simulation in wind turbine vibrations: a data driven analysis

Simulation in Wind Turbine Vibrations:A Data Driven Analysis

Graduate Students: Zijun ZhangPI: Andrew Kusiak

Intelligent Systems LaboratoryThe University of Iowa

OutlineModeling wind turbine vibrationsMulti-objective optimization

modelEvolutionary strategy algorithmSimulation results and discussion

Modeling wind turbine vibrations

Parameter description Parameter Description

y1(t) Average Drive Train Acceleration at time t

y2(t) Tower Acceleration at time t

y3(t) Generated Power at time t

y1(t-1) Average Drive Train Acceleration at time t-1

y2(t-1) Tower Acceleration at time t-1

x1(t) Generator Torque at time t

x1(t-1) Generator Torque at time t-1

x2(t) Average Blade Pitch Angle at time t

x2(t-1) Average Blade Pitch Angle at time t-1

v1(t) Wind Speed at time t

v1(t-1) Wind Speed at time t-1

Models of wind turbine vibrations

Wind turbine vibration models:

1 1 1 1 1 1 1 2 2( ) ( ( 1), ( ), ( 1), ( ), ( 1), ( ), ( 1))y t f y t v t v t x t x t x t x t

Data-derived model to predict drive train acceleration

Data-derived model to predict tower acceleration

2 2 2 1 1 1 1 2 2( ) ( ( 1), ( ), ( 1), ( ), ( 1), ( ), ( 1))y t f y t v t v t x t x t x t x t

Models of wind turbine vibrations

Parametric model of power output:

2 31 ( , )2 pP R C v

Data-derived model of power output:

3 3 1 1 1 1 2 2( ) ( ( ), ( 1), ( ), ( 1), ( ), ( 1))y t f v t v t x t x t x t x t

Power Curve

Power curve of a 1.5 MW turbine

0

200

400

600

800

1000

1200

1400

1600

0 5 10 15 20

Pow

er (k

W)

Wind speed (m/s)

Sample datasets10-s dataset

1-min dataset

Time Torque value Torque Value(T-1) Wind Speed Wind Speed(T-

1) ……. Drive Train Acc Drive Train Acc(T-1)

19/10/08 3:01:10 PM 42.1586 36.6630 9.0259 8.2568 ……. 63.2651 61.5034

19/10/08 3:01:20 PM 45.5093 42.1586 8.9973 9.0259 ……. 59.9151 63.2651

……. ……. ……. ……. ……. ……. ……. …….

Time Torque value Torque Value(T-1) Wind Speed Wind Speed(T-

1) ……. Drive Train Acc Drive Train Acc(T-1)

10/19/08 3:01 PM 40.7994 38.9933 8.3853 8.0945 ……. 59.7646 59.5475

10/19/08 3:02 PM 36.9941 38.0203 8.1524 8.1375 ……. 54.8406 56.1781

……. ……. ……. ……. ……. ……. ……. …….

Validation of data-driven modelsFour metrics to assess the performance of data driven models:

Mean absolute error:Standard deviation of the mean absolute error:

1

1 ˆ| |n

i ii

y yn

2

1 1

1 1 ˆ ˆ( | | | |)n n

i i i ii i

y y y yn n

Mean absolute percentage error:

Standard deviation of Mean absolute percentage error:

1

ˆ1 (| | 100%)n

i i

i i

y yn y

2

1 1

ˆ ˆ1 1( (| | 100%) | | 100%)n n

i i i i

i i i i

y y y yn n y y

Validation of data-driven modelsin 10-s dataset

Test results of the NN models for 10-s data

Predicted Parameter MAE Std of MAE MAPE Std of MAPE

Drive train acceleration 1.27 1.27 0.02 0.03

Tower acceleration 4.73 8.92 0.06 0.10

Generated power 9.86 9.86 0.03 0.08

Validation of data-driven models in 10-s dataset

The first 50 test points of the drive train acceleration for 10-s data

115120125130135140145150155160

1 3 5 7 9 11 13151719212325272931333537394143454749

Driv

e tra

in a

ccel

erat

ion

Time (10-s interval)

