theoretical calculation of inrush currents in three an five legged core transformers

986 IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 22, NO. 2, APRIL 2007

Theoretical Calculation of Inrush Currents inThree- and Five-Legged Core Transformers

Luis Sáinz, Felipe Córcoles, Joaquín Pedra, Member, IEEE, and Luis Guasch

Abstract—In this paper, a theoretical study of the three-phasetransformer behavior in the presence of a sag is presented by con-sidering that the fault clearing is produced simultaneously in allphases. Analytical expressions of the magnetic flux and the inrushcurrent after voltage recovery are obtained, and the results pre-sented in the literature are analytically justified. The influence ofdepth, duration, and the initial point-on-wave on the peak value ofthe inrush current are studied for all sag types. Simple expressionsto obtain the value of the current peaks are presented.

Index Terms—Inrush current, transformer model, voltage sag.

I. INTRODUCTION

AVOLTAGE SAG is a short-duration reduction inroot-mean-square (rms) voltage. The most severe voltage

sags are produced by faults in power systems, and it can leadto malfunctioning equipment [1]. In the case of power trans-formers, the sudden voltage recovery, when a fault is cleared,can saturate the core transformer. This transformer saturationcan involve high inrush currents [2], which are very sensitiveto the voltage recovery instant. Therefore, a precise study ofthe problem must take into account that this instant can onlyhave discrete values, since fault clearing is produced in thenatural current zeros [3], [4]. Nevertheless, as in the previousstudy of [5], the consideration that fault clearing is producedsimultaneously in all phases simplifies the study and providesa preliminary estimation of the inrush currents. The currentpeaks calculated by assuming that voltage recovery can onlybe produced in the current zeros are lower than those obtainedwhen this assumption is not made. In the studies of [4] and[5], the three-phase transformer model is implemented andsimulated in PSpice.

In this paper, the inrush currents caused by symmetrical andunsymmetrical voltage sags are analytically studied in detail byconsidering that fault clearing is produced simultaneously in allof the phases. This simplification allows analytical expressionsof the magnetic flux and inrush current after a voltage sag to be

Manuscript received February 23, 2006; revised May 5, 2006. This work wassupported by Grant DPI2000-0994. Paper no. TPWRD-00081-2006.

L. Sáinz, F. Córcoles, and J. Pedra are with the Department of Electrical Engi-neering, Universitat Politècnica de Catalunya, Barcelona 08028, Spain (e-mail:[email protected]; [email protected]; [email protected]).

L. Guasch is with the Department of Electronic, Electrical, and AutomaticEngineering, Universitat Rovira i Virgili, Tarragona 43007, Spain (e-mail:[email protected]).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TPWRD.2006.881428

Fig. 1. Electric equivalent circuit of a Wye G-Wye G three-phase transformer.

obtained. Inrush current dependence on sag type, depth, dura-tion, and the initial point on wave is characterized. These ex-pressions allow the transformer behavior presented in [5] to beexplained and an approximate value for the peak of the inrushcurrent to be analytically calculated. This can be useful in powersystem studies, such as relay coordination, transformer stress,etc.

II. TRANSFORMER MODEL

Two types of three-phase transformers are studied in thispaper, namely three- and five-legged core transformers. Themodel proposed in [4] and [5] is used to study the three-leggedtransformer, whereas a particular case of this model is used tostudy the five-legged core transformer.

A. Electric- and Magnetic-Circuit Models

The electric and magnetic relations of the Wye G-Wye Gthree-phase transformer are shown in Figs. 1 and 2

(1)

In this model, each leg is viewed as a separate magnetic el-ement, and the expression used to represent the core nonlinearbehavior is a function that relates the magnetic potentialin the leg and the flux through it

(2)

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SÁINZ et al.: THEORETICAL CALCULATION OF INRUSH CURRENTS IN CORE TRANSFORMERS 987

Fig. 2. Magnetic equivalent circuit of a three-legged three-phase transformer.

where , , , and are experimental parameterswhich allow this single-valued function to be fitted to the

transformer saturation curve [5].In Fig. 2, it can be observed that the reluctance has been

considered constant because it represents the air path in a three-legged transformer.

