ATP Transformer Model

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Sung-Don Cho 303

3. Duality-Based Model

Detailed models incorporating core nonlinearities can bederived by applying the principle of duality on topology-based magnetic models. This approach is very useful forcreating models accurate enough for low-frequencytransients. The mesh and node equations of the magneticcircuit are the duals of the electrical equivalents node andmesh equations respectively.

The structure of a three-phase shell-form transformer isshown in Fig. 1. The fluxes in the core are 1 = A /2, 2= B /2, 3 = C /2, 4= A - B , 5 = B - C ,.

The lumped magnetic circuit representing the three-phase shell-form transformer is presented in Fig. 2. Thewindings are represented by MMF sources. Reluctances x, o, m signify the portions of the core with a cross section

that is about 50% that of the core inside the windings.Reluctances x, o, m represent the parallelcombinations of two reluctances for upper and lower coresections (the core structure in Fig. 1 was horizontallyfolded due to symmetry, simplifying the resulting lumpedmagnetic circuit). These portions of the core thus have thesame conditions of saturation as the core inside thewindings.

A shell-form transformer is designed so that the middlelimbs ( y) can carry two fluxes, permitting economy inthe core construction and lower losses. The mean turnlength is usually longer than for a comparable core-formdesign, while the iron path is shorter.

Fig. 3 indicates the equivalent electric circuit for theshell-form transformer.

5

4

A B C

1

2

3

Fig. 1. Shell-form Transformer Structure

Fig. 2. Magnetic Circuit for Shell-form Transformer

Fig. 3. Equivalent Electric Circuit for Sell-form Transformer

4. Parameter Estimation for Transformer Model

4.1 Leakage Inductance

The leakage reactances X SC and X CT in Fig. 3 arecalculated from the test data and scaled to per unit based onthe voltage across each winding. The leakage reactanceXTL is related to the leakage channel between the tertiary

winding and the core.Parameter k=a 1 /a 2 is typically used with a 1, which is the

width of the leakage channel between the tertiary windingand the core and a 2 is the width of the leakage channelbetween the T-winding and the C-winding. If the leakagereactance X CT for a three winding transformer are giventhen the leakage reactance X TL to the core winding areassumed to be X TL =k* XCT The leakage reactancesaccording to the H-X-Y sequence are now sorted accordingto the S-C-T sequence and the leakage reactance X TL related to the artificial core winding is established.


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304 Three-phase Transformer Model and Parameter Estimation for ATP

4.2 Core Saturation

For a shell-form transformer, cross-sectional area ratiosin Fig. 4 were assumed on the basis that the flux density isthe same for all paths. The portions of the core thus havethe same conditions of saturation as the core inside thewindings. Lengths were chosen on the basis that since mostcoil types are pancake type, then legs and yokes are longerthan the limbs. Reluctances 4 ~ 10 in Fig. 5 represent theparallel combinations of two reluctances for upper andlower core sections.

L1

A1

L2

A2 L 3 A 3

L4

A4 L 5 A 5 L 6 A 6

L9

A9

L10

A10

L7

A7

L8

A8

Notation: A: Area, L: Length, Typical Area ratio: A 1 ~A3 =1, A4 ~A6 =0.5, A 7 ~A8 =0.87, A 9 ~A10 =0.5, Typical Length ratio: L1 ~ L6 =1, L 7 ~L10 =0.67

Fig. 4. Dimension of Shell-form Transformer

1 2 3

Fig. 5. Magnetic Equivalent Circuit

i 1= 1(R1+R 4+R 9 )+R 7 ( 1- 2 ) (1)

i 2= 2(R2+R 5 )+R 7 ( 2- 1 )+ R 8 ( 2- 3 ) \ (2)

i 3= 3(R3+R 6 +R 10 )+R 8 ( 3- 2 ) (3)

where, R 1=L 1 /( 1 A1 ), 1=f( 1 /A1 ),R2=L 2 /( 2 A2 ), 2=f( 2 /A2 ), R3 =L 3 /( 3 A3 ), 3=f( 3 /A3 ), R 4=L 4 /(2 4 A4 ), 4=f( 1 /2/A4 ),

R5=L 5 /(2 5 A5 ), 5=f( 2 /2/A5 ), R 6 =L 6 /(2 6 A6 ), 6 =f( 3 /2/A6 ), R7 =L 7 /(2 7 A7 ), 7 =f(( 1 - 2 )/2/A7 ), R8 =L 8

