  The optimal power flow (OPF) problem is one of the most fundamental problems
in power system operations. The non-linear alternating current (AC) power flow
equations that model different physical laws (together with operational
constraints) lay the foundation for the feasibility region of the OPF problem.
While significant research has focused on convex relaxations, which are
approaches to solve an OPF problem by enlarging the true feasibility region,
the opposite approach of convex restrictions offers valuable insights as well.
Convex restrictions, including polyhedral restrictions, reduce the true
feasible region to a convex region, ensuring that it contains only feasible
points. In this work, we develop a sequential optimization method that offers a
scalable way to obtain (bounds on) solutions to OPF problems for distribution
networks. To do so, we first develop sufficient conditions for the existence of
feasible power flow solutions in the neighborhood of a specific (feasible)
operating point in distribution networks, and second, based on these
conditions, we construct a polyhedral restriction of the feasibility region.
Our numerical results demonstrate the efficacy of the sequential optimization
method as an alternative to existing approaches to obtain (bounds on) solutions
to OPF problems for distribution networks. By construction, the optimization
problems can be solved in polynomial time and are guaranteed to have feasible
solutions.
