NREL is a national laboratory of the U.S. Department of Energy, Office of Energy Efficiency and Renewable Energy, operated by the Alliance for Sustainable Energy, LLC.

Wind Power Forecasting Error 

Distributions over Multiple 

Timescales

IEEE Power and Energy Society 

General Meeting

July 27, 2011

Bri-Mathias Hodge

Michael Milligan

National Renewable Energy Laboratory

Golden, Colorado USA

NREL/PR-5500-52294

PIX 06588

PIX 10926
Topics

• Statistical background;

• Dataset background;

• Histograms;

• Q-Q plots;

• Kurtosis and skewness;

• Distribution fitting;

• Confidence intervals.

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PIX 14820
Statistical Background

• Skewness: measure of the asymmetry of the distribution, the 

3rd standardized moment.

• Negative skew – longer tail on left, but more mass on right.

• Positive skew – longer tail on right, but more mass on left.

• Kurtosis: measure of the peakedness of the distributions, the 

4th standardized moment.

• Leptokurtic – more peaked.

• Platykurtic – more flat. 

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













 



 



 − =

3

σ

μ γ

X E

4

4 ( )

σ

ε

κ

E

=
Parametric Statistical Distributions

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0.0 0.2 0.4 0.6 0.8 1.0

0 1 2

3

4

5

6

x

Density

• Normal – Blue: 

μ = 0.5, σ = 0.2

• Beta – Red: 

α = β = 5

• Weibull – Green: 

κ = 5, λ = 0.5

• Cauchy – Black: 

x0 = 0.5, γ = 0.05
ERCOT Dataset

• One-minute power output 

data. 

• 10 ERCOT plants.

• Range in size from 30 MW to 

215 MW.

• Also examined aggregation 

of all ten wind plants – total 

capacity ~ 940 MW.

• Two four month periods:

- Winter (January – April);

- Summer (June – September).

• Persistence model utilized:

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)()|(ˆ =+ tPtktP

PIX 16112 & 18207
Forecasting Results − Histograms

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Wind Plant # 3 - 5 minute - Winter : γ = -2.14, κ = 151.25 
Wind Plant # 8 – One Hour

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Summer: γ = -0.02, κ = 6.18 
Normal Quantile-Quantile Plot – Plant #1

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1 minute - Summer : γ = -0.45, κ = 1717.81 
Wind Plant # 6 – One Hour

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Summer: γ = -0.08, κ = 8.04 
Combined ERCOT – Three Hours

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Summer: γ = 0.02, κ = 0.82 
Kurtosis Values

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Skewness Values

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Distribution Fitting

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ERCOT Data - 15 Minute 

x

Density

0.0 0.2 0.4 0.6 0.8 1.0

0 10 20 30 40

• Beta – Blue: 

α = 82.08 

β = 82.14

• Weibull – Green: 

k = 9.44 

λ = 0.52

• Cauchy – Red: 

xo = 0.4999 

γ = 0.0069

Cauchy produces more 

accurate log-likelihood 

values than the beta and 

Weibull distributions in 89% 

and 95% of the cases. 
Comparison of Confidence Intervals

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• Black – Point forecast.

• Blue – Normal 

distribution.

• Red – Cauchy 

distribution.

• Lighter shade: 

• 90% confidence interval.

• Darker shade: 

• 95% confidence interval.

0 10 20 30 40 50

0 200 400 600 800 1000

ERCOT Power Forecasts

Time Period (15 Minute Intervals)

Power (MW)
Questions?

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PIX 17731 & 16105