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Power circle diagram of interconnected electric power transmission system
Ogushi, Koji
Memoirs of the Faculty of Engineering, Hokkaido University = 北海道大學工學部紀要, 8(3-1): 104115
1950-12-20
http://hdl.handle.net/2115/37758
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8(3-1)_104-115.pdf
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Hokkaido University Collection of Scholarly and Academic Papers : HUSCAP
Peewetf Cfircfie Dfiagifftrvgg
efi
gwafreifcoeewaeckedi EEectyfic Pifwer
TifansgERfissiowa Syseewa
''` KojiO(}(]s}]t''
t
1.t
,-･'(ReceivedDec.26,1949.)
'
Outline
:
Expressing power distribution in interconnected electric transmission
system with many power stations and substations of different voltages
by using matrix, it is shown that circle diagram of this system is given
the main'diagonal tegms of matrix as centers and the other terms as
tadii., A humerical example is taken from the data of the main high
'
tension
transmission ,lipe of Hol<kaido distriet.
. 1. General Eq"atipn Or Matrix ef Power Bistrihtitioft ln
I･', lnterconneGtedTransAiissiollSystem''･/':. i
'
Generally speaking, any l<ip,d of.electric power transmission,nftwork ca,n be
transfomed into an equivalent cir-
ig //'/Y'iiigi'i7it,Xx,,
b,..,,,"
cuit as shown
by ,'g.".r,e ,i,,.W,h.'.C
has many
acting'asreal )K￡4`I'AI'Lff. ypt' /t7 >N'Z'vl･itYIt,t.2.... 2Y2 ',
mutually ,connected,
generating stations or transfomihg
stations. In order to solve such
au
/
NN ' ''
connects stations with each other,
lj' '
and,the value of every transformer
shoued be converted into the value
of standard voltage,'and the value:
of equivalent T circuits between
every two terminals should be
obtained separately.
The equivalent n circuit bet-
ween terminals can be described
t
N 't,
a network,,the,value of impedance Qf everybranch line, which,
lt
YR"i', n;
L
× 'i7"lekhlz,.;lilyrl
I 'i
' .e
2! 1
･-i 1,
Nt
1> "'
/,
3 33
pmw '
NN N
X<,i,
';-',,, '5s･s
-×
,
lt ptN<r-･
tt
V"""'N
'Ns
di
,Y ;:･'
-taN. N
d'
,
lit3 1,
.11.l.pt"l N
'Y,
fv,,
,2 z- ly
s.- rcr+1NS
/.
r"t
kllrlLti..f:s'!t./･/
× uarrn
×
circuit
The
Fig.:. equivalent
transmlssom
cted
networl<.
1-r mean sending
Terminal
ends.
1) recrevmg
-n
{r-
as follows when general electric transmlsslon constants are used :
Admittance on both legs of the circuit,
of lnterconne-
ends, terminal
･
KojiOGusia , i05
-D1 IM,iww-B
A･1
.t
,i,,,,. CIYi=-zsinB,
Y2==L
Series impedance of the circuit,
Z=:B.
--
.t
The forming of these equivalent n circuits is for convenience sake of inducing
admittance between temiinals and short-eircuited abmittance at each terminal will
be related later.' However, it is not the only way.
Out from these equivalent n circuits, we can get yi,,, the driving abmittance
between i anb j terminals, as it is equivalent to the reciprocal of value of series
impedance of'the rr circuit, and yii also, the admittance at terminal i, will be got
by adding up all supplement values belonged to the terminal only: That is-.
'"'yij=i2} "vi,="2v(ilil--.yis'-),Gi-4----]i}).
Besides, there are loads or generating powers to be tal<en into consideration.
ThereforeitwillbecomeasshownbyFigure1. '
'' Getting vei, .re2, ･･･-･･fl, stand for' voltage of each tenninal, terminal suffix 1･t･･`･
r for sending elements and their currents for +Ii, I2, ･-･+' fr; terrninal suffix(r+i)
ri for receiving elements and their currents --Ir.i -.Z..2, ･････--I,,respectivelY,
･･･
. law.
