Optimisation of wave energy extraction with the Archimedes ... · Ocean Engineering 34 (2007)...

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Ocean Engineering 34 (2007) 2330–2344

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Optimisation of wave energy extraction with the ArchimedesWave Swing

Duarte Valerioa,�, Pedro Beiraob, Jose Sa da Costaa

aIDMEC/IST, TULisbon, Av. Rovisco Pais, 1, 1049-001 Lisboa, PortugalbDepartment of Mechanical Engineering, Instituto Superior de Engenharia de Coimbra, R. Pedro Nunes, 3030-199 Coimbra, Portugal

Received 29 November 2006; accepted 21 May 2007

Available online 2 June 2007

Abstract

This paper addresses the Archimedes Wave Swing (an offshore wave energy converter, which produces electricity from sea waves). It

compares the performances of latching control (a discrete, highly non-linear, intrinsically sub-optimum control strategy), of reactive

control, of phase and amplitude control (two optimum control strategies that involve non-causal transfer functions, which have to be

implemented with approximations, thus rending the control sub-optimum), and of feedback linearisation control (a non-linear control

strategy). From extensive simulations it is concluded that the latter performs clearly better irrespective of the sea state, and leads to a

significant increase of absorbed wave power.

r 2007 Elsevier Ltd. All rights reserved.

Keywords: Wave energy; Renewable energy; Archimedes Wave Swing; Reactive control; Phase and amplitude control; Latching control; Feedback

linearisation control

1. Introduction

Due both to the inexorable rise of oil prices and toenvironmental concerns, renewable energies are receivingincreased attention. Among them, the energy of sea wavesmay play an important role in the near future. Wave energyconverters (WECs) for turning it into electricity are notcommercially competitive—yet. But it is expected that theywill soon be, as sufficiently efficient WECs are developed.Control engineering plays an important role towards thatobjective. This paper contributes to that effort by compar-ing the performance of several control strategies suitablefor the Archimedes Wave Swing (AWS), a WEC of which a2MW prototype (Fig. 1) has already been built, tested atthe Portuguese northern coast during 2004, and thendecommissioned. The aim is to see, by means of simula-tions, which one leads to a better efficiency of thatprototype—and of the ones that will follow.

The paper is organised as follows: Section 2 describes theAWS; the identification of a linearised model thereof is

front matter r 2007 Elsevier Ltd. All rights reserved.

eaneng.2007.05.009

ng author. Tel.: +351 218419119; fax: +351 218498097.

ss: [email protected] (D. Valerio).

addressed in Section 3; control strategies are summarised inSection 4; simulation results are given in Section 5;conclusions are drawn in Section 6.Due to industrial secrecy reasons several parameters of

the AWS have been modified in the models given andemployed.

2. The AWS

The AWS is an offshore, fully submerged (43m deepunderwater), point absorber (that is to say, of neglectablesize compared to the wavelength) WEC. Its two main partsare the silo (a bottom-fixed air-filled cylindrical chamber)and the floater (a movable upper cylinder). Due to changesin wave pressure, the floater heaves (Fig. 1). When theAWS is under a wave top, the floater moves downcompressing the air inside the AWS. When the AWS isunder a wave trough, pressure decreases and consequentlythe air expands and the floater moves up (Beirao et al.,2006).Within the air-filled space formed by the silo and the

floater there are several components needed for thefunctioning of the AWS, among which an electric linear

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Fig. 1. The AWS prototype and its working principle.

D. Valerio et al. / Ocean Engineering 34 (2007) 2330–2344 2331

generator (ELG) to convert the floater’s heave motion intoelectricity. The electric energy produced by the ELG istransferred to shore via a 6 km long undersea cable; afterthat it passes through a converter, and following thisconversion is supplied to the electric grid. The AWS alsohas water dampers that are actuated when the floaterapproaches the mechanical end-stops, to reduce its velocityand avoid a strong collision with them; they are alsoactuated together with the ELG when the latter does notsuffice to supply the force required to adequately controlthe movement of the AWS. These water dampers areotherwise inactive, so as not to hinder wave energyextraction.

It is the force exerted by these two components, the ELGand the water dampers, that we can control. Actually it isalso possible to change the mean value of air pressurewithin the AWS, by means of pumps that add or removesea water (thus changing the volume available for air); butthis is a slow process that takes several minutes, and so it isused to cope with the variations of sea level caused by thetides (that are slower still). To control the AWS copingwith different incoming waves and sea states, it is the ELGand the water dampers we have to resort to. This paperdeals with this latter type of control. It will be assumed thatthere will be no changes in tide, and no changes in the meanpressure of the air inside the AWS.

From the description above it is seen that the AWScan be reasonably expected to behave much like amass–spring–damper system, though with relevant non-linearities.

The AWS was submerged 5 km offshore Leixoes,Portugal. Data for wave climate in several locations inPortugal may be found with the ONDATLAS software(Pontes et al., 2005). The nearest location available is theLeixoes-buoy location ð41�12:20N; 9�5:30WÞ. The corre-sponding data on significant wave height Hs (from troughto crest) and on maximum and minimum values of thewave energy period Te is found in Table 1.

Throughout this paper, simulations are performed usingregular and irregular waves. The first are sinusoidal; valuescongruent with Table 1 for the amplitude and the period ofthe sinusoids are used. For the latter, 12 waves (one foreach month of the year) satisfying the Pierson–Mosko-witz’s spectrum, which accurately models the behaviour ofreal sea waves (Falnes, 2002), were used. This spectrum is

given by

SðoÞ ¼A

o5exp �

B

o4

� �, (1)

where S is the wave energy spectrum (a function such thatRþ10 SðoÞdo is the mean-square value of the waveelevation). The numerical values A ¼ 0:780 (SI) and B ¼

3:11=H2s were used. Values for Hs and for T e (from which

the limits of the frequency range were then found) wereprovided by Table 1.A model of the AWS in the time domain is based on

Newton’s law applied to the floater’s vertical acceleration€x. The equation of motion of the floater is

f pi � f pe � wf � f n � f v � f m � f wd � f lg ¼ ðmf þmwtÞ€x.

