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RESONANCE ANDFERRORESONANCE

IN POWER

NETWORKWG C4.307


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Resonance and ferroresonance in power network

Page 1

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



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)


20/175


Page 19

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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Page 20

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


24/175


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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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Page 25

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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Page 26

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


28/175


Page 27

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


29/175


Page 28

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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Page 29

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]:


32/175


Page 31

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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Page 33

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



35/175


Page 34

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


A B C

De-energised phase

Energised phase

A

B

C

De-energised phase

Substation-A

Substation-B


Pole fails to close

Pole fails to close


36/175


37/175


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Page 37

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


39/175


Page 38

Figure 3-7 Double-Circuit Line and Transformer Tertiary Shunt Reactors

a)

b)


40/175


Page 39

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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Page 40

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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Page 41

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


43/175


Page 42

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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Page 43

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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Page 44

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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Page 45

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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Page 46

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

combined r&fr v7

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