The modular concept of the Oscillating Wave Surge Converter

Renewable Energy 85 (2016) 484e497

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The modular concept of the Oscillating Wave Surge Converter Dripta Sarkar a, *, Kenneth Doherty b, Frederic Dias a, c a

UCD School of Mathematical Sciences, University College Dublin, Belfield, Dublin-4, Ireland Aquamarine Power Limited, Elder House, 24 Elder Street, Edinburgh EH1 3DX, UK c Centre de Mathematiques et de Leurs Applications (CMLA), Ecole Normale Superieure de Cachan, 94235 Cachan, France b

a r t i c l e i n f o

a b s t r a c t

Article history: Received 20 October 2014 Received in revised form 27 May 2015 Accepted 7 June 2015 Available online xxx

In this study, we discuss the hydrodynamics of the modular concept of a well known wave energy device - the Oscillating Wave Surge Converter. Such a concept has emerged to address some of the shortcomings in the original design of the device. A mathematical model is presented to analyze the effect of the interactions of the system. The analysis is performed with a modular system comprising of six identical modules of total combined width 24 m, reminiscent of the Oscillating Wave Surge Converter - Oyster800 developed by Aquamarine Power. Various design strategies are explored. It is shown that such a closely packed system of modules results in multiple resonances which can potentially be exploited to capture more power. It is also observed that the modules lying at the center of the system capture more energy than those lying at the edges. An optimization of power take-off system shows that at lower wave periods it is possible to capture the levels of power similar to those of an equivalent size rigid flap while at higher periods, the modular system has the potential to capture more energy due to the occurrence of multiple resonances. © 2015 Elsevier Ltd. All rights reserved.

Keywords: Wave-energy Oyster OWSC Resonance Hydrodynamics

1. Introduction The Oscillating Wave Surge Converter (OWSC) is already recognised as a robust and efficient wave energy conversion device. By nature of its operating principles, the OWSC concepts are largely nearshore based as they try to exploit the amplification in the horizontal surge motion of the water particles in shallow waters [1]. One of the best known OWSC is the Oyster device developed by Aquamarine Power. It is a wide flap which captures energy by performing pitching motion about a horizontal hinge axis located at some distance above the sea bed and ideally located in water depths of 10e15 m. The design of this wave energy converter (WEC) has evolved significantly since its inception and a lot of research is still focussed on modifications which can address some of its shortcomings. One of the disadvantages of having a single flap of such large width is the large wave loads acting on the common foundation at the bottom especially in extreme wave conditions which have been observed on the Oyster800 prototype installed at the European Marine Energy Centre test site, Orkney, Scotland. A possible mechanism to mitigate such destructive

* Corresponding author. E-mail address: [email protected] (D. Sarkar). http://dx.doi.org/10.1016/j.renene.2015.06.012 0960-1481/© 2015 Elsevier Ltd. All rights reserved.

effects is to divide the flap into smaller components. A new concept that has emerged based on the above philosophy is a modular form of the OWSC (see [2]), which is analyzed in this paper (see Fig. 1). In fact, experimental results presented in Ref. [2] show a reduction in the parasitic foundation loads such as the yaw and roll twisting moments. In addition to this, the breakdown of the structure can potentially help in its fabrication and installation. However, it is not yet understood how such a design alteration would impact the hydrodynamics and performance of the OWSC system. The studies on Venice gates were probably the first investigations on the behavior of ocean based modular systems (see e.g. Refs. [3e5]). The purpose of these barriers is to control the flooding of the Venice lagoon and they are at present under construction. The research work on the gates was mostly focussed on understanding the subharmonic resonance of the system of gates which resulted in large out-of-phase oscillations of the coupled gates. A linear theory was developed in Ref. [3] to explain the resonant phenomenon which occurred at half the frequency of the incident wave and was verified with experimental findings as well. Later the method was extended in Ref. [6] to examine inclined gates using a hybrid element method. In Ref. [7], the natural modes of a long barrier comprising of a sequence of discrete gates were determined, while in Ref. [8] the gate system was analyzed in a

D. Sarkar et al. / Renewable Energy 85 (2016) 484e497

485

2. Mathematical model The modular flap-type WEC is considered to be situated in an 0 ocean of constant water depth h , and comprises of a total of M modules. The incoming waves of amplitude A0I are considered to be 0 obliquely incident making an angle j with the negative x -axis. Each of the modules independently performs oscillatory motion 0 about a horizontal hinge which is located at a distance c above the sea bed. The fluid is considered to be inviscid and incompressible, and the flow irrotational. Therefore there exists a velocity potential 0 F which satisfies the Laplace equation in the fluid domain. The scalar potential also satisfies the linearized kinematic-dynamic free surface boundary condition

F0;t 0 t0 þ gF0;z0 ¼ 0;

z0 ¼ 0;

(1)

where g is the acceleration due to gravity, and the no-flux boundary condition on the sea bed Fig. 1. A 3D graphical illustration of a rigid flap and a modular flap-type WEC.

semi-channel open to the sea. Recently, the potential of exploiting the subharmonic resonant mechanisms in harnessing energy was explored in Ref. [9]. However, the resonant phenomenon depends strongly on the parameters e.g. the inertia of the gates, incident wave frequency, water depth. This sensitivity limits the application of the system to a real ocean environment. The key differences between the modular gate system and the modular flap system discussed in this study include their purpose, their dimensions (width of each modular gate similar to a rigid OWSC) and their application in different layouts (gates in a channel, OWSC in the open ocean). The hydrodynamics of a single wide OWSC has been studied extensively since the initial works of Refs. [10e13], and is well understood now. An abundant theoretical literature now exists on it, starting from understanding its behavior in a channel [14], in the open ocean [15], along a straight coast [16], to that in arrays [17,18] and in a wave farm [19]. The analysis is based on approximating the wide flap as a thin-rigid plate. Such a hypothesis is based on the assumption that the thickness of the flap is much smaller than its width. However, when the flap is divided into modules, each of them has a width comparable to its thickness and therefore the thin-plate approximation can no longer be applied in such circumstances. To analyze the concept of modular flaps, we approximate each of the components as cylinders. The hydrodynamics of an isolated large cylindrical bottom-hinged flap type WEC was already studied in Ref. [20], where use was made of the relative velocity Morison equation, and force coefficients were obtained from radiation and diffraction theory. In the case of the modular OWSC, an appropriate modeling of the interactions within the closely packed system needs to be undertaken. The general philosophy of the present analysis is based on the multiple body interaction theory of [21], which uses the addition theorems of Bessel functions. A similar technique had previously been adopted to model interactions among heaving truncated cylindrical systems [see e.g. 22, 23]. In this study, first we develop a mathematical model for the analysis of the system shown in Fig. 2. In x3, computations are performed for some possible power take-off strategies, along with a discussion of the general hydrodynamics and the multiple resonant characteristics of the modular system. In x4, an optimization of the power takeoff damping coefficients is performed using genetic algorithm. And lastly, in x5, the same modular system with a gap underneath is considered and analyzed by using the mathematical model of [22].

