Ferroresonance in Shunt Reactors during Single-Phase Switching

Introduction and Theoretical Framework of Non-Linear Phenomena in High-Voltage Systems

The integration of Ultra-High-Voltage AC (UHVAC) and Extra-High-Voltage AC (EHVAC) transmission systems introduces significant operational complexities associated with managing the capacitive reactive power generated by transmission lines under no-load or light-load conditions. To mitigate power-frequency overvoltages and control the seasonal voltage profile, the use of shunt reactors—whether iron-core with distributed air gaps or shielded air-core reactors—is an indispensable standard practice. However, the interaction between the non-linear magnetic core inductance of the reactor and the positive- and zero-sequence capacitance of the transmission line creates a resonant LC circuit exposed to ferroresonance phenomena and severe transient magnetic saturation during unbalanced operating maneuvers, particularly single-phase switching.

From the perspective of advanced electromagnetic circuit theory, ferroresonance is not a classical linear resonance condition, but rather a non-linear oscillation mode characterized by the coexistence of multiple stable states (multi-stability) for the same network parameters. The system can abruptly jump from a low-voltage, quasi-sinusoidal current state to a high-voltage ferroresonant state characterized by sustained temporary overvoltages (TOV), massive harmonic distortion, extremely high excitation currents, and extreme thermal and dielectric stresses on the reactor windings and associated substation equipment.

The formal analysis of this phenomenon requires the rigorous mathematical formulation of the magnetic flux in the reactor core coupled with the stochastic and deterministic differential equations of the three-phase electrical system with imbalances introduced by the sequential opening or closing of poles in high-voltage circuit breakers (GIS or AIS).

Governing Equations and Non-Linear Modeling of the Magnetic Core

To accurately model magnetic saturation and ferroresonance, it is essential to represent the non-linear magnetization characteristic of the iron core of the shunt reactor. The relationship between the linked magnetic flux \lambda(t) and the excitation current i(t) is expressed via advanced analytical functions that incorporate hysteresis and eddy current effects (core losses). A classical approximation used in electromagnetic transient studies (EMTP) is based on the modified Fröhlich function or odd power series expansions to avoid discontinuities:

i(t) = a \lambda(t) + b \lambda(t)^{2n+1}

Where $a$ represents the inverse of the differential inductance in the linear zone ( $L_0^{-1}$ ), $b$ is the core saturation coefficient, and $n$ is a positive integer determining the sharpness of the saturation knee. The differential equation governing the equivalent series circuit composed of the line resistance $R$ , the stray inductance or that of the reactor itself with its non-linear characteristic, and the effective system capacitance $C$ seen from the reactor terminals is formulated as follows:

\frac{d^2\lambda}{dt^2} + \frac{R}{L(\lambda)} \frac{d\lambda}{dt} + \frac{1}{C L(\lambda)} \lambda = \frac{1}{C} v_s(t)

Where v_s(t) = V_m \sin(\omega t + \phi) represents the sinusoidal system voltage source, and the apparent or differential inductance is defined as L(\lambda) = \frac{d\lambda}{di}. When the core enters deep saturation, L(\lambda) drops drastically toward values close to the winding air inductance ( $Lair$ ), which abruptly alters the natural resonant frequency of the circuit:

\omega_0(\lambda) = \frac{1}{\sqrt{C L(\lambda)}}

During a single-phase switching operation (for example, the opening of a single phase of a circuit breaker due to single-pole autoreclosure operation or a cleared single-line-to-ground fault), the healthy phases remain capacitively coupled to the open phase. The reactor connected to the open phase floats through a capacitive network coupled to the other two energized phases and to ground. This establishes a tuned circuit where the equivalent capacitance $Ceq$ and the non-linear inductance L(\lambda) satisfy the conditions for the excitation of subharmonic (\omega_0 = \omega/2, \omega/3, \dots), harmonic (3\omega, 5\omega, etc.), or quasi-periodic modes.

