Uniqueness of power flow solutions using monotonicity and network topology

This article establishes sufficient conditions for the uniqueness of AC power flow solutions via the monotonic relationship between real power flow and the phase angle difference. More specifically, we prove that the P-Θ power flow problem has at most one solution for any acyclic or GSP graph. In addition, for arbitrary cyclic power networks, we show that multiple distinct solutions cannot exist under the assumption that angle differences across the lines are bounded by some limit related to the maximal girth of the network. In these cases, a vector of voltage phase angles can be uniquely determined (up to an absolute phase shift) given a vector of real power injections within the realizable range. The implication of this result for the classical power flow analysis is that under the conditions specified above, the problem has a unique physically realizable solution if the phasor voltage magnitudes are fixed. We also introduce a series-parallel operator and show that this operator obtains a reduced and easier-to-analyze model for the power system without changing the uniqueness of power flow solutions.