On the dynamics of an array of spar-buoy oscillating water column devices with inter-body mooring connections

Journal Pre-proof On the dynamics of an array of spar-buoy oscillating water column devices with interbody mooring connections C.L.G. Oikonomou, R.P.F. Gomes, L.M.C. Gato, A.F.O. Falcão PII:

S0960-1481(19)31790-2

DOI:

https://doi.org/10.1016/j.renene.2019.11.097

Reference:

RENE 12645

To appear in:

Renewable Energy

Received Date: 29 May 2018 Revised Date:

11 October 2019

Accepted Date: 17 November 2019

Please cite this article as: Oikonomou CLG, Gomes RPF, Gato LMC, Falcão AFO, On the dynamics of an array of spar-buoy oscillating water column devices with inter-body mooring connections, Renewable Energy (2019), doi: https://doi.org/10.1016/j.renene.2019.11.097. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2019 Published by Elsevier Ltd.

Renewable Energy Journal Revised Submission, Title: On the dynamics of an array of spar-buoy oscillating water column devices with inter-body mooring connections Keywords: Wave energy, Floating oscillating water column, Spar-buoy OWC, Mooring system, Array, Irregular waves List of Authors: C. L. G. Oikonomou, PhD IDMEC, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais, 1049-001 Lisboa, Portugal [email protected] R. P. F. Gomes, PhD IDMEC, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais, 1049-001 Lisboa, Portugal [email protected] L. M. C. Gato, PhD IDMEC, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais, 1049-001 Lisboa, Portugal [email protected] Α. F. O. Falcão, PhD IDMEC, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais, 1049-001 Lisboa, Portugal [email protected] Corresponding author: C. L. G. Oikonomou Declarations of interest: none

On the dynamics of an array of spar-buoy oscillating water column devices with inter-body mooring connections C. L. G. Oikonomou, R. P. F. Gomes, L. M. C. Gato, A. F. O. Falc˜ao IDMEC, Instituto Superior T´ ecnico, Universidade de Lisboa, Av. Rovisco Pais, 1049-001 Lisboa, Portugal

Abstract The performance of an array of floating Oscillating Water Column (OWC) devices, known as spar-buoy OWC, is analysed for a configuration with inter-body mooring connections. This configuration has the potential of being a more economically viable solution due to drastic reductions in the amount of mooring cables, when compared to independently moored configurations. Numerical simulations for an array of independently moored devices and for an unmoored array are presented and compared. The frequencydomain model considered throughout this paper uses linear hydrodynamic and hydrostatic forces; real fluid viscous effects are accounted for by using a linear approximation, while the mooring connections are linearised using perturbation theory. For regular waves, by including an inter-body mooring system, the average heave response amplitude of the array’s three buoys decreases by approximately 7.2% at the peak frequency, while the average capture width of the three buoys remains approximately the same. The influence of the wave incidence angle on the array performance is evaluated. A stochastic analysis is conducted to assess the behaviour of the array with mooring connections. A comparative analysis between the performance of an array with inter-body mooring connections and an isolated device suggests a positive park effect for a realistic wave climate. Keywords: Wave energy, Floating oscillating water column, Spar-buoy OWC, Mooring system, Array, Irregular waves

Nomenclature Abbreviations BBDB BEM CoB CoG CWR DoF OWC RAO WEC

backward-bent duct buoy boundary element method centre of buoyancy centre of gravity capture width ratio degrees-of-freedom oscillating water column response amplitude operator wave energy converter

Latin symbols ∗ Corresponding

author Email address: [email protected] (C. L. G. Oikonomou)

Preprint submitted to Journal of LATEX Templates

November 20, 2019

acw Ad,i Aij Aw Bij ca cd,i C Cbm d1 d2 f¯ f F g G GM h Hs ı i, j Iij k kt l L m M Mij N n p Pt Pw q r¯ r s Sb So Sω Te u V

clump weight radius drag area or drag area moment added mass coefficient wave amplitude radiation damping coefficient speed of sound in air drag coefficient hydrostatic restoring term bottom mooring line stiffness buoy diameter OWC diameter bottom mooring line force under calm sea conditions perturbation to bottom mooring line force f¯ force vector acceleration of gravity length of the cable between each buoy and the clump weight metacentric height water depth significant wave height √ imaginary unit −1 modes of the bodies moments of inertia wavenumber turbine coefficient draught distance between buoys and centre of the array mass total number of sea states inertia coefficient total number of bodies index of the drag surface pressure difference between the air chamber and the atmosphere average power available to the turbine time-averaged power transported by a wave per unit wave crest length gain factor inter-body mooring force under calm sea conditions perturbation to inter-body mooring force r¯ line designation buoy waterplane area OWC waterplane area wave energy density energy period velocity buoy’s displaced water volume 2

V0 vcw xi x, y, z Z zb zg zcw

air chamber volume under calm sea conditions clump weight volume complex amplitudes of displacement position coordinates heave complex amplitude of displacement z-coordinate of centre of buoyancy z-coordinate of centre of gravity z-coordinate of clump weight position

Subscripts ann b bm cw d exc hr hst im irr iso p r rad t tb

annual buoy bottom mooring clump weight drag excitation horizontal hydrostatic inter-body mooring irregular isolated pressure relative radiation turbine tube

Superscripts (e) ∗

estimated value dimensionless value

Greek symbols α β Γ δ, ε ∆ η θ ρa ρw

angle of inter-body mooring force angle of bottom mooring force wave excitation coefficient perturbations to inter-body mooring angles interval free-surface elevation angle of wave incidence air density water density 3

σ φ φm ω ωn

standard deviation bottom mooring line extension frequency of occurrence of sea state wave frequency natural frequency

1. Introduction

5

10

15

20

25

30

35

40

The concept of Oscillating Water Column (OWC) devices was first introduced in the 1940s by Yoshio Masuda [1], and is one of the most studied types of Wave Energy Converter (WEC), with a large number of proposed prototypes tested in the sea [2]. The spar-buoy OWC consists of a floater attached to a tail tube, which is open to the sea at its submerged bottom, and has an air chamber above the inner free surface that is connected the atmosphere at its top through an air turbine. Due to wave action, an oscillatory motion of the water column relative to the floater-tube set is established, which produces alternately compression and decompression of the air inside the chamber above the inner free-surface and drives the air turbine coupled to the electrical generator. Except if check valves are employed to rectify the air flow, which is widely considered impractical except in small devices like navigation buoys, self-rectifying turbines are used in which the rotational direction remains unchanged regardless of the air flow direction [2]. Besides floating devices, OWC technology can also be used in fixed structure plants as is the case of the shoreline bottomstanding Pico power plant in the Azores, Portugal [3] and the breakwater-integrated Mutriku Power Plant in the Basque Country, Spain [4]. The space availability and the more energetic resource at offshore locations explain the growing interest in floating versions of the OWC. Examples include the BBDB [5], the spar-buoy OWC [6], and the UGEN [7]. This study is focused on an analysis of the dynamics of an array of spar-buoy OWC devices with inter-body mooring connections. The use of inter-body connections aims the reduction of the total line length, and therefore its costs. We use a numerical model to account for the dynamic interaction between the inter-body mooring system and the converters. Theoretical and numerical studies on the dynamics of the OWC were pioneered by McCormick [8] in the 1970s. With the advances in Boundary Element Methods (BEM) for wave-body interactions, more detailed simulations have been conducted to account for the hydrodynamic interactions between the incident waves and the buoy and OWC motions [9]. Gomes et al. [6] developed a geometry optimisation model for an axisymmetric floating spar-buoy OWC, with a non-uniform tail tube cross-section. Gomes et al. [10] investigated the dynamics of a heaving spar-buoy OWC in a wave channel, providing both numerical and experimental results for regular and irregular waves. This device will be the focus of this work. Mooring connections are needed to keep the device on station, especially in the free surface plane, where no hydrostatic restoring forces are present. Mooring systems for wave energy converters should be designed to avoid negatively impacting the extracted wave energy [11], while being able to withstand extreme events. Significant knowledge has been acquired in the operation of moored offshore platforms, although there are several consequences of directly applying this knowledge to WECs [12]. For example, the effect of the mooring system on the wave energy absorption capability of the WEC should be examined [13]. General guidelines for the design of WEC mooring systems can be found in [14]. A quasi-static analysis of a generic WEC for various mooring configurations was presented in [15], and a review on the mathematical tools for mooring line dynamics was reported in [11]. A theoretical and experimental study on the effect of damping on the dynamics of a moored WEC can be found in [16]. There is growing scientific interest in the impacts of structural loading and extreme events on the moorings of floating structures, and, more specifically, on those of WECs [17, 18, 19, 20]. These advances even include the determination of snap loads in the mooring lines [21]. The need for extraction of large amounts of energy from the ocean and the limitation in coastal space has led to the deployment of floating WECs into arrays, with relatively small spacing between devices [22, 23]. By as early as 1977, Budal [24] raised awareness to the wave interactions that WECs in array configurations can present, especially point absorbers - bodies whose characteristic length is small compared to typical 4

