The results demonstrate the importance of tuning the WEC system for specific wave environments to harvest most energy and to avoid potential capsize due to hurricanes etc.

1. 1
Directional Spreading Effect on a Wave Energy Converter
Elliot Hanlin Song NA520 Project
University of Michigan
Abstract
Directional spreading effect on a wave energy converter (WEC) is studied to further optimize the
design under particular sea conditions where directionality becomes important, such as hurricanes.
The hydrodynamic coefficients are obtained and the equation of motion is solved in frequency
domain to obtain response amplitude operators (RAOs) for different motions. The response
spectrum and subsequently the expected response are calculated from RAOs and input spectra.
The resulting power is shown to be the same as that modelled by one-directional spectrum as the
energy of the waves must be the same for both models. The results also demonstrate the importance
of tuning the WEC system for specific wave environments to harvest most energy and to avoid
potential capsize due to hurricanes etc.
Keywords: Directional Wave Spectrum, Wave Energy Converter, Renewable Energy
1. Introduction
1.1 Motivation
Wave-energy converters (WECs) harvest
ocean wave energy to produce electricity in
an environmentally friendly manner. In order
to maximize power harvest efficiency, the
design natural frequency of the WEC system
is matched to the peak frequency of the wave
spectrum to obtain resonance. Bachynski etc.
(2012) analyzed the behavior of a tethered
WEC in irregular waves and optimized the
configuration of the WEC. The directionality
of waves is mostly neglected in current
design of WECs in the literature because of
the axisymmetric geometry of WECs. In
realistic sea conditions, however, the effect
of disturbances such as local winds or even
hurricanes must be considered as an
additional so called directional spreading
term in the wave spectra (Latheef & Swan,
2013).
1.2 Objectives
The objective of this project is to investigate
how directional spreading effect influences
the design of WECs. In order to do so, an
input spectrum must be constructed as a
function of both frequency and angle. This
study derived cosine-fourth angular
spreading function and used the modified
input spectrum to obtain response of the
WEC system. The power takeoff from the
WEC also needs to be studied to provide an
estimated power for potential investors.
2. Methodology
2.1 Assumptions
The WEC is treated as a smooth vertical
cylinder with free motions in surge, heave,
and pitch. Figure 1 shows the simplified
model of WEC by Bachynski etc. Based on
linear assumptions, the equation of motion is
formulated and solved in frequency domain.
The fluid is considered to be incompressible,
inviscid, irrotational, and subject to
linearized free surface and body boundary

2. 2
conditions. Motion caps are applied to
response amplitude operators (RAOs) to
prevent excessive motion near resonance.
Bachynski etc. (2012) showed that the
probability of mooring line breaking during
normal operation of WECs is negligible and
does not constrain the design, since the
mooring line system introduces an additional
surge-pitch resonance with such low
frequency that it is unlikely to be excited by
wind-generated waves. Therefore in this
project the WEC is considered to be
unmoored to simplify the solution, with an
emphasis on the directional spreading effect
on the WEC only.
Figure 1 Wave energy converter (Bachynski,
Young, & Yeung, 2012).
2.2 Mathematical Formulation
2.2.1 Equation of motion
Bachynski etc. have derived the equation of
motion for the WEC in frequency domain as
Figure 2 Equation of motion derived by
Bachynski etc.
In this study, the only difference is the
absence of terms that are related to mooring
lines such as Mt, It, TMt, K11, K33, and K55.
The free-floating cylinder experiences
resonance at two frequencies. Resonance
frequency in the decoupled heave motion is
33
3
33 33
C
M
ω
µ
=
+
(1)
And the coupled surge-pitch resonance
frequency can be found by
( )( ) ( )
55 11 11
51 2
11 11 55 55 51 51
(M )C
M M M
µ
ω
µ µ µ
+
=
+ + − +
(2)
2.2.2 Response amplitude operators (RAOs)
RAOs are defined as the response magnitude
in a particular motion due to unit amplitude
waves. RAOs are equivalent to transfer
functions in linear systems. In this project,
because of the axisymmetric geometry of the
WEC, RAOs in all three motions are
independent of wave directions.

