x₁, x₂, …, x_n) into k clusters by minimizing the sum of squared Euclidean distances between each point and its assigned cluster center as:

$$\mathrm{a}\mathrm{r}\mathrm{g} \mathrm{m}\mathrm{i}\mathrm{n}={\sum }_{i=1}^k{\sum }_{j=1}^n{w}_{ji}{‖X_j-R_i‖}^2,$$

(15)

where $X_j$ is the jth observation, $R_i$ is the ith cluster center, $w_{ji}$ is the weight, and $‖•‖$is the Euclidean distance. $R_i$ and $w_{ji}$ can be expressed individually as follows:

$$R_i={\displaystyle \frac{\sum _{j=1}^n{w}_{ji}{X}_j}{\sum _{j=1}^n{w}_{ji}}},$$

(16)

$$w_{ji}=\left\{\begin{array}{cc}1,& if ‖X_j-R_i‖ \le ‖X_j-R_m‖, \forall m\ne i\\ 0,& otherwise\end{array}\right.,$$

(17)

After data compensation, abnormal data removal, data smoothing, feature extraction, and data classification, a three-diode power generation model can be established based on the manufacturer’s parameters and historical data, including PV power, solar irradiance, and module temperature. Since the manufacturer typically provides only four parameters ($I_{sc}$, $V_{oc}$, $V_m$, and $I_m$), the remaining unknown parameters must be estimated using Newton–Raphson method. Once all nine parameters are obtained, an optimization algorithm can be applied to perform a more comprehensive search for parameter solutions. The following section introduces five optimization algorithms with distinct characteristics.

3.2. Optimization Algorithms

As shown in Figure 4, five optimization algorithms are used to produce individual outputs, including the Newton–Raphson method, particle swarm optimization (PSO), multiverse optimizer (MVO), evolutionary strategy (ES), and atom search optimization (ASO). These algorithms were selected for their diverse characteristics, which provide complementary estimates, thereby enhancing ensemble effectiveness and reducing the risk of overfitting. Specifically, the Newton–Raphson method was employed as a single-agent approach for parameter searching, while PSO is categorized under swarm intelligence (SI), MVO and ASO under population-based (PB) methods, and ES under evolutionary computation (EC). This diversity ensures that the proposed framework can be generalized to a wide range of scenarios by leveraging the strengths of different algorithmic classes. A detailed description of the five optimization algorithms is provided in the following subsection.

3.2.1. Newton–Raphson Method

The Newton–Raphson method [20] employs a single agent to search for parameter solutions. Let the parameter vector be defined as

$$\omega =\left[I_{scr}, { V}_{ocr},{ I}_{mr}, { V}_{mr},{ I}_{sat1},{ n}_{idl1},{ n}_{idl2},{ n}_{idl3},{G_r, T_r, {\beta }_{I_{SC}}, {\beta }_{V_{OC}}, R}_s, R_{sh}\right],$$

(18)

The solution procedure is outlined as follows:

1.

Parameter initialization: Define the admissible range for each parameter.

2.

Fitness definition: Let $x_1=\omega$, and formulate the objective function $f\left(V_{pv},I_{pv},\omega \right)$as:

$$\begin{array}{c}f\left(V_{pv},I_{pv},\omega \right)={I_{pv}-I}_{ph}^{\prime }-\{I_{sat1}^{\prime }\left[\mathit{exp}\left({\displaystyle \frac{V_{pv}+I_{pv}{R}_s^{\prime }}{n_{idl1}^{\prime }{V}_T}}\right)-1\right]-N_p{I}_{sat2}\left[\mathit{exp}\left({\displaystyle \frac{V_{pv}+I_{pv}{R}_s^{\prime }}{n_{idl2}^{\prime }{V}_T}}\right)-1\right]-\hfill \\ N_p{I}_{sat3}\left[\mathit{exp}\left({\displaystyle \frac{V_{pv}+I_{pv}{R}_s^{\prime }}{n_{idl3}^{\prime }{V}_T}}\right)-1\right]-{\displaystyle \frac{V_{pv}+I_{pv}{R}_s^{\prime }}{R_h}}\}=0,\hfill \end{array}$$

(19)

where $f\left(V_{pv},I_{pv},\omega \right)=0$is a nonlinear recursive function, and its fitness value is expressed as:

$$f_{MAE}={\displaystyle \frac{1}{N}}{\sum }_{i=1}^N\left|f\left(V_{pv},I_{pv},\omega \right)\right|,$$

(20)

where N is the number of data points, MAE is the mean absolute error, which is less sensitive to outliers.

3.

Initialization: Randomly generate an initial value $x_1^{\left(0\right)}$, and evaluate both ${f(x}_1^{\left(0\right)})$ and its derivative ${f\prime (x}_1^{\left(0\right)}).$

4.

Parameter update: Update the parameter values according to Equation (21):

$$x_1^{\left(1\right)}=x_1^{\left(0\right)}-{\displaystyle \frac{{f(x}_1^{\left(0\right)})}{{f\prime (x}_1^{\left(0\right)})}} ,$$

(21)

5.

