A Novel Type of Wave Energy Converter with Five Degrees of Freedom and Preliminary Investigations on Power-Generating Capacity

Abstract

In order to further improve the power-generating capacity of the wave energy converter (WEC) of oscillating buoy type, this paper puts forward a novel type where the WEC can move and extract power in five degrees of freedom. We make a detailed hydrodynamic analysis of such WECs. Each buoy is modeled as a floating truncated cylinder with five degrees of freedom: surge, sway, heave, roll, and pitch, and there are relative motions among buoys in the array. Linear power take-off (PTO) characteristics are considered for simplicity. Under the linear wave theory, a semi-analytical method based on the eigenfunction expansion and Graf’s addition theorem for Bessel functions is proposed to analyze the hydrodynamic interactions among the WEC array under the action of incident waves, and the amplitude response and power extraction of the WEC array are then solved. After verifying the accuracy of hydrodynamic analysis and calculation, we make preliminary case studies, successively investigating the power-generating capacity of a single WEC, an array of two WECs, and an array of five WEC; then, we compare their results with the conventional heaving WECs. The results show that the WEC with five degrees of freedom can significantly improve the power extraction performance.

1. Introduction

With the gradual consumption of traditional fossil energy and the enhancement of environmental awareness, the development of clean and renewable energy has attracted more and more attention all over the world. Among all kinds of new energy, ocean wave energy shows great potential, and the global wave power resource is about 2.1 TW [1]. Research on wave energy has been carried out for decades, and a variety of wave energy converters (WECs) have been proposed [2], such as oscillating water columns, oscillating buoys, overtopping systems, and bottom-hinged systems. We focus here on the wave energy converter (WEC) of oscillating buoy type, which is characterized by converting the kinetic energy of the oscillating buoy into electricity and has the advantages of wide applicability and high conversion efficiency. Such WECs are easy to be clustered into an array, so as to reduce production cost and achieve commercial-level wave energy extraction [3]. Also, it is particularly attractive that a well-designed WEC array can significantly improve the total power generation over the same number of isolated WECs [4].

Compared with an isolated WEC buoy, each buoy in an array will be affected by the diffracted and radiated waves from other buoys, so it is of great significance to accurately investigate the hydrodynamic interactions within the array. For complex-shaped bodies, the numerical method can be used directly [5,6,7,8], but the computational cost may be huge. Semi-analytical methods are more suitable for providing a quick estimation of buoys with simple geometries, especially for large number cases; they also provide physical insights into the problem [9]. A pioneering study on WEC arrays was carried out by Budal [10]; he presented a point-absorber approximation, which is applicable when the ratio of buoy size to wavelength is small. This method was adopted and developed by Evans [11] and Falnes [12]. Within this method, the diffracted waves are neglected and only the radiated waves are considered due to the relatively large oscillating amplitudes of the buoys. In order to achieve fast calculation, there are other assumptions, such as the plane-wave approximation [13,14,15], which replaces the cylindrical wave with plane waves, which is suitable for the case that the inter-buoy spacing is much larger than the buoy size and the wavelength. The development of WEC array also puts forward higher requirements for modeling accuracy. Kagemoto and Yue [16] derived an exact algebraic method, based on the multiple scattering method [17] (Ohkusu, 1974) and the direct matrix method [18] (Spring and Monkmeyer, 1974). This method can produce accurate solutions within the scope of potential flow theory; as such, it has been widely used and developed in later studies [19,20,21,22]. The idea of the exact algebraic method was adopted in this paper for hydrodynamic interaction analysis.

The hydrodynamic interactions among WECs can be evaluated by the interaction factor q, which is defined as the ratio of the total power generation of the WEC array to that of the same number of isolated WECs [12]; q > 1 means that the interactions are constructive, while q < 1 means that the interactions are destructive. Investigations on how the interaction factor q is influenced by various parameters have become the focus of many studies, such as buoy size [23,24], power take-off (PTO) characteristics [25,26], number of buoys [27], inter-buoy spacing [28], and array layout [29,30,31,32,33]. In addition, some researchers have made efforts on how to reduce costs, such as integrating WECs with other offshore structures [34,35].

Most studies have focused on heaving WECs, in which buoys can only oscillate in heave mode, with other directions completely constrained. Heaving WESs may have simple mechanical structures and are easy to implement in engineering. However, with the development of science and technology, WECs are required to further improve power-generating capacity. Theoretically, the buoy can oscillate and absorb energy in six degrees of freedom [12], so the energy of other degrees of freedom, besides heave, should not be ignored. There are a few studies on surging WEC. McIver [36] investigated the q factor of a line of five WECs absorbing in surge and indicated that there is less sensitivity to the angle changes of wave incidence compared with heaving WECs. Babarit [28] compared the power-generating capacity of two heaving cylinders and two surging barges and pointed out that the power of the surging WEC is much more than that of heaving WEC, whereas the oscillation amplitude is much lower. Ruiz et al. [30] conducted layout optimization research for surging WECs and compared the efficiency of three different layout optimization methods. Kara [37] studied two and four truncated vertical cylinders where each cylinder can move in both heave and sway (surge) modes and concluded that surge mode has a wider frequency bandwidth of power absorption. As for pitching WECs, they are usually bottom-hinged [38], and there is no study on floating buoys to the best of the authors’ knowledge. Recently, a physical scaled experiment platform was constructed by Uppsala University [39] in which the buoy could move with six degrees of freedom, although power was extracted from heaving mode. Inspired by the above works, this paper puts forward the concept that the WEC can move and extract power in five degrees of freedom.

The main purpose of this paper is to carry out hydrodynamic modeling of such a WEC array and preliminarily investigate the power-generating capacity. The paper is structured as follows: In Section 2.1, Section 2.2, Section 2.3, Section 2.4, the analytical model of diffraction and radiation problems of the WECs with five degrees of freedom is established, and the formulas of velocity potential and motion amplitude of each degree of freedom are deduced. The power model is then described in Section 2.5. After validation in Section 3, we conducted simple cases studies in Section 4, where three typical array layouts are considered. Lastly, conclusions are made in Section 5.

