Saturation Curve

1432

IEEE Transactions on Power Delivery. Vol. 10. No. 3, July 1995

Saturation Curves of Delta-Connected Transformersfrom Measurements Washington L. A. Neve; Student Member, IEEE

Hermann W. Dommel Fellow, IEEE

Department of Electrical Engineering University of British Columbia Vancouver, B.C., Canada V6T 124

Abstract An algorithm is described for the computation of

the saturation characteristics of three-~hase deltacoMected transformers from measured V- Z- curves and no load losses at rated frequency. For these transfor", positive sequence excitation tests are usually carried out with the delta connection closed. While triplen harmonic currents are present in the deltaamected windmgs, they do not appear in the line connections to the three-phase source. This paper presents an algorithm which accounts for the missing triplen harmonics on the line side in the derivation of the saturation chancteristics.

For three-phase transformers, the standard excitation test data - Z- curves and no-load available are the positive sequence Vlosses. Figure 1 shows a symmetrical three-phase voltage source supplying a no-load delta-connected transformer. The delta branches consist of nonlinear elements. In general, excitation tests are carried out with a closed delta. In that case, ammeters, placed in series with the line, will not contain the triplen harmonic currents because these circulate in the delta connection. This paper presents a method for generathe piecewise linear saturation curves (nonlinear resistance and nonlinear inductance), which accounts for the fact that triplen harmonics circulate in the closed delta, but do not appear in the measured line currents.

Key words: Transformer, delta connection, saturation, harmonics. 1 Introduction insulation coordination studies are needed to specify the insulation levels of power apparatus and systems. These studies are usually done with digital computer programs, such as the EMTP (Electromagnetic Transients Program). They require mathematical models for the various power system components. The type of model depends on the type of study. For inrush current and ferroresonance studies, models for transformers and reactors must represent saturation enects reasonably well. For this, the instantaneous saturation characteristic, which gives flux llnkage R as a function of current i , is neded. Numerical methods have been used for some time to produce these peak fluxcurrent curves from curves[lJ]. measured Vms- -I A simple circuit consisting of a nonlinear resistance in parallel with a nonlinear inductance represents the transformer core reasonably we11[3,4]. An algorithm to obtain the nonlinear resistance (piecewise linear v - i, curve) and the nonlinear inductance (piecewise linear L il curve) from the measured V- Z- curve and no-load losses was presented in [SI. This algorithm assumed all cdd harmonic current components to be present in the measured values.

Figure 1: D ella-con n e c l e d transform erPosit i v e seq u en c e e: c i t at i on test.

2 Basic Considerations In the circuit of Figure 1, the three branch elements of the delta connection are assumed to be nonlinear and identical. The branch currents can be Written as a Fourier time series containing odd harmonic components only. Then:

- -

On leave from Universidade Federal da Paraiba, Campina Grande PB Brazil.

The triplen harmonic currents (Z3. Zg...) are in phase (zero sequence harmonics). The nns current in each branch is (4)

94 SM 459-8 PWRD

A paper recommended and approved by the IEEE Transmission and Distribution Committee of the IEEE Power Engineering Society f o r presentation at the IF,EE/PES 1994 Summer Meeting, San Francisco, CA, July 24 - 28, 1994. Manuscript submitted December 30, 1992; made available for printing April 20, 1994.

Thelinecurrentsare:

0885-8977/95/$04.00 0 1994 IEEE

ia(r)= id(r)-ic,,(r) ib(r) = ik(r)-id(t)

i c ( r ) = ic,,(r)-t&)

.

1433

a

From equations (1) to (8) one can make the following observations: 0 triplen harmonic currents, although present in each branch, are not present in the line currents. if triplen harmonic currents in each delta branch are removed from their nns values (equation (4)) and scaled by 4 , the nns line currents (equation (8)) ax obtained. This information is the basis of the algorithm developed next.

