Hydrodynamic Performance Assessment of a Hybrid Wave Energy Converter Array–Floating Breakwater System Under Irregular Waves

Abstract

A hybrid system combining wave energy converters (WECs) and a floating breakwater presents significant potential for developing commercial-scale wave power operations. The assessment of the hydrodynamic characteristics of a WEC array–floating breakwater system under irregular waves remains in the early stages and requires further investigation. Based on the linear potential theory, a time-domain numerical model is established to evaluate the performance of a hybrid WEC array–floating breakwater system in a target sea area. The interaction between the WECs and the floating breakwater is analyzed. Results show that for the hybrid system with a triangular-baffle-type WEC array under irregular waves, the annual average wave power is 1.16 MW and the annual energy production is 10.16 × 10³ MW·h, representing a 241.2% improvement compared with that of the isolated WEC array. The standard deviations of the mooring forces for the hybrid system with the triangular-baffle-type WEC array are reduced by 13.8% in the surge direction and 26.9% in the pitch direction, while increasing by 90.0% in the heave direction. Similar conclusions are obtained for the motion of the floating breakwater. The findings and data reported in this study provide guidance for the engineering application of a hybrid WEC array–floating breakwater system.

1. Introduction

Wave energy, with its abundant reserves, broad distribution, and high predictability, offers significant potential as a renewable and clean energy source [1,2]. To fully exploit and develop this energy, it is essential to create efficient, reliable, and cost-effective wave energy converters (WECs). A hybrid or integrated system of WECs and breakwaters has garnered considerable attention from researchers due to several benefits, such as enhanced wave energy capture, reduced costs, optimized spatial utilization, and integrated functionality [3,4,5]. These systems are commonly categorized into three main types: Oscillating Water Columns (OWCs) combined with breakwaters [6,7,8,9], Overtopping WECs as breakwater [10], and Oscillating Buoys (OBs) integrated with floating breakwaters [11,12,13]. Compared to the OWC, OB-type WECs are more compact, efficient, and easier to integrate with existing breakwaters [14,15]. As a result, OB-type WECs integrated with breakwaters have become a focal point of research.

The study of OB-type WECs integrated with breakwaters has evolved through three key stages. The first stage primarily focused on the analysis, simulation, and experimental investigation of integrated WEC–breakwater systems using a single-body model. For instance, Zhao et al. [16] and Chen & Zang [17] explored single-body systems with a rectangular bottom, while Madhi et al. [18], Zhang et al. [19], and Zhou et al. [20] examined asymmetric single-body systems with Berkeley-Wedge and triangular-baffle bottoms. These studies demonstrated that the single-body WEC–breakwater system can effectively achieve both energy extraction and wave attenuation. Additionally, asymmetric configurations performed better than symmetric ones in motion response, energy conversion, and wave attenuation, with conversion efficiencies of triangular-baffle-type WECs exceeding 92% during the resonant period. For the triangular-baffle-type WEC, Zhang et al. [19] indicated that decreasing the length of the upper vertical side can increase the heave motion response of the WEC and, thus, improve conversion efficiency, and the maximum heave motion and conversion efficiency appeared at a lower frequency when increasing the angle of the triangular part. A smaller ratio of breadth to draft led to a larger heave motion response and larger optimal power take-off (PTO) damping. However, this efficiency declined sharply when the incident wave period deviated from the resonant period, thus limiting the operational frequency range of the system for optimal energy capture.

