Dynamic Properties and Energy Conversion Efficiency of A Floating Multi-Body Wave Energy Converter

The present study proposed a floating multi-body wave energy converter composed of a floating central platform, multiple oscillating bodies and multiple actuating arms. The relative motions between the oscillating bodies and the floating central platform capture multi-point wave energy simultaneously. The converter was simplified as a forced vibration system with three degrees of freedom, namely two heave motions and one rotational motion. The expressions of the amplitude-frequency response and the wave energy capture width were deduced from the motion equations of the converter. Based on the built mathematical model, the effects of the PTO damping coefficient, the PTO elastic coefficient, the connection length between the oscillating body and central platform, and the total number of oscillating bodies on the performance of the wave energy converter were investigated. Numerical results indicate that the dynamical properties and the energy conversion efficiency are related not only to the incident wave circle frequency but also to the converter's physical parameters and interior PTO coefficients. By adjusting the connection length, higher wave energy absorption efficiencies can be obtained. More oscillating bodies installed result in more stable floating central platform and higher wave energy conversion efficiency.


Introduction
Ocean wave energy has been studied for more than 200 years because it is renewable, pollution-free, and abundantly and widely available. Extracting wave energy from the seas is a promising way to help achieve sustainable development around the world (de O Falcão, 2010). There are three classified categories of wave energy converters (WECs) according to their horizontal dimension versus ocean wave length. When the horizontal dimension of the device is much smaller than the wavelength of an incident wave, the device is known as point absorption WEC, or is called attenuator absorption or terminator absorption WEC otherwise (Amiri et al., 2016). Generally, the point absorption WECs can be subdivided into three types: single-body WEC, double-body WEC and multi-body WEC, according to the number of the floating oscillating bodies.
A single-body WEC has only one oscillating body float-ing on the sea surface to collect wave energy. The oscillating body is generally connected to a fixed reference (the seabed) through a Power Take-off (PTO) system. Many single-body WEC concepts, such as G-1T (Hirohisa, 1982), L-10 (Waters et al., 2007) and AWS (Prado and Polinder, 2011), have been proposed and studied. Budal and Falnes (1975) and Evans (1976) demonstrated respectively an important theoretical result that the maximum wave energy absorption efficiency is λ/(2π), where λ is the wave length, for a single-body WEC with the vertical axis of symmetry oscillating in heave. Zhang et al. (2016) obtained the optimization hydrodynamic parameters such as the added mass, radiation damping and wave excitation force of a single-body absorber by using the linear wave theory. Single-body WEC is low-cost because of its simple structure. But it cannot well adapt to the changing tidal range, so that it is not widely used (López et al., 2013).
A double-body WEC consists of two oscillating bodies which are cylindrical in general cases. One body floats on the water surface to absorb wave energy, while the other one submerges in seawater, working as a damping body to provide an almost motionless platform for the (whole) device. One PTO system connects the two bodies, transforming wave energy to mechanical energy or hydraulic energy. Obviously, as a damping vibration system with linear motion, the geometric factors, natural vibration characteristics and PTO coefficients are important parameters to be taken into consideration in designing such a kind of wave energy device. Williams et al. (2000) investigated the hydrodynamic coefficients and wave excitation forces for double cylindrical bodies of the same radius. Wu et al. (2014) simplified the two-body device as a forced vibration system with viscous damping in two degrees of freedom, and deduced the expressions of the conversion efficiency and response of the device from the equations of motion. Goggins and Finnegan (2014) studied the influence of three geometric factors, including the diameter, shape and draft of the floating component, to maximize the conversion efficiency. Zhang et al. (2017) experimentally analyzed a double-body WEC in a wave flume. Experimental results show that these factors such as the wave height, wave period, PTO damping, and mass ratio between the oscillator and carrier greatly affect the energy conversion process. Many double-body WECs such as American Powerbuoy (Weber et al., 2009) and Irish Wavebob (Prudell et al., 2010) have been well developed. The performance of the double-body WEC is not influenced by tidal range because of its floating structure and its ability of absorbing wave energy from different directions, making it a kind of WEC with good prospects. Nevertheless, the energy capturing ability of the double-body converter is restricted by the fact that, similar to the single-body converter, only one oscillating body can be used to collect wave energy.
In recent years, a type of multi-body WEC has attracted more and more attention, which primarily consists of three parts: a central platform, multiple oscillating bodies, and multiple actuating arms. A typical example is the Wave-Star WEC (Hansen et al., 2014, Hansen et al., 2013 developed by Aalborg University in Denmark, in which 20 oscillating bodies are around the central platform. The central platform needed to be fixed at the bottom of the sea, so that the converter must be installed in the coastal sea, causing a very high cost. Another example is the McCabe Wave Pump (McCormick et al., 1998) consisting of three floating rectangular steel pontoons joined by hinges, where the central pontoon, i.e. the central platform, was damped by a submerged horizontal plate, and the other two pontoons were used as the oscillating bodies to absorb wave energy. The whole converter floated on the sea surface, and the PTO systems on the two actuating arms converted the relative rotational motions into useful energy (McCormick et al., 1998). The M4M reported by Sheng and Lewis (2016) had conceptual similarities to McCabe Wave Pump, which captured wave energy by at most two oscillating bodies and the energy capturing ability was still restricted. An array-raft wave energy converter (AR WEC) integrating multiplepoint wave energy absorption and raft-type wave energy capturing technologies was proposed and experimentally investigated by the authors (Yang et al., 2016). The AR WEC has such advantages as moveable structure and the ability of capturing wave energy by more than two oscillating bodies, and a three-month real sea trial proved the feasibility and effectiveness of the system.
Inspired by the previous work on WECs mentioned above, we proposed and studied a novel multi-body WEC with floating central platform in this paper. Compared with other WECs, the proposed WEC is easier to be installed in pelagic environment due to the floating structure and greater ability to absorb wave energy because multiple oscillating bodies can be used simultaneously. The energy collecting motions of the single-body WECs and double-body WECs are usually considered to have one degree of freedom and two degrees of freedom respectively, while those of the floating multi-body WEC have three degrees of freedom, and relevant studies on the latter have not been reported so far in the literature. The three degrees of freedom motions are the damping heavy motion of each oscillating body, the damping heavy motion of the floating central platform and the rotational motion of each actuating arm. The equations of these motions were established first in this paper based on the linear wave theory and potential flow theory, and then the expressions of the damping vibration characteristics with three degrees of freedom were built. Finally, the dynamic properties and conversion efficiency of the floating multi-body converter were analyzed.

