Specialization in Ocean Energy

MODELLING OF WAVE ENERGY CONVERSION

António F.O. FalcãoInstituto Superior Técnico,

Universidade de Lisboa2014

PART 5STOCHASTIC MODELLING OF WAVE ENERGY CONVERSION

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 and CFD 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

LINEARSYSTEM

Input signal

Ouput signal

• Random

• Gaussian

• Given spectral distribution

• Root-mean-square (rms)

• Random

• Gaussian

• Spectral distribution

• Root-mean-square (rms)

Input signal

Ouput signal

)( aveIncident w t

)(n oscillatio pressureAir tpc

ofdeviation standardor rms cp p ofdeviation standardor rms

0

2 d)( Variance pp S

0

2 d)( Variance S

2

2

2exp

2

1)(

offunction density y Probabilit

2

2

2exp

2

1)(

offunction density y Probabilit

p

c

pcp

c

pp

p

)(on distributi Spectral S )(on distributi Spectral pS

Input signal

Ouput signal

)( aveIncident w t

)(n oscillatio pressureAir tpc

)(

)()(

w

e

A

Q

10i)(

B

p

VG

KD

aa

2

2

2exp

2

1)( offunction density y Probabilit

p

c

pcpc

ppp

d)()()( 2

0

22

Sp

Linear air turbine (Wells turbine)

)( less)(dimension head pressure usPower vers Pf

5322 and

D

P

D

p

a

t

a

c

22a

53at

D

pfDP cP

2

2

2exp

2

1)(

p

c

pcp

pp

c

cP

p

c

p

pD

pf

pDP d

2exp

2 22a

2

253a

t

cctcp ppPpP d)()(t

Linear air turbine (Wells turbine)

c

cP

p

c

p

pD

pf

pDP d

2exp

2 22a

2

253a

t

d)(

2exp

2

12

2

Pf

0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.0005

0.0000

0.0005

0.0010

0.0015

0.0020

0.0025

,

,

)(

)(

Average power output

Linear air turbine (Wells turbine)

K 2

K

Average turbine efficiency

0.02 0.04 0.06 0.08 0.10 0.12 0.140.0

0.2

0.4

0.6

0.8

,

,

)(

)(

Maximum energy production

and maximum profit

as alternative criteria for

wave power equipment optimization

Application of stochastic modelling

The problem

When designing the power equipment for a wave energy

plant, a decision has to be made about the

size and rated power capacity of the equipment.

Which criterion to adopt for optimization?

Maximum annual production of energy,leading to larger, more powerful, more costly equipment

Maximum annual profit,leading to smaller, less powerful, cheaper equipment

or

How to optimize? How different are the results

from these two optimization criteria?

How to model the energy conversion chain

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 AIRPRESSURE TURBINE

TURBINE SHAFT POWER

Random,Gaussian

Linear system.Known hydrodynamic

coefficients

Knownperformance

curves

Time-averaged

GENERATORELECTRICALPOWER OUTPUT

Time-averaged

Random,Gaussian

rms: p

Electricalefficiency

The costs

Operation & maintenanceannual costs

otherelecmechstruc CCCCC Capital costs

Annual repayment nr

CrA

)1(1

cap

M&OA

Income uAPI annuale,8760

Annual profit M&Ocap AAIE

(years) lifetime splant'rate,discount nr

priceenergy

yavailabilt

outputpower annuale,

u

A

P

Calculation example

VALVE

OWC

AIR

TURBINE

WAVES12m

Pico OWC plant

Computed hydrodynamic coefficients

OWC cross section:12m 12m

Wells turbine

0 0.05 0.1 0.15 0.2 0.25 0.3Y , s Y

0

0.2

0.4

0.6

0.8

h,h

Efficiency vs. pressure head

Instantaneous

Averaged with valve

Averaged no valve

0.0

0.2

0.4

0.6

0.8

Calculation example

Dimensionless performance curves

Turbine geometric shape: fixedTurbine size (D): 1.6 m < D < 3.8 m

Equipped with relief valve

InterCalculation method:

• Stochastic modelling of energy conversion process

• 720 combinations

Calculation example

Wave climate: set of sea states

Each sea state:• random Gaussian process, with given spectrum

• Hs, Te, frequency of occurrence

values)(9 m8.3m6.1

values),(8 s14s7

values),(10 m5m5.0

D

T

H

e

s

Three-dimensional interpolation for given wave climate and turbine size

Calculation example

Turbine rotational speed optimally controlled.