Convergence check: Evaluate ${f(x}_1^{\left(1\right)})$. If the value falls below a predefined threshold, the algorithm terminates and the converged $x_1$ is recorded. Otherwise, repeat Step 4.

6.

Output calculation: Once convergence is achieved, compute the output current, voltage, and power using Equation (2).

3.2.2. Particle Swarm Optimization (PSO)

PSO is a population-based stochastic optimization algorithm introduced by James Kennedy and Russ Eberhart in 1995 [38]. It is inspired by the social behavior of birds flocking or fish schooling. PSO employs a swarm of particles (potential solutions) “fly” through the search space, adjusting their positions based on their own best position (personal experience) and the best positions found by the swarm (social learning) as follows:

$$v_i\left(t+1\right)=w{v}_i\left(t\right)+c_1·r_1\times \left(p_i-x_i\left(t\right)\right)+c_2·r_2\times \left(g\left(t\right)-x_i\left(t\right)\right),$$

(22)

$$x_i\left(t+1\right)=x_i\left(t\right)+v_i\left(t+1\right),$$

(23)

where $x_i\left(t\right)$ is the position vector of the ith particle, $v_i\left(t\right)$is the velocity vector of the ith particle, $g\left(t\right)$ is the global best position, $w$ is an inertia weight, $c_1$ and $c_2$ are the acceleration factors that control particles towards the optimal position. $r_1$ and $r_2\in [0, 1]$ are random numbers used to maintain group diversity.

3.2.3. Multiverse Optimizer (MVO)

MVO is a nature-inspired optimization algorithm introduced by Mirjalili et al. in 2016 [39]. It is inspired by the multiverse theory in physics, which includes three core phenomena: white holes, black holes, and wormholes. The solution steps using MVO are described as follows:

1.

Initialization: Randomly generate the initial population of universes (solutions) as follows:

$${\omega }_i=\left[x_{i1}, x_{i2}, ..., x_{id}\right] \mathrm{f}\mathrm{o}\mathrm{r} i=1, 2, ... N_p,$$

(24)

where $N_p$ is the population size and $d$ is the variable dimension. In this paper, $d=14$, representing the number of parameters to be estimated.

2.

Fitness evaluation: Compute the inflation rate (IR) of each universe, which is the inverse of the fitness value of Equation (20). IR is used for sorting and probability exchange.

3.

White- and black-hole exchange: Exchange variables between universes based on fitness ranking. If the inflation rate of the universe ${\omega }_j$is high, transmit its information to other universes ${\omega }_i$ through white holes. Information exchange is performed using the roulette wheel selection method, as follows:

$$x_{ik}^{new}=\left\{\begin{array}{cc}{x}_{jk},& if rand<{IR}_j\\ x_{ik},& otherwise\end{array}\right.,$$

(25)

where $rand\in [0, 1]$ is a random number, ${IR}_j$is the selected inflation rate, and $x_{ik}$ is the parameter value of the kth dimension of the ith universe.

4.

Wormhole tunneling: Adjust each universe’s position toward the best-known universe with a certain probability. All universes have the opportunity to approach the optimal solution ${\omega }^{*}$ through the wormhole. Equation (25) is updated as follows:

$$x_{ik}^{new}=\left\{\begin{array}{cc}{x}_{jk}+T_{DR}·({ub}_k-{lb}_k)·rand,& if rand<0.5\\ x_{ik},-T_{DR}·({ub}_k-{lb}_k)·rand,& otherwise\end{array}\right.,$$

(26)

where ${ub}_k$ and ${lb}_k$ are the upper and lower bounds of the k dimension, respectively, $T_{DR}$is the traveling distance rate, controlling the magnitude of perturbation.

5.

Termination: Repeat step 2 until a maximum number of iterations or convergence criteria are met.

3.2.4. Evolution Strategies (ES)

ES is a class of black-box optimization algorithm inspired by the concept of natural evolution [40]. It is widely used in non-convex, non-differentiable, or noisy optimization problems, especially in high-dimensional continuous domains. The evolving steps using ES are described as follows:

1.

Initialization: Start with a randomly initialized population of individuals (solutions) as:

$$P^{\left(0\right)}=\left\{x_1^{\left(0\right)}, x_2^{\left(0\right)}, ..., x_N^{\left(0\right)}\right\}\subset R^d,$$

(27)

where individual $x_1^{\left(0\right)}\in R^d$ is a $d$-dimensional vector.

2.

Mutation: Each individual is perturbed by adding a normally distributed noise to generate offspring:

$$x_j^{\prime }=x_i+\rho ·{ϵ}_j, {ϵ}_j~N(0,\sigma ),$$

(28)

where $\rho$ is the step size of mutation, and $N(0,\sigma )$ is a standard normal distribution.

3.

Fitness evaluation: Calculate the fitness value of each offspring using Equation (20), as follows:

$$f\left(x_j^{\prime }\right), for all j\in \left\{1, 2, ...,N \right\}$$

(29)

4.

Selection: Sort all populations and select the top-performing individuals (based on fitness) to form the next generation, as follows:

$$P^{(t+1)}={SelectTop}_{\mu }\left({\left\{x_j^{\prime }\right\}}_{j=1}^N\right).$$

(30)

After the best performing individuals are selected, half of them ($\mu$) remain in the population.

