an experimental study of closely spaced point absorber arrays

Performance of closely spaced point absorbers withconstrained floater motion

G. De Backer1, M. Vantorre1, C. Beels1, J. De Rouck1 and P. Frigaard2

1Department of Civil Engineering,Ghent University,

Technologiepark Zwijnaarde, 9052 Zwijnaarde, BelgiumE-mail: [email protected]

2Department of Civil Engineering,Aalborg University,

Sohngaardsholmsvej 57, 9000 Aalborg, DenmarkE-mail: [email protected]

AbstractThe performance of a wave energy converter array

of twelve heaving point absorbers has been assessednumerically in a frequency domain model. Each pointabsorber is assumed to have its own linear powertake-off. The impact of slamming, stroke and forcerestrictions on the power absorption is evaluated andoptimal power take-off parameters are determined. Formultiple bodies optimal control parameters are notonly dependent on the incoming waves, but also on theposition and behaviour of the other buoys. Applying theoptimal control values for one buoy to multiple closelyspaced buoys results in a suboptimal solution, as will beillustrated. Other ways to determine the power take-offparameters are diagonal optimization and individualoptimization. The latter method is found to increase thepower absorption with about 14% compared to diagonaloptimization.

Keywords: array, constraints, multiple bodies, point absorber,wave energy

Nomenclaturebext = external damping coefficient [kg/s]Bext = external damping matrix (NxN) [kg/s]Bhyd = hydrodynamic damping matrix (NxN) [kg/s]d = buoy draft [m]D = buoy waterline diameter [m]f = wave frequency [Hz]F = force [N]I = identity matrix (NxN) [-]K = stiffness matrix (NxN) [kg/s2]H = wave height [m]m = buoy mass [kg]msup = supplementary mass [kg]M = buoy mass matrix (NxN) [kg]

c© Proceedings of the 8th European Wave and Tidal EnergyConference, Uppsala, Sweden, 2009

Ma = added mass matrix (NxN) [kg]Msup = supplementary mass matrix (NxN) [kg]N = number of buoys [-]n f = number of frequencies [-]q = ratio of maximum power absorption by

N interacting bodies to N isolated bodies [-]q = power absorption ratio of N interacting bodies

and N isolated bodies in suboptimal conditions [-]S = wave spectrum [m2s]t = time [s]T = wave period [s]z = buoy position [m]Z = buoy position vector (Nx1)[m]γ = peak enhancement factor [-]ζ = wave elevation [m]σ = spectral width parameter [-]ω = angular frequency [rad−1]

Subscripts

A = amplitudeabs = absorbedex = excitingf = frequencymax = maximum valuep = peak values, sign = significant

1 IntroductionPoint absorbers are oscillating wave energy

converters with dimensions that are small comparedto the incident wave lengths. In order to absorb aconsiderable amount of power, several point absorbersare grouped in one or more arrays. Some point absorberdevices under development consist of a large structurecontaining multiple closely spaced oscillating bodies.Examples are Wave Star [1], Manchester Bobber [2] andFO3 [3].

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Several theoretical models dealing with interactingbodies have been developed. Budal [4], Evans [5] andFalnes [6] adopted the ‘point absorber approximation’to derive expressions for the maximum power an arraycan absorb. The approximation relies on the assumptionthat the bodies are small compared to the incidentwave lengths, so the wave scattering within the arraycan be neglected while calculating the interactions.A theory accounting more accurately for the bodyinteractions is the ‘plane-wave approximation’ [7–9],which is based on the assumption that the bodies arewidely spaced relative to the incident wavelengths, sothe radiated and circular scattered waves can be locallyapproximated by plane waves. For closely spacedbodies, which is the focus of this paper, this theory is notsuitable as the wide spacing requirement is not fulfilled,except for very short wave lengths. In the majorityof studies, both theories have been applied to arraysfor unconstrained conditions. Contrary to the pointabsorber approximation, the plane wave approximationis also suitable to study the power absorption of anarray in suboptimal conditions, as scattering mightbe relatively important in that case [7, 10]. Heavingpoint absorbers with constraint displacements havebeen studied by McIver [10, 11] by means of the planewave approximation and by Thomas and Evans [12]with the point absorber approximation, although lesssuitable. Motion restrictions appeared to significantlyreduce the power absorption capability of the array forlonger wave lengths [10]. Apart from regular waves,McIver et al. [11] also studied array interactions inirregular unidirectional and directionally spread seasfor a varying number of oscillators between 5 and20, arranged in one or two lines. For unconstrainedmotions constructive interactions between the pointabsorbers were observed for unidirectional irregularnormal waves. When constraints were implementedthis effect disappeared and in random seas it is almostnon-existing in both unidirectional irregular waves andrandom seas.