we get the following equation according to the Kirchhoff's
' + L = Yii rei + Yi2(rci " eqi)l"''''''''''''''''''''''''''''''+ Yin (-sci pt -rch)
+ Jli = Y2i(pt2 - sci) + Y22-E2 +''･'''i'''''''''''''-'''''''-'+ Y2,i(-lf2 --'rcn)
+ I,, = y,,1(ve,, - Isc1) -l-- ････-････i･･････--････i･･L-･ii･･i･･･i･･-l- Y,,,}(rc,, - rc7i)
h I,+i = Y(r+i)i (j(Z7r-T -'-,'ani) + Y(v+i) (r+o ZDr+i'''''' + Yo･+i)" (-Zag･-i - En)
ki L･t2 ::: Y<o･.2)i(]ll,･+2 - Wi) "F '''''''''''-'''''''''''''-'+?Y(?･･eo-)"(Z4i-i-2--eq,,)
tt
tlt
-Lz--Yni(Z4i-Yi)+'-''''''''''''''''''''''''''''''''''''`''''''''""J''+Yoi･n,Iii, (l)
The nature of transmission network will be explained more clearly when we
describe the above equation by using matrix as follows:
"
t..-,il" -be,Y,iE--.",r,.liiiilli.ili".TY.lfi.'tl'..111111ili:Zlll
"[Il'l"i.')･
i"
i-------------------i---------S---------O-"---------J-------l / -l--][;" -!I,'i''''J''''-''''''''Yr,'i･''''''`''''''''''''''nv3V)･,n rZl,,
=Dv""n=e"ii-tn-ulldRqq"p-n""uDa===w=-g=["mn==]=-M""a=,:tdt-t"md"!ll-gnnueed=d=Nmvq"vhvt"ecnstitm"onEmnDsed-× -i-(2)
lh
I;,+1
--
- L,.f
"
y(,･+i)i･･ny-"･･･-･･･-････-i':Y(.+1)(..)
i
･･･
.ZIJr+1
"-' Y(r+i)n
IEIr ･f L'
-Y(r+L))1'''-''''''"'''''''':'''':r,'''･･･i
r---T--4-i-----T--l------+-----------------
- L,
ke
But
-4i------4-Tt-
:- :,
l
Y e?.1 '''''''''''''''''' '''I''''''''''''' --t----t--t ISil
?tOl,
l
pt "
Yii == yii + ey,i + zl,L,-･･J････-････-
f
''`'''''''''' + ?lioi
:
pa't I
I06' Po'werCircleDiagramofInterconnectedElectricPowerTransmissionSystem
'
The elements in the above equation will be described also in the form of so--
called power circle diagram constants by using generl circuit constants between
temiinals as follows: '
11 111
'
Yi2 = -B-,,, 3Yis == hZ/7I:.,''''''Zlm == B,,= llil'E, Y3i=: IZ'},,'''"'
Pi2+ -Bi4
.Z)13
b14 Pl,,
-WnF= Bi2 + B!s
+ -･･-･-- + Bin s
Aip.
D23 Lb24 -D2n
Y22= Brl';1, + 2,B + B24 + '''''''''''' + ･ii}I,,
iy,,,,,, ,,,.,, -= "jll-i,, ii:,)- + i.`--;--ij:til --･ ･,････ii-li･i-g,li:)l- + l-ll--i:i'i E2-i-- + -････････
D(rt])n
'
''`''''''''' + J7'l(T+1)n ･
In the matrix (2), XIsi, the terms upon a main diagonal, mean the shortcirguited abmittance at the terminal, that is the admittance in'case of all other
terminals being short--circuited, excepting only theterm of i order, and of which the
value will never become zero. Other admittances excepting that upon the diagonal
mean the driving admittance between terminals, and of which the value between
only noncombinated terminals become zero. However, terms located at symetrical
positions of the diagonal are equal with each other, and terms in the upper side of
the diagonal mean the sending driving abmittance and that of the lower side ,mean
the receiving driving admittance.
Equation (2) is shown with the current upit, but if we want it with the power
unit we must multiply each terminal current, sh6wn by a horizontal rank of the
matrix, by E,A,, the conjugated value of each terminal voltage; that is- .--
'
+P,,+iQ,,･･･････i･･
･
tt
+ P,, -F j Q.････-･･･-
+ Prr +j Q,'r'''''''''
]z--"plltiU=-enEn-t=#"-aDudp=d=="-PaP=e-q=e--d-
"'(Po･+i)(o･+i)j Qo･+i)(,-+i))
---------------------t-t---
- (Pnn +j Q7i")''''''
l
TIYn[eeii2,haYnM2.Ev"'''''''''I'''Y(r-i-i)-iee(r-ia)lscik,"'Yin-)UrozYi
--Y21Y,Xlk,'Sl221.if2i2,･･････-･.I...........,.......,...,............
i"
--------i--------- . -----p------l-.:- - ----;-t------------- e-±---------
i
:
== -:I..e.1[.:l.ll.1..i.li.i.1:'kll.1.liul..li.II..IY..,,.liA･n=:llZTLItli-lh[]IEIIildll-]-liulnlt:lalgl-1[ta1"liiltliti[-liulvl:i"lu
'''(3)
KojiOGusH; ･ .. 107
- 11r-f-i)i ewilElr(,'-i-i)k, '''''''''"''l}f'(r･I-i) (rH-i) lll'(rii)IE, ''''''''''''
------t--E-----t----tj-Jt---,--------:1:-i---l-----l---.-----.--t-----b-1-.-
s
-y,,ilfflriM.z,･･････････････････I･･･････････････Y,,.isc.i2.-･･j･･･-･.
If the network is based on toe constant voltage transmission system, namely,
the absolute value of every terminal voltage is constant, then every term upon the
main diagonal of the above power matrix gets constant value and other terms are
to have different phase angles individually. Accordingly, the vector locus forms
circle with changes of load value, and we will have nany circle diagrams corres-
ponding to each term. The number of the circles will be the twice of that of
ointing transmission Iines between terminals.
