LOAD-FLOW CALCULATIONS IN MESHED SYSTEMS Node voltage method A system part with the node k and its direct neighbour m
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1 LOAD-FLOW CALCLATIONS IN MESHED SYSTEMS Node oltage method A system part wth the ode ad ts dret eghbor m Î Îm Î m m
2 Crrets Î m m m Î Î m m m m Î m m m m m m m Let s dee the ode sel-admttae (adm. matr dagoal elemet (, m m m m m m
3 Node mtal admttae (o-dagoal elemet (,m m Hee or th ode rret Î (, m m m m (, m (,m m m Î m m (,m Matr epresso Î 3 Î Reglar admttae matr (there s at least oe o-zero elemet Î Ẑ Î
4 Sglar admttae matr ode oltage ( - deed y D T B B A y Î Î Hee y D T B y y B A Î Î Let s allate y, Î T B D y D y Î
5 Gass-Sedel method - terate method or o-lear eqatos - ot always a good oergee (! Bas dea Rewrtte g( I s the estmato th step, the et terato s g( We ote tl two ollowg teratos deree s smaller tha the presrbed preso ε (
6 Sometmes the oergee a be mproed by the aelerato ator α (α < or α > ( ( (,,., g( - The system o eqatos wth ows,,,,,,
7 Eah ow s epressed rom oe eqato g (,,,. g g ( (,,,,,, Gass: th terato rom (- th appromato g (,,,,,, m m m Gass-Sedel: or th terato allato also th appromatos rom preos eqatos are sed m- - - g (,,,,,, m m m Coergee s tested or eah arable separately. m- m m
8 Newto-Raphso method - the most ote method or o-lear eqatos - t ses Taylor polyomal - t oerts o-lear eqatos solto to lear eqatos solto, gradally hgher preso o the estmato ( Bas dea I s the tal estmato ad Δ s the deree rom the rght solto, the (
9 Taylor seres d d ( -! Epaso to the Taylor seres o d d ( d! d Hgher orders egletg (learzato d d where Δ s alled deet. ( (
10 Addg Δ to the tal estmato ges the seod appromato ( Δ d d (Note: mpossble the derate eqals zero The same relatos the et steps ge the method algorthm: Δ Δ Δ d d ( (
11 The system o eqatos wth ows,,, (.,,, (,,, ( Epaso to the Taylor seres ( (
12 ( Matr epresso : : : : :. ( : ( ( short X J C
13 Hee X J C The method algorthm: C ( ( : ( X J C X X X
14 ( : ( ( C ( ( ( ( where Δ : Δ Δ ΔX : : : :. J (J Jaob matr, reglarty assmpto
15 Load Flow solto -I eqatos system a be eteded to oltage-power depedee Î Ŝ Ŝ m 3Ŝ m m 3 (,m m Î (,m Ŝ dag 3 m m (,m
16 Ŝ Ŝ Ŝ (, (, (, (, (, (, (, - powers deed olearty Am: to allate,,, odes ad brahes Note: Assmpto o symmetral system ad ts loadg sgle phase models.
17 Node types Node power Node oltage phasor ompoets deed to be allated deed to be allated,,,, sla (swg bs balae ode, balae, or losses, as a hge system, large geerato loads geerators, otrolled oltage attes - ed reqremets (, or loads; or geerators - state depedet arables (, or loads; or geerators - otrol here o hages ( or sla ad geerators, they hage optmzato proedres
18 Callatos relate ales Deomated ales Î 3 Î 3 Ŝ Î Ẑ Î 3Ẑ Base ales Î 3 Ŝ Ŝ 3 Ŝ 3 3Î Ẑ Relate ales I î û 3 S ŝ I î Z ẑ 3 û î û ŝ î ẑ û
19 Node rret (sgle phase m m m m m m m Î û ŷ ŷ û î Node power î û q p û q p î hee û ŷ ŷ û û q p
20 Gass-Sedel ower Flow Solto Solto or, : p û û q (ote: or loads, < Solto or : p ( Reû û ŷ ŷ ŷ û ŷ û
21 Solto or : q ( Imû û ŷ ŷ û Admttae matr elemets ŷ ( ŷ, ŷ (, ŷ :, sla ow (- ow û (p, q, û : q (û, û û (p, q, û magary part tae, real part to be allated
22 e û e û Newto-Raphso ower Flow Solto Ŝ (, (, (, Ŝ Epoetal orm Ŝ Ŝ e Y (, e e (, (, Y(, e (,
23 ower separated to the real ad magary part (, (, os Y (, (, s Y eqatos or eah ode, eqato or eah ode The power hages are epressed (learzato Ŝ Ŝ Ŝ
24 Complete eqato desrpto
25 More ompat eqatos orm 4 3 J J J J J Eqatos mber or odes, s slas, m odes, p odes ( = s + m + p: Δ (-s, Δ (-s-m
26 Y (, os (, Y s (, (, Y Y Y (, (, (, s os os (, (, Y(, os (, (,
27 Y (, s (, Y os (, (, Y Y (, Y (, (, s s os (, (, Y(, s (, (,
28 Iterate solto dea deet J
29 Deopled ower Flow Solto Trasmsso system: hgher rato X/R or power les Coplgs Δ~Δδ, Δ~Δ stroger tha Δ~Δ, Δ~Δδ. Thereore the Jaob matr a be smpled: J J 4 So alled Deopled problem eeds sally less tme or allato. More teratos bt qer matr allatos. (Nmber o operato or l. eqatos system solto reases qer tha learly. systems are soled seqetally eah step. Coergee prese, oly the hage o Jaob matr,.e. terato steps. Appromate solto oly ase o smpled relatos or,.
30 Ideal power le (R =, G = s X X X os s For lttle loaded les ( X X X X os s X os B deoplg prese eogh. X
31 The et smplatos rede allatg J ad J 4 eah terato. Fast Deopled ower Flow Solto p q p p y s (, (, sally y q B (, s y y (, (, s s (, y s (, (, q B(, Im (, y (, s (, y (, B(, q ad (, (,
32 p B (, p y (, s sally, p y(, s p B (, (, (, B Im (, y (, s (, y (,
33 q q q q y (, y B (, y y s (, (, s (, y(, s (, s s (, (, y(, s (, (, q ( y s Im y q sally, (, (, (, q B (,
34 (, (, s y q sally (, (, s y q (, B q B B q p q p B B B ad B are magary parts o the adm. matr ( p.., ther erso s allated oly oe. (Note: Dso by oltages elemet by elemet.
35 DC ower Flow Relate ales. Assmptos: s b s X s Z S p p s p
36 Matr p p b (, p b b (, Oly logtdal reataes b sglar. ode as a reeree wth δ = matr b smaller by oe order. (DC model does t allate losses, ths sla ot eeded bt a agle reeree yes. b p ( g
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