# Isolating the impact of trading on grid frequency fluctuations

Benjamin Schäfer1 Marc Timme1 and Dirk Witthaut2 This work is supported through the German Science Foundation (DFG) by a grant toward the Cluster of Excellence “Center for Advancing Electronics Dresden” (cfaed), the Helmholtz Association (via the joint initiative “Energy System 2050—a Contribution of the Research Field Energy” and grant no. VH-NG-1025) and the Federal Ministry of Education and Research (BMBF Grant No 03SF0472F).1Benjamin Schäfer and Marc Timme are with Chair for Network Dynamics, Center for Advancing Electronics Dresden (cfaed), Institute for Theoretical Physics, Technical University of Dresden, Dresden and Network Dynamics, MPIDS, Göttingen, Germany<EMAIL_ADDRESS>|<EMAIL_ADDRESS>Witthaut is with the Forschungszentrum Jülich, Institute of Energy and Climate Research-Systems Analysis and Technology Evaluation, Jülich and Institute for Theoretical Physics, University of Cologne, Köln, Germany<EMAIL_ADDRESS>
978-1-5386-4505-5/18/$31.00 ©2018 IEEE

###### Abstract

To ensure reliable operation of power grids, their frequency shall stay within
strict bounds. Multiple sources of disturbances cause fluctuations of the grid
frequency, ranging from changing demand over volatile feed-in to energy
trading. Here, we analyze frequency time series from the continental European
grid in 2011 and 2017 as a case study to isolate the impact of trading. We
find that trading at typical trading intervals such as full hours modifies the
frequency fluctuation statistics. While particularly large frequency
deviations in 2017 are not as frequent as in 2011, large deviations are more
likely to occur shortly after the trading instances. A comparison between the
two years indicates that trading at shorter intervals might be beneficial for
frequency quality and grid stability, because particularly large fluctuations
are substantially diminished. Furthermore, we observe that the statistics of
the frequency fluctuations do not follow Gaussian distributions but are better
described using heavy-tailed and asymmetric distributions, for example Lévy-
stable distributions. Comparing intervals without trading to those with
trading instances indicates that frequency deviations near the trading times
are distributed more widely and thus extreme deviations are orders of
magnitude more likely. Finally, we briefly review a stochastic analysis that
allows a quantitative description of power grid frequency fluctuations.

##  I Introduction

Power generation and consumption have to be balanced to allow the power grid
to operate close to its reference frequency (e.g., $f_{R}=50~{}\text{Hz}$ in
Europe) and thereby ensure robust distribution among generators and consumers
[1]. However, the grid frequency does not stay at precisely
$f_{R}=50~{}\text{Hz}$ during operation but is subject to multiple sources of
power fluctuations, ranging from demand fluctuations [2] over stochastic feed-
in by renewable sources [3, 4] to effects caused by energy trading [5].
Fluctuations of the grid frequency must not exceed certain security limits [6,
7]. Whereas renewable energy sources undoubtedly challenge the system due to
their distributed and variable nature [8, 9, 10], an analysis on the German
system has shown that renewables only put light stress on primary control
reserves [11]. In contrast, impacts due to energy trading seem to be
substantial, as discussed in [12, 13] and recently in [14].

When setting up a smart grid [15, 16, 17, 18], it is crucial to know the
underlying systemic dynamics and potential vulnerabilities of the system.
Liberalizing the energy market [5] may have economic upsides yet it is not
fully clear to date how it impacts a grid’s stability [19]. Including active
consumers, e.g., via demand control schemes [20, 21], will also influence the
frequency statistics. So one key question is: Can we isolate effects of
trading in power grid frequency recordings and quantify their impact?

Here, we analyze frequency data from Continental Europe from 2011 and 2017 as
a case study. We first observe the impact of trading based on hourly mean
frequency trajectories and investigate the aggregated distribution of
frequency values. To isolate effects by trading, we split the given data set
into time windows surrounding the 15-minutes trading intervals. Evaluating
measures such as standard deviations and kurtosis, we quantify differences
between trading and non-trading time intervals of the time series. Finally, we
briefly review important stochastic results on quantifying frequency
distributions [14].

