PDE methods for stochastic dynamic optimisation: an application to wind power generation with energy storage

A mixed financial/physical Partial Differential Equation (PDE) can optimise the joint earnings of a single wind power generator (WPG) and a generic energy storage device (ESD). Physically, the PDE includes constraints on the ESD’s capacity, efficiency and maximum speeds of charge and discharge. There is a mean-reverting daily stochastic cycle for WPG power output. Physically, energy can only be produced or delivered at finite rates. All suppliers must commit hourly to a finite rate of delivery C, which is a continuous control variable that
is changed hourly. Financially, we assume heavy ‘system balancing’ penalties in continuous time, for deviations of output rate from the commitment C. Also, the electricity spot price follows a meanreverting stochastic cycle with a strong evening peak, when system balancing penalties also peak. Hence the economic goal of the WPG plus ESD, at each decision point, is to maximise expected Net Present Value (NPV) of all earnings (arbitrage) minus the NPV of all expected system balancing penalties, along all financially/physically feasible future paths through state space. Given the capital costs for the various combinations of the physical parameters, the design and operating rules for a WPG plus ESD in a finite market may be jointly optimisable.