1880 Edison starts full-scale manufacture of DC generators and incandescent lamps.

Transcription

1 1.1 Leture 1 The Eletri ower System Overview. The symmetril three-se power system. Eletril power i AC iruits. Mesuremet of three-se power er-uit vlues of eletril qutities. Overview The moder eletri power system is vst iteroeted etwork of ompoets rgig from geertors, trsformers, trsmissio lies d wide vriety of lods. t is importt to lyse oth the stedy-stte d trsiet ehviour of suh system i order to mke vlid deisios out the desig, opertio d protetio of suh system. Also, tody s power system is iresigly eomig mixture of power d ommuitios, with itelliget etworks eig desiged d istlled ll over the world. Comied with the drive towrds reewle eergy, suh overgee of tehologies hs rejuveted the re of power systems. Erly History of ower Systems 1880 Ediso strts full-sle mufture of DC geertors d ideset lmps. Erly history of power systems 188 First power sttios uilt y Ediso, usig 30 kw 110 DC geertors drive y stem egies: Jury: Holor idut, Lodo. Septemer: erl St, New York City. Supplied 59 ustomers. Due to the low voltge, the mximum trsmissio diste ws limited to out 1.6 km Sprgue strts mufture of DC motors Stley uilds first prtil trsformer First AC system i opertio i Gret Brrigto, Msshusetts. Stem egie drivig Siemes 6 kw 500 sigle-se ltertor. ower trsmitted over diste of 1. km, the trsformed dow to Tesl uilds idutio d syhroous mhies. (3-se?) 1889 First AC lie i USA (4 k, 1 km). (3-se?) 1891 First 3-se lie i Germy (1 k, 179 km). ower Ciruit Theory 011

2 1. reset-dy ower Systems reset-dy systems re uiverslly three-se AC (resos to e disussed lter), ut some lrge systems iorporte high voltge DC trsmissio liks i speil irumstes. Where DC is required for idustril proess, retifier plt is used. Geertor effiiey improves with size. With preset tehology the optimum prtil output of stem turie geertors for lrge-sle geertio is proly roud 600 to 800 MW, d the optimum voltge out 3 k. Trsmissio voltges (log diste) rge from 66 k to 750 k d higher. High voltge distriutio (short to medium diste) voltges rge from. k to 13 k, overlppig the trsmissio rge. Most (exept very lrge) osumers reeive power t 400 / 30 (i Austrli). Frequey for puli systems hs ee stdrdised to 50 Hz i most of the world, d 60 Hz i the USA. The erly power supply systems typilly hd sigle power (geertig) sttio supplyig multitude of lods. Some smll isolted systems tody re similr, ut lrge sle preset-dy puli power supply etworks hve my iteroeted power sttios, d re therefore more diffiult to lyse. ower Ciruit Theory 011

3 1.3 Nottio As, more ofte th ot, we will e delig with omplex, rther th rel vlues, we will use orml upper se letters to deote omplex qutities (sors, impedes, et.). The mgitudes, or solute vlues, of the omplex qutities will geerlly e idited y elosig the upper se letter etwee vertil rs. There re some exeptios to this rule. The most otle exeptios re: = RMS mgitude of the se voltge lie = RMS mgitude of the lie voltge = RMS mgitude of the se urret lie = RMS mgitude of the lie urret (1.1) (1.1) (1.1) (1.1d) Stik to the ottio i the exm, or otherwise idite lerly whether you refer to sors, pek vlue, RMS vlues, istteous vlues, et. You will lose mrks for miguous ottio. lues of AC voltge d urret (solute vlues, s well s rel d imgiry ompoets) will e ssumed to e RMS (ot pek), uless stted otherwise. For istteous vlues of wveform (voltge, urret, power, et.) we will use lower se letters. Speil mrgi otes drw ttetio to some importt poits. ower Ciruit Theory 011

4 1.4 The Symmetril Three-hse ower System The hsor Opertor h Defie: The sor opertor h defied h 1 3 j 3 j e 110 (1.) Though h is defied s omplex ostt, it lso e regrded s opertor whih rottes sor y 10 ti-lokwise. Similrly, j 1 is imgiry ostt, ut e regrded s opertor whih rottes sor y 90. Altertive symols for h, foud i textooks, re d. From the defiitio of h, we esily derive the iteger powers of h, d some of their omitios: h 1 3 h 110 j j h 1 h 1 10 j h h 3 h h h j h h h 4 h, h 5 h h (1.3) The most importt reltio otiig h is: 1 h h 0 (1.4) We lso hve: 1 h h 1 h 160 h h j 3 1 h 3 3 j (1.5) ower Ciruit Theory 011

