Equivalent representation of reduced branches

When a network reduction collapses a group of branches into one, the reduced network must carry a single equivalent branch in place of the group. This page explains how the equivalent's electrical parameters and ratings are formed, and — the subtle part — why the rating policies differ while the impedance aggregation does not.

The aggregated-branch types themselves (BranchesParallel, MixedBranchesParallel, BranchesSeries, the internal ThreeWindingTransformerWinding, and the resulting EquivalentBranch) are documented by their docstrings in the full public API under the internal (non-exported) symbols.

Two kinds of group

Reductions produce two topological groupings:

• Parallel groups (AbstractBranchesParallel): several branches sharing the same pair of buses. Physically they are alternative paths between the same two nodes.
• Series chains (BranchesSeries): a run of branches through intermediate degree-two buses that carry no injection, so the chain behaves as a single branch between its endpoints.

A three-winding transformer contributes its own wye-to-star structure, handled by the internal ThreeWindingTransformerWinding.

Impedance aggregation is physics — one answer

The equivalent electrical parameters are not chosen; they are derived from the requirement that the reduced branch present the same terminal behavior as the group it replaces. PowerNetworkMatrices does this by building the group's equivalent admittance (populate_equivalent_ybus!) and reading the physical parameters back off it (get_equivalent_physical_branch_parameters, src/common.jl), yielding an EquivalentBranch with series r/x, shunt g/b at each end, tap, and phase shift.

The combining rules follow directly:

• Parallel: admittances add. The group's series susceptance is the sum of the members', b = Σ bᵢ — more parallel paths means a stronger (lower-impedance) connection.
• Series: impedances add, so susceptances combine reciprocally, b = 1 / Σ (1/bᵢ) — a chain is weaker than its strongest link.

There is a single correct answer here because the electrical behavior is fixed by Kirchhoff's laws.^[circuits] The equivalent is exact for the linear (DC) model.

Rating aggregation is policy — several answers

A rating is not an electrical quantity the way impedance is. It is a limit imposed by the study, and "the limit of the group" is a genuinely ambiguous question whose answer depends on what the study is protecting against. That is why PowerNetworkMatrices exposes several rating strategies for a parallel group rather than one (src/BranchesParallel.jl):

• get_sum_of_max_rating — nominal capacity.Σ Sᵢ, treating every circuit as independently loadable to its own thermal limit. This is the least conservative aggregate; it assumes flow can be steered freely across the group so that each circuit reaches its limit at once. It answers "how much could this corridor carry in the best case?"
• get_single_element_contingency_rating — N-1 security.Σ Sᵢ − maxᵢ Sᵢ, the capacity that survives when the largest circuit in the group trips. It answers "what can I still rely on after losing one element?" For a group of one it is zero, correctly, because there is nothing left after the sole circuit trips.
• get_impedance_averaged_rating — realistic DC loading. The susceptance-weighted average Σ fᵢ Sᵢ with fᵢ = bᵢ / Σ bₖ. This reflects how DC flow actually divides across a parallel group: current follows the path of least impedance, so the low-impedance (high-susceptance) circuit carries the larger share and reaches its limit first. Because flow cannot in fact be steered arbitrarily, the sum-of-max is optimistic and this weighted figure is closer to the binding constraint. It requires a finite, non-zero total susceptance and throws an ArgumentError otherwise.

The three are ordered from least to most physically constrained: sum_of_max ≥ impedance_averaged, and the N-1 figure answers a different (security) question entirely. Which one is "right" is a modeling decision, not a computation the package can make for you — hence three named policies instead of a silent default.

Series chains and emergency ratings

Ratings propagate through the two group types differently, again for physical reasons:

• Series chain rating is the weakest link: min over the chain's members, because a series path can carry no more than its most limited segment (get_equivalent_rating on a BranchesSeries). When a member of the chain is itself a parallel group, it contributes its single-element-contingency rating — the conservative N-1 figure — rather than its optimistic sum.
• Emergency ratings follow the same shapes: a parallel group sums the members' emergency ratings, a series chain takes the minimum (weakest link). Where a branch has no distinct emergency (rating_b) value, its normal rating is used as the post-contingency limit.

Availability

A group is available only if all of its members are: parallel and series groups both require every branch present and in service. Losing any one member makes the equivalent unavailable, which keeps the reduced model consistent with outages of the underlying branches.

Why this separation matters

The clean split — impedance is derived, rating is chosen — is the key idea. Impedance aggregation has a unique physical answer and PowerNetworkMatrices computes it once. Rating aggregation encodes an operator's risk posture, so PowerNetworkMatrices refuses to pick for you and instead names the policies (sum_of_max, single_element_contingency, impedance_averaged) so a study selects the one matching its purpose: raw capacity, N-1 security, or realistic DC loading.

References