Power Circle Diagram1

7/27/2019 Power Circle Diagram1

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KojiOGusia , i05-D1 A1.t,i,,,,. CIYi=-zsinB, Y2==LIM,iww-B.t--Series impedance of the circuit, Z=:B.The forming of these equivalent n circuits is for convenience sake of inducingadmittance between temiinals and short-eircuited abmittance at each terminal will

be related later.' However, it is not the only way.Out from these equivalent n circuits, we can get yi,,, the driving abmittance

between i anb j terminals, as it is equivalent to the reciprocal of value of seriesimpedance of'the rr circuit, and yii also, the admittance at terminal i, will be gotby adding up all supplement values belonged to the terminal only: That is-.

'"'yij=i2} "vi,="2v(ilil--.yis'-),Gi-4----]i}).Besides, there are loads or generating powers to be tal


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I06' Po'werCircleDiagramofInterconnectedElectricPowerTransmissionSystem'The elements in the above equation will be described also in the form of so--

called power circle diagram constants by using generl circuit constants between

temiinals as follows: '' 11 111Yi2 = -B-,,, 3Yis == hZ/7I:.,''''''Zlm == B,,= llil'E, Y3i=: IZ'},,'''"'Pi2 .Z)13 b14 Pl,,-WnF= Bi2 + B!s + -Bi4 + ---- +Bin sAip. D23 Lb24 -D2nY22= Brl';1, + 2,B + B24 + '''''''''''' + ii}I,,

iy,,,,,, ,,,.,, -= "jll-i,, ii:,)- + i.`--;--ij:til -- ,ii-lii-g,li:)l- + l-ll--i:i'i E2-i-- + -

'D(rt])n''`''''''''' + J7'l(T+1)n In the matrix (2), XIsi, the terms upon a main diagonal, mean the short-

cirguited abmittance at the terminal, that is the admittance in'case of all otherterminals being short--circuited, excepting only theterm of i order, and of which thevalue will never become zero. Other admittances excepting that upon the diagonalmean the driving admittance between terminals, and of which the value betweenonly noncombinated terminals become zero. However, terms located at symetricalpositions of the diagonal are equal with each other, and terms in the upper side ofthe diagonal mean the sending driving abmittance and that of the lower side ,meanthe receiving driving admittance.

Equation (2) is shown with the current upit, but if we want it with the powerunit we must multiply each terminal current, sh6wn by a horizontal rank of thematrix, by E,A,, the conjugated value of each terminal voltage; that is- .--

' tt+P,,+iQ,,i + P,, -F j Q.--+ Prr +j Q,'r'''''''''

]z--"plltiU=-enEn-t=#"-aDudp=d=="-PaP=e-q=e--d-"'(Po+i)(o+i)j Qo+i)(,-+i))---------------------t-t---- (Pnn +j Q7i")''''''

lTIYn[eeii2,haYnM2.Ev"'''''''''I'''Y(r-i-i)-iee(r-ia)lscik,"'Yin-)UrozYi--Y21Y,Xlk,'Sl221.if2i2,-.I...........,.......,...,............i"--------i--------- . -----p------l-.:- - ----;-t------------- e----------i:== -:I..e.1[.:l.ll.1..i.li.i.1:'kll.1.liul..li.II..IY..,,.liAn=:llZTLItli-lh[]IEIIildll-]-liulnlt:lalgl-1[ta1"liiltliti[-liulvl:i"lu '''(3)


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KojiOGusH; .. 107- 11r-f-i)i ewilElr(,'-i-i)k, '''''''''"''l}f'(rI-i) (rH-i) lll'(rii)IE, ''''''''''''------t--E-----t----tj-Jt---,--------:1:-i---l-----l---.-----.--t-----b-1-.-

s-y,,ilfflriM.z,IY,,.isc.i2.-j-.If the network is based on toe constant voltage transmission system, namely,

the absolute value of every terminal voltage is constant, then every term upon themain diagonal of the above power matrix gets constant value and other terms areto have different phase angles individually. Accordingly, the vector locus formscircle with changes of load value, and we will have nany circle diagrams corres-ponding to each term. The number of the circles will be the twice of that ofointing transmission Iines between terminals.

