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Zita MAROSSY Corvinus University of Budapest zita.marossy () uni-corvinus.hu

The behaviour of day-ahead electricity prices Analysis of spot electricity prices using statistical, econometric, and econophysical methods. Zita MAROSSY Corvinus University of Budapest zita.marossy () uni-corvinus.hu Workshop on Deregulated European Energy Market Collegium Budapest

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Zita MAROSSY Corvinus University of Budapest zita.marossy () uni-corvinus.hu

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  1. The behaviour ofday-ahead electricity pricesAnalysis of spot electricity prices using statistical, econometric, and econophysical methods Zita MAROSSY Corvinus University of Budapest zita.marossy () uni-corvinus.hu Workshop on Deregulated European Energy Market Collegium Budapest September 24-25, 2009

  2. 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

  3. 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 Cournty European Energy Exhange Germany Powernext France APX Power NL Netherlands APX Power UK UK Energy Exchange Austria Austria Prague Energy Exchange Czech Republic Opcom Rumania Polish Power Exchange Poland Nord Pool Norway Borzen Slovenia Italian Power Exchange Italy OMEL Madrid Spain Belpex Belgium Source: RMR Áramár Portál. (March 30, 2009) Power exchanges Day-ahead (spot) market Balancing market Futures markets Balancing market Day-ahead (spot) market Futures markets Source: Geman [2005].

  4. 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].

  5. 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

  6. 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”

  7. Modelling approaches • Stochastic model calibration, time series analysis • Find a suitable model, calibrate, use it for forecasting • Fundamental models • Driving factors of supply and demand are modelled • Price behaviour is derived from market equilibrium • Agent-based models • Description of market players’ actions (e.g. simulation) • Statistical models • Directly investigate the distribution • No prior knowledge about the driving factors & market players’ behaviour is needed • Artificial intelligence-based models • E.g. neural networks, SVM • Black box

  8. 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])

  9. Stylized facts 2/7 The time series exhibits seasonality. (Plot: EEX data) • Annual • Plot: 4-month MA-filtered data • Weekly • Daily • Plot: mean of hourly prices

  10. Stylized facts 3/7 Stable autocorrelation structure with high autocorrelations (Plot: EEX data) • High autocorrelation coefficients • Slowly decreasing autocorrelation function (persistency) • Periodicity (seasonality)

  11. 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)

  12. 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

  13. 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)

  14. 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

  15. 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

  16. 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]) m: drift (usually: mean reversion) s: volatility qt: jump (driven by e.g. a Poisson process) Jump diffusion

  17. 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

  18. 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 Time series models

  19. Modelling price spikes • Price spikes are very important in risk management • Definition varies: mean + constant * standard deviation Zhao, 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.”

  20. Own research results • Fractal feature „Detailed analysis of the fractal feature of day-ahead electricity prices” • Distribution of power prices „Extreme value theory discovers electricity price distribution” • Deterministic regime switching and filtering „Deterministic regime-switching, spike behaviour, and seasonality filtering of electricity spot prices”

  21. Own research results • Fractal feature „Detailed analysis of the fractal feature of day-ahead electricity prices” • Distribution of power prices „Extreme value theory discovers electricity price distribution” • Deterministic regime switching and filtering „Deterministic regime-switching, spike behaviour, and seasonality filtering of electricity spot prices”

  22. 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 • 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) • For stationary processes H = 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)

  23. fractional Wiener process (fractional Brownian motion) values and increments (H = 0.25, 0.4, 0.5, 0.6, 0.75 ) Persistency – example

  24. 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

  25. 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

  26. 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).

  27. Multifractality test • Jiang, Zhou [2007] H0: monofractal H1: 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

  28. „There are no spikes” The separate statistical modelling of price spikes is impossible as price spikes can not be distinguished in the price process. • Price spikes behave the same way regarding the correlations as prices at average level do. • 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.

  29. Own research results • Fractal feature „Detailed analysis of the fractal feature of day-ahead electricity prices” • Distribution of power prices „Extreme value theory discovers electricity price distribution” • Deterministic regime switching and filtering „Deterministic regime-switching, spike behaviour, and seasonality filtering of electricity spot prices”

  30. 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

  31. Generalized extreme value (GEV) distribution • 3 parameters: • scale (k) • Fréchet (k>0) • Weibull (k<0) • Gumbel (k=0) • location (m) • scale (s)

  32. pdf cdf Q-Q plot Estimates Statistical test GEV (Fréchet) fits the empirical dataData: EEX daily prices

  33. 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

  34. 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

  35. Own research results • Fractal feature „Detailed analysis of the fractal feature of day-ahead electricity prices” • Distribution of power prices „Extreme value theory discovers electricity price distribution” • Deterministic regime switching and filtering „Deterministic regime-switching, spike behaviour, and seasonality filtering of electricity spot prices”

  36. 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

  37. Estimated GEV parameters For 168 segments of the time series Data: EEX 2 regimes: Different hours of week behave differently There are a few hours with fatter tails These are more sensitive to price spikes Deterministic regime switching Explains deterministic heteroscedasticity and changing spike intensity

  38. Changing distributions (EEX) • Upper plot Vertical axes left: mean of the hour; right: shape parameter k (dotted data) • Lower plot Vetical axes left: mean of the hour; right: regime (0 or 1 – normal or risky) (data with marker)

  39. Deterministic regime switching model in risk management • Probability of exceeding a threshold tr (=cdf) • Data: EEX Line: theoretical probabilities. Dotted line: empirical probabilities (frequencies).

  40. 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.

  41. Suggested filter‘GEV filter’ • Transformation: x original price Fln-1 inverse of the lognormal cdf Fi GEV cdf for hour i y 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.

  42. Empirical results • Figures: periodogram of • ACF (orig prices) • ACF (filtered data) • Intraweekly filtering • successful

  43. 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

  44. 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

  45. Thank you for your attention! • Questions • zita.marossy () uni-corvinus.hu

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