Computational Considerations

Sparsity

Power networks are sparse — most buses connect to only a few others — and this is exploited via sparse linear solvers, so the connectivity matrices (IncidenceMatrix and Ybus) are very sparse. The sensitivity matrices (PTDF, LODF), however, are obviously dense: e.g., one entry per bus in every PTDF column, most of them negligible because a branch is nearly insensitive to an injection electrically far away.

Sparsification and tolerance

Those negligible entries can be dropped to recover sparsity. The tolerance is the cutoff below which an entry is set to zero. The default AutoTolerance picks it from the data as a relative per-row drop — an entry is dropped when |x| < α · max|row| — which keeps large matrices sparse while leaving small systems and the dense constructors exact. The exact rule (including the bus-count gate that makes it a no-op on small systems and the Float64tol alternative) is in the AutoTolerance docstring. For very large studies prefer the VirtualPTDF/VirtualLODF variants, which compute rows on demand and store each one sparsely.

Accuracy and limitations

Sparsification trades exactness for memory. Because each dropped entry is below α · max|row| (α ≤ 1e-2, typically α ≈ 5e-4), the dominant sensitivities are never touched — but two properties are worth understanding before relying on a sparsified matrix for a sensitive calculation:

• The error is one-signed, not zero-mean. A dropped entry becomes exactly zero, so a row's total mass strictly decreases. Summing many small entries of a row — e.g. aggregating a transfer's flow contribution across a subsystem's buses — accumulates that truncation in one direction instead of cancelling. The bias is bounded by (number of dropped entries) · α · max|row|.
• The cutoff is per row, so cross-row/column invariants are not preserved. Each row is sparsified against its own peak, so quantities that couple different rows or columns — Kirchhoff's current law, a column sum, the difference of two entries — are not conserved. Two buses j, k both far from branch i can have similar PTDF[i,j], PTDF[i,k] land on opposite sides of the cutoff: each absolute error stays under threshold, yet the relative error on their tiny difference can approach 100%.

Contingency corrections do not compound this: in VirtualMODF the cutoff is applied to the final post-contingency row, after the exact Woodbury solve, so the error stays bounded by the cutoff however near-critical the contingency.

When you need an exact result — to preserve KCL, to difference two small sensitivities, or to validate against a reference — pass a tol::Float64 (tol = eps() for an unsparsified matrix, or a deliberate fixed cutoff).

Matrix sizes and complexity

Dimensions, storage form, and build cost for every matrix type are tabulated in Matrix overview & indexing. The headline is the split that motivates this page: the structural matrices assemble in a single pass over the branches, while the dense sensitivity matrices require a factorization and a solve per arc — which is why PTDF/LODF dominate both the build time and the memory of a large study.