Abstract
The increasing penetration of renewable energy sources has reduced system inertia by displacing conventional synchronous generators, creating challenges for frequency stability in modern power grids. Existing approaches for virtual inertia provision and battery energy storage system (BESS) sizing rely on deterministic and static assumptions and therefore fail to capture the stochastic and time-varying nature of renewable generation, load demand, and operating conditions. To address these shortcomings, this study proposes a stochastic, reliability-constrained optimization framework for the hourly quantification of virtual inertia requirements and corresponding BESS reserves. Uncertainty in renewable generation and load demand is modeled using Monte Carlo simulation, and an empirical cumulative distribution function (ECDF) based assessment is employed to ensure compliance with grid operator-defined frequency nadir and rate-of-change-of-frequency (RoCoF) constraints at a specified reliability level. The proposed framework is validated on a modified IEEE 39-bus test system with 50% renewable energy penetration. The results show strong hourly variation in the required BESS virtual inertia support, expressed in terms of the equivalent inertia constant, which increases from 0.33-0.62 s during high-inertia night-time hours (21:00-05:00) to 3.25-3.76 s during low-inertia solar hours (11:00-15:00) at a 90% reliability level, with a maximum requirement of 3.76 s at 13:00. As the reliability target increases from 50% to 90%, the BESS inertia reserves determined by the proposed framework progressively reduce frequency nadir and RoCoF violations caused by the loss of the largest generating unit, reaching approximately 9% of scenarios at 90% reliability, at the expense of higher reserve requirements and increased capital expenditure. The proposed framework explicitly captures this reliability-cost interaction, thereby quantifying a clear trade-off between frequency reliability and economic performance and providing a reliability-informed framework for balancing frequency security and investment decisions in low-inertia power systems.