# Nonlinear Maximum Power Point Tracking Control Method for Wind Turbines Considering Dynamics
## Design of the Improved MPPT Control Method
### Feed-Forward Control with the PI Gain Scheduling  
Previous research [11; 12] has already focused on the error feed-forward control based on aerodynamic torque and generator torque. However, they didn't figure out the method to design the proportional gain in the correction path. Meanwhile, the feed-forward control with the fixed gain certainly haa a good performsnce at a few working points due to the nonlinear dynamic performance of VSWTs under various wind conditions. The nonlinear problem has been solved in this paper. Gain scheduling control, feed-forward control, and the conventional OT control are combined to endow VSWTs with a good dynamic performance at most of the working points. To get better performance under small wind variations or nearly steady wind conditions, the integral control is added to the feed-forward path. Thus the new electromagnetic torque setting value is:  
\[T_{\mathrm{en}}^{\star}=k_{\mathrm{opt}}\hat{\omega}_{\mathrm{g}}^{2}-k_{ \mathrm{p}}(\hat{T}_{m}^{\prime}-k_{\mathrm{opt}}\hat{\omega}_{\mathrm{g}}^{2} )-k_{\mathrm{i}}\!\int(\hat{T}_{m}^{\prime}-k_{\mathrm{opt}}\hat{\omega}_{ \mathrm{g}}^{2})\mathrm{d}t \tag{37}\]  
where \(k_{\mathrm{p}}\), \(k_{\mathrm{i}}\) represent the proportional and integral gain in the feed-forward path, respectively. Under assumptions 6-8, using Taylor expansion mentioned in Equation (24) produces:  
\[\frac{\delta\hat{\omega}_{\mathrm{g}}(s)}{\delta v(s)}=\frac{K_{\mathrm{g}}}{ \tau}(1+k_{\mathrm{p}})\frac{s+k_{\mathrm{i}}/(1+k_{\mathrm{p}})}{s^{2}+ \left((1+k_{\mathrm{p}})/\tau+2c_{\mathrm{eq}}/(J_{\mathrm{r}}/n^{2}+J_{ \mathrm{g}})\right)\!s+k_{\mathrm{i}}/\tau} \tag{38}\]  
Substituting Equation (29) into Equation (38) produces:  
\[\frac{\delta\hat{\omega}_{\mathrm{g}}(s)}{\delta v(s)}=2K_{\mathrm{g}}\zeta \omega_{\mathrm{n}}\frac{s+\omega_{\mathrm{n}}/\zeta}{s^{2}+2\zeta\omega_where:  
\[\omega_{\rm n}=\sqrt{\frac{k_{\rm i}}{\tau}}\quad\zeta=\frac{(1+k_{\rm p})}{2} \sqrt{\frac{1}{\tau k_{\rm i}}} \tag{40}\]  
Amplitude-frequency characteristic of the system described by Equation (39) is:  
\[M(\omega)=K_{\rm g}\sqrt{\frac{4\zeta^{2}\omega_{\rm n}^{2}\omega^{2}+4\omega_{ \rm n}^{4}}{\omega^{4}+4\zeta^{2}\omega^{2}\omega_{\rm n}^{2}-2\omega^{2}\omega_ {\rm n}^{2}+\omega_{\rm n}^{4}}} \tag{41}\]  
Subsequently, the bandwidth \(\omega_{\rm bn}\) is:  
\[\omega_{\rm bn}^{2}=\bigg{(}\sqrt{\zeta^{4}-2\zeta^{2}+2}+1-\zeta^{2}\bigg{)} \omega_{\rm n}^{2}=k_{\rm bn}\omega_{\rm n}^{2} \tag{42}\]  
where \(k_{\rm bn}\) is defined as the transfer coefficient from \(\omega_{\rm bn}^{2}\) to \(\omega_{\rm n}^{2}\). The wind turbine nonlinear system is able to work as the second-order linear system does by regulating the gain \(k_{\rm p}\), \(k_{\rm i}\) nonlinearly to maintain the bandwidth \(\omega_{\rm bn}\) and the damping ratio \(\zeta\) to constant. The reference constant of \(\omega_{\rm bn}\) is supposed to be assigned considering the typical wind spectrum and the capacity of wind turbines. \(\omega_{\rm bn}\) is assigned to 0.6 rad s\({}^{-1}\) for a 5 MW wind turbine model [23]. The damping ratio \(\zeta\) is assigned to 0.707. The gains \(k_{\rm p}\), \(k_{\rm i}\) are:  
\[k_{\rm i}=\frac{\tau\omega_{\rm bn}^{2}}{k_{\rm bn}}\quad k_{\rm p}=\frac{2 \zeta\omega_{\rm bn}\tau}{\sqrt{k_{\rm bn}}}-1 \tag{43}\]  
The regulating shame can is also shown in Figure 6.  
When wind turbines start or large wind variations occur, the integral value in the feed-forward path will accumulate, which may result in high overshoot and system oscillations. Integral separation Algorithm 1 is adopted to solve that problem. Here list the rules of integral separation Algorithm 1:  
* If \(\big{|}\hat{T}_{m}^{\gamma}-k_{\rm opt}\hat{\omega}_{\rm g}^{2}\big{|}>\varepsilon\): the integral part will be separated. * If \(\big{|}\hat{T}_{m}^{\gamma}-k_{\rm opt}\hat{\omega}_{\rm g}^{2}\big{|}<\varepsilon\): the integral part will work.