p1 admittance model ybus

CHAPTER ONEADMITTANCE MODEL

Warayut Kampeerawat

EEPW0425 Electrical Power System Analysis

1EEPW0425 Electric Power System Analysis

ตอบไดไหมทําไมตองYbus

ตอบใหชัด Ybus เปนแบบไหน

อยากจะสราง Ybus ตองทําไง

ตอบไมได ใหรีบถอน กอน F เอย.

Why? What? How?


Contents

Admittance Diagram

Node Admittance Matrix

Bus Admittance Matrix (Ybus)

Ybus with Mutually Coupled Branch 4

1

2

3

Ybus Modification

Network Calculation with Ybus6

5


Contents

Admittance Diagram




1

2

3

Ybus Modification


5


Admittance Diagram

Single line/One line diagram


Admittance Diagram

Impedance diagram

o01.25∠ o450.85 −∠


Admittance Diagram

Admittance diagram

o091.00 −∠ o3510.68 −∠


Admittance Diagram

Admittance diagram

o091.00 −∠ o3510.68 −∠

0

0

Bus Admittance Matrix(Ybus)


o01.25∠ o450.85 −∠

o091.00 −∠ o3510.68 −∠

0

0

Admittance Diagram


Contents

Admittance Diagram




1

2

3

Ybus Modification


5


Node Analysis


Node Analysis

1 11 1

2 22 2

31 1 4 3 3

42 3 2 5 3

1 10 0

1 10 0

1 1 1 1 10 0

1 1 1 1 1 00

V IZ Z

V IZ Z

VZ Z Z Z Z

VZ Z Z Z Z

⎡ ⎤− ⎡ ⎤ ⎡ ⎤⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥

− ⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ =⎢ ⎥ ⎢ ⎥⎢ ⎥⎛ ⎞⎢ ⎥− + + − ⎢ ⎥⎢ ⎥⎜ ⎟⎢ ⎥ ⎢ ⎥⎢ ⎥⎝ ⎠⎢ ⎥ ⎢ ⎥⎢ ⎥⎛ ⎞ ⎢ ⎥ ⎢ ⎥⎢ ⎥− − + +⎜ ⎟ ⎣ ⎦⎣ ⎦⎢ ⎥⎝ ⎠⎣ ⎦


Node Admittance Equation

node node nodeY V I=

11 12 13 14 1 1

21 22 23 24 2 2

31 32 33 34 3 3

41 42 43 44 4 4

Y Y Y Y V IY Y Y Y V IY Y Y Y V IY Y Y Y V I

⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥=⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥

⎣ ⎦ ⎣ ⎦⎣ ⎦



11 12 13 14

21 22 23 24

31 32 33 34

41 42 43 44

node

Y Y Y YY Y Y Y

YY Y Y YY Y Y Y

⎡ ⎤⎢ ⎥⎢ ⎥=⎢ ⎥⎢ ⎥⎣ ⎦

(1) (2) (3) (4)(1)

(2)

(3)

(4)


Contents

Admittance Diagram




1

2

3

Ybus Modification


5


0

0

1) Symmetrical

Bus Admittance MatrixCharacteristics

Upper Triangle

Lower TriangleDiagonal


2) Simple

o09− 0.68∠

0

0


(1) (2) (3) (4)

(1)

(2)

(3)

(4)


3) Sparse (For Large Power System)



V

aZ

sE

I

krowteN

VsI

I

krowteN

aY

aa Z

Y 1=

VIZE as += aa

ss VYI

ZE

I +==

1) Synchronous Machine

Equivalent Circuit


2) Transformer/Impedance Branch

Equivalent Circuit


3) Transmission line

Equivalent Circuit


Example 1‐Form Ybus

From To R X B

1 2 0 0.125 0

1 3 0 0.25 0

1 4 0 0.4 0

2 3 0 0.25 0

2 4 0 0.2 0

Transformer : xT = 0.10 pu.

Generator : xd = 1.15 pu.Ed1 = 1.25 /_0Ed2 = 0.85 /_-45

Transmission Line Parameters (pu.)

Ex.1 Equivalent Circuit


j0.25 j0.2j0.125

j0.25

j0.1 j0.1

j1.15j1.15

+

-o01.25∠ o450.85 −∠

+

-

14

3

2

0

j0.4

Impedance/Reactance Diagram



Admittance Diagram


ANS.24EEPW0425 Electric Power System Analysis

1) Short‐cut (Inspection) Method

Ybus Calculations

2) Building Block Method

3) Network Incidence Matrix

3 Methods for Ybus Calculations


Method 1 Short‐Cut

ii ijj i

Y y∈

=∑

( )ij ji ijY Y y= = −

DIAGONAL ELEMENTS

OFF‐DIAGONAL ELEMENTS


Ex.2 Short‐Cut Method

Admittance Diagram27EEPW0425 Electric Power System Analysis



(1) (2) (3) (4)(1) 14 5 8.0 4.0 2 5(2) 8.0 17.0 4.0 5 0(3) 4.0 4.0 5 80 0(4) 2 5 5 00 0 8 30

bus

j . j j j .j j j j .

Yj j j .j . j . j .

−⎡ ⎤⎢ ⎥−⎢ ⎥=⎢ ⎥−⎢ ⎥−⎣ ⎦


)()(

1111

)()(

nm

nm

⎥⎦

⎤⎢⎣

⎡−

−Building block Matrix

Method 2 Building Block

( ) ( )( ) 1 1

( )( ) 1 1 b

m nm

yn

−⎡ ⎤⎢ ⎥−⎣ ⎦

[ ]( )

( ) 1 ( )b

mm y

NORMAL BRANCH

SHUNT BRANCH


-j4.0 -j5.0-j8.0

-j4.0

-j0.8 -j0.8o091.00 −∠ o3510.68 −∠

1

43

2

0

-j2.5

Va

Vb

Vc

Vd

Ve

Vf

Vg

Ia Ig

IeIf

Id

Ib Ic

Admittance Diagram

Ex.3 Building Block Method


1 aY11

− 11−

bY11

− 11−

cY 1 gY

11

− 11−

dY11

− 11−

eY11

− 11−

fY

Storage Matrix



)( fdc YYY ++

dY−

cY−

fY−

bY−

eY−

dY−

)( edb YYY ++ bY−

cY− fY−

eY−

)( cba YYY ++

)( gfe YYY ++

0

0

Nodal Admittance Matrix



Contents

Admittance Diagram




1

2

3

Ybus Modification


5


⎥⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡

b

a

bM

Ma

b

a

II

ZZZZ

VV

⎥⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡

b

a

bM

Ma

b

a

VV

YYYY

II

n

Im

In

Ip

Iq

Za Zb

ZM

Va Vb

Ia Ib

q

pm

n

Im

In

Ip

Iq

Ya Yb

YM

Va Vb

Ia Ib

q

pm

Mutually Couple Branch


⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡−

−−

=⎥⎦

⎤⎢⎣

⎡−

bM

Ma

aM

Mb

MbabM

Ma

YYYY

ZZZZ

ZZZZZZZ

2

11

Mutually Couple Admittance


⎥⎦

⎤⎢⎣

⎡−=⎥

⎦

⎤⎢⎣

⎡−

25.675.375.325.6

25.015.015.025.0 1

jjjj

jjjj

3

j0.25

1

2

j0.15

j0.25Impedance Diagram Admittance Diagram

Ex.4 Mutually Couple Admittance


Ybus Calculations with Mutually Couple Branch

2) Short‐cut Method

1) Building Block Method

3) Network Incidence Matrix

3 Methods for Ybus Calculations


Method 1 Building Block

aY⎥⎦

⎤⎢⎣

⎡−

−1111

MY⎥⎦

⎤⎢⎣

⎡−

−1111

MY⎥⎦

⎤⎢⎣

⎡−

−1111

bY⎥⎦

⎤⎢⎣

⎡−

−1111


3

-j6.25

1 2

j3.75-j6.25

Ex.5 Building Block


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

+−+−−+−−−−

)75.3(2)25.6(225.675.375.325.625.675.325.675.375.325.675.325.6

jjjjjjjjjjjjjj

( )25.61111

j−⎥⎦

⎤⎢⎣

⎡−

− ( )75.31111

j⎥⎦

⎤⎢⎣

⎡−

−

( )75.31111

j⎥⎦

⎤⎢⎣

⎡−

− ( )25.61111

j−⎥⎦

⎤⎢⎣

⎡−

−

Ex.5 Building Block

ANS.40EEPW0425 Electric Power System

Analysis

Equivalent Admittance Network

Mutually Couple Branch Equivalent Circuit







Ex.6 Short‐Cut

-j6.25

-j6.253

2

1

j3.75

j3.75-j3.75


-j6.25

-j6.253

2

1

j3.75

j3.75-j3.75

Ex.6 Short‐Cut


(1) (2) (3) (4)(1) 16.75 11.75 2.50 2.50(2) 11.75 19.25 2.50 5.00(3) 2.50 2.50 5.80 0(4) 2.50 5.00 0 8.30

bus

j j j jj j j j

Yj j jj j j

−⎡ ⎤⎢ ⎥−⎢ ⎥=⎢ ⎥−⎢ ⎥−⎣ ⎦

Ybus-Short Cut

Ex.6 Short‐Cut


⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡−

cMM

MbM

MMa

cM

bM

MMa

YYYYYYYYY

ZZZZ

ZZZ

32

31

211

2

1

21

00

Mutually Couple 3 Branch


Contents

Admittance Diagram




1

2

3

Ybus Modification


5


Ybus Ybus,newAdd branch

Remove branch

Ybus Modification


)()(

)()(

nm

YYYY

nm

Yaa

aabus ⎥

⎦

⎤⎢⎣

⎡−

−=Δ

busbusnewbus YYY Δ+=, busbusnewbus YYY Δ−=,

Add Branch Remove Branch

Add/Remove Normal Branch


)( )( )( )(

)()()()(

qpnm

YYYYYYYYYYYYYYYY

qpnm

Y

bbMM

bbMM

MMaa

MMaa

bus

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

−−−−

−−−−

=Δ

busbusnewbus YYY Δ+=,

busbusnewbus YYY Δ−=,

ADD BRANCH

REMOVE BRANCH

Add/Remove Mutual Branch


Ex.7 Ybus Modification




(4) (3) (2) (1)

30.8000.550.2080.550.250.2

00.550.225.1975.1150.250.275.1175.16

)4()3()2()1(

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

−−

−−

=

jjjjjj

jjjjjjjj

Ybus

(4) (3) (2) (1)

3.800.55.208.80.40.4

0.50.40.170.85.20.40.85.14

)4()3()2()1(

)(

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

−−

−−

=

jjjjjj

jjjjjjjj

Y newbus



Contents

Admittance Diagram




1

2

3

Ybus Modification


5


1. Vnode, Inode Calculations

2. Node Elimination

3. Impedance Matrix (Zbus) Calculation

4. Power Flow Calculations

Network Calculation with Ybus


IVYbus =

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

4

3

2

1

4

3

2

1

44434241

34333231

24232221

14131211

IIII

VVVV

YYYYYYYYYYYYYYYY

?,,, 4321 =VVVV

Vnode, Inode Calculations


IVYbus =

IYV bus1)( −=

Gaussian Elimination

Node Reduction (Kron Reduction)

1

2

3

Triangular Factorization4

Inverse Matrix

Nodal Admittance Equation


Solving Method


⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

°−∠°−∠

=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

−−

−−

13568.09000.1

00

30.8000.550.2080.550.250.2

00.550.225.1975.1150.250.275.1175.16

4

3

2

1

VVVV

jjjjjj

jjjjjjjj

IVYbus =

?,,, 4321 =VVVV

Nodal Admittance Equation



⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

°−∠°−∠

=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

−−

−−

13568.09000.1

00

30.8000.550.2080.550.250.2

00.550.225.1975.1150.250.275.1175.16

4

3

2

1

VVVV

jjjjjj

jjjjjjjj

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

°−∠°−∠

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

13568.09000.1

00

69890.055110.064178.063677.055110.069890.060822.061323.064178.060822.071966.069140.063677.061323.069140.073128.0

4

3

2

1

jjjjjjjjjjjjjjjj

VVVV

Ex.8 Inverse Matrix


⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

°−∠°−∠

=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

−−

−−

13568.09000.1

00

30.800.000.550.200.080.550.250.200.550.225.1975.1150.250.275.1175.16

4

3

2

1

VVVV

jjjjjj

jjjjjjjj

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

dcba

VVVV

xxxxxx

4

3

2

1

1000100

101

Row Operations

Ex.8 Gaussian Eliminations


⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

−−

−−−−

92686.737313.075373.6037313.042686.525373.4075373.625373.400746.11014925.014925.070149.01

jjjjjjjjj

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

4

3

2

1

VVVV

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

°−∠°−∠

13568.09000.1

00

=

STEP 1


⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

−−

−−−−−

78305.398305.20098305.278305.30061356.038644.01014925.014925.070149.01

jjjj

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

4

3

2

1

VVVV

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

°−∠°−∠

13568.09000.1

00

=

STEP 2


⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

−−−−−−−

43082.100078853.010061356.038644.01014925.014925.070149.01

j ⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

4

3

2

1

VVVV

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

°−∠°∠7466.11035738.1

026434.000

=

STEP 3


⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

−−−−−−

100078853.010061356.038644.01014925.014925.070149.01

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

4

3

2

1

VVVV

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

°−∠°∠7466.2094867.0

026434.000

=

30618.091939.04189.1896903.01 jV −=°−∠=

30859.091680.06030.1896734.02 jV −=°−∠=

26499.096388.03716.1599964.03 jV −=°−∠=

33605.088715.07466.2094867.04 jV −=°−∠=

STEP 4


pp

pkjpjknewjk Y

YYYY −=)(

(4) (3) (2) (1)

)4()3()2()1(

44434241

34333231

24232221

14131211

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

YYYYYYYYYYYYYYYY

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

4

3

2

1

VVVV

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

4

3

2

0

III

=

(4) (3) (2)

)4()3()2(

)1(44

)1(43

)1(42

)1(34

)1(33

)1(32

)1(24

)1(23

)1(22

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

YYYYYYYYY

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

4

3

2

VVV

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

4

3

2

III

=

j และ k มีคาตั้งแต 1 ถึง N ยกเวน p

N คือ ขนาดของ matrix เดิม

p คือ node ที่กําจัด

Node Elimination (Kron Reduction)


⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

°−∠°−∠

=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

−−

−−

13568.09000.1

00

30.800.000.550.200.080.550.250.200.550.225.1975.1150.250.275.1175.16

4

3

2

1

VVVV

jjjjjj

jjjjjjjj

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−∠−∠=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−−

−

13568.09000.1

0

001.7649.0552.5649.0475.5026.4552.5026.4578.9

4

3

1

VVV

jjjjjjjjj

Kron Reduction of Node 2

Ex.9 Node Reduction


1( )bus busZ Y −=

Zbus Calculations

16.75 11.75 2.50 2.5011.75 19.25 2.50 5.002.50 2.50 5.80 02.50 5.00 0 8.30

j j j jj j j jj j jj j j

−⎡ ⎤⎢ ⎥−⎢ ⎥⎢ ⎥−⎢ ⎥−⎣ ⎦

0.73128 0.69140 0.61323 0.636770.69140 0.71966 0.60822 0.641780.61323 0.60822 0.69890 0.551100.63677 0.64178 0.55110 0.69890

j j j jj j j jj j j jj j j j

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

busY


Power Flow Calculations

Coming Soon…


END OF CHAPTER ONE


p1 admittance model ybus

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