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 

CONTROL TECHNIQUES FOR WAVE 

ENERGY CONVERTERS 

António F. de O. Falcão 

Instituto Superior Técnico, Lisbon, Portugal

How far have we gone in 30+ yrs Some milestones: 

1974 - Salter & the duck 

1976 – Masuda 

& Kaimei 

1975- …The early theoreticians 

1985-91 

The early OWCs 

Early 1980s 

Point absorbers 

in Scandinavia 

1975-82 - The 

British Program 

Goal: 2 GW plant 

1991: EU 

backs up 

wave energy 

1996 

EURATLAS 

1999-2000 

OWCs in 

Europe 

Since 2004 

The “new” 

offshore devices

Introduction 

Technology 

challenge

Introduction 

Oscillating 

Water Column 

(with air turbine) 

Fixed 

structure 

Isolated: Pico, LIMPET, Oceanlinx 

In breakwater: Sakata, Mutriku 

Floating: Mighty Whale, BBDB 

Oscillating body 

(hydraulic motor, hydraulic 

turbine, linear 

electric generator) 

Floating 

Submerged 

Heaving: Aquabuoy, IPS Buoy, Wavebob, 

PowerBuoy, FO3 

Pitching: Pelamis, PS Frog, Searev 

Heaving: AWS 

Bottom-hinged: Oyster, Waveroller 

Overtopping 

(low head 

water turbine) 

Fixed 

structure 

Shoreline (with concentration): TAPCHAN 

In breakwater (without concentration): SSG 

Floating structure (with concentration): Wave Dragon

Introduction 

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 waveenergy 

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.

Absorbed power 

Spectral power density (m 2 s) 

m 

Introduction 

Large body 

Small body 

Wave energy absorption is widerbanded 

for a large body than for a 

“point-absorber”. 

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). 

Wave frequency 

1.5 

1 

0.5 

0 

0.5 

1 

100 

200 

150 200 250 3 

t s 

0.2 

0.15 

0.1 

0.05 

T e 

H s 

10 s 

2m 

0 

0 0.5 1 1.5 

Frequency (rad/s)

Oscillating-body dynamics 

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

mechanical systems with several degrees of freedom. 

Buoy 

We consider first the simplest case: 

• A single floating body. 

• One degree of freedom: oscillation in heave 

(vertical oscilation). 

Spring 

PTO 

Damper


x 

Basic equation (Newton): 

m 

m 

x 

f 

h 

( t) 

f ( t) 

m 

on wetted 

surface 

PTO 

PTO 

f 

h 

f 

f 

f 

d 

r 

hs 

excitation force (incident wave) 

radiation force (body motion) 

gS x 

= hydrostatic force (body position) 

Spring 

Damper 

S 

mx 

f 

d 

f 

r 

gSx 

f 

m 

Cross-section


Frequency-domain analysis 

• Sinusoidal monochromatic waves 

• Linear system


x 

mx 

f r 

f 

d 

f 

r 

Ax 

A 

gSx 

Bx 

f 

m 

f m 

B radiation damping 

added mass 

Kx 

Linear 

spring 

Cx 

Linear 

damper 

Spring 

m 

PTO 

Damper 

A and B to be computed (commercial codes WAMIT, 

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

( m A) 

x ( B C) 

x 

( gS K) 

x 

f d 

mass 

added 

buoyancy 

mass radiation PTO 

PTO 

damping 

damping 

spring 

Excitation 

force


( m A) 

x ( B C) 

x 

( gS K) 

x 

f d 

Method of solution: ( e i t cos t isin 

t ) 

• Regular waves 

• Linear system 

x( t) 

Re X0e 

i 

t 

, 

f 

d 

Re 

F 

d 

e 

i 

t 

or simply 

x t) 

X e , 

( 0 

i 

t 

f 

d 

F 

d 

e 

i 

t 

Note : X0, 

Fd 

are in general 

complex 

amplitudes 

F d 

wave amplitude 

( 

) 

to be computed for given 

and body geometry 

X 

0 

2 

( m 

A) 

i 

F 

d 

( B 

C) 

gS 

K


X 

0 

2 

( m 

A) 

i 

F 

d 

( B 

C) 

gS 

K 

Power = force velocity 

Time-averaged power absorbed from the waves : 

P 

1 

8B 

F 

d 

2 

B 

2 

i 

= 0 

X 

0 

Fd 

2B 

2 

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

are fixed. 

