1815IEEE Transactions on Power Apparatus and Systems, Vol. PAS-101, No. 6 June 1982

A METHOD FOR ANALYZING HARMONIC DISTRIBUTION IN A.C. POWER SYSTEMS

Aly A. Mahmoud, Senior Member IEEEDepartment of Electrical Engineering

Iowa State UniversityAmes, Iowa

Abstract - This paper presents a method for ana-lyzing'charActeristic harmonic current propagation intoan AC system. Frequency dependent models of AC networkelements and loads have been developed for the appropri-ate range of frequencies. This methodology is appliedto two different systems and the responsesof individualelements and the system as a whole were observed. Theharmonic current flows in the systems appear to be re-lated-to certain characteristics of the system and itselements. Input impedance at harmonic frequencies, themagnitude.of shunt capacitances on transmission lines,the size and location of the converters, and the repre-sentation of system loads are all found to be factorsin cdetermining the magnitude of harmonic current flowsin system elements.

INTRODUCTION

Power system harmonics are known to be generatedby a number of sources in the power network. Harmonicsgenerated.by large synchronous machines are limited byexisting standards and are negligible. In the past,load generated harmonics were neglected, and in somecases fed from special substations or through filters.Loads were mostly leaner with very limited nonlinearcharacteristics. Recently, however, the nature of theload has changed drastically due to energy conservationmethods and the application of solid state devices inmotor control. Loads are becoming highly nonlinear incharacteristics with.a significant increase in primarilycurrent harmonics. Cogeneration and the new nationalrequirement for interconnecting small wind and solarunits to the system is another source of serious currentharmonics. 'Since most.of the solid state devices usedin.conjunction with loads yields periodic and even cur-rent waveforms, one can-assume that the harmonic gener-,ated by these sources are of the characteristic type.Characteristic current harmonics are also known to begenerated f.rom DC link converters. However, such har-monics are usually suppressed by the use of special fil-ters at each converter terminal.

With the increase in cogeneration and solid stateapplication, larger quantities of harmonic currents areexpected to be injected into the AC distribution' andtransmission network. The effects of these harmonicson.cables, generators, transformers and other loads re-present very important.and serious problems.

To determine the impact of these harmonic flows onthe distribivtion and transmission network, the magnitudeof harmonic currents flowing in all the elements of thepower network must be calculated. In this paper, amethod for determining the propagation of characteristicharmonic currents into the power network is developed.

81 TD 700-4 A paper recommended and approved by theIEEE Power System Engineering Committee of the IEEEPower Engineering Society for presentation at theIEEE PES 1981 Transmission and Distribution Confer-ence and Exposition, Minneapolis, Minnesota, Septem-ber 20-25, 1981. Manuscript submitted March 31,1981; made available for printing July 14, 1981.

Richard D. Shultz, Member IEEEDepartment of Electrical Engieering

University of IllinoisUrbana, Illinois

METHOD OF APPROACH

During steady state operation the harmonic currentsentering the AC network can be considered as beinggenerated from ideal current sources. The entire systemcan then be modeled as an assemblage of passive impe-dance elements with currents being injected at sources'locations.

The impedances representing the network elementsmust, of course,. be modified for each harmonic fre-quency. Once this is accomplished, they may be assem-bled 'into a bus impedance matrix. The current beinginjected into the passive system at the converters maythen be used to determine the.harmonic currents flow-ing on each element. This approach requires a knowl-edge of which harmonics are being generated and themagnitude of the currents being injected.

DEVELOPMENT OF MODELS FOR NETWORK ELEMENTS

To assemble the elements of a system into a busimpedance matrix at each harmonic, a frequency depen-dent model for each element must be developed.

Transformers

A good model for. determining resonance in atransformer would be one which represents every turnof the winding and included all turn to turn inductan-ces and capacitances. Complete representations ofevery turn is not practical and cannot be justified.McNutt et al. [1] suggested a much simpler model bylumping 'of successive elements. This model is shownin Figure l(a) and includes sections of leakage induc-tances with interwinding capacitances and capacitanceto ground for each section. If a steady state AC sig-nal is applied to the model in Figure l(a), the cir-cuit can be simplified as shown in Figure l(b).

_ .

*,(a)

A 8 C D E-

A

(b)

III

(c)

Fig. 1.(a)(b)(c)(d)

(d)

Transformer models.Sectionalized model of transformer.Steady state simplification.First resonance simplification.Final model with all elements.

0018-9510/82/0600-1815$00.75 © 1982 IEEE

1816

Knowing the impedances shown in the model ofFigure 1(b), the resonant frequencies of the transformercould be calculated. Reference [1] reports test re-sults on resonances in transformers which have been ob-served in industry. For a typical extra high voltagetransformer, the first resonance occurred in a rangefrom 7kHz to 15kHz. This study is focused on investi-gating harmonics up to the fortieth term. Since thefortieth harmonic is only 2.4kHz, only thefirst reson-ance of a transformer is of interest in this analysis.

For the frequency range which is of interest,less than 3kHz, the input impedance has a positivelysloped linear relationship with frequency. To increasethe accuracy of the transformer model a magnetizationbranch is added.

