Ground, neutral, and earth return

Three different things called "ground", one experiment across floating, impedance-grounded, and solidly-grounded neutrals — then the mathematics: null spaces, the Kron-reduction assumption, and what the Fortescue transform does and does not buy you.

Distribution modelling uses one word — ground — for three different objects, and most four-wire modelling errors trace back to conflating them. The terminals primer makes the conceptual case for explicit neutrals; the grounding spec defines the data model. This tutorial supplies the missing chain between them: the same feeder solved under three grounding assumptions, the null-space view of what "floating" means, a live demonstration of exactly when eliminating the neutral (Kron reduction) is legitimate, the single-wire earth-return system as the limiting case, and the precise distinction between the Fortescue transform (always available) and sequence-network decoupling (an assumption). Every block runs at build time.

1. Three things called "ground"

• The ground reference is a modelling convention: the single 0 V "copper plate" all node voltages are measured against. It is not a matrix row, not a conductor, and nothing flows "through" it (grounding spec).
• The neutral conductor is a physical wire with resistance, reactance, and mutual coupling to the phases. It has its own voltage variable at every bus, and that voltage is generally not zero.
• The grounding electrode (rod, mat, counterpoise) is the tie between the two: a physical, imperfect connection from a conductor — usually the neutral — to the local earth.

The classic conflation is assuming the neutral is the reference: that its voltage is zero everywhere because it is "grounded". Whether that holds is a property of the electrodes, not of the word — §3 measures it.

2. The engine's vocabulary

Four data-model pieces express everything in this tutorial:

• a case-level terminal_conventions block declares which terminal labels are phases, neutrals, or (rare, for genuine earth wires) earth conductors. Absent the block, roles are inferred from naming and flagged W.CONV.TERMINAL_ROLES_INFERRED;
• perfectly_grounded_terminals on a bus pins those terminal voltages to exactly 0 V and adds a free earth-injection current — the ideal electrode;
• an impedance grounding is not a special field at all: it is an ordinary shunt from the neutral terminal to the reference — the real electrode;
• vn_max bounds how far a floating neutral may drift in an OPF.

Look at both on real cases — the LV feeder used across these tutorials, and the SWER feeder whose grounding is its return circuit:

using BMOPFTools, JuMP, Ipopt, LinearAlgebra

lv1  = from_dss(joinpath(ENV["BMOPF_RESTRICTED_DATA"], "LV", "LV1_14bus", "Master.dss"))
swer = from_dss(joinpath(pkgdir(BMOPFTools), "test", "data", "SWER", "Master.dss"))

println("conventions (both cases): ", lv1["terminal_conventions"])
println("SWER perfectly grounded : ",
        [(b, bd["perfectly_grounded_terminals"]) for (b, bd) in swer["bus"]
         if !isempty(get(bd, "perfectly_grounded_terminals", String[]))])
println("SWER electrodes (shunts): ",
        sort([(id, s["bus"], round(Float64(s["G_1_1"]); digits = 0)) for (id, s) in swer["shunt"]]))

One perfectly-grounded source neutral, and five 1000 S (1 mΩ — effectively solid) electrode shunts along the feeder: the SWER system's earth-return path, written as ordinary data. §6 returns to it.

3. Floating, impedance-grounded, solid: one experiment

Build the smallest network where the three assumptions differ: a 500 m four-wire line (mutually coupled phase and neutral conductors), a single-phase 3 kW + 1 kvar customer between phase a and neutral, and the source neutral solidly grounded. Vary only the customer-end neutral: floating, a realistic 10 Ω electrode, or perfect. Watch three quantities — the neutral-to-earth voltage (NEV), the phase-to-neutral voltage the customer actually receives, and how the return current splits between the neutral conductor and the earth:

OPT = optimizer_with_attributes(Ipopt.Optimizer, "print_level" => 0)

lc4 = Dict{String,Any}()
for i in 1:4, j in 1:4
    lc4["R_series_$(i)_$(j)"] = (i == j ? 0.5 : 0.02) / 1000   # Ω/m
    lc4["X_series_$(i)_$(j)"] = (i == j ? 0.2 : 0.05) / 1000
end

