Unified Diagnostics for Quantifying AC Operating-Point Robustness Under Injection and Topological Uncertainties with Regime Changes

In the presence of uncertainties in load, generation, and network topology, power system planning must reflect operational conditions, while operations require situational awareness over credible uncertainty sets. Existing methods screen, analyze, embed, and propagate uncertainty in power flow and optimal power flow settings, but provide only partial insight into how physical constraints, controls, and economic interactions shape steady-state operating-point robustness. By formulating operating-point robustness as a post-solution physical response problem around a solved AC optimal power flow (AC-OPF) equilibrium, this paper presents a unified framework for assessing robustness under injection and topological uncertainty without re-optimization. We construct a primal physical response mapping that accounts for connectivity changes, active power redistribution, generator saturation including $PV \rightarrow PQ$ transitions, and AC network propagation, and introduce quasi-duals that provide a geometric interpretation of shadow prices for off-optimal equilibria. Using these mappings, we develop deterministic screening procedures that generalize $N-k$ contingency analysis to include cost vulnerability $C-k$, and local analogs $N+\delta(k)$ and $C+\delta(k)$ defined through sensitivity-normalized margins and risk tolerances. The framework is extended to probabilistic screening for distribution- and moment-based uncertainties, with sequentially-pruned mixture modeling and $\alpha$-stressed regime constructions to manage combinatorial branching. A case study on the Puerto Rican bulk power system demonstrates integration with geospatial data to enhance operational and planning awareness.