WISSENSCHAFTLICHE PUBLIKATIONEN (gesamt 2014-2017: >60)
Highlight Paper: Propagation of wind power induced fluctuations in power grids H. Haehne, K. Schmietendorf, S. Tamrakar, J. Peinke, S. Kettemann Phys. Rev. E Rapid Communications 99, 050301 (2019). Stability of Synchrony against Local Intermittent Fluctuations in Tree-like Power Grids S. Auer, F. Hellmann, M. Krause, J. Kurths Chaos 27, 127003 (2017). - 2019 - A Generalized Linear Response Theory of Complex Networks with an Application to Renewable Fluctuations in Microgrids. A. Plietzsch, S. Auer, J. Kurths, F. Hellmann arXiv:1903.09585 (2019). Abstract: We study the stability of deterministic systems, given sequences of large, jump-like perturbations. In this work we study the general linear response theory for the distribution of energy fluctuations through complex networks. We develop the response equations for oscillators coupled on arbitrary, directed and weighted networks, when subjected to stationary fluctuations with arbitrary power spectra. Guided by the case study of network models for the distributed control and stabilization of turbulent renewable energy fluctuations in power grids, we then develop approximations that capture the most impactful interactions between intrinsic network modes and typical fluctuations found in renewable energies. These cover an intermediate resonant regime where the fluctuations are neither slow enough to cause a homogeneous response of the whole system, nor fast enough to be localized on the network. Applying these analytic approximations to the question which nodes in a microgrid are particularly vulnerable to fluctuations, we are able to give analytic explanations and expressions for the previously numerically observed network patterns in vulnerability. We see that these effects can only be explained by taking the losses on the lines, and the resulting asymmetry in the effective weighted graph Laplacian, into account. These structural asymmetries give rise to a dynamical asymmetry between nodes that cause a strong response when perturbed (troublemaker nodes), and nodes that always respond strongly whenever the network is somewhere perturbed (excitable nodes). For the important special case of tree-like networks we derive a simple relation for troublemaker nodes stating that fluctuations are enhanced when going upstream. The general theory also opens the door to future investigations into the stabilization of networks under correlated distributed fluctuations. Counter-intuitive behaviour of energy system models under CO2 caps and prices. J. Weber, H. Heinrichs, B. Gillessen, T. Brown, J. Hörsch, D. Maybe, D. Witthaut Energy, Volume 170, Pages 22-30, https://doi.org/10.1016/j.energy.2018.12.052 (2019). Abstract: The mitigation of climate change requires a fundamental transition of the energy system. Affordability, reliability and the reduction of greenhouse gas emissions constitute central but often conflicting targets for this energy transition. Against this context, we reveal limitations and counter-intuitive results in the model-based optimization of energy systems, which are often applied for policy advice. When system costs are minimized in the presence of a CO2 cap, efficiency gains free a part of the CO2 cap, allowing cheap technologies to replace expensive low-emission technologies. Even more striking results are observed in a setup where emissions are minimized in the presence of a budget constraint. Increasing CO2 prices can oust clean, but expensive technologies out of the system, and eventually lead to higher emissions. These effects robustly occur in models of different scope and complexity. Hence, extreme care is necessary in the application of energy system optimization models to avoid misleading policy advice. Dynamic Vulnerability in Oscillatory Networks and Power Grids. X. Zhang, C. Ma, M. Timme arXiv:1908.00957 (2019). Abstract: Recent work found distributed resonances in driven oscillator networks and AC power grids. The emerging dynamic resonance patterns are highly heterogeneous and nontrivial, depending jointly on the driving frequency, the interaction topology of the network and the node or nodes driven. Identifying which nodes are most susceptible to dynamic driving and may thus make the system as a whole vulnerable to external input signals, however, remains a challenge. Here we propose an easy-to-compute Dynamic Vulnerability Index (DVI) for identifying those nodes that exhibit largest amplitude responses to dynamic driving signals with given power spectra and thus are most vulnerable. The DVI is based on linear response theory, as such generic, and enables robust predictions. It thus shows potential for a wide range of applications across dynamically driven networks, for instance for identifying the vulnerable nodes in power grids driven by fluctuating inputs from renewable energy sources and fluctuating power output to households. Fluctuation-induced distributed resonances in oscillatory networks. X. Zhang, S. Hallerberg, M. Matthiae, D. Witthaut, M. Timme Science Advances 31 Jul 2019: Vol. 5, no. 7, eaav1027, DOI: 10.1126/sciadv.aav1027 (2019). Abstract: Across physics, biology, and engineering, the collective dynamics of oscillatory networks often evolve into self-organized operating states. How such networks respond to external fluctuating signals fundamentally underlies their function, yet is not well understood. Here, we present a theory of dynamic network response patterns and reveal how distributed resonance patterns emerge in oscillatory networks once the dynamics of the oscillatory units become more than one-dimensional. The network resonances are topology specific and emerge at an intermediate frequency content of the input signals, between global yet homogeneous responses at low frequencies and localized responses at high frequencies. Our analysis reveals why these patterns arise and where in the network they are most prominent. These results may thus provide general theoretical insights into how fluctuating signals induce response patterns in networked systems and simultaneously help to develop practical guiding principles for real-world network design and control. Impact of Network Topology on the Stability of DC Power Grids. J. F. Wienand, D. Eidmann, J. Kremers, J. Heitzig, F. Hellmann, J. Kurths arXiv:1905.06429 (2019). Abstract: We probe the stability of Watts-Strogatz DC power grids, in which droop-controlled producers, constant power load consumers and power lines obey Kirchhoff's circuit laws. The concept of survivability is employed to evaluate the system's response to voltage perturbations in dependence on the network topology. Following a fixed point analysis of the power grid model, we extract three main indicators of stability through numerical studies: the share of producers in the network, the node degree and the magnitude of the perturbation. Based on our findings, we investigate the local dynamics of the perturbed system and derive explicit guidelines for the design of resilient DC power grids. Depending on the imposed voltage and current limits, the stability is optimized for low node degrees or a specific share of producers. Non-Local Impact of Link Failures in Linear Flow Networks J. Strake, F. Kaiser, F. Basiri, H. Ronellenfitsch, D. Witthaut New J. Phys. 21 053009 (2019). Abstract: The failure of a single link can degrade the operation of a supply network up to the point of complete collapse. Yet, the interplay between network topology and locality of the response to such damage is poorly understood. Here, we study how topology affects the redistribution of flow after the failure of a single link in linear flow networks with a special focus on power grids. In particular, we analyze the decay of flow changes with distance after a link failure and map it to the field of an electrical dipole for lattice-like networks. The corresponding inverse-square law is shown to hold for all regular tilings. For sparse networks, a long-range response is found instead. In the case of more realistic topologies, we introduce a rerouting distance, which captures the decay of flow changes better than the traditional geodesic distance. Finally, we are able to derive rigorous bounds on the strength of the decay for arbitrary topologies that we verify through extensive numerical simulations. Our results show that it is possible to forecast flow rerouting after link failures to a large extent based on purely topological measures and that these effects generally decay with distance from the failing link. They might be used to predict links prone to failure in supply networks such as power grids and thus help to construct grids providing a more robust and reliable power supply. Propagation of wind power induced fluctuations in power grids. H. Haehne, K. Schmietendorf, S. Tamrakar, J. Peinke, S. Kettemann Phys. Rev. E Rapid Communications 99, 050301 (2019). Abstract: We probe the stability of Watts-Strogatz DC power grids, in which droop-controlled producers, constant power load consumers and power lines obey Kirchhoff's circuit laws. Renewable generators perturb the electric power grid with heavily non-Gaussian and time correlated fluctuations. While changes in generated power on timescales of minutes and hours are compensated by frequency control measures, we report subsecond distribution grid frequency measurements with local non-Gaussian fluctuations which depend on the magnitude of wind power generation in the grid. Motivated by such experimental findings, we simulate the subsecond grid frequency dynamics by perturbing the power grid, as modeled by a network of phase coupled nonlinear oscillators, with synthetically generated wind power feed-in time series. We derive a linear response theory and obtain analytical results for the variance of frequency increment distributions. We find that the variance of short-term fluctuations decays, for large inertia, exponentially with distance to the feed-in node, in agreement with numerical results both for a linear chain of nodes and the German transmission grid topology. In sharp contrast, the kurtosis of frequency increments is numerically found to decay only slowly, not exponentially, in both systems, indicating that the non-Gaussian shape of frequency fluctuations persists over long ranges. The winner takes it all -- How to win network globalization C. Han, D. Witthaut, M. Timme, M.Schröder https://arxiv.org/abs/1906.06092, Submitted to PLOS ONE (2019). Abstract: Quantifying the importance and power of individual nodes depending on their position in socio-economic networks constitutes a problem across a variety of applications. Examples include the reach of individuals in (online) social networks, the importance of individual banks or loans in financial networks, the relevance of individual companies in supply networks, and the role of traffic hubs in transport networks. Which features characterize the importance of a node in a trade network during the emergence of a globalized, connected market? Here we analyze a model that maps the evolution of trade networks to a percolation problem. In particular, we focus on the influence of topological features of the node within the trade network. Our results reveal that an advantageous position with respect to different length scales determines the success of a node at different stages of globalization and depending on the speed of globalization. - 2018 - Bounding the first exit from the basin: Independence times and finite-time basin stability. P. Schultz, F. Hellmann, K. N. Webster, J. Kurths Chaos: An Interdisciplinary Journal of Nonlinear Science, Volume 28, Issue 4, 10.1063/1.5013127 (2018). Abstract: We study the stability of deterministic systems, given sequences of large, jump-like perturbations. Our main result is the derivation of a lower bound for the probability of the system to remain in the basin, given that perturbations are rare enough. This bound is efficient to evaluate numerically. To quantify rare enough, we define the notion of the independence time of such a system. This is the time after which a perturbed state has probably returned close to the attractor, meaning that subsequent perturbations can be considered separately. The effect of jump-like perturbations that occur at least the independence time apart is thus well described by a fixed probability to exit the basin at each jump, allowing us to obtain the bound. To determine the independence time, we introduce the concept of finite-time basin stability, which corresponds to the probability that a perturbed trajectory returns to an attractor within a given time. The independence time can then be determined as the time scale at which the finite-time basin stability reaches its asymptotic value. Besides that, finite-time basin stability is a novel probabilistic stability measure on its own, with potential broad applications in complex systems. Cost optimal scenarios of a future highly renewable European electricity system: Exploring the in uence of weather data, cost parameters and policy constraints. D. Schlachtberger, T. Brown, M. Schafer, S. Schramm, M. Greiner Energy, Volume 163, Pages 100-114, https://doi.org/10.1016/j.energy.2018.08.070 (2018). Abstract: Cost optimal scenarios derived from models of a highly renewable electricity system depend on the specific input data, cost assumptions and system constraints. Here this influence is studied using a techno-economic optimisation model for a networked system of 30 European countries, taking into account the capacity investment and operation of wind, solar, hydroelectricity, natural gas power generation, transmission, and different storage options. A considerable robustness of total system costs to the input weather data and to moderate changes in the cost assumptions is observed. Flat directions in the optimisation landscape around cost-optimal configurations often allow system planners to choose between different technology options without a significant increase in total costs, for instance by replacing onshore with offshore wind power capacity in case of public acceptance issues. Exploring a range of carbon dioxide emission limits shows that for scenarios with moderate transmission expansion, a reduction of around 57% compared to 1990 levels is already cost optimal. For stricter carbon dioxide limits, power generated from gas turbines is at first replaced by generation from increasing renewable capacities. Non-hydro storage capacities are only built for low-emission scenarios, in order to provide the necessary flexibility to meet peaks in the residual load. Curing Braess' Paradox by Secondary Control in Power Grids: E. B. Tchawou Tchuisseu, D. Gomila, P. Colet, D. Witthaut, M. Timme, B. Schaefer https://doi.org/10.1088/1367-2630/aad490; New J. Phys. 20, 083005 (2018). Abstract: The 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 destabilizing 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 the grid topology to ensure stable operation of the grid and reveal a new functional benefit of secondary control. Dynamically induced cascading failures in supply networks. B. Schaefer, D. Witthaut, M. Timme, V. Latora https://doi.org/10.1038/s41467-018-04287-5, Nature Communications 9, 1975 (2018). Abstract: Reliable functioning of infrastructure networks is essential for our modern society. Cascading failures are the cause of most large-scale network outages. Although cascading failures often exhibit dynamical transients, the modeling of cascades has so far mainly focused on the analysis of sequences of steady states. In this