To address this limitation, research shifted to the second stage, focusing on dual-floater hybrid systems consisting of a WEC and a floating breakwater. Studies by Zhao & Ning [21], Ning et al. [22,23], Zhang et al. [15,24], and Zhou et al. [25] involved two-dimensional models to investigate the interaction between the WEC and the breakwater. Their findings revealed that a floating breakwater can concentrate waves, thereby enhancing the energy harvesting performance of the WEC in low-frequency waves. This improvement occurs due to the significant increase in wave height in the focused regions [26,27,28,29], which boosts the efficiency of energy extraction. Furthermore, both the wave transmission coefficient and the wave forces on the system were reduced due to the energy harvesting effect of the WEC. Zhang et al. [15,24] found narrow gap resonance, which has been studied by Gao et al. [30], Mi et al. [31], and Gong et al. [32] when it occurs between two fixed floaters, clearly influencing WEC conversion efficiency. At this time, the wave elevation between the WEC and the breakwater reached the maximum value. When the width of the narrow gap increased, the maximum wave elevation and narrow gap resonance frequency decreased. A decreasing WEC draft led to a reduction in the wave elevation between the WEC and the breakwater. The increase in the WEC breadth had a slight influence on the narrow gap resonance frequency, but reduced the wave elevation.

Typically, WECs are arranged in an array on the upwind side of the floating breakwater to maximize wave energy absorption. This WEC array–floating breakwater hybrid system involves complex three-dimensional multi-floater interactions, constrained motion, and multi-degrees-of-freedom behavior, making the hydrodynamic characteristics more intricate and warranting further study. As a result, the third stage of research began. Assuming the breakwater remains fixed and the WECs are unconstrained, Ning et al. [33] and Zhao et al. [34] studied the performance of a WEC array–floating breakwater system using a numerical model and experimental model, respectively. Cheng et al. [35] analyzed a moonpool-type floating breakwater combined with an OB-type WEC array. Their findings showed that the hybrid system outperformed a single WEC array in terms of energy capture. Additionally, with the inclusion of multi-floater constraints [36,37] and multi-degrees-of-freedom motion [38,39,40], Zhang et al. [41] designed and optimized a hybrid WEC array–floating breakwater system, conducting numerical analyses to investigate its energy harvesting capabilities and the interactions between the WECs and the breakwaters. The results highlighted that wave focusing occurs more frequently near the breakwater, enabling higher wave power when the WECs are placed in these areas; these results were also identified by Zhou et al. [42]. Although the vertical forces on the breakwater increased due to the presence of the WECs, horizontal forces were reduced. Zhou et al. [43] experimentally explored the impact of WEC shapes and the nonlinearities of incident waves on the hybrid system, as well as the interactions between the floaters. They observed a 324.6% increase in wave power, while mooring forces decreased by 19.8% in the pitch direction and 47.4% in the surge direction. Additionally, both wave power and the transmission coefficient of the hybrid system increased with higher incident wave amplitudes.

Previous studies on the three stages have analyzed and optimized the hydrodynamic performance of the hybrid WEC arrays–floating breakwater system in regular waves, providing a reference for practical application. However, to the best of the authors’ knowledge, the assessment on the performance of a WEC array–floating breakwater system in irregular waves, which could provide results closer to real sea conditions for practical engineering applications, is still lacking. Therefore, this study aims to fill this gap.

The innovations and intentions of this paper are as follows: Firstly, we aim to compare the wave energy harvesting performance of a hybrid WEC array–floating breakwater system in irregular and regular wave conditions. Secondly, we aim to evaluate the energy harvesting characteristics and the changes in mooring forces of the hybrid system when deployed in the South China Sea, with the floating breakwater subject to six-degrees-of-freedom (DoF) motion in irregular wave environments.

Other sections are organized as shown below. In Section 2, the linear numerical model of the hybrid WEC array–breakwater system is briefly introduced. In Section 3, the numerical model is verified. In Section 4, the wave power of WECs, the motion of the floating breakwater, and the mooring forces of the hybrid system are assessed. Finally, conclusions are drawn in Section 5.

2. Numerical Models

2.1. Hybrid WEC–Breakwater System

This section illustrates the configuration of the hybrid WEC array–floating breakwater system, as shown in Figure 1. The mooring system allows for 6-degrees-of-freedom (6-DoF) motion in the breakwater. The WECs are arranged at regular intervals along the length of the breakwater, with each WEC properly labeled, as shown in Figure 2. The WEC labeled “0” is positioned at the center of the floating breakwater. The distance between adjacent WECs is equal to their width in the y-direction. Each WEC is connected to a power take-off (PTO) system and can achieve heave motion related to the breakwater, enabling the harvesting of wave energy.

