Abstract
In this study, we investigated the hydrodynamic and energy conversion performance of a doublefloat wave energy converter (WEC) based on the linear theory of water waves. The generator power takeoff (PTO) system is modeled as a combination of a linear viscous damping and a linear spring. Using the frequency domain method, the optimal damping coefficient of the generator PTO system is derived to achieve the optimal conversion efficiency (capture width ratio). Based on the potential flow theory and the higherorder boundary element method (HOBEM), we constructed a threedimensional model of doublefloat WEC to study its hydrodynamic performance and response in the time domain. Only the heave motion of the twobody system is considered and a virtual function is introduced to decouple the motions of the floats. The energy conversion character of the doublefloat WEC is also evaluated. The investigation is carried out over a wide range of incident wave frequency. By analyzing the effects of the incident wave frequency, we derive the PTO’s damping coefficient for the doublefloat WEC’s capture width ratio and the relationships between the capture width ratio and the natural frequencies of the lower and upper floats. In addition, it is capable to modify the natural frequencies of the two floats by changing the stiffness coefficients of the PTO and mooring systems. We found that the natural frequencies of the device can directly influence the peak frequency of the capture width, which may provide an important reference for the design of WECs.
Introduction
Wave energy is an ecofriendly energy that attracts extensive research interest due to its broad global distribution and high energy density (Isaacs and Seymour 1973; Falnes 2007). Energy conversion in WECs is categorized into oscillating water column (OWC) and point absorber (Falcao 2010) types. The doublefloat WEC, which is in the point absorber category, has high conversion efficiency and simple construction, and has become a hot spot in the study of wave energy extraction.
In recent years, much work has been performed with respect to energy conversion theories and the hydrodynamics of WECs. Evans (1976) introduced the first efficiency equation of a onebody WEC. Twobody WECs are also studied in various literature (Yu and Li 2013; Kim et al. 2016; Son et al. 2016; Dai et al. 2017; Son and Yeung 2017; Cho and Kim 2017; Liang and Zuo 2017). Falnes (1999) used the frequency domain method to treat a twobody WEC as a onebody WEC. The maximum efficiency can be achieved with an optimal damping coefficient and optimal stiffness coefficient. Eriksson et al. (2005) studied the damped oscillation and efficiency of a cylindrical WEC in both regular waves and real marine conditions. Wu et al. (2013) studied the radiation and diffraction problems of both floatingcylinder and fixedcylinder WECs using an analytical approach and determined the relationships between efficiency, the stiffness coefficient, and the geometry of a WEC, although the relationship between efficiency and the natural frequencies of floats were not clearly explained. Zhang et al. (2016) used an analytical method to study the relationship between the natural frequencies of floats and the conversion efficiency of the WEC, but focused on the effects of the damping plate, PTO stiffness coefficient, and mooring stiffness coefficient. Neither Wu et al. (2013) nor Zhang et al. (2016) proposed an expression for the optimal PTO damping coefficient.
In this paper, we first derive the power absorption and conversion efficiency of the doublefloat WEC. Considering the PTO damping coefficient, stiffness coefficient, and mooring stiffness, we derive the optimal damping coefficient of the doublefloat WEC using the frequency domain method. Based on the potential flow theory, we apply the higherorder boundary element method (HOBEM) to construct a 3D model of the doublefloat WEC in the linear time domain. To decouple the motion of the floats, we introduce a virtual function. We then validate the proposed model by comparing it with the optimal damping coefficient derived from frequency domain method. We investigate the influence of incident wave frequency, PTO damping coefficient, stiffness coefficient, and mooring stiffness coefficient on the motion response of the floats and the conversion efficiency.
