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Scientific Reports volume 15, Article number: 41807 (2025) Cite this article
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This paper introduces a novel, self-tuning equivalent circuit model for polycrystalline photovoltaic (PV) modules to overcome the accuracy limitations of conventional fixed-parameter models under dynamic climatic conditions. The proposed model features dynamically adjustable parameters–the diode quality factor, series resistance, and shunt resistance–that evolve with changes in solar irradiance, temperature, and wind speed. The design of this adaptive model is achieved through a dedicated two-stage hybrid methodology. First, an artificial neural network (ANN) trained via the Back-Propagation Neural Network (BPNN) technique on 300 experimental samples establishes an initial mapping from climatic inputs to the optimal electrical parameters. The primary force of the proposed hybridization is that the optimal biases and weights from this ANN-BPNN model are then used to meticulously initialize the Particle Swarm Optimization (PSO) algorithm. In the second stage, this PSO algorithm, leveraging its superior initial population, performs a refined minimization of the Mean Squared Error (MSE) between the predicted and measured PV currents to yield the final, high-precision model. For validation, this model is compared against a conventional fixed-parameter model optimized with the same PSO parameters. The results demonstrate a dramatic accuracy improvement: the proposed dynamic model achieves an MSE of 0.0387, a 90.82% reduction compared to the conventional model’s MSE of 0.4226. The key novelty lies in this effective ANN-PSO hybridization, which ensures robust and accurate dynamic modeling, providing a superior foundation for advanced PV system applications like maximum power point tracking (MPPT).
The success of solar energy extraction from PV array largely depends on the synthesis method of MPPT controllers where the corresponding control loop must provide smooth and rapid power responses regardless of varying in climatic variations^1,2. This success is grounded in the initial phase of preparing the controller synthesis, which involves modeling the actual behavior of PV cells. The current and voltage responses of PV cells vary according to three climatic conditions: the outdoor temperature, solar irradiance, and wind speed. This modeling heavily relies on adequate selecting an equivalent electrical circuit that includes three key unknown parameters to be optimized: the series resistance, shunt resistance, and the ideally diode factor. These parameters are often calculated using either analytical methods^3,4 or numerical methods^5,6,7,8,9.
In analytical methods, the manufacturer’s data-sheet including some specific parameters such as open-circuit voltage, short-circuit current, maximum power voltage, and maximum power current, is essential in model design. These parameters are used to formulate a system of nonlinear equations, whose optimal resolution allows defining the exact value of each preceding parameter³. Unfortunately, solving nonlinear problem is often not straightforward due to a cross-coupling and a strong interaction occurring among these parameters¹⁰. This complexity necessitates the exploration of numerical methods in which the constrained optimization problems must be solved by adequate optimization tools. Accordingly, several scholars have focused on optimizing the parameters of PV models with fixed parameters using heuristic and metaheuristic algorithms. El-Naggar et al., in 2012, applied the simulated annealing (SA) optimizer for model parameter identification employing equivalent electrical circuits based on either single or double diodes⁵. Ismail et al., in 2013, used the global optimization-based genetic algorithm (GA)⁶. Askarzadeh and Leandro dos Santos, in 2015, applied the simplified bird mating optimizer (SBMO)⁷. Alam et al., in 2015, applied the flower pollination (FP) algorithm⁸, and Bechouat et al., in 2017, applied the PSO algorithm and GA in parameter identification of PV model with fixed parameters using experiment tests⁹.
However, modeling a PV model based on equivalent electrical circuits with fixed parameters has several significant drawbacks, as this model type fails to account for sudden variations in climatic conditions as well as its random behavior¹⁰. Therefore, modeling based on fixed parameters is insufficient to develop a PV model that accurately reflects the real-world behavior of PV cells over time. This results in a significant discrepancy between their actual and predicted currents, which can lead to the failure of the MPPT controller synthesis method, particularly under specific conditions such as potential shading during the operation of a PV array. To address these challenges, it is crucial to implement an innovative equivalent electrical circuit with time-varying parameters. One solution, proposed by Kahla et al., in 2022, uses time-varying parameters in the PV model. This model includes two thermal resistances, metallic and thermistor, which change with environmental temperature¹⁰. Consequently, the optimization problem size increases, and the optimization algorithm may get trapped in local minima during fitness function minimization¹⁰.
