Circulating Sheath Currents in Single-Core Power Cables

Theoretical Introduction and Regulatory Framework of Metallic Shields in Single-Core Cables

In high and extra-high-voltage power transmission systems, the deployment of single-core (monopolar) cables is imperative due to the high ampacity requirements, the manufacturing limitations of massive solid conductors, and the necessity to manage the dielectric potential gradient. However, the intrinsic geometry of a single-core cable—comprising an energized central conductor concentrically surrounded by a metallic shield—induces a fundamental electromagnetic phenomenon: the linkage of time-varying magnetic flux generated by the load current of the main conductor, which embraces the closed loop formed by the metallic shield and the reference ground or earth return path.

In accordance with Faraday's law of electromagnetic induction and Ampère's circuital law, any alternating current $I$ flowing through the central conductor induces a longitudinal electromotive force (EMF) along the metallic shield. If these shields are electrically interconnected and grounded at multiple points (continuous or looped grounding system), a closed low-impedance path is established through which induced currents known as circulating shield currents ($I_s$) flow. These currents contribute no active power to the final load; on the contrary, they generate direct Joule heating losses ($I^2R$) within the shield, drastically reducing the current-carrying capacity (ampacity) of the cable circuit—a phenomenon analytically categorized in the international standard IEC 60287-1-1.

The foundational global reference standard for cable current-carrying capacity calculations is the IEC 60287 series, complemented by IEC 60287-2-1 for thermal resistance calculations and IEC 60840 / IEC 62067 for cross-linked polyethylene (XLPE) insulated cables and their accessories. Concurrently, IEEE Std 575 and IEEE Std 690 provide design and analysis methodologies for shield grounding systems in substations and extensive underground routes. The rigorous analysis of these phenomena demands simultaneous mastery of applied electromagnetic field theory, the electrodynamics of inductively coupled circuits, and the thermodynamics of porous media (the soil surrounding cable trenches).

Electromagnetic Fundamentals and Mutual Impedance Modeling

To quantify with millimetric precision the circulating currents and induced voltages in the shields of a three-phase single-core cable system, it is essential to model the electrical system as a magnetically coupled multiconductor network. Consider a three-phase system comprising three single-core cables laid in typical geometric configurations: trefoil formation or flat horizontal formation.

Let the central conductors carry symmetrical three-phase currents:

The alternating magnetic field generated by these currents traverses space and links the concentric metallic shields. The longitudinal impedance matrix of the complete system, including both the central conductors, the shields, and the earth return path (utilizing Carson's corrections for earth return with finite resistivity \rho_s), is expressed via the Carson-Pollaczek formulation:

For each conductor and shield, the self and mutual impedance per unit length is defined by Carson's general equation:

Where $r_i$ is the AC resistance of the conductor or shield at power frequency $f$, \omega = 2\pi f is the angular frequency, \mu_0 is the magnetic permeability of vacuum, $dij$ is the geometric center-to-center distance between element $i$ and element $j$, and $D_e$ is the equivalent earth return penetration depth, expressed as:

Where \rho_s is the soil resistivity in \Omega\cdot m . Applying the boundary conditions imposed by the shield bonding method (solidly bonded at both ends), the longitudinal induced voltage $E_s$ in the shield of a unit cable due to the central conductor current and neighboring shield currents is calculated via the mutual reactance $X_m$:

Where $s$ is the axis-to-axis cable spacing and $r_m$ is the mean radius of the metallic shield. The circulating induced current $I_s$ in the shield rigidly grounded at multiple points is determined by solving the equivalent electrical mesh:

Where $R_s$ is the AC electrical resistance of the shield and $X_s$ is the shield self-reactance. The circulating current loss factor (\lambda_1), formally defined in standard IEC 60287-1-1, quantifies the increase in thermal losses within the shield relative to the ohmic losses of the central conductor:

This formulation conclusively demonstrates that if the shield resistance $R_s$ is extremely low (typical of large cross-section copper shields), the ratio $X_m/R_s$ can rise significantly, causing circulating current losses to reach unacceptable magnitudes (often exceeding 50% to 100% of the conductor losses), thereby collapsing the thermal capacity of the system.

