Slack distribution & reference-bus conventions

A PTDF answers "if I inject one unit of power at bus j, how much flows on branch i?" But power is conserved: an injection somewhere must be balanced by a withdrawal somewhere else. Where that balancing withdrawal goes is the slack convention, and it changes the PTDF.^[ptdf] This page explains single versus distributed slack, the role of the reference bus, and how the choice shows up in the sensitivities.

For the mechanics of configuring a distributed slack — the dist_slack keyword and its per-matrix input type (Dict{Int, Float64} for PTDF/VirtualPTDF versus Vector{Float64} for VirtualLODF/VirtualMODF) — see each constructor's docstring in the public API reference.

Why a reference is needed at all

The DC power flow solves θ = ABA⁻¹ · p for bus angles from injections. ABA is a grounded graph Laplacian: without grounding it is singular (angles are only defined up to a constant, and injections must sum to zero). Grounding it means designating a reference bus whose angle is fixed and whose row/column is removed from the system. That reference bus is also, by construction, the bus that absorbs the network's power imbalance — the single slack.

So every PTDF is implicitly relative to a slack. PTDF[i, j] is the flow on branch i caused by injecting at bus jand withdrawing at the slack. There is no such thing as a slack-free PTDF.

Single slack (the default)

By default dist_slack is empty, and the matrix uses a single reference bus as the slack. This is the standard convention and is what you want when the model has one designated slack generator absorbing imbalance.

The reference bus is excluded from the solve, so its own column in the PTDF is zero — injecting at the slack and withdrawing at the slack moves nothing.

Effect of the reference-bus choice. Moving the slack to a different bus shifts each PTDF row by a constant: every sensitivity is measured against a different balancing point. The choice is therefore not arbitrary for interpreting a single PTDF[i, j] value. It does not, however, change physically meaningful flows: if the injections you apply already sum to zero (a real transfer, generation minus load), the resulting branch flows are independent of which bus was chosen as slack. LODF values are likewise invariant to the slack choice.

Distributed slack

A single slack is an idealization — real imbalance is picked up by many generators according to participation factors, not one bus. A distributed slack spreads the balancing withdrawal across several buses by weight. Instead of the whole compensating withdrawal landing on one reference bus, each participating bus k absorbs a share wₖ / Σ w.

Concretely, the distributed-slack PTDF is the single-slack PTDF with each row's weighted average subtracted:

PTDF_dist[i, j] = PTDF[i, j] − Σₖ (wₖ / Σ w) · PTDF[i, k]

so the sensitivities are now measured relative to the weighted set of slack buses rather than one. The weights are normalized internally, so only their ratios matter.

Two structural requirements come with distributed slack (enforced by the constructors):

• There must be exactly one reference bus in the system when a non-empty dist_slack is supplied — distributed slack redistributes the imbalance, but the grounding of ABA is still a single reference.
• The weight vector must cover every bus (length equal to the bus count); buses that do not participate simply carry weight zero.

The type differs by matrix

The way you pass the weights depends on the matrix, and this catches people out:

For PTDF/VirtualPTDF you supply a dictionary keyed by bus number, which is convenient because you name only the participating buses; internally it is normalized and expanded to a per-bus vector. For VirtualLODF/VirtualMODF you supply the positional vector directly. In both cases the default is empty, meaning single-reference-bus slack.

Why the LODF/MODF forms take a raw vector rather than a dict is an interface detail, not a semantic difference — the underlying meaning (weighted balancing withdrawal) is identical. Just match the type to the matrix you are building.

Choosing a convention

• Use the default single slack for most analyses, and for validating against references that assume one slack bus.
• Use a distributed slack when you want sensitivities consistent with how generation actually rebalances — for economic dispatch, participation-factor studies, or comparison with an AC solution whose losses and response are spread across machines.

The distributed slack does not make the DC model more or less exact; it changes the question the PTDF answers, from "balanced at one bus" to "balanced across a weighted fleet."

References