Extreme wind speed estimation - Windpower · EMD 9/3/2009 1 Extreme wind speed estimation Outline...

EMD9/3/2009

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Extreme wind speed estimation

Outline• Definition

• Data extraction

• Choice of asymptote

• Convergence to asymptote (bias I)

• Plotting positions (bias II)

• Bias I & Bias II – A Monte Carlo experiment

• The plot

• Choice of fit?

• Conclusion/Recommendations

Lasse Svenningsen, EMD

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Extreme wind speed estimation

Outline• Definition

• Data extraction

• Choice of asymptote

• Convergence to asymptote (bias I)

• Plotting positions (bias II)

• Bias I & Bias II – A Monte Carlo experiment

• The plot

• Choice of fit?

• Conclusion/Recommendations

A matter of choices!

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Definition of “50 year” wind?

Annual risk of exceedance R=2%

Or T = 1/R = 50y

The real task is:

• Estimate the CDF for annual extremes, FA(u)

• Given FA(u) find: u | FA(u) =1-R=98%

• Or reduced variate: u | y=-ln(-ln(0.98))=3.9

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Data extraction

“Annual Max” or “Peak-over-Threshold”?

• AM:

• Only use the max of each year

• POT:

• Use all independent storms above uthreshold

• Stormrate: λ

• Parent:

• For known N & Weibull CDF: FA(u)= WCDF(A,k)N

• ….

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LS2

LS2 Largest / n'th order statitsticLasse Svenningsen, 01/09/2009

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Data extraction

“Annual Max”

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0 2000 4000 6000 8000 10000 120000

5

10

15

20

25

30

days from 1st sample

u[m

/s]

1800 2000 2200 2400 2600 2800 3000 3200 3400

16

18

20

22

24

26

28


u[m

/s]

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Data extraction

“Peak-over-Threshold”

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0 2000 4000 6000 8000 10000 120000

5

10

15

20

25

30


u[m

/s]

1800 2000 2200 2400 2600 2800 3000 3200

18

20

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24

26

28


u[m

/s]

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Choice of asymptote

GEV distribution type 1, 2 or 3??

Type 1: “Gumbel”

• Parent PDF: exponential tail

• E.g.: Weibull, Gauss, …

Type 2: “Fréchet”

• Parent PDF: power-law tail (heavy tailed)

• E.g.:Log-normal, Pareto

Type 3: “Reverse Weibull”

• Parent PDF: bounded above

• E.g.: Beta

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Choice of asymptote


From Palutikof (1999)Ty

pe2

Type 1

Type 3

u

y

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Choice of asymptote


If Weibull…..

….Then Gumbel!!

Common observations:

- Type 3 tends to fit better?

- Type 1 too conservative?

Why?

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Convergence to asymptote

Extreme distributions are only asymptotic!

• Assumes: extremes of N=∞ independent samples

• In reality N<<∞

• Nyear=52560 x 10min but…

• Only ~ 2300 are independent (Bergström,1992)

• The consequence?

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BiasI

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Rayleigh type Weibulls (k=2) converge slowly

From Cook (1982)

Type

3

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BiasI

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Exponential Weibulls (k=1) converge extremely fast

From Cook (1982)

u

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BiasI

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The fix is “preconditioning”:• Transform z(u) to make WCDF(z) exponential

• Transform result back to u after the analysis

• Optimum transform: z=uk

•Weibull CDF:

−−= k

k

CDF AuAkuW exp1),,(

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BiasI

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Plotting positions

Probability plotting positions, Fest.?

• The choices:

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BiasII

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Bias I & Bias II….Extr

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A Monte Carlo experiment - setup:

• 1000 random realizations generated for:

Weibull parent with:Nsamples = 2300 �(Bergström, 1992)A = 10 m/sk = 1.7

⇒ V50, exact= 42.4 m/s (~ IEC class I/II limit)

Nyears = 1….20

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A Monte Carlo experiment - setup:

AM: Pick annual maxima

POT: Select storm threshold / fixed number of storms(independence assured)

Fit: LSQ

Bias estimate: V50, ensemble mean / V50, exact

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A Monte Carlo experiment – Bias II:

• Choice of plotting position: AM

2 4 6 8 10 12 14 16 18 200.98

1

1.02

1.04

1.06

1.08

1.1

1.12

1.14

Number of years

Bia

s: m

ean/

exac

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WeibullHazenGringorton

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• Effect of k-factor preconditioning: AM

