Linear model identification of the Archimedes Wave Swing

Transcription

1 Linear model identification of the Archimedes Wave Swing Pedro Beirão Instituto Superior de Engenharia de Coimbra Department of Mechanical Engineering Rua Pedro Nunes, Coimbra, Portugal Duarte Valério,JoséSádaCosta Technical Univ. of Lisbon, Instituto Superior Técnico Department of Mechanical Engineering GCAR Av. Rovisco Pais, 49- Lisboa, Portugal Abstract This paper uses Levy s identification method to build linear, second-order models for the Archimedes Wave Swing (AWS), an off-shore, fully-submerged, point absorber wave energy converter, expected to behave much like a massspring-damper system, though with relevant non-linearities. Since very few experimental data is available, data from an accurate non-linear simulator of the AWS was used. One of the identified models yields a satisfactory performance, and can now be used for the development of control strategies for the AWS. I. INTRODUCTION Sea waves may become in a near future an important source of renewable energy only if devices capable of competing with other proven technologies (like wind energy or solar energy) are developed. In the current stage of development such devices require yet a great deal of research. This paper about the Archimedes Wave Swing (AWS) a wave energy converter (WEC) of which a MW prototype has already been built (Fig. ), tested at the Portuguese northern coast during 4, and then decommissioned intends to be a contribution towards that objective, by identifying (using a so-called wave frequency-amplitude analysis) an approximate linear model for this WEC, fit for conceiving and testing control strategies. The paper is organised as follows: section II briefly presents the AWS; section III introduces Levy s identification 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 deep underwater), point absorber (that is to say, of neglectable size compared to the wavelength) WEC. Its main two parts are the silo (a bottom-fixed air-filled cylindrical chamber) and the floater (a movable upper cylinder). Due to changes in wave pressure, the floater heaves (Fig. ). When the AWS is under a wave top, the floater moves down compressing the air inside the AWS. When the AWS is under a wave trough, pressure decreases and consequently the air expands and the floater moves up []. The floater s heave motion is converted into electricity by an electric linear generator (ELG). The AWS can hence be expected 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 AWS Time Domain Model (TDM), has already been developed and implemented in Matlab [4], [6], [7]. But before thinking about the extensive use of a non-linear model of the AWS for control purposes, a linear model approximation of that same WEC should be identified in the first place. System identification deals with the construction of mathematical models of dynamical systems using measured data. In the particular situation of the AWS, due to operational problems of the prototype mentioned above, very few experimental data is available. Thus, a different approach has been Fig.. The MW AWS prototype Pedro Beirão was partially supported by the Programa do FSE-UE, PRODEP III, acção 5.3, III QCA. Duarte Valério was partially supported by grant SFRH/BPD/636/4 of FCT, funded by POCI, POS C, FSE and MCTES. Research for this paper was partially supported by POCTI-SFA--46- IDMEC. Fig.. AWS working principle

