Content uploaded by Zhen Gao
Author content
All content in this area was uploaded by Zhen Gao on Nov 12, 2014
Content may be subject to copyright.
COMPARISON OF MOORING LOADS IN SURVIVABILITY MODE
ON THE WAVE DRAGON WAVE ENERGY CONVERTER
OBTAINED BY A NUMERICAL MODEL AND EXPERIMENTAL DATA
Stefano Parmeggiani
Wave Dragon Ltd./Aalborg University
London, UK/Aalborg, Denmark
Made Jaya Muliawan
CeSOS, NTNU
Trondheim, Norway
Zhen Gao
CeSOS, NTNU
Trondheim, Norway
Torgeir Moan
CeSOS, NTNU
Trondheim, Norway
Erik Friis-Madsen
Wave Dragon Aps,
Copenhagen, Denmark
ABSTRACT
The Wave Dragon Wave Energy Converter is ready to be
up-scaled to commercial size. The design and feasibility
analysis of a 1.5 MW pre-commercial unit to be deployed at the
DanWEC test center in Hanstholm, Denmark, is currently
ongoing. With regard to the mooring system, the design has to
be carried out numerically, through coupled analyses of
alternative solutions. The present study deals with the
preliminary hydrodynamic characterization of Wave Dragon
needed in order to calibrate the numerical model to be used for
the mooring design. A hydrodynamic analysis of the small scale
model in the frequency domain is performed by the software
HydroD, which uses WAMIT as core software. The quadratic
damping term, accounting for the viscous effect, is determined
through an iterative procedure aimed at matching numerical
predictions on the mooring tension, derived through time
domain coupled analysis, with experimental results derived
from tank tests of a small scale model. Due to the complex
geometry of the device, a sensitivity analysis is performed to
discuss the influence of the mean position on the quality of the
numerical predictions. Good correspondence is achieved
between the experimental and numerical model. The numerical
model is hence considered reliable for future design
applications.
INTRODUCTION
The Wave Dragon is a floating, offshore Wave Energy
Converter of the overtopping type. Incoming waves are focused
by two wing reflectors towards a doubly-curved ramp, by which
they surge up into a reservoir placed above the mean water
level. The power production takes place as the stored water is
led back to the sea through a set of low-head hydro-turbines
coupled to permanent magnet generators (Figure 1). A
commercial unit of Wave Dragon suitable for North Sea
conditions (yearly average wave power of 24 kW/m) is a 22,000
tons reinforced concrete structure, occupying an area of around
150x260 m2. With 4 MW installed power, it can produce up to
12 GWh per year. Scale ratios used in the following are all
referred to this North Sea size.
Figure 1. Working principle of Wave Dragon.
Wave Dragon has been developed during more than 10
years, following a Technology Readiness Assessment (TRA)
approach in which each new phase of the development is
justified by the good results achieved in the previous one. The
related research and development has been mainly carried out
through physical tests, as suggested by the geometrical
complexity of the device, which made it difficult to establish
reliable ad-hoc numerical models. Since the early phases the
development of Wave Dragon was therefore mainly based on
wave tank testing of a 1:51.8 scale model; among these were the
proof of concept, geometry optimization, hydrodynamic
characterization and preliminary power production assessment
[1].
Based on these experimental results, in 2003 a 1:4.5 scale
prototype was built and tested in Nissum Bredning, a benign
location in Northern Denmark. The extended sea trials program
allowed acquiring valuable operational experience in many
aspects, as well as validating the analytical models resulting
Proceedings of the ASME 2012 31st International Conference on Ocean, Offshore and Arctic Engineering
OMAE2012
July 1-6, 2012, Rio de Janeiro, Brazil
OMAE2012-8
1
Copyright © 2012 by ASME
from the previous development phase and testing the Power
Take-Off system and control strategy [2].