The limitations mentioned above highlight a clear research gap: although the need for models with time-varying parameters is widely recognized, existing approaches often result in complex, high-dimensional optimization problems that are prone to converging to local minima¹⁰. Consequently, a robust and efficient methodology for dynamic parameter adaptation that circumvents these pitfalls remains lacking. The principal novelty of this work lies in proposing a two-stage hybrid ANN-PSO framework to address this gap. The core innovation involves using an ANN structure, initially trained via the BPNN algorithm, to generate an optimal starting point for the PSO algorithm. This intelligent hybridization ensures that PSO commences its search from a near-optimal solution derived from the ANN’s pattern recognition capabilities, rather than from a random population initialization. This strategy significantly improves the speed of convergence and the accuracy of the modeling by effectively guiding the PSO and mitigating premature convergence to local optima. Therefore, the objective of this paper is to propose a photovoltaic (PV) model with auto-adjustable parameters that vary temporally, supervised by the novel ANN-PSO framework. The central research question addresses how to ensure an effective adaptation mechanism for the correct self-adjustment of these parameters under random fluctuations in the three climatic conditions.
This adaptive mechanism must significantly minimize the deviation at each measurement point of the actual current produced by the real PV network. Such a model provides an MPPT controller with auto-adjusting parameters, leading to improved maximum power extraction and increased solar energy production. This MPPT controller synthesis will refer other studies for future candidates interested in this research field.
To achieve this aim, the paper employs an ANN model. Initially, weights and biases are set using BPNN technique and then updated with the PSO algorithm using experimental data. The ANN model design follows these steps. First, an initial dataset is used for off-line parameter identification of the ANN model. Its three outputs are chosen randomly within user-defined limits. These outputs, along with measurement inputs, are used to calculate the predicted global PV current, generating initial modeling errors when compared to the actual global PV current. The BPNN technique minimizes these errors, providing initial weights of the ANN model. The PSO algorithm then updates these weights, reducing modeling errors further. The updated weights rebuild the ANN model, providing the best one with three optimal parameters. Finally, a second dataset is used for ANN model validation, confirming the proposed PV model’s reliability and improved accuracy compared to the standard fixed-parameter PV model.
The paper is structured as follows: Sect. “Design of standard PV model with fixed parameters” details the modeling of the standard PV model with three fixed parameters. Parameter determination is based on an analytic solution using the LambertW function and optimization by solving a constrained optimization problem. Sect. “Design of the proposed PV model” introduces the proposed PV model with three adjustable parameters. These parameters are evolved using an ANN model, initially trained with a BPNN technique and further refined by the PSO algorithm. The paper concludes with an experimental analysis assessing the performance of both the standard and proposed PV models. It also suggests future work for researchers interested in PV array modeling with adjustable parameters using advanced supervisory techniques.
The PV model for a single polycrystalline PV panel can be designed using an equivalent electrical circuit with three fixed parameters: series resistance (R_s^1), the shunt resistance (R_p^1) and the diode ideality factor (n) (see Fig. 1). This circuit includes a diode (D) in parallel with the photo-current source (I_{ph}^1). The series resistance (R_s^1) represents the internal and contact resistances of the PV cell, while the shunt resistance (R_p^1) accounts for leakage currents within the cell. The predicted current (I_m^1) from the PV model is a function of the PV voltage (V_m^1), as given by Eq. 1¹⁰:
(V_T^1) is the thermal voltage generated by (N_c) number of series-connected PV cells in a single PV panel, as given by^11,12:
where (q = 1.602176 times 10^{-19} , text {C}) is the electronic charge and (k = 1.3806503 times 10^{-23} , text {J/K}) is the Boltzmann constant.
Equivalent electrical-circuit used for a single PV panel.
(T_c) is the cell temperature depending on three climatic conditions: outdoor temperature T, solar irradiance G and wind speed (omega). They are interrelated through an empirical relationship, as given by¹³:
(K_r) is a positive parameter provided in the manufacturer’s data sheet. Additionally, (I_{ph}^1) for a single PV panel is defined by¹⁴:
(T_{text {stc}}), where (T_{text {stc}} = 25,^{circ }text {C}), and (G_{text {stc}}), where (G_{text {stc}} = 1000,text {W}cdot text {m}^{-2}), are respectively the nominal values of the outdoor temperature and solar irradiance recorded under Standard Test Conditions (STC). The parameter (alpha) is the temperature coefficient of the nominal short-circuit current (I_{text {sh,0}}^1) of a single PV panel. Additionally, the reverse saturation current (I_{od}^1) of the diode can be derived from the data-sheet information as¹⁴:
where (E_g) is the band-gap energy of the semiconductor measured across the diode (D) and (I_{sr}^{1}) is the corresponding diode current given by¹⁴
(V_{oc}^{1}) is the open-circuit voltage and (beta) is the temperature coefficient for the specific PV panel.
