Conference PaperPDF Available

Code-to-Code Nonlinear Hydrodynamic Modelling Verification for Wave Energy Converters: WEC-Sim vs. NLFK4ALL


Abstract and Figures

In the wave energy conversion field, simulation tools are crucial for effective converter and controller design, but are often prone to become very case-specific, in both structure and parameter selection. This is due to majorly different working principles and diverse importance of nonlinear effects, at times requiring ad-hoc modelling approaches. To tackle this challenge, WEC-Sim (Wave Energy Converter SIMulator) was born from the National Renewable Energy Laboratory (NREL) and Sandia National Laboratories, providing a unique simulation platform for all WECs. Nonlinearities related to time-varying wetted surface, especially important in floating WECs, are included in WEC-Sim through a mesh-based computation of nonlinear Froude-Krylov forces. Virtually arbitrary geometries can be considered, thanks to the discretized representation of wetted surfaces, at the price of a significant increase in computational burden. This paper considers a time-effective alternative, implemented in the open-source toolbox called NLFK4ALL, applicable to the popular and wide family of axisymmetric floaters. The Spar-buoy floating oscillating water column device is considered , particularly challenging due to a submerged volume composed of several different sections. The accuracy of WEC-Sim and NLFK4ALL is verified by a preliminary cross-comparison, using independent methods to compute virtually same effects. Fixed-body numerical experiments are used to quantify nonlinearities and compare not only the accuracy, but also the computation burden. Results show that both methods provide almost identical results, although WEC-Sim doubles computational requirements.
Content may be subject to copyright.
Code-to-Code Nonlinear Hydrodynamic
Modelling Verification for Wave Energy
Converters: WEC-Sim vs. NLFK4ALL
Giuseppe Giorgi, Markel Penalba, and Rui P.F. Gomes
Abstract—In the wave energy conversion field, simu-
lation tools are crucial for effective converter and con-
troller design, but are often prone to become very case-
specific, in both structure and parameter selection. This is
due to majorly different working principles and diverse
importance of nonlinear effects, at times requiring ad-
hoc modelling approaches. To tackle this challenge, WEC-
Sim (Wave Energy Converter SIMulator) was born from
the National Renewable Energy Laboratory (NREL) and
Sandia National Laboratories, providing a unique simula-
tion platform for all WECs. Nonlinearities related to time-
varying wetted surface, especially important in floating
WECs, are included in WEC-Sim through a mesh-based
computation of nonlinear Froude-Krylov forces. Virtually
arbitrary geometries can be considered, thanks to the
discretized representation of wetted surfaces, at the price
of a significant increase in computational burden. This
paper considers a time-effective alternative, implemented
in the open-source toolbox called NLFK4ALL, applicable to
the popular and wide family of axisymmetric floaters. The
Spar-buoy floating oscillating water column device is con-
sidered, particularly challenging due to a submerged vol-
ume composed of several different sections. The accuracy
of WEC-Sim and NLFK4ALL is verified by a preliminary
cross-comparison, using independent methods to compute
virtually same effects. Fixed-body numerical experiments
are used to quantify nonlinearities and compare not only
the accuracy, but also the computation burden. Results
show that both methods provide almost identical results,
although WEC-Sim doubles computational requirements.
Index Terms—Wave Energy Converters Nonlinear
Froude-Krylov force, Spar-buoy oscillating OWC, WEC-
SIMULATION tools are essential for effective design
and development of power conversion systems.