Currently, Wave Dragon is being up-scaled to commercial
size. With regard to this, the structural design of a 1.5 MW pre-
commercial unit to be deployed at the DanWEC test site in
Hanstholm, Northern Denmark, [3] is being carried out together
with the related feasibility analysis. Respect to a 4 MW North
Sea Wave Dragon the scale of the envisaged unit is 1:1.5. The
up-scaling is largely based on the measurement campaign
carried out on the prototype during the sea trials and
experimental data acquired in previous phases of development.
For a floating offshore structure such as Wave Dragon, an
important part of the design process deals with the mooring
system. The conceptual mooring system for Wave Dragon is a
circular spread of slack chains anchored to a central buoy, to
which the main platform and wave reflectors are connected
(Figure 2). This system allows de-coupling the vertical motions
of Wave Dragon, which are actively controlled to optimize the
power production, from the other modes of motions, which are
restrained by the mooring system instead. The catenary solution
is a well-known technology capable of absorbing peak loads,
hence well adequate for a large floating structure such as Wave
Dragon. Other elements of the mooring system are a rear
mooring line to prevent excessive rotations and cables between
the reflectors to prevent them from opening or closing too
much.
Figure 2. Conceptual mooring system of Wave Dragon.
This general layout needs to be adapted to the local
conditions of the DanWEC deployment site and a detailed
mooring system has to be designed. Although being a site-
specific solution, the result of the design operations would also
provide a sound basis for a mooring system suitable for any
future deployment, since this is not expected to change
substantially.
Unlike with other components, the mooring design cannot
be based on direct experience, as both during the tank testing
and the prototype trials the relatively shallow waters did not
allow a proper catenary system to be tested. In these cases the
device and reflectors were connected to a vertical pile instead.
Therefore the mooring design for the up-scaled pre-commercial
unit has to be carried out numerically, through systematic
analyses of alternative configurations aiming at finding the best
solution in terms of reliability and economic feasibility.
In order to do so, coupled analysis of the response of Wave
Dragon to mooring and environmental loads typical of the
deployment location will be carried out. This will be done by
means of the software SIMO/Riflex [4], which has been
extensively used in the past for coupled analysis of motions and
loads on moored offshore structures and ships. However,
preliminary to this, hydrodynamic and mooring models of Wave
Dragon have yet to be established.
A thorough investigation aimed at the hydrodynamic
characterization of Wave Dragon has been carried out and is
presented here. The technique used is to validate a numerical
model for hydrodynamic analysis with experimental data
relative to previous tank tests. A hydrodynamic analysis of the
device is first performed in the frequency domain; the value of
the quadratic damping coefficient is then determined in order to
achieve the best fit on the tension in the main mooring line
between the numerical predictions, deriving from a time domain
coupled analysis, and the experimental data, obtained after the
tank testing of a 1:51.8 scale model of Wave Dragon. The
comparison is made at the model scale, however the results can
be used for the pre-commercial demonstrator design, as a
geometrical similarity is maintained.
In the following the numerical model and the method used
for its calibration are described in detail. The results of the
calibration process are the hydrodynamic parameters of Wave
Dragon, which will be used in the design of the mooring system
for the 1.5 MW pre-commercial unit.
A sensitivity analysis is finally also carried out with regard
to the influence of the geometric configuration of the device on
the hydrodynamic behavior, since it is found that its
hydrostatics and hydrodynamics are very dependent on the
mean configuration.
Finally, main conclusions are summarized and further work
is addressed.
EXPERIMENTAL DATA
In November 2010 a series of tests have been carried out in
order to assess the response of Wave Dragon to extreme wave
conditions, in terms of extreme forces in the main mooring line
and extreme motions. These have been carried out at the
Hydraulic and Coastal Laboratories of Aalborg University
(AAU) on a 1:51.8 scale model of Wave Dragon [5]. The scaled
model used is 5 m wide (from tip to tip of the reflectors), 2.9 m
long in its longitudinal cross-section and 0.28 m high (from the
bottom base to the crest freeboard). Irregular waves have been
tested, representing extreme conditions typical of the Danish
part of the North Sea (with return periods of 10, 50 and 100
back mooring
chain
front mooring
chain spread
wave
reflector
main
platform
main
mooring
line
central buoy
anchor point
2
Copyright © 2012 by ASME
years, see Table II). The mooring system was represented
through linear springs; the main mooring line was horizontally
connected to a fixed vertical pile, where the tension was
measured through a force transducer (Figure 3a). Different
drafts have been tested. Motions of the model in heave, pitch
and surge were measured. Experimental data are available as 30
min long time series, acquired at 20 Hz. This same setup has
been analyzed numerically.