For (N_s) panels connected in series, the corresponding PV voltage is amplified to (V_m^g) where (V_m^g = N_s cdot V_m^1) while the corresponding PV current (I_m^g) remains unchanged, i.e., (I_m^g = I_m^1). This configuration leads to the formation of a single PV string. On the other hand, for (N_p) panels connected in parallel, the resulting PV current (I_m^g) is amplified to (I_m^g = N_p cdot I_m^1) but the corresponding PV voltage (V_m^g) remains unchanged, i.e., (V_m^g = V_m^1). In the case of (N_p) strings connected in parallel, each containing (N_s) panels connected in series, the resulting PV array will have a global PV current (I_m^g) amplified to (I_m^g = N_p cdot I_m^1) and a global PV voltage (V_m^g) also amplified to (V_m^g = N_s cdot V_m^1). Therefore, the corresponding electrical circuit of a global PV array to be modeled is shown in Fig. 2.
Equivalent electrical-circuit used for global PV array.
where the predicted global PV current (I_m^g) from the global PV model is a function of the global PV voltage (V_m^g), as given by Eq. 1¹⁰:
It is essential to mention that Eq. 7 can be rewritten in a general form (f(I^g_m) = 0) as below
where f is a nonlinear transcendental function depending implicitly on climatic conditions, as well as on the preceding three key parameters to be optimized. These parameters can be set as fixed and invariant over time, or adjustable and variable over time. The root vector of (fleft( I^g_mright) = 0) is often found using iterative methods, which provide the predicted global current (I^g_m). This last one must be as close as possible to the actual global PV current (I^g_{pv}), measured previously in the experimental test.
Reaching an adequate root vector is challenging due to random changes in meteorological factors, the initialization of the iterative method, the maximum number of iterations, and so on. These lead the occurrence of modeling errors that can dramatically increase, especially when using the adjustable version of the parameters, as the convergence speed to a desired solution must be fast before the transcendental function changes in form due to sudden variations in climatic conditions. The best way to provide the desired (hat{I}^g_m) is to use an analytical solution based on the LambertW function, which ensures rapid convergence with no cumulative computation errors.
To reach this aim, consider (V^g_{pv}) and (I^g_{pv}) as the actual global PV voltage and global current respectively where each one was recorded during the experiment test. Assume (V^g_{pv} = frac{V^g}{m}) and reformulate Eq. 8 in a general form as:
Z is the argument of the real function W(X), whose exponential function (e^{W(X)}) includes the real vector X. This vector thus presents the solution of Eq. 8, which is given in a general form, defined by
From mathematical developments carried on Eq. 1 in accordance with the last general form, we can obtain
Then, from Eqs. 10, 11 and 12, the predicted global PV current (hat{I}^g_m) is given by^10,15
Once the analytical determination of (hat{I}^g_m) is completed, it is necessary to compare it with the corresponding measured current vector (I^g_{pv}) at each sample (k). The model errors resulting from this comparison allow to formulate an optimization problem under constraints, where the fitness function to be minimized can be given in either form: Mean Squared Error (MSE) or Integral Time Absolute Error (ITAE), and so on. Since the choice of one of these criteria does not significantly impact the optimization results and is not part of the contribution of this paper, the MSE criterion will be chosen as the fitness form to be minimized. Furthermore, as there are several heuristic and meta-heuristic algorithms, each one is capable to perfectly solve this modeling problem, the PSO among those available in the literature will be chosen as an optimization tool for solving where this choice is also not part of the contribution of this paper. Indeed, the standard formulation of the modeling problem is generally given by¹⁰:
where N is the number of total samples collected during the experimental test. Finally, the optimal resulting three fixed parameters (n^*), (R^{g*}_s), and (R^{g*}_p) also lead to determine the global diode voltage (V^g_d) where (V^g_d = V^g_m + R^{g*}_s cdot I^g_m) (see Fig. 2).
In real-world applications, modeling PV behavior based on fixed parameters is often insufficient due to the random evolution of climatic conditions. Consequently, it becomes necessary to use varying parameters to accurately capture these dynamic changes. Therefore, employing a smart supervisor mechanism is indispensable when designing PV models with adjustable parameters. Among various mechanisms, the one employing the ANN learned by the BPNN technique and then enhanced by the PSO algorithm, has proven to be highly effective. However, users can replace this mechanism with other ones, as an appropriate choice among them remains a perspective of this work. This will certainly open new opportunities for future researchers interested in this area of research. Therefore, Fig. 3 highlighted the proposed idea to design the proposed PV model.
Block diagram detailing the proposed idea for ANN design model.
