Frequency- and time-domain analysis of a multi-degree-of-freedom point absorber wave energy converter

Abstract

This study proposes a multi-degree-of-freedom wave energy converter and introduces a 6-PSS (prismatic pair–spherical pair–spherical pair) power take-off mechanism. The frequency- and time-domain dynamic models of a 3-degree of freedom wave energy converter are established. The time-domain model based on frequency-domain data is usually built upon the Cummins equation. The convolution terms in the Cummins equation are substituted by a state-space system by the frequency-domain identification method in this work. The power absorption of the wave energy converter is studied in both regular and irregular waves. It is shown that the advantages with the converter in resonance with waves are evident in regular waves and not so obvious in irregular waves. The appropriate value of power take-off damping terms has relative border range in irregular waves, which can be chosen according to the trends. Through optimization, the total power absorption in the three directions is close to 126 kW. The results indicate that the multi-degree-of-freedom wave energy converter can extract more energy.

Introduction

Offshore wind and wave energy are clean and renewable,^1,2 and the wave resource is more predictable and less variable than the wind resource. Ocean wave energy converters (WECs) have made great development since the 1970s. The development status of WECs has been summarized in detail. ³ Currently, there are no many successful commercial applications and some issues still exist, such as the low power absorption efficiency.

Oscillating body systems are a large class of WECs, typically in water depths between 40 and 100 m. They generally extract power through relative motion between two bodies. AquaBuOY ⁴ and Wavebob ⁵ oscillate in the heave motion. SEAREV^6,7 and Pelamis ⁸ oscillate in the pitch motion. Some attractive device is invented to oscillate in the heave and pitch motions.^9–11 Some of the WECs have been in the commercial stage. Ocean power technology (OPT) is currently deploying OPT arrays at a number of sites, including 1.4- and 2.0-MW commercial projects off the coasts of Spain and Oregon, respectively. In 2008, Pelamis deployed three attenuators with a combined rating of 2.25 MW off the coast of Portugal. ¹² As we all know, the free oscillating body has 6-degree of freedom (DOF) motions in the waves. So, it is necessary to invent new multi-DOF power take-off (PTO) mechanisms to absorb the energy of the oscillating body to improve the power absorption.

Parallel mechanism possesses the advantages of high stiffness and carrying capacity, ¹³ which can be considered to work as the PTO mechanism. The 6-PSS (prismatic pair–spherical pair–spherical pair) mechanism is a variation type of the Stewart platform. This mechanism configuration can be used in compliant wrist ¹⁴ and machine tool. ¹⁵ In most cases, the actuating components are placed in prismatic pairs and the platform works as the manipulator. The idea that the platform works as the driving component is presented here, which has 6-DOF induced by the ocean waves. However, the concrete structure of the WEC is not provided here.

This article studies the power absorption performance of a 3-DOF WEC oscillating in three directions (surge, heave, and pitch). Numerical studies of waves interacting with a cylindrical point absorber have been made. ¹⁶ The effects of the geometry, the mooring system, and the mass distribution of a tethered wave energy absorber based on a vertical cylinder have been studied. ¹⁷ The influences of mooring cables of a WEC through the frequency-domain analysis have been given, ¹⁸ which also selected the cylinder as the oscillating body. Therefore, the vertical cylinder is a starting point because hydrodynamic coefficients can be obtained by analytical expressions, ¹⁹ which used variable separation and eigenfunction expansion of the potential in cylindrical coordinates. In addition, the 3D boundary element method (BEM) such as WAMIT, WADAM, and AQWA-LINE can be applied to obtain the hydrodynamic coefficients.

Wave energy absorption by means of oscillating bodies can be analyzed by frequency or time-domain models. Because of its simplicity, frequency-domain models have been widely used. Another approach is the time-domain equations proposed by Cummins. ²⁰ In these equations, the wave forces including the incident, diffracted, and radiated can be obtained by frequency models. The convolution terms in Cummins equation are not convenient for numerical calculation, and the analysis and design of motion control systems. A method to approximate the convolution with a state-space system has been proposed. ²¹ Perez et al. ²² have compared different methods for replacing the convolution in terms of complexity and performance. Perez et al. not only have compared the quality of estimated models but also on the ease of implement. Cheng et al. ²³ modeled a point absorber WEC with direct-drive linear PTO based on both frequency- and time-domain models. J Pastor and Y Liu ²⁴ also presented, assessed, and optimized a point absorber WEC through numerical modeling, simulation, and analysis in both frequency- and time-domain. This article employs the frequency-domain identification method, which has been considered the simplest method to implement.