5.

Termination: Repeat step 2 until convergence or a maximum number of iterations are met.

3.2.5. Atom Search Optimization (ASO)

ASO is a metaheuristic optimization algorithm introduced by M. Ghaffari Hadigheh et al. in 2020 [41]. It is inspired by the motion and interaction of atoms based on principles from molecular dynamics. Described below are the steps using ASO method:

1.

Initialization: Randomly initialize the positions $x_i\in R^D$ and velocity $v_i\in R^D$ of all atoms as:

$$x_i^d={lb}^d+rand·\left({ub}^d-{lb}^d\right),i=1, 2, ..., N$$

(31)

2.

Fitness evaluation: Evaluate the fitness of each atom ($f_i$) using Equation (20).

3.

Mass calculation: Convert fitness to mass as follows:

$$m_i={\displaystyle \frac{f_{worst}-f_i}{f_{worst}-f_{best}-ϵ}},$$

(32)

where $ϵ$ is the well depth of potential used to control attraction strength.

4.

Force calculation: Use the Lennard-Jones (LJ) potential to calculate forces between atoms i and j:

$$F_{ij}=24·ϵ\left[2{\left({\displaystyle \frac{\delta }{r_{ij}}}\right)}^{13}-{\left({\displaystyle \frac{\delta }{r_{ij}}}\right)}^7\right]·{\displaystyle \frac{x_i-x_j}{r_{ij}}},$$

(33)

$$F_i={\sum }_{j\in K_{best, j\ne i}}{F}_{ij},$$

(34)

where $r_{ij}=‖x_i-x_j‖$ is the interatomic distances and $\delta$ is a distance constant where the potential is zero if there is a balance between attraction and repulsion.

5.

Acceleration calculation: Apply Newton–Raphson’s second law to evaluate acceleration as follows:

$$a_i={\displaystyle \frac{F_i}{M_i+ϵ}},$$

(35)

where $M_i$is the normalization of mass.

6.

Velocity and position update: Update velocity and position as follows:

$$v_i(t+1)=rand·v_i(t+1)+a_i,$$

(36)

$$x_i(t+1)=x_i\left(t\right)+v_i(t+1),$$

(37)

7.

Boundary handling: Ensure that all position values remain within the defined search space:

$$x_i^d=min\left(max\right(x_i^d,{lb}^d),{ub}^d),$$

(38)

8.

Termination: Repeat step 2 until convergence or a maximum number of iterations are met.

Table 2. Comparison between the Newton–Raphson method, MVO, ES, PSO, and ASO.

Method	Characteristics	Advantages	Disadvantages
Newton–Raphson Method [20]	Gradient-based optimization. Quadratic convergence near optimum.	Very fast convergence if initial guess is close. Accurate for smooth, differentiable problems.	Sensitive to initial guess. Not suitable for non-differentiable or highly non-convex problems.
PSO [38]	Population-based approach inspired by swarm intelligence. No gradient required.	Fast convergence in early iterations. Simple to implement, few parameters.	May trap in local optima. Performance degrades in very high-dimensional search spaces.
MVO [39]	Nature-inspired metaheuristic approach. Balances exploration and exploitation via inflation rate.	Good exploration ability in complex search spaces. Few control parameters.	Convergence may be slow for high-dimensional problems. Lacks strong theoretical convergence guarantee.
ES [40]	Population-based approach inspired by biological evolution. No gradient needed.	Can escape local optima. Works well for continuous optimization.	High computational cost. Parameter tuning affects performance.
ASO [41]	Metaheuristic approach inspired by the motion and interaction of atoms. Balance between global and local search.	Strong global search ability. Effective on high-dimensional problems.	Parameter tuning required for best performance. May require more computation than simpler heuristics.

shows the structured comparison between the Newton–Raphson method, PSO, MVO, ES, and ASO. After generating individual initial estimates, the proposed approach employs a hybrid CNN–LSTM model to integrate the outputs of the five aforementioned methods. Since these optimization algorithms exhibit different characteristics, CNNs are used to extract their features, while the LSTM network is employed to capture temporal dependencies, thereby providing more accurate and stable estimates.

3.3. Deep Learning-Based Ensemble Method

The proposed hybrid CNN–LSTM operates as an ensemble framework that aggregates the outputs of various optimization algorithms to generate the final estimates. Specifically, it serves a dual role: functioning as a feature extractor and as a weight allocator, thereby assigning appropriate significance to each optimization algorithm. CNNs are utilized to extract informative temporal features, while LSTM is incorporated to mitigate the gradient vanishing issue in sequence learning and to capture long-term dependencies. Introduced below are the CNNs and LSTM networks.

3.3.1. CNNs

The concept of CNNs was first introduced by Yann LeCun et al. in 1998 [42] for handwritten digit recognition. A CNN typically consists of convolutional layers, pooling layers, and fully connected layers, as illustrated in Figure 5. A brief description of each layer is provided below:

(i)

Convolutional layers:

• Serve as the fundamental building blocks of CNNs.
• Utilize learnable filters (kernels) to convolve across the input, capturing local spatial or temporal dependencies.
• Early layers primarily extract low-level features, while deeper layers capture more abstract and high-level representations.