(2)

The total mass comprises the mass of the floater mf and thewater trapped inside the floater mwt (these two masses areknown to oscillate together). The total force acting on thefloater is the sum of the forces due to external waterpressure f pe, to internal air pressure f pi, to the weight ofthe floater wf , to a nitrogen cylinder extant inside the AWSf n, to the hydrodynamic viscous drag f v, to mechanicalfriction f m, to the water dampers f wd, and to the ELG f lg,the last two being damping forces, and the ones we cancontrol.Notice that the convention of signs assumes that positive

values are given to the most natural direction—henceamong all forces the only one pointing upwards is f pi. Alsonotice that lower-case letters are being used for variables inthe time-domain; their Laplace transforms (in the fre-quency domain) will be denoted using the correspondingcapitals. (Hence XðsÞ ¼def L½xðtÞ�, F lgðsÞ ¼

defL½f lgðtÞ�, and so

on.) Both capitals and lower-case letters are used forconstants.Since the damping forces f wd and f lg are, as mentioned

above, the ones we can control, let us call f u (control force)to their sum:

f u ¼ f lg þ f wd. (3)

The external pressure force can be decomposed thus:

f pe ¼ f hs þ f rad � f exc. (4)

In (4), f exc (wave excitation force) is the force exerted onthe AWS by the incident sea waves assuming that the

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Table 1

Characteristics of several irregular waves according to ONDATLAS

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

Hs (m) 3.2 3.0 2.6 2.5 1.8 1.7 1.5 1.6 1.9 2.3 2.8 3.1

Te;min (s) 5.8 5.8 5.2 5.5 5.0 4.7 4.6 5.0 5.2 5.3 5.5 5.3

Te;max (s) 16.1 14.5 13.7 14.8 12.2 9.7 11.1 10.5 12.0 12.6 13.3 14.2

300 310 320 330 340 350

-2

-1

0

1

2

3

4

ξ /m

0.5m

2.0m

time / s

Fig. 2. Output of the AWS TDM for regular waves with 10 s of period

and amplitudes of 0.5, 1.0, 1.5, and 2.0m.

D. Valerio et al. / Ocean Engineering 34 (2007) 2330–23442332

floater is not moving, f hs is the hydrostatic force, and f rad isthe force exerted in the AWS by the wave that the floatercreates by its movement. We will assume that the forces canbe superimposed on each other; actually, when waveamplitudes are large compared with the wavelength, (4) isno longer valid and a non-linear expression must be used,but this approximation is suitable for our case.

Complete explicit expressions of all these terms cannotbe given here for lack of space. They may be found forinstance in Pinto (2004); Sa da Costa et al. (2003, 2005).These references also describe an accurate, non-linear,Simulink-based simulator of the AWS, the AWS Time-Domain Model (TDM), that has already been developedimplementing the expressions above. The AWS TDM wasused for the simulations presented in this paper.

3. AWS linear dynamic model identification

Before thinking about the extensive use of a non-linearmodel of the AWS (such as the AWS TDM) for controlpurposes, a linear model approximation of that same WECshould be identified in the first place (Beirao et al., 2007b).This is possible because, even though the AWS is a non-linear system, a sinusoidal input causes a fairly sinusoidaloutput, with an amplitude practically proportional to thatof the input, for all wave periods and amplitudes expectedto occur. Fig. 2 shows this for several regular waves ofdifferent amplitudes.

3.1. Identification procedure

System identification deals with the construction ofmathematical models of dynamical systems using measureddata. In the particular situation of the AWS, very fewexperimental data are available. Thus, a different approachhas been followed. The AWS TDM was used as anemulator of the real non-linear AWS WEC; no controllerwas employed, and hence both control forces (f lg and f wd)were assumed to have only a minimum, unavoidableresidual value. Based on simulation results from theAWS TDM, a linear dynamic model of the AWS wasestimated. Since sea waves are periodic oscillations (even ifnot sinusoidal), an identification method in the frequencydomain, such as the classical method of Levy (1959), is theobvious choice.

Levy’s identification method provides us a transferfunction relating the output data with the input data.The method receives frequency data for the input and the

output of the system and the orders of the numerator andthe denominator of the function that it will provide.In what the input and the output of the identified linear

model are concerned, two possibilities were explored forthe AWS.The first one considers the wave excitation force F exc as

the input and the floater’s vertical velocity _X as the output.F exc and _X data provided by the AWS TDM for regular(sinusoidal) waves with periods from 8 to 14 s (the rangethe AWS was conceived for (Sa da Costa et al., 2003)) wasused. Following the ONDATLAS software, the mostfrequent significant wave height Hs (from trough to crest)is admitted as being equal to 2m. Hence several waves witha 1m amplitude (half of Hs) and different periods wereassumed for the simulations. (Notice that an approxima-tion is involved here, since these waves used for identifica-tion are regular, while those addressed by ONDATLAS arereal, irregular waves.) To apply the Levy identificationmethod, Matlab’s function levy was used (Valerio and Sada Costa, 2007). The data found in Table 2 was used in thatprocess.All combinations of values for the numerator and

denominator orders m and n from 0 to 5 were tried. Onlyidentified models with two poles or more and one zero ormore reproduced the wave frequency behaviour correctly.The identified model structure

_XðsÞF excðsÞ

¼2:171� 10�6s� 6:759� 10�7

0:967s2 þ 0:5874sþ 1(5)

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Table 2

Data used in the identification

Period (s)

4 5 6 7 8 9 10 11 12 13 14

F exc ampl. (kN) 31.88 108.57 202.64 290.46 365.29 423.35 467.48 501.30 527.62 548.44 565.10

X amplitude (m) 0.1195 1.1359 1.1905 1.1939 1.2972 1.3573 1.5558 1.4766 1.4132 1.3803 1.4023

X gain (dB) �108.53 �99.61 �104.62 �107.72 �109.20 �109.88 �109.56 �110.62 �111.44 �111.98 �112.11

X phase ð�Þ �160.20 �85.44 �26.80 �15.77 �14.10 �27.20 �30.72 �16.08 �9.00 �8.31 �11.31_X ampl. ðms�1Þ – – – – 1.0341 1.0535 1.0686 1.0071 0.8813 0.7999 0.7488

_X gain (dB) – – – – �110.96 �112.08 �112.82 �113.94 �115.54 �116.72 �117.56

_Xphase ð�Þ – – – – �111.60 �112.00 �97.92 �81.82 �78.60 �78.09 �76.11

10-1 100 101

-90

-120

-150

gain

/ d

B

10-1 100 101

-90

0

90

180

phase /

°

ω / rads-1

0

1

0

0.5

1

1.5

Imag

Real

-0.5 0.5

-0.5

-1.5

Fig. 3. Model (5); left: Bode diagram (dots mark data used for identification); right: pole-zero map.


with one (non-minimum phase) zero and two (stable,complex conjugate) poles is the one that reproduces theAWS TDM responses making use of as few parameters aspossible. By adding an extra pole at the origin, model

XðsÞF excðsÞ

¼2:171� 10�6s� 6:759� 10�7

sð0:967s2 þ 0:5874sþ 1Þ(6)

relating the wave excitation force to the floater’s verticalposition X is found. Fig. 3 shows the Bode diagram and thepole-zero map of model (5). This model’s drawbacks are itsnon-minimum phase zero (something that may be undesir-able) and its unnecessary complexity (since a simpler onecan be got). So another solution was looked for.