F0;z0 ¼ 0;

z0 ¼ h0 :

(2)

The individual modules of the WEC are modeled as cylinders and the kinematic boundary condition on them yields

F0;r0 ¼ qj;t 0 ðz0 þ h0  c0 ÞHðz0 þ h0  c0 Þcos xj ; j

rj0 ¼ a0j ; 0 < x0j < 2p; (3)

where H is the Heaviside step function. The non-dimensional system of variables is chosen as

.
  h0 ; x; y; z; rj ¼ x0 ; y0 ; z0 ; rj0

rffiffiffiffi g 0 t¼ t; h0

F0 F ¼ pffiffiffiffiffiffiffi ; gh0 A0I

εq ¼ q0 ; (4)

where ε ¼ A0I =h0 is the small parameter of the problem. Assuming the motion to be simple harmonic in nature, we obtain

o n qj ¼ Re Qj expiut ;

n
o  F ¼ Re f rj ; xj ; z expiut ;

(5)

pffiffiffiffiffiffiffiffiffi where u ¼ u0 h0 =g and Qj are respectively, the angular frequency and amplitude of oscillation of the j-th module, while ɸ(rj,xj,z) is the complex spatial velocity potential in the co-ordinate system of the j-th module. In order to analyze the modular WEC, we first determine the scattering matrices of an isolated module and then use them to obtain the solution for an array of such bodies. The methodology is similar to that in Ref. [22], with a system of heaving truncated cylinders. 2.1. Isolated module For an isolated module, the coordinate system is located at its center. The spatial velocity potential f is decomposed into the scattering potential ɸ(S) and radiation potential per unit velocity ɸ(R) as follows

  f ¼ fðSÞ þ VfðRÞ ¼ fðIÞ þ fðDÞ þ VfðRÞ ;

(6)

where, ɸ(I) is the incident wave potential, V ¼ iuQ0 is the complex angular velocity of the isolated module, and ɸ(D) is the diffracted wave potential. In our description, the subscript index j ¼ 0 will be used to indicate the behavior of the module in isolation (i.e. no other module is present). The general form of the spatial potential for the radiation (R) and the scattering (S) problem can be written as

486

D. Sarkar et al. / Renewable Energy 85 (2016) 484e497

Fig. 2. Geometry of the physical system a) top view and b) cross-section of the j-th module.

(

fðRÞ

)

ðSÞ

f

9 8 ð1Þ < aðRÞ = H ðk r Þ n 0 0ln l eilx0 Zn ðzÞ4 ð1Þ ¼ ; Hl ðkn a0 Þ : aðDÞ n¼0 l¼∞ 0ln ( )3 J ðkn r0 Þ 0 5; þ l ðIÞ Jl ðkn a0 Þ b0ln ∞ X

∞ X

2

ðDÞ

M Al a0l þ M Bl b0l ¼ 0; (7)

which gives the solution to the unknown coefficients

1  ðDÞ ðIÞ a0l ¼  M Al M Bl b0l ¼ Bl b0l

ð1Þ

For the radiation problem, the boundary condition on the module for the radiation potential per unit velocity is ðRÞ

AI ðIÞ ðiÞlþ1 Jl ðka0 Þeilj d0n b0ln ¼  uZ0 ð0Þ

(15)

ðIÞ

where Hl is the Hankel function of first kind and order l, Jl is the ðR;DÞ Bessel function of the first kind and order l, a0ln are the unknown scattering coefficients,

(8)

f;r0 ¼ ðz þ h  cÞHðz þ h  cÞcos x0 ;

r0 ¼ a0 ;

(16)

which on application of the orthogonality of eilx0 and Zn(z) gives

are the coefficients of the ambient incident wave potential, and

pffiffiffi 2cosh kn ðz þ hÞ Zn ðzÞ ¼
1=2 ; h þ u2 sinh2 kn h

(14)

n ¼ 0; /; ∞;

(9)

0

1

ð1Þ0 H ðkn a0 Þ ðRÞ A a0ln @kn lð1Þ Hl ðkn a0 Þ

¼

1 fn d ; 2 1jlj

(17)

where are the normalized vertical eigenmodes satisfying the orthogonality relation

Z0 Zn ðzÞZm ðzÞdz ¼ dnm ;

(10)

h

fn ¼

pffiffiffi 2½kn ðh  cÞsinhðkn hÞ þ coshðkn cÞ  coshðkn hÞ : 1=2
k2n h þ u2 sinh2 ðkn hÞ

Expressing (17) in a matrix form ðRÞ

where dnm is the Kronecker delta. In (9), k0 ¼ k and kn ¼ ikn are the solutions of the dispersion relations

u2 ¼ k tanh kh;

u2 ¼ kn tan kn h;

n ¼ 1; 2; /;

(11)

respectively. For the scattering problem, the spatial potential needs to satisfy ðSÞ

f;r0 ¼ 0;

r0 ¼ a0 :

0

1

M Al a0l ¼ a+ 0l ;

(19)

gives the solution

1  ðRÞ a0l ¼ M Al a+ 0l

(20)

¼ Dl The matrices Bl and Dl (see Eqs. (15) and (20)) will now be used to solve the modular system of the OWSC.

