Oscillating Body Design for A 3-DOF Wave Energy Converter

Ocean wave energy converters (WECs) are obtaining more and more attentions in the world. So far, many types of converters have been invented. Oscillating body systems are a major class of WECs, which typically have one degree of freedom (DOF), and the power absorption efficiency is not quite satisfactory. In this paper, a 3-DOF WEC is proposed and a simplified frequency-domain dynamic model of the WEC depending on the linear potential theory is conducted. The performances of three geometries of the oscillating body including the cone, the cylinder and the hemisphere have been compared, and the results show that the hemisphere is more suitable for the 3-DOF WEC. Subsequently, the relationship among the parameters of the hemisphere is established based on the equal natural frequencies of the heave and pitch (or roll) motions, and the results show that lowering the center of gravity leads to the better power absorption in the pitch (or roll) motion. In the end, the power matrixes of different sizes of the hemispheres under different irregular waves are obtained, which can give a size design reference for engineers.


Introduction
Wave power is originated from wind, so it is an indirect form of solar energy. It is well known that wave power has the advantages of being clean, renewable, abundant, more predictable and so on (Zhang et al., 2009;Li, 2015, 2017;. There exist three WEC systems classified by the working principle (de Falcão, 2010) including the oscillating water column system (Brito-Melo et al., 2008;Masuda, 1986), the oscillating body system (Albert et al., 2017;Jakobsen et al., 2016;Muliawan et al., 2013;Ruehl et al., 2010;Shi et al., 2015;Vissio et al., 2017;Zhang et al., 2013Zhang et al., , 2017 and the overtopping converter system (Russell et al., 2016). Among various technologies, the oscillating body system has been researched widely in recent years. Many problems exist in the oscillating body system, such as the mooring, the maintenance, the reliability to resist extreme environments, the high capital investment, and so on. As a consequence, only some systems have reached the full-scale stage, such as OPT and Pelamis.
The oscillating body systems generally extract wave power through relative motion between two bodies, almost all of which oscillate in the heave (OPT (Ruehl et al., 2010) and WaveBob (Weber et al., 2009)) or pitch (ISWEC (Bracco et al., 2011) and SEAREV (Babarit et al., 2006)). When the oscillating body is axisymmetric, only the heave motion, the pitch motion and the roll motion of the body have restoring force and moments, which facilitate the power generation, because the restoring force and moments can make the body itself return to the equilibrium position without external force and moments.
The power absorption performance of the oscillating body system mainly depends on the dynamic response of the oscillating body, which is influenced significantly by the body's geometry. Axisymmetric geometry body is commonly studied before, as it absorbs energy without the constraint of wave directions. The search for optimal geometries of wave absorbers has been conducted for many years (Babarit et al., 2005;Bachynski et al., 2012;Kurniawan and Moan, 2013;Mavrakos et al., 2009). And the well-known theoretical results for the maximum capture width by an axisymmetric converter oscillating in the heave and pitch in regular waves have been obtained by Falnes (2002). However, there is no comparison and design of the body's geometry for the 3-DOF WEC. Here we aim at studying the numerical results for the maximum efficiency for the axisymmetric converter oscillating in 3-DOF in irregular waves.
This work here focuses on the geometry, dynamic parameters and size design of the oscillating body for the 3-DOF WEC. First, the 3-DOF WEC is introduced. Then a simplified dynamic model of the WEC is presented in the frequency domain. Relying on the mathematical model, we compare the performances of three axisymmetric geometries including the cone, cylinder and hemisphere based on the criteria: they have the same waterplane, same draft and same rotary inertia.
When the geometry of the oscillating body is determined, the dynamic parameters of the body need to be investigated. The natural frequency of the oscillating body in the heave motion is mainly dependent on the body geometry. While in the pitch motion, the natural frequency relies on the center of gravity and the radius of gyration, besides the geometry. It is beneficial for the power absorption when the natural frequencies of the oscillating body in the heave and pitch motions are approximate. Thus, we design the dynamic parameters on account of the condition.
When the geometry and dynamic parameters of the oscillating body are determined, the size of the geometry needs to be confirmed. In the end, the power matrices of different sizes of the hemispheres under different irregular waves are presented, which can give a size design reference for engineers.