3 Saiuration Curves Each delta branch in Figure 1 is represented by a nonlinear inductance in parallel with a nonlinear resistance (Figure 2). Their nonlinear characteristicsare computed with the following assumptions' :

-

-

the v irA and A i[A c w e s (Figures 2b and 2c) are symmetric with respect to the ongin (Rk and & are the slopes of segment k of the v - ij-A and 2 i / A curves, respectively); the transformer winding resistances and leakage inductances are ignored. The algorithm works then as follows: 1. For the constructionof the v - irA curve (Section 3.1): compute the peak values of the branch current irAl, i r A p . . point by point from the n*load losses. 2. For the " c t i o n

of the A - i/A curve (Section 3.2):

0 from the v - irA c w e , compute the nns values ZrA-m, remove the triplen harmonic currents and obtain the resistive h e current Ir-nns;

obtain the nns values I/-nodinear inductance fkom I,---, the applied voltage V ;

Figure 2: ( a ) N on lin ear elem en Is; (b) y - i,, curve; (c) 1- i,,curve.

I

of the line current due to the the total line current It--, end

compute the peak values of the inductive current i/Al, i/&, ... point by point iteratively.

3.1 Computali'onof the v - irA curve Let us assume that the three-phase ndoad losses Pi, P2, ..., P m are available as a function of the branch voltages VmSl, Vm2, ...,V-,, (Figure 3). If we assume that the applied voltage is sinusoidal, the conversion of rms voltages to peak values (vertical axis of Figure 2b) is simply vk =

for k = L Z . * * , m .

fi

(9)

Figure 3: V r m r - p o w e r loss curve.

1434

Due to symmetry reasons, voltage and current waveforms need only be evaluated over 114 of a cycle. For a sinusoidal voltage v(6) = vk sin e, pk can be written in the form2:

f

fl

-

6. if the absolute value of the difference id = I 1 - w is less than the specified tolerance, convergence is achieved. Otherwise, the residue id is added to ig and the iterative process is repeated fitnu step 2 onwards.

In general, convergence is achieved in less than 20 iteration steps with ilA1 and initial guess a tolerance of For the f i linear segment in the v-rd curve, the current is sinusoidal. The computation of the frrst peak current frAl is therefore straightforward. Since P 1 = 3 v m ~ , I m ,in the linear case,

for every k 2 2. In some cases as many as 40 iteration steps may be n-==Y.

From the second segment onwards (t2 2), equation (10) is evaluated at each segment A , with only i r k being unknown, as explained in more detail in Section 2.1 of (51. The computation of the peak current i r b is done segment by segment, starting with i.4 and endug with the last point i r b . Whenever a point i r 4 is found, its mu value is calculated as well. A Fourier program (Appendix A) is used to campute the triplen harmonics (13. 19,...). They are removed from I r A m s to obtain Ir-ms, which is needed later for the construction of the peak flux peak current II ilA curve.

-

‘T

.c o r r e c t c u r v e

-

8

x

3.2 Computation of the

T

curve

The conversion of the peak branch voltages vk to flux 4 is again a rescaling procedure. Hence, for each linear segment in the II - i/A me, A&+

81

ZT 2

(12)

Let us now compute the peak values of the currents il4 through the nonlinear inductance. First, the mu values of the line currents II+,,,~ are evaluated with

8 F i g u r e 4 : G e n e r a t i n g c u r r e n t waveform from sinusoidal f l u x .

where the line current It+,,,, is available fhm the measurements, and has already been computed h m the previous section. where For the first linear segment, the computation of iiA, is straightforward since there are no harmonics yet. Therefore,

From the second segment onwards (k 2 2) the algorithm works iteratively as follows (see R ilA curye in Figure 4):

-

1. guess ig;

2. with A((B)=Aksine, find 1/4 of a cycle of the distorted current d*cally,

4 Casestudy Consider a 50 Hz three-phase five-legged core type transformer. The following infomation is known[6]: 1. rated power - 750 MVA (three-phase); 2. rated voltages - 420 kV/27 kV (line to line values); 3. wye connection on 420 kV side, delta connection on 27 kV side.

Table I shows the data from the positive sequence excitation tests measured at the closed delta 27 kV side.

