Fernanda Strozzi, Eugénio Gutiérrez Tenrreiro and José-Manuel Zaldívar Comenges

1. APPLICATION OF NON-LINEAR TIME SERIES ANALYSIS TECHNIQUES TO THE NORDIC SPOT ELECTRICITY MARKET DATA Fernanda Strozzi, Eugénio Gutiérrez Tenrreiro and José-Manuel Zaldívar Comenges Update on WP5: Task 5.2 Deliverable 5.3

2. DATA Hourly spot prices in the Nordic electricity market (Nord Pool) from May 1992 until December 1998. (NOK/MWh) Hourly spot prices in the Nordic electricity market (Nord Pool) from January 1997 until January 2007. (EUR/MWh) Norway Statnett Market Sweden Nord Pool Finland W Denmark E Denmark Kontek 1996- EU Electricity Directive starts to have impact: EU countries open their electricity markets to competition ( high consumers can choose their provider). 1993- Nord Pool (Nordic Electricity Market) was created by Norway. …… 2005-KT area. (Kontek cable connection Zealand-Germany). A competition starts between Nord Pool and European Energy Exchange (EEX)

3. Electricity Price • Participants trade power contract one day for each hours of the next day • At the end of the trade all possible congestions or insufficient capacities are checked. • If some congestion or insufficient capacities occurs with these flows the market system established different “prices area” and TSOs ask to generators to increase (reduce) production or to buyer to increase (decrease) demand. • Then sometimes prices are the price of all Nordic region (“system price”) sometimes different “areas prices” exist. • Nordic Market has a single price less than half of the time (due to congestions) • In this work we will consider “System price”: the price if no transmission constrains are considered

4. Electricity Price: dependencies • The variation of the prices in the Nord Pool system is well correlated with the variations in precipitations because of its dependence from hydropower generation. • In the “dry” periods the price and its volatility increase due to the dependence from other source of energy (petrol)

5. Electricity Price

6. Goals 1. Characterizing time series spot price dynamics comparing them with - Linear Gaussian dynamic with the same fft testing nonlinearity - Time series with the same probability distribution but not temporally correlated (shuffled) 2. Detect important events using nonlinear time series analysis

7. Data treatment Hourly logarithmic return

8. Hurst exponent: H H provides a measure of whether the data are a pure random walk or have some trend (i.e. some degree of correlation exists) H is a tool for studying long-term memory and fractality of a time series A long memory process is a process with a random component, where a past event has a slow decaying effect on future events.

9. Hurst exponent: R/S analysis A variety of techniques exist for estimating H but the accuracy of estimation is a complicate issue Hurst measures how the range of cumulative deviations from the mean of the series is changing with the time . (R/S)n→(R/S)0 nH as n → log2(R/S)n (R/S)0is a constant, n is the number of years to calculate the mean value H is the Hurst exponent log2n

10. Hurst exponent: properties 0  H 1 H=0.5 for random walk time series fractional Brownian motion (fBm) is a random walk with H≠0.5 H < 0.5 for anticorrelated time series H > 0.5 for positively correlated series .

11. Power Spectral Density PSD describes how the power of the signal is distributed with frequency The PSD is the Fourier transform of the autocorrelation function R(t) of the signal s(t) if the signal con be treated as a stationary random process

12. Stable Distribution A stable probability distribution is defined by the Fourier transform of its characteristic function a(0,2], b[-1,1] g[0,) d(-,) Gaussian : Chauchy Levy

13. Stable Distribution

14. Stable distribution Nord Pool data fitted parameters using STABLE (Nolan, 1999). Fitted density plot for the NordPool Norwegian Krone and Euro time series data (blue line): a/Original time series, first difference; b/ without zero values (23962 values).

15. Surrogate data Surrogate data is an ensemble of data sets similar (same mean, variance, etc) to the observed data and consistent with some null hypothesis A discriminating statistic Q is chosen If only a single realization exists for the original data we cannot compare the distributions of Q then we define a level of “significance” If the null hypothesis is not true for k=19 surrogate data sets we can reject it with a probability of a=0.05 (95% level of significance)

16. Surrogate data: Null hypothesis 1. Temporally uncorrelated noise. The null hypothesis: any correlation at all. Surrogate data are generated by a random shuffling of the original time series. 2. Linearly correlated noise. The null hypothesis : the time series are originated by a linear random process with the same autocorrelation function or, equivalently, with the same Fourier Power Spectrum. etis un uncorrelated Gaussian noise of unit variance s is chosen so that the variance of the surrogates matches with the one of original data ak contain information on Correlation function

