UNCERTAINTY
The Classical Approach
ALAN DERRICK
SENIOR TECHNICAL MANAGER
September 2009
1
Overview
•
•
•
•
•
Uncertainty Basics
Energy Yield Uncertainty Components
Some Key Components
Discussion on Suitability of Method
Recommendations
2
What Measurement?
In the context of Resource Assessment Measurement is interpreted as:
• the process through which a set of Input Quantities is transformed into a
single output quantity – The Result.
The Input Quantities include:
• Wind speed measurements
• Long Term Prediction Process
• Terrain and surface roughness model
• Wind flow model
• Turbine layout
• Turbine power curve
• Loss Factors
The Result is:
• Best Estimate of the Annual Energy Production (AEP)
• Standard Uncertainty in AEP
3
Uncertainty Basics - What is Uncertainty?
Aim: to evaluate the combined standard uncertainty in energy yield
resulting from the individual uncertainty components associated with
each of the inputs to the energy yield analysis.
Application: use the above result directly to derive the energy yield with a
certain confidence level expressed as a probability: P50, P75, P90, P99,
etc.
Problem: identifying, quantifying and combining the individual uncertainty
components and distributions in a way that is not over-conservative.
Confidence level: indicates the degree of belief that the true value lies
within the specified uncertainty range.
4
Uncertainty Basics - Definitions and Terminology
Dispersion for a
given standard
uncertainty s
2.58SD
99%
1.64SD
90%
Dispersion
increases with
increasing s
5
Uncertainty Basics – Combined Standard Uncertainty
N N

E


U c2  
   i , j ci c j ui u j
 E 
i 1 j 1
2
Uc
Combined Standard Uncertainty
E Uncertainty in AEP
E AEP
 i, j Correlation Coefficient  1   
ci Sensitivity Coefficients
u Standard Uncertainty Components
1
i
6
Uncertainty Basics - Correlations
Negatively Correlated
 1  i, j  0
Uncorrelated
i, j  0
N
 E 
2
2 2
Uc  
   ci ui
 E 
j 1
2
Uncorrelated/independent
Positively correlated
0   i , j  1
 E  N
Uc  
   ci ui
 E  j 1
Fully correlated
Example
x  5MWh
y  10MWh
c 1
U  x2  y 2
U  11.2MWh
U  x y
U  15MWh
7
Uncertainty Basics – Sensitivity Coefficient
f
f
ci 

xi xi
xi  x1x N
Evaluated either by partial
differentiation or by perturbation of
the model inputs
240%
e.g. Sensitivity of
energy yield to wind
speed.
200%
180%
160%
Vref
(E+dE)/E
E
c1 
V
220%
140%
120%
100%
80%
V Vref
E
Slope
evaluated
at Vref
60%
40%
20%
Ranges from 1.25 to 2.00
typically.
0%
50%
60%
70%
80%
90%
100% 110% 120% 130% 140% 150% 160%
(v+dv)/v
8
Uncertainty Basics - Definitions and Terminology
Central Limit Theorem
Combined Standard Uncertainty PDF tends to Normal Distribution if:
Contributing uncertainty components are independent, random variables.
The convergence will occur more rapidly:
i. the larger the number of input distributions
ii. the closer to being equal are the individual contributions
iii. the closer to being normal distributions are the individual input
distributions .
The Result of the Annual Energy Production Uncertainty Evaluation could
be expected therefore to exhibit the characteristics of a normal
distribution.
May not be a valid assumption if:
•a few uncertainty components dominate
•model is neither linear nor nearly linear
•uncertainty components are not independent
9
Uncertainty Basic - Definitions and Terminology
Central Limit Theorem
*
*
=
*
=
Tending towards normal distribution already
10
Energy Yield Uncertainty Components Overview
Power Curve
Loss Factors
Shear
Extrapolation
Air Density
Wake
High Shear
Wind Flow
Model
Wind Speed
Sub Station
Metering
Tree Growth
Energy
Yield
Other
11
Wind Speed Uncertainty Sub-Model
Inter-annual
Variation
(Past)
MCP
Correlation
Inter-annual
Variation
(Future)
Measurement
Annual
Average
Estimate
Seasonal
Coverage
Wind
Speed
Other
12
Energy Yield Uncertainty Components Overview
Number of Turbines: 81
Hub Height: 80m
Met Masts: 4
Wind Measurement Height: 50m 80m 80m 80m
10 Year Wind Speed
Prediction Uncertainty 3.60% 3.50% 4.30% 4.80%
Component
Wind:
Terrain and Roughness:
Wake:
Note on Uncertainty Contribution at each Turbine
Nearest Mast Wind Uncertainty or Combination
of Masts
From Flow Model: Proportional to speed up
and distance to mast
From Wake Model: Proportional to Wake Loss
Assumed ±2 degree C uncertainty in annual
Air Dens.:
average over wind farm lifetime.
