Generic Existence of Unique Lagrange Multipliers in AC Optimal Power Flow

TL;DR

This paper demonstrates that for AC optimal power flow problems, the linear independence constraint qualification is almost always satisfied, ensuring the existence and uniqueness of Lagrange multipliers at local optima, which simplifies analysis and solution methods.

Contribution

The paper proves that the linear independence constraint qualification holds generically for AC OPF problems, guaranteeing unique Lagrange multipliers without needing specific constraint qualifications.

Findings

Lagrange multipliers exist and are unique for almost all AC OPF solutions.

The linear independence constraint qualification is generically satisfied in AC OPF.

This result simplifies the theoretical understanding of optimality conditions in power systems.

Abstract

Solutions to nonlinear, nonconvex optimization problems can fail to satisfy the KKT optimality conditions even when they are optimal. This is due to the fact that unless constraint qualifications (CQ) are satisfied, Lagrange multipliers may fail to exist. Even if the KKT conditions are applicable, the multipliers may not be unique. These possibilities also affect AC optimal power flow (OPF) problems which are routinely solved in power systems planning, scheduling and operations. The complex structure -- in particular the presence of the nonlinear power flow equations which naturally exhibit a structural degeneracy -- make any attempt to establish CQs for the entire class of problems very challenging. In this paper, we resort to tools from differential topology to show that for AC OPF problems in various contexts the linear independence constraint qualification is satisfied almost certainly, thus effectively obviating the usual assumption on CQs. Consequently, for any local optimizer there generically exists a unique set of multipliers that satisfy the KKT conditions.