Nonsmooth optimization of hydrothermal problems

In this paper the authors present a necessary condition for minimum of a functional J(z)≔∫0TL(t,z(t),z′(t))dt in the case in which the function L is continuous but not of class C1. This situation arises in problems of optimization of hydrothermal systems with pumped-storage plants. In such problems, the function Lz′(t,z,·) is discontinuous in z′=0, which is the borderline point between the power generation zone (z′>0) and the pumping zone (z′<0). The problem can be naturally formulated in the framework of nonsmooth analysis, using the generalized (or Clarke's) gradient.