Optimisation of wave energy extraction with the Archimedes Wave Swing

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1 Ocean Engineering 34 (27) Optimisation of wave energy extraction with the Archimedes Wave Swing Duarte Vale rio a,, Pedro Beira o b, Jose Sá da Costa a a IDMEC/IST, TULisbon, Av. Rovisco Pais,, 49- Lisboa, Portugal b Department of Mechanical Engineering, Instituto Superior de Engenharia de Coimbra, R. Pedro Nunes, Coimbra, Portugal Received 29 November 26; accepted 2 May 27 Available online 2 June 27 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 27 Elsevier Ltd. All rights reserved. Keywords: Wave energy; Renewable energy; Archimedes Wave Swing; Reactive control; Phase and amplitude control; Latching control; Feedback linearisation control. Introduction Due both to the inexorable rise of oil prices and to environmental concerns, renewable energies are receiving increased attention. Among them, the energy of sea waves may play an important role in the near future. Wave energy converters (WECs) for turning it into electricity are not commercially competitive yet. But it is expected that they will soon be, as sufficiently efficient WECs are developed. Control engineering plays an important role towards that objective. This paper contributes to that effort by comparing the performance of several control strategies suitable for the Archimedes Wave Swing (AWS), a WEC of which a 2 MW prototype (Fig. ) has already been built, tested at the Portuguese northern coast during 24, and then decommissioned. The aim is to see, by means of simulations, which one leads to a better efficiency of that prototype and of the ones that will follow. The paper is organised as follows: Section 2 describes the AWS; the identification of a linearised model thereof is Corresponding author. Tel.: ; fax: address: dvalerio@dem.ist.utl.pt (D. Valério). addressed in Section 3; control strategies are summarised in Section 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 and employed. 2. The AWS The AWS is an offshore, fully submerged (43 m deep underwater), point absorber (that is to say, of neglectable size compared to the wavelength) WEC. Its two main 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 (Beira o et al., 26). Within the air-filled space formed by the silo and the floater there are several components needed for the functioning of the AWS, among which an electric linear 29-88/$ - see front matter r 27 Elsevier Ltd. All rights reserved. doi:.6/j.oceaneng

2 D. Valério et al. / Ocean Engineering 34 (27) Fig.. The AWS prototype and its working principle. generator (ELG) to convert the floater s heave motion into electricity. The electric energy produced by the ELG is transferred to shore via a 6 km long undersea cable; after that it passes through a converter, and following this conversion is supplied to the electric grid. The AWS also has water dampers that are actuated when the floater approaches the mechanical end-stops, to reduce its velocity and avoid a strong collision with them; they are also actuated together with the ELG when the latter does not suffice to supply the force required to adequately control the movement of the AWS. These water dampers are otherwise inactive, so as not to hinder wave energy extraction. It is the force exerted by these two components, the ELG and the water dampers, that we can control. Actually it is also possible to change the mean value of air pressure within the AWS, by means of pumps that add or remove sea water (thus changing the volume available for air); but this is a slow process that takes several minutes, and so it is used to cope with the variations of sea level caused by the tides (that are slower still). To control the AWS coping with different incoming waves and sea states, it is the ELG and the water dampers we have to resort to. This paper deals with this latter type of control. It will be assumed that there will be no changes in tide, and no changes in the mean pressure of the air inside the AWS. From the description above it is seen that the AWS can be reasonably expected to behave much like a mass spring damper system, though with relevant nonlinearities. The AWS was submerged 5 km offshore Leixo es, Portugal. Data for wave climate in several locations in Portugal may be found with the ONDATLAS software (Pontes et al., 25). The nearest location available is the Leixo es-buoy location ð4 2:2 N; 9 5:3 WÞ. The corresponding data on significant wave height H s (from trough to crest) and on maximum and minimum values of the wave energy period T e is found in Table. Throughout this paper, simulations are performed using regular and irregular waves. The first are sinusoidal; values congruent with Table for the amplitude and the period of the sinusoids are used. For the latter, 2 waves (one for each month of the year) satisfying the Pierson Moskowitz s spectrum, which accurately models the behaviour of real sea waves (Falnes, 22), were used. This spectrum is given by SðoÞ ¼ A o 5 exp B o 4, () where S is the wave energy spectrum (a function such that R þ SðoÞ do is the mean-square value of the wave elevation). The numerical values A ¼ :78 (SI) and B ¼ 3:=H 2 s were used. Values for H s and for T e (from which the limits of the frequency range were then found) were provided by Table. 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 w f f n f v f m f wd f lg ¼ðm f þ m wt Þ x. (2) The total mass comprises the mass of the floater m f and the water trapped inside the floater m wt (these two masses are known to oscillate together). The total force acting on the floater is the sum of the forces due to external water pressure f pe, to internal air pressure f pi, to the weight of the floater w f, to a nitrogen cylinder extant inside the AWS f n, to the hydrodynamic viscous drag f v, to mechanical friction f m, to the water dampers f wd, and to the ELG f lg, the last two being damping forces, and the ones we can control. Notice that the convention of signs assumes that positive values are given to the most natural direction hence among all forces the only one pointing upwards is f pi. Also notice that lower-case letters are being used for variables in the time-domain; their Laplace transforms (in the frequency domain) will be denoted using the corresponding capitals. (Hence XðsÞ def ¼L½xðtÞŠ, F lg ðsþ def ¼L½f lg ðtþš, and so on.) Both capitals and lower-case letters are used for constants. 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 on the AWS by the incident sea waves assuming that the

