G S Power Flow Solution

Transcription

1 G S Power Flow Soluto P Q I y y * 0 1, Y y Y 0 y Y Y 1, P Q ( k) ( k) * ( k 1) 1, Y Y PQ buses * 1 P Q Y ( k1) *( k) ( k) Q Im[ Y ] 1 P buses & Slack bus ( k 1) *( k) ( k) Y 1 P Re[ ] Slack bus 17

2 Calculato for PQ Buses ad are ukow P ad Q are scheduled (geerato/load), deoted by P sch ad Q sch x (k+1) =g(x (k) ) Uder ormal operatg codtos: sch sch * ( k 1) 1, Y Slack bus: 0 0 (typcally 10 o ) Other buses: s close to 1pu or 0. For most of cases, there are: Geerator buses: > 0, > 0, Load buses: < 0, < 0 Ital guess could be =10 o f a better estmate s uavalable. P Q Y 18

3 Calculato for P Buses P =P sch ad are specfed Startg from a tal estmate of = ( k 1) * Q Im[ Y ] 1 ( k 1) c P sch Q * ( k 1) Y 1, Y Sce s specfed, oly I, (k+1)= Im[ c (k+1) ] s retaed ( ) ( k1) 2 ( k1) R, I, 2 Update (k+1) = R, (k+1) + I, (k+1) Cotue the teratos utl ( k1) ( k) ( k1) ( k) R, R, I, I, or, the power msmatch,.e. the largest elemet P ad Q < Usg accelerato factor =1.3 to 1.7 P Q sch ( k 1) *( ) Y k ( k 1) ( k) 1, ( k) c Y ( ) 19

4 Calculato for Slack Bus Calculato of Le Flows ad Losses At bus : ( k1) ( k1) *( k1) ( k1) Y 1 P Q I I I y ( ) y l 0 0 S I At bus : I I I y ( ) y S l 0 0 I Power loss le : S S S L y 20

5 Example 6.7 (slack bus + 2 P Q buses) Usg the G-S method to fd the power flow soluto: (a) Determe the voltage phasors at P-Q buses 2 ad 3 accurate to 4 decmal places (b) Fd the slack bus real ad reactve power (c) Determe the le flows ad losses. Show le flow drectos a power-flow dagram (Solve P 1, Q 1, 2, 2, 3, 3, S ad S l ) Step 1. Check what are kow P 1, Q 1 P y 23 =10-20 y 13 =10-30 y 23 =16-32 Q sch sch * ( k 1) 1, Y P-Q 3, 3 Y P-Q 2, 2 = Step 2. Set tal estmates ad start to terate 21

6 Step 3. Calculate P ad Q of the slack bus P Q * Y 1 Step 4. Calculate le currets, flows ad losses I y ( ) y 0 I y ( ) y 0 S I S I S L S S 22

7 Example 6.8 (slack bus + P Q bus + P bus) Le chargg susceptaces are eglected. Obta the power flow soluto by the G-S method cludg le flows ad le losses (Solve P 1, Q 1, 2, 2, Q 3, 3, S ad S l ) P 1, Q 1 y 13 =10-30 Q 3, 3 y 23 =10-20 P- y 23 =16-32 P-Q 2, 2 Step 1. Check what are kow Step 2. Set tal estmates ad start to terate Note: c3 (1) = = 3 P Q sch sch * ( k 1) 1, Y Y ( k 1) * Q Im[ Y ] 1 (1) R,3 23

8 Bus 2 (P Q): Solve 2, 2 Bus 3 (P ): Solve Q 3, 3 P Q sch sch ( k) *( k) Y ( k 1) * ( k1) Q 1, Y 1 Y sch ( k1) P Q ( k) Im[ ] *( ) Y k k1) 2 ( k1) 3 ] 1.0 Im[ ( k1) 1, c3 c Y Re[ 4 { ]} ( 2 (3) c3 (4) c3 (5) c3 (6) c3 (7) c * 1 P Q Y Bus 1 ( ): P 1, Q 1 24

9 Tap Chagg Trasformers a s the per ut off-omal tap posto (usually, a = 0.9 to 1.1) Complex umber for phase shftg trasformers S T = x I * = - I * x 1 a t I y ( ) I t x 1 a y y a y a t y a I t t * * 2 * I a I Equvalet crcut f a s real (gorg phase shftg) No-tap sde Tap sde yt yt I a I y t y t * 2 a a Y bus s ot symmetrcal f a phase shftg trasformer exsts the system 25

10 Newto Raphso Method Based o Taylor s seres expaso at a tal estmate of the soluto f ( x) c f x ( x ) c f 2 df 1 d f 2 ( x ) ( ) x ( ) ( x ) c 2 2! Igore all terms wth orders 2 Comparso: G-S method gores all dfferetal terms (orders 1) df ( ) x c f ( x ) = c ( 0) x ( 0) c df ( ) x x x (1) 26

11 Iterato 1: (1) x c c f( x ) =x +x x =x df df ( ) ( ) (1) ( 0) Iterato k+1: (k) (k+1) ( k) ( k) ( k1) ( k) c ( k) c f( x ) x =x +x x =x df df ( ) ( ) x c df ( ) Utl ( k1) ( k) x x f(x)=c s actually approxmated by ts taget le at x=x (k). df ( ) ( x x ) f ( x ) c 27

12 Example 6.4 x ( k1) ( k) ( x ) =x c f df ( ) Let x =6 df ( x) df 2 3x 12x 9 2 ( ) 3(6) 12(6) c cf x ( ) 0 [(6) 6(6) 9(6) 4] 50 c x df ( ) (1) x x x c x x x (2) (1) (1) x x x (3) (2) (2) x (2) x (1) x x x x x (4) (3) (3) x x x (5) (4) (4)

13 N dmesoal System f ( x) c 1 ( k1) ( k) ( k) df x x +x x ( ) c ( k) ( k) c c f( x ) f ( x, x,, x ) c f ( x, x,, x ) c f ( x, x,, x ) c 1 2 Jacoba Matrx: J ( k1) ( k) ( k 1) ( k) 1 X X X X J C f f f ( ) ( ) ( ) x1 x2 x f f f ( ) ( ) ( ) x1 x2 x f f f ( ) ( ) ( ) x1 x2 x 1 ( k) 1 ( k) 1 ( k) 2 ( k) 2 ( k) 2 ( k) ( k) ( k) ( k) X C x x x 1 2 c c c ( f ) 1 1 ( f ) 2 2 ( f ) 29

14 Example 6.5 Use the N-R method to fd the tersectos of the curves x e x2 4 x 1 x 2 1 J 2x 2x 1 e x J (k) tells the fastest drecto (gradet) see from pot k o the path toward a soluto If x 1 =2, x 2 = -2: k C J x x

15 Compared to the Gauss Sedel Method Sce hgher-order terms are gored, the N-R method also eeds the tal estmato to be suffcetly close to the actual soluto The N-R method coverges much faster N-R method: quadratc covergece (gorg the 2 d ad hgher orders) G-S method: lear covergece (gorg the 1 st ad hgher orders) The N-R method has more computatoal complexty: Requres [J (k) ] -1 durg each terato, whch s computatoally tese 31

16 Dealg wth [J (k) ] 1 ( k1) ( k) 1 ( k) X J C Try ot to update J (k) so ofte (at least ot every terato) Apply LU decomposto (tragular factorzato): Istead of calculatg X drectly by J -1, we frst solve (UX) ad the solve X J X L U X C ( k) ( k 1) ( k) ( k 1) I MATLAB, the soluto of JX= C ca be obtaed by X= J \ C 32