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

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.


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 r 2 + K t 2 + 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.

Formulas
As shown in Figure 1, a pontoon-type structure (i.e., breakwater) with a width of B = 2a and a draft d 1 is situated in the water with a uniform depth h 1 .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 (=2ρad 1 ) 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.
Figure 1.Sketch of the floating structure with the power take-off (PTO) system (the structure restrained with the vertical pile moves in heave and the PTO system is used to capture wave energy, detailed description see [17]).
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: where t denotes time, i = √−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: With heave motion being the only concern, the velocity potential Φ can be expressed as: where ΦI is the incident velocity potential, ΦD denotes the diffraction potential, and ΦR represents the radiation potential due to heave motion.The spatial velocity potential for the incident waves can be written as: [ ] (4) where k is the wavenumber, which satisfies the dispersion relation ω 2 = 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: D outgoing; finite value, Φ → ∞ x (9) For the radiation problem, the body is forced to heave with the amplitude AR and the angular frequency ω; thus, the radiation potential ΦR can be written as: Sketch of the floating structure with the power take-off (PTO) system (the structure restrained with the vertical pile moves in heave and the PTO system is used to capture wave energy, detailed description see [17]).
As indicated in Figure 1, the fluid domain is divided into three subdomains-i.e., Ω 1 , Ω 2 , and Ω 3 .The fluid motion in the whole domain can be described by the velocity potential: where t denotes time, i = √ −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: With heave motion being the only concern, the velocity potential Φ can be expressed as: 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: where k is the wavenumber, which satisfies the dispersion relation ω 2 = gk tanh (kh 1 ).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: 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: For the complex spatial velocity potential ϕ R , the governing equation is Laplace equation, and its boundary conditions can be written as follows: 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: where S 0 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: 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 λ optimal = (K/ω − ω(M + µ)) 2 + λ 2 according to [19].Correspondingly, the heave RAO is defined as ξ = ζ/A.The incident wave power can be theoretically calculated as: The absorbed power of the device with the PTO damping λ PTO can be written as (see [19]): 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:

Dimensional Analysis
The dimensional parameters include wavenumber k, water density ρ, gravitational acceleration g, water depth h 1 , incident wave amplitude A, breadth B, draft d 1 , 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).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 kh 1 , relative breadth B/h 1 , relative draft d 1 /h 1 and relative PTO damping λ PTO /λ optimal .(24) Table 1.Outline of dimensional analysis.

Dimensional Variables Physical Unit Nondimensional Variables
Water density,

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 kh 1 obtained by the present analytical model and the corresponding numerical results [22].The geometrical parameters are a = h 1 , d 1 = 0.5h 1 , and h 1 =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.
Energies 2017, 10, 712 6 of 16 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 λPTO 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 Kt 2 + Kr 2 + η = 1 [24].The detailed validation is described in Figure 4.As shown, the aforementioned condition is accurately satisfied.The accuracy of the heave RAO ξ is then verified against the numerical results obtained by Isaacson et al. [23] for B = 3d 1 and d 1 = 0.2h 1 .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.
Energies 2017, 10, 712 6 of 16 results for the reflection and transmission coefficients are 5% and 2.5%, respectively.As shown, a good agreement can be achieved.

Figure 2. Comparison of transmission coefficient
Kr and reflection coefficient Kt obtained by the present approach with the results of [22].
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 λPTO 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 Kt 2 + Kr 2 + η = 1 [24].The detailed validation is described in Figure 4.As shown, the aforementioned condition is accurately satisfied.Lastly, the CWR η is verified by using the relation of K t 2 + K r 2 + η = 1 [24].The detailed validation is described in Figure 4.As shown, the aforementioned condition is accurately satisfied.

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 Sections of 4.2-4.4,the relationships between the reflection coefficient Kr, transmission coefficient Kt, and CWR η for the proposed integrated device with the optimal PTO damping λoptimal 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.Figures 4-6 present a comparison of the three cases with respect to the reflection coefficient Kr, transmission coefficient Kt, 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 heavemotion 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 Kt, Kr, η, and Kt 2 + Kr 2 + 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., ηmax = 50%) at resonance.Interestingly, there exists a cross-point for the three curves at which η is 50%, and both Kr and Kt 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 Kt < 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 Variations of the reflection coefficient K r , transmission coefficient K t , capture width ratio (CWR) η, and K t 2 + K r 2 + η with the dimensionless wavenumber kh 1 for the breakwater with the optimal PTO damping.

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 Sections 4.2-4.4 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 d 1 = 2.5 m, B = 8 m, and h 1 = 10 m (i.e., B/h 1 = 0.8 and d 1 /h 1 = 0.25).The incident wave amplitude is A = 1.0 m.Figures 4-6 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 t 2 + K r 2 + η with the dimensionless wavenumber kh 1 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.
coefficients corresponding to analytical and experimental results are similar [23].Thus, the analytical results may have directive significance for the practical engineering.

