wave energy converter control
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António F. de O. FalcãoInstituto Superior Técnico, Lisbon, Portugal
CONTROL TECHNIQUES FOR WAVE
ENERGY CONVERTERS
SUPERGEN MARINE 7th DOCTORAL TRAINING PROGRAMME WORKSHOP
Control of Wave and Tidal Converters
22-26 February 2010, Lancaster University
How far have we gone in 30+ yrs ? Some milestones:
1974 - Salter & the duck
1976 – Masuda
& Kaimei 1975- …The early theoreticians
1975-82 - The
British Program
Goal: 2 GW plant
1991: EU backs up wave energy
1996
EURATLAS
1999-2000
OWCs in
Europe
Early 1980s
Point absorbers
in Scandinavia
1985-91
The early OWCs
Since 2004
The “new”
offshore devices
Technology
challenge
Introduction
Introduction
Oscillating
Water Column(with air turbine)
Oscillating body(hydraulic motor, hy-
draulic turbine, linear
electric generator)
Overtopping(low head
water turbine)
Fixed
structure
Floating: Mighty Whale, BBDB
Isolated: Pico, LIMPET, Oceanlinx
In breakwater: Sakata, Mutriku
Floating
Submerged
Heaving: Aquabuoy, IPS Buoy, Wavebob,
PowerBuoy, FO3
Pitching: Pelamis, PS Frog, Searev
Heaving: AWS
Bottom-hinged: Oyster, Waveroller
Fixed
structure
Shoreline (with concentration): TAPCHAN
In breakwater (without concentration): SSG
Floating structure (with concentration): Wave Dragon
The size
While, in other renewables, the power is
more or less proportional to size/area,
…
… the power-versus-size relationship is much more
complex for wave energy converters.
The concept of “point absorber” was introduced in
Scandinavia around 1980 to describe efficient wave-
energy absorption by well-tuned small devices.
Theoretically (in linear wave theory), energy from a regular wave of given
frequency can be absorbed by a large oscillating body as well as from a small
one, provided both are tuned.
The oscillation amplitude is larger for the smaller body.
Introduction
Wave frequency
Ab
so
rbe
d p
ow
er
Large body
Small body
Wave energy absorption is wider-
banded for a large body than for a
“point-absorber”.
Introduction
This is relevant for real polychromatic
multi-frequency waves.
Here smaller oscillating-bodies are less
efficient than larger ones.
This can be (partially) overcome by
control (phase control).
100 150 200 250 300
t s
1
0.5
0
0.5
1
1.5
m
0 0.5 1 1.5 20
0.05
0.1
0.15
0.2
s10eT
m2sH
Frequency (rad/s)Sp
ectr
al p
ow
er d
ensi
ty (
m2s)
200
Most wave energy converters are complex (possibly multi-body)
mechanical systems with several degrees of freedom.
We consider first the simplest case:
• A single floating body.
• One degree of freedom: oscillation in heave
(vertical oscilation).DamperSpring
Buoy
PTO
Oscillating-body dynamics
DamperSpring
PTO
m
x
Basic equation (Newton):
)()( tftfxm mh
PTOon wetted
surface
Oscillating-body dynamics
excitation force (incident wave)
radiation force (body motion)
= hydrostatic force (body position)
mrd fgSxffxm
xgSf
f
f
f
hs
r
d
h
S
Cross-section
Frequency-domain analysis
Oscillating-body dynamics
• Sinusoidal monochromatic waves
• Linear system
A and B to be computed (commercial codes WAMIT,
AQUADYN, ...) for given ω and body geometry.