In this article, the 6-PSS mechanism is introduced first and the number of DOFs of the mechanism is calculated. Then, the power absorption performance of the 3-DOF WEC under the one-directional waves in frequency- and time-domain considering both regular and irregular waves is studied. The results indicate that the multi-DOF WEC can extract more energy. In the end, the optimum PTO damping terms are investigated in the specified irregular wave, which can provide the foundation to design the generators.

Model description

The 6-PSS mechanism is shown in Figure 1, of which the platform has 6-DOFs including surge, sway, heave, roll, pitch, and yaw. The number of DOFs for the 6-PSS parallel mechanism can be calculated by using the amended Kutzbach–Grubler equation ²⁵

$$F=\sigma (\tau -s-1)+{\displaystyle \sum _{i=1}{e}_i}+v-u$$

(1)

where $\sigma$ is the degree of the space ( $\sigma =6$ ), $\tau$ is the number of links, $s$ is the number of joints, $e_i$ is the DOF of the ith joint, $v$ is the redundancy DOF, and $u$ is the passive DOF. By inspecting the whole mechanism, $\tau =14$ for 1 moving platform, 1 base, and 12 links; $s=18$ for 6 prismatic pairs and 12 spherical joints; $e_1=1$ for the prismatic pair; $e_2=3$ for the spherical joint, no redundancy DOF, six passive DOF corresponding to the rotation of each leg about its axis. So, the DOF of this 6-PSS mechanism is

$$F=6\times (14-18-1)+1\times 6+3\times 12-6\mathrm{=}6$$

Figure 1.

6-PSS mechanism.

The platform can be considered to perform as the oscillating body; the PTO system can be placed in prismatic pairs which connect to a fixed device. The kinematics of the 6-PSS mechanism has been derived, and the equation between the platform motion and the heave motion of the joint at each leg has been established. ¹⁴ Numerous PTO systems for WECs have been applied in previous works, ²⁶ such as rotary generators, hydraulic generators, and linear generators. Hydraulic generators or linear generators can be adopted in this mechanism.

The oscillating body in real waves has 6-DOF motions. In order to facilitate the research, here, we only study the power absorption performances of the WEC based on one-directional waves; nonlinear characteristics and the effects of other parts, such as the prismatic pairs and spherical joints, are not considered. Due to the symmetry, there are only 3-DOF motions consisting of surge, heave, and pitch. Here, the generators are supposed to be viscous damping terms which connect to the earth.

The simplified 3-DOF WEC is illustrated in Figure 2. The 3-DOF motions are surge ( ${\eta }_1$ ), heave ( ${\eta }_3$ ), and pitch ( ${\eta }_5$ ) which are all defined about the center of gravity of the cylinder. $D$ is the diameter of the cylinder, $L$ is the draft of the cylinder, $H$ is the water depth, $Z_G$ is the still sea level relative to the center of gravity, and $c_1,c_3,c_5$ are the idealized linear PTO damping terms.

Figure 2.

A simplified 3-DOF WEC to be considered for the dynamic models.

Numerical model

Frequency-domain analysis

$$\begin{array}{l}-{\omega }^2\left[\begin{array}{ccc}{M}_{11}+{\mu }_{11}& 0& {\mu }_{15}\\ 0& M_{33}+{\mu }_{33}& 0\\ {\mu }_{51}& 0& M_{55}+{\mu }_{55}\end{array}\right]\left[\begin{array}{c}{\widehat{\eta }}_1\\ {\widehat{\eta }}_3\\ {\widehat{\eta }}_5\end{array}\right]\\ +i\omega \left[\begin{array}{ccc}{\lambda }_{11}+c_1& 0& {\lambda }_{15}\\ 0& {\lambda }_{33}+c_3& 0\\ {\lambda }_{51}& 0& {\lambda }_{55}+c_5\end{array}\right]\left[\begin{array}{c}{\widehat{\eta }}_1\\ {\widehat{\eta }}_3\\ {\widehat{\eta }}_5\end{array}\right]\\ +\left[\begin{array}{ccc}{k}_{11}& 0& 0\\ 0& k_{33}& 0\\ 0& 0& k_{55}\end{array}\right]\left[\begin{array}{c}{\widehat{\eta }}_1\\ {\widehat{\eta }}_3\\ {\widehat{\eta }}_5\end{array}\right]=\left[\begin{array}{c}{\widehat{F}}_{e1}\\ {\widehat{F}}_{e3}\\ {\widehat{F}}_{e5}\end{array}\right]\end{array}$$

