  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.
