The footprint of atmospheric turbulence in power grid frequency measurements

Wind energy is one of the core elements of renewable power production with increasing feed-in to the central European power grid: In 2016, an installed capacity of 153.7 GW in the EU generated almost 300 TWh and covered 10.4% of the EU electricity demand [1]. In the first days of 2018, even more than 20% (60%) of the EU (German) daily electricity demand was covered [2].

A stable and reliable supply with electrical power is essential for both society and economy. The power grid frequency reflects the transient ratio of production to demand in the grid and thus serves as an instantaneous and locally inferable stability parameter. Mismatch of production and consumption causes frequency deviations from the nominal frequency [3]. Load frequency control of the grid operator restores the frequency after perturbations: The fastest control ("primary control") sets in seconds after a deviation from the nominal frequency to stabilize, but not yet restore, the frequency. Restoration is achieved by secondary control which operates on time scales of 30 seconds and beyond [4].

Wind energy feed-in is known to be highly volatile. Fluctuations of a process x(t) on a time scale τ are often characterized by means of increments $\Delta_\tau x:= x(t) - x(t+ \tau)$. Traditional analysis and prediction of wind speed considers variations in 15 minutes and longer [5,6]. However, recent findings in the analysis of short-term increments of renewable power generation reveal strongly non-Gaussian fluctuations even on scales of one second [6,7]. But, where does this short-term behavior result from?

The atmospheric boundary layer is known to be non-stationary and turbulent [8–10]. Turbulent flows show scale-dependent increment statistics: In a hierarchical cascade process, kinetic energy is transferred from large- to small-scale structures [11]. Specifically, this implies pronounced tails in short-term increment statistics; an effect termed intermittency in turbulence research [12]. Due to the intermittent increment statistics, severe wind-speed fluctuations are much more likely than expected from a normal distribution.

A wind turbine transforms the kinetic energy of the wind to electric power. Even though ac-dc-ac converters decouple wind speed from power output dynamics, turbine controllers maximize the power output and follow wind speed variations [7]. Hence, atmospheric fluctuations even within a second propagate into the power output of wind farms and are fed to the grid. In fact, intermittency is found in production time series of wind (and also photovoltaic) power plants [6,7]. Consequently, power quality is a major challenge for the grid integration of renewable generators [13].

The transient short-term reaction of power grids to perturbations has attracted great attention in theoretical physics: Simple models of high-voltage ac-grids correspond to complex networks of Kuramoto-like, phase-coupled oscillators [14,15]. Such models have been used to analyze aspects of synchronization [16] and the interplay of stability and topology [17], as well as relaxation after singular [18] and stochastic [19,20] perturbations. The impact of intermittent feed-in on power grids has been addressed in [20,21]: Numerical results indicate that intermittency propagates in a power grid and affects the frequency increment distributions of nodes distant to the feed-in. Stochastic models for non-Gaussian frequency fluctuations are presented in [22]. However, none of the prior results relates the intermittent feed-in to transient stochastic properties of the grid frequency.

It is often believed that intermittency vanishes when power time series of many turbines are averaged. To support this hypothesis, usually the central limit theorem is referred to. Velocity time series v(t) are, however, highly correlated [23] and so are the resulting power time series [6]. The lacking statistical independence makes this theorem inapplicable. Intermittency is, in fact, observed in power outputs of entire wind farms [6] and withstands even country-wide averaging [24]. We show, as an example, the increment distribution for the mean power output of twelve turbines in comparison to the one of a single turbine in fig. 1. But, what exactly is the impact of wind power feed-in on the grid frequency?

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In this letter, we complement the previous numeric and analytic work and show that the feed-in of intermittent wind power has a measurable effect on the increment statistics of the frequency measured in the distribution grid. Instead of focusing on the frequency response to singular, large-deviation events, as, for example, in [25], we use the full statistical information encoded in the increment statistics of the grid frequency and focus on time scales that lie below the activation of primary control.

This paper continues as follows: We first introduce our measurement and data processing techniques. Subsequently, we show our stochastic analysis and its interpretation. Finally, we conclude and give an outlook on further research.

Publicly available measurements of the frequency of the public power grid have, to our knowledge, only a time resolution of 1 second or above. We, however, want to observe the self-organized, transient behavior of the grid and thus need a higher time resolution.

We took 10 kHz voltage samplings u(t) of a single phase of the distribution grid in our lab in Oldenburg, northern Germany, from November 8, 2016, till March 23, 2017. Subsequently, we applied the method of Instantaneous Frequency (IF) [27] to estimate the frequency time series f(t) from the sinusoidal voltage signal u(t).

