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Outdoor air pollution is estimated to cause a huge number of premature deaths worldwide, it catalyses many diseases on a variety of time scales, and it has a detrimental effect on the environment. In light of these impacts it is necessary to obtain a better understanding of the dynamics and statistics of measured air pollution concentrations, including temporal fluctuations of observed concentrations and spatial heterogeneities. Here we present an extensive analysis for measured data from Europe. The observed probability density functions (PDFs) of air pollution concentrations depend very much on the spatial location and on the pollutant substance. We analyse a large number of time series data from 3544 different European monitoring sites and show that the PDFs of nitric oxide ($NO$), nitrogen dioxide ($NO_{2}$) and particulate matter ($PM_{10}$ and $PM_{2.5}$) concentrations generically exhibit heavy tails. These are asymptotically well approximated by $q$-exponential distributions with a given entropic index $q$ and width parameter $\lambda$. We observe that the power-law parameter $q$ and the width parameter $\lambda$ vary widely for the different spatial locations. We present the results of our data analysis in the form of a map that shows which parameters $q$ and $\lambda$ are most relevant in a given region. A variety of interesting spatial patterns is observed that correlate to properties of the geographical region. We also present results on typical time scales associated with the dynamical behaviour., Comment: arXiv admin note: substantial text overlap with arXiv:2203.04296

With economic development, the complexity of infrastructure has increased drastically. Similarly, with the shift from fossil fuels to renewable sources of energy, there is a dire need for such systems that not only predict and forecast with accuracy but also help in understanding the process of predictions. Artificial intelligence and machine learning techniques have helped in finding out wellperforming solutions to different problems in the energy sector. However, the usage of state-of-the-art techniques like reinforcement learning is not surprisingly convincing. This paper discusses the application of reinforcement techniques in energy systems and how explanations of these models can be helpful

To reduce the heavy computational burden of reactive power optimization of distribution networks, machine learning models are receiving increasing attention. However, most machine learning models (e.g., neural networks) are usually considered as black boxes, making it challenging for power system operators to identify and comprehend potential biases or errors in the decision-making process of machine learning models. To address this issue, an explainable machine-learning framework is proposed to optimize the reactive power in distribution networks. Firstly, a Shapley additive explanation framework is presented to measure the contribution of each input feature to the solution of reactive power optimizations generated from machine learning models. Secondly, a model-agnostic approximation method is developed to estimate Shapley values, so as to avoid the heavy computational burden associated with direct calculations of Shapley values. The simulation results show that the proposed explainable framework can accurately explain the solution of the machine learning model-based reactive power optimization by using visual analytics, from both global and instance perspectives. Moreover, the proposed explainable framework is model-agnostic, and thus applicable to various models (e.g., neural networks)., Comment: It was submitted to the 23rd Power Systems Computation Conference (PSCC 2024) on Sept.2023

Accurate forecasts of electricity prices are crucial for the management of electric power systems and the development of smart applications. European electricity prices have risen substantially and became highly volatile after the Russian invasion of Ukraine, challenging established forecasting methods. Here, we present a Long Short-Term Memory (LSTM) model for the German-Luxembourg day-ahead electricity prices addressing these challenges. The recurrent structure of the LSTM allows the model to adapt to trends, while the joint prediction of both mean and standard deviation enables a probabilistic prediction. Using a physics-inspired approach - superstatistics - to derive an explanation for the statistics of prices, we show that the LSTM model faithfully reproduces both prices and their volatility., Comment: 5 pages, 1 figure

In the smart grid of the future, accurate load forecasts on the level of individual clients can help to balance supply and demand locally and to prevent grid outages. While the number of monitored clients will increase with the ongoing smart meter rollout, the amount of data per client will always be limited. We evaluate whether a Transformer load forecasting model benefits from a transfer learning strategy, where a global univariate model is trained on the load time series from multiple clients. In experiments with two datasets containing load time series from several hundred clients, we find that the global training strategy is superior to the multivariate and local training strategies used in related work. On average, the global training strategy results in 21.8% and 12.8% lower forecasting errors than the two other strategies, measured across forecasting horizons from one day to one month into the future. A comparison to linear models, multi-layer perceptrons and LSTMs shows that Transformers are effective for load forecasting when they are trained with the global training strategy.