##  II Analyzing frequency time series

Figure 1: The mean frequency of the 2017 time series is mostly close to the
reference frequency of $f_{R}=50~{}\text{Hz}$ but displays noticeable peaks at
fixed intervals, in particular at full hours. Plotted is the mean frequency
across every day, based on the _RTE_ data set [22].  Figure 2: The mean hourly
frequency of both the 2011 and 2017 data sets displays large deviations and
jumps at the beginning and end of every hour (minutes 0 and 60) as well as
approximately 30 minutes after full hours. We plot the mean frequency for all
hours considered in our 2011 [23] and 2017 [22] data set.  Figure 3: Large
deviations from the reference frequency were more likely in 2011. Recently,
large deviations are likely to occur a few minutes after the trading actions
every 15 minutes. We plot the mean number of measurement points within one
hour for which the frequency deviations surpass a threshold of
$100~{}m\text{Hz}$. To this end, we aggregated all data into hour-blocks, as
in Fig. 2. Within each hour, we counted on a minute-by-minute basis how often
the frequency deviates from the reference of $50~{}\text{Hz}$ by more than the
threshold of $100~{}m\text{Hz}$. We normalized for the number of days and the
different time resolution in the two data sets, assuming a 1 second
resolution.  Figure 4: The overall frequency distribution is not well
described by a Gaussian distribution (red line) but has heavier tails. Shown
is the histogram of the aggregated 2017 data [22]. The Gaussian fit was
obtained using maximum a likelihood analysis. The data displays a kurtosis of
$\kappa\approx 4.2$ in contrast to the Gaussian $\kappa^{\text{Gauss}}=3$.

We analyze frequency statistics using recordings provided by _Réseau de
Transport d’Electricité (RTE)_ [22] and _50Hertz_ [23], describing the
Continental European synchronous zone. _RTE_ provides very recent data so that
we analyze the year 2017, while _50Hertz_ offers also older data so that we
analyze time series from 2011. The _RTE_ data has a resolution of one
measurement every 10 seconds while the _50Hertz_ data has one measurement
every 4 seconds. Unfortunately, both data sets contain gaps or invalid values.
Hence, we restrict our analysis for 2011 to January for a total of 31 days and
for 2017 we include January, February, May, August, September, November and
December (consisting only of 30 days of measurement) for a total of 211 days
of complete data which we use without further processing.

For both 2011 and 2017, the mean frequency per day is computed, i.e., all
measurements taken at a specific time are averaged over all available days,
e.g. all measurements taken at 0:00 on all days are averaged to obtain the
mean frequency at 0:00. We observe that the mean frequency is generally close
to the reference frequency of $f_{R}=50~{}\text{Hz}$ but shows regular large
deviations throughout the day (Fig. 1). An earlier study [12] found that
deviations tend to be towards negative frequencies during the night and noon
and towards positive values during morning and afternoon hours and are more
pronounced every full hour, which is consistent with our observations. The
direction of the jumps could be determined by the ramps of the demand curve.
To highlight the jumps, we aggregate the full data into one hour blocks and
average recordings, e.g. measurements at 0:00, 1:00, etc. are averaged to
obtain the hourly mean for 0 min. Analyzing the result, we indeed observe
regular spikes of the mean frequency at one hour and a smaller one at 30
minutes (Fig. 2).

To further explore how larger frequency deviations are temporally distributed,
we record every instance when the frequency deviation $\Delta f=|f-f_{R}|$
surpassed a threshold of $f_{\text{Threshold}}=100~{}m\text{Hz}$. To do so, we
aggregate the data into hour-blocks and analyze them on a minute-by-minute
basis. 2011 showed more of these large frequency deviations, while recently
the violations mainly occur shortly after the trading interval every 15
minutes (Fig 3).

Next, we investigate the distribution of the frequency values, aggregated over
the 2017 data set in Fig. 4. We notice that the best Gaussian fit
substantially underestimates the tails of the distribution. We quantify these
heavy tails by displaying the kurtosis of the grid, which is approximately
$\kappa\approx 4.2$, compared to the Gaussian $\kappa^{\text{Gauss}}=3$. This
means that large disturbances are much more likely than expected based on a
Gaussian assumption. In particular, the fraction of instances, where $\Delta
f>f_{\text{Threshold}}$, increases approximately by a factor of 160.
Therefore, non-Gaussian distributions, for example Lévy-stable [24] or
q-Gaussian [25], should rather be considered to model the frequency
distribution [14].

