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Comparison of frequency estimation methods for

wave energy control

Paula B. Garcia-Rosa, Member, IEEE, John V. Ringwood, Senior Member, IEEE, Olav B. Fosso, Senior

Member, IEEE, and Marta Molinas, Member, IEEE

Abstract—A number of wave energy controllers tune the power

take-off (PTO) system to the frequency of incident waves. Since

real ocean waves are non-stationary by nature and not deﬁned

by a single frequency component, the PTO can be either tuned

at a constant frequency characterized by the local spectrum,

or continuously tuned to a representative wave frequency. In

either case, a time-frequency representation of the waves is

expected since the wave proﬁle changes over time. This paper

discusses about the PTO tuning problem for passive and reactive

controllers, in real waves, by comparing different methods

for time-varying frequency estimation: the extended Kalman

ﬁlter (EKF), frequency-locked loop (FLL), and Hilbert-Huang

transform (HHT). The aim is to verify the impact of such methods

on the absorbed and reactive powers, and the PTO rating. It is

shown that the mean estimated frequency of the EKF, and FLL,

converges respectively to the mean centroid frequency, and energy

frequency, of the excitation force spectrum. Moreover, the HHT

mean frequency has no correlation with the spectral statistical

properties. A comparison of the energy absorbed shows that up

to 37% more energy is obtained with the HHT over the other

estimation methods. Numerical simulations are performed with

sea elevation data from the Irish coast.

Index Terms—Energy harvesting, wave power, frequency esti-

mation, control systems.

I. INTRODUCTION

IN order to optimize the energy extracted from waves, a

number of control algorithms for wave energy converters

(WECs) tune the power take-off (PTO) system to the frequency

of incident waves, see, e.g., [1]–[6]. For regular waves, the

PTO tuning is well deﬁned since the waves are characterized

by a constant frequency sinusoidal motion. Nevertheless, real

ocean waves are non-stationary by nature and not deﬁned by

a single frequency. Thus, the PTO can be either tuned to a

constant representative frequency of the local wave spectrum,

or continuously tuned to a time-varying frequency.

The PTO tuning consists of adjusting the damping of the

system, if a passive control method is adopted, or it may re-

quire adjustment of both the system damping and the stiffness

for a reactive control method. The tuning strategies usually

require knowledge of the incident wave frequency [1]–[5] or

the excitation force frequency [7], [8]. Since these frequencies

change with time for non-stationary signals, a time-frequency

Paula B. Garcia-Rosa and Marta Molinas are with the Dept. of Eng.

Cybernetics, Norwegian Univer. of Science and Technology, Trondheim,

Norway (emails: p.b.garcia-rosa@ieee.org; marta.molinas@ntnu.no).

John V. Ringwood is with the Centre for Ocean Energy Research, Maynooth

University, Maynooth, Ireland (e-mail:john.ringwood@nuim.ie).

Olav B. Fosso is with the Dept.of Electric Power Eng., Norwegian Univ. of

Science and Technology, Trondheim, Norway (email: olav.fosso@ntnu.no).

representation for the waves is expected – i.e. the frequency

is a function of time.

By adopting different methods to estimate the time-varying

frequency for irregular waves and real ocean waves, some

studies show that continuously tuning the PTO result in greater

energy absorption than tuning the PTO to a constant frequency

of the wave spectrum [2], [8]. In [2], two methods have been

adopted to estimate the on-line dominant wave frequency. One

is based on the sliding discrete Fourier transform, while the

other uses an analysis of the low-pass ﬁltered incident wave,

where wave characteristics, such as the zero-up crossing and

crest-crest periods are calculated. In [8], an estimation of the

instantaneous frequency of the excitation force is obtained

by means of the Hilbert-Huang transform (HHT), and such

information is used for tuning the PTO damping of a WEC

on a wave-by-wave basis. Other methods have adopted the

extremum-seeking approach, where knowledge of the wave

frequency is not needed for tuning purposes [9], [10]. In such

cases, the PTO parameters are adapted on an hourly basis

(according to sea states variations) rather than a wave-by-wave

basis.