Effect of the Relative Breadth B/h1
The breadth and the draft of the pontoon are important parameters to design such a system.In Sections 4.3 and 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 Energies 2017, 10, 712 8 of 16 coefficients corresponding to analytical and experimental results are similar [23].Thus, the analytical results may have directive significance for the practical engineering.

Effect of the Relative Breadth B/h1
The breadth and the draft of the pontoon are important parameters to design such a system.In Sections 4.3 and 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 Energies 2017, 10, 712 8 of 16 coefficients corresponding to analytical and experimental results are similar [23].Thus, the analytical results may have directive significance for the practical engineering.

Effect of the Relative Breadth B/h1
The breadth and the draft of the pontoon are important parameters to design such a system.In Sections 4.3 and 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

Effect of the Relative Breadth B/h 1
The breadth and the draft of the pontoon are important parameters to design such a system.In Sections 4.3 and 4.4, the effects of the dimensionless breadth (i.e., relative breadth B/h 1 ) and dimensionless draft (i.e., relative draft d 1 /h 1 ) on the hydrodynamic properties of the system are conducted, respectively.Figures 8-11 show the variations of the reflection coefficient K r , transmission coefficient K t , CWR η and heave RAO ξ with the dimensionless wavenumber kh 1 for different relative breadths B/h 1 (=0.2, 0.5, 0.8, 1.1 and 1.4).The other geometrical parameters are d 1 = 2.5 m and h 1 = 10 m (i.e., d 1 /h 1 = 0.25).The incident wave amplitude A is 1.0 m.Optimal damping is used in this subsection.From Figures 8 and 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 kh 1 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.From Figures 8 and 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.From Figures 8 and 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.

Effect of the Relative Draft d1/h1
Figures 12-15 show the variations of the reflection coefficient Kr, transmission coefficient Kt, 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 Figures 12 and 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.

Effect of the Relative Draft d1/h1
Figures 12-15 show the variations of the reflection coefficient Kr, transmission coefficient Kt, 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 Figures 12 and 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.The incident wave amplitude A is 1.0 m.From Figures 12 and 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 kh 1 corresponding to the maximum CWR and heave RAO decreases accordingly.

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 Sections 4.2-4.4.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 Kr, transmission coefficient Kt, 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 λPTO = 0.8λoptimal, 1.0λoptimal, 1.5λoptimal, 2.0λoptimal, 5.0λoptimal, and λPTO = 10000λoptimal (i.e., the case of the fixed breakwater).
Figure 16 shows the variations of the reflection coefficient Kr with the dimensionless wavenumber kh1.As observed, the reflection coefficient increases with an increase in the PTO damping.Figures 17 and 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 ωnat of the pontoon in heave mode can be expressed as ωnat

=
, 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 λPTO = 1.5λoptimal (or λPTO = 2λoptimal) 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 Kt < 0.5 and η > 20%, the available frequency region is 1.925 < kh1 < 3.075 when λPTO = 1λoptimal; 1.723 < kh1 < 3.02 when λPTO = 1.5λoptimal, and 1.625 < kh1 < 2.92 when λPTO = 2λoptimal.That is, the effective frequency bandwidth is broadened when λPTO = 1.5-2λoptimal.

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 Sections 4.2-4.4.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 d 1 = 2.5 m, B = 8 m and h 1 = 10 m (i.e., d 1 /h 1 = 0.25, B/h 1 = 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 kh 1 .As observed, the reflection coefficient increases with an increase in the PTO damping.Figures 17  and 18 show the variations of the transmission coefficient and heave RAO against kh 1 .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 < kh 1 < 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 kh 1 .Since the natural frequency ω nat of the pontoon in heave mode can be expressed as ω nat = K M+µ , changes in the PTO damping do not affect the natural frequency of the system [26].Thus, the locations (i.e., kh 1 ) 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 < kh 1 < 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 < kh 1 < 3.075 when λ PTO = 1λ optimal ; 1.723 < kh 1 < 3.02 when λ PTO = 1.5λ optimal , and 1.625 < kh 1 < 2.92 when λ PTO = 2λ optimal .That is, the effective frequency bandwidth is broadened when λ PTO = 1.5-2λ optimal .

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 λPTO = 1.5-2λoptimal may give a broader frequency bandwidth with Kt < 0.5 and η > 20%.Fortunately, the transmission coefficient corresponding to the case with λPTO = 2λoptimal 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 pilerestrained 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.

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/h 1 and relative draft d 1 /h 1 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.

Figure 1 .
Figure1.Sketch of the floating structure with the power take-off (PTO) system (the structure restrained with the vertical pile moves in heave and the PTO system is used to capture wave energy, detailed description see[17]).

Figure 2 .
Figure 2. Comparison of transmission coefficient Kr and reflection coefficient Kt obtained by the present approach with the results of [22].

Figure 3 .
Figure 3.Comparison of the heave response amplitude operator (RAO) obtained using the present approach with the results of Isaacson et al. [23].(B/L represents the dimensionless wavelength).

Figure 2 .
Figure 2.Comparison of transmission coefficient K r and reflection coefficient K t obtained by the present approach with the results of[22].

