Performance analysis of the oscillating-body wave energy converter with a novel bistable mechanism under irregular wave conditions

Abstract

The present study investigates the application of a novel nonlinear bistable mechanism to the oscillating-body wave energy converter (WEC) under irregular wave conditions. The numerical simulations of the linear and nonlinear WEC devices are performed with the hydrodynamic analysis software AQWA and coding program MATLAB. Firstly, the hydrodynamic model of the WEC buoy is validated. Then, a parametric analysis on the energy conversion of the nonlinear WEC is conducted concerning the properties of power take-off (PTO) devices and wave parameters. The findings reveal that each parameter has a significant influence on the nonlinear WEC’s performance under irregular waves. Therefore, the parametric study should be carried out to determine the device parameters, which ensures that the nonlinear WEC outperforms the linear WEC under the specific environmental conditions. Moreover, it is found that the WEC with the proposed bistable system could absorb more power at the low-frequency region, which addresses the issue of efficiently capturing low-frequency wave energy.

Introduction

Because of serious environmental pollution and huge consumption of fossil fuels, the world is paying increasing attention to the sustainable development. ¹ For sustainable development, the exploitation of ocean renewable energy has become a key issue. Wave energy is considered to have more potential than wind, tidal and solar energy. ² This is because the waves have a higher energy density and the all-day availability. The WEC devices have been developed in various types, such as oscillating water columns, overtopping devices, oscillating-body WECs, bottom-hinged WECs and multi-module interconnected WECs. ³ Among these WEC devices, the oscillating-body WEC is more promising due to its low construction and installation costs, and simple energy conversion mechanism. The nature of the energy conversion for an oscillating-body WEC is to drive the PTO system by relative motion about a fixed or floating reference. Generally, a direct drive or a hydraulic device is utilized to fabricate the PTO system for an oscillating-body WEC. For this case, a linear system with one spring and one damper could be considered as the PTO device.

The excitation frequency of waves has a significant influence on the linear oscillating systems, so the desired power capture efficiency is not achieved in real sea conditions. Therefore, to optimize the absorption performance of the oscillating-body WECs, some researchers have investigated the geometrical and physical properties. Ruezga and Canedo ⁴ conducted an investigation on the improvement of linear WEC’s performance by adjusting the shape of a reference cylindrical buoy. The essence of the study was to make the buoy’s resonant frequency same with the dominant frequency of incident wave. Berenjkoob et al. ⁵ conducted an analysis on how the physical properties of two mooring configurations impact the energy conversion of a two-body oscillating WEC. They found that the tension leg system increased the relative motion of this two-body device, which resulted in the enhancement of energy conversion of the WEC. Al Shami et al. ⁶ assessed the effects of degrees of freedom (DoFs) of an oscillating-body WEC on its capture absorption. The focus was to strengthen the power harvesting efficiency of WEC at low frequencies of waves. In addition, some other researchers also conducted the analysis on the control strategies to improve the WEC capture performance. Different control strategies were developed, including declutching control, ⁷ latching control, ⁸ and model predictive control. ⁹ However, as implied in the reference, ¹⁰ their effectiveness for enhancing the energy conversion efficiency of WEC was dependent on a precise forecasting of wave elevation. The control performance was unsatisfactory if there existed a prediction deviation.

In the past years, the nonlinear mechanisms have gradually been explored to optimize the power absorption efficiency of oscillating-body WECs. Compared with the linear systems, multi-stable mechanism is the feature of the nonlinear systems. The vibration energy harvesting was initially investigated based on the stability mechanism by Tseng and Dugundji. ¹¹ Currently, one widely used stability mechanism applied to WEC devices is the bistable mechanism. It is also called the negative stiffness or snap-through mechanism, which bears some unique features: (1) its potential function has two wells; (2) its forms of motions include inter-well, chaotic, and intra-well oscillations; and (3) negative stiffness is the result of the inter-well motion. Zhang and Yang ¹² adopted a bistable system to evaluate the absorption performance of a heaving WEC and concluded that it could enable the WEC device to absorb more power under low-frequency excitation forces. Xiao et al. ¹³ adopted magnets and coils to implement a bistable device for an oscillating-body WEC, exploring the effects of the mass ratio, spring parameter, and stable equilibrium position on the power absorption efficiency. After a bistable PTO system was installed in an oscillating-body WEC, Wu et al. ¹⁴ evaluated the effects of the PTO damping, the geometry dimensions, and the system stiffness. Zhang et al. ¹⁵ introduced the compact magnetic bistable device aimed at enhancing the power harvesting of an oscillating-body WEC, which was composed of two permanent coaxial magnetic rings with identical field direction. Li et al. ¹⁶ introduced an innovative snap-through mechanism, including four oblique springs and a damper, and implemented it in an oscillating-body WEC. Schubert et al. ¹⁷ explored a three-DoF submerged oscillating-body WEC with the bistable structure under irregular and regular wave conditions. The snap-through system was implemented through an adjustable magnetic device, which allowed for the examination and parameterization of a variety of potential profiles. Liu et al. ¹⁸ developed an improved nonlinear snap-through mechanism with a lower potential well, which enhanced the frequency bandwidth and the capture width ratio (CWR) of an oscillating-body WEC. In the study, ¹⁹ the objective was to elucidate the impact of the bistable mechanism on the point-absorber WEC’s effective bandwidth with the perturbation methods, which were employed to explore its complex motion subjected to harmonic wave forces. Jin et al. ²⁰ conducted an investigation into an array of buoys with a nonlinear snap-through mechanism, of which the aim was to achieve both energy extraction and attenuation of ocean waves at the low-frequency region. To improve the power harvesting of an oscillating multi-body WEC device, a nonlinear bistable structure was proposed by Wang et al. ²¹ Zhou et al. ²² applied an adaptive bistable PTO system (rack-pinion-lever-spring) to an oscillating-body WEC, which addressed the issues of low absorption performance and narrow bandwidth of conventional WECs. Lu et al. ²³ designed an innovative oscillating-body WEC with a mechanical motion rectifier (MMR) and a bistable device to investigate its power absorption performance under regular wave conditions.

