[Paper Review] Probabilistic Forecasting for Daily Electricity Loads and Quantiles for Curve-to-Curve Regression

Xiuqin Xu, Ying Chen+2 more|arXiv (Cornell University)|Sep 3, 2020

Energy Load and Power Forecasting26 references4 citations

TL;DR

This paper proposes a novel probabilistic forecasting framework for daily electricity load curves using curve-to-curve linear regression, introducing predictive sets, bands, and quantile curves with proper probability interpretation. It achieves superior forecasting accuracy, higher coverage rates, and narrower predictive bands compared to state-of-the-art methods like GAM, SAR, and SARX, with a MAPE of 1.10% on French load data.

ABSTRACT

Probabilistic forecasting of electricity load curves is of fundamental importance for effective scheduling and decision making in the increasingly volatile and competitive energy markets. We propose a novel approach to construct probabilistic predictors for curves (PPC), which leads to a natural and new definition of quantiles in the context of curve-to-curve linear regression. There are three types of PPC: a predictive set, a predictive band and a predictive quantile, all of which are defined at a pre-specified nominal probability level. In the simulation study, the PPC achieve promising coverage probabilities under a variety of data generating mechanisms. When applying to one day ahead forecasting for the French daily electricity load curves, PPC outperform several state-of-the-art predictive methods in terms of forecasting accuracy, coverage rate and average length of the predictive bands. The predictive quantile curves provide insightful information which is highly relevant to hedging risks in electricity supply management.

Motivation & Objective

• To address the limitations of deterministic and pointwise probabilistic forecasting in electricity load management.
• To develop a joint probabilistic forecasting framework for entire daily load curves with proper probability interpretation.
• To improve forecasting accuracy and reliability by embedding non-stationary daily patterns into a stationary functional framework.
• To provide meaningful predictive intervals and quantile curves for risk hedging in electricity supply systems.
• To overcome the shortcomings of individual interval forecasting, such as loss of joint probability interpretation and poor coverage under dependence.

Proposed method

• Proposes a curve-to-curve linear regression model that treats daily load curves as functional data in a Hilbert space.
• Employs dimension reduction via singular value decomposition (SVD) to handle high-dimensional load curves.
• Defines three types of probabilistic predictors for curves (PPC): predictive set, predictive band, and predictive quantile, all at a nominal probability level.
• Uses chi-squared approximation and empirical cumulative distribution function (ECDF-R) to construct predictive bands with calibrated coverage.
• Introduces a new definition of curve quantiles in the context of functional linear regression, enabling probabilistic interpretation of extreme load scenarios.
• Applies the method to one-day-ahead forecasting of French half-hourly electricity load curves using real data.

Experimental results

Research questions

• RQ1 Can a unified probabilistic forecasting framework be developed for entire daily load curves with proper probability interpretation?
• RQ2 How can predictive bands for curves maintain joint coverage probability while remaining narrow and informative?
• RQ3 To what extent does curve-to-curve regression improve forecasting accuracy and coverage compared to pointwise or marginal methods?
• RQ4 How do predictive quantile curves contribute to risk assessment in electricity supply management?
• RQ5 Can the proposed method outperform existing state-of-the-art models like GAM, SAR, and SARX in accuracy, coverage, and interval width?

Key findings

• The proposed method achieves a MAPE of 1.10% on French daily load curves, representing a 33.3% reduction compared to SARX and a 45.8% reduction compared to SAR.
• The predictive band based on chi-squared approximation with K=5,000 achieves a coverage rate 0.221–0.325 higher than GAM, SAR, and SARX, with a 48–937 unit reduction in average length.
• The predictive quantile curves at 99% and 40% confidence levels effectively visualize extreme and typical load scenarios, supporting risk-aware decision-making.
• The method maintains coverage rates close to the nominal 90% level across diverse data-generating mechanisms in simulation, indicating robustness.
• Forecasting performance is more accurate in summer and on weekdays, with lower MAPE and AvL, suggesting seasonal and weekly patterns are well-captured.
• The predictive band based on ECDF-R offers comparable pointwise coverage rates to GAM but with significantly narrower average length, improving precision.