Thermodynamic Analysis and Ampacity under IEC 60287

The direct impact of circulating currents and eddy currents in the shield translates into internal heat generation within the cable. Standard IEC 60287 establishes the thermal network analogous to Ohm's law to calculate the maximum permissible conductor temperature (\theta_c), which is strictly limited by the operating temperature of the insulation (e.g., $90 °C$ for conventional XLPE and $105 °C$ for high-resistance XLPE or EPR).

The general heat transfer equation for determining steady-state permissible current $I$ is stated as:

Where:

• \Delta\theta: Permissible temperature gradient between the conductor and ambient temperature (\theta_c - \theta_a).
• $W_d$: Dielectric losses in the cable insulation.
• $T_1$: Thermal resistance per unit length between the conductor and the shield.
• $T_2$: Thermal resistance of the bedding layer under the armor (if applicable).
• $T_3$: Thermal resistance of the outer serving/jacket of the cable.
• $T_4$: Thermal resistance of the surrounding medium (soil or air).
• $n$: Number of load-carrying conductors in the circuit (typically 3 for three-phase systems).
• \lambda_1: Loss factor for the metallic shield (circulating currents and eddy currents).
• \lambda_2: Loss factor for the metallic armor.

When the grounding method is defective or shields are bonded at both ends without considering optimal geometric configuration, the term \lambda_1 surges. This drastically reduces the denominator of the ampacity equation, forcing the operator to derate the nominal circuit load current to prevent thermal thermo-oxidative degradation of the polymeric insulation. Thermal degradation accelerates water treeing formation and diminishes the expected service life of the cable, calculated via the modified Arrhenius equation for polymers.

Shield Bonding Methods

To mitigate circulating current losses without compromising personnel safety or insulation integrity against transient overvoltages, power systems engineering implements three fundamental shield grounding architectures:

Solid Bonding (Both Ends Grounded)

This method consists of connecting the metallic shields of all three single-core cables to the substation or manhole ground bus at both ends of the run, and frequently at intermediate points. It is mechanically the simplest method and ensures that the shields maintain a potential close to zero volts throughout the entire route, eliminating dangerous touch voltage hazards for personnel under normal operating conditions.

However, as demonstrated in previous sections, this method permits the free flow of circulating currents comparable in magnitude to the central conductor current (depending on mutual impedance and shield resistance). For this reason, its application is restricted to low-capacity circuits, short runs, or cables with high electrical resistance shields (such as lead or thin aluminum), where additional thermal losses are tolerable.

Single-End Bonding

In this scheme, the shields of the single-core cables are solidly grounded at a single end of the circuit (typically at the source substation) and left open and insulated at the opposite end (and at intermediate points). By breaking the closed loop, the circulating current $I_s$ is completely nullified (\lambda_1 \approx 0), eliminating Joule losses in the shield and recovering 100% of the thermal ampacity of the central conductor.

Nevertheless, the physics of this method introduces a critical challenge: since it is not grounded at the open end, the shield acts as an open capacitive electrode exposed to the electromagnetic field of the central conductor. As the cable run length increases, the electrostatic and electromagnetic induced voltage at the open end grows linearly with distance. To prevent puncturing the outer jacket (over-sheath) insulation and to guarantee touch voltage safety, IEEE Std 575 and international practice impose strict limits: the open-circuit induced voltage under nominal load current must typically not exceed 65\, V to 100\, V under normal operating conditions, limiting maximum section lengths to values on the order of 500\, m to 1000\, m .

Cross-Bonding

For long runs of high and extra-high-voltage cables where single-end bonding proves unfeasible due to excessive induced voltages, cross-bonding is implemented. This sophisticated technique divides the total circuit route into three main sections (or multiples of three) of strictly equal electrical and physical lengths.

