WAVE ENERGY UTILIZATION - labima.unifi.it Slides 3_PhD.pdf · Università degli Studi di Firenze,...

Università degli Studi di Firenze, 18-19 April 2012

WAVE ENERGY UTILIZATION

António F. O. Falcão

Instituto Superior Técnico,

Universidade Técnica de Lisboa

INTERNATIONAL PhD COURSE XXVII° Cycle UNIVERSITY OF FLORENCE - TU-BRAUNSCHWEIG Processes, Materials and Constructions in Civil and Environmental Engineering Florence 18-19 April 2012

http://en.wikipedia.org/wiki/File:Unifi.gif

Part 3

Wave Energy Conversion

Modelling

• Introduction.

• Oscillating-body

dynamics.

• Oscillating-Water-Column

(OWC) dynamics.

• Model testing in wave

tank.


Steps in the development of a wave energy converter:

1. Basic conception of device

• inventor(s)

• new patent or from previous concept

2. Theoretical modelling (hydrodynamics,

PTO, control,…)

• evaluation (is device promising or not?)

• optimization, control studies, …

• requires high degree of specialization

(universities, etc.)

X

Y

Z

Introduction


3. Model testing in wave tank

• to complement and validate the

theoretical/numerical modelling

• scales 1:100 (in small tanks) to 1:10 (in very

large tanks)

• essential before full-sized testing in real sea

4. Technical demonstration: design, construction

and testing of a large model (~1/4th scale) or full-

sized prototype in real sea:

• the real proof of technical viability of the system

• cost: up to tens of M$ 1/4 scale

5. Commercial demonstration: several-MW plant

in the open sea (normally a wave farm) with

permanent connection to the electrical grid.

Introduction


Theoretical/numerical hydrodynamic modelling

• Frequency-domain

• Time-domain

• Stochastic

In all cases, linear water wave theory is assumed:

• small amplitude waves and small body-motions

• real viscous fluid effects neglected

Non-linear water wave theory may be used at a later stage to

investigate some water flow details.

Introduction


Frequency domain model

Basic assumptions:

• Monochromatic (sinusoidal) waves

• The system (input output) is linear

Advantages: • Easy to model and to run

• First step in optimization process

• Provides insight into device’s behaviour

Disadvantages: • Poor representation of real waves (may be overcome by superposition)

• Only a few WECs are approximately linear systems (OWC

with Wells turbine)

• Historically the first model

• The starting point for the other models

Introduction


Time-domain model

Basic assumptions:

• In a given sea state, the waves are represented by a spectral distribution

Advantages:

• Fairly good representation of real waves

• Applicable to all systems (linear and non-linear)

• Yields time-series of variables

• Adequate for control studies

Disadvantages:

• Computationally demanding and slow to run

Essential at an advanced stage of theoretical modelling

Introduction


Stochastic model

Basic assumptions:

• In a given sea state, the waves are represented by a spectral distribution

• The waves are a Gaussian process

• The system is linear

Advantages:

• Fairly good representation of real waves

• Very fast to run in computer

• Yields directly probability density distributions

Disadvantages:

• Restricted to approximately linear systems (e.g. OWCs with Wells turbines)

• Does not yield time-series of variables

Introduction


Most wave energy converters are complex (possibly multi-body)

mechanical systems with several degrees of freedom.

We consider the simplest case:

• A single floating body.

• One degree of freedom: oscillation in heave

(vertical oscilation). Damper Spring

Buoy

PTO

Oscillating-body dynamics


Damper Spring

PTO

m

x

Basic equation (Newton):

)()( tftfxm mh

PTO on wetted

surface

excitation force (incident wave)

radiation force (body motion)

= hydrostatic restoring force (position)

mrd fgSxffxm

xgSf

f

f

f

hs

r

d

h

S

Cross-section



Frequency-domain analysis


• Sinusoidal monochromatic waves, frequency

• Linear system


A and B to be computed (commercial codes WAMIT,

AQUADYN, ...) for given ω and body geometry.

