Analysis of day-ahead electricity data

Zita Marossy & Márk Szenes (ColBud)

MANMADE workshop

January 21, 2008

Topics

Stylized facts of electricity price data Modeling variable: price Autocorrelation structure Persistence Price distribution Seasonality

Time series modeling Neural network SETAR

Main results

Persistence analysis Underlying variable: price, not price change Results: H = 0.7-0.97 (0.8)

Price distribution Generalized extreme value distribution vs. Lévy distribution

Design of a seasonal filter Filtering the intra-weekly seasonality

Performance evaluation of an ANN model Reasonable for short-run forecasts

SETAR model for determining price spikes

Data: EEX, hourly day-ahead prices

Autocorrelation structure

Seasonality Effect of intra-weekly seasonality is strong AC decays slowly

0 50 100 150 200 250 300 350-0.2

0

0.2

0.4

0.6

0.8

Lag

Sam

ple

Par

tial A

utoc

orre

latio

ns

Sample Partial Autocorrelation Function

0 50 100 150 200 250 300 350-0.2

0

0.2

0.4

0.6

0.8

Lag

Sam

ple

Aut

ocor

rela

tion

Sample Autocorrelation Function (ACF)

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

x 104

-0.2

0

0.2

0.4

0.6

0.8

Lag

Sam

ple

Aut

ocor

rela

tion

Sample Autocorrelation Function (ACF)

Modeling prices, not price changes1. The price process has no unit root, there is

no need to differentiate the time series

2. Electricity can not be stored: ‘return’ has no direct meaning

3. By differencing we cause spurious patterns in ACF:

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

1 13 25 37 49 61 73 85 97

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1 13 25 37 49 61 73 85 97

Persistence analysis

Calculating the Hurst exponent of prices Without differencing the time series Hurst exponent – classical usage

(with differencing the time series first): > 0.5 persistent process

High ‘return’ shock followed by high ‘return’ = 0.5 random walk

‘Return’ is white noise < 0.5 antipersistent process (mean reversion)

Hurst exponent – without differencing > 0.5 persistent process

High price followed by high price = Are high prices persistent? = 0.5 white noise < 0.5 antipersistent process

Hurst exponent: estimation resultsMethod Estimated Hurst exponent

Aggregated variance 0.872

Differenced aggregated variance 0.702

Aggregated absolute values/means 0.924

Fractal dimension (Higuchi) 0.967

Residuals of regression (Peng) 0.811

R/S 0.835

Periodogram 0.891

Modified periodogram 0.770

Wavelet 0.839

Price distribution

Two estimated distributions: Lévy

Generalized extreme value

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 1000 2000 3000 4000 5000 6000 70000

0.5

1

1.5x 10

-3

Data

Den

sity

EEX (daily)

GEV (EEX)

0 1000 2000 3000 4000 5000 6000 70000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Data

Cum

ulat

ive

prob

abili

ty

EEX (daily)

GEV (EEX)

k

k

xkxF

/1

,, 1exp

Comparison

Kolmogorov test:

Test statistic: Lévy: 0.0141 GEV: 0.0262

Mean of absolute differences: Lévy: 8.07*10-4

GEV: 7.18*10-4

0 100 200 300 400 500 600-0.03

-0.02

-0.01

0

0.01

0.02

0.03

price (EUR)F

emp-

F

Differences in CDF

Lévy

GEV

estFFD sup

Seasonality

Seasonality: intradaily Weekly

Spectral decomposition Periodogram of prices Periodogram of ACF

Filtering Median or average week Differencing Moving average technique

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-30

-20

-10

0

10

20

30

40

50

60

70

Normalized Frequency ( rad/sample)

Pow

er/f

requ

ency

(dB

/rad

/sam

ple)

Power Spectral Density Estimate via Periodogram (Variable: price)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-70

-60

-50

-40

-30

-20

-10

0

10

20

Normalized Frequency ( rad/sample)

Pow

er/f

requ

ency

(dB

/rad

/sam

ple)

Power Spectral Density Estimate via Periodogram (Variable: ACF)

0 20 40 60 80 100 120 140 160 18010

20

30

40

50

60

70

time

pric

e (E

UR

)

Sample week

Need for new seasonal filter

The type of distribution changes

23 24 25 26 27 28 29 30 31 32 3370

75

80

85

90

95

100

105

110

mean

high

qua

ntile

Quantile dependence

Suggested filter

‘GEV filter’1. Separately estimate a GEV distribution for each

hour and day i: F1(i)2. Transform the prices:

F2-1F1,i(x)

F2: lognormal cdf (parameters: entire distribution)

3. Model the prices of filtered data4. Forecast5. Transform the forecasts back into GEV

Empirical results

Figures: periodogram of ACF (orig prices) ACF (filtered data)

