Proceedings of the Institution of Mechanical Engineers, Part a- Journal of Power and Energy-2006-De Sousa Prado-855-68

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DOI: 10.1243/09576509JPE284

2006 220: 855Proceedings of the Institution of Mechanical Engineers, Part A: Journal of Power and EnergyM G de Sousa Prado, F Gardner, M Damen and H Polinder

Modelling and test results of the Archimedes wave swing

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Modelling and test results of the Archimedeswave swingM G de Sousa Prado1, F Gardner1, M Damen1,2, and H Polinder3�

1Teamwork Technology, Zijdewind, The Netherlands2Rijswijk University of Professional Technical Education, Rijswijk, The Netherlands3Electrical Power Processing, Delft University of Technology, Delft, The Netherlands

The manuscript was received on 7 February 2006 and was acepted after revision for publication on 17 August 2006.

DOI: 10.1243/09576509JPE284

Abstract: The objective of this paper is to describe the modelling and some of the offshore testresults of the full-scale model of Archimedes wave swing (AWS) at Portugal in 2004. The AWS is asystem that converts ocean wave energy into electric energy. A pilot plant of this system hasbeen built. The paper starts with the derivation of a model for the floater motion. Next, thegenerator system is described and modelled. The generator system consists of a permanent-magnet linear synchronous generator with a current source inverter. Subsequently, test resultsare presented to validate the models for the floater motion and the generator. Finally, someconclusions are drawn.

Keywords: ocean wave energy, modelling, permanent-magnet generator, linear generator,Archimedes wave swing

1 INTRODUCTION

Important factors that have stimulated the use ofrenewable energy are energy cost, energy indepen-dence, and mainly environmental protection.Therefore, great efforts are made in the fields ofwind energy, solar energy, hydro power, and so on.Ocean wave energy is a renewable energy sourcewith a huge potential, but further from commercialviability. Different wave energy conversion systemshave been proposed [1–13]. This paper is about asystem that converts ocean wave energy into electricenergy, namely the Archimedes wave swing (AWS).The idea behind the system comes from FredGardner and Hans van Breugel.

Figure 1 illustrates the principle of operation.Basically, the AWS is a cylindrical air-filled chamber.The waves move the lid of this chamber, called thefloater, in vertical direction with respect to thebottom part, which is fixed to the sea-bottom.

When a wave is above the AWS, the AWS volume isreduced by the high water pressure. When a wavetrough is above the AWS, the volume increasesbecause of the air pressure inside the AWS. Fromthis linear motion, energy can be extracted andconverted into electrical energy. By tuning thesystem frequency to the average wave frequency,the stroke of the linear motion can be made largerthan the wave height.

The AWS is a unique wave energy conversionsystem because it is completely submerged. This isimportant, because this makes the system lessvulnerable in storms. Besides, it is not visible, sothat the public acceptance is not such a problem asfor, for example, wind farms.

To prove the principle of operation behind thisidea, a few small models have been developed(scale 1:20 and 1:50 to the final system) [14, 15].These models showed that the system worked andvalidated the models predicting the hydrodynamicforces on and the hydrodynamic damping of thefloater.

As a next step, a pilot plant of the AWS was built atthe Portuguese coast in 2001. The main objective ofthis pilot plant was to prove that the complete

�Corresponding author: Electrical Engineering, Mathematics

and Computer Science, Delft University of Technology, Mekelweg

4, Delft 2628 CD, The Netherlands. email: [email protected]

855

JPE284 # IMechE 2006 Proc. IMechE Vol. 220 Part A: J. Power and Energy



system works and can survive. Predictions of theperformance of the system, the annual energy yield,and the design of the generator system have beenpresented in references [16] to [20].

Figure 2 depicts a photograph of the pilot plantbefore submersion. The centre part is the floaterwith a diameter of 9.5 m. The rated stroke is 7 m andthe rated velocity is 2.2 m/s. The maximum peakpower is 2 MW. The power is delivered to the gridthrough a current source inverter or dumped into aresistor bank in case the inverter is not available. Toimprove the power factor of the generator, a 1.2 mFcapacitor bank was used in parallel with the inverter(or resistor). Although the linear generator is able tobrake with a 1 MN force, a braking system had to beincluded in the device to damp the motion in caseof generator’s limitation for higher waves or failure.This system consists of two cylinders sliding intoeach other, forcing the water trapped inside to flowthrough an orifice. By changing the area of the orifice,the braking force can be adjusted to the desired level.

