ARTICLE IN PRESS Ocean Engineering 35 (2008) 358–366 www.elsevier.com/locate/oceaneng Phase control through load control of oscillating-body wave energy converters with hydraulic PTO system Anto´ nio F. de O. Falca˜ oÃ IDMEC, Instituto Superior Te´cnico, Technical University of Lisbon, Av. Rovisco Pais, 1049-001 Lisbon, Portugal Received 24 July 2007; accepted 15 October 2007 Available online 18 October 2007 Abstract Oscillating bodies constitute an important class of wave energy converters, especially for offshore deployment. Phase control by latching has been proposed in the 1970s to enhance the wave energy absorption by oscillating bodies (especially the so-called point absorbers). Although this has been shown to be potentially capable of substantially increasing the amount of absorbed energy, the practical implementation in real irregular waves of optimum phase control has met with theoretical and practical difﬁculties that have not been satisfactorily overcome. The present paper addresses the case of oscillating-body converters equipped with a high-pressure hydraulic power take-off mechanism (PTO) that provides a natural way of achieving latching: the body remains stationary for as long as the hydrodynamic forces on its wetted surface are unable to overcome the resisting force (gas pressure difference times cross-sectional area of the ram) introduced by the hydraulic PTO system. A method of achieving sub-optimal phase-control is developed, based on the theoretical time-domain modelling of a single-degree of freedom oscillating body in regular and irregular waves, by adequately delaying the release of the body in order to approximately bring into phase the body velocity and the diffraction (or excitation) force on the body, and in this way get closer to the well-known optimal condition derived from frequency-domain analysis for an oscillating body in regular waves. r 2007 Elsevier Ltd. All rights reserved. Keywords: Phase control; Point absorber; Power take-off; Wave energy 1. Introduction Oscillating bodies constitute an important class of wave energy converters, especially for offshore deployment. Phase control by latching has been proposed by Budal and Falnes (1980) to enhance the wave energy absorption by oscillating bodies (namely the so-called point absorbers) whose natural frequency is above the range of frequencies within which most of the incident wave energy ﬂux is concentrated. This has been conﬁrmed experimentally for the ﬁrst time by Budal et al. (1981). Phase control by latching was the object of other theoretical and experimental investigations in the last few years (Korde, 2002; Babarit et al., 2004; Babarit and Cle´ ment, 2006; Bjarte-Larsson and Falnes, 2006; Vale´ rio et al., 2007; Hals et al., 2007). Reviews can be found in Falnes (2002a, b). ÃTel.: +351 21 8417273. E-mail address: antonio.falcao@ist.utl.pt 0029-8018/$ - see front matter r 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.oceaneng.2007.10.005 Although phase control by latching has been shown to be potentially capable of substantially increasing the amount of absorbed energy, the practical implementation in real irregular waves of optimum phase control has met with theoretical and practical difﬁculties that have not been satisfactorily overcome. Sub-optimal control methods have been devised and proposed by several research teams to circumvent such difﬁculties. In a large class of devices, the oscillating (rectilinear or angular) motion of a ﬂoating body (or the relative motion between two moving bodies) is converted into the ﬂow of a liquid (water or oil) at high pressure by means of a system of hydraulic rams (or equivalent devices). At the other end of the hydraulic circuit, there is a hydraulic motor (or a high-head water turbine) that drives an electric generator. The highly ﬂuctuating hydraulic power produced by the reciprocating piston (or pistons) may be smoothed by the use of a gas accumulator system, which allows a more regular production of electrical energy. Naturally, the ARTICLE