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8/12/2019 Combined R&Fr V7
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RESONANCE ANDFERRORESONANCE
IN POWER
NETWORKWG C4.307
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Members
Type Members Names with country of origin (two letter code)
X. XXX, Convenor(XX), A. XXXX, Secretary(XX),
XXXXX (XX) B. XXXXX (XX)
Copyright 2011
Ownership of a CIGRE publication, whether in paper form or on electronic support only infersright of use for personal purposes. Are prohibited, except i f explicitly agreed by CIGRE, total or
partial reproduction of the publication for use other than personal and transfer to a third party;hence circulation on any intranet or other company network is forbidden.
Disclaimer notice
CIGRE gives no warranty or assurance about the contents of this publication, nor does itaccept any responsibility, as to the accuracy or exhaustiveness of the information. All impliedwarranties and conditions are excluded to the maximum extent permitted by law.
ISBN : (To be completed by CIGRE)
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ISBN : (To be completed by CIGRE)
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3.2.1 Uneven Phase Operation in Sigle-Circuit or Multi-Circuit
Corridors ........................................................................................ 313.2.2 Three-Phase switching in Multi-Circuit Corridors ......................... 34
CHAPTER 4 RESONANCE IN SHUNT COMPENSATED TRANSMISSIONCIRCUITS ............................................................................ 39
4.1 Background..................................................................................... 39
4.2 Line Resonance in Uneven Open-Phase Conditions ......................... 40
4.2.1 Physical description ...................................................................... 40
4.2.2 Steady State Approximate Analytical Solution............................. 42
4.2.3 Mixed Overhead Line and Cable Circuits ....................................... 46
4.2.4 Effect of Neutral Reactors ............................................................. 47
4.2.5 Effect of Reactor Core Construction .............................................. 51
4.3 Detailed Analysis of Line Resonance in Uneven Open-Phase
conditions using Time-Domain Simulation ..................................... 52
4.3.1 Steady State Analysis .................................................................... 52
4.3.2 TOV Analysis ................................................................................ 55
4.3.3 Summary of Parameters Affecting Line Resonance in Open-
Phase Conditions ............................................................................ 60
4.4 Line Resonance in Multiple-Circuit Corridors .................................. 62
4.4.1 Background .................................................................................. 62
4.4.2 Physical description ...................................................................... 624.4.3 Approximate Analytical Solution ................................................... 62
4.4.4 Case Study .................................................................................... 64
4.4.5 Summary of resonance issues associated with parallel shunt-
compensated circuits ...................................................................... 72
4.5 Practical Consequences of Line Resonance ..................................... 72
4.6 Mitigation Options .......................................................................... 72
CHAPTER 5 NETWORK CONFIGURATIONS LEADING TO FERRORESONANCE745.1 Ferroresonance in voltage transformers (VT)................................... 74
5.1.1 VT and Circuit Breaker Grading Capacitors ................................... 75
5.1.2 VT and Double Circuit Configuration ............................................ 76
5.1.3 VT in Ungrounded Neutral Systems with Low Zero-Sequence
Capacitance .................................................................................... 76
5.2 Ferroresonance in power transformers ........................................... 79
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5.2.1 Transformer Terminated Transmission Line in Multi-Circuit
Right of Way ................................................................................... 795.2.2 Lightly Loaded Transformer Energized via Cable or Long Line
from a Low Short-Circuit Capacity Network .................................... 80
5.2.3 Transformer energized in one or two phases ................................ 81
5.2.4 Transformer connected to a series compensated line. .................. 83
CHAPTER 6 MODELLING AND STUDYING ................................................ 846.1 Analytical Solution Methods ............................................................ 84
6.2 Digital Simulation Methods ............................................................. 85
6.3 Modelling of Network Components ................................................. 86
6.3.1 Extent of the Network Model ........................................................ 87
6.3.2 Overhead Line Model .................................................................... 87
6.3.3 Transformers ................................................................................ 87
6.3.4 Shunt Reactors.............................................................................. 88
6.3.5 Other Substation Equipment ......................................................... 88
6.4 Sensitivity to Parameters ................................................................. 89
6.4.1 Effect of Magnetising Curve .......................................................... 89
6.4.2 Influence of Circuit Breaker Closing Times .................................... 90
6.4.3 Influence of the Damping in the Circuit ........................................ 91
CHAPTER 7 MITIGATION OF FERRORESONANCE ...................................... 927.1 Mitigation of VT Ferroresonance ..................................................... 92
7.1.1 Secondary Open Delta Resistor ..................................................... 92
7.1.2 Secondary Wye Resistor.............................................................. 93
7.1.3 Secondary Wye Resistor in Series with a Saturable Reactor......... 95
7.1.4 Other Mitigation Options .............................................................. 95
7.1.5 Mitigation of VT Ferroresonance in Ungrounded Neutral Systems . 96
7.2 Mitigation of Power Transformer Ferroresonance ............................ 99
ANNEX A RESONANCE EXAMPLES ...................................................... 108A. 1 Resonance Associated with Single-phase Autoreclose Switching of
275 kV Shunt Reactor ................................................................... 108
A. 2 Line Resonance experienced in 275 kV Double Circuit as a result of
System Expansion ......................................................................... 112
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EXECUTIVE SUMMARYLorem ipsum dolor sit amet, consectetuer adipiscing elit. Maecenas porttitor congue massa. Fusce posuere,
magna sed pulvinar ultricies, purus lectus malesuada libero, sit amet commodo magna eros quis urna.
Explain the technical reasons for conducting the study (system/component failures,
industrial/manufacturer needs for technical improvement, inadequateness of present standards, etc...).
Include reference, if any, to previous CIGRE work on the subject. A limited number of technical or numerical
data may be included, only if strictly necessary.
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CHAPTER 1 INTRODUCTION TO RESONANCE ANDFERRORESONANCE
Ambitious targets for CO2 emissions reductions and integration of renewable generation in power systems are
driving the need for significant reinforcement of existing transmission grids worldwide, in particular new high
capacity corridors are required to transfer large amounts of power from remote areas with high natural resources
(i.e. wind, wave, tidal, etc) to the demand centres. At the same time, increasing public opposition to the
construction of new overhead transmission infrastructure is driving the need for new pylon designs that minimise
visual impact resulting, in many cases, in smaller structures with reduced clearances. Where possible, existing
corridors are being upgraded and operated at higher voltage levels with minimum modifications to the towers, thus
increasing its transfer capability. Furthermore, the use of underground cable circuits at HV and EHV transmission
levels is steadily increasing, not only in congested urban areas, but also in remote rural locations in order to reduce
the environmental impact of new circuits in specific designated zones and to accelerate the connections of wind
farms to the transmission grids. These fundamental changes in the design and technology used for new
transmission circuits are resulting in an increased system capacitance that is shifting the natural resonant
frequencies closer to the power frequency (50/60 Hz).
Generally, resonance occurs in electric circuits that are able to periodically transform energy from an electric field
into a magnetic field and vice versa. It is the characteristic of such a circuit that if some single energy is delivered
into it (either of electric or magnetic type), the circuit then starts to oscillate with the so called free oscillations.
Generally, electric circuits are more complex, consisting of many capacitances and inductances that can exchange
energy between them via various paths and their free oscillations are composed from several frequencies.
It is important to note that resonance referred to in this document applies to fundamental frequency resonance only
and that if harmonics are present, either due to saturation of transformers or reactors, the resonance conditions
may change significantly.
Carlsson originally suggested in 1974[62] that the installation of shunt reactors could increase recovery voltage on
a disconnected phase during single-pole reclosing and that resonance could occur at high degrees of shunt-
compensation on transmission lines ( 90%). However, this publication did not provide any insight into thephenomenon and it was mostly concerned with the extinction of secondary arc current. Reference [65] (1982)
presented field measurements and simulations of open-phase over-voltages in a 750 kV transmission line between
Hungary and USSR. The measured values (1.3 pu) were lower than those predicted by simulation (2.5 pu) and it
was concluded that the discrepancy was due to the limitation effect of corona losses.
The first publication providing a physical description of resonance on shunt-compensated transmission lines during
open phase conditions was [66], in 1984. where a detailed study of over-voltages induced during open-phase
condition in HV lines equipped with shunt-reactors, as may occur in conjunction with single phase reclosure and
stuck circuit breaker poles was presented. This work emphasised, again, the over-voltage limiting effect provided
by corona losses.
