Wave Energy Converter Dynamics Modeling
Presented by: Kelley RuehlDate: 3/3/2011
WEC Device Modeling Research ReviewSandia National Laboratories Albuquerque, NM
Presentation Outline
• Background and Motivation• Wave Energy Converters (WECs)• Equations of Motion (EOM)• ANSYS-AQWA WEC Model• Wave Surface Elevation Definition• Matlab/Simulink WEC Dynamics Model• Sample WEC Dynamics Model Output• Conclusions and Future Work
Background and Motivation
• WECs have been conceptualized for over a century but technology is largely in R&D• Some full scale deployments• Mostly scale model testing
• Knowledge gap WEC device performance• Wind Industry
• Generic Turbine Models• Estimate power output for given wind resource
• Research Goal: Develop Generic WEC Models
Wave Energy Converters
Oscillating Bodies
η (t) [m]
H[m]
L[m]T[s]
m1
PTO
m2
x1
x2
mooring
h[m]
Generic 2 Body Point Absorber (PA) WEC
• PA WEC Model consists of• Heaving Buoy and Damping Plate• Power Take-Off (PTO)• Mooring
Heaving Buoy
Damping Plate
Time-Domain EOM for 2 Body Point Absorber WEC
• Heaving PA WEC EOM, including PTO & Mooring• Buoy EOM
• Plate EOM
Excitation Force
Radiation Forces
PTO Force
Buoyancy Force
Buoyancy Force
Radiation Forces
Excitation Force
PTO Force
Mooring Force
Drag Force
Hydrostatic Force
Added Mass
Added Mass
Impulse Response Functions (IRFs) in Hydrodynamics:Excitation and Radiation Force Calculations
• Excitation Force: • Force imparted on body due to incident wave
• Radiation Force: • Force of radiated waves created by body’s motion
• Coupled Radiation Forces:
Buoy Excitation IRF Plate Excitation IRF
Buoy Radiation IRF Plate Radiation IRF
Coupled Radiation IRF k12 = k21
Time-Domain Impulse Response Functions (IRFs)
• Need 5 IRFs for Time-Domain PA WEC Model• Excitation Impulse Response Function
• Buoy • Damping Plate
• Radiation Impulse Response Function• Buoy • Damping Plate
• Coupled Radiation Impulse Response Function• ANSYS AQWA (-LINE) gives Frequency-Domain response need
Time-Domain IRFs
AQWA: Eidsmoen WEC (From Eidsmoen, 1996)
Frequency-Domain Excitation Force Magnitude
AQWA Max Mesh = 3 m
AQWA Frequency-Domain Excitation
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-5
0
5
10x 10
5
Am
plitu
de
AQWA F-Domain Excitation
BuoyPlate
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-30
-20
-10
0
10
Pha
se [r
ad]
[rad/s]
BuoyPlate
Frequency-Domain Excitation Force, with linear interpolation and
extrapolation
Time-Domain Excitation Impulse Response Function
-25 -20 -15 -10 -5 0 5 10 15 20 25-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5x 10
5 Excitation IRF
Time [s]
Exc
itatio
n Fo
rce
Ker
nel
AQWA BuoyAQWA PlateEidsmoen BuoyEidsmoen Plate
AQWA Frequency-Domain Radiation
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-5
0
5
10x 10
4
dam
ping
AQWA F-domain Radiation
buoyplatecoupled
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20
2
4
6
8x 10
6
[rad/s]
adde
d m
ass
[kg]
buoyplate
Frequency-Domain Radiation Force, with linear interpolation and
extrapolation
Time-Domain Radiation Impulse Response Function
0 5 10 15 20 25-2
-1.5
-1
-0.5
0
0.5
1
1.5
2x 10
4 Radiation IRF
Time [s]
Rad
iatio
n Fo
rce
Ker
nel
AQWA BuoyAQWA PlateAQWA CoupledEidsmoen BuoyEidsmoen PlateEidsmoen Coupled
Time-Series Wave Surface Elevation (η)
