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arxiv: 2604.03133 · v1 · submitted 2026-04-03 · ⚛️ physics.soc-ph

Recognition: 2 theorem links

· Lean Theorem

Understanding the complexity of frequency and phase angle fluctuations in power grids

Alessandro Lonardi , Jacques M. Maritz , Leonardo Rydin Gorj\~ao , Christian Beck
Authors on Pith no claims yet

Pith reviewed 2026-05-13 18:39 UTC · model grok-4.3

classification ⚛️ physics.soc-ph
keywords power grid frequencysuperstatisticsphase angle fluctuationsheavy-tailed distributionsmarket-driven fluctuationsnonlinear frequency controlUK gridSouth Africa grid
0
0 comments X

The pith

Analytical model driven by slow market fluctuations reproduces multimodal grid frequency distributions and heavy tails

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper gathers nearly a billion high-resolution measurements of frequency and phase angles from the UK and South African grids. It models market-driven power changes as a slowly varying parameter and folds in nonlinear frequency control to obtain closed-form statistical distributions. The resulting expressions match the multimodal frequency histograms, heavy-tailed deviations, and double-exponential autocorrelation decays recorded in the data. The same framework also yields a low-dimensional effective description that fits observed spatial phase-angle fluctuations, revealing clear differences between the two national systems.

Core claim

Treating market-driven power fluctuations as a slowly varying parameter and incorporating nonlinear frequency control yields an analytical superstatistical model whose predictions for frequency distributions, heavy tails, and autocorrelation functions agree quantitatively with large-scale measurements from both the United Kingdom and South Africa; the same low-dimensional effective model also accounts for the spatial structure of phase-angle fluctuations.

What carries the argument

Superstatistical modeling that treats market-driven power fluctuations as a slowly varying parameter driving nonlinear frequency control dynamics

If this is right

  • Grid operators can use the closed-form distributions to forecast the statistical impact of different market-trading rules on frequency stability.
  • Phase-angle fluctuations in complex grids can be approximated by low-dimensional effective models without resolving every transmission line.
  • Differences in observed statistics between national grids trace directly to contrasting control policies and infrastructure maturity.
  • The separation of market and control timescales supplies a route to include growing renewable variability while retaining analytical tractability.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same timescale-separation idea could be tested on frequency data from other regions with different market designs to check whether the multimodal pattern persists.
  • Extending the model to include explicit renewable-generation stochasticity would show whether the current heavy-tail predictions remain robust under higher renewable shares.
  • If the low-dimensional phase-angle description holds across more grids, it could simplify real-time monitoring tools that currently rely on full network topology.

Load-bearing premise

Market-driven power fluctuations vary slowly enough relative to grid control dynamics that they can be treated as a fixed parameter when deriving the frequency distributions.

What would settle it

New high-resolution frequency time series from a third grid whose measured distributions deviate systematically from the closed-form expressions predicted by the superstatistical model.

Figures

Figures reproduced from arXiv: 2604.03133 by Alessandro Lonardi, Christian Beck, Jacques M. Maritz, Leonardo Rydin Gorj\~ao.

Figure 1
Figure 1. Figure 1: FIG. 1. Power grids in the UK and SA. Grid nodes include [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Frequency measurements in SA and the UK. We conventionally use data from Stellenbosch and London to represent their [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Phase angle measurements in SA and the UK. (A) [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
read the original abstract

Power grids must modernize to meet climate goals while maintaining reliable and stable operating conditions. Yet progress is hindered by a limited understanding of the stochastic processes underlying grid frequency and phase-angle fluctuations, which are induced by the growing penetration of renewable generation, consumer demand fluctuations, and market trading. This issue is particularly acute in Africa, where grids often face weak investment. Here, we present results from a newly collected, large-scale, high-resolution dataset of grid frequency and phase angles for the United Kingdom and South Africa, comprising close to one billion data points. Using superstatistical modeling, we treat market-driven power fluctuations as a slowly varying parameter driving grid dynamics and incorporate nonlinear frequency control. As a result, we derive an analytical model that reproduces multimodal frequency distributions previously obtained from numerical simulations, as well as heavy-tailed fluctuations and double-exponential frequency autocorrelation decays, all in excellent agreement with experimental measurements. Beyond frequency, we also address the so far largely overlooked problem of characterizing spatial phase-angle fluctuations. By comparing our predictions with measurement data, we demonstrate that a low-dimensional effective grid model accurately fits South African data despite the grid's complexity. We also highlight significant differences between the grids of South Africa and the United Kingdom. Our results clarify how energy markets and control policies shape grid dynamics across countries with contrasting infrastructure maturity.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 2 minor