2. Theoretical Analysis

2.1. Mathematical Description

In our analysis, the linearized potential flow theory is adopted. Each WEC buoy is modeled as a truncated floating circular cylinder and assumed to absorb power through the linear power take-off (PTO) system, as shown in Figure 1. Each cylinder can oscillate in surge, sway, heave, roll, and pitch mode; yaw motion is not considered because it does not produce hydrodynamic response in the ideal fluid. The concept sketch of the PTO system is used to facilitate understanding. The constraints of the PTO system can be simplified as linear spring ${\delta }_s$ and damping ${\lambda }_s$ in each direction of the cylinder. s = 1, 2, 3 represent the three translational modes, whilst s = 4, 5 represent the two rotational modes.

As shown in Figure 2a, for an array with N WECs, we introduce N + 1 coordinate systems (e.g., Zeng et al. [20]): a global Cartesian coordinate system OXYZ with the (X, Y)-plane fixed on the undisturbed free surface and OZ pointing vertically upwards; for each cylinder, we introduce a local cylindrical coordinate system o_ir_iθ_iz_i, with the origin located at the intersection of the cylindrical axis and the undisturbed free surface and o_iz_i pointing vertically upwards.

With the linearized potential flow theory, there is velocity potential $\mathsf{\Phi }$:

$$\mathsf{\Phi }\left(X,\hspace{0.33em}Y,\hspace{0.33em}Z,\hspace{0.33em}t\right)=\mathrm{Re}\left\{\phi \left(X,\hspace{0.33em}Y,\hspace{0.33em}Z\right)e^{-\mathrm{i}{\omega }_{0}t}\right\},$$

(1)

where ${\omega }_0$ is the angular frequency of the incident wave, $\phi$ is time independent, and $\mathrm{i}=\sqrt{-1}$.

The fluid domain in the vicinity of each cylinder is divided into the exterior region and the core region, as shown in Figure 2b, C.G. denotes the center of gravity with coordinates (0, 0, $\overline{z}$); for simplicity, it is also taken as the rotational center. The total velocity potential ${\phi }^{\left(j\right)}$ around cylinder j can be divided into total velocity potential in the exterior region ${\phi }_E^{\left(j\right)}$ and total velocity potential in the core region ${\phi }_C^{\left(j\right)}$. ${\phi }_E^{\left(j\right)}$ can be further expressed as

$${\phi }_E^{\left(j\right)}={\phi }_I+{\displaystyle \sum _{i=1}^N{\phi }_{D-E}^{\left(i\right)}}+{\displaystyle \sum _{i=1}^N{\displaystyle \sum _{s=1}^5{\phi }_{Rs-E}^{\left(i\right)}}},$$

(2)

where ${\phi }_I$ denotes the velocity potential of the ambient incident wave, ${\phi }_{D-E}^{\left(i\right)}$ denotes the diffracted potential in the exterior region of cylinder i, and ${\phi }_{Rs-E}^{\left(i\right)}$ denotes the radiated potential of cylinder i oscillating in mode s. ${\phi }_{D-E}^{\left(i\right)}$ can be further divided into two parts: ${\phi }_{D0-E}^{\left(i\right)}$ and ${\phi }_{D1-E}^{\left(i\right)}$, where ${\phi }_{D0-E}^{\left(i\right)}$ is the diffracted potential of cylinder i resulting from ambient incident waves acting on the fixed array and ${\phi }_{D1-E}^{\left(i\right)}$ is the diffracted potential of cylinder i in the situation of each cylinder oscillating in five degrees of freedom without incident waves, caused by the radiated and diffracted waves of other cylinders except cylinder i. The first part potential is independent of array motion, and the second part is related to the amplitude of each cylinder. Equation (2) can be further expressed as

$${\phi }_E^{\left(j\right)}={\phi }_I+{\displaystyle \sum _{i=1}^N{\phi }_{D0-E}^{\left(i\right)}}+{\displaystyle \sum _{i=1}^N{\displaystyle \sum _{s=1}^5{\phi }_{Rs-E}^{\left(i\right)}}}+{\displaystyle \sum _{i=1}^N{\phi }_{D1-E}^{\left(i\right)}}.$$

(3)

Let

$$\begin{array}{l}{\phi }_{ID-E}^{\left(j\right)}={\phi }_I+{\displaystyle \sum _{i=1}^N{\phi }_{D0-E}^{\left(i\right)}},\\ {\phi }_{RD-E}^{\left(j\right)}={\displaystyle \sum _{i=1}^N{\displaystyle \sum _{s=1}^5{\phi }_{Rs-E}^{\left(i\right)}}}+{\displaystyle \sum _{i=1}^N{\phi }_{D1-E}^{\left(i\right)}},\end{array}$$

(4)

Then

$${\phi }_E^{\left(j\right)}={\phi }_{ID-E}^{\left(j\right)}+{\phi }_{RD-E}^{\left(j\right)},$$

(5)

In this paper, ${\phi }_{ID-E}^{\left(j\right)}$ corresponds to the solution of the array diffraction problem, and ${\phi }_{RD-E}^{\left(j\right)}$ corresponds to the solution of the array radiation–diffraction problem.

Similarly, the velocity potential in the core region can be divided in two parts: ${\phi }_{ID-C}^{\left(j\right)}$ and ${\phi }_{RD-C}^{\left(j\right)}$, corresponding to the solution of the array diffraction problem and the radiation–diffraction problem, respectively:

$${\phi }_C^{\left(j\right)}={\phi }_{ID-C}^{\left(j\right)}+{\phi }_{RD-C}^{\left(j\right)},$$

(6)

Then, the total velocity potential of the truncated cylinder array is divided into two parts: the velocity potential of the diffraction problem in which a fixed array is subjected to incident waves, and the velocity potential of radiation–diffraction problem in which cylinders oscillate with different amplitudes. The two parts will be solved in Section 2.2 and Section 2.3, respectively.