A stable supply of electrical energy is essential for the functioning of our society. Therefore, the electrical power grid's operation and energy and balancing markets are subject to strict regulations. As the external technical, economic, or social influences on the power grid change, these regulations must also be constantly adapted. However, whether these regulatory changes lead to the intended results is not easy to assess. Could eXplainable Artificial Intelligence (XAI) models distinguish regulatory settings and support the understanding of the effects of these changes? In this article, we explore two examples of regulatory changes in the German energy markets for bulk electricity and for reserve power. We explore the splitting of the German-Austrian bidding zone and changes in the pricing schemes of the German balancing energy market. We find that boosted tree models and feedforward neural networks before and after a regulatory change differ in their respective parametrizations. Using Shapley additive explanations, we reveal model differences, e.g. in terms of feature importances, and identify key features of these distinct models. With this study, we demonstrate how XAI can be applied to investigate system changes in power systems., Comment: 7 pages, 3 figures

Quantifying the stability of an equilibrium is central in the theory of dynamical systems as well as in engineering and control. A comprehensive picture must include the response to both small and large perturbations, leading to the concepts of local (linear) and global stability. Here, we show how systems displaying Hopf bifurcations show contrarian results on these two aspects of stability: Global stability is large close to the point where the system loses its local stability altogether. We demonstrate this effect for an elementary model system, an anharmonic oscillator and a realistic model of power system dynamics with delayed control. Detailed investigations of the bifurcation explain the seeming paradox in terms of the location of the attractors relative to the equilibrium., Comment: 29 pages, 9 figures

Mitigating climate change requires a transition away from fossil fuels towards renewable energy. As a result, power generation becomes more volatile and options for microgrids and islanded power-grid operation are being broadly discussed. Therefore, studying the power grids of physical islands, as a model for islanded microgrids, is of particular interest when it comes to enhancing our understanding of power-grid stability. In the present paper, we investigate the statistical properties of the power-grid frequency of three island systems: Iceland, Ireland, and the Balearic Islands. We utilise a Fokker-Planck approach to construct stochastic differential equations that describe market activities, control, and noise acting on power-grid dynamics. Using the obtained parameters we create synthetic time series of the frequency dynamics. Our main contribution is to propose two extensions of stochastic power-grid frequency models and showcase the applicability of these new models to non-Gaussian statistics, as encountered in islands.

Electricity prices in liberalized markets are determined by the supply and demand for electric power, which are in turn driven by various external influences that vary strongly in time. In perfect competition, the merit order principle describes that dispatchable power plants enter the market in the order of their marginal costs to meet the residual load, i.e. the difference of load and renewable generation. Many market models implement this principle to predict electricity prices but typically require certain assumptions and simplifications. In this article, we present an explainable machine learning model for the prices on the German day-ahead market, which substantially outperforms a benchmark model based on the merit order principle. Our model is designed for the ex-post analysis of prices and thus builds on various external features. Using Shapley Additive exPlanation (SHAP) values, we can disentangle the role of the different features and quantify their importance from empiric data. Load, wind and solar generation are most important, as expected, but wind power appears to affect prices stronger than solar power does. Fuel prices also rank highly and show nontrivial dependencies, including strong interactions with other features revealed by a SHAP interaction analysis. Large generation ramps are correlated with high prices, again with strong feature interactions, due to the limited flexibility of nuclear and lignite plants. Our results further contribute to model development by providing quantitative insights directly from data., Comment: 13 pages, 6 figures

The operation of power systems is affected by diverse technical, economic and social factors. Social behaviour determines load patterns, electricity markets regulate the generation and weather-dependent renewables introduce power fluctuations. Thus, power system dynamics must be regarded as a non-autonomous system whose parameters vary strongly with time. However, the external driving factors are usually only available on coarse scales and the actual dependencies of the dynamic system parameters are generally unknown. Here, we propose a physics-inspired machine learning model that bridges the gap between large-scale drivers and short-term dynamics of the power system. Integrating stochastic differential equations and artificial neural networks, we construct a probabilistic model of the power grid frequency dynamics in Continental Europe. Its probabilistic prediction outperforms the daily average profile, which is an important benchmark. Using the integrated model, we identify and explain the parameters of the dynamical system from the data, which reveals their strong time-dependence and their relation to external drivers such as wind power feed-in and fast generation ramps. Finally, we generate synthetic time series from the model, which successfully reproduce central characteristics of the grid frequency such as their heavy-tailed distribution. All in all, our work emphasises the importance of modelling power system dynamics as a stochastic non-autonomous system with both intrinsic dynamics and external drivers., Comment: 21 pages, 5 figures

