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Real-Time Passive Control of Wave Energy Converters Using the Hilbert-Huang Transform

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Passive loading is a suboptimal method of control for wave energy converters (WECs) that usually consists of tuning the power take-o� (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 pro�les with di�erent spectral distribution occur in a speci�c 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 o� the west coast of Ireland are adopted in the simulations.
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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 xRis the vertical position of the body, mR+
is the body mass, fs=Sx is the restoring force, SR+
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(tτ) ˙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(tτ) 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
ωR+is the wave frequency. Thus, the vertical motion of
the floating body (1) becomes
M¨x(t)+
t
Z
0
hr(tτ) ˙x(τ)+Sx(t) = fe(t)+fp(t),(4)
with M=[m+mr()]. The excitation force is given by
fe(t) =
Z
−∞
he(tτ)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 (BpR+) 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))ωS/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= log2Ns1 (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)
tτdτ , (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) = d(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=4m0, and ωe=m0/m1, where mn=R
0ωnS(ω)
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.
(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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... enablers of advanced WEC technology. Examples of this class of controllers include [14,15,16,17,18,19]. ...
... In the limit case m = N , i.e. where we can effectively control every DoF of the device via a specific control input, the result of (21) (and the associated optimal control input in (19)), coincide with the theory presented in [25,Chapter 6], for fully actuated multi-DoF WEC devices. 290 We note that, having derived the corresponding optimal closed-loop response (20)- (21), the phase and amplitude conditions for the case presented in this section can be obtained analogously to those presented in Section 3.1, and hence we avoid a detailed discussion for economy of space. ...
... Note that we leave K tr out of the following discussion, since such a controller can be synthesised based upon a plethora of standard techniques arising both from classical, and modern control theory. Examples of controller adopting the feedforward structure (c), for single-DoF WECs, are [15,19]. ...
Article
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In recent years, the fundamental principle of impedance-matching (IM) has inspired a number of sophisticated, yet simple, control solutions for wave energy converters (WEC). Such controllers have the capability of maximising energy absorption from incoming waves with mild computational requirements, being often intuitive in their design, hence especially appealing for real-time industrial applications. Nonetheless, these control solutions are, to date, almost exclusively developed for single degree-of-freedom (DoF) (and hence fully actuated) WEC systems, hindering their application to realistic underactuated multi-DoF devices, i.e. harvesting systems where energy is extracted from only a handful of its total set of modes of motion. Motivated by this, we present, in this paper, a comprehensive derivation and discussion of the IM conditions for maximum energy absorption in underactuated multi-DoF WEC systems. In particular, we show that the IM principle for single-DoF devices can be effectively extended to underactuated multi-DoF systems, and that a set of optimality conditions can be explicitly derived. In addition, we discuss both the impact and use of this set of optimal conditions for control design and synthesis, hence effectively taking a fundamental step towards the general extension of current IM-based techniques to the case of underactuated multi-DoF devices.
... R ECENT studies have shown that tuning the power take-off (PTO) damping of a wave energy converter (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 provides the instantaneous wave-to-wave frequency of the oscillation modes present in a wave profile. ...
... 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 heave motion x(t) of the floating cylinder is described by: (1) with the kernel of convolution term given by [11] ...
... whereω d (t) is the instantaneous frequency of the dominant IMF component of the excitation force [1]. ...
Conference Paper
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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.
... A widely studied approach to avoid the difficulties in the implementation of the feedback control of the WECs is known in the literature as linear damping of the PTO, also called passive loading [35] or resistive [36], a suboptimal approach where the instantaneous value of the PTO force is linearly proportional to the oscillating body speed, that is to say ...
... Passive damping control is analyzed and compared in [35] with a real-time passive control (PC) based on the Hilbert-Huang transform (HHT). For this solution the damping coefficient is time-varying and tuned instantaneously, based on the frequency of the excitation force. ...
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Wave energy's path towards commercialization requires maximizing reliability, survivability, an improvement in energy harvested from the wave and efficiency of the wave to wire conversion. In this sense, control strategies directly impact the survivability and safe operation of the device, as well as the ability to harness the energy from the wave. For example, tuning the device's natural frequency to the incoming wave allows resonance mode operation and amplifies the velocity, which has a quadratic proportionality to the extracted energy. In this article, a review of the main control strategies applied in wave energy conversion is presented along their corresponding power takeoff (PTO) systems.
... The linear damping of the PTO, also known as passive loading or resistive damping, is a well-researched way of avoiding issues in the implementation of the feedback control of the WECs [225,226]. It can be said that, sub-optimally, the instantaneous PTO force is proportional linearly to the oscillating body speed: ...
