Ferroresonance 3 Sneider

Merlin Gerin Modicon Square D Telemecanique

..........................................................................Collection Technique

Cahier technique n° 190

Ferroresonance

Ph. Ferracci

“Cahiers Techniques” are a collection of documents intended for engineersand technicians people in the industry who are looking for information ingreater depth in order to complement that given in display productcatalogues.

These “Cahiers Techniques” go beyond this stage and constitute praticaltraining tools.They contain data allowing to design and implement electrical equipement,industrial electronics and electrical transmission and distribution.Each “Cahier Technique” provides an in-depth study of a precise subject inthe fields of electrical networks, protection devices, monitoring and controland industrial automation systems.

The latest publications can be downloaded on Internet from theSchneider server.code: http://www.schneiderelectric.comsection: mastering electrical power

Please contact your Schneider representative if you want either a “CahierTechnique” or the list of available titles.

The “Cahiers Techniques” collection is part of the Groupe Schneider’s“Collection Technique”.

ForewordThe author disclaims all responsibility further to incorrect use of informationor diagrams reproduced in this document, and cannot be held responsiblefor any errors or oversights, or for the consequences of using informationand diagrams contained in this document.

Reproduction of all or part of a “Cahier Technique” is authorised with theprior consent of the Scientific and Technical Division. The statement“Extracted from Schneider “Cahier Technique” no..... (please specify)” iscompulsory.

Philippe FERRACCI

Graduated from the Ecole Supérieure d’Electricité in 1991, where hedefended a PhD on the resonant earthed system in cooperation withEDF - Direction des Etudes et Recherches. He joined the GroupeSchneider in 1996, and is currently conducting advanced studies inthe electrotechnical and electrical power system fields.

n° 190Ferroresonance

ECT190, first issued March 1998

Cahier Technique Schneider n° 190 / p.2

Lexicon

CVT: Capacitor Voltage Transformer (IEC 186)Voltage transformer comprising a capacitordivider unit and an electromagnetic unit designedand interconnected so that the secondaryvoltage of the electromagnetic unit issubstantially proportional to and in phase withthe primary voltage applied to the capacitordivider unit.

Irrational number (not fractional)Number which cannot be expressed as thequotient of two integers ( 2 , 3 , π ...).

PIM: Permanent Insulation MonitorA device designed to indicate (by an audible orvisual signal) the appearance of the first earthfault of a live part.

VT: Voltage transformer (IEC 50)Instrument transformer in which the secondaryvoltage, in normal conditions of use, issubstantially proportional to the primary voltageand differs in phase from it by an angle which isapproximately zero for an appropriate directionof the connections.


Ferroresonance

Ferroresonance is a non-linear resonance phenomenon that can affectpower networks. The abnormal rates of harmonics and transient or steadystate overvoltages and overcurrents that it causes are often dangerous forelectrical equipment. Some unexplained breakdowns can be ascribed tothis rare, non-linear phenomenon.

The purpose of this “Cahier Technique” is to help readers understandferroresonance. The methods described ensure credible prediction andevaluation of the risk of ferroresonance in existing and future installations.Practical solutions designed to avoid or eliminate ferroresonance are alsogiven.

Contents1 Introduction p. 4

2 Understanding ferroresonance 2.1 Resonance p. 5

2.2 Ferroresonance p. 5

3 Identifying ferroresonance 3.1 Diagnosis elements p. 10

3.2 Examples of electrical power system situations favourableto ferroresonance p. 10

4 Preventing or damping ferroresonance 4.1 Practical solutions p. 16

4.2 Practical evaluation criteria p. 19

5 Studies for predicting and understanding 5.1 A study example p. 20

5.2 Modelling, a mathematical approach p. 23

5.3 Summary p. 25

6 Conclusion p. 26

Appendix 1: Bibliography p. 27

Appendix 2: Selection guide for VT load resistances p. 28


1 Introduction

The term “Ferro-résonance ”, which appeared inthe literature for the first time in 1920, refers toall oscillating phenomena occurring in an electriccircuit which must contain at least:c a non-linear inductance (ferromagnetic andsaturable),c a capacitor,c a voltage source (generally sinusoidal),c low losses.

Power networks are made up of a large numberof saturable inductances (power transformers,voltage measurement inductive transformers(VT), shunt reactors), as well as capacitors(cables, long lines, capacitor voltagetransformers, series or shunt capacitor banks,voltage grading capacitors in circuit-breakers,metalclad substations). They thus presentscenarios under which ferroresonance canoccur.

The main feature of this phenomenon is thatmore than one stable steady state response ispossible for the same set of the networkparameters. Transients, lightning overvoltages,energizing or deenergizing transformers orloads, occurrence or removal of faults, liveworks, etc...may initiate ferroresonance. Theresponse can suddenly jump from one normalsteady state response (sinusoidal at the samefrequency as the source) to an anotherferroresonant steady state responsecharacterised by high overvoltages andharmonic levels which can lead to seriousdamage to the equipment.

A practical example of such behaviour(surprising for the uninitiated) is thedeenergization of a voltage transformer by theopening of a circuit-breaker. As the transformeris still fed through grading capacitors accross thecircuit-breaker, this may lead either to zerovoltage at the transformer terminals or to

permanent highly distorted voltage of anamplitude well over normal voltage.

To prevent the consequences of ferroresonance(untimely tripping of protection devices,destruction of equipment such as powertransformers or voltage transformers, productionlosses,...), it is necessary to:c understand the phenomenon,c predict it,c identify it andc avoid or eliminate it.

Little is known about this complex phenomenonas it is rare and cannot be analysed or predictedby the computation methods (based on linearapproximation) normally used by electricalengineers. This lack of knowledge means that itis readily considered responsible for a number ofunexplained destructions or malfunctionings ofequipment.

A distinction drawn between resonance andferroresonance will highlight the specific andsometimes disconcerting characteristics offerroresonance.

Practical examples of electrical power systemconfigurations at risk from ferroresonance areused to identify and emphasise the variety ofpotentially dangerous configurations.Well-informed system designers avoid puttingthemselves in such risky situations.

A predictive study should be undertaken byspecialists should doubts persist concerninglimit, inevitable configurations. Appropriatenumerical analysis tools enable prediction andevaluation of the risk of ferroresonance in apower system for all possible values of thissystem’s parameters in normal and downgradedconditions. Practical solutions are available toprevent or provide protection againstferroresonance.


Resonance is a phenomenon encountered onelectrical power systems of all voltage levels.

It can be observed for example in a resonantearthed system (Petersen coil) used to minimiseMV single phase to earth fault currents, or it canbe responsible for dielectric or thermaldestruction or premature ageing of electricalequipment due to overvoltage or overcurrent(harmonic resonance,...).

As regards series resonance and in sinusoidalconditions ( U = E cos(ωnt) ), the vectorialrelationship between voltages can be written as:

r r r rU = U U UR L C + + (see fig. 1 ).

In the specific case of resonance, the voltages atthe terminals of the capacitor and theinductance are compensated and the circuit issaid to be in a resonance situation. Thepulsation ωn for which this resonance appears issuch that

L C ωn2 = 1 .

