On the Fokker-Planck Equation for Stochastic Hybrid Systems: Application to a Wind Turbine Model

This paper presents some recent results concerning a class of continuous-time Markov processes called "stochastic hybrid systems". These processes describe the evolution of a multidimensional hybrid-state dynamical system subject to Gaussian white noise inputs. After a brief recall of the formalism, we state the generalized Fokker-Planck equation, which is a partial differential equation satisfied by the probability density function of the system. As an illustration, we consider a variable-speed wind turbine, with a switching controller that combines stall regulation and pitch control. For a given value of the mean wind speed, the stationary distribution of the state variables is computed numerically. This truly dynamical analysis of the system yields a complete probabilistic characterization of the uncertain power output, which is much more accurate than the usual static analysis