IEEE Transactions on Power Systems, Vol. PWRD-1, No. 2, April 1986

HARMONICS FROM TRANSFORMER SATURATION

H.W. Dommel, Fellow, IEEEThe University of British Columbia

Vancouver, B.C., Canada

A. Yan, Member, IEEEUniversity of Idaho

Moscow, Idaho, U.S.A.

Shi WeiXian Jiaotong University

Xian, People's Republic of China

ABSTRACT

Saturation effects in transformers and shuntreactors can produce harmonics in power systems. Theirmagnitude can sometimes be found with an electro-magnetic transients program, by going from an approxi-mate linear ac steady-state solution directly into atransient simulation in which the nonlinear effects areincluded. In lightly damped systems, such simulationscan take a long time, however, before the distortedsteady state is reached. Therefore, another method wasdeveloped which uses superposition of steady-statephasor solutions at the fundamental frequency and atthe most important harmonic frequencies, with nonlinearinductances represented as harmonic current sources.This method can either be used by itself, or as animproved initialization procedure for electromagnetictransients programs.

1. INTRODUCTION

Saturation effects in transformers and shuntreactors can produce steady-state harmonics, as well astransient harmonics and temporary overvoltages. Thesteady-state harmonics are usually not as high as thoseproduced by converters, but they are somewhat moredifficult to calculate. In converters, the magnitudeof harmonics is reasonably well known, and these har-monics can therefore be represented as given currentsources or voltage sources behind impedances in har-monic power flow programs. In contrast, harmonicsgenerated by saturation effects of transformers dependcritically on the peak magnitude and waveform of thevoltages at the transformer terminals, which in turnare influenced by the harmonic currents and thefrequency-dependent network impedances.

A simple, iterative procedure is described here,which can be used to obtain the magnitude of harmonicsfrom transformer saturation with sufficient accuracy.Other harmonic sources, whose magnitude is known, areeasily included in the solution as additional currentor voltage sources.

The method was primarily developed as an initial-ization procedure for the Electromagnetic TransientsProgram (EMTP), but it could be incorporated into otherprograms as well. Since EMTP studies are usually donewith three-phase representations, the method was testedfor single-phase as well as three-phase cases. Single-phase studies may not always be accurate enough. Forexample, the dominant third harmonic zero sequencecurrent in transformer saturation depends on voltages

85 SMI 381-9 A oaoer recrernended and aoorcveoby the IEEE Trarns.r15ission arnd Distr ibut ior Commit teeof tte 1Q Power Enl Qiireerinq Societv for oresenta-tion at the IEEE/PES 1985 Summer Meetinc. Vancouver.B.C., Canada, July 14 - 19, i9t4t. Manuscriot sub-mitted January 29, 1985; made availanie for orintino+Or: a .30, 1 9,g,>

which are mostly positive sequence. It is thereforebest to use three-phase representations, becausecoupling effects are then automatically included.

2. OBTAINING HARKDNICS WITH THE EMTP

A simple method for obtaining saturatiorr-generatedharmonics is to perform a transient simulation with theEMTP which starts from approximate linear ac steady-state conditions. For the initial ac steady-statesolution, the magnetizing inductances of transformersare represented by their unsaturated values (linearpart of the nonlinear flux/current-curve). In thetransient simulation, the only disturbances will thenbe the deviations between the linear and nonlinearmagnetizing inductance representations. The transientscaused by these deviations will often settle down tothe distorted steady state within a few cycles.

An example for this approach is described in [1].In that example, a power plant was feeding an unloaded345 kV-line of 398 km length. The saturation wascaused by the step-up transformers in the power plant.For the simulation, the generator representation was asimple, though not quite accurate, voltage source Ebehind Xd", which was combined with the transformershort-circuit reactances into a three-phase Theveninequivalent circuit seen from the 345 kV-side (positiveand zero sequence values converted to self and mutualvalues with Eq. (6) of [2] to obtain the phase values).The nonlinear magnetizing inductances were connectedphase-to-ground on the high side of the step-up trans-formers. Again, this is not quite correct. As pointedout in [2], the nonlinear inductances should be corr-nected between the terminals of those windings whichare closest to the iron core, at least in designs withcylindrical winding construction. Construction detailswere not available, however, and placing it anywhereelse would have made it impossible to use the simpleThevenin equivalent circuit for the power plant. Moreimportantly, the chosen representation brought simula-tion results close enough to field test results, sothat a more detailed model was really not justified.Fig. 1 shows the simulation results for the voltages atboth ends of the line, with the simulation startingfrom the approximate linear ac steady-state solutionwith unsaturated magnetizing inductances. In thisparticular case, the final distorted steady state wasreached fairly quickly within approximately 3 cycles.The simulation results compare reasonably well with thefield test results (Fig. 2).