(12)

Using the orthogonality of eilx0 and Zn(z) over the range 0x02p and hz0 respectively, (12) yields ð1Þ0 H ðkn a0 Þ ðDÞ A a0ln @kn lð1Þ Hl ðkn a0 Þ

(18)

  0 J ðkn a0 Þ ðIÞ ¼ 0: þ b0ln kn l Jl ðkn a0 Þ

This Eq. (13) can then be written in a matrix form as

(13)

2.2. Array of modules The WEC analyzed in this study comprises of a finite number of modules placed adjacent to each other. Since it is a multi-body system, there are as many radiation problems as the number of modules. The spatial velocity potential in this case can be decomposed into

D. Sarkar et al. / Renewable Energy 85 (2016) 484e497

ðSÞ

f¼f

ðRÞ

þf



¼ f

ðIÞ

þf

ðDÞ



þ

M X

Vm f

ðmÞ

(21)

m¼1

:

where ∞
 iA cosh kðz þ hÞ ikxcj cos jþikycj sin j X e fI ¼  I eilðxj þjÞ Jl krj ðiÞl cosh kh u l¼∞

(22) is the incident wave potential in the co-ordinate system of the j-th module. In (22), xcj and ycj are the horizontal coordinates of the center of the j-th module with respect to the global coordinate system. In (21), ɸD is the diffracted wave potential, ɸ(m) is the radiation potential per unit velocity induced by the motion of the mth module when all the other modules are held fixed and Vm ¼ iuQm is the complex angular velocity of the moving flap. To solve the multi-body interaction problem, we use the approach introduced by Ref. [21]. The waves scattered by the ith body can be transformed as an incident wave on the jth body using the addition theorem for the Bessel function

Hp ðkn ri Þeipðxi εij Þ ¼ ð1Þ

∞ X


 ð1Þ
Hpþl kn Lij Jl kn rj eilðpxj þεij Þ ;

(23)

l¼∞ ðm;DÞ

and therefore the scattering coefficients (ailn ) can be expressed ðm;DÞ as incident coefficients (bjln ) on the jth body as ðm;DÞ bjln

¼

M X

∞ X

i¼1 isj

p¼∞

ðm;DÞ ij Tpln aipn ;

(24)

 ð1Þ
HðplÞ kn Lij
 ¼ Jl kn aj eiðplÞεij : ð1Þ Hp ðkn ai Þ

∞ X

ðm;DÞ

where bjln

ilxj

2

∞ X

ð1Þ


Hl

(26)

are the incident coefficients on the jth module due to ðIÞ

the scattering by the other bodies and bjnl are the coefficients of the ambient incident wave potential in the coordinate system of the jth module expressed as c AI ðIÞ eikxj bjln ¼  uZ0 ð0Þ

cos jþikycj sin j


 ðiÞlþ1 Jl kaj eilj d0n

(27)

Note, the superscripts (m,D) are used to denote the coefficients for either the (m)-th radiation or (D) diffraction problem. Let a(m,D) ðm;DÞ

be the vector of the scattering coefficients ajln

, where j ¼ 1,2,…M,

b be the vector of coefficients of the ambient incident wave (bðIÞ ), jln and b(D) be the incident coefficients on the jth body due to the (I)

scattering by the other modules. We denote by Barray and Darray the m array form of the matrix Bl and Dl respectively. Therefore, for the scattering problem, the incident and the scattering coefficients can be written as

  aðDÞ ¼ Barray bðIÞ þ bðDÞ :

(28)

Making use of the transformation bðDÞ ¼ T aðDÞ , the solution to the unknown coefficients a(D) is obtained as

aðDÞ ¼ ðI  Barray T Þ

1 array ðIÞ

B

b ;

(29)

where T is the coordinate transfer matrix and I is the identity matrix. For the radiation problems, the relations are

where

ij Tpln

 8 ðmÞ 9 kn rj < ajln = e Zn ðzÞ4 ð1Þ
¼  ðDÞ ; Hl kn aj : ajln ; n¼0 l¼∞ 3
( ðmÞ ) bjln Jl kn rj 5;  þ
ðDÞ ðIÞ Jl kn aj bjln þ bjln

8 9 < fðmÞ = j ðSÞ fj

487

(25)

Note, the terms in (23) are described in Fig. 3, with εij being the angle made by the line joining the centers of the i-th and j-th module with the positive x0i coordinate axis. The general form of the spatial potential for the mth radiation and the scattering (S) problem can be written in the coordinate system of the jth module as

aðmÞ ¼ Barray bðmÞ þ Dm

:

ðmÞ

gives

array

Using b

¼T

aðmÞ

aðmÞ ¼ ðI  Barray T Þ

1

array

Dm

(30)

:

(31)

To obtain the solutions, the infinite series in the vertical eigenmodes is truncated at a finite number N, while for the harmonics 2Lþ1 terms are considered (i.e. l from L to L). For the range of wave periods considered, it was found that convergence with a relative error of O(103) was obtained with the first 6 vertical eigenmodes and 17 harmonics. The solutions to the unknown coefficients a(m,D) (m ¼ 1,2,…,M indicate the radiation modes) are then utilized to obtain the hydrodynamic coefficients of the problem, which are later used to solve for the body equation of motion of the modules.

2.3. Hydrodynamic parameters Suppose for the j-th module, Ij ¼ Ij0 =ðrh05 Þ is the second moment of inertia and Cj ¼ Cj0 =ðrgh04 Þ is the coefficient of the flap restoring

Fig. 3. Top view of two modules and the coordinate systems. The terminologies shown here are utilized to change from the i-th coordinate system to the j-th one.

buoyancy torque. Note, r is the density of water. The magnitudes Ij and Cj are chosen such that their cumulative magnitudes are the same as that of a rigid flap, with the value per unit-width of every module being constant. Then non-dimensional equation of motion of the j-th module can be expressed as shown in Ref. [18].

