Ferroresonance in Transformers Fed by Long Underground Cables

Phenomenology of Harmonic Resonance and Ferroresonance in Cable-Transformer Systems

In the design and operation of medium-voltage and high-voltage (MV/HV) distribution and subtransmission systems, the substitution of overhead lines with extruded insulated underground cables—primarily cross-linked polyethylene (XLPE)—introduces substantial modifications to the distributed parameters of the grid. The shunt capacitive reactance of an underground cable is orders of magnitude lower than that of an equivalent overhead line, owing to the high relative permittivity of the dielectric material ($\varepsilon_r \approx 2.3 - 3.0$) and the significantly reduced distance between the phase conductor and the metallic screen. When an extensive section of underground cable feeds a power transformer operating under no-load or highly reduced load conditions, a high-inductance, high-capacitance oscillating circuit is established. This circuit is highly susceptible to exhibiting both linear resonance and ferroresonance phenomena.

Linear resonance manifests when the equivalent inductive reactance of the network ($X_L$) exactly equals the capacitive reactance of the cable ($X_C$) at a specific frequency, which may be either the fundamental power frequency or a specific harmonic frequency generated by non-linear loads. However, the most destructive and complex phenomenon in these topologies is ferroresonance: a non-linear, non-sinusoidal, subharmonic, or quasi-periodic dynamic phenomenon that arises from the interaction between the distributed capacitance of the underground cable ($C_c$) and the non-linear, saturable magnetizing inductance ($L_m(i)$) of the transformer's ferromagnetic core.

Unlike pure linear RLC resonance, ferroresonance does not exhibit a single, well-defined resonant frequency governed by the classical equation $f_0 = \frac{1}{2\pi \sqrt{LC}}$. Because the transformer core is a highly non-linear element subject to magnetic hysteresis and saturation, the magnetizing inductance varies drastically as a function of the instantaneous flux linkage $\phi(t)$ or the magnetizing current $i_m(t)$:

Where $\lambda$ is the total flux linkage, $N$ is the number of turns in the winding, and $\Phi$ is the magnetic flux in the core. In the unsaturated state, the magnetizing inductance is extremely high ($L_m \approx 10 - 100 H$). However, when the core enters deep saturation, the relative permeability of the ferromagnetic material ($\mu_r$) drops drastically toward values close to the permeability of vacuum ($\mu_0$), thereby reducing the incremental inductance to values equivalent to the leakage inductance of the winding ($L_k \approx mH$).

This abrupt variation in inductance allows the circuit to satisfy the capacitive-inductive resonance condition across multiple voltage and frequency values. This behavior gives rise to state transitions (jump resonance), bistability, sustained harmonic overvoltages with highly distorted peaks, and severe thermal-dielectric transients that threaten the operational integrity of the electrical assets.

Excitation Mechanisms and Critical Configurations

Ferroresonance in underground cables feeding transformers is commonly triggered by non-simultaneous switching operations of primary switchgear, single-phase conductor breakages, or asymmetrical blowing of medium-voltage fuses. The three primary excitation configurations are detailed below:

• Single-Phase Switchgear Operation (Asynchronous Opening or Closing): During the pole-by-pole closing or opening of disconnectors or single-phase circuit breakers, or in the event of a single-pole failure in a three-pole circuit breaker, one or two phases of the cable become de-energized on the source side. However, they remain capacitively coupled to the energized phases through the inter-phase ($C_m$) and phase-to-ground ($C_g$) capacitances of the underground cable. This coupling injects a capacitive current into the energized winding of the transformer, shifting the operating point into the magnetic saturation region.
• Broken Conductor with Unloaded Cable: If a phase conductor breaks along the cable feed section upstream of the transformer, the isolated section of the cable remains galvanically energized through the impedance of the transformer winding and the capacitance of the other phases. This configuration forms a resonant return path with the floating or grounded neutral.
• Capacitive Coupling between Parallel Circuits: Multi-circuit underground cables routed within the same duct bank or trench can transfer significant capacitive energy to an out-of-service, ungrounded transformer, inducing ferroresonance via parasitic capacitive coupling.

