Profit-maximizing operation and valuation of hydroelectric plant: A new solution to the Koopmans problem

Koopmans's approach to hydroelectric storage is reformulated for competitive profit maximization, with electricity priced by time of use. Duality methods of linear programming are applied to the problems of operation and rental valuation of a hydro plant and its river. Both problems are approached by using time-dependent shadow pricing of water, and if the given market price for electricity (p) is a continuous function of time, then the shadow price function for water (ψ) is shown to be unique. The two prices (ψ and p) determine not only the optimal water storage policy but also the marginal values of the plant's capacities, defined as derivatives of the operating profit. In particular, the unit reservoir rent equals the total positive variation of the water value (ψ) over the cycle. Profit-imputed values of the river flow and of the hydro capacities (reservoir and turbine) are therefore definite - unlike Koopmans's cost-imputed values, derived from fuel savings in a mixed hydrothermal system. The marginal capacity values can be used to determine the optimum investment. The model applies also to other natural flows, e.g., water supply and geothermal or tidal energy.