1 Introduction

Since wave energy became a worldwide concern, considerable efforts have been invested in the development of wave energy converters (WECs) (Cruz, 2008; Drew et al., 2009; Falcão, 2010; Truong and Ahn, 2014; Diamond et al., 2015). Among various WECs, oscillating buoy attracts great attention because of its high efficiency, relaxed water-depth limitation, and multiform array deployment (Falnes, 2002; Larsson and Falnes, 2006; Falcão, 2008; Choi et al., 2012; Zhang et al., 2014).

Various oscillating buoy WECs have been developed since the 1980s. G-1T, proposed in Japan, was tested in Tokyo Bay in 1980 (Tanaka, 1984). Wave-power buoy, which consists of a spherical buoy and a pillar attached to the seabed, was developed by a Norwegian and tested in Trondheim in 1983 (Budal and Falnes, 1982). Swedish heaving buoy, which consists of a cylinder buoy and a linear motor, was developed by Uppsala University and tested in Lysekil in 2006–2017 (Ulvgård, 2017). PowerBuoy, which consists of a buoy, a standpipe, and a damping flap, was developed by Ocean Power Technologies and tested in Scotland in 2011 (Ma, 2013). Other types of WECs are Wavebob (Weber et al., 2009), AquaBuoy (Previsic et al., 2004), CETO (Carnegie Clean Energy, 2018), and StingRAY (Columbia Power Technology, 2018).

In buoy development, efforts such as geometry optimi-zation, power take-off (PTO) system improvement, control strategy selection, and others have been conducted to ensure high efficiency and stable output. A common approach is increasing degrees of freedom (DOFs). One of the best-known WECs with multiple DOFs is Pelamis (Falcão, 2010), a snake-like slack-moored articulated structure that captures energy from pitch and roll motions with a hydraulic system. Albatern Wave Energy (2017) developed WaveNET, an array-based converter with many 6-DOF cylindrical buoys. CorPower Ocean AB (2018) developed a single-buoy WEC anchored to the seabed with a tensioned cable and uses a gearbox to capture energy from surge and heave motions. Wello (2018) developed Penguin, an asymmetrical boat-like converter with an interior rotator, which also captures energy from pitch and roll motions. Iijima and Taya (2011) developed Heave and Pitch Buoy, a miniature WEC with two buoys.

Experimental study is a direct and efficient way to obtain real data and observe phenomena. Zhang et al. (2015) proposed a multi-axis WEC and conducted a physical test to analyze its hydrodynamic and power capture performance. Do et al. (2017) proposed a sliding angle self-tuning WEC with hydraulic PTO and verified the sliding angle performance and efficiency. Ning et al. (2016) and Zhao and Ning (2018) studied a breakwater-type WEC with a model test, investigating the influences between a vertical pile and double floaters. Cho and Kim (2017) proposed a two-concentric-cylindrical-buoy WEC and conducted a systematic parametric study through analytical and physical analyses. Chandrasekaran and Raghavi (2015) proposed a buoy-type WEC to address the operational energy demands of an offshore platform and physically tested its design parameters with power output.

The motion of every DOF is assumed to interact with the motions of other DOFs. To determine if this assumption is true, the present study investigates a multi-DOF buoy WEC by physical model experiments. The buoy is a rigid cylinder that adopts itself to all wave directions. It has three DOFs, which are surge, heave, and pitch. The unidirectional and coupling motions are obtained by guide bars and locking rings. The Coulomb damping model is applied as the PTO system (Babarit et al., 2012). A magnetic powder brake and a current controller are used to produce the Coulomb damping force. Experiments without and with PTO damping are conducted, in which the hydrodynamic characteristics and power capture performance are compared.

This paper is organized as follows. In Section 2, a physical experiment is described with a model setup and case illustrations. Data acquisition and analysis methods are also introduced. In Section 3, experiment results are summarized. Coupling effects, PTO damping effects, and power absorption performance are also analyzed. Conclusions are drawn in Section 4.

2 Methodologies 2.1 Experimental Setup

The experiments are conducted in a wave tank at Shandong Provincial Key Laboratory of Ocean Engineering, Ocean University of China. The wave tank has the following dimensions: length of 60 m, width of 36 m, and depth of 1.5 m. A piston-type wavemaker is installed at the fore-end and a wave absorberis located at the rear.