Furthermore, a number of control algorithms also rely on the

estimation of a time-varying frequency or the energy period of

the waves [6], [7], [11]–[14]. An on-line estimate of the exci-

tation force frequency is obtained with the extended Kalman

ﬁlter (EKF) in [7]. In the proposed controller, the velocity

reference is set as the ratio between the excitation force and the

radiation damping, which is tuned to the estimated frequency

[7]. An adaptive vectorial control approach is proposed in [12],

where a frequency-locked loop (FLL) is adopted to estimate

the frequency of the WEC velocity.

The EKF-based method is based on a harmonic model

with one variable frequency [7]. Then, the EKF tracks only

a single dominant frequency. In contrast, the FLL method is

based on an adaptive ﬁlter structure and an integral controller,

where the selectivity of the ﬁlter and the tracking frequency

dynamics can be adjusted [15]. Moreover, the instantaneous

frequency, deﬁned by the derivative of a phase function,

is a time-varying parameter which identiﬁes the location of

the spectral peak of the signal as it varies with time [16].

However, the instantaneous frequency has physical meaning

only for mono-component signals, i.e., signals with a single

frequency or a narrow range of frequencies varying as a

function of time [16]. In order to calculate the instantaneous

frequency of multi-component signals, the HHT method [17]

ﬁrstly decomposes the original signal into a number of mono-

component signals through the empirical mode decomposition

arXiv:1804.02247v1 [cs.SY] 6 Apr 2018

2

(EMD). Such a decomposition has an adaptive basis and relies

on the local characteristics of the signal. Thus, the EMD can

extract different oscillation modes present in a wave proﬁle.

The aim of this paper to verify the impact on the WEC

performance of adopting different frequency estimation meth-

ods for tuning purposes in real ocean waves. Three methods

are adopted for frequency estimation: EKF, FLL, and HHT.

The methods are conceptually different and vary from es-

timating a single dominant frequency (EKF) to estimating

the instantaneous wave-to-wave frequency of the oscillation

modes present in a wave proﬁle (HHT).

This study considers both passive and reactive controllers.

For reactive control, the total power consists of a combination

of active and reactive powers. Therefore, it is fundamental

to individually determine these two power components to

indicate the actual absorbed power, and the power that has

to be supplied by the PTO during the conversion process.

The performance of the WEC is then measured in terms of

absorbed power, peak-to-average power ratio (PTO rating), and

ratio of average reactive power and absorbed power. Numerical

simulations with real sea elevation data from the Irish west

coast are presented.

II. MODELING AND CONTROL OF WECS

A. Equations of Motion

This study considers a single oscillating-body represented

as a truncated vertical cylinder constrained to move in heave.

With the assumption of linear hydrodynamic theory, and

neglecting friction and viscous forces, the motion of the

ﬂoating cylinder is described by the superposition of the wave

excitation force (fe), radiation force (fr), restoring force (fs)

and the force produced by the PTO mechanism (fp):

m¨x(t) = fe(t) + fr(t) + fs(t) + fp(t),(1)

where x∈Ris the vertical position of the body, and m∈

R+is the body mass. The restoring force is given by fs=

−Sx, where S∈R+is the buoyancy stiffness. From [18], the

radiation force is calculated as

−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. From (1) and (2),

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, i.e., the force

due to the incident waves is given by

fe(t) =

∞

Z

−∞

he(t−τ)ζ(τ)dτ , (5)

he(t) = 1

2π

∞

Z

−∞

He(ω)eiωt dω . (6)

heis the inverse Fourier transform of the excitation force trans-

fer function He(ω), which has low-pass ﬁlter characteristics

for ﬂoating WECs, and ζis the wave elevation. Notice that

(6) is non-causal, since in fact, the pressure distribution is the

cause of the force and not the incident waves [19].

B. Passive Control

A generic PTO mechanism, with a damper varying in time

(Bp∈R+), is considered for the passive control (PC). Thus,

fp(t) = −Bp(t) ˙x(t).(7)

For the case when Bp(t) = Bp, for any time t, and for

monochromatic waves, the maximum absorption is obtained

when [19]:

Bp=p(Br(ω))2+ (ω(m+mr(ω)) −S/ω)2,(8)

where mr(ω)∈Ris the added mass. Equation (8) is frequency

dependent, and indicates that there is an optimal damping for

each frequency when the WEC is submitted to real ocean

waves, or irregular waves with a mixture of frequencies.

