Capture Performance of A Multi-Freedom Wave Energy Converter with Different Power Take-off Systems

Among the wave energy converters (WECs), oscillating buoy is a promising type for wave energy development in offshore area. Conventional single-freedom oscillating buoy WECs with linear power take-off (PTO) system are less efficient under off-resonance conditions and have a narrow power capture bandwidth. Thus, a multi-freedom WEC with a nonlinear PTO system is proposed. This study examines a multi-freedom WEC with 3 degrees of freedom: surge, heave and pitch. Three different PTO systems (velocity-square, snap through, and constant PTO systems) and a traditional linear PTO system are applied to the WEC. A time-domain model is established using linear potential theory and Cummins equation. The kinematic equation is numerically calculated with the fourth-order Runge-Kutta method. The optimal average output power of the PTO systems in all degrees of freedom are obtained and compared. Other parameters of snap through PTO are also discussed in detail. Results show that according to the power capture performance, the order of the PTO systems from the best to worst is snap through PTO, constant PTO, linear PTO and velocity-square PTO. The resonant frequency of the WEC can be adjusted to the incident wave frequency by choosing specific parameters of the snap through PTO. Adding more DOFs can make the WEC get a better power performance in more wave frequencies. Both the above two methods can raise the WEC’s power capture performance significantly.


Introduction
Wave energy converters with various working principles have been proposed and studied within recent decades, including oscillating water column (OWC), overtopping and oscillating buoy. Among them, the oscillating buoy is considered as a promising one for its applicability in offshore regions. It absorbs energy by the buoy's motion with incident waves and converts energy to electricity by different power take-off systems. However, the power capture performance of oscillating buoy is not satisfactory. Mean capture width ratios of the heaving device, the fixed and the floating oscillating wave surge converters are 16%, 37% and 12%, respectively (Babarit, 2015). The expected ideal capture efficiency can only be obtained under stable and narrowband wave condition near the WEC's natural resonant frequencies.
To solve the problems, many pieces of research have been carried out. One solution is to develop a multi-free-dom device to capture more energy. Albatern Ltd (Albatern Wave Energy, 2017) in Scotland developed WaveNET, an offshore array-based converter that absorbs the wave energy from multi-freedom motions of the buoyancy floats arranged in the hexagonal shape. Heikkinen et al. (2013) proposed a bottom-supported energy converter with a horizontally orientated cylinder which can oscillate in horizontal and vertical directions along an ellipse track. Wave-cylinder interaction and the converter's efficiency under the influence of phase shift, cylinder radius and wave condition were studied analytically. Zhang et al. (2015) proposed and tested a floating multi-axis WEC which moves in 3 degrees of freedom (DOF) (heave, surge and pitch) with a multi-axis mechanical system. One buoy structure in this study was the achievement of McCabe et al. (2010). They proposed a symmetrical shape for a surge-pitch wave energy collector based on bi-cubic B-spline surfaces and optimized it using genetic algorithm method. Osborne et al. (2015) andBhatt et al. (2016) proposed TALOS, another multi-axis point absorber which has an internal mass PTO composed of an inertial mass and hydraulic cylinders in a solid outer hull. A series of studies including optimization and wave tank experiments were carried out at Lancaster University. Iijima et al. (2000) and Bai et al. (2002) proposed the Heave and Pitch Buoy which consists of two floats and an oil-hydraulic power transmission to convert the oscillation energy. Laboratory experiments were carried out with the 1/15 scale model of 10 kW prototype to investigate the PTO performance. Ye and Chen (2017) proposed a 6 degrees of freedom point absorber WEC with a 6-PSS (prismatic pair-spherical pair-spherical pair) power take-off mechanism. The power absorption was studied in both regular and irregular waves, and such conclusions are drawn that the advantages with the converter in resonance with waves are evident in regular waves and not so obvious in irregular waves.
Various studies of the oscillating buoy WECs are investigated with linear PTO damping, which is derived by multiplying the buoy's velocity with a damping coefficient, or additionally a linear spring (Mei, 2012;Shi et al., 2016). The linear PTO damping is a simplified method. But the consideration of nonlinear PTO damping is necessary. de O. Falcão (2007de O. Falcão ( , 2008 adopted the coulomb damping force to simulate hydraulic-pump-type PTO of the hemi-sphere heaving-body WECs. The damping force is proportional to the product of the gas pressure difference and the cross-sectional area of the ram.  2017) adopted a PTO system consists of a drum, two gearboxes and two electrical machines for the Wave Pioneer. The PTO force is expressed as a formula with the parameters of the gearbox ratio, drum radius and the torque on the machine shaft (related to the differential of the shaft's rotational speed). Wang (2015) used the similar theory of hydraulic turbine to derive a new PTO damping which is proportional to the square of the oscillating buoy's velocity. Zhang and Yang (2015) used a nonlinear snap through PTO system consisting of two symmetrically oblique springs and a linear damper. Bozzi et al. (2013) adopted an electromagnetic PTO force which is obtained through Faraday's Law and the Maxwell equation. The electromagnetic PTO force is pro-portional to the electric voltage and current which are related to both the buoy's velocity and its displacement.
The PTO damping of an oscillating buoy WEC is mainly related to parameters including the buoy's velocity and displacement, or can be considered as a constant coefficient. To give advice to the proper selection of the PTO system, this paper chooses four types of PTO damping: linear, velocity-square, snap through, and constant. The oscillating buoy is assumed to be a rigid cylinder in order to neglect the influence of wave incident direction and has 3 DOFs: surge, heave and pitch. A time-domain method using Cummins equation (Cummins, 1962) is adopted to calculate the average output power and compare the power capture performance.
In the following, Section 2 introduces the 3-DOFs timedomain model. Section 3 compares the optimal average output power among four PTO systems in every DOF. Section 4 discusses the snap through PTO system in detail. Section 5 draws the conclusions.

