How To Calculate Electricity Load Data

Transcription

1 Combining Forecasts for Short Term Electricity Load Forecasting Weight Step M. Devaine, P. Gaillard, Y.Goude, G. Stoltz ENS Paris - EDF R&D - CNRS, INRIA (ENS Paris - EDF R&D - CNRS, INRIA) 11/02/ / 28

2 Motivation of Electricity Load Forecasting Electricity can not be stored, thus forecasting elec. consumption: to avoid blackouts on the grid to avoid financial penalties to optimize the management of production units and electricity trading Managing a wild variety of production units: nuclear plants fuel, coal and gas plants renewable energy: water dams, wind farms, solar panels... (ENS Paris - EDF R&D - CNRS, INRIA) 11/02/ / 28

3 Motivation of Electricity Load Forecasting Short-term load forecasting: from 1 day to a few hours horizon (ENS Paris - EDF R&D - CNRS, INRIA) 11/02/ / 28

4 Application to Electricity Load Data Electricity Data Trend 1/9/ /1/ /5/2003 9/10/ /2/2004 4/7/ /11/ /3/ /8/ /12/2005 8/5/ /9/2006 1/2/ /6/ /10/ /3/ /7/2008 4/12/ /4/ /8/ (ENS Paris - EDF R&D - CNRS, INRIA) 11/02/ / 28

5 Application to Electricity Load Data Electricity Data Yearly Pattern 1/1/ /1/2006 8/2/ /2/ /3/2006 7/4/ /4/ /5/2006 3/6/ /6/ /7/ /7/ /8/2006 7/9/ /9/ /10/2006 4/11/ /11/ /12/ /12/ (ENS Paris - EDF R&D - CNRS, INRIA) 11/02/ / 28

6 Application to Electricity Load Data Electricity Data Weekly Pattern 1/6/2006 2/6/2006 4/6/2006 5/6/2006 7/6/2006 8/6/ /6/ /6/ /6/ /6/ /6/ /6/ /6/ /6/ /6/ /6/ /6/ /6/ /6/ /6/ (ENS Paris - EDF R&D - CNRS, INRIA) 11/02/ / 28

7 Application to Electricity Load Data Electricity Data Daily Pattern Load Mo Tu We Th Fr Sa Su Instant (ENS Paris - EDF R&D - CNRS, INRIA) 11/02/ / 28

8 Application to Electricity Load Data Electricity Data Special Days Load (MW) Normal Special Tariff Instant 20/12/ /12/ /12/ /12/ /12/ /12/ /12/ /12/ /12/ /12/ /12/ /12/ /12/ /12/ /12/2007 1/1/2008 2/1/2008 3/1/2008 4/1/2008 4/1/2008 (ENS Paris - EDF R&D - CNRS, INRIA) 11/02/ / 28

9 Application to Electricity Load Data Electricity Data Load-Temperature (ENS Paris - EDF R&D - CNRS, INRIA) 11/02/ / 28

10 Application to Electricity Load Data Electricity Data Load-Cloud Cover Load (MW) Cloud cover (Octets) Instant Instant (ENS Paris - EDF R&D - CNRS, INRIA) 11/02/ / 28

11 Application to Electricity Load Data Parametric Models Operational Forecasts: a high dimensional non-linear regression model, see [Bruhns et al.(2005)] Metehore Model Separate the Weather dep. and the Weather ind. Load: L WD t : L WI t : L t = L WD t + L WI t + ε t Cooling and Heating effect Felt temperature (expo. smoothing of the real temperature...) Trend Daily, Weekly and Yearly cycles (Regression, Fourier basis) Trend ε t: AR(1 Week) process (ENS Paris - EDF R&D - CNRS, INRIA) 11/02/ / 28

12 Application to Electricity Load Data Parametric Models Metehore Model f(t ) Load (MW) Mo Th We Tu Fr Sa Su T Hour Saturday Shape Monday Shape Hour Hour (ENS Paris - EDF R&D - CNRS, INRIA) 11/02/ / 28

13 Application to Electricity Load Data Semi-Parametric Models In used at EDF R& D, see [Pierrot and Goude (2011)], and [Wood (2006)] for a in depth presentation of the statistical method. GAM Model L t = 6 j=1 f j(hour t) IDayType t =j +f 7(Toy t, I t) + f 8(t) + g 1(T t, Time t) + g 2(T t 48, Time t) + g 3(Cloud t) + h(l t 24h ) + ε t f j s: Weather Independant Load (shapes of days,yearly cycle, trend) g j s: Weather Dependant Load h: Lagged effects (ENS Paris - EDF R&D - CNRS, INRIA) 11/02/ / 28