3 2332 ARTICLE IN PRESS D. Valério et al. / Ocean Engineering 34 (27) Table 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) floater is not moving, f hs is the hydrostatic force, and f rad is the force exerted in the AWS by the wave that the floater creates by its movement. We will assume that the forces can be superimposed on each other; actually, when wave amplitudes are large compared with the wavelength, (4) is no 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 cannot be given here for lack of space. They may be found for instance in Pinto (24); Sa da Costa et al. (23, 25). These references also describe an accurate, non-linear, Simulink-based simulator of the AWS, the AWS Time- Domain Model (TDM), that has already been developed implementing the expressions above. The AWS TDM was used for the simulations presented in this paper. ξ /m 4 2.m 3 2.5m AWS linear dynamic model identification Before thinking about the extensive use of a non-linear model of the AWS (such as the AWS TDM) for control purposes, a linear model approximation of that same WEC should be identified in the first place (Beira o et al., 27b). This is possible because, even though the AWS is a nonlinear system, a sinusoidal input causes a fairly sinusoidal output, with an amplitude practically proportional to that of the input, for all wave periods and amplitudes expected to occur. Fig. 2 shows this for several regular waves of different amplitudes. 3.. Identification procedure System identification deals with the construction of mathematical models of dynamical systems using measured data. In the particular situation of the AWS, very few experimental data are available. Thus, a different approach has been followed. The AWS TDM was used as an emulator of the real non-linear AWS WEC; no controller was employed, and hence both control forces (f lg and f wd ) were assumed to have only a minimum, unavoidable residual value. 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 (959), is the obvious choice. Levy s identification method provides us a transfer function relating the output data with the input data. The method receives frequency data for the input and the Fig. 2. Output of the AWS TDM for regular waves with s of period and amplitudes of.5,.,.5, and 2. m. output of the system and the orders of the numerator and the 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 for the 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 4 s (the range the AWS was conceived for (Sa da Costa et al., 23)) was used. Following the ONDATLAS software, the most frequent significant wave height H s (from trough to crest) is admitted as being equal to 2 m. 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 (Vale rio and Sa da Costa, 27). The data found in Table 2 was used in that process. 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 _XðsÞ F exc ðsþ ¼ 2:7 6 s 6:759 7 :967s 2 þ :5874s þ (5)

4 D. Valério et al. / Ocean Engineering 34 (27) Table 2 Data used in the identification Period (s) F exc ampl. (kn) X amplitude (m) X gain (db) X phase ð Þ _X ampl. ðms Þ _X gain (db) _Xphase ð Þ gain / db phase / ω / rads Imag Real 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 the AWS TDM responses making use of as few parameters as possible. By adding an extra pole at the origin, model XðsÞ F exc ðsþ ¼ 2:7 6 s 6:759 7 sð:967s 2 (6) þ :5874s þ Þ relating the wave excitation force to the floater s vertical position X is found. Fig. 3 shows the Bode diagram and the pole-zero map of model (5). This model s drawbacks are its non-minimum phase zero (something that may be undesirable) and its unnecessary complexity (since a simpler one can be got). So another solution was looked for. The second possibility for input and output choice was to consider the wave excitation force as the input and the floater s vertical position as the output and provide there data (see Table 2 ) 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 In that table, a phase lead of 9 could be expected between the phases of _X and X. Non-linearities, however, prevent this. transfer function, was XðsÞ F exc ðsþ ¼ 2:259 6 :6324s 2 þ :733s þ. (7) Fig. 4 shows the Bode diagram and the pole-zero map of this last identified model. Even though the input of both transfer functions (5) and (7) is the wave excitation force F exc, the outputs take into account the effects of the radiated force F rad as well, since this latter force was included in the AWS TDM Validation of identification results Six hundred seconds ( min) long simulations were carried out, employing the AWS TDM (for the non-linear case) and Simulink implementations of (5) and (7) (for the linear cases). The root mean-square errors, given by sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z 6 RMS ¼ ðx ~XÞ 2 dt (8) 6 ( ~X being the estimate of the floater s vertical position), are given, for several significant simulations, in Tables 3 and 4.

