sympathetic inrush current

Analysis of Sympathetic Inrush Phenomena inTransformers Using Coupled Field-Circuit Approach

G. B. Kumbhar and S. V. Kulkarni, Member, IEEE

Abstract— Sympathetic inrush current phenomenon occurswhen a transformer is switched on in a power system networkcontaining other transformers which are already energized. Inthis paper, the phenomenon of sympathetic inrush current isinvestigated using nonlinear-transient field-circuit coupled finiteelement formulation. The cases of the transformers connected inparallel and in series have been considered. The results obtainedfor a 31.5 MVA, three-phase transformer are presented anddiscussed in detail. This work also includes the investigation ofparameters affecting the magnitude and duration of sympatheticinrush phenomenon, such as series resistance, switching-on angle,residual flux density and load conditions. These analyses providea sound theoretical basis for a thorough understanding of thephenomenon.

Index Terms— Sympathetic inrush phenomenon, finite elementmethod, field-circuit coupling, nonlinear transient.

I. INTRODUCTION

MAGNETIZING inrush current occurs during the ener-gization of a single transformer connected to a power

system network with no other transformers. However, theenergization of a transformer connected to a network in thepresence of other transformers, as shown in Fig. 1, which arealready in operation, leads to the phenomenon of sympatheticinrush current. Although, severity of the inrush current ishigher during single transformer energization, the sympatheticinrush current is of special importance due to its unusualcharacteristics. The inrush current in a transformer decays,usually, within a few cycles, but the sympathetic inrush currentpersists in the network for a relatively longer duration. Thisposes an additional threat to the reliability and security of thepower system. It may lead to a false operation of transformerdifferential relays and may prolong temporary harmonic over-voltages. Finally, it may lead to an increase in the noise levelof other transformers connected in the network [1].

The available literature includes numerous investigations onthe inrush current phenomenon in power transformers and itsimpact on the design and operation of protection schemes[2–7]. Most of the approaches are based on the electricalequivalent circuit [5, 7–9] or magnetic equivalent circuit [10–13]. The sympathetic inrush current has also been analyzed ina few references [14–16]. However, the models reported donot provide sufficient accuracy in the case of asymmetrical ortransient operations involving magnetic nonlinearities. During

Authors thankfully acknowledge the financial support given by Depart-ment of Science and Technology, Government of India, under the projectSR/FTP/ET-165/2001.

G. B. Kumbhar is Research Scholar and S. V. Kulkarni is AssociateProfessor in Department of Electrical Engineering, Indian Institute ofTechnology-Bombay, Mumbai-400076, India (e-mails: [email protected] [email protected]).

the saturation of core, a large amount of flux is present inthe air, which is not considered accurately in the equivalentnetwork models. Also, it is difficult to take into accountactual geometrical effects. In case of such phenomenon, fluxdistribution should be examined by solving the phenomenonelectromagnetically to understand it exactly. The strong cou-pling between field and electric circuit, and the presenceof nonlinear elements necessitate the use of direct solutionapproach.

Vs

S B

C

A

Fig. 1. Sympathetic interaction between transformers.

In this work, nonlinear-transient field-circuit coupled 2-D axisymmetric simulations are performed to investigate thesympathetic inrush phenomenon in parallel and series con-nected transformers. Coupled field-circuit analysis for thephenomenon has not been reported in the existing literature tothe best of the authors’ knowledge. Nonlinearity is taken intoaccount by using the actual B-H curve of the core material.This paper describes a detailed modeling methodology andanalysis of the sympathetic inrush phenomenon. The study offactors affecting the magnitude and duration of the sympatheticinrush current are also presented.

II. SYMPATHETIC INTERACTION BETWEEN

TRANSFORMERS

A transformer already connected to supply system canexperience unexpected saturation during the inrush transient ofan incoming transformer. This saturation, which is establishedby an asymmetrical voltage drop across the system resistancecaused by the inrush current in the incoming transformer,demands offset magnetizing currents of high magnitude inthe already connected transformers. As shown in Fig. 1,when transformer B is switched on to the network wheretransformers A and C are already feeding loads, the transientinrush currents not only flow through B, but also throughtransformers A and C. This sharing of the transient inrushcurrent is called as sympathetic inrush phenomenon.

The normal inrush current in the transformers decay, usu-ally, within a few cycles, but the sympathetic inrush currentpersists in the circuit for a relatively longer duration [17].