2.2. Diffraction Problem

It is considered that the ambient incident plane wave is of amplitude A, frequency ω₀, and heading angle β with the positive X-direction. The velocity potential has the following form [18], expressed in the cylindrical coordinate system relative to cylinder j:

$${\phi }_I^{}=-\frac{\mathrm{i}gA}{{\omega }_0}{\displaystyle \sum _{m=-\infty }^{\infty }{I}_j{e}^{\mathrm{i}m\left(\pi /2-\beta \right)}\cdot Z_0\left(z\right)J_m\left(k_0{r}_j\right)e^{\mathrm{i}m{\theta }_j}},$$

(7)

where g denotes the acceleration of gravity,$I_j=e^{\mathrm{i}{k}_0(x_j\cos \beta +y_j\sin \beta )}$ denotes the phase factor ((x_j, y_j) is the coordinate of the axis of cylinder j in the global coordinate system),$Z_0=\cosh k_0(z+d)/\cosh k_{0}d$ is the characteristic function in the z direction, J_m denotes the mth-order Bessel function of the first kind, and the wavenumber k₀ satisfies the dispersion relationship $k_0\tanh k_{0}d={\omega }_0^2/g$.

For the diffracted waves of cylinder i, the exterior velocity potential has the following form [19], expressed in the cylindrical coordinate system relative to cylinder i:

$${\phi }_{D0-E}^{\left(i\right)}=-\frac{\mathrm{i}gA}{{\omega }_0}{\displaystyle \sum _{m=-\infty }^{\infty }\left[A_{0m}^{\left(i\right)}{Z}_0\left(z\right)H_m\left(k_0{r}_i\right)+{\displaystyle \sum _{n=1}^{\infty }{A}_{nm}^{\left(i\right)}{Z}_n\left(z\right)K_m\left(k_n{r}_i\right)}\right]e^{\mathrm{i}m{\theta }_i}},$$

(8)

where $A_{nm}^{\left(i\right)}$ denotes the undetermined complex coefficients, $H_m$ denotes the mth-order Hankel function of the first kind, and $K_m$ denotes the mth-order modified Bessel function of the second kind. $Z_n=\cos k_n(z+d)$ is the characteristic function in the z direction for $n\ge 1$, and k_n is given by positive real roots of $k_n\tan k_{n}d=-{\omega }_0^2/g$.

Equations (7) and (8) can be written in matrix form:

$${\phi }_I^{}=-\frac{\mathrm{i}gA}{{\omega }_0}\cdot a_j{}^T{\mathsf{\psi }}_j^I,$$

(9)

$${\phi }_{D0-E}^{\left(i\right)}=-\frac{\mathrm{i}gA}{{\omega }_0}\cdot A_i{}^T{\mathsf{\psi }}_i^{D-E},$$

(10)

where the superscript T is the transpose operator. Elements of each vector are defined as follows:

$$\begin{array}{l}{a}_j\left(n,\hspace{0.33em}m\right)=\left\{\begin{array}{l}{I}_j{e}^{\mathrm{i}m\left(\pi /2-\beta \right)},\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}n=0,\\ 0,\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}n\ge 1,\end{array}\right.\\ {\mathsf{\psi }}_j^I\left(n,\hspace{0.33em}m\right)=\left\{\begin{array}{l}{Z}_0\left(z\right)J_m\left(k_0{r}_j\right)e^{\mathrm{i}m{\theta }_j},\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}n=0,\\ Z_n\left(z\right)I_m\left(k_n{r}_j\right)e^{\mathrm{i}m{\theta }_j},\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}n\ge 1,\end{array}\right.\end{array}$$

(11)

$${\mathsf{\psi }}_i^{D-E}\left(n,\hspace{0.33em}m\right)=\left\{\begin{array}{l}{Z}_0\left(z\right)H_m\left(k_0{r}_i\right)e^{\mathrm{i}m{\theta }_i},\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}n=0,\\ Z_n\left(z\right)K_m\left(k_n{r}_i\right)e^{\mathrm{i}m{\theta }_i},\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}n\ge 1,\end{array}\right.$$

(12)

$A_i{}^T$ is composed of the undetermined complex coefficients $A_{nm}^{\left(i\right)}$. Equation (12) can be written within the cylindrical coordinate system relative to cylinder j as follows by using Graf’s addition formulae for Bessel functions:

$${\mathsf{\psi }}_i^{D-E}\left(n,\hspace{0.33em}m\right)=\left\{\begin{array}{l}{\displaystyle \sum _{l=-\infty }^{\infty }{H}_{m-l}\left(k_0{R}_{ij}\right)e^{\mathrm{i}{\alpha }_{ij}\left(m-l\right)}\cdot Z_0\left(z\right)J_l\left(k_0{r}_j\right)e^{\mathrm{i}l{\theta }_j}},\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}n=0,\\ {\displaystyle \sum _{l=-\infty }^{\infty }{K}_{m-l}\left(k_n{R}_{ij}\right)e^{\mathrm{i}{\alpha }_{ij}\left(m-l\right)}{\left(-1\right)}^l\cdot Z_n\left(z\right)I_l\left(k_n{r}_j\right)e^{\mathrm{i}l{\theta }_j}},\hspace{1em}\hspace{1em}n\ge 1,\end{array}\right.$$

(13)

where $R_{ij}$ denotes the distance between centers of cylinder i and j, and ${\alpha }_{ij}$ denotes the angle at cylinder i between the positive X-direction and the line joining the center of i to that of j in an anti-clockwise direction.

