Frequency measurements indicate the state of a power grid. In fact, deviations from the nominal frequency determine whether the grid is stable or in a critical situation. We aim to understand the fluctuations of the frequency on multiple time scales with a recently proposed method based on detrended fluctuation analysis. It enables us to infer characteristic time scales and generate stochastic models. We capture and quantify known features of the fluctuations like periodicity due to the trading market, response to variations by control systems, and stability of the long time average. We discuss similarities and differences between the British grid and the continental European grid.

1.
B. H.
Obama
, “
Presidential policy directive 21: Critical infrastructure security and resilience,
” see https://obamawhitehouse.archives.gov/the-press-office/2013/02/12/presidential-policy-directive-critical-infrastructure-security-and-resil (
2013
).
2.
P.
Kundur
, Power System Stability and Control, EPRI Power System Engineering Series (McGraw-Hill, 1994), see https://books.google.de/books?id=v3RxH_GkwmsC.
3.
A. J.
Wood
,
B. F.
Wollenberg
, and
G. B.
Sheblé
,
Power Generation, Operation and Control
(
John Wiley & Sons
,
New York
,
2013
).
4.
A.
Einfalt
et al., ENERGIE DER ZUKUNFT Publizierbarer Endbericht (2012), see https://www.ea.tuwien.ac.at/fileadmin/t/ea/projekte/ADRES_Concept/PublizierbarerEndberichtADRES_815674.pdf.
5.
T.
Tjaden
,
J.
Bergner
,
J.
Weniger
, and
V.
Quaschning
, “Representative electrical load profiles of residential buildings in Germany with a temporal resolution of one second” (2015), see .
6.
T.
Weißbach
and
E.
Welfonder
, “
High frequency deviations within the European power system—Origins and proposals for improvement
,” in
2009 IEEE/PES Power Systems Conference and Exposition, Seattle, WA, 15–18 March 2009
(IEEE,
2009
).
7.
B.
Schäfer
,
C.
Beck
,
K.
Aihara
,
D.
Witthaut
, and
M.
Timme
, “
Non-Gaussian power grid frequency fluctuations characterized by Lévy-stable laws and superstatistics
,”
Nat. Energy
3
,
119
126
(
2018
).
8.
M.
Anvari
et al., “Stochastic analysis of power grid frequency and of a synthetic model,” arXiv:1909.09110 (2019).
9.
P.
Milan
,
M.
Wächter
, and
J.
Peinke
, “
Turbulent character of wind energy
,”
Phys. Rev. Lett.
110
,
138701
(
2013
).
10.
K.
Schmietendorf
,
J.
Peinke
, and
O.
Kamps
, “
The impact of turbulent renewable energy production on power grid stability and quality
,”
Eur. Phys. J. B
90
,
222
(
2017
).
11.
H.
Haehne
,
J.
Schottler
,
M.
Waechter
,
J.
Peinke
, and
O.
Kamps
, “
The footprint of atmospheric turbulence in power grid frequency measurements
,”
Europhys. Lett.
121
,
30001
(
2018
).
12.
F.
Milano
,
F.
Dörfler
,
G.
Hug
,
D. J.
Hill
, and
G.
Verbic
, “Foundations and challenges of low-inertia systems,” in 2018 Power Systems Computation Conference (PSCC) (IEEE, 2018).
13.
ENTSO-E, see https://www.entsoe.eu/publications/major-publications/Pages/default.aspx for “Statistical Factsheet” (2014).
14.
Ofgem, see https://www.ofgem.gov.uk/publications-and-updates/investigation-9-august-2019-power-outage for more information about Investigation into 9 August 2019 power outage (2019).
15.
S.
Auer
,
F.
Hellmann
,
M.
Krause
, and
J.
Kurths
, “
Stability of synchrony against local intermittent fluctuations in tree-like power grids
,”
Chaos
27
,
127003
(
2017
).
16.
M.
Rohden
,
D.
Jung
,
S.
Tamrakar
, and
S.
Kettemann
, “
Cascading failures in ac electricity grids
,”
Phys. Rev. E
94
,
032209
(
2016
).
17.
B.
Schäfer
,
M.
Timme
, and
D.
Witthaut
, “Isolating the impact of trading on grid frequency fluctuations,” in 2018 IEEE PES Innovative Smart Grid Technologies Conference Europe (ISGT-Europe) (IEEE, 2018), pp. 1–5.
18.
L. R.
Gorjão
et al., “Modelling the dynamics of power grid frequency fluctuations from data,” arXiv:1909.08346 (2019).
19.
T.
Graves
,
R.
Gramacy
,
N.
Watkins
, and
C.
Franzke
, “
A brief history of long memory: Hurst, Mandelbrot and the Road to ARFIMA, 1951–1980
,”
Entropy
19
(
9
),
437
(
2017
).
20.
K.
Fraedrich
and
R.
Blender
, “
Scaling of atmosphere and ocean temperature correlations in observations and climate models
,”
Phys. Rev. Lett.
90
,
108501
(
2003
).
21.
K. Y.
Wan
and
R. E.
Goldstein
, “
Rhythmicity, recurrence, and recovery of flagellar beating
,”
Phys. Rev. Lett.
113
,
238103
(
2014
).
22.
J.
Echeverrıa
et al., “
Interpretation of heart rate variability via detrended fluctuation analysis and αβ filter
,”
Chaos
13
,
467
475
(
2003
).
23.
