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ADR-0003: Distributed conductor model
Status Accepted
Date 2026-05-04
Deciders Project maintainers
Scope groundfield

Context

The previous step in the conductor stack added a lumped finite-impedance branch model: a Conductor between two electrodes collapses to a single \(R = \rho_\text{mat} L / A\) branch in the nodal-analysis system. That model is sufficient when the conductor is short compared to its electrical size — for a few-metre run between two rod electrodes the lumped approximation matches the distributed solution to within a fraction of a percent.

For TN distribution networks several questions fall outside that regime:

  1. PEN-strand spanning the village. A 100 – 200 m PEN section between the transformer station and the cable cabinets does not leak its current at one end — every house tap, every joint, every buried section adds a leakage path. The current distribution along the conductor is non-uniform and depends on the local soil conditions.
  2. Bare-copper interconnects. Inside MV substations and at industrial sites, bare-copper conductors are deliberately laid into the soil and act as longitudinal earth electrodes. They are essentially strip electrodes with a finite length — not lumped branches.
  3. Cable shields with a continuous earth path. For shielded MV cables the metallic screen is grounded at multiple points; the shield itself is a distributed earth conductor whose leakage and longitudinal current are coupled through Carson-type effects below 1 kHz.
  4. Inductive coupling between conductor sections. Analysing the coupling between the measurement lead and the current injection requires the geometry of the leads to be resolved spatially — Neumann integrals are point-pair integrals; lumped branches do not capture them.

The lumped model is therefore insufficient for typical cases in its current form. ADR-0003 settles how to extend the conductor stack so that it remains compatible with the eight existing backends, the finite-impedance branches added in the previous step, and the upcoming inductive-coupling and Carson extensions.

Decision

Conductor becomes a distributable wire. Two new fields drive the model:

field type meaning
discretize_segment_length float (m), default None Maximum segment length used to discretise the conductor. None keeps the conductor lumped (single-segment, equivalent to today's behaviour).
coupling_to_soil Literal["isolated", "galvanic"], default "isolated" Selects whether the conductor exchanges current with the soil along its length. "isolated" (cable, PEN inside an insulating jacket) keeps the conductor purely longitudinal. "galvanic" (bare copper, exposed shield) lets every segment leak current into the soil through the same Green's-function kernel as the electrode segments.

The discretiser produces \(n\) collinear sub-segments along the conductor's axis, each carrying:

  • a midpoint-leakage current \(I_\text{leak}^{(k)}\) (only when coupling_to_soil == "galvanic"),
  • a longitudinal current \(I_\text{long}^{(k)}\) flowing along the conductor axis from node \(K_{k-1}\) to node \(K_k\).

The conductor introduces \(n+1\) nodes: $K_0 = $ start-electrode cluster, $K_n = $ end-electrode cluster, and \(n-1\) interior nodes at the segment midpoints. Each longitudinal segment is one branch in the nodal-analysis system, with series impedance

\[ Z_\text{long}^{(k)} \;=\; R^{(k)} \;+\; j\omega\, L^{(k)} \]

where \(R^{(k)} = \rho_\text{mat}\, \ell_k / A\) and the inductance \(L^{(k)}\) is deferred to ADR-0004 (inductive coupling). For the present step we keep \(L^{(k)} = 0\), so the longitudinal impedance remains purely resistive. The system stays real for f < 1 kHz.

Augmented linear system

Let:

  • \(N_e\) = number of electrode segments (as today),
  • \(N_c\) = number of conductor segments (new, only the coupling_to_soil == "galvanic" ones contribute to the Z-matrix),
  • \(K\) = number of cluster nodes (electrode clusters merged by ideal conductors, plus the interior nodes introduced by distributed conductors),
  • \(M\) = number of longitudinal branches (one per conductor segment).

The combined unknown vector is

\[ \mathbf{x} \;=\; \begin{pmatrix} \mathbf{I}_\text{leak} \\ \boldsymbol{\varphi}_n \\ \mathbf{I}_\text{long} \end{pmatrix} \in \mathbb{C}^{(N_e + N_c) + K + M}. \]

The augmented system reads

\[ \begin{bmatrix} Z & -C_s & 0 \\ C_s^{\top} & 0 & B^{\top} \\ 0 & B & -Z_b \end{bmatrix} \begin{bmatrix} \mathbf{I}_\text{leak} \\ \boldsymbol{\varphi}_n \\ \mathbf{I}_\text{long} \end{bmatrix} \;=\; \begin{bmatrix} \mathbf{0} \\ \mathbf{I}_\text{in} \\ \mathbf{0} \end{bmatrix}, \]

where

  • \(Z \in \mathbb{R}^{(N_e + N_c) \times (N_e + N_c)}\) is the enlarged multi-port grounding matrix, built from the same Green's-function kernel as before but evaluated on all segments (electrode + galvanic conductor segments).
  • \(C_s \in \{0,1\}^{(N_e+N_c) \times K}\) is the segment-to-node incidence matrix. Electrode segments map to the cluster of their owning electrode; conductor segments map to their midpoint node.
  • \(B \in \{-1,0,+1\}^{M \times K}\) is the branch-to-node incidence matrix (+1 at branch start, −1 at branch end). The branch endpoints are the conductor's interior midpoint nodes plus the start / end cluster nodes.
  • \(Z_b \in \mathbb{R}^{M \times M}\) is the diagonal matrix of longitudinal segment impedances \(R^{(k)}\) (purely resistive in this ADR; complex from ADR-0004 on).

