Electromagnetic transients and overvoltages in shunt reactors during switching operations

Electromagnetic Fundamentals and High-Frequency Modeling of Shunt Reactors

Shunt reactors are critical inductive components deployed in high-voltage (HV) and extra-high-voltage (EHV) transmission systems to compensate for the capacitive reactive power generated by long transmission lines under light-load or no-load conditions. From the perspective of electromagnetic field theory and high-voltage engineering, modeling a shunt reactor during transient regimes differs radically from its representation at industrial frequency ( $50$ or $60 Hz$ ).

At transient frequencies, ranging from a few kilohertz to several megahertz, the physical dimensions of the reactor windings are no longer negligible compared to the wavelength of the incident electromagnetic waves. Therefore, the reactor must be modeled as a distributed parameter system. Inter-turn capacitance, layer-to-layer capacitance, stray capacitance to ground, and leakage inductance interact to generate complex oscillatory behavior when subjected to steep wavefronts.

The fundamental transient behavior is governed by Maxwell's equations, which, when simplified to the circuit domain for an equivalent multiconductor transmission line representing the winding, are expressed by the modified telegrapher's equations to include core eddy-current losses and skin effect in the conductors:

-\frac{\partial v(x,t)}{\partial x} = R(s)i(x,t) + L \frac{\partial i(x,t)}{\partial t}

-\frac{\partial i(x,t)}{\partial x} = G v(x,t) + C \frac{\partial v(x,t)}{\partial t}

Where v(x,t) and i(x,t) represent the voltage and current along the spatial coordinate of the winding $x$ at instant $t$ . The resistance R(s) is heavily dependent on the frequency $s$ due to the skin effect and proximity effect in continuously transposed copper conductors (CTC strands). The equivalent longitudinal capacitance $C_s$ and transversal capacitance to ground $C_g$ determine the propagation velocity of the electromagnetic wave through the winding:

v_p = \frac{1}{\sqrt{L C_s}}

When a circuit breaker opens the circuit of a shunt reactor, the interruption of inductive current triggers severe transient phenomena. The energy stored in the reactor's magnetic field, expressed as:

W_m = \frac{1}{2} L I_{ peak }^2

must be instantaneously transferred to the electric field of the system's stray capacitances:

W_e = \frac{1}{2} C V_{ peak }^2

This energy conversion gives rise to a high-frequency transient oscillation whose natural frequency is analytically defined as:

f_n = \frac{1}{2\pi \sqrt{L C_{ eq }}}

Where $C_{ eq }$ comprises the combined equivalent capacitance of the reactor, bushings, adjacent busbars, and the circuit breaker capacitance to ground.

Opening Dynamics and Re-ignition Mechanisms in High-Voltage Circuit Breakers

The disconnection of shunt reactors is one of the most severe switching operations in electric power networks due to the highly inductive nature of the load and current chopping phenomena. During the opening of SF6 (sulfur hexafluoride) or vacuum-technology breaker contacts, the electric arc is stretched and cooled by the extinguishing gas flow until the instantaneous value of the current approaches zero.

Due to the dynamic regime of the arc, the current does not decay in a purely sinusoidal manner down to zero, but extinguishes abruptly prior to the natural zero crossing. This phenomenon, known as current chopping, forces the remnant energy from the magnetic core into the stray capacitance of the system within an extremely short time interval (\Delta t), generating a high-frequency transient overvoltage characterized by:

V_{ chop } = I_{ chop } \sqrt{\frac{L}{C_{ eq }}}

Where $I_{ chop }$ is the abruptly chopped current value just prior to the definitive extinction of the arc. In modern circuit breakers, $I_{ chop }$ typically varies between $3 A$ and $30 A$ depending on the breaker's quenching technology.

As the circuit breaker contacts separate, the dielectric strength of the medium between the contacts increases as a function of time $t_{ sep }$ . However, the Transient Recovery Voltage (TRV) appearing across the breaker terminals oscillates at the natural frequency of the circuit. If the TRV exceeds the instantaneous dielectric strength of the contact gap, a dielectric breakdown occurs, known as re-ignition (or re-strike if it happens after a quarter-cycle following arc extinction).

