Use of Dampers to Improve the Overspeed Control System with Movable Arms for Butterfly Wind Turbines

Abstract

To reduce the cost of small wind turbines, a prototype of a butterfly wind turbine (6.92 m in diameter), a small vertical-axis type, was developed with many parts made of extruded aluminum suitable for mass production. An overspeed control system with movable arms that operated using centrifugal and aerodynamic forces was installed for further cost reduction. Introducing this mechanism eliminates the need for large active brakes and expands the operating wind speed range of the wind turbine. However, although the mechanism involving the use of only bearings is simple, the violent movement of the movable arms can be a challenge. To address this in the present study, dampers were introduced on the movable arm rotation axes to improve the movement of the movable arms. To predict the behavior of a movable arm and the performance of the wind turbine with the mechanism, a simulation method was developed based on the blade element momentum theory and the equation of motion of the movable arm system. A comparison of experiments and predictions with and without dampers demonstrated qualitative agreement. In the case with dampers, measurements confirmed the predicted increase in the rotor rotational speed when the shorter ailerons installed perpendicularly to the movable arms were used to achieve the inclination. Field experiments of the generated power at a wind speed of 6 m/s (10 min average) showed relative performance improvements of 11.4% by installing dampers, 91.3% by shortening the aileron length, and 57.6% by changing the control target data. The movable arm system with dampers is expected to be a useful device for vertical-axis wind turbines that are difficult to control.

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 [2,3], VAWTs are difficult to control.

When the wind speed exceeds the rated wind speed, most wind turbines must either stop rotating or limit their speed to prevent structural or electrical damage. Mechanical brakes [4], such as disc brakes [5], or electric brakes [6] that use dump resistance [7] are generally used to control excessive speed. However, these systems, which require external power (power supply), may become unusable when power outages due to lightning strikes, etc., occur; moreover, they are expensive and can thus increase the costs of small VAWTs.

For example, to prevent the overspeed 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 [8]. However, even during low-speed operation, losses occur because the pump is driven. Recently, Solomin proposed a passive mechanical brake that utilizes centrifugal force to drive pendulums to which brake pads are attached [9]. This system is a governor-type brake equipped with weights and springs. Solomin conducted an operational analysis assuming that the system was applied to a 3 kW VAWT. Pitch control is possible even in VAWT, and Noda et al., realized overspeed suppression mechanisms for gyromill-type VAWTs [10] using centrifugal force and springs [11,12]. 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 [13,14]. A more common and simple method for operating aerodynamic brakes in VAWTs is to install flaps [15] or spoilers [16,17] on certain portions of the blades and apply centrifugal force to move them. 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 [18]. With the same operating principle as Tanzawa et al.’s device, Hara et al., developed an overspeed control system that inclines the VAWT blades by the centrifugal force and synchronizes the movement of the blades by a link mechanism [19]. The overspeed control system proposed by Hara et al., is a mechanism that changes the swept area of a wind turbine. It can be categorized as the well-known Musgrove rotor [20,21] developed in the 1980s, which can transform vertical blades into an arrow shape using hydraulic power. Whitehouse et al., proposed a VG-VAWT [22], in which the swept area of a Darrieus-type floating offshore wind turbine was varied to reduce structural and maintenance costs. A customized VAWT [23] that applies the Magnus force [21] was developed to drive the rotor that can actively control the rotational speed of the cylindrical blades to avoid overspeeding. However, most of the overspeed 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. Table 1 summarizes the overspeed-suppression devices in the VAWT. The information provided in this table is subjective; however, the evaluations were performed based on five viewpoints: cost, controllability, simplicity, driving power, and durability. For example, in terms of driving power, if the driving force is centrifugal force, then it is evaluated as “good”, whereas if electric power or hydraulic drive is required, then it is evaluated as “unsatisfactory”.

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 overspeed control system with movable arms that operates using centrifugal and aerodynamic forces without springs or link mechanisms is 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 [24], 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 a 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.

Table 1. Overspeed control system for VAWTs.

