CHAPTER 3 NETWORK ADMITTANCE AND IMPEDANCE MATRICES

Size: px

Start display at page:

Download "CHAPTER 3 NETWORK ADMITTANCE AND IMPEDANCE MATRICES"

Garey Carson
6 years ago
Views:

1 CHAPTER NETWORK ADTTANCE AND PEDANCE ATRCES As we hve see i Chter tht ower system etwor c e coverted ito equivlet imedce digrm. This digrm forms the sis of ower flow (or lod flow) studies d short circuit lysis. this chter we shll discuss the formtio of us dmittce mtrix (lso ow s us mtrix) d us imedce mtrix (lso ow s us mtrix). These two mtrices re relted y (.) us us We shll discuss the formtio of the us mtrix first. This will e followed y the discussio of the formtio of the us mtrix.. FORATON OF BUS ADTTANCE ATRX Cosider the voltge source S with source (series) imedce of S s show i Fig.. (). Usig Norto s theorem this circuit c e relced y curret source S with rllel dmittce of S s show i Fig.. (). The reltios etwee the origil system d the Norto equivlet re S S d S (.) S S We shll use this Norto s theorem for the formultio of the us mtrix. Fig.. () oltge source with source imedce d () its Norto equivlet. For the time eig we shll ssume the short lie roximtio for the formultio of the us dmittce mtrix. We shll therefter relx this ssumtio d use the π- reresettio of the etwor for ower flow studies. Cosider the 4-us ower system show i Fig... This cotis two geertors G d G tht re coected through trsformers T d T to uses d. et us deote the sychroous rectces of G d G y X G d X G resectively d the lege rectces of T d T y X T d X T resectively. et i, i,, 4 d,, 4 deote the lie imedce etwee uses i d. The the system imedce digrm is s show i Fig.. where (X G X T ) d (X G X T ). this figure the odes with the ode voltges of to 4 idicte the

2 .48 uses to 4 resectively. Bus idictes the referece ode tht is usully the eutrl of the -coected system. The imedce digrm is coverted ito equivlet dmittce digrm show i Fig..4. this digrm i / i, i,, 4 d,, 4. The voltge sources E G d E G re coverted ito the equivlet curret sources d resectively usig the Norto s theorem discussed efore. Fig.. Sigle-lie digrm of simle ower etwor. Fig.. medce digrm of the ower etwor of Fig... Fig..4 Equivlet dmittce digrm of the imedce of Fig...

3 .49 We would lie to determie the voltge-curret reltioshis of the etwor show i Fig..4. t is to e oted tht this reltio c e writte i terms of the ode (us) voltges to 4 d iected currets d s follows or us (.) 4 us (.4) 4 t c e esily see tht we get (.) from (.) d (.4). Cosider ode (us) tht is coected to the odes d. The lyig KC t this ode we get ( ) ( ) ( ) (.5) similr wy lictio of KC t odes, d 4 results i the followig equtios ( ) ( ) 4 ( 4 ) ( 4 ) 44 (.6) 4 ( ) ( ) 4( 4 ) ( 4 ) 44 ( 4 ) 4( 4 ) 4 4 ( 4 4 ) 4 (.7) (.8) Comiig (.5) to (.8) we get (.9) Comrig (.9) with (.) we c write

4 us (.) geerl the formt of the us mtrix for -us ower system is s follows us O (.) where (.) t is to e oted tht us is symmetric mtrix i which the sum of ll the elemets of the th colum is. Exmle.: Cosider the imedce digrm of Fig.. i which the system rmeters re give i er uit y.5,.,.5, 4.4 d 4.5 The system dmittce c the e writte i er uit s 4, 5, 4, 4.5 d 4 The us is the give from (.) s us er uit Cosequetly the us imedce mtrix is give y us er uit t c e see tht lie the us mtrix the us mtrix is lso symmetric.

