Operating unbalance in long-distancetransmission

Prof. W. Derek Humpage, B.Sc, Ph.D., K.P. Wong, M.Sc, Ph.D.,and T.T. Nguyen, B.E., Ph.D.

Indexing terms: Transmission lines, Transmission and distribution plant, Power transmission and distribution

Abstract: Unbalance in steady-state operation is of general concern in power transmission, but it can be espe-cially emphasised when transmission is over long distances, owing to the increasing influence of line parameterasymmetries on operating conditions as transmission-line lengths increase. Specifically in the context of long-distance transmission, the paper first develops procedures for evaluating the dependence of operating unbalanceon the precise sets of connections when conductor transpositions are made at discrete points along the length ofa transmission line. These are included in detailed phase-variable models for transmission-line sections, and onthem is based an overall form of Newton-Raphson solution. The methods are applied to a representativelong-distance transmission interconnection operating at 220 kV and having two separated points of conductortransposition. The inherent operating unbalance in the interconnection is evaluated for a range of power-transfer conditions, and the extent to which the unbalance can be lowered by discrete transposition is quanti-fied. The interaction of load unbalance and transmission-line parameter asymmetry is investigated, and loaddistributions are identified which lead to the greatest increase and the greatest reduction in unbalance in trans-mission. For an optimal choice of conductor transposition from a total of 35 possible sets of connectionsinvolving the two transposition points, the paper then investigates the further lowering of operating unbalancewhich saturated-reactor forms of shunt compensation at selected locations can additionally offer. In total, thepaper provides a comprehensive investigation of operating unbalance in long-distance transmission from whichseveral conclusions are drawn.

XN

List of principal symbols

if/ = propagation coefficient matrixZo = surge impedance matrixC = connection matrix at a point of transpositionP = matrix defining the assignment of phases to con-

ductorsV, I = vectors of conductor voltages and currents,

respectivelyVN ' Av = vectors of network nodal voltages and currents

respectively= network nodal admittance matrix= vector formed from real- and imaginary-parts of

network nodal voltages in VN

/N(XN) = vector of nodal residual functions

JN(XN) = Jacobian matrix

/ = length of transmission line section between trans-position points

p = number of transposition pointsn = number of network nodes

Subscripts S and R identify variables at the sending- andreceiving-ends, respectively, of transmission-line sections.Variables in the a, b, and c phases are denoted by super-scripts a, b, and c, respectively.

1 Introduction

Asymmetries in the series-impedance and shunt-admittance parameters of multiconductor power-transmission lines introduce unbalance into steady-stateoperation in a form which increases as transmission dis-tances increase. Operating unbalance is therefore often ofspecial concern in the planning and design of long-distancepower-transmission interconnections. Conductor transpo-sitions at discrete points along the length of transmissioncircuits might then be considered as a first design measureby which the level of operating unbalance might belowered, for asymmetries in one line section betweenpoints of transposition can then offset those of another.

Paper 3689C (P7, P9), received 3rd July 1984

The authors are with the Department of Electrical & Electronic Engineering, Uni-versity of Western Australia, Crawley, W. Australia 6009, Australia

While it has increasingly been the preference intransmission-system practice to avoid conductor transposi-tions as much as possible [1-3], they provide basic meansof countering the effect of line parameter asymmetries incircuits of increasing length, and it is specifically long-distance transmission to which the developments and find-ings of the present paper relate.

On this basis, the paper first investigates the dependenceof operating unbalance on the precise interconnectionsbetween conductors at nominated points of transposition.For a single-circuit transmission line and a single interme-diate point of transposition, there are five possible and dif-ferent sets of connections between conductors. Thisnumber increases to 35 where conductors are transposedat two separated points, and, more generally, toYJ= I P' 5P/*'! (p — i)\ different cases for a total of p points oftransposition. Systematic means are therefore required forfinding, from amongst all possible sets of connections, theone which achieves the greatest reduction in operatingunbalance.

For any chosen scheme of conductor transposition, thereactive-power compensation that long-distance transmis-sion invariably involves, depending on its type, can alsocontribute to lowering the level of unbalance in steady-state operation. Unbalances in voltage and current vari-ables are interrelated, and while reactive-powercompensators most directly reduce unbalance in voltages,they contribute also to lowering conductor current unbal-ance. Conductor transposition also contributes to reducingboth current and voltage unbalances.

In these terms, the present paper has three main pur-poses. The first is that of investigating the level of unbal-ance in a representative long-distance power-transmissioninterconnection for a typical range of steady-state oper-ating conditions. The second is one of developing ananalysis formulation and procedure for examining all pos-sible transposition arrangements for any given number andlocation of transposition points in finding the schemewhich can lead to the greatest reduction in operatingunbalance. The third purpose is that of evaluating thefurther lowering of unbalance which reactive-power com-pensators of particular form at given locations can addi-tionally offer. Together, these form a comprehensive

IEE PROCEEDINGS, Vol. 132, Pt. C, No. 2, MARCH 1985 67

'SI

'si

VR1

' R ,

*S2

! S2

Sj 'Rj Sk 'Rk ' 5 ( p * R(p

l i

pi ck

lk

c,

p * 1

P p + 1

Fig. 1 Discrete transposition in long-distance transmission

C2t Ctj C(p+1)(7are connection matricesP, Pj, Pk P ( p + , , are phase assignment matrices

/, lj, lt hp+u a r e s e c t ' o n lengthsV and / are vectors of conductor voltages and currents

investigation from which conclusions are drawn at the endof the paper in relation to lowering the operating unbal-ance in long-distance power transmission.

