Ferroresonance and Magnetic Saturation in Shunt Reactors during Single-Phase Switching

Introduction and Electromagnetic Fundamentals of Shunt Reactors

Shunt reactors connected in extra-high voltage (EHV) and ultra-high voltage (UHV) transmission systems play a critical role in compensating for the capacitive reactive power generated by long transmission lines under light-load or no-load conditions. Electromechanically, these devices consist of ferromagnetic cores with distributed air gaps, designed to maintain a linear magnetization characteristic under nominal and moderate temporary overvoltage conditions. However, when single-phase switching operations are performed, or when asymmetrical openings and transient faults occur, the interaction between the distributed line capacitance and the non-linear inductance of the shunt reactor creates a highly complex resonant circuit.

From the perspective of electromagnetic field theory and coupled circuits, the reactor inductance ceases to be a time-invariant parameter and instead becomes a function dependent on the concatenated magnetic flux \lambda and the current $i$ . The behavior of the iron core is governed by the constitutive equation:

i(t) = f(\lambda(t) = \sum_{n=1, 3, 5}^{\infty} c_n \lambda(t)^n

where the coefficients $c_n$ reflect severe non-linearity due to magnetic saturation. During a single-phase opening, the healthy poles of the transmission line continue to induce electrostatic and electromagnetic voltages onto the disconnected phase through interphase mutual capacitances $Cm$ and earth capacitances $Cs$ . If the shunt reactor inductance $L$ forms a resonant circuit with the equivalent capacitance of the open phase $Ceq$ , the conditions for series or parallel ferroresonance are fully established, triggering sustained low-frequency overvoltages, sub-synchronous order harmonics, and excitation currents that exceed the equipment's rated current by several orders of magnitude.

Differential Equations and Non-Linear Modeling of the Saturable Core

Rigorous mathematical modeling of ferroresonance in shunt reactors requires the formulation of systems of ordinary and partial differential equations that couple the electrostatic phenomena of the power grid with the hysteresis and saturation of the magnetic core. Considering a simplified single-phase equivalent circuit representing a disconnected phase of a transmission line with a shunt reactor connected at the terminals, the system dynamics are described by the following integro-differential equation:

\frac{d^2 \lambda}{dt^2} + \frac{R}{L_0} \frac{d\lambda}{dt} + \omega_0^2 f(\lambda) = \omega_0^2 V_m \sin(\omega t + \theta)

Where \lambda is the concatenated flux in the reactor, $R$ is the equivalent copper and core loss resistance, $L_0$ is the inductance under the linear regime, \omega_0 is the fundamental system frequency, and V_m \sin(\omega t + \theta) represents the voltage induced from the adjacent energized phases through capacitive coupling. The non-linearity function of the magnetic core f(\lambda) is commonly modeled using the modified Fröhlich approximation or analytical saturation curves based on hyperbolic arctangent functions:

i(\lambda) = I_0 \left[ \frac{\lambda}{\lambda_s} + M \sinh\left( N \left(\frac{\lambda}{\lambda_s}\right)^k \right) \right]

In this system, the presence of saturation causes the apparent inductance of the reactor to vary dynamically during each cycle of the applied voltage. When the core enters the deep saturation region, the differential inductance of the reactor Ldiff = d\lambda / di drops drastically, approaching the values of the winding leakage inductance. This sudden reduction shifts the circuit resonance frequency toward values close to or lower than the fundamental frequency, inducing subharmonic oscillation modes (typically of order 1/2, 1/3, or 1/5) and quasi-periodic modes.

Lyapunov stability analysis and bifurcation maps demonstrate that the dynamical system of the ferroresonant reactor possesses multiple stable states (multi-stability). Depending on the initial conditions (such as the circuit breaker opening angle and the instantaneous value of the remanent flux \lambda_r in the core), the system can converge toward a normal low-voltage state or suddenly jump to a ferroresonant state characterized by distorted currents and elevated magnetic fluxes that exceed the knee point of the saturation curve by more than 200%.