xCKxfm
Linear
spring
Linear
damperDamperSpring
PTO
m
x
DamperSpring
PTO
m
xOscillating-body dynamics
mrd fgSxffxm
xBxAfr
mass addedA
dampingradiation B
dfxKgSxCBxAm )()()(
mass added
mass radiation
damping
PTO
damping
buoyancyPTO
spring
Excitation
force
Oscillating-body dynamics
KgSCBiAm
FX d
)()(20
• Regular waves
• Linear system
tidd
ti eFfeXtx ,)( 0
tidd
ti eFfeXtx Re,Re)( 0
or simply
dfxKgSxCBxAm )()()(
Method of solution: )sincos( tite ti
amplitudescomplex generalin are ,:Note 0 dFX
geometrybody and givenfor computed be to)(amplitude wave
dF
Power = force velocity
Time-averaged power absorbed from the waves :
2
02
228
1
B
FXi
BF
BP d
d
Oscillating-body dynamics
KgSCBiAm
FX d
)()(20
Note: for given body and given wave amplitude and frequency ω, B and
are fixed.dF
P
B
FXi d
20
Am
KgS
CB
Resonance condition
Radiation damping = PTO damping
m
K
m
K
Then, the absorbed power will be maximum when :
= 0
Capture width L : measures the power absorbing capability
of device (like power coefficient of wind turbines)
E
PL
= absorbed powerP
E = energy flux of incident wave per unit crest length
Oscillating-body dynamics
For an axisymmetric body oscillating in heave (vertical
oscillations), it can be shown (1976) that
2max
EP or
2maxL
593.0PCFor wind turbines, Betz’s limit is
Note: may be larger
than width of bodymaxL
Oscillating-body dynamics
2
Max. capture
width
Axisymmetric
heaving body
Axisymmetric
surging body
Incident
waves
Incident
waves
wavewave
5 7.5 10 12.5 15 17.5 20
T
0
0.2
0.4
0.6
0.8
1
PP
xam
21
*a
gTT
Example: hemi-spherical heaving buoy of radius a
5.0*C
2*C
5*C
Oscillating-body dynamics
maxP
P 0.2*2125 ga
CC
Dimensionless
PTO damping
Dimensionless
wave period
No spring, no reactive control, K = 0
for maxPP
6*21
a
gTT
If T = 9 s
m22opta
Too large !
Oscillating-body dynamics
How to decrease the
resonance frequency of a
given floater, without
affecting the excitation and
radiation forces ?
Am
KgS
CB
Resonance condition
Radiation damping = PTO damping
PTO
system
Body 1
Body 2
WAVEBOB
Time-domain analysis
Oscillating-body dynamics
• Regular or irregular waves
• Linear or non-linear PTO
• May require significant computing-time
• Yields time-series
• Essential for control studies
Time domain
added mass
),,()()()()()()( txxfdxtLtxgStftxAm m
t
d
PTOradiationhydrostatic
excitation
forces
from ( ) and spectral distribution (Pierson-Moskowitz, …)n
ndd tftf )()( ,
Equation (1) to be numerically integrated
(1)
dtB
tL sin)(
2
1)(
0
memory function
DamperSpring
PTO
m
x
DamperSpring
PTO
m
x
Oscillating-body dynamics
From Fourier transform techniques:
Oscillating-body dynamics
HP gas
accumulator
LP gas
accumulator
Cylinder
Valve
B
A
Buoy
Motor
Example: Heaving buoy with hydraulic PTO (oil)
• Hydraulic cylinder (ram)
• HP and LP gas accumulator
• Hydraulic motor
PTO force:Coulomb type (imposed by
pressure in accumulator, piston
area and rectifying valve system)
One of the three
power modules
of a Pelamis
PTO Equipment
High-pressure-
oil PTO
Pelamis
Peniche shipyard,
Portugal, 2006
PTO Equipment
High-pressure-oil PTO
Pelamis
Hydraulic ram
HP accumulators LP
accumulators
Inside power
module
LP accumulators
PTO Equipment
High-pressure-oil PTO
High-pressure accumulators
• Commercially available
• Bladder or piston types
• Gas: Nitrogen
• Max. working pressure up to ~ 500 bar
• Banks of unit required for full-sized WECs
Thermodynamics of gas in accumulator (isentropic process):
• pressure-volume
• pressure-temperature
• energy storage (internal energy)
constantpV 4.1 for air and Nitrogen
TCU v
)1(constant Tp
PTO Equipment
High-pressure-oil PTO
Hydraulic motor
Pistons
Swashplate
Bent axis, variable
displacement
β
PTO Equipment
High-pressure-oil PTO
Hydraulic motor
• Positive displacement machine.
• Max. power up to ~ 300 – 500 kW at > 1000 rpm.
• Direct drive of electric generator.
• Relatively compact.
• Variable displacement (double flow control capability).
• Fairly good efficiency at maximum flow.
• Reversible (as pump).
• Available from a few manufacturers.
• Not “too expensive”.
