State-Space Identification of a Nonlinear Photovoltaic System Using a Bidirectional Long Short-Term Memory (BiLSTM) Recurrent Neural Network

As photovoltaic (PV) systems become increasingly central to the global energy transition, ensuring they operate efficiently and reliably is a top priority. We need accurate models to maximize the power output of these systems, identify faults before they become serious problems, and employ more intelligent control methods. Yet, building these models is not straightforward. A PV system's performance is complex and nonlinear, changing with factors like solar irradiance (G), temperature (T), and its physical condition. Traditional physics-based approaches, such as the single-diode model, often fail to provide accurate results. Their parameters are difficult to pin down, and they struggle to replicate the system's full dynamic behavior. This is where black-box system identification offers a practical solution. It enables us to build models directly from input-output data, without requiring an understanding of the complex physics within [1].

Deep learning provides the right tools for this challenge. Recurrent neural networks (RNNs), for example, are excellent for approximating nonlinear dynamic systems because they are specifically designed to process sequential data [2]. In recent years, Transformer models have become a key tool for working with sequential data. This is because they are very good at finding complex connections between data points that are far apart, an area where they often outperform traditional RNNs [3]. Researchers have successfully used them for specific PV tasks, like predicting the maximum power point or forecasting short-term power output [4, 5]. The field is also advancing with new designs like Mamba, which cleverly combines state-space ideas to be more efficient [6]. However, one major drawback is that these models typically require very large amounts of data to learn properly. For our work on I-V curve modeling, we are using a well-structured but moderately sized dataset. In this situation, we found that a recurrent model like the BiLSTM is still a very practical and effective choice. Within this context of neural network modeling for PV systems, research has largely followed two distinct paths. The most common path involves input-output models, which have been primarily applied to two tasks. For power output forecasting, numerous studies have demonstrated the effectiveness of hybrid architectures, such as CNN-LSTMs [7-9]. For static I-V curve characterization, other works use ANNs or neuro-fuzzy systems to estimate key electrical parameters for applications like MPPT optimization [10, 11]. The main drawback of these input-output models is their narrow focus. While successful in forecasting, their utility is restricted because they only describe a single output or a static state. They therefore fail to capture the system's internal dynamics, which limits their application in physical analysis and diagnostics.

In contrast, the state-space framework provides a more structured and powerful representation. Foundational work established the advantages of using neural networks within this framework, highlighting benefits like model parsimony and applicability to robust control [12, 13], and recent advances with deep learning have revitalized its use. What makes the state-space approach so powerful is that it tries to understand how the system works on the inside, instead of just memorizing the relationship between inputs and outputs. A major 2024 survey by Pillonetto et al. [14] points out that because these models learn the internal dynamics, they are much better at generalizing to new situations and are more reliable for simulations. On top of that, work by Gedon et al. [15] shows that deep state-space models are built to track complex patterns over long periods, which is something simpler input-output models often struggle with. This is a hot topic in research right now, with ongoing work focused on making these models even better for tricky, real-world systems with noise and uncertainty [16]. Even within physics-based modeling of PV systems, a state-space formulation is considered essential for capturing full dynamic behaviors [17]. Despite these parallel advances, a critical gap persists in the literature: few studies apply a data-driven state-space formalism to identify the complete I-V curve of a PV system, and even fewer validate the physical relevance of the learned state representation.

Our work aims to fill this gap. We introduce a method for data-driven state-space identification that uses a bidirectional long short-term memory (BiLSTM) network to model the entire current-voltage (I-V) curve as a dynamic sequence. We frame this as a sequence-to-sequence task where the network learns to predict the current sequence (I) from a voltage sweep (V) and its context (G, T, fault state). The network's internal hidden state serves as an estimator of the unobservable physical state, implicitly learning the system's underlying functions. The key contribution here is proving that this learned representation is physically meaningful. We demonstrate this by successfully extracting key electrical parameters (Pmax, Vmpp, Voc, Isc) from the model's predicted I-V curves. In doing so, we create a robust and interpretable surrogate model—a true "digital twin" of the PV system.

