Energy 296 (2024) 130888
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Energy
journal homepage: www.elsevier.com/locate/energy
High-fidelity numerical modelling of a two-WEC array with accurate
implementation of the PTO system and control strategy using DualSPHysics
Nicolas Quartier a ,∗, Timothy Vervaet a , Gael Verao Fernandez a , José M. Domínguez b ,
Alejandro J.C. Crespo b , Vasiliki Stratigaki a , Peter Troch a
a
Ghent University, Technologiepark 60, 9052 Ghent, Belgium
b Environmental Physics Laboratory (EPhysLab), CIM-UVIGO, Universidade de Vigo, Spain
ARTICLE
INFO
Keywords:
Smoothed particle hydrodynamics
DualSPHysics
Power take-off system
Project chrono
WECfarm project
Heaving wave energy converter
ABSTRACT
In order for Wave Energy Converters (WECs) to become economically viable they will have to be installed
closely together in so-called WEC arrays. When WECs are placed closely together they influence each other’s
motion, which is challenging to implement in a numerical model. The accurate modelling of WECs in WEC
arrays is crucial for their further development into cost-effective renewable energy converters. This paper
presents the study of numerical modelling of an array of two heaving WECs (a two-WEC array) placed in close
proximity of each other in order to increase their interaction effect. The numerical model uses DualSPHysics,
a Smoothed Particle Hydrodynamics (SPH) based software to model wave-WEC interaction, coupled to Project
Chrono for the modelling of the Power Take-Off (PTO) system and the friction. First, a series of numerical
system identification tests are simulated: a free-decay test of an isolated WEC, radiation tests and an excitation
test. During the radiation test one WEC is moved by an external force with a chirp-up noise signal, while
the other WEC is kept still. This allows the computation of the impedance matrix, and thereby also the
hydrodynamic coefficients of the two-WEC array, which has not been done before in SPH. Results were
validated with experimental data from the WECfarm test campaign. Secondly, a series of numerical dynamic
tests are simulated of the two-WEC array in regular and irregular waves. In the experimental test campaign a
friction force caused by the motor and gearbox was counteracting the WEC’s motion, so a friction compensation
(FC) force was added to cancel out this friction. This friction and FC force was numerically modelled by adding
a novel implementation in Project Chrono. Furthermore, a novel global control strategy was implemented using
Project Chrono, in which the PTO force on each WEC depends on the velocity of both WECs. Overall, good to
very good agreement was found in terms of heave and free surface elevation when comparing numerical to
experimental results.
1. Introduction
Wave energy is a potential source of clean electricity that can make
a significant contribution to the de-carbonization of the world’s energy
supply. Wave energy is considered as the most concentrated and least
variable form of renewable energy that can generate power up to 90%
of the time, compared to 25% for solar and wind devices [1]. A wide
variety of devices to harvest wave energy, known as wave energy
converters (WECs), have been designed over the last decades [1].
However, none of them have yet achieved the level of technological
development required to become economically viable and generate
electricity commercially [2,3]. In the current research focus is laid
on the heaving point absorber, one of the most investigated types
of WECs [4]. Such a heaving point absorber typically consist of a
floating buoy moved by the waves and connected to a Power TakeOff (PTO) system, which converts the WEC’s movement into electricity.
In order to assess the performance and survivability of WECs, which
are necessary for exploiting wave energy, the related wave-induced
hydrodynamic forces and WEC motions have to be examined. Physical
experiments are essential and very useful, nevertheless, they have very
high costs and require a high level of expertise [5]. On the other hand,
numerical methods have become very popular in last years [6], mainly
due to the unprecedented growth of the computational resources available. A complete review on the numerical methods to simulate the
hydrodynamic response of point absorbers can be found in [7].
∗ Corresponding author.
E-mail addresses: Nicolas.Quartier@UGent.be (N. Quartier), Timothy.Vervaet@UGent.be (T. Vervaet), Gael.VeraoFernandez@UGent.be (G.V. Fernandez),
jmdominguez@uvigo.es (J.M. Domínguez), alexbexe@UVigo.es (A.J.C. Crespo), Vicky.Stratigaki@UGent.be (V. Stratigaki), Peter.Troch@UGent.be (P. Troch).
https://doi.org/10.1016/j.energy.2024.130888
Received 25 April 2023; Received in revised form 22 January 2024; Accepted 1 March 2024
Available online 14 March 2024
0360-5442/© 2024 Elsevier Ltd. All rights reserved.
Energy 296 (2024) 130888
N. Quartier et al.
revealing significant hydrodynamic interaction between the WECs and
the floating wind turbine. In the work of [46] the impact of long-term
variations in wave resource around Iceland on the power generation of
a WEC array of 45 WECs was studied. In all of these studies however,
the WEC’s motion and power absorption is computed based on linear
potential flow theory, thereby assuming low amplitude motions.
In this work a two-WEC array is modelled in DualSPHysics, where
the two WECs are placed in close proximity to each other. As validation, data from the experimental test campaign in Aalborg University
is used, resulting from the work of [47]. Furthermore, an accurate
implementation for the representation of the Coulomb and viscous
based drivetrain (motor, gearbox and rack and pinion) friction model
present in the experimental tests is added using the DualSPHysics–
Project Chrono coupling [22]. The novelty of this work lies in the
simulation of a WEC array in SPH, which to our knowledge has not been
done before. Furthermore, an accurate model of the friction present
in the experiments is implemented, and a new global control strategy
to maximize the power absorption of the WEC array was numerically
added. The benefit of using SPH, and specifically DualSPHysics, for
WEC array modelling lies in the fact that this method does not assume
small amplitudes, which leads to underestimations of the induced force
and overestimations of the average absorbed power. Also, nonlinear
friction forces, PTO system forces or global control strategies can be implemented in DualSPHysics in a straightforward way. The simulations
of the WEC array allow the calculation of the impedance matrix, which
has not yet been performed before in SPH. Moreover, this work shows
that DualSPHysics has the ability to not only accurately compute the
WEC’s motion, but also the free surface elevation near the WEC array.
The paper is organized as follows: Section 2 describes the basic
theoretical principles of SPH and its implementation in DualSPHysics;
Section 3 gives an overview of the experimental campaign carried out
at the University of Aalborg (AAU) and used for validation in the
current research; Sections 4 and 5 show the main results obtained
using DualSPHysics and compared to experimental data from [47], and
finally Section 6 presents an overview of the conclusions.
Most studies concerning WECs employ potential flow theory based
on the linearized form of the Navier–Stokes equations. Applying linear
potential flow theory allows numerical modelling of WECs in the time
or frequency domain [8], enabling fast calculations of the WEC’s motion. This theory however needs to assume small amplitude oscillations
of the WEC and the fluid to be incompressible, inviscid and with
an irrotational motion. These assumptions lead to an underestimation of the wave-induced forces on WECs under highly-nonlinear sea
states [9]. These restrictive assumptions are not applied in high-fidelity
models such as the CFD (computational fluid dynamics) methods.
The most commonly used CFD methods in hydrodynamics are meshbased. Different point absorbers have been studied numerically using
these methods in [10,11]. Despite being very robust mathematically
and computationally, mesh-based methods face important challenges
when capturing the free surface and the rapidly-evolving nonlinearities. Due to their ability to overcome these drawbacks, meshless CFD
methods have gained attention during the last years, the smoothed
particle hydrodynamics (SPH) method being the most widely used [12–
14]. In contrast with the mesh-based methods, in SPH the fluid is
discretized in a set of points, named particles, that move with the velocity calculated from the Navier–Stokes equations carrying all physical
properties with them [15]. Its meshless lagrangian approach makes
the SPH method a very interesting alternative when simulating freesurface flows with wave–structure interaction, such as the case at
hand. A complete overview of the use of SPH in ocean and coastal
engineering is given in [16]. In [17] the hydrodynamic response of a
point absorber was computed using SPH and a finite volume solver.
