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Lesson 8 Symmetrical Components 8 Symmetrical Components Notes on Power System Analysis 1 Symmetrical Components • Due to C. L. Fortescue (1918): a set of n unbalanced phasors in an n-phase system can be resolved into n balanced phasors by a linear transformation – The n sets are called symmetrical components – One of the n sets is a single-phase set and the others are n-phase balanced sets – Here n = 3 which gives the following case: 8 Symmetrical Components Notes on Power System Analysis 2 Symmetrical component definition • Three-phase voltages Va, Vb, and Vc (not necessarily balanced, with phase sequence a-b-c) can be resolved into three sets of sequence components: Zero sequence Va0=Vb0=Vc0 Positive sequence Va1, Vb1, Vc1 balanced with phase sequence a-b-c Negative sequence Va2, Vb2, Vc2 balanced with phase sequence c-b-a 8 Symmetrical Components Notes on Power System Analysis 3 Z e ro S e q u e n ce Vc c Va a P o sitiv e S eq u e n ce Vb b a c N eg ativ e S e q u e n ce b 8 Symmetrical Components Notes on Power System Analysis 4 Va Vb Vc = 1 1 1 V0 1 a2 a V1 1 a a2 V2 where a = 1/120° = (-1 + j 3)/2 a2 = 1/240° = 1/-120° a3 = 1/360° = 1/0 ° 8 Symmetrical Components Notes on Power System Analysis 5 1 1 1 A= 1 a2 a 1 a a2 V0 Va Vb Vc Vp = Vp = A Vs 8 Symmetrical Components Vs = V1 V2 Vs = A-1 Vp Notes on Power System Analysis 6 1 1 1 A-1 = (1/3) 1 a a2 1 a2 a • We used voltages for example, but the result applies to current or any other phasor quantity Vp = A Vs Vs = A-1 Vp Ip = A Is Is = A-1 Ip 8 Symmetrical Components Notes on Power System Analysis 7 Va = V0 + V1 + V2 Vb = V0 + a2V1 + aV2 Vc = V0 + aV1 + a2V2 V0 = (Va + Vb + Vc)/3 V1 = (Va + aVb + a2Vc)/3 V2 = (Va + a2Vb + aVc)/3 These are the phase a symmetrical (or sequence) components. The other phases follow since the sequences are balanced. 8 Symmetrical Components Notes on Power System Analysis 8 Sequence networks – A balanced Y-connected load has three impedances Zy connected line to neutral and one impedance Zn connected neutral to ground a Zy b Zy c Zn g 8 Symmetrical Components Notes on Power System Analysis 9 Sequence networks Vag Zy+Zn Zn Zn Ia Vbg = Zn Zy+Zn Zn Ib Vcg Zn Zn Zy+Zn Ic or in more compact notation Vp = Zp Ip 8 Symmetrical Components Notes on Power System Analysis 10 a b c Zy Vp = Zp Ip Zy Zy Zn g n Vp = AVs = Zp Ip = ZpAIs AVs = ZpAIs Vs = (A-1ZpA) Is Vs = Zs Is where Zs = A-1ZpA 8 Symmetrical Components Notes on Power System Analysis 11 Zs = Zy+3Zn 0 0 0 Zy 0 0 0 Zy V0 = (Zy + 3Zn) I0 = Z0 I0 V1 = Zy I1 = Z1 I1 V2 = Zy I2 = Z2 I2 8 Symmetrical Components Notes on Power System Analysis 12 a V0 g I0 Zy 3 Zn Zerosequence network n a V1 n I1 Zy Positivesequence network I2 a V2 n Zy Negativesequence network Sequence networks for Y-connected load impedances 8 Symmetrical Components Notes on Power System Analysis 13 a V0 g I0 ZD/3 Zerosequence network n a V1 n I1 ZD/3 Positivesequence network I2 a V2 n ZD/3 Negativesequence network Sequence networks for D-connected load impedances. Note that these are equivalent Y circuits. 8 Symmetrical Components Notes on Power System Analysis 14 Remarks – Positive-sequence impedance is equal to negative-sequence impedance for symmetrical impedance loads and lines – Rotating machines can have different positive and negative sequence impedances – Zero-sequence impedance is usually different than the other two sequence impedances – Zero-sequence current can circulate in a delta but the line current (at the terminals of the delta) is zero in that sequence 8 Symmetrical Components Notes on Power System Analysis 15 • General case unsymmetrical impedances Zp = Zs=A-1ZpA 8 Symmetrical Components = Zaa Zab Zca Zab Zbb Zbc Zca Zbc Zcc Z0 Z10 Z20 Z01 Z1 Z21 Notes on Power System Analysis Z02 Z12 Z2 16 Z0 = (Zaa+Zbb+Zcc+2Zab+2Zbc+2Zca)/3 Z1 = Z2 = (Zaa+Zbb +Zcc–Zab–Zbc–Zca)/3 Z01 = Z20 = (Zaa+a2Zbb+aZcc–aZab–Zbc–a2Zca)/3 Z02 = Z10 = (Zaa+aZbb+a2Zcc–a2Zab–Zbc–aZca)/3 Z12 = (Zaa+a2Zbb+aZcc+2aZab+2Zbc+2a2Zca)/3 Z21 = (Zaa+aZbb+a2Zcc+2a2Zab+2Zbc+2aZca)/3 8 Symmetrical Components Notes on Power System Analysis 17 • Special case