Summary. The manuscript presents a superstatistical model for power-grid frequency and phase-angle fluctuations. Market-driven power fluctuations are treated as a slowly varying parameter that drives the grid dynamics together with nonlinear frequency control. From this separation of timescales the authors derive closed-form expressions for multimodal frequency distributions, heavy-tailed fluctuations, and double-exponential autocorrelation decays. These predictions are compared with a new high-resolution dataset (nearly one billion points) from the UK and South African grids and are reported to be in excellent visual agreement. A low-dimensional effective model is also shown to capture spatial phase-angle statistics in the South African grid.

Significance. If the slow-parameter distribution can be shown to follow from market dynamics rather than being chosen to match the observed histograms, the work would supply an analytically tractable framework linking energy-market statistics to grid stability metrics. The scale of the empirical dataset and the explicit comparison between two grids with different infrastructure maturity are clear strengths. The approach could inform renewable-integration studies in regions with limited grid investment.

major comments (3)
  1. [§3] §3 (Superstatistical closure): The manuscript invokes a specific distribution for the slowly varying power-fluctuation parameter to obtain the closed-form multimodal frequency distribution. It is not demonstrated that this distribution is independently derived from market data or first-principles trading dynamics; if it is calibrated to the same histograms the model is claimed to predict, the agreement reduces to a consistency check rather than an independent derivation.
  2. [§4.2] §4.2 (Autocorrelation and heavy tails): The double-exponential decay of the frequency autocorrelation is stated to follow from the model, yet the derivation appears to rely on additional assumptions about the correlation time of the slow parameter. The text should explicitly show the steps that produce the exact functional form without post-hoc adjustment of the correlation time.
  3. [§5] §5 (Phase-angle model): The claim that a low-dimensional effective grid model accurately reproduces South African phase-angle statistics despite the grid’s topological complexity requires quantitative goodness-of-fit measures (e.g., Kolmogorov-Smirnov statistics or residual distributions) and a clear statement of how many effective parameters are fitted versus predicted.
minor comments (2)
  1. The abstract states 'excellent agreement' with data; the main text should report the quantitative metrics (R², RMSE, or distribution distances) used to substantiate this statement.
  2. Figure captions should explicitly state whether error bars represent standard deviation across days or across grid nodes.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the constructive and detailed review. We address each major comment point by point below. Where revisions are feasible, we have updated the manuscript to improve clarity and add requested details.

read point-by-point responses

  1. Referee: [§3] §3 (Superstatistical closure): The manuscript invokes a specific distribution for the slowly varying power-fluctuation parameter to obtain the closed-form multimodal frequency distribution. It is not demonstrated that this distribution is independently derived from market data or first-principles trading dynamics; if it is calibrated to the same histograms the model is claimed to predict, the agreement reduces to a consistency check rather than an independent derivation.

    Authors: We agree that the specific form of the slow-parameter distribution is chosen to reproduce the observed multimodal frequency histograms, consistent with standard superstatistical practice where the fluctuating parameter's distribution is inferred from data. The manuscript motivates this choice by reference to market-driven power fluctuations but does not derive the distribution from first-principles trading dynamics. In the revised version we will explicitly state that the agreement constitutes a consistency check under this phenomenological choice and will add a brief discussion of possible future links to market data. revision: partial

  2. Referee: [§4.2] §4.2 (Autocorrelation and heavy tails): The double-exponential decay of the frequency autocorrelation is stated to follow from the model, yet the derivation appears to rely on additional assumptions about the correlation time of the slow parameter. The text should explicitly show the steps that produce the exact functional form without post-hoc adjustment of the correlation time.

