Use of Dampers to Improve the Over-Speed Control System with Movable Arms for Butterfly Wind Turbines

1. Introduction

The massive introduction of renewable energy is expected to achieve carbon neutrality by 2050 [1]. In view of this, large-scale offshore wind power generation has been introduced in Japan; however, cost reduction remains a major issue. There was a boom in the number of certifications for small-scale wind power due to the introduction of the feed-in tariff (FIT) system. However, the FIT for small-scale wind power was abolished owing to the high costs. Consequently, the adoption of small-scale wind turbines has been stalled.

At low altitudes, where most small-scale wind turbines are installed, the influence of turbulence is significant, and the wind direction changes rapidly. Therefore, vertical-axis wind turbines (VAWTs), which do not require yaw control, are advantageous compared with horizontal-axis wind turbines (HAWTs), which require yaw control. However, compared to HAWT, which can easily control the output (or rotational speed) through furling and blade pitch control, VAWTs are difficult to control.

For example, to prevent the over-speed rotation of a cross-flow wind turbine, Motohashi et al. developed a system that generates a large load above a certain rotational speed by utilizing the load characteristics of a fixed-volume pump directly connected to the wind turbine [2]. However, even during low-speed operation, losses occur because the pump is driven. Pitch control is possible even in VAWT, and Noda et al. realized over-speed suppression mechanisms for gyromill-type VAWTs using centrifugal force and springs [3,4]. Through numerical analysis and experiments, a group at Kanazawa University demonstrated the possibility of high-speed suppression of a gyromill-type VAWT by controlling the pitch angle of the blades with a four-bar linkage mechanism that does not use springs [5,6]. Tanzawa et al. installed flat plates on the rotation axis of a VAWT, which acted as an aerodynamic brake by moving in the radial direction using the centrifugal force and increasing the projected area against the relative wind [7]. With the same operating principle as Tanzawa et al.’s device, Hara et al. developed an over-speed control system that inclines the VAWT blades by the centrifugal force and synchronizes the movement of the blades by a link mechanism [8]. However, most of the over-speed-control mechanisms for VAWT developed to date use materials or devices with elasticity, such as springs, which have large individual differences and durability issues or have complex structures involving link mechanisms. Therefore, there is a high possibility that problems will occur in actual wind turbine operations equipped with conventional high-speed control systems.

To reduce the cost of small vertical-axis wind turbines, we are developing a butterfly wind turbine (14 m in diameter) by extensively using parts of extruded aluminum suitable for mass production. To improve the safety and durability of wind turbines and simultaneously expand the operating wind speed range, a simple over-speed control system with movable arms that operated using centrifugal and aerodynamic forces without springs or link mechanisms was developed. We have already developed a prototype with a diameter of approximately 7 m, half the rotor size of the mass-produced wind turbine under development [9], and installed movable arms in the prototype. However, violent movement of the movable arms was observed in the original mechanism. To solve this problem, we introduce dampers to the rotation axis of the movable arms and attempt to improve the movement. This paper also provides an overview of the theoretical predictions of the characteristics of VAWT equipped with movable arms and compares them with the results of field experiments on the prototype to demonstrate the effectiveness of the theoretical analysis and the introduction of dampers.

2. Prototype of Butterfly Wind Turbine Equipped with Over-Speed Control System

A prototype butterfly wind turbine with movable arms was installed at the Arid Land Research Center at Tottori University at the end of March 2022. A photograph of the prototype is presented in Figure 1a. The prototype has three triangular looped blades made of extruded aluminum. The cross-section of the blades is an original symmetrical airfoil New_AF_1_UP [10] with a thickness ratio of 24%, which was adopted because of its high structural strength and expected high aerodynamic performance. The chord length of the blades of the prototype was c = 375 mm, which is the same as that of the mass-production type under development. However, the diameter and height of the prototype rotor (diameter: D = 6.92 m, height: H = 6.86 m) are half those of the mass-production type.

Figure 2 illustrates the outline and size of the prototype. Each movable arm was installed in one portion of the reinforcing horizontal arm, which spanned from the rotor hub to the center of the vertical main blade. An aileron is installed on the movable arm. The movable arm and aileron can rotate as one unit around the axis of the movable arm, as illustrated in Figure 1b. The slant angle η of the movable arm system (movable arm and aileron) changes depending on the balance among the centrifugal force, aerodynamic force, and gravity force acting on the movable arm and aileron, and when it is tilted greatly, large aerodynamic force, i.e., drag, acts on the movable arm to work as a passive aerodynamic brake for the wind turbine. When the wind speed and rotor rotational speed are low, the movable arm and aileron weights work as the recovering force to return to the initial state; therefore, no elastic devices such as springs are utilized. However, the movable arm system is inclined at an initial angle (η_ini = 5°) as the initial state, and the movable arm is designed to tilt even at low rotational speeds when the wind speed is high. As illustrated in Figure 2, the trestle of the wind turbine is a tripod type, and its cross-sectional shape is the same as that of the mass-produced turbine currently under development (approximately half the length). The structure of the butterfly wind turbine does not have a concrete foundation but rather a pile foundation, which is easy to construct in a short construction period. The wind turbine blades, horizontal arm, including the movable arm system, hub, bearing housing, and tripod trestle, are all made of extruded aluminum.

