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Power absorption modelling and analysis of a multi‐axis wave energy converter

2021; Institution of Engineering and Technology; Volume: 15; Issue: 14 Linguagem: Inglês

10.1049/rpg2.12277

1752-1424

Ming Tan, Yuhao Cen, Yuxuan Yang, Xiaodong Liu, Yulin Si, Peng Qian, Dahai Zhang,

Mechanical stress and fatigue analysis

IET Renewable Power GenerationVolume 15, Issue 14 p. 3368-3384 ORIGINAL RESEARCH PAPEROpen Access Power absorption modelling and analysis of a multi-axis wave energy converter Ming Tan, Ming Tan [email protected] orcid.org/0000-0001-7209-4083 Ocean College, Zhejiang University, Zhoushan, 316021 ChinaSearch for more papers by this authorYuhao Cen, Yuhao Cen Ocean College, Zhejiang University, Zhoushan, 316021 ChinaSearch for more papers by this authorYuxuan Yang, Yuxuan Yang Ocean College, Zhejiang University, Zhoushan, 316021 ChinaSearch for more papers by this authorXiaodong Liu, Xiaodong Liu Ocean College, Zhejiang University, Zhoushan, 316021 ChinaSearch for more papers by this authorYulin Si, Yulin Si Ocean College, Zhejiang University, Zhoushan, 316021 ChinaSearch for more papers by this authorPeng Qian, Peng Qian Ocean College, Zhejiang University, Zhoushan, 316021 ChinaSearch for more papers by this authorDahai Zhang, Corresponding Author Dahai Zhang [email protected] Ocean College, Zhejiang University, Zhoushan, 316021 China State Key Laboratory of Fluid Power and Mechatronic Systems, Zhejiang University, Hangzhou, 310027 China Hainan Institute of Zhejiang University, Sanya, 572025 P. R. China The Engineering Research Center of Oceanic Sensing Technology and Equipment, Ministry of Education, Zhoushan, China Correspondence Dahai Zhang, Institute of Ocean Engineering and Technology, Ocean College, Zhejiang University, Zhoushan, 316021, China. Email: [email protected]Search for more papers by this author Ming Tan, Ming Tan [email protected] orcid.org/0000-0001-7209-4083 Ocean College, Zhejiang University, Zhoushan, 316021 ChinaSearch for more papers by this authorYuhao Cen, Yuhao Cen Ocean College, Zhejiang University, Zhoushan, 316021 ChinaSearch for more papers by this authorYuxuan Yang, Yuxuan Yang Ocean College, Zhejiang University, Zhoushan, 316021 ChinaSearch for more papers by this authorXiaodong Liu, Xiaodong Liu Ocean College, Zhejiang University, Zhoushan, 316021 ChinaSearch for more papers by this authorYulin Si, Yulin Si Ocean College, Zhejiang University, Zhoushan, 316021 ChinaSearch for more papers by this authorPeng Qian, Peng Qian Ocean College, Zhejiang University, Zhoushan, 316021 ChinaSearch for more papers by this authorDahai Zhang, Corresponding Author Dahai Zhang [email protected] Ocean College, Zhejiang University, Zhoushan, 316021 China State Key Laboratory of Fluid Power and Mechatronic Systems, Zhejiang University, Hangzhou, 310027 China Hainan Institute of Zhejiang University, Sanya, 572025 P. R. China The Engineering Research Center of Oceanic Sensing Technology and Equipment, Ministry of Education, Zhoushan, China Correspondence Dahai Zhang, Institute of Ocean Engineering and Technology, Ocean College, Zhejiang University, Zhoushan, 316021, China. Email: [email protected]Search for more papers by this author First published: 28 August 2021 https://doi.org/10.1049/rpg2.12277AboutSectionsPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat Abstract A point absorber wave energy converter (WEC) which converts wave energy into electrical energy with a multi-axis power-take-off (PTO) system is considered here. Previous wave tank trials have proved the ability of the multi-axis WEC to absorb wave power. In this study, the boundary element method (BEM) based software Ansys@ AQWATM was applied to model and analyze the power absorption performance of the multi-axis WEC for a wider range of regular and irregular waves. To verify the numerical model, a laboratory-scale physical model was manufactured and tested. Results show that the multi-axis WEC can absorb more power from the incident regular wave power compared to single-axis WECs and the efficiency reaches up to 45%. It is found that the wave frequency and incident angle significantly influence the amount of absorbed power from different motion modes and PTO axes. Then the numerical model is simulated at 12 nearshore locations in East and South China Sea. The results indicate that the multi-axis WEC can absorb up to 5 kW and the efficiency can reach up to 29% at the most energetic site Shengshan. In addition, considerable differences of mean absorbed power efficiency can be found between sea states. 