More exact results on array hydrodynamics can beobtained by the ‘multiple scattering’ theory of Mavrakos[13]. This theory has been extensively compared withthe above mentioned approximate theories (the pointabsorber approximation and plane wave approximation)[14, 15].

With current computer capacity, Boundary ElementMethods (BEM) are becoming more and more importantto investigate interacting point absorbers. Ricci etal. [16] compared results obtained with a BEM codeto the point absorber approximation and optimized thepoint absorber geometry and interbody distance of twoarray configurations, each consisting of 5 floaters inirregular waves with and without directional spreading.Taghipour et al. [3] investigated the interaction of 21heaving point absorbers in a floating platform, known asthe FO3 device, in unconstrained conditions. By meansof a mode expansion method, he was able to reduce thecomputation time to calculate the body responses by

a factor of 10 tot 15. Recently, numerical simulationspredicting the response amplitude of the floatermotions of interacting hemispherical floaters usingthe BEM package WAMIT [17] have been validatedwith experimental tests on line array configurations byThomas et al. [12].

The impact of constraints on a single point absorberhas been studied by De Backer et al. [18], showing thatstroke and force restrictions might have a significantimpact on the power absorption. In that paper theinfluence of slamming and stroke restrictions aswell as limitations on the power take-off (PTO)forces are investigated for an array in a frequencydomain model with WAMIT. The results are comparedwith the performance of an isolated buoy underthe same restrictions. Apart from the total amountof power absorbed by multiple bodies, attentionis paid to the individual power absorption and thedifference in performance between the buoys withina certain configuration. The control parameters areoptimized with different strategies, among themdiagonal optimization and individual optimization.

2 ConfigurationA configuration of 12 buoys in a staggered grid is

considered, as shown in Fig. 1. The buoys are placed ina square, fixed structure with four supporting columnsat the edges. However, the effects of diffraction andreflection of waves on the columns will be furtherneglected in this study. The interdistance between twosuccessive rows is 6.5 m. The incoming waves propagatein the direction of the x-axis, as indicated in Fig. 1. Thebuoys are assumed to oscillate in heave mode only. InFig. 2 the buoy geometry is presented. A buoy consistsof a cone shape with apex angle 90◦ and a cylindricalupper part with a diameter of 5 m. The equilibrium draftof the buoy is 3 m.

Figure 1: Multiple body configuration, dimensions in m.

3 Numerical modelling3.1 Equation of motion

The equation of motion of an oscillating pointabsorber can be described by Newton’s second law:

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Figure 2: Point absorber layout, dimensions in m.

mz = Fex +Frad +Fres +Fdamp +Ftun (1)

where m is the mass of the buoy and z its acceleration.Fex is the exciting force, Frad the radiation force, Fres thehydrostatic restoring force, Fdamp the external dampingforce to extract power and Ftun the tuning force forphase-controlling the buoy. The latter two forces are tobe delivered by the power take-off (PTO) system. Forsimplicity, the PTO is assumed linear.

The equation of motion of N multiple bodies,oscillating in heave in a regular wave with angularfrequency ω , can be expressed as follows with lineartheory in the frequency domain:

−ω2(M+Ma+Msup) Z+ jω(Bext+Bhyd) Z+KZ = Fex(2)

where Z is the complex amplitude of the buoypositions, M the mass matrix of the buoys, and Kthe matrix with hydrostatic restoring coefficientsor stiffness matrix. The added mass matrix andhydrodynamic damping matrix are denoted by Ma andBhyd, respectively. They are both symmetric N x Nmatrices with the hydrodynamic interaction coefficientson the non-diagonal positions. The vector Fex containsthe complex amplitudes of the heave exciting forces.The hydrodynamic parameters Ma, Bhyd and Fex arecalculated with the Boundary Element Method (BEM)software WAMIT [17]. Since the natural frequencyof the buoys is generally smaller than the incidentwave frequencies, the buoys are often tuned towardsthe characteristics of the incident waves to augmentpower absorption. This can be effectuated by latchingtechniques [19], by introducing a supplementary springterm [16] or a supplementary mass term [20]. In thispaper, a tuning force proportional to the acceleration hasbeen implemented by means of a supplementary massmatrix, Msup = m( j)

