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Mooring system influence on the efficiency of wave energy converters
International Journal of Marine Energy 3–4 (2013) 65–81
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International Journal of Marine Energy journal homepage: www.elsevier.com/locate/ijome
Mooring system influence on the efficiency of wave energy converters Frederico Cerveira a, Nuno Fonseca a,⇑, Ricardo Pascoal b a Centre for Marine Technology and Engineering (CENTEC), University of Lisbon, Instituto Superior Técnico, Av. Rovisco Pais, 1049-001 Lisbon, Portugal b Institute of Electronics and Telematics Engineering of Aveiro (IEETA), University of Aveiro, 3810-193 Aveiro, Portugal
a r t i c l e
i n f o
Keywords: Floating wave energy converter Catenary mooring system Identification method for mooring system dynamics Mooring system effects on absorbed power Annual absorbed wave energy
a b s t r a c t The paper presents an analysis of the mooring system effects on the dynamics of an arbitrary floating wave energy converter (WEC) and on the efficiency of the device. The mooring system dynamics are calculated by a finite differences method, which accounts for the lines’ physical characteristics such as mass and stiffness, as well as for the hydrodynamic inertial and damping characteristics. A method is proposed and applied to consider the inertial, damping and stiffness effects of the mooring system on the WEC’s linear dynamics. An identification method is used to determine the linear mooring system coefficients from pre-calculated nonlinear time domain forces of the mooring system on the floater (the importance of nonlinear effects is assessed as well). Finally the mooring system dynamics is represented by additional inertial, damping and stiffness matrices which are added to the ones of the WEC. The influence of the mooring system on the WEC dynamics and efficiency is assessed in terms of wave induced motions and absorbed power. Results are presented for the transfer functions, statistics in selected stationary sea states and the expected annual absorbed energy. A wave scatter diagram representative of the Portuguese Pilot Zone is used for the annual predictions. Ó 2013 Elsevier Ltd. All rights reserved.
Introduction Mooring design is a critical part of a floating wave energy converter project (WEC). The devices are generally thought to be employed in areas of demanding environmental loads due to waves, current ⇑ Corresponding author. E-mail address: [email protected] (N. Fonseca). 2214-1669/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijome.2013.11.006
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and wind. These survivability issues are addressed in existing offshore standards, such as the DNV-OSE301 [1] and numerical methods can be applied to estimate extreme loads on WECs [2]. However, the mooring arrangement can have a direct influence on the WEC operation and on the economic viability of the project. Fitzgerald et al. [3] present an analysis of different mooring configurations, in terms of the impact on the device motion, array efficiency, weight and cost. Fonseca et al. [4] present a design methodology for WEC moorings, based on a real case study for the Portuguese coast. The mooring system may have a particular effect on the power capture of devices which depend on the motion of the moored body. The energy dissipation associated with catenary mooring line damping was studied by Johanning et al. [5] through a series of experimental tests. It is concluded that mooring line damping increases in a non-linear manner as the line becomes increasingly taut. The energy dissipation is considered to arise primarily from the drag force, which is proportional to the velocity squared of the motion perpendicular to the line axis. When the line becomes nearly taut a small motion at the top-end, originates large movements and velocities around the midpoint and a corresponding large drag induced damping. On the other hand, when the mooring line is quite slack the same topend motion, will result in small perpendicular movement and velocity around the line midpoint and a corresponding small drag induced damping. These experiments also evaluated the effect of the top-end motion frequency on the mooring damping. The results show a proportional increase in energy dissipation with the frequency, which becomes more relevant as the pretension increases. A frequency domain methodology is presented by Fitzgerald et al. [6] to evaluate the mooring effect on WEC power extraction. The mooring non-linear characteristics are estimated numerically for different conditions with a time domain software and then approximated to the linear order. The approximation is considered to be valid for motion amplitudes small relative to the water depth and load induced amplitudes small relative to the equilibrium pretension. The procedure is applied to a generic device unmoored and moored with different configurations. The results indicate that moorings can have important effects, particularly on heave motion, usually detrimental to the energy production. The present work investigates the mooring system influence on the power production in realistic wave conditions. Use is made of accurate climate data, for the Portuguese west coast, to define the energetic sea states and estimate annual production. Particular attention is given to the mooring dynamics, identified by a non-linear code described in Pascoal et al. [7] and applied to the mooring system oscillating with a pre-defined irregular forced motion. The oscillations are imposed around a horizontal mean displaced position due to steady wave drift forces. A representative sea state is selected for the analysis. During power production conditions the floater motions are of relatively small amplitude, therefore it is possible to approximate the non-linear mooring characteristics to the linear order. The related mass, damping and stiffness coefficients can then be added to the floater linear equations of motion and then solved in the frequency domain. The power production of an arbitrary device is estimated in three different conditions: unmoored, and moored with slack and moderately slack catenaries. The mooring effect on the device dynamics and efficiency under operational conditions is concluded from the results.
Theory Responses and absorbed power in harmonic wave The harmonic wave induced hydrodynamic forces and motions are represented on a Cartesian coordinate system with origin at the calm waterline. The x-axis points to the direction of wave propagation, the z-axis is vertical, positive upwards and passes through the floater center of gravity and the y-axis is perpendicular to the former according to the right hand rule. All degrees of freedom, fj, j = 1, . . ., 6, are sequentially numbered according to standard convention. The floater hydrodynamic coefficients and wave exciting forces in incident harmonic waves are calculated by a standard 3D linear radiation-diffraction flat panel method, which has been applied in the form of the commercial WAMIT package. The method assumes potential flow, which satisfies the Laplace equation in the fluid domain. A linear boundary value problem is formulated for the wave body
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interactions in incident harmonic waves. Green’s theorem is used to derive integral equations with unknown velocity potential on the mean wetted body surface. The body boundary is discretized into a set of panels with constant potential on each panel, which results in a set of linear simultaneous equations in the unknown potentials. Lee and Newman [8], give details of the formulation. Once the velocity potentials are determined, the hydrodynamic pressures can be calculated, as well as the related wave exciting forces and radiation forces. Equating the external forces (hydrodynamic and mooring forces) to the inertial forces gives the equations of motion. In the present work we consider an axisymmetric floater and mooring system (about the vertical axis), therefore the only modes of motion are surge, heave and pitch. The equations of motion are: 6 n o X f pto m x2 M fkj þ M m þ C fkj þ C m fj ¼ F Ek ; kj þ Akj þ ix Bkj þ Bkj þ Bkj kj
k; j ¼ 1; 3; 5
ð1Þ
j¼1
where x represents the angular frequency and fj the displacement for mode j. M fkj , Bfkj and C fkj represent the floater hydrodynamic added masses, damping coefficients and hydrostatic restoring coefficients. m m Mm kj , Bkj and C kj are the mass, damping and restoring coefficients of the mooring system and they pto represent the linear coupling effects of the mooring system on the floater dynamics. Bpto 22 and B22 represent the power take off (PTO) linear damping coefficients. The solution for the motions in the system of Eq. (1) may be represented in the form of a transfer function from wave to motion and comprising harmonic motion amplitudes and phase angles as function of the wave frequency. The present work deals with a point absorber, meaning the floater dimensions are much smaller than the incident wavelength. It is an idealized device and the wave energy is extracted by two PTO systems fixed to the body and to a hypothetical fixed reference frame. The wave energy is extracted from the surge and heave modes of motion and it is assumed that the PTOs can be represented by linear dampers. Fig. 1 illustrates the idealized concept. The PTO forces are: pto _ F pto 2 ¼ B22 n2 ðtÞ pto F ¼ Bpto n_ 3 ðtÞ 3
ð2Þ
33
Fig. 1. Idealized concept of the wave energy converter.
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and the extracted wave power is: pto _ _ PðtÞ ¼ F pto 2 n2 ðtÞ þ F 3 n3 ðtÞ 2 2 pto _ _ ¼ Bpto 22 ½n2 ðtÞ þ B33 ½n3 ðtÞ
ð3Þ
In harmonic incident waves Eq. (3) is equivalent to:
a 2 PðtÞ ¼ Bpto þ 22 xn2 cosðxt /2 Þ a 2 pto B33 xn3 cosðxt /3 Þ
ð4Þ
where fa2 ; fa3 ; /2 ; /3 are the surge and heave harmonic amplitudes and phase angles, each dot over the symbol stands for a derivative with respect to time, and x is the wave frequency. Time integration over one wave cycle, divided by the wave period, results on the mean power absorbed in harmonic waves: pto
a 2
pto
a 2
B ðxn2 Þ B ðxn3 Þ Phar ðxÞ ¼ 22 þ 33 2 2
ð5Þ
Responses and absorbed power in irregular waves Real sea states are well described by the superposition of an infinite number of regular waves, which together define its spectrum, Sw(x). In the present work, the deep water JONSWAP spectral shape is used to describe the sea states. Long crested irregular waves are considered. The floater motion response spectrum depends on the wave spectrum and the motion transfer function amplitude meaning motion amplitude divided by the wave amplitude, faj =fa :
SR ðxÞ ¼ Sw ðxÞ½faj ðxÞ=fa
2
ð6Þ
while the time-average power in irregular waves can be calculated by the following formula:
Pirr ¼ 2
Z
1
Sw ðxÞ½P har ðxÞ=fa dx
ð7Þ
0
Mooring system dynamics The main interest of the moored equipment under study is to convert wave energy into electric energy and as such it is supposed to move in an optimal manner in the wave frequency range, which means close to resonance conditions. One must take into account that cable dynamics are influenced by both the low frequency motions and the wave frequency motions, they are in fact coupled. Wave frequency motions have dynamic influence on the slowly varying response, because for instance the wave frequency motion increases damping forces which make it more difficult to slowly move the cable. On the other hand, there is a parametric influence of the slowly varying motion into the wave frequency motion because the slowly varying shape of the catenary influences the achieved damping and added mass in the wave frequency range. This has been discussed for instance by Triantafyllou [9] and Webster [10]. In this work, cable dynamics have been modeled numerically by using a central finite-difference scheme in arc length and regressive Euler in time for the velocities, as described in Pascoal et al. [7], and, for sake of completeness, is briefly described herein. The algorithm involves solving the equilibrium of the cable as represented in Fig. 2, which results in the governing Eqs. (8–11). The set of equations must be supplemented with a material constitutive relation and initial and boundary conditions.
m
@~ v @~S ~ ¼ þ f ð1 þ cÞ @s @t
ð8Þ
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69
Fig. 2. Equilibrium of an infinitesimal cable segment [7].