Assumed 2% based on evaluation of
Loss Factors.:
production statistics.
5%. In this example assumed uncorrelated
Power Curve:
between turbines.
Subs Met.:
±1%.
Related to difference between hub and mast
Shear Extrapolation to Hub Height: heights (varies on this site)
Derived from turbine dimensions and power
High Shear:
curve at 2%
Tree Growth:
Proportional to predicted tree growth loss.
None.
Other:
Correlation
Across Turbines Distribution
Sensitivity
Combined
(Across Turbines)
Uncertainty
[%]
Correlated
Normal
2.15
8.7%
Correlated
Normal
1.00
6.1%
Correlated
Normal
1.00
1.7%
Correlated
Triangular
1.00
0.3%
Correlated
Triangular
1.00
0.8%
Uncorrelated
Correlated
Normal
Triangular
1.00
1.00
0.6%
0.4%
Correlated
Normal
2.15
1.0%
Correlated
Correlated
Normal
Normal
2.15
2.15
2.2%
0.2%
0.0%
11.0%
Combined (Uncorrelated):
13
Uncertainty Model Summary
• Individual uncertainty components can themselves be based on a complex
model
• Where standard uncertainty component values are “estimated”, tempting
to always round to the nearest “simple” number e.g. 0.5%, 1%, etc.
– The more components and the more detail in the model, the greater the
conservatism or over-prediction in the combined standard uncertainty can
be.
• Consider if the conservative, fully correlated option is ever appropriate
before using to combine uncertainty components.
• Consider if there is any double counting of components, especially in
complex models, and eliminate.
• Conservatism should come in the choice of coverage factor “k” and not in
the estimate of the standard uncertainty which should be realistic.
14
How Universal are these Assumptions?
Based on foregoing example, the largest uncertainty components are
typically:
–
–
–
–
Wind Speed Prediction
Wind Flow Modelling
Shear Extrapolation
Shear Effect on Power Curve (if in high shear flow)
However result is very sensitive to the following:
– Number of Met Masts
– Inter Turbine Power Curve Correlations
15
Wind Flow Modelling & Met Mast Dependency
• Plot error in yield vs. predicted terrain effect
11 turbines predicted to be less windy than mast
• Error correlates with the predicted terrain effect.
1 Turbine windier
than mast
Ref. P. Stuart (RES) BWEA 2008
16
Symmetric Hill Test Case
• Consider 9 turbines on a symmetric hill.
• Assume a 1.7% change in yield for a 1% change in wind speed.
• Assume Uncertainty in Yield ≈ Terrain Effect (as on example wind farm).
• Placing the mast at the
base of the hill results in
7 turbines being underpredicted.
• Placing the mast at the top
of the hill results in 8
turbines
being
overpredicted.
• Placing mast half way up
the hill minimises the
error.
Ref. P. Stuart (RES) BWEA 2008
17
Multiple Mast Uncertainty Reduction
Combined Wind flow
uncertainty versus
number of met masts
deployed
Very large, complex
terrain, forested wind
farm
Uncertainty
proportional to Flow
Correction and
Distance to Mast
• Flow correction term does not necessarily
decrease if mast to turbine ratio small (81
turbines here).
