Ocean Engineering 59 (2013) 20–36 Contents lists available at SciVerse ScienceDirect Ocean Engineering journal homepage: www.elsevier.com/locate/oceaneng Nonlinear dynamics of a tightly moored point-absorber wave energy converter Pedro C. Vicente a,n, Anto´ nio F.O. Falca~ o a, Paulo A.P. Justino b a IDMEC, Instituto Superior Te´cnico, Technical University of Lisbon, Lisbon 1049-001, Portugal b Laborato´rio Nacional de Energia e Geologia, Estrada Pac- o do Lumiar, Lisbon 1649-038, Portugal article info Article history: Received 29 June 2012 Accepted 1 December 2012 Available online 2 January 2013 Keywords: Wave energy Wave power Mooring Tight mooring Nonlinear effects abstract Tightly moored single-body ﬂoating devices are an important class of offshore wave energy converters. Examples are the devices under development at the University of Uppsala, Sweden, and Oregon State University, USA, prototypes of which were recently tested. These devices are equipped with a linear electrical generator. The mooring system consists of a cable that is kept tight by a spring or equivalent device. This cable also prevents the buoy from drifting away by providing a horizontal restoring force. The horizontal and (to a lesser extent) the vertical restoring forces are nonlinear functions of the horizontal and vertical displacements of the buoy, which makes the system a nonlinear one (even if the spring and damper are linear), whose modelling requires a time-domain analysis. Such an analysis is presented, preceded, for comparison purposes, by a simpler frequency-domain approach. Numerical results (motions and absorbed power) are shown for a system consisting of a hemispherical buoy in regular and irregular waves and a tight mooring cable. The power take-off is modelled by a simpliﬁed system of a linear spring and a linear damper and also by a system formed by a hydraulic piston and spring. Different scenarios are analysed. & 2012 Elsevier Ltd. All rights reserved. 1. Introduction Floating oscillating-body devices are a large class of wave energy converters (WECs) for deployment offshore, typically in water depths between 40 and 100 m (Falca~o and de, 2010). Among these, the simplest device is the single body reacting against the sea bottom. An early example is the Norwegian buoy, consisting of a spherical ﬂoater which could perform heaving oscillations relative to a strut connected to an anchor on the sea bed through a universal joint (Budal et al., 1982). A model (buoy diameter¼1 m) was tested (including latching control) in the Trondheim Fjord in 1983. An alternative design is a buoy connected to a bottom-ﬁxed structure by a cable which is kept tight by a spring or similar device. The motion of the wave-actuated ﬂoat on the sea surface activates a power take-off (PTO) system. Such a device was investigated in Norway in the late 1970s (Falnes and Budal, 1978; Budal and Falnes, 1980), but later abandoned (Falnes and Lillebekken, 2003). In the device that was tested in Denmark in the 1990s, the PTO (housed in a bottom-ﬁxed structure) consisted in a piston pump supplying high-pressure water to a hydraulic turbine (Nielsen and Smed, 1998). n Corresponding author. Tel.: þ351 218417519; fax: þ351 218417398. E-mail address: pedro.cabral.vicente@ist.utl.pt (P.C. Vicente). 0029-8018/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.oceaneng.2012.12.008 The taut-moored buoy being developed at Uppsala University, Sweden uses a linear electrical generator (rather than a piston pump) placed on the ocean ﬂoor (Waters et al., 2007). A line from the top of the generator translator is connected to a buoy located at the ocean surface, and in this way acts as a PTO transmission line. Springs attached to the translator bottom, store energy during half a wave cycle and simultaneously act as a restoring force in the wave troughs. Sea tests off the western coast of Sweden of a 3 m diameter cylindrical buoy are reported in Waters et al. (2007). Another system with a heaving buoy driving a linear electrical generator was recently developed at Oregon State University, USA (Elwood et al., 2009). It consists of a deep-draught spar and an annular saucer-shaped buoy. The spar is taut-moored to the sea bed by a cable. The buoy is free to heave relative to the spar, but is constrained in all other degrees of freedom by a linear bearing system. The forces imposed on the spar by the relative velocity of the two bodies is converted into electricity by a permanent magnet linear generator. The spar is designed to provide sufﬁcient buoyancy to resist the generator force in the down direction. A 10 kW prototype was deployed off Newport, Oregon, in September 2008, and tested (Elwood et al., 2009). The mooring system in these devices consists of a cable, that connects the buoy to a sea-bottom-ﬁxed structure and that is kept tight by a spring or equivalent device, or, alternately (as in the Norwegian buoy) is a strut connected to the sea bed by a universal joint. P.C. Vicente et al. / Ocean Engineering 59 (2013) 20–36 21 Nomenclature a radius of ﬂoater A added mass Aw wave amplitude B radiation damping coefﬁcient C damping coefﬁcient for the linear PTO f PTO force fd diffraction or excitation force F pre-tension(linear PTO)or piston force (hydraulic PTO) FM spring force in the hydraulic PTO system FP piston force in the hydraulic PTO system g acceleration of gravity L initial cable length Hs signiﬁcant height of irregular waves K stiffness of the