Two-mode WEC-type attachment for wave energy extraction and reduction of hydroelastic response of pontoon-type VLFS

Ocean Engineering 197 (2020) 106875

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Two-mode WEC-type attachment for wave energy extraction and reduction of hydroelastic response of pontoon-type VLFS H.P. Nguyen a, C.M. Wang a, *, V.H. Luong b a b

School of Civil Engineering, The University of Queensland, St Lucia, Queensland, 4072, Australia Department of Civil Engineering, Ho Chi Minh City University of Technology, Vietnam National University Ho Chi Minh City, Viet Nam

A R T I C L E I N F O

A B S T R A C T

Keywords: Very large floating structure Hydroelastic response Raft wave energy converter Oscillating wave surge converter

Proposed herein is a two-mode wave energy converter-type attachment to the fore edge of pontoon-type very large floating structures (VLFSs) for wave energy extraction and reduction of hydroelastic response. This attachment consists of floating horizontal and submerged vertical plates that are connected to the VLFS with hinges and linear power take-off systems. The use of both floating horizontal and submerged vertical plates combines the superior performances of floating plates in extracting wave energy from short waves (T < 5 s) and of submerged vertical plates in extracting wave energy from intermediate and long waves (T > 7 s). When deployed in sites where 5 s � T � 7 s, the two-mode attachment yields a larger power production than the sole usage of floating plates or the sole usage of submerged vertical plates. Interestingly, the two-mode attachment also combines the superior performance of submerged vertical plates in reducing hydroelastic response for relatively short waves (T < 7 s) and the superior performance of floating plates in reducing hydroelastic response for relatively long waves (T � 9 s). Moreover, this two-mode attachment furnishes a good balance between extracting wave energy and reducing hydroelastic response of VLFS for the same wave period.

1. Introduction Since wave energy converters (WECs) were first studied by Yoshio Masuda in 1940s, many techniques for extracting wave energy have been proposed, and by 1980, more than 1000 WEC-related patents had ~o, 2010). Nowadays, although the number of available been filed (Falca techniques for extracting wave energy remains large, only a few of them have been adopted for wave energy projects that include model tests in real sea conditions. According to CSIRO (2012), only about six WECs had been tested in real sea conditions. Among these WECs, the Pelamis and Oyster WECs have received a great worldwide attention from en­ gineers and scientists (Whittaker and Folley, 2012; Yemm et al., 2012). The Pelamis WEC, which may be also called a raft WEC (Haren and Mei, 1979), consists of pontoons connected to one another with hinges and Power Take-Off (PTO) systems. The Oyster WEC, which may be also called the oscillating wave surge converter OWSC (Whittaker and Folley, 2012), comprises a submerged vertical plate connected to a base struc­ ture at the seabed with a hinge and PTO system. The main advantage of the raft WEC lies in the high wave energy conversion efficiency when compared to other WEC concepts (Yemm et al., 2012). On the other hand, the main advantage of the OWSC is its wide efficiency bandwidth

(Flocard and Finnigan, 2010), and hence it may be effectively used for extracting wave energy for a wide range of wave conditions which in­ cludes a relatively long wave condition. Although a great deal of effort has been invested in the development of WECs for the last few decades, WEC technology remains at an early implementation stage as compared to other renewable energy technol­ ogies as well as the conventional energy technologies such as coal and fossil fuel energy technologies (Falc~ ao, 2010; Mustapa et al., 2017). This makes the electricity generated from ocean waves currently much more expensive than the electricity generated from other sources of energy (IRENA, 2014). One possible way to reduce the cost of wave power is to improve the power capture factor by innovative designs of wave energy converters. Another promising way is to reduce costs for installation, mooring/foundation, operation and maintenance. According to the In­ ternational Renewable Energy Agency report (IRENA, 2014), these costs may account for about 41% of wave energy project life cost. These costs may be reduced through cost sharing between WECs and other purpose marine structures when integrated (IRENA, 2014; Mustapa et al., 2017). Other potential benefits of integration of WECs with other purpose marine structures include:

* Corresponding author. E-mail address: [email protected] (C.M. Wang). https://doi.org/10.1016/j.oceaneng.2019.106875 Received 24 May 2019; Received in revised form 22 November 2019; Accepted 22 December 2019 Available online 9 January 2020 0029-8018/© 2020 Elsevier Ltd. All rights reserved.

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(i) limiting the negative environmental impact (IRENA, 2014); (ii) reducing hydrodynamic motions of marine structures under wave actions as WECs are wave energy absorbers; (iii) improving the ability of marine structures in breaking waves if the marine structures are partly/fully functioned as breakwaters; (iv) providing power supply for operations on marine structures; (v) increasing the reliability, and hence lifetime of WECs by providing additional strength to WECs to withstand strong waves and by allowing some components of WECs to be placed on the marine structures (Mustapa et al., 2017) which are usually more stable under wave action as compared to stand-alone WECs.

the fore edge of VLFS with hinges and linear PTO systems. These auxiliary plates are of the same dimensions, and are connected to different portions of the fore edge of VLFS. This WEC-type attachment allows wave energy to be extracted by two different modes, i.e. from using floating auxiliary plates and from using submerged vertical plates. In this study, the performance of the two-mode WEC-type attachment is examined for various mode configurations which are specified by the numbers and locations of the floating horizontal auxiliary plates and the submerged vertical auxiliary plates. Here, the performance of a WECtype attachment is assessed via the power production, reduction in hydroelastic response of VLFS, and horizontal linear wave forces acting on submerged vertical plates which are then transmitted to the mooring system via the VLFS. For the hydroelastic analysis, VLFS is modelled as a plate (Wang and Wang, 2008), while the linear wave theory (Faltinsen, 1990; Sarpkaya and Isaacson, 1981) is used for modelling fluid motions. The hydroelastic analysis is performed in the frequency domain by using the finite element – boundary element (FE-BE) method (Kim et al., 2013; Wang and Wang, 2008).

Examples of the integration between WECs and marine structures are the WEC-type breakwater (Mustapa et al., 2017), and the combined wind-wave energy converter (P� erez-Collazo et al., 2015). On another front of ocean utilization, pontoon-type very large floating structures (VLFSs) have been shown to be one promising solu­ tion for creating artificial land on the ocean (Lamas-Pardo et al., 2015; Suzuki, 2005; Watanabe et al., 2004). VLFS usually has large horizontal dimensions as compared to wavelengths and relatively small flexural rigidity (Wang et al., 2007). Because of these two properties, the dy­ namic response of VLFS under wave action is dominated by elastic de­ formations, and hence may be called hydroelastic response. There are many methods proposed for reducing the hydroelastic response of VLFS under wave action, as described in a review paper by Wang et al. (2010). These methods include the use of bottom-founded breakwaters (Ohmatsu, 2000) or floating breakwaters (Tay et al., 2009); floating auxiliary structures attached to the fore of VLFS (Kim et al., 2005); submerged vertical/horizontal plates or submerged box-shaped structures rigidly connected to the fore of VLFS (Ohta et al., 1999; Pham et al., 2008; Takagi et al., 2000); and semi-rigid connections between VLFS modules (Riyansyah et al., 2010). Other more recent methods are the use of dual inclined perforated submerged plates (Cheng et al., 2016); vertical mooring lines (Nguyen et al., 2018); and vertical porous barriers (Singla et al., 2018). Motivated by the benefits of the integration between WECs and other purpose marine structures, and the necessity to reduce hydroelastic response of VLFS, Tay (2019) recently proposed a raft WEC-type attachment for extracting wave energy while reducing hydroelastic response of VLFS. This attachment consists of a single wide pontoon connected to the fore edge of VLFS with hinges and linear Power Take-Off (PTO) systems. By using the same wave energy conversion concept, Ren et al. (2019) studied the use of PTO systems placed at the outermost hinge connection of multi-purpose floating structures. By adopting the modular concept for the OWSC (Sarkar et al., 2016; Wil­ kinson et al., 2017), Nguyen et al. (2019a) proposed a modular raft WEC-type attachment where the single wide pontoon of the raft WEC-type attachment (Tay, 2019) is replaced by multiple independent narrow pontoons. This replacement brings several advantages, including: (i) flexibility for downsizing/expansion of the overall width of the attachment; (ii) increasing reliability of the attachment as the system has higher redundancy; (iii) lower costs for manufacturing and deployment owing to the modularity in design; and (iv) more power produced in oblique wave conditions if the width of the whole device is comparable to the wavelength (Nguyen et al., 2019a; Wilkinson et al., 2017). However, the effectiveness of the modular raft WEC-type attachment in extracting wave energy is rather limited for relatively long waves (Nguyen et al., 2019b). To overcome this limitation of the modular raft WEC-type attach­ ment, and motivated by the advantages of the OWSC and the raft WEC, this study presents a novel two-mode WEC-type attachment comprising floating horizontal plates and submerged vertical plates connected to