##  III Impacts of trading

Figure 5: Illustration of how we extract the trading effects: Around every 15
minute interval, we extract a 5 minute frequency window of available frequency
measurements to islolate the impacts of trading activities on frequency
quality. Displayed is the mean frequency of the 2017 data [22]. Figure 6:  The
frequency distributions at the trading interval is broader and displays
heavier tails than the data outside the trading intervals. a: Histogram of the
aggregated 2017 data, using only data within a trading window (5 minutes
centered around the 15 minute trading intervals). b: Histogram of the
aggregated 2017 data, using only data outside of the trading window. The
kurtosis of the data taken at the trading interval is much larger than the
kurtosis outside the trading intervals. In contrast, the standard deviation
between the data sets differs only by approximately $5-10\%$.

Does trading significantly impact the width and the heavy tails of the
distribution? Following the observation that the frequency displays jumps
predominantly at the trading intervals every 15 minutes (see Figs. 1 and
[14]), we isolate those time periods and compare histograms only consisting of
data around the trading intervals with the remaining distributions for all
other intervals without trading. To this end we assume that the trading takes
place every full hour, at 15 minutes, 30 minutes and 45 minutes after every
full hour. Around those time stamps, we cut a window of 5 minutes of the
frequency recordings to be analyzed as the frequency at the trading interval
while the remaining data is left for comparison. See Fig. 5 for an
illustration, using frequency recordings from _RTE_ [22].

Trading causes the distribution to be only slightly wider, yet implies
substantially more pronounced heavy tails (Fig. 6). Similar to the full data
set, we observe significant deviations from Gaussianity, especially for the
trading time windows (panel a), but also during non-trading intervals (panel
b). To quantify the difference between the two data sets, we compute the width
of the distributions, measured via sample standard deviation and the strength
of the heavy-tails of the distributions, quantified via the kurtosis in Fig.
6. While the standard deviation only differs by less than $10\%$, the trading
interval data is much more heavy-tailed ($\kappa=4.8$) than the non-trading
interval ($\kappa=3.8$), implying a high chance for large deviations (extreme
events).

##  IV A stochastic model for power grids

We observed non-Gaussian statistics for power grid frequency fluctuations in
the histograms of the aggregated data (Figs. 4 and 6). To anticipate the risks
due to large frequency excursions, we require a stochastic theory that links
power fluctuations with frequency fluctuations. This would allow us to
anticipate the magnitude of fluctuations when designing a new system, such as
a microgrid or smart grid. We will briefly review some corner stones of the
results recently obtained in [14] on this question.

First, we model the power grid as a network of (virtual) synchronous machines
[26]. A simple dynamical description for such a network applies the swing
equation [1, 6], which gives the dynamics of the voltage phase angle
$\theta_{i}\left(t\right)$ and the angular velocity $\omega_{i}\left(t\right)$
at each node $i\in\left\\{1,...,N\right\\}$ as

$\displaystyle\frac{\text{d}}{\text{d}t}\theta_{i}$ $\displaystyle=$
$\displaystyle\omega_{i},$ (1) $\displaystyle
M_{i}\frac{\text{d}}{\text{d}t}\omega_{i}$ $\displaystyle=$ $\displaystyle
P_{i}+\Gamma_{i}(t)-D_{i}\omega_{i}+\sum_{j=1}^{N}K_{ij}\sin\left(\theta_{j}-\theta_{i}\right),$

with inertia $M_{i}$, active power $P_{i}$, random fluctuations
$\Gamma_{i}(t)$, damping $D_{i}$ and coupling matrix $K_{ij}$. To convert from
frequencies to angular velocities one uses the following conversion

$\omega=2\pi\left(f-f_{R}\right).$ (2)

To allow further analysis of the grid dynamics, we simplify this system by
assuming symmetric coupling $K_{ij}=K_{ji}$, homogeneous damping to inertia
ratio [27] $\gamma=D_{i}/M_{i}$ and balanced power _on average_
$\sum_{i=1}^{N}P_{i}=0$. Then, the dynamics of the bulk angular velocity
$\bar{\omega}=\sum_{i=1}^{N}\omega_{i}M_{i}/\sum_{i=1}^{N}M_{i}$ is given by