3-DOFs time-domain model
A cylinder is adopted as the geometry of the oscillating buoy which has 3 DOFs: surge, heave and pitch. The fluid is assumed to be non-viscous and irrotational which makes the potential theory applicable. Fig. 1 is the schematic figure of the WEC. The PTO damping is added respectively in every DOF and cannot affect each other.
With the nonlinear PTO forces, the multi-freedom buoy's motion can be analyzed numerically by the time-domain model based on the following matrix equation: is the mass matrix, is the added mass matrix at infinite frequency, is the retardation function matrix in time domain, is the hydrodynamic stiffness matrix, is the displacement matrix, is the hydrodynamic excitation force matrix and is the PTO force.
where is the sea water density, is the buoy's cross-sectional area, , , is the buoy's volume.
can be calculated as (Ogilvie, 1964): (2) and are the added mass matrix and radiation damping matrix in frequency domain. F e can be calculated via inverse Fourier transform as: where is the wave excitation impulse response function, is the wave surface function and is the wave excitation force frequency response function.
Four types of PTO damping are selected. They are linear, velocity-square, snap through and constant PTO damping. The expressions of the corresponding PTO force are as follows, where is the damping coefficient derived by the nature of the PTO systems. Fig. 2 shows of linear PTO, velocity-square PTO and constant PTO as a function of . Fig. 3 shows the stiffness coefficient of snap through PTO as a function of .
Snap through PTO force: (The first part is the energy absorption while the second part has no contribution to the energy absorption.) The average output power can be calculated by the integration: T where is the operation time of the buoy.
The integrations of the above equations are calculated by trapezoidal integration method. A fourth-order Runge-Kutta method is selected to solve Eq. (1). The parameters of frequency domain , and are ana- The results fit very well (shown in Fig. 4). In Fig. 4a, the radius of the cylinder is 20.27 m and the draft is 198.1 m according to Shan (2013). The vertical axis is the retardation function of surge unidirectional motion . In Fig. 4b, the radius of the cylinder is 0.4 m and the draft is 0.07 m. The vertical axis is the heave displacement without PTO damping.