5 2334 ARTICLE IN PRESS D. Valério et al. / Ocean Engineering 34 (27) gain / db ω / rads - Imag phase / Real 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) (5) (7) (5) (7) (5) (7) (5) (7) Table 4 Root mean-square errors for the simulations with irregular waves Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Model (5) Model (7) A 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) reproduces the AWS TDM behaviour more accurately; it also requires less parameters than (5), and its structure is similar to the one normally assumed in the literature (Falnes, 22, e.g.). Actually, (5) performs slightly better than (7) for regular waves of low period and high amplitude. But these cases are a minority, and simulations with irregular waves (with which (7) 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 (7), 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 vertical velocity: RðoÞ ¼Re F excðjoþ, (9) _XðjoÞ R may be frequency dependent, but it is physically impossible that it should be negative (Falnes, 22). Indeed, R is always positive for (7), and actually it does not even depend on o, since, by definition (9), R ¼ Re :6324ðjoÞ2 þ :733jo þ 2:259 6 jo :733 ¼ 2:259 6 ¼ 7: ðþ

6 D. Valério et al. / Ocean Engineering 34 (27) position / m Irregular wave for March practice, they can only be implemented with approximations that cause a decrease in their efficiency to an extent that justifies considering other control strategies, that are intrinsically sub-optimum because they can never (even under the most favourable conditions) attain the efficiency that optimum control has in theory. The two sub-optimum strategies considered below are latching control and feedback linearisation control. 4.. Optimum control -.5 non-linear force-position force-velocity Fig. 5. Floater s vertical position for AWS TDM and identified linear models ( s long period out of 6 s). This section closely follows Falnes (22). Let us rewrite (7) as XðsÞ F exc ðsþ ¼ ms 2 þ Rs þ S, (2) where mass m, resistance R, and stiffness S are given by m ¼ 2:87 5, R ¼ 7:675 4,andS ¼ 4: (in SI units). The transfer function in (2) is the same as R / N s m - 5 x 5 model (7) model (5) -2 2 ω / rad s - Fig. 6. Evolution of R for both models. m xðtþþr xðtþþsxðtþ _ ¼f exc ðtþ. (3) Defining complex-valued phasors f ^ exc and x ^_ for f exc and _x, respectively, f f exc ðtþ ¼ ^ exc f 2 eiot þ ^ exc 2 e iot, ð4þ ^_x ^_x _xðtþ ¼ 2 eiot þ 2 e iot, ð5þ we can rewrite (3) as e iot f ^ exc R þ iom þ S ^_ x io þ e iot f ^ exc S R iom ^_ x ¼, ð6þ io where o is the frequency. Defining an impedance Z Z ¼ R þ i om S, (7) o But, for some frequencies, (5) leads to a negative value of R, since :967ðjoÞ 2 þ :5874jo þ RðoÞ ¼Re 2:7 6 jo 6:759 7 ¼ ð4:222o2 :4795Þ 6 :37o 2. ðþ þ Both () and () are plotted in Fig. 6. This seems to denote an inappropriate structure of the model identified in the 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 sense that they maximise (in theory) the absorption of energy. In it is possible to rewrite (6) as e iot f ^ exc Z^_ x þ e iot f ^ exc ^_ Z x ¼. (8) To satisfy (8) for all values of time t the following condition must be verified: f ^_x ¼ ^ exc Z )j^_ jf ^ xj¼ exc j jzj. (9) Definition (7) can be rewritten as Z ¼ R þ ix. The real part R ¼ Re½ZŠ is called resistance and was already mentioned above in (9); the imaginary part X ¼ Im½ZŠ ¼om ðs=oþ is called reactance. As was already mentioned above, Z takes into account both the mechanical and the radiation effects (since the radiation force was included in the AWS TDM). Hence these R and X are the intrinsic resistance and the intrinsic reactance, respectively. Thus they shall henceforth be

7 2336 ARTICLE IN PRESS D. Valério et al. / Ocean Engineering 34 (27) denoted R i and X i, and Z will be similarly denoted Z i (intrinsic impedance). Suppose now that the control force f u is applied to the AWS, to ensure that the conditions leading to maximum wave energy absorption are met (or at least approached). Then Z i x _ ¼ f exc þ f u 3 Z i ðoþ _XðoÞ ¼F exc ðoþþf u ðoþ. (2) Let us define a control impedance Z u by f^ u ¼ Z ux ^_ 3 F u ðoþ ¼ Z u ðoþ _XðoÞ. (2) The real and imaginary parts of Z u are, respectively, termed control resistance R u ¼ Re½Z u Š and control reactance X u ¼ Im½Z u Š. The time-averaged absorbed wave power P u ðtþ can be given by P u ðtþ ¼ f u ðtþ xðtþ _ (22) and the absorbed wave energy W u can be obtained integrating (22): W u ¼ Z þ f u ðtþ _ xðtþ dt. (23) Considering that f u and x _ are real functions, i.e., F u ðoþ ¼F uð oþ and _X ðoþ ¼ _Xð oþ, W u can, by applying Parseval s theorem, be given by W u ¼ Z þ F u ðoþ _X ðoþ do. (24) 2p Knowing that W u is real, F u ðoþ _X ðoþ ¼Re½F u ðoþ _X ðoþš ¼ 2 ½F uðoþ _X ðoþþf u ðoþ _XðoÞŠ, ð25þ it is possible to rewrite (24) as W u ¼ Z þ ½ F u ðoþ _X ðoþ F u 2p ðoþ _XðoÞŠ dt. (26) Since R i is positive, it will be convenient to add and subtract the term F exc ðoþf exc ðoþ=2r i to the integrand of (26), and finally W u is now given by (omitting the frequency argument in order to simplify the notation) W u ¼ Z þ jf exc j 2 jf excj 2 F u _X F u X 2p 2R i 2R _ do i ¼ Z þ jf exc j 2 a do. ð27þ 2p 2R i 2R i In (27), aðoþ is the so-called optimum condition coefficient, given by aðoþ ¼F exc ðoþf exc ðoþþ2r i F u ðoþ _X ðoþþf u ðoþ _XðoÞ. (28) There are two strategies reactive control (also called complex conjugate control) and phase and amplitude control to find the optimum conditions. Both of them are based on the same requisite, which is to prove that a is never negative, i.e., aðoþx. Hence, it is when a ¼ that W u is maximal Reactive control A first way of proving that a is never negative is as follows. Solving (2) in order to F exc, multiplying it by its complex conjugate, F exc ðoþ ¼Z i ðoþ _XðoÞ F uðoþ, and omitting the frequency argument, the following expression is obtained: F exc ¼ Z i _X F u ) F exc F exc ¼ F u F u þ Z iz i _X _X F u Z i _X F u Z i X. _ ð29þ From (7) and its complex conjugate, one possible expression for R i is 2R i ¼ Z i þ Z i 3 R i ¼ Z i þ Z i. (3) 2 Multiplying both sides by F u _X þ F _ u X 2R i ðf u _X þ F _ u XÞ¼Z i F u _X þ Z i F _ u X þ Z i F u X _ þ Z i F _ u X. (3) Inserting (29) and (3) in (28) and simplifying the symmetric terms a ¼ F u F u þ Z iz i _X _X F u Z i _X F u Z i X _ þ Z i F u _X þ Z i F _ u X þ Z i F u X _ þ Z i F _ u X ¼ðF u þ Z i _X ÞðF u þ Z i X _ Þ. ð32þ To complete the proof about the non-negativity of a, (32) can be rewritten recovering again the frequency argument aðoþ ¼jF u ðoþþz i ðoþ _XðoÞj 2 X. (33) From (33), one particular optimum condition attained when aðoþ ¼is F u ðoþ _XðoÞ ¼ Z i ðoþ. (34) Comparing the optimum condition in (34) with (2), it is possible to see that Z u is equal to the complex conjugate of the intrinsic impedance Z i, i.e., Z u ¼ Z i. Another additional conclusion is that under the optimum condition in (34) the reactive component X i ðoþþx u ðoþ of the total impedance Z i ðoþþz u ðoþ is cancelled. This is an inherent consequence of resonance. Reactive control is represented by (34), since it is related to the fact that X u (the imaginary part of Z u ) will cancel X i (the imaginary part of Z i ). Alternatively, this control strategy is also called complex conjugate control since it is related to the fact that the optimum control impedance Z u;opt equals the complex conjugate of the intrinsic impedance Z i, i.e., Z u;opt ¼ Z i. (35) Consequently, from (27), the theoretical maximum absorbed wave energy W u;max is W u;max ¼ Z þ jf exc ðoþj 2 do. (36) 2p 2R i

8 D. Valério et al. / Ocean Engineering 34 (27) Notice that the integrand of (36) is positive since R i is also positive Phase and Amplitude control Alternatively to the previous one, another proof that a is never negative is as follows. We now solve (2) in order to F u, instead of F exc. Inserting the obtained result and its complex conjugate, F u ðoþ ¼Z i ðoþ _X ðoþ F excðoþ, in (28), and omitting once more the frequency argument, in order to simplify the notation, a ¼ F exc F exc þ 2R iðz i _X _X F exc _X þ Z i _X _X F _ exc XÞ ¼ F exc F exc þ 2R iðz i þ Z i Þ _X _X 2R i F exc _X 2R i F _ exc X. ð37þ Inserting (3) in (37) a ¼ F exc F exc ð2r iþ 2 _X _X 2R i F exc _X 2R i F _ exc X ¼ðF exc 2R i _XÞðF exc 2R i X _ Þ. ð38þ To complete the proof about the non-negativity of a, (38) can be rewritten recovering again the frequency argument aðoþ ¼jF exc ðoþ 2R i _XðoÞj 2 X. (39) Since R i is positive, from (39) an alternative optimum condition can be written as XðoÞ F exc ðoþ ¼2R i _XðoÞ 3 _ F exc ðoþ ¼. (4) 2R i The optimum condition in (4) is called phase and amplitude control since it means that _X must be in phase with F exc, and also that the amplitude of the floater s vertical velocity j _Xj must be equal to F exc =2R i Feasibility of optimum control strategies A serious problem with both optimum conditions is that they include (in the general case) non-causal transfer functions: Z i ðoþ in (34), and =2R i in (4) (recall that in the general case R i may vary with frequency). This last optimum condition even requires foreknowledge of the wave excitation force, which in practice is available by means of predictions based on data measured by a buoy (or buoys) placed near the AWS. To make things worse, both conditions often lead to control forces that assist the wave excitation force; in other words, instead of extracting energy from the waves, we will be supplying energy to the waves. This will of course happen only in a small fraction of the time and is necessary to extract the maximum possible wave energy during the remaining time. But if it is impossible to do so we will be limited to a sub-optimum solution. Actually all approximations indulged in to make transfer functions in (34) and (4) causal will also decrease wave energy extraction and place us in a sub-optimum solution. But being sub-optimum should not be seen as a major drawback. At least it can be realisable, something which optimum solutions cannot. Both control strategies can also be applied to a nonlinear WEC, provided that a valid linear model thereof, similar to (2), exists. The controller will be designed using the linear model and then applied to the non-linear WEC. Results cannot, of course, be expected to be