Embed Size (px)
Citation preview
Instructions for use
Title Electrial characteristics of interconnected power transmission systems
Author(s) Ogushi, Koji; Miura, Goro
Citation Memoirs of the Faculty of Engineering, Hokkaido University, 9(3), 231-312
Issue Date 1953-09-10
Doc URL http://hdl.handle.net/2115/37784
Type bulletin (article)
File Information 9(3)_231-312.pdf
Hokkaido University Collection of Scholarly and Academic Papers : HUSCAP
Electficaa Characteristics of ffnterconnectedi
Power Transrnission Systems.
By
Koji OGUSHI /t (Faculty of Engineering, Hokkaido University)
Goro MIURA
(Electrieal Department, Muroran University of Engineeving)
'
CONTENTS.
Abstraet. .........................' 1. Introduetion. ...................... 1i. Genera} CireuitConstants of InteveonnectedTranstnissionSystems, .
3. Power Cirele Diagrams and Apparent Current Cirele Diagrams of
Interconneeted Transmission Systems. . . . . . . . . . . .
4. Mutual Relations of Admittances on Sending-ends and on Reeeiving-
ends of lltterconnected Transmission Systems, . . . . . . . .
5. Admittanee Cirele Dia.crrams and Power-faetor Circle Dia.crram$ of
Interconnezted Transmission Systems. . . . . . . . . . . .
6, Fundarnental Circle Dja.fframs, Namely, O. p. £ and l. p,f. Circle
Diag'rams of lntereonneeted Transmission Systems. . . . . . .
7. Effe:tive Power Circle Diagrams of Intereonnected Transmis$ion
Systeuas....................... S. Transmission Power Ratio Circle Dia.crrams Among Terminals, Syn-
thetie Transinisslon EMeieney Cirele Diagram3, and Apposite
Terminal Voltages of Iiiterconne]ted Transmission Systems. ･ .
9. Reac'tive Power Cirele Diagrams of Interconneeted Transmission
Systerns. .....................10. Effe]tive Conductance and Effea..tive Susceptanee Power Circle Dia-
. .grams of lnterconneeted Transmission Systems. . . . . . . .
11. Resista.nce Cire!e Diagrams and.Reactanee Circle Diagrams of In-
tereonnectedTransmissionSystems. ..........!?. Apparent Admittanee and Apparent Irnpedance Circle Diagrams of
Intere'onneeted Transmission Systems. . . . . . . . . . . '13. Currene An.crle and Voltag'e Angle Circle Dia' grams of Intereonneeted
Systerns... '. ....................14. Caleulation of Voltages and Currents for (Inbalanced Faults of In-
' tereonnectedTransmissionSystems............. 'l5. Conclusion. .............･..･･.････
noi}"o"nf)o"e-233
24o.
247
250
259
269
.n,so
2S6
292
295
300
303
304
30S
1
2･32 Koji OGvsJn and Goro MiuitA
Abstract.
A method o£ analyzing an i'nterconnected transmission system isdiscussed. Usual transmission circuit eonstants, A, B, C, Z]b are ex-
pressed here in matriees and proper calculatio'ns are applied. If there
is a eondition of constant voltages and powers maintained in all stations
except very two stations under consideration, the system can be con-
sidered as an aetive network and be represented by ordinary circuitconstants. Under such cases, the above calculations are more expanded,
leading to many of appTopriate eircle diagrams be drawn, such as the
effective power cirele diagrams, the transmission efficieney cirele dia-
grams, etc. Lastly, an example of the numeric caleu[ation is shownon 60 KV transmission lines.
1. Introdtiction.
' As a power system contains ordinarily a good many generating andtransformi'ng statl{ons, it is quite desirable to obtain the general txans-
mission cireui't constan;ts of this system and to caleulate the sending-
end operations from the receiving-end conditions, or viee-versa.
It is a well-known faet that on a system eomposed of -one sending-
end and one reeeiving-end terminal the caleulations of sending-end'
operations £rom reeeiving-end eonditions, and vice-versa, or of the bests;ttuation of trap-smission eMciencies of the system, e'tc. are exeeuted
by the use of circuit constants, A, ll, C, 7), olr by the drawing ofeircle diagrams under certain changeable loads.
Now, the present authors make such circuit eonstants to be ap-
plicable for a multi-terminal power system by dint o£ showipg themi'n matriee$, Voltages a.nd culrr.e. nts. of a m,ulti-temninal power system
are shown by one equation with admittance matrices. However, ac-cording to kinds. of power sys'tems, 'the number of sending-e'nd term,i-
nals and of recieving'-end terminals will usually be dfferent, a'nd some
of the braneh lines between stations may be Iaek!ng.
The admittance matrix is always constituted in symmetrical asa whole. A sectio'nal Matrix of a sending-end or a reeieving-end will
be, however, usually an unsymmetrical one, of which the theoreticaltreatment is considerably difficul't, Aecoz'dingly as it is the most simple
way to express circqit constants of a system by matrices, there must
be imagined to exist several severe conditions; such as that the number
EleetticalCharacteristiesoflnterconneetedPowerTyansmissionSystems. 233
of sending-end terminals and of recieving-end terminals are equal, thatat･1elast one branch line connecting eaeh sending and receiving station
'
As it is practically rare to find a system whieh satisfies all such
conditions, so appyopriate treatment must be made befoTe startinganalyses to obtain actually these censtants; sueh as the exclusion ･or
the modification o£ lines which are not aeceptable, the equalization ofthe total number of terminals between the two types by presumingsending or reeeiving ends as receiving or sending ends, ete. There are,
howevey, a few examples in practice which can be applieable in that
¢ondition without eonsidering any of the above restyiction. Anyhow, though such superfluous conside-ration･must be needed,this analytic method has a great advantage fo,r catching up the bestconditions of voltages or eflicieneies of an interconnected power trans-
mission syst,em as i£ an ordinary one-terminal network were handled.
Seve℃al kinds of cixxcle diagrams will be studied as, the practical ap-plications.
Z. General Circuit Constants of Interconnected
Transmission Systems,
..
..Fig. 1 shows an equivalent eircuit of a multi-terminal inter.eonnected
//oZt' illlle' isEtaoChbteerzM4,ittl,.V.9kteii.;eegl]llt is corrected by a standard re£exence
terminal current to be Z, 4･･-, Z.,
' .Hg･ti, maE-+2･･･-Z,, admittance be-
tween terminals, yi.j.; and a sel£admittanceofaterminal,yii. Cur-rents with negative sign meanpower recep'tion.
yw is equal praetieally to a re-
cipx'oeal of th･e sum of all series
impedances o£ a line eontainingtxans£Qrmers too, but aeeuratelyequal to 11B, that is a drivin'g-
poinb impedanee between tescmin-als,'Out Likewise, yii praetieally
is equal to exeiting admittances of
trarisEermers in both terminals plus
2
s
g
N
#
Fig. 1.
vask
igmaiii(l
ig
=
ls
ew,･/,
eepme
ge
la
-Conn'Oction diagram of a general
transmission･power neeworl<,
l?,34 Koji OGusrit/and Goro MiunA '
half the sum o£ all capacitive and 'conductive admi'ttaces in a Iine, how-ever, aecurately to X(AIB--1!B) or X ]iP/B-11B), that is the sum of
differenee o£ shor't-circuit admittances and driving-point admittances ofa braneh line.
Then, the following matrix equation is constructed Eor the network.of Fig. 1.
L i5
Zi
-Z･+J
HZ-+2
1･
H 4z
Where
.IEill -Y,,, -Y4:3
-!SL]l X2-Y!:S
-Yri ''''''･････....,mm mu t" rm mm" tmt Vt- meP tu
'Y(r+])l ''''''''''''
-Y(.+2)1 ''''7''v'''
7Yni '''''''''''''''
''''''] -Yl(r+I) "-Yl(r+L')'''-Yln H .'t'''l'''`''''''''''''''''''''' ;Y27t
l
-'' Z'rl'''''''''''''`'''''''''' h-Yrn
r::'.:.T.:ptii}Q:.T,,:..:,T.:::-ca-:U:..:,1,
l-l----t-- 1 ------------F-ti----- .--- r--t-+
--ny--t--i --;l--tiny ----------------i----.........I .,.....-.t..'''''-'''''' '' )'r7t71
'
E, Mc
.Z(J=- -- 'ts'
i -
eq･+1
n,
p･･ (1)
IYIi=:Zfii+Yii+3Yi,i+:･･i･･･-････
=' i!:tuttrm'sum of short-circuit admittances.
This equation is obtained from the faet that the total currents sum of
each terminal is zero, for an example, Zl == ?yiiza + yi,･(jEL --]ilj) +yi3(za -
.ZII.,)+･･･+yi.(.IZ4･-ZtL,). In this admittanee matrix which is asymmetrical
,6ne with rows and columns interehangeable, main diagonal terms, are
the short-cireuir admittances and all other terms are the' qyiving-point
admittances, and also lIYI # O.
If terminals 1, 2,･･･r are to be sending-ends and (r+1), (T+2),-･-n
to be r'eceiving-ends, voltages and eurrents athe expressed by m'atrices
[-Iltsi], [uZLV,], [1sl, [Z},]. ThenEq. (1) is shown by use o£ sectional 'matrices
'
[- I-ziiidi]] =`' [Ig,S [g,i]]]I[.4zq.]]l
or , ' [.eq -- [s;,] [-!il,}]+ [sSX,,] [-Zili2] l ........,,..,........,".1,.I.... (2)
- [.ii},] :== [,Sl-,,] [.lfsl+[Sb,] [U,,i] S
ttwhere [exi] and [S,,.]] mean the short-eircuit actmittances of sending-end
and of receiving-end terminals of the system, SimilaTly, [a2] and ['S-,i]
mean the drivng admittanees. , , Compared with Eq. (1), it is cleakly seen that [Sli] and [S-,,)] are
EleetricalCharaeteristiesoflntereonneetedPowerTransmissionSystems. 235
syrnmetrical- matrices and always IS,il#zO and lg,,,}.l,7E:e. While, [S,,]
and [ASShi] are generally asymmetrical matrices, and s'ometimes IS,,)l=::O,
ISiil==O. However, [Si,i] and [S2i] are transPosed matrices with eaeh
' [,S],,]*=[iS,i] -･･---･･･-･-･-･･･-･･-･`･･････-･d････-･･-･･････････････t･ny(3)
isalways'obtained. ', ･ , 'taineFdrOaMs fEoiqi'ow(2s):. t4,e VOitage and eurrent of the sending-end are obm
[-Zilsl - -[Jsci]-][&2] [-ZilR]-[sSlii]-'[-!}e]
[eq={e[,Srii][S2i]-i[Sl]2]+[S,･]][-ZZIie]-[S]j][S!i]-i[LZI)i].
Now, in order to consider the general circuit constan'ts of the intereon-
nected power system, we put
[A] = -[S[2]]-1[Jsk,,]
[B] = -[S2i]-i ......･････t･･････t-''r (4) [Ct]=-[,f}fll][sSl]-l[rw2,,]+[,9fl,,] ･ ･ [b･]=. -[ex,][S-,,]-i
/t [:.lil]':,[gl[,liil･lt'Ig.1[,:,k,], )･････-････････-････i･-･･(s)
are obtained in the same manner as in an ordinary transmissi6n Iine.
Since the above equations contain an inverse matrix [S],,]-i, thefollowing co'nditions must･be taken into eonsideration for the formati6n
ofEq,(5). , (a) lS,ni,40 or lg,,i4e.
.(b) s[aSfui]e' g/d'2]e'r,!S2i] and [S2i] are,all square matz･ices of, the
Condition (a) means that there is at Ieast one branch line directly con-
necting a sending-end and a receiving-end, and condition (b) means that
the number o£ sending-ends equals that o£ the keeeiving-ends. From Eq. (4), one a]so obtains the following relations:
' ,[Z)]*[A.]-[B]*[C]=[1] ' ･ ,, [A][.b];k -[B][C]* in [1] ･.'or,. , .･--･-･------･-･･(4a) [A]*[.b]-[C]*[]l]---u[1]- ･･ ･ '･ [.Z])][A]:i{-[C(]["]Z}]*==[1] ' ･
236 '' KojiOGusim'andGor6MiuRA'･'''
i ,I, I[A]CB];':}i･'([C][Z)]"`]ii, ([A]'[C]] and {[B]"[iD]]
aiesymmetricalmatrices. t･･･-･･-4･-･T-･････-･･-･･-･･i･･-(4b),., The voltages and currents of the reeeiving-ends are shown by 6on-i
ditions pf the sending-ends as £ollows: . [lil,,,]-[.Zi}]*[ieqgll-[-B]'[-ZM l ' [-iPi}]=..-[C]:'[E',1+[A]]k[uzl,1 I '''''''''''''''･･･t･････(6)-
That is, general eireuit e6nstants o£ a p6wer system must be repre-
sentedi?At]h,,f's..ces[es/,]y[gf,]a-p,spgsedmatriceso£EqJ(4). ,
[B]:k--[,Sl,,]-i･ .･' '' ''[C]"j=-[See][,S',,,]-i[,sr,,]+[,sr,.,] '''''''`''''-''''''''''''(4c)
[.21i)]*=-[,Sifn]-i[g,] 'As p]reviously shown, the gene,ral circuit constants of an interleonnected
power system ean be expressed in the same matnner as those oiff an
ordinary,transmissionline, ,, ･ In the following, the applications are to be shown. ,
As an example of a simplified transmission system, the followingsehematic diagram is eo'nsidered. The value of kilo-watts written near
stations with arrows means effeetive powers of those lines, read
.tt ' [l]ABI,E l. 'Generating Stations
No.
1
2'
3
4
Admittance
(mho)
Yls = O.OO186 - o'O.O177yn :- ,.o,10,?Zg6
Y,3 = O.O04'15 ,- O'O.O093:tl26 == O.OOI18
- O'O.OJ35
Y3:]=ye3
Y37 = O.OO021S - J'O.Ol25
Y4s == O.O021 - j'O.O15Y4s =: O.OO0420 - O'O.O0805
Voltage
<KV)
69.5
65.Q
63.0
59.5
Efreetive
power(KW)
Pir)= -l- tw
Pi7 :== I3 400
n1= + ..
1)e3 =: - ev
P:)f{=,:+tv
Ph2= 4,200
l]ts2 == + t-
Ph7 ti - rv
l]b,== O
PPsF+NP4sF+-vkl == 25,600
Transforming' Stations･
No.
5
6
7'
8
Ad;rl//llXt,a)nce
Y57=Cti15 .Y54 = ?l45Vs(; = O,O03,3S
p2' O. .Ok)75
?ffM = Y26
:tles = Ys6
Y71 == Y17
Y73 '= ?.IS7
Ys4 : Yas
Vgltagg
(KV)
57,5
57.0
6112'
-
59.0
Effeetiye
power･ <KW)
PstT･ 'NPs4 H- ' N
Psfl=-A.1),-,s =,T ny,
Pe2 Fi.: ' tv ･.
Rss=+rv2)(s, = T4,900
el -- H .v
P73 == t- tv
Pw= -g,soo
1ig,f -?,OoO
Pss= A"
EIectrical: Charaeteristics of' 'interdohnec'ted ?ower Transmission Systems. e37
usually from the power distribution diagrams of the stations. Eaehvalue of admit'tanees, and voltages also ai7e shown in Table 1, Sign +tw
means power sending and --tw means power reception, of which thevalues are obtained by drawing ciyele diagrams.
4 OOKW
Eb 2s '･
Bi m%es
-g9oO,gKW
gestt
O KW
Ka B
xe
-- 98oOKW sp
# U.
ag
-An
2S,6OQKW ･Fig. Z. Conneetions applicable to example I.
The diagram of Fig. 2 is expressedThose values are correeted to the
powex and 60 Kv base voltage, by zi,,,,., = zi,.,., . (610o)L' .
T'hen, Fig. 2 and Table 1 take th6
,ln'Fig. 3 and Table 2, each branch'
gether with its reactive power, which '
a power eircle diagi'am in each case.
a.re ofii well-known, they may be omitted
calculation is by s'tarting at s'tation 1 by
on to stations 3, 2, 6, 5; on the othene
passing on 4, 5; and lastly between
A$ a result, the
3 oy Table 2.
13･"OOI<W
izz.
As 7SooO I<W
daigram of a transmission network
by aetual mhos and kilo-watts.
per-unit values with 10MVa, base ･ 1 11 /･/. '''1･1･"1L ,' 1 V,,u.) =: 'P'1.i<v) × 61o '
form e£ Flg. 3'and Table 2. (effectiv6) powex is shown to-
xs u-sually obtained by drawing Since the de'tails of the methods
here. The order, howgver, o£ utilizing Pi, = 1,34, and passing
hand starting from the station 8 $ta'tions 5 and 1 are eompleted.effective powers are obtained whieh az'e shown in Fig,
23,8,.･Koji.
Oe42
ue
o.
,OGusm and･-Goro .MiuR4
-o.4g
O.2+j'alles -
CXi9-j(tsO
on1 tl q6 6
q29+J,ee8
3 O.31+jonS
ct31+jO,16
OBI+J O.14l
xe IZ9-jO.49
'4
2S6
O.2'+ja13
s.
2.12-Ja6Z
3.36--i･aqs
3.56ri1.9,5
z
1.34-jQ84
2.2(F-Jq3S
-xO.98
Fig. 3. Diagram of Fig. 2
by the per-unit values.
TA.lsT,E
8. .Z
. -.-nyO,3
T o.36+iqo･iq3os--jo,o3
with powers shown
2,
Generating Stations
No.
1
2
3
4
Admittance
Y (p u.)
Yis==6.41-s4o
Yi7 =5.67[ -SIE
' = 'Y23=3.681 -66o
Y!6= 4.S7i -sso
Y32 == Y23
Y37 = 4.5 l -sgo
ZtAfi=5.45i -s2o
Yas=2.91-szO
Volt.
(p･ ti･>
1.16
l.OS
1.05
O,09
Effective
power (p.u.)
Rs==3.56
ll7=1.34
P]i=4.90
P23==-O.29
ll,,==o.71
Pli,=O.42
P3･, =O.31
P37 = - O. tt)1
P:,3= O
Ls=:2.255
Pas==O.305
A4:=2.56
l Transforming Stations
Fi No.:i
5
6
7
8
Admittanee
?l (p. u,)
Y51 l Yi5
Y54 = Y45 '
Yst`=9.9Si -830
y62 = y2e
'Y65 = YC6
Y.1 := Yli
Y'J3 == Y3'1
Zfsa !tdS
Volt.
(p. u.)
O.96'
O.95
1,02
O.9S
Effective
power , (p. u.) ･
Pbl == =- .g.s'6
Its4= T?.l2.
I%6=-O.20l)bs= -5.6S
At = - O.6g
Rfi == o.2o
Pb6 == - O.49
P7i= --1.0.9
lV3= O.31
A7=-O.9S
.Ps4==TO,3e
l]ks==Ps4 '
･･ ･1･1 tl･
ElectriealCharaeteristics'of'Interconnected･?owe'rTransmissionSystems. 2g9,
su'
e Ta9..?{. TP,10/g.1/Le,i// l-"ik;.11sa,,,,i,e aiSli,:.,ee/1,fr,i/i//l.:,..¥i///.r.frge.:i;£e.aff/Lg.m.?, which re-
･ '',. '･ ･. ･ ,･-i". ･5,-6, ･= 27057f.,･ ,-. ,.. i.･,. ,.,.,.,, ,1, ,.
ilfe. ,il.lil'it,1?1!1･a. l,/l, 1111i, ii,Ii2aliikeil.'1 12ill,g,,31j,g,i,'&illl/611, 111il.?, IF,i.?11i r. ;. 1.igl.lrr,firlclar.l,1111ii･aff.II
Magnitudes of voltages in stations, E7i, E,,, E,i･･･E,, are shown in Table 2.
at ev"eSryPOpWa?rt gYr =evPe'ry'Qbraatngh"eiiYneSt/)egOC5igaSz}eOdWnb"･ TFI2gifiEaj91"rw"hee"SeS
]z'k is yhr{.:gz,{･.s･Esl,;lj/i/.-,l}･e of LEt///lleglill･lifl-l･/'g.li-$tl)･ Th?I..ii 11i,, ,.,/.,.
'' -l;==O.3e61-5017t .Z; = 2.61 l-11025, Furthermore, by Eq. (4), ci.rcuit con,slants [f#],.IB], [CJ, [.D] are
obtained as follQws:
3.4010058, 1.56i1810 O e.851-1780 3,o6Ilo2et o o 2.04 l-Ii.820[A] - 1.97IT9mO 2.25[4028i 1,07 l loO 4.30 I=1710
O o ,･i･O'-,1.00,LO[) O O O.293I-911.Q.,., =, :,Il,5gii:･lli.f･?ioi.i-s8,;:･- [.i.?..//./giigggo ･･･sliz,IOz,7: ,i,.
,b･ ,''ig,,:siii2211" ll,Eoiii･g,i, i4:I]･g, i,eilg.i.2,l ,i ,g,,i,Ei2･Zs;il////il.lii' i//[g,S, li..".i.i.S,..i.illlill..5,..ii.
5.451...98.0..-.- O O 5.451=8?.O.,･-
?401 Koji O(svsJ-a'and Goro MiuRA,
-1.88110251 O O 3.531-173040t.S [p]== 2IZ3,li-2f,O,.,,, t･:3,II?O,i,O.' g･,Ei,611.i,23,Z,- EIggll?,i3,06,'
. o -oLUux1 oi' 2.88L3015' ..........'...................,......:........,........... (E 3)
Thus, circuit constants are obtained.' If terminal4'voltages Ei, E2,･･･
4ge maintained as constan.ts, voltages behind synehronous impedances,
need not be considered for the later calculations, However, if constant,
voltages behind synchronous,impedances are assumed, that is, if ex-citors' currents are maintainled.constants, these voltqges musL be cal-
culated together with their pdwer angles too. In those eases, impedance
drops due to genescators' synchronous impedanees and transformers'
leakage reactances must be idded to the circuit constants o£ Eq. (E 3),
Namelyi
[[j.,E,ig]]-[[Zi foBi] [[r.,i,,E`iii,,]]
whe,re
[jEL] ==
[-iEI,,] =:
Generatbr
[.iZlsi]'+ [Xb] [ist
[Zil] + [ac] [Zh]
impedanees
}
[Xb] ==
Tr'ansformer impedances [X}]='
...･.,･..･-.･-････:･`･･････-･･･`･･- (E 4a)
0'Mi O""'"""'-"'J""'"""""'O
O'"X-"..