The amplitude of current I is then equal to:

This current can be very high.

Voltage amplitude at the capacitor (andinductance) terminals is equal to k.E .

The quality factor k is expressed as:

k = L

R =

1R C

n

n

ωω

2 Understanding ferroresonance

2.1 Resonance

Fig.1 : series resonance.

According to the value of k, voltage amplitudeUL ( = UC ) can be less than or greater thanamplitude E of excitation voltage U.

Harmonic resonance occurs when the pulsationωn coincides with an harmonic pulsation n ω0(ω0 is the system pulsation) generated bycertain machines (variable speed drives, staticrectifiers...). Harmonic resonance can also haveharmful effects on electrical equipment and, assuch, needs to be controlled [6] , [7] .

2.2 Ferroresonance

The main differences between a ferroresonantcircuit and a linear resonant circuit (§ 2.1) are fora given ω:c its resonance possibility in a wide range ofvalues of C,c the frequency of the voltage and current waveswhich may be different from that of the sinusoidalvoltage source,c the existence of several stable steady stateresponses for a given configuration and values

of parameters. One of these states is theexpected “normal” state (in the linearassumption), whereas the other unexpected“abnormal” states are often dangerous forequipment.

Initial conditions (initial charge on capacitors,remanent flux in the core of transformers,switching instant) determine which steady stateresponse will result.

URULUC

UI

UL UC

UR= U

I

I ER

=


Fig. 2 : free oscillations of a series ferroresonant circuit

Physical approach

A study of the free oscillations of the circuit infigure 2a illustrates this specific behaviour.Losses are assumed negligible and thesimplified magnetization curve φ(i) of the iron-core coil is that represented in figure 2b .Despite these simplifying assumptions, thecorresponding waveforms (see fig. 2c ) aretypical of a periodic ferroresonance.

Originally, voltage at the capacitance terminals isassumed equal to V0.

c At the instant t0 switch K closes. A current i iscreated, and oscillates at the pulsation

ω1 = 1/ LCThe flux in the coil, and voltage V at thecapacitor terminals are then expressed as:

φ ω ω = V sin0 1 1/( ) t ; v c t = V os0 1 ω .

c If V0 1/ ω φ> sat , at the end of time t1, theflux φ reaches the saturation flux φsat , voltage vis equal to V1 and the inductance of thesaturated coil becomes Ls. As Ls is very smallcompared with L, the capacitor suddenly“discharges” across the coil in the form of an

oscillation of pulsation ω2 = 1/ L Cs . Thecurrent and flux peak when the electromagneticenergy stored in the coil is equivalent to the

electrostatic energy 1 2 12/ CV restored by the

capacitor.

c At instant t2 the flux returns to φsat , theinductance reassumes the value L and, since thelosses have been ignored, voltage v, which hasbeen reversed, is equal to -V1.

c At instant t3 the flux reaches - φsat andvoltage v is equal to -V2.As ω1 is in practice very small we can considerV V V2 1 0 ≈ ≈ .

Consequently period T of the oscillation isincluded between 2π LC in the non-saturatedcase, and 2 2 3 2π L C t ts + −( ) in the

saturated case (where t t sat V3 2 02 /− ≈ φ ).

The corresponding frequency f (f = 1/T) is thus

such that: 1

2 LC

1

2 L Csπ π < <f

This initial frequency depends on φsat i.e. on thenon-linearity and the initial condition V0.

In practice, due to the losses R i2 in theresistance R (mainly at each voltage reversal),the amplitude of voltage V decreases (V2 < V1 < V0). Since the flux variation ∆φduring the non-saturated period (t3 - t2) is such

that ∆φ φ = vdtt

t

2

3= ∫2 sat , decrease of v

results in a reduction in frequency. If the energylosses (joule losses, core losses, ...) aresupplied by the voltage source in the system,

the frequency of the oscillations, as itdecreases, can lock at the frequency of thesource (if the initial frequency is greater than thepower frequency) or even at a sub-multiplefrequency of the source frequency (if theinitial frequency is smaller than the powerfrequency).

This shows that unlike linear resonance (§ 2.1),for a given inductance, resonance can occur nolonger for a specific value of C but for a widerange of values of C.

K

R

Cv

i

LS

Li

φ

φsat

φmax

Imax

-φsat

v

tt0 t2t1 t3

V0

-V1

V2

t

t

φ

φsat

φmax

iImax

c - Voltage v, current i and flux φ as a function of time

a - Schematic diagram b - Simplifiedcharacteristic φ(i)


Fig. 3 : illustration of ferroresonance characteristics.

c The main differences between a ferroresonantcircuit and a linear resonant circuit are:v possibility of resonance in a wide range ofparameter values,v the frequency of the voltage and current wavesmay vary from the frequency of the sinusoidalsource,v the existence of several stable steady statesresponses for a given configuration andparameter values.

c Ferroresonance can be single-phase (asabove). It is said to be three-phase if there isnon-linear magnetic coupling between phases,and three-single phase if there is no couplingbetween the three non-linearities.Ferroresonance can be series or parallel.

Main characteristics

Thanks to the appropriate methods definedhereafter in this document, a study of the circuitin figure 3a enables the curves shown infigures 3b and 3c to be plotted. These curvesillustrate the characteristics of ferroresonance:

c Sensitivity to system parameter values, jumpphenomenon

The curve in figure 3b describes the peakvoltage VL at the terminals of the non-linearinductance as a function of peak amplitude E ofthe sinusoidal voltage source.By gradually building up peak amplitude E fromzero, the curve in figure 3b shows that thereare three possible different types of behaviouraccording to the value or E, as well as the jumpphenomenon:v For E = E1, the solution (M1n) is unique, andcorresponds to the normal state (obtained in thelinear assumption),v For E = E2, there are three solutions (M2n,M2i, M2f), two of which are stable (M2n and M2f).M2n corresponds to the normal state, whereasM2f corresponds to the ferroresonant state. Thedotted part of the curve (which cannot beobtained in practice) corresponds to unstablestates.v For E = E’2, voltage VL suddenly moves fromthe point M2 to the point M’2 (the jumpphenomenon). The point M2 is known as a limitpoint,v For E = E3, only the ferroresonant state (M3f) ispossible.v When the value of E decreases from E3, thesolution suddenly moves from the point M1(second limit point) to the point M’1.

The jump phenomenon, characteristic offerroresonance, can also be obtained byconsidering another system parameter (forexample, resistance R or capacitance C).

