Inverse Carson reconstruction

Kind: Problem specification · Maturity: prototype · Direction: inverse · Temporal: single observation · Scope: overhead lines

solve_inverse_carson reconstructs plausible overhead-line construction data from diagonal zero- and positive-sequence impedance, with optional shunt susceptance. It follows Tam, Geth & Mithulananthan, The Inverse Carson's Equations Problem: Definition, Implementation and Numerical Experiments (arXiv:2404.08210), but deliberately reports all compatible candidates instead of forcing a unique construction.

What is—and is not—observed

Diagonal sequence data discard the off-diagonal sequence coupling of an untransposed line. Many physical constructions can therefore produce the same four series values (R0, X0, R1, X1). The result is a physically plausible completion of missing construction data, not a uniquely observed primitive matrix.

The prototype uses a catalog-first workflow:

Supply discrete conductor and geometry candidates.
For each candidate, optimize bounded geometry and temperature variables.
Rank by standardized measurement residual.
Retain every candidate within the stated uncertainty.
Report the singular values and rank of the local prediction Jacobian.

The forward primitive Z and C matrices come from BMOPFTools.overhead_line_constants. A neutral is Kron-reduced only for the sequence-data comparison; the returned and materialized model keeps that neutral explicit for downstream power flow and state estimation.

Example

using PowerOptLab

mars_horizontal = OverheadCarsonCandidate(
    id = "mars-horizontal-3",
    geometry = :horizontal_3,
    r_ac_ref = fill(4.5e-4, 3),       # Ω/m at 20 °C
    gmr = fill(4.0e-3, 3),            # m
    radius = fill(5.0e-3, 3),         # m
    lower = [0.30, 6.0, 20.0],        # half-span [m], height [m], temp [°C]
    upper = [0.80, 12.0, 90.0],
)

obs = SequenceLineObservation(
    z0 = 0.5952 + 1.5873im,           # Ω/km
    z1 = 0.4472 + 0.3692im,           # Ω/km
    frequency = 50.0,
    sigma = [0.006, 0.016, 0.0045, 0.0037],
)

result = solve_inverse_carson(obs, [mars_horizontal]; starts=16)
result.compatible_candidates
result.fits[1].parameters
result.fits[1].jacobian_singular_values

# Produce BMOPF-ready wire_data/line_geometry blocks without mutating a network.
construction = materialize_inverse_carson(result.fits[1], mars_horizontal)

Interpreting identifiability

With modified Carson series data alone, translating every overhead conductor vertically does not change the series impedance. Absolute height therefore has zero sensitivity and should be expected to be rank-deficient. Reliable shunt data can constrain height, but the paper shows it is highly sensitive to small susceptance errors.

Similarly, three-wire horizontal and triangular families can match the same diagonal sequence series components by changing their span. In that case both candidates appear in compatible_candidates; this is the intended result.

Each fit also returns the full reconstructed Z_sequence, including the off-diagonal sequence coupling absent from the observation, plus B_sequence when shunt data were supplied. These entries are model-derived completions and must not be interpreted as separately measured quantities.

sigma controls both weighting and compatibility. When it is omitted, the constructor supplies a descriptive 1% tolerance with an absolute floor in the declared input units. Scores should only be treated as statistical quantities when sigma genuinely represents measurement and source-model uncertainty.

A full covariance matrix may be supplied instead of sigma, in the ordering (R0, X0, R1, X1[, B0, B1]) and in the declared input units. The objective then uses the Mahalanobis residual obtained from a Cholesky factor. Candidate compatibility still uses marginal standardized residuals, so it remains readable and does not depend on the arbitrary ordering used by a triangular whitening factor. Covariance must be symmetric positive definite; singular covariance is not silently regularized.

Confidence and profile intervals

For a locally full-rank fit, local_parameter_covariance is the inverse Fisher approximation (J'J)^-1, mapped back from normalized variables to physical units. It is evaluated from the Jacobian SVD rather than by forming and inverting the more ill-conditioned normal matrix. local_confidence_intervals contains its two-sided normal intervals at confidence_level. Both fields are nothing when the Jacobian is rank deficient: using a pseudoinverse there would misleadingly assign zero variance to an unidentified direction.

profile_inverse_carson provides the more defensible nonlinear check. It fixes one parameter at a time, reoptimizes all others, and traces the connected set below the one-degree-of-freedom likelihood-ratio threshold. Endpoint status distinguishes a threshold crossing from a candidate bound or solver failure. Profiles can reveal asymmetric, bound-limited, and flat directions that the local covariance cannot. They remain local connected profiles; deterministic multistart or a global method is needed to rule out disconnected regions. Each continuation point is warm-started from its predecessor; a failed solve is retried from deterministic fitted/candidate starts before being reported as a solver-limited endpoint.