Conceptual design of the floating multi-body WEC
The floating multi-body WEC is composed of a floating central platform, multiple oscillating bodies, multiple actuating arms and multiple PTO systems. Floating on the water surface, the central platform and the oscillating bodies are cylindrical and connected by the actuating arms. On each actuating arm, a PTO system as an energy transition system is installed to convert the captured wave energy to hydraulic energy. The simplified structure of the system is shown in Fig. 1, where R 0 and R are the radiuses of the floating central platform and the oscillating bodies, respectively, and L 2 is the connection length between the central platform and each oscillating body. As shown in Figs. 1 and 2, the floating central platform provides a stable platform for the (whole) converter. The oscillating bodies are distributed symmetrically around the platform. The number of the oscillating bodies can be regulated to meet the requirement of wave energy absorption. Because of the great difference between the oscillating bodies and the central platform in the volume and weight, the oscillation amplitude of the central platform is much smaller than that of the oscillating buoys, leading to relative motions between them. The relative motions are then transformed into hydraulic energy by compressing or stretching the PTO systems installed on the actuating arms and eventually into electric energy for output.

Assumptions for calculation
To simplify the calculation, the following reasonable assumptions are made: (1) the floating central platform and all the oscillating bodies are cylindrical; (2) the PTO system is linear; (3) the size, damping coefficient, elastic coefficient and mass of all the oscillating bodies are the same.