The second possibility for input and output choice wasto consider the wave excitation force as the input and thefloater’s vertical position as the output and provide theredata (see Table 21) to Levy’s identification method. Since itwas found that the former period range had insufficientdata to allow a good identification, it had to be enlarged to4 s to 14 s in order to obtain an acceptable model. Underthis new assumption, the identified model, a second-order

1In that table, a phase lead of 90� could be expected between the phases

of _X and X. Non-linearities, however, prevent this.

transfer function, was

XðsÞF excðsÞ

¼2:259� 10�6

0:6324s2 þ 0:1733sþ 1. (7)

Fig. 4 shows the Bode diagram and the pole-zero map ofthis last identified model.Even though the input of both transfer functions (5) and

(7) is the wave excitation force F exc, the outputs take intoaccount the effects of the radiated force F rad as well, sincethis latter force was included in the AWS TDM.

3.2. Validation of identification results

Six hundred seconds (10min) long simulations werecarried out, employing the AWS TDM (for the non-linearcase) and Simulink implementations of (5) and (7) (for thelinear cases). The root mean-square errors, given by

RMS ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

600

Z 600

0

ðX� ~XÞ2 dt

s(8)

( ~X being the estimate of the floater’s vertical position), aregiven, for several significant simulations, in Tables 3 and 4.

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10-1 100 101

-90

-120

-150

gain

/ d

B

10-1 100 101

-180

-90

0

phase /

°

ω / rads-1

0

1

0

0.5

1

1.5

Imag

Real

-0.5 0.5

-0.5

-1.5

Fig. 4. Model (7); left: Bode diagram (dots mark data used for identification); right: pole-zero map.

Table 3

Root mean-square errors for the simulations with regular waves

Wave amplitude (m) Model Wave period (s)

8.0 10.0 12.0 14.0

0.5 (5) 0.2001 0.2417 0.3528 0.4005

(7) 0.0886 0.1203 0.0674 0.0604

0.75 (5) 0.2717 0.3376 0.4912 0.5642

(7) 0.1919 0.2523 0.1476 0.1326

1.0 (5) 0.3380 0.4293 0.6100 0.7059

(7) 0.3319 0.4211 0.2639 0.2400

1.25 (5) 0.4105 0.5206 0.7139 0.8283

(7) 0.5181 0.6376 0.4259 0.3911

Table 4

Root mean-square errors for the simulations with irregular waves


Model (5) 0.5490 0.5039 0.5096 0.5316 0.3677 0.4161 0.3296 0.3283 0.3776 0.4576 0.4490 0.6602

Model (7) 0.4227 0.3472 0.2622 0.2386 0.1357 0.1375 0.1177 0.1229 0.1423 0.1991 0.2986 0.3828


A 100 s slice corresponding to March (a significant month)is highlighted for illustration purposes in Fig. 5.

From these results, it is seen that model (7) reproducesthe AWS TDM behaviour more accurately; it alsorequires less parameters than (5), and its structure issimilar to the one normally assumed in the literature(Falnes, 2002, e.g.). Actually, (5) performs slightly betterthan (7) for regular waves of low period and highamplitude. But these cases are a minority, and simulationswith irregular waves (with which (7) is systematicallybetter) are deemed more important since they are expectedto reproduce the behaviour of real sea waves moreaccurately.

There is an additional reason to prefer model (7), relatedto the resistance R, which is the real part of the inverse of

the transfer function from the wave excitation force to thefloater’s vertical velocity:

RðoÞ ¼ ReF excðjoÞ_XðjoÞ

� �, (9)

R may be frequency dependent, but it is physicallyimpossible that it should be negative (Falnes, 2002).Indeed, R is always positive for (7), and actually it doesnot even depend on o, since, by definition (9),

R ¼ Re0:6324ðjoÞ2 þ 0:1733joþ 1

2:259� 10�6jo

� �

¼0:1733

2:259� 10�6¼ 7:6715� 104. ð10Þ

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10-2 100 102

-15

-10

-5

0

5x 105

R / N

·s·m

-1

model (7)

model (5)

ω / rad·s-1

Fig. 6. Evolution of R for both models.

300 320 340 360 380 400

-2

-1.5

1

-0.5

0

0.5

1

1.5

2

positio

n / m

Irregular wave for March

non-linear force-position force-velocity

time / s

Fig. 5. Floater’s vertical position for AWS TDM and identified linear

models (100 s long period out of 600 s).


But, for some frequencies, (5) leads to a negative value ofR, since

RðoÞ ¼ Re0:967ðjoÞ2 þ 0:5874joþ 1

2:171� 10�6jo� 6:759� 10�7

� �

¼ð4:2221o2 � 1:4795Þ106

10:3170o2 þ 1. ð11Þ

Both (10) and (11) are plotted in Fig. 6. This seems todenote an inappropriate structure of the model identified inthe case of (5).

For these reasons, model (7) was the one chosen.

4. Control strategies for maximising absorbed wave energy

Four control strategies will be addressed in this paper.The first two are optimum control strategies, in the sensethat they maximise (in theory) the absorption of energy. In

practice, they can only be implemented with approxima-tions that cause a decrease in their efficiency—to an extentthat justifies considering other control strategies, that areintrinsically sub-optimum because they can never (evenunder the most favourable conditions) attain the efficiencythat optimum control has in theory. The two sub-optimumstrategies considered below are latching control andfeedback linearisation control.

