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Switching sequences for non-predictive declutching control of wave energy converters

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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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Switching sequences for non-predictive
declutching control of wave energy
converters
Paula B. Garcia-Rosa ,∗∗ Olav B. Fosso ∗∗ Marta Molinas ∗∗∗
SINTEF Energy Research, 7465 Trondheim, Norway
(e-mail: paula.garcia-rosa@sintef.no).
∗∗ Department of Electric Power Engineering, Norwegian University of
Science and Technology, NO-7491 Trondheim, Norway
(e-mail: olav.fosso@ntnu.no).
∗∗∗ Department of Engineering Cybernetics, Norwegian University of
Science and Technology, NO-7491 Trondheim, Norway
(e-mail: marta.molinas@ntnu.no)
Abstract: 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.
Keywords: Renewable energy systems, wave energy, declutching control, on-off actions,
numerical simulation.
1. INTRODUCTION
In order to absorb maximum power from waves, con-
trol systems require wave forecasting and bi-directional
power take-off (PTO) systems. In this framework, con-
trol schemes based on model-predictive control are widely
studied in the literature, see, e.g., Faedo et al. (2017)
for a review. Alternatively, passive control methods are
sub-optimal solutions that avoid the need for the PTO to
supply power (and hence are named passive) while still in-
creasing the absorption when compared to passive loading
(PL). In contrast to PL, where the PTO damping is con-
stant, passive control methods can modify the damping on
a timely basis, either by tuning it to the frequency of waves
(Garcia-Rosa et al., 2019) or by solving an optimization
problem to determine the optimal damping profile, which
results in on/off sequences (Tom and Yeung, 2013) – a
characteristic of the declutching control (DLC) method.
Declutching is a phase-control method that consists of
disengaging the PTO system from the oscillating body
during some parts of the body motion cycle. The principle
is to engage and disengage the PTO in order to allow
the oscillating body to “catch up” to the excitation force,
and then, bring the body velocity into phase with the
excitation force.
The switching sequences for DLC are usually determined
by optimization procedures that require the knowledge
of future excitation force (Babarit et al., 2009; Teillant
et al., 2010; Cl´ement and Babarit, 2012), which remains
an open challenge for practical implementation. Babarit
et al. (2009) have used an optimization procedure based
on Pontryagin principle to determine the instants the PTO
system is either on or off. It is shown that changes in
the controller state, from off to on, are followed by zero-
crossings of the velocity, but a heuristic criterion regarding
the instants to switch back the controller state to off has
not been identified.
Furthermore, aiming at determining the best damping
profile in terms of energy absorption for regular waves,
Teillant et al. (2010) have used a general parametrization
of the damping force that allowed for two phase-control
methods: latching and declutching. While latching locks
the body motion for specific intervals of time, declucthing
control modifies the body dynamics without locking it.
Teillant et al. (2010) have shown that declutching (latch-
ing) control is optimal when the wave frequency is higher
(lower) than the body resonant frequency. For declutching,
the damping changes from upper to lower boundary values
when the body velocity vanishes. The duration of time for
lower damping, as well as the upper and lower values, were
determined through an optimization process with a genetic
algorithm and the Nelder-Mead algorithm.
In (Feng and Kerrigan, 2013), the PTO is clutched when
the body velocity is zero, and a derivative-free optimiza-
tion algorithm that reduces the number of function eval-
uations is proposed to determine for how long the PTO
should be active. A comparison of the wave energy con-
verter (WEC) performance for optimization formulations
based on past and future wave data is also presented.
To avoid the prediction of future waves, a few studies on
declutching have used the threshold unlatching strategy
(Hals et al., 2011; Garcia-Rosa and Ringwood, 2016). As
suggested by the name, the strategy was initially proposed
for latching control. It consists of unlatching the body at
the instant when the excitation force (or other reference
variable) passes a chosen threshold (Lopes et al., 2009).