Then, the absorbed power P will be maximum when : 

F d 

i 

X 

0 

Fd 

2B 

B 

C 

gS 

m 

K 

Resonance condition 

A 

Radiation damping = PTO damping 

m 

K 

K 

m


Capture width L : measures the power absorbing capability 

of device (like power coefficient of wind turbines) 

L 

P 

E 

P 

= absorbed power 

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 

P 

max 

E 

2 

or 

L max 

2 

Note: L max 

may be larger 

than width of body 

For wind turbines, Betz’s limit is 

C P 

0.593


Incident 

waves 

2 

Axisymmetric 

heaving body 

Max. capture 

width 

Incident 

waves 

Axisymmetric 

surging body 

wave

ma 


Example: hemi-spherical heaving buoy of radius a 

No spring, no reactive control, K = 0 

P 

T* 

P max 

T 

g 

a 

1 

for 

2 

If T = 9 s 

a opt 

22m 

Too large ! 

6 

P max 

P P x 

1 

P 

C 

0.8 C* 0.5 C* 

2. 0 

5 2 1 2 

a g 

0.6 

0.4 

0.2 

C* 

5 

C* 

2 

0 

5 7.5 10 12.5 15 17.5 

1 2 

Dimensionless g T 

wave period 

T* 

T 

a 

Dimensionless 

PTO damping


How to decrease the 

resonance frequency of a 

given floater, without 

affecting the excitation and 

radiation forces 

gS 

m 

K 

A 

Resonance condition 

B 

C 

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


x 

m 

Time domain 

From Fourier transform techniques: 

Spring 

PTO 

Damper 

( m A ) x( 

t) 

f ( t) 

gS x( 

t) 

L( 

t ) x( 

) d f ( x, 

x 

, t) 

d 

t 

m 

(1) 

added mass 

excitation 

hydrostatic 

radiation 

PTO 

L( 

t) 

1 

2 

0 

B( 

) 

sin 

t d 

memory function 

forces 

f ( t) 

d ( t) 

f d, 

n 

n 

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

Equation (1) to be numerically integrated


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) 

Buoy 

LP gas HP gas 

accumulator accumulator 

Motor 

A 

Valve 

Cylinder 

B

PTO Equipment 

High-pressureoil 

PTO 

Pelamis 

One of the three 

power modules 

of a Pelamis 

Peniche shipyard, 

Portugal, 2006

PTO Equipment 

High-pressure-oil PTO 

Inside power 

module 

Pelamis 

LP accumulators 

Hydraulic ram 

HP accumulators 

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 

pV constant . 4 

p constant T 

1 for air and Nitrogen 

( 1) 

• energy storage (internal energy) 

U 

C 

v 

T

PTO Equipment 


Hydraulic motor 

β 

Pistons 

Bent axis, variable 

displacement 

Swashplate

PTO Equipment 



• 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”.

B 

L 


Example: 

• Hemispherical buoy, radius = a 

B * 

Dimensionless radiation 

damping coefficient 

0.35 

0.3 

0.25 

0.2 

0.15 

0.1 

Analytical 

Hulme 1982 

L * 

Dimensionless memory function 

0.04 

0.03 

0.02 

0.01 

0.05 

0 

0 

0 1 2 3 4 5 

ka 

ka 

Dimensionless radius 

0.01 

0 2 4 6 8 10 12 14 

g t 

Dimensionless time t* 

t 

a


Buoy 

LP gas HP gas 


External PTO force: 

Coulomb type (imposed by 

pressure in accumulator, 

piston area and rectifying 

valve system) 

Cylinder 

A 

Motor 

Valve 

B 

Irregular waves with H s , T e and Pierson-Moskowitz spectral distribution 

S ( ) 

263H 

2 

s 

T 

e 

4 5 

exp( 

1054T 

e 

4 

4 

)

Sphere radius a 5m 

Sphere radius a 5m Sea state H s 3m, T e 11s 

Sea state H s 3m, T e 11s 

x (m) 


x (m) 

x x (m) (m) 

P (kW) 

P (kW) 

P P 

(kW) 

Under damped 

Externalforce 200 kN 

Optimally damped 

Externalforce 647 kN 

Over damped 

Externalforce 

1000 kN 

P 

83.1kW 

P 

178.4 kW 

P 

97.0 kW

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 

Algorithm: 

q 

p 

constant 

Buoy 

LP gas HP gas 


Δp 

q 

q G S 

c 2 

Δp 

control 

piston 

area 

parameter 

Cylinder 

A 

B 

Valve

How to control the instantaneous flow rate of oil 

• Control the rotational speed 

and/or 

• control the angle β (displacement) 