Transmission Lines

Transmission lines should be represented by thelong line model with modifications for tise at frequenciesgreater than 60 Hz. The long line model uses the fol-lowing series impedances and shunt admittances.

sinh yQZw = Z, = )

Yy yi yQZ 2tanh2 Z 2

(1)

(2)

where

Y = rErZ = zQ

= length of transmission in milesz = series impedanceof line/miley = shunt admittanceof line/mile

As frequency increases, the skin effect beginsto dominate the resistance. Stevenson [5] presents a

method of determining the AC resistance of a roundconductor as a ratio to its DC resistance.

R ,mr ber(mr)bei' (mr)-bei(mr)ber' (mr)

Ro 2 (ber'(mr))2 + (bei'(mr))2mr 0.0636

whereRo = DC resistance of conductor.

Thus, the AC resistance is obtained by equations(1) and (2) as a function of the DC resistance andfrequency.

The self inductance of a conductor is composedof an internal inductance and an external inductance.The internal inductance is dependent on the currentdistribution in the conductor. The internal andexternal self-inductances of a conductor are definedby Anderson [61 as

Li = - (3)8w

Z, = ra + ju2xlO17 l(r- + lnr 079L

where

(6)

ra = frequency dependent line resistance tophase current

Deq = cube root of the product of the distancebetween all conductors

If complete transposition is assumed, the negativesequence impedance also equals Zl-

Generators

When harmonic currents flow from the networkinto the stator windings of a generator, they create aflux rotating at a speed greater than the speed of therotor. Thus the harmonic currents react with both thedirect and quadrature axis inductances. This is verysimilar to the action of negative sequence currents ina synchronous machine. In reference [6], the averageinductance seen by negative sequence currents and,therefore' harmonic currents can be approximated by

where

L$I + LiL22

Ld = direct axis subtrasient inductance

(7)

Lq = quadrature axis subtransient inductance

Loads

Loads are difficult to accurately represent;however, they can be approximately modeled by makingthe proper assumptions. Berg [7] proposed thatparallel induction motors could be combined into anequivalent single machine for sixty cycle analysis.This representation was modified to more accuratelymodel the motor with harmonic current being injected.The equivalent motor with modifications and a parallelshunt resistance was used to represent loading at abus. All the reactive power was assumed to be con-sumed in the reactances of the machine. The realpower was divided between the shunt esistor and theresistance of the equivalent motor. The division ofthe power was determined from a knoi: dge of the typesof load present. Knowing the amount of power flowinginto a bus and the percentage of that power which canbe represented as purely resistive, the value of theshunt resistor can be determined.

An equivalent circuit of the equivalent inductionmotor is shown in Figure 2(a). With a 60 Hz source themachine can be represented fairly accurately by the ap-proximate equivalent circuit in Figure 2(b). The hy-steresis and eddy current resistance is frequency de-pendent and must be corrected for each harmonic.

r xl x2 rl xl x2

r, xi X2 22 L K

rL Km r rL2xm 2

_ .~~~~~~~~~~~~~~~~~Le = (ln 2t -1-

2Th \r

.em~ is given by Stevenson [6] as

(4)

Li 4 /ber(mr)ber'(mr)+bei(mr)beI'(mr)cc~~~_(5)Lio mr (ber?(mr))2+(beil(mr))2

whereLi0 internal DC inductance

Deriving the positive sequence impedance permile of a transmission line as shown in reference [6],the frequency dependent positive sequence impedance is

(a)

rL Xm x2

(b)KR

I(c) _

Fig. 2. Load models. (d)(a) Model of equivalent induction motor.(b) Approximate model of equivalent induction

motor.(c) High frequency model of equivalent induction

motor.(d) Final load model.

1817

As frequency increases, the slip of the machineincreases causing rl and r21s to become negligiblecompared to xl and x2 in Figure 2(a). Therefore, athigher frequencies Figure 2(c) can be used to repre-sent the induction motor. It is also reasonable tochoose xl and x2 to be approximately equal. xm can beapproximated to equal 35xl. At sixty hertz, one canestimate the magnetizing current to be thirty per centof the total current entering the motor. It is alsoassumed that at sixty cycles, the current flowing inrL is negligible compared to the currents flowing inxm and x2 of Figure 2(a) and 2(c). Therefore, thereactive power consumed by the motor at 60 Hz is

_ = I2X1 + (.3I)2Xm + (.7I)2X23

- I2 X1 + (.3I)2 35X, + (.7I)2 X1

I2 X1 (1 + 3.15 + .49)

where= 4.64 I2 X1 (8)

I = current entering motor,Qm = reactive power consumed by motor

Relating V, I, Pm, and Qm, XI becomes

QmX m= '-

4.64(p,2 + (% 2)(9)

Since in Figure 2(c) Xm equals 35X2, the parallel com-bination of these two approximately equals Xl. Thefinal representation of loads is shown in Figure 2(d)with sixty cycle impedances defined above.

Other Systems Elements

All other elements, such as line inductor andcapacitor banks are assumed to be pure elements withconstant inductance or capacitance. That is, theywill vary directly or inversely with frequencydepending on the element.

THE IMPEDANCE MATRIX

Several well known algorithms are in use todayto assemble an impedance matrix of a known system. Ineach. of these algorithms, the system is assembledstarting with a single impedance and adding one impe-dance line at a time. The impedance matrix ismodified. for each additional impedance9 Since theimpedances of. a power system change for eachfrequency, it is necessary to assemble an impedancematrix for each harmonic frequency present.

