Capacitive Charging Currents and Ferranti Effect in Long-Distance MV/HV Cables

Electromagnetic Fundamentals and Dielectric Geometry in Medium and High-Voltage Systems

The exhaustive study of underground and submarine power cables requires a rigorous understanding of their fundamental electromagnetic architecture. Unlike overhead transmission lines, where conductors are separated primarily by air and spatial arrangement geometry yields moderate values of distributed inductance and capacitance, medium and high-voltage (MV/HV) cables confine electric fields within compact cylindrical geometries utilizing solid or synthetic dielectrics of high relative permittivity.

From the perspective of electromagnetic field theory, a single-core coaxial cable consists of a central conductor of internal radius $r_i$ , surrounded by a homogeneous or composite insulating layer with an external radius (corresponding to the semiconductor or metallic screen) of value $r_o$ . The electric field intensity \vec{E} at any point at a radial distance $r$ from the center of the conductor, assuming a linear charge density \lambda and an absolute permittivity of the medium \varepsilon = \varepsilon_0 \varepsilon_r, is determined via the direct application of Gauss's law in cylindrical coordinates:

E(r) = \frac{\lambda}{2 \pi \varepsilon_0 \varepsilon_r r}

Where \varepsilon_0 = 8.854 \times 10^{-12} \, F/m is the vacuum permittivity and \varepsilon_r is the relative permittivity (dielectric constant) of the insulating material, which commonly adopts values between $2.3$ and $3.0$ for cross-linked polyethylene (XLPE) and can reach values from $3.5$ to $4.5$ in mass-impregnated paper-insulated cables (MIND). The electrical potential difference $V$ between the central conductor and the grounded metallic screen is obtained by integrating the electric field intensity across the dielectric radius:

V = \int_{r_i}^{r_o} E(r) \, dr = \frac{\lambda}{2 \pi \varepsilon_0 \varepsilon_r} \ln\left(\frac{r_o}{r_i}\right)

From the constitutive relation linking linear charge density with capacitance per unit length $C_s$ (\lambda = C_s V), the fundamental analytical expression for the capacitance of a coaxial cable is derived:

C_s = \frac{2 \pi \varepsilon_0 \varepsilon_r}{\ln\left(\frac{r_o}{r_i}\right)} \quad [ F/m ]

In three-phase systems, the geometric configuration of the cables drastically influences the capacitance matrix. For single-core cables arranged in a trefoil (triangular) formation or in a horizontal flat plane with interconnected metallic screens, the effective phase capacitance $C$ incorporates both the self-capacitance relative to the screen/earth and the mutual capacitances between phases. In a three-core cable with a common screen or individual shielding, the electric field is distributed non-symmetrically, requiring conformal transformations or finite element analysis (FEA) to accurately determine local electrical gradients.

Voltage Gradients and Non-Uniform Electric Field Distribution

The distribution of the electric field within a cable's insulation is non-uniform; it exhibits a hyperbolic profile where the maximum intensity invariably occurs at the surface of the central conductor ( $r = r_i$ ) and the minimum intensity at the metallic screen ( $r = r_o$ ). The maximum value of the electric field is calculated as:

E_{\max} = \frac{V}{r_i \ln\left(\frac{r_o}{r_i}\right)}

For a given outer insulation diameter and a fixed nominal operating voltage, there exists an optimum conductor radius $r_{i, opt }$ that minimizes the maximum dielectric stress, obtained by differentiating E_{\max} with respect to $r_i$ and equating to zero:

\frac{r_o}{r_i} = e \approx 2.71828 \implies r_{i, opt } = \frac{r_o}{e}

This condition implies that the optimum ratio between the outer radius and the inner radius of the insulation must equal Euler's number. When cables operate at extra-high voltages (EHV), this geometric limitation mandates the use of segmented or hollow-core conductors (Milliken) to increase the effective conductor radius without disproportionately increasing the cross-sectional area of copper or aluminum, thereby controlling tangential and radial stresses that could trigger partial discharges (PD).

Constitutive Equations and Capacitive Charging Current Dynamics

When a power cable of length $L$ is energized by a sinusoidal alternating voltage source of angular frequency \omega = 2 \pi f, the cable's dielectric acts as a distributed capacitor along its entire path. The total capacitive charging current $I_c$ flowing from the conductor toward the metallic screen and the grounding system is derived directly from the equivalent capacitive reactance of the circuit:

I_c = \omega \cdot Ctotal \cdot V_{ phase } = 2 \pi f \cdot (C_s \cdot L) \cdot \left(\frac{VLL}{\sqrt{3}}\right)

Where $VLL$ is the rated line-to-line voltage of the system and $f$ is the power frequency ( $50$ or 60 \, Hz ). As the system's nominal voltage and the length of the underground link increase, the charging current ceases to be a negligible parameter and becomes a dominant component of the total current flowing through the cable conductor.

