Forecasting electricity prices as a connected day

Why forecast the hours together?

For an electricity trader or battery operator, the question is not just what each hour might cost. It is also which combinations of prices could occur across the day. Forecasts that describe individual hours well can still misrepresent that wider pattern.

In a preprint, Simon Hirsch develops an algorithm for forecasting the joint probability distribution of 24 hourly electricity prices. Rather than producing one predicted price per hour, it describes uncertainty across possible daily price trajectories, including how the hours move together.

The distinction matters for prediction bands. A band intended to contain the entire day’s price trajectory 95% of the time is a different promise from 24 separate intervals, each intended to contain its own hour’s price 95% of the time.

Learning uncertainty without starting over

The method uses distributional regression: explanatory variables help predict not only expected prices, but also their variability and dependence. Inputs include past prices, calendar information, forecasts of demand remaining after wind and solar production, and fuel and emissions allowance prices.

Modeling all those connections can become expensive. Hirsch combines two ways of controlling complexity. LASSO-type penalties shrink regression coefficients and can remove some explanatory variables. A separate procedure starts with a simple dependence structure and progressively adds connections, stopping when further complexity is not justified by the fitting criterion.

For time-ordered hours, this can mean first allowing connections between neighboring hours, then extending the structure farther across the day. The selected structural complexity remains fixed during subsequent online updates.

Here, “online” means updating the fitted model when new observations arrive—not using an internet service. The algorithm retains mathematical summaries of earlier data and incorporates the new day, instead of repeatedly fitting against the entire historical dataset. Those summaries preserve historical information, although the update is an approximation to full batch fitting.

A German market test

The study evaluates forecasts over 1,831 days, from late December 2018 through December 2023, including the volatile years 2021–2023. Comparisons include simpler price regressions, models with changing volatility, and approaches that forecast individual hours before joining them through a statistical dependence model called a copula.

Some versions of the proposed method produced joint prediction bands relatively close to their intended coverage. For example, one version using a Cholesky decomposition—a structured way to represent the dependence matrix—had a nominal 95% band that contained the full observed daily trajectory about 93.3% of the time. A simpler regression benchmark with correlated Gaussian errors achieved about 78.9% coverage.

Coverage alone is not enough: extremely wide bands can cover many outcomes without being useful. The study therefore also compares band widths and several scoring rules, which evaluate different aspects of probabilistic forecasts.

The results do not identify a universal winner. Modified-Cholesky versions of the new model performed best on the full-sample log score and Dawid–Sebastiani score, measures sensitive to the joint forecast distribution and its mean and covariance. Simpler benchmarks performed better on point-price errors and on CRPS, which evaluates individual-hour distributions. The new method also did not lead on the energy or variogram scores across the full sample.

Speed helps, but does not settle accuracy

Daily updates were fast on the study’s laptop. The Cholesky version in the coverage example took about 132 seconds for its initial fit and averaged 0.20 seconds per update. More heavily regularized versions took longer.

A separate comparison on the earlier test period found that online updating offered a favorable accuracy–computation trade-off. With sufficiently large computation budgets, however, repeated expanding-window batch fitting achieved better log scores. Faster updating was not equivalent to the best attainable accuracy.

What remains uncertain

This is a preprint and a historical study of one market, not evidence of improved trading profits. Assumed distribution families may restrict how well unusual price extremes can be represented, and coverage deteriorated in some stressed-market conditions. The full study dataset cannot be redistributed by the author, although implementation and reproduction code are available.

The practical finding is narrower than “better electricity forecasts”: modeling connections between hours can improve important descriptions of whole-day uncertainty, while sacrificing accuracy elsewhere. Which trade-off is useful depends on the decision the forecast must support.