sup I, with m( j)sup the supplementary

mass for buoy j and I the NxN identity matrix. Inearlier experimental research the supplementary masshas been realized by adding masses on both sides of arotating belt connected to the heaving point absorber[20, 21]. In that way the inertia of the system can bevaried without changing the draft of the buoy. Forpractical applications, the tuning force, associated withthe supplementary mass, is more likely to be realizedby a control mechanism, e.g. by the PTO, rather thanby adding real masses to the device. A linear externaldamping matrix, Bext = b( j)

ext I, has been applied, enablingpower extraction.

The superposition principle is used to obtain the time-averaged power absorption in irregular waves [16]:

Pabs =n f

∑i=1

12

ω2i Z∗i BextZi (3)

where n f is the number of frequencies and Z∗i thecomplex conjugate transpose of Zi. The calculations areperformed for 40 frequencies, ranging between 0.035and 0.300 Hz. All buoys are assumed to be equippedwith their own power take-off system.

3.2 Constraints

Slamming constraint

Slamming is a phenomenon that occurs when thebuoy re-enters the water, after having lost contact withthe water surface. The buoy experiences a slam, whichmay result in very high hydrodynamic pressures andloads. These impacts have a very short duration, with atypical order of magnitude of milliseconds. However,they may cause local plastic deformation of the material[22]. Fatigue by repetitive slamming pressures canbe responsible for structural damage of the material.For this reason, a restriction has been implemented,requiring that the significant amplitude of each buoyposition relative to the free water surface elevationshould be smaller than a fraction α of the draft d of thebuoy:

(z( j)−ζ ( j))A,sign ≤ α ·d (4)

where z( j) is the position of buoy j, ζ ( j) the waterelevation at the center of buoy j and α a parameter that isarbitrarily chosen equal to one. The water elevation hasbeen determined with the incident wave only, therebyneglecting the radiated and diffracted waves from thebuoys.

This slamming restriction might require a decreaseof the tuning parameter msup and/or an increase ofthe external damping coefficient bext . Not only theoccurrence probability of slamming will be reduced bythis measure, but also the magnitude of the associatedimpact pressures and loads will drop, since they aredependent on the impact velocity of the body relative tothe water particle velocity and this impact velocity willdecrease when the control parameters of the buoy areadapted according to the restriction imposed.

Stroke constraint

In practice, many point absorber devices are verylikely to have restrictions on the buoy motion, e.g.imposed by the limited height of the rams in case ofa hydraulic conversion system or by the limited heightof a platform structure enclosing the oscillating pointabsorbers. Therefore a stroke constraint is implemented,imposing a maximum value on the significant amplitudeof the body motion:

z( j)A,sign ≤ z( j)

A,sign,max (5)

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With the position spectrum defined as:S( j)

zi = z( j)2A,i /2∆ f , the significant amplitude of the buoy

motion can be obtained from zA,sign = 2√∫ n f

i=1 S( j)z,i ·d f .

Hence, the constraint on the significant amplitude ofthe buoy motion is not an absolute constraint, buta restriction associated with a statistical exceedanceprobability. In the examples that will be presented in thispaper, a maximum value of the significant amplitude of2.00 m is chosen. Assuming Rayleigh distribution ofthe buoy motions, this restriction means that a strokeof 4.90 m is exceeded for 5.0 % of the waves. Theimplementation of constraints on the body motionincreases the reliability of the linear model, which isbased on the assumption of small body motions.