45

50

55

60

65

70

75

80

wavelengths of incoming waves. A theory for power absorption for an array of a certain number of interacting devices was first described by Evans (1976) [25] and later by Falnes (1980) [26]. Calculation of the absorbed power requires information on the wave excitation forces, radiation coefficients, and device motions. By using the point absorber approximation for an array of heaving devices, these authors derived simple analytic expressions for the aforementioned quantities, and their results showed significant hydrodynamic interaction between devices. These interactions were constructive for certain wave frequencies and destructive for others. Constructive interference leads to an increase in the amount of power generated by an array compared to an equal number of isolated devices. For point absorber type devices that are to be deployed in arrays, it is essential that their inter-body dynamics are assessed and incorporated in the early stages of the design process. The hydrodynamic interaction within arrays of WECs has been presented and assessed throughout various studies, including those of Mavrakos and McIver [23], Justino and Cl´ement [22], Babarit [27], and Borgarino et al. [28]. Methods for designing efficient array configurations can be found in Child and Venugopal [29]. Various studies have emphasised the significance of the wave climate directionality on the array performance [30, 31]. The majority of existing array studies are numerical or theoretical; few experiments have been conducted. Stratigaki et al. [32] performed an experiment of 25 heaving buoys in order to investigate the near-field effects of an array, which they then compared with numerical results of a wave propagation model. The power extracted by an array of WECs depends on its configuration and the separation distance between its devices, but also on the mooring system itself. Inter-body mooring connections (originally implemented in the aquaculture sector [33]) are regarded as a way of reducing the total number of bottom mooring lines, thereby reducing the overall cost of the mooring system. Previous studies [34, 35, 36, 37] have emphasised the importance of considering different mooring configurations when modelling numerically an array of WECs. Vicente et al. [37] developed a numerical analysis of a triangular array of spherical bodies oscillating in three Degrees-of-Freedom (DoF), with bottom and inter-body mooring connections. Their analysis was performed in the frequency-domain, and was later expanded upon by employing a stochastic model for an irregular wave analysis. An analysis of an array of FO3 buoys with moorings was performed in [38]. Since then, the idea of inter-body mooring connections has been the subject of two EU funded experimental projects [39, 40], applied to compact arrays of spar-buoy OWCs. Correia da Fonseca et al. [40] performed experiments on an array of spar-buoy OWCs with shared mooring connections, under moderate and extreme wave conditions. Nevertheless, to the authors’ knowledge, no study has been focused on the detailed analysis of the dynamics of arrays of floating OWC devices with inter-body mooring connections. The spar-buoy OWC geometry of the studied device does not permit the use of analytical methods previously mentioned, as their applicability is limited to arbitrary WEC geometries. Moreover, all DoF need to be considered for studying array interactions. This paper investigates the dynamic effects of an inter-body mooring system with a clump weight on the performance of a triangular array of spar-buoy OWC devices, under regular and irregular wave conditions. A numerical formulation, based on the frequency-domain, is outlined in detail. The results analyse the effect of the inter-body moorings based on the design proposed in [37], by comparing them to the cases of unmoored and independently moored devices. The major findings are summarised in the conclusions. 2. Numerical modelling

85

90

The formulation considered here is applicable to axisymmetric floating OWC devices with an inner free surface that moves predominantly in heave. The formulation assumes incompressible and irrotational water flow. Wave amplitudes and body motions are assumed to be much smaller than the incident wavelength. Linear hydrodynamic forces, accounting for the array effects, are applied. Linear approximations of quadratic drag forces are considered. The inter-body connection studied here consists of mooring lines connected to a clump weight in the centre of the array. The mooring lines are assumed weightless and inelastic, and linear forces are considered by applying perturbation theory, which accounts for the first-order loads due to the tension oscillation and due to the angles oscillation. A frequency-domain formulation is presented to solve 5

the equation of motion. A stochastic analysis is applied to provide motion and power absorption results under more realistic sea conditions. 95

2.1. Equations of motion A top view of the array configuration and its mooring lines is shown in Fig. 1. Fig. 2 displays a threedimensional representation of the array. The devices are labelled as Body 1 (Buoy 1), Body 2 (Buoy 2), and Body 3 (Buoy 3), while the clump weight is Body 4. The origin of the fixed reference frame is located at the free surface, in the middle of the array. The axis shows the positive x, y, and z direction. Regular waves propagate with the positive x-axis direction with amplitude Aw and frequency ω. The motion of each device is defined by six DoF: three translational (surge, sway, and heave) and three rotational (roll, pitch, and yaw). The complex amplitudes of the DoF for Body 1 are denoted by x1 to x6 , for Body 2 by x7 to x12 , and for Body 3 by x13 to x18 (as indicated in Fig. 1). These motions are measured relative to a reference frame whose origin is located at the free surface in the centre of each device, with the same orientation of the fixed reference frame. The complex amplitude of displacements x19 , x20 , and x21 refer to the heave oscillation of the OWCs of each device. The equation of motion for the three bodies and OWCs in the frequency-domain can be written as 7N X


Mij −ω 2 xj =

(1)

j=1

Fexc,i + Frad,i +Fhst,i + Ft,i + Fd,i + Fbm,i + Fim,i , where N represents the total number of bodies (in this case, N = 3) and Mij represents the inertia matrix. The term −ω 2 xj corresponds to the acceleration complex amplitude of mode j. The terms on the right hand side of Eq. 1 represent the complex amplitudes of the forces acting on mode i, namely the wave excitation force Fexc,i , the radiation force Frad,i , the restoring hydrostatic force Fhst,i , the force due to the air pressure variations inside the air chamber Ft,i , the drag force Fd,i , the force from the bottom mooring lines Fbm,i , and the force from the inter-body mooring lines Fim,i . The inertia matrix Mij contains the following non-zero terms, Mii = mb , for i = 1, 2, 3, 7, 8, 9, 13, 14, 15 ,

(2)

Mii = I44 = I55 , for i = 4, 5, 10, 11, 16, 17 ,

(3)

Mii = I66 , for i = 6, 12, 18 ,

(4)

Mij = Mji = mb zg , for (i, j) = (1, 5), (7, 11), (13, 17) ,

(5)

Mij = Mji = −mb zg , for (i, j) = (2, 4), (8, 10), (14, 16) ,

(6)

where mb is the buoy mass and zg is the z−coordinate of the centre of gravity. The terms I44 , I55 , I66 are the moments of inertia of the buoy about the x, y, and z axes of each buoy referential, aligned with the fixed reference frame.

100

105

110

2.2. Hydrodynamic radiation and excitation coefficients The hydrodynamic radiation and excitation coefficients were calculated using WAMIT, a BEM used for radiation and diffraction wave-structure interaction problems [41]. The model considers six DoF (three translational and three rotational) for each floating structure’s motion and one DoF (heave) for each OWC’s motion. The computations make use of the higher order discretisation method used in [6, 10], for a triangular array configuration. The grid used in the computations is shown in Fig. 3. The top of the OWC device is modelled by a rigid piston of finite thickness, with a mass given by the product of the sea water density and its volume (as in [6, 10]). An alternative modelling approach is to use a weightless piston at the free surface. However, this approach is prone to numerical errors at high frequencies. The finite length piston approach solves this problem and provides equivalent results. In Fig. 3, the bottom surface of the piston is presented in red, while the blue grid at the free surface is used for the removal of irregular frequencies [41]. Since the simulation considers three devices (N = 3), the total number of modes used in computations is 21. 6

Figure 1: Top view of the triangular array with inter-body mooring connections. The spar-buoy OWC devices are represented by indices 1-3 and the clump weight is represented by index 4.

Figure 2: Three-dimensional view of the triangular array with inter-body mooring connections.

The radiation force complex amplitude Frad,i is   Frad,i (ω) = − −ω 2 Aij (ω) + ıωBij (ω) xj ,

(7)

where Aij is the frequency-dependent added mass coefficient due to the mass of fluid accelerated in mode i by the oscillatory motion of mode j, and Bij is the frequency-dependent radiation damping coefficient due to the damping affecting mode i caused by the generation of radiated waves from mode j. The symbol ı √ represents the imaginary unit (ı = −1). The wave excitation force complex amplitude is Fexc,i (ω) = Aw Γi (ω) . 115

(8)

Here Aw is the incident wave amplitude and Γi represents the frequency-dependent wave excitation coefficient (force per unit wave amplitude). 7

Figure 3: Grid of an array of three spar-buoy OWC devices, as modelled in WAMIT (higher order discretisation method).

The hydrodynamic coefficients Aij , Bij , and Γi (real and imaginary part) are computed using the BEM for a finite number of wave frequencies ω. 2.3. Hydrostatic coefficients The linearised form of the hydrostatic force is considered, Fhst,i = −Cij xj ,

(9)

with Cij denoting the linear hydrostatic restoring coefficients. For the rotational modes, we assume small amplitude angles so that sin xj ≈ xj for j = 4, 5, 6. For an axisymmetric device, Cij has non-zero values for heave (buoy and OWC), roll, and pitch,

120

Cii = ρw gSb , for i = 3, 9, 15 ,

(10)

Cii = ρw gSo , for i = 19, 20, 21 ,

(11)

Cii = ρw gV GM , for i = 4, 5, 10, 11, 16, 17 ,

(12)

where ρw is the water density, g is the acceleration due to gravity, Sb is the waterplane area of the buoy, So is the waterplane area of the OWC, V is the buoy’s displaced water volume, and GM is the buoy’s metacentric height. 2.4. Power absorption and air compressibility effects Considering air as an ideal gas and the air compression and expansion cycle inside the chamber as an isentropic process, as proposed in [42], the forces acting on the device due to the pressure variations can be linearised by assuming that variations in air chamber volume are small compared to the air chamber volume at rest (V0 ). The force due to pressure variation in the air chamber can be represented by a damping and a restoring term Ft,i (ω) = − [ıωBt,ij + Ct,ij ] xj .

125

(13)

Here the damping terms Bt,ij represent the damping introduced by the extraction of energy from the turbine-generator set, and the restoring terms Ct,ij represent the air compressibility effect [43]. 8

Both of these effects depend only on the relative heave motion between the buoy and OWC. The Bt,ij matrix consists of the following non-zero terms, Bt,ii = Bt , for i = 3, 9, 15, 19, 20, 21 ,

(14)

Bt,ij = Bt,ji = −Bt , for (i, j) = (3, 19), (9, 20), (15, 21) ,

(15)

where the turbine damping coefficient is Bt =

ρa So2 1 . ω 2 V02 kt 1+ 2 4 kt ca

(16)

Here ρa is the average air density inside the chamber, V0 is the air chamber volume at equilibrium, and ca is the speed of sound at atmospheric conditions. The linear turbine coefficient kt , representing the turbine mass flow rate to pressure drop ratio, depends on the turbine’s geometry, size, and rotational speed. Similarly, the non-zero terms of Ct,ij are Ct,ii = Ct , for i = 3, 9, 15, 19, 20, 21 ,

(17)

Ct,ij = Ct,ji = −Ct , for (i, j) = (3, 19), (9, 20), (15, 21) ,

(18)

where the linear restoring coefficient related to the compressibility of air trapped inside the air chamber is ρa So2 Ct = ω 2 V02 1+ 2 4 kt ca



ω 2 V0 kt2 c2a

 .

