Mathematical Study of very High Voltage Power Networks III: The Optimal AC Power Flow Problem
Résumé
This paper shows how to apply the perturbation theory for nonlinear programming problems to the study of the optimal power flow problem. The latter is the problem of minimizing losses of active power over a very high voltage power networks. In this paper, the inverse of the reference voltage of the network is viewed as a small parameter. We call this scheme the very high voltage approximation. After some proper scaling, it is possible to formulate a limiting problem, that does not satisfy the Mangasarian-Fromovitz qualification hypothesis. Nevertheless, it is possible to obtain under natural hypotheses the second order expansion of losses and first order expansion of solutions. The latter is such that the computation of the active and reactive parts are decoupled. We also obtain the high order expansion of the value function, solution and Lagrange multiplier, assuming that interactions with the ground are small enough. Finally we show that the classical direct current approximation may be justified using the framework of very high voltage approximation.