NUMERICAL METHOD
In this study a panel model of Wave Dragon (Figure 3b) is
created through the software GeniE [6], reproducing the same
model geometry and mass distribution used in the tank testing.
This panel model is the input for the software HydroD [7], used
for the hydrodynamic analysis in the frequency domain; this
includes the calculation of hydrostatic stiffness, potential
damping, added mass and 1st and 2nd order wave excitation
forces on the body. Once determined, the hydrodynamic
properties will be used in the coupled mooring analysis in time
domain, performed through the software SIMO/Riflex.
Additionally, the viscous term is introduced according to the
Morison model. This has been estimated by calibrating the
numerical model with data derived from the tank testing of the
scaled model of Wave Dragon. The goal is to determine an
entry value for the quadratic damping term so that the numerical
model is able to reproduce the observed experimental response.
An iterative approach has been used to find the best
correspondence in terms of tension in the main line of the
mooring system, the one connecting the main platform to the
central buoy.
Other non-linear features, such as the overtopping, are
disregarded in the analysis; this is not considered a major
problem at this stage, as previous studies have shown that in
extreme conditions the device should be set to the lowest
floating position possible to increase its survivability, a
condition in which it is almost totally submerged and the
nonlinearities due to the overtopping are very much limited [5].
Three drafts have been considered in the present analysis;
these are referred to as low, medium and high draft and
corresponds respectively to 0.2 m, 0.24 m and 0.26 m at model
scale. Since the hydrodynamic analysis is based on linear
theory, for each draft a mean position had to be assumed. In
general, this was maintained as close as possible to the average
position experienced during the tank tests. More details on this
are discussed in the following.
The calibration process was carried out by varying the
quadratic damping of the system in surge, in order to reproduce
in SIMO/Riflex the response observed in the tank tests. Initially
free-decaying tests have been considered; this part of the
analysis was mainly aimed at identifying the right mean position
to be used. However the core of the analysis was represented by
the comparison of the results in irregular waves, from which the
quadratic damping has been estimated. For each of the 6
experimental test cases considered, 12 different numerical
simulations are performed, using 12 different realizations of the
same wave condition. From these the distribution of the extreme
response in the mooring line could be produced for each case.
(a)
(b)
Figure 3. The experimental setup used in the tank tests (a)
and the panel model used in the hydrodynamic analysis (b).
Irregular wave analysis
Once the mean position to be used in the analysis has been
assessed based on free-decay tests, the comparison between
numerical and experimental response to irregular wave has been
carried out in terms of:
• Statistical analysis of the whole time series of the
tension in the main mooring line (1 experimental and 12
numerical realizations for each case), 30 min long. The
comparison considered mean value and standard deviation of
the time series.
3
Copyright © 2012 by ASME
• Spectral analysis, based on: the experimental incident
wave component, as derived from 3D wave analysis of the wave
measured during the tank tests; the experimental mooring
tension recorded in the tank tests; the numerical mooring
tension, considering the aggregated time series given by the
succession of the 12 simulations for each case considered. A
comparison in the wave frequency region aims at assessing the
correct representation of the linear part of the system, while the
comparison in the low frequency region aims at assessing the
behavior in the resonant part of the system.
• Extreme value analysis of the tension. The comparison
is based on the mean of the extreme value distribution obtained
from the numerical simulations and from the experimental
results.