Article
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As the global interest in renewable energy generation continues, the need to develop new and innovative solutions is being explored every day throughout the world by researchers and innovators. Hybrid renewable energy innovations are gaining progressive interest not only because of the threat of climate change but also due to the technological advancements seen in renewables. Ocean waves have immense potential as a renewable energy source, and related technologies have advanced continuously over the past few decades. In response, this paper extensively studies wave energy converters (WECs) based on the power take-off (PTO) technique, and presents a novel hybrid wave-plus-photon energy (HWPE) harvester called Wavevoltaics, based on wave and solar energy capture systems for coastal communities’ power needs, in line with decarbonization measures. The HWPE harvester uses a simple rack-and-pinion mechanism in combination with solar cell technology to convert the wave energy into usable electrical energy in a water column structural design. This novel HWPE device can be used to provide power for lighting and gadgets for coastal communities that rely heavily on fossil fuels for their lighting and electrical needs. Later in the paper, the challenges faced in hybrid wave energy development are presented.
... For a detailed discussion about the synthesis of stochastic processes, the interested reader is referred to[35].7 An alternative to the use of an EKF can be found in[37], where estimates of andω are computed based on the Hilbert-Huang transform[38]. ...
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The design of controllers for wave energy devices has evolved from early monochromatic impedance-matching methods to complex numerical algorithms that can handle panchromatic seas, constraints, and nonlinearity. However, the potential high performance of such numerical controller comes at a computational cost, with some algorithms struggling to implement in real-time, and issues surround convergence of numerical optimisers. Within the broader area of control engineering, practitioners have always displayed a fondness for simple and intuitive controllers, as evidenced by the continued popularity of the ubiquitous PID controller. Recently, a number of energy-maximising wave energy controllers have been developed based on relatively simple strategies, stemming from the fundamentals behind impedance-matching. This paper documents this set of (5) controllers, which have been developed over the period 2010-2020, and compares and contrasts their characteristics, in terms of energy-maximising performance, the handling of physical constraints, and computational complexity. The comparison is carried out both analytically and numerically, including a detailed case study, when considering a state-of-the-art CorPower-like device.
... In such a way, the effect of the PTO damping on the control strategy is also investigated. In what follows, numerical simulations are performed considering the same heaving cylinder adopted in (Garcia-Rosa et al., 2017). The cylinder has a radius of 5 m, draught of 4 m, mass m = 3 × 10 5 kg and resonant frequency ω r = 1.2 rad/s. ...
Conference Paper
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Aiming at improving the energy absorption from waves, a number of studies have considered declutching control-a phase-control method that consists of disengaging the power takeoff (PTO) system from the oscillating body at specific intervals of time. The on/off sequences with the instants to engage/disengage the PTO are usually determined by optimization procedures that require the knowledge of future excitation force, which remains an open challenge for practical implementation. This paper presents a comprehensive numerical study with different PTO damping coefficients for declutching control. It is shown that the value of the damping plays an important role on the efficacy of the control method and on the optimal time to engage (or disengage) the PTO. Then, two switching sequences that use current information of the body motion are proposed, and compared with the threshold unlatching strategy. When the body velocity vanishes, the PTO is clutched (declutched) if the current estimation of the mean excitation force frequency is lower (higher) than the body resonant frequency. The instant to declutch (clutch) again depends on the damping coefficient. The resultant PTO force profiles are not optimal, but act in an effective way to improve the energy absorption, while not requiring wave short-term predictions and numerical optimization solutions that can be time-consuming depending on the fidelity of numerical models and the prediction horizon. Numerical simulations consider real ocean waves and synthetic waves.
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Utilizing ocean wave energy as a renewable energy source has become the object of rapid research. Energy conversion technology continues to evolve to seek more efficient, cheaper forms of investment, operation, and maintenance and are environmentally friendly. The converter type and PTO hold the key to the efficiency of the whole system. This literature review paper examines various general concepts and innovations of wave activated body converters and commonly used and innovative power take-off systems with a focus on controlling efforts in maximizing the generated power, challenges and efforts to develop a PTO control system as well as various research conducted by various parties.
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Lower efficiencies induce higher energy costs and pose a barrier to wave energy devices’ commercial applications. Therefore, the efficiency enhancement of wave energy converters has received much attention in recent decades. The reported research presents the double snap-through mechanism applied to a hemispheric point absorber type wave energy converter (WEC) to improve the energy absorption performance. The double snap-through mechanism comprises four oblique springs mounted in an X-configuration. This provides the WEC with different dynamic stability behaviors depending on the particular geometric and physical parameters employed. The efficiency of these different WEC behaviors (linear, bistable, and tristable) was initially evaluated under the action of regular waves. The results for bistable or tristable responses indicated significant improvements in the WEC’s energy capture efficiency. Furthermore, the WEC frequency bandwidth was shown to be significantly enlarged when the tristable mode was in operation. However, the corresponding tristable trajectory showed intra-well behavior in the middle potential well, which induced a more severe low-energy absorption when a small wave amplitude acted on the WEC compared to when the bistable WEC was employed. Nevertheless, positive effects were observed when appropriate initial conditions were imposed. The results also showed that for bistable or tristable responses, a suitable spring stiffness may cause the buoy to oscillate in high energy modes.
