This paper proposes a methodology using a tool based on the branch-parameter continuation power flow (BCPF) in order to restore the power flow solvability in unsolvable contingencies. A specified contingency from a set of transmission line contingencies is modeled, considering the transient analysis and practice in the Korean power system. This tool traces a solution path that satisfies the power flow equations with respect to the variation of the branch parameter. At a critical point, in which the branch parameter can move on to a maximum value, a sensitivity analysis with a normal vector is performed to identify the most effective compensation. With the sensitivity information, the location of the reactive power compensation is determined and the effectiveness of the sensitivity information is verified to restore the solvability. In the simulation, the proposed framework is then applied to the Korean power system.