Line impedance models: fidelity, symmetry, and OPF decisions

Why the impedance model you choose changes the symmetry of the problem — and what that does, and does not, mean for the feasibility of an operating point, the decision an OPF makes, and how well-posed the optimization is.

This tutorial makes a four-step argument, and is deliberately careful about how far each step is supported by evidence — including evidence generated in this repository, which does not always flatter the strong version of the claim.

More granularity and better data ⇒ more realism. A geometry- or measurement-derived four-wire impedance carries per-conductor self and mutual terms, the neutral, and the earth return that a sequence/balanced/transposed model averages away. Well established.
More realism ⇒ less symmetry ⇒ natural symmetry breaking. A balanced, transposed line has a circulant phase matrix; with balanced operation it is a fixed point of the phase-rotation symmetry — identically zero neutral current and voltage unbalance. A real untransposed line breaks that symmetry by construction, with no load asymmetry required. Well established, and demonstrated below as an exact structural difference.
Less symmetry ⇒ a better-posed optimization. Symmetry is a potential source of degeneracy — coincident eigenvalues, orbits of equivalent optima, ill-conditioned KKT systems. This is the weakest step: the mechanisms are textbook, but "physical asymmetry speeds up AC-OPF" is not a theorem, and the measurements below show no solver-effort difference on small problems. Treated as motivated analogy, never asserted.
⇒ Better decisions. At fundamental frequency, on a single line segment, the magnitude of the decision difference is small — a percent or two, within impedance parameter uncertainty, and its direction is context-dependent. What is not small or ambiguous is that the balanced model reports unbalance, neutral shift, and inter-sequence coupling as identically zero, so any decision gated on those quantities is made blind; and the per-segment error compounds on real feeders to the 1–4 % that the literature measures.

The demonstrations below are presented in the order step 2 → step 4 → step 3, not 1 → 4. We first establish the structural symmetry break as an exact fact (step 2), then read off its decision consequence and honest magnitude (step 4), and only then return to the most heavily hedged claim — the solver-degeneracy argument (step 3) — so the careful caveats land after the evidence they qualify.

Every code block runs at build time; the numbers are real. Prerequisites: BMOPFTools, JuMP, Ipopt, and the line-geometry workflow. This is the "why it matters" companion to that "how to" page, and extends the impedance fidelity ladder.

The idealisation and its symmetry

Distribution lines are not transposed (Kersting, Distribution System Modeling and Analysis, §4.1.2), so their 3×3 phase matrices have genuinely distinct self and mutual entries. The balanced/transposed idealisation replaces those entries with their averages, producing a circulant matrix (all self terms equal, all mutual terms equal). Circulance is the whole story:

A circulant matrix is diagonalised by the discrete Fourier transform (Gray 2006). For a reciprocal three-phase line the Fortescue / symmetrical- component transform is that DFT — A = (1/√3)[1 1 1; 1 a a²; 1 a² a], a = e^{j2π/3}, is the (conjugate/inverse) 3-point unitary DFT matrix.
Diagonalising decouples the network into independent zero/positive/negative- sequence circuits, and — for a symmetric (reciprocal) circulant — the positive- and negative-sequence eigenvalues coincide, Z₁ = Z₂ = Z_s − Z_m. That spectral degeneracy, and the resulting decoupled identical per-phase subproblems, is the mathematical content of the phase-rotation symmetry.
An untransposed matrix is not circulant, the DFT does not diagonalise it, the sequences stay coupled (Z₀₁, Z₁₂, … ≠ 0), and the degeneracy is lifted (Glover, Power System Analysis and Design, §8.4: "In general Z_S is not diagonal. However, if the line is completely transposed, [it is]").

BMOPFTools already detects the idealisation: the provenance analysis flags a circulant linecode as I.PROV.SEQ_DERIVED (exactly balanced ⇒ sequence-derived or transposed) and a diagonal one as I.PROV.DECOUPLED_PHASES. Those are precisely "this linecode carries the symmetry-inducing idealisation."