During the tests, the area of the orifice was alwayskept at the minimum (water brakes were closed)and, therefore, the damping was quite high. Themoving part of the device has a mass of around0.4 Mkg (fromwhich 0.04 Mkg is due to the translatorof the generator), and the total mass of the device,including the pontoon, is �7 Mkg (from which5 Mkg are just due to the sandballast tanks). The over-all dimensions of the device are 48 m � 28 m � 38 m(L � W � H),wheremost of the volume is reserved forthe ballasting tanks necessary for the submerge/emerge operation. The air volume of the device atmidposition is �3000 m3 and can be changed bypumping in/out water. The total volume of waterthat canbepumped is�1500 m3andpermits the con-troller to tune the natural period of the device in therange of 7–13 s.

The objective of this paper is to derive a model forthe floater motion and the generator system and tovalidate this model by means of the first offshoretest results of the full-scale pilot plant. This modelfloater motion has not been published in so muchdetail before. Also the model validation by mean ofoffshore test results is an important contribution ofthis paper. The generator was tested with the inverteras a load and also with the resistor bank as a load. Inthis paper, the test results presented are only for theresistor load. In reference [21], test results arepresented where the inverter is the generator load.

This paper starts with the derivation of a model forthe motion for the floater of the pilot plant. Next, themain design choices for the generator are discussedand a model of the generator system is presented.Subsequently, some measurement results from thepilot plant are discussed to validate the models andsome conclusions are presented.

2 MECHANICAL MODEL OF AWS

This section starts with a rather complete equationfor the motion of the floater. Then, this equation issimplified to an equation that is used in this paper.

2.1 Non-linear time domain model

The model of the unconstrained heave motion of thefloater can be described in time domain by thesecond law of Newton

mf €x ¼ FRAD|ffl{zffl}Damping andInertia Force

þFBEAR þ FDRAG þ FGEN þ FWB|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}Damping Forces

þ FAIR þ FNITRO þ FGRAV þ FHS|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}Spring Forces

þ FWAVE|fflffl{zfflffl}Exciting Force

(1)

Fig. 1 Sketch illustrating the principle of operation of

the AWS

Fig. 2 Photograph of the pilot plant before

submersion

856 M G de Sousa Prado, F Gardner, M Damen, and H Polinder

Proc. IMechE Vol. 220 Part A: J. Power and Energy JPE284 # IMechE 2006



This equation contains the following quantitiesand forces.

x is the vertical displacement of the generator andmf the mass of the floater (including all moving partsof the device).

FRAD is the radiation force due to the motion of thefloater inside water, responsible for generating theradiated wave. This force includes inertia forcesresponsible for the acceleration of the water anddamping forces due to the radiated energy. Forsmall motions, it can be approximated by

FRAD ¼ �madd1 €x �

ðt0

R(t � t) _x(t)dt (2)

where madd1 is the added mass at infinite frequencyand R(t) the retardation function that expresses thememory of the fluid. The radiation force does notdepend only on the instantaneous acceleration andvelocity of the floater but on the history of themotion of the floater.

The parameters madd1 and R(t) can only be calcu-lated analytically for very simple geometries. For ageneric body, numerical computation is required [22].

FBEAR is the force due to friction of the bearings.This force can be described by

FBEAR ¼ �mBEARFHOR sign( _x) (3)

where mBEAR is the bearing friction coefficient andFHOR the horizontal force that the waves and currentapply to the floater. If the diameter of the device issmall when compared with the wavelength of theincident waves, then the horizontal force may be cal-culated from the Morison’s equation combined withstrip theory [22].

FDRAG is the drag force applied by the water to thefloater and is given by

FDRAG ¼�1=2rSf _xj _xjCDUP, _x 5 0�1=2rSf _xj _xjCDDW, _x , 0

�(4)

where SF is the surface area of the floater, and CDUP

and CDDW are the drag coefficients when the floateris moving upwards or downwards.

FGEN is the force applied by the generator. Whenthe generator is connected to the grid via a powerelectronic converter (such as a voltage source inver-ter), this converter determines the force applied bythe generator within the limitations of the generatorand the converter. Other forms of loading the genera-tor, e.g. resistive loads, are also possible.

