POWERENG 2007, April 12-14, 2007, Setubal, Portugal

Linear model identification of the Archimedes Wave Swing

Pedro BeirdolInstituto Superior de Engenharia de Coimbra

Department of Mechanical EngineeringRua Pedro Nunes, 3030-199 Coimbra, Portugal

pbeirao@isec .pt

Abstract- This paper uses Levy's identification method tobuild linear, second-order models for the Archimedes WaveSwing (AWS), an off-shore, fully-submerged, point absorberwave energy converter, expected to behave much like a mass-spring-damper system, though with relevant non-linearities.Since very few experimental data is available, data from anaccurate non-linear simulator of the AWS was used. One ofthe identified models yields a satisfactory performance, andcan now be used for the development of control strategies forthe AWS.

I. INTRODUCTION

Sea waves may become in a near future an importantsource of renewable energy only if devices capable ofcompeting with other proven technologies (like wind energyor solar energy) are developed. In the current stage of devel-opment such devices require yet a great deal of research. Thispaper about the Archimedes Wave Swing (AWS) a waveenergy converter (WEC) of which a 2 MW prototype hasalready been built (Fig. 1), tested at the Portuguese northerncoast during 2004, and then decommissioned intends tobe a contribution towards that objective, by identifying(using a so-called wave frequency-amplitude analysis) anapproximate linear model for this WEC, fit for conceivingand testing control strategies.

Fig. 1. The 2 MW AWS prototype

1 Pedro Beirao was partially supported by the "Programa do FSE-UE,PRODEP III, accao 5.3, III QCA".

2 Duarte Valerio was partially supported by grant SFRH/BPD/20636/2004of FCT, funded by POCI 2010, POS C, FSE and MCTES.

Research for this paper was partially supported by POCTI-SFA-10-46-IDMEC.

Duarte Valerio2, Jose Sa' da CostaTechnical Univ. of Lisbon, Instituto Superior TecnicoDepartment of Mechanical Engineering - GCARAv. Rovisco Pais 1, 1049-001 Lisboa, Portugal

{dvalerio,sadacosta}@dem.ist.utl.pt

The paper is organised as follows: section II brieflypresents the AWS; section III introduces Levy's identifica-tion method, of which the results are given in section IV;conclusions are drawn in section V.

II. THE AWS

The AWS is an off-shore, fully-submerged (43 m deepunderwater), point absorber (that is to say, of neglectablesize compared to the wavelength) WEC. Its main two 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. 2). When the AWSis under a wave top, the floater moves down compressing theair inside the AWS. When the AWS is under a wave trough,pressure decreases and consequently the air expands and thefloater moves up [1].The floater's heave motion is converted into electricity by

an electric linear generator (ELG). The AWS can hence beexpected to behave much like a mass-spring-damper system,though with relevant non-linearities.

III. AWS LINEAR DYNAMIC MODEL IDENTIFICATION

An accurate non-linear simulator of the AWS, the AWSTime Domain Model (TDM), has already been developedand implemented in Matlab [4], [6], [7]. But before thinkingabout the extensive use of a non-linear model of the AWSfor control purposes, a linear model approximation of thatsame WEC should be identified in the first place.

System identification deals with the construction of math-ematical models of dynamical systems using measured data.In the particular situation of the AWS, due to operationalproblems of the prototype mentioned above, very few exper-imental data is available. Thus, a different approach has been

Fig. 2. AWS working principle

1-4244-0895-4/07/$20.00 C 2007 IEEE 660

-90

- -120

-90

- -120

-1501o-0

180

° 90a)

Q- 0-

-90 -1

10

100co / rads-1

100

101

101

-1501o-0

0O

0

U) -900Q

-180 -

10

100co / rads- 1

100

101

101

Fig. 3. Bode diagram of model (22); dots mark data used for identification Fig. 5. Bode diagram of model (24); dots mark data used for identification

1.5r 1.5

0.50 0.5

E U

-0.50 -0.5

0Real

0.5 0Real

0.5

Fig. 4. Pole-zero map of model (22)

followed. The AWS TDM was used as an emulator of thereal non-linear AWS WEC. Based on simulation results fromthe AWS TDM, a linear dynamic model of the AWS wasestimated. Since sea waves are periodic oscillations (evenif not sinusoidal), an identification method in the frequencydomain, such as the classical method of Levy, is the obviouschoice. It should be noticed, however, that due to industrialsecrecy reasons several parameters of the AWS TDM havebeen modified.

