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On The Dynamics and Design of a Two-body Wave Energy Converter

A two-body wave energy converter oscillating in heave is studied in this paper. The energy is extracted through the relative motion between the floating and submerged bodies. A linearized model in the frequency domain is adopted to study the dynamics of such a two-body system with consideration of both the viscous damping and the hydrodynamic damping. The closed form solution of the maximum absorption power and corresponding power take-off parameters are obtained. The suboptimal and optimal designs for a two-body system are proposed based on the closed form solution. The physical insight of the optimal design is to have one of the damped natural frequencies of the two body system the same as, or as close as possible to, the excitation frequency. A case study is conducted to investigate the influence of the submerged body on the absorption power of a two-body system subjected to suboptimal and optimal design under regular and irregular wave excitations. It is found that the absorption power of the two-body system can be significantly higher than that of the single body system with the same floating buoy in both regular and irregular waves. In regular waves, it is found that the mass of the submerged body should be designed with an optimal value in order to achieve the maximum absorption power for the given floating buoy. The viscous damping on the submerged body should be as small as possible for a given mass in both regular and irregular waves.

relative motion of the two bodies. There are two advantages to add an additional submerged body under the floating buoy. Firstly, the second body can act as a reaction body, which makes mooring easier in deep water, compared with directly connecting the floating buoy to the seabed. Secondly, the dynamics of second body can be utilized to improve the performance of the overall system. Two typical designs of two-body wave energy converters are the Powerbuoy (Figure 1 right), which was developed by Ocean Power Technology in the USA [21], and Wavebob (Figure 1 left), which was developed in Ireland [22]. Both devices use the relative motion between the floating buoy and submerged body to extract energy from ocean waves. However, it is found in Figure 1 that the design philosophy of the submerged body is quite different between these two devices. Wavebob used a second body with a streamline configuration and large mass while Powerbuoy used a resistive heave plate.
Early studies of two-body wave energy converters mainly focused on regular wave excitation. Falnes [23,24] investigated the concept of using relative motion between a floating buoy and a submerged body to harvest energy. He found that it is possible to achieve the optimum power = 3 2 4 3 for a two-body wave energy converter oscillating in heave in regular waves. Korde [25] explored the concept of two-body wave energy converter and compared the performance of a submerged reaction mass with a reaction mass out of water under regular wave excitation. He found that the submerged mass appears to perform better overall. Bijun Wu et al. [26] also studied the response and conversion efficiency of a two-body wave energy converter under regular wave excitation and they found that the performance of a two-body wave energy converter highly depends on the system parameters, like the physical properties of the buoy and incident wave frequency. The results obtained by the previous literature helped people to understand that it is possible to design a submerged body to achieve the theoretical maximum absorption power under regular wave excitation by neglecting the viscous damping. In real application, however, it is necessary to investigate the design of a two-body wave energy converter under irregular wave excitation as well.
Recently, an experiment was conducted by Beatty et al. [27] to compare the hydrodynamic performance of the submerged body for Wavebob and Powerbuoy. The authors found that a streamline submerged body has a smaller added mass compared with a resistive heave plate. The experimental results of added mass and excitation force on the submerged body agreed fairly well with the results obtained using boundary element method (BEM) for both prototypes. However, the damping on the second body is much larger in comparison with radiation damping obtained with BEM code. Therefore, it is necessary to consider the viscous effect while studying the two-body wave energy converter, especially for the submerged body. Based on Beatty's experimental study, in this paper, linearized damping is considered to account for the viscous effect. In the case study, the viscous effect on the floating buoy was neglected [27]. Based on the aforementioned literature review, the first objective of this paper is to study the dynamics of the two-body wave energy converter, including the viscous effect. In addition to radiation damping, a linearized damping due to viscous effect was considered to build the frequency domain model. The closed form solution of the absorption power for a two-body wave energy converter was obtained in terms of system parameters and wave information under regular wave excitation. The suboptimal and optimal power take-off designs are defined based on the obtained closed form solutions.
