Sébastien Olaya
email: olaya@enib.fr

Jean-Matthieu Bourgeot
email: bourgeot@enib.fr

Senior Member, IEEE Mohamed E H Benbouzid
email: mohamed.benbouzid@univ-brest.fr

J.-M Bourgeot

Hydrodynamic Coefficient Computation for a Partially Submerged Wave Energy Converter

Keywords: Wave energy converter, potential theory, eigenfunction expansion, wave-loads, heaving mode, scattering and radiation problem. I

This paper deals with the hydrodynamic parameter computation of a wave energy converter that consists of a cylindrical buoy sliding along a partially submerged platform made up of a plate and a column. The computed parameters are especially needed for the development of a simple hydrodynamic time-dependant model, based on the Cummins' formulation. This model is intended for WEC control purposes. A semi-analytical approach is proposed for the computation of the hydrodynamic coefficients and the excitation forces. The boundary value problem is solved by using variable separation and matched eigenfunction expansion methods. Analytical expressions for the velocity potential are then obtained for each sub-domain. The hydrodynamic coefficients and the excitation force can then be computed by using these expressions. Numerical results are given for different buoy, column, and plate radiuses and clearly the bearing surface of the plate has a significant influence on the wave excitation force applied to the submerged platform.

introduced in [START_REF] Olaya | Hydrodynamic Coefficients and Wave Loads for a WEC Device in Heaving Mode[END_REF] in which only part of the problem has been presented. Due to the simplicity of the geometry and in view of the model purposes (i.e. optimization and control) using a semi-analytical approach is probably the fastest and most reliable method to evaluate these parameters.

From the literature review, it has been noticed that in [START_REF] Berggren | Hydrodynamic coefficients of a wave energy device consisting of a buoy and a submerged plate[END_REF], [START_REF] Berggren | Hydrodynamic coefficients of a wave energy device consisting of a buoy and a submerged plate[END_REF] have studied the radiation problem of a wave energy device consisting of a buoy connected to a submerged plate by an elastomer hose. The buoy and plate have the same radius. They presented the results for added mass and potential damping for the two bodies and included cross terms only for the heaving mode. Under the assumption of linear wave theory, the radiation problem is solved using the matched eigenfunction expansion method. A comparison is made between their solution and a solution proposed by [START_REF] Yeung | Added mass and damping of a vertical cylinder in finite-depth waters[END_REF] [START_REF] Yeung | Added mass and damping of a vertical cylinder in finite-depth waters[END_REF] who studied the whole radiation problem of a floating vertical cylinder following a method similar to that presented by [START_REF] Garrett | Wave forces on a circular dock[END_REF] in [START_REF] Garrett | Wave forces on a circular dock[END_REF] who solves the scattering problem for a circular dock. In the case of a single cylinder, it is worth noting that a summary of both problems, i.e. radiation and scattering, can be found in [START_REF] Bhatta | On scattering and radiation problem for a cylinder in water of finite depth[END_REF] published by [START_REF] Bhatta | On scattering and radiation problem for a cylinder in water of finite depth[END_REF] and for which numerical computation aspects and results are shown in [START_REF] Bhatta | Computations of hydrodynamic coefficients, displacement-amplitude ratios and forces for a floating cylinder due to wave diffraction and radiation[END_REF]. [START_REF] Eidsmoen | Hydrodynamic parameters for a two-body axisymmetric system[END_REF] [START_REF] Eidsmoen | Hydrodynamic parameters for a two-body axisymmetric system[END_REF], studied the case where the buoy is rigidly connected to a submerged vertical tube in which a plate acts as a piston. In addition to the radiation problem, the author has presented results for the excitation forces obtained through the Haskind's theorem. In [START_REF] Zheng | Hydrodynamic properties of two vertical truncated cylinders in waves[END_REF], Zheng et al.

(2005) have extended the solution for the radiation problem of a two-body coaxial axisymmetric system by adding the sway/surge and roll/pitch modes of motion as well as the solution to the scattering of water waves using the velocity potential. As in [START_REF] Eidsmoen | Hydrodynamic parameters for a two-body axisymmetric system[END_REF] authors used Haskind's relation to check the correctness of their diffraction results and used the symmetry of the cross terms in the matrices for the added mass and radiation damping coefficients. In term of theoretical results, Willams et al. (2000) investigated in [START_REF] Williams | Water wave interaction with a floating porous cylinder[END_REF] the case for which the floating cylinder has a side wall which is porous over a portion of its draught. Fluid flow passing through this wall is assumed to follow Darcy's law. As previously mentioned, the buoy and plate radiuses are strictly the same. [START_REF] Wu | On diffraction and radiation problem for a cylinder over a caisson in water of finite depth[END_REF] investigated in [START_REF] Wu | On diffraction and radiation problem for a cylinder over a caisson in water of finite depth[END_REF] hydrodynamic properties of a similar WEC where the buoy floating on waves is directly connected to a caisson fixed on the seabed by a rigid rope. Wave energy is harnessed by a liquid pump in the caisson through the rope.

In this paper the caisson radius is greater than or equal to that of the buoy. In [START_REF] Wu | Response Amplitude and Hydrodynamic Force for a Buoy over a Convex[END_REF], the same authors extended the July 16, 2014 DRAFT solution to the case where the caisson radius is smaller than that of the buoy. They presented analytical results for the responding amplitudes and hydrodynamic forces for different radius sizes of the submerged structure and the gap between them. It is shown that the buoy hydrodynamic behaviour becomes more complex when the caisson radius is large and only at a relatively low frequency. Also it seems that the resonance period does not vary regardless of the increase or decrease of the radius ratio and/or the gap. In [START_REF] Wu | On diffraction and radiation problem for two cylinders in water of finite depth[END_REF], a similar study was performed by the same authors, with a two-body moving structure, but their analysis was restricted to the case where the submerged cylinder radius is larger than or equal to that of the floating one. A set of theoretical hydrodynamic parameters is presented and the behaviour is found to be similar to that of the previous study. More recently, some hydrodynamic results have been reported in [START_REF] Chau | Inertia, Damping, and Wave Excitation of Heaving Coaxial Cylinders[END_REF] by [START_REF] Chau | Inertia, Damping, and Wave Excitation of Heaving Coaxial Cylinders[END_REF] and also in [START_REF] Cochet | Two-Component Axisymmetric Wave-Energy Absorber -Analysis of Dynamics and Geometric Proportions[END_REF] by [START_REF] Cochet | Two-Component Axisymmetric Wave-Energy Absorber -Analysis of Dynamics and Geometric Proportions[END_REF], for the UC Berkeley wave energy device. The main idea is to extract energy due to the relative heaving motion between two concentric surface-piercing truncated cylinders. A similar study was first performed by [START_REF] Mavrakos | Hydrodynamic coefficients in heave of two concentric surface-piercing truncated circular cylinders[END_REF] in [START_REF] Mavrakos | Hydrodynamic coefficients in heave of two concentric surface-piercing truncated circular cylinders[END_REF].

Throughout the literature review, it is worth noting that there are no analytical results for the particular WEC configuration given in Fig. 1.

II. MATHEMATICAL FORMULATION OF THE PROBLEM

Following the established procedure for this kind of resolution, we divided the whole fluid domain into three or four sub-domains depending on the plate radius size as indicated in Fig. 1. Considering a cylindrical coordinate system (r, θ, z) with its origin O located at the intersection of the undisturbed free surface level with cylinder axes and the z-axis is positive upwards, it follows

• region Ω 1 : r ≥ R x ; -h ≤ z ≤ 0 where R x = max(R b , R p ) • region Ω 2 -For R p > R b : R b ≤ r ≤ R p ; -e 1 ≤ z ≤ 0 -For R p = R b : does not exist -For R p < R b : R p ≤ r ≤ R b ; -h ≤ z ≤ -b • region Ω 3 : R c ≤ r ≤ R x ; -e 1 ≤ z ≤ -b where R x = min(R b , R p ) • region Ω 4 : r ≤ R p ; -h ≤ z ≤ -e 2
R b is the radius of the buoy, R p is the radius of the plate, R c is the radius of the column and h is the not necessarily infinite depth of the water. Other geometrical parameters are given in Fig. 1.

Assuming linear wave theory and supposing that the fluid is inviscid, incompressible, and that its motion is irrotational, the fluid flow can be described, using a complex representation, by the velocity potential

Φ(r, θ, z, t) = Re{φ(r, θ, z)e -iωt } (1)
where Re{ } denotes the real part of the complex expression, ω is the angular frequency and t is the time dependency.

Also let us define the free surface elevation as

η(r, θ, z, t) = Re{η(r, θ, z)e -iωt } (2)
which is related to the velocity potential by the linearised Free Surface Kinematic Boundary Condition (FSKBC)

∂φ ∂z = ∂η ∂t , on z = 0 (3)
and the linearised Free Surface Dynamic Boundary Condition (FSDBC)

∂φ ∂t + gη = 0, on z = 0 (4)
The spatial part of the velocity potential φ(r, θ, z) has to satisfy the following boundary value problem

• Governing equation ∆φ = 0, in the whole fluid domain (5)

• Free surface boundary condition from ( 3) and ( 4)

ω 2 φ -g ∂φ ∂z z=0 = 0 (6)
• Seabed boundary condition ∂φ ∂z z=-h = 0

• Body surface boundary condition

∇φ • n = U k • n, on S k , k = 1, 2 (8)
where g is the gravity acceleration, n the unit normal vector on wet body surfaces S k directed into the fluid domain, and U k is the body velocity. k = {1, 2} stands for the buoy and the plate respectively. Finally, the velocity potential φ must satisfy the conditions which specify that the wave propagates away from the structure. This is given by the Sommerfeld radiation condition expressed in Ω 1 as

lim r→+∞ √ r( ∂φ ∂r -ik e φ) = 0 (9)
where k e is the wave number.

Based on the linear water wave theory and according to [START_REF] Wehausen | The Motion of Floating Bodies[END_REF], the velocity potential can be broken down as follows

φ(r, θ, z) = φ 0 (r, θ, z) + φ 7 (r, θ, z) + 2 k=1 6 q=1 φ k q (r, θ, z) (10)
where φ 0 is the incident waves potential, φ 7 is the diffracted potential, and φ k q is the radiated potential due to the motion of the body k in the direction q, with q = {1, 3, 5} standing respectively for the surge, heave, and pitch mode of motion.

In the solution procedure, analytical expressions for velocity potentials are obtained using the variable separation method. Assuming angular independence due to the axisymmetric configuration, velocity potentials are expressed as infinite series of orthogonal functions into each sub-domain fulfilling all boundary conditions. More details can be found on mathematical formulations in [START_REF] Wehausen | The Motion of Floating Bodies[END_REF], [START_REF] Linton | Handbook of Mathematical Techniques for Wave/Structure Interactions[END_REF], [START_REF] Falnes | Ocean Waves and Oscillating Systems -Linear Interactions Including Wave-Energy Extraction[END_REF]. [START_REF] Newman | The exciting forces on fixed bodies in waves[END_REF] DRAFT

A. Radiated Potential for Each Sub-domain in Heaving Mode

The radiation problem corresponds to the case where the structure is forced to oscillate in the absence of incident waves. Following a procedure similar to the one presented in [START_REF] Mavrakos | Hydrodynamic coefficients in heave of two concentric surface-piercing truncated circular cylinders[END_REF], we can break down the velocity potential in each sub-domain, φ l,k q (r, θ, z) as follows

φ l,k q (r, θ, z) = ∞ m=0
-iωζ q ϕ l,k q,m (r, z)cos(mθ) [START_REF] Williams | Water wave interaction with a floating porous cylinder[END_REF] where l denotes the fluid domain and ζ q is the complex amplitude corresponding to the motion mode. In heaving mode (q = 3) only expressions for m = 0 gave a contribution [START_REF] Bhatta | On scattering and radiation problem for a cylinder in water of finite depth[END_REF]. We are then looking for solutions that can be expressed as

ϕ l,k 3,0 (r, z) = ϕ l,k 3,h (r, z) + ϕ l,k 3,p (r, z)
where ϕ l,k 3,p (r, z) represents a particular solution of the velocity potential in heaving mode that fulfils the inhomogeneous boundary conditions and ϕ l,k 3,h (r, z) is the homogeneous part of the solution to the boundary value problem. In region Ω 1 , the velocity potential can be expressed as

ϕ 1,k 3,h (r, z) = A k 0 H (1) 0 (k e r) H (1) 0 (k e R x ) Z 1 0 (z) + ∞ i=1 A k i K 0 (λ i r) K 0 (λ i R x ) Z 1 i (z) (12)
where

R x = max(R b , R p ). A k i are the unknown Fourier coefficients, H (1)
0 () are Hankel functions and K

(1) 0 () are modified Bessel functions of the second kind.