Based on the optimization performed by Zhang et al. [41], the parameters of the hybrid system in this paper are as follows. The size of the breakwater is L₂ = 150 m, B₂ = 20 m, and D₂ = 10 m. There is a total of n = 19 WECs. The breadths between the WEC and breakwater, as well as between individual WECs, are B_g = 0.5 m and B₃ =3.56 m, respectively. The length of the WEC is L₁ = 3.56 m, and its breadth is B₁ = 4.44 m, as shown in Figure 2. The drafts of the WECs with different bottoms are shown in Figure 3. The influence of the WEC bottom configuration on the WEC motion response and free surface elevation, which evolves between the WECs and the floating breakwater, has been thoroughly investigated by Zhang et al. [15,19,24,41] and Gao et al. [44]. Therefore, it is not repeatedly discussed in this paper.

Table 1. Joint probability distribution of annual average wave height and wave period for a certain location in the South China Sea.

		Tj	1.5 s	2.5 s	3.5 s	4.5 s	5.5 s	6.5 s	7.5 s	8.5 s
	Sj
Hj
0.25 m			0.007	2.171	4.506	1.831	0.945	0.038	0	0
0.75 m			0	0.021	7.347	13.590	6.734	3.892	0.938	0.014
1.25 m			0	0	0.003	4.345	11.567	4.701	2.756	1.102
1.75 m			0	0	0	0.007	2.420	7.946	1.852	0.582
2.25 m			0	0	0	0	0.021	3.888	4.546	0.418
2.75 m			0	0	0	0	0	0.133	5.357	1.078
3.25 m			0	0	0	0	0	0	0.774	2.896
3.75 m			0	0	0	0	0	0	0.014	1.418
4.25 m			0	0	0	0	0	0	0	0.133
Sum			0.007	2.192	11.856	19.782	21.687	20.598	16.237	7.641

presents the joint probability distribution of the annual average wave height and wave period for a specific location in the South China Sea, as reported by Zhang et al. [41].

2.2. Motion Equation of Floaters

Based on the Cummins method, the motion equation in the time domain for a floating breakwater hybrid system with WEC arrays can be written as follows [45]:

$$\begin{array}{l}\left[\begin{array}{cc}{\left(M+a_{\infty }\right)}_{6\left(n+1\right)\times 6\left(n+1\right)}& D_{6\left(n+1\right)\times 5n}^{\mathrm{T}}\\ D_{5n\times 6\left(n+1\right)}& 0_{5n\times 5n}\end{array}\right]\left[\begin{array}{c}\stackrel{··}{\xi }{(t)}_{6(n+1)\times 1}\\ F_{\mathrm{L},5n\times 1}\end{array}\right]\\ =\left[\begin{array}{c}{\left[F(t)-{\displaystyle {\int }_{-\infty }^t\left[\kappa \left(t-\tau \right)\right]\stackrel{·}{\xi }(t)d\tau }-C\xi (t)\right]}_{6\left(n+1\right)\times 1}\\ 0_{5n\times 1}\end{array}\right]\end{array}$$

(1)

where n is the number of WECs. M is the floater mass matrix, and C is the restoring force matrix for n + 1 bodies. a_∞ is the additional mass matrix with infinite frequency. ξ(t) is the displacement vector; $\dot{\mathsf{\xi }}$(t) is the velocity vector; and $\ddot{\mathsf{\xi }}$(t) is the acceleration vector. D is the constraint matrix, and F_L,5n×1 is the constraint force and moment between floaters. F(t) is the total force matrix, and κ(t − τ) is the radiation pulse response function matrix.