The results presented in this paper are further used to study the influence of submerged body on the absorption power of the two-body system.
The second objective of this paper is to study the design of the submerged body on the overall performance of a two-body wave energy converter under regular and irregular wave excitations. When the submerged body is deployed far enough from the floating buoy and free surface, the hydrodynamic interaction between these two bodies, as well as the radiation damping on the submerged body, can be neglected. Numerical simulations were conducted to study the influence of the submerged body on the power take-off design and absorption power under regular wave excitation. The suboptimal and optimal designs are modified and investigated in irregular waves. The absorption power based on these two design principles are also compared with the maximum absorption power of a single body system with the same floating buoy. It is found that if the submerged body is designed properly, the absorption power of a two-body wave energy converter can be significantly higher than that of a single buoy system with the same floating buoy under regular and irregular wave excitation, even in the presence of a large viscous damping on the submerged body.
The remaining of this paper is organized as follows: In Section 2, the linearized model of a twobody wave energy converter under regular wave excitation is established and the closed form solution of absorption power and corresponding power take-off parameters are obtained. In Section 3, suboptimal and optimum power take-off design and the corresponding absorption power are studied in regular waves. In Section 4, the suboptimal and optimal design are implemented in irregular waves and compared with the single body system with the same floating buoy. The constrained simplex method is used to find the maximum absorption power of a two-body system under irregular wave excitation. Conclusions are given in Section 5. Figure 2 is the schematic of the two-body wave energy converter considered in this paper. A floating buoy is located on the water surface and is connected to a submerged body through a power take-off (PTO) system. The submerged body is suspended in the water. The energy is extracted through the relative motion of the floating and submerged bodies. One should notice that the floating and submerged bodies considered in this paper are axisymmetric but they do not need to be cylindrical as shown in Figure 2. The power take-off system which is installed between the floating buoy and submerged body is assumed to be linear as in most literature, and the take-off force is:

Dynamics Analysis and Design of Two-body Wave Energy Converter in Regular Waves
One should notice that the submerged body is assumed to be suspended in the water. Therefore, we assume ≥ 0 since negative spring stiffness makes the system unstable. The equation of motion of a two-body wave energy converter shown in Figure 2 is: (3) where, 1 and 2 are the respective heave displacements of the floating and submerged bodies; 1 and 2 are the respective masses of the floating and submerged bodies; ( , = 1,2) is the frequency dependent added mass which accounts for the mass of the water moving with the -th body, induced by the motion of -th body (1 and 2 represent the floating and submerged bodies, respectively); ( , = 1,2) is the frequency dependent radiation damping coefficient, which accounts for the energy dissipation by the radiated wave due to the motion of the -th body, induced by the motion of -th body; _ ( , = 1,2) is the linearized viscous damping coefficients on the floating and submerged bodies; and are the stiffness and damping coefficient of the power take-off system; is the hydrostatic stiffness of the floating buoy, which is the spring effect due to the difference of gravity and buoyancy; 1 and 2 are the exciting forces on the floating and submerged bodies, induced by the incident wave.