symmetrical impedances Zp = Zs = 8 Symmetrical Components Zaa Zab Zab Zab Zaa Zab Zab Zab Zaa Z0 0 0 0 Z1 0 0 0 Z2 Notes on Power System Analysis 18 Z0 = Zaa + 2Zab Z1 = Z2 = Zaa – Zab Z01=Z20=Z02=Z10=Z12=Z21= 0 Vp = Zp Ip Vs = Zs Is • This applies to impedance loads and to series impedances (the voltage is the drop across the series impedances) 8 Symmetrical Components Notes on Power System Analysis 19 Power in sequence networks Sp = Vag Ia* + Vbg Ib* + Vcg Ic* Sp = [Vag Vbg Vcg] [Ia* Ib* Ic*]T Sp = VpT Ip* = (AVs)T (AIs)* = VsT ATA* Is* 8 Symmetrical Components Notes on Power System Analysis 20 Power in sequence networks Sp = VpT Ip* = VsT ATA* Is* 1 ATA* = 1 1 1 1 1 1 a2 a 1 a a2 = 0 3 0 1 a2 1 a2 a a 3 0 0 0 0 3 Sp = 3 VsT Is* 8 Symmetrical Components Notes on Power System Analysis 21 Sp = 3 (V0 I0* + V1 I1* +V2 I2*) = 3 Ss In words, the sum of the power calculated in the three sequence networks must be multiplied by 3 to obtain the total power. This is an artifact of the constants in the transformation. Some authors divide A by 3 to produce a power-invariant transformation. Most of the industry uses the form that we do. 8 Symmetrical Components Notes on Power System Analysis 22 Sequence networks for power apparatus • Slides that follow show sequence networks for generators, loads, and transformers • Pay attention to zero-sequence networks, as all three phase currents are equal in magnitude and phase angle 8 Symmetrical Components Notes on Power System Analysis 23 N E V1 I1 Positive Z1 N V2 I2 Zn Negative Z2 Y generator G 3Zn N 8 Symmetrical Components I0 Z0 Notes on Power System Analysis V0 Zero 24 N V1 I 1 Positive Z Ungrounded Y load N V2 I 2 Negative Z G V0 8 Symmetrical Components I0 Notes on Power System Analysis Z Zero N 25 Zero-sequence networks for loads G V0 I0 Z 3Zn N Y-connected load grounded through Zn G V0 8 Symmetrical Components Z D-connected load ungrounded Notes on Power System Analysis 26 Y-Y transformer H1 A X1 a B b C c N n ZN 8 Symmetrical Components Zn Zeq+3(ZN+Zn) A a Va0 VA0 I0 g Zero-sequence network (per unit) Notes on Power System Analysis 27 Y-Y transformer H1 A X1 B a b C c N n ZN 8 Symmetrical Components Zn A Zeq VA1 I1 a Va1 n Positive-sequence network (per unit) Negative sequence is same network Notes on Power System Analysis 28 D-Y transformer A H1 X1 a B b C c n Zn 8 Symmetrical Components A VA0 Zeq+3Zn I0 a Va0 g Zero-sequence network (per unit) Notes on Power System Analysis 29 D-Y transformer A H1 X1 a B b C c n Zn 8 Symmetrical Components Zeq A VA1 I1 a Va1 n Positive-sequence network (per unit) Delta side leads wye side by 30 degrees Notes on Power System Analysis 30 D-Y transformer A H1 X1 a B b C c n Zn 8 Symmetrical Components Zeq A VA2 I2 a Va2 n Negative-sequence network (per unit) Delta side lags wye side by 30 degrees Notes on Power System Analysis 31 Three-winding (three-phase) transformers Y-Y-D H and X in grounded Y and T in delta Zero sequence Positive and negative Neutral Ground ZT H ZH ZX X H X ZT T 8 Symmetrical Components ZX ZH T Notes on Power System Analysis 32 Three-winding transformer data: Windings Z Base MVA H-X 5.39% 150 H-T 6.44% 56.6 X-T 4.00% 56.6 Convert all Z's to the system base of 100 MVA: Zhx = 5.39% (100/150) = 3.59% ZhT = 6.44% (100/56.6) = 11.38% ZxT = 4.00% (100/56.6) = 7.07% 8 Symmetrical Components Notes on Power System Analysis 33 Calculate the equivalent circuit parameters: Solving: ZHX = ZH + ZX ZHT = ZH + ZT ZXT = ZX +ZT Gives: ZH = (ZHX + ZHT - ZXT)/2 = 3.95% ZX = (ZHX + ZXT - ZHT)/2 = -0.359% ZT = (ZHT + ZXT - ZHX)/2 = 7.43% 8 Symmetrical Components Notes on Power System Analysis 34 Typical relative sizes of sequence impedance values • Balanced three-phase lines: Z0 > Z1 = Z2 • Balanced three-phase transformers (usually): Z1 = Z2 = Z0 • Rotating machines: Z1 Z2 > Z0 8 Symmetrical Components Notes on Power System Analysis 35 Unbalanced Short Circuits • Procedure: – Set up all three sequence networks – Interconnect networks at point of the fault to simulate a short circuit – Calculate the sequence I and V – Transform to ABC currents and voltages 8 Symmetrical Components Notes on Power System Analysis 36