The definitions of the components of the additional mass matrix with infinite frequency a_ij,∞ and the radiation pulse response function matrix components κ_ij(t) are as follows:

$$a_{ij,\infty }={\displaystyle \frac{1}{N}}{\displaystyle {\sum }_{n=1}^N{a}_{ij}\left({\omega }_n\right)+\frac{1}{{\omega }_n}{\displaystyle {\int }_0^{\infty }{\kappa }_{ij}\left(t\right)\sin {\omega }_{n}tdt}}$$

(2)

$${\kappa }_{ij}\left(t\right)={\displaystyle \frac{2}{\pi }}{\displaystyle {\int }_0^{\infty }{b}_{ij}\left(\omega \right)\cos \omega td\omega }$$

(3)

where a_ij(ω_n) is the a_ij(ω) at the n-th incident frequency ω, and N is the total number of incidents. b_ij(ω) is radiation damping.

The total force matrix F(t) is defined as

$$F(t)=F_{\mathrm{wave}}(t)+F_{\mathrm{PTO}}(t)+F_{\mathrm{vis}}(t)+F_{\mathrm{stiff}}(t)$$

(4)

where F_wave(t), F_PTO(t), F_vis(t), and F_stiff(t) are the wave force, PTO force, viscous force, and mooring force, respectively, and are defined as

$$\left\{\begin{array}{l}{F}_{\mathrm{PTO}}(t)=b_{\mathrm{PTO}}\stackrel{·}{\xi }\left(t\right)\\ F_{\mathrm{vis}}(t)=b_{\mathrm{vis}}\stackrel{·}{\xi }\left(t\right)\\ F_{\mathrm{stiff}}(t)=k_{\mathrm{stiff}}\xi \left(t\right)\\ F_{\mathrm{wave}}(t)=R{\displaystyle {\sum }_{i=1}^N{F}_{\mathrm{wave}}({\omega }_i)A_i\cos \left({\omega }_{i}t+{\theta }_{\mathrm{wave}}\left({\omega }_i\right)+{\theta }_{\mathrm{rand},i}\right)}\end{array}\right.$$

(5)

where b_PTO, b_vis, and b_stiff are the PTO damping, viscous damping, and stiffness damping. A_i, ω_i, θ_wave(ω_i) and θ_rand,i are the wave amplitude, frequency, initial phase, and random phase of the i-th cosine wave, respectively. θ_rand,i represent a uniform distribution within the range of [0, 2π]. F_wave(ω_i) is the wave excitation force when A_i = 1.0 m at ω_i. R is the buffer function to ensure the wave force slowly loads onto the floater. In this paper, N = 85, ω_i ∈ [0.05, 4.25] rad/s, and Δω = 0.05 rad/s. b_vis is calculated from the free decay curve obtained from Star-CCM+ [13]. The viscous damping b_vis for the triangular-baffle-type WEC, box-type WEC, and floating breakwater in the heave motion direction are 6.02 × 10³ kg/s, 7.52 × 10³ kg/s, and 7.9 × 10⁶ kg/s, respectively. If b_vis is not considered, the annual power, annual energy production, and mooring loads are overestimated. b_stiff represents the equivalent stiffness matrix used to model the mooring system of the floating breakwater. The mooring system adopts a taut-leg configuration, as illustrated in Figure 1. The mooring lines are made of steel, with a diameter of 0.1 m, a mass per unit length of 113.3 kg/m, and an elastic modulus of 7.5 × 10⁹ N. The equivalent stiffness matrix of the taut-leg mooring system is presented as follows [41]:

$$\left[\begin{array}{cccccc}6.08\times {10}^8& -0.29& -3.70& 36.00& -5.51\times {10}^9& 1.65\times {10}^2\\ 3.76& 1.67\times {10}^8& -2.05& 2.95\times {10}^9& -17.60& -2.24\times {10}^2\\ -1.17& 0.15& 2.62\times {10}^8& 25.00& -21.40& -1.08\times {10}^2\\ -1.79& 2.95\times {10}^9& -0.36& 8.21\times {10}^8& -18.30& 18.60\\ -5.51\times {10}^9& 0.14& -0.61& 94.90& 2.06\times {10}^8& -4.70\times {10}^2\\ 3.99& -0.28& -1.84& 60.90& 11.10& 9.36\times {10}^8\end{array}\right]$$

