Instructions for use

Title Power circle diagram of interconnected electric power transmission system

Author(s) Ogushi, Koji

Citation Memoirs of the Faculty of Engineering, Hokkaido University, 8(3-1), 104-115

Issue Date 1950-12-20

Doc URL http://hdl.handle.net/2115/37758

Type bulletin (article)

File Information 8(3-1)_104-115.pdf

Hokkaido University Collection of Scholarly and Academic Papers : HUSCAP

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TifansgERfissiowa Syseewa

Outline :

Generally

transfomed into an equivalent cir-

cuit as shown by

has many

mutually ,connected,

generating stations or transfomihg

stations. In order to solve such

a network,,the,value of impe-

dance Qf everybranch line, which,

connects stations with each other, lj' 'and,the value of every transformer

shoued be converted into the value

of standard voltage,'and the value:

of equivalent T circuits between

every two terminals should be

obtained separately.

The equivalent n circuit bet-

ween terminals can be described

as follows when general electric

Admittance on both legs of

''` KojiO(}(]s}]t'' ' .t t ' 1 -･ , (ReceivedDec.26,1949.)

Expressing power distribution in interconnected electric transmission

system with many power stations and substations of different voltages

by using matrix, it is shown that circle diagram of this system is given

the main'diagonal tegms of matrix as centers and the other terms as

tadii., A humerical example is taken from the data of the main high

'tension transmission ,lipe of Hol<kaido distriet.

. 1. General Eq"atipn Or Matrix ef Power Bistrihtitioft ln

I･', lnterconneGtedTransAiissiollSystem''･/':. i ' speaking, any l<ip,d of.electric power transmission,nftwork ca,n be

b,..,,,",'g.".r,e ,i,,.W,h.'.Cig //'/Y'iiigi'i7it,Xx,,

acting'asreal )K£4`I'AI'Lff. ypt' /t7 >N'Z'vl･itYIt,t.2.... 2Y2 ',

au YR"i', n;/

I 'i'Y,

di,Y ;:･'

d' X<,i,

-×

lt

NN ' '' t

N 't, × 'i7"lekhlz,.

Nt 1> "' lt ptN<r-･.11.l.pt"l N

ttV"""'N 'Ns

-taN. N NN N ';-',,, '5s･s NS s.- rcr+1

× uarrn

× :. The transmlssom 1-r 1) -n

are

L;lilyrl ･-i 1,

Fig. equivalent cted networl<. Terminal mean {r- recrevmgtransmlsslon constants

the circuit,

' .e2! 1lit3 1,

fv,,,2 z- ly

3 33

pmw '

./

,

,

/,

･

r"tkllrlLti..f:s'!t./･/

circuit of lnterconne-

sending ends, terminalends.

used :

KojiOGusia , i05

-D1 A･1 .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 inducing

admittance 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 series

impedance of'the rr circuit, and yii also, the admittance at terminal i, will be got

by 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<en into consideration.

ThereforeitwillbecomeasshownbyFigure1. ' '' Getting vei, .re2, ･･･-･･fl, stand for' voltage of each tenninal, terminal suffix 1･t･･`･

r for sending elements and their currents for +Ii, I2, ･-･+' fr; terrninal suffix(r+i)

･･･

ri for receiving elements and their currents --Ir.i -.Z..2, ･････--I,,respectivelY,

.we get the following equation according to the Kirchhoff's law.

' + L = Yii rei + Yi2(rci " eqi)l"''''''''''''''''''''''''''''''+ Yin (-sci pt -rch)

+ Jli = Y2i(pt2 - sci) + Y22-E2 +''･'''i'''''''''''''-'''''''-'+ Y2,i(-lf2 --'rcn)

+ I,, = y,,1(ve,, - Isc1) -l-- ････-････i･･････--････i･･L-･ii･･i･･･i･･-l- Y,,,}(rc,, - rc7i)

h I,+i = Y(r+i)i (j(Z7r-T -'-,'ani) + Y(v+i) (r+o ZDr+i'''''' + Yo･+i)" (-Zag･-i - En)

ki L･t2 ::: Y<o･.2)i(]ll,･+2 - Wi) "F '''''''''''-'''''''''''''-'+?Y(?･･eo-)"(Z4i-i-2--eq,,)

tt tlt

-Lz--Yni(Z4i-Yi)+'-''''''''''''''''''''''''''''''''''''`''''''''""J''+Yoi･n,Iii, (l)

