Specialization in Ocean Energy
MODELLING OF WAVE
ENERGY CONVERSION
António F.O. Falcão
Instituto Superior Técnico,
Universidade de Lisboa
2017
PART 5
STOCHASTIC MODELLING OF
WAVE ENERGY CONVERSION
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 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
• Historically the first model
• The starting point for the other models
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)
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
Gaussian process
Physical random variables that are expected to be the sum of many independent
processes have distributions that are nearly Gaussian.
This is the case of the free surface elevation of real irregular waves
Spectral distributi on S ( )
1.5
   rms or standard deviation of 
1

0.5
m
Variance  2   S ( ) d
0
0
Probabilit y density function of 
 2
 ( ) 
exp   2
 2
2  


1




0.5
1
100
150
200
t s
250
300
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
Input
signal
LINEAR
SYSTEM
Ouput
signal
• Random
• Random
• Gaussian
• Gaussian
• Given spectral distribution
• Spectral distribution
• Root-mean-square (rms)
• Root-mean-square (rms)
Ouput
signal
Input
signal
LINEAR
SYSTEM
Air pressure oscillatio n pc (t )
Incident w ave  (t )
Spectral distributi on S ( )
Spectral distributi on S p ( )
   rms or standard deviation of 
 p  rms or standard deviation of pc


Variance  2   S ( ) d
Variance  2p   S p ( ) d
0
0
Probabilit y density function of 
Probabilit y density function of pc
 2
 ( ) 
exp   2
 2
2  


 p2 
 p ( pc ) 
exp   c2 
 2 
2  p
p

1




1
Ouput
signal
Input
signal
LINEAR
SYSTEM
Air pressure oscillatio n pc (t )
Incident w ave  (t )
2
S p ()  2 () () S ()
 (t )
pc (t )
( ) 
Qe ( )
Aw ( )
 ( ) 
 V

Pc ( )  KD

 G  i  0  B 
Qe ( )   a 
  pa

1
Ouput
signal
Input
signal
LINEAR
SYSTEM
Air pressure oscillatio n pc (t )
Incident w ave  (t )
 2p
( ) 

  S ( )  2 ( )  ( ) 2d
Qe ( )
Aw ( )
0
 KD
  V0

 ( )  
 G  i 
 B 
  a 
  pa

1
 p2 
Probabilit y density function of pc   p ( pc ) 
exp   c2 
 2 
2  p
p

1
Linear air turbine (Wells turbine)
Power vers us pressure head (dimension less)   f P ()

pc
 a 2 D 2
and  
Pt
 a 3 D5


p
c

Pt   a  D f P 
  2 D2 
 a

3
5

Pt    p ( pc ) Pt ( pc ) dpc

 p2 
 p ( pc ) 
exp   c2 
 2 
2  p
p

1
 p2  

 a 3 D 5
p
c 
c

 dpc
Pt 
exp  2 f P 

2
2
 2     D 
2  p 
p
 a



Linear air turbine (Wells turbine)
 p2  

 a 3 D 5
p
c 
c

 dpc
Pt 
exp  2 f P 

2
2
 2     D 
2  p 
p
 a




1
2  
 2 
 exp   2 2  f P () d



0.0025
0.0020
 ( )
0.0015
,

Average
power
output
0.0010
 (  )
0.0005
0.0000
0.0005
0.00
0.02
0.04
0.06
0.08
,
0.10
0.12
0.14
Linear air turbine (Wells turbine)
 
  K
Average
turbine
efficiency

2
K 
0.8
 ( )
0.6
,
 (  )
0.4
0.2
0.0
0.02
0.04
0.06
0.08
,
0.10
0.12
0.14
AIR TURBINE AND
ELECTRICAL EQUIPMENT
FOR THE PICO OWC PLANT
The Pico plant
N
WAVES
PLANT
20 m
RELIEF VALVE
AIR TURBINE
12 m
Simplified version of the wave climate (in deep water)
9 sea states, and their frequency of occurrence
1
2
i
Te,i (s)
H s,i (s)
i 
1
9.0
0.8
0.25
2
3
9.5
10.0
1.2
1.6
0.2
0.177
4
10.5
2.0
0.145
5
11.0
2.4
0.10
6
11.5
2.9
0.07
7
12.0
3.4
0.045
8
9
12.5
13.0
4.0
4.5
0.007
0.006
Pwave,i (kW/m)
2.82
6.70
12.5
20.6
31.0
47.4
68.0
98.0
129.0
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
Random,
Gaussian
Linear system.
Known hydrodynamic
coefficients
AIR
PRESSURE
Random,
Gaussian
rms: p
ELECTRICAL
POWER OUTPUT
GENERATOR
Time-averaged
Electrical
efficiency
TURBINE
Known
performance
curves
TURBINE
SHAFT
POWER
Time-averaged
Pico plant
Compare two types of air turbines
Biradial turbine
Inlet/outlet ducts
guide vanes
rotor
Wells turbine
Turbine performance curves
versus pressure head
(dimensionless)
Wells
,  
Pressure head