The extreme value distributions considered the maxima of
time series 5 minutes long (i.e. 6 sub-time series considered for
each case) as well as 30 minutes long (i.e. one time series for
each case). When 5 minutes long time series are considered, 72
points are available for each numerical case (6 maxima for each
of the 12 realizations) and 6 for each experimental case; when
30 minutes long time series are considered, 12 points are
available for each numerical case (one for each realization) and
1 point only for each experimental case, corresponding to the
absolute maximum. The analysis based on the 5 min long time
series is used in order to reduce the statistical uncertainty
related to the extreme value estimate based on the experimental
data.
RESULTS
Free-decay tests
The hydrodynamic analysis is performed in the frequency
domain. This is based on the linear theory, under the assumption
of small motions, where a mean position representing the static
configuration of the model has to be used as reference in the
simulations. Deriving such condition from irregular wave
records can be challenging. A first indication on the mean
position to use was drawn by referring to the free-decay tests,
preliminary carried out in absence of waves. Experimental
decay data were available only for the low draft case. The mean
trim which better reproduced the free-decay motion time series
(in surge, heave and pitch) at that draft was found to be 3.8°,
confirming experimental findings which showed a tendency of
the model to trim backwards.
To further assess the validity of this floating position, the
resulting damping ratio, ζ (-), and natural frequency, ω0 (rad/s),
are also compared with those experimentally determined. In the
experimental tests three free-decay realizations have been
carried out for each mode of motion. ω0 and ζ have been
derived for each of them and the average value between the
three is considered to reduce the uncertainties. With regard to
the numerical results, the outcome of the hydrodynamic analysis
in HydroD including only potential damping is assumed as
reference case; from here various possibilities have been tested,
such as adding more linear or quadratic damping to account for
the drag effect. The best correspondence was achieved by
adding some quadratic damping in surge and some linear
damping in pitch respect to the reference case. Results are
shown in Table I, where also the case of trim = 0° was
considered as an alternative.
Table I. Damping ratio and natural frequency in the three
modes of motions, from decay tests with trim equal to 0° and
3.8°.
natural frequency (rad/s)
,
ω
0
surge
heave
pitch
experimental
1
0.702
3.314
3.329
2
0.697
3.117
2.884
3
0.749
3.264
4.287
mean value
0.716
3.232
3.500
numerical,
trim = 0°
potential damping only
0.5956
6.2
6
pote
ntial + viscous
term
1
0.728
6.2
6.02
numerical,
trim = 3.8°
potential damping only
0.676
2.67
3.11
potential + viscous
term
1
0.73
3.19
3.32
damping ratio (
-
)
,
ζ
surge
heave
pitch
experimental
1
0.484
0.194
0.306
2 0.358 0.196 0.64
3
0.395
0.
196
0.317
mean value
0.412
0.195
0.421
numerical,
trim = 0°
potential damping only
0.0159
0.197
0.146
potential + viscous
term
1
0.09
0.197
0.48
numerical,
trim = 3.8°
potential damping only
0.09
0.017
0.21
potential + viscous
term
1
0.15
0.2
0.31
1Quadratic damping for surge, linear for pitch
The case with trim = 3.8° matches well the experimental
results, showing substantial differences only on the surge ζ (see
the discussion for more details). Overall, it is considered as the
best alternative between the two considered and therefore is
taken as first attempt mean position. Nevertheless, the following
analysis based on irregular wave tests showed that by assuming
3.8° trim the mooring line tension was significantly
overestimated. This is in agreement with the experimental
findings. As the main objective of this comparative analysis is
to well represent the mooring line tension, it is finally decided
to use trim equal to 0° as mean position, which determined
more comparable results in terms of tension for all the cases
considered (when referring to other drafts too). When assuming
0° trim as mean position a good correspondence is kept on the ζ
in heave and pitch, as well as on the ω0 in surge. ω0 in heave
and pitch get almost the double of the ones experimentally
4
Copyright © 2012 by ASME
determined instead, while the ζ in surge is still lower compared
to the one experimentally determined. This point is further
discussed in the sensitivity analysis and discussion below.
