Sensitivity analysis of nonlinear programs and differentiability properties of metric projections
SIAM Journal on Control and Optimization
Directional behaviour of optimal solutions in nonlinear mathematical programming
Mathematics of Operations Research
Pseudopower expansion of solutions of generalized equations and constrained optimization problems
Mathematical Programming: Series A and B
Mathematical study of very high voltage power networks II: the AC power flow problem
SIAM Journal on Applied Mathematics
Mathematical Study of Very High Voltage Power Networks I: The Optimal DC Power Flow Problem
SIAM Journal on Optimization
Using mixed-integer programming to solve power grid blackout problems
Discrete Optimization
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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 square root 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 and improved using the framework of very high voltage approximation.