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Discussion on the mode mixing in wave energy control systems using the Hilbert-Huang transform

Authors:

Abstract

A great improvement in the absorption of energy of a wave energy converter (WEC) is obtained with a time-varying power takeoff (PTO) damping over a constant damping. In a passive control scheme based on the Hilbert-Huang transform (HHT), the PTO damping is time-varying and tuned to the instantaneous frequency of the wave excitation force. The HHT method relies on the use of the empirical mode decomposition (EMD) method to decompose the wave signal into a number of components (IMFs) from the highest to the lowest frequency component. However, the decomposition process is not always perfect and may result in mode mixing, where an IMF will consist of signals of widely disparate frequency scales, or different IMFs will consist of signals with similar frequency scales. Mode mixing can be caused by intermittent/noisy signals, and by specific amplitude and frequency relations of the original modes in the signal. The aim of this paper is to extend the studies on the use of the HHT for WEC tuning purposes by revealing how the EMD mode mixing problem affects the WEC performance. A comprehensive study using firstly synthetic two-tone waves (i.e., superposition of two sinusoidal waves) is performed. Then, the observations from the two-tone studies are used to further improve the energy absorbed by WECs using the HHT in real ocean wave scenarios resembling the analytic scenario.
Discussion on the mode mixing in
wave energy control systems using the
Hilbert-Huang transform
Paula B. Garcia-Rosa, Marta Molinas, and Olav B. Fosso
Abstract—A great improvement in the absorption of
energy of a wave energy converter (WEC) is obtained
with a time-varying power take-off (PTO) damping over
a constant damping. In a passive control scheme based on
the Hilbert-Huang transform (HHT), the PTO damping is
time-varying and tuned to the instantaneous frequency of
the wave excitation force. The HHT method relies on the
use of the empirical mode decomposition (EMD) method
to decompose the wave signal into a number of compo-
nents (IMFs) from the highest to the lowest frequency
component. However, the decomposition process is not
always perfect and may result in mode mixing, where an
IMF will consist of signals of widely disparate frequency
scales, or different IMFs will consist of signals with
similar frequency scales. Mode mixing can be caused by
intermittent/noisy signals, and by specific amplitude and
frequency relations of the original modes in the signal. The
aim of this paper is to extend the studies on the use of the
HHT for WEC tuning purposes by revealing how the EMD
mode mixing problem affects the WEC performance. A
comprehensive study using firstly synthetic two-tone waves
(i.e., superposition of two sinusoidal waves) is performed.
Then, the observations from the two-tone studies are used
to further improve the energy absorbed by WECs using the
HHT in real ocean wave scenarios resembling the analytic
scenario.
Index Terms—wave energy, Hilbert-Huang transform,
mode mixing.
I. INTRODUCTION
RECENT studies have shown that tuning the power
take-off (PTO) damping of a wave energy con-
verter (WEC) to time-frequency estimations obtained
from the Hilbert-Huang transform (HHT) results in
greater energy absorption than tuning the PTO to a
constant frequency of the wave spectrum [1], or to
time-frequency estimations from the extended Kalman
filter (EKF), and frequency-locked loop (FLL) method
[2]. Both the EKF and FLL methods provide single
dominant frequency estimates, whereas the HHT pro-
vides the instantaneous wave-to-wave frequency of the
oscillation modes present in a wave profile. In addition,
by adopting other methods to, e.g., estimate the on-
line dominant wave frequency [3], or determine the
optimal time-varying PTO damping [4], other studies
have also shown that continuously tuning the PTO
ID 1275 track GPC.
P. B. Garcia-Rosa and O. B. Fosso are with the Department
of Electric Power Engineering, Norwegian University of Science
Technology, Trondheim, Norway (e-mails: p.b.garcia-rosa@ieee.org,
olav.fosso@ntnu.no).
M. Molinas is with the Department of Engineering Cybernetics,
Norwegian University of Science Technology, Trondheim, Norway
(e-mail: marta.molinas@ntnu.no).
result in greater energy absorption than tuning it to
a constant frequency of the wave spectrum.
The HHT method [5] relies on the use of the em-
pirical mode decomposition (EMD) to decompose the
wave signal into a number of components, named
intrinsic mode functions (IMFs), from the highest to
the lowest frequency component. However, the decom-
position process may result in mode mixing, and an
IMF will consist of signals of widely disparate fre-
quency scales, or different IMFs will consist of signals
with similar frequency scales [6]. Mode mixing can
be caused by an intermittent/noisy signal, by spe-
cific amplitude and frequency relations of the original
modes in the signal, and by a combination of both
cases. Additionally, the mode mixing effects can be
attenuated by applying, e.g., masking signals prior to
the EMD procedure [7] [8], or by using the ensemble
EMD (EEMD), a white noise-assisted EMD method [9].
