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 1

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

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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 + σ · φ(µ

σ

)

/Φ(µ

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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Spatial Positions of 4 Sites

Site Locations

GH

KW

SHVS

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

Vrt, Kr

t, Grt, and Sr

t.

• Dt = d0 + d1 sin(

2πt24

)

+ d2 cos(

2πt24

)

+ d3 sin(

4πt24

)

+ d4 cos(

4πt24

)

for t = 1, 2, . . . , 24.

5 10 15 20

67

89

Fitted Diurnal Component

Hour

Spee

d (m/

s)

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M. Hering

The TDD Parameters

• The predictive center is modeled by

µt+2 = DVt+2 + µr

t+2

where DVt+2 is the fitted diurnal component at Vansycle and

µrt+2 = a0 + a1V

rt + a2V

rt−1 + a3K

rt + a4K

rt−1 + a5G

rt

+a6sin(θrV,t) + a7cos(θr

V,t) + a8sin(θrK,t)

+a9cos(θrK,t) + a10sin(θr

G,t) + a11cos(θrG,t)

• Frequent changes in volatility are modeled by regressing σt+2 as a

linear function of the volatility value,

vt =

1

6

1X

i=0

“

(Vrt−i − V

rt−i−1)

2 + (Krt−i − K

rt−i−1)

2 + (Grt−i − G

rt−i−1)

2”

!1/2

• Then, Vt+2 = µ+

t+2 = µt+2 + σt+2 · φ(

µt+2

σt+2

)

/Φ(

µt+2

σt+2

)

is the

forecast for the mean.

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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 10 15 20 25 30

0.0

0.5

1.0

1.5

GE 1.5 MW Power Curve

Wind Speed (m/s)

MW

Zone 1 Zone 2 Zone 3 Zone 4

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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.

L(y, y) =

p · (g(y) − g(y)), y ≤ y

(1 − p) · (g(y) − g(y)), y ≥ y, (1)

Then the Power Curve Error, or PCE = 1n

∑ni=1 L(yi, yi).

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M. Hering

Optimizing Forecasts

The optimal forecast that minimizes a particular loss function is

y = 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 5 10 15 20 25 30

0.0

00.0

50.1

00.1

50.2

00.2

5

Wind Speed (m/s)

Densi

ty

RSTD Predictive Distribution

TDD Predictive Distribution

BST Predictive Distribution

TDD Model Produces Best Forecast

RSTD Forecast

TDD Forecast

BST Forecast

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M. Hering

Influence of Penalty on PCE LossSmall p → smaller penalty on underestimates.

Large p → larger penalty on underestimates.

p Forecast Overall

RST 5.49

0.01 TDD 5.48

BST 6.05

RST 30.40

0.10 TDD 30.27

BST 30.10

RST 70.00

0.50 TDD 69.46

BST 70.35

RST 37.56

0.90 TDD 36.67

BST 40.14

RST 14.90

0.99 TDD 14.43

BST 15.24

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M. Hering

Model Comparison Summary

Forecast evaluation on 2006 testing data for horizon t + 2.

Mod VS KW GH SH

PER 2.30 2.46 1.92 2.07

RMSE 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

MAE 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

PCE RST 64.2 74.3 72.5 67.5

TDD 62.9 73.5 72.4 66.8

CRPS RST 1.07 1.21 0.95 1.04

TDD 1.07 1.21 0.95 1.05

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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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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?

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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....

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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

Increase 16.95% 2.31% 13.11%

(1485) (202) (1148)

Forecast Same 2.98% 27.91% 1.35%

Change (261) (2445) (118)

Decrease 11.62% 4.05% 19.72%

(1018) (355) (1727)

Total Correct Classifications: 64.59%

Total Mis-Classifications: 34.42%

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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 17

M. Hering

Varying Coefficient Statistical Model (VCVAR)

5 10 15 20

24

68

1012 Bin0

Bin45

Bin90

Bin135

Bin180

Bin225

Bin270

Bin315

VS Hourly Mean Speed Binned by KW Dir

Hour

Spe

ed (m

/s)

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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

wt+1 =

p∑

i=1

Ai(θt) wt−i+1 + ǫt,

where θt is the wind direction at the current time at on off-site

location.

NCAR; 05.12.10 19

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.

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M. Hering

Varying-Coefficient Statistical Model

FP1FP3

A1(fp3)

FP2FP2

FP4

Example for p=1

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 21

M. Hering

Thank you!

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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

crps(F, x) =

∫

∞

−∞

(F(y) − 1(y ≥ x))2

dy.

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

CRPS =1

n

n∑

i=1

crps(Fi, xi).

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