    Authors: The original text condensed the derivation of the double-exponential autocorrelation. We will expand §4.2 in the revision to present the full step-by-step derivation: starting from the separation of timescales, the assumed exponential correlation of the slow parameter, and the resulting integral expression for the frequency autocorrelation, which yields the exact double-exponential form with no additional fitting of the correlation time beyond the model's intrinsic parameters. revision: yes

  3. Referee: [§5] §5 (Phase-angle model): The claim that a low-dimensional effective grid model accurately reproduces South African phase-angle statistics despite the grid’s topological complexity requires quantitative goodness-of-fit measures (e.g., Kolmogorov-Smirnov statistics or residual distributions) and a clear statement of how many effective parameters are fitted versus predicted.

    Authors: We will add quantitative goodness-of-fit measures, including Kolmogorov-Smirnov statistics and residual-distribution analysis, to the revised §5. We will also explicitly state the number of effective parameters in the low-dimensional model and distinguish those fitted to the South African data from those predicted by the model structure. revision: yes

Circularity Check

1 steps flagged

Superstatistical slow-parameter distribution chosen to reproduce observed multimodal frequency histograms

specific steps
  1. fitted input called prediction [Abstract; superstatistical modeling section]
    "Using superstatistical modeling, we treat market-driven power fluctuations as a slowly varying parameter driving grid dynamics and incorporate nonlinear frequency control. As a result, we derive an analytical model that reproduces multimodal frequency distributions previously obtained from numerical simulations, as well as heavy-tailed fluctuations and double-exponential frequency autocorrelation decays, all in excellent agreement with experimental measurements."

    Superstatistics yields closed-form marginal distributions only for particular choices of the fluctuating-parameter pdf. The quoted passage presents the multimodal, heavy-tailed, and double-exponential forms as derived predictions, yet these forms appear only after the slow-parameter distribution is selected to match the measured frequency histograms. The agreement is therefore enforced by the choice of input distribution rather than obtained from independent market data.

full rationale

The derivation chain begins with the superstatistical ansatz that market-driven power fluctuations act as a slowly varying parameter whose distribution is not derived from first-principles market dynamics. The closed-form multimodal frequency distribution, heavy tails, and double-exponential autocorrelations are obtained only after selecting a specific functional form (typically gamma or inverse-gamma) for that parameter. The manuscript presents the resulting expressions as an analytical prediction that matches data, yet the parameter distribution itself is calibrated to the very histograms it is claimed to predict. This reduces the central claim to a consistency check rather than an independent derivation. No self-citation chain or uniqueness theorem is invoked, so the circularity is confined to the fitted-input step.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The central claim rests on treating market fluctuations as a slowly varying external parameter and on a low-dimensional effective description of the grid; several distribution parameters must be fitted to data.

free parameters (1)
  • superstatistical fluctuation parameters
    Chosen to reproduce the observed multimodal frequency histograms and autocorrelation shapes
axioms (2)
  • domain assumption market-driven power fluctuations vary slowly compared with grid dynamics
    Invoked to justify treating them as a fixed parameter inside the superstatistical average
  • domain assumption nonlinear frequency control can be represented by a simple effective equation
    Required to close the analytical derivation

pith-pipeline@v0.9.0 · 5542 in / 1362 out tokens · 23994 ms · 2026-05-13T18:39:34.538893+00:00 · methodology

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Stochastic Modeling of Power-Grid Frequency Fluctuations in Low-Inertia Systems via a Gaussian-Core Potential and Superstatistics

    physics.soc-ph 2026-05 unverdicted novelty 6.0

    A Gaussian-core potential plus superstatistics reproduces the bimodal frequency distributions, central suppression, and heavy tails observed in Great Britain grid data as inertia declines.

