Theoretical and Experimental Study of A Coaxial Double-Buoy Wave Energy Converter

The double-body heave wave energy converter (WEC) is one of the most conducive devices to absorb the wave energy from relative motion while the law of which is not well understood. This paper makes an in-depth study on this wave energy converter, by means of the combination of theoretical analysis and physical model experiment. The hydrodynamic characteristics and energy capture of the double-buoy under constant and linear Power Take-Off (PTO) damping are investigated. Influences of absolute mass and mass ratio are discussed in the theoretical model. Relative displacement amplitude and average power output are tested in the experiment to analyze the effect of the wave period and outer buoy’s mass, while the capture width ratio (CWR) is also calculated. Results show that the wave period and mass of the buoys have a significant effect on the converter. Different forms of PTO damping have no influence on the optimal wave period and mass ratio of this device. It is recommended to select the double-buoy converter with a mass ratio of 0.80 and to place it in an area with the frequent wave period close to the natural period of the outer buoy to achieve the optimal energy capture.


Introduction
In order to solve the problems of declining fossil fuels and increasing environmental pollution, wave energy resources attract more attention as a kind of clean energy (López-Ruiz et al., 2018). Various forms of WECs have been developed because of the focus on the wave energy (Ji et al., 2020). Among the WECs, the oscillating buoy device has become one of the key research objects due to the advantages of simple manufacture and wide application range (Zhang et al., 2014). PowerBuoy, developed by Ocean Power Technologies (OPT) in the United States, is a relatively mature two-body heaving WEC (van Rij et al., 2017), which utilizes relative motion between a float and a spar with a heave plate to drive the push rod to convert wave energy. Other different functions two-body WECs are invented, which are used to power mobile converter (Shi et al., 2019). Wavebob, a two-body heaving WEC developed by Ireland, includes a streamlined submerged buoy to achieve higher power absorption (Windt et al., 2018). In addition, Liu et al. (2018) evaluated and studied the effect of the shape of the buoy on the average power output. Beatty et al. (2019Beatty et al. ( , 2015 researched the influence of different shapes of the submerged floats on Response Amplitude Operator (RAO).
The mass ratio of the double-buoy is an important aspect which affects the energy absorption (Liang and Zuo, 2017). The radius and draught also have an influence on hydrodynamic characteristics (Amiri et al., 2016). Wu et al. (2006) studied the effects of the radius ratio on the hydrodynamic coefficient and excitation force, obtaining the optimal damping coefficient of the converter. Berenjkoob et al. (2019) analyzed the relationship between draught of the device and energy capture efficiency. PTO is widely recognized as the most critical component to WEC (Barstow et al., 2008). Electromechanical PTO utilizes the rack and pinion system combined generator to realize energy conver-sion. The electrical resistance of the generator has an important influence on the PTO damping. Castro and Chiang (2020) applied electromechanical PTO to the device to study the motion response and energy capture. Hernández et al. (2017) proposed that the rotational inertia and inertia disc radius should be considered while some mechanical PTO systems would assemble a variable inertia flywheel. A widely adopted concept is a direct drive PTO implemented with a linear generator (Tan et al., 2020). Linear generators, which are directly linked to the wave, are significantly affected by several key design parameters, such as the width of the coil and the radius of the central shaft. Elwood et al. (2010) developed a two-body WEC with a specially designed linear generator PTO and tested it in the ocean. With its high load capacity and maturity, the hydraulic PTO, which is typically composed of hydraulic pumps, high-pressure pipes, pressure accumulators, hydraulic motors, and electrical generators, has been a popular type of PTO (Lin et al., 2015;Xu et al., 2019). The PTO damping is determined by the coefficient of hydraulic cylinder and pressure drop in the hydraulic PTO system, while the PTO stiffness is defined by the piston displacement and accumulator (Negahdari et al., 2018). Both of the magnitudes of PTO damping and PTO stiffness have effect on power output (Ma et al., 2020).
The optimal PTO damping is found by different methods, subsequently. Jin et al. (2019) controlled the PTO damping and stiffness by using a linear frequency domain model. The results show that under optimal shape and PTO designs, two-buoy heaving WEC has better power characteristics than single-buoy one. Nevertheless, the energy efficiency is not acceptable under the passive motion. Martin (2017) studied the active and passive control by the closedform solutions. With the development of the control strategy, passive loading, equivalent saturation control and maximum stroke control are applied in the PTO control strategy, which is beneficial to higher average power output (Kim et al., 2016). Abdelkhalik andZou (2019), andAnderlini et al. (2018) investigated the Multi resonant control and Q-learning algorithm to improve the energy capture.
In this paper, the hydrodynamic characteristic of the coaxial double-buoy converter is studied, as well as the PTO and CWR under different wave periods and buoy's mass. The PTO force decides the dynamic performance primarily. The linear PTO damping force has been applied in the previous studies by various methods. The linear PTO damping is common to be applied and researched, but constant PTO force is also another major damping form, which is not easy to be applied in the experiment. For the comprehensiveness of this research, the linear PTO damping is applied into the theoretical model to discuss the hydrodynamic performance of the model. Besides, the physical model test is conducted to explore the power output and CWR of the device, and constant PTO damping force is applied to the device through a designed hydraulic system. The conclusion can provide useful reference for the engineering application of this wave energy converter.