(ii)

Pooling layers:

• Reduce the dimensionality of spatial or temporal features while preserving essential information.
• Enhance translation invariance and decrease computational complexity.

(iii)

Fully connected layers:

• Transform the extracted features from convolutional and pooling layers into a one-dimensional vector.
• Enable high-level reasoning at the final stage, supporting tasks such as classification, regression, or other predictions.

CNNs exhibit strong capabilities in automatic feature extraction and translation invariance, enabling the recognition of features irrespective of their position [43]. However, CNNs alone are limited in modeling long-term temporal dependencies. To address this limitation, LSTM is integrated in this work to complement CNNs and enhance sequence learning.

3.3.2. LSTM

LSTM was first introduced by Sepp Hochreiter and Jürgen Schmidhuber in 1997 [44]. It was designed as a solution to improve the vanishing gradient problem in recurrent neural networks (RNNs). LSTM has been extensively applied to time-series forecasting, speech recognition, and anomaly detection. The overall structure of an LSTM network is illustrated in Figure 6, where the inputs are derived from the outputs of CNNs.

An LSTM unit comprises four key components: input gate ($i_t$), memory cell (${\tilde{c}}_t$), forget gate ($f_t$), and output gate ($O_t$). These elements collectively regulate the flow of information, which can be mathematically formulated as follows:

$$i_t=\sigma (W_i{x}_t+U_i{h}_{t-1}+b_i),$$

(39)

$${\tilde{c}}_t=tanh(W_c{x}_t+U_c{h}_{t-1}+b_c),$$

(40)

$$f_t=\sigma (W_f{x}_t+U_f{h}_{t-1}+b_f),$$

(41)

$$o_t=\sigma (W_o{x}_t+U_o{h}_{t-1}+b_o),$$

(42)

At each time step, these gates determine how much of the past information should be retained, how much new information should be incorporated, and how much should be output. The sigmoid ($\sigma$) and tanh ($\phi$) activation functions are used to modulate these flows. In this study, LSTM is utilized to enhance temporal modeling capabilities of CNNs.

4. Numerical Results

4.1. Data Preprocessing Results

The proposed method was evaluated using a 733 kW PV power generation system. Data were collected from July 2021 to July 2022 at an hourly resolution, including historical PV power output, solar irradiance and module temperature. Outliers were removed using the Z-score method, and the remaining dataset was divided into 75% for training and 25% for testing. The choice of a 75%/25% split was based on the availability of data for each weather condition. Although an 80%/20% split is commonly adopted, the testing data corresponding to rainy days were relatively limited after classification in this study. Therefore, a 75%/25% split was employed to ensure sufficient representation for both training and testing.

During training, solar irradiance and module temperature were used as input variables, while PV power generation served as the output. In the testing stage, predicted global horizontal irradiance (GHI) and air temperature were employed as inputs, as they exhibit stronger correlation with PV output based on the PCC method. The forecasted weather data were obtained from the SOLCAST platform [45]. All experiments were implemented in Python 3.9 on a Windows 11 platform equipped with an Intel Core i7-10700 CPU, 32 GB RAM. Estimation accuracy was evaluated using the mean relative error (MRE), mean absolute Error (MAE), normalized mean absolute error (NMAE), and symmetric mean absolute percentage error (sMAPE), defined as:

$$MRE={\displaystyle \frac{1}{N_m}}{\sum }_{i=1}^{N_m}\left|{\displaystyle \frac{P_i-{\widehat{P}}_i}{P_{cap}}}\right|\times 100\%,$$

(43)

$$MAE={\displaystyle \frac{1}{N_m}}{\sum }_{i=1}^{N_m}\left|P_i-{\widehat{P}}_i\right|,$$

(44)

$$NMAE={\displaystyle \frac{MAE}{\overline{P}}},$$

(45)

$$sMAPE={\displaystyle \frac{1}{N_m}}{\sum }_{i=1}^{N_m}{\displaystyle \frac{\left|P_i-{\widehat{P}}_i\right|}{\frac{\left|P_i\right|+\left|{\widehat{P}}_i\right|}{2}}}\times 100\%,$$

(46)

where $P_{cap}$ is the rated PV capacity, $P_i$ is the actual power, ${\widehat{P}}_i$ is the estimated power, $\overline{P}$ is the mean of actual power, and $N_m$ denotes the number of testing samples. As indicated in (43)–(46), MRE reduces the influence of low-power generation levels on the error percentage. MAE quantifies the average magnitude of the errors regardless of sign. NMAE scales MAE by the mean value, making the error relative to the data scale. The sMAPE metric is a percentage-based symmetric error that mitigates the issue of inflated error values when actual power approaches zero.

The Z-score method was adopted to eliminate outliers. After normalization, the mean value ($\mu$) was set to 0 and the standard deviation ($\sigma$) to 1. Data points with Z-scores greater than 1.5 were considered outliers and subsequently removed.