3. compute the nnr inductive branch current whose peak is ig ; 4. use a Fourier program to find the triplen harmonic inductive currents in the delta branch (Appendix A);

5. remove the triplen harmonics from the estimated nns branch current. Scale the estimated result Ilest by Js and compare it to Il-in equation (13);

I

Table I: Three-phase transformer test data I It-m I P

vm

orv) 22.76 24.29 25.64 27.00 27.50 28.47 29.10 32.50

I

(A)

8.20 11.35 15.50 21.16 24.68 31.63 38.30 80.97

I

orw

206.21 240.26 270.13 311.00 323.03 355.48 385.41 560.00

1435

Vms is the m line to line excitation voltage, Ir-ms is the nns excitation current (three-phase average) and P are the ndoad losses (three-phasevalues). Figure 5 shows two computed 2 - jib, c w e s (points connected by straight line segments). One of them assumes that all odd harmonic components of the current are present in the measured values and it is therefore incorrect. The other curve is the correct one; it has been produced with the algorithm of Section 3.2. Figure 6 shows the piecewise linear v - irA curve. 1.2

-

1.0 -

0.8 -

o'2

1 0.002

0.004

0.006

0.008 0.010

0.012

C u r r e n t ( p. U .) Figure 5:

A-

Figure 5 shows that the correct curve goes deeper into saturation. For the flu value of 1.2 P.u., there is a difference of approximately 14% between the peak currents of the incorrect curve and the wrmt one. In facf for transient studies, it is often neceSSafy to know peak fluxpeak current curves beyond that point. The usual way is to extend the curve up to a value necessary for the study (this extension is sometimes done with a straight line passing through the previous to the last and the last point in the peak flux-peak current curve). This may lead to larger errors for the peak values of the cun-ent since the curves diverge as the flux goes up towards deep saturation. The transformer magnetizing current would always be underedmated if triplen harmonics inside the delta windings were not taken into accoullt Errors can also affect the air core inductance value. A parametric study was done considering typical values for the air core reactances (0.2 p.u to 0.5 P.u.). For the study, a straight line segment was connected to the last point of the correct curve of Figure 5 . Errors on the slope of the saturation characteristics,for this case, are between 18% and 25% when the magnetizing current reaches the transformer rated current. These differences may be important in ferroresonance or inrush current studies. The proper way to represent saturation effects is to look at the transformer magnetic circuit. Saturation is related to fluxes in the wre and tank. There tire several types of core construction. To model the core reasonably well, one should know what kind of geometry the transformer has. One should h o w if the core is three-legged, five legged or shell-type. The knowledge of the zero sequence magnetizing impedance is also required. It can be estimated for each type of core coIlstruction [A. The saturation curve must be obtained for each leg of the transformer and placed across the lowest voltage terminal. In the iteration scheme of Section 3.2, harmonics up to the 99* order were included. An average of 23.86 iteration steps was necessary (the "um number of iterations was 34). In order to check the numerical accuracy of the method, the m line currents were errom recomputed back from v irb, and 1 - jib, Curves. were found to be very small (less than 0.001%).

-

iiacurvc.

5 Uncertainties

I

0.0005

Current (P.u.) Figure 6 : V - i f A c u r v e .

0.0010

To model transformers exactly is very complicated, if not impossible. The ferromagnetic properties of steel laminations are not well understood yet. There is no theory available today that could predict losses accurately as a function of frequency, even for low kequencies [8,9,10]. There is a big discrepancy between calculated and measured losses due to the very complicated domain structure of these materials, There is also the further discrepancy #at transformer core loss pa kilogram is always greater than the nominal loss of the steel as measured in standard testers. The ratio between the transformer per unit loss and the nominal or standard unit loss is the so-called building factor of the core. Building factors usually range from 1.1 to 2.0. The extra loss is due to factors such as [l 11: 0 Non-uniform flux distribution due to differencein path lengths among magnetic circuits; 0 Distortion of fluxwaveform due to magnetic saturation; 0 Circulating flux due to magnetic anisotropy of m a w , 0 Flux directed out of the rolling direction; 0 Transverse flux between layers due to joints.

ne flux distribution in transformalaminationsisn0tl"eVtn

at low

fkqueames. For a Sinusoidal applied flux,the flux m each lamhation is, in genaal,notsinwidal although the fluxcomponents addup to produce the

sinusoidal total flux [12. 131. These at-Mcertaintiesmironm modelling. Amther major problem is the availabilityof data

1436

The core nonlinear elements, in this paper, are obtained from one single frequency. According to this model. transformer no-load losses would then be underestimated for frequencies below the rated frequency and overestimated for frequencies above the rated frequency. However, ferroresonance and inrush current studies show that the power system is much more sensitive to variations of the saturation curves than to variations of core losses. Figure 7 is the measured positive sequence excitation characteristic of a three-phase five-legged 75 kVA distribution transformer (secondary side 120 V). The x-axis is the ratio of the no-load exciting current I , (measured in each phase) to the no-load current at rated voltage I-,a#d (average value of rated current in all phases) for the same transformer. The solid line represents the average curve.