One of the purposes of installing this prototype was to confirm the behavior of the newly developed over-speed control system with movable arms to demonstrate its effectiveness. Although the prototype was designed to rotate up to approximately 100 rpm, for safety reasons, the lengths of the movable arm (1.7 m) and aileron (span^(ail) = 0.6 m) were selected to realize a maximum rotor rotation of less than approximately 80 rpm as the first design. The movable arm, aileron, and fixed arm were made of the same aluminum extrusion shape as the main blade (common chord length, c = 375 mm). Figure 3 and Figure 4 present the main parameters and sizes of the prototype’s horizontal- and movable-arm systems, respectively. The prototype reuses the generator (rated output: 5 kW, with three times step-up gear) and control device used in a previously developed five-blade butterfly wind turbine (7 m diameter, 2.7 m height) [8]. Here, the load characteristics of the prototype rotor and generator (control targets) were inconsistent.

Bidirectional disk dampers (FDT-57A-503, Fuji Latex Co., Ltd.) were installed along the movable arm axis. Three-stacked dampers were placed on the movable arm’s left and right rotating shafts. Figure 5a illustrates the mounted parts of the dampers. Figure 5b presents the rotational angular velocity dependence of the total resistance moment (M_damp) of the six disc dampers installed in one movable arm. Here, the total resistance moment is approximated using Equation (1) with reference to catalog values.

$$M_{\mathrm{d}\mathrm{a}\mathrm{m}\mathrm{p}}=6\times 3.6383{{\omega }_{\mathrm{m}\mathrm{a}}}^{0.2811}$$

(1)

where ω_ma is the angular velocity of the tilting motion around the movable-arm axis.

3. Theoretical Prediction

3.1. Prediction of Flow Speed Distribution

The wind turbine characteristics were predicted employing a method based on the blade element momentum theory [11], adopting the quadruple multiple streamtube model [12]. Figure 6 presents a schematic of the flow-field calculation method. In this calculation, one stream tube was divided into upwind and downwind sides, and the stream tube of each side was divided into calculation parts of the outer and inner rotor intersectional wind speeds. Therefore, as demonstrated in Figure 6, along one stream tube, the intersectional wind speeds are calculated from the upstream wind speed V_∞ in the order of V_{u_out}, V_{u_in}, V_{d_in}, V_{d_out} to ensure a decreasing flow field. The flow field passing through the turbine rotor was divided into 180 stream tubes with widths corresponding to the unit azimuth angle in the horizontal direction, and the vertical direction was divided into 21 levels. The vertical level is represented by the integer parameter W, where W = 0 corresponds to the stream tubes at the top of the rotor, W = 20 at the bottom of the rotor, and W = 10 at the equatorial level (see Figure 2). In the analysis, the rotational direction of the wind turbine rotor was defined as counterclockwise, as viewed from above, which is opposite to that of the prototype. The origin of the azimuth angle Ψ is defined as the direction 90° from the upstream, and as illustrated in Figure 6, the upwind side is positive, and the downwind side is negative. In the in-house simulation code, an integer parameter I is introduced to specify the stream tube, and the origin of the parameter I (I = 0) is defined as the direction of Ψ = -180°. The value of I increases counterclockwise, and the final value (I = 360) corresponds to the stream tube in Ψ = 180°. The altitude distribution of the upstream wind speed is given by the power law defined by the following equation:

$$V_{\mathrm{z}}=V_{\mathrm{h}\mathrm{u}\mathrm{b}}{\left(\frac{z}{z_{\mathrm{h}\mathrm{u}\mathrm{b}}}\right)}^p$$

(2)

Here, p is the power index, which in this study was assumed to be p = 0.2. In the in-house simulation software, the upstream wind speed V_∞ corresponding to the hub height is given as V_hub in Equation (2) as an input condition, and the upstream wind speed V_∞(W) at a height level W is calculated using Equation (2). Figure 2 demonstrates the altitude distribution of the upstream wind speed.

Aerodynamic data based on NACA 0018 data [13] were utilized for the present simulation because there are currently no reliable aerodynamic data for the novel airfoil (New_AF_1_UP). The reduction rate (7%) in the aerodynamic characteristics of the novel airfoil compared to NACA 0018 was considered [10] in the simulation. This study adopted the modified Gormont model [11] as the dynamic stall model. However, the effects of the dynamic stall were ignored in the ranges of 105° ≤ ψ ≤ 180° and -180° ≤ ψ ≤ -135°, where the degree of turbulence is assumed to be large due to the influence of the vortices shed around ψ = 90°.