Under regular wave (sinusoidal) excitation, the exciting force can be expressed as a harmonic function with the following equation: The solutions of Eqs. (2) and (3) can be expressed as, Eqs. (6) and (7) can be rewritten as: Define ( ) = − 2 + + = [ 11 12 21 22 ] where ( ) is the impedance matrix. The elements in ( ) can be written as the following equations: The solution of Eq. (8) can be expressed as, The solution of Eqs. (6) and (7) can be obtained as, The average absorption power of such a two-body wave energy converter is: By combining Eqs. (10), (12), (13) and (14), the closed-form solution of average absorption power can be expressed as:

If
= 0, by taking the derivative of Eq. (15) with respect to , the corresponding suboptimal damping is obtained as, The corresponding maximum absorption power is: Eq. (17) and (18) are called suboptimal condition of a two body wave energy converter in this paper. If > 0, by taking the partial derivative of Eq. (15) with respect to and , respectively, the optimal conditions of ( ) and ( ) are obtained as, By substituting ( ) into ( ) , one can get, By substituting ( ) of Eq. (19) and ( ) of Eq. (20) into Eq. (15), the maximum absorption power for such a two-body wave energy converter can be found as: Eqs. (19), (20) and (21) are called optimal condition of a two body wave energy converter in this paper. They are identical to the results obtained by Fanles [23] and Recci [28]. The physical meaning of ( ) in Eq. (19) is to match one of the damped natural frequency with the incident wave frequency, in a similar way as the resonance condition for single body system. Based on the closed form solutions obtained above, one can find that the power take-off parameters are determined by the system parameter for the maximum power absorption. Therefore, the design principles of a two-body wave energy converter are proposed based on the optimal and suboptimal conditions as follows; Suboptimal design: For a given two-body system and incident wave, the PTO stiffness is zero while the damping coefficient is defined as Eq. (17), which is determined by buoy and incident wave parameters. The maximum power is defined as Eq. (18) for suboptimal design.
Optimal design: For a given two-body system and incident wave, the spring stiffness and damping of the power take-off system are defined as Eq. (19) for ( ) ≥ 0. If ( ) < 0 for the given system, is set to zero and the damping is defined as Eq. (17). The maximum power is defined as Eq. (21) for ( ) ≥ 0 and Eq. (18) for = 0.

Simulation Results in Regular Waves
In this section, the closed-form solutions of suboptimal and optimal conditions obtained in Section 2 are numerically studied first. The influence of the submerged body on the absorption power of a two-body system is also investigated. Since this part is mainly focused on the design of the submerged body under regular wave excitation, the floating buoy is chosen as a cylindrical buoy with a diameter of 10m and draft of 3.5m. The submerged body is assumed to be axisymmetric with an undefined shape. The added mass and radiation damping of the floating buoy were calculated with matched eigenfuction method [31][32][33][34] and are shown in Figure 3. For an axisymmetric body, based on Haskind relation [24], the exciting force can be written as: where, is water density; is the standard gravity; is the wave frequency; is the radiation damping of the floating and submerged bodies; is the wave amplitude, which is chosen as 1m. In addition, the submerged body is assumed to be deployed far enough from the floating buoy and free surface (this assumption is widely used for different devices [27, 29, and 30]). As a result, the cross coupling added mass ( 12 , 21 ) and radiation damping ( 12 , 21 ) as well as radiation damping ( 22 ) on the submerged body can be neglected. With this simplification, the submerged body can be characterized with a total mass ( 2 + 22 ) and a linearized damping ( _2 ) in Eqs. (2) and (3). The suboptimal and optimal design discussed in Section 2 under the influence of the submerged body are numerically studied based on this simplified model.
Based on the experiment results and analysis in [27], it is believed that the damping on the floating buoy is dominated by radiation damping. Therefore, the viscous damping on the floating buoy is neglected here, i.e. _1 = 0. Moreover, since the floating buoy is the same for all the systems considered in this paper, neglecting the viscous damping on the floating buoy does not have any influence on the results. The viscous damping on the submerged body _2 is dimensionless with the following equation, where, 2 is the dimensionless viscous damping on the submerged body; _2 is the linearized viscous damping coefficient on the submerged body; 1 is the mass of the floating buoy; = √ / 1 is the natural frequency of the floating buoy in still water. One should be aware that 2 = 0.1 means the damping on the submerged body is in the same order of the largest radiation damping on the floating buoy in Figure 3(b), which is a large damping. Figure 4 shows the optimal and suboptimal designs described in Section 2 with a different total mass ( 2 + 22 ) of the submerged body for 2 = 0. The excitation wave periods are chosen as 8s, 10s and 12s. In Figure 4(a), the optimum PTO stiffness becomes negative (and the system will be unstable) when the total mass of the submerged body is larger than a critical mass . This critical mass can be obtained by setting ( ) = 0 in Eq. (19), which yields,  (25) where, 1 + 11 = 1 , 2 + 22 = 2 . The critical mass, = 2 / 1 , can be found by solving Eq. (25) for the floating buoy described in this Section. When 2 + 22 > , the PTO stiffness is set  to zero in Figure 4(a) as described in Section 2. The damping coefficients of the suboptimal and optimal designs are shown in Figure 4(b), the damping of the optimal design is the same with the suboptimal design when the mass is larger than the critical mass.   