Thermal-Security Robustness Radii for Power Systems

Abstract

This paper presents a first-order robustness radius for thermal security of alternating-current power-system operating points. The radius is intended for post-dispatch screening in control-room studies, planning studies, and dispatch comparison, where engineers need to know which monitored line is closest to overload and which injection pattern causes that risk. The radius is the largest metric ball of admissible injection changes that remains inside all monitored linearized thermal limits. The admissible changes may include balance rules, participation response, or other response constraints. For alternating-current networks, each line is checked at both ends using apparent-power limits. The required sensitivities are computed by sparse transposed-Jacobian solves, and the chunked implementation can avoid materializing the complete dense line-end sensitivity matrix. The framework covers Euclidean, covariance-weighted, and standard-deviation-scaled radii. It also gives the worst injection direction for each limiting line for nonlinear replay and dispatch screening. We prove exact maximality for the selected affine model, slack-reference invariance for direct-current analysis, and equivalence between covariance metrics and standard-deviation radii. The radius is exact for the selected affine surrogate, but only a local nonlinear indicator for the original alternating-current equations. Zero-flow ends are treated through the two-dimensional active-reactive flow operator. Independent automatic differentiation and multi-case finite differences verify the implemented branch gradients to numerical precision. On original lossy public benchmarks, nonlinear replay gives a median nonlinear-to-affine violation-distance ratio of 1.000, with a minimum of 0.813. Line-wise variance calibration across two matched response mechanisms gives a median analytical-to-empirical standard-deviation ratio of 1.000. These results support rapid post-dispatch ranking and replay; they also show why nonlinear validation remains necessary before operational use.