5 .5 et us ow ssume tht the voltges E G d E G re give y E G u d EG u The curret sources d re the give y E G E G We the get the ode voltges from (.4) s 4 6 u u Solvig the ove equtio we get the ode voltges s Node Elimitio y trix Prtitioig er uit Sometimes it is desirle to reduce the etwor y elimitig the odes i which the curret do ot eter or leve. et (.) e writte s A x K T A x (.) the ove equtio A is vector cotiig the currets tht re iected, x is ull vector d the us mtrix is ortioed with the mtrices K, d. Note tht the us mtrix cotis oth d T due to its symmetric ture. We get the followig two sets of equtios from (.) A KA x (.4) T T (.5) x A x x A Sustitutig (.5) i (.4) we get

6 .5 A T ( K ) A (.6) Therefore we oti the followig reduced us dmittce mtrix reduced us T K (.7) Exmle.: et us cosider the system of Exmle.. Sice there is o curret iectio i either us or us 4, from the us comuted we c write K 5 5 ' d We the hve reduced T us K er uit Sustitutig 4 6 er uit d 4 9 er uit we shll get the sme vlues of d s give i Exmle.. sectig the reduced us mtrix we c stte tht the dmittce etwee uses d is Therefore the self dmittce (the dmittce tht is coected i shut) of the uses d is 4 er uit ( ). The reduced dmittce digrm otied y elimitig odes d 4 is show i Fig..5. t is to e oted tht the imedce etwee uses d is the Thevei imedce etwee these two uses. The vlue of this imedce is /(6.8978).45 er uit. Fig..5 Reduced dmittce digrm fter the elimitio of uses d 4... Node Elimitio y Kro Reductio Cosider equtio of the form Ax (.8) where A is ( ) rel or comlex vlued mtrix, x d re vectors i either R or C. ssume tht the vector hs zero elemet i the th row such tht (.8) is give s

7 .5,, O, x x x x (.9) We c the elimite the th row d th colum to oti reduced ( ) umer of equtios of the form,, O,,, ew x x x (.) The elimitio is erformed usig the followig elemetry oertios ew (.) Exmle.: et us cosider the sme system of Exmle.. We would lie to elimite the lst two rows d colums. et us first elimite the lst row d lst colum. Some of the vlues re give elow ew 44 ew 44 5, ew 44.5 ew , similr wy we c clculte the other elemets. Filly elimitig the lst row d lst colum, s ll these elemets re zero, we get the ew us mtrix s 44 ew us Further reducig the lst row d the lst colum of the ove mtrix usig (.), we oti the reduced us mtrix give i Exmle.... clusio of ie Chrgig Ccitors So fr we hve ssumed tht the trsmissio lies re modeled with lumed series imedces without the shut ccitces. However i rctice, the us mtrix cotis the shut dmittces for lod flow lysis i which the trsmissio lies re rereseted y its

8 .54 π-equivlet. Note tht whether the lie is ssumed to e of medium legth or log legth is irrelevt s we hve see i Chter how oth of them c e rereseted i π- equivlet. Cosider ow the ower system of Fig... et us ssume tht ll the lies re rereseted i equivlet-π with the shut dmittce etwee the lie i d eig deoted y chi. The the equivlet dmittce t the two ed of this lie will e chi /. For exmle the shut ccitce t the two eds of the lie oiig uses d will e ch /. We c the modify the dmittce digrm Fig..4 s show i Fig..6. The us mtrix of (.) is the modified s us ch 4 4 ch 4 4 ch ch4 (.) where ch ch ch ch4 ch ch ch ch4 ch ch ch ch 4 ch4 ch4 (.) Fig..6 Admittce digrm of the ower system Fig.. with lie chrgig ccitors.

9 .55. EEENTS OF THE BUS PEDANCE AND ADTTANCE ATRCES Equtio (.) idictes tht the us imedce d dmittce mtrices re iverses of ech other. Also sice us is symmetric mtrix, us is lso symmetric mtrix. Cosider 4-us system for which the voltge-curret reltios re give i terms of the us mtrix s (.4) We c the write (.5) 4 This imlies tht is the dmittce mesured t us- whe uses, d 4 re short circuited. The dmittce is defied s the self dmittce t us-. similr wy the self dmittces of uses, d 4 c lso e defied tht re the digol elemets of the us mtrix. The off digol elemets re deoted s the mutul dmittces. For exmle the mutul dmittce etwee uses d is defied s (.6) 4 The mutul dmittce is otied s the rtio of the curret iected i us- to the voltge of us- whe uses, d 4 re short circuited. This is otied y lyig voltge t us- while shortig the other three uses. The voltge-curret reltio c e writte i terms of the us mtrix s (.7) The drivig oit imedce t us- is the defied s (.8) 4 i.e., the drivig oit imedce is otied y iectig curret t us- while eeig uses, d 4 oe-circuited. Comrig (.6) d (.8) we c coclude tht is ot the recirocl of. The trsfer imedce etwee uses d c e otied y iectig curret t us- while oe-circuitig uses, d 4 s