Irregular wave tests
From the available experimental database 6 cases have
been selected for comparison, including different wave
conditions and drafts. All waves have been generated from a
JONSWAP spectrum with peak-enhancement factor of 3.3, and
cos2s spreading function with s = 20, representative of long
crested waves. All 3 values of drafts have been considered here.
From the numerical simulations 12 different time series
each 30 minutes long have been obtained for every case; the
experimental results refer to only one realization for each case,
also 30 min long. They all refer to steady state conditions. All
data are acquired at 20 Hz.
All cases examined refer to a target stiffness of the main
mooring line of 460 N/m, as derived from the spring
calibration, with a pre-tension of 12 N. However for case 1 the
actual stiffness during the tests, derived by plotting the recorded
forces against the deformation, was revealed to be 300 N/m. In
this case the numerical stiffness was set accordingly to the
actually tested value.
As stated above, in all numerical cases the mean pitch is
assumed to be zero. During the tank tests the draft actually
tested differed from the target value, due to the fact that in
dynamic conditions the mean heave was in general lower than
zero (especially in the case of large drafts). This phenomenon
has also been confirmed in the numerical simulations, and could
be explained by the fact that at high and medium drafts the
water-plane area becomes very small, providing almost no
hydrostatic restoring force in heave; therefore the device tends
to get submerged, even though maintaining enough buoyancy to
float. In order to reproduce the experimental conditions as much
as possible, the draft used in the numerical setup always
corresponds to the draft actually measured during the tank
testing rather than to the target value.
Table II gives an overview of the test cases, showing
incident wave conditions (Hs and Tp) and return period (TR), as
well as values of the target draft (dr) and the one actually tested.
The best agreement in the mooring tension between
numerical results and experimental data is pursued by varying
the quadratic damping of the system in surge, in order to
represent the viscous/drag term in addition to the potential
damping estimated in hydrodynamic analysis. This is the one
significant phenomenon which cannot be accounted for with
linear theory indeed. The best correspondence is found for a
quadratic damping coefficient Cd* = 500 kg/m for cases 3 to 6,
in which the body is totally submerged in its mean dynamic
position. The quadratic damping coefficient can be described by
the formula
Cd* = ½ ρ Cd Ac (1)
where ρ (kg/m3) is the water density and Ac (m2) is the measured
projected wet surface normal to the surge direction, which is
proportional to the draft of the device (dr).
Using Eq. 1, Cd (-), which is the surge non-dimensional
drag coefficient, can be calculated for cases 3 to 6 (for which
Cd* has been previously determined). Due to this the surge Cd
for the WD model is found to be 0.726. Based on this value,
Cd* can be estimated also for cases 1 and 2, considering the
respective projected wet surface area, resulting to be 357 kg/m.
Table III and Figure 4 are overviews of the results of the
statistical and extreme values analyses for all the cases
considered, shown respectively in absolute and relative terms.
The latter shows the ratio between numerical and experimental
results for each case, related respectively to the mean and
standard deviation of the tension and to the extreme values of
the tension. All results are shown at model scale.
Figure 5 shows the spectral analysis for 3 representative
cases, in which the numerical extreme values resulted to be
respectively the lowest (a), the highest (b) or very similar (c)
with respect to the experimental ones.
Sensitivity analysis
As indicated above the mean floating position is subjected
to significant uncertainties, influencing the hydrodynamic
parameters. Therefore a sensitivity analysis has been carried out
with respect to the draft and trim of the device.
The influence of the mean trim position has already been
discussed above. By adding quadratic damping to the system, as
representative of viscous term, ω0 in surge gets closer to the
experimental value, being 0.728 rad/s. The surge ζ is also
increased to the value of 0.09, however still quite distant from
the one experimentally determined during the free-decay test.
With respect to the draft, Figure 6 shows how the
normalized hydrostatic restoring force in heave and water plane
area vary with it; Figure 7 shows their variation for different
trim mean positions; Figure 8 and Figure 9 show respectively
variations of the normalized added mass and normalized
potential damping in the three modes of motions for different
drafts in the frequency domain. The normalizing factors used
here are (ρ · V · g)/L for the heave hydrostatic restoring force, ρ
· V and ρ · V · L2 respectively for translational and rotational
added mass, ρ · V · (g/L)0.5 and ρ · V · L · (g/L)0.5 for the
translational and rotational potential damping; here g (m/s2) is
the gravity acceleration, V (m3) is the displaced volume and L
(m) is a characteristic length, set to 100 m.