3. 3
To account for viscous effects in realistic sea
conditions, motion caps are applied to the
RAOs to prevent excessive motions near
resonance. In current design, motion caps are
taken as constants: 5 m/m in surge and 10
deg./m in pitch. For heave motion, however,
the motion cap is assumed to be T/Hs, which
increases linearly with draft.
2.2.3 Directional wave spectrum
For the purpose of directional spread, the
directional wave spectrum density function is
assumed to be (Gilloteaux & Ringwood,
2009)
( ) ( ) ( ), ,S S Dω θ ω ω θ= (3)
, where ( )S ω is the one-directional energy
spectral density function. Since the
directional spreading effect is due to locally
generated wind or hurricane, the waves are
not fully developed. JONSWAP spectrum is
selected in this study. ( ),D ω θ is the
parametric angular spreading function. The
formulation of ( ),D ω θ requires that
( ) ( ) ( )
0 0
,S D d d S d
π
π
ω ω θ ω θ ω ω
∞ ∞
−
=∫ ∫ ∫ (4)
hence ( ),D ω θ must satisfy
( ), 1D d
π
π
ω θ θ
−
=∫ (5)
so that the total energy in the directional
spectrum is the same as the total energy in the
corresponding one-directional spectrum.
There are several idealized spreading
functions such as cosine-squared model
(Hughes, 1985), cosine-2s model (Gilloteaux
& Ringwood, 2009) (Hogben & Cobb, 1986),
etc. In this study, the cosine-fourth spreading
function which is recommended by CERC
(Costal Engineering Research Center) to
represent locally generated waves is derived
and applied. It is simple because it is
independent of frequency and it is more
accurate than cosine-squared model (Hughes,
1985). Cosine-fourth spreading function is
constructed as
( )
( )4
0 00
8
cos
2 23
0
D
otherwise
π π
θ θ θθ θ
θ π
⎧
− + < < +−⎪
= ⎨
⎪
⎩
(6)
, where 0θ is the mean wave direction in
radians. The coefficient 8/3π is determined
from Eq. (5) by simple trigonometry
identities.
2.2.4 Spectral response
From input spectrum and RAO of the WEC,
the spectral response can be calculated as
( ) ( ) ( )
2
, ,RS RAO Sω θ ω ω θ= ⎡ ⎤⎣ ⎦ (7)
And the expected amplitude for a particular
motion can be found as
( )exp
0
,RS d d
π
π
ζ ω θ ω θ
∞
−
= ∫ ∫ (8)
Similarly, the expected power takeoff can be
calculated as
( ) ( )
2
exp
0
,pP RAO S d d
π
π
ω ω θ ω θ
∞
−
⎡ ⎤= ⎣ ⎦∫ ∫ (9)
, where the power takeoff RAO is calculated
by Fitzgerald & Bergdahl’s method (2008) as
( )
( )2
0
1
2
PTO
p
B
RAO
ζ ω
ω
ζ
= (10)

4. 4
2.3 Numerical Implementation
2.3.1 Wave environment input
Station 41004 from National Data Buoy
Center near South Carolina is chosen as the
design location (NDBC, 2015). The averaged
wave environment data is tabulated in Table
1.
Table 1 Wave environment for input spectrum
Water depth (H) 38.4 m
Significant wave height (Hs) 1.3 m
Mean wave direction ( 0θ ) 120 deg. S.E.
Wind speed (U10) 12.7 m/s
Fetch (LF) 100000 m
The directional wave spectrum ( ),S ω θ
matrix can therefore be obtained from Eq. (3)
for each frequency and angle.
2.3.1 Dimensional input
Bachynski etc. (2012) proved that for a given
sea environment, the mean annual power
increases with diameter and decrease with
draft within design criteria Hs ≤ T ≤ H/2 and
0.1 ≤D/T ≤0.8. In this study, the ratios are
D/H=0.3 and D/T=0.8 for optimum output.
Fitzgerald and Bergdahl (2008) provided a
solution to obtain the center of gravity and
radius of gyration for pitch motion. The
power takeoff, which is treated as a linear
damping term in the equation of motion, is
tuned to be equal to the resonant heave
hydrodynamic damping (Yeung, Peiffer,
Tom, & Matlak, 2011) but without
consideration of viscous effect so it is
essentially heave radiation damping B33PTO.
The power takeoff damping in surge and
pitch is neglected. The proposed dimensional
input of the WEC is specified in Table 2.
Table 2 Dimensional Input
Diameter (D) 11.5 m
Draft (T) 15.3 m
Center of gravity (ZG) -8.1045 m
Radius of gyration (RP) 11.1423 m
Power takeoff damping (B33PTO) 1100 Ns/m
2.3.2 Hydrodynamic coefficients calculation
Non-dimensional hydrodynamic coefficients
and exciting forces in surge, heave, and pitch
are calculated using the method by Yeung
(1981) and Johansson (1986) for each
frequency with interval of 0.01 rad/s over 0
to 2.5 rad/s. Linear interpolation is used to
obtain finer hydrodynamics and forcing
coefficients. At the end, all non-dimensional
coefficients are dimensionalized for the
calculation of response.
2.3.3 Solution of equation of motion
The equation of motion is solved by iteration
because wave radiation forcing coefficients
are frequency dependent. The solution is a 3-
dimension complex vector. The magnitude
and phase of the responses are calculated by
MATLAB built-in functions abs and atan2,
respectively. Similarly, the solution for each
frequency is obtained in a loop.
3. Results and validation
3.1 Directional wave spectrum
The directional wave spectrum based on
JONSWAP one-directional wave spectrum
and cosine-fourth parametric angular
spreading function is shown in Figure 3. The
directional wave spectrum peaks at frequency