FWB is the damping force applied by the waterbrakes. This force results from the pressure buildup inside the water brakes, due to the forced flowof water through three orifices (one of them

adjustable) and can be expressed by a quadraticfunction of the velocity

FWB ¼ �bWB _xj _xj; bWB ¼ rSWBSWB

CVSO

� �2

(5)

where bWB is the damping coefficient of the waterbrakes that can be deduced from the hydraulichead loss equation of the flow through an orifice, rthe water mass density, SWB the cross area of thewater brakes at which the breaking pressure isacting, CV the discharge coefficient of the orifice,and SO the orifice area through which the water canflow; by varying the area of the orifice it is possibleto adjust the amount of damping.

FAIR is the force due to the air pressure inside AWS.This force depends on the displacement of the floaterdue to the compression and decompression of the airand can be expressed by

FAIR(x) ¼ Sfpa(x), pa(x) ¼ �pa

�V a

Va(x)

� �g

¼ �pa

�V a

Va0 þ Sfx � VW

� �g

(6)

where Sf is the floater area, pa the air pressure insideAWS, g is a coefficient between 1 (isothermal beha-viour) and 1.4 (adiabatic behaviour), Va(x) the airvolume inside AWS for a certain water volume, Va0

the volume for the floater at midposition withoutwater inside AWS, VW the water volume inside theAWS, and �pa and �V a the equilibrium air pressureand volume at the equilibrium position �x.

FNITRO is the force due to the nitrogen cylinder.This force depends on the displacement of the floaterdue to the compression and decompression of thenitrogen and can be expressed by

FNITRO(x) ¼ �Snpn(x), pn(x) ¼ �pn

�V n

Vn(x)

� �g

¼ �pn

�V n

Vn0 � Snx

� �g

(7)

where Sn is the surface area of the nitrogen cylinder, pnthe nitrogen pressure inside the nitrogen cylinder, gcoefficient between 1 (isothermal behaviour) and 1.4(adiabatic behaviour), Vn(x) the nitrogen volumeinside the cylinder for a certain position of the floaterx, Vn0 the nitrogen volume for the floater in the mid-position, and �pn and �V n the equilibrium nitrogenpressure and volume at the equilibrium position �x.

FGRAV is the gravity force applied to the floater,given by

FGRAV ¼ �mfg (8)

where g is the gravity acceleration.

Modelling and test results of the AWS 857




FHS is the force due to the hydrostatic pressureacting outside AWS. This force depends on the dis-placement of the floater and can be expressed by

FHS(x) ¼ �Sfbrg(df þ hT � x)þ pambc (9)

where df is midposition floater’s depth relative tothe Hydrographic zero, hT the tide level relative tothe Hydrographic zero, and pamb the ambientpressure.

FWAVE is the force due to the total dynamicpressure field acting on the floater resulting fromthe incoming wave and the diffracted wave from thedevice. This is the force responsible for exciting theAWS dynamics. Considering linear wave theory andinfinitesimal motions of the device, the wave forcecan be calculated by

FWAVE(v) ¼ H(v)A(v) (10)

where H(v) is the transfer function that relates thewave force with its amplitude A. Owing to the com-plexity of the diffraction phenomena, the transferfunction H(v) has to be generally computed numeri-cally [22] and may be only calculated analytically forvery simple geometries. For devices much smallerthan the wavelength of the exciting wave, the dif-fracted wave may be neglected, and the wave forceis mainly due to the pressure field of the incidentwave, which permits us to write the wave force inthe form

FWAVE(v) � �rgSfKP(v,dZH þ hT,df þ hT � �x)|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}H(v)

A(v)

(11)

where dZH is the water depth with respect to thehydrographic zero level. Kp represents the pressuredepth factor at depth d from the surface, for a totalwater depth of h and wave number k(v) [22] and isgiven by

KP(v, h, d) ¼cosh [k(v)(h� d)]

cosh [k(v)h](12)

2.2 Equilibrium condition, stability, andresonance

In a static situation, with no wave force exciting thedevice, there is one equilibrium position �x where allthe spring forces cancel each other, and may be

given by the following implicit relation

�pa( �x) ¼mfg

Sfþ rg(df þ hT � �x)þ pamb

þSnSf

�pn( �x) (13)

Depending on the parameters of equation (13) (airpressure, nitrogen pressure, mass of the floater, tide,etc.), the authors may have a solution for the equili-brium position within or outside the stroke. In thelater case, the real equilibrium position will belocated at the limit of the stroke became of the endstops.

