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)