The design of small seabed-mounted bottom-hinged wave
energy converters
M. Folley, T.W.T. Whittaker and J. van´t Hoff
Wave Energy Research Group, School of Planning Architecture and Civil Engineering,
Queen`s University Belfast, Belfast BT9 5AG, UK
E-mail: m.folley@qub.ac.uk
Abstract
A linearised frequency domain numerical model of
small seabed-mounted bottom-hinged wave energy
converters is developed that accounts for vortex shedding at
body edges and decoupling at large angles of rotation. The
numerical model is verified and calibrated using data from
wave-tank experiments. It is found that in general the
device capture factor increases with both the device width
and wave frequency due to increasing wave force. The
model also indicates that for typical flap dimensions and
incident wave amplitudes the peak in capture factor at the
body’s natural pitching frequency is suppressed due to
viscous losses and motion constraints. The effect of viscous
losses and motion constraints are also responsible for
limiting the increase in performance that is obtainable with
phase control. Three cost functions, power per unit
displaced volume, power per unit structural task and power
per unit surge force are produced and applied to the results
of a parametric analysis. Three distinct regions of the
design space are identified; EB Frond and BioWave are
found to sit in one region, WaveRoller in another region
and Oyster in the final region. Characteristics are identified
for each region and related to the distinct designs of the
commercial systems identified.
Keywords: design, flap, optimisation, shallow water,
surging, natural amplification
Nomenclature
B
Bv
I
Ia
I∞
kp
K(t)
P
Tw(t)
Tw
Λ
θ
θmax
Θ
ω
ωn
Radiation damping coefficient
Viscous damping coefficient
Body moment of inertia
Added moment of inertia
Infinite freq. added moment of inertia
Pitch stiffness of body
Impulse response function
Power capture
Wave torque at time, t
Vector of wave torque
Applied damping coefficient
Angular rotation of body
Maximum rotation of body
Vector of body rotation
wave frequency
natural pitching frequency of body
© Proceedings of the 7th European Wave and Tidal
Energy Conference, Porto, Portugal, 2007
Introduction
Wave energy is an abundant and promising source of
clean, renewable energy that is yet to be exploited on a
large scale; the technology remains in its infancy. As a
consequence a large range of different technological
solutions exist which exploit different characteristics of the
incident wave energy and are based on differing
assumptions regarding operation and cost. Ultimately, the
only method of identifying the best solutions is by
comparing the full life-cycle performance and costs of the
different technologies, whilst ensuring that the
performances and costs are not skewed by the particular
embodiments of the technologies used for the analysis.
Indeed, this may be the only way to reliably compare
devices as diverse as the 54,000 Tonne Wave Dragon, rated
at 11 MW [1] and a 10 Tonne heaving buoy, rated at 10
kW [2]. However, where device concepts have
hydrodynamic and dynamic similarities it may be possible
to make instructive comparisons.
From the current range of wave energy converter
technologies being commercially developed it is possible to
describe four of these as small bodies that rotate about a
hinge located closer to the seabed than the bulk of the body,
with the seabed providing the reaction to the wave force
and with power extracted from the surge component of the
wave motion. These are termed as small seabed-mounted
bottom-hinged wave energy converters. This group is also
distinct because it extracts power in surge, whilst the
majority of studies have focused on heaving devices. It has
been shown [3], and will be further illustrated in this paper
that these devices have very different characteristics to
heaving devices, which influences the identification of
optimal designs. The four wave energy converters are; EB
Frond [4], WaveRoller [5], Oyster [6] and BioWave [7],
which are illustrated in Figure 1. The basic concept is not
new and can be traced back to at least 1954 when a patent
for this idea was filed [8], with further detailed work being
done by Scher [9]. However, each device has developed
within the conceptual framework depending on what is
perceived as critical in the hydrodynamics or costs. It is
instructive to consider how the commercial companies
describe their devices since it provides an indication of
their perceived mode of operation.