mooring line m mass of ﬂoater n nth harmonic in irregular waves N number of harmonics in irregular waves P power P time-averaged power PTO power take-off mechanism DL cable length variation S SB(o) t Te U X, z X, Z a y G f r o pa2 power spectral distribution time energy period of irregular waves complex amplitude of velocity displacements of body centre (Fig. 1) complex amplitudes of body centre displacement angle between the mooring cable and vertical direction irregular wave harmonic phase excitation force coefﬁcients mooring cable force density radian frequency Subscripts irr irregular wave x, z directions of x, z axes Superscript n dimensionless value Some studies have already been made on the dynamics of a taut-moored buoy (all or most of which without any type of energy conversion) in a ﬂuid ﬂow or in waves. Some early initial results in a theoretical and experimental analysis (Harleman and Shapiro, 1960) of the dynamics of a single-moored hemispherical buoy in shallow water indicated that the sphere motion and mooring line forces were related to the sphere size, weight, submergence and also to the wave frequency, wave height and water depth. Also, some numerical simulations on the motion of buoys in different wave climates (Carpenter et al., 1995), later supported by experimental results, indicated that the surge response of a spherical buoy was highly dependent on the system natural oscillation frequency. The effects of ﬂoaters geometry on the hydrodynamics and performance of a tightly moored ﬂoating single-body WEC was studied by Mavrakos et al. (2009). The work here presented focuses on the nonlinear dynamics introduced by the mooring system of these wave energy converters and on the analysis of the inﬂuence of the mooring Fig. 1. Buoy, PTO and mooring line: (a) Linear PTO model, (b) Hydraulic PTO model. 22 P.C. Vicente et al. / Ocean Engineering 59 (2013) 20–36 parameters upon the motion behaviour of the system. To this end, a proper time-domain model is used and compared with a simpliﬁed frequency-domain one. This mooring system prevents the buoy from drifting away by providing a horizontal restoring force whose value is equal to the tension force times sin a, where a is the angle with the vertical direction. Since this restoring force is in general much smaller than the buoyancy force, resonant horizontal motions are expected to occur at considerably lower frequency, with possibly larger amplitude, as compared with heaving oscillations. Naturally the heaving and surge oscillations are coupled through the wave ﬁelds they generate and through the mooring cable (it should be pointed out that the horizontal resonant motion is usually not excited by wave frequency loads but by second-order slowly varying forces). A linearized version based on a frequency-domain analysis is considered ﬁrst. The fully nonlinear modelling of mooring forces is addressed by employing a time-domain analysis. Numerical results from the linearized and nonlinear theories are shown for motions and absorbed power, in regular and irregular waves, for a system consisting of a hemispherical buoy and a tight mooring cable. The power take-off is modelled by a simpliﬁed system of a linear spring and a linear damper (Fig. 1a) and also by a more realistic system of a hydraulic piston and spring (Fig. 1b), assuming large gas reservoirs for a constant force piston. These two different models are compared. The system non-linear characteristics are brieﬂy analyzed. 2. The model We assume one-directional waves and adopt a hemisphere as the geometry for the converter, which allows the analysis to be 1 restricted to the translational modes of heave and surge. More realistic shapes (namely vertical cylinders) would introduce also pitch oscillations. The ﬂoater is tightly moored to the sea bottom by a single cable (Fig. 1). A linear damper (located at the sea bottom, as shown in Fig. 1, or alternately inside the ﬂoater) provides the conversion of the energy absorbed from the waves. The cable is kept tight by a pre-tensioning spring (that, in Fig. 1, pulls the rod of a double-rod bottom-ﬁxed hydraulic cylinder). Its mass and the viscous forces on it are neglected, and so the cable may be modelled as astraight line. The cable is supposed to be attached to the ﬂoater’s centre, which allows the pitch oscillations and the rotational inertia to be disregarded. The effects of very slow changes in sea surface level (namely tidal oscillations) are ignored. It should also be pointed out that, neither the elasticity of the cable, nor the frictions inside the PTO systems, are modelled. Let x and z be, respectively, the horizontal and vertical time- varying displacements of the ﬂoater’s centre from its rest position located on the undisturbed free-surface (Fig. 1), with z increasing upwards and the x-coordinate aligned with the wave direction. We denote by LþDL(t) the cable length at each moment between attachment points and by L the corresponding value under calm sea conditions. We easily ﬁnd qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ DL ¼ x2 þ ðz þ LÞ2ÀL ð1Þ and that the angle a between the cable and the vertical direction (Fig. 1) is given by sin a ¼ x=ðL þ DLÞ. As stated before, the power take-off is modelled both by a simpliﬁed system of a linear spring and a linear damper and also by a more realistic system of a hydraulic piston and spring. In the simpliﬁed linear model of the PTO we assume that the cable is tensioned by a linear