Advanced Mathematical Formulation and State Bifurcation

The dynamic behavior of ferroresonance is modeled analytically using non-linear ordinary differential equations. By simplifying the topology to a single-phase equivalent circuit supplied by a sinusoidal voltage source $v(t) = V_m \cos(\omega t + \theta)$ through a cable capacitance $C_c$, and incorporating series/parallel loss resistance $R$, the relationship between the magnetic flux linkage $\phi(t)$ and the magnetizing current $i(t)$ of the transformer can be approximated by an odd polynomial of degree $n$ (where typically $n = 7, 9, or 11$):

Applying Kirchhoff's Voltage Law to the series loop consisting of the voltage source, the cable capacitance, and the saturable magnetizing inductance of the transformer, the system's governing differential equation matches a forced Duffing-type equation:

Where $R_s$ represents the equivalent losses due to the Joule effect and core losses (eddy currents and hysteresis losses in the ferromagnetic core). The non-linearity introduced by the term $b \cdot \phi^n$ invalidates the application of the superposition principle and generates a multi-furcated phase-space map.

State-Space Analysis and Bifurcation Diagrams

For a rigorous three-phase analysis, the system is formulated in the vector state-space:

Where the state vector $\mathbf{x}(t) = [\phi_a, \phi_b, \phi_c, vca, vcb, vcc]^T$ contains the magnetic flux linkages in the cores of each phase and the voltages across the cable capacitances; $\mathbf{u}(t)$ represents the external voltage excitation vector; and $\boldsymbol{\mu}$ represents the vector of physical parameters ($L_c, C_c, R_{ core }, cable length$).

As the control parameter (e.g., cable length or system voltage $V_{ rms }$) is continuously varied, the steady-state solutions of the system undergo topological bifurcations. The resulting ferroresonant modes, categorized according to dynamical systems theory, are:

• Fundamental Mode (Symmetric 1/1): The current and voltage responses are periodic, sharing the same period $T = 2\pi/\omega$ as the electrical grid. This mode is characterized by extremely high peak currents and repeated hyper-saturation during each half-cycle.
• Subharmonic Mode (1/n): The response is periodic, with a period that is an integer multiple of the source period ($T_s = n \cdot T$, typically $n=3 or 5$). The magnetic flux oscillates at frequencies such as $16.6 Hz$ or $20 Hz$ (for 50/60 Hz grids), generating a characteristic low-frequency acoustic hum and severe thermal saturation.
• Quasi-Periodic Mode: The frequencies of the response are incommensurable with the grid frequency. The trajectory in the phase space forms a torus and does not repeat in the time domain.
• Chaotic Mode: The trajectory in the state-space is deterministic but highly sensitive to initial conditions (the Butterfly Effect). The frequency spectrum is continuous and flat (broadband noise), and the overvoltages exhibit unpredictable, stochastic peaks of devastating magnitude.

Forensic Analysis of Dielectric and Thermal Failure Modes

The occurrence of undetected or long-duration ferroresonant events in cable-transformer subsystems inevitably results in catastrophic failures. Forensic electrical engineering classifies these failures into four primary degradation and collapse mechanisms:

Thermal Collapse due to Magnetic Overfluxing ($V/f$)

During fundamental or subharmonic ferroresonance, the magnetic flux in the core significantly exceeds the nominal saturation flux density ($B_s \approx 1.7 - 2.0 Tesla$), reaching excursion values of up to $2.3 Tesla$. This extreme overfluxing condition satisfies the relation:

Because the ferromagnetic core is fully saturated, the magnetic flux lines cannot be confined within the cold-rolled grain-oriented (CRGO) silicon steel laminations. Consequently, they spill over into the surrounding media of lower magnetic reluctance: the dielectric oil, mechanical clamping structures (yokes and tie rods), magnetic shields, and the steel walls of the transformer tank. This leakage flux induces massive eddy currents in the external metallic structures and clamping bolts. The power density dissipated per unit volume scales according to:

