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Real-Time Passive Control of
Wave Energy Converters Using the
Hilbert-Huang Transform ?
Paula B. Garcia-Rosa ∗Geir Kulia ∗∗
John V. Ringwoo d ∗∗∗ Marta Molinas ∗
∗Department of Engineering Cybernetics, Norwegian University of
Science and Technology, Trondheim 7491, Norway (e-mails:
p.b.garcia-rosa@ieee.org; marta.molinas@ntnu.no).
∗∗ Signal Analysis Lab AS, Olav Tryggvasons gate 27, Trondheim 7011,
Norway (e-mail: geir.kulia@signalanalysislab.com)
∗∗∗ Centre for Ocean Energy Research, Department of Electronic
Engineering, Maynooth University, Co. Kildare, Ireland, (e-mail:
john.ringwood@eeng.nuim.ie)
Abstract: Passive loading is a suboptimal method of control for wave energy converters (WECs)
that usually consists of tuning the power take-off (PTO) damping of the WEC to either the
energy or the peak frequency of the local wave spectrum. Such approach results in a good
solution for waves characterized by one-peak narrowband spectra. Nonetheless, real ocean waves
are non-stationary by nature, and sea wave profiles with different spectral distribution occur
in a specific location over time. Thus, the average energy absorption of passively controlled
WECs tends to be low. In this paper, we propose a real-time passive control (PC) based on
the Hilbert-Huang transform (HHT), where the PTO damping is time-varying and tuned to the
instantaneous frequency of the wave excitation force. The instantaneous frequency is calculated
by using the HHT, an analysis method for nonlinear and non-stationary signals that relies on
the local characteristic time-scale of the signal. A performance comparison (in terms of energy
absorption) of the proposed solution with the passive loading method is presented for a heaving
system, in a variety of wave spectra. It is shown that a performance improvement of up to 21%,
or 65%, is obtained for the proposed PC scheme, when it is compared to passive loading tuned
to the energy, or the peak frequency of the spectrum, respectively. Real ocean waves off the west
coast of Ireland are adopted in the simulations.
Keywords: Wave energy, renewable energy systems, control applications, suboptimal control,
Hilbert-Huang transform.
1. INTRODUCTION
Currently, a number of studies have been done on the ap-
plication of control technology to maximize energy capture
of wave energy converters (WECs), see, e.g., Ringwood
et al. (2014) for a literature overview of WEC control
algorithms. Regardless of the strategy adopted, optimal
hydrodynamic control requires a power take-off (PTO)
system able to implement bidirectional power flow, since
power has to be injected back into the WEC for some
parts of the cycle. Passive loading, latching control and
declutching are alternative methods of suboptimal control
that avoid the need for the PTO to supply power.
Passive loading consists of tuning the PTO damping of
the WEC to the predominant wave frequency of the local
sea wave profile. Usually, the frequency selected is either
the peak or the energy frequency of the wave spectrum
(Yavuz et al., 2007; Garcia-Rosa et al., 2015). Further-
more, other studies optimize the damping by performing
?This work was partially supported by CNPq-Brazil under grant
number 201773/2015-5.
simulations with a range of possible values for each sea
state adopted (Oskamp and ¨
Ozkan Haller, 2012; Sjolte
et al., 2013). Nonetheless, for irregular waves, the energy
absorbed by passively controlled WECs are lower than
other suboptimal control strategies like latching control
(Bjarte-Larsson and Falnes, 2006; Hals et al., 2011; Garcia-
Rosa et al., 2015). Conversely, Tom and Yeung (2014) have
shown that for a passive control (PC) method where the
PTO damping is time-varying, and obtained through a
nonlinear model predictive controller, a great improvement
in the energy absorption can be obtained over constant
damping.
In this paper, we adopt a passive control approach where
the PTO damping is time-varying and tuned to the in-
stantaneous frequency of the wave excitation force. Due to
the non-stationary nature of real ocean waves, we propose
the Hilbert-Huang transform (HHT) (Huang et al., 1998)
to calculate the instantaneous frequency. The HHT is an
analysis method for nonlinear and non-stationary signals
based on the Empirical Mode Decomposition (EMD).