1 INTRODUCTION In the last few decades, a wide variety of wave energy technologies based on different working principles was proposed and developed. Recent reviews can be found in [1]. Among them, point absorber WECs that possess small dimensions relative to the incident wavelength are particularly popular due to their simpler structure, easier maintenance, and better suitability for deep-sea deployment. An early example was the device named Norwegian buoy developed by Norwegian researchers which was tested in the Trondheim Fjord in 1983 [2]. There are several other famous devices like Wavebob [3], PowerBuoy [4], Archimedes Wave Swing (AWS [5]), PS Frog [6] and so on. The power in ocean waves is composed of kinetic and potential energy moving in multiple directions. Unfortunately, most of the point absorber WECs are single-axis devices and extract energy from one direction of motion. In the ideal case, a point absorber WEC can convert all the usable energy of ocean waves to electricity by oscillating freely in six motion modes meaning that a multi-axis device should be able to generate more electricity than most single-axis devices. The present work aims at studying a point absorber WEC which can convert wave energy into electrical energy through a three-axis PTO system composed of one translational axis and two rotational axes (Figure 1). The capability of the device to harness wave energy has been demonstrated through previous wave tank series and its ability to absorb up to 40% of the incident wave energy was proved [7, 8]. At present, this work aims to design the certain scaled multi-axis WEC specially tailored for the Zhejiang and Guangdong nearshore sea states in East and South China Sea and it is promising to advance it to the next stage. FIGURE 1Open in figure viewerPowerPoint Concept of the multi-axis WEC Compared to the traditional single-axis PTO point absorber WECs, studies on the multi-axis WECs are relatively insufficient and less systematic. It is worth noting that the most important reason for the wide use of single-axis WECs rather than multi-axis WECs is that increasing number of degree of freedom (DOF) of systems would result in the interaction between modes of buoy's motion and power absorption along different PTO axes which introduce difficulties into modelling and simulation studies. A multi-axis PTO system means a more complicated mechanism and the responses of every PTO axis always interact with each other. The time-varying displacement along each PTO axis is different and the wave force that acts on each PTO axis is also not the same. Power output along one axis can even become the load of the others. Some of these complexities have been reported in [9], where the control strategies and power optimization of a 2-DOF WEC were addressed. Focusing on the power absorption in each PTO axis is a key issue aiming to illustrate the interference between each PTO axis. As a result, before taking the multi-axis WEC into next stage of research and application, inherent power absorption characteristics of multi-axis WECs compared to single-axis devices need to be deeply studied. Considering these issues, numerical modelling becomes necessary to simulate the effects of additional PTO axes and investigate the power absorption performance of a multi-axis WEC. The major advantages of using numerical modelling are to test several WEC configurations and wave conditions at a lower cost than with experimental tests, though a simplified physical model test is needed to validate the results obtained with numerical models. The methods on how to build numerical models of the WECs have been a hot issue of research so far. In recent years, an increasing number of numerical modelling techniques have been developed. Each of the WEC numerical modelling methods has a certain set of characteristics that makes it more or less suitable for a particular WEC. In recent years, BEM-based numerical codes such as the well-known WAMIT, Aquaplus and AQWA are commonly used to model the interactions between WECs and the incident waves. [10] studied the power performance of a two-body heave converter using time domain models. [11] investigated the influence of the shape, draft and diameter of the model on the power absorption by building frequency domain models with hydrodynamic parameters calculated in AQWA. [12] investigated the hydrodynamic performance of an oscillating water column WEC by using a higher order BEM. [13] carried out an extensive study of the influence of the PTO characteristics on the performance of CECO wave energy converter and concluded that the CECO WEC can absorb between 10% and 40% of the incident waves. Later, [14] presented a study on the effects of the direction of the translation motion on the power capture of the CECO WEC in AQWA. Although it brings the loss