Pricing Discrete and Nonlinear Markets With Semidefinite Relaxations

TL;DR

This paper introduces a semidefinite relaxation-based pricing scheme for nonconvex markets with discrete and nonlinear constraints, notably improving price signals and reducing opportunity costs in electricity market problems.

Contribution

It develops a novel SDP relaxation approach for pricing in nonconvex markets, extending duality and envelope theorem to derive demand prices, and demonstrates improved performance over traditional methods.

Findings

SDP relaxations are often tight in practice.

The proposed pricing scheme reduces opportunity costs by 46% on average.

The framework applies effectively to both DC and AC unit commitment problems.

Abstract

Nonconvexities in markets with discrete decisions and nonlinear constraints make efficient pricing challenging, often necessitating subsidies. A prime example is the unit commitment (UC) problem in electricity markets, where costly subsidies are commonly required. We propose a new pricing scheme for nonconvex markets with both discreteness and nonlinearity, by convexifying nonconvex structures through a semidefinite programming (SDP) relaxation and deriving prices from the relaxation's dual variables. When the choice set is bounded, we establish strong duality for the SDP, which allows us to extend the envelope theorem to the value function of the relaxation. This extension yields a marginal price signal for demand, which we use as our pricing mechanism. We demonstrate that under certain conditions-for instance, when the relaxation's right hand sides are linear in demand-the resulting lost opportunity cost is bounded by the relaxation's optimality gap. This result highlights the importance of achieving tight relaxations. The proposed framework applies to nonconvex electricity market problems, including for both direct current and alternating current UC. Our numerical experiments indicate that the SDP relaxations are often tight, reinforcing the effectiveness of the proposed pricing scheme. Across a suite of IEEE benchmark instances, the lost opportunity cost under our pricing scheme is, on average, 46% lower than that of the commonly used fixed-binary pricing scheme.