However, the definition of a reliable and representa-
tive mathematical model for wave energy converters
(WECs) is especially challenging. In fact, there is a wide
variety of devices, based on substantially different
working principles and mounting various components,
often making the model quite case-specific. Moreover,
since an appropriate representation of nonlinear effects
This research was funded by the European Research Council (ERC)
under the European Unions Horizon 2020 research and innovation
programme under Grant Agreement No. 832140
Marine Offshore Renewable Energy Lab (MOREnergy Lab),
DIMEAS, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129,
Turin, Italy (e-mail:
Department of Mechanical and Manufacturing, Mondragon Uni-
versity, Loramendi 4 Apdo. 23, 20500, Arrasate, Spain (e-mail: mpe-
IDMEC, Instituto Superior T´
ecnico, Universidade de Lis-
boa, Av. Rovisco Pais 1, 1049-001 Lisboa, Portugal (e-mail:
is likely to be required for achieving acceptably accu-
rate results, the complexity of mathematical models
rapidly escalates, at the expense of its transparency
and flexibility [1]. In order to address such issues,
the National Renewable Energy Laboratory (NREL)
and Sandia National Laboratories developed the open
source high-level software in 2014 WEC-Sim (Wave
Energy Converter SIMulator) [2]. WEC-Sim is gaining
increasing popularity in the wave energy community,
both academic and industrial, since it provides a frame-
work for simulating virtually any WEC [3], including
different subsystems of the wave energy conversion
chain: from mooring systems [4] to power take-off
[5], including various nonlinear hydrodynamic effects
[6]. This paper focuses especially on the computation
of the nonlinear Froude-Krylov (NLFK) force, which
is one of the major and most impactful nonlinearity
in floating WECs [7]–[9], with effects on parametric
resonance [10], [11], mooring design [12] and model-
based controller design [13], [14]. NLFK forces are
due to the undisturbed pressure field acting on the
instantaneous wetted surface of the floater, whereas
the linear approximation considers the mean wetted
surface as constant. Geometries of arbitrary complexity
require a mesh-based discretization of the wetted sur-
face, that can be handled in two alternative ways: either
a re-meshing routine is used, re-computing the mesh
discretization at each time step according to the instan-
taneous position of the floater with respect to the free
surface elevation; or a constant but very fine mesh is
implemented, with a boolean decision on which panel
is either fully wet or fully dry [15]. Both approaches
may slow down the simulation significantly, depend-
ing on the size of the spatial discretization. WEC-Sim,
with the purpose of being flexible on the floater shape,
implements a mesh-based computation of NLFK forces
[16]. However, for axisymmetric geometries there is
an equivalent but computationally more efficient ap-
proach based on an analytical formulation of the in-
stantaneous wetted surface, hence getting rid of the
computational bottleneck of the meshed wetted surface
[17]. An additional frequency-domain (FD) approach
was also suggested in [18], where nonlinear effects are
included using a projection of the dynamical equations
onto a basis of trigonometric functions.
Note that the popular family of point absorber WECs
is mainly composed of axisymmetric devices, mak-
ing such an assumption not too restrictive. Therefore,
the analytical approach, available in the open source
Matlab demonstration toolbox NLFK4ALL [19] and
Proceedings of the 14th European Wave and Tidal Energy Conference 5-9th Sept 2021, Plymouth, UK
ISSN 2309-1983 Copyright © European Wave and Tidal Energy Conference 2021
validated with experimental wave tank data [20], is
implemented in this paper, and compared with results
obtained with the nonlinear hydrodynamics features
of WEC-Sim. The existing Sparbuoy floating oscillating
water column (OWC) is used as a case study [21].
The reminder of the paper is organized as follows:
Sect. II introduces the device considered in this paper,
used to quantify the comparison between the two pro-
posed approaches for the NLFK calculation, WEC-Sim
and NLFK4ALL, respectively discussed in Sect. II-B
and Sect. II-C. Section III shows results, while some
final remarks and conclusions are provided in Sect. IV.
The Spar-buoy OWC is a floating oscillating water
column, composed of a hollow floater opened at the
bottom, hence containing a water column within its
structure. As waves arrive, a relative motion between
the floater and the water column is generated, driving
the air pocket above the water column through a tur-
bine, extracting useful energy. Figure 1 shows a picture
of the prototype [22], tested at 1:16 scale. This device,
being of the spar buoy type, is prone to experience
the highly nonlinear phenomenon of parametric reso-
nance: when the excitation frequency is twice that of
the natural frequency of the pitching degree of freedom
(DoF), the system becomes parametrically unstable and
diverts part of the incoming energy to the rolling
DoF [23]. Such a behaviour, potentially detrimental for
power extraction efficiency and hindering survivability,
is due to time-variations of the wetted surface, so that
it can be successfully articulated by NLFK models. For
this reason, the Spar-buoy is an appropriate choice for
the case study presented in this paper.
Fig. 1. 1:16 scale prototype of the Spar-buoy OWC is a floating
oscillating water column [22].
A. Nonlinear Froude-Krylov force
Froude-Krylov forces are defined as the integral of
the undisturbed pressure field (p) over the wetted
surface of the floater. In the linear approximation, it
is assumed that the relative motion between the body
and the free surface is small, so that FK forces are com-
puted with respect to the mean wetted surface. On the
contrary, nonlinear FK force calculations are performed
with respect to the actual instantaneous wetted surface
fF K =fg+ZZ
pndS, (1a)
τF K =rg×fg+ZZ
pr×ndS, (1b)
where fF K are nonlinear forces, τF K are torques, fg
is the gravity force, nis the unity vector normal to
the surface, ris the generic position vector, and rgis
the position vector of the centre of gravity. The undis-
turbed incident pressure field of an uni-directional
regular wave travelling in the x-direction is defined
p(x, z, t) = ρg z+acos (ωt kx)cosh (k(z+h))
where a,ω, and kare the wave amplitude, frequency,
and wavenumber, respectively, ρthe water density, g
the acceleration of gravity, hthe water depth, and zthe
vertical coordinate zmodified according to Wheeler’s
stretching [24]:
where ηis the free surface elevation.
