Pump-turbine characteristics greatly affect the operational stability of pumped-storage plants. In particular, the S-shaped region of the characteristic curves leads to severe instability during runaway conditions with servomotor failure. Thus, this paper aims to investigate the runaway stability criterion by considering all of the important effects in the hydromechanical system. The criterion also helps to judge the S-characteristics of pump-turbines and can provide a guide for plant design and turbine optimization. First, the pump-turbine characteristic curves are locally linearized to obtain formulae for the relative changes of discharge and torque, which depend on the relative changes of rotational speed and water head. Control theory is then applied to analyze the high-order system, by importing the transfer function of the conduits in the elastic mode. Two different kinds of oscillation are found, associated with water inertia and elasticity, based on the established theoretical mathematical model. New stability criteria for the inertia wave in both rigid and elastic modes are developed and compared. The comparison reveals the effect of the water elasticity on runaway instability, which has often been neglected in the previous work. Other effects, such as friction loss and the timescales of water flow and machinery, are also discussed. Furthermore, the elastic wave, which often has a higher frequency than the inertia wave, is also studied. The stability criterion is deduced with analyses of its effects. Based on the stability criteria for the inertia wave and elastic wave, the unstable regions for two waves of the S-shaped curves are plotted. The results are applied to explain the development from inertia wave to elastic wave during transient behavior at runaway conditions. Model tests of runaway conditions were conducted on a model pumped storage station and the experimental data show good agreement with the theoretical analyses regarding the instability of the inertia wave. Further analyses and validations are made based on transient simulations. The simulation software topsys, which uses the method of characteristics (MOC) and a unit boundary represented by a spatial pump-turbine characteristic surface, was applied to analyze the elastic wave. This also supports the conclusions of the theoretical research.