Dynamic Modelling for Wind Prediction · IntroductionDataMethodologyResultsAnalysisConclusionsFurther ResearchThank you Dynamic Modelling for Wind Prediction Rachael Gri ths { Ben

Introduction Data Methodology Results Analysis Conclusions Further Research Thank you

Dynamic Modelling for Wind Prediction

Rachael Griffiths – Ben Taylor

Lancaster University

2nd September 2010

Rachael Griffiths – Ben Taylor Lancaster University

Wind Prediction


Introduction

Why do we need to predict wind speed?

How can we predict wind speed?

Aim

To develop a dynamic model, capable of predicting the wind speedat a new station, using wind speed data observed at a near byreference station.


Wind Prediction


Introduction



Aim



Wind Prediction


Introduction



Aim



Wind Prediction


The Data

The data used came from the British Atmospheric Data Centreand was recorded by The UK Met Office [BADC, 2006].

Hourly wind speed observations from stations in the UK, from1985 to date.

Reference Station - Coningsby

New Station - Nottingham

Observation Period - 1st March 00:00 - 31st March 23:50,

Prediction Period - 1st April 00:00 - 2nd April 23:50, 1985.


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The Data-Time Series

(a) (b)

Figure: a) Map of Station Locations and b) Time Series of Observed Wind

Speeds throughout March, at both stationsRachael Griffiths – Ben Taylor Lancaster University

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The State-Space Model

Θt = Θt−1 + Wt (1)

Yt = Θt + Vt (2)

where,

Wt ∼ MVN(0,ΣW),Vt ∼ MVN(0,ΣV),

ΣW =

[σ2new γγ σ2

ref

],ΣV =

[σ2Y 00 σ2

Y

].

[Chatfield, 1996]


Wind Prediction



Θt = Θt−1 + Wt (1)

Yt = Θt + Vt (2)

where,


ΣW =

[σ2new γγ σ2

ref

],ΣV =

[σ2Y 00 σ2

Y

].

[Chatfield, 1996]


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Θt = Θt−1 + Wt (1)

Yt = Θt + Vt (2)

where,


ΣW =

[σ2new γγ σ2

ref

],ΣV =

[σ2Y 00 σ2

Y

].

[Chatfield, 1996]


Wind Prediction


Modelling Process

Our main interest is in the posterior distribution, π(ΘT |Y1:t),where T > t.

If π(Θs |Y1, ...,Ys) at time s is Normal, then

1 π(Θs+1|Y1, ...,Ys) is Normal, and

2 since π(Yr |Θr ) is Normal, for any r , thenπ(Θs+1|Y1, ...,Ys+1) is also Normal.


Wind Prediction


Modelling Process

Our main interest is in the posterior distribution, π(ΘT |Y1:t),where T > t.

If π(Θs |Y1, ...,Ys) at time s is Normal, then

1 π(Θs+1|Y1, ...,Ys) is Normal, and

2 since π(Yr |Θr ) is Normal, for any r , thenπ(Θs+1|Y1, ...,Ys+1) is also Normal.


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Modelling Process Outline

1 Uninformed Normal prior for observation period.

2 Estimate fixed parameters and posterior, using the observationperiod.

3 Use posterior from observation period as prior for predictionperiod.

4 Predict wind speeds at new station in prediction period usingdata from only the reference station, and the estimatedparameters.

5 Assess the model performance by comparison with existingmodels.


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Kalman Filter

Uninformed Prior Density

Θ0 ∼ MVN

([88

],

[50 0

0 50

])(3)

Kalman Filter

Set of recursive equations which estimate π(Θt |Y1, ...,Yt)[Chatfield, 1996].

Gives the marginal likelihood in closed form.

Gives the recursions necessary to compute the new mean andcovariance for each π(Θt |Y1, ...,Yt), given the old values fromπ(Θt−1|Y1, ...,Yt−1).

Able to cope with irregular data.


Wind Prediction


Kalman Filter

Uninformed Prior Density

Θ0 ∼ MVN

([88

],

[50 0

0 50

])(3)

Kalman Filter

Set of recursive equations which estimate π(Θt |Y1, ...,Yt)[Chatfield, 1996].

Gives the marginal likelihood in closed form.

Gives the recursions necessary to compute the new mean andcovariance for each π(Θt |Y1, ...,Yt), given the old values fromπ(Θt−1|Y1, ...,Yt−1).

Able to cope with irregular data.