2 Discrete transposition

2.1 Conductor transposition and phase assignmentsA single-circuit power-transmission line is subdivided inFig. 1 into a total of p + 1 sections by p points of conduc-tor transposition. Series- and shunt-path parameter setsare formed for each section for any given transmission-linetower design and conductor specifications. In Fig. 2, the

section j

1

section k

\

\

| \\

transpositionpoint

Fig. 2 Transposition between sections j and k

Conductors are identified in their position on transmission line towers by 1, 2,and 3Phase assignments to conductors are identified by a, b, and c

positions of conductors in the tower configuration aredenoted by the identifiers 1, 2, and 3. Parameter asym-metries may be identified with respect to this identificationof conductor positions, or, when phases are assigned toconductors as in Fig. 2, they may be specified with respectto the individual phases. By cross-connecting conductorsat points of transposition, phase assignments are re-ordered, and parameter asymmetries for the line section onone side of the transposition point are altered with respectto those for the line section on the other. It is required intransposition to find the particular combination of inter-connections at successive transposition points for whichasymmetries in the parameters of line sections betweenthem subtract from one another to leave the lowest overallasymmetry for the complete transmission line.

At each transposition point, interconnections betweenline conductors on each side of it are represented inanalysis by a connection matrix C. In the conventionadopted here, the rows of matrix C relate to the conduc-tors on the right-hand side of the transposition point, andthe columns to conductors on the left-hand side. Eachelement of C is then associated with an interconnectionbetween the two sides. C(x, y) = 1 when the conductoridentified in the tower configuration on the right-hand sideof the transposition point by x is connected to the conduc-tor identified on the left-hand side by y. Otherwise, if there

123

1"0

10

J2001

3100

is no connection, C(x, y) — 0. For a single-circuit line,x = 1, 2, 3 and y = 1, 2, 3. For the conductor intercon-nections between line sections j and k in Fig. 2, the connec-tion matrix Ckj is given by:

(1)

Phase assignments to conductors are now specified foreach line section by matrix P where P(x, y) = 1 whenphase y is assigned to the conductor identified in its posi-tion by x. Otherwise, P(x, y) = 0. Here, x = 1, 2, 3; andy = a, b, c. For the phase-assignments in Fig. 2.

(2)

From the separate definition of connection and phase-assignment matrices:

a100

b001

c0"10

1Pk= 2

3

a010

b100

c001

— P Pl(3)

While connection matrices are required in specifying con-ductor connections at transposition points, it is often pref-erable in analysis to work with phase-assignment matricesinstead. Once a given scheme of transposition has been sel-ected following analysis based on phase-assignmentmatrices, the transposition connections themselves arefound from connection matrices formed as in eqn. 3.

2.2 Phase-assignment matrices in transmission-linemodel

For any transmission-line section) of length /,, as in Fig. 1,voltage and current variables at one end of the section invectors VRj and IRj are related to corresponding variablesat the other end in vectors VSj and ISJ by

cosh (\f/j lj)

- Yooj

j IJ

— sinh

cosh (if/m J usj(4)

Expressions for the propagation coefficient matrix if/j andfor the phase-variable characteristic impedance and admit-tance matrices Zoj and Yoj are given in Appendix 8.1.

An overall transposition scheme is started from a givenphase assignment to conductors for the first line section.As this assignment is at choice, it is convenient in analysisto begin with the case of a — 1, b — 2 and c — 3. On thisbasis, the phase-assignment matrix for the first line sectionPj is a unit matrix, and variables in the vectors VS1, Isl,VR1, and IR1 are in the sequence a, b, c, so that

I/i _ I/a l/fc ]/cSI

t T/aRl — '• R

(5)

(6)

(7)

68 IEE PROCEEDINGS, Vol. 132, Pt. C, No. 2, MARCH 1985

V = I" Ib Ic (8)

At the first point of transposition

VS2 = C2lVRl (9)

IS2 = C2lIRl (10)

From eqn. 3, C2l=P2P\, and so the transpositionrelationships of eqns. 9 and 10 may be expressed equiva-lently in the form

P' V — P' V n nr2 yS2 — r 1 vRl K1 l)pi 1 — P< I (\1\r 2 iS2 ~ r ix RX \lA)

For any given transposition, the multiplications C21VRl

and C2 1/R 1 in eqns. 9 and 10 lead to the original order orsequence of phase variables in VR1 and /R 1 being changedin vectors VS2 and IS2. In eqns. 11 and 12, on the otherhand, the products P'2 VS2 and P\ IS2 recover in the resultsthe initial sequence of phase quantities in vectors VRl andIRl. While this relates specifically to the second section, itwill be seen that it also applies generally at any linesection, irrespective of the particular transpositions preced-ing it. Using equivalences of the form of those in eqns. 11and 12, any given transposition can therefore be represent-ed in analysis in terms of the phase assignments to conduc-tors on each side of the transposition point. In Figs. 1 and2, phase conductors are transposed between line sectionsidentified by j and k, and for which the phase-assignmentmatrices are Pj and Pk, respectively. In this case

