Fig. 4: EP-LTP framework is accurate in sensitivity analysis.

a Case 1: to time-invariant control parameters of the dominant equipment. The color-coded circles represent the practical change of the concern mode λ^c1₁, with the color indicating the parameter variation from -6% to 6%. Sensitivity analysis is performed based on a common point (0%), also marked as the origin of all red arrows. With the parameter variation of 2%, the sensitivity analytical results (end of red arrows) closely approximate the eigenvalue changes. b Case2: to time-varying voltage vector u^s(t) of Jijihu station. Here, the variation form⁶⁰ of u^s(t) is e ^jωt and e^-jωt. The intersection of the white dashed lines denotes the two concerned modes (i.e., λ^c2₁ and λ^c2₂) with the different voltage conditions of Jijihu station. Two gradient lines (gray solid lines) are marked based on the fitted surfaces. With a variation of 1/30 along the gradient direction, the sensitivity analytical results (red solid arrows) accurately capture the damping changes (blue dashed arrows). Supplementary Notes 6 provides more details. Source data are provided as a Source Data file.