Probabilistic Forecasting of the Wind Power Generation ... · Introduction Modelling Application case References Probabilistic Forecasting of the Wind Power Generation Using Kernel

IntroductionModelling

Application caseReferences

Probabilistic Forecasting of the Wind PowerGeneration Using Kernel Density Estimation

Soraida Aguilar1,Reinaldo Castro Souza1 and Jose Francisco Pessanha2

1Pontifical Catholic University of Rio de JaneiroPUC-Rio

2Rio de Janeiro State UniversityUERJ

34th International Symposium on Forecasting 2014Rotterdam, The Netherlands

Aguilar, Souza, Pessanha Probabilistic Forecasting



Summary

1 Introduction

2 Modelling

3 Application case

4 References




Summary

1 Introduction

2 Modelling

3 Application case

4 References




Motivation

Wind power probability density function instead of a pointforecast.

The random nature of wind makes its prediction a very com-plex task.

The operations planning of power systems are affected byrandom nature of wind speed.

Accurate predictions are obtained in order to minimize tech-nical and financial risks.




Motivation








Motivation








Motivation








Motivation








Wind Energy Forecasting

Usually wind power is predicted in two stages:

i) Fitting a model: physical, statistical, computational in-telligence or hybrid to forecast wind speed.

ii) Using the power curve provided by the manufacturer ofthe turbine and the wind speed forecasted in the 1st stageto obtain the wind power generation.

























Objective

Our aim is developing a full probabilistic density forecast forthe wind power split in two stages; on the first the wind speedforecasting are generated by traditional univariate time seriesmethods for each lead time. Such forecasts are then taken intothe second stage to generate the wind energy density forecastingdistribution.




Probabilistic Forecasting

Work on probabilistic forecasting

Quantile regressionBremnes (2004), Nielsen et al.(2006), Bremnes(2006), Juban et al. (2007a),Møller et al. (2008), Bessa et al. (2011a, 2011b), Pritchard (2011), Bessa etal. (2012a), Liu et al. (2012), Anastasiades and McSharry (2013), Jonsson etal. (2013).

Conditional kernel density estimationJuban et al. (2007b), Bessa et al.(2012a, 2012b), Jeon and Taylor(2012).

Ensemble weather predictionTaylor et al. (2009), Pinson and Madsen (2009), Pinson et al. (2009b),Sloughter et al. (2010), Thorarinsdottir and Gneiting (2010) and Al-Yahyaiet al. (2012).




























Summary

1 Introduction

2 Modelling

3 Application case

4 References




Kernel Density Estimation

The Nadaraya-Watson Estimator

Conditional density estimation provides the assessment of the pdf of a random vari-able Y, given an explanatory variable X with the value of x known to be predictedfor time t + k given the information available at time t :

fY (Yt+k|Xt) =fY,X(Yt+k, Xt)

fX(Xt)(1)

f(y|X = x) =N∑i+1

Khy (y − Y ) · wi(x) (2)

wi(x) =Khx (x−Xi)∑Ni=1Khx (x−Xi)

(3)





Figure 1: Conditional density estimation of wind power on wind speed.





Figure 2: Example of wind power forecast pdffor a wind speed of 7 m/s.

Figure 3: Example of wind power forecast pdffor a wind speed of 12 m/s.




Modelling Wind Speed

Wind speed presents seasonal patterns and a high variability.

