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Comparative Evaluation of SDP, SOCP, and QC Convex Relaxations for Large-Scale Market-Based AC Optimal Power Flow

Ata Keskin

TL;DR

This paper implements DCOPF, Shor's SDP relaxation (complex and real-valued forms), chordal SDP, Jabr's SOCP relaxation, and QC relaxations in a unified, solver-native framework using the MOSEK Fusion API, eliminating modeling overhead present in high-level frameworks such as CVXPY.

Abstract

The alternating current optimal power flow (ACOPF) problem is central to modern power system operations, determining how electricity is generated and transmitted to maximize social welfare while respecting physical and operational constraints. However, the nonlinear and non-convex nature of AC power flow equations makes finding globally optimal solutions computationally intractable for large networks. Convex relaxations - including semidefinite programming (SDP), second-order cone programming (SOCP), and quadratic convex (QC) formulations - provide tractable alternatives that can yield provably optimal or near-optimal solutions under appropriate conditions. This paper presents a comprehensive comparative study of multiple ACOPF relaxations applied to market-based welfare maximization. We implement DCOPF, Shor's SDP relaxation (complex and real-valued forms), chordal SDP, Jabr's SOCP relaxation, and QC relaxations in a unified, solver-native framework using the MOSEK Fusion API, eliminating modeling overhead present in high-level frameworks such as CVXPY. To address the practical challenge of missing or overly conservative angle difference bounds required by QC relaxations, we employ quasi-Monte Carlo sampling with Sobol sequences to empirically estimate tighter bounds. We evaluate these relaxations on subnetworks of varying sizes derived from the ARPA-E dataset, systematically comparing solution quality, runtime, and memory consumption. Our results demonstrate the trade-offs between relaxation tightness and computational efficiency, providing practical guidance for selecting appropriate formulations based on network scale and solution requirements.

Comparative Evaluation of SDP, SOCP, and QC Convex Relaxations for Large-Scale Market-Based AC Optimal Power Flow

TL;DR

This paper implements DCOPF, Shor's SDP relaxation (complex and real-valued forms), chordal SDP, Jabr's SOCP relaxation, and QC relaxations in a unified, solver-native framework using the MOSEK Fusion API, eliminating modeling overhead present in high-level frameworks such as CVXPY.

Abstract

The alternating current optimal power flow (ACOPF) problem is central to modern power system operations, determining how electricity is generated and transmitted to maximize social welfare while respecting physical and operational constraints. However, the nonlinear and non-convex nature of AC power flow equations makes finding globally optimal solutions computationally intractable for large networks. Convex relaxations - including semidefinite programming (SDP), second-order cone programming (SOCP), and quadratic convex (QC) formulations - provide tractable alternatives that can yield provably optimal or near-optimal solutions under appropriate conditions. This paper presents a comprehensive comparative study of multiple ACOPF relaxations applied to market-based welfare maximization. We implement DCOPF, Shor's SDP relaxation (complex and real-valued forms), chordal SDP, Jabr's SOCP relaxation, and QC relaxations in a unified, solver-native framework using the MOSEK Fusion API, eliminating modeling overhead present in high-level frameworks such as CVXPY. To address the practical challenge of missing or overly conservative angle difference bounds required by QC relaxations, we employ quasi-Monte Carlo sampling with Sobol sequences to empirically estimate tighter bounds. We evaluate these relaxations on subnetworks of varying sizes derived from the ARPA-E dataset, systematically comparing solution quality, runtime, and memory consumption. Our results demonstrate the trade-offs between relaxation tightness and computational efficiency, providing practical guidance for selecting appropriate formulations based on network scale and solution requirements.
Paper Structure (36 sections, 1 theorem, 77 equations, 6 figures, 2 algorithms)

This paper contains 36 sections, 1 theorem, 77 equations, 6 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Related Work
  3. Problem Formulation
  4. Convex Relaxations of ACOPF
  5. DC Approximation
  6. Shor's SDP Relaxation
  7. Shor's SDP Relaxation (Real-Valued Version)
  8. Chordal SDP Relaxation
  9. Jabr's Relaxation (SOCP)
  10. Quadratic Convex (QC) Relaxation
  11. Global Sampling
  12. Local Sampling
  13. Implementation
  14. Results and Discussion
  15. Dataset and Experimental Design
  16. ...and 21 more sections

Key Result

Theorem 1

Let $G = (\mathcal{V}, \mathcal{E})$ be a chordal graph. Let $\mathcal{C} = \{C_1, \dots, C_k\}$ be the collection of maximal cliques of $G$. Let $\mathbf{W} = (W_{vw})_{v,w\in\mathcal{V}} \in \mathbb{C}^{|\mathcal{V}|\times|\mathcal{V}|}$ be a matrix with rows and columns corresponding to vertices

Figures (6)

  • Figure 1: An example of an unfavorable decomposition. Here two large blocks overlap almost entirely. Decomposing them into separate cliques would introduce a large number of redundant linking constraints. Instead, one should perform the initial chordal decomposition, detect the excessive overlap, and merge the cliques back together.
  • Figure 2: Solve time and overhead (total runtime minus solve time).
  • Figure 3: Peak memory usage and economic welfare.
  • Figure 4: Current phasor error and thermal limit violations (RMS).
  • Figure 5: Impact of the number of QMC samples ($2^4$–$2^{12}$) on QC relaxation performance, showing changes in market welfare and current phasor accuracy.
  • ...and 1 more figures

Theorems & Definitions (1)

  • Theorem 1: fukuda2001