The development of innovative WECs capable of efficiently capturing low-frequency wave energy holds significant importance. In the present study, a novel bistable PTO system for the oscillating-body WEC is introduced with the aim of effectively harnessing wave energy at the low-frequency region. Then, a parametric study of the WEC with the snap-through system is conducted, regarding the significant wave heights, as well as the parameters of the springs and damper in the PTO system. The reminder of this paper is organized as: Section 2 develops the mathematical models of the oscillating-body WEC with the linear and nonlinear PTO systems under irregular waves. The hydrodynamic model of the WEC buoy is validated in Section 3. Section 4 presents the results and discussion of the parametric study of the nonlinear WEC, regarding the power absorption performance. Some key findings are summarized in Section 5.

Theoretical background

Figure 1 illustrates a linear and nonlinear oscillating-body hemispherical WEC. The radius $(R)$ of the WEC buoy is 5 m. The PTO device is assumed to be a combination of one linear damper (with the damping of $C$ ) and one linear spring (with the stiffness of $K$ ). Unlike the linear one, the bistable PTO device is composed of four symmetric oblique springs (with the stiffness of $k_0$ , and $K=4k_0$ ) and one linear damper (with the damping of $C$ ). Among the nonlinear WEC, each spring has an original length $l_0$ of 1 m at the absence of compression and tension, and its oblique distance $h$ is 0.1 m. In addition, the projected length of the oblique spring in the horizontal direction is assumed to be variable $l$ .

Figure 1.

Schematic diagram of an oscillating-body hemispherical WEC with: (a) a linear PTO device and (b) a nonlinear PTO device.

The oscillating-body WEC only has the heave DoF $(z)$ . Based on Figure 1, the restoring force and potential energy of the snap-through system are written as respectively,

$$\begin{array}{c}\multicolumn{1}{c}{F=2*\frac{K}{4}(\sqrt{{(z-h)}^2+l^2}-l_0)\frac{z-h}{\sqrt{{(z-h)}^2+l^2}}}\\ +2*\frac{K}{4}(\sqrt{{(z+h)}^2+l^2}-l_0)\frac{z+h}{\sqrt{{(z+h)}^2+l^2}}\hfill \\ =\frac{K}{2}(1-\frac{l_0}{\sqrt{{(z-h)}^2+l^2}})(z-h)\hfill \\ +\frac{K}{2}(1-\frac{l_0}{\sqrt{{(z+h)}^2+l^2}})(z+h)\hfill \end{array}$$

(1)

$$\begin{array}{c}\multicolumn{1}{c}{E=2*\frac{1}{2}*\frac{K}{4}{[\sqrt{{(z-h)}^2+l^2}-l_0]}^2}\\ +2*\frac{1}{2}*\frac{K}{4}{[\sqrt{{(z+h)}^2+l^2}-l_0]}^2\hfill \\ =\frac{K}{4}\{{[\sqrt{{(z-h)}^2+l^2}-l_0]}^2+{[\sqrt{{(z+h)}^2+l^2}-l_0]}^2\}\hfill \end{array}$$

(2)

In the present study, a nonlinear parameter $\alpha$ is assumed to be $\sqrt{l^2+h^2}/l_0$ , which represents the initial condition of the spring system. $\alpha$ is substituted into equations (1) and (2), and they are transformed into a dimensionless form, that is:

$$\begin{array}{c}\multicolumn{1}{c}{F^{*}=2z^{*}-\frac{z^{*}-h^{*}}{\sqrt{{(z^{*}-h^{*})}^2+{\alpha }^2-{(h^{*})}^2}}}\\ -\frac{z^{*}+h^{*}}{\sqrt{{(z^{*}+h^{*})}^2+{\alpha }^2-{(h^{*})}^2}}\hfill \end{array}$$

(3)

$$\begin{array}{c}\multicolumn{1}{c}{E^{*}={(\sqrt{{(z^{*}-h^{*})}^2+{\alpha }^2-{(h^{*})}^2}-1)}^2}\\ +{(\sqrt{{(z^{*}+h^{*})}^2+{\alpha }^2-{(h^{*})}^2}-1)}^2\hfill \end{array}$$

(4)

where $F^{*}=F/(K{l}_0/2)$ , $E^{*}=E/(K{l_0}^2/4)$ , and $z^{*}=z/l_0$ .