One important issue is the stability of thedynamics at the equilibrium point. The local stabilityat the equilibrium point can be determined from thesign of total spring coefficient k, given by

k ¼ �d(FAIR þ FNITRO þ FGEN þ FHS)

dx

��x¼ �x

¼ ka þ kn þ kh (14)

where ka, kn, and kh are the air, nitrogen, and hydro-static spring coefficients in the equilibrium position

ka ¼ �dFAIR

dx

��x¼ �x

¼ gS2f�pa

�V a

(15)

kn ¼ �dFNITRO

dx

��x¼ �x

¼ gS2n�pn

�V n

(16)

kh ¼ �dFHS

dx

��x¼ �x

¼ �rgSf (17)

If k is positive then the equilibrium point is stableand if k is negative the equilibrium point will beunstable. From this expression, it can be seen thatthe gas spring coefficients (air and nitrogen) arealways positive and therefore contribute to stabilizethe dynamics. In contrast the hydrostatic spring isalways negative, having a destabilizing effect on thedynamics. The stability occurs whenever the gassprings are ‘stiffer’ than the hydrostatic spring.

An important aspect of the gas springs is thedependence of their stiffness with frequency, whichis taken into account by the factor g. For highfrequency (fast motions), g should be close to 1.4(adiabatic behaviour) and therefore a stiffer springis expected. For low frequency (slow motions), g

should be close to 1 (isothermal behaviour) andtherefore a softer spring is expected. The classifi-cation of slow/fast motion depends on whether theperiod of the motion is much higher/smaller thanthe thermal constant of AWS, which is �0.5 h. Forthe range of wave periods from 5 to 15 s, wheremost of the wave energy is expected to be present,





the air and nitrogen spring behaviour can beconsidered as adiabatic (g ¼ 1:4), which means thatthe motion will be fast enough to prevent heatexchange between the gas and the exterior duringeach wave cycle.

Owing to the existence of a spring, the device willhave a natural period given by

Tn ¼ 2p

ffiffiffiffiffim

k

r(18)

where m represents the total inertia of the device. Byadjusting the stiffness of the spring, it is possible totune the natural period of the device and make it res-onate with the incoming waves. At resonance, thepower absorbed by the device from the waves willbe maximized. This is the basic principle of AWSand it is the same principle behind the tuning of areceiver to a desired radio station, where a capacitoris adjusted in order tomatch the natural period of thereceiver with the base frequency of the radio station.It should be noted that the adjustment of the airspring is a slow process since it is done by pumpingwater in/out of the device. Therefore, the tuning ofthe natural period by air spring adjustment cannotbe made on a wave by wave basis, but only for theaverage wave period of the sea spectrum thatchanges slowly with time.

For low frequency motions, the gas springs aresofter and may be not stiff enough to compensatethe destabilizing hydrostatic spring, resulting in anegative k. Therefore, the average motion of thedevice may be unstable and drift away from theequilibrium position. To compensate this instability,control of the position will be needed in order to keepthe average position around the desired equilibriumposition.

2.3 Simplified model for tests evaluation

To make a first evaluation of the measured data, asimplifiedmodel of the heave dynamics of the floaterwas considered. The main simplifying assumptionsmade are as follows.

1. The drag and bearing dampings are neglectedsince generator and water brakes were the mainsources of damping. This is mainly because theresistor load of the generator was quite lowduring the test and the water brakes were alwayson.

2. The spring forces are linearized around the equili-brium position.

3. The radiation force is reduced only to the inertiaforce due to the added mass at infinity frequency(the radiation damping and the dependence ofadded mass with frequency is neglected).

The resulting simplified model can be expressed inthe form

(mf þmadd1) €x þ bGEN _x þ bWB _xj _xj þ kx

¼ FWAVE (19)

where mf is �0.4 Mkg. This value resulted fromadding up all the weights of the separate parts thatmake up the moving part of AWS (including thetranslator of the generator). These weights weremeasured separately during the construction ofAWS. madd1 is �0.2 Mkg. This value was computedfrom linear hydrodynamic software AQUADYN [14].bGEN, is given by equation (35), see section 3.4 fordetails. bWB, is calculated from equation (5), usingthe geometric dimensions of the water brakesand standard hydraulic discharge coefficient, result-ing in 1:42� 106Ns2=m2 . k is calculated fromequations (14), (15), (16), and (17), using the geo-metric dimensions of the floater and the nitrogencylinder and the measured average gas pressures(air and nitrogen).

