{
  "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.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.15496",
  "author_total": 5,
  "authors": [
    {
      "name": "Abdellah Bulaich Mehamdi"
    },
    {
      "name": "Wim van Ackooij"
    },
    {
      "name": "Luce Brotcorne"
    },
    {
      "name": "Stéphane Gaubert"
    },
    {
      "name": "Quentin Jacquet"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9f1qvpx4bvy6j.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.15496",
  "fulltext_url": null,
  "id": "9f1qvpx4bvy6j",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.15496",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.OC"
    ],
    [
      "openalex",
      "2208|Smart Grid Energy Management"
    ]
  ],
  "openalex": "W7213341962",
  "pdf_url": "https://arxiv.org/pdf/2609.15496",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-14",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9f1qvpx4bvy6j.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:29Z",
      "native_id": "2609.15496",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.15496"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "Joint Design of Time-of-Use Schedules and Price Levels: a Bilevel Model with Client Participation and Load Flexibility",
  "title_text": "Joint Design of Time-of-Use Schedules and Price Levels: a Bilevel Model with Client Participation and Load Flexibility",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:29Z",
  "url": "https://research.draconitic.com/paper/9f1qvpx4bvy6j",
  "venue": "arXiv",
  "versions": []
}