ISSN 1751-8687 Frequency domain analysis of a power ...

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Published in IET Generation, Transmission & DistributionReceived on 1st April 2010Revised on 23rd September 2010doi: 10.1049/iet-gtd.2010.0233

ISSN 1751-8687

Frequency domain analysis of a power transformerexperiencing sustained ferroresonanceC.A. Charalambous1,* Z.D. Wang1 P. Jarman2 J.P. Sturgess3

1School of Electrical and Electronic Engineering, University of Manchester, Manchester M13 9PL, UK2National Grid, Warwick Technology Park, Warwick CV34 6DA, UK3Alstom Grid Research & Technology, Stafford ST17 4LX, UK*Currently with School of Electrical and Computer Engineering, University of Cyprus, Nicosia, CyprusE-mail: [email protected]

Abstract: Most transformer finite-element electromagnetic analysis is performed in the frequency domain (with an implicitsinusoidal variation of flux, current and voltage), since the calculation of the transient time-domain solution, particularly ofmultiple cases or conditions, is currently restricted by the conventional computational power (especially in three dimensions).At high levels of saturation, though, core flux can be very non-sinusoidal. Where sinusoidal conditions do not hold, forexample ferroresonant currents, a time-domain solution is normally indicated; however, this article discusses an alternativemethodology for periodic modes of sustained ferroresonance, which can be facilitated in the frequency domain. Specifically,the problem is approached by considering the ferroresonance excitation waveforms as a harmonic series. It is demonstratedthat the basic frequency of a sustained resonance condition not only contributes to the overall saturation but also controls theway the core moves into and out of saturation. All the other frequencies contribute to the extreme of core saturation. Theproposed method is validated through ferroresonance field test recordings.

1 Introduction

Ferroresonance is attributed to the interaction between the non-linear characteristics of the transformer core and the system’scapacitance. It is a low-frequency phenomenon that may occurin a power transformer when one side of a double circuittransmission line, connected to the transformer, is switchedout. The transfer of power from the adjacent energised linethrough capacitive coupling into the de-energised line canexcite the unloaded transformer. Ferroresonance is primarilyassessed by simulation practices based on analytical techniques,and which describe the interaction of the power systemcomponents with the transformer model. A comprehensiveoverview of the ferroresonance phenomenon with the associatedsimulation models is given in [1–3].

Over-fluxing of the core may cause localised flux leakage thatcan lead to heating of the tank, clamping frame, tie rods and/orflux collectors of the transformer. Therefore the determinationof the susceptibility of transformers’ components towardsferroresonance requires a detailed model that has the directbenefit of providing visually the excessive losses (quantitativelyand topologically) attributed to flux redistribution during coresaturation. Certainly, the heating of any component has to beconsidered when addressing ferroresonance-related effects inpower transformers.

The finite element (FE) method is applicable to this type ofcalculation and steady progress has been made over the yearsin its application to problems of practical complexity [4].One concern when dealing with FE modelling is that there isno way of satisfactorily representing all the leakage and stray

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fields within a loaded transformer in 2D (at least, not withoutmaking many, difficult to calibrate, approximations). This isbecause the flux in a 2D model can only be in the plane ofthe model and this is not a realistic scenario. Nevertheless, acombination of 2D models can be adequate to assess apessimistic scenario in terms of component relative lossesbecause of ferroresonance, as presented in [5].

However, it is widely accepted that only three-dimensional(3D) models can represent realistically the detailed topologicalarrangement of transformer components and consequentlytheir electromagnetic behaviour. Such 3D models can bebuilt within FEA softwares, but the calculation of thetransient time-domain solution, particularly of multiple casesor conditions, is restricted by the current computationalpower (although 2D calculations are made routinely forsimulations of normal operation).

Bearing in mind the restrictions imposed by the complexityof time-domain simulations in three dimensions, a solutionin the frequency domain is generally preferred; particularlyin the case where the excitation waveform is sinusoidal andthe variation of core saturation through the time cycle issmall [6, 7]. However, where sinusoidal conditions do nothold, for example ferroresonant currents, a set of modellingapproximations need to be incorporated if the frequencydomain is to realistically assess the ferroresonant conditions.