Multiplicative ARIMA Model

A process multiplicative ARIMA (Box and Jenkins, 1976) can be represented bythe equation:

φp(B)ΦP (Bs)∆d∆Ds yt = c+ θq(B)ΘQ(Bs)εt (4)

Double Seasonal Holt-Winters Exponential Smoothing methods

This is an adaptation of the Holt-Winters method to incorporate two cycles ratherthan just one (Taylor, 2003), which is represented by:

Level→ St = α

(Xt

Dt−S1Wt−S2

)+ (1− α)(St−1 + Tt−1) (5)

Trend→ Tt = γ(St − St−1) + (1− γ)Tt−1 (6)




Modelling Wind Speed

Seasonality 1→ Dt = δ

(Xt

StWt−S2

)+ (1− δ)Dt−S1 (7)

Seasonality 2→Wt = ω

(Xt

StDt−S1

)+ (1− ω)Wt−S2

(8)

Forecasting→ Xt(k) = (St + kTt)Dt−S1+k ∗Wt−S2+k (9)




Summary

1 Introduction

2 Modelling

3 Application case

4 References




Sample Time Series

Figure 4: Wind speed (m/s).




Sample Time Series

Figure 5: Power curve.




Fitting Models

Using SARIMA model we need first analyse:

Figure 6: Autocorrelation function (ACF) and partial autocorrelation func-tion(PACF) of Wind Speed time series.




Fitting Models

Fitting SARIMA(2, 0, 0)x(1, 1, 1)24Coefficiente s.e. z p-value

φ1 0.7995 0.01083 73.84 0.0000 ***φ1 -0.0256 0.01062 -2.41 0.0159 **Φ1 0.0679 0.01161 5.85 4.80e-09 ***Θ1 -0.9411 0.00429 -219.5 0.0000 ***

Best goodness-of-fit testLog likelihood = -16706.96Akaike criterion = 33423.92

Schwarz criterion = 33459.30Hannan-Quinn criterion = 33435.97

Fitting Double Seasonal Holt-Winters model

Coefficiente Valueα 0.01795544γ 0.00145423δ 1.8891e-07ω 0.23845561




Wind Speed Forecasting

After fitting the SARIMA and Double Seasonal Holt-Winters models for windspeed, the next step is forecasting 24 hours ahead.

Figure 7: Wind speed (m/s) forecasting 24 hours ahead.




Wind Power Forecasting

Figure 8: Wind power (m/s) forecasting 24 hours ahead.




Error Measures - Wind Speed/Wind Power Generation

SARIMA(2, 0, 0)x(1, 1, 1)24 model

In-Sample Out-of-SampleRMSE (m/s) 1.6397 1.5629MAE (m/s) 1.2358 1.2693UTHEIL 0.3694 0.6933

Double Seasonal Holt-Winter

In-Sample Out-of-SampleRMSE (m/s) 2.5713 1.8467MAE (m/s) 1.9741 1.4500UTHEIL 1.2662 0.8191

Error Measures - Wind Power Generation

SARIMA Double SHWRMSE (kW ) 359.0157 464.7256MAE (kW ) 259.0089 330.4373UTHEIL 0.722674 0.935461




Wind Power Forecasting

Figure 9: Wind power output of the Nadaraya-Watson estimator.




Conclusions

The results show that the process of probabilistic forecastsof wind generation is consistent.

This density has a good fit because provide reasonable pointforecasting of the wind power output, which is validated withthe error measures that were reasonably good.

The nonlinear nature of the wind speed series is an indicationthat other models should be tested.

Others methodologies of conditional density estimationshould be estimated in order to improve the results whenthe mean is provide.




Summary

1 Introduction

2 Modelling

3 Application case

4 References




References

Al-Yahyai, S., Gastli, A., and Charabi, Y. (2012). Probabilistic wind speedforecast for wind power prediction using pseudo ensemble approach.In 2012 IEEE International Conference on Power and Energy (PECon). KotaKinabalu Sabah, Malaysia, (pp. 127-132).

Anastasiades, G and McSharry, P. (2013). Quantile Forecasting of WindPower Using Variability Indices. Energies, 6, 662-695.

Bessa, R. J., Sumaili, J., Miranda, V., Botterud, A., Wang, J., and Con-stantinescu, E. (2011a). Time-adaptive kernel density forecast: A newmethod for wind power uncertainty modeling. In 17th Power SystemsComputation Conference. Stockholm, Sweden.