Different from methods that are based on the Fourier
expansion, where the decomposition has a priori basis
(trigonometric functions) and a global sense, the EMD
has an adaptive basis and relies on the local character-
istic time-scale of the signal. Thus, the EMD can extract
the different oscillation modes (named as Intrinsic Mode
Functions, IMFs) present in a wave profile.
The aim is to verify the energy content of the IMF compo-
nents of the wave excitation force, identify the dominant
component and use the information of its instantaneous
frequency in the PC approach. A performance comparison
(in terms of energy absorption) of the proposed solution
with passive loading is presented for a heaving system with
one degree of freedom, in a variety of real wave profiles.
Real ocean waves off the west coast of Ireland are adopted
in the simulations. A comparison of the energy absorption
for constant PTO damping tuned at different frequencies,
namely, the energy and peak frequency of the spectrum,
is also presented.
2. DYNAMIC MODELING OF THE WEC
Figure 1 illustrates the WEC considered in this paper.
The WEC is a single oscillating-body represented as a
truncated vertical cylinder with a generic PTO system.
The wetted surface of the cylinder is defined by a draught
dand a radius r.
Fig. 1. Schematic of the generic heaving floating body.
2.1 Equation of Motion
Here, we assume linear hydrodynamic theory and heave
oscillatory motion of the WEC. In such a case, the motion
of the floating body is described by
m¨x(t) = fe(t) + fr(t) + fs(t) + fp(t),(1)
where x∈Ris the vertical position of the body, m∈R+
is the body mass, fs=Sx is the restoring force, S∈R+
is the buoyancy stiffness, fpis the force applied by the
PTO mechanism, feis the excitation force on the body
held fixed in incident waves, and fris the radiation force
due to the body oscillation in the absence of waves.
From Cummins (1962),
−fr(t) = mr(∞) ¨x+
t
Z
0
hr(t−τ) ˙x(τ)dτ , (2)
where mr(∞)∈R+is the infinite-frequency added mass
coefficient, defined with the asymptotic values of the added
masses at infinite frequency. The kernel of the convolution
term hr(t−τ) is known as the fluid memory term:
hr(t) = 2
π
∞
Z
0
Br(ω) cos(ωt −τ)dω , (3)
where Br(ω)∈R+is the radiation damping coefficient, and
ω∈R+is the wave frequency. Thus, the vertical motion of
the floating body (1) becomes
M¨x(t)+
t
Z
0
hr(t−τ) ˙x(τ)dτ +Sx(t) = fe(t)+fp(t),(4)
with M=[m+mr(∞)]. The excitation force is given by
fe(t) =
∞
Z
−∞
he(t−τ)ζ(τ)dτ , (5)
where
he(t) = 1
2π
∞
Z
−∞
He(ω)eiωt dω , (6)
is the inverse Fourier transform of the excitation force
transfer function He(ω), and ζis the wave elevation. He(ω)
is a property of the floating body and has low-pass filter
characteristics. Notice that (6) is non-causal, since in fact,
the pressure distribution is the cause of the force and
not the incident waves (Falnes, 2002). Furthermore, real
ocean waves are usually characterized by their energy
density spectrum S(ω). Following the linear approach, the
spectrum of the excitation force is
Sfe(ω) = |He(ω)|2S(ω).(7)
A generic PTO system with a damper (Bp∈R+) varying
in time is adopted. The PTO force is parameterized as a
function of the body velocity ˙x(t):
fp(t) = Bp(t) ˙x(t).(8)
The extracted energy, and the mean extracted power by
the WEC over a time range Tare, respectively,
Ea=−ZT
0
˙x(t)fp(t)dt , (9)
Pa=Ea
T.(10)
2.2 Optimal Conditions for Maximum Wave Energy
Extraction
For regular wave regime (waves defined by a single fre-
quency) and for Bp(t) = Bpfor any time t, Falnes (2002)
has shown that maximum absorption is obtained when
Bp=p(Br(ω))2+ (ω(m+mr(ω)) −S/ω)2,(11)
where mr(ω)∈Ris the added mass. Furthermore, if mand
Scan be chosen such that
(m+mr(ω))ω−S/ω = 0 ,(12)
then (11) becomes
Bp=Br(ω).(13)
Equation (11) is referred as optimum amplitude condition
and (12) is referred as optimum phase condition (Falnes,
2002). The greatest wave energy absorption is obtained
when both conditions are satisfied, and then the PTO
damping is given by (13). In such a case, the velocity of the
floating body is in phase with the excitation force (Falnes,
2002), and the PTO system should be able to implement
bidirectional power flow.