14 Application to Electricity Load Data z z Semi-Parametric Models GAM Model Temperature Effect week.temp week.ind Load (MW) Mo we Fr Sa Su Hour Yearly Cycle Trend Posan Instant t (ENS Paris - EDF R&D - CNRS, INRIA) 11/02/ / 28

15 Application to Electricity Load Data Non-Parametric Models Similarity models based on wavelets decomposition proposed in [Antoniadis et al. (2006)], [Antoniadis et al. (2010)], results presented in [Cugliary (2011)]. Functional Model Partitioning the load into blocks of load curves Z i (t) Classify this curves into clusters according to calendar informations In each cluster find similarity W i,j [0, 1] between curve i and j with a wavelets based distance Forecast tomorrow s curve Z n+1(t): n 1 Ẑ n+1(t) = W n,mz m+1(t) Tomorrow will look like the days following days similar than today in the past J. Cugliari at the EDF R & D center, Clamart, 2011 m=1 (ENS Paris - EDF R&D - CNRS, INRIA) 11/02/ / 28

16 Application to Electricity Load Data Non-Parametric Models Functional Model Load (MW) Time (ENS Paris - EDF R&D - CNRS, INRIA) 11/02/ / 28

17 Sequential Combination of Specialized Experts Framework and Algorithms This framework was introduced in [Blum (1997)] and further studied in [Freund et al. (1997)] On-line Sequential Aggregation At each time t [1, T ], we have access to Y t 1 = (y 1,..., y t 1), y i [0, B], and the past experts (e.g. GAM, Metehore or functional models) then the environment generates y t and the individual predictors (experts) ( f j,t ) 1 j N the forecaster builds his combined forecast ŷ t the environment reveals y t to the forecaster the experts incur loss l : R + R + R +, l(f j,t, y t) Individual Sequence: worst case bounds no assumption on an underlying stochastic process a general framework to embed all kind of base forecasters The square loss will be used in our experiments, thus l(x, y) = (x y) 2. (ENS Paris - EDF R&D - CNRS, INRIA) 11/02/ / 28

18 Sequential Combination of Specialized Experts Framework and Algorithms Each time t the experts can be active (produce a forecast) or inactive (do not produce any forecast) We denote E t {1,..., N} the set of active experts at time index t Aggregation consists in convex aggregation rule: p t = (p 1,t,..., p N,t ) X ŷ t = N p j,t f j,t j=1 X : {p t R N, p j,t 0, p 1,t p N,t = 1} is the set of convex weight vectors over N elements the weights are produced sequentially with an algorithm based on the concept of regret (ENS Paris - EDF R&D - CNRS, INRIA) 11/02/ / 28

19 Sequential Combination of Specialized Experts Framework and Algorithms Supposing weights p are produced by the algorithm A, the regret with respect to the expert j up to time T is: R t(a, j) = t=1,...,t (l t(p t) l t(δ j )) Where l t(p) is the loss of the combined forecast based on weights p t, δ j the dirac mass of the expert j. R t(a, q) = t=1,...,t (l t(p t) l t(q)) Where q X. Goal: find an algorithm A that minimizes the regret, e.g. that obtains a minimal regret in o(t ) min j R t(a, j): as well or better than the best expert E η min q R t(a, q): as well or better than the best convex combination E grad η (ENS Paris - EDF R&D - CNRS, INRIA) 11/02/ / 28

20 Sequential Combination of Specialized Experts Framework and Algorithms E η, Exponential Weight Aggregation Input: η > 0 Initialisation: w 1 = (1/N,..., 1/N) For t from 1 to T do: End Do -Forecast ŷ t = 1 i Et w i,t j E t w j,t f j,t -Observe y t -For expert i from 1 to N update the weights: End For w i,t+1 = eηr t 1(E η,j) I {j Et } k E t e ηr t 1(E η,k) Eη grad : same algorithm, replacing the loss l t by l t such that l t(p t) l t(q) l t(p t) (p t q ) = l t(p t) l t(q), (ENS Paris - EDF R&D - CNRS, INRIA) 11/02/ / 28

21 Sequential Combination of Specialized Experts Framework and Algorithms Compound Experts: j T 1 = (j 1,..., j T ) size ( j1 T ) T = I {jt 1 j t } and size ( q T ) T 1 = t=2 The regrets are ( ) R T A, j T ( 1 = T ( ) ) t=1 l t(p t) l t δjt F η,α ( ) R T A, q T 1 = T ( t=1 lt(p t) l ) t(q t) Fη,α grad t=2 I {qt 1 q t } (ENS Paris - EDF R&D - CNRS, INRIA) 11/02/ / 28