Force constraint

In some cases, the optimal control parameters formaximum power absorption, result in very large controlforces. The tuning force, in particular, might becomevery high and can even be a multiple of the dampingforce. If this tuning force is to be delivered by thePTO, it might result in a very uneconomic design ofthe PTO system. For this reason it is interesting tostudy the response of the floaters in case the total controlforce is restricted. If the force spectrum is expressedas: S( j)

F,i = F( j)2A,i /2∆ f and the significant amplitude of

the force is defined as F( j)A,sign = 2

√∫ n fi=1 S( j)

F,i ·d f , then thesignificant amplitude of the damping and tuning force,respectively, for buoy j are given by:

F( j)bext,A,sign = 2

√√√√∞∫

0

S( j)Fbext,A

( f )d f (6)

F( j)msup,A,sign = 2

√√√√∞∫

0

S( j)Fmsup,A

( f )d f (7)

A restriction will be imposed on the significantamplitude of the total force, expressed as:

F( j)tot,A,sign = 2

√√√√∞∫

0

(S( j)

Fbext,A( f )+S( j)

Fmsup,A( f ))

d f (8)

Simulation results will be presented where F( j)tot,A,sign

is limited to 100 kN and 200 kN.

3.3 Wave climate

Scatter diagrams based on buoy measurements atWesthinder, on the Belgian Continental Shelf, have beenused to define nine sea states with their correspondingoccurrence probabilities (OP), see Table 1. Westhinderis located 32 km from shore and has a mean water depthof 28.8 m. The majority of calculations will be carriedout for the fifth sea state, characterized by a significantwave height, Hs = 2.25 m and peak period, Tp = 7.22 s.The generated spectrum is based on a parameterizedJONSWAP spectrum [23].

Table 1: Sea states and occurrence probabilities at Westhinderbased on measurements from 1-7-1990 until 30-6-2004(Source of original scatter diagram: Flemish Ministry ofTransport and Public Works (Agency for Maritime and CoastalServices Coastal Division))

Sea state Hs [m] Tp [s] OP [%]1 0.25 5.24 21.582 0.75 5.45 37.253 1.25 5.98 22.024 1.75 6.59 10.655 2.25 7.22 5.146 2.75 7.78 2.277 3.25 8.29 0.798 3.75 8.85 0.219 4.25 9.10 0.07

4 Optimization strategiesThe relative performance of an array is often

expressed by means of the ‘q-factor’, defined as themaximum possible time-averaged total power absorbedby the N bodies in the array devided by N times themaximum time-averaged power absorption by a singlepoint absorber.

q =Pabs,max by array ofNfloaters

N ·Pabs,max by an isolated floater(9)

Since the control matrices are diagonal matrices andthe control parameters are optimized for a certain seastate, rather than for a particular frequency, the powerabsorption will be suboptimal, even in unconstrainedconditions. Hence, for the current purpose, a measureis needed to express and compare the efficiency ofdifferent optimization strategies applied to a given arrayin suboptimal conditions. Therefore a gain factor q isdefined as the ratio of the total power absorbed by thearray to the power absorbed by the point absorbers inisolation, subjected to the same constraints.

Three strategies to determine the control parametersfor multiple bodies will be compared: optimal controlparameters from a single body, diagonal optimizationand individual optimization.

Optimal control parameters from a single body

In the first strategy, the optimal control parametersfrom a single body (OPSB) are applied to all the bodiesin the array. It should be kept in mind that optimalparameters in this case means that these parameters leadto the maximum possible power absorption within theimposed constraints and with the described linear PTOand hence they do not necessarily give the absolutemaximum power absorption of the array. With thismethod all bodies have the same control coefficients.However, the control forces, MsupZ and Bext Z, are notsimilar for the different buoys, since the buoy velocitiesand accelerations differ in amplitude and phase. Ifmsup and bext are the single body optimal parameters

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for a specific sea state and a certain combination ofrestrictions, then the absorbed power for the array isobtained with the following power take-off matrices:

Msup = msup I and Bext = bext I (10)

Diagonal optimization

With the second technique all the buoys still get thesame parameters, but they are optimized with a simplexsearch (SS) method for unconstrained conditions anda sequential quadratic programming (SQP) method forconstrained conditions. The methods are validated withan exhaustive searching (ES) method. The latter methodgives the same results as SS and SQP, however, SS andSQP are much faster if very accurate results need to beobtained. The supplementary mass matrix and externaldamping matrix are similar as those in Eq. 10, but msupand bext are determined so that the total absorbed poweris maximal for the specific array. This technique is calleddiagonal optimization (DO) [16].

