# Nonlinear Maximum Power Point Tracking Control Method for Wind Turbines Considering Dynamics
## 3 Wind Turbine Inertial Response Time  
The response time of using electronic components to control current and electromagnetic torque is far less than that of wind turbine drive-train dynamics. Both the moment of inertia and the wind variations are the main factors leading to the power loss of MPPT. The wind turbine inertial response time \(\tau\) is defined as the time interval between two steady-state speeds when the shaft speed reaches to 63.2% of the total change, which reveals the effect of wind turbine inertia on power loss of MPPT. The inertial response time \(\tau\) is able to be obtained by the numerical method and analytical method. The numerical method refers to perturbing the wind speed at each operating point and measuring the resulting variations in the generator speed. The analytical method refers to using mathematics to obtain the theoretical formula of \(\tau\). The analytical method is based on assumptions 6-8:  
* The response time of realizing the expected electromagnetic torque is neglected. * The equivalent stiffness is infinite. * The equivalent damping \(c_{\rm eq}\) is far less than \(J^{\prime}{}_{\rm r}+J_{\rm g}\). Under assumption 7, the equivalent rotor speed will be synchronous with the generator speed: \[\omega_{\rm r}^{\prime}=\omega_{\rm g}\] (18) Under assumption 6-8, substituting Equation (18) into Equation (9) produces: \[T_{\rm m}^{\prime}-T_{\rm e}=(J_{\rm r}^{\prime}+J_{\rm g})\frac{\mathrm{d} \omega_{\rm g}}{\mathrm{d}t}\] (19) In the conventional MPPT method, the optimal steady electromagnetic torque is [11]: \[T_{\rm e}=k_{\rm opt}\omega_{\rm g}^{2}\] (20)\[k_{\rm opt}=\frac{\pi\rho R^{5}C_{\rm pmax}}{2\lambda_{\rm opt}^{3}n^{3}} \tag{21}\]  
where \(C_{\rm pmax}\) is the maximum power coefficient; \(\lambda_{\rm opt}\) is the optimal TSR. Then:  
\[\frac{\rho\pi R^{2}v^{3}C_{\rm p}(\lambda,\beta)}{2}-k_{\rm opt}\omega_{\rm g}^ {3}=(\frac{J_{\rm r}}{n^{2}}+J_{\rm g})\omega_{\rm g}\frac{\mathrm{d}\omega_{ \rm g}}{\mathrm{d}t} \tag{22}\]  
Equation (22) can be written as:  
\[\frac{\mathrm{d}\omega_{\rm g}}{\mathrm{d}t}=g(\omega_{\rm g},v)=\frac{\rho\pi R ^{2}v^{3}C_{\rm p}(\lambda,\beta)-2k_{\rm opt}n^{2}\omega_{\rm g}^{3}}{2(J_{ \rm r}+n^{2}J_{\rm g})\omega_{\rm g}} \tag{23}\]  
Linearized model of Equation (22) can be obtained using the Taylor expansion [21]:  
\[\delta\bigg{(}\frac{\mathrm{d}\omega_{\rm g}}{\mathrm{d}t}\bigg{)}=\frac{ \partial g(\omega_{\rm g},v)}{\partial v}\delta v+\frac{\partial g(\omega_{ \rm g},v)}{\partial\omega_{\rm g}}\delta\omega_{\rm g} \tag{24}\]  
where \(\delta\) and \(\partial\) is the total differential symbol and the partial differential symbol, respectively. Under assumptions 6-8, taking the Laplace transform of Equation (24) produces a first-order model at the working point \((\omega_{0},\ v_{0},\ \lambda_{0}=\omega_{0}R/v_{0})\):  
\[\frac{\delta\omega_{\rm g}(s)}{\delta v(s)}=\frac{K_{\rm g}}{1+\tau s} \tag{25}\]  
where:  
\[K_{\rm g}=\frac{\rho\pi R^{2}\omega_{0}v_{0}^{2}\bigg{(}3C_{\rm P}(\lambda_{0 })-\lambda_{0}\frac{\partial C_{\rm P}}{\partial\lambda}\bigg{|}_{\lambda= \lambda_{0}}\bigg{)}}{\rho\pi R^{2}v_{0}^{3}\bigg{(}C_{\rm P}(\lambda_{0})- \lambda_{0}\frac{\partial C_{\rm P}}{\partial\lambda}\bigg{|}_{\lambda= \lambda_{0}}\bigg{)}+4k_{\rm opt}\omega_{0}^{3}} \tag{26}\]  
\[\tau=\frac{2\big{(}\frac{J_{\rm r}}{n^{2}}+J_{\rm g}\big{)}\omega_{0}^{2}}{ \rho\pi R^{2}v_{0}^{3}\bigg{(}C_{\rm P}(\lambda_{0})-\lambda_{0}\frac{ \partial C_{\rm P}}{\partial\lambda}\bigg{|}_{\lambda=\lambda_{0}}\bigg{)}+4k_ {\rm opt}\omega_{0}^{3}} \tag{27}\]  
where \(K_{\rm g}\) is also the partial of \(\omega_{\rm g}\) with respect to \(v\).