B. The WEC-Sim mathematical model
The WEC-Sim mathematical model is based on
a Simulink implementation [2], with specifically de-
signed blocks for typical components of wave energy
converters. Figure 2 shows a snapshot of the block
diagram for the Spar-buoy, with two hydrodynamic
bodies, respectively the floater and the OWC, between
which there is a translational power take-off (PTO)
unit converting the relative motion into electricity. The
system is then referenced to the global-frame through
a constraint block, which can be set to free motion (6-
DoF), planar motion (3-DoF), vertical motion (1-DoF),
or no motion (fixed).
The PTO system of an OWC is an air turbine, which
converts the bidirectional air flow induced by the OWC
motion inside the floater. The pressure drop across the
turbine can be simulated at model scale using an orifice
plate [21]:
FP T O =8ρaA3
where ρais the air density, Aais the cross-sectional area
of the air chamber, Cdis the discharge coefficient (Cd=
Fig. 2. Simulink block diagram in WEC-Sim.
0.6466 for the orifice plate used in the experiments), d0
is the diameter of the orifice plate, and ˙zris the relative
velocity between the water column and the buoy. Note
that FP T O acts on both the buoy and the water column,
but with opposite sign.
The NLFK calculations in WEC-Sim (NLFKs) are
performed based on a panel mesh, summing up all
the contribution of each mesh panel. The mesh used
in this study, shown in Fig. 3, has been used for both
the NLFK calculation and computation of FD hydro-
dynamic coefficients via the boundary element method
(BEM) software WAMIT [25]. The computational time
and accuracy of the NLFKsapproach are highly de-
pendent on the number and size of panels, respectively.
The mesh considered in this study is composed of 5376
C. The NLFK4ALL mathematical model
While solving the integrals in (1) requires, in general,
mesh-based approaches, a computationally efficient ap-
proach is available for axisymmetric bodies, exploiting
cylindrical coordinates (̺, ϑ)to achieve an analytical
representation of the wetted surface:
ˆx(̺, ϑ) = f(̺) cos ϑ
ˆy(̺, ϑ) = f(̺) sin ϑ
ˆz(̺, ϑ) = ̺
, ϑ [π, π)̺[̺1, ̺2](5)
where f(̺)is a generic function of the vertical coor-
dinate ̺, describing the profile of revolution of the
axisymmetric body. Since it is convenient to define the
FK integrals in the body-fixed frame of reference, the
pressure field must be mapped from the global to the
body-fixed frame. After the required adjustments, the
integral in (1a), for example, becomes:
fF K =RT
Px, ˆy, ˆz)ndS =
P(̺, ϑ) (e̺×eϑ)d̺ dϑ,
Fig. 3. Mesh discretization of the Spar-buoy device, used for both
the BEM code for the linear hydrodynamic curves, and for the NLFK
computation in WEC-Sim.
where RΘis the rotation matrix from body- to world-
frame, e̺and eϑare the unit vector along ̺and ϑ,
respectively. Note that when internal patches (facing
the water column) are considered, the sign of the nor-
mal vector in (6) should be reversed. The integral in (6)
is solved numerically, using a 2D-quadrature scheme
for trapezoidal integration [26]. An open source Matlab
demonstration toolbox for definition and computation
of nonlinear FK forces for axisymmetric floaters is
available at [19].
Fig. 4 shows, for an arbitrary displacement of the
buoy and wave field, the configuration in both the
world-frame (on the left) and the body-frame (on the
right), and the corresponding mapping of the free sur-
face elevation. Note that the mesh-like representation
in Fig. 4 has a mere visualization purpose, since the
surfaces are described analytically and no mesh is
Note that the considered geometry is rather complex,
with several changes of cross-sectional area. Twelve
different patches can be identified, namely 5 cylindrical
sections, 4 conical sections, 2 quarters of torus, and
a disk for the inner piston. This increases the over-
all computational time, since each patch requires an
independent formulation, hence raising the number
of integrals to be computed. However, it is worth
remarking that equally considering all patches is likely
to be unnecessary, since some patches are relatively
Fig. 4. Example of displaced buoy, in the world frame (on the left) and body-fixed frame (on the right), with corresponding mapped wave
field and its intersection with the buoy.
small and/or so deep that the dynamic pressure has
already significantly decayed. Nevertheless, since the
purpose of this paper is comparison rather than com-
putational time minimization, no simplifying assump-
tion has been investigated.
While WEC-Sim naturally provides a framework for
time-simulations, the analytical-based NLFK computa-
tion algorithm (NLFKa) has been included into an in-
house code. The correctness of implementation is ver-
ified via comparison of time-domain simulations (TD)
for small (linear) waves, independently computed with
WEC-Sim and the in-house code, compared with linear
frequency domain (FD) simulations. Figure 5 shows
the resulting response amplitude operator (RAO) in
surge, heave, and pitch, for a representative set of
wave periods (Tw) at a wave height (Hw) of 0.25m. A
perfect overlap is obtained between all models, giving
confidence on the flawless implementation of the time-
domain models under linear conditions.