The aim of this paper is twofold. Firstly, to investigate
how the PTO damping coefficient affects the optimal de-
clutching/clutching duration and the efficacy of the control
strategy in terms of improving the power absorption in
regular wave regimes. Rather than optimizing the PTO
damping, the purpose is to identify for which values of
damping, declutching control will not represent an efficient
solution to improve the energy absorption, when compared
to a simpler strategy as passive loading. Then, the goal
is to propose switching sequences that do not require
estimation of future incident waves. Two strategies that
use current information of the body motion are proposed,
and compared with the threshold unlatching strategy. Fur-
thermore, in all switching sequences, the decision about
clutching (declutching), i.e. engaging (disengaging) the
PTO, depends on the current estimation of the dominant
frequency of the wave excitation force and on the body
resonant frequency. The sequences differ on defining the
instants to declutch (clutch) again the PTO.
2. DYNAMIC MODELING OF THE WEC
2.1 Equation 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 the-
ory, and neglecting friction and viscous forces, the body
motion is described by the superposition of the wave
excitation force (fe), radiation and restoring forces, and
the force produced by the PTO mechanism (fp):
M¨x(t)+
t
Z
0
hr(tτ) ˙x(τ)+Sx(t) = fe(t)+fp(t),(1)
where xRis the vertical position of the body, M=
[m+mr()], mR+is the body mass, mr()R+
is the infinite-frequency added mass coefficient, defined
with the asymptotic values of the added masses at infinite
frequency, SR+is the buoyancy stiffness, and the kernel
of the convolution term hr(tτ) is known as fluid memory
term (Cummins, 1962),
hr(t) = 2
π
Z
0
Br(ω) cos(ωt τ)dω , (2)
where Br(ω)R+is the radiation damping coefficient, and
ωR+is the wave frequency.
The excitation force, i.e., the force due to the incident
waves is given by
fe(t) =
Z
−∞
he(tτ)ζ(τ)dτ , (3)
where heis the inverse Fourier transform of the excitation
force transfer function He(ω), which has low-pass filter
characteristics for floating WECs, and ζis the wave
elevation. Notice that (3) is non-causal, since in fact, the
pressure distribution is the cause of the force and not the
incident waves (Falnes, 2002).
The mean absorbed power over a time range Tis
Pa=1
T
T
Z
0
fp(t) ˙x(t)dt , (4)
where ˙x(t) is the velocity of the body. By considering a
generic PTO system and PL, the PTO force is defined as
fp(t) = Bc˙x(t),(5)
where BcR+is the PTO damping.
2.2 Steady-state Sinusoidal Motion
Under passive loading (5), the steady-state response of
the body velocity to fe(t) = Fe(ω) cos ωt is given by
˙xss (t) = |H()|Fe(ω) cos(ωt +φ(ω)), where
H() = 1
(Bc+Br(ω)) + j(ω(m+mr(ω)) S/ω),(6)
φ(ω) = arctan ω(m+mr(ω)) S/ω
Bc+Br(ω),(7)
and Fe(ω)R+is the excitation force coefficient.
By assuming the body resonant frequency is ωr
pS/(m+mr(ω)), the phase (7) between the velocity and
the excitation force can be rewritten as
φ(ω) = arctan (m+mr(ω))(ω2ω2
r)
ω(Bc+Br(ω)) .(8)
From (8), it can be noted that when
ω < ωr, then φ(ω)>0, and the velocity leads the
excitation force;
ω > ωr, then φ(ω)<0, and the velocity lags the
excitation force;
ω=ωr, then φ(ω) = 0, and the velocity is in phase
with the excitation force.
Furthermore, for regular wave regime and PL, the optimal
constant damping and the average power absorbed by the
WEC are, respectively, calculated as (Falnes, 2002):
Bc,opt(ω) = p(Br(ω))2+ (ω(m+mr(ω)) S/ω)2,(9)
Pc,opt =Bc,opt(ω)Fe(ω)2
2(Bc,opt(ω) + Br(ω))2.(10)
3. SWITCHING SEQUENCES FOR DECLUTCHING
CONTROL
Here we assume a generic PTO system, where the PTO
force is expressed as
fp(t) = Bp˙x(t)u(t),(11)
BpR+is the PTO damping, and the control signal u(t)
has two states: on (u=1) or off (u= 0).
