Mathematical study of very high voltage power networks II: The AC power flow problem

This paper is the second in a series dedicated to the mathematical study of very high voltage power networks, based on an asymptotic analysis, in which the square root of one of the small parameters is the inverse of the reference voltage of the network. We call this scheme the very high voltage approximation. We show that the active and reactive parts of the limit problem are decoupled and therefore justify the so-called direct current approximation. We give an analytic expansion for the solution of the problem and prove the convergence of two decoupled numerical algorithms. In some specific cases we show that the solutions of the equations may be interpreted as critical points of some potentials.