Here, the PTO damping is continuously modiﬁed, and tuned

to the excitation force frequency. From (8),

Bp(t) = p(Br(ˆω))2+ (ˆω(m+mr(ˆω)) −S/ˆω)2,(9)

where ˆω(t)is the estimated time-domain frequency of the

wave excitation force. In order to examine the impact of the

time-varying frequency on the performance of the WEC, three

methods are adopted for the frequency estimation: EKF, FLL,

and HHT.

C. Reactive Control

For reactive control (RC), the PTO force consists of a

damping term and a spring term:

fp(t) = −Bp(t) ˙x(t)−Sp(t)x(t),(10)

where Sp∈Ris the stiffness coefﬁcient. For incident regular

waves, if Sp(t)= Spfor any time t, and

Sp=ω2(m+mr(ω)) −S , (11)

then the reactive part of the total impedance

[Rp+Rr(ω)] + jω[(m+mr(ω)−(S+Sp)/ω2)]

is cancelled and the velocity of the ﬂoating body is in phase

with the excitation force [19]. In such a case, the PTO damping

(8) becomes

Bp=Br(ω),(12)

and the greatest wave energy absorption is obtained.

3

Following the same procedure for PC, Spand Bpare tuned

to the excitation force frequency. From (11) and (12),

Sp(t) = ˆω2(t)(m+mr(ˆω(t))) −S , (13)

Bp(t) = Br(ˆω(t)) .(14)

Notice that, for practical application studies, the physical

limits of the body excursion and the PTO should be taken

into account. Here, a PTO force constraint is implemented as a

saturation. However, this is only a theoretical approach, since

the body motion is also a function of the excitation force,

which is an external force that cannot be manipulated. The

implementation of saturation on the force signals of a real

WEC is not physically possible.

D. Energy and Power

The mean power and energy absorbed by the WEC over a

time range Tare, respectively,

¯

P=E

T,and E=−ZT

0

Bp(t) ˙x2(t)dt . (15)

Notice that, for the reactive control, the delivered power has

two components: the absorbed power (or active power, that

is the power delivered to the damping Bp), and the reactive

power (the power delivered to the spring Sp) [19]. The mean

reactive power, and reactive energy, are respectively,

¯

Pr=Er

T,and Er=−ZT

0

Sp(t)x(t) ˙x(t)dt . (16)

Since the spring term in the PTO force (10) and the body

velocity can have opposite signs in (16), the PTO has to return

energy for some parts of the wave cycle. Then, the PTO system

should be able to implement bidirectional power ﬂow for RC.

In this study, the performance of the WEC in terms of mean

absorbed power ( ¯

P) is measured by the capture width ratio,

CWR =¯

P

2rPζ

,(17)

where ris the cylinder radius and Pζis the transported wave

power per unit width of the wave front. In deep water [19],

Pζ=ρg

2Z∞

0

Sζ(ω)

ωdω , (18)

where Sζis the wave spectrum, ρis the sea water density, and

gis the gravitational acceleration.

III. EST IM ATIO N OF T HE WAVE EX CI TATION FORCE

FR EQUENCY

A. Extended Kalman Filter

In order to estimate the frequency by means of the EKF,

we assume a harmonic model with a single time-varying

frequency component and amplitude, as proposed in [20].

Thus, fe(t)can be expressed in discrete-time (t=kTs) as:

fe[k] = A[k] cos (ω[k]kTs+ϕ[k]) + η[k],(19)

where Ais the amplitude of the wave excitation force, ϕis

the phase, and Tsis the sampling time. Following the cyclical

structural model from [21], (19) can be written in a recursive

non-linear state space form as

υ[k+ 1] = f(υ[k]) + $[k]

fe[k] = h(υ[k]) + η[k](20)

where υ∈R3is the state vector deﬁned as υ=ψ ψ∗ωT,

$∈R3and η∈Rare zero-mean independent random pro-

cesses with covariance matrices deﬁned, respectively, as

E[$$T] = Rand E[ηηT] =Q. Functions f(·)and h(·)are,

f(υ[k]) =

cos (ω[k]Tsk) sin (ω[k]Tsk) 0

−sin (ω[k]Tsk) cos (ω[k]Tsk) 0

0 0 1

υ[k],

and h(υ[k])=100υ[k], respectively [20].