$\frac{\text{d}}{\text{d}t}\bar{\omega}=-\gamma\bar{\omega}+\frac{\sum_{i=1}^{N}\Gamma_{i}(t)}{\sum_{i=1}^{N}M_{i}}.$
(3)

Assuming the noise $\Gamma_{i}$ at each node $i$ is following a Gaussian
distribution with standard deviation $\sigma^{P}_{i}$, then we may apply a
Fokker-Planck equation[14, 28, 29] to compute the probability distribution of
the angular velocity $\omega$. It is also a Gaussian distribution with
standard deviation

$\bar{\sigma}^{\omega}=\frac{1}{\sum_{i=1}^{N}M_{i}}\sqrt{\frac{\sum_{i=1}^{N}\left(\sigma_{i}^{P}\right)^{2}}{2\gamma}}.$
(4)

To estimate the damping to inertia ration $\gamma$ in Eq. 4, we take advantage
of the autocorrelation function of the frequency signal. Assuming
approximately Gaussian noise, stochastic theory [28] predicts the
autocorrelation $c\left(\Delta t\right)$ to decay as a function of the time
delay $\Delta t$ as an exponential function like

$c\left(\Delta t\right)=\exp\left(-\gamma\Delta t\right).$ (5)

Finally, to capture non-Gaussian effects of the statistics, as displayed, e.g.
in Fig. 4, we use for example Lévy-stable distributions [24] as discussed in
[14]. These distributions are characterized by their scale parameter
$\sigma_{S}$, similar to a standard deviation, and their stability parameter
$\alpha_{S}$ that gives the tails of the distribution. Stable distributions
with $\alpha_{S}=2$ are Gaussian distributions while $\alpha_{S}<2$ indicates
heavy tails. For the full 2017 data, we found that $\alpha_{S}\approx 1.9$,
i.e. well below $\alpha_{S}=2$. Analogue to the Gaussian approach, we may now
formulate a generalized Fokker-Planck equation [14, 30] to obtain the
probability distribution of the angular velocity $\omega$. We require that the
noise at each node in the system follows a stable distribution with one
stability parameter $\alpha_{S}$ but arbitrary scale parameter, which we set
as $\sqrt{2}\sigma_{S,i}^{P}$, where the square root of 2 is necessary to
resemble Gaussian results for $\alpha_{S}=2$. Then, the resulting distribution
of the angular velocity is also a stable distribution, with stability
parameter $\alpha_{S}$ and scale parameter given as [14]

$\bar{\sigma}_{S}^{\omega}=\frac{1}{\sqrt{2}\sum_{i=1}^{N}M_{i}}\left[\frac{1}{\gamma\alpha_{S}}\sum_{i=1}^{N}\left(\sigma_{S,i}^{P}\right)^{\alpha_{S}}\right]^{1/\alpha_{S}}.$
(6)

One important finding of equations (4) and (6) is the scaling with respect to
the size of the grid. Both the scale parameter $\bar{\sigma}_{S}^{\omega}$ and
the standard deviation $\bar{\sigma}^{\omega}$ increase with decreasing total
inertia $\sum_{i}M_{i}$. Hence, splitting large synchronous grids into smaller
ones or replacing conventional generators by inverter technology is expected
to increase frequency deviations, unless the primary control, which determines
$\gamma$, is also increased.

How does the previous analysis of trading intervals influence these stochastic
results? The analytical approach in [14] does not include effects of trading
at all but assumes random and uncorrelated noise to account for all frequency
disturbances. One important finding of Fig. 6 is that the width of the
distribution, i.e. the standard deviation or scale parameter of the
probability distribution does not change dramatically when comparing trading
and non-trading intervals. Only the heavy-tails are easily attributed to the
trading. Hence, the stochastic theory should be further developed to include
non-Gaussian effects due to deterministic trading actions, instead of solely
arising from random fluctuations. However, the current form of the theory will
still be able to predict the approximate width of a distribution if the noise
amplitudes in the system are known.