Reference[67] provides a very good review of aspects associated with single phase tripping and reclosing, namely
transient stability, extinction of secondary arc current, resonance, protection and operational issues. Reference[68]
deals with non-optimum phase and neutral reactor schemes for single-phase reclosing in single and double circuit
transmission lines. The effect of incomplete phase transposition is also considered. Studies and considerations
taken for a 500kV AC circuit in the south western USA in the early nineties examining the application of neutral
reactors to reduce over-voltages due to resonance is covered in Error! Reference source not found.and Error!Reference source not found.. Simulation studies carried out for a 500kV shunt compensated transmission line in
Vietnam where temporary over-voltages up to 1.74 pu following the single-phase opening of the circuit have been
identified is described in [73]. A review of the basic resonant circuit formed during one or two open-phase(s)
conditions can be found in Error! Reference source not found. where the impact of various circuit design
parameters are examined. The same publication presents a case study related to a system expansion with
incomplete line transposition.
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In its simplest terms ferroresonance can be described as a non-linear oscillation due to the interaction of an iron
core inductance with a capacitance. Ferroresonance is a harmful low frequency oscillation where a non-linear
reactance can be driven into saturation and oscillate with the circuit capacitance giving rise to severe overvoltages,with almost no damping when the amplitude is moderate, and in some circumstances, excessive overcurrents. If
enough energy provided by the source is coupled to compensate for the circuit losses, this oscillation can be
sustained indefinitely.
The phenomenon came to light in 1920 when it was first reported by P. Boucherot [1] to describe an oscillation
between a power transformer and a capacitance. Ferroresonance became a problem in the early part of the
century when small isolated systems were interconnected by long transmission lines [2][3],but at that time the
cause of the problem was not understood. In the 1940's and 1950's the phenomenon recurred as the electricity
supply industry expanded and longer overhead distribution systems were introduced into service. The terms
neutral instability [4] and voltage displacement [5] were also used in the 1940s referring to the same or very
similar phenomenon, although the term ferroresonance has prevailed. In 1966 it was discovered that, for cable
connected transformers, ferroresonance can occur even on circuits as short as 200 metres[6][7].Since that time
many studies and investigations have been carried out and a number of papers have been published on the
subject.
Ferroresonance has focussed the attention of numerous researchers over the years with the outcome of extensive
literature addressing the subject, proposing analysis methods and reporting cases experienced by various utilities.
However, despite the vast amount of research and technical documentation available, it still remains widely
unknown today and is somehow misunderstood by many power network utilities. It is especially feared by power
systems operators, as it seems to occur randomly, normally resulting in the catastrophic destruction of electrical
equipment and the consequent adverse effect on the reliability of power network. This general lack of awareness
means that ferroresonance is, by and large, overlooked at the planning and design stages or, at the other extreme,
held responsible for inexplicable equipment failures [8]. However, use of non linear tools enabled a better
understanding of the behaviour and these networks and the determination of the different solutions (harmonic,
pseudo-periodicand even chaotic) along with the importance of the magnetic flux as a crucial state variable, even if
some areas have to be investigated further, especially when transformers are highly non linear.
Sustained overvoltages seen under ferroresonance conditions could stress equipment such as transformers and
breakers, and would cause surge arresters to conduct over extended period of time exceeding their energy
dissipation capabilities. A catastrophic failure of a surge arrester for example could damage other key equipment in
a substation and could also cause injury to personnel if they happen to be around at the time. Therefore
ferroresonance primarily poses a health and safety hazard to the substation personnel due to the risk of explosion
in the work place. An example of such threat is reported in [9], where a 230 kV voltage transformer failed
catastrophically causing damage to equipment up to 33 meters away. Nobody was injured in this instance but the
experience illustrates the danger that site operators are exposed to.
Many examples of plant equipment destruction caused by ferroresonance have been documented in the literature.
A very interesting case is reported in [10] where 72 voltage transformers were destroyed in a 50 kV network in
Norway. An investigation revealed that all the damaged voltage transformers were from the same manufacturer
whereas voltage transformers from other two manufacturers which were also in service survived the incident. The
catastrophic destruction of a 230 kV voltage transformer in a cogeneration substation is reported in[11].The failure
of a 275 kV voltage transformer in UK is reported in[12].Other typical examples include the explosive failure of a
115 kV voltage transformer in Canada[13],the explosive failure of voltage transformers in France[8] and the total
destruction or partial damage of six 345 kV voltage transformers as reported by a USA utili ty[14].
From an operational point of view, ferroresonant oscillations represent a potential threat to power network plantintegrity. The large current pulses caused by transformer saturation may overheat the transformer primary winding
and might, eventually, cause insulation damage. The large voltage oscillations, temporary or sustained, can also
cause severe stresses on the insulation of all the equipment connected to the same circuit. Surge arresters are
normally the most vulnerable apparatus in substations due to their low TOV withstand capabilities[15].
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Ferroresonance can also have an adverse effect on the reliability of the power network. The forced outage of part
of a substation due to an equipment failure can cause severe overloading in other parts of the network that could
evolve into a cascade tripping[16] or result in extended outage of major power network assets.
From an economic perspective, ferroresonance could represent unaccounted costs to electric power utilities. The
cost of ferroresonance could be twofold: on the one hand, there is an explicit cost associated with the replacement
of damaged or destroyed electrical plant, and on the other hand, there are high or perhaps even severe costs
associated with a reduced network security and possible disconnection of some customers. Quantification of the
latter is not a straightforward task and could only be fully quantified if performed on an individual case basis.
Ferroresonant waveforms are highly distorted, with a large content of harmonics and sub-harmonics. This in turn
results in a degraded power quality and possible misoperation of some protection relays [17]. Transformer
overheating may also occur under Ferroresonant conditions due to excessive flux densities. Since the core is
saturated repeatedly, the magnetic flux finds its way into the tank and other metallic parts. This can cause charring
or bubbling of paint in the tank[18].
In general, it is possible to distinguish temporary overvoltages from ferroresonance; in the former, the amplitude
may be very high initially but decreases rapidly in most cases. As harmonics are involved, the fluxes circulating in
the iron core may lead to overheatings in the core, and especially affecting the insulation between laminations.
These points are not covered by the IEC 60071-1, describing the standard tests to be performed, when addressing
stresses linked to insulation coordination issues. IEC 60071-1 enables the specification and subsequent purchase
of transformers for new installations, but does not address particular aspects related to the behaviour of the
equipment under operating conditions such as transformer energization.
As ferroresonance may induce a long duration phenomena, the overvoltages may affect the aging of the insulation,
but may not lead to the insulation breakdown of the bushing, as an example, in the case when the amplitude of the
overvoltages are moderate.
It is interesting to note that ferroresonance is normally accompanied by a very loud and characteristic noise caused
by magnetostriction of the steel and vibrations of the core laminations. This noise has been described in [18] as
the shaking of a bucket of bolts or a chorus of thousand hammers pounding on the transformer from within.
Although difficult to describe, the noise is definitely different from and louder than that heard under normal
operating conditions at rated voltage and frequency.
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CHAPTER 2 UNDERSTANDING RESONANCE ANDFERRORESONANCE
2.1 Introducing ResonanceFor every combination of L and C in a linear circuit, there is only one frequency (in both series and parallel circuits)
that causes XL to exactly equal XC; this frequency is known as the natural or resonant frequency. When the
resonant frequency is fed to a series or parallel circuit, X L becomes equal to XC, and the circuit is said to be
resonant to that frequency.
The simplest oscillatory circuit consisting of one capacitor C and one inductor L is lossless (ideal) and the
frequency of its free oscillation is given by the well known formula of
Eq. 2-1
Free oscillations are also called natural oscillations because their frequency is given by passive parameters of a
circuit. For example the circuit of Figure 2-1 with C = 100 nF and L = 100 H starts to oscillate in an undamped
fashion following switching with a frequency of free oscillation being fn= 50,33 Hz.
Figure 2-1 Undamped inductance voltage (red) and current (green) oscillations
However, in reality, these free oscillations are typically damped as shown inFigure 2-2 (a resistor R value of 1 k
has been used in this example). Mostly, this damping comes in the form of resistive components and hence the
transformation of electric or magnetic energy into thermal energy. Losses, provided by the resistive components
could be high enough to dampen the oscilations within a couple of cycles (Figure 2-3a) or they could be too high
and create an aperiodical transient where all available energy is transformed to losses in the just first cycle of
oscillation ((Figure 2-3b).
There are two types of resonance: series and parallel resonance. Basic schematic circuits for series and parallel
resonance are given inFigure 2-4.In the case of series resonance all elements are in one branch with commoncurrent, resonance being excited by an alternating voltage source. Voltages ULS and UCS reach high amplitudes
but have opposing phase angles. In steady state the combined impedance introduced by L and C is zero and the
circuit current is limited only by the resistor R. In the case of parallel resonance, all elements are in parallel and
they have the same voltage, resonance being excited by an alternating current. Currents ICP and ILP reach high
amplitudes but have opposing phase angles. In steady state the combined impedance introduced by L and C is
infinite and the resonant voltage is limited only by the conductance G.
