Distributionally Robust Scheduling of Electrified Heating Under Heat Demand Forecast Uncertainty

TL;DR

The paper tackles heat-demand forecast uncertainty in day-ahead scheduling for multi-MW electric boilers with storage by applying a Wasserstein-ball distributionally robust chance-constrained framework, anchored on the empirical forecast-error distribution with radius $\theta$. It develops single-stage and two-stage DRCCP formulations, yielding tractable LP reformulations via CVaR under optimal transport duality, and demonstrates substantial reductions in heat-demand violations and real-time gas activations while balancing electricity costs. A key finding is the existence of a critical Wasserstein radius (around $\theta \simeq 0.05$) that closely matches oracle performance with limited historical data; the two-stage model further mitigates price volatility by incorporating rebound costs into scheduling. Practically, the approach offers a data-driven, risk-aware method for BRPs to reliably monetize thermal flexibility under market dynamics and forecast errors.

Abstract

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.

Figures (6)

• Figure 1: Heat demand forecasts (blue line) versus actual heat demand realizations (red line). The red shaded area indicates upward deviations (i.e., realizations exceeding forecasts), while the blue shaded area indicates downward deviations (i.e., realizations falling below forecasts).
• Figure 2: Illustration of the scheduling cost (right) vs. the real-time average violation of the heat demand balance constraint (left axis). The figure shows that increasing values of robustness, depicted by $\theta$ lead to more robust operations, with fewer violations and higher costs.
• Figure 3: Boiler state of energy comparison under deterministic and DRCCP with $\theta=0.01$
• Figure 4: Total out-of-sample operation cost as a function of the Wasserstein radius $\theta$ for different numbers of training samples $N \in \{5, 10, 50, 100\}$. The bottom dashed line represents the oracle cost under perfect forecast information, while the top dashed line denotes the realized cost obtained without accounting for uncertainty. The intermediate curves show the DRCC-based scheduling performance, with a minimum cost around $\theta \simeq 0.05$ that is largely independent of $N$.
• Figure 5: Comparison of average electricity procurement and gas rebound costs for the deterministic, single-stage, and two-stage day-ahead scheduling policies as a function of the Wasserstein radius $\theta$.

Theorems & Definitions (2)

• Definition 1: Conditional Value at Risk rockafellar2000optimization
• Definition 2: Wasserstein Distance shafieezadeh2019regularization