Links (Bus Couplers / Sectionalizers)

Concept

Links represent impedance-less topological connections between buses.

Typical use cases:

• Busbar couplers
• Sectionalizers
• Node splitting / merging in CIM imports

A link is not a physical branch:

• It has no impedance
• It is not stamped into the Y-bus
• It enforces voltage equality constraints

Mathematical interpretation

A closed link between bus i and j imposes:

\[[ V_i = V_j ]\]

This introduces a topological constraint, not an admittance.

Relation to KCL

Since links are not part of the Y-bus, Kirchhoff’s Current Law (KCL) is not enforced via admittance equations at the link.

Instead:

• KCL is enforced per bus after topology processing
• Link flows are reconstructed after solving via power balancing

Zero-impedance loops (critical case)

If multiple links form a loop, the system contains a zero-impedance cycle.

Example:

Bus1 ──link── Bus2
  │             │
 link          link
  │             │
Bus3 ───────────

This leads to:

• No voltage drop in the loop
• Underdetermined current distribution
• Singular system if treated electrically

Resolution via Pseudoinverse

To compute link flows in such cases, Sparlectra uses a minimum-norm solution.

Let:

• $A$ = incidence matrix of link graph
• $f$ = unknown link flows
• $b$ = nodal power imbalance

We solve:

\[[ A f = b ]\]

Since the system is rank-deficient:

\[[ f = A^{+} b ]\]

where $A^{+}$ is the Moore–Penrose pseudoinverse.

Interpretation of the solution

The pseudoinverse yields:

• A consistent KCL solution

• The minimum 2-norm flow distribution

• Physically equivalent to:

  • uniform distribution of flows in symmetric loops
  • no artificial circulation currents

Practical implications

• Link flows in loops are not unique
• Sparlectra returns the minimum-energy solution
• Results are stable and deterministic

Modeling guidelines

• Do not connect links to slack buses
• Prefer identical bus types for linked buses
• Use links only for topology, not impedance modeling
• Avoid large link-only subgraphs without measurements (SE context)

Example

linkNr = addLink!(net = net, fromBus = "Bus1", toBus = "Bus1a", status = 1)

Summary