Oscillation and Conversion Performance of Double-Float Wave Energy Converter
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In this study, we investigated the hydrodynamic and energy conversion performance of a double-float wave energy converter (WEC) based on the linear theory of water waves. The generator power take-off (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 higher-order boundary element method (HOBEM), we constructed a three-dimensional model of double-float WEC to study its hydrodynamic performance and response in the time domain. Only the heave motion of the two-body system is considered and a virtual function is introduced to decouple the motions of the floats. The energy conversion character of the double-float 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 double-float 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.
KeywordsDouble-float WEC Energy conversion Capture width ratio Optimal damping Resonance
Wave energy is an eco-friendly 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 double-float 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 one-body WEC. Two-body 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 two-body WEC as a one-body 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 floating-cylinder and fixed-cylinder 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 double-float WEC. Considering the PTO damping coefficient, stiffness coefficient, and mooring stiffness, we derive the optimal damping coefficient of the double-float WEC using the frequency domain method. Based on the potential flow theory, we apply the higher-order boundary element method (HOBEM) to construct a 3D model of the double-float 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.
2 Mathematical Model
3 Energy Conversion Efficiency
4 Optimal PTO Damping
5 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:
When using the acceleration potential method (Tanizawa 1995) to calculate ϕt 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 ϕt. The calculation of ϕt 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.
6 Numerical Results
Main parameters of device
6.2 Effect of B pto on Conversion Efficiency
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 Bopt at ω = 0.75 is much larger than Bopt 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.
When Kpto = 4 × 105 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 Kpto 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 Kpto and mooring stiffness coefficient Km to modify the natural frequencies of the floats to investigate their influence on the conversion efficiency of the WEC.
6.3 Effect of K pto on Conversion Efficiency
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.
6.4 Effect of K m on Response and Conversion Efficiency
When the PTO damping coefficient is at its optimal value Bopt, 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.
The PTO stiffness coefficient Kpto greatly influences the natural frequency of the lower float. As Kpto increases, the natural frequency of the lower float increases and the peak value of the absorption power moves to a low-frequency regime. In a high-frequency regime, as Kpto increases, the conversion efficiency decreases. Thus, a proper Kpto is needed to confine the conversion efficiency to a low-frequency regime and to avoid reducing a high-frequency regime too much.
The mooring stiffness coefficient Km 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, Km can be used to adjust the natural frequency of the lower float.
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).
- Basco RD (1984) Water wave mechanics for engineers and scientists. Prentice-Hall, New Jersey, 245–255. https://doi.org/10.1142/1232
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