Theoretical model establishment
The motion state of the coaxial double-buoy in the wave field is shown in Fig. 1. Under the action of both incident wave and PTO damping force, the response of the outer and inner buoys is investigated to obtain a two-dimensional solution of the relative motion.
The outer buoy is a cylinder with the annulus warplane area and draft , whose vertical centroid displacement to the equilibrium position is . The inner buoy is a cylinder with the basal area and draft , whose vertical centroid displacement to the equilibrium position is . It can be seen that the buoy is subjected to the combined effects of wave exciting force, PTO damping force and hydrostatic restoring force.
The method of Morison equation is introduced to calculate the vertical wave force. As the draft area of the device is smaller than that of the cross section, the vertical drag force caused by the friction is small enough to be ignored. Therefore, the vertical wave force is proportional to the product of the mass of water discharged by the buoys and the acceleration of water particles which vertically deliver the normal momentum to the buoys. ] . (1) where F 1 and F 2 are the vertical wave forces of the outer and the inner buoys, respectively; η(t) is the wavefront equation.
The acceleration is solved using linear deep-water wave theory. sin ωt ≈ Aω 2 sin ωt; where A is the wave amplitude, ω is the wave frequency, is the density of the seawater, is the inertia force coefficient, and are the velocity of water particles at the bottom of the outer and inner buoys, respectively.
Assuming that the PTO damping forces on the outer (F PTO,1 ) and inner (F PTO,2 ) buoys are linear resistances, which can be expressed as: C where is the damping coefficient.
Hydrostatic restoring forces (F HS,1 and F HS,2 ) are caused by the change of hydrostatic pressure and wet surface when the buoys move, and can be expressed as: ] . (4) So, the kinematic equations of outer and inner buoys are where m 1 and and m 2 are the masses of the outer and the inner buoys, respectively.
Introduce a set of motion , mass , and restoring coefficient K, which can be expressed as: The kinematic equations can be re-written as: with the assumption of The solutions are expressed as: The motions of outer and inner buoys are respectively derived The relative motion is The relative velocity is The total average power output is As the incident wave power of unit width is the CWR is where R is the outer radius of outer buoy.

Similarity principle
The model test is designed according to the gravity similarity principle. The Froude number of the converter model is equal to that of the prototype, and the relative scales are determined after the geometric one is decided. By considering the wave-making ability of the tank, water depth and the size of the coaxial double-buoy converter, the model length scale n is selected to be 25. Other scales are shown in Table 1. In the model test, wave height ranges from 0.04 m to 0.20 m, and the wave period varies from 1.00 s to 2.00 s.

Facility
The model test is conducted in a wave tank with the length of 60 m, width of 36 m, and depth of 1.50 m at Shandong Provincial Key Laboratory of Ocean Engineering. A piston-type wave maker is installed at the front end of the tank, and a wave-absorbing beach located at the rear and both sides to minimize the wave reflection, and the converter is placed 35 m away from the wavemaker. Fig. 2 shows a sketch of the experimental layout. respectively. The coaxial double-buoy only has the heave motion, which is a non-mooring system. As shown in Fig. 4, the outer buoy moves along three guide rods (①②③), while the inner one moves along the other two rods (④⑤). The water depth is fixed at 1.10 m. Wave gauges are used to measure the fluctuation of the water surface, and an NDI Optotrack Certus is used to record the motions of the buoys.  As shown in Fig. 5, the constant PTO damping force is applied to the double-buoy through a hydraulic system. The diameter of the hydraulic cylinder is designed to be 20 mm, while that of the hydraulic rod is 18 mm. The device is pressurized with the hydraulic oil circuit. In the hydraulic system, relief valve (5) is used to keep steady pressure. Check valve (6) is used to exhaust gas from the hydraulic circuit system. Proportional valve (10) is intended to control the hydraulic pressure of the system.

