Computational Considerations

Matrix Construction

All matrices in PowerNetworkMatrices.jl are derived from the Ybus matrix (i.e. building any matrix starts with building the Ybus). Additionally, all network reductions are applied to the Ybus matrix prior to computing the downstream matrices. This design choice is key for enabling high performance and code maintainability: Looping through the system objects is required only when building the Ybus (slow) and subsequent operations are built on fast matrix operations on (often sparse) matrices. In addition, network reductions are only defined for the Ybus but can be applied uniformly across all matrices.

Sparsity

Power networks are sparse; most buses connect to only a few others. This sparsity is exploited for computational efficiency via sparse linear solvers:

• Incidence and admittance matrices are very sparse.
• Common sensitivity matrices (e.g. PTDF and LODF) are dense.

Matrix Sizes

A system with $N_b$ buses and $N_a$ arcs has matrix dimensions:

• Incidence: $N_a × N_b$ (sparse)
• Admittance: $N_b × N_b$ (sparse)
• PTDF: $N_a × N_b$ (dense)
• LODF: $N_a × N_a$ (dense)

Computational Complexity