zita marossy corvinus university of budapest zita.marossy () uni-corvinus.hu

The behaviour ofday-ahead electricity pricesAnalysis of spot electricity prices using statistical, econometric, and econophysical methods

Zita MAROSSYCorvinus University of Budapestzita.marossy () uni-corvinus.hu

Workshop on Deregulated European Energy MarketCollegium BudapestSeptember 24-25, 2009

Zita Marossy: The behaviour of day-ahead electricity prices

3Workshop on Deregulated European Energy market

Topics covered Power exchanges, spot power prices „Stylized facts” of power price fluctuation Power price models

Time series models Distribution of spot prices

Own research results Detailed analysis of Hurst exponents Decomposition of multifractal feature of power prices Distribution of power prices: Fréchet distribution Deterministic regime switching model Intra-week seasonality filtering: GEV filter



Power exchanges Actors:

– Power plants;– Power consumers;– Electricity trading companies.

Products:– Power supplied during a given time period

Organized markets Markets:

– Futures markets– Day-ahead (spot) markets– Balancing markets

Power price: P(t,T)

European power exchanges

Exchange CourntyEuropean Energy Exhange GermanyPowernext FranceAPX Power NL NetherlandsAPX Power UK UKEnergy Exchange Austria AustriaPrague Energy Exchange Czech

RepublicOpcom RumaniaPolish Power Exchange PolandNord Pool NorwayBorzen SloveniaItalian Power Exchange ItalyOMEL Madrid SpainBelpex Belgium

Source: RMR Áramár Portál. (March 30, 2009)

tHatáridős piac

Day-ahead (spot) piac

Kiegyenlítőpiac

T tHatáridős piac

Day-ahead (spot) piac

Kiegyenlítőpiac

T

Tt Day-ahead (spot) piac

Kiegyenlítőpiac

Határidős piac

Tt Day-ahead (spot) piac

Kiegyenlítőpiac

Határidős piac

Source: Geman [2005].

Futures markets

Futures markets

Day-ahead (spot) market

Day-ahead (spot) market

Balancing market

Balancing market



Market prices

Double auction for each hour of the next day Market price:

– Aggregated demand– Aggregated supply– Market clearing price– Transmission congestions:

• Nodal/Zonal prices

Source: Rules for the Operation of the Electricity Market, Borzen [2003].



Motivation for power price modelling

Future power prices are risky Power price forecasts help to

– determine the timing of buying/selling of power products– work out bidding strategies– price derivative products– manage risks

Therefore: the distribution of future prices are in the center of attention



Spot time series

Hourly day-ahead prices One price for each hour Data:

– EEX hourly prices from June 16, 2000 to April 19, 2007

Time series of different products (apples & oranges)– Electricity can not be stored at reasonable cost

Stable correlation structure: existence of a data generating process

Daily prices: sum of 24 hourly prices for the given day („Phelix”: avg) Returns: hourly „log return”

0 1 2 3 4 5 6

x 104

0

500

1000

1500

2000

2500

Óra

Ár

(EU

R)

EEX árak



Modelling approaches

1. Stochastic model calibration, time series analysis Find a suitable model, calibrate, use it for forecasting

2. Fundamental models Driving factors of supply and demand are modelled Price behaviour is derived from market equilibrium

3. Agent-based models Description of market players’ actions (e.g. simulation)

4. Statistical models Directly investigate the distribution No prior knowledge about the driving factors & market players’ behaviour is needed

5. Artificial intelligence-based models E.g. neural networks, SVM Black box



Stylized facts 1/7 High prices (price spikes) in the time series

The volatility is extremely high (Weron[2006]):– T-note (<0.5%)– Equity (1-1.5%, risky: 4%)– Commodities (1.5-4%)– Electricity (50%)

The intensity of spikes changes in time, and it is higher in peak hours (Simonsen, Weron, Mo[2004]).

