Wave energy converter Control

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

http://www.euro.cauce.org/images/flags/eu-flag.gif

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

http://www.solarnavigator.net/solar_cells/solar_cell_manufacturers.htm

http://windeis.anl.gov/guide/photos/photo8.html

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


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


• 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


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


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


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


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


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 !


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


• 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


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)

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 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


Hydraulic motor

Pistons

Swashplate

Bent axis, variable

displacement

β

PTO Equipment


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


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)( 44542ees 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


(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 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 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


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 !


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.


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


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.


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.


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


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


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


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

qq

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

http://hidrox.ist.utl.pt/doc_fct/Lancaster_pres.ppt

http://hidrox.ist.utl.pt/doc_fct/Lancaster_pres.pptThis

THANK YOU FOR

YOUR ATTENTION

J.M.W.Turner 1775 - 1851

Documents

Wave energy converter Control