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**Unformatted text preview: **6.1.2 Performance Equations 2 = R + ij = series impedance per unit length/phase
y = G + ij = shunt admittance per unit length/phase
Z = length of the line - [Zigzag VR—zciR _ .. — z -
V = TBVHTG? Y: at the receiving end (x=0)
- - 20' = zb’ characteristic impedance
f“ VR ZC+IReTx_ VRC-IZ fRe'fx
2 2 'Y : Jy? : a +j|3 propagation constant. attenuation constant a: phase constant [3. x_ (a+')x_ an: -°
:3" - 8 ﬂ - e (cosl3x+}sm|3x) If a line is tenninated in its characteristic impedance ZC, VR is equal to ZCIR,
e -rx = e “‘"(cosBx—jsian) and there is no reﬂected wave. Such a line is called a ﬂat line or an inﬁnite line (since
a line of inﬁnite length cannot have a reﬂected wave). Power lines, unlike
connnunication lines, are not usually terminated in their characteristic impedances. For typical power lines G is practically zero and R<<coL. Therefore, R+ij= [11.
jcoC 2:1,] = «Rewojwc ij[1—j—] 2w]. Fora leaﬂess Zine, Equations 6.8 and 6.9 simplify to
l7 = f’RcosBxﬁZCfRsian I = chosﬁx+j(I7’R/ZC)sinl3x 6.1.3 Natural or Surge Impedance Loading Since G is negligible and R is small, high-voltage lines are assumed to be
lossless when we are dealing with lightning and switching surges. Hence, the
chancteristic impedance 26 with losses neglected is commonly referred to as the
surge impedance. It is equal to m and has the dimension of a pure resistance. The power delivered by a transmission line when it is terminated by its surge
impedance is known as the natural load or surge impedance load (SIL): Vi
SIL=— W Zc where Va is the rated voltage of the line. If V0 is the line-to-neutral voltage, SIL given
by the above equation is the per-phase value; if V0 is the line-to-line value, then SIL
is the three-phase value. the voltage and current along the length of a lossless line at SIL are given by Au- =I' All _ at":
V V33 where 'y =jp ﬂurry/LC.
I [Re At SIL, transmission. lines (lossless) exhibit the following special
characteristics: ' . I7 and f have constant amplitude along the line.
' F'and f are in phase throughout the length of the line.
' The phase angle between the sending end and receiving end voltages (currents) is equal to [31 (see Figure 6.2). 9 = [31
l = line length Figure 6.2 Sending end and receiving end voltage and
current relationships of a lossless line at SIL Table 6.1 Typical overhead transmission line parameters “min“! 230 W 345 w 500 w 765 w 1,100 In;
Voltage R (0.!ka 0.050 0.03? [1005
was: (new) 0.433 0.367 0.202
gc=mC (usfkm) 3.321 4.513 5 544 Elsie—5&5? éqiiai {S {209104 {1 (nepersfkm) 0.00006? 0.000066 0.00005 '1" 0.000025 0.000012 B (radlkm) 0.00123 0.00129 0.00130 0.00128 0.0012?
20 (Q) 380 235 250 25’}If 230
SIL MW) 140 420 1000 2280 5260 = V0200
Notes: 1. Rated frequency is assumed to be 60 Hz.
2. Bundled eenduetors used far all lines listed, except fer the 230 kV line. 3. R, xL, and BIG are per-phase values.
4- SIL and charging WA are three-phase values. $1.6 Performance Requirements of Power Transmission Lines If the power line is very long (greater than 500 km), terminating close to the
characteristic impedance becomes imperative. To increase power levels that can be
transmitted, either the characteristic impedance has to be reduced (b3F adding
compensation) or the transmission voltage has to be increased. Voitage regulation, thermal limits, and system stability are the factors that
determine the power transmission capability of power lines. In what follows, we will
discuss these aspects of power transmission line performance. Wherever appropriate,
we will consider a losslcss line, as it offers considerable simplicity and a better insight
into‘the performance characteristics of transmission lines. 6.1.7 Voltage and Current Profile under lilo-Load - if} PR
V: —e"‘+—e"‘”‘ ~ ~ .. s ..