The nature of transmission network will be explained more clearly when we

describe the above equation by using matrix as follows:

" t..-,il" -be,Y,iE--.",r,.liiiilli.ili".TY.lfi.'tl'..111111ili:Zlll "[Il'l"i.')･

i" i-------------------i---------S---------O-"---------J-------l / -

-l--][;" -!I,'i''''J''''-''''''''Yr,'i･''''''`''''''''''''''nv3V)･,n rZl,,

=Dv""n=e"ii-tn-ulldRqq"p-n""uDa===w=-g=["mn==]=-M""a=,:tdt-t"md"!ll-gnnueed=d=Nmvq"vhvt"ecnstitm"onEmnDsed-× -i-(2)

lh I;,+1

- L,.f

- L,

But

"--

y(,･+i)i･･ny-"･･･-･･･-････-i':Y(.+1)(..)

i-Y(r+L))1'''-''''''"'''''''':'''':r,'''･･･i

---+-----------------r---T--4-i-----T--l----

l pt "ke Y e?.1 '''''''''''''''''' '''I'''''''''''''

lYii == yii + ey,i + zl,L,-･･J････-････-

･･･ "-' Y(r+i)n

-4i------4-Tt- :- :,

--t----t--t ISil ?tOl, f

''`'''''''''' + ?lioi

.ZIJr+1

IEIr ･f L'

: pa't I

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: ' ' 11 111 Yi2 = -B-,,, 3Yis == hZ/7I:.,''''''Zlm == B,,= llil'E, Y3i=: IZ'},,'''"'

Pi2 .Z)13 b14 Pl,, -WnF= Bi2 + B!s + -Bi4 + -･･-･-- + Bin s

Aip. D23 Lb24 -D2n Y22= Brl';1, + 2,B + B24 + '''''''''''' + ･ii}I,,

iy,,,,,, ,,,.,, -= "jll-i,, ii:,)- + i.`--;--ij:til --･ ･,････ii-li･i-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 other

terminals being short--circuited, excepting only theterm of i order, and of which the

value will never become zero. Other admittances excepting that upon the diagonal

mean the driving admittance between terminals, and of which the value between

only noncombinated terminals become zero. However, terms located at symetrical

positions of the diagonal are equal with each other, and terms in the upper side of

the diagonal mean the sending driving abmittance and that of the lower side ,mean

the receiving driving admittance.

Equation (2) is shown with the current upit, but if we want it with the power

unit we must multiply each terminal current, sh6wn by a horizontal rank of the

matrix, 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")''''''

l TIYn[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..,,.liA･n=:llZTLItli-lh[]IEIIildll-]-liulnlt:lalgl-1[ta1"liiltliti[-liulvl:i"lu '''(3)

KojiOGusH; ･ .. 107

- 11r-f-i)i ewilElr(,'-i-i)k, '''''''''"''l}f'(r･I-i) (rH-i) lll'(rii)IE, ''''''''''''

------t--E-----t----tj-Jt---,--------:1:-i---l-----l---.-----.--t-----b-1-.-

s -y,,ilfflriM.z,･･････････････････I･･･････････････Y,,.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 the

main diagonal of the above power matrix gets constant value and other terms are

to have different phase angles individually. Accordingly, the vector locus forms

circle 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 of

ointing transmission Iines between terminals.

The sending power circle diagrams are made･with the upper terms of the rnain

diagonal, and the receiving power circle diagrams are with the lower terms of the

diagonal.

