prévision de consommation électrique avec adaptive gam

GAM models for Day-Ahead and Intra-Day Electricity Consumption Forecasts

week.temp

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20

week.

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45000

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Temperature Effect

Yannig Goude

EDF R&D - Clamart

( EDF R&D - Clamart) March 15, 2012 1 / 24

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 ofproduction units and electricitytrading

Managing a wild variety of productionunits:

nuclear plants

fuel, coal and gas plants

renewable energy: water dams, windfarms, solar panels...


Motivation of Electricity Load Forecasting

Short-term load forecasting: from 1 day to a few hours horizon


(Generalized) Additive smooth models

consider a univariate response y and corresponding predictors x1, ..., xp

an additive smooth model has the following structure:

yi = Xiβ + f1(x1,i ) + f2(x2,i ) + f3(x3,i , x4,i ) + ...+ εi

Xiβ is the linear part of the model

functions fj are supposed to be smooth

εi :

iidE (εi ) = 0,V (εi ) = σ2

normality if needed (tests...)

More precisely, we want to solve the following pb:

minβ,fj ||y − Xβ − f1(x1)− f2(x2) + ...)2 + λ1

∫f′′1 (x)2dx + λ2

∫f′′2 (x)2dx + ...



Estimation of the fj : basis expansion in a spline basis

fj(x) =

kj∑q=1

aj,q(x)βj,q

Then the additive model becomes

yi = Xiβ +

k1∑q=1

a1,q(x)β1,q +

k2∑q=1

a2,q(x)β2,q + ...+ εi

Unknowns:

choice of the spline basis, number-position of knots kj

β and aj,q

⇒ take large kj and proceed to penalized regression (ridge)



Then the initial problem

minβ,fj ||y − Xβ − f1(x1)− f2(x2) + ...)2 + λ1

∫f′′1 (x)2dx + λ2

∫f′′2 (x)2dx + ...

becomes a linear regression problem:

minβ ||y − Xβ||2 +∑

λjβTSjβ

as∫f′′j (x)2dx can be written as βTSjβ

absorbing aj,q(xi ) into Xi

Solution:

β̂λ = (XTX +∑

λjSj)−1XT y



How to choose the penalization parameter λ?

without any penalisation: β̂0 = (XTX )−1XT y

regularised: β̂λ = (XTX +∑λjSj)

−1XT y

β̂λ = Fλβ̂0

WhereFλ = (XTX +

∑λjSj)

−1XTX

tr(Fλ): estimated degrees of freedom



How to choose the penalization parameter λ?



Ordinary Cross Validation

leave one observation yiestimate a model µ̂−i on the new data setforecast yi with µ̂

−ii

do that for all ichoose the λ that minimizes the OCV score:

V0(λ) =n∑

i=1

(yi − µ̂−ii )2/n

# Pb: calculation time

Generalized Cross Validation [Craven and Wahba (1979)]

Vg (λ) = n‖y − X β̂λ|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)])


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 = QT y and denote ||r ||2 = ||y ||2 − ||f ||2

Q orth. matrix, R upper triang.

then we have:

Vg (λ) =n||f − Rβ̂λ||2 + ||r ||2

(n − tr(Fλ))2

where Fλ is now (RTR +∑λjSj)

−1RTR

⇒ Once we have R, f and ||r ||2, X plays no further part


From GAM to BAM

⇒ Application for large data sets:

X is too big and has to be split:

(X0

X1

), similarly y =

(y0y1

)form QR dec. X0 = Q0X0 and

(R0

X1

)= Q1R see section 12.5 of

[Golub and Van Loan (1996)]

then X = QR with Q =

(Q0 00 I

)Q1 and QT y = QT

1

(QT

0 y0y1

)⇒ 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 thenumerical optimization)


Application to Electricity Load Data Electricity Data

Trend

1/9/

2002

13/1

/200

328

/5/2

003

9/10

/200

321

/2/2

004

4/7/

2004

16/1

1/20

0431

/3/2

005

12/8

/200

525

/12/

2005

8/5/

2006

20/9

/200

61/

2/20

0716

/6/2

007

28/1

0/20

0710

/3/2

008

23/7

/200

84/

12/2

008

18/4

/200

931

/8/2

009

4000

050

000

6000

070

000

8000

090

000



Yearly Pattern

1/1/

2006

20/1

/200

68/

2/20

0627

/2/2

006

18/3

/200

67/

4/20

0626

/4/2

006

15/5

/200

63/

6/20

0622

/6/2

006

12/7

/200

631

/7/2

006

19/8

/200

67/

9/20

0626

/9/2

006

16/1

0/20

064/

11/2

006

23/1

1/20

0612

/12/

2006

31/1

2/20

06

3000

040

000

5000

060

000

7000

080

000



Weekly Pattern

1/6/

2006

2/6/

2006

4/6/

2006

5/6/

2006

7/6/

2006

8/6/

2006

10/6

/200

612

/6/2

006

13/6

/200

615

/6/2

006

16/6

/200

618

/6/2

006

19/6

/200

621

/6/2

006

23/6

/200

624

/6/2

006

26/6

/200

627

/6/2

006

29/6

/200

630

/6/2

006

3500

040

000

4500

050

000

5500

0



Daily Pattern

0 10 20 30 40

4000

045

000

5000

055

000

6000

065

000

7000

0

Instant

Load

MoTuWeTh

FrSaSu



Special Days

0 10 20 30 40

6000

065

000

7000

075

000

8000

0

Instant

Load

(M

W)