2 9 9 gain / db gain / db 5 ω / rad s 8 5 ω / rad s phase / o 9 phase / o Fig. 3. Bode diagram of model (); dots mark data used for identification Fig. 5. Bode diagram of model (4); dots mark data used for identification Imag Imag Real Fig. 4. Pole-zero map of model () Real Fig. 6. Pole-zero map of model (4) followed. The AWS TDM was used as an emulator of the real non-linear AWS WEC. Based on simulation results from the AWS TDM, a linear dynamic model of the AWS was estimated. Since sea waves are periodic oscillations (even if not sinusoidal), an identification method in the frequency domain, such as the classical method of Levy, is the obvious choice. It should be noticed, however, that due to industrial secrecy reasons several parameters of the AWS TDM have been modified. Levy s identification method [3] is as follows. Let us suppose we have a plant G with a known frequency behaviour. Let us suppose we want to model it using a transfer function Ĝ(s) = b + b s + b s b m s m +a s + a s a n s n = The frequency response of () is given by m k= Ĝ(jω)= b k(jω) k m k= b ks k + n k= a ks k () + n k= a k(jω) = N(ω) k Dω = α(ω)+jβ(ω) σ(ω)+jτ(ω) () where N and D are complex-valued and α, β, σ and τ (the real and imaginary parts thereof) are real-valued. The error between model and plant, for a given frequency ω, will be ɛ(ω) =G(jω) N(ω) Dω (3) Minimising this norm (or its square) would be an obvious but difficult way of adjusting the parameters of (). Instead of this, Levy s method minimises the square of the norm of ɛ(ω)d(ω) =G(jω)D(ω) N(ω) (4) (which is easier). Let us call this new variable E (and omit the frequency argument ω to simplify the notation); we will have E = GD N = [Re(G)+jIm(G)] (σ + jτ) (α + jβ) = [Re(G)σ Im(G)τ α] + j [Re(G)τ + Im(G)σ β] (5) The square of the norm of E is E = [Re(G)σ Im(G)τ α] From () we see that α(ω) = β(ω) = +[Re(G)τ + Im(G)σ β] (6) m b k Re [ (jω) k] (7) k= m b k Im [ (jω) k] (8) k=

3 .5 Regular wave, amplitude.5 m, period s force position force velocity non linear.5 Regular wave, amplitude. m, period 8 s Regular wave, amplitude.75 m, period s Regular wave, amplitude. m, period s Regular wave, amplitude. m, period s Regular wave, amplitude. m, period 4 s Regular wave, amplitude.5 m, period s σ(ω) = + τ(ω) = n a k Re [ (jω) k] (9) k= n a k Im [ (jω) k] () k= Fig. 7. Floater s position for AWS TDM and identified linear models ( s long period out of 6 s) If we differentiate E with respect to one of the coefficients b k and equal the derivative to zero, we shall have E = b k [Re(G)σ Im(G)τ α] Re [ (jω) k] +[Re(G)τ + Im(G)σ β] Im [ (jω) k] = () If we differentiate E with respect to one of the coefficients a k and equal the derivative to zero, we shall have E = a k σ {[Im(G)] +[Re(G)] } Re [ (jω) k] + τ {[Im(G)] +[Re(G)] } Im [ (jω) k]