The IF reveals the dominant frequency component at each time instant t and is thus suited for signals composed of one major frequency component. The method makes use of the fact that real-valued signals, such as the voltage signal u(t), have conjugate symmetric Fourier representations, $\mathcal{F}[u](-\omega) = \mathcal{F}[u](\omega)^*$. Here, $\mathcal{F}$ denotes Fourier transform. The complex-valued analytic signal z(t) is the inverse Fourier transform of the positive frequencies $\omega > 0$. Discarding the redundant negative frequency components makes the IF accessible. It is defined as the time derivative of the phase $\Phi(t)$ of the analytic signal z(t):

$$\begin{equation} f(t) = \frac{1}{2\pi}\frac{\mathrm{d}}{\mathrm{d}t}\Phi(t) = \frac{1}{2\pi}\frac{\mathrm{d}}{\mathrm{d}t}\mathrm{arg}(z(t)). \end{equation} \tag{ 1 }$$

In practice, z(t) is obtained from the Hilbert transform $\mathcal{H}[u](t)$ of the original signal: $z(t):= u(t) + i\mathcal{H}[u](t)$. The Hilbert transform can be obtained from $\mathcal{H}[u](t) = (u * 1/\pi t' ) (t),$ where "$*$" denotes convolution.

To estimate the derivative in eq. (1) numerically, the phase $\Phi(t)$ was calculated for every time step in the voltage signal. Subsequently, the time derivative was estimated by linear fits of $\Phi(t)$ in disjoint blocks of 2000 samples. This procedure gives a frequency time series f(t) with a time resolution of 200 ms. The $2\sigma$-confidence bounds of the linear fits are, in average, of size $\pm 1\ \text{mHz}$. We show one example hour from our measurements in fig. 2.

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Frequency measurements of the public power grid are influenced by many factors that overlay the influence of renewable generation. The signal shows severe deviations from the nominal frequency each full and half hour caused by power trading. Further, it is influenced by long-term correlations in demand and production. Thus, we applied kernel detrending to isolate the short-term behavior of the frequency signal f(t): A kernel-smoothed signal f_ks(t) was subtracted from the original signal to obtain the detrended signal $f_d(t) = f(t) - f_{ks}(t)$. We used a 30-seconds Gaussian kernel. The kernel-smoothed signal is obtained from convolving the original signal with a Gaussian curve $g_\sigma(t)$ with standard deviation $\sigma =$ 30 s and zero mean:

$$\begin{equation} f_d(t) = f(t) - f_{ks}(t) = f(t) - (f * g_\sigma)(t). \end{equation} \tag{ 2 }$$

Again, "$*$" denotes convolution. We illustrate the detrending in fig. 2. In the following, we use the detrended data and drop the index d.

To evaluate the fluctuations of the grid frequency we focus on probability density functions (PDFs) of increments, $p(\Delta_\tau f)$. As shown in fig. 3(a), the increment distribution $p(\Delta_\tau f)$ is, for a time scale of $\tau = 200\ \text{ms}$, not Gaussian. Its tails cause strong deviations from the normal distribution. This means that large increments occur much more frequently than expected from a normal distribution.

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The kurtosis $k(\tau) = \langle(\Delta_\tau f - \langle\Delta_\tau f\rangle)^4\rangle / \sigma^4$ measures how heavy-tailed a distribution is. Here, σ denotes the standard deviation of $p(\Delta_\tau f)$. The kurtosis takes the value k = 3 for a Gaussian distribution and increases for more heavy-tailed shapes. We observe that the increment PDFs $p(\Delta_\tau f)$ deform to less heavy-tailed shapes for increasing time lags τ (fig. 3(b)). On time scales below one second, we find the most extreme tails. Even though this effect is similar to turbulent intermittency, can we —at all— relate such short-term fluctuations to wind power injection?

The grid frequency measurements are, obviously, influenced by many possibly non-Gaussian and/or correlated processes. To investigate a possible dependence of grid fluctuations on wind energy injection, we extract statistical properties of the detrended frequency signal f(t) which we condition on the amount of onshore wind energy P_w(t) that is fed to the European grid in Germany. We use time series provided by the ENTSO-E Transparency Platform [28], specifically the dataset "actual generation per production type" for Germany and production type "onshore wind". This data set has a time resolution of 15 minutes. Hence, it does not allow for an analysis of the short-term behavior of the feed-in but still enables us to condition our high-frequency measurements on the amount of wind power fed to the grid. We focus on the generation in Germany because, first, it had, in 2016, by far the highest installed capacity of wind power [1] and, second, the provided data of other countries have an even lower time resolution.

