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SEAREV: A FULLY INTEGRATED WAVE ENERGY CONVERTER

Aurlien Babarit(1), Alain Clment(1), Jacques Ruer(2), Catherine Tartivel (2)

(1) Laboratoire de Mcanique des Fluides, CNRS UMR6598 ,Ecole Centrale de Nantes , 1, Rue de la Noe, 44300Nantes France : [email protected]

(2) SAIPEM SA, 1 Av. San Fernando, 78884 St Quentin en Yvelinnes Cedex France : [email protected]

ABSTRACT

A new wave energy converter (WEC), called SEAREV1, is presented. The mathematical models and numerical simulatorsthat have been developped, both in frequency and time domains, to assess the electricity production on a given test site aredescribed. Genetic algorithm using these numerical models have then been implemented in order to optimize the shape andthe mechanical parameters of the device. A discrete real-time control of bang-bang type (latching control) permits toenhance the response of the system to the waves, and therefore the annual production of energy.

INTRODUCTION

The SEAREV wave energy converter is afloating device enclosing a heavy horizontal axiswheel serving as an internal gravity reference(see Fig.2). The centre of gravity of the wheelbeing off-centered, this component behavesmechanically like a pendulum. The rotationalmotion of this pendular wheel relative to the hullactivates an hydraulic Power Take Off (PTO)which, in turn, set an electric generator intomotion. Two major advantages of thisarrangement are that, first: all the moving parts(mechanic, hydraulic, electric, components) aresheltered from the action of the sea inside aclosed, waterproof shell; and secundly that thechoice of a wheel working as a penduluminvolve neither endstop nor any security systemlimiting the stroke.

This freely floating device is kept on site by aslack single line mooring which enable self alignment of the device in the dominant wave direction due to the general shapeof the hull. This allows the device complying easily with tide

The system being based on the strong coupling of two nonlinear mechanical oscillators, an analytical study of its behavior inregular and irregular waves has been performed. A complete mechanical model of the system has been written, includinghydrodynamics of wave device interaction and nonlinear mechanics of the internal PTO. Only the linear version of the

equations will be presented here. We will show how the shape and all the mechanical parameters of the system have beenoptimized using a general purpose multi-parameter multi-criterion software, based on genetic algorithms.

A specific real time control has been developed in order to enhance the production in low energetic sea states. Numericaltime-domain simulations are presented to exemplify the benefit that latching control can bring to the system.

MATHEMATICAL MODELLING

The floating body is assumed to have two vertical planes of symmetry. We assume that the rotation axis of the internalcylinder is perpendicular to the main symmetry plane of the floating body. We suppose also the direction of wavepropagation being parallel to this plane (self alignement).

Under these assumptions, in a 2D incident wave train, the floating body will move only in surge, heave and pitch mode in the(xOz) plane (Fig.2).

1Systme lectrique Autonome de Rcupration dnergie des Vagues

Figure 1 The SEAREV device: external view


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LetGx the surge motion, Gz the heave motion of the centre of gravity G of the floating body, the pitch motion and

the relative motion between the floating body and the inner pendulum. The Power Take Off system (PTO) then works this

relative motion . Let bm the floating body mass, bI its

inertia ,pm the mass of the inner cylindrical pendulum, yI

its moment of inertia. Let APl = the distance between thecenter of rotationA of the inner pendulum and its own centerof gravity P, GAd= the distance between the center ofgravity of the floating body and the center of rotation of thependulum.

Wave amplitude and resulting body motion are assumed tobe sufficiently small to justify working in the classical linearpotential flow theory.

Let us define the four components position vector of the

system ( )tGG zx ,,,=Z , and the matrices M , B andK as:

+

+

=

0

0)(

0

0

lm

ldm

mm

mm

p

p

pb

pb

M

++

+++

22

22)(

00

)(

lp

my

Ildp

mlp

my

I

ldp

mlp

my

Ildp

my

Ib

I

lp

mldp

m

(1)

=

glmglm

glmldgm

k

pp

pp

x

00

)(00

0000

000

K (2)

=

PTO

B000

0000

0000

0000

B (3)

where M is the proper mass and inertia matrix of the system, K the restoring matrix, and B the external damping matrixwhere the only non zero term is a linear term representing the moment exerted by the PTO system proportional to the

rotational velocity, namely: )(tBPTO& .