The elbow curve shown in Figure 7 was derived using k-means clustering, with the sum of squared errors (SSE) expressed as:

$$SSE={\sum }_{i=1}^{N_T}{\left(P_i-{\widehat{P}}_i\right)}^2,$$

(47)

where $N_T$ is the number of training samples. When outliers are removed, the optimal number of clusters was determined using SSE. As illustrated in Figure 7, SSE decreases rapidly until K = 5, after which the reduction becomes marginal. Therefore, the data were classified into five clusters (Figure 8), corresponding to sunny, partly cloudy, overcast, heavily overcast, and rainy conditions. Note that when the data were divided into five classes, the continuity of the monthly time series was broken, and only daily time series could be preserved. The advantages of this classification are threefold: (i) it avoids dominance by the majority class (e.g., sunny days) while maintaining performance across rare but critical conditions (e.g., rainy days); (ii) it enables operators to identify which weather conditions cause the larger prediction errors; and (iii) it facilitates decision-making in energy management and fault diagnosis by clarifying the class membership of each data segment.

4.2. Estimation Results of Different Diode Models

Table 3. The range of parameter settings used for optimization process.

Parameter	Range	Parameter	Range
Short-circuit current, Iscr (A)	3.60~5.40	Current temperature coefficient, βIsc (A/K)	0.000670~0.000690
Open-circuit voltage, Vocr (V)	61~69	Voltage temperature coefficient, βVoc (V/K)	−0.166~−0.05
Current at MPP, Imr (A)	50~55	Ideality factor (1st diode), nidl1	1.0~1.2
Voltage at MPP, Vmr (V)	3.30~3.65	Ideality factor (2nd diode), nidl2	1.2~2.0
Irradiance under STC, Gr (W/m2)	1000~1050	Ideality factor (3rd diode), nidl3	1.2~3.0
Temperature under STC, Tr (K)	290~310	Saturation current (1st diode), Isat1 (A)	1.16${\times 10}^{-15}$~1.16${\times 10}^{-7}$
Parallel resistance under STC, Rshr (Ω)	60~180	Saturation current (2nd diode), Isat2 (A)	1.16${\times 10}^{-15}$~1.16${\times 10}^{-7}$
Series resistance under STC, Rsr (Ω)	2.0~6.0	Saturation current (3rd diode), Isat3 (A)	1.16$\times {10}^{-15}$~1.16${\times 10}^{-7}$

summarizes the parameter ranges used in the optimization process. Figure 9 presents the optimization performance of various diode models using the PSO method, with the maximum number of iterations and population size set to 100 and 50, respectively. Across 100 iterations, the three-diode model achieves a lower mean absolute error (MAE) than the single- and two-diode models. Figure 10 further demonstrates the convergence behavior of PSO over 10 independent runs, showing stable and consistent convergence.

Table 4. Comparison of the three different diode models using the PSO method.

Models	Single-Diode	Two-Diode	Three-Diode
MRE	4.049%	4.005%	3.976%
Calculation time (s)	460.16	683.77	743.76

compares the estimation performance of the three diode models using PSO, where the average estimation errors for the single-, two-, and three-diode models are 4.049%, 4.005%, and 3.976%, respectively. Although the accuracy improvement of the three-diode model over the two-diode model is relatively small (~0.03%), even such a minor reduction can result in meaningful energy prediction gains for large-scale PV systems. For instance, in a 733 kW PV plant, a 0.03% improvement corresponds to approximately 550 kWh annually (based on 2500 h per year). Such an improvement is non-negligible for long-term PV deployment and energy yield estimation. It should also be noted that, while the three-diode model provides slightly higher estimation accuracy, it incurs longer computation time after 100 iterations compared with the single- and two-diode models.

4.3. Preliminary Estimation Results

Building upon the comparative analysis of diode models in Section 4.2, preliminary estimation results are obtained using five optimization algorithms. The parameter settings of these algorithms, which are primarily determined based on prior studies [38,39,40,41] and refined through manual step-by-step adjustment to achieve stable convergence and reliable estimation results under the given dataset, are summarized in

Table 5. Parameter settings of different optimization algorithms.

Algorithm	Parameter	Value
PSO	$\mathrm{Inertia} \mathrm{weight} (w$)	0.4
	$\mathrm{Acceleration} \mathrm{factor} (c_1$)	0.5
	$\mathrm{Acceleration} \mathrm{factor} (c_2$)	0.55
MVO	$\mathrm{Traveling} \mathrm{distance} \mathrm{rate} (T_{DR}$)	[1, 0]
ES	$\mathrm{Step} \mathrm{size} \mathrm{of} \mathrm{mutation} (\rho$)	0.15
ASO	$\mathrm{Well} \mathrm{depth} \mathrm{of} \mathrm{potential} (ϵ$)	1.0
	$\mathrm{Distance} \mathrm{at} \mathrm{zero} \mathrm{potential} (\delta$)	1.0

Note: Maximum number of iterations and population size are set at 100 and 50, respectively.

. These choices were guided by commonly adopted ranges in the literature to ensure fairness and reproducibility. The internal parameter values obtained for each algorithm are reported in

Table 6. The internal parameter values obtained by each algorithm.

