# Phase and amplitude synchronisation in power-grid frequency fluctuations in
the Nordic Grid

Leonardo Rydin Gorjão<EMAIL_ADDRESS>Forschungszentrum Jülich,
Institute for Energy and Climate Research - Systems Analysis and Technology
Evaluation (IEK-STE), 52428 Jülich, Germany  Institute for Theoretical
Physics, University of Cologne, 50937 Köln, Germany  Luigi Vanfretti
Electrical, Computer, and Systems Engineering, Rensselaer Polytechnic
Institute, 8024 Troy, New York, United States  Dirk Witthaut
Forschungszentrum Jülich, Institute for Energy and Climate Research - Systems
Analysis and Technology Evaluation (IEK-STE), 52428 Jülich, Germany  Institute
for Theoretical Physics, University of Cologne, 50937 Köln, Germany  Christian
Beck  School of Mathematical Sciences, Queen Mary University of London, United
Kingdom  Benjamin Schäfer<EMAIL_ADDRESS>School of Mathematical
Sciences, Queen Mary University of London, United Kingdom

###### Abstract

Monitoring and modelling the power grid frequency is key to ensuring stability
in the electrical power system. Many tools exist to investigate the detailed
deterministic dynamics and especially the bulk behaviour of the frequency.
However, far less attention has been paid to its stochastic properties, and
there is a need for a cohesive framework that couples both short-time scale
fluctuations and bulk behaviour. Moreover, commonly assumed uncorrelated
stochastic noise is predominantly employed in modelling in energy systems. In
this publication, we examine the stochastic properties of six synchronous
power-grid frequency recording with high-temporal resolution of the Nordic
Grid from September 2013, focusing on the increments of the frequency
recordings. We show that these increments follow non-Gaussian statistics and
display spatial and temporal correlations. Furthermore, we report two
different physical synchronisation phenomena: a very short timescale phase
synchronisation ($<2\,$s) followed by a slightly larger timescale amplitude
synchronisation ($2\,$s–$5\,$s). Overall, these results provide guidance on
how to model fluctuations in power systems.

Figure 1: Overview of the data. The six power-grid frequency recordings in the
Nordic Grid from 2013 show very distinct increment statistics. a Approximate
locations of the recordings across the Nordic Grid: CTH, LTH, KTH, LTU, TTY,
and AU. b Excerpts of the recordings in a $30$ minutes time scale, displaced
vertically. c Zoom into the arrows of panel b of a total length of
$2\,$seconds, displaced vertically. d-f display the probability density
functions (PDF) of the increments $\Delta f_{\tau}$ at the shortest increment
lag $\tau=0.02\,$s and at $\tau=1\,$s, in a vertical logarithmic scale,
alongside a normal distribution (which is an inverted parabola in a vertical
logarithmic scale) with equal variance for the first recording at
$\tau=0.02\,$s, for comparison. PDFs are vertically displaced for clarity. All
recordings are synchronous and have a sampling time of $0.02~{}\\!$s.

##  I Introduction

The power-grid frequency is a key indicator of the stability of electric power
systems. It weighs in the balance of power generation and consumption as well
as the operation of each component of the grid and thus serves as the main
signal used for balancing [1]. In theory, during steady operation, the
frequency is the same throughout the grid and all generators have a fixed
phase difference that essentially determines the real power flows. Yet,
perfect phase locking is only an approximation to the operation of real power
grids subject to numerous external perturbations. For instance, perturbations,
such as the tripping of important lines or the loss of a major generating
unit, can cause inter-area oscillations corresponding to the normal modes of
the grid around a quasi-equilibrium [2]. Large perturbations can even lead to
the complete loss of synchrony between different parts of the power grid,
eventually leading to blackouts [3]. While luckily these large contingencies
occur seldom, small fluctuations—ambient noise—are always present. These,
nuisance fluctuations, are ubiquitous, and can help us understand a wide range
of characteristics of power-grid frequency, such as the role of inertia or
renewable energy intermittency [4, 5]. This article focuses on analysing,
characterising, and explaining deviations caused by these ambient
perturbations and the ability of the grid to relax to synchrony afterwards.

Frequency dynamics and synchronisation are essential aspects of power system
operation and thus intensively studied in the literature. The ongoing energy
transition strengthened the interest in these topics, as synchronous
generators are replaced by inverter-based power sources [6, 7]. For instance,
recent years saw enormous progress in the mathematical theory of
synchronisation [8], leading to the derivation of a variety of rigorous
stability conditions [9, 10]. An important practical research topic is the
dynamics and design of inverter-based power grids lacking the inertia provided
by large synchronous machines [11, 12, 13, 14]. Another key area is the
development of detailed simulation models to study frequency dynamics and
synchronisation for actual grid layouts and contingency situations [15, 16].
Power hardware-in-the-loop then allows to investigate and test actual
equipment coupled with real-time simulations [17, 18]. A common scenario in
such simulation-based studies is the dynamics after a sudden large
perturbation, modelling a fault (see, e.g. Ref. [6]). Does the grid remain
stable? And how long does it take to relax to a steady synchronous operation?
The current manuscript adopts a very different approach to this
synchronisation in power-grid, focusing on the analysis of synchronous
frequency measurements in the presence of ambient noise. We will extract the
essential scales of synchronisation in space and time from frequency time
series in a model-free way.

