Hydrodynamic Investigation of an Oscillating Buoy Wave Energy Converter Integrated into a Pile-Restrained Floating Breakwater

Abstract

An analytical model is developed based on linear potential flow theory and matching eigenfunction expansion technique to investigate the hydrodynamics of a two-dimensional floating structure. This structure is an integration system consisting of a breakwater and an oscillating buoy wave energy converter (WEC). It is constrained to heave motion, and linear power take-off (PTO) damping is used to calculate the absorbed power. The proposed model is verified against the published results. The proposed integrated structure is compared with the fixed structure and free heave-motion structure, respectively. The hydrodynamic properties of the integrated structure with the optimal PTO damping i.e., the transmission coefficient, reflection coefficient, capture width ratio (CWR), and heave response amplitude operator (RAO), are investigated. The effect of the PTO damping on the performance of the integrated system is also evaluated. Results indicate that with the proper adjustment of the PTO damping, the proposed integrated system can produce power efficiently. Meanwhile, the function of coastal protection can be compared with that of the fixed structure.

1. Introduction

As environmental concerns gain importance, more studies are being conducted on energy extraction from renewable energy resources. Ocean wave energy is a huge, largely untapped renewable energy resource, with the potential to attract researchers and engineers [1]. To date, a wide range of wave energy converters (WECs) has been developed [2]. However, the high construction cost still significantly impedes the industrial application of wave energy utilization [3]. Thus, an increasing number of developers have focused on reducing construction costs. Meanwhile, the greenness of the seaport has accordingly drawn considerable attention. Renewable energy utilization is regarded as one of the representative factors in evaluating the greenness of a seaport [4,5]. Combining the wave energy converter with the existing breakwater structures can simultaneously achieve wave energy utilization and wave attenuation. Cost-sharing between them can naturally lead to cost reduction. The benefits obtained from the integration of breakwaters and WECs over the isolated WECs can be seen in Mustapa et al. [6].

The breakwater–WEC integration includes two categories: fixed bottom-mounted structures and floating structures. Integrated systems, such as overtopping [7,8], oscillating water column-type (OWC type) [9,10] and piston-type WEC-breakwater integration [11], belong to the former category (for detailed descriptions see [6]). It is well understood that the floating breakwaters are favorable for their relatively low costs [12]. Recently, many attempts associated with integrations of WECs and floating breakwaters (including truncated surface-piecing breakwaters) have been made. He et al. [13] integrated OWC-type devices into slack-moored floating breakwaters with pneumatic chambers. Experimental results showed that functions of coastal protection and the wave energy utilization can be satisfied simultaneously for the proposed concept. Mendoza et al. [14] investigated the shore protection function provided by WEC farms and found that the multi-purpose use of WECs can be achieved. Chen et al. [15] numerically investigated the hydrodynamic performance of floating horizontal cylinders as both WECs and floating breakwaters; results indicated that a configuration with small cylinders in groups may achieve the two functions simultaneously. Martinelli et al. [16] experimentally investigated the performance of a hybrid structure consisting of an “active” floating breakwater and a WEC (named ShoWED); results showed that the hybrid structure can successfully generate both electrical energy and coastal protection. Ning et al. [17] proposed a novel integrated system of a vertical pile-restrained floating breakwater that operates under the principle of an oscillating buoy WEC. The integrated system has a simple structural configuration. The experimental study showed that acceptable wave attenuation performance and energy-conversion efficiency can be obtained if the appropriate structure dimensions and power take-off (PTO) damping force are obtained. It is understood that the floating breakwaters are often used in areas characterized by low wave energy. Thus, the high performance of the device in energy conversion can determine the engineering interest.