:oo --------------------::: O Ox.
o'i ..i O･-
O ".,..
O "---""-.
ttttttttt.....
'tt -x
-t-----------T
-------------- o
k.' 'O-".-.-r::-,O 3;,V.n
/
From 'those equations, are obtained the following circuit
cluding tl e generator and transformer ,reactances.
' - r[A] + [Xb] [C], E-B] + t,2Yb] [D] + {[A] + [Xl] [()]}[X}]t[-lil]1
L[iMd- L [O] , t])]+[a][x}] ･･ We assume, [.Xb], [X}] as the following per 'unit values
eonstants ill-
][[i-ztz}Ii]]](E 4)
with lo MVa
Eleetrical Charaeteristies of Intereohnected PoWer Transmission Systems,
TAI3I,E 3.
241
anGenerating,stations
1. Uy.
2. Eb.
3. Ka.
4. An.
O.157
O.44
1.34
O.333
Transforming stations
5. Su6. .Bi.
7. Sp.
8. As.
.O.05
O.05
O.05
'O.05
' ' ' ', Then, circuit eonstants with constant voltages behind synchronous
reaetances and with ineluding transformeys' reactances at' receivinge
ends are' obtained by Eq. (E 4) as follows :･ . ･,' ･' ･. ,'i'/ .,
'' iki '-: ''' i'i8 1,iji. Iii,i,f.,3/l.;. ..l, 1i, 1, 1ii il. 1?,i., Iiil.gi9-ile'i '. ,9111i,6. ?1.'12,i8:/fl. lgi. .,11 ,iil･11g･ ili:#･,6,3ii･l,sl ･･
O.89)-9.-2-O17' .O,174j=82039, O.e45I-810 O.966I--85045t
O.25I153040' IA2I95036, O,5421-650 O,678I-59030, ･[B]ii････4.331mu7solo, o.61s[-64o ,3.64I99015,'.4.951106030'
1 .l 1l ''' ''' VO.09I-820 ''. ･..O.･ . ''i,9..'. .1.44l91057'
ii ,.,3,60I50.. O.9371188e25t O.2831A89e 4,05.l1950e5, ,rDl= O･57165040' .2,441 8015' 1.23I2o4oso' 1,s6j211o2o, ..･t･."..,2.9tltI195045,..･O.53I203045/ 2,46I13010,.3.40l18040t .
O.2721 1889 ･ . i. O .. .,,-. O..･ .3,151 3040' ' ny･･･--･･･--･-･･-･･････････ny･････-･･･････････-･-･--･ [E 5]
tt tt /t[e] is the same as that of Eq. (E 3),. ･ . .･.If there need not be ineluded tJransformers' reactances, [X}]=:O is useq
in Eq. (E 4).
As these'have been diseussed in both cases where terminal voltages
are eonstant･ and where generator nominal voltages are constant, ifgi.?giiii/'Z-lie8a.tTsceeaUg//ajie,a,r/jewOa,n,wuEjqi E",211,2isJkti:`//,ee2r:,il//:OSpslb?g/11g,/L.lsl ///////7,.
the network is fust redueed by ring-star conversions to a general polygo-
nal diagrani and is represented only by generator nominal voltages or
reeeivingvoltages. /.,-'･,･'.// / '･ ,･. /-'･-- t tt From Eq. (E4a), generator nominal voltages (induced air-gap vol-
242'i Koji OG[Ssr-ii -and Gor'o 'MiuRA:
tages) and receiving-end voltages (behind transformer reactance) are
calculated using Eq, (E 2), resulting as follows:
eq=1,67l23020, -EL:r-1.04I--44030, Illii,!',,il4,Z,E.l:g:,51'i -n,,,-:glggI?glg!I t,-･････i･(E6)
T/ ZL=-1.411un3.40404,,n=O,98l-100um ''
In Eq, (E6), ltT,, sc,･-･should be noted as Ei,, li,,, differing from the
formeT values. However, by renewing notations again, ZL, Xt･･･ areused･here too. Currents, li,Irt,･･･ are held as beiEore.
From the above values, we ean also obtain the values of terminaladmiteances or, powers at generators or transformers by such relationsas zifi=:Llen, zffn=r-.Zl,!eq, and eqi==!.Z]"ZZLk=aifiEi, PP17='Z]7itLk=:zffi7E?, etc.
i 3. Power Circle Diagrams and Apparent Current
, CircleDiagramsoflnterconnected , Transmission Systems.
Power circle diagrams of an ordinary transmission line with con-
stant voltage's ,are generally shown by eircles which have centers at
points of shorVeireuie powers and have radii equal to drSving powers
into a sending power eo-ordinate or a receiving power co-ordinate, If
this definition is applied to the case of an interconnected powev system,
the centers of short-circuit powers and the radii o£ driving powers are
shown by matrices for receiving-end ci',trcles as £ollows: ･-' ,/
Center': '-[.B]-i[2g.] [LZIL,z] [-]ZIA-:]k ±'`[S2le] [-eqit] [-Ztlu]k 1
Ah'd"f6inR,a,fd"i:g..,:[dtBII',[,]/ilSl,!,['iftI'"]kl=l[S2'][FEsl['Zi're]kl J''1.(7)//,i,.,
' '.,,i, ･,l.lgn,i.Sgi.fes,"igltt]11[,-Zapiig,}.l,-/grei/r".,iliZ'lill,iii,]u,,[,,l,Il,lilk.,l l･･･.(7.&> .
In ll,ke manner, apparent current eirele diagrams of an interconnected
power system are shown as follows: ,For receiving-end cireles,
' Center:' -[ZP]-i[(]][-Zlln][-EITe]k:･={[iSi][Sii]-`[,SLi]h[S2t]}[-Elre][-Eii]'kl
,,.,,,,.I..g,diusii･[]]']Ii.['IE.i!,..!i4.`ILr..):1.ii.i.![..S.rtlg.tS].i]..r!il.is.[.[rl.[llft..1fti...L....',,..,,'si "'
EiectriealCbaraeteristjcs.oflntereonneetedPowerTransmissionSystenis. 243
A"d for sending-end circles,
center: [A]:i:-i[e]'k[-ee,g[.scsl,=={[S,,]-[G,i][S,,]]"[g,,]}[2ZLg][-rp.],,
Radius: i[A]:'`mi[Zh][Iilsi]k1='i[,Yi.)][S:･,･]-`[-Z}t][Jeqsi]kl
These results are obtained from Eq. (5) by relatively simple caleulations.
The equations represent the total of power circlg.diagrams or
apparent current cinele diagrams o£ the system, between terminal an.dte,rmina!, Since these equations do not contain an inverse matrix [Si2]-i 'kor [Sut]-i, they can be applied for a system of which the number ofsending-ends and recieving-ends differ, on disregarding' the condition
(a) and (b).
RefeyMing to saturation'points, however, the shert-eircuit powey
matrix which defines the center contains the driving powers which are
transmitted only among sending stations or only among receiving thusieonstituting a pair of power sending and reception statiQns. [E]herefor.e.,
there are some eincles among the receiving circles group, though somehave' a'etually sending characteristics; and likewise some circles are
seen amo'ng the sending circles group, of whieh the characteristics are
those of power reception.
Aecordingly, the center, the short cireuit power matrix, does notoccupy definite points but eontains changeable terms dependent upon
Ioads, resulting someWhat in uneven forms, though the effeets is almose
caneel eaeh othex yesulting only in a sligh't fiuctuation or transitlo'n o£points. The above considera'Cion is applicable to the apparent current
cirCiXsdisaugcrhailtS lkOuOs't be noted, the circi6 diagrams are dz'awn using
terms, radii and centers, in t'he matrix, and they are bnly a collection
o£ ordinary power circle diagrams between terminal and terminal. Besides, there az'e some othesc methods to draw differe'nt kinds of
power circle diagrams or apparent eurrent circle diagrams accordingto the kinds of selected assumptions.
tt ..Ip this example of whieh the outlines a,ee explained in Example1, the ahalyses wM be advanced to obtain the poWer circle diagrams
discussed in this seetion, It is already related in Example 1' that if
analys'es aye started firs't firom eonsidering syne-hronous reaetances and
transformer reactances with eons' tan't nominai voltages of generators,
the network should be reduced to a polygonal diagram of a ring-form
by the network reduction, That is to say, i'n this case the network
244 - ･ KojiOGus}iiandGoroMtuRA
reduction is less laborious than the former proeedures. However, i£
the reactances of generaters or trans£ormers need not be eonsidered,nor the admittance matriees, Eq (1), which is instantly obtained from
$,a,t26dd.i:e.C,tl.Yf aESqSilS(t7S) t.h.eddSaqW, i(n7g.).O£ tlie POw,.,gr circle diagrams by the
In this example, the case which includes such reactances' will beanalyzed, Then, we have two methods o'E proceedi'ng: onQ them of is
the tuming back to draw a new network that includes synchronous
and trans#ormer reactanees, to reduce it to a ring-£orm, and to calculateEq. (1); tne other method is the calculating of the admittanee matrixll ,P,i,ei:,er,i,iel]kast{8.t,a,fis`l,II?',oa.ni,Ill,a¥'i?ibizrif,fr8,:,n･gEd･!ilsct,:)c,//.e4s: kTs,ekfis'gt/ll.l,2g:o,g
ecejtWw,//･i)ll-:Sxh.g,tilil:.s.Xeg:,:･,///g.iSk,,g'iini/"itg.X,2',g.X,a.d:za3,' i:,e//'¥a/･]gda}iay'g.hoS,9ii.g,rr3,SanE,e N
. -6 Q/''",,,,''' g･
2f 51
g 3r
'' 7t zn.
aj
4'
gt
'tv
8
c;;i1
.Ses/ g-
Z
" tttt/
xe rk
.
g･ttlt g
tt.,z.Z･'
t'17z
t ttttt/tttt/tttlt/t tt,Flg' `' 11/Rc/gi:idli.ial{.iS.tdiSlle.X,:,t."krgsi.#sP.:n.ei',3an.ki{si /,.･Figi,, 9.i .l/eg'luig,ei6n/i?,fpXiLe,r･:gP,Pta':
' ' In Fig. 5, all lines that connect mutual terminalS have definite
admittance values defined by the reduction. ･ Also, the synchronousreactanees and transformer reactanees are 'included in line constants, '･･' - .The seeond method is to caleulate Eq. (E 5) by the £ollowing proeess.
Eieetrieal Characteristics of'Interconneeted Power Transmission Systems. 245
As the constants [A], [B], [C], [LD], of course, include those machine
reactances, we only ealculate from the constants the original admit-
tancg values, that is by an inverese calculation of Eq. (4).
[S,,] = [D] [B]--i
[S,]=[C]-[LD][B]--i[A] ....................,......,.(Etz)
[S,,] = -[B]-i
[S,,] = [.B]-i[A]
are obtained, which can be used in Eq. (7) or (7a).
Since there need not be carried through two such calculations atthe same time, we make it usable by selecting only one, which is more
easy, among the two, If we start by the first method, then we mustcalculate･[A], [B], [C], [D] by use of the admittanees obtained. If we
start,by the second, we must caleulate the admittance matrices by
,b",StehOf., .Etqh',(dE,73igYdO"tdhe,r ,t.O.d.ra,W,,t.hi2,Powgr eircie diagrams. of course,
Theresults,areshownbythenextequa'tions. ･
4.35 I-88032t
'- O,1081-82b50,
-O.04551-86OO6,
-O.1771-81e50,
Lg,,･] ==
ttttt t -1,02 l-88052t
-O.5781-86018,
-O.1061-77o2ot,g,,i･%.?. iiifsi,,oi,1,,
/1 ' 5,92 I-8500f!l,,
-1,04 l=86040, '
-O.646[-･71O30t
-O,956i -840
[SLn] =T- [&,･]*
-O.108i-82050,
I--85o26t 1.60
-o.lo41-soosot
-O.05851-81035,
-O.1391-85o4ot
-O.2331-9204ot
-O.0361:-8004ot
-e,084･I-8401o,
-O,04551-86006,
-O,104I-8005o,
O,8691-88029,
-O,02 I-81020,
l-86040C
-1.69 I -90.
-O.48 l-･83015t
-O,3561=.-95010rm,
-O.65 l-79o3st
:-O,1771=81osot
-O.0585j-81035,
-O.02 l-slo2o,
2.82 I -s6o
-O,1151-8oo3ot
-e.0441-soolo, tttt-o･oosl izggo?ot
-1.04 ' ./' 1.83 ' fL86o22'
-O.224,I-830101
-- O,081I-79o5ot
JO.8321
-O.646j-71030,
-e.224] --83010,
4･.09 l- 8402o,
-O.101i-750o5,
-O.956
-,O.081
-=O,1O1
2.13
gI
l
-84o
--79050'
-75005tI-85otlLst
,..,.".".......m..."....,....."."."..,,.,...i-･･･ (E 8)
wher6
･.046 '' I<oji OGusrn and Goro ･MIulta
[[Xi:IIgt:-i]=:[,[2i]'[[.',Z`ir,ii]-` . (Eg)
As the total generating power of say, texminal 1, iS gener'ally re-
'presented by '// ''" " i:"'' ' ' ' /t t tP]]+3'Qii==(Pit+.dQgL･)+(P]3+7Qi3)+'''+(Pin+O'Qin)
= ((y]n+ yi2) E'i-y;,ZII-,-Elh] +[(y"3+ yi:3) E7;-y]3fiIiEikl +
---t+tb-ny--------.4-t-------t---t-------4t------i------t-- I(Zfii7t + Yin) E¥ pm 'Yin 2Zllt, -Elk]
' tttt t =(lii;:. 2ijiLBi. 1thiinh),+(IXIi 'k7?- I£i,, ZCsjqh) +
''+(i.B)i.'tE:--Bl,.paIinik)' ' (E10)
the pquatiop itsel£,shows the power eircle diagram of the terminal 1,
w,hieh is a colleetion o£ many civcle diqgrams, As for the ,first term,
(Pi,+jQ,,i) ::::･'(yil,+･y,,,) -ZiElL!iLy,,Elll,-ZII,, ;･-････････..････.,..,.･, ,･(E 11)
it shows the power which is transmit-ted from terminal 1 to terminal2:'･ The terM,(yiis+'zfi,,)uZil? shows the center point aiid lyiD]2eqteI, shows
the radius of the circle respQ.ctively, which.iq a well-knovLrn $ending-
jQli
o v,1O.2O.3O.4QSP
t/...
tt/tttC]13,
1,
tt
tt/
t
t-.'
Flg. 6. Power eirele diagrams between terminals on the co-ordinate of 'the 'siehditig-eh'd '1, 'liil- o'QJi.
Circles, Ci･fi and Ci･7, are excluded
here beeause of the seale- out.
11
jQ77
tt
･,C774 t/t
C77Ct720
･g,71a
6'2s"''
1Q2Q3Q4Q5/t/t tttt/
Fig. 7. Power eircle diagvams between terminals on the co-ordinate of the'reeeiving-end 7, lV7+3'Qn･ 9.",e..Cg'gCS2' ,C,Z'J･,,iZfax.g.'kd, ed herg
EIeetrieai Charaeteristics of Intereonneeted ?ower 'Jrransmission Systems. 2'47
end power circle ctia,gram, The saMe proeedures are earz'ied through
for ma'ny other terms in like mannex and many eirele diagrams aredrawn. In our example, y",,==zfi]3=-T･･･ =zfn.==O is taken, The results
are shown in Y"ig. 6 and Fig, 7, for the Stations Ur. and Sp.
4. Mutuak Relations o'f' Adimittances on Senditig.
ends andi Receiving"ends og interconnected
Transmission Sys'tems.
The ratio o£ a current to a voltage of ea,eh terminal means anadmitta'nce of i;C. Sinee soine terminals have b.r.aneh 1ines, the vatio of
total currents to the voltage is to be named as the terminal admittaynce
and the ratio of a cux'rent iwa a branch line to the vol'tage as the braneh
admittanee, Though the relations oC these voltages and cuz'rents are precisely
shown by Eqs, (5) a:,id (6), some explan,ati.ons shall be made because
of the matrix equations. For an example, the voltag- e Ei of the sending
terminal 1 is expressed by the sum o£ terms o£ the first low o£ Eq.(5), that is, it is composed of terminal voltages and of total terminai
eurrents at receiving-ends eovered from the terminal (T+1) to the ter-
minal n and also of general eireuit eo'nstants deserved, If it is analyzed
to each vol'tage eomponentof each recieving-end tez'mi'nal, as
rEl = -EZII(?,.t.1) +"E)l(,.+Li) +.･b･..1fiIi?t
='(tsu(rti)Jsc;-+i+Bi(r+b-Zl･+i)+(/g-i(?･+Li)-scr+t+Bj<r+L,)Zl･+il)+-+'
'-' (Aineen+B:nln) '''''''''''i (5a)
Sr!m.tiarly, the voltage -Ei.,i o£ the reeeiving' 'terminal (7'+1) is expres･sed
by sending-ends quantities from .Eq. (6), as
sc?'+} ::=: eq?'i'])i -F ']eq1,'+i)Li +''''''-]iCllv･+ip･
='` (-2L)](,･+b-iitIi'fi-B](.+i)-L)+(-:er-Pil(7･+blCe --B2(t･+i)Il･)+'''
''' (-kZ)r<r+i)-Il･-Br<r+blr) ''' (6a)
Also, for terininal 'to'tal eu.rren'ts,
'iif1 =- 'Zl(r.i) + Zl<r+L') + '''''' + 'E]n
=(q<r+i)ZIL･+i+'iLbi(r+])2]･+i)+(Ci(,'+2)uer+i+IJj(,'+i)LZ]'+2)+'''
's' (Cinven+-EL)in-iZ;,) ''''''''''t' (5b)
ti
248 K, oji OGv$m and Goro MiuRA
1;,., = ia,,.,), ---Zl,,.,),) + ･･････-Zl,,,i),,
,=(rmCl(,'+i)"ZIIi+/gi(rt-i)L)+(-C2(r+J)Lsc2+M2(v'+t)Z-')+'-'
''' (- C);･o･+i) eq･ + A'r<r+i')'Z;･) ''' (6b)
Other voltages and currents, namely, sce, usc,.･･･.!ll,+,i, 1;, LZi,･-･2j,+i, e'tc･,
ean be estima'ted in the same way f.rom Eq, (5), (6).
First, relations between sendiitg-ends admittances, zvi =--Z-- , ev, =
vei ttt-tr. ,r･･ny･･ and receiving-ends admi'ttanees, ztp.+i, 'av.+,}, t･-are elari'fied.
" For an example, o'n the sending-end 1, from Eqs. (5aJ), (5b),
?v, = ...T4i... = g{.1)(ig.r+.i)T'ZZJ, +)=I--pt-i{.l7L+.l-?-'lt -vi.-}iCIT,+E)L:'.:iJ-.(C. in.eq. t±BJnZ-V-.pa,,)
-ZII.+i-l-Bi(,.+r)ZV･rt-iuerti)'''+('`{g'sn'ifn eq (A'i(r+i) --Bh,zv,,scn)
..............,...........................･t･,･･･=･･････････ (9a)
is derived, where zcs.,i := tg'-,'.'.`;, v･=-･･L'eep. = -t/ISII";- . I't is known ,Srom
the above equatlon that in the casa. of an in'eei"eonnec'tedi t,ra;nsm:tsslon
sys'tem a eertain admi'ttanee on sencting-ends is affeeted by all the
admittances on )receiving-ends and also by z'elative va].ues o£ the vol'tageon each reeeiv.ing-end. As alreadystated, coeifieients, xd, "]S, (;'and s')
ean not be detez'mined unless the nusmber of sending-end,s and o.ff re-
eeiving-ends is equal. Relations between 'the above equation and t'ie
power or the power fae'tor are shown later.
Ifi all se'nding-e'nds admit'tanees ax'e ca}evda'ted by a matrix,
tzv, o:.-..l.----------------9) ..I....o,.,,......-.---･･o
ege,. "･･-.... [/-e .kli "'---...... I [Ily]---o'･-."'･･-..i i'"･--..i "C'tt"UM l"'"'-･･---...lil)･-t-･..1)1)-6･= 9...........Jtt.J .,/1 i''(9)
kb------------:'･o･ev" b---------------o-211111JY
of whieh the ele}nents are easily calculated from Eqs, (5), (6). Seeoncl,
the adrnittanee on ioeeeiving-ends is
ze･?,..1Ol:----------------O Z,ii
[.z},] -
[.Zilir.]
Zif,,+1
e･l'o---------
i i
o
O Zif?L
-Zl7r+I
o
l
o ------------
O ･･-..
g,.,
Y"'Li .. ...
o
o
eLflt
Izit';
(10)
"
For an
aff r+t
Electrical Charaeteristics of Interconiieeted Power Transmission Systems.
exarcple, zv..i is
= --Zi ., ,,. (-- C)',(,,.,)ZZT, + flg.,o..,,?lg,Ul.)-･･+(- (Jl.(,,.,,-]li,+ zg.,(,,.,>
1･"49
Theseeonneetlion
Third,current ft
CZId
[-ze,,]
zif.g. .)
eS'7i(r+ b := (ig.i<. + bue, +i + Bi<, d-i)II,+i) l'` '''(-Srot-1ff;e + B, nlle)
' 'Z?r+1'}1 qr+I)2 ?'if(,･･t-1)'1 egeL.+I)t ･ti･･-･･･ --attlt-- ･ LptIJ,ue1.J, [Sll.>," -- 'eifw+1)1 'SCI(,,･+,,)L, ･-''･'`'' -I!l!ti･!/･)-i ,..,..,.....,..,...,.