Knowing that data normalization is essential to ensure efficient and effective training of ANN models. It helps to improve the convergence and avoid saturation of activation functions where numerical stability in computation is always ensured. In this paper, the actual input vector (X = (G, T_c, V^{g}_m)^T) is a prior normalized to provide the normalized input vector (X^aleph = (G^aleph , T_c^aleph , V^{galeph }_m)^T) using Eq. 15, given by¹⁶:
where (max (X)) and (min (X)) are upper and lower bounds limiting the actual measurement X. As the predicted global current vector (hat{I}^g_m) is computed by LambertW function using both actual measurement input vector X and denormalized predicted ANN output vectors (Y^mathfrak {D} = (hat{n}, R^g_s, R^g_p)^T) where (Y^mathfrak {D}) is computed from (Y^aleph) using¹⁶:
where (Y^aleph = (n^aleph , R^{galeph }_p, R^{galeph }_s)^T) is the resulting normalized output vector, provided by the ANN model. In addition, (max (Y) = (n_{max }, R^g_{smax }, R^g_{pmax })^T) and (min (Y) = (n_{min }, R^g_{smin }, R^g_{pmin })^T) are upper and lower bounds, defined previously in Eq. 14. They are used to compute the denormalized vector (Y^mathfrak {D}) using Eq. 16.
This section details the integrated methodology of the Backpropagation Neural Network (BPNN) and Particle Swarm Optimization (PSO) algorithm, which forms the core of the proposed hybrid framework for adaptive PV parameter identification. The overall architecture, as depicted in Fig. 5, follows a sequential two-stage process: a BPNN pre-training phase followed by a PSO refinement phase.
Stage 1: BPNN Pre-training and Initial Estimation The first stage involves training the ANN using the Backpropagation algorithm. As shown in Fig. 5, this BPNN acts as an intelligent pattern recognition system. It learns the complex, nonlinear relationships between the normalized input variables (irradiance, cell temperature and measured PV voltage) and the corresponding optimal PV model parameters from the historical dataset. The output of this phase is a set of pre-optimized network weights and biases ((W_1^*), (b_1^*), (W_2^*), (b_2^*)). These values represent an informed initial estimation of the parameters, positioning the solution close to the optimal region in the search space.
Stage 2: PSO-based Fine-Tuning and Optimization
The second stage, illustrated in the refinement loop of Fig. 5, utilizes the PSO algorithm for precise optimization. The optimized weights and biases from the BPNN are flattened and concatenated to form the basis for the initial PSO particle position, (X_b^N). Instead of a purely random initialization, the remaining particles in the swarm are generated by adding bounded randomness to this BPNN-derived solution. This strategic initialization ensures the PSO begins its search from a high-quality, promising starting point.
The critical integration and the complete, adaptive workflow are encapsulated in Fig. 6. The process operates as follows:
Input: The normalized input vector (X^aleph =(G^aleph ,T_c^aleph ,V_m^{aleph g})) is fed into the system.
Stage 1 (BPNN Forward Propagation): The pre-trained BPNN processes the inputs to obtain an initial set of normalized PV model parameters, (Y^aleph).
Stage 2 (PSO Refinement Loop): The PSO population is initialized as described in Fig. 5. The algorithm then enters its iterative loop, as per Eq. 31 in the paper:
For each particle, the position vector (X_b^{aleph (l)}) is denormalized and reshaped into the corresponding weight matrices ((W_1),(W_2)) and bias vectors ((b_1),(b_2)).
The ANN is reconstructed with these parameters to compute the denormalized output (Y^D).
The fitness function (MSE) is evaluated by comparing the predicted global PV current (I_m^g) (calculated using the LambertW function with (Y^D)) against the measured current (I_{pv}^g).
The personal best ((X_{b_j}^{aleph ,text {best}})) and global best ((X_{text {swarm}}^{aleph ,text {best}})) positions are updated.
Particle velocities and positions are updated for the next iteration.
Termination: Once the maximum number of iterations is reached, the algorithm terminates. The best solution, (X_{text {swarm}}^{aleph ,text {best}}), is denormalized and reshaped to yield the optimal weights and biases ((W_1^{text {best}}), (W_2^{text {best}}), (b_1^{text {best}}), (b_2^{text {best}})) for the final ANN model.
Output: The reconstructed ANN model with these optimal parameters provides the adjusted PV model parameters for the given climatic conditions.
This integrated strategy, visually supported by Figs. 5 and 6, offers several key benefits:
Accelerated Convergence: The BPNN-provided initial solution significantly reduces the number of PSO iterations required to reach the optimum compared to a random initialization.
Enhanced Solution Quality: By starting the search near the global optimum region, the PSO is less likely to converge prematurely to local minima.
Robust Adaptation: The synergy between the BPNN’s pattern recognition capability and PSO’s global search strength ensures effective and reliable parameter adjustment under dynamic environmental conditions.