IN PRESS A.F.O. Falca˜o / Ocean Engineering 35 (2008) 358–366 359 smoothing effect increases with the accumulator volume and working pressure. This kind of power take-off (PTO) system is employed, e.g., in the Pelamis wave energy converter (Pizer et al., 2005) and the SEAREV (Josset et al., 2007). The use of such a PTO system provides a natural way of achieving latching: the body remains stationary for as long as the hydrodynamic forces on its wetted surface are unable to overcome the resisting force (gas pressure difference Dp times cross-sectional area SC of the ram) introduced by the hydraulic PTO system. In the paper, which is a follow-up of Falca˜ o (2007), a method of achieving sub-optimal phase-control is developed, and is applied, in a time-domain simulation, to a singledegree of freedom oscillating body. Regular waves are considered ﬁrst. Random irregular waves are then considered (each sea state is characterized by its signiﬁcant wave height HS, its energy period Te, and a discretized Pierson–Moskowitz spectrum). A simple geometry (a hemisphere in deep water oscillating in heave) is adopted for the buoy. Phase control by latching is implemented by adequately delaying the release of the body in order to approximately bring into phase the body velocity and the diffraction (or excitation) force on the body, and in this way get closer to the well-known optimal condition derived from frequencydomain analysis for an oscillating body in regular waves, with linear PTO damping. The proposed control algorithm is simple and easy to implement, and includes (i) a proportionality relationship qm ¼ C1 Dp between the ﬂuid ﬂow rate qm through the hydraulic motor (or water turbine) and the accumulator gas pressure difference Dp, and (ii) a proportionality relationship F ¼ C2 Dp between the release force F and Dp (which regulates the release delay). 2. Governing equations 2.1. The hydrodynamics of wave energy absorption We consider the simple case of a body with a singledegree of freedom (Fig. 1), oscillating in heave (coordinate x, with x ¼ 0 in the absence of waves). The governing equation for the body oscillations is mx€ ¼ f hðtÞ þ f mðtÞ, where m is the mass of the buoy, x€ is the acceleration, fh is the vertical component of the force due to water pressure on the wetted surface of the body (we assume fh ¼ 0 for the motionless body at x ¼ 0 in calm water), and fm is the vertical component of the force applied on the buoy by the PTO mechanism. If the amplitudes of the waves and of the body motions are small (linear system from the wave hydrodynamics viewpoint), we may introduce the usual decomposition fh ¼ fd+fr+fhs, where fd is the vertical force produced by the incident waves on the assumedly ﬁxed body (excitation or diffraction force), fr is the hydrodynamic force due to the body oscillation in otherwise calm water (radiation Buoy LP accumulator HP accumulator A Cylinder B Motor D E Valve Fig. 1. Schematic representation of the wave energy converter. force), and fhs is the hydrostatic force (in the case of a ﬂoating body; we assume fhs ¼ 0 for x ¼ 0). In a linearized version, it is fhs ¼ ÀrgSx, where r is water density, g is acceleration of gravity, and S is the buoy cross-sectional area deﬁned by the undisturbed free water surface. 2.2. Linear system in regular waves: frequency-domain analysis In the case of regular waves of frequency o, the diffraction or excitation force is a simple-harmonic function of time t. In addition, we assume a linear PTO, which allows us to write f m ¼ ÀKx À Cx_, (K and C are constants), where ÀKx represents a spring effect (which may exist or not) and ÀCx_ (C40) is the damping effect associated with the energy extraction. In this case, the system is completely linear: if the incident waves are regular, the coordinate x and the forces are simple-time-harmonic functions, and it is convenient to write xðtÞ ¼ ReðX 0 eiotÞ, f dðtÞ ¼ ReðF d eiotÞ, where X0, Fd are complex amplitudes, and Re( Á ) means real part of (a notation we will be omitting in what