The ongoing energy transition requires power grid extensions to connect renewable generators to consumers and to transfer power among distant areas. The process of grid extension requires a large investment of resources and is supposed to make grid operation more robust. Yet, counter-intuitively, increasing the capacity of existing lines or adding new lines may also reduce the overall system performance and even promote blackouts due to Braess' paradox. Braess' paradox was theoretically modeled but not yet proven in realistically scaled power grids. Here, we present an experimental setup demonstrating Braess' paradox in an AC power grid and show how it constrains ongoing large-scale grid extension projects. We present a topological theory that reveals the key mechanism and predicts Braessian grid extensions from the network structure. These results offer a theoretical method to understand and practical guidelines in support of preventing unsuitable infrastructures and the systemic planning of grid extensions., Comment: 12 pages

Modelling, forecasting and overall understanding of the dynamics of the power grid and its frequency are essential for the safe operation of existing and future power grids. Much previous research was focused on large continental areas, while small systems, such as islands are less well-studied. These natural island systems are ideal testing environments for microgrid proposals and artificially islanded grid operation. In the present paper, we utilize measurements of the power grid frequency obtained in European islands: the Faroe Islands, Ireland, the Balearic Islands and Iceland and investigate how their frequency can be predicted, compared to the Nordic power system, acting as a reference. The Balearic islands are found to be particularly deterministic and easy to predict in contrast to hard-to-predict Iceland. Furthermore, we show that typically 2-4 weeks of data are needed to improve prediction performance beyond simple benchmarks., Comment: 17 pages

Complex systems, such as the power grid, are essential for our daily lives. Many complex systems display (multi-)fractal behavior, correlated fluctuations and power laws. Whether the power-grid frequency, an indicator about the balance on supply and demand in the electricity grid, also displays such complexity remains a mostly open question. Within the present article, we utilize highly resolved measurements to quantify the properties of the power-grid frequency. We show that below 1 second, the dynamics may fundamentally change, including a suddenly increasing power spectral density, emergence of multifractality and a change of correlation behavior. We provide a simplified stochastic model involving positively correlated noise to reproduce the observed dynamics, possibly linked to frequency dependent loads. Finally, we stress the need for high-quality measurements and discuss how we obtained the data analyzed here., Comment: 13 pages

The energy transition introduces more volatile energy sources into the power grids. In this context, power transfer between different synchronous areas through High Voltage Direct Current (HVDC) links becomes increasingly important. Such links can balance volatile generation by enabling long-distance transport or by leveraging their fast control behavior. Here, we investigate the interaction of power imbalances - represented through the power grid frequency - and power flows on HVDC links between synchronous areas in Europe. We use explainable machine learning to identify key dependencies and disentangle the interaction of critical features. Our results show that market-based HVDC flows introduce deterministic frequency deviations, which however can be mitigated through strict ramping limits. Moreover, varying HVDC operation modes strongly affect the interaction with the grid. In particular, we show that load-frequency control via HVDC links can both have control-like or disturbance-like impacts on frequency stability., Comment: 14 pages, 8 figures; submitted version

When Russia invaded Ukraine on the 24\textsuperscript{th} of February 2022, this led to many acts of solidarity with Ukraine, including support for its electricity system. Just 20 days after the invasion started, the Ukrainian and Moldovan power grids were synchronized to the Continental European power grid to provide stability to these grids. Here, we present an initial analysis of how this synchronization affected the statistics of the power grid frequency and cross-border flows of electric power within Continental Europe. We observe faster inter-area oscillations, an increase in fluctuations and changes in the cross-border flows in and out of Ukraine and surrounding countries as an effect of the synchronization with Continental Europe. Overall these changes are small such that the now connected system can be considered as stable as before the synchronization., Comment: 7 pages, 6 figures

Air pollution is one of the leading causes of death globally, and continues to have a detrimental effect on our health. In light of these impacts, an extensive range of statistical modelling approaches has been devised in order to better understand air pollution statistics. However, the time-varying statistics of different types of air pollutants are far from being fully understood. The observed probability density functions (PDFs) of concentrations depend very much on the spatial location and on the pollutant substance. In this paper, we analyse a large variety of data from 3544 different European monitoring sites and show that the PDFs of nitric oxide ($NO$), nitrogen dioxide ($NO2$) and particulate matter ($PM10$ and $PM2.5$) concentrations generically exhibit heavy tails and are asymptotically well approximated by $q$-exponential distributions with a given width parameter $\lambda$. We observe that the power-law parameter $q$ and the width parameter $\lambda$ vary widely for the different spatial locations. For each substance, we find different patterns of parameter clouds in the $(q, \lambda)$ plane. These depend on the type of pollutants and on the environmental characteristics (urban/suburban/rural/traffic/industrial/background). This means the effective statistical physics description of air pollution exhibits a strong degree of spatial heterogeneity., Comment: 13 pages, 5 figures