Model validation
In order to verify the theoretical model derivation above, validations with experimental data are carried out (the experiment is described in detail in Section 3). The wave height is set to 0.10 m with the periods of 1.30, 1.55, and 1.80 s. The masses of outer and inner buoys are 58 and 71 kg, respectively. Fig. 6 compares the theoretical and experimental relative motion results without PTO damping LI De-min et al. China Ocean Eng., 2021, Vol. 35, No. 3, P. 454-464 force, which gives a good agreement. The double-buoy makes a periodic sinusoidal movement, where the amplitude in theoretical model is a little larger than that in experiment. The reason is that the viscosity is neglected in Eq. (5), and the mechanical friction in the experiment leads to some energy loss. In summary, the fitness of the theoretical model is verified. Fig. 7 compares the results of theoretical and experimental relative motion amplitudes without PTO damping force. Regular wave cases are shown in Table 2. The value in the theoretical model is , while that in the experimental test is . The value of the viscosity and mechanical friction is expressed by the difference between theoretical data and experimental data, and the proportion of it is approximately 20% in this device through Eq. (16).

Non-PTO test
A non-PTO test is conducted to explore the influences of different parameters, such as wave period and buoy's mass, on the motion response.

H T
The waves are regular with the height =0.125 m, and the period of =1.30, 1.55, 1.80, 2.05 and 2.30 s. The masses of the outer and inner buoys are 58 and 71 kg, respectively.
In order to present the effect of the wave period, cases without PTO damping force are calculated, as shown in Fig. 8, resulting in the heave motion with the change of the wave period. By comparison, it is drawn that the relative motion amplitude first rises and then declines with the increase of the wave period. Since the period of 1.80 s is the natural period of the inner buoy, the motion amplitude reaches the peak value at the resonant period. It shows that the amplitude of the inner buoy has a similar regularity with that of the relative motion. Compared with the inner buoy, the amplitude of the outer buoy does not change significantly with the change of the period.
(2) mass of the outer buoy The waves are regular with the height =0.10 m and the period =1.30 s. The outer buoy's mass varies in the sequence of =58, 66, 73, 81, 89, and 92 kg, while the inner buoy's mass is fixed to =71 kg. Fig. 9 describes the effects of the outer buoy's mass. It can be seen that the amplitudes of the relative motion and outer buoy decrease with the increasing mass of the outer buoy. The relative motion amplitude reaches the maximum value as the mass ratio of outer-to-inner buoy is 0.80. Compared with the outer buoy, the amplitude of the inner buoy does not change significantly. It is obvious that the amplitude of the outer buoy is much larger than that of the inner buoy. The reason is that the response of the water particles caused by wave motion decreases with the increase of the water depth, and the draft difference between the outer and inner buoys is large. Compared with the inner buoy which is immersed mostly in the water, the outer buoy close to the  water surface is affected significantly by wave force, leading to the larger motion amplitude of the outer buoy.