Category	Subcategory	Cost	Controllability	Simplicity	Driving Power	Durability	Reference
Mechanical	Disc brake	×	✓	×	×	−	[4]
	Pump load	−	×	×	✓	✓	[8]
	Centrifugal force + spring	✓	−	−	✓	×	[9]
Electrical	Load resistance	×	✓	−	×	✓	[6]
Pitch control	Centrifugal force + spring	−	−	−	✓	×	[11,12]
	Link mechanism	×	−	×	✓	×	[10,13,14,25]
Aerodynamic	Flap/Spoiler	✓	−	−	✓	×	[15,16,17]
	Flat plate upon rotational shaft	−	−	−	✓	×	[18]
	Variable arm	✓	−	✓	✓	−	[24]
Variable geometry	Tilting blade to tangential	×	×	×	✓	×	[19]
	Musgrove rotor	×	✓	×	×	−	[20,21]
	Variable rotor height	−	−	−	−	−	[22]
Others	Magnus force	×	✓	−	×	−	[21,23]

good = ✓, fair = −, unsatisfactory = ×.

Table 1. Overspeed control system for VAWTs.

Category	Subcategory	Cost	Controllability	Simplicity	Driving Power	Durability	Reference
Mechanical	Disc brake	×	✓	×	×	−	[4]
	Pump load	−	×	×	✓	✓	[8]
	Centrifugal force + spring	✓	−	−	✓	×	[9]
Electrical	Load resistance	×	✓	−	×	✓	[6]
Pitch control	Centrifugal force + spring	−	−	−	✓	×	[11,12]
	Link mechanism	×	−	×	✓	×	[10,13,14,25]
Aerodynamic	Flap/Spoiler	✓	−	−	✓	×	[15,16,17]
	Flat plate upon rotational shaft	−	−	−	✓	×	[18]
	Variable arm	✓	−	✓	✓	−	[24]
Variable geometry	Tilting blade to tangential	×	×	×	✓	×	[19]
	Musgrove rotor	×	✓	×	×	−	[20,21]
	Variable rotor height	−	−	−	−	−	[22]
Others	Magnus force	×	✓	−	×	−	[21,23]

good = ✓, fair = −, unsatisfactory = ×.

2. Prototype of Butterfly Wind Turbine Equipped with Overspeed 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 [26] 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 is c = 375 mm, which is the same as that of the mass-produced 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-produced type.

Figure 2 illustrates the outline and size of the prototype. Each movable arm is installed in one portion of the reinforcing horizontal arm, which spans 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 gravitational force acting on the movable arm and aileron, and when it is tilted greatly, a 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 overspeed 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) [19]. Here, the load characteristics of the prototype rotor and generator (control targets) were inconsistent.

Bidirectional disk dampers (FDT-57A-503, Fuji Latex Co., Ltd., Tokyo, Japan) 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 [27], adopting the quadruple-multiple streamtube model [28]. Figure 6 presents a schematic of the flow-field calculation method. In this calculation, one streamtube was divided into upwind and downwind sides, and the streamtube 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 streamtube, the intersectional wind speeds were 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 streamtubes 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 streamtubes 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 was 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 was introduced to specify the streamtube, and the origin of the parameter I (I = 0) was defined as the direction of Ψ = −180°. The value of I increases counterclockwise, and the final value (I = 360) corresponds to the streamtube 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. This value is stipulated for small wind turbines under normal wind conditions in the Japanese Industrial Standards, JIS C 1400-2 [29]. 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. Because small wind turbines are generally installed at lower heights, the average wind speed profile is affected significantly by ground friction.

Aerodynamic data based on NACA 0018 data [30] 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 [26] in the simulation. This study adopted the modified Gormont model [27] 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 was 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) was 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 converted 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).

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

(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.