EB Froond has been deeveloped by Lanncaster Universiity
and The Engineering
E
Busiiness Ltd. From their report to thhe
DTI [4] the EB Frond is deescribed as
Figure 1((a): EB Frond
EB Fronnd is a wave gennerator with a coollector vane at
the top of an arm pivooted near the seeabed. The arm
oscillatees like an inverrted pendulum, driven by the
water particle
p
motionn in the wavees. EB Frond
incorporates devices which allow the pendulum
motion to be tuned to the dominant frrequency of the
h
cylindders connected
waves, and a set of hydraulic
betweenn the arm and thhe structure whicch deliver highpressuree oil to a hydrraulic motor coonnected to an
electricaal generator. Thhe structure is held rigidly on
the seaabed in a nearrshore locationn and remains
submergged at all times.
Roller has been developed
d
by AW
W-Energy OY annd
WaveR
is describeed on its websitee [5] as
A WaveeRoller device iss a plate anchoored on the sea
bottom by
b its lower partt. The back and forth
f
movement
of bottoom waves movees the plate, and
a
the kinetic
energy produced
p
is colllected by a pistton pump. This
energy can be convertted to electricitty by a closed
c
witth a hydraulic
hydraullic system in combination
motor/ggenerator system
m.
Figure 1(b
b): WaveRollerr
Oyster has been devveloped by Quueen’s Universiity
Belfast and Aquamarine power
p
Ltd. Oyster is described on
o
its websitee [6] as
Oyster™
™ is a near-shore bottom-m
mounted wave
energy converter designned to interact efficiently with
the doominant surge forces in shallow
s
water
waves…
…The Oyster™ module
m
naturallly swings away
from laarge waves, shhedding load, enabling it to
continuee generating power
p
even during
d
extreme
conditioons. The modulle interacts dirrectly with the
amplifieed surge component in near shore waves;
which means
m
it also caaptures power effficiently in the
smallestt of seas
Figure 1(c): Oyster
Finally,, BioWave has been
b
developed by the Universiity
of Sydney and BioPower Systems
S
Pty. Ltdd, which describes
the system
m on its website [7]
[ as
The wave energy conversion system, bioWAVE™,
b
is
based on
o the swayingg motion of seaa plants in the
presence of ocean waves. The hydrodynamic
interacttion of the bladees with the oscilllating flow field
is desiggned for maxim
mum energy abbsorption… In
extremee wave conditioons, including hurricanes,
h
the
bioWAV
VE™ is autom
matically triggeered to cease
operatinng and assume a safe position lyying flat against
the seabbed.
Figure 1((d): BioWave
The deescriptions provvided for the different devices
indicate thhat each developper emphasises slightly differeent
aspects of the device operation,
o
whicch unsurprisinggly
results in the distinct solutions seen. To understand how
distinct solutions exist, even with such a simple concept, it
is necessary to first consider the basic dynamics of this type
of device.
and magnitude of hydrodynamic heave forces. As an
alternative it is possible to define a maximum angle of
rotation that ensures that decoupling effects are minimised,
e.g.
1
θ max ≤ 30°
Dynamic analysis
The dynamics of small seabed-mounted bottom-hinged
wave energy converters can be modelled and subsequently
analysed in a large number of different ways, with each
method having particular characteristics that make it more
or less suitable for particular purposes. For example, wavetank modelling can provide accurate performance data for
determining the expected productivity of a particular
prototype; however it is less useful in identifying how the
prototype design may be improved. Conversely, inviscid
irrotational analytical models are good at identifying the
maximum power capture of a particular system, though
they may be less useful in determining what the actual
performance will be due to the effect of motion constraints
and viscous losses.
To understand the dynamics of a particular device the
model must be sufficiently clear that the key elements that
influence performance can be identified and the effect of
changes in these elements understood. It is more important
with this type of model that comparative performances are
reasonably accurately modelled, whilst predictive
modelling of performance is of a lesser importance.
It is suggested that the dynamics of small seabedmounted bottom-hinged wave energy converters can be
usefully modelled using a single degree-of-freedom system.
Equation 1 is a common method of representation for this
type of system, where the hydrodynamics are assumed to
be linear.
t
T (t ) = ( I + I )θ&& + θ&(τ ) K (t − τ )dτ + k θ + Λθ&
(1)
∞
w
∫0
(3)
Clearly this does not capture the complexity of the
decoupling wave torque, especially since the constraint is
imposed on the hydrodynamics rather than a consequence
of the hydrodynamics; however they are likely to have
similar comparative effects, which makes the
representation suitable for developing an understanding of
this device type.