-[Je't'i7 = u.+v, zifvvi et.p.:)･.-･･･-･･t･.･e.c:,,,.) Ttt"'P'5''''''''''''''''''''''-tit`'::'i
et?L･7･ti)i - -(ie,,u,.,,･.]ig,l -,iig,,ii:i,-tr''tt..1-/tiJ-<--l･･･l･'L-:t;E-l);Z,eq1 -,-tt./,-J,-:;J,-;.,;.)
nyrii-----------ii----.---------------------L---b (11)
Sinee voltages in denominators axoe fojL'ms of diagonal matriees,ealeulatio'n.c' are easy. As 'the above eq'erat2o'ns show ]t'elations betweell
tersninal admittances a'nd bvaneh admitta/i)ees, b.ra'neh line cui're'n'tsin t'i:-e equatio'ns aire values whieh gre obtainect merely by a caleulatio'n
ancl zitot by aetual data of the system.
Besides, from braneh curvon'ts and voltages,
E,II:l'i t17I,t]i.l1'.1111..1.?t)
[.ik] - ･･･････-･-･･････-･･･････--･･t･1 r["EZI//il- -rm fLZIri(r+i) Mi<r+L･)'''''' an,,h ''li''''''''''''''''''''''''(12)
(scL'(r+1)YLi(1,-+Li)''''''''''''1 .
eq,+i (fif･`"i(,,+i)-El`--ESi(.+i)egei-Zl7i)'''+(yff-].(.+i)-ZIL.-V9,<,.+i)aif.eq,)
.....................................-L4-,-..... (10a)
values of tescminal admi'ttanees a/re practieally obtained. in
with the power and 'the power faetor o£ the terminal.
'tb.e ad.mittance zv,j for the branch current J,j o£ 'terminalis taken i'nto eoi)siClez'a'tion.
eifl(r+1) eif1(r+L))'''''"''' !1(.?-'+b !i!:r:tll')-h .,.......
zl zq egeL,(.-.1)aot(,.l.,,)-.･-･.･･. -Zl}(.+1) -]ZII(,,+L]> .........
-- -ve, -sc･,
--f-t.-------t----------t----- ' -----ny-----------4---t-4-l---l--- '4,7e
N'eVrcr+1) aV･r(,･+L,)'''''"ege･rn X pan C,...,,ll]gl7,..., + Y･),,,..i,Z,.i
250 Koji OGusm alld Gero Mius'{A
is derived. Frorr} it, an actmittanee, such as, Zij =-zifiji,re is tal<enout,
uehkof whieh the trea'tment, howevei', is rather eomplieated as 2't is not of
a diagonal form.
rcxample 3.
Terminal admittances a}'e caleulated by w,=u!}1llEIL or 'eg7･ij--.!}j!.ZItli as
jus't described in Example 1), They are as £ollows as a result of
ag7,, :=-L 3.37 i 1.50t12, Ilec:l/EI- :,1･lislL,tLsZ>:･l,iiiz7･ g･x':.,: i, g::z,l- :g::tll-gi3-;-
Ws =5,74[ 23ot;Tst Ill.l,I,:: i:, 8,Jtg2g･l, l,,3-/li:Jilg･i,' g:f,:: ,. g:g`g･w:E't?g..r･. zg';
Ill/li://･i/Ji':t35:blg-'lllil/I:'ki/l-･aun-(,- illlll'esgi.tig,?7`ifl:i6'i/.gO,//-¥,,l
Ili'ill'l,r'2i.:,li,I-ll'umri//o.rmmli':ll' 'li3'g-i:-emp'l'L'll.3?9ali.il.illi.Zo.g
' ....,",-･-,･･･････-･･--･-････-･-･････ (E rL2)
Of course, kiterrelations sueh as E(:{. (10a) oy (11) a]e nzairtia,{ned among･
these admittances, ux).der the s'teady ol)e.irat-i'ng phe'nomena g]LveB in 'thc)
fiyst,
5. Admaittaaice Circcge g])iageg･sms 2mt'! Power factor
Circle g]>ign'rams of ixntex'coemneeted .v Tra!ismission Systetws, '
Tez'minal admittanees o£ an interconneete,d p<)wer sysAv'em a,re shown
by Eqs. (9a), (10a) as dis¢assed a,bove, l.i.owever, 'i;he nttmbex o.ff vaJtrla-bles is considerably many. If o'ne termkial in each two sides is tal,renas a variable and all oth.e.v t;e:t'"niin.als as eonstants, the equ.a't'.!ons") tal<e
the £oym 'v =' ((e--bp)1(c+¢ep) whrtc'n' atre s:tmpte veetoy equatlons showing .s.ener'al eiz'cles.
'
ElectricalCharapvteristiesoflnterconneeted3PowerTransmissiQnSystems. 2'51
As sendlng,eRd adniittances a]re given by Eq, (7a), t},ie yela'tion be-
tween terminal i a,nd tererminal (r+l), witl} the assumption that other
receiving-end volacages a]re constants, is sliown by tl)e next equatioi}
wi'th Eq. (7a) transfeyred.
¢7 + I," W,,+: ?'V`-= AI,,+Istrmi'i',''IJ;7 ''''`''`''-･･･L･･････････t-･･-･･･t･･.....,.,,.,.... <-]3)
where (i' :'::7 9.`iXi::.-,SX):1:i,'e9.';.2 Ii'"II ff1lof6111in )
A! : A.i(,,+i){--t{gJ<.+.･,)e,.it -････- --Ai･,,(-',,i.
-FBi(,+ll,eeS,+.-e,mit･"'''- +])>ineVne,,in.
-sc sc't ei.tL, =-t -Jmbllili'¥i- s ''-''''''''''''''' et･I71 = Mt.J-it1
ty ,..,..,........,...･･,..･i･.-･-t-････････-･･- (13a)
e,,t. is s eoeflie.Zent oJff a eomplex numbe.r g]rown by the diffe,rence of
volta,ges among yeeeivinf?-end$. ze7i 'thus eb'tai'ned on terminal 1 has
s,Zmilar ii'elat.Zo'ns also w2th vf7.+i, ev.+st･･?･ei. o.£ other receiving-ends, and
moireover ee72, w,i,･･･ on se'nding' 'terminal 2, 3, ･･･ have similarity with
itV.-.l, 'eV.+D, s-･ On z'eee':lv;','Ilg-ellds.
The eirele eq'ga't.Zon (ll) cnn be treuted as quiLte identieal to a case
of one tez'minal trnnsiniss.Ion ]in.e, i.f C' anCl A' aye respeetively assumed
as eoiista'nts, X£ the c;tyele dSa.crram i.s dyaw'rs in a rectan.tyuiar admit-tanee eo-ox'dinate, e･tpi:.= Gi+,tiBi, 'th,e var2able bic.h.comes a coirnple]< numbejr,
as age..i:=G.+E+p'B..ir=l?fi.+ilsj'V"]. [I'he]rofolre, .{'t beeomes an admitta:nee
eSrele dilag.,.;]ram as a, i"Tk>ole., but also bc-eomes a power faetor eiitele dia-
gram iZ".L e is eozistant, an appa,iren't powe.r c,Z?."cle clie'tgiram :/f I'e,{p,..,I eon-
stant, and ,an effeetive eonduetanee or a,n effeetlve suseeptanee eircle
d:tag. ram iiC G..i o'r B..i i･s eonsi;an't. 'Yhese ctiag.xrams are dTawn like to
cas･.a.,s of an ordinary t:,"am.smission system line s7gxith, one 'temnitnal on every
'two sides.
If a poweM £aetor clrcle ctiagram is developed it is sh,own by a groupo.f e.ireles, all passln,{.} tivrou,o,;h a s'ftoirt eircuit po,irnt, 7o..i== oo and an open
eircuit polnt, zv..i=:O, wh.(.ty:re
aeSl. =: t;i ･･-･･i･-･･････････････J･･･-･･･r･･････････-････････････.･-J.. (13b)
'W'iO i=':u -.ev<a'kl'7 '''''''''''''''''''''''''''''''''''''･-･-"-･--･-･･J･･--iJ････ (13c)'
2･52 Koji OGusHil'and Goro MiuRA
The sh.ort eircuit point defined in Eq, (13b) is quit,e unre],ated to all
other receiving-ends, not only in thi's case bu't when expifessed ona eonduetanee or a suseepdJance eireie diagra,m too. Howeveue, the open
eireuit poi.n't l'}as interrelatrtons with otl}er rece:',viii.g-end admit'tanees
or volta.cr,es as shown byEq. (13a), whiel.) is one o/ff t'he mos'LL, dis'tinet;.lve
eharacte]rs itn {ihe ease o£ an in{;ercon'nected transmission system to bestudicd now. If an admit'tance unit is eorireetecl to a poxve:r uv.fit in
the eoordinate, t}.ae z"esu],'ts a]fe s2milar.
Centers o.ff tl],e el,reles are loeated on a peirpendieular biseetor ollg
-these two points, xnThere the eeiiteffs show power factors, awad angles
intersected by cireles show the difference of powe.℃ fae'tors. Standai d
dlagrams among.' these power fac'tor eirele dlagrams are the O. p.£. ciyc)le
diagram arid 'the 1.p.f. eirele diagram o£ which the cen'ters anct radii
are analyzed later as they are qu:ite impovtant, ' Regarding the eonstants C' an6t A,', dterivect up to zioviT, 2t is an
espeeial}y 2mportant fac}t that voltages, adnii'tta,nees aRd line consta,nts
of other reeei'ving-ends are all eontai(ned in the constants as eons'tants.
There is, howe'ver, one more method £or assuming constants, namely,as will be seen in Ilict, (15a) derived' below, av.+i=: --tpt(l)ilC-+ttli(-g'/4-li' contains
at sendin.cr-end adiniteanee ?ifi and a reeeiv2ng-end admittanee zv.,i, wheire
O'/, and L-" are assumed as constants vgrhieh cot'itain voltages axict admit-
tanees and Iine eons'tants now on sending-ends. Frozn Eq. (15), how-ever, the followi'ng equa'ciova is induced.
?V' = Si.Z,g'v']i:'ISIalS,Y#IL ''''''s-''''-''''''''''''''''`'''''''''''"'''''' (14)
It is equal to Eq, (13) :lf 'the system is composed of liL,kes eve]ryone o£which has one termilnal. .A.s it is in quite similax irela'tion wi'iJh Eq,(13),
a proeedure in detail is to be omi't'ted hejre. C" and D" a.Tre der,ived
iff.rom Ect.(15a). [g]he short el]reuit and open eireui.t polnts are irespGet･ive]y
zff,..=--:LW-L ･-t･s･･･-=･-･･-･･-･-･･--･･i･･i･･1･･fb･･p･･-(･･････-･･･-･t.(14a)
ett ･･+･-･･････t,.･･.............,...........,.,,...,....,... (14b) aeqlo :=:: A
i'n which, howeve:r, both the values a]re ehangeab]e aceoxdiing to loads,
dissimllar to th,e preeeding ease.
Simila]rly, a reeeivi'ng-end adm:Lttance has the foBowi,ng interrelatio7i
with a sendlng a,dmi'ttanee.
`
Electrieal Charaeteristies of Tnteyeonnected Power Transmission Systems. 253
where
aff.÷1 =
-CtlH--/g.Zifi
ulL)t/-Baifi
If powereew,,+i == G,. g t-' jB,･+i
£ollowi'ng
ThoughPOi. 11't iS
1. p. f. cirele
laiter.
In steadis derived
In this
in other
presumed,get
Next,'tween
.....-v."..-...."......""...,"...".-- (15)
ev1 = -qcr･vl)rmC2(r+1)e,y,, '''''''"'･ -C.(.+1)esh･
l-Zfi!o･+i)Zgezies,, ''''''''L'''-'' +A-r(?t+i･)2e9.esiro
-Z;t/ = -Di(7･+b+-Y'Pn(o･--i)(i,g,.p '-'''''-' "l"-Zib.<.+r)eg,,.
m-BL}<,,+i)?M-ies,.-''''''''''''''''i' -B,･(.+beee,.eseg..
es = 'I]] , ･=-･･--････i･-........., esh,= FZI)･,
..............･t.･･･-=-･････-･･････-･･････-･･ny･- (15a)
eirele diagrams a]re dscawn on an admit'tanee eo-ordinate,
,the cireles are shown as a g.roup passing upon the through sho.vt eireuit and open circuit poin'ts.
.4.Zf9c7･+i)oo == --Iz-}-- ････････-･･-･･･-･･i･･････-････････`-････････t-････` (15b)
' ･crl/?･V(,i+bo'= - z},, '''''b'''''''-''''''''''''''''''''･''･'･･'-････-･-･ (15e)
the shoz't circuit poftniris a, eonstant one, the open cireuit
changeable aecording to loads. The e.p.f, eircle diagram and
diag,ram whieh are standard diagrams here, are diseussed
' o£ Eq. (l.5), 'the followihig equation can be used too, which from Eq. (13).
-- (]Yt + A./1t71 Z"7i+1=:: b-Briii-UL '''''''i'-'''-L''''･-.f･･=･･J,･･..･4-･.･.,..,+･.(16)
case, however, an assuihption that C' and A' aire constan't, or
words, vo!tatL.;es and powe.rs o/f reeeiving-ends are eonstant is
as directly seen in Eq,(13a). Like 'the ease oiE Eq. (15), we
/., Zifcr+i).. =-r ndr .fi' ''''''"'"''''''･ -･･'t･･-･･-････L･･-i･･･････t･･±･-･･J･ (16a)
up(.y1-)o = - Sli- ･･････-･---･t･･d-･T･-･=t･･t･････-..J.....,.....,...... (16b)
' ' a, b.ran.eh aclmit'ta,nee, ?fgi(,.+i> is "taken fo.r a braneh Iine be-
the sending-end 1 to the .veeeitving-end (r-E-1,), and the inter--
254 ' ecojiOGvsi{ian,dGoroMiuRA
relation of ?･if..i to a receiving-end admittanee is exainined,
e + G)zv,.+i e'ge1(,',"+1) = /gl+.BthII/T/ '''''''''''''''-''''''''-"'''t-''''''L･･-･iJs4･. (17)
Po"Ter £aetor cireie diagrams n,re shown 'by' eireles passing' the fo}lowing
one each short circuit and open el cuit polirts.
D- 1･tPl<.+1>.. =' -B' E"''''`''-iL''''`''`'･i･`･'nyiL-･･･-,--･･t･J･･･-･･..,,.. (1'7a)
tt ' e ?Vi(.r+i)O = zg.t '`'''''='''''''E''-'t''''`'`'li'''-'''''''''''''''''''ny (17b)
Though the short eircuit poin't is a cons'tant one, the open eircuit point
On the contrary, if a' reeeiving-end a,dmittanee is shown fromsendi'ng conditions as a func'tion of one of sendi'ng-end admittance,
-C-+Azv, aV<,･+1)l=: Lisz]}nrBwwenEbJ '-'''L4･･-････t･-･･dt--±･･iL･i,.,.....,,,....(18)
where, eireles passing bo'th as shorl; eircuit and a,'n open ci.rcuit po.in･t
UV<r+i)ae ir' r' iilli '''''L'`''''-'''''''''T-''''i'''･･-･b･･i･･=･･t････････ (zi-sa)
C ?if(?,+1･)o ::= e ff]}i7t -JL-,,･.,L.,-......i`.-･J,-,L･,...,,･-,,,.=..,--:i･･J. (181I})
are power faetoz' circle diagrams. O.p,f, and 1,p.£. eirele diagramsare explained later.
While, the relations between sending or reeeiving admittances and
b: anch admittanees are shown also as follows, on the contrary to 'the
above expression:
zifl .. (]!+-Z)t/?'ffC.r+1>1 ,.....,..,.,..,.,......,,.,,.,.,,..,.,..,...,.., (19)
A+Bzv<.+i)i 'and ?q,,.. -'Cl+Z{'ZViCr+b ,,,....,,,.,,.,.,･.･･ ･････'-i'''''''''' (20)
D-B?.ifl(,,.1)
Shoul; ei]rcuit ap.d open eireuit points are respee'tively
.z])tt afsloo == B .,..,.....,,.,.,,.. ....,.,....,,.....,,....,..,,...., (1.9a) C ZifJt} = A
s'
Eleetrieal Charaeteristics of lnteveonneeted Power Transrnission Systems, 255
and
A., 'e･V(,,+1)oe = - -Bmu
..".,-".,..--.H.-.".･･･--･･--･･････ (20a) ･ c et"(.+1)o =- -?) 'resulting in constant valuesll' o£ the open eircuit poiil tsl
The correcton of coordinate units to the power <li'+3'Q) w!ll be
made merely by multiplication of the square of every terminai voltageto the original admittances; such as lscil2 for `etsi of Eq,(13), 1.ll7..ii-L'.,for
zt7..i of Eq. (15), etc･ On the other hand, the corre.eticn to the.current
units may be made by multiplica'tion o£ vel'tages. IE power output is 'expressed by matrices, transposed matriees are used such as [P+oQ]=[llzi],r-uz'] ==- [-zl!j,[[ w] [-Izr]]*, In Eqs. (9) and (10), every term ･element isto be multiplied by 'the square of voltage if the power units are desired.
Example 4.
The examples of admittance eircle diagram will be explained byreference to the Ur. Generating Station for the Sp, and the Su, [{'.raps-
forming statio'ns. 'rerminal admittances are already shown in Exam' ple
3). Selecting every one terlninal as a parametez' and assuming o'therE2g5Il)liialS aS COnStaiits, cireuit eonstants are obsained by Eqs, (13a) and
･"p, -. .0,+.Dti.:b,., ,,,,
'calc'nihtions show circuit eonstants as follows:
/ A, = 4.20 l-470EYtllLi
' .B := o.sgl92017' ..,,.,,..,..,........･･･ (E13)i
C,=:16.13j212027, ･' ･ . TZ), = 3.60 L 50
Accordingly,
1(,Il]gl::J,:,SI'l,i:::,8gli,Zi ･･i.i･iiji･･.････.ii,･･･J･id.<rci4)
E
256 Koji OGusHz and Goro MiuRA,
b,)
ev,= 0+D2v7 ･ A+Bzv,Circuit' eonstants are
A, == 1,73 l3701IZL' ' Br=o.o4461-81om.. ., , .. .... ..0･=5.22l-1605?ri. ... P==O.283l-8g' ,.･ili1.'.i･' ･･
And ..,,........./,, w,..E:==iiZEr=17･7lwh900 ...,
zv,, Er? =- -S-C,- E7 := 8,42 l--S4009, i /
c.)
,･ ･ 'zus :=: -=..tt.t,+A!.if"i.
.i..... , ,.. ...,,, .f(.tt,,-,eszv, ..
'wheJre .,,'- ..,.,,, ... ...,･...
･.,,i･ A..･=8.80l5048t i･･.････':･ 1'Blr=O.89l92017,7 ･･････ 'i･ CY" =- 21.9 l-4t7o4ot 4-･t .D" == 2.46 i 25020,.
z{ssJoit = - --A-BLEg- = io.7 l9' 3fi3o3!.d
zeg,,Eg' = -::-)-1}E;= g.601 lo7o
tttd.)
,/･ zif .. -CL'-+A2vi. T bn-Bzviwhere
/t =o,891I 18P..P.-.
B =:O,0446i.--810 /' ･,. ･i,... ,cr, -: 3.22 123.IO-39(･
B,,-=O.591-490 ･.i･i･,
---T--i' 44------
･･･ny (E15)
.... (E16)
,.,. (E 17)
.･,.......,...,.,... (E 1.8)
.,.,.,.,.,,,.,..:... (E 19)
'
EIeetrSealCharacteristiesoflntereonneetedPowerTransmissionSystems, 257
I'f,:: .'E7'lc ::l 'ZI2, l-gO-sX･-,n,- l ･･･s.･･････････-･･･4･･-:' (E 2o)
'e.)
' ol -. Dl!?.vs IVi=i= A+n?eos
Sinee th/Is equation is derived .ffJtrorn that o.E e.) by transfo.vmi'n.g inverse}y,
the eircui't consta"ts are qui'te equal to Eq, (E 17).
And
zv,..Ei=!B-1'Ei'=:=7,72L=.66os7, ' ,
,,,1.,.,....,....･･ (E 21) efs,, Ei =-- -Slil'ilE;' =- 6.gsl--s3o2st .
f')
, et･, == Cili'l,;;.i,1-li-L'z
The circuit constants are equal to Eq, (E19), for th,e same reason as e.
?if,..ZCii= iiBl2-!' E?=: 36,g l 32o
･,･･･ ･･ ･･ ･･-･-･･ (E 22) ?tg,, Elr' == -(IIi E? == lo,os l-sgoeoor
g･)
?e, =. -C'+4Lg'i ,?]') -- B ?tz
This is transfo.rmed from a.), and circuit constants are sh,own by Eq.
(El3).
z(,,..E]g' = - tti}-' Eg' = 5.10 l 4ooo3,
" li=-･･･-･t-p'-''-'' (E 23) o, Eg .-nv- 4,85 l 27027, zosoEg = -- ])
h,)
2v, = .L- (]L±-4-'7!,L,
j) - .B w,
Cijreuit eons{;ants are showii by Eq, (E15).
