# Analyse dynamique et contrôle linéaire quadratique Gaussien d'une éolienne

• Claire LINART

Student thesis: Master typesMaster in Mathematics

### Abstract

Several theoretical parts are presented in this thesis: the wind turbine's description, the stochastic processes and the linear quadratic gaussian control (LQG). In addition, a linearized model of the wind turbine is derived and its dynamical properties are analyzed. The last chapter is devoted to numerical simulations. To obtain the linear system of the wind turbine, we must study the wind model. The wind speed is a signal that can be defined by the sum of a mean value and a turbulent component. We obtain a continuum of systems depending on the wind speed, which may vary between 4 and 25 m/s. Each system is made of a mechanical part and a second part representing the disturbance (the wind). The inputs are the optimal pitch, the generator torque and the disturbance. The system has six state components: the angular speed of the wind turbine, the generator angular speed, the torque in the shaft, the pitch, the wind speed and the wind acceleration. These states represent the differences between their actual values and their steady-state values. The output for each system is the generator angular speed. These systems are completely observable but not completely controllable. We can only control the mechanical part of the wind turbine. Moreover, each system is stable, whatever the wind speed. To analyze the wind turbine's behavior, the state trajectories are numerically shown to tend to zero. In order to improve this stability property, we use LQG control for computing an LQ-optimal state feedback control law K and a optimal state estimator fain L, given by a Kalman filter. The weighing matrices are chosen to improve the stabilization. In the last section, a heuristic method is described and numerically implemented for computing the state feedback and the input injection gain matrices. The latter are obtained by computing average values of LQG gain matrices. One advantage of this method is that the matrices K and L are applicable in an interval of wind speeds.
Date of Award 2009 French Joseph Winkin (Supervisor), Vincent Piefort (Jury) & Michel KINNAERT (Jury)

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