follows). Since the system is linear, Fd is proportional to the incident wave amplitude Aw (assumed here real and positive) and we may write jF dj ¼ GðoÞAw, where G(o) is an (real positive) excitation force coefﬁcient. It is convenient to decompose the radiation force as f rðtÞ ¼ ÀAðoÞx€ À BðoÞx_. Here A(o) is the added mass and B(o) is the radiation damping coefﬁcient (for physical reasons, B cannot be negative). For bodies with a vertical axis of symmetry oscillating in heave (as assumed here), it may be found (Falnes, 2002b) that the excitation force coefﬁcient and the radiation damping coefﬁcient are related to each other by 2g3rBðoÞ1=2 GðoÞ ¼ o3 . (1) ARTICLE IN PRESS 360 A.F.O. Falca˜o / Ocean Engineering 35 (2008) 358–366 The coefﬁcients A(o) and B(o) depend on the geometry (and on the frequency o). If the system is linear, the governing equation may be written as ðm þ AÞx€ þ ðB þ CÞx_ þ ðrgS þ KÞx ¼ F d eiot. (2) Replacing x ¼ X0 eiot, we ﬁnd X0 ¼ Ào2ðm þ AÞ þ Fd ioðB þ CÞ þ rgS þ K . (3) The time-averaged absorbed power is (see, e.g., Falnes, 2002b): P¯ ¼ 1 2 Co2jX 0j2 ¼ 1 8B jF dj2 À B 2 U 0 À 2FBd2, (4) where U0 ¼ ioX 0 is the complex amplitude of the velocity x_. This shows that, for a given body and given incident wave (i.e., ﬁxed o, B, and Fd), the time-averaged absorbed power is maximum when the values of K and C are such that F d ¼ 2BU 0, (5) in which case the diffraction force fd is in phase with the velocity x_. By combining (4) and (5), we ﬁnd o2ðm þ AÞ ¼ rgS þ K, (6) B ¼ C. (7) Eq. (6) is a resonance condition, whereas (7) is a damping condition for maximum power. Typically, ocean wave energy is concentrated in the range 0.5ooo0.9 rad/s. In this range and for most point absorbers, (6) would imply negative values for the PTO spring stiffness K, which is a condition difﬁcult to implement in practice. Phase control by latching is a method, initially proposed by Budal and Falnes (1980), for overcoming this problem. 2.3. Irregular waves Real irregular waves may be represented with good approximation as a superposition of regular waves, by deﬁning a spectrum. We adopt a discretized Pierson– Moskowitz spectrum. Since linear water wave theory is assumed, the resulting diffraction force is simply obtained by linear superposition X f dðtÞ ¼ f d;nðtÞ. (8) n 2.4. Time-domain analysis The PTO system of most wave energy converters of the oscillating-body type is strongly non-linear, and so the frequency-domain analysis, outlined above, is not applicable. This can be dealt with by the time-domain analysis (see Falnes, 2002b). The governing equation takes the form Zt ðm þ A1Þx€ðtÞ þ rgS xðtÞ þ Lðt À tÞx€ðtÞ dt À1 ¼ f dðtÞ þ f mðx; x_; tÞ, ð9Þ where AN is the limiting value of the added mass A(o) for o ¼ N. The convolution integral in Eq. (9) represents the memory effect in the radiation force. The force fm applied on the body by the energy conversion mechanism is a linear or non-linear function of x and x_, and may also be an explicit function of time if the system is actively controlled. The memory function L can be calculated from the radiation damping coefﬁcient B(o) by (see Falnes, 2002b): Z 2 1 BðoÞ LðtÞ ¼ sin ot do. (10) p0 o Eq. (9) can easily be integrated numerically, from initial conditions for x and x_. 2.5. The power take-off mechanism The hydraulic circuit includes a hydraulic cylinder or ram, a high-pressure (HP) gas accumulator, a low-pressure (LP) gas accumulator and a hydraulic motor (or a water turbine) (Fig. 1). A rectifying valve system prevents liquid from leaving the HP accumulator at E and from entering the LP accumulator at D. The hydraulic machine is driven by the ﬂow resulting from the pressure difference between the HP and LP accumulators. We denote by m1 and m2 the masses of gas inside the HP and LP accumulators, respectively, which are supposed to remain unchanged during operation. Assuming the duct and accumulator walls to be rigid and the liquid incompressible, the total volume of gas remains constant, i.e., m1v1(t)+m2v2(t) ¼ V0 ¼ constant (vi, i ¼ 1, 