The sending power circle diagrams are made･with the upper terms of the rnain
diagonal, and the receiving power circle diagrams are with the lower terms of the
diagonal.
It the terminal of i orber is tal<en for an example, the totai generating power
Pn +j Qn = Yii1Yij2-" Yi2 ]r2 -iecik " Yi:i Y:i -ifu:-'''''''''''- Yin ]rn IMIJin･･
which is shown on the first line of the equation (3). However, frorr) this terminai,
power is branched out to the temiinals connected, so the above equation can be
shown also as follows, havlng divided Yi,T to each coinponent. (Pi2 + j Qi2) + (Pi3 + j Qi:J +''''''`''''''-'''''''''''''''''''t+ '(Pin + j Qin)
={(Yii2 + yi2)i]ilii/2 - y,2 -Zl3 Zl,n,} '
' +{(Yi,,, + Yi3)1Mil?' - .sc,,, ]l,t, Zir,k}
+{ (.?/i!n -P ?)gin)lJEI]il2 im Yin rcn -Zt]}･n}
If this equation is shown with circle diagram constants which are used in most
cases,
(Pi2 +j Qi2) +(Pi:i +j Qif:) +"'''';'''+ (Pint +j Qin) ' i
s B, rc,,) +･･･.･･･-･･i･- ('a"tir,1EiL?' - Ii'l{s. sc?, -E'iA) -- (tJ-],';,i-gDii2 -･]Ll{,,
････････････-F ([ft--?-1'/tl .sc,I? - tbe,. .!ll,,, .gi,in),
The right side and the left side o'f this equation contain many terms contrasted
with Gach other, and adinittances appeared on the rjght side are obtained by using
equivalent T 'circuit constants as was related before. As for the first term, it is
explained as follows. ' ' '
(P,, + j Q,,) =: (y,,,,, + y,,) i, ]rc,I2 - y,, ve,) pa,,.
'
This shows power which is sent to terminal 2 from terminal l. As is based on
IOg PowereircleDiagramoflnterconnectedElectricPowerTransmissionSystem
'
the constant voltage transmission system, the factor changeable due to the load change
is only phase angle or "power angle" of voltage. And (yii2 +yi2)IIeii2,!yi2sc2Yi l,
show center and radius oi circle respectively, which means the well-known sending
end circle. The same discourse can be made for many other terms 'in like manner,
andmanycirclediagramsaremade. '
However', we are discussing these theories always basing upon the transmission
network, and it will mal<e us notice thefollowing fact.Namely, the circle diagrams
of the terms located on symetrical positions of the main diagonal of matrix (3) have
equal radius value and contrasted power angles. For an example, -yi2Y2veikand
--
yL)i Ei E2,: have equal radius value but contrasted with,each other as to their power
angles. The adove fact shows us that a half of the total number of circles
will be.enough to determine all diagrams we want. This can be utilized in case
we map out the diagrams altogether. While, power and voltage in the transmission
networl< are recorded usually on the power distribution diagrams in generating sta-
tions or transformimg stations. These recorded voltages and line constants which
are obtained separately will supply enough values needed for equation (3) to draw
every power diagram. And points of action on the diagrams can be determined by
effective sending power or otherwise by effective receiving power obtained from
power distribution diagrams. Instead of receving voltage, power factor or load adm-
ittance may be applied. .
The above explanation related up to now is the same with that of the ordinary
power diagram theories, however, the use of matrix will explain mutual relations
'
as an interconnected
transmission system. 'In next chaPter, will be related method
of getting the total sending power and total receiving power. ･
'2. Total Sefieling and Receiviiig Power a"rl Power Circie OiagrarR
'
of lnterconwected Traftsmisson System
'
Total sending power is the' sum of power from 1 order to r order of sending
end terminal shown by equation (3); and
Pi +j Qi == (Pn +j Qii) + (P22 +j Q2L,) +''''''''''''+ (P"- +j Qrr)
== (ynlZV"L' + Er22Ipt2i2 +･-･･････････-････-+ Y,,,,lsw,,l2)
+ (- Yi?. Zt]L, reu; - X3rc:i M7vc''''-'''''''-'' -"- Yir JZIr rctk)
+ (-Y2i ewi itL･ - Y:nMi sc:{k'''''''''''''''in ?)f,･iIZIi.ZZ7rA･)
+(hYi(r-n)JE,tpilllii,･''''''''''''''t'''-YinliZlrnrcu: '
ny ?12 (r+i) -scr+1 ve2in'''''''''''''''''''''--' = ?1in, -pmn -rcrk) '''"'' ( 4 )
The first term of the right side of the equation means short circuited
power at each sending end. The second and the third term means driving power
which is relayed and sent or received between each sending end, and every corres-
Koji OGvsul ]C9
'
'
ponding terms have contrasted power angle and the terms will become zero by
being cancelled if sending loss being negligible. The fourth term means real driving
power actually sent from sending ends towards receiving ends. As was already
related,we may be able to attain the object by drawing each circle individually
between terminals, but it is pracially imposs' ible to draw a lot of circles on the s
ame paper. So, author introduce here a different method that claims to draw deve
loped circlesas shown by Figure 3. based on the equation (4). The total powera
lso can be obtained easily if we have a gance at it. Basically speaking, howeve it
is just an application of the orbinary circle diagram char.
