As the flexible-controlled power source in the power system, the doubly fed-based variable speed pumped storage unit (DF-VSPSU) has a significant role in the power rebalance and the reactive power support. However, under remote grid faults, the DF-VSPSU with the conventional low-voltage ride through control employed may lose the stability due to the absence of equilibrium point. Thus, taking one DF-VSPSU infinite-bus system as a case, the quasi-steady models of the DF-VSPSU generator-side characteristic and the grid-side characteristic are established. The stable boundary and the available region of DF-VSPSU are fully descripted with different grid conditions. Then, together with the sag depth of the remote voltage, the grid short circuit ratio, the grid impedance angle, and the reactive support coefficient, the existence of the equilibrium point of the DF-VSPSU is analyzed. Finally, the MATLAB/Simulink demonstrates availability of the theoretical analysis.