258 Koji OGusBi and Goro MiuRA
e"TooE7; = fi A.ise! E? = 38･1 lth61043'
..,..-.,..,...,.... (E24) ZV,, E; = -- X' 2Z]Tij' = 18,1 l-- 25o52i
Nex't, bra'neh adiinittances are to be studied, foir examples, only with
the Sp. Statiop. (reeeivizag-end) and th,e Ur. Sta'tion (sending-end),
'i')' 'i ' ze'Aij ;- S.,t?)nv?,-{itZ "== rS'maT:
A' = 1.73 l 37olrt"", ,
.B = o.o4461 -slo ..-.,,."" ･･L-･･ (E25) c -s,6i7 l-r9.El.1
I) == O.283 1 18gO
?IFi,..IZ]i' = X-1!]X' == 1.7.tl7 I -9oo
...,,,.., .,,....･.･･ (rc 26) af,i,n E7 =-[2iE'i= g,ls L6g.-ou43,
j')
1<.)
[I]hi.c, ;･s
u-ct+A.w, -. f,1 ?e･7,t=-:: f.1>-r:'?l}Lil,,'U- T'rc,,
A =O,891[ 1890 'B=e.o446i---812-"-, ' ,..,.,.....,.･･･J････ (E 27)c -5,67 l.-u9.S.ne?--
7-"=-O.59 1--490
?t'･nooE; = -- -AIB' IZ7; := 19,6 i 900rmny
' c ･-･･-･･ ･-･･････(E28)?{7vio ll7,i = -- rfiii, Zl]; = 9･421=e2Ja..9-
C11- .ZL}t/ ･l･v,, ?'{"= 1'tiJ-'Ill;sgi]1'-
't,iraRsi'ormed freoin. Cirevii,'t constfl,nts are shovk]n by IIIq. (]Il 27).
2t'i-Ei=Jiill.i.YE?i='::'36'91-u3'-uz2"9'fi'ru ,..,,.,.......-･:'ny''(E2g)
E'ii ==- 17.7 1--900e,tjE, E/r =
A
t
ElectricalCharaeter'istiesof,lriterconyieetedPowerTrans!nissionSystems. '2-59
l,)
-C+Atwn i' ' i"''' eff[r=" mp-IiE}'w""'i,-- '''' '' ' 'i'i"i
'Cireuiteonsta'n'tsa,regivenbyEq.(E25).'''' i'i'' evv)o E; = rm an"-ffi3,l'1 E//' = 38･1 l-61o43J
e{ rio E; =::- llli; ew,-' ::= lg,6Igoo (ti] 30)
From th,e prev.ious data, admittanee circle diagrams ax'e dJrawn whichpass throug"o. ･the si/].ort eiz'eu,!t and the open ci･z'eui't points. Those points
are s"aown oi} N-g. 8 t･o Ii'ig. 19 together wSth f'um{la]rnen'tal cirele dia-
grarns shownbelow. ', ' ' '' '' 6. Fundamentaft Clr¢uit Diagrams, Namely, O.p.f. .and/t]..p.f.Ci.rcleDiagran.)sof.,, ,,.,... 'gltterconnectedTransmissioguS.yqtepas. .' ....,,..,,.,.
' ' These are two eS.reles i'ntev,suJt.ing at xight angle eaeh other and
are standard e:/.rcles on which ma,ny so]rts o.ff eirele dia.crrams are based.
fThese a.tre stmilar to th,e c.I.rele :,n a ease of an ordiT}ary transmission
line witli one terminal on ea:h oE' the Cwo ends. Namely{ poines Qf
inteMseetion o£ the two cire}es are one eaeh short eireuit and opencircuit poin't, wh2ch are respect'ive]y a eenter o£ power eirele diagrams2,"8 ,a,ze"ii%".?,Ik:u.'r2"tst.gltl,e.i.e gi6a.g, ggms,･,.T,b･e cs."S,e,",s,O.f.,i,?.･ft t,W,O,,zir:･l9･g
circles are the axes which aye the projected axes of a sendi,ng or areeeiving rectangular coordinate by the method oE equiangular projection.
I£ P#Ii, the powey df the sending-end 1, is shown by reeeiving-endconditions with an a,ctmittanee o.E the terminal (r +1) to be a parame'ter
assuming all other receiving-end admittances as eonstants,
-･ PPCi="Pii+a'Q]i=-avi[-ieell]---ACi-'7rtiirDJ-]Ei}--Zz':tf:.IEil''1' ･i
=' -ICa`1' di 'i-kYg'"((-GGi'i`, i7SBB' ilt liS)n IEil' (2i)
'If powev factoif is zero he]re, G..i='=O is put used..
g･P(ii ==-:- -X','.OiXttl-,f,i 1-Z`J,IL'i
260 . Koji OGus!iT and Goro MiuRA
Therefore, wh,en B.,i is changed £rom O to oo, a locus thus obtainedindieates the O.p.f. circle diagram. Its eenter is derived by sub-sti･tuting a conjugated value of B..i----A7o'/B which makes V'gyli= Do,
that is (B.,i)k= -o'A'IB, into the above equation, resulting in
q)ii- .0,Bll.k,+.b.l,IX i-ELi2 ････････････-････-･･････-.......･......... (22)
I'ts radius is derived by the difference of the length from a eeza.ter point
to an open circuit point, that is by q,,,- Sl l.ZqlL' ,
' Rpii= rrl'il.f'E'.2A, it,'"'BD,-.-BAY12,Ei' ''''''''''''''t''''-''i･･･''yi･･L･i (22a)
'Sinee it can be prec:tsely verified that
M X+jY - X+jY M1-2+ju M z+ju7
where M:=x+ju and x,y, X, Y, Zand U ai'e respectivery real numbex's,the.ve.Eoi'e Eq. (22a) is equal to
Rpii =i-'ii,Xli.BdizC.B'nE]l' '''''''''`･'e･･･････････J･･L････i+･--tny (22b)
Since ALb-BC' f:1 on sueh an intereonnected transmission system,calculations beeo.rr).e somewhat eomplicated compai'ed to that of anordinary }ine with one terminal on every end. Nextly, a eenter and a radius of the 1. p. f, c.ircle diag.t'am is dez'ived
from Eq. (21) putting o'B..i==O, as follows:
qii == SB]tiz--Hb.kAX Ei･
...･.a..･･･-････i･･･Lt････-''''''''t (23) ftqii= =k4". -"."-pe0- E',;
These £undamen'tal cireles, C.ii, and C,ii, are the p,roJ'eetion of vect-angular eoordinate axes of l]lir,.+i=w..illl.a.ilL', the power of the receiving-
end (r+1) shown on the coordinate l?Vli.
On the other hand, we can obtain a diffeyent egpression wl).erevoltages, powers, and line eonstants on sending-ends are assumed 'tobe constant, tha't is, from Eq. (14),
,. um 0t+-Z)ettzif..1 ]L,=tffi1-pa[l - A+-Bmu･eff,.+i iM,i2 ny-･･･-･･･････-･････-･･･ (24)
e
Eleetrieal'.Characteristies of 1'nterconneeted ?owe'r Transmission Systems, 26j
Then in this case,
Ci)" = g.ajZl-:i-i,{-.b,giJ'i]Z-,g'k l.E,ILi･
Ls･-･d･･t.･±i･`'''-.i''`'''''' (25) rk..sc,i-BO' Rpii"= rmllk'Lu']B",+A,B Ei"
' ' X,iii l:2/t'-""l':',I ':'i <26'
where eonstants in equations inelude condl'tionp of all senaing:end
terminals,
Next, i.f coordinate axes oE, the sending-end 1 are pvoj'eeted into
the coordinate oE the i'eeeiving-end (r-{-1), fttndamental cirele diagrams
onthesending'-end1areob g.ainedbyKq.<15>.,'..,., ･,,
.'e'P7Q,･+b(.r+L)==])(r+i)<ribii-jQLr+ixr+i)=Zt7o'"1'scI'+]I'i' ,
b9-,2"..`,S.G,t,'tstrtg･-isi'Lik･-it ･･･････-････t--･-･･･t'"(27)
gf, g,fi,9./,the ,;e'Ieta-ij/dxes-,:f.".i,,tgl ;.p` 6ix6'ie dil" gram axe 1,z:ived
And if jB..i==O, those o£ 1.p,,f. cirele diagz'am are derived:
tt q(.r+i)cr+i) == .rO,.'F,?C.A'.S,tl. E･?･+i'','
Rqc･7'"")o'･")==,.t"z"),fk'1.J'1'L'tt'tl･'ff'se･i'+'
Instead of }I]q, (27), i.E Eq. (16) i,g us-.d
on receiving"ends are assumed constant,
Vop,.i,,,.,, - -.-bO--+S-('.G(//,ikF)i>. E
Cen'ters and radii o£ O.p.f, and 1. p,f,
;･ -"
:･;･ii･･tJ･t･･･L--･-ri･i (29)
wheye voltag'es
clrcle diagrams
and powelrs
(30)
ax'e resJpectively
e
2)62
q)(,･+1)(,+1) == - liti:BBkh ++2i2alk-)B[)k E;o,.1
Rp･･･+i)o･+i) =' Eil'i iA..',i.'iil'ilillliXlE;-s-i
and
eeBk-A!-Dk El.i q(r+1)co･+o =' --L)JB,+LZ),B -A,.b+jBC, E;",, Rq`.r+txr+i) = -bB,+D,B
are derived ifrom Eq, (17),
VPii(r+i') ::: Pi(r+i) + O'Qi(r+i) ::rT'
' epi..+i) =- -t-'-,"wh".-].'-%'- Ei
4.t-1]) -BC.-- Ei re1)Cr+') -mrm
A,B,+AkB '' 'C{ii(r+i) == :Ci`)Xk -th:411}- Ei
kk Rqi(r+f)== -nv.,,'i//i'/1-'.'Bz.C Ei
)Plr÷bi = P<r+bi + J'Q(r+i)t =
,1,,- ., q,..,,,, - - Sza2-icA.",tr D;..,
Ri)i･･+iy = ..U,,fl'i}ig11i'iJ"."k,tt E],7･+･i
(CB, - At-Zl)t',' .- E;.., (Z?',･+i)i "' -.Pt,B,+.JD,IB Rqo･+s･i:-=: :[-iZi9,lli'4'.tsR,,9LBiz",r･+t
capaeity, from Eqs, (19), (20),
Koji QGvsm and Goro MruRJs
".,......,,.... (31)
....,,........ (32)
In the next, fo.ir sending branch capacity, the £ollowing equations
e + .b zff At+Bti,".l,Ei ''''''''-'''r'''' (33)
.,.....･L-r･･･t'' (34)
.........,.....,.,. ..., (35)
Reeeiving b/ anch power is sh,own from Eq. (18) as follows:
' -C+A--/G-i+j,Bi) u;,., ･pi-･･ (36) rL'"--B(G[+0Bi) , / tt '
(37)
.,......,. (38)
On t'h,e contra]ry i.fi the terminal capacity is expressed by the braneh
s
The
The
qii- gtt.EDzl'."kE;
-x4b't+BC E/r Rq" = A-BkmuAk-B 1/ q<r+L)<･-+i> -= -!Sili`{kEpt,
tt t/ Rq(r+i)(r+i)= tt.D.A,'.'.z",,C.E;"'+i
Example 5.
Fundamental circle diagrams willby the data given under Example 4)1
!owing corresponds in each case to that
' '･- '--･･･ 1 nl,=O+b!-, . i At+Bzv, 'Circuit constants are shown in a).The O. p.f. circle diagram is given by
- C,B,+.DA.k, qJi - 4,B,+ALB
Eleetrical Characteristies･of Intereonnected Power Transmission Systems.
' }eq= C+.D"W<re)i E:.･-････''''''''''''''''''''''''-''''-'''''
A+-B't(2(,.+bl ' eq..,=:: -O+A'?'Vt(r+.ntnvE;., ........,........,...,..1..,.,......
D-,Bzvl<,,.,)
/O. p.f. cirele diagram is defined as
qii-- C.Bi,+.DilAifo E]]
AD,t-BO E,T; Rp" = AB,+A,.B (JV{r+b<r+,b ==,; .t.,i",'.ASI?i E,7･"
Rp(e'+i)(r+i)=-,."ii?.'.r",.CE;i.,. ''''''''''''''''''''''i'
1.p.£. circle diagrams are also
be calculated The heading
o£ Exampld
E;. , .
Example 4), that is by
Eli ;-o.g42-j'il,9 >
, } ------ltr-l--
(39)
(40)
(41)
(42)
263
. , ., .,..,････････ (43)
l'
.....･･.･･.･･･-･l･･ny････ (44)
in this Example letter in the fol-
4).
Eq, <E 13),
(E 31)
264 -,'･ ,KojiOGus}]iand･GoroMivRA･
' A,D-.BC, E; =1,63 Rvii = A.iB, + A.k.BThe t..p.f. circle･ diagram is by
.S:/rt.･t2:.irlA".:"ti.,ES,101?,3,7H-o9･4s ..............(p,.2)
,,, .. .A'-]B,-AZ-B
(J!'+.Dzts 7 E/r 'Pagi'= , A.t+-Bet?, . t tttggrmCX. iie9nC,tOsnSat,aen. tiamaree, ShOWn //nder b･) Example'4), by Eq. (E ls). These
The O. p.f. circle diagrain is:
qii - .0,:i:H.llf.Z E,r -i2.6-ji6.g
I AtD-Bcr E; ..12,6 ..,Rpsis= A!B,+AZBi
is The 1.p.f. cirele diagram is:
(%ii - SBi,iDAts 4,7f -o;4sl -j lo,gs
'Rqss ! ''..",lli2-".E,.IZ',E,r == 6.76
c.)
'' '' pez,== '0Llt±A'eifi EgTheo･p･f.cir6ie'idiag'ram: P"-Bl{i ''
qss = - .Dcr,IB.B:++.ADbzli Eg"= -3.s7+j i2,s
' -A.Dt,+BOt RpssT- pt,-iB,+pxBEg ==3･37
The i.p.f, circie diagram:
,(IUssi -ev,.':,li2if.",S.l,,'i E,7--i.4i±jg,s>}
''''''''････ (E 33)
.･･-･･･ny････ (E 34)
.･･･`･･･････ (E 35)
････････････ (E 36)
Eleetrical Characteristies of'-Intereonne:tgd Power Transmi.ssion Systems.
Rqss=-m-i-il;9i'/li,'.Bb2B'Eg-i･43 )
'd.)
gV,, = Lev'+-Alvi E,; .Dt!-.Bevi ･, The O.p.f. circle diagram; ･
2,,:'1-IH'/TZ/i/¥+//,/7,i'l,E,.tiZZ.+jii･･s･ .,..,........-,,,,,
The 1,p.f. eircle diagramH:'
crttB,-ADZ' X,11i'-i"s",IDi'/3'i"tlSo',ikifl/1;i,Zlll'j5'50' ...,........,･//,,ggs
e,)
nc.,= 0'+D"Zifs E/7 A+-Bwp The O, p.£. circle diagram:
' ' qi)"･:=!llli'ill:-i:7ilEli!!'B !Bk,-!i.lilSkEtr=is･gsv'is･s .
Ri)ii- A.Dilli･{,l.C,k' E,7 .. ,,., ･････････-･･ (E3g)
The 1.p.f. circle diagram;
()hti == Eii'."BBk,EAb,'iAkE?i 3,s3-j6.3o
･･.･･.,･･････(E 4b) Rqii = =ilillkD,ll".,gS. i.cr'Ef' = o.g4
f')
v.q .,. 0'+b"trp7 Er A + .Bzu,The O, p,f, circle diagram:
265.
266･
g,)
h.)
Koji OGvsm and･Goro MiuRA･
q,, -= gi'i:+.l21'flk E;- .. ..
R,,,= AAjBDi･'+-AB,CBY' E:- =oo
Thq 1.p.f. circle diagram;
qii = eviBBk,--jzit igk E}' =- 18.4 +js.sl
-AD,X+ Bcr' E? = 19.1 Rqii = A-B,-A,.iB
- cPt -. A,zv, Eg- PKs = D-Bw,Th.e O.p.f. ciycle diagram:
' C{pss = - (DYllBif.l +ttll?i- Eg = -6.go-j1.42
Rpss=: maDAJB'2D++jDB,BC'Eg'i=11.g
'The 1.p.f. circle diagram:
qss== :i!,{IX,i.Ai81kE,r--4,13+j2.76 l
Rqss=iSiel++DB,0BEt?=o,s64 1
l/t .- C? + A, ?ff, pg, == E;- .b - B tff,
The e. p.f. ,circle diagram:
()lo77==-0DBBi++iSIDBkE;･=.. , l
1 --AtD+.BCt E; - oo Rp77 =: TDB,+.L),BThe 1,p.f. circle diagram:
････････-･･ (E 41)
････････････ (E 42)
......･･････ (E 43)
････ .･･･-････ (E 44)
･･･････････- (E 45)
Eleetrical Characteristies of rnterconnected Power Transmission Systems.
.q,1:ww-ft"s}iilt'.D,xtlEb5';7as208] (E,,)
i`)
Nex℃, braneh power is handled the same as in i.) Example 4).
eq,.. e+Deff, E? At + Bl{p, The O. p.L eircle diagram:
.Cl.':::/lii31tiSii,ItiA../lotk.Eg===26,,iE3886 1 (,,,)
The 1.p,f. eircle diagram:
CUi7==Ili.Z,Bk,-HDAfl"BlEr---4.76-j3.6s' }
Rq`7:IiliA,lili2.-+AB,,IS"E';=.i4.s j '(E48)
j')
k,)
-a+Azvi E;- V'ifi = Dtt -- BwlThe O,p.f. circle diagram:
q)7i == -b!ll,liBk,+.Aii:/Z,,Zi E;-11.7+jg.sl
-AD,,+BC Rpn == E;- =15.2 D,,B,+ DktB 'The 1. p.f. circle diagram:
()tB.-ADU q,1 = E;- = -15.7 +o' O,88 -rZ>t,B,+Dk'B -.DnA+BO E; .,,,24.7 Rq7i = HubttB,+.bXB
,.,.......･･ (E 49)
........,..- (E 50)
267
',268 'Koji OGtisHi and Goro M!upA"
pmi･, -' C.'.Z."iP,,I,Ii Er-
The O.p.f, cireleg'diagram: L olo. =- CBth+]])"Ath E;:= oo
R.,,=.,rmAABri.XAitLE?=,., ' (Esl)
A.B, + A,B
The 1. p.f. circle diagram:
#,ill rigt/if,t?l,it2i.Eb; {.5 :,'I088 cE s2)
l.)
"a= -C'+A'zffi7 E; .)L)-.Btei7
:The O.p.f. cirele aiagram:
qb,, == - .X"i,:,,'P/s E;- -'oo
..........･･ (E53) g, Rp7f= IfoAillll".'.i,?BC E; - oo
' The 1. p.f. cirele diagram:
q77= H9DBiiwAIS. l,i} Eij=,g･o4Tj6･g3 '･ ,,
Rq77== iiDjD:.i++jBD.SE;=2s'i (E54)
Regarding branch' power obtained now, if some explanations are
made circles i) are expressed on the bxa'nch coordinate, Pif -FjQiT ;-6itcIes
j) are on-PTi+oQ,i;cirslgs.k) are concerned with brapeh admit.Yanee zv,,
expressed on Pii+o'Q]i,;; circles 1) are concerned with aesi7 on'Pw+o'Q,,.
TheSe fuhdamental circle diagrams are shown･in Fig. 8 to Fig. 19 to-
gether with short eirel it poi'nts, open cirguit polnts, anq with'effective
power circle diagrams and o'thers.
Electrieal Charaeteristies of. rntereonneeted Power. Transmission Systems. ,269
7. E'ffective Power Circle Diagrams of Inter'connected
/ '' TrarismissionSyste'ms. .
First, the effective power on a sending-end will be expressed byquantities on receiving-ends. The effeetive power on terminal -Z" is
2P. = ap,,L+･4Lk
t''I' '' ''''' ''' ''wheire za and .]rl are given by Eqs. (5a) and (5b). As is .cle.arly seen
from'Eqs. (5a) and (5b), eq or uZ] varies with the load voltages of Tnany
receiyin'g-ends. Ther£ore, it is assumed that only,a load on the re-'eeiVing-end (T+1) varies with other load voltages remaining constant,
iS.XtipCoesstjhbeie?OnSideratiOn of every .voitage at any given timg is obviousiy
,Then,.,this effective power can be shown in the power coordinatesby methods similar to power factor eircle diagr4rps previously digcussed.
2Pii::=(AluiEIo･+i)k+BkXr+i)k)(0-EC'+i+DZ'+i) ' ' ' ' ' '''1(A-'-E).+i+-iBZ･+i)((i)1;-ELL?･+i)k+Dkllr+i)k) ''''''`''(45)
where.A' and C' are constants shown in Eq･ (13a)･ If we put n.+i)k-IE.+i)t/=･
-P(r+i)(r+i)+O'Q(r+i><?･+i), ,qryi)-Z?r+i)k = (P("'r+ixr+b+Q:'r+i)(r+i))!IEo･-1!, eq･+i7IV?r+J)k
.== IIil..i12 substituted into Eq. (45), then
' 2Pii=:(AZ-D+A'Dk+Bl'C'+BCk)P(r+ixr+i)
+(Akj[)--A,P,--B,C'+Bq)o'Q,,,.,×,.. '' / '･+ P:'?'")`rfb,,+..,QIg"r'")(･r"i) (B,D+.B.D,)+(Ake,-Atck)iEi..,[2.