488

h

D. Sarkar et al. / Renewable Energy 85 (2016) 484e497

M h i i  X
 pto u2 mij þ iunij Qi  u Ij þ mjj þ Cj  iu njj þ nj Qj  i¼1 isj 2

   2 1 Q ; Picaptured ¼ u2 npto i  i 2

(38)

and the cumulative power extracted by all the modules together then becomes

¼ FjS ; (32) where

  M 1 X pto  2 ¼ u2 ni Qi  : 2 i¼1

captured

FjS ¼ iuaj pf0

iAI h
 ðDÞ J M j0  kaj ðiÞ1 eij uZ0 ð0Þ 1 i


 þ J1 kaj ðiÞ1 eij ;

Ptotal (33)

In order to quantify the performance of each module, we use the capture factor CiF term defined as the ratio of the power captured by a particular module i, to that incident across its diameter:

is the excitation torque, while

( mij ¼ paj Re

∞ X

captured

) ðiÞ

fn M jn

(39)

(34)

CiF ¼

Pi

2ai P incident

;

(40)

n¼0

where Pincident is the incident wave power per unit wave front. Also, F to understand the dynamics of the system as a whole, a term Ctotal is introduced

is the added inertia and

( nij ¼ upaj Im

∞ X

) ðiÞ

fn M jn

(35)

is the radiation damping, where

2

 ð1Þ
H k L
 iðpþ1Þε ði;DÞ ði;DÞ ði;DÞ ðpþ1Þ n qj qj 4 M jn ¼ ajð1Þn þ aqpn  J1 kn aj e ð1Þ
a H k p¼∞ n q p q¼1 qsj 3  ð1Þ
M ∞ H L k X X
 n qj ðp1Þ ði;DÞ ði;DÞ iðp1Þεqj 5 þ aj1n þ aqpn  J1 kn aj e ð1Þ
H a k p¼∞ n q p q¼1 qsj M X

∞ X

pffiffiffiffiffiffiffiffiffi npto ¼ npto0 =ðrh05 g=h0 Þ j j

(36)

In (32), is the power take-off (PTO) damping coefficient of the j-th module. In the literature, the PTO damping of an oscillating device in an array is usually set to the optimal damping of the same device in isolation (see e.g. Refs. [18,19,23e25]). In monochromatic waves, the optimal PTO damping employed to extract maximum power at a particular wave period in the case of an isolated module is given by

npto1 j

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  i2  uh u open   u2 t Cj  Ij þ mj open 2 ¼ þ n ; j u2 open

(37)

open

where mj and nj are respectively the added moment of inertia and radiation damping of the j-th module isolated in the open ocean. We will refer to this PTO strategy as PTO1. However, for closely packed systems, like the one considered here, such an approximation can be non-optimal as the mutual interaction terms contribute significantly. Another possible option is to consider the PTO damping per unit width of the modules to be same as that of a rigid flap (extracting maximum power) of width equivalent to the combined width of all the modules. We will denote the magnitude of the applied PTO damping on the j-th module in this case as

captured

F Ctotal ¼

n¼0

npto2 j

and call it the PTO2 strategy. Since one

of the main objectives of this study is to compare the performance of the modular system with that of a rigid flap, most of the com. putations will be performed using npto2 j According to the theory of damped oscillating systems (see Ref. [26]), the average extracted power by the i-th module at a particular wave period is

Ptotal : P incident 2 M i¼1 ai P

(41)

The latter is the ratio of the net power captured by all the modules to the wave power incident across a width equivalent to the sum of the diameters of all the modules. 3. Results The computations are performed here for a single rigid flap of 0 width w ¼24 m divided into six modules of equal dimensions. So the radius of each of the cylindrical components is aj0 ¼a0 ¼2 m. For practical reasons, a separation of 0.1 m between the edges of the consecutive modules is introduced. However the net moment of inertia and buoyancy restoring torque of all the modules combined are kept the same as that of the rigid flap of equivalent total width. The modular OWSC is considered to be located at a water depth 0 h ¼ 13 m, with the normally incident waves having an amplitude A0I ¼ 0.4 m. 3.1. Excitation torque The excitation torque of the modules is plotted versus wave period in Fig. 4. As one moves towards the central modules (module 3 and 4) of the system, the excitation torque increases. In fact, the excitation torque is maximum for module 3, while lowest for module 1. The behavior is similar to that of a thin rigid flap: where the difference in pressure across its two sides, which produces the excitation torque, is maximum in the central region, while at the edge of the thin-rigid flap, the difference in the velocity potential tends to zero resulting in low excitation torque at such locations. The variation of the excitation torque is also qualitatively similar to that of a rigid OWSC of equivalent width (see Ref. [15]). 3.2. Radiation parameters Fig. 5 shows the variation of the added inertia and of the radiation damping of the third module. The contributions of the mutual interaction terms on a particular flap are quite significant compared to that due to its own motion. Such a behavior is expected given the closely packed nature of the system. Therefore the mutual interaction terms play a pivotal role in determining the performance of the devices, as shown below.

D. Sarkar et al. / Renewable Energy 85 (2016) 484e497

Fig. 4. Variation of excitation torque versus incident wave period for a six-module flap system with A0I ¼ 0:4 m and normal wave incidence. Module 1 is located at the edge while module 3 is one of the central modules.

489

Fig. 6. Variation of the amplitude of rotation versus wave period of the respective modules for a six module flap-type WEC using npto2.

3.3. Amplitude of rotation Fig. 6 plots the variation of the amplitude of rotation of the three modules versus the incident wave period with npto2 as the PTO damping. Again the nature of the variation is similar to that observed for a rigid flap, increasing monotonically as one moves towards higher wave periods. In the operating range of the device (5e15 s), there is an increase in the amplitude of rotation of the modules as one moves from the edge towards the center. 3.4. Capture factor Fig. 7 plots the variation of the capture factor CF of the individual modules versus the incident wave period using the PTO2 strategy. Again in the operating range of the device, the trend is similar to that of the excitation torque with the capture factor of the modules increasing from the edge towards the center. The behavior suggests that in the case of modular flaps of large widths, it may be wise to use the modules at the edge as dummy/stationary structures as they capture the least amount of power. 3.5. PTO strategy One of the challenges of modeling this particular converter is to determine an optimal PTO strategy. The device is a strongly

Fig. 7. Behavior of the capture factor CF of the individual modules versus wave period.

interacting multi-body system, yet as a whole it is treated as a single entity. At the microlevel, the individual modules have their own natural frequency of resonance, and at the macro level at the macro level the behavior of the device is similar to a rigid flap (with the PTO2 damping strategy). In the computations so far, we have applied

Fig. 5. Variation of (a) added inertia (m0ij ) and (b) radiation damping (n0ij ) versus wave period of the third module of a six module flap-type WEC. The terms (m0ij ) and (n0ij ) indicate the effect of the mutual interaction on the j-th module when the i-th module is oscillating and all the others held fixed.

