  The solvability set of a power network - the set of all power injection
vectors for which the corresponding Power Flow equations admit a solution - is
central to power systems stability and security, as well as to the tightness of
Optimal Power Flow relaxations. Whenever the solvability set is convex, this
allows for substantial simplifications of various optimization and risk
assessment algorithms. In this paper we focus on the solvability set of power
distribution networks and prove convexity of the full solvability set (real and
reactive powers) for tree homogeneous networks with the same r/x ratio for all
elements. We also show this result can not be improved: once the network is not
homogeneous, the convexity is immediately lost. It is nevertheless the case
that if the network is almost homogeneous, a substantial practically-important
part of the solvability set is still convex. Finally, we prove convexity of
real solvability set (only real powers) for any tree network as well as for
purely resistive networks with arbitrary topology.
