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

1. 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@imm.dtu.dk

2. 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) Improving wind power forecasts by considering spatio-temporal effects -Slide 2 / 12

3. 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 Improving wind power forecasts by considering spatio-temporal effects -Slide 3 / 12

4. 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 Improving wind power forecasts by considering spatio-temporal effects -Slide 4 / 12

5. 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. Improving wind power forecasts by considering spatio-temporal effects -Slide 5 / 12

6. 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 Improving wind power forecasts by considering spatio-temporal effects -Slide 6 / 12

7. 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/ Improving wind power forecasts by considering spatio-temporal effects -Slide 7 / 12

8. 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 Improving wind power forecasts by considering spatio-temporal effects -Slide 8 / 12

9. 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) Improving wind power forecasts by considering spatio-temporal effects -Slide 9 / 12

10. 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. Improving wind power forecasts by considering spatio-temporal effects -Slide 10 / 12

11. 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 Improving wind power forecasts by considering spatio-temporal effects -Slide 11 / 12

12. Thank you for your attention! Contact information: Julija Tastu PhD student Informatics and Mathematical Modelling, Technical University of Denmark Email to: jvl@imm.dtu.dk Improving wind power forecasts by considering spatio-temporal effects -Slide 12 / 12