The sending power circle diagrams are madewith the upper terms of the rnaindiagonal, and the receiving power circle diagrams are with the lower terms of thediagonal.

It the terminal of i orber is tal


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Koji OGvsul ]C9''ponding terms have contrasted power angle and the terms will become zero by

being cancelled if sending loss being negligible. The fourth term means real drivingpower actually sent from sending ends towards receiving ends. As was alreadyrelated,we may be able to attain the object by drawing each circle individuallybetween terminals, but it is pracially imposs' ible to draw a lot of circles on the same paper. So, author introduce here a different method that claims to draw developed circlesas shown by Figure 3. based on the equation (4). The total poweralso can be obtained easily if we have a gance at it. Basically speaking, howeve itis just an application of the orbinary circle diagram char.

Now, constant value of vector sum (yiiIZIii2 + y2?[.!ljT2 i2+ blrrlverl2)is drawn on a co-ordinate of total sending power (Pi +j Q,),and from the end of the vector all other vector tem].s ()12 sc2 Y!A,......etc.) are sub-tracted in the vectorical manner, and the end finally got will show the total sendingpower as shown by the equation (4). However, as power angles of Vector yi2E2Eik......etc. are unknown, they should be obtained by utilizing effective powersgiven by the power distribution diagrams. For instance, in case of yiL,E2E!lt, theequation of the circle between ] and 2 is (piL) l- j Qi2) == (yii2 + )-'i2)l.21il2 - yi2 Y2 -Zt]iA,and the circle center andthe radius are, respectively, (yii2 + yiL,)l-rciiL', (yi2 rc2 sciA)If we apply value P]!to the above equation, we can getjQiL, or power angle imm-ediately by figure. Utilizing this reason, we subtract the value -y]2.pa2 .Zt]iin ;::: (-yiiL,+yit) 1,ZDil2 +.iPi! + j QiL,. from the end of the sum ofshort-circuited power (yiil/ Ei:2---+y,,l"gV,iL'). Nain.ely, wesubtract a constant value(YiiL, + Yit)l.21ii2 first, vec--torically, and secondly, from the end of the vector toward a horizontal axis wemeasure Pi2, the effective power given by distribution diagram; thirdly, passingthis point draw a venical line then catch a point intersected by the line and thecircle centered on an end of (yiiliViLL' +-;+ yrrircri2) with radius yiL) -rc2Zilits Inthe same way we subtract yi:} Y,, .Zliic starting from the intersecting point found by theabove method now. Drawing of circles with radius lyi2 ZZI2 uZIiLI, lyi3 -lf3 Uihl,.-etcwill show ordinary power circle diagrams between terminals. If we utilize calcul-ation example, Figure 3a, in order to explain it in detail, the sum of short-circuitedpower of sending end is shown No.12345678917 and point 17 is the end.From this point, we subtract yg, i7sci7ue,:, namely, subtract vector r7r9; (yg, i7 ZC!7 +IYb, i7)l.2E]"l2, and get a point 9 first, and mal


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tlO PowerCircleDiagramofInterconnectedElectricPowerTransmissionSystemfrom other ends, even though it is actually a sending end. The same method isrepeated with all terms of the equation until the final point showing the totai sen--ding power is found. The final circle diagi'ams contain ordinary circle diagrams,and this shows not only the total sum but also the nature of their mutua! relations.So, distributing or mutual relations of power, voltage, power factor, etc. in inter-connected transfrtission system can be glanced at instantly by this. The second andthe third teum of equation (4) have only a difference of the power angle being con-trasted, and so one was determined the other wiil be found naturally.

Likewise, the total receiving power is the sum of power from (r +1) order to norder ef terminals in euqatiou (3); and its value is-

P2 + j Q2 :] (P(ri) (r+i) + j Q(r+i) (r+i) )'''''' + (Pnn + j Qnn)= (Y(r+i) (r+i)lswr+ii2''''''''''''''''''-'''''''''''-'''''tl- ]L,,tlswnb2)+(h-Zf(r+1)(r+2)-gg-i)+2lf(?i+1)ic'''''''''''"'"'''''-Y(r+1)Yn(r+1)in L