The representation can be significantly simplified if it is
assumed that the wave force is sinusoidal and that the
effects of vortex-shedding can be linearised whilst
maintaining constant the amount of energy dissipated [10].
Assuming sinusoidal motion of the body means that the
convolution integral can be reduced to frequency dependent
added mass and damping coefficients. This allows
equations (2) and (3) to be re-written using vector notation
as;
[
]
Τw = ( I + I a )(ωn2 − ω 2 ) + jω( B + Λ + 38π Bvω Θ ) Θ
(4)
Θ ≤ θ max
(5)
For this system, the power capture is given by
P=
1
2
Λω 2 Θ
2
(6)
p
In this representation the hydrodynamic radiation torque
is decomposed into an infinite frequency added inertia term
and a convolution integral, which represents a memory
function due to the body’s motion. It also uses a small
angle approximation for pitch stiffness due to the excess
buoyancy of the flap so that the restoring torque is
proportional to the angle of rotation. Finally, the applied
damping is assumed to be proportional to the angular
velocity of rotation so that the equation remains linear.
Unfortunately, such a representation omits two important
characteristics; the torque induced due to vortex-shedding
and the decoupling of the wave torque at large angles of
rotation. A common approximation of the torque induced
by vortex-shedding is that it is proportional to the
instantaneous velocity squared (as estimated by Morrison’s
equation); thus by including this term, the representative
equation becomes
Tw (t ) =
t
( I + I ∞ )θ&& + ∫0 θ&(τ ) K (t − τ )dτ + k pθ + Λθ& + Bv θ& θ&
(2)
Including the effect of wave torque decoupling within
the representative equation is significantly more complex
since the effect is likely to depend on the phase relationship
between the wave and body rotation as well as the phase
Although the dynamics of the model have been
linearised, the optimisation of the power capture remains
non-linear and complex due to the combination of the
viscous damping term and the possible motion constraints.
The solution requires at least for a quartic, or biquadratic,
equation to be solved; the analytical solution of which is far
too complex to provide any insight. Consequently, the
optimum applied damping coefficient and maximum power
capture are determined iteratively using a golden section
search together with parabolic interpolation (implemented
in Matlab using the fminbnd function), which provides a
robust and simple solution for maximum power capture.
For any particular configuration and incident wave, the
optimum applied damping coefficient and maximum power
capture depends on the relative magnitudes of the
components in equations (4) and (5). These forces can be
usefully represented using an Argand diagram, which plots
the real and imaginary parts of equation (4) as vectors as
illustrated in Figure 2. By graphically illustrating the
relative magnitudes of the forces and the phase relationship
between wave torque and flap motion, the performance can
be more easily understood.
decoupling from motion in surge as well as the increased
tendency for overtopping of the flap; an angle of 30º has
been chosen as being reasonable.
Λω Θ
WAMIT
Experiments Nov-06
35
Tw
30
25
Wave torque (Nm)
Bvω 2|Θ|Θ
Bω Θ
(kp - Iω 2)Θ
15
10
Θ
5
0
Figure 2: Example of an Argand diagram of the
forces
An analysis of equations (4) and (5) shows that
maximum power capture is achieved when;
•
the flap is tuned (ωn = ω),
•
the motion is unconstrained (|Θopt| < θmax),
•
the viscous damping coefficient is zero (Bv = 0).
Unfortunately, the analysis does not provide any
information on how this may be achieved. Moreover, it is
not possible to directly link any one of these factors to a
single dimension of the device as changing any device
dimensions will influence all of these factors so that their
influences on the maximum power capture may change.
To understand the performance and therefore design of a
small seabed-mounted bottom-hinged wave energy
converter, it is necessary to identify what are the key
influences in the body dynamics for that particular
configuration, recognising that these may differ from other
configurations in this classification. This is further
complicated by the fact that the key influences may also
change with wave frequency and amplitude.
2
20
Model verification and calibration
The wave torque and hydrodynamic coefficients, added
inertia and radiation damping are obtained for a wide range
of flap configurations using WAMIT, a linear frequencydomain hydrodynamic analysis software package. WAMIT
has previously been found to provide good estimates of the
wave forces and hydrodynamic coefficients. Facilities for
confirmation of the hydrodynamic coefficients are not yet
available; however comparison of WAMIT data with the
wave torque measured on a 20th scale model of a flap is
shown in Figure 3. The oscillations of the experimental
wave torque are thought to be due to longitudinal
reflections in the wave-tank; however, it can be seen that
the WAMIT data provides a good estimation for the wave
torque.