For convenience, we define λ 0 = -ik e then the dispersion relation is given in terms of the eigenvalues λ i by

ω 2 = k e gtanh(k e h) = -λ i gtan(λ i h) (13)
and the depth dependency function

Z 1 i (z) is Z 1 i (z) = N -1/2 λi cos(λ i (z + h)) (14)
where

N λi = 1 2 1 + sin(2λ i h) 2λ i h (15)
for i = 0, 1, • • • , ∞
If region Ω 2 is defined, for the velocity potential we have

ϕ 2,k 3,h (r, z) = ∞ l=0 B k l S l (r) + C k l Sl (r) Z 2 l (z) (16)
Expressions for the functions S l (r), Sl (r) are defined in Appendix A in terms of modified Bessel functions of the first and second kind, and B k l , C k l are the unknown Fourier coefficients. The depth dependency function Z 2 l (z) is defined as

• In the case where R p > R b Z 2 l (z) = N -1/2 γ l cos(γ l (z + e 1 )) (17)
where the dispersion relation is given in terms of eigenvalues γ l by

ω 2 = k i gtanh(k i e 1 ) = -γ l gtan(γ l e 1 ) (18)
and

N γ l = 1 2 1 + sin(2γ l e 1 ) 2γ l e 1 ( 19
)
for l = 0, 1, • • • , ∞
From the body surface condition (8) we have

∂ϕ 2,k 3 ∂z z=-e1 = δ 2k , for R b ≤ r ≤ R p ( 20
)
and then

ϕ 2,k 3,p (r, z) = (z + g ω 2 )δ 2k (21)
where δ ij is the Kronecker's delta defined as δ ij = 1 if i = j, and δ ij = 0 otherwise.

• In the case where

R p < R b Z 2 l (z) = 1, for l = 0 √ 2 cos(γ l (z + h)), for l ≥ 1 with γ l = lπ h -b (22) for l = 1, 2, • • • , ∞.
From the body surface condition (8) we have

∂ϕ 2,k 3 ∂z z=-b = δ 1k , for R p ≤ r ≤ R b ( 23
)
and then

ϕ 2,k 3,p (r, z) = 1 2(h -b) (z + h) 2 - r 2 2 δ 1k (24)
In region Ω 3 , the velocity potential can be expressed as

ϕ 3,k 3,h (r, z) = ∞ n=0 D k n T n (r) + E k n Tn (r) Z 3 n (z) (25)
Expressions for the functions T n (r), Tn (r) are defined in Appendix B in terms of modified Bessel functions of the first and second kind, and D k n , E k n are the unknown Fourier coefficients. The depth dependency function Z 3 n (z) is defined as

Z 3 n (z) = 1, for n = 0 √ 2 cos(α n (z + e 1 )), for n ≥ 1 with α n = nπ e 1 -b 1 (26) July 16, 2014 DRAFT for n = 1, 2, • • • , ∞.
From the body surface condition (8) we have

∂ϕ 3,k 3 ∂z z=-b = δ 1k , for R c ≤ r ≤ R x (27) ∂ϕ 3,k 3 ∂z z=-e1 = δ 2k , for R c ≤ r ≤ R x ( 28
)
where

R x = min(R b , R p ).
Then the particular solution is given by

ϕ 3,k 3,p (r, z) = 1 2(e 1 -b) (z + e 1 ) 2 - r 2 2 δ 1k - 1 2(e 1 -b) (z + b) 2 - r 2 2 δ 2k (29)
Finally for region Ω 4 we have

ϕ 4,k 3,h (r, z) = F k 0 + ∞ j=1 F k j I 0 (β j r) I 0 (β j R p ) Z 4 j (z) (30)
where I

(1) 0 () are modified Bessel functions of the first kind and F k j are the unknown Fourier coefficients. The depth dependency function Z 4 j (z) is defined as

Z 4 j (z) = 1, for j = 0 √ 2 cos(β j (z + e 2 )), for j ≥ 1 with β j = jπ h -e 2 (31) for j = 1, 2, • • • , ∞.
From the body surface condition (8) we have

∂ϕ 4,k 3 ∂z z=-e2 = δ 2k , for 0 ≤ r ≤ R p (32)
and then the particular solution is expressed as

ϕ 4,k 3,p (r, z) = 1 2(h -e2) (z + h) 2 - r 2 2 δ 2k (33)
All the depth dependency functions Z l τ (z) are constructed in order to form an orthonormal set of eigenfunctions in the corresponding region l.

< Z l i (z), Z l j (z) > = δ ij (34)
where δ ij is the Kronecker delta which has been already defined.

B. Expression of the Scattering Potential in Each Sub-domain

The scattering potential represents the solution for which the structure is considered fixed in presence of incident waves [START_REF] Linton | Handbook of Mathematical Techniques for Wave/Structure Interactions[END_REF] and is defined as

φ s (r, θ, z) = φ 0 (r, θ, z) + φ 7 (r, θ, z) (35)
Considering a linear wave propagating in the positive direction x in water of constant depth h and decribed by a small amplitude A at frequency ω, incident potential φ 0 can be defined as

φ 0 (r, θ, z) = B ∞ m=0 ǫ m i m J m (k e r)Z 1 0 (z)cos(mθ) (36)
where ǫ m is the Neuman symbol defined as ǫ m = 1 for m = 0 and ǫ m = 2 otherwise, J m () are Bessel functions of the first kind and i is the complex number. The coefficient B is defined by

B = -iA g ω 1 Z 1 0 (0) (37)
Following the same approach as for the radiation problem, we are looking for unknown functions expressed as

φ l s (r, θ, z) = ∞ m=0 ϕ l s,m (r, z)cos(mθ) (38)
where only expressions for m = 0 contribute in the vertical direction. From the gradient condition ( 8) at all body surfaces we have [START_REF] Wehausen | The Motion of Floating Bodies[END_REF] ∂ϕ s ∂n = 0

It then follows that the scattering velocity potential in region Ω 1 can be expressed as

ϕ 1 s,0 (r, z) = BJ 0 (k e r)Z 1 0 (z) + A 0 0 H (1) 0 (k e r) H (1) 0 (k e R x ) Z 1 0 (z) + ∞ i=1 A 0 i K 0 (λ i r) K 0 (λ i R x ) Z 1 i (z) (40)
where

R x = max(R b , R p ).
One can note that the expression given above has the same form as expression [START_REF] Wu | On diffraction and radiation problem for a cylinder over a caisson in water of finite depth[END_REF] obtained for the radiation problem. For the purpose of the analytical resolution procedure purpose, let us introduce the notation k = 0, which means that none of the body is moving. Using notations given for the radiation problem we can write

ϕ 1 s,0 (r, z) = ϕ 1,0 3,h (r, z) + ϕ 1,0 3,p (r, z) (41)
where we define

ϕ 1,0 3,h (r, z) = A 0 0 H (1) 0 (k e r) H (1) 0 (k e R x ) Z 1 0 (z) + ∞ i=1 A 0 i K 0 (λ i r) K 0 (λ i R x ) Z 1 i (z) ϕ 1,0 3,p (r, z) = BJ 0 (k e r)Z 1 0 (z)
For others regions, unknown functions have the same forms as those developed for the homogeneous part of the radiation problem, that is

ϕ l s,0 (r, z) = ϕ l,0 3,h (r, z) (42)
for l > 1.

It remains to determine the unknown Fourier coefficients

A k i , B k l , C k l , D k n , E k n
, and F k j in the infinite series of orthogonal functions using the matching eigenfunction expansion method. The coefficients can be determined by applying conditions of pressure and normal velocity continuity at the different imaginary interfaces (i.e. at r = R b and r = R p ) as well as the body surface boundary condition (8) at the body vertical walls.

III. SOLUTION TO THE BOUNDARY VALUE PROBLEM

A. Determination of the unknown coefficients for R p = R b

In the case where the buoy radius and plate radius are equal, i.e. R p = R b , the requirement for pressure and normal velocity continuity has to be fulfilled at the vertical imaginary interface r = R b , as well as the boundary conditions on the body vertical surfaces. In heaving mode, the latter are expressed as

∂ϕ 1,k 3 ∂r r=Rp = 0 , for -e 2 ≤ z ≤ -e 1 ( 43
)
∂ϕ 1,k 3 ∂r r=R b = 0 , for -b ≤ z ≤ 0 (44) ∂ϕ 3,k 3 ∂r r=Rc = 0 , for -e 1 ≤ z ≤ -b (45)
for k = 0, 1 and 2.

At the imaginary interface r = R b , velocity potential continuity conditions are

• for -e 1 ≤ z ≤ -b ϕ 1,k 3 (R b , z) = ϕ 3,k 3 (R b , z) (46) ∂ϕ 1,k 3 ∂r r=R b = ∂ϕ 3,k 3 ∂r r=R b (47) • for -h ≤ z ≤ -e 2 ϕ 1,k 3 (R b , z) = ϕ 4,k 3 (R b , z) (48) ∂ϕ 1,k 3 ∂r r=R b = ∂ϕ 4,k 3 ∂r r=R b (49)
Due to the orthogonal properties of the function Z 4 j (z) valid for -h ≤ z ≤ -e 2 , it follows that

F k j = 1 h -e 2 -e2 -h ϕ 4,k 3,h (R b , z)Z 4 j (z)dz -Q k j3 + O k j3 ( 50
)
where

Q k j3 =      0, for k = 0 1 h -e 2 -e2 -h ϕ 4,k 3,p (R b , z)Z 4 j (z)dz, for k = 1, 2 O k j3 =      1 h -e 2 -e2 -h ϕ 1,0 3,p (R b , z)Z 4 j (z)dz, for k = 0 0, for k = 1, 2
By applying continuity condition (48), Fourier coefficients F k j that describe velocity potential in Ω 4 can be expressed in terms of coefficients A k i as follows

F k j = ∞ i=0 A k i L ji -Q k j3 -O k j3 (51)
with

L ji = 1 h -e 2 -e2 -h Z 4 j (z)Z 1 i (z)dz (52)
Using the same procedure for interval -e 1 ≤ z ≤ -b, orthogonal properties of the function

Z 3 n (z) lead to coefficients D k n D k n = 1 e 1 -b -b -e1 ϕ 3,k 3,h (R b , z)Z 3 n (z)dz -P k n3 + O k n3 ( 53
)
where

P k n3 =      0, for k = 0 1 e 1 -b -b -e1 ϕ 3,k 3,p (R b , z)Z 3 n (z)dz, for k = 1, 2 O k n3 =      1 e 1 -b -b -e1 ϕ 1,k 3,p (R b , z)Z 3 n (z)dz, for k = 0 0, for k = 1, 2
By applying continuity condition (46), Fourier coefficients D k n that describe velocity potential in Ω 3 can be expressed in terms of coefficients A k i as follows

D k n = ∞ i=0 A k i M ni -P k n3 -O k n3 (54)
with

M ni = 1 e 1 -b -b -e1 Z 3 n (z)Z 1 i (z)dz (55)
By multiplying both sides of equations ( 43), (44), ( 47) and (49) by

Z 1 τ (z)/h (for τ = 0, 1, • • • , i, • • • )
, integrating over the corresponding interval of validity and adding resulting expressions, we obtain a complete set of equations.

