Power System Fundamentals · Training Guide

Resonance and Ferroresonance in Power Systems Final Part — Conclusions and Practical Lessons

Reading time ≈ 25 min

Purpose of this guide

Objective

This study guide has introduced the main concepts, tools, and modelling strategies used to identify, analyse, and mitigate power-frequency resonance and ferroresonance in power networks.

The main objective is practical: to understand when resonance or ferroresonance may create temporary overvoltages that can stress equipment, damage insulation, overload surge arresters, affect protection operation, or create safety risks during switching and maintenance.

Resonance and ferroresonance are different phenomena, but both are strongly influenced by network topology, capacitance, inductance, damping, switching condition, and equipment nonlinearity.

What this page brings together

the core conclusions on linear resonance;
the core conclusions on ferroresonance;
why VTs and power transformers behave differently;
which modelling method suits which problem;
the role of sensitivity studies and verification;
the practical mitigation strategy that ties the series together;
the questions an engineer must answer before closing a study.

The single message of this series

Resonance and ferroresonance should be assessed as topology-dependent, switching-dependent and damping-dependent phenomena — never judged from a single component value in isolation.

Table 1 — Key questions to answer before closing a resonance or ferroresonance study.

Engineering Question	Why It Matters
What capacitance and inductance form the resonant loop?	Defines whether a resonant condition can exist at all.
At what frequency does resonance occur relative to system frequency?	Determines how close the circuit is to a dangerous condition.
Is a nonlinear magnetic element present?	Distinguishes linear resonance from possible ferroresonance.
What switching events or faults could trigger it?	Most events are triggered by a specific operation, not steady state.
How much damping is available?	Decides whether an oscillation decays or sustains.
Has the worst credible case been simulated?	Screening finds risk; EMT proves severity and duration.
Is the proposed mitigation valid for that exact case?	A measure is only valid for the topology and switching studied.

Linear resonance

Resonance: main conclusions

Power-system resonance starts from a simple concept: energy exchange between inductance and capacitance.

In the ideal case, this can be explained using simple LC circuits. In real power systems, the behaviour is more complex because the network includes practical combinations of series capacitance, shunt capacitance, inductance, resistance, transformers, reactors, cables, overhead lines, circuit breakers, and stray capacitances.

A useful practical topology is the combination of series capacitance, inductance, and parallel capacitance. This type of arrangement is important because it explains how a circuit may shift between series-type and parallel-type resonance depending on source impedance and circuit configuration.

A small series capacitance can make the upstream voltage source and capacitance behave more like a current source. This can promote parallel resonance with the rest of the circuit. This is why the same network may behave differently depending on how capacitance and inductance are connected.

In shunt-compensated transmission lines, resonance is especially important. Long EHV lines have significant capacitance, and shunt reactors are installed to absorb reactive power and control voltage. However, these same reactors can create resonance risks under open-phase, out-of-service, or coupled-circuit conditions.

One or two open-phase conditions are particularly important. These may occur during single-phase auto-reclosing, stuck breaker poles, breaker failure, protection operation, or incomplete switching. In such cases, an open phase may still be capacitively coupled to energised phases and may resonate with grounded shunt reactors.

Multiple-circuit rights of way introduce additional resonance possibilities. An unearthed out-of-service circuit can still be energised through capacitive coupling from a neighbouring live circuit. If the out-of-service circuit is shunt compensated, resonance may occur even though the circuit appears disconnected from the source.

Resonance can occur in both faulted and non-faulted conditions. Faults on the energised circuit can increase the amplitude of induced overvoltages on the out-of-service circuit. Faults on the out-of-service circuit may change the effective resonance condition and can sometimes sustain abnormal voltages through capacitive coupling.

The most important point is that a line or phase should not be assumed harmless simply because it is open or out of service. If it is unearthed and capacitively coupled to an energised circuit, it may still develop dangerous voltages.

Study methodology

Analytical and EMT assessment of resonance

Analytical equations are useful for initial screening. They can help estimate critical compensation levels, identify possible resonance ranges, and calculate approximate temporary overvoltages. These methods are valuable because they explain the mechanism and help identify which system parameters are important.

However, analytical methods usually simplify the network. They may neglect losses, trapped charge, nonlinear saturation, circuit-breaker restrikes, surge arrester operation, detailed phase asymmetry, and realistic switching sequences.

For this reason, analytical calculations should not normally be used alone for final design decisions.

EMT simulation is required when the system is close to a critical compensation range, when overvoltages may stress equipment, or when the study involves switching transients, trapped charge, saturation, pole discrepancy, arrester duty, or multi-circuit coupling.

The most reliable engineering approach is to use analytical calculations for screening and EMT simulation for confirmation. The analytical model identifies where the risk may exist. The EMT model checks the actual voltage magnitude, waveform, duration, equipment stress, and sensitivity to switching conditions.

