stat.MLSep 30, 2026
SaveLearning to Price Electricity for Optimal Demand Response
Organizations: Stanford University · National Laboratory of the Rockies · Prime Coalition · Eaton Research Labs
Abstract
There is considerable interest in using time-varying electricity prices to shape consumer demand response, and better align energy demand with renewable production. However, optimal prices generally vary over time in response to complex signals such as weather forecasts, sunrise/sunset times, and day-of-week patterns; and existing methods are not able to make efficient use of such rich contextual information. Here, we propose a neural-network-based algorithm for contextual energy pricing, modeling pricing as a Stackelberg game and leveraging a mean-field solution representation from Mehrabi et al.~(2024). The approach learns constrained mappings from contextual features to feasible price signals. We validate our approach by simulating the energy grid in several US cities, and show that incorporating contextual information can considerably increase the value of the demand response programs.
Figures & tables
Figure 1 : Illustration of the value of context-aware pricing. Left: Optimal price profiles for sunny and cloudy weather conditions obtained by minimizing a grid-risk objective based on net demand. The objective combines the norm and the sum of squared differences between consecutive periods, capturing overall demand magnitude and temporal fluctuations. Right: Net demand on a single (sunny) day when both price profiles from the left panel are applied. The sunny-day optimal price produces a more stable demand profile, while the cloudy-day price, applied to the same sunny day, yields poor performance with a pronounced duck curve.
Figure 2 : Sequential contextual price learning. The operator announces the day-ahead price using the forecasted context and the current pricing rule. After operation, HEMS responses, realized consumption, and renewable supply reveal the actual net-demand signal, which is used to update the network weights and price prototypes. Better context forecasts lead to prices that are better aligned with realized system conditions and improve demand management.
Algorithm 1 Day-Ahead Contextual Prototype-Based Price Learning
Figure 3 : Role of weather forecasts in contextual price learning. Forecasted weather variables are used to generate context-dependent price signals prior to real-time operation. Households respond by optimizing device schedules (e.g., HVAC, BESS) based on the announced prices and anticipated conditions, yielding an operation plan. Forecast errors may lead to discrepancies between planned and realized consumption, as devices operate under realized conditions rather than the planned consumption trajectory. This introduces forecast-induced variability into demand feedback and subsequent price updates.
Figure 4 : Comparison of price signals generated by the feedback-only and contextual learning algorithms under an alternating weather regime. The cluster-based soft classifier algorithm (with the number of clusters ) learns distinct price profiles corresponding to the two day types. In contrast, the feedback-only algorithm converges to a single price signal that is approximately the average of the two contextual prices.
Figure 5 : Comparison of peak demand shaving (PDS) achieved by the feedback-only and contextual pricing algorithms under an alternating weather regime. The shaded area represents the PDS gain achieved by contextual pricing over the feedback-only benchmark. The contextual pricing algorithm consistently attains greater peak demand reduction.
Figure 6 : Comparison of peak demand shaving (PDS) across different contextual learning algorithms trained over 90 days of weather data in Denver, CO. Both the projection layer and interior point methods show high volatility, with occasional negative PDS values. The proximal gradient method performs moderately well, while the cluster-based soft classifier is the only algorithm that consistently achieves positive PDS.
Figure 7 : Violin plot comparing peak demand shaving (PDS) across the four contextual learning algorithms. The cluster-based soft classifier exhibits the highest average PDS with the smallest spread.
Figure 8 : Weather profiles of San Francisco, Atlanta, and Phoenix in July 2022. Panel (a) shows the daily temperature profiles; panel (b) shows the daily solar irradiance profiles. Solid lines represent the mean daily profiles across the month, while the shaded regions indicate the corresponding day-to-day variability ( standard deviation).
Figure 9 : Daily peak demand shaving achieved by the cluster-based soft classifier pricing algorithm for San Francisco, Atlanta, and Phoenix in summer 2022.
Figure 10 : Average hourly price signals generated by the cluster-based soft classifier algorithm for San Francisco, Atlanta, and Phoenix in July 2022. Solid lines represent the mean hourly price profiles, and shaded regions represent standard deviation across days.
Appendix figures & tables4 assets
Supplementary material from the paper’s appendix.
Appendix
Figure 11 : Constrained output learning architecture with cluster-based soft classifier. Network maps input features to a probability assignment over clusters, with each cluster representing a archetypal system condition. Constraints are applied directly to the cluster-level outputs, ensuring that the final price signal—computed as a convex combination of the cluster-level outputs—remains feasible.
Figure 12 : Constrained output learning architecture with a projection layer. Unlike projected gradient descent, which adjusts training variables iteratively, the projection layer approach enforces constraints directly on the network outputs by projecting them onto the feasible set.
| Method | Convergence | Adaptiveness | Efficiency |
|---|---|---|---|
| Projection layer | x Leads to over-parametrization | x Requires differentiable projection operator | ✓Utilizes various first-order methods |
| Interior point | ✓Imposes constraints actively | x Requires differentiable norm as constraints | x Causes numerical instability near boundary |
| Proximal gradient | x Provides only lower bound via weak dual | ✓Allows for constraints to be non-differentiable | ✓Combines gradient descent ascent (GDA) with proximal operators |
| Soft classifier | ✓Imposes constraints via projected gradient descent; Admits universal approximation | ✓Allows for constraints to be non-differentiable | ✓Utilizes various first-order methods |
Table 1 : Advantages and Limitations to Different Methods
Figure 13 : Comparison of net peak demand under flat-rate and context-aware demand response pricing across cities in 2022. The 45-degree line indicates identical net peak demand under the two pricing schemes. Points below the line correspond to days on which the proposed learning framework achieves lower net peak demand than the flat-rate scheme.