spring of stiffness K and transmits the (linear) damper force, and therefore we may write f ¼ KDLþ C dDL=dt (the spring and damper are mounted in parallel.) The vertical pulling force 1 0.8 0.8 0.6 0.6 X* Z* 0.4 0.4 0.2 0.2 0 0 0.5 1 1.5 2 ω* 0 0 0.5 1 1.5 2 ω* C*=0.3825 C*=0.51 C*=0.6375 1 0.8 0.6 P* 0.4 0.2 0 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 ω* Fig. 2. Dimensionless results from frequency-domain analysis: heave Zn and surge Xn oscillation amplitudes, and power absorption, versus wave frequency. Regular waves, with Ln ¼8, Fn ¼ 0.05, Kn ¼0.1, and for different values of Cn. P.C. Vicente et al. / Ocean Engineering 59 (2013) 20–36 23 is then fz ¼ ðF þf Þcosa and the horizontal restoring force provided by the cable is fx ¼ ðF þf Þsina where F is the pre-tension in the cable under calm sea conditions. For the hydraulic PTO, we consider that the wave energy converter resorts to a hydraulic mechanism that uses the oscillating motions of the ﬂoating buoy to drive a piston inside a cylinder placed on the bottom, which contains a ﬂuid at high pressure (here we assume a constant pressure and therefore a constant force). This process drives a hydraulic motor that in turn drives an electrical generator to produce electrical energy. In parallel with this piston there is a linear spring of stiffness K whose objective is to apply a restoring force to the ﬂoater, proportional to the cable length variation, in order to restore it to the original equilibrium position (see Fig. 1). Since, in calm sea, the centre of the hemispherical buoy is assumed to lie on the horizontal free-surface plane (z¼0), the buoy mass m must be m¼ 2 pa3, 3 rÀ F g , ð2Þ where a is the buoy radius and r is the water density. Taking into account the spherical shape of the ﬂoater, and recalling that the mooring line is supposed to be attached to the centre of the buoy, it follows that the only signiﬁcant modes of oscillation are heave and surge, and also that these modes are hydrodynamically uncoupled. 3. Frequency-domain analysis We assume here that the displacements x, z are much smaller than L, so that a is a small angle and cos a ﬃ1. Besides, considering the linear PTO system, we take KDLþC dDL/dt 5F and sin a ﬃ x=L. Then we may write, in a linearized form, fx ¼Fx/L 1 and fz ¼ F þ Kz þ Cz_. The vertical component of the force on the buoy due to the PTO damper becomes ÀCz_ and the horizontal component may be neglected. 3.1. Regular waves We start by considering regular waves of angular frequency o. By following the usual linear decomposition of the hydrodynamics forces (Falnes, 2002), we can write the governing linear equations, valid after the transients related to initial conditions have died out, as ðm þ Ax Þx€ þ Bx x_ ¼ f dx ÀF x L , ð3Þ ðm þ AzÞz€ þ Bzz_ þ r gSz ¼ f dzÀCz_ÀKz ð4Þ here, Ax, Az and Bx, Bz are the hydrodynamic coefﬁcients of added mass and radiation damping for surge (subscript x) and heave (subscript z) motions. The forces fdx and fdz are the horizontal and vertical components of the wave-induced excitation force on the buoy. Finally, it is S ¼pa2. Since the system is entirely linear, the displacements x and z and the excitation force components time and so we may write { x, are simple-haÀrmonic z, fdx, fdz}¼ Re X, Z, F dfux,ncFtdizoneisotoÁf, where the complex amplitudes X, Z, Fdx and Fdz are proportional to the amplitude Aw of the incident wave. The moduli of Fdx and Fdz may be written as {9Fdx9,9Fdz9}¼ {GxAw, Gz Aw}, where Gx(o) and Gz(o) are (real positive) excitation force coefﬁcients. Tabulated values (together with asymptotic expressions) can be found in Hulme, 1982 (in dimensionless form) for the added masses Ax, Az and for the radiation damping coefﬁcients Bx, Bz of a ﬂoating hemisphere oscillating in heave and surge in deep water. These values were adopted in our calculations. For Gz(o)we use Haskind’s relation (valid for an axisymmetric body in deep water 1 0.8 0.8 0.6 0.6 X* Z* 0.4 0.4 0.2 0.2 0 0 0.5 1 1.5 2 ω* 0 0 0.5 1 1.5 2 ω* F*=0.03 F*=0.05 F*=0.08 1 0.8 0.6 P* 0.4 0.2 0 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 ω* Fig. 3. Dimensionless results from frequency-domain analysis. Heave Zn and surge Xn oscillation amplitudes, and power absorption, versus wave frequency. Regular waves, with Ln ¼ 8, Cn ¼0.501, Kn ¼0.1, and for different values of Fn. 24 P.C. Vicente et al. / Ocean Engineering 59 (2013) 20–36 (Falnes, 2002; Newman, 1962) Gz ðoÞ ¼ &2g 3 rBz o3 ðoÞ'1=2 : ð5Þ Numerical values for Gx(o) and argðFdzðoÞ=FdxðoÞÞ were obtained with the aid of the boundary element code WAMIT. It should be pointed out that the excitation forces are calculated for the mean position of the buoy and this may be another source of non-linearity if the buoy moves substantially around this position. From Eqs. (3) and (4), we obtain X ¼ F dx Ào2ðm þ AxÞ þ ioBx þ F=L ð6Þ Z ¼ F dz Ào2ðm þ AzÞ þ io ðBz þCÞ þrgS þ K ð7Þ The time-averaged power absorbed by the ﬂoater is (Evans, 1979, 1980) P ¼ 1 Co29Z92 2 ¼ 1 8Bz 9F dz 92ÀBz 2 UÀ2FBdzz2 ð8Þ where U ¼io Z is the complex amplitude of the velocity z_. For a given body and ﬁxed wave amplitude and frequency (i.e., ﬁxed Bz and Fd z), P is maximum for U ¼ Fdz/2Bz, which, together with Eq. (7), gives the optimal conditions o ¼ rgS þ K 1=2 m þAzðoÞ ; ð9aÞ C ¼ BzðoÞ ð9bÞ Eq. (9a) is a resonance condition, whereas Eq. (9b) shows that the optimal PTO damping should equal the radiation damping (from heave oscillations). Now we deﬁne dimensionless values, denoted