P' V — P'Vt L r e t — I :V a :kVSk (13)

P'k I Sk = P) I Rj (14)

It is convenient to denote P)VRj in eqn. 13 by VaRj as pre-

multiplying VRj by P) reorders variables in the result to thea, b, c sequence. The remaining variables for section jarranged in a, b, c sequence are then formed from

(15)

(16)

(17)

Substituting VSj, VRj, ISj and IRj from these relationshipsinto the 2-port phase-variable equations for line section jin eqn. 4 then gives

jabc

jabc

Yoj si

- s i n h Wj Ij)Z

cosh (,/,}/,.) •]

or

' #

(18)

(19)

where

cosh ljlPj (20)

j Ij)ZojPj (21)

sinh WJIJ)PJ (22)

j (23)

This procedure and, in particular, the forms of eqns. 20-23apply for each line section in turn. Specified transpositionconnections are embedded in the A, B, C and D matricesfor each line section in terms of phase-assignment matrices.When the equations of each section into which a complete

Aj=

Bj= -P) s inh

Cj= - J O J

Dj = Pj cosh

transmission line is subdivided by points of transpositionare of the form of those in eqn. 19, they may be combineddirectly. In the voltage and current vectors for each linesection, phase variables are always in the positionalsequence nominated for the first one.

2.3 Network formulationFor inclusion in a nodal voltage formulation for thenetwork of which the subdivided transmission line of Fig.1 is part, eqn. 19 for any section; of the line rearranges tothe form

yYss YYSR ~\rvbc~\Y v vbc

rRS rRRJL V Rj J

(24)

Relationships between the partitions Yss, YSR, YRS andYRR in eqn. 24 are summarised in Appendix 8.2. Node setsare identified at each transposition point, following whichall branch equations assemble to the nodal form

IN = YN VN

If there are n triple-phase node sets in total, then

1 N — ' i> l 2 J • • • > 1n

and

vlN=v\, v2,...,vn

where

/'. = I". lb. lc. j = 1 2 n1 j I j , l j , l j J — 1 , Z., . . . , f(

and

(25)

(26)

(27)

(28)

V)= V°, V»p V) j=l,2,...,n (29)

YN is the phase-variable network nodal admittance matrix.

2.4 Ne wton - Raphson analysisReal and imaginary parts of network nodal voltagesarranged in vector xN are found in Newton-Raphsonphase-variable network analysis from the iterativesequence

— IJ

P~l (30)

in which p is the iteration count. The vector of residualfunctions fN(xN) is formed from load and generator con-straints [4, 5] at network nodes, and the Jacobian matrix,JN(xN), is given by

JN{xN) = dfN(xN)/dxN (31)

The representation of reactive-power compensators hasbeen developed in earlier work [6, 7], and, following this, acompanion paper* shows how saturated-reactor-type com-pensators may be included in phase-variable forms ofNewton-Raphson analysis.

3 Lowering operating unbalance by conductortransposition

Newton-Raphson analysis in which conductor transposi-tions are taken into account using the transmission-linemodel of Sections 2.1 and 2.2 allows unbalance in steady-state operation to be found for any given load and gener-ating conditions. Repeat studies are required for eachpossible set of transposition connections in evaluating thelevel of operating unbalance to which each gives rise.

* HUMPAGE, W.D., WONG, K.P., NGUYEN, T.T., and McLOUGHLIN, J.M.:'Phase-variable modelling for saturated-reactor compensators in long-distancetransmission', Paper submitted to IEE Proc. C Gen., Trans. & Distrib.

IEE PROCEEDINGS, Vol. 132, Pt. C, No. 2, MARCH 1985 69

While analysis is necessarily of phase-variable formthroughout, the negative- and zero-phase-sequence com-ponents of voltage- and current-variables formed fromnumerical solutions for phase components provide a directindication of operating unbalance. Loading in the negativephase sequence is of particular concern in relation to thethermal limits of synchronous generator units supportinglong-distance transmission. A typical requirement in prac-tice is that of finding the transposition scheme which willminimise the negative-phase-sequence component of con-ductor currents at the generation end of long-distancetransmission interconnections, and to thereby reduce asmuch as possible the thermal stressing in synchronous gen-erator units supplying them. Successive evaluations ofunbalance using the analysis methods of Section 2 for arange of steady-state loading conditions are required, inwhich each possible transposition scheme is considered inturn. Comparisons among the levels of operating unbal-ance then allow the transposition scheme to be selected forwhich the unbalance is lowest.

A representative long-distance transmission intercon-nection is shown in Fig. 3. The principal data for the

When transpositions are made at both C and D, the lowestlocus of those for all 25 sets of connections is that of CD 11

Fig. 3 Long-distance transmission interconnectionNode A connects to a 132 kV network including synchronous generationNodes H and I connect to 132 kV networks from which load is supplied.Intermediate loads at C, F and G are zero throughout the studies of Sections 3and 4

system is summarised in Appendix 8.3. There are twopoints of conductor transposition: one at C in Fig. 3 andone at D. If transposition is confined to location C, thereare five possible transposition arrangements. Beginningwith a phase assignment of a — 1, b — 2 and c — 3, thepossible reassignments at the transposition point are thoseof

( 1 ) 6 - 1 ( 2 ) c - l ( 3 ) a - l ( 4 ) 6 - 1 ( 5 ) c - lc - 2 a - 2 c - 2 a-2 b - 2a-3 b-3 b-3 c - 3 a-3

There are similarly five sets of possible connections whentranspositions are made at D. When transpositions aremade at both C and D, there are 25 possible sets of con-nections. In total, there are therefore 35 distinct cases forwhich operating conditions are to be evaluated.