Prompted by [17], the current study aims at theoretically revealing the relationship among the reflection coefficient, transmission coefficient, and capture width ratio (CWR) of the WEC-type breakwater. The hydrodynamics of the integrated system with the optimal PTO damping, fixed breakwater, and free heave-motion breakwater are also compared. In addition, the effect of the PTO damping on the hydrodynamic performance of the integrated device is evaluated. The present study is conducted based on potential flow theory and the assumption of breakwater undergoing heave motion with small response amplitude. The matching eigenfunction expansion method is used to solve diffraction and radiation problems. The exciting force and hydrodynamic coefficients in heave mode are computed based on [18]. The reflection coefficient K_r, transmission coefficient K_t, response amplitude operator (RAO) ξ in heave mode, and CWR η can then be derived. Note that the equation of K_r² + K_t² + η = 1 is satisfied based on the rule of energy conservation.

This study is organized as follows: Section 2 describes the formulas. Section 3 gives the dimensional analysis. Section 4 presents the validation, results, and discussions. Section 5 provides the conclusions.

2. Formulas

As shown in Figure 1, a pontoon-type structure (i.e., breakwater) with a width of B = 2a and a draft d₁ is situated in the water with a uniform depth h₁. Similar to the description in [18], a 2-dimensional Cartesian coordinate (o-xz) system is employed, and the center of origin is located at the cross-point of the still water plane and medial axis of the breakwater. Correspondingly, the mass term and stiffness term of the breakwater in heave mode can be expressed as M (=2ρad₁) and K (=2ρga), where ρ denotes the density of water, and g represents the gravitational acceleration. The structure is subjected to a train of regular waves traveling in the positive x-direction. A is the incident wave amplitude, which is the maximum distance of a water particle from its equilibrium position during a period, and L is the wavelength, which is the distance that the wave travels during a wave period. The structure is assumed to respond only in heave mode.

As indicated in Figure 1, the fluid domain is divided into three subdomains—i.e., Ω₁, Ω₂, and Ω₃. The fluid motion in the whole domain can be described by the velocity potential:

$$\varphi \left(x,z,t\right)=\mathrm{Re}\left[\mathsf{\Phi }\left(x,z\right)e^{-\mathrm{i}\omega t}\right]$$

(1)

where t denotes time, i = $\sqrt{-1}$, ω represents the angular frequency, Re [ ] denotes the real part of a complex expression and Φ is a complex spatial velocity potential, which satisfies Laplace’s equation:

$$\frac{{\partial }^2\mathsf{\Phi }}{\partial x^2}+\frac{{\partial }^2\mathsf{\Phi }}{\partial z^2}=0$$

(2)

With heave motion being the only concern, the velocity potential Φ can be expressed as:

$$\mathsf{\Phi }={\mathsf{\Phi }}_{\mathrm{I}}+{\mathsf{\Phi }}_{\mathrm{D}}+{\mathsf{\Phi }}_{\mathrm{R}}$$

(3)

where Φ_I is the incident velocity potential, Φ_D denotes the diffraction potential, and Φ_R represents the radiation potential due to heave motion.

The spatial velocity potential for the incident waves can be written as:

$${\mathsf{\Phi }}_{\mathrm{I}}=-\frac{\mathrm{i}gA}{\omega }\frac{\cosh \left[k(z+h_1)\right]}{\cosh (k{h}_1)}{e}^{ikx}$$

(4)

where k is the wavenumber, which satisfies the dispersion relation ω² = gk tanh (kh₁). Correspondingly, the wavelength L = 2π/k.

For the wave diffraction problem, the governing equation is Laplace’s equation, and its boundary conditions for the diffracted spatial potential can be written as follows:

$$\frac{\partial {\mathsf{\Phi }}_{\mathrm{D}}}{\partial z}-\frac{{\omega }^2}{g}{\mathsf{\Phi }}_{\mathrm{D}}=0(z=0,x>\left|a\right|)$$

(5)

$$\frac{\partial {\mathsf{\Phi }}_{\mathrm{D}}}{\partial z}=0(z=-h_1)$$

(6)

$$\frac{\partial {\mathsf{\Phi }}_{\mathrm{D}}}{\partial z}=-\frac{\partial {\mathsf{\Phi }}_{\mathrm{I}}}{\partial z}(z=-d_1,\left|x\right|\le a)$$