2, is speciﬁc volume of gas). We may also write qðtÞ À qmðtÞ ¼ Àm1 dv1ðtÞ dt , where q(t) is the volume ﬂow rate of liquid (oil or water) displaced by the piston and qm(t) is the volume ﬂow rate through the hydraulic motor (or water turbine). The speciﬁc entropy s1 of the gas inside the HP accumulator (and s2 for the LP accumulator) will change due essentially to heat transfer. This may be connected to changes in seawater temperature and surrounding air temperature, and also to changes in the power dissipated (viscous losses and electrical losses) inside the converter. Such changes are likely not to be signiﬁcant over time intervals less than, say, 1 h, and so it is reasonable to consider that the gas compression/expansion process inside the accumulator is approximately isentropic (s1 and s2 are constant) during a sea state (this means that, although the changes in gas temperature may be signiﬁcant during the compression/expansion cycle, the corresponding changes in entropy may be neglected). ARTICLE IN PRESS A.F.O. Falca˜o / Ocean Engineering 35 (2008) 358–366 361 We note that the force SCDp required to pump ﬂuid into the HP accumulator is to be overcome by the action of the buoy upon the piston. For details of modelling, see Falca˜ o (2007). 3. Floating converter with gas accumulator We consider a hemispherical buoy oscillating in heave, driving a hydraulic cylinder or ram that pumps highpressure liquid (oil or water) into a hydraulic circuit (Fig. 1). The rectifying valve is controlled in such a way that the liquid is pumped from the cylinder into the HP accumulator and sucked from the LP accumulator into the opposite side of the cylinder. The hydraulic machine is driven by the ﬂow resulting from the pressure difference between the HP accumulator and the LP accumulator. The time variation of the gas pressure difference Dp ¼ p1(t)Àp2(t) between the HP and LP accumulators results from (i) the action of the buoy upon the piston, and (ii) the ﬂow of liquid through the turbine or hydraulic motor. The diffraction force fd(t) that appears in (9) can be obtained from the spectrum characterizing the sea state under consideration and from the excitation force coefﬁcient G(o). While the body is moving, the governing equation is (9), with f m ¼ Àsignðx_ÞF, where F ¼ SC(p1Àp2) ¼ SC Dp and SC is the cylinder cross-sectional area (the piston rod cross-section is neglected). At some time, the time-varying body velocity will become zero. From then on, the body will remain stationary unless, or until, the hydrodynamic force on the body Zt f dðtÞ À rgS xðtÞ À Lðt À tÞx€ðtÞ dt À1 overcomes the resisting force F ¼ SC(p1Àp2) (or, if phase control by latching is implemented, until the controller allows the piston to be released) and ﬂuid is again pumped into the HP accumulator (and sucked from the LP accumulator). The hydrodynamic coefﬁcients A(o) and B(o) for a ﬂoating hemisphere oscillating in heave in deep water were obtained analytically by Hulme (1982). The numerical results presented below are for a hemisphere of radius a ¼ 5 m, r ¼ 1025 kg/m3, g ¼ 1.4 (nitrogen), in irregular waves with a spectral distribution of Pierson–Moskowitz type deﬁned by (SI units, see Goda, 2000) SzðoÞ ¼ 131:5H 2s T À4 e oÀ5 expðÀ1054T À4 e oÀ4Þ, (11) where Hs is signiﬁcant wave height and Te is energy period. The spectrum was discretized into 225 equally spaced 0(pD:1ﬃoﬃp¼ﬃ6ﬃp0.o0p1 r0a:d1/ps)ﬃ6ﬃ sinusoidal þ 2:24 rad/s. harmonics in the range (The irrational number 6 ensures the non-periodicity in the time-series of fd(t).) The phases at t ¼ 0 were made equal to random numbers in the interval (0, 2p). The integral-differential equation (9) was numerically integrated in the time domain with a time step size equal to 0.1 s. The instantaneous power absorbed by the converter is PðtÞ P¯ ¼ ¼DtÀF1jxR_ðtt0tf ÞPj,ðtÞadntd. its The time-average in t0ptptf value of P¯ naturally depends is on the magnitude of the time interval Dt ¼ tfÀt0. The numerical values presented here were obtained for Dt ¼ 1800 s. 4. Coulomb damping It is