2. Problem definition, assumptions and formulation 2.1. Problem definition and assumptions Consider a rectangular VLFS which is modelled as a floating rect­ angular plate with length L, width B and thickness h, as shown in Fig. 1. The plate has Young’s modulus E, Poisson’s ratio ν, and mass density ρ. There are Np number of thin and rigid auxiliary plates connected to the fore edge of the VLFS with hinges. For convenience, these plates are numbered from 1 to Np, starting from the plate with the smallest y-co­ ordinate. Each auxiliary plate is either a floating (horizontal) plate or submerged vertical plate, and has length Lp, width Bp, thickness hp and mass density ρp1 for floating auxiliary plates or ρp2 for submerged ver­ tical plates. Note that B ¼ Np � Bp. The gap between auxiliary plates and the fore edge of VLFS is denoted by s. The zero gap between auxiliary plates is considered by referring to the studies on the modular OWSC (Michele et al., 2016) and on the Venice storm gate (Adamo and Mei, 2005). It is assumed that the horizontal motions of the VLFS are completely constrained by the mooring dolphin-rubber fender system. For simplicity, the draft of VLFS and the floating auxiliary plates are assumed to be zero (Gao et al., 2011). Power Take-Off (PTO) systems are installed in between auxiliary plates and the fore edge of VLFS (see Fig. 1). The behaviour of PTO systems is assumed to be linear, and hence the ith PTO system (i 2 {1, 2, …, Np}) may be modelled as a rotational linear damper (Zheng et al., 2015) with damping coefficient cpi. In Fig. 1b, the subscript j 6¼ i and j 2 {1, 2, …, Np}. The VLFS with the two-mode WEC-type attachment floats on an ideal fluid domain, and is subjected to regular waves. A regular wave is characterized by wave period T (with the corresponding wavelength λ) and wave amplitude A. The wave amplitude is assumed to be small where the linear wave theory may be used for modelling fluid motions. The head sea condition is considered. The seabed is assumed to be flat and the water depth is denoted by H. The mode configuration MC of a two-mode WEC-type attachment may be represented by a string of letters involving H and V. For example, if MC ¼ VHV, it means that the first and the third auxiliary plates are submerged vertical plates while the second auxiliary plate is a floating (horizontal) plate.

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Fig. 1. VLFS with two-mode WEC-type attachment: (a) plan view, (b) side view.

2.2. Equation of motion for VLFS with two-mode WEC-type attachment

�

Similar to a conventional floating structure, the displacement vector utn of an arbitrary point within the VLFS with two-mode WEC-type attachment consists of six components, i.e. utn ¼ [ut1 ut2 ut3 ψ 1 t ψ 2 t ψ 3 t]T where ut1, ut2 and ut3 are, respectively, the displacements along the x-, yand z-axes and ψ 1 t, ψ 2 t and ψ 3 t are, respectively, the rotations about the x-, y- and z-axes; the subscript T denotes the transpose; and t denotes the time. In the frequency domain, the displacement vector may be expressed as � � (1) utn ðx; y; z; tÞ ¼ Re un ðx; y; zÞe iωt ;

ω2 ½M þ Ma ðωÞ�

� �� � � u ¼ Fexc ; iω Cp þ Cd ðωÞ þ K þ Krf b

(2)

where M is the global mass matrix, Ma the global matrix of added mass, K the global stiffness matrix, Krf the global matrix of restoring stiffness which results from the action of the hydrostatic force and the gravita­ tional force (Senjanovi�c et al., 2008), Cp the global matrix of PTO damping, Cd the global matrix of hydrodynamic damping, Fexc the global vector that consists of frequency-dependent complex amplitudes of nodal excitation wave forces, b u the vector of the frequency-dependent b complex amplitudes of displacements of the structure. The vector u comprises the vectors b u n of all nodes of the discretised model. The determination of the matrices Ma, Cd and the vector Fexc (called hy­ drodynamic matrices/vector) may be done by using the boundary element method as given in Sec. 2.3 and in Appendix A. The matrices K and M may be obtained using the numerical pro­ cedures for shell elements (Fu et al., 2007; Liu and Quek, 2003). The hinge connections between the auxiliary plates and the VLFS are taken into account by modifying the stiffness matrix K using the penalty method (Liu and Quek, 2003). The constraints that the horizontal mo­ tions (i.e. the motions along x- and y-axes) as well as the rotation about z-axis at any points within the VLFS are zero because of the presence of the mooring system may be imposed by using the same approach as for the hinge connections. If the kth node and the lth node are connected via the ith PTO system, the non-zero components of the matrix Cp are (Nguyen et al., 2019a; Zheng et al., 2015): Cp(l5,l5) ¼ Cp(k5,k5) ¼ cpi, Cp(l5,k5) ¼ Cp(k5,l5) ¼ -cpi where the subscript 5 denotes the degree of freedom corresponding to the rotation about y-axis. A general calculation approach for Krf has been given by Kim et al. (2013). However, because the drafts of VLFS and floating auxiliary plates are zero, and the submerged vertical auxiliary plates are thin and rigid, the restoring stiffness matrix may be calculated in a simpler manner. In sum, the restoring stiffness matrix Krf comprises the restoring

pffiffiffiffiffiffi where Re [] implies the real part, i ¼ 1 , ω is the incident wave frequency, un ¼ [u1 u2 u3 ψ 1 ψ 2 ψ 3]T is the vector of the frequencydependent complex amplitudes of displacements. The fluid-structure interaction analysis of VLFS with two-mode WECtype attachment in the frequency domain may be carried out by using the hybrid finite element-higher order boundary element method (Kim et al., 2013; Wang and Wang, 2008). In this method, the VLFS and the auxiliary plates are first discretised into elements. Each element has ne nodes, and each node has six degrees of freedom that correspond to the six components of the nodal displacement vector. This vector is given by b1ψ b2ψ b 3 �T where the “^” sign above each variable in­ bn ¼ ½b u u1 b u2 b u3ψ dicates the nodal variable. The vector un of an arbitrary point in the b e where Ne is structure may be approximated by the formula un ¼ Ne u the matrix containing the interpolation function values of the eth element where the considered point is in, and b u e is the vector containing b n of all nodes of the eth element (Liu and Quek, 2003). the vectors u By following the finite element procedure (Kim et al., 2013; Wang and Wang, 2008), the equation of motion for VLFS with two-mode WEC-type attachment may be written in the following matrix form:

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stiffness matrix of the VLFS and the floating auxiliary plates Khrf , and that of the submerged vertical plates Kvrf , and given by # " Khrf 0 Krf ¼ : (3) 0 Kvrf

∂φ ¼ 0 on ​ the ​ seabed; ∂z lim

φin ¼

φin Þ

¼ 0 on the infinite surfaces;

(11)

gA cosh kðz þ HÞ ikx e : cosh kH ω

(12)

On the wetted surface of the structure SHB, the velocity of the fluid that are normal to the wetted surface must be equal to the velocity of the structure, as follows (Faltinsen, 1990; Sarpkaya and Isaacson, 1981):

∂φ ¼ ∂n

iωuj nj on SHB;

(13)

where ∂/∂n indicates the differentiation along the unit normal vector outward from the structure to the fluid, the subscript j varying from 1 to 3 denotes the x-, y-, z-components of the unit normal vector and the displacement vector. The hydrodynamic pressure acting on the structure Ptd may be expressed as � � Ptd ðx; y; z; tÞ ¼ Re Pd ðx; y; zÞe iωt ; (14) where Pd is the frequency-dependent complex amplitude of hydrody­ namic pressure acting on the structure, and is given by (Faltinsen, 1990; Sarpkaya and Isaacson, 1981) Pd ¼

(15)

iωρw φj nj ;

where φ1 ¼ φ2 ¼ φ3 ¼ φ.

where S ¼ {S1, S2, …, SNe}, ς ¼ {ς1, ς2, …, ςNe}T, and

2.3.2. Boundary integral equation By following the Green’s second identity procedure, the Laplace equation and the boundary conditions on the boundary surfaces (i.e. Eqs. (8)–(11), and Eq. (13)) may be transformed into the following boundary integral equation (Kim et al., 2013; Wang and Wang, 2008): � Z � ∂Gðx; ξÞ αφðxÞ ¼ φin ðxÞ þ (16) φðξÞ iωGðx; ξÞuj ðξÞnj ðξÞ dSHB ; ∂nðξÞ

(6)

where Sq and ςq (q ¼ {1, 2, …, Ne}) are respectively the eigenvector and the amplitude corresponding to the qth degree of freedom, Ne is the number of eigenvectors for accurate approximation. While the eigen­ vector may be obtained from the free vibration analysis of the structure, the amplitude vector ς is unknown and may be obtained by solving Eq. (5). The vector of frequency-dependent complex amplitudes of nodal displacements of the structure is then obtained by using Eq. (6).

SHB

where x ¼ (x, y, z) is the source point, ξ ¼ (ξ, η, ζ) is the field point, α is the solid angle coefficient that is equal to 1/2 if the source point x is within the wetted surfaces of the submerged vertical plates, and is equal to 1 if x is within the wetted surfaces of the floating (horizontal) auxiliary plates and the VLFS, G is the free surface Green function that satisfies the Laplace equation and the boundary conditions on the seabed, the free water surface as well as the infinite surfaces. The free surface Green function is given by (Linton, 1999)

2.3. Formulation of the fluid 2.3.1. Hydrodynamic equations The fluid motion may be represented in terms of the velocity po­ tential φt (Sarpkaya and Isaacson, 1981). In the frequency domain, the velocity potential is given by � � (7) φt ðx; y; z; tÞ ¼ Re φðx; y; zÞe iωt ;

∞ X

Gðx; ξÞ ¼

where φ(x, y, z) is time-independent and is usually called the spatial velocity potential. This spatial velocity potential must satisfy the following Laplace equation (Faltinsen, 1990; Sarpkaya and Isaacson, 1981):

K ðk RÞ � 0 m �coskm ðz þ HÞcoskm ðζ þ HÞ; m¼0 π H 1 þ sin 2km H 2km H

(17)

where R is the horizontal distance between the source point x and the qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi field point ξ and this distance is defined as R ¼ ðx ξÞ2 þ ðy ηÞ2 , km

is a positive root number that satisfies the equation kmtan (kmH) ¼ ω2/g if m � 1, k0 ¼ -ik, K0 is the modified Bessel function of the second kind of order zero.