Fig. 4 shows the negative-phase-sequence component ofconductor currents at B in Fig. 3 for a range of balancedactive- and reactive-power loading conditions at E. Theintermediate loads at C and at D are zero, and, initially,the contribution of the compensators at D and E is dis-counted. Without compensation, voltage distributionsalong the main transmission circuit B-C-D-E will notmeet typical requirements, but initial operating unbalancesare established which allow the contribution of com-pensators to lowering unbalance later to be quantified.Locus U in Fig. 4 is that for which transposition is notattempted either at C or D. Locus Cl is the lowest one ofthose for the five possible transposition schemes whentransposition is confined to location C. Similarly, locus Dlis the lowest one when transposition is made only at D.

20.0

,15.0

10.0

5.0

Cl

D1

CD0.0 0.2 0.4 0.6 0.8

active - power transfer, pu.1.0

Fig. 4 Dependence of conductor current unbalance on power transfer

I2 is the negative-phase-sequence component of transmission line current at B. Udenotes the untransposed case. Cl, Dl and CD 11 denote transposition schemesinvolving location C, location D and both locations C and D, respectively. I2 is inper cent and power—transfer values are in per unit on 100 MVA base. Balancedload conditions at E

in Fig. 4. For this latter transposition scheme, the oneidentified by CD11, the negative-phase sequence com-ponent of current at B in the system of Fig. 3 is reduced toabout 30% of that when transposition is not attempted.The greatest unbalance is that when the active-powertransfer is zero. In this case, the unbalance in current vari-ables is related directly to transmission-line charging cur-rents. To the extent that, without compensation, thetransmission-line voltage profile will exceed typicalvoltage-control requirements, the charging-current unbal-ance in Fig. 4 without transposition is greater than thatcalculated at nominal voltage using the line susceptancedata of Appendix 8.3.

Corresponding loci for the zero-phase sequence com-ponent of transmission-line current at B in Fig. 3 areshown in Fig. 5. Delta-connected windings on the trans-formers at A, D and E help to lower the level of unbalance

2.5r

2.0

1.5

1.0CDir

o.o 0.2 0.4 0.6active - power transfer, p.u.

0.8 1.0

Fig. 5 Dependence of conductor current unbalance on power transfer/„ is the zero-phase sequence component of transmission line current at B inpercent on a 100 MVA base. U: untransposed; Cl transposition at C; Dl transpo-sition at D; CD11 transposition at C and D. Balanced load conditions at E

in the zero-phase sequence, and this is significantly lessthan that in the negative-phase sequence. The order of theloci for the Cl and Dl transposition schemes is reversed inFig. 5 when compared with that in Fig. 4. However, thetransposition scheme which minimises the negative-phase-sequence component of current at the generator end of thetransmission line is also that for which the zero-phase-sequence component of current is reduced to a minimum.

Turning now to voltage unbalance, voltage distributionsalong the interconnection of Fig. 3 in the negative-phasesequence are shown in Fig. 6 for the Cl, Dl and CD 11transposition schemes. Compensator operation at D and Eis still discounted. Without transposition, the negative-phase-sequence component of voltage rises from about 5%at the generation end of the line to about 6.5% at the load-

70 IEE PROCEEDINGS, Vol. 132, Pt. C, No. 2, MARCH 1985

supply end. The CD 11 transposition scheme reduces theselevels to about 1.6% and 4.7%, respectively. At B, the

7.0

6.0

o5.0

CM

^ 3 . 0

2.0

1.00.0

0.0 0.25 0.50 075fractional position on line

1.0

Fig. 6 Voltage distribution in the negative phase sequence

Total load demand at E in Fig. 3 0.7 + y'0.175 puV2 is the negative-phase sequence component of conductor voltageU untransposed; Cl transposition at C; Dl transposition at D; CD11 transpositionat C and D; balanced load conditions at E; 100 MVA base

CD 11 scheme leads to a lower negative-phase sequencevoltage than that for the Dl scheme, but the loci for thesecross at a distance from B corresponding to about 20% ofthe line length. However, the difference between the two issmall, and compensation, as in Section 4, reduces it further.The preferred transposition is therefore likely to remainthe one identified by CD11. For completeness, the voltagedistribution in the zero-phase sequence is shown in Fig. 7.

0.25fractional

0.50position on line

Fig. 7 Voltage distribution in the zero phase sequence

Total load demand at E in Fig. 3 0.7 + ./0.175 puVo is the zero-phase-sequence component voltageU untransposed; Cl transposition at C; Dl transposition at D; CD11 transposi-tion at C and D, balanced load conditions at E; 100 MVA base

Depending on its precise form, load unbalance cancombine with transmission-line parameter unbalanceeither to increase or to decrease voltage and current unbal-ances in operation. By simulating a range of unbalancedload-supply conditions, particular cases can be found forwhich the effect of load unbalance is greatest. The effect ofload unbalances on the negative-phase sequence com-ponent of current at B is shown in Fig. 8. In Figs. 4-7, the

10.0

8.0

6 0

A.O

2.0

0.00.0 0.2 1.00.4 0.6 0.8

active- power t ransfer ,pu.

