Open database analysis of scaling and spatio-temporal properties of power grid frequencies

Electricity systems are rapidly changing due to large-scale integration of renewable energy sources, bringing increased variability and reduced inertia that challenge frequency stability. Frequency reflects the instantaneous balance of supply and demand and is regulated to remain near 50/60 Hz. While numerous theoretical works predict scaling laws for fluctuations, propagation of disturbances, and effects of control and topology, empirical validation across different grids is hindered by lack of open data. This study addresses that gap by analysing an open database of GPS-synchronised power-grid frequency measurements from 17 locations in 12 synchronous areas on three continents. The goals are to characterise statistical properties across grids, test a conjectured fluctuation scaling law with system size, quantify how short-term local fluctuations transition to long-term bulk behaviour, and identify inter-area oscillations within the Continental European synchronous area.

Prior research has developed dynamical models for grid synchronisation and stability (e.g., swing/Kuramoto-like models), contrasted centralised versus decentralised topologies, studied impacts of stochastic fluctuations and inertia, and analysed propagation of disturbances, pricing schemes, virtual inertia placement, and cascading failures. However, connections to multi-grid empirical data are limited, and existing frequency databases like GridEye/FNET or GridRadar are not open. Earlier empirical studies observed differences among European grids (e.g., larger variance in Nordic and GB than Continental Europe) and regular dispatch-induced patterns (15-min and hourly). The present work builds on these insights by providing comparative, open-data-driven analyses across diverse synchronous areas and by validating scaling predictions.

Data: GPS-synchronised frequency measurements obtained using an Electrical Data Recorder (EDR) at local power sockets (validated against transmission-grid PMU-like recordings) across 17 sites in 12 synchronous areas spanning Europe, Africa, and North America; plus a 1-week dataset from the Hungarian TSO for Békéscsaba and Győr. Data span 2017–2020; some sites have short runs, others months/years. Unreliable entries (NaNs, GPS loss) were removed for statistics; for autocorrelation and synchronised CE analysis, the longest contiguous NaN-free segments were used. ES-GC analysis used March 2018 data. Analyses: (1) Statistical characterisation via histograms of frequency deviations (assessing width, skewness, kurtosis) and autocorrelation functions to identify intrinsic timescales and dispatch regularities (e.g., 15-min cycles). (2) Scaling law test: Derivation from aggregated swing equation M dω/dt = −Mγω + ΔP(t), with ω = 2π(f − f_ref), ΔP zero-mean power imbalance, and damping-to-inertia ratio γ. The conjecture is ε ∼ 1/√N for aggregated noise amplitude versus effective system size N; assessed empirically across grids of different sizes. (3) Increment analysis: Compute frequency increments Δf(t; τ) = f(t+τ) − f(t) for lags τ (1 s, 10 s, etc.); quantify non-Gaussianity by excess kurtosis κ−3 across areas and lags to study intermittency and approach to Gaussianity with increasing τ. (4) Synchronised multi-site CE analysis: Four sites (Oldenburg, Karlsruhe, Lisbon, Istanbul) plus two Hungarian cities; evaluate spatial correlations at short times (increment scatter at τ=1 s and larger τ) and long times via rate of change of frequency (RoCoF) computed by linear fits over 25 s windows centred around each full hour to capture dispatch-induced surges. (5) Detrended Fluctuation Analysis (DFA): Compute fluctuation function F^2(ℓ) across timescales to observe transition from local to bulk behaviour; compare sites. (6) Time-to-bulk metric: Define relative DFA η(ℓ) = [F^2_location(ℓ) − F^2_Karlsruhe(ℓ)]/F^2_Karlsruhe(ℓ); define time-to-bulk as the smallest ℓ where η(ℓ) < 0.1. Use geographic distances (OSM/OSRM foot routing) from Karlsruhe to each site; fit linear vs quadratic dependence of time-to-bulk on distance and compare RMSEs. (7) Principal Component Analysis (PCA): Decompose synchronised CE dynamics into spatial modes (PCs), quantify explained variance λ_m, map mode spatial patterns (loadings at each site), and analyse temporal spectra (squared Fourier amplitudes) to identify inter-area oscillation periods. Code: Publicly available (https://github.com/LRydin). Data repositories: https://osf.io/by5hu/ and https://osf.io/m43tg/.