4.1. Optimum control

This section closely follows Falnes (2002).Let us rewrite (7) as

XðsÞF excðsÞ

¼1

ms2 þ Rsþ S, (12)

where mass m, resistance R, and stiffness S are given bym ¼ 2:8017� 105, R ¼ 7:6715� 104, and S ¼ 4:4267� 105

(in SI units). The transfer function in (12) is the same as

m€xðtÞ þ R_xðtÞ þ SxðtÞ ¼ f excðtÞ. (13)

Defining complex-valued phasors f exc and_x for f exc and

_x, respectively,

f excðtÞ ¼f exc

2eiot þ

f�

exc

2e�iot, ð14Þ

_xðtÞ ¼_x2eiot þ

_x�

2e�iot, ð15Þ

we can rewrite (13) as

eiot f exc � Rþ iomþS

io

� �_x

� �

þ e�iot f�

exc � R� iom�S

io

� �_x�

� �¼ 0, ð16Þ

where o is the frequency. Defining an impedance Z

Z ¼ Rþ i om�S

o

� �, (17)

it is possible to rewrite (16) as

eiot f exc � Z _x� �

þ e�iot f�

exc � Z� _x�� ¼ 0. (18)

To satisfy (18) for all values of time t the followingcondition must be verified:

_x ¼f exc

Z) j _xj ¼

jf excj

jZj. (19)

Definition (17) can be rewritten as Z ¼ Rþ iX . Thereal part R ¼ Re½Z� is called resistance and wasalready mentioned above in (9); the imaginary partX ¼ Im½Z� ¼ om� ðS=oÞ is called reactance.As was already mentioned above, Z takes into account

both the mechanical and the radiation effects (since theradiation force was included in the AWS TDM). Hencethese R and X are the intrinsic resistance and the intrinsicreactance, respectively. Thus they shall henceforth be

ARTICLE IN PRESSD. Valerio et al. / Ocean Engineering 34 (2007) 2330–23442336

denoted Ri and X i, and Z will be similarly denoted Zi

(intrinsic impedance).Suppose now that the control force f u is applied to the

AWS, to ensure that the conditions leading to maximumwave energy absorption are met (or at least approached).Then

Zi_x ¼ f exc þ f u 3 ZiðoÞ _XðoÞ ¼ F excðoÞ þ FuðoÞ. (20)

Let us define a control impedance Zu by

f u ¼ �Zu_x 3 FuðoÞ ¼ �ZuðoÞ _XðoÞ. (21)

The real and imaginary parts of Zu are, respectively,termed control resistance Ru ¼ Re½Zu� and control reac-tance X u ¼ Im½Zu�.

The time-averaged absorbed wave power PuðtÞ can begiven by

PuðtÞ ¼ �f uðtÞ_xðtÞ (22)

and the absorbed wave energy Wu can be obtainedintegrating (22):

Wu ¼ �

Z þ10

f uðtÞ_xðtÞdt. (23)

Considering that f u and _x are real functions, i.e.,F�uðoÞ ¼ Fuð�oÞ and _X

�ðoÞ ¼ _Xð�oÞ, Wu can, by applying

Parseval’s theorem, be given by

Wu ¼ �1

2p

Z þ1�1

FuðoÞ _X�ðoÞdo. (24)

Knowing that Wu is real,

FuðoÞ _X�ðoÞ ¼ Re½FuðoÞ _X

�ðoÞ�

¼ 12½FuðoÞ _X

�ðoÞ þ F�uðoÞ _XðoÞ�, ð25Þ

it is possible to rewrite (24) as

Wu ¼1

2p

Z þ10

½�FuðoÞ _X�ðoÞ � F�uðoÞ _XðoÞ�dt. (26)

Since Ri is positive, it will be convenient to add andsubtract the term F excðoÞF�excðoÞ=2Ri to the integrand of(26), and finally Wu is now given by (omitting thefrequency argument in order to simplify the notation)

Wu ¼1

2p

Z þ10

jF excj2

2Ri�jF excj

2

2Ri� Fu

_X�� F�u

_X� �

do

¼1

2p

Z þ10

jF excj2

2Ri�

a2Ri

� �do. ð27Þ

In (27), aðoÞ is the so-called optimum condition coefficient,given by

aðoÞ ¼ F excðoÞF�excðoÞ þ 2Ri FuðoÞ _X�ðoÞ þ F�uðoÞ _XðoÞ

.

(28)

There are two strategies—reactive control (also calledcomplex conjugate control) and phase and amplitudecontrol—to find the optimum conditions. Both of themare based on the same requisite, which is to prove that a is

never negative, i.e., aðoÞX0. Hence, it is when a ¼ 0 thatWu is maximal.

4.1.1. Reactive control

A first way of proving that a is never negative is asfollows. Solving (20) in order to F exc, multiplying it by itscomplex conjugate, F�excðoÞ ¼ Z�i ðoÞ _XðoÞ � F�uðoÞ, andomitting the frequency argument, the following expressionis obtained:

F exc ¼ Zi_X� Fu ) F excF

�exc

¼ FuF�u þ ZiZ�i_X _X�� FuZ�i

_X�� F�uZi

_X. ð29Þ

From (17) and its complex conjugate, one possibleexpression for Ri is

2Ri ¼ Zi þ Z�i 3 Ri ¼Zi þ Z�i

2. (30)

Multiplying both sides by Fu_X�þ F�u

_X

2RiðFu_X�þ F�u

_XÞ ¼ ZiFu_X�þ ZiF

�u_Xþ Z�i Fu

_X�þ Z�i F�u

_X.

(31)

Inserting (29) and (31) in (28) and simplifying thesymmetric terms

a ¼ FuF�u þ ZiZ�i_X _X�� FuZ�i

_X�� F�uZi

_Xþ ZiFu_X�

þ ZiF�u_Xþ Z�i Fu

_X�þ Z�i F�u

_X

¼ ðFu þ Zi_X�ÞðF�u þ Zi

_X�Þ. ð32Þ

To complete the proof about the non-negativity of a, (32)can be rewritten recovering again the frequency argument

aðoÞ ¼ jFuðoÞ þ Z�i ðoÞ _XðoÞj2X0. (33)

From (33), one particular optimum condition attainedwhen aðoÞ ¼ 0 is

FuðoÞ_XðoÞ

¼ �Z�i ðoÞ. (34)

Comparing the optimum condition in (34) with (21), it ispossible to see that Zu is equal to the complex conjugate ofthe intrinsic impedance Z�i , i.e., Zu ¼ Z�i . Another addi-tional conclusion is that under the optimum condition in(34) the reactive component X iðoÞ þ XuðoÞ of the totalimpedance ZiðoÞ þ ZuðoÞ is cancelled. This is an inherentconsequence of resonance. Reactive control is representedby (34), since it is related to the fact that X u (the imaginarypart of Zu) will cancel X i (the imaginary part of Zi).Alternatively, this control strategy is also called complexconjugate control since it is related to the fact that theoptimum control impedance Zu;OPT equals the complexconjugate of the intrinsic impedance Z�i , i.e.,

Zu;OPT ¼ Z�i . (35)

Consequently, from (27), the theoretical maximum ab-sorbed wave energy Wu;MAX is

Wu;MAX ¼1

2p

Z þ10

jF excðoÞj2

2Rido. (36)

ARTICLE IN PRESSD. Valerio et al. / Ocean Engineering 34 (2007) 2330–2344 2337

Notice that the integrand of (36) is positive since Ri is alsopositive.