(2)

The frequency dynamic model is established in equation (2). The hydrodynamic coefficients are calculated by the BEM software AQWA-LINE. ${\mu }_{ij}$ and ${\lambda }_{ij}$ are the frequency-dependent added mass and radiation damping, respectively; $\omega$ is the wave frequency; $M_{mm}$ is the inertia of the cylinder; $k_{mm}$ is the hydrostatic stiffness; and ${\hat{F}}_{ei}$ is the excitation force amplitudes per wave amplitude in the specific wave frequency. Due to the axisymmetric property of cylinder, there are only cross-coupling terms between heave/pitch and heave/surge

$$M_{11}=M_{33}=\rho \pi D^{2}L/4$$

(3)

$$M_{55}=M{r}^2$$

(4)

$$k_{33}=\rho g\pi D^2/4$$

(5)

$$k_{55}=Mg(\frac{D^2}{16L}+z_G-\frac{L}{2})$$

(6)

where $\rho$ is the density of water, $r$ is the radius of gyration, and $g$ is the gravity acceleration.

According to the deduction by J Falnes, ²⁷ the dimension of the c₁, c₃, and c₅ is selected approximately. Based on the cylinder specification assigned in Table 1, the hydrodynamic coefficients can be obtained by the software AQWA-LINE and the mechanical parameters can be calculated by equations (3)–(6). The displacement amplitudes of the oscillating body can be solved by equation (2), and then the absorbed power is calculated by equation (7). The results are shown in Figure 3, from which the resonance frequencies of surge, heave, and pitch motions are around 1, 0.9, and 1.2 rad/s, respectively. Furthermore, the power production in regular waves can be achieved

$${\overline{P}}_i(\omega )=\frac{c_i\Vert \omega {{\stackrel{⌢}{\eta }}_i\Vert }^2}{2}(i=1,3,5)$$

(7)

Table 1.

Cylinder specification.

Name	Quantity	Dimension	Value
Diameter of the cylinder	$D$	m	10
Draft of the cylinder	$L$	m	10
Water depth	$H$	m	50
Still sea level relative to the center of gravity	$Z_G$	m	8
Density of water	$\rho$	Kg/m3	1025
Radius of gyration	$r$	m	2
Gravity acceleration	$g$	m/s 2	9.81
Idealized linear PTO damping term in surge motion	$c_1$	Ns/m	2,000,000
Idealized linear PTO damping term in heave motion	$c_3$	Ns/m	100,000
Idealized linear PTO damping term in pitch motion	$c_5$	Nms/rad	10,000,000

Figure 3.

Power absorption per wave amplitude (W/m) in surge, heave, and pitch motions by frequency-domain analysis.

Time-domain analysis

$$\{\begin{array}{c}\hfill [M_{11}+{\mu }_{11}(\infty )]{\stackrel{··}{\eta }}_1(t)+{\mu }_{15}(\infty ){\stackrel{··}{\eta }}_5(t)+\underset{0}{\overset{t}{\int }}{K}_{11}(t-\tau ){\stackrel{·}{\eta }}_1(t)d\tau +\underset{0}{\overset{t}{\int }}{K}_{15}(t-\tau ){\stackrel{·}{\eta }}_5(t)d\tau +c_1{\stackrel{·}{\eta }}_1(t)=f_{e1}(t)\hfill \\ \multicolumn{1}{c}{[M_{33}+{\mu }_{33}(\infty )]{\stackrel{··}{\eta }}_3(t)+\underset{0}{\overset{t}{\int }}{K}_{33}(t-\tau ){\stackrel{·}{\eta }}_3(t)d\tau +c_3{\stackrel{·}{\eta }}_3(t)+k_{33}{\eta }_3(t)=f_{e3}(t)}\\ \multicolumn{1}{c}{[M_{55}+{\mu }_{55}(\infty )]{\stackrel{··}{\eta }}_5(t)+{\mu }_{51}(\infty ){\stackrel{··}{\eta }}_1(t)+\underset{0}{\overset{t}{\int }}{K}_{55}(t-\tau ){\stackrel{·}{\eta }}_5(t)d\tau +\underset{0}{\overset{t}{\int }}{K}_{51}(t-\tau ){\stackrel{·}{\eta }}_1(t)d\tau +c_5{\stackrel{·}{\eta }}_5(t)+k_{55}{\eta }_5(t)=f_{e5}(t)}\end{array}$$