Computational Efficiency: The two-stage approach balances the computational load, with the BPNN providing a swift initial guess and the PSO performing a focused, fine-tuned optimization.
This hybrid framework effectively addresses the challenge of dynamic parameter adaptation, achieving a balance between computational tractability and solution accuracy, as demonstrated by the workflow in Figs 5 and 6.
Back-propagation is a gradient-based optimization method that ensures the gradient descent of the MSE criterion to update the weights and biases of the ANN model. The process involves calculating the error between the current model output and the actual current output and then propagating this error backward through the network to update the weights and biases. The modeling error is calculated for each neuron in the output layer relative to the expected output. It is then propagated backward, layer by layer, adjusting the weights and biases proportionally to the gradient of the error with respect to each weight. Figure 4 illustrated the update process to provide the output vector of the denormalized model.
Weights and biases update using BPNN technique.
As illustrated in Fig. 4, the denormalized output computation and parameter updates via BPNN technique require the following steps:
Forward pass equations:
Inputs to hidden layer : The weight matrix and the bias vector from the normalized input layer (X^aleph) to the hidden layer are denoted as (W_1 in mathbb {R}^{n_H times n_i}) and (b_1 in mathbb {R}^{n_H}) respectively. The hidden layer activations H are calculated as¹⁷:
where (sigma (x)) is the sigmoid activation function, generally defined for the variable x as: (sigma (x) = frac{1}{1 + e^{-x}}). additionally, the weight matrix (W_1) and bias vector (b_1) are defined respectively as¹⁷
Hidden layer to output layer: The normalized output (Y^aleph) is generated using the weight matrix (W_2 in mathbb {R}^{n_o times n_H}) and the bias vector (b_2 in mathbb {R}^{n_o}) given by¹⁷
where (n_i), (n_o), and (n_H) denote the input size, hidden neurons, and output size, respectively. Also, the normalized vector (Y^aleph) is defined as a linear function, given by¹⁷:
Mean squared error (MSE): For N samples, the J loss is¹⁷
Gradients computations:
Loss gradients (chain rule): Updating (W_1), (W_2), (b_1) and (b_2) requires the gradient computation, given by^17,18,19
First intermediate derivative computations
where (sigma ‘(x) = sigma (x) cdot (1 – sigma (x))) is the derivative of the sigmoid function (sigma (x)).
Second intermediate derivative computations
From Eq. 13, in part of the predicted global PV current has the LambertW function, given by^20,21:
where (Z(Y^{mathfrak {D}})) is expressed by^20,21
Moreover, the derivative of (h(Y^{mathfrak {D}})) with respect to (Y^{mathfrak {D}}) is given by:
where (frac{partial h(Y^{mathfrak {D}})}{partial Z(Y^{mathfrak {D}})}) is defined by:
The derivative of (h(Y^{mathfrak {D}})) with respect to the three parameters of the proposed PV model is given by:
Finally, the derivatives of (Z(Y^{mathfrak {D}})) and (I_m^g(Y^{mathfrak {D}})) with respect to the three parameters of the proposed PV model are presented in Supplementary Information, Section S1: Derivation of Derivatives for the PV Model. Also, see Supplementary Table 1 for a summary of all parameters.
Updating of weights and biases At the iteration m, weights and biases are updated as^18,19:
where (eta>0) the user-defined learning rate.
PSO does not require the calculation of the MSE derivatives, which can be advantageous for non-differentiable functions or complex search spaces^22,23. The design vector of the PSO algorithm (X_b^aleph) is created by flattening the ((n_H times n_i)) variables of the weight matrix (W_1) and flattening the ((n_o times n_H)) variables of the weight matrix (W_2). The two resulting vectors are then concatenated with the ((n_H)) variables of the bias vector (b_1) and finally with the ((n_o)) variables of the bias vector (b_2).This results in a total of (n_H cdot (n_i + n_o + 1) + n_o) variables to be optimized. Additionally, the PSO algorithm is initialized according to Fig. 5, where the upper and lower bounds limiting the weights and biases resulting from the BPNN technique allow denormalization of the resulting PSO solutions each time.
PSO initialization algorithm based on best weights and biases resulting from BPNN technique.
Therefore, finding the optimal ANN model including the best normalized weights (W_1^{aleph ,text {best}}), (W_2^{aleph ,text {best}}) and the best biases (b_1^{aleph ,text {best}}) and (b_2^{aleph ,text {best}}) can be detailed in Fig. 6. Consequently, the update of the normalized particle velocities of (n_p) (V_j^{aleph (l)}) and the normalized particle positions of (n_p) (X_{b_j}^{aleph (l)}), where (j = 1, 2, dots , n_p), can be ensured by²²:
where (c_0), (c_1), and (c_2) are the inertia factor, the cognitive rate, and the social learning rate, respectively. (r_{1,j}^l) and (r_{2,j}^l) are random numbers uniformly distributed in [0, 1].
