Complexity and Persistence of Price Time Series of the European Electricity Spot Market

The large variability of renewable power sources is a central challenge in the transition to a sustainable energy system. Electricity markets are central for the coordination of electric power generation. These markets rely evermore on short-term trading to facilitate the balancing of power generation and demand and to enable systems integration of small producers. Electricity prices in these spot markets show pronounced fluctuations, featuring extreme peaks as well as occasional negative prices. In this article, we analyze electricity price time series from the European Power Exchange market, in particular the hourly day-ahead, hourly intraday, and 15-min intraday market prices. We quantify the fluctuations, correlations, and extreme events and reveal different time scales in the dynamics of the market. The short-term fluctuations show remarkably different characteristics for time scales below and above 12 h. Fluctuations are strongly correlated and persistent below 12 h, which contributes to extreme price events and a strong multifractal behavior. On longer time scales, they get anticorrelated and price time series revert to their mean, witnessed by a stark decrease of the Hurst coefficient after 12 h. The long-term behavior is strongly influenced by the evolution of a large-scale weather pattern with a typical time scale of four days. We elucidate this dependence in detail using a classification into circulation weather types. The separation in time scales enables a superstatistical treatment, which confirms the characteristic time scale of four days, and motivates the use of $q$-Gaussian distributions as the best fit to the empiric distribution of electricity prices.

Popular Summary

Wind and solar power are essential elements of the transition to a sustainable energy supply. Their large temporal variability constitutes a main challenge for the organization and operation of energy and power systems. The coordination of power generation is mainly achieved via trading on electricity markets, which is heavily affected by fluctuations in renewable energy. Electricity prices are getting much more volatile and even become negative occasionally. Here, the authors provide an in-depth statistical analysis of the prices on the main European electricity exchanges, elucidating the role of extremes and correlations and the impacts of weather. Large-scale weather conditions have a pronounced footprint in both the averages and the fluctuations of electricity prices, which persist for approximately four days. Price fluctuations show an intriguing pattern with persistent plateaus and occasional extreme jumps, limiting the possibility of hedging. Extreme price events occur if these jumps and strong weather conditions coincide.

Figure 1

The electricity price strongly depends of the residual load. The figure shows a joint histogram of the price $p(t)$ in the day-ahead hourly market and the residual load, i.e., the difference between the load and renewable generation $D(t)-G_r(t)$ in a colormap plot with a logarithmic scale. Assuming a perfect market equilibrium, prices would be given by the function $\overline{p}(t)\approx G_d^{-1}[D(t)-G_r(t)]$, which can be approximated by a linear function (dotted red line). The fluctuations, i.e., deviations from the line, are evident, as well as occasional extreme events. Data from EPEX, 2015–2019 [13].

Figure 2

The three electricity price time series examined in this study, from January 2015 to December 2019. The average fluctuations around a mean value are visible, as well as occasional jumps into either very large or possibly negative prices. The three black curves display the three slowest intrinsic mode functions (IMFs) obtained via the Hilbert-Huang transform, which are subtracted from the data to remove the long-term nonstationarity. Data from EPEX, 2015–2019 [13]. Figures generated with python’s Matplotlib [56].

Figure 3

Empirical distributions of the detrended price time series. Each empirical distribution is fitted via a maximum likelihood algorithm [69, 70] with a $q$-Gaussian and a symmetric $\alpha$-stable distribution, given by Eqs. (3) and (2), respectively. The $q$-Gaussian distributions yield a better fit than the $\alpha$-stable distributions, yielding $q=1.46$ for the day-ahead, $q=1.50$ for the intraday hourly, and $q=1.46$ for the intraday quarterly prices. The Kullback-Leibler divergence of the empirical and fitted distributions are given in Table 1, where one can see that the $q$-Gaussian distribution is the best at describing the empiric price time series. Mean $\mu$, standard deviation $\sigma$, skewness $s$, and kurtosis $\kappa$ of the empiric data are given in each figure.

Figure 4

Fluctuation function $F_{\hat{q}=2}(r)$ over the scale $r$ of the three price time series on a double-logarithmic plot. By fitting the curves we can extract the Hurst exponent as given in Eq. (9). Noticeable is the change from persistence to antipersistence at the $12$ h mark, which is present for the hourly market but not the quarter-hourly market. The separation into three disjoint time ranges of $t<12\mathrm{h}$, $12<t<48\mathrm{h}$, and $t>48\mathrm{h}$ is discussed in the multifractal analysis (see Fig. 5).

