wave energy...wave energy extracting power from ocean waves some basic principles finn gunnar...

Wave energyExtracting power from ocean wavesSome basic principles

Finn Gunnar Nielsen, Geophysical Institute, University of Bergen

[email protected]

Wave Energy - Basic Principles FGN

14.02.17 1

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

2


Classification of ocean waves

3

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:

Energy flux in a wave spectrum


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


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

6

The potential exceeds the demands.


Source: IPCC SRREN, 2011


Main principles

From: Babarit,

Introduction to Ocean Wave Energy Conversion, 2009

8

Illustration of main principles


Source: Bedard(2006)»overview: EPRI Ocean Energy Program» Presented to Duke Univerisity Global Change Centre.


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’’.

10

The Tapchan project at Toftestallen, Øygarden(finished 1985)


Principle.Test site

How is energy extracted?

Key principles

Linear considerations

Linear oscillator – interaction with waves

Heaving buoy as example



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)


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!


Estimating the energy absorption by a heaving buoy

Assumptions:

Heave motion only

Linearized analysis

No viscous effects

15


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

16


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


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

18


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


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

20

E24 /TU 31.05.15

21


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


The idea of latching (2:2)

23


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)


Main principles

From: Babarit,

Introduction to Ocean Wave Energy Conversion, 2009

24


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


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

26

Wave due to heave Heave + surge

wave0.avi

wave_sym.avi

wave_asym.avi

wave_s_a.avi

wave0_as_s.avi


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

27


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

28


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

29

Example: Gabriell


Figure 2. Illustration of the buoy with a horizontal plate attached underneath.

Toftestallen

In the 1980’ies


..and in 2014


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

32


Oscillating water column (OWC) LIMPET

LIMPET

33


Relative motion device attenuator – Pelamis

(750 kW device)

34


Overtopping – Wave Dragon

(prototype 20kW, 4-7MW demonstrator)

35


New ideas - floating hose

The ANACONDA concept.

Water inside the tubes are propagating in longitudinal direction.

Ill. EPSRC

36

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.


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