Optimal placement of inertia and primary control in high voltage power grids

The energy transition's ultimate goal is to meet energy demand from human activities sustainably. Accordingly, the penetration of new renewable energy sources (RES) such as photovoltaic panels and wind turbines is increasing in most power grids. In their current configuration, RES are essentially inertialess, therefore, low inertia situations are more and more common, in periods of high RES production, making grid stability a high concern in power grids with high share of RES. It has been suggested that the resulting reduction of overall inertia can be compensated to some extent by the deployment of substitution inertia - synthetic inertia, flywheels, synchronous condensers aso. Of particular importance is to optimize the placement of the limited available substitution inertia. Here, we construct a matrix perturbation theory to optimize inertia and primary control placement under the assumption that both are moderately heterogeneous. Armed with that efficient tool, we construct simple but efficient algorithms that independently determine the optimal geographical distribution of inertia and primary control.