As discussed in Section 2.2, when the incident wave
frequency is lower than the body resonant frequency, the
velocity is leading the excitation force by a certain phase
shift. Then, the phase-control method should act to slow
down the natural response of the body in order to force
the velocity to be in phase with the excitation force.
Conversely, the controller should act to speed up the
natural response of the body when the wave frequency
is higher than the WEC resonant frequency.
Following previous studies (Teillant et al., 2010), the
PTO force profile is determined according to the inci-
dent wave frequency and the resonant frequency of the
oscillating body. However, only the clutching concept, i.e.
engage/disengage the PTO, is applied here. Thus, for
ω < ωr: To slow down the device motion, the PTO
is connected, i.e. clutched (u= 1), when the body
velocity vanishes, and disconnected, i.e. declutched
(u=0), in a certain instant of time to be determined.
ω > ωr: To allow the device to move “faster”, the
PTO is disconnected (u= 0) when the body velocity
vanishes, and connected (u= 1) in a certain instant
of time to be determined.
The objective is to propose causal control laws for DLC by
performing a comprehensive numerical study using regular
waves at first. Using different values of damping, optimal
clutching durations are determined for the cases when
ω < ωr, and optimal declutching durations, for the cases
when ω > ωr. 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 ×105kg and resonant frequency ωr=1.2 rad/s.
3.1 Optimal clutching/declutching duration
The optimal clutching/declutching duration in seconds
(∆th,opt) that optimizes the power absorbed by the WEC
with decltuching control is determined via simulations
with a complete set of possible values for the time interval.
Figure 1 (right) illustrates the optimal intervals of time
obtained for different PTO coefficients and regular waves
with amplitude of 1 m and some specific frequencies. The
PTO damping coefficients are defined as
Bp=¯
BpBc,opt(ω),(12)
for each wave frequency ω, where ¯
Bpis a dimensionless
constant, 0.5¯
Bp20, and Bc,opt(ω) is the optimal
-0.6-0.4-0.2 0 0.2 0.4 0.6 0.8 1 1.2
0
1
2
3
4
5
6
-0.6-0.4-0.2 0 0.2 0.4 0.6 0.8 1 1.2
0
1
2
3
4
5
6
-0.6-0.4-0.2 0 0.2 0.4 0.6 0.8 1 1.2
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
-0.6-0.4-0.2 0 0.2 0.4 0.6 0.8 1 1.2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
Fig. 1. Evolution of optimal time (right) and ratio of
power (left) versus PTO damping in regular waves. (a)
Optimal clutching duration, ω < ωr; (b) Optimal de-
clutching duration, ω > ωr. The lines in (a) represent
the latching duration for each one of the frequencies.
constant damping (9). It can be noted that, for ω < ωr, as
the value of the damping increases, the optimal clutching
duration reduces and tends to the latching duration. The
latching duration, denoted as ∆tLin Fig. 1.a (right), is
calculated as half the difference between the wave period
and the body resonant period (Babarit and Cl´ement,
2006). When the damping is too high (20Bc,opt(ω)) and
u=1, the PTO force (11) restricts the motion of the body,
and consequently, declutching control acts in a similar way
as latching control for the cases when ω < ωr. Conversely,
when ω > ωrthe optimal declutching duration increases
with the damping.