The price returns to the original level rapidly (Weron[2006]). Reason of spikes:

– „supply shocks” (electricity can not be stored) (Escribano, Pena, Villaplana[2002])– bidding strategies (Simonsen, Weron, Mo[2004])– long-term trends in the market factors (occurrence can be forecasted) (Zhao, Dong,

Li, Wong [2007])



Stylized facts 2/7

The time series exhibits seasonality.(Plot: EEX data)1. Annual

– Plot: 4-month MA-filtered data

2. Weekly3. Daily

– Plot: mean of hourly prices

0 500 1000 1500 2000 250012

14

16

18

20

22

24

26

28

Nap

Ár

(4 h

ónap

os m

ozgó

átla

g)

0 20 40 60 80 100 120 140 160 1800

10

20

30

40

50

60

70

Óra

Ár



Stylized facts 3/7

Stable autocorrelation structure with high autocorrelations

(Plot: EEX data) High autocorrelation

coefficients Slowly decreasing

autocorrelation function(persistency)

Periodicity (seasonality)



Stylized facts 4/7 Volatility changes in time: heteroscedasticity

Hectic and calm periods GARCH-type models

– High shocks cause high volatility in the next period

– Volatility clustering– Stochastic (autoregressive) conditional

volatility

My arguments for deterministic conditional volatility:

– Volatility shows seasonal patters: it is higher in peak hours (Weron [2000]).

– Plot: Weron [2000] reproduced; hourly mean absolute percentage change (EEX data)

0 20 40 60 80 100 120 140 1600

5

10

15

20

25

30

35

40EEX

Óra

Átla

gos

absz

olút

vál

tozá

s

0 500 1000 1500 2000 2500-500

0

500Innovations

Inno

vatio

n

0 500 1000 1500 2000 25000

100

200Conditional Standard Deviations

Sta

ndar

d D

evia

tion

0 500 1000 1500 2000 25000

200

400Returns

Ret

urn



Stylized facts 5/7

Price distributions have fat tails.

Heavier tails and higher kurtosis than Gaussian

– Plot: Q-Q plot of log EEX price versus Gaussian distribution

– Plot: histogram of EEX daily prices

-5 -4 -3 -2 -1 0 1 2 3 4 5-4

-2

0

2

4

6

8

Standard Normal Quantiles

Qua

ntile

s of

Inp

ut S

ampl

e

QQ Plot of Sample Data versus Standard Normal

0

200

400

600

800

1000

0 1000 2000 3000 4000 5000 6000 7000

Series: PR_D_EEXSample 6/16/2000 4/19/2007Observations 2499

Mean 773.0419Median 687.5300Maximum 7237.000Minimum 74.81000Std. Dev. 442.0329Skewness 3.951179Kurtosis 38.24476

Jarque-Bera 135845.7Probability 0.000000



Stylized facts 6/7 No consensus whether the price process has a unit root.

Eydeland, Wolyniec [2003]: Dickey-Fuller test (no unit root) Atkins, Chen [2002]: ADF (no unit root), KPSS (existence of u.r.)

Bosco, Parisio, Pelagatti, Baldi [2007]: traditional testing procedures can not be used

– additive outliers,– fat tails,– heteroscedasticity,– seasonality

Even robust tests disagree:– Escribano, Peña, Villaplana [2002] : no unit root (on outlier-filtered data)– Parisio, Pelagatti, Baldi [2007]: existence of unit root (weekly median prices)



Stylized facts 7/7 Some authors argue that power prices are anti-persistent and mean reverting;

meanwhile others state that the price time series has long memory.Method: Hurst exponent (H) Mean reversion

– Weron, Przybyłowicz [2000] , Eydeland, Wolyniec [2003], Weron [2006], Norouzzadeh et al. [2007], Erzgräber et al. [2008],

Long memory– Carnero, Koopman, Ooms [2003] , Sapio [2004], Serletis, Andreadis [2004], Haldrup,

Nielsen [2006]

Large price changes behave differently: multifractality.Method: generalized Hurst exponent Multifractal property

– Resta [2004] , Norouzzadeh et al. [2007], Erzgräber et al. [2008] Monofractal property

– Serletis, Andreadis [2004] – other methodology



Reduced-form models

Geometric Brownian motion (GBM) GBM with mean reversion Stochastic volatility models:

– Constant Elasticity of Volatility (CEV)– Local volatility models– Hull-White model– Heston model