Non-linear time series analysis has been applied to investigate various
aspects of power-grid systems, including using some stochastic analysis
methods [19], particularly the application of the Hilbert–Huang transform
(empirical mode decomposition) [20], wavelet-based analysis [21], or simply
investigating the spectral density [22]. The spatio-temporal dynamics of the
power system have been studied in more detail, e.g. by revealing inter- and
intra-area oscillations and eigenfrequencies. In the Nordic Grid, in
particular, eigenfrequencies [23] and eigenmodes [24] have been estimated and
inter-area oscillations [25], as well as power oscillation damping [26], have
been observed. Nevertheless, stochastic elements, more commonly denoted as
ambient noise [27], are not in the focus of most power system studies. Only
some studies include noise, e.g. when investigating power generation [28].
Some recent papers have provided a detailed stochastic analysis and modelling
of the bulk frequency without addressing spatial aspects of the dynamics [29,
30, 31]. Meanwhile, other works exist, highlighting the importance of short-
term volatility in renewables [32] and even first spatio-temporal
considerations in Continental Europe [33].

Both spatio-temporal or general time series analysis require access to high-
quality, spatially distributed frequency recordings with phasor measurement
units (PMUs) or PMU-like devices. Unfortunately, most data sets are not shared
openly [34]. A discontinued university-based PMU network in the Nordic area
was briefly in place between 2012 and 2014, which provides some of the
datasets that are examined in this work [35]. More recent initiatives only
cover spatial measurements in Continental Europe [30, 36].

Within this article, we present, analyse, characterise, and explain the
aforementioned data of the Nordic synchronous area and study grid
synchronisation in a data-centred model-free approach. We focus on the
increment statistics of the frequency time series which carries essential
information on the fluctuations and the synchronisation of the frequency. We
show that non-Gaussian increment statistics are ubiquitous and that the
variance on the increments scales faster than Brownian-like motions. Next, we
show that stochastic fluctuations exhibit spatial correlations between
locations even at vanishing time differences and that there exist temporal
correlation within the same incremental time series. Finally, we examine phase
and amplitude synchronisation separately. To this end, we firstly observe
phase synchronisation propagating linearly through the network, which
contrasts the second finding of an diffusive-like coupling in amplitude
synchronisation.

##  II Synchronisation phenomena and increment statistics

In this article, we analyse power-grid frequency recordings in the Nordic Grid
from the 9th to the 11th of September 2013, with a sampling time of $0.02$
seconds. Note the approximately 36 hours of data available here already allow
the analysis of the fluctuations on the time scale of seconds. In addition to
the spatially well-defined measurements from 2013, we also evaluate the same
September days from 2020 measured at an unknown location in central Sweden,
provided by the Swedish TSO, obtaining qualitatively similar results (not
shown).

We illustrate the locations of the recording sites on a map of the Nordic Grid
synchronous area in Fig. 1a. The recordings were taken synchronously at six
universities in Sweden and Finland, here abbreviated to acronyms: Chalmers
University of Technology Gothenburg (CTH); Faculty of Engineering, Lund
University (LTH); Royal Institute of Technology Stockholm (KTH); Luleå
University of Technology (LTU); Tampere University of Technology (TTY); Aalto
University (AU) [35]. An excerpt of $30$ minutes of recordings is displayed in
b, vertically displaced for clarity. Panel c shows a snapshot of $2$ seconds
length, around the time point indicated by an arrow in panel b. The
fluctuations in TTY and AU show a smaller standard deviation than the
remaining sites due to the larger number of generators in the Northern part of
the synchronous area, compared to the mostly consumer-dominated south of the
grid, see also Refs. [29, 37].

Inspecting the time series, one immediately notices the most common properties
of power-grid frequency: On course scales, all recordings are synchronous and
seem perfectly identical at first sight. If one examines them at a time scale
of hundreds of milliseconds (in Fig. 1c) the synchronous behaviour is still
present, but small fluctuations occur, each seemingly with their own dynamics.
We will focus on these small fluctuations on time scales of a few seconds and
introduce here the rather intuitive idea of phase and amplitude
synchronisation.

###  II.1 Phase and amplitude synchronisation

A coupled dynamical system, say for simplicity, a two-area system, with one
area exporting and the other area importing power, can be represented by a
simplified model of two generators, each with given inertia, exchanging power,
which displays some frequency dynamics around an equilibrium [38]. Three
different regimes of the collective dynamics exist, while the generators
synchronise (typically towards $50/60\,$Hz), as depicted in Fig. 2 for two
sinusoidal signals (e.g. a phase voltage at two different nodes of the power
grid). If both signals are only weakly interacting and independent, we would
expect these signals to be uncorrelated, i.e., with a different phase and
displaying different amplitudes (panel a). Enforcing synchronised dynamics,
e.g. via coupling, will match their phases but not yet their amplitude. Hence,
we observe a correlation between both signals (panel b). Lastly, given
sufficient coupling, we also obtain amplitude synchronisation, i.e., in this
case, a full synchronisation, as both phase and amplitude are equal (panel c).
Note that the amplitude discussed here is not the voltage amplitude but the
amplitude of frequency fluctuations.

