2nd sdwed symposium advances in modelling of wave energy ... · integration of non-linear forces in...

Motivation Mathematical model Structural design Outlook

2nd SDWED Symposium

Advances in Modelling of

Wave Energy Devices - WP4

Andrew Zurkinden

Wave Energy Research Group,

Civil Engineering Department,

Aalborg University, Denmark

April 26, 2012

2nd

SDWED Symposium at DTU Lyngby Aalborg University, Denmark


Structural Design of Wave Energy Devices

2nd



Outline

Motivation

Mathematical model

Structural design

Outlook

2nd



Motivation I

◮ Cost of energy from renewable resources in general, andespecially from waves, is still too high compared with otherforms of power generation.

◮ WEC’s are affected by demands for structural efficiency, fastproduction time and cost effectiveness.

◮ The overall safety requirement of a WEC can be lowered dueto the relatively high acceptance in losses. It is unlikely that asystem failure will lead to unacceptable consequences such asloss of life or uncontrolled outflow of oil and gas.

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Problems

◮ Numerical models are needed to accurately predict themotions of a floating wave energy converter, in order to i)estimate the structural response of the system ii) calculate theannual energy output.

◮ Advanced control strategies are needed in order to optimizethe power absorbtion.

◮ A general characteristic of numerical models is the assumptionof linear-fluid structure interaction. Thus it is important toknow within what amplitude of the waves the system has alinear response? - experiments can be very revealing.

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Mathematical model

◮ Wave loads and responses are calculated based on linearpotential flow.

◮ Time domain models are used to perform structural analysis.

◮ Case study on a point absorber, Wavestar laboratory model:

(M44 + a∞44)φ4(t) +

∫ t

0

K44(t − τ)φ4(τ)dτ + C44φ4(t) +Mc(t)

=

∫∞

−∞

heτ (t − τ)η(t)dτ (1)

Mc(t) = mc φ4(t) + cc φ4(t) + kcφ4(t) +

∫∞

−∞

hcφ4

(t − τ)φ4(τ) dτ

(2)

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Non-linear hydrostatic restoring force

◮ Experimental tests on a hemisphere have been carried out atAAU to approximate the non-linear hydrostatic behavior.

◮ Trilinear piecewise approximation of the non-linear hydrostaticbehavior is introduced in the numerical model.

-0.3 -0.2 -0.1 0 0.1 0.2 0.3-20

-10

0

10

20

R44

[Nm

]

φ4

[rad]

Hydrostatic experiments

Piecewiese linear approx.

A

B

C

H

D

R44H

D

H

A B

C

D

R44

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Approximation of the hydrodynamic parameters

◮ Marinecontrol toolbox, Matlab control toolbox.◮ 4th order approximation is accurate enough.

Hr φ4

(s) = B(ω) + iω(A(ω)− A∞) (3)

Hr φ4

(s) ≈P(s)

Q(s)=

p0sm + p1s

m−1 + ...+ pm−1s + pm

sn + q1sn−1 + ...+ qn−1s + qn(4)

0 5 10 15 20 25

0.4

0.5

0.6

0.7

0.8

0.9

1

ω [rad/s]

Imag

(Hrφ

4)

Approx. 3th orderApprox. 4th orderApprox. 5th orderFD calculation

0 5 10 15 20 250

0.5

1

1.5

2

2.5

3

ω [rad/s]

Re(

Hrφ

4)

Approx. 3th orderApprox. 4th orderApprox. 5th orderFD calculation

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Replacement of the convolution integral

∫ t

0

K44(t − τ)φ4(τ)dτ ≈[p0 p1 ... pn−1

]I (t) (5)

◮ ∆ǫ =φ− φFD

φFD

· 100

◮ ∆ǫ = 0.1− 0.6%

16 17 18 19 20 21 22 23 24 25

−0.1

−0.05

0

0.05

0.1

t [s]

φ 4(t)

[rad

]

Linear modelNonlinear modelExperiments

15 20 25−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

t [s]

φ 4(t)

[rad

]


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Interfaces with WP1, WP2, WP3, WP5

◮ The equation of motion is rewritten in a ODE form and solvedwith the ode-package in Matlab.