In medium-voltage distribution networks (13.8 \, kV , 23 \, kV , 34.5 \, kV ), charging currents are typically on the order of a few amperes per kilometer, allowing their integration into conventional load flows without major operational alterations. However, in high-voltage (110 \, kV , 220 \, kV ) and extra-high-voltage (400 \, kV and above) systems, specific capacitance values range between 0.2 \, \mu F/km and 0.4 \, \mu F/km . This generates charging currents that can exceed 10 \, A to 30 \, A per kilometer of circuit.

The thermal impact of this capacitive current is direct and cumulative. The nominal current-carrying capacity (ampacity) of the conductor is reduced because the metal cross-section must share its permissible thermal capacity ( $I^2 R$ ) between the active load current and the reactive capacitive charging current:

I_{ max\_available } = \sqrt{I_{ thermal\_limit }^2 - I_c^2}

In extremely long circuits (for example, submarine interconnections exceeding 30 \, km at 132 \, kV ), the capacitive charging current can equal or even exceed the nominal design load current. Under these conditions, the cable operates at full thermal capacity carrying exclusively its own no-load capacitive charging current, precluding useful power transfer to the receiving end unless compensation measures via shunt reactors are implemented.

Forensic Failure Analysis, Transient Phenomena, and Dielectric Degradation

The continuous presence of intense electric fields and elevated capacitive charging currents exposes power cable systems to severe degradation mechanisms that, if not adequately mitigated, lead to catastrophic insulation failures, terminal destruction, MV/HV switchgear explosions, and irreparable damage to power transformers.

Ferranti Effect in Long Cable Links

The Ferranti effect describes the voltage rise at the receiving end of a transmission line or power cable when operating at no-load or under very light loads. In underground cables, owing to their high distributed parallel capacitance and relatively low series inductance compared to overhead lines, this phenomenon manifests very aggressively even over moderate distances.

Mathematically, the relationship between the sending-end voltage $V_s$ and the receiving-end voltage $V_r$ of a cable of length $L$ with a complex propagation constant \gamma = \alpha + j\beta and characteristic impedance Z_c = \sqrt{\frac{R+j\omega L}{G+j\omega C}} is expressed through the long-line cable equations:

V_s = V_r \cosh(\gamma L) + I_r Z_c \sinh(\gamma L)

For a cable operating strictly at no-load ( $I_r = 0$ ), the equation reduces to:

V_s = V_r \cosh(\gamma L) \implies V_r = \frac{V_s}{\cosh(\gamma L)}

Given that for a high-quality dielectric, conductive losses are negligible (G \approx 0) and conductor resistance is small (R \ll \omega L), the propagation constant approximates \gamma \approx j\omega\sqrt{LC} = j\beta. Using hyperbolic and trigonometric identities, the overvoltage at the open end approximates:

V_r \approx \frac{V_s}{\cos(\beta L)} = \frac{V_s}{\cos(\omega \sqrt{LC} \cdot L)}

As the product of frequency, inductance, capacitance, and the square of the length increases, the denominator decreases drastically, causing $V_r$ to significantly exceed $V_s$ . This permanent overvoltage subjects the solid insulation (XLPE) to a higher dielectric stress than that for which it was designed, accelerating chemical aging and the onset of partial discharges.

Ferroresonance and Switching Transients

The interaction between the massive capacitance of the cable and non-linear inductive elements present in the system (such as inductive potential transformers, distribution transformers operating at no-load, or reactors with ferromagnetic iron cores) can trigger ferroresonance phenomena. This non-linear oscillatory regime is characterized by sustained overvoltages of high amplitudes ( $2.0$ to $4.5$ times the rated voltage) and the appearance of severe harmonic and subharmonic components.

During the opening of vacuum circuit breakers operating on purely capacitive circuits, multiple arc re-ignitions (restriking) occur. Each re-ignition abruptly discharges the energy stored in the cable's capacitance through the network's parasitic inductance, generating high-frequency transients (in the kilohertz to megahertz range) with abrupt wave fronts (extreme $dv/dt$ ). These wave fronts strike the windings of connected transformers and cable shields, causing non-linear voltage distributions along the transformer turns and provoking inter-turn dielectric breakdowns or puncture of the main insulation.

