Convexity of Solvability Set of Power Distribution Networks

TL;DR

This paper proves the convexity of the solvability set in certain power distribution networks, simplifying optimization tasks, and identifies conditions under which convexity is preserved or lost.

Contribution

It establishes convexity of the full solvability set for homogeneous tree networks and of the real solvability set for all tree and resistive networks, highlighting the impact of network homogeneity.

Findings

Convexity holds for homogeneous tree networks with uniform r/x ratio.

Convexity is lost when the network is non-homogeneous.

Almost homogeneous networks retain a large convex subset of the solvability set.

Abstract

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.