Modern high-voltage transformers with grair-oriented steel cores saturate typically somewhere above1.0 to 1.2 times rated flux, with a sharply definedknee (Fig. 3). Often, a two-slope piecewise linearinductance is sufficient to model such curves. Theslope in the saturated region above the knee is theair-core inductance; it is almost linear and fairly lowcompared with the slope in the unsaturated region. Inthe study resulting in Fig. 1 and 2, a two-slope piece-wise linear inductance was accurate enough; going to amore detailed 5-slope inductance changed the resultsvery little.

0885-8977/86/0004-0209$01.00XO1986 IEEE

209

210

Frequently, the saturation curve is not suppliedSENDING END flux/current-curve *=f(i), required for trar-

sient simulations, but as an RMS-voltage/RMS-current-400 _ \ _,><curve VRMS-f(IRMS) instead. A simple conversion tech-B nique is described in Appendix 1, which is based on the

assumption that the influence of hysteresis and eddyc) 0°~ / - § -- - 4 ( ) current losses and of winding resistances on the¢ / V \ \/20\\// 40(nms) saturation curve can be ignored. The contains a

H X<support routine CONVERT which this conversion

o method.> ~~~~~~~~~~~A-400-

The simple method described in this section forRECEIVING END obtaining harmonics works only well if the final dis-torted steady state is reached quickly in a few cycles.

For lightly damped systems, it may take a long timebefore the final steady state is reached. Fig. 4 shows

400 the voltages at both ends of a 500 kV-line with shuntB reactors which go into saturation at 0.92 p.u. of rated

flux at the sending end and at 1.05 p.u. of rated fluxat the receiving end. Because of low damping, the\40(ms) steady state is reached only after a long time. It isfor such cases why the steady-state solution method0°

A described in the next section has been developed.-400

SENDING ENDFig. 1. Voltages with harmonic distortion on a 345 kV

line (simulation starts from approximate Blinear steady state). 400

400- 0

SENDING END -40200O> - ,> ~~~~~~~~~~~~~~~-400

7 ~~~~~~~~~~~~~~~~~~~~~~~A0

-200X> ---- computation RECEIVING END

field tests-400 B

400

RECEIVING END0004 0 40(ms)

> 2~~~~~~~~~~~~~

0-~~~~~~~~~~~0 ~~~~ ~ ~ ~ ~ ~ ~ ~ ~ ~~~-0

H ~~~~~~~~~~~~~~~~~~~~~~~~~A-200 --computation

field tests

-4001 ~~~~~~~~~~Fig.4. Voltages with harmonic distortion on a 500 kVline (simulation starts from approximatelinear steady state).

Fig. 2. Comparison between simulation and field testresults. 3. HARNDNICS WITH STEADY-STATE SOLUTIONS

To obtain the steady-state solution with harmonicsdirectly from phasor equations, the nonlinear induc-

krated tances are replaced by voltage-dependent current1.1e sources at the fundamental frequency and at the har-

monic frequencies (Fig. 5). The network itself is then

linear, and the voltages at any frequency are thenI ~~~~~~~~easily found by solving a system of nodal equations ofi/lrated the form

-1.1[Y][V] = [I] (11)

where the nonlinear effects are represented as currentsFig. 3. Typical saturation curve, in the vector [I]. The solution is found with two

211

iterative loops. First, "power flow" iterations areused to obtain an approximate solution at fundamentalfrequency, while the second "distortion" iterationstake the higher harmonics into account.

node m

(a)

node m

In these "power flow" iterations at fundamentalfrequency, the VRMS/IRMS-curve is used as an approxi-mation to the curve relating the fundamental frequencycurrent I1 to the fundamental frequency voltage V1.While VRMSis equal to V1, IRM5 contains all harmonics.This approximation does provide a good starting point,however, for the following "distortion" iterations, inwhich harmonics are included.