Mathematical Model
The doublefloat WEC includes upper and lower floats (Fig. 1) connected by a flexible string and the rotor of the linear permanent magnet generator. The upper float is a circular ring with an outer radius R_{1} and draught H_{1}. The central lower float, which has a radius R_{2}, is equipped with a generator coil and a cylindrical damping plate with a radius R_{3} and height H_{3} attached underneath. The damping plate is moored to the seabed. The distance from the bottom of the upper float and the top of the damping plate is H_{2}. The water depth is d. When excited by waves, the relative motion between the two floats generates electric power. The two floats are connected by a sping with an elasity coefficient of K_{pto}. The magnet force from the generator provides a damping effect with a damping coefficient of B_{pto}. Wave power is absorbed by this PTO system. We use the Cartesian coordinate system OXYZ. The Zaxis rises vertically upward and coincides with the central line of the WEC. The X and Yaxes are on the calm water surface.
To describe the motion of the WEC in this paper, we assume a linear wave motion, and assume that the fluid is inviscid, incompressible, and irrotational. This simplifies the effect of the PTO system as a linear damper and a spring, and also simplifies the mooring line as a linear spring K_{m}. Figure 2 shows a diagram of the floats. As the absorption of wave energy mainly depends on the relative heave motion of the floats, the motion in the other five degrees of freedom is not significant and is not considered. The motion functions of the two floats are as follows:
where m_{1} and m_{2} are the mass, and z_{1} and z_{2} are the vertical displacements of the floats, \( {\dot{\boldsymbol{z}}}_1 \) and \( {\dot{\boldsymbol{z}}}_2 \) are their velocities in the vertical direction, and \( {\ddot{\boldsymbol{z}}}_1 \) and \( {\ddot{\boldsymbol{z}}}_2 \) are their accelerations. f_{1} and f_{2} are the body forces on the floats.
Energy Conversion Efficiency
The heave displacement of the float is written as follows:
where j = 1 for the upper float, j = 2 for the lower float, ω is the incident wave frequency, and Z_{1} and Z_{2} are the complex amplitudes of the upper and lower floats with respect to the heave motion, respectively.
The energy absorption rate of the WEC in a unit of time is defined as the work done by the fluid in the WEC. The average power absorption efficiency of the WEC (Evans 1976) is as follows:
The capture width ratio of a WEC is defined as the ratio of the wave energy captured by the WEC to the total input wave energy within the width of the WEC, as follows:
where E_{w} is the incident wave power per unit width, i.e.,
where D is the diameter of the float, E_{w} is the incident wave power per unit width (Basco 1984), ρ is the water density, A is the incident wave amplitude, and k_{0} is the wave number.
Optimal PTO Damping
Based on frequency domain theory, Eqs. (1) and (2) can be written as follows:
where F_{d1} and F_{d2} are the excitation forces and C_{1} and C_{2} are the restoration forces on the upper and lower floats, respectively. μ_{ij} and λ_{ij} are the added mass and radiation damping coefficients, respectively, exerted on the ith float by the jth float. We calculated these coefficients using HydroStar software. Let F_{d1} = Re (f_{d1}e^{iωt}) and F_{d2} = Re (f_{d2}e^{iωt}), where f_{d1} and f_{d2} are the complex amplitudes of the upper and lower floats, respectively. By substituting them with Eq. (3) into Eqs. (11) and (12), we have the following:
Let
where Y_{1} is the hydrodynamic coefficient associated with the selfmotion of the upper float, Y_{2} is the hydrodynamic coefficient associated with the selfmotion of the lower float, Y_{12} is the hydrodynamic coefficient of the upper float caused by the lower float, and Y_{21} is the hydrodynamic coefficient of the lower float caused by the upper float. Equation (9) can be rewritten as follows:
Let
By solving the equations, we have the following:
Thus:
By representing Y_{1}, Y_{2}, Y_{12}, and Y_{21} with complex numbers and substituting them into A, we have:
The determinant of A is as follows:
By substituting Eqs. (17) and (19) into Eq. (4), we have
where N is the square of the determinant of A.
Let
Then, we have
When:
E_{pto} reaches its maximum value:
and this damping coefficient is denoted as B_{opt}.