It can be further written in matrix form:

$${\mathsf{\psi }}_i^{D-E}=T_{ij}{\mathsf{\psi }}_j^I,$$

(14)

where $T_{ij}$ denotes a coordinate transformation matrix between two local cylindrical coordinate systems with the form

$$T_{ij}\left(n,\hspace{0.33em}m,\hspace{0.33em}l\right)=\left\{\begin{array}{l}{H}_{m-l}\left(k_0{R}_{ij}\right)e^{\mathrm{i}{\alpha }_{ij}\left(m-l\right)},\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}n=0,\\ K_{m-l}\left(k_n{R}_{ij}\right)e^{\mathrm{i}{\alpha }_{ij}\left(m-l\right)}{\left(-1\right)}^l,\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}n\ge 1.\hspace{0.33em}\end{array}\right.$$

(15)

Therefore, the exterior diffracted potential of cylinder i represented by Equation (10) can be rewritten in coordinate system relative to cylinder j:

$${\phi }_{D0-E}^{\left(i\right)}{|}_j=-\frac{\mathrm{i}gA}{{\omega }_0}{A}_i{}^T{T}_{ij}{\mathsf{\psi }}_j^I.$$

(16)

The total incident waves of cylinder j are composed of the ambient incident wave and the diffracted waves from other cylinders:

$${\phi }_I^{\left(j\right)}={\phi }_I+{\displaystyle \sum _{i=1,i\ne j}^N{\phi }_{D0-E}^{\left(i\right)}{|}_j}=-\frac{\mathrm{i}gA}{{\omega }_0}\left(a_j{}^T+{\displaystyle \sum _{i=1,i\ne j}^N{A}_i{}^T{T}_{ij}}\right){\mathsf{\psi }}_j^I.$$

(17)

The total incident waves and total diffracted waves of cylinder j can be matched by its inherent diffraction transfer matrix $B_j^E$ [16]:

$$A_j=B_j^E\left(a_j+{\displaystyle \sum _{i=1,i\ne j}^N{T}_{ij}{}^T{A}_i}\right),\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}j=1,2,\dots ,N,$$

(18)

where $B_j^E$ are given in Appendix A.

The coefficient vector $A_j$ can be obtained by solving Equation (18), and the total velocity potential of cylinder j in exterior region can be written as follows:

$${\phi }_{ID-E}^{\left(j\right)}=-\frac{\mathrm{i}gA}{{\omega }_0}\left[A_j{}^T{\mathsf{\psi }}_j^{D-E}+\left(a_j{}^T+{\displaystyle \sum _{i=1,i\ne j}^N{A}_i{}^T{T}_{ij}}\right){\mathsf{\psi }}_j^I\right].$$

(19)

Similarly, the total velocity potential in the core region of cylinder j has the form

$${\phi }_{ID-C}^{\left(j\right)}=-\frac{\mathrm{i}gA}{{\omega }_0}\left(a_j{}^T+{\displaystyle \sum _{i=1,i\ne j}^N{A}_i{}^T{T}_{ij}}\right){\left(B_j^C\right)}^T{\mathsf{\psi }}_j^{D-C},$$

(20)

where $B_j^C$ are given in Appendix A, and the elements of the component wave vector in the core region are defined as follows:

$${\mathsf{\psi }}_j^{D-C}\left(p,m\right)=\left\{\begin{array}{l}{r}_j^{\left|m\right|}{e}^{\mathrm{i}m{\theta }_j},\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}p=0,\\ I_m\left(\frac{p\pi r_j}{d-h_j}\right)e^{\mathrm{i}m{\theta }_j},\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}p\ge 1.\end{array}\right.$$

(21)

2.3. Radiation–Diffraction Problem

Each cylinder oscillates sinusoidally with angular frequency ${\omega }_0$:

$${\Xi }_s^{\left(i\right)}(t)=\mathrm{Re}\left\{{\zeta }_s^{\left(i\right)}{e}^{-\mathrm{i}{\omega }_{0}t}\right\},$$

(22)

where ${\zeta }_s^{\left(i\right)}$ represents complex amplitude of cylinder i in mode s. The corresponding velocity should be

$${\dot{\Xi }}_s^i\left(t\right)=\mathrm{Re}\left\{-\mathrm{i}{\omega }_0{\zeta }_s^{\left(i\right)}{e}^{-\mathrm{i}{\omega }_{0}t}\right\}.$$

(23)

The velocity potential of waves radiated by cylinder i oscillating in mode s of complex amplitude ${\zeta }_s^{\left(i\right)}$ has the following form, with the cylindrical coordinate system of cylinder i:

$${\phi }_{Rs-E}^{\left(i\right)}=-\mathrm{i}{\omega }_0{\zeta }_s^{\left(i\right)}{\displaystyle \sum _{m=-\infty }^{\infty }\left[R_{0ms}^{\left(i\right)}{Z}_0\left(z\right)H_m\left(k_0{r}_i\right)+{\displaystyle \sum _{n=1}^{\infty }{R}_{nms}^{\left(i\right)}{Z}_n\left(z\right)K_m\left(k_n{r}_i\right)}\right]e^{\mathrm{i}m{\theta }_i}},$$

(24)

where $R_{nms}^{\left(i\right)}$ represents the radiation characteristics of a single cylinder:

$$R_{nms}^{\left(i\right)}=\left\{\begin{array}{l}\frac{D_R^{}{}_{0m}^s\cosh k_{0}d}{{H^{\prime }}_m\left(k_0{a}_i\right)N_0^{1/2}},\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}n=0,\\ \frac{D_R^{}{}_{nm}^s}{{K^{\prime }}_m\left(k_n{a}_i\right)N_n^{1/2}},\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}n>0,\end{array}\right.$$

(25)

where coefficients $D_R^{}{}_{nm}^s$ are given in Appendix A, and $N_0$ and $N_n$ are defined as

$$N_0=\frac{1}{2}\left(1+\frac{\sinh 2k_{0}d}{2k_{0}d}\right),\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}{N}_n=\frac{1}{2}\left(1+\frac{\sin 2k_{n}d}{2k_{n}d}\right).$$

(26)