Since there were no direct measurements ofthe wave force or of the amplitude of the incidentwaves on top of the device, the wave force had tobe estimated from a water pressure measurementpW made above the outer structure of the device.The wave force was obtained by simply scalingthe measured pressure variation taking into accountthe pressure depth factor of the pressure sensor andthe equilibrium position of the device, at theobserved average wave frequency, according to

FW � �Sf (pW � �pW)KP(vAV, dZH þ hT , df þ hT � �x)

KP(vAV, dZH þ hT , dP þ hT )

(20)

where Kp represents the pressure depth factorintroduced in equation (12), dp the depth of thepressure sensor relative to the hydrographic zero,and vAV the average frequency of the incomingwaves.

3 GENERATOR SYSTEM

This section describes the generator system. It startswith a discussion of a number of design consider-ations. Then a generator model will be derived. Sub-sequently, the parameters of the model are derivedfrom the design. The last part of this section gives aspecific model for the generator with the load usedin the experiments.





3.1 Design considerations

The main requirements for the generator system ofthe pilot plant of the AWS are the following:

(a) maximum stroke: 7 m;(b) maximum speed: 2.2 m/s;(c) maximum force: 1 MN;(d) robust;(e) maintenance as low as possible;(f) efficient;(g) cheap.

During the design process, the following choiceswere made in order to meet the requirements asgood as possible [19].

1. Probably, a generator system consisting of agearbox that converts the linear floater motioninto rotating motion and a standard rotatinggenerator would be a cheap and rather efficientsolution. However, it appears to be extremelydifficult to build a robust, maintenance-freegear. Therefore, a linear generator is used, asalso proposed in references [9] to [13].

2. It is nearly impossible and extremely expensiveto make the generator large enough to take allpossible forces generated by waves. Therefore,the AWS also has water dampers that can makevery high forces. This implies that the generatorcan be designed as a compromise betweenenergy yield and cost.

3. The linear generator that converts the mechan-ical energy into electrical energy is a perma-nent-magnet (PM) generator because it has arather high force density, and it has a relativelyhigh efficiency at low speeds when comparedwith other machine types [19]. That the effi-ciency is relatively high appears from the factthat the annual dissipation in the PM generatorwith stator iron is the lowest from the fivemachine types compared in reference [19].

4. The magnets are on the translator that moves upand down, so that there is no electrical contactbetween the moving part (the translator) andthe stator, which is important because such anelectrical contact suffers from wear.

5. The generator is flat. Maybe, round generatorconstructions fit better in the construction ofthe AWS. However, for a single generator for apilot plant, it was much cheaper to remainclose to existing production technology andbuild a flat generator.

6. The number of slots per pole per phase is1. Increasing this number would lead to largepole pitches, resulting in thicker yokes and ahigher risk of demagnetization. Decreasing thenumber of slots per pole per phase (using

fractional pitch windings) would lead toadditional eddy-current losses due to additionalspace harmonics.

7. The magnets are skewed to reduce cogging.8. The translator with the magnets is only a few

metres longer than the stator in order to reducecost. This means that in the central position,the magnets of the translator are completelyoverlapping the stator so that maximum forcescan be made, but in the extreme positions, themagnets only partly overlap the stator.

9. To balance the attractive forces between statorand translator, the generator is double sided, asdepicted in Fig. 3. The attractive force density is�200 kN/m2.

10. For cooling the stator of the generator, a watercooling system was implemented.

11. The power electronic converter for the gridconnection is placed on shore so that possibleproblems with the power electronics and thecontrol could easily be solved. A 6 km longcable connects the generator terminals to theconverter on shore.

12. A current source inverter on the shore is used forthe utility grid connection. A voltage sourceinverter would have advantages of better controlcharacteristics, better power factor, bettergenerator efficiency, and higher forces (andenergy yield) [18]. However, it appeared to beeasier and cheaper to make a current sourceinverter available.

Fig. 3 Section of the four pole pitches of the linear PM

generator. The middle part is the stator with

stator iron and coils in the stator slots in

between. The left and right parts are the

translators with the magnets with arrows

indicating the magnetization direction





13. Coatings are used to protect the generatoragainst the aggressive environment.

Figure 3 depicts a cross-section of the generator.When the translator with the magnets moves upand down, voltages are induced in the coils in thestator slots. In reference [20], a more detaileddescription of the generator topology may befound. Figure 4 depicts a photograph of a part ofthe stator.