The ultimate aim of our programme of work is to show thatthe leakage fluxes calculated from a 3D frequency-domainsolution can be used to find the profile and distribution ofthe consequent losses in components such as the tank,frames, tie rods and collectors under ferroresonance

IET Gener. Transm. Distrib., 2011, Vol. 5, Iss. 6, pp. 640–649doi: 10.1049/iet-gtd.2010.0233

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conditions. The purpose of the work reported here is morerestricted: to verify that a frequency-domain solution canaccurately model the terminal voltage of a transformerundergoing ferroresonance. This is a calculation that can beperformed with reasonable accuracy in two dimensions,since it principally concerns the core where the field is,essentially, 2D.

2 FE modelling and solver

This paper has utilised the SLIM FEA software [8]. Sturgess[9] discusses the state of the art of this modelling endeavourwith particular reference to power transformers. The mainanalysis in this paper uses the 2D frequency-domain solver,slim-ne, which models time-varying magnetic fields usingthe complex magnetic vector potential (A) in Cartesiancoordinates. The relationship between winding flux linkageand average vector potential is given in [10]. All currents,potentials and fields are assumed to vary sinusoidally in time.

As the analysis is limited to 2D, the transverse view model(Fig. 1) is used as it can establish the level of core saturationunder ferroresonance conditions. This is because thetransverse view can correctly represent the flow of fluxfrom limb to limb and from phase to phase. The 2D modelincorporates the ‘core depth’ factor (defined as the ratio ofthe core’s area to its diameter), to ensure that there is thecorrect amount of flux in the model. The core depth is usedas a multiplying factor in the voltage calculation. (The FEcalculation in 2D gives flux linkage in terms of volts permetre in the third dimension. The core depth value is usedto convert the V/m to V.)

The windings of the transformer are treated as currentsources and are assumed to be sufficiently finely strandedand transposed so that no eddy currents are induced inthem. Directional anisotropy and the laminated nature of thecore are modelled via stacking factors [11].

3 Transformer description

The modelling is based on a 400/275/13 kV autotransformer,for which ferroresonance field test recordings are available.The core is made out of Unisil steel 27M4. Fig. 1 illustratesthe FE mesh used and Table 1 gives the principal coredimensions.

4 Ferroresonance currents imposed

A number of switching tests have been performed on a circuitconfiguration which was known to exhibit ferroresonance.These tests have yielded field recordings of two periodicsustained modes of ferroresonance [12], namely, the

Fig. 1 2D FE mesh for transformer


fundamental (50 Hz) and the sub-harmonic (162/3 Hz).Existing experience suggests that the sustained fundamentalmode (50 Hz) of ferroresonance is considered the moresevere case because of the continuous high energytransferred to the transformer. With a sub-harmonicresonance of 162/3 Hz, the induced voltages and currentsare relatively low and therefore the consequent risk to thetransformer is reduced, but not eliminated. The specificdetails of the tests and the subsequent modelling work aregiven in [13].

Typical waveforms of the three-phase ferroresonant currenthave been selected; Fig. 2 illustrates snapshots (two periods)of the field current recordings of fundamental mode and sub-harmonic mode ferroresonance, respectively. It should benoted that these currents are not balanced, that is they donot sum to zero at every instant in time.

Inspection of the snapshots also shows that the waveformsare not truly periodic. However, to enable a frequency-domain solution, it has been assumed that the currentsare periodic, which is valid since the magnitudes of theadditional harmonics that arise from the lack of periodicityare negligible in comparison with the principal ones.

5 Problem

Specifically, the problem is approached by considering theexcitation (current) waveforms as formed by a harmonicseries, derived from the measured currents by Fourieranalysis. For each Fourier frequency, the harmonic currentat that frequency is applied to the FE model, taking accountof the phase of the current as well as the magnitude.

Following this, the harmonic voltage at each frequencyis calculated from the flux linkages deduced from the FEmodel, taking account of the phase of the voltage as wellas the magnitude. Finally, the overall voltage waveform isreconstructed by summing the time-domain voltages foreach frequency. The overall waveform deduced is validatedagainst field test recordings.

It is worth mentioning that, since ferroresonance is a low-frequency effect, the iron losses in the laminations have onlya secondary effect in comparison to other losses presentduring ferroresonance, such as eddy currents. In the ‘IEEERecommended Practice for Establishing TransformerCapability When Supplying Non-sinusoidal Load Currents’[14], iron losses are in essence ignored, because of theirrelatively small magnitude in comparison to losses in thecoil windings.