According to equations (3) and (4), Figure 2 shows a diagram of the potential energy and restoring force of the bistable PTO device versus the heave displacement with a dimensionless form. Since the focus of the present study is to explore the energy conversion of the WEC with a bistable device, the nonlinear parameter $\alpha$ is chosen to be in the range of (0,1). Given the original length $l_0$ and the oblique distance $h$ of the spring above, five case studies are chosen ( $\alpha$  = 0.15, 0.3, 0.5, 0.7, and 0.9).

Figure 2.

Dimensionless analysis with the heave displacement: (a) restoring force and (b) potential energy.

From Figure 2(a), it can be observed that for 0 <  $\alpha$  < 1, there exists a displacement region that makes the equivalent stiffness $K^{*}=d{F}^{*}/d{z}^{*}$ of the nonlinear snap-through mechanism negative, which implies that the combined force generated by the deformation of the four springs is in the same direction with the heave displacement. Moreover, the value of $\alpha$ determines the magnitude of the equivalent negative stiffness. The larger value means the smaller negative stiffness. From Figure 2(b), it can be found that the snap-through system is characterized by the bistable condition, as it has two potential wells. For the three equilibrium positions, it is obvious that the first and third ones are stable and the middle one is unstable. Therefore, two forms of motion may occur in this nonlinear system, which are the local motion and global motion. For the local motion, the buoy only moves within either of the wells. This implies that the external excitation force is relatively small and it is difficult for the buoy to jump from one well to the other. The global motion occurs when the external excitation force is large, which means the buoy is not trapped inside either of the wells, and it can cross the wells back and forth.

For an oscillating-body WEC with a linear PTO device, its motion equation can be defined as,

$$\begin{array}{c}\{-{\omega }^2[m+A_z(\omega )]+i\omega [B_z(\omega )+C_{\mathrm{pto}}]+\rho \mathrm{gS}+K_{\mathrm{pto}}\}\hfill \\ Z(i\omega )=a(\omega )\Gamma (i\omega )\hfill \end{array}$$

(5)

where $m=\frac{2\pi }{3}\rho R^3$ is the mass, where $\rho$ is the density of sea water (1025 kg.m⁻³); $C_{\mathrm{pto}}$ and $K_{\mathrm{pto}}$ are the PTO equivalent damping and stiffness coefficients, respectively; $B_z(\omega )$ and $A_z(\omega )$ are the radiation damping and added mass coefficients, respectively; $\rho \mathrm{gS}$ is the hydrostatic stiffness in the heave DoF, where $g$ is the gravity acceleration (9.81 m.s⁻²), and $S=\pi R^2$ is the waterline area of the WEC buoy; $a(\omega )$ and $\Gamma (i\omega )$ refer to the wave amplitude and wave excitation force coefficient, respectively. By solving equation (5), the heave displacement of the WEC buoy is,

$$Z(i\omega )=\frac{a(\omega )\Gamma (i\omega )}{-{\omega }^2[m+A_z(\omega )]+i\omega [B_z(\omega )+C_{\mathrm{pto}}]+\rho \mathrm{gS}+K_{\mathrm{pto}}}$$

(6)

Evans ²⁴ derived the mean captured power of the linear WEC at the circular frequency of $\omega$ ,

$$\begin{array}{c}\multicolumn{1}{c}{P=\frac{1}{2}{\omega }^2{C}_{\mathrm{pto}}{|Z(i\omega )|}^2=\frac{1}{8B_z(\omega )}{|a(\omega )\Gamma (i\omega )|}^2}\\ -\frac{B_z(\omega )}{2}{|V-\frac{a(\omega )\Gamma (i\omega )}{2B_z(\omega )}|}^2\hfill \end{array}$$

(7)

where $V=i\omega Z(i\omega )$ refers to the velocity complex amplitude. What can be inferred is that when $V=a(\omega )\Gamma (i\omega )/2B_z(\omega )$ , the WEC captures the maximum power. Then it is substituted into equation (6),

$$\omega ={\left[\frac{\rho \mathrm{gS}+K_{\mathrm{pto}}}{m+A_z(\omega )}\right]}^{\frac{1}{2}}$$

(8)

$$C_{\mathrm{pto}}=B_z(\omega )$$

(9)

Equation (8) indicates that the optimal power absorption occurs at the resonant condition. Equation (9) suggests that to absorb the maximum power, $C_{\mathrm{pto}}$ needs to be same with the radiation damping at the natural frequency. Furthermore, Vicente et al., ²⁵ indicated that the PTO stiffness of a linear WEC device was generally taken as 10% of the hydrostatic stiffness of the buoy. Therefore, $K_{\mathrm{pto}}$ is set to be 78,973 kg/s² in this study. As for the optimal PTO damping $C_{\mathrm{opt}}$ , it will be given in the next section.