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

(12)

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

(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. In Equations (10)–(13), to simplify the description, the azimuth dependence (pseudo time dependence) of the inclination angle η and the angle of attack α_i are not explicitly shown. Similar to the calculation for the main blades, the aerodynamic coefficients for the movable arms and ailerons were calculated using the Gormont model [27] modified using $\mathrm{M}\mathrm{a}\mathrm{s}\mathrm{s}\acute{\mathrm{e}}$s’ method. In this study, the empirical constant A_M used in $\mathrm{M}\mathrm{a}\mathrm{s}\mathrm{s}\acute{\mathrm{e}}$s’ method was set to A_M = 2.5.

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).

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

(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 were 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) was numerically integrated to determine the inclination angle η(n, I) at each azimuth of the n-th rotation. As initial values, the azimuth angle was set as Ψ = −180°, the inclination angle was set as η(0,0) = η_ini, and the inclination angular velocity was 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 Δη_ave of the inclination angle was checked with Equation (24) each time the calculation for one rotation was completed.

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

(24)

If Δη_ave becomes smaller than a certain constant value ε (ε = 0.2 here), the variation in 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 it reaches 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 had three movable arms; however, a tilt sensor (M21-0455, Kyowa Electronic Instruments Co., Ltd., Tokyo, Japan) was 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., Yokohama, Japan) 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., Kyoto, Japan) 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., Tokyo, Japan), 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 were some differences between the data from the different measuring instruments and loggers, it was confirmed that the values for the same physical quantities were 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 the 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 16 April 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 20 January 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 were 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; 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 varied by one period during one rotation. The measured slant angle exceeded 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 [24]. 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. The control target, i.e., the load torque, is changed in accordance with the wind turbine rotation speed by adjusting the output current of the generator via pulse-width modulation [31], which serves as a pseudo-variable electrical resistance. The load torque independent of the generator’s output current, arises due to the frictional resistance of the speed increaser, bearings, etc., and is added to the current-dependent load torque when calculating the control target. Meanwhile, the current-independent load torque was obtained from preliminary experiments conducted by the generator manufacturer.

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 h period from 12:00 on 5 September 2022 to 12:00 on 6 September 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 h period from 12:00 on 24 January 2023 to 12:00 on 25 January 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. However, the difference between the rotation speed prediction, assuming a uniform and constant wind speed, and the actual wind turbine rotation speed was affected by the unsteady wind speed, the turbine inertia, and limitations in the rotation speed measurement. This issue is explained in Appendix C.

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 is 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 h period from 00:00 on 15 August 2023 to 00:00 on 16 August 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 were impossible. In high-wind conditions, demonstrating the effectiveness of both the movable arm with dampers and predicting the behavior are topics for future studies. The behavior of wind turbines under high wind speeds must be investigated and their safety and reliability clarified. Hence, we developed an aeroelastic analysis system for butterfly wind turbines and performed fatigue analysis of a 14 m turbine rotor.

5.4. Prediction and Experimental Data after Changing Control Target Data

Without changing the rotor structure, i.e., under the condition with dampers and the aileron length of span^(ail) = 400 mm, only the table data (control target) of the load torque in the control device were changed, as shown by the solid black line in Figure 21. The predicted torque curves shown in Figure 21 are the same as those shown in Figure 19. In this case, the control target curve was set to pass through the maximum power-coefficient point of each wind speed below 6 m/s. At a wind speed of 7 m/s, the control target intersected the torque curve of 7 m/s at approximately 63 rpm, which was lower than the start condition to tilt the movable arms. However, because of the maximum current limit of the generator, the maximum torque was set to 587.6 N$·$m.