Passive loading consists of setting the PTO damping Bpto
a constant value. Equation (11) represents optimal linear
damping when the floating body is subjected to incident
regular waves. Since irregular waves and real ocean waves
are not defined by a single frequency in the time do-
main, a common approach is to select a frequency that
characterizes the wave spectrum for tuning the damping.
Usually, the frequency selected is either the peak (ωp) or
the energy frequency (ωe) of the wave spectrum. Different
time scales can be applied for tuning the PTO damping:
hourly basis (according to sea states variations), monthly
basis (according to seasonal variations), or annually basis
(Oskamp and ¨
Ozkan Haller, 2012).
An alternative approach where the PTO damping (11) is
modified on a wave-by-wave basis is presented next.
3. REAL-TIME PASSIVE CONTROL
3.1 Overview of the proposed control method
Since some of the high frequency content of the wave
elevation is filtered by He(ω), we consider the instanta-
neous frequency of the excitation force in our real-time
PC approach, rather than the instantaneous frequency of
the waves.
The calculation of the instantaneous frequency by applying
the Hilbert transform (HT) directly to fe(t) results in
negative local frequencies, as the instantaneous frequency
is not well defined for multi-component signals (Boashash,
1992), i.e. signals with more than one local extrema for
each zero crossing. Thus, fe(t) is decomposed into mono-
component signals (IMFs) using the EMD.
Here, we assume that the excitation force is known com-
pletely over the interval T. By applying the HHT ap-
proach, fe(t) can be expressed as
fe(t) = R
N
X
i=1
ˆai(t)eiRˆωi(t)dt ,(14)
where Nis the total number of IMF components defined
here as N= log2Ns−1 (Wu and Huang, 2004), Nsis the
data length, ˆaiand ˆωiare respectively the amplitude and
the instantaneous frequency of the i-th IMF component.
Equation (14) is considered as a generalized form of the
Fourier expansion, with both amplitude and frequency
represented as functions of time (Huang et al., 1998).
The aim is to identify the dominant IMF component (in
terms of the energy of the signal), and use the information
of its instantaneous frequency for tuning the PTO damp-
ing. From (11),
Bp(t) = p(Br(ˆωd))2+ (ˆωd(m+mr(ˆωd)) −S/ˆωd)2.
(15)
where ˆωdis the instantaneous frequency of the dominant
IMF component. Figure 2 illustrates the block diagram of
the proposed real-time PC based on the HHT approach.
The procedure to calculate ˆωdis described next.
3.2 Instantaneous frequency of the dominant IMF
component
In order to determine ˆωd(t), firstly we decompose fe(t) into
IMF components using the EMD. The EMD identifies local
Fig. 2. Block diagram of the proposed PC.
maxima and minima of the signal, and calculates upper
and lower envelopes for these points by using cubic splines.
The mean values of the envelopes are used to decompose
the original signal into frequency components in a sequence
from the highest frequency to the lowest one. The EMD
procedure is summarized in the following 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.
After the EMD, the dominant IMF is identified through
the comparison of the energy of the IMF signals (Eci) with
the energy of the excitation force signal (Ef e),
Eci =ZT
0|ci(t)|2dt , (16)
Efe =ZT
0|fe(t)|2dt , (17)
where ci(t) is the i-th IMF component. The IMF with the
highest ratio of energy content Eci/Efe is the dominant
component cd(t).
Additionally, in order to avoid other limitations of the
HT (Huang, 2005), the dominant IMF is normalized by
dividing it by a spline envelope defined through all the
maxima of the IMF, as described in Huang (2005).