(8)

The boundary conditions are expressed as (Faltinsen, 1990; Sarpkaya and Isaacson, 1981)

∂φ ω2 ¼ φ on ​ the ​ free ​ surface; ∂z g

� ikðφ

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where jxj is given by jxj ¼ x2 þ y2 , the wave number k must satisfy the dispersion relation k tanh (kH) ¼ ω2/g, the spatial velocity potential of the incident wave φin is given by

where rp is the vertical distance between the considered point within the submerged vertical plates to the hinge connection. Eq. (4) indicates that the (rotational) restoring stiffness at an arbitrary point ðρp2 ρw Þghp rp is only positive if the mass density of the submerged vertical plate is larger than the water mass density. This condition is in contrast to the corre­ sponding condition for the OWSC. This is because the (rotational) restoring stiffness at an arbitrary point within the submerged vertical flap of the OWSC should be ðρw ρp2 Þghp rp (Noad and Porter, 2015), which is only positive if the mass density of the submerged vertical plate is smaller than the water mass density. In order to reduce the size of the system of linear equations in Eq. (2) so as to speed up the computation, the modal expansion method (Kim et al., 2013; Nguyen et al., 2019b) is employed. Following the modal expansion method, Eq. (2) may be rewritten as � � �� � � ST ω2 ½M þ Ma ðωÞ� iω Cp þ Cd ðωÞ þ K þ Krf Sς ¼ ST Fexc ; (5)

r2 φðx; y; zÞ ¼ 0:

φin Þ ∂jxj

jxj→∞

The restoring stiffness matrix of the VLFS and the floating auxiliary plates Khrf may be calculated in the same manner as for the case where the VLFS and the floating auxiliary plates resting on a Winkler founda­ tion with stiffness ρwg where g is the gravitational acceleration and ρw the water mass density (Wang and Wang, 2008). The restoring stiffness matrix of the submerged vertical plates may be obtained based on the following formulation of the frequency-dependent complex amplitude f vrf of the restoring moment that acts on an arbitrary point within the submerged vertical plates: � � � f vrf ¼ ρp2 ρw ghp rp ψ 2 ; (4)

b ¼ ς1 S1 þ ς2 S2 þ … þ ςNe SNe ¼ Sς; u

� pffiffi ∂ðφ x

(10)

2.3.3. Calculation of hydrodynamic matrices/vector The matrices Ma, Cd and the vector Fexc in Eq. (2) are related to the vector of frequency-dependent complex amplitudes of nodal

(9)

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H.P. Nguyen et al.

hydrodynamic force Pg , as follows: � u ¼ Pg : Fexc þ ω2 Ma þ iωCd b

submerged vertical plates. (18)

3. Verification

The vector Pg may be obtained by applying the boundary element procedure for Eq. (15) and Eq. (16). The details of the boundary element procedure are given in Appendix A.

Because numerical/experimental results of VLFS with two-mode WEC-type attachment are currently not available, the present formula­ tion and computer code are verified for two related problems, including: (i) VLFS without the WEC-type attachment and (ii) OWSC comprising a submerged vertical plate connected to a base structure fixed at the seabed. For the first problem, the input for verification is given in Table 1. Fig. 2 shows that the numerical results obtained by using the present formulation and computer code agree very well with the pub­ lished numerical results (i.e. “Ref. Num.“) (Fu et al., 2007) as well as the published experimental results (i.e. “Ref. Exp.“) (Yago and Endo, 1996). Note that, for VLFS consisting of two interconnected modules, the nu­ merical results obtained by using the present formulation and computer code also agree very well with the published numerical results (Fu et al., 2007), as presented in our previous study (Nguyen et al., 2019a). For the second problem, the present numerical results are first compared with the published numerical results (Yu et al., 2016) for the OWSC defined in Fig. 3a. The mass density of the submerged plate is 300 kg/m3. The present numerical results are also compared with the experimental results given by Henry (2008) for the OWSC defined in Fig. 4a. The oscillating submerged plate of OWSC in Fig. 4a is not con­ nected directly to the seabed with a hinge (like so of OWSC in Fig. 3a), but it is connected to a base structure fixed at the seabed, and the hinge connection between the base structure and the oscillating plate is located at 1.5 m above the seabed. The comparison with experimental results is only carried out for the excitation wave torque Mexc acting on the oscillating submerged plate because of its availability (Henry, 2008). As the excitation wave torque is independent with respect to the mate­ rial properties of the submerged plate as well as the properties of PTO system, these parameters are not specified. Although the experiment was carried out in a wave flume, the measured excitation wave torque may still be used for comparison with the numerically estimated exci­ tation torque in an open sea which is considered in the present formu­ lation. This is because the width of the wave flume is about 5 times larger than the width of the scaled model (Henry, 2008), which may make the measured excitation wave torque negligibly affected by the wave flume walls (Chakrabarti, 1999). Figs. 3b and 4b show that the present numerical results show a very good agreement with published numerical/experimental results for all considered cases; thereby con­ firming the validity and accuracy of the present formulation and com­ puter code.

2.4. Performance indices The performance of two-mode WEC-type attachment is assessed by the power capture factor as well as the hydroelastic response reduction. In addition, as the two-mode WEC-type attachment consists of several submerged vertical plates, the horizontal wave force acting on these submerged plates is much larger than that acting on the floating hori­ zontal auxiliary plates which have a shallow draft (assumed to be zero in this study for simplicity). Thus, the total horizontal (linear) wave force acting on the submerged vertical plates is also considered in the per­ formance assessment of the two-mode WEC-type attachment. The power capture factor is the ratio between the total power pro­ duced Pav and the incident wave power Pin, and is written as (19)

CF ¼ Pav/Pin.

The determination of the power produced Pav may be done using the following formula (Zheng et al., 2015): N

p 1 X Pav ¼ ω2 cp jΔψ 2i j2 ; 2 i¼1 i

(20)

where jΔψ 2i j is the amplitude of the relative rotation about the y-axis between the two nodes that are connected to each other via the ith PTO system. The power of incident wave Pin is defined as (Falnes, 2002) � � 1 ρ gA2 ω 2kH B; (21) Pin ¼ w 1þ 4 sinh 2 kH k The hydroelastic response reduction HR may be defined as the reduction in the amplitude of the maximum vertical displacement over the entire VLFS, as follows: (22)

HR ¼ 1- |u3|max/|u03|max,

where |u03|max is the maximum vertical displacement amplitude of the VLFS without the two-mode WEC-type attachment, and |u3|max the maximum vertical displacement amplitude of the VLFS with the attachment. The frequency-dependent complex amplitude of the total horizontal wave force acting on submerged vertical plates is given by 0 1 Z Z vþ v Fx ¼ iωρw @ φdS (23) φdS A: Svþ

4. Performance of two-mode WEC-type attachment The VLFS with physical and geometric parameters given in Table 1, which was used in previous research studies (Fu et al., 2007; Nguyen et al., 2019a; Yago and Endo, 1996), is again adopted for numerical studies in this section. The number of auxiliary plates and their di­ mensions are selected as in Table 2 by referring to the designs of Langlee WEC and the Oyster WEC (Babarit et al., 2012). The auxiliary plate thickness is 1 m which is relatively similar to the thickness of the OWSCs in Sec. 3. The mass density of floating auxiliary plates and that of sub­ merged vertical plates are chosen as in Table 2. The gap between the auxiliary plates and the fore edge of VLFS is 1 m (Liu et al., 2017; Yu et al., 2016; Zheng et al., 2015). With regard to properties of sea sites, the water depth H ¼ 20 m and wave period T ranging from 4 s to 16 s are considered. As CF and HR are independent on the wave amplitude A, the wave amplitude A is not specified for numerical studies. However, the wave amplitude A needs to be small to ensure that the adopted linear wave theory is valid. For simplicity, the mode configuration of the attachment is assumed to be symmetrical with respect to Ox. The performance of two-mode WEC-type attachment is assessed for all 8 possible mode

Sv

where Svþ indicates the right wetted surface (perpendicular to Ox) of the submerged vertical plates and Sv indicates the left wetted surface of the Table 1 Input data for verification. Parameter

Magnitude

Unit

L B h

300 60 2 256.25 11.9E9 0.13 58.5 {0.2, 0.6}

m m m kg/m3 N/m2 – m –

ρ

E

ν

H λ/L

5

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Fig. 2. Normalized vertical displacement amplitude along the longitudinal centerline of the VLFS: (a) λ ¼ 60 m, (b) λ ¼ 180 m.

Fig. 3. (a) OWSC considered for comparison with numerical results (unit of dimensions and wave amplitude: m), (b) The amplitudes of the rotation about y-axis for various cases.

Fig. 4. (a) OWSC considered for comparison with experimental results (unit of dimensions and wave amplitude: m), (b) Amplitudes of excitation wave torque acting on oscillating submerged plate for various wave periods.

configurations, including: (1) HHHHHH (or only the first three letters HHH for brevity because of the symmetry); (2) VVV; (3) HVV; (4) VHH; (5) VHV; (6) HVH; (7) HHV; (8) VVH. The PTO damping coefficients are optimally selected within [1E41E8] Nms/rad for maximizing either CF or HR for each wave period and each mode configuration. In Sections 4.1, 4.2 and 4.3, the objective function is maximizing CF whereas in Sections 4.4 and 4.5, the objective function is either maximizing HR or maximizing CF. The upper bound of

PTO damping coefficients is set to 1E8 Nms/rad by referring to the design of Oyster WEC (Babarit et al., 2012). Unless specified hereafter, the PTO damping coefficient of each PTO system is considered as an independent design variable. This unequal PTO damping strategy was adopted for the modular OWSC (Sarkar et al., 2016). The mathematical statement of the optimization problem is given in Appendix B. The optimization process to obtain the optimal PTO damping co­ efficients is based on differential evolution (DE) (Price et al., 2005). DE 6

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generally outperforms the genetic algorithms (GAs), which have been used in optimization studies of WEC and VLFS (Gao et al., 2013; Michailides and Angelides, 2015), with respect to the convergence rate and the quality of the optimal solution (Hegerty et al., 2009). DE has recently been used for research studies on VLFS and WEC (Nguyen et al., 2019b, 2018). The details of the optimization process based on DE may be found in Appendix B.