The design and operation of modern energy systems are heavily influenced by time-dependent and uncertain parameters, e.g., renewable electricity generation, load-demand, and electricity prices. These are typically represented by a set of discrete realizations known as scenarios. A popular scenario generation approach uses deep generative models (DGM) that allow scenario generation without prior assumptions about the data distribution. However, the validation of generated scenarios is difficult, and a comprehensive discussion about appropriate validation methods is currently lacking. To start this discussion, we provide a critical assessment of the currently used validation methods in the energy scenario generation literature. In particular, we assess validation methods based on probability density, auto-correlation, and power spectral density. Furthermore, we propose using the multifractal detrended fluctuation analysis (MFDFA) as an additional validation method for non-trivial features like peaks, bursts, and plateaus. As representative examples, we train generative adversarial networks (GANs), Wasserstein GANs (WGANs), and variational autoencoders (VAEs) on two renewable power generation time series (photovoltaic and wind from Germany in 2013 to 2015) and an intra-day electricity price time series form the European Energy Exchange in 2017 to 2019. We apply the four validation methods to both the historical and the generated data and discuss the interpretation of validation results as well as common mistakes, pitfalls, and limitations of the validation methods. Our assessment shows that no single method sufficiently characterizes a scenario but ideally validation should include multiple methods and be interpreted carefully in the context of scenarios over short time periods., Comment: 20 pages, 8 figures, 2 tables

The transition to a renewable energy system poses challenges for power grid operation and stability. Secondary control is key in restoring the power system to its reference following a disturbance. Underestimating the necessary control capacity may require emergency measures, such as load shedding. Hence, a solid understanding of the emerging risks and the driving factors of control is needed. In this contribution, we establish an explainable machine learning model for the activation of secondary control power in Germany. Training gradient boosted trees, we obtain an accurate description of control activation. Using SHapely Additive exPlanation (SHAP) values, we investigate the dependency between control activation and external features such as the generation mix, forecasting errors, and electricity market data. Thereby, our analysis reveals drivers that lead to high reserve requirements in the German power system. Our transparent approach, utilizing open data and making machine learning models interpretable, opens new scientific discovery avenues., Comment: 8 pages, 6 figures

The ongoing energy transition challenges the stability of the electrical power system. Stable operation of the electrical power grid requires both the voltage (amplitude) and the frequency to stay within operational bounds. While much research has focused on frequency dynamics and stability, the voltage dynamics has been neglected. Here, we study frequency and voltage stability in the case of simple networks via linear stability and bulk analysis. In particular, our linear stability analysis of the network shows that the frequency secondary control guarantees the stability of a particular electric network. Even more interesting, while we only consider secondary frequency control, we observe a stabilizing effect on the voltage dynamics, especially in our numerical bulk analysis., Comment: 10 pages, 7 figures

Superstatistics is a general method from nonequilibrium statistical physics which has been applied to a variety of complex systems, ranging from hydrodynamic turbulence to traffic delays and air pollution dynamics. Here, we investigate water quality time series (such as dissolved oxygen concentrations and electrical conductivity) as measured in rivers, and provide evidence that they exhibit superstatistical behaviour. Our main example are time series as recorded in the river Chess in South East England. Specifically, we use seasonal detrending and empirical mode decomposition (EMD) to separate trends from fluctuations for the measured data. With either detrending method, we observe heavy-tailed fluctuation distributions, which are well described by a log-normal superstatistics for dissolved oxygen. Contrarily, we find a double peaked non-standard superstatistics for the electrical conductivity data, which we model using two combined $\chi^2$-distributions.

Deterministic frequency deviations (DFDs) critically affect power grid frequency quality and power system stability. A better understanding of these events is urgently needed as frequency deviations have been growing in the European grid in recent years. DFDs are partially explained by the rapid adjustment of power generation following the intervals of electricity trading, but this intuitive picture fails especially before and around noonday. In this article, we provide a detailed analysis of DFDs and their relation to external features using methods from explainable Artificial Intelligence. We establish a machine learning model that well describes the daily cycle of DFDs and elucidate key interdependencies using SHapley Additive exPlanations (SHAP). Thereby, we identify solar ramps as critical to explain patterns in the Rate of Change of Frequency (RoCoF)., Comment: 7 pages, 4 figures

