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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