Naturally, several questions arise from this depiction: What are the usual
timescales for each synchronisation to take effect after an external
perturbation? What are the causes of this synchronisation? How do distances
between elements impact synchronisation (especially in networked systems)?
Lastly, are there properties from one synchronisation phenomenon that affect
the other?

We will address these questions based on measured time series of power-grid
frequency. Phase synchronisation is analysed in terms of the signals’
increments, i.e., the change of the signal between two points in time. If two
signals are perfectly phase synchronised, their increments will be perfectly
correlated. In contrast, amplitude synchronisation must be studied in terms of
the original time series.

Figure 2: Synchronisation phenomena. Distinct frequency signals first become
correlated and then converge to identical amplitudes. We illustrate this here:
a: Initially, signals are initially uncorrelated and have differing
amplitudes. b: After some time, signals which are phase synchronised but still
differ in amplitude. c: Finally, signals are synchronised both in phase and
amplitude.

###  II.2 Increment statistics

To study the synchronisation, including correlations of each time series, we
investigate the increment statistics of the recordings. Synchronous power-grid
systems operate at a set frequency. In the Nordic Grid, and all of Europe,
this is the nominal frequency of $50~{}\\!$Hz. Many of the deterministic
properties are studied directly from the time series themselves, by either
studying their deterministic properties, i.e., as a dynamical system [39], or
as a stochastic process [40, 41]. Yet, here we take another approach and focus
on fluctuations of the time series—in particular their increments—and quantify
their stochastic properties and correlations.

Increments $\Delta f_{\tau}(t)$ are defined as

$\Delta f_{\tau}(t)=f(t+\tau)-f(t),$ (1)

where the incremental lag $\tau$ gives the temporal difference of the two
points within a time series $f(t)$. Studying the incremental properties of a
time series focuses on the shortest time scales of the underlying processes,
thus excludes the deterministic trends and deals solely with the stochastic
characteristics of the fluctuations themselves. Note that we move away from
considering the recordings in their time domain and study the difference
between two points separated by a temporal lag $\tau$.

There is an intrinsic relation between increments and inertia in the system.
The ability of a power grid to maintain itself at its nominal frequency
($50\,$Hz or $60\,$Hz) is largely determined by the energy that can be stored
and exchanged by the masses of the coupled rotating generators. Short term
fluctuations in the power system—i.e., precisely the increments—are
conjectured to grow in amplitude and frequency when more inertia is removed as
renewable generators replace fossil-fuelled ones [6, 7]. These fluctuations
are of great importance as they can lead to large angle variations and power
flow fluctuations, putting additional strain on generators and transmission
system components.

After obtaining the incremental time series for all six sites, we investigate
their statistics for the shortest incremental lag $\tau=0.02\,$s and the
longer lag $\tau=1\,$s in Fig. 1d-f. One observes considerable differences in
the empirical probability distributions (PDFs) between the six sites. As a
base-line, we might expect Gaussian distributions, i.e., inverted parabola in
the vertical logarithmic plot (dotted line). Indeed, inspecting the
incremental distributions at a delay $\tau=1\,$s, some sites, such as CTH,
approximately follow such a Gaussian distribution. In contrast, most sites
display clear deviations from the expected Gaussian behavior for the short
delay $\tau=0.02\,$s, i.e. they display heavy-tailed distributions.

The heavy tails in these distributions indicate that uncommonly large
fluctuations take place in the increments, i.e., the difference from one time-
point to the next is abnormally large (compared to normally distributed
noise). Furthermore, we note that these increments seem rather distinct at
each location, in particular, compared to the much more homogeneous power-grid
frequency recordings. This tells us straightforwardly that, although there is
strong synchrony in the system—the $50\,$Hz of operation—each location varies
ever-so-slightly and each in its own manner. What we observe here are the
local properties of the recordings. An in-depth analysis of the underlying
mechanism giving rise to these distribution of the increments and their local
properties is addressed in Ref. [42].

To best quantify both the deviations from Gaussian distributions as well as
the differences between sites, we investigate two statistical parameters: the
variance and the kurtosis of each distribution, as a function of time step
$\tau$.

###  II.3 Statistical properties of incremental time series

Figure 3:  Variance $\sigma(\Delta f_{\tau})^{2}$ and kurtosis $\kappa(\Delta
f_{\tau})$ of the increment $\Delta f_{\tau}$ display clear deviations from
Gaussianity. a A power-law scaling is observed, with exponent $>1$. For
comparison, the linear-like scaling of an uncorrelated Brownian motion is
shown. LTU and LTH seem to display the presence of microscopic noise,
uniformly increasing the variance of their increments, irrespective of $\tau$.
b Kurtosis $\kappa(\Delta f_{\tau})$ of the increment $\Delta f_{\tau}$ in a
vertical logarithmic scale. The increments statistics are always leptokurtic,
i.e., $\kappa>3$. All increment statistics $\kappa(\Delta f_{\tau})$ converge
to $\kappa(\Delta f_{\tau\gg 0})\approx 3.35$ at $\tau\gtrsim 2$.