~ @M þ ^t ~ Sð1 þ cÞ ¼ 0 @s
ð9Þ
@~ v @ ^ ¼ ðtð1 þ cÞÞ @s @t
ð10Þ
R @~ d~ R ¼ dt @t
ð11Þ
~ v¼
The dynamic simulation starts from a relaxation solution, where the initial conditions become zero velocities and the geometric configuration is determined for the mean drift displacement corresponding to the slowly varying drift loads. The external loading along the arc length is of the form: 2~ ~ f ð1 þ cÞ ¼ ma @@t2R ð1 þ cÞ pffiffiffiffiffiffiffiffiffiffiffiffi V rn k~ V rn D 1 þ cC Dn 12 qf k~ pffiffiffiffiffiffiffiffiffiffiffiffi f sb þ ~ f cd V rt k~ V rt pD 1 þ cC Dt þ mw~ 12 qf k~ g þ~
ð12Þ
where ma is the sectional added mass, ~ R is the Lagrangian coordinate along the centreline, c is V rn is the fluid velocity normal engineering strain, qf the specific mass of the medium (1025 kg m3), ~ to the local cable (chain) centreline, CDn is the normal drag coefficient, CDt is the tangential drag coefficient, mw is the equivalent sectional mass in water accounting for hydrostatic effects,~ g is the f c d concentrated loading and ~ f sb the interaction with the bottom. acceleration of gravity, ~ Interaction with the seabed is complex to model and its effects greatly depend on the soil type and the cable profile. As proposed by Webster [10], a distributed elastic support is used. In this case the thickness of the elastic support was assumed constant (0.5 m) and critical damping, mainly meant to stabilize the numerical solution, acts only as the cable penetrates into the undisturbed floor level. This mismodeling becomes more important as cables become slack and overall damping will tend to be greater than reported herein for that situation. The discrete form of the equations is solved in a non-iterative time marching scheme. This noniterative property is achieved by using regressive Euler to eliminate the non-linearity on the next time step due to damping and the geometric average of the position vector to eliminate the nonlinearity due to the strain equation. This approximation results in the algorithm schematically represented in Fig. 3. The ‘‘Updated Kinematics’’ block corresponds to using LU matrix factorization solving for velocities and positions.
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Fig. 3. Cable dynamics simulation loop [7].
Case study Wave climatology Figueira da Foz wave climate scatter diagram is used in the present investigation to assess the wave power absorbed in realistic conditions, as well as the annual absorbed energy. The choice of this site is of particular interest because of its proximity to the 320 km2 Portuguese pilot zone, located at São Pedro de Moel (West coast of Portugal). Fig. 4 presents the scatter diagram, namely the annual joint probability of occurrence of Hs Tz for (Hs is the significant wave height and Tz is the average wave period between zero upcrossings). The scatter diagram is the result of the statistical analysis of wave records obtained from a wave rider station located 20 km from the shoreline (40°110 0800 N and 9°080 4400 W) at 92 m water depth (Portuguese Hydrographic Institute). The data acquisition was made for periods between 20 and 30 min every 3 h. The acquisition was continuous when the significant wave height was larger than 5 m. The scatter diagram results from the analysis of 11,193 valid records obtained between July 1990 and January 1996. The wave rider buoy was located at a distance of around 30 km North of the Portuguese pilot zone. Point absorber characteristics The floater geometry is a generic vertical cylinder with rounded bottom edge. An advantage of the cylindrical shape is that it can be easily reproduced numerically. Fig. 5 presents the profile of the Hs(m) 1.75 3.75 4.25 4.75 5.25 5.75 6.25 6.75 7.25 7.75 8.25 0.25 0.01 0.08 0.11 0.11 0.11 0.03 0.06 0.03 0.02 0.75 0.17 0.51 1.22 1.79 2.17 2.00 1.77 1.14 0.68 0.36 1.25 0.03 0.18 0.87 2.28 3.16 3.72 3.27 2.36 1.95 1.46 1.75 0.10 1.01 2.61 3.30 3.03 2.57 2.13 2.02 2.25 0.05 0.85 1.77 2.23 1.79 1.86 1.58 2.75 0.04 0.64 1.09 0.96 0.92 1.29 3.25 0.06 0.28 0.40 0.53 0.66 3.75 0.05 0.12 0.29 0.37 4.25 0.01 0.10 0.17 percentage 4.75 0.01 0.08 0.00-0.10 5.25 0.01 0.10-0.50 5.75 0.50-1.00 6.25 1.00-2.00 6.75 2.00-3.00 7.25 3.00-4.00 7.75 4.008.25
Tz(s) 8.75 9.25 9.75 10.25 10.75 11.25 11.75 12.25 12.75 13.25 13.75 14.50 16.25 0.28 1.22 1.44 1.31 1.03 0.65 0.51 0.26 0.13 0.03
0.15 0.92 1.15 1.10 1.00 0.66 0.48 0.42 0.23 0.04 0.03 0.01 0.01
0.08 0.72 1.03 0.88 0.82 0.76 0.47 0.34 0.26 0.08 0.05 0.04 0.01
0.01 0.41 0.67 0.69 0.72 0.65 0.57 0.36 0.18 0.13 0.05 0.06 0.01 0.01
0.13 0.36 0.47 0.54 0.58 0.37 0.21 0.21 0.13 0.08 0.03 0.01 0.02 0.01
0.01 0.06 0.19 0.39 0.41 0.41 0.32 0.18 0.15 0.10 0.11 0.04 0.01 0.04
0.07 0.10 0.19 0.22 0.36 0.15 0.15 0.20 0.07 0.06 0.03 0.02 0.01 0.02
0.01 0.03 0.06 0.20 0.17 0.18 0.15 0.11 0.12 0.08 0.05 0.03 0.01 0.04 0.02
Fig. 4. Scatter diagram for Figueira da Foz, West cost of Portugal.
0.02 0.05 0.09 0.17 0.08 0.11 0.06 0.06 0.06 0.08 0.03 0.03 0.01 0.01
0.01 0.03 0.06 0.05 0.06 0.04 0.04 0.05 0.03 0.03 0.03 0.01
0.01 0.01 0.01 0.01 0.04 0.01 0.02 0.02 0.03 0.03 0.03 0.03 0.03 0.04 0.03 0.01 0.01 0.02 0.02 0.03 0.02 0.06 0.01 0.01 0.01 0.01 0.01 0.01
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Fig. 5. Axisymmetric buoy main dimensions.
Table 1 Buoy main dimensions. Dimensions Cylinder diameter Draft Mass Vert. position of CoG Pitch radius of gyration
D (m) T (m) D (t) Zg (m) r (m)
15 4.5 786.5 0.7 4.5
Table 2 Mooring lines dimensions. Dimensions Length Stiffness Chain diameter Tangential drag coefficient Normal drag coefficient Mass per unit length (air) Top-end tension slack mooring Top-end tension moderately slack mooring
L (m) K (N/m/m) Dchain (mm) CDt () CDn () mchain (kg/m) Tslack (N) Tmoderately (N)
350 4.12E8 64 1.0E-2 1.2 90 1.15E5 2.22E5
axisymmetric body with the respective main dimensions given in the drawing and in Table 1. The body added mass and damping coefficients for the considered directions were evaluated by the WAMIT software and are given in Table 3 for the respective motion natural frequency. The floater hydrodynamic coefficients and wave exciting forces in incident harmonic waves are calculated by a standard 3D linear radiation-diffraction flat panel method, which has been applied in the form of the commercial WAMIT package. In the present case the floater is axisymmetric, therefore only one quadrant of the cylindrical floater has been represented by 270 low order panels. This means the body is represented by flat panels and the solution for the potentials are approximated by piecewise constant values on each panel (higher order panels could have been used as well, however, the
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Direction
Surge Heave – Surge Pitch – Surge Heave Surge – Heave Pitch – Heave Pitch Surge – Pitch Heave – Pitch
Buoy
PTO
Weakly slack mooring
Mass
Added mass
Damping
Damping
Mass
Damping
Stiffness
Mass
Moderately slack mooring Damping
Stiffness
7.86 105 [kg] 0 5.5 105 [kg.m] 7.86 105 [kg] 0 0 1.6 107 [kg.m2] 5.5 105 [kg.m] 0
3.38 105 [kg] – – 5.56 105 [kg] – – 2.47 106 [kg.m2] – –
5.20 104 [N.s/m] – – 2.27 105 N.s/m – – 6.62 105 [N.m.s] – –
9.00 105 [N.s/m] 0 0 2.00 106 [N.s/m] 0 0 0 0 0
9284 [kg] 2060 [kg] 40850 [kg.m] 33166 [kg] 2266 [kg] 9969 [kg.m] 1.90 105 [kg.m2] 42170 [kg.m] 9600 [kg.m]
7712 [N.s/m] 1480 [N.s/m] 33933 [N.s] 17954 [N.s/m] 1395 [N.s/m] 6138 [N.s] 130010 [N.m.s] 28883 [N.s] 5610 [N.s]
7755 [N/m] 664 [N/m] 34123 [N] 5940 [N/m] 747 [N/m] 3285 [N] 1.57 105 [N.m] 35850 [N] 5920 [N]
42111 [kg] 2538 [kg] 1.85 105 [kg.m] 53214 [kg] 2442 [kg] 10746 [kg.m] 8.72 105 [kg.m2] 1.94 105 [kg.m] 3240 [kg.m]
25684 [N.s/m] 5359 [N.s/m] 113010 [N.s] 35084 [N.s/m] 1559 [N.s/m] 6859 [N.s] 4.14 105 [N.s.m] 92173 [N.s] 20985 [N.s]
29311 [N/m] 639.3 [N/m] 1.29 105 [N] 11331 [N/m] 931.2 [N/m] 4097 [N] 6.64 105 [N.m] 1.49 105 [N] 5020 [N]
F. Cerveira et al. / International Journal of Marine Energy 3–4 (2013) 65–81
Table 3 Floater hydrodynamic added masses and damping coefficients, PTO damping coefficients and the identified linear dynamic coefficients of the mooring system.