Phenomenon Topology During Single-Phase Switching Operations

Single-pole opening and closing maneuvers in high-voltage circuit breakers are standard operational procedures for clearing transient faults on transmission lines. However, during the single-phase opening interval, the affected phase is isolated from the main power supply source but remains electromagnetically coupled to the rest of the energized system. Mutual capacitances between phases $Cab, Cbc, Cca$ and earth capacitances $Can, Cbn, Ccn$ form a complex capacitive network that feeds the open phase with an electrostatically originated voltage.

When a shunt reactor is permanently connected to the line terminals, the inductance of this reactor is placed in parallel with the equivalent zero-sequence and negative-sequence capacitance seen from the open terminals of the phase. The Thévenin equivalent circuit seen from the reactor terminals consists of an equivalent voltage source $Vth$ driven by the healthy phase voltages through the mutual coupling capacitors, and an equivalent impedance $Zth$ dominated by the equivalent capacitance $Ceq$ .

Switching Parameter	Normal / Operational Condition	Critical Ferroresonance Condition	Electromechanical Consequence
Circuit Breaker State	Symmetrical three-phase closure	Asymmetrical single-phase opening	Severe voltage unbalance and zero-sequence currents.
Equivalent Capacitance ( $Ceq$ )	Low line capacitive impedance	Optimal coupling with $Lreactor$	Tuning of the resonant circuit to subharmonic frequencies.
Remanent Flux (\lambda_r)	Low or symmetrically distributed	High post-interruption remanent flux	Asymmetry in the hysteresis loop and immediate entry into saturation.
System Losses ( $Req$ )	Normal damping via load	Low damping under no-load / light-load	Inability of the system to dissipate accumulated oscillatory energy.

During the opening of a phase pole, the transient electric arc within the circuit breaker can generate high-frequency re-ignitions (re-strikes), injecting high-frequency transient components that excite the natural modes of the reactor-line assembly. If the remanent flux in the reactor column corresponding to the open phase is high and coincides in polarity with the initial half-cycle of the induced voltage, the core reaches deep magnetic saturation within the first half-cycle following arc extinction, initiating the trajectory toward ferroresonant instability.

Forensic Failure Analysis and Consequences on Substation Assets

Ferroresonance and extreme magnetic saturation events in shunt reactors generate severe thermal, dielectric, and mechanical stresses that compromise the integrity of substation assets. Forensic analysis of failures occurring in EHV systems reveals characteristic damage patterns across various components of the electrical system.

Mechanical Deformation and Electrodynamic Stresses in Windings

Ferroresonant excitation currents are massively composed of odd-order harmonics (primarily third and fifth harmonics) and subharmonic components. These currents can reach magnitudes of 3 to 5 times the reactor's rated design current. Electromechanical forces on winding conductors are proportional to the square of the instantaneous current (F \propto i(t)^2). Cyclical radial and axial forces induce mechanical fatigue in insulation supports, causing loosening of the winding clamping blocks, abrasion of kraft paper insulation, and eventually turn-to-turn short circuits.

Supplementary Heating by Eddy Currents and Core Losses

The operation of the magnetic core in the deep saturation region alters the magnetic flux distribution, forcing flux out of the grain-oriented silicon steel laminations and into the clamping structures, magnetic shields, and the metal reactor tank. Eddy current losses in the shielding plates and tank walls increase in a non-linear manner:

Peddy = \frac{k_f \cdot d^2 \cdot f^2 \cdot Bmax^2}{\rho}

Where $d$ is the lamination thickness, $f$ is the effective frequency (including harmonics), and $Bmax$ is the peak magnetic flux density. This massive increase in losses generates local hot spots in the insulating oil, exceeding permissible thermal limits in accordance with IEEE C57.109 and IEC 60076-6 standards, causing thermal degradation of cellulosic insulation, generation of dissolved gases in the oil (hydrogen, methane, acetylene), and tripping of sudden pressure (Buchholz) protections.