(8)

The time-domain dynamic model is established in equation (8) based on Cummins equation, where $K(t)$ is the memory function of the radiation forces, which is caused by radiating waves, and $f_{ei}(t)$ is the excitation force in time series. The relationship between the parameters of the frequency- and time-domain models is found via direct application of Fourier transform under a sinusoidal regime ²⁸

$$\mu (\omega )=\mu -\frac{1}{\omega }\underset{0}{\overset{\infty }{\int }}K(t)\sin (\omega t)dt$$

(9)

$$\lambda (\omega )=\underset{0}{\overset{\infty }{\int }}K(t)\cos (\omega t)dt$$

(10)

$$\mu (\infty )=\underset{\omega \to \infty }{lim}\mu (\omega )$$

(11)

$$K(t)=\frac{2}{\pi }\underset{0}{\overset{\infty }{\int }}\lambda (\omega )\cos (\omega t)d\omega$$

(12)

$$K(j\omega )=\lambda (\omega )+j\omega [\mu (\omega )-\mu (\infty )]$$

(13)

Due to this, these terms in time-domain model are not convenient for analysis and design of motion control systems. The complexity and performance of different methods for replacing the convolution have been compared. ²⁹ The retardation frequency response method is chosen in this article. Consider the parametric model of $K(j\omega )$ , then

$$\tilde{K}(s,\theta )=\frac{P(s,\theta )}{Q(s,\theta )}=\frac{p_m{s}^m+p_{m-1}{s}^{m-1}+\cdots +p_{1}s+p_0}{q_n{s}^n+q_{n-1}{s}^{n-1}+\cdots +1}$$

(14)

The properties of retardation function are shown in Table 2. For detailed proof, refer to the appendix in the study of Perez and Fossen. ²² According to Table 2, the type of rational function can be simplified as

$$\tilde{K}(s,\theta )=\frac{P(s,\theta )}{Q(s,\theta )}=\frac{p_{n-1}{s}^{n-1}+p_{n-2}{s}^{n-2}+\cdots +p_{1}s}{q_n{s}^n+q_{n-1}{s}^{n-1}+\cdots +1}$$

(15)

Table 2.

Properties of retardation function.

Property	Implication on $\tilde{K}(s,\theta )$
$\underset{\omega \to 0}{lim}K(j\omega )=0$	$p_0=0$
$\underset{\omega \to \infty }{lim}K(j\omega )=0$	$n>m$
$\underset{t\to 0^{+}}{lim}K(t)\ne 0$	$m-n=1$
$\underset{t\to \infty }{lim}K(t)\ne 0$	BIBO stable

The corresponding frequency response

$$\begin{array}{c}\tilde{K}(j\omega ,\theta )=\frac{P(s,\theta )}{Q(s,\theta )}\hfill \\ =\frac{(-p_2{\omega }^2+p_4{\omega }^4+\cdots )+j\omega (p_1-p_3{\omega }^2+\cdots )}{(1-q_2{\omega }^2+\cdots )+j\omega (q_1-q_3{\omega }^2+\cdots )}\hfill \\ =\frac{\alpha (\omega )+j\beta (\omega )}{\sigma (\omega )+j\tau (\omega )}=\frac{N(j\omega )}{D(j\omega )}\hfill \end{array}$$

(16)

The unknown coefficients in equation (16) are marked as

$$\theta =[p_{n-1},...,p_1,q_n,...,q_1]$$

(16)

According to the data from AQWA-LINE

$$K(j{\omega }_i)=\lambda ({\omega }_i)+j\omega [\mu ({\omega }_i)-\mu (\infty )]=P_i+j{Q}_i,i=1~N$$

(17)

The error between $K(j{\omega }_i)$ and $\tilde{K}(j{\omega }_i,\theta )$ can be described as

$${\epsilon }_i=K(j{\omega }_i)-\tilde{K}(j{\omega }_i,\theta )=P_i+j{Q}_i-\frac{N(j{\omega }_i)}{D(j{\omega }_i)}$$

(18)

With the least-squares method

$$J={\displaystyle \sum }_{i=1}^N|\epsilon (j{\omega }_i{)|}^2=min$$

(19)

According to the error criterion proposed by ³⁰

$$\begin{array}{c}\multicolumn{1}{c}{J^{*}={\displaystyle \sum }_{i=1}^N|D(j{\omega }_i)\epsilon (j{\omega }_i{)|}^2}\\ ={\displaystyle \sum }_{i=1}^N|(P_i+j{Q}_i)D(j{\omega }_i)-N(j{\omega }_i{)|}^2\hfill \\ ={\displaystyle \sum }_{i=1}^N|U({\omega }_i)+jV({\omega }_i{)|}^2\hfill \end{array}$$

