The human brain, power grids, arrays of coupled lasersand, the Amazon rainforest are all characterized by multistability. The likelihood that these systems will remain in the most desirable of their many stable states depends on their stability against significant perturbations, particularly in a state space populated by undesirable states. Here we claim that the traditional linearization-based approach to stability is too local to adequately assess how stable a state is. Instead, we quantify it in terms of basin stability, a new measure related to the volume of the basin of attraction. Basin stability is non-local, nonlinear and easily applicable, even to high-dimensional systems. It provides a long-sought-after explanation for the surprisingly regular topologies of neural networks and power grids, which have eluded theoretical description based solely on linear stability. We anticipate that basin stability will provide a powerful tool for complex systems studies, including the assessment of multistable climatic tipping elements.
Specifically, we employ a novel component-wise version of basin stability, a nonlinear inspection scheme, to investigate how a grid's degree of stability is influenced by certain patterns in the wiring topology. Various statistics from our ensemble simulations all support one main finding: The widespread and cheapest of all connection schemes, namely dead ends and dead trees, strongly diminish stability. For the Northern European power system we demonstrate that the inverse is also true: `Healing' dead ends by addition of transmission lines substantially enhances stability. This indicates a crucial smart-design principle for tomorrow's sustainable power grids: add just a few more lines to avoid dead ends.
Moreover, we investigate in detail how stable one singlenode or the whole power grid is against large perturbations. Specifically, we start by illustrating basin stability using the simple mode with one generator and one load, and then extend it to the IEEE Reliability Test System 30. A first-order transition in the plot of basin stability versus coupling strengths is observed in not only the simple model but also the whole network.
Reference
P. Menck, J. Heitzig, N. Marwan, and J. Kurths, Nature Physics 9, 89 (2013)
P. Menck, J. Heitzig, J. Kurths, and H. Schellnhuber, Nature Communication 5, 3969 (2014)