5 1.5 Some of these reltioships re show elow: h-h h -h 1- h -1 1=h 3 h -h Figure 1.1 ower Ciruit Theory 011

6 1.6 Str-Coeted oltge Soure Cosider simple symmetril three-se system, osistig of three voltge soures s show: Figure 1. The termils,, re the lie termils. The termil is the eutrl termil. The voltges re the lie-to-eutrl voltges, whih re (i the str-oeted system) the sme s se voltges. Let these e, i sor ottio: (1.6) = RMS mgitude of the se voltge (positive rel vlue) The sors ( ) re drw i lokwise order. Rottig i the tilokwise diretio the sors will the preset themselves i the orret sequee to sttiory oserver. ower Ciruit Theory 011

7 1.7 Usig the opertor h: h h (1.7) The sor digrm i Figure 1. is drw for 0. The orrespodig istteous voltges re: v v v v t Figure 1.3 with: v os t (1.8) v v ost ost (1.8) (1.8) The system is sid to hve se sequee. The ses my lso e idetified y three olours, e.g. red, white, lue. tht se red-white-lue (formerly red-yellow-lue) is the stdrd se sequee. Differet olours d symols (e.g. RST) might e used i other prts of the world. ower Ciruit Theory 011

8 1.8 By ovetio the omil voltge of the three-se system refers to the lie-to-lie voltge The lie-to-lie voltges, mesured etwee the three pirs of lie termils, re: 1 h 330 hh h h h (1.9) (1.9) (1.9) oltge mgitudes quoted without qulifitio usully re lie-to-lie voltges, or simply lie voltges d: 3 3 lie lie = RMS mgitude of the lie-to-lie voltge, usully lled (somewht miguously), the lie voltge. (1.10) Thus the lie-to-lie voltges ( ) hve the sme sequee s the lieto-eutrl voltges ( ), ut their mgitudes re multiplied y 3, d they led the orrespodig se voltges y 30. Similrly: 1 h 3 30 h h h1 h 390 h (1.11) (1.11) (1.11) Thus the lie-to-lie voltges ( ) lso hve the sme sequee s the lie-to-eutrl voltges ( ), ut i this se their se gles lg the se voltges y 30. ower Ciruit Theory 011

9 1.9 Str-Coeted Lod Cosider simple symmetril three-se system, with three equl lod impedes oeted i str: = 0 Figure 1.4 The idepedet voltge soures re s efore: h h (1.1) d the three equl lod impedes re: R jx (1.13) ower Ciruit Theory 011

10 1.10 Oviously, the se urrets (idetil to lie urrets i this se) re: h h (1.14) Thus, oe the se urret hs ee lulted, the other two se urrets e otied y rottig the sor y 10 iremets. Sie 1 h h 0, the se urrets sum to zero. Therefore, the eutrl urret 0, d the eutrl oetio is redudt. prtie, for resos to e overed lter, it my e either omitted or oeted, d we tlk of 3-wire d 4-wire systems. We lso hve: lie str oetio oly (1.15) = RMS mgitude of the se urret lie = RMS mgitude of the lie urret ower Ciruit Theory 011

11 1.11 Delt Coetio Either the soures or the lods (or oth) my lso e oeted i loop or delt formtio. This is illustrted elow for delt oeted lod: ' ' ' Delt-oeted lod Equivlet str lod Figure 1.5 For the delt iruit the lie-to-lie voltges ( ) re lso the se. voltges. Thus lie delt The se urrets re ( urrets re ( ) with mgitudes of delt 3. ), with mgitudes of lie delt. The lie For every delt oeted iruit there is equivlet str iruit. Beuse delt iruit hs o eutrl oetios, the reverse is ot eessrily true. Let ( ) e the equivlet str se voltges. The:,, therefore 3 3 (1.16) We ssume oth d hve the sme gle, therefore the equivlet str impede is: 3 (1.17) To void ofusio, it is geerlly est to overt ll delt iruits to equivlet str iruits for etwork ower Ciruit Theory 011