The ratio of power, calculated as r=Pa/Pc,opt, is also
shown in Figure 1 (left). For ω < ωrand Bp<2Bc,opt(ω),
DLC will result in less power absorption than the optimal
PL power, Pc,opt (10), or about the same value. Thus,
the value of damping is also important for the efficacy of
the strategy. This is further evidenced by Figure 2, which
shows the evolution of the ratio of power for different wave
frequencies, PTO damping coefficients, optimal clutch-
ing duration (Fig. 2.a) and optimal declutching duration
(Fig. 2.b). For ω > ωrand the WEC considered in this
study, the damping for the declutching strategy should
be higher than 2Bc,opt(ω) to allow an energy absorption
greater than optimal passive loading.
Fig. 2. Ratio of power as a function of PTO damping and
wave frequencies in regular waves with (a) optimal
clutching duration (b) optimal declutching duration.
To illustrate how the behaviour of system variables
changes according to PTO damping for ω < ωr, Figure 3
shows the time-series of normalized variables ( ¯
fe,¯x, ¯
˙x, x ˙x)
and the control signal (u). Constants ¯
Bp= 1.5 (Fig 3.a),
and ¯
Bp= 5 (Fig. 3.b), are used with their corresponding
optimal clutching duration. The instants uchange from
on to off occur very close to the instants the position is
zero, or equivalently, when the signal x˙xhas a positive
zero crossing. Nonetheless, the same conclusion cannot be
drawn when ¯
Bp= 5 (Fig. 3.b). In such a case, the high
damping force employed after the velocity zero crossing
causes the velocity to remain around zero, almost stopping
the body motion, and thus, neither xnor x˙xprovide a good
indication for disengaging the PTO.
For ω > ωr, the instants when ˙x= 0 define the instants to
disengage the PTO. Figure 4 illustrates the time-series of
normalized variables and the control signal when ¯
Bp=2.5
(Fig 4.a), and ¯
Bp= 5 (Fig. 4.b), with their corresponding
optimal declutching duration. It is noted that the intervals
while u=1 coincide with intervals the body is speeding up,
i.e. acceleration and velocity are in the same direction, and
the optimal instant the PTO is switched on occurs when
the device starts to slow down (Fig.4.a), i.e. ˙x¨x < 0, or a
few seconds after it (Fig. 4.b) depending on the damping.
3.2 Switching sequences without prediction of future waves
This section proposes two strategies to determine the in-
stants the control signal uis either on or off, based on the
100 105 110 115
-1
0
1
100 105 110 115
-1
0
1
100 105 110 115
-1
0
1
100 105 110 115
-1
0
1
Fig. 3. Time-series of normalized variables and control
signal for (a) Bp= 1.5Bc,opt, and (b) Bp= 5Bc,opt .
(ω=0.75 rad/s, optimal clutching duration).
100 105 110 115
-1
0
1
100 105 110 115
-1
0
1
100 105 110 115
-1
0
1
100 105 110 115
-1
0
1
Fig. 4. Time-series of normalized variables and control
signal for (a) Bp= 2.5Bc,opt, and (b) Bp= 5Bc,opt .
(ω=1.65 rad/s, optimal declutching duration).
observations from Section 3.1. The strategies are evaluated
through a comparison with the unlatching threshold strat-
egy, when the WEC is submitted to synthetic irregular
waves and real ocean waves. Since such waves are not
defined by a single frequency, and the oscillating body has
low-pass filter characteristics, the mean centroid frequency
of the excitation force (ω1,fe) is adopted in replacement to
the wave frequency ωof regular wave cases.
The mean centroid frequency of the excitation force can
be estimated, e.g., by the extended Kalman filter (EKF)
as shown in (Garcia-Rosa et al., 2019). In such a case,
fe(t) has to be estimated as well. Note that the switching
sequences proposed here rely only on the estimation of the
frequency. Thus, an alternative scheme using real wave
measurements and He(ω) could be adopted to obtain an
estimation of the mean frequency of the excitation force.