Bessa, R. J., Mendes, J., Miranda, V., Botterud, a., Wang, J., and Zhou, Z.(2011b). Quantile-copula density forecast for wind power uncertaintymodeling. In 2011 IEEE Trondheim PowerTech, Trondheim, Norway, (pp.1–8).

Bessa, R. J., Miranda, V., Botterud, A., Zhou, Z., Wang, J. (2012). Time-adaptive quantile-copula for wind power probabilistic forecasting.Renewable Energy, 40(1), 29–39.




References

Bessa, R. J., Miranda, V., Botterud, A., Wang, J., and Constantinescu, E.M. (2012b). Time adaptive conditional kernel density estimation forwind power forecasting. IEEE Transactions on Sustainable Energy, 3(4),660–669.

Box, G. E. P., and Jenkins, G. M. (1970). Time Series Analysis: Fore-casting and Control, Holden–Day Inc., San Francisco.

Bremnes, J.B. (2004). Probabilistic Wind Power Forecasts using LocalQuantile Regression. Wind Energy, 7(1), 47-54.

Bremnes, J.B. (2006). A Comparison of a Few Statistical Models forMaking Quantile Wind Power Forecasts. Wind Energy, 9(1-2), 3-11.

Jeon, J., and Taylor, J. W. (2012). Using Conditional Kernel Density Es-timation for Wind Power Density Forecasting. Journal of the AmericanStatistical Association, 107(497), 66–79.

Jonsson, T., Pinson P., Madsen, H., and Nielsen, H. A. (2013). PredictiveDensities for Day-Ahead Electricity Prices Using Time-AdaptiveQuantile Regression. Working Paper, Preprint submitted to Applied En-ergy.




References

Juban, J., Siebert, N., and Kariniotakis, G. N. (2007a). Probabilistic Short-term Wind Power Forecasting for the Optimal Management of WindGeneration. In 2007 IEEE Lausanne Power Tech (pp. 683–688).

Juban, J., Fugon, L., and Kariniotakis, G. (2007b). Probabilistic short-term wind power forecasting based on kernel density estimators.In Probabilistic wind power forecasting - European Wind Energy Conference.Milan, Italy, (pp. 1–11).

Liu, Y., Yan, J., Han, S., and Peng, Y. (2012). Uncertainty Analysisof Wind Power Prediction Based on Quantile Regression. In Powerand Energy Engineering Conference (APPEEC), 2012 Asia-Pacific (pp. 1–4).Shanghai.

Møller, J.K., Nielsen, H.A., and Madsen, H. (2008). Time-adaptive quan-tile regression. Computational Statistics and Data Analysis, 52(3),1292–1303.

Pritchard, G. (2011). Short-term variations in wind power Somequantile-type models for probabilistic forecasting. Wind Energy, 14(2),255-269.




References

Pinson, P., and Madsen, H. (2009), Ensemble-based probabilistic fore-casting at Horns Rev. Wind Energy, 12(2), 137-155.

Sloughter, J. M., Gneiting, T., and Raftery, A. E. (2010). Probabilistic WindSpeed Forecasting using Ensembles and Bayesian Model Averaging.Journal of the American Statistical Association, 105(489), 25-35.

Taylor, J. W., McSharry, P. E., and Buizza, R. (2009). Wind Power DensityForecasting using Ensemble Predictions and Time Series Models.IEEE Transactions on Energy Conversion, 24, 775-782.

Thorarinsdottir, T. L., and Gneiting, T. (2010). Probabilistic forecasts ofwind speed: ensemble model output statistics by using heteroscedas-tic censored regression. Journal of the Royal Statistical Society: Series A(Statistics in Society), 173(2), 371–388.




Acknowledgment

The authors would like to thank CAPES PEC-PG for theirfinancial supporting.


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Probabilistic Forecasting of the Wind Power Generation ... · Introduction Modelling Application case References Probabilistic Forecasting of the Wind Power Generation Using Kernel