12 1.1 ower Ciruit Theory 011 Couplig Betwee hses the previous setios we ssumed o ouplig etwee the three se impedes (or i se of delt-oeted impedes). Now, let us ssume tht we hve symmetril pssive iruit, so tht the self-impedes of the three ses re equl, d the mutul impedes etwee ses re lso equl. Let: ses mutul impede etwee y pir of eh se - impede of self m s (1.18) the: m s m s m s s m m m s m m m s (1.19) For symmetril urrets, 0, therefore: m s (1.0) where m s is the effetive impede per se, or simply impede per se. The effetive impede per se e mesured diretly, usig three-se urrets, or e lulted, if we kow the self d mutul impedes. Whe the term impede, without further qulifitio, is used i threese work, the effetive impede per se is geerlly implied. This impede e used for lultios i the ompletely symmetril se oly. Eq. (1.19) is more geerl: lthough it ssumes symmetril impedes, it does ot ssume symmetril urrets. Effetive impede defied

13 1.13 Eletril ower i AC Ciruits ower i Sigle-hse Ciruit Cosider the sigle-se etwork show elow: i v v 1-se etwork i t Figure 1.6 Let: v i os t os t (1.1) d defie: (1.) The the istteous power iput to the etwork is: p vi os os os t ost ost ost (1.3) ower Ciruit Theory 011

14 1.14 is the verge vlue of p, ot the RMS vlue, whih hs o useful meig i this se, despite the widespred (mis)use of the term RMS power i the hi-fi idustry The seod term osilltes t doule the supply frequey, d otriutes othig to the verge power, whih is: os Further expdig Eq. (1.3), usig osa B os Aos B si Asi B get: (1.4), we p os os os ost ost os sit si 1 ost si sit (1.5) Now defie the retive power: Retive power defied Q si (1.6) the: p 1 ost Qsit (1.7) The istteous power ssoited with the rel d retive power ompoets is show elow: p ssoited with pssoited with Q Q 0 t 0 t Figure 1.7 ower Ciruit Theory 011

15 1.15 Usig sors, we defie omplex power: S (1.8) Complex power defied Correspodig to Eq. (1.1), we hve: (1.9) d we hve: os jsi (1.30) Hee the omplex power is: S S jq (omplex A) (1.31) where: S ppret power (A) Compoets of omplex power Q S S os rel power si retivepower (W) (vr) (1.3) Rel power is lso kow s tive power d verge power = gle y whih the voltge leds the urret S Q ower Ciruit Theory 011

16 1.16 These reltioships re illustrted elow: Q S Figure 1.8 f the etwork (lod) hs impede, the, d: S (1.33) f the etwork (lod) hs dmitte Y, the Y, d: S Y Y (1.34) t e show tht the totl omplex power S jq osumed y etwork is the sum of the omplex powers osumed y ll the ompoet prts of the etwork. This oservtio property is ot true of the ppret power S. ower Ciruit Theory 011

17 1.17 ower Ftor d Dissiptio Ftor We defie: ower ftor (p.f.) S os (1.35) ower ftor defied d: Dissiptio ftor d Q t Dissiptio ftor defied (1.36) where 90 = loss gle ower ftor is ormlly pplied to useful power system lods where the idel is to mximise rel power. Dissiptio ftor is used with idutors, pitors, d isultig mterils, where the idel is to miimise rel power dissiptio. Qulity ftor = retive power, Q. 1 d, ommoly lled Q, is ot to e ofused with the ower Ciruit Theory 011

18 1.18 ower i Three-hse Ciruit Complex ower Beuse the omplex power S jq i y iruit is equl to the sum of the omplex powers i prts of the iruit, the omplex power of symmetril three-se iruit must e equl to the sum of the powers i eh se. the se of the symmetril system this is three times the power i eh se: Three-se power for symmetril iruit usig se qutities 3 Q 3 S 3 os si (1.37) where d re RMS mgitudes of voltge d urret per se, d is the gle y whih the se voltge leds the se urret. Eqs. (1.37) overted to use lie urret d lie voltge red: Three-se power for symmetril iruit usig lie qutities Q 3 3 lie lie lie lie os si S 3 lie lie (1.38) The ppret power is S Q s for the sigle-se se. ower Ciruit Theory 011

19 1.19 stteous ower Cosider the simple symmetril three-se system show elow: = 0 Figure 1.9 Usig se voltge s the time referee, i.e. settig 0 i Eq. (1.1), we get: v v v os os os t i ost t 3 i ost 3 t 4 3 i ost 4 3 (1.39) The istteous power i eh se is: p p p v i v i v i os os os t ost t 3ost 3 t 4 3ost 4 3 (1.40) Usig os Aos B osa B osa B, we expd: p p p v i v i v i os os t os os t 4 3 os os t 3 (1.41) ower Ciruit Theory 011

