  The energy transition towards high shares of renewable energy will affect the
stability of electricity grids in many ways. Here, we aim to study its impact
on propagation of disturbances by solving nonlinear swing equations describing
coupled rotating masses of synchronous generators and motors on different grid
topologies. We consider a tree, a square grid and as a real grid topology, the
German transmission grid. We identify ranges of parameters with different
transient dynamics: the disturbance decays exponentially in time, superimposed
by oscillations with the fast decay rate of a single node, or with a smaller
decay rate without oscillations. Most remarkably, as the grid inertia is
lowered, nodes may become correlated, slowing down the propagation from
ballistic to diffusive motion, decaying with a power law in time. Applying
linear response theory, we show that tree grids have a spectral gap leading to
exponential relaxation as protected by topology and independent on grid size.
Meshed grids are found to have a spectral gap which decreases with increasing
grid size, leading to slow power law relaxation and collective diffusive
propagation of disturbances. We conclude by discussing consequences if no
measures are undertaken to preserve the grid inertia in the energy transition.