Parameter	Newton	PSO	MVO	ES	ASO
Short-circuit current, Iscr (A)	3.66	4.50	4.08	3.96	3.84
Open-circuit voltage, Vocr (V)	66.4	63.5	66.3	64.5	61.0
Current at MPP, Imr (A)	3.51	3.48	3.31	3.54	3.53
Voltage at MPP, Vmr (V)	52	50.2	50.0	52.4	51.2
Irradiance under STC, Gr (W/m2)	1000	1046.3	900	1040.0	1011.9
Temperature under STC, Tr (K)	298	290.03	290	305.89	300.52
Parallel resistance under STC, Rshr (Ω)	150	61.55	60.0	121.07	123.58
Series resistance under STC, Rsr (Ω)	2.4	5.99	6.0	4.16	5.52
Current temperature coefficient, βIsc (A/K)	0.000681	0.000672	0.000670	0.000685	0.000684
Voltage temperature coefficient, βVoc (V/K)	−0.1660	−0.0379	−0.1000	−0.0639	−0.0942
Ideality factor (1st diode), nidl1	1.2	1.0	1.01	1.17	1.16
Ideality factor (2nd diode), nidl2	1.8609	1.2	1.98	1.79	1.38
Ideality factor (3rd diode), nidl3	1.8609	1.2	3.0	1.87	1.64
Saturation current (1st diode), Isat1 (A)	1.16${\times 10}^{-15}$	9.75${\times 10}^{-13}$	1.00${\times 10}^{-15}$	5.03${\times 10}^{-8}$	4.29${\times 10}^{-10}$
Saturation current (2nd diode), Isat2 (A)	1.16${\times 10}^{-15}$	9.75${\times 10}^{-13}$	1.00${\times 10}^{-15}$	5.03${\times 10}^{-8}$	4.29${\times 10}^{-10}$
Saturation current (3rd diode), Isat3 (A)	1.16$\times {10}^{-15}$	9.75${\times 10}^{-13}$	1.00${\times 10}^{-15}$	5.03${\times 10}^{-8}$	4.29${\times 10}^{-10}$

. The estimation curves under various weather conditions are illustrated in Figure 11. Across all methods, estimation errors are generally higher on rainy days due to increased fluctuations in weather conditions. Overall, the Newton–Raphson method exhibits consistently lower accuracy than the metaheuristic algorithms. A comparative summary of the five optimization approaches is provided in

Table 7. Comparison of the five different methods (MRE%).

Weather Conditions	Newton	PSO	MVO	ES	ASO
Rainy	3.53 w	2.64	2.63 b	2.70	2.68
Heavily overcast	5.49 w	4.62	4.60b	4.65	4.67 w
Overcast	4.60 w	4.52	4.54	4.49 b	4.50
Partly cloudy	4.30 b	4.57	4.59	4.62 w	4.57
Sunny	3.56	3.53	3.53	3.58 w	3.52 b
Average	4.295	3.976	3.978	4.008	3.988

^b: Best estimation; ^W: Worst estimation.

, highlighting their respective strengths and weaknesses under different weather conditions. Specifically, the MVO achieves the best estimation performance on rainy and heavily overcast days while avoiding the worst performance under any weather condition. This robustness can be attributed to the inherent mechanism of MVO, which preserves population diversity through multi-universe exploration. Such a mechanism mitigates the risk of premature convergence and enhances resilience against random noise in the input data.

ES delivers the most accurate results on overcast days but performs poorly on partly cloudy and sunny days. ASO achieves the best accuracy on sunny days but produces the least reliable results on heavily overcast days. By contrast, the Newton–Raphson method demonstrates the most unstable performance overall, producing the best results on partly cloudy days but ranking worst under three of the four weather conditions. Newton–Raphson relies heavily on precise derivative information. In noisy or degraded PV conditions, inaccurate gradient estimation amplifies errors during iterative updates, leading to unstable convergence. Hence, its sensitivity is fundamentally associated with noise amplification rather than PV degradation itself.

4.4. Final Estimation Results

For the final estimation stage, an ensemble architecture integrating CNNs and LSTM networks is employed, aiming to enhance both feature extraction and temporal dependency modeling, thereby improving overall prediction accuracy. The parameter settings for CNNs and LSTM are provided in

Table 8. Parameter settings of CNNs and LSTM.

Method	Layer	Parameter	Value	Input Shape	Output Shape
CNNs	Conv2D	filter	16	(N 1, 5, 1, 1, 1)	(N, 5, 1, 1, 16)
		kernel size	(1, 1)
		activation	Relu 2
	MaxPooling2D	pool size	(1, 1)	(N, 5, 1, 1, 16)	(N, 5, 1, 1, 1)
	Flatten			(N, 5, 1, 1, 16)	(N, 5, 16)
LSTM		units	64	(N, 5, 16)	(N, 64)
		activation	Relu
	Dense	units	32	(N, 64)	(N, 32)
		activation	Relu
	Dense	units	1	(N, 32)	(N, 1)
		activation	linear

¹: N: the number of samples; ²: Relu: rectified linear unit.

