J.Arrillaga W. Enright N. R . Watson A.R. Wood

Indexing terms: Electromagnetic transient, Converter transformer, Simulation

~

Abstract: In electromagnetic transient simulation HVDC converter transformers are normally represented as magnetically independent single- phase units. However, three-limb converter transformers are often used in HVDC schemes. Single-phase and three-limb three-phase transformer models are derived using a unified magnetic equivalent circuit concept. The new models are verified with laboratory and field data. Also, a comparison is made between the transient performance of HVDC converters with conventional and proposed transformer models.

1 Introduction

The conventional representation of HVDC converter transformers in electromagnetic transient programs assumes a uniform flux throughout the core legs and yokes, each phase modelled as a magnetically inde- pendent unit. Moreover, the primary and secondary winding leakages are combined and the magnetising current placed on one side of the resultant series leak- age reactance. Among the proposed alternatives for calculating core flux are the use of the core closest to the core [l] and an equal division of the magnetising current between the primary and secondary windings.

However, some HVDC schemes use three-limb core transformers and their representation as independent phase units is likely to affect the accuracy of the solution, particularly when simulating unbalanced conditions.

A three-limb three-phase transformer model has recently been proposed [2] which derives an electric circuit dual from a magnetic circuit equivalent that includes air and steel flux paths. Although the core leg, yoke and zero-sequence paths are individually represented, the individual winding leakages are still lumped together and the question of the subdivision of the magnetising components between the windings is not resolved. 0 IEE, 1997 IEE Pvoceedings online no. 19970849 Paper first received 20th May and in revised form 20th August 1996 W. Enright is with the Electricity Corporation of New Zealand J. Arrdlaga, N.R. Watson and A.R. Wood are with the University of Canterbury, Christchurch, New Zealand

This paper describes a new equivalent circuit based on unified magnetic theory, which is generally applica- ble to three-phase multi-limb transformers. The new model is incorporated in the electromagnetic transients (EMTDC) program [3] and its response to HVDC dis- turbances is compared with that of the conventional converter transformer model.

2 Single-phase model

In power system studies transformer core and leakage fluxes are often considered separately, as shown in Fig. 1. This approximation permits the use of superpo- sition and thus combines the linear leakage reactance with a linearised Norton equivalent of the magnetising current. This combination has been used to derive har- monic domain models of the single-phase [4] and three- limb [5] three-phase transformers, respectively; in these cases the primary and secondary voltage average was used to calculate the core flux, while the magnetising current was halved and placed on both sides of the transformer leakage admittance network.

Fig. 1 Single-phase transformer flwc paths, separate magnetic paths

2.1 Unified magnetic equivalent circuit The magnetic circuit shown in Fig. 2, is based on Steinmetz’s ‘exact’ model, and leads to the unified magnetic equivalent circuit (UMEC) of Fig. 3. In this alternative there is no need to specify in advance the distribution of magnetising current components which, as will be shown later, is determined by the transformer internal and external circuit parameters. Moreover, all the elements of the equivalent circuit can be derived purely from the conventional manufacturers specifications.

The two MMF (magnetomotive forces) sources in Fig. 3, i.e. N1i,(t) and N2i2(t), represent the windings individually, and the winding voltages vl(t) and v2(t) are used to calculate winding-limb fluxes # l ( t ) and q52(t), respectively.

IEE Proc-Gener. Transm. Distrib., Vol. 144, No. 2, March 1997 100

I I Eqn. 1 can be rewritten as

At each node the flux must sum to zero, stated as

Application of the branch-node connection matrix to the vector of nodal MMF gives the branch MMF

Multiplying eqn. 2 by [AIT and substituting in eqns. 3 and 4 gives

Solving eqn. 5 for enode and multiplying both sides by [A] gives

Substitution of eqn. 4 into eqn. 6 allows eqn. 2 to be written as

where

If the vector of branch flux is partitioned into the set that contains the branches associated with each trans- former winding then eqn. 7 becomes

where q?& is the vector containing the winding flux $1

and $2, is is the vector containing the winding current il and iz, [M,,] a square permeance matrix and [NsJ a diagonal matrix of winding turns.