Where $t_d$ represents the thickness of the conducting material and $K$ is a structural constant. This phenomenon produces localized hot-spots with temperatures rapidly exceeding $300^\circ C$ to $500^\circ C$ within minutes. This extreme heat pyrolyzes the insulating oil (generating dissolved gases such as ethylene $C_2H_4$, hydrogen $H_2$, and methane $CH_4$) and irreversibly degrades the solid cellulose insulation (Kraft paper) via rapid thermal depolymerization.

Dielectric Breakdown due to Neutral Shift and Harmonic Overvoltages

Overvoltages originating from ferroresonance differ significantly from lightning impulse (BIL) or switching impulse (SIL) overvoltages. They are sustained in the time domain (lasting from seconds to hours) with highly distorted waveforms that exhibit high rates of voltage change over time ($dV/dt$). In transformers with isolated neutrals or neutrals grounded through high impedance, the displacement of the neutral point during single-phase switching elevates the voltage of the healthy phases to phase-to-phase values ($\sqrt{3} p.u.$) or higher (up to $3.5 - 4.5 p.u.$).

The voltage stress applied to the XLPE cable dielectric and the inter-turn/inter-layer insulation of the transformer windings exceeds the critical dielectric strength gradient ($E_{ crit }$). This triggers:

• Water Treeing and Electrical Treeing in XLPE: Harmonic overvoltages accelerate aging due to partial discharges (PD) within microscopic voids in the polyethylene dielectric, eroding the material through continuous dendritic discharges until total puncture to ground occurs.
• Inter-Turn Insulation Failure in Transformers: High-frequency harmonics impose a highly non-uniform voltage distribution along the primary winding. The first few turns closest to the line terminal support up to $70\%$ of the peak applied overvoltage, causing dielectric puncture of the conductor enamel or paper insulation and resulting in an inter-turn short circuit.

Thermal-Destructive Effects on Surge Arresters (MOV)

Metal Oxide Varistors (MOVs) installed at the cable-to-transformer transition are sized to absorb short-duration transient energy impulses (microseconds to milliseconds). During a sustained ferroresonant event, the terminal voltage continuously exceeds the Maximum Continuous Operating Voltage (MCOV) of the arrester.

The MOV enters a state of continuous active conduction of high current. The energy dissipated within the ZnO blocks is calculated as:

Because the duration $t_{ ferro }$ is prolonged, the accumulated energy exceeds the rated thermal energy absorption capacity of the varistor (typically expressed in $kJ/kV$ of MCOV, ranging from $4 to 12 kJ/kV$). Once this thermal limit is exceeded, the ZnO blocks undergo thermal runaway, reducing their intrinsic resistance to zero and causing a violent phase-to-ground short circuit accompanied by mechanical explosion and the projection of porcelain or polymeric fragments.

International Standards and Criteria Framework (IEEE / IEC)

The design of integrated cable-transformer subsystems requires strict compliance with international standards to ensure immunity to ferroresonant phenomena and maintain operational stability.

The primary applicable international standards are:

• IEEE C57.105: IEEE Guide for Application of Transformer Connections in Three-Phase Distribution Systems. This standard provides detailed guidelines on transformer connections susceptible to ferroresonance and defines practical limits for the critical length of underground cables.
• IEC 60071-1 / IEEE Std 1313.1: Insulation Coordination - Part 1: Definitions, principles and rules. This standard defines standardized insulation levels, temporary overvoltage (TOV) factors, and evaluation criteria for sustained non-linear overvoltages.
• IEC 60076-1 / IEEE C57.12.00: Power Transformers - Part 1: General. This standard specifies the allowable continuous magnetic overfluxing limits ($V/f \le 1.10 p.u.$ at full load, $V/f \le 1.05 p.u.$ at no-load).
• IEEE 519: IEEE Standard for Harmonic Control in Electric Power Systems. This standard establishes the allowable limits for total harmonic voltage distortion ($THD_v \le 5\%$ in MV, $\le 1.5 - 2.5\%$ in HV) and harmonic current injection.