If it is defined
E,"-,., R2)r = .....................,.,.....:.,..,....,･･･i･ (46) B,.D+B-D, t .t A!D - Bcr R2)(o･+i)(r+b = E,":.,･ B,D + BD, q)(･ +b(･ +i) -=- 9,#,k,+. f'.D,k leq .,i, <3i)
itthenfollowsthat ', '2?ii,Rpr+R;)cr+ixr+i)=P<"'r+ixr+i)+QZ'r+i)(r+i)
' +Pge･+i)(r+b(Ak'D+A!Pk+BkC'+BC'k)R.,. +jQ(r+i)(r+i)(AZuD-A'-Z)kmmBkC'+JBeZ)R,,:
･270 ･,' ' KojiOGusmandGoroMiuRA-
= ( BO,BD' ++AB'rMDL)7 Ertil2+ P(r+i)(rk>+ o'Q(r+ixr+i) )
× (-BCITB,++tk',IDD IEr+i12+Pcr.+ixr+i)=rjQcr+i)<r+i) ]
/ If scalars are converted to vectors, ' '
rprEj¢'= -()li)(?i+ixr+i)+(P(r+i)(r+i)+5'Q(r+i)(rTi)) ''''''''''''''' (47)
Eq. <47) is the effective power circle diagram drawn on a coordinaeeof the receiving-end (r+1), where C.(.+i)(.+i) is a definite point obtainqd
as before and IT..1 is changeable only by an angle ¢. under the.ptidmise
o£ the effective power Pii being constant, Radius lr..l is
. 1rprl = V24iRpv+R;cr+i)(,･+i) '''''''''''''''''''''''''''''''''''' (48a),
Then, the effective power P]i becomes ' ' P,,= ("pr+Rp(r+iXr+2i});}iT,.pr-Rp(･'+i)(r+i).) ,....'.,,........,....(4s)
When r..=R,(.÷ix..>, then P"=O, that is to say, the sending power be-comes zero, iand ,the ci'rcle eoincides to th,e O. p.f, circle diagram･di' s-
eussed in Eq. (31). In the area outside the O,p.f. circle, Pii is Positive,
and in the inside Pii is negative. It is a maximum value when r,..=O,
when the value is ' ,,?iingax " R}/'i`)`1''t" :(U"'rZ'5f:1:;.),(.-.".:'f?f'""q) [E.+il2 (49)
In the former case voltages or powers of receiving-ends are assumed
constant. If those of sending-ends except end 1 are assumed constant,
then from Eqs. .(6a), (6b),
4.., = - g,]l4 +AZ
E.., = -D,LII4 -BZ
Y:':A,t.'･ilSi･,l.,==th`ff,X,E;.L'.,, (so)
are derived just as in the former ease. If these equations are used
for ealculating the effective power, P]i, corresponding to Eq. (45) be-
'comesasfollows: '･ ･,,.
Electriea}Charaeteristicsoflntereo'nnectedPowerTransmissionSystems. 271
2PiiAtik=(Ak'EZfr+i)k+Bk'qr+i)k)(C"jEtd+i+mD"Z･+i) ' +(AEr+i+TiBZ･+i)(CL'M(r+i)k+rD"qo-+i)k''''''''''''''''''(51)
As an intermediate calculation is 'the same as before, it is here
neglected. Then the effective power circle diagram of the terminal(r+1) drawn on the coordinate P<..b(,.+b+j'Q(.+ix..i) with assumpti6n that
P"isconstant,is ･, ' t/ ' 'The 6entTe'
;EI.g"' = -{,);<""i'(""i'+(P(''i'('"i'+0'Q(''`)(' ")) ' '' '' '' '('s2)
' C;(r+i)(r+i) t IH Iifl/llill)lk, .' A-i}l,l,'Ll,}, iEr+il2 ''''''''i'',''''''''''''''1' (28)
'and the radius is ' rS. = 'N12PiiaakRS..+R}',(.+ix.+i) 'r'''''''''''''''''''''''''-''''''･: (53)
' ,R}" = iB,p4,/'.'"iB)' k･ '''t'''''''''''-'''''''''''''''''''''''=''''- (s4)
'
Rp(o･+ixr+i)= S,9)',',-+B-B:iii'k,E,":+i ,'''''',''''''''''''-''･'･･'-'(28)
' The effective power of terminal 1 is then
p,,= (rSr+Rp(7'+')s'2la'l(Rtli:,HRp(r+i>(r+i>) ･･････-･･････`････-･- (ss)
When P" == O, the radius beeomes T..= R.(..i)<..i), and the circle coincides
with the O.p.f, circle diagram discussed in Eq. (28). The maximumsending power is aequirect when r..=O, as
' P"max==t";2('ia"iiC';･"il'=='2(B,.Sf51';"t]ii.DkJ)=tr"'''''''''(56)
The method will be applicable for all other terminals of sending-ends.
Next, an effective power of receiving-ends will be drawn on powereoordinates of sending-ends. For an bxample, P<..i,c..i) of the receiving-
end (r+1) is drawn on the eoordinate Pii+dQii o£ the sending-end 1.
' 2P(r+b(r+b=:LZZI.r+t)k4r+i)+Z!lr+"-Zl?'+i)k
Y,.+i) and -Zl,.+i) are derived from Eqs. (6a), (6b) and (15a) ･under the
premise of assuming each sending voltage or power as coestant except
272 ,.liKoji''QGusm and'Goro ･MmR'A"
'the sending-end'1, as shown ih Eq. (15a). '
2Pc;･¥,s(4..,s =' (-Dk'za,--B,L-,)'( -'atu]EI, + A.L)
+(D'LMihrBZ)(-Ck'E]k+Ak.Lk) '''''''''''･`''',t'''':'''(57)
While, P<.±ix.+i) ls expressgd too by assuming receiving'poivers 'or
voltages as constants 'except 'the receiving-end 1, namely by use of Eq.s.
(5a) and (5b), Ei =A'JE7..i+Bl..i, Ti =C'E..i+Dl,,.,, whieh 'are modified
asfollows: ･, .,,' HDjM,LBI, 1 Z"+i -- rA,fib-Bc, =-2il, (iDEi-BL)
Z･+i =' '-it'-.-]Z4-'."6)iii..,L2i/(.;b,uic,.k,) ･''''''''"'r''. (58)
whenee eonsequently, . .t. tt 2P(r+r)(oT+i> = ti,1tik [(jDk-iwrik--BkLk)(-([>'Ei+A.I"Zi)
+ (bE, -BL) (- Ck.sc,,+ Al.L,) ] -･t････ ny ･･･-･･--･-･･-･ ny･ (59)
Therfore the effeetive power circle diagram on the receiving co-ordinate is obtained like the above on the sending ceordinate. Ftoni
Eq, (57), obtained , ., Tp,ejips=-Ci,ii+(Pii+o`Qii) ････････････-･････-z･･･t･･･J･--････････ (6o)
' tt 'From Eq, (59), obtained ' '
rS,£j¢g = -C!1. +(P,,+o'Q,,) -･････････････ny-････････････ny･･--･･･ny･･ (61)
,whence
7"ps == 4A'R;)ii`'-2P(t･+i)(r+i)Rps '''''''''''''''"'''''"'''"1'iL','r''.(,62)
' ' E; H' Rps ='･ AB,+A..B ''S''''''"''''''''''-''''''-'`"'''''1'.1',r: (63)
t/ tt A.D,,-Bctt ,Rpii''=AB,+A,.B'E?', ''' ''''1''' ''itil,','tC'/Bk+-Z)"A,E; 'i,''`''''i'''r''"T''''''r''･1:(25)..
, ,,dB,+A,B..･,..., ,-･-,. ' Tl)s = "/RloZ'-'`2P(r;ix?･+i)d'tikRl)s' '''''''''''''''''''''''''''(64)
R;)s = x,B,E+;AkB .-････i･;････d･･･ny･･･････････････III･･t･. (6s)
Electrieal Charaeteristies of･Intereonneeted Power Transmission Systems. 273
t' Ri)ii-'.A.l,i,ll-.."'kei ･E7r ', ･ L･
qii -L- .`Y,B.k,'.".it?k E;- (22)
,Froip the radius o,f the circle of ,an effective power on the x'eceiving-
end (r+1),
P(,..i)c..,)== (Tps-Rpn)(rps"-Rpii) .............................. (66)
2Rps ''or
p,..,,,..,)==: (Tl"s-RiintL(ig:,'Rl'i') ･-･･･-･･J････-･･t･････････････ (67)
If P<.+ix,i+i(==O, it coincides to the O. p.f. cirele diagram which has the
radius of RS" or R.ii shown in Eq. (22) or Eq. (25), and in the inside
of the circle, the power is positive, and on the outside, the power is
negative. Of these tb,e maximum value is gained when r.,==rl,,=O.
' ･･.p,,;.,･,,,;.,,.,. =;- ,Ri);; i,- (""'t-,"(.(]i'),(,A.fi{,llXs "A;q:) zl]'i (6s)
P(f+ix?+i)max=2d4diO:i,,=='2(A,-B?iAkzr):::=RE)S (69)
Next, the power, Pi(..b, whieh, is transmitted out from total effeetive
power Pii of･the sending-end 1 to the receiving-end (r+1), is drawn
on the coordinate of ehe total power o£ the receiving-end (T+1), P(.+ix.+i)+0'Q<,i+i)(,･+i);'From Eq. (11), it is obtained
2Pi(r+i) Hpt' (A2-iEILr+i)k + -Bkqr+i)k) (CL](IL･+i + -DZ･+i) +
(A'n･+i+BZ･+i)(CteM(r+i)k+-Dkqr+i)k) ''''''`''''''''''' (70)
In a siMilar way by which Eq, (47) is derived from Eq. (45),
rp',･ad'fi;"' = -qc.+ix.+i)+(Pc,-÷ix,･+i)+e'Q(r+i)(f+b) ''''''''''''''' (71)
is obtaiped as the effective power circle diagram.
Wbile, op the contrary, the power 'P(..i)i which is a part of thetotal PoWer of the receiving-end (r+1) and is the one transmitted from
the sending-end 1, is drawn on the coordinate 1. FroM Eq. (11),
2P(..,), = (Dk'.]Vl, - B,Zl,)(- CM, + AL)
+ (Dt,E,-BL-qE,,+A,L,) ･･･････････-･･･････････-･･･ (72)
is obtained, and then the effeetive power circle diagram is shown by
,
,
274 Koji Oqvsflr and Goro MruRA'
whe're
ttri;sejpts .,, - C';;+(Pi,+3'Q,,) ･･･-････-･･･t･･･････}･･-･･･････････････ (73)
TSI･ :'= V 2Pi<r+i)R;r + RS'(7'+i)cr+i)
RS'rh B,IDEi"kLDk' ''''''''''''''
A,D-BCR}'(,t+i)(r+i) =' E?.., j[)B,+ -D,BCfl,,,.,,,,,.,, - - Siiliiik,+.Ail6Illili; E;',.,
' tt
...,.............･.･.,･-･･････ (74)
･････････････-･･･････････;･･･････ (46)
............,.........･.,･i (42)
,
(76) 2P,, (A.Dt,-BC)(A,])1'-B,()Z)
where D" is a constant eontaining receiving-end powers or voltages to
be assumed as eonstants. In the same way, also P(..i)(..ih, the effective power' of the receivi'ng-
end (T+1) can be shown on the coordinate ofi Pic.+i)+o'Qi(.+i) which,is
the branch power transmitted £,rom the sepding-end 1. Us:Lng' £romEq. (11) that -EL:=:A'n..i+BZ,.i, and Ii(.+i):r-CLjEll,+i+-DZ-+i,
(D,za,-B,.L,,..,,,)(-CLzq+AtJrl(..,))+･ (Dve,-Bq,...)(-qn,,+AIL,,..,,,) ･･･ (77) 2Pcr+b<r+i) (A,.D-BC)(AkD.-B,q)
, TS`,:r=VRI;,2-2P,,,.,,,Ri;, ･･T･t･････････-････････t･･･････････････(75)
Ri)'s== ArB,E+'!A.,B '''''''''''''''''''''''''''''''.'r''"'`'･''････' (63)
Ri;i'= f,9,'t-AB,ZE7
.....,..,+.･t････et･･'･''-' (41) cl,',,:=-T CA/B.Bk;.D-'IAii}k Er-
If Pi(,-+i) or P(.+i)i becomes zero, the above coincides with the O. p,f.
eircle diagram of Eqs. (41) or (42).
Viewing from another point, the effeetive power Pii of the sending-
end 1 wil] be drawn.on the eoordinate of P(.+i)i+o'Q(.+i)i, which is the
braneh power of the reeeiving-end (r+1). As from Eq.(11), is obtained
eq..,= b"-ZZL --Bq, IC..,), = -- CM, + Aq, the expression o£ Pii becomes
(Ak-E"t+i)k+Bk'Ikr+i)ik)(ewu'r+i+D"ZIr+i)i)+
= (Ajli,+i+BX.+i,i)((ZeM,･+nk+DkZr+Dik)....
EleetriealCharacteristicsof･Interconnected･PowerTransmissionSystems. X'75
is obtained, Circle diagram equatio'ns are derived in the same wayas before,
tt ' rlo';.ejP;" = wq(.+i)i + (P<.X+i)t +o'Qc.+i)i) '''''''''''''''J'''"''-'''･' (78)
ttl toy, . ttt Tfo'slEjhS =-q}i(r+i)+(?i(r+i)+S'Qi<r+i)) ''''''''''`''''`''''''''(79)
whey, e･･
/-l)1 -Bk()tt) R"; + R;)<r+i) '''''' (80)rp. == V2Pii(A-Z)"-LBO)(AkDli
mm, orl,+ADkt9`';'l.]l 7I [ibnB,+Dk'B E;"+i
R}･,i..･･ 7 ..,･,.4ir'+lbz･B
'ii
ge,.,.f,,, == S,tkl'.-,B.t,,.E;-+i
' t/tt t ,.,r;,' =r- 'v! R;i,(.+i),
tt, Cpi(r+i) =' .C,",.kii".Akk' E?
R;,G = A,.iBjlll+l:'A.2B
A!uD-BC Rpi(r+i) := E? ,,,,. A,,B,+A2B,
The circle of Eq. (78) concerns the O,p.f.and the circ}b of Eq. (79) concerns Phat of
Lastly, each branch effeetive power of
a coordinate of each branch effective powerc
terminals. However,' '
circlediagrarpof ' ''omitted here. As power circle diagramsdrawn on each brarrch power coordinate, jtbranch effective Power cirele
these coordinates simultaneously. Those areconcerned with ordinary transmission lines.,
.....,..,......･.･ (37)
-2P(r+i)cr-i)(A'DHBC)(AIDk -B,C,) Rfokl. 1,(81)
ttt -tt t
.................. (34)
circle diagram of Eq. (37), Eq. (34). terminals can be shown on of sending or receiving it is the same as the case of drawing'ail 'ordinary
an one-terminal transmission line, so explanations are
diseussed in Section 3 are may be convenient to draw diagrams or eMeieney circle diagxams on nothing but the methods Namely, a fundamental E;- .,circle on the reeeivingaend coordinate is with a radius, R.. := DB, + D,B
276 ･.･' ･･-KojiOGusmandGoro･MtuRA･
anda･eenter, C.l ==- (CDBBk,++DA,DBk) E?,.{; and that on the se,n,d}' pg-enq
eoording,te is with R.,=- zB.E+J'!A,B 4nd C., = SBBZ++DA.$,E;; from
which an effective power or a txansmission efficiency are derivid out.
In this case, however, it is quite noteworthy to see that the voltage"
on terminals which are interconnected by braneh lines with each other,
should coincide to a eertain value obtained from certain mutual relation-
ships, This is not an arbitrary one on effective power circle diagtaM's'
but is restricted by all other terminal voltages o£ the.systerp.
Example 6.
From the da'ta previously derived for the main power transmission
system in Hokkaido, now effective circle diagramS will be shown, Asthe calqulation is made only be equations ,is this Section, it is done by
a routine manner as in the former case. The heading letters of thefollowing also corresponq to those of Examples 5) and 4), The valueof effective power, P, which is to be maintained as a eonstant in the
diagram, is assumed as a value under the normal operation .,which is
obtained £rom zvEL' or IE]k by use of Eqs, (El), (E2) 'or (E12), as is just
mentionedinthelastogExample1), '',･ ,. .
' '' -･-･･ rSiej¢'=-q,,i+Pii+O'Q. , '･, '
' R;)i:::: A,B,Ei2AkB =o･4ss ･-･ny･･-･---･･･-･-････-････-･･･(Ess)
(Zl,,, -- -O.P42-g`11.9, Rl,,,-1.63 ･････`･-･I･･･.･････････. (Egl) ,ill''l.
P,,･=5,68 , ,..-. '1 ' '''/1'/''''i'' ,',,lre,l=･viRfo,?+2L,ti'dkRi.,=8.02 ,, ,,.,,,,.., tt t tt ,IA,l==:IAtmZ])-.iBCTI=3.33 ･' . ･' '. ,i, .., i'ssq,p. = RE" ==･ O.244, which shows the minirfiuni power, i' , '','
' ,Jl.i' ..rLiej¢'=-qii+P"+o'Qii,.,,, - .,. ,,..,,.,.where
---
and ,･･
c.)
where
;-'i
and
d.)
where
and
te
"
Electrical Characteriseies of･lntereohnected Power Transrnission Systems.
Eii Rfo, = = 38.2 ･･･-･･･-･:･i-F-････-･･L･････i････ (E 56) AtB,+AZ-B C(1)ii=12･6-0'16.9, RI,ii:-T12.6 ''''''''`'･'''''･'-'･'･''･-'･' (E33)
' P,,=0.985 -1･･ ''' .. Irl,,l=･v'Il{ll,,?--2ttlEtt-'d-,'Rlr=12.g ''
' ld'lr-O.33 '' ' . . '･, ' ' 'b,,'Ak.'-'='tt''s'':--ig.i'i'' '' i
2 ' ' ' r;,sejb' = - q,ss + Pss+j'Qss
' t tt/ ttl Rlos = B.D,FlB.jDz =o･63 -･･-･t+････････････-････････.- (Es7)
Cfo,, -- -3.57+j12.5,' RS,, t:3,37 -･･i･-･-･･･-･-･････････-･･- (E35)
lrfosl =:-r Vr 2PiiuakRfos+RSsg = 13,7
1id ,i e 5･35
t tttP,,,.,. =- =!lirifi-5 = O.315 which shows the minimum p,owgr.
I ,. rS7EjP' = - (Zl)77 + P77+0'Qn
R;)7=: -fi,D,Fk'B-D-:,T == 22･o ･･･････-･････-････t････････`･.･.. (Ess)
q},. =8.78+p'11,5, R;,,=11.9 -･･････････････････t･･･i････-･ (E37)
L,:4.88. . .. I/fll=-'. ol:(s422P"ddkRlo7+Rl'v2' =: 14･3 , .'. , .',J.
277
278
e .)
where
-e-
and
£･)
where
ve-
and
g･)
where'
Kojj,OGvsm and･Goro Mivm.
Ptl.,. = :!ilirl!7)7 =-::- n.o
2
Tpiej¢ = - qii +Pn +j'Qii
Rpi ='r AiB,ti'.di,.B ::-=-2･90 '････････--･･･--･･-･･････････････(Es9i
Cpii = 15･95W'15･5, Rpii = 15}3 ''''''''''''''''`'''''''''' (E39)
F,lTpsl = iV' R;iiww2PIifiR.i = 14･2
Phsmax= R}'" == 40.4 2Rpi
, r.]edP==-qil+Pii+o'Qi`
'R.i= AB.l+i"2
A,B =:p.Q ････!･･･r･･････t･････-････････････l,･,'(E6o)
' t ':C,ii = R.i, = co t･････････t･･-i-････--･-･･･････--･･････-･･-･･･i･.: (E 41)
' ttt ttt1dipail=VR;)ti H2LvRpi =I oo ･'
PT7tlla: ::= !ilZ/1!)u :oo
2Rpi
''i.sEjip=--q,ss-Lpss+j'Q,s /'x
.t ttt ttt ' 'Rps == B.bt5:'BD, == 3･55 ''r'''''-''･･････- ････････････ (E61)
(](,ss == -6,90-o'1.42, R,,,, =11.8s ..,..,........,........, (E4b)"
,g
t--
an'd
h.)
where
---
and
i')
where
t-b
j')
where
Eleetrical Characteristics of' Interconnected Power Transmission Systems.
IT.,1=V2P',7R,,,+Ri'llgll,,=13.2 ' ' ' ･
Pmt}fix == IIIIii35., = 19･7
rp,ej'¢ = -Cv77 YP77+p'Q,7
tt ' Eij' RpT= JB.D+.iB]zl/ =OO ････--････････-･･･ny･･･････････b･･･-･(E62)
(L)w= RpTT = Oll' '''''''''''''''''tb'''''''''''''''･''･''･''････････ (E45)
lr.,1 = oo
tt P"max = oo
tt t" ･r,';;Ej"'" == - qn+Pi7+0'Qi7
Rl)G' =' A.,.B,t'"'.,,tkB =:38･2 ････････････--`-･･･t･･････････ny･･ (E63)
' q,,, = 26.1-o'8,86, R.,1 == 27.6 ･･ny････-･･････････--･･････････ (E 47)
1ri;ii = V R:.'nr2P,,zi'"dZ"Rlo'I = 26.8
1tir/,l-p,72EI ' ' /t 1 r;;7'ejP"'=-C.,i+Pn+g'Q,, ' ,,
tt . ' ' ' RS'; -T z)'me,E+7;'bk･1?' = 21･O ''''''''''''''''''''''''''''''''' (E64)
qn = 11,7+j"9.81, R.,i = 15.2 ････--･･･-････i････-････････-･ (E49)
P,, = 4.88
･279
2SO
k.)
KojiOGusHi and Goro MtuRA
le-
where
IrS;I -
iA"'l =
･vi 2P,,A,,tdk"Rlo'V
O.694
7'1;,E
+R?," =r- 18,3
M == -C ,',i+l'ii+g'Qii
R;;, oo ･･ ･-･･-ny･･･-･-･･-････-･･･････-･･･････'-････-･･･････-･･(65)
(ZSI, Rfo',, oo ････････-･･････････････--･･･････････････-･････････････(51)
･:sg-'ITlo',I oo
1.)