Focusing on the case when mode mixing is caused
by specific amplitude and frequency relations of the
original modes in a signal, [10] presents a rigorous
mathematical analysis that shows how the EMD sepa-
rates the original modes in signals with two frequency
components. Three different domains are identified by
the authors depending on the frequency and amplitude
ratios of the modes. After the EMD procedure, the
components can be separated and correctly identified
(domain 1), considered as a single wave-form (domain
2), or the EMD does something else (domain 3) [10].
In this paper, the focus is also on mode mixing
caused by specific amplitude/frequency relations of
original modes in a wave signal. In this framework,
the aim is to extend the studies on the use of the
HHT for WEC tuning purposes with a passive control
(PC) strategy, by revealing how the EMD mode mixing
problem affects the WEC performance. A comprehen-
sive study using initially synthetic two-tone waves
(superposition of two sinusoidal waves) is performed.
Then, the mode mixing conditions observed for the
two-tone wave studies are used to further improve
the energy absorbed by WECs using the HHT in real
ocean wave scenarios resembling the synthetic two-
tone wave scenario.
II. PASS IV E CON TR OL U SI NG T HE HHT
Here, we assume linear hydrodynamic theory, and
consider a single oscillating-body represented as a
truncated vertical cylinder constrained to move in
heave. By neglecting friction and viscous forces, the
GARCIA-ROSA et al.: DISCUSSION ON THE MODE MIXING IN WAVE ENERGY CONTROL SYSTEMS USING THE HILBERT-HUANG TRANSFORM
heave motion x(t)of the floating cylinder is described
by:
M¨x(t)+
t
Z
0
hr(tτ) ˙x(τ)+Sx(t) = fe(t)+fp(t),(1)
with the kernel of convolution term given by [11]
hr(t) = 2
π
Z
0
Br(ω) cos(ωt τ)dω , (2)
where M= [m+mr()],mis the body mass, mr()
is the infinite-frequency added mass coefficient, Sis
the buoyancy stiffness, Br(ω)is the radiation damp-
ing coefficient, ω, is the wave frequency, fe(t)is the
wave excitation force, and fpis the force of the PTO
mechanism.
The excitation force is calculated as
fe(t) =
Z
−∞
he(tτ)ζ(τ)dτ , (3)
where ζis the wave elevation, and heis the inverse
Fourier transform of the excitation force transfer func-
tion He(ω),
he(t) = 1
2π
Z
−∞
He(ω)eiωt dω , (4)
which has low-pass filter characteristics for floating
WECs. Equation (3) is non-causal, since in fact, the
pressure distribution is the cause of the excitation force
and not the incident waves [12].
For the passive control, the PTO force is parameter-
ized as a function of the damping Bp,
fp(t) = Bp(t) ˙x(t),(5)
where BpR+is continuously modified, and tuned to
the excitation force frequency. Here, the PTO damping
is calculated as
Bp(t) = p(Br(ˆωd))2+ (ˆωd(m+mr(ˆωd)) S/ˆωd)2,
(6)
where ˆωd(t)is the instantaneous frequency of the domi-
nant IMF component of the excitation force [1].
Prior to calculating the instantaneous frequency, the
excitation force is decomposed into a number of IMF
components by applying the EMD. Therefore, the
dominant IMF is identified through the comparison of
the energy of each IMF signal (Eci) with the energy of
the excitation force signal (Efe) [1],
Eci=ZT
0
|ci(t)|2dt , Efe=ZT
0
|fe(t)|2dt , (7)
where ci(t)is the i-th IMF component. The dominant
component cd(t)is the IMF with the highest Eci/Efe
ratio. The Hilbert transform (HT) is then applied to
cd(t)[5]:
υd(t) = 1
πPZ
−∞
cd(τ)
tτdτ , (8)
where Pindicates the Cauchy principal value. Thus,
the dominant IMF is represented as an analytic signal,
zd(t) = cd(t) + d(t),(9)
with amplitude ˆ
Ad, phase ˆ
φd, and instantaneous fre-
quency ˆωd, respectively estimated as
ˆ
Ad(t) = qc2
d(t) + υ2
d(t),(10)
ˆ
φd(t) = arctan υd(t)
cd(t),(11)
ˆωd(t) = ˙
φd(t).(12)
Furthermore, the extracted energy by the WEC over
a time range Tis calculated as
Ea(t) = ZT
0
˙x(t)fp(t)dt , (13)
where ˙xis the body velocity.