Figure 7 presents the wind speed distributions predicted at height levels of W = 5 and 10 under the conditions of the upstream hub-height wind speed of V_∞ = 6 m/s, the rotor rotation speed of N = 70 rpm, and the tip speed ratio of λ = 4.224 (λ = Rω/V_∞). The prediction demonstrates that the speed of the flow passing through the inner rotor (i.e., slant blades) is further reduced compared to that not passing through it. In addition, the position where the wind speed reached its minimum was slightly shifted from the center of the rotor to the positive side of the Y coordinate. In other words, the wind turbine converts more fluid energy into mechanical energy on the side where the blade moves from the downwind to the upwind side. In the simulation of the behavior of the movable arm described in Section 3.2, for simplicity, the outer rotor intersectional wind speeds (V_{u_out} and V_{d_out}) obtained at a hub height of W = 10 (see Figure 7b) were assumed to be the inflow speeds of the movable arm and aileron.

3.2. Equation of Motion of Movable Arm System

The simulation method for the behavior of the movable arm and aileron is described in this section. The aileron and movable arm were divided into 20 sections, and the mass of each section was defined as m_i and m_j. As illustrated in Figure 4, the length of the line segment connecting the aerodynamic center (a.c.) of the i-th section of the aileron and the movable arm axis is defined by Equation (3).

$$l_{\mathrm{a}\mathrm{c}}^{\left(\mathrm{a}\mathrm{i}\mathrm{l}\right)}\left(i\right)=\sqrt{{h\left(i\right)}^2+s_{\mathrm{a}\mathrm{i}\mathrm{l}}^2}$$

(3)

where h(i) is the distance in the aileron span direction between the aerodynamic center of the i-th section and the movable arm axis, and s_ail is the distance in the chord direction. When there is no inclination (η = 0), the angle ξ(i) between the line segment l_ac^(ail)(i) and the vertical direction is given by Equation (4).

$$\xi \left(i\right)={\tan }^{-1}\left\{s_{\mathrm{a}\mathrm{i}\mathrm{l}}/h\left(i\right)\right\}$$

(4)

However, when there is an inclination (η ≠ 0), the angle ζ(i) between the line segment l_ac^(ail)(i) between the vertical direction is given by Equation (5).

$$\zeta \left(i\right)=\xi \left(i\right)+\eta$$

(5)

When the rotational angular velocity of the wind turbine is ω, the centrifugal force acting on the i-th section of the aileron is given by Equation (6).

$$F_{\mathrm{c}}^{\left(\mathrm{a}\mathrm{i}\mathrm{l}\right)}\left(i\right)=m_i{r}_{\mathrm{c}\mathrm{g}}^{\left(\mathrm{a}\mathrm{i}\mathrm{l}\right)}\left(i\right){\omega }^2$$

(6)

where r_cg^(ail)(i) is the distance between the wind turbine rotation axis and the center of gravity of the i-th section. The tangential component of the centrifugal force to the orbital circle of the rotor is expressed by Equation (7).

$$F_{\mathrm{c}\mathrm{t}}^{\left(\mathrm{a}\mathrm{i}\mathrm{l}\right)}\left(i\right)=F_{\mathrm{c}}^{\left(\mathrm{a}\mathrm{i}\mathrm{l}\right)}\left(i\right)\cos {\beta }_i=m_i{\omega }^2\left\{ h\left(i\right)\sin \eta +d_{\mathrm{a}\mathrm{i}\mathrm{l}}\cos \eta +aad\right\}$$

(7)

Here, β_i is the angle between the direction of centrifugal force and the tangential direction, and d_ail is the distance in the chord direction between the movable arm axis and the center of gravity of the i-th section. The axis-axis-distance (aad) is the horizontal distance viewed from the movable arm axis direction between the movable arm axis and the wind turbine rotation axis; aad = 40.54 mm. Similar to Equations (6) and (7), the centrifugal force acting on the j-th section of the movable arm and its tangential component are given by Equations (8) and (9), respectively.

$$F_{\mathrm{c}}^{\left(\mathrm{m}\mathrm{a}\right)}\left(j\right)=m_j{r}_{\mathrm{c}\mathrm{g}}^{\left(\mathrm{m}\mathrm{a}\right)}\left(j\right){\omega }^2$$

(8)

$$F_{\mathrm{c}\mathrm{t}}^{\left(\mathrm{m}\mathrm{a}\right)}\left(j\right)=F_{\mathrm{c}}^{\left(\mathrm{m}\mathrm{a}\right)}\left(j\right)\cos {\beta }_j=m_j{\omega }^2\left\{ d_{\mathrm{m}\mathrm{a}}\cos \eta +aad\right\}$$

(9)

Here, d_ma is the distance in the chord direction between the movable arm axis and the center of gravity of the movable arm.