The expressions for the buffer function R and wave amplitude A_i are

$$R=\left\{\begin{array}{cc}{\displaystyle \frac{1}{2}}\left[1-\cos \left({\displaystyle \frac{\pi t}{t_{\mathrm{bf}}}}\right)\right],& \left(t<t_{\mathrm{bf}}\right)\\ 1,& \left(t\ge t_{\mathrm{bf}}\right)\end{array}\right.$$

(6)

$$A_i=\sqrt{2S\left({\omega }_i\right)\Delta \omega }$$

(7)

where t_bf is the blocking time. S(ω_i) is the spectral density function of the Joint North Sea Wave Project (JONSWAP spectrum).

The constraint matrix D is expressed as follows [46]:

$${\mathit{D}}_{5n\times 6\left(n+1\right)}=\left[\begin{array}{cccccc}{D}_1^1& \dots & 0& \dots & 0& D_{n+1}^1\\ ⋮& \ddots & ⋮& \ddots & ⋮& ⋮\\ 0& \dots & D_i^i& \dots & 0& D_{n+1}^i\\ ⋮& \ddots & ⋮& \ddots & ⋮& ⋮\\ 0& \dots & 0& \dots & D_n^n& D_{n+1}^n\end{array}\right]$$

(8)

where

$$D_i^i=\left[\begin{array}{cccccc}1& 0& 0& 0& z_i-z_{\mathrm{c}i}& -\left(y_i-y_{\mathrm{c}i}\right)\\ 0& 1& 0& -\left(z_i-z_{\mathrm{c}i}\right)& 0& x_i-x_{\mathrm{c}i}\\ 0& 0& 0& 1& 0& 0\\ 0& 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 0& 1\end{array}\right],\left(i=1,n\right)$$

(9)

$$D_{n+1}^i=\left[\begin{array}{cccccc}-1& 0& 0& 0& -(z_i-z_{\mathrm{c}(n+1)})& y_i-y_{\mathrm{c}(n+1)}\\ 0& -1& 0& z_i-z_{\mathrm{c}(n+1)}& 0& -(x_i-x_{\mathrm{c}(n+1)})\\ 0& 0& 0& -1& 0& 0\\ 0& 0& 0& 0& -1& 0\\ 0& 0& 0& 0& 0& -1\end{array}\right],\left(i=1,n\right)$$

(10)

where (x_i, y_i, z_i) is the global coordinates of the connection point between the WEC and breakwater, and (x_ci, y_ci, z_ci) is the global rotation center of the floaters.

2.3. Wave Power of WECs

The wave power of the system at peak period T_p, including the total power P_total (T_p) and total power per unit mass P_ave (T_p) of P_total (T_p) to the mass of the WEC array m_total, is expressed as follows [41]:

$$\left\{\begin{array}{l}{P}_{\mathrm{total}}(T_{\mathrm{p}})={\displaystyle \sum _{i=-\left(n-1\right)/2}^{\left(n-1\right)/2}{P}_i}(T_{\mathrm{p}})\\ P_{\mathrm{ave}}(T_{\mathrm{p}})={\displaystyle \frac{P_{\mathrm{total}}(T_{\mathrm{p}})}{m_{\mathrm{total}}}}\end{array}\right.$$

(11)

where n is the total number of WECs. P_i (T_p) is the wave power of the i-th WEC, which is defined as

$$P_i(T_{\mathrm{p}})={\displaystyle \frac{b_{\mathrm{pto}}}{\Delta t}}{\displaystyle \underset{t}{\overset{t+\Delta t}{\int }}{v}_{3i}{(t)}^2\mathrm{d}t}$$

(12)

where v_3i(t) is the heave motion velocity of the i-th WEC, and b_opt is the optimal PTO damping, which is calculated through numerical optimization [47].