• Created a Matlab Function for defining η • Regular Waves:
• Defined by Tp and Hs
• Irregular Waves:• Import time-series directly from NDBC/CDIP data buoy data• Create time-series based on PM Spectrum by defining Tp and Hs
• Create time-series from NDBC data buoy spectra• Create time-series from a location’s representative spectra
• NOTE: time-series created using random amplitude and phase according to Tucker(1984)
Time-Series Wave Surface Elevation (η): Irregular Waves Based on Tucker (1984)
• Time-series based on PM Spectrum by defining Tp and Hs
• Let Tp =9 [s] and Hs= 1.5 [m]
0 0.1 0.2 0.3 0.4 0.5 0.6 0.70
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
Frequency [Hz]
Sf [m
2 /Hz]
PM Spectrum for Hs = 1.5 [m] and Tp = 9 [s]
0 0.1 0.2 0.3 0.4 0.5 0.6 0.70
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
Frequency [Hz]
Sf [m
2 /Hz]
PM Spectrum for Hs = 1.5 [m] and Tp = 9 [s]
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50
0.05
0.1
n
Standard Deviation
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5-0.2
-0.1
0
0.1
0.2Normal RV
a n
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5-0.4
-0.2
0
0.2
0.4
b n
Frequency [Hz]
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50
0.05
0.1
n
Standard Deviation
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5-0.2
-0.1
0
0.1
0.2Normal RV
a n
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5-0.4
-0.2
0
0.2
0.4
b n
Frequency [Hz]
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50
0.05
0.1
0.15
0.2
0.25
0.3
0.35
Am
plitu
de, c
n
Rayleigh RV
randomRMS aka deterministic
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50
1
2
3
4
5
6
Pha
se,
Uniform RV
Frequency [Hz]
0 20 40 60 80 100 120 140 160-1.5
-1
-0.5
0
0.5
1Wave Surface Elevation
Time [s]
[m
]
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50
0.05
0.1
0.15
0.2
0.25
0.3
0.35
Am
plitu
de, c
n
Rayleigh RV
randomRMS aka deterministic
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50
1
2
3
4
5
6
Pha
se,
Uniform RV
Frequency [Hz]
Simulink WEC Dynamics Model:Top Level
time
Pout
Power
xdot2
Plate Velocity
x2
PlateDisplacement
xdotFpto
Pout
PTO and Control
Fe1
Fe2
Fpto
Fm
x1
xdot1
x2
xdot2
PA WEC Dynamics
xdot2
x2Fm
Mooring Force Determination
eta eta
Fe1
Fe2
Excitation Force Determination
Clock
xdot1
Buoy Velocity
x1
Buoy Displacement
Buoy & PlateExcitation Impulse
Response Functions
Model Input Model Outputs
4xdot2
3x2
2xdot1
1x1
Fr_12
Fpto
Fm
Fb2
Fe2
xdotdot2
xdot2
x2
Plate Dynamics
Fb2
PlateBuoyancy
xdotdot1
xdot1
xdot2
xdotdot2
Fr_21
Fr_12
Coupling Radiation Damping Force
Fe1
Fb1
Fpto
Fr_21
x1
xdot1
xdotdot1
Buoy Dynamics
Fb1
Buoy Bouyancy
4Fm
3Fpto
2Fe2
1Fe1
Simulink WEC Dynamics Model: PA WEC Subsystem
PA WEC Subsystem
Inputs
PA WEC Subsystem
Outputs
Buoy Radiation Impulse Response
Function
Plate Radiation Impulse Response
Function
Coupling Radiation Impulse Response
Function
Simulink WEC Dynamics Model: Buoy Dynamics Subsystem
xdotFr_11
Hydrostatic Stiffness
3xdotdot12
xdot11x1
k1
Spring Constant
Rate Transition1 Rate Transition
1s
Integrator1
1s
Integrator
Hydrostatic Force
rho*g*A1
num(z)
1
Discrete FIR Fil ter1
b1
Damping
BuoyRadiation Force
BuoyDisplacement
1/(m1+A_11)
1/mass
4Fr_12
3
Fpto
2Fb1
1Fe1
xdot
xdotxddotx
Buoy Radiation Impulse Response
Function
Buoy Radiation Force Calculation