(19)

2.5. Drag force for regular waves The calculation of the drag forces due to viscous fluid effects on the buoy is based on the drag-term of the Morison equation [44]. For consistency with the linear model, the non-linear viscous drag force under regular waves (proportional to the square of the velocity) is approximated with a linear force by using only the first harmonic of the Fourier series [10, 45, 46]. The complex amplitudes of the linearised drag forces in surge (i = 1, 7, 13) and sway (i = 2, 8, 14), and of the linearised drag moments in roll (i = 4, 10, 16) and pitch (i = 5, 11, 17) are given by Fd,i = −ıωBd,i xi ,

(20)

where the linear damping coefficient is   1 3π (e) Bd,i = ρw Ad,i cd,i ıωxi . 2 8 130

135

(21)

Here Ad,i is the drag area (for i = 1, 2, 7, 8, 13, 14) or drag area moment (for i = 4, 5, 10, 11, 16, 17) for the normal direction, and cd,i is the drag coefficient for the normal direction in mode i. The term in parenthesis (e) in Eq. 21 represents the linear approximation of the quadratic drag force. The term xi represents the estimated value of xi . Since xi is the solution of the system of equations presented in Eq. 1, these values require an iterative calculation. For the computation of the drag forces in heave, two sources of viscous drag effects are taken into account, as presented in [10]. The first one applies to the buoy’s drag inducing surfaces, and is proportional to the relative velocity between the buoy and the water flow. In this case, the Morison equation for a moving body in an oscillatory flow is considered [47]. The second drag force, which accounts for the flow inside the OWC tube, is proportional to the relative heave velocity between the buoy and the OWC. 9

For the drag force induced by the buoy motion, the buoy is simplified into various drag sections, representing the dominant sources of drag effects. The heave drag force, due to the buoy’s heave motion, is given by [45, 10] X Fd,z,i = − Bd,z,n (ıωxi − uz (zn )) , for i = 3, 9, 15 . (22) n=a,b,...

where n is the index of the drag surface, zn is the z-coordinate of the drag surface, uz is the complex amplitude of the velocity component aligned with the z-axis, uz =

ıgkAw sinh[k(h + z)] [−ık(x cos θ+y sin θ)] e , ω cosh kh

(23)

where k is the wavenumber and h is the water depth. The velocity uz is taken at the position (x, y) of the axis of the corresponding device at rest. The heave drag damping term is given by the sum of damping terms from each drag inducing surface   1 3π (e) (24) Bd,z,n = ρw cd,3 Ad,z,n ıωxi − uz (zn ) , 2 8 140

where cd,3 is the heave drag coefficient of the buoy and Ad,z,n is the area normal to the drag inducing surface. With regards to the drag force accounting for the drag effect inside the tube, its magnitude depends on the relative heave motion between the buoy and the OWC, Fd,tb = −ıωBd,tb (xi − xj ) .

(25)

Here (i, j) = (3, 19), (9, 20), (15, 21) and Bd,tb is the linearised damping coefficient of the flow inside the tube,   1 3π (e) (e) Bd,tb = ρw cd,tb Ad,tb (26) ıω(xi − xj ) . 2 8 Here cd,tb is the drag coefficient of the flow inside the tube and Ad,tb is the drag area, which we consider to be equal to So . The determination of the drag coefficients (cd,3 , cd,tb ) was based on experimental results [10]. The total drag force in heave for the buoy is Fd,i = (Fd,z,i + Fd,tb ) , for i = 3, 9, 15 ,

(27)

and the drag force experienced by the OWC is Fd,i = −Fd,tb , for i = 19, 20, 21 .

145

150

155

(28)

2.6. Clump weight and inter-body mooring connections The moorings consist of a three line system (1-4, 2-4, 3-4), connecting the three devices to a centrally placed clump weight. This weight pulls the buoys towards the centre of the triangle. The inter-body mooring cables are assumed inelastic and of negligible mass. As was the case in [37], the damping on the cables due to real fluid effects is ignored. All lines are attached to the centres of the buoys. In the absence of waves, the centres√of the buoys lie on the free surface plane, a distance L from the centre of the triangle, and a distance 3L apart from each other. Under calm √ sea conditions, the centre of the clump weight (also defined as Body 4) is at a vertical position zcw = − G2 − L2 below the free surface, where G is the length of cables 1-4, 2-4, 3-4. The wave incidence angle θ is the angle between the direction of propagation of incident waves and the x-axis. Variables x22 , x23 , and x24 correspond to the surge, sway, and heave oscillation amplitudes of Body 4. The coordinates x22 , x23 , x24 of Body 4 depend on the instantaneous values of xi for i = {1, 2, 3, 7, 8, 9, 13, 14, 15} through linearised geometric relationships [37]. Small wave amplitudes and 10

small body motions are assumed. The displacements of the three bodies in surge, sway, and heave are small compared to the distance L between each buoy and the clump weight, and to the water depth h. By neglecting products of small quantities, the following relations apply, based on the principle of virtual works [36, 37]: √

2|zcw | (x3 − x24 ) = 0 , L √ 2|zcw | x22 − x7 − 3(x8 − x23 ) + (x9 − x24 ) = 0 , L |zcw | x22 − x13 + (x24 − x15 ) = 0. L

x22 − x1 +

160

3(x2 − x23 ) +

(29) (30) (31)

The tension force vectors r¯1 , r¯2 , r¯3 correspond to the lines 1-4, 2-4, 3-4 and to the angles α1 , α2 , α3 with the horizontal surface plane, and depend on the instantaneous positions of Buoys 1, 2, 3. In the absence of waves, α1 = α2 = α3 = α = arctan(|zcw |/L) ,

(32)

r¯1 = r¯2 = r¯3 = r¯ .

(33)

The tension force is r¯ = (g/3)(mcw − ρw vcw ) csc α .

165

(34)

Here g is the acceleration due to gravity, mcw is the mass of the weight, and vcw is the volume of the weight. The parameters rj , εj , are the perturbations to the calm sea values r¯, α: εj =

xj − x24 sec α . G

(35)

In this case j = {3, 9, 15}. The projections of the lines 1-4, 2-4, 3-4, relative to each body, onto the horizontal plane form the angles δ1 − π/3, δ2 + π/3, δ3 + π [37]: √ 3 1 δ1 = (x22 − x1 ) + (x23 − x2 ), (36) 2L 2L √ 3 1 δ2 = − (x22 − x7 ) + (x23 − x8 ), (37) 2L 2L 1 δ3 = (x14 − x23 ). (38) L The weight oscillates only in three DoF: surge, sway, and heave. The equations of motion for the clump weight are included in the formulation to account for its dynamic behaviour. Therefore, it is subject to the inertia of the weight mcw , the frequency-dependent hydrodynamic added masses Acw,hr and Acw,z , the radiation damping coefficients Bcw,hr and Bcw,z , the damping coefficient due to drag induced by the fluid on the weight, Bcw,d , but also to the inter-body mooring force Fim,i for i = {22, 23, 24}. The clump weight is assumed to be submerged at a depth where the excitation force can be neglected. The parameters Acw,hr and Acw,z represent the horizontal and heave added mass coefficients of the clump weight, and are denoted by subscripts hr and z, respectively. The clump weight is assumed to have spherical geometry, therefore the added mass is the same for the horizontal and vertical modes, and was calculated as Acw = (2/3)ρw πa3cw , where acw is the radius of the sphere. This corresponds to the added mass of an accelerating sphere in an unbounded fluid [48]. The equation of motion for the clump weight (Body 4) in the three modes, in the frequency-domain, is 24 X

[−ω 2 (mcw + Acw ) + ıωBcw ]xj = Fim,i + Fd,cw,i ,

j=22

11

(39)

where Fd,cw,i represents the amplitude of the drag force that the clump weight experiences from the motion in mode i. The forces Fim,j are the forces the clump weight experiences due to the inter-body mooring connections with the three buoys, and are written as √ sin α 3 (δ2 − δ1 ) cos α + r¯ (ε1 + ε2 ) Fim,22 = r¯ 2 2 (40) cos α − (r1 + r2 ) − r¯ε3 sin α + r3 cos α, 2

Fim,23

√ 3 cos α = −¯ r(δ1 + δ2 ) + r¯ (ε2 − ε1 ) sin α 2 2 √ 3 + (r1 − r2 ) cos α + r¯δ3 cos α, 2

Fim,24 = r¯(ε1 + ε2 + ε3 ) cos α + (r1 + r2 + r3 ) sin α.

(41)

(42)

The individual terms representing the forces associated with the inter-body mooring connections for the three buoys are associated with the three modes x22 , x23 , x24 , and are expressed as √ 3 sin α cos α + r¯ δ1 cos α − r¯ε1 , (43) Fim,1 = r1 2 2 2 √ √ 3 cos α 3 Fim,2 = −r1 cos α + r¯δ1 + r¯ ε1 sin α, (44) 2 2 2 Fim,3 = −¯ rε1 cos α − r1 sin α, (45) √ cos α 3 sin α Fim,7 = r2 − r¯ δ2 cos α − r¯ε2 , (46) 2 2 2 √ √ 3 3 cos α cos α + Rδ2 − r¯ ε2 sin α, (47) Fim,8 = r2 2 2 2 Fim,9 = −¯ rε2 cos α − r2 sin α, (48) √ 3 Fim,13 = −r3 cos α + r¯ δ3 cos α + r¯ε3 sin α, (49) 2 Fim,14 = −¯ rδ3 cos α, (50) Fim,15 = −¯ rε3 cos α − r3 sin α.