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The insight into this behavior can be obtained by comparingFig. 2(a), which shows the inrush current in an isolated trans-former connected to the power system network, with Fig. 2(b),which shows both the inrush current and the sympatheticinrush current in the incoming and the already energizedtransformer respectively. Longer duration of the sympatheticinrush current may mal-operate differential relays. It may alsoprolong harmonic over-voltages in the system and increasenoise level in the already connected transformers.

We have followed the convention of representing the inrushcurrent in blue and the sympathetic inrush current in red inall the figures in this paper.

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Cur

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Inrush current in single transformer

Sympathetic Inrush current Inrush current in incoming transformer

Fig. 2. Typical waveforms of (a) inrush and (b) sympathetic inrush current.

III. COUPLED FIELD-CIRCUIT FORMULATION

A. Electromagnetic Model

For a 2-D axisymmetric magnetic analysis, consider theclassical Poisson’s equation [18]:

∂

∂r

(1r

1µ

∂AR

∂r

)+

∂

∂z

(1r

1µ

∂AR

∂z

)= − Jφ (1)

where AR (= r Aφ) and Jφ denote the magnetic vectorpotential and the free or the source current density respectively,and µ is the material permeability.

It is assumed that the windings are made up of thin strandedconductors. These thin stranded conductors can be modeledas carrying uniform current densities, i.e., the induction termin the magnetic field equations representing eddy currentsis neglected. Assuming that each turn carries I amperes ofcurrent, the magnitude of current density in a winding is:

Jφ = dnI

Sc(2)

where n and Sc are the number of turns and the cross-sectionalarea of the winding respectively, and d is the polarity (+1 or -1)to represent the forward or the return path in φ-direction. Afterspace discretization of Eq. (1) using Galerkin method [19], wehave,

[K] AR + [D] I = 0 (3)

where

[K] =∑Ω

1r

1µ

∫∫∆ e

(∂NT

e

∂r

∂Ne

∂r+

∂NTe

∂z

∂Ne

∂z

)drdz

[D] =∑Ω

n

Sc

∫∫∆ e

NTe drdz

where Ne and ∆e are the element shape function and theelement area respectively, and Ω indicates the domain underconsideration.

B. External Circuit Equations

The matrix form of the external circuit equations can bewritten as:

[U ] =

dΦdt

+ [R] I + [L]

dI

dt

(4)

If flux linkages are expressed in terms of the magnetic vec-tor potential, the matrix form of the external circuit equationscan be written as:

[U ] = [G]

dAR

dt

+ [R] I + [L]

dI

dt

(5)

where U is the vector of input voltages, [L ] and [R ] arethe matrices of leakage inductances and winding resistancesrespectively, and G is a matrix similar to D and it depends onthe geometrical features of the windings.

C. Nonlinear-Transient Field-Circuit Coupled Model

From (3) and (5), the field-circuit coupled global system ofequations can be written as:

[0 0G L

]AR

I

+

[K D0 R

]AR

I

=

0U

(6)

In this system of equations, the unknowns are the nodal valuesof the vector potential (AR) and the currents in the externalcircuits (I). To solve this time dependent system of equations,a numerical integration scheme such as Euler Backward orCrank-Nicholson algorithm can be used. The nonlinearitiesin the modeled system are taken into account by using theNewton-Raphson iterative procedure. The resulting system ofequations is,

[β (Kn + Kn

nl) βDG∆t βR + L

∆t

] ∆AR

∆I

n+1

t+∆t

=

0βVt+∆t + (1 − β)Vt

+[ − (1 − β)K − (1 − β) D

G∆t − (1 − β)R + L

∆t

] AR

I

t

−[

β (Kn) βDG∆t βR + L

∆t

] AR

I

n

t+∆t

(7)


where the term Knnl accounts for the nonlinearity of the core

material. The value of β determines the nature of the time-stepping scheme [19].

IV. COMPUTATIONAL METHODOLOGY

A 31.5 MVA, 132/33 kV, star/delta, three-phase transformeris analyzed to investigate the sympathetic inrush phenomenon.The design details and name-plate data of the transformerunder consideration are given in Table I. The 2-D axisym-metric field-circuit coupled model is developed based onthe methodology described in section III. The transformer isanalyzed on a per-phase basis with excitation of LV winding.Three-phase analysis is also done subsequently to compare theinrush currents in all the three phases.

