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

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.This work has been partially supported by the European Union Seventh Framework Programme [FP7-PEOPLE-2010-ITN] under Grant agreement No. 64735-SADCO, project FCT PTDC/EEA-CRO/116014/2009 and project FCOMP-01-0124-FEDER-028894, FCT PTDC/EEI-AUT/1450/2012. The authors are deeply grateful to the anonymous referee whose remarks were very helpful to improve the quality of the paper. Also our gratitude to our colleagues M.d.R. de Pinho and F.A.C.C. Fontes, for their suggestions and revision of the writing