Individual optimization

With the last technique the floaters are individuallyoptimized (IO), i.e. for every floater seperate valuesof m( j)

sup and b( j)ext are determined. IO has only been

successfully applied in constrained conditions. Theoptimization is carried out with a SQP method only,since a simple exhaustive searching method wouldrequire too much CPU-time. If nmsup and nbext values of

m( j)sup and b( j)

ext , respectively, are to be evaluated, the totalpower absorption need to be calculated (nmsup · nbext )

12

times. If e.g. 40 values for each control parameter areto be assessed, 2.8 · 1038 power absorption calculationswould be required, which is not feasible. A drawback ofthe SQP algorithm for individual tuning is that it mightconverge to a local maximum, depending on the initialconditions. Hence, the choice of the initial conditionsis very important. Here, the control parameters fromdiagonal optimization have been used as initial valuesfrom which individual tuning parameters are obtained.If the number of buoys were smaller (preferably smallerthan 6), the SQP algorithm could be executed with aset of initial conditions for m( j)

sup and b( j)ext (multistart

algorithm) to increase the chance of reaching theabsolute maximum value. Unfortunately, a multistartapplication is much too time-consuming for twelvebuoys, even if only two initial values per parameter areselected.

5 Results5.1 Unconstrained

In this section, simulations are presented forunconstrained point absorber motions and controlforces. Fig. 3(a) shows the time-averaged powerabsorption for each floater. As expected, the diagonaloptimization (DO) method is more efficient thanapplying the optimal parameters of a single buoy to

the array (OPSB). The power absorption is distributedvery unequally between the buoys: the front buoys (inparticular Nos 3 and 8) absorb about 2.7 times moreenergy than the buoy No 7 in the back. Rather largebuoy motions are observed in Fig. 3(b), showing thesignificant amplitudes of the position of the buoys.Figs. 4(c) and 4(d) present the external dampingcoefficients and supplementary mass coefficients,respectively.

If diagonal optimization is applied, the total powerabsorption is ca 400 kW. This corresponds to an increasewith almost 50 kW compared to OPSB, as can beobserved in Table 2. Although the power absorptionof the array is large, the gain factor q is rather small(46%), since the absorbed power of an isolated buoyis quite high. It must be stressed, however, that thisunconstrained case leads to unpractically large controlforces (up to a value of Ftot,A,sign of 400 kN) and floatermotions, violating the assumptions behind linear theory.Consequently, these power absorption figures will mostlikely not be achieved in practical cases. Furthermore thepower absorption figures do not take into account lossesdue to mechanical friction, turbulence losses, turbine andgenerator losses or any other losses in the conversionsystem and are for that reason not equal to the producedelectrical power.

In unconstrained conditions, an individual tuning(IO) did not lead to any realistic solutions. Some buoysmay be totally tuned towards resonance, oscillating withextremely large amplitudes, while other buoys are keptstill. This is of course not a desired situation at all.Moreover, in unconstrained conditions the optimizationalgorithm has a large chance to find a local maximuminstead of the absolute maximum, sometimes evenwithout finding a solution which is symmetric withrespect to the x-axis. More realistic, constrained caseswill be addressed in the next section.

5.2 Constrained

5.2.1 Slamming and Stroke constraint

The power absorption and gain factor are determinedfor the three optimization strategies, taking into accountthe slamming and stroke restrictions, as explained insection 3.2. Table 2 shows that diagonal optimizationperforms slightly better than the optimal parameters of asingle body: the total power absorption is 381 kW withOPBS and 389 kW with DO. A significant benefit can bemade by individually optimizing the control parametersof the floaters. The power absorption becomes 443 kWwith IO, which is an increase of about 14 % comparedto DO or an increase of 54 kW. This figure correspondswith the amount of power produced by an isolated buoy,and illustrates the large profit than can be made byindividually tuning the buoys in case they are closelyspaced. For the configurations evaluated in [16], thebenefit of IO compared to DO was found to be lessthan one percent. This is most probably due to the factthat the floaters are wider spread from each other in

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1 2 3 4 5 6 7 8 9 10 11 120

10

20

30

40

50

60

70

Floater number

Pow

er a

bsor

ptio

n [k

W]

Unconstrained

OPSBDO

(a) Time-averaged power absorption.

1 2 3 4 5 6 7 8 9 10 11 120

0.5

1

1.5

2

2.5

3

3.5

4

4.5

Floater number

Sig

nific

ant a

mpl

itude

of m

otio

n (z

A, s

ign)[

m]

Unconstrained

OPSBDO

(b) Significant amplitude of motion.

