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Reduction of Wind Turbine Blade Model

M. Khorsand Vakilzadeh

Department of Applied Mechanics,

Chalmers University of Technology,

Gothenburg, Sweden.

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Outline of the ThesisModel validation of wind turbine rotor blade will be considered in two levels:

1- A detailed structural dynamics model which has good correlation with

experimental of wind turbine testing.

A FEM model motivated by its connection to the observed physical

phenomena, such as wing twisting, nonlinear effects, spatial ortemporal load variations.

2- A low-order model needs to be validated by a good model-to-model

correlation.

It gives a correct representation of thestimuli-to-response

characteristics of the system in an efficient

simulation environment.

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Outline of the ThesisHigh-order blade model basedon 5-MW reference wind

turbine defined by NationalRenewable Energy Laboratory.

Apply developed modalreduction method based on

input-output relationpreservation

Low-order model

Low-order beam model basedon 5-MW reference windturbine defined by National

Renewable Energy Laboratory.

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High-order model of NREL bladeA master thesis has been defined to build a high-order blade model based on 5-MW reference windturbine defined by National Renewable Energy

Laboratory.

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High-order model of NREL blade

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An Improved Modal Approach For Model Reduction Based On Input-output Relation-Presented in ISMA2012 and would be submitted as journal paper

Structural dynamicsmodel

Model reduction Modal truncation

Complex structures

Large FE models

Time consuming

and computationally

intensive

Simple low-order

models

Keep important

features

Approximation

error

A tradeoff between

accuracy and

simplicity

Can handle very

large models with

lightly damped or

unstable modes

Keep the dominant

eigenvalue subset ofthe full system

Lack of a general

modal dominancy

analysis

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An Improved Modal Approach For Model Reduction Based On Input-output Relation-Presented in ISMA2012 and would be submitted as journal paperAn improved modal based model reduction containing dominant eigensolutions of thefull order model

Dominancy metric: the squared norm contribution of each eigensolution to the system

output.

Numerical example: FE based model of an aluminum plate

Advantages

Computationally efficient

Detection and elimination of unobservable and uncontrollable modes

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Modal TruncationConsider a continuous, linear, time invariant, asymptotically stable dynamic system

wherex(t) , u(t) , y(t) .

Lets assume that system is diagonalizable as

( ) ( ) ( )

( ) ( ) ( )

x t Ax t Bu t

y t Cx t Du t

( ) ( ) ( )

( ) ( ) ( )

z t z t Bu t

y t Cz t Du t

1

1 2

1

1 2

1 2

( , ,..., ) ,

,

.

n

n

n

diag A

b

b

B B

b

C C c c c

xn un yn

where

A

( ) ( )x t z t

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Modal Truncation( ) ( ) ( )

( ) ( ) ( )

z t z t Bu t

y t Cz t Du t

1 1 1 1

2 2 2 2

1

1 2

2

00

z z B uz z B

zy C C Du

z

1 1 1 ( , , , )r B C D

Truncated system

Properties

.

H-norm of error system is upper bounded

1

1

( )

( ) ( ) sup ( ( ) ( ))Re( )

n

i i

r rs i i k i

c bG s G s G s G s

where

z1 k

Challenge

What are the dominant eigenvalues!?!

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Modal Dominancy Analysis

i i i i

i i i

z z b u

y c z

1 ( )i i i iY c s bU

0( )Hi i iM y y dt

0( ( ) ( ))Hi i iM Y Y d

0

1 1

H H

i i i i i

ii

M b c c b djj

( )

1

arctan( )Re( ) Re( )

i

H H

i i i i i

i i Im

M b c c b

( ) ( ) ( )( ) ( ) ( )

z t z t Bu t

y t Cz t Du t

Define dominancy metric as

Extract ith modal

coordinate

Laplace

transform

Parsevals

Theorem u(t) is a zero mean white noise withunit intensity, U(s)=1

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Modal Dominancy Analysis

( )

1

arctan( )Re( ) Re( )

i

H H

i i i i i

i i Im

M b c c b

Properties

Computationally efficient,

Able to detect the unobservable and uncontrollable modes,

Applicable for both real and complex eigenvalues,

Independent from the other eigensolutions.

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Improved Modal Truncation Algorithm

Given the stable system (A, B, C, D).

Solve eigenvalue problem ( ) for A.

Transfer the system to modal decomposed form.

Find multiple eigenvalues with dimension of multiplicity larger than the dimension of the

loading, nu.

Do QR decomposition of to make the eigenvectors found in step 4 orthogonal to load.

Compute the metric correspond to each modal coordinate.

Rearrange the modal coordinates as they appear in order:

Set the limit for error resulted from truncation and compute the low-order model order.