20 1.0 The totl istteous power is: stteous threese power for symmetril iruit is equl to the verge power, d is ostt (does ot vry with time)! p p 3 p p os (1.4) Eq. (1.4) shows tht the symmetril three-se istteous power is ostt d equl to the rel power. There is o osilltory term s i siglese iruit. The property of ostt istteous power is ot uique to the three-se system, ut pplies to poly-se systems i geerl. The ostt istteous power is gret sset for eletril mhies, filittig smooth power oversio without virtio. With sigle-se eletromehil power oversio virtio t doule the supply frequey is ievitle. ower Ciruit Theory 011

21 1.1 Mesuremet of Three-hse ower Usig Three Wttmeters Cosider the mesuremet of the totl rel power i the etwork show elow: i v 3-se supply i v 3-se uled lod i v o v i o Figure 1.10 Let: p = totl 3-se istteous power supplied to the lod p m = totl istteous power see y the three wttmeters (1.43) the: p p v i v i v i m v v i v v i v v p v o i o o v i v i v i v i o o i (1.44) Therefore, the sum of the three wttmeter redigs is the true verge power,, of the lod oly if v 0. The method lwys mesures orretly if oi poits o d re oded, ut if i 0 (led lod, or lod eutrl ot oeted), the poit o my e left flotig. ower Ciruit Theory 011

22 1. Usig Two Wttmeters From the previous setio, if i 0, the v o e y ritrry vlue. So oet poit o i Figure 1.10 to poit. Now the se wttmeter eomes superfluous, s it would lwys red zero. See the etwork elow: A i v 3-se supply C i v 3-se uled lod i v Figure 1.11 The sum of the remiig two wttmeters ow gives the true verge power,. Therefore A for led or uled oditios, if there is o eutrl urret. C The sor digrm, drw for led oditios, is show elow: Figure 1.1 ower Ciruit Theory 011

23 1.3 The two wttmeter redigs re: A C os 30 os 30 (1.45) ut: 3 (1.46) hee: A C 3 3 os 30 os 30 (1.47) d we oti: A C C A 3 3 os (1.48) si Q 3 Therefore: Q A W 3 vr C C A (1.49) For mesuremet of Q the method requires led oditios. The sigs of the wttmeter redigs re importt. Either A or C my e egtive uder some oditios. ower Ciruit Theory 011

24 1.4 Usig Oe Wttmeter d Swith this se, we hve oe wttmeter whose voltge oil oetio is swithed etwee two ses: i 3-se supply SW 1 i v v 3-se uled lod i v Figure d re wttmeter redigs with swith SW i positios 1 d respetively. The sme equtios s for the two wttmeters method pply. This method oly e used whe the lod is stedy. Q 1 W 3 vr 1 (1.50) ower Ciruit Theory 011

25 1.5 ower i No-Lier Lods f v d i represet periodi wveforms d urret, the they e writte usig ompt trigoometri Fourier Series: v DC k 1 os t i DC k 1 os t (1.51) Note: d here re RMS vlues, ot sors. Sie: 0 os x osmx dx 0 os if m if m (1.5) the: 1 T 0 T 0 0 vidt DC k 1 (1.53) where: os power i the - th hrmoi (1.54) f the supply voltge is siusoidl, d the supply impede is low, so tht the hrmoi urret tke y the o-lier lod does ot use sigifit distortio of the supply voltge, the the oly power pplied to the lod is the first hrmoi (fudmetl frequey). i.e. oly the fudmetl ompoet of the urret is relevt to the power iput to the lod. Higher hrmoi powers do ot exist. f oth voltge d urret re distorted, the Eq. (1.53) must e used to oti the totl power. ower Ciruit Theory 011

