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DC OPF Bundle Schema

powerio dcopf <case>.m -o <out> (or opf_pipeline::write_dcopf_bundle) writes <out>/<case>_dcopf/: a set of Matrix Market files plus dcopf_meta.json. Everything is a pure function of the case. The files and conventions are below.

Conventions

  • Format. Matrix Market. Matrices are coordinate real; square symmetric ones (L, L_grounded) use the symmetric header and store the lower triangle only. Vectors are array real general, one value per line.
  • Index base. .mtx row/column indices are 1-based (Matrix Market standard). reference_buses in the manifest are 0-based dense bus indices.
  • Sign convention. The Laplacians are the positive (M-matrix) form: diagonal \(> 0\), off-diagonal \(< 0\), with \(L_{ii} = \sum_j \lvert L_{ij} \rvert\) for \(L\). An off-diagonal entry is \(L_{ij} = -b_e\) for the branch between \(i\) and \(j\), so a consumer recovers the edge weight as \(-L_{ij} > 0\).
  • Units. PerUnit by default: power divided by base_mva, cost scaled so it is a function of per unit power: \(q \leftarrow 2c_2 \cdot \mathrm{base}^2\) and \(c \leftarrow c_1 \cdot \mathrm{base}\). Native keeps MW / native cost. The choice is recorded in the manifest.
  • Generator costs. The default DC OPF export policy is require: an in-service generator without cost data is an error. Use --missing-gen-cost to explicitly fill missing rows for feasibility tests.
  • Reference buses. reference_buses in the manifest lists every grounded bus as a 0-based dense index. Each in-service island needs at least one reference. If several references lie in one island, the bundle fixes all of those voltage angles to zero; it is not a participation factor slack model.
  • DC convention. PaperPure by default (\(b_e = 1/x\), taps and phase shifts ignored). Matpower uses \(b_e = 1/(x \tau)\) plus the phase shift injection p_shift. Recorded in the manifest.

Matrices

fileshapewhat
A.mtx\(n \times m\)signed incidence; column \(e\) has \(+1\) at from-bus, \(-1\) at to-bus
L.mtx\(n \times n\)generic Laplacian \(L = A \operatorname{diag}(b) A^\mathsf{T}\), singular with \(\operatorname{rank}(L) = n - 1\), \(\mathbf{1} \in \ker L\)
L_grounded.mtx\((n-k) \times (n-k)\)\(L\) with \(k\) reference rows and columns removed; SPD when every island is grounded
BAt.mtx\(m \times n\)flow map \(B A^\mathsf{T}\), where \(f = B A^\mathsf{T} \theta\)
Cg.mtx\(n \times n_{\mathrm{gen}}\)generator-to-bus incidence, one \(1\) per column

Vectors

Bus-indexed (length \(n\)): pd (load), q/c (cost diag/linear), pmax/pmin (generation bounds), e_r (reference indicator: \(1\) at every reference bus, else \(0\)), p_shift (phase shift injection, all zero unless Matpower + shifters). Branch-indexed (length \(m\)): b (susceptances), fmax (thermal limits; \(0\) means unlimited per MATPOWER). Generator-space provenance (length \(n_{\mathrm{gen}}\)): q_gen, c_gen, pmax_gen, pmin_gen.

Manifest (dcopf_meta.json)

Schema powerio.dcopf version 0.1.0 writes Matrix Market files plus structured metadata:

  • dimensions: n_buses, n_source_branches, n_branch_columns, n_generators, n_reference_buses, and n_grounded_buses.
  • index_base: dense = 0 for manifest bus, branch, generator, and reference indices; matrix_market = 1 for .mtx coordinates.
  • dc_convention, units, build_options, and zero_impedance. The zero impedance block records the skip flag, denominator rule, skipped count, and skipped source branch rows.
  • grounding: reference buses, removed rows and columns, the grounded operator (L_grounded), and the reference selector (e_r).
  • operators[]: one entry per emitted operator with name, file, kind, rows, cols, index_space, and units.

The legacy aliases n, m, n_gen, reference_buses, and convention remain for current readers. cost_policy, synthesized_gen_costs, patched_gen_costs, files[], and powerio_version remain top level fields.

Solving with it

The grounded system is the one to factor: L_grounded is SPD when every island has a reference. For DC power flow \(L\theta = p\) with net injection \(p = g - d\), drop all reference_buses entries from \(p\), solve \(L_{\mathrm{grounded}}\theta_{\mathrm{red}} = p_{\mathrm{red}}\), and set each reference angle to \(0\). e_r identifies the grounded buses without parsing the manifest. The full singular \(L\) can be used instead with a consistent zero-sum RHS.

An interior point DC OPF solver builds reweighted Laplacians each Newton step from the same A and b (only the edge weights change), so A is the durable operator to hand over.