• A hierarchical framework links internal states to tie-line transfer capacity. • Multi-objective FS-FUC balances flexibility and cost with full frequency security. • Robust TCR is derived via a systematic vertex search and MILP reformulation. • Uncertainty-induced capacity loss is elucidated and validated by simulations. The low-carbon transition of High Voltage Direct Current (HVDC)-infeed receiving-end grids drives a shift toward low-inertia conditions, precipitating critical frequency stability vulnerabilities. Existing scheduling approaches prioritize internal cost minimization but overlook how Frequency-Secured Unit Commitment (FS-UC) strategies constrain the external HVDC infeed Transfer Capacity Region (TCR). To bridge this gap, this paper proposes a hierarchical framework linking internal states to external transfer capability. First, a multi-objective Frequency-Secured Flexibility-enhanced UC (FS-FUC) model is developed. Incorporating primary frequency response and secondary regulation limits under DC monopole blocking, this model balances economic cost with bidirectional flexibility. The ideal point method is then employed to identify the best compromise solution. Second, based on committed resources, a systematic two-stage vertex search method is proposed to determine the Nominal TCR (N-TCR) under deterministic conditions. Furthermore, a robust projection algorithm reformulated as a Mixed-Integer Linear Programming (MILP) problem is derived to calculate the Robust TCR (R-TCR) under uncertainty. Case studies demonstrate that the proposed framework expands bidirectional flexibility by over 537% with a marginal 2.45% cost increase. Crucially, it limits uncertainty-induced capacity loss to 5.93%, preventing the complete feasibility collapse observed in conventional methods. Moreover, while high renewable penetration drives a significant contraction or even geometric degeneracy of the R-TCR due to combined inertia scarcity and severe reserve depletion, the proposed algorithm ensures robust feasibility. Furthermore, sensitivity analysis indicates that confidence levels can modulate the feasible region size by approximately 20% to 28%. Finally, time-domain simulations verify the frequency security of the proposed strategy.
Wang et al. (Wed,) studied this question.