Operating / Dielectric Parameter	Reference Standard	Critical Criteria Threshold	Incipient Failure Mechanism	Thermal/Dielectric Consequence on the Asset
Overfluxing Factor ($V/f$)	IEC 60076-1 / IEEE C57.12.00	$> 1.10 p.u.$ (Continuous) $> 1.25 p.u.$ ($t > 10 s$)	Magnetic flux leakage into non-laminated structural components.	Oil pyrolysis, hot-spots in yokes ($>250^\circ C$), and degradation of paper insulation.
Temporary Overvoltage (TOV)	IEC 60071-1	$> 1.50 p.u.$ ($t > 1 s$) $> 2.50 p.u.$ ($t > 0.1 s$)	Exceeding the power-frequency dielectric strength.	Partial discharges in XLPE, inter-layer/inter-turn puncture in windings.
Energy Absorption in MOV	IEC 60099-4 / IEEE C62.11	$E_{ absorbed } > E_{ thermal }$ ($4 - 10 kJ/kV$)	Excessive power dissipation during continuous conduction at MCOV.	Thermal runaway, internal short circuit, and catastrophic explosion of the arrester.
Relative Cable Length ($L_{ cable }$)	IEEE C57.105	Equivalent cable capacitance exceeds the critical capacitance $C_{ crit }$.	Formation of a series resonant loop between cable capacitance and saturated inductance.	Self-sustained subharmonic or chaotic bifurcation during single-phase switching.
Voltage Harmonic Distortion ($THD_v$)	IEEE 519 / IEC 61000-3-6	$THD_v > 5.0\%$ (MV) $THD_v > 2.5\%$ (HV)	Harmonic injection coinciding with natural parallel resonance frequencies.	Additional eddy current losses in winding conductors and severe parallel resonance.

Design, Mitigation, and Network Topology Strategies

Preventing and suppressing ferroresonance in systems fed by long underground cables requires a combination of robust network topologies, strict switching protocols, and passive or active damping elements.

Elimination of Single-Phase Operation (Three-Pole Switching Systems)

The most effective preventive measure at the primary design level is to prohibit the installation of single-phase fuses or pole-by-pole disconnect switches at the sending end of underground lines feeding distribution or subtransmission transformers. Instead, the exclusive use of three-pole gang-operated circuit breakers or load-break switches is required. By ensuring that all three phases open or close with an inter-pole time dispersion of less than $2 ms$, the injection of capacitive voltage through open phases is eliminated.

Selection of Transformer Core Topology and Construction

The physical construction of the magnetic core significantly influences susceptibility to ferroresonance:

• Three-Leg Cores (3-Leg Core): In a three-phase, three-leg coplanar core, zero-sequence magnetic flux does not have a closed ferromagnetic path for return. The zero-sequence flux must exit the core and return through the oil and the metallic tank wall, resulting in an extremely low zero-sequence magnetizing inductance ($L_0 \approx 0.1 - 0.2 p.u.$). This low inductance provides a powerful natural damping effect against ferroresonant modes involving the neutral, making these transformers highly resilient to the phenomenon.
• Five-Leg Cores or Single-Phase Transformer Banks: In five-leg transformers (or banks of three independent single-phase units), a very low-reluctance path exists for zero-sequence flux through the outer return legs. Consequently, $L_0 \approx L_d$ (high zero-sequence magnetizing inductance), which exponentially increases the risk of ferroresonance during single-phase switching. This topology should be avoided when feeding through long underground cables without three-pole circuit breakers.

Damping Resistors in Open-Delta Tertiary Windings

When the transformer features a secondary or tertiary winding connected in an open-delta (broken delta) configuration, a damping resistor ($R_d$) can be installed across the open corner of the delta. Under balanced, normal three-phase operating conditions, the vector sum of the voltages across the delta is zero ($V_{\Delta} = 0 V$), meaning the resistor dissipates no power in steady state.