We work with one line throughout: a 50 Hz, four-wire LV overhead feeder (a small AAC phase conductor on a 0.3 m cross-arm at 8 m, an offset neutral — an untransposed arrangement), and its transposed idealisation built by averaging. Same conductors, same line; the only difference is the symmetry.

using BMOPFTools, JuMP, Ipopt
mm = 1e-3

net0 = Dict{String,Any}(
    "wire_data" => Dict{String,Any}(
        "ph" => Dict{String,Any}("kind"=>"overhead","r_ac"=>0.9e-3,"radius"=>3.75mm,"gmr"=>2.9mm,"i_max"=>150.0),
        "nt" => Dict{String,Any}("kind"=>"overhead","r_ac"=>0.9e-3,"radius"=>3.75mm,"gmr"=>2.9mm,"i_max"=>150.0)),
    "line_geometry" => Dict{String,Any}("g" => Dict{String,Any}(
        "frequency"=>50.0, "earth_model"=>"modified_carson", "earth_resistivity"=>100.0,
        "conductors" => Any[
            Dict{String,Any}("wire_data"=>"ph","x"=>0.0,"y"=>8.0,"terminal"=>"a"),
            Dict{String,Any}("wire_data"=>"ph","x"=>0.3,"y"=>8.0,"terminal"=>"b"),
            Dict{String,Any}("wire_data"=>"ph","x"=>0.6,"y"=>8.0,"terminal"=>"c"),
            Dict{String,Any}("wire_data"=>"nt","x"=>0.3,"y"=>7.7,"terminal"=>"n")])))
compile_linecode(net0, "g")
lc_geo = net0["linecode"]["g"]
R = BMOPFTools._pattern_keys_to_matrix(lc_geo, "R_series_")
X = BMOPFTools._pattern_keys_to_matrix(lc_geo, "X_series_")

# transposed idealisation: average self, average phase-mutual, average phase-neutral
function transposed_of(R, X)
    avg(M, S) = sum(M[i, j] for (i, j) in S) / length(S)
    d3, m3, pn = [(1,1),(2,2),(3,3)], [(1,2),(1,3),(2,3)], [(1,4),(2,4),(3,4)]
    sR, sX = avg(R, d3), avg(X, d3); mR, mX = avg(R, m3), avg(X, m3); pR, pX = avg(R, pn), avg(X, pn)
    lc = Dict{String,Any}()
    for i in 1:4, j in 1:4
        lc["R_series_$(i)_$(j)"] = i==j ? (i<=3 ? sR : R[4,4]) : (i<=3 && j<=3 ? mR : pR)
        lc["X_series_$(i)_$(j)"] = i==j ? (i<=3 ? sX : X[4,4]) : (i<=3 && j<=3 ? mX : pX)
    end
    lc
end
lc_bal = transposed_of(R, X)

# the untransposed asymmetry lives in the phase-mutual reactances (Ω/km):
(X_ab = round(X[1,2]*1e3, digits=4), X_ac = round(X[1,3]*1e3, digits=4), X_bc = round(X[2,3]*1e3, digits=4))

(X_ab = 0.5052, X_ac = 0.4617, X_bc = 0.5052)

The outer phase pair (a–c) is more widely spaced than the adjacent pairs, so its mutual reactance is genuinely smaller — the physical asymmetry a transposed model erases.

Step 2, demonstrated: the physics breaks the symmetry, exactly

Solve the same perfectly balanced load on each model, with the load-bus neutral floating (grounded only at the source — the common European/Australian four-wire configuration):

