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Facet-Defining Inequalities for the Angle-Based DC Optimal Transmission Switching Formulation

Behnam Jabbari-Marand, Adolfo R. Escobedo

TL;DR

This work tackles DC-OTS by tightening angle-difference bounds in angle-based big-$M$ formulations, a key factor in solution tractability for topology decisions. It develops an extended formulation to describe the convex hull of an angle-based relaxation over a cycle and derives cycle-induced path-based valid inequalities (C-PVIs) that are facet-defining when projected onto the original variables. The approach leverages a lifted, disjunctive representation with variables $\underline{z}_{mn}, \overline{z}_{mn}, \overline{\zeta}_{mn}$ to capture shorter and longer paths $\underline{\rho}_{mn}, \overline{\rho}_{mn}$ and tightens angle-difference bounds beyond traditional cycle-based or path-based inequalities. The resulting convex-hull descriptions enable tighter LP relaxations and offer a polynomial-time separation route, with future work planned to map CVIs into $(\theta,y)$-space and benchmark performance on large-scale DC-OTS instances.

Abstract

The switching of transmission lines can significantly improve the economic and operational efficiency of power systems. The Direct-Current Optimal Transmission Switching (DC-OTS) problem provides a formal framework for minimizing power generation costs by reconfiguring the transmission network topology under a linearized power flow model. DC-OTS is typically formulated as a mixed-integer linear program that incorporates disjunctive constraints to capture the required relationships between certain variables via big-M parameters. More specifically, these parameters represent upper bounds on voltage angle differences across non-operational transmission lines. In practice, overly conservative (and arbitrary) bounds tend to be used. The belief is that tightening these values requires the solution of the computationally intractable longest path problem. This work challenges that view through a novel polyhedral analysis of the angle-based DC-OTS formulation. We construct an extended formulation for the convex hull of an angle-based relaxation and derive facet-defining inequalities that tighten angle-difference bounds.

Facet-Defining Inequalities for the Angle-Based DC Optimal Transmission Switching Formulation

TL;DR

This work tackles DC-OTS by tightening angle-difference bounds in angle-based big- formulations, a key factor in solution tractability for topology decisions. It develops an extended formulation to describe the convex hull of an angle-based relaxation over a cycle and derives cycle-induced path-based valid inequalities (C-PVIs) that are facet-defining when projected onto the original variables. The approach leverages a lifted, disjunctive representation with variables to capture shorter and longer paths and tightens angle-difference bounds beyond traditional cycle-based or path-based inequalities. The resulting convex-hull descriptions enable tighter LP relaxations and offer a polynomial-time separation route, with future work planned to map CVIs into -space and benchmark performance on large-scale DC-OTS instances.

Abstract

The switching of transmission lines can significantly improve the economic and operational efficiency of power systems. The Direct-Current Optimal Transmission Switching (DC-OTS) problem provides a formal framework for minimizing power generation costs by reconfiguring the transmission network topology under a linearized power flow model. DC-OTS is typically formulated as a mixed-integer linear program that incorporates disjunctive constraints to capture the required relationships between certain variables via big-M parameters. More specifically, these parameters represent upper bounds on voltage angle differences across non-operational transmission lines. In practice, overly conservative (and arbitrary) bounds tend to be used. The belief is that tightening these values requires the solution of the computationally intractable longest path problem. This work challenges that view through a novel polyhedral analysis of the angle-based DC-OTS formulation. We construct an extended formulation for the convex hull of an angle-based relaxation and derive facet-defining inequalities that tighten angle-difference bounds.
Paper Structure (3 sections, 4 equations, 1 figure)

This paper contains 3 sections, 4 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Model and related work
  3. Preliminaries

Figures (1)

  • Figure 1: Cycle partitions used by each set of VIs: (a) Cycle-based VIs: solid edges form subset $S$ whose total weight exceeds half of the cycle weight; dashed edges are $C \setminus S$; (b) Path- and cycle-based VIs: solid edges are the longer path on the cycle by total weight (subset $S$); dashed edges are the shorter path (subset $C \setminus S$).