M. Hering
Space-Time Wind Speed Modeling Techniques
Amanda S. Hering
Colorado School of Mines
Department of Mathematical and Computer Sciences
Wind Energy Prediction R& D Workshop
May 11-12, 2010
NCAR; 05.12.10
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M. Hering
Outline of This Talk
• Past Projects
– Space-time statistical models and evaluation tools
Joint with Marc G. Genton of TAMU
• Current Projects
– SIParCS Climate Model Project
Joint with Steve Sain and Doug Nychka of GSP
– Forecasting Categorical Changes in Wind Power
Joint with Megan Yoder of CSM
– Varying-Coefficient Statistical Models
Joint with Marc G. Genton of TAMU and Pierre Pinson of DTU
NCAR; 05.12.10
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M. Hering
Space-Time Statistical Model
• The Trigonometric Direction Diurnal model grew out of some prior
work by Gneiting et al. (2006) to predict hourly average wind speed
at Vansycle, OR two hours ahead.
• The wind speed at Vansycle is modeled by a truncated normal
distribution which has two parameters, µ and σ.
• The mean of the truncated normal distribution is
µ
µ
+
µ =µ+σ·φ
/Φ
.
σ
σ
• The key is in modeling µ and σ appropriately.
Hourly speed and direction data was provided by Bonneville Power
Administration and Energy Resources Research Laboratory at Oregon State
University.
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M. Hering
Spatial Positions of 4 Sites
Site Locations
KW
SH
GH
VS
SH=Sevenmile Hill, GH=Goodnoe Hills, KW=Kennewick, and
VS=Vansycle
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M. Hering
The TDD Model Notation
• Let the wind speed at time t be denoted by Vt , Kt , Gt , and St for
the Vansycle, Kennewick, Goodnoe Hills, and Sevenmile Hill sites.
• A diurnal component in the wind speeds is removed by subtracting
the least squares fit of the hourly means of each wind speed time
series regressed on a pair of harmonics, resulting in residual series
Vtr , Krt , Grt , and Srt .
2πt
4πt
4πt
2πt
• Dt = d0 + d1 sin 24 + d2 cos 24 + d3 sin 24 + d4 cos 24
for t = 1, 2, . . . , 24.
8
7
6
Speed (m/s)
9
Fitted Diurnal Component
5
10
15
20
Hour
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M. Hering
The TDD Parameters
• The predictive center is modeled by
µt+2 = DVt+2 + µrt+2
where DVt+2 is the fitted diurnal component at Vansycle and
µrt+2
r
= a0 + a1 Vtr + a2 Vt−1
+ a3 Krt + a4 Krt−1 + a5 Grt
r
r
r
)
) + a8 sin(θK,t
) + a7 cos(θV,t
+a6 sin(θV,t
r
r
r
+a9 cos(θK,t
) + a10 sin(θG,t
) + a11 cos(θG,t
)
• Frequent changes in volatility are modeled by regressing σt+2 as a
linear function of the volatility value,
vt =
1
”
1 X“ r
r
2
r
r
2
r
r
2
(Vt−i − Vt−i−1 ) + (Kt−i − Kt−i−1 ) + (Gt−i − Gt−i−1 )
6 i=0
• Then, V̂t+2 = µ̂+
t+2 = µ̂t+2 + σ̂t+2 · φ
forecast for the mean.
µ̂t+2
σ̂t+2
/Φ
µ̂t+2
σ̂t+2
!1/2
is the
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M. Hering
Comparing the Speed Predictions
• Speed predictions are commonly compared with Root Mean Squared
Error (RMSE) and Mean Absolute Error (MAE).
• However,we want to incoporate the basic relationship between wind
speed and wind power:
0.5
MW
1.0
1.5
GE 1.5 MW Power Curve
Zone 3
Zone 2
Zone 4
0.0
Zone 1
0
5
10
15
20
25
30
Wind Speed (m/s)
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M. Hering
Power Curve Error Measure
• Estimates of the true power are made using the observed wind
speeds, and estimates of the predicted power are made using the
predicted wind speeds.