article, we focus on electrical transmission networks and introduce a framework that takes into account both the event-based nature of cascades and the essentials of the network dynamics. We find that transients of the order of seconds in the flows of a power grid play a crucial role in the emergence of collective behaviors. We finally propose a forecasting method to identify critical lines and components in advance or during operation. Overall, our work highlights the relevance of dynamically induced failures on the synchronization dynamics of national power grids of different European countries and provides methods to predict and model cascading failures. Extreme prices in electricity balancing markets from an approach of statistical physics. M. Mureddu, H. Meyer-Ortmanns Physica A 490, 1324 (2018). Abstract: An increase in energy production from renewable energy sources is viewed as a crucial achievement in most industrialized countries. The higher variability of power production via renewables leads to a rise in ancillary service costs over the power system, in particular costs within the electricity balancing markets, mainly due to an increased number of extreme price spikes. This study analyzes the impact of an increased share of renewable energy sources on the behavior of price and volumes of the Italian balancing market. Starting from configurations of load and power production, which guarantee a stable performance, we implement fluctuations in the load and in renewables; in particular we artificially increase the contribution of renewables as compared to conventional power sources to cover the total load. We then determine the amount of requested energy in the balancing market and its fluctuations, which are induced by production and consumption. Within an approach of agent-based modeling we estimate the resulting energy prices and costs. While their average values turn out to be only slightly affected by an increased contribution from renewables, the probability for extreme price events is shown to increase along with undesired peaks in the costs. Our methodology provides a tool for estimating outliers in prices obtained in the energy balancing market, once data of consumption, production and their typical fluctuations are provided. How Decentral Smart Grid Control Limits Non-Gaussian Power Grid Frequency fluctuations. B. Schaefer, M. Timme, D. Witthaut http://dx.doi.org/10.1109/CCTA.2018.8511400, Proceedings of the 2018 IEEE Conference on Control Technology and Applications (CCTA) Copenhagen, 32-39 (2018). Abstract: 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. Hysteretic percolation from locally optimal individual decisions. M. Schröder, J. Nagler, M. Timme, D. Witthaut https://doi.org/10.1103/PhysRevLett.120.248302, Phys. Rev. Lett. 120, 248302 (2018). Abstract: The emergence of large-scale connectivity underlies the proper functioning of many networked systems, ranging from social networks and technological infrastructure to global trade networks. Percolation theory characterizes network formation following stochastic local rules, while optimization models of network formation assume a single controlling authority or one global objective function. In socioeconomic networks, however, network formation is often driven by individual, locally optimal decisions. How such decisions impact connectivity is only poorly understood to date. Here, we study how large-scale connectivity emerges from decisions made by rational agents that individually minimize costs for satisfying their demand. We establish that the solution of the resulting nonlinear optimization model is exactly given by the final state of a local percolation process. This allows us to systematically analyze how locally optimal decisions on the microlevel define the structure of networks on the macroscopic scale. Impact of climate change on backup and storage needs in highly renewable power systems in Europe. J. Weber, J. Wohland, M. Reyers, J. Moemken, C. Hoppe, J. G. Pinto, D. Witthaut, https://doi.org/10.1371/journal.pone.0201457, PLoS ONE 13, e0201457 (2018). Abstract: The high temporal variability of wind power generation represents a major challenge for the realization of a sustainable energy supply. Large backup and storage facilities are necessary to secure the supply in periods of low renewable generation, especially in countries with a high share of renewables. We show that strong climate change is likely to impede the system integration of intermittent wind energy. To this end, we analyze the temporal characteristics of wind power generation based on high-resolution climate projections for Europe and uncover a robust increase of backup energy and storage needs in most of Central, Northern and North-Western Europe. This effect can be traced back to an increase of the likelihood for long periods of low wind generation and an increase in the seasonal wind variability. Impact of strong climate change on the statistics of wind power generation in Europe. J. Weber, F. Gotzens, D. Witthaut Proceedings of the 5th International Conference on Energy and Environment Research (ICEER 2018) in Prague, https://doi.org/10.1016/j.egypro.2018.10.004, Energy Proceedia 153, 22-28 (2018). Abstract: Variable renewable energy sources highly rely on weather and climate variability. Therefore, their power output may become subject to climate change. We analyze how strong climate change may affect wind power resources in Europe, based on the outcome of high-resolution climate simulations. In particular, we evaluate the probability and persistence of low, medium and high wind regimes and the seasonal variability of wind speeds. For many parts in Europe we find a shift in the wind speed distribution: from higher to smaller wind velocities. Thus, the occurrence of wind velocities smaller than the cut-in velocity becomes more likely, which may result in lower wind power output. We further observe an increasing seasonal wind variability over most of Central and North-Western Europe. This may enhance curtailment in the winter months and backup energy needs in summer. Inferring power-grid topology in the face of uncertainties. F. Basiri, J. Casadiego, M. Timme, D. Witthaut https://doi.org/10.1103/PhysRevE.98.012305, Phys. Rev. E 98, 012305 (2018). Abstract: We develop methods to efficiently reconstruct the topology and line parameters of a power grid from the measurement of nodal variables. We propose two compressed sensing algorithms that minimize the amount of necessary measurement resources by exploiting network sparsity, symmetry of connections, and potential prior knowledge about the connectivity. The algorithms are reciprocal to established state estimation methods, where nodal variables are estimated from few measurements given the network structure. Hence, they enable an advanced grid monitoring where both state and structure of a grid are subject to uncertainties or missing information. Isolating the impact of trading on grid frequency fluctuations B. Schaefer, M. Timme, D. Witthaut http://dx.doi.org/10.1109/ISGTEurope.2018.8571793, Proceedings of the 8th IEEE PES Innovative Smart Grid Technologies Europe (ISGT Europe 2018) Sarajevo (2018). Abstract: 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 Levy-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. Linear Optimal Power Flow Using Cycle Flows J. Hoersch, H. Ronellenfitsch, D. Witthaut, T. Brown https://doi.org/10.1016/j.epsr.2017.12.034, Electric Power Systems Research 158, 126 (2018). Abstract: Linear optimal power flow (LOPF) algorithms use a linearization of the alternating current (AC) load flow equations to optimize generator dispatch in a network subject to the loading constraints of the network branches. Common algorithms use the voltage angles at the buses as optimization variables, but alternatives can be computationally advantageous. In this article we provide a review of existing methods and describe a new formulation that expresses the loading constraints directly in terms of the flows themselves, using a decomposition of the network graph into a spanning tree and closed cycles. We provide a comprehensive study of the computational performance of the various formulations, in settings that include computationally challenging applications such as multi-period LOPF with storage dispatch and generation capacity expansion. We show that the new formulation of the LOPF solves up to 7 times faster than the angle formulation using a commercial linear programming solver, while another existing cycle-base formulation solves up to 20 times faster, with an average speed-up of factor 3 for the standard networks considered here. If generation capacities are also optimized, the average speed-up rises to a factor of 12, reaching up to factor 213 in a particular instance. The speed-up is largest for networks with many buses and decentral generators throughout the network, which is highly relevant given the rise of distributed renewable generation and the computational challenge of operation and planning in such networks. Network-induced multistability: Lossy coupling and exotic solitary states. F. Hellmann, P. Schultz, P. Jaros, R. Levchenko, T. Kapitaniak, J. Kurths, Y. Maistrenko arXiv:1811.11518v1 (2018). Abstract: The stability of synchronised networked systems is a multi-faceted challenge for many natural and technological fields, from cardiac and neuronal tissue pacemakers to power grids. In the latter case, the ongoing transition to distributed renewable energy sources is leading to a proliferation of dynamical actors. The desynchronization of a few or even one of those would likely result in a substantial blackout. Thus the dynamical stability of the synchronous state has become a focus of power grid research in recent years. In this letter we uncover that the non-linear stability against large perturbations is dominated and threatened by the presence of \textit{solitary states} in which individual actors desynchronise. Remarkably, when taking physical losses in the network into account, the back-reaction of the network induces new {\it exotic} solitary states in the individual actors, and the stability characteristics of the synchronous state are dramatically altered. These novel effects will have to be explicitly taken into account in the design of future power grids, and their existence poses a challenge for control. While this letter focuses on power grids, the form of the coupling we explore here is generic, and the presence of new states is very robust. We thus strongly expect the results presented here to transfer to other systems of coupled heterogeneous Newtonian oscillators. . Modeling long correlation times using additive binary Markov chains: Applications to wind generation time series. J. Weber, C. Zachow, D.Witthaut https://doi.org/10.1103/PhysRevE.97.032138, Phys. Rev. E 97, 032138 (2018). Abstract: Wind power generation exhibits a strong temporal variability, which is crucial for system integration in highly renewable power systems. Different methods exist to simulate wind power generation but they often cannot represent the crucial temporal fluctuations properly. We apply the concept of additive binary Markov chains to model a wind generation time series consisting of two states: periods of high and low wind generation. The only input parameter for this model is the empirical autocorrelation function. The two-state model is readily extended to stochastically reproduce the actual generation per period. To evaluate the additive binary Markov chain method, we introduce a coarse model of the electric power system to derive backup and storage needs. We find that the temporal correlations of wind power generation, the backup need as a function of the storage capacity, and the resting time distribution of high and low wind events for different shares of wind generation can be reconstructed. Non-Gaussian power grid frequency fluctuations characterized by Levy-stable laws and superstatistics. B. Schaefer, C. Beck, K. Aihara, D. Witthaut, M. Timme https://doi.org/10.1038/s41560-017-0058-z, Nature Energy 3, 119 (2018). Abstract: 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 analyse 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 strong deviation from Gaussianity best described as Lévy-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 that 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. Principal ow patterns across renewable electricity networks. F. Hofmann, M. Schafer, T. Brown, J. Hörsch, M. Greiner, S. Schramm EPL (Europhysics Letters), Volume 124, Number 1 (2018). Abstract: Using Principal Component Analysis (PCA), the nodal injection and line flow patterns in a network model of a future highly renewable European electricity system are investigated. It is shown that the number of principal components needed to describe 95% of the nodal power injection variance first increases with the spatial resolution of the system representation. The number of relevant components then saturates at around 76 components for network sizes larger than 512 nodes, which can be related to the correlation length of wind patterns over Europe. Remarkably, the application of PCA to the transmission line power flow statistics shows that irrespectively of the spatial scale of the system representation a very low number of only 8 principal flow patterns is sufficient to capture 95% of the corresponding spatio-temporal variance. This result can be theoretically explained by a particular alignment of some principal injection patterns with topological patterns inherent to the network structure of the European transmission system. Propagation of Disturbances in AC Electricity Grids. S. R. Tamrakar, M. Conrath, S. Kettemann Scientific Reports 8, 6459 (2018). Abstract: The energy transition towards high shares of renewable energy will affect the stability of electricity grids in many ways. Here, we aim to study its impact on propagation of disturbances by solving nonlinear swing equations describing coupled rotating masses of synchronous generators and motors on different grid topologies. We consider a tree, a square grid and as a real grid topology, the german transmission grid. We identify ranges of parameters with different transient dynamics: the disturbance decays exponentially in time, superimposed by oscillations with the fast decay rate of a single node, or with a smaller decay rate without oscillations. Most remarkably, as the grid inertia is lowered, nodes may become correlated, slowing down the propagation from ballistic to diffusive motion, decaying with a power law in time. Applying linear response theory we show that tree grids have a spectral gap leading to exponential relaxation as protected by topology and independent on grid size. Meshed grids are found to have a spectral gap which decreases with increasing grid size, leading to slow power law relaxation and collective diffusive propagation of disturbances. We conclude by discussing consequences if no measures are undertaken to preserve the grid inertia in the energy transition. PyPSA-Eur: An open optimization model of the European transmission system. J. Hörsch, F. Hofmann, D. Schlachtberger, T. Brown Energy Strategy Reviews, https://doi.org/10.1016/j.esr.2018.08.012 (2018). Abstract: PyPSA-Eur, the first open model dataset of the European power system at the transmission network level to cover the full ENTSO-E area, is presented. It contains 6001 lines (alternating current lines at and above 220 kV voltage level and all high voltage direct current lines, 3657 substations, a