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73
advantages in terms of computational efficiency are not relevant for the present body). The interior free surface was meshed as well to remove numerical problems associated with irregular frequencies. Hydrodynamic calculations were carried out for 70 m water depth. The numerical model results were compared with results from experimental tank testes in waves [11], and identified the need to add extra damping to the numerical model in the surge and pitch modes, since potential flow damping underestimates the real floater damping; these values are given in Table 3. Regarding the wave power absorption, the device has two liner PTOs, one operating in heave and the other in surge (see Section II. A). The respective damping coefficient magnitudes were optimized according to the site climatology for maximum annual production. The method consists on systematically varying the damping coefficient and calculating the annual absorbed energy. The PTO damping coefficient is selected as the one which absorbs the largest amount of energy. The damping coefficients are optimized separately for the surge and heave modes, which is valid because these modes are weakly coupled. A very weak coupling exists due to the mooring system. The PTO damping values in surge and heave are presented in Table 3. It is known that a heaving point absorber maximizes the wave energy absorption when the wave frequency is equal to the heave natural frequency and, in this case, the optimum PTO damping is equal to the sum of the other damping coefficients (hydrodynamic and mooring). It is possible to conclude that the heave damping coefficient of Table 3 is larger than the optimum for the natural frequency. This is because the PTO is optimized for the case study wave climatology, which is characterized by sea states with peak periods larger than the heave natural frequency. The average wave peak period for Figueira da Foz is around 9.6 s, while the floater heave natural frequency is 5.5 s. Mooring system design and specification Two mooring systems have been designed using a quasi-static method, with corresponding safety factors according to existing Classification Society rules. The mooring lines extreme loads were obtained from nonlinear time domain simulations for a design storm representative of the ocean area being considered. Finally the selected systems consist of catenary (64 mm chain) spread mooring configurations with three lines at 120° to each other. The incident wave direction is aligned with one mooring line, Fig. 6. The respective static geometries are illustrated in Fig. 7, where the slack mooring system has 2/3 of the lines length on the sea bottom, while the moderately slack system has 1/2 of the lines length lying on the bottom. Both mooring systems are made of chain lines with the same length and mechanical properties, presented in Table 2, but different horizontal distance from the anchor to the top-end in order to produce different catenary shapes and pretensions. The water depth is 70 m.
Fig. 6. Mooring geometry sketch from top view.
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Fig. 7. Slack and moderately slack mooring geometries for the equilibrium condition, side view.
Identifications of the mooring system dynamics Identification method In light of what has been stated in the previous sections, it is important, even during the preliminary design stage, to take the mooring system dynamics into account. However, using the full nonlinear mooring dynamics code is a highly prohibitive procedure because it is extremely time consuming (assuming it does not collapse due to numerical instabilities). One possible solution to this deadlock is to use procedures which allow for reduction in model complexity. In this work, the complexity of the mooring system model has been reduced by using a system identification procedure. This procedure is the generalized version of the one which has been described in Pascoal et al. [7], and includes all terms (even and odd) up to third order of the Taylor expansion of the forces at the mooring point. The need to include even terms stems from the fact that the present mooring system is not symmetric with respect to the y–z plane, as can be observed in see Fig. 6 (although, as referred in Section II.A, the mooring system is axisymmetric with respect to the z-axis). In order to successfully indentify the model, the special Schroeder phased signal is also used herein. The signal represents forced irregular motions imposed at the fairleads connecting the mooring lines to the floater. The mooring dynamics is calculated for the forced motion and the important result are the mooring system forces at the connecting point. The second part of the analysis consists on identifying the Taylor expansion coefficients of an approximated model for the mooring line forces. Care has been exercised to perform the Taylor expansion about the average slow drift position of a representative sea state (wave drift forces and motions for the present floater geometry have been analyzed in Pessoa et al. [11]). In this case the representative sea state has been chosen as the one with more annual energy (which is related to the average wave power and the probability of occurrence). Notice that the Schroeder phased signal is chosen to have approximately the same energy as the wave frequency motion and the slow drift motion is chosen to encompass that which will actually occur. By this we mean that, during identification, there is no intention to replicate the motions which actually occur but they must be highly representative. Producing data for the identification procedure is highly time consuming since it consists on using the numerical model for the mooring system dynamics, but, once it has been performed, the top end force is a simple, albeit nonlinear, function of kinematic parameters. The identification procedure presents a nonlinear model which is not applicable in a consistent manner for frequency domain analysis but is mandatory to prevent strong bias on the linear coefficients due to unmodeled higher order terms. The linear terms, possibly corrected by using statistical averaging of nonlinear terms, are then used for further frequency domain analysis. Approximate model The approximate model of the mooring arrangement used for first order frequency domain analysis is necessarily a linear one. Therefore, once the system identification procedure has been applied to successfully recover the nonlinear model, the model must be linearized according to some criteria. Inspection of the contributions from the identified components determines the need to linearize the contribution of cubic damping because it has a major contribution to the response, especially in the case of the moderately slack mooring. Since we are dealing with energy extraction and average
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values, it has been considered plausible to perform matching of the absorbed energy, i.e., to find a correction to the linear coefficient such that the absorbed energy remains the same as with cubic damping. This additive correction is of the form:
B ¼ Bcubic
X
X x_ 2
_4
x
N
!1 ð13Þ
N
found from the energy balance for the same simulation time series, with the N time instants used in the identification of the nonlinear model. Since the expected value of energy remains the same, this is a form of statistical linearization. Finally, the approximate linear model is:
o Xn m linear x2 Mm þ Bm þ Cm kj þ ix Bkj kj kj fj
Fm k ¼
ð14Þ
j¼1;3;5
and has frequency independent coefficients. The linear mass, damping and stiffness coefficients obtained for the slack and moderately slack mooring systems are given in Table 3 (right columns). For comparative purposes, Table 3 includes also the floater hydrodynamic coefficients in surge and heave corresponding to their natural frequencies (left columns), and the PTOs damping coefficients. It is clear that the mooring system effects on the floater dynamics are small: the inertial and damping mooring coefficients are up to around 7% and 5% of the floater wave frequency hydrodynamic coefficients. These contributions correspond to the moderately slack mooring system – the contributions are smaller for the slack mooring. Since the
80 60
Force [kN]
40 20 0 -20 -40 -60
Linear force Non-Linear Force
-80 400
420
440
460 time [s]
480
500
520
Fig. 8. Sample of the heave mooring forces at the fairleads due to forced heave motion for the moderately slack mooring system – comparison between nonlinear results and the linear model results.
Table 4 Standard deviation of the linear and non-linear mooring forces. Direction
Surge [kN] Heave [kN] Pitch [kN.m]
Slack
Moderately Slack
Non-linear Model
Linear Model
Non-linear Model
Linear Model
21.3 25.7 20.4
20.3 25.0 18.4
83.4 44.4 75.2
76.4 42.3 69.5
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F. Cerveira et al. / International Journal of Marine Energy 3–4 (2013) 65–81 0.7
Surge - Amotion/Awave [ ]
0.6
0.5
0.4
0.3
0.2
Buoy+PTO Buoy+Weakly Slack Mooring+PTO Buoy+Moderatly Slack Mooring+PTO
0.1
0
0
0.5
1
1.5
2
0
0.5
1
1.5
2
0
0.5
1
1.5
2
1
Heave - Amotion/Awave [ ]
0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
Pitch - Amotion/Awave [rad/m]
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
Frequency [rad/s] Fig. 9. Motion transfer function amplitude for surge, heave and pitch.
PTO damping coefficients are of the same order as the hydrodynamic floater coefficients, we conclude that the mooring damping coupling effects are even smaller than the number given above. It should be noted that wave frequency optimal damping coefficients are not those which are applicable to the low frequency dynamics. In the low frequency range, potential damping becomes very small, and, as discussed by Brown and Mavrakos [12], other effects dominate and mooring line damping is of primary importance in reducing peak excursions, providing up to 80% of the total damping.