Finally, the Hilbert transform is applied to the normalized
dominant component ¯cd(t) (Huang et al., 1998):
¯υd(t) = 1
πPZ∞
−∞
¯cd(τ)
t−τdτ , (18)
where Pindicates the Cauchy principal value, and ¯υd
is the HT of ¯cd. Then, the dominant IMF component is
represented as an analytic signal
¯zd(t) = ¯cd(t) + j¯υd(t) = ˆad(t)eiRˆωd(t)dt ,(19)
with amplitude ˆad(t) and instantaneous frequency ˆωd(t)
calculated by
ˆad(t) = q¯c2
d(t) + ¯υ2
d(t),(20)
ˆωd(t) = dφd(t)
dt ,(21)
where φd(t) = arctan(¯υd(t)/¯cd(t)).
4. SIMULATIONS
4.1 Hydrodynamic parameters
Here, we consider a heaving cylinder with radius r=
5 m, draught d= 4m, and mass m= 3.2×105kg. The
hydrodynamic coefficients of the cylinder were computed
using the boundary element solver WAMIT, Inc. (1998-
2006). Figure 3 illustrates the added mass and radiation
damping coefficients, and the frequency response of the
excitation force.
0 1 2 3
2
2.5
3
3.5
x 105
mr(ω), (kg)
(a)
0 1 2 3
0
2
4
6
8
x 104
ω, (rad/s)
Br(ω), (kg/s)
(b)
0 1 2 3
0
2
4
6
8
x 105
|He(ω)|, (N/m)
(c)
0 1 2 3
0
1
2
3
4
ω, (rad/s)
6He(ω), (rad)
(d)
Fig. 3. Hydrodynamic data. (a) Added mass mr(ω), (b)
Radiation damping Br(ω), (c) Magnitude |He(ω)|and
(d) phase ∠He(ω) of the excitation force frequency
response.
4.2 Real Wave Data
Real wave data provided by the Irish Marine Institute are
adopted for the simulations. The data consists of wave
elevation records of 30 minutes, sampled at 1.28 Hz, and
it was collected in 2010 from a data buoy in the Belmullet
wave energy test site, off the west coast of Ireland.
In order to verify the performance of the proposed con-
troller, nine wave elevation records with different spectral
distributions have been selected for our study. The selected
records are referred as sea states S1-S9. Figure 4 illustrates
the wave spectra of the sea states, and Table 1 shows
the significant wave height Hs, the energy frequency ωe,
and the peak frequency ωpof the spectra. The statisti-
cal parameters Hsand ωeare respectively calculated as:
Hs=4√m0, and ωe=m0/m−1, where mn=R∞
0ωnS(ω)dω
is the spectral moment of order n.ωpis the frequency at
which the wave spectrum is maximum.
4.3 Simulation Results
Figure 5 shows the spectral density of the excitation force
for the selected sea states. It can be noted that some of
0 1 2 3
0
0.5
1
0 1 2 3
0
0.2
0.4
S(ω), (m2s/rad)
0 1 2 3
0
0.5
1
0 1 2 3
0
0.1
0.2
0 1 2 3
0
0.2
0.4
0 1 2 3
0
0.2
0.4
ω, (rad/s)
0 1 2 3
0
0.5
1
0 1 2 3
0
0.5
1
0 1 2 3
0
0.5
1
S1
S2
S3 S6 S9
S8
S5
S4 S7
Fig. 4. Wave spectra of real wave data from Belmullet.
Table 1. Significant wave height Hs(m), en-
ergy frequency ωe(rad/s), and peak frequency
ωp(rad/s) of the selected sea states.
S1 S2 S3 S4 S5 S6 S7 S8 S9
Hs1.26 1.43 1.80 1.06 1.06 1.35 1.80 1.88 1.90
ωe0.59 0.94 0.94 0.84 0.75 0.85 0.79 0.70 0.75
ωp0.52 1.22 0.90 0.42 0.46 0.50 1.01 0.57 0.44
the high frequency waves in Figure 4 are filtered out by
the transfer function He(ω), that is defined by the shape
of the floating body. Thus, the excitation force spectra are
characteristic of the cylinder adopted in this study.