3.2 Generator modelling

The general form of a voltage equation of a wire isgiven by Faraday’s law

u ¼ Ri þdl

dt(21)

where u is the voltage between the terminals, R theresistance of the wire, i the current through thewire, and l is the flux linkage of the wire.

Within the generator, there are four sources thatcontribute to the flux linkage, namely the PMsand the three stator currents. If it is assumed that(a) the system is linear (saturation is negligible);(b) the mutual inductances between the differentphases are equal because of symmetry; (c) the sumof the three stator currents is zero; and (d) the pos-ition of the translator is chosen zero when the fluxlinkage of phase a is zero, then the flux linkage ofstator phase a can be written as

la ¼ Lia þMib þMic þ lpma

¼ Lsia þ l pm sinp

tpx

� �(22)

where L is the self inductance of a phase, M the

mutual inductance between two stator phases, Lsthe synchronous inductance of a phase if the otherphases are also conducting (Ls ¼ L�M), lpm is theflux linkage due to the PMs, and tp is the pole pitch.

If this is substituted in the voltage equation and thesame is done for the other phases, the result is

ua ¼ Rsia þ Lsdiadt

þ lpmp

tp

dx

dtsin

p

tpx

� �

¼ Rsia þ Lsdiadt

þ ep sinp

tpx

� �

ub ¼ Rsib þ Lsdibdt

þ lpmp

tp

dx

dtsin

p

tpx �

2

3p

� �

¼ Rsib þ Lsdibdt

þ ep sinp

tpx �

2

3p

� �

uc ¼ Rsic þ Lsdicdt

þ lpmp

tp

dx

dtsin

p

tpx �

4

3p

� �

¼ Rsic þ Lsdicdt

þ ep sinp

tpx �

4

3p

� �(23)

3.3 Parameter determination from the design

The parameters used in these equations are now cal-culated from the design of the machine, as describedmore extensively in reference [20]. The followingassumptions are used in the calculations.

1. Space harmonics of the magnetic flux densitydistribution in the air gap are negligible, only thefundamental is considered.

2. The magnetic flux density crosses the air-gapperpendicularly.

3. The magnetic permeability of iron is assumed tobe infinite.

The amplitude of the no-load voltage induced bythe magnets can be calculated as [20]

ep ¼ lpmp

tp

dx

dt¼ 2NskwlsBg

dx

dt(24)

where Ns is the number of turns of the phase wind-ing, kw the winding factor, ls the stack length of themachine perpendicular to the plane of the drawing,and Bg the amplitude of the fundamental space har-monic of the magnetic flux density in the air gap dueto the magnets.

The amplitude of the fundamental space harmonicof the magnetic flux density in the air gap due to themagnets can be calculated as [20]

Bg ¼lm

mrmgeffBrm

4

psin

pbp

2tp

� �(25)

where lm is the magnet length in the direction of theFig. 4 Photograph of a stator part in the AWS





magnetization, mrm the relative recoil permeability ofthe magnets, Brm the remanent flux density of themagnets, bp the width of the magnet, and geff theeffective air gap.

The effective air gap of themachine is calculated as

geff ¼ kCg1

g1 ¼ g þlmmrm

kC ¼ts

ts � g1g

g ¼4

p

bs

2g1arctan

bs

2g1

� �� log

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ

bs

2g1

� �2s2

435

(26)

where kC is the Carter factor [20, 23], g the mechan-ical air gap, ts the slot pitch, and bs the slot openingwidth.

The main part of the synchronous inductance canbe calculated as

Lsm ¼6m0lstp(kwNs)

2

pgeffp2(27)

where p is the number of pole pairs.Slot, air-gap, and end-winding leakage induc-

tances are calculated as in reference [23].The stator phase resistance is calculated from the

dimensions of the machine, the number of turns ina slot, and the cross-section of a slot

Rs ¼ rCulCusACus

(28)

where

lCus ¼ 2Ns(ls þ 2tp) (29)

ACus ¼pksfilbhs

Ns(30)

ksfil is the slot fill factor, bs the slot width, and hs is theslot height.

3.4 Generator with resistive load

In the tests presented in this paper, the generator isconnected to a load consisting of a parallel con-nection of a resistor and a capacitor, as depicted inFig. 5.