Replacing coefficients F k j and D k n by their respective definitions (51) and ( 54) we obtain

h k τ = ∞ i=0 d τ i A k i + ∞ n=0 d τ n E k n ( 56
)
July 16, 2014 DRAFT with

h 0 τ = B   Υ 0 δ 0τ -J 0 (k e R b )   h -e 2 h ∞ j=0 Γ j L j0 L jτ + e 1 -b h ∞ n=0 T ′ n (R b )M n0 M nτ (57) h 1 τ = R b 2h M 0τ + e 1 -b h ∞ n=0 T ′ n (R b )P 1 n3 M nτ (58) h 2 τ = R b 2h (L 0τ -M 0τ ) + h -e 2 h ∞ j=0 Γ j Q 2 j3 L jτ + e 1 -b h ∞ n=0 T ′ n (R b )P 2 n3 M nτ ( 59
)
d τ i = ∆ i δ iτ + h -e 2 h ∞ j=0 Γ j L ji L jτ + e 1 -b h ∞ n=0 T ′ n (R b )M ni M nτ ( 60
)
d τ n = e 1 -b h ∞ n=0 T ′ n (R b )M nτ (61)
where we define

Γ 0 = 0 Γ j = β j I 1 (β j R b ) I 0 (β j R b ) (62)
∆ 0 = k e H (1) 1 (k e R b ) H (1) 0 (k e R b ) ∆ i = λ i K 1 (λ i R b ) K 0 (λ i R b ) (63)
for i, j ≥ 1 and

Υ 0 = -k e J 1 (k e R b ) (64)
Finally, multiplying the body surface condition (45), expressed on the column, by

Z 3 ν (z)/(e 1 -b) (for ν = 0, 1, • • • , n, • • • ), integrating with respect to z, i.e.
-e 1 ≤ z ≤ -b, and using the definition of D k n , we obtain the last set of equations at the vertical boundary r = R c .

h k ν = ∞ i=0 d νi A k i + ∞ n=0 d νn E k n ( 65
)
where

h 0 ν = -BJ 0 (k e R c ) ∞ n=0 T ′ n (R c )M n0 δ nν (66) h 1 ν = ∞ n=0 T ′ n (R c )P 1 n3 δ nν + R c 2(e 1 -b) δ 0ν (67) h 2 ν = ∞ n=0 T ′ n (R c )P 2 n3 δ nν - R c 2(e 1 -b) δ 0ν (68) d νi = ∞ n=0 T ′ n (R c )M ni δ nν (69) d νn = T ′ n (R c )δ nν (70)
In order to find a solution to unknown Fourier coefficients A k i and E k n , we have to truncate the infinite series to the first N terms. By introducing matrix notation, the unknown coefficients can be determined, solving simultaneously the two sets of equations ( 56) and (65) which can be expressed by the following linear system

     d τ i d τ n d νi d νn           A k i1 E k n1      =      h k τ 1 h k ν1      (71)
Note that the matrix on the left hand side of (71) does not depend on the boundary value conditions of the BVP.

The matrix term values would only change if we modify the geometrical configuration. This can be an advantage in terms of reducing the numerical computation time. Finally, the remaining Fourier coefficients F k j and D k n can be computed respectively from equations ( 51) and (54).

B. Determination of the unknown coefficients for R p > R b

In the case where the plate radius is greater than the buoy radius, i.e. R p > R b , the requirement for pressure and normal velocity continuity has to be fulfilled at the two vertical imaginary interfaces r = R p and r = R b , as well as the boundary value conditions on the body vertical surfaces. In heaving mode the latter are expressed as

Specifying velocity potential continuity conditions at the imaginary interface r = R p we have (76)

• for -h ≤ z ≤ -e 2 ϕ 1,k 3 (R p , z) = ϕ 4,k 3 (R p , z) (75
• for -e 1 ≤ z ≤ 0 July 16, 2014 DRAFT ϕ 1,k 3 (R p , z) = ϕ 2,k 3 (R p , z) (77) ∂ϕ 1,k 3 ∂r r=Rp = ∂ϕ 2,k 3 ∂r r=Rp (78)
Using the orthogonal properties of the trigonometric function Z 4 j (z) valid for -h ≤ z ≤ -e 2 , it follows that for r = R p

F k j = 1 h -e 2 -e2 -h ϕ 4,k 3,p (R p , z)Z 4 j (z)dz -Q k j3 + O k j3 ( 79
)
where

Q k j3 =      0, for k = 0 1 h -e 2 -e2 -h ϕ 4,k 3,p (R p , z)Z 4 j (z)dz, for k = 1, 2 O k j3 =      1 h -e 2 -e2 -h ϕ 1,0 3,p (R p , z)Z 4 j (z)dz, for k = 0 0, for k = 1, 2
From the normal velocity continuity condition (75) we can expressed F k j as

F k j = ∞ i=0 A k i L ji -(Q k j3 -O k j3 ) (80)
with

L ji = 1 h -e 2 -e2 -h Z 4 j (z)Z 1 i (z)dz (81)
In a similar way, by using orthogonal characteristics of the set Z 2 l (z) valid in -e 1 ≤ z ≤ 0 and from (77), we can express B k l as

B k l = ∞ i=0 K li A k i -(R k l3 -O k l3 ) (82)
with

K li = 1 e 1 0 -e1 Z 2 l (z)Z 1 i (z)dz (83)
and

R k l3 =      0, for k = 0 1 e 1 0 -e1 ϕ 2,k 3,p (R p , z)Z 2 l (z)dz, for k = 1, 2 O k l3 =      1 e 1 0 -e1 ϕ 1,0 3,p (R p , z)Z 2 l (z)dz, for k = 0 0, for k = 1, 2
By multiplying both sides of equations ( 76), ( 72) and (78) by

Z 1 τ (z)/h (for τ = 0, 1, • • • , i, • • • )
, integrating over the corresponding domain of validity and adding resulting expressions, a complete set of equations is formed.

Replacing coefficients F k j and B k l by their respective definitions (80) and (82), we obtain

h k τ = ∞ i=0 d τ i A k i + ∞ l=0 d τ l C k l ( 84
)
July 16, 2014 DRAFT where

h 0 τ = B   Υ 0 δ 0τ -J 0 (k e R p )   h -e 2 h ∞ j=0 Γ j L j0 L jτ + e 1 h ∞ l=0 S ′ l (R p )K l0 K lτ (85) h 1 τ = 0 (86) h 2 τ = R p 2h L 0τ + h -e 2 h ∞ j=0 Γ j Q 2 j3 L jτ + e 1 h ∞ l=0 S ′ l (R p )R 2 l3 K lτ ( 87
)
d τ i = ∆ i δ iτ + h -e 2 h ∞ j=0 Γ j L ji L jτ + e 1 h ∞ l=0 S ′ l (R p )K li K lτ ( 88
)
d τ l = e 1 h S′ l (R p )K lτ (89)
The same procedure as the one used above is applied at the imaginary interfaces r = R b . From the normal velocity continuity condition (90)

ϕ 2,k 3 (R b , z) = ϕ 3,k 3 (R b , z) (90)
the Fourier coefficients D k n can be expressed as

D k n = ∞ l=0 M nl C k l -(P k n3 -R k n3 ) (91)
where

M nl = 1 e 1 -b -b -e1 Z 3 n (z)Z 2 l (z)dz (92)
and as well as the body surface boundary condition (73) on the buoy have to be fulfilled. Multiplying both sides of these two conditions by Z 2 ν (z)/e 1 (for ν = 0, 1, • • • , l, • • • ), integrating over the corresponding interval and adding resulting expressions, a complete set of equations is formed.

P k n3 = 1 e 1 -b -b -e1 ϕ 3,k 3,p (R b , z)Z 3 n (z)dz (93) R k n3 = 1 e 1 -b -b -e1 ϕ 2,k 3,p (R b , z)Z 3 n (z)dz (
h k ν = ∞ i=0 d νi A k i + ∞ l=0 d νl C k l + ∞ n=0 d νn E k n ( 96
)
July 16, 2014 DRAFT where

h 0 ν = BJ 0 (k e R p ) ∞ l=0 S ′ l (R b )K l0 δ lν (97) h 1 ν = ∞ n=0 T ′ n (R b )P 1 n3 δ nν + R b 2e 1 M 0ν (98)
h 2 ν = - ∞ l=0 S ′ l (R b )R 2 l3 δ lν + ∞ n=0 T ′ n (R b )(P 2 n3 -S 2 n3 )δ nν - R b 2e 1 M 0ν (99)
d νi = - ∞ l=0 S ′ l (R b )K li δ lν ( 100
)
d νl = -S ′ l (R b )δ lν + e 1 -b e 1 ∞ n=0 T ′ n (R b )M nl M nν (101)
d νn = e 1 -b e 1 T ′ n (R b )M nν (102)
Finally, multiplying the body surface condition (74), expressed on the column, by

Z 3 σ (z)/(e 1 -b) (for σ = 0, 1, • • • , n, • • •
) and integrating with respect to z, i.e. -e 1 ≤ z ≤ -b, the last set of equations at the imaginary interface r = R c is obtained

h k σ = ∞ l=0 d σl C k l + ∞ n=0 d σn E k n (103)
where

h 0 σ = 0 (104) h 1 σ = ∞ n=0 T ′ n (R c )P 1 n3 δ nσ + R c 2(e 1 -b) δ 0σ (105) h 2 σ = ∞ n=0 T ′ n (R c )P 2 n3 δ nσ - R c 2(e 1 -b) δ 0σ (106)
d σl = ∞ n=0 T ′ n (R c )M nl δ nσ (107)
d σn = T ′ n (R c )δ nσ (108)
In order to find a solution to unknown Fourier coefficients A k i , C k l , and E k n we have to truncate the infinite series to the first N terms. The unknown coefficients can be determined by solving the linear system (109) and the remaining coefficients B k l , D k n , and F k j can be obtained respectively from their expressions (82), (91), and (80). In the case where the plate radius is smaller than the buoy radius, i.e. R p < R b , the requirement for pressure and normal velocity continuity has to be fulfilled at the two vertical imaginary interfaces r = R p and r = R b , as well as the boundary conditions on the body vertical surfaces. In heaving mode the latter are expressed as

           d τ i d τ l 0 d νi d νl d νn 0 d σl d σn                       A k i1 C k l1 E k n1            =            h k τ 1 h k ν1 h k σ1            (
Specifying the normal velocity continuity condition at the imaginary interface r = R b , we have

ϕ 1,k 3 (R b , z) = ϕ 2,k 3 (R b , z) (113)
for -h ≤ z ≤ -b and unknown Fourier coefficients B k l that describe velocity potential in sub-domain Ω 2 can then be expressed as

B k l = ∞ i=0 K li A k i -(R k l3 -O k l3 ) (114)
where

K li = 1 h -b -b -h Z 2 l (z)Z 1 i (z)dz (115)
and

R k l3 =      0, for k = 0 1 h -b -b -h ϕ 2,k 3,p (R b , z)Z 2 l (z)dz, for k = 1, 2 O k l3 =      1 h -b -b -h ϕ 1,0 3,p (R b , z)Z 2 l (z)dz, for k = 0 0, for k = 1, 2
From gradient condition (110) and applying the pressure continuity condition

∂ϕ 1,k 3 ∂r r=R b = ∂ϕ 2,k 3 ∂r r=R b (116)
a complete set of equations is formed by multiplying both sides of these last conditions by Z 1 τ (z)/h (for τ = 0, 1, • • • , i, • • • ); following a similar procedure as described above. Using expressions (114) for coefficients B k l , we obtain

h k τ = ∞ i=0 d τ i A k i + ∞ l=0 d τ l C k l ( 117
)
July 16, 2014 DRAFT for which we define

h 0 τ = B Υ 0 δ 0τ -J 0 (k e R b ) h -b h ∞ l=0 S ′ l (R b )K l0 K lτ (118) h 1 τ = R b 2h K 0τ + h -b h ∞ l=0 S ′ l (R b )R 1 l3 K lτ (119) h 2 τ = 0 ( 120
)
d τ i = ∆ i δ iτ + h -b h ∞ l=0 S ′ l (R b )K li K lτ ( 121
)
d τ l = h -b h S′ l (R b )K lτ (122)
At the imaginary interface r = R p , for continuity conditions we have (124) From condition (123) we can express F k j as

• for -h ≤ z ≤ -e 2 ϕ 2,k 3 (R p , z) = ϕ 4,k 3 (R p , z) (123
• for -e 1 ≤ z ≤ -b ϕ 2,k 3 (R p , z) = ϕ 3,k 3 (R p , z) (125
F k j = ∞ l=0 C k l L jl -Q k j3 -R k j3 ( 127
)
where

L jl = 1 h -e 2 -e2 -h Z 4 j (z)Z 2 l (z)dz (128)
and

Q k j3 = 1 h -e 2 -e2 -h ϕ 4,k 3,p (R p , z)Z 4 j (z)dz (129) R k j3 = 1 h -e 2 -e2 -h ϕ 2,k 3,p (R p , z)Z 4 j (z)dz (130)
for k = 1, 2 and 0 otherwise.