P.
Gopikrishnan
et al., “
Scaling and correlation in financial time series
,”
Physica A
287
,
362
373
(
2000
).
24.
K.
Hu
,
P. C.
Ivanov
,
Z.
Chen
,
P.
Carpena
, and
H. E.
Stanley
, “
Effect of trends on detrended fluctuation analysis
,”
Phys. Rev. E
64
,
011114
(
2001
).
25.
K.
Willson
and
D. P.
Francis
, “
A direct analytical demonstration of the essential equivalence of detrended fluctuation analysis and spectral analysis of RR interval variability
,”
Physiol. Meas.
24
,
N1
(
2002
).
26.
R.
Weron
, “
Estimating long-range dependence: Finite sample properties and confidence intervals
,”
Physica A
312
,
285
299
(
2002
).
27.
D.
Maraun
,
H. W.
Rust
, and
J.
Timmer
, “
Tempting long-memory—On the interpretation of DFA results
,”
Nonlinear Process. Geophys.
11
,
495
(
2004
).
28.
Z.
Chen
et al., “
Effect of nonlinear filters on detrended fluctuation analysis
,”
Phys. Rev. E
71
,
011104
(
2005
).
29.
R.
Bryce
and
K.
Sprague
, “
Revisiting detrended fluctuation analysis
,”
Sci. Rep.
2
,
315
(
2012
).
30.
Q.
Zhang
,
Y.
Zhou
, and
V. P.
Singh
, “
Detrending methods for fluctuation analysis in hydrology: Amendments and comparisons of methodologies
,”
Hydrol. Process.
28
,
753
763
(
2014
).
31.
M. S.
Taqqu
,
V.
Teverovsky
, and
W.
Willinger
, “
Estimators for long-range dependence: An empirical study
,”
Fractals
3
,
785
798
(
1995
).
32.
C.
Heneghan
and
G.
McDarby
, “
Establishing the relation between detrended fluctuation analysis and power spectral density analysis for stochastic processes
,”
Phys. Rev. E
62
,
6103
(
2000
).
33.
J.-M.
Bardet
and
I.
Kammoun
, “
Asymptotic properties of the detrended fluctuation analysis of long-range-dependent processes
,”
IEEE Trans. Inf. Theory
54
,
2041
2052
(
2008
).
34.
N.
Crato
,
R.
Linhares
, and
S.
Lopes
, “
Statistical properties of detrended fluctuation analysis
,”
J. Stat. Comput. Simul.
80
,
625
641
(
2010
).
35.
M.
Hoell
and
H.
Kantz
, “
The fluctuation function of the detrended fluctuation analysis—Investigation on the AR(1) process
,”
Eur. Phys. J. B
88
,
126
(
2015
).
36.
Z.
Czechowski
and
L.
Telesca
, “
Detrended fluctuation analysis of the Ornstein-Uhlenbeck process: Stationarity versus nonstationarity
,”
Chaos
26
,
113109
(
2016
).
37.
P. G.
Meyer
and
H.
Kantz
, “
Inferring characteristic timescales from the effect of autoregressive dynamics on detrended fluctuation analysis
,”
New J. Phys.
21
,
033022
(
2019
).
38.
M.
Hoell
and
H.
Kantz
, “
The relationship between the detrendend fluctuation analysis and the autocorrelation function of a signal
,”
Eur. Phys. J. B
88
,
327
(
2015
).
39.
P. G.
Meyer
and
H.
Kantz
, “
A simple decomposition of European temperature variability capturing the variance from days to a decade
,”
Clim. Dyn.
53
,
6909
6917
(
2019
).
40.
M.
Höll
,
K.
Kiyono
, and
H.
Kantz
, “
Theoretical foundation of detrending methods for fluctuation analysis such as detrended fluctuation analysis and detrending moving average
,”
Phys. Rev. E
99
,
033305
(
2019
).
41.
C. K.
Peng
et al., “
Mosaic organization of DNA nucleotides
,”
Phys. Rev. E
49
,
1685
(
1994
).
42.
J. W.
Kantelhardt
,
E.
Koscielny-Bunde
,
H. H. A.
Rego
,
S.
Havlin
, and
A.
Bunde
, “
Detecting long-range correlations with detrended fluctuation analysis
,”
Physica A
295
,
441
(
2001
).
43.
W.
Palma
,
Long-Memory Time Series: Theory and Methods
(
John Wiley & Sons
,
2007
), Vol. 662.
44.
J.
Beran
,
Statistics for Long-Memory Processes
(
Routledge
,
2017
).
45.
G. E.
Box
,
G. M.
Jenkins
,
G. C.
Reinsel
, and
G. M.
Ljung
,
Time Series Analysis: Forecasting and Control
(
John Wiley & Sons
,
2015
).
46.
European Parliamentary Research Service, “Understanding electricity markets in the EU (2016), see http://www.europarl.europa.eu/RegData/etudes/BRIE/2016/593519/EPRS_BRI(2016)593519_EN.pdf.
47.
Q.
Deng
,
D.
Nian
, and
Z.
Fu
, “
The impact of inter-annual variability of annual cycle on long-term persistence of surface air temperature in long historical records
,”
Clim. Dyn.
50
,
1091
(
2018
).
48.
“From BBC TV series: Britain from above,” see https://www.youtube.com/watch?v=slDAvewWfrA for “Britain Peak Power Demand” (2010).
You do not currently have access to this content.