10 .56 4 (.9) t c lso e see tht is ot the recirocl of.. ODFCATON OF BUS PEDANCE ATRX Equtio (.) gives the reltio etwee the us imedce d dmittce mtrices. However it my e ossile tht the toology of the ower system chges y the iclusio of ew us or lie. tht cse it is ot ecessry to recomute the us mtrix gi for the formtio of us mtrix. We shll discuss four ossile cses y which existig us imedce mtrix c e modified. et us ssume tht -us ower system exists i which the voltge-curret reltios re give i terms of the us imedce mtrix s orig (.) The im is to modify the mtrix orig whe ew us or lie is coected to the ower system... Addig New Bus to the Referece Bus t is ssumed tht ew us ( > ) is dded to the referece us through imedce. The schemtic digrm for this cse is show i Fig..7. Sice this us is oly coected to the referece us, the voltge-curret reltios the ew system re ew orig (.) Fig..7 A ew us is dded to the referece us.

11 .57.. Addig New Bus to Existig Bus through medce This is the cse whe us, which hs ot ee rt of the origil etwor, is dded to existig us through trsmissio lie with imedce of. et us ssume tht ( > ) is the ew us tht is coected to us ( < ) through. The the schemtic digrm of the circuit is s show i Fig..8. Note from this figure tht the curret flowig from us will lter the voltge of the us. We shll the hve ( ) (.) similr wy the curret will lso lter the voltges of ll the other uses s ( ) i i i i i i, (.) Furthermore the voltge of the us is give y ( ) (.4) Therefore the ew voltge curret reltios re ew orig (.5) t c e oticed tht the ew us mtrix is lso symmetric. Fig..8 A ew us is dded to existig us through imedce... Addig medce to the Referece Bus from Existig Bus To ccomlish this we first ssume tht imedce is dded from ew us to existig us. This c e ccomlished usig the method discussed i Sectio... The to dd this us to the referece us through, we set the voltge of the ew us to zero. However ow we hve ( ) ( ) us mtrix isted of mtrix. We c the remove the lst row d lst colum of the ew us mtrix usig the Kro s reductio give i (.).

12 Addig medce etwee two Existig Buses et us ssume tht we dd imedce etwee two existig uses d s show i Fig..9. Therefore the curret iected ito the etwor from the us side will e isted of. Similrly the curret iected ito the etwor from the us side will e isted of. Cosequetly the voltge of the i th us will e ( ) ( ) ( ) i i i i i i i i i i i i i (.6) Similrly we hve ( ) (.7) d ( ) (.8) Fig..9 A imedce is dded etwee two existig uses. We shll ow hve to elimite from the ove equtios. To do tht we ote from Fig..9 tht (.9) Sustitutig (.7) d (.8) i (.9) we get ( ) ( ) ( ) ( ) ( ) (.4) We c the write the voltge curret reltios s ew orig (.4) where

13 .59 (.4) We c ow elimite the lst row d lst colum usig the Kro s reductio give i (.)...5 Direct Determitio of us trix We shll ow use the methods give i Sectios.. to..4 for the direct determitio of the us mtrix without formig the us mtrix first. To ccomlish this we shll cosider the system of Fig.. d shll use the system dt give i Exmle.. Note tht for the costructio of the us mtrix we first elimite ll the voltge sources from the system. Ste-: Strt with us-. Assume tht o other uses or lies exist i the system. We dd this us to the referece us with the imedce of.5 er uit. The the us mtrix is us.5 (.4), Ste-: We ow dd us- to the referece us usig (.). The system imedce digrm is show i Fig... We the c modify (.4) s.5 us, (.44).5 Fig.. Networ of ste-. Ste-: We ow dd imedce of. er uit etwee uses d s show i Fig... The iterim us mtrix is the otied y lyig (.4) o (.44) s i us, Elimitig the lst row d lst colum usig the Kro s reductio of (.) we get us, (.45) Ste-4: We ow dd us- to us- through imedce of.5 er uit s show i Fig... The lictio of (.5) o (.45) will the result i the followig mtrix