Specifically, this paper contributes the following:

1. A new way to model the PV I-V curve using a state-space framework solved by a bidirectional RNN.

2. Proof that the network's hidden state learns a physically relevant representation of the system.

3. Validation of this concept by accurately extracting derived electrical parameters.

The rest of this paper is organized to walk through our research process and findings. Section II is dedicated to our complete methodology, covering everything from the state-space formulation to the model's architecture and training protocol. In Section III, we present the experimental results, beginning with the final model configuration and its training convergence, before moving on to a complete evaluation of its performance and a detailed analysis of the learned dynamics. Sections IV and V then provide a deeper discussion of these results, where we compare our state-space model against a static input-output approach and further validate its physical relevance by checking it against experimental dynamic behavior. Finally, Section VI concludes the paper with a summary of our contributions and suggests directions for future work.

In this section, we present the findings from our state-space identification performed using the proposed BiLSTM model. The model was trained with optimal hyperparameters determined by Bayesian optimisation, and the final training process achieved a validation RMSE of 0.0475 A.

Our analysis begins by illustrating the complexity of the identification challenge itself. A look at the I-V curves in Figure 2, which shows a sample from our test set, immediately reveals how dramatically the system's behaviour changes across its different operational and fault conditions. The system's state has a profound impact on the shape of the I-V curve. For example, under healthy conditions, the curve is typically smooth and concave. A fault like partial shading, however, can drastically alter this shape, introducing significant distortions and often creating multiple local power maxima. The true test for any system identification approach, therefore, is its ability to accurately model this full range of complex, nonlinear behaviors, rather than just fitting a single curve shape [34].

Following this, we will assess the model's overall predictive performance using global metrics and conclude with a detailed investigation into its physical relevance and learned internal dynamics.

3.1 Model configuration and training convergence

3.1.1 Optimal model parameters

We used a process to find the best model settings, which is described in the methodology.

The final choice was made based on one key factor: minimizing the error on the validation data. This method ensures a good balance between the model's ability to learn and the need to prevent overfitting, as detailed in Table 2.

Table 2. Final hyperparameters of the proposed BiLSTM model

3.1.2 Training convergence

The training progress of the final model is shown in Figure 7. The plot shows how the Root Mean Square Error (RMSE) and the loss function changed during the training for both the training and validation sets.

At first, both the loss and RMSE dropped quickly, then the values became stable as the training continued. The process stopped automatically after 23 epochs because of the early stopping rule, which helped prevent the model from learning the training data too well (overfitting). This early stopping ensured that the model would perform well on new, unseen data. The final RMSE on the validation data at the point of convergence was 0.014278, showing that the training was successful and stable.

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Figure 7. Training progress of the BiLSTM network

3.2 Overall identification performance

After training the model and confirming its convergence, we tested it with a separate set of 1630 I-V curves. The predictions were then converted back to their original units (Amperes). The model performed well and showed it could generalize to new data. The performance metrics on the test set were as follows:

Root Mean Square Error (RMSE): 0.0475 A

Coefficient of Determination (R²): 0.9974

The following figures further illustrate this high level of accuracy:

The correlation plot (Figure 8) compares predicted and actual current values. The data points form a very dense and narrow cluster around the y=x line, which represents an ideal prediction. This tight alignment visually confirms the model's robustness and its ability to make highly accurate predictions, consistent with the high R² score [35].

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Figure 8. Correlation between predicted and actual values

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Figure 9. Distribution of prediction errors (I_{actual} - I_{predicted})

The error distribution histogram (Figure 9) provides further insight into the model's performance.