We examine two statistical moments of the incremental distributions as a
function of the incremental lag $\tau$, namely, the variance and kurtosis. The
variance (i.e., the second moment with subtracted mean) indicates the average
displacement of each recording from their mean, i.e., how far the point of the
recordings are spread out from their mean. Note that we have a mean zero here,
as we investigate frequency increments. On the other hand, the kurtosis
(fourth normalised moment) roughly indicates how often rather large deviations
happen, i.e., deviations much larger than those that fall inside the spread of
the variance [43].

Note that we handed-picked two distributions at $\tau=0.02\,$s and $\tau=1\,$s
for Fig. 1d-f. Now, we systematically investigate how the variance and
kurtosis of each increment distribution change as we increase the incremental
lag $\tau$. Let us examine the variance $\sigma(\Delta f_{\tau})^{2}$ (Fig.
3a) and the kurtosis $\kappa(\Delta f_{\tau})$ (Fig. 3b) for the first five
seconds of incremental lags $\tau\in[0.02\,$s$,5\,$s$]$. The first observation
is that the variance of the increments $\Delta f_{\tau}$ increases in a power-
like relation. In particular, we observe a scaling of the variance as

$\sigma(\Delta f_{\tau})^{2}\sim\tau^{2H},~{}\mathrm{with}~{}1.4<2H<1.9,$ (2)

which we can quantify using the Hurst exponent $H$ [44]. We know that the
increments of a time series are intrinsically related to the nature of the
fluctuations, i.e., the ambient noise. For example, classical uncorrelated
Brownian motion with mean zero—often assumed when simulating noisy processes,
e.g. as in Ref. [27]—scales with
$\sigma(\Delta_{\mathrm{Bm}}f_{\tau})^{2}\sim\tau$, i.e., $H=0.5$, which is
given by the dotted line in Fig. 3a.

What we observe here is a strong diffusion scaling of the fluctuations. It is
particularly relevant to take this scaling into account when performing any
from of simulations of the power-grid frequency and thus should be
incorporated accordingly [41]. Importantly, the observation of this correlated
and non-white noise is ubiquitous and uniform across locations and evidenced
from a data analysis, i.e., without imposing any model. We come back to this
observation in Sec. II.5, precise the value of the Hurst coefficient $H$.

As a last remark, one observes similarly that in both LTU and LTH additional
microscopic noise is present, possibly with some regular properties (notice
the curves do not approach zero for decreasing $\tau$). Whether these are
artefacts or fundamental physical property of each site’s local stochastic
properties is left to future analyses.

Let us now turn our attention to how the kurtosis changes with increasing
$\tau$ (Fig. 3b). As a reference, we provide the kurtosis of a normal
distribution as $\kappa_{\mathcal{N}}=3$ in the plot. Notably, the kurtosis of
the increments decreases with growing incremental lags $\tau$, that is to say,
as $\tau$ increases the distributions become more and more like normal
distributions. One should note that this is not completely unexpected
behaviour. Recent observations by Hähne et al. [33] and Schmietendorf et al.
[32] point towards an increase of this phenomena generated by volatile wind-
energy input. This also fundamentally links properties of power-grid frequency
with, e.g turbulent flows [45, 46, 47] and stock market prices [48]. In
particular, nontrivial scaling of the probability densities depending on the
time lag parameter $\tau$ and involving a fractional Brownian motion parameter
$H$ was experimentally found in turbulent Taylor–Couette flow [49]. In that
sense, the frequency difference fluctuations on a small scale have
similarities with those of small-scale velocity differences in turbulent
flows, at least in a statistical sense. What is relevant for our analysis is
to understand this as a synchronisation phenomenon. As depicted in Fig. 2c, if
one is to achieve amplitude synchronisation, then obviously two recordings
must end up with identical distributions. This does not mean that the kurtosis
of all incremental time series must become normally distributed but does imply
they must be identical.

With these insights of the increment statistics at each location, we can now
address the first guiding question formulated in the introduction: Can we
unravel the physics of phase synchronisation and its timescale? To do so we
must study the smallest timescales of the incremental time series and focus on
their correlations.

###  II.4 Phase synchronisation

Figure 4: Correlations of the increments increases with delay $\tau$. (a-c)
display the correlations between CTH and AU at a $\tau=0.02\,$s, b
$\tau=0.40\,$s, and c $\tau=2.00\,$s. (d) Displays the Pearson correlation
$c(\Delta f_{\tau}^{X},\Delta f_{\tau}^{\mathrm{CTH}})$ in
$\tau\in[0.02\,$s$,20\,$s$]$. Generally correlations are zero or small with
all other recordings at $\tau=0.02s$, yet do not vanish for the closest
locations to CTH: LTH and KTH. For larger increments, the correlation
approaches one: $\lim_{\tau\rightarrow\infty}c=1$.