Levy's identification method [3] is as follows. Let us sup-pose we have a plant G with a known frequency behaviour.Let us suppose we want to model it using a transfer function

bo+ bis + 2S2+ ...+bms m 2obkSkG(s) 1+i+ 2s2+ +a~ Sn 1 inakSk~The frequency response of (1) is given by

I k 0 bk~(j)k N(w) a (w) +j/3(w)(G ) + k=la)gkz ( )dGKIw)

+ l:=1ak(w) Dwo O Ow) +jT(w)(2)

where N and D are complex-valued and a, /3, o and T (thereal and imaginary parts thereof) are real-valued. The errorbetween model and plant, for a given frequency w, will be

=(w) G(jw) N(w)d Dwo (3)

Fig. 6. Pole-zero map of model (24)

Minimising this norm (or its square) would be an obviousbut difficult way of adjusting the parameters of (1). Insteadof this, Levy's method minimises the square of the norm of

c(w)D(w) = G(j)D() -N(w) (4)

(which is easier). Let us call this new variable E (and omitthe frequency argument w to simplify the notation); we willhave

E = GD-N

[Re(G) + jlm(G)] (a + T)[Re(G)(-Im(G)T -a]+j [Re(G)T+ Im(G)(J-]

(a +j/3)

(5)

The square of the norm of E is

El2 = [Re(G)( -Im(G)T _a]2+ [Re(G)T + Im(G)( _ /3]2

From (2) we see thatm

a (w) = ZbkRe [(j)k]k=O

3(bi) = :bklM [(jk=o

(6)

(7)

(8)

661

1 k

-1

1.o 5

Regular wave, amplitude 0.5 m, period 10 s1.5

--- force-position force-velocity non-linear2

Regular wave, amplitude 1.0 m, period 8 s

1.5

E 0.5

0

Q 0 Li

E 0.53.2o 0 u0~

-0.5

-0.5FX

r00 320 340 360 380 400time / s

Regular wave, amplitude 0.75 m, period 10 s2

1.50

E

0.5o0

0

10-0.5

-16 83Y00 320 340 360 380 400

time / sRegular wave, amplitude 1.0 m, period 12 s

2.5

2e

1.5-S

E 1 !I4

0

-0.5

300 320 340 360 380 400time / s

Regular wave, amplitude 1.0 m, period 10 s2.5

2-

1.5-

0~~~~~~~~~00~~~~~~~~~~~~i,0t i sI

-1 .1 ,1- X

300 320 340 360 380 400time / s

Regular wave, amplitude 1.0 m, period 14 s3

E 1

0

-0.5 i

300 320 340 360 380 400time / s

Regular wave, amplitude 1.25 m, period 10 s3

300 320 340 360 380 400time / s

Fig. 7. Floater's position for AWS TDM and identified linear models (100 slong period out of 600 s)

If we differentiate E 2 with respect to one of the coefficientsbk and equal the derivative to zero, we shall have

E 1L

,--0

0

300 320 340 360 380 400time / s

n

or(X() = I + 1: akRe [(jO)k ] (9)

&bkX [Re(G)u -Im(G)T- a] Re [(j)k]+ [Re(G)T + Im(G)J-3] Im [(j)k] = 0 (11)

If we differentiate E 2 with respect to one of the coefficientsak and equal the derivative to zero, we shall have

OakE = 0 XOak

nk=l CfaIlm(G)]2 +[Re(G)]2 Re [(jw)k]T3akJm [(iw)k1 (I 0)T(Wn) ) +T {[lm(G)]2 + [Re(G)]2}Im 0(jw)k]

662

Re(G)Re [(jw)k] }

Re(G)lm [(jw)k] }

(12)

