Probabilistic energy forecasting through quantile regression in reproducing kernel Hilbert spaces

TL;DR

This paper addresses probabilistic energy forecasting under climate variability by using kernel quantile regression in reproducing kernel Hilbert spaces ($RKHS$). The authors formulate a nonparametric KQR model that minimizes the pinball loss with a regularization term, producing calibrated quantile forecasts through a kernel expansion whose coefficients are found via quadratic programming, with an open-source implementation released. The method is evaluated on diverse energy datasets—the Energy Charts data for Switzerland and Germany, the SECURES-Met dataset, and the GEFCom2014 benchmark for load and price—demonstrating reliable calibration, sharpness, and competitive performance relative to state-of-the-art quantile regression approaches. The results highlight robustness to different kernels (notably Absolute Laplacian and Gaussian $RBF$) and predictor sets, and support open-source reproducibility through a public Python implementation. The work provides a practical uncertainty quantification framework for energy-system planning under climate shocks, with potential to inform TSO decision-making and policy during the transition to sustainable energy.

Abstract

Accurate energy demand forecasting is crucial for sustainable and resilient energy development. To meet the Net Zero Representative Concentration Pathways (RCP) $4.5$ scenario in the DACH countries, increased renewable energy production, energy storage, and reduced commercial building consumption are needed. This scenario's success depends on hydroelectric capacity and climatic factors. Informed decisions require quantifying uncertainty in forecasts. This study explores a non-parametric method based on \emph{reproducing kernel Hilbert spaces (RKHS)}, known as kernel quantile regression, for energy prediction. Our experiments demonstrate its reliability and sharpness, and we benchmark it against state-of-the-art methods in load and price forecasting for the DACH region. We offer our implementation in conjunction with additional scripts to ensure the reproducibility of our research.

Figures (9)

• Figure 1: Pinball loss function at the $q$ quantile--the lower the pinball loss, the more accurate the quantile forecast is.
• Figure 2: Load 90% confidence interval for Switzerland Energy charts data using KQR Absolute Laplacian: Electric load probabilistic forecast for Switzerland 2022. The black line is the observed path for the load. The 90% confidence interval bands are plotted in green. Lower and upper red lines denote the 95% and 5% quantile forecast respectively.
• Figure 3: Load 90% confidence interval for Germany Energy charts data using KQR Absolute Laplacian: Electric load probabilistic forecast Germany 2022. The black line is the observed path for the load. The 90% confidence interval bands are plotted in green. Lower and upper red lines denote the 95% and 5% quantile forecast respectively.
• Figure 4: Load 90% confidence interval task 9 using KQR Absolute Laplacian: Electric load probabilistic forecast for June 2011. The black line is the observed path for the load. The 90% confidence interval bands are plotted in green. Lower and upper red lines denote the 95% and 5% quantile forecast respectively. The prediction out-of confidence interval is denoted in red.
• Figure 5: Price 90% confidence interval task 6: Electricity price probabilistic forecast for the $13^{\text{th}}$ July $2013$. The black line is the observed path for the price. The $90\%$ confidence interval bands are plotted in green. Lower and upper red lines denote the $95\%$ and $5\%$ quantile forecast respectively.

Theorems & Definitions (1)

• definition 1