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Parameters

Parameter Estimates

σnew = 1.202

σref = 1.531

γ = 1.150

σY = 1.298

Prediction Model Variables

Θt =

[Θnew

t

Θreft

]Yt =

[Y reft

]


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Parameters

Parameter Estimates

σnew = 1.202

σref = 1.531

γ = 1.150

σY = 1.298

Prediction Model Variables

Θt =

[Θnew

t

Θreft

]Yt =

[Y reft

]Rachael Griffiths – Ben Taylor Lancaster University

Wind Prediction


The Prediction Model

Θt = Θt−1 + Wt (4)

Yt =[

0 1]

Θt +[

0 1]

Vt (5)

where,

Wt ∼ MVN(0, ΣW),Vt ∼ MVN(0, ΣV),

ΣW =

[(1.202)2 1.150

1.150 (1.531)2

],ΣV =

[(1.298)2 0

0 (1.298)2

].


Wind Prediction


The Prediction Model

Θt = Θt−1 + Wt (4)

Yt =[

0 1]

Θt +[

0 1]

Vt (5)

where,

Wt ∼ MVN(0, ΣW),Vt ∼ MVN(0, ΣV),

ΣW =

[(1.202)2 1.150

1.150 (1.531)2

],ΣV =

[(1.298)2 0

0 (1.298)2

].


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Fitted Model

Figure: Coningsby - Observed and Fitted Wind Speeds, with 95% CI

Figure: Nottingham - Observed and Fitted Wind Speeds, with 95% CIRachael Griffiths – Ben Taylor Lancaster University

Wind Prediction


Prediction

Figure: Predicted Wind Speeds, with 95 % CI, at New Station (Nottingham)-

April 1st to 3rd


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Fitted for Reference Station

Figure: Fitted Wind Speed, with 95% CI, at Reference Station (Coningsby) -

April 1st to 3rd


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The Derrick(1) Model

Ynew = A + BYref (6)

Derrick(1)

Linear regression between wind speed observations atreference station and wind speed observations at new station,in the observation period.

Use coefficients to predict wind speed at new station inprediction period, using wind speed observations at referencestation.

[Derrick, 1992]


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The Derrick(2) Model

Ynew = Ai + BiYref (7)

where i = 1, ..., 12, indicates the direction bin.

Derrick(2)

Using wind directions recorded at the reference station.

Divide observation data into 12 30-degree direction bins.

Linear regression to fit 12 separate models (one for each bin)in the observation period.

Predict wind speeds at new station in the prediction period,using reference station data divided into the 12 bins.

[Derrick, 1992]


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Derrick Model Results

Figure: Nottingham - Observed and Predicted Wind Speeds in April for all

models, plus 95% CI for KF predictions


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Derrick Error Analysis

Model Month Station MSE MAE

KF April (predicted) Nottingham 6.510 2.102Derrick(1) 8.281 2.418Derrick(2) 9.056 2.331

Table: MSEs and MAEs of the Predictions from all models


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Conclusions

The KF model appears to perform better than both the Derrickmodels.

MAE of 2.102 is much smaller than other models (around10% less than both).

MSE of 6.510 is between 20% and 30% less than the MSEvalues for both other models.

The Derrick model does not use the time dependance of the windspeeds.


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Further Research Possibilities

Further Testing

Evaluate model using data from different stations and duringdifferent time periods.

Assess model against other types of models.

Extend Model

More reference stations.

More covariates, such as wind direction or elevation.

Extend the wind speed model, e.g. use AR(2), ARMA...


Wind Prediction


Further Research Possibilities

Further Testing

Evaluate model using data from different stations and duringdifferent time periods.

Assess model against other types of models.

Extend Model

More reference stations.

More covariates, such as wind direction or elevation.

Extend the wind speed model, e.g. use AR(2), ARMA...


Wind Prediction


Thank you

Any questions?

References

Chatfield, C. (1996).

The Analysis of Time Series; An Introduction.

British Atmospheric Data Centre. (2006).

UK Meteorological Office. MIDAS Land Surface Stations data(1853-current). Available from http://badc.nerc.ac.uk/view/badc.

nerc.ac.uk__ATOM__dataent__ukmo-midas.

Derrick, A. (1992).

Development of the Measure-Correlate-Predict Strategy for SiteAssessment.


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http://badc.nerc.ac.uk/view/badc.nerc.ac.uk__ATOM__dataent__ukmo-midas

http://badc.nerc.ac.uk/view/badc.nerc.ac.uk__ATOM__dataent__ukmo-midas

Documents

Dynamic Modelling for Wind Prediction · IntroductionDataMethodologyResultsAnalysisConclusionsFurther ResearchThank you Dynamic Modelling for Wind Prediction Rachael Gri ths { Ben