Comparation of the PTO systems
To investigate which type of PTO has the best power capture performance, the optimal average output power is calculated in every DOF. In the calculation, linear wave is assumed, and the wave amplitude is 0.125 m. Wave frequency is selected from 0.2 rad/s to 5 rad/s. The radius of  A4 in the appendix. For snap through PTO, is 1 m and is 0.5. The spring stiffness coefficient is in heave and surge and in pitch. The effects of , and on the buoy's average output power are discussed in Section 4. Fig. 5 shows the optimal average output power of unidirectional motions with four PTO types. The average power with linear PTO is regarded as a control. The velocity-square PTO has similar power performance with the linear one. The optimal average power at every wave frequency in surge and pitch is almost the same. In heave, the average power of velocity-square PTO is equal to that of linear PTO with wave frequencies higher than 2 rad/s, while is lower with wave frequencies lower than 2 rad/s. The snap through PTO has a good power capture performance in every DOF, especially in surge. The average output power in surge peaks on wave frequency 2.2 rad/s and the maximum value is 2 times the maximum value of linear PTO. The constant PTO has a larger average power than the linear PTO in heave and pitch while there is an opposite result in surge.
By comparing the results in Figs. 5a-5c, it is drawn that the buoy has the maximum power capture in heave, then in surge. The wave frequency of peak value in heave is smaller than that in surge and pitch except for one case. As the power limits of surge and pitch are twice the power limit of heave (Budar and Falnes, 1975;Evans, 1976;Newman, 1976), the geometry of the buoy can be optimized to get larger power absorption in surge and pitch. Fig. 6 shows the optimal average output power of coupling motions. It is obvious that multi-freedom WEC has a better power capture performance than the single-freedom WEC. The power of multi-freedom WEC is approximately the sum of the power in every DOF component. This is because, as the geometry of symmetric cylinder, heave motion has no effect on surge and pitch and the influence between surge and pitch motions is negligible. As heave is the maximum power capture DOF, it is selected to compare the wave frequency range. The mean values of the optimal average power output of heave motion with the four PTO types are listed in Table 1. The wave frequency ranges with the average power are larger than the corresponding mean value of heave and 3-DOF buoys are also listed. It is indicated that the multi-freedom buoy can get a better power absorption in more wave frequencies than the single-freedom one, which means that the multi-freedom WEC can adapt to more wave conditions. From Figs. 5 and 6, it can be seen that in most conditions, WEC with snap through PTO has the best power capture performance, while the constant one is the second best. But when the wave frequency is 2-4 rad/s, the constant PTO is the best and the snap through PTO is the second best for heave and heave-pitch motions. The velocity-square PTO is the worst for most conditions, but is the best for surge or surge-pitch motions with the wave frequency lower than 2 rad/s. The output power can be increased significantly both by using nonlinear PTO (snap through PTO or constant  HUANG Shu-ting et al. China Ocean Eng., 2019, Vol. 33, No. 3, P. 288-297 291 PTO) and adding DOFs.

Discussion of the snap through PTO
The expression of snap through PTO force has four parameters: , , and . In this section, is 1000 for heave, 100 for surge and 50 for pitch.  Fig. 7 shows the average output power varying with . is 0.5 and is 1 m. The average power increases firstly and decreases after certain point, with the increasing wave frequency in every DOF. As shown in Fig. 7a, for surge motion, both the resonant frequency and the corresponding average power increase with the increase of . And the average power with bigger begins to increase later. So, should be small in low frequency waves and large in high frequency waves. Fig. 7b indicates that for heave motion, the average output power has an obvious growth with the increase of and the resonant frequency deceases slightly. The average power of pitch motion also has a small increase with the increase of (shown in Fig. 7c).  Fig. 8 shows the effect of . is 1 m. is for surge and heave and is for pitch. It is drawn that the increase of weakens the power capture. The decrease of the average power in surge and heave is significant. But for surge motion in low wave frequency, higher should be better. also changes resonant frequency. The resonant frequency reduces in surge and increases a little in heave and pitch.  Fig. 9 shows the effect of . is 0.5. is for surge and heave and is for pitch. Other than and , the influence of on the average power is limited. Raising only increases the average power in surge obviously when is very small. When is larger than 0.6 m, the average power in surge begins to remain constant. The average power of heave and pitch has small changes with the varying .

Conclusions
The paper aims to contrast the power capture with four PTO types to find the best one and to investigate the performance of the multi-freedom oscillating buoy WEC.
An oscillating buoy WEC with 3 DOFs (surge, heave and pitch) is adopted and the buoy is assumed as a cylinder. Four PTO systems are applied to the WEC. A time-domain model is established and calculated numerically. The average output power of the WECs with four PTO systems are calculated and compared in every DOF. The following conclusions are drawn: (1) Generally, the rank of the PTO types from the best to the worst is snap through, constant, linear and velocitysquare PTO according to the power capture performance. It should be noted that for pitch motion, constant PTO is the worst. (2) For WEC with snap through, in surge, a small value of and a large value of should be selected in waves with low frequency while a large value of and a small value of should be selected in waves with high frequency. For WEC with snap through PTO in heave and pitch, increasing and decreasing can raise the power capture.
(3) Changing the parameters of the snap through PTO can change the resonant frequency of the WEC. Adding more DOFs can make the WEC get a better power performance in more wave frequencies. This means that the multifreedom WEC can be applied to more wave conditions and more sea areas.

Appendix:
The optimal PTO coefficients of cases in Figs. 5 and 6 are listed below.