In the present study, the irregular waves are simulated through the JONSWAP spectrum, which is expressed as, ²⁶

$$S(\omega )=\epsilon H_s^2\frac{{\omega }^{-5}}{{\omega }_p^{-4}}\exp [-1.25{(\omega /{\omega }_p)}^{-4}]{\gamma }^{\exp [-\frac{{(\omega -{\omega }_p)}^2}{2{\sigma }^2{\omega }_p^2}]}$$

(10)

where ${\omega }_p$ and $H_s$ refer to the spectral peak frequency and significant wave height, respectively; $\gamma$ is the enhancement factor; $\sigma$ denotes the shape parameter, which can be defined as,

$$\sigma =\{\begin{array}{c}0.07,\omega >{\omega }_p\hfill \\ \multicolumn{1}{c}{0.09,\omega <{\omega }_p}\end{array}$$

(11)

And $\epsilon$ is the spectral parameter, and its value is dependent on $\gamma$ ,

$$\epsilon =\frac{0.0624}{0.230+0.0336\gamma -0.185(1.9+\gamma )}$$

(12)

Furthermore, the irregular wave consists of multiple regular components. Wherein, the amplitude of the $j$ -th component can be expressed as,

$$A({\omega }_j)=\sqrt{2S({\omega }_j)\mathrm{\Delta }\omega }$$

(13)

where $\mathrm{\Delta }\omega$ is the interval between two successive frequencies.

Therefore, the wave excitation forces of the WEC buoy under irregular wave condition are expressed as:

$$F_{\mathrm{ex}}(t)=\sum _{j=1}^{N}A({\omega }_j)\Gamma ({\omega }_j)\cos ({\omega }_{j}t+{\phi }_j+{\theta }_j)$$

(14)

where $\Gamma ({\omega }_j)$ denotes the wave excitation force coefficient, and ${\phi }_j$ is the corresponding phase; ${\theta }_j$ is the random phase in the range of $[0,2\pi ]$ . For an axisymmetric buoy under the deepwater condition, $\Gamma ({\omega }_j)$ can be obtained based on the Haskind’s relation, ²⁷

$${\Gamma }_z(\omega )={\left(\frac{2\rho g^3{B}_z(\omega )}{{\omega }^3}\right)}^{1/2}$$

(15)

The frequency-domain simulation is not appropriate for the analysis of the nonlinear systems under irregular wave conditions. Therefore, the time-domain simulation is utilized to analyze the performance of an oscillating-body WEC with the snap-through device. The heave motion equations of the nonlinear and linear WECs can be formulated as equations (16) and (17), respectively ²⁸ :

$$\begin{array}{c}[m+A_z(\infty )]\stackrel{··}{z}(t)+\underset{0}{\overset{t}{\int }}{k}_z(t-\tau )\stackrel{·}{z}(\tau )d\tau +C_{\mathrm{pto}}\stackrel{·}{z}(t)\hfill \\ +\rho \mathrm{gSz}(t)+\frac{K_{\mathrm{pto}}}{2}[1-\frac{l_0}{\sqrt{{(z(t)-h)}^2+l^2}}]\hfill \\ [z(t)-h]+\frac{K_{\mathrm{pto}}}{2}[1-\frac{l_0}{\sqrt{{(z(t)+h)}^2+l^2}}]\hfill \\ [z(t)+h]=F_{\mathrm{ex}}(t)\hfill \end{array}$$

(16)

$$\begin{array}{c}[m+A_z(\infty )]\stackrel{··}{z}(t)+\underset{0}{\overset{t}{\int }}{k}_z(t-\tau )\stackrel{·}{z}(\tau )d\tau +C_{\mathrm{pto}}\stackrel{·}{z}(t)\hfill \\ +(\rho \mathrm{gS}+K_{\mathrm{pto}})z(t)=F_{\mathrm{ex}}(t)\hfill \end{array}$$

(17)

where $k_z(t)$ is the retardation function, which refers to the fluid memory effects,

$$k_z(t)=\frac{2}{\pi }\underset{0}{\overset{\infty }{\int }}{B}_z(\omega )\cos \omega \mathrm{td}\omega$$

(18)

Before solving equations (16) and (17), the trapezoidal method is adopted to discretize the integral part,

$$\begin{array}{c}\multicolumn{1}{c}{\underset{0}{\overset{t}{\int }}{k}_z(t-\tau )\stackrel{·}{z}(\tau )d\tau =\frac{1}{2}{k}_z(t)\stackrel{·}{z}(0)\mathrm{\Delta }\tau +\frac{1}{2}{k}_z(0)\stackrel{·}{z}(t)\mathrm{\Delta }\tau }\\ +\sum _{i=1}^{n-1}[k_z(t-i·\mathrm{\Delta }\tau )\stackrel{·}{z}(i·\mathrm{\Delta }\tau )·\mathrm{\Delta }\tau ]\hfill \end{array}$$

(19)

where $t=n\mathrm{\Delta }\tau$ .