by an asterisk, as {Xn, Zn} ¼ {9X9,9Z9}/Aw, Cn ¼ CrÀ1aÀ5/2gÀ1/2, o* ¼ o (a/g)1/2, Fn ¼ F/(2pa3rg/3), Kn ¼K/(rgS) and L* ¼L/a. For the case of an unmoored hemispherical buoy (K¼0), the (optimal) solution of Eq. (9a) and (9b)) is (in dimensionless form) o*¼1.027, C n opt ¼ 0:510. We deﬁne a dimensionless (time-averaged) power absorption coefﬁcient as Pn ¼ P=Pmax, where Pmax is the theore- tical maximum limit of the (time-averaged) power that an axisymmetric heaving wave energy converter can absorb from breeg(uFlaalrnwesa,v2e0s0o2f)fPremqaux e¼ngc3yroA2wan=Àd4aom3Ápl(ictuordreesApwo,nadnidngistkoncoawptnutroe width l/2p, where l¼wavelength). Numerical results are presented, in dimensionless form, in Figs. 2–4. In these Figures, the nominal values of K* ¼0.1, F* ¼0.05, Ln ¼ 8 and C * ¼ C n opt ¼ 0:510 are used and the inﬂuence of a 25% variation in each parameter Cn, Fn, Kn is shown. The curves show the heave and surge oscillation amplitudes, Zn and Xn, and the absorbed power, Pn, versus wave frequency on. In Fig. 2 it can be seen that absorbed power is maximum for on ﬃ 1.1 using a value of C*¼ 0.510. For the other values of Cn the peak values are slightly smaller. Note that the peak value for Cn ¼0.510, although the highest, is still less than unity because the value C*¼0.510 is the one that maximizes the power absorption by the unmoored (not the moored) buoy. Close to the peak frequency on ﬃ1.1, the heave oscillation amplitude Zn is more than twice as large as Xn. The curve for surge oscillation amplitude Xn has an Fig. 4. Dimensionless results from frequency-domain analysis. Heave Zn and surge Xn oscillation amplitudes, and power absorption, versus wave frequency. Regular waves, with Ln ¼8, Cn ¼0.501, Fn ¼ 0.05, and for different values of Kn. P.C. Vicente et al. / Ocean Engineering 59 (2013) 20–36 25 (unrealistically large) peak at on ﬃ 0.08 indicating a low frequency resonance characteristic of moored ﬂoating structures. (An identical phenomenon was found to occur in spread-moored wave energy converters, see Vicente et al., 2009). It should be noticed however that close to the low frequency resonance the damping would result mostly from horizontal drag forces not accounted for in the model. In Figs. 3 and 4 it can be seen that changing the values of F and K does not change the dimensionless maximum power, changing the frequency in which the power peak occurs, and also changing mainly the heave amplitudes (for F in the peak frequency area and for K in the frequencies lower than the peak frequency). The variation in the initial cable length revealed almost no inﬂuence in the curves. Numerical results are plotted in Figs. 5 and 6, for regular waves of Aw ¼ 1 m, T ¼ 10 s, for a¼ 7.5 m, L ¼ 60 m, K¼ 1.8 Â 105 kg/s2, C¼ 2.5 Â 105 kg/s, F ¼ 1 MN, r¼ 1025 kg/m3 and g ¼ 9.8 m/s2. It should be point out that Figs. 5 and 6 are for illustration purpose, since they represent the frequency-domain solution for a speciﬁc value of Aw and T. 1 0.5 z [m] 0 −0.5 1 −1 0 1 0.5 10 20 30 40 50 60 t [s] 0 z [m] −0.5 0.5 −1 −1 0 1 0 x [m] x [m] −0.5 −1 0 10 20 30 40 50 60 t [s] Fig. 5. Results from frequency-domain analysis: time series of heave z and surge x displacements. Trajectory of buoy centre is also shown. Regular waves of Aw¼ 1 m, T ¼ 10 s, with a¼ 7.5 m, L¼60 m, K ¼ 1.8 Â 105 kg/s2, C ¼2.5 Â 105 kg/s, F ¼1 MN. 1.5 1 F*PTO 0.5 0 0 0.2 10 20 30 40 50 60 t [s] 0.15 P*reg 0.1 0.05 0 0 10 20 30 40 50 60 t [s] Fig. 6. Results from frequency-domain analysis: time series of the cable force and absorbed power. Regular waves as in Fig. 5. 26 P.C. Vicente et al. / Ocean Engineering 59 (2013) 20–36 In Fig. 5 the heave and surge oscillations can be seen, having similar amplitudes. In Fig. 6 the oscillations of the force applied by the mooring cable (divided by F) and the oscillations of the absorbed power (in dimensionless form) are shown. It can be seen that for the simulated conﬁguration the cable force exceeds in about 25% the pre-tension force and that the power absorbed is about 15% of the possible optimum power take-off. 3.2. Irregular waves. Real irregular waves may be represented, in a fairly good approximation, as a superposition of regular waves, by deﬁning a spectrum. Since our wave energy converter is axisymmetric and insensitive to wave direction, it is reasonable to assume the spectrum to be one-dimensional. We adopt the Pierson–Moskowitz spectral z [m] 3 2 1 0 −1 −2 −3 0 2 1 0 50 100 150 t [s] z [m] 3 2 1 0 −1 −2 −3 −2 0 2 x [m] x [m] −1 −2 −3 0 50 100 150 t [s] Fig. 7. Results from frequency-domain analysis: time series of heave z and surge x displacements. Trajectory of buoy centre is also shown. Irregular waves of HS¼ 2 m, Te¼ 10 s, with a¼ 7.5 m, L¼ 60 m, K ¼1.8 Â 105 kg/s2, C ¼2.5 Â 105 kg/s, F ¼1 MN. 4 3 F*PTO 2 1 0 0 0.8 20 40 60 80 100 120 140 160 180 t [s] 0.6 P*irr 0.4 0.2 0 0 20 40 60 80 100 120 140 160 180 t [s] Fig. 8. Results from frequency-domain analysis: time series of the cable force and absorbed power. Irregular waves as in Fig. 7. P.C. Vicente et al. / Ocean Engineering 59 (2013) 20–36 27 distribution, deﬁned by (SI units, Goda, 2000) SBðoÞ ¼ 263 H2s T À4 e oÀ5exp À1054 T o À4 À4 e , ð10Þ where Hs is the signiﬁcant wave height and Te is the energy period. This is known as frequency spectral density function, or simply frequency spectrum, and has unit m2/s. The time-averaged power output in irregular waves is Z1 PirrðHs,TeÞ ¼ 2 P1ðoÞSBðo do, 0 ð11Þ where P1ðoÞ is the power absorbed