It the terminal of i orber is tal<en for an example, the totai generating power

Pn +j Qn = Yii1Yij2-" Yi2 ]r2 -iecik " Yi:i Y:i -ifu:-'''''''''''- Yin ]rn IMIJin･･

which is shown on the first line of the equation (3). However, frorr) this terminai,

power is branched out to the temiinals connected, so the above equation can be

shown also as follows, havlng divided Yi,T to each coinponent. -

(Pi2 + j Qi2) + (Pi3 + j Qi:J +''''''`''''''-'''''''''''''''''''t+ '(Pin + j Qin)

={(Yii2 + yi2)i]ilii/2 - y,2 -Zl3 Zl,n,} '

' +{(Yi,,, + Yi3)1Mil?' - .sc,,, ]l,t, Zir,k}

+{ (.?/i!n -P ?)gin)lJEI]il2 im Yin rcn -Zt]}･n}

If this equation is shown with circle diagram constants which are used in most

cases,

i (Pi2 +j Qi2) +(Pi:i +j Qif:) +"'''';'''+ (Pint +j Qin) ' s - ('a"tir,1EiL?' - Ii'l{s. sc?, -E'iA) -- (tJ-],';,i-gDii2 -･]Ll{,, B, rc,,) +･･･.･･･-･･i･-

････････････-F ([ft--?-1'/tl .sc,I? - tbe,. .!ll,,, .gi,in),

The right side and the left side o'f this equation contain many terms contrasted

with Gach other, and adinittances appeared on the rjght side are obtained by using

equivalent T 'circuit constants as was related before. As for the first term, it is

explained as follows. ' ' ' ' (P,, + j Q,,) =: (y,,,,, + y,,) i, ]rc,I2 - y,, ve,) pa,,.

This shows power which is sent to terminal 2 from terminal l. As is based on

IOg PowereircleDiagramoflnterconnectedElectricPowerTransmissionSystem

'the constant voltage transmission system, the factor changeable due to the load change

is only phase angle or "power angle" of voltage. And (yii2 +yi2)IIeii2,!yi2sc2Yi l,

show center and radius oi circle respectively, which means the well-known sending

end circle. The same discourse can be made for many other terms 'in like manner,

andmanycirclediagramsaremade. ' However', we are discussing these theories always basing upon the transmissionnetwork, and it will mal<e us notice thefollowing fact.Namely, the circle diagrams

of the terms located on symetrical positions of the main diagonal of matrix (3) have

equal radius value and contrasted power angles. For an example, -yi2Y2veikand

-- yL)i Ei E2,: have equal radius value but contrasted with,each other as to their power

angles. The adove fact shows us that a half of the total number of circles

will be.enough to determine all diagrams we want. This can be utilized in case

we map out the diagrams altogether. While, power and voltage in the transmission

networl< are recorded usually on the power distribution diagrams in generating sta-

tions or transformimg stations. These recorded voltages and line constants which

are obtained separately will supply enough values needed for equation (3) to draw

every power diagram. And points of action on the diagrams can be determined by

effective sending power or otherwise by effective receiving power obtained from

power distribution diagrams. Instead of receving voltage, power factor or load adm-

ittance may be applied. . The above explanation related up to now is the same with that of the ordinary

power diagram theories, however, the use of matrix will explain mutual relations

'as an interconnected transmission system. 'In next chaPter, will be related method

of getting the total sending power and total receiving power. ･

' 2. Total Sefieling and Receiviiig Power a"rl Power Circie OiagrarR

' of lnterconwected Traftsmisson System

' Total sending power is the' sum of power from 1 order to r order of sending

end terminal shown by equation (3); and

Pi +j Qi == (Pn +j Qii) + (P22 +j Q2L,) +''''''''''''+ (P"- +j Qrr)

== (ynlZV"L' + Er22Ipt2i2 +･-･･････････-････-+ Y,,,,lsw,,l2)

+ (- Yi?. Zt]L, reu; - X3rc:i M7vc''''-'''''''-'' -"- Yir JZIr rctk)

+ (-Y2i ewi itL･ - Y:nMi sc:{k'''''''''''''''in ?)f,･iIZIi.ZZ7rA･)

+(hYi(r-n)JE,tpilllii,･''''''''''''''t'''-YinliZlrnrcu: '

ny ?12 (r+i) -scr+1 ve2in'''''''''''''''''''''--' = ?1in, -pmn -rcrk) '''"'' ( 4 )

The first term of the right side of the equation means short circuited

power at each sending end. The second and the third term means driving power

which is relayed and sent or received between each sending end, and every corres-

