going to extremes: a parametric study on peak-over-threshold and other methods wiebke langreder...
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Going to Extremes: A parametric study on Peak-Over-Threshold
and other methods
Wiebke Langreder
Jørgen Højstrup
Suzlon Energy A/S
Source: Wind Power Monthly
Nightmare... Extreme Winds...
Contents
Introduction
Objective
Methodology
Results and Conclusions
Importance of Extreme Wind
The 50-year maximum 10-minute average wind speed Vref is one of the important factors to classify a site according to IEC 61400-1.
Source: IEC 61400-1 ed 3
General Problem
Extreme winds are not related with mean wind speed.
Example: Vave Vref
Site 2 7.9 m/s 34 m/s
Site 3 4.6 m/s 36 m/s
IEC 61400-1?
Vref = 5 · Vave
Where do we get the information from?
Source: IEC 61400-1 ed 2
0
0.02
0.04
0.06
0.08
0.1
0.12
0 10 20 30
Wind Speed [m/s]
Fre
qu
ency
[%
]
1.25
1.5
1.75
2
2.25
Where do we get the information from?
EWTS (European Wind Turbine Standard)?
connection between Weibull k factor and extreme winds
Vave=8m/s
decreasing k
Vref= factor · Vave
Source: EWTS
EWTSV
ref/V
ave
Weibull shape parameter k
Gumbel Distribution?
• Extreme events in nature can frequently be described by a Gumbel distribution
• Measured maximum wind speeds are fitted to Gumbel distribution
• Gumbel distribution is extrapolated to 50-year recurrence time
Where do we get the information from?
The objective
Ideal:
Long-term data available with several occurances of 50-year event
Real world:Only short term data available (1 year or more)
Task:
How well can we estimate Vref?Compare different methods using short-term data• IEC• EWTS• Gumbel
Method
Long-time series are split in shorter sub-sets, each method is applied to each sub-set.
LT
Sub-set 1 → Vref
Sub-set 2 → Vref
Sub-set 3 → Vref
Sub-set 4 → Vref
Sub-set 5 → Vref
We need a ”true” reference value for comparison!
”True” Reference Value
Assumption
The “true” Vref is determined applying :
• Gumbel distribution
• FULL data set
• POT (Peak-over-Threshold)
Method
Results from all methods have been normalised with this ”true” value.
N subsets → N results per method
→ Standard deviation
→ Bias
POT: LT → ”True” Vref
Sub-set 1 → Vref
Sub-set 2 → Vref
Sub-set 3 → Vref
Sub-set 4 → Vref
Sub-set 5 → Vref
Test Data
Where Period Mean wind speed [m/s]
Weibull shape factor
Site 1 South of Spain 5 years 7.7 2.04Site 2 North of France 5 years 7.9 2.08Site 3 Colorado 10 years 4.6 1.34Site 4 Denmark 10 years 7.3 2.05Site 5 Netherlands 10 years 4.7 1.68Site 6 Minnesota 10 years 7.8 2.25Site 7 Korea 10 years 3.3 1.96
The objective
Compare different methods
• IEC:
– Determine mean wind speed of each sub-set
– Multiply with factor 5
– Normalise result with ”true” value
• EWTS
• Gumbel
Findings - IEC
R2 = 0.6221
50%
60%
70%
80%
90%
100%
110%
120%
130%
140%
150%
160%
1.00 1.50 2.00 2.50 3.00
Weibull k factor
Nor
mal
ised
50-
year
max
win
d sp
eed
Findings - IEC
• IEC is dependent on Weibull k factor
• Standard Deviation is 26%!!!
• Average of all results fits the “true” value bias = 0%
The objective
Compare different methods
• IEC
• EWTS:
– Identify k factor of each sub-set
– Determine corresponding factor to multiply Vave with
– Normalise result with “true” value
• Gumbel
EWTS does not specify:
• Shall we use the 360 degree k factor?• Shall we use a sector-specific k factor?
EWTS
Findings EWTS
360 degree
• Not dependent on k factor
• Negative bias of 9%
EWTS predicts less than our assumed ”true” reference value
• Standard deviation is 16%
Sector
• Not dependent on k factor
• Positive bias of 7%
EWTS predicts more than our assumed ”true” reference value
• Standard deviation is 16%
The objective
Compare different methods
• IEC
• EWTS
• Gumbel
How to identify maxima?
Methods to identify maximum wind speeds
Two commonly used methods:
• POT Peak-over-Threshold (using WindPRO)
• PM Periodical Maximum
POT Peak-over-Threshold
• Pick a threshold wind speed and identify all wind speeds above
• Introduce independency criteria
• Two options:
wind speed
dynamic pressure (square of wind speed)
• Every result has been normalised with the reference value.
• The average of all results and their standard deviation has been calculated.
Ideal Gumbel Plot
POT-Problems start...several slopes
POT: Influence of threshold
Two sub-sets from one site
Findings Gumbel - POT
• deviations from the Gumbel distribution lead to dependency of result from threshold
• strong variations between individual sub-sets
• inconclusive regarding how threshold influences result
POT – Wind
• Positive bias of 4%
• Standard deviation is 12%.
POT – Dynamic Pressure
• Negative bias of 4%
• Standard deviation is 11%
Methods to identify maximum wind speeds
Two commonly used methods:
POT Peak-over-Threshold
PM Periodical Maximum:
• Cut data set in sub-sections
• Identify maximum wind speed in each sub-section
• Ensure statistic independence between samples
Findings Gumbel - PM
Findings Gumbel - PM
POT vref= 35m/s
PM vref= 40m/s
Findings Gumbel - PM
• Seasonal bias problematic but can be avoided choosing periods carefully
• Smallest recommended period is 6 months
• Method cannot be applied to the same sub-sets as the other methods because of seasonal bias
• Thus statistics cannot be compared with the other results
Summary Findings
+/- 1 std dev
70%
80%
90%
100%
110%
120%
130%
IEC EWTS360 degr
EWTSsector
POTwind
POTpressure
No
rmal
ised
50-
year
max
win
d s
pee
d
Summary Findings
Methoddependend on Weibull k factor
bias Std Dev
IEC yes none 26%EWTS 360 degr no - 9 % 16%EWTS sector no + 7 % 16%POT wind no +4 % 12%POT pressure no - 4 % 11%
Methoddependend on Weibull k factor
bias Std Dev
IEC yes none 26%EWTS 360 degr no - 9 % 16%EWTS sector no + 7 % 16%POT wind no +4 % 12%POT pressure no - 4 % 11%
Brute Force?
When added
Combined EWTS no - 1% 13%
Methoddependend on Weibull k factor
bias Std Dev
EWTS 360 degr no - 9 % 16%EWTS sector no + 7 % 16%
Conclusion
• IEC (factor 5) is not working
• PM not suitable for short-term data sets (<5 years)
• Always standard deviation >10%
• Squared wind speed (dynamic pressure) results in lower Vref than wind data
• Combination of methods possible, leading to a small bias and standard deviation comparable to Gumbel
Acknowledgement
We would like to thank www.winddata.com for providing data.