A Matrix Perturbation-Based Sequential Optimization Method for Virtual Inertia in Grid-Following and Grid-Forming Hybrid Systems

The increasing integration of renewable energy sources has transformed modern power systems from conventional synchronous generator-dominated dynamics to control-driven synchronization governed by grid-following (GFL) and grid-forming (GFM) converters. In such GFL/GFM hybrid systems, the allocation of GFM virtual inertia critically affects small-signal stability. This paper proposes a matrix perturbation-based sequential optimization method for systematically tuning GFM virtual inertia. The method decomposes the global nonconvex optimization problem into a sequence of linearized subproblems and employs first- and second-order eigenvalue sensitivities to capture the impact of parameter variations on system eigenvalues. Curvature constraints derived from the second-order eigenvalue sensitivities are then explicitly imposed to limit the nonlinear contribution of parameter perturbations. An adaptive backtracking strategy dynamically adjusts perturbation magnitudes, ensuring monotonic improvement of small-signal stability metrics. The proposed approach is validated on the IEEE 39-bus system, where it significantly reduces the spectral abscissa and improves the minimum damping ratio. Comparative studies with Monte Carlo sampling, particle swarm optimization, and first-order sensitivity methods demonstrate that the proposed algorithm achieves superior small-signal stability enhancement with lower computational effort, offering a practical framework for virtual inertia allocation in GFL/GFM hybrid systems.