Reference graph

Works this paper leans on

78 extracted references · 78 canonical work pages · cited by 1 Pith paper

  1. [1]

    UnitedNationsClimateChangeConference,COP29Global Energy Storage and Grids Pledge (2024)

    work page 2024

  2. [2]

    World Economic Forum, Why scaling clean energy in the Global South is a three-legged balancing act (2025)

    work page 2025

  3. [3]

    International Energy Agency, World Energy Outlook 2024 (2024)

    work page 2024

  4. [4]

    Weissbach and E

    T. Weissbach and E. Welfonder, High frequency deviations within the European Power System: Origins and proposals forimprovement,in2009IEEE/PESPowerSystemsConfer- ence and Exposition(2009) pp. 1–6

    work page 2009

  5. [5]

    Rohden, A

    M. Rohden, A. Sorge, M. Timme, and D. Witthaut, Self- organized synchronization in decentralized power grids, Phys. Rev. Lett.109, 064101 (2012)

    work page 2012

  6. [6]

    A. E. Motter, S. A. Myers, M. Anghel, and T. Nishikawa, Spontaneous synchrony in power-grid networks, Nature Physics9, 191 (2013)

    work page 2013

  7. [7]

    Manik, D

    D. Manik, D. Witthaut, B. Schäfer, M. Matthiae, A. Sorge, M. Rohden, E. Katifori, and M. Timme, Supply networks: Instabilities without overload, The European Physical Jour- nal Special Topics223, 2527 (2014)

    work page 2014

  8. [8]

    F. M. Mele, Á. Ortega, R. Zárate-Miñano, and F. Milano, Impact of variability, uncertainty and frequency regulation onpowersystemfrequencydistribution,in2016PowerSys- tems Computation Conference (PSCC)(2016) pp. 1–8

    work page 2016

  9. [9]

    Anvari, G

    M. Anvari, G. Lohmann, M. Wächter, P. Milan, E. Lorenz, D. Heinemann, M. R. R. Tabar, and J. Peinke, Short term fluctuations of wind and solar power systems, New Journal of Physics18, 063027 (2016)

    work page 2016

  10. [10]

    B.Schäfer,M.Timme,andD.Witthaut,IsolatingtheImpact of Trading on Grid Frequency Fluctuations, in2018 IEEE PESInnovativeSmartGridTechnologiesConferenceEurope (ISGT-Europe)(2018) pp. 1–5

    work page 2018

  11. [11]

    Schäfer, D

    B. Schäfer, D. Witthaut, M. Timme, and V. Latora, Dy- namicallyinducedcascadingfailuresinpowergrids,Nature Communications9, 1975 (2018)

    work page 1975

  12. [12]

    B.Schäfer,C.Beck,K.Aihara,D.Witthaut,andM.Timme, Non-Gaussian power grid frequency fluctuations character- ized by Lévy-stable laws and superstatistics, Nature Energy 3, 119 (2018)

    work page 2018

  13. [13]

    X. Deng, H. Li, W. Yu, W. Weikang, and Y. Liu, Fre- quency Observations and Statistic Analysis of Worldwide Main Power Grids Using FNET/GridEye, in2019 IEEE Power&EnergySocietyGeneralMeeting(PESGM)(2019) pp. 1–5

    work page 2019

  14. [14]

    Haehne, K

    H. Haehne, K. Schmietendorf, S. Tamrakar, J. Peinke, and S. Kettemann, Propagation of wind-power-induced fluctua- tions in power grids, Phys. Rev. E99, 050301 (2019)

    work page 2019

  15. [15]

    Vorobev, D

    P. Vorobev, D. M. Greenwood, J. H. Bell, J. W. Bialek, P. C. Taylor, and K. Turitsyn, Deadbands, Droop, and Iner- tia Impact on Power System Frequency Distribution, IEEE Transactions on Power Systems34, 3098 (2019)

    work page 2019

  16. [16]

    Yalcin, J

    L.RydinGorjão,R.Jumar,H.Maass,V.Hagenmeyer,G.C. Yalcin, J. Kruse, M. Timme, C. Beck, D. Witthaut, and B. Schäfer, Open database analysis of scaling and spatio- temporalpropertiesofpowergridfrequencies,NatureCom- munications11, 6362 (2020)

    work page 2020

  17. [17]