Let us examine the emergence of phase synchronisation of the incremental time
series (from uncorrelated to correlated time series). Power grids are designed
to ensure that all power generators work synchronously across the power grid.
Naturally—and as we have seen so far—this means that the frequency recordings
themselves are almost identical, i.e., highly correlated with each other, yet
this does not directly translate into fluctuations at each location acting
similarly. Indeed, both models and theory treating the power-grid frequency
often assume uncorrelated fluctuations. Again without relying on any model, we
will show that spatial correlations of the fluctuations are ubiquitous, and
especially prevalent below a certain distance—similar to a scaling law in
long-range fluctuations in power-grid frequency presented in Ref. [37].

To study the correlations of the increments between two locations, we examine
the Pearson correlation of two recordings $c(\Delta f_{\tau}^{X},\Delta
f_{\tau}^{Y})$

$c(\Delta f_{\tau}^{X},\Delta f_{\tau}^{Y})=\frac{\mathrm{Cov}\\!\left(\Delta
f_{\tau}^{X},\Delta f_{\tau}^{Y}\right)}{\sigma(\Delta
f_{\tau}^{X})\sigma(\Delta f_{\tau}^{Y})}\in[-1,1],$ (3)

with $X$ and $Y$ the two locations or recordings, $\sigma(\cdot)$ their
individual standard deviation and $\mathrm{Cov}(\cdot,\cdot)$ their
covariance. A value of $c=1$ indicates total correlation, $c=-1$ total anti-
correlation, and $c=0$ no correlation.

Let us take the site CTH as an example. The correlation $c(\Delta
f_{\tau}^{X},\Delta f_{\tau}^{\mathrm{CTH}})$ with all other locations is
displayed in Fig. 4, for $0.02\,$s $<\tau<20\,$s. Panels a-c show how large
incremental lags $\tau$ lead to highly aligned, i.e., correlated increments,
while short incremental lags $\tau$ display no correlation. The de-correlation
of the increments for very small time differences $\tau$ is clear, yet this
does not seem to be the case for the two closest locations to CTH: LTH and
KTH. In this case, the Pearson correlations are $c(\Delta
f_{\tau}^{\mathrm{LTH}},\Delta f_{\tau}^{\mathrm{CTH}})\sim 0.25$ and
$c(\Delta f_{\tau}^{\mathrm{KTH}},\Delta f_{\tau}^{\mathrm{CTH}})\sim 0.25$ at
the lower limit of $\tau\to 0$.

Figure 5:  Time to reach a correlated state increases approximately linearly
with distance. (a) displays the Pearson correlation $c(\Delta
f_{\tau}^{X},\Delta f_{\tau}^{Y})$ of two locations at the temporal lag
$\tau=0.1\,$s in relation to their driving distance. The presence of non-
vanishing correlations at small distances can be observed, following an
exponential-like function exp($ax^{k}$), with parameters $a=-0.10$ and
$k=0.47$. (b) displays the time at which the Pearson correlations between two
locations become greater than $0.5$, $\tau_{c>0.5}$, plotted against the
distance between the locations. In similar fashion, we see that the time it
takes for the increments to become correlated is approximately linearly
proportional to the distance between two locations. Shaded area indicates the
standard deviation of the polynomial fitting. Three couplings are discarded:
AU–TTY and AU–LTU, due to their seemingly small anti-correlation at
$\tau=0.02\,$, which quickly becomes correlated at $\tau>0.1s$, and the
LTH–KTH coupling, which as seen in Fig. 3a presents microscopic noise, mooting
possible correlations.

This gives rise to the following question: Are fluctuations in the increments
correlated for shorter distances, i.e., are there spatial correlations in the
increments themselves? We examine this by studying the Pearson correlations
against the geographical distance between the locations in two manners: (1)
find the correlation at the lowest temporal lag $\tau=0.1\,$s and the distance
between the locations; (2) find how much time it takes for the locations to
surpass a given threshold of correlation and the distance between the
locations.

Firstly we note that all sea cables in the Baltic Sea or the Gulf of Bothnia
are direct current (DC) cables, they transmit power asynchronously. That being
the case, driving/walking distances between two locations serve as a proxy to
actual electric distance, i.e., the equivalent AC impedance between two
locations [50].

In Fig. 5a, we display the fifteen distance pairs between the six locations
and their respective Pearson correlation at temporal lag $\tau=0.1\,$s are
displayed. The Pearson correlations $c(\Delta f_{\tau}^{X},\Delta
f_{\tau}^{Y})$ for all locations at $\tau=0.1\,$s are plotted against the
distances between the points. A pattern of correlations emerges for short
distances with non-vanishing Pearson correlations. Three couplings are
discarded: (1) AU–TTY and AU–LTU, due to their small anti-correlation at very
small $\tau=0.02\,$s–$0.1\,$s. (2) the LTU–LTH coupling, where the present
microscopic noise moots evaluating correlations (see e.g. Fig. 3a).