If trapezoidal integration is used to relate the trans- former winding voltage to the winding flux, the deriva- tion will be suited for use in an electromagnetic transient program [6]. The winding voltage is related to the winding flux by

4 = [P]([N]I - 8) ( 2 )

[AIT$ = O (3)

[~IJnude = J (4)

( 5 )

[ ~ ] 8 n u d e = [AI([A]~[P][AI)-’[A]T[PI[NIS (6)

fj = [ M ] [ N ] l (7)

0 = [AlT[P][N]; - [A]TIP][A]e,o,,

[MI = ([I1 - ~ ~ I ~ ~ I ( ~ A I T ~ ~ I ~ ~ I ~ ~ ’ ~ ~ l T ~ ~ ~ l

4 s = [ ~ s s l [ ~ s s l ~ s ( 8 )

At &(t) = &(t - At) + ,[Ns.]-’(Os(t) + &(t - At)) (9)

where At is the simulation time step and ;S the vector of winding voltage v1 and v2. Each winding voltage is used to calculate the corresponding winding-limb flux.

Combining eqns. 8 and 9 gives the Norton equivalent

I s = [yss]as(t) + Ins (10) where

at [ Y S S ] = ([M,s][~s~])-’~[~.”l-l

and

Fig. 2 Single-phase transformer flux paths, unified magnetic path

N I

Fig. 3 model

Unified magnetic equivalent circuit for single-phase transformer

PI and Pz represent the permeance of the transformer winding limbs and P, the permeance of the transformer yokes. Core permeance is calculated using the hyper- bola B-H characteristic representation described in the Appendix. If the total length of core surrounded by windings L, has a uniform cross-sectional area A,, then UMEC branches 1 and 2 have length Ll = L2 = LJ2 and cross-sectional area A I = A2 = A,. The upper and lower yokes are assumed to have the same length L, and cross-sectional area A,. Both yokes are repre- sented by a single UMEC branch of length L, = 2Ly and area A, = Ay. Leakage permeances P4 and P, are obtained from open and short-circuit tests and there- fore the effective lengths and cross-sectional areas of their leakage paths are not required.

I 4 It) -y12 19(t) I

v z l t )

Fig. 4 Single-phase transformer model, UMEC PSCAD-EMTDC model, electric circuit

The magnetic circuit of Fig. 3 can be represented by the Norton equivalent of Fig. 4, which is suitable for electromagnetic transient implementation. The MMF across each branch of the circuit in Fig. 3 can be written in vector form as

e = [NI5 - [RI4 (1) IEE Prm-Gener. Transm. Distrib., Vol. 144, Nu. 2, March 1997

2.2 Experimental verification The test system shown in Fig. 5 is used for the experi- mental verification of the new model. A lOkW sine- wave generator suplies the test transformer, which is loaded with a series resonant circuit tuned to the fifth harmonic. The fifth harmonic impedances of the test circuit are ZSOUlce = 0.3 + j180.64Q Rprim = 0.SQ jXIprim = jl8.54 Q jXl,,, = j18.54 Q R,,, = 0.5Q Zroad a = 1

= 0.364 - j0.8301 Q

101

series sine wave resonant generator- - - I . t e s t t rans fo rmer - 1 load ,c-

Zsource Rpr im j x i p r l m

I I I

A 7 -

Fig.5 Laboratory experimental set up

+ Fig.5 Laboratory experimental set up

In the absence of other harmonic sources the magnetis- ing current fifth harmonic component will divide between the primary and secondary windings according to Steinmetz's exact ratio, i.e.

which for the numerical values of the experimental sys- tem yields K = 11.2. Alternatively, the current division constant obtained by Steinmetz approximate equivalent is

K S t m z ( a p p T 0 X ) - - I z s o u T c e + Rprzm I I j X l p r i m + a2 ( j x l s e c + R s e c + zload)i

= 4.97 I (1-2)

Measured and simulated ratios of fifth harmonic cur- rent between the primary and secondary windings are shown in Table 1.

P7* to P1; are leakage permeances (obtained from open and short-circuit tests) P13* and P1; are the permeances of the yokes PI5* to P17* are zero-sequence permeances (obtained from in-phase excitation of all three primary or second- ary windings) [2] . As for the single-phase case there is no need to specify in advance the distribution of magnetising current amongst the windings.

011

'15

01 2

Fig. 6 UMEC EMTDC three-phase three-limb transformer model

Table 1: Measured and simulated primary and secondary fifth harmonic magnetising current

Experimental UMEC Conventional Fig.7 Magnetic equivalent circuit for UMEC EMTDC three-phase three-limb transformer model 4.00 (mA) 2.42 (mA) 1.11 (mA)

The simulation results, obtained with the EMTDC program for the UMEC and conventional transformer models are close to those predicted by eqns. 11 and 12. Because hysteresis and core loss have not been included, the actual levels of fifth harmonic predicted by the UMEC model do not exactly match the experi- mental results. However, for modern HVDC converter transformers with highly efficient core material, the inaccuracies caused by excluding hysteresis and core loss representation should be less significant.

