Experimental investigation on the hydrodynamic performance of a wave energy converter

Wave energy is an important type of marine renewable energy. A wave energy converter (WEC) moored with two floating bodies was developed in the present study. To analyze the dynamic performance of the WEC, an experimental device was designed and tested in a tank. The experiment focused on the factors which impact the motion and energy conversion performance of the WEC. Dynamic performance was evaluated by the relative displacements and velocities of the oscillator and carrier which served as the floating bodies of WEC. Four factors were tested, i.e. wave height, wave period, power take-off (PTO) damping, and mass ratio (RM) of the oscillator and carrier. Experimental results show that these factors greatly affect the energy conversion performance, especially when the wave period matches RM and PTO damping. According to the results, we conclude that: (a) the maximization of the relative displacements and velocities leads to the maximization of the energy conversion efficiency; (b) the larger the wave height, the higher the energy conversion efficiency will be; (c) the relationships of energy conversion efficiency with wave period, PTO damping, and RM are nonlinear, but the maximum efficiency is obtained when these three factors are optimally matched. Experimental results demonstrated that the energy conversion efficiency reached the peak at 28.62% when the wave height was 120 mm, wave period was 1.0 s, RM was 0.21, and the PTO damping was corresponding to the resistance of 100 Ω.


Introduction
Energy is the essential driving force of social development and economic growth, as well as the foundation of human survival. At present, fossil fuels constitute the main part of energy source, and around 74% of the world's total energy consumption comes from fossil energy such as coal, petroleum, and natural gas. However, with the development of economy and population growth, fossil energy resource is declining rapidly. Additionally, utilization of fossil energy leads to more emissions of greenhouse gases and is harmful to the environment. Owing to the problem of resource exhaustion and environmental degradation, the international communities have been more concerned about developing renewable energy. Ocean is often regarded as the last resource treasury that stores abundant energy. Theoretically, the global storage of ocean energy is 7.66×10 5 GW. Be-sides, the utilization of ocean energy has little negative influence on the environment and does not occupy land resource. Wave energy is one type of ocean renewable energy which is widely spread all over the world with high energy density. Hence, developing wave energy is promising (Falnes and Løvseth, 1991;Zheng et al., 2014;Zhang et al., 2013).
AWEC is the device which absorbs wave energy and converts it into other forms of energy. Many types of the WEC have been invented, such as oscillating water column (OWC), raft, buoy, pendulum, and duck. Since the 1970s, many scholars have researched the motion and energy conversion performance of the WEC to optimize its efficiency. Evans (1976) showed that if the PTO damping matched with the radiation damping, the theoretical extraction efficiency of the WEC could attain 100% in an in-viscid fluid. Tom and Yeung (2013) investigated the hydrodynamic effects of two floater shapes to increase motion and decrease viscous losses. The paper showed that a rounded bottom shape made motion increase by approximately 64% at resonance. Mueller (2002), Ivanova et al. (2004) and Rhinefrank et al. (2006) studied how the PTO damping influenced WEC's motion and extraction efficiency through mathematical simulation and model experiment. Eriksson et al. (2005) carried on the theory research on the motion and extraction efficiency of a direct drive WEC under regular wave and real sea. Recently, a design of cycloidal WEC has been investigated by using potential-flow theory and experiments. The device consists of foils rotating around a shaft perpendicular to the direction of the incident waves. Madhi et al. (2014) used the potential-flow code TRWYADMXA to predict the hydrodynamic behavior of the asymmetric floater in heave. The theoretical predictions suggested that the wave energy capturing efficiency could reach 96.34% if the floater was operated at resonance and the absorption damping of the generator was matched with the radiation damping of the floater.
The motion of the WEC forced by the wave is an excited oscillating system. The system's oscillating characteristics greatly affect the energy conversion efficiency. Mathematical description of motion for some types of the WEC has been developed. Falnes (2002a) illustrated the wave energy absorption by oscillating bodies. He proposed that a one-mode oscillating system happened to have the optimum phase condition if the wave frequency was the same as the natural frequency of the oscillating system. Because in that case, the oscillatory velocity of the system was in phase with the wave's excitation force. Falcão (2007) studied the phase control method of the oscillating-body WECs with hydraulic PTO system based on analyzing the oscillating characteristics of the oscillating body in the frequency domain. Through the phase control by latching, the body velocity got in the phase with the diffraction force on the body. To maximize the power absorption of a point absorber under unconstrained conditions, De Backer (2009) developed a numerical model in the frequency and time domain to describe the hydrodynamic behavior of a heaving point absorber in regular and irregular waves. Based on the numerical calculation, he studied the oscillating characteristics of the point absorber, and analyzed how buoy's shape, diameter, draft PTO damping and other conditions affected the energy absorption. It is difficult to analyze the motion and energy conversion of the WEC with multi-bodies. Cochet and Yeung (2012) developed configuration design and dynamic analysis of a two-component wave-energy absorber. According to the oscillating floater WEC, Tom and Yeung (2013) and Madhi (2012) designed a wave-energy extractor to test the device's performance.
This paper investigates the motion and energy conversion of a WEC based on the experiments in a tank. The WEC have two floating bodies which are concentric cylinders. One floating body is an oscillator to absorb wave energy; the other is a carrier to load PTO system and other equipments, which is anchored at the bottom of ocean. Fig. 1 shows the geometric sketch of the WEC. The wave energy absorbed by oscillator is converted into other types of energy through the PTO system. Zheng et al. (2014) and Guo et al. (2014) constructed the motion and energy conversion model of the WEC and analyzed the hydrodynamic performance in the frequency and time domain. In above investigation, the model experiment has been carried out in a tank. In this paper, the beneficial effect on the motion and energy conversion of the WEC is analyzed after experiment. Experimental results show that the energy conversion efficiency of the WEC model can reach 28.62% under the optimized condition.