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77
140
Surge - Preg/A2wave [kw/m 2]
120
100
80
60
40
Buoy+PTO Buoy+Weakly Slack Mooring+PTO Buoy+Moderatly Slack Mooring+PTO
20
0
0
0.5
1
1.5
2
180
Heave - Preg/A2wave [kw/m 2]
160 140 120 100 80 60 40 20 0
0
0.5
1
1.5
2
Frequency [rad/s] Fig. 10. Time averaged wave power absorbed by the surge and heave modes of motion (power normalized by the wave amplitude squared.
Johanning et al. [5] verified experimentally that the mooring line damping increases with the pretension, which is in agreement with the results obtained for the considered mooring systems. It is of interest to assess how well the identification method and the related linear model are able to represent the nonlinear mooring system dynamic forces acting on the floater. Fig. 8 presents a sample from the moderately slack mooring system heave force time history associated to forced vertical irregular motion of the fairleads. The black line represents the nonlinear force calculated by the time domain mooring system dynamic model (see Section II. C), while the green line represents the force calculated by the linear mooring system model (Eq. (14)). It is clear that the linear approximation closely follows the nonlinear force time history. Similar results are obtained for forces in all directions, due to both surge and heave forced motions. Notice that the asymmetry of the force found between low and high loads is lost in the linear model. The standard deviation of the force time history can also be used to compare the linear and nonlinear mooring system models. Table 4 presents the standard deviations for the two mooring systems, calculated from the linear and nonlinear force time histories, for forced irregular motions of the fairleads in the surge, heave and pitch directions. One observes that the linear model always underestimates the standard deviations, although by a small margin. The difference is larger for the moderately slack mooring system, since nonlinear effects on the mooring lines increase in this case. From the previous analysis, it is possible to conclude that the identification method and related linear model for the mooring forces at the fairleads is accurate and can be used for a qualitative assessment of the mooring system effects on the WEC dynamics.
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F. Cerveira et al. / International Journal of Marine Energy 3–4 (2013) 65–81 0.7 Buoy+PTO Buoy+Weakly Slack Mooring+PTO Buoy+Moderatly Slack Mooring+PTO
Surge - Amotion/Awave [ ]
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.5
0
0.5
1
1.5
2
1
1.5
2
1.4
Heave - Amotion/Awave [ ]
1.2
1
0.8
0.6
0.4
0.2
0
Frequency [rad/s] Fig. 11. Response spectra for an annual high energetic sea state (Hs = 2.75 m and Tp = 10.6 s).
Analysis of the mooring system effects Results in regular waves According to the previous specifications, three systems are studied (PTO operates in surge and heave motion modes): Buoy and PTO (no mooring). Buoy, slack mooring and PTO. Buoy, moderately slack mooring and PTO. The equations of motion are solved in the frequency domain. Fig. 9 presents the motion transfer function amplitudes for surge, heave, and pitch. The motion amplitudes are normalized by the wave amplitude and presented as function of the wave frequency. The results show that the mooring systems do not affect significantly the floater motions. Only the pitch motion resonance amplification is clearly reduced when mooring system effects are included. This is because pitch motion is lightly damped (small hydrodynamic damping and no PTO for this mode), therefore the mooring system damping contribution is relevant. Surge and heave are almost unaffected by the mooring system. This was expected since the identified mooring system coefficients are small compared to the floater and PTOs coefficients (see Table 3).
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The time-averaged power absorption in regular waves for the surge and heave directions is presented in Fig. 10. The results are normalized by the wave amplitude squared. In the surge direction, the slack mooring has no significant influence, while the moderately slack mooring decreases slightly the power absorption for frequencies above 1 rad/s. The power interference due to the surge couplings with pitch at natural frequencies is reduced by the mooring systems. The power absorption in the heave direction is also not significantly affected by both mooring system, although the moderately slack system reduces slightly the absorbed power for frequencies above around 0.5 rad/s. Results in irregular waves This Section presents the analysis of mooring system effects on the floater motion responses in irregular waves, as well as on the absorbed power. It ends with the estimation of wave energy absorbed during one year. Long crested stationary sea states are considered, represented by JONSWAP spectra with a peakedness parameter of 3.3. Combining the average wave power in irregular waves, which can be calculated for all scatter diagram classes, with their annual probability of occurrence, it is possible to determine the scatter diagram class with more energy during the year. The corresponding significant wave height and peak period are: Hs = 2.75 m and Tp = 10.6 s. The floater motion response spectrum was calculated for this sea state and the result is presented in Fig. 11. One observes no influence from the mooring system on the body motion apart from the small surge–pitch coupling effect around the pitch natural frequency.
14 Buoy+PTO Buoy+Weakly Slack Mooring+PTO Buoy+Moderatly Slack Mooring+PTO
Surge - Pir/ Hs2 [kw/m 2]
12
10
8
6
4
2
0
0
5
10
15
20
10
15
20
20
Heave - Pir/ Hs2 [kw/m 2]
18 16 14 12 10 8 6 4 2 0
0
5
Tz [s] Fig. 12. Time-averaged power absorption per significant height squared.
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Table 5 Annual wave energy captured according to the mooring configuration. Direction
Unmoored
Slack
Moderately slack
Surge [MW.h] Heave [MW.h] Total [MW.h]
300.9 857.1 1158.0
300.9 852.1 1153.0
302.4 844.4 1146.8
Fig. 12 presents the time-averaged power absorbed in irregular waves, normalized by the significant wave height squared, as function of the zero up-crossing wave period (Tz). The graphs show, again, no significant mooring system influence. In the surge direction the mooring effect can only be identified at the higher production periods, corresponding to a power decrease of about 0.5% with the slack and 1% with the moderately slack mooring. A slightly larger effect is found on the heave direction where the moderately slack mooring decreases the power absorption by 1.5% at the higher production periods. The total annual wave energy absorbed by the wave energy converter is estimated by summing the average absorbed power for all sea states of the scatter diagram, weighted by their probability of occurrence in one year. Table 5 presents the annual captured wave energy for the no mooring system configuration, the slack and the moderately slack mooring system configurations. The moderately slack mooring system reduces around 1% the annual captured wave energy, while the slack mooring system influence is even smaller. Conclusions The main conclusion from the present study is: a realistic mooring system consisting of catenary chain lines has a negligible influence on the floating wave energy dynamics and captured wave energy. The mooring system reduced the annual captured energy by, at most, 1%. The conclusion is valid for the investigated point absorber with an idealized power take off system hypothetically reacting against a fixed reference frame. The investigation must be generalized for more realistic devices and additional mooring system configurations. The paper contributes with a proposed method to account for the mooring system dynamics into the linear frequency domain floater equations of motion and captured wave power. The mooring line dynamics is nonlinear, therefore an identification method was applied to represent the dynamics in terms of inertial, damping and stiffness coefficients. It was demonstrated that the approximation is accurate, within the typical wave frequency motion amplitudes, if the linearization is carried out around the mean mooring system horizontal excursion. The influence of the mooring system on the WEC motions may be considered since the very initial stages of design. Acknowledgment The authors would like to acknowledge the Portuguese Foundation for Science and Technology to support the present work under the project MOORWEC, contracts – PTDC/EME –MFE/103524/2008. References [1] Det Norske Veritas offshore standard DNV-OS-E301, Position Mooring, October 2010. [2] N. Fonseca, J. Pessoa, R. Pascoal, T. Morais, R. Dias, Design pressure distributions on the hull of the FLOW wave energy converter, Ocean Engineering 37 (2010) 611–625. [3] J. Ftzgerald, L. Bergdahl, Considering mooring cables for ofshore wave energy converters, in: Proc. 7th EWTEC, 2007. [4] N. Fonseca, R. Pascoal, T. Morais, R. Dias, Design of a mooring system with synthetic ropes for the FLOW wave energy converter, in: Proc. 28th OMAE, 2009, paper 80223. [5] L. Johanning, G. Smith, J. Wolfram, Measurements of static and dynamic mooring line damping and their importance for floating WEC devices, Ocean Engineering 43 (2007) 1918–1934. [6] J. Fitzgerald, L. Bergdahl, Including moorings in the assessment of a generic offshore wave energy converter: a frequency domain approach, Marine Structures 21 (2008) 23–46.
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[7] R. Pascoal, S. Huang, N. Barltrop, C. Guedes Soares, Equivalent force model for the effect of mooring systems on the horizontal motions, Applications of Oceanic Engineering 27 (2005) 165–172. [8] C.H. Lee, J.N. Newman, Computation of wave effects using the panel method, in: S. Chakrabarti (Ed.), Numerical Models in Fluid–Structure Interaction, WIT Press, Southampton, 2004. Preprint. [9] M.S. Triantafyllou, Cable Mechanics for Moored Floating systems, in: Proc. of the 7th International Conference on the Behavior of Offshore Structures (BOSS’94), vol. 2, 1995, pp. 57–77. [10] W.C. Webster, Mooring induced damping, Ocean Engineering 22 (6) (1995) 571–591. [11] J. Pessoa, N. Fonseca, C. Guedes Soares, Analysis of the first order and slowly varying motions of an axisymmetric floating body in bichromatic waves, Journal of Offshore Mechanics and Arctic Engineering 135 (1) (2013). [12] D.T. Brown, S. Mavrakos, Comparative study on mooring line dynamic loading, Marine Structures 12 (1999) 131–151.