important to control the device in order to maximize the produced energy. This should take into account the sea state, characterized by Hs and Te. Since the system is assumed linear from the hydrodynamic point of view, then, for ﬁxed Te, the values of P¯ =H 2 s and q¯ =H s will depend only on the ratio F/Hs (we assume here the force F to be approximately constant over the sea state under considera- tion, and, as before, q ¼ SCjx_j is the ﬂow rate pumped by the piston). If phase control by latching is introduced (see Section 5), then those values will depend also on the control strategy and algorithm. We consider ﬁrst that the ﬂoater (and hence the piston) is unable to move for as long as the resultant hydrodynamic force on its wet surface is less than F ¼ SCDp. This kind of damping (simple Coulomb damping) does not involve any phase control strategy and was analysed in detail in Falca˜ o (2007). For this situation, the proposed control algorithm consists in establishing a proportionality relationship between the instantaneous liquid-ﬂow rate through the hydraulic motor (or water turbine), qm, and the instantaneous pressure difference Dp between the HP and LP accumulators: qm ¼ S2CG Dp, (12) where G is a constant. The power available to the hydraulic machine is Pm ¼ qm Dp ¼ G(SC Dp)2 ¼ GF2. We note that, over a sufﬁciently long time, the time-averaged values of P and Pm are equal (no energy losses are assumed to occur in the hydraulic circuit). 5. Phase control by latching It was found in Falca˜ o (2007) (for a hemi-spherical ﬂoater of 5 m radius in irregular waves), that, if G is optimized for Te, the simple Coulomb damping (produced by a hydraulic circuit as described above) allows approximately the same level of energy absorption as a PTO with an optimized linear damper (and no spring, i.e., K ¼ 0). It is known that a linear damper may be combined (in what is often called ‘‘phase control’’) with a ‘‘spring’’ (possibly with negative stiffness, Ko0, in the case of ‘‘small’’ bodies or point absorbers) to enhance the wave energy absorption. This suggests that the use of a hydraulic circuit as described above, combined with a control algorithm like (12), may be extended, by some kind of phase control, to achieve an increase in the amount of absorbed wave energy. In the important case of a ‘‘point absorber’’ (which is the object of the present paper), such modiﬁcation could ARTICLE IN PRESS 362 A.F.O. Falca˜o / Ocean Engineering 35 (2008) 358–366 possibly consist in extending the period of time during which the buoy is kept ﬁxed, in the following way. When the body is moving, its velocity will, at some time, come to zero, as a result of the hydrodynamic forces on its wetted surface and the PTO forces. The body will then remain ﬁxed until the hydrodynamic force |fh| exceeds RF ¼ R(SC Dp), where R41. It is to be noted: (i) that the force that has to be overcome (if the body is to restart moving) is now larger (by a factor R) as compared with the simple Coulomb damping (i.e., compared with SCDp); (ii) that the acceleration of the ﬂoater (unlike in the case of R ¼ 1) is discontinuous when the body is released. There is now a new parameter, R, to be optimized, jointly with parameter G. The optimization of this pair of parameters in phase control by latching may, in some way, be regarded as corresponding to the pair of optimization conditions (6 and 7) in the case of reactive phase control of a linear system in regular waves. Numerical simulations (30 min each) were carried out, based on this procedure and algorithm, for a hemispheric ﬂoater of radius a=5 m, in deep water, in regular waves (see Section 5.1) and irregular waves (Section 5.2). Piston area was SC ¼ 0.0314 m2. The masses of gas (nitrogen) in the HP and LP accumulators were m1 ¼ 100 kg and m2 ¼ 20 kg. In each simulation, the values of gas entropies s1 and s2 were taken such that the time-averaged gastemperatures in the HP and LP accumulators remained close to environmental temperature (ﬃ300 K). 