Power Circle Diagram

SUPPLY SECTION

TRANSMISSION-LINE ESTIMATIONS BY COMBINED

POWER CIRCLE DIAGRAMS

By F. DE LA C. CHARD, M.Sc, Member.

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

SUMMARY

// was published

Power circle diagrams drawn for either sending or receiving ends of a transmission line do not have a common centre for the voltage circles, nor are the power axes within the same semicircle. There is also a possibility of confusion over the sign of reactive power when conditions at both ends of the line are estimated by means of separate diagrams.

A quadrant diagram with current taken as reference, which com-plies with the convention of the American Institute of Electrical Engineers for reactive power, is shown to be a satisfactory base on which 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 sufficient accuracy for the estimation of system load conditions.

Examples of the use of the calculating board are given. LIST OF PRINCIPAL SYMBOLS A,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 the

receiving end of the line.

WATTS OUT, Po — Power supplied or produced by the source at

the sending end of the line.

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

VARS OUT, Qo — Reactive power produced at the sending or

receiving ends of the line by the source or load, i.e. that due to an inductive source or a capacitive load.

(1) INTRODUCTION

The power circle diagrams for sending and receiving ends of a line are derived from the corresponding voltage phasor diagrams (Figs. 1 and 2). These phasor diagrams are usually drawn with the reference quantities in the same direction, although they refer in one case to a source and in the other to a load. The derived power circle diagrams have different centres for the voltage circles, with a common active- and reactive-power axis. They can, however, use the same set of

Correspondence on Monographs is invited for consideration with a view to publication.

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 the sending-end diagram, are rotated through 180°. This axis reversal might be objectionable were it not justified by con-siderations of usage.

Power-system metering has long made use of the terms

"WATTS I N , " "WATTS OUT," "VARS I N " and "VARS OUT."

Decrease _Decrease

\

Increase

VArs, out

Fig. 3.—Quadrant diagram, in accordance with the current American I.E.E. sign convention for reactive power. These terms imply a quadrant diagram (see Fig. 3) in which current sent or received is shown on the horizontal axis and sending- and receiving-end voltages appear in two diagonally

opposite quadrants. Thus an inductive load demands WATTS IN

and VARS IN, which necessitates WATTS OUT and VARS OUT

from the source. A capacitive load absorbs active power and produces reactive power, which necessitates the production of active power and the absorption of reactive power by the

source. The use of the terms VARS IN and VARS OUT avoids

an explicit sign convention, but the quadrant diagram shows an implicit convention by relating the American Institute of Electrical Engineers sign convention for reactive power to the Argand diagram. Thus, reactive power taken by an inductive load circuit (VARS IN) is in the +y direction and reactive power given by a capacitive load circuit (VARS OUT)

is in the —j direction. Similarly, WATTS IN may be considered positive and WATTS OUT negative. It follows that in calculating the in-phase and quadrature components of apparent power, in order to obtain reactive power with the correct sign, the current term must be conjugated as follows:

P + jQ= vl

An advantage inherent in the quadrant diagram and the use of the terms, VARS IN and VARS OUT, is that the behaviour of the synchronous machine as a generator or motor is seen to be consistent. In either case, increase of excitation produces

VARS OUT and decrease of excitation demands VARS IN.

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

(2) THEORY

The phasor diagrams of Figs. 1 and 2 are drawn for current in phase with the reference voltage, from the equations

Es = 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 OQ are at right-angles and that an increase in the power component of current moves the extremities of Es or Er along OP, while

increase in quadrature current moves them parallel to OQ. Thus, on a power diagram, OrP, OSP form the power axis and

OrQ, OSQ the reactive-power axis.

(2.1) Receiving-end Power Circle Diagram

From 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 particular values of the sending-end voltage Esl and the transmission

angle dt. It will be seen, by comparing Figs. 1 and 4, that the

BIr phasor lies along the active-power axis but that the

reactive-power axis is in the opposite direction to that which might have been expected from Fig. 1. This change results from con-jugating Ir in order to comply with the convention that the

reactive power due to an inductive load circuit (VARS IN) be considered positive. Sufficient accuracy for a diagram is obtained by the use of a nominal-n- arrangement which modifies eqn. (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} and

the diagram is normally drawn by first setting up this distance from the power origin OR. The Es ( ~Er) circle then passes

through F.