Now, constant value of vector sum (yiiIZIii2 + y2?[.!ljT2 i2･･･+ blrrlverl2)
is drawn on a co-ordinate of total sending power (Pi +j Q,),
and from the end of the vector all other vector tem].s ()12 sc2 Y!A,......etc.) are sub-
tracted in the vectorical manner, and the end finally got will show the total sending
power as shown by the equation･ (4･). However, as power angles of Vector yi2E2
Eik......etc. are unknown, they should be obtained by utilizing effective powers
given by the power distribution diagrams. For instance, in case of yiL,E2E!lt, the
equation of the circle between ] and 2 is (piL) l- j Qi2) == (yii2 + )-'i2)l.21il2 - yi2 Y2 -Zt]iA,
and the circle center andthe radius are, respectively, (yii2 + yiL,)l-rciiL', (yi2 rc2 sciA･)･
If we apply value P]!to the above equation, we can getjQiL, or power angle immediately by figure. Utilizing this reason, we subtract the value -y]2.pa2 .Zt]iin ;::: (-yiiL,
+yit) × 1,ZDil2 +.iPi! + j QiL,. from the end of the sum ofshort-circuited power (yiil/ Ei:2
--･････-+y,,l"gV,iL'). Nain.ely, wesubtract a constant value(YiiL, + Yit)l.21ii2 first, vec--
torically, and secondly, from the end of the vector toward a horizontal axis we
measure Pi2, the effective power given by distribution diagram; thirdly, passing
this point draw a venical line then catch a point intersected by the line and the
circle centered on an end of (yiiliViLL' +･-･･;･+ yrrircri2) with radius yiL) -rc2Zilits･ In
the same way we subtract yi:} Y,, .Zliic starting from the intersecting point found by the
above method now. Drawing of circles with radius lyi2 ZZI2 uZIiLI, lyi3 -lf3 Uihl,.-････etc･
will show ordinary power circle diagrams between terminals. If we utilize calculation example, Figure 3a, in order to explain it in detail, the sum of short-circuited
power of sending end is shown No.12345678917 and point 17 is the end.
From this point, we subtract yg, i7sci7ue,:, namely, subtract vector r7r9･; (yg, i7 ZC!7 +
IYb, i7)l.2E]"l2, and get a point 9 first, and mal<e a vertical line at a place where the
effective power is (Pg,i7) which is obtained from distribution diagrams, with the
point 9 as its starting, point. The vertical line then intersects with circie (9, 17) of
radius (yg, i7 .Zl7g vei7) at the point 17, which shows the point of action. In the same
way we draw a circle (9, 7) by starting from the point 17. If effective power P
on distribution diagrams takes a negative sign, it means reiayed power reception
tlO PowerCircleDiagramofInterconnectedElectricPowerTransmissionSystem
from other ends, even though it is actually a sending end. The same method is
repeated with all terms of the equation until the final point showing the totai sen--
ding power is found. The final circle diagi'ams contain ordinary circle diagrams,
and this shows not only the total sum but also the nature of their mutua! relations.
So, distributing or mutual relations of power, voltage, power factor, etc. in inter-
connected transfrtission system can be glanced at instantly by this. The second and
the third teum of equation (4) have only a difference of the power angle being contrasted, and so one was determined the other wiil be found naturally.
Likewise, the total receiving power is the sum of power from (r +1) order to n
order ef terminals in euqatiou (3); and its value isP2 + j Q2 :] (P(r÷i) (r+i) + j Q(r+i) (r+i) )'''''' + (Pnn + j Qnn)
= (Y(r+i) (r+i)lswr+ii2''''''''''''''''''-'''''''''''-'''''tl- ]¥L,,tlswnb2)
+(h-Zf(r+1)(r+2)-gg-i)+2lf(?i+1)ic'''''''''''"'"'''''-Y(r+1)Yn(r+1)in L
+ (-Y(r+2) (r+i) -scr+i 21Y(r+L)) ts-'''''''''`'''''''''' ny Zfn (r+i) Yr+i -Zl7nk)
-l- (- Y(r+1) 1 Jrc1 pt(1+1) IL'''''''''''''''''''-' mx Ynl -ZVI venic
ny Y(r+1) 2 'rc2 'rc(r+1) ts'-''''`'''''''''`'''' lo Yn?' ver -rcnii)''`'''( 5 )
The first tenn means the sum of receiving ends short-circuited power which
is constant value, the second and the third terms mean driving power which is sent
or received between receiving ends by relay and has contrasted power angle having nature to cancel each other, and the fourth term means driving power actually
sent from sending ends to receiving ends. The ,fourth term is determined of itself
if the fourth term of equation (4) is given.
