Key parameter
Degree of shunt compensation
The degree of compensation describes how much of the line capacitance is offset by the shunt reactor. It is normally expressed as a percentage of the line positive-sequence susceptance.
If the compensation is low, the line remains mainly capacitive. If the compensation is high, the reactor can dominate the behaviour of the open-phase or out-of-service circuit.
A compensation level of 100% means that the shunt reactor theoretically cancels the full positive-sequence capacitive effect of the line. This may sound ideal from a voltage-control point of view, but it can create resonance risk in some operating conditions.
For open-phase resonance, a useful screening idea is based on the ratio between zero-sequence capacitance and positive-sequence capacitance. If the compensation level becomes higher than this ratio, the open-phase circuit may become inductive and can resonate with the surrounding capacitance.
However, this should not be treated as one universal trip point. The critical compensation level depends on whether one phase or two phases are open, the reactor connection, the line geometry, the degree of transposition, and the zero-sequence behaviour of the circuit. The \(C_0 / C^+\) ratio is therefore a useful screening anchor, not a single fixed answer for all open-phase cases.
\[ \frac{C_0}{C^+} \]
- \(C_0\)
- zero-sequence capacitance
- \(C^+\)
- positive-sequence capacitance
Generic worked illustration using capacitance ratio
For a typical EHV overhead line, assume the following capacitance values:
- \(C^+ = 11.59 \text{ nF/km}\)
- \(C_0 = 7.77 \text{ nF/km}\)
The ratio is therefore:
\[ \frac{C_0}{C^+} = \frac{7.77}{11.59} \approx 0.67 \]
This means the open-phase resonance risk may start to become important when the degree of shunt compensation approaches or exceeds approximately 67%.
This does not mean that 67% is always the exact resonant point. It is only a screening indication. One-open-phase and two-open-phase conditions can resonate at different compensation levels. In general, two open phases may produce a more severe condition and may resonate at a lower compensation level than one open phase.
The important engineering message is that the danger does not exist only at one exact compensation value. A wider compensation range can be near-resonant, and open-phase voltages may still exceed acceptable values even when perfect resonance is not reached.
Switching scenarios
Uneven open-phase conditions
One important resonance scenario occurs when one or two phases are open while the remaining phase or phases stay energised.
This may happen during single-phase tripping, single-phase auto-reclosing, breaker pole failure, a faulted phase with autoreclose delay, or incomplete switching.
In this condition, the open phase is not simply dead. It may still be coupled to the energised phases through capacitance. The circuit seen by the open phase can include inter-phase capacitance, phase-to-ground capacitance, and a grounded shunt reactor.
If these elements form a resonant path, high overvoltages can appear on the open phase.
The behaviour of the open phase can be understood through the equivalent phase-to-ground impedance. If the compensation level is below the critical range, the equivalent impedance is mainly capacitive. If it is close to the critical range, the equivalent impedance can become very high and the system may be near a borderline resonance condition. If it is above the critical range, the equivalent impedance can become inductive and may resonate with other capacitances in the circuit.
This condition is important in single-phase auto-reclosing, stuck-pole situations, maintenance switching, and any operation where one phase is temporarily separated from the rest of the system.
Open-phase conditions are not normal balanced operation
During open-phase conditions the network becomes unbalanced and sequence coupling becomes important. A line that looks acceptable in balanced load-flow can still produce high temporary overvoltages during single-pole switching or autoreclose.
Parallel-line risk
Three-phase disconnection in double-circuit lines
Another important resonance scenario occurs when one complete circuit is disconnected while a nearby parallel circuit remains energised.
This is common in double-circuit transmission corridors or multiple-circuit rights of way. Even if the disconnected circuit is open at both ends, it can still receive energy through circuit-to-circuit capacitance from the energised circuit.
If the disconnected circuit is shunt compensated, the line capacitance and shunt reactor can form a resonant circuit. This is sometimes called parallel line resonance.
The result is that a line that appears to be out of service can still experience significant voltage. In some cases, the voltage may become much higher than a normal induced voltage.
This is particularly important for maintenance planning, outage safety, grounding operations, protection design, and shunt-reactor switching philosophy.