Table 2 Input data for numerical studies. Parameter

Magnitude

Unit

Lp Np hp

8.5 6 1 512.5 1025 1 [1E4-1E8] 20 [4–16]

m – m kg/m3 kg/m3 m Nms/rad m s

ρp1 ρp2 s cpi H T

4.1. Power production Fig. 5 shows the variations of the power capture factor and the corresponding power produced with respect to wave period for all 8 mode configurations of the two-mode WEC-type attachment. The optimal PTO damping coefficients for maximizing CF are given in Fig. 6.

Fig. 5. Variations of (a) power capture factor and (b) power produced with respect to wave period for all 8 mode configurations and for the case where PTO damping coefficients are optimized for maximizing CF.

Fig. 6. Optimal PTO damping coefficients for maximizing CF: (a) cp1,6, (b) cp2,5, (c) cp3,4. 7

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Fig. 5 shows that for MC ¼ HHH, CF is largest when T ¼ 4 s and then it decreases with respect to wave period. The largest CF is observed for T ¼ 4 s because for this wave period, the ratio between the length of the auxiliary plates and the wavelength is close to 1/3 which is the ideal ratio for using floating auxiliary plates in wave energy extraction (Nguyen et al., 2019a). The corresponding ideal ratio for the raft WEC is about 1/2 (Zheng et al., 2015). For MC ¼ VVV, Fig. 5 shows that CF first increases with respect to wave period, and reaches its maximum value at a certain wave period (T ¼ 7 s). Thereafter, CF decreases with increasing wave period. This variation for MC ¼ VVV is relatively similar to the corresponding vari­ ation for the OWSC (Renzi and Dias, 2013). Fig. 5 shows that among the 8 mode configurations, preferable mode configurations for wave energy extraction are changed significantly for three different ranges of wave periods, which are indicated by the three different shaded areas. For T < 5 s, a mode configuration with a larger number of floating (horizontal) auxiliary plates shows larger power production. This means that the use of the floating auxiliary plates is preferable to submerged vertical plates. This may be because in this wave condition, the ratio between the length of the auxiliary plates and the wavelength is close to the ideal ratio (i.e. 1/3) for extracting wave energy using floating auxiliary plates. Note that the corresponding ideal ratio for the use of the submerged vertical plates is smaller, about 0.12 based on the data in Fig. 5 for MC ¼ VVV (because CF is largest when T ¼ 7 s or when λ ¼ 72 m). For T > 7 s, a mode configuration with a larger number of submerged vertical plates yields larger power production. The use of the submerged vertical plates is preferable because for this wave condition, the ratio between the length of auxiliary plates and the wavelength is closer to the ideal ratio for submerged vertical plates than that for floating auxiliary plates (because the former ideal ratio is smaller than the latter one). For 5 s � T � 7 s, preferable mode configurations for extracting wave energy are generally changed from mode configurations having a larger number of floating auxiliary plates to mode configurations having a larger number of submerged vertical plates as the wave period increases from 5 s to 7 s. Interestingly, in this wave condition, power production for mode configurations with both submerged vertical plates and floating auxiliary plates may be larger than that for the HHH and VVV mode configurations. This observation implies that the interaction be­ tween submerged vertical plates and floating auxiliary plates of the twomode WEC-type attachment may result in enhancing the power pro­ duction of the attachment. Thus, for this wave condition, the use of the two-mode WEC-type attachment consisting of both floating and sub­ merged vertical plates may be preferable to the use of the attachment comprising solely floating auxiliary plates or solely submerged vertical

plates. Fig. 7 shows the variations of CFi (i ¼ 1, 2, 3) with respect to the wave period for MC ¼ {HHH, VVV}. Here, CFi is the power capture factor associated with the ith auxiliary plate, and is defined as CFi ¼ Pav,i/(Pin/ Np) where Pav,i is the power produced by the ith PTO system (see Eq. (20)). Fig. 7a shows that the power capture factors associated with different auxiliary plates in HHH configuration are almost the same for all wave periods considered. This observation is expected because wave forces acting on different floating auxiliary plates should be almost the same for the head sea condition. Fig. 7b shows that the power capture factor associated with an auxiliary plate CFi in VVV configuration is generally lowest for the outermost plate (1st plate), and highest for the central plate (3rd plate). In other words, CFi increases as one moves from the edge plate to the central plate. This variation is expected because the wave force acting on a submerged plate is generally minimum for the outermost submerged plate, and maximum for the central plate (Sarkar et al., 2016; Wilkinson et al., 2017). Fig. 8 shows the variations of CFi (i ¼ 1, 2, 3) with respect to the wave period for MC ¼ {HVV, VHH, VHV, HVH, HHV, VVH}. Among the mode configurations with the same number of submerged plates, a mode configuration allowing a larger number of submerged plates placed next to one another generally shows a higher power capture factor associated with submerged plates. For example, among the three mode configu­ rations {HVV, VHV, VVH}, the power capture factors associated with submerged plates are generally highest for MC ¼ HVV, and smallest for MC ¼ VHV. This observation may be explained by the fact that when submerged plates are placed next to one another, the difference in water pressures acting on the two surfaces of submerged plates should be not zero along the edges adjacent to those of other submerged plates. Hence, by placing a larger number of submerged vertical plates next to one another, the wave force acting on submerged plates should increase (see Sec. 4.3); thereby leading to an increased power capture factor from the submerged plates. This arrangement of submerged plates next to one another also produces higher power capture factor (CF) for intermediate and long waves (T � 7 s), as evident in Fig. 5. Fig. 8 shows that the power capture factors associated with hori­ zontal plates for MC ¼ {HVV, VHH, VHV, HVH, HHV, VVH} are higher than those for MC ¼ HHH for most wave periods (T > 5 s). Thus, it is beneficial to have some submerged vertical plates to improve the power capture factors associated with horizontal plates. This finding agrees with that for WECs with three hinged bodies (Yu et al., 2016). The improvement in the power capture factors associated with horizontal plates may be due to the interference between the incident wave and the

Fig. 7. Variations of CFi (i ¼ {1, 2, 3}) with respect to wave period: (a) MC ¼ HHH, (b) MC ¼ VVV.

8

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Fig. 8. Variations of CF and CFi (i ¼ {1, 2, 3}) with respect to wave period: (a) MC ¼ HVV, (b) MC ¼ VHH, (c) MC ¼ VHV, (d) MC ¼ HVH, (e) MC ¼ HHV, (f) MC ¼ VVH. Note that ‘CF (HHH)’ indicates CF for MC ¼ HHH.

Fig. 10. Variations of hydroelastic response reduction with respect to wave period for all 8 mode configurations and for the case where PTO damping co­ efficients are optimized for maximizing CF.

Fig. 9. Normalized vertical displacement amplitudes along the longitudinal centerline of the 2nd auxiliary plate (i.e. the horizontal plate) for T ¼ 6 s and MC ¼ {HHH, VHV, VHH, HHV}.

9

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(T � 13 s), it was also claimed that the use of the submerged vertical plate may result in increasing the hydroelastic response of the VLFS (Ohta et al., 1999). For 7 s � T < 9 s, preferable mode configurations for hydroelastic response mitigation are generally changed from mode configurations having a larger number of submerged vertical plates to mode configu­ rations having a larger number of floating auxiliary plates as the wave period increases from 7 s to 9 s. In this wave condition, the worst mode configuration for hydroelastic response mitigation is not necessarily HHH or VVV, but it may be another mode configuration. This indicates that the interaction between the submerged vertical plates and the floating auxiliary plates may result in decreasing the hydroelastic response reduction.

reflected wave from submerged vertical plates that increases the vertical displacements of horizontal plates. The increase in vertical displace­ ments of horizontal plates due to the presence of submerged vertical plates can be seen in Fig. 9, by comparing the vertical displacement amplitude of the second auxiliary plate for MC ¼ HHH with that for other mode configurations. 4.2. Hydroelastic response reduction Fig. 10 shows the variations of the hydroelastic response reduction with respect to wave period for all 8 mode configurations of the twomode WEC-type attachment. Fig. 10 shows that, for all mode configu­ rations, the hydroelastic response reduction is largest when T ¼ 4 s and it decreases as the wave period increases, as expected. Fig. 10 shows that, similar to preferable mode configurations for wave energy extraction, preferable mode configurations for hydroelastic response mitigation are changed considerably for three different ranges of wave periods, which are indicated by the three different shaded areas in Fig. 10. Interestingly, these three shaded areas are in reverse order as compared to those in Fig. 5. For T < 7 s, Fig. 10 shows that a mode configuration with a larger number of submerged vertical plates generally shows larger HR. This result may also be seen in Fig. 11 which shows the hydroelastic response of the VLFS without the attachment and with the attachment for several mode configurations for T ¼ 5 s. The use of submerged vertical plates for reducing hydroelastic response of VLFS is preferable because for this wave condition, the length of the submerged vertical plates is relatively close to the depth of the region where the water particle motion is nonzero (or where the hydrodynamic pressure is non-zero). This depth is about λ/2 for the short wave condition which corresponds to the deep sea condition (Dean and Dalrymple, 1991). Thus, a large portion of incident wave should be reflected from the submerged vertical plates, resulting in decreasing significantly the amount of incident wave energy absorbed by the VLFS and hence reducing considerably the hydroelastic response of the VLFS. For T � 9 s, a mode configuration with a larger number of floating auxiliary plates shows larger HR (see Fig. 10). This observation may also be seen in Fig. 12 which shows the hydroelastic response of the VLFS without the attachment and with the attachment for several mode configurations for T ¼ 10 s. This observation implies that the use of submerged vertical plates is not preferable for reducing hydroelastic response of the VLFS under action of long waves. This finding agrees with the finding for a submerged vertical plate rigidly connected to the fore edge of a VLFS (Masanobu et al., 2003; Ohta et al., 1999) where the submerged vertical plate was claimed to be not effective in reducing the hydroelastic response of the VLFS for long waves. For very long waves

