Multiple stability problems in angle, voltage and frequency may occur under severe faults in large power grids. In this paper, the impacts of the relevant factors such as the fault severity on the interactive evolvement of multiple stability problems are studied and quantified. In addition, a matrix quantitative evaluation index and a corresponding method of multiple stability problems based on the improvement of transient kinetic energy and control sensitivity are proposed, which fully take the effects of single stability and comprehensive stability control, negative effects of control and overshoot control and so on into account. Based on the index, an AC/DC emergency control strategy under the circumstance of the coexistence of multiple stability problems is proposed to enhance the applicability of emergency control algorithms to deal with complex stable situations. Simulation results based on the actual electric network demonstrate the effectiveness of the proposed AC/DC emergency control strategy.