Activate the Dual Cones: A Tight Reformulation of Conic ACOPF Constraints

TL;DR

This paper introduces a reformulation of the dual conic ACOPF problem that eliminates conic constraints by leveraging their tightness at optimality, enabling more efficient optimization and bounding.

Contribution

It presents a tight reformulation of the dual conic ACOPF constraints into a non-conic form, simplifying the problem and facilitating future GPU-accelerated methods.

Findings

Reformulation recovers the same dual objective as conic solvers.

Numerical experiments on benchmark systems validate the approach.

The new formulation offers performance benefits over traditional conic methods.

Abstract

By exploiting the observed tightness of dual rotated second-order cone (RSOC) constraints, this paper transforms the dual of a conic ACOPF relaxation into an equivalent, non-conic problem where dual constraints are implicitly enforced through eliminated dual RSOC variables. To accomplish this, we apply the RSOC-based Jabr relaxation of ACOPF, pose its dual, and then show that all dual RSOC constraints must be tight (i.e., active) at optimality. We then construct a reduced dual maximization problem with only non-negativity constraints, avoiding the explicit RSOC inequality constraints. Numerical experiments confirm that the tight formulation recovers the same dual objective values as a mature conic solver (e.g., MOSEK via PowerModels) on various PGLib benchmark test systems (ranging from 3- to 1354-buses). The proposed formulation has useful performance benefits, compared with its conic counterpart, and it allows us to define a bounding function which provides a guaranteed lower bound on system cost. While this paper focuses on demonstrating the correctness and validity of the proposed structural simplification, it lays the groundwork for future GPU-accelerated first-order optimization methods which can exploit the unconstrained nature of the proposed formulation.