Time Domain Model
The wave–structure interaction model in open water is established using the incident wave and diffraction separation method based on potential flow theory (Zhou et al. 2015; Zhou and Wu 2015). The diffraction potential satisfies the Laplace equation and the following boundary conditions:
The boundary conditions on the free surface are:
where ϕ_{b} and η_{b} are the incident wave potential and wave height, respectively. According to the linear wave theory, ϕ_{a} and η_{a} are the diffraction potential and diffraction wave height, respectively; and δ(r) is a function for absorbing a diffraction wave on the outer boundary. V is the velocity of the body, and n is the normal vector pointing outside the body surface.
Based on the time domain theory, the motion equations of the floats are as follows:
where f_{1}^{’} and f_{2}^{’} are the forces exerted on the abovewater and lower floats, which can be derived using a pressure integral on the average wet surfaces S_{B1} and S_{B2}:
When using the acceleration potential method (Tanizawa 1995) to calculate ϕ_{t} in Eqs. (35) and (36), the acceleration of the body at that constant is needed. At the same time, the acceleration of the body must be calculated by the motion equations, which require a known ϕ_{t}. The calculation of ϕ_{t} and the acceleration of the body is therefore a process of iterative inductions. On the other hand, the problem in this study involves two floats, which means that two unknown accelerations are in the equation. Zhou et al. (2016) proposed a virtual function method to solve these two problems.
We introduce the virtual function ѱ_{3} that satisfies the Laplace equation. By Green’s second identity ѱ_{3} satisfies the following:
When calculating the forces on the upper float, let the boundary condition of ѱ_{3} be the following:
When calculating the forces on the lower float, let the boundary condition of ѱ_{3} be as follows:
By substituting the virtual function boundary conditions of Eqs. (42) and (43) into Eq. (41) and calculating the pressure integral Eqs. (39) and (40), we obtain Eq. (44) for the two accelerations \( {\ddot{z}}_1 \) and \( {\ddot{z}}_2 \). Then, we can decouple the abovewater and lower floats by solving the equation. After the accelerations are computed, we use the 4thorder RungeKutta method to update the velocity and displacement at the next constant.
where:
Numerical Results
Validation
To verify the accuracy of our proposed numerical model, we performed simulations of the doublefloat WEC with different PTO damping coefficients in linear waves. Table 1 lists the main parameters of the WEC. Figure 3 shows the relationship between the absorption power E_{pto} and the PTO damping coefficient B_{pto} with the incident wave period T = 7 s, PTO stiffness coefficient K_{pto} = 2 × 10^{4} N/m, and mooring stiffness coefficient K_{m} = 0. As B_{pto} increases, the absorption power of the WEC first increases and then decreases. When B_{pto} = 1.92 × 10^{6} N · s/m, the absorption power reaches its peak value. This result agrees well with the optimal PTO damping coefficient B_{opt} = 1.90 × 10^{6} N s/m calculated by Eq. (32), based on the frequency domain theory. Therefore, our proposed model is validated.
Effect of B _{pto} on Conversion Efficiency
In this section, we analyze the effect of B_{pto} on the conversion efficiency of the WEC. Here, we denote B_{opt} as the optimal PTO damping coefficient that makes the absorption power reach its peak value. We study the influence of the incident wave frequency on the capture width ratio and motion response of the floats at v = B_{pto} / B_{opt} equal to 0.5, 1, and 1.5 with K_{pto} = 4 × 10^{5} N/m and K_{m} = 0 (Fig. 4). In Fig. 4a, the capture width ratio at B_{pto} / B_{opt} = 1 is larger than that at B_{pto} / B_{opt} = 0.5 or 1.5. The difference is even greater for the peak value. On the other hand, the capture width ratio has two peak values with incident frequencies of ω = 0.48 and 0.75.