In Fig. 5b we examine the speed at which correlations between two locations
become larger than $0.5$ ($50\%$), i.e., we find the time $\tau_{c>0.5}$ at
which the correlations $c(\Delta f_{\tau}^{X},\Delta f_{\tau}^{Y})>0.5$ in
relation to the distance between the locations. We observe that locations
closer to each other see their increments becoming correlated faster than
locations that are further apart. While this general trend is expected, note
the approximately linear relation between phase-synchronisation and
geographical distance between the sites. We will have cause to contrast this
with amplitude synchronisation in the following section. Furthermore, notice
that this phenomenon is strictly bound to be taking place at a time scale
$<2\,$s, i.e., phase synchronisation takes place at very short temporal
scales. These results point towards a potential linear phase-synchronisation
phenomenon in power-grid frequency recordings, which should be carefully
examined with more spatially-extended data.

We now move forward to examine the emergence of amplitude synchronisation.

###  II.5 Amplitude synchronisation

We have seen that phases of increments synchronised within $<2\,$s, so let us
now move to amplitude synchronisation. As we have seen in Fig. 3a the variance
of the increment statistics increases in a power-law relation with the
incremental lag $\tau$. In order to best describe this—and to retrieve the
Hurst exponent $H$—we employ Detrended Fluctuation Analysis (DFA) of the
power-grid frequency recordings [51, 52, 53]. Recall that DFA studies the
scaling of the fluctuations of a time series by studying the local properties
of the data, similarly to what increment statistics does. With DFA we extract
the fluctuation function $F(r)$ over a scale $r$ of a time series, in a much
similar fashion to our incremental lag $\tau$. We will keep $\tau$ and $r$
distinct since the DFA is applied directly to the power-grid frequency
recordings, not the incremental time series. Still, DFA and increment analysis
are intrinsically related and studying the fluctuation function $F(r)$ as a
function of the scale $r$ will help us uncover the scaling parameter $\alpha$,
as also observed in Fig. 3a.

The DFA procedure, applied to the power-grid frequency recordings, is the
following: Take non-overlapping segments of the power-grid frequency, fit a
polynomial function (of order one, in this case), subtract the fit from the
segment of data, extract the variance and average over each detrended segment.
By increasing the segment size one can study the change of the variances,
i.e., fluctuation function $F(r)$, as a function of the segment size (the
scale $r$). If the time series follows a power-law, which we show in Fig. 3a
it does, one can evaluate the plots of the fluctuation function $F(r)$ in the
scale $r$ in a double logarithmic scale. The reason to compute the fluctuation
function $F(r)$ instead of examining the original time series is simple:
Actual power-grid frequency recordings have trends, such as short-terms jumps
due to dispatch and market activity or a permanent small mismatch of power
generation and consumption. We wish to disentangle these trends from the true
underlying fluctuation dynamics, which DFA is capable of.

For our purpose here, we have already established that the incremental time
series follows a diffusion scale with a Hurst exponent $H>0.5$ given by Eq.
(2), which is manifestly larger in exponent than an uncorrelated Brownian
motion. We plot the fluctuation function $F(r)$ of the six time series in a
scale $r\in[1\,$s$,20\,$s$]$ in Fig. 6a, normalised by the scaling of a
Brownian motion, which has a scaling power $\alpha_{\mathrm{Bm}}=1.5$ (i.e.,
$\alpha=H+1$). For illustrative purposes, we show the difference in scaling,
represented by $\alpha^{\prime}=\alpha-\alpha^{\mathrm{Bm}}$. We find that
$\alpha^{\prime}\approx 0.376$, i.e., $\alpha=1.876$, for the range of
$r\in[5\,$s$,10\,$s$]$. Noticeable is also the distinct separation of the
curves at timescales $r<5\,$s, which has been identified in Ref. [37] as a
possible method to study the synchronisation of fluctuation over spatial
distances. In order to compare the fluctuation functions $F(r)$ an each
location, i.e., to fundamentally compare the amplitudes of the fluctuations at
each locations, we compute the relative DFA function $\eta(r)$, as introduced
in Ref. [37], with CTH as a reference in Fig. 6b. This function represents the
relative variations of the fluctuations to a reference point, given by

$\eta(r)^{Y}=\frac{F(r)^{X}-F(r)^{Y}}{F(r)^{Y}},$ (4)

with $Y$ the reference location, CTH in our case, and $X$ the remaining
locations. The synchronous nature of power-grids ensures all fluctuations
eventually collapse into a single function. That means all frequency increment
amplitudes become identical.