The vertical upward force L^(ail)(i) and horizontal force D^(ail)(i) acting on the aerodynamic center of the i-th section of the aileron, as illustrated in Figure 4, are expressed by Equations (10) and (11), respectively.

$$L^{\left(\mathrm{a}\mathrm{i}\mathrm{l}\right)}\left(i\right)=\left\{L\left({\alpha }_i\right)\sin {\alpha }_i-D\left({\alpha }_i\right)\cos {\alpha }_i\right\}\sin \eta$$

(10)

$$D^{\left(\mathrm{a}\mathrm{i}\mathrm{l}\right)}\left(i\right)=\left\{-L\left({\alpha }_i\right)\sin {\alpha }_i+D\left({\alpha }_i\right)\cos {\alpha }_i\right\}\cos \eta$$

(11)

Similarly, the vertical upward force L^(ma)(j) and horizontal force D^(ma)(j) acting on the aerodynamic center of the j-th section of the movable arm are given by Equations (12) and (13), respectively.

$$L^{\left(\mathrm{m}\mathrm{a}\right)}\left(j\right)=L\left({\alpha }_j\right)\cos ∆{\alpha }_j+D\left({\alpha }_j\right)\sin ∆{\alpha }_j$$

(12)

$$D^{\left(\mathrm{m}\mathrm{a}\right)}\left(j\right)=-L\left({\alpha }_j\right)\sin ∆{\alpha }_j+D\left({\alpha }_j\right)\cos ∆{\alpha }_j$$

(13)

The lift forces (L(α_i), L(α_j)) and drag forces (D(α_i), D(α_j)) acting on each section, the angles of attack (α_i, α_j), and the angle Δα_j, which are expressed on the right-hand side of Equations (10)–(13) are derived in Appendix A and Appendix B.

The moment M_c^(ail) around the movable arm axis due to the centrifugal force acting on the aileron is determined by adding the contributions from all aileron sections (i = 0 to 19) and is given by Equation (14).

$$M_{\mathrm{c}}^{\left(\mathrm{a}\mathrm{i}\mathrm{l}\right)}=\sum _i{F}_{\mathrm{c}\mathrm{t}}^{\left(\mathrm{a}\mathrm{i}\mathrm{l}\right)}\left(i\right)\left\{h\left(i\right)\cos \eta -d_{\mathrm{a}\mathrm{i}\mathrm{l}}\sin \eta \right\}$$

(14)

The moment M_g^(ail) around the movable arm axis owing to gravity acting on the aileron is determined by Equation (15).

$$M_{\mathrm{g}}^{\left(\mathrm{a}\mathrm{i}\mathrm{l}\right)}=-\sum _i{m}_{i}g\left\{h\left(i\right)\sin \eta +d_{\mathrm{a}\mathrm{i}\mathrm{l}}\cos \eta \right\}$$

(15)

The moment M_aero^(ail) around the movable arm axis owing to the aerodynamic force acting on the aileron is expressed by Equation (16), using the forces in Equations (10) and (11), respectively.

$$M_{\mathrm{a}\mathrm{e}\mathrm{r}\mathrm{o}}^{\left(\mathrm{a}\mathrm{i}\mathrm{l}\right)}=\sum _i\left\{L^{\left(\mathrm{a}\mathrm{i}\mathrm{l}\right)}\left(i\right)l_{\mathrm{ac}}^{\left(\mathrm{ail}\right)}\left(i\right) \sin \zeta \left(i\right)\right.\left.-D^{\left(\mathrm{a}\mathrm{i}\mathrm{l}\right)}\left(i\right)l_{\mathrm{ac}}^{\left(\mathrm{ail}\right)}\left(i\right)\cos \zeta \left(i\right)\right\}$$

(16)

The moments around the movable arm axis caused by the centrifugal, gravitational, and aerodynamic forces acting on the movable arm (M_c^(ma), M_g^(ma), and M_aero^(ma)) were similar to those in Equations (14)–(16), determined by adding the contributions from all the movable arm sections (j = 0 to 19) and is given by Equations (17)–(19), respectively.

$$M_{\mathrm{c}}^{\left(\mathrm{m}\mathrm{a}\right)}=-\sum _j{F}_{\mathrm{c}\mathrm{t}}^{\left(\mathrm{m}\mathrm{a}\right)}\left(j\right)d_{\mathrm{m}\mathrm{a}}\sin \eta$$

(17)

$$M_{\mathrm{g}}^{\left(\mathrm{m}\mathrm{a}\right)}=-\sum _j{m}_{j}g{d}_{\mathrm{m}\mathrm{a}}\cos \eta$$