(20)

$$U({\omega }_i)=P_i\sigma ({\omega }_i)-Q_i\tau ({\omega }_i)-\alpha ({\omega }_i)$$

(21)

$$V({\omega }_i)=Q_i\sigma ({\omega }_i)+P_i\tau ({\omega }_i)-\beta ({\omega }_i)$$

(22)

$$\frac{\partial J^{*}}{\partial \theta }=0$$

(23)

When the coefficients are found, the matrix and vectors of state-space which approximate the convolution integral can be written as

$$\begin{array}{c}A=\left[\begin{array}{ccccc}\hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \cdots \hfill & \hfill -1/q_n\hfill \\ \multicolumn{1}{c}{1}& \hfill 0\hfill & \hfill 0\hfill & \hfill \cdots \hfill & \hfill -q_1/q_n\hfill \\ \multicolumn{1}{c}{0}& \hfill 1\hfill & \hfill 0\hfill & \hfill \cdots \hfill & \hfill \hfill \\ \multicolumn{1}{c}{⋮}& \hfill ⋮\hfill & \hfill \ddots \hfill & \hfill \hfill & \hfill ⋮\hfill \\ \multicolumn{1}{c}{0}& \hfill 0\hfill & \hfill \cdots \hfill & \hfill 1\hfill & \hfill -q_{n-1}/q_n\hfill \end{array}\right],\hfill \\ B=\left[\begin{array}{c}\hfill 0\hfill \\ \multicolumn{1}{c}{p_1/q_n}\\ \multicolumn{1}{c}{⋮}\\ \multicolumn{1}{c}{⋮}\\ \multicolumn{1}{c}{p_{n-1}/q_n}\end{array}\right],C=\left[\begin{array}{ccccc}\hfill 0\hfill & \hfill 0\hfill & \hfill \cdots \hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right]\hfill \end{array}$$

(24)

$$\stackrel{·}{X}(t)=AX(t)+Bu(t)$$

(25)

$$Y(t)=CX(t)=\underset{0}{\overset{t}{\int }}K(t-\tau )\stackrel{·}{\eta }(\tau )d\tau$$

(26)

With the method above, we obtain the following transfer function setting the order of the models to 3

$${\tilde{K}}_{11}(s)={10}^5\frac{3.802s^2+2.031s}{0.513s^3+1.063s^2+1.461s+1}$$

(27)

$${\tilde{K}}_{33}(s)={10}^4\frac{2.733s^2+4.879s}{2.387s^3+1.278s^2+3.051s+1}$$

(28)

$${\tilde{K}}_{55}(s)={10}^7\frac{0.735s^2+0.512s}{0.288s^3+0.855s^2+1.143s+1}$$

(29)

$${\tilde{K}}_{15}(s)={10}^6\frac{1.623s^2+1.006s}{0.38s^3+0.948s^2+1.268s+1}$$

(30)

$K(j\omega )$ and $\tilde{K}(j\omega )$ based on the frequency response curve fitting is compared in Figure 4, which matches well. The dynamic model is implemented in MATLAB/Simulink. Ode45 is the fourth-order Runge–Kutta method to solve the ordinary differential equation, which is chosen to solve the differential equations above. Power absorption in regular waves ( $\zeta =2\mathrm{m}$ and $T=5.236\mathrm{s}(\omega =1.2\mathrm{rad}/\mathrm{s})$ ) by time-domain analysis is plotted in Figure 5. Power absorption in regular waves ( $\zeta =2\mathrm{m}$ and $T=6.98\mathrm{s}$ ( $\omega =0.9\mathrm{rad}/\mathrm{s}$ )) by time-domain analysis is plotted in Figure 6. $\zeta$ is the incoming wave height and $T$ is the wave period. From the results below compared with the results in frequency domain, we can see that because $\zeta =2\mathrm{m}$ , the mean power absorptions in time domain in Figures 5 and 6 are both double than the power absorption in frequency domain at corresponding frequency. So, the convolution replaced by state-space is correct and the time-domain approach can be validated with frequency-domain approach.

Figure 4.

$K(j\omega )$ and $\tilde{K}(j\omega )$ based on frequency response curve fitting.

Figure 5.