The following switching sequences, denoted respectively as
D1 and ¬D2, are proposed for the cases when ω1,fe< ωr:
u(t) = 1, x ˙x(t)0,
0, x ˙x(t)>0,(13)
and for the cases when ω1,fe>ωr:
u(t) = 0,˙x¨x(t)0,
1,˙x¨x(t)<0.(14)
For threshold unlatching strategies, denoted here as D3
and D4, ˙x=0 triggers the control state until fe(t) passes a
chosen threshold (Lopes et al., 2009). Table 1 summarizes
the switching sequences, where D1–D4 are applied when
ω1,fe< ωr,¬D1–¬D4 are applied when ω1,fe> ωr. The
symbol ¬represents logical negation.
Three wave elevation records off the west coast of Ireland
(as in Garcia-Rosa et al. (2019)), referred as sea states
S1–S3, and six irregular waves (Ir1–Ir6) are adopted as
inputs to the WEC. For each real sea state, two irregular
waves are generated by modifying the parameters of Ochi
spectral distributions (Ochi, 1998). Figure 5 illustrates the
wave spectra of sea states and synthetic waves (top), and
the corresponding excitation force spectra (bottom). The
significant wave height Hsof the wave spectra, and the
mean centroid frequency for both the wave (ω1) and the
excitation force spectra (ω1,fe) are shown in Table 2.
The simulation interval is about 30 min for each case. A
comparison in terms of power absorption for the switching
sequences (Table 1) with PL power is shown in Figure6.
The ratio rcis calculated as Pa/Pc, where Pais the power
absorbed by the WEC when DLC is applied, and Pcis
the PL power with damping tuned at frequency ω1,fe.
As suggested by the analysis of Fig. 3, D1 results in the
greatest energy absorption when the damping is around
1.5 to 2.5Bc(ω1,fe) for waves Ir1–Ir3 and Ir5. Such waves
have mean centroid frequencies much lower than the body
resonant frequency (1.2 rad/s). As the damping increases,
the best strategies become the threshold unlatching (D3,
D4). Such strategies do not rely on the dynamic motion of
the WEC to switch off the control signal. Furthermore, the
benefit of applying DLC using D1–D4 in Ir4 is lower than
the other cases, since ω1,feis very close to ωr. Finally, when
ω1,fe>ωr(Ir6) and Bp<4Bc(ω1,fe), sequence ¬D2 is the
best choice. ¬D2 defines the time interval in which the
WEC is speeding up (slowing down) as the time interval
the control signal is off (on), and thus, it allows the body
to gain momentum.
In order to compare the strategies with best performances
in terms of power absorption for ω1,fe< ωr, Table 3
summarizes the obtained results for S1–S3 with D1 ( ¯
Bp=
2) and D4 ( ¯
Bp= 5). A comparison of the peak-to-average
power ratio (PTO rating), maximum PTO force, and
maximum displacement of the body are included with the
ratio rc. Although the strategy D4 with ¯
Bp= 5 results in
the greatest power in most of the cases, it requires higher
PTO forces and larger displacements of the body than D1.
It is important to note that constraints on the PTO force
and body motion are not considered here, but for practical
Table 1. Switching sequences
Condition u(t) Str. Condition u(t)
D1 x˙x0 1 ¬D1 x˙x0 0
x˙x > 0 0 x˙x > 0 1
D2 ˙x¨x0 1 ¬D2 ˙x¨x0 0
˙x¨x < 0 0 ˙x¨x < 0 1
D3 trigger: ˙x=0 1 ¬D3 trigger: ˙x=0 0
thres: ˙
fe=0 0 thres: ˙
fe=0 1
D4 trigger: ˙x=0 1 ¬D4 trigger: ˙x=0 0
thres: fe=0 0 thres: fe= 0 1
0 1 2 3
0
0.2
0.4
0.6
0.8
1
0 1 2 3
0
1
2
3
4
51011
0123
0
0.1
0.2
0.3
0.4
0123
0
5
10
15 1010
0123
0
0.2
0.4
0.6
0123
0
0.5
1
1.5
2
2.5 1011
Fig. 5. Wave spectra of real data and synthetic irregular
waves (top), and excitation force spectra (bottom).