A small variation in the value of a systemparameter or a transient can cause a suddenjump between two very different stable steadystates.

c Sensitivity to initial conditions

Whether M2n or M2f is obtained depends on theinitial conditions. Figure 3c illustrates thetrajectories of the transient of pairs (φ,Vc) as afunction of time for different initial conditions(M01 and M02). Curve C describes a boundary. Ifthe initial conditions (residual flux, voltage atcapacitor terminals) are on one side of theboundary, the solution converges to M2n. If theinitial conditions are on the other side, thesolution converges to M2f. As the point M2i

R C

VL

VC

E

VL

E

M1n

M2n

M1

M'1

M2i

M2fM'2

M3f

E1 E"2 E3E2 E'2

M2

φ

VC

M2n

M2f

M01

M02

C t

t

c - Sensitivity to initial conditions

a - Basic series ferroresonance circuit

b - Sensitivity to system parameters and the jumpphenomenon


Fig. 4 : illustration of ferroresonance characteristics.

c - Quasi-periodic mode

a - Fundamental mode

b - Subharmonic mode

v(t)

t

v

if

V(f)

nf03f0f0

Normal mode

Ferroresonant mode(1 point)

T

v(t)

t

v

if

V(f)

f0f0/3f0/n

(n points)

nT

v(t)

t

v

if

V(f)

3f1-f2f2-f1

(Closed curve)

f1 f2 nf1+mf2

d - Chaotic mode

v(t)

t

v

if

V(f)

Strange attractor


belongs to the boundary, the steady stateeffectively reached around this point is extremelysensitive to the initial conditions.

Classification of ferroresonant modes

Experience of waveforms appearing on powersystems, experiments conducted on reducedsystem models, together with numericalsimulations, enable classification offerroresonance states into four different types.

This classification corresponds to the steadystate condition, i.e. once the transient state isover, as it is difficult for a ferroresonant circuit todistinguish the normal transient state fromferroresonant transient states. However, this inno way implies that transient ferroresonancephenomena do not present a risk for electricalequipment. Dangerous transient overvoltagescan occur several system periods after an event(for example following energizing of an unloadedtransformer) and persist for several powersystem cycles.

The four different ferroresonance types are:

c fundamental mode,

c subharmonic mode,

c quasi-periodic mode,

c chaotic mode.

The type of ferroresonance [13] can beidentified:

c either by the spectrum of the current andvoltage signals,

c or by a stroboscopic image obtained bymeasuring current i and voltage v at a givenpoint of the system and by plotting in plane v, ithe instantaneous values at instants separatedby a system period.

The characteristics of each type offerroresonance are defined below.

c Fundamental mode (see fig. 4a )Voltages and currents are periodic with aperiod T equal to the system period, and cancontain a varying rate of harmonics. The signalspectrum is a discontinuous spectrum made upof the fundamental f0 of the power system and ofits harmonics (2f0, 3f0 ...). The stroboscopicimage is reduced to a point far removed fromthe point representing the normal state.

c Subharmonic mode (see fig. 4b )

The signals are periodic with a period nT whichis a multiple of the source period. This state isknown as subharmonic n or harmonic 1/n.Subharmonic ferroresonant states are normallyof odd order. The spectrum presents afundamental equal to f0/n (where f0 is thesource frequency and n is an integer) and itsharmonics (frequency f0 is thus part of thespectrum).A stroboscopic plotted line reveals n points.

c Quasi-periodic mode (see fig. 4c )

This mode (also called pseudo-periodic) is notperiodic. The spectrum is a discontinuousspectrum whose frequencies are expressed inthe form: nf1+mf2 (where n and m are integersand f1/f2 an irrational real number). Thestroboscopic image shows a closed curve.

c Chaotic mode (see fig. 4d )

The corresponding spectrum is continuous, i.e. itis not cancelled for any frequency. Thestroboscopic image is made up of completelyseparate points occupying an area in plane v, iknown as the strange attractor.

To conclude:

c Ferroresonance is a complex phenomenon inwhich:

v there is several steady states for a givencircuit,

v the appearance of these states is highlysensitive to system parameter values,

v the appearance of these states is highlysensitive to initial conditions.

c Small variations in the value of a systemparameter or a transient may cause a suddenjump between two very different steady statesand initiate one of the four ferroresonance types.The modes most commonly encountered are thefundamental and subharmonic ones.

c Abnormal rates of harmonics, overvoltages/currents, either as stable oscillation or astransients caused by ferroresonance, oftenrepresent a risk for electrical equipment.

c Steady state ferroresonance is sustained bythe energy supplied by the power systemvoltage.


3 Identifying ferroresonance

3.1 Diagnosis elements

system configuration while the symptoms arepresent, together with the events preceding them(transformer energizing, industrial processspecific operating phase, load rejection...) whichmight initiate the phenomenon.

The next step is to determine whether the threeconditions necessary (but not sufficient) forferroresonance to be present, are united:

c simultaneous presence of capacitances withnon-linear inductances,

c existence in the system of at least one pointwhose potential is not fixed (isolated neutral,single fuse blowing, single phase switching....),

c lightly loaded system components (unloadedpower or instrument voltage transformers...) orlow short-circuit power sources (generators).

If any one of these conditions is not verified,ferroresonance is highly unlikely. Otherwisemore extensive investigations are required.

A predictive study may be carried out byspecialists, which will require implementation ofmethods defined later on in this document.

A comparison with examples of typical powersystem situations favourable to ferroresonancemay simplify identification of configurations atrisk.

Ferroresonance is frequently accompanied bysome of the symptoms described below:

c high permanent overvoltages of differentialmode (phase-to-phase) and/or common mode(phase-to-earth),

c high permanent overcurrents,

c high permanent distortions of voltage andcurrent waveforms,

c displacement of the neutral point voltage,

c transformer heating (in no-load operation),

c continuous, excessively loud noise intransformers and reactances,

c damage of electrical equipment (capacitorbanks, VT, CVT,...) due to thermal effect orinsulation breakdown.A characteristic symptom of VT destruction byferroresonance is a destroyed primary windingand an intact secondary winding.

c apparent untimely tripping of protectiondevices.

Some of these symptoms are not specific to theferroresonance phenomenon. For examplepermanent displacement of the neutral point ofan unearthed neutral system may be theconsequence of a single phase-to-earth fault.

An initial diagnosis is simplified by comparing thecurves of any recordings taken with typicalferroresonance waveforms, specified in theabove paragraph (see fig. 4 ).

Faced with the difficulty of diagnosis (norecordings, a number of possible interpretationsof the symptoms) the first reflex is to analyse

3.2 Examples of electrical power system situations favourable to ferroresonance

Due to the multitude of various sources ofcapacitances and non linear inductances in areal power network and the wide range ofoperating conditions, configurations under whichferroresonance can occur are endless.Experience has, however, made it possible to listthe main typical configurations that may lead toferroresonance. A few standard examples aregiven below.

Voltage transformer energized throughgrading capacitance of one (or more) opencircuit-breaker(s).

In EHV, certain switching operations (padlockinga bus coupler or switched busbar circuit-breaker,removal of a fault on a busbar section...) candrive voltage transformers (VT) connectedbetween phases and earth into ferroresonance.


VTC

Opencircuit

Cd

En

Cd

Circuit-breaker D

Fig. 6 : ferroresonance of a VT between phase and earth in an isolated neutral system.

VTCo

Co : system zero-sequence capacitance

VTCo

VTCo

Fig. 5 : : ferroresonance of a voltage transformerconnected in series with an open circuit-breaker.