Dielectric Overvoltages and Insulation Collapse

Sustained ferroresonant voltages can reach values of up to 2.5 to 3.5 p.u. with highly distorted waveforms featuring abrupt wave fronts. These overvoltages act continuously on the main insulation system (oil-paper or dry solid insulation). Metal-oxide surge arresters (MOA) located at the reactor terminals are subjected to excessive thermal energy absorption while attempting to limit these low-frequency overvoltages, which can lead to thermal failure and catastrophic explosion of the arresters if their energy dissipation capacity (line discharge class) is exceeded.

Advanced Mitigation Strategies and Design Criteria

Effective mitigation of ferroresonance and magnetic saturation in shunt reactors requires the implementation of countermeasures at the electromagnetic design level, switching equipment selection, and advanced protection system configuration.

Core Design and Air Gap Optimization

To prevent rapid entry into saturation under temporary overvoltage and capacitive coupling conditions, modern reactors are designed with shell-type cores or core-type designs featuring multiple non-magnetic air gaps uniformly distributed along the height of the columns. This linearizes the magnetization characteristic up to 140% - 160% of the rated voltage. The linearized inductance is expressed by:

L = \frac{\mu_0 N^2 A}{\sum lcore/\mu_r + \sum lgap} \approx \frac{\mu_0 N^2 A}{\sum lgap}

By making the total length of the air gaps \sum lgap the dominant term in the denominator, the total magnetic reluctance becomes independent of the relative permeability of the steel \mu_r, suppressing non-linearity and preventing ferroresonance.

Implementation of Pre-insertion Resistors and Circuit Breaker Synchronization

The use of circuit breakers equipped with phase-angle-controlled mechanisms (Point-on-Wave Switching) eliminates severe connection and disconnection transients. By opening or closing the circuit breaker poles at current zero-crossings or optimal voltage instants, the generation of asymmetrical remanent fluxes is minimized. Additionally, the use of pre-insertion resistors in EHV circuit breakers dampens high- and medium-frequency transient oscillations during switching operations.

Damping Schemes and Passive Filters

In substations where the ferroresonant risk is structurally high due to complex network topologies featuring long no-load lines, damping networks are implemented based on high-resistance circuits connected in series with auxiliary capacitor banks or through the addition of tertiary windings in reactors connected in a closed delta with an external damping resistor. This resistor dissipates the oscillatory energy of the subharmonic modes, preventing the growth of the ferroresonant oscillation amplitude.

Practical Application and Analysis with the Vexten Suite

To illustrate the sizing and technical validation of a reactive compensation system subject to transient and harmonic regimes, a case study developed through the analytical modules of the Vexten Suite platform is presented, applying international standards IEC 60909, IEEE 141, and IEC 60287.

Power System Specifications

Nominal system voltage ( $U_n$ ): 400 kV (Three-phase network, 50 Hz).
Three-phase network short-circuit power ( $Sk3$ ): 25 GVA.
Three-phase shunt reactor: 150 MVAR, 400/√3 kV per phase, solidly grounded star connection.
Equivalent zero-sequence capacitance of the line ( $C_0$ ): 12.5 µF per phase.
Equivalent reactor winding resistance ( $Rcu$ ): 0.15 \Omega per phase at 75 °C.

Short-Circuit Calculation and Transient Parameters (IEC 60909 / IEEE 141)

The equivalent network impedance seen from the reactor terminals is calculated from the short-circuit power:

Znet = \frac{U_n^2}{Sk3} = \frac{(400 \times 10^3)^2}{25 \times 10^9} = 6.4 \ \Omega

The equivalent network inductive reactance ( $Xnet$ ), assuming an $X/R = 20$ ratio:

Rnet = \frac{Znet}{\sqrt{1 + (X/R)^2}} = \frac{6.4}{\sqrt{1 + 400}} = 0.319 \ \Omega

Xnet = Rnet \times 20 = 0.319 \times 20 = 6.38 \ \Omega

The rated inductance of the shunt reactor per phase is determined using the rated reactance of the equipment:

