Optimal deterministic reservoir operations in continuous time

In 1946, Masse discussed the optimal operation of a single reservoir for hydroelectric energy production. He obtained his results by both economic reasoning and rigorous mathematical derivations using a generalized form of the calculus of variations. In this article, the procedure of Masse is generalized to cases where: (1) the benefits from release are not related directly to the release but rather to the discharge at a downstream point; (2) there may be several points downstream where an objective is to be achieved; and (3) there are several reservoirs. Masse and Varlet found that the optimal strategy is the one that maintains the marginal value of the release constant in time whenever the reservoir is neither full nor empty. It is shown rigorously here that, in the general case, for a strategy to be optimal, the "memory-integrated future marginal value" of the release must be constant. This result is generalized to a system of several reservoirs and illustrated on a simplified description of the Seine river basin upstream of Paris, France.