Finding the best ANN model using the PSO-based weights and biases update.
For simplicity, the termination PSO algorithm is set as a maximum number of iterations (l_{max}), chosen by user^23,24. Additionally, the computation of the denormalized best particle positions (X_{b_j}^{mathfrak {D},text {best}(l)}) and (X_{swarm}^{mathfrak {D},text {best}(l)}) can be ensured by²²:
where (X_{b_j}^{mathfrak {D},text {best}(l)}) is the personal best particle position that is updated if the new particle position yields a better minimization of the MSE value. (X_{text {swarm}}^{mathfrak {D},text {best}(l)}) is the global best particle position that is updated if any particle’s new position yields a better minimization of the MSE value.
This section details the integrated methodology of the BPNN and the PSO algorithm, which forms the core of the proposed hybrid framework for adaptive PV parameter identification. The overall architecture, as depicted in Fig. 5, follows a sequential two-stage process: a BPNN pre-training phase followed by a PSO refinement phase.
a. Stage 1: BPNN Pre-training and Initial Estimation
The first stage involves training the ANN model using the Backpropagation algorithm. As shown in Fig. 5, this BPNN acts as an intelligent pattern recognition system. It learns the complex, nonlinear relationships between the normalized input variables (irradiance, temperature, etc.) and the corresponding optimal PV model parameters from the historical dataset. The output of this phase is a set of pre-optimized network weights and biases ((W_1^*), (b_1^*), (W_2^*), (b_2^*)). These values represent an informed initial estimation of the parameters, positioning the solution close to the optimal region in the search space.
b. Stage 2: PSO-based Fine-Tuning and Optimization
The second stage, illustrated in the refinement loop of Fig. 5, utilizes the PSO algorithm for precise optimization. The optimized weights and biases from the BPNN are flattened and concatenated to form the basis for the initial PSO particle position, (X_b^N). Instead of a purely random initialization, the remaining particles in the swarm are generated by adding bounded randomness to this BPNN-derived solution. This strategic initialization ensures the PSO begins its search from a high-quality, promising starting point.
The critical integration and the complete, adaptive workflow are encapsulated in Fig. 6. The process operates as follows:
Input: The normalized input vector (X^N = (G^N, T_c^N, V_m^{gN})^T) is fed into the system.
Stage 1 (BPNN Forward Propagation): The pre-trained BPNN processes the inputs to obtain an initial set of normalized PV model parameters, (Y^N).
Stage 2 (PSO Refinement Loop): The PSO population is initialized as described in Fig. 5. The algorithm then enters its iterative loop, as per Eq. (31):
For each particle, the position vector (X_b^{N(l)}) is denormalized and reshaped into the corresponding weight matrices ((W_1), (W_2)) and bias vectors ((b_1), (b_2)).
The ANN is reconstructed with these parameters to compute the denormalized output (Y^mathfrak {D}).
The fitness function (MSE) is evaluated by comparing the predicted global PV current (hat{I}_m^g) (calculated using the LambertW function with (Y^mathfrak {D})) against the measured current (I_{pv}^g).
The personal best ((X_{b_j}^{N,text {best}})) and global best ((X_{text {swarm}}^{N,text {best}})) positions are updated.
Particle velocities and positions are updated for the next iteration.
Termination: Once the maximum number of iterations is reached, the algorithm terminates. The best solution, (X_{text {swarm}}^{N,text {best}}), is denormalized and reshaped to yield the optimal weights and biases ((W_1^{text {best}}), (b_1^{text {best}}), (W_2^{text {best}}), (b_2^{text {best}})) for the final ANN model.
Output: The reconstructed ANN model with these optimal parameters provides the adjusted PV model parameters for the given climatic conditions.
This integrated strategy, visually supported by Figs. 5 and 6, offers several key benefits:
Accelerated Convergence: The BPNN-provided initial solution significantly reduces the number of PSO iterations required to reach the optimum compared to a random initialization.
Enhanced Solution Quality: By starting the search near the global optimum region, the PSO is less likely to converge prematurely to local minima.
Robust Adaptation: The synergy between the BPNN’s pattern recognition capability and PSO’s global search strength ensures effective and reliable parameter adjustment under dynamic environmental conditions.
Computational Efficiency: The two-stage approach balances the computational load, with the BPNN providing a swift initial guess and the PSO performing a focused, fine-tuned optimization.
This hybrid framework effectively addresses the challenge of dynamic parameter adaptation, achieving a balance between computational tractability and solution accuracy, as demonstrated by the workflow in Figs. 5 and 6.