5.1. Regular waves Results of simulations in regular waves (wave amplitude Aw ¼ 0.667 m and period T ¼ 9 s) are shown in Figs. 2–6. In Fig. 2, the solid line shows the numerically optimized values (that maximize P¯ ) of control parameter G for several values of the latching control parameter R (note that R ¼ 1 means simple Coulomb damping). In the same ﬁgure, the dashed line represents the amplitude of oscillation xmax. Figs. 3 and 4 represent, for the same situations, the time- 14 12 G×106 (s/kg), xmax/Aw 10 8 6 4 2 0 0 5 10 15 20 25 30 R Fig. 2. Regular waves, Aw ¼ 0.667 m, T ¼ 9 s: optimized control parameter G Â 106 (solid line) and dimensionless oscillation amplitude, xmax/ Aw (dashed line), versus latching control parameter R. 200 150 P (kW) 100 50 0 0 5 10 15 20 25 30 R Fig. 3. As in Fig. 2, time-averaged absorbed power P¯ (kW) versus R (G optimized for each R). 100 80 p1 (bar) 60 40 20 0 0 5 10 15 20 25 30 R Fig. 4. As in Fig. 2, time-averaged gas pressure in HP accumulator versus R (G optimized for each R). averaged absorbed power P¯ and the time-averaged gas pressure (in the HP accumulator) p¯1, respectively, versus R. It may be seen that, by increasing R above unity and (for each R) suitably optimizing G, a substantial increase (by a factor up to about 3.8) in the time-averaged absorbed power P¯ can be achieved. The maximum power attained in this way, about 206 kW for R=16, should be compared with the theoretical maximum power ð1=4Þg3rA2woÀ3 ¼ 315 kW absorbed by an axisymmetric body with a linear PTO damper oscillating in heave. It is to be noted that this increase in absorbed power results mostly from larger ﬂoater oscillations xmax (and hence greater liquid ﬂow through the hydraulic motor) rather than from greater pressure levels in the HP hydraulic circuit (Fig. 4). Figs. 5 and 6 represent the time variation of the diffraction force fd(t), the ﬂoater velocity dx/dt and its displacement x(t) for two situations optimized with respect to G (same incident wave time series, for easier comparison): R=1, G=0.86 Â 10À6 s/kg (simple Coulomb damping; Fig. 5) and R ¼ 16, G ¼ 7.7 Â 10À6 s/kg (latching control; Fig. 6). It is to be noted that, in Fig. 6 (but not in Fig. 5), the velocity and the diffraction force are (very approximately) ARTICLE IN PRESS A.F.O. Falca˜o / Ocean Engineering 35 (2008) 358–366 363 dx/dt (m/s) 10 × fd (MN) 4 2 0 −2 −4 600 602 604 606 608 610 612 614 t (s) 4 dx/dt (m/s) 10 × fd (MN) 4 2 0 −2 −4 600 602 604 606 608 610 612 614 t (s) 4 2 2 x(m) x(m) 0 0 −2 −2 −4 600 602 604 606 608 610 612 614 t (s) Fig. 5. Performance of a hemispherical ﬂoater in regular waves for R ¼ 1, G ¼ 0.86 Â 10À6 s/kg. Above, dx/dt: solid line, fd(t): broken line. Below, x(t). Absorbed power: P¯ ¼ 55:0 kW. in phase with each other (in agreement with optimal condition (5) for linear PTO). 5.2. Irregular waves Optimal phase control in random irregular waves is known to require the prediction of the incoming waves (theoretically over the inﬁnite future, in practice over a few tens of seconds, see Falnes, 2002a; Naito and Nakamura, 1986). In addition to this difﬁculty, the theoretical determination of the wave-to-wave optimal latching period requires heavy computation (this seems to have been done for the ﬁrst time, in wave energy applications, by Hoskin et al. (1986), who used the Principle of Maximum of Pontryagin), which makes it inappropriate for implementation in real time. Therefore, it is particularly interesting to investigate whether the simple control strategy outlined above, and tested in Section 5.1 in regular waves, can be applied successfully to irregular waves. One ought to bear in mind that this should be regarded, at best, as a sub-optimal strategy, and that the achievable results should not be expected to be close to the theoretical maximum. Numerical simulations, identical to those presented in Section 5.1 for regular waves, were performed for irregular −4 600 602 604 606 608 610 612 614 t (s) Fig. 6. As in Fig. 5, for R ¼ 16, G ¼ 7.7 Â 10À6 s/kg. P¯ ¼ 206:1 kW. waves as modelled by a Pierson–Moskowitz