The Optimal Power Flow Operator: Theory and Computation

TL;DR

This paper introduces an operator-theoretic framework for optimal power flow problems, enabling new insights into sensitivity, robustness, and solution uniqueness without auxiliary variables.

Contribution

It formalizes the OPF as an operator, characterizes conditions for unique and differentiable solutions, and derives a closed-form Jacobian expression, advancing theoretical understanding.

Findings

Operator mapping has singleton output under certain parameters.

Derived a closed-form Jacobian for the OPF operator.

Provided a scheme for computing derivatives using homogenous self-dual embedding.

Abstract

Optimal power flow problems (OPFs) are mathematical programs used to determine how to distribute power over networks subject to network operation constraints and the physics of power flows. In this work, we take the view of treating an OPF problem as an operator which maps user demand to generated power, and allow the network parameters (such as generator and power flow limits) to take values in some admissible set. The contributions of this paper are to 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. In contrast to related results in the optimization literature, we do not rely on introducing auxiliary slack variables. Indeed, our approach provides results that have a clear interpretation with respect to the power network under study. 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. Our framework of treating a mathematical program as an operator allows us to pose sensitivity and robustness questions from a completely different mathematical perspective and provide new insights into well studied problems.