## 3 Numerical Results
### Power Performance and Load Simulation  
Since a wind turbine is rarely working with a large yaw angle, we define scenarios for the control study according to Case I in the validation. The specifics of the four different scenarios are described below:  
1. No yaw, no shear;
2. 10\({}^{\circ}\) yaw, no shear;
3. No yaw, 0.3 wind shear power law exponent;
4. 10\({}^{\circ}\) yaw, 0.3 wind shear power law exponent. Figure 6 illustrates the streamwise velocity and pressure field for Case 1 and Case 4. Horizontal axis x and vertical axis y are non-dimensional with the rotor radius of R = 40 m, which means the real values of them should equal to the axis values times R. The rotor center is located at x = 1.5 and y = 1.5. In Figure 6a, a velocity slowdown (induction effect) created by the rotor when exacting wind energy can be observed in the areas both in front of and behind the rotor, where the impact of induction effect is larger in the wake. When no shear is included, no wind speed stratification above the ground can be observed. A significant wind speed stratification above the ground can be observed in Figure 6b, while the wake is also mitigated by the lower longitudinal wind speed due to the shear. Similarly, Figure 6c,d illustrate the pressure field for Case 1 and Case 4, respectively. In general, the pressure in front of the rotor is larger than the pressure behind the rotor. By introducing the wind shear and yaw, the pressure both in front of and behind the rotor is changed, especially for the pressure in front of the rotor, which becomes more homogeneous. Figure 5: Forces normal and tangential to the local chord at 33%, 48%, 76% and 92% rotor radius for Case II (a) Fn33; (b) F33; (c) Fn48; (d) F48; (e) Fn76; (f) F76; (g) Fn92; (h) Ft92. The angle of attack (AoA) at 90% length of the blade for the four scenarios with no control is investigated. The comparison of the AoA can be found in Figure 7. In general, it can be observed that the fluctuations of AoA become much larger by introducing yaw or shear. In this study, the 0.3 wind shear power law exponent leads to a more significant amplitude change than the 10-degree yaw angle. Figure 6: Streamwise velocity field and pressure field for Case 1 and Case 4: (**a**) streamwise velocity for Case 1; (**b**) streamwise velocity for Case 2; (**c**) pressure for Case 1; (**d**) pressure for Case 2. Figure 7: Angle of attack in a cross-section of 35.24 (90%R) without control strategy. Figure 8 illustrates the comparisons of AoA under pitch control with respect to different scenarios. It is noted that the pitch angle control in this study is performed using the AoA at 90%R where the wind turbine noise source center is located [37]. In the beginning, due to inexact interpolation values, the interpolated pitch control strategy results in more throbbing of AoA than the full step control strategy. However, when it is converged, both control techniques lead to a significant reduction of AoA fluctuations in all the four scenarios, while the mean of AoA remains the same. Figure 9 shows the pitch input values based on the results of AoA mentioned above. Pitch angles for two control strategies are calculated by using the angle of attack at 90%R of the no control case. It can be observed that the pitch is constant with 0.15\({}^{{}^{\circ}}\) pitch offset when no control is performed. In terms of pitch in all the four cases, no significant difference can be found between control strategies I and II. Figure 8: Angle of attack in section 35.24 m (90%R) with control strategy: (**a**) Case 1; (**b**) Case 2; (**c**) Case 3; (**d**) Case 4. Figure 9: _Cont._  
In Figure 10, power and thrust coefficients, noted as \(C_{p}\) and \(C_{T}\), respectively, are plotted for the four scenarios.