# Monotonicity Between Phase Angles and Power Flow and Its Implications for the Uniqueness of Solutions

- SangWoo Park, IEOR, UC Berkeley, Berkeley, California, United States
- Richard Zhang, IEOR, UC Berkeley, Berkeley, California, United States
- Javad Lavaei, IEOR, UC Berkeley, Berkeley, California, United States
- Ross Baldick, Department of Electrical and Computer Engineering, The University of Texas at Austin, Austin, Texas, United States

**Abstract**This paper establishes sufficient conditions for the uniqueness of power flow solutions in an AC power system via the monotonic relationship between real power flow and the phase angle difference. More specifically, we prove that strict monotonicity holds if the angle difference is bounded by the steady-state stability limit in a power system with a series-parallel topology, or if transmission losses are sufficiently low. In both cases, a vector of voltage phase angles can be uniquely determined (up to an absolute phase shift) given a vector of active power injections within the realizable range. The implication of this result for 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 tightly controlled.