## 2 Explanation of Variable Pitch Concept
### Aerodynamics in Traditional VP-Technology  
The VAWT blade is subject to cyclic variations in wind speed and the AoA, which are efficiency improvements that can result from corresponding variations of the pitch angle of the blade to optimize the AoA at each point in its cycle [20]. Any improvement made to the AoA can be of a cyclic nature, as shown in Figure 3[23]. Thus, a typical curve of the pitch angle in traditional VP-technology is expressed by Equations (5) and (6):  
Figure 1: Illustration of Darrieus rotor. \[\gamma = A_{\rm c}\sin(\theta) \tag{5}\] \[A_{\rm c} = A_{\rm s}\ -\ \left(A_{\rm s}\frac{\lambda}{X_{0}}\right)\sin(\theta) \tag{6}\]  
where \(\gamma\) is the variable value of the pitch angle changed with the azimuth, \(A_{\rm c}\) is the maximum pitch angle at a certain TSR, \(A_{\rm s}\) is the maximum pitch angle at all of the TSRs, \(\lambda\) is the TSR, \(X_{0}\) is the TSR of max \(C_{\rm P}\) at zero \(A_{\rm c}\), and \(\theta\) is the azimuthal angle. In another conventional VP-technology of VAWT, the blades have an oscillatory motion around their own axis, which was superimposed on the uniform rotation motion. Considering a cross-section of the turbine, the lines normal to the profile chord for every blade position intersect a given point  
Figure 3: Illustration of cyclical variable pitch. Figure 2: Relative wind speed vectors in the upstream and downstream section. \"\(P\)\" during a complete revolution (Figure 4) [24]. The variation of the pitch angle with respect to the azimuthal angle is expressed as:  
\[\gamma=A_{\mathrm{c}}\sin\theta-\arctan\left(\frac{\sin\theta}{\cos\theta+\lambda}\right) \tag{7}\]  
Two conventional VP-approaches have one point in common: the azimuthal angle with a larger geometric AoA will be given a larger pitch angle. Affected by the performance, as expressed by the lift coefficient (\(C_{\mathrm{L}}-\alpha\)) of the airfoil, the blade has a stall static AoA of about 10\({}^{\circ}\), as shown in Figure 5a [18]. At the start-up and small TSRs, the maximum AoA is much greater than the stall angle, the lift of the blade will decrease, and the drag will greatly increase after the stall occurs. In such cases, the VP-technology will change the pitch angle in the direction of the decreased AoA to avoid the stall. Two methods were used to achieve this goal. In the first method, the pitch angle changes only in the positions where the local geometric AoA exceeds the stall value, and the final local effective AoA is kept below the static stall angle [7; 25], as illustrated in Figure 5b. The advantage of this method is that it makes the blade rotate with the optimum AoA in a much wider azimuthal zone. Its disadvantage is that the change of the pitch angle (represented by a dotted line in Figure 5b) is discontinuous at some points, which may not be practical to implement, and also causes early fatigue due to the abrupt dynamical loads [7]. The other method continuously changes the pitch angle following a sinusoidal function (Figure 5c). Under this control, the maximum amplitude of the sinusoidal function for the pitch angle is set to be equal to the maximum difference between the geometric AoA in the FP-VAWT and the effective AoA in the VP-VAWT. The advantage of this method is that the pitch angle as well as the forces changes smoothly and continuously, and then the method is physically and mechanically feasible to implement; the implementation procedures are presented in Figures 2 and 3. However, the power efficiency, although significantly larger than that of the FP-VAWT, is much lower than that of the other method illustrated in Figure 5b, because the final local effective AoA curve also follows a sinusoidal function in which the blade experiences the maximum performance only at two inflection points. At an optimum or larger TSR, the pitch angle should change toward the direction of the increased AoA for an improved performance in terms of the increased peak power coefficient of the VAWT.