Successively, in order to ease the comparison be-
tween NLFKaand NLFKs, results are generated for a
numerical excitation force experiment, i.e. by fixing the
floater at the rest position, and sending monochromatic
waves, at different Twand Hw; the total excitation force
is measured, composed of NLFK and linear diffraction
components. While the proportionality between linear
forces and Hwis constant for all the the range of Hw,
nonlinear forces may show differences as Hwincreases.
A. Computational time
Quantifying the computational time has important
consequences on the applicability of either of the two
models for specific simulation purposes, ranging from
power production, response and load assessment, to
control and optimization. However, since both accu-
racy and computational effort depend on different
setup parameters for each model, a comprehensive
sensitivity analysis should be performed in order to
express a definitive, robust and solid statement on the
comparative performance.
Calculations are performed on a single core of a
standard laptop, with processor Intel(R) Core (TM)
i7-7500U CPU @ 2.70GHz and 8GB RAM. Since the
overall computational effort depends on the time-
advancing scheme and time step size (δt), the compu-
tational time is hereafter referred to a single time step
execution of the numerical excitation force experiment
(tCP U ). WEC-Sim results in a required 6 ms per δt,
which seems satisfactory. However, the excitation force
experiment is favourable to mesh-based approaches,
since the mesh is not displaced at each time step,
which may increase the computational time. Moreover,
with the current mesh discretization, the accuracy of
NLFKsis questionable in surge and pitch for longer
waves, as further discussed in Sect. III-B; therefore, a
more refined mesh should be tested, implying higher
computational burden. Nevertheless, it can be assumed
with confidence that the computational time for NLFKs
remains, at worst, below the tens of milliseconds per δt,
which is faster than previous mesh-based approaches
suggested in the literature, e.g. [15].
Conversely to the NLFKsapproach, the considered
case study of the Spar-buoy is particularly disadvan-
tageous to the NLFKaapproach. In fact, the compu-
tational time is linearly proportional to the number
of geometrical sections considered; therefore, NLFK
force computation for the Spar-buoy, composed of 12
patches, is likely to be about 3 times longer than the
time required for a simpler floater composed of just 4
patches (for example 2 cylinders connected via a trun-
cated cone). Nevertheless, the resulting computational
time is of about 3 ms per δt.
B. NLFK results
Figure 6 shows a representative time trace of the
total excitation force (Froude-Krylov and diffraction),
using the linear model (LFK) as a reference benchmark,
and the nonlinear models for comparison. Since they
5 10 15
0.6 WAMIT (FD)
5 10 15
5 10 15
Fig. 5. Response amplitude operators (RAO) in surge (left), heave (middle), and pitch (right), using linear frequency domain (FD) data, and
time-domain (TD) models with a small wave (height of 0.25m).
0 0.5 1
0 0.5 1
1.5 106
0 0.5 1
1.5 108
Fig. 6. Time trace over a single wave period of the total excitation force at Twof 7 s and Hwof 4 m.
are different approaches to compute the same quantity,
they should overlap. In Fig. 6 a short wave Twof 7 s
is considered, with Hwof 4 m, hence very steep and
nonlinear. It can be remarked that NLFKsand NLFKa
agree well, particularly in heave. Differences between
linear and nonlinear forces are negligible in surge and
pitch, while are remarkable in heave.
Figure 7 shows a long wave, Twof 14 s, at the same
height of 4 m. On the one hand, the perfect match
between NLFK approaches is preserved in heave; on
the other hand, NLFKspresents irregular oscillation
in surge and pitch due to numerical inaccuracies,
while NLFKaremains smooth. This can be ascribed
to a static mesh-discretization of the wetted surface,
not sufficiently refined to describe variations of the
nonlinear forces smoothly. In fact, although very wob-
bly, the moving average of NLFKsfollows the NLFKa
trend closely. The numerical inaccuracies of NLFKs
are not present in heave because the portion of the
buoy intersecting the free surface elevation, i.e. where
the extension of wetted surface is varying in time, is
cylindrical, while all other sections are either fully-
submerged or fully-dry: the normals of a cylinder with
vertical axis are horizontal, hence perpendicular to the
heave direction, so they do no contribute to the build-
up of the total heave force; it follows that the total
vertical force depends on portions of the buoy that
are always fully-submerged, hence not affected by im-
proper definition of a mesh discretization. Conversely,
surge and pitch forces depend also on the top cylinder,
whose mesh quality influences the smoothness and
accuracy of the results.
