Statistical Post-Processing of Statistical Post-Processing of General Time Series Data - With General Time Series Data - With Wind Turbine ApplicationsWind Turbine Applications
LeRoy Fitzwater, Lance Manuel, Steven Winterstein
Implementation/Interpretation of Implementation/Interpretation of Standards: IEC & IS0Standards: IEC & IS0 Issues:
– How to “Fill In”/Extrapolate Load Spectra for Ultimate & Fatigue Loads:
US wind consultants (e.g. Kamzin) National Labs (e.g. RISO-Denmark, ECN-Netherlands,
NREL/Sandia-United States) Academic Research (e.g. RMS)
– Design Bases for Ultimate Loads: Series of Design Gust Scenarios Full Turbulence Simulation
Implementation/Interpretation of Implementation/Interpretation of Standards: IEC & IS0Standards: IEC & IS0 Issues: (cont’d)
– How Much Data? How Uncertain Given the Imperfect Information
– Limited Data from Prototype Machines– Imperfect Analysis Models (e.g. Cd Uncertainty)
Cover with Appropriate “Safety” Factor
Loads: A Bottom-Up ApproachLoads: A Bottom-Up Approach
Short-term Problem (Given a Stationary Wind/Sea State)– Have loads data {L1, …, Ln}, (e.g., rainflow ranges) for a
given wind condition model statisitical moments i: 1 = Average (Mean) Load
2 = Normalized second-moment (Coefficient of Variation):
3 =Normalized third-moment (Coefficient of Skewness):
1 =Normalized fourth-moment(Coefficient of Kurtosis):
– Algorithm: FITS estimates load distribution from i
2 1
; 2 (L L)2
n
iiLn 1
1
1
3 L L 3
3
4 L L 4
4
Loads: A Bottom-Up ApproachLoads: A Bottom-Up Approach Long-term Problem
– Across multiple wind conditions: Model load moments mi vs. wind parameters V and I:
– Where Power -law flexible form; permits:
– Linear dependence (b,c = 1)– Superlinear Dependence (b,c > 1)– Sublinear Dependence (b,c < 1)– No dependence (b,c = 0)
a,b,c estimated by linear regression (and their uncertainties) Vref, Iref = central V, I values (geometric means)
– Algorithm: PRECYCLES estimates a, b, c, and their uncertanties; provides input to reliability analysis routine CYCLES (FAROW)
i aV
Vref
b
I
Iref
c
Moment-Based Models of Dynamic Moment-Based Models of Dynamic Loads & ResponseLoads & Response
Moment-Based Models of Dynamic Moment-Based Models of Dynamic Loads & Response - Two OptionsLoads & Response - Two Options Option 1- Model Process
Two-Sided Distribution X=C0+C1N+C2N2+C3N3
– N=Normal– Ci’s depend on the 4 Statistical
Moments of X 3= skewness (right vs. left tail)
4=Kurtosis (“heaviness” of both tails)
Option 2- Model Ranges/Peaks
One-Sided Distribution Y=C0+C1W+C2W2
– W=Weibull– Ci’s depend on the 3 Statistical
Moments of Y
Moment-Based Models of Dynamic Loads & Moment-Based Models of Dynamic Loads & Response - Critical Issues & TradeoffsResponse - Critical Issues & Tradeoffs
Option 1- Model Process
– Only Need Original History No Peak Counting
– Must Approximate Peaks Narrow Band Approximation
– Can Model Fatigue and Extremes
Option 2 - Model Ranges/Peaks
– Can use Stats of Rainflow Ranges Directly (often stored)
– Fewer Moments Needed; Simpler Fitting
– May Need to Filter Small/Uninteresting Ranges
– Can Model Fatigue and Extremes
Data Analysis Algorithm: Data Analysis Algorithm: FITSFITS (Stanford University/Sandia National Laboratory)
Other Routines– FITTING: 4-Moment Distortions of Normal and Gumbel Distributions
– FAROW/CYCLES: Fatigue Reliability Analysis (Given Moment Based Loads)– PRECYCLES: Fits Moments vs. V, I Input to FAROW/CYCLES
Data Sets
Raw Data Probabilistic
Histograms DistributionMoments Select from among: Function fits
Normal to dataLognormalExponentialWeibullGumbelShifted ExponentialShifted WeibullQuadratic WeibullShifted Quadratic Weibull
FITS
HAWT Data SetHAWT Data Set Description:
– Horizontal Axis Wind Turbine (HAWT)– 101 Data Sets; each of Ten-Minute Duration– Wind Speed: 15 to 19m/sec
Subset of Collected Data– Turbulence Intensity: 10 to 23 percent– Rainflow-counted cycles or ranges available– Flap(Beam) and Edge(Chord) Bending Moment ranges available– Data were gathered as counts of ranges exceeding specific levels of a bending moment range.