Figs. 1 and 2 show the physical model of the multi-DOF buoy. The radius of the buoy is 0.4 m and the height is 0.4 m. The buoy has three DOFs: surge, heave, and pitch. The 3-DOF buoy is installed in a braced frame with height of 1.78 m, width of 1.5 m along the wave direction, and width of 1.5 m perpendicular to the wave direction. The motions along the three DOFs are accomplished by a horizontal bar, a vertical guide bar, and a rolling bearing. A sliding frame connects the horizontal and vertical guide bars. A spring is assembled to the sliding frame to provide restoring force for surge motion. The PTO system of every DOF consists of a magnetic powder brake and a current controller. The brake provides bidirectional constant torque, which is used to model the Coulomb damping. The controller controls the torque magnitude by adjusting the current of the magnetic powder brake. For surge and heave, each PTO system is connected to the buoy with a gear-and-rack transmission mechanism, which transfers the rectilinear motion of the buoy to the rotation of the brake shaft. For pitch, the PTO system is connected to a synchronous belt-and-pulley transmission mechanism, which transfers the rotation of the buoy to the rotation of the brake shaft. Fig.3 shows a sketch of the experimental setup. Two wave gauges are placed in the front and at the rear of the model for wave monitoring. A position sensor (NDI Optotrak Certus 3D investigator) is placed next to the model to measure the motion of the buoy.

In all cases, the still-water depth is fixed at 1 m and the wave height is fixed at 0.2 m. The draft of the buoy is 0.12 m, which is also unchanged during the experiment. The stiffness coefficient of the spring is 85 N m⁻¹. The test cases of the motions without and with PTO damping are listed in Tables 1 and 2, where T denotes the wave period. In the tests without PTO damping, the mechanical friction in heave is large when the wave period is small. In the test cases with PTO damping, the wave period is fixed at 2.0 s to minimize the influence of the mechanical friction on the results. All the cases are tested at least twice to meet the accuracy requirement.

2.2 Data Acquisition Analysis

The data acquisition setup includes a set of wave sensors used for wave height and period and an NDI Optotrak Certus 3D investigator for sensing the buoy displacement with sampling frequency of 100 Hz.

The PTO damping of surge and heave are calculated by dividing the torque of the brake by the gear radius (3.6 cm). The PTO damping torque of pitch is calculated by multiplying the torque of the brake by the radius ratio of the pulleys (6.4 cm/1.55 cm). The real-time output power is calculated by multiplying the PTO damping by the velocity of the buoy. The time-averaged power $\bar P$ is averaged by the real-time output power. In this study, $\bar P$ is calculated with the data of 20 s. The incident wave power per unit width is expressed as

${P_{{\rm{wave}}}} = \frac{1}{4}\frac{{\rho g{A^2}\omega }}{k}\left({1 + \frac{{2kd}}{{\sinh 2kd}}} \right), $

(1)

where ρ is water density, g is gravitational acceleration, A is wave amplitude, ω is wave frequency, k is wave number, and d is the water depth.

Capture width ratio (CWR) is

$\eta = \frac{{\bar P}}{{{P_{{\rm{wave}}}} \cdot 2R}}, $

(2)

where R is the buoy radius.

3 Results and Discussion 3.1 Coupling Effects

A none-PTO test is conducted to investigate the hydrodynamic performance and coupling effects on the buoy movements.

Fig.4 shows the time histories of the buoy displacements of every motion mode. Test cases are those in which the wave period is 2.0 s, as shown in Table 1. The buoy periodically and sinusoidally (or approximately sinusoidally) moves in every DOF in the case of unidirectional and coupling motions. The coupling effect between the DOFs affects the movement.

Fig.5 shows the amplitudes of the buoy under the test cases in Table 1. Figs. 5(a) and 5(b) illustrate that with the raising wave period, the amplitudes of surge and heave increase in the unidirectional and coupling motions but decrease in pitch (Fig.5(c)). In the surge and heave coupling motion, compared with the surge unidirectional motion, the amplitudes increase in a smaller wave period and decrease in a larger one (Fig.5(a)) but exhibit minor differences. Compared with the heave unidirectional motion, the amplitudes increase in all wave periods (Fig.5(b)). As a result of the horizontal wave force on the buoy, a large moment acts on the contact surface of the vertical guide bar and rolling bearing, thereby leading to mechanical friction. Adding surge DOF reduces the friction considerably. This phenomenon is obvious in a small wave period in which the horizontal wave force is large. For the surge and pitch coupling motion, compared with the corresponding unidirectional motions, the surge amplitude increases slightly in all wave periods (Fig.5(a)), whereas the pitch amplitude decreases slightly (Fig.5(c)). For the heave and pitch coupling motion, compared with the corresponding unidirectional motions, the heave amplitude decreases slightly in all wave periods (Fig.5(b)), whereas the pitch amplitude increases slightly (Fig.5(c)). In other words, the coupling effect can cause the amplitude of one DOF to increase slightly and the others to decrease except when the surge DOF is added, which reduces the mechanical friction to benefit the heave motion.