4.3. Horizontal wave force acting on submerged vertical plates Fig. 13a shows the variations of the amplitude of the total horizontal linear wave force acting on submerged vertical plates with respect to wave period for several mode configurations. Fig. 13b shows the vari­ ations of the horizontal force reduction FR with respect to wave period. Here, FR ¼ 1 - |Fx|i/|Fx|0 where |Fx|0 is the amplitude of the horizontal force for the VVV (or 2nd) mode configuration and |Fx|i is the amplitude of the horizontal force for the ith mode configuration (i 2 {3, 4, …, 8}). Fig. 13 shows that the wave force decreases as the number of sub­ merged vertical plates decreases. The horizontal force reduction for the mode configurations HVV, VHV and VVH is about 30%–60%. This figure is about 65%–85% for the mode configurations VHH, HVH, HHV. Fig. 13 also shows that among mode configurations with the same number of submerged vertical plates, a mode configuration allowing a larger number of the submerged vertical plates placed next to one another generally shows a larger horizontal wave force. For example, the horizontal wave force for HVV is larger than that for VVH because the arrangement of the six auxiliary plates for HVV is HVVVVH and that for VVH is VVHHVV. Similarly, it may be seen that the horizontal wave force for VVH is larger than that for VHV. 4.4. Wave energy generation versus hydroelastic response mitigation Fig. 14 shows the variations of CF and HR with respect to wave period for the case where PTO damping coefficients are optimized for maximizing HR. The optimal PTO damping coefficients for maximizing HR are given in Fig. 15. In general, Fig. 14 shows that both the variations of CF and HR with respect to wave period for various mode configurations are rather similar to the corresponding variations for the case where PTO damping coefficients are optimized for maximizing CF. Thus, the findings in Sec. 4.1 and Sec. 4.2 are generally still valid for the case where PTO damping coefficients are optimized for maximizing HR. Fig. 16 shows the variations of ΔCF and ΔHR with respect to wave period. Here, ΔCF ¼ CF2 – CF1, and ΔHR ¼ HR2 – HR1, where the su­ perscript 1 indicates the case where PTO damping coefficients are optimized for maximizing CF, and the superscript 2 indicates the case where PTO damping coefficients are optimized for maximizing HR. Fig. 16 shows that for MC ¼ HHH, both ΔCF and ΔHR are small (about �0.025) for all wave periods. This indicates that, for MC ¼ HHH, if PTO damping coefficients are designed for maximizing CF (or HR), the resulting HR (or CF) obtained is almost equal to its maximum attainable value. This finding agrees with that given by Nguyen et al. (2019b). For MC ¼ VVV, the absolute values of ΔCF and ΔHR are large, up to about �0.15, when T < 7 s. For longer waves, ΔCF and ΔHR are small (about �0.025). This finding implies that, for MC ¼ VVV, if PTO damping coefficients are designed for maximizing only CF (or HR), the resulting HR (or CF) is generally much smaller than its maximum attainable value for T < 7 s, but the resulting HR (or CF) is almost equal to its maximum attainable value for longer waves. The large absolute values of ΔCF and ΔHR for short waves, and the

Fig. 11. Normalized vertical displacement amplitudes along the longitudinal centerline of VLFS for various cases, T ¼ 5 s.

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H.P. Nguyen et al.

Fig. 12. Displacement amplitudes of VLFS without attachment (blue), VLFS with attachment (red), floating auxiliary plates (cyan) and submerged vertical plates (green) for A ¼ 0.5 m and T ¼ 10 s: (a) MC ¼ HHH, (b) MC ¼ VHH, (c) MC ¼ HVV, (d) MC ¼ VVV. Note that x, y, z in this figure are the original coordinates of the VLFS and auxiliary plates. (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.)

Fig. 13. Variations of (a) horizontal linear wave force amplitude acting on submerged vertical plates, and (b) FR with respect to wave period.

small absolute values for longer waves for MC ¼ VVV may be explained by the power produced and the amount of incident wave reflected from the submerged vertical plates for these two different wave conditions. In detail, the incident wave energy is mainly transformed into: (i) the en­ ergy of the reflected wave from the submerged vertical plates, (ii) the useable energy, and (iii) the energy restored in the motions of the VLFS and the submerged vertical plates. Thus, in general, if a larger amount of power is produced or a larger amount of incident wave is reflected, HR

should be larger. For relatively short wave conditions (T < 7 s), because a large portion of incident wave should be reflected and the amount of power produced is not large (as discussed in Sec. 4.1 and Sec. 4.2), HR may be mainly dependent on the amount of incident waves reflected. As the amount of reflected wave may increase by limiting the motion of the submerged vertical plates, HR may increase by generally increasing the PTO damping coefficients. This makes the optimal PTO damping coefficients 11

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Fig. 14. Variations of (a) CF, and (b) HR with respect to wave period for various mode configurations and for the case where PTO damping coefficients are optimized for maximizing HR.

Fig. 15. Optimal PTO damping coefficients for maximizing HR: (a) cp1,6, (b) cp2,5, (c) cp3,4. Note that, for MC ¼ HVH and T ¼ 4 s, cp1,6 ¼ 1E4 Nms/rad.

for maximizing HR large as compared to the optimal PTO damping co­ efficients for maximizing CF (see Figs. 6 and 15 for T < 7 s and MC ¼ VVV). However, for longer waves (T � 7 s), because the amount of reflected wave should be smaller than that for short waves and the amount of power produced is generally larger than that for short waves, HR may mainly depend on the amount power produced. Thus, for this wave condition, increasing the power produced generally results in increasing HR. This makes the optimal PTO damping coefficients for maximizing HR close to, and not necessarily larger than, those for maximizing CF (see Figs. 6 and 15 for T � 7 s and MC ¼ VVV).

For other mode configurations, Fig. 16 shows that the variations of ΔCF and ΔHR are quite similar to those for MC ¼ VVV. However, the absolute values of ΔCF and ΔHR may be larger than those for MC ¼ VVV, which may result from the interaction between submerged vertical plates and floating auxiliary plates. 4.5. Unequal damping strategy versus equal damping strategy Fig. 17 shows the variations of (CF1unequal - CF1equal) and (HR1unequal HR1equal) with respect to the wave period for various mode configura­ tions while Fig. 18 shows the variations of (CF2unequal – CF2equal) and 12

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Fig. 16. Variations of (a) ΔCF and (b) ΔHR with respect to wave period for various mode configurations.

Fig. 17. Variations of (a) CF1unequal - CF1equal, and (b) HR1unequal - HR1equal with respect to wave period for various mode configurations.

Fig. 18. Variations of (a) CF2unequal – CF2equal, and (b) HR2unequal – HR2equal with respect to wave period for various mode configurations.

(HR2unequal – HR2equal). Here, the superscript 1 indicates the case where the objective is to maximize CF, while the superscript 2 denotes the case where the objective is to maximize HR. The subscripts unequal and equal denote the corresponding PTO damping strategies. For the case where the equal PTO damping strategy is adopted (i.e. damping coefficients are the same for all PTO systems), the optimal PTO damping coefficients for various wave periods and mode configurations are given in Fig. 19. For MC ¼ {HHH, VVV}, Figs. 17 and 18 show that both CF and HR are almost insensitive to PTO damping strategy. This is expected because

all auxiliary plates for MC ¼ HHH (or MC ¼ VVV) have the same di­ mensions and the same operation mode. The insensitivity of CF and HR to PTO damping strategy implies that the equal PTO damping strategy should be adopted for these mode configurations to facilitate selection and control of PTO systems, but without compromising CF and HR. For other mode configurations, Figs. 17 and 18 show that the effects of PTO damping strategy on CF and HR appear to be not significant, except for a special case where the objective is to maximize HR and for short waves (T < 6 s). For this special case (see Fig. 18 for T < 6 s), by 13

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Fig. 19. Optimal PTO damping coefficients cpi (i ¼ {1, 2, 3, 4, 5, 6}) where the equal PTO damping strategy is adopted, and for the case: (a) where the objective is to maximize CF, (b) where the objective is to maximize HR.

(iv) Among mode configurations with the same number of submerged vertical plates, a mode configuration involving a larger number of submerged vertical plates placed next to one another results in a larger horizontal force.

altering the PTO damping strategy from the unequal damping strategy to the equal damping strategy, HR decreases by up to about 0.14. Thus, while the equal damping strategy can be adopted to facilitate selection and control of PTO system, the maximum attainable hydroelastic response reduction may be comprised considerably. Note that, for the special case where the objective is to maximize HR and for short waves (T < 6 s), if the unequal PTO damping strategy is adopted, the optimal PTO damping coefficients for submerged plates are much larger than those for horizontal plates (e.g. see Fig. 15 for MC ¼ HVH).