+ (-Y(r+2) (r+i) -scr+i 21Y(r+L)) ts-'''''''''`'''''''''' ny Zfn (r+i) Yr+i -Zl7nk)-l- (- Y(r+1) 1 Jrc1 pt(1+1) IL'''''''''''''''''''-' mx Ynl -ZVI venic

ny Y(r+1) 2 'rc2 'rc(r+1) ts'-''''`'''''''''`'''' lo Yn?' ver -rcnii)''`'''( 5 )The first tenn means the sum of receiving ends short-circuited power which

is constant value, the second and the third terms mean driving power which is sentor received between receiving ends by relay and has contrasted power angle hav-ing nature to cancel each other, and the fourth term means driving power actuallysent from sending ends to receiving ends. The ,fourth term is determined of itselfif the fourth term of equation (4) is given.

According to an usual practice, a receiving end power circle diagram is drawnwith its negative power value locating at positive side of coordinate. Accordingly,total sending power and receiving power can be' drawn on a same coordinate asshown by Figure 3a, b, and so it seems that an interconnected transmission systemwith many terminals works in a similar way with the well-1


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Koji O usmFig. Z. Power Distribution Diagram.

It shows tertninal voetage of each sJtation and effectivepower at each lipe, when the standard voltage is66kv.

65KV63KV nof


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l12 PowerCircleDiagramofdrawing as related before.Without receiving voltage, vol-tage can be determined alsoby using loads or power factor.However, this method, after all,is euqal with the ordinary cir-cle diagram method, so it maybe neglected here. On the otherhand, when the value of load- l-varles, lts vector moves on tothe circumference of the circleas the system is based on aconstant voltage system, andthe situation of linking ofcircles will vary introducingchanges of mutual relations ofelectric power distribution. Themethod shown here give us agreat convenience to clazify thedistribution of power and volt-age, but, it necesSitates rathercomplicated troubles because itis the same in principle with themethod of ordinary circle dia-gram. Hower, this method seemsto be an only method if we wantto show the variation conditionof load together with other cha-racters.'- .',

e Figure 3 a, b, are constru-cted only by ordinary circle-diagrams that'are developed by

Fig, 3, a. Power Circle Diagramat Sending end.Fig 3 a, b, indicate pow6r matrixfon

Tab. 1. Ma'in diagonal terms of matrixare indicated by indented ljne on .theleft side, oth.er terms by raflii of circlesarid power angles with.'obliqud li'nes,

Interconnected

o

Electric

1

10litol

L'OC

[llih

ge/111/ g


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'"'A

No. l 2 3 4 5 6' 7 8 9 lt 12 '13 14 15 16 l7 t1

({llili:il:+' kil)IE"2 i1- DIIt5oEioEt-i 1rljIIt5E13EIK

2 (:t3+:tlli9;)1E2I2 1"-D5TtE3E2K .

3 ' 1'rbllTtE-oE:iK, ({l:ti322f:tii+ue5:l33)E3l2 l--li5T4E4E3K ' 1-i;igltll3E13E:K

4 1-B43Eil3E4K (rkt3+ftt5s).i4Ep --iilligEsE4K.

5 --B-1 s4E4BsK As4-at.,IEsig ';

6 D616EEITtiE612 ( -Pllltg6E16E6x

7D7g-R-iT,lE,12 1l-E,EgE7K

I

8Ds17'at,,[Es:g

'I '--Pgltl7EwEsK

9110[1

' i'''1--E!EloKBIol'

1-q7E7EgK

IH'H"H'Dg17DgieAg7(ii-g7+E51t5o+Bgl7.-...t...1"ettigEgEioKiIEgi21'-BiioEioEgKI)---.---..--

l,(l{:i'gl:lfSi':le,9,tlllg9ilil)jEl,eli2'

-!tttttttttttttt"ttttt

i-B-ioi2Ei2EieK.t...t+..t.---t.....ttt...t.ttt-...... tttttttt

.

1--figlt17E17Eg

ttttt-tt

1'

Dni2i']IItli12IEIIi2 '-t"n2Ei!Eiik"rkti2E2E12K1

'

li'-B12ioEieEi2K/; -B-11z.EilE'?K; litli,+B.-lii,:iA.li-li)liE,,i21

"1 A131+ttss+D.t214)IE13i2l3 l--EtE13KBIsln

rt.

