621.315.09:621.3.012.2 Monograph No. 89SUPPLY SECTION

TRANSMISSION-LINE ESTIMATIONS BY COMBINEDPOWER CIRCLE DIAGRAMS

By F. DE LA C. CHARD, M.Sc, Member.(The paper was first received 24th June, and in revised form \2th October, 1953.

as an INSTITUTION MONOGRAPH, \5th January, 1954.)

SUMMARY

// was published

Power circle diagrams drawn for either sending or receiving ends ofa transmission line do not have a common centre for the voltagecircles, nor are the power axes within the same semicircle. There isalso a possibility of confusion over the sign of reactive power whenconditions at both ends of the line are estimated by means of separatediagrams.

A quadrant diagram with current taken as reference, which com-plies with the convention of the American Institute of ElectricalEngineers for reactive power, is shown to be a satisfactory base onwhich a combined send-receive power circle diagram may be con-structed. If such a diagram is built as a calculating board, all sending-and receiving-end quantities may be simply read with sufficientaccuracy for the estimation of system load conditions.

Examples of the use of the calculating board are given.

LIST OF PRINCIPAL SYMBOLSA,B,C,D— General network constants.

Es, Is — Sending-end line voltage and current.Er, lr — Receiving-end line voltage and current.Zt/_P — Line impedance and angle, per phase.

Y = Line admittance, phase to neutral.Zt,/_h = Transfer impedance and angle, per phase.Zs,/_Ps = Sending-end driving-point impedance and angle,

per phase.Zr/Pr — Receiving-end driving-point impedance and angle,

per phase.8 — Transmission angle (angle between Es and Er).

Ps,Pr ~ Sending- and receiving-end powers.Qs,Qr ~ Sending- and receiving-end reactive powers.

WATTS IN, Pt — Power received or absorbed by the load at thereceiving end of the line.

WATTS OUT, Po — Power supplied or produced by the source atthe sending end of the line.

VARS IN, Qj -~ Reactive power absorbed at the sending orreceiving ends of line, by source or load, i.e.that due to a capacitive source or an inductiveload.

VARS OUT, Qo — Reactive power produced at the sending orreceiving ends of the line by the source or load,i.e. that due to an inductive source or acapacitive load.

(1) INTRODUCTIONThe power circle diagrams for sending and receiving ends of

a line are derived from the corresponding voltage phasordiagrams (Figs. 1 and 2). These phasor diagrams are usuallydrawn with the reference quantities in the same direction,although they refer in one case to a source and in the other toa load. The derived power circle diagrams have differentcentres for the voltage circles, with a common active- andreactive-power axis. They can, however, use the same set of

Correspondence on Monographs is invited for consideration with a view topublication.

Mr. Chard is in the Department of Electrical Engineering, University of Bristol.

Fig. 1.—Receiving-end voltage phasor diagrams.

DE, 0 .

r2

Fig. 2.—Sending-end voltage phasor diagram.

circles, within the same semicircle, if the axes, in respect of thesending-end diagram, are rotated through 180°. This axisreversal might be objectionable were it not justified by con-siderations of usage.

Power-system metering has long made use of the terms"WATTS IN," "WATTS OUT," "VARS I N " and "VARS OUT."

Decrease Decrease

\

Increase

VArs, out

Fig. 3.—Quadrant diagram, in accordance with the currentAmerican I.E.E. sign convention for reactive power.

These terms imply a quadrant diagram (see Fig. 3) in whichcurrent sent or received is shown on the horizontal axis andsending- and receiving-end voltages appear in two diagonally

[204]

CHARD: TRANSMISSION-LINE ESTIMATIONS BY COMBINED POWER CIRCLE DIAGRAMS 205

opposite quadrants. Thus an inductive load demands WATTS INand VARS IN, which necessitates WATTS OUT and VARS OUTfrom the source. A capacitive load absorbs active power andproduces reactive power, which necessitates the production ofactive power and the absorption of reactive power by thesource. The use of the terms VARS IN and VARS OUT avoidsan explicit sign convention, but the quadrant diagramshows an implicit convention by relating the AmericanInstitute of Electrical Engineers sign convention for reactivepower to the Argand diagram. Thus, reactive power takenby an inductive load circuit (VARS IN) is in the +y directionand reactive power given by a capacitive load circuit (VARS OUT)is in the —j direction. Similarly, WATTS IN may be consideredpositive and WATTS OUT negative. It follows that in calculatingthe in-phase and quadrature components of apparent power, inorder to obtain reactive power with the correct sign, the currentterm must be conjugated as follows:

P + jQ= vl

An advantage inherent in the quadrant diagram and the useof the terms, VARS IN and VARS OUT, is that the behaviour ofthe synchronous machine as a generator or motor is seen to beconsistent. In either case, increase of excitation producesVARS OUT and decrease of excitation demands VARS IN.