of accuracy and the increase in computational cost, the BEM is still the most recommended method for investigating the performance of large-scale WECs, especially the multi-DOF systems. It has been reported by [15, 16] that the time domain models are more suitable for studies on irregular waves and related non-linear forces and oscillations than frequency-domain models which suit regular waves and systems with linear approximations. In order to realistically investigate the performance of the multi-axis WEC, any further simulation works in this study will be carried out in time domain. This work aims to obtain a detailed knowledge of the behaviour of the multi-axis WEC in terms of the following performance characteristics: the interaction between translation and rotation modes, the absorbed power and efficiency of the device and the differences of performance between modes. In order to achieve this, the performance of multi-axis WEC was simulated first in the time domain with AQWA for a wider number of regular wave conditions. In this study, the multi-axis WEC with a cylindrical buoy were considered, and two kinds of single-axis WEC were also analysed for comparison. Later, as an application case study, the mean power absorbed by the multi-axis WEC at the 12 Zhejiang and Guangdong nearshore locations in the East and South China Sea is estimated, as well as the power absorption efficiency. This paper is organized as follows: In Section 2, a description of the multi-axis WEC design concepts, the mathematic model and the BEM model in AQWA are presented; Simulations for regular waves with different periods and heights are shown in Section 3, while the results are detailedly discussed including the analysis of the WEC performance characteristics; then, the application to real sea states in East and South China Sea is presented in Section 4; Finally, the main conclusion and prospective work is mentioned in Section 5. 2 MODEL AND METHODS Mathematic models The simplified scheme of the multi-axis point absorber WEC is shown in Figure 2. A vertical cylinder is adopted as the geometry of the floating buoy, which introduces heave, pitch and roll oscillations. The proposed multi-axis PTO structure consists of a vertical shaft (part 3 in Figure 2) and a cylindrical linear bearing (part 2 in Figure 2) linked to a rectangular frame (part 1 in Figure 2) by means of two pairs of joints, allowing the buoy to move anywhere within the limits of the PTO system. For a multi-axis structure, there are three relative motions, that is, three modes of energy absorption, namely translation mode, two rotation modes (defined as rotational1 mode and rotational2 mode, respectively). Compared to traditional translational/rotational axis devices, two more axis PTO units were added making it possible to absorb both kinetic and potential energy. A linear damper in each joint of the device provides the conversion of the energy absorbed from waves. A major problem with this kind of application in real sea states is an offshore platform where the multi-axis PTO system being installed needs to be built. Another potential problem is that the complexity of supporting structure and PTO system mean higher costs and risks. A full discussion of reliability and economics problems lies beyond the scope of this study and will not be discussed here. FIGURE 2Open in figure viewerPowerPoint The schematic of WECs: (a) translation-axis WEC, (b) rotation-axis WEC, (c) multi-axis WEC Two kinds of single-axis WECs are also demonstrated in Figure 2 as a comparison group. These single-axis WECs, commonly referred to as single translational axis or single rotational axis WECs, have been extensively studied [17, 18]. In this study, the same floating cylindrical buoy was also applied to the single-axis WECs. Such a choice has been made in order to compare the performance of single-axis WECs explicitly with the performance of the multi-axis WEC. The dynamic analysis of the multi-axis device needs to be divided into two parts: the hydrodynamic interaction between floating buoy and waves, and the multi-body dynamics of multi-axis structure. For the hydrodynamic analysis, the equation of motion could be expressed in a convolution integral form: { m + A ∞ } X ̈ ( t ) + c X ̇ ( t ) + KX ( t ) + ∫ 0 t R ( t − τ ) X ̇ ( τ ) d τ = F ( t ) , (1)where X ( t ) is the motion response of the buoy, m is the structure mass matrix, A ∞ is the fluid added mass matrix at infinite frequency, c is the damping matrix except the linear radiation damping effects due to diffraction panels, K is the total stiffness matrix, R ( t ) is the velocity impulse function matrix, and F ( t ) is the external force acting on the centre of gravity of the