Theoretical background
The oscillator and carrier move in the heave under the action of wave. Because of the different structures, the oscillator and carrier have different hydrodynamic performances, which cause the relative movement between the oscillator and carrier in the heave motion. The relative movement drives the PTO system and converts the wave energy into electricity or other types of energy. Firstly, we will construct the motion and energy conversion model in the frequency domain. Falnes (2002b) discussed the dynamical model and energy conversion model of the oscillating system with one oscillator. In this paper, we assume that the oscillator and carrier are limited to move in the vertical direction. In addition, the PTO damping and spring coefficients are all assumed to be linear. Based on the above assumption, in the vertical direction, the oscillator and carrier are forced by the wave force, hydrostatic restoring force, the PTO damping force and spring force. The converter constitutes a linear oscillating system. To study the energy conversion performance of wave energy converter, we simplify the converter system into a mass-spring-damping system model shown in Fig. 2. ZHENG Xiong-bo et al. China Ocean Eng., 2017, Vol. 31, No. 3, P. 370-377 371 In Fig. 2, M j (j=1, 2) represent equivalent masses of the oscillator and carrier, respectively; K j and C j (j=1, 2) represent the spring and damping coefficients that force to oscillator and carrier, respectively. These factors constitute an energy converting system with the mass, spring and damping.
Assume that the vertical displacements of the oscillator and carrier are x 1 and x 2 , respectively, and the wave forces that put on them are f 1 and f 2 . Let where, X j (j=1, 2) are the complex motion amplitudes of the oscillator and carrier.
ρ n z where, i is the imaginary unit, is the density of water, S j is the wet surface of structure, is the normal vector, φ I is the incident potential, and φ D is the diffraction potential.
are the added mass and wave damping coefficients of the oscillator and carrie, and φ R is the radiation potentialr.
Assume that the masses and restoring force coefficients of the oscillator and carrier are m j and . Let c is the linear damping coefficient of the PTO system, k is the spring coefficient. Then Suppose that , and then the motion equations of the oscillating system can be written as: Thus, the complex amplitude of the relative displacement between the oscillator and carrier can be written as: The output power of PTO system is (9) According to the above calculation and assumption, the output power can also be written as: (10) P pto P pto From Eqs. (8) and (10) we can see that the output power and relative displacement between the oscillator and carrier are influenced by the wave frequency, spring coefficient, PTO system damping coefficient, the masses and hydrodynamic coefficients of the oscillator and carrier. According to the WEC, is hoped to be maximized. Many scholars have proposed some optimal methods to maximize the under the same wave condition (Bachynski et al., 2012;Price et al., 2009;Abraham and Kerrigan, 2012;Count and Evans, 1984).