III. EMD AND MODE MIXING
The calculation of the instantaneous frequency by
applying the HT (8) directly to the excitation force
signal results in negative local frequencies, as the in-
stantaneous frequency is not well defined for multi-
component signals [13], i.e. signals with more than
one local extrema for each zero crossing. Thus, fe(t)
is decomposed into monocomponent signals (IMFs)
using the EMD. An intrinsic mode function is a signal
that is symmetric with respect to the local zero mean,
and has numbers of zero crossings and extrema dif-
fering at most by one. Therefore, an IMF satisfies the
necessary conditions for a meaningful interpretation of
the instantaneous frequency obtained from the HT [5].
The EMD algorithm identifies local maxima and
minima of fe(t), and calculates upper and lower en-
velopes for such extrema using cubic splines. The
mean values of the envelopes are used to decompose
the original signal into IMFs, in a sequence from the
highest frequency component to the lowest frequency
component. The EMD algorithm is summarized in
Table I. The steps 1 to 5 are known as sifting process.
TABLE I
EMD ALGORITHM.
Step 0: Set i=1;r(t)=fe(t);
Step 1: Identify the local maxima and minima in r(t);
Step 2: Calculate the upper envelope defined by the maxima,
and the lower envelope defined by the minima;
Step 3: Calculate the mean envelope m(t);
Step 4: Set h(t)=r(t)m(t);
Step 5: If h(t)is an IMF, go to next step. Otherwise, set r(t)= h(t)
and go back to step 1;
Step 6: Set ci(t)=h(t);r(t)=r(t)ci(t);
Step 7: If i=N, define the IMF components as c1(t),...,cN(t),
and the residue as r(t). Otherwise, set i=i+ 1 and
go back to step 1.
The decomposition process may result in mode mix-
ing, and an IMF will consist of signals of widely
disparate frequency scales, or different IMFs will con-
sist of signals with similar frequency scales. Mode
GARCIA-ROSA et al.: DISCUSSION ON THE MODE MIXING IN WAVE ENERGY CONTROL SYSTEMS USING THE HILBERT-HUANG TRANSFORM
mixing can be caused by an intermittent/noisy signal,
by specific amplitude and frequency relations of the
original modes in the signal, or by a combination of
both cases. The second case is considered in this study.
In this framework, [10] presents a mathematical
analysis that shows how EMD separates the original
modes in signals with two frequency components. Fig-
ure 1 illustrates a summary of how EMD behaves for
two-tone signals. Here, a two-tone signal represents the
normalized excitation force for two wave components,
¯
fe(t) = acos (ω1t+ϕ) + cos ω2t , (14)
a=a1|He(ω1)|(a2|He(ω2)|)1,(15)
where ω1is the low-frequency (LF) component, ω2
is the high-frequency (HF) component, ϕ=ϕ1ϕ2,
ϕ1=He(ω1),ϕ2=He(ω2), and a1,a2, and ϕ1,
ϕ2are, respectively, the amplitudes and phases of LF
and HF components. Following [10], the frequencies
{ω1, ω2}and phases {ϕ1, ϕ2}are much lower than the
sampling frequency and then, the EMD behaviour is
only sensitive to the relative parameters ω12and ϕ.
If the two components are correctly separated by
the EMD, then the first IMF (c1(t)) matches the high-
frequency component of the signal (cos ω2t). Two well-
defined regions (Fig. 1.a) are observed depending on
the amplitude and frequency ratios of original modes:
the white region indicates a perfect separation of the
components (p10), and the darkest region indicates
the separation is not perfect (p11). In such a case, the
EMD either consider the signal as a single modulated
component or it does something else. The separation of
the components is not possible if either the frequency
ratio is above the cutoff frequency ωc= 0.67 (dotted
line) or the frequency and amplitude ratios are in the
area above aω1
ω22=1 (dashed line) [10].
In addition, it is important to distinguish between
the single modulated component area and the area
which EMD does something else. Notice that if the
signal is a single modulated component, then the first
IMF should be the original signal ( ¯
fe(t)). Figure 1.b
illustrates a comparison of the first IMF with the orig-
inal signal for the areas ω1
ω2ωcand aω1
ω221(cases
when the separation of components is not perfect).