(file Fig_2-1.pl4; x-var t) v:U_L - c:U_C -U_L
0.08 0.10 0.12 0.14 0.16 0.18 0.20[s]-10.0
-7.5
-5.0
-2.5
0.0
2.5
5.0
7.5
10.0
[kV]
-1.0
-0.6
-0.2
0.2
0.6
1.0
[A]
Red waveform: Voltage across inductor L (plotted in left Y axis)Green waveform: Current in inductor L (plotted in right Y axis)
UL IL
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Figure 2-2 Damped inductance voltage (red) and current (green) oscillations
Figure 2-3 Example of damped oscillations
Red waveform: Voltage across inductor L (plotted in left Y axis)
Green waveform: Current in inductor L (plotted in right Y axis)
(file Fig_2-2.pl4; x-v ar t) v:U_L - c:U_R -U_L
0.08 0.10 0.12 0.14 0.16 0.18 0.20[s]-10.0
-7.5
-5.0
-2.5
0.0
2.5
5.0
7.5
10.0
[kV]
-1.0
-0.6
-0.2
0.2
0.6
1.0
[A]
UL IL
(file f ig_2-3a.pl4; x-var t) v:U_L - c:U_R -U_L
0.08 0.10 0.12 0.14 0.16 0.18 0.20[s]-7.0
-3.6
-0.2
3.2
6.6
10.0
[kV]
-0.70
-0.36
-0.02
0.32
0.66
1.00
[A]
UL IL
(file f ig_2-3b.pl4; x-var t) v:U_L - c:U_R -U_L
0.08 0.10 0.12 0.14 0.16 0.18 0.20[s]-2
0
2
4
6
8
10
[kV]
-0.2
0.0
0.2
0.4
0.6
0.8
1.0
[A]
UL IL
Red waveform: Voltage across inductor L (plotted in left Y axis)Green waveform: Current in inductor L (plotted in right Y axis)
Red waveform: Voltage across inductor L (plotted in left Y axis)Green waveform: Current in inductor L (plotted in right Y axis)
a) strongly damped with R = 10 k b) aperiodical transient with R = 10 k
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Figure 2-4 Series and parallel resonant circuits
One important characteristic of the parallel resonant circuit is that the excitation can be realized only by a current
source. In power networks, the possibility of forming parallel resonant circuits are higher than that of series
resonant circuits but with no current sources to excite them, parallel resonance at power frequency is encountered
less frequently. Hence most of this publication deals with series resonant circuits.
The following sections introduce various concepts of series resonance in its transient form from zero initial
conditions to the final resonant state, rather than straight into the steady state form, as the former is of more
concern in power networks. In understanding series resonance it is appropriate to choose either the voltage across
the inductor (UL) or the capacitor (UC) as the circuit parameter to monitor. Both are of equal magnitude but with
phase angle shift of 180 between them. ULhas been selected in this document.
2.1.1 Ideal Series Resonant CircuitFigure 2-5 shows the transition of an ideal lossless series oscillatory circuit with natural frequency fn= 50 Hz to
resonance following the connection of a 50 Hz voltage source. From the start the phase angle between the voltage
across the inductor (UL) and the source voltage (US) is kept at 90 giving maximum increase to the resonant
voltage with amplitude rising linearly proportional to US. for every period. As a function of time, the rise of theresonant voltage amplitude UL(as an envelope) is given by
Eq. 2-2If we consider the 50/60 Hz network frequency as constant, the rise time of resonant voltage on this basic circuit is
independent of the circuit parameters, except for the magnitude of the excitation voltage U S. In this particular
example the resonant voltage rate of rise is 1570.8 kV/s, based on USvalue of 10 kV. The capacitor and inductor
values used in this resonant circuit were 101.32 nF and 100 H respectively, but the same result could be obtained
for different combinations of capacitor and inductor values with the same product, such as 1013.2 nF and 10 H
respectively.
a) Series b) Parallel
Comment [mve1]: I dont think this sentencevery clear. In fact, Im not sure what exactly are
trying to say here and how is the 90deg shift
relevant to the maximum increase of resonant
voltage or rate of rise of the voltage in the indu
or capacitor. In fact, simulations show that the
phase shift is higher than 90 deg for a few cycles
after the switch is closed.
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polarity). Integral value of the pulsed power gives the accumulated energy in the oscillatory circuit (green) which
periodically reaches a maximum and then returns back to zero.
Figure 2-6 Ideal series resonance oscillation with nS
Figure 2-7 Energy exchange between source and resonant circuit
2.1.2 Damped Series Resonant CircuitIn real power networks resonant circuits are never lossless and hence it is important to visualise the effect of losses
on the resonant cases explained in the previous section.
In a damped resonant circuit with fn= fs, the resonant voltage will not increase above all limits as in ideal lossless
circuits, because the resonant current is limited by the resistance R, the maximum resonant voltage being given by:
(file Fig_2-6-a.pl4; x-var t) v:U_L - v:U_S
0.0 0.1 0.2 0.3 0.4 0.5 0.6[s]-150
-100
-50
0
50
100
150
[kV]
Red waveform: Voltage across inductor L (114.632H)Green waveform: Source Voltage
Red waveform: Voltage across inductor L (88.0H)Green waveform: Source Voltage
(file fig_2-6-b.pl4; x-var t) v:U_L - v:U_S
0.0 0.1 0.2 0.3 0.4 0.5 0.6[s]
-150
-100
-50
0
50
100
150
[kV]
(fs+fn)/2 =
48.35 Hz
(fs+fn)/2 =
51.65 Hz
(fs-fn) = 3.3 Hz (fn-fs) = 3.3 Hz
a) fn= 46,7 Hz < fs b) fn= 53,3 Hz > fs
(file fig_2-6-b.pl4; x-var t) p:U_S -XX0001 e:U_S -XX0001
0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45[s]-40
-30
-20
-10
0
10
20
30
40
[kW]
0
300
600
900
1200
1500
[J]
Power Energy
Red waveform: Power exchange (plotted in left Y axis)
Green waveform: Energy exchange (plotted in right Y axis)
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Eq. 2-7
An example of this is given inFigure 2-8 obtained for a series resonant circuit with a 101.32 nF capacitor and a
100 H inductor with a source voltage of 10 kV, for two different resistance values (300 and 1000 ).
Introducing a small difference between the excitation frequency fsand the frequency of free oscillation fnresults to a
modulated wave. There is a transient after source switching, which for small damping seems to look as a
modulated wave of the ideal lossless circuit but with a difference. In the higher loss case the modulating waves are
damped step by step until the transition to steady state where the resonant voltage reaches a constant amplitude
and a fixed phase angle difference to source voltage, as shown in Figure 2-9 (L = 114.63 H, C = 101.32 F, Us=
10kV, fn= 46.70 Hz).
If fn fs, the angle between phasors moves from 90 to 0 and then, due to losses, it cant reach 270 but settles toa value between 0and 90 following the non-zero minimum point of the modulation as seen in Figure 2-10.
Similarly for fn fs the angle between phasors moves from 90 to 180 and then to a value between 0and 90following the non-zero minimum point of the modulation.
It is worth remembering that there is always some stray capacitance involved with inductors and this should always
form part of a circuit that can be re-configured by use of Thevenin theory to a series LC circuit.
Figure 2-8 Resonant voltage with damping and n=S
Red waveform: Voltage across inductor L - Resistance 300 UL(max) = (10 * w* 100)/300 = 1047.2 kVGreen waveform: Voltage across inductor LResistance 1000 UL(max) = (10 * w* 100)/1000 = 314.16 kV
0 1 2 3 4 5 6 7 8[s]-1.2
-0.8
-0.4
0.0
0.4
0.8
1.2
[MV]
Comment [mve3]: maybe we should explainwhat is this difference? In my simulations, the o
difference that I can see is the amplitude of the
oscillation and a very smal phase shift.