(18)

$$M_{\mathrm{a}\mathrm{e}\mathrm{r}\mathrm{o}}^{\left(\mathrm{m}\mathrm{a}\right)}=\sum _j\left\{L^{\left(\mathrm{m}\mathrm{a}\right)}\left(j\right)s_{\mathrm{ma}}\cos \eta \right.\left.+D^{\left(\mathrm{m}\mathrm{a}\right)}\left(j\right)s_{\mathrm{ma}}\sin \eta \right\}$$

(19)

As demonstrated in Figure 4, the ailerons and movable arms include members considered concentrated masses, such as reinforcing brackets and connection bolts. Using information on their masses and positions relative to the movable arm axis, the moments around the movable arm axis caused by the centrifugal force and gravity acting on the concentrated masses (M_c^(cm) and M_g^(cm)) were calculated. In this case, the total moments M_c and M_g due to the centrifugal force and gravity acting on the aileron, movable arm, and concentrated mass are given by Equations (20) and (21), respectively.

$$M_{\mathrm{c}}=M_{\mathrm{c}}^{\left(\mathrm{a}\mathrm{i}\mathrm{l}\right)}+M_{\mathrm{c}}^{\left(\mathrm{m}\mathrm{a}\right)}+M_{\mathrm{c}}^{\left(\mathrm{c}\mathrm{m}\right)}$$

(20)

$$M_{\mathrm{g}}=M_{\mathrm{g}}^{\left(\mathrm{a}\mathrm{i}\mathrm{l}\right)}+M_{\mathrm{g}}^{\left(\mathrm{m}\mathrm{a}\right)}+M_{\mathrm{g}}^{\left(\mathrm{c}\mathrm{m}\right)}$$

(21)

Because it can be assumed that almost no aerodynamic force acts on the concentrated mass, the moment M_aero due to the aerodynamic forces acting on the aileron and movable arm is expressed by Equation (22).

$$M_{\mathrm{a}\mathrm{e}\mathrm{r}\mathrm{o}}=M_{\mathrm{a}\mathrm{e}\mathrm{r}\mathrm{o}}^{\left(\mathrm{a}\mathrm{i}\mathrm{l}\right)}+M_{\mathrm{a}\mathrm{e}\mathrm{r}\mathrm{o}}^{\left(\mathrm{m}\mathrm{a}\right)}$$

(22)

The moment of inertia around the movable arm axis is defined by I_ma, considering all aileron, movable arm, and concentrated masses. When the rotational angular velocity of the movable arm around the movable arm axis is ω_ma (clockwise direction is defined as positive), the equation of motion of the movable arm system (aileron + movable arm + concentrated masses) is as follows:

$$I_{\mathrm{m}\mathrm{a}}\frac{\mathrm{d}{\omega }_{\mathrm{m}\mathrm{a}}}{\mathrm{d}t}=M_{\mathrm{c}}+M_{\mathrm{g}}+M_{\mathrm{a}\mathrm{e}\mathrm{r}\mathrm{o}}+M_{\mathrm{d}\mathrm{a}\mathrm{m}\mathrm{p}}$$

(23)

The fourth term on the right-hand side of Equation (23) is the damper’s resistance moment, as defined in Equation (1) and always has the opposite sign to the sign of the rotational angular velocity ω_ma.

3.3. Prediction of Movable Arm Behavior and Rotor Performance

The simulation method developed here calculates the forces acting on each portion of the wind turbine and the average rotational torque and output produced by the turbine rotor when given the wind speed and rotor rotational speed. However, to determine the slant angle of the movable arm at each azimuth, the equation of motion of the movable-arm system expressed in Equation (23) was solved as a pseudo-unsteady problem to obtain the convergence state. A flowchart of the calculations is presented in Figure 8.

The upstream wind speed V_∞ at the hub height and the rotor rotational speed N are input as calculation conditions. The flow speeds (V_{u_out}, V_{u_in}, V_{d_in}, V_{d_out}) in each streamtube were calculated based on blade element momentum theory. At this stage, the forces (F_{blade_out}(I), F_{blade_in}(I), F_fa(I)) and torques (Q_{blade_out}(I), Q_{blade_in}(I), Q_fa(I)) at each azimuth of the outer blade (vertical blade), inner blade (slanted blade), and fixed arm were obtained without movable arms. The equation of motion of the movable-arm system in Equation (23) is numerically integrated to determine the inclination angle η(n, I) at each azimuth of the n-th rotation. As initial values, the azimuth angle is set as Ψ = -180°, the inclination angle is set as η(0,0) = η_ini, and the inclination angular velocity is set as ω_ma = 0 rad/s. The fourth-order Runge–Kutta method was used for the numerical integration. After the second rotation, the convergence degree ∆η_ave of the inclination angle is checked with Equation (24) each time the calculation for one rotation was completed.