Therefore, three domains are defined [10]: (D1) the
components are separated and correctly identified (p1
0); (D2) the components are considered as a single
waveform (p20); (D3) the EMD does something else
(the darkest areas in Fig. 1.b, i.e., p21). In domain D3,
the components are either halfway between domains
D1 and D2 or contain fake oscillations, i.e., oscillations
derived from the decomposition process and not from
the original modes.
IV. EMD M OD E MI XI NG E FFE CT O N TH E EN ER GY
ABSORBED BY WECS
In this section, we investigate the effect of EMD
mode mixing on the energy absorbed by WECs using
the HHT for tuning purposes in a passive control
strategy. The analysis will initially focus on synthetic
two-tone waves and the domains to which the mode
Fig. 1. EMD behavior for two tones signal (a) performance measure
of separation: p1=||c1(t)cos ω2t||
||acos ω1t|| , (b) distance measure of the first
IMF to the original signal: p2=||c1(t)¯
fe(t)||
|| cos ω2t|| . Dotted line: ωc=0.67,
dashed line: aω1
ω22= 1. (The plots are reproduced from [10] with
10 sifting iterations and sampling period of 1ms).
mixing belongs (D2 and D3), and then it is extended
to real ocean waves that exhibit similar properties as
the synthetic two-tone waves. The floating body is a
heaving cylinder with mass m= 3.2×105kg, radius of
5meters, draught of 4meters, and resonance frequency
1.2rad/s [1].
A. Two-tone waves
By using two-tone waves in domains D2, and D3, as
incident waves, Figures 2, and 3, compare the energy
absorbed by the WEC over T= 10 min for the cases
when HHT is used for tuning purposes (Ea) with the
cases when a constant dominant frequency is used
(Ea,D). The constant dominant frequency is either ω1
or ω2depending on which component has the highest
excitation force amplitude. Only domains D2 and D3
are considered, as in domain D1 the components are
well separated and identified by the EMD. Four differ-
ent frequencies are considered for the HF components:
0.75, 0.9, 1.2, and 1.5 rad/s.
The effect of the EMD mode mixing on the energy
absorbed depends on both the domain (D2 or D3) and
the frequency of the components, which is related to
the frequency response of the floating body.
GARCIA-ROSA et al.: DISCUSSION ON THE MODE MIXING IN WAVE ENERGY CONTROL SYSTEMS USING THE HILBERT-HUANG TRANSFORM
For D2 and the cases in which HF components are
below the resonance frequency of the body (1.2rad/s),
the dominant IMFs are single modulated components
resulting in greater absorption of energy than tuning
to the constant dominant frequency (Fig. 2, top). The
benefit of using the IMF modulated components is
higher for the cases when the amplitudes of the ex-
citation force are close to each other (regions around
log10 a0). However, when the resonance frequency
is the HF or the LF component (Fig. 2, bottom), the
single modulated IMFs can either have a positive or
negative effect on the energy absorbed by the WEC.
Figure 4 shows the IMF instantaneous frequency, and
the PTO damping, for the case when ω2= 1.2rad/s
and Ea/Ea,D = 0.94. It can be observed that the IMF
frequency varies from 1.1 rad/s to 1.7 rad/s, but since
the constant dominant frequency is the resonance fre-
quency of the WEC, the energy absorption with HHT
is less than with the constant frequency. Conversely,
when the dominant frequency is 0.96 rad/s (ω1=0.8ω2
and a1|He(ω1)|>a2|He(ω2)|), the HHT is superior. Table
II summarizes the results for domain D2.
TABLE II
SUMMARY OF RESULTS FOR DOMAIN D2 (ω12=0.8).
COMPONENTS ARE DECOMPOSED AS SINGLE WAVE-FO RM S.