According to an usual practice, a receiving end power circle diagram is･ drawn
with its negative power value locating at positive side of coordinate. Accordingly,
total sending power and receiving power can be' drawn on a same coordinate as
shown by Figure 3a, b, and so it seems that an interconnected transmission system
with many terminals works in a similar way with the well-1<nown diagram of
simple transmission line. It immediately shows the total sending efficiency and the
total loss.
3. Example of Calc"Iation
The main part of power ･transmission system ot high tension in Hokkaido Dis-
trict is simplified as shown by Fig. 2. Terminal 1 --9 mean generating stations
and 10-17 mean transforming stations. The power matrix of this system is shown
by Table I. when calculated using the equation (3). ･･
Short-circuited admittance, driving admittance and line constants of equivalent
ic circuit between terminals of this system are shown by Table II together with
voltage and power of distribution diagram of Fig. 2.
t
Koji O usm
Nl
Fig. Z. Power Distribution Diagram.
It shows tertninal voetage of each sJtation and effective
power at each lipe, when the standard voltage is66kv.
65KV
63KV n
of<w.'
S7-.EiVoooKla,
61.2KV
43oo 12 14oooKw
gliO,Z08i,X,S3 ] Kw .i/f@6oKv
'
S160KW
6st<v
R
e'"'>
-2300K'
'
'
10 57SKV
'
""
Xx. t8700KW ,
i' i3tl-OOKW
",,i 'h"'kh >---4 6g.sKv
W
t" tin"'
""ss..:"--S"[ltli3s"oo'KEwo926iLoKoVKw
63KV 14
-40ool<w
62Kv 1 S
' 17 sgKv
-2SOOKw x-
63K
v' lo5oOKW
e 6tKV
-16 '
19060KW
e 15800KW '
66KV (SOKV
The number'n.eans
generatirig station
and transformingstations locoted at
the places shown below
GeneratingStation･
1. Uryu.
2. Ebetsu･
3. Kariki,
4. Moiwa.
5. Jozankei･
6.Kombu. t
7. I<amikawa.
8. Antaruma.
9. Eoroshi.
Transforrning Station.
10. Sunagawa.
Il. Coal mines.
12. Bibai.
I3. Sapporo. '
14. 0taru.
15. Nagahama. i
16. Yoichi.
IZ. Asahigawa.
'
tt' t
Using this table, each term of the power matrix of Table I can be calculated,
and so sending power circle diagram by equation (4) and receiving power circie
diagram by equation (5) can be
drawn as Figure 3 a, b, by the method related
before. The drawing of circle diagrams needs value of admittance of every rr circuit between terminals, absolute valne of voltages, and also effective powers
numbering
as much as branch lines which connect terminals. The effective power is unnecessary in case of drawing a simple transmission line system, but becomes necessary
when interconncted system is drawn by･the above manner. In this calculation exa-
mple, there are 17 branch lines, so 17 effective power will be necessary as
described on the power distribution diagram. The points of action of other effective
power and reactive power will be got spontaneously on their diagrams by this
･
l12 PowerCircleDiagramof
Interconnected Electric Power Transmission System
drawing as related before.
I? K;4!s
Without receiving voltage, vol-
･loexlO uoo
vttmp-'MdiiT-pmMpmfop"-"utfTN.
o
tage can be determined also
1
by using loads or power factor.
However, this method, after all,
is euqal with the ordinary cir1
cle diagram method, so it may
hand, when the value of load
- l-lts
varles,
10
xio
be neglected here. On the other
10
litol
lx
x
vector moves on to
the circumference of the circle
,Y
as the system is based on a
constant voltage system, and
the situation of linking of
electric power distribution. The
method shown here give us a
great convenience to clazify the
1-IO
13
N
z
l
jQ, ,3
L'OC
K.,
2i
N
Sh
$g.
ig
circles will vary introducing
changes of mutual relations of
muptP,J
1
-
x
,
ge･/111･/･ ･･
1if1 J
le
[llih
Z･1
3
g
3
z.i
1.
X5,
x.
,
3
x
.2
<ll; 3oo
･ Z3
2X3 ･2?
:;t?2
3
distribution of power and voltage, but, it necesSitates rather
3X
complicated troubles because it
j')
"
3･1 .3
is the same in principle with the
N
method of ordinary circle dia-
gram. Hower, this method seems
IS
3
lk3
･IOO
3
si s
to be an only method if we want
4
s4
'
7 16
of load together with other cha-
racters.'- ･･ ･･ .',
Figure 3 a, b, are constru-
cted only by ordinary circle-
5
G
, 49/,,'
soo-
B
2S!Jr t
p '17
z ..w
8･1 7
7Z
91O'
Fig, 3, a. Power Circle Diagram
at Sending end.