The EKF obtains an estimate of the state vector υ[k]based

on observations of fe[k], and on the ﬁrst-order linearization of

model (20) around the last state estimate. The EKF algorithm

is summarized in Table I, where Iis the identity matrix of

order 3, Jfand Jhare the Jacobian matrices of f(.)and h(.),

denoted respectively by Jf[k] = ∇f|ˆυ[k|k]and Jh[k+ 1] =

∇h|ˆυ[k+1|k].

TABLE I

EKF ALGORITHM.

Prediction step:

ˆυ[k+ 1|k] = f(ˆυ[k|k])

P[k+ 1|k] = Jf[k]P[k|k]Jf[k]T+Q[k]

Innovation step:

ˆυ[k+ 1|k+ 1] = ˆυ[k+ 1|k]

+K[k+ 1](fe[k+ 1] −h(ˆυ[k+ 1|k]))

K[k+ 1] = P(k+ 1|k)Jh[k+ 1]T

(Jh[k+ 1]P[k+ 1|k]Jh[k+ 1]T+R[k+ 1])−1

P[k+ 1|k+ 1] = (I−K[k+ 1]Jh[k+ 1])P[k+ 1|k]

Once an estimate of the state vector ˆυ[k|k]is available from

the EKF (Table I), the amplitude and frequency of the wave

excitation force are, respectively, obtained as:

ˆ

AEKF [k|k] = qˆ

ψ[k|k]2+ˆ

ψ∗[k|k]2,(21)

ˆωEKF [k|k] = ˆω[k|k].(22)

B. Frequency-Locked Loop

An adaptive ﬁlter is implemented by means of a second-

order generalized integrator (SOGI), where the FLL estimates

the frequency of the input signal. Such a scheme is termed

SOGI-FLL, and has been proposed for grid synchronization of

power converters [15]. Figure 1 illustrates the block diagram

of the SOGI-FLL, consisting of the SOGI-QSG (SOGI quadra-

ture signal generator) and the FLL structure. The method

was ﬁrst adopted, within the wave energy control context, to

estimate the frequency components of the WEC velocity [12].

In this study, the method is adopted for estimating the wave

excitation force frequency.

The SOGI-QSG acts as an adaptive bandpass ﬁlter with

two in-quadrature output signals (ξ,ξ∗), where ξ∗lags ξ

by 90◦. The bandwidth of the ﬁlter is exclusively set by

the gain κ[15]. The FLL is responsible for estimating the

frequency of the input signal. Notice that the in the nonlinear

4

X

X

X

+

+

+

+

−

−

fen(t)

ˆωF LL

ξ(t)

ξ∗(t)

εω

ωe

R

R

R

κ

−γ

1

Fig. 1. Frequency estimation by the SOGI-FLL method.

frequency adaptation loop, εω=εξ∗can be interpreted as

a frequency error variable, and the parameter γrepresents

the gain of the integral controller [15]. The selectivity of the

adaptive bandpass ﬁlter and the tracking frequency dynamics

are respectively determined by the tuning parameters κand γ.

From Fig. 1, the state-space equations of the SOGI-FLL are

˙

ξ

˙ν=−κˆωFLL −ˆω2

FLL

0 1 ξ

ν+κˆωFLL

0fen

ξ

ξ∗=1 0

0 ˆωFLL ξ

ν(23)

˙

ˆωFLL =−γ(fen −ξ)ξ∗ˆωFLL ,(24)

where (ξ, ν)and (ξ, ξ∗)are, respectively, the state and output

vectors of the SOGI, and fen is the normalized excitation

force. The FLL state equation is represented by (24).

C. Hilbert-Huang Transform

In the HHT method, the wave excitation force fe(t)is ﬁrstly

decomposed into Nmono-component signals (IMFs) by the

EMD. Then, the instantaneous frequency of the dominant IMF

is adopted for tuning purposes [8]. Figure 2 illustrates the

block diagram of the frequency estimation by this method.

HHT method

Hilbert

transform

EMD identify

dominant

IMF

fe(t)

c1(t)

c2(t)

.

.

.

cn(t)

cd(t) ˆωHHT (t)

1

Fig. 2. Frequency estimation by the HHT method.

The EMD identiﬁes local maxima and minima of fe(t), and

calculates upper and lower envelopes for such extrema 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 Table II.