Pressure head
,  
Pressure head
Turbine performance curves
versus pressure head
(dimensionless)
0.14
0.12
0.10
0.08
0.06
0.04
rotor
0.02
0.00
0.0
guide vanes
Inlet/outlet ducts
Biradial
0.2
0.4
0.6
,  
0.8
1.0
0.8
1.0
Pressure head
0.20
0.80
linearize
0.15
0.75

0.70
0.10
0.65
0.05
0.00
0.0
0.60
0.2
0.4
0.6
,  
Pressure head
0.8
1.0
0.55
0.0
0.2
0.4
0.6
,  
Pressure head
Constraint: blade tip velocity of the turbine
rotor should not exceed 180 m/s
D/2 < 180 m/s
Why?
Centrifugal stresses, shock waves
Wells turbine: single stage and two-stages
were considered
For each turbine size D and each sea state
(Hs,Te), the rotational speed  was
numerically optimized for maximum
averaged power output of the turbine.
The annual-averaged power output was
computed.
Turbine type and size optimization
D (m)
Rotor diameter
Turbine efficiency

5
50
10
Pwave (kW/m)
Incident wave power
100
Le,in  a   1
Rotational speed control
Points fairly well
aligned along lines
Pt  a  
BIRADIAL
TURBINE
But, in average values
Pe, in  Pt
Control algorithm
(instantaneous values):
Pe, in  a  
Electromagnetic torque:
Le,in  a   1
Rotational speed
Effect efficiency and rated power of electrical equipment
The efficiency of the electrical equipment (generator and power
electronics) decays markedly for load factor less than about 30 – 35%
The electrical rated power
should not be exceeded at any
time. The power electronics is
very sensitive to overheating.
This affects the plant performance and the
rotational speed control
5.5
5.0
4.5
Le (kN)
4.0
Prated
Le 
e (1) 
Le  a   1
3.5
3.0
2.5
160
180
200
220
 (rad/s)
240
260
i 9
1400
Pe,in  1053 kW
1200
A
1000
D
800
i7
Pe,in  632 kW
B
E
600
G
400
Pe,in  316 kW
i5
0
50
100
150
 (rad/s)
guide vanes
Turbine size D = 2 m
Inlet/outlet ducts
rotor
C
F
H
200
200
250
Average rotational speed  versus electrical rated power
Prated in the most energetic sea state i = 9, for different
turbine sizes D = 2.0 to 3.0 m
The annual production of electrical energy depends on
turbine size and electrical rated power
D  2.5 m
D  2.75 m
D  2.25 m
D  3.0 m
D  2.0 m
220
200
D  1.75 m
180
D  1.5 m
160
140
200
400
600
Prated (kW)
800
1000
The final choice of turbine size and electrical
rated power will also depend energy tariff and
equipment costs.
Maximum profit versus maximum energy
production.
Maximum energy production
and maximum profit
as alternative criteria for
wave power equipment optimization
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
or
Maximum annual profit,
leading to smaller, less powerful, cheaper equipment
How to optimize? How different are the results
from these two optimization criteria?
The costs
C  C
Capital costs
struc
 C
Acap 
Annual repayment
mech
 C
elec
 C
other
Cr
1  (1  r )  n
r  discount rate, n  plant's lifetime (years)
Operation & maintenance
annual costs
Income
AO & M
I  8760 Pe,annual A u
Pe,annual  power output
A  availabilty
u  energy price
Annual profit
E  I  Acap  AO&M
Calculation example
Pico OWC plant
OWC cross section:
12m 12m
VALVE
TURBINE
AIR
OWC
12m
WAVES
Computed hydrodynamic coefficients
Calculation example
Wells turbine
Dimensionless performance curves
Turbine geometric shape: fixed
Turbine size (D): 1.6 m < D < 3.8 m
Equipped with relief valve
Calculation example
Wave climate: set of sea states
Each sea state:
• random Gaussian process, with given spectrum
• Hs, Te, frequency of occurrence
Calculation method:
Inter
• Stochastic modelling of energy conversion process
0.5 m  H s  5 m (10 values),
• 720 combinations 
7 s  Te  14 s (8 values),
1.6 m  D  3.8 m (9 values)
Three-dimensional interpolation for given
wave climate and turbine size
Calculation example
0.6
800
0.55
700
Rated power (kW)
Dimensionless power output
Turbine size range 1.6m < D < 3.8m
0.5
D =1.6m
0.45
D =2.3m
0.4
0.35
400
200
0.25
100
500
300
D =3.8m
0.3
600
150
200
250
D (m/s)
300
350
Turbine rotational speed 
optimally controlled.
Max tip speed = 170 m/s
1.5
2
2.5
D (m)
3
3.5
Plant rated power
(for Hs = 5m, Te=14s)
4
Calculation example
Wave climates
Wave climate 3: 29 kW/m
Reference climate:
Wave climate 2: 14.5 kW/m
• measurements at Pico site
• 44 sea states
• 14.5 kW/m
Wave climate 1: 7.3 kW/m
Calculation example
0.6
Utilization factor
0.5
wave climate 3
0.4
0.3
wave climate 2
Wind plant
average
0.2
wave climate 1
0.1
0
1.5
2
2.5
3
D (m)
Utilization factor
3.5
4
Calculation example
Annual averaged net power
(kW)
300
250
wave climate 3
200
wave climate 2
150
100
wave climate 1
50
0
1.5
2
2.5
3
3.5
4
D (m)
Annual averaged net power (electrical)
Calculation example
Costs
Capital costs
Mechanical equipment Cmech  Bmech D 2
Electrical equipment
0.7
Celec  Belec Prated
 D  2.3 m  Bmech  62
Reference : Prototype in 2003 
 Prated  400 kW  Belec  3.3
Structure  others : Cstruc  Coth  0
Operation & maintenance
Availability
A  0.95
AO&M  0.03(Cmech  Celec )
Calculation example
400
Bmech  30, Belec  2.0
350
discount rate r  0.1,
lifetime n  20 years
Annual profit (kEuro)
300
250
wave climate 3: 29 kW/m
wave climate 2: 14.5 kW/m
wave climate 1: 7.3 kW/m
200
150
u  0.225 €/kWh
u  0.1 €/kWh
100
u  0.05 €/kWh
50
0
-50
1.5
2
2.5
D (m)
3
3.5
4
Influence of
wave climate
and energy price
Calculation example
Bmech  30, Belec  2.0
Annual profit (kEuro)
150
lifetime n  20 years,
125
u  0.1 €kWh
100
wave climate 3: 29 kW/m
wave climate 2: 14.5 kW/m
wave climate 1: 7.3 kW/m
75
50
25
r  0.1
0
r  0.15
-25
1.5
2
2.5
3
3.5
D (m)
Influence of wave climate
and discount rate r
4
Calculation example
Belec  2.0,
discount rate r  0.1,
lifetime n  20 years
u  0.1 €/kWh
50
Annual profit (kEuro)
150
125
Annual profit (kEuro)
Bmech  20
wave climate 3: 29 kW/m
wave climate 2: 14.5 kW/m
wave climate 1: 7.3 kW/m
100
75
50
25
0
Bmech  30
Bmech  45
u  0.05 €/kWh
40
30
20
10
0
-10
-20
-25
1.5
2
3
2.5
D (m)
3.5
4
1.5
2
2.5
3
3.5
D (m)
Influence of wave climate & mech. equip. cost
4
Calculation example
Annual profit (kEuro)
150
Bmech  30, Belec  2.0,
discount rate r  0.1,
125
u  0.1 €/kWh
100
75
29 kW/m
14.5 kW/m
7.3 kW/m
50
25
0
n  10 years
n  20 years
-25
1.5
2
2.5
3
3.5
D (m)
Influence of wave climate
and lifetime n
4
CONCLUSIONS
1. Stochastic modelling is a powerful tool in basic studies
and preliminary design
2. Maximum profit criterion yields smaller size and rated
power for equipment, compared with maximum produced
energy criterion
3. Optimized equipment size and rated power found to be
sensitive to:
 Wave climate
 Produced energy price
 Equipment basic cost level
 Discount rate
 Equipment lifetime
4. Equipment cost reduction by standardization and series
production should be considered (even if negatively
affecting 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 5
STOCHASTIC MODELLING OF
WAVE ENERGY CONVERSION