This work is devoted to the analysis of a mathematical model of hydropower unit, consisting of synchronous generator, hydraulic turbine, and speed governor. It is motivated by the accident happened on the Sayano-Shushenskaya hydropower plant in 2009. In the analysis we follow the line of classical mathematical control theory approach developed in the works of famous scientists J.C. Maxwell, I.A. Vishnegradsky, A.A. Andronov, and F. Tricomi for the study of centrifugal turbine governor and electrical machines limit-load problem. It is shown that the occurrence of vibrations in the Sayano-Shushenskaya hydropower plant can be caused by nonlinear dynamics of the closed-form model. (C) 2020 Elsevier B.V. All rights reserved.