## 2 Explanation of Variable Pitch Concept  
### Oscillating AoA  
The Darrieus-type VAWT is a cross-flow rotor whose axis of rotation meets the flow of the working fluid at right angles. The blades perform the work twice in one cycle: i.e., the upwind half-cycle (azimuthal angle from 0\({}^{\circ}\) to 180\({}^{\circ}\), where the 0\({}^{\circ}\), 90\({}^{\circ}\), 180\({}^{\circ}\), and 270\({}^{\circ}\) positions are defined in Figure 1) and the downwind half-cycle (ranging from 180\({}^{\circ}\) to 360\({}^{\circ}\)). The azimuth angle of 0\({}^{\circ}\) is defined as a position where the directions of the local wind speed \({}^{\prime}V_{\rm w}\) and the tangential flow speed \({}^{\prime}U=r\circ^{\prime}\) are the same; the azimuthal angle of 180\({}^{\circ}\) is the position where the directions of the two speeds are opposite; as its anticlockwise rotation, the azimuthal angle of 90\({}^{\circ}\) is the position moving anticlockwise on a quarter circle from 0\({}^{\circ}\), which is the fixed reference position. The resultant flow, which is the vector sum of the local wind speed \({}^{\prime}V_{\rm w}\)\({}^{\prime}\) and the tangential flow speed \({}^{\prime}U=\sigma r^{\prime}\), always comes from the upwind side of the blade: i.e., the outer side on the upwind half-cycle and the inner side on the downwind half-cycle [22]. Accordingly, as illustrated in Figure 1, the rotating blade experiences a changing resultant flow, including its magnitude and direction. The changing direction of the resultant velocity and chord line of the airfoil causes the oscillating AoA between the negative and positive values following a sinusoidal function of the azimuthal angle.  
As shown in Figure 2, the AoA is determined by:  
\[\alpha=\tan^{-1}\biggl{[}\frac{a\sin\theta}{\lambda_{0}+a\cos\theta}\biggr{]} \tag{1}\]  
where \(\alpha\) is the AoA, \(a\) is the axial induction factor defined as \(a=V/V_{\rm w}\) (\(V\) is the velocity of the flow in the stream tube, \(V_{\rm w}\) is the velocity of the free wind), \(\theta\) is the azimuthal angle, and represents the local TSR, where \(\omega\) is the rotational speed, \(R\) is the rotational radius. The 0\({}^{\circ}\) and 180\({}^{\circ}\) positions are transformation points of the AoA between the negative and positive values, which determine their zero AoAs. Affected by the oscillating AoA, the blade of the VAWT inevitably produces fluctuating torque, even in steady conditions. For a given airfoil section, the non-dimensional normal and tangential forces are written as:  
\[C_{\rm N} =C_{\rm L}\cos\alpha+C_{\rm D}\sin\alpha \tag{2}\] \[C_{\rm T} =C_{\rm L}\sin\alpha-C_{\rm D}\cos\alpha \tag{3}\]  
where \(C_{\rm N}\) and \(C_{\rm T}\) are the normal force and tangential force coefficients, respectively, and \(C_{\rm L}\) and \(C_{\rm D}\) are the lift and drag coefficients, respectively. According to Equations (2) and (3), the blade in the 0\({}^{\circ}\) and 180\({}^{\circ}\) azimuths will generate a zero-normal force and a negative tangential force, which cause a negative torque, according to Equation (4):  
\[M=\frac{1}{2}\rho cR(2H)C_{\rm T}V_{\rm r}^{2} \tag{4}\]  
where \(M\) presents the torque of the blade, \(\rho\) is the air density, \(c\) is the chord length, \(2H\) is the height of the blade, \(V_{\rm r}\) is the local resultant velocity, and \(R\) is the rotation diameter. Due to the continuity of the blade torque variation, produced by the 0\({}^{\circ}\) and 180\({}^{\circ}\) azimuths, the torque in a zone around the 0\({}^{\circ}\) or 180\({}^{\circ}\) azimuth remains negative, and a wider zone may have a positive torque, but is still not big enough. In contrast, the 0\({}^{\circ}\) and 180\({}^{\circ}\), and 90\({}^{\circ}\) and 270\({}^{\circ}\) angles produce the maximum AoAs. This is because the directions of the upwind flow and tangential flow are vertical to each other (Figure 1), and furthermore, the 90\({}^{\circ}\) azimuth having the larger upwind flow speed normally shows better aerodynamics than the 270\({}^{\circ}\) azimuth.