# Nonlinear Maximum Power Point Tracking Control Method for Wind Turbines Considering Dynamics
## 6 Performance Validation
### Dynamic Response of Sinusoidal Wind Containing Several Frequencies  
In order to test the bandwidth of the conventional MPPT method and the novel MPPT method, a sinusoidal wind profile containing several frequencies is used for simulation, as shown in Figure 8a. The wind profile is defined by:  
\[v=8+2\sin(0.05t)+1.5\sin(0.2t)+\sin(0.6t) \tag{46}\]  
From Figure 3, the rotating parts of wind turbines absorb or release kinetic energy while converting wind energy into electric energy. The mechanical energy extracted from the inflow wind can't be accurately measured. The electric energy or the electric power is usually used to represent the wind turbine power performance. However, the change in kinetic energy stored in rotating parts will disturb the results if we use electric energy to represent the energy extracted from the wind. To solve that problem, kinetic energy stored in rotating parts is expected to maintain the same at the beginning and the end of the analysis period. Correspondingly, the wind speed in the period of [0 s-50 s] and [200 s-250 s] will be both set as the constant, 8 m/s. The analysis period is [50 s 200 s]. The black line and red line represent the simulating wind and the estimated wind, respectively, in Figure 8a. Notably, the UKF-based estimator and the Newton-Raphson method work well to estimate the turbulent wind speed with high accuracy.  
Figure 8: _Cont._  
Figure 8b depicts the blade-pitch angle. Obviously, while using the novel MPPT method, the blade-pitch angle varies around \(-1^{\circ}\), corresponding to the value of using the conventional MPPT method.  
Figure 8c demonstrates the dynamic response of rotor speed. While using the conventional MPPT method, wind turbines show better performance in tracking the wind variations under high-speed wind conditions during the period of [50 s-100 s], and relatively worse performance under the low-speed wind conditions during the period of [130 s-180 s]. That affirms the nonlinear property of VSWTs while using the conventional MPPT method.  
Figure 8d depicts the cumulative electric energy increment between using novel MPPT method and using the conventional MPPT method. Obviously, the electric energy extracted from wind has significantly increased by 0.59 kW-h from 50 s to 250 s. That affirms the energy improvement of the novel MPPT method.  
Figure 8e shows the frequency spectrum of rotor speed corresponding to the time-domain graphics in Figure 8c. From Equation (30), the bandwidth is supposed to vary from 0.05 rad/s to 0.2 rad/s while using the conventional MPPT method and is designed to maintain around 0.6 rad/s while using the novel MPPT method. Thus, both the conventional method and the novel method are able to track the wind components of the frequency, 0.05 rad/s, as shown in Figure 8e. The amplitude response to the wind components of the frequencies, 0.2 rad/s, and 0.6 rad/s, shows that wind turbines significantly track the wind variations better using the novel MPPT method than that using the conventional MPPT method. That affirms the improved tracking bandwidth of the novel MPPT method.