Finally, a synthetic but comprehensive view on the
differences between linear and nonlinear excitation
force coefficient is provided in Fig. 8, where ˆ
Fex is
defined as the ratio between the force and the wave
height: in a linear case, all curves should overlap,
while differences may arise as nonlinearities increase.
In the plot, both the amplitude and the mean are
presented, where the amplitude is defined as the semi-
difference between peak and trough. In the linear case,
the amplitude is the same as the peak value, while
the mean is zero. Figure 8 shows that, as Hwincreases
(colour code), the mean diverges from zero, especially
in heave. Conversely, differences in the amplitude seem
0 0.5 1
0 0.5 1
0 0.5 1
Fig. 7. Time trace over a single wave period of the total excitation force at Twof 14 s and Hwof 4 m.
5 10 15
2.5 106
5 10 15
5 10 15
Fig. 8. Amplitude and mean values of the nonlinear excitation force at different wave heights. Dashed line corresponds to the linear case.
This paper implements two alternative methods for
the computation of nonlinear Froude-Krylov forces,
one using a mesh-based approach in the popular WEC-
Sim software, the other using a mesh-less analytical
formulation of the wetted surface and numerical inte-
gration. As a case study, the Spar-buoy floating oscillat-
ing water column is used. An overall good agreement
is found between the two alternative methods, cross-
checking the underlying mathematical frameworks and
implementations. However, the mesh-based approach
shows inaccuracies in surge and pitch force computa-
tion, likely due to an improper selection of the mesh
discretization. The computational time, although de-
pending on various parameters and simulation setups,
appear to be in favour of the analytical formulation
(twice as fast as the mesh-based approach), despite of a
complex shape definition, also providing accurate and
smooth results.
In the future, this trends should be evaluated with
a more thorough sensitivity analysis for the mesh and
incorporating the dynamics of floating buoys with, at
least, 3 degrees-of-freedom.
This research has received funding from the Eu-
ropean Research Council (ERC) under the European
Union’s Horizon 2020 research and innovation pro-
gram under Grant No. 832140.
[1] M. Penalba, G. Giorgi, and J. V. Ringwood, “Mathematical
modelling of wave energy converters: a review of nonlinear
approaches,” Renewable and Sustainable Energy Reviews, vol. 78,
pp. 1188–1207, 2017.
[2] Y.-H. Yu, K. Ruehl, J. V. Rij, N. Tom, D. Forbush, D. Ogden,
A. Keester, and J. Leon, “WEC-Sim v4.2,” 2020.
[3] J. Ringwood, F. Ferri, N. Tom, K. Ruehl, N. Faedo,
G. Bacelli, Y. H. Yu, and R. G. Coe, “The wave energy
converter control competition: Overview,” in Proceedings
of the International Conference on Offshore Mechanics and
Arctic Engineering - OMAE, vol. 10. American Society
of Mechanical Engineers (ASME), nov 2019. [Online].
[4] S. Sirnivas, Y. H. Yu, M. Hall, and B. Bosma, “Coupled
mooring analyses for the wec-sim wave energy converter
design tool,” in Proceedings of the International Conference on
Offshore Mechanics and Arctic Engineering - OMAE, vol. 6.
American Society of Mechanical Engineers (ASME), oct 2016.
[Online]. Available:
[5] Y. H. Yu, N. Tom, and D. Jenne, “Numerical analysis
on hydraulic power take-off for wave energy converter
and power smoothing methods,” in Proceedings of
the International Conference on Offshore Mechanics and
Arctic Engineering - OMAE, vol. 10. American Society
of Mechanical Engineers (ASME), sep 2018. [Online].
[6] J. Van Rij, Y. H. Yu, and R. G. Coe, “Design load analysis
for wave energy converters,” Proceedings of the International
Conference on Offshore Mechanics and Arctic Engineering - OMAE,
vol. 10, no. November 2019, 2018.
[7] A. M ´
erigaud, J.-C. Gilloteaux, and J. V. Ringwood, “A
nonlinear extension for linear boundary element methods
in wave energy device modelling,” in ASME 2012
31st International Conference on Ocean, Offshore and Arctic
Engineering, Rio de Janeiro, 2012, pp. 615–621. [Online].
[8] G. Giorgi and J. V. Ringwood, “Analytical representation of
nonlinear Froude-Krylov forces for 3-DoF point absorbing wave
energy devices,” Ocean Engineering, vol. 164, no. 2018, pp. 749–
759, 2018.
[9] H. Wang, A. Somayajula, J. Falzarano, and Z. Xie, “Develop-
ment of a Blended Time-Domain Program for Predicting the
Motions of a Wave Energy Structure,” Journal of Marine Science
and Engineering, vol. 8, no. 1, p. 1, dec 2019.