Harmonic Current Flows in Elements

The magnitudes of the harmonic currentsgenerated and the impedance matrices for the variousfrequencies present are obtainable from system andconverter data which is normally known. For theseharmonics, the reference node for the impedance matrixwill be at ground potential. The harmonic voltages ateach node in the system may now be calculated. If aconverter were placed at node m of an n node system,the matrix equation for the harmonic voltages would be

V1

Vm

Vnw

where

Zll

Zml

Znl

Z12.eeeeZln* -

Zm2......Zmn

Zn2Z**0*nn

IL-(10)'

Zij = transfer impedance between nodes i and jZii = driving point impedance at node iIL = magnitude-of harmonic current entering

the system at node mVi =harmonic voltage at node i

Therefore the harmonic voltages are

Vi = IL Zim (11)

where i is an integer from one to n. Since the impe-dance matrix and IL change for each frequency, theharmonic voltages will also vary. Knowing -the, har-monic voltages at every node, it is possible to deter-mine the harmonic current flow on any link.

Vj - V (2Ijk = (12)

Zikwhere

Ijk 2 harmonic current flowing on link betweennodes j and h

V - harmonic voltage at node jVk = harmonic voltage at node k

Zjk impedance of link between nodes j and k

If there. is more than one converter in thesystem, the harmonic current in each element is calcu-

lated for each source separately. All the currents as-

sociated with a particular element are then summed vec-

torially. In order to obtain the proper phase angle forthe currents, it is necessary to know which phase rela-tionship of the harmonic sources.

APPLICATIONS

The first system which was selected for study was

a small test system with a single largeharmonicsource.The second system was a much larger system with two i-dentical medium sized harmonic sources. The response ofthe large and small systems may be compared for simi-larities.

SYSTEM I

System I contains nineteen busses with twentytransmission lines and six transformers. Five gener-ating units connected at two busses were also included.A one line diagram of System I is shown in Figure 3.Busses one through six of Figure 1 are 345 kV. Theharmonic source was simulated by a single six pulseconverter. The converter was rated at 1500 MW at ±

400 kVDC. It operated at a delay angle of 15 degrees.When loads were included in the analysis, they were

added at eleven busses and totaled 2,133 KW and 537 KVAR.

1818

0 0

X3Fig. 3. One line diagram of system I.

Harmonic Current Flow

System I was first studied with the converterlocated at bus one and without loads being represent-ed. It was observed that the four 345 kV lines con-nected directly to bus one carried the largest har-monic currents at all frequencies. The sixty cycledata for these lines is given in Table 1. The magni-tude of the harmonic current flowing on each of theselines varied significantly as frequency changed.Table 2 lists the current at the middle of these fourlines for the varied frequencies generated.

Table 2 shows that for some cases, the magnitudeof currents flowing on some of the lines exceed thevalue of harmonic current injected by the converter.This amplification of injected harmonic current may bemore readily seen in Figure 4. In this graph theamplification of injected current on the four linesconnect to bus one are plotted versus harmonic order.An oscillatory pattern of amplification for variousharmonic orders is seen. All four of the lines shownon the graph seem to follow approximately the samepattern through the nineteenth harmonic. Although themagnitudes of the peaks vary, the frequency of the

TABLE ITransmission Line Data for System I.

F- Izw i

z 25 _

° 2.0_ '

1.5-~ ~ ~ 1

< .8 \266

57 8i13 1719 2325N 29-31357HAR,MONIC ORDER

Fig. 4. Amplification of current injected into systemI at bus 1 on lines from bus 1 to the busindicated with the system not loaded.

four graphs are very close for this range of harmon-ics. Beyond that range, however, the amplificationfactors seem to be much more individualistic of eachline. The peaks no longer appear at the same harmonicand the frequencies for each graph are different.

Loads were then added to system I and the har-monic current flows determined again. Table 3 givesthe currents on the four 345 kV lines connected to bus1 with the system loaded. Figure 5 shows the graph ofinjected current amplification with the system loaded.With system I loaded, the oscillations of the amplifi-cation graph are significantly damped for harmonicorders less than nineteen. However, the same generaltrends of increases and decreases can be observed.For both of these cases, the lines seem to responddifferently at low and high order harmonics.

Lower Order Harmonic

.~~~(3

Since the current flowing on a line is functionof the impedance of that line and the voltage acrossthat impedance, it is important to observe the voltageacross these four lines and the series impedances of

Length NumberVoltage in of Sub-

T evel(kV) Miles conductors

1-2 .140 1.803 30.432 345 36.23 21-3 .080 1.035 17 .436 345 20.78 21-4 .165 2.272 39.353 345 46.16 21-6 .176 2.506 43.249 345 50.8 2

1. All impedances are on a lOOMVA base.

TABLE I IHarmonic Current in Per Unit Flowing on Lines Connectedto 3us 1 of System I without Loads Being Represented.