Contents lists available at ScienceDirect
International Journal of Marine Energy journal homepage: www.elsevier.com/locate/ijome
Mooring system influence on the efficiency of wave energy converters Frederico Cerveira a, Nuno Fonseca a,⇑, Ricardo Pascoal b a Centre for Marine Technology and Engineering (CENTEC), University of Lisbon, Instituto Superior Técnico, Av. Rovisco Pais, 1049-001 Lisbon, Portugal b Institute of Electronics and Telematics Engineering of Aveiro (IEETA), University of Aveiro, 3810-193 Aveiro, Portugal
a r t i c l e
i n f o
Keywords: Floating wave energy converter Catenary mooring system Identification method for mooring system dynamics Mooring system effects on absorbed power Annual absorbed wave energy
a b s t r a c t The paper presents an analysis of the mooring system effects on the dynamics of an arbitrary floating wave energy converter (WEC) and on the efficiency of the device. The mooring system dynamics are calculated by a finite differences method, which accounts for the lines’ physical characteristics such as mass and stiffness, as well as for the hydrodynamic inertial and damping characteristics. A method is proposed and applied to consider the inertial, damping and stiffness effects of the mooring system on the WEC’s linear dynamics. An identification method is used to determine the linear mooring system coefficients from pre-calculated nonlinear time domain forces of the mooring system on the floater (the importance of nonlinear effects is assessed as well). Finally the mooring system dynamics is represented by additional inertial, damping and stiffness matrices which are added to the ones of the WEC. The influence of the mooring system on the WEC dynamics and efficiency is assessed in terms of wave induced motions and absorbed power. Results are presented for the transfer functions, statistics in selected stationary sea states and the expected annual absorbed energy. A wave scatter diagram representative of the Portuguese Pilot Zone is used for the annual predictions. Ó 2013 Elsevier Ltd. All rights reserved.
Introduction Mooring design is a critical part of a floating wave energy converter project (WEC). The devices are generally thought to be employed in areas of demanding environmental loads due to waves, current ⇑ Corresponding author. E-mail address: [email protected] (N. Fonseca). 2214-1669/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijome.2013.11.006
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and wind. These survivability issues are addressed in existing offshore standards, such as the DNV-OSE301 [1] and numerical methods can be applied to estimate extreme loads on WECs [2]. However, the mooring arrangement can have a direct influence on the WEC operation and on the economic viability of the project. Fitzgerald et al. [3] present an analysis of different mooring configurations, in terms of the impact on the device motion, array efficiency, weight and cost. Fonseca et al. [4] present a design methodology for WEC moorings, based on a real case study for the Portuguese coast. The mooring system may have a particular effect on the power capture of devices which depend on the motion of the moored body. The energy dissipation associated with catenary mooring line damping was studied by Johanning et al. [5] through a series of experimental tests. It is concluded that mooring line damping increases in a non-linear manner as the line becomes increasingly taut. The energy dissipation is considered to arise primarily from the drag force, which is proportional to the velocity squared of the motion perpendicular to the line axis. When the line becomes nearly taut a small motion at the top-end, originates large movements and velocities around the midpoint and a corresponding large drag induced damping. On the other hand, when the mooring line is quite slack the same topend motion, will result in small perpendicular movement and velocity around the line midpoint and a corresponding small drag induced damping. These experiments also evaluated the effect of the top-end motion frequency on the mooring damping. The results show a proportional increase in energy dissipation with the frequency, which becomes more relevant as the pretension increases. A frequency domain methodology is presented by Fitzgerald et al. [6] to evaluate the mooring effect on WEC power extraction. The mooring non-linear characteristics are estimated numerically for different conditions with a time domain software and then approximated to the linear order. The approximation is considered to be valid for motion amplitudes small relative to the water depth and load induced amplitudes small relative to the equilibrium pretension. The procedure is applied to a generic device unmoored and moored with different configurations. The results indicate that moorings can have important effects, particularly on heave motion, usually detrimental to the energy production. The present work investigates the mooring system influence on the power production in realistic wave conditions. Use is made of accurate climate data, for the Portuguese west coast, to define the energetic sea states and estimate annual production. Particular attention is given to the mooring dynamics, identified by a non-linear code described in Pascoal et al. [7] and applied to the mooring system oscillating with a pre-defined irregular forced motion. The oscillations are imposed around a horizontal mean displaced position due to steady wave drift forces. A representative sea state is selected for the analysis. During power production conditions the floater motions are of relatively small amplitude, therefore it is possible to approximate the non-linear mooring characteristics to the linear order. The related mass, damping and stiffness coefficients can then be added to the floater linear equations of motion and then solved in the frequency domain. The power production of an arbitrary device is estimated in three different conditions: unmoored, and moored with slack and moderately slack catenaries. The mooring effect on the device dynamics and efficiency under operational conditions is concluded from the results.
Theory Responses and absorbed power in harmonic wave The harmonic wave induced hydrodynamic forces and motions are represented on a Cartesian coordinate system with origin at the calm waterline. The x-axis points to the direction of wave propagation, the z-axis is vertical, positive upwards and passes through the floater center of gravity and the y-axis is perpendicular to the former according to the right hand rule. All degrees of freedom, fj, j = 1, . . ., 6, are sequentially numbered according to standard convention. The floater hydrodynamic coefficients and wave exciting forces in incident harmonic waves are calculated by a standard 3D linear radiation-diffraction flat panel method, which has been applied in the form of the commercial WAMIT package. The method assumes potential flow, which satisfies the Laplace equation in the fluid domain. A linear boundary value problem is formulated for the wave body
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67
interactions in incident harmonic waves. Green’s theorem is used to derive integral equations with unknown velocity potential on the mean wetted body surface. The body boundary is discretized into a set of panels with constant potential on each panel, which results in a set of linear simultaneous equations in the unknown potentials. Lee and Newman [8], give details of the formulation. Once the velocity potentials are determined, the hydrodynamic pressures can be calculated, as well as the related wave exciting forces and radiation forces. Equating the external forces (hydrodynamic and mooring forces) to the inertial forces gives the equations of motion. In the present work we consider an axisymmetric floater and mooring system (about the vertical axis), therefore the only modes of motion are surge, heave and pitch. The equations of motion are: 6 n o X f pto m x2 M fkj þ M m þ C fkj þ C m fj ¼ F Ek ; kj þ Akj þ ix Bkj þ Bkj þ Bkj kj
k; j ¼ 1; 3; 5
ð1Þ
j¼1
where x represents the angular frequency and fj the displacement for mode j. M fkj , Bfkj and C fkj represent the floater hydrodynamic added masses, damping coefficients and hydrostatic restoring coefficients. m m Mm kj , Bkj and C kj are the mass, damping and restoring coefficients of the mooring system and they pto represent the linear coupling effects of the mooring system on the floater dynamics. Bpto 22 and B22 represent the power take off (PTO) linear damping coefficients. The solution for the motions in the system of Eq. (1) may be represented in the form of a transfer function from wave to motion and comprising harmonic motion amplitudes and phase angles as function of the wave frequency. The present work deals with a point absorber, meaning the floater dimensions are much smaller than the incident wavelength. It is an idealized device and the wave energy is extracted by two PTO systems fixed to the body and to a hypothetical fixed reference frame. The wave energy is extracted from the surge and heave modes of motion and it is assumed that the PTOs can be represented by linear dampers. Fig. 1 illustrates the idealized concept. The PTO forces are: pto _ F pto 2 ¼ B22 n2 ðtÞ pto F ¼ Bpto n_ 3 ðtÞ 3
ð2Þ
33
Fig. 1. Idealized concept of the wave energy converter.
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and the extracted wave power is: pto _ _ PðtÞ ¼ F pto 2 n2 ðtÞ þ F 3 n3 ðtÞ 2 2 pto _ _ ¼ Bpto 22 ½n2 ðtÞ þ B33 ½n3 ðtÞ
ð3Þ
In harmonic incident waves Eq. (3) is equivalent to:
a 2 PðtÞ ¼ Bpto þ 22 xn2 cosðxt /2 Þ a 2 pto B33 xn3 cosðxt /3 Þ
ð4Þ
where fa2 ; fa3 ; /2 ; /3 are the surge and heave harmonic amplitudes and phase angles, each dot over the symbol stands for a derivative with respect to time, and x is the wave frequency. Time integration over one wave cycle, divided by the wave period, results on the mean power absorbed in harmonic waves: pto
a 2
pto
a 2
B ðxn2 Þ B ðxn3 Þ Phar ðxÞ ¼ 22 þ 33 2 2
ð5Þ
Responses and absorbed power in irregular waves Real sea states are well described by the superposition of an infinite number of regular waves, which together define its spectrum, Sw(x). In the present work, the deep water JONSWAP spectral shape is used to describe the sea states. Long crested irregular waves are considered. The floater motion response spectrum depends on the wave spectrum and the motion transfer function amplitude meaning motion amplitude divided by the wave amplitude, faj =fa :
SR ðxÞ ¼ Sw ðxÞ½faj ðxÞ=fa
2
ð6Þ
while the time-average power in irregular waves can be calculated by the following formula:
Pirr ¼ 2
Z
1
Sw ðxÞ½P har ðxÞ=fa dx
ð7Þ
0
Mooring system dynamics The main interest of the moored equipment under study is to convert wave energy into electric energy and as such it is supposed to move in an optimal manner in the wave frequency range, which means close to resonance conditions. One must take into account that cable dynamics are influenced by both the low frequency motions and the wave frequency motions, they are in fact coupled. Wave frequency motions have dynamic influence on the slowly varying response, because for instance the wave frequency motion increases damping forces which make it more difficult to slowly move the cable. On the other hand, there is a parametric influence of the slowly varying motion into the wave frequency motion because the slowly varying shape of the catenary influences the achieved damping and added mass in the wave frequency range. This has been discussed for instance by Triantafyllou [9] and Webster [10]. In this work, cable dynamics have been modeled numerically by using a central finite-difference scheme in arc length and regressive Euler in time for the velocities, as described in Pascoal et al. [7], and, for sake of completeness, is briefly described herein. The algorithm involves solving the equilibrium of the cable as represented in Fig. 2, which results in the governing Eqs. (8–11). The set of equations must be supplemented with a material constitutive relation and initial and boundary conditions.
m
@~ v @~S ~ ¼ þ f ð1 þ cÞ @s @t
ð8Þ
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69
Fig. 2. Equilibrium of an infinitesimal cable segment [7].