1 2 3 4 5 6 7 8 9 10 11 120

5

10

15

20

25

30

35

40

45

50

55

Floater number

Bex

t [ton

/s]

Unconstrained

OPSBDO

(c) External damping coefficients.

1 2 3 4 5 6 7 8 9 10 11 120

50

100

150

200

250

Floater number

Msu

p [ton

]

Unconstrained

OPSBDO

(d) Supplementary mass coefficients.

Figure 3: Simulation results in unconstrained conditions, with optimal control parameters for a single body and diagonaloptimization.

[16]. The interdistance (center-center spacing) betweentwo successive rows, for which the two methods wereevaluated, varied between 3 and 4 times the diameter D,whereas in the present configuration the interdistance isabout 1.3 D. The larger the interdistance, the more thebuoy behaviour resembles that of an isolated buoy andthe less difference will be found between the differenttechniques. Also the number of floaters is much smallerin [16]: five floaters are considered compared to twelvefloaters in the present configuration. Particularly,the buoys in the back benefit considerably from anindividual tuning, as is shown in Fig. 4(a), presentingthe power absorption for each buoy. With IO, thepower absorption is much better distributed among thefloaters. A maximum factor of 2.6 is found betweenthe individual power absorption of the front buoys andthe rear buoys if DO is used, whereas only a factor of1.9 is observed when IO is utilized. With individualoptimisation the buoys in the front absorb less powerthan with OPSB or DO, however, the buoys in the backbecome more efficient. This is realized by detuning thefront buoys (small value of msup) and increasing theirdamping (larger value of bext ) (Figs. 4(c)-4(d)). Therear buoys, on the other hand, are tuned very well toincrease their velocity amplitude and power absorption.Consequently, the power that is not absorbed anymore

by the front buoys can be absorbed by the buoys inthe rows behind them. This makes the influence ofrestrictions less drastic for an array than for a singlebuoy. These conclusions will be even more pronouncedwhen the constraints are stricter, as they will be in thenext sections.

All the buoys reach the maximum stroke value, so thestroke restriction is dominant (Fig. 4(b)). The slammingconstraint will generally not be critical in the presentedexamples for the given buoy dimensions and sea state.

5.2.2 Slamming, stroke and mild force constraint

In this section the slamming and stroke restrictionsare included together with a force constraint of 200 kNon the significant amplitude of the total control force.Figs. 5(a)-5(d) give the power absorption, significantamplitude of the motion and the control parameters ofeach buoy. The significant amplitude of the dampingforce, tuning force and total force are presented inFigs. 5(e)-5(g) for the three optimization methods. Theforce constraint is clearly the most critical constraint.The buoy motions are considerably smaller than thestroke limit. The force limit on the other hand, is reachedby the front buoys with the OPSB and DO methodsand by all the buoys with IO. The front buoys evenexceed the limit of 200 kN in case of OPSB. If certain

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1 2 3 4 5 6 7 8 9 10 11 120

10

20

30

40

50

60

70

Floater number

Pow

er a

bsor

ptio

n [k

W]

Slamming & stroke constraint

OPSBDOIO


1 2 3 4 5 6 7 8 9 10 11 120

0.5

1

1.5

2

2.5

3

Floater number

Sig

nific

ant a

mpl

itude

of m

otio

n (z

A, s

ign)

[m]


OPSBDOIO


1 2 3 4 5 6 7 8 9 10 11 120

10

20

30

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50

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100

Floater number

Bex

t [ton

/s]


OPSBDOIO


1 2 3 4 5 6 7 8 9 10 11 120

50

100

150

200

250

300

350

Floater number

Msu

p [ton

]


OPSBDOIO


Figure 4: Simulation results in constrained conditions (slamming and stroke restriction), with optimal control parameters for asingle body, diagonal and individual optimization.

control parameters meet the constraints for an isolatedbuoy, they do not necessarily satisfy the same restrictionsapplied to an array. This explains why more poweris absorbed with OPSB (336 kW) than with the DOmethod (332 kW) in this case (see Table 2), althoughthe DO method was found to be more efficient. Again,a large benefit can be made with the IO technique: thepower absorption becomes 379 kW, corresponding withan increase of 46 kW compared to DO, which is evenmore than the power absorbed by an isolated buoy underthe same constraints. Due to the extra force constraint,the power distribution among the floaters has again beenimproved compared to the previous case; the buoys inthe front absorb about 50% more than those in the backwith IO and about double as much than the rear buoysif DO is employed. The gain factors q have risen to0.69 and 0.79 for this restriction case, although the totalpower absorption of the array has decreased due to theextra force constraint. This rise of q can be attributedto the considerable power drop by an isolated buoyunder these restrictions. An isolated buoy loses about25 % of the power absorption, whereas the array losesonly 12 to 14 % compared to the slamming and strokerestriction case. Since the array suffers a bit less fromthe constraints than the isolated buoy, it has an increasedrelative performance, although the absolute performance

is decreased.