Power capture performance with different PTO models
The average power output and CWR are used as indicators to investigate the relations between wave periods, buoy masses and energy capture, which reveals the law of the hydrodynamic performance.
(1) wave period In order to explore the effect of the wave period under linear PTO damping, data from theoretical model results are given in Figs. 10 and 11. The heights of regular incident wave are 0.10, 0.12, 0.14, 0.16, and 0.18 m. The linear PTO damping force is the coefficient of 350 N•s/m, and the masses of the outer and inner buoys are fixed at 58 and 71 kg, respectively. In respect of linear PTO damping force condition, the captured power is proportional to the wave height, while the CWR at different wave heights are coincident. The wave period corresponding to the optimal power output is coincident with the increasing wave height, similarly. The results show that the device has a good motion response under wave period from 1.30 s to 1.50 s, which is between the resonance periods of the two buoys. The en-ergy capture depends on the coupling property of the two buoys, and reaches the optimal power output within the resonance range which especially approaches the outer buoy's natural period of 1.30 s. The power capture becomes worse when the wave period exceeds the optimal range, which is markedly affected by the wave period. As a result, a larger separation between two natural periods is beneficial, which leads to a wider energy capture band with higher value. T Based on the results of the theoretical model, the effects of the period on the amplitudes with different constant PTO damping forces are further explored in the physical experiment. The waves are regular with the height of 0.10 m, and the periods of = 1.30, 1.55 and 1.80 s. As shown in Figs. 12 and 13, the amplitudes of the outer and inner buoys increase with the increasing wave period as the PTO damping force is fixed. The period of 1.80 s is the natural period of the inner buoy, which resonates with the wave motion, leading to the maximum motion response. As shown in Fig. 14, the relative motion amplitude rises with the increasing wave period at smaller PTO damping force while declines at larger PTO damping force. As shown in Fig. 12, the amplitude of the outer buoy increases with the rise of the PTO damping force but reveals minor differences. Compared with the outer buoy, the amplitude of the inner buoy has the same regularity with that of the outer buoy except in special cases with long wave period, as shown in Fig. 13. The reason for the exception is that with the increasing PTO damping force, the movement state of the converter changes,    LI De-min et al. China Ocean Eng., 2021, Vol. 35, No. 3, P. 454-464 causing the direction change of the PTO damping force to keep it consistent with those of the outer and inner buoys, thereby promoting the vertical movement.
Figs. 15 and 16 show the average power output and CWRs of the converter with constant PTO damping force. Fig. 15 illustrates that with the increasing wave period, the power output of the converter decreases, which reaches the peak value at the wave period of 1.30 s. The reason is that the period of 1.30 s is closer to the natural period of the outer buoy, which forms resonance with the wave motion. Moreover, the draft of the outer buoy is small and the velocity of the water particles at its bottom is large, which causes the outer buoy responding violently. On the other hand, the condition of the inner buoy is opposite to that of the outer buoy, leading to smaller amplitude of the inner buoy. Hence, the relative motion amplitude and power output increase, respectively. With the increase of the PTO damping force, the power output under different periods shows a trend of first increasing and then decreasing. Fig. 16 presents that the CWR under short periods is much larger than that under long periods. The reason is that the average power output rises with the increase of the wave period but exhibits minor differences with a small PTO damping force, leading to the lowest CWR under a long wave period. On the contrary, the CWR decreases significantly with the increase of the wave period, causing an obvious disparity in the CWR under different wave periods, which leads to the maximum CWR under a short period. This phenomenon implies that the converter proposed in this paper is suitable for China's offshore areas under short wave period , achieving the goal of highefficiency energy capture. It also shows that the period has a significant effect on optimal PTO damping force, which presents that the PTO damping force should be adjusted substantially with the change of the wave periods. Coincident with linear PTO damping force, the optimal power capture under constant PTO damping is fixed at the same period which is around the outer buoy's natural period.
(2) mass of the outer buoy The outer buoy has less draft, and is regarded more sensitive to the wave excitation than the inner one. Here, considering the outer buoy's mass varies in the sequence of =43, 51, 58, and 73 kg, while the inner buoy mass is fixed to =71 kg. Calculation results could be obtained under the conditions of the regular incident wave with the height H=0.175 m, and the linear PTO damping force with the coefficient of 350 N•s/m. As shown in Fig. 17, the average power output varies with the wave frequency . It shows that with the rise of wave frequency, the power output of the double-buoy increases first and then decreases. The same trend is found between the wave frequency and CWR, as shown in Fig. 18. The maximum power output also increases first and then decreases with the increasing   mass of the outer buoy. The wave frequency corresponding to the optimal power output gradually moves to the low frequency area as the mass of the outer buoy increases. By comparation, it is drawn that the CWR is almost not affected by the mass of the outer buoy at low wave frequency area. With the mass of the outer buoy increasing, the frequency band where the optimal CWR is located gradually becomes narrower, which leads to that the larger mass of outer buoy is not suggested to be an optimal power absorption.