5. 5
around 0.95 rad/s and at angle of -1.04 rad
and 2.1 rad, corresponding to the mean wave
direction 120 deg. S.E. and 180 deg. after
that, i.e., 300 deg. N.W., respectively. These
two directions are essentially the same as
they are both along the main wave direction.
As a pictorial verification, Figure 3 has the
same trend as Figure 4 by Hogben & Cobb
(1986), which is based on cosine-2s model.
The difference between Figure 3 and Figure
4 is due to the different directional spreading
function models, but the peak frequencies are
located at the same frequencies and angles.
Figure 3 Directional wave spectrum
Figure 4 Directional spectrum based on cosine-
2s model from buoy DB1 (48°43’ N 8°58’ W) in
1979 (Hogben & Cobb, 1986)
3.2 RAOs and spectral responses
The resulting RAOs with motion caps based
on the solution of the equation of motion are
shown in Figure 5. As discussed in 2.2.2,
RAOs are independent of angles. As a
verification, the RAOs in heave and pitch are
both 1 at zero frequency and 0 at infinite
frequency. For surge motion, however, the
RAO at zero frequency can be very large
because there is no restoring force and little
system damping in surge for an unmoored
WEC. This peak is not due to resonance, and
it is not applicable to the design of WECs.
Figure 5 RAOs with motion caps
The spectral responses results are shown in
Figure 6, which indicates that the input
spectra are significantly modified by RAOs
in heave motion and less modified in surge
and pitch motions. This is because the peaks
of input spectrum and RAO in heave motion

6. 6
are located at very close frequencies (0.95
rad/s and 0.72 rad/s, respectively), whereas
the RAOs of surge and pitch climax at around
0.3 rad/s. Since the input spectrum is based
on JONSWAP one-directional spectrum,
significantly large values of are present only
at frequencies close to 0.95 rad/s.
Figure 6 Spectral responses
3.3 Power takeoff RAO and power response
spectrum in heave
Since only heave motion results in significant
response from the waves, the power takeoff
estimation is based on heave motion only.
The resulting power takeoff based on Eq.
(10) is shown in Figure 7. It can be shown
that the power takeoff RAO peaks at 0.72
rad/s, same as heave resonance frequency
because from frequency domain analysis,
velocity is ( )3iωζ ω .The maximum velocity
also locates at the resonance frequency. The
resulting response spectrum is shown in
Figure 8, which is similar to heave response
spectrum in Figure 6 with concentrated
response near the resonance frequency. From
the two plots we know that maximum power
RAO is about 20 kW/m and the maximum
power response spectrum is 35 kW2
s, both of
which are reasonable values.
Figure 7 Power takeoff RAO in heave motion
Figure 8 Power response spectrum in heave
3.4 Expected power and energy harvest
The expected power of the WEC is calculated
from Eq. (9). The double integral is
essentially the volume under the power

7. 7
response surface shown in Figure 8. It is
calculated in MATLAB by numerical
interpolation and integration. The resulting
expected power for the current configuration
and wave environment is 0.9204 kW.
As a verification, this power must agree with
the results from one-directional wave
spectrum since the energy of the waves must
be the same for both models. The results from
JONSWAP spectrum shows the agreement.
The expected annual energy harvested from
the WEC is therefore 29.02 GJ, if the power
is assumed to be constant throughout the
year. It must be noticed that in reality, the
WEC can only harvest few percentage of this
amount. The significance of its expected
power and annual energy harvest, shown in
Table 4, can be compared to the results from
Bachynski etc. (2010). in Table 3.
Table 4 Expected power and energy
Draft (T) 15.3 m
Expected power (P3exp) 0.9204 kW
Expected energy (J3) 29.02 GJ
Table 3 Heave energy extraction for a WEC
located near NDBC BUOY 46026, Northern
California (Bachynski, Young, & Yeung, 2010)
4. Conclusion and discussion
The directional spectrum was developed
based on the derivation of cosine-fourth
spreading function and JONSWAP spectrum.
RAOs and response spectra are obtained and
plotted with respect to both wave frequency
and direction. The response spectrum is
shown to peak along the mean wave
direction.
The results show that the expected power
takeoff and annual energy harvest are in good
match to the literature. As a verification for
the directional wave spectrum model, an
agreement in energy has been found between
both directional spectrum and one-directional
JONSWAP spectrum. From the response
spectrum we can draw the conclusion that in
order to harvest the most energy, the WECs
in a wind farm should be collectively placed
along the main wave direction, for those non-
axisymmetric WEC models. To prevent
hurricane attack, on the other hand, WECs or
other offshore structures must be placed
perpendicularly to the incoming hurricane
direction.
For future students, the time domain
simulation of the response of the WEC is
preferred, since locally generated wind and
hurricanes may cause large responses which
violate linear approximations. Furthermore,
viscous damping is not considered in this
project due to its non-linear nature.
However, it may be even exceed radiation
damping in reality.

8. 8
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