Analysis of Single-Phase Switching Operations and the Origin of Imbalance

Single-pole opening and closing operations in EHVAC and UHVAC systems are operational or protective in nature (e.g., single-pole automatic reclosing following temporary single-line-to-ground faults). During the time interval when a phase is open at the circuit breaker while the reactor remains connected on the line side, the network topology changes drastically. Mutual capacitances between phases ( $Cab, Cbc, Cca$ ) and capacitances to ground ( $Can, Cbn, Ccn$ ) play a critical role.

Considering a transposed transmission line of length $l$ , the capacitance matrix per unit length transforms, upon opening phase $A$ , into an equivalent Thévenin network seen from the phase $A$ reactor terminals. The electrostatic and electromagnetically induced voltage from healthy phases $B$ and $C$ energizes the resonant circuit. If the inductive reactance of the reactor at power frequency approaches or crosses the value of the equivalent capacitive reactance of the unbalanced network, the system enters a zone of severe instability.

Operation / Network Parameter	Normal Condition (Symmetrical)	Single-Phase Switching Condition	Normative Limit (IEC / IEEE)
Zero-Sequence Voltage ( $V_0$ )	Practically 0% to 1% of $V_1$	Can reach up to 30% - 50% of $V_1$	IEEE Std 1036: Max 5% under asymmetrical conditions
Reactor Excitation Current	Rated design current (I_n \le 1.0 p.u. )	Peaks up to 8 - 15 p.u. due to deep saturation	IEC 60289: Transient overcurrent thermal tolerance
Total Harmonic Voltage Distortion (THDv)	< 1.5\% at EHV buses	15\% - 40\% with predominance of 2nd, 3rd, and 5th harmonics	IEEE Std 519: Strict limit of 1.5% to 2.0% at EHV
Temporary Overvoltage Factor (TOV)	$1.0 p.u.$	$1.8 p.u. - 2.6 p.u.$ sustained for several cycles	IEC 60071-2: Temporary withstand curves at 10s - 30s

Forensic Failure Analysis and Consequences on Substation Equipment

When a shunt reactor experiences sustained ferroresonance induced by single-phase switching, the physical and dielectric consequences on substation assets are catastrophic if protections do not operate with adequate speed. Below is a breakdown of the forensic failure mechanisms for each equipment type:

Failures in Shunt Reactor Windings

Highly distorted saturation currents generate massive radial and axial electrodynamic forces on the winding conductors. These forces exceed the elastic limits of copper and mechanical supports, causing physical displacement of turns, deformation of cooling ducts, and abrasion of cellulosic paper insulation. Likewise, supplementary losses due to eddy currents in the windings and metallic structural elements of the tank caused by high harmonic content (h \ge 3) produce local hotspots with temperatures exceeding 250 °C, rapidly degrading insulating oil and generating characteristic combustible gases ( $CO, CO₂, CH_4, C_2H_2$ ).

Dielectric Deterioration in High-Voltage Cables and Bushings

Ferroresonant temporary overvoltages (TOV) with high-frequency components subject the polymeric (XLPE) or oil-impregnated insulation of connection cables and bushings to repetitive dielectric stresses. These stresses accelerate internal partial discharge (PD) phenomena, eroding the molecular structure of the insulation until dielectric perforation occurs in the form of electrical treeing, followed by a catastrophic ground fault and bushing explosion.

Anomalous Behavior in Circuit Breakers

During the interruption of currents with high harmonic content and a decaying direct current (DC) component generated by the ferroresonant regime, SF6 circuit breakers face severe difficulties at current zero-crossings. This increases the probability of reignitions or restrikes, injecting additional high-frequency transients that aggravate the general dielectric collapse of the network.

Practical Design Strategies and Advanced Mitigation

Effective mitigation of ferroresonance instability and magnetic saturation in shunt reactors requires the implementation of multidisciplinary design criteria in both network engineering and equipment specification. Critical design formulas and factors are detailed below:

Selection of Core Linearity Characteristics

To avoid premature saturation under single-phase imbalances, the reactor design must ensure that the knee of the magnetic saturation curve is located at least between 140% and 160% of the continuous rated operating voltage:

V_{ knee } \ge 1.45 \cdot V_{n\_phase}

Additionally, the use of shielded air-core reactors completely eliminates the non-linearity introduced by ferromagnetic cores, removing the risk of ferroresonance. However, their larger size, weight, and stray magnetic field require detailed electromagnetic compatibility (EMC) studies.