Q･7
Fig 3 a, b, indicate pow6r matrixfon
left side, oth.er terms by raflii of circles
arid power angles with.'obliqud li'nes,
g6L16
%
lolo
t2 97
11
9･17
iCJi
soo
:L..
lt'tupv
4 4･S
i6 s･4
6,
1/.
diagrams that'･are developed by
Tab. 1. Ma'in diagonal terms of matrix
are indicated by indented ljne on .the
4･3
's
to show the variation condition
e
S･"
4
tt
7･9
!
Tab. 1. Power Matrix
'
It shoyvs power distribution of network shown by Fig. 2. The main diagonal terms mean short-circuited powe
No.
l
5
8
7
6'
9
lt
'13
12
15
14
l7
16
t
({llili:il:+'
1
i1-
kil)IE"2
1rljIIt5E13EIK
DIIt5oEioEt-i
(:t3+:tlli9;)1E2I2
2
1"-D5TtE3E2K
.
1'rbllTtE-oE:iK, ({l:ti322f:tii+ue5:l33)･E3l2
3
't"'A
4
3
2
{) other terms driving power o{ sending or recieving.
1-i;igltll3E13E:K
l--li5T4E4E3K
'
'
1-B43Eil3E4K
4
(rkt3+ftt5s).i4Ep
--
iilligEsE4K
.
B-1
--
5
s4E4BsK
As4-at.,IEsig
'
;
D616
-Pllltg6E･16E6x
EEITtiE612
6
(
1l-E,EgE7K
D7g-R-
iT,lE,12
7
I
Ds17'
at,,[Es:g
8
I
'
IEgi21'-BiioEioEgKI)---.---･..--
1-q7E7EgK
Dg17DgieAg7(ii-g7+E51t5o+Bgl7.-...t...1"ettigEgEioKi
'-- Pgltl7EwEsK
-!
1--figlt17E17Eg
9110[1
''
IH'H"H'
i'''
1--E!EloKBIol'
tttttttttttttt"ttttt
l,(l{:i'gl:lfSi':le,9,tlllg9ilil)jEl,eli2'
Dni2i'
i]IItli12IEIIi2
li
1
"rkti2E2E12K1
.t...t+..t.---t.....ttt...t.ttt-......
ttttt-tt
tttttttt
i-B-
ioi2Ei2EieK
l'-B12ioEieEi2K/;
.
'-t"n2Ei!Eiik
'
-B-11z.EilE'?K;
litli,+B.-lii,:iA.li-li)liE,,i2
'
"1
l3
A131+ttss+D.t214)IE13i2
1
l--EtE13KBIsln
1-'RIIIItl4E14E13K
B131B133B1314s
l･il[il
rt.
14
({lllili`,i33t::,:i:`i,5stD-bu':l:NyEi4:2
1-i}-IItlilsE13E14K
1-glitill14E14ElsK
15
･1-
.t
i}IItlilsElsE14K
1'i]IItlE6E16E14K
Als14'
ent,4IEIsl2
s
l.li-1
1-blgTtE6E16K
16
1'
-glFtgEsE17K
!
'
1-GEtfi1E14E16K
'
(Sill:,flll-X(/',)iEi6[2
A17sA17g
(st,-st,7)IEitrj2･
Tab. II.
Terminal
Order
EffectivePower'
'Voltage
Admittancemho
Figures for Drawing Circle Diagrams.
kw
kv
Ei=:69.5
Terminal
Order
1=O.O0203-jO.O176Blol1=O.OO127-jO.Oli04Blogl=O.O0336-iO.02T5Blo12`
P;io=18700
EffectivePower
Voltage
kw
kv
Esol=57.s
'10
Pn3=13400
l
Admittancemho
Plol==-tv
'PIOg=-th.
Pn=32100'
Plo12=-.v
Alol=O.O023-jO.Oi752BlolAlog=O.OO127-jO.OI079
PIOIO=--rv
B23
1
E2=65
.=O.O042-jO.OO?3
Blog
P23==-tv
DioR
BIo12
P212--t'nv
eO.OO12--jO.O135B212D23=O.O042-1O.O0931B23L
2
.
1Bin2=O.OO05g-jo.oos4
P22=43))
DBilli,=o.oo]2-jo.o36s'
11
-L1'
B32B23
.3'
=O.O0336-:/O.02?44
Ell=60
Pmz==-"v
Din2=o.oooss-jo.oojgs]3Bni2
Pim=-tv
'
E?,=r63
P32-H---v
P34=--v
P313=hnv
12
P33--O
E4==64
B43B34'
P43=+AJ
P4s=-N
4
E12=57
P122=--.v
P121e=:tN
P12n=14)OO
P1212==-5)OO
-----nm---
ll=
11=
11=
Blo.loBio12
11=
Bi2iiEm2
Bio..pB2i2
li･:fii,[glggg?i:lglgo,6,lsiA32=･O.O042-jO.O0931B32P34=O.OOOP5-jO.O063B34D313==O.OOO14-jO.O1241B313
11=
B131Bl13
ll=
B13BB313
v[...