Force analysis
The force diagrams in the vertical direction for the oscillating body and the floating central platform are shown in Fig. 3. The heave motion of the oscillating body results from the applied forces including the gravity M n g, the buoyancy force F fn , the wave excitation force F en , the wave radiation force F rn caused by the body itself, the wave radiation force caused by the central platform, the hydrostatic restoring force F sn , the hydrodynamic damping force F cn , and the hydrodynamic elastic force F kn caused by the seawater, where n is the sequence number of the oscillating bodies and n=1, 2, …, m (m stands for the total number of the oscillating bodies).
The heave motion of the floating central platform results from the gravity M 0 g, the buoyancy force F f0 , the wave excitation force F e0 , the wave radiation force F r0 caused by the platform itself, the wave radiation force caused by the n-th oscillating body, the hydrostatic restoring force F s0 , the hydrodynamic damping force F c0 , and the hydrodynamic elastic force F k0 caused by the seawater. The vertical damping force and the vertical elastic force, F ptocn and F ptokn , caused by the PTO systems, act on the n-th oscillating body and the central platform at the same time, as shown in Fig. 3. Under hydrostatic equilibrium conditions, the gravities M n g and M 0 g are equal to the buoyancy forces F fn and F f0 , respectively.

Heave motion equations of the oscillating bodies and
floating central platform According to Newton's second law, the heave motion equations of the n-th oscillating body and the floating central platform can be expressed as: where x n is the displacement of the n-th oscillating body, and x 0 is that of the central platform. The relative heave dis-  YANG Shao-hui et al. China Ocean Eng., 2018, Vol. 32, No. 3, P. 347-357 349 placement between the two is The displacement equations in the complex form for the oscillating bodies and central platform are expressed as: where A n (n=1, 2, ..., m) and A 0 are the complex amplitudes of the oscillating bodies and central platform, t is the time, ω is the circular frequency of sea wave, and Re[ ] is the real part of the complex expression. Then the hydrodynamic damping force and elastic force can be expressed as: , n = 1, 2, . . . , m; The hydrostatic restoring force can be expressed as: where g is the acceleration of gravity, and ρ is the density of seawater.
The solution of the vertical wave excitation forces F en (n=1, 2, …, m) and F e0 needs to use the potential flow theory and linear plane wave theory. The wave excitation forces produced by the velocity potentials are given as: where is the incident velocity potential, is the diffraction velocity potential, and are the average wetted surface of the n-th oscillating body and the floating central platform, respectively, and N z is the unit vector in the vertical direction.
The vertical wave radiation forces acting on the oscillating bodies and the central platform are caused by the velocity potential due to the heave motion of the buoys and central platform, being defined as: , n = 1, 2, . . . , m; (14) , n = 1, 2, . . . , m; where M nn , M n0 , M 00 and M 0n are the added mass, and c nn , c n0 , c 00 and c 0n are the added damping coefficients. The detailed calculation process of these coefficients was given by Ricci (2012).

Rotational motion equations of the actuating arms
When the oscillating bodies move up and down, the relative heave motion between the bodies and central platform makes the actuating arms rotate, and then compresses or stretches the PTO systems on the actuating arms, thus realizing multi-point absorption of wave energy at the same time.
As shown in Fig. 2, the angle β between L 1 and L 2 is a right angle when the device is in still water, where L 1 is the distance between the hinge joints B and C, and L 2 is the distance between the hinge joints A and C. L 2 is also the connection length between each oscillating body and the central platform. The length L 0 of the PTO system is calculated by (18) Driven by the incident waves, the n-th actuating arm is in the shape as shown in Fig. 4 because of the difference between x 0 (the displacement of the floating central platform) and x n (the displacement of the n-th oscillating body). According to the law of Cosines, the new length of the PTO system can be calculated by (19) When L 2 is rather large and then the change of β is significantly small, β can be approximately expressed as: Substituting Eqs, (3), (18) and (20) into Eq. (19), we can obtain and then the increment ΔL of the PTO length can be given by Hence, when the damping coefficient and elastic coefficient of the PTO are linear, the vertical damping force and vertical elastic force can be calculated by 3.5 Conversion efficiency of the converterĒ For regular waves, the average wave energy density, , per unit surface area over a wave period T is given bȳ