5
Copyright © 2012 by ASME
Table II. Overview of the tested conditions in each case considered (values at model scale).
case TR (y) Hs (m) Tp (s) target dr (m) actual dr (m)
1
test06_k300_
Cd*357
50
0.173
1.896
0.2
0.2
2 test11_k460_ Cd*357 100 0.201 1.969 0.2 0.2
3 test17+_k460_ Cd*500 10 0.145 1.829 0.24 0.29
4 test27#_k460_ Cd*500 100 0.184 1.896 0.24 0.29
5 test30+_k460_ Cd*500 10 0.167 1.829 0.26 0.34
6 test35+_k460_ Cd*500 50 0.168 1.969 0.26 0.34
Table III. Overview of the results from the statistical and extreme values analyses, in absolute terms (values at model scale).
case
TS mean (N) TS std (N) max (5 min dist) (N) max (30 min dist) (N)
1 test06_k300_ Cd*357 num. 15.60 12.64 58.59 65.25
exp. 17.21 8.68 67.24 83.39
2 test11_k460_ Cd*357 num. 16.40 18.00 88.24 101.18
exp. 21.68 16.8 98.95 113.9
3 test17+_k460_ Cd*500 num. 13.48 13.8 68.08 81.01
exp. 18.98 10.9 67.43 77.79
4 test27#_k460_ Cd*500 num. 13.4 16.4 83.46 97.7
exp. 24.74 13.8 79.41 88.83
5 test30+_k460_ Cd*500 num. 11.95 12.4 57.31 63.4
exp. 17.67 10.3 55.89 67.79
6 test35+_k460_ Cd*500 num. 12.24 13.3 61.99 69.05
exp. 17.45 11.5 65.15 71.36
(a) (b)
Figure 4. Comparison of mooring line tension obtained as (a) response statistics and (b) extreme values,
shown as ratio between numerical and experimental values.
6
Copyright © 2012 by ASME
(a)
(b)
(c)
Figure 5. Results from the spectral analysis showing:
experimental incident wave spectrum (η), spectrum of the
numerical mooring tension (F) as simulated, spectrum of the
experimental mooring tension (F) as measured.
Figure 6. Normalized hydrostatic restoring force in heave
and water plane (WP) area for different mean drafts.
Figure 7. Water plane (WP) area and normalized
hydrostatic restoring force in heave for different mean trim
static positions.
7
Copyright © 2012 by ASME
(a)
(b)
(c)
Figure 8. Normalized added mass in surge (a), heave (b),
pitch (c) in the frequency domain, for different drafts.
(a)
(b)
(c)
Figure 9. Normalized potential damping in surge (a), heave
(b), pitch (c) in the frequency domain, for different drafts.
8
Copyright © 2012 by ASME
DISCUSSION
It should be noted that the numerical model is based on the
1st and 2nd order wave forces, while other non-linearities might
have been experienced in the tank testing of the model, such as
the afore-mentioned water plane area variations.
The statistical analysis shows that in general the mean
value is numerically underestimated, while the standard
deviation is slightly overestimated, respect to the experimental
results. The former can be explained as the increased draft
experienced in dynamic conditions, due to the low heave
restoring force, might have added some extra pre-tension to the
system, whereas in the numerical simulation this did not happen
since the target draft was already set as mean position. The
mean tension values are however relatively small in absolute
terms, so that even small differences from experimental and
numerical values are reflected by a large differences in terms of
ratio (Table III).