With WAMIT providing data for the wave torque &
hydrodynamic coefficients and the flap inertia and pitch
stiffness derived from the flap dimensions (see section
below), the only remaining parameters requiring definition
are the maximum angular rotation and the viscous damping
coefficient. Definition of the maximum angular rotation is
complex since it needs to encompass both the tendency for
0
1
2
3
4
5
6
Period (s)
Figure 3: Comparison of WAMIT wave torque and
experimental data
The viscous damping coefficient has been estimated
using data from wave-tank testing of a flap in a set of
polychromatic seas. The value that provided the best
estimate of power capture using equation (4) in a timeseries simulation of the test is used. To estimate the viscous
damping coefficient for other flap configurations it is
assumed that the viscous damping torque is proportional to
the square of the amplitude of flap motion at the water
surface, the flap facing area and inversely proportional to
the flap thickness, using previously identified relationships
of body dimensions and the viscous loss coefficient [11].
There are clearly many weaknesses with this method when
compared to a more thorough investigation of non-linear
effects, which could be used to estimate the viscous
damping coefficient. However, the method used is expected
to be relatively robust and provides an estimate of the
viscous damping coefficient considered to be sufficiently
accurate for the intended analysis and subsequent
conclusions. Further studies into this coefficient are ongoing.
3
Parametric analysis
A parametric analysis of all the possible configurations
of small seabed-mounted bottom-hinged wave energy
converters is not practicable due to the number of potential
configurations, even if they could be adequately defined.
Consequently, the analysis is restricted to flaps that are
surface piercing and extend the full depth of the water
except for a 1.0 metre high base that represents the
foundations of the device. It is assumed that in all cases the
flap’s density is 300 kg/m3 and that its centre of mass is in
the centre of its submerged volume. These values are
believed to be reasonable for this type of wave energy
converter. Within this configuration, the effects of water
depth, flap width and flap thickness are studied.
Equations (4) – (6) are used to estimate maximum power
capture for each parametric configuration for a
representative range of monochromatic wave frequencies
and amplitudes to provide indicative performances of the
configurations. This type of monochromatic analysis is
together with 4 measures of performance (capture factor,
power per unit volume, power per unit structural task and
power capture per unit surge force). Moreover, there is the
analysis of the hypothetical cases where one of the key
factors is removed to determine its influence on maximum
power capture. It is clearly impracticable to illustrate and
describe the interaction of all the parameters and measures
of performance, but the presentation is limited to showing
results that illustrate important relationships and can lead to
insightful conclusions.
As a precursor to the calculation of power capture and to
provide insight into the potential significance of tuning on
device performance it is considered beneficial to calculate
the variation in natural pitching period with water depth,
flap width and flap thickness. Figure 4 shows the variation
of natural pitching period, derived from the pitch stiffness
and total moment of inertia, for a flap thickness of 2.0
metres. This illustrates that only relatively small flaps in
shallow water have natural pitching periods in the region of
the incident wave periods (7 – 13 seconds) and outside of
this region, in the absence of additional tuning forces, the
flap will be poorly tuned.
20
30
25
20
18
30
25
14
20
Water depth (m)
16
15
25
12
20
15
10
8
20
15
10
commonly used in the design of wave energy converters
because it allows a large number of configurations to be
investigated in a reasonable length of time [12]; simulations
using polychromatic waves and a time-domain analysis are
not considered practicable.
In addition the maximum power capture is also
calculated for the following hypothetical cases:
•
The vortex damping coefficient is zero.
•
There are no motion constraints.
These cases clearly do not represent realistic conditions,
but can help to indicate to what extent the particular
factors, vortex shedding and motion constraints
respectively, influence the actual power capture. The
hypothetical case of perfect tuning is also analysed to
identify the potential increase in performance if phase
control were available.