From condition (125) we can express D k n as

D k n = ∞ l=0 M nl C k l -(P k n3 -R k n3 ) (131)
with

M nl = 1 e 1 -b -b -e1 Z 3 n (z)Z 2 l (z)dz (132)
and

P k n3 = 1 e 1 -b -b -e1 ϕ 3,k 3,p (R p , z)Z 3 n (z)dz (133) R k n3 = 1 e 1 -b -b -e1 ϕ 2,k 3,p (R p , z)Z 3 n (z)dz (134)
By multiplying both sides of equations ( 124), (111), and (126) by

Z 1 ν (z)/(h-b) (for ν = 0, 1, • • • , l, • • • )
, integrating over the corresponding interval of validity and adding resulting expressions, a complete set of equations is obtained.

Replacing coefficients B k l , F k j , and D k n by their respective definitions (114), (127), and (131) we obtain

h k ν = ∞ i=0 d νi A k i + ∞ l=0 d νl C k l + ∞ n=0 d νn E k n ( 135
)
where

h 0 ν = BJ 0 (k e R b ) ∞ l=0 S ′ l (R p )K l0 δ lν (136) h 1 ν = R p 2(h -b) (M 0ν -δ 0ν ) - ∞ l=0 S ′ l (R p )R 1 l3 δ lν + e 1 -b h -b ∞ n=0 T ′ n (R p ) P 1 n3 -R 1 n3 M nν - h -e 2 h -b ∞ j=0 Γ j R 1 j3 L jν (137) h 2 ν = R p 2(h -b) (L 0ν -M 0ν ) + h -e 2 h -b ∞ j=0 Γ j Q 2 j3 L jν + e 1 -b h -b ∞ n=0 T ′ n (R p )P 2 n3 M nν ( 138
)
d νi = - ∞ l=0 S ′ l (R p )K li δ lν ( 139
)
d νl = -S′ l (R p )δ lν + h -e 2 h -b ∞ j=0 Γ j L jl L jν + e 1 -b h -b ∞ n=0 T ′ n (R p )M nl M nν ( 140
)
d νn = e 1 -b h -b T ′ n (R p )M nν (141)
Finally, multiplying the body surface condition (112), expressed on the column, by

Z 3 σ (z)/(e 1 -b) (for σ = 0, 1, • • • , n, • • •
) and integrating with respect to z, i.e. -e 1 ≤ z ≤ -b, we obtain the last set of equations at the imaginary interface r = R c .

h k σ = ∞ l=0 d σl C k l + ∞ n=0 d σn E k n ( 142
)
July 16, 2014 DRAFT where

h 0 σ = 0 (143) h 1 σ = ∞ n=0 T ′ n (R c ) P 1 n3 -S 1 n3 δ nσ + R c 2(e 1 -b) δ 0σ (144) h 2 σ = ∞ n=0 T ′ n (R c )P 2 n3 δ nσ - R c 2(e 1 -b) δ 0σ ( 145
)
d σl = ∞ n=0 T ′ n (R c )M nl δ nσ ( 146
)
d σn = T ′ n (R c )δ nσ (147)
Using matrix notation, the unknown Fourier coefficients A k i , C k l , and E k n can be determined by solving the following linear system.

           d τ i d τ l 0 d νi d νl d νn 0 d σl d σn                       A k i1 C k l1 E k n1            =            h k τ 1 h k ν1 h k σ1            (148)
The remaining coefficients B k l , D k n , and F k j can be obtained respectively from their expressions (114), (131), and (127).

IV. EXCITATION FORCES AND HYDRODYNAMIC COEFFICIENTS

Once the scattering and/or radiation problems are solved, which means that we determined the unknown Fourier coefficients for the orthogonal series, the velocity potential is known in the whole fluid domain. Waves exciting forces and/or hydrodynamic coefficients can then be determined by integrating the hydrodynamic pressure, given by the Bernoulli equation, over the wet surface of the body under consideration. Based on the linear wave theory, the pressure is given by

p(r, θ, z, t) = -ρ ∂Φ ∂t = iωρφ(r, θ, z)e -iωt (149)

A. Added Mass and Radiation Damping Coefficients

When the structure is moving, a radiation force acts on it and can be expressed as

F k,i q,j (ω) = iωρ Si φ k q (r, θ, z)n j dS (150)
where F k,i q,j (ω) is the complex representation of the radiation force acting on the body i in the direction j due to the motion of the body k in the direction q (here q = 3). It is conventional to break down this radiation force into two components: one proportional to the acceleration of body and the other proportional to his velocity as follows

F k,i q,j (ω) = -ω 2 ζ q µ k,i q,j + i ω λ k,i q,j (151)
July 16, 2014 DRAFT where hydrodynamic coefficients µ k,i q,j and λ k,i q,j are referenced in the literature as the added mass and the radiation or potential damping, respectively. Finally, using expression [START_REF] Williams | Water wave interaction with a floating porous cylinder[END_REF] in heaving mode we obtain

µ k,i 3,3 + i ω λ k,i 3,3 = 2πρ L l=1 Si φ l,k 3 (r, z)n 3 rdr (152)

B. Wave Excitation Forces in z Direction

When the structure is considered as fixed in the presence of an incident wave, an excitation force acts on the bodies. Using a complex representation, excitation forces in the vertical direction can be expressed in terms of scattering potential, see (35), as

f k z (ω) = iωρ S k φ s (r, θ, z)n 3 dS (153)
where k indicates which body is under consideration and n 3 is the component of the generalised normal vector in z direction. The Haskind's relation is another approach for evaluating wave excitation forces [START_REF] Haskind | The Exciting Forces and Wetting of Ships in Waves[END_REF], [START_REF] Newman | The exciting forces on fixed bodies in waves[END_REF], [START_REF] Falnes | Ocean Waves and Oscillating Systems -Linear Interactions Including Wave-Energy Extraction[END_REF]. They can be evaluated either at the wet body surface S k by (154)

f k z (ω) = iωρ S k φ 0 ∂φ 3 ∂n -φ 3 ∂φ 0 ∂n dS (154)
or at the control surface S ∞ by (155).

f k z (ω) = -iωρ lim r→∞ 2π 0 0 -h φ 0 ∂ϕ 1,k 3,0 ∂r -ϕ 1,k 3,0 ∂φ 0 ∂r rdzdθ (155)
where we take, as control surface S ∞ , a vertical cylinder about the z axis of large radius r, as recommended in [START_REF] Newman | The exciting forces on fixed bodies in waves[END_REF].

The main advantage of these two relations is that we only need to know solutions for the incident and radiated potential. However, as already mentioned in [START_REF] Wu | On diffraction and radiation problem for a cylinder over a caisson in water of finite depth[END_REF], evaluating expressions obtained by (154) is complicated since potentials have to be integrated on the whole of the wet structure. Therefore, the use of the far-field behaviour of the radiation solution given by (155), which simplifies the analysis is used here. After integral evaluation, we find that the wave excitation force for the heaving mode is (see [START_REF] Chau | Inertia, Damping, and Wave Excitation of Heaving Coaxial Cylinders[END_REF] for details)

f k z (ω) = -4iρgh N λ0 A k 0 cosh(k e h)H (1)
0 (k e R x ) (156)

V. NUMERICAL RESULTS AND DISCUSSION

A specific code based on the above formulation has been developed in order to solve problems (71) , (109) and

(148). To carry out numerical computations, the infinite series in the expressions of the radiated and scattering potentials have to be truncated to a finite number of terms. According to the literature [START_REF] Berggren | Hydrodynamic coefficients of a wave energy device consisting of a buoy and a submerged plate[END_REF], [START_REF] Yeung | Added mass and damping of a vertical cylinder in finite-depth waters[END_REF], [START_REF] Wu | On diffraction and radiation problem for a cylinder over a caisson in water of finite depth[END_REF], it seems that considering only the first 30 terms shows good truncation characteristics. We have therefore chosen N = 30 for potential expressions in all sub-domains.

In order to validate the analytical expressions obtained for velocity potentials, several checks are performed on numerical results and then different case studies have to be considered. In and Yeung [START_REF] Chau | Inertia, Damping, and Wave Excitation of Heaving Coaxial Cylinders[END_REF]. For all numerical computations, we have used h = 1. For the scattering problem, solutions have been obtained based on the assumption of a unit amplitude incident wave, A = 1.

A. Vertical boundary condition matching

We first look at the matching of the velocity potential and its first derivative in the radial direction along the imaginary interfaces between sub-domains. In Fig. 2, results are presented for the structure depicted in Section I for the case study No.3, where the plate radius is smaller than that of the buoy, α = .75, and where the buoy is moving and the platform is fixed. Excellent matching is achieved on velocity potential using the above numerical truncation. However, for gradient visualisation purposes, we have used more terms in expressions to reduce the oscillations due to Bessel functions (N = 80 for regions Ω 1 , Ω 2 , and N = 40 for regions Ω 3 , Ω 4 ). The well-known singularity should be noted in Fig. 2 (b), at the corner due to the discontinuous boundary condition (see [START_REF] Linton | Handbook of Mathematical Techniques for Wave/Structure Interactions[END_REF] for more details). Also looking at external domain curves, we can notice that the gradient conditions (110) and (111) are met.

B. Wave excitation force comparison -velocity potential vs Haskind's relation method

A comparison is made between the two approaches presented in Section IV to evaluate the excitation forces. The first approach is based on the solution of the scattering problem and given by equation ( 153) and the other one is based on the Haskind's relation (156). Fig. 3, shows numerical results for case study No.2 with α = 2 1 .51 .

Excitation forces are non-dimensionalised dividing them by hydrostatic stiffness i.e. ρgπ(R 2 b -R 2 c ) for the buoy, and ρgπR 2 c for the submerged platform. Evidently, the two approaches obtain very well matched results, which implies that the correctness of the scattering potential seems to be confirmed.

C. Asymptotic behaviour comparison with existing models

Also one could use asymptotic behaviours of the presented model to compare results with existing models found in the literature, which indirectly verify the proposed expressions for potentials. for hydrodynamic parameters when the column radius tends towards zero, case study No.1 for α = 8 2 1 .

This configuration is similar to the model presented by Wu et al. in [START_REF] Wu | On diffraction and radiation problem for two cylinders in water of finite depth[END_REF] (not shown). Same coefficients as those given in [START_REF] Wu | On diffraction and radiation problem for two cylinders in water of finite depth[END_REF] have been used to non-dimensionnalise excitation forces in Fig. 4 and hydrodynamics coefficients in Figures 5 to 8, and excellent correlation is found. Also parameters are shown for the case where the radius of the column is half of the radius of the buoy that corresponds to case study No.2. Looking at the added mass and radiation damping, we note that only the amplitude of the coefficients changes with the increase and decrease of the wet body surface but not the behaviour. If we look at the excitation forces on the buoy, the presence of the column does not seem to have an effect on it. However, we notice that for the platform, the behaviour is totally different at low frequency and we can observe a rebound which varies with the radius of the plate except for relatively big sizes. It seems that a radius ratio between the plate and column exists for which the column no longer influences the behaviour of the wave excitation force on the platform. In addition, a last check is performed on the correctness of non-dimensionnalised excitation forces. For hydrodynamic coefficients, we divided by factor ρπR 2 b for the buoy and by factor ρπR 2 c for the platform. Here again figures show very good correlation between our model and the literature. Looking at the excitation forces on the platform, Fig. 10 (b), one can notice that the behaviour is not as previously described. It seems that a radius ratio between the plate and column exists for which the plate radius is not big enough for the rebound to be observed. Fig. 12 shows variation of the excitation force on the platform for different column and plate radiuses. Also we compare results with the case of a heaving single cylinder (dash-dotted lines) presented by Battha and Rahman in [START_REF] Bhatta | On scattering and radiation problem for a cylinder in water of finite depth[END_REF] with the same geometrical parameters as the column. When the radius ratio between plate and column is relatively small and the bearing surface of the plate is not big enough, it can be noted that the platform behaves in a similar way as a single cylinder. As already mentioned, for a relatively large plate radius compared to that of the column, it is found that the platform mainly behaves as the plate without the column. Between these two extremes, we have a rebound phenomenon for which properties such as amplitude and peak amplitude frequency vary with bearing surface.

