Geometry of feasible injection region of power networks

We investigate the problem of power flow and its implications to the optimization in power networks. To understand how to solve these optimization problems, we look at the injection region of power networks. The injection region of a network 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, we show that the injection region with voltage magnitude, line loss constraints, line flow constraints and certain bus power constraints has the same set of Pareto optimal points as its convex hull. The set of Pareto-optimal points are of interest since these are the the optimal solutions to the minimization of a increasing convex function 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, a lossless cycle with a chord and certain combinations of these networks. The convex hull is of interest since they correspond to optimizing linear functions.