Thermal Effects and Degradation via Electrical Trees (Water Trees and Electrical Trees)

The combination of continuous electrical stress derived from the capacitive field and the presence of microscopic moisture in the extruded polymer insulation generates the phenomenon known as water treeing. These branched microchannels grow slowly over years driven by the force of the alternating electric field acting on water molecules trapped in discontinuities or impurities within the XLPE. Although water trees by themselves do not lead to immediate failure, they drastically reduce the material's dielectric strength.

Under the influence of transient overvoltages or lightning impulses, these water trees transform into electrical trees, which grow exponentially and rapidly, carbonizing the polymer until total insulation perforation and the consequent phase-to-earth short circuit occur.

Electrical Parameters and Regulatory Limits (IEEE / IEC)

The design, sizing, and safe operation of medium and high-voltage cables are strictly governed by established international standards. The following tables outline the critical electrical parameters, international regulatory limits according to IEEE and IEC, and the operational consequences of their non-compliance.

Electrical / Physical Parameter	Typical Range (Medium Voltage 15-35 kV)	Typical Range (High Voltage 110-400 kV)	Main Reference Standard
Specific Capacitance ( $C_s$ )	$0.15$ to 0.30 \, \mu F/km	$0.10$ to 0.25 \, \mu F/km	IEC 60502-2 / ICEA S-94-649
Capacitive Charging Current ( $I_c$ )	$0.5$ to 3.5 \, A/km	$8.0$ to 35.0 \, A/km	IEEE Std 400 / IEC 60287
Maximum Electric Gradient (E_{\max})	$3.0$ to 5.0 \, kV/mm	$8.0$ to 15.0 \, kV/mm	IEC 60840 / IEC 62067
Dielectric Dissipation Factor (\tan \delta)	\le 0.001 (New XLPE)	\le 0.0005 (Extra-clean XLPE)	IEEE Std 400.2 / IEC 60502

Critical Failure Condition	Regulatory Limit (IEEE / IEC)	Operational and Dielectric Consequence
Ferranti Effect Overvoltage	V_r \le 1.10 \times V_n (Continuous operation)	Acceleration of XLPE aging, sustained partial discharges, overvoltage protection tripping.
Induced Screen Current	Limited to $300$ A (Acceptable losses per IEC 60287)	Excessive heating of metallic screens, drastic reduction of main conductor ampacity.
Partial Discharge (PD) Level	\le 5 pC at 1.5 \times V_0 (Factory)	Progressive erosion of internal cavities, formation of discharge channels, and premature disruptive failure.
Dielectric Strength under Impulse	Compliant with specific BIL (Basic Insulation Level)	Instantaneous insulation puncture under atmospheric discharges or severe switchgear switching.

Advanced Mitigation and Reactive Compensation Strategies

To counteract adverse effects derived from elevated capacitive charging currents and the Ferranti effect in medium and high-voltage cable systems, power systems engineering implements various reactive compensation and topological management strategies.

Compensation with Shunt Reactors

The most effective methodology to neutralize the capacitive reactive power generated by a long cable involves the installation of inductive reactors connected in parallel (shunt) at the circuit terminals (typically at substation terminals). The capacitive reactive power generated by the cable $Q_c$ is calculated as:

Q_c = 3 \cdot \omega \cdot C_{ total } \cdot V_{ phase }^2 = \omega \cdot C_{ total } \cdot VLL^2

The shunt reactor is designed to provide an inductive reactive power $Q_L$ that compensates a specified percentage (typically between 50\% and 100\%) of the capacitive power generated at rated frequency:

Q_L = \frac{3 \cdot V_{ phase }^2}{\omega L_{ reactor }}

By balancing or matching $Q_L$ with $Q_c$ , the power factor seen from the substation busbar approaches unity, eliminating the voltage rise due to the Ferranti effect and freeing up thermal capacity in network transformers and conductors.

Design of Shield Grounding and Transposition Systems

The metallic screens of single-core cables (generally consisting of copper wires or tapes) act as a conductive armor confining the electric field. However, due to the alternating current flowing through the central conductor, a longitudinal electromotive force (EMF) is induced on the metallic screen via electromagnetic induction. If screens are rigidly grounded at both ends of the circuit, induced circulating currents will flow, generating significant Joule losses in the screen and reducing the cable's current-carrying capacity (ampacity).

To mitigate this problem, advanced grounding schemes are employed:

Single-Point Bonding: The screen is grounded at a single extremity and insulated at the other via an outer jacket surge arrester (link box). It is used in short circuits to completely eliminate circulating currents, although it requires limiting the cable length so that the induced voltage at the open end does not exceed outer jacket safety limits (typically V_{ induced } \le 50 \, V under normal load conditions).
Cross-Bonding: In long circuits, the cable route is divided into three equal sections (or multiples). The metallic screens are interrupted and interconnected via cyclic transpositions within joint bays (cross-bonding link boxes), such that the induced voltages in each section third cancel out vectorially, reducing net screen current to nearly zero and allowing operation over much greater circuit lengths without penalizing ampacity.