Field experience has shown that analytical methods may overpredict or underpredict actual voltages depending on assumptions. Even when measured or EMT-simulated voltages are lower than simplified analytical predictions, they may still be high enough to damage surge arresters, reactors, VTs, circuit breakers, or other equipment.

Nonlinear resonance

Ferroresonance: main conclusions

Ferroresonance is a nonlinear form of resonance. It occurs when capacitance interacts with a nonlinear inductance, usually a saturable magnetic core.

The fundamental ferroresonant circuit contains a nonlinear inductance and capacitance. The nonlinear inductance may be a VT, power transformer, or reactor. The capacitance may come from lines, cables, busbars, GIS, transformer bushings, grading capacitors, open circuit breakers, or stray capacitance.

Ferroresonance is more difficult to predict than linear resonance because the inductance changes when the magnetic core saturates. This can create multiple possible operating states for the same network condition.

A circuit may remain in normal operation, move into fundamental-frequency ferroresonance, enter a sub-harmonic mode, or become chaotic or quasi-periodic. The final state may depend on initial conditions, switching instant, trapped charge, residual flux, damping, and system topology.

Typical ferroresonance topologies include VTs energised through circuit-breaker grading capacitors, VTs connected in unearthed or low zero-sequence capacitance systems, transformer-terminated lines coupled to live parallel circuits, single-phase or two-phase transformer energisation, weak-source transformer energisation, and transformers connected to series-compensated lines.

The most dangerous cases often occur when equipment appears isolated, lightly energised, or only partially connected, but is still supplied through capacitance.

VT-specific mechanisms

Voltage transformer ferroresonance

VT ferroresonance is often associated with weak capacitive energisation paths.

Open circuit breakers with grading capacitors are a key example. Although the breaker is open, grading capacitors can still pass a small current. If an electromagnetic VT is connected on the isolated side, the VT may be energised through the grading capacitor and may form a nonlinear LC circuit.

Busbar VTs, line VTs, and VTs in double-circuit arrangements may all be exposed to this mechanism. In double-circuit configurations, capacitive coupling from a live circuit can energise equipment connected to the out-of-service circuit.

VTs in unearthed or arc-suppressed systems are also vulnerable because the neutral point is not strongly fixed to earth. Neutral displacement, switching transients, or earth faults may drive the VT core into saturation.

CVTs normally include ferroresonance-suppression circuits, but they should not be considered immune to all resonance concerns. CVTs have their own internal ferroresonance mechanism involving the capacitor divider and intermediate transformer. The suppression circuit is installed specifically to damp that behaviour.

Optical or electronic VTs remove the ferroresonance mechanism associated with the VT's own magnetic core. However, they do not make the whole bay or busbar immune to ferroresonance from other magnetic devices.

Power transformer risk

Power transformer ferroresonance

Power transformer ferroresonance can occur when a transformer is connected to capacitance under low-loss or weak-source conditions.

Transformer-terminated transmission lines are important examples. If one circuit is out of service while a parallel circuit remains energised, capacitive coupling may supply energy to the transformer and connected line capacitance. The transformer magnetising branch provides the nonlinear inductance.

Lightly loaded transformer energisation through a long line or cable from a weak source is another risk condition. The source may not provide strong damping, and the line or cable capacitance may interact with the transformer magnetising inductance.

Single-phase or two-phase transformer energisation is also important, especially in distribution networks where fuses or single-pole switching may leave only part of the transformer energised. This can create abnormal resonance paths and significant overvoltage.

Transformers connected to series-compensated lines can experience ferroresonance during load rejection or abnormal breaker operation. The series capacitor and transformer nonlinear inductance may form a dangerous resonant condition.

The possible consequences include transformer overfluxing, insulation stress, overheating, harmonic distortion, protection malfunction, audible noise, accelerated ageing, and equipment damage.

Study techniques

Modelling methods

Ferroresonance can be studied using analytical methods and digital simulation.

Analytical methods include harmonic balance, Galerkin methods, incremental describing functions, continuation methods, and energy-transfer approaches. These methods are useful for understanding mechanisms and periodic solutions, but they are limited for non-periodic, sub-harmonic, quasi-periodic, or chaotic behaviour.

EMT simulation is normally required for practical engineering studies because it can represent nonlinear magnetic saturation, realistic switching sequences, trapped charge, residual flux, grading capacitors, line and cable capacitance, damping, arrester operation, and equipment connections.

Because ferroresonance is highly sensitive to initial conditions, multiple simulation runs are often required. A single switching case may not be enough. Different point-on-wave switching instants, residual flux levels, trapped charge conditions, and damping assumptions may lead to different outcomes.

Nonlinear dynamics tools can support deeper understanding. Phase-space plots, Poincaré sections, and bifurcation diagrams can show how the system changes from normal operation to periodic, sub-harmonic, quasi-periodic, or chaotic behaviour.