(7)

$$\frac{\partial {\mathsf{\Phi }}_{\mathrm{D}}}{\partial x}=-\frac{\partial {\mathsf{\Phi }}_{\mathrm{I}}}{\partial x}(-d_1<z<0,x=\pm a)$$

(8)

$${\mathsf{\Phi }}_{\mathrm{D}}\mathrm{outgoing};\mathrm{finite}\mathrm{value},\left|x\right|\to \infty$$

(9)

For the radiation problem, the body is forced to heave with the amplitude A_R and the angular frequency ω; thus, the radiation potential Φ_R can be written as:

$${\mathsf{\Phi }}_{\mathrm{R}}=-\mathrm{i}\omega A_{\mathrm{R}}{\phi }_{\mathrm{R}}\left(x,z\right)$$

(10)

For the complex spatial velocity potential φ_R, the governing equation is Laplace equation, and its boundary conditions can be written as follows:

$$\frac{\partial {\phi }_{\mathrm{R}}}{\partial z}-\frac{{\omega }^2}{g}{\phi }_{\mathrm{R}}=0(z=0,x>\left|a\right|)$$

(11)

$$\frac{\partial {\phi }_{\mathrm{R}}}{\partial z}=0(z=-h_1)$$

(12)

$$\frac{\partial {\phi }_{\mathrm{R}}}{\partial z}=1(z=-d_1,\left|x\right|\le a)$$

(13)

$$\frac{\partial {\phi }_{\mathrm{R}}}{\partial x}=0(-d_1<z<0,x=\pm a)$$

(14)

$${\phi }_{\mathrm{R}}\mathrm{outgoing};\mathrm{finite}\mathrm{value},\left|x\right|\to \infty$$

(15)

The frequency-dependent expressions of the diffraction potential and the radiation potential can be obtained from [18]. The vertical exciting force F_z, added mass μ, and radiation damping coefficient λ in heave mode can then be computed using the following expressions, respectively:

$$F_z=\rho \mathrm{i}\omega {\displaystyle {\int }_{S_0}\left({\mathsf{\Phi }}_{\mathrm{I}}+{\mathsf{\Phi }}_{\mathrm{D}}\right)}\mathrm{d}s$$

(16)

$$\mu =\rho {\displaystyle {\int }_{S_0}\mathrm{Re}\left[{\phi }_{\mathrm{R}}\right]n_z}\mathrm{d}s$$

(17)

$$\lambda =\rho {\displaystyle {\int }_{S_0}\mathrm{Im}\left[{\phi }_{\mathrm{R}}\right]n_z}\mathrm{d}s$$

(18)

where S₀ is the bottom area of the structure, Re and Im denote the real and imaginary parts of a complex number, and n_z is the unit normal vector along the positive z-axis. The detailed expressions of F_z, μ, and λ can be found in [18].

On the basis of the motion equation in the frequency domain, the heave response amplitude ζ can be expressed as:

$$\zeta =\frac{F_z}{-{\omega }^2(M+\mu )-\mathrm{i}\omega (\lambda +{\lambda }_{\mathrm{PTO}})+K}$$

(19)

where λ_PTO denotes the PTO damping. For λ_PTO = 0 and λ_PTO = ∞, free heave-motion and fixed types, can be defined respectively. The optimal PTO damping can be defined as ${\lambda }_{\mathrm{optimal}}=\sqrt{{\left(K/\omega -\omega \left(M+\mu \right)\right)}^2+{\lambda }^2}$ according to [19]. Correspondingly, the heave RAO is defined as ξ = ζ/A.