α = 8 α = 2 α = 1 0 0.5 1 1.5 2 2.5 3 -1.5 -1 -0.5 0 0.5 •10 -1 k e R b λk,i 3,3 (a)
(k, i) = (1, 2) (k, i) = (2, 1) (b)

VI. CONCLUSION

Based on the potential theory, a semi-analytical method has been presented in order to solve the radiation and scattering problems which provides hydrodynamic parameters in heaving mode for a specific wave energy converter. These parameters are particularly needed for designers to analyse the WEC dynamics in irregular waves. A specific code has been developed based on the mathematical formulations presented in this paper. For validation purposes, several numerical simulations have been carried out for different buoy, column, and plate radiuses. The obtained results have been compared to well-known and available models in the literature. These results clearly confirm the appropriateness of the proposed semi-analytical approach. Finally, a set of theoretical results for the wave excitation forces, added mass and radiation damping have been presented for different submerged plate to buoy radius ratio as well as column to buoy radius ratio. These results have obviously shown that excitation forces applied on the submerged platform have a remarkable behaviour that depends on the plate bearing surface for a given column radius. Regarding the available literature, this behaviour is the first reported for that kind of structure and future investigations will be carried out on the studied structure for wave energy extraction. T n = I 0 (γ l r)K 0 (γ l R c ) -I 0 (γ l R c )K 0 (γ l r) I 0 (γ l R x )K 0 (γ l R c ) -I 0 (γ l R c )K 0 (γ l R x ) Tn = I 0 (γ l R x )K 0 (γ l r) -I 0 (γ l r)K 0 (γ l R x ) I 0 (γ l R x )K 0 (γ l R c ) -I 0 (γ l R c )K 0 (γ l R x )

where R x = min(R b , R p ).

July 16, 2014 DRAFT

Fig. 1 .

 1 Fig. 1. Definition sketch of the wave energy converter.

  for -e 2 ≤ z ≤ -e 1 for -e 1 ≤ z ≤ -b

) ∂ϕ 1

 1

  of the unknown coefficients for R p < R b

  for -e 1 ≤ z ≤ -b

) ∂ϕ 2

 2

) ∂ϕ 2

 2

Figures 4 to 9

 9

Fig. 2 .

 2 Fig. 2. Matching of the velocity potential (a) and its first derivative (b) and (c) for case study No.3, α = .75, at the imaginary interface r = R b and r = Rp for ke = 1.

Fig. 3 .

 3 Fig. 3. Dimensionless vertical forces, (a) on buoy (b) on platform, for case study No.2.

Fig. 4 . 2 Fig. 5 .

 425 Fig. 4. Dimensionless vertical forces, (a) on buoy (b) on platform.

Fig. 6 . 2 Fig. 7 . 2 Fig. 8 .

 62728 Fig. 6. Dimensionless added mass (a) and radiation damping (b) for the platform.

Fig. 9 .

 9 Fig. 9. Non-diagonal elements in the matrices for the added mass (a) and radiation damping (b) coefficients divide by 2 3 ρgπR 3 p for case study No.2.

Fig. 10 .

 10 Fig. 10. Dimensionless vertical forces on the buoy (a), on the platform (b).

Fig. 11 .Fig. 12 .

 1112 Fig. 11. Dimensionless added mass (a) and radiation damping (b) for the buoy.

  S l (r), Sl (r)• In the case where R b < R p i R p )H

  i R b ) -H (1) 0 (k i R b )H

  i R p ) for l ≥ 1 S l = I 0 (γ l r)K 0 (γ l R b ) -I 0 (γ l R b )K 0 (γ l r) I 0 (γ l R p )K 0 (γ l R b ) -I 0 (γ l R b )K 0 (γ l R p ) Sl = I 0 (γ l R p )K 0 (γ l r) -I 0 (γ l r)K m (γ l R p ) I 0 (γ l R p )K 0 (γ l R b ) -I 0 (γ l R b )K 0 (γ l R p ) • In the case where R b > R p for l = 0 S 0 = ln(r/R p ) ln(R b /R p ) S00 = ln(R b /r) ln(R b /R p ) for l ≥ 1 S l = I 0 (γ l r)K 0 (γ l R p ) -I 0 (γ l R p )K 0 (γ l r) I 0 (γ l R b )K 0 (γ l R p ) -I 0 (γ l R p )K 0 (γ l R b ) Sl = I 0 (γ l R b )K 0 (γ l r) -I 0 (γ l r)K 0 (γ l R b ) I 0 (γ l R b )K 0 (γ l R p ) -I 0 (γ l R p )K 0 (γ l R b ) APPENDIX B EXPRESSION FOR T n (r), Tn (r) for n = 0 T 0 = ln(r/R c ) ln(R x /R c ) T0 = ln(R x /r) ln(R x /R c )for n ≥ 1

  Table I, we have summarised July 16, 2014 DRAFT geometrical configurations for each case study where α and β are two variable parameters defined as α ∈ 8 2 1 .75 .65 .51 and β ∈ .5 .35 .25 .15 . They represent respectively the submerged plate to buoy radius ratio and the column to buoy radius ratio and they are used to cover all the test scenarios. Geometrical parameters for case studies no.1 and no.3 are taken respectively from models developed by Wu et al. [14] and Chau

[START_REF] Newman | The exciting forces on fixed bodies in waves[END_REF] DRAFT

ACKNOWLEDGMENT

This work was supported in part by the Fonds Unique Interministériel (France) -Project "EM Bilboquet", in part by Région Bretagne, and in part by Conseil Général du Finistère.


A Lebsir

A Bentounsi

M E H Benbouzid

H Mangel

Electric Generators Fitted to Wind Turbine Systems: An Up-to-Date Comparative Study

Keywords: Wind turbine, electric generators, comparison. Wind Energy Conversion Systems Fundamentals, Topologies, and Control ISIE 2012 19 19 Rotor Diameter Generator Rotor blade Tower Nacelle Gearbox Brake Rotor Diameter Gearbox Rotor Shaft Generator Rotor blade Base Brake Rotor

This paper describes a comparative study allowing the selection of the most appropriate innovative structures for electrical machines for a wind turbine system. This study is based on an exhaustive review of the state of the art and on an effective comparison of the performances of the three main conventional electric generator in wind energy application system that are the Doubly-Fed Induction Generator (DFIG), the Squirrel-Cage Induction Generator (SCIG), the Permanent Magnet Synchronous Generator (PMSG), and an innovative machine that is the Switched Reluctance Generator (SRG). The main conclusion drawn by the proposed comparative study is that the innovative switched reluctance generator is able fulfills the major requirements of a wind energy system.

Introduction

In recent years, the wind energy know a rapid growth because, it plays a more important role in global electricity generating, the use of wind is discussed in [START_REF] Dang | Introduction, history, and theory of wind power[END_REF], it is becoming more and more popular with the passage of time; the global wind energy capacity has grown rapidly and has become the fastest growing renewable energy technology [START_REF] Cheng | Review of the wind energy generating system[END_REF][START_REF] Jamil | A Review of power converter topology used with PMSG based wind power generation[END_REF]. Various wind turbine concepts have been developed and different wind generators have been used in researching and marketing, so as to efficiently utilize the wind power [START_REF] Cheng | Review of the wind energy generating system[END_REF][START_REF] Jamil | A Review of power converter topology used with PMSG based wind power generation[END_REF][START_REF] Hansen | Wind turbine concept market penetration over 10 years[END_REF], [START_REF] Amirat | Generators for wind energy conversion systems: State of the art and coming attractions[END_REF].

The choice of the type of aero-generators is however difficult. In fact, the choice of electric generator for wind power mainly depends on several criteria: structure, converter topology, environment (location where the turbine is installed), performances, and cost. Therefore, selecting the most appropriate electric generator for a wind power is a challenging task.

In an industrial point, the major types of electric generators adopted or under serious consideration for wind power include the early aero-generator concept is using a standard Squirrel-Cage Induction Generator (SCIG) and a multi-stage gearbox, directly connected to the grid, the Doubly-Fed Induction Generator (DFIG), the Permanent Magnet Synchronous Generator (PMSG), and an innovative machine that is the Switched Reluctance Generator (SRG). Cross-sections of each of these four generator types are provided in Fig. 1.

Moreover, according to an exhaustive review of the state of the art related to wind power, it is observed that investigations on doubly-fed induction generators are highly dominant, whereas those on permanent magnet synchronous generator are used increasing continuously in small wind power, those on cage induction are dropping while those on switching reluctance generators are gaining much interest [START_REF] Cheng | Review of the wind energy generating system[END_REF][START_REF] Jamil | A Review of power converter topology used with PMSG based wind power generation[END_REF], [START_REF] Mueller | Design and performance of a 20kW, 100rpm, switched reluctance generator for a direct drive wind energy converter[END_REF][START_REF] Jebaseeli | Performance Analysis of various configurations of switched reluctance machine for wind energy applications[END_REF].

In this paper, potential candidates in wind power are presented and evaluated according to the major requirements of a wind generating system. Conclusions are then drawn to identify the most potential candidate of wind energy. Fig. 1. Wind power generators.

Wind power requirement

Wind power configuration

Modern wind turbines divided into two basic groups; the horizontal-axis variety (HAWT), (Fig. 2), like the traditional farm windmills used for pumping water, and the vertical-axis design (VAWT), (Fig. 2), like the eggbeater Darrieus model or Savonius model, most large modern wind turbines are horizontal-axis turbines. The proposed comparative study has been done on the horizontal-axis wind turbine (HAWT) configuration (Fig. 3). In fact, the difference between them is at the level of the shaft, where the mounting of the shaft determines the direction and shape of the propeller blades, which means that horizontal and vertical wind generators can often vary in appearance. Both designs have their advantages and disadvantages [START_REF] Dang | Introduction, history, and theory of wind power[END_REF]. Historically, horizontal axis wind generators have been more popular and widely used which means its design and technologies are more widely understood and developed. Vertical axis wind generators are still a large unknown to many wind power users and technicians. Its technologies and designs are still in the process of being more widely understood.

Wind Energy Conversion Systems Fundamentals, Topologies, and Control ISIE 2012 22 22 The nacelle

Conversion methods of wind energy

The first requirement of the system is the existence of a wind flow sufficiently strong, a typical wind profile has been calculated in [START_REF] Amirat | Generators for wind energy conversion systems: State of the art and coming attractions[END_REF], [START_REF] Mueller | Design and performance of a 20kW, 100rpm, switched reluctance generator for a direct drive wind energy converter[END_REF] and is shown in Fig 4 . Where the aerodynamic power is related with the wind speed w v by equation (1). Finally, the torque developed on the rotor of the turbine is determined by (3):

= ρ λ

= 2 w T kv (3)
In a direct-drive power the generator rotates at the same speed as the turbine, therefore the same torque. Fig. 4. Typical wind profile [START_REF] Mueller | Design and performance of a 20kW, 100rpm, switched reluctance generator for a direct drive wind energy converter[END_REF].

Efficiency as a function of tip speed

Wind Turbines Fundamentals, Technologies, Application, Economics E. Hau, Springer Tip speed ratio: Speed at blade tip divided by incoming wind speed.

At low tip speed ratio, losses from wake.

A high tip speed, losses from airfoil drag.

4 th blade buys ~1-2% power

Generator characteristics versus wind power

The major requirements of generators for wind turbine, as mentioned in past literature, are summarized as follows [START_REF] Dang | Introduction, history, and theory of wind power[END_REF][START_REF] Cheng | Review of the wind energy generating system[END_REF][START_REF] Jamil | A Review of power converter topology used with PMSG based wind power generation[END_REF][START_REF] Hansen | Wind turbine concept market penetration over 10 years[END_REF][START_REF] Amirat | Generators for wind energy conversion systems: State of the art and coming attractions[END_REF][START_REF] Mueller | Design and performance of a 20kW, 100rpm, switched reluctance generator for a direct drive wind energy converter[END_REF].

-High torque and high power density; -Reduction of the system parts; -High efficiency; -Lowest maintenance; -High reliability and robustness for various wind operating conditions; -Reasonable cost.

Moreover, in the extreme environments, the electric generator should be fault-tolerant [START_REF] Cardenas | Switched reluctance generators for wind energy applications[END_REF][START_REF] Torrey | Switched reluctance generators and their control[END_REF]. Finally, in an industrial point of view, an additional selection criterion is the market acceptance degree of each generator type, which is closely associated with the comparative availability and cost of its associated power converter technology, in [START_REF] Baroudi | A review of power converter topologies for wind generators[END_REF] many different generator-converter combinations are compared on the basis of topology, cost, efficiency, power consumption and control complexity.