β

CONTROL OF WAVE ENERGY CONVERTER 

Note: hydrodynamically the system is linear 

2 

(kW/m 

Performance curves, radius a = 5m 

12 

Te=5s 

5s 7s 9s 11s 

10 

Te=7s 

Te=9s 

8 

P 

Te=11s 

13s 

2 

Te=13s 

H 6 s 

4 

) 

2 

0 

0 50 100 150 200 250 300 350 

S p PTO force 

(kN/m) 

c 

H s


Control algorithm 

q( 

t) 

Sc 2 G 

p 

P 

S c p piston force 

2 

q p G 

motor 

P 

motor G 

2 

2 

Hs 2 

H s 

(kW/m 

2 

) 

piston area 

12 

10 

8 

6 

4 

2 

0 

Control 

parameter 

G 1 

G 2 G 3 

0 50 100 150 200 

H s 

Regulation curves 

(kN/m) 

parabolae


Control algorithm 

qm ( t) 

S 

c 

G 

p 

piston area 

control parameter 

12 

G 1 G 2 G 3 

10 

(kW/m 

P 

2 

H s 

2 

) 

8 

6 

4 

2 

0 

5s 

7s 

9s 

11s 

13s 

Te=5s 

Te=7s 

Te=9s 

Te=11s 

Te=13s 

G=G1 

G=G2 

G=G3 

0 50 100 150 200 250 

H s 

(kN/m)

P Hs 2 , Pm Hs 2 k W m 2 

k W m 2 


Buoy 

radius 5m 

LINEAR DAMPER 

H 2 s 

H 2 s 

(kW/m 2 ) 

P 

(kW/m 2 ) 

H s 

12 

10 

(kN/m) 8 

6 

T e 

5s 

7 

9 

(KN 8 

4 

4 

effective in irregular waves (NO PHASE CONTROL). 

2 

2 

G4 

G3 

G2 

13 

G1 

11 

P H s 

2 

10 

6 

7 

9 11 

13 

A hydraulic PTO and a linear damper may be almost equally 

P 

C 

s/m)


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. 

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 multimode 

oscillations and for multibody 

systems. 

Control should be regarded as an open problem and a 

major challenge in the development of wave energy 

conversion.


Phase-control by latching 

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 freeoscillations, 

and achieving resonance. 

Phase-control by latching was introduced by 

Falnes and Budal 

J. Falnes, K. Budal, Wave-power conversion by 

power absorbers. Norwegian Maritime 

Research, vol. 6, p. 2-11, 1978. 

Johannes 

Falnes 

Kjell Budall 

(1933-89)

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 

Sphere radius 

5 m 

Gas (Nitrogen): 

• accumulator: 100 kg 

• turbine casing: 20 kg 

qm ( t) 

G S 

c 

p 

S c 

0.0314 

m 

2 

(diameter 200 

mm) 

Phase-control by latching: body is released when 

hydrodynam 

ic 

force on body exceeds 

R( 

S 

c 

p) 

( R 

1) 

Control parameters: 

G controls oil flow rate through hydraulic motor 

R controls latching (release of body)

x m 

d x d t m s , 1 0 fd M N 

PHASE CONTROL 

How to achieve phase-control by latching in a 

floating body with a hydraulic power-take-off 

mechanism 

4 

2 

0 

2 

excit. force 

velocity 

Introduce a delay in the release of the 

latched body. 

4 

608 

600 602 604 606 608 610 612 614 

4 

t s 

2 

0 

2 

No phase-control: 

optimal G, R = 1 

displacement 

4 

600 602 604 608 606 608 610 612 614 

t (s) 

t s 

How 

Increase the resisting force the 

hydrodynamic forces have to 

overcome to restart the body 

motion.