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 driving

power actually sent from sending ends towards receiving ends. As was already

related,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 s

ame paper. So, author introduce here a different method that claims to draw deve

loped circlesas shown by Figure 3. based on the equation (4). The total powera

lso can be obtained easily if we have a gance at it. Basically speaking, howeve it

is 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 sending

power as shown by the equation･ (4･). However, as power angles of Vector yi2E2

Eik......etc. are unknown, they should be obtained by utilizing effective powers

given by the power distribution diagrams. For instance, in case of yiL,E2E!lt, the

equation 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 we

measure Pi2, the effective power given by distribution diagram; thirdly, passing

this point draw a venical line then catch a point intersected by the line and the

circle centered on an end of (yiiliViLL' +･-･･;･+ yrrircri2) with radius yiL) -rc2Zilits･ In

the same way we subtract yi:} Y,, .Zliic starting from the intersecting point found by the

above method now. Drawing of circles with radius lyi2 ZZI2 uZIiLI, lyi3 -lf3 Uihl,.-････etc･

will 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-circuited

power 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<e a vertical line at a place where the

effective power is (Pg,i7) which is obtained from distribution diagrams, with the

point 9 as its starting, point. The vertical line then intersects with circie (9, 17) of

radius (yg, i7 .Zl7g vei7) at the point 17, which shows the point of action. In the same

way we draw a circle (9, 7) by starting from the point 17. If effective power P

on distribution diagrams takes a negative sign, it means reiayed power reception

tlO PowerCircleDiagramofInterconnectedElectricPowerTransmissionSystem

from other ends, even though it is actually a sending end. The same method is

repeated 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 and

the 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 n

order ef terminals in euqatiou (3); and its value is-

P2 + j Q2 :] (P(r÷i) (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 sent

or 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 actually

sent from sending ends to receiving ends. The ,fourth term is determined of itself

if the fourth term of equation (4) is given.

According to an usual practice, a receiving end power circle diagram is･ drawn

with its negative power value locating at positive side of coordinate. Accordingly,

total sending power and receiving power can be' drawn on a same coordinate as

shown by Figure 3a, b, and so it seems that an interconnected transmission system

with many terminals works in a similar way with the well-1<nown diagram of

simple transmission line. It immediately shows the total sending efficiency and the

total loss.

3. Example of Calc"Iation

The main part of power ･transmission system ot high tension in Hokkaido Dis-

trict is simplified as shown by Fig. 2. Terminal 1 --9 mean generating stations

and 10-17 mean transforming stations. The power matrix of this system is shown

by Table I. when calculated using the equation (3). ･･ Short-circuited admittance, driving admittance and line constants of equivalent

ic circuit between terminals of this system are shown by Table II together with

voltage and power of distribution diagram of Fig. 2.

t

Koji O usm

Fig. Z. Power Distribution Diagram.

It shows tertninal voetage of each sJtation and effective

power at each lipe, when the standard voltage is66kv.

65KV 63KV n of<w.' S7-.EiVoooKla, 61.2KV 43oo 12 14oooKwgliO,Z08i,X,S3 ] Kw .i/f@6oKv

' R S160KW6st<ve'"'-2300K'

W63KV

>

-40ool<w

62Kv 1 S

Generating 1. Uryu. 2. Ebetsu･ 3. Kariki, 4. Moiwa. 5. Jozankei･

6.Kombu. t 7. I<amikawa. 8. Antaruma. 9. Eoroshi.

Using this table, each term of

and so sending power circle diagram

diagram by equation (5) can be

before. The drawing of circle diagrams

cuit between terminals, absolute valne of voltages, and also effective powers

as much as branch lines which connect

ssary in case of drawing a simple

when interconncted system is drawn

mple, there are 17 branch lines,

described on the power distribution

power and reactive power will

14

-2SOOKw x- -16 ' 63K 19060KW

The number and transforming the places

Station･

Nl

' 10 57SKV "" ' ' Xx. t8700KW , ",,i 'h"'kh >---4 6g.sKv

i' i3tl-OOKW t" tin"' ""ss..:"--S"[ltli3s"oo'KEwo926iLoKoVKw

' 17 sgKv

v' lo5oOKW e 6tKV e 15800KW ' 66KV (SOKV 'n.eans generatirig station

stations locoted at

shown below

Transforrning Station.

