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Induced voltage on single system (IND)

We introduced a new postprocessing plot called "IND" for calculation of induced voltages in the metallic sheaths of a single system as a result of the currents flowing in the conductors. The method is based on the standard IEEE 575-2014 and available for single-core cables only.

Posted 2026-02-23
Categories: New feature , User guides

Sheath voltage and current calculation for single-core cable systems

High-voltage single-core power cables require careful management of the voltages and currents induced in their metallic sheaths. Depending on the bonding arrangement chosen — single-point bonding, both-ends (solid) bonding, or cross-bonding — the standing sheath voltage profile and the circulating sheath currents change significantly. An optional Earth/Ground Continuity Conductor (ECC) can be included, either untransposed or transposed, to provide a return path for fault currents while influencing the induced voltage distribution (only available in electrical module, not in the rating modules).

This new module implements a unified solver that computes the induced sheath-to-ground voltage along the cable route and the sheath (and ECC) currents for all three bonding types. The calculation is based on the multi-conductor impedance approach described in IEEE Std 575-2014, Appendixes D, E and F.

Supported bonding types

  • Single-point bonding — the sheaths are grounded at one end only. The standing voltage rises linearly from zero at the grounded end to a maximum at the open end. An ECC, if present, carries the return current and modifies the voltage profile.
  • Both-ends (solid) bonding — the sheaths are grounded at both ends. Circulating currents flow in the sheaths, and the standing voltage is zero along the entire route for a uniform section without a transposed ECC. With a transposed ECC, a voltage peak appears at the midpoint.
  • Cross-bonding — the sheaths are transposed at two intermediate joints, dividing the route into three minor sections. The sheaths are grounded at both ends. For equal minor-section lengths (perfect cross-bonding), the residual voltage at the grounded ends approaches zero. For unequal lengths (imperfect cross-bonding), a residual voltage ΔU remains, which is shown on the plot as a double-headed arrow. An optional cables_transposed flag allows the phase conductors to also change physical positions at each joint.

Flexible cable geometry

Cable and ECC positions are specified as (x, y) coordinates in millimetres, allowing any formation to be modelled — flat horizontal, trefoil, rectangular, or fully arbitrary arrangements. Convenience functions flat_formation and trefoil_formation are provided for the most common layouts which perfectly match the IEEE standard.

ECC transposition

When the ECC is flagged as transposed, it changes its physical position (mirrored about the centroid of the phase formation) at the midpoint of each section. For single-point and both-ends bonding this creates two sub-sections; for cross-bonding it creates two sub-sections per minor section (six sub-sections total for one major section).


Input parameters

Parameter Meaning Typical input guidance
f System frequency [Hz] 50 or 60
I_c Phase current magnitude [A] Balanced positive-sequence current, e.g. 1675
d_sh Mean sheath diameter [mm] From cable datasheet, e.g. 104
R_s Sheath resistance [Ω/m] Scalar (same for all phases) or array of length Nc (per-phase values), e.g.
5.42e-5 or [0.00073902, 0.00073905, 0.00074047]
xy List of 3 tuples of phase positions (x_mm, y_mm) Physical coordinates of each phase cable.
Index 0 = Phase 1/R/A, 1 = Phase 2/S/B, 2 = Phase 3/T/C.
Example flat: [(-250, 0), (0, 0), (250, 0)]
L_sys Route length [m] e.g. 1000 or 2100
bonding Bonding type 'single-point', 'both-ends', or 'cross-bond'
a_S1, a_S2, a_S3 List of 3 minor-section lengths [m] Cross-bond only. None = equal thirds. e.g. [420, 630, 1050]
transposed Phase conductors change position at joints (phase transposition) True or False (cross-bond only)
ecc Earth Continuity Conductor parameters (not available in the rating modules) None to omit, or a dict:
{'R': 0.000173, 'D_mm': 11.68, 'alpha': 0.726, 'pos_mm': (500, 0), 'transposed': True}