Goal: – Long Data Sets - “True” Long Run Statistics– Fit to Subsets - Assess:
Accuracy (Bias) Uncertainty
HAWT - HAWT - Turbulence vs. Wind SpeedTurbulence vs. Wind Speed
HAWT Wind DataMean Wind Speed 15m/sec to 19m/sec
0.00
0.05
0.10
0.15
0.20
0.25
15.0 15.5 16.0 16.5 17.0 17.5 18.0 18.5 19.0
Mean Wind Speed
Tu
rbu
len
ce I
nte
nsi
ty
HAWT - Typical HistogramsHAWT - Typical HistogramsHistogram
0
500
1000
1500
2000
2500
0 10 20 30 40 50 60 70
Beam Bending Moment Range
Num
ber of O
ccurr
ences
15.026m/sec, 7133 Data Points 18.964m/sec 7612 Data Points
15.026m/sec, maximum bending = 38.6718.964m/sec, maximum bending = 63.95
HAWT - Fitted DistributionHAWT - Fitted DistributionQuadartic Weibull Model - FITS
Weibull Scale
0.1
1.0
10.0
100.0
0.1 1.0 10.0 100.0 1000.0
Beam Bending Moment Range
-log(
1-P
[X>x])
15.026m/sec QW Fit 18.964m/sec QW Fit
HAWT - Shifted DataHAWT - Shifted DataQuadratic Weibull Fit to Shifted Data
Weibull Scale, X shift = 12 units
0.01
0.10
1.00
10.00
100.00
0.01 0.10 1.00 10.00 100.00 1000.00
Beam Bending, X-12
-log(1
-P[X
<x])
15.026m/sec QW Fit 18.964m/sec QW Fit
HAWT - Damage ReductionHAWT - Damage ReductionHAWT Data - Effect of BM Range Shift of 11.5 on Damage
-12
-10
-8
-6
-4
-2
0
15.0 15.5 16.0 16.5 17.0 17.5 18.0 18.5 19.0
Wind speed (m/s)
Per
cent R
educt
ion in
Dam
age
due
to S
hift
of D
ata
b = 3
b = 6
b = 9
Fatigue Exponent, b
HAWT - Data vs. Fit, Range 1HAWT - Data vs. Fit, Range 1
Quadratic Weibull Model Bin 1mean windspeed 15.026 - 16.188(m/sec)
1
10
100
10.00 100.00
Beam Bending, X-12
-log(1
-P[X
<x])
Data average
Data mean+sigma
Data mean-sigma
FITS average
FITS mean+sigma
FITS mean-sigma
HAWT - Data vs. Fit, Range 1HAWT - Data vs. Fit, Range 1
Quadratic Weibull Model Bin 1mean windspeed 15.026 - 16.188(m/sec)
1
3
5
7
9
11
13
15
10.00 20.00 30.00 40.00 50.00 60.00 70.00
Beam Bending, X-12
-log(1
-P[X
<x])
Data average
Data mean+sigma
Data mean-sigma
FITS average
FITS mean+sigma
FITS mean-sigma
HAWT - Data vs. Fit, Range 2HAWT - Data vs. Fit, Range 2
Quadratic Weibull Model Bin 2mean windspeed 16.124 - 17.937(m/sec)
1.00
10.00
100.00
10.00 100.00
Beam Bending, X-12
-log(1
-P[X
<x])
FITS Average
FITS mean+sigma
FITS mean-sigma
Data Average
Data mean+sigma
Data mean-sigma
HAWT - Data vs. Fit, Range 2HAWT - Data vs. Fit, Range 2
Quadratic Weibull Model Bin 2mean windspeed 16.124 - 17.937(m/sec)
1.00
3.00
5.00
7.00
9.00
11.00
13.00
15.00
10.00 20.00 30.00 40.00 50.00 60.00 70.00
Beam Bending, X-12
-log(1
-P[X
<x])
FITS Average
FITS mean+sigma
FITS mean-sigma
Data Average
Data mean+sigma
Data mean-sigma
HAWT - Data vs. Fit, Range 3HAWT - Data vs. Fit, Range 3
Quadratic Weibull Model Bin 3mean windspeed 18.114 - 18.964(m/sec)
1.00
10.00
100.00
10.00 100.00
Beam Bending, X-12
-log(1
-P[X
<x])
FITS Average
FITS mean+sigma
FITS mean-sigma
Data Average
Data mean+sigma
Data mean-sigma
HAWT - Data vs. Fit, Range 3HAWT - Data vs. Fit, Range 3
Quadratic Weibull Model Bin 3mean windspeed 18.114 - 18.964(m/sec)
1.00
3.00
5.00
7.00
9.00
11.00
13.00
15.00
10.00 20.00 30.00 40.00 50.00 60.00 70.00
Beam Bending, X-12
-log(1
-P[X
<x])
FITS Average
FITS mean+sigma
FITS mean-sigma
Data Average
Data mean+sigma
Data mean-sigma
SummarySummary
I. Estimating Load Distributions (Spectra) From Statistical Moments– Fairly Mature (2nd Generation)– Special Issues:
Fit Process or Ranges/Peaks Periodicity Response Events
II. Uncertainty/Confidence Bands From Limited Data– Methods Available - Simulation vs. Bootstrap (e.g. MAXFITS)– Tests Needed to Validate (via Long Data Sets)
SummarySummary (cont’d)
I + II Statistical Load Characterization– Combine with Reliability Analysis
Pf (case specific)
– Proposed Guidelines/Standards Implied Pf Across Cases
– Target Pf
Consistent Safety Factors (information sensitive)
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