Matlab/Simulink Model Demonstration
Overview
• Ran AQWA Simulation for Eidsmoen WEC• Max mesh = 3m is sufficient for IRF calculations
• Calculated IRFs similar in Magnitude and Time-scale to those in Eidsmoen (1996)• Don’t match 100%, which is okay because Eidsmoen’s response is based on modeling buoy and plate
with same diameter and combining the response, AQWA model doesn’t make this assumption• Have not found a way to get coupled radiation terms from AQWA, so they are approximated by
negative average of buoy/plate response in frequency-domain
• Created a Matlab script that defines η(t)• Regular Waves: • Irregular Waves:
• Import time-series directly from NDBC/CDIP data buoy data• Create time-series based on PM Spectrum by defining Tp and Hs
• Create time-series from NDBC data buoy spectra• Create time-series from a location’s representative spectra
Conclusions
Established a 2-Body PA WEC Modeling Methodology:
1. Create WEC Geometry File2. Run AQWA Frequency-Domain Simulation for WEC3. Calculate IRFs via Excitation and Radiation IRF Matlab functions4. Define WEC Properties in Matlab/Simulink WEC Model5. Run Matlab/Simulink WEC Model
Future Work
• Model Validation:• OSU’s L-10 sea trial data• NREL’s StarCCM OPT-like WEC simulation
• Refine mooring system coefficients and plate drag coefficient, current values:• Mooring Stiffness, Km = 8083.44 [N/m] based on
Loukogeorgaki (2005)• Mooring Damping, Bm = 400 [Ns/m] based on
Fitzgerland (2008) Fig 9b• Plate Drag Coeff, Cd = 15 [] based on Vengatesan (2000)
Thank You
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Simulink Model Output for Irregular Waves using AQWA IRFs
0 20 40 60 80 100 120 140 160-1.5
-1
-0.5
0
0.5
1
1.5Wave Surface Elevation
Time [s]
[m
]
0 20 40 60 80 100 120 140 160-1.5
-1
-0.5
0
0.5
1
1.5
time [s]
disp
lace
men
t [m
]
buoyplate
0 20 40 60 80 100 120 140 160-1
-0.5
0
0.5
1
time [s]
rel.
velo
city
[m/s
]
0 20 40 60 80 100 120 140 160-6
-4
-2
0
2
4
6
8x 10
5 Excitation Force
time [s]
forc
e [N
]
buoyplate
0 20 40 60 80 100 120 140 160-150
-100
-50
0
50
100
150Radiation Force
time [s]
forc
e [N
]
buoyplate
0 20 40 60 80 100 120 140 160-3
-2
-1
0
1
2
3x 10
5 Coupled Radiation Force
time [s]
forc
e [N
]
on plate by buoyon buoy by plate
0 20 40 60 80 100 120 140 160-1
0
1
2x 10
6 Hydrostatic Force
time [s]
forc
e [N
]
0 20 40 60 80 100 120 140 160-2
-1
0
1x 10
5 Drag Force
time [s]
forc
e [N
]
0 20 40 60 80 100 120 140 160-2000
0
2000
4000Mooring Force
time [s]
forc
e [N
]
Ongoing Research (not presented)
• Single Body WEC Dynamics Model• Created Matlab/Simulink Model
• Diameter = 11m, Draft = 10.34m• Modeled a Heaving Cylinder in AQWA
• Assessed the influence of mesh sizing: max = 3m, vs. max = 0.5 m• Refined mesh didn’t yield better results, just took longer
to solve, stick with max = 3m• Calculated IRFs for Heaving Cylinder and compared
them to Falnes(2002) pg. 142• Ran simulations in Matlab/Simulink for various
conditions
-25 -20 -15 -10 -5 0 5 10 15 20 25-0.5
0
0.5
1
1.5
2
2.5x 10
5 Excitation IRF
time
exci
tatio
n fo
rce
kern
el
AQWA max mesh = 3mAQWA max mesh = 0.5mFalnes
0 5 10 15 20 25-1
-0.5
0
0.5
1
1.5
2x 10
4 Radiation IRF
time
radi
atio
n fo
rce
kern
el
AQWA max mesh = 3mAQWA max mesh = 0.5mFalnes
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