Considerations

Technical

  • The solver uses a multi-conductor circuit approach: for each sub-section, the mutual and self impedance matrices are built from the physical geometry, and the system of equations is solved for the unknown sheath (and ECC) currents subject to the boundary conditions imposed by the bonding type.
  • The solver is built around a unified sub-section approach. Every bonding configuration — regardless of cable transposition or ECC transposition — is decomposed into a sequence of sub-sections, each with its own conductor permutation, sheath permutation, and ECC position. The impedance matrices are assembled per sub-section and then combined according to the boundary conditions imposed by the bonding type.
  • For cross-bonding, the constraint equations enforce equal voltage at both grounded ends for each sheath pair, plus zero net sheath current (Kirchhoff). The residual voltage ΔU at the ends is the difference between the highest and lowest sheath-to-ground voltage at x = L.
    Note: Cross-bonding is calculated without ECC or with parallel non-transposed ECC in the IEEE standard, no example is shown with transposed ECC.
  • The R_s parameter now supports per-phase values. When an array is provided, each diagonal entry of the sheath impedance matrix uses the resistance of the sheath that physically occupies that position (tracked through the permutation index). A scalar value is broadcast to all three phases for backward compatibility.
  • All impedance formulas use the natural logarithm (ln), consistent with IEEE 575-2014 Appendix F equations (F.11), (F.13), and (F.14).
  • For both-ends bonding of a single uniform section (no ECC or untransposed ECC), the standing voltage is identically zero. Floating-point residuals below 10⁻⁶ V are cleaned to exactly zero for plotting clarity.
    Note: Both-ends bonding is not considered in the IEEE standard.
  • The IEEE standard considers the same resistance for all three metallic sheaths in the single-core cables. However, in reality the resistances will be different if the loading is not identical which is the case in flat formation. Cableizer does consider this and in that case a difference in residual voltage between the phases will remain.

Plotting features

The plot function produces publication-quality figures with the following conventions:

Element Colour Line style
Sheath A Blue Solid
Sheath B Blue-violet Dashed
Sheath C Crimson Dash-dot
ECC Forest green Dotted
Residual voltage ΔU Black Double-headed arrow (cross-bond only)

Sub-section decomposition

Bonding Cables transposed ECC transposed Sub-sections
Single-point / Both-ends N/A No or None 1 uniform section
Single-point / Both-ends N/A Yes 2 halves (ECC mirrors at midpoint)
Cross-bond No No or None 3 (one per minor section)
Cross-bond Yes No or None 3 (conductors + sheaths both permuted)
Cross-bond No Yes 6 (each minor section split at midpoint for ECC)
Cross-bond Yes Yes 6 (both effects combined)

Permutation logic

Three independent permutation mechanisms control how conductors, sheaths, and the ECC are mapped to physical positions in each sub-section:

Element Rule
sheath_perm Always cycles at cross-bond joints:
[0, 1, 2] → [1, 2, 0] → [2, 0, 1]
cond_perm Stays [0, 1, 2] if cables_transposed = False.
Follows the same cycle as sheath_perm if cables_transposed = True.
ECC position Fixed at ecc['pos_mm'] if ecc['transposed'] = False.
Mirrors about the formation centroid at the midpoint of each (minor) section if ecc['transposed'] = True.

Key physics results

The following table summarises the expected sheath voltage profiles for each combination of bonding type and transposition options:

Case Expected voltage profile
Both-ends, no ECC V = 0 everywhere.
Single uniform section — Kirchhoff's voltage law forces zero net gradient.
Both-ends, untransposed ECC V = 0 everywhere.
Still a single uniform section with a 4×4 system; the ECC current is constant and does not break the symmetry.
Both-ends, transposed ECC V peaks at midpoint, returns to 0 at both ends.
Two different half-sections with different mutual impedances create a non-zero voltage that must vanish at the grounded boundaries.
Cross-bond, not transposed Sawtooth pattern.
Standard IEEE 575 result — voltage rises within each minor section and drops at the cross-bond joints.
Cross-bond, transposed Reduced sawtooth.
Better voltage balance — transposing the phase conductors together with the sheaths reduces the residual voltage ΔU.