X_L = \frac{(U_n / \sqrt{3})^2}{Q_3 / 3} = \frac{(400 / \sqrt{3})^2}{50} = \frac{53333.33}{50} = 1066.67 \ \Omega

L = \frac{X_L}{2 \pi f} = \frac{1066.67}{2 \pi (50)} = 3.395 \ H

Evaluation of the Resonant Frequency Factor

During single-phase opening, the resonance frequency of the circuit formed by the reactor inductance $L$ and the equivalent capacitance of the open phase Ceq \approx C_0 is calculated by:

fres = \frac{1}{2 \pi \sqrt{L \cdot Ceq}} = \frac{1}{2 \pi \sqrt{3.395 \times (12.5 \times 10^{-6})}}

fres = \frac{1}{2 \pi \sqrt{4.2437 \times 10^{-5}}} = \frac{1}{2 \pi (6.514 \times 10^{-3})} = 24.45 \ Hz

The value of fres = 24.45 \ Hz lies squarely in the sub-synchronous region (approximately half the 50 Hz fundamental frequency), confirming the system's vulnerability to 1/2-order subharmonic excitation under asymmetrical switching conditions.

Harmonic Derating and Conductor Sizing (IEC 60287 / NEC 310)

As a consequence of magnetic saturation and ferroresonant transient regimes, the currents flowing through connection cables and windings contain significant harmonic content. The Vexten Suite evaluates the supplementary loss factor and applies the Harmonic Derating Factor (HDF) in accordance with IEC 60287:

I_{max\_derated} = Irated \cdot \sqrt{\frac{1 + \Delta_1}{1 + \sum_{h=2}^{n} M_h^2 \cdot (1 + Y_h)}}

Where $M_h$ is the harmonic distortion rate of order $h$ , $Y_h$ is the conductor eddy current loss factor for harmonic $h$ , and \Delta_1 represents the increased hysteresis and eddy loss increment in the core. For a third harmonic content of 25% and a fifth harmonic content of 15% derived from a severe transient event, the derating factor calculated by the Vexten Suite tool indicates that the allowable current-carrying capacity of the cable must be reduced by 18.5% to prevent thermal degradation of the polymeric insulation (XLPE).

Vexten Suite Module	Reference Standard	Main Input Parameter	Analytical Result / Design Criterion
Vexten Transient Studio	IEC 60076-6 / IEEE C57.109	Non-linear inductance matrix and remanent fluxes	Identification of bifurcation points and 2.8 p.u. ferroresonant overvoltages.
Vexten Short-Circuit Pro	IEC 60909 / IEEE 141	Short-circuit power ( $Sk3$ ) and sequence impedances	Determination of peak electrodynamic stresses and asymmetrical currents.
Vexten Thermal Cable Sizer	IEC 60287 / NEC 310	Current harmonic spectrum and ambient temperature	Calculation of the HDF factor and thermal oversizing of power cables.
Vexten Protection Coordination	IEC 60255 / IEEE C37.112	Excitation characteristics and temporary overcurrent	Setting of timed overcurrent relays and residual overvoltage protection.

Conclusions and Normative Guidelines

Ferroresonance instability and magnetic saturation in shunt reactors during single-phase switching operations represent one of the most complex and destructive electromagnetic phenomena in high-voltage electric power systems. Rigorous analysis requires moving away from traditional linear models and adopting non-linear simulation approaches that incorporate hysteresis, core saturation, and the distributed capacitive electromagnetic coupling of transmission lines.

The implementation of core designs featuring multiple distributed air gaps, the use of phase-angle-controlled synchronized circuit breakers, and validation via advanced engineering tools such as the Vexten Suite ensure that reactive compensation installations operate within safety margins established by international IEC and IEEE standards. Strict compliance with these criteria prevents catastrophic failures, optimizes investment in substation assets, and guarantees the operational reliability and stability of large-scale electrical power transmission networks.