In order to eliminate the winding turns, the above relationsare rewritten according to [4] as follows:

(3)

where is the winding turn ratio andis the core magnetic flux linked by the primary windings. Therelation between the magnetic potential in the leg and the fluxthrough it is rewritten as

In this paper, only the primary voltages, currents, and totalfluxes are of interest. Thus, these primary variables are renamedas follows for clarity purposes:

(4)

B. Transformer Data

The study of the three-legged model has been performed witha 60-kVA, 380/220-V Wye G-Wye G three-phase transformerwhose linear and nonlinear parameters have been obtained fromexperimental measurements [4].

The study of the three-phase five-legged transformer is alsoperformed with the same data as the three-legged model butconsidering:

• reluctance values of legs and are forced to be equal tothe values of leg ;

• value of the reluctance has been considered null be-cause, during transformer saturation, it will not be signifi-cant in comparison with the leg reluctances.

Fig. 3. Magnetic fluxes in a symmetrical voltage sag of characteristics h =0.4, �t = 4.375T , and = 0 .

The above considerations are also true for the transformerbank. Thus, the study of the five-legged transformer can be ex-tended to the transformer bank.

In all of the cases studied in this paper, the secondary suppliesa rated resistive load of 0.8 per phase. In fact, load magnitudeand load character (resistive or inductive) have no influence onthe transformer behavior [5].

III. ANALYTICAL CHARACTERIZATION OF THE CURRENT PEAKS

References [4] and [5] show that the main reason why trans-former current peaks appear is found in the dc component ofthe total magnetic flux after the sag; for example, with

in Fig. 3. Transformer saturation occurs when thisdc component is not null, causing the current to be high. Thissection presents an analytical study of the total magnetic fluxand the inrush current when the voltage sag ends andwith . This study is conducted in a Wye G-Wye Gthree-phase five-legged transformer (or a transformer bank).

A. Total Flux Peak Determination

Considering the transformer model (3), the total magneticflux linked by the primary windings can be calculated as

(5)

where the primary total magnetic flux is , andthe voltage drop in the primary winding resistances (with ) has been neglected.

The total magnetic flux is derived from (5) and gives

(6)



TABLE IMAGNITUDE AND ANGLE OF VOLTAGE SAG PHASORS

When the supply voltage iswith , and the transformer is in steady state, the

magnetic flux is expressed as

(7)As the mean value of the flux is zero in steady state, the next

condition must be true for the initial flux

(8)

and the final expression of the flux is

(9)

The supply voltages assumed when a sag is produced are

.(10)

The pre and postsag voltage is defined as ,, .

The magnitude and angle values of the voltage phasors duringthe sag ( and ) are shown in Table I. They are determinedfrom the sag expressions of Table II in [4], which were obtainedfrom the sag classification of [1].

Flux continuity implies that flux can be calculated at any in-stant by means of (6), even if the voltage changes during the sag.The flux after the sag (Fig. 3) can be calculated as

(11)

where the sag is produced between times and .

TABLE IIVALUES OF AND �t TO AVOID OBTAINING FLUX AND CURRENT PEAKS.

VALUES OF TO OBTAIN THE MAXIMUM FLUX AND CURRENT PEAKS

WHEN �t IS FIXED, � AND i

Taking into account that

(12)

the postsag flux (11) can be rewritten as

(13)

where the two first terms are identical to the terms of the casewithout a sag (9).

Then, the final expression of the flux when voltage is recov-ered is

(14)

where

(15)

This term is constant and its physical meaning is a dcmagnetic flux. The Appendix shows the expressions ofobtained from (15) for the different sag types.

The current peak is produced when there is a flux peak be-cause they are related by the magnetic curve (3). There-fore, the peak values of the total magnetic fluxes must be calcu-lated to determine the current peaks.



The flux peak produced by the sag depends on the depth ,the duration , and the initial point-on-wave . Thus, the fol-lowing function must be studied:

(16)

where

(17)

In the case shown in Fig. 3, the flux peak is produced in phase, and its value is .Then, must be analyzed to study the influence

of sag type, depth, duration, and the initial point-on-wave on themaximum and minimum values of the postsag magnetic flux.