These configurations can be illustrated by thecircuit in figure 5 . Opening of circuit-breaker Dinitiates the phenomenon by causingcapacitance C to discharge through the VTwhich is then driven into saturation [11] . Thesource delivers enough energy through thecircuit-breaker grading capacitance Cd tomaintain the oscillation.

Capacitance C corresponds to all thecapacitances to earth of the VT and the

connection supplied by means of the gradingcapacitances of the open circuit-breaker(s).Ferroresonance is of the subharmonic type.

Voltage transformers (VT) connected to anisolated neutral system

This earthing system can be chosen, can resultfrom the coupling of an isolated neutralemergency source or from a loss of systemearthing.Transient overvoltages or overcurrents due toswitching operations on the power system (loadrejection, fault-clearing ...) or to an earth fault,can initiate the phenomenon by driving intosaturation the iron core of one or two of the VTsof the parallel ferroresonant circuit in figure 6 .Ferroresonance is then observed both on thephase-to-earth voltages and on the neutral pointvoltage (VN).The neutral point is displaced and the potentialof one or two phases rises with respect to earth,which may give the impression of a singlephase-to-earth fault in the system.Overvoltage values may exceed normal phase-to-phase voltage under steady state condition,and cause dielectric destruction of the electricalequipment.Depending on the relative values of themagnetizing inductance of the VT and thecapacitance C0, ferroresonance is fundamental,subharmonic or quasi-periodic.


Transformer accidentally energized in onlyone or two phases

A few examples of configurations at risk are givenin figure 7 . These configurations can occurwhen one or two of the source phases are lostwhile the transformer is unloaded or lightly loaded,as a result of a fuse blowing on an MV powersystem, of conductor rupture or of live works, forexample when commissioning a remote controlledbreaking cubicle (ACT). The capacitances can bein the form of capacitance of underground cableor an overhead line supplying a transformerwhose primary windings are wye connected with

Fig. 7 : examples of unbalanced systems at risk from ferroresonance.

isolated or earthed neutral, or delta connected.For example the series ferroresonant circuit ismade up of the connection in series of the phaseto earth capacitance (between circuit breaker andtransformer) of the open phase and themagnetizing impedance of the transformer.The modes are fundamental, subharmonic orchaotic.

The phase-to-phase and phase-to-earthcapacitances, the primary and secondary windingsconnections, the core configuration (three single-phase, free flux or forced flux), the voltage sourcesystem neutral earthing (solidly earthed, earthed,


isolated) and the supply mode (one or two phasesenergized) are all factors involved in theestablishment of a given state. Isolated primaryneutral is more susceptible to ferroresonance.

To avoid such risks, use of multi-pole breakingswitchgear is recommended.

Voltage transformers and HV/MVtransformers with isolated neutral

Ferroresonance may occur when the HV and MVneutrals are isolated, and unloaded VTs areconnected on the MV side between phase andearth (see fig. 8a ).

Fig. 8 : ferroresonance of a VT between phase and earth with an isolated neutrals source transformer.

When an earth fault occurs on the HV sideupstream from the substation transformer, the HVneutral rises to a high potential. By capacitive effectbetween the primary and secondary, overvoltagesappear on the MV side, and may triggerferroresonance of the circuit made up of thevoltage source E0, the capacitances Ce and C0 andthe magnetizing inductance of a VT (see fig. 8b ).

Once the HV fault has been removed, thevoltage of the HV neutral due to a naturalunbalance of the system, may be enough tosustain the phenomenon.Ferroresonance is fundamental.

VT

HV

Ce

Co Co Co

MV

VT VT

Ce

Eo

Co Co Co

a - Faulty system

b - Equivalent diagram

E0 : zero-sequence voltage on the HV sideCe : capacitance between HV and MV windingC0 : zero-sequence capacitance of the MV power system


Fig. 9 : ferroresonance in the case of a system earthed through a reactor.

Power system earthed through a reactor

The two configurations in figure 9 aresusceptible to ferroresonance between aninductance connected between neutral andearth, and the capacitances to earth of thenetwork.

c As regards LV power systems with isolatedneutral, standards recommend (IEC 364) orstipulate (NF C 15-100) implementation of aPermanent Insulation Monitor (PIM).Some PIMs measure insulation impedance of apower system by injecting direct current betweenthe system and the earth. Their impedance ismainly inductive (low impedance for the directcurrent, and high impedance at powerfrequency). They can be a factor contributing toferroresonance.Overvoltages may cause sufficient potential riseof the neutral point to initiate ferroresonancebetween the inductance of the PIM and thecapacitances to earth of the network(see fig. 9a ).

c In MV, in order to limit earth-fault currents andhelp the fault to self-extinghish, a coil ofinductance L (such that 3 L C0 ω0

2 = 1 where C0is the zero-sequence capacitance of the MVpower system and ω0 the power pulsation) isconnected between the MV neutral of an HV/MVtransformer and earth (Petersen coil).Excitation and start of resonance of the circuitconsisting of series connection of inductance Land capacitance 3 C0 may occur in the followingcases:

v HV neutral of the HV/MV transformer earthed,and HV fault flowing through the earth conductorof the substation,v iron core saturation of the HV/MV transformer,v transformer design dissymmetry,v natural dissymmetry of the capacitances (C1,C2, C3 in figure 9b ) to earth.This may result in saturation of the iron coil, thusinitiating or sustaining ferroresonance.

Transformer supplied by a highly capacitivepower system with low short-circuit power

Ferroresonance may occur when an unloadedpower transformer is suddenly connected to alow short-circuit power source compared withtransformer rated power through an undergroundcable or a long overhead line. This is the case,for example, on return to service in an MV(underground cable) urban or industrialpower networks, but also in very extended ruralpublic MV power networks (see fig. 10 ) orwhere underground cables are increasingly used(reliability and aesthetics).

This parallel ferroresonance (capacitanceparallel-connected on the transformer’smagnetizing inductance) is normally three-phase, of the fundamental or of the quasi-periodic type.

In short:

c Configurations under which ferroresonancecan occur are endless.

c There are many different types offerroresonance: single-phase, three-phase,common mode, differential mode.

C1PIM

PIM = permanent insulation monitor

C2 C3

L

C1 C2 C3

b - Resonant earthing systema - Internal inductance of a PIM of an impedance between neutraland earth


L

Source Unloaded power transformerCapacitive connection(long line or cable)

C

Fig. 10 : equivalent diagram of unloaded power transformer supplied by a capacitive system.

c Experience, however, makes it possible toidentify some configurations at risk whichcommand some attention. These are:v voltage transformer connected between phaseand earth on an isolated neutral system,v transformer fed through long and/or capacitivelines,v fuse protection, blowing of which results innon-multi pole breaking,

v unloaded or lightly loaded power or voltageinstrument transformer.

c The phenomena most likely to triggerferroresonance are:v switching operation of capacitor banks andunloaded lines,v insulation faults,v lightning,v switching operation of unloaded transformers.


