# Robust Control Examples Applied to a Wind Turbine Simulated Model
## 2 Wind Turbine Simulated Model
Note that the benchmark simulator considered in this work does not include possible nonlinear gearbox dynamics, as addressed e.g., in [26; 27; 28]. However, the data-driven approach proposed in this study could be able to include this further nonlinearity for the design of the control solutions. Moreover, the generator/converter dynamics are described as a first-order transfer function, as highlighted by Equation (7):  
\[\frac{\tau_{g}(s)}{\tau_{r}(s)}=\frac{\alpha_{gc}}{s+\alpha_{gc}} \tag{7}\]  
\(s\) being the Laplace operator, \(1/\alpha_{gc}\) is the time constant of the generator/converter, whilst the power \(P_{g}\) produced by the generator is given by Equation (8):\[P_{g}(t)=\eta_{g}\,\omega_{g}\,\tau_{g}(t) \tag{8}\]  
with \(\eta_{g}\) denoting the efficiency of the generator. More details regarding the considered simulator are in [25]. Under these assumptions, the complete state-space description of the wind turbine model has the form of Equation (9):  
\[\left\{ \begin{array}{lcl}\dot{x}_{c}(t)&=&f_{c}\left(x_{c}(t),u(t)\right)\\ y(t)&=&x_{c}(t)\end{array} \right. \tag{9}\]  
where \(u(t)=\big{[}\beta(t),\,\tau_{g}(t)\big{]}^{T},y(t)=\big{[}P_{g}(t),\,\omega_{g }(t)\big{]}^{T}\), and \(x_{c}(t)=\big{[}P_{g}(t),\,\omega_{g}(t)\big{]}^{T}\) are the control inputs, the monitored output measurements, and the state vector, respectively, as shown in Figure 1. \(P_{g}(t)\) is the generator power measurement, whilst \(f_{c}\left(\cdot\right)\) represents the continuous-time nonlinear function that will be approximated via discrete-time models from \(N\) sampled data \(u_{k}\) and \(y_{k}\), with the sample index \(k=1,\,2,\,\ldots\,N\), as presented in Section 3. The model parameters, and the map \(C_{p}\left(\beta,\,\lambda\right)\) are chosen in order to represent a realistic wind turbine plant [25]. Note that Section 4.2 will analyze the reliability and robustness properties of the developed controllers when parameter variations and measurement errors are considered. This investigation will rely on the Monte Carlo tool, since the control behavior and the tracking capabilities depend on both the model-reality mismatch effects and the input-output uncertainty levels. Therefore, this analysis will be performed by describing the parameters of the wind turbine model simulator as Gaussian stochastic processes with average values corresponding to the nominal ones summarized in Table 1[25]. Moreover, the input and output measurements available from the wind turbine simulator are assumed to be acquired via sensors [25] that introduce additive Gaussian noise processes with zero mean and standard deviation values summarized in Table 2.  
Note that the parameters summarized in Table 2 are the values implemented in the wind turbine benchmark simulator that has been used in this work for the validation of the proposed control solutions. These values represent the uncertainty and disturbance effects of a realistic wind turbine deployment, as modeled by the wind turbine benchmark simulator presented and developed in [25]. On the other hand, this benchmark model set-up uses a predefined wind speed sequence \(v(t)\) consisting of real measured wind data of a wind park from 0 to 4400 s. Figure 2 highlights that this wind speed covers the range from 5 to 20 m/s, with a few spikes at 25 m/s, which is a good coverage of normal operational for a wind turbine. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Variable** & \(v(t)\) & \(\omega_{r}\) & \(\omega_{g}\) & \(\tau_{g}\) & \(P_{g}\) & \(\beta\) \\ \hline
**Std. Dev. Value** & 0.5 m/s & 0.025 rad/s & 0.05 rad/s & 90 Nm & \(10^{3}\) W & 0.2 deg \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Sensor standard deviation values used in the wind turbine simulator [25].