##  V Discussion

Overall, we found that the power grid frequency is impacted by trading
actions, as visible already from the average frequency trajectory in Fig. 1.
In particular, the mean frequency shows consistent jumps every hour and less
pronounced effects every 30 minutes. When investigating large deviations from
the reference frequency we found that the occurrence of these deviations has
been reduced when comparing data from 2011 with data from 2017. Between those
years, additional short-time trading products have been established in Europe
[5] so that a larger number of smaller trading actions seems to be beneficial
for the frequency quality.

Analyzing the full frequency distribution, we found that it is not well
described by a Gaussian distribution but displays heavy tails. Therefore, more
sophisticated distributions like Lévy-stable [24] or q-Gaussian distributions
[25] should be considered when describing frequency distributions. Choosing a
small time interval around the trading actions every 15 minutes allowed us to
reveal that trading has only little impact on the width (standard deviation)
of the distribution but much larger impact on the heavy-tails. Nevertheless,
even outside trading intervals, the frequency is not well-described by a
Gaussian distribution. A stochastic theory that neglects the effects of
trading will therefore overestimate the heavy tails of the noise as those are
significantly impacted by trading actions. However, such a theory will give a
good approximation for the width of the distribution since the standard
deviation does not depend as strongly on the trading actions. We briefly
reviewed some core results obtained in [14] that allow to quantify the impact
of power fluctuations on the power grid frequency even in non-Gaussian
settings.

While we based our analysis on the equations of motion of synchronoues
machines, our results should also hold true if conventional generators were
replaced by devices with small or only virtual inertia [31], as the equations
of motions do not change.

Our current analysis emphasizes that future markets have to consider their
impact on frequency quality. Based on the comparison between 2011 and 2017,
splitting trading actions into smaller packages may reduce the overall impact
of trading on the frequency quality, especially with respect to large
deviations. This will have to be taken into account in addition to network
transmission constraints to balance the system. Otherwise, additional and
costly control actions will be necessary [12, 13].

Future research should explore models for grids without any inertia and
include more systematic studies of different grids, e.g. the US grids with
different trading intervals or even real time pricing would be very
interesting to analyze. However, frequency data are rarely available publicly.

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|  Benjamin Schäfer received his Diploma (MSc) degree in Physics from the
University of Magdeburg, Germany in 2013. Persuing his Ph.D. in Göttingen
(Germany), London (United Kingdom) and Tokyo (Japan), he received his Ph.D.
degree in physics in 2017 from the University of Göttingen. Since 2017 he is
Postdoctoral Researcher at the Max Planck Institute for Dynamics and Self-
Organization, Göttingen, Germany and the Technical University Dresden,
Germany.
---|---
|  Marc Timme studied Physics and Mathematics at the University of Würzburg,
Germany, and the State University of New York at Stony Brook, USA. He gained
his doctorate in Theoretical Physics at the University of Göttingen. After
research stays at the Max Planck Institute for Flow Research and the Center
for Applied Mathematics, Cornell University (USA), he established and lead a
broadly cross-disciplinary research group on Network Dynamics at the Max
Planck Institute for Dynamics and Self-Organization, became an Adjunct
Professor at the University of Göttingen, was Visiting Professor at TU
Darmstadt, visiting faculty at the ETH Zurich Risk Center (Switzerland) and
recently assumed a Chair for Network Dynamics as a Strategic Professor of the
Cluster of Excellence Center for Advancing Electronics Dresden (cfaed) at TU
Dresden, Germany. Marc Timme is Co-Chair of the Division of Socio-Economic
Physics of the German Physical Society and received a research award of the
Berliner Ungewitter Foundation, the Otto Hahn Medal of the Max Planck Society,
and a Research Fellow position of the National Research Center of Italy.
---|---
|  Dirk Witthaut received his Diploma (MSc) and PhD in Physics from the
Technical University of Kaiserslautern, Germany, in 2004 and 2007,
respectively. He has been a Postdoctoral Researcher at the Niels Bohr
Institute in Copenhagen, Denmark and at the Max Planck Institute for Dynamics
and Self-Organization in Göttingen, Germany and a Guest Lecturer at the Kigali
Institute for Science and Technology in Rwanda. Currently, he is leading a
Research Group at Forschungszentrum Jülich, Germany and he is a junior
professor at the University of Cologne.
---|---