Jump diffusion Markov regime switching models

ydWydtdy dWdtydy



Jump diffusion

Empirical findings: High mean reversion rate

– Positive jumps followed by a negative jump (Weron, Simonsen, Wilman [2004])– Mean reversion rate depends on jump size (Weron, Bierbrauer, Trück [2004])– „Regime jump model”: 3 regimes: normal, jump, return (Huisman, Mahieu

[2001])– „Signed jump model”: sign of a jump depends on the price (Geman, Roncoroni

[2006]) The intensity of jumps changes

– Intensity depends on the price (Eydeland, Geman [1999])– Non-homogeneous Poisson process with time-dependent jump intensity (Weron

[2008b])

: drift (usually: mean reversion): volatilityqt: jump (driven by e.g. a Poisson process)

),(),(),( tydqdWtydttydy ttttt



Regime-switching models 2 regimes with different price dynamics

Transition matrix: probability of changing regime

Weron[2006]: RS models do not outperform JD models with log prices Weron [2008a]: RS model provides better results than JD models with prices De Jong [2006] compares RS and JD models. Best fit: 2-state RS model. Haldrup, Nielsen [2006]: ARFIMA and RS models have similar forecasting power

UUU0

LLL

dWdt

dWdt

S

dS

2222

1111

1

1

qq

qqQ



Time series models ARMA, ARIMA

– SARIMA• Seasonality + ARIMA

– ARFIMA• ARMA+fractional integration

– TAR (threshold AR)• Different price dynamics under and

above threshold– PAR (periodic AR)

• AR coefficients are different for each hour

GARCH– Stochastic volatility

Regime switching models– Different time series models in the

regimes

Exogenous variables:– (Forecasted) consumption– Seasonality variables– Weather– Coal, gas… prices– Capacities– …

Empirical findings:– Good fit for fractional models– RS models provide poor forecasting

performance



Modelling price spikes Price spikes are very important in risk management Definition varies:

mean + constant * standard deviationZhao, Dong, Li, Wong [2007]: constant depends on market, season, and time

Filtering:– „similar day”: mean of the hour– „limit”: threshold (T)– „damped”: T + Tlog10(P/T)

Adding to the model: jump diffusion, regime-switching models

Separate spike forecasting models– Zhao, Dong, Li, Wong [2007]:

„„An effective method of predicting the occurrence of spikes has not yet been observed in the literature so far.”



Own research results

I. Fractal feature„Detailed analysis of the fractal feature of day-ahead electricity prices”

II. Distribution of power prices„Extreme value theory discovers electricity price distribution”

III. Deterministic regime switching and filtering„Deterministic regime-switching, spike behaviour, and seasonality filtering of electricity spot prices”



Persistency: Hurst exponent (H) H:

– A measure for self similar (self affine) processes– The increments b(t0,t) and r-Hb(t0,rt) r>0 are statistically indistinguishable– The process scales at a rate of H

0<H<1 1. For integrated processes (widely-used definition)

H = 0.5 the increments have no autocorrelation (e.g. Wiener-process)H > 0.5 persistent (the increments have a positive autocorrelation)H < 0.5 antipersistent (the increments have a negative autocorrelation)

2. For stationary processesH = 0.5 the process values have no autocorrelation (e.g. Gaussian white noise)H > 0.5 persistent (the process values have a positive autocorrelation)H < 0.5 antipersistent (the process values have a negative autocorrelation)



Persistency – example

fractional Wiener process (fractional Brownian motion)values and increments

(H = 0.25, 0.4, 0.5, 0.6, 0.75 )



Estimates on H in the case of EEX

Power prices have an H of 0.8-0.9 (1). Parentheses: „multiscaling” H = 1: pink noise

EEX

Price Log price Log return Price difference

R/S 0.88 0.77 0.26(0.77)* 0.30(0.71)*

Aggregated Variance 0.86 0.88 -0.03 -0.03

Differenced Variance 0.79 0.70 0.11 -0.02

Periodogram regression 0.83 1.06 0.22 -0.08

AWC 0.85 0.94 0.11 0.05

DFA2 0.84 0.87 0.06 0.08

hmod(2) 0.83 0.86 0.06 0.08



Multiscaling? MF-DFA(2) Data: EEX Tangents:0.76 ;0.11 ;0.03 Cut-off points:ln(44.7) ≈ 3.8