Equation (24) can be written in matrix form and further expressed in the cylindrical coordinate system of cylinder j by the coordinate transformation matrix $T_{ij}$:

$${\phi }_{Rs-E}^{\left(i\right)}=-\mathrm{i}{\omega }_0{\zeta }_s^{\left(i\right)}{R}_{is}{}^T{\mathsf{\psi }}_i^{D-E}=-\mathrm{i}{\omega }_0{\zeta }_s^{\left(i\right)}{R}_{is}{}^T{T}_{ij}{\mathsf{\psi }}_j^I.$$

(27)

The diffracted waves in the exterior region of cylinder i caused by other cylinders’ radiated and diffracted waves have the following form of velocity potential [20]:

$${\phi }_{D1-E}^{\left(i\right)}={\displaystyle \sum _{m=-\infty }^{\infty }\left[A_{R0m}^{\left(i\right)}{Z}_0\left(z\right)H_m\left(k_0{r}_i\right)+{\displaystyle \sum _{n=1}^{\infty }{A}_{Rnm}^{\left(i\right)}{Z}_n\left(z\right)K_m\left(k_n{r}_i\right)}\right]e^{\mathrm{i}m{\theta }_i}}.$$

(28)

Similar to Equation (27), the above formula can be rewritten as

$${\phi }_{D1-E}^{\left(i\right)}=A_{Ri}^T{\mathsf{\psi }}_i^{D-E}=A_{Ri}^T{T}_{ij}{\mathsf{\psi }}_j^I,$$

(29)

with the coefficients vectors $A_{Ri}^T$ to be determined.

The total incident waves of cylinder j are composed of waves radiated and diffracted from other cylinders:

$$\begin{array}{l}\hspace{1em}{\displaystyle \sum _{i=1,i\ne j}^N{\phi }_{Rs-E}^{\left(i\right)}{|}_j}+{\displaystyle \sum _{i=1,i\ne j}^N{\phi }_{D1-E}^{\left(i\right)}{|}_j}\\ ={\displaystyle \sum _{i=1,i\ne j}^N{\displaystyle \sum _{s=1}^5\left(-\mathrm{i}{\omega }_0{\zeta }_s^{\left(i\right)}{R}_{is}{}^T\right)}{T}_{ij}{\mathsf{\psi }}_j^I}+{\displaystyle \sum _{i=1,i\ne j}^N{A}_{Ri}{}^T{T}_{ij}{\mathsf{\psi }}_j^I}\\ ={\displaystyle \sum _{i=1,i\ne j}^N\left[{\displaystyle \sum _{s=1}^5\left(-\mathrm{i}{\omega }_0{\zeta }_s^{\left(i\right)}{R}_{is}{}^T\right)}+A_{Ri}{}^T\right]T_{ij}{\mathsf{\psi }}_j^I}.\end{array}$$

(30)

Similar to Equation (18), linear system of equations are obtained:

$$A_{Rj}=B_j^E{\displaystyle \sum _{i=1,i\ne j}^N{{\rm T}}_{ij}{}^T\left[{\displaystyle \sum _{s=1}^5\left(-\mathrm{i}{\omega }_0{\zeta }_s^{\left(i\right)}{R}_{is}\right)}+A_{Ri}\right]},\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}j=1,\hspace{0.33em}2,\hspace{0.33em}\dots \hspace{0.33em},\hspace{0.33em}N.$$

(31)

The total velocity potentials in the exterior and core region of cylinder j in the array radiation–diffraction problem can be written as follows:

$$\begin{array}{l}{\phi }_{RD-E}^{(j)}=\left[{\displaystyle \sum _{s=1}^5\left(-\mathrm{i}{\omega }_0{\zeta }_s^{(j)}{R}_{js}{}^T\right)}+A_{Rj}{}^T\right]{\mathsf{\psi }}_j^{D-E}\\ \hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}+{\displaystyle \sum _{i=1,i\ne j}^N\left[{\displaystyle \sum _{s=1}^5\left(-\mathrm{i}{\omega }_0{\zeta }_s^{(i)}{R}_{is}{}^T\right)}+A_{Ri}{}^T\right]T_{ij}{\mathsf{\psi }}_j^I},\end{array}$$

(32)

$$\begin{array}{l}{\phi }_{RD-C}^{(j)}={\displaystyle \sum _{s=1}^5\left[-\mathrm{i}{\omega }_0{\zeta }_s^{(j)}{\phi }_{Rs-C}^j(r_j,{\theta }_j,z)\right]}+\\ \hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\left\{{\displaystyle \sum _{i=1,i\ne j}^N\left[{\displaystyle \sum _{s=1}^5\left(-\mathrm{i}{\omega }_0{\zeta }_s^{(i)}{R}_{is}{}^T\right)}+A_{Ri}{}^T\right]T_{ij}}\right\}{\left(B_j^C\right)}^T{\mathsf{\psi }}_j^{D-C},\end{array}$$

(33)

where ${\phi }_{RD-C}^{(j)}$ denotes the core region situation corresponding to ${\phi }_{Rs-E}^{(i)}$ and has the following form:

$${\phi }_{Rs-C}^{(j)}(r_j,\hspace{0.33em}{\theta }_j,\hspace{0.33em}z)={\displaystyle \sum _{m=-\infty }^{\infty }\left\{\begin{array}{l}\frac{C_{R0m}^s}{2}{\left(\frac{r_j}{a_j}\right)}^{\left|m\right|}+\\ {\displaystyle \sum _{p=1}^{\infty }{C}_{Rpm}^s\frac{I_m\left(\frac{p\pi r_j}{d-h_j}\right)}{I_m\left(\frac{p\pi a_j}{d-h_j}\right)}\cos \left[\frac{p\pi \left(z+d\right)}{d-h_j}\right]+{\mathsf{\Lambda }}_s{\lambda }_{ms}}\end{array}\right\}{e}^{\mathrm{i}m{\theta }_j}},$$

(34)

where $C_{\mathrm{R}pm}^s$,${\mathsf{\Lambda }}_s$ and ${\lambda }_{ms}$ are defined in detail in Appendix A.

