8 symmetrical componentsnotes on power system analysis1 lesson 8 symmetrical components
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Notes on Power System Analysis 18 Symmetrical Components
Lesson 8
Symmetrical Components
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:
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
Notes on Power System Analysis 48 Symmetrical Components
Zero Sequence
Positive Sequence
Negative Sequence
a
b
c
a
c
b
Va
Vb
Vc
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
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
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
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)/3
V2 = (Va + a2Vb + aVc)/3These are the phase a symmetrical (or sequence) components. The other phases follow since the sequences are balanced.
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
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
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
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
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
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
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
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
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
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
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)
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*
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*
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.
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
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
Notes on Power System Analysis 258 Symmetrical Components
Ungrounded Y load
Zero
I1V1
Z
Z
I2V2
ZI0V0
N
G
NNegative
NPositive
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
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
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
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
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
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
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
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%
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%
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
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