4 Preventing or damping ferroresonance

A number of practical measures can be taken toprevent ferroresonance, whose overvoltages,overcurrents and distortions wave forms result inthermal and dielectric stresses which may bedangerous for electrical equipment (failure,reduction in performance and lifetime ofinsulators...).

The various methods used are based on thefollowing principles:

c avoid, by proper design and/or switchingoperations, configurations susceptible toferroresonance.This may involve prohibiting certain systemconfigurations, and/or certain power systemswitching operations and /or certainswitchgear;

c ensure that system parameter values are notincluded (even temporarily) in an area at risk andif possible provide a safety margin with respectto danger areas;

c ensure that the energy supplied by the sourceis not sufficient to sustain the phenomenon. Thistechnique normally consists of introducing losseswhich damp out ferroresonance when it occurs.

Publication 71 of the IEC [2] states thattemporary ferroresonance (and resonance)overvoltages “shall be prevented or limited” (byone of the above means). “They shall notnormally be considered as the basis for thesurge arrester rated voltage, or for the insulationdesign unless these remedial measures are notsufficient”.This means that the insulation co-ordinationprocedure does not normally take into accountthe overvoltage levels due to ferroresonance,and that, consequently, surge arresters (whoseresidual voltage is usually higher than theovervoltages due to ferroresonance) do nottheoretically provide protection againstferroresonance.

4.1 Practical solutions

Application of these principles results in therecommendation of practical solutions, some ofwhich are defined below for a few typicalconfigurations susceptible to ferroresonancequoted in paragraph 3.2.

c In well-designed VTs and CVTs, suitabledesign measures are taken in order to neutralisethe phenonemon.

The case of (single pole) VTs connectedbetween phase and earth on an isolated neutralsystem is considered, in practice, as the casemost favourable to ferroresonance (caused, forexample, by overvoltages between soundphases and earth as a result of a singlephase-to-earth fault).A fact which justifies in this case theimplementation of special ferroresonanceprotection measures.

The case of (two poles) VTs connected betweenphases can also be the source of ferroresonancephenomena when one of these VTs is likely tobe supplied, even momentarily, on a singlephase.This can occur, for example, during live work,during non-simultaneous operations on all threephases, on non-multi-pole breaking by fuseblowing or conductor rupture on one phase.

The practical solutions are:v in isolated neutral systems, avoid wye-connection of VT primaries with earthed(primary) neutral either by leaving the neutral ofthe VT primaries unearthed or using delta-connection for the VTs,

v if wye-connection of primaries with earthedneutral is used (for example to measure zero-sequence voltage) in an isolated neutral systemor on a system whose earthing system cannot beanticipated:- use design measures to make magnetic corework at lower induction value (around 0.4 to0.7 T) so that overvoltages are unable to initiateferroresonance, with at least a ratio of 2 betweenthe saturation bend voltage and rated voltage,- introduce losses by means of one or more loadresistances whose value is sufficiently low toeffectively damp the phenomenon, while yetensuring that total power consumption complieswith required precision conditions.

The following method can be used to computeload resistances values. It should be applied toeach case individually:

v VTs with one secondary winding:A damping resistor R is connected to the


secondary of each VT (see fig. 11 ), ifconsumption downstream of the VT is notsufficient. In this case the resistors continuouslyabsorb power as soon as the VTs areenergized.The recommended minimum values for theresistance R and power PR of this resistanceare:

R = U

k Ps2

t −Pm, P =

UR

s2

R

where:Us : rated secondary voltage (V),k : factor between 0.25 and 1 such that errorsand service conditions remain within the limitsspecified by standard IEC 186 [1] (k Pt is forexample around 30 W for a 50 VA rated output).Pt : VT rated output (VA),Pm : power required for measurement (VA).

v In the case of VTs with two secondary windings(one secondary winding for measurement, andone residual voltage secondary winding alsoknown as a tertiary winding), it is advisable toconnect a resistance to the terminals of the opendelta connected tertiary windings of the threetransformers (see fig. 12 ). The advantage ofthis damping device is that it does not affectmeasurement accuracy or introduce losses innormal (balanced) operating conditions, but onlyin unbalanced conditions in order to damp thephenomenon.

The recommended minimum values for theresistance R and power PR of this resistanceare:

R = 3 3 U

Ps

e

2, P =

3 U

RRs( )2

where:Us : rated voltage of the VT secondary,connected to the resistance (V)Pe : rated thermal burden of the VT secondarywinding concerned by the resistance (VA).

The rated thermal burden (in VA) is the apparentpower that can be supplied by the VT to thesecondary without exceeding the limits of normaltemperature rise, without precision requirements.Resistance R must be chosen to ensurepermanent dissipation of power PR.

For example:

TT = 10000: 3 - 100: 3 - 100: 3 V ,Pe = 100 VA

(Us = 100/3)

R = 3 3 100/3

100 = 57.7

( )2

Ω ,

P = 3 100/3 /57.7 = 173 WR2×( )

(standardised value immediately above57.7 Ω : 2 120 //× Ω , 2 140 W× )

Secondary

N N N

A A A

n n n

a a a

R

R

R

1

2

3

Fig. 12 : protection device against the risk offerroresonance for VTs with two secondaries.

Secondary

Residual voltagesecondary

R R : Damping resistor

1

2

3

N N N

A A A

n n n

a a a

dn dn dn

da da da

Fig. 11 : protection device against the risk offerroresonance for VTs with one secondary.


c To prevent ferroresonance occurring on atransformer accidentally energized in only one ortwo phases (see fig. 7 ), practical solutionsconsist of:v lowering the value of the capacitance betweenthe circuit breaker and transformer below itscritical value by using, for example, a circuitbreaker cubicle closer to the transformer orplacing circuit breakers just upstream of thetransformers and closing them only when voltagehas been restored to all three phases,v avoiding use of transformers delivering anactive power which is lower by 10% than itsrated apparent power,v avoiding no-load energizing,v prohibiting single-phase operations or fuseprotection, blowing of which results in single-polebreaking,v prohibiting live work on a cable-transformerassembly when the cable length exceeds acertain critical length,v resistance-earthing of the supply substationneutral,v solidly earthing of the neutral (permanently oronly during energizing and deenergizingoperations) of a transformer whose primary iswye-connected (available neutral).

c Case of isolated neutral systemsTo avoid risk of ferroresonance withover-inductive PIMs, an impedance can beinstalled between the transformer neutral andearth.This is the “impedant neutral” solution.An impedance whose purely resistive value

Fig. 13 : PIM auxiliaries.

at 50 Hz is around 1500 Ω is recommended for ashort LV power system supplied by an MV/LVsubstation [4] .In MV, DC injection PIMs are connected,according to voltage, either to a VT loaded by aresistance (see fig. 13 ) or to a resistanceconnected in series with the PIM (Un < 5.5 kV).