2 2 V = Viv-Unﬁt) as = P’chﬂl = rﬂcose
f = Eel“ _ is --.u: f = KF’RIZCJ views) Is I ﬂVercJ Bil-13 = ﬂEslzcﬂanﬂ
2 c 22c where 9=ﬁi. The angle 3 is referred to as the electrical length or the fine angle,
- - cosﬂx .. IE3 sinﬂx' _ The crib-2 line parameter; other tkan line length that aﬁcts the results of
V : Es cosﬁ I = IZ— cosB Figure 6.5 is 6. Since 8 is practically the same for overhead tines ofait voltage levels
c (see Table 6.1), the results are universally aﬂnticabie, notjast for a 500 itV tine. 53:1.0 pu thu)
' 1.10 i x LDS
—y — _1-n_- 13' = 0.0013
0 = [31 = 22.3m
1} ltltl Elli) 3190 The rise in voltage at the receiving end on open-circuit is due to the ﬂow of y {km}
line changing (capacitive) current through line inductance. This phenomenon one ﬁrst {11} Voltage proﬁle
nﬂﬁcﬁd by Ferrenti on overhead lines supplying a lightly loaded {and hence highly
capacitive} cable network; it is therefore referred to as the Ferranti' eﬁizer. I (Pu) line is assumed to be losslees with ﬂ=ﬂ.Dﬂ13 radfkrn and Zc=250 ft. {1.5 1.031 \ 1.0512 coe[ﬂ.tl[l13.r) I=3ﬂ0 km lJlﬂ (3}3che1nat'ic diagram 0 = 300x00013 = 0.39 rad = 22.3" 0.4
l 1 [)3] 0 l.ﬂﬂIEain{ﬂ.Wle]
=' = _ .3
R 00522.3“ 1’“ = HESIZC) tenﬂ is [L]
Ir =ﬁ Extent! pu= 1.0tan22.3°= 11411 P11
1.ﬂm(ﬂ.ﬂ013r) 0 100 200 300
V - — = 1.0811008 ﬂﬂﬂl3x
were“ { ) P" ﬂkm}
001E223“ Figure 6.5 Voltage and current proﬁles for a Elli] knr losslese
line with receiving end open-circuited (1;) Line connected to sources at both ends For simplicity, let us assume that the line is symmetrical; i.e., it is connected
to identical sources at the two ends. Let ES and ER denote the voltages at the sending end and receiving end, respectively. From Equations 6.8 and 6.9, with x=£ and 3:62, we have
' — " “v ' s- " s —s 3-?! " its
V: ﬂe?x+§L'E-Se-Tx f: —S_S__eTI_Le_Tx
ell—3"” t':""i-¢=3‘"'I ZC(eT‘-e"”) ZC(eVi—e‘*i) For a lossless line, 7:}13. With 9=Bl, we have cosﬁUﬂ -x) f _ .155 mean —x) -— _ -—}_——
cos(6f2) 2:: cos(B/2) The voltage and current proﬁles are shown in Figure 6.6 for a 400 km line with
ES=ER=L0 pu. The generators at the sending end and receiving end should be capable
of absorbing the reactive power due to line charging. If this exceeds the underexcited
reactive power capabilityr of the connected generators, compensation may have to be
provided. If E3 and ER are not equal, the voltage and current proﬁles are not symmetrical
and the highest voltage is not at midpoint, but is nearer to the end with higher voltage. Hr V=E", E5511] pu I x ER=LU
P3=ﬁ| IPR=ﬂ B = ﬂ.ﬂﬂ13 Iadfkm
= {1.52 rad = 29.3” 1" {nu} 1.0 (pH) 1? 100 2m} 3m] 4;][1 km
Sending end Raceiving and
(a) Voltag: proﬁle
I {p11}
Base em = LWZC
[1.4 $256 '(b) Ewen: proﬁle Figure I16 Vﬂlfﬂge and current proﬁle of a 400 km
10551355 line under nth-load 5.1.9 Power Transfer and Stabilinlr Considerations E E
Pa = Z '2th sin?) let 5 be the angle by which E? leads ER, i.e., the load angle 01' the transmission “”313-
c
For a short line, sinB can be replaced by 9 in radians. Hence,
zcsme = 266 : l/LlCmJL—Cl = {oLl
= X , the series inductive reactance
E E
PR = 3 Rsinﬁ
XL
If E 3:13 R=Vm the rated voltage, then the natural load is
P : EsEa
0 ZC
Po _
PR = _ 31116
51116 With the voltage magnitudes ﬁxed, the power transmitted is a function of only
the transmission angle 5. When PR is equal to the natural load (P0), 5:9. 1 1.2.8 Principles of Transmission System Compensation xii
be uniformly distributed fated series and shunt compensation Wt a = series inductance per unit length
C =shunt capacitance per unit length series inductive reactance per unit length
shunt capacitive susceptanee per unit length XL = total series inductive teactance
BC = total shunt susceptance
= line length i shunt compensation 5% = E’s—baa = bcﬂ‘kss) is be It is- for inductive shunt compensation and is - fer capacitive ShUIlt CUIﬂDEﬂSﬂtIDI’L where k5}: is the degree of shunt compensation deﬁned as follows: k3,; = The effective values cf the characteristic impedance and phase constant with
shunt compensation are related to the uncompensated values as follows: 3:: Z
26- L: c bé l—gﬂ 3’: Bvl—ka Shunt capacitive compensation in effect whereas shlmt
inductive ccmpensaticn . series capacitive compensation of C 33 03C” 17x05;- = xL(1—kse) where kw is the degree of series capacitive compensarion deﬁned as follows: 'k _ sze . . . . . . .