Figure 4b shows the relationship between the motion responses of the floats and the incident wave frequencies. When ω = 0.48, the displacement of the lower float is much larger than that of the upper float; when ω = 0.75, the displacements of the two floats are similar; when ω is far away from the peak frequency, the displacement of the upper float is much larger than that of the lower float. The displacements of the floats at ω = 0.48 are much larger than those at ω = 0.75. One reason for this is that B_{opt} at ω = 0.75 is much larger than B_{opt} at ω = 0.48, and the relative motion between the two floats is small. The other reason is that the lower float is resonant at ω = 0.48, whereas the upper float is resonant at ω = 0.75. The restoration force of the upper float is much larger than that of the lower float, which means that the motion of the lower float is large and the motion of the upper float is small. When ω = 0.48, the resonant lower float has a small restoration force coefficient; therefore, it oscillates with large amplitude. The upper float with a large restoration coefficient will not move identically with the lower float. When ω = 0.75, the restoration force coefficient of the lower float is small, and it will move with the upper float.
From the above analysis, we find that the two peaks on the conversion efficiency curve are caused by the respective resonances of the two floats. Next, we analyze the influence of the resonant frequencies of the floats on the capture width ratio of the WEC. Here, the coupled effects are not considered. The natural frequencies of the two floats are derived from the undamped linear equation for free motion in the frequency domain, as follows:
where μ_{11} and μ_{12} are the added masses of the two floats as they move in the heave direction, which we calculated using HydroStar.
When K_{pto} = 4 × 10^{5} N/m, the natural frequency of the upper float is 2.07 and that of the underwater float is 0.58. The natural frequency of the lower float is near the first peak frequency of 0.48 in Fig. 4a, but the natural frequency of the upper float is much larger than the 0.75 value. This is because K_{pto} is a coupled term associated with the relative motion between the two floats, which can be seen from Eq. (47). If the coupled effect is neglected, the numerator in Eq. (47) is larger than its exact value. Therefore, the calculated natural frequency is larger than the peak frequency in Fig. 4. On the other hand, the mass and added mass of the upper float in the denominator is much smaller than those of the lower float, so the influence of the coupled effects is much greater. From Eq. (47), we find that the natural frequency is related to the mass, added mass, restoration coefficient, PTO stiffness coefficient, and mooring stiffness coefficient. Zhang et al. (2016) studied the influence of the geometry and mass of the WEC on conversion efficiency. In the following, we vary the PTO stiffness coefficient K_{pto} and mooring stiffness coefficient K_{m} to modify the natural frequencies of the floats to investigate their influence on the conversion efficiency of the WEC.
Effect of K _{pto} on Conversion Efficiency
In this section, we analyze the effect of K_{pto} on the conversion efficiency of the WEC. If we let ε = K_{pto} / (4 × 10^{5} N/m), when B_{pto} = 4 × 10^{5} N · s/m and K_{m} = 0, and consider ε = 0, 1, and 2, in Fig. 5, it shows the corresponding absorption powers, conversion efficiencies, and motion responses of the two floats. When ε = 1 or 2, a sharp peak appears in the lowfrequency regime (Fig. 5a). When ε = 0, a gentle peak appears in the highfrequency regime (Fig. 5c). In Fig. 5c, the displacements of the upper and lower floats have no peak but their relative displacements are large. The reason for this is that when ε = 0, no apparent resonance occurs and the peaks of the absorption power and capture width ratio occur when the relative motion between the floats is large; when ε = 1 or 2, a large K_{pto} leads to a large resonant frequency of the lower float, and sharp peaks in the absorption power and corresponding displacements in the two floats occur. The lower float is resonant in the lowfrequency regime, which leads to sharp peaks in the absorption power and corresponding upper and lowerfloat displacements. The absorption power curves in Fig. 5a and the capture width ratio in Fig. 5b exhibit similar trends. When ε = 1, the conversion efficiency in the lowfrequency regime is greater than that at ε = 0, but the peak of the conversion efficiency still occurs in the highfrequency regime. This is because the conversion efficiency is related to the incident wave power in addition to the absorption power.