To compare configurations the maximum power capture
for each configuration should be divided by a reasonable
measure of the configuration’s cost; providing an inversed
measure of the cost per unit kilowatt-hour. Capture factor is
defined as the device power capture divided by the wave
power incident on a representative device length (in this
case we have chosen flap width). Capture factor has often
been used as a measure of a wave energy converter’s
potential and effectively assumes that the cost is
proportional to device width. This may be valid if only
width is varying, but is clearly inadequate to account for
variations in water depth or flap thickness. A second
measure that is often used in maximum power capture per
unit displaced volume. This implies that costs are
proportional to volume, which would appear to be more a
more reasonable assumption (though consideration of ship
costs indicate that a better measure may be to use the
volume to the power 2/3; however this is not used in this
study to limit the number of measures calculated).
Alternatively, it could be considered that the costs depend
primarily on the structural task, defined in this case as
surge wave force multiplied by the distance from this force
to the axis squared divided by the body thickness and
providing a measure of the structural material required in
the device to transmit the wave loads to the seabed [13].
This leads to the performance measure of power per unit
structural task. Finally, it may be considered that the cost
depends primarily on the surge load experienced by the
foundations; thus providing a final measure of power
capture per unit wave surge force.
Without detailed engineering studies it is not possible to
say which of the measures described will provide the best
measure of device cost. Indeed, the legitimacy of the
measures may vary across the design space so that in one
region one particular measure is more appropriate, whilst in
another region this is replaced by an alternative measure
and in general the final cost will undoubtedly depend on
device volume, structural task and foundation loads
(amongst other factors not considered here). Thus all of the
measures of performance are calculated and used to
compare difference configurations.
Although the parametric analysis has been kept to a
minimum, there are still 5 parameters (water depth, flap
width, flap thickness, wave frequency and wave amplitude)
6
5
10
15
20
Width of oscillator (m)
25
30
Figure 4: Variation of natural pitching period with
flap width and water depth
The variation of capture factor with device width, for a
water depth of 8.0 metres, a flap thickness of 2.5 metres, a
wave period of 12.0 seconds and wave amplitude of 1.0
metres is calculated using the linearised frequency-domain
representation and is shown in Figure 5. The capture factor
of an ideal point absorber and the ideal linear model (no
viscous losses or motion constraints) are shown for
reference. This has two surprising features; firstly, the
capture factor increases monotonically from the minimum
width modelled of 4.0 metres to the maximum width of
30.0 metres and secondly, the peak in capture factor that
would be expected when the flap’s natural period equals
12.0 seconds, which occurs at a flap width of
approximately 10.0 metres, is not apparent.
2
1.8
0.1
0.4
0.2
1.8
1.6
1.6
1.4
Wave amplitude (m)
25
30
Figure 5: Variation of capture factor with flap width
The increase in capture factor with flap width is due to
the approximately quadratic increase in wave torque with
flap width. The increasing wave torque means that the
motion required to capture the same amount of power is
reduced, thus reducing the effect of viscous losses and
motion constraints (though in this case it is the viscous
losses that are dominating the hydrodynamics). For the set
of parameters modelled the increase in wave torque with
flap width more than compensates for the deterioration in
flap tuning with increasing flap width.
The absence of the peak in capture factor when the flap
is tuned occurs because the flap rotation required for the
increase in capture factor can not be obtained due to
viscous losses and motion constraints. Figure 5 shows that
the actual capture factor is primarily limited by the viscous
losses, but that even if they could be eliminated the motion
constraints would similarly limit the capture factor when
the flap is tuned. A small improvement in performance
does occur due to the improved phase relationship between
the wave torque and flap rotation, but this is relatively
insignificant compared to the other influences on
performance.
The increase in capture factor that exists close to the
natural period of the flap can be seen in Figure 6, where the
capture factor for a single flap configuration is shown for
difference wave periods and amplitudes. For the flap
illustrated (water depth = 8 metres, flap width = 8 metres,
flap thickness = 2.5 metres) the natural pitching period is
approximately 11.5 seconds. The increase in capture factor
can clearly be seen at this wave period; significantly
however this is only clearly evident for small wave
amplitudes. For this configuration, the peak in capture
factor at the flap’s natural period only occurs at wave
amplitudes of less than approximately 0.4 metres.
Figure 6 also shows that away from the natural pitching
period, the capture factor decreases with the wave period;
the highest capture factors occur in the shortest wave
periods. The increase in capture factor with reduction in
wave period is beneficial because the reduction in incident
wave power is typically accompanied by a reduction in
wave period. Thus, the reduction in wave power may be
partially offset by an increase in capture factor, which leads
to a more consistent power capture for the device.