E
where H is the wave height. The average rate of the energy flux across a fixed control surface, i.e., the wave power per unit wave-front length, is the product of the energy density and wave group velocity V g : dĒ dt = V gĒ (26) The wave group velocity can be calculated by where i is the wave number and h is the seawater depth. In deep water where hλ (λ is the wave length), Eq. (26) can be simplified to .
(28) So the average wave power P wave in deep water is (29) The average power extracted by the PTO system is calculated bȳ where v pton denotes the time-dependent velocity, and * denotes the conjugate. The wave energy capture width, η, of each oscillating body can be defined as: 4 Results and discussion

Oscillation properties analysis
The modeling and simulation were carried out using the mathematical model developed above. Under specified geometric dimensions, the main factors influencing the dynamic properties of the floating multi-body WEC are the PTO damping coefficient c pto , the PTO elastic coefficient k pto , the connection length L 2 , and the total number of oscillating bodies m. The reference geometric parameters used in the simulation are shown in Table 1.
Figs. 5a-5c show the RAO curves, where RAO is the ratio of the maximum heave in response to the amplitude of the incident waves, of the heave motions for different PTO damping coefficients c pto when the PTO elastic coefficient k pto =0 kN/m, the connection length L 2 =20 m and the number of oscillating bodies m=4. Fig. 5a reveals that, when the wave frequency ω is very low, i.e. the wave period is very long, the RAOs equal to 1. When ω=0.76 rad/s, the RAOs begin to scatter, and increase gradually. After reaching the peak value, the RAOs begin to decrease as the wave frequency increasing. When the PTO damping coefficient is smaller, the peak value is greater and the wave frequency corresponding to the peak value is higher. After ω>1.72 rad/s, the RAOs come to be zero. As shown in Fig. 5b, the variation tendency of the RAO curves for the central platform is similar to that of the oscillating body, but the RAOs are obviously lower at the same wave frequencies and the resonant frequencies corresponding to the RAO peak values are also obviously lower than those of the oscillating body.
It is revealed in Fig. 5c that there is almost no relative motion between the oscillating bodies and the floating central platform at lower and higher wave frequencies, and when the wave frequency is larger than 1 Hz or smaller than 1.4 Hz, the relative motion is comparatively remarkable, with the largest relative motion vibration amplitude appear- YANG Shao-hui et al. China Ocean Eng., 2018, Vol. 32, No. 3, P. 347-357 ing at ω=1.19 rad/s when c pto =200 kN·s/m. As shown in Fig. 5d, the maximal wave energy capture width appears when c pto = 900 kN·s/m, which indicates that neither smaller nor larger damping coefficients can generate satisfactory results. Actually there always exists an optimal damping coefficient which leads to the highest energy capture efficiency. Figs. 6a-6c show the RAO curves of the oscillating body, the central platform and the relative motion between them for different elastic coefficients k pto when c pto =900 kN·s/m, m=4 and L 2 =20 m. It can be found that the resonance peak value continually shifts to the right with the increasing elastic coefficient. For the oscillating body, the elastic coefficient considerably changes its dynamic properties. As shown in Fig. 6a, the RAOs increase at first, and then decrease as the wave frequency increases. When the elastic coefficient is positive, there exists a secondary resonance peak value in the heave motion of the central platform and the relative motion, as shown in Figs. 6b and 6c. The existence of the secondary induced resonance peak value in the central platform dynamics is interesting, but is not associated with high energy absorption because of the very small wave frequencies involved.
As to the wave capture width, the effect of the secondary peak value is not fairly visible, as shown in Fig. 6d. When the elastic coefficient increases, the wave capture width decreases at lower wave frequencies, but increases when the wave frequency is relatively high. That is to say, when the wave period is short, the converter should have higher elastic coefficient. The maximum capture width ap-pears at ω=0.99 rad/s when the elastic coefficient is -100 kN/m. It can be found from the formula deduction process in Section 3 that the oscillation properties of the converter is influenced by the connection length L 2 between each oscillating body and the central platform when L 1 is fixed. Figs. 7a-7c show the heave RAOs of the oscillating body, the central platform and the relative motion between them for different connection lengths.
In Figs. 7a-7c, the heave response resonance peak values of the oscillating body, the central platform and the relative motion continually shift to the left when L 2 increases. This is very similar to the effect of the damping coefficient. A longer connection length leads to a larger heave motion amplitude in lower wave frequencies, though the maximal response at the resonant frequency is reduced. On the contrary, a longer connection length leads to smaller heave motion amplitude in higher wave frequencies. Fig. 7c reveals that, compared with the relative motion RAO under shorter connection length, the RAO under longer connection length is higher at a lower wave frequency, and lower at a higher wave frequency. With the increase of ω, the RAO of the relative motion under L 2 =60 m is the earliest to be higher than zero (at ω=0.39 rad/s). For all the considered connection lengths, the RAOs come to be zero again at ω=1.71 rad/s. It is thus clear that the response wave frequency band is the widest when L 2 =60 m.
From the energy capture width curves shown in Fig. 7d, it can be seen that the energy capture width is not always higher when the connection length gets longer, with the change rule not the same as that of the relative motion responses. Although a longer connection length still causes a better capture capacity in lower wave circle frequencies, i.e. longer wave periods, the maximum capture width appears at L 2 =40 m when the wave frequency is 0.84 Hz. This indicates that the energy conversion efficiency of the floating multi-body WEC can be enhanced by adjusting the connection length of the actuating arm according to the incident wave frequency.
The effects of the total number of oscillating bodies on the dynamic properties are shown in Fig. 8. Without considering the mutual influences among the oscillating bodies, Fig. 6. RAO and energy capture width curves for different k pto when c pto =900 kN·s/m, m=4 and L 2 =20 m. YANG Shao-hui et al. China Ocean Eng., 2018, Vol. 32, No. 3, P. 347-357 353 the heave motion of each oscillating body keeps unchanged when the number of oscillating bodies is different, as shown in Fig. 8a. According to the mathematical model, when the number of the oscillating bodies installed on the central platform changes, the hydrodynamic damping force and hydrodynamic elastic force acting on the central platform change correspondingly, so do the heave RAOs of the central platform and the relative motion. Fig. 8b shows that the heave motion of the floating central platform drops off slightly when the number of oscillating bodies increases, indicating that the central platform is more stable when there are more oscillating bodies. Fig. 8c shows the relative motion response curve against different number of oscillating bodies, where the resonance frequency is almost invariable when the number of oscillating bodies varies. Although the change of the heave RAOs of relative motion is not significant, more oscillating bodies still lead to higher heave RAOs of relative motion. Similar but more notable change of the energy capture width can be seen in Fig. 8d. It reveals that more beneficial wave energy capture can be obtained when more oscillating bodies are installed on the floating central platform.

