  A major transition in the operation of electric power grids is the
replacement of synchronous machines by distributed generation connected via
power electronic converters. The accompanying "loss of rotational inertia" and
the fluctuations by renewable sources jeopardize the system stability, as
testified by the ever-growing number of frequency incidents. As a remedy,
numerous studies demonstrate how virtual inertia can be emulated through
various devices, but few of them address the question of "where" to place this
inertia. It is however strongly believed that the placement of virtual inertia
hugely impacts system efficiency, as demonstrated by recent case studies. In
this article, we carry out a comprehensive analysis in an attempt to address
the optimal inertia placement problem. We consider a linear network-reduced
power system model along with an H2 performance metric accounting for the
network coherency. The optimal inertia placement problem turns out to be
non-convex, yet we provide a set of closed-form global optimality results for
particular problem instances as well as a computational approach resulting in
locally optimal solutions. Further, we also consider the robust inertia
allocation problem, wherein the optimization is carried out accounting for the
worst-case disturbance location. We illustrate our results with a three-region
power grid case study and compare our locally optimal solution with different
placement heuristics in terms of different performance metrics.