The mass of outer buoy has a significant influence on the energy capture of the device through the results of theoretical analysis. The influence of the outer buoy's mass under constant PTO damping force is further explored in the experiment. Figs. 19, 20 and 21 indicate the effects of the outer buoy's mass on the amplitudes with different constant PTO damping force. The waves are regular with the height of 0.175 m and the period of 1.30 s, the mass of the outer buoy varies in the sequence of 58, 81 and 92 kg, while the inner buoy mass is fixed to 71 kg. It can be drawn that the amplitude of the relative motion reaches the maximum value when the mass ratio of the outer-to-inner buoys is 0.80. As shown in Figs. 19 and 20, the amplitude of the outer and inner buoys increases with the increasing PTO damping force but the change trend is not obvious, whereas the opposite phenomenon is shown with the relative motion. The amplitudes of the outer and inner buoys decline with the increasing mass of the outer buoy. The reason is that the draft of the outer buoy rises with the increasing mass, thereby leading to increasing wave resistance. On the other hand, the motion of the converter changes as the mass of the outer buoy increases, making the acting direction of PTO damping force be opposite to those of the outer and inner buoys, which causes the decreasing motion amplitude.
The average power output and the CWRs of the converter under different masses of the outer buoy with different constant PTO damping force are shown in Figs. 22 and 23. It shows that the power output reaches the peak value when the mass ratio of the converter is 0.80. The same trend is found in the theoretical model which is expressed with the assumption of the linear PTO damping force. The reason is that the velocity of the water particles at the bottom is larger with the smaller mass of the outer buoy. Under the action of waves, the amplitude of the outer buoy is large while that of the inner buoy is small, causing the increasing relative motion amplitude and wave energy capture. The larger the outer buoy's mass is, the larger the rising resistance is. Besides, the inertia increases with the rising mass of the out buoy, which keeps the small amplitude of the outer buoy, declining the relative motion amplitude and the energy capture. With the increase of the PTO damping force, the power    output and CWR under different masses of the outer buoy show a trend of first increasing and then decreasing. In Fig. 23, it is shown that the mass of the outer buoy has significant effects on the CWR and optimal damping of the converter. The phenomenon illustrates that it is recommended to select a coaxial double-buoy converter with the mass ratio of 0.80, and the load of the converter should be adjusted within a wide range with the changing mass of the outer buoy.
(3) mass of the inner buoy m 1 m 2 Here, considering the outer buoy with the mass = 58 kg, which has the highest PTO above, and the inner buoy with the mass range of =58, 71, and 81 kg. The relations between the wave frequency and average power output are illustrated in Fig. 24. It is seen that the power output rises first and then declines with the increase of wave frequency. Compared with the trend of the changing outer buoy's mass, the wave frequency corresponding to the optimal power output does not change significantly with the increasing mass of the inner buoy. The power output increases slightly with the rising mass of the inner buoy at low wave frequency area while decreases significantly at high wave frequency area. The relations between the wave frequency and CWR are presented in Fig. 25. It shows that the CWR has a similar regularity with the average power output.
(4) mass ratio of the outer-to-inner buoys Here, considering all conditions of the outer and inner buoy mass above, the variation curves of the average power output with wave frequency under different mass ratios are shown in Fig. 26. It can be seen that the average power output of the device at different mass ratios increases first and then decreases with the increase of wave frequency. The optimal power output reaches the peak value with the smaller mass of the inner buoy. The inner buoy is designed to be a     slender structure, which is not sensitive to the change of the wave field, thereby leading to an optimal mass ratio of the converter. By comparing the CWRs under different mass ratios, as shown in Fig. 27, it is obvious that the mass ratio does not have significant effect on the CWR at low wave frequency area while it has an opposite trend at high wave frequency area. The CWR has the peak value with the mass ratio of 0.80 at high wave frequency area, and the power absorption frequency band is the widest under the same condition, which leads to the optimal energy capture. By comparison, different forms of PTO damping have no influence on the optimal mass ratio of this device, and the optimal mass ratio under linear PTO damping force is consistent with it under constant one with the value being 0.80.

Conclusions
The hydrodynamic characteristics of the coaxial doublebuoy is systematically investigated in this paper. By starting from the potential flow theory, the response equation of the device under the assumption of linear PTO damping force is solved. On the basis of theoretical research, a model test is carried out.
(1) The wave period has a significant impact on the hydrodynamic performance of the double-buoy device. The relative motion amplitude reaches the peak value at the resonant period of inner buoy under non-PTO condition, while the CWR identically reaches the optimal value when the wave approximately meets the resonant period of the outer buoy as constant or linear PTO damping is employed. A larger separation between two natural periods is beneficial, which leads to a wider energy capture band with higher value. The double-buoy WEC behaves linearly for different wave heights under the linear PTO damping.
(2) The mass ratio of the outer-to-inner buoys has a greater effect on the CWR. Different forms of PTO damping have no influence on the optimal mass ratio of this device, and the double-buoy with mass ratio of 0.80 is suggested to achieve the goal of optimal energy capture.