Local minima of quadratic functionals and control of hydro-electric power stations

Abstract

We consider a control problem for a cascade of hydro-electric power stations, where some of the stations have reversible turbines. Our aim was to optimize the profit of power production satisfying restrictions on the water level in the reservoirs. From the mathematical point of view, this consists in minimizing an infinite-dimensional quadratic functional subject to cone constraints. Sufficient conditions of optimality for the abstract problem are derived and are then specialized for our problem. Noteworthy, the restrictions imposed on the power stations problem are in the form of control constraints and pure state constraints. The particular case of one power station is analyzed in detail, showing that reversible turbines always improve the profit.