For junior engineers, the practical message is that ferroresonance cannot be fully understood from one steady-state calculation. Time-domain behaviour and sensitivity to initial conditions are essential.

Model construction

Modelling practices

The model should be built around the ferroresonant loop. It is not always necessary to model the whole power system in detail. A Thevenin equivalent may be sufficient for the external network if it correctly represents source strength and damping. However, the local circuit containing capacitance, nonlinear inductance, switching devices, and damping must be represented carefully.

Overhead lines should be modelled with accurate capacitance and phase transposition information. Geometry, conductor spacing, tower structure, height above ground, and adjacent circuits can all influence coupling.

Frequency-dependent line models may be needed if the switching transient itself is important. For power-frequency ferroresonance, simpler distributed-parameter models may be sufficient in many cases, provided the relevant capacitance and coupling are represented correctly.

Line-model passivity should be considered. A non-passive model may artificially create energy and produce false resonance behaviour. The model should not behave as an artificial energy source over the frequency range relevant to the study.

Transformer modelling is critical. The magnetic saturation curve, core losses, hysteresis representation, winding connection, phase coupling, and damping must be selected carefully. Where detailed data is unavailable, simplified nonlinear inductors and calibrated damping resistors may be used, but the limitations should be understood.

Shunt reactor modelling should include the appropriate core construction and zero-sequence behaviour. Three-leg core reactors can behave differently from shell-type, four-leg, five-leg, air-core, or single-phase reactors because magnetic coupling affects resonance behaviour.

Substation equipment should be included when its capacitance is part of the ferroresonant loop. Relevant capacitances may include busbars, disconnectors, VTs, surge arresters, circuit breakers, transformer bushings, grading capacitors, GIS components, and shunt capacitor banks. Current transformers and line traps are often not important unless they are directly involved in the ferroresonant loop.

Study rigour

Sensitivity and study verification

Ferroresonance is highly sensitive to parameters.

The magnetising curve can strongly change the result. A VT or transformer with a low saturation point may enter ferroresonance easily. A higher saturation point may reduce the probability of ferroresonance but may allow higher overvoltage before saturation limits the response.

The way the magnetising curve is represented also matters. Piecewise-linear, polynomial, hysteretic, and simplified nonlinear models can produce different results. Therefore, saturation modelling should not be treated as a minor detail.

Circuit-breaker closing and opening times can influence ferroresonance. Asynchronous pole operation may either initiate or suppress ferroresonance depending on switching instant and residual flux.

Damping is one of the most important parameters. It controls whether the oscillation decays or continues. Damping may come from winding resistance, core losses, connected load, VT burden, damping resistors, neutral resistors, tertiary loads, or reactor losses.

Because the result depends on many uncertain parameters, sensitivity studies are essential. A good ferroresonance study should vary the important parameters and check whether the mitigation remains effective across realistic operating conditions.

Prevention and response

Mitigation strategies

Mitigation can be passive or active.

Passive mitigation aims to avoid risky configurations. This may include better substation layout, avoiding VTs trapped between open switching devices, avoiding unnecessary grading capacitors, choosing suitable VT types, using appropriate reactor and transformer designs, avoiding critical line lengths, and applying interlocking or safe switching sequences.

Active mitigation introduces damping or changes the circuit condition after a risk appears. This may include open-delta resistors, wye-connected resistors, saturable reactor and resistor combinations, grounding switches, neutral resistors, loaded delta tertiary windings, series resistors, reactor switching, or special protection schemes.

Open-delta resistors are commonly used for VT ferroresonance because they provide damping under zero-sequence conditions while having little effect during balanced normal operation.

Wye-connected resistors can provide direct phase damping but may introduce continuous burden and affect metering or protection accuracy.

Saturable reactor and resistor combinations can provide selective damping. They remain high impedance during normal operation and insert damping when overvoltage causes saturation.

In unearthed or arc-suppressed systems, adding parallel capacitance may detune one fundamental-frequency ferroresonance mode. However, this should never be treated as a general rule that more capacitance is safer. Added capacitance may suppress one mode while moving the system closer to another, including a sub-harmonic mode. Study verification is required.

For power transformers, mitigation may include limiting line length, avoiding long unearthed transformer-terminated lines, reducing inter-circuit coupling, loading delta tertiary windings, avoiding single-phase switching, installing circuit breakers close to transformer terminals, or disconnecting the energy source.

Earth switches and special protection schemes can be effective but must be checked against equipment duty, fault current rating, insulation coordination, system security, and operational requirements.

Restoration

Black-start and weak-network energisation

Black-start conditions deserve special attention because the source is weak and the network may be lightly loaded.

Weak sources provide less damping and may allow nonlinear oscillations to continue. Large transformers, long lines, shunt reactors, and low load can create difficult energisation conditions.