0

The m+1 equations given by (11) and the n equations givenby (12) form a linear system that may be solved so as to findthe coefficients of (1). Usually the frequency behaviour ofthe plant is known in more than one frequency (otherwiseit is likely that the identified model will be rather poor).Let us suppose that it is known at f frequencies. Then thesystem to solve, given by (11) and (12) written explicitly on

coefficients a and b, is

C D eawhere

f{-Re [(jWp)] Re [(jWp)c]

p=1

Am [( O.p). I Im I[( 0.P },I = 0...mAc= 0 ...m

f

Bl,c = E {Re [(jWp)1] Re [(jwp)c] Re [G(jwp)]p=1

+Jm [(jWp)l] Re [(jwp)c] Im [G(jwp)]-Re [(JWp)1] Im [(jwp)c] Im [G(Jwp)]+ Im [(jWp)1] Im [(jWp)c] Re [G(jp)]},

I 0...mAc= in

Cl c

f{-Re [(jWp)1] Re [(jop)c] Re [G(w)p)]

p=1

+Jm [(jWp)1] Re [(jwgp)] Im [G(jwp)]-Re [(jWp)1] Im [(jwgp)] Im [G(jwp)]-Im [(jWp)1] Im [(jwp)c] Re [G(jwp)] },

nAc= 0 mT

Dl ,c = , [({Re [G(jw)]} + {Im [G(jp)]1}2)p=1

{Re [(jwp)1] Re [(jgp)c]+Im [Ujop)ll Im [(jop)c]}2

I I ... nAc l ...n

bo

b

bm

a

a

fel,i -Re [(jWp)1] Re [G(jwp)]

p=1

-Im [(jWp)1] Im [G(jwp)]},I = 0

f91,l {-Re [(jWp)1]

p=l

({Re [G(jwp)]}2 + {Jm [G(jp)1}2) },1 in (21)

The input and the output of the system must be chosen inadvance. Two possibilities were explored for the AWS.The first one considers the wave excitation force Fexc as

the input and the floater's velocity as the output. Fexc and

data provided by the AWS TDM for regular (sinusoidal)

waves with periods from 8 s to 14 s (this is the range theAWS was conceived for [6]) was used. According to wave

data provided by ONDATLAS software [5] for the Leix6es-buoy location (41012.2' N, 905.3' W), near the test site wherethe AWS prototype was submerged (at the Portuguese coast,5 km offshore Leix6es), the most frequent significant wave

height H, (from trough to crest) is admitted as being equalto 2 metres. Hence several waves with a 1 m amplitude(half of Hs) and different periods were assumed for thesimulations. (Notice that an approximation is involved here,since these waves used for identification are regular, whilethose addressed by ONDATLAS are real, irregular waves.)To apply the Levy identification method, Matlab's functionlevy was used [8]. The data found in Table I was used inthat process.

Regarding Levy's identification method, all combinationsof values for the numerator and denominator orders m and n

from 0 to 5 were tried. Only identified models with two polesor more and one zero or more reproduced the wave frequencybehaviour correctly. The identified model structure

B(s) 2.171 x 10-6- 6.759 x 10-7Fexc(s) 0.967s2 + 0.5874s + 1

with one (non-minimum phase) zero and two (stable, com-

plex conjugate) poles is the one that reproduces the AWSTDM responses making use of as few parameters as possible.By adding an extra pole at the origin, model

(s)

Fexc (S)

2.171 x 10-6S - 6.759 x 10-7s (0.967s2 + 0.5874s + 1)

(23)

relating the wave excitation force to the floater's position is

17) found. Figures 3 and 4 show, respectively, the Bode diagramand the pole-zero map of model (22). This model's majordrawback is its complexity. So another solution was looked

(18) for.The second possibility is to consider the wave excitation

force Fexc as the input and the floater's position as the

output and provide this data (see Table I) to Levy's identi-(19) fication method. Since it was found that the former period

range had insufficient data to allow a good identification,it had to be enlarged to 4 s to 14 s in order to obtain an

acceptable model. Under this new assumption, the identifiedmodel, a second order transfer function, was