new open database of conventional power plants, time series for electrical demand and variable renewable generator availability, and geographic potentials for the expansion of wind and solar power. The model is suitable both for operational studies and generation and transmission expansion planning studies. The continental scope and highly resolved spatial scale enables a proper description of the long-range smoothing effects for renewable power generation and their varying resource availability. The restriction to freely available and open data encourages the open exchange of model data developments and eases the comparison of model results. A further novelty of the dataset is the publication of the full, automated software pipeline to assemble the load-flow-ready model from the original datasets, which enables easy replacement and improvement of the individual parts. This paper focuses on the description of the network topology, the compilation of a European power plant database and a top-down load time-series regionalisation. It summarises the derivation of renewable wind and solar availability time-series from re-analysis weather datasets and the estimation of renewable capacity potentials restricted by land-use. Finally, validations of the dataset are presented, including a new methodology to compare geo-referenced network datasets to one another. PyPSA: Python for Power System Analysis. T. Brown, J. Hörsch, D. Schlachtberger Journal of Open Research Software, 6(1) (2018). Abstract: Python for Power System Analysis (PyPSA) is a free software toolbox for simulating and optimising modern electrical power systems over multiple periods. PyPSA includes models for conventional generators with unit commitment, variable renewable generation, storage units, coupling to other energy sectors, and mixed alternating and direct current networks. It is designed to be easily extensible and to scale well with large networks and long time series. In this paper the basic functionality of PyPSA is described, including the formulation of the full power flow equations and the multi-period optimisation of operation and investment with linear power flow equations. PyPSA is positioned in the existing free software landscape as a bridge between traditional power flow analysis tools for steady-state analysis and full multi-period energy system models. The functionality is demonstrated on two open datasets of the transmission system in Germany (based on SciGRID) and Europe (based on GridKit). Response to `Burden of proof: A comprehensive review of the feasibility of 100% renewable-electricity systems'. T.W. Brown, T. Bischof-Niemz, K. Blok, C. Breyer, H. Lund, B.V. Mathiesen Renewable and Sustainable Energy Reviews 92, 834-847, https://doi.org/10.1016/j.rser.2018.04.113 (2018). Abstract: A recent article ‘Burden of proof: A comprehensive review of the feasibility of 100% renewable-electricity systems’ claims that many studies of 100% renewable electricity systems do not demonstrate sufficient technical feasibility, according to the criteria of the article's authors (henceforth ‘the authors’). Here we analyse the authors’ methodology and find it problematic. The feasibility criteria chosen by the authors are important, but are also easily addressed at low economic cost, while not affecting the main conclusions of the reviewed studies and certainly not affecting their technical feasibility. A more thorough review reveals that all of the issues have already been addressed in the engineering and modelling literature. Nuclear power, which the authors have evaluated positively elsewhere, faces other, genuine feasibility problems, such as the finiteness of uranium resources and a reliance on unproven technologies in the medium- to long-term. Energy systems based on renewables, on the other hand, are not only feasible, but already economically viable and decreasing in cost every year. Rotor-angle versus voltage instability in the third-order model for synchronous generators. K. Sharafutdinov, L. R. Gorjão, M. Matthiae, T. Faulwasser, D. Witthaut https://doi.org/10.1063/1.5002889, Chaos 28, 033117 (2018). Abstract: We investigate the interplay of rotor-angle and voltage stability in electric power systems. To this end, we carry out a local stability analysis of the third-order model which entails the classical power-swing equations and the voltage dynamics. We provide necessary and sufficient stability conditions and investigate different routes to instability. For the special case of a two-bus system, we analytically derive a global stability map. A reliable supply of electric power requires a stable operation of the electric power grid. Thousands of generators must run in a synchronous state with fixed voltage magnitudes and fixed relative phases. The ongoing transition to a renewable power system challenges the stability as line loads and temporal fluctuations increase. Maintaining a secure supply thus requires a detailed understanding of power system dynamics and stability. Among various models describing the dynamics of synchronous generators, analytic results are available mainly for the simplest second-order model which describes only the dynamics of nodal frequencies and voltage phase angles. In this article, we analyze the stability of the third order model including the transient dynamics of voltage magnitudes. Within this model we provide analytical insights into the interplay of voltage and rotor-angle dynamics and characterize possible sources of instability. We provide novel stability criteria and support our studies with the analysis of a network of two coupled nodes, where a full analytic solution for the equilibria is obtained and a bifurcation analysis is performed. Stochastic basins of attraction and generalized committor functions. M. Lindner, F. Hellmann arXiv:1803.06372 (2018). Abstract: We generalize the concept of basin of attraction of a stable state in order to facilitate the analysis of dynamical systems with noise and to assess stability properties of metastable states and long transients. To this end we examine the notions of mean sojourn times and absorption probabilities for Markov chains and study their relation to the basins of attraction. Our approach is applicable to a large variety of problems since in most cases the transfer operator associated to a dynamical system can be approximated by a Markov chain. Synergies of sector coupling and transmission reinforcement in a cost-optimised, highly renewable European energy system. T. Brown, D. Schlachtberger, A. Kies, S. Schramm, M. Greiner Energy 160, 720-739, https://doi.org/10.1016/j.energy.2018.06.222 (2018). Abstract: There are two competing concepts in the literature for the integration of high shares of renewable energy: the coupling of electricity to other energy sectors, such as transport and heating, and the reinforcement of continent-wide transmission networks. In this paper both cross-sector and cross-border integration are considered in the model PyPSA-Eur-Sec-30, the first open, spatially-resolved, temporally-resolved and sector-coupled energy model of Europe. Using a simplified network with one node per country, the cost-optimal system is calculated for a 95% reduction in carbon dioxide emissions compared to 1990, incorporating electricity, transport and heat demand. Flexibility from battery electric vehicles (BEV), power-to-gas units (P2G) and long-term thermal energy storage (LTES) make a significant contribution to the smoothing of variability from wind and solar and to the reduction of total system costs. The cost-minimising integration of BEV pairs well with the daily variations of solar power, while P2G and LTES balance the synoptic and seasonal variations of demand and renewables. In all scenarios, an expansion of cross-border transmission reduces system costs, but the more tightly the energy sectors are coupled, the weaker the benefit of transmission reinforcement becomes. The Impact of Climate Change on a Cost-Optimal Highly Renewable European Electricity Network. M. Schlott, A. Kies, T. Brown, S. Schramm, M. Greiner Applied Energy, Volume 230, Pages 1645-1659, https://doi.org/10.1016/j.apenergy.2018.09.084 (2018). Abstract: We use three ensemble members of the EURO-CORDEX project and their data on surface wind speeds, solar irradiaton as well as water runoff with a spatial resolution of 12 km and a temporal resolution of 3 h under representative concentration pathway 8.5 (associated with a strong climate change and a temperature increase of 2.6 – 4,8 °C until the end of the century) until 2100 to investigate the impact of climate change on wind, solar and hydro resources and consequently on a highly renewable and cost-optimal European power system. The weather data is transformed into power, different aspects such as capacity factors and correlation lengths are investigated and the resulting implications for the European power system are discussed. In addition, we compare a 30-node model of Europe with historical and climate change-affected data, where investments in generation, transmission and storage facilities are optimised. Differences in capacity factors among European countries are more strongly emphasized at the end of the century compared to historic data. This results in a significantly increased photovoltaic share in the cost-optimal power system. In addition, annual hydro inflow patterns of major hydro producers change considerably. System costs increase by 5% until the end of the century and the impact of climate change on these costs is of similar magnitude as differences between the ensemble members. The results show that including climate affected-weather data in power system simulations of the future has an observable effect. - 2017 - A Dual Method for Computing Power Transfer Distribution Factors. Ronellenfitsch, H., Timme, M., Witthaut, D. IEEE Transactions on Power Systems 32 (2), 1007-1015 (2017), https://doi.org/10.1109/TPWRS.2016.2589464. Abstract: Power Transfer Distribution Factors (PTDFs) play a crucial role in power grid security analysis, planning, and redispatch. Fast calculation of the PTDFs is therefore of great importance. In this paper, we present a non-approximative dual method of computing PTDFs. It uses power flows along topological cycles of the network but still relies on simple matrix algebra. At the core, our method changes the size of the matrix that needs to be inverted to calculate the PTDFs from N×N, where N is the number of buses, to (L-N+1)×(L-N+1), where L is the number of lines and L-N+1 is the number of independent cycles (closed loops) in the network while remaining mathematically fully equivalent. For power grids containing a relatively small number of cycles, the method can offer a speedup of numerical calculations. A universal order parameter for synchrony in networks of limit cycle oscillators. Schröder, M., Timme, M., Witthaut, D. http://dx.doi.org/10.1063/1.4995963, Chaos 27, 073119 (2017). Abstract: We analyze the properties of order parameters measuring synchronization and phase locking in complex oscillator networks. First, we review network order parameters previously introduced and reveal several shortcomings: none of the introduced order parameters capture all transitions from incoherence over phase locking to full synchrony for arbitrary, finite networks. We then introduce an alternative, universal order parameter that accurately tracks the degree of partial phase locking and synchronization, adapting the traditional definition to account for the network topology and its influence on the phase coherence of the oscillators. We rigorously proof that this order parameter is strictly monotonously increasing with the coupling strength in the phase locked state, directly reflecting the dynamic stability of the network. Furthermore, it indicates the onset of full phase locking by a diverging slope at the critical coupling strength. The order parameter may find applications across systems where different types of synchrony are possible, including biological networks and power grids. Balancing between competition and coordination in smart grids – a Common Information Platform (CIP). Brandstätt, C., Brunekreeft, G., Buchmann, M., Friedrichsen, N. accepted for publication in Economics of Energy & Environmental Policy (2017). Can Distribution Grids Significantly Contribute to Transmission Grids’ Voltage Management? Auer, S. Steinke, F. Chunsen, W. Szabo, A. Sollacher, R. IEEE ISGT (2017), DOI: 10.1109/ISGTEurope.2016.7856194. Abstract: Power generation in Germany is currently transitioning from a system based on large, central, thermal power plants to one that heavily relies on small, decentral, mostly renewable power generators. This development poses the question how transmission grids’ reactive power demand for voltage management, covered by central power plants today, can be supplied in the future. In this work, we estimate the future technical potential of such an approach for the whole of Germany. For a 100% renewable electricity scenario we set the possible reactive power supply in comparison with the reactive power requirements that are needed to realize the simulated future transmission grid power flows. Since an exact calculation of distribution grids’ reactive power potential is difficult due to the unavailability of detailed grid models on such scale, we optimistically estimate the potential by assuming a scaled, averaged distribution grid model connected to each of the transmission grid nodes. We find that for all except a few transmission grid nodes, the required reactive power can be fully supplied from the modelled distribution grids. This implies that – even if our estimate is overly optimistic – distributed reactive power provisioning will be a technical solution for many future reactive power challenges. Comparison of electricity market designs for stable decentralized power grids. Heitzig, J., Meyer-Ortmanns, H. arXiv:1704.04644 (2017). Curing critical links in oscillator networks as power flow models. Rohden, M., Witthaut, D., Timme, M., Meyer-Ortmanns, H. New J. Phys. 19, 013002 (2017). Abstract: Modern societies crucially depend on the robust supply with electric energy so that blackouts of power grids can have far reaching consequences. Typically, large scale blackouts take place after a cascade of failures: the failure of a single infrastructure component, such as a critical transmission line, results in several subsequent failures that spread across large parts of the network. Improving the robustness of a network to prevent such secondary failures is thus key for assuring a reliable power supply. In this article we analyze the nonlocal rerouting of power flows after transmission line failures for a simplified AC power grid model and compare different strategies to improve network robustness. We identify critical links in the grid and compute alternative pathways to quantify the grid's redundant capacity and to find bottlenecks along the pathways. Different strategies are developed and tested to increase transmission capacities to restore stability with respect to transmission line failures. We show that local and nonlocal strategies typically perform alike: one can equally well cure critical links by providing backup capacities locally or by extending the capacities of bottleneck links at remote locations. Cycle flows and multistabilty in oscillatory networks: an overview. Manik, D., Timme, M., Witthaut, D. http://dx.doi.org/10.1063/1.4994177, Chaos 27, 083123 (2017). Abstract: The functions of many networked systems in