The experimental test was conducted using a set of PV panels installed on the roof of the faculty at Yildiz University, Istanbul, Turkey^14,15. The PV panels included three types: thin-film, mono-crystalline, and poly-crystalline. This study focuses on polycrystalline panels, as the other types yielded similar results. The PV array consisted of four (04) panels connected in series to achieve the desired current and voltage levels.
Initially, modeling was performed using an equivalent electrical circuit with fixed parameters optimized by the PSO algorithm. Subsequently, an equivalent electrical circuit with adjustable parameters was employed, where parameters were supervised by an ANN using the BPNN technique and a second PSO algorithm. The modeling phase utilized a subset of 300 samples, while the validation phase used a separate subset of 200 samples.
Data were collected hourly over a one-month period, starting at 9 a.m. on September 9th, 2024, resulting in 500 samples of total current (I_{pv}^g) and voltage (V_{pv}^g), solar irradiance G (measured using a Kipp-Zonen CMP21 pyranometer), outdoor temperature T (measured using a Campbell CS215 thermocouple), and wind speed (omega) (measured using a Kipp-Zonen CMP21 pyranometer). Figure 7 illustrates the experimental setup, including the various measuring instruments^14,15.
PV panels and measuring equipment used in the experimental test.
The minimum and maximum values recorded for each measurement during the one-month period are summarized in Table 1.
From the upper and lower limits summarized in Table 1, it is evident that STC conditions were not achieved during the specified period. Since the performance assessment of any control strategy must be conducted under STC, the goal is to design an efficient equivalent electrical circuit that can ensure accurate predictions not only under STC conditions but also for any random variations in the aforementioned conditions.
The data-sheet for a single PV panel based on polycrystalline technology is summarized in Table 2 below
The primary objective is to design a model that accurately describes the behavior of the PV array utilizing polycrystalline technology. Due to the stochastic nature of the PSO algorithm, it is run multiple times to optimally determine the three key fixed parameters of the equivalent electrical circuit. The first dataset, comprising 300 samples, was used for parameter identification. The PSO tuning parameters are summarized in Table 3 below.
The search space for the constrained optimization problem was selected after several PSO runs. Extensions were are performed when the solution became stuck in saturated constraints. The most appropriate choice to avoid saturation is as follows:
The best minimization of the fitness function is (J_{text {min}} = 0.4226), providing the optimal parameters: ideal diode factor (n^* = 1.301), global series resistance (R_s^{*g} = 0.42 , Omega), and global shunt resistance (R_p^{*g} = 3183.08 , Omega).
In this design, the ANN configuration adopted to initialize the PSO algorithm consists of three input layers ((n_i = 3)), three output layers ((n_o = 3)), and five hidden layers ((n_H = 5)). The sigmoid activation function was chosen for the hidden layers, while a linear activation function was used for the output layers.
The three predicted ANN model outputs were initially assigned randomly within their upper and lower bounds, as previously defined in Eq. 33. The network was trained using 300 normalized real measurements corresponding to the PV cell temperature (T_c), solar irradiance vector G, and the global PV voltage vector (V_{text {pv}}^g). After training, the network produced three predicted normalized outputs, which were then denormalized to calculate the predicted global PV current vector (hat{I}_m^g).
The BPNN-based learning process yielded the best updates for the optimal weights and biases, as follows:
Input to hidden layer weight matrix:
Hidden layer bias vector:
Hidden to output layer weight matrix:
Output layer bias vector:
The PSO tuning parameters used for the adjustable-parameters PV model are identical to those employed in the fixed-parameter PV model design. The initial normalized vector, as previously generated from flattening the two weights and then the concatenating with the two biases, is augmented with 29 additional vectors, each comprising 38 variables. Each variable is randomly selected within the range]0, 1].
The objective is to identify the optimal normalized vector among the (n_p=30) candidates, where vector component has own normalized position and velocity. The PSO algorithm is configured with a maximum of 380 iterations and terminates if the best denormalized solution achieves an MSE below the user-imposed tolerance.
The minimization process produces an MSE of 0.0387219 (see Fig. 8), which presents a significant improvement compared to the MSE obtained by the fixed-parameter PV model. This corresponds to an improvement rate of approximately 90.82(%), underscoring the effectiveness and value of the proposed approach.
Best MSE minimization using the PSO algorithm.
The optimal solution obtained by the PSO algorithm provides the best prediction of the global PV current, which is compared with the actual current for the 300 samples. Figure 9 illustrates the comparison between the predicted and actual global PV currents, while Fig. 10 depicts the corresponding modeling accuracy achieved during the parameter identification process for the PV model with adjustable parameters.
Actual and predicted global PV currents.