It is not the purpose of the paper to justify the quadrantdiagram which is already in general use, but to show that acombined send-receive power circle diagram may be based uponit. Previous "universal" power circle diagrams have had touse the entire circle, but the method described in the paperplaces both sending- and receiving-end regions within the samesemicircle. This compactness has made possible the con-struction of a line calculating board based on the combineddiagram.

(2) THEORYThe phasor diagrams of Figs. 1 and 2 are drawn for current

in phase with the reference voltage, from the equationsEs = AEr + BIr and Er = DES — BJS. Taking A and D as(1 + j Z r / 9 0 + p), it can be shown that the lines OP and OQare at right-angles and that an increase in the power componentof current moves the extremities of Es or Er along OP, whileincrease in quadrature current moves them parallel to OQ.Thus, on a power diagram, OrP, OSP form the power axis andOrQ, OSQ the reactive-power axis.

(2.1) Receiving-end Power Circle DiagramFrom the above equations, the components of apparent power

are obtained in terms of voltages and impedances.

6=0

E2

(1)

From eqn. (1), the diagram of Fig. 4 is drawn for particularvalues of the sending-end voltage Esl and the transmissionangle dt. It will be seen, by comparing Figs. 1 and 4, that theBIr phasor lies along the active-power axis but that the reactive-power axis is in the opposite direction to that which mighthave been expected from Fig. 1. This change results from con-jugating Ir in order to comply with the convention that thereactive power due to an inductive load circuit (VARS IN) beconsidered positive. Sufficient accuracy for a diagram isobtained by the use of a nominal-n- arrangement which modifieseqn. (1) to

Watts in

Fig. 4.—Receiving-end power circle diagram.OF = Er, ON ~ En

In this equation Z is the transfer impedance Zt, with its angle p,,and the line admittance Y is assumed to be a pure susceptance.Then the intercept ORF on the reactive-power axis is \YE} andthe diagram is normally drawn by first setting up this distancefrom the power origin OR. The Es ( ~Er) circle then passesthrough F.

(2.2) Sending-end Power Circle DiagramThe sending-end components of apparent power are

(3)

or using the nominal-77- arrangement

Ps+jQs = EsIs^El/P-

From either of these equations, Fig. 5 is drawn for particularvalues of the receiving-end voltage Erl and the transmission

Wattsout

• (2) Fig. 5.—Sending-end power circle diagram.

206 CHARD: TRANSMISSION-LINE ESTIMATIONS BY COMBINED POWER CIRCLE DIAGRAMS

angle 6{. Comparison of Figs. 2 and 5 shows that if the phasordiagram is superimposed on the power circle diagram, the Blsphasor lies along the active-power axis. If a leading quadraturecomponent of current Isq is added in Fig. 2, the correspondingreceiving-end voltage is Er2 which shows VARS IN in the —jdirection on Fig. 5. This is the opposite direction to thatpreviously obtained for VARS IN in Fig. 4.

(2.3) The Combined Send-Receive Power Circle DiagramRotating the sending-end diagram (see Fig. 5) through 180°

will make the reactive-power axis correspond with that of thereceiving-end diagram (see Fig. 4) and will give WATTS OUT inthe negative direction on the power axis. The axes for bothsending- and receiving-end diagrams are now as shown in thequadrant diagram (Fig. 3). Eqns. (1) and (3) have a term ofcommon magnitude, Efir\Zt. Rotation of the sending-enddiagram through 180° also gives this common term the same angleto the reference (power) axis in both diagrams, since

E Fs r, if 0 = 0

This equality is the basis of a combined send-receive diagramhaving common circles and a common direction for zero trans-mission angle.

(3) THE COMBINED SEND-RECEIVE DIAGRAMThe combined diagram starts from the origin O (Fig. 6),

which is the same point as O in Figs. 4 and 5. The line OFG,

0=0,

Pr \in n^circlen circle

O

Fig. 6.—Combined send-receive diagram.For Er — UNIT voltage, n — Ef\\ErFor E, = UNIT voltage, n = Er\IEsE,i and Er\ are particular values of E, and Er

at an angle /Pt to the reference axis, gives the zero of angle onthe transmission-angle scale. The voltage circle through F maybe taken as UNIT volts and is the Es (=Er) circle of the receiving-end diagram. The voltage circle through G is then of radius(UNIT x EjJE}), where Esl is the chosen or determined valueof Es, and the intermediate circle is of radius (UNIT X Esl/Er).A number of circles whose radii are proportional to differentvalues of Es and E} are drawn. Alternatively, the circle throughG may be chosen to represent UNIT voltage, in which case thecircle through F represents (UNIT X E}JEj) and the inter-mediate circle is (UNIT X Erl/Es), where Erl is a particular valueof£"r.