4.1.2. Phase and Amplitude control

Alternatively to the previous one, another proof that a isnever negative is as follows. We now solve (20) in order toFu, instead of F exc. Inserting the obtained result and itscomplex conjugate, F�uðoÞ ¼ Z�i ðoÞ _X

�ðoÞ � F�excðoÞ, in (28),

and omitting once more the frequency argument, in orderto simplify the notation,

a ¼ F excF�exc þ 2RiðZi

_X _X�� F exc

_X�þ Z�i

_X� _X� F�exc

_XÞ

¼ F excF�exc þ 2RiðZi þ Z�i Þ

_X _X�� 2RiF exc

_X�� 2RiF

�exc_X.

ð37Þ

Inserting (30) in (37)

a ¼ F excF�excð2RiÞ

2 _X _X�� 2RiF exc

_X�� 2RiF

�exc_X

¼ ðF exc � 2Ri_XÞðF�exc � 2Ri

_X�Þ. ð38Þ

To complete the proof about the non-negativity of a, (38)can be rewritten recovering again the frequency argument

aðoÞ ¼ jF excðoÞ � 2Ri_XðoÞj2X0. (39)

Since Ri is positive, from (39) an alternative optimumcondition can be written as

F excðoÞ ¼ 2Ri_XðoÞ 3

_XðoÞF excðoÞ

¼1

2Ri. (40)

The optimum condition in (40) is called phase andamplitude control since it means that _X must be in phasewith F exc, and also that the amplitude of the floater’svertical velocity j _Xj must be equal to F exc=2Ri.

4.1.3. Feasibility of optimum control strategies

A serious problem with both optimum conditions is thatthey include (in the general case) non-causal transferfunctions: �Z�i ðoÞ in (34), and 1=2Ri in (40) (recall that inthe general case Ri may vary with frequency). This lastoptimum condition even requires foreknowledge of thewave excitation force, which in practice is available bymeans of predictions based on data measured by a buoy (orbuoys) placed near the AWS. To make things worse, bothconditions often lead to control forces that assist the waveexcitation force; in other words, instead of extractingenergy from the waves, we will be supplying energy to thewaves. This will of course happen only in a small fractionof the time and is necessary to extract the maximumpossible wave energy during the remaining time. But if it isimpossible to do so we will be limited to a sub-optimumsolution. Actually all approximations indulged in to maketransfer functions in (34) and (40) causal will also decreasewave energy extraction and place us in a sub-optimumsolution. But being sub-optimum should not be seen as amajor drawback. At least it can be realisable, somethingwhich optimum solutions cannot.

Both control strategies can also be applied to a non-linear WEC, provided that a valid linear model thereof,

similar to (12), exists. The controller will be designed usingthe linear model and then applied to the non-linear WEC.Results cannot, of course, be expected to be as good as theywould with a linear plant.

4.1.4. Implementation

The material in this subsection allows conceiving twodifferent optimum control strategies suited to the AWS:reactive control (or complex conjugated control) and phaseand amplitude control (Valerio et al., 2007b).In what the application of optimum control to the linear

second-order model of the AWS (7) is concerned, aSimulink block diagram comprising the two controlstrategies mentioned above is shown in Fig. 7. The oneused can be chosen by means of switch 1 (set for reactivecontrol in Fig. 7). The second switch allows choosing themodel of the AWS (the linear model (12) or the non-linearAWS TDM).In that diagram, reactive control is implemented repla-

cing the non-causal transfer function �Z�i ðsÞ with�Z�i ðsÞ=ðsþ 1Þ, the extra pole placed at �1 ensuringcausality. Several locations have been tested for the pole,and the one leading to a larger absorption of wave energywas kept. An alternative procedure would have been toidentify from the frequency response of �Z�i a causal,stable, approximate transfer function with a similarresponse in the frequency range of interest; this approachwas pursued, but led to no acceptable results.A proportional controller is used together with phase

and amplitude control to drive the floater’s verticalvelocity to the optimum value reckoned by 1=2Ri. Noticethat in our case this transfer function is constant(and hence causal). The controller was obtained maximis-ing the absorbed wave energy with the MatLab functionfminsearch (simplex direct search method), the opti-mum being 5:1348� 104. Integral and derivative terms(forming a PID controller) did not improve results.Absorbed wave energy is obtained with a variation of(23), because f u corresponds to the force exerted both bythe ELG and the water dampers, but the energy absorbedby the latter is wasted. Only the energy absorbed by theELG is used; hence we will have

Wu ¼ �

Z þ10

f lgðtÞ_xðtÞdt. (41)

The implementation of optimum control together withthe non-linear AWS TDM is again that of Fig. 7. Absorbedwave power is obtained with a Simulink block simulatingthe ELG (with non-linear dynamics and saturations).

4.2. Phase control by latching

Because of the difficulties associated with putting thefloater’s vertical velocity in phase with the wave excitationforce, a sub-optimum alternative we can resort to is to latchthe floater during some periods of its oscillation andunlatch it so that it will be (as nearly as possible) in phase

ARTICLE IN PRESS

Fig. 7. Block diagram for optimum control.

Table 5

Latching force in kN

Wave period (s) Wave amplitude (m)

0.25 0.5 1.0 1.5 2.0

8 95 285 570 855 1425

10 95 285 760 1520 1995

12 190 380 1235 1900 2185

14 190 380 1520 2185 1995


with the wave excitation force; more precisely, the floater willbe latched when its velocity vanishes, and released when it ispredicted that its maximum (or minimum) will coincide (intime) with the maximum (or minimum) of the wave excitationforce (Falnes, 1993, 2002; Greenhow and White, 1997;Babarit et al., 2004). This latching control is sub-optimumsince it can never achieve the wave energy absorptionefficiency that the optimum control would achieve.

Latching is clearly a discrete, highly non-linear form ofcontrol. In what concerns the AWS, latching is achieved byactuating the water dampers so as to prevent (as much aspossible) the floater from moving; unlatching meansturning the water dampers off to let the floater go about(as much as possible) freely.