Figure 6 illustrates one of the major factors affecting the performance of a wind power is its power response to different wind speeds. This is usually given by the ideal power curve of the wind turbine, which reflects the aerodynamic, transmission and generation efficiencies of the system in an integrated form. In this example, the rated power of the turbine is 2 MW. The performance characteristic depending on the wind speed comprises three distinct zones:

-Zone I: where = 0 t P , the turbine does not produce power. -Zone II: speed of a turbine is the minimum wind speed at which the system begins to produce power. The min w v speed varies from turbine to turbine, depending on its design features. However, in general, most of the commercial wind turbines min w v is at velocities between 3 to 5 [ / m s]. -Zone III: the turbine is restricted to produce constant power r P .

Power curve

Cut in Rated Cut-out

Pitch control

Wind Turbines Fundamentals, Technologies, Application, Economics E. Hau, Springer Fig. 6. Typical power curve of a wind turbine [START_REF] Hau | Wind-Turbines Fundamentals, Technologies, Application[END_REF].

Comparative study

Generator is one of the most important components of a wind energy conversion system (second important component), generator of a wind turbine has to work under fluctuating power levels. Different types of generators are being used with wind turbines. These generators can either be induction (asynchronous) generators or synchronous generators and recently appeared innovative machine.

Squirrel-Cage Induction Generator (SCIG)

The first production of electrical energy with wind power was 1887 by Charles Brush in Cleveland, Ohio. The rated Power of the used dc-generator was 12kW and was designed to charge batteries. The induction machine was used at the first time in 1951 [START_REF] Soter | Development of induction machines in wind power technology[END_REF]. But, in wind power using squirrel-cage induction generators (see Fig. 7), must be operated at a constant speed, which is not favored at the varied wind speed application. However, SCIG drives have bulky construction, low efficiency, low reliability and need of maintenance, also the existing of slip ring, brush and three-stage gearbox increases the system mass and cost, also electrical and mechanical loss. Recently, squirrel-cage induction generators are dropping in this application. Typical torque speed curve of the machine is shown in Fig. 8. Hence, the rotor always rotates at a speed slightly lower than the synchronous speed. The difference between the synchronous speed s N and the rotor speed r N is termed as the slip of the motor. Thus, the slip (S) is given by:

- = s r s N N S N (4)
The synchronous speed of the induction motor is given by:

= 120 s f N p ( 5
)
Where f is the frequency and p is the number of poles.

When we couple this machine with a grid integrated wind turbine, initially it draws current from the grid as in case of a motor. The speed picks up and the rotation of the wind turbine causes the system to exceed the synchronous limit s N . Thus, rotor moves faster than the rotating magnetic field. At speeds higher than s N , the torque is negative as seen from Fig. 8. Thus, current flows in the opposite direction that is from the system to the grid. Thus the machine functions as a generator when it is driven by an external prime mover, like the wind turbine in our case [START_REF] Mathew | Wind Energy, Fundamentals, Resource Analysis and Economics[END_REF][START_REF] Soter | Development of induction machines in wind power technology[END_REF][START_REF] Kinnares | Characteristic requirements of a small scale squirrel cage induction generator for effective electricity generation from wind energy[END_REF].

Moreover, the development of rugged solid-state power semiconductors made it increasingly practical to introduce DF induction and synchronous generators drives that are mature to replace IG drive in wind power [2-15-17]. For illustration, Fig. 9 shows industrial wind power induction generators [18-19]. Fig. 8. Characteristics of a typical induction machine [START_REF] Mathew | Wind Energy, Fundamentals, Resource Analysis and Economics[END_REF]. Fig. 9. Industrial induction generator for wind turbine.

Doubly-Fed Induction Generator (DFIG)

Today over 70% of the wind turbines are build up with Doubly-Fed Induction Generator (DFIG) [START_REF] Soter | Development of induction machines in wind power technology[END_REF]. Many manufacturers, such as Vestas, Gamesa, GE and Repower, have provided the wind turbine system with this concept [2] Fig. 10.

In fact, DFIG has great improvement comparing with the SCIG concept, in [START_REF] Hansen | Wind turbine concept market penetration over 10 years[END_REF], [START_REF] Takahashi | Efficiency calculation of wind turbine generation system with doubly-fed induction generator[END_REF] there are summary of DFIG. This system (Fig. 11) consists of a wind turbine with DFIG. This means that the stator is directly connected to the grid while the rotor winding is connected via slip rings to a converter. As also seen in the Fig. 12 the DFIG can operate both in motor and generator operation with a rotor speed range of max r w ±Δ around the synchronous speed, 1 w . The dynamic behavior model and their control system of the DFIG is considered in [START_REF] Yongbo | Aggregating DFIGS in power system online analysis[END_REF][START_REF] Polinder | Comparison of direct-drive and geared generator concepts for wind turbines[END_REF][START_REF] Bhukya | Modeling and analysis of double fed induction generator for Variable Speed Wind Turbine[END_REF].

Moreover, new types of generators which may change the configuration of the wind energy are being developed, where this new concept eliminates most of the mechanical parts, such as brush, slip ring and gearbox, that are considered drawbacks to this concept, as a result, reducing the mass and cost of the system and achieving high reliability and availability. Finally, it should be noticed that most research works tend to use the DFIGs in wind energy, as they have excellent performance [START_REF] Takahashi | Efficiency calculation of wind turbine generation system with doubly-fed induction generator[END_REF][21][START_REF] Petersson | Analysis, modeling and control of doubly-fed induction generators for wind turbines[END_REF][START_REF] Yongbo | Aggregating DFIGS in power system online analysis[END_REF][START_REF] Polinder | Comparison of direct-drive and geared generator concepts for wind turbines[END_REF][START_REF] Bhukya | Modeling and analysis of double fed induction generator for Variable Speed Wind Turbine[END_REF]. Such as in [START_REF] Muller | Doubly fed induction generator systems for wind turbines[END_REF] adjustable speed operation of the DFIG offers many advantages to reduce cost and has the potential to be built economically at power levels above 1.5 MW for off-shore applications. A typical application, as mentioned earlier, for DFIG is wind turbines, since they operate in a limited speed range of approximately ±30%. Other applications, besides wind turbines, for the DFIG systems are, for example, flywheel energy storage system [START_REF] Hansen | Wind turbine concept market penetration over 10 years[END_REF], stand-alone diesel systems [78], pumped storage power plants [START_REF] Mueller | Design and performance of a 20kW, 100rpm, switched reluctance generator for a direct drive wind energy converter[END_REF]43], or rotating converters feeding a railway grid from a constant frequency public grid [61].

Equivalent Circuit of the Doubly-Fed Induction Generator

The equivalent circuit of the doubly-fed induction generator, with inclusion of the magnetizing losses, can be seen in Fig. 2.11. This equivalent circuit is valid for one equivalent Y phase and for steady-state calculations. In the case that the DFIG is ∆-connected the machine can still be represented by this equivalent Y representation. In this section the jω-method is adopted for calculations. Note, that if the rotor voltage, V r , in Fig. 2.11, is short circuited

+ + - - R s jω 1 L sλ jω 1 L m R m R r /s jω 1 L rλ I s I r V s I Rm V r s
V s = R s I s + jω 1 L sλ I s + jω 1 L m (I s + I r + I Rm ) (2.7) V r s = R r s I r + jω 1 L rλ I r + jω 1 L m (I s + I r + I Rm ) (2.8) 0 = R m I Rm + jω 1 L m (I s + I r + I Rm ) (2.9)
Fig. 12. Torque-speed characteristic of a DFIG [START_REF] Petersson | Analysis, modeling and control of doubly-fed induction generators for wind turbines[END_REF].

Permanent Magnet Synchronous Generator (PMSG)

Permanent magnet synchronous generators (PMSG)s consists of a rotor and a threephase stator similar to an induction generator, is shown in Fig. 1(c), are most capable of competing with induction generators for the wind power applications. In fact, they are adopted by well-known small wind turbine (Fig. 13). These generators have a number of advantages, which are:

(a) Simple and more effective configuration in the rotor with permanent magnet. (b) Overall weight and volume significantly reduced for a given output power (high power density).

(c) Higher efficiency and self excited.

(d) Heat efficiently dissipated to surroundings. In the PMSG the air-gap flux density can be increased by increasing the thickness of the magnets. The air-gap flux density in the PMSG machine is approximated by the following equation [START_REF] Duan | Multi-objective design optimization of surface mount permanent magnet machine with particle swarm intelligence[END_REF]:

= µ + * * 1 leak m r r carter m k B B g k h ( 6
)
Where: g : minimum air-gap length, µ r : relative recoil permeability for the magnet. For NdFe35, it's 1.09981. leak k : leakage factor. Typically in the range of 0.9 to 1.0 for surface-mounted PM machine. Although the PM machine can achieve a larger air-gap flux density and the stator bore diameter is smaller. Moreover, the advantage of the PM machine design is that it has a much higher efficiency (97 %) than the induction machine (85 %).

Due to their excellent performance especially, efficiency and reliability, the general trend in wind industry is to go for higher powers, which is especially relevant with harsh environment. it has become more and more popular during this year's. where several companies have been tried this concept: Jeumont (0.75 MW), Vensys (1.5 MW), Leitner (1.5 MW), Harakosan (2 MW), Mitsubishi (2 MW), Siemens (3.6 MW), and TheSwitch (4.25 MW) [START_REF] Duan | Multi-objective design optimization of surface mount permanent magnet machine with particle swarm intelligence[END_REF]. In [START_REF] Baroudi | A review of power converter topologies for wind generators[END_REF] work, there are summary of commercial wind turbine with synchronous machine (Large than 1MW).

For illustration, Fig. 13 shows industrial PMSG 1650-6300 kW, 11-17 rpm. TheSwitch company wind power generators [29]. Many authors examine the benefits and the physical and economic limitations of PMSGs and consider their appropriateness as a key piece in the overall wind turbine system design [START_REF] Zhang | Review of modeling methods in electromagnetic and thermal design of permanent magnet generators for wind turbines[END_REF][29][START_REF] Scott Semken | Direct-drive permanent magnet generators for high-power wind turbines: benefits and limiting factors[END_REF][START_REF] Baroudi | A review of power converter topologies for wind generators[END_REF][START_REF] Yang | Permanent magnet generator design and control for large wind turbines[END_REF][START_REF] Chan | Permanent-magnet machines for distributed power generation: A review[END_REF]. The reported results are promising.

From analysis of the commercially available wind turbine generators, the elimination of the gearboxes and the power electronic converters will significantly increase the system reliability. The overall system efficiency will increase because the losses in the gearbox and power electronic converters are eliminated, it is concluded that direct drive, grid connected generators indicate a future trend in the wind generation. Finally, the reported results are promising, however, the requirement of PM materials restricts the applications of PMSG, either for high cost or potential to demagnetization in harsh environment.

Switched Reluctance Generator (SRG)

An innovative Switched Reluctance Generators are gaining much interest and are recognized to have potential for wind power applications, the structure of a 6/4 SRG, as shown in Fig. 1 (d). These generators have the definite advantages of simple and rugged construction, fault-tolerant operation, simple control, and outstanding torque-speed characteristics (Fig. 14). SRG can inherently operate with extremely long constant power range. There are, however, several disadvantages, which for many applications outweigh the advantages. Among these disadvantages, acoustic noise generation, torque ripple, special converter topology, excessive bus current ripple, electromagnetic interference (EMI) noise generation. All of the above advantages as well as the disadvantages are quite critical for wind power applications. Nevertheless, SRG is a solution that is actually envisaged in the future for wind energy applications (Fig. 15) [START_REF] Mueller | Design and performance of a 20kW, 100rpm, switched reluctance generator for a direct drive wind energy converter[END_REF]. Jebaseeli has highlighted the performances of the machine for this particular application in [START_REF] Jebaseeli | Performance Analysis of various configurations of switched reluctance machine for wind energy applications[END_REF]. This paper has presented simulate process of the SRG with three different winding constructions in advanced variable-speed wind turbines, showing the SRG systems offer improved efficiency and reduced cost over the conventional generator. H. Chen has proven that the wind turbine generating system can be easily controlled and implemented with SRG, which has the control simplicity. In addition, the SRG can offer high efficiency over a very wide speed range with a rotor structure that is compatible with high speeds and tolerant of extreme environments [START_REF] Torrey | Switched reluctance generators and their control[END_REF], [START_REF] Chen | Research on the switched reluctance wind generator system[END_REF]. In an industrial point of view, the application of the SRM in some of the industrialist and companies is very limited, where many of the applications are categorized as low-, medium-, and high-power and high speed drives for rotary motor drives. Some emerging applications and underwater applications are given in [START_REF] Krishnan | Switched Reluctance Motor Drives: Modeling, Simulation, Analysis, Design, and Applications[END_REF]. Finally, if reluctance machines are combined with direct drive features, the machine would be extremely large and heavy, making them less favorable in wind power applications.