B(s) = 2.259 x 10-6 (24)

(20) Fexc(s) 0.6324s2 + 0.1733s + 1

663

+ a {Jm(G)lm [(jw)k]+ { J-lm(G)Re [(jw)k]

Al,c

TABLE I

DATA USED IN THE IDENTIFICATION

TABLE II

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 3.1Te,min / s 5.8 5.8 5.2 5.5 5.0 4.7 4.6 5.0 5.2 5.3 5.5 5.3Te,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

Figures 5 and 6 show, respectively, the Bode diagram andthe pole-zero map of this last identified model.Even though the input of both transfer functions (22) and

(24) is the wave excitation force Fexc, the outputs take intoaccount the effects of the radiated force as well, since it wasincluded in the AWS TDM.

IV. RESULTS

600 s (10 min) long simulations were carried out, employ-ing the AWS TDM (for the non-linear case) and Simulink im-plementations of (22) and (24) (for the linear cases). Modelswere submitted to several regular (sinusoidal, with differentperiods and amplitudes) and irregular incident waves withamplitudes and periods within the ranges expected to occur,

based in data supplied by ONDATLAS software for theLeix6es-buoy location.

In what concerns irregular waves, Pierson-Moskowitz'sspectrum, that accurately models the behaviour of real sea

waves [2], was used. This is given by

S(W)=A

exp (4) (25)

where S is the wave energy spectrum (a function suchthat f0o" S(w)dw is the mean-square value of the wave

elevation). The numerical values A = 0.780 (SI) and B =

3.11/H,2 were used. Values for the significant wave heightHs (from trough to crest) and for the limits of the frequencyrange (corresponding to the maximum and minimum valuesof the wave energy period Te) were those provided byONDATLAS for the twelve months of the year (see Table II).From these simulations, 100 s slices corresponding to

seven regular waves and two significant months are high-lighted in Figures 7 and 8; the root mean-square errors, givenby