The incident wave power can be theoretically calculated as:

$$P_{\mathrm{incident}}=\frac{1}{4}\frac{\rho g{A}^2\omega }{k}\left(1+\frac{2h_{1}k}{\sinh 2h_{1}k}\right)$$

(20)

The absorbed power of the device with the PTO damping λ_PTO can be written as (see [19]):

$$P_{\mathrm{capture}}=\frac{1}{2}{\lambda }_{\mathrm{PTO}}{\omega }^2\frac{{\left|F_z\right|}^2}{{\left(K-{\omega }^2(M+\mu )\right)}^2+{\left(\omega ({\lambda }_{\mathrm{PTO}}+\lambda )\right)}^2}$$

(21)

The CWR η is an important indicator to evaluate the hydrodynamic efficiency of WECs and can be calculated as η = P_capture/P_incident.

As indicators of breakwater performance, the reflection coefficient K_r and transmission coefficient K_t can be written as:

$$K_{\mathrm{t}}=\left|{\frac{{\mathsf{\Phi }}_{\mathrm{I}}+{\mathsf{\Phi }}_{\mathrm{D}}-\mathrm{i}\omega \zeta {\phi }_{\mathrm{R}}}{{\mathsf{\Phi }}_{\mathrm{I}}}|}_{x=+\infty }\right|$$

(22)

$$K_{\mathrm{r}}=\left|{\frac{{\mathsf{\Phi }}_{\mathrm{D}}-\mathrm{i}\omega \zeta {\phi }_{\mathrm{R}}}{{\mathsf{\Phi }}_{\mathrm{I}}}|}_{x=-\infty }\right|$$

(23)

3. Dimensional Analysis

The dimensional parameters include wavenumber k, water density ρ, gravitational acceleration g, water depth h₁, incident wave amplitude A, breadth B, draft d₁, PTO damping λ_PTO and heave response amplitude in heave ζ. The objective parameter includes the outputted power P_capture, reflected wave amplitude A_r and transmitted wave amplitude A_t. The incident wave amplitude, water depth, water density and gravitational acceleration are fixed parameters in this analysis. P_ref denotes an arbitrary reference power level and λ_ref an arbitrary reference damping level. According to the Buckingham's theorem [20], the dimensionless variables can be determined (details see

Table 1. Outline of dimensional analysis.

Dimensional Variables	Physical Unit	Nondimensional Variables
Water density, ρ	kg·m−3	-
Gravitational acceleration, g	m·s−2	-
Water depth, h1	m	-
Incident wave amplitude, A	m	-
Wavenumber, k	m–1	π1 = kh1
Breadth, B	m	π2 = B/h1
Draft, d1	m	π3 = d1/h1
PTO damping, λPTO	kg·s−1	π4 = λPTO/λref
Response amplitude in heave, ζ	m	π5 = ζ/A
Outputted power, Pcapture	kg·m2·s−3	πa = Pcapture/Pref
Reflected wave amplitude, Ar	m	πb = Ar/A
Transmitted wave amplitude, At,	m	πc = At/A

). Here, the incident wave power (i.e., P_incident) was chosen as P_ref and the optimal PTO damping λ_optimal as λ_ref [21]. Thus, the dimensionless outputted power can be expressed as the CWR η (=P_capture/P_incident). The reflected wave and the transmitted wave are dimensionalized by the incident wave amplitude A. Correspondingly, the dimensionless reflected wave and transmitted wave can be expressed as the reflection coefficient K_r (=A_r/A) and the transmission coefficient K_t (=A_t/A). Then, we can write the CWR, reflection coefficient and transmission coefficient as function of the dimensionless wavenumber kh₁, relative breadth B/h₁, relative draft d₁/h₁ and relative PTO damping λ_PTO/λ_optimal.

$$\begin{array}{l}\eta =f\left(B/h_1,d_1/h_1,k{h}_1,{\lambda }_{\mathrm{PTO}}/{\lambda }_{\mathrm{optimal}}\right)\\ K_{\mathrm{r}}=f\left(B/h_1,d_1/h_1,k{h}_1,{\lambda }_{\mathrm{PTO}}/{\lambda }_{\mathrm{optimal}}\right)\\ K_{\mathrm{t}}=f\left(B/h_1,d_1/h_1,k{h}_1,{\lambda }_{\mathrm{PTO}}/{\lambda }_{\mathrm{optimal}}\right)\end{array}$$