5.1 Fourier decomposition of excitationwaveforms

Fourier decomposition has been performed on a single cycleof each phase of the selected current waveforms of both thefundamental and the sub-harmonic mode ferroresonance. Asfar as the fundamental mode case is concerned the timeperiod chosen was 0.02 s (50 Hz), whereas the time period

Table 1 Core dimensions

core diameter 800 mm

distance between phase centres 1694 mm

unwound phase centres 1115 mm

leg length 2720 mm

yoke diameter 480 mm

outer yoke diameter 320 mm

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Fig. 2 Recorded three phase – ferroresonance currents

a Fundamental mode ferroresonanceb Sub-harmonic mode ferroresonance

chosen for the sub-harmonic case was 0.06 s (162/3 Hz). Theresulting harmonic series is given by

I = I1 cos(vt + w1) + I2 cos(2vt + w2) + · · ·+ In cos(nvt + wn) (1)

Similarly, both the magnetic field (H ) waveform (arisingfrom the current) and the flux density (B) waveform (givingrise to the flux linkage) can be described as harmonic series

H = H1 cos(vt + a1) + H2 cos(2vt + a2) + · · ·+ Hn cos(nvt + an) (2)

B = m1H1 cos(vt + a1) + m2H2 cos(2vt + a2) + · · ·+ mnHn cos(nvt + an) (3)

To reduce the number of calculations required, it is necessaryto determine the ranking of the harmonic orders so as to selectfor computation only those frequencies that significantlycontribute to the voltage calculation and the losses. As far

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as the ranking is concerned, it should be kept in mind thatflux is proportional to current whereas the voltage isproportional to the rate of change of flux. Thus, in asinusoidally varying system, voltage is related tothe inductance by the product of frequency and current asgiven by

V = − dF

dt= − d(LI)

dt= 2pfLI cos(2pft + g) (4)

The ranking order was deduced by multiplying the resultantmagnitude of the Fourier coefficients of the current by theharmonic order, as a surrogate for the frequency. Theranking by harmonic order can be determined from Fig. 3,which shows that all even harmonics and the oddharmonics beyond the 41st and 21st (fundamental andsub-harmonic, respectively) are negligible. The dominantharmonics of the fundamental mode ferroresonance are,in order, the odd harmonics 3rd (150 Hz), 5th (250 Hz),fundamental (1st) (50 Hz), 7th (350 Hz) etc.


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Fig. 3 Fourier decomposition – resultant current coefficients

a Fundamental modeb Sub-harmonic mode

5.2 Degree of harmonic contribution to coresaturation

Fig. 4 illustrates one phase of the recorded voltage and currentwaveforms for the fundamental mode (50 Hz) and sub-harmonic mode (162/3 Hz) ferroresonance. The subplotsshow that there is a quadrature between the current andvoltage, a peak current occurring at a voltage zero, which isalso the time of maximum rate of change of voltage. Thus,a peak in the current waveform is associated with a rapidchange in the polarity of the voltage.

For each frequency-domain solution, it is necessary tospecify the permeability distribution that will be used tolink the magnetic field and the flux density in the FEcalculation [i.e. the constants m1 to mn in (3)]. Followingthe observations made, it was decided to perform a set ofnon-linear time-domain tests to determine the degree ofeach harmonic’s contribution to the core saturation. Thequantification of their contribution, in the light of the core’snon-linearity, is important.

Fig. 5 illustrates a number of time-domain currentwaveforms reconstructed from the summation of specificfrequency-domain coefficients as determined from the


Fourier decomposition, see Fig. 3. These time-domainwaveforms, one for each of the three phases, were used asinputs to the SLIM non-linear time-domain solver, utilisingthe 2D transformer mesh, of Fig. 1. Magnetic non-linearitywas treated in the normal manner, through interpolation ofthe B/H curve and a Newton–Raphson iterative scheme tohandle convergence to a solution at each time step [15].

The reconstructed time-domain waveforms have beenutilised to study the permeability distribution (m), therebyenabling the assessment of the harmonic’s contributiontowards core saturation. To do this, the concept of thespace-averaged permeability was devised.

