Geometry of power flows in tree networks

We investigate the problem of power flow and its relationship to optimization in tree networks. We show that due to the tree topology of the network, the general optimal power flow problem simplifies greatly. Our approach is to look at the injection region of the power network. The injection region is simply the set of all vectors of bus power injections that satisfy the network and operation constraints. The geometrical object of interest is the set of Pareto-optimal points of the injection region, since they are the solutions to the minimization of increasing functions. We view the injection region as a linear transformation of the higher dimensional power flow region, which is the set of all feasible power flows, one for each direction of each line. We show that if the voltage magnitudes are fixed, then the injection region becomes a product of two-bus power flow regions, one for each line in the network. Using this decomposition, we show that under the practical condition that the angle difference across each line is not too large, the set of Pareto-optimal points of the injection region remains unchanged by taking the convex hull. Therefore, the optimal power flow problem can be convexified and efficiently solved. This result improves upon earlier works since it does not make any assumptions about the active bus power constraints. We also obtain some partial results for the variable voltage magnitude case.