Prévision de consommation électrique avec adaptive GAM
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The document discusses generalized additive models (GAM) for short-term electricity load forecasting. GAMs are smooth additive models that decompose a response variable into additive components like trends, cyclic patterns, and nonlinear effects. They summarize how GAMs can model various drivers of electricity consumption, including temperature effects, day-of-week patterns, and lagged load values. Big additive models (BAM) allow applying GAMs to large electricity load datasets. BAMs use QR decomposition and online updating to efficiently estimate high-dimensional additive models.
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Prévision de consommation électrique avec adaptive GAM
- 1. GAM models for Day-Ahead and Intra-Day Electricity Consumption Forecasts Temperature Effect 65000 60000 55000 z 50000 45000 50 40 0 30 d .in ek we 10 20 we ek.t em p 20 10 Yannig Goude EDF R&D - Clamart ( EDF R&D - Clamart) March 15, 2012 1 / 24
- 2. Motivation of Electricity Load Forecasting Electricity can not be stored, thus forecast- ing 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... ( EDF R&D - Clamart) March 15, 2012 2 / 24
- 3. Motivation of Electricity Load Forecasting Short-term load forecasting: from 1 day to a few hours horizon ( EDF R&D - Clamart) March 15, 2012 3 / 24
- 4. (Generalized) Additive smooth models consider a univariate response y and corresponding predictors x1 , ..., xp an additive smooth model has the following structure: yi = Xi β + f1 (x1,i ) + f2 (x2,i ) + f3 (x3,i , x4,i ) + ... + εi Xi β is the linear part of the model functions fj are supposed to be smooth εi : iid E (εi ) = 0,V (εi ) = σ 2 normality if needed (tests...) More precisely, we want to solve the following pb: minβ,fj ||y − X β − f1 (x1 ) − f2 (x2 ) + ...)2 + λ1 f1 (x)2 dx + λ2 f2 (x)2 dx + ... ( EDF R&D - Clamart) March 15, 2012 4 / 24
- 5. (Generalized) Additive smooth models Estimation of the fj : basis expansion in a spline basis kj fj (x) = aj,q (x)βj,q q=1 Then the additive model becomes k1 k2 yi = Xi β + a1,q (x)β1,q + a2,q (x)β2,q + ... + εi q=1 q=1 Unknowns: choice of the spline basis, number-position of knots kj β and aj,q ⇒ take large kj and proceed to penalized regression (ridge) ( EDF R&D - Clamart) March 15, 2012 5 / 24
- 6. (Generalized) Additive smooth models Then the initial problem minβ,fj ||y − X β − f1 (x1 ) − f2 (x2 ) + ...)2 + λ1 f1 (x)2 dx + λ2 f2 (x)2 dx + ... becomes a linear regression problem: minβ ||y − X β||2 + λ j β T Sj β as fj (x)2 dx can be written as β T Sj β absorbing aj,q (xi ) into Xi Solution: βλ = (X T X + λj Sj )−1 X T y ( EDF R&D - Clamart) March 15, 2012 6 / 24
- 7. (Generalized) Additive smooth models How to choose the penalization parameter λ? without any penalisation: β0 = (X T X )−1 X T y regularised: βλ = (X T X + λj Sj )−1 X T y β λ = Fλ β 0 Where Fλ = (X T X + λj Sj )−1 X T X tr (Fλ ): estimated degrees of freedom ( EDF R&D - Clamart) March 15, 2012 7 / 24
- 8. (Generalized) Additive smooth models How to choose the penalization parameter λ? ( EDF R&D - Clamart) March 15, 2012 8 / 24
- 9. (Generalized) Additive smooth models Ordinary Cross Validation leave one observation yi estimate a model µ−i on the new data set forecast yi with µ−i i do that for all i choose the λ that minimizes the OCV score: n V0 (λ) = (yi − µ−i )2 /n i i=1 Pb: calculation time Generalized Cross Validation [Craven and Wahba (1979)] Vg (λ) = n y − X βλ |2 /(n − tr (Fλ ))2 Advantages of GCV: λ is obtained by numerical minimization of Vg (few comp. cost) Vg (λ) is invariant when doing useful transf. of the data (on-line update, big data) ⇒ Software: R, package mgcv (see[Wood (2001)] and [Wood (2006)]) ( EDF R&D - Clamart) March 15, 2012 9 / 24