Soft energisation, also called controlled voltage build-up, can reduce risk. In this method, the generator is brought up to speed without excitation. The breaker and excitation are then closed, and the voltage is gradually increased using the automatic voltage regulator.

This allows the transformer core to be fluxed progressively instead of being exposed to a sudden voltage step. It reduces the risk of saturation, inrush current, and temporary harmonic overvoltage.

For black-start studies, the total MVA being energised should be limited, AVR voltage set-points should be controlled carefully, and the voltage ramp should be slow enough to avoid abrupt saturation.

Observed in practice

Practical lessons from field experience

Field experience confirms that resonance and ferroresonance are not only theoretical phenomena.

Recovery voltage resonance during single-phase autoreclosing can be strongly affected by reactor size and neutral earthing arrangement. A reactor change that appears acceptable for voltage control can move the system into or out of a narrow resonance zone.
Network expansion can unintentionally create resonance. Changing line length, adding a substation, modifying compensation, or altering switching arrangements can shift the system closer to a resonant point.
Out-of-service lines in double-circuit corridors can develop significant voltage due to coupling from live circuits. EMT simulations and field measurements have confirmed that de-energised or open circuits may still show high voltages if they are unearthed and capacitively coupled.
Series resonance in ungrounded HV systems can produce very high temporary overvoltages when an independent power producer or generator remains connected to an isolated faulted network. Direct transfer trip and correct breaker sequencing may be required to prevent the generator from sustaining the resonance.
Transformer-terminated line ferroresonance can occur in real multi-circuit systems. Both sub-harmonic and fundamental modes may appear, and the mode that occurs can depend on point-on-wave switching and residual flux. Detection schemes may not capture every mode.
Disconnector operation during ferroresonance can be hazardous. Some disconnectors may quench the phenomenon, but arcing and contact damage are possible. Equipment capability must be verified before relying on this as a mitigation method.
VT ferroresonance through grading capacitors has been observed during commissioning. Damping resistors may not always provide consistent suppression unless properly designed and verified. Relocation of VTs, interlocking, air-core reactors, or alternative VT types may be more effective in some arrangements.
Series-compensated lines can create severe ferroresonance during load rejection or stuck-breaker conditions. Bypassing series capacitors can be an effective mitigation, while surge arresters alone may not be sufficient for sustained or high-energy events.
Black-start energisation can initiate pseudo-periodic ferroresonance even when voltage ramping is used. Safe energisation may require limiting the total transformer MVA, maintaining a low AVR set-point, and using a slow controlled voltage ramp.

Overall lesson

Final overall takeaway

Closing statement

Resonance is mainly topology-driven and condition-sensitive. It depends on how inductance and capacitance are connected, how the system is switched, and whether an apparently disconnected circuit is still capacitively coupled to an energised source.
Ferroresonance is local, nonlinear, and highly sensitive to initial conditions. It depends on the interaction between capacitance, nonlinear magnetic inductance, low damping, and an energy source.
EMT simulation is indispensable for accurate assessment because it can represent switching transients, nonlinear saturation, trapped charge, residual flux, arrester behaviour, damping, and waveform evolution.
Practical mitigation normally requires a combination of passive design and active intervention. Good layout, correct equipment selection, safe switching sequences, damping resistors, neutral components, earth switches, special protection schemes, and controlled energisation may all be required.
The most important engineering lesson is this: do not judge resonance risk only from the single-line diagram. Equipment that appears open, isolated, lightly loaded, or out of service may still be energised through capacitance. If nonlinear magnetic equipment and low damping are also present, resonance or ferroresonance can produce dangerous voltages.

The closing principle

A frequency scan finds the risk; an EMT simulation proves the consequence. One identifies where resonance can occur; the other shows how severe and how long the overvoltage will be — and both are needed before a result can be trusted.

Do not close the study until

A resonance or ferroresonance assessment is not complete until:

the resonant loop and its capacitance and inductance sources are identified;
the resonant frequency is compared against system frequency;
any nonlinear magnetic element has been represented with its true saturation curve;
the worst credible switching event, fault, or open-phase condition has been included;
trapped charge, residual flux, and available damping have been accounted for;
the worst case has been confirmed by EMT simulation, not by screening alone;
any mitigation has been verified for that exact topology and switching condition;
the study has been re-checked against the equipment and network actually installed.

Part Five Reading now

Conclusions and Practical Lessons

Series synthesis — resonance and ferroresonance assessment methodology, EMT modelling practices, mitigation decision frameworks, and practical field lessons drawn across all five parts of this series.

Series progress 5 of 5

Part Four

Mitigation of Ferroresonance

Practical mitigation methods for VTs and power transformers — passive and active damping approaches, open-delta resistors, wye resistors, saturable reactors, and black-start energisation strategies.