The choice of a common origin for both sending- and receiving-end diagrams makes it inevitable that the active- and reactive-power axes, although parallel, are displaced for the sending- andreceiving-end quantities. The receiving-end reactive-power axisis the vertical through F, and the power-axis intersection is ata distance \YE} on the power scale, below F. Similarly, thesending-end reactive-power axis passes through G and the poweraxis intersects it at a distance \YE} vertically below G. Theload angle 9, which is common for both sending and receivingends, is measured clockwise from the zero line for the receivingend and counter-clockwise for the sending end.

(3.1) Method of Using the Combined DiagramGiven the receiving-end quantities, Er, Pr and Qri the diagram

(Fig. 6) is entered at OR, and Pr and Qr are set off along theappropriate axes. This gives Esl in terms of Er, which has beentaken as UNIT volts, n and also the load angle 6{ are noted,and a second circle of radius n2 times the unit value is chosen.The intersection of this circle with the 6 = 0 line gives the pointG, and a distance ^YE} is measured vertically below G to givethe sending-end power origin Os. From the intersection of then-circle and the line at an angle 6X to the load-angle zero in thecounter-clockwise direction (point K), perpendiculars aredropped to the active- and reactive-power axes, from which Psand Qs are read off. Alternatively, if the sending-end quantities,Es, Ps and Qs, are known, Es is taken as UNIT volts and thesending-end power scale is positioned by setting the origin Osat a distance WEjx vertically below the point G. Measuringoff the sending-end active and reactive power along theirrespective axes gives some point K on an intermediate circle,whose radius is nEs where n = ErlEs. This circle gives thevalue of Er, corresponding to the sending-end conditions, andthe transmission angle 62 is read from the direction of the lineOK. To estimate the receiving-end power conditions, the

771(

Reactive / Power' [\

11 ft\7 6 / 5 4 3 2 loutOli . Power i

Fig. 7.—Calculating-board settings for sending-end voltagedetermination.

CHARD: TRANSMISSION-LINE ESTIMATIONS BY COMBINED POWER CIRCLE DIAGRAMS 207

active-power axes must be repositioned according to the receiving-end diagram. This is done by selecting a circle correspondingto E}, which is the UNIT value times n2, The intersection of the6— O line with this circle gives the point F. Vertically belowF, at a distance %YE}, is the active-reactive-power origin OR.Setting the transmission angle 6 to 6{ gives the point N fromwhich the receiving-end active power Pr and reactive power Qrcan be read.

(3.2) Long-Line Calculating BoardHaving secured the advantages of common concentric circles

for both sending- and receiving-end diagrams, a calculatingboard can be devised in which the active- and reactive-powerscales can be positioned by engraving them on Perspex cursorswhich can be moved horizontally and vertically respectively. Athird cursor, pivoted at O, is set according to the line constantsfor the 6 ~ 0 position, and it reads transmission angles in thecounter-clockwise direction for "send," and in the clockwisedirection for "receive." The only calculation involved is thatof the vertical distances FOR = \YE2 and GOS = \YEl andthe power scale in terms of unit voltage. For a given transmissionline, tables of values can be prepared for these quantities, andthe reactive-power cursor can be offset by the required amount,according to its own scale. Examples of typical measurementsare given in Section 5, together with Fig. 7 on which the appro-priate calculating-board settings are shown.

(4) BIBLIOGRAPHYThe following selected bibliography is confined to those

references which deal with fundamental considerations relatingto the solution of transmission-line problems by diagrammaticmethods. The numerous references in which such methods areused have not been included.

(1) THIELEMANS, P.: "Calculs, diagrammes et regulation deslignes de transport d'energie a longue distance," RevueGenerate de UElectricite, 1920, 8, pp. 403, 435, 475 and515; and ibid., 1921, 9, pp. 451, 599, 675, 878 and 929.

(2) THIELEMANS, M. L.: "Calculs et diagrammes des lignes detransport de force a longue distance," Comptes Rendus,170, p. 1170.

(3) EVANS, R. D., and SELS, H. K.: "Circle Diagrams forTransmission Systems," Electric Journal, 1921,18, p. 530.

(4) HOLLADAY, C. H.: "A Graphic Method for the ExactSolution of Transmission Lines," Transactions of theAmerican I.E.E., 1922, 41, pp. 785.

(5) DAHL, O. G. C : "Electric Circuits: Theory and Applica-tions" (McGraw-Hill, New York, 1928), Vol. 1, Chap. X.

(6) WOODRUFF, L. F.: "Principles of Electric Power Trans-mission" (John Wiley, New York, 1938), Chap. VI.

