Inverse Carson reconstruction: a modelling tutorial

Audience: power-system researchers · Scope: reconstructing plausible overhead-line constructions from diagonal sequence impedance data.

Inverse Carson is an inverse construction problem, not a magic conversion from sequence impedance to a unique geometry. Diagonal zero- and positive-sequence data discard coupling information; several conductor catalogs, spacings, and neutral arrangements can fit the same reported values. This tutorial shows how to frame the problem honestly, fit a bounded catalog, assess ambiguity, and materialize models without erasing their assumptions.

1. Start from the observation model

SequenceLineObservation represents the reported diagonal sequence data. Series observations are (R0, X0, R1, X1); optional shunt observations are (B0, B1). State the units, frequency, soil resistivity, earth-return approximation, transposition convention, neutral treatment, and sequence convention before fitting anything.

using PowerOptLab

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],
)

sigma is in the declared input units and defines both residual weighting and candidate compatibility. If measurement errors are correlated, provide a positive-definite full covariance in the ordering (R0, X0, R1, X1[, B0, B1]). The objective then uses a Mahalanobis residual; compatibility remains reported as readable marginal standardized residuals.

Pitfall: interpreting published rounding as metrology

When a paper reports only rounded sequence values, its last digit is not a calibrated measurement uncertainty. A nominal one-percent sigma is useful for descriptive screening, but it does not give likelihood-ratio intervals a frequentist interpretation. Include catalog tolerance, earth-model mismatch, and rounding/source uncertainty in the uncertainty model—or explicitly label the result as a tolerance study.

2. Represent a construction catalog, not an unconstrained line

The method enumerates discrete conductor/geometry families outside the NLP and optimizes only bounded continuous geometry and temperature variables inside each candidate. For example:

candidate = 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),
    radius = fill(5.0e-3, 3),
    lower = [0.30, 6.0, 20.0],   # half-span [m], height [m], temperature [°C]
    upper = [0.80, 12.0, 90.0],
)

The supported canonical families are :horizontal_3, :triangle_3, :horizontal_4, and :neutral_under_4. Physical bounds matter: they keep conductor separation positive, prevent a nonlinear solver from exploring meaningless geometry, and encode the actual construction domain under study.

Pitfall: treating bounds as harmless optimizer settings

Candidate bounds are prior physical knowledge. A fit at a span, height, or temperature bound says the data would prefer to leave the candidate domain; it does not validate that construction. Report bound-active fits and test whether a different catalog family explains the observation more credibly.

3. Fit every credible candidate and retain ambiguity

result = solve_inverse_carson(obs, [candidate_a, candidate_b];
                              starts=16, acceptance_sigma=3.0)

result.compatible_candidates
result.fits[1].parameters
result.fits[1].standardized_residual

The solver uses deterministic multistart and bounded smooth weighted least squares. acceptance_sigma is a componentwise compatibility rule: every standardized residual must lie within the threshold. It is intentionally not a selection rule that collapses all plausible candidates to a single winner.

Pitfall: reporting only the minimum objective candidate

The numerically best candidate can be practically indistinguishable from other compatible constructions. Choosing it alone turns catalog ambiguity into false precision. Report all compatible ids, their residual vectors, parameter bounds, and local rank diagnostics. If a downstream study needs one model, describe the external engineering criterion used to select it.

4. Know what sequence data discard

The forward model constructs a primitive phase/neutral impedance and capacitance matrix, applies the stated neutral-elimination convention for comparison, then forms the observed diagonal sequence quantities. The returned fit includes a full reconstructed Z_sequence (and B_sequence when applicable), including off-diagonal sequence coupling absent from the observation.

Those off-diagonal entries are model-derived completions, not new measurements. Likewise, for four-wire candidates a neutral may be Kron-reduced only for the sequence comparison; the materialized operational model keeps the neutral explicit.