17. Surrogate data: discriminating statistic Q The dynamic is chaotic? Q = Correlation dimension Q = Lyapunov exponent Q = Forecasting error The structures of recurrence plots? Q = RQA measures

18. State Space reconstruction dE=embedding dimension t=time delay

19. Embedding Parameters First minimum of the Average Mutual Information t=time delay EUR/MWh time series for t=13. NOK/MWh time series occurs at t=15 Cao (1997) dE=embedding dimension NOK/MWh dE=10 EUR/MWH dE=10 E2=another way to measure relative increment. E1=relative increment of the mean distance between nearest points

20. Stationarity test: space time separationplot The reconstruction parameters do not change in the time if the series is stationary Stp=Contour plot of Probability Density Function in the space: separation in space versus separation in time in the reconstructed space Stationary = flat profile Random walk Sn=Random walk Rn=Gaussian Random Number Separation in time Provenzale et al. (1992)

21. Stationarity test: space time separationplot Space-time separation plot of the Nord Pool spot prices (NOK/MWh). Space-time separation plot of Australian-US dollar foreign exchange time series. Space-time separation plot of the Nord Pool spot prices (EUR/MWh). Space-time separation plot Belgium Franc-US dollar foreign exchange time series.

22. Stationarity test: space time separationplot

23. Recurrence Plots: Introduction Real data Recurrence: Henri Poincaré 1890. when a system recurs many times as close as one wish to its initial state Recurrence Plots (RPs): Eckmann, 1987. A trajectory of a system in its phase space can produce a RP i.e. a matrix given by:

24. Recurrence Plots EUR/KMh t =13, dE =10, e=10 NOK/MWh t=15, dE=10, e=40

25. Recurrence Plots: Surrogate linearly correlated NOK/MWh t=15, dE=10, e=40 EUR/KMh t =13, dE =10, e=10

26. Recurrence Plots: Surrogate Temporally uncorrelated NOK/MWh t=15, dE=10, e=40 EUR/KMh t =13, dE =10, e=10

27. Recurrence Quantification Analysis a/Measures based on recurrence density %recurrence where is one if the state of the system at time i and the one at time j have a distance less than and zero otherwise. b/ Measures based on diagonal lines %determinism (DET) is then the percentage of recurrent points forming diagonal line structures lmin=100 the length of the longest diagonal line found in the RP, or its inverse, the divergence (DIV) Shannon entropy of the probability to find a diagonal line of length l in RP.

28. Recurrence Quantification Analysis c/ Measures based on vertical lines % vertical lines of length n in RP The average length of vertical structures is given by

29. Surrogate: linear surrogate

30. Surrogate: linear surrogate

31. Surrogate: temporally uncorrelated

32. RQE analysis RQE=RQA quantities on moving windows 720 point window (one month), data are shifted 720 points NOK/MWh EUR/MWh Finland W Denmark Norway E Denmark Sweden Kontek RQE analysis is able to detect important changes?

33. Volatility Inverse of standard deviation and %determinism (top) and %laminarity (bottom) for EUR/MWh Inverse of standard deviation and %determinism (top) and %laminarity (bottom) for NOK/MWh

34. Volatility: EUR/MWh RQA measures of EUR/MWh: Values are computed from a 720 point window (one month), shifted of 720 points. RQA parameters: t =13, dE=10, distance cutoff: max. distance between points/10, line definition: 100 points (~4 days). Vertical lines correspond to the following dates: 1st October 2000, 5th October 2005 (see historical background).

35. Volatility: NOK/MWh Nonlinear metrics of the Nord Pool spot prices time series in NOK/MWh: Values are computed from a 720 point window (one month), data are shifted 720 points. RQA parameters: t =15, dE=10, distance cutoff: max. distance between points/10, line definition: 100 points (~4 days). Vertical lines correspond to the following dates: 1st January 1993, 1st January 1996, 29th December 1997 and 1st July 1999 (see historical background).

36. Conclusions and Future Developments Measuring long term correlation of the time series: • H < 0.5 anticorrelation • Stable distribution Characterize the underlying dynamics using surrogates: • Space Time Separation Plot: the data seems stationary • RQA Measures (%determ, %Lam) are able to distinguish between data: - surrogate linearly correlated - temporally uncorrelated Detect important changes in the time series not evident from their time representation RQE: (RQA measures on moving windows) are able to detect changes RQE→%determ, %Lam give a new measure of volatility New Developments: analyse the possible correlation between the volatility of energy market and blackouts

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