B131B133B1314s1-'RIIIItl4E14E13K

14 lil[il 1-i}-IItlilsE13E14K ({lllili`,i33t::,:i:`i,5stD-bu':l:NyEi4:2 .t1- i}IItlilsElsE14K 1'i]IItlE6E16E14K15

s

1-glitill14E14ElsK Als14'ent,4IEIsl2

16 1-blgTtE6E16K l.li-1 ' 1-GEtfi1E14E16K ' (Sill:,flll-X(/',)iEi6[2A17sA17g

!1'-glFtgEsE17K (st,-st,7)IEitrj2

Tab. 1. Power Matrix't shoyvs power distribution of network shown by Fig. 2. The main diagonal terms mean short-circuited powe {) other terms driving power o{ sending or recieving.


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Tab. II. Figures for Drawing Circle Diagrams.TerminalOrder Admittancemho 'Voltagekv EffectivePower'kw TerminalOrder Admittancemho Voltagekv EffectivePowerkw

l

Ei=:69.5 P;io=18700Pn3=13400Pn=32100'

'101=O.O0203-jO.O176Blol1=O.OO127-jO.Oli04Blogl=O.O0336-iO.02T5Blo12`

Alol=O.O023-jO.Oi752BlolAlog=O.OO127-jO.OI079

=O.O0336-:/O.02?44

Esol=57.s Plol==-tv'PIOg=-th.

Plo12=-.vPIOIO=--rv

B231 E2=65.

BlogDioRBIo12

2

.=O.O042-jO.OO?3eO.OO12--jO.O135B212D23=O.O042-1O.O0931B23L

DBilli,=o.oo]2-jo.o36s'

P23==-tvP212--t'nvP22=43))

'

111Bin2=O.OO05g-jo.oos4

Din2=o.oooss-jo.oojgs]3Bni2Ell=60 Pmz==-"vPim=-tv

.3' -L1'32B23i:fii,[glggg?i:lglgo,6,lsiA32=O. -jOO93B2P4 O.OP5j .O6B3D13 =.OO4- O.14133 E?,=r63 P32-H---vP34=--vP313=hnvP33--O

-----nm---

12

11=Bio..pB2i211=Blo.loBio1211=Bi2iiEm2

E12=57 P122=--.vP121e=:tNP12n=14)OOP1212==-5)OO

4

ll=B43B34' E4==64 P43=+AJP4s=-NP44=9000

v[...

/'-tl'.'

i f v " ' '

13

11=B131Bl13ll=B13BB3131B13,4=OOO185-jo.ols4A131B,3,=OOC254-jo.ols3gtl133=O.OOO14-.jO.O1239

=O.aO185--jO.O1844

E13=6I.2

'

P131==-.vP133=+nvP1314-----.v

P1313=L120005

ilt=Bs4B4sAs4=O.OO045-jO.O0652.Bs4 Er,=65lPs4==5160Pss=Ps4

B!33D1314B1314

'

61=O.OO135-jO.O048B616:i,i:-----O.OO135-jO.oot17s EG=66 P616=!9060

Pfi6=P616Bl"3E1314 E4==63 P1413="+t-v.'L

P1415=+.vP1416=-NP14!4=-2300

7'1=O.OO036-jO.O0884B7g

Zll:=OOO036-jo.oos713E7=60 P7g=]5800

P77=P7g14'

==C).OO047-jO.O085

=o.eol2-jo.oos2B1416A1413=O.OO185-jO.O1844B1413D141s'=O.OO047-jO.O0851B141sD1416=O.OO12-jO.O0521

Est=61 Bl"68 =O.OO0312-JO.O040BsnDs17=O.OO0312--jO.O0397 Ps17=I0500.Pss=Psl7 ]5 il=.BlslsBl"sAls14.= EIi,=62.. Pls14=-400Plsls=-Pls14Eg=59.6

9

' =Bg7B7g1v-O.OO)21-jO.Oll04Bgie1=O.OO041-jO.O0267Bg17Ag7=o.ooo36-j'o.oogsg6Bg7iBJ:l:Lo.oo127-.jo.oo:o7g