To assess the wave energy harvesting performance of the WEC, annual average wave power P_year and annual energy production W_AEP are adopted. They are defined as

$$\left\{\begin{array}{l}{P}_{\mathrm{year}}={\displaystyle \sum _{j=1}^N[}{({\displaystyle \frac{H_{\mathrm{i}}}{2}})}^2\times P_{\mathrm{total}}(T_j)\times S_j]\\ W_{\mathrm{AEP}}=P_{\mathrm{year}}\times t_{\mathrm{year}}\end{array}\right.$$

(13)

where T_j is the wave period, H_j is the wave height, and S_j is the occurrence probability for the j-th component in Table 1. N is the total number of components in the joint probability distribution table. t_year is the total seconds of one year.

The annual average wave power per unit mass, P_year/mass, and annual energy production per unit mass, W_AEP/mass, are defined as

$$\left\{\begin{array}{l}{P}_{\mathrm{year}/\mathrm{mass}}={\displaystyle \frac{P_{\mathrm{year}}}{m_{\mathrm{total}}}}\\ W_{\mathrm{AEP}/\mathrm{mass}}={\displaystyle \frac{W_{\mathrm{AEP}}}{m_{\mathrm{total}}}}\end{array}\right.$$

(14)

3. Validation

To demonstrate the accuracy of the present numerical model, the spectra of the surge, heave, and pitch motions of an OC3 Hywind [48] platform obtained through Fast Fourier Transform (FFT) are compared with the numerical results of Zhou et al. [49] in Figure 4. The constraint matrix and coupled hydrodynamic coefficients of the present model were verified in a previous study [41,50]. The OC3 Hywind platform can move in six DoFs. The JONSWAP spectrum is used with a spectral peak period of T_p = 8.54 s and significant wave height H_s = 3.42 m. The simulation length is 600 s, and the time step is 0.1 s. The parameters of the OC3 Hywind platform were introduced by Zhou et al. [49]. The maximum difference between these two results for Figure 4a–c is less than 5%, and the root mean square errors for surge, heave, and pitch motion are 1.8 × 10⁻⁸, 2.25 × 10⁻⁹, and 2.01 × 10⁻⁹, respectively. This demonstrates the accuracy of numerical simulation.

4. Numerical Results and Discussion

4.1. Evaluation of Wave Power for WEC

This section evaluates the wave energy harvesting performance of the hybrid system under irregular waves in the target sea area. Figure 5 illustrates the time history of wave power for both the hybrid system and the standalone WEC under irregular wave conditions with T_p = 6.96 s and H_1/3 = 2.17 m.

Figure 5 indicates that the wave power of the hybrid system with a triangular-baffle-type WEC array is larger than that of the corresponding isolated WEC array most of the time. This is because incident waves superimpose on the reflected waves by the breakwater, forming wave-focusing areas with higher wave height on the upward side of the breakwater [41]. Heave motion response of the WECs placed in these areas increases and leads to a larger wave power.

Figure 6a shows that the annual average wave power of the hybrid system under irregular waves is 1.16 MW. This is a 241.2% improvement compared to the wave power of 0.34 MW for the isolated WEC array in irregular waves. The annual energy production of the hybrid system under irregular waves reaches 10.16 × 10³ MW·h, which is 3.41 times the P_year of the isolated WEC array, as shown in Figure 6b. Under regular waves, the annual average wave power and annual energy production of the hybrid system are 1.05 MW and 9.20 × 10³ MW·h, respectively, which are smaller than those under irregular waves. Compared with the isolated WEC array, the P_year and W_APE of the hybrid system both increase by 238.7% under regular waves, similar to that under irregular waves, just as Huang et al. [51] concluded in their study on the integrated system of wave energy converters and breakwater.

Figure 7 illustrates the annual average wave power and total annual energy production for WECs numbered 0 to 9, for both the hybrid system and the isolated WEC array, due to the symmetrical configuration of the WECs. In the hybrid system with triangular–baffle–type WECs, WEC number 2 achieves the highest annual average wave power, whereas WEC number 9 generates the least power. When compared to the isolated WEC array, the annual average wave power of each WEC in the hybrid system increases, with WEC number 2 exhibiting the greatest increase of 283.1%, while WEC number 9 shows the smallest increase at 125.3%. A noticeable drop in the annual average wave power occurs as the WECs move closer to the end of the breakwater in the hybrid system. Figure 7b indicates that the variation in annual energy production follows a similar trend to that of the annual average wave power.