The works of [18,19] exploited the capabilities of the SPH to model
nonlinearities to study the interaction between a point absorber WEC
and extreme waves. In this research, the open-source software DualSPHysics [20] (available at www.dual.sphysics.org) is employed to
model the motion of two heaving WECs placed in close proximity
to each other. Furthermore, DualSPHysics has been coupled [21,22]
with Project Chrono [23], enabling the effect of the PTO system to be
included in the simulations. This software has proven to be a valuable
tool in the modelling of wave–structure interaction in general and of
floating WECs in particular — see [3,19,24–27].
The application of SPH to industrial problems is one of the SPHERIC
Grand Challenges [28] and, in particular, the simulation of WECs is
one of the most promising application fields [29]. The studies of [30–
32] were pioneers presenting SPH simulations of oscillating wave surge
devices. The SPH method has also been applied for the modelling of
Oscillating Water Column (OWC) WECs [33–37]. The works of [18,38]
were the first ones to deal with extreme waves interacting with pointabsorbers using SPH. More recently, heaving WECs were simulated
in DualSPHysics in [3,19,27]. In [39] a linear damping and Coulomb
damping PTO system was applied to a heaving WEC, in [3] a linear
spring–damper system was used as PTO system for a heaving WEC,
in [36] a PTO system of an OWC WEC was modelled, and in [40] a
moored heaving WEC with a linear spring–damper PTO system was
modelled.
In these cases however, only a single WEC was considered. In order
for WECs to become economically viable they will have to be installed
in large numbers grouped together in WEC arrays [2,41]. When WECs
are positioned close to each other they can affect each other’s motion
due to the modified wave field surrounding an oscillating WEC. It is
known that the type of PTO system also has an influence on these
surrounding modified wave fields, as studied in [42]. Furthermore,
the optimal PTO system configuration can depend on the motion of
a neighbouring WEC and therefore be different than the optimal PTO
configuration for an isolated WEC [43].
WEC arrays and the control of WEC arrays have been numerically
studied before: in the work of [44] the performance of an array of two
and three WECs was studied, showing an increased efficiency when
motion constraints are applied. Zhou et al. [45] studied multiple WECs
placed closely together around a floating wind turbine are studied,
2. Smoothed particle hydrodynamics—DualSPHysics
The DualSPHysics solver applies the SPH method, which is a meshless lagrangian method in which the fluid is discretized in a set of
particles for which the position, velocity, density and pressure are computed by solving the Navier–Stokes equations and by interpolation of
the values of neighbouring particles. The contribution depends on the
distance between the particle of interest and the neighbouring particles,
with 𝑑𝑝 the initial inter-particle distance, and on the Kernel function
𝑊 . This Kernel has an area of influence defined
by the smoothing
√
length ℎ, which is a function of 𝑑𝑝 (ℎ = 𝑐𝑜𝑒𝑓ℎ 3𝑑𝑝 in 3D) — see [20].
The mathematical description of the SPH method and DualSPHysics
specifically is based on the description given in [20].
2.1. Governing equations
The Navier–Stokes equations written in their SPH notation are
solved each timestep for each of the particles. The SPH-based continuity
and momentum equations are given in Eqs. (1) and (2), respectively.
In the following equations the velocity and density of particle 𝑎 is
calculated, with 𝑏 each of its neighbouring particles:
∑
𝑚
𝑑𝜌𝑎 ∑
=
𝑚𝑏 (𝒗𝒂 − 𝒗𝒃 )∇𝑎 𝑊𝑎𝑏 + 𝛿ℎ𝑐𝑎
𝜓𝑎𝑏 ⋅ ∇𝑎 𝑊𝑎𝑏 𝑏
(1)
𝑑𝑡
𝜌𝑏
𝑏
𝑏
)
(
∑
𝑑𝒗𝒂
𝑃𝑏 + 𝑃𝑎
=−
𝑚𝑏
∇𝑎 𝑊𝑎𝑏 + 𝜞 + 𝒈
(2)
𝑑𝑡
𝜌𝑏 𝜌𝑎
𝑏
with 𝑚𝑏 the mass of particle 𝑏, 𝒗𝒃 the velocity vector of particle 𝑏, 𝑃𝑏
the pressure of particle 𝑏, 𝜌𝑏 the density of particle 𝑏, 𝜞 the dissipation
term and 𝒈 the gravity acceleration. 𝑊𝑎𝑏 represents the Kernel function
and depends on the distance between particles 𝑎 and 𝑏. In this work a
2
Energy 296 (2024) 130888
N. Quartier et al.
Quintic Kernel [48] is applied, since this type of Kernel is well suited for
general free surface problems — see [49]. One of the main advantages
of the Quintic Kernel is that the tensile instability that appears using
other kernels such as the Cubic Spline, is avoided using the kernel
adopted in this work. More information can be found in the work
of [49].
The diffusion term introduced in the right hand side of the continuity equation (Eq. (1)) acts as a numerical noise filter thereby improving
the numerical stability and smoothing the density and pressure field —
see [50–52]. In this paper the recently proposed density diffusion term
of [53] is applied, since it has proven to produce more accurate results
for the pressure field near boundaries while keeping the computational
cost limited. The magnitude of the diffusion term is controlled with the
parameter 𝛿, set to 0.1 in this study. The artificial dissipation term 𝜓𝑎𝑏
takes the following form:
𝐷 𝒓𝒂𝒃
(3)
𝜓𝑎𝑏 = 2(𝜌𝐷
𝑏 − 𝜌𝑎 )‖
‖
‖𝒓𝒂𝒃 ‖
in [58]. For each boundary particle of mDBC a 𝑔ℎ𝑜𝑠𝑡 𝑛𝑜𝑑𝑒 is mirrored
into the fluid. Boundary particles then receive the properties of the
fluid at the position of the ghost node. The density is calculated using a
first order consistent SPH interpolation, as proposed in [59]. The mDBC
method has proven to show results with more realistic physical values
for the pressure at the boundaries and a significant reduction in the
size of the gap between fluid and boundary without a significant extra
computational cost. Throughout this paper mDBC will be applied for
the modelling of the WECs.
2.3. Floating bodies
DualSPHyics has the capability to accurately simulate fluid-driven
objects as described and validated in [26,60], which will be used
extensively in this paper. The force per unit mass acting on one boundary particle 𝑘 of the floating body is calculated by summing up the
contributing forces exerted on this boundary particle 𝑘 by fluid particles
𝑎 within the compact support of the Kernel:
∑
𝒇 𝑘𝑎
(8)
𝒇𝑘 =
where 𝒓𝒂𝒃 = 𝒓𝒂 −𝒓𝒃 with 𝒓𝒌 the position of particle 𝑘 and 𝜌𝐷 the dynamic
density, equal to the difference of the total (𝜌𝑇 ) and hydrostatic (𝜌𝐻 )
density: 𝜌𝐷 = 𝜌𝑇 − 𝜌𝐻 .