TABLE I

THREE-PHASE THREE-LIMB TRANSFORMER DATA

Rated LV current 318.2 A

Rated HV current 137.78 A

Number LV turns 433

Number HV turns 1000

Percentage impedance 14.33

Core steel type M3H

Fig. 3 shows the circuit connection diagram used in thesimulation. It consists of two field domains correspondingto the two identical transformers, which are modeled usingFEM. The resistances R1 and R2 correspond to the source andthe transmission line resistances respectively. The equationsof the field domains are then coupled with the electricalcircuit equations of the associated power system network (asdescribed in section III). The advantage with using the FEMbased model is that the actual geometry and material propertiesare considered, which increases the accuracy of the obtainedresults as compared with other models, which are based onequivalent circuit and reluctance network approaches. Stronglycoupled field-circuit model considers the dynamics of the cir-cuit and field parts accurately. In this work, 2-D axisymmetricmodel is used because it is sufficiently accurate, and alsotransient nonlinear problems require the simulation to run fora longer time, thereby, making 3-D models computationallyexpensive.

RR

R sys

1 2

r1

r2

FEM domains

T1 T2

S

p

V

p

Fig. 3. Electrical circuit connection for sympathetic inrush current analysis.

V. RESULTS AND DISCUSSION

A. Parallel Connected Transformers

Parallel configuration of two transformers is shown inFig. 3. The switch S is closed after one second to allowinitial circuit transient currents to reach their respective steadystate values. For the configuration of the parallel-connectedtransformers, the influence of the factors such as system re-sistance, switching-on angle, residual magnetism of incomingtransformer and load current, is investigated.

• Effect of system resistance: Inrush current in trans-former T2 and sympathetic inrush current in transformerT1 are shown in Fig. 4(a), Fig. 4(b) and Fig. 4(c) forsystem resistances of 1 Ω, 5 Ω and 20 Ω respectively.Rsys reduces the magnitudes of the initial peaks of theinrush current in T2 considerably. The sympathetic inrushcurrent experienced by the already energized transformerT1 is due to the coupling between the two transformers onaccount of the asymmetrical voltage drop in the systemresistance Rsys of the transmission line feeding them.Hence, the higher the Rsys, the higher is the sympatheticinrush [17] as evident from the figure. However, themagnitude of Rsys has very little effect on the duration ofthe sympathetic inrush current in T1. From Fig. 4, it canbe observed that the magnitude of the current at around1.8 s is almost same for all values of Rsys.

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2000

(b)

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rent

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0

1000

2000

(c) Time in seconds

Cur

rent

in A

mpe

res

Rsys = 1 Ohm

Rsys = 5 Ohm

Rsys = 20 Ohm

Fig. 4. Effect of system resistance (R sy s ) on sympathetic inrush currentphenomenon.

• Effect of switching-on angle: Fig. 5 shows the effect ofswitching-on angle on the magnitude of the sympatheticinrush current. Similar to the single transformer case,the magnitude of the sympathetic inrush current reduceswith an increase in the switching-on angle up to 90,and then increases in the negative direction up to 180.The magnitudes of the inrush currents are minimum whenthe switching-on angle is 90 (because the inductivereactance of the transformer is much larger than theresistance). Figs. 5(a), 5(b) and 5(c) show the sympatheticinrush current patterns for switching angles of 36, 90

and 180 respectively.


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−1000

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rent

in A

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Switching−on angle = 36 degree



Fig. 5. Effect of switching-on angle on sympathetic inrush current phe-nomenon.

• Effect of residual magnetism: Similar to the singletransformer case, residual magnetism of the incomingtransformer plays a key role in the sympathetic inrushphenomenon. Fig. 6 shows the inrush current in T2 andsympathetic inrush current in T1 at different residualmagnetization conditions. It can be seen that the mag-nitudes of these currents largely depend on the polarityof the residual flux density. If the transformer is switched-on at angle zero with positively increasing voltage wave,positive polarity of flux density leads to higher values ofinrush currents and vice-versa.

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Residual flux density = 0 pu

Residual flux density = 0.8 pu

Residual flux density = −0.8 pu

Fig. 6. Effect of residual flux on sympathetic inrush phenomena.

• Effect of load current: When the transformer isswitched on to a heavy load with power factor close tounity, the peak values of both inrush and sympatheticinrush currents are slightly smaller as compared to the no-load case (see Fig. 7). The peak of inrush current in T2

as well as the sympathetic inrush current in T1 slightlyincrease with reduction in power factor as evident fromFig. 8, which shows these currents for different powerfactors of the load. Thus, the load current of the switched

transformer produces small effect on the magnitude andduration of the sympathetic inrush current.