Example:
• Hemispherical buoy, radius = a
0 1 2 3 4 5
ka
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
B
Analytical
Hulme 1982
*B
ka0 2 4 6 8 10 12 14
t
0.01
0
0.01
0.02
0.03
0.04
L
a
gtt*
*L
Oscillating-body dynamics
Dimensionless radiation
damping coefficient
Dimensionless radius Dimensionless time
Dimensionless memory function
External PTO force:
Coulomb type (imposed by
pressure in accumulator,
piston area and rectifying
valve system)
HP gas
accumulator
LP gas
accumulator
Cylinder
Valve
B
A
Buoy
Motor
Oscillating-body dynamics
Irregular waves with Hs, Te and Pierson-Moskowitz spectral distribution
)1054exp(263)( 44542ees TTHS
s11 m,3 state Sea
m5 radius Sphere
es TH
a
kW4.178
kN 647 force External
dampedOptimally
PkW1.83
kN 200 force External
dampedUnder
P kW0.97
kN 1000 force External
dampedOver
P
s11 m,3 state Sea
m5 radius Sphere
es TH
a
Oscillating-body dynamics
(m) x
(kW) P
(m) x (m) x
(kW) P (kW) P(kW) P
(m) x
Avoid overdamping and underdamping. Recall that
accumulator size is finite.
How to control the damping level (PTO force or accumulator
pressure) to the current sea state (or wave group) ?
Answer: Control the oil flow rate q through hydraulic motor
as function of pressure difference Δp
HP gas
accumulator
LP gas
accumulator
Cylinder
Valve
B
A
Buoy
qΔp
Algorithm:
constantpq
piston area
control
parameter
ΔpSGq c2
Algorithm:
constantpq
piston area
control
parameter
ΔpSGq c2
How to control the instantaneous flow rate of oil?
β
• Control the rotational speed
and/or
• control the angle β (displacement)
CONTROL OF WAVE ENERGY CONVERTER
0
2
4
6
8
10
12
0 50 100 150 200 250 300 350
Te=5s
Te=7s
Te=9s
Te=11s
Te=13s
Note: hydrodynamically the system is linear
)kW/m( 2
2sH
P
(kN/m)sH
Performance curves, radius a = 5m
force PTOpSc
5s 7s 9s 11s
13s
CONTROL OF WAVE ENERGY CONVERTER
Control algorithm
0
2
4
6
8
10
12
0 50 100 150 200
(kN/m)sH
G1
G2 G3
Regulation curves
piston areaControl
parameter
pGStq c2)(
force pistonpSc
2motor GpqP
)kW/m( 2
2
2
2motor
ss H
G
H
P
parabolae
0
2
4
6
8
10
12
0 50 100 150 200 250
Te=5s
Te=7s
Te=9s
Te=11s
Te=13s
G=G1
G=G2
G=G3
CONTROL OF WAVE ENERGY CONVERTER
Control algorithm
piston area control parameter
(kN/m)sH
)kW/m( 2
2sH
P
G1 G2 G3
5s 7s
9s
11s
13s
pGStq cm )(
0 250 500 750 1000 1250 1500
C kNs m
0
2
4
6
8
10
PH
s2
Wk
m2
Te 5s
79
11
13
0 50 100 150 200
Hs kN m
0
2
4
6
8
10
12
PH
s2,
Pm
Hs2
Wk
m2 Te 5s
7
9
11
13
G1G2G3G4
Oscillating-body dynamics
LINEAR DAMPER
A hydraulic PTO and a linear damper may be almost equally
effective in irregular waves (NO PHASE CONTROL).
Buoy
radius 5m
)(k
W/m
22 s
HP
)(k
W/m
22 s
HP
(kN/m)sH s/m)(KNC
For point absorbers (relatively small bodies) the resonance
frequency of the body is in general much larger than the
typical wave frequency of sea waves:
• No resonance can be achieved.
• Poor energy absorption.
How to increase energy absorption?
Phase control !
Oscillating-body dynamics
Phase control, i.e. wave-to-wave control in radom waves, is one of
the main issues in wave energy conversion.
Control should be regarded as an open problem and a
major challenge in the development of wave energy
conversion.
Optimal control is a difficult theoretical control problem, that has
been under investigation since the late 1970s.
Control is made difficult by the randomness of the waves and by the
wave-device interaction being a process with memory.
The difficulty increases for multi-
mode oscillations and for multi-
body systems.
Oscillating-body dynamics
Whenever the body velocity comes down to zero, keep the body
fixed for an appropriate perid of time.