(24)

4. Results and Discussion

4.1. Validation

First, the correctness of the present formulation for the reflection coefficient K_r and transmission coefficient K_t is considered. Figure 2 shows the variations of K_r and K_t with the dimensionless wavenumber kh₁ obtained by the present analytical model and the corresponding numerical results [22]. The geometrical parameters are a = h₁, d₁ = 0.5h₁, and h₁ =1 m; λ_PTO = 10000 λ_optimal is used to solve for the motion equation in the present study, in order to have the pontoon fixed as in [22]. The incident wave amplitude is A = 0.1 m. The maximum difference between the present and reference results for the reflection and transmission coefficients are 5% and 2.5%, respectively. As shown, a good agreement can be achieved.

The accuracy of the heave RAO ξ is then verified against the numerical results obtained by Isaacson et al. [23] for B = 3d₁ and d₁ = 0.2h₁. Since the breakwater is constrained to heave motion, but no PTO damping was considered in [23], the value chosen for the PTO damping λ_PTO is 0 in the present study. Figure 3 shows the comparison of the heave RAO obtained using the present approach with the results obtained in [23].The maximum difference between the present and reference results for the heave RAO is 4.5%. It can be seen that a good agreement can be obtained.

Lastly, the CWR η is verified by using the relation of K_t² + K_r² + η = 1 [24]. The detailed validation is described in Figure 4. As shown, the aforementioned condition is accurately satisfied.

4.2. Comparison of the Different Breakwater Systems

From the standpoint of the WEC, the optimal PTO damping is often used to assess the performance of the devices. In the Section 4.2, Section 4.3 and Section 4.4 the relationships between the reflection coefficient K_r, transmission coefficient K_t, and CWR η for the proposed integrated device with the optimal PTO damping λ_optimal are critical in determining the frequency region in which the acceptable wave attenuation performance and the efficient wave energy conversion can be obtained.

To illustrate the features of the integrated system with the optimal PTO damping, the hydrodynamic performance corresponding to the breakwater with the optimal PTO damping (case 1), free heave-motion breakwater (case 2), and fixed breakwater (case 3) are compared. The geometrical parameters are d₁ = 2.5 m, B = 8 m, and h₁ = 10 m (i.e., B/h₁ = 0.8 and d₁/h₁ = 0.25). The incident wave amplitude is A = 1.0 m. Figure 4, Figure 5 and Figure 6 present a comparison of the three cases with respect to the reflection coefficient K_r, transmission coefficient K_t, and heave RAO ξ, respectively.

As shown in Figure 5, the reflection coefficient of fixed breakwater is larger than those of the others. Figure 6 shows that the transmission coefficient of the breakwater with the optimal PTO damping is near to that of the fixed breakwater; they are markedly smaller than that of the free heave-motion breakwater. The heave RAO of the fixed breakwater is null; thus, it is not plotted in Figure 7. As intuitively expected, the heave RAO of the PTO damping-controlled breakwater is markedly smaller than that of the free heave-motion breakwater, which may be beneficial to improve the stability of the breakwater. By introducing the PTO damping, the breakwater performance can be improved significantly and the heave RAO can be reduced.

Figure 4 shows the variations of K_t, K_r, η, and K_t² + K_r² + η with the dimensionless wavenumber kh₁ for the breakwater with the optimal PTO damping. The curve of η exhibits a parabolic trend and reaches the maximum (i.e., η_max = 50%) at resonance. Interestingly, there exists a cross-point for the three curves at which η is 50%, and both K_r and K_t are 0.5. This observation indicates that 25% of the incident wave energy is reflected toward the left, 25% is transmitted toward the right, and the remaining 50% is absorbed. This is a consequence of wave energy theory (see [19] (p. 198)). The condition K_t < 0.5 shall be satisfied for a qualified breakwater [25]. Therefore, the ideal frequency region with a lower threshold corresponding to the natural frequency is of interest for the integrated system with the optimal PTO damping. Since the study is conducted under the context of small amplitude assumption within linear potential theory neglecting viscous and nonlinear effects, the heave RAO of the device may not be totally in accordance with the corresponding experimental results. From the literature, it can be seen that the whole variation trends of the hydrodynamic coefficients corresponding to analytical and experimental results are similar [23]. Thus, the analytical results may have directive significance for the practical engineering.