by the ﬂoater from regular waves of frequency o and unit amplitude. In dimensionless form, we write, for irregular waves, Pnirr ¼ Pirr=Pmax,irr, where Pmax,irr ¼ g3r Z 1 20 oÀ3SBðoÞdo ¼ 149:5H2s T 3 e ð12Þ (SI units) is the maximum (time-averaged) power that can be extracted by an axisymmetric body oscillating in heave in a sea state represented by the spectral distribution SB(o). For time-series calculations, the spectral distribution (10) is discretized as the sum of a large number N of regular waves of frequency on¼o0þnDo, where o0 is the lowest frequency considered (o0/Do should be an irrational number in order to ensure the non-periodicity in the time-series), Do is a small frequency interval, n¼0, 1, 2,y, NÀ 1, and the spectrum is supposed not to contain a signiﬁcant amount of energy outside the frequency range o0ro ro0þ(NÀ 1)Do. The (deterministic) amplitude of the wave component of order n is Awn¼ (2SB(on)Do)1/2. (Alternatively, random amplitudes could have been used, see Tucker et al., 1984; for a comparison of both approaches see Saulnier et al., 2009). The excitation force may be written as X X f djðtÞ ¼ f djnðtÞ ¼ AwnReFdjðonÞeiðont þ ynÞ ð13Þ n n pﬃﬃﬃ (j¼x, z). In the simulations we adopted o0 ¼ 0:05 þ 6 rad/s, Do ¼0.01 rad/s and N¼200. The phase yn of each component was chosen as a random real number in the interval (0, 2p). Numerical results are plotted in Figs. 7 and 8, for irregular waves of Hs ¼4 m, Te¼10 s, and again for a ¼7.5 m, L ¼60 m, K¼ 1.8 Â 105 kg/s2, C ¼ 2.5 Â 105 kg/s, F ¼ 1 MN. The curves show the wave grouping effects on the body oscillations (Fig. 7) and on the absorbed power (Fig. 8). In Fig. 8 it can be seen that for this set of irregular waves of Hs ¼4 m and Te¼10 s, the cable force can get almost four times higher than the pre-tension force. It should be pointed out again that Figs. 7 and 8 are for illustration purpose, since they represent the solution for a speciﬁc value of Hs and Te and a given set of random phases. 4. Time-domain analysis In Section 3, linearizing approximations were introduced to allow the frequency-domain analysis to be applicable. More precisely, the horizontal and vertical components of the forces applied on the ﬂoater by the mooring cable are in fact not linear functions of the displacements x and z and their derivatives. The dynamics of ﬂoating bodies in waves with nonlinear mooring systems were theoretically studied in connection with moored ships and offshore platforms without wave energy absorption. Analyses can be found on the inﬂuence of the mooring lines on the horizontal motion in Gottlieb and Yim (1997), Aranha and Pinto (2001), Ellermann et al. (2002), Umar and Datta (2003), Jorda´ nand Beltra´ n-Aguedo (2004), Rosales and Filipich (2006), Ellermann (2009) and on the nonlinear dynamics of moored vessels in Idris et al. (1997), Pascoal et al. (2005, 2006). Studies on the instability and reliability of these systems can be found in Yilmaz and Incecik (1996), Shah et al. (2005), and Radhakrishnan et al. (2007). Some similar vertical (1D) time domain non-linear simulations of wave-energy devices were performed and published, e.g., by Greenhow (1982) and Eidsmoen (1998). The nonlinear effects are in general much more signiﬁcant in the case of slack moorings (with catenary mooring lines) than for tightly moored ﬂoaters (the case analysed here). If nonlinearities are to be taken into account, then a time-domain (rather a 1 z [m] z˙ [m/s] 0.5 0 −0.5 1 −1 0 0.5 10 20 30 40 50 60 t [s] 0 z [m] 1 −0.5 x [m] x˙ [m/s] 0.5 −1 −1 0 1 x [m] 0 −0.5 −1 0 10 20 30 40 50 60 t [s] Fig. 9. Results from the time-domain analysis with linear PTO. Buoy heave (z) and surge (x) oscillations (and respective speeds). Trajectory of buoy centre is also shown. Regular waves of Aw ¼1 m, T ¼ 10 s, for a ¼7.5 m, L¼60 m, K ¼ 1.8 Â 105 kg/s2, C ¼ 2.5 Â 105 kg/s, F ¼1 MN. 28 P.C. Vicente et al. / Ocean Engineering 59 (2013) 20–36 frequency-domain) analysis is to be employed. This approach was ﬁrst applied to ships in wavy seas (Cummins, 1962) and later extended to oscillating-body WECs (Jefferys, 1980). In our case, the dynamic equations are then Zt ðm þ A1xÞx€ ðtÞ þ KxðtÀtÞx_ ðt dt ¼ f dxÀfX :, À1 ð14Þ Zt ðmþ A1zÞz€ðtÞ þ rgSz tÞ þ KzðtÀtÞz_ðt dt ¼ f dzÀfZ : À1 ð15Þ Here, AN j(j ¼x,z) are the limiting values of the added masses Aj(o)for o ¼N. For a hemispherical ﬂoater, it is ANz ¼ m/2 and ANx ¼0.2732 m, where m¼2pa3r/3 (see Hulme, 1982).The diffrac- tion forces fdx and fdz for regular and irregular waves are deﬁned as in Section 3 fx and fz, are, respectively, the horizontal and vertical components of the mooring force. The convolution integrals in Eqs. (14) and (15) represent the memory effect in the radiation forces. Their kernels can be written as K j ðtÞ ¼ 2Z 1 p0 BjðoÞcos ot do ðj ¼ x,zÞ: ð16Þ They decay rapidly and may be neglected after a few tens of seconds, which means that the inﬁnite interval of integration in Eqs. (14) and (15) may be replaced by a ﬁnite one in the numerical calculations (a 20 s interval was adopted since it has been seen, for the buoy used in the simulations, that after this time period the kernels are very close or equal to zero, and therefore may be neglected). The integral-differential Eqs. (14) and (15) were numerically integrated from initial values of x, z, x_ and z_, equal to zero, with an integration time step of 0.02 s. The accelerations in the two modes of motion in Eqs. (14) and (15) at the current time step are estimated using a Runge–Kutta 2nd order approximation method and, from these, the velocities and displacements are calculated accordingly. 