26 1.6 Summry Three-se AC power systems developed rpidly d re ow stdrd roud the world. A symmetril three-se iruit e lysed s sigle-se iruit. Delt oetios e overted to str oetios to filitte this lysis. Rel power i three-se AC iruit is give y os. 3 Retive power i three-se AC iruit is give y Q si. 3 Complex power i three-se AC iruit is give y 3 S. ower ftor is mesure of how losely lod delivers mximum rel power. Dissiptio ftor is mesure of how losely lod presets itself s lossless. The istteous power i symmetril three-se system is equl to the verge power, p. This filittes smooth power oversio. Three-se rel power e mesured usig just two wttmeters, regrdless of lod ule, soure ule, d the wveform of the periodi soure. The mesuremet of retive power requires the system to e led. f voltge d / or urret re ot siusoidl, we lulte the totl power y osiderig the power i eh hrmoi. Referees Crmo, J.: ower Ciruit Theory Notes, UTS, Truupold, E.: ower Ciruit Theory Notes, UTS, ower Ciruit Theory 011

27 1.7 Exerises 1. Evlute the followig expressios i polr form: The symmetril three-se power system () h 1 () h () h 3 h 1 h (d) jh. For the delt oeted lod i Figure 1.5, prove tht: d fid the vlue of. Drw sor digrm of the six urrets d six voltges, ssumig resistive lod. 3. For the iruit elow, lulte: (i) Mgitudes of ll urrets. (ii) Lie voltge t termils A, B, C. 1 A 3 1 B 4 C se led supply 1 = j = + j3 4 = 3 - j 6 ower Ciruit Theory 011

28 1.8 Eletril power i AC iruits 4. A pitor is required to geerte 50 kvr t 6.35 k, 50 Hz. Clulte: (i) The vlue of the pite (ii) Mximum eergy stored i the pitor. 5. Show tht for etwork of two impedes the omplex power iput to the etwork equls the sum of the omplex powers sored y eh impede, whe: (i) the impedes re i series (ii) the impedes re i prllel 6. A high voltge withstd test is to e pplied to the sttor widig to groud isultio of 660 MW turo-geertor. The pite to erth of the widig (the test ojet ) is 73 F. The speified test voltge is 48 k, 50 Hz. The dissiptio ftor of the isultio vries with voltge, ut is expeted to e 40 mw/vr t 48 k. () Clulte the rel d retive power output of the H testig trsformer used to provide the test voltge. () A ltertive test iruit uses vrile H retor L i series with the test ojet C. The retor is djusted for resoe t 50 Hz. The dissiptio ftor of the retor is estimted to e 30 mw/vr. Determie: (i) The rete of the retor. 1 L C 48 k (ii) ower (rel d retive) sored y the retor. (iii) ower iput to the iruit. (iv) put voltge 1 required. ower Ciruit Theory 011

29 Two sigle-se geertors re oeted k to k vi idutor (rete = 5 ). The emfs of the geertors re s show, d their iterl impedes re osidered egligile. j5 e 1 e e1 100 os 100t e 100 si 100t 3 Determie the wtts d vrs supplied or reeived y eh ompoet of the etwork. 8. For the iruit i Exerise 3, lulte the omplex power supplied to the totl lod from the termils A, B, C. 9. A 3-se trsmissio iruit hs impede per se of 5 j 35. The lod t the reeivig ed osumes 600 kw t uity p.f. d 13. k (lie voltge). Clulte sedig ed voltge mgitude, rel, retive d ppret power. 10. A 3-se trsmissio lie hs impede per se of 5 j 60. At the sedig ed the iput is 10 MW, 30 Mvr t 0 k. Fid the power d voltge t the reeivig ed. ower Ciruit Theory 011

30 A 3-se geertor supplies lod vi two prllel iruits A d B, with impedes j 0.7 d j 1.3 respetively. The lod o the geertor is 30 MA t 0.8 lggig power ftor, d termil voltge of 11 k. Fid the omplex power iput d output for eh iruit A d B. A resistor R is oeted to lies d of symmetril 3-se supply. t is suggested tht the lodig of the supply e led y oetig retor jx ross lies d, d pitor jx ross d. () erify tht the suggestio is vlid, d fid the required rtio X R () Drw equivlet str iruit (with ompoet vlues) for the omposite lod. Mesuremet of three-se power 1. For the oe wttmeter d swith method of mesurig three-se power, drw sor digrm d prove tht 1 the totl wtts d vrs respetively. d Q 3 1. Give 13. t is possile to use sigle wttmeter to mesure vrs i symmetri threese iruit. Drw iruit to show how this e doe, d fid the vlue of the lirtio ostt y whih the wttmeter redig hs to e multiplied to oti totl vrs. (Hit: spet the sor digrm, d oet the wttmeter so tht is would red zero whe the p.f. = 1.) ower Ciruit Theory 011