# Adaptive Pitch Control of Variable-Pitch PMSG Based Wind Turbine
## 2 Model and Problem Formulation
### Aerodynamic Model  
The wind power extracted by a WT is represented as [34; 35]  
\[P_{\mathrm{w}}=\frac{1}{2}\rho\pi R^{2}V^{3}C_{\mathrm{p}}(\beta,\lambda) \tag{1}\] \[\lambda=\frac{R\omega_{\mathrm{m}}}{V} \tag{2}\]  
where \(\beta\) is the pitch angle, \(\rho\) is the air density, \(V\) is the wind speed, \(R\) is the radius of WT, \(C_{\mathrm{p}}\) is the power coefficient, \(\lambda\) is the tip speed ratio, and \(\omega_{\mathrm{m}}\) is the mechanical rotation speed. The \(C_{\mathrm{p}}\) can be defined as a function of \(\beta\) and \(\lambda\)  
\[C_{\mathrm{p}}=0.22(\frac{116}{\lambda_{i}}-0.4\beta-5)e^{\frac{- 12.5}{\lambda_{i}}} \tag{3}\] \[\frac{1}{\lambda_{i}}=\frac{1}{\lambda+0.08\beta}-\frac{0.035}{ \beta^{3}+1} \tag{4}\]  
A hydraulic/mechanical actuator can vary the blade pitch. The following first order linear model represents a simplified model of the dynamics:  
Figure 2: Configuration of a PMSG-WT.  
\[\dot{\beta}=-\frac{\beta}{\tau_{\beta}}+\frac{\beta_{\rm r}}{\tau_{\beta}} \tag{5}\]  
where \(\beta_{\rm r}\) is required pitch angle, and \(\tau_{\beta}\) is the actuator time constant.  
The state-space model of the PMSG-WT is given as [35]:  
\[\dot{x}=f(x)+g_{1}(x)u_{1}+g_{2}(x)u_{2}+g_{3}(x)u_{3} \tag{6}\]  
where  
\[f(x) = \left[\begin{array}{ccc}-\frac{\beta}{\tau_{\beta}}\\ -\frac{R_{\rm s}}{L_{\rm md}}i_{\rm md}+\frac{\omega_{\rm e}L_{\rm md}}{L_{\rm md }}i_{\rm mq}\\ -\frac{R_{\rm s}}{L_{\rm mq}}i_{\rm mq}-\frac{1}{L_{\rm mq}}\omega_{\rm e}(L_{ \rm md}i_{\rm md}+K_{\rm e})\\ \frac{1}{I_{\rm tot}}(T_{\rm e}-T_{\rm m}-T_{\rm f}-B\omega_{\rm m}) \end{array}\right],\] \[g_{1}(x) = [-\frac{\beta}{\tau_{\beta}}\quad 0\quad 0\quad 0]^{\rm T},\] \[g_{2}(x) = [0\quad\frac{1}{L_{\rm md}}\quad 0\quad 0]^{\rm T},\] \[g_{3}(x) = [0\quad\frac{1}{L_{\rm md}}\quad 0]^{\rm T},\] \[x = [\beta\quad i_{\rm md}\quad i_{\rm mq}\quad\omega_{\rm m}]^{\rm T},\] \[u = [u_{1},u_{2},u_{3}]^{T}=[\beta_{\rm r},V_{\rm md},V_{\rm mq}]^{ \rm T},\] \[y = [y_{1},y_{2},y_{3}]^{T}=[h_{1}(x),h_{2}(x),h_{3}(x)]^{T}=[\omega_ {\rm m},i_{\rm md},i_{\rm mq}]^{T}\]  
where \(x\in R^{4}\), \(u\in R^{3}\) and \(y\in R^{3}\) are state vector, input vector and output vector, respectively; \(f(x)\), \(g(x)\) and \(h(x)\) are smooth vector fields. \(V_{\rm md}\) and \(V_{\rm mq}\) are the \(\rm d,q\) axis stator voltages, \(i_{\rm md}\) and \(i_{\rm mq}\) are the \(\rm d,q\) axis stator currents, \(L_{\rm md}\) and \(L_{\rm mq}\) are \(\rm d,q\) axis stator inductances, \(R_{\rm s}\) is the stator resistance, \(p\) is the number of pole pairs, \(K_{\rm e}\) is the field flux given by the magnet, \(I_{\rm tot}\) is the total inertia of the drive train, \(B\) is the friction coefficient of the PMSG, \(\omega_{\rm e}(=p\omega_{\rm m})\) is the electrical generator rotation speed, and \(T_{\rm m}\), \(T_{\rm f}\) and \(T_{\rm e}\) are the WT mechanical torque, static friction torque and electromagnetic torque, respectively.  
The electromagnetic torque is expressed as:  
\[T_{\rm e}=p[(L_{\rm md}-L_{\rm md})i_{\rm md}i_{\rm mq}+i_{\rm mq}K_{\rm e}] \tag{7}\]