7-1;,Ej¢" = - Cl; +P,,A--jQ,,
where
R;;, ::':- oo ･････-････-･･･････-･-･-･･t--････-･----･t･--･-･･ (66)
Oi;,, = Ril =- oo ････-･･･ny･･`･･････--･･･,･.･･....................l..... (53)
(･ :. i7'lo'71=- oo
Those circle diagrams are shown in Fig. 8 to Fig. 19. It is clearly seenthat all the cireles pass through their operating points of power"bn
their eo-ordinat.es, the reason for which is that the,effective powers used
in calculations are of the normal steady values given in Fig. 3 by draw-
ing ordinary pgwer circle diagrams. .
8. Transmission Power Ratio Circle Diagrams Between Terminals, Synthetic Transmission Edicieney Ciccle,
Diagrams, and Apposite Terminal Voltgg.es gf, [.
Interconnected Transmission Systems..
t .. Effective power circle diagrams discussed formerly can easily betransformed into circle diagrams which show'the ratio of a sendingeffective power to a reeeiving effective power. This diagram becomes
the eMcieney circle diagram when a system is composed of an ordinary
transmission lindS each with tWo terminals. In the case of an inter-:g.n:,ees･2d,,,t.ra."ew,tsfilo,:,sKgtgge,･ ,i`6 2g,s,oggied,･.d,oe,i ",i `,,';h.?･.w, igh,e..vsr,g
ElectriealCharacteristies･oflntereonneetedPowerTransmissionSystems. 281
'' However, it can be completed in the same way as that of an
ordinary transmission line. Namely, on a eertain O,p.f. circle diagram
discussed before, a group of circles that intersect in reetangular with
the circle whieh interseets the fundamental eircles in rectangular and
which centers at an interseeting point of a vertical axis of the co-ordinate
with a horizontal line passing a center of the O. p. f. ciycle diagram, and
that haye centers on this horizontal line, alre all supplying a relation
that ratios of the effeetive powers between terminals are constant,These circle diagrams on sending power co-ordinate are differed fromthose on receiving power co-ordinates with only the directions of 'signs,
If an example is shown with the sending power ratio circle dia-gram, which is derived from the effective power circle diagram, upon9,8,e".d,tnS.g,O.W,ge.,gO,-Oggi¥a,gei,figQh,.a,S･,if..E,q,g.･ ,(47)･ (52)) (7i)･ (78) and
V2PiitidkRpr+R;)(･r+ixr+t)=-Ci)cr+b<r+i)+(P(r+i)(r+i)+OQ(r+i)(r+i)) ''
whereitisassumed -- '' '.(82) qJcr+b(r+1) ==-ap<r+])-O'Pp(r+b '
Ifalsoaratioofpoweris s, , z7=P{?'+'"r+'i ......."...,.....,.............,...................,l,,..(s3)
. 2PO''t>(r'1>･ti`clkRp.+Ri,o,4Tx,.+i'):--(P(.+ixr+i)LFap(r+i))2+(Qcr+i)(r+{)-'l?p(r+b)L'
' As it is modified to the .form ' ' 'I/'(''R;'}(r+ixr+b-a;kr+i)+,(ti4LvR""Tatp(r+i)(?･+i))2･. ,.'.,..1'','''
''･'
'' '-'::::
(p(.l,)<.ItsH(id4sRtt' mcta)<,a-s)(,+t>))2+(Q<･}'+i>L'+{)+'Bpt<'r+i))2 (84) i
ttacenterandaradiusoftheequationbecomeasfollows: . ''
rv,･ = VR:p'<r+i)(,･+i>-a;)crti>+ (tiAkvR""',t--at.c,.,.i))2 =-r Vra'2i.d-a'ij<,..b
･ (85) cv., = (EIEtlfovRp7, - a.<,..i) ) -j3,(, .i) == av(p+i) rms(?p(, +i)
'Ifthisequationisexprgssedinvectors, ',･,･ rvr ::= Cfvr+(P(r+b(r+D+3'Q(r+i)(r+t) '''''''''''''''''''`''.'E''-''`''' (84a)
2S2 ･ KojjOGvstnandGoroMiuR,x
On 'the contrary, let us expres's a sending power ratio cirele diagram
on a sending-end coordinate from Eqs. (58), (59), (70), (77) etc, which
are the effective power circle diag.rams on sending-end co-ordinates.
For.example,.if an effective power circle diagram is shown by the fol-
lowmg equations,
l 21Il,IIilll'2i`i.1.?lll';ptxi'dehR}'s==-C"ii'(Pii'jQti)l.....:...(s6).
then, if power ratio, Eq. (83) is substituted
R;"+2vP]id'zikR.,=(Po-a.i)2+(Qii-P.i)2･
By modification it will be as follows:
R2.ii-a2.i+(at.i+rpR.,A'Ak)2=[Pi]-(a.:+vR.,A'idZ))+(Q,i-B.t)2 (87)
or ' r2v,= (Pii-av,)2+(Qii-B.i)2 -･･････t････････--･･-･･････････････ (87a)
where
r,,V,i, i.i II..Fi'ihH.stfo,',l!i,la:i,B.T,R;'s:1",//",ti;H;,," -aZs+a2o:l･.-... (s,)
Aeeordigly, the sending eMeiency eircle diagram is scepresented bY
a group o£ circles whieh have centers on a line passing a point, C,mthat is the eenter o£ a fundame,ntal circle diagram, Rfoii.
Example 7.
This is the drawing of the sending power ratio eiacle diagrams in
Fig. 8 to Fig. 19, to show theix synthetic transmission eMciencies.Calculations are due to the equaeions given in this Section;the eMcieney
rp is to be taken as that of the normal steady values obtained from the
ratio of the receiving to the sending power. '
a.)
'rvj= Cvi+P"+o'Qii
- i7 ==P,,IP,,==116%
.q,e,,li,=-:.-,a.lo6gjBl}i==-O･942-jii･9 ) .....................(,,,) ･i
ElectricalCharaeteristiesoflnterconneetedPowerTransmissionSystems, '2s'3
Rfo, == O,488 ･--･････-･･････-･･･-･･･-･--･-････-･-･-･･-･･-･--･ (E 55)
1ift1= 3.33
' .'. ()'et == (al,i+vRl,id'dL)+o'Bl,i =5.34-o'11.9
' ITvil =:: V Rloi-al,l'+(ct;,i+vRSiA'dfe)2 :=: 5.50 -････-･･i･･･--･ (E67)
b.)
rvi = C!vi+Pgi+o'Q"
V =P,,IP,,==2o.2%
i qm=a;,,+o'3fo,=-12.6w'16.9 i R;,,,=12.6 f ''''･'･･･-･･t-････-･-(E33)
R{,, = 38.2 ････････ny･ny`ny･･ny･････-･･･-･･･J-･･･････････････････t･･-･･･ (E56)
ltit1 =::: O.33
:･ ()?i :=(a'.i+rpRl,idAk)+s'3",=13.44-d16.g
lrv]1=VRfoit'-al,:+(aSi+rpR;,iA'din)L'==13.44 ny･････････-･(E68)
'
Tvs = ()os+P6s+0'Qss
V ==Ps,IP,,=116%
where
Rq.,g,s,-,.-313/1)sV'BSs--L3･57+0'12･5 ) ,...........,.....,.,.(II]3s)
l RS, =O.63 +･･-･-･-･-･-･-･-･････----･-･-･･-･-･-･---･･ (E57)
1dl =r- 5.35
.'. (],, =( uakvRli'ELS -al,,) -o'Pi,, == 11.93+3'12.5
Irnbl ::-r V R;,,g-al,g'+( Ad"pR;'s -ai,)2 =: 11,ss ･････`･･････ (E 6g)
Also we can obtain a,s=:1.18 whieh is the radius of a commoninterseeted eircle in rectangular with the cirele C,,s6 and with all thesending rower ratio circle diagrams.
284 Koji OGvsg! and Goro MivRA,
d.)
r,, = Ci,,+P.+o'Q.
v ==P.IP]i=20.2%where
Sbll,-.,'Illts,)7HO'BS7=8･78+0'11･5 ) .......,......,......(E37)
Rfo, = 22.0 ････-･･･････････････---･･･-･--････-････-･---･･-- (E58)
iAl =-7 O.542
:. (Y,,=(dd;Rl'7-a;,,)-o'Bl,7=40.68+jll･5 ' ''
lrv7[= VR;,,;-a;,?+(.4.i.sleml.,i -ae,)2 - 4i.4 -･±･･-･･,･･･ (E7o)
'e.)
Tvi = Cei+P:i+jQ"
V ==P,,IP,,=116%where
.`i:::.ifg･'.i-j3p'==i5'95Vi5'5 l (E3g)
R.i = 2i90 -･････-････-････-･･･=･･････････-････t･-･-ny･-t････････････ (E59)
･'･ Coi =(api--rpR.i)+.iP,,i =- 12.58-3'15.5 '･
ITvil== V R;,ii-a"ii+(a.i-uR.i)2 == 11.7 ･･･-･-･･･････-･････-･ (E 71)
f')
Tm :'- Cm+P"+OQ" rp =P.IP,,=20.2%where tt ' ...gili'LR"ii,i==Rlllli==oo ) (E72)
Ae'eordingly,the circle coincides with the effeetive power'cirele diagram.
Electrical CharaGteristics of Intereonnected Power'Transmission Systems.
g･)
r,s = C,s+Pss+e'Qss
V == P,,IP,, =-L 116%
RCT.p:s,.-.'fi.as2gs-S-Pps==--6･90-O'1･42 ) ........,.,..........(E43)
Rps =3･55 ''"''''''''''"`''''-'''''''''''''""''-'"''"-"'' (E61)
:- Tvs = iVfR}ss-a2.s+(-:liv}0Lr -at.,)! == 10.4 (E73)
h.)
rv, == Cv7+ P,7,+p'Q,,
V ==P,,/P,,=20,2%
where . (rpv7 =r Rpn,== Rp7 =' Oo ''''''''''''''''''''''''''''''''' (E 45), (E 62)
.'. (l]r,7 ==1r,,l= co ･････-････-･･･-･--･･t-･･･---････-･･･-･･･-･･-･･ (E74)
Eq. (E74) coineides with the effeetive power circle diagram.
i.)
' rvn = Cvi7+Pu+0'Qi7
'･' i7::=P･nlPn=-77% ' '. .1 R(JIp.r,i.aiex6i=jBim=26･1-j'8･86 l ･･･････-.....,..,...(E47)
Rl;,' = 38.2 ･･･-･･････--･･-･･･-････-ny･･i:-･･････････････････-･･･････- (E63)
･ICvi7==(api7-vR;;i'd"riZ")･+o'B.i7=10,8-3'8.86 . ,
lrvi71= VR;,i7-a;,i･r+ (at.i7-vRl;;ri"'Ala'')2 == 12,O ････････t･-･ (E 7s)
j')
r?p =:: Cmi +Piri +o'Q7i
V ==P,JIP,,:= 26.3%
285
･2S6
where
--"
Koji OGuF.m and Goro･MivRA
q,71 = - ap71 -otpp,1
Rpn == 15.2
Rl;,' = 22.0
q, ,,, - ( ti"'tai,"Rfo?'
== 11.7 +o' 9.81
-atp71) -0'PpTl ==
lJ' ...........,..･･･････ (E 49)
51.8+j9, .81
ITv7il= VR;n-a;7i + (A"'ta;"Rlo'; H a..)2 = sl.3
k,)
Tv7i =: Cvn+Pis+OQii
V =P7i/P,,=26.3%
where
(';;ii == Rlo1i-R5'i -= oo } ". Cwl :T- IT,,,I=co IIt coincides with the effeetive power circle diagram.
r,, == C,,+P77+o'Qw
V =P,,IP,,=77%
where
Cth = Rl;,, == R;;,= oo ･-･･･････････････････ny･････-･････
.". Cn7=IT?71 =- oo ,･"ny･ny･"''''"'"''''''''''"'"'"''''
It coincides with the effective power･circle diagram.
in Fig. 8 to IFig. 19) together with the previous
any other diagram is drawn under a eertainthe same procedure as in this example.
9. Transmission .Systems.
The method of proeedure is quite similar to
f'
......... (E 76)
t+t
･.･･･････ (E 77)
(E 53), (E 66)
･･--･････ (E 78)
These sending power ratio circle diagrams just considered are shown
diagrams. O£eourse, value of eMcieney v, by
Reactive ?ower circle Diagrams of Interconneeted
that for effeetivepower circle diagrams, from whieh the only difference is that of treating
Eleetrieal Characteristies'of Intereonhedted'Power Transmission Systems. '287
reactive power in this case. Therefore, successive equations will be
derived heye as in the former eases.
If, first, a load of the receiving-end (r+1) is only changeable with
other reeeiving-end voltages or poWers to be assumed constants, thereactive power of the sending-end 1 is expressed in the following way.
2Q,, = (ZljEL,--Z,.El) = (0Y,.., +- D.Z;..,) (Ak.E,,,.,,, + B.Z..i,,)
-(ClfiEir+i)k+bre'eqr+i)k)(A'mZl7r+i+BZ,+> '''''''''''''''<89)
A' and C' are conseants defined by Eq. (13a). Substituting relations EJ(.+i)kT.+i=P(･r+i)cr+i)+O'Q(?･+i)(r+i), iEr+i)i(r+i)k ::=
(P7'r+ixr+i)+Q:r+i><?'+b)!lEr+il2, and LZIkr+bk-Erlr+i)=EEr+il2,
-Z4,･ 89 = (AkD -A"Dk -Bek + BLO) P(r+ixr+i)
+ (AkuD + A'Dk-Bq -Bthev) O'Q(r+i)(r+i)
+ P2'r+`×'ffo,,!,IQ..?'+'×""`'(B,Z])-B.D,)+IE..,12(AkC'-A!CL),
If it is defined
Rq'=HDBE,;"i]D,B ' ''･ (go)
-k,b+BO Rg(r+i)(r+i) = E;-.., -DB,+b,B Cq(?+i)<･+b -- -`Y."x."iSl}Ek, (32)
also with attention to <R,.)k=-R,., it followes that
2QiiRqr + R;(r+i)(r+i) == (P("'r+i)(r+i) + Q:r+b(r+i))
+ P(r+i)(r+i) <Ak'D mu A'Dk -Bq + uBkcr) Rq.
+jQ(r+ix,'+i)(AZD+A"Dk-riBCk-BkC')Rq.
+ {(-DBh- DkB) (Akcr ri A'Ck>' -(L A'D + BO')(- AIDte + -BSk)) R:.
='-(-"iiill;;',d.BiiS9.E?･+i+(P(r+i)(r+b+jQ(r-pi)cr+i))(r.A-ieH-.B,2'E;'-+t
+ (P(r+i)(r+i)-jQ(r+ixr+i)))
I£ sehalars are eonverted to veetors, a vector equation is
TqrEjPr :' -Cq(g･+b(r+t)+(P(r--ixr+i)+3'Qcr+b(o'+i)) '''':'''''''''' (91)
where
e
288 Koji OGvsm･and Goro MiuRA
lTq,･l=V2Qj'iRqot'+R:cr+ix,-+i) '''''''''''''''''''''''''''''''''(91a)
, The center point of Eq. (91) is equal to the eenter of a 1.p.f. circle
diagram.of a receiving-end coordinate. The equation expresses thereactive power circle diagram studied now. Also from Eq. (91a), the
reaetive power o£ terminal 1 is ' ' ' ' Q,, ., H m.(.r-qr-PJ..Rmg-(r+i)l(e:.t.t`)R) ,(l.q!:.:.:.4tt'"i'("'b･- '･''･'''':''''''' (92)
Inside of the circle, R,c,,.ix..i> represe'nts positive (capacitive) reactive
power, and outside represents negative (inductive) reaetive power. The
maximum sending power is ob'tained when r,.=O as
Q]i-ax = R/?/'f`iif'",'i== (-A'u"'2"(Iillb/1-.iilli'klXk'Bkq) IE7r+ii2 (93)
, Though the diseussion up till now assumes receiving-end voltagesand powers as eonstants excep't the terminal (r+1), nextly the case
which assumes sending-end voltages or powers except the terminal 1aS constants is to be treated, In this case from T,..i==-C"E]s+AL,E.+i=D"Yi-BZ, we get, as previously derived
Mi -= Att";,i,'m]iZc'"` == {lirA'rcr+i+'BZ'+t)
....･.-･････ (50) L - 0'#b'i,;haDB'bf"'i - -}(e"JEL..,+b"Ib.,.,
By using those,
2Q,,dA. == (cr,eq.., + rZ])t'.iny.,,) (A,M.,.b, + B, ZZ,..,>,)
- (CL'-jEl..,,, + DZI(,,.,,.) (Aleq ., + BZ,.,) ･･････････-･ (g4)
Thereforee, the reactive power eirele diagram on the eoordinate ofthe terrninal (T+1), Pc,..ix.+i)+o'Q(,,.i)(,,+i), with assuming Pii as a constant,
is shown by the following veetor equation.like as before.
' ' Te.E"W" = -( 11(r+t)(.+b+(P(o･+tx7tg)+j'Q(r+i)cr+b) ''''''''''''''' (95)
t. ' ' q(..ix,･.)--Ci",B,x-ttii-tkl"l-E,+i ･････････+-･･･-････････････+･･ (2g)
ra,･:== AL/2Q,,dd,RG.+R;,t",,.,×,,.i) ･･････････+･･････････････････････ (96)
and
EleetricalCharaeteristiesofIntereonnected?ower･TransmissionSystems. 289
R4･･ == -b,,9//r,'+tI)k,B ････････J-･････--J････････････-･････-･･-･･ (g7)
' Ra(･･+'i)cr+i) = -t--L･.･:4--;-;-!ll-lzil･1'･¥-1-;i-->!,lo'-E;"･.i ･･･････-･･･-s･･･-,･･-ny･･･i･･ (2g)
The reactive power of the terminal 1 is,then
- (r6.+Re(,t+fx.+i>)(r6runRG(r+b(r+i>) Q,r - ････-･･････････-･: (98) 2,dd,RG,.
If Qii=O, we get Tp.=Rl,".+b(.+i), and the above cix'cle coincides with
the 1.p.f. circle diagram. The rnaximum sending power is reachedwhen r..:=O, that is on the eenter of the circle,
Qih.,, == -lliiii'ittS,;:-ib'- =- 2(.,l.E,"ii`:sla,.)- -getgx"- ･b･.;･i･･･････-･･ (gg)
' The above analyses are app]icable to every terminal of the inter- 'eonnectedpowersystem.., ''･ Second, a receiving-end reactive power will be shown on a powercoordinate of a sending-end. For an example, a reactive power Q(,,+ix.+b
of the receiving-end (r+1) is to be shown on a power coordinate ofthe sending-end 1, Pii+2'Qii･
2Qcr+ixr+i) = Kr+o'ZIQr+t)k--ZQr+bhswgr+i)
If za.+i) and q,.+i) are so expressed that all sending vo!tages or powers
except the terminal 1 are assumed to be constant, being containedto their line constants, we get
2Q<,,.,x,,.,) := (-C'tM, +AZ,)(.Z]>Z'uitzI,,-.B,.L,)
-- (-q'-j[IL,+A,X,)(Z),,"IZ7,-BZ) ･-･････-･･-････ (100)
Or if n.+i) and q,.+i) are sueh that all the reeeiving-end voltages or
powers except the terminal (r+1) are assumed to be constant, thenusing -swi == A'n..i+ -BZ..i, Zi == C'eq,.i+.DZ,.t,
-Du]EI,-BZ 1 Je;･+i - A,D inBo = -2ir, (Dma{ -Bi;)･
T"p' ::: :Iiii(,)q-i]ii4sv+7£bL=t, ('-cfn,+A,z) (58)
we get 2Q(?'+i)(r+';) = AIAs ((-(iY"Ed + A']l)('bk"Ziik--Bki']k)
' -(-qn,+AiZI,)(iD-U,-jBX)} -･･-･･･････i･-･(lol)
"
290 'Koji OGusiii and Goro.Miuf{A
From the first, Eq. (100)
TqsEjPS=HC/qii+(l'ii+3'Q") ''''''''''''''''''''''''''''''''''''(102)
and from the second, Eq. (101)
' r8,e'Ws=-(l)[li,+(P"+o'Q,,)････-･･-･･･････････-t･･････････････････(lo3) '
are respectively･･obtained, where
thrqs = "/ Rbu-2Q(r+ixr+i)Rqs ''''''''''''-'''''''''''''''''''''`' (104)
Rqs= A-B,tiSA,B ''''''''''･`･･････････････････-t･･････････ny･･(los)
-ADrt+Bo, Rqit= E? AB.--A,B ..M.".,..-....".････ (26) eu,,-: {ll'i'.B.:H-l7ii:k E･:
ofes :=: 'N/ ltts'f-2QL,.+ix,,+i)d'dER6, ･･ny･･･････････････ny･･･････････ (106)
Ras : A,-B,-'rAIllB ny･･-･･････････････-････-･････-･･･-･･･････････ (lo7)
-A,LD+Bcr RG,,= E7 AitiB,-AkB C6ii== I:l,BBk,ijADI,XEtr) ''''''''･････=･････-････ (23).
From the radius of the circles, the reactive power on the receiving-
end eoordinate (T+1), is
. Q(..ix..b=-(rqs+Rqi`R)(,Iqs-Rqii) ..･.....t.....,..,.........,(los)
or
Q,..,,,,..,,--(r6s'S21;2)S,tt:,iRe'i) -･･･i･･････････････-･･･････(iog)
If Q(,,+i)(.+i)=O, it respectively coincides with the 1.P,f. circle diagram
which has a radius, R,ii or Reii, and inside of the cierle shows the pos-
sitive (capacitive) power and outside shows the negative (inductive)
power. The maximum power is reaehed when T,,=ra,=O, of whichthe value is
'QLv'+ixr+j.n}ax == lliRI:ii, ''''''''''''''''':'''''''''''L'''''L''''''''''･' (110)
ElectricalCharacteristiesoflntereonneetedPovijerTransmissionSystems. 291
or
Q(r+')(r+i)max= 2id4,,iii･/'l G, := tq'S .''''-'''''''''''`'''''-''･････ ('xl.1)
Third, the reactive braneh power Qi(,,.t) which is transmitted from
the total reactive power Qi[ of terminal'1 to the receiving-end (r+1),
will be shown on a coordinate o£ the total receiving power R.+ix.+i)+O'Q(r+b(,-+b of the terminal (r+ 1).
