Management of a hydropower system via convex duality

We consider a stochastic hydroelectric power plant management problem in discrete time with arbitrary scenario space. The inflow to the system is some stochastic process, representing the precipitation to each dam. The manager can control how much water to turbine from each dam at each time. She would like to choose this in a way which maximizes the total profit from the initial time 0 to some terminal time T. The total profit 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, we derive a dual problem to the management problem. This dual problem turns out to be simple to solve in the case where the profit rate process is a martingale or submartingale with respect to the filtration modeling the information of the dam manager. In the case where we only consider a finite number of scenarios, solving the dual problem is computationally more efficient than the primal problem.