Simulation in Wind Turbine Vibrations: A Data Driven Analysis
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Simulation in Wind Turbine Vibrations:A Data Driven Analysis
Graduate Students: Zijun ZhangPI: Andrew Kusiak
Intelligent Systems LaboratoryThe University of Iowa
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OutlineModeling wind turbine vibrationsMulti-objective optimization
modelEvolutionary strategy algorithmSimulation results and discussion
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Modeling wind turbine vibrations
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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
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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
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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
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Power Curve
Power curve of a 1.5 MW turbine
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200
400
600
800
1000
1200
1400
1600
0 5 10 15 20
Pow
er (k
W)
Wind speed (m/s)
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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
……. ……. ……. ……. ……. ……. ……. …….
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Validation of data-driven modelsFour metrics to assess the performance of data driven models:
Mean absolute error:Standard deviation of the mean absolute error:
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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:
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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
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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
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Validation of data-driven models in 10-s dataset
The first 50 test points of the drive train acceleration for 10-s data
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Driv
e tra
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ccel
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Time (10-s interval)
Observed value Predicted value
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Validation of data-driven modelsin 10-s dataset
The first 50 test points of the tower accelerations for 10-s data
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1 3 5 7 9 11 1315 171921 2325 272931 333537394143454749
Tow
er a
ccel
erat
ion
Time (10-s interval)
Observed value Predicted value
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Validation of data-driven modelsin 10-s dataset
The first 50 test points of the power output for 10-s data
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Gene
rate
d po
wer
Time (10-s interval)
Observed value Predicted value
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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
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Validation of data-driven modelsin 1-min dataset
The first 50 test points of the drive train accelerations for 1-min data
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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
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Time (1 minute interval)
Observed value Predicted value
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Validation of data-driven modelsin 1-min dataset
The first 50 test points of the tower acceleration for 1-min data
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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
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Time (1 minute interval)
Observed value Predicted value
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Validation of data driven modelsin 1-min dataset
The first 50 test points of the power output 1-min data
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1 3 5 7 9 11 13151719212325272931333537394143454749
Gene
rate
d po
wer
Time (1 minute interval)
Observed value Predicted value
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Multi-objective optimization model
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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
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Evolutionary strategy algorithm
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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.
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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
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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
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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
Ratio2 (2parents/4offsprings) 637 478 1170 761.6667
Ratio3 (2parents/6offsprings) 97 96 974 389.0000
Ratio4 (2parents/8offsprings) 97 96 974 389.0000
Ratio5 (2parents/10offsprings) 134 59 419 204.0000
Ratio6 (2parents/12offsprings) 108 60 736 301.3333
Ratio7 (2parents/14offsprings) 110 35 277 140.6667
Ratio8 (2parents/16offsprings) 87 47 214 116.0000
Ratio9 (2parents/18offsprings) 106 41 180 109.0000
Ratio10 (2parents/20offsprings) 171 15 306 164.0000
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Tuning parameters of SPEA
Convergence for 10 values of the selection pressure in experiment 2
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) 190 253 190 211.0000
Ratio2 (2parents/4offsprings) 466 31 466 321.0000
Ratio3 (2parents/6offsprings) 258 178 258 231.3333
Ratio4 (2parents/8offsprings) 35 80 35 50.0000
Ratio5 (2parents/10offsprings) 99 52 99 83.3333
Ratio6 (2parents/12offsprings) 292 48 292 210.6667
Ratio7 (2parents/14offsprings) 149 28 149 108.6667
Ratio8 (2parents/16offsprings) 83 41 83 69.0000
Ratio9 (2parents/18offsprings) 15 118 15 49.3333
Ratio10 (2parents/20offsprings) 36 55 36 42.3333
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Tuning parameters of SPEAConvergence of the ES algorithm for two populations of experiment 1Population Sizes
Converge speed ofaverage drive train
acceleration(generations)
Converge speed ofinverse of power
output(generations)
Converge speed oftower acceleration
(generations)
Averageconverge speed(generations)
PS1(2parents/18offsprings) 106 41 180 109.0000
PS2(10parents/90offsprings) 18 1 51 23.3333
Convergence of the ES algorithm for two populations of experiment 2
Population Size
Converge speed ofaverage drive train
acceleration(generations)
Converge speed ofinverse of power
output(generations)
Converge speed oftower acceleration
(generations)
Averageconverge speed(generations)
PS1(2parents/20offsprings) 36 55 36 42.3333
PS2(10parents/100offsprings) 49 137 49 78.3333
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Simulation results and discussion
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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%
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Simulation Results of Single Point Optimization
Solution of the elite set in a 3-dimensional space
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Pow
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utpu
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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%
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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)
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Driv
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Time (10-s interval)
Optimized value Original value
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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)
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Torq
ue v
alue
Time (10-s interval)
Computed value Original value
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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
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Blad
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tch
angl
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Time (10-s interval)
Computed value Original value
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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
10-s data set 87.22 127.82 31.76%1-min data set 106.26 130.32 18.46%
Maximize Power Output Optimized Power Output Original Power Output Gain
10-s data set 1497.99 1481.72 1.10%1-min data set 1497.79 1482.57 1.03%
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Thank You !