Fig. 8 Effect of load unbalance

CD11 transposition scheme. balanced load 2% load unbalance4% load unbalance. 100 MVA base

1EE PROCEEDINGS, Vol. 132, Pt. C, No. 2, MARCH 1985

active- and reactive-power demand at E is balancedbetween the separate phases irrespective of the voltageunbalance at the point of load supply. In considering loadunbalance in Fig. 8, if P is the active-power demand in onephase, it is P + AP in another and P — AP in the third.Cases are shown in Fig. 8 for which AP is first 2% of Pand then 4% of P. Load distributions at E in Fig. 3 whichlead to the greatest increase in operating unbalance at B,and also to the greatest reduction in unbalance, are sum-marised in Table 1. Fig. 9 shows the effect of load unbal-ance on the voltage distribution in the negative phase

0.25 0.50 0.75fractional position on line

Fig. 9 Voltage distribution in the negative phase sequence with loadunbalance

CD11 transposition scheme, a Zero power-transfer b 0.5 pu power-transfer c 1.00 pupower-transfer 2% load unbalance at 0.5 pu transfer - • 2% loadunbalance at 1.00 pu transfer. 100 MVA base

Table 1: Load unbalance distributions

distribution whichincreases operatingunbalance

distribution whichdecreases operatingunbalance

phase a P - APphase b Pphase c P + AP

P + APP-APP

sequence for the CD 11 transposition scheme. In bothcurrent and voltage unbalances, the effect increases pro-gressively as load transfers increase.

In the zero phase sequence, voltage and current levelsare not affected significantly by load unbalance, and thesedependencies are therefore not separately, shown.

4 Operating unbalance and reactive-powercompensation

4.1 Compensator typeFor a range of typical operating conditions in the intercon-nection of Fig. 3, voltage control requires a reactive-powerabsorption at D and E. Different types of compensator canoffer different characteristics, but the present investigationsrelate to compensators of the saturated reactor kind [8, 9].For this form of compensator, the level of unbalance whichconductor transposition can achieve is unlikely to lead tonegative-phase-sequence suppression being a compensatordesign constraint. Where a supply of reactive power isrequired to meet voltage control requirements, additionalshunt-connected capacitors complement the compensatorsshown in Fig. 3. The delta-connected windings on thecoupling transformers at D and E to which compensatorsare connected isolate their circuits in the zero phasesequence from those of the primary transmission system.Where two compensators operate in close parallel connec-tion at one location, the parallel connections between com-

71

pensators are made between their slope-compensatingcircuits.

4.2 Compensation at intermediate pointWhen compensation is confined to location D in Fig. 3,the total reactive-power requirements of voltage controlmight most typically be met by a twin compensator con-figuration as shown. Voltage distributions in the positive-phase sequence which are within a control band of +10%can then be maintained for a substantial range of loadingconditions. The negative- and zero-phase-sequence com-ponent of current at B do not now change significantlywith changing load-transfer conditions. Negative-phasesequence current distributions are therefore shown in Fig.10 for the zero-load-transfer condition. The near-linear dis-tribution without transposition or compensation is first

0.00.25 0.50 0.75fractional position on line

1.0

Fig. 10 Negative-phase-sequence current distribution

untransposed without compensation —••— untransposed with com-pensation at D in Fig. 3 CD 11 transposition without compensation- • CD 11 with compensation at D. 100 MVA baseZero power-transfer conditions

reduced by the CD11 transposition scheme for the line sec-tions up to the transposition location at D. Compensationat D then both lowers the distribution substantially andreduces its variations along the transmission-line length.At the generation end of the line, the negative-phase-sequence component of current is reduced to about 10% ofits value without compensation and when conductor trans-position is not attempted. In the untransposed case, com-pensation at D reduces the negative-phase-sequencecurrent at B to about 50% of its value when compensationis not applied, but this level is 4-5 times greater than thatwith transposition. The current distribution in the zerophase sequence is of a more discontinuous form, as in Fig.11, but the CD 11 transposition scheme, together with com-pensation at D, lowers the zero-phase-sequence com-ponent of transmission-line current at B to about 25% ofits value in the untransposed case with compensation.

2.5r

0.00.25 050 0.75

fractional position on line1.0

Fig. 11 Zero-phase-sequence current distribution

untransposed without compensation — untransposed with com-pensation at D in Fig. 3 CD11 transposition without compensation- • CD11 with compensation at D. 100 MVA baseZero power-transfer conditions

The direct effect of compensator operation on thevoltage profile for the transmission line leads to com-pensation at D reducing substantially the negative-phase-sequence component of line voltage throughout its length,as in Fig. 12. Here the voltage in the negative phase

0.0 0.25 0.50 0.75fractional position on line

1.0

Fig. 12 Effect of intermediate compensation on negative-phase-sequencevoltage distribution

untransposed without compensation— untransposed with compensation at D in Fig. 3

CD11 transposition without compensation- • CD 11 transposition with compensation at D in Fig. 3Zero power-transfer conditions

sequence does not exceed 1% at any point along thelength of the primary transmission circuit. The corre-sponding voltage distribution in the zero-phase sequence,and the lowering of it that compensation at D provides, isshown in Fig. 13.