Then, the wave excitation force can be expressed as

fe(t) =

N

X

i=1

ci(t) + r(t),(25)

TABLE II

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.

where Nis the total number of IMFs, which is deﬁned here as

log2Ns−1[22], Nsis the data length, and r(t)is the residue.

The dominant IMF is identiﬁed through the comparison of

the energy of the IMF signals (Eci) with the energy of the

excitation force signal (Efe),

Eci=ZT

0|ci(t)|2dt , Efe=ZT

0|fe(t)|2dt , (26)

where ci(t)is the i-th IMF component. The dominant com-

ponent cd(t)is the IMF with the highest Eci/Eferatio.

Finally, the Hilbert transform (HT) is applied to cd(t)[17]:

υd(t) = 1

πPZ∞

−∞

cd(τ)

t−τdτ , (27)

where Pindicates the Cauchy principal value. Then, the

dominant IMF is represented as an analytic signal,

zd(t) = cd(t) + jυd(t),(28)

with amplitude ˆ

AHHT , phase ˆ

φHHT , and instantaneous frequency

ˆωHHT , respectively estimated as

ˆ

AHHT (t) = qc2

d(t) + υ2

d(t),ˆ

φHHT (t) = arctan υd(t)

cd(t),(29)

ˆωHHT (t) = ˙

φd(t).(30)

IV. SIMULATION RESULTS

A. Hydrodynamic parameters

The same heaving cylinder adopted in [8] is considered here.

The cylinder has a radius of r= 5 m, draught d= 4 m, mass

m= 3.2×105kg and resonance frequency 1.2rad/s. The hy-

drodynamic coefﬁcients of the cylinder were computed using

the boundary element solver WAMIT [23]. The added mass,

radiation damping coefﬁcients, and the frequency response of

the excitation force are shown in [8].

B. Real sea elevation data

The wave data was collected in 2010 from a data buoy in the

Belmullet wave energy test site, off the west coast of Ireland.

The wave data, provided by the Irish Marine Institute, consists

of wave elevation records of 30 minutes sampled at 1.28 Hz.

Six wave elevation records (referred as sea states S1-S6),

with different spectral distribution, were selected for our study.

Figure 3 illustrates the wave spectra of the sea states, and

5

Table III shows the signiﬁcant wave height (Hs), the peak

frequency (ωp), the energy frequency (ωe), and the mean cen-

troid frequency (ω1) of the spectra. The statistical parameters

Hs,ωeand ω1are respectively calculated as: Hs= 4√m0,

ωe=m0/m−1,ω1=m0/m1, where mn=R∞

0ωnS(ω)dω is

the spectral moment of order n.ωpis the frequency at which

the wave spectrum is maximum.

Figure 4 shows the spectral density of the excitation force

for the selected sea states. Some of the high frequency waves

are ﬁltered out by the transfer function He(ω), as can be noted

from Figure 3. The ﬁltering characteristics are deﬁned by the

shape of the ﬂoating body, so that the excitation force spectra

are characteristic of the cylinder adopted in this study.

Fig. 3. Wave spectra of real wave data from Belmullet.

TABLE III

SIGNIFICANT WAVE HEIGHT Hs(M), PEAK FREQUENCY ωp(R AD/S) ,

ENERGY FREQUENCY ωe(RAD/S), A ND ME AN C ENT ROI D FR EQU ENC Y

ω1(RAD/S)OF T HE SE LEC TE D SEA S TATES .

S1 S2 S3 S4 S5 S6

Hs1.26 1.43 1.18 1.39 1.42 1.62

ωp0.52 1.22 0.52 0.57 0.74 0.93

ωe0.59 0.94 0.80 0.80 0.90 0.97

ω10.70 1.18 1.13 1.18 1.06 1.08

C. Time-frequency estimation by EKF, FLL and HHT

1) Superposition of two regular waves: In order to illustrate

how the estimated excitation force frequency differs according

to the method adopted, ﬁrstly we consider a simple incident

wave deﬁned as the superposition of two regular waves: ζ(t)=

2 cos (2π/6t) + cos (2π/8t). The energy frequency and the

mean centroid frequency of the excitation force spectra are,

respectively, ωe,fe=0.94 rad/s and ω1,fe=0.96 rad/s.