What needs to be checked
For a shunt-compensated transmission circuit, two questions are essential.
The first question is: what degree of compensation causes resonance?
The second question is: what overvoltage appears in the open phase or out-of-service circuit?
Simplified equations can be used for initial screening. These usually rely on LC resonance conditions and Thevenin-equivalent models to estimate induced voltage. However, these methods often neglect losses, saturation, detailed switching transients, trapped charge, arrester behaviour, breaker restrikes, and network asymmetry.
Screening calculations are useful because they show where the resonance risk may exist. However, they may overpredict or underpredict the actual field voltage depending on the assumptions used. Even when measured or simulated voltages are lower than a simplified analytical prediction, they may still be high enough to damage surge arresters, reactors, instrument transformers, or circuit breakers.
For practical engineering, simplified calculations should therefore be treated as screening tools only. Time-domain EMT simulation is normally required when the system is close to a critical compensation range or when the expected overvoltages may affect insulation, surge arresters, reactors, or circuit breakers.
System factors
Design parameters that influence resonance
Influence of mixed overhead line and cable sections
Cables have much higher capacitance than overhead lines. A cable section can introduce approximately 20 to 30 times more capacitance than an equivalent overhead line section.
When cable is added to an overhead line, the resonance behaviour changes. Even a relatively short cable section can shift the resonance peak significantly.
In mixed overhead-line and cable circuits, resonance may move toward very high compensation levels, close to 100%. Cable screens may also reduce inter-phase coupling, which can lower induced voltages. However, the overall resonance risk may still remain important because the location of the resonance point changes.
This is important for modern transmission systems where overhead lines are sometimes combined with underground cable sections near substations, urban areas, environmentally sensitive areas, or cable sealing ends.
Effect of neutral reactors
Neutral reactors can be used to detune the system during unbalanced open-phase conditions.
They modify the zero-sequence path and can shift resonance away from normal operating compensation levels. This can reduce the probability that the system will resonate during single-phase tripping, stuck-pole operation, or open-phase conditions.
However, neutral reactor selection is a compromise. A stronger neutral reactor may improve resonance suppression, but it can also increase the insulation requirement at the reactor neutral point. A weaker neutral reactor may be cheaper and easier to insulate, but it may not provide enough detuning or may not support effective arc extinction after faults.
Therefore, the neutral reactor should not be selected only from a voltage-control viewpoint. It should also be checked against resonance, single-phase auto-reclosing performance, insulation requirements, and temporary overvoltage behaviour.
Effect of reactor core construction
The magnetic construction of a shunt reactor affects phase coupling and resonance behaviour.
Shell-type reactors or four-leg and five-leg core reactors normally have minimal zero-sequence coupling. In this case, the phases behave more independently.
Three-leg core reactors can have stronger magnetic coupling between phases. This can reduce the effective zero-sequence impedance and move resonance to lower compensation levels.
This means that two reactors with the same positive-sequence rating may not behave the same during unbalanced open-phase conditions. The reactor core construction and the zero-sequence reactance ratio must be considered in resonance studies.
Effect of tower design
Tower geometry affects capacitance values. Different tower configurations produce different relationships between phase-to-ground capacitance, inter-phase capacitance, and zero-sequence capacitance.
Flat configurations, vertical configurations, and inverted delta configurations can therefore produce different resonance margins.
Tower geometry can influence both the resonant compensation range and the temporary overvoltage magnitude. Compact designs may be more sensitive because the phase spacing and conductor arrangement strongly influence capacitance.
This means that resonance assessment should not rely only on line length and voltage level. The actual line geometry should be represented when possible.
Effect of line transposition
Line transposition affects phase symmetry.
In a fully transposed line, each phase sees the same average capacitance and impedance over the full line length. This gives more symmetrical resonance behaviour.
In an untransposed line, each phase can behave differently. Multiple resonance peaks may appear, especially under two open-phase conditions. The overvoltage may depend on which phase is open and where the open condition occurs.
For practical studies, this means that untransposed or partially transposed lines need phase-specific analysis. It may not be safe to study only one representative phase.
