Management of a hydropower system via convex duality

We consider the problem of managing a hydroelectric power plant system. The system consists of N hydropower dams, which all have some maximum production capacity. The inflow to the system is some stochastic process, representing the precipitation to each dam. The manager can control how much water to release from each dam at each time. She would like to choose this in a way which maximizes the total revenue from the initial time 0 to some terminal time T. The total revenue of the hydropower dam system depends on the price of electricity, which is also a stochastic process. The manager must take this price process into account when controlling the draining process. However, we assume that the manager only has partial information of how the price process is formed. She can observe the price, but not the underlying processes determining it. By using the conjugate duality framework of Rockafellar, we derive a dual problem to the problem of the manager. This dual problem turns out to be simple to solve in the case where the price process is a martingale or submartingale with respect to the filtration modelling the information of the dam manager.