The flux is obtained from the analytical ex-pressions of in the Appendix. For example, the dc com-ponent of the total magnetic fluxes in a type B sag (Appendix)is

(18)

and, according to (16), the flux peak gives

(19)

Fig. 4(b) shows the curve for a type B sag ofcharacteristics 0.73 and 5.5 obtained from (19).

B. Current Peak Determination

The transformer total flux and the primary currents can be re-lated if some simplifications are made in the transformer model.

• Transformer is at no load (or the secondary currents are ofless importance than the primary currents when the trans-former is saturated) (or ).

• Reluctance is null, as the five-legged transformer isconsidered (the magnetic potential is also null).

• Core-loss resistances are neglected .

In this case, the magnetic equation of (3) gives

(20)

This expression allows the relation to be calculated

(21)

and the use of the nonlinear expression in (3) yields

(22)

Fig. 4(a) shows the influence of the nonlinear curve param-eters on the shape of the nonlinear saturationcurve. The above nonlinear relation can be approximated by apiecewise linear saturation curve. If only two pieces are usedFig. 4(a), their parameters can be related easily to the parame-ters of the nonlinear function (22). The two-piece linear satura-tion curve is defined as

(23)

where is the saturation flux. Thecurrent peak can be easily determined from (23).

Fig. 4(b) shows the current peak determination from the sat-uration curve for a type B sag of , , and

ranging from 0 to . Fig. 4(b) shows that the difference be-tween the use of the nonlinear saturation curve (continuous line)and the two-piece approximation (dashed line) is small.

The expression for calculating the current peak whenis derived from (23)

(24)

where is the phase with the greatest flux peak.

IV. BEHAVIOR STUDY OF THE THREE-PHASE TRANSFORMER

The expressions of the previous section allow the transformercurrent peak to be analytically calculated. Thus, the influence ofsag type,duration ,depth ,and the initialpoint-on-wave onthe current peaks can be studied. It must be noted that in the studyof the previous section, the source has no internal impedance.This leads to an overestimation of the transformer current peaksbecause the source impedance damps these peaks [5].

All of the analytical calculations in this paper have beenobtained by programming the analytical expressions fromSection III and the Appendix in MATLAB [6].

To analyze the duration, initial point-on-wave and depth in-fluence on the current peaks, the following series of sags havebeen studied for the seven sag types, A to G.

• Duration influence: Four series of sags of depthhave been studied

.



Fig. 4. (a) Influence of the parameters on the shape of the saturation curve. (b) Current peak determination from the saturation curve.

• Initial point-on-wave influence: Four series of sags ofdepth have been studied

.

(25)

• Depth influence: One series of sags with the most unfa-vorable duration and the most unfavorable initial point-on-wave and , obtained from Table II, have beenstudied

(26)

A. Three-Phase Five-Legged Transformer

In order to study this transformer analytically, the data of [4]have been considered imposing that the three legs are identical(the reluctance values of legs and are set equal to the valuesof leg ) and the air-path linear reluctance is null. Thus, thisstudy can also be extended to the transformer bank.

Figs. 5 and 6 show the sag duration and the initial point-on-wave influence on the current peak, respectively (series toand to ).

By comparing these figures with the corresponding onesin [5], it can be noted that the analytical study predicts thefive-legged transformer behavior (and the transformer bankbehavior) accurately. Nevertheless, the analytical results areslightly lower than the simulation results in [5] because theresistance voltage drop has been neglected.

1) Duration and Initial Point-on-Wave Influence: The valueof and required to avoid obtaining current peaks (whichcorresponds to minimum flux values) when voltage is recoveredcan be calculated by imposing in the expressions ofthe Appendix. These values are shown in Table II.