as good as they would with a linear plant Implementation The material in this subsection allows conceiving two different optimum control strategies suited to the AWS: reactive control (or complex conjugated control) and phase and amplitude control (Vale rio et al., 27b). In what the application of optimum control to the linear second-order model of the AWS (7) is concerned, a Simulink block diagram comprising the two control strategies mentioned above is shown in Fig. 7. The one used can be chosen by means of switch (set for reactive control in Fig. 7). The second switch allows choosing the model of the AWS (the linear model (2) or the non-linear AWS TDM). In that diagram, reactive control is implemented replacing the non-causal transfer function Z i ðsþ with Z i ðsþ=ðs þ Þ, the extra pole placed at ensuring causality. Several locations have been tested for the pole, and the one leading to a larger absorption of wave energy was kept. An alternative procedure would have been to identify from the frequency response of Z i a causal, stable, approximate transfer function with a similar response in the frequency range of interest; this approach was pursued, but led to no acceptable results. A proportional controller is used together with phase and amplitude control to drive the floater s vertical velocity to the optimum value reckoned by =2R i. Notice that in our case this transfer function is constant (and hence causal). The controller was obtained maximising the absorbed wave energy with the MatLab function fminsearch (simplex direct search method), the optimum being 5: 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 by the ELG and the water dampers, but the energy absorbed by the latter is wasted. Only the energy absorbed by the ELG is used; hence we will have W u ¼ Z þ f lg ðtþ _ xðtþ dt. (4) The implementation of optimum control together with the non-linear AWS TDM is again that of Fig. 7. Absorbed wave power is obtained with a Simulink block simulating the ELG (with non-linear dynamics and saturations) Phase control by latching Because of the difficulties associated with putting the floater s vertical velocity in phase with the wave excitation force, a sub-optimum alternative we can resort to is to latch the floater during some periods of its oscillation and unlatch it so that it will be (as nearly as possible) in phase

9 2338 ARTICLE IN PRESS D. Valério et al. / Ocean Engineering 34 (27) Fig. 7. Block diagram for optimum control. withthewaveexcitationforce;moreprecisely,thefloaterwill be latched when its velocity vanishes, and released when it is predicted that its maximum (or minimum) will coincide (in time) with the maximum (or minimum) of the wave excitation force (Falnes, 993, 22; Greenhow and White, 997; Babarit et al., 24). This latching control is sub-optimum since it can never achieve the wave energy absorption efficiency that the optimum control would achieve. Latching is clearly a discrete, highly non-linear form of control. In what concerns the AWS, latching is achieved by actuating the water dampers so as to prevent (as much as possible) the floater from moving; unlatching means turning the water dampers off to let the floater go about (as much as possible) freely. The following unlatching strategies were implemented (Beira o et al., 26): () The latching time is constant. This is only reasonable when the incident wave is regular (sinusoidal). Hence this strategy was used for testing only, and will not be addressed further. (2) When the floater is latched, the duration of the last unlatched period is obtained. The next unlatched period is assumed to be going to last the same as the previous one. The floater s vertical velocity is assumed to have its maximum (or minimum) precisely at the middle of that time interval. So the latching time is reckoned for that velocity maximum (or minimum) to coincide in time with the next maximum (or minimum) of the wave excitation force. The force required from the water dampers to latch the floater is constant. (3) The same as above, say that the force required from the water dampers to latch the floater depends on the Table 5 Latching force in kn Wave period (s) Wave amplitude (m) amplitude and period of the incoming wave, larger waves requiring a larger force and smaller waves requiring a smaller force. The forces for each wave amplitude and period are those necessary to latch effectively the floater when the incident wave is regular and has the required amplitude and period. Values are given in Table 5 and are interpolated and extrapolated as needed. (4) Same as above, but the duration of the next unlatched period is assumed to be equal to the last one corrected according to the ratio between the next wave amplitude and the previous. (5) Same as above, but the duration of the next unlatched period is assumed to be equal to half of the floater s natural period of oscillation, which is s. Thus, the floater is unlatched 2.75 s (one quarter of the natural period) before the next maximum (or minimum) of the wave excitation force (Falnes, 993, 997; Eidsmoen, 998) Feedback linearisation control The dynamics described by (2) are far from being linear. The expressions for nearly all the forces involved show that

