Power System Sparse Matrix Statistics

This paper provides practice-oriented statistics on the scalability and the growth of power system sparse matrix computational complexity, with the results based on models of real and synthetic electric grids, including very large grids with up to 110,195 buses. The statistics include how the computational effort of factorizing a Jacobian matrix and the factorization path length scale with the system size $n$, which shows the number of buses. The study shows the number of nonzeros in the Jacobian matrix after factorization grows as $n^{1.07}$, the time to factor the matrix grows as $n^{1.38}$, and Forward (F) /Backward (B) substitution time grows as $n^{1.17}$. In addition, applying sparse vector methods, the fast forward/fast backward substitution (FF/FB) grows as $n^{0.45}$, which shows an improvement in the computational effort. Taking advantage of the statistics mentioned in this paper, the trend, scaling, and computation complexity of factorization steps can be easily predicted.