Experimental investigation on WEC
Based on the above theoretical background, we designed an experimental device and the scheme to investigate the motion and energy performance of the WEC. The device is floating type. It is fixed by the mooring system. The mooring line is replaced by the spring in Fig. 1. And the mooring's stiffness coefficient is represented by the spring coefficient. The wave energy absorbed by the device was converted into electrical energy by a permanent magnet linear generator. Thus the PTO damping is placed by the permanent magnet linear generator's electromagnetic damping.

Tank
The experiment was carried out in a tank which is 108 m in length, 7 m in width and 4 m in depth. There are a wave maker and wave absorbing revetment at the ends of the tank. A trailer which is used to place test instruments and the install experiment device is stretched over the tank.

Rock flap wave generator
A rock flap wave generator can generate regular wave in the tank. The wave period can be changed from 0.4 s to 4 s. The largest wave height that can be generated is 0.4 m. The generator can also generate irregular wave with ITTC single-parameter and double-parameter spectrum, JON-SWAP spectrum, P-M spectrum, the actual wave sampling spectrum, and so on. The largest significant wave height that can be generated is 0.32 m.

Non-contact motion measurement system
The system includes two HD cameras, signal transmission ware, and Qualisys track manager software. Using the system, we can record and analyze the moving body's motion state in six degrees of freedom without contacting the moving body. The measure precision of the system is 0.02 mm.

Data collection and analysis instruments
Because the current outputted by the WEC device is alternating current, we designed an AC/DC conversion module to convert the alternating current into direct current which can be collected by the data collection system (DH5920). The data collection system can collect the motion and current data.

Installation of the experimental device
The experiment device was installed under the trailer. The trailer was located at the middle of the tank. The device was floating on the water. It was located by three-point mooring positioning system. Measurement instruments such as the data collection system and AC/DC conversion module were placed on the trailer. The electric current generated by the device was transmitted to the data collection system through the electric cable. Non-contact motion measurement system was fixed on the trailer to collect the dis-λ= 1:10 placement data of the oscillator and carrier. The experimental pictures are shown in Fig. 3. The parameters of the device are shown in Table 1. This kind of wave energy device is designed to be installed in the deep ocean. The water depth is about 30-50 m. According to the size of the tank, the scale of the experiment is valued as . During the experiment, we generated 22 groups of wave with two different heights and twelve different periods. The two wave heights are 0.08 m and 0.12 m. And the twelve periods are between 1.0 s and 3.0 s with the step of 0.2 s. We measured the heaving displacements of the oscillator and carrier, and collected the current and voltage outputted by the device under these different wave conditions.