490

D. Sarkar et al. / Renewable Energy 85 (2016) 484e497

the PTO2 strategy. Now, we will consider the other extreme of the possible PTO strategy (i.e. PTO1) and will also assess the behavior with some other values of the PTO damping which lie in between the two extremes, as shown in Fig. 8. It can be seen that in the operating range of the device, PTO1 has the lowest magnitude of damping, while PTO2 has the maximum. Fig. 9 plots the amplitude of rotation of the three modules for the various PTO strategies versus the incident wave period. There is a general increase in the amplitude of rotation of all the modules as the magnitude of the PTO damping is reduced from npto2. In the case of PTO1, large peaks can be observed in the amplitude of rotation of the modules located at the edges in particular. We now look more closely into the variation of the amplitude of rotation of the modules using PTO1 (see Fig. 10) and we note the occurrence of another peak at a higher period. To look for a possible explanation, let us consider the unforced, undamped case of the system (i.e. njj ¼ 0, npto ¼ 0, FjS ¼ 0 in Eq. (32) j for j ¼ 1,2,..M). The objective is to identify the natural modes of the system which are relevant to the problem investigated. Since normal wave incidence is considered in this study, the modular system exhibit a symmetrical behavior about the centerline y ¼ ðyc3 þ yc4 Þ=2 which is halfway between the nearest edges of the third and the fourth module, and so only the even natural modes of the modular system are analyzed. This implies that Q1¼Q6, Q2¼Q5 and Q3¼Q4, and the number of independent equations of motion the system reduces (from 6) to 3. It is well known that a single isolated oscillating system has a pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi unique natural frequency given by u ¼ C=ðI þ mÞ, where C is the coefficient of the buoyancy restoring torque, I is the moment of inertia and m is the coefficient of added inertia of the rotating body. However, for non-trivial solutions to the system of equations for a 6-module flap system, we require

  u2 ðI þ m11 þ m16 Þ þ C   u2 ðm12 þ m62 Þ   u2 ðm13 þ m63 Þ

u2 ðm12 þ m15 Þ u ðI þ m22 þ m52 Þ þ C u2 ðm23 þ m53 Þ 2

expected to resonate and in fact large spikes in the amplitude of rotation (see Fig. 11) are observed. A similar resonant behavior at multiple frequencies has also been reported in the case of three inline OWSCs (see Ref. [27]). Note that the amplitude of rotation plotted in Fig. 11(b) is for the case with no applied PTO damping. Interestingly, there is an increase in the bandwidth of the resonances as one moves towards higher wave periods. Nevertheless, in reality viscous (and PTO) damping will reduce the extent of the peaks depending on their magnitude. The modal profile of the modular system corresponding to the natural frequencies are shown in Fig. 12. In the operating range of the device, the magnitude of npto1 is the lowest (see Fig. 8) and resonant peaks are manifested in the variation of the amplitude of rotation (see Fig. 9). On the other extreme, the resonant peaks are invisible when the PTO2 strategy is employed where the magnitude of the PTO damping is the largest. F Fig. 13 plots the total capture factor Ctotal of the system versus the incident wave period. The efficiency levels for all the PTO strategies are similar except for PTO1, where a huge reduction in the net power captured is observed at intermediate wave periods, and it can be attributed to the low magnitudes of the PTO damping coefficients (see Fig. 8). 3.6. Larger number of modules Some computations are performed with a larger number of modules to understand the effect of the width of the system. The geometry of the modules and the separation between them are 0 0 kept the same i.e. a ¼2 m and s ¼0.1 m along with PTO2 (npto2) strategy. Fig. 14(a) plots the excitation torque of the respective modules of a system comprising of twenty modules. Module 1 or rather the module located at the edge, has the lowest magnitude of

  u2 ðm13 þ m14 Þ  2 ¼0 u ðm32 þ m42 Þ  2 u ðI þ m33 þ m43 Þ þ C 

Let us denote the matrix in (42) as E. The variation of Det[E] is plotted in Fig. 11(a) with respect to the incident wave period. The plot shows three distinct positive values of the wave period for which relation (42) is satisfied. At these periods, the system is

(42)

torque across the entire range of wave periods. Although some variations are observed in the excitation torque of the other modules at low periods, beyond a period of 7 s there is a clear increasing trend in torque as one moves from the edge towards the center. A similar behavior is also observed in the variation of the capture factor CF of the individual modules at higher periods. However, there appears to be a saturation in both the parameters jF'j and CF as one moves towards the central module for longer waves. F Fig. 15 plots the total capture factor Ctotal for systems with a larger number of modules. There is a general decrease in the efficiency with an increase of the number of modules. For longer F waves, there is a decrease in Ctotal as the number of modules is increased. This is because at higher periods, there is a significant reduction in the power captured by the modules at the edges, which deteriorates the performance of the system as a whole. Therefore, for practical applications, the modules at the edges may be considered as dummy structures. 4. Optimal PTO damping

Fig. 8. Variation of the different applied PTO strategies versus incident wave period.

One of the major difficulties in modeling a multiple oscillating body system is to obtain the optimal PTO damping of the modules which maximizes the power captured by the system as a whole. In order to obtain the optimal PTO damping coefficients, we use a

D. Sarkar et al. / Renewable Energy 85 (2016) 484e497

491

Fig. 9. Behavior of the amplitude of rotation of the modules of a six module flap-type WEC for the various applied PTO strategies.