• A nonparametric regression estimate is used to predict powers in
Zone 2.
• Let g(·) be the nondecreasing function that yields power estimates.
• We form a loss function that can penalize underestimates differently
than overestimates.


p · (g(y) − g(ŷ)),
ŷ ≤ y
L(y, ŷ) =
,
(1)
 (1 − p) · (g(ŷ) − g(y)), ŷ ≥ y
Then the Power Curve Error, or PCE =
1
n
Pn
i=1
L(yi , ŷi ).
NCAR; 05.12.10
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M. Hering
Optimizing Forecasts
The optimal forecast that minimizes a particular loss function is
ŷ = arg miny EF [Li (y, Y)].
• For a quadratic loss (MSE), the optimal forecast is the mean of the
predictive distribution, F.
• For an absolute loss (MAE), the optimal forecast is the median of
the predictive distribution, F.
• For the Power Curve Error loss, the optimal forecast is the pth
quantile of the predictive distribution (Gneiting, 2010) since this loss
is of the Generalized Piecewise Linear form.
The TDD model’s predictive distribution is the truncated normal.
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M. Hering
The Predictive Distribution
0.20
0.25
TDD Model Produces Best Forecast
RSTD Predictive Distribution
0.15
TDD Predictive Distribution
Density
BST Predictive Distribution
RSTD Forecast
0.10
TDD Forecast
0.00
0.05
BST Forecast
0
5
10
15
20
25
30
Wind Speed (m/s)
NCAR; 05.12.10
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M. Hering
Influence of Penalty on PCE Loss
Small p → smaller penalty on underestimates.
Large p → larger penalty on underestimates.
p
0.01
0.10
0.50
0.90
0.99
Forecast
Overall
RST
5.49
TDD
5.48
BST
6.05
RST
30.40
TDD
30.27
BST
30.10
RST
70.00
TDD
69.46
BST
70.35
RST
37.56
TDD
36.67
BST
40.14
RST
14.90
TDD
14.43
BST
15.24
NCAR; 05.12.10
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M. Hering
Model Comparison Summary
Forecast evaluation on 2006 testing data for horizon t + 2.
RMSE
MAE
PCE
CRPS
Mod
VS
KW
GH
SH
PER
2.30
2.46
1.92
2.07
RST
2.03
2.30
1.75
1.95
TDD
2.01
2.30
1.74
1.93
PER
1.67
1.77
1.44
1.53
RST
1.49
1.69
1.32
1.45
TDD
1.48
1.68
1.33
1.45
PER
83.4
88.4
90.4
78.4
RST
64.2
74.3
72.5
67.5
TDD
62.9
73.5
72.4
66.8
RST
1.07
1.21
0.95
1.04
TDD
1.07
1.21
0.95
1.05
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M. Hering
Climate Change Impacts on Wind Resource
NARCCAP–North American Regional Climate Change Assessment
Program, spatial resolution of 50 km, 3 hourly wind fields
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M. Hering
Climate Change Impacts on Wind Resource
Three Main Focus Areas:
• Verification of NCEP-Driven Runs:
– 1980-2004, 25 years, 2-D U & V Components
– Does the RCM adequately duplicate the observed wind resource?
• Creation of Hub Height Wind Fields on GFDL-Driven Runs:
– 1971-2000 (Current, 30 years) and 2041-2070 (Future, 30 years),
3-D U &V Components
– Are there any spatial and/or temporal shifts in the wind resource
comparing the current to the future?
• Validation of Atmospheric Wind Processes:
– Are subfeatures such as the North American Monsoon, Pineapple
Express, or San Francisco Delta Breezes recreated by the RCM?
NCAR; 05.12.10
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M. Hering
Forecasting Categorical Wind Power Changes
For short-term forecasts of winds (1-3 hours), utility operators plan for
ramping events, scheduling, and transmission.