1 ∆L [m] ∆˙ L [m/s ] 0.5 0 −0.5 −1 0 1 10 20 30 40 50 60 t [s] 0.5 α [°] 0 −0.5 −1 0 10 20 30 40 50 60 t [s] Fig. 10. Results from the time-domain analysis with linear PTO. Time series of the cable length variation and cable length variation speed and cable vertical angle. Regular waves as in Fig. 9. φ [MN] Z 0.02 1.3 1.2 0.01 φ [MN] X 1.1 1 0 0.9 0.8 −0.01 0.7 −0.02 −1 −0.5 0 0.5 1 −1 −0.5 0 0.5 1 z [m] x [m] Fig. 11. Results from the time-domain analysis with linear PTO. Vertical and horizontal cable force components versus the heave and surge displacements. Regular waves as in Fig. 9. P.C. Vicente et al. / Ocean Engineering 59 (2013) 20–36 29 4.1. Linear PTO As stated before, in the simpliﬁed linear model of the PTO we assume that f¼KDLþC dDL/dt since the spring and damper are mounted in parallel. The vertical pulling force is then fz ¼ ðF þ f Þcosa and the horizontal restoring force provided by the cable is fx ¼ ðF þf Þsina. Numerical results (for the same sea state and properties of the device as earlier: a¼ 7.5 m, L¼ 60 m, K¼ 1.8 Â 105 kg/s2, C¼ 2.5 Â 105 kg/s, F¼1 MN) are plotted for regular waves with Aw¼1 m, T¼10 s in Figs. 9–13. The oscillations presented in the ﬁgures are after an initial time period after the beginning of the numerical simulation, long enough to allow the initial transient response to die out and (if applicable) the periodic oscillation to be established. φ [MN] 1.3 1.2 1.1 1 0.9 0.8 0.7 0 80 10 20 30 40 50 60 t [s] 60 P [kW] 40 20 0 0 10 20 30 40 50 60 t [s] Fig. 12. Results from the time-domain analysis with linear PTO. Time series of the cable force and absorbed power. Regular waves as in Fig. 9. 1 z [m] ∆L [m] φ∗Z 0.5 0 0.5 1 0 1 0.5 10 20 30 40 50 60 t [s] x [m] α[ ] φ∗X 0 0.5 1 0 10 20 30 40 50 60 t [s] Fig. 13. Results from the time-domain analysis with linear PTO. Representative comparison between the motion, forces and cable length variation and vertical angle oscillations. Regular waves as in Fig. 9. 30 P.C. Vicente et al. / Ocean Engineering 59 (2013) 20–36 Fig. 14. Results from the time-domain analysis with linear PTO. Comparison between horizontal motion and mooring force and the vertical angle. On top: H¼ 50 m, below H¼ 70 m. Regular waves as in Fig. 9. fr [Hz] fr [Hz] 1 0.022 x [m] x˙ [m/s] 0.5 0.02 0 0.5 1 0 100 200 300 400 500 t [s] Fig. 15. Results from the time-domain analysis with linear PTO. Buoy surge (x) free-oscillations (no incident waves) with initial conditions z ¼ z_ ¼ x_ ¼ 0 and x¼ 1 m. Buoy and moorings as in Fig. 9. 0.018 0.016 0.014 0.7 0.8 0.9 1 1.1 1.2 1.3 F [MN] x 106 Fig. 17. Results from the time-domain analysis with linear PTO. Frequency of the free oscillations, for different values of F. Buoy and mooring parameters as in Fig. 9, except where otherwise stated. 0.035 0.03 0.025 0.02 0.015 0.01 5 6 7 8 9 10 a [m] Fig. 16. Results from the time-domain analysis with linear PTO. Frequency of the free oscillations, for different values of a. Buoy and mooring parameters as in Fig. 9, except where otherwise stated. Fig. 9 shows time series for the heave z(t) and surge x(t) oscillations and Fig. 10 shows the cable length variation and cable length variation speed as well as the cable vertical angle. It can be seen that for regular waves of 1 m amplitude the variations are of the same order. The surge oscillations are higher than those predicted by the frequency domain method. Fig. 11 shows, respectively the vertical and horizontal cable force components versus the heave and surge displacements and Fig. 12 shows the oscillations of the force applied by the mooring cable and the oscillations of the absorbed power. Finally in Fig. 13 it can be seen that the heave oscillation is in a way correspondent to the cable length variation and the surge oscillation is related to the cable vertical angle. The vertical and horizontal cable force (in dimensionless form divided by the pre-tension force) is also represented. In Fig. 14 similar results for the horizontal component can be seen but for H¼50 m and H¼70 m, which show that the agreement, between surge amplitude and cable vertical angle, P.C. Vicente et al. / Ocean Engineering 59 (2013) 20–36 31 in terms of amplitude (not phase), depends mainly of the water depth. The next Fig. 15–17 concern the free oscillations (in the absence of incident waves) that occur when the buoy, displaced from its static position is released, in order to analyse the system natural oscillation frequency. In Fig. 15 the time series for x show (in this case a slow) exponentially decaying damped oscillation when the buoy is horizontally displaced 1 m. A FFT analysis reveals a low frequency peak at around 0.0185 Hz. In Figs. 16 and 17 it can be seen the inﬂuence of 25% variation of the parameters a and Fin the natural oscillation frequency. It can be seen that the natural frequency decreases with increasing buoy radius a, and with decreasing initial pre-tension F. 4 z [m] z˙ [m/s] 2 0 2 4 4 2 0 20 40 60 80 100 120 t [s] 0 z [m] 4 2 x [m] x˙ [m/s] 2 4 42024 x [m] 0 2 4 0 20 40 60 80 100 120 t [s] Fig. 18. Results from the time-domain analysis with linear PTO. Buoy heave (z) and surge (x) oscillations (and respective speeds). Trajectory of buoy centre is also shown. Irregular waves of HS ¼2 m, Te¼ 10 s, with a¼ 7.5 m, L¼ 60 m, K ¼1.8 Â 105 kg/s2, C ¼2.5 Â 105 kg/s, F ¼1 MN. 2 1.5 φ [MN] 1 0.5 0 0 800 20 40 60 80 100 120 t [s] 600 P [kW] 400 200 0 0 20 40 60 80 100 120 t [s] Fig. 19. Results from the time-domain analysis with linear PTO. Time series of the cable force and absorbed power. Irregular waves as in Fig. 17. 