R1 600

~~~RMS= (600

A\ 2dt

( being the estimate of the floater's position) are given,for these simulations and others similar thereto, in Tables IIIand IV.From these results, it is seen that model (24) reproduces

the AWS TDM behaviour more accurately; it also requiresless parameters than (22), and its structure is similar to theone normally assumed in the literature (e.g. [2]). Actually,(22) performs slightly better than (24) for regular waves

of low period and high amplitude. But these cases are a

minority, and simulations with irregular waves (with which(24) is systematically better) are deemed more importantsince they are expected to reproduce the behaviour of realsea waves more accurately.

There is an additional reason to prefer model (24), relatedto the resistance R, which is the real part of the inverse ofthe transfer function from the wave excitation force to thefloater's velocity:

R(w) = Re F (27)

R may be frequency dependent, but it is physically impos-sible that it be negative [2]. Indeed, R is always positivefor (24) (actually in this particular case it does not even

depend on w). But, for some frequencies, (22) leads to a

negative value of R, as seen in Fig. 9. This seems to denotean inaccurate identification in the case of (22).

For these reasons, model (24) was the one chosen.

V. CONCLUSIONS

From the last section it can be seen that the identifiedsecond order linear model approximation (24) yields a sat-isfactory performance. This model is a first step towards thedevelopment of control strategies for the AWS. This wouldbe difficult with a non-linear model only. Now this simplermodel can be used for controller design and testing, and thenon-linear model for validation.

664

[period/s [ 4 5 6 7 8 9 10 I 11 12 13 14Feic ampl. I kN 31.88 108.57 202.64 290.46 365.29 423.35 467.48 501.30 527.62 548.44 565.10

amplitude I m 0.1195 1.1359 1.1905 1.1939 1.2972 1.3573 1.5558 1.4766 1.4132 1.3803 1.4023Egain / dB 108.53 99.61 104.62 107.72 109.20 109.88 109.56 110.62 111.44 111.98 112.11

phase / 0 -160.20 -85.44 -26.80 -15.77 -14.10 -27.20 -30.72 -16.08 -9.00 -8.31 -11.31ampl. / ms 1 1.0341 1.0535 1.0686 1.0071 0.8813 0.7999 0.7488gain / dB -110.96 -112.08 -112.82 -113.94 -115.54 -116.72 -117.56phase / ° -111.60 -112.00 -97.92 -81.82 -78.60 -78.09 -76.11

Irregular wave for March

0.5 X

E

.° 0 !

00 .

-0.5

non-linear---force-position -- force-velocity

320 340 360time / s

380 400 )00 320 340 360time / s

Fig. 8. Floater's position for AWS TDM and identified linear models (100 s long period out of 600 s)

TABLE III

ROOT MEAN-SQUARE ERRORS FOR THE SIMULATIONS IN FIG. 7 AND OTHERS SIMILAR THERETO

Wave Wave period/s l_1amplitude / m Model 8.0 [ 10.0 [ 12.0 14.0

0.5 (22) 0.2001 0.2417 0.3528 0.40050.5 (24) 0.0886 0.1203 0.0674 0.0604

0.75 (22) 0.2717 0.3376 0.4912 0.5642(24) 0.1919 0.2523 0.1476 0.1326

1.0 (22) 0.3380 0.4293 0.6100 0.7059.___________ (24) 0.3319 0.4211 0.2639 0.2400

1.25 (22) 0.4105 0.5206 0.7139 0.8283.___________ (24) 0.5181 0.6376 0.4259 0.3911

TABLE IV

ROOT MEAN SQUARE ERRORS FOR THE SIMULATIONS IN FIG. 8 AND OTHERS SIMILAR THERETO

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Decmodel (22) 0.5490 0.5039 0.5096 0.5316 0.3677 0.4161 0.3296 0.3283 0.3776 0.4576 0.4490 0.6602model (24) 0.4227 0.3472 0.2622 0.2386 0.1357 0.1375 0.1177 0.1229 0.1423 0.1991 0.2986 0.3828

ACKNOWLEDGMENT

The authors acknowledge the very useful comments andsuggestions made by Professor Johannes Falnes (ProfessorEmeritus at the Norwegian University of Science and Tech-nology) on the identification process.

10515

10

.z 5-z

model (24)

0-

model (22)

10-2 100co / rads-1

102

REFERENCES

[1] P. Beirao, D. Valerio, and J. Sa da Costa. Phase control by latchingapplied to the Archimedes Wave Swing. In Proceedings of the 7thPortuguese Conference on Automatic Control, Lisbon, 2006.

[2] J. Falnes. Ocean waves and oscillating systems. Cambridge UniversityPress, Cambridge, 2002.

[3] E. Levy. Complex curve fitting. IRE transactions on automatic control,4:37-44, 1959.

[4] P. Pinto. Time domain simulation of the AWS. Master's thesis,Technical University of Lisbon, IST, Lisbon, 2004.

[5] M. T. Pontes, R. Aguiar, and H. Oliveira Pires. A nearshore wave

energy atlas for Portugal. Journal of Offshore Mechanics and ArcticEngineering, 127:249-255, August 2005.

[6] J. Sa da Costa, P. Pinto, A. Sarmento, and F. Gardner. Modelling andsimulation of AWS: a wave energy extractor. In Proceedings of the4th IMACS Symposium on Mathematical Modelling, pages 161-170,Vienna, 2003. Agersin-Verlag.

[7] J. Sa da Costa, A. Sarmento, F. Gardner, P. Beirao, and A. Brito-Melo. Time domain model of the Archimedes Wave Swing wave energyconverter. In Proceedings of the 6th European Wave and Tidal EnergyConference, pages 91-97, Glasgow, 2005.

[8] D. Valerio and J. Sa da Costa. Identification of fractional models fromfrequency data. In J. Tenreiro Machado, J. Sabatier, and 0. Agrawal,editors, Advances in Fractional Calculus: theoretical developments andapplications in Physics and Engineering. Springer-Verlag, 2006.

Fig. 9. Evolution of R for both models

665

2

1.5

0.5

0.

-0.5-

E

0

00-

-1 .5

300 380 400

Irregular wave for June