Fig. 10. Variation of the amplitude of rotation of the modules of a six module flap-type WEC using npto1 as the applied PTO damping coefficient (PTO1 strategy).

genetic algorithm (GA) (see e.g. Ref. [28]), which is general enough to be used for a large number of modules. It is a heuristic approach based on the theory of evolution of species. The function one wants to optimize is known as the fitness function which in this particular problem is the total power captured by all the modules together. An individual is a point (vector) where the fitness function is evaluated and in our case it comprises of the PTO damping coefficients. And the magnitude of the fitness function of a particular individual is known as its score. An array of such individuals is known as population. Individuals in the current population having higher fitness values, known as parents, are selected and then used to produce children for the next generation. Two major operations are performed to arrive at the new generation - crossover and mutation. The crossover operation is analogous to reproduction and two parents are combined to produce new children, while the mutation operation involves the alteration in the set of parameters which define a possible solution and are essential in maintaining

Fig. 11. (a) Plot of Det[E] (see 42) and (b) amplitude of rotation of the modules with radiation damping but no applied PTO damping (npto ¼ 0; j ¼ 1,2,…,M) versus incident j wave period for a six module flap-type WEC.

heterogeneity. We will consider two cases, one in which the PTO damping coefficient is taken to be the same for all the modules, and the other where they can vary for each module. However, an upper and lower bound is set on the PTO damping coefficients to discard unrealistic values. Fig. 16 plots the optimized total capture factor for such a

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D. Sarkar et al. / Renewable Energy 85 (2016) 484e497

Fig. 12. Modal profiles of the modular system corresponding to the natural frequencies of Fig. 11. The values around the modules are that of the angular rotation Q for the respective modules.

can now be treated as truncated cylinders. The region underneath each truncated cylinder is identified as interior region and that outside as exterior region specific to that cylinder (see Fig. 18). Arrays of truncated cylinders have been modeled in quite a few studies using the interaction theory of [21]. The diffraction problem was solved in Ref. [29] for a group of vertical truncated cylinders. The same approach was then adopted in Ref. [30] to solve for the radiation problem of a group of cylinders oscillating as a single unit. In recent years, a similar methodology was adopted in Ref. [22] to analyze the interaction in an array of truncated cylinders oscillating independently, however, without the application of any external forcing. The same method was used in Ref. [23] to obtain optimal configurations of heaving wave energy converters approximated as truncated cylinders. In our analysis, we follow the approach of [22]. The decomposition of the problem into scattering and radiation problems is the same as in x2 and will not be described here again to avoid redundancy. The general form of the spatial velocity potential in the exterior region of the j-th cylinder is the same as that of the previous problem (see (26)) while, in the interior region it can be expressed as.

8 9
;

∞ X

¼

l¼∞

2 1 rj eilxj 4 2 aj

þ

!jlj 8 ðmÞ 9
: gðDÞ ; jl0  

∞ Il X s¼1

Il

8 93 < gðmÞ = sp jls 5  cos ðz þ hÞ : gðDÞ ; c aj sp rj sp c

jls

c

! 9 8 r3 > > > = < 1 ðz þ hÞr 2  j cos xj djm > j 4  2c : > > > > ; : 0 (43)

F Fig. 13. Variation of total capture factor Ctotal versus wave period of a six module flaptype WEC.

system with the upper and lower limit on nPTO set at 0.002 and 0.05 respectively. Note that nPTO is a non-dimensional quantity. The best performance is achieved in the case where the PTO damping coefficients vary for all the modules (red dashed line), with multiple peaks in its variation. While in the case of fixed damping (black solid line), where the damping is same for every module, a single peak is observed at a higher period. Note that this peak exactly coincides with the third resonant period of the oscillating system (see Fig. 11(b)). A comparison with an equivalent rigid OWSC shows that even at low wave periods, away from the resonant frequencies, the modular system performs equally well. While at higher periods, multiple resonances occur for a system of modules (see again Fig. 11(b)) and this phenomenon enables a properly tuned modular system to capture more power. 5. Modules with gap In this section, we perform the analysis of the modular flaps with the bottom foundation beneath the oscillating flaps absent. The whole modular system is then assumed to be supported centrally or at the edges through structural components which are of negligible dimensions compared to the radius of the cylinders (see Fig. 17). For convenience of mathematical modeling, the modules

where Il is the modified Bessel function of first kind and order l. These solutions have been extensively used in modeling truncated cylinders. The last term inside curly brackets in (43) is the inhomogeneous component of the velocity potential (when the module performs pitching motion in the absence of incident waves), while the other part (before the minus sign) in (43) is the homogeneous potential which satisfies the boundary condition on the bottom surface of the truncated cylinder when it is at rest (see e.g. Refs. [31,22]). The matching conditions for the continuity of potential and velocity underneath the truncated module respectively are ðm;SÞ

fj

ðm;SÞ

~ ¼f j

ðm;SÞ

vfj

vrj

;

rj ¼ aj ;

h  z  h þ c;

(44)

ðm;SÞ

¼

~ vf j

vrj

;

rj ¼ aj ;

h  z  h þ c;

(45)

while the kinematic boundary condition on the sides of the modules require ðSÞ

vfj

vrj

¼ 0;

rj ¼ aj ;

h þ c  z  0

(46)

for the scattering problem and ðmÞ

vfj

vrj

¼ ðz þ h  cÞcos xj ;

rj ¼ aj ;

h þ c  z  0;

(47)

D. Sarkar et al. / Renewable Energy 85 (2016) 484e497

493

0

Fig. 14. Variation of (a) excitation torque jF'j and (b) capture factor CF versus wave period of a twenty module flap-type WEC. The radius of each of the modules is a ¼2 m with a 0 distance of separation s ¼0.1 m between their edges.

array of modules (see Appendix B). With these solutions the hydrodynamic parameters of the individual modules are evaluated. These hydrodynamic coefficients can then be employed to solve for the body equation of motion of the modules (see (32)). The excitation torque is expressed as

" FjS ¼ iuaj p

∞ X

ðDÞ

fn M jn 

n¼0

h
 iAI f0 J1 kaj ðiÞ1 eij uZ0 ð0Þ i
 þ J1 kaj ðiÞ1 eij þ N

# ðDÞ jn

(48) while the radiation parameters are given by

( mij ¼ paj Re

∞ X

ðiÞ

fn M jn þ N

ðiÞ jn



n¼0 F Fig. 15. Comparison of the variation of the capture factor Ctotal versus incident wave period with different number of modules.

(49)

and

( for the radiation problem. Following the methodology of [22], the solution to the unknown coefficients of the scattered and radiated potentials due to an isolated truncated module is first obtained (see Appendix A), which are then used to obtain the solutions for an

! ) aj 3 c aj 5  d 48c ij 8

nij ¼ upaj Im

∞ X n¼0

where

ðiÞ fn M jn

þN

ðiÞ jn



! ) aj 3 c aj 5 d  48c ij 8

(50)

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D. Sarkar et al. / Renewable Energy 85 (2016) 484e497

N

F Fig. 16. Comparison of the total capture factor Ctotal obtained using PTO coefficients determined using a genetic algorithm and that for the other cases of a six module WEC. The black line denotes the case where the PTO damping of all the modules are considered to be same while the red dashed line is for the case when the PTO damping coefficients of the modules are allowed to vary, enabling the system to capture larger power. The green line shows the capture factor of a rigid OWSC and is obtained using the mathematical model of [15]. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Fig. 17. A 3D graphical illustration of a modular flap-type WEC with a gap in the bottom.

