ower Flow uses wth ether or both Geerator Load G G G D D 4 5 D4 D5 ecto G Net Comple ower ecto - D D ecto s egatve sg at load bus = _ G D mlarl Curret ecto = G _ D At each bus coservato of comple power curret holds: 5 G = + + 5 D G D
ower Flow Equatos Geerators Load L Node = = bus = = = comple power ecto = = = = Let = = = = = + - =
Eample: Use = equvalet crcuts for trasmsso les 00 all capactors c Z = 0 L 0 L bus = -998 0 0-998 0-998 Comple ower ectos: = 998 0 0 0 998 0 0 0 9 98 uadratc fucto wth respect to voltages Nolear equatos that ca ot be easl solved closed form Now cosder the power flow equatos
= = cos s = cos F s Full load Flow equatos Aother Formulato: Let G { cos G s } [cos cos s s ] = G cos = = s F [s cos cos s ] = cos G s = = t 4
mplfed Forms: Al Neglect resstaces trasmsso les G 0 Trasmsso losses represet about % of the power geerated s = F cos = reactace b / susceptace z b b b z b s s cos cos mall A le agles 0 less tha 0 0 s rada cos The J = become does ot deped o δ ad δ F 5
& decoupled A A A or costat for all buses the DC load flow Lear equatos uows gve = = Zero f ot coected b a le dag : off dag : f coected b a le 6
7 DC load flow soluto Fd the verse of The do the multplcato - Fast: ormall for o-le computato ad plag purpose accurac ma suffer Ca be used to redspatch geerato for cogesto maagemet F
ower Flow roblem - For bus there are 4 etwor varables comple power comple voltage Ge pecfed uattes Load Uow arables Load bus Ge bus lac bus Ref bus swg bus all scalars tead-state etwor relatoshp -bus cl Ref Real power balace F 0 Reactve power balace G 0 4 varables { specfed { uows 8
- equatos olear wat comple voltages ca be solved to fd the uows - The soluto ma ot est Eve f a soluto ests t ma ot be uque - For ormal operato codtos of a power sstem a soluto ests the eghborhood of = =0 zero flow for all les rated voltages at all buses - Closed form soluto mpossble - Numercal methods - Gauss-edel - Fast Decoupled Load Flow - Newto-Raphso - DC load Flow Eample: F e relato of F Real power flow relatoshp ol assume all voltage magtudes = p set 0 Gve wat to fd les are assumed lossless hece 0 9
0 s / s / s / s / s / s / + 0 = 0 equatos are depedet o eed ol equatos from fd to wat gve s s s s
E eqs doe load eqs for geerator slac bus bus gve Noe for slac 0 load ge 0 swg slac bus -bus wat ad comple voltages wll be ow ca fd ad all le flows Form bus = Ge: equato Load: & equatos F cos cos s s cos s cos cos s eqs wth uows
g of r bus elemets z r z r r0 0 r r r r pos real part 0 0 r r r eg real part 0 0 A ver log but complete eample: Le Wat bus & power flow formulato Le Le Le : 90 m : 95 m : 5 m Les: R= 0075 /m ωl= 05 /m = 0 /m c slac
semes semes semes z z z 0 5 5 0 06 96 0 008 48 688 975 5 05 0075 4855 75 95 05 0075 9709 45 90 05 0075 huts: 6 0 5 / 0 c m M c Le : 0 0054 0 054 90 5 6 semes half le legth Le : 0 057 95 5 semes Le : 0 000 5 5 semes
4 0 5 5 5 5 0 96 5 5 49 7 0 48 00 96 0 48 98 444 ] [ 0 008 48 ] [ 0 98 444 0 057] 0054 06 96 008 [48 semes semes bus shuts F s cos = Total of 6 eqs tep comple voltages Fd Wrte
tep ar ect of ge ecto of slac bus Oce are foud teratve method ca calculate s s s ad cos cos cos s s s TE : Le Flows R X R X 5
6 currets short b dffer ower Flow tep 4 c ' R X ' c ' ' ' ' geeral c c c c
7 f 0 G For all les the bus reduces to 5 5 0 5 49 0 0 0 98 5 5 0 5 49 0 0 0 98 bus F s 5 s 0 s s cos s ad Tr
8 F4 5 0 49 5 0 for smlarl F5 DC load Flow: 5 5 0 5 49 0 0 0 98 bus
9 lac 0 Has to be true for loseless sstem Needs ol F5 too 0 referece 0 0 98 fromst row of bus Note the matr for F5 s dfferet from the bus for F4! 54 0 0 0 huts have o real power flow 0 5 = 0