The electrical energy system has attracted much attention from an increasingly diverse research community. Many theoretical predictions have been made, from scaling laws of fluctuations to propagation velocities of disturbances. However, to validate any theory, empirical data from large-scale power systems are necessary but are rarely shared openly. Here, we analyse an open database of measurements of electric power grid frequencies across 17 locations in 12 synchronous areas on three continents. The power grid frequency is of particular interest, as it indicates the balance of supply and demand and carries information on deterministic, stochastic, and control influences. We perform a broad analysis of the recorded data, compare different synchronous areas and validate a previously conjectured scaling law. Furthermore, we show how fluctuations change from local independent oscillations to a homogeneous bulk behaviour. Overall, the presented open database and analyses constitute a step towards more shared, collaborative energy research.

Cosmic ray energy spectra exhibit power law distributions over many orders of magnitude that are very well described by the predictions of q-generalized statistical mechanics, based on a q-generalized Hagedorn theory for transverse momentum spectra and hard QCD scattering processes. QCD at largest center of mass energies predicts the entropic index to be $$q=\frac{{\bf{13}}}{{\bf{11}}}$$q = 13 11 . Here we show that the escort duality of the nonextensive thermodynamic formalism predicts an energy split of effective temperature given by Δ $${\boldsymbol{k}}{\boldsymbol{T}}{\boldsymbol{=}}{\boldsymbol{\pm }}\frac{{\bf{1}}}{{\bf{10}}}{\boldsymbol{k}}{{\boldsymbol{T}}}_{{\boldsymbol{H}}}{\boldsymbol{\approx }}{\boldsymbol{\pm }}{\bf{18}}$$k T = ± 1 10 k T H ≈ ± 18 MeV, where T _H is the Hagedorn temperature. We carefully analyse the measured data of the AMS-02 collaboration and provide evidence that the predicted temperature split is indeed observed, leading to a different energy dependence of the e⁺ and e⁻ spectral indices. We also observe a distinguished energy scale E^* ≈ 50 GeV where the e⁺ and e⁻ spectral indices differ the most. Linear combinations of the escort and non-escort q-generalized canonical distributions yield excellent agreement with the measured AMS-02 data in the entire energy range.

We analyse the probability densities of daily rainfall amounts at a variety of locations on Earth. The observed distributions of the amount of rainfall fit well to a q -exponential distribution with exponent q close to . We discuss possible reasons for the emergence of this power law. In contrast, the waiting time distribution between rainy days is observed to follow a near-exponential distribution. A careful investigation shows that a q -exponential with yields the best fit of the data. A Poisson process where the rate fluctuates slightly in a superstatistical way is discussed as a possible model for this. We discuss the extreme value statistics for extreme daily rainfall, which can potentially lead to flooding. This is described by Fréchet distributions as the corresponding distributions of the amount of daily rainfall decay with a power law. Looking at extreme event statistics of waiting times between rainy days (leading to droughts for very long dry periods) we obtain from the observed near-exponential decay of waiting times extreme event statistics close to Gumbel distributions. We discuss superstatistical dynamical systems as simple models in this context.

A reliable supply of electricity is critical for our modern society, and any large-scale disturbance of the electrical system causes substantial costs. In 2015, one overloaded transmission line caused a cascading failure in the Turkish power grid, affecting about $75\times {10}^6$ people. Here, we analyze the Turkish power grid and its dynamical and statistical properties. Specifically, we propose, for the first time, a model that incorporates the dynamical properties and the complex network topology of the Turkish power grid to investigate cascading failures. We find that the network damage depends on the load and generation distribution in the network with centralized generation being more susceptible to failures than a decentralized one. Furthermore, economic considerations on transmission line capacity are shown to conflict with stability.

We show that size-rank distributions with power-law decay (often only over a limited extent) observed in a vast number of instances in a widespread family of systems obey Tsallis statistics. The theoretical framework for these distributions is analogous to that of a nonlinear iterated map near a tangent bifurcation for which the Lyapunov exponent is negligible or vanishes. The relevant statistical-mechanical expressions associated with these distributions are derived from a maximum entropy principle with the use of two different constraints, and the resulting duality of entropy indexes is seen to portray physically relevant information. Whereas the value of the index a fixes the distribution's power-law exponent, that for the dual index 2 - alpha ensures the extensivity of the deformed entropy.