Then, equation (19) is substituted into equations (16) and (17), respectively,

$$\begin{array}{c}[m+A_z(\infty )]\stackrel{··}{z}(t)+[\frac{1}{2}{k}_z(0)\mathrm{\Delta }\tau +C_{\mathrm{pto}}]\stackrel{·}{z}(t)+\rho \mathrm{gSz}(t)\hfill \\ +\frac{K_{\mathrm{pto}}}{2}[1-\frac{l_0}{\sqrt{{(z(t)-h)}^2+l^2}}][z(t)-h]\hfill \\ +\frac{K_{\mathrm{pto}}}{2}[-\frac{l_0}{\sqrt{{(z(t)+h)}^2+l^2}}][z(t)+h]=F_{\mathrm{ex}}(t)\hfill \\ -\frac{1}{2}{k}_z(t)\stackrel{·}{z}(0)\mathrm{\Delta }\tau -\sum _{i=1}^{n-1}[k_z(t-i·\mathrm{\Delta }\tau )\stackrel{·}{z}(i·\mathrm{\Delta }\tau )·\mathrm{\Delta }\tau ]\hfill \end{array}$$

(20)

$$\begin{array}{c}[m+A_z(\infty )]\stackrel{··}{z}(t)+[\frac{1}{2}{k}_z(0)\mathrm{\Delta }\tau +C_{\mathrm{pto}}]\stackrel{·}{z}(t)\hfill \\ +(\rho \mathrm{gS}+K_{\mathrm{pto}})z(t)=F_{\mathrm{ex}}(t)-\frac{1}{2}{k}_z(t)\stackrel{·}{z}(0)\mathrm{\Delta }\tau \hfill \\ -\sum _{i=1}^{n-1}[k_z(t-i·\mathrm{\Delta }\tau )\stackrel{·}{z}(i·\mathrm{\Delta }\tau )·\mathrm{\Delta }\tau ]\hfill \end{array}$$

(21)

For the convenience of solution, equations (20) and (21) are organized as:

$$\begin{array}{c}\stackrel{··}{z}(t)=\tilde{A}\stackrel{·}{z}(t)+{\tilde{B}}_{1}z(t)+\tilde{C}\hfill \\ \{[1-\frac{l_0}{\sqrt{{(z(t)-h)}^2+l^2}}][z(t)-h]+[1-\frac{l_0}{\sqrt{{(z(t)+h)}^2+l^2}}][z(t)+h]\}\hfill \\ +\tilde{D}(t)\hfill \end{array}$$

(22)

$$\stackrel{··}{z}(t)=\tilde{A}\stackrel{·}{z}(t)+{\tilde{B}}_{2}z(t)+\tilde{D}(t)$$

(23)

where,

$$\tilde{A}=-\frac{1}{[m+A_z(\infty )]}[\frac{1}{2}{k}_z(0)\mathrm{\Delta }\tau +C_{\mathrm{pto}}]$$

(24)

$${\tilde{B}}_1=-\frac{1}{[m+A_z(\infty )]}·\rho \mathrm{gS}$$

(25)

$${\tilde{B}}_2=-\frac{1}{[m+A_z(\infty )]}·(\rho \mathrm{gS}+K_{\mathrm{pto}})$$

(26)

$$\tilde{C}=-\frac{1}{[m+A_z(\infty )]}·\frac{K_{\mathrm{pto}}}{2}$$

(27)

$$\begin{array}{c}\tilde{D}=-\frac{1}{[m+A_z(\infty )]}·\hfill \\ \{F_{\mathrm{ex}}(t)-\frac{1}{2}{k}_z(t)\stackrel{·}{z}(0)\mathrm{\Delta }\tau -\sum _{i=1}^{n-1}[k_z(t-i·\mathrm{\Delta }\tau )\stackrel{·}{z}(i·\mathrm{\Delta }\tau )·\mathrm{\Delta }\tau ]\}\hfill \end{array}$$

(28)

Furthermore, assuming $X={\left[\begin{array}{cc}{X}_1\hfill & X_2\hfill \end{array}\right]}^T={\left[\begin{array}{cc}z\hfill & \stackrel{·}{z}\hfill \end{array}\right]}^T$ , equations (22) and (23) can be written as:

$$\{\begin{array}{l}{\stackrel{·}{X}}_1=X_2\\ {\stackrel{·}{X}}_2=\tilde{A}{X}_2+{\tilde{B}}_1{X}_1+\tilde{C}\left\{\left[1-\frac{l_0}{\sqrt{{(X_1-h)}^2+l^2}}\right][X_1-h]+\left[1-\frac{l_0}{\sqrt{{(X_1+h)}^2+l^2}}\right][X_1+h]\right\}+\tilde{D}(t)\end{array}$$

(29)

$$\{\begin{array}{c}{\stackrel{·}{X}}_1=X_2\hfill \\ {\stackrel{·}{X}}_2=\tilde{A}{X}_2+{\tilde{B}}_2{X}_1+\tilde{D}(t)\hfill \end{array}$$

(30)

Finally, due to the fact that equations (28) and (29) are the first-order differential equations, they could be resolved with the fourth order Runge-Kutta method. After solving the motion equations of linear and nonlinear WECs, the mean captured power can be derived as,

$$P=\frac{1}{T}\underset{0}{\overset{T}{\int }}{C}_{\mathrm{pto}}{\stackrel{·}{z}}^2(t)\mathrm{dt}$$

(31)

where $T$ is the simulating time.

The present study is conducted with AQWA and MATLAB. In AQWA, the linear potential theory ²⁹ and boundary element method (BEM) ³⁰ are utilized to calculated the hydrodynamic coefficients of the buoy. In MATLAB, the motion equations of the linear and nonlinear WECs are firstly resolved, and then their captured power is obtained.