float. In which R ( t ) and A ∞ are defined by: R t = 2 π ∫ 0 ∞ B ( ω ) cos ω t d ω , (2) A ∞ = A ω + 1 ω ∫ 0 ∞ R ( t ) sin ω t d ω , (3)where A ( ω ) and B ( ω ) are the added mass and hydrodynamic damping matrices, respectively. The external force comprises three components: F t = F e ( t ) + F PTO ( t ) + F cons ( t ) , (4)where the first term represents the wave excitation force on the body, the second refers to the reaction force from the PTO system and the third to the constraint force caused by the supporting frame to hold the WEC in position. The coordinate system and several key parameters of the system need to be defined before carrying out and dynamic analysis [7]. Here, same definition of the coordinate system and method to derive the dynamic equation of motion is adopted, as shown in Figure 2. The global coordinates of the model are ( X , Y , Z ) and the local coordinates ( x 1 , y 1 , z 1 ) , ( x 2 , y 2 , z 2 ) , ( x 3 , y 3 , z 3 ) are the coordinates for the individual motion components of the multi-axis WEC model, respectively—a rotational about the rotational1-axis (part 1), a rotational about the rotational2-axis (part 2) and a translation along the translational-axis (part 3), as shown in Figure 2. The input wave forces [ F , M ] transmitted from the buoy act at the lower end of part 3 which is rigidly connected to the floating body. The dynamic analysis of the multi-axis structure is similar to that of a gyroscope and thus the motion equation can be formulated by applying the Newton–Euler equations. F M = m I − m c m c J c − m c c r ̈ ω ̈ + m ω × ω × c ω × J c − m c c ω , (5)where F is the total force acting on the centre of gravity; M is the total moment acting about the centre of gravity; m is the mass of the body; r ̈ and ω ̈ are the linear and angular acceleration of the system; J c is the mass moment of inertia about the centre of mass; I is the identity matrix; c is the location of the centre of mass of the system in the global coordinate. The dynamic equation of the multi-axis structure is obtained by applying Equation (5) to the system: 0 0 F z − k 3 z t − c 3 z ̇ t − m 3 g c o s γ t c o s θ t − 2 k 1 γ t − 2 c 1 γ ̇ t − m 3 g z g s i n γ t c o s θ t − M y z t + z i + 1 − F y z t + z i − 2 k 2 θ t − 2 c 1 θ ̇ t − m 3 g z g s i n θ t + M x z t + z i + 1 + F x z t + z i 0 = m 3 0 0 0 m 3 z g 0 0 m 3 0 m 3 z g 0 0 0 0 m 3 0 0 0 0 − m 3 z g 0 I x x + m 3 z g 2 0 0 − m 3 z g 0 0 0 I y y + m 3 z g 2 0 0 0 0 0 0 I z z 0 0 z ̈ t γ ̈ t θ ̈ t 0 + 0 0 γ ̇ 2 t m 3 z g + θ ̇ 2 t m 3 z g 0 0 γ ̇ t θ ̇ t I x x + m 3 z g 2 − γ ̇ t θ ̇ t I y y + m 3 z g 2 (6)where k 1 , k 2 and k 3 are the stiffness of each joint and c 1 , c 2 and c 3 are the damping constants of each joint, respectively. I x x , I y y are the moments of inertia of the system about the axes, z g is the distance from the origin of the global coordinate to the centre of gravity of the system. γ, θ and z are the rotational displacements about the X-axis, Y-axis and linear displacement along the Z-axis, respectively, z i is the distance between the origin of the global coordinate system and the buoy, and m 3 g is the weight of the part 3. Each component in [ F x F y F z , M x M y M z ] is decomposed by the input wave forces [ F , M ] . The time-averaged absorbed power in each PTO axis can be calculated as: P γ = 1 T ∫ 0 T c 1 γ ̇ 2 t dt P θ = 1 T ∫ 0 T c 2 θ ̇ 2 t dt P z = 1 T ∫ 0 T c 3 z 2 t dt . (7) The time-averaged total absorbed power of the system is P total = P γ + P θ + P z . (8) A common parameter used to represent the performance of a point absorber WEC is the capture factor, which can be expressed as C f = P PTO P W × D , (9)where P PTO is the power absorbed by the PTO system of the WEC which equals P total here, P W is the incident wave power and D is the physics width of the WEC. However, the definition of capture factor C f to some extent as it is defined only in relation to the total absorbed power and does not sufficiently represent the difference of absorbed power along each PTO axis. Mode capture ratio C γ , C θ , C z accounting for different PTO axes are defined as: C γ = P γ P total , C θ = P θ P total , C z = P z P total (10) Numerical model description Developed to analyze the wave interaction of marine structures and ships, the BEM based numerical modelling software AQWA provides an engineering toolset for the investigation of the effects of waves on the renewable energy systems, especially the wave energy conversion devices. The relatively high efficiency in computation and low hardware requirements make the further optimization work of WECs possible. The simulations on power absorption performance of all the single and multi-axis WECs mentioned above are carried out with AQWA. The specific simulation settings are as follows. 