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

(3)

or using the nominal-77- arrangement

P

+jQ

= E

I

^

l/P-From either of these equations, Fig. 5 is drawn for particular values of the receiving-end voltage Erl and the transmission

Watts out

angle 6{. Comparison of Figs. 2 and 5 shows that if the phasor

diagram is superimposed on the power circle diagram, the Bls

phasor lies along the active-power axis. If a leading quadrature component of current Isq is added in Fig. 2, the corresponding

receiving-end voltage is Er2 which shows VARS IN in the —j

direction on Fig. 5. This is the opposite direction to that

previously obtained for VARS IN in Fig. 4.

(2.3) The Combined Send-Receive Power Circle Diagram Rotating the sending-end diagram (see Fig. 5) through 180° will make the reactive-power axis correspond with that of the receiving-end diagram (see Fig. 4) and will give WATTS OUT in the negative direction on the power axis. The axes for both sending- and receiving-end diagrams are now as shown in the quadrant diagram (Fig. 3). Eqns. (1) and (3) have a term of common magnitude, Efir\Zt. Rotation of the sending-end

diagram through 180° also gives this common term the same angle to the reference (power) axis in both diagrams, since

E Fs r

, if 0 = 0 This equality is the basis of a combined send-receive diagram having common circles and a common direction for zero trans-mission angle.

(3) THE COMBINED SEND-RECEIVE DIAGRAM The 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^circle n circle

O

Fig. 6.—Combined send-receive diagram.

For Er — UNIT voltage, n — Ef\\Er

For E, = UNIT voltage, n = Er\IEs E,i and Er\ are particular values of E, and Er

at an angle /Pt to the reference axis, gives the zero of angle on the transmission-angle scale. The voltage circle through F may be 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 value

of Es, and the intermediate circle is of radius (UNIT X Esl/Er).

A number of circles whose radii are proportional to different values of Es and E} are drawn. Alternatively, the circle through

G may be chosen to represent UNIT voltage, in which case the circle through F represents (UNIT X E}JEj) and the inter-mediate circle is (UNIT X Erl/Es), where Erl is a particular value

of£"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- and receiving-end quantities. The receiving-end reactive-power axis is the vertical through F, and the power-axis intersection is at a distance \YE} on the power scale, below F. Similarly, the sending-end reactive-power axis passes through G and the power axis intersects it at a distance \YE} vertically below G. The load angle 9, which is common for both sending and receiving ends, is measured clockwise from the zero line for the receiving end and counter-clockwise for the sending end.

(3.1) Method of Using the Combined Diagram

Given the receiving-end quantities, Er, Pr and Qri the diagram

(Fig. 6) is entered at OR, and Pr and Qr are set off along the

appropriate axes. This gives Esl in terms of Er, which has been

taken 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 point G, and a distance ^YE} is measured vertically below G to give the sending-end power origin Os. From the intersection of the

n-circle and the line at an angle 6X to the load-angle zero in the

counter-clockwise direction (point K), perpendiculars are dropped to the active- and reactive-power axes, from which Ps

and Qs are read off. Alternatively, if the sending-end quantities,

Es, Ps and Qs, are known, Es is taken as UNIT volts and the

sending-end power scale is positioned by setting the origin Os

at a distance WEjx vertically below the point G. Measuring off the sending-end active and reactive power along their respective axes gives some point K on an intermediate circle, whose radius is nEs where n = ErlEs. This circle gives the

value of Er, corresponding to the sending-end conditions, and

the transmission angle 62 is read from the direction of the line

OK. To estimate the receiving-end power conditions, the 771(

Reactive / Power' [\

11 ft

\7 6 / 5 4 3 2 loutOl i . Power i

Fig. 7.—Calculating-board settings for sending-end voltage determination.