The distribution of prediction errors (I_actual - I_predicted) is sharply peaked and centered almost perfectly at zero, indicating that the model is unbiased and that the vast majority of errors are very close to zero. This narrow spread, with most errors falling within a ±0.1 A range, reinforces the reliability of the model and is consistent with the low overall RMSE.

A visual check of the model's performance in Figure 10 confirms its high accuracy. Here, we've overlaid the predicted I-V curves on the actual curves for several random test samples. The two curves track each other very closely, even across different voltage and current ranges, which is a strong visual testament to the low RMSE and high R² scores [36].

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(a) Test sample # 1323

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(b) Test sample # 1524

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(c) Test sample # 71

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(d) Test sample # 317

Figure 10. Comparison of actual and predicted I-V curves for randomly selected test samples

3.3 Detailed analysis and validation of the identified state-space model

Despite the confirmation of the model's statistical accuracy through global metrics, additional analysis is essential to confirm that the model has grasped a valid state-space representation. To validate our identified state-space model beyond global metrics, we carry out a detailed, three-part analysis. First, we check how well the model follows a typical example of the system's behavior. Then, we take a closer look at the internal state dynamics and their physical meaning by using time-based plots and t-SNE visualizations. Finally, we test how well the model can predict critical physical parameters.

3.3.1 Dynamic identification performance

To visually assess the model's performance on a dynamic trajectory, a random sample from the test set was selected for detailed analysis. Figure 11 illustrates the identification process for this sample.

The figure's subplots show that (a) the model was subjected to a standard voltage sweep input. In response, (b) the model's predicted output ŷ(t) almost perfectly tracks the actual system output y(t), indicating a very high tracking accuracy. This is further confirmed in (c), where the prediction error e(t) remains small, stable, and centred around zero, demonstrating that the model's predictions are consistent and unbiased throughout the sequence.

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Figure 11. Dynamic identification performance of the state-space model for a random test sample. Subplots show: (a) input voltage u(t); (b) comparison between actual y(t) and predicted ŷ(t) current; (c) prediction error e(t)

3.3.2 Analysis of the learned internal state

The main idea behind our state-space approach is that the hidden states of the BiLSTM can serve as estimators for the unobservable system state x(k).To check this, we first examined how the hidden states evolved for the same test sample (Figure 11). This analysis validates that the model has learned a coherent internal state-transition function.

$\mathrm{h}(k)=f(h(k-1), u(k))$.          

This is the most insightful part of the analysis, as it shows how the first three components of the learned hidden state vector h(t) evolve. Figure 12 clearly shows that the hidden states don't change randomly. Instead, they follow a smooth and organized pattern that lines up perfectly with the input signal. What this tells us is that the network has successfully learned a stable and coherent set of internal rules—a transition function—which is precisely what a good state-space model should do.

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Figure 12. Evolution of the first three components (h1, h2, h3) of the learned hidden state vector h(t) for the test sample shown in Figure 11

To further validate that this learned state is physically meaningful, we visualized the final hidden state h(t) for all test samples using the t-SNE technique. The results, presented in Figure 13, offer a direct visualization of the learned latent space.

The t-SNE visualizations provide a robust validation of our state-space approach, revealing that the model has learned to organize its latent space based on a clear physical hierarchy. The primary organizing factor is irradiance (a), followed by a secondary organization based on temperature (b) within each irradiance cluster, and a final, local organization based on fault categories (c), demonstrating that the model has learned to distinguish the unique dynamic signature of each health state. This automatically discovered structure confirms that the final hidden state h(t) is a physically meaningful embedding, not just a random vector. It demonstrates that our model has gone beyond simple curve-fitting to construct a true, interpretable internal state-space model of the PV system.

3.3.3 Physical parameter validation

This analysis provides strong evidence that the BiLSTM has successfully identified a meaningful internal state-space representation of the PV system. The learned hidden states are not arbitrary, but follow a clear dynamic structure, confirming that the model has learned an internal representation of the system's evolution, not just a static input-output mapping.

