# Robust Control Examples Applied to a Wind Turbine Simulated Model
## 2 Wind Turbine Simulated Model
In particular, the main signals available from the wind turbine simulator and used for control purpose are listed in the following:  
\(\beta\): blade pitch angle (deg);  
\(\omega_{g}\): generator/converter speed (rad/s);  
\(P_{g}\): generator power \(P_{g}\) (W);  
\(\omega_{r}\): rotor speed (rad/s);  
\(\tau_{r}\): requested torque (N m);  
\(\tau_{g}\): generator torque (N m);  
\(\tau_{zero}\): aerodynamic torque (N m);  
\(v\): wind speed (m/s);  
The aerodynamic model defining the aerodynamic torque provides the \(\tau_{zero}(t)\) signal, which is a nonlinear function of the wind speed \(v(t)\), and it is not highlighted in Figure 1 since included in the 'Aerodynamic model' block. This measurement is very difficult to be acquired correctly, as described in [25]. The aerodynamic model reported in Figure 1 is described by the following relation:  
\[\tau_{zero}(t)=C_{p}\left(\beta_{r}(t),\,\lambda(t)\right)\,\frac{\rho\,A\,v^ {3}(t)}{2\,\omega_{r}(t)} \tag{1}\]  
where the variable \(\rho\) represents the air density, whilst \(A\) is the effective rotor area. Figure 1: Scheme of the wind turbine process. The pitch system implemented in the wind turbine benchmark as depicted in Figure 1 uses a second-order transfer function, represented by Equation (2):  
\[\frac{\beta_{r}(s)}{\beta(s)}=\frac{\omega_{n}^{2}}{s^{2}+2\,\chi\,\omega_{n}\,s +\omega_{n}^{2}} \tag{2}\]  
where \(\beta_{r}\) is the actuated pitch angle, which is exploited in Equation (1), whilst \(\beta\) is the controlled pitch angle. \(\omega_{n}\) is the natural frequency of the pitch actuator model, and \(\chi\) its damping ratio. Another important variable is represented by the so-called tip-speed ratio, which is defined as:  
\[\lambda(t)=\frac{\omega_{r}(t)\,R}{v(t)} \tag{3}\]  
with \(R\) the rotor radius. \(C_{p}(\cdot)\) represents the power coefficient, that is normally represented via a two-dimensional map [25]. The expression of Equation (1) allows the computation of the signal \(\tau_{zero}(t)\) (highlighted in Figure 1) by means of the estimated wind speed \(v(t)\) and the signals \(\beta(t)\) and \(\omega_{r}(t)\). Due to the uncertainty of the wind speed, the estimate of \(\tau_{zero}(t)\) is considered affected by an unknown measurement error, which justifies the robust approaches described in Section 3. Moreover, the nonlinearity represented by the expressions of Equations (1) and (3) motivates the required reliable and robust control approaches suggested in this work. The wind turbine simulator includes a two-mass model that is exploited to describe the drive-train system depicted in Figure 1, as shown by the following linear state-space representation [22]:  
\[\left[ \begin{array}{c}\dot{\omega}_{r}(t)\\ \dot{\omega}_{g}(t)\\ \dot{\theta}_{\Delta}(t)\end{array} \right]=\text{A}_{dt}\,\left[ \begin{array}[ ]{c}\omega_{r}(t)\\ \omega_{g}(t)\\ \theta_{\Delta}(t)\end{array} \right]+\text{B}_{dt}\,\left[ \begin{array}{c} \tau_{zero}(t)\\ \tau_{g}(t)\end{array} \right] \tag{4}\]  
where the matrices \(\text{A}_{dt}\) and \(\text{B}_{dt}\) are defined as:  
\[\text{A}_{dt}=\left[ \begin{array}{ccc}-\frac{\text{B}_{dt}-\text{B}_{r}}{I _{r}}&\frac{\text{B}_{dt}}{N_{g}\,I_{r}}&-\frac{\text{K}_{dt}}{I_{r}}\\ \frac{\text{q}_{tt}\,B_{dt}}{N_{g}\,I_{g}}&\frac{-\frac{\text{q}_{tt}\,B_{dt} }{N_{g}\,I_{g}}-\text{B}_{g}}{\frac{\text{q}_{tt}}{N_{g}\,I_{g}}}&\frac{\text{ q}_{tt}\,K_{dt}}{N_{g}\,I_{g}}\\ 1&-\frac{1}{N_{g}}&0\end{array} \right] \tag{5}\]  
and:  
\[\text{B}_{dt}=\left[ \begin{array}{ccc}-\frac{1}{I_{r}}&0\\ 0&-\frac{1}{I_{g}}\\ 0&0\end{array} \right] \tag{6}\]  
where \(J_{r}\) is the momentum of inertia of rotor shaft, \(K_{dt}\) is the torsion stiffness of the drive-train, \(B_{dt}\) is the torsion damping coefficient of the drive-train, \(B_{g}\) is the viscous friction of the generator shaft, \(N_{g}\) is the gear ratio, \(J_{g}\) is the moment of inertia of the generator shaft, \(\eta_{dt}\) is the efficiency of the drive-train, and \(\theta_{\Delta}\) is the torsion angle of the drive train.