(7) RISSIK, H.: "Power System Interconnection" (Pitman,1940), Chap. II.

(8) SCHWAGER, A. C, and WANG, P. Y.: "New Transmission-Line Diagrams," Transactions of the American I.E.E.,1945, 64, p. 610.

(9) American I.E.E. Subcommittee: "The Sign of ReactivePower," Electrical Engineering, 1946, 65, p. 512; andibid., 1948, 67, p. 49.

(10) KIMBARK, E. W.: "Electrical Transmission of Power andSignals" (John Wiley, New York, 1950), p. 205.

(11) GOODRICH, R. D.: "A Universal Power Circle Diagram,"Transactions of the American I.E.E., 1951, 70, p. 2042.

(12) MORTLOCK, J. R., and HUMPHREY DAVIES, M. W.: "PowerSystem Analysis" (Chapman and Hall, London, 1952),p. 247.

(5) APPENDIX

(5.1) Calculating-Board Estimations based on Section 3.1

The calculating board has three sets of circles. Black circlesare used for voltage determination and have radii from 10 to12-5 in power-scale units. For receiving-end determinationsthe inner black circle represents UNIT voltage, and so the circlesare labelled 1-0, 1 02, 1 04, etc., up to 1-25. For sending-enddeterminations the same set of black circles are used, but theouter circle is now UNIT voltage and the remaining circles arelabelled 0-992, 0-976, 0-96, etc., down to 0-8.

Red circles are used for «2-circles in sending-end power deter-minations. The first red circle coincides with the inner blackcircle, and the red circles are labelled 1 0,1 02,1-04, etc., up to1 • 25, as are the black circles. Green circles are used for w:-circlesin receiving-end power determinations. The outer green circlecoincides with the outer black circle, and they are labelled 1-0to 0-8 as are the black circles when used with a sending-endvoltage as UNIT volts. In the estimations which follow, theparticular circles used will therefore be distinguished by a colourand a number.

(5.2) Given Er, Pr and Qr, determine Es and Angle 6

Line Data: ZT= \(A/11'\ ohm.Yr=0 001083/ 90 mho.

Receiving-end quantities: Er = 220 kV = UNIT voltage.Pr = 1 3 0 M W I N .Qr = 42-7MVAriN.

Since the radius of the UNIT black circle is 10 power-scale£2

divisions, one division on the power scale = jj.r- — 29-5 MW.

} = 26-6, and so the VAr cursor must be set down adistance 26-6/29-5 = 0-9 division.

Procedure (see Fig. 7):(a) Set the angle cursor with O RECEIVE against 77 1° on the

/_P scale.(b) Find the intersection of the 6 — 0 line on the angle

cursor and the UNIT black circle (point F).(c) Set the active-power cursor with O IN line passing

through F.(d) Set the reactive-power cursor with the zero line 0-9

division below F (point OR).(e) Scale Pr = 4-41 divisions and Qr ~ 1-45 divisions on

their respective cursors (point N).if) Read results: 6 = 20°.

£ , = 1-23 Er= 270 kV.

(5.3) Determination of Ps and Qs under these ConditionsThe determination of Ps and Qs involves repositioning the

watt and VAr cursors and the use of the red circle whoseradius is 1 • 232 times the UNIT value. The power scale remainsas 1 division = 29-5 MW.

Procedure:(g) Set the angle cursor with O SEND against 77-1° on the £P

scale.Qi) Find the intersection of the 1-23 red circle and the

6 = 0 line (point G). Reset the watt cursor so that the zeroline passes through G.

0) \YEj = 40 = 40/29-5 =1-36 divisions on the powerscale. Set the VAr cursor with the zero line 1-36 divisionsbelow G (point Os).

208 CHARD: TRANSMISSION-LINE ESTIMATIONS BY COMBINED POWER CIRCLE DIAGRAMS

(k) Find the intersection of the 9 = 20 line and 1-23 blackcircle (point k).

(m) Read results: Ps = 4-9 = 144-5 MW OUT.QS = i • 1 = 32-4 MVAr OUT.

(5.4) Given ESi P, and Q,, determine Eri Pf, Qr and 6As in the previous examples, this is done in two stages by the

method given in Section 3.1. The first stage, corresponding toSection 5.1, gives ET and the angle 9, by using the outer blackcircle (radius 12-5 power scale divisions) as UNIT voltage, equal

to Es. Thus one division on the power scale becomes E}\\2-5Z.Repositioning the watt and VAr cursors and using the appro-priate green circle gives Pr and Qr.

As an indication of the possible accuracy of reading, after fourseparate determinations involving repositioning of the variouscursors, the receiving-end quantities were given as

Er = 221 kVP r = 131 M W I NQr = 41 MVAr IN

These figures compare with the initial values given in Section 5.1.