3 Three-phase three-limb transformers I I G ( t

3. I UMEC generalisation An extension of the single-phase UMEC concept to the three-phase transformer, shown in Fig. 6, leads to the three-limb three-phase representation shown in Fig. 7, where

Y55fY15+Y25 ''35 'y45 ' y56

* to '6* are the permeances Of the transformer Fi 8 Electric equivalent circuit for UMEC EMTDC three-phase three-

winding-limbs lim? tmnsjhmer model

102 1FF P m r . C : o n r r T v n n r r n nia t r ih KnI idd hln 7 l l n v c h 7007

MMF sources Nlil(t) to N&(t) represent each trans- former winding individually, and winding voltages v1 ( t ) to v6(t) are used to calculate winding-limb flux $l(t) to #6(t), respectively. The permeances of the UMEC of Fig. 7 are used to derive the elements of the admittance matrix in the electric Norton equivalent, i.e.

- Y 1 1 Yl2 Y13 Y14 Yl5 Y16

Y2l Y22 Y23 Y24 Y25 Y26

Y31 Y32 Y33 Y34 y35 y36

Y41 Y42 Y43 y44 y45 y46

Y51 Y52 Y53 Y54 Y55 y56

-Y61 Y62 Y63 Y64 Y65 Y66

t

Matrix [Y,] is symmetric and this Norton equivalent can be implemented in the EMTDC program as shown in Fig. 8, where only the blue phase network of a star- grounded transformer is shown.

Trapezoidal integration is now applied to the six transformer windings to calculate the winding-limb flux vector $s(t - At) . Once the previous time step winding current vectoj i,(t - At) is formed, the flux leakage elements of q$(t - At) can be calculated using 47(t - At) = P7(Niii(t - At) - $l( t - &)PI)

f&(t - At) = P,(N222(t - At) - $&(t - At)P2)

4 g ( t - At) = Pg(Nii3(t - At) - 4 3 ( t - At)P3)

$ l o ( t - At) = Plo(N2i4(t - At) - 44(t - At)P4)

4ll(t - At) = Pii(NlZs(t - At) - 45(t - At)P5)

(14)

(15)

(16)

(17)

(18)

412(t - At) = P12(N2'&3(t - At) - 46(t - At)P6) (19) The zero-sequence elements of &(t - At) are calculated using the MMF loop sum around the primary to sec- ondary winding-limb and zero-sequence branch, stated as 415(t - At) =Pi5(Nlil(t - At) + N2i2(t - At)

- 4l(t - At)Pl - 42(t - 4 P 2 ) (20)

- 43(t - At)P, - 4 4 ( t - At)P4) (21)

417(t - At) =Pl7(Nli5(t - At) + N2ie(t - At)

$ 1 6 ( t - At) 'P16(Nli3(t - At) + N223(t - At)

- 4 5 ( t - at)p5 - 4 6 ( t - at)P6) (22) Finally, the yoke flux is obtained using the flux sum- mation at nodes N I and N2, stated as 413(t-At) = 4l( t -At)-47(t-At)-4,5( t -at) (23)

414(t-at) = 45(t-at) -dil(t-At)-417(t-At) (24)

3.2 Application to HVDC converters To demonstrate the influence of the transformer core representation in electromagnetic AC/DC studies the proposed UMEC model is used to compare the tran- sient behaviour of HVDC converters with three single- phase and multi-limb transformers. For a realistic com- parison the alternative transformer models must draw similar magnetising currents in the steady state. How- ever, it is difficult to achieve a perfect match because three-limb transformers use less core steel and their phases are unbalanced. Accordingly the parameters of the single-phase transformers of the Benmore convert- ers have been modified to derive an equivalent three- limb UMEC model of similar parameters, as shown in

IEE Proc.-Gener. Transm. Distrib., Vol. 144, No. 2, March 1997

the Appendix. A comparison of the steady-state mag- netising currents of both cases (using the UMEC model) is shown in Figs. 9 and 10 with ten percent overvoltage. The three-limb red, yellow and blue mag- netising currents are 0.078 kA, 0.067kA and 0.078kA, respectively as compared with 0.086 kA for the three phases of the independent transformer banks.