Harmonic I~ ~Order 5 7 11 13 17 19 23 25 29 31 35

Line 1-2 .281 1.37 .079 .061 .016 .039 .115 .130 .179 .Ii8 .031 rLine 1-3 .202 1.22 .101 .049 .014 .012 .040 .010 .042 .046 .011 .008Line 1-4 .268 .769 .202 .174 .044 .067 .058 057 .215X 266 123 041

Line 1-6 .468 2.69 .351 .382 .064 .076 .039 .102 .635 1.12 .036 .022cuirante 2.97 1.88 .791 .487 .144 .103 .141 .143 .109 .082 .033 .028

generated

TABl .E HIXHaronic. sur-nt in Per Unit Flnwing on Lines tonsn-ed

to Bus i of Systen with Loads Represented

Ord er | 5 17 11 13 17 19 23 S25 29 35 37

Line 1-2 .195 .148 .151) .070 .013 OIi .103 .1IB .140 .144 .060 .045Line 1-3 .325' .267 .155 .055 .007 .004 .009 .004 .021 .030 .020 .016Lse 1-4 .582 |.454 .176 .074 .017 .043 .074 .043 .147 .324 .207 .057

Line 1-6 .174 .126 .319 .189 .090 .078 .079 .068 .449 1.28 .063 .029

generate2.d l| Wi .791 .487 .144 .103 .141 .143 .109 .082 .033 .028

the lines as they change with harmonic order. Examin-ing Figure 4 for harmonic orders less than nineteen,it is observed that for the unloaded case the largestamplification of harmonic current occurs at theseventh order. Figure 6 shows that for these con-ditions the voltage across the terminals of the fourtransmission lines connected to bus one are maximumsat the seventh harmonic. This could explain the largeamplification of currents on some of the lines at thisharmonic. It is noted that the voltage curves alsofollow the same patterns as the current amplificationcurves up to the nineteenth harmonic.

IShtintResist- React- Admit-

Node ance(%)l ance(%) tamiuc (%Z)

1819

z

ELi00

0

z0

7.0fL

157

430

4.0

30

2.5

2.0

1.5

1.0.6.6.4.2

6

E~~~V 4

I5 G * * *; tXi5 7 11 l3 17 rH Z3 2 29 31 it 37

HARMONIC, ORDERFig. 5. Amplification of current injected into system

I at bus 1 on lines from bus 1 to the busindicated with the system loaded.

The magnitude of harmonic voltage across thefour 345 kV lines are shown in Figure 7 for the loadedcase. The maximum magnitudes of the votlages are muchless for the case with load representation. ComparingFigures 5 and 7 however, it is observed that theamplification curves for the four lines follow thesame pattern as the corresponding voltage curves forthe lower order harmonics.

It appears that the harmonic voltage across theterminals of a transmission line is an important fac-tor in calculating the current amplification at thelower harmonics. From Figures 4 and 5, it was learnedthat the major amplification of injected current forfrequencies less than the nineteenth harmonic occursat the unloaded seventh harmonic. These are also theconditions where the largest voltage appears acrossthe lines observed.

It is also interesting to note that thereappears to be a correlation between the input impe-dance of the system at the converter and the voltagesacross the transmission lines. The input impedancesof the system at the harmonic frequencies are plottedin Figure 8 for both the loaded and unloaded cases.All the impedances fall into a relatively closecluster with the exception of the seventh harmonic forthe unloaded case. The magnitude of the impedance forthis harmonic is well over three times the nextlargest impedance.

zooL2200

1601140p 5

120 i

80

60[d 2.

401200n 259 v23

o H-20

Z -401-

-60L'-803

100

-120--140

OSF

00 0.10 6CCRESiSTANCE .2

37 i5

I, 13 17 24 23

H0-2MON!c OROER

Fig. 6. MagnitudeterminalsThe linesand loads

of harmonic voltages between theof transmission lines in system I.are between 1-2, 1-3, 1-4, and 1-6are not represented.

Fig. 8. Input impedances at the various harmonics ofsystem I with converter at bus 1 both with(2) and without (X) loads.

:iCL

Ct)ui .1z

z0cn0..-E ((nzcrF- .(

EnCf)0 .(f-Yu

LLIJ(DCfti0

-3 __t*ii I-, R7 9 232?

HARMONIC ORDER

MagnitudeterminalsThe linesand loads

of harmonic voltages between theof transmission lines in system I.are between 1-2, 1-3, 1-4, and 1-6are represented.

Higher Order Harmonics

For harmonic orders of less than nineteen, itappears that the harmonic voltage is a primary factorin determining the harmonic current flow. However, itwas observed that beyond that order, the voltagesacross the observed lines play a lesser role in con-trolling the current. As previously mentioned, thecurrent flow is a function of both the harmonicvoltage and the impedance of the line. Therefore itmay be interesting to investigate the series impedanceof the lines as a funciton of frequency.

The reactive part of the impedance begins todecrease in magnitude after the seventeenth harmonic.This continues until between the twenty-ninth andthirty-first harmonics, the reactance goes to zeroand then goes negative. When this happens the har-monic current amplification on that line is verylarge. The amplification factor for line 1-6 in theloaded case approaches sixteen. The amplificationfactors for those lines which reach a zero reactancefrequency are significantly larger for the loaded casethan the unloaded at those harmonics. This is theopposite of conditions at the lower orders.

Fig. 7.

z

-iz

'nrGe

v)z

Oi3 il

11IU.,

1820Also of much interest is the current flow on

those elements of system 1 which are connected to busone for the characteristic harmonic closest to theresonant frequency of line 1-6. Figure 9 shows themagitudes of these harmonic current flows with thesystem loaded. In Figure 9, line 1-6 is very nearits resonant frequency. The current flow at thecenter of the line is approximately 15.6 times thecurrent being injected by the converter. However thecurrent at each terminal of that line is about one-tenth of the injected current.