ði;DÞ jn

∞   2 X   ði;DÞ ði;DÞ aj ði;DÞ ði;DÞ g þ ¼ gjð1Þ0 þ gj10 þ gj1s 8 s¼1 jð1Þs   a sp I2 jc aj c  ð1Þs :  sp a sp I1 jc

(51)

These expressions (48),(49) and (50) are then utilized to solve for the body equation of motion of the modules, and thereafter the performance indicators are evaluated. Fig. 19 plots the variation of the excitation torque of the individual modules versus incident wave period for the system with gap underneath and comparison is also made with the modular system with rigid foundation. At low wave periods, the excitation torques are similar for both systems, however, for longer waves, there is a clear reduction in the excitation torque on all the modules for the case. In Fig. 20, a comparison of the total capture F factor Ctotal is made for two different applied power take-off mechanisms e PTO1 and PTO2. In general, in both cases, the modular system with rigid foundation captures more power except in short waves, where the behaviors of both systems are similar. In the case of PTO2 (see Fig. 20(b)), there is almost a steady decrease in the net capture factor with increasing period. This has a lot of practical implications, because in the PTO2 mechanism, the net PTO damping is the same as that of the rigid flap and consequently the qualitative behaviors of the performance of the two systems (modular and rigid OWSC with PTO2 strategy) are similar as well (see Fig. 16). From this, it can be reasonably inferred that there would be a reduction in the efficiency of a rigid OWSC as well, with the absence of the bottom foundation. Such a behavior can also be expected on observation of the variation of the excitation torque (Fig. 19). Most of the theoretical work on rigid OWSC includes a bottom foundation which makes it easier to model the system. And it is well known that the dynamics of a rigid OWSC is primarily governed by the diffraction phenomenon. Therefore a reduction in the excitation torque is associated with a decrease in the performance as well. The presence of a gap beneath the structure provides a free passage to the waves. Although the bottom foundation is static and doesn't contribute directly to wave power extraction, its presence provides a blockage to the waves which consequently produces a larger difference in potential across the two sides of the device resulting in a greater excitation torque than in the case without blockage. 6. Conclusion

Fig. 18. Geometry of the j-th module of system with a gap underneath a) cross-section and b) top view.

The hydrodynamics of the modular concept of the OWSC is analyzed in this work. The idea is to decompose a rigid flap into a series of modules. The possible advantage of the breakdown is that it enables the distribution of wave loads acting on the common foundation, reduction in the twisting wave forces and flexibility in the operation of the system. It is shown that the modular-flap WEC is equally good in capturing power as well and can be a very effective and promising concept. The behavior of the system is found to be strongly dependant on the applied power take-off mechanism. When the magnitude of the applied PTO-damping is low, strong resonating effects are observed which are attributed to the closely packed nature of the system resulting in strong mutual interactions. On the other hand, the variation of total capture F factor Ctotal of a six module flap system of total width 24 m is found to be similar to that of a rigid flap when the applied PTO damping per unit width is the same for both the cases (see Fig. 16). In such a case, the modules at the center have higher amplitudes of oscillations and capture more power than those at the edges. So for

D. Sarkar et al. / Renewable Energy 85 (2016) 484e497

495

0

Fig. 19. Variation of excitation torque jF'j versus wave period module flap-type WEC with gap underneath it. The radius of each of the modules is a ¼2 m with a distance of 0 0 separation s ¼0.1 m between their edges and the gap underneath is c ¼4 m.

F Fig. 20. Comparison of the net capture factor Ctotal of a modular flap with rigid foundation and that with bottom gap for two different applied power take-off damping mechanisms (a) npto1 and (b) npto2.

very large systems, one may envisage having the modules at the edges as dummy or stationary structures, however, they would alter the hydrodynamics of the system. In order to maximize the net power captured by the modular system, an optimization of the power take-off damping coefficients is performed using a genetic algorithm. At lower periods, away from the resonant frequencies, similar levels of efficiency as that of a rigid flap are obtained using the optimized coefficients, while at higher periods, the occurrence of multiple resonances enables the modular system to capture more power than that of a rigid flap. An analysis is also performed for a modular system with gap underneath. It is shown that such a system captures less power than a system with rigid foundation. The mathematical model used in this paper is based on linear potential flow theory and gives a valuable understanding and insight into the fundamental dynamics of the modular OWSC. However, in the real ocean, viscous, nonlinear and other effects would modify the behavior of the system, especially at high wave periods and at the locations of resonance where large amplitudes of oscillations of the modules occur.

Acknowledgments This work is supported by Science Foundation Ireland (SFI) under the research project Highend computational modelling for wave energy systems (Grant number SFI/10/IN.1/12996) in collaboration with Marine Renewable Energy Ireland (MaREI), the SFI Centre for Marine Renewable Energy Research e (12/RC/2302). Appendix A. Isolated module with bottom gap Consider the matching condition for the continuity of potential across the interface between the interior and exterior regions (44) and the boundary condition on the edge of the module for the scattering problem (46). Applying the orthogonality of cos(sp/ c(zþh)) in the domain h
g0ls ¼

 ∞   2 X ðDÞ ðIÞ Csn a0ln þ b0ln c n¼0

(A.1)

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D. Sarkar et al. / Renewable Energy 85 (2016) 484e497

For the radiation problem, the incident wave field is absent and the matching conditions at the interface (44)e(45) and boundary condition (47) yield

where

8 > > > > > <

Csn

pffiffiffi 2



2

1=2

2

sinhðkn cÞ kn

h þ u sinh kn h ¼ pffiffiffi > > kn ð1Þs sinhðkn cÞ 2 > > > 1=2 : ðsp=cÞ2 þ k2n h þ u2 sinh2 kn h

s ¼ 0;

ðRÞ

s  0:

ðRÞ

(A.2) Again using the orthogonality of Zn(z) and e on the matching condition for velocity at the interface (45), yields

