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Probabilistic Time Series Forecasting of Residential Loads -- A Copula Approach

Marco Jeschke, Timm Faulwasser, Roland Fried

TL;DR

This work tackles probabilistic forecasting of residential electricity loads by modeling time-varying marginals $f_t$ with kernel density estimation and capturing temporal dependencies with Vine Copulas, enabling quantile-based predictions and synthetic load generation. It clusters daily load densities into homogeneous time blocks using the squared Hellinger distance and applies vine-based dependencies within each block to produce realistic, non-Gaussian load trajectories. Validation on the Ausgrid dataset via a permutation test indicates that the generated profiles closely resemble real consumption patterns, including tail behavior. The framework offers a practical approach for uncertainty quantification in distribution grids and supports scenario-based planning under high penetration of renewables.

Abstract

Predicting the time series of future evolutions of renewable injections and demands is of utmost importance for the operation of power systems. However, the current state of the art is mostly focused on mean-value time series predictions and only very few methods provide probabilistic forecasts. In this paper, we rely on kernel density estimation and vine copulas to construct probabilistic models for individual load profiles of private households. Our approach allows the quantification of variability of individual energy consumption in general and of daily peak loads in particular. We draw upon an Australian distribution grid dataset to illustrate our findings. We generate synthetic loads that follow the distribution of the real data.

Probabilistic Time Series Forecasting of Residential Loads -- A Copula Approach

TL;DR

This work tackles probabilistic forecasting of residential electricity loads by modeling time-varying marginals with kernel density estimation and capturing temporal dependencies with Vine Copulas, enabling quantile-based predictions and synthetic load generation. It clusters daily load densities into homogeneous time blocks using the squared Hellinger distance and applies vine-based dependencies within each block to produce realistic, non-Gaussian load trajectories. Validation on the Ausgrid dataset via a permutation test indicates that the generated profiles closely resemble real consumption patterns, including tail behavior. The framework offers a practical approach for uncertainty quantification in distribution grids and supports scenario-based planning under high penetration of renewables.

Abstract

Predicting the time series of future evolutions of renewable injections and demands is of utmost importance for the operation of power systems. However, the current state of the art is mostly focused on mean-value time series predictions and only very few methods provide probabilistic forecasts. In this paper, we rely on kernel density estimation and vine copulas to construct probabilistic models for individual load profiles of private households. Our approach allows the quantification of variability of individual energy consumption in general and of daily peak loads in particular. We draw upon an Australian distribution grid dataset to illustrate our findings. We generate synthetic loads that follow the distribution of the real data.
Paper Structure (10 sections, 1 theorem, 9 equations, 6 figures, 2 tables, 1 algorithm)

This paper contains 10 sections, 1 theorem, 9 equations, 6 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem statement and preliminaries
  3. Data
  4. Statistical methods
  5. Kernel density estimation
  6. Clustering
  7. Vine Copulas
  8. Permutation test
  9. Case study on Ausgrid Data
  10. Conclusions and Outlook

Key Result

Theorem 1

Let $F$ be an $n$-dimensional distribution function with marginal distributions $F_{1},...,F_{n}$. Then, there exists an $n$-dimensional copula $C:[0,1]^{n}\rightarrow [0,1]$ such that If the multivariate distribution has a density $f$ we have for all $x_{i}\in\mathbb{R}\cup\{-\infty,\infty\}$ with the copula density $c$.

Figures (6)

  • Figure 1: 3-dimensional vine structure for active power
  • Figure 2: Two examples for (non-normalized) densities of different clusters. The blue density as an example for cluster 1 and the red density for cluster 2.
  • Figure 3: Average silhouette coefficient for different numbers of clusters. The optimal choice is two clusters with a coefficient of 0.61 indicating high homogeneity within the clusters.
  • Figure 4: Silhouette for every density
  • Figure 5: Histogram of p values for household #1
  • ...and 1 more figures

Theorems & Definitions (6)

  • Definition 1: Kernel function
  • Definition 2: Kernel density estimator b2
  • Definition 3: Squared Hellinger distance Vaart
  • Definition 4: Silhouette coefficientSil
  • Theorem 1: Sklarb4
  • Definition 5: Permutation test