At each joint point between sections (cross-bonding or transposition chambers), the shields of the three cables are physically disconnected and cyclically transposed in phase order (e.g., Phase A moves to position B, B to C, and C to A). Upon completing the cycle of three consecutive sections, the vector sum of the induced electromotive forces along the entire closed path is theoretically canceled (\sum E_s = 0), nullifying circulating currents while keeping the shields grounded at the extremities of the complete system and at each transposition chamber via Sheath Voltage Limiters (SVLs).

Transposition Chamber Architecture and Sheath Voltage Limiters (SVL)

The design of cross-bonding chambers and the selection of Sheath Voltage Limiters (SVLs) constitute the core of shield protection engineering. An SVL is essentially a high-energy non-linear zinc oxide (ZnO) varistor specifically designed to protect the cable outer jacket and accessories against hazardous transient overvoltages.

During normal transposed steady-state operation, the voltage across the SVL is practically zero (or equal to the minor residual voltage caused by length imbalance or geometric asymmetry). However, under symmetrical or asymmetrical short-circuit conditions in the main cable central conductor, a massive fault current traverses the system. This current generates a high-intensity transient magnetic field that induces an extreme overvoltage in the metallic shield.

Without SVLs, this transient overvoltage would instantly puncture the outer polymeric jacket insulation of the cable (HDPE or MDPE), triggering catastrophic secondary ground faults. The SVL acts by diverting the surge current to the chamber ground, clamping the voltage peak below the Basic Insulation Level (BIL) of the cable jacket (which typically withstands DC tests of 10\, kV to 20\, kV ).

The insulation coordination of SVLs requires evaluating the maximum transient induced voltage during a short circuit via transient impedance matrix calculations:

Where $Msc$ is the mutual inductance between conductor and shield, $Isc$ is the symmetrical short-circuit current, and $Lsection$ is the length of the transposition section.

Forensic Failure Analysis and Operational Consequences

Errors in the design, installation, or maintenance of shield grounding systems trigger catastrophic failures in power networks. Typical failure modes and their forensic etiology are detailed below:

• Outer Jacket Puncture due to SVL Failure: If an SVL suffers thermal degradation from repeated energy absorption (ZnO block aging) or fails open-circuit, a lightning or short-circuit overvoltage generates an electrical arc that punctures the outer polymeric cable jacket, allowing moisture ingress and initiating galvanic corrosion in the metallic shield.
• Uncontrolled Circulation of Zero-Sequence and Harmonic Currents: In systems with multiple solid grounding, the presence of zero-sequence harmonic currents (such as the 3rd harmonic and its multiples in unbalanced networks or power electronics converter feeds) induces high-frequency circulating currents in the shields. These currents exponentially increase skin effect losses in the shield, generating local hotspots that degrade adjacent XLPE insulation.
• Circulating Currents from Geometric Imbalance: If distances between cable centers in a flat horizontal three-phase installation are not maintained strictly symmetric (or physical transpositions are omitted), mutual reactances differ between phases, producing permanent circulating currents even with supposedly balanced grounding systems.
• Failures in Cross-Bonding Links: Mechanical loosening of bolts inside cross-bonding link boxes causes high contact resistance. This generates localized Joule heating, fusion of bimetallic terminals, opening of the shield circuit, and the consequent appearance of dangerous voltages posing risks to operations and maintenance personnel.

Comparative Matrix of Shield Grounding Methods

The following technical matrix summarizes the operational, electrical, and normative characteristics of the different shield grounding methods according to IEC 60287 and IEEE Std 575 standards:

Parameter / Characteristic	Solid Bonding (Both Ends)	Single-End Bonding	Cross-Bonding
Shield Losses (\lambda_1)	Very high (dependent on $R_s$ and $X_m$)	Negligible (\approx 0)	Practically negligible (compensated per section)
Cable Ampacity	Reduced (severe thermal penalty)	Maximum (100% of nominal capacity)	High (approaching 98% - 100%)
Maximum Section Length	No strict limit imposed by induced voltage	Limited (500\, m - 1000\, m max.)	Indefinite (segmented into 3-section cycles)
Open-Circuit Induced Voltage	Zero (maintained at earth potential)	Elevated (grows linearly with length)	Very low (vectorially compensated per section)
Protective Devices	None required for steady-state operation	Optional SVLs at the open end	Mandatory SVLs at each linking chamber
Typical Failure Consequence	Global circuit heating and hotspots	Outer jacket puncture by overvoltage	Electrical arc in link box and SVL damage