As compared to the WEC-type attachment comprising either solely submerged vertical plates or solely floating horizontal plates, the twomode WEC attachment possesses several positive features, including: (i) good wave energy extraction performance for a wide range of wave periods owing to the combination of the superior perfor­ mance of floating horizontal plates in extracting wave energy for short waves and the superior performance of submerged vertical plates in extracting wave energy for intermediate and long waves; (ii) good hydroelastic response mitigation performance for a wide range of wave periods owing to the combination of the superior performance of submerged vertical plates in reducing hydroe­ lastic response for relatively short waves and the superior per­ formance of floating horizontal plates in reducing hydroelastic response for relatively long waves; (iii) higher CF for ocean waves with 5 s � T � 7 s owing to the interaction between the vertical plates the horizontal plates; (iv) better balance between extracting wave energy and reducing hydroelastic response of VLFS for the same wave period, where although CF (or HR) for a certain wave period is smaller than its maximum attainable value, the resulting HR (or CF) for the same wave period is closer to its maximum attainable value; (v) smaller horizontal wave force acting on submerged vertical plates and therefore on the mooring system (as compared to the VVV mode configuration).

5. Conclusions and recommendations for future studies Presented herein is a two-mode wave energy converter (WEC)-type attachment to pontoon-type VLFS for wave energy extraction and reduction in hydroelastic response. The two-mode WEC-type attachment comprises floating horizontal auxiliary plates and submerged vertical auxiliary plates connected to the fore edge of the VLFS with hinges and linear PTO systems. The following key findings may be drawn from this study: (i) A two-mode WEC-type attachment with a larger number of floating auxiliary plates (and hence a smaller number of sub­ merged vertical plates) generally shows larger power production for short waves (T < 5 s), but smaller power production for in­ termediate and long waves (T > 7 s). Thus, while the use of floating auxiliary plates for wave energy extraction is preferable in short waves, the use of submerged vertical plates is preferable in intermediate and long waves. (ii) A two-mode WEC-type attachment with a larger number of sub­ merged vertical plates (and hence a smaller number of floating auxiliary plates) yields larger hydroelastic response reduction in relatively short waves (T < 7 s), but smaller hydroelastic response reduction in relatively long waves (T � 9 s). Thus, while the use of submerged vertical plates is preferable for hydroelastic response mitigation in relatively short waves, the use of floating auxiliary plates is preferable in relatively long waves. (iii) For intermediate and long waves (T � 7 s), when PTO damping coefficients are designed for maximum CF (or HR), the resulting HR (or CF) is also almost maximized. In the case of short waves, the same finding is observed for the WEC-type attachment with solely floating horizontal plates. However, if submerged vertical plates are introduced, designing PTO damping coefficients for maximizing CF (or HR) poses a problem that the resulting HR (or CF) is generally much smaller than its maximum attainable value.

This study shows that the mode configuration may be tuned to the incident wave period for maximum wave energy production or hydroelastic response reduction, and for reducing horizontal wave forces acting on submerged vertical plates (and hence a reduced hori­ zontal force on the mooring system). In fact, a floating plate may be submerged by increasing the mass density of the plate (e.g. filling its compartments with water). In addition, a submerged vertical plate may become a floating auxiliary plate by ballasting. However, if the mode configuration is allowed to be tuned to the incident wave period, a more detailed design of PTO systems is necessary. Some recommendations for future studies are given below: (i) The present study is based on the linear wave theory where the wave amplitude is small. For strong waves where nonlinearity is 14

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important, CFD (computational fluid dynamics) analyses and physical model tests should be carried out. (ii) Hydroelastic analysis and optimization of VLFS with two-mode WEC-type attachment are usually time-consuming because of the VLFS large dimensions. This problem becomes more serious when the number of design variables or the number of objective functions increases. In order to speed up the hydroelastic analysis and optimization, future research studies should focus on using faster optimization techniques such as parallel differential evo­ lution (Pedroso et al., 2017); and using faster evaluation methods for the matrices that are related to the Green function such as the fast multipole method (Utsunomiya and Watanabe, 2006) or the pre-corrected FFT method (Jiang et al., 2012). (iii) The behaviour of PTO systems is assumed to be linear in this study. This assumption has been widely adopted when analysing conceptual designs of WECs (Renzi and Dias, 2013; Yu et al., 2016). However, the actual behaviour of PTO systems may be nonlinear (Liu et al., 2017; Zheng et al., 2015). Thus, the nonlinearity of PTO systems should be considered in future studies for more accurate estimation of power production. (iv) This study focuses on estimating power production. However, to evaluate WEC designs, a number of other indirect technoeconomic indices (Babarit et al., 2012) are needed. Thus, techno-economic assessments for the two mode WEC attachment should be carried out in future studies.

H.P.N.; writing-review and editing, H.P.N., C.M.W. and V.H.L.; visuali­ zation, H.P.N.; funding acquisition, C.M.W. Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Fig. B.1. Optimization process to obtain optimal PTO damping coefficients for either maximizing CF or maximizing HR

Author contributions Conceptualization, H.P.N., C.M.W. and V.H.L.; methodology, H.P.N.; formal analysis, H.P.N. and C.M.W.; writing-original draft preparation,

Table B.1 Comparison between the conventional optimization approach (‘Con.‘) and DE approach for the case where the objective is to maximize CF T

MC

s

–

7

HVV

Con. DE

VVV 10

HVV VVV

Approach

CF

HR

cp1,6

cp2,5

cp3,4

NA

topt

–

–

Nms/rad

Nms/rad

Nms/rad

–

s

0.4956 0.4999

0.3385 0.3398

1.00E7 1.32E7

2.00E7 1.85E7

2.00E7 2.06E7

50653 1500

17127 603 (4487)

Con. DE Con. DE

0.5017 0.5055 0.2597 0.2632

0.3839 0.3739 0.1343 0.1365

2.00E7 1.51E7 2.00E7 1.95E7

2.00E7 1.81E7 1.00E7 1.29E7

2.00E7 1.85E7 2.00E7 1.57E7

50653 1470 50653 1650

22892 519 20567 911

Con. DE

0.3376 0.3429

0.0694 0.0786

1.00E7 1.21E7

2.00E7 1.54E7

2.00E7 1.66E7

50653 1680

20813 631

Table B.2 Comparison between the conventional optimization approach (‘Con.‘) and DE approach for the case where the objective is to maximize HR T

MC

s

–

7

HVV

10

Approach

CF

HR

cp1,6

cp2,5

cp3,4

NA

topt

–

–

Nms/rad

Nms/rad

Nms/rad

–

s

Con. DE

0.4788 0.4885

0.3448 0.3473

1.00E7 1.21E7

3.00E7 2.51E7

3.00E7 2.93E7

50653 1470

17127 713

VVV

Con. DE

0.4737 0.4667

0.3991 0.3995

2.00E7 2.23E7

3.00E7 2.91E7

3.00E7 3.29E7

50653 1260

22892 1508

HVV

Con. DE

0.2461 0.2491

0.1485 0.1488

3.00E7 3.10E7

1.00E7 9.80E6

1.00E7 1.13E7

50653 1440

20567 547

VVV

Con. DE

0.3175 0.3281

0.0874 0.0878

9.00E6 9.60E6

1.00E7 1.09E7

1.00E7 1.15E7

50653 1380

20813 775

15

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Acknowledgements

from Delft University of Technology for assistance in Differential Evo­ lution, Dr. Zhi Yung Tay from the Singapore Institute of Technology for discussions on hydroelastic analysis, and Professor Tom Baldock from The University of Queensland for discussions on ways to extract wave energy while reducing hydroelastic response of VLFS.

The authors are grateful for the financial support provided by the Australian Research Council under the Discovery Project scheme (DP170104546). The first author would like to thank Mr. V. Ho-Huu

Appendix A. Boundary element procedure First of all, we weight a test function φðxÞ to both sides of Eq. (16), and then integrate the both sides over the entire wetted surface SHB, as follows (Kim et al., 2013; Wang and Wang, 2008): Z Z α φðxÞφðxÞdx ¼ φin ðxÞφðxÞdx SHB

Z Z �

þ

SHB

∂Gðx; ξÞ φðξÞ ∂nðξÞ

(A.1)

� iωGðx; ξÞuj ðξÞnj ðξÞ dξφðxÞdx:

SHB SHB

Because the submerged vertical plates are assumed to be thin, the wetted surfaces of each submerged plate that are perpendicular to Oy and Oz may be neglected in the numerical model, and only the wetted surfaces that are perpendicular to Ox are considered. Because of this simplification, an artificial continuity condition is imposed along the submerged edges of the considered wetted surfaces to ensure that the difference in the velocity potential between these surfaces are zero along the submerged edges. This artificial continuity condition has been used for fluid-structure interaction analysis of submerged thin plates (Noad and Porter, 2015; Renzi and Dias, 2013). The wetted surface of the entire structure SHB may be decomposed into three component surfaces, including: (i) Sh that is the wetted surface of the VLFS and the floating auxiliary plates; (ii) Svþ that is the right wetted surface (perpendicular to Ox) of the submerged vertical plates; (iii) Sv that is the left wetted surface of the submerged vertical plates. Following the boundary element procedure, these wetted surfaces are discretised into elements that possess the same properties (i.e. the inter­ polation functions and the number of nodes in each element) of the elements used for the FEM formulation in Sec. 2.2. However, different from the nodes in the FEM formulation, each node in the BEM formulation has only one degree of freedom that corresponds to the nodal spatial velocity potential φ b . The spatial velocity potential of an arbitrary point may be approximated from the nodal spatial velocity potentials by using the formula b e , where the subscript e denotes the element where the considered point is in, φ b e is the vector containing the nodal spatial velocity potentials φ ¼ N*e φ of the element and is obtained by assembling the vectors φ b of all nodes of the element, N*e the vector containing the interpolation function values (Liu

and Quek, 2003). Note that while the size of the vector N*e is 1 � ne, the size of the vector Ne in Sec. 2.2 is 1 � 6ne. We first consider the case where the source point is in Sh (i.e. x 2 Sh). Eq. (A.1) may be rewritten as � Z Z Z Z � 2 ω φðxÞφðxÞdx ¼ φin ðxÞφðxÞdx þ Gðx; ξÞφðξÞ þ iωGðx; ξÞu3 ðξÞ dξφðxÞdx g Sh