5.2.3 Slamming, stroke and strong force constraint

In this case, the same slamming and stroke constraintsare applied, but the force constraint has been increasedtill a maximum value of 100 kN. Figs. 6(a)-6(g) show thesimulation results of this test case. The power absorptionseems to be considerably decreased with this restriction,but it is more equally distributed among the floaters.The largest difference between the power absorption ofthe front and rear buoys is a only factor of 1.7 and 1.4with DO and IO, respectively. In order to satisfy thisstrong force restriction, phase control of the buoys isvery limited. The tuning forces are considerably smallin this case, even smaller than the damping forces.

Again, the restriction is not perfectly fulfilled with theOPSB method. With IO, all floaters have reached themaximum value of the total force, so it is very likelythat the IO method has given the correct, most optimalsolution. This strong force restriction has a harmfulinfluence on the total power absorption, as displayed inTable 2. Compared to the previous case with the mildforce constraint, the absorbed power drops with morethan 40% for the single buoy and with 31% to 33% forthe array.

The results in Table 2 for the different restriction

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combinations illustrate the benefits of phase control onpower absorption. However, an economic optimumneeds to be found between the power absorption profitof phase control and the costs to realise this tuning andto allow e.g. a larger stroke. Since the power absorptionfor an isolated buoy is very small, the gain factor reachesquite large values, up to 0.88. In other words the arrayhas a good relative performance, since the rear buoysare still able to absorb an important share in the power,compared to an isolated buoy.

5.2.4 Angle of wave incidence and effects ofmistuning

So far, irregular waves propagating along the x-axis have been considered. It might be of interest toinvestigate the array behaviour for a different angleof wave incidence βi, e.g. βi = 45◦. In that case, theconfiguration in Fig. 1 can be considered as an alignedgrid. The power absorption values and gain factors arepresented in Table 2 for unconstrained motion as well asfor the combination of slamming, stroke and mild forcerestrictions. It appears that the array performs slightlybetter when the incident waves propagate in the directionof the diagonal of the array (aligned grid), comparedto normal incidence (staggered grid), however, thedifference is not significant. The performance of anarray is strongly dependent on the wave frequency andtherefore the power absorption is calculated for the othersea states at Westhinder (as defined in Table 1). In Fig. 7the gain factor q for DO and IO is shown versus thesea state. A steep rise in q-factor can be noticed fromsea state 4 onwards. This might give the impressionthat the gain factor rises for longer waves. However,the increase is most probably caused by the constraints,which become important for the more energetic waveclasses. Since a single body is relatively more affectedby the constraints, the gain factor rises in larger waves.

In Fig. 7 the difference between an angle of incidenceof 0◦ and 45◦ is very minor for all sea states and nofinal conclusion can be made about the best angle ofincidence. The difference in optimization strategy, onthe other hand is very clear: for all sea states and bothangles of incidence, individual optimization outperformsdiagonal optimization.

In practise, there will be uncertainties on thecharacteristics of the sea state and the real sea statemight not perfectly correspond to the values used in thenumerical calculations. E.g. the real spectrum mightdiffer from the JONSWAP spectrum that has beenemployed, the current angle of wave incidence mightnot be known exactly, etc. Hence, it is important toknow the sensibility of the optimization techniquesto mistuning effects. An example is given in Table 3for unconstrained motion and one case of constrainedconditions. Simulation results are presented for anangle of incidence of 45◦, but the optimal controlparameters are taken from the simulations with normalwave incidence. It is expected that for the individuallyoptimized parameters the effect of mistuning will be