function feeder(lc; p_nom, len)
    Dict{String,Any}(
        "bus" => Dict{String,Any}(
            "src" => Dict{String,Any}("terminal_names"=>["a","b","c","n"], "perfectly_grounded_terminals"=>["n"]),
            "b1"  => Dict{String,Any}("terminal_names"=>["a","b","c","n"])),
        "voltage_source" => Dict{String,Any}("vs" => Dict{String,Any}("bus"=>"src","terminal_map"=>["a","b","c"],
            "v_magnitude"=>[230.0,230.0,230.0], "v_angle"=>[0.0,-2.0944,2.0944])),
        "linecode" => Dict{String,Any}("lc"=>lc),
        "line" => Dict{String,Any}("l1" => Dict{String,Any}("bus_from"=>"src","bus_to"=>"b1",
            "terminal_map_from"=>["a","b","c","n"], "terminal_map_to"=>["a","b","c","n"],
            "linecode"=>"lc", "length"=>len)),
        "load" => Dict{String,Any}("ld" => Dict{String,Any}("bus"=>"b1","terminal_map"=>["a","b","c","n"],
            "configuration"=>"WYE", "p_nom"=>p_nom, "q_nom"=>0.2 .* p_nom)))
end

# VUF from phase-to-ground phasors (zero-sequence-blind, so a floating neutral is fine)
function vuf(r)
    b = r["bus"]["b1"]; a = exp(im*2pi/3)
    V = [b[t]["vm"]*exp(im*b[t]["va"]) for t in ("a","b","c")]
    Vp = (V[1] + a*V[2] + a^2*V[3])/3
    Vn = (V[1] + a^2*V[2] + a*V[3])/3
    abs(Vn)/abs(Vp)*100
end

rb = solve_pf(feeder(lc_bal; p_nom=[8e3,8e3,8e3], len=300.0))
rg = solve_pf(feeder(lc_geo; p_nom=[8e3,8e3,8e3], len=300.0))
(transposed_VUF_pct = round(vuf(rb), digits=4),
 transposed_Vn_V    = round(rb["bus"]["b1"]["n"]["vm"], digits=3),
 geometry_VUF_pct   = round(vuf(rg), digits=4),
 geometry_Vn_V      = round(rg["bus"]["b1"]["n"]["vm"], digits=3))

(transposed_VUF_pct = 0.0003, transposed_Vn_V = 0.0, geometry_VUF_pct = 0.1471, geometry_Vn_V = 0.204)

The transposed model returns numerically zero voltage unbalance and neutral shift — the residual (≈ 4×10⁻⁴ %) is solver tolerance; in exact arithmetic it is zero, because a balanced load on a circulant matrix is a fixed point of the phase-rotation symmetry (the power flow's determinate high-voltage solution is that symmetric point). The same balanced load on the real geometry produces a VUF two to three orders of magnitude larger and a fraction-of-a-volt neutral-point shift. No load asymmetry was needed; the untransposed geometry generates the unbalance on its own — Survilo (2014) computes exactly this effect, of the order of a couple of percent negative sequence, for an untransposed line under balanced load. This is step 2 as an exact structural fact, independent of any parameter value: the idealisation manufactures a symmetry the real line does not have.

Step 4, demonstrated — and its honest magnitude

Put a single-phase rooftop PV cluster on phase a of a long rural LV feeder and ask each model for the hosting capacity: the maximum export before a 1.10 pu voltage limit binds. The PV carries a negative cost, so the OPF maximises its active injection pg (positive = injection into phase a), and the exported magnitude is that pg.

function lv_opf(lc; len=700.0)
    net = feeder(lc; p_nom=[2.5e3,2.5e3,2.5e3], len=len)
    net["bus"]["b1"]["v_min"] = [0.90*230, 0.90*230, 0.90*230]
    net["bus"]["b1"]["v_max"] = [1.10*230, 1.10*230, 1.10*230]
    net["generator"] = Dict{String,Any}("pv" => Dict{String,Any}("bus"=>"b1","terminal_map"=>["a","n"],
        "configuration"=>"SINGLE_PHASE", "p_min"=>[0.0], "p_max"=>[25e3],
        "q_min"=>[0.0], "q_max"=>[0.0], "cost"=>[-1.0]))   # negative cost ⇒ reward export
    net
end
hosting(lc) = sum(solve_opf(lv_opf(lc))["generator"]["pv"]["a"]["pg"]) / 1e3   # kW injected on phase a