0
0.8
1 1.5
2
10
15
Wave period (secs)
5
20
Figure 6: Variation of capture factor with wave
period and amplitude for a single flap configuration
Of particular interest in the design of wave energy
converters is the ability to increase the power capture for a
particular device by ensuring that the incident wave force
and body velocity remain in phase. It has been suggested
that this phase control can be achieved using latching [14],
stiffness modulation [15] or complex conjugate control
[16]. In the frequency domain model used these are all
assumed to be equivalent to eliminating the net force in
phase with the flap’s acceleration so that flap velocity is inphase with the wave torque; as such this represents an
upper bound to what can be achieved using phase control.
Figure 7 compares the capture factor for the same flap
configuration described above (water depth = 8 metres, flap
width = 8 metres, flap thickness = 2.5 metres) with and
without phase control for a wave amplitude of 1.0 metres.
1
Without phase control
With phase control
0.9
0.8
0.7
Capture factor
15
20
Width of oscillator (m)
0.1
10
6
0.
0.2
5
0.
6
0.4
8
0.
0
0.6
0.2
0.2
0.8
0.4
point absorber
linear ideal
actual
0.1
0.4
0.2
0.6
1
0.4
0.8
1.2
0.6
1
08
Capture factor
1.4
1.2
0.6
0.5
0.4
0.3
0.2
0.1
0
5
10
15
20
Wave period (secs)
Figure 7: Comparison of capture factor for a flap
with and without phase control
As would be expected, the capture factor for the two
control strategies can be seen to coincide at the flap’s
natural pitching period and the capture factor is higher with
phase control at all other wave periods. However, the
improvement in performance due to phase control is
relatively modest with the maximum increase in capture
factor of approximately 25% occurring in the shortest
period waves. The reason that phase control does not
20
5
10
2
18
16
Water depth (m)
improve performance to an extent that has previously been
predicted [17, 18] is that viscous losses and motion
constraints suppress the large amplifications of motion that
are required. From an alternative perspective it can be said
that the optimum level of applied damping, which limits
viscous losses and excessive motion, means the wave
torque and flap angular velocity are already close to being
in-phase. This can be seen in Figure 8, which shows the
Argand diagrams for the cases of with and without phase
control for the same flap configuration and at a wave period
of 7.0 seconds.
5
10
20
14
10
12
20
5
30
20
10
30
40
60 0
7 0
8 0
9
50
8
6
20
1
60 7 0
8090
50
1 00
2
2.5
3
Thickness of Oscillator (m)
40
30
1.5
Λω Θ
T
w
B ω 2|Θ|Θ
v
20
15
Bω Θ
0
(kp - Iω )Θ
250
16
14
25
1.5
0
Figure 9: Power capture per unit displaced volume
(kW/m3)
1
0
24
0
26
0
300
0
28
4
14
0
16
0
0
3.5
24
1.1
1.2
2
2.5
3
Thickness of Oscillator (m)
0
18
0
2 00
2 20
0
1.5
80
6
1.1
10
1
1
1.3
1
12
8
0.9
1
1.1
1.2
28
240
260
10
0.8
0.8
0.9
8
4
220
180
20 0
220
200
180
12
160
0.7
14
140
0.6
0.7
0.8
1
10
16 0
20
120
12 0.9
0
3.5
Figure 11: Power capture per unit surge force
(W/kN)
Water depth (m)
6
0.
0.6
0.7
2
2.5
3
Thickness of Oscillator (m)
16
0.5
0.6
0
30
0
25
1
18
0.5
14 0
.8
35
35
6
0.4
16 0.
7
0
8
0.4
0.5
30
0
10
20
18
0
12
20
As discussed earlier, the capture factor is not an ideal
measure for assessing the cost of a device since it does not
take account of changes in the flap cross-section; power
capture per unit displaced volume, power capture per unit
structural task and power capture per unit surge force have
been proposed as more suitable measures. Figures 9 - 11
show these alternative cost functions for a flap width of
10.0 metres and a wave of 10.0 second period, 1.0 metre
amplitude, whilst Figure 12 simply shows the variation of
power capture with water depth and flap width.