If the network contains nonlinear loads of thepower flow type, e.g., active power Pm and reactivepower Qm specified rather than current Imn then theadjustments to achieve constant power are incorporatedinto this iterative loop by recalculating

P - i Qm V *

m(2)

currentsource i

Fig. 5. Replacingsources.tance, (b)

(b)

nonlinear inductances by current(a) Network with nonlinear induc-network with current source.

at the beginning of each iteration step.

The "distortion" iterations

The "power flow" iterations produce a steady-statesolution at fundamental frequency only, without har-monic distortion. To obtain the harmonics, the RMSvoltages found from the power flow iterations are usedin an initial estimate for the flux. Since v=dp/dt,and assuming that the peak voltage phasor is jVj|e,or

The "power flow" iterations

In the "power flow" iterations, an approximatelinear ac steady-state solution is found which repre-sents the VRMS/IRms-curves of the nonlinear inductancescorrectly, but does not include harmonic distortion.For the nonlinear inductance, say at node m in Fig. 5,the original data may already be in the form of aVRMS/IRMS-curve, as shown in Fig. 6. If not, it isstraightforward to convert the 41/i-curve into a VRMS/IRMS-curve (see Appendix 1). To start the iterationprocess, a guess for the RMS voltage Vm is used to findthe RMS current Im (Fig. 6). This current, with theproper phase shift of 900 with respect to Vm, isinserted into the current vector [I] in Eq. (1), and anew set of voltages is then found by solving the systemof linear equations (1). This solution process isrepeated, until the prescribed error criterion for thecurrent Im is satisfied. Note that the admittancematrix [Y] in Eq. (1) remains constant for all itera-tion steps; therefore, [Y] is only triangularized onceoutside the iteration loop. Inside the iteration loop,the downward operations and backsubstitutions are onlyperformed on the right-hand side [I], by using theinformation contained in the triangularized matrix("repeat solutions").

Vm

Im I

rms

v(t) = IVi cos(w1t + 4) (3)

as a function of time (w1 = angular fundamental fre-quency), it follows that the flux is

4(t) = M sin(lut + *) .

With p(t) known, one full cycle of the distorted cur-rent i(t) is generated point-by-point with the p/i-curve (Fig. 7). If hysteresis is ignored, then it issufficient to produce one quarter of a cycle of i(t),since each half-cycle wave is symmetric, and since thesecond half is the negative of the first half of eachcycle.

The distorted current i(t) in each nonlinear in-ductance is then analyzed with a Fourier AnalysisProgram, which produces the harmonic content expressedby

Fg .... . _/ ---A *~~~~~~~~~~~~~~~~. 0f

_._. ~. ._ . .A

2 _ I _t ,.,.' ,-

t ~ ~ ~ ~ ~~~~~, ;I.'

'.S_

Fig. 7. Generatinlg i(t) from 4(t) .

Fig . 6. VRMS/IRMS characteristic of a nonlinearinductance.

(4)

212

ki(t) = I |In| sin(wut + fd,

n=l

with

wn = n1, (6)

being the angular frequency of the n-th harmonic.Experience has shown that it is usually sufficient toconsider the fundamental and the odd harmonics oforder 3 to 15, and to ignore the other harmonics. Ateach harmonic considered (including the fundamental),the harmonic component from Eq. (5) is entered into [I]with its proper magnitude and angle for all nonlinearinductances, and the voltages at that harmonic fre-quency are then found by solving the system of linearequations (1). Known harmonic current sources fromconverters and other harmonic producing equipment areadded into the vector [I].