spectrum (see Eq. (11)), and Hs ¼ 2 m, Te ¼ 7, 9, and 11 s. The results, in terms of time-averaged absorbed power P¯ divided by H2s , are presented in Figs. 7–9. The ﬁgures show that a large increase (by a factor about 2.3–2.8) in absorbed power (as compared with simple Coulomb damping, R=1) can be achieved by suitably combining the values of the control parameters R (R41) and G. The largest absorbed power occurs for R equal to about 16 and a value of G that depends on R and Te. Curves for the diffraction force fd(t), and the ﬂoater velocity x_ðtÞ and displacement x(t), are given in Figs. 10 and 11, for Te=9 s and control parameter pairs (R=1, G=0.7 Â 10À6 s/kg), (R=16, G=4.2 Â 10À6 s/kg). It is not surprising that those large values of absorbed power occur for relatively large amplitudes of the ﬂoater oscillations, that typically attain nearly twice the value of the signiﬁcant wave height HS, as shown in Fig. 11. Since the whole analysis is (as usual) based on linear hydrodynamic theory (which assumes the amplitude of body oscillations to be small compared with the body size), such oscillations are unrealistically large (except in calm seas, say Hso1 m) and so are the values of absorbed power. Of course, this is also true in general, whenever the theory predicts large oscillation amplitudes as a result of a wave energy converter being tuned (by phase control or otherwise) to the incoming waves. ARTICLE IN PRESS 364 A.F.O. Falca˜o / Ocean Engineering 35 (2008) 358–366 25 3 (dx/dt)/Hs (s-1), 10 × fd/Hs (MN/m) 20 4 15 28 P/Hs2 (kW/m2) 10 8 12 16 20 24 R=1 5 0 0 2.5 5 7.5 10 12.5 15 G×106 (s/kg) Fig. 7. Hemispherical ﬂoater in irregular waves, Te ¼ 7 s. Plot of P¯ =H 2 s versus control parameter G, for several values of latching control parameter R. 30 P/Hs2 (kW/m2) 25 20 4 15 10 5 0 0 28 12 16 20 24 8 R=1 2 4 6 8 10 12 G×106 (s/kg) Fig. 8. As in Fig. 7, for Te ¼ 9 s. P/Hs2 (kW/m2) 30 25 20 4 15 10 5 0 0 8 12 R=1 16 20 2 4 6 8 G×106 (s/kg) Fig. 9. As in Fig. 7, for Te ¼ 11 s. 28 24 10 It should be noted that the increase in absorbed power (achievable from suitable latching) that is apparent in Figs. 7–9, results mostly from the large increase in amplitude of ﬂoater oscillations rather than from an increase in pressure level in the PTO hydraulic circuit, as 2 1 0 −1 −2 −3 700 720 740 760 780 800 t (s) 2 1 x/Hs 0 −1 −2 700 720 740 760 780 800 t (s) Fig. 10. Performance of a hemispherical ﬂoater in irregular waves for Te ¼ 9 s, R ¼ 1, G ¼ 0.7 Â 10À6 s/kg. Above, dx/dt: solid line, fd(t): broken line. Below, x(t). Absorbed power: P¯ =H 2 s ¼ 10:3 kW/m2. shown in Fig. 12. (This conﬁrms the results in Section 5.1 for regular waves.) On the other hand, values of the latching control parameter R much larger than unity (required to maximize P¯ ) may imply very large forces to keep the body ﬁxed prior to its release. Such forces are likely to exceed the practical limits of the ram and remaining hydraulic circuit and would possibly require a special braking system. This is an engineering problem that has to be faced whenever phase control by latching is considered. Fig. 11 shows that the peaks of velocity dx/dt in general (but not in every oscillation) coincide, in time, approxi- mately with the peaks of the diffraction force fd, which matches the optimal condition expressed by Eq. (5). The values of P¯ =H 2 s , x/Hs and x_ =H S plotted in Figs. 7–11 were computed for Hs=2 m. These values would change with HS due to the non-linear response of the gas accumulator, as was analysed in detail in Falca˜ o (2007). Provided the accumulator is appropriately sized, those changes are relatively small within the range of HS in which the linear wave theory is applicable. If this is the case, one can say approximately that the latching control algorithm proposed and numerically optimized here is approximately independent of signiﬁcant wave height. ARTICLE IN PRESS A.F.O. Falca˜o / Ocean Engineering 35 (2008) 358–366 365 (dx/dt)/Hs (s-1), 10 × fd/Hs (MN/m) 3 and other speciﬁcations of the accumulator are dictated by several criteria, namely the maximum allowable working 2 pressure, the desired power output smoothness and equipment costs. 