/'-tl'.'
ifv:"/''
E13=6I.2
P133=+nv
1B13,4=O･OO185-jo.ols4
P44=9000
13
P131==-.v
P1314-----.v
A131
B,3,=O･OC254-jo.ols3g
P1313=L12000･
tl133=O.OOO14-.jO.O1239
ilt=Bs4B4sAs4=O.OO045-jO.O0652.Bs4
Er,=65lPs4==5160
5
B!33
D1314
B1314
Pss=Ps4
=O.aO185--jO.O1844
'
'
1=O.OO135-jO.O048B616:i,i:-----･O.OO135-jO.oot17s
･EG=66
6
==C).OO047-jO.O085
Bl"3E1314
P616=!9060
E4==63
P1415=+.v
Pfi6=P616
P1416=-N
=o.eol2-jo.oos2B1416A1413=O.OO185-jO.O1844B1413D141s'=O.OO047-jO.O0851B141sD1416=O.OO12-jO.O0521
14'
'1=O.OO036-jO.O0884B7g
7
E7=60
Zll:=O･OO036-jo.oos713
P1413="+t-v.'L
P14!4=-2300
P7g=]5800
P77=P7g
Bl"6
Est=61
=O.OO0312-JO.O040BsnDs17=O.OO0312--jO.O0397
Ps17=I0500
il=
8
.Pss=Psl7
'･
Eg=59.6
=Bg7B7g
1v-O.OO)21-jO.Oll04Bgie1=O.OO041-jO.O0267Bg17Ag7=o.ooo36-j'o.oogsg6Bg7iBJ:l:Lo.oo127-.jo.oo:o7g
]5
EIi,=62..
Als14.=
Pgl7=-3500
11=
11=
B166B616-
16
E16=63
P1616=-2500
Dg17
= O.OO041 - j O.O0237
1
Bi7s
1
17
P166==-.v
P1614=+.v
Bi6nBi4i6
Pgg=20400
Bg17
Pls14=-400
Plsls=-Pls14
LPS7=-A.
Pg}o=+.v
9
.BlslsBl"s
B17g
!a178
-1
Bsm
En = 59
1
Bg17
== O.OO03i2- j O.O0402
Bns
P17s ==
P17g ==
P 1717 ==
A17g
== O.OO041 - j O.O0264
B17g
Eoot-Note.
Pno stands for effective sending powe' r from terminal 1 fo terminal 10, Pn for total
means sending, sign-gecieving, sign--that the figuFe is determined by circle diagram.
'
.
t 't. tttvtt
tT .t
tt t'
power on terminal 1,
and it is the same with the following correspondingly.
Sign-
rv
+
N
.Nt
lJ3･
Koji , OGUSHI
power matrix, but some 6xplanations will benecessary to
be given as author fear they
t7,'9
16･o
17J8 ,9e,
are not well-l<nown yet: -
616･14
t7t' 1
(1) Circle Group:
S6 A
i64
Sign (1, 10), (1, 13),
615･14
,,t"
,,,
etc. in Figure 3 a. means a
s
sending end power circle from
m
sending end 1 to receiving
i
end 10. Power angle with
7eL
l:l,l 614,1s
1
14
IS.
rs t4
i-s･i
i3
600
oblique lines means shortcircuited power betwen termeny
13･14
inals, of which･ the one situated
on the left side of the circle
means electric receiving by
4
seor
XX,×.Tj
g
E;, 13 l
lj]ii/`i
relay from other terminals
t
x
:3-.
though is originaliy a sending
ss 133
t
%>x
tc
･1?J
end and that situated on the
tw
right side of the circle means
E'
3'
:
k2
electric sending As for the
-
x
ss
s3
circle (1, 13), for instance,
origin of coordinate (Pi,jQi),
t2 10
1211
E
kst
iz
x
x2
l2
it should be situated on the
i
)
31',
1'.
IZ IO
but the same consequence will
be got if the origin is tran-il,l'
sformed to point 1, of which
the coordinate is named (Pi.
o
li･lZ
12
sz
1Ql .'
tl
k32o,-
,,,j Q,, ,,).
,/('×XXL
El
Horizontal axis takes effec-
'
tive power and vertical axis
i
of9
'
iXOi
-
io :z
twtwlk ,,
SJtl
Fig. 3, b. Power Circle Diagram
at R.ecieving end.
iO l"
io
l
Circles {IO, 1), (I2, 10j, etc. ireans
to terminal 10, etc, Total reci.eving
e 1<ltt
power is shown by effective comp-
foo
thrp-"'- T-r.
'
ving powey by et'l'ective component
ofyector 1,10 on Fig.3 a.
//
xs-tgtw/
71
10,･)'
siOl '
a circle recieved from terminal 1
nent of veetor O.1,ai;d total recie-
x
1'
)o
10Cli llOs
o
ZUL.
r--
t14 PowerCircleDiagramoflnterconnectedElectricPowerTransmissionSystem
'
'
'
reactive power. Centerpoint of the circle is obtained by transitting in parallel the
vector l, 1' 3, a term in the main diagona! of power matrix. Radius of the circle is
'
obtained from the Table II.. '
The point of action 13 is determined by radius of circle and effective power
which is obtained from powerdistribution diagrams or mutual relation ofthis c'ir--
cle group. Circle (2, 3) and circle (3, 2) show that they have'a contrast power
angle each other, and that a sending end terminal 3 is the one receiving power
from
other terminais
by therelay.