Stable operation of the electrical power system requires the power grid frequency to stay within strict operational limits. With millions of consumers and thousands of generators connected to a power grid, detailed human-build models can no longer capture the full dynamics of this complex system. Modern machine learning algorithms provide a powerful alternative for system modelling and prediction, but the intrinsic black-box character of many models impedes scientific insights and poses severe security risks. Here, we show how eXplainable AI (XAI) alleviates these problems by revealing critical dependencies and influences on the power grid frequency. We accurately predict frequency stability indicators (such as RoCoF and Nadir) for three major European synchronous areas and identify key features that determine the power grid stability. Load ramps, specific generation ramps but also prices and forecast errors are central to understand and stabilize the power grid., Comment: 26 pages, 19 figures; updated comments; this is the initially submitted version, for the final version see journal link

Monitoring and modelling the power grid frequency is key to ensuring stability in the electrical power system. Many tools exist to investigate the detailed deterministic dynamics and especially the bulk behaviour of the frequency. However, far less attention has been paid to its stochastic properties, and there is a need for a cohesive framework that couples both short-time scale fluctuations and bulk behaviour. Moreover, commonly assumed uncorrelated stochastic noise is predominantly employed in modelling in energy systems. In this publication, we examine the stochastic properties of six synchronous power-grid frequency recording with high-temporal resolution of the Nordic Grid from September 2013, focusing on the increments of the frequency recordings. We show that these increments follow non-Gaussian statistics and display spatial and temporal correlations. Furthermore, we report two different physical synchronisation phenomena: a very short timescale phase synchronisation ($<2\,$s) followed by a slightly larger timescale amplitude synchronisation ($2\,$s-$5\,$s). Overall, these results provide guidance on how to model fluctuations in power systems., Comment: 11 pages, 6 figures

Power-grid systems constitute one of the most complex man-made spatially extended structures. These operate with strict operational bounds to ensure synchrony across the grid. This is particularly relevant for power-grid frequency, which operates strictly at $50\,$Hz ($60\,$Hz). Nevertheless, small fluctuations around the mean frequency are present at very short time scales $<2$ seconds and can exhibit highly complex spatio-temporal behaviour. Here we apply superstatistical data analysis techniques to measured frequency fluctuations in the Nordic Grid. We study the increment statistics and extract the relevant time scales and superstatistical distribution functions from the data. We show that different synchronous recordings of power-grid frequency have very distinct stochastic fluctuations with different types of superstatistics at different spatial locations, and with transitions from one superstatistics to another when the time lag of the increment statistics is changed., Comment: 11 pages, 6 figures

Frequency fluctuations in power grids, caused by unpredictable renewable energy sources, consumer behavior and trading, need to be balanced to ensure stable grid operation. Standard smart grid solutions to mitigate large frequency excursions are based on centrally collecting data and give rise to security and privacy concerns. Furthermore, control of fluctuations is often tested by employing Gaussian perturbations. Here, we demonstrate that power grid frequency fluctuations are in general non-Gaussian, implying that large excursions are more likely than expected based on Gaussian modeling. We consider real power grid frequency measurements from Continental Europe and compare them to stochastic models and predictions based on Fokker-Planck equations. Furthermore, we review a decentral smart grid control scheme to limit these fluctuations. In particular, we derive a scaling law of how decentralized control actions reduce the magnitude of frequency fluctuations and demonstrate the power of these theoretical predictions using a test grid. Overall, we find that decentral smart grid control may reduce grid frequency excursions due to both Gaussian and non-Gaussian power fluctuations and thus offers an alternative pathway for mitigating fluctuation-induced risks.

To ensure reliable operation of power grids, their frequency shall stay within strict bounds. Multiple sources of disturbances cause fluctuations of the grid frequency, ranging from changing demand over volatile feed-in to energy trading. Here, we analyze frequency time series from the continental European grid in 2011 and 2017 as a case study to isolate the impact of trading. We find that trading at typical trading intervals such as full hours modifies the frequency fluctuation statistics. While particularly large frequency deviations in 2017 are not as frequent as in 2011, large deviations are more likely to occur shortly after the trading instances. A comparison between the two years indicates that trading at shorter intervals might be beneficial for frequency quality and grid stability, because particularly large fluctuations are substantially diminished. Furthermore, we observe that the statistics of the frequency fluctuations do not follow Gaussian distributions but are better described using heavy-tailed and asymmetric distributions, for example L\'evy-stable distributions. Comparing intervals without trading to those with trading instances indicates that frequency deviations near the trading times are distributed more widely and thus extreme deviations are orders of magnitude more likely. Finally, we briefly review a stochastic analysis that allows a quantitative description of power grid frequency fluctuations.