Z Z �

þ Sh Svþ

Sh

∂Gðx; ξÞ φðξÞ ∂xðξÞ

Z Z �

þ Sh Sv

Sh Sh

� iωGðx; ξÞu1 ðξÞ dξφðxÞdx:::

(A.2)

�

∂Gðx; ξÞ φðξÞ þ iωGðx; ξÞu1 ðξÞ dξφðxÞdx; ∂xðξÞ

where the first derivative of the Green function with respect to x is given by

∂Gðx; ξÞ ¼ ∂xðξÞ

∞ X

K ðk RÞ k ðx ξÞ �coskm ðz þ HÞcoskm ðζ þ HÞ m � 1 m ; R sin 2k H m m¼0 π H 1 þ 2km H

(A.3)

where K1 is the modified Bessel function of the second kind of order one. Eq. (A.2) may be transformed in a matrix form, as follows: � 2 � ω h h vþ � b h ¼ φh KI1 φ b hin þ b h þ iωφh G1 b b vþ iωφh G2 b u3 þ u1 φh KI1 φ φ G1 φ φh Gx1 φ g � b v þ iωφh G3 u b v1 ; þ φh Gx2 φ

(A.4)

where each term in Eq. (A.4) is the matrix form of the corresponding term in Eq. (A.2), the superscript h denotes that the vectors are for Sh, the subscript vþ implies that the vectors are for Svþ, the subscript v-indicates that the vectors are for Sv , the matrices G1, G2 and G3 are related to the Green function, the matrices Gx1, Gx2 are related to the first derivative of the Green function with respect to x. By eliminating φh from all terms of Eq. (A.2) and by setting b u vþ u v1 ¼ b u v1 , Eq. (A.4) becomes 1 ¼ b � � � � � � 2 ω v (A.5) b h ¼ KI1 φ b hin þ b h þ iωG1 u b vþ iωG2 b b v þ iωG3 u b h3 þ b v1 KI1 φ u 1 þ Gx2 φ Gx1 φ G1 φ g Let consider another case where x 2 Svþ, Eq. (A.1) may be rewritten as 16

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Z

Ocean Engineering 197 (2020) 106875

1 φðxÞφðxÞdx ¼ 2

Svþ

Z φin ðxÞφðxÞdx Svþ

Z Z �

ω2

þ

g

� Gðx; ξÞφðξÞ þ iωGðx; ξÞu3 ðξÞ dξφðxÞdx

Svþ Sh

Z Z �

∂Gðx; ξÞ φðξÞ ∂xðξÞ

þ Svþ Svþ

(A.6)

� iωGðx; ξÞu1 ðξÞ dξφðxÞdx

Z Z �

�

∂Gðx; ξÞ φðξÞ þ iωGðx; ξÞu1 ðξÞ dξφðxÞdx: ∂xðξÞ

þ Svþ Sv

Similar to the case where x 2 Sh, Eq. (A.6) may be transformed into the following equation: � � � � � � 2 1 ω h v v b vþ iωG5 b b vþ ¼ KI2 φ b vþ b h þ iωG4 b b v þ iωG6 b u3 þ 0φ u 1 þ Gx3 φ u1 ; KI2 φ G4 φ in þ 2 g

(A.7)

b vþ where each term in Eq. (A.7) is the matrix form (after eliminating φvþ ) of the corresponding term in Eq. (A.6). Note that the matrix 0 before φ vþ indicates the fact that for the case where both the source and the field points are in S , the first derivative of the Green function (Eq. (A.3)) is zero. Next, we consider the case where x 2 Sv . Eq. (A.2) may be rewritten as Z Z 1 φðxÞφðxÞdx ¼ φin ðxÞφðxÞdx 2 Sv

Sv

Z Z �

ω2

þ Sv

g Sh

Z Z � þ Sv

� Gðx; ξÞφðξÞ þ iωGðx; ξÞu3 ðξÞ dξφðxÞdx

Svþ

∂Gðx; ξÞ φðξÞ ∂xðξÞ

Z Z � þ Sv

Sv

(A.8)

� iωGðx; ξÞu1 ðξÞ dξφðxÞdx �

∂Gðx; ξÞ φðξÞ þ iωGðx; ξÞu1 ðξÞ dξφðxÞdx: ∂xðξÞ

The matrix form of Eq. (A.8) is � 2 � � 1 ω h b v ¼ KI2 φ b vin þ b h þ iωG7 b KI2 φ u3 þ G7 φ 2 g

� b Gx4 φ

vþ

v

iωG8 b u1

� � v b v þ iωG9 b þ 0φ u1 ;

(A.9)

b v results from the where each term in Eq. (A.9) is the matrix form (after eliminating φv ) of the corresponding term in Eq. (A.8), the matrix 0 before φ fact that the first derivative of the Green function (Eq. (A.3)) is zero when both the source and the field points are in Sv . From Eqs. (A.5), (A.7) and (A.9), we obtain the following system of equations: 2� 3 � ω2 3 2 Gx1 Gx2 7 G1 6 KI1 g 6 72 3 2 3 h 6 7 b 7 φ 0 0 6 KI1 bh 6 7 φ 6 in 7 7 6 76 7 6 76 2 6 7 ω 1 6 76 vþ 7 vþ 7 7 6 76 0 K ¼ 0 b 7 6 7 6 6 b φ K G G φ I2 4 I2 x3 6 74 5 4 56 in 7 2 g 7 6 7 6 7 6 7 v 0 0 KI2 4 φ 6 7 φ b b vin 5 6 7 2 4 5 ω 1 KI2 Gx4 G7 2 g 2 3 2 3 h7 b u G1 G2 G3 6 6 3v 7 7 þiω4 G4 (A.10) G5 G6 56 u1 7 6b 7 4 v G7 G8 G9 5 b u 1

For the sake of brevity, Eq. (A.10) may be rewritten as 3 2 3 2 2 h 3 h b hin b φ u b φ 6 vþ 7 6 3v 7 vþ 5 4 7 6 6 E φ ¼ F4 φ b in 5 þ iωH4 b b u1 7 5; v v bv φ b b u φ in

(A.11)

1

where the matrices E, F and H represents the corresponding matrices in Eq. (A.10). Similar to the continuity conditions for the hinge connections between the VLFS and the auxiliary plates, the artificial continuity condition for the velocity potential along the submerged edges of the submerged vertical plates can be imposed in the numerical model by using the penalty method (Liu and Quek, 2003) (for the matrix E in Eq. (A.11)). After applying the boundary element procedure to transform Eq. (16) into Eq. (A.11), we now discuss the calculation of the vector of frequency17

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dependent complex amplitudes of nodal hydrodynamic force Pg (in Eq. (18)). This vector is related to the nodal velocity potential vector, as follows: 2 h 3 � � φ b KI1 0 0 4φ (A.12) Pg ¼ iωρw Tg b vþ 5; 0 KI2 KI2 bv φ where Tg is the transformation matrix that is used to transform the vector for nodal hydrodynamic forces in the BEM formulation into the corre­ sponding vector for nodal hydrodynamic forces in the FEM formulation. This transformation process is required because while each node in the BEM formulation has only one degree of freedom, that in the FEM formulation has six degrees of freedom, as discussed earlier. The size of the matrix Tg is 6nn � nn where is nn the number of nodes of the entire discretised FEM model. The component vector of the matrix Tg for each node is [0 0 1 0 0 0]T if the node is within the VLFS or the floating auxiliary plates, and is [1 0 0 0 0 0]T if the node is within the submerged vertical plates. Eq. (A.12) may also be expressed as 2 h 3 b φ (A.13) Pg ¼ iωρw Tg L4 φ b vþ 5; bv φ where the matrix L in Eq. (A.13) represents the corresponding matrix in Eq. (A.12). By substituting the vector for nodal velocity potential from Eq. (A.11) into Eq. (A.13), we obtain the following equation: 3 2 " # b hin φ 7 6 b h3 u vþ 7 1 2 Pg ¼ iωρw Tg LE 1 F6 þ ; ω ρ T LE HT b φ g v w 4 in 5 b v1 u b vin φ where Tv 2 I1 Tv ¼ 4 0 0

is the transformation matrix given by 3 0 I2 5; I2

(A.14)

(A.15)

where I1, I2 are the identity matrices, and their sizes are equal to the sizes of KI1 and KI2, respectively. T

By expressing the vector½b u h3 b u , Eq. (A.14) becomes u v1 � in Eq. (A.14) in terms of the global vector b

Pg ¼

(A.16)

b in þ ω2 ρw Tg LE 1 HTv TTg b u: iωρw Tg LE 1 F φ

From Eq. (A.16) and Eq. (18), the matrices Ma, Cd and Fexc may be obtained, as follows: i h Ma ¼ ρw � Re Tg LE 1 HTv TTg ;