. The CNN–LSTM consists of three CNN layers (Conv2D, MaxPooling2D, and Flatten), one LSTM layer with 64 units, and two fully connected Dense layers (32 and 1 units, respectively). Relu was selected for CNN and intermediate Dense layers to mitigate the vanishing-gradient problem, while the output layer employed a linear activation. The CNNs function as a pre-projection layer before the LSTM, receiving five inputs from individual optimization algorithms and linearly projecting 16 features to the LSTM for efficient training. While LSTM models typically adopt the tanh activation function, the Relu activation function was used in this study to achieve better estimation performance.

To evaluate the sensitivity of hyperparameters in the CNN–LSTM, an additional experiment was conducted by varying batch size, optimizer, and learning rate within a reasonable range while keeping the other parameters fixed. The results, illustrated in Figure 12, indicate that the optimal settings for batch size, optimizer, and learning rate are 25, Adam, and 0.001, respectively. Furthermore, the number of timesteps was set to 5. To mitigate overfitting, the dropout rate was set to 0.3 for the CNN layer and 0.5 for the LSTM layer. The patience parameter for early stopping was fixed at 5 epochs, and the L2 regularization coefficient was set to 0.0001.

Table 9. Comparison between the proposed and the other methods (MRE%).

Weather Conditions	Single Optimization Method					Ensemble Method
	Newton	PSO	MOV	ES	ASO	LSTM	CNN-LSTM
Rainy	3.53 w	2.64	2.63	2.70	2.68	2.63	2.58 b
Heavily overcast	5.49 w	4.62	4.60	4.65	4.67	4.87	4.47 b
Overcast	4.60 w	4.52	4.54	4.49 b	4.50	4.64	4.59
Partly cloudy	4.30 b	4.57	4.59	4.62	4.57	4.65 w	4.63
Sunny	3.56	3.53	3.53	3.58 w	3.52	3.49 b	3.57
Average	4.295	3.976	3.978	4.008	3.988	4.056	3.968

^b: Best estimation; ^W: Worst estimation.

compares the proposed ensemble method against several optimization algorithms under various weather conditions. The LSTM alone achieves the best estimation on sunny days but performs worst on partly cloudy days. By contrast, the proposed ensemble method delivers the most accurate predictions on rainy and heavily overcast days, although its performance improvement is relatively small under other weather scenarios. Overall, in terms of average MRE, the proposed approach achieves a lower estimation error compared with all other optimization algorithms. The source code of the proposed CNN–LSTM model and the baseline methods is available in [46,47].

Table 10. Comparison of various methods under the proposed ensemble framework (MRE%).

Weather Conditions	Single Model					Hybrid Model
	XGBoost	LightGBM	GRU	CNN	LSTM	A-LSTM	CNN-A-LSTM	CNN-LSTM
Rainy	2.677	2.637	2.619	2.641	2.587	2.605	2.580	2.577
Heavily overcast	4.596	4.562	4.578	4.821	4.668	4.694	4.558	4.471
Overcast	4.537	4.541	4.660	4.694	4.595	4.569	4.585	4.590
Partly cloudy	4.831	4.880	4.659	4.579	4.572	4.553	4.598	4.631
Sunny	3.845	3.493	3.522	3.528	3.502	3.502	3.536	3.571
Average	4.097	4.023	4.008	4.053	3.985	3.984	3.971	3.968
Training time (s)	0.32	0.2	14.59	28.82	66.62	90.7	114.4	105.8

,

Table 11. Comparison of various methods under the proposed ensemble framework (MAE (kW)).

Weather Conditions	XGBoost	LightGBM	GRU	CNN	LSTM	CNN-LSTM
Rainy	19.62	19.33	19.20	20.72	18.97	19.22
Heavily overcast	33.69	33.44	33.56	37.63	34.16	33.51
Overcast	33.25	33.29	34.16	36.02	33.76	33.61
Partly cloudy	35.41	35.77	34.15	35.83	33.59	33.48
Sunny	28.18	25.60	25.82	27.28	25.67	25.83
Average	30.03	29.49	29.38	31.50	29.23	29.13

,

Table 12. Comparison of various methods under the proposed ensemble framework (NMAE).

Weather Conditions	XGBoost	LightGBM	GRU	CNN	LSTM	CNN-LSTM
Rainy	0.385	0.379	0.376	0.406	0.372	0.377
Heavily overcast	0.241	0.239	0.240	0.269	0.244	0.240
Overcast	0.169	0.169	0.173	0.183	0.171	0.171
Partly cloudy	0.140	0.142	0.135	0.142	0.133	0.133
Sunny	0.099	0.090	0.091	0.096	0.090	0.091
Average	0.207	0.204	0.203	0.219	0.202	0.202

and

Table 13. Comparison of various methods under the proposed ensemble framework (sMAPE%).

