To investigate impact of time delays on the small signal stability of power systems, the least-damped eigenvalues with the smallest damping ratios have been calculated by eigen-analysis methods based on Solution Operator Discretization (SOD) with Pseudo-Spectral collocation (PS) and Implicit Runge-Kutta (IRK) methods. This paper evolves SOD-PS/IRK into their partial counterparts, i.e., PSOD-PS/IRK, with greatly enhanced efficiency and reliability in analyzing large-scale time delayed power systems. Compared with SOD-PS/IRK, PSOD-PS/IRK are characterized in constructing low order discretization matrices of solution operator as well as efficiently and directly solving the embedded Matrix-Inverse-Vector Products (MIVPs). The dimensions of the discretization matrices of solution operator are largely reduced as only the retarded state variables are discretized, rather than all state variables as in SOD-PS/IRK. Meanwhile, the proposed PSOD-PS/IRK optimize the most computationally expensive operations in SOD-PS/IRK by avoiding the iterative solutions to the two embedded MIVPs. PSOD-PS/IRK directly and efficiently compute the MIVPs via factorizing the Kronecker product-like discretization matrices of the solution operator into Schur complements. The Central China-North China ultra-high-voltage power grid with 80577 state variables serves to validate the proposed PSOD-PS/IRK and shows that compared with SOD-PS/IRK, the computational time consumed by PSOD-PS/IRK is cut down by 49.96 times without loss of any accuracy.