Frequency domain model

Let us first consider the case of a regular wave train with circular frequency exciting the system. Adding the couplingterms between the hull and the cylinder in the classical floating body motion equations, one can derive easily the equationsof the system under the matrix form:

[ ] )()()()( 2 exHAM i FZKKCBCM =+++++ (4)

where:

exF is the hydrodynamical excitation vector

Figure 2 notations


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)(MC and )(AC the added mass and wave radiation damping matrices, both function of the hull geometryand wave frequency

HK the hydrostatic restoring matrix

For a typical SEAREV device as shown on Fig.1 (25m length, 15m width, 14m depth, 1000T displacement, 400T pendulummass), the Response Amplitude Operators (RAO) are shown in Fig.3;

Computations of the hydrodynamic coeficients )(MC and )(AC have been performed using a standard seakeepingcomputer code. (AQUADYN. Delhommeau et al. 1992).

The double-peaked curve of pitch RAO on Fig.3 illustrates the advantage of such a multiple resonance system. We are ableto tune the different natural frequencies, (mainly pitch of the floater and pendular wheel rotation) to broaden the bandwidthof the composite system with regard to the wave spectrum.

In this frequency domain approach, the mean power absorbed by the system is simply given by:2

)(2

1)( PTOBp = (5)

where )( is derived from the solution of algebraic system (4).

In figure (4), the function 2/)( Ap is plotted for the samedesign as for the RAO above. For the sake of comparison, wehave also plotted on this figure the theoretical maximum powerabsorbed by the body. Since the SEAREV device works inpitch and surge modes, it can be shown that the maximumabsorbed energy at frequency is given by (e.g Falnes 2002a):

242

2

max

gp = (6)

which is twice the maximum energy absorbed by a heavingpoint aborber.

Time-domain linear model

In the time domain waves are no longer assumed to be regular,and non linear components of forces (like hydraulic PTO for instance) can be introduced. Hydrodynamics will still beconsidered as linear here, implying moderate motions of the hull.

The time domain counterpart of eq.(4) is :

Figure 3: RAOs of the SEAREV device

left: pitch and wheel motion , right: surge

Figure 4 absorbed power in regular waves

for a typical SEAREV device


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)()()( ttexM

ZBFZCM &&& =+ ( ) )()()(0

tdt H

t

ZKKZH + & (7)

where the wave radiation forces are accounted for through a convolution product (Cummins 1962). The computation of theimpulse response functions )(tH has been performed by using a dedicated code ACHIL3D (Clment 1999), but could have

been obtained through a Fourier transform of their frequency domain counterpart )(MC and )(AC (Falnes 2002a).

The mean absorbed power during a time interval T will now be given by

dttBT

p

T

PTO )(1

0

2

= & (8)

Given an excitation vector )(texF and initial conditions, the integro-differential equation (7) can be integrated to simulatethe behaviour of the system in irregular waves. But the presence of the convolution integral in this equation is not well suitedto such numerical integration.

Time domain state equation model

We have therefore developped an alternative formulation by approximating the convolution products in (8). Thisidentification, in the sense of system theory, is based on the approximation of the elements of matrix )(tH by series ofexponentials, following a method proposed by R de Prony (1795) and implemented by Duclos et al. 2001.

Let )(tS denotes a scalar convolution integral

=t

dftgtS0

)()()( (9)

and suppose that we are able by using Pronys method to find an approximation of )(tS under the form:

=

N

i

t

i

i

etg 1

)(

)(

(10)

then )(tS can be written:

==

==

N

i

i

N

i

t

t

i tSdfetSi

11 0

)( )()()( (11)

with:

=

t

t

ii dfetSi

0

)( )()( (12)

which satifies : )()()( tftStS iiii +=& (13)

Given the initial conditions 0)0( =iS , )(tS can now be computed by integrating the set of ordinary differential equations(11) (13) rather than computing the integral (9).