E_bal = hosting(lc_bal)
E_geo = hosting(lc_geo)
(transposed_kW = round(E_bal, digits=2),
 geometry_kW   = round(E_geo, digits=2),
 difference_pct = round((E_bal - E_geo)/E_geo*100, digits=1))

(transposed_kW = 13.28, geometry_kW = 13.47, difference_pct = -1.4)

The two models disagree — but by only about a percent, and note the sign: here the transposed model authorises less than the true untransposed feeder can host (its averaged mutual reactance over-predicts the phase-a voltage rise). So the balanced decision is conservative, not dangerously optimistic. That direction is not universal — Antić et al. (2024) find phase-modelling assumptions swing hosting capacity in either direction by up to tens of percent once optimal single-phase placement is in play; and Claeys, Geth & Deconinck (PSCC 2022), across 128 real LV feeders, found Kron-reduced/balanced dispatch re-evaluated on the true four-wire network violates the phase-to-neutral bound by 1–4 %. The single-segment, fundamental-frequency gap here is at the small end of that range.

Be honest about what this magnitude means. A one-percent gap does not clear the noise floor of the impedance data itself: conductor manufacturing tolerances alone permit ~3 % impedance variation, and temperature adds ~4 % per 10 °C (Urquhart & Thomson 2015). At fundamental frequency, on one segment, the fidelity gap is within parameter uncertainty — a referee is right to ask whether it is distinguishable from noise, and for the headline number the answer is often no. Three things nonetheless make the higher-fidelity model the right one:

Structural blindness. The transposed model computed unbalance and neutral shift as identically zero under balanced operation (previous section). Any operational limit on VUF, neutral-earth voltage, or negative-sequence current is invisible to it — not mis-estimated by 1 %, but absent. (Here the single-phase injection is itself unbalanced, so both four-wire models see a large neutral rise — of order 30 V on this deliberately stressed feeder, a safety concern that a three-wire Kron-reduced model would report as zero entirely; that is the four-wire-vs-Kron gap, quantified by Claeys, Geth & Deconinck.)
Aggregation. The per-segment error is small but systematic, not random; across a feeder of many untransposed segments and single-phase laterals it accumulates to the measured 1–4 %.
Frequency. The gap grows sharply with frequency — Urquhart & Thomson report 16–33 % impedance differences at 450 Hz. This engine is fundamental-frequency (constant-r_ac modified Carson), but only a geometry retains the physical data a frequency-dependent method would need; a frozen 50 Hz matrix cannot be re-derived at harmonics at all (see Where the fidelity actually bites).

Step 3: symmetry, degeneracy, and the algorithm — carefully

It is tempting to argue that because symmetry causes degeneracy, and degeneracy hurts solvers, the asymmetric model must solve better. That argument conflates three distinct objects, and the evidence does not support the strong form.

Spectral degeneracy of the impedance (Z₁ = Z₂ for a symmetric circulant) is a property of the matrix. It does not, by itself, make the optimization degenerate.
Multiplicity of optima requires the symmetry to act on decision variables with a nontrivial orbit — genuinely interchangeable resources. A balanced load (or a single-phase PV) on a symmetric network has a unique symmetric (or determinate) solution; there is no orbit, so nothing for the symmetry to make non-unique. Consistent with that, the barrier-iteration counts are identical whether the impedance is transposed or geometric, balanced load or unbalanced:

const MOI = JuMP.MOI
function barrier_iterations(lc; p_nom)
    mref = Ref{Any}(nothing)
    solve_opf(feeder(lc; p_nom=p_nom, len=300.0); model_hook! = ctx -> (mref[] = ctx.model))
    Int(MOI.get(mref[], MOI.BarrierIterations()))
end
(transposed_balanced = barrier_iterations(lc_bal; p_nom=[8e3,8e3,8e3]),
 geometry_balanced   = barrier_iterations(lc_geo; p_nom=[8e3,8e3,8e3]),
 geometry_unbalanced = barrier_iterations(lc_geo; p_nom=[14e3,5e3,3e3]))