2 4 6 8 10 12 14 16 18 200.98

1

1.02

1.04

1.06

1.08

1.1

1.12

1.14

Number of years

Bia

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• Choice of plotting position: POT (fixed uthreshold = 24m/s)

0 5 10 15 200.97

0.98

0.99

1

1.01

1.02

1.03

1.04

1.05

Number of years

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• Choice of plotting position: POT (fixed NS=20)

0 5 10 15 200.97

0.98

0.99

1

1.01

1.02

1.03

1.04

1.05

Number of years

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ean/

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Weibull

Hazen

Gringorton

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• Effect of k-factor preconditioning: POT (fixed NS=20)

0 5 10 15 200.97

0.98

0.99

1

1.01

1.02

1.03

1.04

1.05

Number of years

Bia

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A Monte Carlo experiment – conclusion:

• Total bias < 1% if…

• AM:

• k-factor preconditioning

• Hazen plotting positions

• POT:

• Fixed “storm” number ~ 20

• k-factor preconditioning

• Weibull plotting positions

(some exceptions!)

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• Scatter around mean of the 1000 realisations??

0 5 10 15 20 250.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

Number of years

Ens

embl

e C

OV

- 1

000

real

isat

ions

POT-20AM

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• Correlation with max? Nyears = 1

POT r = 0.95 AM

25 30 35 40 45 50 5530

35

40

45

50

55

60

65

Max u each period (N years) [m/s]

Res

ultin

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50 f

or p

erio

d [m

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POT r = 0.95 AM r = 0.95

32 34 36 38 40 42 44 46 48 50 5230

35

40

45

50

55

60

65

70


Res

ultin

g V

50 f

or p

erio

d [m

/s]

30 35 40 45 50 55 6030

35

40

45

50

55

60

65

70


Res

ultin

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50 f

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d [m

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POT r = 0.95 AM r = 0.95

30 35 40 45 50 5535

40

45

50

55

60

65


Res

ultin

g V

50 f

or p

erio

d [m

/s]

30 35 40 45 50 55 6035

40

45

50

55

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65


Res

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50 f

or p

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d [m

/s]

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The Plot

Order of the axes?

Which has the greater uncertainty, u or y?

• Hopefully u is measured accurately!

• y uncertainty (FA) is large especially for highest u!

Or

u

y

y

u

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The Plot

Order of the axes?

Which has the greater uncertainty, u or y?

• Hopefully u is measured accurately!

• y uncertainty (FA) is large especially for highest u!

u

y

y

u

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The Plot

Making AM and POT plots directly comparable

• AM:

• Plot: (uk,y)

• Extrapolate to: y = -ln(-ln(1-1/T)) = 3.9

• POT:

• Plot: (uk,y)

• Extrapolate to: y = -ln(-ln( 1-1/λT )) = ?Extr

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Differen

t y-axes!

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The Plot

The problem:

• AM:

• Estimates CDF of annual extremes directly: FA(u)

• Extrapolate to risk: R = 0.02

• POT:

• Estimates CDF of the extracted λ storms/year: FS(u)

• Extrapolate to risk: R = 0.02/λ

The solution:

• Get FA from FS (Cook, 82): FA(u) = FS(u)λ

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The Plot

Making AM and POT plots comparable

• AM:

• Plot: (uk,y)

• Extrapolate to: y = -ln(-ln(1-1/T)) = 3.9

• POT:

• Plot: (uk, y0-ln(λ) )

• Extrapolate to: y = -ln(-ln( 1-1/T )) = 3.9Extr

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Same y-axes!

!!!!

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Choice of fit

Common types of fit used:

• LSQ (L2)

• PWM (probability weighted moments)

• W-LSQ (weighted L2)

• Which weights to use? Harris’? Gumbels?

• LAD (L1)?

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Conclusion/suggestionsExtr

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ices • Choose POT for real life applications (always too few data!)

• Use the GEV Type 1 asymptote (i.e. Gumbel)

• Use k-factor preconditioning (z=uk)

• AM: Use “Hazen” plotting positions

• POT: Use “Weibull” plotting positions (y0), and…

• Use fixed “storm” number ~ 20, not fixed threshold

• Plot: z=uk versus y=y0-ln(λ), extrapolate to y=3.9

• Fit: ???

Extreme wind speed estimation - Windpower · EMD 9/3/2009 1 Extreme wind speed estimation Outline...

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