This identification technique has been applied to all the coefficients of the matrix )(tH in order to replace the integro-differential eq.(7) by an equivalent state equation. Defining the vector )(tS as:

=T

dtt0

)()()( ZHS & (14)

and assuming that all the components have been approximated by series of exponentials we have the system:


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+=

=

= =

= =

= =

)(0

)(

)(

)(

)(3

1 1 3

3

1 1 2

3

1 1 1

3

2

1

tSS

tS

tS

tS

t

jijrijrijrijr

j

N

r jr

j

N

r jr

j

N

r jr

j

j

j

Z

S

&&

(15)

Introducing a vector of radiation states )(tI such that

( )ijNrji

t

rijSt = 1;31;31)(I

we obtain finally a state equation for the global system

CAXX +=& (16)

with

+

+=

0

)(

00

00

0)(0

001

BKK

I

I

CM

I

A H

d

d

M

d

+=

0

0

00

0)(0

001

ext

d

M

d

F

I

CM

I

C

where the new extended state vector X is defined as [ ]tttt IZZX &= The time-domain numerical simulations of the system as those reported here can now be performed very efficiently byintegrating this system of ODE using a 4th order Runge-Kutta algorithm (see e.g Press et al. 1992). For the simulations inirregular waves, the incident wavetrain was obtained by discretizing the ITTC spectrum (Rawson and Tupper 1983):

4

5)( f

B

ef

AfS

= with4

1

231

165

T

HA = and4

1

145

TB = , where 1T is the peak period of the spectrum (sometimes noted

pT ).

The classical method of random phases is then used to synthetized the irregular wavetrain as a superposition of N regular

waves with amplitudes ffSfa = )(2)(

DEVICE OPTIMIZATION

method

Once validated, the above mathematical model was availableto be used for the optimization of the device. Not only theshape of the hull, but also the mechanical parameters such as:mass and inertia of the pendulum cylinder, position of its axisof rotation, position of the centre of gravity of the floater andof the cylinder,must be determined in order to optimize theannual production of energy. For a given site with a

probability of occurrence of sea state ( 131 ,TH ) given by a

function ),( 131 THC , in the above fully linear approach, theannual production to be optimized reads:

2

1131131 ,,2

()(),(31 1

=

=

N

j

j

j

H T

THapTHCE (18)

Figure 5 Scatter diagram at Yeu Island site


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In the present study we have used a reference site located near Yeu Island, a French Atlantic coast island, where the meanannual incident power is about 23 kW/m. The scatter diagram Fig.5 gives a repartition of the sea states statistics.

So in this optimization, we have to maximize E while, indeed, minimizing a variable representing the cost of the device. Atthis early stage of the study a precise cost function was not yet established, so we have used the total displacement of thedevice as a cost function.

The optimization problem is thus multi objective, and the number of parameters to be varied is large: say between 10 and 20.Furthermore, there are several constraints about the design that must be respected during the optimization. For example, the

static stability of the device must be positive in pitch and roll. The inner pendulum must have a realistic density (we havechosen here the density of concrete). The draught of the floating body must be kept reasonable (15 meters max, say), etc...

For such a global optimization problem, one can imagineusing genetic algorithms which are known to be veryefficient in these circumstances (R. Duvigneau 2002).Genetic algorithms are optimization algorithm which are notproperly speaking deterministic. They attempt to reproducethe Darwinian process of species evolution that can befound in the living world. For our problem, a set ofparameters is used to define a particular shape of thefloating body and the mechanical characteristics of the innerpendulum at PTO; this set constitutes the genome of each

individual. From a set of initial individuals, which forms apopulation, genetic algorithms first make a call to numericalcodes which compute the transfer functions, solving eq.4,and the annual energy produced at the reference site (eq.18)for each individual. Then they select the best ones and crosstheir genome to create a new generation, which ispotentially better than the first one (Darwin's hypothesis).Finally, iterating the process leads to the optimal designs(Fig.6). Moreover, to closer mimic living species evolution,genetic algorithms also introduce mutations from time totime in the genome of the population, in order to enhancethe potential and to enlarge the field of possibilities. Forshape optimization problems, genetic algorithms constitutesthe very best choice, but they are known to converge ratherslowly.

In our optimization, each evaluation (computation of theannual energy produced at the reference site) of an

individual is quite CPU time consuming (about 5 mn/individual on a pentium III 1 GHz PC). The computation of thehydrodynamical coefficients was rapidly identified as the most heavy part of these computations. Once it has beenperformed, the calculation of the absorbed energy afterwards is practically instantaneous. So the optimization process hasbeen split in two sub-process, in order to accelerate the global computation.

The idea here was to compute first the hydrodynamical coefficients for a given hull shape. The shape being given, thedisplacement is also known. To maximize the absorbed energy, one have to find the optimal internal pendulum for this givenhull. To do this sub-optimization, the best way is to use a gradient method. Basically, the method consists in computing the

direction , in the parameter space, in which the energy growth rate is maximum, and to move by a step in this direction.Iterating the process leads to the local maximum closest to the initial point. In practice, the algorithm used to do this is theFletcher-Reeves algorithm (Press et al. 1992). Globally, we have divided by 4 the number of optimization parameters byusing this method.

