A steadily increasing fraction of renewable energy sources for electricity production requires a better understanding of how stochastic power generation affects the stability of electricity grids. Here, we assess the resilience of an IEEE test grid against single transmission line overloads under wind power injection based on the dc power flow equations and a quasi-static grid response to wind fluctuations. Thereby we focus on the mutual influence of wind power generation at different nodes. We find that overload probabilities vary strongly between different pairs of nodes and become highly affected by spatial correlations of wind fluctuations. An unexpected behaviour is uncovered: for a large number of node pairs, increasing wind power injection at one node can increase the power threshold at the other node with respect to line overloads in the grid. We find that this seemingly paradoxical behaviour is related to the topological distance of the overloaded line from the shortest path connecting the wind nodes. In the considered test grid, it occurs for all node pairs, where the overloaded line belongs to the shortest path.

Molecular self-assembly on surfaces constitutes a powerful method for creating tailor-made surface structures with dedicated functionalities. Varying the intermolecular interactions allows for tuning the resulting molecular structures in a rational fashion. So far, however, the discussion on the involved intermolecular interactions is often limited to attractive forces only. In real systems, the intermolecular interaction can be composed of both attractive and repulsive forces. Adjusting the balance between these interactions provides a promising strategy for extending the structural variety of molecular self-assembly on surfaces. However, this strategy relies on a method to quantify the involved interactions. Here, we investigate a molecular model system of 3-hydroxybenzoic acid (3-HBA) molecules on calcite (10.4) in a ultrahigh vacuum. This system offers both anisotropic short-range attraction and long-range repulsive dipolar interactions between molecules, resulting in the self-assembly of molecular stripes. We analyze the stripe-to-stripe distance distribution and the stripe length distribution and compare these distributions with analytical expressions from an anisotropic Ising model with additional repulsive interactions. We show that this approach allows the extraction of quantitative information about the strength of the attractive and repulsive interactions. Our work demonstrates how the detailed analysis of the self-assembled structures can be used to obtain quantitative insight into the molecule−molecule interactions.

Mobile molecules on surfaces can arrange into stripes due to directional attractive interactions such as π−π stacking, hydrogen, or covalent bonding. The structural arrangement of the stripes depends on the underlying substrate lattice and omnipresent long-range electrostatic interactions. To model the impact of the interplay of short-range attractive and long-range interactions on the molecular arrangements, we study a coarsegrained theoretical approach, where the attractive interaction is described by an anisotropic Ising model. As for the long-range electrostatic interaction, we focus on repulsive dipole−dipole interactions. An efficient Monte Carlo algorithm is developed by which even stripe patterns with very long stripes can be equilibrated. Using this algorithm, we assess the limits of a previously developed mean-field theory, which provides analytical predictions for stripe-to-stripe distance and stripe length distributions. This theory allows one to extract interaction parameters by fitting respective distributions to experimental data. We determine the limits of the applicability of the mean-field theory and beyond its limits suggest a combined approach of mean-field analysis and simulations. The power of this approach is demonstrated by applying it to experimental observed stripe pattern of 3hydroxybenzoic acid (3-HBA) on the calcite (10.4) surface.

We present a theory for analyzing residence times of single molecules in a fixed detection area of a scanning tunneling microscope (STM). The approach is developed for one-dimensional molecule diffusion and can be extended to two dimensions using the same methodology. Explicit results are derived for an harmonic attractive and repulsive tip-molecule interaction. Applications of the theory allows one to estimate the type and strength of interactions between the STM tip and the molecule. This includes the possibility of an estimation of molecule-molecule interaction when the tip is decorated by a molecule. Despite our focus on STM, this theory can analogously be applied to other experimental probes that monitor single molecules.