Comment [mve4]:
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Figure 2-9 Damped series resonance oscillation with nS
Figure 2-10 First over-swing and steady state of damped resonance (R = 1000 ) with
nS
a) R = 300 b) R = 1000 _ . : _
0.0 0.5 1.0 1.5 2.0 2.5 3.0[s]-150
-100
-50
0
50
100
150
[kV]
_ . : _
0.0 0.5 1.0 1.5 2.0 2.5 3.0[s]-150
-100
-50
0
50
100
150
[kV]
UL UL
(file fig_2-10-a.pl4; x-var t) v:U_L - v:U_S
0.1 0.2 0.3 0.4 0.5 0.6[s]-120
-80
-40
0
40
80
120
[kV]
(file Fig_2-10-b.pl4; x-var t) v:U_L - v:U_S
0.1 0.2 0.3 0.4 0.5 0.6[s]-120
-80
-40
0
40
80
120
[kV]
A) fn< fs= 50 Hz
B) fn> fs= 50 Hz
Red waveform: Voltage across inductor LGreen waveform: Source Voltage
(file fig_2-10-a.pl4; x-var t) v:U_L - v:U_S
2.95 2.96 2.97 2.98 2.99 3.00[s]-80
-60
-40
-20
0
20
40
60
80
[kV]
(file Fig_2-10-b.pl4; x-var t) v:U_L - v:U_S
2.95 2.96 2.97 2.98 2.99 3.00[s]-80
-60
-40
-20
0
20
40
60
80
[kV]
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2.2 Introducing FerroresonanceIn simplest terms, ferroresonance can be described as a non-linear oscillation arising from the interaction betweenan iron core inductance and a capacitor. In this section, the description of ferroresonance follows from the previous
sections with a basic analysis of a series resonant circuit and gradually increases the level of complexity to provide
a comprehensive explanation of the physical mechanism driving the nonlinear oscillation of ferroresonance. In this
initial description, a very simplified model of the magnetic core is used for a better understanding of the basic
mechanisms driving the oscillation.
As with linear resonance, ferroresonant circuits can be either series or parallel, albeit only series configurations are
typically encountered in transmission networks. It should be noted that parallel ferroresonant configurations are
common in distribution systems with ungrounded or resonant neutral connections. For simplicity and better
understanding, the analysis and explanation that follows is based on series resonant and ferroresonant circuits
only.
A basic series R-L-C circuit is shown in Figure 2-11 which includes the series connection of a voltage source US, to
a capacitor C, an inductor L, and a resistor R. All circuit elements are linear.
Making use of phasor analysis, the equation describing the steady-state behaviour of the above circuit expressed
as:
[ ( )] Eq. 2-8where wsis the angular frequency of the voltage source.
Figure 2-11 Linear Series R-L-C circuit
Resonance occurs when the capacitive reactance equals the inductive reactance at the driving frequency. Under
this condition the circuit impedance becomes purely resistive.
Eq. 2-9The most characteristic feature of a linear R-L-C circuit is that there is only one natural frequency, fn, at which the
inductive and capacitive reactances are equal. This frequency is given inEq. 2-1.
A graphical solution of Eq. 2-8 is presented in Figure 2-12 [18]. The circuit resistance has been ignored for
simplicity. The voltage-current representation results in two straight lines with slopes equal to the inductive andcapacitive reactances respectively. The intersection of both lines yields the current in the circuit.Figure 2-12 (a)
shows the operating point for a source frequency fSbelow the circuit natural frequency fn. It can be seen that the
capacitive reactance, XC, exceeds the inductive reactance, XL, resulting in a leading current and a high voltage
across the capacitor. Similarly,Figure 2-12 (c) shows the operating point for a source frequency above the circuit
natural frequency, fn. It can be seen that in this case the inductive reactance, XL, exceeds the capacitive reactance,
XC, resulting in a lagging current and a high voltage across the inductor. Finally,Figure 2-12 (b) shows that, for a
.t)
CL
R
UL UC UR
I
US(t) = US.sin(wS.t)
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source frequency equal to circuit natural frequency fn, the inductive and capacitive reactances are equal and the
two lines become parallel, yielding a solution of infinite current and voltages.
In practice all circuits have some sort of losses, even if in small amounts. These resistive losses have the effect of
limiting the amplitude of current and voltages in resonance as follows:
Eq. 2-10 Eq. 2-11 Eq. 2-12
Q is normally referred as the circuit quality factor, which gives an indication of the resistive losses and the circuit
gain. It becomes apparent that low circuit losses lead to high capacitor and inductor voltages under resonant
conditions.
Figure 2-12 Graphical Solution of Linear Series L-C circuit
Replacing the inductor L of the linear series R-L-C circuit of Figure 2-11 with a saturable magnetic core, a series
ferroresonant circuit can be obtained as shown in Figure 2-13. What differentiates ferroresonance from linear
resonance is that the inductance is not constant; therefore the ferroresonant frequency calculated with Eq. 2-1
becomes a moving target. This means that a range of circuit capacitances can potentially lead to ferroresonance at
a particular source frequency. Another characteristic of ferroresonance is the existence of several solutions. This
distinctive behaviour will be il lustrated next.
I
U
I
U
US
XL
XC
I
U
(a) fS < fn
Capacitive Circuit Resistive Circuit
Inductive Circuit
I
I
(b) fS = fn (c) fS > fn
XL
XC
XL
XCUS US
UC
UC
US US US
UL
UL
UL =-UC =
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Figure 2-13 Series Ferroresonant Circuit
An in-depth analysis of the circuit shown in Figure 2-13 is complex and requires the solution of nonlinear differential
equations. The circuit analysis, however, can be simplified considerably and yet provide a thorough conceptual
description of ferroresonance by limiting the calculations to power frequency and steady state [19]. It should be
noted that the presence of the non-linearity introduces harmonics in the current and voltage waveforms. However,
for simplicity, the description that follows assumes perfect sinusoidal voltage and current waveforms oscillating at
power frequency. Without this assumption, the application of phasor analysis would be invalid. Furthermore, the
resistive losses are also ignored. Under these particular conditions, the equation describing the steady-state circuit
behaviour at power frequency can be expressed as:
Eq. 2-13where XCis the circuit capacitive reactance at power frequency, wS is the source angular frequency and UL(I) is thevoltage of the saturable magnetic core. This voltage across the non-linear inductance is a function of the current,
which is characteristic of the ferromagnetic inductance and is solely dependent on the number of turns and the
dimensions of the iron core.
Eq. 2-13 has been solved graphically inFigure 2-14[18] where the voltage across the non-linear inductance (UL(I))
must always be equal to the sum of the source voltage U S and the voltage across the capacitor, which isproportional to the current. The intersection of the US+ I.XC line withthe non-linear UL(I) curve gives the solution
for the current in the circuit. The first distinctive characteristic of this graphical visualisation is that there are three
possible solutions:
Point 1 represents a normal operating point in which the circuit is working in an inductive mode, withlagging current and low voltages. Voltage and current related by a linear expression. The inductive voltageis greater than the capacitive voltage by the source voltage. This is a stable solution.
Point 3 represents a ferroresonant state in which the circuit is working in a capacitive mode, with leadingcurrent and high voltages. Voltage and current are related by a non-linear expression. The capacitivevoltage is greater than the inductive voltage by the source voltage. This is also a stable solution.
Point 2 is another circuit solution but it represents an unstable state.
The stability of solutions 1 and 3 can be demonstrated with the following considerations: at point 1, a small
increase or decrease of the current will result in a linear change in the capacitor voltage (U C), acting in the samedirection of the source voltage (US). However, the counteracting inductive voltage (UL) changes more intensely with
current due to its steeper slope, therefore the current will return to its original value. Similarly, at point 3, a small
variation in current will result in a small variation in inductive voltage (UL), acting in the same direction of the source
voltage (U0). The counteracting capacitive voltage (UC) changes more intensely due to its steeper slope, and
therefore the current will return to its original value again.
C
R
LI
C
R
I
L
M
UR US(t) = US.sin(wS.t) US(t) = US.sin(wS.t)
UCUL
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The instability of point 2 can be demonstrated by slightly increasing the current, which results in an increase in the
capacitor voltage (UC), acting in the same direction of the source voltage (US). In this case, the steepness of (US+
I.XC) is higher than the opposing voltage (UL), therefore the current will continue increasing away from point 2. Asimilar consideration can be made for a small decrease in current.
Figure 2-14 Graphical Solution of the Series Ferroresonant Circuit
2.2.1 Effect of circuit capacitanceFigure 2-15 illustrates the effect of the circuit capacitance on the onset of ferroresonance. It can be seen that as the
capacitance value is reduced, the slope of the US+UC line increases and the three possible solutions move
towards the vertical axes. Figure 2-15 (a) shows that there is a critical capacitance value, C critical, for which theoperating points 1 and 2 disappear and the only possible solution is a ferroresonant state, point 3. Similarly, Figure
2-15 (b) shows that higher capacitances result in a reduced slope in the US+UC line. It is inferred that, for a large
enough capacitance value, the operating points 2 and 3 disappear and the only possible solution is a normal state,
point 1. This result has practical implications in transmission substations since it suggests that ferroresonance can
be avoided by the connection of a large capacitance.