R/S method:101.5 ≈ 58

The cut-off point is difficult to explain

The log return (and the price increment) is not a self affine process

1 2 3 4 5 6 7 8 9 10 112

3

4

5

6

7

8

9

10

log(s)

log(

F(s

))

DFA2 (EEX ár)

1 2 3 4 5 6 7 8 9 10 11-3

-2

-1

0

1

2

3

4

5

6

7

log(s)

log(

F(s

))

DFA2 (EEX log ár)

1 2 3 4 5 6 7 8 9 10 11-2.5

-2

-1.5

-1

-0.5

0

log(m)

log(

F(s

))

DFA2 (EEX loghozam)

1 2 3 4 5 6 7 8 9 10 112

3

4

5

6

7

8

9

10

11

log(m)

log(

F(s

))

DFA2 (EEX szûrt ár)



Multifractal feature Generalized Hurst exponent: h(q)

– Low q: persistency for small shocks– High q: persistency for large shocks

Sources of multifractality:– Fat tails– Correlations

Shuffling the time series helps to separate the two effects

Modified h(q).

-20 -15 -10 -5 0 5 10 15 200

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

q

Hur

st

MFDFA (EEX ár)

h(q)h

mod(q)

hshuff led

(q)

-20 -15 -10 -5 0 5 10 15 200

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6MFDFA (NordPool ár)

qH

urst

h(q)

hmod

(q)

hshuff led

(q)

5.0mod qhqhqh shuffled



Multifractality test Jiang, Zhou [2007]

H0: monofractalH1: multifractal

EEX: p = 0.36 monofractal NordPool: p = 0.00 multifractal

NordPool:h(q) for each hour of the week

– Upper plot: original h(q)s– Lower plot: modified h(q)s– p < 0.05 for 14 segments– p < 0.01 for 4 segments

The process is monofractal if the segments are separated. The different hours have different distributions:

– The distributions are mixed in the whole time series

-10 -5 0 5 100

0.5

1

1.5

2

q

h(q)

-10 -5 0 5 100

0.5

1

1.5

2

q

h mod

(q)



„There are no spikes”

The separate statistical modelling of price spikes is impossible as price spikes can not be distinguished in the price process.

a. Price spikes behave the same way regarding the correlations as prices at average level do.

b. Price spikes are high realizations of a fat tailed distribution. They constitute no separate regime, and they are not “outlier” from the price process. Giving them a separate name causes confusion in modelling.



Distribuition of power prices

Weron [2006]:– Alfa-stable

– Hyperbolic distribution

– NIG (normal inverse Gaussian)

– Tests: on MA-filtered prices– Best fit (price difference, log prices): alfa-stable distribution

1

21

12

1

)( ln

zz

zizzi

tgz

zizzi

z

xx

H eK

xf22

)(2 221

22

22

221 )(22

x

xKexf x

NIG



Generalized extreme value (GEV) distribution

3 parameters:– scale (k)

• Fréchet (k>0)• Weibull (k<0)• Gumbel (k=0)

– location ()– scale ()

k

k

xkxF

/1

,, 1exp

GEV pdfs

0

0,2

0,4

0,6

0,8

1

1,2

-5,0

0

-4,3

7

-3,7

4

-3,1

1

-2,4

8

-1,8

5

-1,2

2

-0,5

9

0,04

0,67

1,30

1,93

2,56

3,19

3,82

4,45

Fréchet

Gumbel

Weibull

0/1 xk



GEV (Fréchet) fits the empirical dataData: EEX daily prices

pdf

cdf

Q-Q plot

Estimates

Statistical test

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)

Parameter Estimate (EEX)

k 0,12

586,81

258,38

Chi-squared statistics

p-value

APX 39,63 0,112

EEX 141,87 0,075

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



GEV provides better fit than LévyData: EEX daily prices. Marossy, Szenes [2008]