Max tip speed = 170 m/s

Plant rated power

(for Hs = 5m, Te=14s)

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)

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

100 150 200 250 300 350

D (m/s)

Dim

ensi

on

less

po

wer

ou

tpu

t

D=1.6m

D=2.3m

D=3.8m

Calculation example

Reference climate:

• measurements at Pico site

• 44 sea states

• 14.5 kW/m

Wave climates

Wave climate 3: 29 kW/m

Wave climate 2: 14.5 kW/m

Wave climate 1: 7.3 kW/m

0

0.1

0.2

0.3

0.4

0.5

0.6

1.5 2 2.5 3 3.5 4D (m)

Uti

liza

tio

n f

ac

tor

wave climate 3

wave climate 2

wave climate 1

Calculation example

Utilization factor

Wind plantaverage

0

50

100

150

200

250

300

1.5 2 2.5 3 3.5 4D (m)

An

nu

al a

ve

rag

ed

ne

t p

ow

er

(kW

)wave climate 3

wave climate 2

wave climate 1

Calculation example

Annual averaged net power (electrical)

Calculation example

Costs

Capital costs

3.3 kW 400

62 m3.2in

elecrated

mech

BP

BD2003 Prototype :Reference

0.7ratedelecelec

2mechmech

PBC

DBC

equipment Electrical

equipment Mechanical

0: othstruc CCothersStructure

Operation & maintenance

Availability

)(03.0 elecmechM&O CCA

95.0A

-50

0

50

100

150

200

250

300

350

400

1.5 2 2.5 3 3.5 4

D (m)

An

nu

al p

rofi

t (k

Eu

ro)

Calculation example

years 20 lifetime

,1.0 ratediscount

0.2,30 elecmech

n

r

BB

wave climate 3: 29 kW/mwave climate 2: 14.5 kW/mwave climate 1: 7.3 kW/m

€/kWh 0.05

€/kWh 0.1

€/kWh 225.0

u

u

u

Influence ofwave climate

and energy price

-25

0

25

50

75

100

125

150

1.5 2 2.5 3 3.5 4D (m)

An

nu

al p

rofi

t (k

Eu

ro)

Calculation example

Influence of wave climate and discount rate r

€kWh 0.1

years, 20 lifetime

0.2,30 elecmech

u

n

BB

15.0

1.0

r

r

wave climate 3: 29 kW/mwave climate 2: 14.5 kW/mwave climate 1: 7.3 kW/m

-20

-10

0

10

20

30

40

50

1.5 2 2.5 3 3.5 4

D (m)

An

nu

al p

rofi

t (k

Eu

ro)

Calculation example

-25

0

25

50

75

100

125

150

1.5 2 2.5 3 3.5 4

D (m)

An

nu

al p

rofi

t (k

Eu

ro)

Influence of wave climate & mech. equip. cost

years 20 lifetime

,1.0 ratediscount

,0.2elec

n

r

B wave climate 3: 29 kW/mwave climate 2: 14.5 kW/mwave climate 1: 7.3 kW/m 45

30

20

mech

mech

mech

B

B

B

€/kWh 1.0u €/kWh 05.0u

-25

0

25

50

75

100

125

150

1.5 2 2.5 3 3.5 4

D (m)

An

nu

al p

rofi

t (k

Eu

ro)

Calculation example

€/kWh 0.1

,1.0 ratediscount

,0.2,30 elecmech

u

r

BB

29 kW/m14.5 kW/m

7.3 kW/m

Influence of wave climate and lifetime n

years 20

years 10

n

n

CONCLUSIONS

1. Stochastic modelling is a powerful tool in basic studiesand preliminary design

2. Maximum profit criterion yields smaller size and ratedpower for equipment, compared with maximum producedenergy criterion

3. Optimized equipment size and rated power found to besensitive to: Wave climate Produced energy price Equipment basic cost level Discount rate Equipment lifetime

4. Equipment cost reduction by standardization and seriesproduction should be considered (even if negativelyaffecting energy production in different wave climates)

Example: Optimization of an OWC sparbuoy for the wave climate off the western coast of Portugal (31.4 kW/m)

Optimization involved several geometric parameters

Size and rotational speed of air turbine were optimized

R.P.F. Gomes, J.C.C. Henriques, L.M.C. Gato, A.F.O. Falcão. "Hydrodynamic optimization of an axisymmetric floating oscillating water column for wave energy conversion", Renewable Energy, vol. 44, pp. 328-339, 2012.

END OF PART 5STOCHASTIC MODELLING OF WAVE ENERGY CONVERSION