[10] G. Giorgi, R. P. F. Gomes, G. Bracco, and G. Mattiazzo, “Nu-
merical investigation of parametric resonance due to hydrody-
namic coupling in a realistic wave energy converter,” Nonlinear
Dynamics, 2020.
[11] K. R. Tarrant and C. Meskell, “Investigation on parametrically
excited motions of point absorbers in regular waves,” Ocean
Engineering, vol. 111, pp. 67–81, 2016.
[12] G. Giorgi, R. P. F. Gomes, G. Bracco, and G. Mattiazzo, “The
effect of mooring line parameters in inducing parametric reso-
nance on the Spar-buoy oscillating water column wave energy
converter,” Journal of Marine Science and Engineering, vol. 8, no. 1,
pp. 1–20, jan 2020.
[13] M. Penalba, A. M´
erigaud, J. C. Gilloteaux, and J. V. Ringwood,
“Influence of nonlinear Froude–Krylov forces on the perfor-
mance of two wave energy points absorbers,” Journal of Ocean
Engineering and Marine Energy, vol. 3, no. 3, pp. 209–220, 2017.
[14] C. Windt, N. Faedo, M. Penalba, F. Dias, and J. V. Ringwood,
“Reactive control of wave energy devices – the modelling
paradox,” Applied Ocean Research, vol. 109, p. 102574, 2021.
[Online]. Available:
[15] J.-C. Gilloteaux, “Mouvements de grande amplitude d’un corps
flottant en fluide parfait. Application ´
a la recuperation de
l’energie des vagues,” Ph.D. dissertation, Ecole Centrale de
Nantes-ECN, 2007.
[16] M. Lawson, Y. H. Yu, A. Nelessen, K. Ruehl, and C. Michelen,
“Implementing nonlinear buoyancy and excitation forces
in the WEC-SIM wave energy converter modeling tool,”
in Proceedings of the International Conference on Offshore
Mechanics and Arctic Engineering - OMAE, vol. 9B. American
Society of Mechanical Engineers (ASME), oct 2014. [Online].
[17] G. Giorgi and J. V. Ringwood, “Comparing nonlinear hydrody-
namic forces in heaving point absorbers and oscillating wave
surge converters,” Journal of Ocean Engineering and Marine En-
ergy, vol. 4, no. 1, pp. 25–35, 2018.
[18] A. M´
erigaud and J. V. Ringwood, “A nonlinear frequency-
domain approach for numerical simulation of wave energy
converters,” IEEE Transactions on Sustainable Energy, vol. 9, no. 1,
pp. 86–94, 2018.
[19] G. Giorgi, “Nonlinear Froude-Krylov Matlab demonstration
toolbox,” 2019.
[20] G. Giorgi, R. P. Gomes, J. C. Henriques, L. M. Gato, G. Bracco,
and G. Mattiazzo, “Detecting parametric resonance in a floating
oscillating water column device for wave energy conversion:
Numerical simulations and validation with physical model
tests,” Applied Energy, vol. 276, oct 2020.
[21] R. Gomes, J. Henriques, L. Gato, and A. Falc˜
ao, “Time-domain
simulation of a slack-moored floating oscillating water column
and validation with physical model tests,” Renewable Energy,
vol. 149, pp. 165–180, apr 2020.
[22] G. Rinaldi, J. C. Portillo, F. Khalid, J. C. Henriques,
P. R. Thies, L. M. Gato, and L. Johanning, “Multivariate
analysis of the reliability, availability, and maintainability
characterizations of a Spar–Buoy wave energy converter
farm,” Journal of Ocean Engineering and Marine Energy,
vol. 4, no. 3, pp. 199–215, aug 2018. [Online]. Available:
[23] F. Correia da Fonseca, R. Gomes, J. Henriques, L. Gato, and
A. Falc˜
ao, “Model testing of an oscillating water column spar-
buoy wave energy converter isolated and in array: Motions and
mooring forces,” Energy, vol. 112, pp. 1207–1218, 2016.
[24] G. Giorgi and J. V. Ringwood, “Relevance of pressure field
accuracy for nonlinear Froude–Krylov force calculations for
wave energy devices,” Journal of Ocean Engineering and Marine
Energy, vol. 4, no. 1, pp. 57–71, 2018.
[25] I. WAMIT, “WAMIT User Manual,” 2019.
[26] L. F. Shampine, “Matlab program for quadrature in 2D,” Applied
Mathematics and Computation, vol. 202, no. 1, pp. 266–274, 2008.
ResearchGate has not been able to resolve any citations for this publication.
