Co-Optimizing Synthetic and Synchronous Inertia in Frequency-Constrained Unit Commitment: A Case Study on La Palma

Abstract

High penetrations of converter-interfaced renewable energy sources (RES) are eroding synchronous inertia in many power systems, making frequency security a binding constraint in unit commitment (UC), especially for weak island grids. This paper develops an analytical frequency-constrained unit commitment (FCUC) formulation that (i) co-optimises synchronous inertia and synthetic inertia (SI) from wind power plants and (ii) supports both classical DC power-flow and PTDF-based linear sensitivity factor (LSF) network representations within a unified mixed-integer linear programming (MILP) framework. The frequency nadir constraints is enforced via a separable-programming approximation that remains fully MILP-compatible. The model is validated on the real La Palma island system. Results show that, under an unconstrained network (transmission-capacity factor TCF =1.0), varying the emulated inertia constant kem between 0 s and 6 s has negligible impact on total cost, renewable spillage, and frequency-security indicators: RoCoF remains orders of magnitude below its limit and the nadir constraint is numerically binding in all cases. A comparison between DC and LSF formulations confirms that the LSF model closely reproduces the DC dispatch and frequency metrics while achieving smaller optimality gaps. A subsequent TCF evaluation shows that tightening transmission limits only becomes economically material at TCF=0.6, where costs rise and a small amount of RES curtailment appears, without compromising RoCoF or nadir security. Overall, the results demonstrate that SI from wind can be co-optimised with synchronous inertia and LSF-based transmission constraints in a single tractable FCUC model, providing a structured way to assess inertia provision, wind utilisation, and congestion management in low-inertia island grids.