Figure 8 presents a comparison of the P_year/mass and the W_AEP/mass for the hybrid systems with different shapes of WECs and corresponding isolated WEC arrays. The P_year/mass of the hybrid system with box–type WECs is 3.44 W/kg, increasing by 244.0% compared with the isolated WEC array. The P_year/mass of the hybrid system with triangular–baffle–type WECs is 8.98 W/kg, which is 2.61 times that of the system with box–type WECs. This is mainly because much weaker vortices develop near the corner of the triangular–baffle–type WEC bottom than that of the box–type WEC bottom [19], leading to a smaller viscous damping b_vis used in Equation (5). A smaller viscous force is beneficial for the hydrodynamic performance of WEC, and thus, the wave energy harvesting performance can be improved. The ratio of annual energy production to the WEC array mass of the hybrid system with triangular–baffle–type WECs reaches 78.66 W·h/kg, while the hybrid system with box–type WECs is 30.13 W·h/kg. If it is assumed that the cost of energy harvesting in the hybrid system is directly related to the mass of the WEC, the above analysis indicates that the energy harvesting characteristics of the hybrid system with triangular–baffle–type WECs are superior to the hybrid system with box–type WECs under the same cost.

4.2. Evaluation of Motion Response for Breakwater

This section examines the motions of the floating breakwater within WEC array–floating breakwater hybrid systems under irregular wave conditions and compares them to the motions of isolated floating breakwaters, aiming to assess the impact of WECs on the breakwater’s motion.

Figure 9 shows the time history of the floating breakwater’s motion responses in the surge, heave, and pitch directions. It is evident that the surge and pitch motion amplitudes of the breakwater in the hybrid system are generally smaller than those of the isolated breakwater. However, the heave motion response of the breakwater in the hybrid system increases relative to the isolated breakwater, which is attributed to the damping force provided by the PTO system.

Figure 10 provides the standard deviations and maximum values of the breakwater motion responses for the hybrid system and the corresponding isolated breakwater, based on the statistical analysis of time history curves in Figure 9. Compared to the isolated breakwater, the standard deviations of the heave and pitch motion responses of the breakwater in the hybrid system with triangular-baffle-type WECs reduce by 27.5% and 30.0%, respectively, as shown in Figure 10. This is attributed to the harvesting of incident wave energy in WECs, thereby reducing the wave forces acting on the breakwater. However, the standard deviation of the heave motion response of the breakwater in the hybrid system with triangular-baffle-type WECs increases by 92.8% due to the PTO damping force. The variation trend of the maximum values of the motion responses in different directions is consistent with the standard deviations. These findings indicate that the presence of the WEC array reduces the surge and pitch motion of the floating breakwater in the hybrid system, but the impact of the PTO damping force increases the heave motion of the breakwater in the hybrid system significantly.

Figure 11 shows the σ and ζ_max values of the breakwater motion responses for the hybrid systems with triangular–baffle–type and box–type WECs. It can be observed that the standard deviations of the heave and pitch motion responses of the breakwater for the hybrid systems with triangular–baffle–type are similar to those with box–type WECs, with a difference of 28.2% in the standard deviation for the heave direction. The maximum values of the hybrid system with triangular–baffle–type WECs are larger than those with box–type WECs, with increments of 19.0%, 42.0%, and 66.7% in surge, heave, and pitch directions, respectively.

4.3. Evaluation of Mooring Forces

In this section, the evaluations for the mooring forces of the hybrid system with triangular-baffle-type WECs are conducted. Figure 12 and Figure 13 present the time history curves of mooring forces for the hybrid system in the heave, pitch, and surge directions under irregular waves, along with the maximum values and standard deviations of the mooring forces in each direction obtained from these curves.