(.004)

,62IAWS; 2 ") { 008M H

r)(0 650IC.0655

015)

@1 0(.324)

01

Fig. 9. Magnitude of 31st harmonic current on ele-ments connected to bus 1 of system I withconverter at bus 1 and the system loaded.

Variations in System I

Having observed the characteristics of thesystem with the converter at bus one, it is ofinterest to learn the effects of moving the converterto bus seven, a 138kV bus. For most harmonic frequen-cies present, at least one of the 345kV lines carriesas much current as any other line on the system. Thiswas true for eight out of twelve harmonics with loadrepresentation and for ten out of twelve without theloads being represented. Two of the six cases where345kV lines were not carrying the largest harmoniccurrent, lines not directly connected to bus seven

were carrying the largest flow. In some cases theamplification on these lines was greater with the con-verter at bus seven than at bus one. The amplifica-tion curves for lines 1-6 and 1-4 with the converterat bus seven are shown in Figure 10 and 11 for the

,7.5,1- 6

12q 4

z 7.0

24.5

z

2

i 6z .0 ,

° 2.50k

°~~~~~6

2 4

-M -4 iI_I

5 7 11 13 17 19 23 25 29 bI 35 3-7

HAP,AMNIC ORDER

Fig. 10. Amplification of the current injected intosystem I at bus 7 on lines from bus 1 to thebus indicated with the system not loaded.

Fz32w

Z r~00

2).5 -w -zLL 2.0 -

z

o 6

.247 94

5 7 II1 (3 17 19 23 25 29 31 35 37HARMONIC ORDER

Fig. 11. Amplification of the current injected intosystem I at bus 7 on lines from bus 1 to thebus indicated with the system loaded.

unloaded and loaded cases respectively. It is inter-esting to note that loading the system under theseconditions not only damps the amplification curve atthe lower harmonics, but also at the resonantfrequencies. Thus, for the lower order harmonic, thepresence of load seems to significantly damp the peakcurrent flows. With the converter at bus one, theaddition of load increases the harmonic current flowat resonance. With the converter at bus seven theresonant currents were decreasing when loads wereadded to the system.

A system variation which would be of obviousinterest is the effect of removing various transmis-sion lines from the network. With the converter atbus one, lines were removed from several locations inthe system. The removal of lines closest to the con-verter seemed to have the greatest effect. Linesrelatively remote to the converter had much lessimpact. To illustrate this point, Table 4 lists theharmonic current flow on line 1-4 for two variationsin the system. Also shown is the percentage change inthe current flow for the two new operating conditionscompared to the original system I flows. It is notedthat for both variations, the changes in current forthe harmonics with orders higher than nineteen aresignificantly less than most of the lower orderharmonics.

SYSTEM II

System I was a small network with a rather largeharmonic source on it. This made it very useful for ob-serving and analyzing the reaction of network elementsto the characteristic harmonics generated. System IIis a much larger network consisting of ninety-two busses,ninety-six transmission lines, thirty-three transformers,twenty-five generating units, and loads totaling morethan 8000 MW and 4500 MVAR at thirty-five busses. Itcontains both twelve pulse converters of a DC link ratedat 100 MW and ± 100 kVdc as harmonic sources. The sourcesare placed at 138 kV busses, 99 and 100 which are phy-sically close but not directly electrically connected.This system will be helpful in determining the reactionof a large interwoven network to multiple medium sizedharmonic sources.

Influence of Converters Size and Locations

The system was analyzed both with and withoutloads being represented. It was found that only atthe lower order harmonics were the harmonic currentslarge enough to cause concern. The magnitude of

('016)