The star point of the primaries of all otherwye-connected VTs connected to the sameisolated neutral system must also be earthed bymeans of a capacitance (P1 plate). Thismeasure is often overlooked in extensions andsub-switchboards.

c Case of MV power systems earthed through areactor (see fig. 9b )The following measures can be taken inresonant earthed systems:v overcompensate the power frequencycapacitance component of the earth-faultcurrent by detuning the neutral earthingreactance,v add a resistive component to increase earthcoil losses.The measure taken must not, however, affectself-extinguishing of earth faults which is one ofthe aims of the resonant earthing system.

c As regards a transformer fed through acapacitive power system (see fig. 10 ), thebest solution consists of avoiding riskyconfigurations when active power delivery isless than 10 % of the transformer ratedpower.This risk is considerable during low load periods(holidays, night time).

C

1R

VT

R

P1

PIM

VT

C

3R

R

P1

PIM

b - Unavailable neutrala - Available neutral


Lωo

Lsωo

> Lωo

Cωo

nn

n <

Lsωo

Cωo

n

VL (I)

I

V

4.2 Practical evaluation criteria

The equation systems describing the behaviourof ferroresonant circuits cannot generally besolved in analytical manner, but require use ofnumerical methods. However, as regards theseries ferroresonant circuit, it is possible toanalytically predict the existence of periodic,fundamental (at the power pulsationω0 and n-subharmonic ferroresonance (ofpulsation ω0 /n where n is an integer).L is the value of the non-linear inductance inlinear unsaturated state. Ls corresponds to thesaturated state.

Fig. 14 : values of C incompatible with periodicferroresonance.

Synthetic insulated screened cables Overhead lines

Rated voltage Cross- PE EPR 20 kV 90 kV 150 kV 220 kV 400 kVUo/U(Um) (kV) section double circuit

(mm2) line +earth cable

5.8/10 (12) 16 0.17 0.21 5 x 10-3 4.8 x 10-3 5.6 x 10-3 5.5 x 10-3 7.1 x 10-3

240 0.43 0.52

8.7/15 (17.5) 25 0.16 0.19

240 0.34 0.41

12/20 (24) 35 0.15 0.18

240 0.28 0.35

Periodic ferroresonance is impossible if one ofthe criteria below is verified:

c n

C

L

0

0

ωω

>n

where n is the order of the subharmonic.(n equals one in the case of fundamentalferroresonance)

c n

C

L

0

s 0

ωω

<n

Both criteria can be illustrated by figure 14 : themagnetization curve enables knowledge of Land Ls.v The value of Ls can be supplied by themanufacturer,v The value of the magnetizing inductance of aVT or of a single-phase power transformer is:

L = 1

P UmH

00 n

ω

Un

I02 2− ( )

( )

Where:Un : rated voltage (kV),I0 : no-load current under Un (A),P0 : iron losses under Un (kW).

v Typical capacitance valuesThe magnitudes of the zero-sequencecapacitances of synthetic insulated, screenedunderground cables and of overhead lines aregiven in figure 15 .

As regards cables, we advise the reader toconsult the analytical formulae provided by cablemanufacturers or the value tables given instandard NF C 33-220.

v Finally, for ferroresonance to be sustained,source coupling must be able to compensatecircuit losses [10] .

Fig. 15 : zero-sequence capacitances (in µF/km) of cables and lines (typical values).


5 Studies for predicting and understanding

The ideal solution is to predict the risk offerroresonance for all possible power systemparameter values in normal and exceptionaloperating conditions, taking account of futuremodifications to the installation. A safety margincan thus be provided with respect to the dangerous

area, and countermeasures taken. A reliable andcredible solution requires implementation ofnumerical methods suited to the study of certaintypes of ferroresonant states.

5.1 A study example

The procedure, together with implementationand practical application of these methods areillustrated below.

The case studiedIn order to reduce breaking times due tomalfunctions, MV rural public power systemoperators operate switch-disconnectors byremote control.Figure 16 shows the example of a remotecontrolled breaking cubicle (ACT), free standingand connected to an MV overhead power systemby an over-underground connection.A voltage transformer (VT) connected betweentwo phases (phase 1 and phase 3) provides the

Fig. 16 : remote controlled breaking cubicle (ACT), free standing, connected to an overhead MV power system.

independent LV power supply of the SRI typeremote control box (SRI = Switch Remote controlInterface).This study is motivated by explosive failure ofVTs in various installations of the same kindwhile energized for the carrying out of live work(live installation of the jumpers connecting theoverhead line to one of the two over-underground connections).

The explosion of the VT was observed 5 to55 minutes (depending on the case) afterinstalling the jumper for phase 1 of the pole(switch-disconnector closed and pole jumpernot installed).

SRI

ACT

20 km 15 m 15 m

Jumpers Jumpers

Another feeder fromthe same substationor feeder from adifferent substation

Cable

VT(between two phases)

Cable

HV/MV substation

1 2

2

1


Parameters of the power system studied:

TT : 20000/230 V, 100 VA,

HV/MV substation: 63 kV/21 kV, 10 MVA,

Neutral earthing resistance: 40 Ω,20 km overhead line between the HV/MVsubstation and pole ,

ACT-pole connections: 15 m cable (radial field),150 mm2, Aluminium.

At working frequencies (50 Hz andsubharmonics), the overhead line can bemodelled by its longitudinal impedance. Theequivalent diagram is then the one shown infigure 17a .

Where:

e(t): sinusoidal voltage source

e(t) = E cos(100 π t)

where E = 21000 2 3 = 17000 / .

R: neutral earthing resistance + resistance of theHV/MV transformer + longitudinal resistance ofthe overhead line.

L : self-inductance of the HV/MV transformer +longitudinal self-inductance of the overheadline.

C0 : zero-sequence capacitance of the 30 mcable (C0 = 6.7 nF).

Lp : (non-linear) magnetizing inductance of theVT seen from the primary. Its characteristicis determined from voltage-currentmeasurements (magnetization curve) taken onthe no-load VT.

Rp : primary winding resistance.

R2 : resistance equivalent to iron losses andto hysteresis losses. R2 is assumed to beconstant and independent of rms voltage andpeak flux.

This circuit can be simplified as the diagramshown on figure 17b . This circuit is a seriesferroresonance circuit and thus favourable toferroresonance.

Application of appropriate methods allows thepossibility of ferroresonance to be studiedbetween the VT and the capacitance to earth ofthe 30 m length of cable connected to thefree phase (not connected to the source) ofthe VT.

Determining the areas at risk

The bifurcation diagram φ (E) in figure 18shows that for power system parameter values,ferroresonance (point Mf) may occur for nominalphase to neutral system voltage (17 kV).

The area at risk is located above the greenbifurcation line shown in figure 19 in the plane(C,E) (position of points M1 in the bifurcationdiagram in figure 18 ).The capacitance value of 6.7 nF is located amplyin the area at risk.

Fig. 17 : installing the jumper for phase 1 of the pole .

Fig. 18 : bifurcation diagram.

φ(Wb)^

00 5 10 15

17 kV

20 25 30 35

20

40

60

80

100

120

140 M1Mf

Mn

E(kV)

0

17 kV1

2

3

4

5

6

7

8

9

10

E (kV)

C0 (nF)

0 5 10 15 20 25 30 35

Fig. 19 : bifurcation lines.