𝜞 represents the dissipation term, which is either the artificial
viscosity scheme proposed in [15] or the laminar viscosity scheme with
a Sub-Particle Scale (SPS) large eddy simulation model as proposed
in [54,55] (Eq. (5)). In this research the laminar viscosity scheme with
SPS model is used, with the dissipation term 𝜞 in Eq. (2) written as:
𝜞 = 𝜈0 ∇2 𝒗𝒂 +
1
∇ ⋅ 𝜏𝑎𝑖𝑗
𝜌
Eq. (4) contains a laminar viscosity term written as:
∑
4𝜈0 𝒓𝒂𝒃 ⋅ ∇𝑎 𝑊𝑎𝑏
𝑚𝑏
𝜈0 ∇2 𝒗𝒂 =
(𝜌𝑎 + 𝜌𝑏 )(𝑟2𝑎𝑏 + 𝜂 2 )
𝑏
𝑎
with 𝒇 𝑘 the force per unit mass acting on boundary particle 𝑘 of the
floating body and 𝒇 𝑘𝑎 the force per unit mass acting on boundary
particle 𝑘 exerted by fluid particle 𝑎, calculated with Eq. (2). Once the
force acting on the floating body is computed, the body’s motion can
be determined assuming it is rigid:
∑
𝑑𝑽
=
𝑚𝑘 𝒇 𝑘
(9)
𝑀
𝑑𝑡
𝑘
𝑑𝜴 ∑
𝐼
𝑚𝑘 (𝒓𝑘 − 𝑹) × 𝒇 𝑘
(10)
=
𝑑𝑡
𝑘
(4)
(5)
with 𝑀 the body’s total mass, 𝐼 the moment of inertia, 𝑽 the velocity,
𝜴 the rotational velocity, 𝑹 the centre of mass, 𝑚𝑘 the mass of floating
boundary particle 𝑘 and 𝒓𝑘 the position of floating particle 𝑘. Eqs. (9)
and (10) are integrated over time in order to compute the velocity 𝒗𝑘
of the floating particle:
and a Sub-Particle Scale (SPS) model term written as:
∑ ⎛ 𝜏𝑎𝑖𝑗 − 𝜏𝑏𝑖𝑗 ⎞
1
⎟ ∇𝑖 𝑊
∇ ⋅ 𝜏𝑎𝑖𝑗 =
𝑚𝑏 ⎜
⎜ 𝜌𝑎 𝜌𝑏 ⎟ 𝑎 𝑎𝑏
𝜌
𝑏
⎝
⎠
(6)
with 𝒗𝒌 the velocity vector of particle k, 𝜂 2 = 0.1 h2 , and 𝜈0 is the
kinematic viscosity of the fluid (10−6 m2 ∕s for water). Since the fluid
is weakly-compressible in DualSPHysics, an equation of state is used to
calculate the fluid pressure as function of the density instead of solving
a Poisson-like equation:
[( )
]
𝑐 2 𝜌0
𝜌𝑎 𝛾
−1
(7)
𝑃𝑎 = 𝑠
𝛾
𝜌0
𝒗𝑘 = 𝑽 + 𝜴 × (𝒓𝑘 − 𝑹)
(11)
2.4. Coupling DualSPHysics–Project Chrono
Project Chrono is an open-source software package that enables
the numerical modelling of mechanical constraints and collision between objects [23]. It has recently been successfully coupled to DualSPHysics [22] and can be used in this case for the modelling of the
PTO system of a WEC. Already implemented in the coupling was the
linear damping, which is applied here for the simulation with a linear
damping PTO system — see Eq. (12)
with 𝛾 = 7 the polytropic constant, 𝑐𝑠 the numerical speed of sound
and 𝜌0 the reference fluid density. The speed of sound 𝑐𝑠 is artificially
lowered such that a reasonable time step can be employed while
ensuring that density variations are kept lower than 1% during the
simulation.
𝐹𝑃 𝑇 𝑂,𝑙 (𝑡) = −𝐶𝑃 𝑇 𝑂,𝑙 𝑣(𝑡)
(12)
with 𝐶𝑃 𝑇 𝑂,𝑙 the linear PTO system damping coefficient and 𝑣(𝑡) the
WEC’s heave velocity. The DualSPHysics–Project Chrono coupling has
been extended with the Coulomb damping model in [39], applying the
force described in Eq. (13), with 𝐶𝑃 𝑇 𝑂,𝑐 the PTO damping coefficient
for a Coulomb damping PTO system.
2.2. Boundary conditions
In DualSPHysics the boundaries are described by a set of particles
for which the same equations (Eqs. (1) and (2)) as for fluid particles
are solved. However, the particles belonging to the boundary do not
move according to the forces acting on them: boundary particles remain either fixed or move according to a predefined movement. These
boundary conditions are called Dynamic Boundary Conditions (DBC)
and have the advantage of being able to deal with complex geometries
and being computationally efficient [56,57]. However, due to excessive
repulsive forces near the boundary between a structure and the fluid,
a gap appears of the order of magnitude of the smoothing length ℎ —
see [58]. Furthermore at this same boundary unphysical values of the
pressure are observed. These issues are dealt with in the modified DBC
(mDBC) implementation recently added to DualSPHysics, as described
𝐹𝑃 𝑇 𝑂,𝑐 (𝑡) = −𝐶𝑃 𝑇 𝑂,𝑐 𝑠𝑖𝑔𝑛(𝑣(𝑡))
(13)
In this work the DualSPHysics–Project Chrono coupling is further extended by (i) adding an accurate implementation of the drivetrain
friction and (ii) adding a new control strategy scheme.
3. Experimental campaign
During the WECfarm project two heaving WECs were built and
tested in the wave basin of AAU, Denmark [47,61]. A realistic rendering
3
Energy 296 (2024) 130888
N. Quartier et al.
Fig. 1. Rendering of the ‘‘WECfarm’’ WEC.
Source: Figure adopted from [61].
of the WEC model and its components is shown in Fig. 1. Note however
that in the numerical simulations and results the positive heave is
defined opposite as the definition shown in Figs. 1 and 2, where
positive heave is defined when the WEC is below the still water level
(SWL). Among the tests are: system identification tests (radiation and
excitation tests), drivetrain friction assessment tests, free decay tests
and power absorption tests with PTO system and control strategies.
The hydrodynamic part of the WEC is an Acrylonitrile-ButadieneStyrene (ABS) thermofolded truncated cylindrical buoy with a Polymethylmethacrylate PMMA plate on top — see Fig. 1. The WEC buoy
is connected to guide shafts and a rack, the latter being connected to a
pinion. Three OAV (OAV Air Bearings, Princeton, NJ, USA) 40 mm air
bushings are installed to provide zero friction between the guide shafts
and bushings. A PTO system force 𝐹𝑃 𝑇 𝑂 is applied to the WEC buoy
with a Beckhoff Permanent-Magnet Synchronous Motor drive (PMSM)
connected with a rack and pinion via a gearbox with ratio 1:4. The
gearbox break-away torque 𝑇01 is expected to be around 0.20–0.40 N
m, with convergence towards lower values for a longer operational lifetime. The motor static friction 𝑀𝑅 is reported to be equal to 0.02 Nm.
As a result, the total Coulomb friction experienced by the WEC will
be mainly determined by 𝑇01 . Moreover, additional viscous friction
attributed to the drivetrain will occur. The presence of the gearbox
and motor induced a friction force acting on the WEC, therefore in
some tests a friction compensation force was added to compensate this
friction force. This friction compensation is also explained in Section 5.
The experimental test campaign was carried out in a water depth of
1.010 m. Tests were performed for an isolated WEC and for a twoWEC array where the two WECs were placed next to each other with
an interdistance of two diameters from centre to centre, equivalent
to 1.20 m. The motor drive receives real-time position and velocity
values from a single-turn absolute encoder. A detailed explanation of
the experimental test campaign is given in [47,61].
Eight wave gauges (WGs) were placed in the wave basin to measure
the free surface elevation (FSE) — see Fig. 3.
