From Damping Ratios to Waveform Characteristics: A Residue-Based Mechanistic Analysis Framework for Hydropower Mechanical Variables

Conventional stability analysis of hydropower units focuses on damping ratios, which characterize how fast oscillations decay but cannot explain why different mechanical variables exhibit distinct waveforms under the same disturbance. This paper proposes a three-stage framework to interpret the origin of such waveform differences: variable classification via zero-pole structure, parameter sensitivity for coupling paths between variables, and residue analysis for waveform interpretation. Rotor speed is identified as a basic variable governed solely by system poles or damping, while guide vane opening (GVO) and torque are non-basic variables shaped by additional zeros in their numerators. Parameter sensitivity shows that the speed-to-GVO path is governed primarily by the proportional gain and servo time constant, while the GVO-to-torque path is governed primarily by the water-hammer and water inertia time constants. Residue analysis further reveals that the sign relationship between the real-pole and conjugate-pole residues determines the waveform characteristic: same signs yield monotonic decay; opposite signs produce early suppression followed by oscillatory recovery. These findings are verified through nonlinear simulations under various operating conditions. The framework offers a mechanistic explanation for waveform characteristics and provides insights for parameter interpretation in hydropower units.