(A.17)

i h Cd ¼ ωρw � Im Tg LE 1 HTv TTg ;

(A.18)

Fexc ¼

(A.19)

b in ; iωρw Tg LE 1 F φ

where Im [] indicates the imaginary part. Appendix B. Optimization of PTO damping coefficients The optimization problem is defined as follows: (B.1)

maximize ​ f ðXÞ; h

i

(B.2)

subject to cpi 2 cLp ; cUp ;

where f(X) may be either CF(X) or HR(X) (as discussed in Sec. 4); X ¼ {cp1, cp2, …, cp Np } the vector of PTO damping coefficients of Np numbers of PTO

systems; cLp the lower bound of PTO damping coefficients in the search space; and cUp the upper bound of PTO damping coefficients in the search space. The search space for PTO damping coefficients is [1E4 – 1E8] Nms/rad. cpi (i ¼ {1, 2, …, Np) are considered as continuous design variables. The optimization process is presented in Fig. B1. The optimization process starts with the generation of NP numbers of initial solutions Xj,0 ¼ {cp1, j,0, cp2,j,0, …, cp Np ;j } where the subscript 0 denotes the initial generation, and j ¼ {1, 2, …, NP}. NP must be at least 4 to have enough vectors for the ;0

mutation phase (see Eq. (B.4)), and NP needs to be sufficiently large to avoid premature convergence (Price et al., 2005). In this study, NP is set to a large value, NP ¼ 10Np (Storn and Price, 1997). Each component cpi,j,0 (i ¼ {1, 2, …, Np}) of the vector Xj,0 is created as follows (Price et al., 2005): � � cpi;j;0 ¼ cLp þ randi ð0; 1Þ � cUp cLp ; (B.3)

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where randi (0,1) is a number randomly generated within the range [0,1]. After generating the initial solutions, the mutation phase is employed to generate NP numbers of mutant vectors Vj,g where the subscript g denotes the current generation, as follows (Price et al., 2005): � (B.4) Vj;g ¼ Xr1;g þ FM � Xr2;g Xr3;g ; where FM is called a scale factor which is a positive real number and is usually not greater than 1; r1, r2, r3 are randomly selected from {1, 2, …, NP} and r1 6¼ r2 6¼ r3 6¼ j. In the crossover phase, the vector Vj,g and the vector Xj,g are used to create a trial vector Uj,g. The ith component ui;j;g of the vector Uj,g is created as follows (Price et al., 2005): � if randi ð0; 1Þ � CR or i ¼ irand ; vi;j;g ui;j;g ¼ c (B.5) otherwise; p i;j;g

where vi;j;g is the ith component of the vector Vj,g, cp i;j;g is the ith component of the vector Xj,g, CR 2[0,1] is the crossover control value, irand is a random index within 1 to Np. In the selection phase, each vector Xj,gþ1 for the next generation is selected within the vectors Uj,g and Xj,g based on the objective function value, as follows (Price et al., 2005): � � � Uj;g if f Uj;g > f Xj;g ; Xj;gþ1 ¼ (B.6) Xj;g otherwise: If the following stopping condition is satisfied, the optimization process is terminated and the optimal solution is found (Price et al., 2005): jfworst = fbest

(B.7)

1j � toler;

where toler is a user-defined value and is set to 10 5 in the present study, fworst and fbest are, respectively, the smallest and the largest objective function values over the objective function values for NP numbers of possible solutions Xj,gþ1. If the stopping condition is not satisfied, the mutation, crossover, and selection phases are employed again. The selection phase requires the evaluation of the objective function. This evaluation may be done by using directly the input and the numerical analysis framework that is based on the numerical formulation given in Sec. 2. However, this direct evaluation approach is time-consuming (Nguyen et al., 2019b). To speed up the computation, a two-step procedure is proposed to evaluate the objective function, as shown in Fig. B.1. In the first step, the matrices M, Ma, Cd, K, Krf, Fexc and S are precalculated using the numerical analysis framework. In the second step, these matrices and the matrix Cp that is created based on the PTO damping coefficients of a possible solution are used to calculate CF (or HR). The two-step procedure is adopted for evaluating the objective functions because all the matrices M, Ma, Cd, K, Krf, Fexc and S are insensitive to cpi, and hence should be predetermined to avoid unnecessary repeated calculation for every generations in differential evolution. This two-step procedure for evaluating the objective function has also been used for other optimization studies on VLFS and WEC (Nguyen et al., 2019b, 2018). The set {FM, CR} that provides the most satisfactory optimization performance for a given input can be obtained by performing optimization studies for all possible sets {FM, CR}. However, this approach is time-consuming because it requires a large number of analyses (Ho-Huu et al., 2016). Effective techniques for selection of {FM, CR} are being investigated (Ho-Huu et al., 2016; Wang et al., 2011). In this study, for simplicity, FM and CR are selected within the ranges of FM and CR used in previous studies where FM is within [0.4,1] (Ho-Huu et al., 2016; Wang et al., 2011) and CR is close to 1 (Wang et al., 2011) or is within [0.7,1] (Ho-Huu et al., 2016). Based on these recommended ranges, FM and CR are set to 0.8 and 0.9, respectively. The accuracy of the optimal solution obtained by using DE is verified by comparing with the optimal solution obtained by performing a lot of parametric studies (this optimization approach is called ‘conventional optimization approach’). The verification is carried out for 208 cases (for 13 wave periods T ¼ {4, 5, …, 16} s, 8 mode configurations, and 2 objective functions). For the conventional optimization approach, cpi where i ¼ {1, 2, 3} are selected within a set of 37 discrete values {1E4, 2E4, 3E4, …., 1E5, 2E5, 3E5, …., 1E8} Nms/rad. Thus, for each wave period and mode configuration, the optimal solution is obtained from the results of 37Np analyses (i.e. 373 or 50653 analyses, because Np ¼ 3 due to the symmetry of the mode configuration). For all examined cases, the optimal PTO damping coefficients obtained from the conventional optimization approach are almost the same as those obtained from DE approach (given in Figs. 6 and 15). The differences in the objective function values obtained by using the two optimization approaches are negligible. For reference, the details of the comparison between the two optimization approaches for MC ¼ {VVV, HVV} and T ¼ {7, 10} s are presented in Tables B.1 and B.2. As compared with the number of analyses (NA) required for the conventional optimization approach, NA required for DE is much smaller. Thus, DE approach is superior to the conventional optimization approach. Here, NA ¼ NP � (gtotal þ 1) where gtotal is the total number of generations required to obtain the optimal solution (Pedroso et al., 2017). The computational time required for optimization topt is also provided in Tables B1 and B2. The high performance computing (HPC) system (with 20 processors) at The University of Queensland is used. The computational time topt that is underlined in Table B1 is obtained when a desktop computer (64-bit operating system, 16 GB RAM, Intel® Core™ i7-7700 CPU @ 3.6 GHz) is used. For each mode configuration and wave period, topt is generally not linearly proportional to NA. This may be because different jobs (i.e. MATLAB codes) submitted to the HPC system may be assigned to different computer nodes within the HPC system. In addition, the use of the requested multiple processors is not controlled by users, and may be not the same for different runs. Note that topt shown in Tables B1 and B.2 does not include the computational time required to obtain the precalculated matrices {M, Ma, Cd, K, Krf, Fexc, S} which ranges from 2 to 4 h (by using the HPC system with 8 processors).

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Glossary Oxyz: Cartesian coordinate system L, B, h: length, width, thickness of VLFS Lp, Bp, hp: length, width, thickness of an auxiliary plate s: gap between auxiliary plates and the fore edge of VLFS Np: the number of auxiliary plates (¼ the number of PTO systems) MC: mode configuration ρ: mass density of VLFS ρp1, ρp2: mass density of floating auxiliary plates, mass density of submerged vertical plates ρw: water mass density (¼1025 kg/m3) E: Young’s modulus of VLFS ν: Poisson’s ratio of VLFS cpi: PTO damping coefficient of the ith PTO system λ, T, ω: wavelength, wave period, wave frequency A: wave amplitude H: water depth i: imaginary unit (i2 ¼ 1) g: gravitational acceleration (¼ 9.81 m/s2) ut1,2,3, u1,2,3: displacements, and frequency-dependent complex amplitudes of displace­ ments along x, y and z – axes ψ t1,2,3, ψ 1,2,3: rotations, and frequency-dependent complex amplitudes of rotations about x, y and z – axes

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φt, φ: velocity potential, and frequency-dependent complex amplitude of velocity potential (i.e. spatial velocity potential) φin: frequency-dependent complex amplitude of velocity potential of the incident wave CF: power capture factor associated with the two-mode WEC-type attachment CFi: power capture factor associated with the ith auxiliary plate HR: reduction in the amplitude of the maximum vertical displacement over the entire VLFS owing to the use of two-mode WEC-type attachment Fx: frequency-dependent complex amplitude of the total horizontal wave force acting on submerged vertical plates M: global mass matrix K: global stiffness matrix Krf: global restoring stiffness matrix

Cp: global PTO damping matrix Ma: global added mass matrix Cd: global added damping matrix (i.e. hydrodynamic damping matrix) Fexc: global vector that consists of frequency-dependent complex amplitudes of nodal excitation wave forces b u : global vector of the frequency-dependent complex amplitudes of displacements of the structure S: matrix consisting of eigenvectors used for approximating the vector b u in the modal expansion method ς: vector consists of complex amplitudes used for approximating the vector b u in the modal expansion method

21