The aviation industry is of great importance for a globally connected economy. Customer satisfaction with airlines and airport performance is considerably influenced by how much flights are delayed. But how should the delay be quantified with thousands of flights for each airport and airline? Here, we present a statistical analysis of arrival delays at several UK airports between 2018 and 2020. We establish a procedure to compare both mean delay and extreme events among airlines and airports, identifying a power-law decay of large delays. Furthermore, we note drastic changes in plane delay statistics during the COVID-19 pandemic. Finally, we find that delays are described by a superposition of simple distributions, leading to a superstatistics., Comment: 14 pages, 10 figures

The dynamics of power consumption constitutes an essential building block for planning and operating energy systems based on renewable energy supply. Whereas variations in the dynamics of renewable energy generation are reasonably well studied, a deeper understanding of short and long term variations in consumption dynamics is still missing. Here, we analyse highly resolved residential electricity consumption data of Austrian and German households and propose a generally applicable methodology for extracting both the average demand profiles and the demand fluctuations purely from time series data. The analysis reveals that demand fluctuations of individual households are skewed and consistently highly intermittent. We introduce a stochastic model to quantitatively capture such real-world fluctuations. The analysis indicates in how far the broadly used standard load profile (SLP) may be is insufficient to describe the key characteristics observed. These results offer a better understanding of demand dynamics, in particular its fluctuations, and provide general tools for disentangling mean demand and fluctuations for any given system. The insights on the demand dynamics may support planning and operating future-compliant (micro) grids in maintaining supply-demand balance.

Air pollution is among the highest contributors to mortality worldwide, especially in urban areas. During spring 2020, many countries enacted social distancing measures in order to slow down the ongoing COVID-19 pandemic. A particularly drastic measure, the 'lockdown', urged people to stay at home and thereby prevent new COVID-19 infections during the first (2020) and second wave (2021) of the pandemic. In turn, it also reduced traffic and industrial activities. But how much did these lockdown measures improve air quality in large cities, and are there differences in how air quality was affected? Here, we analyse data from two megacities: London as an example for Europe and Delhi as an example for Asia. We consider data during first and second wave lockdowns and compare them to 2019 values. Overall, we find a reduction in almost all air pollutants with intriguing differences between the two cities except Delhi in 2021. In London, despite smaller average concentrations, we still observe high-pollutant states and an increased tendency towards extreme events (a higher kurtosis of the probability density during lockdown) during 2020 and low pollution levels during 2021. For Delhi, we observe a much stronger decrease of pollution concentrations, including high pollution states during 2020 and higher pollution levels in 2021. These results could help to design policies to improve long-term air quality in megacities., Comment: Open code available here https://osf.io/jfw7n/

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 data base 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 data base and analyses constitute a step towards more shared, collaborative, Comment: 11 pages, 9 figures. Supplemental Material. Associated database: https://osf.io/by5hu/

The transformation of the electrical energy system attracts much attention in diverse research communities. Novel approaches for modeling, control, and power grid architectures are widely proposed. Data from actual power system operation are therefore critical to evaluate models and to analyze real-world scenarios. However, they are rarely available. In the present paper, we introduce a precisely time-stamped data set of power grid frequency measurements. It covers twelve synchronous areas of different sizes in one-second resolution. The contribution includes synchronized measurements within the Continental European synchronous area acquired in Portugal, Germany, and Turkey, maximizing the geographical span. Finally, we provide excerpts of the underlying waveform. Data were collected using a self-developed measurement instrument, the Electrical Data Recorder (EDR), connected mostly to conventional power sockets. We close our description with a discussion on measurement error and data quality., Comment: 28 pages, 10 figures. Data associated with this paper are located here: https://osf.io/by5hu/

The power grid frequency is the central observable in power system control, as it measures the balance of electrical supply and demand. A reliable frequency forecast can facilitate rapid control actions and may thus greatly improve power system stability. Here, we develop a weighted-nearest-neighbor (WNN) predictor to investigate how predictable the frequency trajectories are. Our forecasts for up to one hour are more precise than averaged daily profiles and could increase the efficiency of frequency control actions. Furthermore, we gain an increased understanding of the specific properties of different synchronous areas by interpreting the optimal prediction parameters (number of nearest neighbors, the prediction horizon, etc.) in terms of the physical system. Finally, prediction errors indicate the occurrence of exceptional external perturbations. Overall, we provide a diagnostics tool and an accurate predictor of the power grid frequency time series, allowing better understanding of the underlying dynamics., Comment: 12 pages, 8 figures, Supplementary material on data preparation

