Research agenda

Page status: proposal and open-work register.

The current paper-sized dissemination cuts are recorded in the repository's review track. They reuse this agenda's claim and evidence boundaries rather than creating a second roadmap.

Research objective

Develop a theory, reference architecture, and executable toolkit for typed, asset-, constraint-, and provenance-preserving transformations of multiconductor power-network models.

The agenda should deliberately connect formal results to utility-relevant applications. A transformation is valuable when it enables faster or more reliable computation without silently invalidating the decisions being made.

Workstream A: model categories and semantics

1. Define the typed hierarchical port–factor category.
2. Define linked asset/property semantics and stable identities.
3. Specify conductor, phase, neutral, ground, orientation and reference-frame types.
4. Formalize observation contracts and relative expressiveness $\succeq_Q$.
5. Separate physical equivalence, terminal behavioral equivalence, feasible-set equivalence, and approximation.

Candidate result A1. A representation theorem showing that ordinary bus–branch multigraphs, conductor-expanded graphs, hypergraphs/factor graphs, and common component compilations embed into the port–factor kernel.

Status: partial — ARCH-BLOCK-001, ARCH-LOWER-001, ARCH-PORT-001, and the representation taxonomy establish finite typed embeddings and lowering examples; a general representation theorem remains open.

Candidate result A2. Conditions under which a projection between model categories is faithful, conservative, or admits a reconstruction functor on a restricted subcategory.

Status: partial — ARCH-LENS-001, ARCH-RECOVERY-002, and the representation-map query-sufficiency analysis classify scoped faithful, set-identifiable, and non-identifiable cases; a general reconstruction result is open.

Workstream B: normalization calculus

1. Specify typed rewrite rules and negative application conditions.
2. Prove semantic preservation of conductor permutation, ideal-switch contraction, homogeneous line concatenation, grounding extraction, and transformer compilation.
3. Characterize critical pairs among rules.
4. Determine whether useful subsets terminate and are confluent up to typed isomorphism.
5. Define normal forms by purpose rather than one universal form.

Candidate result B1. Necessary and sufficient conditions for degree-two multiconductor bus elimination to remain inside a selected line model class.

Status: partial — TR-SER-001 and TR-SER-002 discharge the guarded uncoupled behavioural rule and show why homogeneous line-class closure is separate; TR-SER-003 now gives a distinct exact rule for a complete mutually coupled section pair, while necessary-and-sufficient closure conditions for broader line libraries remain open.

Candidate result B2. A closure classification for series composition of series-only, nominal-$\pi$, exact distributed-parameter, frequency-dependent, and thermally coupled line models.

Status: partial — the guarded-normalization catalogue and the degree-two series chapter cover series-only, nominal-$\pi$, and distributed-parameter warnings; frequency-dependent and thermally coupled closure are open.

Candidate result B3. A non-existence result showing that no single simple edge with conventional scalar or per-conductor limits can exactly represent the feasible set of general heterogeneous parallel branches.

Status: partial — the parallel decision cases and TR-PAR-001/TR-PAR-002 give scalar counterexamples and explicit outer-relaxation witnesses; a formal non-existence theorem for the general heterogeneous multiconductor class is open.

Candidate result B4. Necessary and sufficient redundancy certificates for multiconductor parallel-member constraint sets, extending scalar quadratic containment to coupled phase, neutral, ground, and terminal-direction models, with explicit guards for topology and control states.

Current partial result. Claim TR-PAR-005 gives a necessary-and-sufficient PSD test for each individual centered linear-current norm implication and a two-end componentwise certificate. Claim TR-PAR-006 adds an exact complex polydisc row-norm test when all component limits of one nonsingular series member jointly imply another member's limits, and exercises it in a reciprocal non-proportional four-wire AC decision case. Claim TR-PAR-007 generalizes the same support-function argument to an invertible stacked terminal-current map and exercises distinct from/to shunts in a nominal-$\pi$ case. Singular shunted maps, implication by several different members, non-Euclidean regions, and state-conditioned models remain open parts of B4.

Workstream C: decision-preserving reduction

1. Treat equations and feasible sets together.
2. Develop recovery maps for internal voltages, currents, losses and thermal states.
3. Classify exact, inner, outer and scenario-approximate constraint maps.
4. Treat certified removal of implied constraints as exact presolve, retaining the asset laws, identities, recovery maps, and all nonredundant constraints.
5. Study preservation for OPF, security-constrained OPF, reconfiguration, expansion planning, dynamic operating envelopes and state estimation.
6. Quantify when reduced models change optimal decisions rather than merely state-variable errors.

Candidate result C1. A general lifting theorem: if eliminated variables are uniquely recoverable and all source constraints are composed with that recovery map, optimization over boundary variables is exactly equivalent.

Status: partial — PRESERVE-001, the recovery-map chapter, and the Kron, parallel, and transformer certificates establish the statement for declared finite linear and decision cases; a general theorem over nonlinear and mixed discrete models remains open.

Candidate result C2. Complexity or representability bounds for projecting branch-wise thermal constraints onto boundary variables.

Status: open — current work provides exact recovery and support-function certificates, but no general complexity or representability bound.

Workstream D: approximate but certified models

1. Define application-specific observation norms.
2. Develop scenario and uncertainty-domain error certificates.
3. Preserve radiality, phase availability, grounding modes and selected physical corridors.
4. Compare Kron, clustering, aggregation, sparsification and learned surrogates under the same contract.
5. Measure errors in feasibility, optimal objective, active constraints and decisions—not voltage alone.

Workstream E: implementation and interoperability

Develop a Julia reference implementation with:

• immutable source identities and explicit generated-object identities;
• typed ports, factors and hierarchy;
• a transformation registry with machine-readable certificates;
• rule tracing and reversible provenance;
• adapters for CIM/CGMES, OpenDSS, PowerModelsDistribution and selected Julia optimization models;
• generated multigraph, simple-graph and sparse-matrix views;
• property-based tests and adversarial counterexamples.

Graph representation should be independent of any one solver. Mathematical models should consume generated views and expose the mapping back to stable source entities.

Workstream F: empirical corpus

Build a deliberately difficult test corpus containing:

• heterogeneous parallel lines with distinct ratings and decisions;
• four-wire feeders with multi-grounded and impedance-grounded neutrals;
• phase discontinuities and conductor permutations;
• same-code and mixed-code degree-two line chains;
• nominal-$\pi$ versus distributed line concatenation;
• physically parallel and endpoint-parallel circuits with full, sequence-only, partial-overlap, different-voltage, open, and grounded coupling states;
• multiwinding and autotransformer/regulator models;
• lossy and controllable switches;
• measurements and protection zones at otherwise eliminable nodes.

Each case should include an expected preservation/failure certificate. Small symbolic cases are as important as large benchmarks because they expose exact semantic errors.

High-value applications

The first demonstrations should target decisions for which structural loss has obvious consequences:

1. Parallel-line OPF and contingency analysis: show incorrect feasible regions from naïve aggregation.
2. Four-wire state estimation: show the effect of grounding-aware versus topology-only normalization.
3. Distribution model cleaning: safely merge genuine line subdivisions while retaining construction and provenance.
4. Multiwinding transformer compilation: prove terminal equivalence and source-level constraint recovery.
5. Feeder reduction for hosting capacity or operating envelopes: compare voltage accuracy with decision accuracy.

Longer-term formalization

A formal methods track could encode the core semantics and selected rewrite proofs in Lean. The initial targets should be finite-dimensional linear relations, incidence conservation, conductor permutations, series composition, parallel feasible sets, and Schur-complement recovery. This should follow a stable mathematical specification rather than precede it.