0
20
140
Figure 8: Comparison of Argand diagrams for a flap
with and without phase control
Water depth (m)
Θ
p
120
Θ
(k - Iω 2)Θ
Water depth (m)
20
18
2
Bω Θ
6
4
0
Bvω 2|Θ|Θ
3.5
Figure 10: Power capture per unit structural task
(W/kJ)
Λω Θ
Tw
40
50
10
1.5
2
2.5
3
Thickness of Oscillator (m)
3.5
4
Figure 12: Variation of power capture (kW)
For the flap width chosen there is a clear ridge of water
depth and flap thickness that maximises the power capture,
which goes approximately from a thickness of 1.0 metres at
a water depth of 18.0 metres to a thickness of 4.0 metres at
a water depth of 10.0 metres as illustrated in Figure 12. In
configurations above this ridge, the power capture reduces
due to poor tuning, whilst for configurations below this
ridge, the power capture is limited because of the maximum
amplitude of rotation. The position of this optimum
changes depending on the incident wave and the flap width.
In general, the wider the flap, the shorter the wave period
and the smaller the wave amplitude the optimum will tend
to move to shallower water and thinner flaps. Power
capture per unit surge force (Figure 11) shows a similar
variation except that the magnitude of the optimum
increases with thickness (whereas it decreases with
thickness for power per unit volume). This is because the
increased thickness has a relatively small influence on the
surge wave force, but increases power capture due to a
reduction in viscous losses and an improvement in device
tuning.
The power capture per unit structural task, shown in
Figure 10, can be seen to increase with decreasing water
depth and increasing flap thickness. As stated above, the
power capture typically increases little with water depth,
whilst the structural task increases approximately
proportionally to the water depth squared, resulting in the
reduction in power capture per unit structural task with
water depth. Meanwhile, increasing the flap thickness
increases the power capture due to a relative reduction in
viscous losses, whilst simultaneously reducing the
structural task due to the deep section available to transmit
the wave torque; both of these lead to an increase in power
capture per unit structural task with flap thickness. Thus,
the highest power captures per unit structural task are
obtained with thick flaps in shallow water.
4
Discussion
It is clear from the analysis of the dynamics of small
seabed-mounted bottom-hinged wave energy converters
developed in this paper that it is difficult to be
unconditional about their design. The relative importance
of factors such as device tuning, viscous losses, motion
constraints and the advantage of phase control all depend
on the dimensions of the device as well as the
characteristics of the incident wave. The conditions that
dictate the optimum design within one part of the design
space may be different from those that dictate it in another;
however it is possible to draw some general conclusions
and relate these to the four commercial systems that
currently exist within the classification of small seabedmounted bottom-hinged wave energy converters. Although
the analysis has only been performed for full-depth surface
piercing flaps, the fundamental representation of the
hydrodynamics is the same for all this device family.
Consequently, the conclusions regarding how different
factors influence performance remain valid in a general
sense, although the exact values may not apply. This
enables the conclusions to be used to assess the design of
EB Frond, WaveRoller, Oyster and BioWave, although
they may not have been represented exactly in the above
analysis.
It is clear that unless large amplitudes of motion can be
achieved and viscous losses can be kept to a minimum the
potential for phase control to increase the power capture
significantly is limited. This implies that such devices need
to have bodies that are relatively thick and be located in
relatively deep water; this part of the design space is
occupied by EB Frond and BioWave. The high structural
task costs of locating this type of device in relatively deep
water and the possible reduction in power capture with
water depth for a device without phase control implies that
locating devices in this depth appears to be sensible only if
phase control is used. Unfortunately, designing a system
with a natural frequency similar to the incident wave
frequency, to minimise the work required of the phase
control system, is likely to be difficult in this water depth
due to the relative magnitudes of added moment of inertia
and pitch stiffness that can be achieved. This leads to the
associated problem of having to cycle large amounts of
energy in the phase control system, with highly efficient
energy transfers required to avoid excessive parasitic
losses. Conversely, the limited increase in power capture
with phase control in shallower water and where viscous
losses are not insignificant means that there is less of an
improvement in power capture to pay for the increased
complexity that the implementation of any kind of phase
control implies; consequently it is likely that a costeffective device of this type would employ dissipative
damping only.
To maximise the power capture per unit structural task
the most important characteristic is to reduce the water
depth, or more accurately the distance from the hinge to the
centre of surge wave pressure. This part of the design space
is occupied by WaveRoller, which does not even pierce the
water surface thereby keeping the structural task small.