Mathematical model
In deep water, the energy flux transmitted by irregular waves per unit crest width can be written as (Falnes, 2007): where ρ is the sea water density, g is the acceleration of gravity, H s is the significant wave height, and T e is the wave energy period. The expression of the wave power shows that it is the square dependence on the significant wave height and the linear dependence on the wave energy period. The oscillating body has six DOFs including the surge, sway, heave, pitch, roll and yaw in the random ocean waves, as shown in Fig. 1. The best case is that the energy of all motions of the oscillating body is absorbed. For an axisymmetric oscillating body, it is beneficial for the power absorption in the heave, roll and pitch motions. Because the restoring forces (moments) in the three directions can enforce the body return to the balance position, when the body oscillates. In consideration of the surge and sway motions, there are no restoring forces. Therefore, some devices should be applied to restrict the drift of the oscillating body and provide the restoring forces to realize reciprocating motions. Especially it is difficult for the power generation when the body is too large. Theoretically, there is no motion in the yaw direction for the axisymmetric body in one-directional waves. Therefore, it is significant to design the 3-DOF WEC.
The 3-DOF WEC is shown in Fig. 2. The oscillating body induced by waves has three motions, including the heave, roll and pitch. Specifically, the body can rotate around Point O and oscillate up and down. The 3-DOF mechanism is proposed to transmit the three motions of the oscillating body to reciprocating rotations (θ, ψ and ϕ) relative to the frame, which have the same rotation axis.
Then the three rotations drive the hydraulic PTO system which is displayed in Fig. 3 in detail. A group of energy conversion devices (ECDs) are driven by each rotation of the mechanism to export high pressure oil stored in accumulators. Finally, the high pressure oil drives the hydraulic motor and electric generator to produce electricity. The gearboxes are utilized to increase the rotational speed.
The 3-DOF mechanism is demonstrated in Fig. 4 in de-  tail, which mainly consists of a four-bar linkage and a 2-DOF spherical joint. The four-bar linkage is used to capture the heave motion and the 2-DOF spherical joint is employed to capture the pitch and roll motions. A RSSR linkage is applied to change the transmission direction of the roll motion from the spherical joint vertically. Then the two parallel rotations are transmitted to the frame by the transmission linkage. Thus, the 3-DOF mechanism achieves the coaxial outputs and decouple the three motions of the oscillating body. Real fluid effects (viscous effects) are expected to be important, especially for larger wave amplitudes (Yeung and Jiang, 2014). In this paper, non-linear characteristics and viscous effects are not considered. Fig. 5 represents a simplified dynamic model based on a hemispherical oscillating body in one-directional waves. D and L are the diameter and draft of the body, respectively, and the body only has two modes of motion in the heave (η 3 ) and pitch (η 5 ). The PTO systems for the WECs are assumed to be linear damping terms here. The origin of the fixed coordinate system is at the center of the waterplane, with the z axis pointing vertically upward and the x direction aligned with the direction of the wave propagation. Due to the axisymmetry of the geometry of the oscillating body, there are no cross coupling terms between the heave and pitch motions. As the system is completely linear, the frequency-domain dynamic equations are given by where M 3 is the mass of the oscillating body, M 5 is the rotary inertia, which is equal to with respect to the axis through the point O, r g is the radius of gyration, F e3 and F e5 are the complex excitation force and moment. F r3 and F r5 are the radiation force and moment. F k3 and F k5 are the restoring force and moment. F g3 and F g5 are the force and moment induced by the PTO systems. F where μ 33 , μ 55 , λ 33 , and λ 55 are the frequency dependent added mass and radiation damping terms, k 3 and k 5 are the restoring stiffness coefficients (Bachynski et al., 2012). k where R is the radius of the waterplane, Z G is the z coordinate of the center of gravity (Point G), Z B is the z coordinate of the centroid of the geometry under the waterline (Point where c 3 and c 5 are the ideal PTO systems. Because where and are the amplitudes of the complex displacement. and are the amplitudes of the complex excitation force and moment, and ω is the wave frequency. Thus, Eq. (2) can be described as: Solve the equations above: Then the power outputs can be written as: We note that p 3 =0 for c 3 =0 and for c 3 =∞, and p 3 >0 for 0<c 3 <∞. Thus, there is the maximum of the absorbed power when , which occurs if (Falnes, 2002) Thus, we have the maximum absorbed power in the heave motion: Similarly, when The maximum power output in the pitch motion: The natural frequencies of the body oscillating in the heave and pitch motions can be written as: To improve the universality of the comparison of the three geometries. The non-dimensional parameters used here are given below: (32) Real irregular wave can be represented as a superposition of regular waves, by defining a spectrum. Pierson-Moskowitz spectral distribution is adopted here, which is used in the previous work (Vicente et al., 2013).
where H s is the significant wave height and T e is the wave energy period. The irregular waves can be deemed as a superposition of N ω regular waves with the amplitudes: (34) The averaged power absorptions in irregular waves can be written as (Babarit, 2010): In real waves, the ocean wave power transmitted across a wave crest length equal to the diameter 2R of the waterplane can be written as: Thus, the power absorption efficiency by the oscillating body system in the heave motion can be expressed as: Similarly, in the pitch motion The total efficiency is

