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A novel algorithm (TSA-SPA) that combines the Two-Sided Arnoldi method (TSA) and the Sensitive Pole Algorithm (SPA) is proposed in this paper for calculation of the most sensitive eigenvalues to control parameters in large power systems. In the proposed method, first, with the shift-invert transformation precondition, TSA builds two Krylov subspaces and obtains a reduced matrix of a much smaller scale, which contains eigenvalues close to the chosen shift point. Second, SPA is adopted to realize the most sensitive eigenvalue computation. TSA-SPA can find the most sensitive eigenvalues of interest, with satisfactory reliability and convergence, in a specified frequency domain. With proper selection of sizes of Krylov subspace and the reduced matrix, the convergence to good eigenvalue approximations is practically guaranteed. Moreover, with the deflation technique, the algorithm is also capable of finding several other dominant eigentriplets which may relate to inter-area and/or local control modes. The efficiency of the proposed algorithm has been validated on small and large-scale power systems. It has been found that compared to other available sensitive pole algorithms, the proposed algorithm has more robust and reliable performance. The proposed algorithm is suitable for practical applications in large-scale power systems.