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![Page 1: Wave energy - Universitetet i Bergen · PDF fileWave energy Extracting power from ocean waves Some basic principles Finn Gunnar Nielsen, Geophysical Institute, University of Bergen](https://reader034.vdocuments.mx/reader034/viewer/2022051720/5a78b3b97f8b9a1f128ef366/html5/thumbnails/1.jpg)
Wave energyExtracting power from ocean wavesSome basic principles
Finn Gunnar Nielsen, Geophysical Institute, University of Bergen
Wave Energy - Basic Principles FGN
14.02.17 1
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Wave Energy - Basic Principles FGN 14.02.17
Issues to be discussed
A reminder on the nature of (deep water) gravity waves
The Global picture
Examples on wave energy converters
Extracting wave energy by an oscillating system
“To absorb wave energy is about generating waves”
Theoretical efficiencies
Two simple numerical examples
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Wave Energy - Basic Principles FGN 14.02.17
Classification of ocean waves
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Wave Energy - Basic Principles FGN 14.02.17 4
Linear, deep water waves (Airy waves)
2
2
2tanh
gc kd
k k
Dispersion relation:
, tanh( ) 1, c , g
d kd kgk
As
21
4 2p v
HE E g
Kinetic = potential energy (per unit surface area):
2 2
1 1 1
2 2 2 4 2g p v
H H gP c E E g g
k k
Energy flux:
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Energy flux in a wave spectrum
Wave Energy - Basic Principles FGN 14.02.17 5
0 0.1 0.2 0.3 0.4 0.50
2
4
6
8
10
12
14
16
18Jonswap spectrum
Frequency [1/sec]
Sf(
f) (
m2se
c)
Mean energy frequency =1/ (6.78 sec)
Zero upcrossing frequency = 1/ (5.88 sec]
Peak frequency
= 1/ (7.47 sec)
Hs = sqrt(m0) = 3.50m
Energy flux:2 2 2 2
1 1 0 1
1 1 1
2 4 64sP g m g T m g T H
n
nm S d
Spectral moments:
Significant wave height
Average period
Zero up-crossing period
Energy mean period
0
001
1
002
2
11
0
4
2
2
2
sH m
mT
m
mT
m
mT
m
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Wave Energy - Basic Principles FGN 14.02.17
RESOURCES Waves(From World Energy Council)
Average energy densities:
20 – 100 kW/m wave front
50 kW/m, 30% efficiency:
130 MWh/ym
100TWh/y: 760km
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The potential exceeds the demands.
Wave Energy - Basic Principles FGN 14.02.17 7
Source: IPCC SRREN, 2011
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Wave Energy - Basic Principles FGN 14.02.17
Main principles
From: Babarit,
Introduction to Ocean Wave Energy Conversion, 2009
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Illustration of main principles
Wave Energy - Basic Principles FGN 14.02.17 9
Source: Bedard(2006)»overview: EPRI Ocean Energy Program» Presented to Duke Univerisity Global Change Centre.
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Wave Energy - Basic Principles FGN 14.02.17
Key principles for extracting wave energy
Remember:
Waves are not only quasi-static change in surface elevation
Energy absorption requires a force working together with a velocity
‘Falnes and Budal (1978):
‘In order for an oscillating system to be a good wave absorber
it should be a good wave generator’’.
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The Tapchan project at Toftestallen, Øygarden(finished 1985)
Wave Energy - Basic Principles FGN 14.02.17 11
Principle.Test site
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How is energy extracted?
Key principles
Linear considerations
Linear oscillator – interaction with waves
Heaving buoy as example
Wave Energy - Basic Principles FGN 14.02.17 12
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Wave Energy - Basic Principles FGN 14.02.17
Linear oscillator
Dynamic equilibrium:
Harmonic oscillation, stationary solution.
Dynamic equation, frequency domain:
2
0 0
( )
2 ( )
Mx Bx Kx F t
x x x F t
cos Re[ ]i t
A AF t F t F e
2M i B K x F
0
0
=2
K
M
B
M
13
M
K
X
B
F(t)
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Wave Energy - Basic Principles FGN 14.02.17
Linear oscillator - solution
2
cos Rei t
A Ax x t x e
Fx
M K i B
0 1 2 3 4 50
1
2
3
4
5
6
/0
ab
s(X
/X0)
0 1 2 3 4 5-180
-160
-140
-120
-100
-80
-60
-40
-20
0
/0
(
De
g)
14
At resonance: Response controlled by damping!
