UNCERTAINTY ANALYSIS OF LONG TERMWIND SPEED PREDICTION ALEX KAPETANOVICMANAGER WIND DATA ANALYSIS

14TH SEPTEMBER 2010

Wind Farm Site

Reference Stn.

Site Measurements

Historic Estimate

Long Term Estimate

Time

Historic Reference Measurements

Concurrent Period Relationship

Wind Speed Prediction Overview

Problem Overview

• Not all predictions are equal…

• The uncertainty in a wind speed prediction depends on:

• The site / reference relationship usually varies by season, yet traditionally this has seemingly not been explicitly included

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IAV

MHE

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Wind Speed Prediction

Uncertainty

Quality of the relationship

Annual variability of the future forecast

period

Annual variability of the

historic & measured data

Extrapolation to hub height

some are more equal than others

Quality of the measured data ?

Problem Overview

• This presentation focuses on:

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Wind Speed Prediction

Uncertainty

Quality of the relationship

Annual variability of the future forecast

period

Annual variability of the

historic & measured data

Extrapolation to hub height

Quality of the measured data

Inter-annual Variation

QUALITY OF THE SITE TO REFERENCE STATION RELATIONSHIP

MCP

TECHNICALLY THOUGH, THE QUALITY OF THE RELATIONSHIP IS DEFINED BY:

• The confidence limits of the estimated model parameters

• The number of data points

Quality of the Relationship

GOOD INDICATORS MIGHT BE:

• A trusted method (indicated by prior studies)

• Good ‘r’ value, but be careful

– ‘r’ increases when averaging over a larger timescale, e.g. r Hourly <= r Monthly

– Even if ‘r’ = 1, the uncertainty in the prediction is not negligible

Slope

Offset

Regression coefficient

Intermediate calcs.Time series of Reference Stn. data (x) and Measured Data (y)

An Example : Least Squares

nn yx

yx

yx

yx

,

.

.

,

,

,

33

22

11

2

2

iyy

ixx

iixy

iy

ix

yS

xS

yxS

yS

xS

2

xxx

yxxy

SSn

SSSnm

xy Sn

mSn

b11

yStdev

xStdevmr

It is “easy” using classical theory to develop the uncertainty in some relationships…

e.g. in the relationship y = m x + b:

An Example : Least Squares Continued

22 )2(

1xxxyyy SSnmSSn

nnStdev

What is the error on m and b?

Standard practice assumes that the number of points ‘n’ is large enough to apply the Central Limit Theorem, which in turn implies that the errors in regression are normally distributed

2

2

xxx SSn

StdevnmStdev

xxSnmStdevbStdev

12

] [ˆ %2 ntbStdevbb

Where tn-2% represents the % quantile of Student’s t-distribution and the confidence level of

the errors

] [ˆ %2 ntmStdevmm

An Example : Least Squares Continued

n-2 90% 95% 99%1 3.08 6.31 31.822 1.89 2.92 6.973 1.64 2.35 4.544 1.53 2.13 3.755 1.48 2.02 3.37

100 1.29 1.66 2.36 1.28 1.65 2.33

Example: •y = 1.1881x + 2.1583•Mean at reference station, x = 5.66m/s•Stdev(m) = 0.01127•Stdev(b) = 0.07025•tn-2 from table is 1.65

•Error

=((0.01127*1.65*5.66)+(0.07025*1.65))/(1.1881*5.66+2.1583)

=2.5%

What does that mean?

Look up table for % quantile of Student’s t-distribution and error confidence level

For most wind predictions one can assume an infinite number of points

Confidence Level

Dat

a P

oint

s m

inus

2

Unfortunately empirical evidence suggests that this calculation underestimates the true error

Uncertainty in Other Methods

• Not so easy to calculate uncertainty in other cases, e.g. the non-linear ‘matrix method’

• In such cases the uncertainty can be evaluated using empirical methods

• RES uses the following relationship to evaluate the uncertainty in all of its predictions [1]

nMCP

375(%)

Derived from a ‘bootstrap’ method

Where ‘n’ is the number of hours used to define the relationship

[1] http://www.res-americas.com/Resources/MCP-Errors.pdf

Multiple masts

Ref StnMast 1Mast 2

Example

Time

•First predict Mast 1 in the normal way using a reference station

•Now compare two possible approaches for predicting Mast 2

•How do we evaluate which method gives the lowest uncertainty for Mast 2?

Ref StnMast 2

Ref StnMast 1Mast 2

Method 1Same method as used for Mast 1

Method 2“Second Step” or “Intra-site” prediction

Evaluating “Second-step” Uncertainty

nndstepMCP

25.187(%)

2

2

3752 nMastMCP

2

2

2

1

22 25.187375212

nnndstepMastMast MCPMCPMCP

• In a similar study we determined the following relationship:

Method 2: Mast 2 has 1 yr, Mast 1 has 1.5 yrs

Method 2: Mast 2 has 1 yr, Mast 1 has 2.0 yrs

Method 2: Mast 2 has 1 yr, Mast 1 has 3.0 yrs

Method 1: Mast 2 has 1 yr of data

EXTRAPOLATION OF WIND SPEED FROM MEASURED HEIGHT TO HUB

HEIGHT

HH

Shear Extrapolation Uncertainty

Shear Exponent α defined by:

The error in the shear exponent is :

The Shear Extrapolation Uncertainty :

is derived from

And yields

The commonly applied rule “1% for 10m of extrapolation” is too generic…

Insufficient vertical separation between anemometer levels leads to higher uncertainty

1

2

1

2

h

h

V

V

1

2

1

2

2

1

1

2

2

22

11

2

22 ln

2

ln hh

hh

VV

VV

VV

VV

inst

HH

HHHH V

V

2

lnh

hV

VV h

HHHH

HH

12

2

/ln

/ln2

hh

hhhinstHH

Hub Height (m)Meas. Heights (m) 80

50/30 2.6% 0.9%50/40 6.0% 2.0%60/40 2.0% 1.0%60/50 4.5% 2.2%

per 10 m of interpolation

ANNUAL VARIABILITY OF WIND SPEED IN THE REGION

(INTER-ANNUAL VARIATION)

IAV

• Annual mean wind speed varies on a yearly basis

• IAV (“Inter-annual Variation”) is defined as the standard deviation of the annual means divided by the overall mean

• More variation requires a longer measurement campaign for a given uncertainty

• Not all regions in the United States have the same amount of variability

• Is the value of 6% that is typically used “representative”?

Annual Variability

• Based on 10 years of NCEP Reanalysis Surface Winds (2000-2009)

Annual Variability

Data are here: http://www.esrl.noaa.gov/psd/data/gridded/data.ncep.reanalysis.derived.surface.html

• Numerical weather prediction model output• Global 2.5 deg grid (~200km in lower 48)• “Surface” wind speed is at sigma level 0.995

• Recent work presented here for the first time shows the variation of IAV across the United States based on over 8000 US surface stations

• Here those with a 10 year record are presented un-filtered (700 stations)

Annual Variability

Data are here: ftp://ftp.ncdc.noaa.gov/pub/data/gsod/

• Simple filters were then applied after which 251 stations remained• Every day had to have a minimum of 22 hours to ‘count’• Each year had to have a minimum of 90% availability over 10 years (2000

– 2009)

Annual Variability

• Problem: Many stations exhibit discontinuities

Annual Variability

After ~7 years the cumulative IAV has settled to less than 3% and remains ~constant out to 16 years

After ~7 years the cumulative IAV has settled to less than 3% and remains ~constant out to 16 years

Add 1 more year and the IAV jumps to ~5.7%

Add 1 more year and the IAV jumps to ~5.7%

Add some more data and ….Add some more data and ….

Thunder Bay, Ontario

• How do we know that these stations were valid over the period without examining them all ‘by hand’?

• Statistical procedure to remove the outliers was used– Calculate the first (Q1) and third (Q3) quartiles of the observed 10-year series,

i.e. the 25th and 75th percentiles – Calculate the Inter-Quartile Range: IQR = Q3 – Q1– Define boundaries above and below which points are considered to be outliers:

– Upper Bound (UB) = Q3 + k * IQR– Lower Bound (LB) = Q1 - k * IQR

– Taking k = 3 (a commonly used value in statistics for extreme outliers) reduced the number of stations to 234

• Using a cumulative sum technique 3 more stations were removed because they had step changes, or changes in the mean level (outside of defined limits)

Annual Variability

Annual Variability

231 Valid Stations

Only a small portion of the US appears to have an IAV of 6% or greater

Only a small portion of the US appears to have an IAV of 6% or greater

• A minor problem with this result is that we know that stations have inconsistencies:o ASOS stations start ~1996-1998 or latero AWOS stations start 2002-2003 or later o ASOS stations switched to Ice Free Instrumentation between

2002-2009

• No stations were left with a 10 year record if filtered using the criterion that the station had not changed

• However, ‘inconsistency’ should tend to increase IAV, so we believe that the map is valid

Annual Variability

Annual Variability

231 Valid Stations

NCEP

CONCLUSIONS

Conclusions

• Seasonality is not accounted for in a classical approach to the Quality of the relationship

• However, seasonality can be accounted for in empirical estimations of the Quality of the relationship.

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Quality of the relationship

Empirical

Conclusions

• The 1% per 10 meter rule of thumb is just that. It needs to be evaluated on a case by case basis

• Insufficient vertical separation between anemometers used for calculating shear leads to higher shear extrapolation uncertainty

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Empirical

Extrapolation to hub height

Conclusions

• The NCEP Reanalysis Surface Winds map indicates that only a few regions might have an annual variability greater than 6%

• Analysis conducted on 231 ground stations shows that much of the US has an IAV closer to 4% and only a very small portion of the US is >=6%

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Empirical

Inter-annual Variation

THANK YOU

ALEXANDRE KAPETANOVICMANAGER, WIND DATA ANALYSIS

RENEWABLE ENERGY SYSTEMS AMERICAS INC.11101 West 120th Avenue, Suite 400

BROOMFIELD, CO 80021(303) 439 4200

With thanks to Gail Hutton, Brian Healer, Andrew Oliver, Dan Ives,Kristofer Zarling, Jerry Bass & Mike Anderson