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1000

2000

(a)

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in A

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2000

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2000

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Cur

rent

in A

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Load = Full load

Load = Half load

Load = No load

Fig. 7. Effect of load current on sympathetic inrush phenomena.

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2000

(a)

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in A

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rent

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0

1000

2000

(c) Time in seconds

Cur

rent

in A

mpe

res

Power factor = Unity

Power factor = 0.8 lagging

Power factor = 0.6 lagging

Fig. 8. Effect of load power factor on sympathetic inrush phenomena.

B. Series Connected Transformers

In the case of transformers connected in series, the primaryLV winding of the first transformer (33/132 kV) is connectedto supply, and its secondary winding feeds the HV winding ofthe transformer (132/33 kV) being energized. The nameplatedetails of both the transformers are same as in Table I. Fig. 9shows T1, the already connected transformer, and T2, thetransformer that is being energized by closing the switch S.The currents in the primaries of transformers T2 and T1 areshown in Fig. 10(a) and Fig. 10(b) respectively. The positivehalf cycles of the primary current of T1 correspond to theinrush current of T2 referred to the primary of T1 since the twotransformers are series connected. The negative half cycles ofthis current correspond to the sympathetic inrush phenomenon.

C. Sympathetic Inrush in Three Phase Transformers

A three-phase model has also been developed, which repre-sents the actual condition at site. Fig. 11 shows the normal


RR

R sys

1 2

r1

r2

FEM domains

T1 T2

S

p p

V

Fig. 9. Series connected transformers.

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−500

0

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1000

1500

2000

(b) Time in seconds

Cur

rent

in A

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res

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0

200

400

600

800

(a)

Cur

rent

in A

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res Inrush current in transformer T2

Sympathetic Inrush current in transformer T1

Fig. 10. Sympathetic inrush current phenomenon in series connectedtransformers.

inrush current and the sympathetic inrush current for theconfiguration of parallel connected transformers. All the three-phase currents in both the transformers are marked.

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800

Time in seconds

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IA in T1

IB in T1

IC in T1

Ia in T2

Ib in T2

Ic in T2

Fig. 11. Inrush current phenomenon in three-phase transformers.

VI. CONCLUSION

In this paper, we have presented a detailed formulation andmodeling for the analysis of sympathetic inrush phenomenon

for the configuration of parallel and series connected trans-formers. Two 31.5 MVA, three-phase, 132/33 kV, transformersare modeled using FEM. The FEM field equations of thetransformers are then solved simultaneously with the circuitequations of the power system network. The analysis is firstdone on a per-phase basis. Subsequently, three-phase simula-tion results are also reported.

It is observed that the sympathetic inrush current persistsin the network for a much longer duration than the inrushcurrent for the singly connected transformer. The parametersaffecting the magnitude and duration of the current, such asseries resistance, switching-on angle, residual flux density andload conditions are discussed in detail. It is observed thateven though an increase in the system resistance increasesthe magnitude of the sympathetic inrush current appreciably,there is little effect on its duration. The change in switching-on angle, and the magnitude and direction of the residual fluxdensity can cause significant variations in the phenomenonof sympathetic inrush currents. Moreover, the magnitude andpower factor of the load currents have only a very small effecton the magnitude as well as duration of the sympathetic inrushcurrents.

REFERENCES

[1] S. V. Kulkarni, “Influence of system operating conditionson magnetizing inrush phenomenon of transformer,” In-ternational Conference on Transformers, TRAFOTECH-94, Bangalore, India, pp. VI 19–23, January 1994.

[2] R. Yacamini and A. A. Nasser, “Calculation of inrushcurrent in single-phase transformers,” IEE Proceedings -Electric Power Applications, vol. 128, no. 6, pp. 327–334, November 1981.

[3] R. Yacamini and A. Abu-Nasser, “Numerical calculationof inrush current in single-phase transformers,” IEE Pro-ceedings - Electric Power Applications, vol. 128, no. 6,pp. 327–334, 1981.

[4] C. Paul, Y. Ling, and A. Basak, “Investigation of magne-tizing inrush current in a single-phase transformer,” IEEETransactions on Magnetics, vol. 24, no. 6, pp. 3217–3222, November 1988.