Intraweekly filtering successful

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-70

-60

-50

-40

-30

-20

-10

0

10

20

30

Normalized Frequency ( rad/sample)

Pow

er/f

requ

ency

(dB

/rad

/sam

ple)

Power Spectral Density Estimate via Periodogram

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-70

-60

-50

-40

-30

-20

-10

0

10

20

Normalized Frequency ( rad/sample)

Pow

er/f

requ

ency

(dB

/rad

/sam

ple)

Power Spectral Density Estimate via Periodogram

Estimated GEV parameters

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.1612

13

14

15

16

17

18

19

k

mu

GEV parameters

Distributions with high scale param

0 20 40 60 80 100 120 140 160 1800

2

4

6

8

10

12

14

16

18

20Distributions with high shape parameters

Hour

mu

Conclusion

Different hours of week behave differently There are a few hours with fatter tails These are more sensitive to price spikes We can model fat tails and forecasting

separately

Performance evaluation of an ANN

Short term price forecasting (few hours to days) ANN: simple but flexible tool Architecture: standard feedforward type Layers: 168 – 15 – 1 Input: historical data Training set: 42 days Prediction horizon:

from 1 hour to 1 week

Performance evaluation of an ANN Measuring error by MAPE

Testing against naive method Averaged over 50 runs:

50 consecutive weeks from Nov. 2005 to Nov. 2006

Results: NN performs well in day-ahead

forecasting But it fails to compete with naive

method in wider time horizon

Improvements: Exogenous variables

TAR (Threshold AR)

SETAR

Aim: Identifying the limit (C)

between high and low prices 2 state SETAR model

On daily price Threshold: 44.26

Cyify

Cyifyy

ttt

tttt

122

111

Dependent Variable: P Method: Least Squares Date: 11/02/07 Time: 17:28 Sample (adjusted): 16 2499 Included observations: 2484 after adjustments

Variable Coefficient Std. Error t-Statistic Prob. IND1 2.027058 0.776760 2.609634 0.0091

IND1*P(-1) 0.473548 0.026531 17.84886 0.0000 IND1*P(-2) -0.057900 0.026815 -2.159224 0.0309 IND1*P(-3) 0.135780 0.027550 4.928506 0.0000 IND1*P(-4) -0.063816 0.027579 -2.313925 0.0208 IND1*P(-5) 0.038770 0.032474 1.193888 0.2326 IND1*P(-6) 0.068898 0.037351 1.844597 0.0652 IND1*P(-7) 0.485232 0.042672 11.37114 0.0000 IND1*P(-8) -0.159181 0.030646 -5.194250 0.0000 IND1*P(-9) -0.050625 0.027950 -1.811270 0.0702 IND1*P(-10) -0.028025 0.029898 -0.937356 0.3487 IND1*P(-11) -0.024351 0.028674 -0.849228 0.3958 IND1*P(-12) -0.031132 0.032490 -0.958191 0.3381 IND1*P(-13) 0.031935 0.029685 1.075779 0.2821 IND1*P(-14) 0.166089 0.032485 5.112824 0.0000 IND1*P(-15) -0.043086 0.025717 -1.675384 0.0940

IND2 6.776519 2.197847 3.083253 0.0021 IND2*P(-1) 0.410020 0.028457 14.40836 0.0000 IND2*P(-2) 0.392799 0.034373 11.42748 0.0000 IND2*P(-3) -0.060909 0.032771 -1.858627 0.0632 IND2*P(-4) 0.177400 0.033091 5.360928 0.0000 IND2*P(-5) -0.073694 0.029019 -2.539503 0.0112 IND2*P(-6) 0.137620 0.026011 5.290871 0.0000 IND2*P(-7) 0.050451 0.028216 1.788034 0.0739 IND2*P(-8) 0.025822 0.031253 0.826232 0.4088 IND2*P(-9) -0.183298 0.035174 -5.211262 0.0000 IND2*P(-10) -0.043402 0.030573 -1.419632 0.1558 IND2*P(-11) 0.000582 0.031882 0.018240 0.9854 IND2*P(-12) -0.033339 0.027918 -1.194172 0.2325 IND2*P(-13) 0.072398 0.031071 2.330104 0.0199 IND2*P(-14) 0.051356 0.030162 1.702676 0.0888 IND2*P(-15) -0.007765 0.030624 -0.253556 0.7999

R-squared 0.677215 Mean dependent var 31.72947 Adjusted R-squared 0.673134 S.D. dependent var 18.46424 S.E. of regression 10.55641 Akaike info criterion 7.564143 Sum squared resid 273245.5 Schwarz criterion 7.639088 Log likelihood -9362.665 Durbin-Watson stat 1.993690