Validation of the WEC buoy’s hydrodynamic model

To make the present study reliable, the validation of the WEC buoy’s hydrodynamic model is carried out. The Buoy’s surface is discretized into 18,024 panel elements, which is then simulated in the frequency-domain module of AQWA. First of all, the mesh convergence of the WEC model can be verified with the Near-field and Far-field methods. ³¹ As shown in Figure 3, the surge steady drift forces agree well under these two methods. Then, Figure 4 illustrates the Buoy’s hydrodynamic coefficients in the heave DoF, including the wave excitation force, radiation damping and added mass. Hulme ³² presented the results of the latter two objects of the heaving hemispherical buoy under the deepwater condition. The results of the wave excitation force are obtained based on the radiation damping and Haskind’s relation. It can be inferred from the figure that the hydrodynamic model of the buoy is validated. Furthermore, based on equations (8) and (9), the frequency-domain simulation obtains the Buoy’s natural frequency and the optimal damping ( $C_{\mathrm{opt}}$ ) of linear PTO device, which are 1.516 rad/s and 87,480 kg/s, respectively.

Figure 3.

Comparison of the surge steady drift forces between the near-field and far-field methods.

Figure 4.

(a) Added mass, (b) radiation damping, and (c) wave excitation force coefficients of the WEC buoy with the heave DoF.

Results and discussions

The time-domain approach is employed to analyze the performance of a linear WEC and a nonlinear WEC with the bistable PTO device. Then a parametric analysis of the energy conversion of the nonlinear WEC is conducted, with respect to the nonlinear parameter $\mathrm{\alpha }$ , the significant wave height $H_s$ , the original length $l_0$ and the oblique distance $h$ of the springs, and the PTO damping coefficient. It is worth noting that if $l_0$ , $h$ , $K_{\mathrm{pto}}$ , and $C_{\mathrm{pto}}$ are not specifically mentioned in the following analysis, they remain consistent with the values given in Sections 2 and 3. In addition, the time length of each simulation is 4600 s, and the mean captured power of the WEC is calculated with the results of the latter 3600 s.

Nonlinear parameter $\alpha$

The nonlinear parameter $\alpha$ is taken as 0.15, 0.3, 0.5, 0.7, and 0.9 for the nonlinear WEC. For the irregular wave, $H_s$ is chosen to 2 m. Meanwhile, ${\omega }_p$ is chosen with an interval of 0.2 rad/s in the range of 0.1–2.3 rad/s. Figure 5 demonstrates the power absorption of both linear and nonlinear WECs under irregular waves. Meanwhile, it also presents the factor $q_{\mathrm{ratio}}$ of the mean harvested power of the nonlinear WEC to that of the linear one.

Figure 5.

Power absorption performance of linear and nonlinear WECs: (a) mean power and (b) $q_{\mathrm{ratio}}$ .

Figure 5(a) shows that the linear WEC captures the most power when ${\omega }_p$  = 1.5 rad/s. And for the nonlinear WEC, it captures the most power when ${\omega }_p$  = 1.3 rad/s. Obviously, the mean power produced by the linear and nonlinear WECs attains the largest at the spectral peak frequency near the system’s resonant frequency. The nonlinear WEC absorbs more power compared with the linear one over a wider frequency range, except at the high-frequency region ( ${\omega }_p$  > 1.5 rad/s) where the linear WEC captures slightly more power compared to the nonlinear one. In addition, the variation of the nonlinear parameter $\alpha$ leads to a significant influence on the mean captured power of the WEC with snap-through system. The energy conversion of the nonlinear WEC weakens with the increase of $\alpha$ , as ${\omega }_p$  < 1.5 rad/s. However, at the high-frequency region ( ${\omega }_p$  > 1.5 rad/s), the nonlinear WEC absorbs more power for a larger value of $\alpha$ . It can be found in Figure 5(b) that the absorbed power of the nonlinear WEC reaches 1.2–1.5 times that captured by the linear one for various values of $\mathrm{\alpha }$ , when ${\omega }_p$  < 1.5 rad/s. However, in the high-frequency range ( ${\omega }_p$  > 1.5 rad/s), the nonlinear WEC only captures 0.4–0.9 times power of the linear one.

Significant wave height $H_s$

When the airy wave theory is utilized to analyze a linear WEC, the captured power has a proportional relationship against the square of wave amplitude or height. The irregular waves are obtained by the superposition of multiple regular components, indicating that the captured power of the linear WEC also shows proportionality against $H_s$ . However, this may not happen in the nonlinear WEC. For $H_s$ , the values are taken with an interval of 0.5 m in the range of 0.5–4 m. Meanwhile, three spectral peak frequencies ( ${\omega }_p$  = 0.3, 1.3, and 2.1 rad/s) are chosen to assess the effects of $H_s$ on the energy conversion of the nonlinear WEC, which are smaller than, identical with, and larger than the optimal frequency, respectively (as presented in Figure 5(a)). Figures 6 –8 illustrate the power absorption of the linear and nonlinear WECs under various values of $H_s$ . Meanwhile, they also give the ratio of the converted energy to the square of $H_s$ based on the case where the value of $H_s$ is 0.5 m.