HF Region Dominant Comparison of
(rad/s) component Eawith Ea,D
0.75 log10 a < 0HF (ω2= 0.75 rad/s) Ea> Ea,D
log10 a > 0LF (ω1= 0.6rad/s) Ea> Ea,D
0.9 log10 a < 0HF (ω2= 0.9rad/s) Ea> Ea,D
log10 a > 0LF (ω1= 0.72 rad/s) Ea> Ea,D
1.2 log10 a < 0HF (ω2= 1.2rad/s) Ea< Ea,D
log10 a > 0LF (ω1= 0.96 rad/s) Ea> Ea,D
1.5 log10 a < 0HF (ω2= 1.5rad/s) Ea> Ea,D
log10 a > 0LF (ω1= 1.2rad/s) Ea< Ea,D
Notice that the damping tuning equation (6) is not
optimal for instantaneous frequency tuning. Such an
equation represents the optimal damping equation for
regular waves [12], i.e., sinusoidal waves with a con-
stant frequency component. Further studies are needed
to determine the best damping tuning equation when
the instantaneous frequency is used as an input. As
an example, if the PTO damping oscillations below
2×105kg/s are filtered out from the damping profile
in Figure 4, the energy absorbed increases 3% when
compared to the original damping obtained directly
from the HHT approach and (6).
For domain D3 (Fig. 3), only the region log10 a > 0
is considered, as indicated by Fig. 1.b. Therefore, the
LF component (ω1) is the dominant component for all
the cases, and the HHT is superior in all such cases.
Nonetheless, in this domain the dominant IMFs are
either halfway between domains D1 and D2 or contain
oscillations derived from the decomposition process.
Figure 5.a shows the instantaneous frequency of the
first and second IMFs for the case when ω2=1.2rad/s
and Ea/Ea,D = 1.18. The dominant IMF clearly has an
oscillation around 250 s that is different from the origi-
nal modes, causing an irregularity in the PTO damping
-1 -0.5 0 0.5 1
1
1.01
1.02
1.03
1.04
1.05
1.06
1.07
-1 -0.5 0 0.5 1
1
1.01
1.02
1.03
1.04
1.05
1.06
1.07
-1 -0.5 0 0.5 1
0.9
0.95
1
1.05
1.1
1.15
1.2
1.25
-1 -0.5 0 0.5 1
0.9
0.95
1
1.05
1.1
1.15
1.2
1.25
Fig. 2. Ratios between Eaand Ea,D in domain D2.
0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
1
1.01
1.02
1.03
1.04
1.05
1.06
0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
1
1.05
1.1
1.15
1.2
1.25
1.3
1.35
1.4
1.45
1.5
Fig. 3. Ratios between Eaand Ea,D in domain D3.
profile (Fig. 5.b). Although the energy absorption is
higher than the constant dominant frequency, such
irregularity decreases the energy absorption in about
8% for the time interval 225 s to 255 s, when compared
to the case the components are a single wave-form.
Figure 6 shows the instantaneous frequency and the
PTO damping for both the dominant IMF derived
from EMD (IMF 1) and the analytic single wave-form
for the case the components are together (i.e., single
modulated components).
GARCIA-ROSA et al.: DISCUSSION ON THE MODE MIXING IN WAVE ENERGY CONTROL SYSTEMS USING THE HILBERT-HUANG TRANSFORM
150 200 250 300 350
0
1
2
3
4
5105
Fig. 4. Time-series for case ω2= 1.2rad/s, Ea/Ea,D = 0.94: (domain
D2, ω12=0.8) (a) Hilbert spectrum for the first IMF, and constant
dominant frequency; (b) PTO damping.
150 200 250 300 350
0
0.5
1
1.5
2
2.5 106
Fig. 5. Time-series for case ω2=1.2rad/s, Ea/Ea,D = 1.18 (domain
D3, ω12=0.62): (a) Hilbert spectrum for the first and second IMFs,
and constant dominant frequency; (b) PTO damping.
B. Real ocean waves
Previous studies have shown that tuning the damp-
ing with frequency estimates from the HHT results
in greater absorption of energy than tuning with es-
timates from EKF and FLL [2]. Here, the aim is to use
the mode mixing conditions observed for the synthetic
two-tone wave studies to further improve the energy
absorbed by the HHT for real ocean waves that fulfill
similar conditions. Therefore, we adopt two sea states
(S1, S2) from the northeast coast of Brazil for the
simulations. The wave elevation data consist of records
of about 20 minutes (T20 min) sampled at 1.28 Hz
225 230 235 240 245 250 255
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
225 230 235 240 245 250 255
0
0.5
1
1.5
2
2.5 106
Fig. 6. (a) Instantaneous frequency, and (b) PTO damping for both
the dominant IMF derived from EMD (solid line) and the analytic
single wave-form (dashed line).