3.2 PTO Damping Effects

Tests with PTO damping are conducted to investigate the hydrodynamic performance and coupling effects of the buoy. In the coupling motions, the PTO damping in one DOF is changed while those of the others are fixed to analyze the amplitude variations.

Fig.6 shows the amplitudes of surge and heave motion with different PTO damping. As shown in Figs. 6(a) and 6(b), the PTO damping force of heave is fixed to 36.1 N and 126.4 N separately and the PTO damping force of surge is changed. In Figs. 6(c) and 6(d), the PTO damping force of surge is fixed to 12.2 N and 39.9 N separately and the heave is changed. Fig.6(a) indicates that adding a surge DOF enlarges the heave amplitude. The heave amplitude increases further when the PTO damping of surge is small. This condition means that the larger the surge amplitude is, the larger is the heave amplitude. The same conclusion can be inferred from Fig.6(c). Figs. 6(b) and 6(d) show that adding heave motion can benefit the surge amplitude with a small PTO damping force of surge, whereas the opposite phenomenon is shown with a large PTO damping force of surge.

Fig.7(a) shows the amplitudes of surge and pitch coupling motion with PTO damping. In Figs. 7(a) and 7(b), the PTO damping force of surge is fixed to 12.2 N and 39.9 N separately and the PTO damping moment of pitch is changed. In Figs. 7(c) and 7(d), the PTO damping moment of pitch is fixed to 0.5 N m and 7 N m separately and that of surge is changed. As shown in Figs. 7(a) and 7(c), adding pitch DOF causes the surge amplitude to decrease except when the PTO damping force of surge is small enough. The pitch amplitude increases except when the PTO damping moment of pitch is large and the PTO damping force of surge is small, as shown in Figs. 7(b) and 7(d). In most cases, the coupling effect makes the surge amplitude decrease and the pitch amplitude increase.

Fig.8 shows the amplitudes of heave and pitch coupling motion with PTO damping. In Figs. 8(a) and 8(b), the PTO damping force of heave is fixed to 36.1 N and 126.4 N and changes that of pitch. In Figs. 8(c) and 8(d), the PTO damping force in pitch is fixed to 0.5 N m and 7 N m and changes that of heave. As shown in Figs. 8(a) and 8(c), compared with the amplitudes of unidirectional motions, the heave amplitude increases except when the heave PTO damping force is small enough. Figs. 8(b) and 8(d) shows that the pitch amplitude decreases except in special cases with small heave PTO damping force. The coupling effect causes the heave amplitude to increase and the pitch amplitude to decrease in most cases.

Compared with the free motions, the coupling effect leads to different variations of the buoy amplitudes. However, in most cases, the coupling effect makes the amplitude of one DOF increase and the other decrease, which is the same as the free motions.

3.3 Power Absorption Performance

CWRs with different PTO damping are calculated to investigate the power absorption performance. Test cases are presented in Table 2 and the wave period is fixed at 2.0 s. The calculation method is introduced in Section 2.2. The CWRs of the coupling motions are compared with the results of the corresponding unidirectional motions. Growth rates are calculated.

Table 3 shows the CWRs of the surge and heave coupling motion. The CWR can exceed 30% with suitable PTO damping. The CWRs of the heave unidirectional motion is much larger than those of the surge unidirectional motion, which means that heave is the main power absorption DOF in the surge and heave coupling motion. Table 4 presents the growth rates of the CWR relative to the heave unidirectional motion. Compared with the unidirectional motions, the coupling effect improves the power absorption in all cases. The growth rate is between 23% and 53%. The growth rate decreases first, and then increases with the increasing PTO damping in heave, while decreasing with the increasing PTO damping in surge in most cases.