2.4. Dynamics Equation

The relationship between spatial pressure and velocity potential in the vicinity of cylinder j is as follows:

$$p^{\left(j\right)}=\mathrm{i}\rho {\omega }_0{\phi }^{\left(j\right)},$$

(35)

The total hydrodynamic force of cylinder $j$ can be obtained by integrating the pressure over its wet surface:

$$F_{Hs}^{\left(j\right)}=\left\{\begin{array}{cc}{\displaystyle \int p^{\left(j\right)}n}dS,\hfill & s\hspace{0.33em}=\hspace{0.33em}1,\hspace{0.33em}2,\hspace{0.33em}3,\hfill \\ {\displaystyle \int p^{\left(j\right)}\left(r\times n\right)dS},\hfill & s\hspace{0.33em}=\hspace{0.33em}4,\hspace{0.33em}5,\hfill \end{array}\right.$$

(36)

where $r$ is the position vector pointing from the center of mass to integrating position and $n$ is the unit normal vector of wet surface pointing into the body, defined as follows:

$$n=\left\{\begin{array}{l}\left(-\cos {\theta }_j,-\sin {\theta }_j,0\right),\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}lateral\hspace{0.33em}surface,\\ \left(0,0,1\right),\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}bottom\hspace{0.33em}surface.\end{array}\right.$$

(37)

Substituting Equations (19), (20), (32), and (33) into Equations (35) and (36) yields

$$\begin{array}{l}{F}_{Hs}^{\left(j\right)}=F_{IDs}^{\left(j\right)}+{\displaystyle \sum _{i=1}^N{\displaystyle \sum _{p=1}^5{F}_{RDsp}^{\left(ji\right)}{\zeta }_p^i}}\\ \hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}=F_{IDs}^{\left(j\right)}+{\displaystyle \sum _{i=1}^N{\displaystyle \sum _{p=1}^5\left(-{\omega }_0{}^2\cdot a_{sp}^{\left(ji\right)}{\zeta }_p^i+{\displaystyle \sum _{i=1}^N{\displaystyle \sum _{p=1}^5\left(-\mathrm{i}{\omega }_0\right)\cdot b_{sp}^{\left(ji\right)}{\zeta }_p^i}}\right)}}\end{array}$$

(38)

where $F_{IDs}^{\left(j\right)}$ denotes the exciting force introduced by the diffraction problem, and $F_{RDsp}^{\left(ji\right)}$ refers to the s-th mode radiation force on the j-th cylinder due to p-th mode motion per unit amplitude of i-th cylinder. $a_{sp}^{\left(ji\right)}$ and $b_{sp}^{\left(ji\right)}$ are added mass and damping, respectively.

The steady-state motion of cylinder j in mode s may be described by applying Newton’s second law at the center of mass:

$$\begin{array}{l}{\displaystyle \sum _{p=1}^5{\displaystyle \sum _{i=1}^N\left[-{\omega }_0{}^2\left(M_s^{\left(j\right)}+a_{sp}^{\left(ji\right)}\right)-\mathrm{i}{\omega }_0\left(b_{sp}^{\left(ji\right)}+{\lambda }_s^{\left(j\right)}\right)+\left({\delta }_s^{\left(j\right)}+k_s^{\left(j\right)}\right)\right]{\zeta }_p^{\left(i\right)}}}=F_{IDs}^{\left(j\right)},\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\\ \hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}j=1,\hspace{0.33em}2,\hspace{0.33em}\dots ,\hspace{0.33em}N;\hspace{0.33em}s=1,\hspace{0.33em}2,\hspace{0.33em}\dots ,\hspace{0.33em}5.\end{array}$$

(39)

with $M_s^{\left(j\right)}$, ${\lambda }_s^{\left(j\right)}$, ${\delta }_s^{\left(j\right)}$, $k_s^{\left(j\right)}$ being non-zero only if $i=j$ and $s=p$. $M_s^{\left(j\right)}$ denotes the mass (s = 1, 2, 3) or moment of inertia (s = 4, 5) of the cylinder j; ${\lambda }_s^j$ and ${\delta }_s^j$ denote the linear PTO damping and spring of cylinder j in s mode, respectively; $k_s^{\left(j\right)}$ denotes the hydrostatic restoring stiffness of cylinder j in direction s.

By solving the linear equations of Equation (39), the amplitude of each cylinder in each direction ${\zeta }_s^{\left(j\right)}$ can be obtained; then, they can be used for power calculation.

2.5. Power Calculation

According to Falnes [40], the mean power extracted by cylinder j oscillating in mode s with amplitude ${\zeta }_s^{\left(j\right)}$ should be

$$P_s^j=\frac{1}{2}{\omega }_0{}^2{\lambda }_s^j{\left|{\zeta }_s^{(j)}\right|}^2.$$

(40)

There are two important dimensionless quantities for describing the performance of a WEC array: capture width w and interaction factor q, shown in Equations (41) and (42), which are used to represent the total power-generating capacity of the array and whether the hydrodynamic interactions within the array are conducive to power extraction.

$$w=\frac{{\displaystyle \sum _{i=1}^N{\displaystyle \sum _{s=1}^5{P}_s^{(i)}}}}{P_I},$$

(41)

$$q=\frac{{\displaystyle \sum _{i=1}^N{\displaystyle \sum _{s=1}^5{P}_s^{(i)}}}}{N\times {\displaystyle \sum _{s=1}^5{P}_s^{iso}}},$$

(42)

where $P^{iso}$ denotes the power extraction of an isolated cylinder and $P_I$ denotes the power flux of the incident wave:

$$P_I=\frac{1}{2}\rho g{A}^2\cdot \frac{{\omega }_0}{2k_0}\left(1+\frac{2k_{0}d}{\sinh 2k_{0}d}\right).$$

(43)

Given the clear physical meaning of capture width w and interaction factor q, the case studies and discussion of this paper will be mainly based on them.