Re-ignition causes a sudden discharge of the network-side capacitance toward the reactor, or vice versa, injecting high-frequency, severe-amplitude current waves that generate multiple reignitions. These overvoltages not only jeopardize the main insulation of the reactor but also produce extreme thermal and mechanical stresses on the breaker contacts and adjacent windings.

Forensic Failure Analysis and Dielectric Consequences in Substation Components

Electromagnetic transients generated during the unmitigated disconnection of shunt reactors trigger a degradation cascade across substation assets. Forensic failure analysis reveals that the most critical failure modes are concentrated within the reactor's cellulosic-oil insulation system, associated power cables, and circuit breaker mechanisms.

Under the action of high-frequency oscillatory overvoltages ( $10 kHz$ to $500 kHz$ ), the voltage distribution along the reactor winding ceases to be linear. Due to the ratio between longitudinal capacitance $C_s$ and capacitance to ground $C_g$ (expressed by the voltage distribution parameter \alpha = \sqrt{C_g / C_s}), the turns located at the winding extremities (near the high-voltage terminals) bear the most severe initial voltage gradient.

This phenomenon provokes continuous Partial Discharges (PD) within microscopic gas bubbles or high electrical stress concentration zones in the insulating oil. Over time, this leads to the carbonization of kraft paper, tracking formation, and ultimately an inter-turn short circuit that catastrophically destroys the reactor core and windings.

Affected Component	Transient Stress Mechanism	Normative Limits (IEEE / IEC)	Forensic Failure Mode	Operational and Dielectric Consequence
Shunt Reactor Winding	High-frequency oscillations, steep wavefronts (high \frac{dv}{dt}).	IEEE Std C57.21 / IEC 60048-6: BIL and BSL specified according to insulation class.	Inter-turn partial discharges, oil degradation, kraft paper perforation.	Massive internal short circuit, catastrophic core failure, tank explosion, and oil fire.
Bushings (OIP/RIP Condenser Bushings)	Switching-type overvoltages and steep TRV wavefronts.	IEC 60137 / IEEE Std 69: Lightning and switching impulse tests.	Capacitive condenser core puncture, gasket failure, SF6/oil leaks.	External flashover or bushing explosion, loss of pressure containment.
Interconnecting Power Cables	Traveling wave reflections, open-end voltage amplification.	IEC 60287 / ICEA S-97-682: Insulation levels according to temporary overvoltage (TOV).	Discharge degradation in XLPE insulation voids (Electrical treeing).	Permanent cable ground fault, tripping of line differential protections.
High-Voltage Circuit Breaker (SF6)	Multiple re-ignitions, high-frequency high currents, current chopping.	IEEE C37.011 / IEC 62271-100: TRV requirements and inductive interruption capability.	Severe PTFE nozzle erosion, contact welding, loss of SF6 pressure.	Inability to interrupt current, opening failure, structural pole damage.

Mitigation Strategies, Design, and Advanced Protection Parameters

To mitigate severe overvoltages arising from shunt reactor disconnection, modern engineering implements a combination of solutions based on switchgear design, surge arrester installation, and synchronous switching technologies.

The installation of gapless metal-oxide surge arresters (MOV, Metal Oxide Varistors) directly at the reactor terminals serves as the first line of defense. These devices must be sized considering the energy absorbed during fault clearing and their capability to withstand temporary overvoltages (TOV). The energy absorbed by the surge arrester is calculated by integrating the discharge current and residual voltage:

W_{ surge } = \int0^{t_{ clear }} v_{ res }(t) i_{ desc }(t) dt

Additionally, snubber networks are deployed, consisting of a resistor $R_s$ and a capacitor $C_s$ connected in parallel across the reactor or breaker terminals. These networks drastically reduce the transient recovery voltage slope (\frac{dv}{dt}) and lower the circuit's natural frequency, thereby preventing breaker re-ignitions. The characteristic impedance of the snubber network is optimized via the relationship:

Z_s = \sqrt{\frac{L}{C_{ eq }}}

The most advanced technology for eliminating switching transients is the use of controlled-switching or synchronized circuit breakers. By precisely measuring the system voltage and current phase angles, the control system actuates the breaker poles at the optimal instant:

Synchronized Opening: Contacts part such that inductive current interruption occurs precisely at the point where the current naturally crosses zero, minimizing or completely eliminating the current chopping phenomenon (I_{ chop } \approx 0).
Synchronized Closing: Closing instances for each phase are programmed to coincide with the network voltage peak value, avoiding the appearance of direct current components (DC offset) and subsequent transient magnetic core saturation.