Pitfall: confusing a data transformation with an operational neutral model

Imposing zero neutral voltage to form a Schur complement is an observation/data convention. It does not say that a downstream LV feeder has a perfectly grounded neutral. Preserve the primitive four-wire representation for unbalanced power flow and state estimation, then use the appropriate neutral termination in those operational models.

5. Assess local identifiability before quoting error bars

Each InverseCarsonFit returns singular values and rank of the ForwardDiff prediction Jacobian at the fitted point. A rank-deficient Jacobian means one or more local parameter directions are not identified. In that case the local covariance and normal confidence intervals are deliberately absent:

fit = result.fits[1]
fit.jacobian_rank
fit.jacobian_singular_values
fit.local_parameter_covariance

For series impedance alone, vertically translating all conductors does not change the series impedance, so absolute height is expected to be weak or unidentified. Shunt data can help, but height sensitivity to shunt uncertainty can make the practical interval broad.

Pitfall: using a pseudoinverse covariance as evidence of certainty

A pseudoinverse can assign a convenient-looking number to an unidentified direction. This implementation instead refuses local covariance when rank is deficient. Treat that as an inference result: additional data, a narrower catalog, or a different question is required.

6. Profile nonlinear and bound-limited directions

Local covariance is only a linear approximation around one fit. Use profile likelihood to fix one parameter, reoptimize the others, and trace the connected region below the one-degree-of-freedom threshold:

profile = profile_inverse_carson(fit, candidate, obs;
                                 parameter=:height,
                                 confidence_level=0.95)

profile.lower_status
profile.upper_status

Profile endpoints distinguish a threshold crossing, a candidate bound, and a solver-limited endpoint. Continuation warm-starts neighboring profile points and retries failures from deterministic fitted/candidate starts.

Pitfall: calling a profile interval global

The profile follows the connected region reached from the selected fit. It can miss disconnected feasible regions or another candidate family. Use deterministic multistart, broader catalog enumeration, or a global method before claiming global confidence regions.

7. Keep earth-return and forward-model mismatch visible

The inverse fit uses power-frequency modified Carson. OpenDSS commonly uses the Deri complex-depth approximation; they are related but not interchangeable. The included OpenDSS benchmark is intentionally rejected under tight uncertainty when the earth-return mismatch is visible, rather than inventing a geometry to absorb it.

Pitfall: calibrating geometry against the wrong forward model

If an observation was produced with a different earth-return model, frequency, soil resistivity, transposition, conductor resistance convention, or neutral elimination than the inverse model, a low residual can be physically misleading and a high residual can be the correct result. Treat forward-model provenance as part of the data, not a hidden software detail.

8. Materialize a candidate without mutating the study network

After selecting or retaining a fit, create BMOPF-ready construction blocks:

construction = materialize_inverse_carson(fit, candidate)
construction["wire_data"]
construction["line_geometry"]

Materialization does not mutate a network. It permits a transparent next step: build one operational network per compatible construction and propagate the structural ambiguity into power flow, DSSE, or DOE scenarios.

Pitfall: converting ambiguity into one deterministic linecode too early

Operational studies often care about voltage/current sensitivity that was not observed in diagonal sequence data. Use all compatible constructions as model scenarios where the decision is sensitive. A single selected linecode is a decision, not an inverse-problem discovery.

9. Minimum research-report checklist

For every study, report:

1. input values, units, frequency, covariance/sigma source, and rounding model;
2. earth-return formulation, soil resistivity, transposition, and sequence convention;
3. neutral-elimination and shunt observation conventions;
4. conductor catalog, geometry family, parameter bounds, temperature/resistance convention, multistart count, and compatibility threshold;
5. all compatible candidates and their standardized residuals, not only the nominal winner;
6. Jacobian rank/singular values, local covariance availability, and profile endpoint statuses;
7. treatment of forward-model mismatch and every downstream use of the materialized primitive model.

With this record, inverse Carson reconstruction is a reproducible constrained inference exercise rather than an overconfident reverse-engineering claim. See Inverse Carson reconstruction for the full API and benchmark conventions.