physics, biology or engineering rely on a coordinated or synchronized dynamics of its constituents. In power grids for example, all generators must synchronize and run at the same frequency and their phases need to appoximately lock to guarantee a steady power flow. Here, we analyze the existence and multitude of such phase-locked states. Focusing on edge and cycle flows instead of the nodal phases we derive rigorous results on the existence and number of such states. Generally, multiple phase-locked states coexist in networks with strong edges, long elementary cycles and a homogeneous distribution of natural frequencies or power injections, respectively. We offer an algorithm to systematically compute multiple phase- locked states and demonstrate some surprising dynamical consequences of multistability. Deciphering the imprint of topology on nonlinear dynamical network stability. Nitzbon, J., Schultz, P., Heitzig, J., Kurths, J., Hellmann, F. New Journal of Physics, 19 033029 (2017). Abstract: Coupled oscillator networks show complex interrelations between topological characteristics of the network and the nonlinear stability of single nodes with respect to large but realistic perturbations. We extend previous results on these relations by incorporating sampling-based measures of the transient behaviour of the system, its survivability, as well as its asymptotic behaviour, its basin stability. By combining basin stability and survivability we uncover novel, previously unknown asymptotic states with solitary, desynchronized oscillators which are rotating with a frequency different from their natural one. They occur almost exclusively after perturbations at nodes with specific topological properties. More generally we confirm and significantly refine the results on the distinguished role tree-shaped appendices play for nonlinear stability. We find a topological classification scheme for nodes located in such appendices, that exactly separates them according to their stability properties, thus establishing a strong link between topology and dynamics. Hence, the results can be used for the identification of vulnerable nodes in power grids or other coupled oscillator networks. From this classification we can derive general design principles for resilient power grids. We find that striving for homogeneous network topologies facilitates a better performance in terms of nonlinear dynamical network stability. While the employed second-order Kuramoto-like model is parametrised to be representative for power grids, we expect these insights to transfer to other critical infrastructure systems or complex network dynamics appearing in various other fields. Dual theory of transmission line outages. Ronellenfitsch, G., Manik, D., Hörsch, J., Brown, T., Witthaut, D. IEEE Transactions on Power Systems, Volume: PP, Issue: 99 (2017). Abstract: A new graph dual formalism is presented for the analysis of line outages in electricity networks. The dual formalism is based on a consideration of the flows around closed cycles in the network. After some exposition of the theory is presented, a new formula for the computation of Line Outage Distribution Factors (LODFs) is derived, which is not only computationally faster than existing methods, but also generalizes easily for multiple line outages and arbitrary changes to line series reactance. In addition, the dual formalism provides new physical insight for how the effects of line outages propagate through the network. For example, in a planar network a single line outage can be shown to induce monotonically decreasing flow changes, which are mathematically equivalent to an electrostatic dipole field. Escape Routes, Weak Links, and Desynchronization in Fluctuation-driven Networks. Schaefer, B., Matthiae, M., Zhang, X., Rohden, M., Timme, M., Witthaut D. https://doi.org/10.1103/PhysRevE.95.060203, Phys. Rev. E 95, 060203 (2017). Abstract: Shifting our electricity generation from fossil fuel to renewable energy sources introduces large fluctuations to the power system. Here, we demonstrate how increased fluctuations, reduced damping and reduced intertia may undermine the dynamical robustness of power grid networks. ^ Focusing on fundamental noise models, we derive analytic insights into which factors limit the dynamic robustness and how fluctuations may induce a system escape from an operating state. Moreover, we identify weak links in the grid that make it particularly vulnerable to fluctuations. These results thereby not only contribute to a theoretical understanding of how fluctuations act on distributed network dynamics, they may also help designing future renewable energy systems to be more robust. Extreme prices in electricity balancing markets from an approach of statistical physics. Mureddu, M., Meyer-Ortmanns, H. submitted (2017). Abstract: An increase in energy production from renewable energy sources is viewed as a crucial achievement in most industrialized countries. The higher variability of power production via renewables leads to a rise in ancillary service costs over the power system, in particular costs within the electricity balancing markets, mainly due to an increased number of extreme price spikes. This study focuses on forecasting the behavior of price and volumes of the Italian balancing market in the presence of an increased share of renewable energy sources. Starting from configurations of load and power production, which guarantee a stable performance, we implement fluctuations in the load and in renewables; in particular we artificially increase the contribution of renewables as compared to conventional power sources to cover the total load. We then forecast the amount of provided energy in the balancing market and its fluctuations, which are induced by production and consumption. Within an approach of agent based modeling we estimate the resulting energy prices and costs. While their average values turn out to be only slightly affected by an increased contribution from renewables, the probability for extreme price events is shown to increase along with undesired peaks in the costs. From State Estimation to Network Reconstruction. Basiri, F., Casadiego, J., Timme, M., Witthaut, M. submitted to Phys. Rev. Applied., preprint at https://arxiv.org/abs/1701.09084 (2017). Abstract: We develop methods to efficiently reconstruct the topology and line parameters of a power grid from the measurement of nodal variables. We propose two compressed sensing algorithms that minimize the amount of necessary measurement resources by exploiting network sparsity, symmetry of connections and potential prior knowledge about the connectivity. The algorithms are reciprocal to established state estimation methods, where nodal variables are estimated from few measurements given the network structure. Hence, they enable an advanced grid monitoring where both state and structure of a grid are subject to uncertainties or missing information. Linear Optimal Power Flow Using Cycle Flows. Hörsch, J., Ronellenfitsch, H., Witthaut, D., Brown, T. Electric Power Systems Research, volume 158, pages 126-135, https://doi.org/10.1016/j.epsr.2017.12.034 (2017). Abstract: Linear optimal power flow (LOPF) algorithms use a linearization of the alternating current (AC) load flow equations to optimize generator dispatch in a network subject to the loading constraints of the network branches. Common algorithms use the voltage angles at the buses as optimization variables, but alternatives can be computationally advantageous. In this article we provide a review of existing methods and describe new formulations, which express the loading constraints directly in terms of the flows themselves, using a decomposition of the graph into a spanning tree and closed cycles. We provide a comprehensive study of the computational performance of the various formulations, showing that one of the new formulations of the LOPF solves up to 20 times faster than the angle formulation using commercial linear programming solvers, with an average speed-up of factor 3 for the standard networks considered here. The speed-up is largest for networks with many nodes and decentral generators throughout the network, which is highly relevant given the rise of distributed renewable generation. Long-term impacts of a coal phase-out in Germany as part of a greenhouse gas mitigation strategy. Heinrichs, h., Markewitz, P. Applied Energy, Volume 192, 15 April 2017, Pages 234-246, ISSN 0306-2619, https://doi.org/10.1016/j.apenergy.2017.01.065. Abstract: Germany appears set to miss its CO2 reduction target in 2020. As a result, ideas for additional political measures have been put forward. One such idea involves an early phase-out of coal-fired power plants. However, the possible impacts of such a phase-out on the energy system have not yet been fully analyzed. We therefore apply a German energy system model to analyze these impacts. To do so, we calculate three different scenarios. The first represents a business-as-usual scenario, while the second takes a coal phase-out into account. The third scenario has to achieve the same CO2 reduction as the second without being forced to implement a coal phase-out. Our three scenarios show that a definitive coal phase-out by 2040 would result in only a relatively small amount of additional CO2. However, an equal CO2 reduction can be obtained using a different strategy and slightly lower costs. In the latter scenario, the additional costs are also distributed more evenly across the sectors. The sensitivities analyzed show the robustness of the conclusions drawn. In summary, this analysis outlines what consequences could arise by excluding several options in parallel from a technology portfolio. More Homogeneous Wind Conditions Under Strong Climate Change Decrease the Potential for Inter-State Balancing of Electricity in Europe. Wohland, J., Reyers, M., Weber, J., Witthaut, D. preprint at: https://doi.org/10.5194/esd-2017-48, 2017. Manuscript under review for Earth System Dynamics. Abstract: Limiting anthropogenic climate change requires the fast decarbonisation of the electricity system. Renewable electricity generation is determined by the weather and is hence subject to climate change. We simulate the operation of a coarse-scale fully-renewable European electricity system based on downscaled high resolution climate data from EURO-CORDEX. Following a high emission pathway (RCP8.5), we find a robust increase of backup needs in Europe until the end of the 21st century. The absolute increase of the backup needs is almost independent of potential grid expansion, leading to the paradoxical effect that relative impacts of climate change increase in a highly interconnected European system. The increase is rooted in more homogeneous wind conditions over Europe resulting in extensive parallel generation shortfalls. Our results are strengthened by comparison with a large CMIP5 ensemble using an approach based on Circulation Weather Types. Network susceptibilities: theory and applications. Manik, D., Rohden, M., Ronellenfitsch, H., Zhang, X., Hallerberg, S., Witthaut, D., Timme, M. Phys. Rev. E 95, 012310 (2017). Abstract: We introduce the concept of network susceptibilities quantifying the response of the collective dynamics of a network to small parameter changes. We distinguish two types of susceptibilities: vertex susceptibilities and edge susceptibilities, measuring the responses due to changes in the properties of units and their interactions, respectively. We derive explicit forms of network susceptibilities for oscillator networks close to steady states and offer example applications for Kuramoto-type phase-oscillator models, power grid models, and generic flow models. Focusing on the role of the network topology implies that these ideas can be easily generalized to other types of networks, in particular those characterizing flow, transport, or spreading phenomena. The concept of network susceptibilities is broadly applicable and may straightforwardly be transferred to all settings where networks responses of the collective dynamics to topological changes are essential. Opening the black box of energy modelling: strategies and lessons learned. S. Pfenninger, L. Hirth, I. Schlecht, E. Schmid, F. Wiese, T. Brown, C. Davis, B. Fais, M. Gidden, H. Heinrichs, C. Heuberger, S. Hilpert, U. Krien, C. Matke, A. Nebel, R. Morrison, B. Müller, G. Pleßmann, M. Reeg, J. C. Richstein, A. Shivakumar, I. Staffell, T. Tröndle, C. Wingenbach Energy Strategy Reviews, Vol. 19, 63-71, https://doi.org/10.1016/j.esr.2017.12.002 (2017). Abstract: The global energy system is undergoing a major transition, and in energy planning and decision-making across governments, industry and academia, models play a crucial role. Because of their policy relevance and contested nature, the transparency and open availability of energy models and data are of particular importance. Here we provide a practical how-to guide based on the collective experience of members of the Open Energy Modelling Initiative (Openmod). We discuss key steps to consider when opening code and data, including determining intellectual property ownership, choosing a licence and appropriate modelling languages, distributing code and data, and providing support and building communities. After illustrating these decisions with examples and lessons learned from the community, we conclude that even though individual researchers' choices are important, institutional changes are still also necessary for more openness and transparency in energy research. Optimal heterogeneity in a simplified highly renewable European electricity system. Eriksen, E., Schwenk-Nebbe, L., Tranberg, B., Brown, T., Greiner, M. Energy, https://doi.org/10.1016/j.energy.2017.05.170 (2017). Abstract: The resource quality and the temporal generation pattern of variable renewable energy sources vary significantly across Europe. In this paper spatial distributions of renewable assets are explored which exploit this heterogeneity to lower the total system costs for a high level of renewable electricity in Europe. Several intuitive heuristic algorithms, optimal portfolio theory and a local search algorithm are used to find optimal distributions of renewable generation capacities that minimise the total costs of backup, transmission and renewable capacity simultaneously. Using current cost projections, an optimal heterogeneous distribution favours onshore wind, particularly in countries bordering the North Sea, which results in average electricity costs that are up to 11% lower than for a homogeneous reference distribution of renewables proportional to each country's mean load. The reduction becomes even larger, namely 18%, once the transmission capacities are put to zero in the homogeneous reference distribution. Heuristic algorithms to distribute renewable capacity based on each country's wind and solar capacity factors are shown to provide a satisfactory approximation to fully optimised renewable distributions, while maintaining the benefits of transparency and comprehensibility. The sensitivities of the results to changing costs of solar generation and gas supply as well as to the possible cross-sectoral usage of unavoidable curtailment energy are also examined. Potentials and limits to basin stability estimation. Schultz, P., Menck P.J., Heitzig, J., Kurths, J. New Journal of Physics, Volume 19 (2017). Abstract: Stability assessment methods for dynamical systems have recently been complemented by basin stability and derived measures, i.e. probabilistic statements whether systems