This is an artificial way of reducing the frequency of the body free-
oscillations, and achieving resonance.
Phase-control by latching
Kjell Budall
(1933-89)
Johannes
Falnes
J. Falnes, K. Budal, Wave-power conversion by
power absorbers. Norwegian Maritime
Research, vol. 6, p. 2-11, 1978.
Phase-control by latching was introduced by
Falnes and Budal
Oscillating-body dynamics
Optimal phase control in random waves requires the
prediction of incoming wave and heavy computing.
Sub-optimal control strategies by latching were devised by
several teams.
Usually, control algorithm determines the duration of time
the oscillator is kept fixed (latched) in each wave cycle.
Alternative strategy is in terms of load (not time
duration):
Opposing force to be overcome before the body is
released.
Numerical simulations of phase control
Gas (Nitrogen):
• accumulator: 100 kg
• turbine casing: 20 kg
m 5 radius Sphere
mm) 200diameter (m 0314.0 2cSpSGtq cm )(
Phase-control by latching: body is released when
)1()( exceedsbody on force ichydrodynam RpSR c
Control parameters:
G controls oil flow rate through hydraulic motor
R controls latching (release of body)
PHASE CONTROL
How to achieve phase-control by latching in a
floating body with a hydraulic power-take-off
mechanism?
Introduce a delay in the release of the
latched body.
How?
Increase the resisting force the
hydrodynamic forces have to
overcome to restart the body
motion.
600 602 604 606 608 610 612 614
t s
4
2
0
2
4
xd
td
ms
,0
1f d
NM
600 602 604 606 608 610 612 614
t s
4
2
0
2
4
xm
No phase-control:
optimal G, R = 1
excit. force
velocity
displacement
608
608t (s)
REGULAR WAVES
Period T = 9 s
Amplitude 0,667 m
600 602 604 606 608 610 612 614
t s
4
2
0
2
4
xd
td
ms
,0
1f d
NM
600 602 604 606 608 610 612 614
t s
4
2
0
2
4
xm
No phase-control:
velocity
Excit.
force
displacement
Regular waves: T = 9 s, amplitude 0.67 m
)s(t )s(t
608 608
kW 0.551
s/kg 1086.0 6
PR
G
600 602 604 606 608 610 612 614
t s
4
2
0
2
4
xd
td
ms
,0
1f d
NM
600 602 604 606 608 610 612 614
t s
4
2
0
2
4
xm
Phase-control:
kW 1.20616
s/kg 107.7 6
PR
G
608t (s)
0
50
100
150
200
250
0 5 10 15 20 25 30
R
)kW(P
G is optimized
for each R
R
IRREGULAR WAVES
Period Te = 7, 9, 11 s
Height Hs = 2 m
0
5
10
15
20
25
0 2 4 6 8 10 12 14 16
2
2
mkW
sH
P
(s/kg) 106G
R = 1
4
8 12 16 20 24
28
m 2 s, 7 se HT
0
10
20
30
0 2 4 6 8 10 12
2
2
mkW
sH
P
(s/kg) 106G
R = 1
4
8 12 16 20 24
28
m 2 s, 9 se HT
0
10
20
30
0 2 4 6 8 10
2
2
mkW
sH
P
(s/kg) 106G
R = 1
4
8 12 16 20 24
28
m 2 s, 11 se HT
Detailed analysis
Te = 9 s
Hs=2 m
700 720 740 760 780 800
t s
3
2
1
0
1
2
3
xd
td
Hs
s1
,0
1f d
Hs
NM
m
700 720 740 760 780 800
t s
2
1
0
1
2
xH
s22
6
kW/m 3.101
s/kg 107.0
sHPR
G
diffr. force
velocity
displacement
700 720 740 760 780 800
t s
3
2
1
0
1
2
3
xd
td
Hs
s1
,0
1f d
Hs
NM
m
700 720 740 760 780 800
t s
2
1
0
1
2
xH
s
22
6
kW/m 5.2816
s/kg 102.4
sHPR
G
760
760t (s)
760
760
t (s)
0
20
40
60
80
100
0 5 10 15 20 25 30
R
The large increase in time-averaged power output
results:
• from a large increase in oil flow rate (increase in
control parameter G), and hence in motor size;
• not from an increase in hydraulic circuit pressure.