Article
Aiming at improving the energy absorption from waves, a number of studies have considered declutching control – a phase-control method that consists of disengaging the power take-off (PTO) system from the oscillating body at specific intervals of time. The on/off sequences with the instants to engage/disengage the PTO are usually determined by optimization procedures that require the knowledge of future excitation force, which remains an open challenge for practical implementation. This paper presents a comprehensive numerical study with different PTO damping coefficients for declutching control. It is shown that the value of the damping plays an important role on the efficacy of the control method and on the optimal time to engage (or disengage) the PTO. Then, two switching sequences that use current information of the body motion are proposed, and compared with the threshold unlatching strategy. When the body velocity vanishes, the PTO is clutched (declutched) if the current estimation of the mean excitation force frequency is lower (higher) than the body resonant frequency. The instant to declutch (clutch) again depends on the damping coefficient. The resultant PTO force profiles are not optimal, but act in an effective way to improve the energy absorption, while not requiring wave short-term predictions and numerical optimization solutions that can be time-consuming depending on the fidelity of numerical models and the prediction horizon. Numerical simulations consider real ocean waves and synthetic waves.
Article
Full-text available
Wave-energy converters of the point-absorbing type (i.e., having small extension compared with the wavelength) are promising for achieving cost reductions and design improvements because of a high power-to-volume ratio and better possibilities for mass production of components and devices as compared with larger converter units. However, their frequency response tends to be narrow banded, which means that the performance in real seas (irregular waves) will be poor unless their motion is actively controlled. Only then the invested equipment can be fully exploited, bringing down the overall energy cost. In this work various control methods for point-absorbing devices are reviewed, and a representative selection of methods is investigated by numerical simulation in irregular waves, based on an idealized example of a heaving semisubmerged sphere. Methods include velocity-proportional control, approximate complex conjugated control, approximate optimal velocity tracking, phase control by latching and clutching, and model-predictive control, all assuming a wave pressure measurement as the only external input to the controller. The methods are applied for a single-degree-of-freedom heaving buoy. Suggestions are given on how to implement the controllers, including how to tune control parameters and handle amplitude constraints. Based on simulation results, comparisons are made on absorbed power, reactive power flow, peak-to-average power ratios, and implementation complexity. Identified strengths and weaknesses of each method are highlighted and explored. It is found that overall improvements in average absorbed power of about 100-330% are achieved for the investigated controllers as compared with a control strategy with velocity-proportional machinery force. One interesting finding is the low peak-to-average ratios resulting from clutching control for wave periods about 1.5 times the resonance period and above.
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Energy-maximising controllers for wave energy devices are normally based on linear hydrodynamic device models. Such models ignore nonlinear effects which typically manifest themselves for large device motion (typical in this application) and may also include other modelling errors. In this paper, we present a methodology for reducing the sensitivity to modelling errors and nonlinear effects by the use of a hierarchical robust controller, which also allows good energy maximisation to be recovered through a passivity-based control approach.
Article
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The energy cost for producing electricity via wave energy converters (WECs) is still not competitive with other renewable energy sources, especially wind energy. It is well known that energy maximising control plays an important role to improve the performance of WECs, allowing the energy conversion to be performed as economically as possible. The control strategies are usually subsequently employed on a device that was designed and optimized in the absence of control for the prevailing sea conditions in a particular location. If an optimal unconstrained control strategy, such as pseudo-spectral optimal control (PSOC), is adopted, an overall optimized system can be obtained no matter whether the control design is incorporated at the geometry optimization stage or not. Nonetheless, strategies, such as latching control (LC), must be incorporated at the optimization design stage of the WEC geometry if an overall optimized system is to be realised. In this paper, the impact of device motion and force constraints in the design of control-informed optimized WEC geometries is addressed. The aim is to verify to what extent the constraints modify the connection between the control and the optimal device design. Intuitively, one might expect that if the constraints are very tight, the optimal device shape is the same regardless of incorporating or not the constrained control at the geometry optimization stage. However, this paper tests the hypothesis that the imposition of constraints will limit the control influence on the optimal device shape. PSOC, LC and passive control (PC) are considered in this study. In addition, constrained versions of LC and PC are presented.
