Tube Loss based Deep Networks For Improving the Probabilistic Forecasting of Wind Speed

TL;DR

This work tackles uncertainty quantification for wind speed forecasting by introducing Tube loss–based probabilistic forecasting within autoregressive deep models. The Tube loss provides a differentiable, distribution-free mechanism to estimate prediction interval bounds with target calibration $1-\alpha$ while explicitly minimizing PI width, controlled by a movement parameter $r$ and a width-penalizing term $\\delta$. The authors extend this approach to LSTM, GRU, and TCN architectures, proposing a simple heuristic to tune $\delta$ and evaluating on three wind datasets (Jaisalmer, Los Angeles, San Francisco) against strong baselines including Quantile, QD, Deep AR, MDN, and Time-GPT. Across all datasets, Tube loss–based models achieve reliable coverage (PICP $\geq 0.95$) and substantially narrower prediction intervals (MPIW) than competing methods, with TCN+Tube, GRU+Tube, and LSTM+Tube often delivering the best performance. The results suggest that distribution-free, width-aware PI estimation via Tube loss can meaningfully improve the reliability and operational usefulness of probabilistic wind speed forecasts.

Abstract

Uncertainty Quantification (UQ) in wind speed forecasting is a critical challenge in wind power production due to the inherently volatile nature of wind. By quantifying the associated risks and returns, UQ supports more effective decision-making for grid operations and participation in the electricity market. In this paper, we design a sequence of deep learning based probabilistic forecasting methods by using the Tube loss function for wind speed forecasting. The Tube loss function is a simple and model agnostic Prediction Interval (PI) estimation approach and can obtain the narrow PI with asymptotical coverage guarantees without any distribution assumption. Our deep probabilistic forecasting models effectively incorporate popular architectures such as LSTM, GRU, and TCN within the Tube loss framework. We further design a simple yet effective heuristic for tuning the $δ$ parameter of the Tube loss function so that our deep forecasting models obtain the narrower PI without compromising its calibration ability. We have considered three wind datasets, containing the hourly recording of the wind speed, collected from three distinct location namely Jaisalmer, Los Angeles and San Fransico. Our numerical results demonstrate that the proposed deep forecasting models produce more reliable and narrower PIs compared to recently developed probabilistic wind forecasting methods.

Figures (5)

• Figure 1: Tube loss function for $1-\alpha =0.9$
• Figure 2: Tube loss methodology: red line represents the convex combination of ${\mu}_{1}(x)$ and ${\mu}_{2}(x)$, i,e., $r{\mu}_{1}(x)+(1-r){\mu}_{2}(x)$, $0<r<1$. Blue dots represent data points $(x_i,y_i)$, $i =1,2,,..,m$. For simplicity, both PI bounds considered as the linear function of $x$.
• Figure 3: Flow Diagram: Probabilistic forecasting with Tube loss based deep learning models
• Figure 4: Average Improvement obtained by the Tube loss model over existing forecasting models
• Figure 5: Plot of GRU-Tube on Jaisalmer Dataset