2.2.1 Simulation configuration There are three steps to build a numerical model in AQWA. At first, the geometry of the WEC (includes several parts) is meshed to create non-diffracting and diffracting elements, respectively, and the simulation parameters need to be fully defined. It should be noted that the 64-bit version of the AQWA solver is limited to 40,000 elements, of which the diffracting elements should not be greater than 30,000. The models are illustrated with their surfaces meshed for hydrodynamic analysis, in Figure 3. There are four components composing of the models which are labelled components 1, 2, 3 and 4, respectively. The multi-axis WEC model consists of components 1, 2, 3 and 4. The single translation-axis WEC model consists of component 3 and 4. The single rotation-axis model consists of component 2, 3 and 4. It should be noted that component 3 is fixed in place in single translation-axis WEC model, as well as the component 2 in single rotation-axis WEC model. Furthermore, component 3 is kept in single rotation-axis WEC model such that the value of the pitching moment of inertia equals that of the multi-axis WEC model and fixed on component 4 to limit the relative translation motion between component 3 and 4. The physical properties of the components and meshing configuration are listed in Table 1, with their values and units. FIGURE 3Open in figure viewerPowerPoint The mesh of the multi-axis WEC TABLE 1. Physical properties of the components Component 1 Component 2 Component 3 Component 4 Length (m) 10.8 3.6 - - Dimensions Width (m) 6.48 3.6 - - Height/draught (m) 0.9 0.9 5.4 5.76/3.76 Diameter (m) - - 1.8 4.14 Connection Multi-axis WEC Fixed Hinged Hinged Free to move Single translation-axis WEC - - Fixed Single translation-axis type Single rotation-axis WEC - Fixed Hinged Single rotation-axis In the second step, the analysis settings were established. First, the AQWA solver was used to solve the wave interaction of the parts containing diffracting elements in the frequency domain. A total of 50 frequency ranging from 0 to 1 Hz were calculated while the wave directions varied from 0° to 360° with an interval 15°. The velocities potentials were first calculated and the hydrodynamic coefficients, including the response amplitude operators, the diffraction forces, the radiation forces and the added mass were obtained. The frequency-domain results are shown in Figure 4. The response amplitude operator (RAO) in heave and surge of the cylindrical buoy without connecting to the multi-axis PTO system was obtained. It can be seen that the nature frequency of the buoy in heave was around 0.2 Hz. As can be seen in the follow-up simulation results, there will be significant differences between the motion responses of the buoy with and without multi-axis PTO system. Subsequently, based on the results obtained in frequency domain, the wave interaction of WEC models was solved in the time domain. The waves can be regular or irregular. Each simulation runs 200 s with a constant time step set as 0.01 s. For each time step in a simulation, the dynamic equation of motion was solved and the position and velocity of each part in the models was computed. As the wetted surfaces of the model vary significantly during a simulation, the hydrostatic and the Froude–Krylov forces were non-linear and calculated for each time step. The radiation force was calculated as the sum of the impulse response function convolution and the inertia force due to the added mass at infinite frequency was calculated. FIGURE 4Open in figure viewerPowerPoint The position RAO of the buoy without multi-axis PTO system 2.2.2 Waves Once the numerical models were built and the parameters were determined, the characteristics of the power absorption performance of the multi-axis WEC with optimized linear PTO damping was examined for a broad range of regular wave conditions. Regular waves were characterized by their wave period (T) and wave height (H). More specifically, a total of 77 simulations were carried out under different wave conditions which are the combination between T = 3, 3.2, 3.4, 3.6, 3.8, 4, 4.2, 4.4, 4.6, 4.8, 5 s and H = 0.2, 0.4, 0.6, 0.8, 1, 1.2 and 1.6 m. For regular waves, the incident wave power P W in W m − 1 can be described as below: P W = 1 8 ρ g H 2 C g , (10) where ρ , g and H present the wave density, gravity, wave height and C g is the wave group velocity, which can be expressed as: C g = 1 2 g k tanh k d 1 + 2 k d sinh 2 k d , (11)where k is wave number and d is water depth. Subsequently, the multi-axis WEC is exposed to twelve different real sea states to test the ability to capture wave power in