Article
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With the recent sharp increases in the price of oil, issues of security of supply, and pressure to honor greenhouse gas emission limits (e.g., the Kyoto protocol), much attention has turned to renewable energy sources to fulfill future increasing energy needs. Wind energy, now a mature technology, has had considerable proliferation, with other sources, such as biomass, solar, and tidal, enjoying somewhat less deployment. Waves provide previously untapped energy potential, and wave energy has been shown to have some favorable variability properties (a perennial issue with many renewables, especially wind), especially when combined with wind energy [1].
Article
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A novel strategy for the real-time control of oscillating wave energy converters (WECs) is proposed. The controller tunes the oscillation of the system such that it is always in phase with the wave excitation force and the amplitude of the oscillation is within given constraints. Based on a nonstationary, harmonic approximation of the wave excitation force, the controller is easily tuned in real-time for performance and constraints handling, through one single parameter of direct physical meaning. The effectiveness of the proposed solution is assessed for a heaving system in one degree of freedom, in a variety of irregular (simulated and real) wave conditions. A performance close to reactive control and to model predictive control is achieved. Additional benefits in terms of simplicity and robustness are obtained.
Chapter
Hilbert-Huang transform (HHT) is the designated name for the result of empirical mode decomposition (EMD) and the Hilbert spectral analysis (HSA) methods, which were both introduced recently by Huang et al. (1996, 1998, 1999, and 2003), specifically for analyzing data from nonlinear and nonstationary processes. Data analysis is an indispensable step in understanding the physical processes, but traditionally the data analysis methods were dominated by Fourier-based analysis. The problems of such an approach were discussed in detail by Huang et al. (1998). As data analysis is important for both theoretical and experimental studies (for data is the only real link between theory and reality), we desperately need new methods in order to gain a deeper insight into the underlying processes that actually generate the data. The method we really need should not be limited to linear and stationary processes, and it should yield physically meaningful results.
Article
This paper evaluates the theoretical application of nonlinear model predictive control (NMPC) to a model-scale point absorber for wave energy conversion. The NMPC strategy will be evaluated against a passive system, which utilizes no controller, using a performance metric based on the absorbed energy. The NMPC strategy was setup as a nonlinear optimization problem utilizing the interior point optimizer (IPOPT) package to obtain a time-varying optimal generator damping from the power-take-off (PTO) unit. This formulation is different from previous investigations in model predictive control, as the current methodology only allows the PTO unit to behave as a generator, thereby unable to return energy to the waves. Each strategy was simulated in the time domain for regular and irregular waves, the latter taken from a modified Pierson-Moskowitz spectrum. In regular waves, the performance advantages over a passive system appear at frequencies near resonance while at the lower and higher frequencies they become nearly equivalent. For irregular waves, the NMPC strategy leads to greater energy absorption than the passive system, though strongly dependent on the prediction horizon. It was found that the ideal NMPC strategy required a generator that could be turned on and off instantaneously, leading to sequences where the generator can be inactive for up to 50% of the wave period.
Article
The Hilbert-Huang transform (HHT) [see the author et al., Proc. R. Soc. Lond., Ser. A, Math. Phys. Eng. Sci. 459, No. 2037, 2317–2345 (2003; Zbl 1045.94503); ibid. 454, No. 1971, 903–995 (1998; Zbl 0945.62093)] is a relatively new method in data analysis. Its power is in the totally adaptive approach that it takes, which results in the adaptive basis, the intrinsic mode functions (IMFs) from which the instantaneous frequency can be defined. This offers a totally new and valuable view of nonstationary and nonlinear data analysis methods. With the recent developments on the normalized Hilbert transform, the confidence limit, and the statistical significance test for the IMFs, the HHT has become a more robust tool for data analysis, and it is now ready for a wide variety of applications. The development of HHT, however, is not over yet. We still need a more rigorous mathematical foundation for the general adaptive methods for data analysis, and the end effects must be improved as well.
Article
A wave-structure interaction model is implemented, and power output estimates are made for a simplified wave energy converter operating in measured spectral wave conditions. In order to estimate power output from a wave energy converter, device response to hydrodynamic forces is computed using a boundary element method potential flow model. A method is outlined for using the hydrodynamic response to estimate power output. This method is demonstrated by considering an idealized non-resonating wave energy converter with one year of measured spectral wave conditions from the Oregon coast. The power calculation is performed in the frequency domain assuming a passive tuning system which is tuned at time scales ranging from hourly to annually. It is found that there is only a 3% gain in productivity by tuning hourly over tuning annually, suggesting that for a non-resonating wave energy converter, power output is not very sensitive to the value of the power take off damping. Interaction between wave energy converters in arrays is also considered, and results for an array of idealized point absorbers suggests that interactions are minimal when devices are placed 10 diameters apart from each other.