A Generalized Linear Response Theory of Complex Networks with an Application to Renewable Fluctuations in Microgrids.

In this work we study the general linear response theory for the distribution of energy fluctuations through complex networks. We develop the response equations for oscillators coupled on arbitrary, directed and weighted networks, when subjected to stationary fluctuations with arbitrary power spectra. Guided by the case study of network models for the distributed control and stabilization of turbulent renewable energy fluctuations in power grids, we then develop approximations that capture the most impactful interactions between intrinsic network modes and typical fluctuations found in renewable energies. These cover an intermediate resonant regime where the fluctuations are neither slow enough to cause a homogeneous response of the whole system, nor fast enough to be localized on the network. Applying these analytic approximations to the question which nodes in a microgrid are particularly vulnerable to fluctuations, we are able to give analytic explanations and expressions for the previously numerically observed network patterns in vulnerability. We see that these effects can only be explained by taking the losses on the lines, and the resulting asymmetry in the effective weighted graph Laplacian, into account. These structural asymmetries give rise to a dynamical asymmetry between nodes that cause a strong response when perturbed (troublemaker nodes), and nodes that always respond strongly whenever the network is somewhere perturbed (excitable nodes). For the important special case of tree-like networks we derive a simple relation for troublemaker nodes stating that fluctuations are enhanced when going upstream. The general theory also opens the door to future investigations into the stabilization of networks under correlated distributed fluctuations.