-

1.4,-

-

-

-

---,- -

the current at any point in between A and B. For instance at point B, the open-circuited capacitancecan be obtained by: Copon =

+. m AB

Table II shows *corrupted*fluxcurrent points obtained using Figure Sa as input data. The following information is needed: peak current at turning point A = 1.240726 A; peak current at tUming point B = 0.123851 A; peak flux at turningpoint B = 0.189066 Vs. Then, the open circuit capacitancereferred to the 120 V side is 1.240726- 0.123851 ~ ~ 4 1 . 5 6 ~ . ( 2 z x 60)2x0.189066 is referred to the primary side (14400 V), Copen =41.56~(120/14400~~2.89~F. Copen =

7 - 7

If.,C

Table II: Corrupted fluxcurrent curve.

2

4

6

8

1

Flux Linkage(V.s) 0.000000

Cment(A) 0.0000000

0.040327

0.426281

0

lexcdexc-rated

Figure 7: V,,

- 1,characteristicfor three-phase transformers

There are discrepancies below the rated voltage, but the curves are very close as the transformer goes into deep satmition. I, curve shown in Figure 8 was obtained from a recently The V , manufactured 10 kVA single-phase distribution transformer . The nns current at 90% of the rated voltage is smaller than the m s current at 50% of the applied voltage. Here, the exciting current in the unsaturated region is very much affected by stray capacitances. As the transformer goes into deep saturation, stray capacitances aEkct very little the computation of saturation curves.

-

0.059046

0.747657

0.075777

0.905318

0.095659

1.1933 16

0.114790

1.240726

0.151553

0.815392

0.169560

0.396883

0.189066

0.123851

0.207823

0.162665

To represent the transformer exactly, one should represent saturation, hysteresis and eddy currents in the core as well as eddy currents and stray capacitances effects in the windings. Modern power transformer cores have very low losses and saturation is the predominant effect. In this paper the main focus is on estimation of the saturation curves of transformers.

4 Conclusions An approach for the computation of instantaneous saturation curves of rms C u n m t (A)

1

(a) (b) Figure 8 a)V, - L curve for a recently manufactured transformer. b) Corrupted fluxcurrent curve (not to scale). A crude estimation of the saturation curves and the open-circuit capacitance can be found iteratively. First, the algorithm generates the peak fluxcurrent characteristic "corrupted" by stray capacitance effects as shown in Figure 8b. The peak current is decreasing in the region between points A and B. The inductance is very high in this region. It can be assumed that it is infinite between turning points A and B. The peak cucrent 1. through the open circuit capacitance C, is given by the difference between the current at turning point A and

deltaannected transformers has been presented. It uses positive sequence excitation test data as input, and is suitable for situations in which the tests are performed with a closed delta.

-

Once the v irA and R - ilA curves have been obtained, they can then be used to model the excitation branch of transformers in transient and harmonic studies.

7 Acknowledgments The authors would like to thank the r e v i m for their valuable suggestions.Also. the authors would like to thank Powatech Labs Inc. for providmg the t"mtest data of Section 5. The financial support of Mr.Washington Nevw from Univmidade Federal da Paraiba, Campina Grande PB Brazil, and from The University of British Columbia is gratefully acknowledged.

-

-

1437

For segment R = 2 ,

8 References 1. S.N. Talukdar, J. K. Dickson. R. C. Dugan, M. J. SprinZen, C. J. Lenda, On Modeling T m f o n n e r and Reactor Satumtion Chamcteristicsfor Digital and Analog Studies, IEEE Trans. on PAS, vol. PAS-94.1975, pp. 612621. 2. S. Prusty d M.V. S.Rm,A Dimxtp'ecewise Linean.zedAppd toGmveli musalumtionchomcrenstr . 'cto Instantaneous Wmtim CUM,IEEE Trans. Mag., vol. Mag-16, NO. 1, J ~ ~ M1980, I Y p ~ 1.5 6 160. 3. L. 0. Chua and K. A. StrO"oe, Lumped Circuit Models for Nonlinear Inductors Exhibiting Hysteresis Loops, IEEETrans. on Circuit Theory, vol. CT-17, NO.4, NOV.1970, pp. 564-574. 4. J. G. Santesnases,J. Ayah, S. H Cachero, Analytical Approximation of Dynamic Hysteresis Loops and its Application to a Series FerroresonanceCircuit, Proc. IEE 117,No. 1, Jan 1970, pp. 234-240. 5. W. L A N e v e H. W. Donrmel, On Modelling Imn Core Nmlinamities. IEEE T d o n s on Power Systems, vol. PWRS-8. h4ay 1993,pp. 417425.