Full-text available
The implementation of energy maximising control systems (EMCSs) in wave energy converter (WEC) devices is an important step towards commercially viable operation of WECs. During the design stage of such EMCSs, linear hy-drodynamic models are commonly used and are, in fact, the most viable option due to the real-time computational requirements of optimisation routines associated with energy-maximising optimal control techniques. However, the objective function of EMCSs, i.e. maximising the generated power by exaggerating WEC motion, inherently violates the underlying assumption of the linear hydrodynamic control design models, i.e. small amplitude device motion (compared to the device dimensions). Consequently, the linear models, used as a basis for EMCSs, in fact conspire to violate the very assumption upon which they were built-hence leading to a modelling paradox. It is important to evaluate WEC controllers in realistic physical or numerical environments, to gain knowledge of the disparity between the performance prediction from the EMCS design and performance evaluation models. This paper presents a comprehensive assessment of the performance prediction by a linear and non-linear hydrodynamic model of three different EMCSs, implemented in two different WEC structures, in an attempt to quantify the severity of this modelling disparity, or paradox.
Full-text available
The wave energy sector has faced enormous technological improvements over the last five decades, however, due to the complexity of the hydrodynamic processes, current numerical models still have limitations in predicting relevant phenomena. In particular, floating spar-type wave energy converters are prone to large undesirable roll and pitch amplitudes caused by a dynamic instability induced by parametric resonance. Detecting this phenomenon accurately is essential as it impacts drastically on power extraction, structural loads and mooring forces. This paper presents the validation of results from a numerical model, capable of detecting parametric resonance, using experimental data. Experiments were carried out for a scaled model of the Spar-buoy OWC (Oscillating Water Column) device at a large ocean basin. The buoy uses a slack-mooring system attached to the basin floor. The scaled turbine damping effect is simulated by a calibrated orifice plate. Two different buoy draft configurations are considered to analyse the effect of different mass distributions. The numerical model considers the nonlinear Froude-Krylov forces, which allows it to capture complex hydro-dynamic phenomena associated with the six-degree-of-freedom motion of the buoy. The mooring system is simulated through a quasi-static inelastic line model. Real fluid effects are accounted for through drag forces based on the Morison's equation and determined from experimental data. The comparison of results from regular wave tests shows good agreement, including when parametric resonance is detected. Numerical results show that parametric resonance can produce a negative impact on power extraction efficiency up to 53%.
Full-text available
Representative models of the nonlinear behavior of floating platforms are essential for their successful design, especially in the emerging field of wave energy conversion where nonlinear dynamics can have substantially detrimental effects on the converter efficiency. The spar buoy, commonly used for deep-water drilling, oil and natural gas extraction and storage, as well as offshore wind and wave energy generation, is known to be prone to experience parametric resonance. In the vast majority of cases, parametric resonance is studied by means of simplified analytical models, considering only two degrees of freedom (DoFs) of archetypical geometries, while neglecting collateral complexity of ancillary systems. On the contrary, this paper implements a representative 7-DoF nonlinear hydrodynamic model of the full complexity of a realistic spar buoy wave energy converter, which is used to verify the likelihood of parametric instability, quantify the severity of the parametrically excited response and evaluate its consequences on power conversion efficiency. It is found that the numerical model agrees with expected conditions for parametric instability from simplified analytical models. The model is then used as a design tool to determine the best ballast configuration, limiting detrimental effects of parametric resonance while maximizing power conversion efficiency.
Full-text available
Although it is widely accepted that accurate modeling of wave energy converters is essential for effective and reliable design, it is often challenging to define an accurate model which is also fast enough to investigate the design space or to perform extensive sensitivity analysis. In fact, the required accuracy is usually brought by the inclusion of nonlinearities, which are often time-consuming to compute. This paper provides a computationally efficient meshless nonlinear Froude-Krylov model, including nonlinear kinematics and an integral formulation of drag forces in six degrees of freedom, which computes almost in real-time. Moreover, a mooring system model with three lines is included, with each line comprising of an anchor, a jumper, and a clump weight. The mathematical model is used to investigate the highly-nonlinear phenomenon of parametric resonance, which has particularly detrimental effects on the energy conversion performance of the spar-buoy oscillating water column (OWC) device. Furthermore, the sensitivity on changes to jumper and clump-weight masses are discussed. It is found that mean drift and peak loads increase with decreasing line pre-tension, eventually leading to a reduction of the operational region. On the other hand, the line pre-tension does not affect power production efficiency, nor is it able to avoid or significantly limit the severity of parametric instability.
Full-text available
Traditional linear time-domain analysis is used widely for predicting the motions of floating structures. When it comes to a wave energy structure, which usually is subjected to larger relative (to their geometric dimensions) wave and motion amplitudes, the nonlinear effects become significant. This paper presents the development of an in-house blended time-domain program (SIMDYN). SIMDYN’s “blend” option improves the linear option by accounting for the nonlinearity of important external forces (e.g., Froude-Krylov). In addition, nonlinearity due to large body rotations (i.e., inertia forces) is addressed in motion predictions of wave energy structures. Forced motion analysis reveals the significance of these nonlinear effects. Finally, the model test correlations examine the simulation results from SIMDYN under the blended option, which has seldom been done for a wave energy structure. It turns out that the blended time-domain method has significant potential to improve the accuracy of motion predictions for a wave energy structure.