Effect of reactor saturation
Shunt reactor saturation can limit open-phase voltage, but it can also introduce nonlinear behaviour.
When the reactor saturates, the voltage may be capped near the saturation knee point. A lower knee point may limit overvoltage more strongly, while a higher knee point allows higher overvoltage before saturation becomes significant.
However, saturation can also lead to ferroresonance. After a switching transient, the system may become trapped in a nonlinear oscillation state. In that case, the voltage may not simply decay to the expected linear steady-state value.
Therefore, saturation should not be treated only as a protective limiting effect. It must also be considered as a possible source of nonlinear resonance behaviour.
Temporary overvoltage behaviour
Temporary overvoltages during switching can exceed the steady-state resonance prediction.
After switching, the system may contain trapped charge, DC offset, and transient oscillations. These transient effects can produce short-duration overvoltages before the system reaches a steady condition.
The magnitude and duration of the TOV depend on the switching instant, line geometry, transposition, reactor saturation, damping, neutral reactor design, and protection clearing time.
In some cases, a neutral reactor can significantly reduce the oscillation and allow the voltage to decay quickly. Without adequate detuning or damping, temporary overvoltages can become severe.
The duration of the overvoltage is as important as the magnitude. Surge arresters, shunt reactors, voltage transformers, and circuit breakers may tolerate short transients but may not tolerate sustained temporary overvoltages.
Sensitivity of resonance to design parameters
The resonance behaviour of a shunt-compensated line is sensitive to several design parameters.
- The compensation percentage is the main driver of resonance risk. The closer the compensation is to a critical resonance value, the higher the risk.
- Line asymmetry and lack of transposition can strongly affect phase-specific resonance behaviour.
- Reactor saturation can strongly influence both the voltage limit and the possibility of ferroresonance.
- The reactor zero-sequence to positive-sequence reactance ratio is important because it affects the location of resonance.
- Neutral reactors and neutral resistors are important because they can detune or damp the resonant circuit.
- Cable sections are also important because they add significant capacitance and can shift resonance toward higher compensation levels.
- Circuit-breaker grading capacitors may be too small to significantly affect some shunt-compensated line-resonance cases. However, open-breaker capacitance and grading capacitors can still be important in other transient, open-breaker, CVT, grading-capacitor, and ferroresonance scenarios. This links back to the parasitic-capacitance concept introduced in Part One: small capacitances may be negligible in one study but decisive in another, depending on the resonance mechanism.
Why EMT simulation is needed
Analytical methods are useful for initial screening, but they cannot fully represent real switching behaviour.
EMT simulation is needed because practical resonance studies may involve circuit asymmetry, line transposition, nonlinear saturation, realistic switching sequences, trapped charge, breaker pole discrepancy, surge arrester operation, breaker restrikes, and complex network topology.
Time-domain EMT studies allow engineers to calculate temporary overvoltages, observe oscillation build-up, check whether damping is sufficient, and assess whether the system may enter a ferroresonant state.
This is also why hand calculations and EMT simulations should be used together rather than treated as competing methods. Hand calculations identify the dangerous compensation ranges. EMT simulation then checks the actual voltage magnitude, duration, waveform shape, arrester duty, and sensitivity to switching conditions.
For resonance studies in shunt-compensated lines, EMT simulation is especially important when the compensation level is close to a critical range or when the system includes mixed overhead-line and cable sections, neutral reactors, reactor saturation, or multi-circuit coupling.
Multiple-circuit corridors
Line resonance in multiple-circuit rights of way
When a shunt-compensated line is fully out of service and unearthed, it can still form a resonant circuit through capacitive coupling with a nearby energised line.
In a fully transposed double-circuit line, several resonance peaks can exist. These peaks are mainly controlled by the zero-sequence and positive-sequence capacitance ratio of the out-of-service line and the zero-sequence and positive-sequence reactance ratio of the shunt reactor.
Some resonant peaks can appear under normal no-fault conditions. Others require faults on either the energised or out-of-service circuit.
A useful principle is that the energised circuit mainly affects the amplitude of the induced overvoltage, while the parameters of the out-of-service circuit mainly determine the location of the resonance peaks.