Power absorption by the time-domain analysis in surge, heave, and pitch motions ( $\zeta =2\mathrm{m}$ , $T=5.236\mathrm{s}$ ).

Figure 6.

Power absorption by the time-domain analysis in surge, heave, and pitch motions ( $\zeta =2\mathrm{m}$ , $T=6.98\mathrm{s}$ ).

Real irregular waves may be represented as a superposition of regular waves. This study adopts the Pierson–Moskowitz spectral distribution which has been used ³¹

$$S(\omega )=\frac{263H_s^2}{T_e^4{\omega }^5}\exp (-\frac{1054}{T_e^4{\omega }^4})$$

(31)

where $H_s$ is the significant wave height and $T_e$ is the energy period. For time-series calculations, the spectral distribution is decomposed as the sum of regular waves of frequency ${\omega }_n={\omega }_0+n\Delta \omega$ , where ${\omega }_0$ is the lowest frequency considered, $\Delta \omega$ is a small frequency interval, and $n$ is the number of regular waves. The amplitude of wave component is $(2S({\omega }_n)\Delta \omega {)}^{0.5}$ . The excitation force in time series can be expressed as

$$f_i(t)=\sum _{n}A({\omega }_n)\mathrm{Re}{F}_{ei}({\omega }_n)e^{i({\omega }_{n}t+{\theta }_n)}$$

(32)

where the phase ${\theta }_n$ is a random real number in the interval $(0,2\pi )$ . The average of the power extracted from the wave by the converter in the time interval $[0,t]$ is given by equation (33). The instantaneous and average power absorption in three directions are plotted in Figure 7 in irregular waves: $H_s=2\mathrm{m},T_e=6.98\mathrm{s}$ and in Figure 8 in irregular waves: $H_s=2\mathrm{m},T_e=6.98\mathrm{s}$ . The results show that the resonance effect in irregular waves exists though not so obviously relative to the regular waves

$$P(t)=\frac{1}{t}\underset{0}{\overset{t}{\int }}c{(\stackrel{·}{\eta })}^{2}dt$$

(33)

Figure 7.

Power absorption by the time-domain analysis ( $H_s=2\mathrm{m},T_e=5.236\mathrm{s}$ ).

Figure 8.

Power absorption by the time-domain analysis ( $H_s=2\mathrm{m},T_e=6.98\mathrm{s}$ ).

The shape and mechanical parameters of the oscillating body and the selected PTO damping have great effects on the power absorption of the WECs. The power absorption varying with the PTO damping terms in each direction is displayed in Figures 9 and 10 on the premise of the specified cylinder and the irregular waves ( $H_s=2\mathrm{m},T_e=6.98\mathrm{s}$ ). The trend in the heave direction is depicted in Figure 9: the optimum PTO damping is around 1.5 × 10⁵ Ns/m and the power absorption is close to 32 kW. The trend of total power absorption in the surge and pitch directions is depicted in Figure 10: the optimum PTO damping of surge direction is about 2 × 10⁶ Ns/m; the optimum PTO damping of pitch direction is about 7.6 × 10⁶ Nms/rad, and the total power absorption in the surge and pitch directions is close to 94 kW. The optimum is not very obvious and the appropriate value of PTO damping terms can be chosen between a relatively large range. Through optimization, the total power absorption in the three directions is close to 126 kW.

Figure 9.

Power absorption varying with the PTO damping in the heave direction ( $H_s=2\mathrm{m},T_e=6.98\mathrm{s}$ ).

Figure 10.

Total power absorption varying with the PTO damping in the surge and pitch directions ( $H_s=2\mathrm{m},T_e=6.98\mathrm{s}$ ).

Conclusion

In this article, the multi-DOF WEC based on the 6-PSS mechanism is proposed. Then, the dynamic analysis of the 3-DOF WEC under one-directional waves is presented in frequency- and time-domain. In the end, the power absorption with the change of the PTO damping terms in each direction is investigated.

Numerical results were obtained in the frequency model; we can predict the power production and the approximate resonance frequency in each DOF of WEC in regular waves. The frequency-domain identification has been used to replace the convolution term in Cummins equation. The instantaneous power absorption has been obtained by time-domain analysis, which indicates that it is significant to invent the multi-DOF WEC. Moreover, it is shown that the advantages with the converter in resonance with waves are evident in regular waves and not so obvious in irregular waves. The appropriate value of PTO damping terms has relative border range in irregular waves, which can be chosen according to the trends. Through optimization, the total power absorption in the three directions is close to 126 kW.