Table 1 gives an overview of the experimental test cases that were
numerically simulated and validated in the current work. The test case
id is kept the same as in the experimental test campaign [47]. Note
that laboratory scale is used in the present simulations, however SPH
can be also applied to full-scale modelling. The considerations about
Fig. 2. Dimensions of the WECfarm heaving WEC [m].
Source: Figure adopted from [61].
scaling and potential scale effects (such as surface tension effects, or
viscous scale effects) will be the same as the ones considered in physical
modelling or any other CFD tool.
4. System identification tests
A series of system identification (SID) tests are carried out in DualSPHysics for which the results are compared to experimental data.
Agreement between numerical and experimental tests is expressed in
terms of an index of agreement 𝑑𝑟 defined in Eq. (14):
⎧
𝑀𝐴𝐸
⎪1 − 𝑐𝑀𝐴𝐷
𝑑𝑟 = ⎨
𝑐𝑀𝐴𝐷
⎪ 𝑀𝐴𝐸 − 1
⎩
if 𝑀𝐴𝐸 ≤ 𝑐𝑀𝐴𝐷
(14)
if 𝑀𝐴𝐸 > 𝑐𝑀𝐴𝐷
with scaling factor c = 2. MAE is the mean absolute error defined by:
1 ∑|
|
𝑀𝐴𝐸 =
(15)
|𝑃𝑖 − 𝑂𝑖 |
𝑁
𝑖
and MAD the mean absolute deviation:
1 ∑|
|
𝑀𝐴𝐷 =
|𝑂𝑖 − 𝑂|
|
|
𝑁
𝑖
4
(16)
Energy 296 (2024) 130888
N. Quartier et al.
Fig. 3. Wave gauges layout during the experimental test campaign — dimensions in cm.
Source: Figure adopted from [47].
Table 1
Description of experimental test cases, numerically simulated in the current work.
Test case id
Test case description
Detailed description
Test 256
Test 254
Test 247
Test 282
Test 224
Test 225
Test 226
Test 238
Test 90
Test 186
Radiation test
Radiation test
Excitation test
Regular wave test
Regular wave test
Regular wave test
Regular wave test
Focussed wave test
Irregular wave test
Irregular wave test
WEC A imposed motion, WEC B fixed
Both WECs imposed motion
𝐻 = 0.16 m, 𝑇 = 1.5 s, WEC A fixed
𝐻 = 0.16 m, 𝑇 = 1.5 s, WEC A without motor
𝐻 = 0.16 m, 𝑇 = 1.5 s, both WECs with friction
𝐻 = 0.16 m, 𝑇 = 1.5 s, both WECs with FC
𝐻 = 0.16 m, 𝑇 = 1.5 s, both WECs with PTO system
𝐻𝑠 = 0.20 m, 𝑇𝑝 = 1.1 s, both WECs without FC and PTO system
𝐻𝑠 = 0.16 m, 𝑇𝑝 = 1.5 s, WEC A with FC and PTO system
𝐻𝑠 = 0.16 m, 𝑇𝑝 = 1.5 s, both WECs with FC and PTO system with global control
Table 2
Rating of index of agreement 𝑑𝑟 [63].
𝑑𝑟
Rating
0.90–1.00
0.80–0.90
0.70–0.80
0.50–0.70
0.30–0.50
(−1.00)–0.30
Excellent
Very good
Good
Reasonable/Fair
Poor
Bad
Table 3
Convergence study for a simulation of the free-decay test, for 16 s physical time with
the initial inter-particle distance 𝑑𝑝, number of particles 𝑛𝑝𝑎𝑟𝑡𝑖𝑐𝑙𝑒𝑠 , runtime and index of
agreement 𝑑𝑟 , run on a NVIDIA A100-SXM4.
with 𝑃𝑖 the predicted SPH output at time 𝑖, and 𝑂𝑖 te experimental
output at time 𝑖 [62]. A classification of index of agreement is given
in Table 2, as defined in [62,63].
Besides the index of agreement, two other non-dimensional error
estimators are given: the linear correlation coefficient 𝑅 and the error
in amplitude 𝐴. The linear correlation coefficient quantifies the error
in phase [26]:
1 ∑
𝑖 (𝑃𝑖 − 𝑃 )(𝑂𝑖 − 𝑂)
𝑁
𝑅=
(17)
𝜎𝑃 𝜎𝑂
√∑
√
√ 𝑖 𝑃2
𝐴 = √∑ 𝑖
(18)
2
𝑖 𝑂𝑖
𝑑𝑝 [m]
𝑛𝑝𝑎𝑟𝑡𝑖𝑐𝑙𝑒𝑠
Runtime [h]
𝑑𝑟
0.05
0.025
0.02
0.38E6
2.64E6
5.02E6
0.12
1.60
3.67
0.83
0.91
0.95
of the water, which means that bottom slamming will occur. This freedecay test with an isolated WEC is performed without attaching the
WEC to the motor. The only possible friction in the experiment could
come from the airbushings, but this friction was minimal. The initial
inter-particle distance 𝑑𝑝 for the free-decay test was 0.025 m, decided
after a convergence study where the agreement between the heave from
DualSPHysics and experiment was compared for three different interparticle distances — see Fig. 4 and Table 3. The index of agreement 𝑑𝑟
increases from 0.91 to 0.95 when reducing 𝑑𝑝 from 0.025 m to 0.02 m,
but to keep the computational runtime within reasonable limits, a 𝑑𝑝
of 0.025 m is chosen.
4.2. Radiation test
The closer R and A are to 1, the better the agreement between
DualSPHysics and experiment. Finally, the relative root square error
(RRSE) is calculated as well as:
√∑
√
√
(𝑃𝑖 − 𝑂𝑖 )2
𝑅𝑅𝑆𝐸 = √ ∑𝑖
(19)
2
𝑖 (𝑃𝑖 − 𝑃 )
During a first radiation test, Test 256 as defined in [47], 𝐹𝑃 𝑇 𝑂 is
imposed on one of the WECs, in this case on WEC A, in a still basin
without waves while WEC B is kept fixed. The numerical basin used
during the radiation tests has a width of 6 metres and has numerical
damping layers on all sides, as is shown in Fig. 5. A 𝑑𝑝 of 0.025 m was
used, based on the convergence test analysis (Fig. 4).
The velocity of WEC A and the force on WEC B caused by the radiated waves surrounding WEC A are measured, allowing the calculation
of the impedance matrix Z describing the relationship between the force
F̂ and velocity v̂ in the frequency domain:
𝑑𝑟 , 𝑅, 𝐴 and 𝑅𝑅𝑆𝐸 are summarized for all studied testcases in Table 5.
4.1. Free-decay test
During the free-decay test a single WEC (WEC A) was lifted 0.214 m
above the water and then released, while its heave motion 𝑧 was
measured. Note that during this test the WEC was lifted completely out
F̂ = Zv̂
5
(20)
Energy 296 (2024) 130888
N. Quartier et al.
Fig. 4. Convergence analysis for a free-decay test of 1 WEC.
Fig. 5. Top view of the numerical wave basin used in DualSPHysics for the radiation tests of the WEC array.
or:
[ ] [
𝐹̂𝑎
𝑍𝑎𝑎
=
𝐹̂𝑏
𝑍𝑏𝑎
][ ]
𝑍𝑎𝑏
𝑣̂ 𝑎
𝑍𝑏𝑏
𝑣̂ 𝑏
frequencies from 0.4 to 1.75 Hz are covered in the input 𝐹𝑃 𝑇 𝑂 . Note
that from Eq. (21) only the coefficient 𝑍𝑎𝑎 and 𝑍𝑏𝑎 can be determined,
since 𝑣̂ 𝑏 equals zero. To determine the remaining coefficients of the
impedance matrix a second test should be carried out during which
WEC A is kept still and WEC B is heaving. However, it is considered that
the impedance matrix is symmetrical, with 𝑍𝑏𝑏 = 𝑍𝑎𝑎 and 𝑍𝑏𝑎 = 𝑍𝑎𝑏 .