Figure 5 illustrates the excitation force frequency estimated

by the EKF, FLL, and HT. The Hilbert spectrum shows that the

instantaneous frequency varies from about 0.92 to 1.4 rad/s,

with the highest energy content (highest amplitude) in the

lowest frequency. The EKF tracks a single frequency (0.96

rad/s) which represents the mean centroid frequency of the

excitation force spectrum, and the mean frequency estimated

Fig. 4. Excitation force spectra for sea states S1-S6.

by the FLL (0.93 rad/s) is close to the energy frequency of

the spectrum. The frequency estimated by the EKF is nearly

constant, while the tracking frequency dynamics of the FLL

depends mainly on the selection of the parameters κand γ.

To ensure high frequency selectivity, and accurate direct and

quadrature components within frequency range 0.6 to 1 rad/s,

κis set to √2and γ=0.16, as discussed in [12].

600 650 700

0.8

0.9

1

1.1

1.2

600 650 700

0.8

1

1.2

1.4

1.6

2.5

5

7.5

10

105

Fig. 5. Frequency estimated (rad/s) by the EKF, FLL (left) and HT (right) for

the wave ζ(t)= 2 cos (2π/6t) +cos (2π/8t). The plot on the right represents

the Hilbert spectrum.

2) Sea Elevation Data: Table IV shows the energy and

mean centroid frequencies of the excitation force spectra, and

the mean frequencies estimated by the studied methods for

sea states S1 to S6. It can be noted that the mean values of

the frequencies estimated by the EKF converge to the mean

centroid frequency of the spectra, while the mean frequency

values of the FLL converge to the energy frequency of the

spectra. However, the mean values estimated by the HHT have

no correlation with the statistical parameters obtained from the

spectra. As has been remarked by Huang et al. [24], frequency

in the Hilbert spectrum has a different meaning from Fourier

spectral analysis. In Fourier spectral analysis, the existence of

energy at a frequency means that a component of a sine (or

a cosine) wave persisted through the entire time range of the

data, whereas in the Hilbert spectrum the wave representation

is local and the exact time of such oscillation is given [24].

The estimated frequencies for sea states S1 and S2 are

illustrated in Figure 6. From the Hilbert spectrum of the ﬁrst

IMF, it can be noted that the frequency ranges 0.5−0.6rad/s,

and 0.6−0.8rad/s, have the highest energy content, respec-

tively, for S1 and S2. Such frequency values coincide with

6

0 300 600 900 1200 1500 1800

0.45

0.5

0.55

0.6

0.65

0 300 600 900 1200 1500 1800

0

0.5

1

1.5

2

2.5

1

2

3

4

5

105

0 300 600 900 1200 1500 1800

0.5

0.6

0.7

0.8

0 300 600 900 1200 1500 1800

0

0.5

1

1.5

2

2.5

1

2

3

4

105

Fig. 6. Frequency estimated (rad/s) by the EKF, FLLt and HHT for (a) S1 and (b) S2. The plots in the bottom represent the Hilbert spectrum of the ﬁrst

IMF.

TABLE IV

ENERGY FREQUENCY (ωe), MEAN CENTROID FREQUENCY (ω1)OF TH E

EXCITATION FORCE SPECTRA,AND M EAN VAL UE S OF TH E ES TIM ATED

FREQUENCIES (¯ωEKF ,¯ωFLL ,¯ωHHT ). FREQUENCIES IN RAD/S.

S1 S2 S3 S4 S5 S6

ωe,fe0.54 0.66 0.59 0.56 0.72 0.82

ω1,fe0.55 0.73 0.63 0.59 0.76 0.87

¯ωEKF 0.56 0.75 0.65 0.60 0.78 0.88

¯ωFLL 0.53 0.63 0.57 0.55 0.70 0.80

¯ωHHT 0.62 1.01 0.80 0.73 0.90 0.97

the estimates from the EKF and FLL, but the HHT method

identiﬁes the time at which such oscillations occur. Notice that

the HHT analysis returns nine IMFs, and the ﬁrst IMF is the

dominant component in all studied cases, as shown in [8].

D. Effect of the estimated frequency on the control strategy

In order to limit the body excursions to 2.5m for the studied

cases, the PTO force of the PC (7) and each term of the RC

in (10) is limited to ±500 kN.

1) Passive Control: The performance of the WEC is illus-

trated in Figures 7 and 8, for the cases when the PC strategy

adopts the EKF, FLL or HHT methods to estimate the excita-

tion force frequency for sea states S1 to S6 (section IV-C2).