ﬂ — It 13 DOSIUVB for capaCItwe 3613165 compensatlon. IL The effective values of charactei'istic impedance and phase constant 1With series
compensation are given by xi. := _
b—— -C1~21i [3 milk“ 36 Series (capacitive) compensation decreases both ZC and B. [3’ : B (14590463.!) 3' = e (l-kshMl-kﬂ)
I 1—gﬁ
1—}: SE Eﬁct of compensation on line voltage: - a ﬂat voltage proﬁle is achieved-
.For example, with k, =1 (100% inductive compensation), 9’ and
P0" are reduced to zero and 25 is increased to inﬁnity; this results in a ﬂat voltage at zero load. 1_ k
a ﬂat voltage can be achieved .t 25 = 2;; 1_ I:
or example, in order to transmit 1.1415,;I with a flat voltage sh
fl, sh t "ti ti fit =-0.96' 'd. I
pm 1 '3 a 1111 CﬂpﬁCl V3 0015111361153. OH O s 15 require [3 : B (1 __ kl!!!) (1 _ k“)
Series capacitive compensation may, in theory, be used instead of shunt 3* = 31“] -1; h)(1_k a)
compensation to give a ﬂat voltage proﬁle, under heavy loading. For example, a ﬂat . 3 E
voltage proﬁle can be achieved at a load of 1.-4|-P[_-I with a distributed series 1—153 1,!
compensation of k5,, =0.49. In practice, lumped series capacitors are not suitable for P5 = Po 1 k
obtaining a smooth voltage proﬁle along the line- Obviously, step changes in voltage _ 3: occur at points Where ﬁe ser1es capae1lors are applied. They do, however, improve voltage regulation at any given point, i.e., voltage changes with load are reduced. Eject of compensation on maximum power: : EsEa Z 6. sinB ’
The maximum power (corresponding to 5:90”) can be increased by decreasing either
ZC’, or 9', or both. The characteristic impedance .can be decreased with ca acitive
compensation, but it is accompanied by an increase in the electrical le _
other hand, inductive shunt compensation decreases 8', but increases ZC’. ntributes to the ecrease o o iii an 9'.
e s ou , owever, recognize that compensation is not required in all cases
to satisfy both objectives: (i) increasing P ', the power level at which the vol e
proﬁle is ﬂat, and (ii) decreasing electrical length in order to improve stability. Short
lines may require voltage support, i.e., an increase in 133', even though the inherent
electrical length is small. This may be achieved by shunt capacitors, provided that 3!
does not become excessive as a result. On the other hand, as we saw in Chapter 6 (See Figure 6.13), lines longer than about 500 km cannot be loaded even up to Pi! because; of excessive B; in such cases, reduction of 9’ is the ﬁrst priority. a, sine Illustrative ammpIe For purposes of illustration, we will consider a lossless 500 kV line having the
following parameters: a : 0.0013 radl’km 20 = 250 9 (Pa = 1,000 MW)
.51 = 0.325 Qa’km b5 = 5.2 nSIkm The line is 600 km long and transfers power between two sources as shown in Figure
11.54. The magnitudes of the source voltages are held at 1.0 pu. Our objective is to
examine the line performance without and with compensation. We will consider shunt
capacitor and series capacitor compensations chosen so as to maintain 1.0 pu midpoint voltage when the power transferred (P) is equal to 1.41130. i 133,13 V’” ER‘ZO as =ER _ 1.0 pu '
l——l—l 0 = a: = 000135600.