Table IVHarmonic Current Flow on Line 1-4 with Variations

in System I

Harm,onic Order 13 1.7 I

l per usi: },.e'93 P.4097i.j502 1606 .u593 0).9Lines current flow1-6 per cent

removel change from

~~~~~~ 3l2?e21ttO1"

3tfuall systemrepresentation

Les per unit

10-13,current flow

I 5 .1159 ..lI .ii5 0.7 l.ii

7-25, per cent'and ctia1nge fro47-12 fulss4nI11 .71l 7.67. 2 7. 6% 4.3

removed representation

Harmonic Order 23 5 29 31 35 37

per unit .0750 05 1.1S4- .3564 .2465 .0660

Lines clurrent flow1-6 per cent

removed clhange fromfull systenm , 1 " i V 1.2 66

representation

tines c~cuPrrenUthftow 0048 u46 u633i02 .2100 .0567

7-25, per centand chanoe 4rom 3

7~12fu11 s >s tem 1 1 , 4 t 9'2 1 1 e' I Y'' 2 2%/7-12 full senstemn 4

rennvedJ representation

currents flowing in the network elements for harmonicfrequencies greater than the twenty-fifth harmonicwere less than one tenth of one per cent for all but a

few exceptions. The current flow for these exceptionswas found to be less than one half of one per cent.This could be a direct result of the smaller conver-

ters used in system II and, therefore, smaller har-monic currents being injected at the converters.Three lines in System II reach resonant frequency atless than 2.4kHz. However, the combination of a smallinjected current and the relative proximity of theactual resonant frequency to the closest harmonicfrequency present prevent the presence of any extremecurrent flows on these lines due to this resonance.

It is important to note the locations of thesources relative to the rest of the system. Theharmonic sources were placed in the middle ofthe 138kV network. Most of the 138kV transmissionlines are relatively close to at least one of theconverters. The EHV transmission lines lie on theoutskirts of system II. These lines are, therefore,relatively remote to the sources. In several ofthe cases studied, however, the largest harmoniccurrent flow in the system was on a 345kV transmissionline. Combining both the loaded and the unloadedcases, a 345kV line carries the largest current in 38%of the cases up through the twenty-fifth harmonic.

Effects of Loading

Also observed is the fact that loading system IIhad no universal effect on the harmonic current flowon the lines. Sometimes it increased the current flowand sometimes it decreased it. In all cases exceptthe eleventh harmonic which will be discussed later,it had no drastic effect. Adding loads caused a

change in all the currents, but the effect was never

enough to raise the current flow to near one per centfor these harmonics.

The largest harmonic current flow for system IIoccurred at the eleventh harmonic with the system notloaded. With these conditions the current flow was

relatively large throughout the system. The largestflow was on a 138kV line which was close to the con-

verter at bus one hundred. This line connected busses

1821

eighty and one hundred sixty six. The flow on theline for the unloaded case is almost fifteen per centat the midpoint and exceeds fifteen per cent at oneterminal. When loads were added to the system, theharmonic current flow on this line dropped to two percent at one terminal and less than two per cent at themidpoint. The current on line 80-166 and on otherlines in the immediate proximity are shown in Figures12 and 13 for the unloaded and loaded cases respec-tively. While the addition of load did not result ina reduction in eleventh harmonic current on all lines,the larger current flows were significantly damped.

0.086(.0619) (.0051 (.0520'

(l0S7) 1 2( I ol (06191 °i0051) 0521 (.1564) (.64) (.1471) .290

(.0242) 151 (.0051) (.0521)

('02(.0619~ .05)1

(24P1L2L (_005') (.0527

(.0619) (.00511 (.0521)(_0_17.1564) )(1564) (.1478) (.1298)

01 0819) (.05id) .0520

Fig. 12. Magnitude of 11th harmonic in a section ofsystem II without load representation.

,00711 10039) (.0091

tour Ol. (Z.003l979 '009i11 i152001j (.0200-) 10)58~'1 0)041-10 i00710 l.00391 (.0091)

E4o1(.071 (.00391' (.00911(@0242).(0t 03} t

1,t~so31K 7 . T(- 00) (.02001 1.0981) (01641 I

')101141 H100?11 .00391) 10095) "

Fig. 13. Magnitude of 11th harmonic in a section ofsystem II with loads represented.

Input Impedance

In system I there appeared to be a relationshipbetween larger harmonic current flow at lower orderharmonics and input impedance. It also appears thatthe same relationship is present in system II. Themagnitude of the input impedance at busses ninety-nineand one hundred for the unloaded system at the elev-enth harmonic are much greater than the input impe-dances at any other harmonic for any conditions.These are the conditions which also resulted in thelargest current flows in system II. The input impe-dances for busses ninety nine and one hundred areplotted in Figure 14 for the lower order harmonicspresent. At both terminals the eleventh loaded, thir-teenth loaded, and thirteenth unloaded impedances haveapproximately the same magnitude. The unloaded elev-enth input impedance, however, is shown to be signfi-ficantly larger. It is interesting to note that whenthe system is changed from unloaded to loaded theinput impedance at bus one hundred, the converter buswhich is very close to line 80-166, decreases by afactor of 7.9 for the eleventh harmonic. The currenton line 80-166 decreased by a factor of 7.7 when loadwas added.

1822

Fig.

20

10 - loL

0 0l10 20 30 40 50 60 I0 20 30 40 50 60

-10 - .13 RESISTANCE- -!, RESISTANCE Qi-20 -20L . ,3

-30 -30'

-40 - -40;-LiiJ Lii

z50 _ z-50<-60 -601-60~ ~ ~Lw701 LL -70r

-830 _-8 0,

-100--_001-

-110-_OL

-120 - - 2C L

-130- - 350L--140- -360I

14. Input impedances of system II with convertersat busses 99 and 100 with (-) and without (X)loads.(a) Input impedances at bus 99.(b) Input impedances at bus 100.

CONCLUSIONS AND DISCUSSIONS