10. Sunagawa. Il. Coal mines. 12. Bibai. ' I3. Sapporo. 14. 0taru. 15. Nagahama. i 16. Yoichi. IZ. Asahigawa.

' tt t ' the power matrix of Table I can be calculated,

by equation (4) and receiving power circie

drawn as Figure 3 a, b, by the method related

needs value of admittance of every rr cir-

numbering

terminals. The effective power is unnece-

transmission line system, but becomes necessary

by･the above manner. In this calculation exa-

so 17 effective power will be necessary as

diagram. The points of action of other effective

be got spontaneously on their diagrams by this

･

l12 PowerCircleDiagramof

drawing as related before.

Without receiving voltage, vol-

tage can be determined also

by using loads or power factor.

However, this method, after all,

is euqal with the ordinary cir-

cle diagram method, so it may

be neglected here. On the other

hand, when the value of load

- l- varles, lts vector moves on to

the circumference of the circle

as the system is based on a

constant voltage system, and

the situation of linking of

circles will vary introducing

changes of mutual relations of

electric power distribution. The

method shown here give us a

great convenience to clazify the

distribution of power and volt-

age, but, it necesSitates rather

complicated troubles because it

is the same in principle with the

method of ordinary circle dia-

gram. Hower, this method seems

to be an only method if we want

to show the variation condition

of 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 Diagram

at Sending end. Fig 3 a, b, indicate pow6r matrixfon

Tab. 1. Ma'in diagonal terms of matrix

are indicated by indented ljne on .the

left side, oth.er terms by raflii of circles

arid power angles with.'obliqud li'nes,

Interconnected

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7Z

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3

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le

x Z･1

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IS

4 ' 16 1/.

, 49/,,'

B2S!Jr t

p z ..w lolo

97 t2 11

lt'tupv

l

x1-IO

1if1 J

･ Z3X3 ･2?

j')"

4

6,

%'17

's

5

3

4

3･1 .3

S･"

3

4･3

g6L16

tt

8･1 7

9･17iCJi

4 4･S

i6 s･4

7･9

!

't"'A

No. l 2 3 4 5 6' 7 8 9 lt 12 '13 14 15 16 l7t

1

({llili:il:+' kil)IE"2i1-

DIIt5oEioEt-i1rljIIt5E13EIK

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

.

3 '

1'rbllTtE-oE:iK, ({l:ti322f:tii+ue5:l33)･E3l2 l--li5T4E4E3K'

1-i;igltll3E13E:K

41-B43Eil3E4K (rkt3+ftt5s).i4Ep

--iilligEsE4K

.

5 --B-1

s4E4BsKAs4-at.,IEsig '

;

6D616EEITtiE612 (

-Pllltg6E･16E6x

7

D7g-R-

iT,lE,121l-E,EgE7K

I

8

Ds17'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'i]IItli12IEIIi2'-t"n2Ei!Eiik

"rkti2E2E12K1

'

lil'-B12ioEieEi2K/;-B-11z.EilE'?K;

litli,+B.-lii,:iA.li-li)liE,,i2

1

"1

A131+ttss+D.t214)IE13i2

l3l--EtE13KBIsln

rt.

B131B133B1314s

1-'RIIIItl4E14E13K

14l･il[il 1-i}-IItlilsE13E14K ({lllili`,i33t::,:i:`i,5stD-bu':l:NyEi4:2

.t

･1-i}IItlilsElsE14K

1'i]IItlE6E16E14K

15s

1-glitill14E14ElsK Als14'ent,4IEIsl2

161-blgTtE6E16K l.li-1 '

1-GEtfi1E14E16K'

(Sill:,flll-X(/',)iEi6[2

A17sA17g

!

1'-glFtgEsE17K (st,-st,7)IEitrj2･

Tab. 1. Power Matrix 'It 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.

Tab. II. Figures for Drawing Circle Diagrams.