4 + α { Im(G)Im [ (jω) k] Re(G)Re [ (jω) k]} () + β { Im(G)Re [ (jω) k] Re(G)Im [ (jω) k]} = The m+ equations given by () and the n equations given by () form a linear system that may be solved so as to find the coefficients of (). Usually the frequency behaviour of the plant is known in more than one frequency (otherwise it is likely that the identified model will be rather poor). Let us suppose that it is known at f frequencies. Then the system to solve, given by () and () written explicitly on coefficients a and b, is [ A B C D where A l,c = B l,c = C l,c = D l,c = b = a = e l, = ][ b a ] = [ e g Re (jωp ) l] Re [(jω p ) c ] p= ] (3) Im [ (jω p ) l] Im [(jω p ) c ] }, l =...m c =...m (4) Re (jωp ) l] Re [(jω p ) c ] Re [G(jω p )] p= +Im [ (jω p ) l] Re [(jω p ) c ] Im [G(jω p )] Re [ (jω p ) l] Im [(jω p ) c ] Im [G(jω p )] + Im [ (jω p ) l] Im [(jω p ) c ] Re [G(jω p )] }, l =...m c =...n (5) Re (jωp ) l] Re [(jω p ) c ] Re [G(jω p )] p= +Im [ (jω p ) l] Re [(jω p ) c ] Im [G(jω p )] Re [ (jω p ) l] Im [(jω p ) c ] Im [G(jω p )] Im [ (jω p ) l] Im [(jω p ) c ] Re [G(jω p )] }, l =...n c =...m (6) f [({Re [G(jω p )]} + {Im [G(jω p )]} ) p= { [ Re (jωp ) l] Re [(jω p ) c ] +Im [ (jω p ) l] Im [(jω p ) c ] }], l =...n c =...n (7) b... (8) b m a.. a n (9) Re (jωp ) l] Re [G(jω p )] p= Im [ (jω p ) l] Im [G(jω p )] }, l =...m () g l, = Re (jωp ) l] p= ( {Re [G(jω p )]} + {Im [G(jω p )]} )}, l =...n () The input and the output of the system must be chosen in advance. Two possibilities were explored for the AWS. The first one considers the wave excitation force F exc as the input and the floater s velocity Ξ as the output. F exc and Ξ data provided by the AWS TDM for regular (sinusoidal) waves with periods from 8 s to 4 s (this is the range the AWS was conceived for [6]) was used. According to wave data provided by ONDATLAS software [5] for the Leixõesbuoy location (4 o. N, 9 o 5.3 W), near the test site where the AWS prototype was submerged (at the Portuguese coast, 5 km offshore Leixões), the most frequent significant wave height H s (from trough to crest) is admitted as being equal to metres. Hence several waves with a m amplitude (half of H s ) and different periods were assumed for the simulations. (Notice that an approximation is involved here, since these waves used for identification are regular, while those addressed by ONDATLAS are real, irregular waves.) To apply the Levy identification method, Matlab s function levy was used [8]. The data found in Table I was used in that process. Regarding Levy s identification method, all combinations of values for the numerator and denominator orders m and n from to 5 were tried. Only identified models with two poles or more and one zero or more reproduced the wave frequency behaviour correctly. The identified model structure Ξ(s) F exc (s) =.7 6 s s () s + with one (non-minimum phase) zero and two (stable, complex conjugate) poles is the one that reproduces the AWS TDM responses making use of as few parameters as possible. By adding an extra pole at the origin, model Ξ(s) F exc (s) =.7 6 s s (.967s (3) s +) relating the wave excitation force to the floater s position Ξ is found. Figures 3 and 4 show, respectively, the Bode diagram and the pole-zero map of model (). This model s major drawback is its complexity. So another solution was looked for. The second possibility is to consider the wave excitation force F exc as the input and the floater s position Ξ as the output and provide this data (see Table I) to Levy s identification 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 4 s in order to obtain an acceptable model. Under this new assumption, the identified model, a second order transfer function, was Ξ(s) F exc (s) = s (4) +.733s +

5 TABLE I DATA USED IN THE IDENTIFICATION period / s F exc ampl. / kn Ξ amplitude / m Ξ gain / db Ξ phase / o Ξ ampl. / ms Ξ gain / db Ξ phase / o TABLE II CHARACTERISTICS OF SEVERAL IRREGULAR WAVES ACCORDING TO ONDATLAS Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec H s / m T e,min / s T e,max / s Figures 5 and 6 show, respectively, the Bode diagram and the pole-zero map of this last identified model. Even though the input of both transfer functions () and (4) is the wave excitation force F exc, the outputs take into account the effects of the radiated force as well, since it was included in the AWS TDM. IV. RESULTS 6 s ( min) long simulations were carried out, employing the AWS TDM (for the non-linear case) and Simulink implementations of () and (4) (for the linear cases). Models were submitted to several regular (sinusoidal, with different periods and amplitudes) and irregular incident waves with amplitudes and periods within the ranges expected to occur, based in data supplied by ONDATLAS software for the Leixões-buoy location. In what concerns irregular waves, Pierson-Moskowitz s spectrum, that accurately models the behaviour of real sea waves [], was used. This is given by S(ω) = A ( ω 5 exp B ) ω 4 (5) where S is the wave energy spectrum (a function such that + S(ω)dω is the mean-square value of the wave elevation). The numerical values A =.78 (SI) and B = 3./Hs were used. Values for the significant wave height H s (from trough to crest) and for the limits of the frequency range (corresponding to the maximum and minimum values of the wave energy period T e ) were those provided by ONDATLAS for the twelve months of the year (see Table II). From these simulations, s slices corresponding to seven regular waves and two significant months are highlighted in Figures 7 and 8; the root mean-square errors, given by RMS = 6 ( ˆΞ) Ξ dt (6) 6 (ˆΞ being the estimate of the floater s position) are given, for these simulations and others similar thereto, in Tables III and IV. From these results, it is seen that model (4) reproduces the AWS TDM behaviour more accurately; it also requires less parameters than (), and its structure is similar to the one normally assumed in the literature (e.g. []). Actually, () performs slightly better than (4) for regular waves of low period and high amplitude. But these cases are a minority, and simulations with irregular waves (with which (4) is systematically better) are deemed more important since they are expected to reproduce the behaviour of real sea waves more accurately. There is an additional reason to prefer model (4), related to the resistance R, which is the real part of the inverse of the transfer function from the wave excitation force to the floater s velocity: [ ] Fexc (jω) R(ω) =Re (7) Ξ(jω) R may be frequency dependent, but it is physically impossible that it be negative []. Indeed, R is always positive for (4) (actually in this particular case it does not even depend on ω). But, for some frequencies, () leads to a negative value of R, as seen in Fig. 9. This seems to denote an inaccurate identification in the case of (). For these reasons, model (4) was the one chosen. V. CONCLUSIONS From the last section it can be seen that the identified second order linear model approximation (4) yields a satisfactory performance. This model is a first step towards the development of control strategies for the AWS. This would be difficult with a non-linear model only. Now this simpler model can be used for controller design and testing, and the non-linear model for validation.

