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

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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.
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1
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-
Sim, NLFK4ALL.
I. INTRODUCTION
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: giuseppe.giorgi@polito.it).
Department of Mechanical and Manufacturing, Mondragon Uni-
versity, Loramendi 4 Apdo. 23, 20500, Arrasate, Spain (e-mail: mpe-
nalba@mondragon.edu).
IDMEC, Instituto Superior T´
ecnico, Universidade de Lis-
boa, Av. Rovisco Pais 1, 1049-001 Lisboa, Portugal (e-mail:
ruigomes@tecnico.ulisboa.pt).
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
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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
2
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.
II. THE SPAR-BUO Y OWC DE VI CE
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
(Sw(t)):
fF K =fg+ZZ
Sw(t)
pndS, (1a)
τF K =rg×fg+ZZ
Sw(t)
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
as:
p(x, z, t) = ρg z+acos (ωt kx)cosh (k(z+h))
cosh(kh),
(2)
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]:
z=hz+h
η+hh(3)
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
a
π2C2
dd4
0
˙zr|˙zr|(4)
where ρais the air density, Aais the cross-sectional area
of the air chamber, Cdis the discharge coefficient (Cd=
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FIRST-AUTHOR-SURNAME et al.: PAPER-TITLE 3
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
panels.
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
Θfg+ZZ
Sw(t)
Px, ˆy, ˆz)ndS =
=RT
Θfg+
π
Z
π
̺2
Z
̺1
P(̺, ϑ) (e̺×eϑ)d̺ dϑ,
(6)
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
needed.
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
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4
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.
III. RESULTS
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
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FIRST-AUTHOR-SURNAME et al.: PAPER-TITLE 5
5 10 15
0
0.2
0.4
0.6 WAMIT (FD)
IN-HOUSE (TD)
WECSIM (TD)
5 10 15
0
1
2
3
5 10 15
0
0.5
1
1.5
2
2.5
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
-5
0
5106
LFK
NLFKs
NLFKa
0 0.5 1
-1.5
-1
-0.5
0
0.5
1
1.5 106
0 0.5 1
-1.5
-1
-0.5
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
negligible.
51971-
6
0 0.5 1
-3
-2
-1
0
1
2
3106
LFK
NLFKs
NLFKa
0 0.5 1
-3
-2
-1
0
1
2
3106
0 0.5 1
-4
-2
0
2
4107
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
-0.5
0
0.5
1
1.5
2
2.5 106
0.5
1
1.5
2
2.5
3
3.5
4
5 10 15
-2
0
2
4
6
8
10
12
105
0.5
1
1.5
2
2.5
3
3.5
4
5 10 15
-1
0
1
2
3
4
5
6107
0.5
1
1.5
2
2.5
3
3.5
4
Fig. 8. Amplitude and mean values of the nonlinear excitation force at different wave heights. Dashed line corresponds to the linear case.
IV. CON CL US IO NS
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.
ACKNOWLEDGEMENT
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.
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