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

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.

Figures (15)

• Figure 1: Power grids in the UK and SA. Grid nodes include power plants and substations, while links represent transmission lines connecting them. Node and link data were retrieved separately from Open Infrastructure Map (https://openinframap.org) and matched by aggregating link endpoints and nodes within 3.33km. After merging, loops and duplicate lines were removed. The final networks consist of a single connected component with 2059 nodes and 2614 links for SA, and 2115 nodes and 2467 links for the UK. Cities where PMUs were installed are marked, along with Johannesburg and Cape Town, which are connected by a main transmission corridor running diagonally across SA and relevant to phase-angle fluctuations.
• Figure 2: Frequency measurements in SA and the UK. We conventionally use data from Stellenbosch and London to represent their countries, as, during regular operations, they are effectively identical to those in Bloemfontein and Glasgow. MLE fits in panels C and G are done with one million points sampled uniformly from the frequency signal in the deadband and tails. (A, D) Average frequency time series aggregated over 24h. Lighter areas are standard deviations. (B, E) Frequency histograms. Vertical lines indicate the control regions of panel F. The peaks at $\pm \omega_1$ are asymmetric due to finite-size effects. The histogram's robustness is validated by splitting the data into monthly segments in SI Appendix, Note S1, Fig. S4, S5. (C) MLE fit in the deadband. Data is fitted in $\Omega_0$ (see panel F) and plotted over a larger interval. The dashed line is the uniform fit that neglects the slow power contribution schaefer2018non. (F) Control function. In the deadband $\Omega_0$ there is no control. In $\Omega_1$ and $\Omega_2$ control is linear with coefficients $\gamma_1 < \gamma_2$. The $y$-axis is arbitrarily scaled. (G) Tails. The left and right tails are aggregated. In the UK, the tail at $|\omega| > \omega_1$ is fitted with a Gaussian and a $q$-Gaussian. See SI Appendix, Note S7, S11, Fig. S9. (H) Autocorrelation at short time lags $\Delta t \leq \qty{20}{min}$ with lighter areas being standard deviations. We fit a single- and a double-decay ansatz. See SI Appendix, Note S8.
• Figure 3: Phase angle measurements in SA and the UK. (A) Phase-locked hexbin distributions. A companion figure showing data processing is in SI Appendix, Fig. S1. (B) MLE fit of the phase-angle difference histogram in SA. Both the potential $V(x)$ and the corresponding distribution $p(x)$ obtained from the MLE are drawn. The fit was performed using one million points sampled uniformly from $x(t)$.
• Figure S1: Trimming the phase-locked bands. Phase angles are phase-locked, as evidenced by the hexbin plots that form diagonal bands. Outliers outside $\Delta_1 \leq x \leq \Delta_2$ are those points lying beyond the red diagonal lines. The upper bands are fitted. To visually verify the goodness of this fit, we plot the upper-band fit on the lower bands by opportunity shifting the line by $2\pi$. (A) Data for the United Kingdom. To improve the visualization, we subsample the phase time series, taking one point every 10. (B) Data for South Africa. To improve the visualization, we subsample the phase time series, taking one point every 200.
• Figure S2: Processing the phase-angle difference for South Africa. (A) A snapshot of the original signal $x(t)$ is shown together with its lower envelope. (B) Panel A is zoomed in to approximately the first $\qty{1.5}{min}$. (C) The distribution of the original signal $x(t)$ and its lower envelope is plotted. The lower envelope is used to fit the tilted washboard potential described in the main text. The distribution mode lies well below $\pi / 2$, which is the boundary of the stability region allowed by the tilted washboard potential (see \ref{['apx-fig: tilted washboard panel']}B and \ref{['apx-sec: Analytical Derivations of Phase Angle Difference Distribution']}). Therefore, MLE yields a good fit.