-2000
-0.06
UL(I)U
I
US+ UC
2
1
3
ULUS
UCU0
UC
US
UL XC
US
UL
UC
I
Solution at
point 1
UL
UC
I
Solution at
point 2
(unstable)
UL
UC
I
Solution at
point 3
US
US
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Figure 2-15 Graphical Solution Illustrating the Effect of Circuit Capacitance
2.2.2 Effect of source voltageThe effect of the source voltage is illustrated inFigure 2-16.As this voltage is increased, the US+UC line moves
upwards to a point in which there is no intersection in the first quadrant. Operating points 1 and 2 disappear and
the only possible solution is point 3, which is a ferroresonant state. Note also that the disconnection of the source
voltage, U, may not result in the elimination of ferroresonance, as illustrated wit h state 3. As U is removed, the
operating point simply slides to the right, but remains in the saturated region. This statement assumes that the
circuit has no losses, which is not true in reality, but it serves to illustrate the fact that, in theory, the ferroresonant
oscillations can be self-sustained.
Figure 2-16 Graphical Solution Illustrating the Effect of the Source Voltage
-2000
-0.06
C1
C2
Ccritical
C1 > C2 > Ccritical
3
Reducing Capacitance
-2000
-0.06
C1
C2
C3
C1 < C2 < C3
1
Increasing Capacitance
(a) Effect of reducing Capacitance (b) Effect of increasing Capacitance
U U
I I
USUS
UL(I) UL(I)
-2000
-0.06
33
Removing Source Voltage
U
IIncreasing Source
Voltage
UL(I)
U1< U2 < UcriticalU1
U2
Ucritical
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Figure 2-17 Graphical Solution Illustrating the Effect of Circuit Resistance
2.3 Physical Description of a Ferroresonant OscillationThe description of ferroresonance presented in the previous section, although good enough as a first
approximation, does not provide a real understanding of the mechanisms driving a ferroresonant oscillation.Various explanations of the physical behaviour of ferroresonant circuits can be found in[8], [13],[19] and[20].A
review of those descriptions, expanded for an enhanced understanding of this complex phenomenon, is presented
next.
Figure 2-18 shows a series R-L-C circuit with a nonlinear inductor and a switch. A two-segment piecewise linear
representation is used for the magnetizing impedance. The circuit losses are initially ignored for simplicity. The
prospective current and voltage waveforms under this simplification are presented in Figure 2-19. Initially, the
capacitor charge is equal to U0. At t = 0 sec the switch is closed and the capacitor C starts discharging through the
inductor working in its linear region, Lunsat. The frequency of this oscillation is:
Eq. 2-17This is a very slow discharge process due to the large value of Lunsat. Nevertheless, the flux linkage slowly builds up
in the magnetic core until saturation is reached. This is shown in Figure 2-19 at t = t 1, when the magnetizing
reactance drops to its saturated value, Lsat.
As Lsatis a few orders of magnitude smaller than Lunsatthe capacitor discharges very rapidly. The frequency of this
new oscillation is w2:
0
220 IRE -
IXV CL -
IC
I
V
2
0
R
E
0E 1
0
220 IRE -
IXV CL -
IC
I
V
1
0
R
E
0E 1 2 3
0
0.00 0.01
IXV CL -
VL(I)VC
I
V
CapacitiveInductive
Zone
XcUC
Inductive
Zone
Capacitive
Zone
IXU CL -
U
(a) First Term of Equation 2.16
IXU CL - IXU CL - 22 IRUs -
USUS 22 IRUs -
(b) Solution of Equation 2.16 with low resistance value R1(b) Solution of Equation 2.16 with high resistance value R2
(US/ R1) (US/ R2)
UU
UL(I)
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Eq. 2-18
Figure 2-18 Basic Ferroresonant Circuit
Between t1 < t < t2all the energy stored in the electric field of the capacitor is transferred into the magnetic field ofthe coil. At t = t2 the voltage has dropped to zero and the current reaches its peak. The magnetic field then
collapses and starts charging the capacitor in the opposite polarity. At t = t3the current through the inductor falls
into the linear region and the capacitor starts charging through L unsat. As Lunsatis a few orders of magnitude higher
than Lsat, the frequency of this oscillation w1 is much lower than the previous one. The current decreases veryslowly and, consequently, very little variation can be appreciated in the capacitor voltage. At t = t 4the voltage in the
capacitor reaches U0and the discharge process starts again. It can be observed that a full ferroresonant period
comprises two full charge-discharge cycles.
Using Faradays law, the flux linkage at any time can be calculated as the area under the voltage -time curve. As
such, the flux linkage from t3to t5is equal to the shaded area in Figure 2-19 (a). This can be expressed as:
Eq. 2-19
Eq. 2-20Eq. 2-20 can be used to calculate the period of the ferroresonant oscillation as follows:
Eq. 2-21 Eq. 2-22
Eq. 2-23
I
Lunsat
Lsatsat
IsatC
L
R
U0
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Figure 2-19 Physical Behaviour of a Ferroresonant Circuit without Losses
Eq. 2-23 indicates that the frequency of a ferroresonant oscillation is directly linked to the circuit capacitance, C, the
initial charge of the capacitor, U0, and the non-linear characteristics of the magnetic core: Lsatand sat.
It has been shown that the basic ferroresonant circuit of Figure 2-18 behaves like a two-state oscillator switching
between two frequencies: low frequency during the unsaturated state and high frequency during the saturated
state. In the absence of losses, this process will repeat indefinitely with a period T ferro. In reality, the circuit losses
will cause the amplitude of the oscillation to decay. It is a direct consequence of Faradays law that, the lower the
voltage amplitude applied to the magnetic core, the longer it will take to reach saturation. As a result, the frequency
of the ferroresonant oscillation will decrease gradually until the process dies out. Figure 2-20 illustrates a
ferroresonant oscillation affected by circuit losses. It is shown that the voltage magnitude decreases with each
transition of polarity. This is due to the high (I2R) losses occurring during the saturated state. These losses are very
low during the unsaturated period due to the low current flow and, hence the voltage remains almost constant.
It has been illustrated that the introduction of losses makes the system dissipative, which causes the amplitude of
the oscillations to decay. In order for the ferroresonant oscillations to be maintained, energy needs to be supplied
externally to counteract the losses. This is shown in Figure 2-21, where a voltage source has been introduced to
represent an external source of energy. It is shown that the combined effect of the source voltage and the
oscillatory trapped charge is to raise the voltage at the reactor terminals just before each transition. If this voltage
rise is enough to compensate for the voltage drop caused by the resistive losses during the transition in polarity,
the oscillations is maintained indefinitely.
(a) Voltage, Flux and Current Waveforms (b) Flux-Current relationship
t
t
t
U0
-U0
sat
- sat
Isat
-Isatt1
t2t5t3 t4
Lsat Lunsat
I
t = 0
t1
t2
t3
t4
t5
Lsat
Lunsat
Charge L
Discharge CCharge C
Discharge L
Charge L
Discharge C
Charge C
Discharge L
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Figure 2-20 Physical Behaviour of a Ferroresonant Circuit with Losses
With regards to the voltage source two situations could arise in a ferroresonant circuit[13]:
1) If the initial ferroresonant frequency calculated with Eq. 2-23 is higher than the source frequency, there is a
chance that the decaying frequency of the oscillations will lock at the source frequency. This will result in
fundamental frequency ferroresonance, as illustrated inFigure 2-22(a) where TL-C=TS, or fL-C=fS.
2) If on the other hand the initial oscillation frequency calculated with Eq. 2-23 is lower than the source
frequency, there is a chance that it will lock at an odd sub-multiple of the power frequency. This will result
in sub-harmonic ferroresonance, as illustrated inFigure 2-22(b) where TL-C=3TS, or fL-C=fS/3.
Figure 2-21 Effect of Coupled Voltage on Ferroresonant Waveform
U0
U
t
U1
U2
U3
U0 > U1 > U2 > U3 > .
T0 < T1 < T2 < .
T0 T1 T2
C
L
US
UCUL
5
0
5
0
U
t
ULUC
US
-5
0
5
0 0.035
-5
0
5
0 0.035
(a) Fundamental Frequency Ferroresonance (b) Sub-Harmonic Ferroresonance
ULUC
USUS
UCUL
TS
TL-C
TS
TL-CU
tt
U
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Figure 2-22 Derivation of Ferroresonant Modes
2.4 Types of Ferroresonance OscillationsFerroresonant waveforms are categorised according to their periodicity. Based on field experience, experimental
observations and extensive numerical simulations, ferroresonance has been categorised into the following modes.
Periodic Ferroresonance Modes
Periodic ferroresonance is characterised by waveforms that repeat themselves. These waveforms are highly
distorted, presenting a dominant frequency that can be either fundamental or sub-harmonic.
In the case of fundamental frequency ferroresonance, the oscillations are mainly at the same frequency as the
driving source. Although the supply frequency is dominant, a large number of harmonics is normally present. In
case of sub-harmonic ferroresonance, the oscillations normally arise at frequencies that are integral odd sub-
multiples of the fundamental frequency. Two examples of typical periodic ferroresonant waveforms are shown in
Figure 2-23.