• Statistical diversity across grids: Islands (e.g., Iceland IS, Faroe FO, Gran Canaria ES-GC, Mallorca ES-PM) and South Africa (ZA) exhibit broader, heavy-tailed frequency distributions with high kurtosis (e.g., κ_IS ≈ 7) and larger deviations from nominal frequency than large continental grids (e.g., DE, EE, RU, US-UT).
• Autocorrelation patterns vary: Some areas show rapid decay (IS), others slow decay indicating long correlations (FO, US-UT); clear 15-min dispatch signatures in DE, GB, ES-PM.
• Scaling law validated: Fluctuation amplitudes decrease with effective system size, consistent with ε ∼ 1/√N predicted by the aggregated swing equation model.
• Increment statistics: Islands and smaller/isolated grids display strong intermittency with large excess kurtosis (κ−3 ≈ 10^1–10^2) persisting up to τ = 10 s (e.g., IS, ES-GC, FO, ES-PM, GB, US-TX, ZA). Larger continental grids (EE, DE, SE, RU, US-UT) show near-Gaussian increments for τ ≥ 1 s (κ−3 ≈ 10^−1 or smaller). Non-Gaussian aggregated distributions do not necessarily imply non-Gaussian increments (e.g., DE).
• Short vs long timescales in CE: At τ = 1 s, increments are highly correlated only for nearby sites (Oldenburg–Karlsruhe) and largely uncorrelated for distant pairs (e.g., Istanbul–Karlsruhe); peripheral sites display wider increment distributions. At hourly times, RoCoF values are highly correlated across all CE sites (R^2 ≥ 0.93), reflecting synchronous response to hourly dispatch.
• DFA transition: Short-timescale fluctuations differ by site (notably larger at Istanbul and Lisbon), but beyond ~10 s the DFA curves converge, indicating homogeneous bulk behaviour across CE.
• Time-to-bulk scaling with distance: Time-to-bulk increases with geographic distance from Karlsruhe; quadratic (diffusive) fit outperforms linear (RMSE 0.5 s vs 1.2 s). Estimated times: ≈0.5–1 s at 100 km, ≈3–5 s at 1000 km.
• Inter-area oscillations in CE: PCA reveals that PC1 represents bulk behaviour and explains λ1 = 99.2% of variance. PC2 (λ2 = 3.93×10^−3) shows a West–East dipole (Western Europe vs Istanbul). PC3 (λ3 = 1.79×10^−3) shows a North–South dipole (Lisbon and Istanbul vs Oldenburg). Spectral peaks correspond to periods τ ≈ 7 s and τ ≈ 4.5 s, consistent with reported inter-area oscillation ranges (≈1.25–8 s). On τ > 12 s, bulk dynamics dominate again, aligning with estimated time-to-bulk of ~12–15 s.

Open, multi-grid frequency data enable empirical validation of theoretical predictions. The confirmed ε ∼ 1/√N scaling supports aggregated swing-equation models linking system size to fluctuation amplitude, informing expectations for volatility in microgrids versus large interconnections. Spatially synchronised CE measurements show a clear separation of timescales: short-term (≈1 s) fluctuations are local and weakly correlated over long distances, while long-term dynamics (hourly dispatch) are coherent across the entire synchronous area. DFA quantifies the transition to bulk behaviour (~10 s typical), and the time-to-bulk dependence on distance is better captured by diffusive-like propagation than by a simple linear model, offering guidance for designing control strategies at remote or weakly coupled locations. PCA uncovers inter-area modes with characteristic periods (≈7 s, 4.5 s) and spatial dipoles, corroborating prior theoretical and empirical findings and clarifying that global resonant modes operate on intermediate timescales between local fluctuations and bulk behaviour. Differences from some earlier theoretical timescale assumptions (e.g., bulk onset at 2–5 s) highlight the need to account for specific grid properties (size, inertia, topology, control) when predicting timescale separations.

This work introduces and analyses an open database of power-grid frequency recordings spanning 17 locations in 12 synchronous areas. It establishes empirical scaling of fluctuation amplitudes with system size, quantifies the transition from localised short-term fluctuations to homogeneous bulk dynamics, and reveals inter-area oscillations and their spatial patterns within Continental Europe. The findings provide actionable insights for grid stability, control design, and understanding of spatio-temporal propagation of disturbances. Future research directions include: combining additional open frequency recordings for broader comparative studies; determining propagation velocities of disturbances and validating with detailed network models; quantifying the influence of specific generators and large industrial loads; systematically assessing market activity impacts on frequency dynamics; and exploring scaling of higher moments (skewness, kurtosis) with time lag and system size.

• Data coverage is uneven: some synchronous areas and continents lack measurements; some sites have only short records (e.g., one week), limiting statistical power.
• Missing topology and operational detail: analyses rely on geographic distance (OSM/OSRM foot routing) as a proxy for network proximity; actual transmission paths may be longer or structurally constrained.
• Data quality issues: occasional GPS losses and NaNs necessitated removal and selection of contiguous segments, which may introduce selection bias.
• Heuristic thresholds: the 10% criterion for time-to-bulk is an arbitrary (though reasonable) choice; alternative definitions might shift estimated times.
• Limited observability of drivers: attribution of non-Gaussian tails and dispatch effects is indirect; detailed market and generation data were not jointly analysed across all areas.
• Generalisability: results (e.g., timescales for bulk behaviour) are specific to the periods and grids studied (notably CE) and may vary with system conditions (inertia, control policies, renewable penetration).