3. Validation

The above theoretical analysis is coded with Fortran. Before carrying out the case studies, we first ensured the accuracy of the modeling and calculation through comparison with published results. As shown in Figure 3, we simplified the present model to a single degree of freedom (heave), and referred to the works by Babarit [28] and Child and Venugopal [29]. Figure 3a gives the relative capture width of each WEC in an array composed of two heaving WECs. Figure 3b gives the interaction factor q of two different arrays composed of five WECs. It can be found that there is good agreement.

We further investigated the hydrodynamic coefficients of rotational modes, as shown in Figure 4, where an equilateral triangle array composed of three truncated cylinders is considered. The calculation results in this paper are in good agreement with those of Mavrakos and Koumoutsakos [41] with the multiple scattering method.

4. Results and Discussion

In this section, we preliminarily investigate the power-generating capacity of WECs with five degrees of freedom. The amount of data from parametric studies may be immense, so only results for an isolated WEC, an array composed of two WECs, and an array composed of five WECs are presented here.

For simplicity, WECs in the array have identical size and PTO characteristics, and we refer to the settings of Child and Venugopal [29] for the sake of comparison; each cylinder is of radius a, draft a, mass ρgπa³, moment of inertia 1.5ρa⁵, and center of rotation (0, 0, −a), with water depth d = 8a. The PTO characteristics are tuned at dimensionless wavenumber k₀a = 0.4, with values shown in

Table 1. Power take-off (PTO) characteristics.

${\delta }_{1,2}^{}/\rho a^3{\omega }_n^2$	${\delta }_3^{}/\rho a^3{\omega }_n^2$	${\delta }_{1,2}^{}/\rho a^5{\omega }_n^2$	${\lambda }_{1,2}^{}/\rho a^3{\omega }_n^{}$	${\lambda }_3^{}/\rho a^3{\omega }_n^{}$	${\lambda }_{4,5}^{}/\rho a^5{\omega }_n^{}$
5.51	0	0	2.50	2.94	3.19

. All calculation results in this paper have been presented in dimensionless form.

4.1. An Isolated WEC

First of all, we investigated the power-generating capacity of an isolated WEC which can move and absorb power in five degrees of freedom (5-DOF WEC). In addition, we also calculated the power of a conventional WEC operating in heave mode only (heaving WEC) for comparison. The incident wave angle is considered as β = 0. The capture widths of the two kinds of WEC are shown in Figure 5a. The amplitudes in each direction of a 5-DOF WEC are shown in Figure 5b.

From Figure 5a, it can be found that compared with a heaving WEC (black line), the power-generating capacity of a 5-DOF WEC (Red line) has been greatly improved and the maximum capture width of a 5-DOF WEC is about 3.9 times that of a heaving WEC.

The reason for this difference is that more power is extracted from the surge and pitch modes for a 5-DOF WEC (the hydrodynamic responses of the sway and roll mode of a single cylinder are zero due to symmetry). For example, at k₀a = 0.4, where the PTO characteristics are tuned, the capture width of a 5-DOF WEC is 3.0. According to Equations (40) and (41), for a single 5-DOF WEC, the capture width is directly proportional to the sum of squares of the amplitude in each degree of freedom multiplied by the corresponding PTO damping. Also, from Figure 5b, it can be found that the dimensionless motion amplitudes in the surge, heave, and pitch mode are 1.54, 0.86, and 0.67, respectively, which indicates that the contributions of the surge, heave, and pitch mode to the whole power absorption are about 62%, 23%, and 15%, respectively. This means that, at k₀a = 0.4, most power is extracted from the surge mode. In fact, because the heave mode of a single buoy is decoupled from other modes in the dynamic equations, the total power of a heaving WEC is equal to the power of the heaving part of the 5-DOF WEC.

When the incident wavelength is small relative to the buoy size (k₀a > 1.2), the difference will become even larger. For example, at k₀a = 1.5, it can be found from Figure 5b that the dimensionless motion amplitudes in the surge, heave, and pitch mode are 0.28, 0.06, and 0.26, respectively, which indicates that the total power extraction of a 5-DOF WEC is mainly attributable to the surge (47%) and pitch (51%) modes, and only 2% to the heave mode. So the capture width of a heaving WEC is near zero and much lower than that of a 5-DOF WEC.

In addition, the corresponding wavefields are also calculated at k₀a = 0.4. The contour plots of the wave amplitude responses (divided by incident wave amplitude A) around a 5-DOF WEC and a heaving WEC are given in Figure 6a,b, respectively.

Identical color levels are used in the two pictures for the sake of comparison; red color represents constructive interference among incident, diffracted, and radiated waves, while blue is destructive. It can be found that compared with the heaving WEC, the 5-DOF WEC has stronger effects on the wave field. Because the 5-DOF WEC absorbs large energy from incident waves, the shielding effect is significant in a large area behind the WEC. It is noteworthy that the biggest and smallest wave amplitudes are located at the front and rear surface of the buoy for a 5-DOF WEC, which is different from that of a heaving WEC.

Next, three typical array layouts are studied.

4.2. Two WECs

Firstly, the simplest interacting unit composed of two WECs is examined. The inter-buoy spacing L = 5a, and the coordinates of the two WECs in the global coordinate system are (0, 0) and (5a, 0), respectively. Figure 7 shows the results of array performance when the incident wave angle β = 0, i.e., when the array is parallel to the direction of the incident wave. When the incident wave angle β = π/2, i.e., the array is perpendicular to the direction of the incident wave; results are shown in Figure 8. Arrays composed of 5-DOF WECs and heaving WECs are compared.

From Figure 7a, it can be found that the capture width of the array of 5-DOF WECs (red line) is much high than that of heaving WECs (Black line) in all ranges of k₀a. The maximum capture width of 5-DOF WECs is about 3.6 times that of heaving WECs.

As shown in Figure 7b, for heaving WECs, the interaction factor fluctuates around q = 1 with the increase of k₀a (black line). Meanwhile, for 5-DOF WECs, it is noteworthy that the interaction factor is always less than 1 (red line), which means that the hydrodynamic interactions within the array are always destructive to the power absorption. This problem may be caused by the downstream WEC undergoing strong shielding effects from the upstream WEC, as shown in Figure 6a. Therefore, 5-DOF WECs should not be arranged along the direction of the incident wave.