The most unfavorable situation is produced when the max-imum flux peak (which corresponds to the maximum currentpeak) is obtained. The maximum flux peak can be calculatedanalytically. At this point, the difference between the maximumflux peak (which takes into account all current peaks corre-sponding to all values of and ), and the maximum fluxpeak when is fixed (which takes into account only the cur-rent peaks corresponding to all values of , but only one valueof ) must be considered

(27)



Fig. 5. Three-phase five-legged transformer: analytical results of the sag dura-tion influence on the current peak for the seven sag types.

The value of required to obtain (for example, inseries , , , or ) is obtained by imposing

in the expressions of Appendix A. These valuesare shown in Table II for the relation .

Fig. 6. Three-phase five-legged transformer: analytical results of the initialpoint-on-wave influence on the current peak for the seven sag types.

Among all of the values of shown in Table II, onlythe highest values produce the most unfavorable situations, thatis, the maximum flux peaks . According to the Appendix,these most unfavorable situations are produced for specific



Fig. 7. Three-phase five-legged transformer: analytical results of the depth in-fluence on the most unfavorable current peak for the seven sag types.

values of (whenwith i.e., series 7) and for the corresponding

relations of Table II.The maximum flux peak can be calculated as

(28)

where

for type A,B,D,E, and F sagsfor type C sagfor type G sag.

(29)

The maximum current peak can be calculated by thesubstitution of (28) in (24). It can be noted that type A, B, D, E,and F sags have the highest values.

2) Depth Influence: According to series , Fig. 7 shows thedepth influence on the maximum current peak.

For a given and , the expressions of the Appendix showthat the relation between and the depth is a straight linewith negative slope (i.e., there is a linear relation). Then

(30)

where does not depend on the sag depth .Considering (23), the relation between the current peaks and

the depth is also linear with two possible negative slopes de-pending on whether the total magnetic flux is above or belowthe saturation flux or .

B. Three-Phase Three-Legged Transformer

In order to study this transformer analytically, the data of [4]for the three legs have been considered.

According to series to and to , Figs. 8 and 9 showthe sag duration and the initial point-on-wave influence on thecurrent peaks, respectively.

By comparing these figures with the corresponding ones in[5], it can be observed that although the analytical model con-siders that the value of the reluctance is null ( inSection III-B), the study predicts the three-legged transformerbehavior accurately. This good agreement is achieved becausethe magnetic potential in the reluctance (the magnetic poten-tial , (3)) is not significant in comparison with the remainingmagnetic potentials when the transformer is saturated [

, which is equivalent to considering , as in (20)]. Bycomparing the five- and three-legged transformer results (Figs. 5

Fig. 8. Three-phase three-legged transformer: analytical results of the sag du-ration influence on the current peak for the seven sag types.

and 6 with Figs. 8 and 9), it can be observed that the main differ-ence is in the symmetry of the waveforms whereas the currentpeaks do not essentially change.



Fig. 9. Three-phase three-legged transformer: analytical results of the initialpoint-on-wave influence on the current peak for the seven sag types.

1) Duration and Initial Point-on-Wave Influence: As in thecase of the five-legged transformer, Table II shows the valuesof and required to avoid obtaining current peaks, and thevalues of to obtain the maximum current peaks when isfixed .

The procedure to calculate the maximum current peaks is thesame as the procedure of the previous section. However, thethree-legged transformer has fewer most unfavorable situationsthan the five-legged transformer for type A, E, and G sags be-cause the central leg is shorter than the outer legs.

2) Depth Influence: The study of the depth influence leadsto the same conclusions as those obtained for the five-leggedtransformer.

V. CONCLUSION

The effects on the current peaks of symmetrical and unsym-metrical voltage sags in Wye G-Wye G three-phase transformershave been analytically studied by considering that fault clearingis produced simultaneously in all of the phases. Analytical ex-pressions for the total magnetic flux and the inrush current whenthe transformer is saturated have been obtained. This saturationis produced when voltage is recovered. This simple model al-lows the behavior of the current of the five- and the three-leggedtransformers to be explained when a sag is produced. These an-alytical expressions also allow the current peak value to be ana-lytically estimated. It can be observed that the obtained currentpeaks are higher than those obtained when it is considered thatvoltage recovery can only be produced in the current zeros [4].Thus, following the analytical process indicated in this paper,this consideration must be taken into account in future studiesto obtain more precise results.