Figure 22 shows a comparison of the wind speed dependence of the wind turbine rotation speed predicted from the intersections between the torque curves and the control target curve in Figure 21, with the rotation speed data measured in the prototype after changing the control target data. The measurement data were the 20 s moving average of the rotor rotational speed when the wind was relatively strong, which were obtained using Logger-1 during the 24 h period from 18:00 on 5 November to 18:00 on 6 November 2023. As shown in Figure 22, the rotational speed was predicted to decrease significantly in the wind speed range of 14–24 m/s. This is because the control target curve in Figure 21 intersects the torque curves at low rotational speeds in the wind -speed range. At a wind speed of 5 m/s for the case presented in Figure 20 before changing the control target, the predicted rotational speed was 64.4 rpm, and the measured data dispersed around it, with a recorded maximum value of approximately 68 rpm. Meanwhile, in the case shown in Figure 22, after changing the control target, the predicted rotational speed at 5 m/s was 52.1 rpm, and most of the measured data at 5 m/s were distributed between 33 and 56 rpm with a maximum value of 60 rpm. This indicates that the rotational speed at the operating point (intersection point) decreased when the control target curve was set to a high state.

5.5. Comparison of Power Generation

Figure 23 shows the wind speed dependence of the generated electric power P_e obtained in the field experiments shown in Figure 16, Figure 18, Figure 20 and Figure 22. The results were processed using the bin method (bin width: 0.5 m/s) for two days of measurement data, including the strong-wind time range for each case, and the relations between the 10 min average wind speed and generated power were plotted as shown in Figure 23. The installation of the dampers tended to increase the generated power. Meanwhile, reducing the length of the ailerons from 600 to 400 mm increased the wind turbine torque at each wind speed, which increased the maximum rotational speed and, thus, power generation. Furthermore, by changing the control target data, the power generation improved at a wind speed of 5.5 m/s or higher. Before changing the control target, the overall load torque was set to a low value. Therefore, the movable arm began tilting from a relatively low wind speed of approximately 6 m/s, and the aerodynamic brake was activated, which significantly decreased the rotational speed temporarily. In this case, even when the wind speed was high, the rotation speed could not immediately return to the optimum value owing to the inertia of the rotor. However, after changing the control target, because of the high-load torque, the initial wind speed of the aerodynamic brake increased, thus improving the generated power. Therefore, if the control target is further improved to obtain stable operating points, where the aerodynamic brake is not activated, at high wind speeds of 8–10 m/s, then a further increase in power generation is expected, similar to the dashed-dotted line shown in Figure 23. The dashed-dotted line represents an extrapolation of the measurement data below 5 m/s after changing the control target.

Table 2. Comparison of electric power and total efficiency at wind speed of 6 m/s.

Wind Turbine Condition	Electric Power: Pe [W]	Total Efficiency: ηt [%]	Relative Improvement [%]
without damper, span(ail) = 600 mm	217	3.52	-
with damper, span(ail) = 600 mm	241	3.92	11.4
with damper, span(ail) = 400 mm	461	7.50	91.3
after changing control data	726	11.82	57.6

shows a comparison of the generated power obtained at a wind speed of $V_{\infty }$ = 6 m/s and the total power generation efficiency η_t which is defined as the ratio of the generated power P_e to the wind energy ($0.5\mathsf{\rho }{V}_{\infty }^{3}A$). Based on Table 2, the step-by-step increase in the power generation efficiency is 11.4% by adding the dampers, 91.3% by shortening the ailerons, and 57.6% by changing the control target.

6. Conclusions

A small vertical-axis-type butterfly wind turbine prototype (6.92 m in diameter) equipped with a novel overspeed 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 in 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, were 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. Furthermore, the generated power improved at a wind speed of 5.5 m/s or higher by changing the control target. At a wind speed of 6 m/s, the power generation increased by 11.4% by installing the dampers, 91.3% by shortening the aileron length, and 57.6% by changing the control target. The overspeed control system using movable arms has been compared with various overspeed-suppression methods investigated or adopted thus far and is considered to be highly favorable in terms of cost, simplicity, and driving force. The movable arms equipped with dampers proposed herein are expected to exhibit increased durability and reliability. The use of this system will eliminate the necessity for large disc brakes or dump electric resistances, which have been primarily used previously, thus reducing costs significantly. Herein, we have described the design and characteristic-prediction method for movable arms with dampers, which improve the controllability of vertical-axis wind turbines and can potentially reduce costs. Therefore, the contents of this paper would benefit researchers of vertical-axis wind turbines.