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The AC power flow equations describe the steady-state behavior of the powergrid. While many algorithms have been developed to compute solutions to thepower flow equations, few theoretical results are available characterizing whensuch solutions exist, or when these algorithms can be guaranteed to converge.In this paper, we derive necessary and sufficient conditions for the existenceand uniqueness of a power flow solution in balanced radial distributionnetworks with homogeneous (uniform $R/X$ ratio) distribution lines. We studythree distinct solution methods: fixed point iterations}, convex relaxations,and energy functions we show that the three algorithms successfully find asolution if and only if a solution exists. Moreover, all three algorithmsalways find the unique high-voltage solution to the power flow equations, theexistence of which we formally establish.}%, which can be formally defined asthe high-voltage solution -- any other solution of the power flow equations hassmaller or equal voltage magnitudes at every bus. At this solution, we provethat (i) voltage magnitudes are increasing functions of the reactive powerinjections, (ii) the solution is a continuous function of the injections, and(iii) the solution is the last one to vanish as the system is loaded past thefeasibility boundary.

High-voltage solution in radial power networks:Existence, properties and equivalent algorithms