At high frequency with ω > 0.75, the absorption power and capture width ratio decrease with increases in ε (Fig. 5a and b). The relative displacements of the two floats decrease with increases in ε (Fig. 5c, d, e). This is because as the PTO stiffness increases, the motions of the two floats are similar, which leads to decreases in the absorption power.
Effect of K _{m} on Response and Conversion Efficiency
In this section, we analyze the effect K_{m} on the conversion efficiency of the WEC. If we define γ = K_{m} / (4 × 10^{5} N/m), when B_{pto} = 4 × 10^{5} N · s/m and K_{pto} = 2 × 10^{4} N/m, and let γ = 0.5, 1, and 1.5, in Fig. 6a and b, it shows the corresponding absorption powers and conversion efficiencies of the WEC. Both the absorption power curve and conversion efficiency curve have two peak values. The first peak value increases as γ increases and the corresponding peak frequency becomes slightly higher. The second peak value increases slightly as γ increases and the corresponding peak frequency does not change. This is because in Eq. (47), an increase in K_{m} directly leads to an increase in the natural frequency of the lower float. Therefore, the first peak frequency becomes higher but the second peak frequency does not change as γ increases. An increase in the K_{m} value leads to an increase in the constraint force exerted on the lower float, which prevents the lower float from moving with the upper float. Therefore, the relative motion between the two floats increases and both peak values increase as γ increases. According to Fig. 5a and c, when K_{pto} is small, the peak value appearing at ω = 1.1 rad/s is generated by the large relative motion between the two floats, whereas the first peak value is generated by the resonance of the lower float. Therefore, as γ increases, the increase in the first peak value is much larger than that of the second peak value. When designing WECs, K_{m} can be made large to increase the natural frequency of the lower float and thus guarantee a greater absorption power over a wider spectrum.
Conclusions

(1)
When the PTO damping coefficient is at its optimal value B_{opt}, there is a relationship between the conversion efficiency of the WEC and the natural frequencies of the two floats. Two peak values occur when the two floats are resonant. When the lower float is resonant, the displacements of both floats are too large for the purposes of engineering. On the other hand, when the upper float is resonant, the displacements of both floats are within 1 m. Therefore, when designing a WEC, it is better to make the upper float resonant or increase the mooring stiffness of the lower float to confine its resonant motion.

(2)
The PTO stiffness coefficient K_{pto} greatly influences the natural frequency of the lower float. As K_{pto} increases, the natural frequency of the lower float increases and the peak value of the absorption power moves to a lowfrequency regime. In a highfrequency regime, as K_{pto} increases, the conversion efficiency decreases. Thus, a proper K_{pto} is needed to confine the conversion efficiency to a lowfrequency regime and to avoid reducing a highfrequency regime too much.

(3)
The mooring stiffness coefficient K_{m} has a direct influence on the natural frequency of the lower float but has no apparent influence on the upper float. When designing a WEC, if the geometric parameters of the upper and lower floats are defined, K_{m} can be used to adjust the natural frequency of the lower float.
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Funding
This work was supported by the National Natural Science Foundation of China (51409066, 51761135013), High Technology Ship Scientific Research Project from the Ministry of Industry and Information Technology of the People’s Republic of China–Floating Security Platform Project (the second stage, 201622), and the Fundamental Research Fund for the Central University (HEUCFJ180104, HEUCFP1809).
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Article Highlights
•The optimal PTO damping is analytically derived for a doublefloat WEC system.
•Through actively controlled generator damping, a doublefloat WEC achieves larger capture width ratio.
•Mooring stiffness is used to tune the natural frequency of the doublefloat WEC.
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Zhang, L., Jin, P., Zhou, B. et al. Oscillation and Conversion Performance of DoubleFloat Wave Energy Converter. J. Marine. Sci. Appl. 18, 54–63 (2019). https://doi.org/10.1007/s11804019000839
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Keywords
 Doublefloat WEC
 Energy conversion
 Capture width ratio
 Optimal damping
 Resonance