The index of agreement for the velocity of WEC A and the heave
force on WEC B is 0.87 and 0.82 respectively. All complex quantities
are obtained by calculating the discrete Fourier transform from their
real time series signal. Fig. 8 shows good agreement between the experimentally and numerically determined magnitude and phase of the
impedance. Furthermore, the magnitude and phase of the impedance
determined with the BEM solver Capytaine [64] (based on NEMOH,
(21)
with 𝐹̂𝑎 the complex PTO force imposed on WEC A, 𝐹̂𝑏 the complex
force acting on WEC B measured with loadcells, 𝑣̂ 𝑎 the complex velocity
of WEC A and 𝑣̂ 𝑏 the complex velocity of WEC B, which is in this specific
case equal to zero since WEC B is kept fixed. The input PTO force
𝐹𝑃 𝑇 𝑂 imposed on WEC A in both the experiment and DualSPHysics is a
chirp-up noise signal, which is a polychromatic sinusoidal signal with
a constant amplitude, covering a range of frequencies — see Figs. 6
and 7. This input 𝐹𝑃 𝑇 𝑂 signal shown in Fig. 6 is the input force before
application of the friction compensation. From Fig. 7 it can be seen that
6
Energy 296 (2024) 130888
N. Quartier et al.
Fig. 6. Radiation test 256 with a noise (chirp-up) signal.
Fig. 7. Frequency of the chirp-up input force signal 𝐹𝑃 𝑇 𝑂 over time in radiation test 256.
[65]) are shown. It proves that DualSPHysics is able to accurately
capture the interaction between both WECs.
From the impedance matrix Z the added mass and hydrodynamic
damping matrices can be deduced by taking the real and imaginary
part of the impedance:
Z(𝜔) = B(𝜔) + 𝑖(𝜔(M + A(𝜔)) − K∕(𝜔))
[
] [
]
𝑍𝑎𝑎 𝑍𝑎𝑏
𝐵𝑎𝑎 𝐵𝑎𝑏
=
𝑍𝑏𝑎 𝑍𝑏𝑏
𝐵𝑏𝑎 𝐵𝑏𝑏
⎡
⎢ [
⎢ ⎛
𝑀𝑎
+𝑖 ⎢𝜔 ⎜
⎢ ⎜ 0
⎢ ⎝
⎢
⎣
] [
0
𝐴𝑎𝑎
+
𝑀𝑏
𝐴𝑏𝑎
]
⎞
𝐴𝑎𝑏 ⎟
−
𝐴𝑏𝑏 ⎟
⎠
two reasons: (1) an additional inertia induced by the motor, gearbox
and rack and pinion, and (2) a significant drag force effect due to
large relative velocity differences between the WEC and the fluid at
resonance, similarly as was found in [66]. Fig. 9 shows that the real
part of 𝑍𝑎𝑎 – which corresponds to the hydrodynamic damping 𝐵𝑎𝑎
in Eq. (22) – is underestimated by Capytaine. This can be related
to the absence of viscosity in the BEM solver, resulting in a lower
hydrodynamic damping. A similar conclusion was made in [3]. The
underestimation of the hydrodynamic damping with the BEM solver
cannot be dedicated to the absence of friction, since in the experiment
and DualSPHysics friction compensation was modelled, cancelling the
friction coming from the motor, gearbox and rack and pinion. The
difference in hydrodynamic damping 𝐵𝑎𝑎 (= 𝑅𝑒𝑍𝐴𝐴 ) between Capytaine
and DualSPHysics is around 30 Ns/m at resonance, which means the
drag coefficient 𝐶𝑑 of the WEC can be estimated at 0.46 using the
method mentioned in [39]. Similarly as was found in [3], there is
an underestimation of the real part of 𝑍𝑎𝑎 in DualSPHysics at low
frequencies, however, this underestimation was not present for 𝑍𝑎𝑏 .
A second radiation test, Test 254, was carried out in which both
WECs were excited by a PTO force. During this test, the velocities of
both WECs were measured. In Fig. 10 very good agreement between
the velocity from DualSPHysics and experiment is shown, with an index
of agreement 𝑑𝑟 of 0.86 and 0.88 for WEC A and WEC B respectively.
Also the free surface elevation (FSE) measured in DualSPHysics at WG7
agrees well with the FSE from the experiment. The non-linear effects
resulting in steeper crests and flatter troughs are well captured in
DualSPHysics.
(22)
[
𝐾𝑎
0
]⎤
0 ⎥
𝐾𝑏 ⎥
⎥
⎥
𝜔
⎥
⎥
⎦
(23)
with 𝐵𝑖𝑗 the hydrodynamic damping, 𝐴𝑖𝑗 the added mass, 𝑀 the WEC’s
mass, 𝐾 the hydrostatic stiffness and 𝑖 and 𝑗 either 𝑎 or 𝑏. In Fig. 9 the
real and imaginary parts of the matrix components of Z are shown,
computed with DualSPHysics and with the BEM solver Capytaine, and
compared to the experimental results. The resonance frequency is the
frequency at which the heave velocity 𝑣 is in phase with the excitation
force 𝐹𝑧 , so at resonance the phase and the imaginary part of 𝑍𝑎𝑎
should be zero. In the experiment and DualSPHysics this happens at
f = 0.93 Hz, corresponding to a resonance period of 1.08 s, whereas in
Capytaine this happens at f = 1.02 Hz, corresponding to a resonance
period of 0.98 s. This shift in resonance frequency can be explained by
7
Energy 296 (2024) 130888
N. Quartier et al.
Fig. 8. Magnitude and phase of the components of the impedance matrix determined in radiation test 256.
Fig. 9. Real and imaginary part of the components of the impedance matrix determined in radiation test 256.
In Fig. 11 a visual comparison between experiment and DualSPHysics is displayed, showing a similar pattern of radiated waves
around the WECs. Furthermore, the vertical velocity of the fluid particles is shown, showing significant interaction between the radiated
waves of both WECs.
good agreement (𝑑𝑟 = 0.89) with the experimental FSE. The WEC is
placed at 4.6 m from the piston, which is the same distance as in the
experiments. This distance is sufficient since it is recommended to place
the WEC at least one wavelength away from the piston.
For the excitation tests again the laminar viscosity model is applied
in DualSPHysics. Excellent agreement (𝑑𝑟 = 0.93) is found when comparing the experimental with the numerical 𝐹𝑒 — see Fig. 13, where the
FSE at the wave gauge between the two WECs (WG7 in experiment, see
Fig. 3) as well as the 𝐹𝑒 on WEC A is shown. The force on only one of
the WECs is shown since the case is symmetrical and therefore 𝐹𝑒 is
identical for both WECs.
4.3. Excitation test
Excitation tests are carried out by keeping the WEC(s) fixed while
a surface elevation is imposed. This allows measurement of the heave
force 𝐹𝑒 acting on the WEC(s). In the experimental campaign this 𝐹𝑒
was measured with loadcells. Similarly as in the experimental campaign, this excitation test is carried out for the two WECs together. The
numerical basin has a width of 3 metre, with periodic boundaries and
numerical damping on the sides reducing the lateral reflection — see
Fig. 12. The numerical basin has a beach with numerical damping to
prevent reflection from the back. Regular waves with a wave height
(𝐻) of 0.16 m and a wave period (𝑇 ) of 1.5 s, resulting in a wavelength
(𝐿) of 3.36 m, are numerically generated with a piston, moving with
the same motion as in the experimental campaign. The wave generation
and propagation was tested in an empty numerical basin, showing very
5. Dynamic tests of the WEC array
During the dynamic tests the WEC array is simulated in regular,
irregular and focussed waves. The same numerical wave basin as shown
in Fig. 12 is applied, while the inter-particle distance 𝑑𝑝 is set to 0.02 m
after a convergence analysis. An overview of the order of simulations
of the two WECs in regular waves is represented in Fig. 14.