In the event of a ferroresonant imbalance or neutral displacement, a zero-sequence voltage ($3V_0$) appears across the open-delta terminals. The resistor then dissipates the oscillating capacitive energy, introducing a damping factor ($\zeta > 1/\sqrt{2}$) that collapses the resonant oscillations. The critical value of the damping resistor, referred to the secondary side, is calculated using the following relationship:

Where $C_c'$ is the equivalent cable capacitance seen from the primary side, and $N_{ tertiary }/N_{ primary }$ is the turns ratio of the windings.

Controlled Switching (Point-on-Wave) and Pre-Insertion Resistors (PIR)

In transmission and subtransmission networks, the use of circuit breakers equipped with synchronous switching controllers (point-on-wave) or pre-insertion resistors (PIR) mitigates peak overvoltages and inrush currents during the energization of underground cables and transformers. By closing the circuit breaker poles at the optimal electrical angle (at the voltage peak for capacitive circuits, or at the voltage zero-crossing for the transformer's magnetizing inductance), the DC component of the magnetic flux is minimized ($\Phi_{ dc } = 0$), preventing the core from entering deep saturation.

Analysis Methodology and Practical Application with Vexten Suite

To illustrate the workflow for a rigorous analysis of ferroresonance and overvoltages in a high-level industrial engineering project, the integrated Vexten Suite platform is utilized. This platform executes a coordinated methodological sequence using calculation modules for short-circuit analysis (IEC 60909), cable sizing (IEC 60287), transient simulation, and frequency response.

Methodological Workflow in Vexten Suite

1. Step 1: Network Characterization and Short-Circuit Power (Vexten Short-Circuit / IEC 60909 Module):

   The equivalent high-voltage network is modeled by calculating the initial symmetrical short-circuit power ($Ssc''$) and the equivalent complex impedance of the system ($Z_k = R_k + j X_k$) at the point of common coupling (PCC).

2. Step 2: High-Frequency Underground Cable Modeling (Vexten Cable Manager / IEC 60287 Module):

   The distributed electrical parameters of the three-phase XLPE underground cable run are calculated at the maximum operating temperature ($90^\circ C$), yielding the AC resistance ($R_c$), distributed inductance ($L_c$), and positive- and zero-sequence capacitances ($C_1, C_0$).

3. Step 3: Output Impedance Calculation and Frequency Sweep (Vexten Frequency Scan):

   The input impedance seen from the transformer terminals is calculated as a function of the harmonic order ($h$), identifying the parallel resonance (anti-resonance) and series resonance points:

   $$Z_{ in }(h) = \frac{R_c + j h \omega_1 L_c + j h \omega_1 L_m(i)}{1 - h^2 \omega_1^2 L_m(i) C_c + j h \omega_1 R_c C_c}$$
4. Step 4: Critical Capacitance Determination and Ferroresonance Risk Assessment (Vexten Transient Engine):

   The total installed cable capacitance ($C_{ total } = C_c \cdot length$) is compared against the critical capacitance threshold ($C_{ crit }$) defined by the empirical-analytical relationship in IEEE C57.105:

   $$C_{ crit } = \frac{S_n}{2\pi f_1 \cdot V_{L-L}^2} \cdot \left( \frac{\%I_m}{100} \right) \cdot K_{ core }$$

   Where $S_n$ is the nominal power of the transformer, $V_{L-L}$ is the nominal line-to-line voltage, $\%I_m$ is the percent magnetizing current, and $K_{ core }$ is a core topology factor ($K_{ core } \approx 0.01$ for 3-leg cores, and $K_{ core } \approx 1.0$ for 5-leg cores or single-phase banks).