(51)

The clump weight experiences drag which depends on the flow velocity in the x-, y- and z-axis direction. The clump weight drag force amplitude Fd,cw,i in mode i is defined as in Eq. 20. The corresponding damping coefficients Bd,cw,j , associated with the drag of the flow on the clump weight is   1 3π (e) Bd,cw,i = ρw Acw cd,cw (52) ıωxi − un (zcw ) , 2 8 where Acw is the projected surface area of the clump weight, cd,cw is the drag coefficient of a sphere, and the flow velocity in the x- and y-axis direction, in the absence of the weight and buoys, is

170

ux =

gkAw cosh[k(h + z)] cos θ e[−ık(x cos θ+y sin θ)] , ω cosh kh

(53)

uy =

gkAw cosh[k(h + z)] sin θ e[−ık(x cos θ+y sin θ)] . ω cosh kh

(54)

The flow velocity in the z-axis direction is given by Eq. 23. 12

2.7. Bottom mooring line forces

175

Bottom mooring lines are attached to each device (Fig. 2). The tensions f¯1 , f¯2 , f¯3 from the bottom mooring lines at the fairlead form an angle denoted by β with the horizontal plane, which is assumed to be unaffected by the bodies’ motions, since the length of the line is assumed to be much larger than their displacements. In the absence of waves, f¯1 = f¯2 = f¯3 = f¯. The bottom mooring tension f¯ is related to the inter-body line tension r¯ (applied on the lines 1-4, 2-4, 3-4) by f¯ = r¯ cos α/ cos β. The perturbation to the calm sea value f¯j is denoted by fj , for j = 1, 2, 3 (for each bottom mooring line). The forces applied on the three devices, due to the presence of the bottom mooring lines, are cos β f1 , Fbm,1 = − √ 2 3 Fbm,2 = cos βf1 , 2 Fbm,3 = − sin βf1 , cos β f2 , Fbm,7 = − √ 2 3 Fbm,8 = − cos βf2 , 2 Fbm,9 = − sin βf2 ,

180

185

(55) (56) (57) (58) (59) (60)

Fbm,13 = cos βf3 ,

(61)

Fbm,14 = 0,

(62)

Fbm,15 = − sin βf3 .

(63)

The equations representing the bottom mooring forces are identical to those presented in [37]. These forces were originally introduced in [15], where the authors explained the linear approximation of such forces, and their dependence on the static condition of the cable, as well as on the frequency and respective amplitude of the device motions. The attachment angle β - the angle formed between the horizontal surface plane and the bottom mooring cables on Bodies 1, 2, 3 - is based on the assumption that it remains constant, which is valid only for the case where the length of each bottom mooring line is much greater than the displacements of the bodies. The amplitudes of the extensions φj of mooring lines j (j = 1, 2, 3) due to the displacements of each body j are √

cos β + x3 sin β, 2 √ cos β φ2 = (x7 + 3x8 ) + x9 sin β, 2 φ3 = −x13 cos β + x15 sin β.

φ1 = (x1 −

3x2 )

(64) (65) (66)

The extension amplitude φj represents the amplitude of change in line length due to the motion of the body j. For the perturbation fj of the mooring force f¯j , the following dynamic formula in the frequency-domain applies  2  −ω Abm + ıωBbm + Cbm φj = fj , (67) 190

195

where the terms −ω 2 Abm φj , ıωBbm φj , and Cbm φj are perturbation forces representing the inertia, linear damping, and spring effect introduced by the bottom mooring cable j (for j = 1, 2, 3), respectively. The forthcoming analysis is expressed in the frequency-domain where the inertia and the damping forces are neglected since the focus of this analysis is centered on the dynamics of the inter-body mooring lines. Inertia and damping forces can be calculated through Morison’s equation [11], where the local fluid motion can be taken into consideration (e.g. [49]).

13

200

205

2.8. Independently moored array The methodology presented previously was also applied for an array where the devices were only moored with bottom mooring lines. Each device was attached to the sea bottom via three bottom mooring lines. In this case, the equation of motion (Eq. 1) still applies but the force amplitude of the inter-body line Fim is zero. Eqs. 1-67 remain the same, but a different set of equations is introduced for the three bottom mooring lines per device. The tensions on the bottom mooring lines are denoted as f¯j,s , where j is the index of the device (j = 1, 2, 3) and s refers to the line designation (s = A, B, C), with A referring to the bottom mooring line from the inter-body mooring configuration, and the others being distributed in a clockwise direction. Based on the works of [15] and [37], it can be easily shown that the forces applied to the three devices due to the presence of the bottom mooring lines can be expressed as 1 1 f1,bm = − cos β( f1,A − f1,B + f1,C ), 2√ 2 3 f2,bm = cos β(f1,A − f1,C ), 2 f3,bm = − sin β(f1,A + f1,B + f1,C ), 1 1 f7,bm = − cos β( f2,A + f2,B − f2,C ), 2√ 2 3 f8,bm = − cos β(f2,A − f2,B ), 2 f9,bm = − sin β(f2,A + f2,B + f2,C ), 1 1 f13,bm = cos β(f3,A − f3,B − f3,C ), 2 2 √ 3 f14,bm = − cos β(f3,B + f3,C ), 2 f15,bm = − sin β(f3,A + f3,B + f3,C ).

(68) (69) (70) (71) (72) (73) (74) (75) (76)

As in the previous case, the extensions φj,n of mooring line s (s = A, B, C) from body j (j = 1, 2, 3) due to displacements of body j can be obtained by linear decomposition,

φ1,C φ2,A φ2,B

√

cos β + x3 sin β, 2 φ1,B = −x1 cos β + x2 sin β, √ cos β = (x1 + 3x2 ) + x3 sin β, 2 √ cos β = (x7 + 3x8 ) + x9 sin β, 2 √ cos β = (x7 − 3x8 ) + x9 sin β, 2 φ2,C = −x7 cos β + x9 sin β,

φ1,A = (x1 −

3x2 )

φ3,A = −x13 cos β + x15 sin β, √ cos β φ3,B = (x13 + 3x14 ) + x15 sin β, 2 √ cos β φ3,C = (x13 − 3x14 ) + x15 sin β. 2

(77) (78) (79) (80) (81) (82) (83) (84) (85)

The dynamics of each mooring line are described by Eq. 67.

210

2.9. Capture width ratio The power available to the turbine is calculated via the following formula Pt =

1 2 2 ω Bt |xi − xj | , 2 14

(86)

for (i, j) = (3, 19), (9, 20), (15, 21). For a regular wave with wavenumber k propagating at a water depth h, the time-averaged power transported by a wave per unit wave crest length is   ρw gωA2w 2kh Pw = 1+ . (87) 4k sinh(2kh) The Capture Width Ratio (CWR) [50, 51] is a measure of the wave energy conversion efficiency, given by CWR =

Pt , P w d1

(88)

where d1 is the buoy’s diameter. 215

220

2.10. Irregular waves The model was further extended in order to compute the performance of the array under irregular wave conditions. For the purposes of the study, a stochastic model was constructed following the methodology presented in Gomes et al. [10], and originally presented in Falc˜ao and Rodrigues [52]. The model assumes that a sea state is represented by the superposition of elementary regular wave components with frequency ω and amplitude Aw . The instantaneous free-surface elevation at a given position can be described as a stationary, ergodic, and Gaussian process. The linear relation between the free-surface elevation and system responses allow statistical estimations of these parameters to be made. Further details on this theory may be found in [53, 54]. The semi-empirical Pierson-Moskowitz energy density spectrum for fully developed seas is given by [55]   1051.97 H2 , (89) Sω (ω) = 262.99 4 s 5 exp − 4 4 Te ω Te ω where Hs is the significant wave height and Te is the energy period. The variance of the system response is given by Z ∞ χ(ω) 2 2 dω . (90) σχ = Sω (ω) Aw 0 Here χ is a generic parameter that represents displacement, velocity, acceleration, or pressure difference. The time-averaged power available to the turbine for a given irregular wave sea state is Pt =

225

230

kt 2 σ , ρα p

(91)

where σp is the standard deviation of the pressure difference between the air chamber and the atmosphere. The CWR can now be calculated for the case of irregular waves, the same way as the case for regular waves presented in Eq. 88. In this case, the time-averaged power transported by a wave per unit wave crest length P w,irr can be computed via the sum of all discrete regular wave components, P w (ω, Aw (ω)) over P the range of frequencies considered in the irregular-wave energy spectrum. This is to say, P w,irr = P w (ω, A pw (ω)), where the regular wave components are a function of the frequency-dependent amplitude Aw (ω) = 2Sω (ω)∆ω, and ∆ω is the frequency interval being considered. For irregular waves, the linearised drag formulation presented in Sec. 2.5 is not applicable since the quadratic dependency with the wave amplitude Aw is not possible to verify. For this reason, the method described in [56] is applied. The method linearises the drag forces using the standard deviation of the oscillatory velocity for a given sea state [53]. This approach considers that a given oscillatory velocity u follows a Gaussian probability distribution with mean zero and standard deviationpσu . A better linear approximation of the quadratic force (proportional to |u|u) is observed for |u|u ≈ 8/πσu u. Therefore,

15

for irregular waves, the linearised damping coefficients in surge (i = 1, 7, 13), sway (i = 2, 8, 14), roll (i = 4, 10, 16), and pitch (i = 5, 11, 17) are expressed as r 8 1 σx˙ , (92) Bd,i = ρw Ad,i cd,i 2 π i where the standard deviation of the velocity ıωxi of mode i can be obtained from Eq. 90. The drag damping coefficient of the buoy in heave presented in Eq. 24 is replaced by r 1 8 Bd,z,n = ρw cd,3 Ad,z,n σ(x˙ −u ) , 2 π 3 z

(93)

where σ(x˙ 3 −uz ) is the standard deviation of the velocity ıωx3 − uz (zn ), and uz (zn ) is the flow velocity at the z-coordinate zn . The drag damping coefficient of the flow inside the tube (replacing Eq. 25) is given by r 8 1 σ(x˙ −x˙ ) . (94) Bd,tb = ρw cd,tb Ad,tb 2 π i j

235

Here σ(x˙ i −x˙ j ) is the standard deviation of the heave velocity difference between the buoy and the OWC (ıωxi − ıωxj ), for (i, j) = (3, 19), (9, 20), (15, 21). Similarly, the corresponding damping coefficients associated with drag effects on the clump weight are r 1 8 Bd,cw,i = ρw Acw cd,cw σ(x˙ i −un ) , (95) 2 π where σ(x˙ i −un ) is the standard deviation of relative velocity between the clump weight in mode i and the flow velocity in the same direction (ıωxi − un ).

