# Nonlinear Maximum Power Point Tracking Control Method for Wind Turbines Considering Dynamics
## 2 Model of Wind Turbine Systems
### Dynamic Drive-Train  
The dynamics of the wind turbine drive-train can be modeled as the two-mass model by making the rotor of wind turbines and the gearbox equivalent to one-mass model on the side of the high-speed shaft, as shown in Figure 1. In this model, \(n\) is the gearbox ratio; \(J_{\rm r}\) is the moment inertia of rotor and \(J_{\rm r}^{\prime}=\frac{I_{\rm r}}{n^{2}}\) is the equivalent value; \(J_{\rm g}\) is the moment inertia of generator; \(T_{\rm m}\) is the aerodynamic torque of wind turbines and \(T_{\rm m}^{\prime}=\frac{T_{\rm m}}{n}\) is the equivalent aerodynamic torque; \(\omega_{\rm r}\) is the rotor speed of wind turbines and \(\omega_{\rm r}^{\prime}=n\omega_{\rm r}\) is the equivalent value; \(c_{\rm eq}\) and \(k_{\rm eq}\) represent the equivalent damping and stiffness of the two-mass model, respectively.  
The damping torque is mainly generated by the friction and is proportional to rotor speed. The stiffness torque is mainly generated by the torsion of the drive-train shafts and is proportional to the torsional angle of equivalent shafts. Under assumptions 4-5, the dynamics of the wind turbine drive-train is:  
\[\left\{\begin{array}{l}T_{\rm m}^{\prime}=J_{\rm r}^{\prime}\frac{{\rm d} \omega_{\rm r}^{\prime}}{{\rm d}t}+c_{\rm eq}\omega_{\rm r}^{\prime}+k_{\rm eq }(\theta_{\rm r}^{\prime}-\theta_{\rm g})\\ \frac{{\rm d}\theta_{\rm g}^{\prime}}{{\rm d}t}=\omega_{\rm r}^{\prime}\\ -T_{\rm e}=J_{\rm g}\frac{{\rm d}\omega_{\rm g}}{{\rm d}t}+c_{\rm eq}\omega_{ \rm g}+k_{\rm eq}(\theta_{\rm g}-\theta_{\rm r}^{\prime})\\ \frac{{\rm d}\theta_{\rm g}}{{\rm d}t}=\omega_{\rm g}\end{array}\right. \tag{9}\]  
where \(\theta_{\rm r}^{\prime}=n\theta_{\rm r}\) is the equivalent rotor angular displacement. \(\theta_{\rm g}\) is the generator angular displacement.  
The mechanical dynamics of the DFIG system is analyzed in the dynamics of the wind turbine drive-train since the shaft of the DFIG system is coupled to the drive-train, as shown in the third formula of Equation (9). The state-space form of Equation (9) is:  
\[\left\{\begin{array}{l}\dot{\mathbf{x}}=\mathbf{A}\mathbf{x}+Bu\\ \mathbf{y}=\mathbf{C}\mathbf{x}\end{array}\right. \tag{10}\]  
where:  
\[\mathbf{x}=\left[\omega_{\rm r}^{\prime},\omega_{\rm g},\theta_{\rm sh},T_{\rm m} ^{\prime}\right]^{\rm T}\!\!;\;u=T_{\rm e};\;\theta_{\rm sh}=\theta_{\rm r}^{ \prime}-\theta_{\rm g} \tag{11}\]  
Figure 1: Two-mass equivalent drive-train model.  
\[\mathbf{A}=\left[\begin{array}{cccc}-\frac{c_{\rm eq}}{J_{\rm r}^{\prime}}&0&- \frac{k_{\rm eq}}{J_{\rm r}^{\prime}}&\frac{1}{J_{\rm r}^{\prime}}\\ 0&-\frac{c_{\rm eq}}{J_{\rm g}}&\frac{k_{\rm eq}}{J_{\rm g}}&0\\ 1&-1&0&0\\ 0&0&0&0\end{array}\right]\mathbf{B}=\left[\begin{array}{c}0\\ -\frac{1}{J_{\rm g}}\\ 0\\ 0\end{array}\right] \tag{12}\]  
\[\mathbf{C}=\left[\begin{array}{cccc}1&0&0&0\\ 0&1&0&0\end{array}\right] \tag{13}\]  
The aerodynamic power extracted from the wind is:  
\[P_{\rm m}=0.5\rho\pi R^{2}v^{3}C_{\rm p}(\lambda,\beta) \tag{14}\]  
where \(\rho\) is the air density; \(R\) is the rotor radius; \(v\) is the wind speed; \(C_{\rm p}\) is the power coefficient; \(\beta\) is the collective blade-pitch angle; \(\lambda\) is the TSR; \(C_{\rm p}(\lambda,\beta)\) is used to show the value of \(C_{\rm p}\) mainly depends on \(\lambda\) and \(\beta\).  
The TSR is:  
\[\lambda=\frac{\omega_{\rm r}R}{v} \tag{15}\]  
The aerodynamic torque is:  
\[T_{\rm m}=\frac{P_{\rm m}}{\omega_{\rm r}}=0.5\rho\pi R^{3}v^{2}C_{\rm T}( \lambda,\beta) \tag{16}\]  
where:  
\[C_{\rm T}(\lambda,\beta)=\frac{C_{\rm p}(\lambda,\beta)}{\lambda} \tag{17}\]