Frequency-constrained robust unit commitment via physics-guided piecewise-linear nadir surrogates and adaptive virtual inertia

The increasing share of inverter-based renewable resources reduces synchronous inertia and complicates frequency control. In such systems, uncertainty, disturbances, and N–k contingencies can violate the rate-of-change-of-frequency (RoCoF), frequency nadir, and Quasi–Steady-State (QSS) constraints. Conventional unit commitment (UC) formulations are not designed to capture these dynamics reliably. In this paper, we develop a frequency-constrained robust UC (FC-RUC) framework that embeds these frequency constraints into the scheduling process under renewable and load uncertainty and N-k contingencies, with a fixed, budgeted contingency criterion k. A physics-guided piecewise-linear surrogate is constructed for the nadir-secure disturbance capacity. It combines analytic RoCoF and QSS bounds with monotonicity enforcement and quantile-based safety calibration to guarantee a conservative envelope. Furthermore, the framework incorporates adaptive virtual inertia and fast frequency support from wind, photovoltaic, and energy storage units via headroom-coupled deliverability constraints, ensuring physically feasible synthetic inertia provision. A column-and-constraint generation algorithm solves the resulting tri-stage robust problem and iteratively generates disturbance-capacity and frequency-security cuts, avoiding full N–k enumeration, while preserving mixed-integer linear programming tractability. Case studies on the IEEE 9-bus and 118-bus systems show that the proposed method satisfies all frequency-security constraints and reduces the total operating cost by 24. 4 and 27. 5%, respectively.