Figure 6 (a) shows the damped natural frequency of the optimal design and suboptimal design for different choices of 2 + 22 when the wave period is 10s and viscous damping 2 is 0.05. Since the power take-off stiffness is zero for the suboptimal design, there is only one vibration frequency ( ) for this design. This damped natural frequency ( ) decreases with the increase of 2 + 22 , and when 2 + 22 is around 6 in Figure 6(a), this damped natural frequency is very close to the wave excitation frequency. As a result, the maximum absorption power of the suboptimal design is achieved, as one can find in Figure 6(b). The corresponding optimal mass can be obtained by take the derivative of Eq. (18) with 2 + 22 and set it to zero,  = 0 (26) where, ( ) is expressed in Eq. (18); For a given wave condition and buoy information, one can solve the optimal mass from Eq. (26). One should notice that optimal mass > critical mass when 2 > 0 and = when 2 = 0. For the optimal design, there are two natural frequencies ( 1 and 2 ) when ( ) > 0, as one can find in Figure 6(a). With the increase of 2 + 22 , one of the natural frequencies ( 1 ) increases while the other one ( 2 ) matches the excitation frequency when 2 + 22 ≤ . When 2 + 22 > , the spring stiffness is zero and the damped natural frequency of this system reduces from two to one. This damped natural frequency becomes smaller than the excitation frequency with the increase of 2 + 22 , which is the same as the suboptimal design. These results show that the optimal condition in Eq. (19) is the resonance condition. In Figure 6(b), when 2 = 0, the maximum absorption power of the optimal design is a constant if 2 + 22 ≤ . This means that one can always achieve an optimal absorption power when 2 + 22 ≤ with the optimal design. When 2 > 0, it is found that the maximum power of both optimal and suboptimal designs is achieved when 2 + 22 = for the given wave condition. This means that 2 + 22 = is the optimal mass of the submerged body in the presence of 2 . Moreover, for the same 2 + 22 , the increase of 2 always results in the decrease of absorption power. Therefore, one should design 2 as small as possible when 2 + 22 is decided. The output power of the single body system with the same floating buoy is also plotted as a comparison. It is found that the absorption power of a two-body system is significantly increased in the presence of a large viscous damping on the submerged body.
The maximum absorption power of the optimal design under different wave conditions is plotted in Figure 7. The results obtained by Fanles [24] (Eq. (27)) when 2 = 0 is also plotted as a comparison.
where, is the maximum absorption power for an axisymmetric two-body system oscillating in heave without consideration of viscous damping; is the density of water; is the standard gravity; is the amplitude of incident wave; is the frequency of incident wave.
In Figure 7, when 2 = 0, the results of Eq. (21) match well with Fanles' results. Based on the previous analysis, for the given 10m floating buoy, this theoretical maximum can be achieved as long as the power take-off is designed as Eq. (19) and 2 + 22 ≤ . When 2 > 0, the design of the  submerged body is unique in order to achieve the maximum absorption power, which is 2 + 22 = for the given 10m floating buoy under certain wave frequency. The maximum power decreases dramatically at low frequency with the increase of 2 , but it remains almost the same at high frequency. The reason is that the increase of 2 dramatically decreases ( ) in Eq. (17) at low frequency, as one can find from Figure 5(b). The displacement of the floating buoy and submerged body decrease while the relative motion remains almost the same in Figure 7(b) when 2 increases from 0.05 to 0.1. In conclusion, based on the results in Figures 4-7, the overall performance of the optimal design is better than the suboptimal design when 2 + 22 ≤ . This is reasonable, since one of the natural frequencies of the system matches with the incident wave frequency by design ( ) with Eq. (19). The natural frequency of the suboptimal design matches with the excitation frequency only when 2 + 22 = . Therefore, when 2 + 22 > , the performance of the optimal design worsens with the increase of 2 + 22 since the natural frequency of the two-body system moves always from the excitation frequency. When 2 > 0 , the absorption power of both the optimal and suboptimal designs decreases dramatically for the same design of 2 + 22 . In this situation, the optimal design of the submerged body is unique, which is 2 + 22 = . In addition, the overall performance of the twobody system is better than the single body system even if one accounts for a large viscous damping on the submerged body.