0.080 0.081 0.088 0.092 0.096 0.100 time,s

Benmore steady-state over voltage transformer energisation, Fig. 9 three-phase bank primary winding current

~ red phase _ _ _ _ yellow phase

._ blue phase

o.20r

-0.201 I I I

0.080 0.081 0.088 0.092 0.096 0.100 time, s

Fig. 10 three-limb three-phase primary winding current

~ red phase _ _ _ _ yellow phase ~. ~. -. ~ blue phase

Benmore steady-state over voltage transformer energisution,

L 0 0.00L 0.008 0.012 0.016 0.020

time,s Fi .11 Benmore group connection generator current waveform, YyO an2 Ydl l three-limb converter transformers, Idc = 405A

~ field measurement simulation

103

> Y IJ 1 0 -

-101

I I I J

0 0.004 0.008 0.012 0.016 0.020 time,s

Fi . I 2 Benmore group connection generator voltage waveform, YvO an% Ydii three-limb converter transformers, Idc = 405A __ field measurement _ simulation

The ability of the EMTDC program and UMEC transformer models to simulate the steady-state per- formance of the Benmore converter when operated as a group connected station are shown in Figs. 11 and 12. The Figures show that the simulated voltage and cur- rent waveforms (dotted line) are very close to the field test results (continuous line); the simulated waveforms of the two alternatives (single banks and multi-limb) were found to be indistinguishable.

Finally, the group-connected converter system was used to demonstrate the use of the multi-limb model under transient conditions. The test system, shown in Fig. 13, was subjected to a single-phase fault at the converter terminals and details of the fault, circuit- breaker, by-pass valve, etc. are given in the Appendix.

, Itvv I I 7 -, I

bypass valves

. ) ‘9 * & red phase-to-ground fault

Fig. 13 system

Conventional converter transformer con$guration transient test

The fault is applied to the 16kV busbar (time = 1 .Os), the converter is blocked and the bypass valves fired two cycles later (time = 1.04s). The circuit- breaker opens two and a half cycles after the fault (time = 1.05s). The circuit-breaker recloses after 0.03s and the converter is deblocked three cycles after the reclose (time = 1.14s).

The three-phase bank and three-limb three-phase Yy0 converter transformer primary currents it,, are shown in Figs. 14 and 15, respectively. In this winding configuration the single-phase bank is not affected by the presence of the fault. However, significant fault currents flow in the primary windings of the three-limb three-phase converter transformer.

Between fault initiation and converter blocking the fault current is superimposed on the converter currents. Once blocking is ordered, and the bypass valve is fired,

104

the red, yellow and blue valves commutate off. After the circuit breaker is opened conduction does not cease until the fault current in each phase passes through zero.

-I0S

- 20 1.06 1.08 0.98 1.00 1.02 l . O &

time,s Fig. 14 ground fault, three-phase bank, YyO primary cuvrent - red phase _ _ - - yellow phase

blue phase

Converter transformer winding currents, I6kV red phase to

* O l

-201 I

0 98 1.00 1.02 1.Od 1.06 1.08 time,s

Fig. 15 ground fault, three-limb three-phase, YyO primary arrent - red phase - - - - yellow phase

blue phase

Converter transformer winding currents, i6kV red phase to

~

Fig. 16 Star-grounded three-limb transformer flux distribution, red phase to ground fault open-circuit core flu

Consider only the transformer primary windings as shown in Fig. 16 and the corresponding phasor diagram of Fig. 17. The reference phaso; is the three- limb primary winding red-phase voltage VI prior to the fault. During the fault, instead of the nominal flux &, the shorted red-phase causes the sum of the yellow and blue-phase flux &3 and J5, respectively to pass into the

IEE Proc.-Gener. Transm. Distrib., Vol. 144, No. 2, March 1997

red-phase winding-limb. This flux 6; is equal in magnitude, and 180” out of phase, to the nominal flux &. Near rated voltage v; is induced in the red-phase winding, and the fault current i; flows from the yellow and blue phases through the mainly inductive transformer primary winding impedance.

p’i”

Fig. 17 Volta e, current and flux phasor diagram for star-grounded three-limb transjrmer flux distribution, red phase to ground fault open-cir- cuit core,flux

1 I /

I I I 1.08 1.10 1 .I2 1 .I1 1.16

time,s Converter transformer winding current, fault recovery, three- Fig. 18

phase bank, YyO primary current ___ red phase _ _ _ _ yellow phase

blue phase

L I /

- 1

1.08 1.10 1 .I2 1.11 1.16 time,s

Converter transformer winding current, fault recovery, three- Fi .19 lirn% three-phase, YyO primary current ~ red phase _ _ _ _ yellow phase

blue phase

The waveforms of Figs. 14 and 15 are continued in Figs. 18 and 19 for the period following the circuit- breaker reclose (time = 1.08s). The three-phase bank in-rush currents are symmetrical (Fig. IS), while the

IEE Proc.-Gener. Transm. Distrib.. Vol. 144, No. 2, March 1997

three-limb three-phase (Fig. 19) in-rush currents are asymmetrical. The maximum in-rush current peak of the YyO three-phase bank is greater than that of the three-limb three-phase equivalent. When the converter is deblocked (time = 1.14s) the in-rush currents are superimposed on the converter currents.