10 D. Valério et al. / Ocean Engineering 34 (27) they depend from variables such as floater s vertical position, velocity, or acceleration. Heavy non-linearities, both smooth (with continuous derivatives of any order) and hard (without continuous derivatives; this refers to such common non-linearities as saturations, dead-zones, or hysteresis), are always present. This makes the AWS a suitable candidate for a nonlinear control strategy called feedback linearisation control (Vale rio et al., 27a). Its aims are to provide a control action judiciously chosen to cancel the non-linear dynamics of the plant, so that the closed-loop dynamics will be (as much as possible) linear (Slotine and Li, 99). This method can be applied to plants that can be put into companion form; this is the case of (2), since it suffices to solve it in order to x. From (2) (4), we have f u ¼ðm f þ m wt Þ x f pi þ f hs þ f rad f exc þ f n þ f v þ f m þ w f. ð42þ Let us provide a control action given by f u ¼ f pi f hs f rad f n f v f m w f þ n. (43) This is possible because there are explicit (non-linear) expressions for all the forces involved in the right-hand member of (43). Here n is an additional term that will be chosen to provide the desired (linear) closed-loop dynamics. Replacing (43) into (42), we will have the following dynamics: n ¼ðm f þ m wt Þ x f exc. (44) Three different values for n were considered First control strategy Let n ¼. (45) Then, from (44), f exc ¼ðm f þ m wt Þ x. (46) In other words, the floater s vertical acceleration will be proportional to the wave excitation force, according to Newton s law, as though there were no other effects at all Second control strategy Let n ¼ ðm f þ m wt Þ x þ x _ max j f excj. (47) 2:2 Replacing this in (44) we get f exc ¼ _ x max j f excj 2:2 3 _ x ¼ 2:2 max j f exc j f exc. (48) In other words, the floater s vertical velocity will be in phase with the wave excitation force. In (47) and (48), constant 2:2 appears because the nominal value for the floater s velocity that the AWS should work with is 2.2 m/s (Polinder et al., 24). Also notice that, since it will be assumed (as seen below) that f exc is known beforehand, max j f exc j will be a constant value. This control law was chosen because, as shown above in Section 4., the optimum control of an oscillating WEC is the one providing a velocity in phase with the wave excitation force Third control strategy The original version of the AWS TDM was implemented together with a simple proportional controller that may easily be replaced by another one to test any desired control strategy. This original controller provides a control force given by jf u j¼j xjk _ p j x _ x _ stp j, 8 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2p j x _ stp j¼ 3:5 x 2 >< if jxjo3:5m; 3:5 >: if jxjx3:5m: ð49þ ð5þ In (49), k p is the adjustable gain of the proportional controller. Constant 3.5 m shows up in (5) because it is the maximum possible amplitude for the floater s vertical oscillations, while constant s shows up because it is a reasonable value for the period of an incoming wave (Pinto, 24). Suppose that we want to follow the set point reference _ x stp for the floater s velocity, using feedback linearisation control. Then we would like to have ðm f þ m wt Þ x ¼ðm f þ m wt Þ d_ x stp. (5) dt By comparison with (44), we see that we must have n ¼ ðm f þ m wt Þ d_ x stp dt þ f exc. (52) Implementation The three control laws (defined by (43) together with one of (45), (47), or (52)) from the previous section were implemented in the AWS TDM. Concerning (3), whenever the ELG was able to exert the control force f u all alone, the water dampers were not resorted to (f wd ¼, f lg ¼ f u ). They were used only when f u was beyond the limits of the ELG. 5. Results 5.. Simulations Six hundred seconds ( min) long simulations were carried out to test these control strategies. As mentioned above, full knowledge of how the incident wave will behave in the future is assumed, as done for instance in Falnes (997); Eidsmoen (998); Babarit et al. (24). This unreal assumption 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 around the WEC (Naito and Nakamura, 985), was not tackled,

11 234 ARTICLE IN PRESS D. Valério et al. / Ocean Engineering 34 (27) Table 6 Power in kw obtained under several regular waves (figurative data) Wave amplitude (m) Wave period (s) AWS TDM, no control Linear model, reactive control % increase from no control AWS TDM, reactive control % increase from no control AWS TDM, phase and amplitude control % increase from no control AWS TDM, latching strategy % increase from no control AWS TDM, latching strategy % increase from no control AWS TDM, latching strategy % increase from no control AWS TDM, latching strategy % increase from no control AWS TDM, feedback linearisation, strategy % increase from no control AWS TDM, feedback linearisation, strategy % increase from no control AWS TDM, feedback linearisation, strategy % increase from no control but postponed to some later opportunity. In all cases, the absorbed wave energy is given by (4) with integration limits from to 6 s. From these simulations, values for the absorbed wave power (time-averaged in the case of irregular waves) are given in Tables 6 and 7. For comparison purposes, absorbed wave power when the AWS has no control at all are also given. 