0.00.0 0.25 0.50 0.75

fractional position on line1.0

Fig. 13 Effect of intermediate compensation on zero-phase-sequencevoltage distribution

untransposed without compensation— untransposed with compensation at D in Fig. 3

CD11 transposition without compensation- - - - - CD 11 transposition with compensation at D in Fig. 3Zero power-transfer conditions

4.3 Compensation at remote line terminationIn Fig. 14 a comparison is made between the voltage dis-tribution in the negative phase sequence when com-pensation is confined to location D and that when thesame compensator capacitance is in operation at E. While

1.6

1.2

0.8

0.4

0.00.0 0.25 0.50 0.75

fractional postion on line1.0

Fig. 14 Comparison between intermediate and remote-end compensation

CD11 transposition. Negative-phase-sequence voltage distribution for zero powertransfer

compensation at D in Fig. 3 compensation at E

72 IEE PROCEEDINGS, Vol. 132, Pt. C, No. 2, MARCH 1985

compensation at D appears to lead to the more uniformdistribution, both are reduced to levels at which the choiceof compensator location is likely to be determined bygeneral voltage control considerations. In Fig. 15 the com-parison is extended to the current distribution in the nega-tive phase sequence.

5.0

4.0

»3.0

"2.0

1.0

0.00.0

Fig. 15

0.75 1.00.25 0.50fractional position on line

Comparison between intermediate and remote-end compensation

CD11 transposition. Negative-phase-sequence current distribution for zero powertransfer. 100 MVA base

compensation at D in Fig. 3 compensation at E

4.4 Intermediate and remote-end compensationIn the representative long-distance transmission system ofFig. 3, the nominal design configuration is most likely tobe that in which there are compensators at both D and E,although it is likely to be required to evaluate other casescorresponding to where compensators at either D or E areout of service. The cases shown in Fig. 16 are where there

0.6

0.4

0.2

0.00.0 0.25

fractional1.00.50 0.75

position on line

Fig. 16 Two-point compensation

CD11 transposition. Negative-phase-sequence voltage distribution for zero powertransfer— single compensator at D, single compensator at E

twin compensators at D, single compensator at Esingle compensator at D, twin compensators at E

is a single compensator at both D and E; where there aretwo compensators at D and one at E; and where there aretwo at E and one at D. Negative-phase-sequence voltagedistributions for these combinations are close together,and lie between those when compensation is either at D orat E.

4.5 Compensator loadingWhen compensation is confined to location D themaximum reactive-power absorption requirement at thatpoint is about 110 MVAr. This reduces to about 85 MVArwhen compensation is located at the remote end of theinterconnection at E. For the most likely design case inwhich compensation is provided both at the intermediatepoint D and at the remote end E, the maximum absorp-tion requirements are about 60 MVAr at D and 45 MVArat E. An arrangement which can give closely balancedreactive-power loading conditions on compensators iswhere there are twin 60 MVAr units at D together with asingle 60 MVAr compensator at E. For the range of all

probable steady-state operating conditions, the negative-phase-sequence component of compensator current is lessthan 2% of rated current in any of the arrangements con-sidered involving compensators separately at D or at E, orwhere they are provided at both locations.

5 Conclusions

From the analysis formulation and evaluation proceduresof the present paper, all possible discrete transpositionschemes in long-distance transmission may be considered,and the combinations of connections at given transposi-tion points which can achieve the greatest reduction inoperating unbalance may be found. The final system modelon which successive evaluations and comparisons arebased is necessarily a detailed phase-variable one, and it isonly that part of the complete model in which any speci-fied conductor transposition is represented in analysiswhich is developed in detail in the present work. Phaseassignment definitions for line sections between transposi-tion points represent a convenient basis for the develop-ment of analysis methods, and they also add considerablyto the ease with which the procedures may be used in par-ticular design studies. When an optimal transpositionscheme has been found from analysis based on phase-assignment matrices, connection matrices by which theconductor connections at points of transposition arefinally specified may be found from eqn. 3. In terms of thecomputing time requirements of analysis, a single evalu-ation for the system of Fig. 3 requires about 120 s of CPUtime on a CDC Cyber 72/73. For two points of transposi-tion, a complete transposition design might require a totalof about 100 individual studies. Quantifying the effects ofoperating unbalance on saturated-reactor compensators inthe present work required a further 30 separate solutions.

In the representative long-distance transmission inter-connection of Fig. 3, an optimal choice of transpositioncan lower the negative-phase-sequence current at the gen-eration end of the line to about 30% of its value whentransposition is not attempted. This reduction represents alowering in the thermal loading of synchronous generatorssupplying the interconnection to about 10% of thatwithout transposition. Load unbalance interacts with lineparameter unbalance, and can either increase or decreasethe level of unbalance when compared with balancedloading conditions depending on its form. For any giveninterconnection and any transposition scheme designed forit, the precise form of load unbalance which increases ordecreases operating unbalance to the greatest extent can befound. In further developing this aspect of the presentwork, composite load models in which their intrinsic char-acteristics in unbalanced operation are represented mightusefully be sought.