1

Co Co

Re(t)

Rp

R2VT

L

Lp

R1

R2

Co

e(t)

Lp

a - Equivalent single-phase diagram

b - Simplified diagram

1


This condition is not sufficient to conclude thatthe phenomenon can really exist.

Time domain digital simulations

The question is then to know whether thepossible initial condition values are responsiblefor initiating ferroresonance. A time domaindigital simulation of the three-phase circuit willprovide an answer.

In the case in hand, the cable capacitance isdischarged and the initial conditions are thusmainly determined by the energizing conditionsof a phase of the over-underground connection.These energizing conditions are directlydependent on the operator, and cannot becompared with the conditions obtained byenergizing by switch.

The waveforms obtained by simulations andillustrated on figures 20a and 20b show thatthe live work procedure described above canresult in a sustained ferroresonance.

Equipment withstand to stresses

We must then determine whether this state isdangerous for equipment and whether it canaccount for VT failure. As the overvoltagescomputed at the terminals of the VT in theferroresonance steady state are lower than VTrated power-frequency short-duration withstandvoltage (50 kVrms/1 min, i.e. 70 kV peak), thepossibility of thermal and/or mechanicaldestruction must be investigated.

Fig. 20 : time domain numerical simulation.

As the rated short-circuit voltage of the VTis 2.5%, it is designed to withstand withoutdamage at least the mechanical and thermaleffects of an external short-circuit current of 40 Infor the duration of 1 sec (IEC 186). This enablescomputation of the thermal withstand time tminunder Irms:

40 tn min I I( ) × = ( ) ×2 21 s rms

hence t = 40

0.13 = 2.4 s

×( )( )100 20000 2

2

Consequently, there is at least one ferroresonantstate resulting in VT failure by thermal effect, andspecial precautions must be taken.

Solutions

In this precise case, the following methods forincreased ferroresonance protection can besuggested:c load the VT secondary windings: appropriatenumerical methods will enable you to determinethe value of this load,c perform switchings when the equipment is de-energized,c modify the energizing procedure. First installthe three jumpers of pole : the switch-disconnector is open. The switch is then closedon the three phases simultaneously supplyingthe two phases of the VT. The three jumpers ofpole can then be installed.

The methods used in this study (and others) aredescribed in the following sections.

2

1

-0,31,8 1,85 1,9 1,95 2

-0,20

-0,10

0

0,10

0,20

0,30

s

A

-750 0,2 0,4 0,6 0,8 1

-50

-25

25

0

50

75

s

kV

b - VT primary voltagea - VT primary current


5.2 Modelling, a mathematical approach

The following means are used to study transientelectromagnetic problems and ferroresonance:

c Analog simulationBased on a reduced model representation ofpower system components, this method has theadvantage of obtaining real time results, but thedisadvantage of finding it hard to represent a realcase with any accuracy.

c Time domain digital simulation in transientstateResolution of the system of mathematicalequations describing power system behaviourrequires use of computer tools. As transientferroresonant states are normally long,simulation times are lengthy and studies costly.

Since ferroresonance is extremely sensitive toparameter values and to initial conditions ofwhich we have little practical knowledge, a studymust be conducted for each possiblecombination. This is not realistic. The methodsdescribed above are therefore not suited to thesearch for an overall view of a power systembehaviour. To guard against the drawbacks ofsuch methods, mathematicians have developedthe following methods:

c Methods for direct computation of steady stateThese methods enable solutions to be computedin steady state without requiring computation ofthe transient state which is normally lengthy inthe case of ferroresonance.

c Continuation methodThe mathematical framework most suited to thegeneral study of dynamic systems behaviour isthe bifurcation theory, the main tool of which isthe continuation method. Used jointly with themethods for direct computation of steady state, itdetermines the areas at risk.

Note that time domain digital simulation hasundeniable advantages:v it confirms the results of another method for agiven configuration and numerical parametervalues,v it specifies waveforms and their correspondingovervoltage/current levels, thanks to finemodelling of power system components.

Numerical methods for computation ofsteady state

Mathematicians have developed frequency andtemporal methods enabling computation ofsolutions in steady state without requiringcomputation of the transient state.

The following methods enable study of periodicferroresonance (fundamental, subharmonic).The main frequency method is the Galerkinemethod. The main temporal methods are thePoincaré map fixed point method and theperturbation method.

c Galerkine methodThis method consists of finding an approximatesolution for the system of non-linear differentialequations describing power system behaviour.The solution is broken down in the form of aFourier series limited to the order k in order tofind periodic solutions. The unknown quantity isreplaced by its expression. This results in asystem of 2k+1 equations with 2k+1 unknownquantities (the Fourier components).

c Poincaré map fixed point methodAs the solution is periodic with a period nT, themethod consists of iteratively searching for aninvariant solution by numerical simulation of thesystem over a period nT.

c Perturbation methodThis method consists of simplifying the system ofequations describing circuit behaviour bycancelling certain parameters such as the lossesand/or amplitude of the voltage source. Once theresulting equation has been solved, thecancelled terms are introduced by a limiteddevelopment around the solution previouslyobtained. This method is of particular interestwhen combined with the continuation method.

Continuation method

This numerical method based on an iterativeprocess allows the study of the influence of aparameter (e.g. voltage source amplitude) on thesolutions (e.g. the flux in a transformer core) ofthe equations describing the system behaviour(e.g. the electrical power system). The points onthe solution curve are obtained step-by-step froma known solution:

With knowledge of a solution x0 corresponding tothe value E0 of the chosen parameter, theneighbouring solution x1 is obtained byinitialising system resolution by the solution x0and the neighbouring value E1 of the parameter.The resulting step-by-step curve is known as abifurcation diagram.The resolution method used for ferroresonantcircuits is one of the steady state directcomputation methods, thus ensuringindependence from initial conditions.


Fig. 21 : bifurcation diagrams and lines (fundamentalferroresonance).

c Bifurcation diagramsAs regards the series ferroresonant circuit infigure 21a , examples of bifurcation diagramsaccording to amplitude E of the voltage sourcefor two values (Ra and Rb) of parallel resistanceR2 are plotted on figure 21b .For a given value of R2, these curves reveal twospecific points (M1 and M2 for R2 equal to Ra)known as limit points for which there is a changein stability (see § 2.2.).

c Bifurcation linesFor each point, M1 and M2, the correspondingvalue of E known as the critical value can bedetermined (E1 for point M1, and E2 forpoint M2).The line plotted in the plane of two parameters,for example E and R2, of pairs (E1, R2) and(E2, R2) corresponding to singular points fordifferent values of R2, results in curves(see fig. 21c ) representing the boundarybetween two different operating conditions,namely normal conditions and ferroresonantconditions. The resulting curves are known asbifurcation lines.