The space-averaged value of the core’s permeability ateach time step in the time-domain solution was found froma simple average of the permeability of each elementrepresenting the core in the 2D mesh at that particularinstant in time. The result is a convenient, single numberthat can represent the degree of saturation of a core undercomplex excitation conditions. For this calculation, the corewas excited by all three phase-currents simultaneously.

Fig. 6 illustrates how the space-averaged permeabilityvalue of the transformer core varies over time. Obviously,the space average permeability of the fundamental (1st)

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Fig. 4 Voltage and current field recordings

a Fundamental mode – phase Rb Sub-harmonic mode – phase R

harmonic repeats at a frequency of 50 Hz. It is interestingto note that the space-averaged permeability of thesuperimposed waveforms of the fundamental (1st) and 7thharmonics (Fig. 6b) is almost identical to the space-averaged permeability of the fundamental (1st) harmonic.

However, this distribution differs slightly in the maximumpermeability value reached; therefore it is suggested that theodd harmonics (2n + 1 for n . 3) would mostly contributeto the overall ferroresonance voltage response at the instantswhere the core is out of peak saturation, that is at itshighest permeability.

Similarly, an almost identical average space permeabilityat the lowest values can be observed for the superimposedwaveform of the fundamental (1st) and 13th harmonics andthat for the fundamental (1st) harmonic. This suggests thatthe fundamental (1st) harmonic dominates the effects of the13th harmonic on the permeability distribution, especially atthe lowest values that is when the core is at its peaksaturation. Therefore the contribution of any harmonic withmagnitude less than that of the 13th harmonic is practicallynegligible. The same principle applies for the evenharmonics, which are of negligible magnitude.

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However, the contribution of the two most dominantharmonics (for the fundamental mode case), that is the 3rd andthe 5th, is evident on the average space permeability of Fig. 6a.

Both cases show that there is a significant contributionof these dominant harmonics at the lower permeabilities,corresponding to a significant contribution to the core’smaximum saturation. Moreover, both harmonics alsocontribute to the maximum permeability that is contributeto the time instants where the core is out of its peak saturation.

The preceding section analysed the phase R of thefundamental mode ferroresonance case. Similar time-domainwaveforms were constructed for the other two phases. Fig. 3illustrates the same pattern in the dominant harmonics of allthe three phases present following Fourier decomposition;therefore as also verified, the same principles apply for all phases.

The analysis was also applied to the sub-harmonic modeof ferroresonance. Fig. 2 illustrates that all three phases ofthe sub-harmonic mode ferroresonance scenario are periodicwith a frequency of 162/3 Hz. In addition, Fig. 3b showsthe same pattern in the dominant harmonics of all threephases present (following Fourier decomposition) for thesub-harmonic mode as was found for the fundamental mode.


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Fig. 5 Equivalent time-domain waveforms for phase R reconstructed from their harmonics

a Effect of 3rd and 5th harmonicb Effect of 7th and 13th harmonic

Therefore the time-domain waveforms for this particularscenario utilise the 0.33rd harmonic (i.e. the mode thatcharacterises the response) and the superimposed harmonicsdescribed by 0.33n (where n ¼ 3, 5, 7, 9 . . . etc.). The previousconclusions for the fundamental mode scenario are the same asthe conclusions obtained for the sub-harmonic scenario.

5.3 Core permeability

As stated earlier, the objective of this study is to consider theexcitation waveforms as formed from a harmonic series, withthe non-linear characteristics of the core modelled by a seriesof permeability coefficients – the constants m1 to mn in (3).The simplest choice of coefficient is to make mm ¼ m1 for allvalues of m. However, this would give a linear solution (i.e. thecore would have constant inductance), which is clearly not thecase. The next simplest approach is to use two coefficients,allocating them to different harmonics in some systematicmanner. The analysis performed earlier is now utilised.

Specifically, the important conclusion of Section 5.2 isthat the current harmonic that describes the mode offerroresonance [e.g. fundamental (1st) for the fundamentalcase or 0.33rd for the sub-harmonic case], is correlated


with the core moving into and out of peak saturation (seeFig. 4). However the dominant harmonic (as determined bythe weighted Fourier decomposition) will contribute themost to the maximum saturation of the core as well ascontributing to the sudden change of the voltage polarity.These two effects need to be weighted accordingly (toaccount for the conclusions documented in Section 5.2).

