  Many problems in power systems involve optimizing a certain objective
function subject to power flow equations and engineering constraints. A
long-standing challenge in solving them is the nonconvexity of their feasible
sets. In this paper, we propose an analytical method to construct the convex
restriction of the feasible set for AC power flows in radial networks. The
construction relies on simple geometrical ideas and is explicit, in the sense
that it does not involve solving other complicated optimization problems. We
also show that the construct restrictions are in some sense maximal, that is,
the best possible ones. Optimization problems constrained to these sets are not
only simpler to solve but also offer feasibility guarantee for the solutions to
the original OPF problem. Furthermore, we present an iterative algorithm to
improve on the solution quality by successively constructing a sequence of
convex restricted sets and solving the optimization on them. The numerical
experiments on the IEEE 123-bus distribution network show that our method finds
good feasible solutions within just a few iterations and works well with
various objective functions, even in situations where traditional methods fail
to return a solution.