300 km 300 km P = 0 75 rad = 44.7., Figure 11.54 (3) With an enmpensnrfnn, the power-angle relationship is E E
P = 3‘3 sine
chmEI
With ES and ER at rated values,
3 = -—1 51115 = . 1 sinﬁ =1.423in6
Pa 31116 sm44.7° Also, considering one half nf the symmetrical line, P mayr be expressed in terms nf
V»: 35
P £st ‘ (6(2)
= —s1n
chiIIOB‘IZ)
V sin :5 2
z P ,,, < 1 ) '3 sin(44.?° [2) Hence, the per unit value of midpnint veltage as a function of P is given by V = 3 0.33
m Pu sin(6/2) (b) With untfermly distributed ﬁxed shunt compensation, to maintain V", at 1.0 pu
when P=1.4P9, we have 1'43: = Pi: = o l'kss Therefore, Egg-0.96. This will, in fact, result in 1.0 pu voltage threugheut the line
length at P=l.4Pﬂ. The eerrespending values ef 2C' and 6’ are HI = a 1-4%,: = 10:92 rad = 52.57“ The pewer transferred is given by Z r .
I: = ass-“‘5 = 1.53am.
ESER P Zésinﬂ' 2,; sinﬂ’ P = sinﬁ The midpoint veltage is new given by V = EESMB’IZ) = i 0.371
P m R} 2'5 sin(6f2) ﬂ sin(5[2) (e) With unffermb’ distributed ﬁred series cameraman”, to maintain V,” at 1.0 pm
when P=1.4Pﬂ, we have 1.4}; = a; = a 1/14” Therefere, 137550.49. The line parameters change to 2,; = 2C 1-ku = 25W1-0.49 = ”3.57 n B’ = 3 1—155: = 0.557 rad = 31.9? The power transfer and midpeint voltage equations new become Z n.
i = Jain—5f = 2.6Ssin6
P") Zr: emf}
and
V : gﬁmam = 3 0.1964
'“ a, C gimme) .ﬂ,ein(6}2) 1 1.2.19 Application at Ta p-Changing Transformers tn Tra nemiaaicn Systems Transfnnners with eff-lead tap—changing facilities can alsc help maintain
satisfacttirjir 1IrtIIltage prcﬁles. 1While transfnrmers with ULTC can be used tn take care
deaﬂy, heurly, and minnte-by-mmute T.r'ariaticns in system ccnditinns, settings nfcf'ﬂ-
Iced tap-changing transfcntiers have tn he careﬁtlly chcsetl depending en Icing-term
variations due tn system expansictt, lead gtewth, cr seasnnal changes. Dpﬂﬂ'lai pnwer-
ﬂew analysis previtles a ccnveniettt method cf detennjning apprnpriate ta]:- settings with either type cf tap-changing facilityr [49-51]. SDI] k‘u’ iﬂﬂ IN 23'!) EU 23!] k‘t” 44 RV ssc'kv ass in! 115 w 13.s tor FT = Fixed tap er eff-lead tap changing
ULTC = Under-lead tap changing Figure 11.?4 Single-line diagram of transmission network
illustrating transfcrmer tap change facilities 11.2.11 Distribution System Voltage Regulation [25.43] 1' Bus regulation at the substation
I Individual feeder regulation in the substation * Supplementary regulation along the feeders Substation has regulatiou A distribution substation transformer is usuallyr equipped with ULTC
equipment that automatically controls the secondary voltage. ltiitlternativelyl the
substation Ina)»r have a separate voltage regulator that regulates the secondar},r Side bus
voltage. Bus regulation generallyr employs three—phase units, although Sil'lglb—phase
regulators could he used in applications where phase voltages have a signiﬁcant
unbalance. Types of feeder regulators P = P‘rirnarj.r (shunt)
vending R = Regulating (series)
winding Figure 11.?5 Schematic of an induction regulator Intﬁtctton voltage regulator: Step mirage regiriamr (EVE): Stewed series R3 = reversing switch Figure 1136 Schematic of a step voltage regulator Typical] , the EVE. has rcvisiun fer ccrrecting the vultagcﬁm.
subunits s skis-v i -
the series winding intc eight equal parts, with each part prcvicling cine-eighth ﬁfth:
10% change in vultage. The cutput terminal is ccnnected tn the centre tap {If the
bridging reactar assu-ciated with the tap-changing mechanism. This in effect further
divides each step intc twc equal parts, giving a tctal cf 115 steps cf 5i3'ii: each. The
reversing switch allcws the regulatcr tc raise as well as lcwer the ctttpur i’ﬁltage "
ccvering a range cf plus cr minus lili’a vultage regulaticn in a tctal cf 32 stepg I Substation R1 R; C 5% r Consumers‘ senfiee Souree Maxhnum limit "M““hQLoed only T— Figure 11.1"? SVR control mechaaﬁsm Length along feeder Figure 1138 lI.I"oll:1age proﬁle of a feeder with a station regulator {R1},
Supplementary regulator (R2) and a shunt capacitor bank (C) ...

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