As power systems grow reliant on large-scale integration of renewable energy sources, pumped-storage power stations are being called on to provide greater adjustment capabilities. This study focuses on the stability and fast-slow dynamics of a pump-turbine governing system (PTGS). First, the time delay τ is considered into the PTGS since a lag in time exists between the signal and response in the hydraulic servo system. Second, six analytical expressions for the transfer coefficients of the guide vane opening in the process of wide-ranging load decrease are obtained utilising the complete characteristic curve of pump turbines. The effects of the time delay τ on system stability for the transient process are explored. Furthermore, considering the dynamic transfer coefficient e is a variable during the operation, the authors introduce the intermediate variable e as a periodic excitation into the PTGS. From the perspective of non-linear dynamics, the bursting oscillation behaviours of the multi-scale coupling system are analysed in detail. On the basis of practise engineering, the stability of the PTGS is studied in depth. All of the above methods and results supply the theoretical basis for maintaining a stable operation of pumped-storage power stations.