xCKxf m

Linear

spring

Linear

damper DamperSpring

PTO

m

x

DamperSpring

PTO

m

x

mrd fgSxffxm

xBxAf r

mass addedA

dampingradiation B

dfxKgSxCBxAm )()()(

mass added

mass radiation

damping

PTO

damping

buoyancy PTO

spring

Excitation

force



KgSCBiAm

FX

d

)()(20

• Regular waves

• Linear system

ti

dd

tieFfeXtx

,)( 0

ti

dd

tieFfeXtx

Re,Re)( 0

or simply

dfxKgSxCBxAm )()()(

Method of solution: )sincos( titeti

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

0

2

228

1

B

FXi

BF

BP

dd

KgSCBiAm

FX

d

)()(20

Note: for given body and given wave amplitude and frequency ω, B and

are fixed. dF


m

K

m

K

B

FXi

d

20

Am

KgS

CB

Resonance condition

Radiation damping = PTO damping

PThen, 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 power P

E = energy flux of incident wave per unit crest length

For an axisymmetric body oscillating in heave (vertical

oscillations), it can be shown (1976) that

2max

EP or

2max L

593.0PCFor wind turbines, Betz’s limit is

Note: may be larger

than width of body maxL




2

Max. capture

width

Axisymmetric

heaving body

Axisymmetric

surging body

Incident

waves

Incident

waves

wave wave


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

If T = 9 s

m22opt a

maxP

P 0.2*2125

ga

CC

Dimensionless

PTO damping

Dimensionless

wave period

No spring, K = 0

for maxPP

6*

21

a

gTT




How to increase the resonance frequency of a

given “small” floater?

Wavebob,

Ireland

Attach a deeply submerged body.

Am

KgS


Time-domain analysis

• Regular or irregular waves

• Linear or non-linear PTO

• A.F. de O. Falcão, “Modelling and control of oscillating-body wave energy converters with

hydraulic power take-off and gas accumulator”, Ocean Engineering, vol. 34, pp. 2021-2032, 2007.

• A.F. de O. Falcão, “Phase control through load control of oscillating-body wave energy

converters with hydraulic PTO system”, Ocean Engineering, vol. 35, pp. 358-366, 2008.

• May be require significant time-computing

• Yields time-series

• Essential for control studies



Time domain

added mass

),,()()()()()()( txxfdxtLtxgStftxAm m

t

d

PTO radiation hydrostatic

excitation

forces

from () and spectral distribution (Pierson-Moskowitz, …)

nndd tftf )()( ,

Equation (1) to be numerically integrated

(1)

dt

BtL sin

)(

2

1)(

0

memory function

DamperSpring

PTO

m

x

DamperSpring

PTO

m

x


From Fourier transform techniques:


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)



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

ka

0 2 4 6 8 10 12 14

t

0.01

0

0.01

0.02

0.03

0.04

L

a

gtt *

*L

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

Irregular waves with Hs, Te and Pierson-Moskowitz spectral distribution

)1054exp(263)(44542

ees TTHS



s11 m,3 state Sea

m5 radius Sphere

es TH

a

kW4.178

kN 647 force External

dampedOptimally

PkW1.83


dampedUnder

P kW0.97


dampedOver

P

s11 m,3 state Sea

m5 radius Sphere

es TH

a

(kW) P

(m) x (m) x

(kW) P (kW) P(kW) P

(m) x



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 !



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, 6, 2-11, 1978.

Phase-control by latching was introduced by

Falnes and Budal



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

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: Phase-control:

velocity

Excit.

force

displacement

kW 0.55P kW 1.206P


)s(t )s(t608 608

Regular waves: T = 9 s, amplitude 0.67 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

s

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

excit. force

velocity

displacement

Irregular waves: Te = 9 s, Hs = 2 m 22

kW/m 2.41sHP 22kW/m 114sHP

No phase control With phase control



Phase control is being investigated by several teams as a

way of enhancing device performance.

Phase by latching may significantly increase the amount of

absobed energy by point absorbers.

Problems with latching phase control:

• Latching forces may be very large.

• Latching control is less effective in two-body WECs.

Apart from latching, there are forms of phase control (reactive,

unlatching, …).



Several degrees of freedom

• Each body has 6 degrees of freedom

• A WEC may consist of n bodies (n >1)

P T O

b o d y 1

b o d y 2

P T O

b o d y 1

b o d y 2

P T O

b o d y 1

b o d y 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 ,




Several degrees of freedom

Example: heaving bodies 1 and 2 reacting

against each other.

1212212

21211111111 )()()(

dfxBxA

xxKxxCxgSxBxAm

2112112

21212222222 )()()(

dfxBxA

xxKxxCxgSxBxAm

21122112 , BBAA Note:

PTO

body 1

body 2

PTO

body 1

body 2


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


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

qq


Air turbine

head pressure

outputpower

diameterrotor

speed rotational

p

P

D

N

t

3ND

m

a

22DN

p

a

53

DN

P

a

t

In dimensionless form:

flow pressure

head

power

)(),( Pw ff

Performance curves of turbine

(dimensionless form):

power flow

pressure head

OWC Dynamics


Frequency domain

)( tq

)( tm

a

)( tq

)( tm

a

ti

rr eQQQMPtqtqtqmtp

excexc ,,,,)(),(),(,),(

Linear air turbine K

esusceptancradiation

econductancradiation )()(

)(

)(

C

BiCB

P

Q r

wAQ )()(exc

excitation

coeff. wave

ampl.

