## 7 Simulation Results and Discussion
### Performance Evaluation under Stochastic Wind Speed Profile  
The performance of the proposed MPPT paradigm has been tested and evaluated, in this section, for a 3 kW PMSG-WECS under a stochastic wind speed profile that comprises rapid and frequent wind speed fluctuations. The maximum power conversion coefficient, \(C_{p_{max}}=0.4762\), computed from Equation (2) occurs at an optimal tip speed ratio, \(\lambda_{opt}=7\). The simulations are performed for a 100 s horizon at an average wind speed of, \(v_{w_{w}}=7\) m/s (also depicted in Figure 2).  
Figure 5: Closed-loop control system for the PMSG–WECS.  
Figure 6 demonstrates the PMSG rotational speed tracking performance. It is evident from the zoomed-in segments of the figure that the FDTSMC is offering a superior tracking performance among its competitors. Moreover, both the FTSMC and FDTSMC offer a finite-time convergence. The superior performance of the FDTSMC to the FTSMC, CSMC, FBLC and PID, as well as the finite-time convergence of both the FTSMC and FDTSMC, are also evident from the speed tracking error of the PMSG, as depicted in Figure 7. The CSMC exhibits slightly better performance than both the FBLC and PID.  
Figure 8 indicates the tip speed ratio performance of the VSWT. The PID has the worst performance, and it offers a lot of oscillations around the \(\lambda_{opt}\). However, both the FTSMC and FDTSMC are accurately tracking the \(\lambda_{opt}\) with FDTSMC being the best MPPT candidate.  
Figure 9 shows the power conversion coefficient of the VSWT. It is clear that both the FTSMC and FDTSMC strategies are tracking the optimal power conversion coefficient, \(C_{pmax}=0.4762\), throughout the selected stochastic wind speed profile, thus guaranteeing the maximum wind power extraction. The zoomed-in segments of the figure clearly depict that, based on the \(C_{pmax}\) tracking, the FDTSMC is the best MPPT candidate, while the PID is the worst.  
Figures 10 and 11 indicate the low-speed shaft mechanical power and the high-speed shaft mechanical power versus the tip speed ratio, respectively. It is evident that both the FTSMC and the FDTSMC strategies maintain the low-speed as well as the high-speed shaft mechanical powers in the close neighborhood of the optimal tip speed ratio, \(\lambda_{opt}\). On the other hand, both the FBLC and PID techniques have very poor performance on account of having a lot of power fluctuations around the optimal tip speed ratio.  
Figure 6: High-speed shaft rotational speed tracking.  
Figure 7: High-speed shaft rotational speed tracking error.  
Figure 8: Tip speed ratio performance.  
Figure 9: Power conversion coefficient.  
Figure 12 represents the low-speed shaft mechanical power versus the low-speed shaft rotational speed. The proposed MPPT controller efficiently maintains the rotational speed of the VSWT in such a manner that the system operates on the ORC (optimal regimes characteristic)--that is, the region where the maximum wind power can be extracted. However, both the FBLC and PID schemes experience a lot of speed as well as mechanical power fluctuations.  
Figure 13 illustrates the PMSG electromagnetic torque versus the tip speed ratio. Since both the FTSMC- and the FDTSMC-based MPPT candidates maintain the torque at the optimal tip speed ratio, \(\lambda_{opt}=7\), it implies that both are capable of extracting the maximum wind power. The FDTSMC has the best, while the PID has the worst performance because of frequent torque fluctuations. Similarly, Figure 14 shows the VSWT mechanical torque versus the low-speed shaft rotational speed. Again, both the FTSMC- and the FDTSMC-based MPPT candidates maintain the torque around the ORC. It signifies that these two MPPT strategies are capable of extracting the maximum wind power, with FDTSMC being superior to the remaining MPPT candidates.  
Figure 11: High-speed shaft mechanical power versus the tip speed ratio.  
Figure 12: Mechanical power versus the low-speed shaft rotational speed.