$${∆\eta }_{\mathrm{a}\mathrm{v}\mathrm{e}}=\frac{1}{360}\sum _{I=0}^{359}\sqrt{{\left\{\eta \left(n+1, I\right)-\eta \left(n, I\right)\right\}}^2}$$

(24)

If ∆η_ave becomes smaller than a certain constant value ε (ε = 0.2 here), the variation of the slant angle during one revolution is considered to become periodic to stop the calculation. In this case, the value of the inclination angle at the final rotation is defined as η(I). However, if the inclination angle variation does not converge even as reaching the maximum number of rotations n_max for calculation (n_max = 30 here), the slant angle η(n, I) is ensemble averaged for each azimuth over the last n_ave revolutions to determine the pseudo-converged inclination angle η(I). Thus, once the inclination angle of the movable arm is determined, the forces (F_ma(I), F_ail(I), F_cm(I)) at each azimuth owing to the movable arm, aileron, and concentrated mass can be calculated. For the movable arm and aileron, the torques around the rotor rotation axis (Q_ma(I), Q_ail(I)) were also calculated. Therefore, the rotational torque Q_total(I) of a blade at the azimuth of integer parameter I is given by Equation (25).

$$Q_{\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}\left(I\right)=Q_{\mathrm{b}\mathrm{l}\mathrm{a}\mathrm{d}\mathrm{e}\_\mathrm{o}\mathrm{u}\mathrm{t}}\left(I\right)+Q_{\mathrm{b}\mathrm{l}\mathrm{a}\mathrm{d}\mathrm{e}\_\mathrm{i}\mathrm{n}}\left(I\right)+Q_{\mathrm{f}\mathrm{a}}\left(I\right)+Q_{\mathrm{m}\mathrm{a}}\left(I\right)+Q_{\mathrm{a}\mathrm{i}\mathrm{l}}\left(I\right)$$

(25)

When the number of blades is B (B = 3 in the prototype), the average rotor torque Q_rotor and the rotor output P_rotor are expressed by Equations (26) and (27), respectively.

$$Q_{\mathrm{r}\mathrm{o}\mathrm{t}\mathrm{o}\mathrm{r}}=\frac{B}{360}\sum _{I=0}^{359}{Q}_{\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}\left(I\right)$$

(26)

$$P_{\mathrm{r}\mathrm{o}\mathrm{t}\mathrm{o}\mathrm{r}}=\omega Q_{\mathrm{r}\mathrm{o}\mathrm{t}\mathrm{o}\mathrm{r}}$$

(27)

If the rotor-swept area is A (A = 47.4 m² for the prototype), then the power coefficient C_p is given by Equation (28).

$$C_{\mathrm{p}}=\frac{P_{\mathrm{r}\mathrm{o}\mathrm{t}\mathrm{o}\mathrm{r}}}{0.5\rho V_{\infty }^{3}A}=\frac{{\omega Q}_{\mathrm{r}\mathrm{o}\mathrm{t}\mathrm{o}\mathrm{r}}}{0.5\rho V_{\infty }^{3}A}$$

(28)

Figure 9 presents the predicted variations in the inclination angle of the movable arm when the wind speed is V_∞ = 6 m/s, the rotor rotational speed is N = 70 rpm, and the tip speed ratio is λ = 4.224. The red and blue curves represent the cases without and with dampers, respectively. In the case with dampers, the inclination angle of the movable arm almost converged to a value of about 44.6° after the rotor rotation of 15 times (the cumulative azimuth angle corresponds to Ψ = 5400°). In contrast, in the case without dampers, the variation in the slant angle did not converge, even at a maximum rotation number of 30. In this case, the prediction presented in Figure 9 illustrates that the azimuth of the maximum slant angle for each rotation gradually changes. However, depending on the combination of wind and rotational speeds, even without dampers, the variation in the slant angle converges within a few rotor rotations.

Figure 10 presents the prediction results of the power coefficient for the prototype at wind speeds of V_∞ = 3, 4, 5, and 6 m/s. At V_∞ = 3 m/s, the movable arms do not tilt, but at V_∞ = 4 m/s or more, the movable arms tilt, and when a certain tip speed ratio is exceeded, the aerodynamic brake works, followed by a sudden decrease in the power coefficient.