Practical Application and Computational Analysis via Vexten Suite

Rigorous sizing and validation of a medium and high-voltage cable system under international standards require the use of advanced calculation platforms. Below is a case study solved through the calculation engines of the Vexten Suite, integrating IEC 60909 / IEEE 141 standards for short-circuit analysis and IEC 60287 for ampacity and thermal regime.

Industrial Case Study Specifications

A High Voltage underground link of 132 \, kV (three-phase system, 50 \, Hz frequency) is analyzed, with length L = 18.5 \, km , constructed using single-core XLPE cables with an 800 \, mm ^2 aluminum cross-section conductor. The physical and electrical parameters obtained from the design catalog are as follows:

Nominal Line-to-Line Voltage ( $VLL$ ): 132 \, kV
Specific Capacitance per phase ( $C_s$ ): 0.21 \, \mu F/km
Total Circuit Capacitance ( $C_{ total }$ ): 0.21 \times 18.5 = 3.885 \, \mu F
Conductor Resistance at $90 °C$ ( $Rac$ ): 0.048 \, \Omega/ km
Self Inductance per phase ( $L_s$ ): 0.38 \, mH/km

Automated Analytical Calculation in Vexten Suite

Using the network analysis module of the Vexten Suite, the total capacitive charging current generated by the link is evaluated first:

I_c = 2 \pi f \cdot C_{ total } \cdot V_{ phase } = 2 \pi (50) \cdot (3.885 \times 10^{-6}) \cdot \left(\frac{132 \times 10^3}{\sqrt{3}}\right)

I_c = 314.16 \times 3.885 \times 10^{-6} \times 76,210.23 = 93.04 \, A

The obtained capacitive charging current is 93.04 \, A . This value represents a significant reactive load that must be accounted for in the cable's thermal design and substation power balance.

Next, the calculation engine evaluates the Ferranti effect under no-load conditions ( $I_r = 0$ ). The phase constant \beta is calculated as:

\beta = \omega \sqrt{L_s C_s} = 314.16 \times \sqrt{(0.38 \times 10^{-3}) \times (0.21 \times 10^{-6})} = 314.16 \times 2.828 \times 10^{-5} = 0.00888 \, rad/km

For the total cable length L = 18.5 \, km :

\beta L = 0.00888 \times 18.5 = 0.16428 \, rad \implies 9.41^\circ

Applying the approximation formula for no-load receiving-end voltage:

V_r = \frac{V_s}{\cos(\beta L)} = \frac{132 \, kV }{\cos(9.41^\circ)} = \frac{132}{0.9866} = 133.80 \, kV

Although the Ferranti increase is moderate for 18.5 \, km (\approx 1.36\%), if the link were extended to 60 \, km , the product \beta L would reach 0.5328 \, rad (30.52^\circ), resulting in a receiving-end voltage of:

V_r = \frac{132}{\cos(30.52^\circ)} = \frac{132}{0.8613} = 153.25 \, kV

This 16.1\% overvoltage exceeds tolerable operational margins for connected equipment, justifying the automatic incorporation of shunt reactors recommended by the Vexten Suite optimization engine.

Thermal and Ampacity Evaluation per IEC 60287

The Vexten Suite ampacity module processes dielectric losses, conductor Joule effect losses, and induced losses in metallic screens to determine maximum allowable current. Dielectric losses $W_d$ per meter of cable are calculated via the expression:

W_d = \omega \cdot C_s \cdot V_{ phase }^2 \cdot \tan\delta

Assuming a dissipation factor \tan\delta = 0.0008 for high-quality XLPE:

W_d = 314.16 \times (0.21 \times 10^{-6}) \times (76,210.23)^2 \times 0.0008 = 0.306 \, W/m per phase

These internal losses generate a continuous heat flux toward the environment which, combined with the thermal dissipation of the 93.04 \, A charging current, reduces the capacity of the 800 \, mm ^2 conductor (whose base ampacity in open terrain is 720 \, A ) to a net available ampacity for active load of:

I_{ net\_util } = \sqrt{720^2 - 93.04^2} = \sqrt{518,400 - 8,656.4} = \sqrt{509,743.6} = 713.96 \, A

The Vexten Suite software automatically generates the technical compliance report, specifying that metallic screens must be configured under a cross-bonding grounding scheme to prevent additional screen losses, ensuring that the system operates within thermal and dielectric limits established by IEC 60287, IEC 60840, and IEEE Std 400.