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Integrating Physics and Data-Driven Approaches: An Explainable and Uncertainty-Aware Hybrid Model for Wind Turbine Power Prediction

Alfonso Gijón, Simone Eiraudo, Antonio Manjavacas, Daniele Salvatore Schiera, Miguel Molina-Solana, Juan Gómez-Romero

TL;DR

This work advances wind turbine power prediction by integrating a physics-based submodel with a data-driven residual component in an additive semi-parametric framework. The physics term uses a neural network to model the power coefficient $C_p$ under the Betz limit, while the residual term leverages additional variables to capture unmodeled effects, trained in two stages. The approach achieves substantial accuracy gains over a purely physics-based model, and its interpretability is enhanced with SHAP analyses, complemented by uncertainty quantification via conformalized quantile regression. The resulting framework provides a flexible, reliable tool for power forecasting, anomaly detection, and potential optimization in wind energy systems, with practical implications for operational decision-making and maintenance planning.

Abstract

The rapid growth of the wind energy sector underscores the urgent need to optimize turbine operations and ensure effective maintenance through early fault detection systems. While traditional empirical and physics-based models offer approximate predictions of power generation based on wind speed, they often fail to capture the complex, non-linear relationships between other input variables and the resulting power output. Data-driven machine learning methods present a promising avenue for improving wind turbine modeling by leveraging large datasets, enhancing prediction accuracy but often at the cost of interpretability. In this study, we propose a hybrid semi-parametric model that combines the strengths of both approaches, applied to a dataset from a wind farm with four turbines. The model integrates a physics-inspired submodel, providing a reasonable approximation of power generation, with a non-parametric submodel that predicts the residuals. This non-parametric submodel is trained on a broader range of variables to account for phenomena not captured by the physics-based component. The hybrid model achieves a 37% improvement in prediction accuracy over the physics-based model. To enhance interpretability, SHAP values are used to analyze the influence of input features on the residual submodel's output. Additionally, prediction uncertainties are quantified using a conformalized quantile regression method. The combination of these techniques, alongside the physics grounding of the parametric submodel, provides a flexible, accurate, and reliable framework. Ultimately, this study opens the door for evaluating the impact of unmodeled variables on wind turbine power generation, offering a basis for potential optimization.

Integrating Physics and Data-Driven Approaches: An Explainable and Uncertainty-Aware Hybrid Model for Wind Turbine Power Prediction

TL;DR

This work advances wind turbine power prediction by integrating a physics-based submodel with a data-driven residual component in an additive semi-parametric framework. The physics term uses a neural network to model the power coefficient under the Betz limit, while the residual term leverages additional variables to capture unmodeled effects, trained in two stages. The approach achieves substantial accuracy gains over a purely physics-based model, and its interpretability is enhanced with SHAP analyses, complemented by uncertainty quantification via conformalized quantile regression. The resulting framework provides a flexible, reliable tool for power forecasting, anomaly detection, and potential optimization in wind energy systems, with practical implications for operational decision-making and maintenance planning.

Abstract

The rapid growth of the wind energy sector underscores the urgent need to optimize turbine operations and ensure effective maintenance through early fault detection systems. While traditional empirical and physics-based models offer approximate predictions of power generation based on wind speed, they often fail to capture the complex, non-linear relationships between other input variables and the resulting power output. Data-driven machine learning methods present a promising avenue for improving wind turbine modeling by leveraging large datasets, enhancing prediction accuracy but often at the cost of interpretability. In this study, we propose a hybrid semi-parametric model that combines the strengths of both approaches, applied to a dataset from a wind farm with four turbines. The model integrates a physics-inspired submodel, providing a reasonable approximation of power generation, with a non-parametric submodel that predicts the residuals. This non-parametric submodel is trained on a broader range of variables to account for phenomena not captured by the physics-based component. The hybrid model achieves a 37% improvement in prediction accuracy over the physics-based model. To enhance interpretability, SHAP values are used to analyze the influence of input features on the residual submodel's output. Additionally, prediction uncertainties are quantified using a conformalized quantile regression method. The combination of these techniques, alongside the physics grounding of the parametric submodel, provides a flexible, accurate, and reliable framework. Ultimately, this study opens the door for evaluating the impact of unmodeled variables on wind turbine power generation, offering a basis for potential optimization.