Relative movement of the oscillator and carrier
The oscillator and carrier moved under the force of wave when the wave was generated by a rock flap wave generator. Then, non-contact motion measurement system was used to record the transient position of both the oscillator and carrier in time. Their relative motions are analyzed as follows. Fig. 4a shows the positions of the oscillator and carrier in the vertical direction when the wave height (H) is 80 mm and wave period (T) is 1.0 s. From Fig. 4a, we can find that under the force of wave, the oscillator and carrier oscillate with the wave. The oscillating periods of the oscillator and carrier are the same as that of wave, even though their oscillating is not regular due to the irregular friction force, damping and mechanical resistance. As shown in Fig. 4a, the displacement amplitude of the oscillator is much larger than that of the carrier during the same period. Thus, the oscillator will slide up and down along the carrier and leads to their relative heave motion which will drive the PTO system. Fig. 4b shows the curves of the relative displacement and velocity of the oscillator and carrier. The periods of the relative displacement and velocity are the same as the wave's period which is 1.0 s while the phase of the displacement is , slower than that of the velocity. The maximum displacement amplitude is 37.7 mm which is 94.25% of the wave amplitude. The average displacement amplitude is 16.2 mm which is 40.5% of the wave amplitude. The maximum velocity amplitude is 268.98 mm/s. And the average velocity amplitude is 112 mm/s. Fig. 5 shows the test results of the relative displacement and relative velocity between the oscillator and carrier in waves. As shown in Figs. 5a and 5b, the periods of the oscillator and carrier's relative movement change with the wave period. The relative displacement synchronizes with the wave. In addition, we can infer that the relative displacement amplitudes will also change with the wave period at the same wave height. From Fig. 5a, in terms of the three conditions, when the wave period is 1.6 s, the amplitudes of the displacement and velocity are both the largest. To analyze the law that the amplitudes of the displacement and velocity change with wave period, we calculated the average amplitudes of the displacement and velocity at different wave periods, and Fig. 5c shows that the amplitudes of the displacement and velocity change with the wave period. When the wave period changes from 1 s to 2.8 s, the amplitudes of the displacement and velocity increase first, and then they reach the maximum values when the wave period is 1.4 s. Afterwards, they decrease as the wave period increases. It is because the device has a natural period, when its natural period is equal to the wave period, they will resonate. The resonance period of the device is 1.4 s under the above conditions. Then we analyze the movement under different wave heights. We generated two different waves with the wave heights of 80 mm and 120 mm, respectively. Fig. 6 shows the relative movement characteristics under these two conditions. We focus on the experimental results when the wave period is 1.4 s. From Figs. 6a and 6b, we can see that the amplitudes of the relative displacement and velocity both grow up when the wave height increases from 80 mm to 120 mm. Fig. 6c shows their ratios under different conditions. We can see that the ratios vary and the average value is 0.667 which is the ratio of the above two wave heights. The average ratio of the displacement is 0.669, variance is 0.0016, and the average ratio of velocity is 0.67, variance is 0.0017. The two average ratios are both near the ratio of the wave heights, and the variances are very small. Consequently, the amplitude of the relative displacement and velocity of the oscillator and carrier change linearly with wave height under some conditions. The result is constricted to the condition of small amplitude waves.
the larger the resistance, the smaller the PTO damping. Fig. 7 shows the amplitudes of the relative displacement and velocity of the oscillator and carrier under different resistance conditions. From Fig. 7, we can conclude that the resistance influences the relative movement of the oscillator and carrier. When the resistance is 100 Ω, the relative displacement and velocity of the oscillator and carrier reach their maximum values. That is because the variation of the resistance value results in the variation of the PTO damping. It should be noted that PTO damping is an important parameter influening the relative movement and power output according to Eq. (12). An optimal damping existed at the movement and the output power of WEC was proved . Therefore, the PTO damping is optimal when the resistance value is 100 Ω.