/(2 8 A8 ), 8=f(( 2 - 3 )/2/A8 ), R 9 =L 9 /(2 9 A9 ), 9=f( 1 /2/A9 ), R10 =L 10 /(2 10 A10 ), 10 =f( 3 /2/A10 )

In order to estimate the core saturation, variables areclassified as known and unknown:

Known values: 1=v 1 /( N), 2=v 2 /( N), 3=v 3 /( N)v=peak-voltage for each phase, =2 f, N = number of

turn Magnetizing Current: I rms,AVG =(I 1,rms +I 2,rms +I 3,rms )/3Core dimensions or ratios (A and L)

Unknown values: a, b fora

) Bb1( H B

== (4)

If the core dimension ratios and average RMSmagnetizing currents at 100% and 110% voltages are given,some optimization techinique can be used to estimate the aand b coefficients for the B-H Frolich Equation (4) [3], [6].

The optimization performed is as follows:

1(k)=v 1 /( N) sin( k/40) and 2(k)=v 2 /( N) sin( k/40-2 /3)

B1(k)=B 4(k)=B 9(k)= 1(k)/A1 and B 7 (k)= ( 1(k)- 2(k))/2/A7 )

a )k ( Bb1

)k ( 11

= anda

)k ( Bb1 )k ( 7 7

=

R1(k) =L 1 /( 1(k) A1 ) and R 7 (k)=L 7 /( 7 (k) A7 )

i 1(k)= 1(k) (R1(k)+R 4(k)+R 9(k))+R 7 (k) ( 1(k)- 2(k))

40

))k ( i( I

21

40

1k RMS ,1

=

= and I RMS,

AVG =(I 1,RMS +I 2,RMS +I 3,RMS )/3

Minimize f(a,b)=

[ ] 2 RMS , AVG RMS , AVG V %100@ I Calculated V %100@ I Measured

[ ] 2 RMS , AVG RMS , AVG V %110@ I Calculated V %110@ I Measured +Subject to inequality constraints 0 < a and 0 < b < 1

From the optimization technique, the results area=3.7651, and b=0.5651. Fig. 6 shows the obtained B-Hcurve. The calculated RMS currents for the three phasesare [99.9829, 107.5387, 107.5386]A at 100% voltage. Thecalculated average RMS current is 105.02A and thedifference from the test report is only 7.9357 10 -5 A. Thecalculated RMS currents for the three phases are [215.8171,228.2594, 228.2594]A at 110% voltage. The calculatedaverage RMS current is 224.11A and the difference fromthe test report is 3.7518 10 -5 A. Fig. 7 shows the -i curvefor each core section of the example transformer. Fig. 8presents the simulated current waveforms of lines at 100%voltage.


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Sung-Don Cho 305

Fig. 6. B-H for Each Section

Fig. 7. -i Magnetization Curves for Each Section

Fig. 8. Current Waveforms for Each Line at 100% Voltage

4.3 Core Loss Model

The modeling of eddy currents and hysteresis has beenapproximate and difficult, because of the lack ofinformation. In the approach developed here, parameters

for the transformer model are estimated using basictransformer test data and optimization techniques.

For the detailed model, separation of the core losses is:PC (core loss) = P H (hysteresis loss) +P E (eddy current

loss)

From test data, average core losses at 100% voltage and110% voltage are given as [297600, 402240]W at =[51.77, 56.95]Wb-t.

The calculated average core loss should be:

nn

10

1n n

nnn

10

1n

nn L A Bb1

Ba L A ) B( P

= ==

(5)

The optimization performed is as follows:

Minimize f(a,b)=2

nn

10

1n n

n

2

nn

10

1n n

n L A Bb1

BaV %110@ P L A

Bb1 Ba

V %100@ P +

==

Subject to inequality constraints 0 < a and 0 < b < 1

Thus, a= 7071.8 and b =0.4272 are calculated for thecore saturation curve by using the optimization technique.In this case, there is only a minor difference [0.01520.0112] W between the given core loss and the calculatedcore loss. If the ratio( ) of P H to total core loss and theratio ( ) of P E to total core loss given to define the

frequency-dependent effects of core loss, the equivalentresistance (R E ) for P E is:

R E = V 2 /(P E A L) = (V @ B)

2 / [ (P C @ B) (A L)] ( )(6)

In [6], the equation for DC hysteresis loss was given asEquation (10). From the core loss separation, aa and bbare obtained.

MAX

MAX MAX E MAX C MAX H

Bbb1 Baa

) B( P ) B( P ) B( P

==

(7)

Fig. 9 shows each core loss curve for the shell-formtransformer.

