Spreading of disturbances in realistic models of transmission grids in dependence on topology, inertia and heterogeneity.

The energy transition towards more renewable energy resources (RER) profoundly affects the frequency dynamics and stability of electrical power networks. Here, we investigate systematically the effect of reduced grid inertia, due to an increase in the magnitude of RER, its heterogeneous distribution and the grid topology on the propagation of disturbances in realistic power grid models. These studies are conducted with the DigSILENT PowerFactory software. By changing the power generation at one central bus in each grid at a specific time, we record the resulting frequency transients at all buses. Plotting the time of arrival (ToA) of the disturbance at each bus versus the distance from the disturbance, we analyse its propagation throughout the grid. While the ToAs are found to be distributed, we confirm a tendency that the ToA increases with geodesic distance linearly. Thereby, we can measure an average velocity of propagation by fitting the data with a ballistic equation. This velocity is found to decay with increasing inertia. Characterising each grid by its meshedness coefficient, we find that the distribution of the ToAs depends in more meshed grids less strongly on the grid inertia. In order to take into account the inhomogeneous distribution of inertia, we introduce an effective distance [Formula: see text], which is weighted with a factor which strongly depends on local inertia. We find that this effective distance is more strongly correlated with the ToAs, for all grids. This is confirmed quantitatively by obtaining a larger Pearson correlation coefficient between ToA and [Formula: see text] than with r. Remarkably, a ballistic equation for the ToA with a velocity, as derived from the swing equation, provides a strict lower bound for all effective distances [Formula: see text] in all power grids. thereby yielding a reliable estimate for the smallest time a disturbance needs to propagate that distance as function of system parameters, in particular inertia. We thereby conclude that in the analysis of contingencies of power grids it may be advisable that system designers and operators use the effective distance [Formula: see text], taking into account inhomogeneous distribution of inertia as introduced in Eq. (12), to locate a disturbance. Moreover, our results provide evidence for the importance of the network topology as quantified by the meshedness coefficient [Formula: see text].

Analysis of the dynamics and topology dependencies of small perturbations in electric transmission grids.

We study the phase dynamics in power grids in response to small disturbances and how this depends on the grid topology. To this end, we consider the swing equations in linear order in phase disturbances and solve the resulting linear wave equation, deriving the eigenmodes of the weighted graph Laplacian. A linear response expression for the deviation of frequency is given in terms of these eigenvalues and eigenvectors, which it is argued to be the basis for future power system stabilizers and other control measures in power systems. As an example, we present results for random networks based on the Watts-Strogatz model, where we observe a transition to localized eigenstates as the randomness in the degree distribution grows. Moreover, it is found that localization leads to faster decay rates. Thereby, disturbances are found to remain localized on a few nodes where they decay faster. Finally, we also consider the German transmission grid topology, where the eigenstate of the lowest eigenfrequency, the Fiedler vector, is found to be extended, with large intensities at the northwestern and southern boundaries.

Time delay effects in the control of synchronous electricity grids.

The expansion of inverter-connected generation facilities (i.e., wind and photovoltaics) and the removal of conventional power plants is necessary to mitigate the impacts of climate change, whereas conventional generation with large rotating generator masses provides stabilizing inertia, inverter-connected generation does not. Since the underlying power system and the control mechanisms that keep it close to a desired reference state were not designed for such a low inertia system, this might make the system vulnerable to disturbances. In this paper, we will investigate whether the currently used control mechanisms are able to keep a low inertia system stable and how this is affected by the time delay between a frequency deviation and the onset of the control action. We integrate the control mechanisms used in Continental Europe into a model of coupled oscillators which resembles the second order Kuramoto model. This model is then used to investigate how the interplay of changing inertia, network topology, and delayed control affects the stability of the interconnected power system. To identify regions in the parameter space that make stable grid operation possible, the linearized system is analyzed to create the system's stability chart. We show that lower and distributed inertia could have a beneficial effect on the stability of the desired synchronous state.

Propagation of wind-power-induced fluctuations in power grids.

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.

Propagation of Disturbances in AC Electricity Grids.

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.

Cascading failures in ac electricity grids.

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.

Long-range response in ac electricity grids.

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)EPJBFY1434-602810.1140/epjb/e2013-40469-4], 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)1951-635510.1140/epjst/e2014-02273-0]. 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.

Delocalization of disturbances and the stability of ac electricity grids.

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 φ_{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 P_{i}. 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.

Nanomechanical detection of itinerant electron spin flip.

Electrons and other fundamental particles have an intrinsic angular momentum called spin. A change in the spin state of such a particle is therefore equivalent to a mechanical torque. This spin-induced torque is central to our understanding of experiments ranging from the measurement of the angular momentum of photons and the g-factor of metals to magnetic resonance and magnetization reversal in magnetic multilayers. When a spin-polarized current passes through a metallic nanowire in which one half is ferromagnetic and the other half is nonmagnetic, the spins of the itinerant electrons are 'flipped' at the interface between the two regions to produces a torque. Here, we report direct measurement of this mechanical torque in an integrated nanoscale torsion oscillator, and measurements of the itinerant electron spin polarization that could yield new information on the itinerancy of the d-band electrons. The unprecedented torque sensitivity of 1 x 10(-22) N-m Hz(-1/2) may have applications in spintronics and precision measurements of charge-parity-violating forces, and might also enable experiments on the untwisting of DNA and torque-generating molecules.