Figure 2-23 Typical Periodic Ferroresonant Voltage Waveforms
Quasi-Periodic Ferroresonance Modes
The quasi-periodic regimes are characterised by non-periodic oscillations having, at least, two main frequencies.
The fundamental frequency is normally present along with lower sub-harmonic frequencies. A distinctivecharacteristic of these waveforms is the presence of a discontinuous frequency spectrum.
This ferroresonant mode has not been reported very frequently as a stable state. It was first observed in France
[26] during a black-start restoration test in a 400 kV system. It has also been referred to as transitional chaos in
[27] to describe a state that has no indication of periodicity but still shows features of fundamental and sub-
harmonic ferroresonance. This behaviour suggests that the operation is continuously shifting between various
periodic modes without stabilising into any particular one. An example of a quasi-periodic waveform is given in
Figure 2-24.
20 or 16.6ms 60 or 50ms
(a) Fundamental Frequency (b) 3rd Sub-Harmonic Frequency
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Figure 2-24 Typical Quasi-Periodic Ferroresonant Voltage Waveform
Chaotic Ferroresonant Modes
Chaotic ferroresonance waveforms show an irregular and apparently unpredictable behaviour and a broadband
power spectrum with a sharp component at system frequency. This ferroresonant mode is characterised by a non-
periodic waveform with a continuous frequency spectrum. Although the possibility of chaotic ferroresonant modes
has been widely described in literature, [26] to [33], this mode has only been predicted in EHV substations for
unrealistic values of source voltage, circuit capacitance or losses[29] to[32].For instance, reference[29] reported
that chaotic ferroresonance could only be obtained for a source voltage in excess of 25.26 pu when realistic values
of transformer losses were employed. It is noteworthy that no practical experience of a sustained chaotic
ferroresonance in an EHV substation has been reported to date. A typical simulated example is shown inFigure
2-25.
Figure 2-25 Typical Simulated Chaotic Ferroresonant Voltage Waveform
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CHAPTER 3 TYPICAL NETWORK TOPOLOGIES LEADING TORESONANCE IN SHUNT COMPENSATED CIRCUITS
3.1 IntroductionPower frequency resonance conditions in shunt-compensated transmission circuits have been described in
literature[62] -[67],and are explained in detail in section4.2.As a rule of thumb, shunt compensation degrees in
excess of 70% can lead to high overvoltages following single-phase switching operations or a result of circuit
breaker malfunctioning. The resonant condition arises from the interaction between the shunt-reactor and the
phase capacitance in the disconnected phase(s), with energy coupled from the energised phases via the inter-
phase capacitances. The key elements required to form a resonant circuit are:
1. Shunt reactors directly connected to a transmission circuit
2. Inter-phase capacitive coupling in the transmission circuit
3. At least one phase is disconnected
4. At least one phase is energised
The phenomenon of power frequency resonance in a shunt-compensated multi-circuit Right-of-Way has been
described in literature[75]-[80],and is explained in detail in section4.4.This resonant condition occurs when a de-
energised shunt-compensated circuit is in close proximity to another energised circuit. As a rule of thumb, shunt
compensation degrees in excess of 60% can lead to high overvoltages for typical inter-circuit capacitive coupling.
The resonant condition arises from the interaction between the shunt-reactors and the line capacitance in the
disconnected circuit, with energy coupled from the nearby parallel circuit(s). The key elements required to form a
resonant circuit are:
1. Shunt reactors directly connected to a de-energised transmission circuit
2. Inter-circuit capacitive coupling with a energised transmission circuit
Typical network topologies with risk of resonance at power frequency are presented in the next subsections. This
list is not exhaustive and additional topologies can also result in resonant circuits during unsusual network
topologies, such as blackstart restoration operations (see AppendixANNEX A A. 3 for an example).
Both, series and parallel, circuit capacitances are important when assessing potential resonances. Parallel
capacitances are due to the phase-to-ground capacitance of the lines or cables, shunt capacitor banks, and to a
lesser extentstray capacitances in all apparatus. Series circuit capacitances appear in the grading capacitors of
circuit-breakers, phase-to-phase capacitances in single-circuit lines and inter-circuit capacitance in multi-circuit
corridors.
3.1.1 Typical transmission circuit capacitancesTypical circuit capacitances reported in various transmission systems are listed below, for illustration purposes:
France : The typical phase-to-ground capacitance (C0) of overhead-lines is in the range of 10-13 nF/km for400 kV lines and 8-9 nF/km for 225 kV and 90 kV lines. The inter-circuit capacitance of 400 kV double
circuit-lines is in the range 0.2-1.2 nF/km. The cable capacitance to ground is in the range of 100-200
nF/km for 400 kV and 225 kV XLPE cables and 150-350 nF/km for 90 kV XLPE cables.
Ireland : 400 kV overhead-line (single circuit): C+= 11.59nF/km, C0= 7.77nF/km
750kV overhead line between Hungary and USSR [65]: (Hungarian section): C+ = 13.25nF/km, C0 =9.72nF/km
Saudi Arabia : 380kV double circuit line[81]:C+= 13.76 nF/km, C0= 7.78 nF/km
500 kV circuits in Thailand[71]:
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o Single circuit construction: Cph-gr= 8.55 nF/km, Cph-ph= 1.64 nF/km (i.e. C+= 13.47 nF/km, C0=
8.55 nF/km
o Double circuit construction:
Configuration Cph-gr Cph-ph
Cckt-ckt
(perfect
transposition)
Cckt-ckt
(like phases
in incomplete
transposition)
Cckt-ckt
(unlike
phases in
incomplete
transposition)
Both circuits in
service5.39 nF/km 1.76 nF/km 1.05 nF/km 0.74 nF/km 1.21 nF/km
One circuit in service
with the other circuit
grounded
8.55 nF/km 1.76 nF/km --- --- ---
400 kV circuit construction in Hungary[72].
Line configuration C0 [nF/km] C+ [nF/km] Cph-ph [nF/km]
Conventional 400 kV flat arrangement 8.235 10.958 0.907
Conventional 400 kV delta arrangement 5.95 8.77 0.94
Compact 400 kV
(2 x 500mm2 phase conductors)7.03 12.55 1.83
Compact 400 kV
(3 x 300mm2 phase conductors)7.46 13.95 2.16
500kV circuit capacitances in China: C+= 13.06 nF/km, C0= 8.5 nF/km [reference ???]
3.2 Potentially Risky Configurations in Shunt CompensatedTransmission Networks3.2.1 Uneven Phase Operation in Sigle-Circuit or Multi-Circuit
CorridorsUneven phase operation in transmission circuits can be:
Desirable: single-phase tripping schemes applied to improve system transient stability, system reliability andavailability, reduce switching overvoltages and/or reduce shaft torsional oscillations in large thermal units[67]
or
Undesirable: mal-operation in circuit breakerso during an opening operation: one (or two poles) may get stuck, resulting in two (or one) phases being de-
energised while one (or two) phase remains energised (seeFigure 3-1A and B).o during a closing operation: one (or two) poles my fail to close, resulting in two (or one) phases being
energised while one (or two) phases remain de-energised (seeFigure 3-1 B and C).
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Figure 3-2 De-energisation of Line and Busbar with shunt-reactors connected to the
Busbar
A B C
De-energised phase
Energised phase
A
B
CStuck Pole
De-energised phase
(A) One stuck circuit breaker pole during Busbar + Line De-Energisation
A B C
De-energised phase
Energised phase
A
B
CStuck Pole
(B) Two stuck circuit breaker poles during Busbar + Line De-Energisation
Stuck Pole
Energised phase
Substation-A
Substation-B
Substation-A
Substation-B
Busbar Shunt-Reactors
Busbar Shunt-Reactors
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Figure 3-3 Energisation of Line and Busbar with shunt-reactors connected to the
Busbar
3.2.2 Three-Phase switching in Multi-Circuit CorridorsFigure 3-4 shows a typical double circuit tower with one circuit in service (I) and another circuit out-of-service (II).
Due to inter-circuit capacitive coupling, voltage is induced in an open (not earthed) line if the parallel circuit is
energized. The normal induced voltage in the de-energized circuit (Ucircuit_II) can be estimated as:
Eq. 3-1where Csis the inter-circuit capacitance between circuits I and II and Cpis the capacitance to ground of circuit II
(seeFigure 3-4).
(A) One stuck circuit breaker pole during Busbar + Line Energisation
(B) Two stuck circuit breaker poles during Busbar + Line Energisation
Pole fails to close
A B C
De-energised phase
Energised phase
A
B
C
Energised phase
Substation-A
Substation-B
Busbar Shunt-Reactors
A B C
De-energised phase
Energised phase
A
B
C
De-energised phase
Substation-A
Substation-B
Busbar Shunt-Reactors
Pole fails to close
Pole fails to close
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This example presents two credible topologies leading to resonance in a double-circuit transmission line due to the
interaction with busbar shunt reactors. The dangerous topology arises when the busbar (with the shunt reactor)
and one of the circuits are de-energized while the parallel circuit remains energized from a remote end, thuscoupling energy to the reactor + de-energized circuit combination.
