lecture6

lecture6 - Power Systems I Example Using the Newton-Raphson...

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Unformatted text preview: Power Systems I Example Using the Newton-Raphson PF, find the power flow solution 0.02 + j 0.04 3 1 2 |V 3 | = 1.04 200 MW 400 MW 250 MVAR Slack Bus V 1 = 1.05 ∠ ° 0.01 + j 0.03 0.0125 + j 0.025 pu P pu j j S pu j y pu j y pu j y sch sch . 2 100 200 5 . 2 . 4 100 250 400 32 16 30 10 20 10 3 2 23 13 12 = = − − = + − = − = − = − = Power Systems I Example ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 3 2 23 23 3 2 22 22 2 2 1 2 21 21 1 2 2 33 33 2 3 2 3 32 32 2 3 1 3 31 31 1 3 3 3 2 23 23 3 2 22 22 2 2 1 2 21 21 1 2 2 sin sin sin cos cos cos cos cos cos 17 . 1 2 . 67 03 . 2 8 . 35 89 . 1 6 . 31 03 . 2 8 . 35 11 . 1 1 . 58 03 . 2 4 . 22 89 . 1 6 . 31 03 . 2 4 . 22 90 . 1 9 . 53 62 26 32 16 30 10 32 16 52 26 20 10 30 10 20 10 50 20 δ δ θ θ δ δ θ θ δ δ θ δ δ θ δ δ θ θ δ δ θ + − − − + − − = + + − + + − = + − + + + − = − ∠ ∠ ∠ ∠ − ∠ ∠ ∠ ∠ − ∠ = − + − + − + − − + − + − + − − = Y V V Y V Y V V Q Y V Y V V Y V V P Y V V Y V Y V V P j j j j j j j j j Y bus angles are in radians Power Systems I Example ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) − − − = − = − = Δ Δ Δ = Δ + − − − − − − − + + − + − + − + − + − = = = ) , , ( ) , , ( ) , , ( 5 . 2 . 2 . 4 ) , , ( ) , , ( ) , , ( 03 . 2 sin 8 . 35 04 . 1 11 . 1 sin 1 . 58 03 . 2 sin 3 . 22 05 . 1 17 . 1 cos 2 . 67 04 . 1 03 . 2 cos 8 . 35 04 . 1 89 . 1 cos 6 . 31 05 . 1 03 . 2 cos 8 . 35 04 . 1 11 . 1 cos 1 . 58 03 . 2 cos 3 . 22 05 . 1 ) , , ( ) , , ( ) , , ( 2 3 2 2 2 3 2 3 2 3 2 2 2 3 2 2 2 3 2 3 2 3 2 2 2 3 2 2 3 2 3 2 2 2 2 2 2 2 2 3 2 3 3 3 2 2 2 2 1 2 2 2 3 2 2 2 3 2 3 2 3 2 2 2 3 2 V Q V P V P V Q V P V P Q P P x f c Q P P c V V V V V V V V V Q V P V P x f V x sch sch sch δ δ δ δ δ δ δ δ δ δ δ δ δ δ δ δ δ δ δ δ δ δ δ δ δ δ δ δ δ Power Systems I Example ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 3 2 2 2 3 2 23 23 2 1 2 21 21 2 22 22 2 3 2 , 1 2 2 2 22 22 2 2 2 3 2 2 3 2 23 23 3 2 3 2 3 2 2 2 2 3 2 23 23 3 2 2 21 21 1 2 3 2 , 1 2 2 2 2 2 2 03 . 2 cos 8 . 35 04 . 1 03 . 2 cos 4 . 22 05 . 1 03 . 2 cos 1 . 58 2 cos cos cos 2 cos cos 2 03 . 2 sin 8 . 35 04 . 1 sin 03 . 2 sin 8 . 35 04 . 1 03 . 2 sin 4 . 22 05 . 1 sin sin sin δ δ δ δ δ θ δ δ θ θ δ δ θ θ δ δ δ δ θ δ δ δ δ δ δ θ δ θ δ δ θ δ + − + − + = + − + + − + = + − + = ∂ ∂ + − − = + − − = ∂ ∂ + − + − = + − + − = + − = ∂ ∂ ∑ ∑ ≠ = ≠ = V Y V Y V Y V Y V Y V V P V Y V V P V V Y V V Y V V Y V V P j j j j j j j j j j...
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lecture6 - Power Systems I Example Using the Newton-Raphson...

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