The y-axis is capped at a maximum of 50 000 V (50 kV). Near-zero voltages (below 10⁻⁶ V) are snapped to exactly zero to avoid misleading floating-point residuals. For cross-bonding cases, a residual voltage annotation with a ΔU label and double-headed arrow appears at the right side of the plot when the residual is above 1 mV.

Example

The following example reproduces the practical cases from IEEE 575-2014, Section F.5. The cable system has a mean sheath diameter of 104 mm and a sheath resistance of 5.42 × 10⁻⁵ Ω/m. Two sets of parameters are used:

  • Set 1 (F.5.2 – F.5.4): f = 50 Hz, I = 1675 A, L = 1000 m, flat formation with 250 mm spacing.
  • Set 2 (F.5.5): f = 60 Hz, I = 900 A, L = 2100 m, flat formation with 190 mm spacing.
F.5.2 — Single-point bonding, no ECC
F.5.2
The sheath voltage rises linearly from the grounded end to the open end. The outer phases see different voltages due to the asymmetric flat formation.
F.5.3 — Single-point bonding, untransposed ECCF.5.4 — Single-point bonding, transposed ECC
F.5.3
Adding an untransposed ECC on one side of the formation reduces the sheath voltage on the nearest phase but introduces asymmetry.		 F.5.4
Transposing the ECC at the midpoint balances the voltage reduction across all three phases.
F.5.5.2 — Perfect cross-bondingF.5.5.3 — Imperfect cross-bonding
F.5.5.2
Three equal minor sections of 700 m each. The sheath voltages rise within each minor section and drop at the cross-bond joints. Even with perfect section lengths, a residual voltage ΔU at the ends remains, shown by the double-headed arrow on the plot.		 F.5.5.3
Minor sections of 420 m, 630 m and 1050 m. The unequal lengths produce a higher induced voltage but an almost equal non-zero residual voltage ΔU at the grounded ends.

The following two cases are not covered in the examples of IEEE 575. In the standard, the cables are considered not to be transposed in cross-bonding. Transposition of the phases has an impact of the symmetry and the level of induced voltage.

F.5.5.2t — Perfect cross-bonding, transposedF.5.5.3t — Imperfect cross-bonding, transposed
F.5.5.2
Three equal minor sections of 700 m each. The sheath voltages of the two outer cables in flat formation now rise and fall symmetrically. With perfect section lengths, the residual voltage ΔU at the ends is essentially zero.		 F.5.5.3
Minor sections of 420 m, 630 m and 1050 m. The unequal lengths produce a small non-zero residual voltage ΔU at the grounded ends, shown by the double-headed arrow on the plot.

The standard IEEE 575 also does not cover any calculations or examples for solid both-sided bonding so we added an additional chapter F.5.6 for both-side bonding cases which will be modelled same as cases F.5.2 to F.5.4 (Set 1).
In the literature these cases are sometimes shown to have a small induced voltage in the middle of the section with zero voltage at the two ends. However, our calculations show that the voltage must remain at 0 V throughout the length of the section if there is no ECC or a non-transposed ECC. Adding transposition to the ECC results in an induced voltage on the cable sheaths.

F.5.6.2/3 — Both-side bonding, none or untransposed ECCF.5.6.3 — Both-side bonding, transposed ECC
F.5.6.3
With both ends grounded and non-transposed ECC (or without ECC), the standing voltage is zero along the entire route. Circulating currents flow in the sheaths and represent the primary output of interest for this bonding type.		 F.5.6.4
The sheath voltages rise symmetrically to a small value.

Reference

IEEE Std 575™-2014 (Revision of IEEE Std 575-1988)
IEEE Guide for Bonding Shields and Sheaths of Single-Conductor Power Cables Rated 5 kV through 500 kV
IEEE Power and Energy Society, New York, USA, 2014.

This standard provides guidance on the design of bonding systems for single-conductor shielded power cables. Appendixes D, E and F detail the mathematical framework for computing induced sheath voltages and circulating currents under various bonding arrangements, including the effects of earth continuity conductors and cable transposition. The practical examples in Section F.5 serve as the validation basis for this module.

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