Full-text available
This document purports to describe in detail the mathematical framework for a computationally efficient computation of nonlinear Froude-Krylov forces (NLFK) in 6 degrees of freedom (DoFs) for axisymmetric floating objects. Additionally, this document also acts as a reference manual to a set of Matlab scripts, forming a demonstration toolbox to show the capabilities of the NLFK approach and provide an easy, operative, and ready-to-use implementation of the method. The toolbox is openly available at [1]. This document and the toolbox are licensed with a Creative-Commons-By-Attribution-Share-Alike (CC-BY-SA) license [2]. Note that this is the first version of the toolbox, so any feedback and potential corrections are welcome. Moreover, the user is highly invited to contact the author for any doubt, ideas or suggestions, deeper investigation, higher-complexity problems, and eventually collaboration. Finally, note that this toolbox is the precursor of an open source software, coded in a lower-level coding language than Matlab, hence much faster, which will be virtually shared by mid 2021.
Full-text available
Conference Paper
This study demonstrates a systematic methodology for establishing the design loads of a wave energy converter. The proposed design load methodology incorporates existing design guidelines, where they exist, and follows a typical design progression; namely, advancing from many, quick, order-of-magnitude accurate, conceptual stage design computations to a few, computationally intensive, high-fidelity, design validation simulations. The goal of the study is to streamline and document this process based on quantitative evaluations of the design loads’ accuracy at each design step and consideration for the computational efficiency of the entire design process. For the wave energy converter, loads, and site conditions considered, this study demonstrates an efficient and accurate methodology of evaluating the design loads.
The development of devices for extracting wave energy from the ocean is largely supported by numerical models, as they allow the simulation of different configurations without the large costs of tank testing. From the different available options, time-domain models offer a very good combination between accuracy, flexibility and computational time. They allow the incorporation of non-linearities from power take-off systems, mooring lines, sophisticated control techniques and other relevant hydrodynamic effects. In this paper, we present a time-domain model to simulate the dynamics and power performance of a slack-moored Spar-buoy OWC (Oscillating Water Column) wave energy converter. The model considers linear hydrodynamics, mean drift forces, viscous drag effects and air compressibility inside the OWC chamber. The mooring system is simulated using a quasi-static approach. The floating structure is defined as a rigid body with six degrees of freedom, whereas the OWC free surface is assumed flat. The converter motion and power extraction from regular and irregular wave simulations are compared with experimental results from small-scale model tests in a wave channel. Numerical results show good agreement with experimental data except when parametric resonance is observed and near the channel cut-off frequencies.
Conference Paper
Over the past two years, a wave energy converter control systems competition (WECCCOMP) has been in progress, with the objective of comparing different wave energy converter (WEC) control paradigms on a standard benchmark problem. The target system is a point absorber, corresponding to a single float with an absolute reference, of the WaveStar WEC prototype. The system was modelled in WEC-Sim, with the hydrodynamic parameters validated against tank test data. Competitors were asked to design and implement a WEC control system for this model, with performance evaluated across six sea states. The evaluation criteria included a weighted combination of average converted power, peak/average power, and the degree to which the system physical constraints were exploited or temporarily exceeded. This paper provides an overview of the competition, which includes a comparative evaluation of the entries and their performance on the simulation model. It is intended that this paper will act as an anchor presentation in a special session on WECCCOMP at OMAE 2019, with other papers in the special session contributed by the competitors, describing in detail the control algorithms and the results achieved over the various sea states.
Conference Paper
One of the primary challenges for wave energy converter (WEC) systems is the fluctuating nature of wave resources, which require the WEC components to be designed to handle loads (i.e., torques, forces, and powers) that are many times greater than the average load. This approach requires a much greater power take-off (PTO) capacity than the average power output and indicates a higher cost for the PTO. Moreover, additional design requirements, such as battery storage, are needed, particularly for practical electrical grid connection, and can be a problem for sensitive equipment (e.g., radar, computing devices, and sensors). Therefore, it is essential to investigate potential methodologies to reduce the overall power fluctuation while trying to optimize the power output from WECs. In this study, a detailed hydraulic PTO model was developed and coupled with a time-domain hydrodynamics model (WEC-Sim) to evaluate the PTO efficiency for WECs and the trade-off between power output and fluctuation using different power smoothing methods, including energy storage, pressure relief mechanism, and a power-based setpoint control method. The study also revealed that the maximum power fluctuation for WECs can be significantly reduced by one order of magnitude when these power smoothing methods are applied.