Advanced Design and Mitigation Strategies

The optimal design of single-core cable systems demands the application of Finite Element Analysis (FEA) numerical simulation tools for coupled electromagnetic and thermal field analysis. Advanced engineering guidelines include:

• Geometric Layout Optimization: Prefer compact trefoil configurations whenever possible, as this drastically reduces the loop area formed by cables and earth, lowering mutual reactance $X_m$ and, consequently, circulating currents in solid bonding configurations.
• Shield Material Selection: Evaluate corrugated aluminum shields or optimized-thickness copper tapes. While a larger cross-section shield reduces resistance $R_s$ and improves short-circuit withstand capability, it also increases the $X_m/R_s$ ratio, exacerbating circulating current losses when solid bonding is used.
• Real-Time Monitoring (DTS & Thermal Rating): Implement Distributed Temperature Sensing (DTS) fiber optic monitoring along the cable jacket, integrated with Dynamic Cable Rating (DCR) calculation engines to supervise in real time the thermal impact of circulating currents under variable load profiles.

Practical Application and Vexten Suite Analysis

Within the Vexten Academy engineering environment, the modeling and validation of single-core cable systems are executed using the advanced module Vexten Suite: PowerCable & Grounding Engine, which integrates the normative algorithms of IEC 60287-1-1, IEC 60909, and IEEE Std 141.

An analytical case study processed through the platform for a high-voltage circuit is presented below:

• System Input Data:
  • Nominal System Voltage ($U_n$): 132\, kV (Frequency: 50\, Hz ).
  • Maximum Conductor Load Current ($I_c$): 800\, A .
  • Conductor: Copper, cross-section 1000\, mm ^2, AC resistance R_c = 0.0224\,\Omega/ km at $90 °C$.
  • Shield: Copper wire screen, cross-section 95\, mm ^2, AC resistance R_s = 0.206\,\Omega/ km at $20 °C$.
  • Geometric layout: Trefoil, outer cable diameter 115\, mm , axis spacing s = 130\, mm .
  • Soil resistivity: \rho_s = 1.0\,\Omega\cdot m .
• Automated Calculation Results in Vexten Suite:
  • Calculated mutual reactance ($X_m$): 0.0784\,\Omega/ km .
  • Open-circuit induced voltage (for single-end bonding, 800\, m section): 49.5\, V (strictly complies with normative limit < 65\, V ).
  • Circulating current loss factor (\lambda_1) under solid bonding at both ends:
    $$\lambda_1 = \left(\frac{0.0784}{0.206}\right)^2 \times \frac{0.206}{0.0224} = 0.1448 \times 9.196 = 1.332$$
    This value indicates that shield losses represent 133.2\% of central conductor losses, collapsing cable ampacity by 38\%.
  • Solution adopted by Vexten Suite: Mandatory implementation of a cross-bonding system across three sections of 800\, m each, reducing \lambda_1 to $0.012$ and recovering 99.4\% of the main conductor nominal ampacity, with specification of 12\, kV discharge voltage SVLs and a nominal energy absorption rating of 40\, kJ .

Conclusions and Engineering Recommendations

Rigorous analysis of circulating currents in single-core power cable shields is an irreplaceable pillar in designing highly reliable power transmission systems. Strict application of IEC 60287-1-1 guidelines combined with IEEE Std 575 criteria allows proper network sizing, prevents catastrophic thermal failures, and guarantees personnel and facility safety. The deployment of advanced cross-bonding methodologies and precise voltage limiter (SVL) selection ensures circuit ampacity maximization and dielectric insulation service life preservation across its entire operational lifecycle.