We begin with a visual comparison of the increments $\Delta_\tau f$ and P_w: In fig. 3(c), we show the time instants at which large increments occur as well as the amount of wind power P_w fed to the grid in Germany. We consider, for the moment, data aggregated for two days. Large increments $\Delta_\tau f$ coincide with high values of P_w. This is a first indication that wind power feed-in affects grid frequency fluctuations. In the following, we will provide a detailed analysis of the impact of wind power feed-in on frequency increment distributions on different time scales.

Instationary stochastic processes are known to potentially produce heavy tails in their probability distributions (e.g., [29]). Wind turbulence, in particular, shows characteristic turbulent behavior only when wind speed increments are conditioned on the absolute wind speed [30]. Motivated by such findings, we pinpoint the impact of wind power injection on the grid frequency by the analysis of conditioned increment PDFs $p(\Delta_\tau f | P_w)$. Thus, we learn how likely an increment $\Delta_\tau f$ is if an amount P_w of wind energy is fed to the grid. We show this PDF for a short ($\tau = 200\ \text{ms}$) and a long ($\tau = 10$ s) time scale for different ranges of P_w in figs. 4(a) and (b). First, we observe that on the short scale, the tails deviate from the normal distribution (gray reference curve), whereas the increment PDF is very close to normal on the long scale. Second, for the long time scale, the PDFs are almost identical, irrespectively of P_w. On the short time scale, however, we observe a broadening of the distribution with increasing P_w.

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We quantify the time-scale–dependent impact of the feed-in P_w on the increment PDF by means of width and shape of the conditioned PDFs. In fig. 4(c), we show that the variance

$$\begin{equation} \sigma^2(\tau,P_w):= \int (\Delta_\tau f - \langle \Delta_\tau f \rangle_{P_w})^2 p(\Delta_\tau f | P_w)\mathrm{d}\Delta_\tau f \end{equation} \tag{ 3 }$$

of the conditioned increment PDF increases with P_w for $\tau = 200$ ms. For increasing time lags τ, this effect quickly diminishes. On time scales of $\tau = 800\ \text{ms}$ and above, the variances show no clear trend with P_w.

An increased probability of large fluctuations may result not only from an increased variance but also from non-Gaussian, heavy-tailed shapes of the PDFs. To quantify the shape development of $p(\Delta_\tau f|P_w)$ with P_w, we use a model which was proposed by Castaing for the characterization of intermittency in turbulence [31].

Castaing uses superimposed Gaussian PDFs with log-normally distributed variances to grasp the tails in intermittent increment PDFs. The standard deviation $\lambda^2$ of the log-normal distribution,

$$\begin{equation} \lambda^2 = \frac{1}{4}\ln\left(\frac{\langle(\Delta_\tau f - \langle\Delta_\tau f\rangle)^4\rangle}{3\cdot\langle(\Delta_\tau f - \langle\Delta_\tau f\rangle)^2\rangle^2}\right) = \frac{1}{4}\ln\left(\frac{k(\tau)}{3}\right), \end{equation} \tag{ 4 }$$

governs the shape of the thus obtained PDF $p_c(\Delta_\tau f | P_w)$ and is called shape parameter [32]. Due to its close relation to the kurtosis $k(\tau)$, it serves as a measure for the heavy-taildness of the Castaing PDF: For a Gaussian PDF, $\lambda^2$ is zero. It increases the larger the deviations of the tails from the Gaussian PDF become. In wind speed measurements, we find $\lambda^2$ in the range 0.2–0.3 for increment PDFs on short time scales [30].

We calculate $\lambda^2$ from our measurements with eq. (4) and follow the steps in [30] to obtain the explicit expression of $p_c(\Delta_\tau f | P_w)$. The results match the data very well as shown in fig. 5(a), where we compare the conditioned increment PDFs for a small and a large P_w on the short scale ($\tau = 200\ \text{ms}$). With this result we are now able to analyze the change in shape as a function of the wind power P_w, see fig. 5(b). In accordance with figs. 4(a) and (b), we observe lower $\lambda^2$-values for $\tau = 10$ s than for $\tau = 200\ \text{ms}$; i.e., the PDFs are closer to the Gaussian distribution on the longer time scale. In contrast to the variance (fig. 4(c)), we do not observe a clear trend of $\lambda^2$ with P_w. This means that wind energy feed-in mainly broadens the conditioned increment PDF on short scales without much affecting its shape.