For all the studied cases, tuning the damping with frequency

estimates from the HHT gives greater energy capture than

tuning with EKF and FLL. The highest improvement is of

a factor of 1.27 when HHT is compared to EKF, or 1.37

when compared to FLL, for sea state S2. Moreover, the lowest

differences in the CWR is obtained for S1. Such behaviour can

be explained by the different energy spectral distributions of

both sea states. S1 is characterized by a narrowband spectrum

with a single dominant swell (low frequency waves generated

in other locations), with the energy concentrated in a narrow

band of frequencies. However, S2 is characterized by a two-

peak spectrum with mixed wind-sea (high frequency waves

generated by the local wind) and swell conditions, with the

energy spread over a wider band of frequencies than S1.

In such a case, a method that calculates the wave-to-wave

frequency is more beneﬁcial for PC than a method that gives a

dominant sea state frequency. Nevertheless, the PTO rating and

the maximum PTO required for the HHT frequencies are also

higher than for the EKF and FLL, especially for S2 (Fig. 8).

S1 S2 S3 S4 S5 S6

0

0.1

0.2

0.3

0.4

EKF

FLL

HHT

Fig. 7. CWR under PC tuned at frequencies from the EKF, FLL and HHT.

S1 S2 S3 S4 S5 S6

5

10

15

20

EKF

FLL

HHT

S1 S2 S3 S4 S5 S6

200

300

400

500

Fig. 8. WEC performance under PC (a) Peak-to-average power ratio (b)

Maximum PTO force required.

2) Reactive Control: Figures 9 and 10 illustrate the perfor-

mance of the WEC, for the cases when the reactive control

strategy adopts the EKF, FLL or HHT methods to estimate

the wave excitation force frequency. For the unconstrained

cases, the CWR is very large, in most situations, and the

HHT obtains an energy improvement of up to 2.64 over

the EKF, or 1.67 over the FLL. However, the body motion

ranges from −10 to 10 m, which is practically impossible for

a WEC with a draught of 4m. Moreover, the PTO rating for

the HHT approaches a factor of 50 and the ratio of average

reactive power and absorbed power is almost 60% in some

cases (Fig. 10.a). Such values would require oversized PTO

equipment, which would not be a rational economic choice.

For the constrained cases, the improvement obtained from

HHT over EKF or FLL is lower, but represents a more

7

S1 S2 S3 S4 S5 S6

0

0.25

0.5

0.75

1

S1 S2 S3 S4 S5 S6

0

1

2

3

4

EKF

FLL

HHT

Fig. 9. CWR under RC tuned at frequencies from the EKF, FLL and HHT

(a) Unconstrained case (b) Constrained case.

S1 S2 S3 S4 S5 S6

5

20

35

50

65

EKF

FLL

HHT

S1 S2 S3 S4 S5 S6

6

8

10

12

0.1 1.1 2.1 3.1

0

0.2

0.4

0.6

0.1 0.3 0.5 0.7

0

0.05

0.1

0.15

Fig. 10. WEC performance under RC: Peak-to-average power ratio (top) and

ratio of average reactive power and absorbed power as a function of CWR

(bottom) (a) Unconstrained case (b) Constrained case.

realistic scenario: an average energy improvement of 1.13 is

obtained when the HHT is compared to the EKF, or 1.23 when

compared to the FLL. Although the PTO rating varies from

a factor of 7to 10 for all frequency estimation methods, the

ratio of average reactive power and absorbed power are much

higher for the HHT, reaching almost 14% (Fig.10.b) for the

studied cases.

In order to illustrate the effect of the frequency estimates on

the variables of the system, for the constrained RC, Figure 11

shows samples of time-series simulation, and Figure 12 shows

the absorbed and the reactive energy over a 30-min simulation

interval for sea states S1-S2. It can be noted that, for sea

state S2, the reactive energy required for the RC, tuned with

HHT frequency estimates is much higher than the FLL or EKF

cases. In such a case, the HHT reactive power represents about

4.2% of the total power, whereas the FLL reactive power is

about 0.7%. Nevertheless, the absorbed power is 18% greater

with the HHT than with the FLL.