Figure 3-6 Double-Circuit Line and Busbar Shunt ReactorsTopology 1:
Figure 3-6 (a) shows a busbar section in substation B with two line feeders and one shunt reactor connected to it.
Ckt ii is energized from substation A and open at substation B. A resonant circuit can be formed upon opening the
parallel Ckt i circuit breaker in Substation A. This topology effectively leaves the busbar shunt reactor directly
connected to the de-energised circuit. Resonance occurs between the busbar shunt reactor and the capacitance of
the de-energised circuit (Ckt i), with energy coupled from Ckt ii, via inter-circuit capacitive coupling.
Topology 2:
Figure 3-6 (b) shows another situation where resonance can occur in a similar network topology. In this case, Ckt iiis energized from substation A and open at substation B while Ckt i is connected to Substation B (without voltage)
but open at Substation A. A resonant circuit can be formed upon closing the shunt-reactor circuit breaker. The
resonant circuit is identical to the previous topology.
3.2.2.2 Power Transformer, Tertiary Shunt Reactors and Double CircuitTransmission Line
This example presents two possible topologies leading to resonance in a double-circuit transmission line due to the
interaction with shunt reactors connected to the tertiary winding of a power transformer. The dangerous topology
arises when the transformer (with the tertiary shunt reactor) and one of the circuits are supposedly de-energized
while the parallel circuit remains energized from a remote end, thus coupling energy to the transformer/reactor +
de-energized circuit combination.
Similarly to the example described in section 3.2.2.1 for busbar shunt reactors, Figure 3-7 shows the network
topology where a resonant circuit can be formed. The description of the switching scenarios and topologies is thesame as in section3.2.2.1,with the circuit reactance arising from the series combination of tertiary reactors and
power transformer reactance.
A
A
B
B
Ckt i
Ckt i
Ckt ii
Ckt ii
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Figure 3-7 Double-Circuit Line and Transformer Tertiary Shunt Reactors
a)
b)
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CHAPTER 4 RESONANCE IN SHUNT COMPENSATEDTRANSMISSION CIRCUITS
4.1 BackgroundThe application of shunt reactors to long transmission circuits has been common practice for many years as a
passive and economical means to compensate for the effect of distributed line capacitance. The shunt reactors
compensate for the reactive power surplus in case of reduced power transfer, load rejection or an open
transmission line end, limiting steady-state over-voltages. Shunt reactors are usually required in EHV overhead
lines longer than 200 km Error Reference source not found..The degree of shunt compensation, k, provided by a reactor bank is quantified as a percentage of the positive
sequence susceptance of the circuit to which it is applied:
100
1
100)(
)(1
100[%] 2
CLC
L
B
B
kss
s
C
L
ww
w
Eq. 4-1
where L+ is the shunt reactor inductance per phase (positive sequence), C + is the positive sequence line
capacitance and wsis the system angular frequency.
Typical degrees of shunt compensation for overhead circuits are in the range of 60%-80%, although higher values
have been reported in literature [81], [82].Shunt compensation degrees close to 100% are normally required for
EHV cable circuits due to their higher capacitance.
Notwithstanding the main objective of limiting steady-state over-voltages in lightly loaded or open transmission
circuits, the installation of shunt reactors can result in induced voltages above nominal values under certain
resonant conditions. A resonant circuit can be formed between the shunt reactors and the line capacitance when
one or more phases are de-energized. Energy is coupled into the resonant circuit via capacitive coupling from
energized conductor(s) in same circuit or from parallel circuits.
The resonant conditions can be the result of:
1. Uneven open-phase conditions in a shunt compensated transmission circuit i.e. at least one phase is
disconnected while the other phase(s) remain energized. This condition can arise from the use of single-phase
tripping and autoreclosing schemes (SPAR) or from the mal-operation of circuit breakers with independent
operating mechanisms on each phase. During line energization, one phase could be left open while the other
two phases are energized due to a stuck pole in the circuit breaker. Similarly, two phases could be left open
while the other phase is still energized as a result of a stuck pole during line de-energization. Energy is coupled
into the resonant circuit via the phase-to-phase capacitances. Reference[66] provides a very good insight into
this resonant condition.
2. Disconnection of one circuit in a shunt compensated double-circuit line, while the parallel circuit remains
energized. Energy is coupled into the resonant circuit via the circuit-to-circuit capacitances. References[75] to
[80] deal with this resonant condition in great level of detail.
These resonant conditions will be analysed in detail in the next subsections.
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4.2 Line Resonance in Uneven Open-Phase Conditions4.2.1 Physical descriptionThe following assumptions and simplifications are made in order to describe the basic mechanisms of line
resonance in a shunt compensated circuit operated with one or two phases open (de-energized):
1. The transmission circuit is fully transposed and without losses.
2. All circuit elements are linear.
3. The circuit series impedance is neglected.
4. Shunt reactors are applied to compensate for kof the circuit capacitance (Eq. 4-1Error Reference sourcenot found.)
5. There is no inter-phase magnetic coupling in the shunt reactors. This is the same as saying that the positive
and zero sequence reactances are equal.
6. The neutral point of the shunt reactors is directly connected to ground.
Given the above assumptions and simplifications, a shunt-compensated transmission circuit, at no load, can be
represented by the parallel combination of a lumped capacitance and inductance, as shown in Figure 4-1. The
lumped parameters representation is adequate because the phenomenon of interest is resonance at power
frequency.
Figure 4-1 Connection of shunt reactors in Transmission Circuit
The equivalent phase-to-ground impedance per phase (Zeq) is given by the following expression:
-
-
100
1
1||
1
002
0
k
C
C
Lj
CL
LjLj
CjZ s
s
s
seq
w
w
w
w
w
Eq. 4-2
where k is the degree of shunt compensation defined in Error Reference source not found., L+ is theshunt reactor inductance per phase (positive sequence), C+is the positive sequence capacitance of the circuit, C0
is the zero sequence capacitance of the circuit1and wSis the angular frequency of the voltage source.
Three situations can occur depending on the degree of shunt compensation (k):
1Note that the zero sequence capacitance of a symmetrical transmission circuit (C0) is the capacitance of the phase conductors
to ground (Cph-gr)
A
B
C
Us
Us
Us
Cph-ph
Cph-phCph-ph
C0 = Cph-grC0C0
Zeq
L+ L+ L+
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1)
C
Ck 0
100The equivalent phase to ground impedance, Zeq, is capacitive.
2)
C
Ck 0
100 The equivalent phase to ground impedance, Zeq, is inductive.
3)
C
Ck 0
100 The equivalent phase to ground impedance, Zeq, is infinite.
Figure 4-2 shows the frequency scan of the equivalent phase-to-ground impedance per phase, Zeq, of a 400 kV
transmission line assuming two degrees of shunt compensation: 60% and 70%. The C 0/C+ ratio of this circuit is
0.67. Figure 4-2 (a) shows that with shunt compensation degree of 60% (i.e. < C0/C+), the phase-to-ground
impedance is capacitive at 50 Hz. Increasing the degree of shunt compensation to 70% (i.e. > C 0/C+),Figure 4-2
(b) shows that the phase-to-ground impedance becomes inductive at power frequency. Although not shown in the
figure, it is clear that a shunt compensation degree of 67% would result in infinite impedance to ground at 50 Hz.
Figure 4-2 Equivalent line-to-ground impedance (Zeq) in a transmission line with
C0/C+=0.67
If we assume that one phase conductor is disconnected while the other two phases remain energized (for example
following a single phase trip), the equivalent phase to ground impedance - Zeq (Eq. 4-2)becomes series connected
with the inter-phase capacitances to the energized phases. This is illustrated in Figure 4-3 below. As previously
discussed, Zeq can be capacitive or inductive depending on the degree of shunt compensation applied to the circuit.
For low degrees of shunt compensation (i.e. k < C0/C+ ) Zeq is capacitive. The series connection of two
capacitances will not give rise to resonance issues. At k = C 0/C+, Zeqbecomes infinite, and there is a potential risk
of parallel resonance. However this parallel resonant mode cannot be excited with a voltage source, therefore k =
C0/C+is not a harmful topology. (How could the circuit be excited by a current source?). Finally, high degrees of
shunt compensation (i.e. k > C0/C+) will result in Zeq becoming inductive. The series connection of inductive and
capacitive elements will result in series resonance if both reactance values become equal. This series resonant
circuit is excited by the voltage source on the energized phases and gives rise to high currents and voltage across
the reactor.