Bifurcation lines can also be plotted in the planeof various parameter pairs such as (R2, C) or(R1, E).

c IsolatsThe bifurcation diagrams in figure 21billustrate the case of fundamentalferroresonance. These curves in fact passthrough the obvious solution (0,0), the point fromwhich the user can also initialise the continuationmethod.However, continuation of isolats, which areclosed curves, is more tricky, as it is necessaryto know a solution belonging to this isolatedcurve (isolat) in order to initialise the continuationmethod. The perturbation method can be used toobtain this solution. The resulting curvescorrespond to a frequency different from that ofthe voltage source, such as for example thesubharmonic isolat shown in figure 22 .For a voltage E between the ends of the isolat,the slightest perturbation may cause the systemto move from the green curve to the black curveand vice versa.

c Practical use of the bifurcation diagramsWith given circuit parameter values (and inparticular R2 equal to Ra), figure 21b showsthat as long as E remains less than E1, afundamental ferroresonance cannot be sustainedunder steady state condition.

c Practical use of the bifurcation linesIf in all power system operating conditions, theamplitude E of the voltage source is less thanEn, figure 21c shows that a resistance value ofR2 less than Rn guarantees that a fundamentalferroresonance cannot occur.

E

VL Stable stateUnstable state

Subharmonicmode(isolat)

Fundamental mode

Fig. 22 : isolat of a subharmonic mode.

CE

R1R2

a - Series ferroresonant circuit

b - Bifurcation diagram according to E

c - Bifurcation lines in the plane R, E

R2 = Rb (< Ra)M'1

E

R2 = Ra

M1

M'2

E1 E'1 E2 E'2

M2

φ

Stable stateUnstable state

EE1 E'1En E2 E'2

R2

M2

M'2

M1

M'1

Ra

Rb

Point of M2 , M'2 ...Point of M1 , M'1 ...

Rn


Given the inaccuracy of the numerical values ofthe power system parameters, the practicalsolution is normally to provide a safety marginwith respect to the areas at risk.

It should be stressed that not all the stateslocated in the areas at risk, whose boundaries

are formed by the bifurcation lines, arenecessarily reached in practice as special initialconditions have to be verified. Time domainnumerical simulations are used to determinewhether these initial conditions are possible inreality.

5.3 Summary

1 - Identifying a configuration at risk

2 - Simplifying the circuit

3 - Determining the power system parameters- Non-linear characteristic of the inductance- Variations and tolerances of R and C.

4 - Determining the areas at risk (overview)- Continuation method- Steady state computation method

5 - In practice, do initial conditions result inan area at risk?Time domain digital simulations.

6 - Consequences on thermal, mechanical anddielectric withstand of equipment.

7 - Proposed solutions.

a

R

Below is a plan for a ferroresonance studyconducted using the methods described above.The various steps are listed in chronologicalorder in the following table.


6 Conclusion

The risk of ferroresonance must be taken intoconsideration as early as the design stage of anelectrical installation. Vigilance is also called forwhen servicing or extending a power system.Basically, risk control requires knowledge ofdangerous configurations and of the conditionsin which the phenomenon can exist. If aconfiguration considered critical is inevitable,only a detailed study will enable an assessmentof the risks and an evaluation of the efficiency ofthe solutions to be provided.This document has provided LV and HV powersystem designers and installers with an insightinto the precautions to be taken to prevent thisodd and often dangerous phenomenon, andshould facilitate their dialogue with specialists.Operators will have found a few elements fordiagnosis enabling them to rightly suspect thatferroresonance is present. They now understandthat not all unexplainable breakdowns can beascribed to this phenomenon!

It seems a good idea at this point to brieflyremind readers of the events initiatingferroresonance and of the configurations at risk:

c A few examples of phenomena likely to causeferroresonance:v capacitor switching,v insulation faults,v lightning,v transformer switching.

c A few configurations at risk deservingparticular attention:v voltage transformer (VT) between phase andearth of an isolated neutral power system,v long and/or capacitive cables or lines supplyinga transformer,v fuse protection where blowing results in a non-multi pole breaking,v unloaded or lightly loaded voltage or powertransformer,v voltage transformer working at saturation limit,v over-powerful voltage transformer.

Readers seeking additional information orinterested in case studies are invited to consultthe extensive bibliography on this subject.


Appendix 1: Bibliography

Standards

[1] IEC 186: Voltage transformers

[2] IEC 71: Insulation co-ordination

Cahiers Techniques

[3] Neutral earthing in an industrial HV network,F. SAUTRIAU,Cahier Technique Merlin Gerin n°62.

[4] Earthing systems in LV, B. LACROIX,R. CALVAS, Cahier Technique Merlin Gerinn°172.

[5] Overvoltages and insulation coordination inMV and HV, D. FULCHIRON, Cahier TechniqueMerlin Gerin n°151.

[6] Harmonics in industrial networks,P. ROCCIA, N. QUILLON, Cahier TechniqueMerlin Gerin n° 152.

[7] Active harmonic conditioners and unity powerfactor rectifiers, E. BETTEGA,J.N. FIORINA, Cahier TechniqueMerlin Gerin n° 183.

Various works

[8] Revue des phénomènes de ferrorésonancedans les réseaux haute tension et présentationd’un modèle de transformateur de tension pourleur prédétermination, N. GERMAY,S. MASTERO, J. VROMAN, CIGRE Session de1974-21-29 août.

[9] Ferroresonance in a transformer switchedwith an EHV line, E.J. DOLAN, D.A. GILLIES,E.W. KIMBARK, IEEE Power Apparatus andSystems, 1972.

[10] Contribution théorique et expérimentale àl’étude des phénomènes de ferrorésonancemonophasée, P. MAHY, SRBE, mars 1972.

[11] Transformateurs de mesure-Généralités.Théorie. Fonctionnement, J.P. DUPRAZ,D 4720, Techniques de l’Ingénieur, traité Génieélectrique.

[12] Ferroresonance study using Galerkin Methodwith pseudo-arclength continuation method,G. KIENY, G. LE ROY, A. SBAI, IEEE PWD,Vol. 6, No.4, October 1991.

[13] Ferrorésonance dans les réseaux,C. KIENY, A. SBAÏ, D 4745, Techniques del’Ingénieur, traité Génie électrique.


Rated Rated thermal Minimum Practical choice (in the 140 W series)secondary burden of the computedvoltage secondary resistance Standardised Resistance

winding concerned (Ω) resistance powerby the resistance (Ω) (W)(VA)

50 115.5 120 83

100 57.7 2 x 120 in // 2 x 83

200 28.9 2 x 100 in // 2 x 100

50 139.7 150 80.7

100 69.9 2 x 150 in // 2 x 80.7

200 34.9 2 x 100 in // 2 x 121

50 346.4 390 77

100 173.2 2 x 390 in // 2 x 77

200 86.6 2 x 220 in // 2 x 1363 x 390 in // 3 x 77

50 419.2 470 77

100 209.6 2 x 470 in // 2 x 77

200 140.8 2 x 390 in // 2 x 93

110

3

1103

1003

100

3

Appendix 2: Selection guide for VT load resistances

Residual voltage windings connected in open delta, closed on a resistance.

Ferroresonance 3 Sneider

Documents

Transcript of Ferroresonance 3 Sneider