Now that we've seen that the model has learned a coherent internal dynamic, the next logical step is to check its practical usefulness. We did this by testing if it could accurately predict key physical parameters from the I-V curves. This test is essential because it confirms that our identified model is not just internally consistent but also physically meaningful.

Table 3. Performance on key physical parameter extraction

We extracted five standard PV parameters from both the predicted and the actual curves for all 1630 samples in the test set. Table 3 summarises the results, comparing the mean of the actual and predicted values and showing the resulting error metrics calculated on these means.

The results show a strong agreement for most parameters. The model predicts the mean maximum power (Pmax), maximum power point voltage (Vmpp), current (Impp), and open-circuit voltage (Voc) with relative errors on the mean of less than 1%. This high level of accuracy on derived physical quantities is strong evidence that the learned state-space representation is physically accurate.

A higher relative error of 16.6% was observed for the short-circuit current (Isc). However, as the error distribution analysis in Figure 14 reveals, this is not due to a model weakness but is primarily a statistical artefact. Looking at the error distribution for Isc, we see a tall, narrow peak right at zero. This shows two things: the model has no systematic bias, and its absolute errors are almost always tiny. The large percentage in the table is a direct result of the math involved, as the relative error formula exaggerates even these minor absolute errors when the actual Isc value is very low (≈0.26 A). This slight initial deviation can be attributed to a "sequence edge effect" occurring at the start of the voltage sweep (V ≈ 0), where the forward LSTM layer has less contextual information.

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Figure 13. t-SNE visualization of the learned latent space, colored by (a) Irradiance (G), (b) Temperature (T), and (c) Fault category

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Figure 14. Probability density functions of the prediction errors for each of the five extracted physical parameters

To get a more complete picture, we visualize the full error distributions for each parameter in Figure 14. In this visualization, we plot the probability density functions (PDFs) derived from the prediction errors, which lets us directly check for any model bias and see how consistent the predictions are.

The distributions in Figure 13 tell a clear story. All five are sharply peaked and centred on zero, a strong sign that our model is unbiased. Looking closer at Isc, its PDF is noticeably the most concentrated of the group. This confirms that the model's absolute performance for this specific parameter is, in fact, excellent. Nevertheless, the overall performance confirms that the model has acquired a physical understanding of the system, far beyond simple curve-fitting.

We can further validate the physical relevance of our identified state-space model by comparing its predictions with experimental results from the literature. For example, the work of Belaout [20] presents the dynamic response of a similar PV system's Maximum Power Point (MPP) to real-time changes in irradiance and temperature. Although our work focuses on static system identification, this comparison allows us to confirm that our model has learned the correct underlying physical behaviors.

5.1 Comparison of physical trends

The experimental data in the study [20] reveal two important physical trends. First, it shows a clear link between irradiance and current. When irradiance goes up, the maximum power point current (Impp) rises with it, leading to a direct increase in maximum power (Pmax). The second is the inverse relationship between temperature and voltage: as temperature increases, the maximum power point voltage (Vmpp) falls, slightly reducing Pmax.

What this means is that our model, despite being trained only on static I-V curves, has still learned these fundamental physical relationships. The highly accurate parameter predictions we show in Table 3 are strong evidence that our data-driven model has implicitly captured the correct dependencies between environmental conditions and key electrical parameters, and that these learned relationships align with the dynamic behavior seen in experiments.

5.2 Complementarity of the approaches

This comparison also highlights the collaboration between system identification and control algorithm validation. An MPPT algorithm, like the one tested in study [20], requires a target—the true Maximum Power Point—to track. Our work provides a direct method for predicting the location of this MPP under any given static condition.

Therefore, our identified BiLSTM model can be seen as a high-fidelity "virtual test bench" or surrogate model. It could be used to generate the reference I-V curves and target MPPs necessary for testing, validating, or even training new MPPT algorithms under thousands of simulated conditions. This task would be impractical to perform on a physical system.