[Paper Review] Probabilistic Load Forecasting of Distribution Power Systems based on Empirical Copulas

Pål Forr Austnes, Celia García‐Pareja+2 more|arXiv (Cornell University)|Oct 5, 2023

Energy Load and Power Forecasting4 citations

TL;DR

This paper proposes a non-parametric probabilistic load forecasting method for distribution power systems using empirical copulas with beta kernel density estimation on the unit hypercube. The approach models dependence between diverse variables—such as weather forecasts and load—without assuming parametric distributions, yielding more accurate, asymmetric prediction intervals than quantile regression, with a 25% reduction in quantile loss and CRPS at low aggregation levels.

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.

Motivation & Objective

• To develop a data-driven probabilistic load forecasting method that does not rely on parametric assumptions for variable distributions or dependence structures.
• To improve forecast reliability at low aggregation levels (e.g., neighborhoods) where traditional statistical methods fail due to limited data.
• To enable accurate uncertainty quantification in distribution system operation by modeling complex, non-linear dependencies between weather and load variables.
• To support Distribution System Operators in optimizing day-ahead market bids and reducing reserve capacity costs through better forecast confidence intervals.

Proposed method

• The method employs empirical copulas to model the joint dependence structure of input variables, transforming them into uniform marginals on the unit hypercube.
• Beta kernel density estimation with bounded support is used to non-parametrically estimate the multivariate density of input variables, preserving unit interval constraints.
• The model uses a multivariate kernel density estimator with bandwidths optimized via cross-validation to minimize integrated squared error (ISE).
• Conditional prediction distributions are derived by kernel smoothing over historical data, allowing full probabilistic forecasts per time step.
• The approach naturally handles variables with different marginal distributions—e.g., temperature forecasts and electricity consumption—without transformation.
• The method supports asymmetric prediction intervals, reflecting the true non-Gaussian uncertainty in low-granularity load data.

Experimental results

Research questions

• RQ1 Can a non-parametric, data-driven approach outperform parametric or semi-parametric models like quantile regression in probabilistic load forecasting at low aggregation levels?
• RQ2 How well can empirical copulas with beta kernels model complex, non-linear dependencies between heterogeneous variables such as weather forecasts and electricity load?
• RQ3 To what extent does the absence of parametric assumptions on marginal distributions or copula families improve forecast reliability in volatile, low-data regimes?
• RQ4 How do the prediction intervals from the empirical copula model compare to those from quantile regression in terms of coverage and sharpness metrics?
• RQ5 Can the method produce meaningful probabilistic forecasts even when aggregated load data are sparse or non-normal?

Key findings

• The empirical copula model achieved a 25% reduction in quantile loss (QL) and CRPS compared to quantile regression across all aggregation levels.
• At the L4 aggregation level, the empirical copula with hand-selected bandwidth (h=0.05) achieved a CRPS of 1587 and QL of 801, outperforming the bandwidth-optimized version (CRPS: 3511, QL: 1773).
• The model produced narrower and more variable confidence intervals than quantile regression, better capturing non-linear ramp-up and ramp-down patterns in load.
• The model demonstrated superior coverage and sharpness, with a 98% PICP (prediction interval coverage probability) at the 5–95% interval and a 0.33 PINAW (prediction interval width) at L4.
• The method produced asymmetric prediction intervals that better reflect the true uncertainty structure in low-granularity load data, unlike symmetric parametric models.
• Even with suboptimal bandwidth selection, the model consistently outperformed quantile regression in all standard probabilistic evaluation metrics.