Consequently, one permeability coefficient is assignedto the harmonic that describes the mode of the periodicsustained ferroresonance, whereas the other coefficient isassociated with the dominant harmonic. It then remains tobe decided which of the two coefficients to use for all theremaining harmonics.

These harmonics principally contribute to the overallmaximum saturation, although their contribution is, to adegree, overshadowed by the dominant harmonics. However,at the point on wave where the dominant harmonic current is(instantaneously) zero, the higher harmonics can have asignificant influence on the core permeability. For this reason,they are assigned the permeability of the dominant harmonic.

5.3.1 Calculation of permeability distribution: Thetwo permeability coefficients are determined from the

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Fig. 6 Average space permeability resulting from the interaction of all three phases

equivalent non-linear 2D time-domain study [5, 16]. (Thefield test results cannot be used as they provide noinformation on the core’s overall permeability.)

SLIM’s time-domain model can produce snapshots ofthe distribution of permeability throughout the core at eachtime step, from which is calculated the space-averagedpermeability of the core as a function of time. As expected,this varies throughout the time cycle from maximum tominimum values. Fig. 6 can, thus, be thought of as anindex of the state of the core saturation at any instant or,alternatively, as a measure of the core’s saturating inductance.

The point in the time cycle when the space-averagedpermeability is a minimum is noted. This can be termed thepoint of maximum saturation, and the space-averagedpermeability at this instant can be denoted as mmin. Thepoint of minimum saturation occurs a quarter of a cyclelater. The space-averaged permeability at this instant can bedenoted as mmax. Then, the model becomes

B = mmaxH1 cos (vt + a1) + mminH2 cos (2vt + a2)

+ · · · + mminHn cos (nvt + an) (5)

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It should be emphasised that this 2D time-domain study,performed to establish the level of core saturation, would beused even when the ultimate aim of the overall calculationis 3D. This is justified because the variation of thepermeability in the core is, essentially, 2D even when theleakage fields are very 3D.

Of course, mmax is a single, scalar value and thepermeability distribution from which it was derived was forone particular instant in the time-domain solution. Thispermeability distribution cannot be used directly in thefrequency domain, as the currents in the two solutions arenot the same, so an equivalent set of frequency domaincurrents must be found that will give the same value of thespace-averaged permeability.

A magnetically non-linear, frequency-domain solution istherefore performed using the fundamental-frequencycurrents from the harmonic analysis (I1), and the spaceaverage permeability is found for this condition: m1. Themagnitudes of the individual phase currents in thisfrequency-domain solution are then adjusted (by a commonfactor, k) until the resulting space-averaged permeability isthe same as that found from the time-domain solution at thepoint of minimum saturation, that is mmax = m1(k I1). The


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core permeability distribution in this condition forms thefirst permeability coefficient of (5); the correspondingpermeability pattern is shown in Fig. 7a (fundamental modescenario). For this calculation, the core is excited by thefundamental Fourier components of the individual phasecurrents (which are not balanced) and this gives rise to apartly saturated core pattern.

It then remains to ensure that all remaining harmonicscontribute to the overall core saturation, by associating themwith the permeability distribution of a highly saturated core.In order to do this, the exercise described above is repeated(still with the fundamental-frequency components ofcurrent) but this time, scaling the excitation of thefrequency-domain solution until the resulting space averagepermeability is equal to mmin. The core permeabilitydistribution in this condition forms the second permeabilitycoefficient of (5), the corresponding permeability pattern isshown in Fig. 7a (fundamental mode scenario).

Consequently, Fig. 7 illustrates the core’s relativepermeability patterns corresponding to the non-linearstudies in the frequency domain, for fundamental andsub-harmonic modes, respectively. For the sub-harmonicscenario, the magnitude of the 0.33rd harmonic was used toset the maximum permeability coefficient, as describedabove, that is mmax ,0.33 = m0.33(k ′ I0.33).