For coupling_to_soil == "isolated" the conductor's segments contribute to \(C_s\) and \(B\) but not to \(Z\) — there are no leakage rows for these segments. The full system collapses gracefully to the lumped finite-branch model when discretize_segment_length is None (single segment, midpoint-only leakage).

Boundary conditions at the conductor ends

The endpoints of a distributed conductor sit on the cluster of the attached electrodes. Both ideal galvanic shorts and finite branches to other conductors are handled by the existing cluster + branch infrastructure — distributed conductors integrate without changing those rules.

If a conductor's start- or end-electrode is None (purely geometric end), the corresponding endpoint becomes a floating node: it participates in the linear system but has no external source, no leakage, and no constraint other than KCL. This case will be useful for future work on partial measurement-lead geometries; for typical cases the ends are always anchored to electrodes.

Implementation strategy

  1. Conductor schema. Add discretize_segment_length, coupling_to_soil. Backwards-compatible: both default to the previous lumped behaviour (None, "isolated").
  2. Discretiser. New _discretize_conductor(c, ds) in solver/image.py produces _Segment instances tagged with kind="conductor" plus the conductor's name and the per-segment node indices (K_in, K_out).
  3. Cluster / node / branch builders. Extend _build_clusters, _build_finite_branches, and the new _build_distributed_branches so that interior conductor nodes become regular nodes in the active set.
  4. Solvers. Each backend that consumes _solve_cluster_currents / _galerkin_solve is updated to:
  5. assemble the enlarged \(Z\) over electrode + galvanic conductor segments,
  6. feed the new branch list into the existing nodal-analysis block.
  7. Frequency loop. As long as \(L^{(k)} = 0\), the linear system is real per frequency and the solution is identical for all entries of engine.frequencies. The frequency dispatch becomes meaningful with ADR-0004.

Validation

  • Limit discretize_segment_length is None → bit-exact match with the lumped finite-branch solution from the previous step (regression test).
  • Limit coupling_to_soil == "isolated" with finite discretize_segment_length → solver behaves like a lumped branch with \(R = \sum_k R^{(k)}\) (the segment chain is a series resistor).
  • Convergence in n_segments → as the conductor is refined the earth current and EPR profile converge to a limit. Test \(\Delta < 1\%\) for \(n \ge 8\) on a 30 m PEN strand.
  • Cross-engine consistency → image, image_2layer, mom, cim, bem agree to ≤ 3 % on a galvanic 30 m bare-copper strip electrode modelled three ways: as a StripElectrode, as a distributed Conductor with coupling_to_soil="galvanic", and as a fine-grained chain of small electrodes connected by ideal conductors. The three formulations should give the same cluster-impedance to within the discretisation tolerance.
  • Strip-equivalence → a Conductor(galvanic, n=20) between two rods reproduces the cluster impedance of an equivalent StripElectrode of the same length placed at the same depth, to within 5 % (the difference reflects the strip's symmetric current feed at both ends vs. the conductor's directed feed).

Consequences

Positive

  • Sets the geometric stage for ADR-0004 (inductive coupling): once the conductor has segments, Neumann integrals between segment pairs become a straightforward addition to the longitudinal impedance.
  • Enables research questions that the lumped model cannot answer: PEN-strand voltage drop with intermediate leakage, bare-copper conductor as a distributed earth element, partial-shield current redistribution.
  • One unified discretisation kernel for both electrodes and conductors — they share _discretize_conductor for line geometries with _discretize_strip (the strip electrode is now a special case).

Negative

  • Larger linear system. For an a typical world with ~100 rods and ~150 m of PEN at \(\Delta s = 1\,\mathrm{m}\), the additional conductor segments push the segment count from ~1 500 (electrodes only) towards ~3 000. The dense \(O(N^2)\) memory and \(O(N^3)\) LU solve still fit in the 32 GB workstation budget, but the runtime grows by ≈ 8×. This motivates the iterative-solver roadmap item.
  • Two-layer / n-layer kernels need to be evaluated on the conductor midpoints in the upper layer. The existing precondition (\(z < h_1\)) extends to all segments — including conductor segments. The user has to make sure that buried conductors stay in the upper layer; otherwise a clear ValueError is raised.

Neutral

  • The default behaviour stays lumped. Distributed mode is opt-in; existing notebooks and tests continue to work.
  • The augmented system has the same block structure as the lumped-finite-branch system from the previous step. The change is quantitative (more rows / columns) rather than qualitative — readers familiar with the previous formulation will recognise the same pattern.

References

  • Sunde, E. D. (1968). Earth Conduction Effects in Transmission Systems. Dover, ch. 7 — wire antenna formulation, distributed current and voltage along buried conductors.
  • Dawalibi, F. P. (1986). Electromagnetic fields generated by overhead and buried short conductors. IEEE Trans. PWRD 1(4) — the canonical reference for distributed conductor segments with a layered Green's function.
  • Meliopoulos, A. P. S., & Moharam, M. G. (1983). Transient analysis of grounding systems. IEEE Trans. PAS 102(2).
  • Colominas, I., Navarrina, F., & Casteleiro, M. (1999). A boundary element formulation for the substation grounding design. Adv. Eng. Soft. 30(9–11).