R
(bar)
1p
Pressure in HP accumulator
Phase control by latching may significantly increase the amount
of absorbed energy by point absorbers.
Problems with latching phase control:
• Latching forces may be very large.
• Latching control is less effective in two-body WECs.
Oscillating-body dynamics
Apart from latching, there are forms of phase control
(reactive, uncluching, …).
Control of load prior to release is an alternative to latch duration
control.
Control parameters (G, R) are practically independent of wave height
and weakly dependent on wave period.
Oscillating-body dynamics
Several degrees of freedom
• Each body has 6 degrees of freedom
• A WEC may consist of n bodies (n >1)
PTO
body 1
body 2
PTO
body 1
body 2
PTO
body 1
body 2
All these modes of oscillation interact with each
other through the wave fields they generate.
Number of dynamic equations = 6n
The interference between modes affects:
• added masses
• radiation damping coefficients
Hydrodynamic coefficients are defined
accordingly.
They can be computed with commercial software
(WAMIT, …).
ijij BA ,
WAVE ENERGY TECHNOLOGIES
Oscillating bodies(with hydraulic motor,
hydraulic turbine,linear
electrical generator)
Floating
Submerged
Essencially translation (heave):
IPS Buoy, WaveBob, PowerBuoy
Essencially rotation: Pelamis, PS Frog, SEAREV
Essencially translation (heave): AWS
Rotation: WaveRoller, Oyster
Floating structure: Mighty Whale, BBDB, Oceanlinx
Oscillating
water column(with air turbine)
Fixed structureIsolated: Pico, LIMPET
In breakwater: Sakata, Douro river
Run up(with low-head hydraulic
turbine)
Fixed
structure
Shoreline (with concentration): TAPCHAN
In breakwater (without concentration): SSG
Floating structure (with concentration): Wave Dragon
TWO-BODY
POINT ABSORBERS
Oscillating-body dynamics
Several degrees of freedom
Example: heaving bodies 1 and 2 reacting
against each other.
1212212
21211111111 )()()(
dfxBxA
xxKxxCxgSxBxAm
2112112
21212222222 )()()(
dfxBxA
xxKxxCxgSxBxAm
21122112 , BBAANote:
PTO
body 1
body 2
PTO
body 1
body 2
IPS Buoy Wave Bob
AquaBuoy
Hose
pumps
Hydraulic
ram
a1
b1
2
Simplifying assumptions for optimization and control
Acceleration tube represented by
body 1b
Inertia of piston and enclosed water
represented by body 2
Buoy represented by body 1a
(hemispherical buoy)
Bodies 1b and 2 are “deeply”
submerged:
• Wave excitation forces neglected
• Radiation forces neglected
Hydrodynamic interference between
bodies 1a, 1b and 2 neglected
a1
b1
2
Two-body motion, linear PTO
Coordinates:
x : body 1 (1a+1b)
y : body 2
21 , MM b include added mass
0)()(2 yxKyxCyM
11111 )()()( dbaa fyxKyxCgSxxBxMAm
damper spring
Regular waves, frequency domain
titi eYtyeXtx 00 )(,)(ti
jwdj eAtf )()(
,)(
)())((
10
1112
0
aw
baa
AKCiY
gSCBiMAmX
.0)()( 22
00 KCiMYKCiX
200
2
2
1YXCP
Time-averaged absorbed power
Theoretical max power (axisymmetric body, heave motion):
3
23
max4
wAgP
Radius of buoy = 7.5 m
Mass of buoy
Hydrostatic restoring force coeff.