Observed value Predicted value


The first 50 test points of the tower accelerations for 10-s data

100110120130140150160170180190200

1 3 5 7 9 11 1315 171921 2325 272931 333537394143454749

Tow

er a

ccel

erat

ion




The first 50 test points of the power output for 10-s data

1460146514701475148014851490149515001505

1 3 5 7 9 1113151719212325272931333537394143454749

Gene

rate

d po

wer



Validation of data-driven modelsin 1-min dataset

Test results of the NN models for 1-min data

Predicted Parameter MAE Std of MAE MAPE Std of MAPE

Drive train acceleration 0.77 1.58 0.01 0.01

Tower acceleration 2.76 7.97 0.03 0.04

Generated power 8.99 13.83 0.03 0.15


The first 50 test points of the drive train accelerations for 1-min data

20

25

30

35

40

45

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49

Driv

e tra

in a

ccel

erat

ion

Time (1 minute interval)



The first 50 test points of the tower acceleration for 1-min data

25

30

35

40

45

50

55

60

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49

Tow

er a

ccel

erat

ion



Validation of data driven modelsin 1-min dataset

The first 50 test points of the power output 1-min data

0

50

100

150

200

250

300

1 3 5 7 9 11 13151719212325272931333537394143454749

Gene

rate

d po

wer



Multi-objective optimization model

Multi-objective optimization

1 2 3

1 1 1 1 1 1 1 2 2

2 2 2 1 1 1 1 2 2

3 3 1 1 1 1 2 2

min( , , )

( ) ( ( 1), ( ), ( 1), ( ), ( 1), ( ), ( 1))

( ) ( ( 1), ( ), ( 1), ( ), ( 1), ( ), ( 1))

( ) ( ( ), ( 1), ( ), ( 1), ( ), ( 1))

m

Obj Obj Objsubject to

y t f y t v t v t x t x t x t x t

y t f y t v t v t x t x t x t x t

y t f v t v t x t x t x t x t

1

2

ax{0,currentSettings 50} ( ) min{100,currentSettings 50}max{ 5,currentSettings 5} ( ) min{15,currentSettings 5}

x tx t

Evolutionary strategy algorithm

Strength Pareto Evolutionary Algorithm

1. Initialize three sets, parent set (Sp ), offspring set ( So) and elite set (Se ). Generate u individuals (solutions) randomly to conduct the first generation of population.2. Repeat until the stopping criteria (number of generation, N) is satisfied 2.1. Search the best non-dominated solutions in So. Copy all non-dominated solutions to Se. 2.2. Search and delete all dominated solutions in Se. 2.3. A clustering technique is applied to reduce size of Se if the size of Se is too large. 2.4. Assign fitness to solutions in Se and So. 2.5. Apply a binary tournament selection to select u parents from the SoUSe to form the population of parents and this population is stored in Sp. 2.6. Recombine two parents from Sp to generate a new population. 2.7. Mutate individuals in So by the mutation operator and assign fitness values to them.3. Check number of generation, if it is equal to N , then stop.

Strength Pareto Evolutionary AlgorithmRecombination of parents in

SPEA

( , )2 2p p

i ij j

j S j S

x

Mutation operator

1 2(0, ) (0, ) (0, ) (0, )[ , ]N N N Ni i e e

(0, )i i ix x N

Tuning parameters of SPEA

Experiment No. Description

1Select an instance from the 10-s data set to tune the selection pressure and population size of the ES

algorithmthat will be implemented in the model extracted from 10-s data set

2Select an instance from the 1-min data set to tune the selection pressure and population size of the ES

algorithmthat will be implemented in the model extracted from 1-min data set

One instance selected from the 10-s data set for experiment 1

Time TV TV(t-1) WS WS(t-1) Power TA TA(t-1) BPA BPA(t-1) DTA DTA(t-1)

10/18/08

10:55:10 PM100.93 100.06 12.32 14.11 1484.47 164.64 167.20 6.77 8.21 147.43 139.09

One instance selected from the 1-min data set for experiment 2

Time TV TV(t-1) WS WS(t-1) Power TA TA(t-1) BPA BPA(t-1) DTA DTA(t-1)

10/18/08

10:55 PM100.43 100.58 14.42 14.96 1481.49 169.72 170.22 10.68 11.41 142.68 144.27

Two experiments for tuning selection pressure and population size

Tuning parameters of SPEAConvergence for 10 values of the selection pressure in experiment 1