I _ _I . ; - :
35 40 45 50 55 60 65 70-100
-75
-50
-25
0
25
50
75
100
I _ _I . ; - :
35 40 45 50 55 60 65 700
30
60
90
120
150
*10 3
Fre uenc
Frequency
Ma
nitude
PhaseZ
51.1
51.1
Inductive Ca acitive
_ _ _ . ; - :
35 40 45 50 55 60 65 700
30
60
90
120
150
*10 3
_ _ _ . ; - :
35 40 45 50 55 60 65 70-100
-75
-50
-25
0
25
50
75
100
Ma
nitude
PhaseZ
47.3
47.3
Inductive
Ca acitive
Frequency
Fre uenc
(a) 60% Shunt Compensation Degree (b) 70% Shunt Compensation Degree
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Figure 4-3 Simplified Equivalent Circuit during Single-Phase Opening
To summarise:
1. Series resonance can occur during open-phase conditions when k > C0/C+. Series resonance arises fromthe parallel combination of the shunt reactor and line-to-ground capacitance in series with the inter-phase
capacitances.2. A parallel resonant circuit between the line-to-ground capacitance and the shunt reactor exists when k =
C0/C+. However, parallel resonance cannot be excited by a voltage source; therefore this is not a harmfultopology.
In practice, typical C0/C+ratios in standard transmission line constructions are in the 0.6 0.7 range. This means
that, under the assumptions made above, there is a risk of series resonance following open-phase conditions when
the degree of shunt compensation exceeds 60-70%. The source of the series resonance is the uneven
compensation of positive and zero sequence capacitance provided by the shunt reactors.
4.2.2 Steady State pproximate nalytical Solution Given the potential damage to line connected equipment, such as surge-arresters, instrument transformers, shunt
reactors and circuit breakers, the circuit configurations leading to excessive over-voltages need to be identified.
The key questions to be resolved for any line construction requiring shunt compensation are:
1. What are the particular reactor sizes that give rise to resonant conditions?
2. What is the induced open-phase voltage for any particular degree of shunt compensation?
A high level answer to those questions can be given using the simple formulae presented in sections4.2.2.1 and
4.2.2.2 next. It should be noted that this is a steady-state analysis and higher temporary over-voltages can be
expected during transient conditions.
For clarity, the analytical equations will be expressed in terms of both, positive and zero, sequence capacitances as
well as phase-to-ground and inter-phase capacitances. The relationship between these magnitudes (assuming
symmetrical line construction) is as follows:
phphgrph CCC -- 3 Eq. 4-3
grphCC
-
0 Eq. 4-4
The equations presented next (sections4.2.2.1 and4.2.2.2)are based on the assumptions made in section4.2.1.
In particular, the assumptions of symmetrical line parameters, equal positive and zero sequence reactance for the
shunt reactors and solidly earthed reactor neutral connection apply (see section 4.2.4 for the effect of a neutral
reactor). Furthermore, it must be emphasised that losses and saturation effects have been ignored at this stage for
A
B
C
Cph-ph
Cph-phCph-ph
C0 = Cph-grC0C0
Zeq
L+ L+ L+
A
B
C
Zeq
Cph-ph
Cph-ph
Us
Us
Us
Us
Us
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simplicity. In practice, the theoretical steady-state over-voltages calculated with this approach may be limited by
corona losses and reactor core saturation.
4.2.2.1 One Open-PhaseIt is assumed that phases B and C are energized while phase A is disconnected (Figure 4-4 (a)). This circuit, as
seen from disconnected phase A, can be simplified asFigure 4-4 (b). By applying the Thevenin theorem, this circuit
can be reduced further as Figure 4-4 (c), which is a common series L-C circuit with a natural frequency of
oscillation equal to fn_(1 open-phase):
)2(2
1)1(
phphgrph
nCCL
phaseopenf
-- -
Eq. 4-5
Figure 4-4 Simplified circuit for the analysis of Line Resonance
Using circuit analysis to the equivalent shown inFigure 4-4 (c), the following expressions are derived:
Shunt compensation degree that causes series resonance at power frequency:
4.2.2.1.1 Eq. 4-6
Induced open-phase voltage for a compensation degree k:
A
B
C
Cph-ph
Cph-phCph-ph
C0 = Cph-grC0C0L+ L+ L+
A
B
C
Cph-ph
Cph-ph
(a)
(b)
L+ C0 = Cph-gr
A
UThev2 Cph-ph+ Cph-ph
(c) L+ s
grphphph
phphThev U
CC
CU
--
-
22
Us
Us
Us
Us
Us
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Eq. 4-7
4.2.2.2 Two Open-PhasesA similar approach can be used with the two open-phases scenario, resulting in another equivalent L-C circuit with
a natural frequency of oscillation equal to fn_(2 open-phases):
)3(2
1)2(
phphgrph
nCCL
phasesopenf
-- -
Eq. 4-8
Similarly to the one open-phase condition, the following expressions are derived:
Shunt compensation degree that causes series resonance at power frequency:
Eq. 4-9
Open-phase voltage for a compensation degree k:
2
1
13
1
2)1(3
1
0
2
-
-
-
-
-
-
-
C
C
kk
C
CU
phph
grph
Eq. 4-10
4.2.2.3 Practical ExampleAs an illustrative example, the analytical method presented above has been used to assess the resonant
conditions in a standard 400 kV transmission line design used in Ireland as a function of the degree of shunt
compensation. For this construction, the circuit capacitances are C +=11.59 nF/km and C0=7.77 nF/km. The line is
assumed to be fully transposed and the neutral point of the shunt reactors is directly connected to ground.
Figure 4-5 shows the natural frequencies of oscillation for one and two open-phase(s) conditions, as a function of
the degree of shunt compensation. It can be seen that the natural frequency increases with the degree of
compensation. These frequencies reach values within 0.5 Hz of power frequency for compensation degrees
between 77% and 79% during operation with two open phases and between 88% and 91% during operation with
one open-phase.
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Figure 4-5 Natural oscillation frequencies of a 400 kV shunt-compensated line under
one and two open phase conditions
Figure 4-6 presents the steady-state open-phase voltages as a function of the shunt compensation degree,
calculated using Eq. 4-7 and Eq. 4-10. These curves clearly show resonant conditions at 50 Hz for shunt
compensation degrees of 78% and 89% for the two open-phases and the one open-phase conditions respectively.
Shunt compensation degrees from 68% to 99% yield near-resonant conditions with steady-state open-phase
voltages in excess of 1 pu.
It should be noted that this illustrative example is based on a number of simplifications and the calculated voltages
refer to steady-state conditions only. In practice, temporary conditions may lead to voltages in excess to those
calculated using this analytical method. On the other hand, saturation or circuit losses may limit these over-
voltages. Notwithstanding its limitations, this method enables the engineer to carry-out a speedy estimation of the
risk of power frequency resonance for a particular circuit configuration and degree of shunt compensation. Furtherdetailed studies are required when it is envisaged to operate close to a resonant peak. This is typically done using
time domain simulation, as shown in section4.3.
0
10
20
30
40
50
60
70
10% 20% 30% 40% 50% 60% 70% 80% 90% 100% 110% 120%
[Hz]
[k]
fn_1open-phase fn_2open_phases
48.0
48.5
49.0
49.5
50.0
50.5
51.0
51.5
52.0
70% 75% 80% 85% 90% 95% 100%
[Hz]
[k]
fn _1 ope n- ph ase f n_ 2o pe n_ ph as es
91%88%79%77%
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
10% 20% 30% 40% 50% 60% 70% 80% 90% 100% 110%
k
V [pu]
Two open-phases
One open-phase10.2510.25
6.0 m
26.0
4 m
4.1 m
78% 89%68% 99%
U [pu]
U2U1
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Figure 4-6 Steady-State open-phase voltage (approximate analytical solution) in a 400
kV line as a function of the Shunt Compensation Degree, k.
To summarise :
1. A symmetrical shunt-compensated transmission circuit exhibits two series resonant peaks: one for oneopen-phase and a second one for two open-phases conditions.
2. The two open-phases condition presents a resonant peak at a lower degree of shunt compensation thanthe one-open-phase condition.
3. Steady-state voltages in excess of 1 pu can be expected for a wide range of shunt compensation degrees.
4.2.3 Mixed Overhead Line and Cable CircuitsThere are two main characteristics of underground cables that have a direct impact on line resonance:
1. The capacitance of an underground cable is typically in the order of 20 30 times the capacitance of an
equivalent overhead line circuit. 2. HV and EHV cables have screens on each phase, therefore there is no inter-phase capacitive coupling. The addition of a section of underground cable to an overhead transmission line increases the overall C0/C+ratio of
the circuit. This ratio changes rapidly from approximately 0.6-0.7 (no cable section) to 1 (no overhead line section).
The main implication of a higher C0/C+ ratio is that the resonant peak