Paper Structure

This paper contains 12 sections, 7 equations, 5 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Related work and main contributions
  3. Methods
  4. Additive semi-parametric hybrid model
  5. Explainability and Uncertainty Quantification
  6. Case study and experimental setup
  7. Wind turbines physical background
  8. Data description and preprocessing
  9. Power hybrid model
  10. Computational details
  11. Results and discussion
  12. Regression analysis

Figures (5)

  • Figure 1: Box plots of wind speed, pitch angle, rotor speed, outdoor temperature and power for the four different wind turbines.
  • Figure 2: Average power curve derived from the data of the four wind turbines.
  • Figure 3: Diagram of the hybrid model, according to the taxonomy presented in vanBekkum2021. Rectangular, rounded, and hexagonal boxes represent data, functions, and models, respectively. Yellow and blue boxes are used for inputs and outputs, respectively.
  • Figure 4: The $C_p$ curves derived from the physics-based model are represented as a function of the tip speed ratio $\lambda$, for various pitch angles in different colors (measured in radians). The observed data are represented with solid lines, while the model predictions are shown with dashed lines.
  • Figure 5: (a) Absolute residual between the true power and the prediction made by the physical model. (b) Predicted vs true residual power. The contributions of the physics-based model (orange) and the residual model (green) to the predicted power are plotted as functions of wind speed, alongside the observed data (blue).To further enhance the interpretation and reliability of the hybrid model's predictions, we apply explainability and uncertainty quantification techniques. As noted in the previous section, the physics-based component is fully interpretable, as it relies on a physical equation and an intermediate variable constrained within a defined range. In contrast, the residual component acts as a black box, taking input variables and generating predictions without explicit interpretability. The variables $(v,\theta,\omega)$ are inputs of both components of the hybrid model. To assess their significance within each submodel, two different instances of these variables are analyzed for the explainability study. The mean absolute SHAP values calculated over a sample of the test dataset and shown in panel (a) of \ref{['fig:SHAP_ranking_sum']}, provide insights into the relative importance of each input feature on the output. As expected, the variables incorporated into the physics-inspired model, $(v_\text{phys},\theta_\text{phys},\omega_\text{phys})$, are the most influential. However, the corresponding instances of these variables in the residual model also exhibit high SHAP values, ranging from 10 to 20 kW. This indicates that the dependence of power on these variables is not fully captured by \ref{['eq:Power-cp']}. Furthermore, the magnitude of the mean SHAP value associated with outdoor temperature indicates a notable influence on the predicted power, making it a good candidate for inclusion in a more sophisticated and accurate physical model. An interesting property of the additive hybrid model is its ability to recover the contributions of its two components by summing the SHAP values associated with each submodel. When the average power plus the sum of the SHAP values is plotted against wind speed, the sigmoid shape characteristic of the physics-based submodel emerges, while the residual component oscillates around zero, exhibiting larger amplitudes at intermediate wind velocities, as shown in panel (b) of \ref{['fig:SHAP_ranking_sum']}. (a) Ranking of mean absolute SHAP values of the hybrid model. (b) Sum of SHAP values regarding physics and residual submodels.SHAP value of the residual power model vs. associated variable for (a) wind speed,(b) rotor speed and (c) outdoor temperature. Linear and quadratic regressions are shown in red color.At this stage, it is natural to question whether the hybrid model can provide insights for developing new, more sophisticated physics-based models. To delve deeper into the contribution of input variables to the residual model's output, scatter plots of SHAP values against their corresponding variables can be analyzed. \ref{['fig:SHAP_correlations']} illustrates the influence of the most significant variables, accompanied by low-order polynomial regression fits to capture underlying trends. The positive correlation between the SHAP value and rotor angular speed indicates that this variable positively contributes to the power residual, suggesting that the physical model underestimates its impact. Conversely, the negative correlation between the SHAP value and outdoor temperature suggests that an increase in temperature should decrease the power output of the physical model. This is logical, as higher temperatures reduce the efficiency of electrical energy conversion, a factor not accounted for by the physical model. Lastly, the correlation between the SHAP value and wind speed implies that the power output of the physical model should be adjusted for both higher and lower velocities than the mean, as the SHAP value is positive in both directions. However, the observed correlations between SHAP values and input variables do not necessarily imply a causal relationship between those input variables and the generated power shap_causal_insights. In fact, attempts to approximate the residual power by the sum of polynomials of the input variables, as suggested by \ref{['fig:SHAP_correlations']}, result in poor regression performance. This is mainly due to the interdependence of input variables and the complex correlations among them. In fact, to fully analyze the dependence of the target variable $P$ on the input variables, it is necessary to consider the sum of all SHAP values, as demonstrated in the bottom panel of \ref{['fig:SHAP_ranking_sum']}. Tools like SHAP are effective in identifying the most informative relationships between input features and the