14 .6 Fig.. Networ of ste us, (.46) Fig.. Networ of ste-4. Ste-5: Coect uses d through imedce of.4 er uit s show i Fig... The iterim us mtrix is the formed from (.4) d (.46) s i us, Fig.. Networ of ste-5. Usig the Kro s reductio we get the followig mtrix us, (.47)

15 .6 Ste-6: We ow dd ew us-4 to us- through imedce of.5 s show i Fig..4. The the lictio of (.5) o (.47) results i the followig mtrix us,6 (.48) Fig..4 Networ of ste-6. Ste-7: Filly we dd uses d 4 through imedce of.4 to oti the etwor of Fig.. mius the voltge sources. The lictio of (.4) o (.48) results i the iterim us mtrix of i us, Elimitig the 5 th row d colum through Kro s reductio we get the fil us s us,7 (.49) The us mtrix give i (.49) is the s tht give i Exmle. which is otied y ivertig the us mtrix..4 THEENN PEDANCE AND us ATRX To estlish reltioshis etwee the elemets of the us mtrix d Thevei equivlet, let us cosider the followig exmle. Exmle.4: Cosider the two us ower system show i Fig..5. t c e see tht the oe-circuit voltges of uses d re d resectively. From (.) we c write the us mtrix of the system s

16 .6 Fig..5 Two-us ower system of Exmle.4. us The determit of the ove mtrix is us Therefore the us mtrix is us us Solvig the lst two equtios we get us us ( ) ( ) (.5) Now cosider the system of Fig..5. The Thevei imedce of looig ito the system t us- is the rllel comitio of d, i.e., ( ) (.5) th, Similrly the Thevei imedce otied y looig ito the system t us- is the rllel comitio of d, i.e., ( ) (.5) th, Hece the drivig oit imedces of the two uses re their Thevei imedces.

17 .6 et us ow cosider the Thevei imedce while looig t the system etwee the uses d. From Fig..5 it is evidet tht this Thevei imedce is the rllel comitio of d, i.e., th, With the vlues give i (.5) we c write ( ) ( ) ( ) [ ] Comrig the lst two equtios we c write th, (.5) As we hve see i the ove exmle i the reltio us, the ode or us voltges i, i,, re the oe circuit voltges. et us ssume tht the currets iected i uses,, d,, re zero whe short circuit occurs t us. The Thevei imedce t us is th, (.54) From (.5), (.5) d (.54) we c surmise tht the drivig oit imedce t ech us is the Thevei imedce. et us ow fid the Thevei imedce etwee two uses d of ower system. et the oe circuit voltges e defied y the voltge vector d corresodig currets e defied y such tht us (.55) Now suose the currets re chged y such tht the voltges re chged y. The ( ) us (.56) Comrig (.55) d (.56) we c write (.57) us

18 .64 et us ow ssume tht dditiol currets d re iected t the uses d resectively while the currets iected t the other uses remi the sme. The from (.57) we c write us (.58) We c therefore write the followig two equtios form (.58) o o o o The ove two equtios c e rewritte s o ( ) ( ) (.59) o ( ) ( ) (.6) Sice, the etwor c e drw s show i Fig..6. By isectio we c see tht the oe circuit voltge etwee the uses d is (.6) o oc, d the short circuit curret through these two uses is sc, (.6) Also durig the short circuit. Therefore comiig (.59) d (.6) we get o o ( ) ( ), (.6) Comiig (.6) to (.6) we fid the Thevei imedce etwee the uses d s oc, th, sc, sc (.64) The ove equtio grees with our erlier derivtio of the two us etwor give i (.5).

19 Fig..6 Thevei equivlet etwee uses d..65

Fig. 1. I a. V ag I c. I n. V cg. Z n Z Y. I b. V bg

Fig. 1. I a. V ag I c. I n. V cg. Z n Z Y. I b. V bg ymmetricl Compoets equece impedces Although the followig focuses o lods, the results pply eqully well to lies, or lies d lods. Red these otes together with sectios.6 d.9 of text. Cosider the -coected lced