◮ Integration of non-linear forces in EQM is possible

◮ Analysis of multiple degree of freedom systems

X

X

Ii (t)

= (6)

X

(M + a∞)−1· [∑

Ii (t)− KHX + FPTO + FM + FH + Fo ]

Ii (t)

(7)

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Validation with experimental tests

◮ Eigenperiod of the laboratory model: Tn = 0.79sec

Wave states Hm0 Tp PwaveHm0

λpν

[m] s] [W /m] [−] [-]

IRA1 0.027 0.8 0.374 0.02 0.53IRA2 0.044 1.25 1.174 0.02 0.37IRA3 0.062 1.3 3.137 0.02 0.20IRA4 0.080 2.5 6.024 0.02 0.19IRA5 0.12 3.0 14.42 0.02 0.13

IRB1 0.055 0.85 1.520 0.04 0.52IRB2 0.090 1.30 5.287 0.04 0.32IRB3 0.115 1.35 10.83 0.04 0.23IRB4 0.155 2.5 22.89 0.04 0.17IRB5 0.232 3.0 51.36 0.04 0.12

2nd



◮ IRA1 and IRA2:

15 20 25−0.04

−0.03

−0.02

−0.01

0

0.01

0.02

0.03

0.04

t [s]

φ 4(t)

[rad

]


15 20 25−0.08

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

0.08

t [s]

φ 4(t)

[rad

]


◮ IRA3 and IRA5:

15 20 25−0.08

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

0.08

0.1

t [s]

φ 4(t)

[rad

]


15 20 25−0.1

−0.05

0

0.05

0.1

0.15

t [s]

φ 4(t)

[rad

]


2nd



◮ IRB1 and IRB2:

15 20 25−0.08

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

0.08

t [s]

φ 4(t)

[rad

]


15 20 25−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

t [s]

φ 4(t)

[rad

]


◮ IRB4 and IRB5:

20 22 24 26 28 30−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

t [s]

φ 4(t)

[rad

]


15 20 25−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25

t [s]

φ 4(t)

[rad

]


2nd



◮ Correlation factors:

0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.60.82

0.84

0.86

0.88

0.9

0.92

0.94

0.96

0.98

1

Tp

Exx

Linear modelNon−linear model

0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.60.965

0.97

0.975

0.98

0.985

0.99

0.995

Tp

Exx

Linear modelNon−linear model

◮ Non dimensional performance index vs.Tp:

1 1.5 2 2.50.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

Tp

ν

ν Experimentsν Linear numerical model

1 1.5 2 2.5 30.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

Tp

ν

ν Experimentsν Linear numerical model

2nd



Conclusion of the mathematical model

◮ A mathematical model based on linear-fluid structureinteraction has been set up and validated by experimentaltests on a point absorber wave energy converter.

◮ Good agreement between numerical and experimental testswas found for mild, regular and irregular wavesHm0/λp ≤ 0.03.

◮ For steep waves, as well as overtopping waves, where the wavesteepness is larger than Hm0/λp ≥ 0.06. the linear modelseems to underestimate the response and thus underestimatethe power absorbtion of the device.

◮ Non-liner effects such as the non-linear hydrostatic restoringforce become important and should be included in thenumerical model.

2nd



References & Acknowledgment

◮ Perez, T. and Fossen, T. I. A Matlab Tool for parametricidentification of radiation-force models of ships and offshorestructures. Modeling, Identification and Control,MIC-30(1):1-15. 2009.

◮ Cummins, W.E. The impulse response function and shipmotions. Schiffstechnik 9:101-109, 1962.

◮ Yu, Z. and Falnes, J. State-space modeling of a verticalcylinder in heave. Applied Ocean Research. 17:265-275, 1995.

◮ Taghipour, R. Efficient Prediciton of Dynamic Response forflexible and Multi-Body Marine Structures. PhD Thesis, 2008.

◮ Morten Kramer and Francesco Ferri for providing theexperimental data.

2nd



Structural design

◮ Floating offshore structures are subjected to high fatigueloading, stress response from wave action shows typically 5mioloading cycles a year.

◮ For the majority of wave energy devices the structural analysiscan be decoupled from the hydrodynamic calculation.

◮ The objective of my work is to produce a model which can beused to estimate the fatigue damage of a component byincorporating the unique features of a WEC (PTO).

2nd



Example: Long-term fatigue damage of a hydraulic

cylinder subjected to internal fluid pressure induced by

wave loads, Limin Yang, NTNU, 2011

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Structural design

◮ Stresses on the structure are obtained numerically for eachshort term condition by solving the mathematical dynamicalmodel.

◮ Rainflow cycle counting will be applied to each obtained timeseries. Long-term distribution of stresses for each short-termcondition are obtained by considering the probability ofoccurrence of each wave state.

◮ Long-term fatigue analysis is performed based on theMiner-Palmgren damage accumulation rule.

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Structural analysis of the Wavestar prototype

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Outlook

Start,2011

Goal, 2014

Today

Thank youfor yourAttention:)

2nd



Acknowledgements

◮ Danish Council for Strategic Research

2nd


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