    Anvari, L

    M. Anvari, L. Rydin Gorjão, M. Timme, D. Witthaut, B. Schäfer, and H. Kantz, Stochastic properties of the fre- quency dynamics in real and synthetic power grids, Phys. Rev. Res.2, 013339 (2020)

    work page 2020

  18. [18]

    L.RydinGorjão,M.Anvari,H.Kantz,C.Beck,D.Witthaut, M.Timme,andB.Schäfer,Data-DrivenModelofthePower- Grid Frequency Dynamics, IEEE Access8, 43082 (2020)

    work page 2020

  19. [19]

    Rydin Gorjão, B

    L. Rydin Gorjão, B. Schäfer, D. Witthaut, and C. Beck, Spatio-temporalcomplexityofpower-gridfrequencyfluctu- ations, New Journal of Physics23, 073016 (2021)

    work page 2021

  20. [20]

    del Giudice, A

    D. del Giudice, A. Brambilla, S. Grillo, and F. Bizzarri, Ef- fects of inertia, load damping and dead-bands on frequency histograms and frequency control of power systems, Inter- nationalJournalofElectricalPower&EnergySystems129, 106842 (2021)

    work page 2021

  21. [21]

    Witthaut, F

    D. Witthaut, F. Hellmann, J. Kurths, S. Kettemann, H. Meyer-Ortmanns, and M. Timme, Collective nonlinear dynamicsandself-organizationindecentralizedpowergrids, Rev. Mod. Phys.94, 015005 (2022)

    work page 2022

  22. [22]

    Anvari, E

    M. Anvari, E. Proedrou, B. Schäfer, C. Beck, H. Kantz, and M. Timme, Data-driven load profiles and the dynamics of residentialelectricityconsumption,NatureCommunications 13, 4593 (2022)

    work page 2022

  23. [23]

    C. Han, H. Hilger, E. Mix, P. C. Böttcher, M. Reyers, C. Beck, D. Witthaut, and L. R. Gorjão, Complexity and PersistenceofPriceTimeSeriesoftheEuropeanElectricity Spot Market, PRX Energy1, 013002 (2022)

    work page 2022

  24. [24]

    Schäfer, Nonstandard power grid frequency statistics across continents, Scientific Reports15, 38470 (2025)

    X.Wen,M.Anvari,L.RydinGorjão,G.C.Yalcin,V.Hagen- meyer, and B. Schäfer, Nonstandard power grid frequency statistics across continents, Scientific Reports15, 38470 (2025)

    work page 2025

  25. [25]

    Kraljic, Towards Realistic Statistical Models of the Grid Frequency, IEEE Transactions on Power Systems38, 256 (2023)

    D. Kraljic, Towards Realistic Statistical Models of the Grid Frequency, IEEE Transactions on Power Systems38, 256 (2023)

    work page 2023

  26. [26]

    Schafer, Non-linear, bivariate stochastic modelling of power-grid frequency applied to islands, in 2023 IEEE Belgrade PowerTech(2023) pp

    U.Oberhofer, L.R.Gorjao, G.C.Yalcin, O.Kamps, V.Ha- genmeyer, and B. Schafer, Non-linear, bivariate stochastic modelling of power-grid frequency applied to islands, in 2023 IEEE Belgrade PowerTech(2023) pp. 1–1

    work page 2023

  27. [27]

    Kruse, E

    J. Kruse, E. Cramer, B. Schäfer, and D. Witthaut, Physics- InformedMachineLearningforPowerGridFrequencyMod- eling, PRX Energy2, 043003 (2023)

    work page 2023

  28. [28]

    Rydin Gorjão, R

    L. Rydin Gorjão, R. Jumar, H. Maass, V. Hagenmeyer, G. C. Yalcin, J. Kruse, M. Timme, C. Beck, D. Witthaut, and B. Schäfer, Open access power-grid frequency database (2025)

    work page 2025

  29. [29]

    Drewnick, X

    T. Drewnick, X. Wen, U. Oberhofer, L. Rydin Gorjão, C. Beck, V. Hagenmeyer, and B. Schäfer, Analyzing deter- ministic and stochastic influences on the power grid fre- quency dynamics with explainable artificial intelligence, Chaos: An Interdisciplinary Journal of Nonlinear Science 35, 033153 (2025)

    work page 2025

  30. [30]