In Figure 5 the low frequency peak in the tension spectrum
corresponds to the surge natural frequency of the system, while
the higher frequency peak corresponds to incoming wave
frequency. The tension spectral analysis shows that the wave
frequency regime is very well represented numerically, while in
the resonant part a slight shift in the low peak frequency can be
observed. However, it is considered the best match achievable
due to the uncertainties related to the numerical modeling and
experimental results. It’s worth mentioning that a great
improvement in the correspondence of the low frequency peak
was observed as quadratic damping was added to the system,
accounting for the 2nd order drag forces (Table I).
The extreme value analysis shows that the numerical
predictions are within 80-110% of the experimental values for
the 30 min long time series distribution, and within 90-105% on
the 5 min long time series distribution (Figure 4b). When 30
min long time series were considered only one experimental
realization was available, only one maximum being therefore
available for the comparison. By basing the extreme value
analysis on 5 min long time series, statistical uncertainty is
reduced. Here the number of realizations available for the
analysis increased by a factor of 6, allowing respectively 72
numerical and 6 experimental maxima to be used for each case.
Moreover, the maxima considered are distributed along the
whole time series, not being locally concentrated due to wave
groups. Overall, the comparison can be therefore considered
more sound.
With regard to the static mean position used, this has
revealed to be critical in the analysis. Due to the complex
geometry of the device - mainly made of opened-bottom
chambers - the distribution of the buoyant elements on the
device plays a key role in the determination of the water-plane
are Aw. Due to this, even small changes in the model floating
position (dr) or trim position had a significant influence on the
water plane area, Aw. As dr was increased to the medium or high
level, for which the water free surface was close to or above the
top of the buoyant element, Aw dramatically reduced indeed
(Figure 6); the same happened when trim occurred (Figure 7),
even for small values barely noticeable to the naked eye (e.g.
around a 30% decrease in Aw for just 2° trim difference).These
variations in Aw in turn affected the hydrostatic stiffness (as c33
= ρ · g · Aw, being c33 the heave hydrostatic stiffness) and, in
cases where respectively the added mass and the potential
damping were almost constant, also the natural frequency of
oscillation ω0 and the damping ratio, ζ. This is what happened
in the case of low draft, which the free-decay tests refer to. In
this case both added mass and potential damping in heave, and
partly in pitch, were almost constant (Figure 8, 9); this,
combined with a significant increase in Aw caused by even small
trim differences, determined ω0 to increase and ζ to decrease.
These effects are particularly evident on ω0 in heave and pitch
(Table I).
It is quite difficult to assume univocally one value for the
experimental static trim position; this was possibly not exactly
0° all the time, any difference affecting the results by
determining an overall lower Aw with the mentioned
consequences on ω0 and ζ.
However, for the sake of mooring tension analysis, the
most important mode of motion is surge, which is far less
affected by variations in Aw. Due to this, 0° trim could be
reasonably assumed since it provided a good overall
correspondence on the mooring tension in all the 6 cases
considered in the analysis.
Here it also emerged that ω0 seems to be more important
than ζ in characterizing the hydrodynamic behavior of the
device; a good correspondence on ω0, such as the one achieved
by adding quadratic damping, allowed to reproduce the
experimental response very well indeed, although ζ was still
quite different from the one experimentally determined. With
regard to this, it has to be mentioned that, being the system
highly damped in surge, only a few oscillations were available
to estimate ζ both in the experimental and numerical case,
increasing the uncertainties related to it.
It has to be mentioned here that a commercial unit of Wave
Dragon would be built differently respect to the small scale
model considered in this study, which was designed for the
purpose of preliminary tank testing only. On a commercial unit
the buoyancy elements would be distributed more uniformly
throughout the main platform, limiting significantly the
hydrodynamic sensitivity to the draft and mean trim position.
It is also worth noting that Wave Dragon is designed so that
the added buoyancy provided by closed compartments is
enough for it not to sink in any condition, always allowing a
minimum Rc. This can be already observed at the model scale,
where in order to test the lowest Rc needed for survivability
mode some external weight had to be added on top of the
platform deck.