- 10. From GAM to BAM BAM: Big Additive Models ⇒ for huge data sets (more than 10 000 observations) we use the QR decomposition: X = QR, f = Q T y and denote ||r ||2 = ||y ||2 − ||f ||2 Q orth. matrix, R upper triang. then we have: n||f − R βλ ||2 + ||r ||2 Vg (λ) = (n − tr (Fλ ))2 where Fλ is now (R T R + λj Sj )−1 R T R ⇒ Once we have R, f and ||r ||2 , X plays no further part ( EDF R&D - Clamart) March 15, 2012 10 / 24
- 11. From GAM to BAM ⇒ Application for large data sets: X0 y0 X is too big and has to be split: , similarly y = X1 y1 R0 form QR dec. X 0 = Q0 X0 and = Q1 R see section 12.5 of X1 [Golub and Van Loan (1996)] T Q0 0 Q0 y0 then X = QR with Q = Q1 and Q T y = Q1 T 0 I y1 ⇒ On-line update X0 , y0 past data, X1 , y1 last observations Use the new data X1 , y1 to update R, f and ||r ||2 Re-estimate λ and βλ (previous values can be used as starting values for the numerical optimization) ( EDF R&D - Clamart) March 15, 2012 11 / 24
- 12. ( EDF R&D - Clamart) 40000 50000 60000 70000 80000 90000 1/9/2002 13/1/2003 28/5/2003 9/10/2003 21/2/2004 4/7/2004 16/11/2004 31/3/2005 Application to Electricity Load Data 12/8/2005 25/12/2005 Trend 8/5/2006 20/9/2006 1/2/2007 16/6/2007 28/10/2007 Electricity Data 10/3/2008 23/7/2008 4/12/2008 18/4/2009 31/8/2009 March 15, 2012 12 / 24
- 13. ( EDF R&D - Clamart) 30000 40000 50000 60000 70000 80000 1/1/2006 20/1/2006 8/2/2006 27/2/2006 18/3/2006 7/4/2006 26/4/2006 15/5/2006 Application to Electricity Load Data 3/6/2006 22/6/2006 12/7/2006 31/7/2006 19/8/2006 Yearly Pattern 7/9/2006 26/9/2006 Electricity Data 16/10/2006 4/11/2006 23/11/2006 12/12/2006 31/12/2006 March 15, 2012 13 / 24
- 14. ( EDF R&D - Clamart) 35000 40000 45000 50000 55000 1/6/2006 2/6/2006 4/6/2006 5/6/2006 7/6/2006 8/6/2006 10/6/2006 12/6/2006 Application to Electricity Load Data 13/6/2006 15/6/2006 16/6/2006 18/6/2006 19/6/2006 Weekly Pattern 21/6/2006 23/6/2006 Electricity Data 24/6/2006 26/6/2006 27/6/2006 29/6/2006 30/6/2006 March 15, 2012 14 / 24
- 15. Application to Electricity Load Data Electricity Data Daily Pattern 70000 Mo Fr Tu Sa 65000 We Su 60000 55000 Th Load 50000 45000 40000 0 10 20 30 40 Instant ( EDF R&D - Clamart) March 15, 2012 15 / 24
- 16. Load (MW) 60000 65000 70000 75000 80000 0 10 Normal ( EDF R&D - Clamart) 20 Instant 30 Special Tariff 40 Application to Electricity Load Data 55000 60000 65000 70000 75000 80000 85000 Special Days 20/12/2007 20/12/2007 21/12/2007 Electricity Data 22/12/2007 23/12/2007 24/12/2007 25/12/2007 25/12/2007 26/12/2007 27/12/2007 28/12/2007 29/12/2007 30/12/2007 30/12/2007 31/12/2007 1/1/2008 2/1/2008 3/1/2008 4/1/2008 4/1/2008 March 15, 2012 16 / 24
- 17. Application to Electricity Load Data Electricity Data Load-Temperature ( EDF R&D - Clamart) March 15, 2012 17 / 24
- 18. Application to Electricity Load Data Electricity Data 75000 Load-Cloud Cover 8 70000 6 Cloud cover (Octets) Load (MW) 65000 4 2 60000 0 0 10 20 30 40 0 10 20 30 40 Instant Instant ( EDF R&D - Clamart) March 15, 2012 18 / 24
- 19. Application to Electricity Load Data Model Lt = f1 (Lt−48 , It ) IHH +f2 (Lt−48 , It ) IHW +f3 (Lt−48 , It ) IWH +f4 (Lt−48 , It ) IWW + g1 (Tt , It ) + g2 (Tt−48 , Tt−96 ) + g3 (Cloudt ) + h(Toyt , It ) 48 + i=1 γi Spec.Tarift 11 + j=1 αj + s(t) + εt fj s: lagged load effects gj s: meteo. effects hs: yearly pattern, Toy is time of year γi : special tariff effect by half-hour of the day αj mean load for: sunday, monday, tuesday,...,saturday, HH,HW,WH and WW days s(t) is the trend Estimation period: september 2002 - august 2008 Forecasting period: september 2008 - august 2009 ( EDF R&D - Clamart) March 15, 2012 19 / 24