LPS7=-A.Pg}o=+.vPgl7=-3500Pgg=20400 16

11=B166B616-1=Bi6nBi4i6 E16=63 P166==-.vP1614=+.vP1616=-2500Dg17= O.OO041 - j O.O0237Bg17

17

1Bi7s1B17g!a178BnsA17gB17g

-1Bsm1Bg17== O.OO03i2- j O.O0402== O.OO041 - j O.O0264

En = 59 P17s ==P17g ==P 1717 ==

+rvN.Nt

Eoot-Note. Pno stands for effective sending powe' r from terminal 1 fo terminal 10, Pn for totalmeans sending, sign-gecieving, sign--that the figuFe is determined by circle diagram.t t. tttvtt'' tttT '. .t t

power on terminal 1, and it is the same with the following correspondingly. Sign-


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t

power matrix, but some 6xpl-anations will benecessary tobe given as author fear theyare not well-l


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t14 PowerCircleDiagramoflnterconnectedElectricPowerTransmissionSystem'' 'reactive power. Centerpoint of the circle is obtained by transitting in parallel the

vector l, 1'3, a term in the main diagona! of power matrix. Radius of the circle is'btained from the Table II.. 'The point of action 13 is determined by radius of circle and effective power

which is obtained from powerdistribution diagrams or mutual relation ofthis c'ir--cle group. Circle (2, 3) and circle (3, 2) show that they have'a contrast powerangle each other, and that a sending end terminal 3 is the one receiving powerfrom other terminais by relay. .'The more transmission system becomes complicated, the more increases thenumber of circles, therefore proper dividing becomes necessary in this case. In theFigure 3 a, b, there exist circle (i, j) and circle (j, i) which have an equal radiusvalue with different center distances, but, these circles respectively correspond toterm (i, j) of the upper side of main diagonal and teim (j, i) of the lower side.

(2) Sum of short-circuited power:It is shown on left side of the Figure 3 bs an indented line, of which com-

poiients are vectors to indicate circle centers, and the component's slant meansthevector angle of short-circuited admittance (yitf + yid). With an increase of line res-istance, the angle of slant becomes wider, which means an increase of line loss.In case of a System which has an ideal, line loss should be consrant in every partof the system and the indented line becomes straight. In Figure 3 a, most partof line is straight except parts 2L, 3; 3, 2J 6, l6. ,(3) Total transmission efficiency:

The effective components of vector O, ld and blrml' in the Figure 3 a, b, arethe total generating power and the total receiving power, hereupon: --

99,OOOKWTotal transmission efficiency: = ntt';bb-oKw 100 =-- 85%(4-) Phase improving condenser: 'The total receiving power is 99,OOO KW and its reactive power is 45,OOO KW

as shown by vector ' O;1. Even if a condenser fitted for this figure was prepared,it is unsufficient because of iregularity of current phases at each terminal,namely,becauseof Iagging or Ieading which differs in reality. we take an example on circle (10,1) of Figure 3, b. Cood efficiency can be got for electricity sending between ter--ininals when the power factor is improved, that is approximately l. Therefore, ifwe place a condenser onto a receiving end, terminal IO, and the voltage on terminal10 is left as it is, sending voltageon terminal 1 is to become low and transmissionefficiency becomes higher, It can be seen also from the Figure 3a, namely, povLierfactor becomes better owjng to sending voltage dropped. However, in case the rec--

fiving voitage on terminal 10 got dropped and is desired to be raised, and voltage


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Koji OGus}i] ,' 115'on terminal 1 is left as it is, receiving voltage on terminal 10 bccomcs high by the

conclenser, and so, power factor, coi]sequently transmission efficiency will be impr-oved. This is understood from circle (10, 1) and (l, IO). On the other hand,however, the above treatment spoils terrr}inal 9 kecause ExcEssive leading currEnt isflowed out irom this terminal, which causcs rcduction of tran.cmission efficicncy asit is julgcd from circle (10, 9). To raise this tcirmiiral x'oltase is :ot casy as thet--eirnjnal is a rE]ayc{] {lcraatii g Etatkn, Ac(ci'c:5r9y,ttuc rrv.ct o;iEt[uiic"tle"valucsof condenser capacity lit for jrr]prov(rrcnt of tlie ro",Er factors. To Eolve thjsdiffic.ulty, we must tal

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Power Circle Diagram1

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