Atomic nuclei appearing in cosmic rays are typically classified as primary or secondary. However, a better understanding of their origin and propagation properties is still necessary. We analyse the flux of primary (He, C, O) and secondary nuclei (Li, Be, B) detected with rigidity (momentum/charge) between 2 GV and 3 TV by the Alpha Magnetic Spectrometer (AMS) on the International Space Station. We show that $q$-exponential distribution functions, as motivated by generalized versions of statistical mechanics with temperature fluctuations, provide excellent fits for the measured flux of all nuclei considered. Primary and secondary fluxes reveal a universal dependence on kinetic energy per nucleon for which the underlying energy distribution functions are solely distinguished by their effective degrees of freedom. All given spectra are characterized by a universal mean temperature parameter $\sim$ 200 MeV which agrees with the Hagedorn temperature. Our analysis suggests that QCD scattering processes together with nonequilibrium temperature fluctuations provide a plausible explanation for the observed universality in cosmic ray energy spectra. Our analysis suggests that QCD scattering processes together with nonequilibrium temperature fluctuations imprint universally onto the measured cosmic ray spectra, and produce a similar shape of energy spectra as high energy collider experiments on the Earth., Comment: 16 pages, 3 figures

Air pollution by Nitrogen Oxides (NOx) is a major concern in large cities as it has severe adverse health effects. However, the statistical properties of air pollutants are not fully understood. Here, we use methods borrowed from nonequilibrium statistical mechanics to construct suitable superstatistical models for air pollution statistics. In particular, we analyze time series of Nitritic Oxide ($NO$) and Nitrogen Dioxide ($NO_2$) concentrations recorded at several locations throughout Greater London. We find that the probability distributions of concentrations have heavy tails and that the dynamics is well-described by $\chi^2$ superstatistics for $NO$ and inverse $\chi^2$ superstatistics for $NO_2$. Our results can be used to give precise risk estimates of high-pollution situations and pave the way to mitigation strategies., Comment: 10 pages, including the appendices

The frequency constitutes a key state variable of electrical power grids. However, as the frequency is subject to several sources of fluctuations, ranging from renewable volatility to demand fluctuations and dispatch, it is strongly dynamic. Yet, the statistical and stochastic properties of the frequency fluctuation dynamics are far from fully understood. Here, we analyse properties of power grid frequency trajectories recorded from different synchronous regions. We highlight the non-Gaussian and still approximately Markovian nature of the frequency statistics. Further, we find that the frequency displays significant fluctuations exactly at the time intervals of regulation and trading, confirming the need of having a regulatory and market design that respects the technical and dynamical constraints in future highly renewable power grids. Finally, employing a recently proposed synthetic model for the frequency dynamics, we combine our statistical and stochastic analysis and analyse in how far dynamically modelled frequency properties match the ones of real trajectories.

The energy system is rapidly changing to accommodate the increasing number of renewable generators and the general transition towards a more sustainable future. Simultaneously, business models and market designs evolve, affecting power-grid operation and power-grid frequency. Problems raised by this ongoing transition are increasingly addressed by transdisciplinary research approaches, ranging from purely mathematical modelling to applied case studies. These approaches require a stochastic description of consumer behaviour, fluctuations by renewables, market rules, and how they influence the stability of the power-grid frequency. Here, we introduce an easy-to-use, data-driven, stochastic model for the power-grid frequency and demonstrate how it reproduces key characteristics of the observed statistics of the Continental European and British power grids. We offer executable code and guidelines on how to use the model on any power grid for various mathematical or engineering applications., Comment: 14 pages, 10 figures, Supplemental Material with code

Due to the emergence of new technologies, the whole electricity system is undergoing transformations on a scale and pace never observed before. The decentralization of energy resources and the smart grid have forced utility services to rethink their relationships with customers. Demand response (DR) seeks to adjust the demand for power instead of adjusting the supply. However, DR business models rely on customer participation and can only be effective when large numbers of customers in close geographic vicinity, e.g., connected to the same transformer, opt in. Here, we introduce a model for the dynamics of service adoption on a two-layer multiplex network: the layer of social interactions among customers and the power-grid layer connecting the households. While the adoption process - based on peer-to-peer communication - runs on the social layer, the time-dependent recovery rate of the nodes depends on the states of their neighbors on the power-grid layer, making an infected node surrounded by infectious ones less keen to recover. Numerical simulations of the model on synthetic and real-world networks show that a strong local influence of the customers' actions leads to a discontinuous transition where either none or all the nodes in the network are infected, depending on the infection rate and social pressure to adopt. We find that clusters of early adopters act as points of high local pressure, helping maintaining adopters, and facilitating the eventual adoption of all nodes. This suggests direct marketing strategies on how to efficiently establish and maintain new technologies such as DR schemes., Comment: 14 pages, 6 figures