a1
b1
2
kg107.905 31am
1mMN776.1gS
abb mMM 11*1
amMM 12*2
max
*P
PP
Dimensionless values
Motion amplitude
Mass of body 1b
Mass of body 2
Damping coefficient
Spring stiffness
Power
)(*
8B
CC
gS
KK*
wAXX 0*
Regular waves
Linear PTO
No spring K = 0
1*maxP
PP
Motion amplitude (dimensionless)
X*
Y*
R* (relative)
Results from optimization
a1
b1
2
0
0,5
1
1,5
2
2,5
3
0 0,25 0,5 0,75 1 1,25 1,5*1bM
*2M
*X
*Y
*R
0
5
10
15
20
25
0 0,25 0,5 0,75 1 1,25 1,5*1bM
*C
s8T
0
1
2
3
4
5
6
0 0,5 1 1,5 2 2,5 3 3,5*1bM
*2M
*X
*Y
*R
0
20
40
60
80
0 0,5 1 1,5 2 2,5 3 3,5*1bM
*C
s10T
0
2
4
6
8
10
0 1 2 3 4 5*1bM
*2M
*X
*Y
*R
0
50
100
150
200
0 1 2 3 4 5*1bM
*C
s12T
Irregular waves(Pierson-Moskowitz)
Linear PTO
No spring K = 0
Results from optimization
a1
b1
2
0
5
10
15
20
25
0 0,5 1 1,5 2 2,5 3 3,5*1bM
*C
0
1
2
3
4
5
0 0,5 1 1,5 2 2,5 3 3,5*1bM
*2M
*irr10 P
s8eT
0
10
20
30
40
50
0 1 2 3 4 5*1bM
*C
0
1
2
3
4
5
0 1 2 3 4 5*1bM
*2M
*irr10 P
s10eT
21 , MM bFix
Results from optimization
a1
b1
2
Reactive phase control
Irregular waves(Pierson-Moskowitz)
Linear PTO
Spring 0K
2*1bM
76.1*2M
2
2,5
3
3,5
4
0 0,05 0,1 0,15 0,2
*C
*K
0,44
0,45
0,46
0,47
0,48
0 0,05 0,1 0,15 0,2
*irrP
*K
s8eT
0
2
4
6
8
0 0,1 0,2 0,3 0,4 0,5
*C
*K
0,25
0,3
0,35
0,4
0,45
0 0,1 0,2 0,3 0,4 0,5
*irrP
*K
s10eT
0,1
0,15
0,2
0,25
0,3
0,35
0,4
0 0,1 0,2 0,3 0,4 0,5 0,6 0,7
*irrP
*K
0
2
4
6
8
0 0,1 0,2 0,3 0,4 0,5 0,6 0,7
*C
*K
s12eT
Phase control by latching
PTO: high pressure oil circuit
Wavebob (Ireland)
Hydraulic motor
Hydraulic ram
Gas accumulator
a1
b1
2
HP gas
accumulator
LP gas
accumulator
Cylinder
Valve
Motor
pSGtq cm2)(
Phase-control by latching: body is released when
)1()( exceedsbody on force ichydrodynam RpSR c
Control parameters:
G controls oil flow rate through hydraulic motor
R controls latching (release of body)
Flow rate control through motor
areapiston cS
difference pressurep
HP gas
accumulator
LP gas
accumulator
Cylinder
Valve
Motor
HP gas
accumulator
LP gas
accumulator
Cylinder
Valve
Motor
Non-linear PTO: time-domain analysis
,)()()(
)()())((
1
111
md
t
baa
ftfdtxtL
tgSxtxMAm
.)(2 mftyM
Continuity equation for oil-flow
Accumulator gas thermodynamics
Te=8s
0,0
0,1
0,2
0,3
0,4
0 1 2 3 4 5
R=1
R=2
R=4
Te=10s
0,00
0,05
0,10
0,15
0,20
0,25
0 1 2 3 4 5
R=1
R=2
R=4
Te=12s
0,00
0,02
0,04
0,06
0,08
0,10
0 1 2 3 4 5
R=1
R=2
R=4
)s/kg(106G
*irrP
*irrP
*irrP
76.1
2
m5.7
*2
*1
M
M
a
b
Te=8s
0,0
0,1
0,2
0,3
0,4
0,0 0,5 1,0 1,5 2,0 2,5 3,0
R=1
R=2
R=4
Te=10s
0,00
0,05
0,10
0,15
0,20
0,25
0,0 0,5 1,0 1,5 2,0 2,5 3,0
R=1
R=2
R=4
Te=12s
0,00
0,04
0,08
0,12
0,0 0,5 1,0 1,5 2,0 2,5 3,0
R=1
R=2
R=4
*irrP
*irrP
*irrP
)s/kg(106G
3
1
m5.7
*2
*1
M
M
a
b
Te=12s
0,00
0,05
0,10
0,15
0,20
0,0 0,1 0,2 0,3 0,4
R=1
R=2
R=4 Te=14s
0,00
0,02
0,04
0,06
0,08
0,10
0,00 0,05 0,10 0,15 0,20 0,25
R=1
R=2
R=4
)s/kg(106G
*irrP
*irrP
)s/kg(106G
Te=8s
0,0
0,1
0,2
0,3
0,4
0,0 0,5 1,0 1,5 2,0
R=1
R=4
R=8
*irrP
Te=10s
0,0
0,1
0,2
0,3
0,0 0,2 0,4 0,6 0,8 1,0
R=1
R=2
R=4
*irrP
6
1
m5.7
*2
*1
M
M
a
b
Te=8s
0,0
0,1
0,2
0,3
0,4
0,0 0,5 1,0 1,5 2,0
R=1
R=4
R=8
T=12s
0,00
0,05
0,10
0,15
0,0 0,2 0,4 0,6
R=1
R=4
R=8
)s/kg(106G
*irrP
*irrP
Te=10s
0,00
0,05
0,10
0,15
0,20
0,25
0,0 0,5 1,0 1,5
R=1
R=4
R=8
*irrP*
2
*1 1
m5.7
M
M
a
b
0
10
20
30
0 2 4 6 8 10
2
2
mkW
sH
P
(s/kg) 106G
R = 1
4
8 12 16 20 24
28
*2
*1 0
m5
M
M
a
b
We are far from:
A. Falcão, Ocean Engineering, 35,
358-366, 2008.