Substituting

r - -!-=

'-4

i,A (@=

2

b,,sinne

R I

for n=l,3,--.,

with

(A. 1)

4

above equation, the

4

+4CO4)1

4-1 -rl~~~n,ef-l~~k+-C~n~j-l

r, = i Lk '

'm,= $of

sin[(n-lP,I

13. A. Basak and AAA Qader, FtI"enta1 and Hanncnic Flux Behaviour In a IOOkVA Disbibutim Tmnr/onner Core, IEEE Trans. onh4ag.,Vol. Mag-l9,NoS,Septanber 1983,pp. 2100-2102.

Although only the computation of triplen harmonics of the current is needed, it is appropriate to show the derivation of all odd harmonic components that may be produced by saturation curves during excitation tests. The equations below are developed for nonlinear inductances. For nonlinear resistances, one needs to replace 1 by V and L by R accordingly. Consider the piecewise nonlinear inductance of Figure 2c. For a sinusoidal flux A(@) = sin(@, the current can be written in a Fourier series form containing odd harmonics only. So,

1 . r, = mto the

4 = r24 + $rI - r2 x+z

-

-

4

fundamentalcurrent(n=l) anditsharmonics(n23)are:

6. ATP Rule Book, Leuven EMTP Center (LEC), Section XIXG, Revision July, 1987. 7. H. W. Dommel, Electromagnetic Tmnsients Progmm Reference Manual, Section 6, Department of Electrical Engineering-The University of British Columbia, Vancouver, 1986. 8. G. E. Fish, So) Magnetic Materials, F'mxdqs of the IEEE, Vol. 78, No. 6, June 1990, pp. 947-972. 9. T. H. O'Dell, Fe"amtoC+"ics, Chapters 1 and 5, Mcmillan PressLtd,LonQns 198L and 10. G. Hener and H. Hilzinger, Recent Developnents in Sop A h p d c b,,=L?{-, Matetials, physlat Scripta,Vol. T24,1988, pp. 22-28. 11. T. Nakata, Numerical Analysis of Flux and Loss Distribution in 1-2 Electrical Machinety, (Invited Paper), IEEE Trans. on Magnetics, Vol. Mag 20, NO. 5 , September, 1984. where 12. F. J. Willcins and A E. Drake, Meann"t and Intmpretation of Ponw Larw in Elecbiml Sheet Steel. Proc. IEE Vol. 112, NO.4, April, 1%5, pp. 71-785. and

Appendix A Computation of HmmOniC Componenis

!!h and

gm(n.w =

qn-1)

-i s i n 20,) - sin[(n+l)811 2(n+l)

- Washington L. A. Neves was bom in Brazil on March I, 1957. He received the B. Sc. and M. Sc. degrees in Electrical Engineaing h m Universidade Federal da Paraiba in 1979 and 1982 respectively. From 1982 to 1985 he was with the Departmat of Electrical Engineerkg of Faculdade de Engenharia de Joinville, Santa Catarina, Brazil. Since November 1985, he is with the Department of Electrical hguleering of Universidade Federal da Paraiba, Campina Grande-PB Brazil. He is currently a Ph. D candidate at the University of British Columbia, Vancouver, Canada.

-

Hennunn K Dommel was born in Germany in 1933. He received the Dip1.-Ing. and Dr.-Ing. degrees in Electrical hgu~eerhg from Technical University, Munich, Germany in 1959 and 1962 respectively. From 1959 to 1966 he was with the Technical University Munich, and from 1966 to 1973 with Bonneville Power Administration, Portland, Oregon. Since July 1973 he has been with the University of British Columbia in Vancouver, Canada. Dr. Dommel is a Fellow of IEEE and a registered professional engineer in British Columbia, Canada.