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Information about the boundary of local bifurcations is important for utilities to guarantee the secure operation of power systems, and therefore local bifurcation analysis is a useful tool in power systems stability analysis. A new method is presented for calculating the multi-parameter singularity-induced, saddle-node and Hopf bifurcation boundary associated with the parameter-dependent differential-algebraic equations (DAE), which are used to model power systems dynamics. This method is based on the idea of the continuation method, which means that these three kinds of local bifurcations of DAE systems are expressed by appropriate nonlinear algebraic equations, which can be used to track the multi-parameter local bifurcation boundary directly by the continuation method from a known one-parameter local bifurcation point on the boundary, and thus it has the advantage of being a direct method as the continuation method itself inherently contains an iteration procedure during tracking the boundary point by point. Another advantage of this method is that it keeps the DAE form of the mathematical model of power systems and thus preserves the sparsity of the data structure. Several example power systems are used to illustrate the proposed method.