    P. C. Böttcher, C. Hartmann, A. Benigni, T. Pesch, and D. Witthaut, Impact of Market Reforms on Deterministic Frequency Deviations in the European Power Grid (2026), arXiv:2602.09645 [physics.soc-ph]

    work page arXiv 2026

  31. [31]

    A.J.Wood,B.F.Wollenberg,andG.B.Sheblé,Powergen- eration, operation, and control(John wiley & sons, 2013)

    work page 2013

  32. [32]

    Machowski, Z

    J. Machowski, Z. Lubosny, J. W. Bialek, and J. R. Bumby, Power system dynamics: stability and control(John Wiley & Sons, 2020)

    work page 2020

  33. [33]

    Kundur, Power system stability, Power system stability and control10, 7 (2007)

    P. Kundur, Power system stability, Power system stability and control10, 7 (2007)

    work page 2007

  34. [34]

    K.MayerandS.Trück,Electricitymarketsaroundtheworld, Journal of Commodity Markets9, 77 (2018)

    work page 2018

  35. [35]

    Kashima, H

    K. Kashima, H. Aoyama, and Y. Ohta, Modeling and lin- earization of systems under heavy-tailed stochastic noise with application to renewable energy assessment, in2015 54th IEEE Conference on Decision and Control (CDC) (2015) pp. 1852–1857. 9

    work page 2015

  36. [36]

    Manik, M

    D. Manik, M. Timme, and D. Witthaut, Cycle flows and multistabilityinoscillatorynetworks,Chaos: AnInterdisci- plinary Journal of Nonlinear Science27, 083123 (2017)

    work page 2017

  37. [37]

    Schäfer, M

    B. Schäfer, M. Matthiae, X. Zhang, M. Rohden, M. Timme, and D. Witthaut, Escape routes, weak links, and desynchro- nization in fluctuation-driven networks, Phys. Rev. E95, 060203 (2017)

    work page 2017

  38. [38]

    Hindes, P

    J. Hindes, P. Jacquod, and I. B. Schwartz, Network desyn- chronizationbynon-gaussianfluctuations,Phys.Rev.E100, 052314 (2019)

    work page 2019

  39. [39]

    Schmietendorf, J

    K. Schmietendorf, J. Peinke, and O. Kamps, The impact of turbulent renewable energy production on power grid stability and quality, The European Physical Journal B90, 222 (2017)

    work page 2017

  40. [40]

    Tumash, S

    L. Tumash, S. Olmi, and E. Schöll, Effect of disorder and noise in shaping the dynamics of power grids, Europhysics Letters123, 20001 (2018)

    work page 2018

  41. [41]

    NESO: National Energy System Operator, Dynamic Ser- vices (DC/DM/DR) (2025)

    work page 2025

  42. [42]

    Ofgem,DecisiononDynamicRegulation,DynamicModer- ation, and Dynamic Containment in relation to an update to the Terms and Conditions related to Balancing (2023)

    work page 2023

  43. [43]

    EPEXSPOT,AnnualTradingResultsof2024-PowerTrad- ing on EPEX SPOT reaches all-time high (2024)

    work page 2024

  44. [44]

    SAPP: Southern African Power Pool, SAPP Statistics 2020 (2020)

    work page 2020

  45. [45]

    S. R. Conradie and L. J. M. Messerschmidt,A Symphony of Power: The Eskom Story(C. van Rensburg Publications, 2000)

    work page 2000

  46. [46]

    CSIR Energy Research Centre, Utility-scale power genera- tion statistics in South Africa (2025)

    work page 2025

  47. [47]

    International Energy Agency, South Africa Energy Mix (2024)

    work page 2024

  48. [48]

    T.JansevanRensburgandK.Morema,Reflectionsonload- shedding and potential GDP, South African Reserve Bank OccasionalBulletinofEconomicNotesOBEN/23/012301 (2023)

    work page 2023

  49. [49]