Figure 6.

Power absorption of linear and nonlinear WECs under various $H_s$ with ${\omega }_p$  = 0.3 rad/s: (a) mean power and (b) $(P/P_{H_s=1/2})/(H_s/\frac{1}{2}{)}^2$ .

Figure 7.

Power absorption of linear and nonlinear WECs under various $H_s$ with ${\omega }_p$  = 1.3 rad/s: (a) mean power and (b) $(P/P_{H_s=1/2})/(H_s/\frac{1}{2}{)}^2$ .

Figure 8.

Power absorption of linear and nonlinear WECs under various $H_s$ with ${\omega }_p$  = 2.1 rad/s: (a) mean power and (b) $(P/P_{H_s=1/2})/(H_s/\frac{1}{2}{)}^2$ .

The relationship between the absorbed power of the WEC with snap-through system and the square of $H_s$ is no longer linear. As seen in Figure 6(b), the value of $(P/P_{H_s=1/2})$ is smaller than that of $(H_s/\frac{1}{2}{)}^2$ for the nonlinear WEC, as ${\omega }_p$  = 0.3 rad/s, which indicates that the increase in $H_s$ has less contributions to the captured power compared to the linear WEC. Furthermore, as $\mathrm{\alpha }$ increases, the value of $(P/P_{H_s=1/2})/(H_s/\frac{1}{2}{)}^2$ tends to 1. This is due to the fact that a larger $\mathrm{\alpha }$ attenuates the bistable characteristic of the nonlinear PTO system due to the lower barrier, which makes it approximate to the linear one. As presented in Figure 7(b), the value of $(P/P_{H_s=1/2})$ is still smaller than that of $(H_s/\frac{1}{2}{)}^2$ for the WEC with snap-through system, as ${\omega }_p$  = 1.3 rad/s. However, for the nonlinear WEC with $\alpha \le 0$ .3, the value of $(P/P_{H_s=1/2})/(H_s/\frac{1}{2}{)}^2$ is larger than that for ${\omega }_p$  = 0.3 rad/s. Figure 8(b) demonstrates that the value of $(P/P_{H_s=1/2})$ is larger than that of $(H_s/\frac{1}{2}{)}^2$ for the nonlinear WEC, as ${\omega }_p$  = 2.1 rad/s, which is opposite from the case as ${\omega }_p$  = 0.3 or 1.3 rad/s. This indicates that the increase in $H_s$ contributes more to the energy conversion of the nonlinear WEC compared to its linear counterpart. However, the mean harvested power of the WEC with snap-through system is still less than that of the linear one, as shown in Figure 8(a). As analyzed above, for nonlinear WEC, its power absorption performance with respect to $H_s$ depends on the value of ${\omega }_p$ .

Original length $l_0$ of the spring

As indicated in Section 2, the negative stiffness of the WEC with snap-through device is dependent on the nonlinear parameter $\alpha$ , which affects its power absorption performance. For a given $\alpha$ , the spring’s original length $l_0$ also affects the performance and the negative stiffness of the WEC with snap-through system. The value of $\alpha$ is taken as 0.15, while that of $l_0$ is chosen with an interval of 1 m in the range of 1–7 m. As for the irregular wave, three various spectral peak frequencies are considered ( ${\omega }_p$  = 0.3, 1.3, and 2.1 rad/s), and the value of $H_s$ is chosen to 2 m. Figure 9 shows the energy conversion of the WEC with snap-through system in the presence of different original lengths of the spring.

Figure 9.

Mean absorbed power of the nonlinear WEC with various $l_0$ of the spring under three different frequencies: (a) ${\omega }_p$  = 0.3, (b) ${\omega }_p$  = 1.3, and (c) ${\omega }_p$  = 2.1 rad/s.

From Figure 9, it can be observed that the variation trend of mean absorbed power of the nonlinear WEC regarding the spring’s original length $l_0$ is different at the three spectral peak frequencies. The mean absorbed power of the WEC with snap-through mechanism is approximate to increase linearly, as ${\omega }_p$  = 0.3 rad/s. This is because a larger $l_0$ leads to a larger negative stiffness, which is equivalent to the impact of decreasing the nonlinear parameter $\alpha$ in the low-frequency range. It can be observed in Figure 5 that, the WEC with negative stiffness harvests more power for a smaller $\alpha$ . When ${\omega }_p$  = 1.3 rad/s, the optimal spring’s original length is 4 m for the power absorption performance. However, the mean captured power of the WEC with negative stiffness has a downward trend with an increase in the value of $l_0$ , as ${\omega }_p$  = 2.1 rad/s, which is equivalent to the case that the decrease in the value of $\alpha$ weakens its power absorption performance in the high-frequency range, as demonstrated in Figure 5.

Oblique distance $h$ of the spring

From Figure 9, it can be inferred that the spring’s original length $l_0$ affects the negative stiffness of the WEC with snap-through system. For a given $\alpha$ , the oblique distance also has an impact on the performance and the negative stiffness of the nonlinear WEC. The values of $\alpha$ and $l_0$ are set to 0.15 and 7 m, respectively. The value of $h$ is chosen with an interval of 0.2 m in the range of 0.1–0.9 m. For the irregular waves, three various spectral peak frequencies are applied ( ${\omega }_p$  = 0.3, 1.3, and 2.1 rad/s). Meanwhile, the significant wave height ( $H_s$  = 2 m) remains constant. Figure 10 presents the energy conversion of the nonlinear WEC with various spring’s oblique distances.