0 1 2 3
0
2
4
6
x 1010
0 1 2 3
0
5
10
15
x 109
Sfe (ω), (N2s/rad)
0 1 2 3
0
1
2
x 1010
0 1 2 3
0
5
10
15
x 109
0 1 2 3
0
1
2
x 1010
0 1 2 3
0
1
2
3
x 1010
ω, (rad/s)
0 1 2 3
0
1
2
3
x 1010
0 1 2 3
0
2
4
x 1010
0 1 2 3
0
2
4
6
x 1010
S1 S7
S4
S2 S5 S8
S3 S6 S9
Fig. 5. Excitation force spectra.
A. Constant damping Firstly, we adopt the passive
loading method, and the PTO damping is tuned either at
ωeor at ωp. The energy absorbed by the WEC is denoted
by Ea,ωeand Ea,ωpfor the two cases, respectively. In order
to compare the energy absorbed in such cases, Table 2
shows the ratio Ea,ωe/Ea,ωp.
For most of the studied cases, tuning the constant damping
at ωegives greater energy capture than tuning at ωp.
Table 2. Ratio of Ea,ωeand Ea,ωp.
S1 S2 S3 S4 S5 S6 S7 S8 S9
1.01 1.36 0.99 1.36 1.16 0.93 1.08 1.18 1.18
The highest differences in the absorption of energy are
obtained for sea states S2 and S4. Both sea states are
characterized by wideband spectra. Nonetheless, for S2
the high frequency peak (ωp= 1.22 rad/s) is filtered out
by the shape of the body, as it is illustrated in Fig. 5.S2,
which can explain tuning Bpat ωeresults in greater energy
absorption than tuning at ωp. In the sea state S4, the
energy is spread from about 0.4 to 1.5 rad/s (Fig. 5.S4),
and the low frequency peak (ωp=0.42 rad/s) is half of the
energy frequency (ωe=0.84 rad/s).
B. Proposed control method The damping is varying in
time and uses the instantaneous frequency information of
the dominant IMF component for tuning purposes, as it
is described in Section 3. The EMD procedure is applied
to the excitation force signals, and the resultant ratios of
the energy of the IMF signals (Eci) and the energy of the
excitation force signal (Efe) are shown in Figure 6. Clearly,
the IMF component c1is the dominant component for all
cases, although in some cases, for instance the wideband
spectra S2, S4, and S7, the signal energy of component c2
is also significant (about 40% of Ef e).
1 2 3 4 5 6 7 8 9
0
0.2
0.4
0.6
0.8
1
1 2 3 4 5 6 7 8 9
0
0.2
0.4
0.6
0.8
1
Eci/Efe
1 2 3 4 5 6 7 8 9
0
0.2
0.4
0.6
0.8
1
1 2 3 4 5 6 7 8 9
0
0.2
0.4
0.6
0.8
1
1 2 3 4 5 6 7 8 9
0
0.2
0.4
0.6
0.8
1
1 2 3 4 5 6 7 8 9
0
0.2
0.4
0.6
0.8
1
ci
1 2 3 4 5 6 7 8 9
0
0.2
0.4
0.6
0.8
1
1 2 3 4 5 6 7 8 9
0
0.2
0.4
0.6
0.8
1
1 2 3 4 5 6 7 8 9
0
0.2
0.4
0.6
0.8
1
S1
S2
S3 S6
S5
S4 S7
S8
S9
Fig. 6. Ratio of Eci and Ef e.
In order to illustrate the effect of the proposed real-
time PC in the variables of the system, Figure 7 shows
samples of time-series simulations for the sea states S1
and S2. For S1, the evolution of the position x, and
the PTO force fpfor the proposed controller are slightly
different from the case when the damping is constant
and tuned at ωe. In such a case, the energy absorbed
over a 30-min simulation (Fig. 8.a) is only 2.9% greater
than the constant damping approach, whereas for S2 the
improvement is more significant, about 21% greater than
constant damping (Fig. 8.b), and the PTO force has higher
peaks than the constant damping case (Fig. 7.b).
