Probabilistic Load Forecasting of Distribution Power Systems based on Empirical Copulas

TL;DR

This work tackles the challenge of probabilistic load forecasting for distribution systems under weather-driven uncertainty. It introduces a data-driven empirical copula (EC) framework that estimates conditional densities via a beta-kernel density estimator on the unit cube, enabling nonparametric modeling of both marginals and dependence without assuming parametric distributions. By independently modeling each time series and aggregating via scenario-based sums, the method yields coherent probabilistic forecasts across aggregation levels and horizons, with objective bandwidth optimization and a practical multi-step prediction workflow. Empirical results on a public Swiss dataset show an approximately 18% reduction in Quantile Loss (QL) compared with Quantile Regression (QR), while maintaining good coverage and sharper prediction intervals, illustrating the method’s practical impact for day-ahead market participation and grid operation.

Abstract

Accurate and reliable electricity load forecasts are becoming increasingly important as the share of intermittent resources in the system increases. Distribution System Operators (DSOs) are called to accurately forecast their production and consumption to place optimal bids in the day-ahead market. Forecasts must account for the volatility of weather-parameters that impacts both the production and consumption of electricity. If DSO-loads are small or lower-granularity forecasts are needed, parametric statistical methods may fail to provide reliable performance since they rely on a priori statistical distributions of the variables to forecast. In this paper, we introduce a Probabilistic Load Forecast (PLF) method based on Empirical Copulas (ECs). The model is datadriven, does not need a priori assumption on parametric distribution for variables, nor the dependence structure (copula). It employs a kernel density estimate of the underlying distribution using beta kernels that have bounded support on the unit hypercube. The method naturally supports variables with widely different distributions, such as weather data (including forecasted ones) and historic electricity consumption, and produces a conditional probability distribution for every time step in the forecast, which allows inferring the quantiles of interest. The proposed non-parametric approach differs significantly from previous forecasting methods based on copulas, which typically uses copulas to model hierarchical dependence. The bandwidth of the beta kernel density estimators is optimized using Integrated Square Error (ISE). We present results from an open dataset and showcase the strength of the model with respect to Quantile Regression (QR) using standard probabilistic evaluation metrics.

Figures (4)

• Figure 1: Overview of the fixed values in the evaluation of the conditional pdf (\ref{['eq:cond-density-estimation']}) for multi-step prediction.
• Figure 2: Aggregation-structure of the data used to evaluate the model.
• Figure 3: Example of day-ahead forecast for a weekend-day. Left: empirical copula. Right: quantile regression. The green and red curve show the actual outcome and the 0.5-quantile of the forecast, respectively. The pink curve shows the expected value. The blue shading shows the different quantiles. The color-gradient goes from darkest blue at the 0.5-quantile to the lightest blue at the 0.01 and 0.99-quantiles.
• Figure 4: Example of day-ahead forecast for a working-day. Left: empirical copula. Right: quantile regression. It is observed that our model provides narrower confidence intervals, especially in the morning ramp-up and the evening ramp-down. Confidence intervals are also more variable over the prediction horizon, highlighting the non-linear, data-driven nature of the model.