0 1 2 3 4
0
0.05
0.1
0.15
0 1 2 3 4
0
1
2
3
4
1010
01234
0
0.05
0.1
0.15
0.2
01234
0
1
2
3
4
1010
Fig. 7. Wave spectra from the northeast coast of Brazil (top) and
excitation force spectra (bottom).
and registered by a wave buoy in the Port of Pec´
em
area. Figure 7 shows two-peak wave spectra and the
excitation force spectra for the cylinder. The frequency
ratio for the peak frequencies (ωp1p2) of the spectra
are, 0.62 and 0.5, respectively, for sea states S1 and S2.
Figure 8 shows samples of time-series simulations
for the excitation forces fe, and the instantaneous
frequencies for IMF 1 ( ˆωd, dominant IMF) and IMF 2.
It can be noted that the IMFs consist of signals with
similar frequency scales (mode mixing), and a simi-
lar behaviour between the real ocean waves and the
synthetic two-tone waves (Fig. 5.a) is observed around
690 s (Fig. 8.a bottom) and 660 s (Fig. 8.b bottom).
The mode mixing could be attenuated by applying
masking signals to the EMD, as done in, e.g., [8].
Conversely, here the mode mixing in domain D2 is
beneficial to the energy absorption. However, the small
oscillations on the PTO profile, caused either by the
mode mixing in domain D3, or by the nature of the
damping tuning equation (6), have a negative impact
on the energy. Therefore, such oscillations are filtered
out. Figure 9 shows the evolution of the PTO damping
Bp(t), and the body motion x(t)for the cases when
GARCIA-ROSA et al.: DISCUSSION ON THE MODE MIXING IN WAVE ENERGY CONTROL SYSTEMS USING THE HILBERT-HUANG TRANSFORM
650 700 750
-2
-1
0
1
2
105
650 700 750
0
1
2
3
4
650 700 750
-2
-1
0
1
2105
650 700 750
0
1
2
3
4
Fig. 8. Time-series of the excitation force, and instantaneous fre-
quency of IMF 1 (dominant IMF) and IMF 2: (a) S1; (b) S2.
650 700 750
0
0.5
1
1.5
2106
650 700 750
-0.5
0
0.5
650 700 750
0
2
4
6106
650 700 750
-0.5
0
0.5
Fig. 9. Time-series of the PTO damping, and body motion: (a) S1;
(b) S2. Solid lines: PC with HHT, dashed lines: PC with HHT and
filtering of small damping oscillations.
the PTO damping is directly obtained from HHT and
(6) (solid lines), and when the small PTO damping
oscillations are filtered out (dashed lines). An energy
improvement of about 2.6%, and 3.9%, is obtained
for sea states S1, and S2, when the small damping
oscillations are filtered out, which agrees with the two-
tone signal studies in Section IV-A.
V. CONCLUSION
This study revealed how the EMD mode mixing
problem affects the performance of WECs with a pas-
sive control strategy that tunes the PTO damping to
frequency estimates from the HHT. The effect of mode
mixing on the energy absorbed depends both on the
type of IMF derived from the EMD procedure and the
frequencies of the components (which is related to the
frequency response of the floating body).
A comprehensive numerical study using synthetic
two-tone waves showed that for mode mixing cases
where components are decomposed as single wave-
forms (domain 2), the effect is beneficial to the absorp-
tion of energy if the constant dominant frequency is not
the resonance frequency of the WEC. Additionally, the
damping tuning law might result in small oscillations
on the PTO profile that will negatively impact the
energy absorption. For cases where the components
are within two IMFs, or contain oscillations derived
from the decomposition process (domain 3), the mode
mixing effects might result in irregularities in the PTO
damping profile that also decrease the absorption of
energy. If the mode mixing causes a high peak in the
instantaneous frequency of an IMF, the PTO damping
will contain a small oscillation that is otherwise not
present in the PTO damping profile.
The studies with synthetic two-tone waves were
used to further improve the HHT absorbed energy for
real ocean waves by filtering out the small oscillations
on the PTO profile. An average energy improvement
of 3.3% was obtained for two studied cases. Such
oscillations in the damping are caused either by high-
frequency oscillations derived from the EMD process
or by the nature of the damping tuning equation (that
is not optimized for the instantaneous frequency). Fur-
ther studies are needed to determine the best damping
tuning equation when the instantaneous frequency of
waves is used as an input for the PTO tuning.
REFERENCES
[1] P. B. Garcia-Rosa, G. Kulia, J. V. Ringwood, and M. Molinas,
“Real-time passive control of wave energy converters using the
Hilbert-Huang transform,” in IFAC-PapersOnLine, vol. 50, no.
1 (Proc. of the 20th IFAC World Congress), Toulouse, France,
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