The AC Power Flow Equations are fundamental to several aspects of power systems operations. However, they constitute a system of multivariate quadratic equations and are hard to deal with in a theoretically sound manner. In this work, we study the power flow equations from a differential perspective. By analyzing regions over which the power flow Jacobian is non-singular, we develop techniques to find operationally relevant solutions to the power flow equations, or certify the non-existence of these solutions (a red flag for the system operator). Further, we obtain conditions on the injections that guarantee the existence of a solution to the power flow equations satisfying operational constraints. We discuss various applications of these results to power systems operations and present numerical results on a set of IEEE benchmark networks.

(Joint work with Steven Low, Michael Chertkov and Konstantin Turitsyn)