Conversion efficiency analysis
This section focuses on the maximum wave energy capture width of the converter. Since we only care about the maximum conversion efficiencies at the resonant frequencies under different operation conditions, the data of the maximum wave energy capture width were picked out from Figs. 5-8, with the results illustrated in Fig. 9.
As shown in Fig. 9a, the maximum energy conversion efficiency is largely affected by the PTO damping coefficient, with the peak value appearing at c pto =900 kN·s/m; the efficiency decreases gradually with increasing damping coefficient when c pto > 900 kN·s/m. Fig. 9b shows the maximum energy conversion efficiency for different elastic coefficients. It can be seen that the curve decreases monotonically with the increase of the elastic coefficient and the peak value is 208.53% when k pto =-100 kN·s/m within the parameters ranges being studied.
It is revealed in Fig. 9c that the maximum energy conversion efficiency first increases and then decreases as the connection length between the oscillating bodies and the central platform increases, indicating that we can always get high absorption efficiency if the automatic adjustment of the connection length can be realized as needed. The best energy conversion efficiency of 271.65% can be obtained when the connection length L 2 =40 m. Under the same conditions, the maximum conversion efficiency is just 171.6% when L 2 =10 m. Thus, a proper connection length is highly beneficial to wave energy absorption.
The total number of oscillating bodies can affect the maximum conversion efficiency of each oscillating body too, although the effect is smaller than that of the other factors. Fig. 9d shows that more oscillating bodies lead to higher conversion efficiency. In the case that the total number is eight, the maximum conversion efficiency can reach up to 206.07%.