Fig. 7 Relative permeability patterns

a Fundamental mode scenariob Sub-harmonic mode scenario


It is evident in subplots B in both fundamental and sub-harmonic scenarios that the core is highly saturated sincethe permeability is mostly low. By contrast, subplots A ofboth scenarios illustrate that the core is only partlysaturated. This is expected in the periodic ferroresonancescenarios: when the line currents of each case of Fig. 2are added up (borrowing the steady-state convention,3I0 ¼ IR + IY + IB), the zero sequence current 3I0 is notequal to zero. This zero sequence current in both scenariosis significantly larger than the magnetisation current anddrives the core into saturation, when the transformer isunloaded.

By this means, the core of the frequency domain model iscalibrated using the space-averaged permeability, settingthe overall core impedance at an appropriate value for thesubsequent leakage field calculation.

5.3.2 Implementation of the proposed method:Having established the saturated permeability distribution

Fig. 8 Comparison – fundamental mode

a Phase Rb Phase Yc Phase B

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using the fundamental-frequency currents, it is then employedin a series of harmonic studies using each of the higherharmonic currents (found in Section 5.1) in turn.

The terminal voltage of a transformer undergoingferroresonance is modelled by summing the weightedcontribution of each of the individual harmoniccomponents. As shown by (4), the voltage is proportionalto the rate of change of flux. Thus, voltage is related to theweighted inductance (which is a function of the weightedpermeability coefficients) by the product of frequency,which in essence, describes the core’s impedance. This isillustrated by

V =∑n

k=1

2p fk Lk Ik (6)

V = Z1IF1 cos (vt + w1) + Z2IF2 cos (2vt + w2)

+ · · · + ZnIFn cos (nvt + wn) (7)

Fig. 9 Comparison – sub-harmonic mode

a Phase Rb Phase Yc Phase B

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Consequently, the periodic voltage waveform is constructedfrom the summation of the individual frequencycomponents, each derived from a finite-element study in thefrequency domain.

6 Validation of the methodology

The validation of the principle described is achieved throughcomparing the summation of the time-domain voltages foreach mode of resonance with the voltage field recordings.Fig. 8 illustrates a comparison between the field test resultsand the equivalent summed time-domain voltages derivedfrom their frequency-domain magnitude, frequency andphase. The figures correspond to the three phases of thefundamental mode ferroresonance. Fig. 9 illustrates thesame comparison for the three phases of the sub-harmonicmode ferroresonance.

The calculated results show a slightly higher harmoniccontent (3rd and 5th harmonics) in the voltage waveformthan the measured ones do. The harmonic voltages arederived using the second permeability coefficient mmin andthe results suggest that there could be modellinguncertainties in the behaviour of the B/H curve at the highestfluxing levels. Unfortunately, there are no measurements atthese highly saturated regions, just extrapolations from lowerfluxing, that we have to depend on. The similarity of thecalculated and measured voltages shows that, by using theconcept of the space-averaged permeability to select twopermeability distributions, the terminal voltage can beaccurately modelled. The first permeability distribution isrelated to the fact that the frequency that describes thesustained resonance condition not only contributes to theoverall saturation but also controls the way the core movesinto and out of saturation. The second permeabilitydistribution is related to all the other frequencies whichcontribute to the extreme of core saturation.

7 Conclusions

This paper discusses a means of facilitating a frequency-domain FE electromagnetic analysis of an autotransformerexperiencing sustained modes of ferroresonance. The needto develop such a methodology arose from the fact thatthe 3D time-domain solution that would normally beconsidered to be necessary is currently not computationallyaffordable because of the problem’s size and complexity.

In this paper, it has been shown that it is possible to usethe measured current waveform to establish two permeabilitydistributions that represent the states of maximum andminimum saturation of the core during the sustainedferroresonant condition. The current waveform is thendecomposed into its frequency components and they areapplied individually to the core, along with the appropriatepermeability distribution, in a series of frequency-domainanalyses. Finally, the frequency-domain solutions aresummed to give the corresponding ferroresonance voltageresponse. This has been achieved through the fact thatthe frequency of a sustained steady-state ferroresonancecondition not only contributes to the overall saturation butalso controls the way the core moves into and out of saturation.

The method was applied to two sustained modes offerroresonance, namely the fundamental (50 Hz) andthe sub-harmonic (162/3 Hz), and the calculated voltagewaveforms were found to be comparable with thosemeasured in field tests. This clearly suggests that the flux inthe model is correctly distributed within the transformer


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core, confirming the validity of the model and theassumptions made.