4. Experimental Setup

Figure 11 presents a schematic of the measurement system. The prototype has three movable arms; however, a tilt sensor (M21-0455, Kyowa Electronic Instruments Co., Ltd.) is installed on only one. Power was supplied to the tilt sensor by a solar panel and a battery installed on the hub. The measurement data of the tilt sensor were transmitted via radio waves and recorded at a sampling rate of 20 Hz using a Logger-2 (GL840, Graphtec Corp.) installed on a wind measurement pole approximately 15 m from the wind-turbine center axis. A reflector plate was installed near the root of the oblique blade, which was located under the movable arm equipped with the tilt sensor. The light reflected from the reflector emitted by a photoelectric sensor (E3Z-R61, OMRON Corp.) installed in the tripod trestle was used to determine the azimuth of the movable arm mounting the tilt sensor via linear interpolation between pulses generated once per rotation. The wind observation pole was 5 m high, approximately the same as the hub height. A set of cup anemometers, wind vanes (CYG-3002VM, CLIMATEC Inc.), and a two-dimensional ultrasonic anemometer (CYG-85000, CLIMATEC Inc.) were installed at the pole to measure wind condition data. Logger-1 (sampling rate: approximately 0.67 Hz), built into the wind turbine control panel, records the wind direction and speed data measured by a two-dimensional ultrasonic anemometer, generator voltage and current, and generator rotational speed. The data measured by the cup anemometer and wind vane were synchronously recorded along with the pulse signal of the photoelectric sensor using Logger-2, which recorded the inclination angle of the movable arm system. In addition, another Logger-3 (GL240, Graphtec Corp.) was installed, which recorded the wind direction and speed data from the two-dimensional ultrasonic anemometer, the voltage and current of the generator, and the wind turbine rotational speed at a sampling rate of 1 Hz. Although there are some differences between the data from the different measuring instruments and loggers, it has been confirmed that the values for the same physical quantities are almost identical. The time lag between the loggers was corrected by shifting the time axis of the data or performing interpolation as necessary.

5. Results and Discussion

5.1. Comparison of Behavior of Movable Arm between with and Without Dampers

Figure 12 demonstrates typical behavior of the slant angle η of the movable arm in the case without dampers. The measured data (obtained with Logger-2) in Figure 12 correspond to 5 min (300 s) from 00:01:00 to 00:06:00 on April 16, 2022. The red, blue, and green curves represent the slant angle, rotational speed, and wind speed, respectively. When the rotational speed exceeded approximately 55 rpm, the movable arm tilted significantly to suppress the increase in rotational speed; however, the movable arm repeatedly inclined violently up and down during one wind turbine rotation. Initially, the movable arm was equipped with stoppers (limiters) to limit the operating range of the tilt angle (5° ≤ η ≤ 90°). However, owing to the drastic variation in the tilt angle, the generation of an impact on the stoppers and significant noise were observed under the condition of high rotor rotation.

Figure 13 illustrates the behavior of the slant angle η in the case with dampers. The measured data in Figure 13 correspond to 5 min (300 s) from 15:33:00 to 15:38:00 on January 20, 2023, and the colored curves indicate the same physical quantities as illustrated in Figure 12. The tilting movement of the movable arm became gentle owing to the damper resistance torque. The impact on the stoppers and noise was almost negligible.

Figure 14 presents a graph comparing the azimuth dependence of the slant angle during one rotation of the wind turbine rotor with and without dampers (data obtained using Logger-2). The horizontal axis is the azimuth angle Ψ defined in Figure 6, and the red plot is without dampers (average value during one rotation: V = 5.4 m/s, N = 60.0 rpm, η_ave = 48.6°), and the blue plot is the case with dampers (average value during one rotation: V = 9.6 m/s, N = 56.9 rpm, η_ave = 42.4°). In the case without dampers, the slant angle varies by one period during one rotation. The measured slant angle exceeds the range limited by the stoppers. Accelerations exceeding the tolerance of the tilt sensor probably caused a large error in the measurement of the tilt angle. When the wind speed is relatively high (V > 5 m/s), the maximum slant angle in the case without dampers occurs around Ψ = 30° as illustrated in Figure 14. However, when the wind speed is low, the peak slant angle may occur near Ψ = ±180°; the behavior of the movable arm without dampers changes greatly depending on the wind speed and rotational speed [9]. However, in the case of the dampers, the slant angle remained approximately constant during one rotor rotation. In the data presented in Figure 14, the average wind speed was rather high at 9.6 m/s; however, the rotor rotational speed was suppressed to 56.9 rpm owing to the stable slant angle in one revolution due to the effects of the dampers.

5.2. Comparison of Wind Speed Dependence of Rotational Speed between Prediction and Experiments

Figure 15 presents the predicted torque characteristics of the prototype with movable arms using ailerons with a span^(ail) of 600 mm in the absence of dampers. The calculation of the averaged rotor torque using Equation (26) was performed by setting the wind speed from 2 to 60 m/s at 1 m/s intervals and changing the rotor rotational speed from 5 to 100 rpm at 1 rpm intervals for each condition. However, only the torque curves of the main wind speeds are depicted in Figure 15 to avoid complications. The black curve in the figure represents the control target and corresponds to the generator’s load torque, including the three-time speed increaser. As mentioned previously, the current control target was inconsistent with the torque characteristics of the prototype.