Taking the fundamental and the odd harmonics 3, 5,7, 9, 11, 13 and 15 into consideration requires 8 solu-tions of that system of equations, with [Y] obviouslybeing different for each of the harmonic frequencies.For lumped inductances L and capacitances C, it isclear that values uwnL and wnC must be used as reac-tances and susceptances in building [Y]. Lines can bemodelled as cascade connections of i-circuits, as longas the number of it-circuits per line is high enough torepresent the line properly at the highest harmonicfrequency. It is safer, however, to define the linedata as distributed parameters, and to generate anexact equivalent i-circuit at each frequency with thewell-known long-line equations with hyperbolic func-tions. For balanced M-phase lines, exact single-phasei-circuits are first found for zero and positivesequence parameters, and the admittances are then con-verted to phase quantity matrices with

[Yphase]

=

[T][Ymode ][T] (7)

where [Ymode] is a diagonal matrix, with the firstdiagonal element being Yzero, and all other diagonalelements being Ypositive' Matrix [T] describes thenormalized transformation from af3O-components to phasequantities, which is well-known for three phases andcan be generalized to any number of phases. For M-

phase untransposed lines, exact single-phase it-circuitsare first found for all modes 1,...M, and Eq. (7) isthen again used to convert to phase quantities. In

that case, [Ymodel is again a diagonal matrix of modequantities, and [T] is the eigenvector matrix of thematrix product [Y'][Zt], with [Y'] and [Z'] being theshunt admittance and series impedance matrices per unitlength of the M-phase line [3].

Once the voltages have been found for the funda-mental and for the harmonics, an improved flux functionq(t) can be calculated for each nonlinear inductance

from the peak voltage phasors IViIej¢l, IV3jej¢3, etc.,

k IV n|(4(t) = -n-- sin(u t + 4n) (8)

n=1 n

With 4(t) known, i(t) is again generated point-by-pointas shown in Fig. 7, and then analyzed with the Fourier

Analysis Program to obtain an improved set of harmonicsexpressed as Eq. (5). These are then again used to

find an improved set of harmonic voltages. This itera-

tive process is repeated until the changes in the har-

monic currents are sufficiently small. Experience has

shown that 3 iterations are usually enough to obtainthe harmonic currents with an accuracy of +5%.

4. DISCREPANCIES BETWEEN STEADY-STATE AND TRANSIENT(5) SOLUTIONS

As mentioned in the introduction, the method ofSection 3 was primarily developed as an improvedinitialization procedure for the EMTP, for cases wherethe initial steady state is already distorted withharmonics. It should be emphasized, however, that itcan also be used by itself as a harmonics power flowprogram.

If it is used as an initialization procedure forthe EMTP, discrepancies can appear between the resultsfrom the steady-state and transient solutions. Thesediscrepancies are caused by the unavoidable discretiza-tion error of the trapezoidal rule, which is used forlumped inductances and capacitances in the EMTP. Inthe steady-state solution for the n-th harmonic, cor-rect reactance values wnL are used for inductances. Inthe transient solution, the differential equationv = L.di/dt is replaced by

v(t) + v(t-At) = L i(t) - i(t-At)2 At (9)

To understand how Eq . (9) will process the n-th har-monic, let us assume that both voltages and currentsare expressed as complex peak phasors V and I. Then

jutt ju tv(t) = Re{Ve n I and i(t) = Re{fIe nI (10)

Inserting Eq. (10) into Eq. (9) and dropping Re{. ... },

we obtain

jw t jw(^)(t-At ) jw - jwIe AtVe n + Ve n =LIe n -lie n

2 L At

which can be rewritten as

-ju At2L 1-e nAt ~ -ju At

l+e n

or

tan(un At

V = juwL At I .

')n 2

(11)

From Eq. (11) it can be seen that the transient simula-tion with the trapezoidal rule of integration does not

see the correct reactance unL, but a somewhat largerreactance which is increased by the error factor

tan(Wn A

n Atwn 2

(12)

It can be shown that the susceptance of a capacitanceseen by the trapezoidal rule is also too large by thesame factor E . While a small At can keep the error

factor of Eq. e12) reasonably close to 1.0 (e.g., At=50Ls leads to an error factor of E7=1.0015, or to an

error of 0.15%, at the 7th harmonic of 60 Hz), it can

never be avoided completely. Even small errors can

shift the resonance frequencies of the network. Fig. 8

compares the impedance at the location of the nonlinear

inductance in the problem of Fig. 1, as it would be

seen by a steady-state phasor solution and by a tran-

sient solution with the error factor of Eq. (12). To

213

7 * > fE A - with error factor E

6 o - without errorfactor E

5

-4

3

0 J FREQUENCY (Hz)400 410 420 430

Fig. 8. Frequency response with and without errorfactor E (At = 200 ps).

emphasize the differences in Fig. 8, the line wasmodelled as a cascade connection of three-phase i-circuits, rather than with distributed parameters.Since the EMTP uses other, more accurate, methods forsolving the equations of distributed-parameter lines,the differences would be much less with distributed-parameter representations.