Industrial applications

Table 1 briefly reviews the top manufacturers recently adopted in the wind power industry. Other examples corresponding more to the generators for wind power concepts are given in [START_REF] Cheng | Review of the wind energy generating system[END_REF], [START_REF] Jebaseeli | Performance Analysis of various configurations of switched reluctance machine for wind energy applications[END_REF], [29], [START_REF] Duan | Present and future trends in wind turbine generator designs[END_REF][START_REF] Polinder | Trends in wind turbine generator systems[END_REF][38][39][40][41][42][43][44][45][46].

Preliminary conclusion and perspectives

The permanent magnet synchronous generator seems to be the most adapted candidate for the wind power systems. In fact, this solution is a consensual one as illustrated by the evaluation summarized in Table 2 and based on the main characteristics of the wind power systems, each of them is graded from 1 to 5 points, where 5 points means the best. In our comparative study we have implicitly given the same weight to all the characteristic factors so as to cover a wide range of wind turbine applications, Otherwise, some of these factors should be weighted according to the application.

For example, in [START_REF] Tiegna | Overview of high power wind turbine generators[END_REF], the generator choice is determined by four criteria: reliability and ease of maintenance; total weight of the nacelle; energy yield and grid integration issues and cost. Another example is nacelle in the harsh and variable environment where reliability and technological maturity are much more important than efficiency. In some cases, this could lead to another classification.

However, among the mentioned wind generator features, the torque density and energy efficiency are the two basic characteristics that are influenced by wind turbine dynamics and system architecture. Therefore, the selection of the electric generation system for wind energy demands special attention on these two characteristics. From this analysis, a conclusion that should be drawn is that permanent magnet synchronous generators are an alternative (Table 2). This is why competition remains hard between the DF induction and PM generators. In this context, some manufacturers try to combine the advantages of these two machines.

Recently, an innovative switched reluctance generator is gaining much interest and is recognized to have potential for wind power applications [START_REF] Lebsir | Comparative study of PMSM and SRM capabilities[END_REF][START_REF] Chan | Implementation of a three-phase switched reluctance generator system for wind power applications[END_REF][START_REF] Koreboina | Modelling and simulation of switched reluctance generator control for variable speed wind energy conversion systems[END_REF]. Other researchers optimize their control and characteristics in this domain. The developed generator can produce excellent performance without using permanent magnet and copper materials, which means reduced manufacturing costs and operation at higher temperatures and higher speed.

Conclusion

In this paper, potential candidates of wind power applications have been presented and evaluated according to major requirements of a wind energy system. The comparative study has revealed that the permanent magnet synchronous generator is the solution that makes the consensus even if competition remains hard with doubly-fed induction generator. Moreover, in this paper there was an innovative generator that is gaining much interest and is recognized to have potential for wind power applications in the near future, whilst cage induction generators are dropping.

Fig. 2 .

 2 Fig. 2. HAWT and VAWT configurations [8-9].

Fig. 3 .

 3 Fig. 3. HAWT subsystems [8-9].

S

  is the surface active of the turbine blade [ 2 m conversion factor for the wind turbine, p C varies with λ the tip speed ratio, which connects the wind speed with the rotor speed as given by equations (2) its behavior in modern turbines is represented inFig 5.

  [ m ] is the blade radius, W is the rotational rotor speed [ / rad s ].

Fig. 5 .

 5 Fig. 5. Power coefficient Cp for different wind turbines [10].

  Fig. 7. Scheme of a SCIG wind generating system [5].

Fig. 10 .

 10 Fig. 10. Industrial DFIG for wind turbine of the REpower manufacturer [21].

Fig. 11 .Fig. 2 .

 112 Fig. 11. Scheme of a DFIG wind generating system [5].

Fig. 2 .

 2 Fig. 2.11. Equivalent circuit of the DFIG.the equivalent circuit for the DFIG becomes the ordinary equivalent circuit for a cage-bar induction machine. Applying Kirchhoff's voltage law to the circuit in Fig.2.11 yields[87]

  flux density for the magnet. For NdFe35, it's 1.23 T.

:

  Carter coefficient. Typically in the range of 1.0 to 1.1 for surface-mounted PM machine.

Fig. 13 .

 13 Fig. 13. Industrial permanent magnet synchronous generator.

Fig. 14 .

 14 Fig. 14. Typical torque-speed characteristic of a switched reluctance motor.

Fig. 15 .

 15 Fig. 15. Stator and rotor of the 20kW switched reluctance machine.

Table 1 .

 1 The top wind turbine manufacturers by market.

Table 2 .

 2 Wind power systems evaluation.


Vincent Buso

Iván González-Díaz

Jenny Benois-Pineau

OBJECT RECOGNITION WITH TOP-DOWN VISUAL ATTENTION MODELING FOR BEHAVIORAL STUDIES

Keywords: Saliency Maps, Object Recognition, Egocentric Vision, Vision Modelling, Image Processing, Video Processing

Behavioural analysis in instrumental activities of daily living has become a powerful tool in clinical studies and rises the question of what objects are manipulated by patients. In this paper we present a top-down probabilistic visual attention model for manipulated object recognition in egocentric video content. Although arms often occlude objects and are usually seen as a burden for many vision systems, they become an asset in our approach, as we extract both global and local features describing their geometric layout and pose, as well as the objects being manipulated. We integrate this information in a probabilistic generative model, provide update equations that automatically compute the model parameters optimizing the likelihood of the data, and design a method to generate maps of visual attention that are later used in an object-recognition framework. This taskdriven assessment reveals that the proposed method outperforms the state of the art in object recognition for egocentric video content.

INTRODUCTION AND PREVIOUS WORK

Egocentric videos recorded by a mono or stereo camera worn by patients are more and more in focus nowadays since they provide a close-up view on actions and allow for an efficient analysis of objects manipulation [START_REF] Borj | Stereovision and augmented reality for closed-loop control of grasping in hand prostheses[END_REF]. Behavioural studies of subjects executing simple Instrumental Activities of Daily Living (IADL) are necessary in different clinical applications that is why the mobile of our research is the recognition of manipulated objects in egocentric video scenes for behavioural studies of patients with dementia. Indeed as studies show [START_REF] Pirsiavash | Detecting activities of daily living in first-person camera views[END_REF], the analysis of behaviour in IADL performance rises the question of what object is manipulated by the patient.

In our application scenario of egocentric videos recorded by a patient with a body worn camera, objects in the focus-of-attention of an observer analysing the visual scene (such as the medical doctor) show very clear correspondence with the objects being manipulated. However, the measurement of the subject fixations with eye-tracker devices is costly that is why the point of view of the medical doctor has to be modelled. Indeed, efficient ways of visual attention prediction have to be proposed for egocentric scenes.

Since the pioneering work of Itti [START_REF] Itti | A model of saliency-based visual attention for rapid scene analysis[END_REF], modelling of visual attention for scene interpretation has become a very popular subject in image and video analysis. The so-called "predicted visual attention maps" used in feature selection and pooling [START_REF] Fathi | Learning to recognize daily actions using gaze[END_REF][START_REF] Ogaki | Coupling eye-motion and ego-motion features for first-person activity recognition[END_REF] in the problem of object recognition, have become a good competitor for heavy sliding window methods such as Deformable Part-Based Models (DPM) [START_REF] Felzenszwalb | Object detection with discriminatively trained part-based models[END_REF]. In all these and many other saliency approaches, the predicted visual attention maps are built accordingly to a "bottomup" approach, which simulates sensitivity of Human Visual System(SVH) to colours, contrasts, orientation, residual motion [START_REF] Brouard | Cartes de Saillance Spatio-Temporelle basées Contrastes de Couleur et Mouvement Relatif[END_REF], [START_REF] Borj | State-of-the-art in visual attention modeling[END_REF], and often incorporates the center-bias hypothesis, which means attraction of human gaze by the center of a frame [START_REF] Tatler | The central fixation bias in scene viewing: Selecting an optimal viewing position independently of motor biases and image feature distributions[END_REF]. In this paper on the contrary, we propose a top-down attention prediction knowing that the target of attention are manipulated objects.

The introduction of top-down factors into the classical bottomup framework by extracting semantic clues (e.g., face, speech and music, camera, motion) was proven to provide impressive results [START_REF] Ma | A generic framework of user attention model and its application in video summarization[END_REF][START_REF] Cerf | Predicting human gaze using low-level saliency combined with face detection[END_REF]. More recent works using machine learning approaches to learn top-down behaviours based on eye-fixation or annotated salient regions, have also proven to be very useful for static images [START_REF] Gao | Discriminant saliency, the detection of suspicious coincidences, and applications to visual recognition[END_REF][START_REF] Kanan | Sun: Top-down saliency using natural statistics[END_REF][START_REF] Torralba | Contextual guidance of eye movements and attention in real-world scenes: the role of global features in object search[END_REF] as well as videos [START_REF] Itti | Computational modelling of visual attention[END_REF][START_REF] Li | Probabilistic multitask learning for visual saliency estimation in video[END_REF]. However, in practice, most videos contain many different attractors. Therefore, with such methods, it is impossible to generate a category-agnostic detector of the object of interest as they are based on a fixed set of pre-defined object categories.

In this paper we propose to use domain specific knowledge to present a new method for saliency maps computation based only on top-down components for egocentric video content. One of the main particularity of egocentric videos is the presence of hands and arms. Actors performing activities of daily living are indeed most-likely manipulating objects bare-handed. Hands are a commonly known burden in egocentric videos as they often obstruct objects resulting in a more challenging recognition process. However, our work consider hands and arms as assets for building top-down-only visual attention maps and without prior training of a fixed number of high semantic cues such as object categories [START_REF] Itti | Computational modelling of visual attention[END_REF][START_REF] Li | Probabilistic multitask learning for visual saliency estimation in video[END_REF]. Thus, our saliency maps aim to measure the likelihood of pixels in the vicinity of hands to belong to a manipulated object. This function depends on the hands relative position with regard to the camera, their relative position with regard to each other, and the particular (learned) pose of manipulated objects under each hands configuration. This map is then used for psycho-visual weighting of frames signature in a supervised learning framework for object recognition.

In this work we completely remodel the approach firstly described in [START_REF] Buso | Goaloriented top-down probabilistic visual attention model for recognition of manipulated objects in egocentric videos[END_REF] with the following novel contributions:

• We totally avoid any segmentation of foreground objects, but instead require some manually annotated bounding boxes around objects of interest present in the training set. • In contrast to our previous developments, in which the values of the model parameters were either set based on previous intuitions or cross-validated, here we propose an integrated data-driven learning framework that uses the Expectation-Maximization (EM) algorithm to automatically estimate parameters optimizing the likelihood of the data. The rest of the paper is organized as follows: in Section 2 we present Goal-Oriented Top-down visual attention model construction. In Section 3 we summarize the object recognition approach with predicted visual attention maps. Experiments, benchmarking and results are presented in Section 4. Section 5 concludes the paper and proposes perspectives of this work.

GOAL-ORIENTED TOP-DOWN VISUAL ATTENTION MODEL

In this section we present our model of visual attention prediction for the task of manipulated object recognition. It relies on extraction of the arms/hands automatically fulfilled for each frame using the approach introduced by Fathi et al. [START_REF] Fathi | Learning to recognize objects in egocentric activities[END_REF].

Defining global and local features

We propose to build our attention model as a combination of two distinct sets of features. The first one is a set of global features that describe the geometric configuration of the segmented arms. The features are then clustered into a pre-defined number of states/configurations. Prior knowledge on saliency is then generated for each of these global states. The second one is based on local features which will allow to adaptively relocate these saliency priors.