, 2Qi(r+i)=(AZeqr+Sk+'Bkqr+bk)(CLieq'+i+rDZ'+i)
-(A'4r+:)+Bq?t+i))((Zeeqr+i)k+Dkqr+i)k) '''''' (112)
whence
tt ' 7AGI･ejPr==-CZI2r+ixr+i)+(P(r+i)(r+i)+j'Q(r+i)<r+i))''''''`'''''(113)
To the eontrary, the branch power Q(..i)i which is transmitted from
terminal 1 and is actually a part of the total receiving reactive power,
Q(.+tx,,+b, of the terminal (r+1), is likewise shown on the coordinate,
Pii+3'Qfi, of the terminal 1 power.
2Q.,.,,, = (Dk'HEI,,-B.Zi,) (- C!-E, + AL)
-(DVM--BL)(-q-EL,+A.L,) ･･････････-･･･････ (114)
whence
relEjWg'==-(J{1;,+(P,,+j'Q,,) ･-･････････････t･････････････････i･･(115)
where on both
re;.= V'2TQIITR,':;.Iill+Rekl.i)(..i) ･･･････････････ny･･-･･･････････ny--･･- (116)
' "E;"+1 L .....".J-..r-"..-･'.'-''''-'''''''"'' (90) Ra;･ = -.D-B,+ D,B
RaQ,,.i,(.+t)=HHJDAk'-D,t'L"2DBICB''E?'"` ...........'.............(44)
ql.+ixr+i)± -UjLB[)E+A.'DD,kBE;"+i '
and
Ta3=･viRd',it'-2Q,,Rak ････････`･･･-･･･････････････-･･････L･--･･･(117)
zil? ==Rqs '''''--''''''''''''b''''''''''''''''(105) R6g - AB,-A,B
292 ･'- KojiOGusJirandGoroMiuRA'
R:,,=- -AA]ii-jD-,1'+A-,BBCE!?
,,'.9Z'ii=tt]iik',Itt.','1rkE,ir･ . '''''''''''''i'････････････(43)
Moreover, similarly to the process of effective powg,e circle dia-
grams diseussed above in Section 7, Qii, the reactive power of thesending-end 1 can'be shown on a coordinate of Pc.+bi+o'Q(,,+i)i which is
a branch power transmitted from terminal 1 to terminal (r+1) andwhich is a part of the total receiving poWer bf th'e terminal (r+1); or
the reactive power, Q(,..i,(a.+i),o£ the terminal (r'+1) can be shown ona coordinate of l'i(.+s)+3'Qi(,.+b which is a part of Pll+o'Q" and is a com-
ponent transmitted from the terminal (T+1); etc, Sinee these theoretical treatments are .quite identi6al to those of
Section7"heywillbeomittedwithdetailshere. ,
, 10. Effeetive Conductance and Effective Susceptance''' '' PowerCircleDiagramsoflriterconneeted ' TransmissionSystems. '' As in the previous secti6n, adniittance eircle diagrams are discussed
assuniihg the power faetor 0 to be 'constant in a formula of w=:G+o'P
= lwleje. In this section, however, cases o£ assuming G or B to be con-,stants are taken up. geirst, G will be constant, calling the effective conductance cirele
diagrams. If sending-end voltages or powers are assumed constant, and, 'an admittance of the sending-end 1 is expressed by receiving ter'minalconditions with taking only the terminal (r-1) among receiviitg-ends as
l,P:e8,E,2,r/l,,.sg･,iinB,:+.IEi-si:EG,f,l･,2i:}Ei ais)
vKi=- [SIII+.iZgi,f,[i+.l.i!iiilli･;i E; ･･････-･･-･i`･d･･･････,t---･････ (iig)
whence the center and radius of a eirele are derived instantly.
qi] == .Cl",,i".'li2･IX+.2,:i.fif,G:ii; E? ,
(120) 'g'' = A.iB,+Aiill･iiif.B20.B'.B,a..,E?' == '2rBri･uiiR"asi!]･g'tdtit'i+b?
EleJtrical Characteristies of Intereonneeted Power Transmission Systems. 293
whgre RS, is given by Eq. (65) and R.,i by Eq. (22b). While, the ef-
fective c,onduetance G..i is given from the above equatiorps as
' G!"'i=-iZl/;i(2;}lo,l 2S,ii`'fo,) '''''''`''''''''''''''''''''''''(121)
In the like manner, if sending-end admittances are assumed as con-
stants instead of receiving-end admittances, from Eq. (14)
vz,= CY'+-D"zv..i Ef. A + BtVr+i
may be used, from which
c;,,= (pu"Bk+'D"4+2'D"BltGr+i El. AB,+A,-B+2BB,G,.., ............... (122) ･ A.b,,-Bc7t -, E? Tall = ABk+Ak-B+2-B'BkGr+i
and
G･+i- {lliii-i (21}.,i 2.lll,,pR",,,) ･･･x･･ ･･･････････ ･i･ a23)
where R., is given by Eq. (63) and R.i] by Eq. (25).
Nextthesewillbeshownonreceiving-endcoordinates. Accordingto wheather one assumes sending admittances as constants
- (JY' + Azv 'PY(r+i)c,+i) ::: thD'i}=Btv,' E,;i+i
or receiving admittanees as constants
-Ct+A.Lev 'PKr+'b(?'")= .b-BwYE;1+'!
there are two wayp of response. Namely, from the former, the centez'or a,radius of the'effective conductance cirele diagrazn is given by ii q(.+tx,･+i)"Li' ilj",,.71Iifrr,A,,,IE?f'`-",2./tXl,k.G,`EJ,7･"
･･･-･-･･･ (124) -ADtt+BCt, E,2. .i 7'acr+ixr+r) =:: .bt,B,+uDZllB-2BB,G,
Gi= t71 (2illo, "--- 2Til,l,"il,",,ii:;R"1,,) '' (i25)
where Rl,. and R,,(.+ix..i) are given by Eq. (54) and Eq. (28). Also from
'
294 - , Koji,OGu.sHiandGoroMiuRA
CI](r+ixoL+i)'=: -D(f.'BJB,`i.bl'Sf'2.2Bttlli/S,G' E,?･+i
rg("'l)("+i)=bLB.U+"llX,ZB)+-{ii'B,G,E;,., -'T'''''(126),
Gi= El/l:'i(2]£lp.± 2TII,,?i'i:il+'i')'i'p.) '''''-''''''''J''''''(1.27)
where R.. and R.<..i)(..i) are given by Eq. (46) and Eq. (31).
and Ci](r+i)i=: il69,iZMBs,k.--A,,ttk''-',2.rk."..k,G,i EZ+i
.･--･･-･･ (130) -.{tDt,+BC E!;., ,ra(,･+1)1 "' uD,,-iB.+Dk'B-2BB,G,
Gi == `IZ'll'i(2illifo.' 2Tlll,f,i'i;,i'R`s.) ''''''''''''''''''''''''''' (i3"
Rpc.+])t, Ri. are given by Eq. (37) and (54).
On the other hand, for a sending-end braneh admittance, £romEq. (33)
. ' C+-Z)ev Vag<?･+i)= A,+.BzElt,Ef" ･`･････`･････････-･･････････t･-････t･･.････(33)
and for a reeeiving-end branch admittance,
PKr+i)i== iC,,!.A-/."i` E,? "''''''-''''''''''''''''''''''-'"''''' (36)
are obtained respectively. Starting from those equations, their con-
duetance eircle diagrams are obtained in the £ollowing way.
q!(,-,i, == .C,#.k.+.-"./,gik･ +.2,X-".k,G.t･jtt., E;
.･･.,.-･- (128) AttD-.BC Efo rgl(r+1) = A'.Bth+Al-iB+2B'BkGr+i
' G!"'` =:: IZIIi (21ii,, '- 2TIII,:iifi;i';,,) '''''''''L''''''-''''-''''' (129)
Rpio,+i), Rb, are given by Eq. (34) and (6s).
Also on the contrary, even if terminal powers are expressed bybranch power as in Eq. (39) or (40), conductance eircle diagrams about
both cases can be induced in similar ways.
Second, an effective susceptance circle diagram is taken. It is quite
Eleetrical Characteristic's of Iiitetieohnec'ted PoWer Transtnission Systems. ･295
identical to that of the effeetive co'nC ixcta'nee circle diagram.just discus-
$ed pxeept for treating B as a constant ins'tead of Gi For an example,
that on al'sending;end coordinate and a receiving-end cgordinate is ex-
pressedasfollows:, / ' ' cz,, = evBz:-Z)Ai+2.d-ZPIS,B,,., Ei.
za,B, - AiB I- 2j .BB,B,,.･, 1<g!b'LBOl',, , ･･J･･t･･･t･･(132) E/7 ' rbl] =' . A!B,-AAB+2iZS-BkB..i ' ' ' 'B'""= lillil (L2Nll'lisi:･:l, ""-' "2'-.lll,fi}i Q,) (i33)
where R,ii and RG, are obtained by Eqs. (23) and (107) respectively.
And
i' ''ift,,i.'.'IIiiilll'Lr" i'l,/i/+l,711,iliE.l/SB,'"･;Iil. lil.lj ･i-iiiLi(i5ii･S
Bi== r`ili"1-'Li(-2]i¥6, ± zrlll,lf(:,l,i,'t',li'Q, ) (135)
where R,o,+ix.+i) and RG. are derjved f)rom Eq. (29) and (97)respectively.
11"Resis,tanceCircleDiagramsandReaetance , , .,CircleDiagramsofliitercpnnected･ , . TransmissionSystems. :,,,,,
In this case a reciproeal 11zif" is used instead of an admittance tv.
First, if receiving-end resistance is expressed on a' sending-end co-
ordinate with assuming only the resistance of the tez'minal (r+1) as
aparanieter, ' ･･ ･, , t t/ ' ' W,,.,. (pt'+!JItlg:,･,ti E?.. g'l-Zrif,･-,tLt･-+-)D7.,E]l2
A'+-BZV.+i A'ltifr+itB,i -- :',((R.:iti+.0,X.･,ti?)l]-.' zifi)-kb,.+.CIRili:l))-.-;-.:,,Y]:IO.,',E,o (i36)
The resistance cLrele diagram which is drawn by assuming R.+ias a constant and Xl,.i as changeable, is defined therefore by the fol-
'lowingcenterandradius. '･ '･
296 Koji OGusi･ii and Goro MiuRA
c;.,= (Y-Bk+-Z])"{ti+2(7A2R.., E;
AtBk+t4[iB+2A'AiRr+i r,.'.. `di'D--BC' E;.. E?R,,, '''(137) A'-Bk+a{ai.B+2A'AiR.+f 2R.+iA!iRSs+Ei'
whence the resistance of the terminal (r+1), R..t is calcqlated by
R""'i=:'-ii'i)']'(2kSs"h2Tli.iii¥'ll;I',) ''''''''''''''''''''''''''''''''',(i38)
where R.ii and R6, are given by Eqs. (22b) and (65). As it is nowassumed receiving-end voltages or powers are constant,.if the nextsending-end voltages or powers are assumed constant,
5` w,,--C.'till.i#-i'.tkfiE?-{llii'iR.･,+.i,+.Sj4.ies+.{l"E; (i3g)
is used. The center and radius of a resistance eircle di4gram is ob-
tained in the samg manner as before.
c;:,, .. U'gSA･+-]P"Ak+2CV'AAR..i Er?
Al-Bk+AkB+2AA'kRr+i ...,-････ (140) , AZ)"--.Bev, i'Z･ii = .AIR2'+A.,.B{-2Alts,ZR,.., El"'
R' "':== gA:' (-2 'k'ill,i 2rl,i)k,,,) (i`i)
where R.t{ and R., are given by Eqs. (25) and (63) respectively.
Then, that like to the previovs ease now on receiving-end coordinates
are as follows:
]iK.,,,(,.,, - --,C,n,,'((.R,t.+,g.'Xll))-+.A ,E;-.,
--,EAi.0.'#,k)sf::lilll,kE:- ･ ･(i42)
or
Pqt+ix･+i) -= i}(5',S¥;klXL+,.A' E,o.,
-(ES-.(.'R.･))P.PiU,',X., lf;･., a43)
The centers and radii o.f Eqs. (142) and (143). are respectively,
Eleetrical Charaeteristlcs of Intereonnected Power Transmission Systems.
q･(7-+1)(r+1) ==-ev!B,-Abk' + Z(Y,bl R,
Tr(r+1xr+1) =:
bttlS,+.kZPZ'.B-2.Zb,tDXR,
, -A.bn+Bot
E/7..,
or
-DUB,+.Z])k'B-2.Z)btiDI.'R,
-evB,-A,Z),+2CVZ),R,
E.1 .,
････-･ (14,4)
297
q:(r+i)(r+i) ""
' rr(r+D("'+1) Z)B,+J[),B-2I)b,R,Resistan6e, Ri is'accordingly,
R, - itZ'i,+i(,;}fo, ±,,,R,,21(i,:,ii`i,'su
ox'
Ri = mttti"'(2]l}.i ' 2Tl'il,oll(.:l-l:ii"tt'pr
iri Eq. (146)j R.c.+i)(.+i) and Rl"･
DB.+D,B-2bZ),R, ALD+-BO
E?,.,
EL? .,
ny･････ (!45)
.,...,.....,,............... (146)
) ., ......･･･(147)
are given by Eqs. (28) and (54); inEq･ (147), R.(..ixi,.i) and R.. are given by Eqs. (31)･and (46) resPectively.
Wi'th xegard to sending stet and a reeeiving-end br,anch line resistances,
the following equations are respee'tively obtained from Eqs. (33) and (36).
'P"i(r+i)= .l,il,t,Iii,:::'.lEir'=: iS,,i."Rif.i.'/Ii;iifi.".D. zi'; -･･･J- (i4s)
' VJeX,,,,,,==it±+i61,ifiE7-=zmuJ,(,][iii-.Fji.KM>)+-",,CE; `･(i4g)
Foti these, eenters or radii are as £ollows:' "' cr,.,,..,,:2.(),(i.iB--4.{ll.,'"-.E'.2,`.iZ,(a,te'.i,l/x}f.kEr"
, ･-･･--(150)･ A,D-BC r"'("i) 7,' z/rB,+AkBi-2A,AzR..,Ei2
' R'"i=+lill/L""(2;};,,i,2Tli.l,ili.:'1",-ii'lilo,) ''''''-''''''''''''''''''''(i5')
where R.i<,..i), R;,, are given by Eqs. (34) and (65),
'''''i' c);･,..-i,i = .s,C,.jB,k.-.:,I,lilk'EX!Il,llXLi,IR.: E,.,
-･-･･･ (152) ,,･ -A.b"+pBC .JOt,B,+.2[bZ.B-2.z),,rz]pL,re,E,f"･+i ' }Tr<r+1)1 =
298 ''-Koji OGusm and Goro MiuRA
R"'= l･Zti']" (mu2Mk'ri,± 2TR.i:[(iii,'k";,ma.) '''' ' ''''''' (i53'
/t t twhere R.i.+i), R}. are given by Eqs. (37) and (54).
This time, a 'sending and a receiving-end of the power will be shown
by the branch powez' by use of Eqs. (19) and (20).
'g･･eri･, = (lli/:ll/?li;',;?:,i'#tl,Elaflll fi:>;,l)'.H2,ii"' iE7lr' ･.･････････････-･････E･--･･･ (is4)
and -l P'itr+`)(r+i) == -zC, ((kRii'.:',','.';,X,ii,Y',i,?))IS' E/+i -'''`'''''`'''' (i55)
R(.+bi and -2Yk.+i)i are respeetively a resistance and a reactanee of a
branch impedanee !f'eif(..i)i, and so forth. If a result regarding Eq.
(154) is shown
･qi',,=:: Si:B2s,'.D.1',A,V,2.Cf,th.R,ttf:? Efo
....,･ (156) ,, monMBC Ei' r,ili l ABk+A-k-B+2AAkRi(r+i)
ft'`'"i' == tl'2 (fRi']V;, i 2ri;li ,,) (i57)
where ltS', and Rl;ii are given by Eqs. (63) anct 41. Thae ,regarding Eq.
(155) is obtained in like manner.
The same courses of analyses are ap'plied also foy reaetance cirele
diagrams, in the cases of which the vaMiables are resistances instead
of z'eacta'nces. F'or examples, 'the expression on the sending-end 1coordina'te with assuming receiving voleages or powers as constants, is
, m,.. (-Z)+jX;･+iO)+Rr+iU Er. (-B+j.2Y;.+iA')+Rr+i A'
' cv,, ,. evBA:-DA-L--2j(]!'"<gl.Xl.., E.,,
, Ar-B,-AZ.-B-2jA!AkX]..i '-,. .. ･ -A!.]D+BV , E? ･--･･･<158)
-,Eg.,-B.-A2nB-2jArA22C,., . ･. 'gnndd Y,itie ia.sSSUMing Sending voltages or powers as constants except the.
tt tt t PYii == (ie(ilii'-!L-f'Klklf.ICIiii),IFiifileii" Ei ･-････-･i･･････････････"･･･ (isg)
Eleetrical CbaracteristiOs of Iロterconneeted Power Tran6mission Systems． 299
α。＿．σ残一の”4－2ゴ。’」・エ・・研 ∠乱み乃一通ん」β一2ゴ∠産浸ん瓦，＋1
・・り・・（160） ， 一∠翻）”十み0’！ 研 偽11＝ 孟」㍑ん一通ゐ盈一2ゴ通．厘乃瓦．＋1
Also，七he expressions on七he receiving－e臓d（7十1）coordina七e wi七h
＆ssuming sending vol七ages or powers and receiving vol七ages or poWers
as colls七an七s， are respec七ively訊s follows：
翫・晦1）一（≦劣畿驚矯篇“Eタ＋1…一・…・…．（・β・）
臨・w）一一綴楽譜雲際篇鑑撫研＋1
・・・… 、（162） 浸■）〃＿．B《アノ 研。， 物（，・＋i）（ゐ十1）＝ 一」θ”一露ガト．置）∬露一2ゴ■）”■）yXl
ahd
肌一一（讐畿鋼籍・1一・・∴…一…（・63）
鯨一・．一越誰書雛篇戯驚研・1
・・… （164） ， 孟ノD一．霧（ア 研。1 γz（？・＋1）（7＋1） ＝＝ ・
一ηみん十．Pんお一2μ）．Z）ん瓦
Next， reg翫rding branch powers，七he react＆nce circle diagr勘ms forsending and receiviRgl branch Iines shown on．sending and塾eceiving
coordinates respec七ively， are as follows：
擁71い＋、）＝三＿（落≧±．Z墨＋1藍∠）．±1～？・＋1む 1i7号 ・・ 一… ＿・噛… ．・．．．・一．∴ （165）
多解，．＋1）1二
（み十3瓦．錘41）十E，．＋認ノ
（・4一ン墨lo）一壷璽
（一謬十ゴ瓦、ハ”）十RljDノノ
面hellce resul七s are：
一1馨＋1 ・・・・・・・・・・・・・… ．・・（166＞
・◎、、州）二紹rη礁一2ゴ畷聾瓦・1研
？4♂正（？・＋1）二
◎（，，＋b1二
∠量’．露ん一ノ亙島」謬一2ゴ且ノ．礁瓦，＋i
7∠丘ノ．D十おσ
ノ4．ノあ㌦一∠翰雰一一2ゴ∠置ノ・4ゐX州
σ8乃一ノ1．D望十2ゴα署Xl
研
一■）”勇㌦十．浅履澱一23．」り”■鰯Xl
」．η”L＿．君σ 研．1一■）”みん十■端B－2ゴ■）ノノ■履Xl
躍．1
一 一・ （167），
’錫（？・．卜1）1二
（168）．
'
?jOO '}<ioji OGvsHi and Goro MiuRA
On the contxary, sending and 'r6ceiving-end yea,ctance eirele dia-
gramsfcan be,exlpressed on eaeh 'branchi.ng coordinate respeetive]y, lil<e
the resistanee eircle diagrams, so any diseussion will be omatted here.
i2' "PPXer"ctie".Dd,ittr'grl.liail.i,il.o.7"gd:,:"2./..'.#'..e.n:tgdMPedanCe
' ' ' FitSt, apparent admit'tance circle diagrams on the receiving-end
COOrd'"" le., f(.liiC ;')'..l,3.(tllllrlilrm, E'?il.ier, iOtt.Sldler:df;(FrOM Eq,,g,3i
While, from Eqs. (16a) and (16b),
Short.circuit POint ?･V(."i)Do=:'- B`
' ･･･ 0 Open elrcUlt POMt ZV(r+bi = -r)m･･
are substituted in Eq. (169), resulting / ,
1.Z"1l==::'klil,lili(,il'.ilO..×' -T-ii3''-- ''--`'1'''''''L.`''''JL`''1;'''1`''``,'`.1.(170)
I£ lzvil is a eonstant value', then a ratio of zv..izv('.+i)o (the length-
#,OII' ,%'6"mei,g."i,Oge,l.8)iriZ"X'`,.P,O,i.nR, t8f iec','i{Zifh("z`/'o,e (;h.g,igX,gX2 £rh9,M, 1･t"'i･g
represented by a eircle centering on a !ine, Tv,,.+b..?･vL.+i)o･ (On a pOwer
cordinate, -Illj.i is multiplied, making a point of w.+i aS W(.+bc,･+i>)
Also if there is a conditionl?･v,l :T- ':ll , this r. atio becomes !, apd
the circle beeomes a perpendieular biseetor' of" a line, Zif(.+i)..7Vk.+i)o･
Generally, a value o.f I?{p,I is calculated by the following may in
quite identical Vo that oin an ordinary one-terminal transmissrton line.