In the transient simulation, the discretizationerror factor of Eq. (12) is unavoidable, and theanswers will therefore be slightly incorrect. In suchsituations, it may be best to introduce the same errorfactor into the initialization with the steady-statesolution method of Section 3, to avoid discrepanciesbetween initial conditions and transient simulations.With this modification, the discrepancies between theinitialization procedure of Section 3 and subsequenttransient simulations of an otherwise undisturbed net-work become practically negligible.

Fig. 9 shows the transient simulation results forthe same case used for Fig. 1, except that the ini-tialization procedure of Section 3 was now used. Itcan be seen that the initial conditions must have con-tained more or less correct harmonics because no dis-turbance is noticeable after t=0. Fig. 10 shows simi-lar results for the case used in Fig. 4, with the ini-tialization procedure of Section 3. The improvementfrom the inclusion of harmonics in the initializationis quite evident in this second example.

5. FERRORESONANCE

An attempt was made to apply the method of Section3 to ferroresonance cases, but with little success. Inferroresonance phenomena, more than one steady-statesolution is possible. It depends very much on theinitial conditions and on the type of disturbance whichone of these possible steady states will be reached.The method of Section 3 is therefore not useful forferroresonance studies. The EMTP can be used for thesimulation of ferroresonance phenomena, however, thoughit will not give any insight into all possible steady-

EH0

SENDING END

400

0:4

H

0

-400

B=

0(ms)20 C

0SE-4

H

A

SENDING END

B

40 (ms)C

A

B

40 (ms)C

RECEIVING END

RECEIVING END

400

0

-400

B

40(ms)C

A

Fig. 9. Same case as in Fig. 1, except simulationstarts from steady state with harmonics.

Fig. 10. Same case as in Fig. 4, except simulationstarts from steady state with harmonics.

state conditions. In that sense, EMTP simulations aresomewhat similar to transient stability simulations,which also do not give global answers about the overallstability of the system.

6. CONCLUSIONS

A phasor solution method for calculating steady-state conditions with transformer-generated harmonicshas been presented. It can be used by itself, or as

part of a harmonic power flow program [4]. It can alsobe used as an improved initialization procedure forelectromagnetic transients programs for cases whereharmonics are already present before the disturbancebeing simulated occurs. In transient simulations,discretization errors are unavoidable. Their influenceon the accuracy of harmonics is briefly discussed.

clD

0-H0

214

APPENDIX 1. CONVERSION OF SATURATION CURVES

Often, saturation curves supplied by manufacturersgive RMS voltages as a function of RMS currents. Con-version from a VRMS/IRms-curve to a flux/current-curve(p=f(i) is easy if

1. hysteresis and eddy current losses in the iron-core are ignored,

2. resistance in the winding is ignored, and if3. the (p/i-curve is to be generated point by point

at such distances that linear interpolation isacceptable in between points.

For the conversion it is necessary to assume that theflux varies sinusoidally at fundamental frequency as afunction of time, because it is most likely that theVRMS/IRMS-curve has been measured with a sinusoidalterminal voltage. With assumption (2), v=d(p/dt.Therefore, the voltage will also be sinusoidal and theconversion of VRMS values to flux values becomes asimple re-scaling:

VRMS/2(p=

Fig. 11. Recursive conversion of a VRMS/IERMS-curveinto a (p/i-curve.