To quantify this synchronisation, we define a ‘time-to-bulk’ $\chi$ as
follows: We record the time it takes each time series’ relative fluctuations
$\eta$ to be reduced to $0.1$ ($10\%$) and thereby become indiscernible from
the reference (indicated by the grey line in Fig 6b). Using CTH as a
reference, we obtain $\chi^{\mathrm{CTH}}$ and observe how the time towards
synchronisation increases with distance in Fig. 6c. What we observe here
contrasts sharply with the linear relation of phase synchronisation in Fig. 5.
Here we observe an approximately diffusive coupling, i.e., the time it takes
for the amplitude of the increments to become identical grows approximately
with the square of the distance, as described in Ref. [37]. Interestingly, the
data suggest a potentially even stronger dependency with distance, which could
be linked to strong temporal correlations discussed earlier: If the dynamic at
one site depends substantially on its own past (high temporal correlation),
then it will take longer to synchronise fully with the remaining network. This
poses many interesting research questions for the future, emphasising the
importance of correlations.

Figure 6:  Correlated, diffusive coupling revealed by DFA, relative DFA
$\eta(r)$, and ‘time-to-bulk’ $\chi$. (a) displays the normalised fluctuation
function $F(r)$ obtained from DFA. normalised by the fluctuation function
$F(r)^{\mathrm{Bm}}$ of a Brownian motion ($\alpha_{\mathrm{Bm}}-1=0.5$). A
fit to the empiric fluctuation function yields the scaling exponents
$\alpha^{\prime}=\alpha-\alpha_{\mathrm{Bm}}=0.376$ or equivalently a Hurst
index $H=\alpha-1=0.876$, i.e., strongly positively correlated noise. (b)
displays the relative fluctuation function $\eta(r)^{\mathrm{CTH}}$, with CTH
as reference. The relative fluctuations of the other five locations decay in
time. The markers indicate the points (in time) where the relative size of the
fluctuations hits 0.1 (10%) of the maximal relative fluctuation. (c) shows the
‘time-to-bulk’ function $\chi^{\mathrm{CTH}}$ against the distances of each
recordings’ location in relation to CTH. A quadratic fit (thus diffusive
coupling with distance) approximates the amplitude synchronisation phenomenon.

We pause here to thoroughly examine what the ‘time-to-bulk’ analysis just
uncovered. First, we confirmed the suspicion raised in Fig. 3a that the power-
grid frequency indeed exhibits temporally correlated noise, i.e., the
assumption that the high-frequency fluctuations in power-grid frequency
recordings are purely white noise is not justified. In fact,
$\alpha>1.5~{}(H>0.5)$ indicates the presence of positively correlated
motions, which show a power-law like diffusion, as seen in Fig. 3a. One finds
that all time series coalesce to a Hurst index of $H=\alpha-1=0.876$, i.e.,
strongly positively correlated noise. This Hurst index can be determined
efficiently for any given power-grid frequency measurement and should be
incorporated in stochastic studies of power systems.

Secondly, we observe that these fluctuations display a spatial relation
between the locations, i.e., locations that are closer see their fluctuation
amplitude synchronise faster than those farther apart. Note that this is a
second effect, different from the (temporal) correlation in the time series,
which we have examined before. Focusing on the new spatial correlations, we
know all power-grid frequency recordings are highly correlated, they
practically follow the same phenomena at timescales $>5\,$s, i.e., their
dynamical oscillations around $50\,$Hz. Moreover, we have seen above that the
correlations of the increments are already large at these long timescales,
while displaying zero or low correlations on short time scales. What we now
observe is a synchronisation of the amplitudes of the fluctuations as a
function of time. If we compare Fig. 6 to Fig. 5, we notice a sharp difference
in the physics of the synchronisation phenomena: The amplitude synchronisation
(see Fig. 6) is achieved approximately via diffusive coupling, i.e., following
a quadratic relation with distance, in sharp contrast to the linear relation
for phase synchronisation (see Fig. 5).

We note that a priori there is no necessary relation between local temporal
properties of the incremental time series and the spatial convergence of all
oscillations into one bulk behaviour. Nevertheless, the fact that local
temporal properties affect the spatial correlations of systems that rely
heavily on synchronisation to operate is plausible. This, however, has not
been described before.

##  III Conclusion

In this article, we have analysed synchronous recordings of the power-grid
frequency from six locations in the Nordic Grid from September 2013. The high
temporal resolution of these recordings of $0.02\,$s allows a detailed
analysis of power system synchronisation via the statistics and correlations
of the increments. Essential insights into the temporal and spatial scales of
synchronisations can be extracted from the ambient fluctuations in a model-
free approach. Our results further emphasise the outstanding importance of
broad availability of high-quality data for research on power system operation
and energy science in general [54].

We investigated the distribution of increments ($\Delta
f_{\tau}(t)=f(t+\tau)-f(t)$) and noted severe deviations from Gaussianity. In
particular, increment distributions are highly leptokurtic, i.e., display
heavy tails. Noticeable differences in the increment statistics at every
location are observed, indicating that the incremental time series reflect
above all the local phenomena of power generation and consumption in the
location of the recording. We also saw a relaxation of the incremental time
series kurtosis for large time lags $\tau$, yet even for very large lags we
observe a leptokurtic distribution with kurtosis of $\kappa(\Delta f_{\tau\gg
0})\approx 3.35$. We continued with a detailed discussion of phase and
amplitude synchronisation of frequency increments.