Weather Conditions	XGBoost	LightGBM	GRU	CNN	LSTM	CNN-LSTM
Rainy	66.15	65.69	66.07	85.88	65.93	65.70
Heavily overcast	49.40	48.87	48.25	63.58	49.35	47.87
Overcast	29.10	30.04	31.34	33.90	30.06	30.80
Partly cloudy	27.50	27.81	27.22	30.81	26.45	26.63
Sunny	13.81	12.57	12.39	13.58	12.26	12.51
Average	37.19	37.00	37.05	45.55	36.81	36.70

present the comparative results of various methods under the proposed ensemble framework in terms of MRE, MAE, NMAE, and sMAPE. Baseline models include single approaches such as extreme gradient boosting (XGBoost), light gradient boosting machine (LightGBM), gated recurrent unit (GRU), CNN, and LSTM, as well as hybrid models including attention LSTM (A–LSTM) and CNN-A-LSTM. As shown in Table 10, the proposed CNN–LSTM method achieves the lowest average error (3.968%), marginally outperforming the other approaches and demonstrating robustness across diverse weather conditions. Nevertheless, the hybrid model requires substantially higher training time compared with tree-based models (XGBoost and LightGBM) and other deep learning models (GRU and CNN), indicating a trade-off between computational efficiency and accuracy. Furthermore, Table 11 further shows that CNN–LSTM attains the best performance with the lowest average MAE (29.13 kW), slightly surpassing the benchmark methods. The results of NMAE in Table 12 are consistent with those of MAE. Moreover, Table 13 indicates that large estimation errors occur on rainy days due to greater weather variability; nevertheless, the proposed CNN–LSTM still outperforms the other baselines.

5. Discussions

The experimental findings lead to the following key observations:

1.

The three-diode model provides superior estimation accuracy compared with the single- and two-diode models, albeit with a longer computation time (~12 min). Nevertheless, this additional computational burden does not hinder its applicability for fault detection in PV systems.

2.

As presented in Table 4, the optimization results indicate a decrease in parallel resistance and an increase in series resistance compared with the Newton–Raphson method. The Newton–Raphson method is employed as a benchmark because the manufacturer typically provides only four parameters ($I_{sc}$, $V_{oc}$, $V_m$, and $I_m$). The remaining unknown parameters are estimated using the Newton–Raphson method. Therefore, the parameters obtained via the Newton–Raphson method are regarded as the baseline solutions derived from the manufacturer’s data. This outcome suggests possible PV module oxidation and degradation [48,49], thereby requiring maintenance to restore performance.

3.

The Newton–Raphson method exhibits instability due to its gradient-based nature and sensitivity to initial values (Table 5). In contrast, multi-agent optimization approaches leverage multi-point search and parallelism to achieve more reliable solutions.

4.

The proposed CNN–LSTM ensemble method consistently achieves stable performance and yields lower MRE compared with standalone LSTM and other individual optimization algorithms (Table 7). Although the difference in mean MRE between CNN–LSTM (3.968%) and the best individual method PSO (3.976%) is relatively small (0.008 p.p.), this improvement translates into meaningful gains in energy prediction accuracy when applied to a 733 kW PV system. In practical deployment, it corresponds to approximately 146.6 kWh annually (based on 2500 h per year).

6. Conclusions

This study proposed a deep learning-based ensemble framework for accurate parameter estimation of solar cells using a three-diode model. By integrating the outputs of diverse optimization algorithms including Newton–Raphson, PSO, MVO, ES, and ASO through a hybrid CNN–LSTM network, the method effectively leverages complementary search behaviors to improve robustness and precision. The main findings are summarized as follows:

1.

Enhanced model accuracy: Compared with single- and two-diode models, the three-diode structure demonstrated superior capability in capturing nonlinear and physical effects, thereby improving estimation reliability.

2.

Robust ensemble performance: The proposed ensemble method consistently outperformed tree-based models (XGBoost and LightGBM) and other deep learning models (GRU, CNN, and LSTM) across all evaluation metrics, including MRE, MAE, NMAE, and sMAPE.

3.

Deep learning integration: The CNN–LSTM architecture successfully combined feature extraction and temporal sequence learning, mitigating overfitting while delivering more stable parameter estimates.

4.

Practical validation: Experimental results on a 733 kW PV power system indicated a decrease in parallel resistance and an increase in series resistance compared with the Newton–Raphson method. The Newton–Raphson method is employed as a benchmark since it relies on manufacturer-provided parameters to generate the estimated results. This outcome suggests possible PV module oxidation and degradation, thereby requiring maintenance to restore performance.

5.

Deployment aspect: Although the difference in mean MRE between the CNN–LSTM and the best-performing individual method (PSO) is relatively small (0.008%), this improvement translates into meaningful practical benefits. In particular, it yields an additional energy prediction gain of approximately 146.6 kWh per year in a 733 kW PV system, underscoring the practical significance of the proposed framework, especially when applied to larger PV systems.

6.

Scalability aspect: The proposed method has been validated on a 733 kW PV system and can be readily extended to larger-scale PV power generation systems. With appropriate parameter tuning, such as adjustments to the number of parallel and series modules, the proposed framework can be adapted to different system configurations, thereby ensuring its applicability to a wide range of deployment scenarios.

In conclusion, the proposed ensemble approach provides a promising solution for solar cell parameter estimation by integrating the strengths of physical modeling, optimization algorithms, and deep learning. Future work will focus on extending the framework to larger PV arrays, incorporating statistical validation (e.g., confidence intervals or Diebold–Mariano tests), analyzing the sensitivity of optimization parameters, and performing a quantitative analysis of deployment costs in practical PV operation and maintenance. Moreover, the adaptive parameter threshold requires further investigation, particularly under long-term PV operation with environmental fluctuations.