Optimal Fast-Acting Frequency Containment for Modern Power Systems with Variable Inertia

This paper proposes an Optimal Fast Frequency Containment (OFFC) approach during loss-of-generation (LoG) events, combining analytical derivation with an optimization framework for resource deployment. The system frequency response is modeled by a second-order power-frequency transfer function, enabling a closed-form characterization of the optimal power injection profile. This profile maximizes the frequency nadir for a given energy input. Building on this, a scalable optimization framework is established to realize the target injection using available generation resources, each represented by a trapezoidal power profile. A novel linear formulation is developed to model the trapezoidal ramping without bilinear terms. This ensures scalability and guarantees global optimality under operational constraints such as ramp limits, delays, and energy capacities. The proposed approach supports real-time deployment decisions while accounting for resource diversity, facilitating greater integration of renewable energy sources.