The rescaling of currents is more complicated, exceptfor point iB at the end of the linear region A-B:

iB = IRMS-B . (14)

The following points ic, iDI ... are found recursively(Fig . 11): Assume that iE is the next value to befound. Assume further that the sinusoidal flux justreaches the value fE at its maximum,

( BE sin ut . (15)

Within each segment of the curve already defined by itsend points, in this case A-B and B-C and C-D, i isknown as a function of ( (namely piecewise linear), andwith Eq. (15) is then also known as a function of time.Only the last segment is undefined inasmuch as iE isstill unknown. Therefore, i=f(t,iE) in the last seg-ment. If the integral needed for RMS-values,

In/ 2F = 2- i2d(t) (16)

is evaluated segment by segment, the result will con-tain iE as an unknown variable. With the trapezoidalrule of integration (reasonable step size = 10), F hasthe form

(13)

If the (/i-curve thus generated is used to re-compute a VRMS/IRMS-curve, it will match the originalVRMS/IRMS-curve, except for possible round-off errors.This conversion procedure is used in the support rou-tine CONVERT of the EMTP. As an example, it wouldconvert the table of per-unit RMS exciting currents asa function of per-unit RMS-voltages,

VRMS (p *U *) IRMS (P *U *)

O 00.9 0.00561.0 0.01501.1 0.0401

with base power = 50 MVA and base voltage = 635.1 kV,into the following flux/current relationship:

(p(Vs) i(A)

0 02144.22 0.62352382.46 2.72382620.71 7.2487

This (q/i-curve is then converted back into a VRMS/IRMS-curve as an accuracy check. In this case, the VRMS andIRms values were identical with the original inputdata.

ACKNOWLEDGEMENTS

The financial assistance of the System EngineeringDivision of B.C. Hydro and Power Authority, Vancouver,Canada, through a Power System Research Agreement, isgratefully acknowledged. The data conversion routinedescribed in Appendix 1 was developed with the assis-tance of C.F. Cunha, CEMIG, Belo Horizonte, Brazil.

The authors are also indebted to the reviewers ofthis paper for their valuable suggestions, which wereincorporated in the final version of this paper.

REFERENCES

[1] C.A.F. Cunha and H.W. Dommel, "Computer simulationof field tests on the 345 kV Jaguara-Taquarilline" (in Portuguese). Paper BH/GSP/12, presentedat "II Seminario Nacional de Producao e Trans-missao de Energia Eletrica" in Belo Horizonte,Brazil, 1973.

[2] V. Brandwajn, H.W. Dommel, and I.I. Dommel,"Matrix representation of three-phase n-windingtransformers for steady-state and transientstudies". IEEE Trans. Power App. Syst., vol.PAS-101, pp. 1369-1378, June 1982.

[3] H.W. Dommel, "Transmission line models for har-monics studies". Int. Conf. on Harmonics in PowerSystems, Worcester, Mass., Oct. 22-23, 1984, pp.127-131.

[4] D. Xia and G.T. Heydt, "Harmonic power flowstudies. Part I - Formulation and solution. PartII - Implementation and Practical Application."IEEE Trans. Power App. Syst., vol. PAS-101, pp.1257-1270, June 1982.

F = a + biE + ciE2 (1 7)2

with a,b,c known. Since F must be equal to IRMS-E bydefinition, Eq. (17) can be solved for the unknownvalue iE' This process is repeated recursively untilthe last point iN has been found.

.- : - -7 . .

215

DiscussionAdam Semlyen (University of Toronto, Ontario, Canada): I would liketo congratulate the authors for their timely paper. I consider it impor-tant and useful that phasor solutions should exist for the periodic steadystate in a network with nonlinear elements that generate harmonics.

I would appreciate clarification on the following problem. In thegeneral step of "distortion" iterations, the output from the nonlinearelement contains the base current Il together with its harmonics. Thusa "power flow" update is implicit in this step. What is the rationale forhaving separate (possibly converged) "power flow" iterations? Wouldit be possible, and perhaps faster, to use only the general iterative pro-cedure described by the authors, starting with Eq. (8) of the paper?

Manuscript received July 24, 1985.

R. A. Walling (General Electric Company, Schenectady, NY): Theauthors are to be congratulated on publishing this very useful methodof analyzing the steady-state effects of transformer nonlinearities.