The following unlatching strategies were implemented(Beirao et al., 2006):

(1)
The latching time is constant. This is only reasonablewhen the incident wave is regular (sinusoidal). Hencethis strategy was used for testing only, and will not beaddressed further.
(2)
When the floater is latched, the duration of the lastunlatched period is obtained. The next unlatchedperiod is assumed to be going to last the same as theprevious one. The floater’s vertical velocity is assumedto have its maximum (or minimum) precisely at themiddle of that time interval. So the latching time isreckoned for that velocity maximum (or minimum) tocoincide in time with the next maximum (or minimum)of the wave excitation force. The force required fromthe water dampers to latch the floater is constant.
(3)
The same as above, say that the force required from thewater dampers to latch the floater depends on the
amplitude and period of the incoming wave, larger wavesrequiring a larger force and smaller waves requiring asmaller force. The forces for each wave amplitude andperiod are those necessary to latch effectively the floaterwhen the incident wave is regular and has the requiredamplitude and period. Values are given in Table 5 and areinterpolated and extrapolated as needed.

(4)
Same as above, but the duration of the next unlatchedperiod is assumed to be equal to the last one correctedaccording to the ratio between the next wave amplitudeand the previous.
(5)
Same as above, but the duration of the next unlatchedperiod is assumed to be equal to half of the floater’snatural period of oscillation, which is 11 s. Thus, the floateris unlatched 2.75 s (one quarter of the natural period)before the next maximum (or minimum) of the waveexcitation force (Falnes, 1993, 1997; Eidsmoen, 1998).
4.3. Feedback linearisation control

The dynamics described by (2) are far from being linear.The expressions for nearly all the forces involved show that

ARTICLE IN PRESSD. Valerio et al. / Ocean Engineering 34 (2007) 2330–2344 2339

they depend from variables such as floater’s verticalposition, velocity, or acceleration. Heavy non-linearities,both smooth (with continuous derivatives of any order)and hard (without continuous derivatives; this refers tosuch common non-linearities as saturations, dead-zones, orhysteresis), are always present.

This makes the AWS a suitable candidate for a non-linear control strategy called feedback linearisation control(Valerio et al., 2007a). Its aims are to provide a controlaction judiciously chosen to cancel the non-linear dynamicsof the plant, so that the closed-loop dynamics will be (asmuch as possible) linear (Slotine and Li, 1991). Thismethod can be applied to plants that can be put intocompanion form; this is the case of (2), since it suffices tosolve it in order to €x.

From (2)–(4), we have

�f u ¼ ðmf þmwtÞ€x� f pi þ f hs þ f rad � f exc þ f n

þ f v þ f m þ wf . ð42Þ

Let us provide a control action given by

f u ¼ f pi � f hs � f rad � f n � f v � f m � wf þ n. (43)

This is possible because there are explicit (non-linear)expressions for all the forces involved in the right-handmember of (43). Here n is an additional term that will bechosen to provide the desired (linear) closed-loop dy-namics. Replacing (43) into (42), we will have the followingdynamics:

�n ¼ ðmf þmwtÞ€x� f exc. (44)

Three different values for n were considered.

4.3.1. First control strategy

Let

n ¼ 0. (45)

Then, from (44),

f exc ¼ ðmf þmwtÞ€x. (46)

In other words, the floater’s vertical acceleration will beproportional to the wave excitation force, according toNewton’s law, as though there were no other effects at all.

4.3.2. Second control strategy

Let

n ¼ �ðmf þmwtÞ€xþ _x

max j f excj

2:2. (47)

Replacing this in (44) we get

f exc ¼_xmax j f excj

2:23 _x ¼

2:2

max j f excjf exc. (48)

In other words, the floater’s vertical velocity will be inphase with the wave excitation force. In (47) and (48),constant 2:2 appears because the nominal value for thefloater’s velocity that the AWS should work with is 2.2m/s(Polinder et al., 2004). Also notice that, since it will be

assumed (as seen below) that f exc is known beforehand,max j f excj will be a constant value.This control law was chosen because, as shown above in

Section 4.1, the optimum control of an oscillating WEC isthe one providing a velocity in phase with the waveexcitation force.

4.3.3. Third control strategy

The original version of the AWS TDM was implementedtogether with a simple proportional controller that may easilybe replaced by another one to test any desired control strategy.This original controller provides a control force given by

jf uj ¼ j_xjkpj

_x� _xstpj, ð49Þ

j_xstpj ¼2p10

3:5

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1�

x3:5

� �2s

if jxjo3:5m;

0 if jxjX3:5m:

8>><>>: ð50Þ

In (49), kp is the adjustable gain of the proportional controller.Constant 3.5m shows up in (50) because it is the maximumpossible amplitude for the floater’s vertical oscillations, whileconstant 10 s shows up because it is a reasonable value for theperiod of an incoming wave (Pinto, 2004). Suppose that wewant to follow the set point reference _xstp for the floater’svelocity, using feedback linearisation control. Then we wouldlike to have

ðmf þmwtÞ€x ¼ ðmf þmwtÞ

d_xstpdt

. (51)

By comparison with (44), we see that we must have

n ¼ �ðmf þmwtÞd_xstpdtþ f exc. (52)

4.3.4. Implementation

The three control laws (defined by (43) together with oneof (45), (47), or (52)) from the previous section wereimplemented in the AWS TDM. Concerning (3), wheneverthe ELG was able to exert the control force f u all alone, thewater dampers were not resorted to (f wd ¼ 0, f lg ¼ f u).They were used only when f u was beyond the limits ofthe ELG.

5. Results

5.1. Simulations

Six hundred seconds (10min) long simulations werecarried out to test these control strategies. As mentionedabove, full knowledge of how the incident wave will behavein the future is assumed, as done for instance in Falnes(1997); Eidsmoen (1998); Babarit et al. (2004). This unrealassumption will have to be dropped in future research, but,for now, the independent problem of wave prediction,either from past data or from measurements done aroundthe WEC (Naito and Nakamura, 1985), was not tackled,

ARTICLE IN PRESS

Table 6

Power in kW obtained under several regular waves (figurative data)