(transposed_balanced = 3, geometry_balanced = 2, geometry_unbalanced = 3)

There is no measured speed-up, and none should be expected here: the solution is unique in every case, and Ipopt's own regularization (it adds δ_w·I to the Hessian and −δ_c·I to the constraint block — a Tikhonov/Levenberg–Marquardt perturbation; Wächter & Biegler 2006) absorbs any ill-conditioning that a near-symmetric point would create.

Where symmetry genuinely produces multiplicity is when it acts on interchangeable decisions: e.g. identical-cost single-phase resources on symmetric phases, whose optimal dispatch split is then a continuum. That is a real degeneracy — and BMOPFTools already flags one instance of it, identical generator cost vectors, as a dispatch-degeneracy finding. Note it is cost symmetry there; asymmetric impedance breaks the network side of such symmetries, giving each phase a distinct sensitivity and a unique optimal split. This is the defensible version of "asymmetry helps": it removes a source of non-uniqueness, not (measurably here) a source of iterations.

The honest summary of step 3: high-fidelity asymmetry is best understood as a free, physically meaningful regulariser — it plays the role that a solver's δ_w·I and a modeller's symmetry-breaking constraints (Liberti 2012; Margot

play artificially. But those results are from adjacent domains (continuous

KKT regularization; discrete branch-and-bound), so the transfer to AC-OPF is by analogy, not theorem. The nearest in-domain precedent, Lavaei & Low (2012), restores their zero-duality-gap sufficient condition by adding a tiny resistance to lossless transformers — but what that repairs is connectivity of the resistive graph Re{Y}, not a phase symmetry; cite it for the idea that a small structural perturbation can restore a well-posedness guarantee, not as a symmetry result. And regardless of symmetry, AC-OPF remains non-convex with genuine local optima from the nonlinear power-flow constraints (Bukhsh et al. 2013; Molzahn 2017) — breaking the impedance symmetry removes one avoidable source of degeneracy; it does not convexify the problem.

Where the fidelity actually bites

Putting the honest pieces together, the case for high-fidelity impedance is not "the fundamental-frequency numbers are far off" — on a single segment they differ by about as much as the data is uncertain. The case is:

Structure: unbalance, neutral shift, and coupled sequences are entirely absent from a balanced model, so any decision or limit involving them is made blind — a qualitative gap, not a 1 % one.
The neutral: retaining it explicitly (four-wire, not Kron-reduced) is where the largest, best-quantified errors live — 1–4 % voltage-limit violations on real feeders (Claeys, Geth & Deconinck 2022), because the neutral is grounded only at the transformer in most LV networks.
Harmonics and estimation: the fidelity gap grows sharply with frequency (Urquhart & Thomson 2015) and materially changes state-estimation and impedance-identification outcomes. This engine does not yet compute harmonic impedances, but only geometry — not a frozen matrix — preserves the data a future frequency-dependent method (or an external tool) would need.

Practical guidance

Prefer geometry (fidelity ladder): its realisability is checkable directly, it recompiles at other frequencies, and it keeps the provenance link the analysis layer cross-checks.
Watch for I.PROV.SEQ_DERIVED / I.PROV.DECOUPLED_PHASES: they mark a linecode carrying the balanced/transposed idealisation — fine if intended, a red flag if a real untransposed feeder was flattened into it.
Keep the neutral explicit and use the IVR solver (both defaults here). For four-wire networks the power–voltage (ACR) formulation admits non-physical multiple solutions — equivalent to spuriously grounding the neutral — and failed to converge within 500 iterations for 35 % of 128 instances, whereas the current–voltage (IVR) formulation BMOPFTools uses is robust (Claeys, Geth & Deconinck 2022). This is a real, in-domain conditioning result — about formulation, not impedance symmetry.
Do not oversell the fundamental-frequency magnitude. State the parameter- uncertainty caveat, and reach for geometry because of structure, the neutral, and frequency-extensibility — not a headline number.

