Geometry of Power Flows and Optimization in Distribution Networks

We investigate the geometry of injection regions and its relationship to optimization of power flows in tree networks. The injection region is 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. If the voltage magnitudes are fixed, the injection region of a tree network can be written as a linear transformation of the product of two-bus injection 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. Moreover, the resulting convexified optimal power flow problem can be efficiently solved via semi-definite programming or second-order cone relaxations. These results improve upon earlier works by removing the assumptions on active power lower bounds. It is also shown that our practical angle assumption guarantees two other properties: 1) the uniqueness of the solution of the power flow problem and 2) the non-negativity of the locational marginal prices. Partial results are presented for the case when the voltage magnitudes are not fixed but can lie within certain bounds.