First, the WECs are simulated without any friction or PTO system
(Project Chrono is not used), which corresponds to the experimental
8
Energy 296 (2024) 130888
N. Quartier et al.
Fig. 10. Radiation test 254 with a noise (chirp-up) signal.
Table 4
Convergence study for a simulation of Test 282, for 50 s physical time with the interparticle distance 𝑑𝑝 , number of particles 𝑛𝑝𝑎𝑟𝑡𝑖𝑐𝑙𝑒𝑠 , runtime and index of agreement 𝑑𝑟 ,
run on a NVIDIA A100-SXM4.
𝑑𝑝 [m]
𝑛𝑝𝑎𝑟𝑡𝑖𝑐𝑙𝑒𝑠
Runtime [h]
𝑑𝑟
0.025
0.020
0.017
1.72E6
3.24E6
5.23E6
5.06
11.88
19.08
0.89
0.92
0.93
which the motor was attached, no friction compensation was applied in
the experiment. This means that the WEC will feel the full friction force.
Initially the friction force added in DualSPHysics was as described in
Eqs. (24) and (25), however it was found that this lead to a heave
motion in DualSPHysics that was significantly lower than in the experiment. Since there was very good agreement between DualSPHysics
and experiment in Test 282, it was concluded that the disagreement
in Test 224 is caused by an incorrect friction force implemented in
DualSPHysics. Therefore it was decided to tune the Coulomb damping
coefficient in the friction force in DualSPHysics until the heave motion
in DualSPHysics matched the experimental heave motion of both WECs
in Test 224. Finally, the following expression was found for the friction
force for WEC A and WEC B:
Test 282, where no motor is present. The only possible friction in
the experiment could come from the airbushings, but this friction was
minimal. After Test 282 the WEC array is simulated with friction in
order to determine the correct Coulomb and linear damping terms of
the friction force caused by the gearbox and motor. This corresponds
to Test 224 in the experiments. Once the correct friction force is
determined, Test 225 is simulated during which friction compensation
(FC) was added. Finally, FC and a PTO system is added on both WECs,
as was done in Test 226. Test 282 was simulated with three different 𝑑𝑝
values, showing an increasing index of agreement with decreasing 𝑑𝑝
— see Table 4. Fig. 15 shows the heave from DualSPHysics with three
different 𝑑𝑝 values and the heave from the experiment. Finally, a 𝑑𝑝 of
0.02 m was chosen, since this resulted in excellent agreement between
numerical and experimental heave for WEC A while still keeping the
computational runtime limited.
An isolated WEC was tested in regular waves of H = 0.16 m and 𝑇
= 1.5 s without PTO system. It was found from the experimental test
campaign that the total friction acting on the WEC can be described as a
combination of Coulomb and linear damping. Note that this description
of the friction force is a simplification since the non-linear friction
shows complex behaviour. From the experimental test campaign however it followed that simplifying the friction force as a combination
of Coulomb and linear damping is a justified simplification. From the
initial experimental tests the following expression was found for the
friction force 𝐹𝑓 ,𝐴 and 𝐹𝑓 ,𝐵 acting on WEC A and WEC B respectively:
(24)
𝐹𝑓 ,𝐵 (𝑡) = −49.6 ⋅ 𝑠𝑖𝑔𝑛(𝑣(𝑡)) − 180 ⋅ 𝑣(𝑡)
(25)
(26)
𝐹𝑓 ,𝐵 (𝑡) = −30 ⋅ 𝑠𝑖𝑔𝑛(𝑣𝐵 (𝑡)) − 180 ⋅ 𝑣𝐵 (𝑡)
(27)
Note that this friction force 𝐹𝑓 is different for WEC A and WEC B
due to small differences in the motors and gearboxes of these WECs, as
was also observed in the experiments. As is clear from Fig. 16, excellent
agreement is found between SPH and experiment with the updated
Coulomb damping coefficient in 𝐹𝑓 — the index of agreement 𝑑𝑟 for
the heave is 0.95 and 0.91 for WEC A and WEC B respectively, while
𝑑𝑟 for the FSE at WG7 equals 0.93. The 𝐹𝑓 described in Eqs. (26) & (27)
is kept throughout the remainder of the simulations.
In a next test, Test 225, the WEC array was tested with friction
compensation. This means that in the experiments an external force
was applied via the motor in order to compensate for the friction. In
the experimental campaign the friction compensation force (𝐹𝐹 𝐶 ) was
as described in Eqs. (28) & (29):
5.1. WECs without PTO system
𝐹𝑓 ,𝐴 (𝑡) = −44.8 ⋅ 𝑠𝑖𝑔𝑛(𝑣(𝑡)) − 180 ⋅ 𝑣(𝑡)
𝐹𝑓 ,𝐴 (𝑡) = −25 ⋅ 𝑠𝑖𝑔𝑛(𝑣𝐴 (𝑡)) − 180 ⋅ 𝑣𝐴 (𝑡)
𝐹𝐹 𝐶,𝐴 (𝑡) = 44.8 ⋅ 𝑠𝑖𝑔𝑛(𝑣𝐴 (𝑡)) + 144 ⋅ 𝑣𝐴 (𝑡)
(28)
𝐹𝐹 𝐶,𝐵 (𝑡) = 49.6 ⋅ 𝑠𝑖𝑔𝑛(𝑣𝐵 (𝑡)) + 144 ⋅ 𝑣𝐵 (𝑡)
(29)
Note that only 80% of the linear friction term was compensated in
the experimental tests, since compensating for more than 80% resulted
in unstable behaviour of the WEC. Test 225 was carried out in DualSPHysics by adding the 𝐹𝐹 𝐶 from Eqs. (28) and (29) to the friction force
𝐹𝑓 from Eqs. (26) and (27), using the DualSPHysics–Project Chrono
coupling. In the experimental test campaign the friction compensation
was only applied when the absolute value of the WEC’s velocity was
higher than 0.0035 m/s, in order to avoid rapid sign changes in the
𝐹𝐹 𝐶 when the WEC’s velocity was around zero. This velocity threshold
The Coulomb and linear damping can be added in DualSPHysics by
using the coupling with Project Chrono. In the simulation of Test 224 in
9
Energy 296 (2024) 130888
N. Quartier et al.
Fig. 11. Pictures from experimental Test 254 (left column), snapshots from numerical Test 254 at the same time instants (middle column) and snapshots of the numerical vertical
fluid velocity (right column).
was also implemented in DualSPHysics by changing the definition of
the Coulomb damping force in Project Chrono. Fig. 17 shows very good
agreement between the FSE (𝑑𝑟 = 0.85), and the WEC’s heave motion
(𝑑𝑟 = 0.89 for WEC A and 𝑑𝑟 = 0.90 for WEC B) from DualSPHysics and
experiment.
The absorbed power 𝑃 is defined in Eq. (32):
𝑃 (𝑡) = −𝐹𝑃 𝑇 𝑂 (𝑡) ⋅ 𝑣(𝑡)
When the absorbed power is calculated, an overestimation is found
in DualSPHysics when the WECs have a negative velocity. A possible
reason for this overestimation could be that the Coulomb damping
value of the friction force of the motor in the experiment has a different
value for negative and positive velocities.
