Representation taxonomy

Page status: foundational taxonomy; categories are analytical, not a standards claim.

This page is a classification card, not a second transformation specification. The authoritative registry of typed views, lowering maps, state-conditioned surgery, and diagnostics is From source graphs to views and graph surgery. The taxonomy below names the families and their typical losses; it does not redefine their object maps or preservation contracts.

The phrase power-network graph covers several mathematical and data structures that retain different facts. This taxonomy prevents a physical asset graph, an equation graph, and a solver sparsity graph from being treated as interchangeable merely because all have vertices and edges.

Five questions before naming a graph

For a representation $M$, ask:

1. Identity: which physical, logical, and generated objects remain distinguishable?
2. Interconnection: what counts as a terminal, junction, edge, hyperedge, or port?
3. Behaviour: where do constitutive equations, controls, limits, and uncertainty live?
4. State: which topology and equipment states are fixed, variable, or omitted?
5. Purpose: which observations, analyses, and decisions is the representation intended to support?

Two structures with identical unlabelled topology can answer these questions differently.

Representation families

Asset and property models

Vertices represent physical or organizational entities; typed relations record containment, ownership, construction, location, protection, measurement, and provenance. Such a model may retain two electrically equivalent parallel assets as distinct while omitting the internal variables of a compiled electrical equivalent.

Terminal-connectivity models

Objects expose ordered terminals or conductor bundles. Junctions express equality and conservation without yet requiring every device to be an ordinary edge. This family is the natural baseline for explicit phases, neutrals, grounding, phase discontinuities, and switchgear.

Node–breaker and bus–branch models

A node–breaker model retains switching equipment and detailed connectivity. A state-resolved bus model quotients nodes connected by closed ideal switches. An oriented bus–branch multigraph adds identified two-terminal branches, where orientation is normally a coordinate choice rather than a physical transfer direction. A simple graph additionally forgets parallel identity.

These are different representations, not interchangeable names for resolution levels. The node–breaker row is specialized in Node–breaker, bus–breaker, and topology processing; its state-resolved quotient is one concrete instance of the general view/surgery contract.

Four principal levels and the orthogonal companions

For the main argument, the families above are organized into four principal levels. The word level refers to the dominant electrical or identity query, not to a universal refinement order:

The first two rows are derived electrical views; the third is the canonical electrical source formalism adopted for this book; and the fourth is an orthogonal companion rather than a more detailed circuit graph. Terminal, node–breaker, bus–branch, equation, and sparsity views are named specializations or computational projections around these four levels. They should not be counted as additional rungs in a single ladder. The rigorous objects and maps are specified in Formal representation frameworks.

The diagram is the classification in one view. The computational projections hang off the electrical source model because their vertices and edges are chosen from a declared equation or matrix support; the asset model is drawn sideways because ownership and maintenance are not a refinement of electrical boundary behaviour.

Port, factor, and hypergraph models

A behavioural factor relates variables on an arbitrary ordered set of ports. Ordinary branches are arity-two special cases. Multiwinding transformers, converters, mutual couplings, shared controls, and measurement relations need not be decomposed into artificial pairwise edges. The formal relation between a two-uniform flag multigraph, an arbitrary-arity incidence structure, a bipartite incidence graph, and a typed port–factor relation is fixed in Multigraphs for expert modelers. This taxonomy names the representation family; it does not use hypergraph and factor graph as interchangeable data models.

Algebraic and equation models

Incidence, admittance, Laplacian, Jacobian, KKT, and constraint matrices induce graphs from their nonzero structure. Their vertices may represent variables, equations, blocks, or matrix rows rather than physical objects. They are indispensable for analysis and computation, but their edges do not automatically carry physical meaning.

Equation-graph choice is coupled to formulation choice: nodal support, modified nodal, sparse tableau, branch-current, and port/factor targets can induce different variable and sparsity graphs from the same source model. The guarded formulation boundary is defined in Circuit formulations and the lowering boundary.

The block- and scalar-support graphs of the compound nodal operator are defined once in Two topology levels and the nodal projection. This taxonomy records their role without introducing a competing support definition. Likewise, a reduced/Kron row in this table names a view family; its elimination and recovery conditions belong to the compiled-view and reduction chapters.

Study-specific compiled models

A power-flow, OPF, state-estimation, fault, or decomposition model selects variables and relations for a task. Compilation may create virtual buses and branches, eliminate explicit currents, fix states, relax constraints, or introduce auxiliary variables. Graph size alone therefore says nothing reliable about physical or decision expressiveness.

Comparison by retained meaning

No row is globally maximal. An asset graph and a compiled equation graph may retain incomparable information.

A simple topology graph must not be identified with a sparsity graph merely because both are simple graphs. Their vertices and adjacency rules differ: one records a quotient of physical incidence, while the other records nonzero algebraic dependence in a declared matrix or equation system.

Comparison by decision support

Axes, not one ladder

Representations should be located along several independent axes:

• physical identity retained or forgotten;
• conductor and terminal resolution;
• factor arity and device vocabulary;
• hierarchy and subsystem boundaries;
• fixed versus variable topology state;
• equations only versus equations plus feasible sets and decisions;
• exact versus approximate behaviour;
• source provenance and recoverability;
• physical versus computational interpretation.

A representation can be simpler on one axis and richer on another. Compiling a transformer into a loss network may increase the number of vertices while reducing the device vocabulary. Eliminating internal buses may reduce the vertex count while creating dense coupling and more complicated constraints.

The general multiconductor baseline

The book begins with arbitrary ordered terminal sets and full conductor coupling. A transmission bus–branch model appears when additional conditions justify several collapses, for example:

• the retained buses have compatible phase sets and ordering;
• the system is sufficiently balanced for the declared observations;
• sequence domains decouple under the element models used;
• neutral and grounding behaviour is absent, externally fixed, or validly reduced;
• equipment can be represented as identified two-terminal branches;
• per-conductor and device-internal constraints do not affect the decisions of interest;
• parallel aggregation retains all relevant member states and all nonredundant limits, with any removed constraint certified as implied, or those distinctions are outside the contract.

This explains why many transmission studies can use compact graphs successfully without treating their assumptions as universal power-network semantics.

Proposed linked reference architecture

The working hypothesis of this book is that two linked structures provide a useful source from which the major representation families can be generated:

1. a typed asset/property model for stable physical and organizational identity;
2. a typed hierarchical port–factor model for electrical interconnection, behaviour, limits, and decisions.

This is a proposal to test. Evidence must include faithful mappings of multiconductor lines, multiwinding transformers, switchgear, grounding, parallel assets, controls, and study-specific models—not only an abstract claim of generality.

A representation card

Every representation chapter and knowledge-base entry should record: