Chapter 10 Sym Comp and Unbalanced Fault(2)

8/10/2019 Chapter 10 Sym Comp and Unbalanced Fault(2)

1/45

C

F

C 10


2/45

F

D

L


3/45

C

C.L F, 1918

B

B

,


4/45

1

aI

1

bI

1

cI

0

aI

0

bI

0cI

2

aI

2

bI

2cI


5/45

2

aI

2

bI

2

cI

1

aI

1

bI

1

cI

0

aI

0

bI

0

cI

=

+ +


6/45

C

210 aaaa IIII ++=

210

bbbb IIII ++=

210

cccc IIII ++=

000

acb III ==

1211 240 aab IaII ==

111 120 aac aIII ==

222

120 aab aIII ==

1212 240 aac IaII ==

I

= 1201a

= 24012a


7/45

012 012

=

2

1

0

2

2

11

111

a

a

a

c

b

a

II

I

aaaa

II

I

=

c

b

a

a

a

a

II

I

aaaa

II

I

2

2

2

1

0

11

111

3

1

012AIIabc

= abc

IAI1012

=

012

AVVabc

=

abc

VAV1012

=


8/45

=== 120200;120230;0200 cba VVV

EG ,

F .

=

c

b

a

a

a

a

VV

V

aaaa

VV

V

2

2

2

1

0

11

111

3

1


9/45

=

c

b

a

a

a

a

V

V

V

aa

aa

V

V

V

2

2

2

1

0

1

1

111

3

1

=++=

12010)(3

10cbaa

VVVV

=++= 12010)(3

1 21cbaa VaaVVV

=++= 0210)(3

1 22cbaa aVVaVV


10/45

=== 180100;00;0150 cba III

EG , ,

,

F .

=

c

b

a

a

a

a

II

I

aaaa

II

I

2

2

2

1

0

11

111

3

1


11/45

=

c

b

a

a

a

a

I

I

I

aa

aa

I

I

I

2

2

2

1

0

1

1

111

3

1

=++= 067.16)(

3

10cbaa IIII

=++= 41.2365.72)(3

1 21cbaa IaaIII

=++= 41.2365.72)(3

1 22cbaa aIIaII


12/45

I

1 : I

.

2 : I

.

0 : I .


13/45

I C

L


14/45

+++

+++

+++

=

c

b

a

nSnmnm

nmnSnm

nmnmnS

c

b

a

I

I

I

ZZZZZZ

ZZZZZZ

ZZZZZZ

V

V

V

abcabcabcIZV =

I ,012012

a

abc

a AIZAV =

012012

0121012

a

a

abc

a

IZ

AIZAV

=

=


15/45

AZAZ

abc1012 =

F

++

=

mS

mS

mnS

ZZ

ZZZZZ

Z

00

00003

012

0 12


16/45

L

21

ZZ =

nXXX 310+=

)ln2.0(2D

DfX n

n

=

,

>3


17/45

I

M

D

''1dXX = '1 dXX = dXX =1

''2

dXX =

lXX =0

=


18/45

I

021

ZZZ ==

,

=

=


19/45

0 = + 3

=0


20/45

0 = + 3

=

I0=0I0=0


21/45

0 = + 3

I0 I0=0


22/45

0 = + 3


23/45

0 = + 3


24/45

Zs

ZsZs

Ea

EbEc

+

+

+

Va

Vb

Vc

Ia

Ib

Ic

In

Zn

+

+

+

=

c

b

a

nsnn

nnsn

nnns

c

b

a

c

b

a

I

I

I

ZZZZ

ZZZZ

ZZZZ

E

E

E

V

V

V

012012012012

0121012012

012012012

aaa

a

abc

aa

a

abc

aa

abcabcabcabc

IZEV

AIZAEV

AIZAEAV

IZEV

=

=

=

=


25/45

=

+

=2

1

0

012

0000

00

0000

003

ZZ

Z

ZZ

ZZ

Z

s

s

ns

=

0

01012

aa EE

G ,

=

2

1

0

2

1

0

1

2

1

0

0000

00

0

0

a

a

a

a

a

a

a

II

I

ZZ

Z

EVV

V


26/45

Z0 0

aI

+

-

0

aV

2

aI

2a

V


27/45

I 1. 2.

3. . ,

4.

5.

6. , . ,

7. 3.

8. . .


28/45

L G F

Zs

ZsZs

Ea

EbEc

+

+

+

Va

Vb

Vc

Ia

Ib

Ic

In

Zn

I = 0

I = 0


29/45

=

0

0

1

1

111

3

1

2

2

2

1

0

a

a

a

a I

aa

aa

I

I

I

0==

=

cb

afa

II

IZV

fa ZZZZ

E

I 302100

+++

=

f

aaaf

ZZZZEIII

333

021

0

+++===

F C


30/45

L L F

Zs

ZsZs

Ea

EbEc

+

+

+

Va

Vb

Vc

Ia

Ib

Ic

In

ZnI

I


31/45

cb

a

bfcb

III

IZVV

=

=

=

0

=

b

b

a

a

a

I

I

aa

aa

I

I

I 0

1

1

111

3

1

2

2

2

1

0

( )

ba

ba

a

IaaI

IaaI

I

)(3

1

3

1

0

22

21

0

=

=

=

21

aa II =


32/45

( ) ( )( )aaaaIZIZZE afaa

=

22

1121 3

f

aa

ZZZEI

++=

21

1

112

1

1

2

2

3)(

0

1

1

111

aacb

a

a

c

b

a

IjIaaII

I

I

aa

aa

I

I

I

===

=

KL

F


33/45

D L G F

Zs

ZsZs

Ea

EbEc

+

+

+

Va

Vb

Vc

Ia

Ib

Ic

In

Zn


34/45

0)(

=

+==

a

cbfcb

I

IIZVV

f

aaa

ZZIZEI

30

110

+

=

2

11

2

ZIZEI aaa =

( )f

f

aa

ZZZZZZZ

EI

33

02

02

1

1

++

++

=

03 acbf IIII =+=

F


35/45

F 3

.

Generators G1 and G2: 1.021 jXX == p.u; 05.00

jX = p.u; Voltage: 11 kV

Transformers T1 and T2: 15.0021 jXXX === p.u; Voltage: 11/275 kV

Line: 05.021 jZZ == p.u, 1.00 jZ = p.u ; Voltage: 275 kV

All per unit values are on a base of 100 MVA, and pre-fault voltage is 1.0 per unit.

C B


36/45

D ,

.

.

j0 1 j0 1j0 15 j0 15j0 05


37/45

j0.1 j0.1j0.15 j0.15j0.05

j0.05 j0.05j0.15 j0.15j0.1

A B


38/45

D

B.

fBBB

a

aaa ZZZZ

E

III 3021210

+++===

)1.015.0//()05.015.01.0(1 jjjjjZB +++= = j0.1364

12BB ZZ = = j0.1364

)15.0//()1.015.0(0 jjjZB += = j0.09375

009375.01364.01364.0

0.1210

+++=== jjjIII aaa = -j2.728 .pu

1844.83)(0

jIFI aB == p.u

B


39/45

D

B.

A B,


40/45

)55.0

25.0)((*)15.01.0( 11 FIjjV BA += = 454.0*728.2*25.0 jj = -0.31

111 )0()( AAA VVFV += = 1.0 0.31 = 0.69 p.u

)55.0

25.0

)((*)05.015.01.0(11

FIjjjV BB ++= = -0.372

111 )0()( BBB VVFV += = 1.0 0.372 = 0.628 p.u

Negative sequence voltage buses A and B,

222 )0()( AAA VVFV += = 0.0 0.31 = - 0.31

222 )0()( BBB VVFV += = 0.0 0.372 = -0.372


41/45

Zero sequence voltage buses A

and B,

)55.0

25.0)((*)15.0(

00

FIjV BA = = - 0.186

000 )0()( AAA VVFV += = 0.0 0.186 = -0.186 p.u

)55.0

25.0

)((*)1.015.0(00

FIjjV BB += = - 0.31 p.u000 )0()( BBB VVFV += = 0.0 0.31 = -0.31 p.u


42/45

=

2

22

1

11

0

00

012

)()(

)()(

)()(

AB

BA

AB

BA

AB

BA

AB

z

FVFV

z

FVFV

z

FVFV

I =

05.0

)372.0(31.0

05.0

628.069.0

1.0

)31.0(186.0

j

j

j

=

24.1

24.1

24.1

j

j

j

p.u

)()( 012 FAIFI ABabcAB = =

24.1

24.1

24.1

1

1

111

2

2

j

j

j

aa

aa

)24.1(3 jIaAB = = -j3.72 p.u

++++= 1209024.12409024.19024.1bABI = 0.0 p.u

++++= 2409024.11209024.19024.1cABI = 0.0 p.u

C


43/45


44/45

F 3

.

Generators G1 and G2: 1.021 jXX == p.u; 05.00

jX = p.u; Voltage: 11 kV

Transformers T1 and T2: 15.0021 jXXX === p.u; Voltage: 11/275 kV

Line: 05.021 jZZ == p.u, 1.00 jZ = p.u ; Voltage: 275 kV

All per unit values are on a base of 100 MVA, and pre-fault voltage is 1.0 per unit.

C B


45/45

D

B.

F ,

12 )( acb IaaII ==ZfZZ

EIa

++=

21

01

upjjj

jIjII acb .725.4)09375.01364.01364.0(

133 1 =

++===

Chapter 10 Sym Comp and Unbalanced Fault(2)

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