remain in a basin of attraction given a distribution of perturbations. Their application requires numerical estimation via Monte Carlo sampling and integration of differential equations. Here, we analyse the applicability of basin stability to systems with basin geometries that are challenging for this numerical method, having fractal basin boundaries and riddled or intermingled basins of attraction. We find that numerical basin stability estimation is still meaningful for fractal boundaries but reaches its limits for riddled basins with holes. Power-functional network. Sun, Y., Kurths, J., Zhan, M. Chaos 27, 083116 (2017); doi: http://dx.doi.org/10.1063/1.4995361 (2017). Abstract: Power grids and their properties have been studied broadly in many aspects. In this paper, we propose a novel concept, power-flow-based power grid, as a typical power-functional network, based on the calculation of power flow distribution from power electrical engineering. We compare it with structural networks based on the shortest path length and effective networks based on the effective electrical distance and study the relationship among these three kinds of networks. We find that they have roughly positive correlations with each other, indicating that in general any close nodes in the topological structure are actually connected in function. However, we do observe some counter-examples that two close nodes in a structural network can have a long distance in a power-functional network, namely, two physically connected nodes can actually be separated in function. In addition, we find that power grids in the structural network tend to be heterogeneous, whereas those in the effective and power-functional networks tend to be homogeneous. These findings are expected to be significant not only for power grids but also for various other complex networks. Unveiling the core structure-function relationship has become a key objective in many disciplines. Borrowing the key concept in brain connectome, functional brain network, here we propose a similar concept, power-functional network (FN), to describe the functional connection in power systems, and especially, the so-called power-flow-based power grid, to describe the functional connection for transporting active power in power systems. The calculation of the power-flow-based power grid is mainly based on the power flow tracing, which has been extensively used in power electrical engineering and power economic market. Its difference with the structural network from the shortest path length in the graph theory and the effective network (EN) for the effective electrical distance is studied here. We expect that this study is helpful for our understanding of the structure-function relationship not only in power grids but also in various other flow complex systems. Stability of meshed DC microgrids using probabilistic analysis. Strenge, L., Kirchhoff, H., Ndow, L. G., Hellmann, F. DC Microgrids (ICDCM), IEEE Second International Conference on DC Microgrids (2017). Abstract: We analyze the stability of meshed DC microgrids using the proposed Probabilistic Analysis of Deterministic Systems (PADeS). By sampling a random distribution of networks as well as disturbances, we make probabilistic statements valid for two- (ring) and three-valent (meshed) networks. The producing nodes are droop controlled sources and the consuming nodes are modeled as constant power loads (CPLs). ^ We introduce basin stability and survivability as probabilistic stability measures within the PADeS framework and illustrate that it has a large potential for understanding the properties of DC microgrids and beyond. To show this, we run numerical experiments with 128-node networks and 1000 samples for the system's probability (a) to return to a stable equilibrium point after a perturbation (basin stability) and (b) to remain within defined operational bounds for disturbances such as sudden load power increase, line outages and short circuits at random nodes or power lines (survivability). Stability of Synchrony against Local Intermittent Fluctuations in Tree-like Power Grids. Auer, S., Hellmann, F., Krause, M., Kurths, J. Chaos 27, 127003 (2017). Abstract: 90% of all Renewable Energy Power in Germany is installed in tree-like distribution grids. Intermittent power fluctuations from such sources introduce new dynamics into the lower grid layers. At the same time, distributed resources will have to contribute to stabilize the grid against these fluctuations in the future. In this paper, we model a system of distributed resources as oscillators on a tree-like, lossy power grid and its ability to withstand desynchronization from localized intermittent renewable infeed. We find a remarkable interplay of the network structure and the position of the node at which the fluctuations are fed in. An important precondition for our findings is the presence of losses in distribution grids. Then, the most network central node splits the network into branches with different influence on network stability. Troublemakers, i.e., nodes at which fluctuations are especially exciting the grid, tend to be downstream branches with high net power outflow. For low coupling strength, we also find branches of nodes vulnerable to fluctuations anywhere in the network. These network regions can be predicted at high confidence using an eigenvector based network measure taking the turbulent nature of perturbations into account. While we focus here on tree-like networks, the observed effects also appear, albeit less pronounced, for weakly meshed grids. On the other hand, the observed effects disappear for lossless power grids often studied in the complex system literature. The Benefits of Cooperation in a Highly Renewable European Electricity Network. Schlachtberger, D., Brown, T., Schramm, S., Greiner, M. Energy 134, 469-481, https://doi.org/10.1016/j.energy.2017.06.004 (2017). Abstract: To reach ambitious European CO2 emission reduction targets, most scenarios of future European electricity systems rely on large shares of wind and solar photovoltaic power generation. We interpolate between two concepts for balancing the variability of these renewable sources: balancing at continental scales using the transmission grid and balancing locally with storage. This interpolation is done by systematically restricting transmission capacities from the optimum level to zero. We run techno-economic cost optimizations for the capacity investment and dispatch of wind, solar, hydroelectricity, natural gas power generation and transmission, as well as storage options such as pumped-hydro, battery, and hydrogen storage. The simulations assume a 95% CO2 emission reduction compared to 1990, and are run over a full historical year of weather and electricity demand for 30 European countries. In the cost-optimal system with high levels of transmission expansion, energy generation is dominated by wind (65%) and hydro (15%), with average system costs comparable to today's system. Restricting transmission shifts the balance in favour of solar and storage, driving up costs by a third. As the restriction is relaxed, 85% of the cost benefits of the optimal grid expansion can be captured already with only 44% of the transmission volume. The Contribution of Different Electric Vehicle Control Strategies to Dynamical Grid Stability. Auer, S., Roos, C., Heitzig, J., Hellmann, F., Kurths, J. arXiv:1708.03531 (2017). Abstract: A major challenge for power grids with a high share of renewable energy systems (RES), such as island grids, is to provide frequency stability in the face of renewable fluctuations. In this work we evaluate the ability of electric vehicles (EV) to provide distributed primary control and to eliminate frequency peaks. To do so we for the first time explicitly model the network structure and incorporate non-Gaussian, strongly intermittent fluctuations typical for RES. We show that EVs can completely eliminate frequency peaks. Using threshold randomization we further demonstrate that demand synchronization effects and battery stresses can be greatly reduced. In contrast, explicit frequency averaging has a strong destabilizing effect, suggesting that the role of delays in distributed control schemes requires further studies. Overall we find that distributed control outperforms central one. The results are robust against a further increase in renewable power production and fluctuations. The relevance of grid expansion under zonal markets. J. Bertsch, T. Brown, S. Hagspiel, L. Just The Energy Journal, Vol 38, No 5, https://doi.org/10.5547/01956574.38.5.jber (2017). Abstract: The European electricity market design is based on zonal markets with uniform prices. Hence, no differentiated locational price signals are provided within these zones. If intra-zonal congestion occurs due to missing grid expansion, this market design reveals its inherent incompleteness, and might lead to severe short and long-term distortions. In this paper, we study these distortions with a focus on the impact of restricted grid expansion under zonal markets. Therefore, we use a long-term model of the European electricity system and restrict the allowed expansion of the transmission grid per decade. We find that the combination of an incomplete market design and restricted grid expansion leads to a misallocation of generation capacities and the inability to transport electricity to where it is needed. This results in an energy imbalance in some regions of up to 2-3% and difficulty when reaching envisaged political targets in the power sector. The role of spatial scale in joint optimisations of generation and transmission for European highly renewable scenarios. Hörsch, J., Brown, T. accepted to 14th International Conference on the European Energy Market - EEM 2017 (2017). Abstract: The effects of the spatial scale on the results of the optimisation of transmission and generation capacity in Europe are quantified under a 95% CO2 reduction compared to 1990 levels, interpolating between one-node-per-country solutions and many-nodes-per-country. The trade-offs that come with higher spatial detail between better exposure of transmission bottlenecks, exploitation of sites with good renewable resources (particularly wind power) and computational limitations are discussed. It is shown that solutions with no grid expansion beyond today's capacities are only around 20% more expensive than with cost-optimal grid expansion. - 2016 - Cascading failures in ac electricity grids. Rohden, M. Jung, D. Tamrakar, S. Kettemann, S. Phys. Rev. E 94, 032209 – Published 9 September 2016. Abstract: Sudden failure of a single transmission element in a power grid can induce a domino effect of cascading failures, which can lead to the isolation of a large number of consumers or even to the failure of the entire grid. Here we present results of the simulation of cascading failures in power grids, using an alternating current (AC) model. We first apply this model to a regular square grid topology. For a random placement of consumers and generators on the grid, the probability to find more than a certain number of unsupplied consumers decays as a power law and obeys a scaling law with respect to system size. Varying the transmitted power threshold above which a transmission line fails does not seem to change the power law exponent q˜1.6. Furthermore, we study the influence of the placement of generators and consumers on the number of affected consumers and demonstrate that large clusters of generators and consumers are especially vulnerable to cascading failures. As a real-world topology we consider the German high-voltage transmission grid. Applying the dynamic AC model and considering a random placement of consumers, we find that the probability to disconnect more than a certain number of consumers depends strongly on the threshold. For large thresholds the decay is clearly exponential, while for small ones the decay is slow, indicating a power law decay. Critical links and nonlocal rerouting in complex supply networks. Witthaut, D., Rohden, M., Zhang, X., Hallerberg, S., Timme, M. Phys. Rev. Lett. 116, 138701 (2016). Abstract: Link failures repeatedly induce large-scale outages in power grids and other supply networks. Yet, it is still not well understood which links are particularly prone to inducing such outages. Here we analyze how the nature and location of each link impact the network’s capability to maintain a stable supply. We propose two criteria to identify critical links on the basis of the topology and the load distribution of the network prior to link failure. They are determined via a link’s redundant capacity and a renormalized linear response theory we derive. These criteria outperform the critical link prediction based on local measures such as loads. The results not only further our understanding of the physics of supply networks in general. As both criteria are available before any outage from the state of normal operation, they may also help real-time monitoring of grid operation, employing countermeasures and support network planning and design. Delocalization of Phase Disturbances and the Stability of AC Electricity Grids. Kettemann, S. Phys. Rev. E 94, 062311 (2016), published December 2016. Abstract: In order to study how local disturbances affect the ac grid stability, we start from nonlinear power balance equations and map them to complex linear wave equations. Having obtained stationary solutions with phases i at generator and consumer nodes i, we next study the dynamics of deviations. Starting with an initially localized perturbation, it is found to spread in a periodic grid diffusively throughout the grid. We find the parametric dependence of diffusion constant D. We apply the same solution strategy to general grid topologies and analyze their stability against local perturbations. The perturbation remains either localized or becomes delocalized, depending on grid topology, power capacity, and distribution of consumers and generator power Pi. Delocalization is found to increase the lifetime of perturbations and thereby their influence on grid stability, whereas localization results in an exponentially fast decay of perturbations at all grid sites. These results may therefore lead to new strategies to control the stability of electricity grids. Deciphering the imprint of topology on nonlinear dynamical network stability. Nitzbon, J., Schultz, P., Heitzig, J., Kurths, J., Hellmann, F. arXiv:1612.03654 (2016). Abstract: Coupled oscillator networks show a complex interrelations between topological characteristics of the network and the nonlinear stability of single nodes with respect to large but realistic perturbations. We extend previous results on these relations by incorporating sampling-based measures of the transient behaviour of the system, its survivability, as well as its asymptotic behaviour, its basin stability. By combining basin stability and survivability we uncover novel, previously unknown asymptotic states with solitary, desynchronized oscillators which are rotating with a frequency different from their natural one. They occur almost exclusively after perturbations at nodes with specific topological properties. More generally we confirm and significantly refine the results on the distinguished role tree-shaped appendices play for nonlinear stability. We find a topological classification scheme for nodes located in such appendices, that exactly separates them according to their stability properties, thus establishing a strong link between topology and