Design of Two-body Wave Energy Converters in Irregular Waves
In this section, the influences of the submerged body on the power take-off design and absorption power are discussed under irregular wave excitation. In the simulation, the floating buoy is the same as described in Section 3 with a diameter of 10m and draft of 3.5m. 12 = 21 = 12 = 21 = 22 = 0. A Pierson-Moskowitz spectra shown in Eq. (28), which is an empirical equation that defines the distribution of energy with wave frequency in the fully developed sea, is used here to describe the irregular waves.
where is the significant wave height and is energy wave period. The average absorption power of a wave energy converter in the irregular waves can be calculated through the spectrum integration with the following equation. where, ̅̅̅̅̅ is the absorption power of a two-body system in irregular waves; ( ) is the absorption power in regular waves described in Eq. (15). When the floating buoy, submerged body and power takeoff system are determined, in Eq. (15) is a function of the excitation wave frequency. ( ) is the wave spectrum described in Eq. (28).
In Section 3, it was found that the maximum absorption power of the optimal design can be achieved if the spring stiffness is designed in such a way that one of the damped natural frequencies matches with the incident wave frequency. In irregular waves, however, it is hard to apply this design principle since the excitation is a spectrum instead of single frequency. Therefore, to implement suboptimal and optimal design defined in Section 2 in irregular waves, the excitation frequency appeared in Eqs. (17), (18), (19) and (21) is chosen as the frequency corresponding to the energy period in Eq. (28), i.e. = 1/ .
A constrained complex method described in [35] is used to find the maximum absorption power for the two-body system described in this section. The problem is formulated as following: for a given wave condition ( ) and submerged body ( 2 + 22 , 2 ), the absorption power of such a two-body wave energy converter can be written as a function of as the following equations, The initial points were chosen randomly from the parameter domain. The details about simplex method can be found in [35]. The maximum absorption power of a single body system with the same floating buoy subjected with passive damping control is also calculated numerically as a comparison. The absorption power is calculated with the following equation. ( , ) is the absorption power of this single body system in regular waves. ( ) is the wave spectrum described in Eq. (28). Figure 8 compares the absorption power of optimal and suboptimal design in irregular waves. The maximum absorption power obtained from constrained complex method was also plotted here as a comparison. The straight line represents the maximum absorption power of single body system with the same floating buoy subjected with passive damping control under the same wave condition. Several very useful conclusions can be obtained for a two-body wave energy converter.   (1) If one compares the maximum absorption power of a two-body system obtained from constrained complex method (black line) with that of a single body system (blue line with square), it is found that the maximum absorption power of a two-body system is more than twice of single-body system in the irregular waves considered here. Even if one considers 2 = 0.05 , the maximum absorption power of the two-body system is still almost twice as that of single body system. This indicates that the absorption power of a single body system can be increased significantly by designing an additional body underneath in the irregular waves.
