Power grids are undergoing major changes from a few large producers to smart grids build upon renewable energies. Mathematical models for power grid dynamics have to be adapted to capture, when dynamic nodes can achieve synchronization to a common grid frequency on complex network topologies. In this paper we study a second-order rotator model in the large network limit. We merge the recent theory of random graph limits for complex small-world networks with approaches to first-order systems on graphons. We prove that there exists a well-posed continuum limit integral equation approximating the large finite-dimensional case power grid network dynamics. Then we analyse the linear stability of synchronized solutions and prove linear stability. However, on small-world networks we demonstrate that there are topological parameters moving the spectrum arbitrarily close to the imaginary axis leading to potential instability on finite time scales.
We show the existence of self-similar solutions with fat tails for Smoluchowski's coagulation equation for homogeneous kernels satisfying C1 x −a y b + x b y −a ≤ K (x, y) ≤ C2 x −a y b + x b y −a with a > 0 and b <
In this paper we consider the long time asymptotics of a linear version of the Smoluchowski equation which describes the evolution of a tagged particle moving at constant speed in a random distribution of fixed particles. The volumes v of the particles are independently distributed according to a probability distribution which decays asymptotically as a power law v −σ . The validity of the equation has been rigorously proved in [19] for values of the exponent σ > 3. The solutions of this equation display a rich structure of different asymptotic behaviours according to the different values of the exponent σ. Here we show that for 5 3 < σ < 2 the linear Smoluchowski equation is well posed and that there exists a unique self-similar profile which is asymptotically stable.
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