Wave amplitude (m) 0.5 0.75 1.0 1.25

Wave period (s) 14.0 12.0 10.0 8.0

AWS TDM, no control 1.0 3.2 10.2 16.7

Linear model, reactive control 6.2 17.6 39.8 78.2

% increase from no control 520 450 290 368

AWS TDM, reactive control 23.8 43.0 93.7 124.1


AWS TDM, phase and amplitude control 17.3 47.8 87.8 98.0


AWS TDM, latching strategy 2 1.9 7.8 7.3 70.6

% increase from no control 90 144 �28 323







AWS TDM, feedback linearisation, strategy 1 19.2 51.7 97.2 122.3







but postponed to some later opportunity. In all cases, theabsorbed wave energy is given by (41) with integrationlimits from 0 to 600 s. From these simulations, values forthe absorbed wave power (time-averaged in the case ofirregular waves) are given in Tables 6 and 7. Forcomparison purposes, absorbed wave power when theAWS has no control at all are also given.2 Notice that thereis only one case, that of reactive control, in which it ispossible to present results from simulations with the linearmodel (7), since the pole added to make the controllercausal leads immediately to a sub-optimum strategy. Whenno control is applied, there is no control force, and henceno energy extraction, in that case. And when phase andamplitude control is employed, everything being linear andno saturations existing, control works too well and valuesobtained have magnitudes absolutely impossible to obtainwith the AWS. Finally, by their very nature, latchingcontrol and feedback linearisation control cannot besimulated with a linear model.

5.2. Comments

The main conclusion to take from the results obtainedwith the AWS TDM is that nearly all control strategies

2Values given in this paper for absorbed wave power when the AWS has

no control follow Valerio et al. (2007a, b); Beirao et al. (2007a) and are

higher than those given in Beirao et al. (2006). This is because a residual

force exerted by the ELG that had been neglected was now taken into

account, which seems to be more correct.

significantly improve the performance of the AWS in whatwave energy absorption is concerned. Since nowadays themajor problem of WECs is their low efficiency, these arevery satisfactory and promising results. But there aresignificant differences between them (Beirao et al., 2007a).Concerning the two optimum control strategies, it is seen

that they do not work equally well in practice: reactivecontrol works best during the whole year, while phase andamplitude control always lags behind, and providesdisappointing results from May to September (when thereis less wave energy). Fig. 8 (showing 20 s highlights, fromthe simulations corresponding to March and June, of thefloater’s velocity, together with the wave excitation force,for the several control strategies) gives an insight into thereason why this is so (similar results are obtained for allmonths). Phase and amplitude control manages to put thefloater’s velocity nearly in phase with the wave excitationforce; reactive control does the same but not so efficiently.Nevertheless, the magnitude of the floater’s velocity isclearly larger in this latter case, and so the higher phasedifference is more than compensated. Results obtained (forirregular waves) with the linear model and those obtainedwith the AWS TDM can be related using a linearregression:

power obtained with the AWS TDM ¼ �27:12 kW

þ 1:514� power obtained with the linear model. ð53Þ

Even though this regression has a significant error (themaximal residual being 14.5 kW), it shows that the trend

ARTICLE IN PRESS

Table 7

Power in kW obtained under several irregular waves (figurative data)


AWS TDM, no control 14.3 11.4 9.1 7.7 4.5 4.5 3.5 3.4 4.5 6.7 9.9 13.6

Linear model, reactive control 60.6 51.9 59.1 42.1 30.1 36.8 28.5 23.8 33.1 43.2 50.9 73.5

% increase from no control 324 355 549 447 569 718 714 600 636 545 414 440

AWS TDM, reactive control 79.1 65.5 50.1 45.3 18.9 14.2 10.8 12.2 21.8 35.5 55.4 73.5


AWS TDM, phase & ampl. ctrl. 77.1 61.6 38.7 35.2 10.2 7.2 4.9 6.0 12.4 24.7 47.3 67.6


AWS TDM, latching strategy 2 42.3 41.0 33.5 32.7 17.2 14.5 11.0 13.9 19.8 27.3 37.7 42.2







% increase from no control 255 207 105 99 �27 �47 �57 �50 �11 52 156 191

AWS TDM, feedback lin., strat. 1 85.2 69.5 47.0 43.5 14.6 10.6 7.4 9.0 17.3 31.8 56.2 75.6






410 415 420 425 430

-2.5

-2

-1.5

1

-0.5

0

0.5

1

1.5

2

velo

city / m

s-1

; fo

rce / M

N

410 415 420 425 430

-1

0

1

velo

city / m

s-1

; fo

rce / M

N

Irregular wave for June

time / s

0.5

-0.5

-1.5

time / s


Fexc

none

reactive

phase and amplitude

latching, strat. 3

feedback lin., strat. 2

Fig. 8. Wave excitation force and floater’s vertical velocity.


obtained with the (much simpler) linear simulations isreliable and allows inferring the behaviour of the non-linear model.

Concerning latching control, results for regular wavesshow that latching strategies 3 and 4 have nearly thesame performance (to no surprise: since the wave is regular,the duration of each heave motion is always the same,and there is no need of correcting the duration of the lastunlatched period; the usefulness of such a correction—ifany—is to be assessed with irregular waves). Resultsfor irregular waves show that winter and summer monthsare rather different. During summer (loosely defined as theMay–September period), when there is less energy available

in waves, strategies 2, 3, and 4 are comparable; 3 isalways the best (with one single exception, and that by anarrow margin). During the rest of the year (theOctober–April period), when there is more energyavailable, strategies 3 and 4 are clearly better than allothers. Strategy 5, though improving energy absorptionover the situation without control, never leads to accep-table results. It is clear that strategy 4 is not a goodimprovement over 3. Its more complicated algorithmseldom leads to a better performance. From this analysis,it is clear that strategy 3 is the one to choose if latchingcontrol is resorted to, and will be the only one addressed infurther analysis.


Concerning feedback linearisation control, the energyabsorption increases observed with all strategies are oftenachieved in spite of the desired linear closed-loop dynamicsnot being obtained. This is because, with strategy 1, thefloater’s acceleration only rarely is in phase with the waveexcitation force, and with strategy 3 the set point for thefloater’s velocity is always very far from being attained(consequently the energy absorption improvement is not ashigh as with the other strategies). Strategy 2, on the otherhand, achieves (as seen in Fig. 8) better results in causingfloater’s velocity and wave excitation force to be in phase(especially in what irregular waves are concerned), andhence the better results achieved. The deviations from thedesired behaviour are likely to derive from the simplifica-tions that had to be admitted when designing the controller,namely assuming that the actuators (the ELG and the waterdampers) respond immediately and have no saturations(when this is not the case, for each has its own internaldynamics, and saturation values; this is especially the case ofthe water dampers, and so f wd will not always follow its setpoint). Since results show that strategy 2 is the best, in whatwave energy absorption is concerned, this is the one tochoose if feedback linearisation control is resorted to, andwill be the only one addressed in further analysis.