Fig. 9. Core Loss Curve using Frolich Equation

The right displacement in the hysteresis loop is linearand the left displacement is nonlinear in [6]. At zero flux,both displacements must be the same and this is a coerciveforce (H C ). The coercive force at each loop should bedetermined to meet the power loss P H at the B max for eachloop.


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306 Three-phase Transformer Model and Parameter Estimation for ATP

The displacements at each loop are shown in Figure 10.

Fig. 10. Left and Right Displacements of Hysteresis Current

5. Conclusion

The overall transformer model for ATP implementationis shown in Fig. 3. Fig. 11 displays the DC hysteresis loopmodeled using a Type-60 current source controlled byTACS. Fig. 12 illustrates the current of the eddy currentloss and the resistive hysteresis current. Fig. 13 shows themagnetizing current modeled with a Type-93 nonlinearinductance. Fig. 14 and 15 show the line-current andwinding-current waveforms.

To verify the transformer model developed using TACS,results from simulated open circuit tests were compared tothe transformer test report. After all models wereimplemented and run with ATP, the results of open-circuit

and short-circuit simulations are very close to the testreport.

(f i le TESTTRS.pl4; x-v ar t : LEG1) t : B1

-4 -2 0 2 4-1.6

-1.2

-0.8

-0.4

0. 0

0. 4

0. 8

1. 2

1. 6

Fig. 11. DC Hysteresis Loop Generated by ATP

( f i l e T E S T TR S . p l 4 ; x - v a r t ) t : E D D Y 2 t : H Y S C 2

0 1 0 2 0 3 0 4 0 5 0*10 -3-4

-2

0

2

4

o: i E : i H

Fig. 12. Eddy Current( i E ) and Hysteresis Current( i H ) Wave-forms at 100%V

The simulated results using the model developed hereare reasonable and more correct than those of theBCTRAN-based model. Simulation accuracy is dependenton the accuracy of the equipment model and its parameters.This work is significant in that it advances existingparameter estimation methods in cases where the availabledata and measurements are incomplete. Theoretical resultsobtained from this work provide a sound foundation for thedevelopment of transformer parameter estimation methodsusing engineering optimization. To further refine anddevelop the models and transformer parameter estimationmethods developed here, iterative full-scale laboratory testsusing high-voltage and high-power three-phase trans-formers would be helpful.

(file TESTTRS.pl4; x-var t) c:LEG2 -N99999 c:LEG7 -BBBBBB0 10 20 30 40 50

-50

-25

0

25

50

*10 -3

o: Leg-1 : Mid Limb A-B at 100%V

Fig. 13. Mangnetizing Current Waveforms of Leg 2 andMid Limb A-B (Leg-7)

(file TESTTRS.pl4; x -var t) c:A138F -A138T c:B138F -B138T c:C138F -C138T0 10 20 30 40 50

-200

-100

0

100

200

*10 -3

o: Line-A : Line-B : Line-C

Fig. 14. Line Current Waveforms for Tertiary at 100% V

(file TESTTRS.pl4; x-var t) c:A138T -XX0103 c:B138T -B-T c:C138T -C-T0 10 20 30 40 50

-200

-100

0

100

200

*10 -3

o: Phase-A : Phase-B : Phase-C

Fig. 15. Winding Current Waveforms for Tertiary at 100% V

References

[1] H.W. Dommel, EMTP Theory Book , Bonneville Power Administration, Portland, USA, August 1986.

[2] G.R. Slemon, Equivalent Circuits for Transformersand Machines Including Non-Linear Effects,Proceedings Institution of Electrical Engineers, Vol.100, Part IV, 1953, pp. 129-143.

[3] Working Group C-5 of the Systems ProtectionSubcommittee, Mathematical models for current,


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Sung-Don Cho 307

voltage, and coupling capacitor voltage transformers ,IEEE Transactions on Power Delivery, Vol. 15 Issue: 1,Jan. 2000, pp. 62 72.

[4] Armco Catalog, "Oriented and TRAN-COR HElectrical Steels", 10th Edition, Jan. 1986.

[5] M. J. Heathcote, The J&P Transformer Book - 12 th Edition , Newnes Ltd., 1998.

[6] Sung Don Cho, Transformer Core Model and Parameter Estimation for ATP , KIEE InternationalTransactions on Power Engineering, vol. 5-A, no. 4,2005.

Sung-Don Cho He received his Ph.D. degree fromMichigan Tech University in 2002. Hisresearch interests are transient analysis,EMTP and ATP application.

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ATP Transformer Model