  The AC power flow equations are fundamental in all aspects of power systems
planning and operations. They are routinely solved using Newton-Raphson like
methods. However, there is little theoretical understanding of when these
algorithms are guaranteed to find a solution of the power flow equations or how
long they may take to converge. Further, it is known that in general these
equations have multiple solutions and can exhibit chaotic behavior. In this
paper, we show that the power flow equations can be solved efficiently provided
that the solution lies in a certain set. We introduce a family of convex
domains, characterized by Linear Matrix Inequalities, in the space of voltages
such that there is at most one power flow solution in each of these domains.
Further, if a solution exists in one of these domains, it can be found
efficiently, and if one does not exist, a certificate of non-existence can also
be obtained efficiently. The approach is based on the theory of monotone
operators and related algorithms for solving variational inequalities involving
monotone operators. We validate our approach on IEEE test networks and show
that practical power flow solutions lie within an appropriately chosen convex
domain.