REGULAR WAVES 

Period T = 9 s 

Amplitude 0,667 m

x m 


x m 


G 

R 

Regular waves: T = 9 s, amplitude 0.67 m 

0.86 10 

6 

s/kg 

G 7.7 10 

1 P 55.0 kW R 16 P 206.1kW 

6 

s/kg 

4 

2 

0 

2 

4 

600 602 604 606 608 610 612 614 

4 

t s 

2 

Excit. 

force 

velocity 

No phase-control: 

displacement 

4 

2 

0 

2 

4 

600 602 604 606 608 610 612 

4 

t s 

2 

Phase-control: 

0 

0 

2 

2 

4 

600 602 604 608 4 

606 608 610 612 614 

608 

600 602 604 606 608 610 612 

t (s) t s 

t t (s(s) 

) 

t s

250 

200 

P(kW) 

150 

100 

50 

G is optimized 

for each R 

0 

0 5 10 15 20 25 30 

R 

R

IRREGULAR WAVES 

Period T e = 7, 9, 11 s 

Height H s = 2 m

T e 7 s, Hs 

2 m 

25 

kW 

P 

2 

H s 

m 

2 

20 

15 

10 

5 

4 

R = 1 

28 

8 12 16 20 24 

0 

0 2 4 6 8 10 12 14 16 

G 

10 6 

(s/kg)

T e 9 s, Hs 

2 m 

30 

kW 

P 

2 

H s 

m 

2 

20 

10 

4 

R = 1 

28 

8 12 16 20 24 

0 

0 2 4 6 8 10 12 

G 10 6 (s/kg)

T e 11 s, Hs 

2 m 

30 

P 

2 

H s 

20 

4 

28 

kW 

m 

2 

10 

R = 1 

8 12 16 20 24 

0 

0 2 4 6 8 10 

G 10 6 (s/kg)

Detailed analysis 

T e = 9 s 

H s =2 m

d x d t Hs s 1 , 1 0 fd Hs M N m 

d x d t Hs s 1 , 1 0 fd Hs M N m 

G 

0.7 

10 

6 

s/kg 

G 

4.2 

10 

6 

s/kg 

R 

1 

P 

2 

H s 

10.3 kW/m 

2 

R 

16 

P 

2 

H s 

28.5 kW/m 

2 

3 

diffr. force 

2 

3 

2 

1 

1 

0 

0 

1 

2 

velocity 

1 

2 

3 

700 720 740 760 760 780 800 

2 

t s 

3 

760 

700 720 740 760 780 80 

2 

t s 

1 

displacement 

1 

x Hs 

0 

x Hs 

0 

1 

1 

2 

700 720 740 760 760 780 800 

t (s) 

t s 

2 

700 720 740 760 

t (s) 760 780 8 

t s

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. 

100 

80 

Pressure in HP accumulator 

p 1 

(bar) 

60 

40 

20 

0 

0 5 10 15 20 25 30 

R


Phase control by latching may significantly increase the amount 

of absorbed energy by point absorbers. 

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. 

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, …).


Several degrees of freedom 

• Each body has 6 degrees of freedom 

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

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 

accordingly. 

A ij , B ij 

are defined 

They can be computed with commercial software 

(WAMIT, …). 

body 2 

PTO 

body 1

WAVE ENERGY TECHNOLOGIES 

Oscillating 

water column 

(with air turbine) 

Isolated: Pico, LIMPET 

Fixed structure 

In breakwater: Sakata, Douro river 

Floating structure: Mighty Whale, BBDB, Oceanlinx 

Oscillating bodies 

(with hydraulic motor, 

hydraulic turbine,linear 

electrical generator) 

Run up 

(with low-head hydraulic 

turbine) 

Floating 

Submerged 

Fixed 

structure 

Essencially translation (heave): 

IPS Buoy, WaveBob, PowerBuoy 

Essencially rotation: Pelamis, PS Frog, SEAREV 

Essencially translation (heave): AWS 

Rotation: WaveRoller, Oyster 

TWO-BODY 

Shoreline (with concentration): TAPCHAN 

POINT ABSORBERS 

In breakwater (without concentration): SSG 

Floating structure (with concentration): Wave Dragon


Several degrees of freedom 

Example: heaving bodies 1 and 2 reacting 

against each other. 