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Wave Energy - Basic Principles FGN 14.02.17
Estimating the energy absorption by a heaving buoy
Assumptions:
Heave motion only
Linearized analysis
No viscous effects
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Wave Energy - Basic Principles FGN 14.02.17
The heaving point absorber, a linearized approach (1:3)
Equation of motion (1DOF):
In frequency domain:
Wave radiation force:
Force due to power off-take:
Heave motion:
ex R Pmx F F F
33R r wlF A x B x k x
2
33 r l wl exm A i B B B k K x F
PF Bx Kx
2 22
33
cos( )Aex
wl r l
F tx
m A k K B B B
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Wave Energy - Basic Principles FGN 14.02.17
The heaving point absorber , a linearized approach (2:3)
Instantaneous power:
Integrated over one wave period:
Maximum power at resonance
P PP t F x Kx Bx x
17
2 22 2
2 22
33
1 1
2 2
AexP A
wl r
F BP Bx
m A k K B B
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Wave Energy - Basic Principles FGN 14.02.17
The heaving point absorber , a linearized approach (3:3)
Optimum damping in power off-take:
Obtained for
Maximum mean power:
/ 0PdP dB
rB B
2
_ max
08
Aexp
r
FP
B
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Wave Energy - Basic Principles FGN 14.02.17
Invoke “Haskind relation” (relates wave excitation force to wave radiation damping.)
3D symmetric, heaving buoy:
I.e. max power:
“Capture width”:
Theoretical limits for power extraction – 3D axisymmetric body (deep water)
2
33
33 3
2
2
ex
r
FB
Hg
19
Falnes and Budal , 1978: ‘‘In order for an
oscillating system to be a good wave absorber it
should be a good wave generator’’.
2
3
2 32 3
_ max 3 3
0
2
8 4 128
Aexp
r
Hg
F gP H T
B
32 3
3
22 2
max power extraction 1281power in wave per meter 2
32
cap
gH T
gL
g TH
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Wave Energy - Basic Principles FGN 14.02.17
Heaving point absorber – Theoretical versus “Budal limit”.- Or the implication of limited physical size.
C0 = = 7.90 kWs/m4
C∞ = = 0.244 kW/(m2s3)
Capture width (PA):
L=λ/2π (Heave only)
L = 3λ/2π (Heave & surge)
From Falnes (2007)
ResonanceSemisubmerged sphere
1
4g
3
3128
g
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E24 /TU 31.05.15
21
Wave Energy - Basic Principles FGN 14.02.17
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The idea of latching (1:2)
22
A. Babarit, G. Duclos, A.H. Clément: Comparison of latching control strategies for a heaving wave energy device in random sea
Wave Energy - Basic Principles FGN 14.02.17
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The idea of latching (2:2)
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Wave Energy - Basic Principles FGN 14.02.17
1DOF system.
Power absorbed in harmonic and first sub-harmonic latching modes, compared with uncontrolled, and ideal modes.
Solid line, without control; squares, latching, Tout=Twa; circles, latching, Tout=3×Tin; dashed dotted line, max power
Max.
W/o control
Tout = Twa
Tout = 3Twa
Source: Clement and Babarit (2012)
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Wave Energy - Basic Principles FGN 14.02.17
Main principles
From: Babarit,
Introduction to Ocean Wave Energy Conversion, 2009
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Wave Energy - Basic Principles FGN 14.02.17
Energy flux in incident wave:
Net absorbed wave power per unit length:
AT: amplitude of transmitted wave
AR: amplitude of reflected wave
Interaction between waves and body - 2D
22
0
1
4g
gJ c E A
22 2 2
2 04
D R T
gP A A A
25
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Wave Energy - Basic Principles FGN 14.02.17
Interaction between waves and body 2D Incident wave
plus heaving and surging body.