32 P.C. Vicente et al. / Ocean Engineering 59 (2013) 20–36 The parameters C and K as well as the initial displacement x(t¼0)were seen to be unimportant for the natural oscillation frequency. This seems to be mainly in accordance with what could be expected (Radhakrishnan et al., 2007): an increase in radius results in a mass or inertia increase and an increase in the pre-tension produces an increase in the elasticity response of the system. Similar numerical results as the ones presented for the regular waves are plotted in Figs. 18–20, for irregular waves of Hs¼4 m, Te¼ 10 s, and again for a¼ 7.5 m, L¼ 60 m, K¼ 1.8 Â 105 kg/s2, F¼1 MN. Fig. 18 shows time series for irregular waves of the heave z(t) and surge x(t) amplitudes and Fig. 19 shows the oscillations of the force applied by the mooring cable and the oscillations of the 3 z [m] ∆L [m] φ∗Z 2 1 0 1 2 3 0 20 40 60 80 100 120 t [s] 2 x [m] α[ ] φ∗X 1 0 1 2 3 4 0 20 40 60 80 100 120 t [s] Fig. 20. Results from the time-domain analysis with linear PTO. Representative comparison between the motion, forces and cable length variation and vertical angle oscillations. Irregular waves as in Fig. 17. 1 z [m] z˙ [m/s ] 0.5 0 0.5 1 0 1.5 1 20 40 60 80 100 120 t [s] x [m] x˙ [m/s ] 0.5 0 0.5 1 1.5 0 20 40 60 80 100 120 t [s] Fig. 21. Results from the time-domain analysis with hydraulic PTO. Buoy heave (z) and surge (x) oscillations (and respective speeds). Regular waves of Aw¼ 1 m , T¼10 s, for a ¼7.5 m, L¼60 m, K ¼ 1.8 Â 105 kg/s2, F ¼1 MN. P.C. Vicente et al. / Ocean Engineering 59 (2013) 20–36 33 absorbed power. Fig. 20 shows again that the heave oscillation is in a way correspondent to the cable length variation and the surge oscillation is related to the cable vertical angle. 4.2. Hydraulic PTO As stated before, for the hydraulic PTO we consider that the wave energy converter resorts to a hydraulic mechanism that uses the oscillating motions of the ﬂoating buoy to drive a piston inside a cylinder placed on the bottom. In parallel with this piston there is linear spring of stiffness K whose objective is to apply a restoring force to the ﬂoater, proportional to the cable length variation, in order to restore it to the original equilibrium position. From the moment on that the time-varying cable velocity inverts its direction (or is zero), the piston (and therefore also the cable length) become stationary, unless or until, the total hydrodynamic 1 ∆L [m] 0.5 ∆˙L [m/s ] 0 0.5 1 0 10 20 30 40 50 60 1 F [MN] M 0.5 F [MN] P 0 0.5 PTO ON OFF 1 0 1.5 1 0.5 0 0.5 0 10 20 30 40 50 60 10 20 30 40 50 60 t [s] Fig. 22. Results from the time-domain analysis with hydraulic PTO. Time series (on top) of the cable length variation and cable length variation speed, (in the middle) of the spring and piston force on the cable and (below) of the moments where the piston is in motion or stopped. Regular waves as in Fig. 20. φ [MN] 1.5 1 0.5 0 0.5 1 1.5 0 800 10 20 30 40 50 60 t [s] 600 P [kW] 400 200 0 0 10 20 30 40 50 60 t [s] Fig. 23. Results from the time-domain analysis with hydraulic PTO. Time series of the cable total force and of the power absorbed by the piston. Regular waves as in Fig. 20. 34 3 2 1 0 1 2 3 0 6 4 P.C. Vicente et al. / Ocean Engineering 59 (2013) 20–36 z [m ] z˙ [ m/s ] 20 40 60 80 100 120 t [s] x [m ] x˙ [ m/s ] 2 0 2 4 0 20 40 60 80 100 120 t [s] Fig. 24. Results from the time-domain analysis with hydraulic PTO. Buoy heave (z) and surge (x) oscillations (and respective speeds). Irregular waves of HS¼ 2 m , Te ¼10 s, with a¼ 7.5 m, L¼ 60 m, K ¼1.8 Â 105 kg/s2, F¼ 1 MN. force on the body, projected in the direction of the cable, over comes trhgeSzpðitsÞtÀonRÀt r1esKisztðitnÀgtÞfzo_ðrtcÞed.tTahnedtoftdaxlÀhRyÀtd1roKdxyðntÀamtÞix_cðtfÞodrtcesfoarrehef dazvÀe and surge, respectively. If the total hydrodynamic force overcomes the piston resisting force, then the forces applied on the body are both the hydrodynamic forces and the mooring forces which result from the action of the piston and the spring. When the cable length varies, the piston exerts a force F (related to the piston section and ﬂuid pressure assumed constant), whose direction is contrary to that of the velocity DL_ of the cable length variation FP ¼ Àsign DL_ F. The spring exerts a force proportional to the cable length variation FM¼ ÀK DL. In this case, the governing equations are (14) and (15) and fx ¼ signðxÞ ðFP þFMÞ sin a and fz ¼ ðFP þ FMÞcos a. If it does not overcome the piston force, then the resulting hydrodynamic force in the direction of the cable is nulliﬁed by the piston inertia and the only resulting force on the body is the projection of the hydrodynamic force on the perpendicular direction of the cable. Numerical results are plotted for regular waves with Aw¼ 1 m, T¼10s in Figs. 21–26, for a¼7.5 m, L¼60 m, K¼1.8 Â 105 kg/s2, F¼1 MN. Fig. 21 shows time series for the heave z(t) and surge x(t) oscillations and respective speeds. It can be seen that the heave position is maintained for a short period while the piston is kept ﬁxed and change of motion direction of the oscillation occurs, and that the surge oscillation presents the superposition of a lower frequency oscillation with the wave frequency oscillation. Fig. 22 shows (on top) the cable