1 0 −1 −2 −3 700 720 740 760 780 800 t (s) 2 1 x/Hs 0 −1 −2 700 720 740 760 780 800 t (s) Fig. 11. As in Fig. 10, for R ¼ 16, G ¼ 4.2 Â 10À6 s/kg. P¯ =H 2 s ¼ 28:5 kW/ m2. 100 80 p1 (bar) 60 40 20 0 0 5 10 15 20 25 30 R Fig. 12. Irregular waves, Hs ¼ 2 m, Te ¼ 9 s. Time-averaged pressure p¯1 in HP accumulator versus latching control parameter R (G is optimized for each R). One should bear in mind that the pressure difference Dp decreases (due to the continuous ﬂow of liquid from the HP to the LP reservoir through the hydraulic machine) whenever the ﬂoater in unable to move (this decrease is faster the smaller the accumulator size); this effect tends to adjust the pressure level Dp to the current sea state and also (in a different measure) to the wave group or even the wave-to-wave succession. Naturally, the choice of the size 6. Conclusions A simple strategy and algorithm were proposed for the phase control by latching of a point absorber in irregular random waves. The device is a heaving ﬂoater equipped with a PTO consisting of a high-pressure hydraulic circuit and a gas accumulator. The control algorithm was found, by numerical simulation, to be a promising and remarkably effective way to increase the amount of absorbed energy from regular as well as irregular waves. The control algorithm (i.e., the pair of parameters R and G) to be implemented is weakly dependent on signiﬁcant wave height, but (for best results) should be adjusted to match the sea state wave period, as illustrated in Figs. 7–9. One should take into consideration that the gain from phase controlling a device is greater if its own resonance frequency is much higher than the typical sea wave frequency. This was the case simulated here: a hemispherical buoy of radius a=5 m, whose natural frequency (as deﬁned by resonance condition (6) with K ¼ 0) is o ¼ 1.44 rad/s (corresponding resonance period: 4.37 s). Naturally, in spite of the gain that phase control may bring, the geometry and size of the ﬂoater should be chosen by the designer, within economic and other constraints, having in view its natural frequency to match the representative sea wave frequency. As is known in general to be the case of phase-controlled point absorbers (or devices whose natural frequency is substantially higher than the sea wave frequency), the great gain in absorbed energy was found here, by theoretical simulations of latching control, to imply ﬂoater oscillations that are unrealistically large (except in sea states of very low energy level). This means that, although the control strategy proposed here is expected to be effective in real conditions, the optimization of the control parameters R and G (and the realistic evaluation of the absorbed energy) should be performed (or validated) otherwise, possibly by prototype testing, or by model testing at a scale large enough to allow the PTO to be properly simulated. In any case, what is proposed here is a control strategy and algorithm that may be implemented very easily in a wide variety of oscillating-body devices (in the case of multiple-mode devices, although conceptually not difﬁcult, the control may be less straightforward in practice). Acknowledgements The work reported here was partly supported by IDMEC, Lisbon (programme POCI2010), by the Portuguese Foundation for Science and Technology (FCT) through contract no. ARTICLE IN PRESS 366 A.F.O. Falca˜o / Ocean Engineering 35 (2008) 358–366 POCTI/ENR/56079/2004, and by the European Commission through contract no. 502701 (SES6). References Babarit, A., Cle´ ment, A.H., 2006. Optimal latching control of a wave energy device in regular and irregular waves. Applied Ocean Research 28, 77–91. Babarit, A., Duclos, G., Cle´ ment, A.H., 2004. Comparison of latching control strategies for a heaving wave energy device in random sea. Applied Ocean Research 26, 227–238. 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