-200 -150 -100 -50 0 50 100 150 200-3
-2
-1
0
1
2
3
x
wave e
levation
Incident waves moving in positive x-direction
-200 -150 -100 -50 0 50 100 150 200-3
-2
-1
0
1
2
3
x
wave e
levation
Radiated waves from a heaving 2D source
-200 -150 -100 -50 0 50 100 150 200-5
-4
-3
-2
-1
0
1
2
3
x
wave e
levation
Radiated waves from a surging 2D source
-200 -150 -100 -50 0 50 100 150 200-5
-4
-3
-2
-1
0
1
2
3Radiated waves from a heaving plus surging source
-200 -150 -100 -50 0 50 100 150 200-5
-4
-3
-2
-1
0
1
2
3
x
wave e
levation
Incident plus radiated waves from a heaving and surging source
Incident wave
Wave due to surge Incident + heave + surge
Note importance of phasing
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Wave due to heave Heave + surge
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Wave Energy - Basic Principles FGN 14.02.17
0 1 2 3 4 5 6 7 8 9 100
1
2
3
4
5
6
7
8
Wave period (sec)
abs(X
3)/
A
Amplitude of motion
gam = 0.1
gam = 0.5
gam = 1
gam = 2
gam = 5
gam = 10
Example – heaving buoy, motions
INPUT
Radius 1 m
Draft 5 m
Wave amplitude 1 m
Mass 12880.5299 kg
Water line stiffness, 31.5787 kN/m
Power offtake stiffness 15.7894 kN/m
Natural period 3.539 sec
Radiation damping at T0 1947.692 N/(m/s)
Theoretical maximum energy absorbtion, at resonance (kW) 43.1835
Gam = B_powerofftake / B_radiation (T0)
B_powerofftake independent of frequency (not optimum)
x1
x3
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Wave Energy - Basic Principles FGN 14.02.17
Heaving buoy - Power production
0 1 2 3 4 5 6 7 8 9 100
5
10
15
20
25
30
35
40
45
Wave period (sec)
Avera
ge p
ow
er
(kW
)
Power production, wave amplitude 1 m
gam = 0.1
gam = 0.5
gam = 1
gam = 2
gam = 5
gam = 10
0 2 4 6 8 10 12 140
20
40
60
80
100
120
140
160
Wave period (sec)
rage p
ow
er
(kW
)
Theoretical maximum versus "Budal limit"
Theoretical max
"Budal limit"
Budal limit with A= 1m
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Wave Energy - Basic Principles FGN 14.02.17
Power production off resonance - passive system
INPUT
Radius 2.5 m
Draft 5 m
Wave amplitude 1 m
Mass 80503.3117 kg
Water line stiffness, 197.367 kN/m
Power offtake stiffness 98.6835 kN/m
Natural period 3.8998 sec
Radiation damping at T0 49277.4514 N/(m/s)
Theoretical maximum energy absorbtion, at resonance (kW) 57.7829
Gam = B_powerofftake / B_radiation (T0)
High damping in power off-take important to extract energy at periods above resonance
0 2 4 6 8 10 12 140
10
20
30
40
50
60
70
Wave period (sec)
Avera
ge p
ow
er
(kW
)
Power production, wave amplitude 1 m
gam = 0.1
gam = 0.5
gam = 1
gam = 2
gam = 5
gam = 10
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Example: Gabriell
Wave Energy - Basic Principles FGN 14.02.17 30
Figure 2. Illustration of the buoy with a horizontal plate attached underneath.
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Toftestallen
In the 1980’ies
Wave Energy - Basic Principles FGN 14.02.17 31
..and in 2014
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Wave Energy - Basic Principles FGN 14.02.17
Wave power converters -Examples on installations in full / reduced scale (2:2)
Bostrøm et al. (Sweden, 2006-)
Heaving buoy. Linear generator
Fred Olsen, “Buldra”. (Norway 2004-)
Array of heaving buoys. Semisubmersible
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Wave Energy - Basic Principles FGN 14.02.17
Oscillating water column (OWC) LIMPET
LIMPET
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Wave Energy - Basic Principles FGN 14.02.17
Relative motion device attenuator – Pelamis
(750 kW device)
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Wave Energy - Basic Principles FGN 14.02.17
Overtopping – Wave Dragon
(prototype 20kW, 4-7MW demonstrator)
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Wave Energy - Basic Principles FGN 14.02.17
New ideas - floating hose
The ANACONDA concept.
Water inside the tubes are propagating in longitudinal direction.
Ill. EPSRC
36
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Summary
- Vaste amount of wave energy available. Technical availability uncertain.
- No convergence on technical solutions
- For an oscillating system to extract energy, it has to generate waves
- Advanced control needed to enhance power offtake.
- Survivability has shown up to be a critical issue.
Wave Energy - Basic Principles FGN 14.02.17 37