[5] C. E. Lin, C. L. Cheng, C. L. Huang, and J. C. Yeh, “In-vestigation of magnetizing inrush current in transformers,Part I - numerical simulation,” IEEE Transactions onPower Delivery, vol. 8, no. 1, pp. 246–254, January 1993.

[6] J. H. Brunke and K. J. Frohlich, “Elimination of trans-former inrush currents by controlled switching, part I:Theoretical considerations,” IEEE Transactions on PowerDelivery, vol. 16, no. 2, pp. 276–280, April 2001.

[7] G. Baoming, A. T. de Almeida, Z. Qionglin, and W. Xi-angheng, “An equivalent instantaneous inductance-basedtechnique for discrimination between inrush current andinternal faults in power transformers,” IEEE Transactionson Power Delivery, vol. 20, no. 4, pp. 2473–2482,October 2005.

[8] A. A. Adly, H. H. Hanafy, and S. E. Abu-Shady, “Utiliz-ing preisach models of hysteresis in the computation ofthree-phase transformer inrush currents,” Electric PowerSystems Research, vol. 65, pp. 233–238, 2003.


[9] S.-D. Chen, R.-L. Lin, and C.-K. Cheng, “Magnetizinginrush model of transformers based on structure param-eters,” IEEE Transactions on Power Delivery, vol. 20,no. 3, pp. 1947–1954, July 2005.

[10] R. Yacamini and H. S. Bronzeado, “Transformer inrushcalculation using a coupled electromagnetic model,” IEEProcedings - Science, Measurement and Technology, vol.141, no. 6, pp. 491–498, November 1994.

[11] M. Elleuch and M. Poloujadoff, “A contribution to themodeling of three phase transformers using reluctances,”IEEE Transactions on Magnetics, vol. 32, no. 2, pp. 335–343, March 1996.

[12] K. S. Smith, L. Ran, and B. Leyman, “Analysis oftransformer inrush transients in offshore electrical sys-tems,” IEE Proceedings - Generation, Transmission andDistribution, vol. 146, pp. 89–95, January 1999.

[13] S. G. Abdulsalam, W. Xu, and V. Dinavahi, “Modellingand simulation of three-phase transformers for inrushcurrent studies,” IEE Proceedings - Generation, Trans-mission and Distribution, vol. 152, no. 3, pp. 328–333,May 2005.

[14] H. S. Bronzeado, P. B. Brogan, and R. Yacamini, “Har-monic analysis of transient currents during sympatheticinteractions,” IEEE Transactions on Power Systems,vol. 11, no. 4, pp. 2051–2056, November 1996.

[15] H. S. Bronzeado and R. Yacamini, “Phenomenon ofsympathetic interaction between transformers caused byinrush transients,” IEE Procedings - Science, Measure-ment and Technology, vol. 142, no. 4, pp. 323–329, July1995.

[16] M. Sengul, B. Alboyaci, S. Ozturk, and H. B. Cetinkaya,“Case study of sympathetic interaction between trans-formers caused by inrush transients,” International Con-ference on Power Systems Transients, Montreal, Canada,June 2005.

[17] S. V. Kulkarni and S. A. Khaparde, Transformer En-gineering: Design and Practice. New York: MarcelDekker, Inc., Taylor & Francis Group, May 2004.

[18] J. Jin, The Finite Element Methods in Electromagnetics.John Wiley and Sons, Inc, 1993.

[19] S. J. Salon, Finite Element Analysis of Electrical Ma-chines. Kluwer Academic Publishers, 2000.

G. B. Kumbhar received his B.E. degree in Electrical Engineering fromGovernment College of Engineering, Karad, Maharashtra, India, in 1999 andM.Tech degree from the Indian Institute of Technology-Madras, Chennai, inJanuary 2002. He is presently a research scholar at the Indian Institute ofTechnology-Bombay, Mumbai, India, working in the area of coupled fieldformulations in transformers.

S. V. Kulkarni (M’99) is Associate Professor, Department of Electrical Engi-neering, Indian Institute of Technology, Bombay, India. Previously, he workedat Crompton Greaves Limited and specialized in the design and developmentof transformers up to 400 kV class. He has authored book ”TransformerEngineering: Design and Practice” published by Marcel Dekker, Inc. He isthe recipient of the Young Engineer Award (2000) from the Indian NationalAcademy of Engineering. His research interests include transformer designand analysis, computational electromagnetics, and distributed generation.


sympathetic inrush current

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