function feeder(ground)
    net = Dict{String,Any}(
        "bus" => Dict{String,Any}(
            "src" => Dict{String,Any}("terminal_names" => ["a","b","c","n"],
                                      "perfectly_grounded_terminals" => ["n"]),
            "lb"  => Dict{String,Any}("terminal_names" => ["a","b","c","n"],
                     "perfectly_grounded_terminals" =>
                         ground == :perfect ? ["n"] : String[])),
        "voltage_source" => Dict{String,Any}("source" => Dict{String,Any}(
            "bus" => "src", "terminal_map" => ["a","b","c","n"],
            "v_magnitude" => [230.0, 230.0, 230.0, 0.0],
            "v_angle" => [0.0, -2π/3, 2π/3, 0.0])),
        "linecode" => Dict{String,Any}("lc4" => deepcopy(lc4)),
        "line" => Dict{String,Any}("l1" => Dict{String,Any}(
            "bus_from" => "src", "bus_to" => "lb", "linecode" => "lc4",
            "length" => 500.0,
            "terminal_map_from" => ["a","b","c","n"],
            "terminal_map_to"   => ["a","b","c","n"])),
        "load" => Dict{String,Any}("ld" => Dict{String,Any}(
            "bus" => "lb", "terminal_map" => ["a","n"],
            "configuration" => "SINGLE_PHASE",
            "p_nom" => [3000.0], "q_nom" => [1000.0])))
    ground == :electrode && (net["shunt"] = Dict{String,Any}(
        "rod" => Dict{String,Any}("bus" => "lb", "terminal_map" => ["n"],
                                  "G_1_1" => 0.1)))        # a 10 Ω rod
    net
end

println("neutral       NEV      V_pn(load)   I_neutral wire")
for g in (:float, :electrode, :perfect)
    r  = solve_pf(feeder(g); optimizer = OPT)
    lb = r["bus"]["lb"]
    vn = abs(lb["n"]["vr"] + im*lb["n"]["vi"])
    va = abs((lb["a"]["vr"] + im*lb["a"]["vi"]) - (lb["n"]["vr"] + im*lb["n"]["vi"]))
    println(rpad(g, 12), lpad(round(vn; digits = 2), 6), " V",
            lpad(round(va; digits = 2), 10), " V",
            lpad(round(r["line"]["l1"]["n"]["cm_fr"]; digits = 2), 10), " A")
end

Three lessons, one table:

• Floating: the full 14 A of return current flows in the neutral conductor, and its impedance drop appears as ≈ 3.6 V of NEV at the customer — stray voltage a person can touch, and a direct bite out of the voltage the load receives.
• A real electrode barely moves it. The 10 Ω rod diverts only ≈ 0.35 A to earth — the neutral conductor's fraction of an ohm is a far lower-impedance return than any single rod. One electrode is bonding, not a return path; this is why multi-grounded systems need many electrodes before the "grounded" idealisation is honest.
• Perfect grounding is not a small step from a good electrode — it is a different circuit. NEV is zero by construction, the customer gains 3.4 V, and — look at the last column — the neutral conductor now carries 1.4 A instead of 14 A: the model quietly rerouted 90 % of the return current through the zero-impedance earth. If your physical feeder does not do that, neither should your model. perfectly_grounded_terminals is the right model for substation mats and dense multi-grounded neutrals, and a silently wrong one for a single rod at the end of a long rural line.

4. What "floating" means mathematically

The system nodal admittance matrix makes the reference story precise. ybus_passive assembles every passive element over explicit (bus, terminal) nodes, with earth as the reference (not a matrix row), grounded terminals collapsed onto it, and — deliberately — no contribution from ideal voltage sources. Whatever that matrix cannot see, something else must pin:

cases = [
    "no grounding anywhere" => (n = feeder(:float);
                                delete!(n["bus"]["src"], "perfectly_grounded_terminals"); n),
    "source neutral grounded" => feeder(:float),
    "both neutrals grounded"  => feeder(:perfect),
]
for (name, net) in cases
    yb = ybus_passive(net)
    sv = svdvals(Matrix(yb.Y))
    println(rpad(name, 26), "nodes: ", size(yb.Y, 1),
            "   near-zero singular values: ", count(s -> s < 1e-9 * sv[1], sv))
end

Read the counts as degrees of freedom the passive network leaves undetermined — rigid voltage translations of conductor groups that have no tie to the reference. With nothing grounded, all four conductor paths float (shift both ends of any conductor pair equally and no current changes); grounding the source neutral removes exactly the neutral's mode; the three phase modes remain in every case because a passive matrix cannot know about sources — it is the voltage source's fixed phasors, a boundary condition, that pin them in an actual solve. This is the graph-Laplacian story of reference nodes, made concrete: sources pin voltages; groundings tie conductors to earth; every conductor group needs one or the other, or the problem is singular — the numerical face of an electrician's "floating".

