Joint Design of Time-of-Use Schedules and Price Levels: a Bilevel Model with Client Participation and Load Flexibility
Abstract
We study the joint design of a time-of-use (ToU) tariff (the partition of the day into pricing blocks and the price level attached to each block) by a profit-maximizing electricity retailer facing flexible clients who may refuse the offer in favour of an outside option. The problem is modeled as a bilevel program in which each client, at the lower level, reshapes its consumption under pointwise bounds and a fixed daily energy total, and simultaneously chooses a participation level penalized by a quadratic regularizer. We prove the client's best response is computable in loglinear time. We derive two equivalent single-level reformulations: a closed-form continuous one tackled via stochastic gradient ascent (using a continuous reparametrization of block start times and durations) and a mixed-integer nonlinear program solved via dedicated solvers. The stochastic ascent scheme provides Clarke-critical accumulation points almost surely, in a scalable way. On a case study from the French retail electricity market with $5000$ simulated households, the gradient method matches a commercial mixed-integer solver on the pointwise problem to within 0.3% of the optimal profit while running two to eight times faster on $100$ representative clients. Restricting unconstrained pointwise prices to admissible ToU schedules yields an estimated profit loss of 1.3%- 13.8% for two-period tariffs and 0.6%-6.6% for three-period tariffs on the instances considered.
Related
- First-Order Bilevel Optimization for Non-truthful Auction Design
- Network Flexibility Design Under Endogenous Uncertainty
- Optimizing Flexibility and Commitments: Partially-Adaptive Multi-Stage Matrix Assembly Layout Design Problem
- The data geometry of masking diffusion: Certified-optimal schedules via unmasking growth complexity
- Accelerated Concatenation of Stepsize Schedules for Smooth Strongly Convex Gradient Descent
BibTeX
@misc{mehamdi2026joint,
title = {{Joint Design of Time-of-Use Schedules and Price Levels: a Bilevel Model with Client Participation and Load Flexibility}},
author = {Abdellah Bulaich Mehamdi and Wim van Ackooij and Luce Brotcorne and Stéphane Gaubert and Quentin Jacquet},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.15496},
eprint = {2609.15496},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.15496},
}
RIS
TY - UNPB
TI - Joint Design of Time-of-Use Schedules and Price Levels: a Bilevel Model with Client Participation and Load Flexibility
AU - Abdellah Bulaich Mehamdi
AU - Wim van Ackooij
AU - Luce Brotcorne
AU - Stéphane Gaubert
AU - Quentin Jacquet
PY - 2026
DA - 2026/09/14
PB - arXiv
DO - 10.48550/arxiv.2609.15496
UR - https://arxiv.org/abs/2609.15496
AB - We study the joint design of a time-of-use (ToU) tariff (the partition of the day into pricing blocks and the price level attached to each block) by a profit-maximizing electricity retailer facing flexible clients who may refuse the offer in favour of an outside option. The problem is modeled as a bilevel program in which each client, at the lower level, reshapes its consumption under pointwise bounds and a fixed daily energy total, and simultaneously chooses a participation level penalized by a quadratic regularizer. We prove the client's best response is computable in loglinear time. We derive two equivalent single-level reformulations: a closed-form continuous one tackled via stochastic gradient ascent (using a continuous reparametrization of block start times and durations) and a mixed-integer nonlinear program solved via dedicated solvers. The stochastic ascent scheme provides Clarke-critical accumulation points almost surely, in a scalable way. On a case study from the French retail electricity market with $5000$ simulated households, the gradient method matches a commercial mixed-integer solver on the pointwise problem to within 0.3% of the optimal profit while running two to eight times faster on $100$ representative clients. Restricting unconstrained pointwise prices to admissible ToU schedules yields an estimated profit loss of 1.3%- 13.8% for two-period tariffs and 0.6%-6.6% for three-period tariffs on the instances considered.
ER -