Figure 3 .
Figure 3.Comparison of the heave response amplitude operator (RAO) obtained using the present approach with the results of Isaacson et al. [23].(B/L represents the dimensionless wavelength).

Figure 3 .
Figure 3.Comparison of the heave response amplitude operator (RAO) obtained using the present approach with the results of Isaacson et al. [23].(B/L represents the dimensionless wavelength).

Figure 4 .
Figure 4. Variations of the reflection coefficient Kr, transmission coefficient Kt, capture width ratio (CWR) η, and Kt 2 + Kr 2 + η with the dimensionless wavenumber kh1 for the breakwater with the optimal PTO damping.

Figure 4 .
Figure 4. Variations of the reflection coefficient K r , transmission coefficient K t , capture width ratio (CWR) η, and K t 2 + K r 2 + η with the dimensionless wavenumber kh 1 for the breakwater with the

Figure 5 .
Figure 5. Variations of the reflection coefficient Kr with the dimensionless wavenumber kh1 for cases 1-3.

Figure 6 .
Figure 6.Variations of the transmission coefficient Kt with the dimensionless wavenumber kh1 for cases 1-3.

Figure 7 .
Figure 7. Variations of the heave RAO ξ with the dimensionless wavenumber kh1 for cases 1 and 2.

kh 1 Figure 5 .
Figure 5. Variations of the reflection coefficient K r with the dimensionless wavenumber kh 1 for cases 1-3.

Figure 5 .
Figure 5. Variations of the reflection coefficient Kr with the dimensionless wavenumber kh1 for cases 1-3.

Figure 6 .
Figure 6.Variations of the transmission coefficient Kt with the dimensionless wavenumber kh1 for cases 1-3.

Figure 7 .
Figure 7. Variations of the heave RAO ξ with the dimensionless wavenumber kh1 for cases 1 and 2.

kh 1 Figure 6 .
Figure 6.Variations of the transmission coefficient K t with the dimensionless wavenumber kh 1 for cases 1-3.

Figure 5 .
Figure 5. Variations of the reflection coefficient Kr with the dimensionless wavenumber kh1 for cases 1-3.

Figure 6 .
Figure 6.Variations of the transmission coefficient Kt with the dimensionless wavenumber kh1 for cases 1-3.

Figure 7 .
Figure 7. Variations of the heave RAO ξ with the dimensionless wavenumber kh1 for cases 1 and 2.

kh 1 Figure 7 .
Figure 7. Variations of the heave RAO ξ with the dimensionless wavenumber kh 1 for cases 1 and 2.

Figure 8 .
Figure 8. Variations of the reflection coefficient Kr with the dimensionless wavenumber kh1 for cases of relative draft d1/h1 = 0.25 and PTO damping λPTO = λoptimal (λoptimal refers to the optimal PTO damping).

Figure 8 .
Figure 8. Variations of the reflection coefficient Kr with the dimensionless wavenumber kh1 for cases of relative draft d1/h1 = 0.25 and PTO damping λPTO = λoptimal (λoptimal refers to the optimal PTO damping).

Figure 12 .
Figure 12.Variations of the reflection coefficient Kr with the dimensionless wavenumber kh1 for cases of relative breadth B/h1 = 0.8 and λPTO = λoptimal.

Figure 13 .
Figure 13.Variations of the transmission coefficient Kt with the dimensionless wavenumber kh1 for cases of B/h1 = 0.8 and λPTO = λoptimal.

dimensionless wavenumber kh 1 capture width ratio η d 1 Figure 12 .
Figure 12.Variations of the reflection coefficient K r with the dimensionless wavenumber kh 1 for cases of relative breadth B/h 1 = 0.8 and λ PTO = λ optimal .

Figure 12 .
Figure 12.Variations of the reflection coefficient Kr with the dimensionless wavenumber kh1 for cases of relative breadth B/h1 = 0.8 and λPTO = λoptimal.

Figure 13 .
Figure 13.Variations of the transmission coefficient Kt with the dimensionless wavenumber kh1 for cases of B/h1 = 0.8 and λPTO = λoptimal.

dimensionless wavenumber kh 1 capture width ratio η d 1 Figure 13 .
Figure 13.Variations of the transmission coefficient K t with the dimensionless wavenumber kh 1 for cases of B/h 1 = 0.8 and λ PTO = λ optimal .

Figure 12 .
Figure 12.Variations of the reflection coefficient Kr with the dimensionless wavenumber kh1 for cases of relative breadth B/h1 = 0.8 and λPTO = λoptimal.

Figure 13 .
Figure 13.Variations of the transmission coefficient Kt with the dimensionless wavenumber kh1 for cases of B/h1 = 0.8 and λPTO = λoptimal.

dimensionless wavenumber kh 1 capture width ratio η d 1 Figure 14 .
Figure 14.Variations of the CWR η with the dimensionless wavenumber kh 1 for cases of B/h 1 = 0.8 and λ PTO = λ optimal .