An useful approximation of the force made by thegenerator can be calculated in the following way.The following assumptions are used:

(a) the changes in speed are so slow that the electri-cal quantities (currents, voltages, and powers)can be calculated for steady state;

(b) iron losses are negligible.

In steady state, the stator current phasor can becalculated as

Is ¼Ep

Z;

Z ¼ Rs þ jvLs þ Rc þ jvLc þRl

1þ jvRlCl

(31)

where v is the electrical angular frequency, given by

v ¼ p_x

tp(32)

Rc is the cable resistance, Lc the cable inductance, Rl

the load resistance, and Cl the capacitor connected inparallel to the load resistance.

The values of the parameters Rs, Ls, Rc, Lc, Rl, and Cl

and the measurement/calculation methodologyused to obtain them are described in reference [20].

From this, the force made by the generator can becalculated as

Fg ¼Pg

_x¼

3EpI�s

_x(33)

where I�s is the complex conjugate of Is.Figure 6 depicts the resulting calculated force as a

function of the velocity.If the speed is not too high, the generator force is

almost linear with speed and can be approximatedby

FGEN ¼ �bGEN _x (34)

where bGEN is the mechanical damping due to the

Fig. 5 Equivalent circuit of the generator, the cable,

and the load





generator given by

bGEN(RAC) ¼@FGEN( _x, x,RAC)

@ _x

��_x¼x¼0

�1:83� 106(Nsm�1V)

Rl þ Rc þ Rs

(35)

4 EXPERIMENTAL RESULTS

4.1 Introduction

The results presented are measurements performedon the 2nd of October of 2004, from 15:57:30 until16:00:00 (�2.5 min). At that time, the followingconditions were observed

1. Sea(a) significant height of the waves Hs � 2:4m;(b) average wave period TAV � 11:5 s;

(c) tide level hT � 2:55m.2. AWS

(a) water volume inside AWS VW � 1180m3;(b) average air pressure inside AWS

pa ¼ 32.35 mwc;(c) water brakes on, the damping of the water

brakes is given by bWB ¼ 1:42� 106 Ns2=m2

3. Landstation(a) electrical load: resistor Rl ¼ 6V

4.2 Currents, voltages, and position

Figure 7 depicts the measured current waveformsthrough the load resistance during one waveperiod.

The total generator current is the sum of the loadcurrent and the current through the load capacitors,and can be calculated as

is ¼ iRlþ RlCl

diRl

dt(36)

Substituting this in equation (23) gives the no-loadvoltages. The products of the no-load voltages andthe generator currents give the generator inputpower. Figure 8 depicts both the power dissipatedin the load resistors and the input power of thegenerator.

From the no-load voltages calculated from themeasurements, the relative position can also becalculated, because these voltages are functions ofthe position [24]. Figure 9 depicts the positionsignal that has been derived in this way.

The speed is obtained by time differentiating theposition signal. The generator force may then beobtained from the generator power by dividing it bythe speed. Figure 10 depicts the generated powerand generator force as a function of the speed. Com-paring with Fig. 6, it appears that the damping of thegenerator determined from the measurements islarger than the theoretical damping calculated from

Fig. 6 Force of the generator with resistive load as a

function of the velocity for load resistances of

1, 3, 6, and 15 V

Fig. 7 Measured current waveforms through the load resistance during one wave period





equation (33). This is probably caused by the factthat the generator temperature is lower than whatit was designed for. A lower temperature leads to ahigher remanent flux density of the magnets, andtherefore to a higher voltage induced by the PMs.From the correlation between the measurementand the calculation, it can be concluded that thegenerator model is acceptable.

4.3 Equation of motion

According to the equilibrium condition (13) andtaking into account the measured tide level and aver-age air pressure, the equilibrium position of thedevice should be �x � 0:7m.

When the water volume inside AWS, the averageposition and average air pressure are taken into

Fig. 8 Power dissipated in the load resistance (solid line) and power delivered by the generator

during one wave period (dashed line)

Fig. 9 Position determined from the no-load voltage

Fig. 10 Generator power and generator force as a function of generator speed





account, the air and nitrogen spring coefficients canbe calculated using equations (15) and (16) as ka �1:16� 106 N/m and kn � 9� 104 N/m.