Geometry comparison
The hydrodynamic parameters are dependent on both the body geometry under the waterline and wave frequency. Obviously, the dynamic response of the WEC is affected significantly by these parameters (see Eq. (17)). Thus, the geometry has a great impact on the power outputs of WEC. Three axisymmetric geometries are taken into account including the cone, cylinder and hemisphere, all extended by a cylindrical part (see Fig. 6). The ocean power energy is proportional to the wave crest length. Therefore, the performances of the three geometries are compared based on the same waterplane. In order to study the geometry influence only, the parameters , and are confirmed equal.
Thus, one size and parameters of the geometries are listed in Table 1. The hydrodynamic parameters are obtained by the software AQWA-LINE. The amplitudes of the wave excitation forces (moments) for the three geometries in the heave and pitch motions per unit wave amplitude is plotted in Fig. 7. It is clear that the wave excitation forces in the heave motion of different geometries are cone>hemisphere>cylinder. The curves of the wave excitation moments for the cone and hemisphere are relatively close to each other, whereas the values for the cylinder are much smaller. Fig. 8 shows the frequency dependent non-dimensional added mass of the three geometries in the heave and pitch motions. And Fig. 9 shows the corresponding non-dimen-sional damping coefficients. As we can see from Fig. 8, the non-dimensional added mass of the cone is bigger than that of the hemisphere and cylinder. The change of the added mass of the hemisphere and cylinder is relatively stable. From Fig. 9, the non-dimensional damping coefficients of the cone are also larger than that of the cylinder and hemisphere.
Substitute the non-dimensional parameters into Eq. (20) and Eq. (22), then Based on Eqs. (40) and (41), and the parameters shown in Figs. 7-9, the maximum power absorptions per unit wave amplitude for the three geometries are drawn in Fig. 10, when the PTO damping terms are best selected. The power output of the cylinder is the largest in specific wave frequency both in the heave and pitch motions. But the range of the frequency for high power absorption is relatively small. The cylinder is the preference in regular low frequency waves. For real waves, the cone and hemisphere are better choices. The performances of the two geometries are similar. Specifically, in the pitch motion, the power output of the hemisphere is higher than that of the cone under the most frequencies. Consequently, the hemisphere may be better to be chosen.
Because the added mass is dependent on the wave frequency, the natural frequencies of the three geometries are also related to the wave frequency according to Eqs. (23) and (24). The crossover points of the solid line and dashed line shown in Fig. 11 are the natural frequencies of the body. In the heave motion, it seems that the cone is benefi-cial to absorb the energy from high frequency waves, while the next are the hemisphere and cylinder. In the pitch motion, the cone is beneficial to absorb the energy from the low frequency waves, and the next are the hemisphere and cylinder. When the 3-DOF WEC are designed, it is important to make the natural frequencies of the three motions approximate. As can be seen from Fig. 11, the natural frequencies of the two motions of the hemisphere are close, and better to be selected. In consideration of the forces (moments), power absorbed spectrum and natural frequencies, the hemisphere is selected here.

Influence of the dynamic parameters
When designing the dynamic parameters, it is necessary to keep the natural frequencies of the heave and pitch motions equal, which is beneficial for the power absorption in the corresponding sea state. Thus, According to Eq. (23)-Eq. (26), then  ( r g Z G Thus, the relationship between and is obtained.
The wave excitation moment in the pitch motion is dependent on and , where is the amplitude of the wave excitation force in the surge motion. For the hemisphere in Table 1, and are shown in Fig. 12. We can draw the relationship beloŵ (45) Then, Eq. (41) can be written as: μ 33 λ 55 Z G When R=10 m, and are determined to be 0.42 and 0.1, respectively according to Fig. 5 and Fig. 6. As we can see from Eq. (46), p 5max is mainly dependent on . The power outputs from the pitch motion affected by the center of gravity are depicted in Fig. 13. Obviously, the power absorption performance is better when the center of gravity is lower.      For the specific size of the geometry, it must adapt to the specific wave. The efficiency of the power absorption of the hemisphere (R=10 m, =0.5 and =0.577) varying with T e =5, 6, 7, 8, 9 and 10 s is displayed in Fig. 15. The results show that the total efficiency is 0.846 when T e =6 s. It is necessary to give a design reference for selecting the size of the hemisphere. According to the computation for the efficiency of the hemisphere (R=10 m) above, the power matrices of different size hemispheres under different T e waves are obtained, including the heave motion, the pitch motion and the heave and pitch motions. As we can see from Fig. 16 to Fig. 18, in the heave motion, the maximum efficiency is over 30%, when the appropriate size of the hemisphere is selected in specific waves. Similarly, the maximum efficiency is over 50% in the pitch motion. Because there are no cross coupling terms between the heave and pitch motions, the power output of the heave and pitch motions is independent. When the 3-DOF WEC is designed, the efficiency can exceed 80% theoretically.

Conclusions
The objective of this paper is to find a better geometry of the oscillating body system and to give a design reference of the wave absorber for the engineers. In this paper, the 3-DOF WEC is presented. A simplified frequency-domain analysis of the WEC depending on the linear potential theory is conducted, and the generators are modeled as viscous damping terms. The performances of three geometries (the cone, cylinder and hemisphere) have been compared and the hemisphere is selected. Subsequently, we reveal the relationship between the dynamic parameters of the hemisphere. The results show that the lower the center of gravity is, the better the power absorption in the pitch motion will be. In the end, the power matrices of different sizes of the hemispheres under different irregular waves are obtained, which can give a size design reference for engineers.
In this paper, there are some shortcomings at present. Only one-directional waves and the simplified linear frequency model are conducted, and the results are achieved only by calculations. In the future works, the experiment should be conducted to verify the results.