Tuning grid-forming converters for optimal allocation of virtual inertia and damping in low-inertia power systems

The increasing penetration of inverter-based resources (IBRs) in power systems progressively erodes the rotational inertia traditionally provided by synchronous generators, threatening frequency stability. Grid-forming (GFM) converters can emulate virtual inertia and damping, but their tuning is strongly topology-dependent and non-intuitive. This article first characterizes how network topology affects key frequency-stability metrics, then proposes an optimization framework for allocating virtual inertia and damping among GFM converters in low-inertia distribution networks. The work exploits a high-order linearized state-space model that retains the full internal dynamics of synchronous generators, GFM and grid-following converters, and passive network elements. Practical frequency-stability metrics (frequency nadir, RoCoF, and active power peaks) are minimized subject to budget and feasibility constraints. The framework is validated on three network topologies (meshed, radial, and random radial). Results show that optimal allocations are strongly topology dependent: a homogeneous distribution is near-optimal in small impedant networks, whereas radial topologies require counter-intuitive asymmetric allocations that cannot be deduced analytically, motivating the formal optimization approach.