Adding the gas springs coefficients to the hydro-static spring coefficient kh � �7� 105 N/m, resultsin a total positive spring coefficient k ¼ 5:5�

105 N/m and a natural period Tn ¼ 6:6 s.Using equation (35), the damping coefficient of the

generator is calculated as bGEN � 275 kNs/m. Compar-ing the generator dampingwith thewater brake damp-ing coefficient (5), it can be seen that for velocitiesapproximately higher than 0:18m/s, the braking forceof the water brakes is higher than that of the generator.

Figure 11 depicts the following threemeasurements.

1. The water pressure on top of the structure of AWS.This pressure signal gives information about theexciting wave force that drives the floater. Whenthe pressure signal increases the wave force actingon the device should also increase and vice-versa.

2. The air pressure inside AWS. This pressure signalgives information about the motion of the devicedue to the compression and decompression ofthe air. When the pressure reaches a maximum,the position reaches a minimum and vice-versa.

3. The electrical current at resistor bank at the land-station is related to the speed of the device. Whenthe current is high the speed is high and vice-versa.

Just from visual inspection of the measurementssome conclusions can already be drawn.

1. The peaks of the air pressure are slightly delayedin relation to the water pressure peaks. Thismeans that the extreme position of the device isalways delayed relative to the extreme values ofthe exciting force. This phase shift (,908) resultsfrom the mass spring behaviour of the device. Ifthe system would be in resonance with thewaves the phase shift would be 908. In this case,

Fig. 11 Measurements of water pressure, air pressure, and resistor currents





the natural period of the mass spring system(6.6 s) is lower than the average period of thewaves (11.5 s) and therefore the spring forcesdominate over the inertia forces, which results ina phase shift below 908.

2. When the air pressure time derivative is high, theelectrical current is also high. This results directly

from the fact that the air pressure time derivativeis directly related to the speed of the device.

A more precise analysis can be made in order tocheck the consistency and correlation between themeasured signals. One way of achieving this is by cal-culating from each measured signal independently

Fig. 12 Absolute value of the velocity as a function of time derived from different measured

signals





one common signal, like the velocity of the device forinstance.

The relation between the velocity of the device andthe water pressure signal is given by the dynamicmass-spring model. By solving the differentialequation (19) excited by the water pressure signal,the velocity signal may be recovered. The resultingsignal will however depend also on the initial con-ditions used for the position and velocity. Since theeffect of initial conditions on the signal will decayrapidly with time (after a few wave cycles it’s negli-gible already), the initial conditions are assumed tobe zero.

The relation between the velocity of the device andthe air pressure signal is simpler and results from thestatic air spring curve of the device. The air pressureis related to the position of the device by

pa ¼ �kaSf

x

Therefore, the velocity can be calculated as

_x ¼ �Sfka

_pa

As explained in section 4.2, also a position signalcan be obtained from themeasured currents and vol-tages. The time derivative of this position signal alsogives a speed signal.

Figure 12 presents the absolute value of the vel-ocity signals calculated independently from each ofthe three measured signals.

From the figure, it can be seen that the signalsmatch quite well, specially the ones calculated fromthe electrical current and the air pressure. Themain reason for this is the simplicity of the corre-sponding models used for the velocity estimation.The mass-spring model is however more complex,especially the wave force estimation from the waterpressure signal. It is a non-linear function of the pos-ition of the generator and dependent on the fre-quency of the waves. Nevertheless even with thesimple model (see section 2.3), where most of thesenon-linearities and higher order dynamics were neg-lected, the estimated velocity still follows the otherestimated velocities signals quite well.

From these results, it can be concluded that thederived equation for the motion of the floater isvalid. Without proper estimates of the wave forces,the spring constants, the masses, and the damping,there would be much larger differences betweenthe different velocity signals.

5 CONCLUSIONS

In this paper, a model for the motion of the floater ofthe AWS has been derived. From the measurementresults, it can be concluded that a 2nd order modelfits reasonably well to the measured data, whichpermit to conclude about the simplicity of the mainstructure of the device dynamics. A higher ordernon-linear model is however needed to achieve abetter fit. Also, a generator model has been derived.From the measurements, it can also be concludedthat this model is valid.

ACKNOWLEDGEMENTS

This research project has been supported by a MarieCurie Early Stage Research Training Fellowshipof the European Community’s Sixth FrameworkProgramme under contract number MRTN-CT-2004-505166.

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Proceedings of the Institution of Mechanical Engineers, Part a- Journal of Power and Energy-2006-De Sousa Prado-855-68