1450 1500 1550

-300

-150

0

150

300

1450 1500 1550

-2.5

-1.25

0

1.25

2.5

1450 1500 1550

-600

-300

0

300

600

1450 1500 1550

-300

-150

0

150

300

1450 1500 1550

-2.5

-1.25

0

1.25

2.5

1450 1500 1550

-600

-300

0

300

600

Fig. 11. Time-series of the excitation force, position, and PTO force for the

constrained RC tuned with the EKF (dashed blue line), the FLL (solid red

line) and the HHT (dashed dotted black line) (a) S1; (b) S2.

0 600 1200 1800

0

2.5

5107

0 600 1200 1800

-10

-5

0

5106

0 600 1200 1800

0

2.5

5107

0 600 1200 1800

-10

-5

0

5106

Fig. 12. Energy absorbed (top) and reactive energy (bottom) over a 30-min

simulation for the constrained RC tuned with the EKF (dashed blue line), the

FLL (solid red line) and the HHT (dashed dotted black line) (a)S1; (b) S2.

E. Discussion

In the EKF, a sinusoidal extrapolation method is used to

model the excitation force as a monochromatic harmonic

process, with varying amplitude and frequency. Thus, the

EKF follows a single dominant frequency. Simulation results

have shown that the estimated EKF frequency tracks the

mean centroid frequency of the excitation force spectrum.

The performance of the SOGI-FLL depends mainly on the

appropriate selection of two design parameters: κand γ. Such

parameters deﬁne the bandwidth of the adaptive ﬁlter and the

FLL tracking frequency dynamics [15]. For the FLL, it has

been shown that the mean estimated frequency converges to

the energy frequency of the excitation force spectrum.

The HHT calculates the instantaneous frequency of the

excitation force by decomposing the signal into a number

of IMF components. Here, we have chosen only the IMF

component with the highest energy content. This component

has a wider bandwidth than the estimates from the EKF or

8

FLL, and the CWR of the WEC is greater when the HHT

frequency estimates are used for PC and RC. Even with a low

energy content, the resonance frequency of the WEC is within

the HHT estimates, which can also explain why tuning the

controllers with these estimates result in the greatest energy

absorption.

As expected, the PTO rating for PC is higher with the HHT

than with the EKF, or FLL, in most of the cases. However, for

the constrained RC, the greatest PTO rating does not indicate

the greatest energy absorption. The average value of the PTO

rating is about the same for all methods. Still, the greatest

energy absorption, obtained with the HHT, also requires the

greatest amount of reactive energy.

Here we have assumed the wave excitation force is known

completely over the simulation interval. Both the EKF and

FLL methods give an online estimate of the frequency, pro-

vided that an estimate of the excitation force is available. In

the HHT method, we have adopted an off-line EMD algorithm,

but a few implementation studies on the HHT have proposed

real-time EMD algorithms, see, e.g., [25].

V. CONCLUSION

This paper has shown how different frequency estimation

methods, used for controller tuning purposes, impact the

energy absorbed by the WEC, the PTO rating and the required

reactive power during the conversion process. The effect of the

estimation methods on the WEC performance depend on the

control strategy employed, the PTO system constraints, and

the local wave spectrum.

For a control strategy that relies on the information of a

dominant frequency component, such as the schemes in [7]

and [13], the EKF or FLL should be adopted. The mean

frequency estimated by such methods converges, respectively,

to the mean centroid frequency or the energy frequency of

the spectrum. Moreover, if the sea state is characterized by a

narrowband spectrum, the beneﬁt of adopting a method that

estimates the wave-to-wave frequency is relatively small.

By adopting a method that estimates the instantaneous

frequency of the excitation force (the HHT), an average

improvement in the energy absorbed of about 18% is obtained

over the EKF and FLL methods, for the constrained reactive

control strategy. For passive control, an average improvement

of 16% is also obtained for the HHT. The greatest improve-

ments of the HHT over the other methods are obtained for

wideband spectra. In contrast to the EKF and FLL methods,

where the bandwidth is narrow and the frequency estimates

oscillate around a dominant frequency component, the HHT

frequency estimates cover a wider range, and the location of

the dominant frequency component is identiﬁed.

In this study, the ﬁrst IMF component is adopted for

the HHT approach. The frequency bandwidth of this IMF

component could be narrowed by applying techniques that

deal with mode mixing in EMD. Such an approach will be

explored in future studies.

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