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The Castaing parametrizations $p_c(\Delta_\tau f | P_w)$ can be used to estimate the impact of P_w on extreme fluctuations of the frequency: In fig. 5(a), we compare the probability of a 5σ-event during high wind energy feed-in P_w to a Gaussian model. We observe a factor 900 between the Castaing model and the Gaussian (black arrow in fig. 5(a)). Note that this probability factor increases further by many orders of magnitude for larger σ-events which for the Gaussian statistics are expected to almost never occur —even though we observe them already in our relatively short data set. This stresses the importance of a correct non-Gaussian modeling of frequency fluctuations in power grids with intermittent feed-in.

We have shown that wind power feed-in impacts the power grid frequency on time scales that lie below one second. The time range up to approximately one second is interesting in two aspects: First, it lies in the range of activation of primary frequency control [4]. This suggests that fluctuations by wind power injection on longer time scales are successfully compensated. Second, in the context of Small Signal Stability Analysis, one second is approximately the time scale that separates local modes, which affect only a localized subset of nodes in the grid, from so-called interarea modes [33]. This suggests that the effect we measure is local; a result which is in accordance with our analysis in so far as the used German wind power data are dominated by the northern region of Germany where our frequency measurements were made.

Power quality is a key challenge for the grid integration of renewable generators [13]. Although the absolute size of the fluctuations we consider is small ($\Delta_\tau f<20\ \text{mHz}$ for $\tau = 200\ \text{ms}$), a precise knowledge of the fluctuation statistics is essential to correctly estimate the probability of large, possibly critical, increments. In future power grids with a high share of renewable energy sources, the amount of rotational inertia will be much lower than today. This will lead to faster frequency dynamics with larger amplitudes [34]. If grid design and control strategies are not properly adapted, such frequency fluctuations may become highly critical for the grid stability [19]. Thus, an explicit expression for increment probabilities is desirable to correctly quantify this risks.

Our analysis offers a new tool to quantify the impact of renewable generation on the frequency increment statistics: The conditioned increment PDFs $p(\Delta_\tau f | P_w)$ are well described by Castaing's parametrization. For a given distribution of the feed-in p(P_w) (fig. 5(c)), the conditioned PDFs may be assumed to follow $p_c(\Delta_\tau f | P_w)$ with, in the simplest model, constant shape parameter (fig. 5(b)) and variance increasing with P_w (fig. 4(c)). If shape parameter and variance evolution are inferred from calibration measurements, the increment PDF

$$\begin{equation} p(\Delta_\tau f) = \int p(P_w)p_c(\Delta_\tau f | P_w) \mathrm{d}P_w \end{equation} \tag{ 5 }$$

describes the overall impact of wind energy feed-in on the fluctuation characteristics of a given grid and may be helpful for the design of new control strategies for grids with a high share of renewable sources.

We want to point out that the non-Gaussian increment statistics $p(\Delta_\tau f)$ may also be fitted with other heavy-tailed distributions, such as q-Gaussians or α-stable distributions, which have successfully been applied to single-point PDFs of grid frequency data [22] as well as to other complex systems like stock markets [35] or biological systems [36]. An important property of such distributions is, besides the stability, the fact that they have diverging moments for wide parameter ranges. We use here the turbulence-like finite-moment approach because, first, we see that power fluctuations are driven by wind turbulence and, second, we observe that frequency increment PDFs are not stable: The sum of two consecutive increments is per definition an increment of a larger scale. However, with increasing time lag, the kurtosis of the increment PDF decreases (fig. 3(b)). Hence, the hypothesis of stability is violated.

Independently of the question about the best model of the increment statistics, our main finding is that we observe a broadening of frequency increment PDFs with increasing share of wind power generation. There remains an open question, i.e., to what extent the shape of the increment PDFs is caused by the turbulent wind statistics¹ or by other collective effects of interacting grid components. We conclude that a deep understanding of the non-Gaussian fluctuations of renewable energy sources and their interaction with the grid is an important field of further research.

This work was funded by the ministry for science and culture of the German federal state of Lower Saxony (grants Nos. ZN3045 and ZN3024, nieders. Vorab). We gratefully acknowledge the technical support from the electronic workshop at Carl von Ossietzky University Oldenburg, especially T. Madena. We would like to thank K. Schmietendorf, M. Anvari, C. Behnken, S. Kettemann, and U. Feudel for valuable discussions.