5. Kron reduction is an assumption, not a simplification

Textbooks reduce the 4×4 series-impedance matrix to 3×3 by eliminating the neutral row — the Schur complement $Z_{\text{red}} = Z_{pp} - Z_{pn} Z_{nn}^{-1} Z_{np}$. The step is algebraically exact iff the eliminated node's voltage is held at zero wherever current enters it — that is, the neutral is perfectly grounded at every connection point. §3 just showed when that premise holds. Test the whole chain: Kron-reduce our line by hand, build the three-wire equivalent (loads become phase-to-ground — the reduced model has no neutral to connect to), and compare against the four-wire truth under each grounding:

Z4  = [(i == j ? 0.5 : 0.02) + im*(i == j ? 0.2 : 0.05) for i in 1:4, j in 1:4] ./ 2 # 500 m, Ω
Zk  = Z4[1:3,1:3] .- Z4[1:3,4:4] * (Z4[4:4,4:4] \ Matrix(transpose(Z4[1:3,4:4])))

lc3 = Dict{String,Any}()
for i in 1:3, j in 1:3
    lc3["R_series_$(i)_$(j)"] = real(Zk[i,j]) / 500.0
    lc3["X_series_$(i)_$(j)"] = imag(Zk[i,j]) / 500.0
end
net3w = Dict{String,Any}(
    "bus" => Dict{String,Any}(
        "src" => Dict{String,Any}("terminal_names" => ["a","b","c"]),
        "lb"  => Dict{String,Any}("terminal_names" => ["a","b","c"])),
    "voltage_source" => Dict{String,Any}("source" => Dict{String,Any}(
        "bus" => "src", "terminal_map" => ["a","b","c"],
        "v_magnitude" => [230.0, 230.0, 230.0], "v_angle" => [0.0, -2π/3, 2π/3])),
    "linecode" => Dict{String,Any}("lc3" => lc3),
    "line" => Dict{String,Any}("l1" => Dict{String,Any}(
        "bus_from" => "src", "bus_to" => "lb", "linecode" => "lc3",
        "length" => 500.0,
        "terminal_map_from" => ["a","b","c"], "terminal_map_to" => ["a","b","c"])),
    "load" => Dict{String,Any}("ld" => Dict{String,Any}(
        "bus" => "lb", "terminal_map" => ["a"], "configuration" => "WYE",
        "p_nom" => [3000.0], "q_nom" => [1000.0])))

va3 = let r = solve_pf(net3w; optimizer = OPT)
    abs(r["bus"]["lb"]["a"]["vr"] + im*r["bus"]["lb"]["a"]["vi"])
end
println("Kron three-wire model:  V_a = ", round(va3; digits = 3), " V, always")
for g in (:perfect, :electrode, :float)
    r  = solve_pf(feeder(g); optimizer = OPT)
    lb = r["bus"]["lb"]
    va = abs((lb["a"]["vr"] + im*lb["a"]["vi"]) - (lb["n"]["vr"] + im*lb["n"]["vi"]))
    println(rpad(g, 12), " four-wire V_pn = ", round(va; digits = 3),
            " V    Kron error = ", round(abs(va - va3); digits = 3), " V")
end

Under perfect grounding at both ends the reduction is exact to the last digit — Kron is not an approximation there, it is algebra. Under a floating or realistically-earthed neutral the "same" model is ≈ 3.3 V optimistic — 1.4 % of nominal, larger than the margins hosting-capacity studies argue over, and invisible from inside the reduced model, which cannot even represent the NEV it is mispredicting. European LV feeders ground the neutral at the transformer and rarely along the way — the wrong regime for Kron. This is precisely why the engine keeps four-wire models un-reduced end to end (impedance spec); its single deliberate exception is collapsing concentric-neutral/tape-shield subconductors held at earth potential inside compile_linecode, recorded in the linecode's derivation.shields_reduced provenance (line geometry tutorial).

