The ongoing transition to renewable energy supply comes with a restructuring of power grids, changing their effective interaction topologies, more and more strongly decentralizing them and substantially modifying their input, output, and response characteristics. All of these changes imply that power grids become increasingly affected by collective, nonlinear dynamic phenomena, structurally and dynamically more distributed and less predictable in space and time, more heterogeneous in its building blocks, and as a consequence less centrally controllable. Here cornerstone aspects of data-driven and mathematical modeling of collective dynamical phenomena emerging in real and model power grid networks by combining theories from nonlinear dynamics, stochastic processes and statistical physics, anomalous statistics, optimization, and graph theory are reviewed. The mathematical background required for adequate modeling and analysis approaches is introduced, an overview of power system models is given, and a range of collective dynamical phenomena are focused on, including synchronization and phase locking, flow (re)routing, Braess’s paradox, geometric frustration, and spreading and localization of perturbations and cascading failures, as well as the nonequilibrium dynamics of power grids, where fluctuations play a pivotal role.

The current understanding of finite-temperature phase transitions in QCD is reviewed. A critical discussion of refined phase-transition criteria in numerical lattice simulations and of analytical tools going beyond the mean-field level in effective continuum models for QCD is presented. Theoretical predictions about the order of the transitions are compared with possible experimental manifestations in heavy-ion collisions. Various places in phenomenological descriptions are pointed out where more reliable data for QCD's equation of state would help in selecting the most realistic scenario among those proposed. Unanswered questions are raised about the relevance of calculations that assume thermodynamic equilibrium. Promising new approaches for implementing nonequilibrium aspects of the thermodynamics of heavy-ion collisions are described. [S0034-6861(96)00302-9]

Self-similar networks with scale-free degree distribution have recently attracted much attention, since these apparently incompatible properties were reconciled in [C. Song, S. Havlin, and H. A. Makse, Nature 433, 392 (2005)] by an appropriate box-counting method that enters the measurement of the fractal dimension. We study two genetic regulatory networks (Saccharomyces cerevisiae [N. M. Luscombe, M. M. Babu, H. Yu, M. Snyder, S. Teichmann, and M. Gerstein, Nature 431, 308 (2004)] and Escherichia coli [http://www.ccg.unam.mx/ComputationaḻGenomics/regulondb/DataSets/RegulonNetDataSets.html and http://www.gbf.de/SystemsBiology]) and show their self-similar and scale-free features, in extension to the datasets studied by [C. Song, S. Havlin, and H. A. Makse, Nature 433, 392 (2005)]. Moreover, by a number of numerical results we support the conjecture that self-similar scale-free networks are not assortative. From our simulations so far these networks seem to be disassortative instead. We also find that the qualitative feature of disassortativity is scale-invariant under renormalization, but it appears as an intrinsic feature of the renormalization prescription, as even assortative networks become disassortative after a sufficient number of renormalization steps.

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.