dynamics. Hence, the results can be used for the identification of vulnerable nodes in power grids or other coupled oscillator networks. From this classification we can derive general design principles for resilient power grids. We find that striving for homogeneous network topologies facilitates a better performance in terms of nonlinear dynamical network stability. While the employed second-order Kuramoto-like model is parametrized to be representative for power grids, we expect these insights to transfer to other critical infrastructure systems or complex network dynamics appearing in various other fields. Flexibility Mechanisms and Pathways to a Highly Renewable US Electricity Future. Frew, B.A., Becker, S., Dvorak, M.J., Andresen, G.B., Jacobson, M.Z. Energy 101 (2016) 65-78. Abstract: This study explores various scenarios and flexibility mechanisms to integrate high penetrations of renewable energy into the US (United States) power grid. A linear programming model – POWER (Power system Optimization With diverse Energy Resources) – is constructed and used to (1) quantify flexibility cost-benefits of geographic aggregation, renewable overgeneration, storage, and flexible electric vehicle charging, and (2) compare pathways to a fully renewable electricity system. Geographic aggregation provides the largest flexibility benefit with ~5–50% cost savings, but each region's contribution to the aggregate RPS (renewable portfolio standard) target is disproportionate, suggesting the need for regional-and-resource-specific RPS targets. Electric vehicle charging yields a lower levelized system cost, revealing the benefits of demand-side flexibility. However, existing demand response price structures may need adjustment to encourage optimal flexible load in highly renewable systems. Two scenarios with RPS targets from 20% to 100% for the US (peak load ~729 GW) and California (peak load ~62 GW) find each RPS target feasible from a planning perspective, but with 2× the cost and 3× the overgeneration at a 100% versus 80% RPS target. Emission reduction cost savings for the aggregated US system with an 80% versus 20% RPS target are roughly $200 billion/year, outweighing the $80 billion/year cost for the same RPS range. Generalised flow tracing for the analysis of networked renewable electricity systems. Hörsch, J., Schäfer, M., Becker, S., Schramm, S., Greiner, M. submitted for review to IEEE Transactions on Power Systems (2016). Abstract: Flow allocation methods represent a valuable tool set to analyze the power flows in networked electricity systems. Based on this flow allocation, the costs associated with the usage of the underlying network infrastructure can be assigned to the users of the electricity system. This paper presents a generalization of the flow tracing method that is applicable to arbitrary compositions of inflow appearing naturally in aggregated networks. The composition of inflow is followed from net-generating sources through the network and assigns corresponding shares of the total power flow as well as of the outflow to the net-consuming sinks. We showcase the analytical power of this method for a scenario based on the IEEE 118 bus network and emphasize the need of appropriate aggregating measures, which allow to integrate over whole time series of fluctuating flow patterns. Growth in Wind and Sun: Integrating Variable Generation in China L. Jiang, C. Wang, Y. Huang, Z. Pei, S. Xin, W. Wang, S. Ma, T. Brown IEEE Power and Energy Magazine, Vol 13, Issue 6 (2016). Abstract: Variable generation (VG) in China is primarily wind and photovoltaic (PV) power. By the end of 2014, the cumulative installed capacity of VG in China reached 123.86 GW, accounting for 9.1% of the country?s total generation capacity. The cumulative installed capacity of wind and PV power were, 95.81 GW and 28.05 GW, respectively. The total amount of energy generated from VG in 2014 was 181 TWh (of which wind power was 156 TWh and PV power 25 TWh), accounting for 3.3% of total electricity generation. The newly installed capacity of wind and PV power in 2014 was 19.5 GW and 10.6 GW, respectively, accounting for 28% of the newly installed capacity of all generation types. Following coal-fired power and hydropower, wind has already become China?s third-largest power source both by capacity and by power generation, after nine years of high-speed growth. A rapid development of PV power generation has been experienced since 2010. The newly installed PV capacity in 2013 and 2014 reached over 10 GW for two consecutive years, accounting for about one-third of newly installed PV capacity worldwide over the same period. By capacity, PVs have become China?s fifth largest generation source after natural gas based generation. Incentives from network charges for self-supply with small scale PV. Brandstätt, C. Konferenzband der IAEE Energy Conference 2016, Bergen, Juni 2016. Interaction Control to Synchronize Non-synchronizable Networks. Schroeder, M., Chakraborty, S., Witthaut, D., Nagler, J., Timme, M. Scientific Reports 6, 37142 (2016). Abstract: Synchronization constitutes one of the most fundamental collective dynamics across networked systems and often underlies their function. Whether a system may synchronize depends on the internal unit dynamics as well as the topology and strength of their interactions. For chaotic units with certain interaction topologies synchronization might be impossible across all interaction strengths, meaning that these networks are non-synchronizable. Here we propose the concept of interaction control, generalizing transient uncoupling, to induce desired collective dynamics in complex networks and apply it to synchronize even such non-synchronizable systems. After highlighting that non-synchronizability prevails for a wide range of networks of arbitrary size, we explain how a simple binary control may localize interactions in state space and thereby synchronize networks. Intriguingly, localizing interactions by a fixed control scheme enables stable synchronization across all connected networks regardless of topological constraints. Interaction control may thus ease the design of desired collective dynamics even without knowledge of the networks’ exact interaction topology and consequently have implications for biological and self-organizing technical systems. Islanding the power grid on the transmission level: less connections for more security. Mureddu, M. Caldarelli, G. Damiano, A. Scala, A. Meyer-Ortmanns, H. Scientific Reports 6, Article number: 34797 (2016). Abstract: Islanding is known as a management procedure of the power system that is implemented at the distribution level to preserve sensible loads from outages and to guarantee the continuity in electricity supply, when a high amount of distributed generation occurs. In this paper we study islanding on the level of the transmission grid and shall show that it is a suitable measure to enhance energy security and grid resilience. We consider the German and Italian transmission grids. We remove links either randomly to mimic random failure events, or according to a topological characteristic, their so-called betweenness centrality, to mimic an intentional attack and test whether the resulting fragments are self-sustainable. We test this option via the tool of optimized DC power flow equations. When transmission lines are removed according to their betweenness centrality, the resulting islands have a higher chance of being dynamically self-sustainable than for a random removal. Less connections may even increase the grid’s stability. These facts should be taken into account in the design of future power grids. Local vs. global redundancy -- trade-offs between resilience against cascading failures and frequency stability. Plietzsch, A., Schultz, P., Heitzig, J., Kurths, J. European Physical Journal-Special Topics, 225(3), 551-568, 2016, DOI: 10.1140/epjst/e2015-50137-4. Abstract: When designing or extending electricity grids, both frequency stability and resilience against cascading failures have to be considered amongst other aspects of energy security and economics such as construction costs due to total line length. Here, we compare an improved simulation model for cascading failures with state-of-the-art simulation models for short-term grid dynamics. Random ensembles of realistic power grid topologies are generated using a recent model that allows for a tuning of global vs local redundancy. The former can be measured by the algebraic connectivity of the network, whereas the latter can be measured by the networks transitivity. We show that, while frequency stability of an electricity grid benefits from a global form of redundancy, resilience against cascading failures rather requires a more local form of redundancy and further analyse the corresponding trade-off. Long-range response in ac electricity grids. Jung, D., Kettemann, S. Journal: Phys. Rev. E 94, 012307 (2016), DOI: 10.1103/PhysRevE.94.012307. Abstract: Local changes in the topology of electricity grids can cause overloads far away from the disturbance [D. Witthaut and M. Timme, Eur. Phys. J. B 86, 377 (2013)], making the prediction of the robustness against changes in the topology—for example, caused by power outages or grid extensions—a challenging task. The impact of single-line additions on the long-range response of dc electricity grids has recently been studied [D. Labavic, R. Suciu, H. Meyer-Ortmanns, and S. Kettemann, Eur. Phys. J.: Spec. Top. 223, 2517 (2014)]. By solving the real part of the static ac load flow equations, we conduct a similar investigation for ac grids. In a regular two-dimensional grid graph with cyclic boundary conditions, we find a power law decay for the change of power flow as a function of distance to the disturbance over a wide range of distances. The power exponent increases and saturates for large system sizes. By applying the same analysis to the German transmission grid topology, we show that also in real-world topologies a long-ranged response can be found. Optimising the European transmission system for 77% renewable electricity by 2030. Brown, T., Schierhorn, P.-P., Tröster, E., Ackermann, T. IET-RPG, Volume 10, Issue 1, p. 3–9 (Jan 2016). Abstract: Local changes in the topology of electricity grids can cause overloads far away from the disturbance [D. Witthaut and M. Timme, Eur. Phys. J. B 86, 377 (2013)], making the prediction of the robustness against changes in the topology—for example, caused by power outages or grid extensions—a challenging task. The impact of single-line additions on the long-range response of dc electricity grids has recently been studied [D. Labavic, R. Suciu, H. Meyer-Ortmanns, and S. Kettemann, Eur. Phys. J.: Spec. Top. 223, 2517 (2014)]. By solving the real part of the static ac load flow equations, we conduct a similar investigation for ac grids. In a regular two-dimensional grid graph with cyclic boundary conditions, we find a power law decay for the change of power flow as a function of distance to the disturbance over a wide range of distances. The power exponent increases and saturates for large system sizes. By applying the same analysis to the German transmission grid topology, we show that also in real-world topologies a long-ranged response can be found. Power flow tracing in complex networks. Schäfer, M., Hempel, S., Hörsch, J., Tranberg, S., Schramm, S. and Greiner M. In: Schramm, S. and Schäfer, M. (Eds.) New Horizons in Fundamental Physics. FIAS Interdisciplinary Science Series (2016). Abstract: The increasing share of decentralized renewable power generation represents a challenge to the current and future energy system. Providing a geographical smoothing effect, long-range power transmission plays a key role for the system integration of these fluctuating resources. However, the build-up and operation of the necessary network infrastructure incur costs which have to be allocated to the users of the system. Flow tracing techniques, which attribute the power flow on a transmission line to the geographical location of its generation and consumption, represent a valuable tool set to design fair usage and thus cost allocation schemes for transmission investments. In this article, we introduce a general formulation of the flow tracing method and apply it to a simplified model of a highly renewable European electricity system. We review a statistical usage measure which allows to integrate network usage information for longer time series, and illustrate this measure using an analytical test case. Representation of the German transmission grid for Renewable Energy Sources. Mureddu, M. arXiv:1612.05532 (2016). Abstract: The increasing impact of fossil energy generation on the Earth ecological balance is pointing to the need of a transition in power generation technology towards the more clean and sustainable Renewable Energy Sources (RES). This transition is leading to new paradigms and technologies useful for the effective energy transmission and distribution, which take into account the RES stochastic power output. In this scenario, the availability of up to date and reliable datasets regarding topological and operative parameters of power systems in presence of RES are needed, for both proposing and testing new solutions. In this spirit, I present here a dataset regarding the German 380 KV grid which contains fully DC Power Flow operative states of the grid in the presence of various amounts of RES share, ranging from realistic up to 60\%, which can be used as reference dataset for both steady state and dynamical analysis. Survivability of Deterministic Dynamical Systems. Hellmann, F., Schultz, P., Grabow, C., Heitzig, J., Kurths, J. Scientific Reports, 6, 29654 (2016), DOI: 10.1038/srep29654. Abstract: The notion of a part of phase space containing desired (or allowed) states of a dynamical system is important in a wide range of complex systems research. It has been called the safe operating space, the viability kernel or the sunny region. In this paper we define the notion of survivability: Given a random initial condition, what is the likelihood that the transient behaviour of a deterministic system does not leave a region of desirable states. We demonstrate the utility of this novel stability measure by considering models from climate science, neuronal networks and power grids. We also show that a semi-analytic lower bound for the survivability of linear systems allows a numerically very efficient survivability analysis in realistic models of power grids. Our numerical and semi-analytic work underlines that the type of stability measured by survivability is not captured by common asymptotic stability measures. Synchronizing noisy nonidentical oscillators by transient uncoupling. Tandon, A., Schröder, M., Mannattil, M., Timme, M., Chakraborty, S. Chaos 26:094817 (2016). Abstract: Synchronization is the process of achieving identical dynamics among coupled identical units. If the units are different from each other, their dynamics cannot become identical; yet, after transients, there may emerge a functional relationship between them—a phenomenon termed “generalized synchronization.” Here, we show that the concept of transient uncoupling, recently introduced for synchronizing identical units, also supports generalized synchronization among nonidentical chaotic units. Generalized synchronization can be achieved by transient uncoupling even when it is impossible by regular coupling. We furthermore demonstrate that transient uncoupling stabilizes synchronization in the presence of common