APPENDIX

DC COMPONENT OF THE TRANSFORMER

TOTAL MAGNETIC FLUXES

The expression of the dc component of the transformer totalmagnetic fluxes when voltage is recovered can be obtained from(15) by considering and

with

(31)

which can be rewritten as follows:

(32)

Equation (32) can be rewritten by using duration and the ini-tial point-on-wave as variables

(33)

Thus, for the different sag types, the dc component of the totalmagnetic flux can be obtained from expression (33) and by con-



sidering and , ,and the values of and of Table I.

• Type A

(34)

• Type B

(35)

• Type C

(36)

Using trigonometric relations, the above expression can berewritten as

(37)

• Type D

(38)


(39)

• Type E

(40)

• Type F

(41)


(42)



• Type G

(43)


(44)

REFERENCES

[1] M. H. J. Bollen, Understanding Power Quality Problems: Voltage Sagsand Interruptions. Piscataway, NJ: IEEE Press, 2000.

[2] E. Styvaktakis, M. H. J. Bollen, and I. Y. H. Gu, “Transformer satu-ration after a voltage dip,” IEEE Power Eng. Rev., vol. 20, no. 4, pp.62–64, Apr. 2000.

[3] M. H. J. Bollen, “Voltage recovery after unbalanced and balancedvoltage dips in three-phase systems,” IEEE Trans. Power Del., vol. 18,no. 4, pp. 1376–1381, Oct. 2003.

[4] J. Pedra, L. Sáinz, F. Córcoles, and L. Guasch, “Symmetrical and un-symmetrical voltage sag effects on three-phase transformers,” IEEETrans. Power Del., vol. 20, no. 2, pt. 2, pp. 1683–1691, Apr. 2005.

[5] L. Guasch, F. Córcoles, J. Pedra, and L. Sáinz, “Effects of symmetricalvoltage sags on three-phase three-legged transformers,” IEEE Trans.Power Del., vol. 19, no. 2, pp. 875–883, Apr. 2004.

[6] The MathWorks, MATLAB 5.3 and Simulink 3.0. Natick, MA, 1999.

Luis Sáinz was born in Barcelona, Spain, in 1965. Hereceived the B.S. degree in industrial engineering andthe Ph.D. degree in engineering from the UniversitatPolitècnica de Catalunya, Barcelona, Spain, in 1990and 1995, respectively.

Since 1991, he has been Professor in the Depart-ment of Electrical Engineering, Universitat Politèc-nica de Catalunya. His main field of research is powersystem quality.

Felipe Córcoles was born in Almansa, Spain, in1964. He received the B.S. degree in industrial en-gineering and the Ph.D. degree in engineering fromthe Universitat Politècnica de Catalunya, Barcelona,Spain, in 1990 and 1998, respectively.

Currently, he is a Professor in the Departmentof Electrical Engineering, Universitat Politècnicade Catalunya, where he has been since 1992. Hisresearch interests are electric machines and powersystem quality.

Joaquín Pedra (S’85–M’88) was born in Barcelona,Spain, in 1957. He received the B.S. degree in indus-trial engineering and the Ph.D. degree in engineeringfrom the Universitat Politècnica de Catalunya,Barcelona, Spain, in 1979 and 1986, respectively.

Since 1985, he has been Professor in the Depart-ment of Electrical Engineering, Universitat Politèc-nica de Catalunya. His research interest lies in theareas of power system quality and electric machines.

Luis Guasch was born in Tarragona, Spain, in 1964.He received the B.S. degree in industrial engineeringand the Ph.D. degree in engineering from the Univer-sitat Politècnica de Catalunya, Barcelona, Spain, in1996 and 2006, respectively.

Currently, he is a Professor in the Department ofElectronic, Electrical, and Automatic Engineering,Universitat Rovira i Virgili, Tarragona, where he hasbeen since 1990. His research interests are electricmachines and power system quality.


theoretical calculation of inrush currents in three an five legged core transformers

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