These intervals only have coverage meaning when the supplied covariance is a credible total error model. Paper rounding, uncertainty in conductor catalogs, and a mismatch between modified Carson and Deri are source-model errors, not meter noise, and should not be disguised as very precise measurement sigma.

Numerical formulation

Discrete candidates are enumerated outside the NLP.
Continuous variables are affinely scaled to [0,1].
The default objective is smooth weighted least squares.
Ipopt uses limited-memory Hessians, exact (unrelaxed) variable bounds, and deterministic multistart. This prevents trial points from escaping the candidate domain validated before optimization.
Alternative optimizers do not receive Ipopt-specific raw attributes and must honor the JuMP variable bounds during nonlinear evaluations.
Distances remain strictly positive through physical candidate bounds.
Candidate compatibility requires every standardized residual to lie within acceptance_sigma (three by default).
A ForwardDiff Jacobian is evaluated at the best local solution for rank and conditioning diagnostics.

The prototype supports modified Carson and four canonical overhead families: :horizontal_3, :triangle_3, :horizontal_4, and :neutral_under_4. Concentric-neutral and tape-shield cables, multi-frequency observations, global confidence regions, and joint operational-data estimation remain future work.

Validation datasets and conventions

The test suite contains two machine-readable benchmark families under test/data/inverse_carson:

paper_table_iv.toml reconstructs all five overhead cases in Table IV of Tam, Geth & Mithulananthan from Tables I, II and V. The published sequence values are reproduced within two units of their last printed decimal. The paper does not explicitly state frequency or earth resistivity; 50 Hz and 100 Ω·m are labelled as inferred from the constants in Table I, not as reported facts.
opendss_mars_triangle.toml and its .dss input retain the primitive matrices from DSS C-API 0.14.3/OpenDSS SVN 3723. It explicitly records Deri earth return, no transposition, no neutral, units, soil resistivity, and the choice Rdc=Rac that prevents an unrelated frequency interpolation.

The independent OpenDSS case is intentionally not forced to agree with modified Carson. With tight uncertainty it is rejected as model mismatch, primarily in zero-sequence reactance. This is a validation success: changing earth-return formulation must not be absorbed into a fictitious conductor geometry.

Every benchmark records frequency, soil resistivity, earth model, transposition, neutral elimination, symmetrical-component convention, units, and the status of measurement uncertainty. Published cases distinguish unavailable measurement uncertainty from a rounding-only surrogate; deterministic OpenDSS output records reproducibility tolerance rather than pretending it is measurement noise.

The experiments additionally cover correlated noise, rounded inputs, a wrong Libra-vs-Mars conductor catalog, incorrect soil resistivity, and loose shunt susceptance. The latter visibly widens the height confidence interval, agreeing with the paper's warning that height is highly sensitive to shunt error.

Literature and modelling context

The electromagnetic starting point is Carson's homogeneous half-space ground return formulation Carson (1926). The implementation uses the familiar power-frequency modified-Carson approximation for inverse fitting. Deri et al.'s complex-depth image plane provides a different, efficient approximation (Deri et al. (1981)); the OpenDSS benchmark makes this model choice observable rather than treating all "Carson-like" calculations as interchangeable. Keshtkar, Solanki & Solanki compare these approximation families and their frequency/soil sensitivity (IEEE TPWRD 2014).

For a four-wire construction, the paper comparison uses the impedance Schur complement obtained by imposing zero neutral voltage. That is a data convention, not a claim that the downstream operational network has a perfectly grounded neutral. The returned primitive remains four-wire. The modelling consequences of impedance transformations under sparse neutral grounding are discussed by Geth, Heidari & Koirala (2022), while Low (2024) treats reverse Kron reduction as a separate network-identification problem. Shunt comparison uses the phase principal block corresponding to a grounded eliminated neutral; another neutral termination requires another observation model.

Finally, Jacobian rank addresses local structural sensitivity, whereas finite noise creates practical identifiability limits. The distinction and the use of profile likelihood are standard in inverse modelling; see Raue et al. (2009). Operational time-series line estimation is complementary rather than equivalent: for example, Vanin et al. estimate impedance matrices jointly with unbalanced network state, whereas inverse Carson maps linecode data to a constrained construction catalog.

Relationship to parameter estimation

Parameter estimation fits line lengths and transformer taps from multiple operational snapshots. Inverse Carson instead starts from a per-length sequence linecode and infers plausible construction data. Keeping the two formulations separate makes their distinct data assumptions and identifiability limits explicit. A future joint estimator can use compatible inverse-Carson candidates as structural priors.