Table 2. Significant wave height Hs(m),
wave mean centroid frequency ω1(rad/s),
and excitation force mean centroid frequency
ω1,fe(rad/s) for S1–S3 and Ir1–Ir6.
S1 Ir1 Ir2 S2 Ir3 Ir4 S3 Ir5 Ir6
Hs1.26 1.50 2.00 1.43 1.85 1.49 1.39 1.67 0.90
ω10.70 0.54 0.91 1.18 1.12 1.37 1.18 1.09 2.18
ω1,fe0.55 0.54 0.61 0.73 0.75 1.18 0.59 0.61 1.74
12345
0.5
1
1.5
2
2.5
12345
0.5
1
1.5
2
2.5
12345
0.5
1
1.5
2
12345
0.5
1
1.5
12345
0.5
1
1.5
2
2.5
12345
0.5
1
1.5
2
2.5
Fig. 6. Ratio rcfor irregular waves (Ir1-Ir6).
application studies, the physical limits of both the body
excursion and PTO have to be taken into account. Figure
7 shows a sample of time-series simulation for S1 with D1
(¯
Bp= 2) when the excitation force frequency is estimated
by the extended Kalman filter.
Table 3. Ratio rc, PTO rating, maximum PTO
force, and maximum body displacement for D1
and D4 in S1–S3.
Sea Strategy rcPmax /Pafp,max (kN) xmax (m)
S1 D1 ( ¯
Bp=2) 2.07 8.93 750.5 1.13
D4 ( ¯
Bp=5) 2.47 8.84 1.29×1032.27
S2 D1 ( ¯
Bp=2) 1.69 14.12 624.5 1.03
D4 ( ¯
Bp=5) 1.68 15.02 1.02×1031.64
S3 D1 ( ¯
Bp=2) 2.02 13.73 808.1 1.33
D4 ( ¯
Bp=5) 2.16 12.33 1.25×1032.16
900 905 910 915 920 925 930 935 940 945 950
0.55
0.552
0.554
900 905 910 915 920 925 930 935 940 945 950
-1
-0.5
0
0.5
1
Fig. 7. Time-series of EKF estimated frequency (top),
normalized excitation force, velocity, position, and
control signal for S1 with D1 and ¯
Bp=2 (bottom).
4. CONCLUSIONS
A comprehensive numerical study with different PTO
damping coefficients showed that the value of damping
plays an important role on the efficacy of the declutching
control method and also on the optimal time to engage
(or disengage) the PTO. Thus, the switching sequences for
non-predictive control should take into account the value
of the damping, or the PTO force applied.
Furthermore, the switching sequences proposed here rely
on current estimations of the mean centroid frequency of
the excitation force and on the body resonant frequency.
The decision about clutching (declutching) the WEC, i.e.
engaging (disengaging) the PTO, is defined according to
current estimations only. When the body velocity van-
ishes, the PTO is clutched (declutched) if the estimated
mean frequency is lower (higher) than the body resonant
frequency. The switching sequences differ on defining the
instants to declutch (clutch) the PTO again.
For a generic PTO system, the results showed that when
the declutching damping is twice PL damping as defined
by (9), the variables of motion (position, velocity, acceler-
ation) can be used for defining both the instants to clutch
and declutch. For instance, when the mean excitation force
frequency is higher than the body resonant frequency,
the PTO is disengaged while the body is speeding up
(i.e. velocity and acceleration are in the same direction
˙x¨x(t)0) to allow the body to gain momentum, and
engaged while the body is slowing down (i.e. ˙x¨x(t)<0).
However, the switching sequences relying on the motion
to define both instants to clutch and declutch are only
meaningful if the body motion is not excessively restricted
by high damping forces during clutching (e.g., when the
damping is five times the passive loading damping). In
such cases, the threshold unlatching strategy is the best
option for defining the switching sequences if wave fore-
casting is to be avoided.
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