240

2.11. Model implementation In this analysis, we consider the geometry of the spar-buoy OWC device presented in [6], with diameter d1 = 12 m, an OWC diameter d2 = 4.82 m and a draught l = 36 m. The geometry was optimised for a wave climate off the western coast of mainland Portugal. The physical properties of the geometry are presented in Table 1. The frequency-dependent hydrodynamic coefficients were calculated, considering a water depth h = 80 m. Table 1: Spar-buoy OWC physical properties. The moments of inertia are measured at the axes of the device reference frame. Parameter

Value

Buoy diameter d1 (at z = 0) [m] OWC diameter d2 (at z = 0) [m] Draught l (unmoored) [m] Mass m [kg] Moment of inertia I55 (= I44 ) [kg m2 ] Moment of inertia I66 [kg m2 ] CoG z-coordinate zg [m] CoB z-coordinate zb [m] Metacentic height GM [m]

245

12.00 4.82 36.00 1217.4×103 778.51×106 29.1×106 -19.29 -18.04 2.08

For regular-wave analysis, the wave amplitude was set to Aw = 1 m. For the formulation of the heave drag force (see Eq. 22), two drag surfaces are considered, as in [10]. The first drag surface (n = A), located at zA = −5 m has an area Ad,z,A = 94.87 m2 . The second surface (n = B), located at the bottom part of the device, at zB = −36 m has an area Ad,z,B = 96.91 m2 . Since the geometry is axisymmetric, surge and sway modes have the same frontal area. Therefore, Ad,i = 282.99 m2 , for i = 1, 2, 7, 8, 13, 14. The same 16

250

255

260

rationale applies for roll and pitch, where Ad,i = 4.1×106 m5 , for i = 4, 5, 10, 11, 16, 17. The drag coefficients were determined using experimental data from small-scale wave flume tests (decay and regular wave tests). The damping forces associated with viscous effects are likely to be over-estimated, since the experimental coefficients also include scale effects. For all buoys, the same drag coefficients were considered. The drag coefficient of the buoy in heave was cd,3 = 0.731, for the tube was cd,tb = 1.965, and for all other modes was cd,i = 2.0. The selection of the turbine coefficient kt was made based on its effect on the buoy heave RAO, |Zb |/Aw (Zb ≡ xi for i = 3, 9, 15), but also on its effect on the heaving RAO between the buoy and the OWC, |Zr |/Aw (Zr ≡ xi − xj for (i, j) = (3, 19), (6, 20), (15, 21)). Different values of the turbine coefficient kt were tested (kt = 0.00023, 0.00150, 0.00500, 0.01, 0.002909 m s). The selected value for the results presented in this study was kt = 0.00150 m s, since it contributed to the optimum CWR. The following computations were undertaken: • Case I: Unmoored array of spar-buoy OWCs; • Case II: Independently moored array of spar-buoy OWCs, with 3 bottom mooring lines each; • Case III: Array of spar-buoy OWCs with bottom and inter-body mooring connections.

265

270

275

280

√ The three spar-buoy OWCs were set in a triangular configuration, separated by a distance of 3L, where L was set to 30 m. The added mass Aij and the radiation damping coefficient Bij for the individual buoys and the corresponding OWCs of the triangular array were calculated using WAMIT, considering a uniform water depth of h = 80 m. For Case II, each device was bottom moored with three mooring connections. The dimensionless stiffness ∗ ∗ of the mooring line Cbm was set to 0.1 (where Cbm = Cbm ρgSb ). The linear damping Bbm and inertia of the mooring cables Abm were neglected in this analysis (as in [37]). The angle formed between the bottom mooring line and the horizontal free surface plane, β, was set to 30o , and the angle of wave incidence θ was set to 0. For Case III, the inter-body connections were modelled as in Fig. 1 and Fig. 2, where the clump weight (Body 4) is introduced in the centre of the array. The clump weight is of spherical shape, and has a density of ρcw = 2500 kg/m3 . Its mass is equal to 5% of the mass of the device. Since the clump weight is deeply submerged, and is located at a significant vertical distance from the free surface (z = zcw ), the frequency-dependent radiation damping coefficients (Bcw,hr and Bcw,z ) were neglected in the equation of motion. Regarding the bottom mooring lines, the linear damping Bbm and inertia of the mooring cables Abm were neglected, as for Case II and in [37]. The angle formed between the bottom mooring line and the horizontal free surface plane, β, was set to 30o . The angle α, formed between the inter-body mooring line and the horizontal free surface plane, was also set to 30o throughout these calculations. The angle of wave incidence θ was set to 0. 3. Results and discussion

285

3.1. Regular waves The current section presents an analysis of the Response Amplitude Operators (RAOs) of the system. The RAOs correspond to the ratio between the amplitudes of the modes of interest (that is, all 6 DoF for the buoy and one additional heaving mode for the OWC) and the incident wave amplitude Aw . Due to the symmetry of the problem for θ = 0o , Buoy 1 and Buoy 2, and their respective OWCs experienced the same RAOs in all modes and the same CWR. Therefore, in the following results, Buoy 1 and Buoy 2 are referred as ”Buoy 1,2”. The results are presented as a function of the incident wave frequency ω (Fig. 4). The vertical lines in Figs. 4d, 4e, and 4f indicate the system’s natural frequencies in heave. The natural frequency of heave mode i is  ωn,i =

ρw gS mii + Aii (ω)

1/2 .

(96) 17

290

295

300

305

310

315

320

325

330

Here i = 3, 9, 15 for the buoy’s heave mode and i = 19, 20, 21 for the OWC’s heave mode. This is the frequency at which maximum power extraction is most likely to be achieved [50]. Since the spar-buoy OWC is a two-body heaving system, it has two natural frequencies, one for the buoy and one for the OWC. The natural frequency of Buoys 1, 2, and 3 was estimated to be 0.693 rad/s (period of 9.1 s), and for their corresponding OWCs it was estimated to be 0.555 rad/s (period of 11.3 s). In Fig. 4, the results on the left-hand side (Figs. 4a, 4d, 4g) correspond to Case I (unmoored array), while the results in the centre refer to Case II (independently moored array) (Figs. 4b, 4e, 4h). The results on right-hand side refer to Case III (array with bottom and inter-body connections) (Figs. 4c, 4f, 4i). The results for a single unmoored device and a single device with three bottom mooring connections are presented for comparison. By introducing bottom mooring lines in a single device (i.e. a horizontal restoring effect), the surge RAO peak appears at low frequencies, instead of tending to infinity at ω = 0 in the unmoored case (Figs. 4a, 4b). The pitch RAO significantly increases at frequencies lower than 0.5 rad/s due to the coupling effect between surge and pitch (Figs. 4g, 4h). The CWR increases by 8.1% at the peak frequency, despite the decrease by approximately 12.2% in the heave RAO at the peak frequency (Figs. 5a, 5b). For the triangular array with unmoored devices (Case I), an oscillatory behaviour is observed for the surge and pitch RAOs over frequency ω (Fig. 4a). This can be seen for wave frequencies higher that 0.6 rad/s, due to the hydrodynamic coupling between devices (array effect). When compared to the isolated unmoored device, the unmoored array shows a better CWR for Buoys 1 and 2 at the peak frequency, but a lower value for Buoy 3. The CWR for the moored device and independently moored array shows the same trend (Fig. 5a). Regarding the array with inter-body connections (Case III), the heave RAO of Buoy 1,2 decreases by 9.0% at the peak frequency (Fig. 4f), compared to the heave RAO of Buoy 1,2 for the unmoored array, while the heave RAO of Buoy 3 decreases by 3.8%, compared to the heave RAO of Buoy 3 of the unmoored array (Fig. 4d). Although this appears to have an adverse effect on the array’s performance, the CWR must also be examined, as it is a function of the relative motion between the buoys and their respective OWCs. When compared to the unmmoored array configuration (Case I), the independently moored array (Case II) shows improvements in CWR of 7.1% for Buoy 1,2 and 7.8% for Buoy 3, at the peak frequency (Fig. 5b). Making the same comparison, Case III shows a similar value for Buoy 1,2 and an increase of 7.1% of Buoy 3, at the peak frequency (Fig. 5c). It is worth mentioning that these findings are strongly dependent on the value of the mooring stiffness Cbm . For example, in Case II, a higher mooring stiffness would contribute to a smaller increase in the CWR. The peaks in the surge and pitch responses of the buoys at very low frequencies (Figs. 4a, 4b, 4c, 4g, 4h, 4i) are likely due to a breakdown in the model’s validity, which is based on linear theory. In this region of operation, the model is not able to account for the non-linearities introduced by the mooring lines. In the above responses, the peaks occur between 0.1 − 0.2 rad/s (wave periods of 31.4 − 62.8 s), which correspond to the long wavelengths seen in stormy seas. Although the model does not accurately predict the performance of the system during storms, it should also be noted that under these circumstances, a WEC would normally be entering its survivability mode of operation. 3.1.1. Effect of wave incidence angle Figs. 6 and 7 present the RAOs of the independently moored array (Case II) and the array with interbody mooring connections array (Case III), for different angles of wave incidence θ. The heave RAO of the three devices appears to remain constant with variations in the angle of wave incidence, and small variations can be observed for the CWR for both cases. For higher values of the angle of wave incidence, the surge and sway RAOs of Buoys 1 and 2 for Case II and Case III are no longer identical, since there is no array symmetry about the direction of wave propagation. 3.2. Irregular Waves

