In longitudinal power systems, there is the possibility that a low-frequency oscillation mode may become unstable because of autoparametric resonance. The resonance occurs through interaction between two oscillation modes. In this paper, we calculate the stable region for the resonance by considering the interaction of the modes. First, we calculate steady-state solutions by the harmonic balance method. The steady-state solutions are stable or unstable. If we decrease the amplitude of one mode in an unstable solution, the modes decay. Conversely, if we increase the amplitude, the modes diverge. Namely, the unstable solution is located on the boundary of the stable solution. The amplitudes of the modes are rarely the same as those of the steady-state solution. However, the amplitudes approach a steady state after some transients. If the steady state is in the stable region, the system is stable. If it is in the unstable region, it diverges. Lastly, we estimate the amount of damping torques necessary to stabilize the system with the newly calculated stable region. We have obtained results similar to those derived from the Mathieu diagram, for example, a certain amount of damping torques can stabilize the system irrespective of its size. AVRs can substantially reduce the amount of damping torques and other quantities.