- 20. Application to Electricity Load Data Model GAM Model 10000 Temperature Effect 5000 70000 65000 Load (MW) 0 L[t] 60000 −5000 55000 50000 40 30 30 −10000 20 20 I[t] T[t 10 Mo Fr Su ] 10 0 we Sa 0 0 10 20 30 40 Hour Yearly Cycle Trend 10000 80000 70000 5000 60000 z 0 50000 40000 −5000 40 0.0 30 0.2 0.4 20 nt Po sta san 0.6 In 10 −10000 0.8 0 120000 140000 160000 180000 200000 220000 240000 t ( EDF R&D - Clamart) March 15, 2012 20 / 24
- 21. Application to Electricity Load Data Model Lagged Load Effect, WW Lagged Load Effect, WH 80000 70000 70000 60000 L[t] 60000 L[t] 50000 50000 40000 40000 40 30000 40 80000 30 80000 30 70000 20 70000 20 60000 60000 ] ] I[t I[t L[t 50000 10 L[t 50000 10 −1 −1 ] 40000 ] 40000 30000 0 30000 0 Lagged Load Effect, HW Lagged Load Effect, HH 30000 20000 80000 L[t] 10000 L[t] 60000 0 −10000 40000 40 40 80000 30 80000 30 70000 20 70000 20 60000 60000 ] ] I[t I[t L[t 50000 10 L[t 50000 10 −1 −1 ] 40000 ] 40000 30000 0 30000 0 ( EDF R&D - Clamart) March 15, 2012 21 / 24
- 22. RMSE (MW) −8000 −4000 0 2000 4000 6000 500 1000 1500 2000 0 9/1/2002 Figure: 12/19/2002 4/8/2003 ( EDF R&D - Clamart) 8/12/2003 10 12/4/2003 3/22/2004 7/26/2004 11/17/2004 20 3/7/2005 7/6/2005 Tu Th Mo We 10/23/2005 Instant 2/18/2006 30 6/20/2006 Fr 10/8/2006 Su Sa 2/3/2007 6/4/2007 40 9/21/2007 1/17/2008 5/6/2008 8/31/2008 Application to Electricity Load Data MAPE (%) −8000 −4000 0 2000 4000 6000 0.5 1.0 1.5 2.0 2.5 3.0 Model 0 0 10 10 20 20 Instant 30 30 Tu Th Mo We 40 40 Fr Su Sa Top: half hourly RMSE (left) and MAPE (right) by type of day. Bottom: residuals March 15, 2012 22 / 24
- 23. Application to Electricity Load Data Model Performances Model RMSE (MW) MAPE (%) RGCV score Estimation set m0 831 1.17 882 m1 1024 1.46 806 Forecasting set m0 1220 1.87 On-line m0 1048 1.49 m1 1156 1.62 On-line m1 1109 1.53 ( EDF R&D - Clamart) March 15, 2012 23 / 24
- 24. Application to Electricity Load Data Model Residuals 0 −100 −200 −300 m0 m1 On−line update −400 −500 −600 9/1/2008 9/17/2008 10/4/2008 10/21/2008 11/15/2008 12/2/2008 12/19/2008 1/13/2009 1/30/2009 2/16/2009 3/4/2009 3/21/2009 4/7/2009 4/28/2009 5/27/2009 6/17/2009 7/4/2009 7/25/2009 8/11/2009 8/31/2009 Figure: Cumulative residuals (right) for models m0 (black), m1 (red), and their on-line updated version (dashed lines). ( EDF R&D - Clamart) March 15, 2012 24 / 24
- 25. Application to Electricity Load Data Model Craven and Wahba (1979) ”Smoothing noisy data with spline functions: estimated the correct degree of smoothing by the method of general cross validation”. Numerische Mathematik 31, 377-403. Golub and Van Loan (1996) ”Matrix Computations, 3rd edition”. John Hopkins Studies in the Mathematical Sciences. Green and Silverman (1994) ”Nonparametric Regression and Generalized Linear Models”. Chapman and Hall. Hastie and Tibshirani (1990) ” Generalized Additive Models”. Chapman and Hall. Pierrot and Goude (2011) ”Short-Term Electricity Load Forecasting With Generalized Additive Models”, Proceedings of ISAP power 2011. Wahba (1990) ”Spline Models of Observational Data”. SIAM Wood (2001) mgcv:GAMs and Generalized Ridge Regression for R. R News 1(2):20-25 Wood and Augustin (2002) ”GAMs with integrated model selection using penalized regression splines and applications to environmental modelling”. Ecological Modelling 157:157-177 Wood (2006)Generalized Additive Models, An Introduction with R (Chapman and Hall, 2006) ( EDF R&D - Clamart) March 15, 2012 24 / 24