noise. Transient uncoupling works best if the units stay uncoupled whenever the driven orbit visits regions that are locally diverging in its phase space. Thus, to select a favorable uncoupling region, we propose an intuitive method that measures the local divergence at the phase points of the driven unit's trajectory by linearizing the flow and subsequently suppresses the divergence by uncoupling. Taming Instabilities in Power Grid Networks by Decentralized Control. Schaefer, B., Grabow, C., Auer, S., Kurths, J., Witthaut, D., Timme, M. Eur. Phys. J. ST 225, 569 (2016). Abstract: Renewables will soon dominate energy production in our electric power system. And yet, how to integrate renewable energy into the grid and the market is still a subject of major debate. Decentral Smart Grid Control (DSGC) was recently proposed as a robust and decentralized approach to balance supply and demand and to guarantee a grid operation that is both economically and dynamically feasible. Here, we analyze the impact of network topology by assessing the stability of essential network motifs using both linear stability analysis and basin volume for delay systems. Our results indicate that if frequency measurements are averaged over sufficiently large time intervals, DSGC enhances the stability of extended power grid systems. We further investigate whether DSGC supports centralized and/or decentralized power production and find it to be applicable to both. However, our results on cycle-like systems suggest that DSGC favors systems with decentralized production. Here, lower line capacities and lower averaging times are required compared to those with centralized production. The impact of model detail on power grid resilience measures. Auer, S., Kleis, K., Schultz, P., Kurths, J., Hellmann, F. European Physical Journal-Special Topics, 225(3), 609-625, 2016, DOI: 10.1140/epjst/e2015-50265-9. Abstract: Extreme events are a challenge to natural as well as man-made systems. For critical infrastructure like power grids, we need to understand their resilience against large disturbances. Recently, new measures of the resilience of dynamical systems have been developed in the complex system literature. Basin stability and survivability respectively assess the asymptotic and transient behavior of a system when subjected to arbitrary, localized but large perturbations in frequency and phase. To employ these methods that assess power grid resilience, we need to choose a certain model detail of the power grid. For the grid topology we considered the Scandinavian grid and an ensemble of power grids generated with a random growth model. So far the most popular model that has been studied is the classical swing equation model for the frequency response of generators and motors. In this paper we study a more sophisticated model of synchronous machines that also takes voltage dynamics into account, and compare it to the previously studied model. This model has been found to give an accurate picture of the long term evolution of synchronous machines in the engineering literature for post fault studies. We find evidence that some stable fix points of the swing equation become unstable when we add voltage dynamics. If this occurs the asymptotic behavior of the system can be dramatically altered, and basin stability estimates obtained with the swing equation can be dramatically wrong. We also find that the survivability does not change significantly when taking the voltage dynamics into account. Further, the limit cycle type asymptotic behaviour is strongly correlated with transient voltages that violate typical operational voltage bounds. Thus, transient voltage bounds are dominated by transient frequency bounds and play no large role for realistic parameters. The relevance of grid expansion under zonal markets. Bertsch, J., Brown, T., Hagspiel, S., Just, L. The Energy Journal, IAEE, Vol. 38, Number 5(2016). Abstract: The European electricity market design is based on zonal markets with uniform prices. Hence, no differentiated locational price signals are provided within these zones. If intra-zonal congestion occurs due to missing grid expansion, this market design reveals its inherent incompleteness, and might lead to severe short and long-term distortions. In this paper, we study these distortions with a focus on the impact of restricted grid expansion under zonal markets. Therefore, we use a long-term model of the European electricity system and restrict the allowed expansion of the transmission grid per decade. We find that the combination of an incomplete market design and restricted grid expansion leads to a misallocation of generation capacities and the inability to transport electricity to where it is needed. This results in an energy imbalance in some regions of up to 2-3% and difficulty when reaching envisaged political targets in the power sector. Tweaking synchronization by connectivity modifications. Schultz, P., Peron, T., Eroglu, D., Stemler, T., Ramirez Avila, G. M., Rodrigues, F. A., Kurths, J. Physical Review E, 93(6), 062211, 2016, DOI: 10.1103/PhysRevE.93.062211. Abstract: Natural and man-made networks often possess locally treelike substructures. Taking such tree networks as our starting point, we show how the addition of links changes the synchronization properties of the network. We focus on two different methods of link addition. The first method adds single links that create cycles of a well-defined length. Following a topological approach, we introduce cycles of varying length and analyze how this feature, as well as the position in the network, alters the synchronous behavior. We show that in particular short cycles can lead to a maximum change of the Laplacian's eigenvalue spectrum, dictating the synchronization properties of such networks. The second method connects a certain proportion of the initially unconnected nodes. We simulate dynamical systems on these network topologies, with the nodes' local dynamics being either discrete or continuous. Here our main result is that a certain number of additional links, with the relative position in the network being crucial, can be beneficial to ensure stable synchronization. Wie verändern stärker leistungsbasierte Netzentgelte die Anreize für Eigenversorgung? Brandstätt, C. Im Tagungsband der OTTI-Veranstaltung Zukünftige Stromnetze für Erneuerbare Energien, Berlin, Januar 2016. - 2015 - Backup flexibility classes in emerging large-scale renewable electricity systems. Schlachtberger, D., Becker, S., Schramm, S., Greiner, M. Energy Conversion and Management 4, 20 (2016). Abstract: High shares of intermittent renewable power generation in a European electricity system will require flexible backup power generation on the dominant diurnal, synoptic, and seasonal weather timescales. The same three timescales are already covered by today’s dispatchable electricity generation facilities, which are able to follow the typical load variations on the intra-day, intra-week, and seasonal timescales. This work aims to quantify the changing demand for those three backup flexibility classes in emerging large-scale electricity systems, as they transform from low to high shares of variable renewable power generation. A weather-driven modelling is used, which aggregates eight years of wind and solar power generation data as well as load data over Germany and Europe, and splits the backup system required to cover the residual load into three flexibility classes distinguished by their respective maximum rates of change of power output. This modelling shows that the slowly flexible backup system is dominant at low renewable shares, but its optimized capacity decreases and drops close to zero once the average renewable power generation exceeds 50% of the mean load. The medium flexible backup capacities increase for modest renewable shares, peak at around a 40% renewable share, and then continuously decrease to almost zero once the average renewable power generation becomes larger than 100% of the mean load. The dispatch capacity of the highly flexible backup system becomes dominant for renewable shares beyond 50%, and reach their maximum around a 70% renewable share. For renewable shares above 70% the highly flexible backup capacity in Germany remains at its maximum, whereas it decreases again for Europe. This indicates that for highly renewable large-scale electricity systems the total required backup capacity can only be reduced if countries share their excess generation and backup power. Cost-optimal design of a simplified, highly renewable pan-European electricity system. Rodriguez, R.A., Becker, S., Greiner, M. Energy 83, p. 658–668 (2015). Abstract: Based on a data-intensive weather-driven modelling approach, technically and economically optimal designs are derived for a simplified, highly renewable pan-European electricity system, which minimise the need for backup energy, backup capacity, transmission capacity and the levelised system cost of delivered electricity. The overall cost-optimal design, based on standard cost assumptions, relies on synchronised backup across the transmission grid and comes with a renewable penetration of 50% with a rather high wind fraction of 94%. Given the current European electricity consumption, this corresponds to 600 GW rated wind power capacities, 60 GW installed solar power capacities, 320 GW conventional backup power capacity, and about five times today's installed transmission capacities. A sensitivity analysis reveals that the design and cost of the optimal system depend mostly on the assumed cost of wind capacity and fuel for backup energy. Lower costs for wind capacity, higher costs for backup energy and usage of otherwise curtailed excess electricity generation lead to a strong increase of the optimal renewable penetration. The sensitivity analysis is also used to find that a CO2 tax of over 100 €/ton would be needed for the economic viability of carbon capture and sequestration. Decentral Smart Grid Control. Schäfer, B., Matthiae, M., Timme, M., Witthaut, D. Journal: New Journal of Physics, 17, 015002 (2015), 10.1088/1367-2630/17/1/015002. Abstract: Stable operation of complex flow and transportation networks requires balanced supply and demand. For the operation of electric power grids—due to their increasing fraction of renewable energy sources —a pressing challenge is to fit the fluctuations in decentralized supply to the distributed and temporally varying demands. To achieve this goal, common smart grid concepts suggest to collect consumer demand data, centrally evaluate them given current supply and send price information back to customers for them to decide about usage. Besides restrictions regarding cyber security, privacy protection and large required investments, it remains unclear how such central smart grid options guarantee overall stability. Here we propose a Decentral Smart Grid Control, where the price is directly linked to the local grid frequency at each customer. The grid frequency provides all necessary information about the current power balance such that it is sufficient to match supply and demand without the need for a centralized IT infrastructure.Weanalyze the performance and the dynamical stability of the power grid with such a control system. Our results suggest that the proposed Decentral Smart Grid Control is feasible independent of effective measurement delays, if frequencies are averaged over sufficiently large time intervals. Delocalization of Phase Perturbations and the Stability of Electricity Grids. Kettemann, S. arXiv:1504.05525v2, subm. to Phys. Rev. Lett. (2015). Abstract: The energy transition towards an increased supply of renewable energy raises concerns that existing electricity grids, built to connect few centralized large power plants with consumers, may become more difficult to control and stabilized with a rising number of decentralized small scale generators. Here, we aim to study therefore, how local phase perturbations which may be caused by local power fluctuations, affect the AC grid stability. To this end, we start from nonlinear power balance equations and map them to complex linear wave equations, yielding stationary solutions with phases fi at generator and consumer sites i. Next, we study deviations from these stationary solutions. Starting with an initially localized perturbation, it is found to spread in a periodic grid diffusively throughout the grid. We derive the parametric dependence of diffusion constant D. We apply the same solution strategy to general grid topologies and analyse their stability against local perturbations. The perturbation remains either localized or becomes delocalized, depending on grid topology, power capacity and distribution of consumers and generators Pi. Delocalization is found to increase the lifetime of perturbations and thereby their influence on grid stability, while localization results in an exponentiallyfast decay of perturbations at all grid sites. These results may therefore lead to new strategies to control the stability of electricity grids. Distribution planning and pricing in view of increasing shares of intermittent, renewable energy in Germany and Japan. Brandstätt, C., Brunekreeft, G., Furusawa, K., Hattori, T. Renewable Energy in Germany and Japan (2015). Abstract: In response to the global climate challenge many countries are faced with increasing shares of energy from renewable sources in their power supply. The integration of RES (renewable energy sources) generation however entails technical as well as institutional challenges for power grids. This study relies on recent experiences of German distribution network operators in network planning and network pricing and looks at their transferability to Japan. Distributed generation may cause problems of voltage variation and asset overloading in conventional power grids. Technical solutions for these problems are available and well-known yet require considerable investments. The study presents regulatory incentives for network operators to take efficient means to maintain supply quality. With distributed generation self-supplying customers may contribute too little to network cost and new generators and flexible consumers may cause significant investment by uncoordinated siting and operation. An adequate pricing scheme can serve to sustainably finance the infrastructure while at the same time giving incentives to coordinate network users. This study points out options for network charging in grids with high shares of distributed generation from renewable sources. Focus on Networks, Energy and the Economy. Timme, M., Kocarev, L., Witthaut, D. New J. Phys. 17, 110201 (2015). Abstract: A sustainable and reliable energy supply constitutes a fundamental prerequisite for the future of our society. The change to renewable sources comes with several systemic changes and includes, among others, smaller and more distributed producers as well as stronger and less predictable fluctuations. Parallel developments such as the transition from conventional producers and consumers to prosumers and the increasing number of electric vehicles add further complications. These changes require to extend and upgrade currently existing power grids. Yet precisely how to achieve an effective, robustly operating (electric) energy system is far from being understood. This focus issue aims to contribute to a number of these upcoming challenges from the perspective of self-organization and the collective nonlinear dynamics of power grids, interacting economic factors as well as technical restrictions