    Makoni, Power cuts and South Africa’s health care, The Lancet401, 1413 (2023)

    M. Makoni, Power cuts and South Africa’s health care, The Lancet401, 1413 (2023)

    work page 2023

  50. [50]

    Greyling, S

    T. Greyling, S. Rossouw, and AFSTEREO, Gross national happiness.today index,http://gnh.today(2019)

    work page 2019

  51. [51]

    NERSA: National Energy Regulator of South Africa, The South African Grid Code: 2022 (2022)

    work page 2022

  52. [52]

    Rydin Gorjão and J

    L. Rydin Gorjão and J. Maritz, The stochastic nature of power-grid frequency in south africa, Journal of Physics: Complexity4, 015007 (2023)

    work page 2023

  53. [53]

    Maritz, L

    J. Maritz, L. Rydin Gorjão, P. A. Bester, N. Esterhuy- sen, S. Erasmus, S. Riekert, R. Immelman, T. Geldenhuys, A. Viljoen, and C. Bodenstein, Data-Driven Modeling of FrequencyDynamicsObservedinOperatingMicrogrids: A SouthAfricanUniversityCampusCaseStudy,IEEEAccess 12, 14466 (2024)

    work page 2024

  54. [54]

    C.Beck,DynamicalFoundationsofNonextensiveStatistical Mechanics, Phys. Rev. Lett.87, 180601 (2001)

    work page 2001

  55. [55]

    Beck and E

    C. Beck and E. G. D. Cohen, Superstatistics, Physica A: Statistical Mechanics and its Applications322, 267 (2003)

    work page 2003

  56. [56]

    C. Beck, E. G. D. Cohen, and H. L. Swinney, From time series to superstatistics, Phys. Rev. E72, 056133 (2005)

    work page 2005

  57. [57]

    Risken, Fokker-planck equation, inThe Fokker-Planck equation: methods of solution and applications(Springer,

    H. Risken, Fokker-planck equation, inThe Fokker-Planck equation: methods of solution and applications(Springer,

    work page

  58. [58]

    Y.G.Rebours,D.S.Kirschen,M.Trotignon,andS.Rossig- nol, A Survey of Frequency and Voltage Control Ancillary Services—Part I: Technical Features, IEEE Transactions on Power Systems22, 350 (2007)

    work page 2007

  59. [59]

    Ulbig, T

    A. Ulbig, T. S. Borsche, and G. Andersson, Impact of low rotational inertia on power system stability and operation, IFAC Proceedings Volumes47, 7290 (2014)

    work page 2014

  60. [60]

    Milan, M

    P. Milan, M. Wächter, and J. Peinke, Turbulent Character of Wind Energy, Phys. Rev. Lett.110, 138701 (2013)

    work page 2013

  61. [61]

    Lamouroux and K

    D. Lamouroux and K. Lehnertz, Kernel-based regression of drift and diffusion coefficients of stochastic processes, Physics Letters A373, 3507 (2009)

    work page 2009

  62. [62]

    L.RydinGorjãoandF.Meirinhos,kramersmoyal: Kramers– Moyalcoefficientsforstochasticprocesses,JournalofOpen Source Software4, 1693 (2019)

    work page 2019

  63. [63]

    Touchette and C

    H. Touchette and C. Beck, Asymptotics of superstatistics, Phys. Rev. E71, 016131 (2005)

    work page 2005

  64. [64]

    22 (Cambridge University Press, 2007)

    R.W.Butler,Saddlepointapproximationswithapplications, Vol. 22 (Cambridge University Press, 2007)

    work page 2007

  65. [65]

    International Energy Agency, United Kindom Energy Mix (2024)

    work page 2024

  66. [66]

    M. S. Almas, L. Vanfretti, R. S. Singh, and G. M. Jonsdot- tir, Vulnerability of synchrophasor-based wampac applica- tions’ to time synchronization spoofing, IEEE Transactions on Smart Grid9, 4601 (2018)

    work page 2018

  67. [67]

    Redner,A guide to first-passage processes(Cambridge University Press, 2001)

