## 4 Results and Discussion
### Weathervane Test in Steady Wind  
An important application of the SPM system for floating offshore wind turbine is the self-aligned yaw control ability, which can reduce the cost of nacelle yaw system. Therefore, a weathervane test of the SPM turbine is performed using FAST in the steady-wind and still-water condition. The wind comes with a rated wind speed 11.4 m/s of the NREL 5 MW baseline turbine, from various directions ranging from \(-20\) degrees to 20 degrees, step by 5 degrees. Simulations are not performed with a too-large wind direction angle because of the limitation in the FAST that it adopts a small-angle approximation for the platform rotations to avoid the sequence of rotation (Equation 2-2 in Reference [21]). The limit threshold angle is 20 degrees exceeding which the FAST simulations will loss considerable accuracy. The wind inflow directions are therefore set within this range to assure the enough accuracy of computation. In the simulations, the input parameter of YawDOF in the ElastoDyn is set to be false for the SPM turbine so that the nacelle yaw is always fixed. As shown by Figure 9, the platform does rotate in accordance with the wind inflow direction. Nevertheless, there are three important points worth to be noted. Above all, the platform does not rotate to an angle exactly the same with the wind direction, e.g., when the wind comes from 20 degrees, the ultimate platform weathervane yaw angle is 10 degrees,  
Figure 8: Platform motion RAOs computed by MoorDyn for the DeepCwind MPM and SPM systems with and without wind: (**a**) in surge; (**b**) in sway; (**c**) in heave; (**d**) in roll; (**e**) in pitch and (**f**) in yaw. having a twice difference. Secondly, the ultimate platform weatherane angles (after the platform reaching a steady state) are not equalized between two image wind directions with respect to the \(x\)-\(z\) plane, e.g., for the wind directions of \(-15\) and \(15\) degrees, the platform weatherane angles are finally \(-12.58\) and \(6.39\) degrees, respectively. Furthermore, the standard deviations for the time-histories of two image wind directions also show a quite obvious asymmetry. This asymmetry of motion response is basically attributed to the aerodynamic asymmetry of the rotor properties. Unless the airfoils are symmetric (having zero camber), the aerodynamic twist angles are zero, the blade-pitch angles are \(90\) degrees, and the rotor is not rotating, the rotor will be aerodynamically asymmetric. In addition, the blade structural twist could also induce asymmetry in the rotor. The phenomena can be understood as well in another way, which are explained in detail as below. Firstly, due to existence of the wind shear distribution, the rotor has a higher side force at its upper part and a lower side force at its lower part. The net side force induces a yaw moment with respect to the tower center, as shown in Figure 10. Secondly, because of the shaft tilt, the \(V_{y}\) component of the relative wind speed to the blade is smaller when the blade is going upward (induction factor \(a\) is larger) and larger when the blade is going downward (induction factor \(a\) is smaller), leading to the imbalance of the rotor thrust force between the left and the right side of the rotor and finally inducing a yaw moment, as shown in Figure 11. Last but not least, due to the presence of blade cone angle, the blade with a larger projection area to the wind experiences a higher thrust force than the one with a smaller projection area, as shown in Figure 12. The deviation of the thrust force causes a yaw moment as well with respect to the tower center. A net summation of all the above three moments (there may be other more) leads to the wind-induced weatherane of the platform yaw motion. Figure 9: Platform weathervane yaw angle against wind inflow direction (results calculated by MAP++ overlapped those by MoorDyn hence only the latter are shown): points and bars stand for mean positions and standard deviations of the platform yaw displacements, respectively.