Lecture EE333 - lecture 13people.clarkson.edu/~lwu/ee333/Lectures/Lecture EE333 - lecture 13.pdf ·...

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Lecture 13 Transmission Lines: Steady-State Operation Reading: 5.1 – 5.5 Homework 4 is due on March 1 st Dr. Lei Wu Department of Electrical and Computer Engineering EE 333 POWER SYSTEMS ENGINEERING

Transcript of Lecture EE333 - lecture 13people.clarkson.edu/~lwu/ee333/Lectures/Lecture EE333 - lecture 13.pdf ·...

Page 1: Lecture EE333 - lecture 13people.clarkson.edu/~lwu/ee333/Lectures/Lecture EE333 - lecture 13.pdf · Surge impedance loading (SIL) The power delivered by a lossless line to a load

Lecture 13

Transmission Lines: Steady-State Operation

Reading: 5.1 – 5.5

Homework 4 is due on March 1st

Dr. Lei Wu

Department of Electrical and Computer Engineering

EE 333

POWER SYSTEMS ENGINEERING

Page 2: Lecture EE333 - lecture 13people.clarkson.edu/~lwu/ee333/Lectures/Lecture EE333 - lecture 13.pdf · Surge impedance loading (SIL) The power delivered by a lossless line to a load

Outline

Distributed line model

ABCD parameters

Equivalent PI model

Lossless line

2

c

-1

zwhere characteristic impedance Z = Ω

y

propagation constaint γ= zy m

( )( )

( ) ( )( ) ( )

( )( )

cosh sinh 01

sinh cosh 0

c

c

x Z xV x V

x xI x IZ

γ γ

γ γ

=

Page 3: Lecture EE333 - lecture 13people.clarkson.edu/~lwu/ee333/Lectures/Lecture EE333 - lecture 13.pdf · Surge impedance loading (SIL) The power delivered by a lossless line to a load

Outline

Distributed line model

ABCD parameters

Equivalent PI model

Lossless line

3

( )( )

( ) ( )( ) ( )

( )( )

cosh sinh 01

sinh cosh 0

c

c

x Z xV x V

x xI x IZ

γ γ

γ γ

=

S R R

S R R

V V VA BT

I C D I I

= =

Page 4: Lecture EE333 - lecture 13people.clarkson.edu/~lwu/ee333/Lectures/Lecture EE333 - lecture 13.pdf · Surge impedance loading (SIL) The power delivered by a lossless line to a load

Outline

Distributed line model

ABCD parameters

Equivalent PI model

Lossless line

4

( )'

2 1'

Z B

AY

B

=−

=

' '1

2'

Y ZA

B Z

= +

=

' '1

2' '

' 14

Y ZD

Y ZY C

+ =

+ =

Page 5: Lecture EE333 - lecture 13people.clarkson.edu/~lwu/ee333/Lectures/Lecture EE333 - lecture 13.pdf · Surge impedance loading (SIL) The power delivered by a lossless line to a load

Lossless Transmission Lines

For a lossless line, R=G=0, thus, z=jωL and y=jwC.

Characteristic (Surge) impedance – pure real number

Propagation constant – pure imaginary number

ABCD parameters

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( )( )zy j L j C j LCγ ω ω ω= = =

c

z j L LZ

y j C C

ωω

= = =

( ) ( ) ( )cosh cosh cos2

j LCl j LCle eA D l j LCl LCl

ω ω

γ ω ω−+= = = = =

( ) ( ) ( )sinh sinh sin2

j LCl j LCl

c

L L e e LB Z l j LCl j LCl

C C C

ω ω

γ ω ω−−= = = =

( ) ( )sinhsin /

c

l LC j LCl

Z C

γω= =

Page 6: Lecture EE333 - lecture 13people.clarkson.edu/~lwu/ee333/Lectures/Lecture EE333 - lecture 13.pdf · Surge impedance loading (SIL) The power delivered by a lossless line to a load

Lossless Transmission Lines

Surge impedance loading (SIL)

The power delivered by a lossless line to a load resistance equal

to the surge impedance.