.
' complicated,
The more transmission system becomes
more increases the
number of circles, therefore proper dividing becomes necessary in this case. In the
Figure 3 a, b, there exist circle (i, j) and circle (j, i) which have an equal radius
value with different center distances, but, these circles respectively correspond to
term (i, j) of the upper side of main diagonal and teim (j, i) of the lower side.
(2) Sum of short-circuited power:
It is shown on left side of the Figure 3 bs an indented line, of which compoiients are vectors to indicate circle centers, and the component's slant meansthe
vector angle of short-circuited admittance (yitf + yid). With an increase of line res-
istance, the angle of slant becomes wider, which means an increase of line loss.
In case of a System which has an ideal, line loss should be consrant in every part
of the system and the indented line becomes straight. In Figure 3 a, most part
, 3; 3, 2J 6, l6. ,
of line is straight except parts 2L
(3) Total transmission efficiency:
The effective components of vector O, ld and blrml' in the Figure 3 a, b, are
the total generating power and the total receiving power, hereupon: --
99,OOOKW
Total transmission efficiency: = ntt';bb-oKw 100 =-- 85%
(4-) Phase improving condenser: '
The total receiving power is 99,OOO KW and its reactive power is 45,OOO KW
as shown by vector ' O;1. Even if a condenser fitted for this figure was prepared,
it is unsufficient because of iregularity of current phases at each terminal,namely,because
of Iagging or Ieading which differs in reality. we take an example on circle (10,
1) of Figure 3, b. Cood efficiency can be got for electricity sending between ter--
ininals when the power factor is improved, that is approximately l. Therefore, if
we place a condenser onto a receiving end, terminal IO, and the voltage on terminal
10 is left as it is, sending voltageon terminal 1 is to become low and transmission
efficiency becomes higher, It can be seen also from the Figure 3a, namely, povLier
factor becomes better owjng to sending voltage dropped. However, in case the rec--
f voltage
eiving voitage on terminal 10 got dropped and is desired to be raised, and
Koji OGus}i] ,' 115
'
on terminal 1 is left as it is, receiving voltage on terminal 10 bccomcs high by the
conclenser, and so, power factor, coi]sequently transmission efficiency will be impr-
oved. This is understood from circle (10, 1) and (l, IO). On the other hand,
however, the above treatment spoils terrr}inal 9 kecause ExcEssive leading currEnt is
flowed out irom this terminal, which causcs rcduction of tran.cmission efficicncy as
it is julgcd from circle (10, 9). To raise this tcirmiiral x'oltase is :ot casy as thet-eirnjnal is a rE]ayc{] {lcraatii g Etatkn, Ac(ci'c:5r9y,ttuc rrv.ct o;iEt[uiic"tle"valucs
of condenser capacity lit for jrr]prov(rrcnt of tlie ro",Er factors. To Eolve thjs
diffic .ulty,
we must tal<e cfforts rraking ca}culaticfis in vaiicvs "ay Gn tigures or
equanons.
(5) Transmission Capacity:
･ Thisisshownasaradiusofeachcircleiflinelossisneglected.Ifthe
capacity is proportional to the radius of circle, irregularity of dimention of each cir-
cle is quite negligible. However, circles (9, 10), (6, l6), etc. which are relative-
ly smail compared with their sending power 'are impossible to send enough
power. As the situation of circle com.bination on Figure 3 is irregular in ord
er, electric power transmitted from a 'sending end to an aparted receiving end mu
st be followed or obtained in accordance with the order of that of the Figure 2, and
not the Figure 3. That is, sending power is not transmitted in order of the circle
order of Figure 3. If sending power is proportional to the radii of circlcs, all power
angles and line losses become equal,and an indented line showing short-circ"ited po
wer becomes straight. Then Figure 3a, b, will become like an ordinal transmission
circlediagramofasingletransmissionline. . . .
'
-･ tlT.Coitcl"sion .
Distribution of power and voltage of an interconn'ected power transmisson system can be converted into an equivalent junction points network, and if it is developed in matrix the main diagonal rneans short circuited power, terms of the upper
side of the diagonal mean sending driving power, and the lower side mcan receiving
driving power. Also, if we divide the systcm into sending ends and ' receivipg ends
and sum thEm up with variable factors of ca.ch of their voltage's phase angle, "power
angte," we can draw a power circle dragiam of the net-work, which is an asscm-
bly of ordinary power circle diagram between terminals. The diagram does show
not merely the total scnding and rGceivjng Fower, but all other kinds of electric
nature of the system. Even though there are some presumptive values in the network taken up as an example and the real situation is different from that calculation, this shows that we can improve the line system furthermorei