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Advanced Cutting-Plane Algorithms for ACOPF

Daniel Bienstock, Matias Villagra

TL;DR

This paper addresses tight bounds for the ACOPF problem by introducing a linear cutting-plane framework that outer-approximates PSD cones, avoiding full SDP solves while enabling warm-start and multi-period solves. It clarifies the relationship between SDP formulations, proving that the real SDP relaxation is at least as strong as the complex one and detailing the (mis)perceived equivalences between SDP-R and SDP-CR, alongside a practical algorithm with clique decompositions and eigenvector/projection cuts. The proposed disciplined approach yields scalable, numerically robust relaxations and demonstrates promising preliminary results on large-scale benchmarks, including robustness to data perturbations. The work has practical significance for scalable, reliable lower-bounding in massive ACOPF problems, potentially enabling tighter bounds in settings where traditional nonlinear solvers struggle.

Abstract

We propose a disciplined, numerically stable, and scalable approach to SDP relaxations of the ACOPF problem based on linear cutting-planes. Our method can be warm-started and, owing to its linear nature, enables the computation of tight and accurate bounds for large-scale multi-period relaxations -- well beyond what nonlinear convex solvers can achieve. Preliminary experiments show promising results when benchmarked against state-of-the-art bounds on PGLIB instances.

Advanced Cutting-Plane Algorithms for ACOPF

TL;DR

This paper addresses tight bounds for the ACOPF problem by introducing a linear cutting-plane framework that outer-approximates PSD cones, avoiding full SDP solves while enabling warm-start and multi-period solves. It clarifies the relationship between SDP formulations, proving that the real SDP relaxation is at least as strong as the complex one and detailing the (mis)perceived equivalences between SDP-R and SDP-CR, alongside a practical algorithm with clique decompositions and eigenvector/projection cuts. The proposed disciplined approach yields scalable, numerically robust relaxations and demonstrates promising preliminary results on large-scale benchmarks, including robustness to data perturbations. The work has practical significance for scalable, reliable lower-bounding in massive ACOPF problems, potentially enabling tighter bounds in settings where traditional nonlinear solvers struggle.

Abstract

We propose a disciplined, numerically stable, and scalable approach to SDP relaxations of the ACOPF problem based on linear cutting-planes. Our method can be warm-started and, owing to its linear nature, enables the computation of tight and accurate bounds for large-scale multi-period relaxations -- well beyond what nonlinear convex solvers can achieve. Preliminary experiments show promising results when benchmarked against state-of-the-art bounds on PGLIB instances.
Paper Structure (20 sections, 8 theorems, 21 equations, 2 tables, 1 algorithm)

This paper contains 20 sections, 8 theorems, 21 equations, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Our contributions
  3. Problem Formulation and SDP Relaxations
  4. ACOPF
  5. SDP relaxations for ACOPF
  6. SDP-C
  7. SDP-R
  8. SDP-CR
  9. Brief review on prior work on SDP relaxations
  10. On the relationship between SDP-R and SDP-CR
  11. Our disciplined approach to SDPs
  12. SDP Cuts
  13. Eigenvector-based cuts
  14. Projection-based cuts
  15. Cut computations
  16. ...and 5 more sections

Key Result

Lemma 2.2

Let $X \in \mathbb{C}^{n}$. Then the quadratic form $\mathbb{C}^{n} \to \mathbb{C}$ defined as $u \mapsto u^* X u$ is real-valued.

Theorems & Definitions (17)

  • Remark 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Definition 3.1
  • Theorem 3.2
  • proof
  • Corollary 3.3
  • proof
  • Theorem 3.4
  • proof
  • ...and 7 more