So, the inner optimization loop computes the optimal pendulum for a given shape of the floating body. Hull shapeoptimization, in the outer loop, is performed by genetic algorithms, using a high performance commercial codeModeFrontier (www.esteco.it).

results

In figure (7), we have plotted an example of results of an optimization run with a population of 40 individuals on 100generations, which represents about 15 days of computation on a pentium III 1GHz PC. The shape of the floating body wasdefined by 15 real parameters. For each optimization run, various constraints were put on shape and mechanical parametersto account for technological or practical criterion. The horizontal axis of the figure corresponds to the total displacement intons, the vertical axis to the mean absorbed power, in kW, at the Yeu island site for this particular run.

Figure 6: genetic algorithm for SEAREV shape optimization


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Here, the best individuals are those which deliver the most energy with the least displacement (under the prescribedconstraints), so the corresponding dots should be located in the left upper part of the plots in figure 7. Naturally, this cornerof the plot is empty, but one can clearly identify a frontier on which optimal individuals tends to accumulate, and abovewhich there will be no solution. This is the so-called Pareto front, which is therefore the locus of the optimal individuals.

The Pareto front that we obtained may be roughly split in two parts. In the first one, for displacements between, say, 0 and500 tons, the absorbed power growing rate is about 0.1 kW/t whereas in the second, for displacement larger than 500 t, it isabout 0.05 kW/t.

In the upper plot in figure (7), the colour field represent the ratiobetween the mass of the pendulum over the total mass of the device.One can notice that on the Pareto front, the ratio of the pendulummass on the floating body mass is about 0.5. In the middle plot ofFig.7, the color indicates the draught T of the device. It appearsclearly that the best designs are those having the largest allowabledraught, which was set to 15 meters in this run. This constraint onthe draught, and all the dimensional constraints on the otherparameters was automatically accounted for by the optimizationsoftware. In the last plot, we show the ratio of the depth of cylindercentre d over the device draught T. Globally it appears that theoptimizer always choose the deeper possible location for the

pendulum cylinder.Several tens of thousand of individuals has been generated andevaluated (i.e full year production at the reference site) in this study.The best design was shown to have a mean annual capture width of4 meters when installed on this 23 kW/m site, resulting in a captureratio of 35% when we reduce the capture width not to the breadth ofthe device, but to the cubic root of its displacement.

LATCHING CONTROL

These results are already promising for a site featuring such amoderate resource level. At this stage, without control, the devicetraps energy only from the wave components whose frequency laysin the band of the natural frequencies of the system, (see Fig.4).Nevertheless, this system is potentially able to catch energy in thewave spectrum outside this band provided it can be dynamicallycontrolled to adapt its bandwidth in real time. We therefore equipped it with a latching control system. Let us first recallhere the principle of this control, introduced by Budal and Falnes in 1980 and 1982 for the control of point absorbers.Condidering a single degree of freedom (DoF) wave energy converter, we introduce a technological system which is able tolock the motion of this productive DoF when its velocity vanishes. While the motion is locked, the wave exciting forcecontinue to evolve. The motion is then released after a while in such a way that during its next oscillation, the velocity andthe excitation force are in phase. (this control is sometimes named: phase control). This kind of control enable to createparametric resonance in the dynamics of the device, and thus can improve its efficiency by a constant adaptation of itsdynamics to the incoming wave.

We have shown in previous publication (A. Babarit et al. 2003b) how it can be applied to a single DoF heaving pointabsorber

regular waves

The problem to be solved to implement latching control in a wave energy device is the in-line determination of the optimalduration of the latching phase. In other word, the system itself says when it should be locked (velocity vanishes), but youhave to decide when to release it to obtain the best performance from its motion afterwards. In regular waves, and providedthe system equation have been expressed in a state equation form as eq.(16), a semi-analytical method has been proposedand validated in A. Babarit et al. 2003b. The method has been applied here to the four degrees of freedom SEAREV device

The potential enhancement brought by latching control to our device is clearly illustrated by Fig.8 where we have plotted theresult of a time-domain simulation of the response of the SEAREV device to a regular incident wavetrain. This was obtained

by integrating eq.16 by a 4th order Runge-Kutta algorithm from the instant 0=t when the body is released in a regularestablished wave field. From 0 to 300 sec, we can see (Fig.8) a period of transients after which the regular oscillating regimeis established. At t=360 s, the latching control is switched on; the pitching motion of the device is practically not affectedwhereas the cylinder motion )(t is dramatically magnified, after a long transient period. The slope of the energy curve

Figure 7 results of an optimization run


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show the gain in term of extracted power. Here the latching delay was determined by the above mentionned semi-analyticalmethod.