4 Conclusions

The proposed unified magnetic equivalent circuit (UMEC) eliminates the need for approximations such as the dependence of core flux on a single transformer winding, the arbitrary placement of magnetising cur- rent, the uniform core flux assumption and lumped leakage reactance. The single-phase UMEC model has been shown to be representative of the Steinmetz exact equivalent circuit. The simulated division of magnetis- ing current between the UMEC transformer model windings has been verified in a laboratory test system.

The steady-state and transient performance of three- phase bank and three-limb three-phase converter trans- formers have been compared. Under steady-state con- ditions both transformer types produced similar waveforms. However, when a single-phase to ground fault is applied to the converter transformer primary busbar significant fault currents flow in all primary phases of the three-limb transformer, yet no fault cur- rents flow in the three-phase bank.

5 Acknowledgments

The authors wish to thank K. Devine, N. Frampton, and the Electricity Corporation of New Zealand for the financial support of this project. They are also grateful for the technical support of Trans Power New Zealand, and especially the help of P. Thompson and of K. Smart of the University of Canterbury.

6 References

1 DOMMEL, H.W.: ‘Transformer models in the simulation of electromagnetic transients’. Proceedings of 5th Power systems computation conference, Cambridge, United Kingdom, 1975, pp. 3.114: 1-1 6 STUEHM, D.L.: ‘Three-phase transformer core modeling’. Tech- nical report, North Dakota State University, 1993

2

3 ‘Electromagnetic transients simulation program manual’. Manitoba HVDC Research Centre, Winnipeg, Canada, 1988

4 ACHA, E., ARRILLAGA, J., MEDINA, A., and SEMLYEN, A.: ‘General frame of reference for analysis of harmonic distor- tion in systems with multiple transformer nonlinearities’, ZEE Proc. C, Gener. Transm. Distrib., 1989, 136, (9, pp. 271-278 MEDINA, A., and ARRILLAGA, J.: ‘Generalised modelling of power transformers in the harmonic domain’, ZEEE Trans. Power Deliv., 1992, 7, (3), pp. 1458-1465

6 DOMMEL, H.W.: ‘Digital computer simulation of electromag- netic transients in single and multiphase networks’, ZEEE Trans. Power Appar. Syst., 1969, PAS-88, (4), pp. 388-399

5

7 Appendix

Benmore generator parameters rating 112.5MVA voltage 16.0kV frequency 50 H z XI’ 0.1820p.u. R a 0.0042p.u. Converter transformer datu type star-star, and star-delta rating 1 87.5 MVA primary voltage 16kV secondary voltage 1 lOkV Xlbvim-ter) 0.113 p.u. (62.5MVA base)

105

[Cl=

UMEC three-limb three-phase converter transformer

-1 0 0 0 0 0 - 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1

-0 -1 0 -1 0 -1-

Ydll winding limb length winding limb area yoke length yoke area primary winding turns number secondary winding turns number

p7,9, 11

p 8 , 10,12

p15,16,17

connection matrix

0 0 1 ‘‘l=Io 0 0

0 0 0 L 0 -1 0

Converter model parameters PLO proportional gain PLO integral gain PLO input reference variable snubber resistance snubber capacitance thyristor on resistance thyristor off resistance forward voltage drop forward break-over voltage minimum extinction time

0 -1 0 1 0 0

Circuit-breaker parameters breaker open resistance breaker closed resistance open possible if current flowing?

Bypass valve parameters snubber resistance snubber capacitance thyristor on resistance thyristor off resistance forward voltage drop forward break-over voltage minimum extinction time

3.59m 0.4536m2 2.656m 0.4536m2 65 turns 780 turns 5 A5e-8 5.85e-8 5.85e-8

0 0 0 0 0 0 0 -1 1 0 0 1

10.0 100.0 0 2400 62 0.29 pF 0.01 Q 1 .Oe6 62 0.001kV 1 .Oe5 kV 0.op

1 .Oe6 Q 0.00562 no

2400 Q 0.01 pF 0.00562 1.Oe662 0.001 kV 1 .Oe5 kV 0 .op

106 IEE Proc.-Gener. Transm. Distrib., Vol. 144, No. 2, March 1997