Such behaviour can be explained by the different energy
spectral distributions of both sea states. S1 is character-
ized by a narrowband spectrum with a single dominant
swell (low frequency waves generated in other locations),
and then the energy is concentrated in a narrow band of
frequencies, mostly around ωe. However, S2 is character-
ized by a two-peak spectrum with mixed wind-sea (high
frequency waves generated by the local wind) and swell
conditions and then, the energy is spread over a wider
band of frequencies than S1.
950 1000 1050
−6
−3
0
3
6
x 105
(a) S1
fe, (N)
950 1000 1050
−1
−0.5
0
0.5
1
x, (m)
950 1000 1050
−4
−2
0
2
4
x 105(b) S2
950 1000 1050
−0.5
0
0.5
1
950 1000 1050
0.5
1
1.5
2
2.5
x 106
Bp, (kg/s)
950 1000 1050
−4
−2
0
2
4
x 105
fp, (N)
t, (s)
950 1000 1050
0
0.5
1
1.5
2
2.5
x 106
950 1000 1050
−2
−1
0
1
2
3
x 105
t, (s)
Fig. 7. Time-series of the excitation force, position, PTO
damping, and PTO force for the proposed PC (solid
line) and the passive loading approach (dashed line)
(a) S1; (b) S2.
0 600 1200 1800
0
1
2
3
x 107
(a) S1
Ea, (J)
t, (s)
0 600 1200 1800
0
1
2
3
x 107(b) S2
t, (s)
Fig. 8. Energy absorbed over a 30-min simulation for
the proposed PC (solid line) and the passive loading
approach (dashed line) (a) S1; (b) S2.
A comparison of the energy absorbed (Ea) by the WEC
when the proposed control scheme is adopted and when
passive loading is adopted is illustrated in Figure 9. For
the passive loading approach, the PTO damping is tuned
either at ωeor at ωp. The energy capture for the proposed
controller is superior to passive loading in all studied cases.
A performance improvement in the absorbed energy from
2.9% to 21% is obtained when it is compared with constant
damping tuned at ωe, and from 3.6% to 65% when the
damping is tuned at ωp.
S1 S2 S3 S4 S5 S6 S7 S8 S9
1
1.05
1.1
1.15
1.2
1.25
1.3
Ea/Ea,ωe
1
1.1
1.2
1.3
1.4
1.5
1.6
1.7
Ea/Ea,ωp
Ea/Ea,ωe
Ea/Ea,ωp
Fig. 9. Ratios between Eaand Ea,ωe, and Eaand Ea,ωp.
5. CONCLUSION
When real ocean waves and the passive loading method
are adopted, tuning the constant damping at the energy
frequency of the wave spectrum usually results in greater
absorption of energy than tuning it at the peak frequency.
The excitation force spectrum filters out some of the high
frequency components of the wave spectrum. The filtering
characteristics depend on the shape of the body.
A real-time passive control based on the Hilbert-Huang
transform has been proposed to improve the energy ab-
sorption of WECs that cannot implement bidirectional
power flow. For the studied sea states, an average en-
ergy improvement of 15%, and 29%, is obtained for the
proposed control when it is compared with the constant
damping respectively tuned at the energy, and the peak
frequency of the wave spectrum. The lowest improvements
are obtained for the sea state characterized by a nar-
rowband spectrum with energy concentrated at a single
dominant swell.
The proposed controller is suboptimal, as it considers only
the dominant IMF component of the excitation force. To
further improve the energy absorption, a scheme adopting
more than one IMF component could be developed. Such
scheme would be specially beneficial for the cases where
the second IMF is also significant, as characterized by some
of the wideband spectra adopted in this study.
Furthermore, the proposed controller requires higher PTO
forces than the constant damping approach. For practical
application studies, constraints on the position of the body
and on the PTO force should be taken into account. A
scheme similar to the constrained control from Fusco and
Ringwood (2013) could be adopted.
ACKNOWLEDGEMENTS
The authors would like to acknowledge the Irish Marine
Institute for providing the wave data from the Belmullet
test site, and Lars Lundheim for the useful discussions
while implementing the HHT software. Special thanks are
in order to Norden Huang for personally giving insight into
his methods. His open-mindedness and generosity have
been highly appreciated.
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