Table 5 shows the CWRs of the surge and pitch coupling motion. The CWRs in all cases are much lower than those of the surge and heave coupling motions. Furthermore, the CWRs of the surge unidirectional motion are larger than those of the pitch unidirectional motion, which means that surge is the main power absorption DOF in the surge and pitch coupling motion. Table 6 reports the growth rates of the CWR relative to the surge unidirectional motion. The CWR decreases with the increasing PTO damping in surge and increases with the increasing PTO damping in pitch. When the PTO damping in surge is large and that in pitch is small, the CWR shows negative growth, which means that the coupling effect canweaken the power absorption in certain cases. When the PTO damping in surge is small and that in pitch is large, the growth rate can exceed 70%, thereby improving the power absorption significantly.

Table 7 shows the CWRs of the heave and pitch coupling motion. Heave is also the main power absorption DOF, as indicated by the comparison of the CWRs of the two unidirectional motions. Table 8 shows the growth rates of the CWR relative to the heave unidirectional motion. The coupling effect raises the CWRs in all cases. The growth rate increases with the increasing PTO damping in pitch and increases with the increasing PTO damping in heave.

Table 9 reports the CWRs of the 3-DOF coupling motion. The PTO damping moment in pitch is 0.5 N m and is changeable in surge and heave. A comparison of the CWRs of the unidirectional motions shows that heave is the main power absorption DOF. Compared with the results in Table 3, the CWRs are smaller than those of the surge and heave coupling motion, except in one case (F_PTO in surge is 39.9 N and F_PTO in heave in 126.4 N). This condition illustrates that adding the pitch DOF can weaken the power absorption of the surge and heave coupling motion. The reason for the exception is that the PTO damping of surge and heave are large and the corresponding amplitudes are small enough, making the coupling effect unapparent. Table 10 shows the CWR growth rate of the 3-DOF coupling motion relative to the heave unidirectional motion. A positive growth of CWR is observed in all cases, which means that the power absorption of the 3-DOF coupling motion is superior to those of the unidirectional motions. The growth rate increases with the increasing PTO damping in heave, and increases first and then decreases with the increasing PTO damping in surge.

Table 11 shows the CWRs of the 3-DOF coupling motion. The PTO damping moment in pitch is 7 N m. A comparison of the CWRs of the unidirectional motions shows that heave is also the main power absorption DOF. The power absorption performance is better than that with 0.5 N m F_PTO in pitch (Table 9) but worse than that of surge and heave coupling motion (Table 3), except in one case (F_PTO in surge is 39.9 N and F_PTO in heave in 126.4 N). Table 12 shows the growth rate of CWR relative to the heave unidirectional motion. All the rates are positive, showing the superiority of the 3-DOF coupling motion in power absorption. The growth rate decreases first, and then increases with the increasing PTO damping in heave. Furthermore, the growth rate increases first, and then decreases with the increasing PTO damping in surge.

The preceding results indicate that adding DOFs can improve the power absorption performance in most cases, but a larger number of DOFs does not necessarily lead to better performance. The improvement depends on the PTO damping in the DOFs. The growth rate decreases first, and then increases with the increasing PTO damping in heave. Furthermore, the growth rate increases first, and then decreases with the increasing PTO damping in surge and pitch.

4 Conclusions

An experimental study on a multi-freedom buoy is conducted in the wave tank. Amplitudes of unidirectional motions and coupling motions under different wave periods and PTO damping are measured to analyze the hydrodynamic performance. CWRs are calculated.

The coupling effect causes the amplitude in one DOF to increase and the amplitudes in other DOFs to decrease, whether without or with PTO damping. This result means that the kinematic energy of the buoy is redistributed to the DOFs.

Adding DOFs improves the buoy power absorption in most cases. However, a larger number of DOFs does not necessarily lead to improved performance. The performance depends on the PTO damping in the DOFs.

Acknowledgements

The authors are grateful for the support of the National Key R & D Program of China (No. 2018YFB1501900), the National Natural Science Fund of China (No. 41706100), the Shandong Provincial Natural Science Key Basic Program (No. ZR2017ZA 0202), the Special Projects for Marine Renewable Energy (No. GHME2016YY02), the Shan-dong Provincial Key Laboratory of Ocean Engineering, and the Qingdao Municipal Key Laboratory of Ocean Renewable Energy.