Management of water resource systems in the presence of uncertainties by nonlinear approximation techniques and deterministic sampling

Two methods of approximate solution are developed for T-stage stochastic optimal control (SOC) problems, aimed at obtaining finite-horizon management policies for water resource systems. The presence of uncertainties, such as river and rain inflows, is considered. Both approaches are based on the use of families of nonlinear functions, called “one-hidden-layer networks” (OHL networks), made up of linear combinations of simple basis functions containing parameters to be optimized. The first method exploits OHL networks to obtain an accurate approximation of the cost-to-go functions in the dynamic programming procedure for SOC problems. The approximation capabilities of OHL networks are combined with the properties of deterministic sampling techniques aimed at obtaining uniform samplings of high-dimensional domains. In the second method, admissible solutions to SOC problems are constrained to take on the form of OHL networks, whose parameters are determined in such a way to minimize the cost functional associated with SOC problems. Exploiting these tools, the two methods are able to cope with the so-called “curse of dimensionality,” which strongly limits the applicability of existing techniques to high-dimensional water resources management in the presence of uncertainties. The theoretical bases of the two approaches are investigated. Simulation results show that the proposed methods are effective for water resource systems of high dimension.