5.2. WECs with PTO system
Next, a linear damping PTO system force 𝐹𝑃 𝑇 𝑂 is added to both
WECs, with 𝐹𝑃 𝑇 𝑂 depending on the WEC’s heave velocity:
𝐹𝑃 𝑇 𝑂,𝐴 (𝑡) = −𝐶𝑃 𝑇 𝑂,𝐴𝐴 ⋅ 𝑣𝐴 (𝑡)
(30)
𝐹𝑃 𝑇 𝑂,𝐵 (𝑡) = −𝐶𝑃 𝑇 𝑂,𝐵𝐵 ⋅ 𝑣𝐵 (𝑡)
(31)
(32)
5.3. Focussed wave test
Before simulating the WEC array in irregular waves, the WEC
array is simulated in a focussed wave. A focussed wave test contains
information in a range of frequencies without requiring a large computational time. Focussed waves have been modelled in DualSPHysics
before [40]. The focussed wave has a significant wave height of 0.2 m
and peak period of 1.1 s, and the WECs are modelled without FC or
PTO system damping so the full friction force is added. The 𝑑𝑝 used
in the simulation was again equal to 0.02 m. Comparison of the heave
In this case, 𝐶𝑃 𝑇 𝑂,𝐴𝐴 and 𝐶𝑃 𝑇 𝑂,𝐵𝐵 both equal 300 Ns/m, as in the
experimental Test 226. Fig. 18 shows very good to excellent agreement
between experimental and DualSPHysics results for the FSE at WG7 (𝑑𝑟
= 0.92) and the heave of WEC A (𝑑𝑟 = 0.93) and WEC B (𝑑𝑟 = 0.89).
10
Energy 296 (2024) 130888
N. Quartier et al.
Fig. 12. Lateral and top view of the numerical wave basin used in DualSPHysics for the excitation and dynamic tests of the WEC array.
Fig. 13. Excitation test (Test 247): FSE at WG7 from experiment and SPH (above), and vertical wave excitation force 𝐹𝑒 on WEC A from experiment and SPH (below).
with experimental results is shown in Fig. 19. An index of agreement
𝑑𝑟 of 0.87 and 0.88 was found for WEC A and B respectively.
is larger. This hints at an error in the estimation of the Coulomb damping coefficient in the friction force. At these low amplitudes the WEC
is often not moving in the experiment, while in DualSPHysics the WEC
keeps moving, so possibly the friction compensation is overestimated
in DualSPHysics for this test. A possible explanation could be that the
friction force present in the experiments was not constant through time,
meaning that the Coulomb friction term could be varying during the
same test or between different tests. The friction force could have been
higher during Test 90 in the experimental test campaign. The absorbed
power shown in Fig. 20 is even more overestimated in DualSPHysics,
since errors in velocity are increased by a power 2 in the absorbed
power. The average absorbed power in the experiment was 3.58 W,
while in DualSPHysics this was 4.46 W.
Next, the two-WEC array is simulated in the same irregular wave
conditions. Contrary to the conventional way of describing the PTO
damping force as proportional to the WEC’s own velocity, the PTO
damping force now depends on both its own velocity and the velocity of
5.4. Irregular wave test
Finally, the WECs are simulated in irregular waves while PTO
damping is applied. A 𝑑𝑝 of 0.02 m was used, as in the regular waves
and focussed wave test. First, an isolated WEC (WEC A) is simulated in
irregular waves with a Jonswap spectrum with 𝐻𝑠 = 0.16 m and 𝑇𝑝 =
1.5 s, with 𝐶𝑃 𝑇 𝑂,𝐴𝐴 = 280 Ns/m, since this is the damping coefficient
that theoretically leads to maximum average absorbed power. This is
the set-up applied in Test 90. The PTO force 𝐹𝑃 𝑇 𝑂,𝐴 is as described in
Eq. (30). An index of agreement 𝑑𝑟 of 0.79 is found when comparing the
heave of the WEC with the experimental heave, which is a lower 𝑑𝑟 than
what was obtained in previous simulations. From Fig. 20 it can be seen
that at low amplitudes the overestimation of the heave in DualSPHysics
11
Energy 296 (2024) 130888
N. Quartier et al.
Fig. 14. Schematic representation of order of simulations.
Fig. 15. Test 282: Heave of WEC A for three different 𝑑𝑝s in DualSPHysics, and from experiment.
the neighbouring WEC (see Eqs. (33) & (34)), since it is known that the
optimal PTO system force leading to maximum average absorbed power
depends on the velocity of both WECs [67–69]. This coupled resistive
control strategy, called global control, was also applied in Test 186 of
the experiments.
𝐹𝑃 𝑇 𝑂,𝐴 (𝑡) = −𝐶𝑃 𝑇 𝑂,𝐴𝐴 ⋅ 𝑣𝐴 (𝑡) − 𝐶𝑃 𝑇 𝑂,𝐴𝐵 ⋅ 𝑣𝐵 (𝑡)
(33)
𝐹𝑃 𝑇 𝑂,𝐵 (𝑡) = −𝐶𝑃 𝑇 𝑂,𝐵𝐵 ⋅ 𝑣𝐵 (𝑡) − 𝐶𝑃 𝑇 𝑂,𝐵𝐴 ⋅ 𝑣𝐴 (𝑡)
(34)
WEC A was 0.75. The average absorbed power in the experiment was
3.90 W, while in DualSPHysics it was 4.81 W (for WEC A). Using the
result of Test 90, a 𝑞-value (defined in Eq. (35)) of 1.09 is obtained
in the experiment and 1.08 in DualSPHysics, meaning that in both the
experiment as well as in DualSPHysics an increase in average absorbed
power of around 8%–9% in the two-WEC array was observed, caused
by constructive interference of the radiated waves of both WECs. The
WEC’s large diameter to draft ratio causes high radiated waves, which
can enlarge the heave amplitude of the neighbouring WEC due to their
close proximity.
𝐶𝑃 𝑇 𝑂,𝐴𝐴 and 𝐶𝑃 𝑇 𝑂,𝐵𝐵 were set to 300 N s/m and 𝐶𝑃 𝑇 𝑂,𝐴𝐵 and
𝐶𝑃 𝑇 𝑂,𝐵𝐴 were set to 35 N s/m. A schematic representation of the
coupling between DualSPHysics and Project Chrono for this control
strategy is shown in Fig. 21, mainly based on the scheme of [20,21]
(see Fig. 22).
Again, the absorbed power seems to be overestimated in DualSPHysics. The index of agreement for the heave of both WEC A and
B was 0.82, while the index of agreement for the absorbed power of
𝑞=
𝑃𝑊 𝐸𝐶𝑎𝑟𝑟𝑎𝑦
𝑛𝑃𝑠𝑖𝑛𝑔𝑙𝑒𝑊 𝐸𝐶
(35)
In Table 5 a summary is given of the agreement between DualSPHysics and experiment for all simulated testcases. This agreement is
expressed with the index of agreement 𝑑𝑟 , the error in amplitude 𝐴, the
correlation coefficient 𝑅 and the relative root square error 𝑅𝑅𝑆𝐸. In
12
Energy 296 (2024) 130888
N. Quartier et al.
Fig. 16. Test 224: FSE at WG7, Heave of WEC A and Heave of WEC B from SPH and experiment.
Fig. 17. Test 225: FSE at WG7, Heave of WEC A and Heave of WEC B from SPH and experiment.
Fig. 18. Test 226: FSE at WG7, Heave of WEC A and Heave of WEC B from SPH and experiment.