6 Irregular wave for March Irregular wave for June non linear force position force velocity Fig. 8. Floater s position for AWS TDM and identified linear models ( s long period out of 6 s) TABLE III ROOT MEAN-SQUARE ERRORS FOR THE SIMULATIONS IN FIG. 7AND OTHERS SIMILAR THERETO Wave Wave period / s amplitude / m Model () (4) () (4) () (4) () (4) TABLE IV ROOT MEAN SQUARE ERRORS FOR THE SIMULATIONS IN FIG. 8AND OTHERS SIMILAR THERETO Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec model () model (4) ACKNOWLEDGMENT The authors acknowledge the very useful comments and suggestions made by Professor Johannes Falnes (Professor Emeritus at the Norwegian University of Science and Technology) on the identification process. R / N s m 5 x Fig. 9. ω / rad s Evolution of R for both models model (4) model () REFERENCES [] P. Beirão, D. Valério, and J. Sá da Costa. Phase control by latching applied to the Archimedes Wave Swing. In Proceedings of the 7th Portuguese Conference on Automatic Control, Lisbon, 6. [] J. Falnes. Ocean waves and oscillating systems. Cambridge University Press, Cambridge,. [3] E. Levy. Complex curve fitting. IRE transactions on automatic control, 4:37 44, 959. [4] P. Pinto. Time domain simulation of the AWS. Master s thesis, Technical University of Lisbon, IST, Lisbon, 4. [5] M. T. Pontes, R. Aguiar, and H. Oliveira Pires. A nearshore wave energy atlas for Portugal. Journal of Offshore Mechanics and Arctic Engineering, 7:49 55, August 5. [6] J. Sá da Costa, P. Pinto, A. Sarmento, and F. Gardner. Modelling and simulation of AWS: a wave energy extractor. In Proceedings of the 4th IMACS Symposium on Mathematical Modelling, pages 6 7, Vienna, 3. Agersin-Verlag. [7] J. Sá da Costa, A. Sarmento, F. Gardner, P. Beirão, and A. Brito- Melo. Time domain model of the Archimedes Wave Swing wave energy converter. In Proceedings of the 6th European Wave and Tidal Energy Conference, pages 9 97, Glasgow, 5. [8] D. Valério and J. Sá da Costa. Identification of fractional models from frequency data. In J. Tenreiro Machado, J. Sabatier, and O. Agrawal, editors, Advances in Fractional Calculus: theoretical developments and applications in Physics and Engineering. Springer-Verlag, 6.