In practical double-circuit systems, more than one resonance peak may exist. One peak may occur at a lower compensation level and produce a moderate voltage rise, while another may occur closer to 100% compensation and produce a more severe voltage rise. The exact values depend on the line capacitance ratios, reactor zero-sequence behaviour, transposition, and fault condition.
Resonant peaks in double-circuit lines
In transposed double-circuit systems, the resonance peaks can be associated with zero-sequence or positive-sequence behaviour.
Zero-sequence resonance can be excited by ground faults and capacitive coupling. Single-line-to-ground faults on the energised circuit can be especially severe because they produce significant zero-sequence voltage, which can couple into the out-of-service circuit.
Faults on the energised circuit usually amplify the induced voltage on the out-of-service line, but they do not necessarily shift the resonance location. The resonance location is mainly governed by the out-of-service circuit parameters.
Faults on the out-of-service circuit can be different. They may change the effective circuit configuration and can therefore shift the resonance condition as well as increase the voltage magnitude.
A single-line-to-ground fault or double-line-to-ground fault on the out-of-service circuit may be sustained by capacitive coupling from the energised circuit. This can be particularly dangerous if the out-of-service circuit remains unearthed and the fault is not cleared.
Positive-sequence resonance can occur near 100% compensation. This can be dangerous because it may be excited under a wider range of operating conditions. For this reason, compensation degrees close to 100% should be treated with caution in multiple-circuit corridors.
Effect of circuit parameters in double-circuit resonance
Two ratios are especially important:
\[ \frac{C_0}{C^+} \qquad \text{and} \qquad \frac{X_0}{X^+} \]
The capacitance ratio depends on the line geometry and whether the circuit is overhead line, cable, or a hybrid line. Overhead lines often have lower zero-sequence to positive-sequence capacitance ratios, while cables may have values closer to one.
The reactor reactance ratio depends on reactor construction and neutral treatment. A three-leg reactor core may produce lower zero-sequence reactance, while shell-type or air-core reactors may behave differently.
Neutral reactors can increase the effective zero-sequence reactance. This can shift zero-sequence resonance points to higher compensation levels and help move the system away from practical operating points.
Low capacitance ratios or low reactance ratios tend to separate resonance peaks, making them easier to avoid. High capacitance ratios, especially in cable or mixed overhead-line/cable systems, can move resonance peaks closer to 100% compensation and increase the risk zone.
Effect of neutral reactors in double-circuit lines
Adding a neutral reactor can shift zero-sequence resonance points toward higher compensation levels.
This does not necessarily reduce the amplitude of every induced voltage, but it can move the resonance away from the normal compensation range. In this way, the probability of resonance during practical operating conditions can be reduced.
The positive-sequence resonance near 100% compensation is not strongly affected by neutral reactor changes, because the neutral reactor mainly influences the zero-sequence path.
Therefore, neutral reactors are useful but not universal mitigation devices. Their effect depends on the resonance mode being targeted.
Equipment and safety
Practical consequences of line resonance
Line resonance can create overvoltages and overcurrents that stress several items of equipment.
Shunt reactors may experience saturation, increased core heating, magnetic noise, vibration, and harmonic distortion. In some reactor designs, zero-sequence currents may cause overheating even for relatively short line sections.
Surge arresters may be exposed to temporary overvoltages beyond their energy or thermal capability. This can lead to arrester failure, especially if the overvoltage is sustained or repeated.
Voltage transformers may experience insulation stress, nonlinear behaviour, or ferroresonance-related overvoltage. The term VT is used here as the main term. In some older or IEEE-style terminology, the same device may also be called a PT, or potential transformer.
Circuit breakers may experience high longitudinal stress during open-contact conditions. In severe cases, this may increase the risk of internal flashover.
Single-phase auto-reclosing performance can also be compromised if high recovery voltage prevents arc extinction.
There are also maintenance hazards. High induced voltages on unearthed out-of-service lines can cause arcing during application of portable earths. Earthing switches may also be overstressed if they are not rated for the resulting current.
For this reason, resonance is not only an equipment design issue. It is also an operational safety and outage-planning issue.
