Geometry of Injection Regions of Power Networks

TL;DR

This paper studies the injection regions in power networks, characterizing their geometric properties and implications for optimal power flow, especially focusing on tree and cycle network topologies.

Contribution

It provides a geometric analysis of injection regions, showing equivalence of injection region and its convex hull Pareto-front in tree networks under various constraints.

Findings

Injection region equals energy conservation set without operation constraints.

In tree networks, injection region and convex hull share the same Pareto-front under certain constraints.

For non-tree networks, the convex hull of the injection region is characterized for specific cycle configurations.

Abstract

We investigate the constraints on power flow in networks and its implications to the optimal power flow problem. The constraints are described by the injection region of a network; this is the set of all vectors of power injections, one at each bus, that can be achieved while satisfying the network and operation constraints. If there are no operation constraints, we show the injection region of a network is the set of all injections satisfying the conservation of energy. If the network has a tree topology, e.g., a distribution network, we show that under voltage magnitude, line loss constraints, line flow constraints and certain bus real and reactive power constraints, the injection region and its convex hull have the same Pareto-front. The Pareto-front is of interest since these are the the optimal solutions to the minimization of increasing functions over the injection region. For non-tree networks, we obtain a weaker result by characterize the convex hull of the voltage constraint injection region for lossless cycles and certain combinations of cycles and trees.