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• A refined non-linear model reflecting the grid-connected operation of a cluster of hydropower units is established. • An improved damping quantification method and parameter sensitivity analysis indices are proposed. • Influence of the operating conditions of a hydropower plant on the hydraulic-mechanical damping (HMD) is studied. • The variation law of HMD and its parameter sensitivity with system parameters is analyzed. Hydropower is probably the largest sustainable energy source in use and plays an important role in maintaining power and frequency stability of the power system. However, in hydropower-dominated power systems (HDPS), hydropower units often exhibit negative damping, increasing the risk of ultra-low frequency oscillations (ULFO). In this paper, the operating characteristics and parameter sensitivities of hydropower unit (HPU) damping in an HDPS are investigated, taking a large hydropower plant in southwest China as an example. To this end, a refined mathematical model of a cluster of HPUs is established, and an improved damping quantification method and parameter sensitivity analysis indices are proposed for the complex energy system. On this basis, the damping of the HPU with different control modes under a ULFO is investigated with the change of operating conditions represented by guide vane opening, water head and the number of parallel-running units. Meanwhile, the sensitivity of the damping to the system parameters is analyzed to reveal the influence mechanism of the parameters on the ULFO. Also, measures to suppress the ULFO are formulated from two levels of operation and control combined with theoretical analysis. [ABSTRACT FROM AUTHOR]