    S. Redner,A guide to first-passage processes(Cambridge University Press, 2001)

    work page 2001

  68. [68]

    B. B. Mandelbrot and J. W. Van Ness, Fractional brownian motions, fractional noises and applications, SIAM review 10, 422 (1968)

    work page 1968

  69. [69]

    EPEX SPOT, Basics of the Power Market (2025)

    work page 2025

  70. [70]

    Samoradnitsky,Stable non-Gaussian random processes: stochastic models with infinite variance(Routledge, 2017)

    G. Samoradnitsky,Stable non-Gaussian random processes: stochastic models with infinite variance(Routledge, 2017)

    work page 2017

  71. [71]

    hard-to-spot

    C. Tsallis, Possible generalization of Boltzmann-Gibbs statistics, Journal of Statistical Physics52, 479 (1988). 10 Supporting Information: Understanding the complexity of frequency and phase angle fluctuations in power grids S1. DATA A. Data collection Phasor Measurement Units (PMUs) are installed in Stellenbosch and Bloemfontein, South Africa, and in Lo...

    work page 1988

  72. [72]

    Contracted generators measure the frequency deviation from the nominal value and inject power proportional to the observed deviation, damping fast frequency fluctuations

    Primary control starts acting within seconds. Contracted generators measure the frequency deviation from the nominal value and inject power proportional to the observed deviation, damping fast frequency fluctuations

    work page

  73. [73]

    It gradually restores the generator rotation to its reference frequency, correcting any steady-state deviation left by primary control

    Secondary control acts on a slower time scale, typically minutes. It gradually restores the generator rotation to its reference frequency, correcting any steady-state deviation left by primary control

    work page

  74. [74]

    Dynamic Regulation

    On even longer time scales of minutes to hours, tertiary control may intervene. This, for instance, adjusts each generator’s target power output and ensures efficient electricity supply while maintaining adequate reserves. We focus on primary control only, which provides the fastest and strongest response to stabilize the grid. We model it as a (nonlinear...

    work page 2025

  75. [75]

    The power grid admits stable working points that are the minima of𝑉(𝑥): 𝑥min,1 =arcsin𝜌 𝑥 min,2 =𝑥 min,1 −2𝜋 𝑥max,1 =−𝑥 min,1 +𝜋 𝑥 max,2 =−𝑥 min,1 −𝜋

    If𝜌 >0and|𝜌|<1, there is a positive power difference between generator1and2. The power grid admits stable working points that are the minima of𝑉(𝑥): 𝑥min,1 =arcsin𝜌 𝑥 min,2 =𝑥 min,1 −2𝜋 𝑥max,1 =−𝑥 min,1 +𝜋 𝑥 max,2 =−𝑥 min,1 −𝜋

    work page

  76. [76]

    If𝜌 <0and|𝜌|<1, the setup is analogous to (1), but with a negative power difference: 𝑥min,1 =arcsin𝜌 𝑥 min,2 =𝑥 min,1 +2𝜋 𝑥max,1 =−𝑥 min,1 −𝜋 𝑥 max,2 =−𝑥 min,1 +𝜋

    work page

  77. [77]

    Physically, this means that both generators balance their own power demand

    If𝜌=0, then𝛿=0. Physically, this means that both generators balance their own power demand. If this happens, the transmission line between1and2becomes idle. In fact, the stable points of the grid are 𝑥min,1 =𝑥 min,2 =0,2𝜋 𝑥max,1 =𝑥 max,2 =±𝜋 , which make the sine term in Eq. 3 (main text) vanish

    work page

  78. [78]

    Maxima and minima collapse into the saddle points: 𝑥saddle,1 =± 𝜋 2 𝑥saddle,2 =∓ 3𝜋 2

    If𝜌=±1, the power exchanged between1and2is the maximum allowed by the transmission line. Maxima and minima collapse into the saddle points: 𝑥saddle,1 =± 𝜋 2 𝑥saddle,2 =∓ 3𝜋 2 . 5.|𝜌|>1would imply a power exchange larger than what is physically allowed by the transmission line. In this case, there is no stable fixed point for the grid to operate, but only ...

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