~ @M þ ^t ~ Sð1 þ cÞ ¼ 0 @s
ð9Þ
@~ v @ ^ ¼ ðtð1 þ cÞÞ @s @t
ð10Þ
R @~ d~ R ¼ dt @t
ð11Þ
~ v¼
The dynamic simulation starts from a relaxation solution, where the initial conditions become zero velocities and the geometric configuration is determined for the mean drift displacement corresponding to the slowly varying drift loads. The external loading along the arc length is of the form: 2~ ~ f ð1 þ cÞ ¼ ma @@t2R ð1 þ cÞ pffiffiffiffiffiffiffiffiffiffiffiffi V rn k~ V rn D 1 þ cC Dn 12 qf k~ pffiffiffiffiffiffiffiffiffiffiffiffi f sb þ ~ f cd V rt k~ V rt pD 1 þ cC Dt þ mw~ 12 qf k~ g þ~
ð12Þ
where ma is the sectional added mass, ~ R is the Lagrangian coordinate along the centreline, c is V rn is the fluid velocity normal engineering strain, qf the specific mass of the medium (1025 kg m3), ~ to the local cable (chain) centreline, CDn is the normal drag coefficient, CDt is the tangential drag coefficient, mw is the equivalent sectional mass in water accounting for hydrostatic effects,~ g is the f c d concentrated loading and ~ f sb the interaction with the bottom. acceleration of gravity, ~ Interaction with the seabed is complex to model and its effects greatly depend on the soil type and the cable profile. As proposed by Webster [10], a distributed elastic support is used. In this case the thickness of the elastic support was assumed constant (0.5 m) and critical damping, mainly meant to stabilize the numerical solution, acts only as the cable penetrates into the undisturbed floor level. This mismodeling becomes more important as cables become slack and overall damping will tend to be greater than reported herein for that situation. The discrete form of the equations is solved in a non-iterative time marching scheme. This noniterative property is achieved by using regressive Euler to eliminate the non-linearity on the next time step due to damping and the geometric average of the position vector to eliminate the nonlinearity due to the strain equation. This approximation results in the algorithm schematically represented in Fig. 3. The ‘‘Updated Kinematics’’ block corresponds to using LU matrix factorization solving for velocities and positions.
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Fig. 3. Cable dynamics simulation loop [7].
Case study Wave climatology Figueira da Foz wave climate scatter diagram is used in the present investigation to assess the wave power absorbed in realistic conditions, as well as the annual absorbed energy. The choice of this site is of particular interest because of its proximity to the 320 km2 Portuguese pilot zone, located at São Pedro de Moel (West coast of Portugal). Fig. 4 presents the scatter diagram, namely the annual joint probability of occurrence of Hs Tz for (Hs is the significant wave height and Tz is the average wave period between zero upcrossings). The scatter diagram is the result of the statistical analysis of wave records obtained from a wave rider station located 20 km from the shoreline (40°110 0800 N and 9°080 4400 W) at 92 m water depth (Portuguese Hydrographic Institute). The data acquisition was made for periods between 20 and 30 min every 3 h. The acquisition was continuous when the significant wave height was larger than 5 m. The scatter diagram results from the analysis of 11,193 valid records obtained between July 1990 and January 1996. The wave rider buoy was located at a distance of around 30 km North of the Portuguese pilot zone. Point absorber characteristics The floater geometry is a generic vertical cylinder with rounded bottom edge. An advantage of the cylindrical shape is that it can be easily reproduced numerically. Fig. 5 presents the profile of the Hs(m) 1.75 3.75 4.25 4.75 5.25 5.75 6.25 6.75 7.25 7.75 8.25 0.25 0.01 0.08 0.11 0.11 0.11 0.03 0.06 0.03 0.02 0.75 0.17 0.51 1.22 1.79 2.17 2.00 1.77 1.14 0.68 0.36 1.25 0.03 0.18 0.87 2.28 3.16 3.72 3.27 2.36 1.95 1.46 1.75 0.10 1.01 2.61 3.30 3.03 2.57 2.13 2.02 2.25 0.05 0.85 1.77 2.23 1.79 1.86 1.58 2.75 0.04 0.64 1.09 0.96 0.92 1.29 3.25 0.06 0.28 0.40 0.53 0.66 3.75 0.05 0.12 0.29 0.37 4.25 0.01 0.10 0.17 percentage 4.75 0.01 0.08 0.00-0.10 5.25 0.01 0.10-0.50 5.75 0.50-1.00 6.25 1.00-2.00 6.75 2.00-3.00 7.25 3.00-4.00 7.75 4.008.25
Tz(s) 8.75 9.25 9.75 10.25 10.75 11.25 11.75 12.25 12.75 13.25 13.75 14.50 16.25 0.28 1.22 1.44 1.31 1.03 0.65 0.51 0.26 0.13 0.03
0.15 0.92 1.15 1.10 1.00 0.66 0.48 0.42 0.23 0.04 0.03 0.01 0.01
0.08 0.72 1.03 0.88 0.82 0.76 0.47 0.34 0.26 0.08 0.05 0.04 0.01
0.01 0.41 0.67 0.69 0.72 0.65 0.57 0.36 0.18 0.13 0.05 0.06 0.01 0.01
0.13 0.36 0.47 0.54 0.58 0.37 0.21 0.21 0.13 0.08 0.03 0.01 0.02 0.01
0.01 0.06 0.19 0.39 0.41 0.41 0.32 0.18 0.15 0.10 0.11 0.04 0.01 0.04
0.07 0.10 0.19 0.22 0.36 0.15 0.15 0.20 0.07 0.06 0.03 0.02 0.01 0.02
0.01 0.03 0.06 0.20 0.17 0.18 0.15 0.11 0.12 0.08 0.05 0.03 0.01 0.04 0.02
Fig. 4. Scatter diagram for Figueira da Foz, West cost of Portugal.
0.02 0.05 0.09 0.17 0.08 0.11 0.06 0.06 0.06 0.08 0.03 0.03 0.01 0.01
0.01 0.03 0.06 0.05 0.06 0.04 0.04 0.05 0.03 0.03 0.03 0.01
0.01 0.01 0.01 0.01 0.04 0.01 0.02 0.02 0.03 0.03 0.03 0.03 0.03 0.04 0.03 0.01 0.01 0.02 0.02 0.03 0.02 0.06 0.01 0.01 0.01 0.01 0.01 0.01
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Fig. 5. Axisymmetric buoy main dimensions.
Table 1 Buoy main dimensions. Dimensions Cylinder diameter Draft Mass Vert. position of CoG Pitch radius of gyration
D (m) T (m) D (t) Zg (m) r (m)
15 4.5 786.5 0.7 4.5
Table 2 Mooring lines dimensions. Dimensions Length Stiffness Chain diameter Tangential drag coefficient Normal drag coefficient Mass per unit length (air) Top-end tension slack mooring Top-end tension moderately slack mooring
L (m) K (N/m/m) Dchain (mm) CDt () CDn () mchain (kg/m) Tslack (N) Tmoderately (N)
350 4.12E8 64 1.0E-2 1.2 90 1.15E5 2.22E5
axisymmetric body with the respective main dimensions given in the drawing and in Table 1. The body added mass and damping coefficients for the considered directions were evaluated by the WAMIT software and are given in Table 3 for the respective motion natural frequency. The floater hydrodynamic coefficients and wave exciting forces in incident harmonic waves are calculated by a standard 3D linear radiation-diffraction flat panel method, which has been applied in the form of the commercial WAMIT package. In the present case the floater is axisymmetric, therefore only one quadrant of the cylindrical floater has been represented by 270 low order panels. This means the body is represented by flat panels and the solution for the potentials are approximated by piecewise constant values on each panel (higher order panels could have been used as well, however, the
72
Direction
Surge Heave – Surge Pitch – Surge Heave Surge – Heave Pitch – Heave Pitch Surge – Pitch Heave – Pitch
Buoy
PTO
Weakly slack mooring
Mass
Added mass
Damping
Damping
Mass
Damping
Stiffness
Mass
Moderately slack mooring Damping
Stiffness
7.86 105 [kg] 0 5.5 105 [kg.m] 7.86 105 [kg] 0 0 1.6 107 [kg.m2] 5.5 105 [kg.m] 0
3.38 105 [kg] – – 5.56 105 [kg] – – 2.47 106 [kg.m2] – –
5.20 104 [N.s/m] – – 2.27 105 N.s/m – – 6.62 105 [N.m.s] – –
9.00 105 [N.s/m] 0 0 2.00 106 [N.s/m] 0 0 0 0 0
9284 [kg] 2060 [kg] 40850 [kg.m] 33166 [kg] 2266 [kg] 9969 [kg.m] 1.90 105 [kg.m2] 42170 [kg.m] 9600 [kg.m]
7712 [N.s/m] 1480 [N.s/m] 33933 [N.s] 17954 [N.s/m] 1395 [N.s/m] 6138 [N.s] 130010 [N.m.s] 28883 [N.s] 5610 [N.s]
7755 [N/m] 664 [N/m] 34123 [N] 5940 [N/m] 747 [N/m] 3285 [N] 1.57 105 [N.m] 35850 [N] 5920 [N]
42111 [kg] 2538 [kg] 1.85 105 [kg.m] 53214 [kg] 2442 [kg] 10746 [kg.m] 8.72 105 [kg.m2] 1.94 105 [kg.m] 3240 [kg.m]
25684 [N.s/m] 5359 [N.s/m] 113010 [N.s] 35084 [N.s/m] 1559 [N.s/m] 6859 [N.s] 4.14 105 [N.s.m] 92173 [N.s] 20985 [N.s]
29311 [N/m] 639.3 [N/m] 1.29 105 [N] 11331 [N/m] 931.2 [N/m] 4097 [N] 6.64 105 [N.m] 1.49 105 [N] 5020 [N]
F. Cerveira et al. / International Journal of Marine Energy 3–4 (2013) 65–81
Table 3 Floater hydrodynamic added masses and damping coefficients, PTO damping coefficients and the identified linear dynamic coefficients of the mooring system.