A single forecast of average wind power expected in the next hour is
insufficient, and perhaps a suite of various types of forecasts is more
useful.
• Forecast of average hourly wind power
• Uncertainty estimate of average hourly wind power forecast
• Forecast of the probability of a ramping event
• Forecast of the probability of an increase, decrease, or no change in
wind power
• Others....
NCAR; 05.12.10
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M. Hering
Forecasting Categorical Wind Power Changes
True Change: Wind speed → wind power → t + 1 power changes
Forecast Change: TDD Model t + 1 wind speed forecasts → power
forecasts → power changes
True Change
Increase
Same
Decrease
16.95%
2.31%
13.11%
(1485)
(202)
(1148)
2.98%
27.91%
1.35%
(261)
(2445)
(118)
11.62%
4.05%
19.72%
(1018)
(355)
(1727)
Increase
Forecast
Same
Change
Decrease
Total Correct Classifications: 64.59%
Total Mis-Classifications: 34.42%
NCAR; 05.12.10
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M. Hering
Forecasting Categorical Wind Power Changes
Why not use quantitative forecasts of wind speed or wind power to
predict an upcoming change in wind power?
• We don’t get an estimated probability for a change in power.
– For example, quantitative forecasts would be coded as
0=decrease, 1=increase, and 2=no change.
– A forecast of “60% chance of an increase, 30% of a decrease,
and 10% of no change” is more informative.
• We don’t get a prediction interval for the probability of a change in
power.
• We may reduce misclassfications by using models targeted to
forecast such a categorical variable, like multicategorical multiple
autologistic regression models.
NCAR; 05.12.10
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M. Hering
Varying Coefficient Statistical Model (VCVAR)
Bin0
Bin180
Bin45
Bin225
Bin90
Bin270
Bin135
Bin315
8
6
4
2
Speed (m/s)
10
12
VS Hourly Mean Speed Binned by KW Dir
5
10
15
20
Hour
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M. Hering
Varying Coefficient Statistical Model (VCVAR)
• Goal of this model: Make short-term forecasts (1-3 hours) of both
speed and direction at a sparse number of spatial locations.
• It makes sense in a wind forecasting application that the
relationships between the u and v components observed at each
location may change based on the wind direction, thus changing the
coefficients. This motivates the following type of model:
• A detrended vector of u & v components at each location is
modeled by
p
X
wt+1 =
Ai (θt ) wt−i+1 + ǫt ,
i=1
where θt is the wind direction at the current time at on off-site
location.
NCAR; 05.12.10
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M. Hering
Estimating Parameters in the VCVAR
In a 45 day window of observations before each forecast,
1. Select several fitting points between 0 and 360 degrees.
2. Estimate the coefficients at each fitting point with weighted least
squares, giving larger weight to u & v components whose wind
direction is closer to the fitting point.
3. To make a forecast given the current wind direction, take a weighted
average of the coefficients for the two closest fitting points.
Extra decisions to make with this model, in addition to the order:
• Selection of the number of fitting points.
• Selection of a nonparametric bandwidth with which to assign the
weights.
NCAR; 05.12.10
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M. Hering
Varying-Coefficient Statistical Model
Example for p=1
FP2
FP3
FP1
A1(fp3)
FP4
A1(fp4)
Results from this model have not been great with the dataset we have,
but this may be due to two factors:
(1) Wind directions are primarily from the west.
(2) Wind direction is more difficult to forecast.
NCAR; 05.12.10
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M. Hering
Thank you!
NCAR; 05.12.10
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M. Hering
What is CRPS Estimation?
If F is the predictive CDF and x is a realization, the continuous ranked
probability score is
Z ∞
2
(F(y) − 1(y ≥ x)) dy.
crps(F, x) =
−∞
The crps for the truncated normal distribution can be written explicitly,
and the parameters in the model are estimated by finding the minimum
value of CRPS where
n
1X
CRPS =
crps(Fi , xi ).
n
i=1
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