The same sequence is reported in Fig.9, but here the optimal response was found to have a period triple of the incident waveperiod. This remarkable non-linear feature was already pointed out for a single DoF system in Babarit et al. 2003a. Themagnification of the productive motion is again very impressive here, and we can now observe the effect of the triple periodmotion of the cylinder on the pitch motion of the hull which exhibits now higher order components. This is due to thecoupling terms between surge pitch and cylinder rotation in mechanical equations (1), (2).

irregular waves

The problem of determining in line the optimal latching delay when waves are irregular, like in the real world, is far moredifficult to solve. The semi-analytical method used in regular waves no longer works here, and alternative approaches shouldbe considered.

Using non-causal approaches, we were able to assess different latching strategies for a single DoF system (heaving pointabsorber) in Babarit et al. 2003a, 2003b.

In order to extend this to the SEAREV device, we have implemented a method of optimal command (Hoskin and Nichols1986) based on Pontryagin maximum principle. Let u be the control variable. We will consider then that the latching of thesystem is performed by adding an very large extra damping term of the form

&)(tGuF = (19)

in the RHS of system (16), now written in the general form

),,( ut XfX =& (20)

Ideally, G should be infinite to obtain a perfect and instantaneous latching. Practically, for numerical stability arguments, it

is given a finite value, very large compared to the PTO damping.

The optimal command problem can then be formulated as follows. Given a long sequence [ ]T,0 of an incident irregularwavetrain, find the sequence )(tu over the interval [ ]T,0 which maximizes the integral of the absorbed power:

Figure 8 Switching latching control on at t=360s in

regular waves

Figure 9 Switching latching control on at t=360s in s;

period tripling of the pendulum response


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dtBuET

PTOu =

0

2)(max & (21)

A Hamiltonian H is then defined: f t

PTOBH +=2& , with )(t

an adjoint state vector, solution of the differential

equation

),,(

XH

uX=& (22)

with the end condition 0)( =T

. If )(* tu is the optimal sequence, the maximum principle states that is should maximize

the Hamiltonian at each time step. This problem (of finding )(* tu given the irregular wave field over [ ]T,0 ) is solvediteratively, integrating alternatively (20) and (22) in, respectively, direct and reverse time direction, and maximizingHby anew sequence ofu(t) during each iteration. This process converges in 50 to 60 iterations, the convergence depending on thevalue ofG.

Applying this method to our device, with irregular wave trains synthetized from the ITTC spectrum mentionned above, wewere able to construct two production matrices for the device: one when no control is applied, the other one with optimallatching control computed by this method. These matrices showing the extracted power normalized by the rated devicepower are shown on Figure 10. Applying both these matrices to the scatter diagram of our test site (Yeu Island) we observedan overall magnification by a factor around 2 for the energy producted yearly. From these figures therefore, it is clear thatreal time control is not simply an improvement for this kind of resonant wave energy converter, but a real breakthrough.

CONCLUSION

A new patented wave energy device has been developed. From a robust principle, optimizations studies has been performedin order to maximize the annual energy production at a reference site (Yeu Island) were the wave energy resource ismoderate. Results are promising especially when latching control of the internal pendulum is active. This system has some

advantages that we can sum up here:

It is completely closed; all the mechanical parts, hydraulic and electric components, etc.. are located inside, thensheltered from sea action. This will certainly increase their lifetime and reduce maintenance costs.

The choice of a wheel as a pendulum makes the internal system very robust due to the absence of any endstop (thetechnical design was done such that a complete turn of the wheel is allowed and will not damage the system)

Use of low technology components available on the shelf, in a floating body (rather than submerged high technology)result in global robustness and cost reduction

Being self-referenced, the system reacts against gravity, instead of moorings. A slack single line mooring is thensufficient to keep it on site, and being freely floating, it can accommodate any tidal range

It is easy to take away and tow to shore for off-site maintenance which will then be easier (then cheaper) than for fixedbottom seated systems.