For the system of transposition which can offer thegreatest reduction in operating unbalance in the intercon-nection of Fig. 3, voltage unbalances can still be as high as5%. Shunt compensators then lower this level to about1%, and they also achieve a near uniform voltagedistribution in the negative and zero phase sequences.Negative- and zero-phase-sequence components oftransmission-line current are then almost independent ofthe power-transfer level. Once the location of com-pensators and their capacitance have been determined bythe general considerations of voltage control, it seems thatthey can guarantee low levels of operating unbalance whenthe optimal transposition scheme is used: Compensators ofthe saturated reactor kind only have been considered in

IEE PROCEEDINGS, Vol. 132, Pi. C, No. 2, MARCH 1985 73

the present work, and further work remains in consideringother forms in relation to their contribution to reducingoperating unbalance.

6 Acknowledgments

The authors are grateful to the West Australian RegionalComputing Centre for their co-operation in running theevaluation programmes of the present work. Dr. Nguyengratefully acknowledges the award of a Research Fellow-ship from the State Energy Commission of Western Aus-tralia.

7 References

1 GROSS, E.T.B., and HESSE, M.H.: 'Electromagnetic unbalance ofuntransposed transmission lines', IEEE Trans., 1953, PAS-73, (7), pp.1323-1332

2 HOLLEY, H, COLEMAN, D., and SHIPLEY, R.B.: 'Untransposedehv line computations', ibid., 1964, PAS-83, (2), pp. 291-296

3 HESSE, M.H.: 'Simplified approach for estimating current unbalancesin e.h.v. loop circuits', Proc. IEE, 1972, 119, (11), pp. 1621-1627

4 WASLEY, R.G., and SHLASH, M.A.: 'Newton-Raphson algorithm for3-phase load flow', ibid., 1974, 121, (7), pp. 630-638

5 BIRT, K.A., GRAFFY, J.J., McDONALD, J.D., and EL-ABIAD,A.H.: 'Three-phase load-flow program', IEEE Trans., 1976, PAS-95,pp. 59-65

6 BARTHOLD, L.O., BECKER, H., CLERICI, A., DALZELL, J.,MORAN, R.J., NORMAN, A.B., PEIXOTO, C.A.O., REICHERT, K.,ROY, J.C., THOREN, B., and WILLIAMS, W.P.: 'Modelling of staticshunt VAR systems for system analysis', Electra, 1977, 51, pp. 45-74

7 COOPER, C.B., and YACAMINI, R.: 'Choice of analytical and mod-elling methods for reactive compensation equipment', IEE Proc. C,Gen., Trans, and Distrib., 1981, 128, (6), pp. 402-406

8 THANAWALA, H.L., KELHAM, W.O., and WILLIAMS, W.P.: 'Theapplication of static shunt reactive compensator in.conjunction withline series capacitors to increase the transmission capabilities of longa.c. lines', CIGRE, 1976, report 31-09

9 FRIEDLANDER, E.B.: 'Transient reactance effects in static shuntreactive compensators for long a.c. lines', IEEE Trans., 1976, PAS-95,(5), pp. 1669-1680

8 Appendixes

8.1 Propagation coefficient and surge impedancematrices

If Z is the series-path impedance matrix per unit length ofa transmission line, and Y is the corresponding shunt-pathadmittance matrix, a matrix )} is formed in which ele-ments are eigenvalues of ZY or YZ. If Cx and C2 aremodal transformation matrices for ZY and YZ, respec-tively, the phase-variable propagation matrix, i/̂ , is givenby

iff = CylC\- 1 (32)

The phase-variable surge-impedance matrix is then givenby

0 = y/ Z (33)

and the surge-admittance matrix by

(34)

8.2 Partitions of nodal admittance matrixBeginning with transmission-line equations in the phase-variable, 2-port form as in eqn. 4:

ra-e a ta (35)

where

A = cosh (ij/l)

B = -sinh(iA0Zo

C= -Yo sinh

and

D = cosh (iA7)

(36)

(37)

(38)

(39)

In eqns. 36-39, / is the length of the transmission-linesection to which eqn. 35 relates.

Eqn. 35 rearranges to the nodal form

YSR~\ [VS~\

YRR] LVR][RJ L J K S

From this rearrangement

Yss= -B~1\_A - U]

— YSSR

(40)

(41)

(42)

(43)

(44)

In eqns. 41 and 44, U is the unit matrix.

8.3 System data

8.3.1 Transmission line: Initially, series-impedance andshunt-path susceptance matrices for the transmission lineof Fig. 3 are formed with the earth conductor included.Using a formal elimination procedure, the earth conductoris eliminated to give the series-impedance and shunt-susceptance matrices of Tables 2 and 3, respectively. In

Table 2: Transmission line series-path impedance matrix

a 0.305 +>1.288b 0.133+y0.691c O.133+yO.6O3

0.133+y0.6910.298 +y1.290.130+/0.62

0.133+y0.6030.130+y0.6200.298+y1.29

per-unit * 10~3/km on a 100 MVA base

Table 3: Transmission-line shunt-path susceptance matrix

a /1.512 -y'0.364 -y0.173b -y0.364 /1.564 -y'0.182c -y0.173 -y'0.182 y 1.495

per-unit * 10" 3 / k r n on a 100 MVA base

particular, mutual coupling between the phase conductorsand the earth conductor in its relationship to the evalu-ation of operating unbalance is taken directly into account,but without the earth conductor being represented inexplicit form. If required, the earth-conductor current caneasily be found once the phase currents have been deter-mined, using the sequences of the procedure by which theearth conductor is eliminated from explicit representation.