335

Calculations were performed for irregular waves using the same model parameters that were used throughout Section 3.1 (i.e. the same turbine damping, angle of wave incidence, stiffness parameter, drag coefficients, wave amplitude, and mooring parameters). The output of these calculations is characterised by statistical 18

5

3 2 1 0.5

1.0

1.5

Frequency [rad/s]

3 2 1 0 0.0

2.0

0.5

Buoy heave RAO [ ]

Buoy heave RAO [ ]

5

3 2 1 0 0.0

0.5

1.0

1.5

Frequency [rad/s]

4

2 1 0.5

Buoy pitch RAO [o /m]

Buoy pitch RAO [o /m]

1 1.0

1.5

Frequency [rad/s] (g)

1.0

1.5

Frequency [rad/s]

5

2

0.5

1 0.5

2.0

2.0

2 1 0.5

1.0

1.5

Frequency [rad/s] (h)

2.0

Case III : Buoy 1, 2 Case III : Buoy 3 Clump weight

4 3 2 1 0 0.0

0.5

1.0

1.5

Frequency [rad/s]

5

3

0 0.0

1.5

2.0

(f)

Case II : Buoy 1, 2 Case II : Buoy 3 Single buoy, moored

4

1.0

Frequency [rad/s]

(e)

3

0 0.0

2

5

3

0 0.0

2.0

Case I : Buoy 1, 2 Case I : Buoy 3 Single buoy, unmoored

4

3

(c)

Case II : Buoy 1, 2 Case II : Buoy 3 Single buoy, moored

(d)

5

4

0 0.0

2.0

Case III : Buoy 1, 2 Case III : Buoy 3 Clump weight

(b)

Case I : Buoy 1, 2 Case I : Buoy 3 Single buoy, unmoored

4

1.5

Frequency [rad/s]

(a)

5

1.0

Buoy heave RAO [ ]

0 0.0

4

Buoy pitch RAO [o /m]

4

5

Case II : Buoy 1, 2 Case II : Buoy 3 Single buoy, moored

Buoy surge RAO [ ]

Case I : Buoy 1, 2 Case I : Buoy 3 Single buoy, unmoored

Buoy surge RAO [ ]

Buoy surge RAO [ ]

5

2.0

Case III : Buoy 1, 2 Case III : Buoy 3

4 3 2 1 0 0.0

0.5

1.0

1.5

Frequency [rad/s]

2.0

(i)

∗ = 0.1, β = 30o , Figure 4: RAOs for surge, heave, and pitch, against the wave frequency ω, for kt = 0.0015 ms, Aw = 1 m, Cbm and θ = 0o . The graphs on the left refer to the unmoored array (Case I) (4a, 4d, 4g), along with the results for the free-floating single buoy. The graphs in the centre correspond to the independently moored array (Case II) (4b, 4e, 4h), and the single moored buoy. The graphs on the right (4c, 4f, 4i) refer to the array with bottom and inter-body mooring connections (Case III) (with α = 30o ), and the clump weight when applicable.

19

0.6

0.6

Case I : Buoy 1, 2 Case I : Buoy 3 Single buoy, unmoored

0.5

0.5

0.2 0.1

0.4

CWR [ ]

0.3

0.3 0.2 0.1

0.0 0.0

0.5

1.0

1.5

Frequency [rad/s] (a)

2.0

0.0 0.0

Case III : Buoy 1, 2 Case III : Buoy 3

0.5

0.4

CWR [ ]

CWR [ ]

0.4

0.6

Case II : Buoy 1, 2 Case II : Buoy 3 Single buoy, moored

0.3 0.2 0.1

0.5

1.0

1.5

Frequency [rad/s] (b)

2.0

0.0 0.0

0.5

1.0

1.5

Frequency [rad/s]

2.0

(c)

∗ Figure 5: CWR against the wave frequency ω, for kt = 0.0015 ms, Aw = 1 m, Cbm = 0.1, β = 30o , and θ = 0o . The graph on the left (5a) refers to the array without mooring connections (Case I) and the free-floating single buoy. The graph in the centre (5b) refers to the array with bottom mooring lines only (Case II) and the single moored buoy. The graph on the right (5c) corresponds to the array with bottom and inter-body mooring connections (Case III) (with α = 30o ).

340

345

350

355

values - standard deviations of the motion response amplitudes. The results shown in Figs. 8 and 9 are given in dimensionless form; the standard deviations of the motion responses are divided by the standard deviation of the free surface elevation ση , for the wave spectrum presented in Eq. 89, ση = Hs /4. Contrary to the case of regular waves, a different linearisation method was applied to model the drag force under irregular wave operation. For regular waves, the relative velocity standard deviation has the same value over all wave frequencies. For irregular waves, it varies with the sea state. Fig. 8 presents the dimensionless standard deviation for surge (Fig. 8a), heave (Fig. 8b), and pitch motions (Fig. 8c), as well as the CWRirr for a single device (Fig. 8d), over a range of energy periods Te , and for a selection of significant wave heights Hs = 0.5, 1.0, 2.0, 3.0, 4.0 m, for an isolated device. Higher values of the significant wave height Hs contribute to lower values of the dimensionless standard deviation for surge, heave, and pitch. The CWRirr also decreases with increasing Hs . The linearisation of the quadratic viscous drag force is still dependent on the standard deviation of the velocity (Eq. 92 -94). Therefore, a higher relative flow velocity will lead to a more damped response, as observed in Fig. 8. Motions with a higher velocity will be subjected to a higher magnitude drag damping. This effect causes a reduction of the CWRirr with the increase of the Hs . The same cases that were studied in Fig. 4, were examined in this section for irregular sea states with Hs = 1.62 m (Fig. 9), where the average values of the studied cases are presented for surge (Fig. 9a), heave (Fig. 9b), pitch (Fig. 9c), and the CWRirr (Fig. 9d). For high energy periods Te , the average standard deviation of surge (Fig. 9a) is slightly higher with mooring lines (Cases II and III), compared to Case I. This is in agreement with the regular waves results. Based on the average CWRirr results (Fig. 9d), the performance of the independently moored array appears to be similar to that of the array with inter-body mooring connections. The annual average gain factor q is presented in Table 2, for various angles of wave incidence θ and ∗ various mooring parameters (kt , β, Cbm , α). The annual average gain factor is defined as PN q=

i=1

P ann,i

N P ann,iso

,

(97)

where P ann,i is the annual average power available to the turbine of device i in the N -device array, and P ann,iso is the annual average power available to the turbine of an unmoored isolated device with similar

20

2 1 1.5

Frequency [rad/s]

4

4

3 2 1 0 0.0

2.0

0.5

1.0

1.5

Frequency [rad/s]

3 2 1 0 0.0

2.0

5

4

4

4

3 2 1 0.5

1.0

1.5

Frequency [rad/s]

3 2 1 0 0.0

2.0

Sway RAO 3 [ ]

5

Sway RAO 2 [ ]

0.5

1.0

1.5

Frequency [rad/s]

5

4

4

4

2 1 0 0.0

0.5

1.0

1.5

Frequency [rad/s]

2 1 0 0.0

2.0

0.5

1.0

1.5

Frequency [rad/s]

0.5

0.5

0.2 0.1 0.0 0.0

0.5

1.0

1.5

Frequency [rad/s]

2.0

CWR Buoy 3 [ ]

0.5

CWR Buoy 2 [ ]

0.6

0.3

0.3 0.2 0.1 0.0 0.0

0.5

1.0

1.5

Frequency [rad/s]

2.0

1.0

1.5

2.0

0.5

1.0

1.5

2.0

0.5

1.0

1.5

2.0

Frequency [rad/s]

1

0.6 0.4

0.5

2

0.6 0.4

2.0

3

0 0.0

2.0

1.5

1

5

3

1.0

Frequency [rad/s]

2

5

3

0.5

3

0 0.0

2.0

Heave RAO 3 [ ]

Sway RAO 1 [ ]

1.0

5

5

0 0.0

Heave RAO 1 [ ]

0.5

5

Surge RAO 3 [ ]

3

Surge RAO 2 [ ]

4

0 0.0

CWR Buoy 1 [ ]

= 0o = 30o = 60o Single buoy

Heave RAO 2 [ ]

Surge RAO 1 [ ]

5

Frequency [rad/s]

0.4 0.3 0.2 0.1 0.0 0.0

Frequency [rad/s]

Figure 6: Buoy RAOs and CWRs, as a function of the wave frequency ω, for Case II (independently moored array), for different ∗ = 0.1. values of the angle wave of incidence θ. Default values were used: kt = 0.0015 ms, Aw = 1 m, β = 30o , and Cbm

21

2 1 1.5

Frequency [rad/s]

4

4

3 2 1 0 0.0

2.0

0.5

1.0

1.5

Frequency [rad/s]

3 2 1 0 0.0

2.0

5

4

4

4

3 2 1 0.5

1.0

1.5

Frequency [rad/s]

3 2 1 0 0.0

2.0

Sway RAO 3 [ ]

5

Sway RAO 2 [ ]

0.5

1.0

1.5

Frequency [rad/s]

5

4

4

4

2 1 0 0.0

0.5

1.0

1.5

Frequency [rad/s]

2 1 0 0.0

2.0

0.5

1.0

1.5

Frequency [rad/s]

0.5

0.5

0.2 0.1 0.0 0.0

0.5

1.0

1.5

Frequency [rad/s]

2.0

CWR Buoy 3 [ ]

0.5

CWR Buoy 2 [ ]

0.6

0.3

0.3 0.2 0.1 0.0 0.0

0.5

1.0

1.5

Frequency [rad/s]

2.0

1.0

1.5

2.0

0.5

1.0

1.5

2.0

0.5

1.0

1.5

2.0

Frequency [rad/s]

1

0.6 0.4

0.5

2

0.6 0.4

2.0

3

0 0.0

2.0

1.5

1

5

3

1.0

Frequency [rad/s]

2

5

3

0.5

3

0 0.0

2.0

Heave RAO 3 [ ]