Zhejiang and Guangdong nearshore locations in East and South China Sea. A JONSWAP spectrum was used to simulate irregular waves characterized by two spectral parameters peak wave period ( T P ) and the significant wave height ( H S ), which can be expressed as S H ω = 319.1 H s 2 T p 4 ω 5 − 1948 T p ω 4 3.3 e x p − 0.159 ω T p − 1 2 2 σ 2 . (12) 2.2.3 Numerical model validation To validate the numerical model above with experimental data, a physical 1/10 scaled model of multi-axis WEC was tested in the wave flume of the Zhejiang University Ocean College, shown in Figure 5. The Froude similitude criterion was used to scale the experiment equipment and conditions. The dimensions of the wave tank are 25 m length, 0.7 m width and 0.7 m depth. The device was placed on the central line of the wave tank at a distance of 12.5 m from the wave-maker, as shown in Figure 6. The buoy's motion along each PTO axis is measured by displacement sensors. The wave condition of wave height 0.1 m and wave period 3 s is considered. Figure 7 compares the numerical and experimental results of the device's displacement along each PTO axis without PTO damping. It can be seen that the numerical and experimental results have a good agreement. FIGURE 5Open in figure viewerPowerPoint Physical model experiment FIGURE 6Open in figure viewerPowerPoint Experimental set-up in the wave flume FIGURE 7Open in figure viewerPowerPoint Validation with experimental data: (a) translation axis, (b) rotation axis However, it should be noted that when the wave height is too high, angular displacement curve with smaller slope and reduced amplitude of the buoy would be found and evident difference was observed between the numerical and experimental results. It may due to undesired non-linear effects caused by the relatively larger wave height. Splashing and submergence can be observed during the experiment when the wave height is relatively high, which would slow the forward return rotation of the buoy. This non-linear phenomenon is very interesting and worth to be studied, but it's not the focus of this paper. In the follow-up simulation study, moderate waves are used to ensure the accuracy of the model. 3 RESULTS AND DISCUSSION Interactions between modes The simulations for single-axis WECs aim to establish a reference point in order to investigate the effects of increasing number of DOFs on the power absorption capabilities of single-axis WECs. When allowing the WEC oscillates in more than one mode of motion, the interactions between modes will exist. As a result, both the total wave force acting on the buoy and the resulting amplitude of the motion in each mode differ from those of single-axis WECs. The interactions are investigated by analysing the translational response ratio and rotational response ratio defined as the ratio of the amplitude of translational or rotational motion to the wave height. Four simulation cases with the wave height 0.4, 0.8, 1.2 and 1.6 m are considered in this study. The incident wave angles are assumed to be 0° in all cases. The results of the translational and rotational response ratios of multi-axis and single-axis WECs versus the wave period are shown in Figure 8. In general, with lower wave height (H = 0.4, 0.8 and 1.2 m), both the translational and rotational response ratios of the buoy increases as the wave period increases, and approaches to the peak value at a certain period, and then decreases as the wave period continues to increase. The wave periods at which the translational and rotational response ratio in each mode reaches the highest point can be defined as the resonance periods in each mode, respectively. It is worth pointing out that there is a difference between the resonance periods in translation and rotation modes, indicating that the dominancy of the modes shifts from translation to rotational as the wave period increases. It can be seen that the translational response ratio in translational mode reaches maximum at around 4.2 s. It is clear from the figure that the effects of wave heights on the peak values of translational and rotational response ratios are small. For the lower and higher wave periods, the differences between the translational and rotational response ratio curves of multi-axis WEC and single-axis ones are almost reduced to zero. Rotational response ratio curves show similar variation tendency to those presented in translation mode while the rotational response ratio reaches maximum at around 3.8 s in all cases. FIGURE 8Open in figure viewerPowerPoint Translational and rotational response ratios versus wave period: (a) H = 0.4 m, (b) H = 0.8 m, (c) H = 1.2 m, (d) H = 1.6 m One can see that in simulations with the highest wave height