and opportunities for distributed systems. Kuramoto dynamics in Hamiltonian systems. Witthaut, D., Timme, M.. Journal: Physical Review E, 90, 032917 (2014). DOI: 10.1103/PhysRevE.90.032917. Abstract: The Kuramoto model constitutes a paradigmatic model for the dissipative collective dynamics of coupled oscillators, characterizing in particular the emergence of synchrony (phase locking). Here we present a classical Hamiltonian (and thus conservative) system with 2N state variables that in its action-angle representation exactly yields Kuramoto dynamics on N-dimensional invariant manifolds. We show that locking of the phase of one oscillator on a Kuramoto manifold to the average phase emerges where the transverse Hamiltonian action dynamics of that specific oscillator becomes unstable. Moreover, the inverse participation ratio of the Hamiltonian dynamics perturbed off the manifold indicates the global synchronization transition point for finite N more precisely than the standard Kuramoto order parameter. The uncovered Kuramoto dynamics in Hamiltonian systems thus distinctly links dissipative to conservative dynamics. Localized vs. synchronized exports across a highly renewable pan-European transmission network. Rodriguez, R.A., Dahl, M., Becker, S., Greiner, M. Energy, Sustainability and Society 5:21 (2015). Abstract: Background: A future, highly renewable electricity system will be largely based on fluctuating renewables. The integration of wind and solar photovoltaics presents a major challenge. Transmission can be used to lower the need for complementary generation, which we term backup in this article. Methods: Generation data based on historical weather data, combined with real load data, determine hourly mismatch timeseries for all European countries, connected by physical power flows. Two localized export schemes determining the power flows are discussed, which export only renewable excess power, but no backup power, and are compared to a synchronized export scheme, which exports renewable excess power and also backup power. Results: Compared to no or very limited power transmission, unconstrained power flows across a highly renewable pan-European electricity network significantly reduce the overall amount of required annual backup energy, but not necessarily the required backup capacities. Conclusions: The reduction of the backup capacities turns out to be sensitive to the choice of export scheme. Results suggest that the synchronized export of local backup power to other countries is important to significantly save on installed backup capacities. Nonlocal effects and counter measures in cascading failures. Witthaut, D. Timme, M. Phys. Rev. E 92, 032809 (2015). Abstract: We study the propagation of cascading failures in complex supply networks with a focus on nonlocal effects occurring far away from the initial failure. It is shown that a high clustering and a small average path length of a network generally suppress nonlocal overloads. These properties are typical for many real-world networks, often called small-world networks, such that cascades propagate mostly locally in these networks. Furthermore, we analyze the spatial aspects of countermeasures based on the intentional removal of additional edges. Nonlocal actions are generally required in networks that have a low redundancy and are thus especially vulnerable to cascades. Power flow tracing in a simplified highly renewable European electricity network. Tranberg, B., Thomsen, A., Rodriguez, R., Andresen, G., Schäfer, M., Greiner, M. New Journal of Physics, 17(10):105002 (2015). Abstract: The increasing transmission capacity needs in a future energy system raise the question of how associated costs should be allocated to the users of a strengthened power grid. In contrast to straightforward oversimplified methods, a flow tracing based approach provides a fair and consistent nodal usage and thus cost assignment of transmission investments. This technique follows the power flow through the network and assigns the link capacity usage to the respective sources or sinks using a diffusion-like process, thus taking into account the underlying network structure and injection pattern. As a showcase, we apply power flow tracing to a simplified model of the European electricity grid with a high share of renewable wind and solar power generation, based on long-term weather and load data with an hourly temporal resolution. Renewable electricity: Generation costs are not system costs. Becker, S., Frew, B.A., Andresen, G.B., Jacobson, M.Z., Schramm, S., Greiner, M. Energy 81, p. 437–445 (2015). preprint available at http://arxiv.org/abs/1412.4934 Abstract: The transition to a future electricity system based primarily on wind and solar PV is examined for all regions in the contiguous US. We present optimized pathways for the build-up of wind and solar power for least backup energy needs as well as for least cost obtained with a simplified, lightweight model based on long-term high resolution weather-determined generation data. In the absence of storage, the pathway which achieves the best match of generation and load, thus resulting in the least backup energy requirements, generally favors a combination of both technologies, with a wind/solar PV (photovoltaics) energy mix of about 80/20 in a fully renewable scenario. The least cost development is seen to start with 100% of the technology with the lowest average generation costs first, but with increasing renewable installations, economically unfavorable excess generation pushes it toward the minimal backup pathway. Surplus generation and the entailed costs can be reduced significantly by combining wind and solar power, and/or absorbing excess generation, for example with storage or transmission, or by coupling the electricity system to other energy sectors. The Dynamics of Coalition Formation on Complex Networks. Auer, S. Heitzig, J. Kornek, U. Schöll, E. Kurths, J. Nature Scientific Reports 5, 13386; (2015). DOI: 10.1038/srep13386. Abstract: Complex networks describe the structure of many socio-economic systems. However, in studies of decision-making processes the evolution of the underlying social relations are disregarded. In this report, we aim to understand the formation of self-organizing domains of cooperation (“coalitions”) on an acquaintance network. We include both the network’s influence on the formation of coalitions and vice versa how the network adapts to the current coalition structure, thus forming a social feedback loop. We increase complexity from simple opinion adaptation processes studied in earlier research to more complex decision-making determined by costs and benefits, and from bilateral to multilateral cooperation. We show how phase transitions emerge from such coevolutionary dynamics, which can be interpreted as processes of great transformations. If the network adaptation rate is high, the social dynamics prevent the formation of a grand coalition and therefore full cooperation. We find some empirical support for our main results: Our model develops a bimodal coalition size distribution over time similar to those found in social structures. Our detection and distinguishing of phase transitions may be exemplary for other models of socio-economic systems with low agent numbers and therefore strong finite-size effects. The relevance of grid expansion under zonal markets. Bertsch, J., Brown, T., Hagspiel, S., Just, L. EWI Working Paper, No 15/07 (2015), accepted pending review in The Energy Journal. Abstract: The European electricity market design is based on zonal markets with uniform prices. Locational price signals within these zones - necessary to ensure long-term efficiency - are not provided. Speci cally, if intra-zonal congestion occurs due to missing grid expansion, the market design is revealed as inherently incomplete. This might lead to severe, unwanted distortions of the electricity market, both in the short- and in the long-term. In this paper, we study these distortions with a speci c focus on the impact of restricted grid expansion under zonal markets. For this, we use a long term fundamental dispatch and investment model of the European electricity system and gradually restrict the allowed expansion of the transmission grid per decade. We nd that the combination of an incomplete market design and restricted grid expansion leads to a misallocation of generation capacities and the inability to transport electricity to where it is needed. Consequences are severe and lead to load curtailment of up to 2-3 %. Moreover, missing grid expansion makes it difficult and costly to reach envisaged energy targets in the power sector. Hence, we argue that in the likely event of restricted grid expansion, either administrative measures or - presumably more efficient - an adaptation of the current market design to include locational signals will become necessary. Transient Uncoupling Induces Synchronization. M. Schröder, M. Mannattil, D. Dutta, S. Chakraborty, M. Timme. Phys. Rev. Lett. 115:054101 (2015). Abstract: Finding conditions that support synchronization is a fertile and active area of research with applications across multiple disciplines. Here we present and analyze a scheme for synchronizing chaotic dynamical systems by transiently uncoupling them. Specifically, systems coupled only in a fraction of their state space may synchronize even if fully coupled they do not. While for many standard systems coupling strengths need to be bounded to ensure synchrony, transient uncoupling removes this bound and thus enables synchronization in an infinite range of effective coupling strengths. The presented coupling scheme therefore opens up the possibility to induce synchrony in (biological or technical) systems whose parameters are fixed and cannot be modified continuously. Value of Lost Load: An Efficient Economic Indicator for Power Supply Security? A Literature Review. Schröder, T., Kuckshinrichs, W. Front. Energy Res., 24 December 2015. Abstract: Security of electricity supply has become a fundamental requirement for well-functioning modern societies. Because of its central position in all sections of society, the present paper considers the economic consequences of a power supply interruption. The value of lost load (VoLL) is a monetary indicator expressing the costs associated with an interruption of electricity supply. This paper reviews different methods for calculating VoLL, provides an overview of recently published studies, and presents suggestions to increase the explanatory power and international comparability of VoLL. Whose line is it anyway? Tracing the flows through Germany’s power grid. Hörsch, J., Schäfer, M., Becker, S., Schramm, S., Greiner, M. submitted for review and published as SDEWES proceedings (Sep 2015). - 2014 - A random growth model for power grids and other spatially embedded infrastructure networks. Schultz, P., Heitzig, J., Kurths, J. European Physical Journal-Special Topics, 223(12), 2593-2610, (2014), DOI: 10.1140/epjst/e2014-02279-6. Abstract: We propose a model to create synthetic networks that may also serve as a narrative of a certain kind of infrastructure network evolution. It consists of an initialization phase with the network extending tree-like for minimum cost and a growth phase with an attachment rule giving a trade-off between cost-optimization and redundancy. Furthermore, we implement the feature of some lines being split during the grid's evolution. We show that the resulting degree distribution has an exponential tail and may show a maximum at degree two, suitable to observations of real-world power grid networks. In particular, the mean degree and the slope of the exponential decay can be controlled in partial independence. To verify to which extent the degree distribution is described by our analytic form, we conduct statistical tests, showing that the hypothesis of an exponential tail is well-accepted for our model data. Detours around basin stability in power networks. Schultz, P., Heitzig, J., Kurths, J. New Journal of Physics, 16, (2014), DOI: 10.1088/1367-2630/16/12/125001. Abstract: To analyse the relationship between stability against large perturbations and topological properties of a power transmission grid, we employ a statistical analysis of a large ensemble of synthetic power grids, looking for significant statistical relationships between the single-node basin stability measure and classical as well as tailormade weighted network characteristics. This method enables us to predict poor values of single-node basin stability for a large extent of the nodes, offering a node-wise stability estimation at low computational cost. Further, we analyse the particular function of certain network motifs to promote or degrade the stability of the system. Here we uncover the impact of so-called detour motifs on the appearance of nodes with a poor stability score and discuss the implications for power grid design. Impact of network topology on synchrony of oscillatory power grids. Rohden, M., Sorge, A., Witthaut, D., Timme, M. Chaos: an Interdisciplinary Journal of Nonlinear Science, 24(1), 013123, (2014). Abstract: Replacing conventional power sources by renewable sources in current power grids drastically alters their structure and functionality. In particular, power generation in the resulting grid will be far more decentralized, with a distinctly different topology. Here, we analyze the impact of grid topologies on spontaneous synchronization, considering regular, random, and small-world topologies and focusing on the influence of decentralization. We model the consumers and sources of the power grid as second order oscillators. First, we analyze the global dynamics of the simplest non-trivial (two-node) network that exhibit a synchronous (normal operation) state, a limit cycle (power outage), and coexistence of both. Second, we estimate stability thresholds for the collective dynamics of small network motifs, in particular, star-like networks and regular grid motifs. For larger networks, we numerically investigate decentralization scenarios finding that decentralization itself may support power grids in exhibiting a stable state for lower transmission line capacities. Decentralization may thus be beneficial for power grids, regardless of the details of their resulting topology. Regular grids show a specific sharper transition not found for random or small-world grids. Supply networks: Instabilities without overload. Manik, D., Witthaut, D., Schaefer, B., Matthiae, M., Sorge, A., Rohden, M., Katifori, E., Timme, M. Journal: European Physical Journal Special Topics, 223, 2527 (2014). DOI: 10.1140/epjst/e2014-02274-y. Abstract: Supply and transport networks support much of our technical infrastructure as well as many biological processes. Their reliable function is thus essential for all aspects of life. Transport processes involving quantities beyond the pure loads exhibit alternative collective dynamical options compared to processes exclusively characterized by loads. Here we analyze the stability and bifurcations in oscillator models describing electric power grids and demonstrate that these networks exhibit instabilities without overloads. This phenomenon may well emerge also in other sufficiently complex supply or transport networks, including biological transport processes. |
ANWENDERPUBLIKATIONEN Volatil und doch stabil - Forschungen zur Förderung der erneuerbaren Energien Zeitschrift Elektropraktiker 06/2016 http://www.elektropraktiker.de/nc/fachinformationen/fachartikel/volatil-und-doch-stabil/ |