Quadratic Regularization of Bilevel Pricing Problems and Application to Electricity Retail Markets

TL;DR

This paper introduces a quadratic regularization approach to bilevel pricing problems in electricity markets, enhancing robustness and computational tractability by leveraging geometric properties and a novel heuristic method.

Contribution

The paper proposes a quadratically regularized model for customer choice in bilevel pricing, leading to a quadratic program with complementarity constraints, improving stability and solution methods.

Findings

The regularized model captures customer behavior with a polyhedral complex structure.

The proposed pivoting heuristic outperforms traditional nonlinear methods.

Numerical results demonstrate the approach's effectiveness in electricity retail scenarios.

Abstract

We consider the profit-maximization problem solved by an electricity retailer who aims at designing a menu of contracts. This is an extension of the unit-demand envy-free pricing problem: customers aim to choose a contract maximizing their utility based on a reservation bill and multiple price coefficients (attributes). A basic approach supposes that the customers have deterministic utilities; then, the response of each customer is highly sensitive to price since it concentrates on the best offer. A second classical approach is to consider logit model to add a probabilistic behavior in the customers' choices. To circumvent the intrinsic instability of the former and the resolution difficulties of the latter, we introduce a quadratically regularized model of customer's response, which leads to a quadratic program under complementarity constraints (QPCC). This allows to robustify the deterministic model, while keeping a strong geometrical structure. In particular, we show that the customer's response is governed by a polyhedral complex, in which every polyhedral cell determines a set of contracts which is effectively chosen. Moreover, the deterministic model is recovered as a limit case of the regularized one. We exploit these geometrical properties to develop a pivoting heuristic, which we compare with implicit or non-linear methods from bilevel programming, showing the effectiveness of the approach. Throughout the paper, the electricity retailer problem is our guideline, and we present a numerical study on this application case.