Namely, a eircle whieh passes" through points i･fe(.+b.. and iif<.+bo and
which eenters at the biseeting point of a line i-fl'(,.$i)..?tpc..{)o, 'is cQnsidered
first{, and the radius o£ this, say a..b is obtained as folloWs:
a.., == -l･i ttL.- :i -F. -l2･- A"iB'-bgS--q ････'･at--･･･--･ (i7i)
ElectricalCharaeteristiesofrntereonneetedPowevTransmissionSystems. 301
Now, let a new eoordinate be considerea o£ which the two axesare such that the bisector obtained now is P(.+i)c,..i) axls, apd a line
perpendicular to it is Q(..bc..i) axis. And an apparent circle diagram
freely selected which is loeated on this new coordinate is to has a radiusgfxigY"i'gib.?.?l?T haTShean,Cew"ete'greeOint Of Which the abscissa with Qt.,,,,..,s
I9;,(r+i) = a?r+i>+T;(r+i) ''`'''-''"'''''''''''''''''''`''''-''''''`''' (172)
tt
Zif,,+iiV(r+i)o .. By(r+i>-ar+i ..,............................･････ (173)
/･ltV.+'11V(,t+'1)oa 7'y(r+b '
A,ccordingly, .., ..,.,.,. Izvi1--Py('r"il,Iii,?r"i･lbBi i･････････`･････-･L･･････`･････････i･a74)`
FromEqs.(172)and(174)also ' ' D? o L.B]' + Zffi
Py(r+D==D`.) ar+1 o ' 171tt-Zvi D 2zif, a,.+1 B ry(?"+1)= b'3 . inmBH2 hwr
are obtained.
-･' , Since receiving-end voltages or
stants up till now, the same process is
sending-end voltaggs or powers as
to enploy Eq. (14) instead of Eq. (13),
In the next, thewill 'be obtained, From Eq. (13),
'i'
.. .,. ZVi--Any A
.- . ZV.+t:=iZ LD7,XB '''` . mztvi+ -･･, .BSince a magnitude e£ iv..i, lw,,.4 is a
.,･..,..･･.･･.････････-････ (175)
powers have been handled as con- repeated in a ease of handling constants, the only difference being
t ttlttt/tttttttttttttlt/ttttttttdiagram projected on the receiving-end 1 coordinate
' '
-･l････--i･･･4-･-･i･･･-･.-.-･,.- (176)
eonstant,
302 'Koji QGusm,a4d.Gorp, tM.ivRA
s izt'+ii-i:iilltillilit-×-AB---:--T-:i::k×i21; `aw)
where'wi.. and wio'are respectively a shor･t circuit and･an;o.pen circuit
point of the coordinate 1 as given by Eqs. (14a),･a,nd･(14b)., If q line ivi..-ili,to and its perpendicular bisect,or are taken respeetivly
as the Qli axis and the PIi axis of a new eo6rdihat'e','and if the radius
and the center of evexy apparent admittance circle diagram are s･･r,f
and ,(o･ //di",,sy.e/:el.II liLY,' /h..9i) W9.F.9P.i..............,..,..,,,.,...v.w- ･(i7s)
where cri is the radius of a eirele intexrpected with pe.rpendieular to
all ci#clp, diagrams.
ai= ?'ifso2eZ'ifio =Lil}- A''Dil'i-iBcr' ･-L:･i-･ai･･･Jt････-:-･it･t q7g)
A2 . { ,,.(?.t.l.r.. IXill'.i`'F ZVi '. a, .Ii,.....,..,,.,,.,.,,.,,,,,,...,,..,,.,,.,,,, (iso)
-lil7/,2 --- ?･f9,:+l
A. i- ,.. 22v..i --.B- ai , '''' ryl=rmrmIlili･r] . -" t,'ii･･･t-･･･････i,･"･･･.･-.･,tJ･Ti=,･-,:..(181)
L-B'r, U?'if7:+I 1 ･As t'hose values are･ derived by assuming sending Powers or vol-
tages to be constant and by indluding them into coeMcients, similarres'ults are derived with/constant reeeiving powers or･voltages. More-over, regardizig to braneh line admittanees, wk.,") or w(,:-k)t; -similar
'apparentadmittanc' ecirclediagramseanbedrawn'' ''/' i' Sinee, nextly, the apparent impedance cJirele ･diagrams musit be
drawn by use of only the reciprocals o£ quantities of the apparentadmittance eircle diagrams, the formere ean be represented by the ap-
parent admittance circle diagrams themselves. As in Eq.(174) or (178),
apparent impedances are obtained from the following equations.
IZil= B,,,T../tt`11'i!i`'..ii '.tt･'' '･'-'･･Ii･･･-･･i'･･･････-･i･`'･･･:････'･-t･･･ as2)
EleetricalCharacteristigsQfi,Interqon.n,.ee.le,dI?Qwer･'I]ranslpissionSysterr)s. tt03
' IZ""il=l?,,TpmY'.,'ilii ''''''''-1;"''''1'''''''''''''''''''''''''(183)
13. Current Angle and Voltage Angle,Circle Diagrams of Interconnected Ttansmigsiop Systems.
o£ thTeherercaeltOinOgfmeandCU(rrr+e?; iXt Of the $endingdend 1 to a eurrent g..,-
gi"' .=.I{/:iff1 e･ier+,-s'fi, .., p,j,s
where tt tt ' p=TLt".i.il and o"=o r+t- 01 II,l , If s is eonstant.'tcq, (ls4) shows d' eirele, kvhieh is the eurrent's
power angle eir.cle diagz'am as follow$:....,., , , . .. ,･ pu,j, .,,, A-z; tev' ;-]],uIL' Apzi-.ifo, ".zzrlr'.-,,. Mi :plC;i/l:g'IL!? k .A
S-''1'Wdif･'l'/･Lidi(Ili,l"li'IZ,?vs ''''''''''''''.b';''''''i'''''''''J,`''''''''''''''rii(lss)
ttlb/,g/･:/･ ,11.Z'.:P.ZW.i:g,.Z g,l2･i".R.,Og,,ililieev.S･W.h,;'?h P.?.f,s. ?pon. both..?".pr.iFln?i
When coeMeients, C' and A' are used instead of ･0' and LD" theyare eircles which pass through both an origina} point and an opencircuitpoi'tnt, tt, E//i'1''''' .'",' ..'.'i'1i'.','.,.,'....,1111i" "
Next, a current Il'and branch current q,,.i)i introdiuce, as in the?,.i.,1 ,i･l 1,.l, ],,g' f,Ilet.iO:2,W, ,11 ZCi 1'llffiii ,je(."i-joi L' ptEjs'
i..,.,, ....ll.i,IZ" . .,.,,.,. ..,,'
11'ii1"1 ''de)'S'･g.1.41iz...c.'.t/･,i..IIIi ･!il+i/1fi,..,1, E.r'.. .k.......1,.1,,.. ,l,',,'
passihg ah' ojigirial poipt"and ah 6pen circuit point of lltll-r . .,. ..,. .
/･ .,, ,In the uext, .voltages on both texminals are take,n instead Q£ cu#rents,
e
,
t
304 Koji OGvsfli and GotoiMiuRA･
' 'theq-iiE3F` =:: -l'Y.E' T'itiL Ej9r+]-J'gl } = gEje
where
q==-l{.lil.-TiL and s-=q.i--opi.
and if e is assumed as a eonstant, it shows a group of ei"e.les, 'called
the voltage's power angle cirele diagram,
q,,, = -LiD"veiiiv-I B.z; .., uilJZ!llL' li:r,PEii .TB (iss)
havin,g a een'ter on a shor't c4reuit point, -fiZB)-L/' M//. If.coefficients C' and
A' are used, a short girguit point becomesibB -PV.,
Nex//tH,.,,li,,r:izl)?)i let-.w,e.melilil.oL'ag-iS l-:D %lge,y, 11]l '.. ,,,,,
, 11/shows a group of eircles centering at a srhot circuit point, £ B?.
As those discussed till now are all shown on the sending'coordi.nate 1,t,h.oss,.a.rfi a,ifio,s,h,o,w,.n,.o,n a,g,y,g:/.n,di,n,g s,ooX,ina.t,e･,,h･gw,e,v.e,ri tk,;,},bn,ee,d,.n,9L.
selvesareinverstigated. .'' ･ ' //..'.... '･.･.,,･･./.-,-, .･'
i4･ CaicuiatV.n.bo.f,.X.o.i5age.s.,a,ed.,Currents for ,,., ;. .
' Interconnected･Transmiss'ion Systems. '-'･ unbaianeed aiternating three phaS'ei 'e'hrr6ht''IS''iqul'l'e 'bM6ciseiy
studied usually by the symmetrieal c`oordinates. In many cases un-balaneed current usually oceurred during the faults, and it effects:R･,".SgyZr,a,b.i .e.%"X,rL"ew.i8,,"e.'gLZs.x2.,g.z"ge?g,,Eus,h,.th`ngs as reigy opera-
Generally,ci.rcuit constants of ah interconnected transmisslon system
are represented by .Sour constants A, B, C,,D, if eveve･y ･t.w. g,.,epds arg,
seleeted among reeeiving-ends and sending-ends respeetively with as:suniing other sendi' ng or receiving voltages or powers to ･be,constant
4
EleetriealCharacteristieso£Inte'reonneetedPowerTransmissionSystems. 305
according to assumptions or conditions given'. The detailed diseuSsions
abou't these points have been t,horoughly worked out in previous sec-
tions. These eonstants now aretransformed respectively tosymmetrical
eomponents by means of the symmetrieal coorctinates ordinarily handled.
Furthermore, unbalanced voltages and currents o£ a sending-end, say n,,, eq,b, -ZZZ,., qnd 4., 4b, q,., are transformed to their symmetrieal
eonmponents, say, LElao, .Z%i, rZt),,, and -iEbo, Ibi, ,Zl)L,. Similarly, those of a
reeeiving-end .swi,,, Y]b', Ei., afid Z., llb, Ii., are ,-eransformed to .ELo, E,i,
IFii'eeaenidvii'tZ;go:ellia'. jC2' A SUfiiX O refers to a sending-end and sufiix i to
' It can be consldered that 'each individual component exists sepa-
rately,.,and the results are supeirposed as a whole after each casecaic"ia`ek,,#:iz,ri,I/r,iS,i,,R,PSVI.9.E9.1].9.1fi9.?P.E.eli]f.?Y?,i.....,,,,,
' Positivesequeneecurrentisquitethesameastheordinarythree phase alternating current, 'therefore in Eq. (190) are nothi'ng but the
fund.amgntal equations of a transmisslon system previously discussed.
If vlltsi is taken as a nominal voltage of a generator, the synehronous
impedance of a generator must be added in series to the coeMcients,
A.,,B,,q,.D,, ' ･ '. Seconder, for the negative phase sequence eomponents:
td.2=-=Ht･-.ft･.+S,lai17] , (igi)
If a sending voltage zam, is taken as a nQminal generator vo!tag, e, ]q),?
is., equaJ,,tg, ,O･ In this case, the negative-sequenee impedanee of the
generator tnust be added in series to the coefflcients, A2, llB,}, a2, b,),
that is, 'the negative sequence impedances of a transmission system,
are quite equal to A.i, rBi, q, DJ respectively･, unless the sys'tem itself
doesnotcontainrotating,machines. . . ' /P･ aStiii,,, /irmiOr...`,Pe,eir//.e,.ri,t,,slLpq]"epi ce componfints: (,,,,
If uZIL, is tal<en as a nominal voltage of a generator, then Eoo=='O.
(a),,A Fault with a line to earth.
External eonditions, Mi.=O, Zb=ny.:=:O directly follow the next
306 ･'.'Koji OGvsGi and Goro,MivRA
'relationships among sequence components.
''' ' ' i' 1' zao+'-]ZITii+-iEL2"za.=O
-Zl,= il,,= -Zl,,
Z' O-M.2q/? ve ,l.iE`1/iilSil,:IiOIitk'in; ee.ua¥;"i sr;-;u,tainel,,,,
-ZIIio='rlziil(il-ii,ili,L'zM,iL't+i'xi' ''' L･･:=･･;････････LL･･4･:L･･(lg4)
- z2jEl}l eqL7 == A,(z,+-z'II'rtrz-iJ,5-,
-'l(ii.,=i,,IZ])liglli:i,i,1,'iZ.oEilll.L'.)l.,-ZZL)i ,,
LZI'o = '･'Af.,A.i(zlltll2!z,+t5.).,' ' , ' ''''=-'''''L''i','-''-li'''- (1･gs)
' ny･L) -= h.;k,tail-l,.vak,,s-'''' ''i 'i'
where ''''"'''' Z, :z:= -illZ.iLi･, `ii, =i-:- -th- -i-, Z.,, - ?.,,'il- ････i･･･ ･i･･i ･･ i･-･t･･･ '(ig6)
(b)' A' Fatt!t .with two'lines to earth, '' '' Fscom terminal coridkt,ions,, Zlii=liti.:=O, L.:=:O there are 6btained the
following relatiQns of sylp,metylcal components.
･1･ ,i pa. io ;･ ,,ZZTI,t u= .Zl]2
' ' .zl,-l-.zl,-t-.zl,, = zi,, ='oBy substituting the eondit' ions 'in Eqs, (190), (191), and (l92) with as-
suming qo::=Ztln:=O just as before, the following results are obtained.
' z,--.(,zif.g.2),eva-,,, ) '. -Zo::=B,(yU,Yi'y'ft,l}Iiv-,) '', I･･･････-t･-i･･i･･I･･1-4,i･-i(lg7)
EleetricalCharae'teristicsoEInterconnectedl'ower'TransmissionSystems. 307
'' Z,f2.,al,t-"n.,i...,) ' ) . t. 4,..{q+bi(y,+y,)}jnts,･ ･,. ･ B,(y,+y,+y,) ･- ' tt '16'== B.Bi(y: e4'bi+!f,,) '''' ' ;'''''"'''''''''1''''l':''' (lgs)
' ' Z62= m-IEIIn . -BiB,) (zfo + zfi + z/a)
where t/t ttt .tt/t-/･ y,-=LIL==rr4･'g ,'･ ･. ..
ttt t ' Yi-';i,--Ii-ili--X-., , 'i i-.･=･･-i･--･-･･･t････-･･i････(igg)
' 1 Ao '･ '' '･ ,', ,-,.Y2'=,z,･=:Bi' ''''''i'''
(c)AFaultamongtwolines. , .,, ' ･.Extemal conditions are eqb-=IL{., Zib= -:Zl. and' 1;.= O. Conditions
converted to the symmetrical eomponents, are and Lu=:O, Zl4o= O, Zli==
･-L,,andlV".=.rcm,'' '･, ' ..･ , . Accord,lngly, the followj.ng resul'ts are obtained in,the･same･ way
-Iii:;=-Il:･=='A,tt'#t'm`t..r) t.,,,..,.,..,････J･･-･････a･(2oo)
I'io=O' .･ J Ib,= ((;Yt±-PiY2)Yth}i A, (Z, + Z,,)
II. -- LgC-q- =-R,irmYrm:)-4L ･-･-･･t･ 4･････-･･i･- --･-･ (2ol)
idi-i (Zi + Z2)
･I)o =O
Up to this present point, three kinds of faults are dealt with.These faults are assumed to have occurred directly, without sueh anoecurrence as an arc, etc. If a]rcs are attended with the faul'ts, arcs
are replaced in their equivalent resistances.
The above discussions assume voltages or curren'ts at a fault poin't
308 ･ Koji OGvsm and'Goiro MiuRA
.･ being known, and calculate unbalanced voltages or currents o£,a sending- end. If unbalanced voltages or eurrents in bther points, such as a load
point, are to be ealculated, they are easlly obtained as voltages or eurrents at a fault point are already known. Namely, first, coeMcients,
A, B, C, D of lines £rom a fault point to ariy other points, such 'as
a load point, are calculated, 4nd seeond, operations are caleulated as
in the previous procedures,
IS. Conclusion.
The electrical characteristics of an interconnected transmission
system are analyzed by use of'charaeterized circuit constants [A], [B],
[ej and [.Z])]. These matrices constants must be all of square and of
. samg, order, as is diseussed in the paper. This assumption, however, is relatively easily accomplished by modifying or correcting the system
oz' by presuming appropriate stations to be certain definite stations,
Examples show the direct details of the theories. According to whether assuming sending-ends to be constant or assuming receiving- ends constant, there are considerable differenees'in 'the results, as
clearly･seen in Figs, 8 to 19, This assumption must be- duely executed
by inspecting'or examining the distribution, the operation and the capacities of actual generators, loads and characteristics of the system,
in oxder to obtain the most suitable and correct approximation of
Eleetrical Charaeteristies of Intereonu.eqted ?ower Transmission Systems. 309
jQ,i
'io
Q
c
qp,
Z468 XO
Wn
Ui
fo ;
ce'Tptt
rzi.
Fig. 9. civcltt t/,it/. tttaipsi isbowih'g''ohe sih6rt
circuit al.l.d, Qne open circuit point,
fundamenta,I''eireles, an effeetive
power eirc}e, and a sending ratio
eMciency eircle, between stations
Ur. and Sp,･
t,,: tt t ,- 1, /t//,./., ,/./-/
t,t . ..,//-. .,t ,/./,
:i i'''' t/t･ ,･･t ,.,,･･･ ;, tt /t ttt.t
Fig. 8.
Circle diagrams showingrone short
circuit and one open eireuit point,
fundamental cireles,9 anlJ'effeetive
power eircle, and a sending ratio
eMciency clrcle, between stations
Ur. and Su.
. l.
.-//･
J'
'
k.
'
jQ,
o s 10 i: .20. Pn
'
.t
ts-
tt'
Wil
rpll''ltzi
'
ooCPJiC'
･z;
;
tt
/,. //. ,/./-/ /･･･ /･1･"･,･,･,t.t-./･,-,
., ,.,
,jQts.
4 rtz,
lCPG"'
l Ci5
Cetss'
Wss･.P,ljs
tt
tt
'5' 10 ･15-2,Ott.
t1.,.
' ,Fig.10.Cirele diagrams showing one short eireUit and one open eircuit
point, fundamenta! cireles, an effeetive pbwer circle, and a ,
selidin.pt ratio eMeieney eirele, between stations Su. and 'Ur.
l
310 Koji OGusin 'and Geio' MvuRA
jQ77
-` 30
tt/'''1''Zdtt..
/--oo
CP77
reT,,.CeT
'r?'7T'
Ob'
t.
3oI)77'
''t
'
jQ,,
o
ACommonperpefidic N
Fig. 11. t tt 1/tt/ttlt/ttCircle diagrams showing one short cireuit and one open g. ircuit /, ,,
point, fundamental cireles, an effeetivg power circle, an,d, a,.
sending ratio eMciency circle, between staV,igns.Sp･ and Ur･,,
ttt s 10 lg･' 2og-pll Fig･ IZ･Cvvl O
oo SMiT 2i
ucircle'
!
CPII
rpl
ft1'
C {1,
Fig. 13.
Cirele diagrams showing one short eircuit
and one open cireuit point, fundamental
eireles, an effective power eircle, and
a sending ratio eMeiency cirele, between
stations Ur. and SP.
Circle diagrams showing one short
circuit and one open eireuit point,
fundamental eivcles, an effeetive
power cirele, and a sending ratio
eMcieney eircle, between stations
Ur. and Su.
jQ,i
..?Q
t.
.oe.
'
1 L/t,.,,l',S'
tt ･//.l],.
'
t/
C4/lt
Po
Wio 20 30.
-o '
tt. tt
Pil
Eleetvieal Charaeterjsties fo Intereonneeted ?ower Transmission Systems. 3rl
jQss
-10
'6'..
ro.s4cx}
'''
' wz CqG3oA
o 24 '
Cpss cbs
Tpstt
-
b.i
'
ags
Fig.
Civcle diagrams showing one
cireuit point, fundamental
circle, and a sending ratio
stations Su. and Ur.
14.
shoyt eircuit and one open
circles, an efreetive power
eficiency cirele, between
jQ･77
p
/
le 20
W77
o
C ttVT
/oo
--29,dg,7
Fig. IS.Circ]e diagrams showing one short eireuit
and one open eircuit point, fundamentalcircles, an effeetive power eircle, and a
$ending ratio eMeieney cirele, betweenstations Sp. and Ur.
j･Q17
o ozo304oPWIT
r,tn
'
oCtlnoo'
A
C,ll .rplCPtl
Fig. 16.
Circle diagrams showing one short
cireuit and one open cireuit point,
fundamental circles, an effeetive
powe" circle, and a sending ratio
eMciency cirele, between stations
Ur. and Sp. 1
312 Koji OGusHi and Goro MIuRA
,
.jQ7iJ/
･t
'
tt.
.
oo"t
rP7
CP7J cz
C(l71
QW7! P71
o 10zt
o
.P----P.""":-ijro
e7i
ZTI
'
Fig. 17.
Cirele diagram showing one short circuit and one open cireuit points,
fundamental circles, effective power circle, and a sending ratio eraciency
cirele, between stations Sp. and Uv.
jQ,1
'
co/
o 5 10 IS o P: Wll
'
o'
'
'
Pll
Fig. 18.
Circle diagrams showing one short
circuit and one open eireuit points,
fundamental eireles an effeetive
power cirele, and a sending ratio
eMeieney eirele, between stations
Ur. and Sp.
,iQ77
'
tt -t
o globr qo
W07 '' '
c.･'
V71
'
'
tt
' oo
e7
Fig. 19.
Circle diagrams shewing one short
cireuit and one open eircuit point,
fundarnental eireles, an effective
power circle, and a sending ratio
eM:inncy circle, between stations
Sp. and Ur.
t