From the results of the two systems studied, it ap-pears that harmonic current flow on system transmissionlines can exceed the current injected by a harmonicsource at any frequency. At lower order harmonics, thecurrent flow seems to be associated with the magnitudeof the input impedance at the harmonic source for thatfrequency. When, graphed, the input impedances tend tocluster in the right half of the impedance plane within3ome boundary of the origin. If a frequency happens tohave an input imnpedance whose magnitude is several timesgreater than this boundary, then large harmonic voltageand currents may be expected on some elements of thesystem.

As the harmonic frequencies increase, the magni-t,ude of the injected current decreases. Therefore, inorder to obtain a large harmonic current flow at thehigher orders, the amplification of injected currentfor a line would have to be very large. It was foundthat this happens when a transmission line reaches aresonant frequency. Only lines with very large shuntcapacitance resonate at less than 2.4kHz. Therefore,cables and EHV lines are the most likely to carrylarger higher order harmonic currents. However, evenif a line does resonate at less than the fortiethharmonic, this is no guarantee that a large harmoniccurrent will flow. If the resonant frequency of theline is not near a characteristic harmonic frequencyof the harmonic sources, then the fact that the line re-sonates at less than 2.4 kHz may have little orno effecton the harmonic current flow.

The relative location of the harmonic source to aparticular line effects the amount of harmonic currentflowing on that line. However, greater distance fromthe source does not always mean less current. This seemsto be particularly true of the higher voltage lines. Insystem I, at some frequencies the current flow on someof the 345 kV line which was relatively distant from theconverter was the largest in the system.

Near resonant frequencies, the addition of loads tothe system effects the harmonic current flow. For in-stance, the location of the harmonic source will influ-ence the changes due to load representation. In systemI, with the source at one location, the addition of loadincreased current flow at these frequencies, while atanother location it resulted in a decrease.

At lower frequencies, which are not near theresonance of system transmission lines, the effects ofadding loads are varied. While the addition of load canresult in either an increase or decrease in harmoniccurrent flow, some general observations can be made.When the current flow on a line was small compared tothe harmonic current being injected at the harmonicsource, loading the system resulted in only a moderatechange in current for that line. However, if the un-loaded current on a line was relatively large, loadingresulted in a significant damping of that current. Thus,the inclusion of load representation as modeled for thisstudy definitely effected the critical harmonic currentflows on transmission lines in the system.

ACKNOWLEDGMENTS

The authors would like to acknowledge the finan-cial support made available by the Power AffiliateResearch Program of the Iowa State UniversityEngineering Research Institute.

The authors also acknowledge the valuablecontribution of Dr. C. Calabrese of ConsolidatedEdison during the progress of this study.

REFERENCES

I1] W. J. McNutt, T. J. Blalock, R. A. Hinton, "Re-sponse of transformer windings to systems trans-ient voltages," IEEE Transactions on Power Appa-ratus and Systems, Vol. PAS-93, March/April 1974.

[21 R. C. Degeneff, "A general method for determin-ing resonances in transformer windings," IEEETransactions on Power Apparatus and Systems,Vol. PAS-96, March7April 1977.

[31 Electrical Transmission and Distribution Refer-ence Book, Westinghouse Electric Corporation,East Pittsburgh, PA, 1974.

[41 W. D. Stevenson, Jr., Elements of Power SystemsAnalysis, Third Edition, McGraw-Hill, New York,1975.

[5] W. D. Stevenson, Jr., Elements of Power SystemsAnalysis, Second Edition, McGraw-Hill, New York,1962.

[6] P. M. Anderson, Analysis of Faulted PowerSystems, Iowa State University Press, Ames, IA,1976.

[7) G. J. Berg, M. M. Abdel Hakim, "Dynamic singleunit representation of induction motor groups,"IEEE Transactions on Power Apparatus and Systems,Vol. PAS-95, January/February, 1976.

[8] A. G. Phadke, J. H. Harlow, "Generation ofAbnormal Harmonics in High-Voltage AC-DC PowerSystems, " IEEE Transactions on Power Apparatusand Systems, Vol. PAS-87, March 1968.

[9] H. Sasaki, T. Machida, "A New Method to EliminateAC Harmonic Currents by Magnetic Flux Compensa-tion-Consideration on Basic Design," IEEETransactions on Power Apparatus and Systems,Vol. 'PAS-90, September/October 1971.

[10] A. Ametani, "Harmonic Reduction in ThyristorConverters by Harmonic Current Injection," IEEETransactions on Power Apparatus and Systems,Vol. PAS-95, March/April 1976.

LIST OF SYMBOLS

aC Delay angleCIL Ratio of internal AC inductance to internal DC

inductanceY Propagation constanth Harmonic orderId DC line currentIh Harmonic rms current of order h in amps

Harmonic rms current in per unitLength of transmission line in milesExternal inductance of a conductorInternal inductance of a conductorInternal DC inductance of a conductorAbsolute magnetic permeabilityPulse numberPer unit power consumed by equivalent motor

Per unit reactive power consumed by equivalent

motorRadius of a conductorLine resistance to phase a current

Core loss resistance of equivalent motor

DC resistance of a conductorResistive load parallel to equivalent motor

Stator resistance of the equivalent motorRotor resistance of the equivalent motor

Overlap angleMagnetizing reactance of the equivalent motor

Stator reactance of equivalent motor

Rotor reactance of the equivalent motor

Shunt admittance of a transmission line per mile

Long line model equivalent shunt admittance

Series impedance of a transmission line per mile

Long line model equivalent series impedance

jnx1 R

IA V VNODE

12V (KV)

R (Q) =

LOAD (MW)

XI = 0.1 R; representingtransformer leakage reactancein series with resistive load.

Fig. A. Load Model A

NODEv2 ( KV )

R(a) =

LOAD (MW) x0 .2

Xl = 0.1 R

v2 (KV)

X2 Dx 0W.)3LOAD(MW)xO.8

Fig. B. Load Model B

Discussion

K. Murotani (Nissin Electric Co., Ltd. Kyoto, Japan): The authors are

