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Linear Stability and the Braess Paradox in Coupled Oscillators Networks and Electric Power Grids

We investigate the influence that adding a new coupling has on the linear stability of the synchronous state in coupled oscillators networks. Using a simple chain model we show that, depending on its location, the new coupling can lead to enhanced or reduced stability. We extend these results to electric power grids where a new line can lead to four different scenarios corresponding to enhanced or reduced grid stability as well as increased or decreased power flows. Our analysis shows that the Braess paradox may occur in any complex coupled system, where the synchronous state may be weakened and sometimes even destroyed by additional couplings. We further note that the addition of couplings which create loops in the network can stabilize new solutions, labeled by winding numbers, which are related to each other by circulating power flows.