Lesson 8 Symmetrical Components
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Transcript of Lesson 8 Symmetrical Components
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Notes on Power System Analysis 18 Symmetrical Components
Lesson 8
Symmetrical Components
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Notes on Power System Analysis 28 Symmetrical Components
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:
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Notes on Power System Analysis 38 Symmetrical Components
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-cNegative sequence Va2, Vb2, Vc2 balanced
with phase sequence c-b-a
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Notes on Power System Analysis 48 Symmetrical Components
Zero Sequence
Positive Sequence
Negative Sequence
a
b
c
ac
b
Va
Vb
Vc
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Notes on Power System Analysis 58 Symmetrical Components
wherea = 1/120° = (-1 + j 3)/2 a2 = 1/240° = 1/-120° a3 = 1/360° = 1/0 °
Va
=
1 1 1 V0
Vb 1 a2 a V1
Vc 1 a a2 V2
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Notes on Power System Analysis 68 Symmetrical Components
Vp = A Vs Vs = A-1 Vp
A =1 1 11 a2 a1 a a2
Vp =Va
Vb
Vc
Vs =V0
V1
V2
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Notes on Power System Analysis 78 Symmetrical Components
A-1 = (1/3)1 1 11 a a2
1 a2 a
Ip = A Is Is = A-1 Ip
• We used voltages for example, but the result applies to current or any other phasor quantity
Vp = A Vs Vs = A-1 Vp
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Notes on Power System Analysis 88 Symmetrical Components
Va = V0 + V1 + V2 Vb = V0 + a2V1 + aV2 Vc = V0 + aV1 + a2V2
V0 = (Va + Vb + Vc)/3 V1 = (Va + aVb + a2Vc)/3V2 = (Va + a2Vb + aVc)/3
These are the phase a symmetrical (or sequence) components. The other phases follow since the sequences are balanced.
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Notes on Power System Analysis 98 Symmetrical Components
Sequence networks– A balanced Y-connected load has three impedances Zy connected line to
neutral and one impedance Zn connected neutral to ground
Zy
Zy
g
cba
Zn
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Notes on Power System Analysis 108 Symmetrical Components
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
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Notes on Power System Analysis 118 Symmetrical Components
Zy
n
Vp = Zp Ip
Vp = AVs = Zp Ip = ZpAIs
AVs = ZpAIs
Vs = (A-1ZpA) Is
Vs = Zs Is where
Zs = A-1ZpA
Zy
Zy
g
c
ba
Zn
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Notes on Power System Analysis 128 Symmetrical Components
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
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Notes on Power System Analysis 138 Symmetrical Components
Zy n
g
a3 ZnV0
I0
Zero-sequencenetwork
Zy
n
aV1
I1
Positive-sequencenetwork
Zy
n
aV2
I2
Negative-sequencenetwork
Sequence networks for Y-connected load impedances
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Notes on Power System Analysis 148 Symmetrical Components
ZD/3
n
aV1
I1
Positive-sequencenetwork
ZD/3
n
aV2
I2
Negative-sequencenetwork
Sequence networks for D-connected load impedances.Note that these are equivalent Y circuits.
ZD/3 n
g
aV0
I0
Zero-sequencenetwork
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Notes on Power System Analysis 15
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
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Notes on Power System Analysis 168 Symmetrical Components
• General case unsymmetrical impedances
Zs=A-1ZpA =Z0 Z01 Z02
Z10 Z1 Z12
Z20 Z21 Z2
Zp =
Zaa Zab Zca
Zab Zbb Zbc
Zca Zbc Zcc
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Notes on Power System Analysis 178 Symmetrical Components
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
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Notes on Power System Analysis 188 Symmetrical Components
• Special case symmetrical impedances
Zs =Z0 0 00 Z1 00 0 Z2
Zp =Zaa Zab Zab
Zab Zaa Zab
Zab Zab Zaa
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Notes on Power System Analysis 198 Symmetrical Components
Z0 = Zaa + 2Zab
Z1 = Z2 = Zaa – Zab
Z01=Z20=Z02=Z10=Z12=Z21= 0Vp = Zp Ip Vs = Zs Is
• This applies to impedance loads and to series impedances (the voltage is the drop across the series impedances)
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Notes on Power System Analysis 208 Symmetrical Components
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*
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Notes on Power System Analysis 218 Symmetrical Components
Power in sequence networks
ATA* =
1 1 1 1 1 1
=
3 0 0
1 a2 a 1 a a2 0 3 0
1 a a2 1 a2 a 0 0 3
Sp = 3 VsT Is*
Sp = VpT Ip* = Vs
T ATA* Is*
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Notes on Power System Analysis 228 Symmetrical Components
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.
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Notes on Power System Analysis 23
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
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Notes on Power System Analysis 248 Symmetrical Components
Y generator
Zero
I1V1
Z1
Z2
I2
Z0I0
V0
N
G
N
Negative
N
Positive
Zn
V2
3Zn
E
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Notes on Power System Analysis 258 Symmetrical Components
Ungrounded Y load
Zero
I1V1
Z
Z
I2V2
ZI0V0
N
G
NNegative
NPositive
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Notes on Power System Analysis 268 Symmetrical Components
Zero-sequence networks for loads
ZI0V0
N
G
3Zn
ZV0
G
Y-connected load grounded through Zn
D-connected load ungrounded
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Notes on Power System Analysis 278 Symmetrical Components
Y-Y transformer
A
B
C
N
H1 X1 a
b
c
nZnZN
Zeq+3(ZN+Zn)
g
AVA0 I0
Zero-sequencenetwork (per unit)
Va0
a
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Notes on Power System Analysis 288 Symmetrical Components
Y-Y transformer
A
B
C
N
H1 X1 a
b
c
nZnZN
Zeq
n
A
VA1 I1
Positive-sequencenetwork (per unit)Negative sequence
is same network
Va1
a
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Notes on Power System Analysis 298 Symmetrical Components
D-Y transformer
A
B
C
H1 X1 a
b
c
nZn
Zeq+3Zn
g
AVA0 I0
Zero-sequencenetwork (per unit)
Va0
a
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Notes on Power System Analysis 308 Symmetrical Components
D-Y transformer
A
B
C
H1 X1 a
b
c
nZn
Zeq
n
AVA1 I1
Positive-sequencenetwork (per unit)
Delta side leads wyeside by 30 degrees
Va1
a
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Notes on Power System Analysis 318 Symmetrical Components
D-Y transformer
A
B
C
H1 X1 a
b
c
nZn
Zeq
n
AVA2 I2
Negative-sequencenetwork (per unit)Delta side lags wyeside by 30 degrees
Va2
a
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Notes on Power System Analysis8 Symmetrical Components 32
Three-winding (three-phase) transformers Y-Y-D
ZX
ZT
ZHH X
Ground
Zero sequence
ZXZHH X
Neutral
ZT
T
Positive and negative
T
H and X in grounded Y and T in delta
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Notes on Power System Analysis 338 Symmetrical Components
Three-winding transformer data:Windings Z Base MVAH-X 5.39% 150H-T 6.44% 56.6X-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%
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Notes on Power System Analysis 348 Symmetrical Components
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%
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Notes on Power System Analysis 358 Symmetrical Components
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
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Notes on Power System Analysis 368 Symmetrical Components
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