4.3. Effect of the Relative Breadth B/h₁

The breadth and the draft of the pontoon are important parameters to design such a system. In Section 4.3 and Section 4.4, the effects of the dimensionless breadth (i.e., relative breadth B/h₁) and dimensionless draft (i.e., relative draft d₁/h₁) on the hydrodynamic properties of the system are conducted, respectively. Figure 8, Figure 9, Figure 10 and Figure 11 show the variations of the reflection coefficient K_r, transmission coefficient K_t, CWR η and heave RAO ξ with the dimensionless wavenumber kh₁ for different relative breadths B/h₁ (=0.2, 0.5, 0.8, 1.1 and 1.4). The other geometrical parameters are d₁ = 2.5 m and h₁ = 10 m (i.e., d₁/h₁ = 0.25). The incident wave amplitude A is 1.0 m. Optimal damping is used in this subsection. From Figure 8 and Figure 9, it can be seen that, for the pontoon-type floating breakwater, the wider the breadth of the breakwater, the more effective is the wave barrier [17,25]. For the CWR, the maximum CWR does not vary with the relative draft. The maximum heave RAO obviously increases with the decreasing of the relative breadth. The dimensionless wavenumber kh₁ corresponding to the maximum heave RAO (or the CWR) decreases with the increasing of the relative breadth. It is due to the fact that the natural frequencies of the pontoon decrease with the increasing of the relative breadth.

4.4. Effect of the Relative Draft d₁/h₁

Figure 12, Figure 13, Figure 14 and Figure 15 show the variations of the reflection coefficient K_r, transmission coefficient K_t, CWR η and heave RAO ξ with the dimensionless wavenumber kh₁ for different relative drafts d₁/h₁ = 0.05, 0.15, 0.25, 0.35 and 0.45. The other geometrical parameters are B = 8 m and h₁ = 10 m (i.e., B/h₁ = 0.8). The incident wave amplitude A is 1.0 m. From Figure 12 and Figure 13, it can be seen that, the deeper the draft of the breakwater, the more effective is the wave barrier. For the CWR, the effective frequency bandwidth (η > 20%) narrows with an increase in the draft. However, the maximum CWR does not vary with the relative draft. The maximum heave RAO increases with the increasing of the relative draft. Note that, since the natural frequencies of the pontoon decreases with the increasing of relative draft, the kh₁ corresponding to the maximum CWR and heave RAO decreases accordingly.

4.5. Effect of the PTO Damping

The integrated system with the optimal PTO damping, which may lead to the optimization of the CWR, is investigated in Section 4.2, Section 4.3 and Section 4.4. Given that both the wave attenuation performance and wave energy extraction efficiency shall be considered simultaneously for the integrated system, the effect of the PTO damping on the reflection coefficient K_r, transmission coefficient K_t, and CWR η are considered. The geometrical parameters are d₁ = 2.5 m, B = 8 m and h₁ = 10 m (i.e., d₁/h₁ = 0.25, B/h₁ = 0.8). The incident wave amplitude A is 1.0 m. The values selected for the tested PTO dampings are λ_PTO = 0.8λ_optimal, 1.0λ_optimal, 1.5λ_optimal, 2.0λ_optimal, 5.0λ_optimal, and λ_PTO = 10000λ_optimal (i.e., the case of the fixed breakwater).

