# Investigation into the Yaw Control of a Twin-Rotor 10 MW Wind Turbine
## 2 Twin-Rotor WT Model  
This section presents the mechanical power and thrust calculation of the WT to design a yaw control for a twin-rotor WT that overcomes the thrust force difference between the two rotors, which affects the system's performance. The MATLAB Simulink(r) tool was used to model and simulate the overall NREL 5 MW WT system's behavior. WT devices convert the kinetic energy in flowing wind into electrical energy through the rotor's shaft, which is connected to a generator, converting rotational mechanical power into electrical power. The mechanical output power of WTs is affected by several factors, as shown in Equation (1) [23]:  
\[P_{m}=\frac{1}{2}\rho AV_{w}{}^{3}C_{p}(\lambda,\beta) \tag{1}\]  
where \(P_{m}\) is the WT's output mechanical power, \(V_{w}\) is the wind velocity, \(\rho\) is the air density, \(A\) is the rotor's swept area, and \(C_{p}\) is the rotor's power coefficient or efficiency, which is the ratio between the actual obtained power and the maximum power available in the wind. The maximum value of \(C_{p}\) represents the maximum power the WT can produce. Theoretically, the maximum value of \(C_{p}\), called the Betz limit, is 0.59, which indicates that the WT produces 59% of the available power in the flowing wind. A WT's power coefficient varies with the tip speed ratio (TSR) of the turbine (\(\lambda\)) and the blades' pitch angles (\(\beta\)), as expressed in Equation (2) [37]:  
\[C_{p}(\lambda,\beta)=0.5176\cdot\left(116\frac{1}{\lambda_{i}}-0.4\beta-5 \right)e^{\frac{21}{\lambda_{i}}}+0.0068\lambda \tag{2}\]  
Figure 1: Proposed TRWT configuration. where  
\[\frac{1}{\lambda_{i}}=\frac{1}{\lambda+0.08\beta}-\frac{0.035}{\beta^{3}+1} \tag{3}\]  
and  
\[\lambda=\frac{\omega_{r}RR}{V_{w}} \tag{4}\]  
where \(\beta\) is the blade pitch angle, \(\omega_{r}\) is the rotational speed of the rotor, and \(RR\) is the rotor's radius. Regarding the thrust load applied to the rotor plane from the flowing wind, the rotor is considered to be a permeable ideal disc that reduces the flowing air from \(V_{0}\) before the rotor to \(V_{1}\) at the rotor disc and to \(V_{2}\) beyond the rotor, which is called the wake velocity, as shown in Figure 2.  
[PERSON]'s equation can be applied to calculate the pressure drop, thus calculating the dynamic axial force applied to the rotor center. Assuming frictionless and incompressible flow and no external forces acting on the fluid, Equation (5) represents [PERSON]'s equation:  
\[\frac{P_{0}}{\rho_{0}}+gz_{1}+\frac{V_{0}^{2}}{2}=\frac{P_{2}}{\rho_{2}}+gz_{2 }+\frac{V_{2}^{2}}{2} \tag{5}\]  
where \(g\) is gravity, \(z\) denotes the tube height, and \(\rho\) is the fluid density. Since the flow is at the same height, the fluid density does not change, and the wind velocity before the rotor plane is greater than that beyond the rotor (\(V_{0}\) > \(V_{1}\)), so the pressure drop along the Betz tube can be calculated according to Equation (6):  
\[\Delta P_{max}=\frac{1}{2}{\rho{V_{0}}^{2}} \tag{6}\]  
Therefore, the maximum axial force acting on the rotor plane can be calculated using Equation (7):  
\[F_{A}=\frac{1}{2}{\rho{A{V_{0}}^{2}}} \tag{7}\]  
Regarding the HAWT, the fraction of the maximum axial force that the turbine actually experiences is the axial thrust force coefficient (\(C_{t}\)). Thus, the thrust force (\(F_{T}\)) is the maximum axial force centered on the rotor's rotational axis.