0 2 4 6 8 100.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

Sea state

q-fa

ctor

βi = 0° − DO

βi = 0° − IO

βi = 45° − DO

βi = 45° − IO

Figure 7: q-factors for diagonal and individual optimizationas a function of the nine sea states on the Belgian ContinentalShelf.

more pronounced than for the diagonally optimizedparameters. This appears to be correct, however, theeffect of mistuning is extremely small. For DO thedifference in power absorption is less than 0.50%compared to the results with the correct controlparameters for βi = 45◦. For IO this difference is 1.76%for the studied case. Obviously, the larger the mistuning,the larger will be the impact on the power absorption, inparticular for IO. It needs to be mentioned that with themistuned parameters, the force constraint is still fulfilledfor DO, but not anymore for IO. The problem is causedby buoy No 12, exceeding the force limit by 20%. Thebuoy is tuned as a rear buoy (βi = 0◦), getting a highvalue of the supplementary mass, and becomes rathera front buoy when βi = 45◦, where it is subjected tohigher incident waves. The combination of an increasedsupplementary mass and large buoy motion parametersleads to larger control forces.

6 ConclusionThe behaviour of twelve closely spaced point

absorbers in unconstrained and constrained conditionshas been analysed in irregular unidirectional waves. Thebuoys are assumed to be equipped with a linear powertake-off, consisting of a linear damping and a lineartuning force. In unconstrained conditions the absorbedpower is found to be very unequally distributed amongthe floaters. For the considered sea state, the front buoysabsorbed 2.7 times more power than the rear buoys inunconstrained conditions. For the strongest constraintsthe difference was reduced to a factor of 1.4 to 1.7. Thetotal power absorption of the array is negatively affectedby the implementation of constraints. However, it isobserved that the relative power loss of the array is lessthan for a single body, since mainly the front buoys areaffected by the constraints and to a lesser extent therear buoys. The power absorption of the array has beendetermined in three different ways. Firstly, the optimalparameters of a single body are applied to the array.This turns out not to be an efficient way. Moreover, theconstraints were not always fulfilled for all the bodies

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Figure 5: Simulation results in constrained conditions (slamming and stroke restriction, and mild force restriction: F( j)tot,A,sign ≤

200 kN), with optimal control parameters for a single body, diagonal and individual optimization.

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Figure 6: Simulation results in constrained conditions (slamming and stroke restriction, and strong force restriction: F( j)tot,A,sign ≤

100 kN), with optimal control parameters for a single body, diagonal and individual optimization.

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Table 2: Absorbed power and gain factor q for unconstrained and constrained cases.Sea state: Single body (SB) Multiple bodies (MB)Hs = 2.25 m, Tp = 7.22 s OPSB DO IOConstraints NPabs [kW] Pabs [kW] q [-] Pabs [kW] q [-] Pabs [kW] q [-]βi = 0◦

Unconstrained 872 354 0.41 399 0.46 - -

Slam, stroke 2.00 m 645 381 0.59 389 0.60 443 0.69Slam, stroke 2.00 m, force 200 kN 482 336 0.70 332 0.69 379 0.79Slam, stroke 2.00 m, force 100 kN 286 232 0.81 225 0.79 252 0.88

βi = 45◦

Unconstrained 872 360 0.41 403 0.46 - -Slam, stroke 2.00 m, force 200 kN 482 337 0.70 334 0.69 379 0.79

Table 3: Effects of mistuning - power absorption for βi = 45◦, while the control parameters (CP) are those obtained for βi = 0◦.Sea state:Hs = 2.25 m, Tp = 7.22 s DO IOConstraints Pabs [kW] q [-] Power loss [%] Pabs [kW] q [-] Power loss [%]βi = 45◦, CP from βi = 0◦

Unconstrained 403 0.46 0.01 - -Slam, stroke 2.00 m, force 200 kN 333 0.69 0.45 372 0.77 1.76

in the array, although they were satisfied for the singlebody case. Secondly, diagonal optimization has beenapplied. With this method all buoys have the samecontrol parameters, but they are optimized for the array.In the third method, the buoys get individually optimizedcontrol parameters. This strategy clearly outperformsthe two other methods. By individually optimizingthe control parameters, the power absorption was onaverage increased by 14 %, compared to diagonallyoptimizing them.

AcknowledgementsResearch funded by Ph.D. grant of the Institute

of Promotion of Innovation through Science andTechnology in Flanders (IWT-Vlaanderen), Belgium.

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