Normal Special Tariff

20/1

2/20

0720

/12/

2007

21/1

2/20

0722

/12/

2007

23/1

2/20

0724

/12/

2007

25/1

2/20

0725

/12/

2007

26/1

2/20

0727

/12/

2007

28/1

2/20

0729

/12/

2007

30/1

2/20

0730

/12/

2007

31/1

2/20

071/

1/20

082/

1/20

083/

1/20

084/

1/20

084/

1/20

08

5500

060

000

6500

070

000

7500

080

000

8500

0



Load-Temperature



Load-Cloud Cover

0 10 20 30 40

6000

065

000

7000

075

000

Instant

Load

(M

W)

0 10 20 30 40

02

46

8Instant

Clo

ud c

over

(O

ctet

s)


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 γiSpec.Tarift+

∑11j=1 αj

+ s(t)+ εt

fjs: lagged load effects

gjs: 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 2008Forecasting period: september 2008 - august 2009



GAM Model

I[t]

0

10

20

30

40

T[t]0

10

2030

L[t]

50000

55000

60000

65000

70000

Temperature Effect

0 10 20 30 40

−10

000

−50

000

5000

1000

0

Hour

Load

(M

W)

Mowe

FrSa

Su

Posan

0.00.2

0.4

0.6

0.8

Inst

ant

0

10

20

30

40

z

40000

50000

60000

70000

80000

Yearly Cycle

120000 140000 160000 180000 200000 220000 240000

−10

000

−50

000

5000

1000

0

Trend

t



I[t]

0

10

20

30

40

L[t − 1]

30000

40000

5000060000

7000080000

L[t]

40000

50000

60000

70000

80000

Lagged Load Effect, WW

I[t]

0

10

20

30

40

L[t − 1]

30000

40000

5000060000

7000080000

L[t]

30000

40000

50000

60000

70000

Lagged Load Effect, WH

I[t]

0

10

20

30

40

L[t − 1]

30000

40000

5000060000

7000080000

L[t]

40000

60000

80000

Lagged Load Effect, HW

I[t]

0

10

20

30

40

L[t − 1]

30000

40000

5000060000

7000080000

L[t]

−10000

0

10000

20000

30000

Lagged Load Effect, HH



Figure: Top: half hourly RMSE (left) and MAPE (right) by type of day. Bottom: residuals

0 10 20 30 40

500

1000

1500

2000

Instant

RM

SE

(M

W)

MoTuWeTh

FrSaSu

0 10 20 30 40

0.5

1.0

1.5

2.0

2.5

3.0

Instant

MA

PE

(%

)

MoTuWeTh

FrSaSu

9/1/

2002

12/1

9/20

02

4/8/

2003

8/12

/200

3

12/4

/200

3

3/22

/200

4

7/26

/200

4

11/1

7/20

04

3/7/

2005

7/6/

2005

10/2

3/20

05

2/18

/200

6

6/20

/200

6

10/8

/200

6

2/3/

2007

6/4/

2007

9/21

/200

7

1/17

/200

8

5/6/

2008

8/31

/200

8

−80

00−

4000

020

0040

0060

00

0 10 20 30 40

−80

00−

4000

020

0040

0060

00



Performances

Model RMSE (MW) MAPE (%) RGCV scoreEstimation set

m0 831 1.17 882m1 1024 1.46 806

Forecasting setm0 1220 1.87

On-line m0 1048 1.49m1 1156 1.62

On-line m1 1109 1.53



Residuals

9/1/

2008

9/17

/200

8

10/4

/200

8

10/2

1/20

08

11/1

5/20

08

12/2

/200

8

12/1

9/20

08

1/13

/200

9

1/30

/200

9

2/16

/200

9

3/4/

2009

3/21

/200

9

4/7/

2009

4/28

/200

9

5/27

/200

9

6/17

/200

9

7/4/

2009

7/25

/200

9

8/11

/200

9

8/31

/200

9

−60

0−

500

−40

0−

300

−20

0−

100

0m0m1On−line update

Figure: Cumulative residuals (right) for models m0 (black), m1 (red), and their on-line updated version (dashed lines).



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)


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