H. Madsen, J.Tastu*, P.Pinson

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Multivariate Conditional Parametric models for
aspatio-temporal analysis of short-term wind
power forecasterrors

• H. Madsen, J.Tastu, P.Pinson

Informatics and Mathematical Modelling, Technical
University of Denmark jvl_at_imm.dtu.dk
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The spatio-temporal effects

• An error made at a certain point in space and
  time will propagate both spatially and
  temporally, conditional to the weather conditions
• The objective is to improve wind power forecasts
  over the region by accounting for such
  spatio-temporal effects
• to provide corrected point forecasts
• accompanied with the estimates of the associated
  uncertainty level (probabilistic forecasts)

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Outline

• Energinet.dk dataset
• Correcting point forecasts based on
  spatio-temporal aspects
• Vector AutoRegressive model (VAR)
• Conditional Parametric-VAR model (CP-VAR)
• Application results and comments
• Probabilistic forecasts
• Parametric approach based on truncated
  multivariate normal distribution
• Assessment of the probabilistic forecasts
• Application results and comments

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The Energinet dataset

• 23 months (2006-2007)
• 15 onshore groups
• wind power forecasts (WPPT) and
• measurements
• meteorological forecasts
• Focus here on 1-hour-ahead forecast errors

- for more details on WPPT see www.enfor.dk
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Point forecasts. Proposed models

• For revealing a linear spatio-temporal
  inter-dependence structure without accounting for
  the effects of the meteorological data a VAR
  model can be employed
• where wt is a vector of the dimension m 1
  showing wind power forecast errors at m groups
  of wind farms obtained for time t, ?t term is
  assumed to be distributed multivariate with zero
  mean, A is a coefficient matrix to be estimated
  from the data.
• The effect of wind direction may be captured with
  a CP-VAR model
• which translates to replace the Ai coefficients
  in VAR model by coefficient functions. The model
  can be fitted to data with zt being an average
  forecasted wind direction for
• time t.

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Application Results 1/2

• Results in terms or RMSE of the 1-hour-ahead
  forecasts, and related improvement of the RMSE (?
  RMSE) as a result of the overall forecast
  correction methodology

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Application Results 2/2

• Predictive performance of the CP-VAR model in
  terms of a percentage reduction in the RMSE of
  the forecast errors.
• The larger improvements correspond to the eastern
  part of the region. This is in line with the fact
  that in Denmark the prevailing wind direction is
  westerly, so the easterly located groups are
  usually situated "down-wind" from the rest of the
  region.
• A new EU project NORSEWInD will include the data
  from the North Sea which can potentially result
  in better improvements for all Denmark area.

Produced using http//maps.google.com/
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Probabilistic forecasts

• The main objective is to estimate the uncertainty
  associated with the previously presented point
  forecasts by providing the probability density
  function of the corresponding random variable
• Parametric approach based on truncated
  MultiVariate Normal (MVN) distribution
• where the mean of the distribution is assumed to
  be equal to the point forecast of the wind power
  forecast errors obtained from the CP-VAR model.
• b(ppt) and a(ppt) are vectors denoting upper and
  lower truncation limits of the distribution. Here
  ppt denotes a vector of power predictions for
  time t.
• ? is a covariance matrix of the distribution
  (conditional on the meteo forecast zt) to be
  estimated from the data

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Assessment of the probabilistic forecasts

• Making probabilistic forecasts for each group
  individually, based on the corresponding
  univariate marginals of the estimated MVN density
• Reliability assessment for a multidimensional
  forecast is more complex and requires additional
  research (future work)

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Application results

• Episodes with forecasts and measurements
  corresponding to a one-week period from 4 a.m.
  2007-08-03 to 4 a.m. 2007-08-10 for Group 5.

The width of the prediction intervals changes in
time the larger is the variation in the WPPT
errors, the wider are the prediction intervals.
The actual value depends on historical values and
forecasted wind direction.
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Conclusions

• Spatio-temporal effects can be employed for
  improving wind power forecasts over the Denmark
  area
• CP-VAR models have been tested and evaluated. The
  correspondingly corrected 1-hour-ahead forecasts
  when evaluated on the test case of western
  Denmark result in a reduction of prediction
  errors up to 18.46 in terms of RMSE.
• The forecast error depends on a number of
  factors
• Most recent errors
• Forecasted wind direction
• Predictive densities are modelled as truncated
  multivariate normal distribution. This results in
  reliable univariate probabilistic forecasts for
  each individual group.
• In the future, explanatory variables from the
  North Sea region should also be integrated in the
  proposed methodologies
• Forecast correction will then be extended to
  further look-ahead times

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Thank you for your attention!
Contact information Julija Tastu PhD
student Informatics and Mathematical Modelling,
Technical University of Denmark Email to
jvl_at_imm.dtu.dk
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