Ch_6_and_11.pdf - 6.1.2 Performance Equations 2 = R ij =...

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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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