F. Cerveira et al. / International Journal of Marine Energy 3–4 (2013) 65–81
73
advantages in terms of computational efficiency are not relevant for the present body). The interior free surface was meshed as well to remove numerical problems associated with irregular frequencies. Hydrodynamic calculations were carried out for 70 m water depth. The numerical model results were compared with results from experimental tank testes in waves [11], and identified the need to add extra damping to the numerical model in the surge and pitch modes, since potential flow damping underestimates the real floater damping; these values are given in Table 3. Regarding the wave power absorption, the device has two liner PTOs, one operating in heave and the other in surge (see Section II. A). The respective damping coefficient magnitudes were optimized according to the site climatology for maximum annual production. The method consists on systematically varying the damping coefficient and calculating the annual absorbed energy. The PTO damping coefficient is selected as the one which absorbs the largest amount of energy. The damping coefficients are optimized separately for the surge and heave modes, which is valid because these modes are weakly coupled. A very weak coupling exists due to the mooring system. The PTO damping values in surge and heave are presented in Table 3. It is known that a heaving point absorber maximizes the wave energy absorption when the wave frequency is equal to the heave natural frequency and, in this case, the optimum PTO damping is equal to the sum of the other damping coefficients (hydrodynamic and mooring). It is possible to conclude that the heave damping coefficient of Table 3 is larger than the optimum for the natural frequency. This is because the PTO is optimized for the case study wave climatology, which is characterized by sea states with peak periods larger than the heave natural frequency. The average wave peak period for Figueira da Foz is around 9.6 s, while the floater heave natural frequency is 5.5 s. Mooring system design and specification Two mooring systems have been designed using a quasi-static method, with corresponding safety factors according to existing Classification Society rules. The mooring lines extreme loads were obtained from nonlinear time domain simulations for a design storm representative of the ocean area being considered. Finally the selected systems consist of catenary (64 mm chain) spread mooring configurations with three lines at 120° to each other. The incident wave direction is aligned with one mooring line, Fig. 6. The respective static geometries are illustrated in Fig. 7, where the slack mooring system has 2/3 of the lines length on the sea bottom, while the moderately slack system has 1/2 of the lines length lying on the bottom. Both mooring systems are made of chain lines with the same length and mechanical properties, presented in Table 2, but different horizontal distance from the anchor to the top-end in order to produce different catenary shapes and pretensions. The water depth is 70 m.
Fig. 6. Mooring geometry sketch from top view.
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Fig. 7. Slack and moderately slack mooring geometries for the equilibrium condition, side view.
Identifications of the mooring system dynamics Identification method In light of what has been stated in the previous sections, it is important, even during the preliminary design stage, to take the mooring system dynamics into account. However, using the full nonlinear mooring dynamics code is a highly prohibitive procedure because it is extremely time consuming (assuming it does not collapse due to numerical instabilities). One possible solution to this deadlock is to use procedures which allow for reduction in model complexity. In this work, the complexity of the mooring system model has been reduced by using a system identification procedure. This procedure is the generalized version of the one which has been described in Pascoal et al. [7], and includes all terms (even and odd) up to third order of the Taylor expansion of the forces at the mooring point. The need to include even terms stems from the fact that the present mooring system is not symmetric with respect to the y–z plane, as can be observed in see Fig. 6 (although, as referred in Section II.A, the mooring system is axisymmetric with respect to the z-axis). In order to successfully indentify the model, the special Schroeder phased signal is also used herein. The signal represents forced irregular motions imposed at the fairleads connecting the mooring lines to the floater. The mooring dynamics is calculated for the forced motion and the important result are the mooring system forces at the connecting point. The second part of the analysis consists on identifying the Taylor expansion coefficients of an approximated model for the mooring line forces. Care has been exercised to perform the Taylor expansion about the average slow drift position of a representative sea state (wave drift forces and motions for the present floater geometry have been analyzed in Pessoa et al. [11]). In this case the representative sea state has been chosen as the one with more annual energy (which is related to the average wave power and the probability of occurrence). Notice that the Schroeder phased signal is chosen to have approximately the same energy as the wave frequency motion and the slow drift motion is chosen to encompass that which will actually occur. By this we mean that, during identification, there is no intention to replicate the motions which actually occur but they must be highly representative. Producing data for the identification procedure is highly time consuming since it consists on using the numerical model for the mooring system dynamics, but, once it has been performed, the top end force is a simple, albeit nonlinear, function of kinematic parameters. The identification procedure presents a nonlinear model which is not applicable in a consistent manner for frequency domain analysis but is mandatory to prevent strong bias on the linear coefficients due to unmodeled higher order terms. The linear terms, possibly corrected by using statistical averaging of nonlinear terms, are then used for further frequency domain analysis. Approximate model The approximate model of the mooring arrangement used for first order frequency domain analysis is necessarily a linear one. Therefore, once the system identification procedure has been applied to successfully recover the nonlinear model, the model must be linearized according to some criteria. Inspection of the contributions from the identified components determines the need to linearize the contribution of cubic damping because it has a major contribution to the response, especially in the case of the moderately slack mooring. Since we are dealing with energy extraction and average
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values, it has been considered plausible to perform matching of the absorbed energy, i.e., to find a correction to the linear coefficient such that the absorbed energy remains the same as with cubic damping. This additive correction is of the form:
B ¼ Bcubic
X
X x_ 2
_4
x
N
!1 ð13Þ
N
found from the energy balance for the same simulation time series, with the N time instants used in the identification of the nonlinear model. Since the expected value of energy remains the same, this is a form of statistical linearization. Finally, the approximate linear model is:
o Xn m linear x2 Mm þ Bm þ Cm kj þ ix Bkj kj kj fj
Fm k ¼
ð14Þ
j¼1;3;5
and has frequency independent coefficients. The linear mass, damping and stiffness coefficients obtained for the slack and moderately slack mooring systems are given in Table 3 (right columns). For comparative purposes, Table 3 includes also the floater hydrodynamic coefficients in surge and heave corresponding to their natural frequencies (left columns), and the PTOs damping coefficients. It is clear that the mooring system effects on the floater dynamics are small: the inertial and damping mooring coefficients are up to around 7% and 5% of the floater wave frequency hydrodynamic coefficients. These contributions correspond to the moderately slack mooring system – the contributions are smaller for the slack mooring. Since the
80 60
Force [kN]
40 20 0 -20 -40 -60
Linear force Non-Linear Force
-80 400
420
440
460 time [s]
480
500
520
Fig. 8. Sample of the heave mooring forces at the fairleads due to forced heave motion for the moderately slack mooring system – comparison between nonlinear results and the linear model results.
Table 4 Standard deviation of the linear and non-linear mooring forces. Direction
Surge [kN] Heave [kN] Pitch [kN.m]
Slack
Moderately Slack
Non-linear Model
Linear Model
Non-linear Model
Linear Model
21.3 25.7 20.4
20.3 25.0 18.4
83.4 44.4 75.2
76.4 42.3 69.5
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F. Cerveira et al. / International Journal of Marine Energy 3–4 (2013) 65–81 0.7
Surge - Amotion/Awave [ ]
0.6
0.5
0.4
0.3
0.2
Buoy+PTO Buoy+Weakly Slack Mooring+PTO Buoy+Moderatly Slack Mooring+PTO
0.1
0
0
0.5
1
1.5
2
0
0.5
1
1.5
2
0
0.5
1
1.5
2
1
Heave - Amotion/Awave [ ]
0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
Pitch - Amotion/Awave [rad/m]
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
Frequency [rad/s] Fig. 9. Motion transfer function amplitude for surge, heave and pitch.
PTO damping coefficients are of the same order as the hydrodynamic floater coefficients, we conclude that the mooring damping coupling effects are even smaller than the number given above. It should be noted that wave frequency optimal damping coefficients are not those which are applicable to the low frequency dynamics. In the low frequency range, potential damping becomes very small, and, as discussed by Brown and Mavrakos [12], other effects dominate and mooring line damping is of primary importance in reducing peak excursions, providing up to 80% of the total damping.