( m 

( m 

 

 

1 A1 

) x1 

B1 

x1 

gS1x1 

C( 

x1 

x2) 

K( 

x1 

x2) 

A12 

x2 

B12x 

2 f d 1 

 

 

2 A2 

) x2 

B2x2 

gS2x2 

C( 

x1 

x2) 

K( 

x1 

x2) 

A12 

x1 

B12x 

1 f d 2 

 

 

 

 

PTO 

body 1 

Note: 

A 

12 A21, B12 

B21 

body 2

IPS Buoy 

Wave Bob 

AquaBuoy 

Hose 

pumps 

Hydraulic 

ram

Simplifying assumptions for optimization and control 

Buoy represented by body 1a 

(hemispherical buoy) 

1a 

Acceleration tube represented by 

body 1b 

Inertia of piston and enclosed water 

represented by body 2 

Bodies 1b and 2 are “deeply” 

submerged: 

• Wave excitation forces neglected 

• Radiation forces neglected 

1b 

Hydrodynamic interference between 

bodies 1a, 1b and 2 neglected 

2

Two-body motion, linear PTO 

Coordinates: 

x : body 1 (1a+1b) 

y : body 2 

1a 

M 1 b , M 2 

include added mass 

 

 

( m1 a A1 

a M1b 

) x B1 

x gSx C( 

x y) 

K( 

x y) 

fd1 

 

 

damper spring 

1b 

M2 

y C( 

x 

y 

) K( 

x y) 

0 

2

Regular waves, frequency domain 

i t 

x( 

t) 

X0 e , y( 

t) 

Y0 

e 

i 

t 

f ( t) 

A ( ) e 

dj 

w 

j 

i 

t 

X 

0 

Y 

2 

0 

( m 

( i 

1a 

C 

A 

1a 

K) 

( 

) 

A 

w 

M 

1b 

1a 

, 

) 

i 

( B 

C) 

gS 

X0( i C K) 

Y0 

( M2 

i C K) 

2 

0 . 

Time-averaged absorbed power 

1 

P 

2 

2 

C X 

0 

Y 

0 

2 

Theoretical max power (axisymmetric body, heave motion): 

P 

max 

g 

3 

4 

2 

A w 

3

Radius of buoy = 7.5 m 

Mass of buoy 

m 1a 

905.7 

10 

kg 

Hydrostatic restoring force coeff. gS 

3 

1.776 MNm 

1 

Dimensionless values 

1a 

Motion amplitude 

X* 

X 0 

A w 

Mass of body 1b 

* 

M1b 

M1b 

m1 

a 

Mass of body 2 

* 

M2 

M2 

m1 

a 

Damping coefficient 

Spring stiffness 

Power 

C* 

K* 

P* 

C 

B( 

K 

gS 

P 

P 

max 

8 

) 

1b 

2

Results from optimization 

Regular waves 

Linear PTO 

No spring K = 0 

1a 

P* 

P 

P max 

1 

1b 

X* 

Motion amplitude (dimensionless) 

Y* 

R* (relative) 

2

T 

8s 

T 

12 10s 

3 

2,5 

2 

1,5 

1 

0,5 

0 

25 

20 

15 

10 

5 

0 

* 

M 2 

X * 

Y * 

R* 

0 0,25 0,5 0,75 1 1,25 1,5 

* 

M 1b 

C * 

0 0,25 0,5 0,75 1 1,25 1,5 

* 

M 1b 

6 

10 

X * 

X * 

5 

8 

4 

Y * Y * 

6 

3 

* * 

4 

MM 2 2 

R* 

2 

R* 

2 

1 

0 

0 

0 0 10,5 12 1,5 * 3 2 42,5 35 

3,5 

M 1b * 

M 1b 

200 

80 

15060 

10040 

C * 

5020 

0 0 

0 0 0,5 1 1 2 1,5 * 

3 2 2,5 4 35 

3,5 

M * 1b M 1b


Irregular waves 

(Pierson-Moskowitz) 

Linear PTO 

No spring K = 0 

1a 

1b 

2

T e 

8s 

T e 

10s 

5 

4 

3 

2 

1 

0 

25 

20 

15 

* 

10 P irr 

Fix 

* 

M 2 

M 1 b , M 2 

0 0,5 1 1,5 2 2,5 3 3,5 

* 

M 1b 

5 

4 

3 

2 

1 

0 

50 

40 

30 

* 

10 P irr 

* 

M 2 

0 1 2 3 4 5 

* 

M 1b 

10 

5 

C * 

20 

10 

C * 

0 

0 0,5 1 1,5 2 2,5 3 3,5 

* 

M 1b 

0 

0 1 2 3 4 5 

* 

M 1b


Reactive phase control 

Irregular waves 

(Pierson-Moskowitz) 

