An Adaptive Threshold Warning Method for Multi-Machine Power System Transient Stability Based on Geometric Algebra

Conventional transient stability assessment in multi-machine power systems relies predominantly on fixed thresholds, which exhibit limited adaptability to varying operating conditions and fail to provide a unified analytical framework for rotor angle and voltage stability. To address these challenges, this paper proposes an adaptive threshold warning method based on geometric algebra. A multi-dimensional unified state vector incorporating generator rotor angles, speeds, electromagnetic powers and bus voltage magnitudes and phases is constructed to map system dynamics onto a high-dimensional geometric trajectory. The second- and third-order wedge products of this trajectory are computed to quantify disturbance severity and volumetric expansion preceding instability. An adaptive threshold mechanism is established utilizing sliding window robust statistics (Median Absolute Deviation) to track the trajectory’s instantaneous dimension in real time. Validation on the IEEE 39-bus system demonstrates that the proposed method issues a warning at t = 4.90 s, achieving a detection advance of 0.30 s relative to the conventional 30° rotor angle separation threshold. The method exhibits strong noise robustness with only 40 ms warning delay under 20 dB SNR conditions, and effectively captures rotor angle–voltage coupling characteristics. The geometric algebra framework offers a unified assessment tool with distinct advantages in computational speed, adaptivity, and interpretability.