22 Sequential Combination of Specialized Experts Framework and Algorithms F η,α, Fixed-Share Input: η > 0, α [0, 1] Initialisation: w 1 = (1/N,..., 1/N) For t from 1 to T do: -Forecast ŷ t = 1 wi,t j w j,tf j,t -Observe y t -For expert i from 1 to N update the weights: End For v i,t = w i,t e ηlt (δ i ) w i,t+1 = (1 α)v i,t + α M 1 j i v j,t End Do (ENS Paris - EDF R&D - CNRS, INRIA) 11/02/ / 28

23 Context: Sequential Combination of Specialized Experts Application produce one day ahead load forecasts of the French load consumption every day at noon (weigths are updated according to that constraint, it induces a modif. of the algorithms) base forecasters are obtained from R& D models: Metehore model: 15 experts GAM model: 8 experts Functional model: 1 expert this experts specialized on winter/summer periods, some are inactive on banking holidays Time intervals Every 30 minutes Number of days D 320 Time indexes T Number of experts N 24 (= ) Median of the y t (GW) Bound B on the y t (GW) Table: Some characteristics of the observations y t of the French data set of operational forecasting. (ENS Paris - EDF R&D - CNRS, INRIA) 11/02/ / 28

24 Sequential Combination of Specialized Experts Application Name of the benchmark procedure Formula Value Uniform convex weight vector rmse ( (1/24,..., 1/24) ) = Best single expert Best convex weight vector min j=1,...,24 min q X rmse(j) = rmse(q) = Best compound expert Size at most m = 50 Size at most m = 100 Size at most m = T 1 = min rmse ( j T ) j1 T 1 C50 min rmse ( j T ) j1 T 1 C100 min j T 1 E1 E2... E T rmse ( j1 T ) = = = Table: Definition and performance of several (possibly off-line) benchmarking procedures on the French data set (GW) of operational forecasting. (ENS Paris - EDF R&D - CNRS, INRIA) 11/02/ / 28

25 Sequential Combination of Specialized Experts Application Optimisation of the aggregation rules parameters: fixeds values Value of η rmse of E η (u) E grad η (u) Table: Performance obtained by the sequential aggregation rules for various choices ofη. Value of η α mse of F η,α F grad η,α Table: Performance obtained by the sequential aggregation rules F η,α and F grad η,α for various choices of η and α. (ENS Paris - EDF R&D - CNRS, INRIA) 11/02/ / 28

26 Sequential Combination of Specialized Experts Application Optimisation of the aggregation rules parameters: online calibration Table: Best constant pair (η, α) Grid rmse of F η,α F grad η,α Performance obtained by the rules F η,α and F grad η,α for the best constant choices of η and α and with the meta-rule selecting sequentially the values of η and α. We obtain a significant improvement of 20% of the RMSE over the best expert. Performance of the fixed-share rule is comparable to the best compound expert with 50 shifts. (ENS Paris - EDF R&D - CNRS, INRIA) 11/02/ / 28

27 Sequential Combination of Specialized Experts Application Example of Weights Weight Weight Half hours Half hours (ENS Paris - EDF R&D - CNRS, INRIA) 11/02/ / 28

28 Conclusion and Future Work Building the specialized experts: for extreme weather conditions, holidays etc... Intraday forecasts Algorithms based on exogenous informations: meteo, calendar data... Density forecasts based on experts advices (ENS Paris - EDF R&D - CNRS, INRIA) 11/02/ / 28

29 Conclusion and Future Work A. Antoniadis, E. Paparoditis, and T. Sapatinas. A functional wavelet kernel approach for time series prediction. Journal of the Royal Statistical Society: Series B, 68(5): , A. Antoniadis, X. Brossat, J. Cugliari, and J.M. Poggi. Clustering functional data using wavelets. In Proceedings of the Nineteenth International Conference on Computational Statistics (COMPSTAT), J. Cugliari, Prévision non paramétrique de processus à valeurs fonctionnelles, Application à la consommation d électricité, PhD Thesis. A. Blum, Empirical support for winnow and weighted-majority algorithms: Results on a calendar scheduling domain. Machine Learning, 26:5-23, Bruhns, A., Deurveilher G., and Roy, J.S. (2005), A non-linear regression model for mid-term load forecasting and improvements in seasonnality, presented at the 15th Power Systems Computation Conference, August 22 26, 2005, Liege, Belgium. Y. Freund, R. Schapire, Y. Singer, and M. Warmuth, Using and combining predictors that specialize, In Proceedings of the Twenty-Ninth Annual ACM Symposium on the Theory of Computing (STOC), pages , A. Pierrot and Y. Goude, Short-Term Electricity Load Forecasting With Generalized Additive Models, Proceedings of ISAP power 2011, Wood, S.N. (2006) Generalized Additive Models: An Introduction with R. CRC/Chapman & Hall. (ENS Paris - EDF R&D - CNRS, INRIA) 11/02/ / 28