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 million 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., Comment: 8 pages, 5 figures

Mitigating climate change demands a transition towards renewable electricity generation, with wind power being a particularly promising technology. Long periods either of high or of low wind therefore essentially define the necessary amount of storage to balance the power system. While the general statistics of wind velocities have been studied extensively, persistence (waiting) time statistics of wind is far from well understood. Here, we investigate the statistics of both high- and low-wind persistence. We find heavy tails and explain them as a superposition of different wind conditions, requiring $q$-exponential distributions instead of exponential distributions. Persistent wind conditions are not necessarily caused by stationary atmospheric circulation patterns nor by recurring individual weather types but may emerge as a combination of multiple weather types and circulation patterns. Understanding wind persistence statistically and synoptically, may help to ensure a reliable and economically feasible future energy system, which uses a high share of wind generation., Comment: 15 pages, 8 figures, Supplemental Material available here: https://static-content.springer.com/esm/art%3A10.1038%2Fs41598-019-56286-1/MediaObjects/41598_2019_56286_MOESM1_ESM.pdf

Multiple types of fluctuations impact the collective dynamics of power grids and thus challenge their robust operation. Fluctuations result from processes as different as dynamically changing demands, energy trading, and an increasing share of renewable power feed-in. Here we analyze principles underlying the dynamics and statistics of power grid frequency fluctuations. Considering frequency time series for a range of power grids, including grids in North America, Japan and Europe, we find a substantial deviation from Gaussianity best described as L\'evy-stable and q-Gaussian distributions. We present a coarse framework to analytically characterize the impact of arbitrary noise distributions as well as a superstatistical approach which systematically interprets heavy tails and skewed distributions. We identify energy trading as a substantial contribution to today's frequency fluctuations and effective damping of the grid as a controlling factor enabling reduction of fluctuation risks, with enhanced effects for small power grids., Comment: Accepted version of Nature Energy article (see doi for original publication)

Robust operation of power transmission grids is essential for most of today's technical infrastructure and our daily life. Adding renewable generation to power grids requires grid extensions and sophisticated control actions on different time scales to cope with short-term fluctuations and long-term power imbalance. Braess' paradox constitutes a counterintuitive collective phenomenon that occurs if adding new transmission line capacity to a network increases loads on other lines, effectively reducing the system's performance and potentially even entirely removing its operating state. Combining simple analytical considerations with numerical investigations on a small sample network, we here study dynamical consequences of secondary control in AC power grid models. We demonstrate that sufficiently strong control not only implies dynamical stability of the system but may also cure Braess' paradox. Our results highlight the importance of demand control in conjunction with grid topology for stable operation and reveal a new functional benefit of secondary control., Comment: 11 pages, 6 figures, under review. Link to final publication will be provided after acceptance

As the energy transition transforms power grids across the globe, it poses several challenges regarding grid design and control. In particular, high levels of intermittent renewable generation complicate the task of continuously balancing power supply and demand, requiring sufficient control actions. Although there exist several proposals to control the grid, most of them have not demonstrated to be cost efficient in terms of optimal control theory. Here, we mathematically formulate an optimal centralized (therefore non-local) control problem for stable operation of power grids and determine the minimal amount of active power necessary to guarantee a stable service within the operational constraints, minimizing a suitable cost function at the same time. This optimal control can be used to benchmark control proposals and we demonstrate this benchmarking process by investigating the performance of three distributed controllers, two of which are fully decentralized, that have been recently studied in the physics and power systems engineering literature. Our results show that cost efficient controllers distribute the controlled response amongst all nodes in the power grid. Additionally, superior performance can be achieved by incorporating sufficient information about the disturbance causing the instability. Overall, our results can help design and benchmark secure and cost-efficient controllers., Comment: Major revision with more descriptive third-order model, updated simulations for a new four-node network, and two more controllers added to the comparison. 14 pages, two columns, 6 figures with 19 subfigures in total. Results from previous version with two-node and five-node networks are still applicable