The emergence of phase synchronisation is revealed by the correlations of the
increments on different temporal and spatial scales. Our analysis has shown
two essential results: (1) If locations are relatively close electrically, the
increments are correlated even at the shortest possible time lag of $0.02\,$s,
indicating that ambient fluctuations of power generation and consumption that
drive the frequency dynamics are correlated. In light of this result, any
assumption of spatially independent noise in power system simulations should
be carefully reviewed. (2) The increments at locations become strongly
correlated on a time scale of one second depending on the electrical distance
of the location. The further two locations are from each other, the longer it
takes before correlations exceed a certain value.

Strong temporal correlations in the fluctuations are revealed by the variance
of the incremental time series. We find that the variance follows a power-law
$\sim\tau^{2H}$ as a function of the incremental lag $\tau$, with an exponent
much higher than for ordinary uncorrelated Brownian motion. This result is
confirmed by detrended fluctuation analysis (DFA), which yields the exponent
$\alpha=1.876$, i.e., a positively correlated Hurst index $H=0.876$. DFA is
further used to study the time scales of amplitude synchronisation by
quantifying the time needed until the fluctuation functions at two locations
become similar up to a certain level. This ‘time-to-bulk’ function increases
with distance approximately following diffusive coupling (quadratic function),
as observed before in Ref. [37]. Critically, we note a systematic deviation
from pure diffusive coupling, which suggests an even stronger scaling with
distance, i.e. far-apart sites take longer to fully synchronise. This finding
is coherent with the observation of temporally correlated noise at each site:
local temporal properties might influence the global synchronisation
phenomena, delaying a fully synchronised power grid [55]. Interestingly, the
linear scaling of phase synchronisation, compared to the approximately
quadratic scaling of amplitude synchronisation implies that locations far
apart will first synchronise in their phase, then amplitude, while the
ordering might be reversed for locations geographically nearby. A definite
answer will require data from more locations, preferably in large synchronous
grids.

We conclude that the implications of these findings are considerable for both
the understanding of the dynamical processes in power grids as well as their
simulation, as the increments provide a proxy for fluctuations of the power
imbalance that drives the frequency dynamics. Foremost, we uncovered that the
temporal stochastic properties at each recording site impact the speed at
which amplitude synchronisation is achieved. This grants an exact measure for
amplitude synchronisation across any power-grid, which can now be uncovered
solely from one single local recording, i.e., since the scaling parameter
$\alpha$ is a local property, a single power-grid frequency measurement in a
synchronous region allows us to determine at which speed an electrically
distant far-away location will be amplitude-synchronised with the rest of the
grid. Furthermore, two distinct and so far practically unaddressed
characteristics are uncovered: First, fluctuations at different locations are
correlated. This implies immediately that any simulation—especially for small
power grids as microgrids—must consider the presence not only of noise, i.e.,
stochastic fluctuations but of spatially correlated noise. We observe that
substantial correlations are seen up to distances of $1000\,$km. Secondly,
each location, each power-grid frequency recordings, shows a distinct strong
temporal correlation within itself. This is a clear indication that temporal
correlations in stochastic fluctuations are present. In fact, these
correlations determine the physics of the aforementioned amplitude
synchronisation. Thus, adequate simulations should extent their analysis far
beyond classical white noise (uncorrelated Brownian noise) and consider
instead spatio-temporally correlated (non-white) noise.

All of these effects significantly impact risk assessment: heavy-tailed noise
leads to larger deviations than expected from Gaussian noise, while correlated
noise continues to push the system in one given direction, instead of randomly
fluctuating around zero. Hence, both correlations and non-Gaussian
distributions induce larger frequency deviations than assumed from white
Gaussian noise and thereby increase the risk of destabilising the system.
Simply adding a larger security margin when carrying out simulations is barely
appropriate: a broad Gaussian noise distribution would imply that medium
deviations occur very often when instead rare large deviations are the
problem. Hence, to properly dimension back-up capacities and design adequate
control options, the non-standard statistics presented here should be taken
into account.

In the future, it would be desirable to study the link between local temporal
correlation and spatial amplitude synchronisation in different data sets [56].
Also, further theoretical work is necessary to offer efficient simulation and
modelling tools for all discussed phenomena.

###### Acknowledgements.

We gratefully acknowledge support from the German Federal Ministry of
Education and Research (grant no. 03EK3055B) and the Helmholtz Association
(via the joint initiative “Energy System 2050 – A Contribution of the Research
Field Energy” and the grant “Uncertainty Quantification – From Data to
Reliable Knowledge (UQ)” with grant no. ZT-I-0029). This work was performed as
part of the Helmholtz School for Data Science in Life, Earth and Energy (HDS-
LEE). This project has received funding from the European Union’s Horizon 2020
research and innovation programme under the Marie Skłodowska-Curie grant
agreement No 840825. Luigi Vanfretti was supported in part by the Engineering
Research Center Program of the National Science Foundation and the Department
of Energy under Award EEC-1041877 and by the CURENT Industry Partnership
Program.

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