• However distance term does always decrease if
masts sited appropriately
18
Power Curve (PC) Correlations
T01
SCADA Power Curve
3
Corrected to Reference Density (1.22 kg/m
)
T02
16/01/02
Date Range:30/06/01 to 21/09/01
Sector: 275 - 360 °N
900.0
800.0
800.0
700.0
700.0
600.0
500.0
Measured
400.0
Warranted
300.0
200.0
Density Corrected Power (kW)
900.0
100.0
SCADA Power Curve
3
Corrected to Reference Density (1.22 kg/m
)
Sector: 265 - 360 °N
600.0
500.0
Measured
400.0
Warranted
300.0
200.0
100.0
0.0
0.0
0.0
5.0
10.0
15.0
20.0
25.0
0.0
-100.0
5.0
10.0
15.0
20.0
Wind Speed (m/s)
D:\Performance\Beenag\SCADA\[BGAll03.xls]NP
D:\Performance\Beenag\SCADA\[BGAll03.xls]NP
SCADA Power Curve
3
Corrected to Reference Density (1.22 kg/m
)
T03
16/01/02
Date Range:30/06/01 to 21/09/01
SCADA Power Curve
3
Corrected to Reference Density (1.22 kg/m
)
T04
16/01/02
Sector: 252 - 360 °N
Date Range:30/06/01 to 30/09/01
900.0
900.0
800.0
800.0
700.0
700.0
600.0
500.0
Measured
400.0
Warranted
300.0
200.0
Density Corrected Power (kW)
Density Corrected Power (kW)
25.0
-100.0
Wind Speed (m/s)
100.0
Sector: 70 - 271 °N
600.0
500.0
Measured
400.0
Warranted
300.0
200.0
100.0
0.0
0.0
0.0
5.0
10.0
15.0
20.0
25.0
0.0
-100.0
5.0
10.0
15.0
20.0
25.0
-100.0
Wind Speed (m/s)
Wind Speed (m/s)
D:\Performance\Beenag\SCADA\[BGAll03.xls]NP
T05
16/01/02
Date Range:30/07/01 to 30/09/01
D:\Performance\Beenag\SCADA\[BGAll03.xls]NP
SCADA Power Curve
3
Corrected to Reference Density (1.22 kg/m
)
T06
16/01/02
Date Range:30/06/01 to 30/09/01
Sector: 0 - 215 °N
900.0
900.0
800.0
800.0
700.0
700.0
600.0
500.0
Measured
Warranted
400.0
300.0
200.0
100.0
Density Corrected Power (kW)
Density Corrected Power (kW)
• Measured PC’s
exhibit certain
common features
• probably
measurement
errors so not
relevant to AEP
prediction
uncertainty
• Likely that turbine
performance is
also influenced in
some correlated
way hence PC
combined
uncertainty
increased.
Density Corrected Power (kW)
16/01/02
Date Range:30/06/01 to 21/09/01
SCADA Power Curve
3
Corrected to Reference Density (1.22 kg/m
)
Sector: 95 - 360 °N
600.0
500.0
Measured
Warranted
400.0
300.0
200.0
100.0
0.0
0.0
0.0
5.0
10.0
15.0
20.0
25.0
-100.0
0.0
5.0
10.0
15.0
20.0
25.0
-100.0
Wind Speed (m/s)
Wind Speed (m/s)
D:\Performance\Beenag\SCADA\[BGAll03.xls]NP
D:\Performance\Beenag\SCADA\[BGAll03.xls]NP
19
Inferring Power Curve Correlations Pre-Construction
• CFD can provide clues.
• In-Flow angle, Turbulence and Shear
• e.g in the 81 turbine example
–
Original assumption all uncorrelated
 Upc = 0.6%
–
Assume adjacent turbines correlated in
groups, uncorrelated between groups
 Upc = 1.9%
–
Assume all turbines correlated
 Upc = 5.0%
20
Summing Up
IS THE METHOD VALID?
21
Weaknesses in Application of GUM to AEP Uncertainty
Weaknesses
• Assumption that resultant combined standard uncertainty is normally
distributed
• Subjectivity in assessment of many component uncertainty values and
distributions.
• Central Limit Theorem Requirements only Approximately Satisfied
• Key Models/Methods for Wind Speed Prediction and Wind Flow Modelling
“highly uncertain” and non-linear in complex terrain
• Usually only practical to make fully correlated or uncorrelated assumptions
• Some key uncertainty components, influences and/or correlations may be
missing from the model
22
Weaknesses in Application of GUM to AEP Uncertainty
Consequences
• Combined Standard Uncertainty Distribution may actually deviate from
normal distribution hence wrong coverage factors are being used.
• “Uncertainty” in the uncertainty could be high.
• Uncorrected errors or biases may be present in the best estimate which
the GUM assumes should be corrected for (and an uncertainty component
in the correction included)
Evidence
• comparison of 53 RES windfarm years of actual
production with RES predictions shows:
 s = 9.7% - close to average of RES prediction
uncertainties
• Garrad Hassan study of 535 wind farm years (right)
also shows good agreement of standard deviation
• Both RES and GH studies exhibit small bias.
23
More Evidence.....
• Same site, same data package
• 4 different, expert consultants
24
Recommendations
• Key focus of future work is to further understand and confirm the
uncertainty contributions:
 Are we overestimating some components but underestimating others
and somehow ending up with the correct result?
 Where are the biases in the best estimates of AEP coming from?
• The extensive production databases now available can help with this
• Simulations can also fill some gaps in our knowledge………….
25
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