* 

M 1b 

* 

M 2 

2 

1.76 

Linear PTO 

Spring K 0 

1a 

1b 

2

T e 

8s 

T e 

10s 12s 

0,48 

0,47 

0,46 

0,45 

0,44 

4 

3,5 

3 

2,5 

* 

P irr 

0 0,05 0,1 0,15 0,2 

C * 

K * 

0,45 

0,35 

0,4 

irr * 

0,3 

P irr 

0,35 0,25 

0,2 

0,3 

0,15 

0,25 0,1 

8 

6 

4 

2 

0 0,1 0,1 0,2 0,20,3 0,4 0,3 0,5 0,4 0,6 0,5 0,7 

K K ** 

C * 

C * 

2 

0 0,05 0,1 0,15 0,2 

K * 

0 

0 0,1 0,1 0,2 0,20,3 0,4 0,3 0,5 0,4 0,6 0,5 0,7 

K *

Phase control by latching 

PTO: high pressure oil circuit 

Hydraulic ram 


Gas accumulator 

Wavebob (Ireland)

1a 

LP gas 

accumulator 

HP gas 

accumulator 

Motor 

1b 

Valve 

Cylinder 

2

Flow rate control through motor 

LP gas 

accumulator 

HP gas 

accumulator 

q 

m 

( t) 

G S 

2 

c 

p 

Motor 

S c 

p 

piston area 

pressuredifference 

Cylinder 

Valve 

Phase-control by latching: body is released when 

hydrodynam 

ic 

force on body exceeds 

R( 

S 

c 

p) 

( R 

1) 

Control parameters: 

G controls oil flow rate through hydraulic motor 

R controls latching (release of body)

Non-linear PTO: time-domain analysis 

( m 

1a 

t 

A 

1a 

L( 

t 

( 

) 

M 

) x( 

t) 

d 

1b 

) x( 

t) 

f 

d1 

( t) 

gSx( 

t) 

f 

m 

, 

M 

y 

( t ) 

f 

2 m 

. 

Continuity equation for oil-flow 

Accumulator gas thermodynamics

0,4 

a 

M 

M 

* 

1b 

* 

2 

7.5m 

2 

1.76 

* 

P irr 

* 

P irr 

0,3 

0,2 

0,1 

0,0 

0,25 

0,20 

0,15 

0,10 

0,05 

0,00 

0,10 

0,08 

0,06 

* 

P irr 

0,04 

0,02 

R=1 

R=2 

Te=8s 

R=4 

0 1 2 3 4 5 

R=1 

R=2 

R=4 

Te=10s 

0 1 2 3 4 5 

R=1 

R=2 

R=4 

Te=12s 

0,00 

0 1 2 3 4 5 

G 

10 6 (s/kg)

0,4 

0,3 

* 

P irr 

0,2 

R=1 

R=2 

R=4 

a 

M 

M 

* 

1b 

* 

2 

7.5m 

1 

3 

0,1 

0,0 

0,25 

0,20 

0,15 

* 

P irr 

0,10 

Te=8s 

0,0 0,5 1,0 1,5 2,0 2,5 3,0 

R=1 

R=2 

R=4 

0,05 

Te=10s 

0,00 

0,0 0,5 1,0 1,5 2,0 2,5 3,0 

0,12 

0,08 

* 

P irr 

R=1 

R=2 

R=4 

0,04 

Te=12s 

0,00 

0,0 0,5 1,0 1,5 2,0 2,5 3,0 

G 10 6 (s/kg)

0,4 

a 

M 

M 

* 

1b 

* 

2 

7.5m 

1 

6 

0,3 

0,3 

0,2 

* 

P irr 

0,2 

0,1 

0,0 

0,20 

Te=8s 

R=1 

R=4 

R=8 

0,0 0,5 1,0 1,5 2,0 

* 

P irr 

0,1 

R=1 

Te=10s 

R=2 

R=4 

0,0 

0,0 0,2 0,4 0,6 0,8 1,0 

0,10 

0,15 

* 

P irr 

0,10 

0,05 

0,00 

Te=12s 

R=1 

0,0 0,1 0,2 0,3 0,4 

G 

10 6 (s/kg) 

* 

P irr 

0,08 

0,06 

0,04 

R=2 

0,02 

R=4 Te=14s 

0,00 

R=1 

R=2 

R=4 

0,00 0,05 0,10 0,15 0,20 0,25 

G 

10 6 (s/kg)