Sway RAO 1 [ ]

1.0

5

5

0 0.0

Heave RAO 1 [ ]

0.5

5

Surge RAO 3 [ ]

3

Surge RAO 2 [ ]

4

0 0.0

CWR Buoy 1 [ ]

= 0o = 30o = 60o Single buoy

Heave RAO 2 [ ]

Surge RAO 1 [ ]

5

Frequency [rad/s]

0.4 0.3 0.2 0.1 0.0 0.0

Frequency [rad/s]

Figure 7: Buoy RAOs and CWRs, as a function of the wave frequency ω, for Case III (array with bottom and inter-body mooring connections), for different values of the angle wave of incidence θ. Default values were used: kt = 0.0015 ms, Aw = 1 m, α = 30o , ∗ = 0.1. β = 30o , and Cbm

22

3.0

[ ]

2.0

2.5

1.5

heave /

[ ]

2.5

surge /

3.0

Hs = 0.5m Hs = 1.0m Hs = 2.0m Hs = 3.0m Hs = 4.0m

1.0 0.5 0.0

2.0 1.5 1.0 0.5

6

8

10

12

14

EnergyPeriod Te [s]

16

0.0

18

6

8

10

(a)

12

14

16

18

0.30

CWRirr [ ]

[o /m] pitch /

18

0.35 0.25 0.20 0.15 0.10

1 0

16

0.40

4

2

14

(b)

5

3

12

EnergyPeriod Te [s]

0.05 6

8

10

12

14

EnergyPeriod Te [s]

16

0.00

18

6

8

10

(c)

Energy Period Te [s] (d)

Figure 8: Standard deviations for surge (8a), heave (8b), pitch (8c), and CWRirr (8d), against the energy period Te , for an isolated device, for kt = 0.0015 ms, θ = 0o , and various values of Hs .

properties. The annual average power available to the turbine of device i is P ann,i =

M X

φm P irr,i ,

(98)

m=1

360

365

370

where φm represents the frequency of occurrence of a given sea state m over a long period of time, and M is the total number of sea states considered for defining the annual average wave climate. In this case, an annual wave climate off the western coast of mainland Portugal was considered, which is characterised by a set of sea states described by a Pierson-Moskowitz wave spectrum (with inputs Hs and Te ). A total of 14 sea states were taken into consideration. The frequency of occurrence φm for these sea states was obtained from [6], which is a reduced version of a larger set of sea states representing the same wave climate, presented originally in [57]. Throughout these calculations, a dominant angle of wave incidence θ is considered. For this triangular configuration, for all cases and for the default turbine coefficient kt , the most beneficial angle of wave incidence θ, appeared to be 30o . Both mooring configurations (Case II and III), appear to be beneficial with the selection of the default turbine coefficient kt = 0.0015 ms and varying mooring parameters α and β (q > 1). A higher turbine coefficient and a higher mooring stiffness ∗ parameter Cbm seem to adversely affect the system performance. For the default values (kt = 0.0015 ms, ∗ Cbm = 0.1, β = 30o , and α = 30o ), Case II appears to perform slightly better than Case III for the studied wave climate. 23

3.0

[ ]

2.0

2.5

1.5

heave /

[ ]

2.5

surge /

3.0

Case I Case II Case III Single buoy, unmoored Single buoy, moored

1.0 0.5 0.0

2.0 1.5 1.0 0.5

6

8

10

12

14

EnergyPeriod Te [s]

16

0.0

18

6

8

10

(a)

12

14

16

18

0.30

CWRirr [ ]

[o /m] pitch /

18

0.35 0.25 0.20 0.15 0.10

1 0

16

0.40

4

2

14

(b)

5

3

12

EnergyPeriod Te [s]

0.05 6

8

10

12

14

EnergyPeriod Te [s]

16

0.00

18

(c)

6

8

10

EnergyPeriod Te [s] (d)

Figure 9: Standard deviations for surge (9a), heave (9b), pitch (9c), and CWRirr (9d), against the energy period Te , for ∗ = 0.1, β = 30o , α = 30o , and θ = 0o . For the array cases, the values presented refer to the kt = 0.0015 ms, Hs =1.62 m, Cbm array-averaged motions.

4. Conclusions

375

380

385

390

This paper presented a numerical analysis of a triangular array of spar-buoy OWCs, with bottom and inter-body mooring connections, for regular and irregular waves. The array performance was compared to an unmoored and an independently moored array. The model considered a linearised approximation of the quadratic viscous drag force, with drag coefficient values based on experimental data. In addition, a stochastic model was applied to evaluate the array performance in irregular wave sea states. For regular waves, the analysis showed that the unmoored array was influenced by the hydrodynamic coupling between the three bodies in surge; no significant difference was observed for the average heave amplitude of the array at the peak frequency. By including the inter-body connections, a coupling effect between surge/sway and heave was observed. The buoys’ average heave amplitude at the peak frequency decreased by 7.2%. The average CWR of all devices slightly increased after including the inter-body mooring lines, at the peak frequency. The simulations with a set of uni-directional irregular wave sea states, characterising a particular wave climate, showed that an angle of wave incidence of 30o provided better results for the different mooring configurations tested. The independently moored array configuration and the array with inter-body connections showed similar results for the different configurations tested, with variations around 2 − 3% of each other’s values. Since the individually moored array is more expensive than the one with interbody mooring lines, the array with inter-body connections seems to be the more economically attractive solution. 24

Table 2: Annual-averaged gain factor q for the studied cases, for three angles of wave incidence θ, and varying model parameters.

Case I θ = 0o

θ = 30o

θ = 60o

kt [ms]

1.014 1.016

1.018 1.016

1.015 1.018

0.0015 0.0050

θ = 0o

θ = 30o

θ = 60o

kt [ms]

β [o ]

∗ Cbm [-]

1.036 1.046 1.034 1.053 1.013 0.832 0.984

1.040 1.090 1.038 1.057 1.018 0.842 0.984

1.037 1.046 1.032 1.054 1.014 0.832 0.985

0.0015 0.0015 0.0015 0.0015 0.0015 0.0015 0.0050

30 30 30 60 60 60 30

0.05 0.10 0.20 0.05 0.10 0.20 0.10

θ = 0o

θ = 30o

θ = 60o

kt [ms]

β [o ]

∗ Cbm [-]

α [o ]

1.012 1.013 1.017 1.010 0.995 1.026 1.027 1.019 1.033 1.036 1.031 1.002 0.991 0.944 0.971

1.017 1.015 1.022 1.019 1.013 1.026 1.032 1.028 1.034 1.043 1.041 1.002 0.992 0.952 0.972

1.013 1.015 1.019 1.015 1.012 1.025 1.030 1.026 1.036 1.041 1.039 1.003 0.993 0.956 0.975

0.0015 0.0015 0.0015 0.0015 0.0015 0.0150 0.0150 0.0150 0.0015 0.0015 0.0015 0.0050 0.0050 0.0050 0.0050

30 30 30 30 30 60 60 60 60 60 60 30 30 30 60

0.05 0.10 0.10 0.10 0.20 0.05 0.05 0.05 0.10 0.10 0.10 0.05 0.10 0.20 0.10

30 20 30 40 30 20 30 40 20 30 40 30 30 30 30

Case II

Case III

Acknowledgments This work was partially funded by the Portuguese Foundation for Science and Technology (FCT), through IDMEC, under LAETA, project UID/EMS/50022/2019, and contract OCEANERA/0008/2016 (CAPTOW project). R. P. F. Gomes was supported by postdoctoral fellowship SFRH/BPD/ 93209/2013 from FCT. 395

400

405

410

References [1] Y. Masuda, An experience of wave power generator through tests and improvement, in: D. V. Evans, A. F. de O. Falc˜ ao (Eds), Hydrodynamics of Ocean Wave Energy Utilization Symposium, Lisbon, Portugal, 1985. [2] A. F. O. Falc˜ ao, J. C. C. Henriques, Oscillating-water-column wave energy converters and air turbines: A review, Renew. Energy 85 (2015) 1391–1424. [3] A. Brito-Melo, T. Hofman, A. J. N. A. Sarmento, A. H. Cl´ ement, G. Delhommeau, Numerical modelling of OWC shoreline devices including the effect of surrounding coastline and non-flat bottom, Int. J. Offshore Polar Eng. 11 (2) (2001) 147–154. [4] Y. Torre-Enciso, I. Ortubia, L. I. L´ opez de Aguileta, J. Marqu´ ez, Mutriku wave power plant: from the thinking out to the reality, in: Proc. 8th European Wave Energy Conf., Uppsala, Sweden, 2009, pp. 319–29. [5] Y. Masuda, M. E. McCormick, Experiences in pneumatic wave energy conversion in Japan, in: McCormick M. E., Kim Y. C., editors. Utilization of ocean waves-wave to energy conversion, New York: ASCE, 1987, pp. 1–33. [6] R. P. F. Gomes, J. C. C. Henriques, L. M. C. Gato, A. F. O. Falc˜ ao, Hydrodynamic optimization of an axisymmetric floating oscillating water column for wave energy conversion, Renew. Energy 44 (2012) 328–339. [7] S. Ribeiro e Silva, R. P. F. Gomes, A. F. O. Falc˜ ao, Hydrodynamic optimization of the UGEN: Wave energy converter with U-shaped interior oscillating water column, Int. J. Mar. Energy 15 (2016) 112–26. [8] M. E. McCormick, Analysis of a wave energy conversion buoy, Journal of Hydronautics 8 (3) (1974) 77–82.