13
Energy 296 (2024) 130888
N. Quartier et al.
Fig. 19. Test 238: Heave of WEC A and Heave of WEC B from SPH and experiment.
Fig. 20. Test 90: Heave of WEC A from DualSPHysics and experiment, and absorbed power 𝑃 of WEC A from SPH and experiment.
Fig. 21. Schematic representation of the coupling procedure between DualSPHysics and Project Chrono for the applied control strategy.
14
Energy 296 (2024) 130888
N. Quartier et al.
Fig. 22. Test 186: FSE at WG7, Heave of WEC A and Heave of WEC B from SPH and experiment, and absorbed power 𝑃 of WEC A and B from SPH and experiment.
most cases, very good–excellent agreement is found, with the exception
of the absorbed power in irregular waves.
Table 5
𝑑𝑟 , 𝑅, 𝐴 and 𝑅𝑅𝑆𝐸 for all testcases, simulated with laminar viscosity, mDBC and
density diffusion.
6. Conclusions
𝑑𝑟
A
R
RRSE
Rating
0.91
0.95
0.99
0.13
Excellent
0.87
0.82
0.91
0.99
0.96
0.93
0.28
0.37
Very good
Very good
0.79
0.85
0.87
1.03
1.15
1.05
0.91
0.97
0.97
0.44
0.29
0.26
Good
Very good
Very good
0.90
0.94
0.84
0.95
0.99
0.99
0.23
0.13
Excellent
Excellent
0.93
0.92
0.99
0.16
Excellent
0.93
0.95
0.91
0.98
0.99
1.03
0.99
1.00
0.98
0.16
0.11
0.19
Excellent
Excellent
Excellent
0.85
0.89
0.90
1.01
0.85
0.90
0.96
0.99
0.98
0.34
0.22
0.21
Very good
Very good
Excellent
0.92
0.93
0.89
1.10
1.10
1.09
0.99
1.00
0.98
0.18
0.16
0.23
Excellent
Excellent
Very good
0.87
0.88
1.08
1.06
0.97
0.98
0.28
0.23
Very good
Very good
0.79
0.72
1.20
1.23
0.94
0.91
0.44
0.54
Good
Good
0.82
0.82
0.75
1.18
1.14
1.20
0.95
0.96
0.91
0.38
0.36
0.52
Very good
Very good
Good
Free-decay test
In this work a WEC array of two heaving cylindrical truncated
WECs is modelled in DualSPHysics and validated with experimental
data provided from the WECfarm test campaign held in the wave basin
of AAU in February 2022 [47]. A series of system identification tests
are performed, leading to the following observations and conclusions:
z WEC A
Radiation test 256
v WEC A
𝐹𝑧 WEC B
Radiation test 254
• A free-decay test with an isolated WEC was carried out in DualSPHysics, showing excellent agreement with the experimental
heave of the WEC when mDBC and laminar viscosity were applied
in DualSPHysics;
• An excitation test with the two-WEC array was carried out in DualSPHysics, showing very good agreement with the experimental
vertical excitation force;
• Two radiation tests with the two-WEC array were carried out in
DualSPHysics, allowing the calculation of the impedance matrix.
In this work the impedance matrix for a two-WEC array is for the
first time determined with SPH. Very good agreement with the
experimental heave velocity was found, and with the free surface
elevation at the location between the two WECs. DualSPHysics is
able to accurately capture the non-linear wave field modifications
surrounding the WEC array.
WG
v WEC A
v WEC B
Excitation test 247
WG
𝐹𝑧 WEC A
Regular wave test 282 without motor
z WEC A
Regular wave test 224 with friction
WG
z WEC A
z WEC B
Regular wave test 225 with FC
WG
z WEC A
z WEC B
In the experiments a motor is present to add a PTO force to the
WEC, which also introduces friction. This friction was cancelled in experimental tests by adding a friction compensation force. It was found
that this friction compensation in the experiments overcompensated
the real friction. In order to add the friction force in DualSPHysics,
a combination of Coulomb damping and linear damping was applied,
using the coupling between DualSPHysics and Project Chrono. Furthermore, a novel implementation was made in Project Chrono that
only applies the friction compensation force when the absolute value
of the WEC’s heave velocity exceeded a certain threshold, exactly as
was done in the experiments. Once the correct friction and friction
(over)compensation was determined, a series of tests in regular and
irregular waves were performed. These showed very good agreement
between the experimental heave and the heave from DualSPHysics. In
irregular waves however the agreement between the absorbed power
from experiment and DualSPHysics was only reasonable: it was found
that the absorbed power in DualSPHysics was overestimated, which
could be due to a mismatch in the estimation of the friction force
at low heave velocities. A novel implementation was added in the
Regular wave test 226 with FC and PTO
WG
z WEC A
z WEC B
Focussed wave test 238
z WEC A
z WEC B
Irregular wave Test 90
z WEC A
P WEC A
Irregular wave Test 186
z WEC A
z WEC B
P WEC A
DualSPHysics–Project Chrono coupling to model the global control
strategy. This allowed the definition of a PTO force that depends
15
Energy 296 (2024) 130888
N. Quartier et al.
on the velocity of both WECs corresponding to a causal impedance
matching formulation for a two-WEC array. Both in the experiments
and in DualSPHysics it was found that an increase in absorbed power
of up to 9% is observed per WEC when the WECs are installed in the
studied array configuration. The good agreement between numerical
and experimental results proves the applicability of DualSPHysics for
the modelling of a WEC array.
Future research following this work will look into larger WEC
arrays, e.g. a 2 × 2 or a 5-WEC array in star configuration. In order
to simulate WEC arrays with more than two WECs a larger numerical basin and thus longer computational runtimes will be required.
Therefore, further research looking into a multi-GPU version of DualSPHysics should be performed. Furthermore, a variable resolution
scheme might be necessary to reduce the required number of particles
of the numerical basin.
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CRediT authorship contribution statement
Nicolas Quartier: Conceptualization, Formal analysis, Funding acquisition, Investigation, Methodology, Validation, Visualization, Writing – original draft, Writing – review & editing. Timothy Vervaet: Data
curation, Funding acquisition, Investigation, Methodology, Validation.
Gael Verao Fernandez: Data curation, Methodology, Writing – review
& editing. José M. Domínguez: Resources, Software, Writing – original
draft, Writing – review & editing. Alejandro J.C. Crespo: Conceptualization, Methodology, Resources, Software, Supervision, Validation,
Visualization, Writing – original draft, Writing – review & editing. Vasiliki Stratigaki: Funding acquisition, Project administration, Supervision, Writing – original draft, Writing – review & editing. Peter Troch:
Funding acquisition, Project administration, Resources, Supervision,
Writing – review & editing.
Declaration of competing interest
The authors declare the following financial interests/personal relationships which may be considered as potential competing interests:
Nicolas Quartier reports financial support was provided by Research
Foundation Flanders.
Data availability
The data that has been used is confidential.
Acknowledgements
This research is being supported by the Research Foundation Flanders (FWO), Belgium - FWO fellowship No. 1SC5421N of Nicolas
Quartier, and FWO fellowship No. 11A6919N of Timothy Vervaet.
In addition, Vasiliki Stratigaki is a postdoctoral researcher (fellowship 1267321N) of the FWO, Belgium. The experimental test campaign was supported by WECANet (COST Action CA17105) funding for the short term scientific mission (STSM) at Aalborg University. The collaboration between UGent and UVigo was supported by
WECANet funding, enabling STSMs of the first author of this work
at EPhysLab, Ourense, Spain. This work was also supported by the
project SURVIWEC PID2020-113245RB-I00 financed by MCIN/ AEI
/10.13039/ 501100011033.
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