Combinations of selection pressure

Converge speed ofaverage drive train

acceleration(generations)

Converge speed ofinverse of power output

(generations)

Converge speed oftower acceleration

(generations)

Averageconverge speed(generations)

Ratio1 (2parents/2offsprings) 968 163 1420 850.3333










Tuning parameters of SPEA

Convergence for 10 values of the selection pressure in experiment 2

Combinations of selection pressure



Converge speed ofinverse of power output

(generations)


(generations)












Tuning parameters of SPEAConvergence of the ES algorithm for two populations of experiment 1Population Sizes



Converge speed ofinverse of power

output(generations)


(generations)


PS1(2parents/18offsprings) 106 41 180 109.0000


Convergence of the ES algorithm for two populations of experiment 2

Population Size



Converge speed ofinverse of power

output(generations)


(generations)




Simulation results and discussion

Simulation Results of Single Point Optimization

Partial solution set generated by the evolutionary strategy algorithmSolution

No.Solution

(TV, BPA)Drive Train Acceleration

Gain in Drive Train

Acceleration

Tower Acceleration

Gain in Tower

AccelerationPower

Gain in Power

1 (90.0, 8.81) 136.86 7.17% 160.61 2.45% 1460.96 -1.58%2 (90.0, 7.34) 136.85 7.18% 164.47 0.10% 1460.80 -1.59%3 (63.9, 15.00) 136.96 7.10% 119.42 27.47% 1007.14 -32.16%4 (67.6, 15.00) 136.71 7.27% 120.34 26.90% 1031.21 -30.53%5 (50.9, -3.23) 122.57 16.86% 356.37 -116.45% 785.72 -47.07%6 (90.0, 8.09) 136.88 7.15% 162.46 1.33% 1462.77 -1.46%7 (63.4, 15.00) 136.98 7.09% 119.41 27.48% 1005.11 -32.29%

Simulation Results of Single Point Optimization

Solution of the elite set in a 3-dimensional space

120

125

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100150

200250

300350

400700

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1300

1400

1500

Average drive train accelerationTower acceleration

Pow

er o

utpu

t

Multi-points Optimization Simulation Results

Gains in vibration reductions of the drive train for Case 1 (10/19/2008 2:43:00 AM - 10/19/2008 2:54:00 AM)

Case 1 (Minimize average drive train

acceleration)Minimum value (mean) Original value (mean) Gain (mean)

Average drive train acceleration 119.61 131.67 9.16%

Simulation ResultsThe optimized and original drive train acceleration of Case 1 for

10-s data (10/19/2008 2:43:00 AM - 10/19/2008 2:54:00 AM)

110

115

120

125

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135

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145

1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46 49 52 55 58 61 64

Driv

e tra

in a

ccel

erat

ion


Optimized value Original value

Simulation ResultsThe computed and original torque value of Case 1 for 10-s data

(10/19/2008 2:43:00 AM - 10/19/2008 2:54:00 AM)

2030405060708090

100110

1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46 49 52 55 58 61 64

Torq

ue v

alue


Computed value Original value

Simulation ResultsThe computed and original average blade pitch angle of Case 1

for 10-s data(10/19/2008 2:43:00 AM - 10/19/2008 2:54:00 AM)

-6

-4

-2

0

2

4

6

8

10

12

1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46 49 52 55 58 61 64

Blad

e pi

tch

angl

e


Computed value Original value

Simulation ResultsComparison of computational results for 10-s data set and 1-min

data set over 10 min horizon

Mean ValueMinimize Drive Train

AccelerationOptimized Drive Train

AccelerationOriginal Drive Train

Acceleration Gain

10-s data set 119.53 131.49 9.10%1-min data set 124.06 131.79 5.87%

Minimize Tower Acceleration

Optimized Tower Acceleration Acceleration Gain


Maximize Power Output Optimized Power Output Original Power Output Gain


Thank You !

simulation in wind turbine vibrations: a data driven analysis

Documents

s data validation of

s datasetthe

drive train accelerations

time ty2ttower acceleration

time tx2t

time ty1t

time tx1t

time tv1t