Extended Ritz method for reservoir management over an infinite horizon

The management problem of water reservoirs can be formulated as a stochastic optimal control (SOC) problem, where the objective function is an aggregated cost that accounts for the interests acting in the water system (e.g. hydropower production, irrigation supply, etc.) and the design variable is the reservoirs release policy. Solving the SOC problem through stochastic dynamic programming is often impossible, since the numerical resolution of the Bellman equation is computationally prohibitive even for small reservoir networks. An approximate solution can be searched for by assuming a priori the family of function to which the control laws belong and replacing the SOC problem with a nonlinear programming one. Recently, a method based on this approach has been proposed in the literature, coupled with the use of nonlinear approximating networks to approximate the optimal control laws. This optimization method, called Extended RItz Method (ERIM), is suited for finite horizon SOC problems. However, management problems for environmental systems are spontaneously formulated over an infinite horizon, since the life time of the system is infinite. This paper thus presents an extension of the ERIM to the infinite horizon case. The algorithm that implements such method is tested on a numerical example where a 10-reservoirs network is optimized for hydropower production and irrigation supply.