# Adaptive Pitch Controller of a Large-Scale Wind Turbine Using Multi-Objective Optimization
## 3 Control Methodology
### Identification of Linear Models of the Wind Turbine  
A representative utility-scale multimegawatt turbine known as the \"NREL offshore 5-MW baseline wind turbine\" has been used in this work. This wind turbine is a conventional three-bladed upwind variable-speed variable-blade-pitch-to-feather-controlled turbine based on the REpower 5 MW commercial model [26]. The NREL 5-MW turbine model has been implemented by using software FAST version 8. Table 1 shows the properties for the NREL 5-MW Baseline Wind Turbine. More information on the parameters of the analyzed turbine can be found in [58; 59; 60].  
The proposed design is based on linear models. Due to the nonlinearity of the system, their approximated linear models vary considerably depending on the operational point. In this study, approximated linear models are identified at the six operation conditions considered according to the wind speed \(\mathrm{v_{w}}\) (12, 14, 16, 18, 20 and 22 m/s) within the nominal region (where \(\mathrm{\omega_{g_{rated}}}\) is 1173.7 rpm).  
The linear models are given by the transfer functions from (13) to (16), where \(\mathrm{G_{\omega g}(s)}\) is the transfer function relating the generator speed (\(\mathrm{\omega_{g}}\) ) in rpm to the pitch angle (\(\mathrm{\beta}\)) in degrees; \(\mathrm{Gd_{\omega g}(s)}\) is the transfer function relating the controlled variable \(\mathrm{\omega_{g}}\) to the wind speed (\(\mathrm{v_{w}}\)) in m/s as disturbance input; \(\mathrm{G_{\mathrm{x}}(s)}\) is the transfer function of the tower f-a displacement (\(\mathrm{x_{t}}\)) in m as output and the pitch angle (\(\mathrm{\beta}\)) as input, and \(\mathrm{Gd_{\mathrm{x}t}(s)}\) is the transfer function relating the controlled variable \(\mathrm{x_{t}}\) to the wind speed \(\mathrm{v_{w}}\).  
\[\mathrm{G_{\omega g}(s)}=\frac{\mathrm{K_{\beta 1}}}{\mathrm{T_{\beta 1}s+1}} \tag{13}\]  
\[\mathrm{Gd_{\omega g}(s)}=\frac{\mathrm{K_{w1}}}{\mathrm{T_{w1}s+1}} \tag{14}\]  
\[\mathrm{G_{\mathrm{x_{t}}}(s)}=\frac{\mathrm{K_{\beta 2}(\mathrm{Tz_{\beta 2}s+1 })}}{\mathrm{(T_{\beta 2}s+1)}\mathrm{(s^{2}+2\sigma_{\beta 2}s+\omega_{\beta 2}^{2})}} \tag{15}\]  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Property** & **Value** \\ \hline Rated Power & 5 MW \\ Cut-In, Rated Rotor Speed & 6.9 rpm, 12.1 rpm \\ Cut-In, Rated, Cut-Out Wind Speed & 3 m/s, 11.4 m/s, 25 m/s \\ Rotor Orientation, Configuration & Upwind, 3 Blades \\ Control & Variable Speed, Collective Pitch \\ Drivetrain & High Speed, Multiple-Stage Gearbox \\ Rated Generator Speed & 1173.7 rpm \\ Gearbox Ratio & 97.1 \\ Electrical Generator Efficiency & 94.4\% \\ Rotor, Hub Diameter & 126 m, 3 m \\ Hub Height & 90 m \\ Rotor Mass & 110,000 kg \\ Nacelle Mass & 240,000 kg \\ Tower Mass & 347,460 kg \\ \hline \hline \end{tabular}
\end{table}
Table 1: Properties of the NREL 5-MW Turbine and Drivetrain.  
\[\text{Gd}_{\text{x}_{\text{t}}}(\text{s})=\frac{\text{K}_{\text{w2}}(\text{T}_{ \text{zw2}}\text{s}+1)}{(\text{T}_{\text{w2}}\text{s}+1)\big{(}\text{s}^{2}+2 \sigma_{\text{w2}}\text{s}+\omega_{\text{w2}}^{2}\big{)}} \tag{16}\]  
Although the model parameters at the different operation points are not shown, the obtained open loop dynamics of all these models are stable.