6. Earth as the return conductor: SWER

Push §3 to its limit: remove the neutral conductor entirely and let the earth be the return path — Single-Wire Earth-Return, thousands of kilometres of which electrify rural Australia. In the data model this needs nothing new: one phase conductor per span, and the grounding machinery of §2 carrying the full load current at every electrode. The earth-return impedance itself lives in the line constants — Carson's frequency- and resistivity-dependent correction, applied when geometry compiles to a linecode (impedance derivation):

report = analyze(swer)
println("SWER zones found: ", report.results[:connectivity]["n_swer_zones"])
for f in report.findings
    startswith(String(f.code), "I.PROV.SWER") && println("  ", f.code, ": ", f.message)
end

analyze recognises the pattern (single-wire, transformer-isolated galvanic zones) and labels it as provenance, not as an error — earthing as a design choice. The SWER case study takes this feeder on to the optimisation question; here the point is conceptual: SWER is not an exotic special case bolted onto the model, it is §1's three objects — a reference, (no) neutral conductor, and electrodes — in their most honest configuration, where getting the earth path wrong is not a 3 V subtlety but the whole circuit.

7. The Fortescue transform vs sequence decoupling

The last conflation in the chain is coordinate change versus physical assumption. The Fortescue matrix $\mathbf{F}$ (a conjugate three-point DFT, notation) can always be applied — it is invertible, no assumptions required. What is not free is the step every sequence-network diagram takes next: treating zero, positive, and negative sequence as three decoupled single-phase circuits. That requires $\mathbf{F}^{-1} \mathbf{Z} \mathbf{F}$ to be diagonal, which holds exactly when $\mathbf{Z}$ is circulant — balanced, transposed construction. Our §5 matrix is circulant by construction (all mutuals equal); perturb the mutuals the way untransposed geometry does and watch the decoupling die:

a = cis(2π/3)
F = (1/sqrt(3)) .* [1 1 1; 1 a a^2; 1 a^2 a]
offdiag_max(M) = maximum(abs(M[i,j]) for i in 1:3, j in 1:3 if i != j)

Zs = inv(F) * Zk * F
println("circulant line:    Z0 = ", round(Zs[1,1]; sigdigits = 3),
        "  Z1 = Z2 = ", round(Zs[2,2]; sigdigits = 3),
        "  max off-diag = ", round(offdiag_max(Zs); sigdigits = 2), " Ω")

Zu = copy(Zk)                          # untransposed: unequal mutual spacings
Zu[1,2] = Zu[2,1] = 0.015 + 0.040im
Zu[2,3] = Zu[3,2] = 0.010 + 0.025im
Zu[1,3] = Zu[3,1] = 0.008 + 0.020im
Zsu = inv(F) * Zu * F
println("untransposed line: Z0 = ", round(Zsu[1,1]; sigdigits = 3),
        "  Z1 = ", round(Zsu[2,2]; sigdigits = 3),
        "  max off-diag = ", round(offdiag_max(Zsu); sigdigits = 2), " Ω")

Two readings tie the whole tutorial together. First, $Z_0 \neq Z_1$ even for the perfectly balanced line — because zero-sequence current needs a return path, and $Z_0$ is where the neutral-and-earth story of §§3–6 hides inside sequence analysis: change the grounding and you change $Z_0$, while $Z_1$ never notices. Second, for the untransposed line the sequence coupling is small but structural — the decoupled diagram is now an approximation with an error you must budget, not a change of coordinates. The engine sidesteps the entire question by solving in phase coordinates and using sequence quantities only as derived outputs; the impedance-models tutorial develops the circulant ⇔ DFT ⇔ $Z_1 = Z_2$ story and its consequences for OPF in full — link, don't duplicate.

8. The checklist

• Say which "ground" you mean. Reference, neutral conductor, or electrode — if a sentence works with all three meanings, it is hiding an assumption.
• perfectly_grounded_terminals is a strong claim: zero NEV and a free zero-impedance return path (§3). Defensible for substation earthing mats and dense multi-grounded systems; not for a rod at the end of a feeder — model that as a shunt and let the solver tell you what it carries.
• A floating neutral is not an error — it is a modelling statement that the only return is metallic. Expect NEV, bound it with vn_max, and expect ybus_passive to show the corresponding null mode if nothing ties the group to earth (§4).
• Kron-reduce only what is genuinely pinned to zero volts (§5). If you ingest three-wire data that was reduced upstream, the assumption arrived with it — I.PROV.SEQ_DERIVED / I.PROV.DECOUPLED_PHASES flag the fingerprints.
• Sequence networks are a symmetry statement, not a coordinate change (§7). Use the transform freely on results; trust decoupled networks only as far as the construction is balanced — and remember $Z_0$ is the grounding story.