Do modal truncation along with residualization.

,

B

1 2 ... nM M M

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Aluminum plate

Dimension: 500 500 1mm

Density: 2700kg/m3

Youngs modulus: 70 GPa

Poissons ratio: 0.33

4-noded shell elements

16 16 elements mesh

Rayleigh viscous damping of

Numerical Example

3 610 , 10

V M K

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Results

-7 -6 -5 -4 -3 -2 -1 0

x 104

0

0.5

1

1.5

2

2.5

3

3.5

4x 10

5

radian/seconds

radian/seconds

N=3468 For the full state space model

Eigenvalue spectrum of full model; eigenvalues with positive imaginary part

N=1710 Guyan reduction is applied to remove massless rotational DOFs

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-800 -600 -400 -200 00

1

2

3

4x 10

4

radian/seconds

radian/seconds

Bal. Trunc.,k=288Prop. Mod. Trunc.,k=288

-800 -600 -400 -200 00

1

2

3

4x 10

4

radian/seconds

radian/seconds

Bal. Trunc., k=250Prop. Mod. Trunc., k=250

-800 -600 -400 -200 00

1

2

3

4x 10

4

radian/seconds

radian/seco

nds

Bal. Trunc., k=200

Prop. Mod. Trunc., k=200

-1200 -1000 -800 -600 -400 -200 00

1

2

3

4x 10

4

radian/seconds

radian/seco

nds

Bal. Trunc., k=150

Prop. Mod. Trunc., k=150

Eigenvalue spectrum of low-order models of plate;

eigenvalues with positive imaginary part

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Numerical statistics for reduced-order models with proposed modal truncation and balanced truncation using unit-impulse input signal; The Frobenius norm of the full system

output is 29.34 and the H-norm of the full system is 28.42.

Proposed Modal Truncation Method Balanced TruncationReduced-Order

ModelRHN of the error

system (%)RFN of the error

system (%)Reduction time

(s)RHN of the error

system (%)RFN of the error

system (%)Reduction time

(s)k = 100 3.6e-4 0.74 9.36 4.7e-5 0.28 68.07k = 50 8.9e-4 1.83 9.53 1.7e-4 1.33 67.99k = 20 1.4e-2 4.66 9.19 2.6e-3 4.96 68.12

Numerical Statistics - Comparison with Balanced TruncationRelative Frobenius norm of error

Time domain analysis

Relative H-norm of error system -

Frequency domain analysis

2

1 1

2

1 1

y

y

n m

ij

r i jF F

n m

F Fij

i j

ee y y

y yy

rG G

G

CHALMERS S C

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Displacement as output, nr=20 Acceleration as output, nr=150

MethodRHN of the error

system (%)

RFN of the error

system (%)

RHN of the error

system (%)

RFN of the error

system (%)

Davison 0.19 16.01 3.08 96.40

Rommes 0.01 5.35 0.88 43.84

Aguirre 0.015 6.79 46.80 31.68

Proposed method 0.02 4.66 16.52 21.90

Numerical Statistics Comparison with other dominancy definitionsRelative Frobenius norm of error

Time domain analysis

Relative H-norm of error system -

Frequency domain analysis

2

1 1

2

1 1

y

y

n m

ij

r i jF F

n m

F Fij

i j

ee y y

y yy

rG G

G

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Conclusion

A new dominancy index is suggested to measure and rank the contribution of modes intoinput-output relation:

- Squared norm of output deviation resulted from deflation of eigensolutions from

the system

The explicit formulation resulted from decomposed metric decreases computation time

needed for model reduction.

Comparison to balanced truncation showed that:

For a given model order, the balanced truncation is superior in approximation accuracy in

frequency domain analysis while the proposed modal truncation is superior in time domain

analysis.

Obtained low-order model includes a subset of the eigenvalues of the full model, same

physical interpretation, while balanced truncation alters the system eigenvalues.

The time required for reduction is decreased due to the explicit form of dominancy index as

compared with balancing reduction techniques.

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Conclusion

Comparison to other dominancy definitions showed that: In a time domain analysis the proposed modal reduction technique yields less

approximation error.

This method can be efficiently applied to very large-scale dynamics systems

while the application of the balanced truncation is restricted for very large-scale

systems inasmuch as the solution of the Lyapunov equation is required in this

method.

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Future Works

Now it is time to apply the described input-output based modal reduction methodon the high-order model of NREL blade to obtain the reduced-order model.

On the other hand, the state-of-the-art beam element based low-order model

would be developed to be compared with the reduced-order model in regard to

their input-output relation preservation.

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THANKS FOR YOUR ATTENTION