(2) The total mass of the submerged body plays a very important role in the two-body system. One can find that if the mass ratio is small (( 2 + 22 )/ 1 < 1.5), the absorption power of the two-body system obtained from the complex method (black line) is smaller than that of the single body system (blue line with square). Therefore, the total mass 2 + 22 of the submerged body should be large enough in order to get a better performance in irregular waves. For the system considered here, ( 2 + 22 )/ 1 >4 can guarantee an increase of absorption power compared with a single body system. In addition, if 2 + 22 → ∞, the two-body system is equivalent to a single body system, and the absorption power of the two-body system approaches that of the single body system as shown in Figure  8. The added mass of the submerged body 22 is usually a shape dependent constant if it is deployed in the deep water. This allows people to relate 2 + 22 with the dimension of the submerged body in the design stage. Take the floating buoy described in this paper as an example, if one wants to design a submerged sphere with 2 + 22 = 8 1 . Based on the linear wave theory [24], the added mass of a submerged sphere is half of its mass, i.e. 22 = 0.5 2 . The diameter of the desired sphere is 14m. For other complicated shapes, one can resort to commercial software for the design of 2 + 22 .
(3) The linearized damping due to viscous effects 2 generally has a bad effect on the power absorption for a two-body system design. For all the absorption power plotted in Figure 8, increase in 2 always results in a decrease of absorption power with the same 2 + 22 . Based on [27], this damping term is mainly from drag force. Therefore, one should try to minimize the drag coefficient while designing the submerged body with desired mass. In this view, the submerged body is recommended to be designed with a streamline to decrease the drag damping when 2 + 22 is determined. Figure 9. The wave spectra and absorption power of optimal and suboptimal design with different wave frequencies. For the results plotted here, the irregular wave has a significant wave height of 2m and energy period of 10s. The floating buoy is a cylinder with a diameter of 10m and draft of 3.5m. 2 + 22 = 4 and 2 = 0.05.
(4) Different from the results in regular waves, the performance of the suboptimal design is better than the optimal design when 2 + 22 ≤ , as shown in Figure 8, especially when 2 < 2 + 22 < . Despite having the peak power of in Figure 9, the bandwidth of optimal design is narrower than that of the suboptimal design and complex optimization method. In addition, one should notice that the absorption power of the suboptimal design gets closer to results obtained by the constrained complex method when we increase 2 from 0.01 to 0.05, especially when 2 + 22 > 10. Therefore, if one designed the submerged body with a large added mass and drag damping, like a heave plate used in the Powerbuoy, the suboptimal design can be used and the absorption power is very close to the maximum power.

Conclusion
A frequency domain model of a two-body wave energy converter is considered in this paper by accounting for the viscous damping. The closed-form solution of the suboptimal and optimal absorption power and corresponding power take-off parameters are obtained. The suboptimal and optimal designs of the two-body wave energy converter are defined based on the obtained closed-form solutions. Despite the linearized drag damping and hydrodynamic simplification considered in the simulation, the results are believed to be useful to understand the dynamics and design of a two-body wave energy converter. A case study is conducted to investigate the influence of the submerged body on the performance of a two-body wave energy converter subjected to suboptimal and optimal designs in regular waves. If 2 = 0, the absorption power of the optimal design can always reach the theoretical maximum power when the total mass of the submerged body is smaller than the optimal mass. This is because the optimal design ensures one of the natural frequencies of the two-body system matches with the incident wave frequency. When the total mass of the submerged body is larger than the optimal mass (i.e. 2 + 22 > ), the absorption power decreases with the increase of the total mass of the submerged body. This is because the frequency match condition does not hold any more. The optimal mass is actually the optimal design for the submerged body when the damping due to viscous effects on the submerged body is large than zero (i.e. 2 > 0). By comparing the absorption power of a two-body system with a single body system with the same floating buoy, it is found that the absorption power can be increased significantly even with a large damping due to viscous effects on the submerged body.
The suboptimal and optimal designs of the two-body wave energy converter are also studied under irregular wave excitation by accommodating the energy frequency of the irregular wave in the closedform solution. It is found that the maximum absorption power of a two-body wave energy converter is more than twice that of a single body system with the same floating buoy for the system considered in this paper. Moreover, the total mass of the submerged buoy needs to be designed large enough in order to achieve a higher absorption power compared with single-body system. The shape dependent drag coefficient should be minimized while designing the submerged body with a desired total mass.