Distributionally Robust Scheduling of Electrified Heating Under Heat Demand Forecast Uncertainty

Electrified heating systems with thermal storage, such as electric boilers and heat pumps, represent a major source of demand-side flexibility. Under current electricity market designs, balance responsible parties (BRPs) operating such assets are required to submit binding day-ahead electricity consumption schedules, and they typically do it based on forecasts of heat demand and electricity prices. Common scheduling approaches implicitly assume that forecast uncertainty can be well characterized using historical forecast errors. In practice, however, the cumulative effect of uncertainty creates significant exposure to imbalance-price risk when the committed schedule cannot be followed. To address this, we propose a distributionally robust chance-constrained optimization framework for the day-ahead scheduling of a multi-MW electric boiler using only limited residual forecast samples. We derive a tractable convex reformulation of the problem and calibrate the ambiguity set directly from historical forecast-error data through an a priori tunable risk parameter. Numerical results show that enforcing performance guarantees on the heat-demand balance constraint reduces demand violations by 40% compared to a deterministic forecast-based scheduler and up to 10% relative to a nominal chance-constrained model with a fixed error distribution. Further, we show that modeling the real-time rebound cost of demand violations as a second-stage term can reduce the overall daily operating cost by up to 34% by hedging against highly volatile day-ahead electricity prices.

💡 Research Summary

This paper tackles the day‑ahead scheduling problem of a large‑scale industrial electric boiler that is coupled with a backup gas boiler, focusing on the uncertainty inherent in heat‑demand forecasts and volatile electricity prices. In current market designs, a balance‑responsible party (BRP) must submit a binding electricity consumption schedule for the next day. If the realized heat demand deviates from the forecast, the BRP faces imbalance‑price penalties, especially when the deviation exceeds the capacity of the backup gas boiler. Traditional approaches either assume that a large historical dataset accurately captures the forecast‑error distribution (sample‑average approximation) or adopt a deterministic robust set that can be overly conservative.

The authors propose a distributionally robust chance‑constrained (DRCC) framework that works with only a limited set of residual forecast errors. They construct a Wasserstein‑ball ambiguity set around the empirical distribution of the errors, with a tunable radius θ that governs the level of distributional robustness. The chance constraint that the available heat (storage plus electric‑boiler output) must meet the uncertain demand with confidence 1 − α is approximated by a Conditional Value‑at‑Risk (CVaR) constraint, which is convex and admits a tractable reformulation. By enforcing the CVaR constraint for all distributions inside the Wasserstein ball, the model hedges against both sampling error and possible shifts in the underlying error distribution.

Two formulations are presented. The first is a single‑stage DRCC model in which the day‑ahead power purchase vector p_da is decided, and any shortfall is immediately covered by the gas boiler (treated as a fixed‑cost recourse). The second is a two‑stage DRCC model that adds a real‑time recourse decision: after the day‑ahead schedule is fixed, the actual electricity price and the realized shortfall determine how much gas boiler capacity to activate, allowing the BRP to hedge against high imbalance prices. Both models remain linear‑objective, convex‑constraint problems and can be solved with standard MILP solvers.

Numerical experiments use data from the Danish day‑ahead market, a 10 MW electric boiler, and a 1 MW gas backup. Only 30 residual samples are used to build the empirical distribution; θ is varied between 0.1 and 0.5, and α is set to 5 %. Results show that the single‑stage DRCC reduces heat‑demand violation frequency by roughly 40 % compared with a deterministic scheduler and by about 10 % relative to a conventional chance‑constrained model that assumes a fixed error distribution. When the second‑stage recourse is included, total daily operating cost drops up to 34 % because the model can strategically shift electric‑boiler charging to low‑price periods and limit expensive gas‑boiler activation during price spikes.

Key contributions are: (i) a Wasserstein‑ball based ambiguity set that enables robust decision‑making with scarce forecast‑error data; (ii) a CVaR‑based convex reformulation of the heat‑balance chance constraint; (iii) integration of real‑time rebound cost into a two‑stage DRCC, demonstrating substantial cost savings under price volatility. Limitations include the assumption of temporally independent errors and a fixed gas‑boiler cost that does not capture fuel‑price dynamics. Future work could extend the ambiguity set to multivariate, time‑correlated errors, incorporate stochastic fuel prices, and develop online updating schemes for real‑time data streams.