Abstract
Increased penetration of renewable energy is depleting grid inertia by replacing synchronous generators, making modern power systems more susceptible to frequency instability during generator outages or large load fluc tuations. Virtual inertia support through battery energy storage systems has emerged as a potential solution to address this stability challenge. However, most existing approaches to virtual inertia allocation and BESS sizing are deterministic, relying on fixed parameters or heuristic assumptions that overlook the time-varying and uncertain nature of renewable generation, load demand, and system contingencies. This paper introduces a reliability-constrained stochastic framework to determine the virtual inertia and corresponding BESS capacity required to maintain frequency stability under such uncertainty. The framework integrates Monte Carlo simula tion with H infinity-norm minimization to ensure that frequency nadir and rate-of-change-of-frequency (RoCoF) limits are satisfied in a predefined proportion of operating scenarios. Reliability is incorporated as a probabilistic con straint, enabling operators to adjust inertia support in line with target reliability levels. A case study based on a modified IEEE 39-bus system demonstrates that the required BESS capacity varies dynamically with renew able output and system inertia, and that higher reliability enhances frequency resilience but with diminishing economic returns. This reliability-driven formulation advances existing practice by linking frequency security, uncertainty, and investment efficiency within a unified planning framework, providing system operators with a practical tool to balance stability assurance and economic viability in low-inertia grids.