CHARD: TRANSMISSION-LINE ESTIMATIONS BY COMBINED POWER CIRCLE DIAGRAMS active-power axes must be repositioned according to the

receiving-end diagram. This is done by selecting a circle corresponding to E}, which is the UNIT value times n2, The intersection of the 6— O line with this circle gives the point F. Vertically below

F, at a distance %YE}, is the active-reactive-power origin OR.

Setting the transmission angle 6 to 6{ gives the point N from

which the receiving-end active power Pr and reactive power Qr

can be read.

(3.2) Long-Line Calculating Board

Having secured the advantages of common concentric circles for both sending- and receiving-end diagrams, a calculating board can be devised in which the active- and reactive-power scales can be positioned by engraving them on Perspex cursors which can be moved horizontally and vertically respectively. A third cursor, pivoted at O, is set according to the line constants for the 6 ~ 0 position, and it reads transmission angles in the counter-clockwise direction for "send," and in the clockwise direction for "receive." The only calculation involved is that of the vertical distances FOR = \YE2 and GOS = \YEl and

the power scale in terms of unit voltage. For a given transmission line, tables of values can be prepared for these quantities, and the reactive-power cursor can be offset by the required amount, according to its own scale. Examples of typical measurements are given in Section 5, together with Fig. 7 on which the appro-priate calculating-board settings are shown.

(4) BIBLIOGRAPHY

The following selected bibliography is confined to those references which deal with fundamental considerations relating to the solution of transmission-line problems by diagrammatic methods. The numerous references in which such methods are used have not been included.

(1) THIELEMANS, P.: "Calculs, diagrammes et regulation des lignes de transport d'energie a longue distance," Revue

Generate de UElectricite, 1920, 8, pp. 403, 435, 475 and

515; and ibid., 1921, 9, pp. 451, 599, 675, 878 and 929. (2) THIELEMANS, M. L.: "Calculs et diagrammes des lignes de

transport de force a longue distance," Comptes Rendus, 170, p. 1170.

(3) EVANS, R. D., and SELS, H. K.: "Circle Diagrams for

Transmission Systems," Electric Journal, 1921,18, p. 530. (4) HOLLADAY, C. H.: "A Graphic Method for the Exact

Solution of Transmission Lines," Transactions of the

American 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 Reactive Power," Electrical Engineering, 1946, 65, p. 512; and

ibid., 1948, 67, p. 49.

(10) KIMBARK, E. W.: "Electrical Transmission of Power and Signals" (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.: "Power System 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 circles are used for voltage determination and have radii from 10 to 12-5 in power-scale units. For receiving-end determinations the inner black circle represents UNIT voltage, and so the circles are labelled 1-0, 1 02, 1 04, etc., up to 1-25. For sending-end determinations the same set of black circles are used, but the outer circle is now UNIT voltage and the remaining circles are labelled 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 black circle, and the red circles are labelled 1 0,1 02,1-04, etc., up to 1 • 25, as are the black circles. Green circles are used for w:-circles in receiving-end power determinations. The outer green circle coincides with the outer black circle, and they are labelled 1-0 to 0-8 as are the black circles when used with a sending-end voltage as UNIT volts. In the estimations which follow, the particular circles used will therefore be distinguished by a colour and 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 a

distance 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 Conditions

The determination of Ps and Qs involves repositioning the

watt and VAr cursors and the use of the red circle whose radius is 1 • 232 times the UNIT value. The power scale remains as 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 zero

line passes through G.

0) \YEj = 40 = 40/29-5 = 1 - 3 6 divisions on the power scale. Set the VAr cursor with the zero line 1-36 divisions below G (point Os).

(k) Find the intersection of the 9 = 20 line and 1-23 black

circle (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 6

As in the previous examples, this is done in two stages by the method given in Section 3.1. The first stage, corresponding to Section 5.1, gives ET and the angle 9, by using the outer black

circle (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 four separate determinations involving repositioning of the various cursors, the receiving-end quantities were given as

Er = 221 kV Pr= 131 M W I N Qr = 41 MVAr IN