The Optimal Power Flow Operator: Theory and Computation

Abstract

Optimal power flow (OPF) problems are mathematical programs to determine how to distribute power over networks subject to power flow and operational constraints. In this article, we treat an OPF problem as an operator that maps user demand to generated power, and allow the problem parameters to take values in some admissible set. We formalize this operator theoretic approach, define and characterize restricted parameter sets under which the mapping has a singleton output, independent binding constraints, and is differentiable. We show that for any power network, these analytical properties hold under almost all operating conditions and can thus be relied upon in applications. We further provide a closed-form expression for the Jacobian matrix of the OPF operator and describe how various derivatives can be computed using a recently proposed scheme based on homogenous self-dual embedding. In contrast to related work in the optimization literature, our results have a clear physical interpretation.

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