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