We introduce global features based on the geometry of arms in the camera field of view correlated with manipulated object size and position. Each arm, from elbow to the hand extremity, is approximated by an elliptic region in the image plane. Hence an ellipse is first fitted to each segmented arm area and, then, the six following global features are extracted:

• Relative location of hands: The magnitude ρ Rel and phase ϕ Rel between hands centres are extracted (see Fig. • Left arm depth and Right arm depth: an object size is correlated with the "depth" of the arms. In this work, the bodyworn cameras do not provide a real depth information. A trivial approximate of the "depth" of an arm is the minor axis length dL and dR) of the fitted ellipse (see figure 1(c)).

A vector g = (ρ Rel , ϕ Rel , ϕL, ϕR, dL, dR) T containing these six geometrical features is computed for each image in the training set. Then, the whole set is clustered into K global appearance models z k , k = 1..K using k-means [START_REF] Lloyd | Least squares quantization in pcm[END_REF]. The aim of this pre-processing stage is to use the models to initialize our probabilistic approach (to be described in the next section). It is worth noting that a Z-score normalization has been performed over the data, in order to prevent out-weighting features with large range over attributes with small ones [START_REF] Al | Normalization as a preprocessing engine for data mining and the approach of preference matrix[END_REF].

Furthermore, we consider some "local" features that help to compute a saliency distribution given by the global arm configuration of a frame. These features are the coordinates of hand centres c, the hand indicator h (left or right), and the candidate pixels x around the hand to belong to the object being manipulated.

A Probabilistic Model for Top-down Visual Attention Prediction

As a human observer would be attracted by hand-manipulated objects, we consider the joint locations of arms/hands and objects as predictors of top-down visual attention. Hence our probabilistic model for top-down visual attention incorporates distributions of both global and local features presented in the previous section. The graphical model of our approach is shown in Fig. 2. Based on this, given a corpus of D training images our objective is, for each image d, to learn the process that chooses a set of N d salient spatial locations x.

To do so, the generative process first randomly picks a global arm model z k from the K candidates. K corresponds to the number of clusters as defined in section 2.1, and remains an open parameter in our model. Then, depending on the selected global model z k , global (g) and local (x, c, h) features are drawn from the particular conditional distributions p(g|z k ) and p(x, c, h|z k ), respectively. Here, h is an index variable with two possible values h = 0, 1 for left and right hands, respectively

In the following paragraphs we will first introduce the distributions modelling both global and local features. Then we will combine them together to build the expression of the corpus likelihood. Next, we will describe the learning process and the update equations that allow us to obtain the optimal model parameters maximizing this likelihood. Finally, we will describe how this model is used to generate saliency maps from data.

Distributions of Global Features: We define the conditional distribution that models the global features given the component z k with a Gaussian pdf p(g|z k ) = N (g; µ g k , Σ g k ), with mean vector µ g k and covariance matrix Σ g k . Distributions of Local Features: Concerning the local features, for each elementary arms model z k , we draw N d points at spatial locations x considered as salient. For that end, we start by picking a hand (left or right) following the distribution p(h|z k ). Next, once the hand is chosen, we randomly locate its centre by drawing its coordinates c using the distribution p(c|h, z k ). Finally, we use the conditional distribution p(x|h, c, z k ) to randomly choose a spatial location x that belongs to the object being manipulated. This distribution models the probability of a pixel to belong to the object being manipulated given the current geometric configuration of arms and hands.

Putting everything together and marginalizing over the variable h, we can expand the distribution involving the local features:

p(x, c, h|z k ) = 1 j=0 p(hj|z k )p(c|hj, z k )p(x|hj, c, z k ) (1)
Now, we can define the particular conditional distributions that model each variable:

1. The selected hand is given by a discrete distribution p(hj|z k ) = α jk , with 1 j=0 α jk = 1. 2. The hand centre c follows a Gaussian distribution p(c|hj, z k ) = N (c; µ c jk , Σ c jk ). 3. The spatial location x is defined with an experimental discrete distribution: p(xi|hj, c, z k ) = β kji , so that L 2 i=0 β kji = 1. This distribution is defined over a square 2D box of size LxL centred at c built by superimposing all accordingly-centred annotated objects from images belonging to the cluster z k . In Fig. 3 we show some empirical examples of this distribution.

Log-Likelihood: From the graph depicted in Fig. 2, the likelihood of the corpus given the model parameters: θ = {π, µ g , Σ g , α, µ c , Σ c , β} can be defined by means of a mixture of K components:

L = p(x, g|θ) = D d=1 p(z d )p(g d |z d ) N d i=1 p(xi, ci, hi|z d ) (2)
where p(z k ) = π k is discrete with parameter π k and stands for the prior distribution of the global arm models (weights of components in the mixture).

If we marginalize over the latent arm models we get:

L = D d=1 K k=1 p(z k )p(g d |z k ) N d i=1 p(xi, ci, hi|z k ) (3)
Taking logarithms and applying the Jensen's inequality one can obtain a lower bound of the log-likelihood:

log L ≥ D,K d,k φ dk log p(z k )p(g d |z k ) • N d i=1 p(xi, ci, hi|z k ) -log φ dk ( 4
)
where we have introduced a new variable φ dk = p(z k |g d , x) which stands for the posterior distribution of the arms model given the observed variables and obeys k φ dk = 1.

In addition, we can also lower-bound the term of log-likelihood related to the local features if we apply again the Jensen's inequality:

log L local = dki φ dk log 1 j=0 p(hj|z k )p(c|hj, z k )p(xi|hj, c, z k ) ≥ dkij γ dkij [log p(hj|z k )p(c|hj, z k )p(xi|hj, c, z k ) -log γ dkij ] (5)
where γ dkij = p(hj|z k , c, xi) is the posterior distribution of the selected hand once the global model, the center and the spatial location are known.

Inference: We aim to learn the set of optimal model parameters θ = {π, µ g , Σ g , α, µ c , Σ c , β} that maximize the log-likelihood. For that end, we have used the Expectation-Maximization (EM). Due to the lack of space, we omit the algebra to obtain the EM update equations.

In the E-Step, the algorithm computes the expected values of the posterior distributions φ dk , γ dkij :

φ dk ∝ p(z k )p(g d |z k ) N d i=1 p(xi, ci, hi|z k ) (6)
γ dkij ∝ p(hj|z k )p(c|hj, z k )p(xi|hj, c, z k ) (7)
In the M-Step, our algorithm updates the values of the model parameters:

π k = 1 D d φ dk (8)
µ g k ∝ d φ dk g d (9)
Σ g k ∝ d φ dk (g d -µ g k )(g d -µ g k ) T (10)
α jk ∝ di φ dk γ dkij (11)
µ c jk ∝ d φ dk c dj i γ dkij (12)
Σ c jk ∝ d φ dk (c dj -µ c jk )(c dj -µ c jk ) T i γ dkij (13)
β kji ∝ d φ dk γ dkij (14)
Building Saliency Maps: Once the optimal parameters have been learned, we can build a saliency map by measuring the saliency of every pixel location. For that end, the saliency value of a pixel S(x) can be defined as its likelihood over the proposed generative model for saliency S(x) = p(xi, g|θ) Fig. 4: Object recognition performances between different paradigms. The results are given in average precision per category and averaged.

OBJECT RECOGNITION APPROACH

In our previous work [START_REF] González Díaz | Modeling instrumental activities of daily living in egocentric vision as sequences of active objects and context for alzheimer disease research[END_REF] we have proposed an object recognition approach in the family of methods which use psycho-visual weighting [START_REF] De Carvalho Soares | Spatial locality weighting of features using saliency map with a bagof-visual-words approach[END_REF] of the conventional Bag-of-Visual-Words (BoVW) [START_REF] Sivic | Video google : A text retrieval approach to object matching in videos[END_REF], [START_REF] Csurka | Visual categorization with bags of keypoints[END_REF] paradigm. Here we compute SURF descriptors [START_REF] Bay | Speeded-up robust features (surf)[END_REF] on a dense grid, then a visual dictionary is built from all descriptors extracted in the training video database. When computing the signatures, the contribution of each descriptor to the histogram is weighted by the maximal predicted saliency value in a small circular patch around it. Once each image is represented by its weighted histogram of visual words, an SVM classifier [START_REF] Cortes | Support-vector networks[END_REF] is used with χ 2 kernel.

EXPERIMENTS AND RESULTS

In the context of this study, developing a model of top-down visual saliency aims at improving object recognition performances in patient-worn egocentric videos. The proposed approach is compared to other well-known paradigms for object recognition applied to this video content.

Dataset and setup

The experiments were conducted on the GTEA dataset [START_REF] Fathi | Learning to recognize objects in egocentric activities[END_REF] since it is a publicly available database of egocentric videos of 4 subjects performing 7 types of instrumental activities of daily living. The segmentations of arms are provided for 17 videos, a subset of which is used for training our distributions. In [START_REF] Fathi | Learning to recognize objects in egocentric activities[END_REF] the frames were annotated with the objects of interest but we manually extend this annotation by drawing bounding boxes on them. The dictionary size for computing the visual dictionary in BoVW was fixed at 4000. Finally, we set the number of global configurations K to the one which gave the maximum recognition performances with cross validation over the training set, that is to say K = 70.

Object recognition performances

In Fig. 4 we present a comparison of object recognition performances between our approach (denoted as 'Ours'), the reference BoVW scheme, and a well-known technique considered as State-ofthe-art in egocentric vision: the discriminatively-trained Deformable Part Model (DPM) [START_REF] Felzenszwalb | Object detection with discriminatively trained part-based models[END_REF], a sliding window technique that has been taken as the object detection paradigm by the authors of the ADL dataset [START_REF] Pirsiavash | Detecting activities of daily living in first-person camera views[END_REF]. Our method outperforms these two famous paradigms for object recognition by achieving absolute improvements of mean Average Precision (mAP) of 13% over BoVW and 10.7% over DPM. Here the "ideal" case BoVW with BB is added for the upper bound estimate. It is a BoVW scheme where descriptors were extracted only in manually annotated bounding boxes around objects of interest. We consider these bounding boxes as "ideal" saliency Table 1: Mean Average Precision (mAP) Results over all categories for all considered object recognition paradigms maps. As can be seen from the mAP score (last set of bars in figure 4, reported in table 1 for clarity), our method not only outperforms any other in this kind of video content but also achieves very close performances to the "ideal" case. All the improvements were backed up by performing Student's t-tests with significance level of 0.05.

CONCLUSIONS AND PERSPECTIVES

In this paper we have presented a top-down task-driven visual attention model for the goal of object recognition in egocentric videos. In this kind of video content we aim at facilitating behavioural studies of activities of daily living for clinical scenarios. However we believe this model of saliency could extend to any scenario willing to detect manipulated objects in egocentric videos, especially with the current development of wearable cameras such as GoPro. Our method only requires prior input from a hand detector [START_REF] Li | Pixel-Level Hand Detection in Ego-centric Videos[END_REF], and annotations of objects in the training set, after that it becomes fully automatic and, as presented in section 4.2, outperforms state of the art results in the GTEA egocentric dataset (close to what we defined as the "ideal" method requiring user inputs). Our main perspective now is to make this visual attention model auto-dependent by adapting and incorporating a state of the art Hand detection scheme in order to review its performances on different publicly available egocentric datasets.

Fig. 1 :

 1 Fig. 1: Illustrations of the 6 global features. 1(a): Relative location of hands, 1(b): Left and right arm orientations, 1(c): Left arm depth and Right arm depth with regard to the camera.

  1(a)) since they are strong indicators of the objects width and holding pose, respectively. • Left arm orientation and Right arm orientation: As illustrated on Fig. 1(b) the orientation of each arm ( ϕL and ϕR) is extracted. The arms are mostly oriented depending on the objects being manipulated.

Fig. 2 :

 2 Fig. 2: Graphical model of our approach Top-down visual attention modelling with manipulated objects. Nodes represent random variables (observed-shaded, latent-unshaded), edges show dependencies among variables, and boxes refer to different instances of the same variable.

Fig. 3 :

 3 Fig. 3: Two examples of the obtained experimental distributions p(xi|hj, c, z k ). Left column: arms segmentation representing the global model. Middle column: left hand distribution. Right column: right hand distribution.

Fig. 5 :

 5 Fig. 5: Illustration of saliency maps obtained with our model as a heat map in alpha-blending with the original frame. The The top left corner shows the corresponding arm segmentation. Simple BoVW DPM Ours BoVW with BB mAP 0.262 0.285 0.392 0.425

ACKNOWLEDGEMENTS

This research is supported by the EU FP7 PI Dem@Care project #288199.