TerminalOrder Admittancemho

'Voltagekv

EffectivePower'kw

TerminalOrder Admittancemho Voltage

kvEffectivePower

kw

l

Ei=:69.5 P;io=18700

Pn3=13400

Pn=32100'

'10

1=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=-.v

PIOIO=--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==-tv

P212--t'nv

P22=43))

'

11

1Bin2=O.OO05g-jo.oos4

Din2=o.oooss-jo.oojgs]3Bni2

Ell=60 Pmz==-"v

Pim=-tv

.3' -L1'B32B23li･:fii,[glggg?i:lglgo,6,lsiA32=･O.O042-jO.O0931B32P34=O.OOOP5-jO.O063B34D313==O.OOO14-jO.O1241B313

E?,=r63 P32-H---v

P34=--v

P313=hnv

P33--O

-----nm---

12

11=Bio..pB2i211=

Blo.loBio1211=

Bi2iiEm2

E12=57 P122=--.v

P121e=:tN

P12n=14)OO

P1212==-5)OO

4

ll=B43B34' E4==64 P43=+AJ

P4s=-NP44=9000

v[...

/'-tl'.'

ifv:"/''

13

11=B131Bl13ll=

B13BB3131B13,4=O･OO185-jo.ols4

A131B,3,=O･OC254-jo.ols3gtl133=O.OOO14-.jO.O1239

=O.aO185--jO.O1844

E13=6I.2

'

P131==-.v

P133=+nv

P1314-----.v

P1313=L12000･

5

ilt=Bs4B4sAs4=O.OO045-jO.O0652.Bs4

Er,=65lPs4==5160Pss=Ps4

B!33D1314

B1314

'

6

1=O.OO135-jO.O048B616:i,i:-----･O.OO135-jO.oot17s

･EG=66 P616=!9060

Pfi6=P616

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

P1415=+.v

P1416=-N

P14!4=-23007

'1=O.OO036-jO.O0884B7g

Zll:=O･OO036-jo.oos713

E7=60 P7g=]5800

P77=P7g

14'

==C).OO047-jO.O085

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

Est=61Bl"6

8

=O.OO0312-JO.O040BsnDs17=O.OO0312--jO.O0397 Ps17=I0500

.Pss=Psl7]5

il=.BlslsBl"sAls14.=

EIi,=62.. Pls14=-400

Plsls=-Pls14

Eg=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=+.v

Pgl7=-3500

Pgg=2040016

11=

B166B616-11=Bi6nBi4i6

E16=63 P166==-.v

P1614=+.v

P1616=-2500

Dg17 = O.OO041 - j O.O0237Bg17

17

1Bi7s 1B17g!a178

BnsA17g

B17g

-1 Bsm 1 Bg17== O.OO03i2- j O.O0402

== O.OO041 - j O.O0264

En = 59 P17s ==

P17g ==

P 1717 ==

+

rv

N.Nt

Eoot-Note. Pno stands for effective sending powe'r from terminal 1 fo terminal 10, Pn for total

means sending, sign-gecieving, sign--that the figuFe is determined by circle diagram.

t t. tttvtt ' ' tt tT ' . .t t t .tt .t '

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

t

power matrix, but some 6xpl-

anations will benecessary to

be given as author fear they

are not well-l<nown yet: -

(1) Circle Group:

Sign (1, 10), (1, 13),

etc. in Figure 3 a. means a

sending end power circle from

sending end 1 to receiving iend 10. Power angle with

oblique lines means short-

circuited power betwen termeny

inals, of which･ the one situated

on the left side of the circle

means electric receiving by

relay from other terminals

though is originaliy a sending

end and that situated on the

right side of the circle means

electric sending As for the

circle (1, 13), for instance,

it should be situated on the

origin of coordinate (Pi,jQi),

but the same consequence will

be got if the origin is tran--

sformed to point 1, of which

the coordinate is named (Pi.

,,,j Q,, ,,).

Horizontal axis takes effec-

tive power and vertical axis

Fig. 3, b. Power Circle Diagram at R.ecieving end.