Implementation of Damping Resistors and Surge Arresters

When iron-core reactors are used, the installation of high-energy-capacity metal-oxide surge arresters directly at the reactor terminals helps limit ferroresonant voltage peaks. The energy absorbed by the arrester during a prolonged transient event is calculated using the Joule integral:

W_{ surge } = \int0^{t_{ clear }} v(t) i(t) dt

Likewise, incorporating a grounding scheme via a damping resistor in the reactor neutral (provided the system configuration permits it) dissipates zero-sequence oscillatory energy, breaking the resonance condition.

Practical Application and Analysis via Vexten Suite

Within the advanced engineering environment of Vexten Academy, the validation of ferroresonance and saturation studies is executed using integrated modules combining international IEC and IEEE standards. The computational workflow in the Vexten Suite platform encompasses the following normative stages:

Short-Circuit Calculation and Contribution Levels per IEC 60909 / IEEE 141

The first step consists of determining the network short-circuit power at the reactor interconnection point to evaluate the system stiffness ratio ( $SCR$ - Short Circuit Ratio):

SCR = \frac{S_{ sc\_3ph }}{Q_{ reactor }}

If the $SCR$ is low ( $SCR < 3$ ), the network is considered weak, which exponentially increases susceptibility to ferroresonant overvoltages. The Vexten Suite calculation engine processes nodal impedance matrices $Zbus$ to extract exact symmetrical components during single-pole circuit breaker opening.

Harmonic Derating and Cable Sizing per IEC 60287 / NEC 310

The presence of elevated harmonic currents induced by reactor magnetic saturation alters Joule effect and eddy current losses in power cables associated with the substation. The harmonic derating factor ( $FHD$ ) is calculated in Vexten Suite by applying the coefficients specified in IEC 60287:

I_{ max\_cable } = I_{ base } \cdot \sqrt{\frac{1}{\sum_{h=1}^{k} I_h^2 \cdot (1 + Y_h)}}

Where $I_h$ is the relative magnitude of harmonic $h$ and $Y_h$ represents the frequency-dependent supplementary loss factor for the conductors and metallic shields of the cable.

Power Factor Mitigation and Harmonic Resonance

To prevent coincidence between the parallel resonance frequency of the reactive compensation system and the characteristic harmonics generated by saturation, Vexten Suite performs a Frequency Scan calculating the input impedance Zin(j\omega) in the range of $0 Hz$ to $3 kHz$ :

Zin(j\omega) = \left( Rsys(\omega) + j\omega Lsys(\omega) \right) \parallel \left( Rreact(\omega) + j\omega L_{)(\lambda)}(\omega) \right) \parallel \left( \frac{1}{j\omega Ceq} \right)

The results of this sweep allow design engineers to adjust harmonic filter tuning parameters or modify the nominal inductance of shunt reactors to ensure impedance poles remain strictly away from dominant harmonic frequencies ( $2^ nd , 3^ rd , 5^ th , 7^ th$ ), thereby securing total operational stability of the EHVAC/UHVAC transmission system under any single-phase switching contingency.

Conclusions and Engineering Recommendations

Ferroresonance instability and magnetic saturation in shunt reactors during single-phase switching operations represent a critical challenge in high-power electrical systems engineering. Rigorous analysis using non-linear magnetic flux-dependent inductance models, combined with the strict application of international standards (IEC 60289, IEC 60071, IEEE Std 1036, and IEEE Std 519), is essential to prevent catastrophic failures in substation assets. The utilization of advanced simulation tools such as the Vexten Suite platform guarantees that design specifications, thermal derating factors, and frequency scan studies effectively mitigate the risk of temporary overvoltages and extreme harmonic distortion, consolidating the reliability and operational safety of future transmission networks.