Figure 5

Singularity spectrum $f(\hat{\alpha })$ and singularity strength $\hat{\alpha }$ for three distinct time scales of the three price time series: $t<12\mathrm{h}$, $12<t<48\mathrm{h}$, and $t>48\mathrm{h}$. The horizontal width, i.e., the multifractal spectrum width $\mathrm{\Delta }\hat{\alpha }$, indicates the strength of the fluctuation and jumps at the indicated time scales. The data show larger $\mathrm{\Delta }\hat{\alpha }$ at short time scales, indicating that within these windows, very large price variations are seen and are reverted back to their mean value. The effects become milder as the time scales increase, telling us that the market shows weaker jumps at large time scales and always reverts back to its mean value. Finally, at scales $t>48\mathrm{h}$, the three markets become indistinguishable, as we had also seen in Fig. 4 at the same scales. Results in Table 2. The singularity spectra are shifted horizontally for better visibility to allow for comparison of the widths and heights of the spectra.

Figure 6

Autocorrelation functions $C(t^{\prime })/C(0)$ of the price time series and their respective volatilities. The short superstatistical times $\tau$ for each price time series is extracted from the initial exponential decay of $C_p(t^{\prime })$. The results are given in Table 3. Also shown is the autocorrelation of the $f$ parameter, discussed in Sec. 3e.

Figure 7

Estimating the long superstatistical time $T$ from the vanishing of the local skewness $s(\delta t)$ given in Eq. (16). The top panel shows the $s(\delta t)$ for increasing segments of time length $\delta t$. At approximately four days, indicated with the circular markers, all price time series show a vanishing local skewness. We define this as marking the long superstatistical time $T$. The lower panel shows the local kurtosis ${\kappa }_p(\delta t)$. We also show the kurtosis of a Gaussian distribution, i.e., $\kappa =3$, for comparison. We see that the time series attain an equilibrium at the vanishing skewness $s$, but they do so with different kurtosis, implying that their equilibrium distributions are not Gaussian distributions, possibly apart from the day-ahead hourly price time series. Each long superstatistical time $T$ can be found in Table 3.

Figure 8

Impact of large-scale weather regimes on the statistics of electricity prices. We sort all time intervals according to the $f$ parameter that represents the strength of the flow over Central Europe, and evaluate the statistics of the resulting subsets of the time series. We observe that low $f$-parameter values (calm wind conditions) are associated with a large mean $\mu$, small standard deviation $\sigma$, positive skewness $s$, and somewhat large kurtosis $\kappa$. These are the weather periods with low renewable generation in Germany, which is dominated by wind generation. When examining the raw prices (bottom plots), there are no periods of negative price (i.e., number of hours of negative prices), and a maximum of “high” price events (where the raw prices $p_r>{\mu }_r+3{\sigma }_r$, $r$ for raw). As the $f$-parameter value increases (windier conditions over Central Europe), the mean $\mu$ and skewness $s$ turn negative, the standard deviation $\sigma$ increases, and the kurtosis $\kappa$ decreases slightly. Likewise, negative (raw) price events increase and “high” (raw) prices vanish. The top four plots utilize the processed price data, and the bottom plots use the actual raw prices, to best showcase the “true” negative price events.

Figure 9

Distributions $f(\beta )$ of the volatilities $\beta$, on a double logarithmic scale, of the three price time series and the two best-fitting log-normal and inverse-Gamma distributions, given in Eqs. (A1) and (A2), minimizing the Kullback-Leibler divergence $D_{\mathrm{KL}}$, as given in Eq. (6). Inset shows a linear scale.

Figure 10

Impact of particular weather types on the statistics of price time series. We condition the price time series to either westerly or anticyclone states for segments longer than 12 h ($g>12$) and for segments longer than 24 h ($g>24$). For each, we calculate the mean $\mu$, standard deviation $\sigma$, and skewness $s$ of the prices. A separation between westerly and anticyclone weather types is in agreement with the price relation with the f parameter in Fig. 8. Westerly weather types are associated with a negative mean, more elevated standard deviation, and slightly negative kurtosis. In opposition, anticyclone weather types are associated with positive mean, smaller standard deviation, and slightly positive skewness.