Transcript Uncertainty Analysis of Long Term Wind Speed

UNCERTAINTY ANALYSIS OF LONG TERM
WIND SPEED PREDICTION
ALEX KAPETANOVIC
MANAGER WIND DATA ANALYSIS
14TH SEPTEMBER 2010
Wind Speed Prediction Overview
Long Term Estimate
Historic Estimate
Site Measurements
Wind Farm Site
Concurrent Period
Relationship
Historic Reference Measurements
Reference Stn.
Time
Problem Overview
• Not all predictions are equal… some are more equal than others
• The uncertainty in a wind speed prediction depends on:
Wind Speed
Prediction
Uncertainty
Annual variability
of the historic &
measured data
   MCP 
2
Quality of the
relationship
 IAV 2
N HE  N M

Extrapolation to
hub height
 IAV 2
n
  HH   inst   Re cov ery   Seasonal
Annual variability
of the future
forecast period
2
2
Quality of the
measured data
• The site / reference relationship usually varies by season, yet
traditionally this has seemingly not been explicitly included
2
?
2
Problem Overview
• This presentation focuses on:
Wind Speed
Prediction
Uncertainty
Annual variability
Inter-annual
of the historic &
Variation
measured
data
   MCP 
2
Quality of the
relationship
 IAV 2
N HE  N M

Extrapolation to
hub height
 IAV 2
n
  HH   inst   Re cov ery   Seasonal
Annual variability
of the future
forecast period
2
2
Quality of the
measured data
2
2
QUALITY OF THE SITE TO REFERENCE
STATION RELATIONSHIP

MCP
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
TECHNICALLY THOUGH, THE QUALITY OF THE
RELATIONSHIP IS DEFINED BY:
• The confidence limits of the estimated model
parameters
• The number of data points
An Example : Least Squares
It is “easy” using classical theory to develop the uncertainty in some relationships…
e.g. in the relationship y = m x + b:
Time series of
Reference Stn. data
(x) and Measured
Data (y)
x1 , y1
x2 , y 2
x3 , y3
.
Intermediate calcs.
Slope
m
S x   xi
S y   yi
Offset
S xy   xi yi
.
S xx   xi
xn , y n
S yy   yi
2
Regression coefficient
2
n S xy  S x S y
n S xx  S x
2
1
1
b  S y  m Sx
n
n
Stdevx 
rm
Stdev y 
An Example : Least Squares Continued
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
Stdev  


1
2
2
n S yy  S y  m n S xx  S x
n (n  2)

n Stdev 
Stdevm  
2
n S xx  S x
2
Stdevb   Stdevm
2
1
S xx
n
ˆ [m  Stdevm tn2% ]
m
%
bˆ [b  Stdevb tn2 ]
Where tn-2% represents the % quantile of Student’s t-distribution and the confidence level of the errors
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
Data Points minus 2
An Example : Least Squares Continued
n-2
1
2
3
4
5
100

Confidence Level
90%
95%
99%
3.08
6.31
31.82
1.89
2.92
6.97
1.64
2.35
4.54
1.53
2.13
3.75
1.48
2.02
3.37
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%
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]
375
 MCP (%) 
n
[1] http://www.res-americas.com/Resources/MCP-Errors.pdf
Derived from a ‘bootstrap’
method
Where ‘n’ is the number of
hours used to define the
relationship
Multiple masts
Example
Ref Stn
Mast 1
Mast 2
Time
• First predict Mast 1 in the normal way using a reference station
• Now compare two possible approaches for predicting Mast 2
Method 1
Same method as used for Mast 1
Method 2
“Second Step” or “Intra-site” prediction
Ref Stn
Mast 2
Ref Stn
Mast 1
Mast 2
• How do we evaluate which method gives the lowest uncertainty for Mast 2?
Evaluating “Second-step” Uncertainty
• In a similar study we determined the following relationship:
 MCP
Mast 2

375
n2
 MCP
(%) 
2 ndstep
Method 1: Mast 2 has 1 yr of data
2
 MCP
Mast 2
187.25
n
  MCPMast 1   MCP2 ndstep
2
2
 375   187.25 
 

 
 n   n 
2
 1 

2
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
EXTRAPOLATION OF WIND SPEED FROM
MEASURED HEIGHT TO HUB HEIGHT

HH
Shear Extrapolation Uncertainty

Shear Exponent α defined by:
V2  h2 
 
V1  h1 
The error in the shear exponent is :
 V2   V1 

  

2
2
V
V
 
  

2 inst
 2   1 
  
V2   
V1  

h
h 
 V2
  V1

ln 2
ln 2 
h1
 h1 
2
The Shear Extrapolation Uncertainty :  HH 
is derived from
And yields
VHH 
2
VHH
VHH
h 
VHH
  VHH ln h 

 h2 
 HH  2 inst
lnhh / h2 
lnh2 / h1 
Meas. Heights (m)
50/30
50/40
60/40
60/50
Hub Height (m)
80
2.6%
6.0%
2.0%
4.5%
per 10 m of interpolation
0.9%
2.0%
1.0%
2.2%
The commonly applied rule “1% for 10m of extrapolation” is too generic…
Insufficient vertical separation between anemometer levels leads to higher uncertainty
ANNUAL VARIABILITY OF WIND SPEED
IN THE REGION
(INTER-ANNUAL VARIATION)

IAV
Annual Variability
• 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)
•
•
•
Numerical weather prediction model output
Global 2.5 deg grid (~200km in lower 48)
“Surface” wind speed is at sigma level 0.995
Data are here: http://www.esrl.noaa.gov/psd/data/gridded/data.ncep.reanalysis.derived.surface.html
Annual Variability
• 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)
Data are here: ftp://ftp.ncdc.noaa.gov/pub/data/gsod/
Annual Variability
• 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
Add some more data and ….
Add 1 more year and the
IAV jumps to ~5.7%
After ~7 years the cumulative IAV
has settled to less than 3% and
remains ~constant out to 16 years
Thunder Bay, Ontario
Annual Variability
• 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
231 Valid Stations
Only a small portion of the US appears
to have an IAV of 6% or greater
Annual Variability
• A minor problem with this result is that we know that stations have
inconsistencies:
o ASOS stations start ~1996-1998 or later
o 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
NCEP
231 Valid Stations
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.
   MCP 
2
Empirical
Quality of the
relationship
 IAV 2
N HE  N M

 IAV 2
n
  HH   inst   Re cov ery   Seasonal
2
2
2
2
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
Extrapolation to
hub height
   MCP 
2
Empirical
 IAV 2
N HE  N M

 IAV 2
n
  HH   inst   Re cov ery
2
2
2
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%
Inter-annual
Variation
   MCP 2 
Empirical
 IAV 2
N HE  N M

 IAV 2
n
  HH   inst   Re cov ery
2
2
2
THANK YOU
ALEXANDRE KAPETANOVIC
MANAGER, 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