Nonsmooth optimization of hydrothermal problems

In this paper the authors present a necessary condition for minimum of a functional J ( z ) ≔ ∫ 0 T L ( t , z ( t ) , z ′ ( t ) ) d t in the case in which the function L is continuous but not of class C 1 . This situation arises in problems of optimization of hydrothermal systems with pumped-storage plants. In such problems, the function L z ′ ( 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 Clarkeʹs) gradient.