Compared to the isolated breakwater, Figure 12 illustrates a decrease in the mooring forces in both the surge and pitch directions for the hybrid system. This is caused by the WECs harvesting a part of the incident wave energy and, thus, reducing the wave energy acting on the breakwater. At this time, the motion responses of the floating breakwater decrease, thereby reducing the mooring forces. However, an increase in the heave direction is observed, which is attributed to the damping force from the PTO system. For a more detailed quantitative analysis, Figure 13 provides the σ and F_max of the mooring forces, derived from a statistical analysis of the time history curves between 180 and 540 s. Compared to the isolated breakwater, the hybrid system shows a reduction in the standard deviations of the mooring forces in the surge and pitch directions by 13.8% and 26.9%, respectively. In the heave direction, the standard deviation increases by 90.9% due to the influence of the PTO damping force, though it remains smaller than standard deviations in the surge and pitch directions. The standard deviation of mooring forces in the pitch direction is the largest; it is 2.55 times larger than the heave direction and 4.05 times larger than the surge direction.

The trends in the maximum values of mooring forces follow a similar pattern to the standard deviations, as shown in Figure 13b. Compared to the isolated breakwater, the maximum mooring forces in the surge and pitch directions for the hybrid system decrease by 9.6% and 21.0%, respectively. In contrast, the maximum mooring force in the heave direction increases by 103.1%. This analysis indicates that while the WEC array reduces mooring forces in the surge and pitch directions for the hybrid system, the PTO damping force significantly increases the mooring forces in the heave direction and, therefore, the vertical strength of the mooring system can be appropriately enhanced during the design stage.

Figure 14 shows the σ and F_max of the mooring forces for the hybrid systems with triangular–baffle–type and box–type WECs. From Figure 14a, the standard deviations of the mooring forces for both hybrid systems are quite similar, with a maximum difference of only 9.5% observed in the heave direction. The maximum mooring forces in the hybrid system with triangular–baffle–type WECs surpass those of the system with box–type WECs, showing increases of 7.9%, 19.8%, and 15.9% in the surge, heave, and pitch directions, respectively. This analysis indicates that the mooring forces for the hybrid system with triangular–baffle–type WECs are larger than those with box–type WECs, leading to more demanding requirements for the mooring system of the former.

5. Conclusions

In this paper, a numerical model based on the linearized potential flow theory and the Cummins method is developed to evaluate the wave energy harvesting characteristics, motion of the floating breakwater, and the mooring forces of an optimized WEC array–floating breakwater hybrid system under irregular wave conditions. The following conclusions can be drawn from this study:

(1)

In irregular waves, the annual average wave power and the annual energy production of the hybrid system with a triangular-baffle-type WEC array are 1.16 MW and 10.16 × 10³ MW·h, respectively, which is a 241.2% improvement compared to those of the isolated WEC array. This configuration is highly recommended for irregular wave conditions to maximize annual energy production and reduce the levelized cost of energy.

(2)

Annual average wave power and annual energy production of the hybrid system under regular wave conditions are smaller than those under irregular wave conditions. Design evaluations must rely on irregular wave inputs rather than regular-wave tests to avoid underestimating actual energy yield.

(3)

Compared to the isolated WEC array, the standard deviations of the mooring forces for the hybrid system with a triangular-baffle-type WEC array in the surge and pitch directions reduce by 13.8% and 26.9%, respectively, while increasing by 90.0% in the heave direction. This indicates that the vertical mooring design must be reinforced.

(4)

The wave energy harvesting characteristics and maximum mooring forces of the hybrid system with triangular-baffle-type WECs are both superior to those of the hybrid system with box-type WECs under the same cost. It means that under equivalent budget constraints, the triangular-baffle-type WEC is the preferred choice for better energy capture, lower peak mooring forces, and higher cost-effectiveness.

The numerical model employed in this study is linear and, therefore, cannot accurately account for the effects of nonlinear factors on the device power performance and mooring system under extreme sea conditions. Ignoring nonlinearities tends to overestimate the power conversion performance while underestimating the mooring forces.