0,4 

a 

M 

M 

* 

2 

7.5m 

* 

1b 

1 

0,3 

* 

P irr 

0,2 

0,1 

0,0 

0,25 

0,20 

0,15 

* 

P irr 

0,10 

0,05 

0,00 

Te=8s 

R=1 

R=4 

R=8 

0,0 0,5 1,0 1,5 2,0 

Te=10s 

R=1 

R=4 

R=8 

0,0 0,5 1,0 1,5 

0,15 

* 

P irr 

0,10 

0,05 

0,00 

T=12s 

R=1 

R=4 

R=8 

0,0 0,2 0,4 0,6 

G 

10 6 (s/kg)

We are far from: 

a 

M 

M 

* 

1b 

0 

* 

2 

5m 

Cylinder 

LP gas 

HP gas 

accumulator 

accumulator 

Motor 

Valve 

30 

P 

2 

H s 

20 

4 

28 

kW 

m 

2 

10 

R = 1 

8 12 16 20 24 

0 

0 2 4 6 8 10 

A. Falcão, Ocean Engineering, 35, 

358-366, 2008. 

G 10 6 (s/kg)

CONCLUSIONS 

• A two-body system with a linear damper can be optimized to 

absorb theoretical maximum energy from regular waves: P* 1 . 

• This drops to typically less than 50% in irregular waves. 

• 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

Power output 

The problem: 

• The performance of self-rectifying air turbines (Wells, 

impulse, …) is strongly dependent on pressure (or on 

flow rate) and on rotational speed. 

constant 

rotational 

speed 

Air pressure Δp 

• How to control the turbine (instantaneous rotational 

speed) to achieve maximum energy production

OWC Dynamics 

Two different approaches to modelling: 

weightless 

piston 

uniform air 

pressure 

Oscillating body (piston) model 

(rigid free surface) 

Uniform pressure model 

(deformable free surface)

OWC Dynamics 

m (t) 

V0 

p(t) 

air volume 

pressure 

q(t) 

m (t) 

volume-flow rate displace by 

free-surface 

mass-flow rate of air 

through turbine 

a 

q(t) 

a air density p(t) 

air pressure 

Conservation of air mass 

(linearized) 

m 

( t) 

a 

q( 

t) 

V 

0 

2 

aca 

dp( 

t) 

dt 

flow rate 

q 

q 

q r 

exc 

excitation 

radiation 

Effect of air compressibility

Air turbine 

N 

D 

P 

t 

p 

rotational speed 

rotor diameter 

power output 

pressure head 

OWC Dynamics 

In dimensionless form: 

m 

p 

3 

aND 

aN 

flow pressure 

head 

2 D 

2 

Performance curves of turbine 

(dimensionless form): 

P 

t 

3 D 5 

aN 

power 

power Π 

flow 

Φ 

f w ( ), fP( 

) 

pressure head

OWC Dynamics 

Frequency domain 

m (t) 

a 

q(t) 

Linear air turbine 

K 

p( t), 

m 

, q( 

t), 

q ( t), 

q 

 

r 

exc ( t) 

P, 

M , Q, 

Qr 

, 

Q 

exc 

e 

i 

t 

Q r ( 

P( 

) 

) 

B( 

) 

iC( 

) 

B 

C 

radiation conductance 

radiation susceptance 

Q ) ( ) A 

exc ( 

excitation 

coeff. 

w 

wave 

ampl. 

P 

KD 

N 

a 

B 

Q 

exc 

i C 

V 

a 

0 

2 

a 

c

OWC Dynamics 

2 

0 

exc 

a 

a 

a 

c 

V 

C 

i 

B 

N 

KD 

Q 

P 

K 

gS 

C 

B 

i 

A 

m 

F 

X 

d 

) 

( 

) 

( 

2 

0 

q(t) 

m (t) 

a 

q(t) 

m (t) 

a 

) 

( 

), 

( P 

f w f 

t 

i 

Pe 

t 

p 

Re 

) 

( 

2 

2 

5 

3 

) 

( 

) 

( 

) 

( 

: 

power output 

D 

N 

t 

p 

f 

D 

N 

t 

P 

t 

a 

P 

a 

t 

Ψ

OWC Dynamics 

m (t) 

Time domain: 

• Linear or non-linear turbine 

a 

q(t) 

V 0 dp( 

t) 

m 

( t) 

g ( t ) p( 

) d qexc 

( ) 

2 

r 

c dt 

t 

a 

a 

a 

turbine flow vs 

memory function 

t 

pressurecurve 

g r 

( t) 

2 

m 

0 

B( 

a 

ND 

) cos 

3 

f 

t d 

w 

a 

N 

p 

2 

D 

2 

To be integrated numerically for p(t) 

power output 

: 