When the array is perpendicular to the incident wave, as shown in Figure 8a, the capture width of 5-DOF WECs (red line) is still much higher than that of heaving WECs (black line) in all ranges of k₀a. As shown in Figure 8b, the interaction factor of 5-DOF WECs is larger than 1 at a relatively large wavenumber range (k₀a > 0.56), and the fluctuation is more moderate. Meanwhile, the interaction factor of heaving WECs reaches 1.73 at k₀a = 1.94.

As shown in Figure 9, we further investigated the effects of more general wave incident angles—in addition to the two cases of parallel and perpendicular to the array—on the interaction factor of the 5-DOF WECs. The variation of interaction factor with dimensionless wavenumber and incident wave angle was plotted, and three different inter-buoy spacings were considered. The three plots are symmetric about β = π/2 due to the arrays being symmetric about the Y-axis. It can be found from Figure 9 that when the incident wave is parallel to the array (β = 0), the interaction factor is almost less than 1 for all wavenumbers, regardless of the inter-buoy spacing. For larger inter-buoy spacing, the interaction factor changes more frequently with wavenumber. In addition, it is noteworthy that for the case of L = 3a, the maximum value of the interaction factor occurs at β = π/2, while for the case of L = 5a and L = 8a, the maximum value of the interaction factor occurs at oblique incident wave angle (near β = π/4).

4.3. Five WECs

Finally, we examined an array composed of five WECs. The array layout is presented by Child and Venugopal [29] with a parabolic intersection method. The coordinates of the five WECs in the global coordinate system are (0, 0), (0, 37.6a), (−14.1a, 18.7a), (−14.1a, −18.7a), and (0, −37.6a), respectively. Results are shown in Figure 10. Arrays composed of 5-DOF WECs and heaving WECs are compared.

From Figure 10a, it can be found that the capture width curve has many small fluctuations for both heaving WECs and 5-DOF WECs, which may be caused by the large inter-buoy spacing. The maximum capture width of 5-DOF WECs is about 3.6 times that of heaving WECs.

As shown in Figure 10b, for heaving WECs, the interaction factor curve (black line) fluctuates more violently, with higher amplitude and frequency, and the biggest value reaches 2.06 at k₀a = 1.77. Meanwhile, for 5-DOF WECs, the interaction factor changes more smoothly with k₀a, which means that when the frequency of the incident wave changes, the array composed of 5-DOF WECs is less affected than heaving WECs. Furthermore, 5-DOF WECs have a wider bandwidth at which the interaction factor is larger than 1, for example, 0.85 < k₀a < 1.19 and 1.45 < k₀a < 1.87.

Figure 11 shows the interaction factor of the array composed of five 5-DOF WECs and its variation with dimensionless wavenumber and incident wave angle. It can be found that when β = 0, the interaction factor of the array is always less than 1, except when the dimensionless wavenumber is close to 2.0. The maximum value of the interaction factor reaches 1.46 and occurs at k₀a = 1.72 and β = 0.47.

In addition, we estimated the impact of the array on the surrounding wave environment, which is an essential step before array construction [42]. The dimensionless wave amplitude responses corresponding to the array composed of five 5-DOF WECs and five heaving WECs are shown in Figure 12a,b, respectively. Identical color levels are used in the two pictures for the sake of comparison; red color means an increase in wave amplitude compared with the amplitude A of the incident wave, whilst blue means a decrease. It can be found that the array composed of five 5-DOF WECs has a stronger impact on the wave environment, especially for the area behind the array. Their shielding ranges are similar in shape, but 5-DOF WECs have stronger shielding effects. Take the point with coordinate (35a, 0) as an example; the wave amplitude caused by 5-DOF WECs is 0.767, while the wave amplitude caused by heaving WECs is 0.937. In the front area of the array, the constructive and destructive interference occurs alternately; the two pictures are similar at this point. In the vicinity of the cylinders, the wave amplitudes of 5-DOF WECs are much larger in the front and much smaller in the rear, which is similar to that of an isolated WEC (shown in Figure 6).

5. Conclusions

In this paper, we present a novel concept for a wave energy converter (WEC) of oscillating buoy type, in which the WEC can move and absorb power with 5 degrees of freedom (5-DOF). We conducted a detailed theoretical analysis and preliminary assessments of such 5-DOF WECs.

Each WEC buoy is modeled as a floating truncated circular cylinder with five degrees of freedom: surge, sway, heave, roll, and pitch. For simplicity, the linear power take-off (PTO) system is considered in each degree of freedom. Hydrodynamic interactions among WECs are solved by a semi-analytic method based on eigenfunction expansion and Graf’s addition theorem for Bessel functions. Then, the amplitude of each buoy in each degree of freedom and the related power absorption can be solved. The model is verified by comparison with published results.

The power-generating capacity of the 5-DOF WECs was investigated by case studies and compared with the conventional heaving WECs. In all cases, the power absorption of 5-DOF WECs is much larger than that of heaving WECs. Especially when the incident wavelength is small relative to buoy size (k₀a > 1.2), the capture width of heaving WECs is almost zero, while the 5-DOF WECs perform well. The interaction factor of 5-DOF WECs is more affected by the array layout, and it is advised to try to avoid arranging them along the direction of the incident wave. Meanwhile, for a fixed array layout, the interaction factor of 5-DOF WECs changes more gently with the variation of k₀a, which is beneficial for suppressing power output fluctuation. The effects of wave incident angle on the interaction factor of the 5-DOF WECs were investigated, revealing that an oblique incident wave will produce a higher interaction factor in most cases. We also found that the shielding effect of an array composed of 5-DOF WECs is stronger.

Preliminary calculations show that the 5-DOF WEC is helpful to further improve the efficiency of power absorption.