References

W. H. Kersting, Distribution System Modeling and Analysis, CRC Press — §4.1 (untransposed lines, Carson, Kron reduction, sequence impedances).
W. H. Kersting & R. K. Green, "The application of Carson's equation to the steady-state analysis of distribution feeders," IEEE PES PSCE, 2011, DOI 10.1109/PSCE.2011.5772579.
J. D. Glover, M. S. Sarma, T. Overbye, Power System Analysis and Design, §8 (transposed ⇒ circulant ⇒ Fortescue-diagonal; "in general Z_S is not diagonal").
R. M. Gray, Toeplitz and Circulant Matrices: A Review, Found. Trends Commun. Inf. Theory 2(3):155–239, 2006, DOI 10.1561/0100000006 (circulant ⇔ DFT-diagonalisable).
A. Ferrero et al., "100 Years of Symmetrical Components," Energies 12(3):450, 2019, DOI 10.3390/en12030450.
J. Survilo, "Impact of Untransposed Power Lines," Power and Electrical Engineering 32, 2014, DOI 10.7250/pee.2014.005 (computes ~2.5 % negative sequence from an untransposed line under balanced load; case-specific).
S. Claeys, F. Geth, G. Deconinck, "Optimal Power Flow in Four-Wire Distribution Networks: Formulation and Benchmarking," PSCC 2022 / EPSR 213:108522, arXiv:2204.08126 (Kron/balanced dispatch violates the true four-wire bound by 1–4 %; IVR vs ACR robustness across 128 LV networks).
F. Geth, R. Heidari, A. Koirala, "Computational Analysis of Impedance Transformations for Four-Wire Power Networks with Sparse Neutral Grounding," ACM e-Energy 2022, arXiv:2206.07274, DOI 10.1145/3538637.3538844.
A. J. Urquhart & M. Thomson, "Series impedance of distribution cables with sector-shaped conductors," IET GTD 9(16):2679–2685, 2015, DOI 10.1049/iet-gtd.2015.0546 (manufacturing/temperature uncertainty ~3 %/~4 % per 10 °C; 16–33 % impedance change at 450 Hz).
T. Antić, A. Keane, T. Capuder, "Impact of Phase Selection on Accuracy and Scalability in Calculating DER Hosting Capacity," 2024, arXiv:2405.20682.
W. A. Bukhsh, A. Grothey, K. I. M. McKinnon, P. A. Trodden, "Local Solutions of the Optimal Power Flow Problem," IEEE TPWRS 28(4):4780–4788, 2013, DOI 10.1109/TPWRS.2013.2274577.
D. K. Molzahn, "Computing the Feasible Spaces of Optimal Power Flow Problems," IEEE TPWRS 32(6):4752–4763, 2017, arXiv:1608.00598.
J. Lavaei & S. H. Low, "Zero Duality Gap in Optimal Power Flow Problem," IEEE TPWRS 27(1):92–107, 2012, DOI 10.1109/TPWRS.2011.2160974 (small transformer resistance restores resistive-graph connectivity and the zero-gap condition).
L. Liberti, "Reformulations in mathematical programming: automatic symmetry detection and exploitation," Math. Prog. 131:273–304, 2012, DOI 10.1007/s10107-010-0351-0.
F. Margot, "Symmetry in Integer Linear Programming," 50 Years of Integer Programming, Springer 2010, pp. 647–686.
A. Wächter & L. T. Biegler, "On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming," Math. Prog. 106:25–57, 2006, DOI 10.1007/s10107-004-0559-y (δ_w/δ_c inertia correction as Tikhonov regularization).
EN 50160: voltage unbalance limit VUF = |V₂|/|V₁| ≤ 2 % (up to ~3 % where single-/two-phase loads dominate); IEC 61000-2-2 sets a 2 % compatibility level.