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Pricing Discrete and Nonlinear Markets With Semidefinite Relaxations

Cheng Guo, Lauren Henderson, Ryan Cory-Wright, Boshi Yang

TL;DR

A new pricing scheme for nonconvex markets with both discreteness and nonlinearity is proposed, by convexifying nonconvex structures through a semidefinite programming (SDP) relaxation and deriving prices from the relaxation's dual variables.

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.

Pricing Discrete and Nonlinear Markets With Semidefinite Relaxations

TL;DR

A new pricing scheme for nonconvex markets with both discreteness and nonlinearity is proposed, by convexifying nonconvex structures through a semidefinite programming (SDP) relaxation and deriving prices from the relaxation's dual variables.

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.
Paper Structure (25 sections, 12 theorems, 46 equations, 3 tables)

This paper contains 25 sections, 12 theorems, 46 equations, 3 tables.

Table of Contents

  1. Introduction
  2. Literature Review
  3. Notation
  4. Background
  5. Semidefinite Relaxations of Mixed Binary Quadratic Problems
  6. Semidefinite Relaxation of DCUC Problems
  7. UC Pricing with Semidefinite Relaxation
  8. Envelope Theorem for Semidefinite Relaxation
  9. Envelope Theorem for DCUC Pricing
  10. Pricing for ACUC
  11. SDP Relaxation for ACUC
  12. Envelope Theorem for ACUC Pricing
  13. Bounding Lost Opportunity Cost
  14. Numerical Experiments
  15. DCUC Case Study
  16. ...and 10 more sections

Key Result

Proposition 1

If $\mathcal{P}^\rm{MBQP}(\theta)$ is feasible and satisfies Assumption ass: bound, then strong duality holds between the SDP relaxation $\mathcal{P}^\rm SDP-bd(\theta)$ and its dual $\mathcal{P}^{\rm SDP-dual}(\theta)$.

Theorems & Definitions (22)

  • Definition 1: Primal value function of $\mathcal{P}^{\rm SDP-bd}(\theta)$
  • Proposition 1: Strong duality for $\mathcal{P}^\rm SDP-bd(\theta)$
  • Definition 2: Dual value function of $\mathcal{P}^{\rm SDP-bd}(\theta)$
  • Theorem 1: Envelope theorem for $\mathcal{P}^\rm SDP-bd$
  • Corollary 1: Vector envelope theorem for $\mathcal{P}^{\rm SDP-bd}$
  • Definition 3: Strengthened DCUC SDP relaxation and its dual
  • Proposition 2: Expression for DCUC pricing
  • Definition 4: SDP-based mechanism for DCUC
  • Remark 1
  • Proposition 3: Finite nondifferentiability of $V^{\rm UC-SDP}(D_{kt})$
  • ...and 12 more