Circles {IO, 1), (I2, 10j, etc. ireans

a circle recieved from terminal 1

to terminal 10, etc, Total reci.eving

power is shown by effective comp-

nent of veetor O.1,ai;d total recie-

ving powey by et'l'ective component

ofyector 1,10 on Fig.3 a.

Koji ,

t7,'9

17J8 ,9e,

t7t' 1

S6 A

4 i6

s

m

g

OGUSHI

16･o

616･14

615･14

7eL,,,

,,t"

l:l,l 614,1s

14IS.

rs t4

600

XX,×.Tj

4

:3-.

･1?J

:

s3

k2

izkst

t2 10

12

seor

l2

E'

x

t

3'

ss

lj]ii/`i

%>x

foo

El

1i-s･i

i3

13･14

E;, 13 l

x

ss 133tc

tw

-i

x2

31',

il,l'

k32o,-

i

'

SJtliOl"

1'

l 10,･)'siOl '

)o

E 1'.

tl

o

1211

thrp-"'-T-r.

'

li･lZ

sz

io :z

x)

IZ IO

1Ql .'

' of9

twtwlk ,,

xio

xs-tgtw/ 71 e 1<ltt 10Cli llO

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iXOi

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so

-

lJ3･

ZUL.

r--

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

'obtained 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 power

angle each other, and that a sending end terminal 3 is the one receiving power

from other terminais by relay. . ' The more transmission system becomes complicated, the more increases the

number of circles, therefore proper dividing becomes necessary in this case. In the

Figure 3 a, b, there exist circle (i, j) and circle (j, i) which have an equal radius

value with different center distances, but, these circles respectively correspond to

term (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 meansthe

vector 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 part

of the system and the indented line becomes straight. In Figure 3 a, most part

of 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, are

the total generating power and the total receiving power, hereupon: --

99,OOOKW Total 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,because

of 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, if

we place a condenser onto a receiving end, terminal IO, and the voltage on terminal

10 is left as it is, sending voltageon terminal 1 is to become low and transmission

efficiency becomes higher, It can be seen also from the Figure 3a, namely, povLier

factor becomes better owjng to sending voltage dropped. However, in case the rec--

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

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 is

flowed out irom this terminal, which causcs rcduction of tran.cmission efficicncy as

it 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"valucs

of condenser capacity lit for jrr]prov(rrcnt of tlie ro",Er factors. To Eolve thjs

diffic.ulty, we must tal<e cfforts rraking ca}culaticfis in vaiicvs "ay Gn tigures or

equanons.

(5) Transmission Capacity:

･ Thisisshownasaradiusofeachcircleiflinelossisneglected.Ifthecapacity is proportional to the radius of circle, irregularity of dimention of each cir-

cle is quite negligible. However, circles (9, 10), (6, l6), etc. which are relative-

ly smail compared with their sending power 'are impossible to send enough

power. As the situation of circle com.bination on Figure 3 is irregular in ord

er, electric power transmitted from a 'sending end to an aparted receiving end mu

st be followed or obtained in accordance with the order of that of the Figure 2, and

not the Figure 3. That is, sending power is not transmitted in order of the circle

order of Figure 3. If sending power is proportional to the radii of circlcs, all power

angles and line losses become equal,and an indented line showing short-circ"ited po

wer becomes straight. Then Figure 3a, b, will become like an ordinal transmission

circlediagramofasingletransmissionline. . . . ' -･ tlT.Coitcl"sion . Distribution of power and voltage of an interconn'ected power transmisson sys-

tem can be converted into an equivalent junction points network, and if it is deve-

loped in matrix the main diagonal rneans short circuited power, terms of the upper

side of the diagonal mean sending driving power, and the lower side mcan receiving

driving power. Also, if we divide the systcm into sending ends and ' receivipg ends

and sum thEm up with variable factors of ca.ch of their voltage's phase angle, "power

angte," we can draw a power circle dragiam of the net-work, which is an asscm-

bly of ordinary power circle diagram between terminals. The diagram does show

not merely the total scnding and rGceivjng Fower, but all other kinds of electric

nature of the system. Even though there are some presumptive values in the net-

work taken up as an example and the real situation is different from that calcula-

tion, this shows that we can improve the line system furthermorei