to be appreciated for their valuable works on the harmonic analysis.The author's model shown in Fig. 2(d) is very informative for us,

because we believe that the most important factor of the harmonicanalysis is how to represent the load characteristics. We would like toknow if this model has already been confirmed by field measurementsin some actual substations.The discussor's procedure for the harmonic distribution calculation is

similar to the author's one except following points;1) The transformer's and generator's impedance Zn at n-th harmonic

frequency is expressed just likeZn = ro (I + 0.1 In1.5) + jnXl

where r O: DC resistance0.1ron1 .5 frequency-dependent resistance term determined from

measurementsXl: leakage reactance for transformers, and negative se-

quence reactance for generators at fundamental fre-quency

The frequency-dependent resistance of the electric machinery isbelieved to be more important than the magnetization branch as shownin Fig. 1(d).

2) Load modelling includes several difficult factors. Some examplescoming from our experiences are shown in Fig. A and Fig. B.

In Fig. A, the load is simulated as a resistor in series with a reactorrepresenting a transformer leakage reactance.

In Fig. B, the load is assumed to be composed of 80% inductionmotor load and 20% resistive load.The induction motor is represented there as the reactance obtained

from a lock test, which is on average 30% based on its rating.Fig. C shows the example of a comparison between the measurementand the calculation, which are in good agreement with each other.The field measurement has been done at the 275kV substations in one

of the Japanese utility systems where 22 generators are included with

the total capacity of 6500MVA in about 30,000 km area.

The total number of nodes, branches, and harmonic current sources

in the analytical model are 97,210, and 53, respectively.The harmonic current sources are assumed to be diverted in the whole

system.

V5(Zoo)

without AC filter

Theoretical values

(i) ' t2>2,~j (3 8~Node Number

Fig. C. 5-th harmonic voltage on 275kv buses.(measured in Oct., 1980)

Manuscript received November 13, 1981.

1823

ILQLeLi1iopPmQm

rrarL

rR

rIr4U

xm

x1x2y

yzrz

zTrr

1824

A. A. Mahmoud and R. D. Shultz: The authors would like to thankMr. Kaneyoshi Murotani (Nissin Electric Co. Ltd., Kyoto, Japan) forhis very informative discussion. As our results indicate, there is nodoubt that the magnitude and order of the harmonics in a power systemare significantly affected by incorporating load models. Thus, har-monic analyses and their accuracy will depend on the load model usedin the simulation.With respect to our load model, we would like to mention that

laboratory experiments on 2 to 5 HP machines appear to justify themodel presented. Grouping of various machines have been dealt with ina number of other publications (1-6). Although we are currently in theprocess of evaluating some actual field substation measurements,analysis of such field data is proving very difficult. This is due to thelarge number of variables that controls a substation load.

Regarding the other remarks stated by Mr. Murotani, we would liketo offer the following observations.

1. We still believe that the transformer and generator modelspresented in our paper are adequate models for the frequencyrange below 2400 Hz. However, the discusser points out correctlythat both the transformer and generator models should containfrequency dependent resistances. We also agree that thefrequency-dependent resistance of electric machinery may be im-portant and should be incorporated in second generation models.We believe, however, that a magnetization branch including eddycurrents and iron losses should be included.

2. The load models used by the discusser appear to be based on ex-perimental data. These models, however, do not appear to be ap-plicable at higher order harmonics. At higher frequencies thetotal load impedance presented in Models A and B will be veryhigh and thus the normal load damping effect would be negligi-ble. Thus while models A and B may very well be applicable atthe 5th and 7th harmonics, they should be modified for higherfrequencies. For instance, the load could still be presented as thediscusser suggests, but parallel R-L combination is used. This willinsure the inclusion of the critically important damping effects of

the load on the harmonics and their propagation.Finally, we would like to add our voice to that of the discusser to em-

phasize that there is a need for improved power system models that areapplicable for harmonic frequencies up to at least the 50th harmonic.There is no doubt that there is a need for a second generationtransformer, machine and load models beyond what we and thediscusser have presented.

REFERENCES

1. F. Roohparvar, A. A. Mohmoud, J. Hanania, "Multi-machinemodeling of a group of three-phase induction miotors," Pro-ceedings of the International Conference on Electrical Energy,Oklahoma City, Oklahoma, April 1981.

2. J. Hanania, A. A. Mahmoud, Al Day, F. Roohparvar, "Inductionmotor group representation by variable speed-dependent reactancemodel," Midwest Power Symposium, Urbana-Champaign, 1981.

3. M. M. Abdel Harkim, G. J. Berg, "Dynamic single unit representa-tion of induction motor groups," IEEE Transacitons on PowerAp-paratus and Systems, Vol. PAS-95, January/February, 1976.

4. G. J. Berg, P. Subramaniam, "Induction motor load representa-tion," Paper A 79 492-2 Presented at the IEEE PES SummerMeeting, Vancouver, British Columbia, Canada, July 15, 1979.

5. C. P. Arnold, E. J. P. Pacheco, "Modelling induction motor start-up in a multi-machine transient-stability programme, Paper A 79492-0 Presented at the IEEE PES Summer Meeting, Vancouver,British Columbia, Canada, July 15, 1979.

6. F. Iliceto, A. Capasso, "Dynamic Equivalents of asynchronousmotor loads in system stability studies," Paper T 74 117-8 Presentedat the IEEE PES Winter Meeting, New York, N.Y,, January 27,1974.

Manuscript received November 13, 1981.