HP gas
accumulator
LP gas
accumulator
Cylinder
Valve
Motor
HP gas
accumulator
LP gas
accumulator
Cylinder
Valve
Motor
CONCLUSIONS
• This drops to typically less than 50% in irregular waves.
• A two-body system with a linear damper can be optimized to
absorb theoretical maximum energy from regular waves: . 1*P
• For fixed masses, a linear PTO with a “negative spring” (reactive
control) can significantly increase the energy absorbed from irregular
waves.
• Simulations were made for high-pressure-oil PTO.
• The performance is slightly poorer than with a linear damper.
• In the simulated situations, latching was unable to improve
the performance, except if mass of body 2 is very large
(approaching a single-body system).
OSCILLATING
WATER COLUMNS
The problem:
• The performance of self-rectifying air turbines (Wells,
impulse, …) is strongly dependent on pressure (or on
flow rate) and on rotational speed.
• How to control the turbine (instantaneous rotational
speed) to achieve maximum energy production ?
Air pressure Δp
Pow
er
outp
ut
constant
rotational
speed
OWC Dynamics
uniform air
pressure
Two different approaches to modelling:
weightless
piston
Oscillating body (piston) model
(rigid free surface)
Uniform pressure model
(deformable free surface)
OWC Dynamics
)(tq
)(tm
a)(tq volume-flow rate displace by
free-surface
)(tm mass-flow rate of air
through turbine
a air density )(tp air pressure
0V air volume
pressure)(tp
dt
tdp
c
Vtq
tm
aaa
)()(
)(2
0
Effect of air compressibility
Conservation of air mass
(linearized)
radiation
excitation rate flow
exc
rq
OWC Dynamics
Air turbine
head pressure
outputpower
diameterrotor
speed rotational
p
P
D
N
t
3ND
m
a
22DN
p
a53DN
P
a
t
In dimensionless form:
flow pressure
head
power
)(),( Pw ff
Performance curves of turbine
(dimensionless form):
power Πflow
Φ
pressure head
OWC Dynamics
Frequency domain )(tq
)(tm
a
)(tq
)(tm
a
tirr eQQQMPtqtqtqmtp excexc ,,,,)(),(),(,),(
Linear air turbine K
esusceptancradiation
econductancradiation )()(
)(
)(
C
BiCB
P
QrwAQ )()(exc
excitation
coeff. wave
ampl.