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77
140
Surge - Preg/A2wave [kw/m 2]
120
100
80
60
40
Buoy+PTO Buoy+Weakly Slack Mooring+PTO Buoy+Moderatly Slack Mooring+PTO
20
0
0
0.5
1
1.5
2
180
Heave - Preg/A2wave [kw/m 2]
160 140 120 100 80 60 40 20 0
0
0.5
1
1.5
2
Frequency [rad/s] Fig. 10. Time averaged wave power absorbed by the surge and heave modes of motion (power normalized by the wave amplitude squared.
Johanning et al. [5] verified experimentally that the mooring line damping increases with the pretension, which is in agreement with the results obtained for the considered mooring systems. It is of interest to assess how well the identification method and the related linear model are able to represent the nonlinear mooring system dynamic forces acting on the floater. Fig. 8 presents a sample from the moderately slack mooring system heave force time history associated to forced vertical irregular motion of the fairleads. The black line represents the nonlinear force calculated by the time domain mooring system dynamic model (see Section II. C), while the green line represents the force calculated by the linear mooring system model (Eq. (14)). It is clear that the linear approximation closely follows the nonlinear force time history. Similar results are obtained for forces in all directions, due to both surge and heave forced motions. Notice that the asymmetry of the force found between low and high loads is lost in the linear model. The standard deviation of the force time history can also be used to compare the linear and nonlinear mooring system models. Table 4 presents the standard deviations for the two mooring systems, calculated from the linear and nonlinear force time histories, for forced irregular motions of the fairleads in the surge, heave and pitch directions. One observes that the linear model always underestimates the standard deviations, although by a small margin. The difference is larger for the moderately slack mooring system, since nonlinear effects on the mooring lines increase in this case. From the previous analysis, it is possible to conclude that the identification method and related linear model for the mooring forces at the fairleads is accurate and can be used for a qualitative assessment of the mooring system effects on the WEC dynamics.
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F. Cerveira et al. / International Journal of Marine Energy 3–4 (2013) 65–81 0.7 Buoy+PTO Buoy+Weakly Slack Mooring+PTO Buoy+Moderatly Slack Mooring+PTO
Surge - Amotion/Awave [ ]
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.5
0
0.5
1
1.5
2
1
1.5
2
1.4
Heave - Amotion/Awave [ ]
1.2
1
0.8
0.6
0.4
0.2
0
Frequency [rad/s] Fig. 11. Response spectra for an annual high energetic sea state (Hs = 2.75 m and Tp = 10.6 s).
Analysis of the mooring system effects Results in regular waves According to the previous specifications, three systems are studied (PTO operates in surge and heave motion modes): Buoy and PTO (no mooring). Buoy, slack mooring and PTO. Buoy, moderately slack mooring and PTO. The equations of motion are solved in the frequency domain. Fig. 9 presents the motion transfer function amplitudes for surge, heave, and pitch. The motion amplitudes are normalized by the wave amplitude and presented as function of the wave frequency. The results show that the mooring systems do not affect significantly the floater motions. Only the pitch motion resonance amplification is clearly reduced when mooring system effects are included. This is because pitch motion is lightly damped (small hydrodynamic damping and no PTO for this mode), therefore the mooring system damping contribution is relevant. Surge and heave are almost unaffected by the mooring system. This was expected since the identified mooring system coefficients are small compared to the floater and PTOs coefficients (see Table 3).
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The time-averaged power absorption in regular waves for the surge and heave directions is presented in Fig. 10. The results are normalized by the wave amplitude squared. In the surge direction, the slack mooring has no significant influence, while the moderately slack mooring decreases slightly the power absorption for frequencies above 1 rad/s. The power interference due to the surge couplings with pitch at natural frequencies is reduced by the mooring systems. The power absorption in the heave direction is also not significantly affected by both mooring system, although the moderately slack system reduces slightly the absorbed power for frequencies above around 0.5 rad/s. Results in irregular waves This Section presents the analysis of mooring system effects on the floater motion responses in irregular waves, as well as on the absorbed power. It ends with the estimation of wave energy absorbed during one year. Long crested stationary sea states are considered, represented by JONSWAP spectra with a peakedness parameter of 3.3. Combining the average wave power in irregular waves, which can be calculated for all scatter diagram classes, with their annual probability of occurrence, it is possible to determine the scatter diagram class with more energy during the year. The corresponding significant wave height and peak period are: Hs = 2.75 m and Tp = 10.6 s. The floater motion response spectrum was calculated for this sea state and the result is presented in Fig. 11. One observes no influence from the mooring system on the body motion apart from the small surge–pitch coupling effect around the pitch natural frequency.
14 Buoy+PTO Buoy+Weakly Slack Mooring+PTO Buoy+Moderatly Slack Mooring+PTO
Surge - Pir/ Hs2 [kw/m 2]
12
10
8
6
4
2
0
0
5
10
15
20
10
15
20
20
Heave - Pir/ Hs2 [kw/m 2]
18 16 14 12 10 8 6 4 2 0
0
5
Tz [s] Fig. 12. Time-averaged power absorption per significant height squared.
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Table 5 Annual wave energy captured according to the mooring configuration. Direction
Unmoored
Slack
Moderately slack
Surge [MW.h] Heave [MW.h] Total [MW.h]
300.9 857.1 1158.0
300.9 852.1 1153.0
302.4 844.4 1146.8
Fig. 12 presents the time-averaged power absorbed in irregular waves, normalized by the significant wave height squared, as function of the zero up-crossing wave period (Tz). The graphs show, again, no significant mooring system influence. In the surge direction the mooring effect can only be identified at the higher production periods, corresponding to a power decrease of about 0.5% with the slack and 1% with the moderately slack mooring. A slightly larger effect is found on the heave direction where the moderately slack mooring decreases the power absorption by 1.5% at the higher production periods. The total annual wave energy absorbed by the wave energy converter is estimated by summing the average absorbed power for all sea states of the scatter diagram, weighted by their probability of occurrence in one year. Table 5 presents the annual captured wave energy for the no mooring system configuration, the slack and the moderately slack mooring system configurations. The moderately slack mooring system reduces around 1% the annual captured wave energy, while the slack mooring system influence is even smaller. Conclusions The main conclusion from the present study is: a realistic mooring system consisting of catenary chain lines has a negligible influence on the floating wave energy dynamics and captured wave energy. The mooring system reduced the annual captured energy by, at most, 1%. The conclusion is valid for the investigated point absorber with an idealized power take off system hypothetically reacting against a fixed reference frame. The investigation must be generalized for more realistic devices and additional mooring system configurations. The paper contributes with a proposed method to account for the mooring system dynamics into the linear frequency domain floater equations of motion and captured wave power. The mooring line dynamics is nonlinear, therefore an identification method was applied to represent the dynamics in terms of inertial, damping and stiffness coefficients. It was demonstrated that the approximation is accurate, within the typical wave frequency motion amplitudes, if the linearization is carried out around the mean mooring system horizontal excursion. The influence of the mooring system on the WEC motions may be considered since the very initial stages of design. Acknowledgment The authors would like to acknowledge the Portuguese Foundation for Science and Technology to support the present work under the project MOORWEC, contracts – PTDC/EME –MFE/103524/2008. References [1] Det Norske Veritas offshore standard DNV-OS-E301, Position Mooring, October 2010. [2] N. Fonseca, J. Pessoa, R. Pascoal, T. Morais, R. Dias, Design pressure distributions on the hull of the FLOW wave energy converter, Ocean Engineering 37 (2010) 611–625. [3] J. Ftzgerald, L. Bergdahl, Considering mooring cables for ofshore wave energy converters, in: Proc. 7th EWTEC, 2007. [4] N. Fonseca, R. Pascoal, T. Morais, R. Dias, Design of a mooring system with synthetic ropes for the FLOW wave energy converter, in: Proc. 28th OMAE, 2009, paper 80223. [5] L. Johanning, G. Smith, J. Wolfram, Measurements of static and dynamic mooring line damping and their importance for floating WEC devices, Ocean Engineering 43 (2007) 1918–1934. [6] J. Fitzgerald, L. Bergdahl, Including moorings in the assessment of a generic offshore wave energy converter: a frequency domain approach, Marine Structures 21 (2008) 23–46.
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[7] R. Pascoal, S. Huang, N. Barltrop, C. Guedes Soares, Equivalent force model for the effect of mooring systems on the horizontal motions, Applications of Oceanic Engineering 27 (2005) 165–172. [8] C.H. Lee, J.N. Newman, Computation of wave effects using the panel method, in: S. Chakrabarti (Ed.), Numerical Models in Fluid–Structure Interaction, WIT Press, Southampton, 2004. Preprint. [9] M.S. Triantafyllou, Cable Mechanics for Moored Floating systems, in: Proc. of the 7th International Conference on the Behavior of Offshore Structures (BOSS’94), vol. 2, 1995, pp. 57–77. [10] W.C. Webster, Mooring induced damping, Ocean Engineering 22 (6) (1995) 571–591. [11] J. Pessoa, N. Fonseca, C. Guedes Soares, Analysis of the first order and slowly varying motions of an axisymmetric floating body in bichromatic waves, Journal of Offshore Mechanics and Arctic Engineering 135 (1) (2013). [12] D.T. Brown, S. Mavrakos, Comparative study on mooring line dynamic loading, Marine Structures 12 (1999) 131–151.