2

0

exc

aaa c

VCiB

N

KD

QP

OWC Dynamics


2

0

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

OWC Dynamics

air pressure


Time domain:

• Linear or non-linear turbine

22

3 curve pressure vsflow turbine

DN

pfNDm

a

wa

dtBtg r cos)(2

)( function memory

0

)()()()()(

2

0tqdptg

tm

dt

tdp

c

V

i

t

r

aaa

)( tq

)( tm

a

)( tq

)( tm

a

To be integrated numerically for p(t)

2253

)()()( :outputpower

DN

tpf

DN

tPt

a

P

a

t

OWC Dynamics


Memory

function

Pico OWC

Numerical application

AQUADYN

Brito-Melo et al. 2001

(rad/s)

X

Y

Z

OWC Dynamics


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

Numerical application

OWC Dynamics


Stochastic modelling

• 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

OWC Dynamics


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 OWC AIR

PRESSURE TURBINE

TURBINE SHAFT POWER

Random, Gaussian

Linear system. Known hydrodynamic

coefficients

Known performance

curves

GENERATOR ELECTRICAL

POWER OUTPUT

Time-averaged

Random, Gaussian rms: p

Electrical efficiency

Stochastic modelling

OWC Dynamics


Stochastic model:

• Linear turbine (Wells turbine)

• Random Gaussian waves

Pierson-Moskowitz spectrum

0222

253

2

exp2

2)()( dp

DN

pf

pDNdppPpfP

a

P

pp

att

).1054exp(263)(44542

ees TTHS

2

2

1

2

02

0

2

2

exp2

1)( pdf and

where)()()(2

ance with variGaussian, random is )( system,linear For

pp

aaa

p

ppf

C

c

ViB

N

KDΛdS

tp

OWC Dynamics


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 4

D (m )

Ra

te

d p

ow

er (

kW

)

0

50

100

150

200

250

300

1.5 2 2 .5 3 3 .5 4

D (m )

An

nu

al

av

era

ge

d n

et

po

we

r

(kW

)

w a ve c lim a te 3

w a ve c lim a te 2

w a ve c lim a te 1

Annual averaged

net power (electrical)

Application of stochastic model

OWC Dynamics


Theoretical/numerical modelling, based on linear water wave

theory, is unable to account for:

• large amplitude waves

• large amplitude motions of bodies and OWCs

• real-fluid effects (viscosity, turbulence, eddies)

• Survival in very energetic seas

Model testing in wave tank is essential to:

• validate theoretical results

• investigate non-linear effects effects

• investigate survival issues

It is an essential step before testing under real sea conditions.

Model Testing


Model Testing

How to scale up model test results, assuming geometric similarity ?

3gL

F

27gL

P

21)( gL

V

L Linear dimension

Time (wave period)

Force

Velocity

Power

2121Lg

T

Dim

en

sio

nle

ss

co

eff

icie

nts


Required testing conditions:

• For power performance: Hs up to 4 - 5 m.

• For survival Hs up to 10 - 15 m (individual wave height 18 – 28 m ?) for

offshore devices.

• Tank depth may be important for simulation of mooring systems.

Testing for survival is usually done at smaller scale than for power

performance, due to limitations in wave generation by wave makers.

Typical scales for power performance testing:

• 1/80 to 1/25 in small to medium tanks.

• Up to 1/10 in very large tanks.

Model Testing


Model testing of BBDB

(floating OWC)

HMRC – National Ocean Energy Test Facility, Cork, Ireland

Mid-sized tank

Scales 1/50 – 1/30

Model Testing


University of Porto, Portugal

Mid-sized tank

Scales 1/50 – 1/30

Model Testing

Floating OWC


Large tank: Ecole Centrale de Nantes, France

Pelamis model testing

Model Testing


27gL

P

What is the power level in the simulation of a 500 kW full-sized prototype ?

Scale Power in model

1/50 0.6 W

1/25 6.4 W

1/15 38 W

1/10 0.16 kW

1/4 3.9 kW

A realistic simulation of the PTO (hydraulic, linear generator, etc.)

with control capability in general requires scale larger than 1/10.

Some technology developers use tests at scales 1/4 to 1/5 in

sheltered sea conditions.

Model Testing


Model Testing

OWC testing

• The spring-like effect of air compressibility in the air chamber is important.

• In testing, the air chamber volume should be scaled as not as .

• This may raise practical problems, especially when model testing floating

OWCs.

• The presence of the air turbine is usually simulated by a pressure drop.

• There are several techniques for doing that (orifice, porous plate, …).

21L

31L


END OF PART 3

WAVE ENERGY CONVERSION

MODELLING


WAVE ENERGY UTILIZATION - labima.unifi.it Slides 3_PhD.pdf · Università degli Studi di Firenze,...

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