( t) 

P ( t) 

a 

p( 

t) 

t f 

f 

w ( ), fP( 

) 

3 5 P 2 2 

D aN 

D 

N

OWC Dynamics 

Numerical application 

Y 

Z 

X 

Pico OWC 

AQUADYN 

Brito-Melo et al. 2001 

Memory 

function 

(rad/s)

OWC Dynamics 

Numerical application 

p(t) 

P t (t) 

Air pressure in chamber 

Power 

Results from time-domain modelling of impulse turbine over 

• Turbine D = 1.5 m, N = 115 rad/s (1100 rpm) 

t = 120 s 

• Sea state H s = 3 m, T e = 11 s 

• Average power output from turbine 97.2 kW

OWC Dynamics 

Stochastic modelling 

• Irregular waves 

• Linear air-turbine 

• Much less time-consuming than time-domain analysis 

• Appropriate for optimization studies 

• 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.

OWC Dynamics 

Stochastic modelling 

Wave climate represented by a set of sea states 

• For each sea state: H s , T e , freq. of occurrence . 

• Incident wave is random, Gaussian, with 

known frequency spectrum. 

WAVES 

OWC 

AIR 

PRESSURE 

TURBINE 

Random, 

Gaussian 

Linear system. 

Known hydrodynamic 

coefficients 

Random, 

Gaussian 

rms: p 

Known 

performance 

curves 

ELECTRICAL 

POWER OUTPUT 

Time-averaged 

GENERATOR 

Electrical 

efficiency 

TURBINE 

SHAFT 

POWER 

Time-averaged

Gaussian process (e.g. surface elevation ζ ) 

Probability density function (pdf) : 

f 

( 

) 

2 

1 

exp 

2 

2 

2 

2 

S ( )d = variance 

spectral 

density 

= standard deviation

OWC Dynamics 

Stochastic model: 

• Linear turbine (Wells turbine) 

• Random Gaussian waves 

Pierson-Moskowitz spectrum 

S 

( 

) 

263H 

2 

s 

T 

e 

4 

5 

exp( 

1054T 

e 

4 

4 

). 

For linear system, 

2 

p 

2 

0 

S 

( ) ( 

and pdf f ( p) 

p( 

t) 

) 

( 

1 

2 

is 

) 

randomGaussian, 

2 

p 

d 

exp 

where 

2 

p 

2 

2 

p 

Λ 

with variance 

KD 

N 

a 

B 

i 

a 

V 

c 

0 

2 

a 

C 

Q exc ( ) ( ) A w 

excitation 

coeff. 

1 

wave 

ampl. 

P 

t 

f 

3 

5 

2 N D p p 

( p) 

P ( p) 

dp 

a 

t 

exp fP 

dp 

2 

2 2 

2 N D 

p 

0 

2 

2 

p 

a

Time-averaged turbine power output : 

In dimensionless form : 

dimensionless 

pressure variance 

dimensionless timeaveraged 

power 

Wells 

turbine 

( 

) 

with relief valve 

( 

) 

without relief valve

Time-averaged turbine power output : 

In dimensionless form : 

dimensionless 

pressure variance 

How to control the rotational speed N for maximum 

dimensionless 

averaged power 

= 0 for maximum energy production

= 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 P t . 

Control algorithm: 

• Set electrical power 

P 

e 

P 

t 

function ( N 

)

Example: Pico OWC plant with 2.3m Wells turbine 

Local wave climate represented by 44 sea states (44 circles) 

P e 

5 

1.583 

10 N 

3.16

Maximum rotational speed may be constrained by: 

• Centrifugal stresses in turbine and electrical generator 

• Mach number effects (shock waves) 

P e 

5 

1.583 

10 N 

3.16 

1500 rpm

Rated power (kW) 

Annual averaged net power 

(kW) 

OWC Dynamics 

Application of stochastic model 

Wells turbine size range 1.6m < D < 3.8m 

800 

300 

700 

600 

250 

200 

wave climate 3 

500 

150 


400 

300 

100 

50 


200 

1.5 2 2.5 3 3.5 4 

D (m) 

Plant rated power 

(for H s = 5m, T e =14s) 

0 

1.5 2 2.5 3 3.5 4 

D (m) 

Annual averaged 

net power (electrical)

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

CONTROL TECHNIQUES FOR WAVE ENERGY CONVERTERS

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