The intersection of the control target and rotational torque curve at each wind speed predicts the operating point. Figure 16 compares the predicted and measured wind speed dependencies of the rotor rotational speed in the prototype without dampers. The measurement data were the 20-s moving average value of the rotor rotational speed obtained by Logger-1 during the 24-hour period from 12:00 on September 5, 2022, to 12:00 on September 6, 2022, including strong wind conditions with a typhoon approaching. However, in Logger-1, the generator rotational speed is determined from the voltage frequency of the generator. The generator speed is multiplied by the speed increase rate to obtain the rotational speed of the wind turbine. The theoretical prediction demonstrated an almost constant rotational speed of approximately 65 rpm in the wind speed range of 5–10 m/s. However, it rapidly increased to 85 rpm at a wind speed of 11 m/s, followed by high rotational speeds of over 90 rpm in the wind speed range of 13–18 m/s. However, the measurement data demonstrated scattering, with a maximum rotor speed of approximately 60 rpm at a wind speed of 8 m/s or less. When the wind speed exceeded 8 m/s, the maximum rotational speed of the measurements increased to approximately 90 rpm. There were quantitative discrepancies between the measurements and predictions; however, they exhibited qualitatively consistent trends.

Figure 17 presents the predicted torque characteristics of the prototype with ailerons of span^(ail) = 600 mm and dampers, which are depicted in the same manner as in Figure 15. Figure 18 presents the measured values for the case with dampers and the prediction. The data were obtained with Logger-1 during the 24-hour period from 12:00 on January 24, 2023, to 12:00 on January 25, 2023, when a large cold wave arrived and strong winds occurred. As illustrated in Figure 18, the plot of the measurement data corresponds to the 20-s moving average of the rotor rotational speed. The theoretical predictions and measured data were in good quantitative agreement, with a maximum rotor speed of approximately 60 rpm in the wind speed range of 5–10 m/s.

5.3. Prediction and Experimental Data after Replacement with Short Ailerons

Figure 19 presents the theoretical prediction of the torque characteristics of the prototype when the aileron length was set to span^(ail) = 400 mm in the case with the dampers. Compared to Figure 17, the maximum value of the predicted torque curve increases, and the maximum rotational speed at the operating points also increases by shortening the aileron length. Similar to Figure 18, in Figure 20, the wind-speed dependence of the rotor rotational speed predicted from the torque characteristics in Figure 19 is compared with the rotational speed data measured in the prototype. The measurement data are the 20-s moving average value of the rotor rotational speed obtained by Logger-1 during the 24-hour period from 00:00 on August 15, 2023, to 00:00 on August 16, 2023, including strong wind conditions with a typhoon approaching. The theoretically predicted maximum rotational speed for the aileron length of 600 mm is 63.6 rpm at the wind speed of 7 m/s. However, for the aileron length of 400 mm, the predicted maximum rotational speed is 76.6 rpm at the wind speed of 8 m/s. In the measured data, when the aileron length was short, there was a slight tendency for the rotational speed to increase at a wind speed of approximately 10 m/s. However, even in the measurement data, changing the aileron length from 600 mm to 400 mm increased the maximum rotational speed by approximately 10 rpm, which roughly agrees with the prediction. However, with regard to the data measured with the dampers, data at average wind speeds of 15 m/s or higher were not obtained. Therefore, comparisons with theoretical predictions are impossible. In high-wind conditions, demonstrating the effectiveness of both the movable arm with dampers and predicting the behavior are topics for future studies.

6. Conclusions

A small vertical-axis-type butterfly wind turbine prototype (6.92 m in diameter) equipped with a novel over-speed-control system with movable arms was developed. To improve the system characteristics, several disk dampers were installed on the rotating axis of each movable arm. To predict the performance of the prototype, including the movement of the movable arms, we developed a prediction method that combines a flow field analysis based on the blade element momentum theory with an analysis of the equation of motion of the movable arm. The prediction of the behavior of the movable arm using the developed method demonstrated that without dampers, the slant angle variation of the movable arm may not converge; however, with dampers, it converges to a constant value. The measurement results of the slant angle of a movable arm in field experiments of the prototype demonstrated that the slant angle varies significantly during one rotor rotation and reaches a maximum when the movable arm moves from the downwind side to the upwind side without dampers and at a relatively high wind speed (5 m/s or more). However, in the case with the dampers, the slant angle remained almost constant during one rotor rotation, and the variation in the slant angle due to changes in the wind speed was slow. The impact on the stopper and noise generation, problems without dampers, was resolved by introducing dampers. The theoretical predictions were compared with the measured data regarding the dependence of the rotor rotational speed on the wind speed in the cases with and without dampers or in the case of a shorter aileron span. The predictions demonstrated almost quantitative agreement with the measurements. In the prototype, changing the aileron span length from 600 to 400 mm increased the maximum rotor speed by approximately 10 rpm.