Figure 16 shows the variations of the reflection coefficient K_r with the dimensionless wavenumber kh₁. As observed, the reflection coefficient increases with an increase in the PTO damping. Figure 17 and Figure 18 show the variations of the transmission coefficient and heave RAO against kh₁. With an increase in the PTO damping, the heave RAO decreases (referring to Figure 18); the transmission coefficient increases in the lower frequencies and, differently, the trend of decreasing firstly and then increasing was found in the middle frequency region (i.e., 1.3 < kh₁ < 2.7). These findings indicate that the wave attenuation performance can be superior to the fixed breakwater by proper adjustment of the PTO damping. Figure 19 shows the variations of the CWR η with kh₁. Since the natural frequency ω_nat of the pontoon in heave mode can be expressed as ω_nat = $\sqrt{\frac{K}{M+\mu }}$, changes in the PTO damping do not affect the natural frequency of the system [26]. Thus, the locations (i.e., kh₁) of the CWR peak value are similar for the different cases. With an increase in the PTO damping, the CWR first increases and then decreases. Notably, the CWR corresponding to the PTO damping of λ_PTO = 1.5λ_optimal (or λ_PTO = 2λ_optimal) is only slightly inferior to the case with the optimal PTO damping in the range of 1.3 < kh₁ < 2.7; under this condition, the transmission coefficient of the former is superior to the latter. Specifically, considering the conditions K_t < 0.5 and η > 20%, the available frequency region is 1.925 < kh₁ < 3.075 when λ_PTO = 1λ_optimal; 1.723 < kh₁ < 3.02 when λ_PTO = 1.5λ_optimal, and 1.625 < kh₁ < 2.92 when λ_PTO = 2λ_optimal. That is, the effective frequency bandwidth is broadened when λ_PTO = 1.5–2λ_optimal.

5. Conclusions

The hydrodynamic properties of a WEC-type floating breakwater system is investigated theoretically based on linear potential flow theory. The breakwater is constrained in heave motion. The linear PTO damping is used to calculate the absorbed power. The hydrodynamic properties of the breakwater with the optimal PTO damping, fixed breakwater, and free heave-motion breakwater are compared. The effect of the PTO damping on the performance of the integrated system is particularly evaluated in this study.

The following conclusions can be drawn from this study:

(1)

Compared with that of the free heave-motion breakwater, the wave attenuation performance of the breakwater is improved for the proposed integrated system.

(2)

For the system with the optimal PTO damping, the low threshold of the practical frequency region corresponds to the natural frequency.

(3)

With a decrease in the heave RAO of the breakwater, the transmission coefficient increases in the lower-frequency region, although a decreasing trend is initially observed, followed by an increasing trend in the middle-frequency region.

(4)

Due to the changing of the natural frequency, the effect of the relative breadth B/h₁ and relative draft d₁/h₁ of the pontoon affect the performance of the system significantly. This shall be paid attention while such a system is designed.

(5)

The breakwater with the PTO damping of λ_PTO = 1.5–2λ_optimal may give a broader frequency bandwidth with K_t < 0.5 and η > 20%. Fortunately, the transmission coefficient corresponding to the case with λ_PTO = 2λ_optimal is slightly superior to that of the fixed breakwater.

(6)

The proposed system is theoretically proved to produce power effectively and, at the same time, the function of coastal protection can be comparable to that of the fixed breakwater.

From the point of engineering application, the proposed scheme is more applicable for the pile-restrained floating breakwater, for which the pontoon moves in heave motion under the control of the vertical pile [17,23]. Heave-type floating bodies are often used to capture wave energy [24]. Thus, the effects of the non-heave motions are not considered. This preliminary investigation is performed under frame of linear potential theory in frequency domain. The linear damping is adopted to calculate the produced power and conduct the parametric study. In practice, the nonlinear PTO damping (such as Coulomb damping) is often used for the hydraulic PTO system [27]. Despite this, the theoretical results predict the potential application. We can design the breakwater system based on the sea state (such as dominating wave length L) at the deployment site. It is well understood that the disadvantage of the pontoon type breakwater is the bad breakwater performance in long waves [23]. In the future research, we will focus on how to improve the wave attenuation performance of the proposed system.