predicted outcome, providing valuable insights into the model's behavior. However, to develop causal or physical models, it is necessary to make assumptions and leverage the methodologies of causal analysis. Power curve comparing model predictions with uncertainty intervals (blue), data (solid black line) and manufacturer specifications (dash black line). The data curve is nearly indistinguishable to the human eye, as it closely overlaps with the model curve, lying beneath it.Finally, to evaluate the reliability of the predictions, uncertainties are calibrated using a calibration set comprising half of the test set. The model's performance is then assessed using prediction intervals on the remaining portion of the test set. This procedure does not require re-training the hybrid model, but instead involves training the upper (95%) and lower (5%) quantile estimations of the predictions. The conformalized quantile regression method is then used to provide conformal predictions, generating target predictions for new samples with associated confidence intervals. The effective coverage is determined by estimating the fraction of true labels that fall within the prediction intervals. In this case, the coverage is 86%, and the prediction intervals have a mean length of 49 kW. The predicted power curve with uncertainty estimation is shown in \ref{['fig:PvsV_uncertainty']}, alongside the mean curve derived from the test data and the manufacturer's specifications. As observed, the uncertainty intervals are negligible at low wind speeds and become wider at mid to high speeds, providing an indication of the regions where the power model is more or less accurate. Furthermore, these results are consistent with previous research Gijon2023, which found larger uncertainties in the same wind speed range when using physics-informed neural network models. Discrepancies with the theoretical manufacturer curve are typical in large datasets from real wind farms, and often does not accurately reflect the actual performance of WTs WT_power_curvePAGNINI2015YAN2019.In this study, we design and validate a hybrid semi-parametric model comprising a physics-inspired submodel, $P_\text{phys}$ and a non-parametric submodel for predicting the residual power of the physical term, denoted as $P_\text{res}$. The proposed model, trained on real historical data from four turbines at 'La Haute Borne' wind farm, demonstrates an improvement of a 37% in predicting generated power. The physics-inspired submodel offers inherent interpretability of the power coefficient, as it is built upon a physical equation that captures the relationships among the system’s most critical variables. In contrast, the non-parametric residual submodel, which accounts for the unexplained components of the system, requires an additional analysis to interpret its behavior, which is accomplished using the SHAP explainability technique. Moreover, the conformalized quantile regression method provides conformal predictions along with associated confidence intervals, consistent with the data and previous results. The integration of these components results in a flexible, accurate and reliable framework. Once deployed, this hybrid model can serve as a robust regression-based anomaly detection tool, quantifying the probability of new data being classified as normal or anomalous. Furthermore, all models presented in this study are fully differentiable, making them suitable for developing optimal operation controllers. These controllers have the potential to further enhance power generation efficiency across varying wind speed conditions. Using SHAP values, the model also provides insights into the relative importance of input features and their influence on the predicted power output. The analysis indicates that incorporating outdoor temperature into future, more advanced physical models could enhance prediction accuracy. Additionally, it highlights potential adjustments in the relationships between power and wind or rotor speed. Future research should focus on a more in-depth analysis of correlations and the application of causal analysis tools, which would be valuable for the development of improved physics-based causal models. Although the hybrid model shows significant potential, further research is required to assess its robustness and scalability across different wind farm datasets. Nevertheless, the model can be readily adapted to different manufacturers by fine-tuning the parameters of the pre-trained residual submodel and updating the turbine-specific parameters of the physics-based submodel. Furthermore, the proposed methodology is highly versatile and applicable to a wide range of scenarios where a physics-based model approximates key dynamics. By incorporating additional data through an additive non-parametric submodel, the framework effectively captures residual components and integrates previously unknown physical phenomena, all without the need to retrain the physics-based model.A. Gijón: Conceptualization, Methodology, Software, Data Curation, Formal Analysis, Investigation, Validation, Visualization, Writing - Original Draft, S. Eiraudo: Conceptualization, Methodology, Software, Investigation, Writing - Review & Editing, A. Manjavacas: Methodology, Software, Investigation, Writing - Review & Editing, D.S. Schiera Supervision, Writing - Review & Editing, M. Molina-Solana: Supervision, Funding acquisition, Project administration, Writing - Review & Editing, J. Gomez-Romero: Supervision, Funding acquisition, Project administration, Writing - Review & Editing.The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.This work was primarily funded by the Spanish Ministry of Economic Affairs and Digital Transformation (NextGenerationEU funds) within the project IA4TES MIA.2021.M04.0008. It was also partially funded by ERDF/Junta de Andalucía (D3S project P21.00247, and SE2021 UGR IFMIF-DONES), and MICIU/AEI/ 10.13039/501100011033 and EU ERDF (SINERGY, PID2021.125537NA.I00). 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