length variation and cable length variation speed, (on the middle) the piston and spring force and (below) the time when the piston is moving (stated as ‘1’) or not (stated as ‘0’). We can again see the cable maintaining its length and the piston stopping, when change of motion direction occurs. Fig. 23 shows (on top) the oscillations of the total force applied by the mooring cable, where the contributions of both the spring and the piston can be seen and (below) the oscillations of the absorbed power, where it can be seen the moments when the piston is stopped and the power is zero. Similar numerical results as the ones presented for the regular waves are plotted in Figs. 24–26, for irregular waves of Hs¼4 m, Te¼ 10 s, and again for a¼ 7.5 m, L¼ 60 m, K¼ 1.8 Â 105 kg/s2, F¼1 MN. Fig. 24 shows time series for the heave z(t) and surge x(t) oscillations and respective speeds where it can also be seen that, in some moments, the heave position is maintained for a short period while the piston is kept ﬁxed when change of motion direction occurs. It can also be seen that the surge oscillations are higher in amplitude than the ones of heave. Fig. 25 shows (on top) the cable length variation and cable length variation speed, (below) the piston and spring force and (below) the moments when the piston is moving or not. Finally, Fig. 26 shows (on top) the oscillations of the total force applied by the mooring cable and (below) the oscillations of the absorbed power. 5. Conclusions A theoretical analysis of the wave-induced heave and surge oscillations of a tightly moored wave energy converter was presented. The work focuses on the nonlinear dynamics introduced by the mooring system of these wave energy converters and on the analysis of the inﬂuence of the mooring parameters upon the motion P.C. Vicente et al. / Ocean Engineering 59 (2013) 20–36 35 4 ∆L[m] 2 ∆˙L[m /s ] 0 2 4 0 1 0.5 0 0.5 1 0 20 40 60 80 100 120 t [s] F [MN] M F [MN] P 20 40 60 80 100 120 t [s] PTO ON OFF 1.5 1 0.5 0 0.5 0 20 40 60 80 100 120 t [s] Fig. 25. Results from the time-domain analysis with hydraulic PTO. Time series (on top) of the cable length variation and cable length variation speed, (in the middle) of the spring and piston force on the cable and (below) of the moments where the piston is in motion or stopped. Irregular waves as in Fig. 23. 1.5 1 0.5 0 0.5 1 1.5 0 20 40 60 80 100 120 t [s] 1500 1000 500 0 0 20 40 60 80 100 120 t [s] Fig. 26. Results from the time-domain analysis with hydraulic PTO. Time series of the cable total force and of the power absorbed by the piston. Irregular waves as in Fig. 23. 36 P.C. Vicente et al. / Ocean Engineering 59 (2013) 20–36 behaviour of the system. To this end, a proper time-domain model is used (and compared) with a simpliﬁed frequency-domain one. Numerical results were obtained for a hemispherical buoy whose PTO consists of either a linear spring and a linear damper or a hydraulic piston and a spring. In the simpliﬁed linear model of the PTO we assume that the cable is tensioned by a linear spring of stiffness K and transmits the (linear) damper force. For the hydraulic PTO, we consider that the wave energy converter resorts to a hydraulic mechanism that uses the oscillating motions of the ﬂoating buoy to drive a piston inside a cylinder placed on the bottom, which contains a ﬂuid at high pressure (here we assume a constant pressure and therefore constant force). First, a frequency-domain analysis was applied to a linearized representation of the horizontal and vertical mooring forces. The results showed that the amplitudes of the heave and surge oscillations depend very differently on the wave frequency. The inﬂuence of the PTO damping coefﬁcient and the mooring parameters were also analyzed and seen to have a limited inﬂuence on the system performance. Then a time-domain analysis was applied to investigate the nonlinear effects of the mooring forces in waves of moderate amplitude, for the two proposed PTO models. For the linear PTO model it was seen that for regular waves of 1 m amplitude the variations are of the same order. The surge oscillations were slightly higher than those predicted by the frequency domain method, although power and mooring tension are similar. It was also seen that the heave oscillations were in a way correspondent to the cable length variation and the surge oscillations were related to the cable vertical angle, but for this latter case, the accordance in terms of amplitude (not phase) was seen to depend mainly of the water depth. Free oscillations (in the absence of incident waves) simulations, which occur when the buoy displaced from its static position is released, were also performed, in order to analyse the system natural oscillation frequency. From the comparison of the results between the frequency domain and the time domain (for a linear PTO), it seems that there are mostly some differences in terms of surge motion amplitude. For the hydraulic PTO, it was seen that both for the regular and irregular waves, the heave position is maintained for a short period while the piston is kept ﬁxed at its extreme position when change of motion direction occurs. The nonlinearities were found to affect more the horizontal oscillations (rather than the vertical ones), where a second low order frequency can be seen, which makes the surge motion amplitude to be higher than the ones predicted by the previous models. 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