Whilst it would be possible to tune this type of device to
the incident wave frequency the potential benefits are
unlikely to be large because the device is likely to operate a
large amount of the time with highly constrained motion
either due to large angular rotations or large viscous losses
or both. Fundamentally devices in this part of the design
space will have a relatively small power output, probably
rated at less than 50 kW (WaveRoller is stated to have a
‘nominal power output’ of 13 kW), due to the limited swept
volume of their motion. Traditionally, the cost per unit
kilowatt of power generated has decreased with increasing
installed capacity, which may limit the potential of devices
in this part of the design space, though this could be overshadowed by lower fundamental construction costs.
Maximisation of power capture per unit displaced
volume leads to the specification of relatively wide devices
in relatively shallow water. This part of the design space is
occupied by Oyster, which is 18 metres wide and located in
a water depth of approximately 10 metres. The analysis
would tend to imply that such devices should be located in
water shallow than 10 metres; however in shallower water
the tidal variation becomes a larger proportion of the total
water depth, which tends to limit the minimum deployable
water depth for a surface piercing device such as Oyster.
Because the natural pitching period of the device is
typically much greater than the incident wave period,
tuning has a minimal influence on performance and the
power capture per unit displaced volume is increased by
increasing the wave torque, which increases approximately
proportionally to the width squared; this leads to the
specification of relatively wide devices. It also implies that
devices should be surface piercing to ensure that the wave
torque generated is maximised, which is consistent with
this part of the design space. Finally, the water depth
should be relatively shallow since this increases the
horizontal amplitude of water particle motion and
consequently the wave force [3]. That is, the maximisation
of power capture per unit displaced volume approximates
to maximisation of wave torque per unit displaced volume.
Maximising the wave torque has the effect of increasing the
structural cost of the device, though a distinction may be
made between the structural loads during normal operation
and the structural loads in extreme sea-states, where the
natural decoupling of the surge wave force may limit the
extreme loads experienced.
By using this basic analysis of the dynamics of this
family of devices it has been possible to identify at least
three distinct regions in the design space that may contain a
locally optimal solutions; each region is occupied by at
least one commercial system implying that each region is
considered to have potential for a cost-effective wave
energy converter. However, it has not been the purpose of
this study, nor would it be possible, to attempt to rank the
commercial systems. Although they are all part of the same
family the assumptions inherent in their manifestations
mean that it is impossible to compare the systems sensibly;
one system may assume that the wave forces can be
resisted economically, whilst another system may assume
that phase control can be implemented economically and
operate effectively. However, what the analysis has shown
is that within each region there are a consistent set of
design requirements associated with the optimal solution
within that region and these are typically different from the
design requirements in the other regions.
Finally it is important to highlight that this analysis
indicates that results of analyses using purely linear
dynamics should be viewed with caution. Whilst there are
undoubted cases where linear hydrodynamics and dynamics
are entirely adequate to both predict performance and guide
the design process, this may not always be the case and
surging flaps are a case in point.
5
Further work
The numerical model developed within this paper is
based on a relatively small amount of experimental
verification/calibration. Further experimental work is
required to increase confidence in the numerical model and
the conclusions made subsequently based on this model.
An area that needs to be particularly targeted is the effect of
vortex shedding on the hydrodynamics and its variation
with body shape and motion. Another area is how best to
model the effect of decoupling of the body at large
amplitude of motion, including the effect of over-topping.
6
Conclusions
It is possible to draw four general conclusions from this
analysis.
• Three distinct regions of the design space for small
seabed-mounted bottom-hinged wave energy converters
have been identified, with an individual set of design
requirements and optimum configuration
• The capture factor of a surging flap-type wave energy
converter typically increases with both flap width and
wave frequency
• The potential increase in power capture at the natural
pitching frequency of a flap is often suppressed by
viscous losses and motion constraints
• Phase control may not produce dramatic increases in
power capture where viscous losses and motion
constraints influence the flap dynamics significantly
Acknowledgements
The authors acknowledge the financial support of:
• Aquamarine Power Ltd.
• EU Marie Curie Actions (WAVETRAIN RTN)
• Department of Trade and Industry (DTI)
• Engineering and Physical Sciences Research Council
(EPSRC)
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