Modeling error provided during ANN model design.
Also, the optimal solution obtained from the PSO algorithm yields the optimal normalized vector (X_{swarm}^{aleph ,text {best}}). This vector is subsequently denormalized and reshaped to provide the best weights and biases, given by
Input to hidden layer weight matrix:
Hidden layer bias vector:
Hidden to output layer weight matrix:
Output layer bias vector:
The ANN model is reconstructed using the optimal weights and biases derived from the PSO algorithm. To evaluate its accuracy and generalization capability, the model is tested on a second dataset comprising 200 additional samples. During this validation phase, the performance of the PV model with adjustable parameters is assessed to ensure its ability to accurately predict the behavior of the PV array under varying conditions.
Figure 11 illustrates the evolution of each model parameter during the validation phase. The results reveal significant fluctuations in the model parameters, demonstrating that the actual behavior of the PV array cannot be effectively captured by a model with fixed parameters. In contrast, the proposed PV model, supervised by the ANN with PSO-based updates, provides a more accurate representation of the PV array’s performance. This is clearly demonstrated in Fig. 12, where the predicted global PV current closely aligns the actual measurements, even for new data points not included in the parameter identification step.
Evolution of three adjustable parameters during ANN model validation.
Comparison of the predicted global PV currents during ANN model validation.
Finally, Fig. 13 presents a comparison of the modeling errors between the two PV models: one with fixed parameters and the other with adjustable parameters. The results clearly demonstrate that the proposed PV model with adjustable parameters achieves the lowest error, confirming its superiority in addressing this type of modeling problem.
Comparison of modeling errors during model validations.
This paper has presented an innovative hybrid strategy for PV modeling that effectively addresses the limitations of conventional fixed-parameter approaches through the synergistic integration of ANN model trained via BPNN and PSO algorithm. The proposed methodology demonstrates significant improvements in modeling accuracy, achieving approximately 91% reduction in MSE criterion compared to conventional fixed-parameter models by enabling dynamic parameter adjustment in response to fluctuating environmental conditions. The strategic two-stage architecture, where BPNN provides an intelligent initial estimation and PSO performs refined optimization, represents a substantial advancement in capturing the real-world behavior of polycrystalline PV arrays. However, the proposed approach is not without limitations; the computational complexity of the hybrid strategy may pose challenges for real-time implementation on low-cost hardware, and the requirement for extensive training data could limit its applicability in scenarios with insufficient historical measurements. Additionally, the performance remains dependent on proper selection of neural network architecture and PSO parameters, which may require domain expertise. Looking forward, this research opens several promising avenues for future investigation, including the exploration of alternative supervisory mechanisms such as fuzzy logic or neuro-fuzzy models that could offer improved interpretability and potentially reduced computational demands. A particularly compelling direction involves the synthesis of MPPT controllers with auto-adjustable parameters that can dynamically co-evolve with the varying parameters of the proposed PV model, creating a fully adaptive solar energy conversion system. Further research should also focus on extending this framework to other PV technologies, investigating long-term degradation modeling, and optimizing the computational efficiency for practical real-time deployment. This work ultimately establishes a robust foundation for next-generation PV system design, contributing significantly to enhanced energy harvesting efficiency and reliability in renewable energy applications.
The datasets used and/or analyzed during the current study are available from the corresponding author upon reasonable request.
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All authors contributed equally to the research, analysis, writing, and editing of this article. Each author has approved the final version of the manuscript and agrees to its submission.
This research did not receive specific grants from funding agencies in the public, commercial or non-profit sectors.
LEER Laboratory, University of Mohamed-Cherif Messaadia, Souk Ahras, Algeria
Zouhir Boumous & Samira Boumous
Laboratory of Inverse Problems, Modeling, Information and Systems (PI:MIS), University 8 Mai 1945 Guelma, Guelma, Algeria
Moussa Sedraoui
Department of Automation and Electromechanics, Faculty of Science and Technology, University of Ghardaia, Ghardaia, Algeria
Mohcene Bechouat
School of Engineering, University of Eldoret, Eldoret, Kenya
Cyrus Wabuge Wekesa
Department of Electrical Engineering, Yildiz Technical University, Istanbul, 34220, Turkey
Ramazan Ayaz
Clean Energy Technologies Institute, Yildiz Technical University, Istanbul, 34220, Turkey
Ramazan Ayaz
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Boumous, Z., Boumous, S., Sedraoui, M. et al. A hybrid ANN-PSO approach for self-tuning parameters of polycrystalline photovoltaic arrays. Sci Rep 15, 41807 (2025). https://doi.org/10.1038/s41598-025-25778-8
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DOI: https://doi.org/10.1038/s41598-025-25778-8
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