Overall, feedback linearisation is clearly the best controlstrategy, in what absorbed wave energy is concerned, andthis happens all over the year, for all sea states. (The case ofthe regular wave of 0.5m of amplitude and period of 14 sdoes not seem to be relevant.) This is clear from theevolution of wave energy absorption with time for theseveral control strategies given in Fig. 9.

The analysis of the plots in Fig. 8 provides an insightinto the reason why feedback linearsation is the best. Therequirement that wave excitation force and floater’svelocity be in phase, as seen in (40), is reasonably fulfilledin the case of feedback linearisation, of latching, and ofphase and amplitude control, but no so well accomplishedby reactive control. (Notice that the latching of the floater

0 100 200 300 400 500 6000

1

2

3

4

5x 107

ab

so

rbe

d e

ne

rgy /

J


none

phaseand

amplitude

latching, strategy 3

feedback linearisation, strategy 2

reactive

time / s

Fig. 9. Evolution of absorbed wave energy with time (figurative data).

is not perfect. The ELG and the water dampers do notsuffice to effectively prevent the floater from moving, butthey can hinder it well enough.) When phase and amplitudecontrol is employed, the amplitude of the floater’soscillations is relatively small (as can be seen from thesmall values of the velocity), and this accounts for the poorperformance. At a first glance, latching control might seemto have to be better than feedback linearisation control,since the floater’s vertical oscillations have a largeramplitude. The reason why this is not so is seen in theplots of Fig. 10. The absorbed wave energy (41) iscomputed as the integral of the product of two oscillatingvariables (the force exerted by the ELG and the floater’svertical velocity). If both these oscillations have the sameamplitude, the absorbed wave energy will increase as thephase difference decreases, since the time period when theproduct is negative will diminish. For the same phasedifference between the two variables, absorbed waveenergy will increase with the increase of the magnitude ofeither of the variables. And, indeed, the force exerted bythe ELG is clearly much larger with feedback linearisationthan with latching control. That is why absorbed waveenergy attains its maximum with feedback linearisation.Also notice that latching and feedback linearisation are

the control strategies requiring the larger efforts from thewater dampers, that nearly are not needed with reactivecontrol or phase and amplitude control. The energy theyabsorb is dissipated, but nevertheless what the ELGreceives is more than if they were not used.The two (supposedly) optimum control strategies are not

the best performing ones, likely because they are sub-optimally implemented. The approximations used toensure causality prevent them from performing as well asthey should in the ideal case (the only one really optimum).

5.3. Variations around the year

Table 1 shows that the energetic content of waves is not thesame around the year. This is reflected in the variations ofwave power Pu absorbed by the AWS, whatever the controlstrategy employed. Fig. 11 attempts to explain thesevariations as functions of Hs, that suffices to explain muchof the monthly variations (the maximum and minimumvalues of the wave energy period T e have also been tested aspossible explanations but seem to be irrelevant). It is possibleto use the least-squares method to find relations between Pu

and Hs. These can be given by linear expressions, but it ismore reasonable (as can be seen by visual inspection of theplots) to use quadratic expressions, as follows:

No control:

Pu ¼ 2:71H2s � 6:55Hs þ 7:30. (54)

Reactive control:

Pu ¼ 8:59H2 � 0:34Hs � 8:78. (55)
s

ARTICLE IN PRESS

410 415 420 425 430

-1

-0.5

0

0.5

1

1.5Reactive control

time / s

velo

city / m

·s-1

; fo

rce / M

N

velo

city / m

·s-1

; fo

rce / M

N

velo

city / m

·s-1

; fo

rce / M

N

velo

city / m

·s-1

; fo

rce / M

N

Fwd

Flg

velocity

410 415 420 425 430

-1

-5

0

1Phase and amplitude control

410 415 420 425 430

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2Control by latching

410 415 420 425 430

-6

-4

-2

0

2

4

6Control by feedback linearisation

time / s

time / stime / s

0.5

Fig. 10. Evolution of the control forces with time (figurative data) for a March irregular wave.

1 1.5 2 2.5 3 3.5

0

20

40

60

80

100

120

Feb

absorb

ed p

ow

er

/ kW

May

Jan

Mar

Jun

Dec

Nov

Oct

Sep

Aug

Jul

Apr

none

reactive

phase and amplitude

latching, strat. 3

feedback lin., strat. 2

Hs / m

Fig. 11. Evolution of the absorbed wave power with Hs.


Phase and amplitude control:

Pu ¼ 18:41H2s � 45:01Hs þ 31:06. (56)

Latching control, strategy 3:

Pu ¼ 5:21H2s þ 30:07Hs � 47:65. (57)

Feedback linearisation control, strategy 2:

Pu ¼ 9:45H2s þ 14:05Hs � 28:46. (58)

6. Conclusions

The main conclusion to take is that phase and amplitudecontrol are better than no control, reactive control is betterthan phase and amplitude control, latching control (withthe best strategy conceived) is better than reactive control,and feedback linearisation control (with the best strategyconceived) is the best. This is true for all months of theyear, and also for most regular waves (though not for all).Nevertheless, irregular waves are expected to be closer tothe real sea states the AWS will have to work in, and sotheir results are deemed more significant.In principle, reactive control and phase and amplitude

control should perform better. But these two optimum


(non-causal) control strategies cannot be implemented inpractice without approximations and modifications (tomake sure that all transfer functions are causal) that makethem sub-optimum. And so latching control and feedbacklinearisation control (that are intrinsically sub-optimum)are not in disadvantage.

Further refinements in control algorithms may bepossible. The influence of the control algorithm on theperformance of the converter that supplies the grid is stillto be investigated. Algorithms for estimating the incidentwave and the consequent wave excitation force from datacollected by buoys placed around the WEC or by someother means have to be studied and improved. But themain future task will certainly be the application of thesecontrol strategies to the second generation of AWSprototypes—first in simulation, then in hardware, to helpmaking them economically viable alternative sources ofelectrical power.

Acknowledgements

Duarte Valerio was partially supported by grant SFRH/BPD/20636/2004 of FCT, funded by POCI 2010, POS C,FSE, and MCTES. Pedro Beirao was partially supportedby the ‘‘Programa do FSE-UE, PRODEP III, acc- ao 5.3,III QCA’’. Research for this paper was partially supportedby grant PTDC/EME-CRO/70341/2006 of FCT, fundedby POCI 2010, POS C, FSE and MCTES.

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