It is worth noting that the modelling/validation procedurethat has been described for the 2D case can, in principle,work equally well without need for alteration for a 3Dcalculation. Consequently, it is therefore possible toconstruct a 3D frequency-domain model that can provideinformation about the distribution of energy and lossesbecause of the high flux leakage in a transformerundergoing periodic ferroresonance, avoiding the need fora very time-consuming time-domain analysis.

8 Acknowledgments

The contribution of Dr. A. Sitzia of the Alstom Grid Researchand Technology, Stafford, is also acknowledged. Thanks arealso due to AREVA T&D Power Transformers for supplyingdetails of the autotransformer design used in the modelling.

9 References

1 Mukerjee, R.N., Tanggaselu, B., Ariffin, A.E., Balakrishnan, M.:‘Indices for ferroresonance performance assessment in powerdistribution network’. IPST 2003, New Orleans, 2003, pp. 1–5

2 Jacobson, D.A.N., Menzies, R.W.: ‘Investigation of station servicetransformer ferroresonance in Manitoba hydro’s 230 kV Dorseyconverter station’. IPST 2001, Rio de Janeiro, 24–28 June 2001, pp. 1–6

3 Tong, Y.K.: ‘NGC experience on ferroresonance in power transformers andvoltage transformers on HV transmission systems, warning! ferroresonancecan damage your plant’. IEE Colloquium, 12 November 1997

4 Lefevre, A., Miegeville, L., Fouladgar, J., Olivier, G.: ‘3-D computationof transformers overheating under nonlinear loads’, IEEE Trans. Magn.,2005, 41, (5), pp. 1564–1567


5 Charalambous, C.A., Wang, Z., Osborne, M., Jarman, P.: ‘Two-dimensionalfinite element electromagnetic analysis of an auto transformer experiencingferroresonance’. doi: 10.1109/TPWRD.2009.2016629

6 Guerin, C., Meunier, G.: ‘Surface impedance for 3D non-linear eddycurrent problems – application to loss computation in transformers’,IEEE Trans. Magn., 1996, 32, (3), pp. 808–811

7 Guerin, C., Tanneau, G., Meunier, G.: ‘3D Eddy current lossescalculation in transformer tanks using the finite element method’,IEEE Trans. Magn., 1993, 29, (2), pp. 1419–1422

8 SLIM version 3.9.1, # Alstom Grid, October 2006,[email protected]

9 Sturgess, J.P.: ‘Advanced electromagnetic analysis of powertransformers and inductors’. UK Magnetics Society Seminar –Electromagnetics in Power Systems Applications, Alstom Grid UK,Stafford, 18 June 2008, pp. 1–4

10 Sitzia, A.: ‘The relationship between winding flux linkage andaverage vector potential (Amean)’. Technical Note – Slim TechnicalManual Report (Alstom Grid Research and Technology, Stafford, UK,22 July 2003

11 Sitzia, A.: ‘Stacking factors and directional permeabilities inSLIM magnetic solvers’. Technical Note – Slim Technical ManualReport (Alstom Grid Research and Technology, Stafford, UK, 13September 2005

12 Milicevic, K., Emin, Z.: ‘Impact of initial conditions on the initiationof ferroresonance’, Int. J. Electr. Power Energy Syst., 2009, 31, (4),pp. 146–152

13 Charalambous, C., Wang, Z., Osborne, M., Jarman, P.: ‘Sensitivitystudies on power transformer ferroresonance of a 400 kV doublecircuit’, IET Proc. Gener. Transm. Distrib., 2008, 2, (2), pp. 159–166

14 ‘IEEE recommended practice for establishing transformer capabilitywhen supplying non-sinusoidal load currents’. IEEE STD C57.110-1998, The Institute of Electrical Electronics Engineering Inc.,30 March 1999

15 Salon, S.J.: ‘Finite element analysis of electrical machines’ (KluwerAcademic Publishers, Boston, 1995), Chap. 2

16 Charalambous, C.A., Wang, Z., Sturgess, J.P., Jarman, P.: ‘Finiteelement techniques of a power transformer under ferroresonance- atime domain approach’, in preparation

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