25

415

420

425

430

435

440

445

450

455

460

465

470

475

[9] M. A. Alves, I. R. Costa, A. J. N. A. Sarmento, J. F. Chozas, Performance evaluation of an axisymmetric floating OWC, in: Proc. 20th Int. Offshore and Polar Engineering Conference, Beijing, China, 2010. [10] R. P. F. Gomes, J. C. C. Henriques, L. M. C. Gato, A. F. O. Falc˜ ao, Wave power extraction of a heaving floating oscillating water column in a wave channel, Renew. Energy 99 (2016) 1262–1275. [11] J. Davidson, J. Ringwood, Mathematical Modelling of Mooring Systems for Wave Energy Converters: A Review, Energies 10 (2017) 46 pages. [12] G. M. Paredes, L. Bergdahl, J. Palm, C. Eskilsson, F. T. Pinto, Station keeping design for floating wave energy devices compared to floating offshore oil and gas platforms, in: Proc. 10th European Wave and Tidal Energy Conference, Aalborg, Denmark, 2013. [13] P. C. Vicente, A. F. O. Falc˜ ao, P. A. P. Justino, Optimization of mooring configuration parameters of floating wave energy converters, in: Proc. of the ASME 2011 30th Int. Conf. Ocean Offshore Arctic Eng., Rotterdam, The Netherlands, 2011. [14] L. Bergdahl, Mooring design for WECs, in: Pecher A., Kofoed J. (eds) Handbook of Ocean Wave Energy, Springer, 2017, pp. 159–202. [15] J. Fitzgerald, L. Bergdahl, Including moorings in the assessment of a generic offshore energy converter: A frequency domain approach, Mar. Struct. 21 (1) (2008) 23–46. [16] L. Johanning, G. H. Smith, J. Wolfram, Measurement of static and dynamic mooring line damping and their importance for floating WEC devices, Ocean Eng. 34 (2007) 1918–1934. [17] Y. Liu, L. Bergdahl, Extreme mooring cable tensions due to wave-frequency excitations, Appl. Ocean Res. 20 (4) (1998) 237–249. [18] D. T. Brown, S. Mavrakos, Comparative study on mooring line dynamic loading, Mar. Struc. 12 (3) (1999) 131–151. [19] S. Parmeggiani, J. P. Kofoed, E. Friis-Madsen, Experimental study related to the mooring design for the 1.5 MW Wave Dragon WEC demonstrator at DanWEC, Energies 6 (2013) 1863–1886. [20] V. Harnois, P. R. Thies, L. Johanning, On peak mooring loads and the influence of environmental conditions for marine energy converters, J. Mar. Sci. Eng. 4 (2016) 24 pages. [21] J. Palm, C. Eskilsson, L. Bergdahl, An hp-adaptive discontinuous Galerkin method for modelling snap loads in mooring cables, Ocean Eng. 144 (2017) 266–276. [22] P. A. P. Justino, A. H. Cl´ ement, Hydrodynamic performance for small arrays of submerged spheres, in: 5th European Wave Energy Conference, Cork, Ireland, 2003, pp. 266–273. [23] S. A. Mavrakos, P. McIver, Comparison of methods for computing hydrodynamic characteristics of arrays of wave power devices, Appl. Ocean Res. 19 (1997) 283–291. [24] K. Budal, Theory for absorption of wave power by a system of interacting bodies, J. Ship Res. 21 (1977) 248–253. [25] D. V. Evans, A theory for wave-power absorption by oscillating bodies, Journal of Fluid Mechanics 77 (1976) 1–25. [26] J. Falnes, Radiation impedance matrix and optimum power absorption for interacting oscillators in surface waves, Appl. Ocean Res. 2 (1980) 75–80. [27] A. Babarit, Impact of long separating distances on the energy production of two interacting wave energy converters, J. Ocean Eng. 37 (2010) 718–729. [28] B. Borgarino, A. Babarit, P. Ferrant, Impact of wave interactions effects on energy absorption of large arrays of wave energy converters, Ocean Eng. 41 (2012) 79–88. [29] B. M. F. Child, V. Venugopal, Optimal configurations of wave energy device arrays, Ocean Eng. 37 (2011) 1402–17. [30] P. McIver, Some hydrodynamic aspects of arrays of wave energy devices, Appl. Ocean Res. 19 (1994) 283–291. [31] J. P. L. McGuinness, G. Thomas, Hydrodynamic optimisation of small arrays of heaving point absorbers, J. Ocean Eng. Mar. Energy 2 (2016) 439–457. [32] V. Stratigaki, P. Troch, T. Stallard, D. Forehand, J. P. Kofoed, M. Folley, M. Benoit, A. Babarit, J. Kirkegaard, Wave basin experiments with large wave energy converter arrays to study interactions between the converters and effects on other users in the sea and the coastal area, Energies 7 (2014) 701–734. [33] D. W. Fredriksson, J. DeCew, M. Robinson Swift, I. Tsukrov, M. D. Chambers, B. Celikkol, The design and analysis of a four-cage grid mooring for open ocean aquaculture, Aquacult. Eng. 32 (2004) 77–94. [34] L. Johanning, G. H. Smith, J. Wolfram, Towards design standards for WEC moorings, in: 6th European Wave and Tidal Conference, Glasgow, UK, 2005. [35] L. Johanning, G. H. Smith, J. Wolfram, Mooring design approach for wave energy converters, in: Proc. Inst. Mech. Eng. Part M-J Eng. Marit. Environ., Vol. 220, 2006, pp. 159–74. [36] P. C. Vicente, A. F. de O. Falc˜ ao, L. M. C. Gato, P. A. P. Justino, Hydrodynamics of multiple floating point-absorber wave energy systems with inter-body and bottom slack-mooring connections, in: Proc. 28th Int. Conf. Ocean Offshore Arctic Eng., Honolulu, Hawai, USA, 2009. [37] P. C. Vicente, A. F. de O. Falc˜ ao, L. M. C. Gato, P. A. P. Justino, Dynamics of arrays of floating point-absorber wave energy converters with inter-body and bottom slack-mooring connections, Appl. Ocean Res. 31 (4) (2009) 267–281. [38] Z. Gao, T. Moan, Mooring system analysis of multiple wave energy converters in a farm configuration, in: 8th European Wave and Tidal Energy Conference, Uppsala, Sweden, 2009. [39] B. Howey, G. Iglesias, K. M. Collins, M. Hann, D. Greaves, R. Gomes, J. C. C. Henriques, Wave energy converter arrays: Motion response of inter-connected array, in: Advances in Renewable Energies Offshore - Guedes Soares (Ed.), Taylor and Francis Group, London, 2019, pp. 625–632. [40] F. X. Correia da Fonseca, R. P. F. Gomes, J. C. C. Henriques, L. M. C. Gato, A. F. O. Falc˜ ao, Model testing of an oscillating water column spar-buoy wave energy converter isolated and in array: Motions and mooring forces, Energy 112 (2016) 1207–1218. [41] WAMIT Inc., Available online at www.wamit.com, WAMIT, User’s manual, Version 6.3 (2006).

26

480

485

490

495

500

[42] A. F. de O. Falc˜ ao, P. A. P. Justino, OWC wave energy devices with air flow control, Ocean. Eng. 26 (12) (1999) 1275–1295. [43] A. J. N. A. Sarmento, A. F. de O. Falc˜ ao, Wave generation by an oscillating surface-pressure and its application in wave energy extraction, J. Fluid. Mech. 150 (1985) 467–85. [44] J. R. Morison, M. P. O’Brien, J. W. Johnson, S. A. Schaaf, The force exerted by surface waves on piles, Pet. Trans., Amer. Inst. Min. Eng. 189 (1950) 149–154. [45] C. K. Chakrabarti, Offshore Structure Modeling, World Scientific, Singapore, 1984. [46] G. Terra, W. Jan van de Berg, L. R. M. Maas, Experimental verification of Lorentz linearisation procedure of quadratic friction, Fluid Dyn. Res. 35 (2005) 175–188. [47] C. K. Chakrabarti, Handbook of Offshore Engineering, Elsevier, London, 2005. [48] S. M. Richardson, Fluid mechanics, New York: Hemisphere Pub. Corp., 1989. [49] Y. Liu, L. Bergdahl, Influence of current and seabed friction on mooring cable response: Comparison between time-domain and frequency-domain analysis, Eng. Struct. 19 (1997) 945–953. [50] J. Falnes, Ocean waves and oscillating systems: Linear interactions including wave-energy extraction, Cambridge University Press, Cambridge, 2002. [51] A. Babarit, A database of capture width ratio of wave energy converters, Ren. Energy 80 (2015) 610–628. [52] A. F. de O. Falc˜ ao, R. J. A. Rodrigues, Stochastic modelling of OWC wave power plant performance, Appl. Ocean Res. 24 (2004) 59–71. [53] T. Sarpkaya, M. Isaacson, Mechanics of Wave Forces on Offshore Structures, Van Nostrand Reinhold, New York, USA, 1981. [54] Y. Goda, Random Seas and Design of Maritime Structures, third ed., World Scientific, Singapore, 2010. [55] W. J. Pierson, L. Moskowitz, A proposed spectral form for fully developed wind seas based on the similarity theory of S. A. Kitaigorodskii, J. Geophys. Res. 69 (24) (1964) 5181–5190. [56] L. S. Borgman, Spectral analysis of ocean wave forces on piling, J. Waterw. Harb. Div. 93 (1967) 129–156. [57] R. P. F. Gomes, J. C. C. Henriques, L. M. C. Gato, A. F. O. Falc˜ ao, IPS 2-body wave energy converter: acceleration tube optimisation, J. Offshore and Polar Eng. 20 (4) (2010) 247–255.

27

Manuscript highlights

    

A hydrodynamic analysis of a moored array of spar-buoy OWCs is presented Two different mooring configurations are compared A stochastic analysis is undertaken for irregular waves Independent and inter-body moored arrays showed efficiency differences within 2-3%. The average heave RAO of the array decreases by 7.2% after including the moorings

Declaration of Interest Statement 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. The authors declare no conflict of interest. The work presented in the revised manuscript RENED-18-02101R2 has not been published previously, and it is not under consideration for publication elsewhere. We have all agreed on the author list. All authors and responsible authorities have approved the publication of this research. If accepted, the work will not be published elsewhere in the same form, in English or any other language, without the written consent of the copyright holder.

Charikleia L. G. Oikonomou (corresponding author)

09.10.2019