Difference in empirical and estimated cdfs

See Kolmogorov-Smirnov statistic

KS statistic:– Lévy: 0.0141, GEV: 0.0262, critical value: 0.068

Mean of the differences:– Lévy: 8.07*10-4, GEV: 7.18*10-4

GEV is better at the tails of the distribution

empFFD sup

0 100 200 300 400 500 600-0.03

-0.02

-0.01

0

0.01

0.02

0.03

price (EUR)

Fem

p-F

Differences in CDF

Lévy

GEV



A theoretical model

Explaining why power prices have GEV distribution Background: extreme value theory

– Fisher-Tippett Theorem

Reason for Fréchet:– The price has to be an exponential function of the

quantity on the market supply curve– Empirical „supply stack”: exponential



Distributions changing their shapes

The time series is divided into 168 segments The distributions differ not only in means but in shapes Plot: EEX data

0 10 20 30 40 50 60 700

50

100

150

200

250

300

350

400

450

500

mean

99th

per

cent

ile



Estimated GEV parameters

For 168 segments of the time series Data: EEX 2 regimes:

– Different hours of weekbehave differently

– There are a few hourswith fatter tails

– These are more sensitiveto price spikes

Deterministic regime switchingExplains deterministic heteroscedasticity and changing spike

intensity

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



Changing distributions (EEX)

Upper plotVertical axes

left: mean of the hour;right: shape parameter k(dotted data)

Lower plotVetical axes

left: mean of the hour;right: regime (0 or 1 – normal or risky)(data with marker)

0 20 40 60 80 100 120 140 160 18020

25

30

35

óra0 20 40 60 80 100 120 140 160 180

0

0.05

0.1

0.15

0 20 40 60 80 100 120 140 160 18023

24

25

26

27

28

29

30

31

32

33

0 20 40 60 80 100 120 140 160 180

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

óra



Deterministic regime switching model in risk management Probability of exceeding a threshold tr (=cdf) Data: EEX

Line: theoretical probabilities.Dotted line: empirical probabilities (frequencies).

0 20 40 60 80 100 120 140 160 1800

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

óra

p

tr=110



Seasonality filtering (intra-week) Methods (Weron [2006]):

– Differencing (alters the correlation structure)– Median or average week (negative values)– Moving average– Spectral decomposition– Wavelet decomposition

Approaches:

Data = periodic component + stochastic part

Assume that distributions differ only in means.This is not true for the power prices.



Suggested filter‘GEV filter’

Transformation:x original priceFln

-1 inverse of the lognormal cdfFi GEV cdf for hour iy filtered price

Properties:– If the prices have a GEV distribution, filtered prices have lognormal distribution– The transformation is always well-defined.– Risky distributions: heavy tails disappear (outlier filtering)– Time series models can be applied to filtered (log) prices– An inverse filter is defined accordingly.– Separate time series modelling and (outlier, seasonality, heteroscedasticity) filtering.

0 500 1000 1500 2000 2500 30000

0.5

1

1.5x 10

-3

x

pdf

a

GEV

lognormal

0 500 1000 1500 2000 2500 30000

0.5

1

1.5x 10

-3

x

pdf

b

GEV

lognormal

GEV with high expected value

2500 2550 2600 2650 2700 2750 2800 2850 2900 2950 30002

3

4

5

6

7

8

9

10x 10

-6

x

pdf

GEV

lognormal

xFFy i1

ln



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



Price spikes and seasonality

Trück, Weron, Wolff [2007]– Price spikes influence the calculations during seasonality filtering.– With seasonality present, spikes are difficult to identify– The two filtering procedures are interconnected– Suggestion: iterative procedure

(seasonality -> spike - > seasonality)

My result: GEV filter– Filters fat tails and seasonality at the same time



Conclusions

Prices have long memory Price spikes constitute no separate regime

(monofractal property) Price spikes are high realizations of GEV (Fréchet)

distribution Deterministic regime switching causes time-

dependent jump intensity, heteroscedasticity and seasonality

It can be removed by the GEV filter



Thank you for your attention!

Questions zita.marossy () uni-corvinus.hu

zita marossy corvinus university of budapest zita.marossy () uni-corvinus.hu

Documents

power productswork

daymarket price

processdaily prices

eex hourly prices

spot marketday

electricity market

riskypower price forecasts

distribution of future