Developed Numerical Case Study

Consider an industrial distribution system with the following field parameters:

• Nominal system voltage: $V_{L-L} = 33 kV$ ($50 Hz$).
• Short-circuit power at the grid node: $Ssc'' = 500 MVA$ ($X/R = 10$).
• Underground feeder cable: Three-phase group of single-core XLPE cables, $1 \times 240 mm ^2$ Copper, length $L = 6.5 km$.
  • Cable capacitance per unit length: $C_c = 0.26\ \mu F/km$.
  • Total link capacitance: $C_{ total } = 6.5 \times 0.26 = 1.69\ \mu F$.
• Receiving power transformer: $S_n = 5.0 MVA$, $33/11 kV$, Dy11 connection.
  • Core type: 5-leg core (high zero-sequence magnetizing inductance).
  • No-load magnetizing current: $\%I_m = 0.35\%$.
  • Leakage reactance: $e_k = 7.5\%$.
• Evaluated contingency: Accidental single-phase opening of a fuse or circuit breaker pole at the sending end of the underground cable.

Analytical Resolution and Vexten Suite Simulation

1. Calculation of the Total Cable Capacitance:

The capacitive reactance at the fundamental frequency ($50 Hz$) is calculated as:

2. Calculation of the Unsaturated Magnetizing Inductance of the Transformer:

The nominal magnetizing current per phase (referred to the $33 kV$ primary side) is:

The equivalent unsaturated magnetizing reactance per phase ($X_m$) is approximated by:

3. Determination of the Critical Capacitance ($C_{ crit }$) for a 5-Leg Core:

Applying the IEEE C57.105 standard criteria for a $33 kV$ network with a 5-leg core ($K_{ core } = 1.0$):

4. Forensic Risk Assessment:

The installed capacitance of the underground cable exceeds the critical ferroresonance threshold by more than 33 times. The relationship between the capacitive reactance of the cable and the magnetizing reactance in the partial saturation zone ($X_m \to X_{ sat } \approx 200 - 800\ \Omega$) directly intersects the transformer's hysteresis loop.

Simulating the single-phase opening of Phase A in Vexten Suite reveals that the voltage on the open phase escalates into a Fundamental Mode ferroresonant response, with steady-state peak overvoltages reaching:

This is accompanied by a highly saturated peak current in the primary winding of $I_{ peak } = 14.2 A$ ($46 times$ the nominal magnetizing current), with a total harmonic voltage distortion of $THD_v = 42.8\%$, dominated by the 3rd and 5th harmonics.

Corrective Actions and Redesign Implemented in Vexten Suite

To permanently mitigate the imminent risk of dielectric failure and thermal collapse identified during the Vexten Suite analysis, the following two technical solutions are prescribed and integrated:

1. Replacement of the Primary Switchgear: Complete elimination of single-phase disconnectors/fuses and the installation of a motorized three-pole circuit breaker with simultaneous pole operation (inter-pole dispersion time $\Delta t < 1.5 ms$).
2. Installation of a Damping Resistor in an Open-Delta Tertiary Winding: An auxiliary voltage/power transformer with a $110 V$ open-delta secondary winding is integrated. The value of the required damping resistor $R_d$, sized using the Vexten Suite simulation engine to dissipate the capacitive oscillation energy, is calculated as:
   $$R_d = \frac{1}{3 \cdot \omega \cdot C_{ total } \cdot \left(\frac{110}{33000/\sqrt{3}}\right)^2} = \frac{1}{3 \cdot 314.16 \cdot 1.69 \times 10^{-6} \cdot (0.005774)^2} = \frac{1}{5.309 \times 10^{-1}} = 18.83\ \Omega$$

   A commercially standard $18\ \Omega$ nickel-chromium power resistor bank is selected, rated to continuously dissipate $5 kW$ during temporary unbalance transients.

Upon re-running the dynamic analysis in Vexten Suite with the topological modifications applied, the transient response following asymmetrical disturbances decays exponentially to zero in less than $t_{ damp } < 40 ms$ (2 power frequency cycles). This returns the voltage and magnetic flux to stable nominal parameters without entering saturation regions or inducing hazardous dielectric overvoltages.