A real time (wave to wave) sophisticated control can bring a considerable increase in energy production (this systemhaving the ability to lock the pendulum will be used also for security procedures).

Figure 10: SEAREV_01 Production matrix; without (left) and with (right) control


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As other recent wave energy systems, it is modular and will be deployed offshore in wave farms of several tenth ofindividuals, (representing typically 20 to 30 MW/km2 installed power density) in water depth of (typically) 50m,inducing therefore practically no visual impact from the shore.

Tests with a model at 1/12 scale are being performed in the

multidirectional 50m by 30m wave tank of Ecole Centrale de Nantes,and a full scale prototype (1000T, 500kW) to be tested at sea in 2009is currently at the pre-design stage.

ACKNOWLEDGMENT

This study was financially supported by the CNRS, the RegionalCouncil of Pays de la Loire and the French Energy Agency ADEME.

REFERENCES

A. Babarit, G. Duclos, and A.H. Clement. Benefit of latching controlfor heaving wave energy device in random sea. Proc. 13th IntOffshore and Polar Engineering Conf.ISOPE2003, 1:341348,2003. ISBN 1-880653-60-5.

A. Babarit, G. Duclos and A.H. Clment: Comparison of LatchingControl Strategies for a heaving wave energy Device in randomsea. Proceedings of the fifth EuropeanWave Energy Conference,Cork, 2003, pp.245-257.

A. Babarit, A.H. Clment: Optimal Latching Control of a WaveEnergy Converter. Proceedings of the sixth EuropeanWave andTidal Energy Conference, Glasgow, 2005.

A. Babarit, (2005), Optimisation hydrodynamique et contrle optimal dun rcuprateur d nergie des vagues. PhD thesisreport, Ecole Centrale de Nantes.

K. Budal and J. Falnes. Interacting point absorbers with controlled motion, in Power from Sea Waves. BM Count, AcademicPress, 1980.

K. Budal, J. Falnes, L.C. Iversen, P.M. Lillebeken, G. Oltedal, T. Hals, T. Onshus, and A.S. Hoy. The Norwegian wavepower buoy project. The Second International Symposium on Wave Energy Utilization, pages 323344, June 1982.

A.H. Clment. Using differential properties of the green function in seakeeping computational codes. Proc. 7th Intern. Conf.Numer. Ship Hydrod., 6(5):115, 1999.

W.E. Cummins. The impulse response function and ship motions. Schiffstechnik, p. 491502, 1962.

Riche de Prony. Essai exprimental et analytique sur les lois de la dilatabilit des fluides lastiques et sur celles de la forceexpansive de la vapeur de leau et de la vapeur de lalcool diffrentes tempratures. Ecole polytechnique 1, cahier 2,pages 2476, 1795.

G. Delhommeau, P. Ferrant, and M. Guilbaud. Calculation and measurement of forces on a high speed vehicle in forcedpitch and heave.Applied Ocean Research, 14(2):119126, 1992.

G. Duclos, A.H. Clment, and G. Chatry. Absorption of outgoing waves in a numerical wave tank using a self-adaptiveboundary condition.Int. Journal of Offshore andPolar Engineering, 11(3):168175, 2001.

R. Duvigneau. Contribution loptimisation de formes pour des coulements forts nombres de Reynolds autour degomtries complexes. PhD thesis, Ecole Centrale de Nantes et Universit de Nantes, 2002.

J. Falnes. Ocean Waves and Oscillating Systems, Linear interactions including wave-energy extraction. CambridgeUniversity Press, 1 edition, 2002.

J. Falnes. Optimum control of oscillation of wave-energy converters. International Journal of Offshore and PolarEngineering, 12(2):147155, 2002.

Figure 11: the 1/12 scale PTO model being

tested on moving table


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R.E. Hoskin and N.K. Nichols. Optimal strategies for phase control of wave energy devices. In Michael E. McCormick andYoung C. Kim, editors, Utilization of Ocean Waves :Wave to Energy Conversion, pages 184199, 345 East 47th StreetNew York,New york 10017-2398, 1986. American Society of Civil Engineers.

W.H. Press, S.A. Teukolsky ,W.T. Vetterling, and B.P. Flannery. Numerical Recipes in Fortran 77. Cambridge UniversityPress, 1992.

K.J. Rawson and F.C. Tupper.Basic ship theory, volume 1. Longman, 3 edition, 1983.

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