This discusser has independently developed a program that closelyparallels their method. It has been used in a stand-alone basis, indepen-dent of a time domain solution, to calculate harmonic overvoltages oc-curring as a result of transformer saturation, particularly in low-frequencyresonant circuits. This program has been verified against time domainsimulations and has been found to be useful and efficient as a screeningtechnique to indicate critical cases for transient studies. My experienceand observations engender the following comments and questions.1. I have also found the representation of the transformer saturation

curve by two slopes to be quite adequate. In fact, the unsaturatedmagnetizing impedance can be considered infinite in most cases asit is many orders of magnitude greater than the air core impedance.The use of more than two saturation curve slopes will primarily in-crease the accuracy of the higher-frequency harmonics as the two sloperepresentation generators a slightly excessive amount of these com-ponents due to the sharp transition between the slopes.

2. I have obtained adequate results with truncation of the computationto the ninth harmonic and below. Excellent results have been obtain-ed by including up to the 15th harmonic, with negligible improve-ment beyond this frequency. Have the authors established any criteriafor determining the range of harmonics to be calculated?

3. Is there any particular advantage to calculating an RMS saturationcurve and using this for the initial fundamental frequency solution?Would not use of the same algorithm and instantaneous flux-currentcurve as the "distortion" solution, but limited to fundamental fre-quency, be a more straightforward way to calculate an initial guessfor the distortion iterations? My program uses this second approachfor fundamental frequency and then adds one harmonic at a time,iterating until convergence to coarse tolerances are reached for all har-monics that have been included to that point. When the last desiredharmonic is added, iterations are performed until the specified finaltolerances are achieved. This method tends to achieve convergencein less total iterations than the method offered by the authors, par-ticularly when the system is resonant near a low-order harmonic.

4. Have the authors experienced cases where this algorithm fails to con-verge? I have found that the use of the last calculated voltage spectrato calculate the flux for the next iteration can lead to instability, par-ticularly if the system has a poorly damped resonance near a harmonicinjected by the transformer and the transformer is greatly overexcited.I use a weighted average between the latest and preceding voltagespectra.

5. This technique is also extendable to the quasi-steady-state harmonicsinjected by a transformer with offset flux. In poorly damped systems,flux offsets caused by energization, etc., decay very slowly. Steady-state solution techniques can be used to calculate the distortedwaveshapes produced by asymmetric saturation, provided the evenharmonics are also calculated.It is hoped that these questions and comments are useful to the authors

and others.

Manuscript received August 5, 1985.

Hermann W. Dommel and Andrew Yan: We would like to thank thediscussers for their valuable comments and questions.We fully agree with Mr. Walling's comments on the adequacy of the

two-slope representation of transformer saturation, and we are happyto see that he reached more or less the same conclusions with respectto the number of harmonics that must be considered. We did not establishclearly defined criteria for determining the range of harmonics to becalculated, but such criteria clearly would be desirable. Ignoring har-monics above order 15 is simply based on experience with a limitednumber of test cases.

Dr. Semlyen and Mr. Walling both ask whether the initial "powerflow" iterations at fundamental frequency are really necessary. We usethem only as a starting algorithm to obtain a somewhat better intial guessfor the "distortion" iterations with the flux-current loop. In a typicalcase, the initial power flow converged in three iteration steps, and reducedthe number of iterations in the flux-current loop from three to two. Oneiteration in the power flow loop requires one solution of [Y1 [V1' = [1];whereas one iteration in the flux-current loop involves the generationof the ,6(t) and i(t) curves, the Fourier analysis of i(t), and a solutionof [Y] [V] = [1] eight times if odd harmonics of order up to 15 are re-tained. Three iterations in the power-flow loop therefore take less timethan one iteration in the flux-current loop. Whether this saving is largeenough to justify the initial power-flow loop depends on many factors.In our program, not much extra code was needed for this initial loop.

In our limited experience, we have not come across convergence pro-blems, even though we used the last calculated voltage spectra to calculatethe flux for the next iteration. Maybe the initial power-flow loop im-proves convergence through a better initial guess, or maybe we just havenot yet come across a case where convergence is more difficult.Once again, we thank the discussers for sharing their valuable ex-

perience with us.

Manuscript received September 3, 1985.