0 1 0  ð1Þ0 ðkn a0 Þ Jl ðkn a0 Þ ðDÞ @Hl A þ bðIÞ k a0 ln k n n 0 ln J ðk a Þ ð1Þ l n 0 Hl ðkn a0 Þ   !   X ∞ Il0 a0csp sp jlj ðDÞ ðDÞ Csn C0n þ gjls ¼ gjl0   c 2a0 Il a0csp s¼1

ðDÞ

ðIÞ

ðRÞ

(A.15)

where

! 8 a0 c a0 3 > > >  d1jlj  < 6 8c g0l ¼ > > > :  a0 c ð1Þs d1jlj p2 s2

if

s ¼ 0; (A.16)

if

s > 0;

and

(A.3)

The two above equations (A.1) and (A.2) can then be written in a matrix form as

g0l ¼ M 1 a0l þ M 1 b0l

(A.14)

M 2l a0l ¼ M 3l g0l þ a0l

ilx0

ðDÞ

ðRÞ

M 1 a0l ¼ g0l þ g0l

0 a0l

B 1 ¼@ 4c

hþc Z "

# 3a0 2 Zn ðzÞdz ðz þ hÞ  4 2

h

1 þ 2

(A.4)

Z0

1

(A.17)

C ðz þ h  cÞZn ðzÞdzAd1jlj :

hþc ðDÞ M 2l a0l

þ

ðIÞ M 3l b0l

¼

ðDÞ M 4l g0l

(A.5) Again solving the simultaneous equations, we obtain the unknown vectors

where

 M 1 ðs; nÞ ¼

2 Csn c



ðRÞ

(A.6)

8 ð1Þ0 > H ðkn a0 Þ > < l kn 2 M l ðn1 ; n2 Þ ¼ H ð1Þ ðkn a0 Þ l > > : 0 8 0 > < Jl ðkn a0 Þ kn M 3l ðn1 ; n2 Þ ¼ Jl ðkn a0 Þ > : 0

g0l ¼ ℰ l ;

(A.18)

ðRÞ

a0l ¼ D l if

n1 ¼ n2 ;

if

n1 sn2 :

if

n1 ¼ n2 ;

if

n1 sn2 :

(A.7)

(A.8)

(A.19)

where

   1   1  M 1 M 2l ℰ l ¼ I  M 1 M 2l M 3l a0l  g0l ;

(A.20)

1    M 3l ℰ l þ a0l D l ¼ M 2l

(A.21)

and

jlj C 2a0 0n   M 4l ðn; sÞ ¼ I 0 a0 sp   > l sp > > : a csp Csn 0 c Il c 8 > > > <

if

Appendix B. Array of modules with bottom gap

s ¼ 0; (A.9)

if

s > 0:

Using the solution to the isolated case, the solution to the system of multiple modules can be expressed in a matrix form as



Solving the simultaneous equations we obtain the expressions ðDÞ ðDÞ for the unknown vectors g0l and a0l as follows, ðDÞ

ðIÞ

g0l ¼ C l b0l ; ðDÞ

(A.10)

ðIÞ

a0l ¼ B l b0l

(A.11)

where

   1   1  M 1  M 1 M 2l C l ¼ I  M 1 M 2l M 3l M 4l ; 

B l ¼ M 2l

1 

M 3l C l  M 4l



(A.12) (A.13)

aðDÞ gðDÞ

 9 8 < B array bðDÞ þ bðIÞ =   ¼ : C array bðDÞ þ bðIÞ ;

(B.1)

for the diffraction problem and



aðmÞ gðmÞ

¼

B array bðmÞ þ D array m array C array bðmÞ þ ℰ m

(B.2)

for the radiation problem. In (B.1) and (B.2), the matrices B array , C array , D array and ℰ array are defined as follows: is a square matrix of size B array Mð2L þ 1ÞðN þ 1Þ  Mð2L þ 1ÞðN þ 1Þ. If j1, l1, n1 are the j, l, n index of ajln and j2, l2, n2 are that of bjln, then the matrix B array can be expressed as

D. Sarkar et al. / Renewable Energy 85 (2016) 484e497

B array ¼

B l1 ðs1 ; n2 Þ 0

j1 ¼ j2

if

l1 ¼ l2 ;

and else:

(B.3)

C array is a rectangular matrix of size Mð2L þ 1ÞðS þ 1Þ  Mð2L þ 1ÞðN þ 1Þ. If j1, l1, s1 are the j, l, s index of gjls and j2, l2, n2 are that of bjln, then the matrix C array can be expressed as



C l1 ðs1 ; n2 Þ 0

C array ¼

j1 ¼ j2

if

l1 ¼ l2 ;

and else:

(B.4)

For the mth radiation problem, D array is a column vector m comprising of Mð2L þ 1ÞðN þ 1Þ elements and is given by array

Dm

¼

D l ðnÞ 0

if

j ¼ m; else:

(B.5)

While, ℰ array is a column vector comprising of Mð2L þ 1ÞðS þ 1Þ m elements and is written as



ℰ array ¼ m

ℰ l ðsÞ 0

if

j ¼ m; else;

(B.6)

where the infinite system of the modes in the interior region, s is terminated at S. Solving the simultaneous Equation (B.1) and (B.2), the expressions for the unknown coefficients are finally obtained as



aðDÞ gðDÞ

(

¼

1

ðI  B array T Þ B array 1 C array T ðI  B array T Þ

) bðIÞ

(B.7)

and



aðmÞ gðmÞ

(

¼

1

ðI  B array T Þ D array m 1 array array C array T ðI  B array T Þ D m þ ℰ m

) (B.8)

where T is the coordinate transfer matrix while relates the incident wave coefficients b(D,m) to a(D,m) using the relation (24). T is a square matrix of size Mð2L þ 1ÞðN þ 1Þ  Mð2L þ 1ÞðN þ 1Þ. If j1, l1, n1 are the j, l, n index of bjln and j2, l2, n2 are that of ajln, then the matrix T can be expressed as

( T ¼

Tlj2;l;j1;n 1 2 1 0

if

j1 sj2

and else;

n1 ¼ n2 ;

(B.9)

j ;j

where Tl 2;l 1;n is defined in (25). 1 2

1

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