20
exc
aaa c
VCiB
N
KD
QP
OWC Dynamics
20
exc
aaa c
VCiB
N
KD
QP
KgSCBiAm
FX d
)()(20
)(tq
)(tm
a
)(tq
)(tm
a
)(),( Pw ff
tiePtp Re)(
2253
)()()( :outputpower
DN
tpf
DN
tPt
a
P
a
t
Ψ
Time domain:
• Linear or non-linear turbine
22
3 curve pressure vsflow turbineDN
pfNDm
a
wa
dtBtgr cos)(2
)( function memory 0
)(tq
)(tm
a
)(tq
)(tm
a
OWC Dynamics
)()()()()(
exc20 tqdptg
tm
dt
tdp
c
Vt
raaa
To be integrated numerically for p(t)
2253
)()()( :outputpower
DN
tpf
DN
tPt
a
P
a
t )(),( Pw ff
OWC Dynamics
Memory
function
Pico OWC
Numerical application
AQUADYN
Brito-Melo et al. 2001
(rad/s)
XY
Z
Results from time-domain modelling of impulse turbine over t = 120 s
• Turbine D = 1.5 m, N = 115 rad/s (1100 rpm)
• Sea state Hs = 3 m, Te = 11 s
• Average power output from turbine 97.2 kW
p(t) Pt (t)
Air pressure in chamber Power
OWC Dynamics
Numerical application
Stochastic modelling
OWC Dynamics
• Irregular waves
• Linear air-turbine
• A.F. de O. Falcão, R.J.A. Rodrigues, “Stochastic modelling of OWC wave power
performance”, Applied Ocean Research, Vol. 24, pp. 59-71, 2002.
• A.F. de O. Falcão, “Control of an oscillating water column wave power plant for
maximum energy production”, Applied Ocean Research, Vol. 24, pp. 73-82, 2002.
• A.F. de O. Falcão, "Stochastic modelling in wave power-equipment optimization:
maximum energy production versus maximum profit". Ocean Engineering, Vol. 31,
pp. 1407-1421, 2004.
• Much less time-consuming than time-domain analysis
• Appropriate for optimization studies
Time-averaged
Wave climate represented by a set of sea states
• For each sea state: Hs, Te, freq. of occurrence .
• Incident wave is random, Gaussian, with
known frequency spectrum.
WAVES OWCAIR
PRESSURE TURBINE
TURBINE SHAFT POWER
Random,Gaussian
Linear system.Known hydrodynamic
coefficients
Knownperformance
curves
GENERATORELECTRICAL
POWER OUTPUT
Time-averaged
Random,Gaussianrms: p
Electricalefficiency
OWC Dynamics
Stochastic modelling
Gaussian process (e.g. surface elevation ζ )
Probability density function (pdf) :
2
2
2exp
2
1)(f
dS )(2 = variance
spectral
density
= standard deviation
Stochastic model:
• Linear turbine (Wells turbine)
• Random Gaussian waves
Pierson-Moskowitz spectrum
0222
253
2exp
2
2)()( dp
DN
pf
pDNdppPpfP
a
P
pp
att
).1054exp(263)( 44542ees TTHS
2
2
1
202
0
2
2exp
2
1)( pdf and
where)()()(2
ance with variGaussian, random is )( system,linear For
pp
aaap
ppf
Cc
ViB
N
KDΛdS
tp
OWC Dynamics
wAQ )()(exc
excitation
coeff.
wave
ampl.
Time-averaged turbine power output :
In dimensionless form : dimensionless
pressure variance
)(
)(
with relief valve
without relief valve
Wells
turbine
dimensionless time-
averaged power
Time-averaged turbine power output :
In dimensionless form :
How to control the rotational speed N for maximum ?
dimensionless
pressure variance
= 0 for maximum energy production
dimensionless
averaged power
= 0 for maximum energy production
For given turbine is
function of For given OWC, turbine and
sea state, is function of N
We obtain optimal N and maximum .tP
)(function NPP te
Control algorithm:
• Set electrical power
Example: Pico OWC plant with 2.3m Wells turbine
Local wave climate represented by 44 sea states (44 circles)
16.3510583.1 NPe
Maximum rotational speed may be constrained by:
• Centrifugal stresses in turbine and electrical generator
• Mach number effects (shock waves)
1500 rpm
16.3510583.1 NPe
Plant rated power
(for Hs = 5m, Te=14s)
Wells turbine size range 1.6m < D < 3.8m
200
300
400
500
600
700
800
1.5 2 2.5 3 3.5 4D (m)
Ra
ted
po
we
r (k
W)
0
50
100
150
200
250
300
1.5 2 2.5 3 3.5 4
D (m)
An
nu
al a
ve
rag
ed
ne
t p
ow
er
(kW
)
wave climate 3
wave climate 2
wave climate 1
Annual averaged
net power (electrical)
Application of stochastic model
OWC Dynamics
This presentation can be downloaded from:
http://hidrox.ist.utl.pt/doc_fct/Lancaster_pres.ppt
THANK YOU FOR
YOUR ATTENTION
J.M.W.Turner 1775 - 1851
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