Implementing General-Order Frequency Dynamic Response Model and Frequency Excursion Duration Criterion in Unit Commitment Problem

This paper introduces a novel approach for incorporating frequency dynamics into the unit commitment (UC) problem through a general-order differential equation model, solved using Bernstein polynomial approximation. Traditional frequency-constrained UC (FCUC) models typically rely on simplified first-order assumptions or scalar frequency metrics, such as frequency nadir, to indirectly enforce dynamic behavior. In contrast, our formulation explicitly models time-domain frequency response using second-order dynamics, enabling a more accurate and flexible representation of generator behavior. The resulting differential equations are approximated with high fidelity using Bernstein polynomials, leading to a mixed-integer linear programming (MILP) formulation that remains computationally tractable for small-scale power systems. Additionally, we introduce a new constraint based on the duration of frequency excursions below a critical threshold, motivated by practical concerns such as relay operation and equipment protection. A data-driven method is employed to relate the area under this threshold-computed as the integral of the Bernstein approximation-to the duration of frequency deviation. The proposed framework is validated using real-world data from an island system in Spain, demonstrating enhanced frequency security with a moderate increase in operational cost. These results suggest the method's strong potential for application in low-inertia, small-scale power systems.