2 Notice that there is only one case, that of reactive control, in which it is possible to present results from simulations with the linear model (7), since the pole added to make the controller causal leads immediately to a sub-optimum strategy. When no control is applied, there is no control force, and hence no energy extraction, in that case. And when phase and amplitude control is employed, everything being linear and no saturations existing, control works too well and values obtained have magnitudes absolutely impossible to obtain with the AWS. Finally, by their very nature, latching control and feedback linearisation control cannot be simulated with a linear model Comments The main conclusion to take from the results obtained with the AWS TDM is that nearly all control strategies 2 Values given in this paper for absorbed wave power when the AWS has no control follow Valério et al. (27a, b); Beirão et al. (27a) and are higher than those given in Beirão et al. (26). 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 what wave energy absorption is concerned. Since nowadays the major problem of WECs is their low efficiency, these are very satisfactory and promising results. But there are significant differences between them (Beira o et al., 27a). Concerning the two optimum control strategies, it is seen that they do not work equally well in practice: reactive control works best during the whole year, while phase and amplitude control always lags behind, and provides disappointing results from May to September (when there is less wave energy). Fig. 8 (showing 2 s highlights, from the simulations corresponding to March and June, of the floater s velocity, together with the wave excitation force, for the several control strategies) gives an insight into the reason why this is so (similar results are obtained for all months). Phase and amplitude control manages to put the floater s velocity nearly in phase with the wave excitation force; reactive control does the same but not so efficiently. Nevertheless, the magnitude of the floater s velocity is clearly larger in this latter case, and so the higher phase difference is more than compensated. Results obtained (for irregular waves) with the linear model and those obtained with the AWS TDM can be related using a linear regression: power obtained with the AWS TDM ¼ 27:2 kw þ :54 power obtained with the linear model. ð53þ Even though this regression has a significant error (the maximal residual being 4.5 kw), it shows that the trend

12 D. Valério et al. / Ocean Engineering 34 (27) Table 7 Power in kw obtained under several irregular waves (figurative data) Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec AWS TDM, no control Linear model, reactive control % increase from no control AWS TDM, reactive control % increase from no control AWS TDM, phase & ampl. ctrl % increase from no control AWS TDM, latching strategy % increase from no control AWS TDM, latching strategy % increase from no control AWS TDM, latching strategy % increase from no control AWS TDM, latching strategy % increase from no control AWS TDM, feedback lin., strat % increase from no control AWS TDM, feedback lin., strat % increase from no control AWS TDM, feedback lin., strat % increase from no control Irregular wave for March Irregular wave for June velocity / m s - ; force / MN F exc none reactive phase and amplitude latching, strat. 3 feedback lin., strat velocity / m s - ; force / MN Fig. 8. Wave excitation force and floater s vertical velocity. obtained with the (much simpler) linear simulations is reliable and allows inferring the behaviour of the nonlinear model. Concerning latching control, results for regular waves show that latching strategies 3 and 4 have nearly the same 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 last unlatched period; the usefulness of such a correction if any is to be assessed with irregular waves). Results for irregular waves show that winter and summer months are rather different. During summer (loosely defined as the May September period), when there is less energy available in waves, strategies 2, 3, and 4 are comparable; 3 is always the best (with one single exception, and that by a narrow margin). During the rest of the year (the October April period), when there is more energy available, strategies 3 and 4 are clearly better than all others. Strategy 5, though improving energy absorption over the situation without control, never leads to acceptable results. It is clear that strategy 4 is not a good improvement over 3. Its more complicated algorithm seldom leads to a better performance. From this analysis, it is clear that strategy 3 is the one to choose if latching control is resorted to, and will be the only one addressed in further analysis.