Figure 10.

Mean power absorbed by the nonlinear WEC with various $h$ of the spring under three different frequencies: (a) ${\omega }_p$  = 0.3, (b) ${\omega }_p$  = 1.3, and (c) ${\omega }_p$  = 2.1 rad/s.

From Figure 10, it can be seen that regarding these three spectral peak frequencies, the oblique distance affects the power absorption performance in different ways for the WEC with negative stiffness. The mean captured power of the WEC has a decreasing trend with the value of $h$ , as ${\omega }_p$  = 0.3 rad/s. This is because a larger $h$ makes the oblique angle of the spring larger, which then reduces its negative stiffness, similar to increasing the nonlinear parameter $\alpha$ in the low-frequency range. As presented in Figure 5, the nonlinear WEC with a larger $\alpha$ captures less power. For ${\omega }_p$  = 1.3 rad/s, the optimal oblique distance is found to be 0.7 m. However, the mean absorbed power of the WEC with snap-through system has an upward tendency with the value of $h$ for the case ( ${\omega }_p$  = 2.1 rad/s), which is similar with the phenomenon that the increase in the value of $\alpha$ enhances its captured energy performance at the high-frequency region, as shown in Figure 5.

PTO damping coefficient

The PTO damping coefficient adopted in the above analysis is the optimal for the linear WEC under the regular waves. However, its impact on the nonlinear WEC possibly differs from those on the linear one. Therefore, multiple damping coefficients (0.5 $C_{\mathrm{opt}}$ , 0.8 $C_{\mathrm{opt}}$ , 1.5 $C_{\mathrm{opt}}$ , 2 $C_{\mathrm{opt}}$ , 3 $C_{\mathrm{opt}}$ , and 4 $C_{\mathrm{opt}}$ ) are taken to explore the energy conversion of the linear and nonlinear WECs under irregular wave conditions. The value of $H_s$ is chosen to 2 m. Meanwhile, the spectral peak frequency is chosen with an interval of 0.2 rad/s in the range of 0.1–2.3 rad/s. The nonlinear parameter $\alpha$ is configured to a value of 0.15 for the WEC with snap-through system. Figure 11 demonstrates the mean absorbed power of the linear and nonlinear WECs with various PTO damping coefficients.

Figure 11.

Mean absorbed power of the linear and nonlinear WECs with various PTO damping coefficients: (a) linear and (b) $\alpha$  = 0.15.

From Figure 11, it is inferred that the variation of PTO damping coefficient results in a similar impact on the power absorption of the linear and nonlinear WECs. The optimal frequency remains constant with a value of 1.5 rad/s, when the PTO damping coefficient is not larger than $C_{\mathrm{opt}}$ . The optimal frequency gradually becomes small with the PTO damping coefficient increasing from $C_{\mathrm{opt}}$ . Especially, the optimal frequency decreases to 0.9 rad/s, when the PTO damping coefficient is 4 $C_{\mathrm{opt}}$ . Furthermore, for the maximum power of the WEC, it has an upward trend across the PTO damping coefficient.

Conclusions

The present study focuses on the energy conversion performance of the oscillating WEC with a novel snap-through device under irregular waves. A parametric study on the nonlinear WEC is conducted concerning the nonlinear parameter $\alpha$ , the significant wave height $H_s$ , the original length $l_0$ , and the oblique distance $h$ of the spring, and the PTO damping coefficient $C_{\mathrm{pto}}$ . The hydrodynamic coefficients of the WEC buoy are calculated with AQWA, and MATLAB is used to develop the motion equations of the WECs with linear and nonlinear mechanism. Based on the results, some key findings are summarized:

• (1) The WEC with the snap-through system shows a better power harvesting performance than the linear WEC, under the irregular waves with ${\omega }_p$ smaller than the resonant frequency of the system. However, when ${\omega }_p$ is larger than the resonant frequency, the situation is opposite.

• (2) The nonlinear WEC with a smaller nonlinear parameter $\alpha$ captures more power in the low-frequency range. However, at the high-frequency region, the situation is opposite.

• (3) In terms of ${\omega }_p$ and $\alpha$ , the contribution of $H_s$ to the power absorption of the nonlinear WEC varies significantly under irregular waves.

• (4) The original length $l_0$ and the oblique distance $h$ of the spring affect the negative stiffness and the power harvesting efficiency of the nonlinear WEC. Moreover, they have the opposite influence on the nonlinear WEC either in the low or high-frequency ranges.

• (5) The impact of the PTO damping coefficient on the power harvesting of the WEC with snap-through system mirrors that on the linear WEC. With the increase in the PTO damping coefficient, the maximum power of the WEC becomes larger and the optimal frequency decreases.

Finally, there is no experimental data available to compare with the numerical results of this study. Therefore, some experimental tests will be conducted on the proposed bistable WEC in the near future. Meanwhile, the WEC shape will also be optimized to improve the power output.