9
Copyright © 2012 by ASME
Therefore, the only case in which the device can get
significantly submerged is by consciously entering the
survivability mode in extreme waves. This would be due to the
combination of extremely low Rc and dynamic loads originated
from very high waves (resulting overall in a low heave restoring
force) rather than to a lack of buoyancy. By effect of this, as
operational conditions are re-established the device would
naturally float back to the desired Rc.
Finally, in figure 8 it can be observed how the added mass
in heave and pitch is negative for low frequencies in the case of
dr = 0.29 m, while the same is positive for either higher or
lower drafts. This phenomenon takes place only as the body has
a very low submergence (the total height of the model is 0.28
m) and has been previously described by many hydrodynamic
researchers. For a more detailed description, which is out of the
scope of this paper, the reader should refer to [8].
CONCLUSIONS AND FURTHER WORK
The hydrodynamic characterization of the Wave Dragon
has been carried out. A numerical model has been developed
and validated with experimental results derived from tank tests
of a small scale model, being ready for future use. Added mass,
linear and quadratic damping, hydrostatic stiffness and
excitation force have been derived for different drafts.
The numerical results obtained for the tension in the main
mooring line are matching very well the experimental ones. The
numerical model can be therefore considered well calibrated
and reliable for future use.
The validity of these parameters can be extended to larger
scale devices as long as a geometrical similarity is maintained;
therefore they will be used in the future coupled analysis aimed
at designing a suitable mooring for a 1.5 MW Wave Dragon to
be deployed in Hanstholm, at the DanWEC test site.
A deeper understanding on the hydrodynamic behavior of
Wave Dragon is also a result of this study, especially with
regard to the influence of its static mean position on the natural
frequency of oscillation, the damping ratio and hence the
mooring line tension. Results are in agreement with previous
experimental findings, and will be very valuable for the
implementation of the control strategy on large-scale devices.
Future work will include the use of the software
SIMO/Riflex to analyze the response of the Wave Dragon to
environmental and mooring loads for extreme wave conditions
at the DanWEC deployment site. Different mooring layouts will
be assessed and the best option chosen based on criteria of
reliability and economic feasibility.
ACKNOWLEDGMENTS
This work has been carried out at CeSOS, which personnel
the first author acknowledges for their availability and support.
He also acknowledges Aalborg University for financial support.
The first two authors gratefully acknowledge the financial
support from the European Commission through the 7th
Framework Programme (the Marie Curie Initial Training
Network WaveTrain2 project, Grant agreement number
215414).
The researchers from CeSOS also acknowledge the support
from FP7 Marina Platform.
REFERENCES
[1] Frigaard P., Hald T. Forces and Overtopping on 2nd
generation Wave Dragon for Nissum Bredning. Hydraulic and
Coastal Engineering Laboratory, Aalborg University. Technical
Report. Phase 3 project, Danish Energy Agency. Project no:
ENS-51191/00-0067
[2] Soerensen H. C., Friis-Madsen E. Sea Testing and
Optimisation of Power Production on a Scale 1:4.5 Test Rig of
the Offshore Wave Energy Converter Wave Dragon. Final
Technical Report, PROJECT N°: NNE5-2001-00444,
CONTRACT N°: ENK5-CT-2002-00603.
[3] DanWEC – Danish Wave Energy Center, Hanstholm
c/o Hanstholm Havneforum, http://www.danwec.com
[4] Riflex Manual.
http://www.sintef.no/Home/Marine/MARINTEK/Software-
developed-at MARINTEK/RIFLEX/
[5] Parmeggiani S., Kofoed J.P., Friis-Madsen E. Extreme
Loads on the Mooring Lines and Survivability Mode for the
Wave Dragon Wave Energy Converter. Proceedings from the
World Renewable Energy Congress, 2011.
[6] DNV GeniE User Manual, program version 5.3.
[7] DNV HydroD user manual – program version 1.1-
01,2004.
[8] Newman J.N., Sortland B., Vinje T. Added Mass and
Damping of Rectangular Bodies Close to the Free Surface.
Journal of Ship Research, Vol. 28, No. 4, Dec. 1984, pp 219-
225.
10
Copyright © 2012 by ASME