Comparison with the AR WEC
To evaluate the performance of the floating multi-body WEC, a comparison with other type WECs should be performed. The problem is that the existing study on similar WECs is very limited. Except the AR WEC (Yang et al., 2016) mentioned in Section 1, the information of other type WECs is not sufficient for the comparison. To evaluate the proposed WEC to a certain extent, a comparison with the AR WEC is done below.
An AR WEC with ten oscillating buoys is shown in Fig.  10. The buoys are divided into two linear arrays and arranged evenly on both sides of the floating wedge-shaped central platform. Every five buoys form one array. Clearly, the AR WEC is also a floating type WEC with multiple buoys. The main differences between the two WECs are: in the proposed WEC, the floating central platform is circular and the oscillation buoys are distributed symmetrically around it, while in the AR WEC, the floating platform is wedge-shaped, and the oscillation buoys are divided into two columns and arranged on both sides of it. The authors studied the performance of the AR WEC using a single degree of freedom mathematical model, and carried out a real sea trial for more than 3 months in 2014 (Yang et al., 2016).
It is revealed in Fig. 11a that the floating multi-body WEC has higher energy conversion efficiency than AR WEC at relatively low wave frequencies when c pto =900 kN·s/m, k pto =100 kN/m, L 2 =10 m and m=5. The difference in the shape of the central platform and the distribution of the oscillating bodies leads to distinctly different resonance frequency of the two WECs. The floating multi-body WEC is more adaptive to the open sea where the wave periods are longer.
For the floating multi-body WEC, more oscillating bodies lead to higher conversion efficiency, as shown in Fig.  11b. The phenomenon is the same as that for the results shown in Fig. 9d, although some calculation parameters are  YANG Shao-hui et al. China Ocean Eng., 2018, Vol. 32, No. 3, P. 347-357 355 changed. It is worth noting that, contrary to the proposed WEC, the energy conversion efficiency of the AR WEC gets lower when the number of the oscillating bodies increases. As shown in Fig. 11, the variation tendency of the energy conversion efficiency in real sea trials of the AR WEC is the same as the simulation result, but the values are obviously lower, because the incident waves in the real sea are random and irregular, and some frictional losses were ignored in the simulation model.

Conclusions
A novel floating multi-body WEC was proposed, modeled and analyzed theoretically in this paper. The equations for two heave motions and one rotational motion of the oscillating body, the floating central platform and the actuating arm were deduced respectively, based on the forced oscillation theory for three degrees of freedom systems, and the vibration properties and energy conversion efficiency of the converter were investigated.
The numerical calculation results show that the vibration amplitude-frequency characteristics and the energy conversion efficiency of the converter are related not only to the incident wave circle frequency but also to its own physical dimensional parameters and interior PTO coefficients, mainly including the connection length between the bodies and the central platform, the total number of the oscillating bodies, the PTO damping coefficient, and the PTO elastic coefficient.
When the elastic coefficient is positive, there exists a secondary crest in the heave motion of the central platform and the relative motion. The effect of the connection length between the oscillating body and floating central platform on the dynamic performance of the converter is similar to that of the PTO damping coefficient. Although longer connection length leads to a wider wave response frequency band, the maximum capture width appears at a medium connection length. The converter can always get high absorption efficiency if the automatic adjustment of the connection length can be realized as needed. Although the total number of the oscillating bodies has only marginal influ-ence on the relative motion, more oscillating bodies lead to obviously improved wave energy conversion efficiency.
The characteristics studied above provide theoretical basis for optimizing the energy absorption capacity of the floating multi-point WEC. In the future, we will study the maximum power tracking control strategy based on varying connection lengths between the oscillating bodies and the central platform, develop pony-size prototype for the real sea trail test, and conduct more numerical and experimental studies for the optimization of this novel WEC.