Output power of the experimental device
In the experiment, we measured the voltage and current of the resistance which is connected to the generator. We calculated the average output power in a time period based on Eq. (11) and calculated energy conversion efficiency according to Eq. (12).
where, T is the wave period, t is time, V(t) and I(t) are the voltage and current at time t, and is the wave power in the range of the oscillator width which is calculated as follows: ρ where, is the density of water, H is the wave height, and D is the diameter of the oscillator (Li and Yu, 2012;Cruz, 2008). Fig. 8 shows the energy conversion efficiency curves of the experimental device which connects different resistances, 10 Ω, 20 Ω, 50 Ω, 100 Ω and 500 Ω. In Eq. (12), the PTO damping is regarded as the parameter of the power output function. Then as shown in the figure, the energy conversion efficiencies are different when the device connects different resistance values under the same wave condition. According to the figure, we can also deduce that the efficiency curves will have similar tendency. The efficiency increases with the increment of the wave period, and decreases with the wave period further increasing after reaching the maximum value. Whether the conversion efficiency can be the maximum is dependent on the resonance period. The resonance periods and maximum efficiencies corresponding to resistance are shown in Table 2. We can con-   clude that the devices with different resistances have different resonance periods as well as different maximum efficiencies. For instance, the resistance value increases with the increment of the resonance period while the maximum efficiency varies. The maximum efficiency value reaches the peak value when the resistance value is 100 Ω. Additionally, in the cases of most wave periods, the conversion efficiency is larger than those of many other resistance values when the resistance value is 100 Ω. Thus, we can infer that the PTO damping under the condition of 100 Ω is more optimal than those under other conditions. The PTO damping can be controlled through changing resistance values or other components, such as the generator characteristics or oil hydraulic pump characteristics. In a certain ocean environment, the PTO damping should be controlled to match the wave period. According to Eq. (12), we can conclude that both the mass of the oscillator and carrier are the influence factors that affect the device's output power. Here, a non-dimensional parameter is used, i.e. the mass ratio (R M ) of the oscillator and carrier. It is described as follows: where, m 1 and m 2 are the masses of the oscillator and carrier. We change the value of R M to study the relationship between R M and the power output. Four values of R M are used, i.e. 0.15, 0.17, 0.19 and 0.21. In the preceding paragraphs, the values of R M are all 0.15. Fig. 9 shows the efficiency curves at different values of R M . From Fig. 9 we can infer that the wave period corresponding to the maximum efficiency decreases with the increase of R M . When R M is 0.21, the maximum efficiency reaches 28.62%. Meanwhile, the corresponding wave period is 1 s. The maximum efficiency is larger than that under other conditions. The maximum efficiency and corresponding wave period are shown in Table 3.
The curves in Fig. 10 indicate the energy conversion efficiency with two different wave heights (80 mm and 120 mm), the resistance value of 100 Ω, and the mass ratio of 0.15. The results show that the energy conversion efficiency of the device is higher when the wave height is 120 mm. The average ratio of the two efficiency values is 1.13. So we can conclude that the energy conversion efficiency increases with the increasing wave height instead of being constant.

Conclusions
The WEC introduced in this paper is a model with two floating bodies which are concentric cylinders and mooring system. The PTO system is a permanent magnet linear generator. The heaving motion of the oscillator and carrier is analyzed. Wave conditions and parameters of the wave energy converter and their effects on the capability of the energy extraction are also analyzed through experiments in a tank.
The experimental results show that the maximization of the motion leads to the maximization of the energy conversion efficiency. We observed that the mass of the oscillator and carrier, wave period, wave height, and PTO damping are important factors that influence the wave energy converter's motion and energy extraction capability. According to the same experimental results, there exists an optimal wave energy period that the energy conversion efficiency will be the highest. When the wave period is close to the optimal period, the energy conversion efficiency will also be higher than those of other periods which are far away from the optimal period. Under the same wave condition, the device's energy conversion efficiency curves are different from the PTO damping coefficients. Under the condition of a certain wave period, there also exists an optimal PTO damping. But the optimal PTO damping is not a constant when the wave period changes. The experimental results show that different mass ratios of the oscillator and carrier lead to different efficiency curves when other factors are all the same. In real ocean environment, the wave is irregular. Wave energy converter is hoped to have higher energy conversion efficiency in a wide period range. The PTO damping and R M can be controlled to adapt to the period of ocean wave. And through the experimental result, when the resist-   100 Ω ance is and the mass ratio is 0.19, the experimental device has higher energy conversion efficiency in the period range of 1.0-2.0 s.