13 2342 ARTICLE IN PRESS D. Valério et al. / Ocean Engineering 34 (27) Concerning feedback linearisation control, the energy absorption increases observed with all strategies are often achieved in spite of the desired linear closed-loop dynamics not being obtained. This is because, with strategy, the floater s acceleration only rarely is in phase with the wave excitation force, and with strategy 3 the set point for the floater s velocity is always very far from being attained (consequently the energy absorption improvement is not as high as with the other strategies). Strategy 2, on the other hand, achieves (as seen in Fig. 8) better results in causing floater s velocity and wave excitation force to be in phase (especially in what irregular waves are concerned), and hence the better results achieved. The deviations from the desired behaviour are likely to derive from the simplifications that had to be admitted when designing the controller, namely assuming that the actuators (the ELG and the water dampers) respond immediately and have no saturations (when this is not the case, for each has its own internal dynamics, and saturation values; this is especially the case of the water dampers, and so f wd will not always follow its set point). Since results show that strategy 2 is the best, in what wave energy absorption is concerned, this is the one to choose if feedback linearisation control is resorted to, and will be the only one addressed in further analysis. Overall, feedback linearisation is clearly the best control strategy, in what absorbed wave energy is concerned, and this happens all over the year, for all sea states. (The case of the regular wave of.5 m of amplitude and period of 4 s does not seem to be relevant.) This is clear from the evolution of wave energy absorption with time for the several control strategies given in Fig. 9. The analysis of the plots in Fig. 8 provides an insight into the reason why feedback linearsation is the best. The requirement that wave excitation force and floater s velocity be in phase, as seen in (4), is reasonably fulfilled in the case of feedback linearisation, of latching, and of phase and amplitude control, but no so well accomplished by reactive control. (Notice that the latching of the floater absorbed energy / J 5 x Irregular wave for March feedback linearisation, strategy latching, strategy 3 reactive phase and amplitude none Fig. 9. Evolution of absorbed wave energy with time (figurative data). is not perfect. The ELG and the water dampers do not suffice to effectively prevent the floater from moving, but they can hinder it well enough.) When phase and amplitude control is employed, the amplitude of the floater s oscillations is relatively small (as can be seen from the small values of the velocity), and this accounts for the poor performance. At a first glance, latching control might seem to have to be better than feedback linearisation control, since the floater s vertical oscillations have a larger amplitude. The reason why this is not so is seen in the plots of Fig.. The absorbed wave energy (4) is computed as the integral of the product of two oscillating variables (the force exerted by the ELG and the floater s vertical velocity). If both these oscillations have the same amplitude, the absorbed wave energy will increase as the phase difference decreases, since the time period when the product is negative will diminish. For the same phase difference between the two variables, absorbed wave energy will increase with the increase of the magnitude of either of the variables. And, indeed, the force exerted by the ELG is clearly much larger with feedback linearisation than with latching control. That is why absorbed wave energy attains its maximum with feedback linearisation. Also notice that latching and feedback linearisation are the control strategies requiring the larger efforts from the water dampers, that nearly are not needed with reactive control or phase and amplitude control. The energy they absorb is dissipated, but nevertheless what the ELG receives is more than if they were not used. The two (supposedly) optimum control strategies are not the best performing ones, likely because they are suboptimally implemented. The approximations used to ensure causality prevent them from performing as well as they should in the ideal case (the only one really optimum) Variations around the year Table shows that the energetic content of waves is not the same around the year. This is reflected in the variations of wave power P u absorbed by the AWS, whatever the control strategy employed. Fig. attempts to explain these variations as functions of H s, that suffices to explain much of the monthly variations (the maximum and minimum valuesofthewaveenergyperiodt e have also been tested as possible explanations but seem to be irrelevant). It is possible to use the least-squares method to find relations between P u and H s. These can be given by linear expressions, but it is more reasonable (as can be seen by visual inspection of the plots) to use quadratic expressions, as follows: No control: P u ¼ 2:7H 2 s 6:55H s þ 7:3. (54) Reactive control: P u ¼ 8:59H 2 s :34H s 8:78. (55)

14 D. Valério et al. / Ocean Engineering 34 (27) Reactive control Phase and amplitude control F wd velocity / m s - ; force / MN velocity / m s - ; force / MN F lg velocity Control by latching velocity / m s - ; force / MN velocity / m s - ; force / MN Control by feedback linearisation Fig.. Evolution of the control forces with time (figurative data) for a March irregular wave. absorbed power / kw none reactive phase and amplitude latching, strat. 3 feedback lin., strat. 2 Aug Jul Jun May Sep Phase and amplitude control: Apr Mar P u ¼ 8:4H 2 s 45:H s þ 3:6. (56) Oct H s / m Nov Feb Dec Jan Fig.. Evolution of the absorbed wave power with H s. Latching control, strategy 3: P u ¼ 5:2H 2 s þ 3:7H s 47:65. (57) Feedback linearisation control, strategy 2: P u ¼ 9:45H 2 s þ 4:5H s 28:46. (58) 6. Conclusions The main conclusion to take is that phase and amplitude control are better than no control, reactive control is better than phase and amplitude control, latching control (with the best strategy conceived) is better than reactive control, and feedback linearisation control (with the best strategy conceived) is the best. This is true for all months of the year, and also for most regular waves (though not for all). Nevertheless, irregular waves are expected to be closer to the real sea states the AWS will have to work in, and so their results are deemed more significant. In principle, reactive control and phase and amplitude control should perform better. But these two optimum