Practical Application and Performance Analysis via the Vexten Suite

To illustrate the technical rigor required in extra-high-voltage electrical systems engineering, computational calculation and simulation analysis under international standards using the Vexten Suite engineering environment are presented below. The following calculation modules demonstrate the analytical validation of shunt reactor installations in $400 kV$ networks.

Short-Circuit Current Calculation and Thermal Stresses according to IEC 60909 / IEEE 141

In designing the substation associated with the shunt reactor, it is imperative to calculate the initial symmetrical short-circuit current $I''_{k}$ and peak current $i_{ peak }$ to ensure switchgear withstands electrodynamic stresses. The Vexten Suite short-circuit module applies the standard IEC 60909 formulation:

I''_{k} = c \frac{U_n}{\sqrt{3} Z_k }

Where $c$ is the voltage factor taking a value of $1.1$ for high-voltage systems with maximum operating voltage, $U_n$ is the nominal system voltage ( $400 kV$ ), and $Z_k$ is the positive-sequence equivalent impedance viewed from the interconnection node. For a $100 MVAR$ shunt reactor connected to a network with a short-circuit power of $25 GVA$ , the system impedance is:

Z_{ sys } = \frac{U_n^2}{S_{ sc }} = \frac{(400\times 10^3)^2}{25 \times 10^9} = 6.4\,\Omega

The inductive reactance of the shunt reactor is calculated as:

X_r = \frac{U_n^2}{Q_r} = \frac{(400\times 10^3)^2}{100 \times 10^6} = 1600\,\Omega

Upon performing simulation within the Vexten Suite, the dynamic shock factor for calculating the maximum peak current under a severe switching transient is determined via the equivalent circuit's $R/X$ attenuation ratio, yielding a peak current value of:

i_{ peak } = \kappa \sqrt{2} I''_{k} = 1.8 \times \sqrt{2} \times 144.33 kA = 367.5 kA

Power Cable Sizing and Harmonic Derating according to IEC 60287 / NEC 310

Underground cables or gas-insulated switchgear (GIS) connections feeding shunt reactors are exposed to high-frequency harmonic currents generated during switching operations and continuous system harmonic distortion. The Vexten Suite ampacity module calculates current-carrying capacity considering the Harmonic Derating Factor (HDF) based on IEC 60287:

HDF = \sqrt{\frac{1}{1 + \sum_{h=2}^{n} (\frac{I_h}{I_1})^2 \cdot \left(\frac{R_h}{R_1}\right)}}

Where $I_h / I_1$ is the harmonic current ratio of order $h$ with respect to the fundamental, and $R_h / R_1$ represents the increase in effective conductor resistance due to high-frequency skin and proximity effects. For a $1200 mm ^2$ copper XLPE cable operating in a network with a 12\% fifth-harmonic and 8\% seventh-harmonic content, the Vexten Suite processes thermal coefficients and determines that the cable nominal current rating must be reduced by 14.5\% to prevent premature thermal degradation of the polymeric insulation.

Resonances Mitigation and Power Factor Correction

Coupling between transmission line capacitance and shunt reactor inductance can create parallel resonance conditions at frequencies close to characteristic network harmonics (typically 5th, 7th, or 11th harmonics). The Vexten Suite executes a Frequency Scan Analysis to identify Thévenin impedances viewed from the reactor terminals.

The total harmonic impedance as a function of angular frequency \omega is modeled as:

Z(\omega) = \frac{j \omega L \cdot \frac{1}{j \omega C_{ line }}}{j \omega L + \frac{1}{j \omega C_{ line }}} + R_{ system }

If the frequency scan detects an impedance peak coinciding with the fifth harmonic ( $250 Hz$ in $50 Hz$ systems), the Vexten Suite software automatically sizes the incorporation of passive-damped filters or modifies reactor reactance via adjustment taps, shifting the resonant frequency out of the harmonic excitation spectrum and ensuring operational stability of the EHV transmission system against severe electromagnetic transients.