发表机构
École polytechnique, Institut Polytechnique de Paris; EDF Lab Paris-Saclay; Institut National de Recherche en Informatique et en Automatique (Inria)（巴黎综合理工学院，巴黎理工学院; 法国电力集团萨克雷实验室; 法国国家信息与自动化研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本研究提出双层模型联合优化分时电价时段与价格，考虑客户灵活负荷和参与选择，证明客户响应可对数线性求解，并通过随机梯度上升法高效逼近最优利润，案例显示与商业求解器误差小于0.3%。
AI 中文摘要
我们研究了一个利润最大化的电力零售商如何联合设计分时电价（ToU）方案（即一天中定价时段的划分以及每个时段对应的价格水平），该零售商面对的是灵活的客户，这些客户可能拒绝该方案而选择外部选项。该问题被建模为一个双层规划，其中每个客户在底层根据逐点约束和固定的日总用电量重新调整其消费，同时通过一个受二次正则化项惩罚的参与水平来选择是否参与。我们证明了客户的最优响应可以在对数线性时间内计算。我们推导出两个等价的单层重构：一个闭式连续形式，通过随机梯度上升法求解（使用时段开始时间和持续时间的连续重新参数化），以及一个通过专用求解器求解的混合整数非线性规划。随机上升方案几乎必然地提供Clarke临界累积点，且具有可扩展性。在一个来自法国零售电力市场的案例研究中，包含5000个模拟家庭，梯度方法在逐点问题上与商业混合整数求解器的结果匹配，最优利润差异在0.3%以内，而在100个代表性客户上运行速度快2到8倍。将无约束的逐点价格限制为可行的ToU方案，在所考虑的实例中，对于两时段电价，估计利润损失为1.3%-13.8%，对于三时段电价，利润损失为0.6%-6.6%。
英文摘要
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.
