lecture16 - three-phase single-line to ground double-line...

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Power Systems I Fault Analysis Fault types: balanced faults Percentage of total faults three-phase <5% unbalanced faults single-line to ground 60-75% double-line to ground 15-25% line-to-line faults 5-15% Unbalance fault analysis requires new tools symmetrical components augmented component models
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Power Systems I Symmetrical Components Allow unbalanced three-phase phasor quantities to be replaced by the sum of three separate but balanced symmetrical components applicable to current and voltages permits modeling of unbalanced systems and networks Representative symmetrical components I a1 I b1 I c1 120 ° 120 ° 120 ° 0 ° I a0 I b0 I c0 I a2 I b2 I c2 120 ° 120 ° 120 ° a c abc sequence positive sequence acb sequence negative sequence zero sequence
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Power Systems I Symmetrical Components ( ) ( ) ( ) 0 1 0 1 0 1 866 . 0 5 . 0 240 1 866 . 0 5 . 0 120 1 120 240 0 2 3 2 1 1 1 1 2 1 1 1 1 1 = + + + = ° = = ° = + = ° = = ° + = = ° + = = ° + = a a j a j a j a I a I I I a I I I I I a a c a a b a a a δ δ δ Positive sequence phasors Operator a identities
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Power Systems I Symmetrical Components ( ) ( ) ( ) ( ) ( ) ( ) 0 0 0 0 0 0 0 0 0 2 2 2 2 2 2 2 2 2 2 0 0 0 240 120 0 a a c a a b a a a a a c a a b a a a I I I I I I I I I I a I I I a I I I I I = ° + = = ° + = = ° + = = ° + = = ° + = = ° + = δ δ δ δ δ δ Negative sequence phasors Zero sequence phasors
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Power Systems I Relating unbalanced phasors to symmetrical components In matrix notation Symmetrical Components 2 2 1 0 2 1 0 2 1 2 0 2 1 0 2 1 0 2 1 0 a a a c c c c a a a b b b b a a a a a a a I a I a I I I I I I a I a I I I I I I I I I I I I + + = + + = + + = + + = + + = + + = Υ੟ Υ੟ Υ੟ Φ੯ Τ੏ ΢ਯ ΢ਯ ΢ਯ Σਿ Ρਟ Υ੟ Υ੟ Υ੟ Φ੯ Τ੏ ΢ਯ ΢ਯ ΢ਯ Σਿ Ρਟ = Υ੟ Υ੟ Υ੟ Φ੯ Τ੏ ΢ਯ ΢ਯ ΢ਯ Σਿ Ρਟ 2 1 0 2 2 1 1 1 1 1 a a a c b a I I I a a a a I I I
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Power Systems I [A] is known as the symmetrical components transformation matrix Solving for the symmetrical components leads to Symmetrical Components Υ੟ Υ੟ Υ੟ Φ੯ Τ੏ ΢ਯ ΢ਯ ΢ਯ Σਿ Ρਟ = = 2 2 012 1 1 1 1 1 a a a a abc A I A I * 2 2 1 1 012 3 1 1 1 1 1 1 3 1 A A I A I = Υ੟ Υ੟ Υ੟ Φ੯ Τ੏ ΢ਯ ΢ਯ ΢ਯ Σਿ Ρਟ = = a a a a abc
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