8.3.2 Transformers: For each of the transformers at A, Dand E in Fig. 3, winding voltages are 220 kV, 132 kV, and30 kV. Measurements between pairs of windings with thethird on open circuit give the following supply-frequencyreactances in per unit on a base of 100 MVA:

ZHz.=;0.0525pu

ZHT =70.145 pu

ZLT =;0.0916pu

(45)

(46)

(47)

74 IEE PROCEEDINGS, Vol. 132, Pt. C, No. 2, MARCH 1985

A model suitable for use in phase-variable networkanalysis is based on these and included in the nodal-voltage network formulation of Section 2.3. Tap-changecontrol is provided on the 220 kV winding. Throughoutthe studies of the present paper, the off-nominal tap-position is —8%.

8.3.3 Compensators: Compensator units of Fig. 3 areidentical and have the principal data given in Table 4.

Table 4: Principal compensator data

Nominal operating voltage of main windings 30 kVIntercept voltage in each phase 30.8 kVPositive-phase-sequence reactance at 50 Hz y'0.229 p.u.Negative-phase-sequence reactance at 50 Hz /0.458 p.u.Slope compensating susceptance at 50 Hz y'1.328 p.u.

Reactance and susceptance values in Table 4 are in perunit on a base of 100 MVA. From the winding reactancesin the separate sequences, it is required to form corre-sponding data for use in a phase-variable model. For thispurposeformed where

a matrix of sequence parameters, Zseq, is first

ZM, = d i a g { Z 0 , Z 1 , Z 2 } (48)

Subscripts 0, 1, and 2 denote the zero, positive, and nega-tive phase sequences, respectively. This transforms to thephase-variable form Zs using

Zs = QZseqQ-i (49)

where Q is the symmetrical-phase sequence component tophase-variable transformation matrix.

Abstracts of papers published in other Parts of the IEE PROCEEDINGSThe following papers of interest to readers of IEE Proceedings Part C, Generation, Transmission & Distribution haveappeared in other Parts of the IEE Proceedings:

Marx circuit modified by adding a tail sphere gap to gener-ate lightning impulses lasting a few microsecondsA. CARRUS

IEE Proc. A, 1985, 132, (1), pp. 40-44

A substantial decrease in efficiency compared with thestandard lightning impulse occurs in the production ofvery short-tailed lightning impulses by means of the Marxgenerator. A modification of one of the typical Marxcircuit configurations, i.e. the series connection of a spheregap and an inductance to the tail resistance, is suggestedhere to overcome this drawback. Digital simulation of thephenomena involved and extensive experimental workprove this solution to be suitable for the purpose.

Electrification current and voltage on a cylindrical object inthe presence of an ion flow field in an air streamM. HARA, M. YASHIMA, T. TSUTSUMI and M.AKAZAKI

IEE Proc. A, 1985, 132, (1), pp. 59-66

The paper is concerned with the assessment of the electri-cal environment in the vicinity of HVDC transmissionfacilities; the ion flow electrification phenomenon has beenrecognised as the subject of extensive research. The deter-mination of current and electric field at ground level underthe facilities and the estimation of electrification quantitieson objects placed in an ion flow field is the basis of thedesign of conductor geometries. It has been found frommeasurements of current and field using the model-scale

and full-scale transmission lines, that the current distribu-tion at ground level is strongly influenced by wind,although the effect of wind on electrification quantities onan object placed in the ion-flow field is not understood indetail. In the present study, the electrification current andvoltage on a cylindrical object placed in an air stream con-taining unipolar ions are estimated from analytical andnumerical calculations, and the influences of wind, theleakage resistance and the object arrangement on the iontrajectory and electrification quantities are discussed.

Analysis of dielectric measurements on switchgear bushingsin British Rail 25 kV electrification switching stationsA. BRADWELL and G.A. BATES

IEE Proc. B, Electr. Power Appi, 1985, 132, (1), pp. 1-17

Diagnostic measurements have been carried out in the 25kV electrification trackside switching stations on BritishRailways since 1972, following the discovery of audible dis-charge in the synthetic resin-bonded paper bushings.Laboratory measurements of discharge magnitude, capac-itance, dielectric loss and ultrasonic noise were correlatedwith physical damage at the bushing foils and criteria forbushing rejection were established. A batch of inferior-quality bushings was identified which were spread between16 switching stations. Further service experience, includingbushing punctures, led to a revision of the rejection criteriaand prompted diagnostic testing of 2500 bushings in all 92switching stations. The results are discussed in terms of therate of deterioration, critical discharge levels for stabilityand the merits of various diagnostic techniques.

IEE PROCEEDINGS, Vol. 132, Pt. C, No. 2, MARCH 1985 75