[en] Highlights: • A universal model of multi-hydro-turbine governing systems are established. • We have put the systems to the theory frame of generalized Hamiltonian system. • The Hamiltonian function of the system is discussed in detail. • Dynamic characteristics are analyzed under a shock load with randomness. • The variable law of the output power is presented with the change of shock load. - Abstract: This paper focuses on the Hamiltonian mathematical modeling and dynamic characteristics of multi-hydro-turbine governing systems with sharing common penstock under the excitation of stochastic and shock load. Considering the hydraulic-coupling problem in the common penstock, we propose a universal dynamic mathematical model of the multi-hydro-turbine governing system. Then, the proposed model is fitted into the theoretical framework of the generalized Hamiltonian system, utilizing the method of orthogonal decomposition. The dissipation energy, the produced energy and the energy supplied from the external sources are derived from the Hamiltonian model and compared with the physical energy flow. Furthermore, numerical experiments based on a real hydropower station demonstrate that the Hamiltonian function can describe accurately the energy variation of the hydro-turbine system in the transient process and in the stable process. Moreover, in order to deal with the randomness and mutability of the electrical load, we introduce a Gaussian function and a jump function to the control signal of the PID controller to analyze the dynamic characteristics. In addition, the intensity of the shock load is discussed when the system loses its stability. The proposed approach can be used for improving the stability of hydropower stations.