The voltage profile remains constant

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( ) ( )

( ) ( )

cos sin

cos sin

RS R R R

j LClR R

VLV AV BI LCl V j LCl

C L C

LCl j LCl V e Vω

ω ω

ω ω

= + = +

= + =

S RV V=

Page 7: Lecture EE333 - lecture 13people.clarkson.edu/~lwu/ee333/Lectures/Lecture EE333 - lecture 13.pdf · Surge impedance loading (SIL) The power delivered by a lossless line to a load

Lossless Transmission Lines

Surge impedance loading (SIL)

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( ) ( )

( ) ( )

sinos

cos sin

RS R R R

j LClR R

j LCl VI CV DI V c LCl

L C L C

V VLCl j LCl e

L C L Cω

ωω

ω ω

= + = +

= + =

( )* 2

** Rj LCl j LCl RS S S R

VVS V I e V e

L C L Cω ω

= = =

No reactive power flow across the line

The real power flow along a lossless line at SIL remains constant

Page 8: Lecture EE333 - lecture 13people.clarkson.edu/~lwu/ee333/Lectures/Lecture EE333 - lecture 13.pdf · Surge impedance loading (SIL) The power delivered by a lossless line to a load

Power Transfer in Lines

V1 V2

+ +

- -

I1 I1TransmissionLine with

ABCDS12 S21

( )

1 1 1 2 2 2

1 22

2* 1 2 2

1 2 '

'

2 2 ' 12

c

with ,

osh ; ' '

Z

Z

A

A

Z

A

V V V V

V AVI

B

V V AV

x A B Z

Z

Z

S V IZ

γθ θ

θ θ

θ

θ θ

θ

== ∠ = ∠

−=

= ∠ = = ∠

= = ∠ − − ∠ −

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Often we want to know the maximum power that could be

transferred through a transmission line

Page 9: Lecture EE333 - lecture 13people.clarkson.edu/~lwu/ee333/Lectures/Lecture EE333 - lecture 13.pdf · Surge impedance loading (SIL) The power delivered by a lossless line to a load

Power Transfer in Lossless Lines

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If we assume a line is lossless (i.e., with the impedance of jX )

and are just interested in real power transfer.

Thus, the maximum real power transfer is

'

21 2 2

12 12 12

0 ; ' , 90

90 90

A ZB Z jX

V V AVP jQ

Z Z

θ θ

θ

= ° = = = °

+ = ∠ ° − − ∠ °

( ) ( )1 2 1 212 12 12cos 90 sin

V V V VP

Z Zθ θ= − ° − =

1 212Max V V

PX

=

Page 10: Lecture EE333 - lecture 13people.clarkson.edu/~lwu/ee333/Lectures/Lecture EE333 - lecture 13.pdf · Surge impedance loading (SIL) The power delivered by a lossless line to a load

Lossless Line Example

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For a 765kV lossless transmission line with receiving end line-

to-line voltage of 765kV and surge impedance loading .

z=j0.535 Ω/mile, y=j 7.75* 10-6 mho/mile. Calculate the ABCD

parameter, the equivalent Π circuit, and the theoretical

maximum real power that a 200-mile line can deliver.

( ) ( )( ) ( )

( ) ( )( ) ( )

-3 oc

200

-3 -3

-3 -3

Z = 262.74Ω γ= =2.036*10 90

cosh sinh

1sinh cosh

cosh 2.036*10 * 200 262.74sinh 2.036*10 * 2000

1sinh 2.036*10 * 200 cosh 2.036*10 * 200

262.74

c

c x

zzy

y

x Z xA B

Tx xC D

Z

j j

j j

γ γ

γ γ=

= ∠

= =

= =

.9182 104.06

0.0015 0.9182

j

j

Page 11: Lecture EE333 - lecture 13people.clarkson.edu/~lwu/ee333/Lectures/Lecture EE333 - lecture 13.pdf · Surge impedance loading (SIL) The power delivered by a lossless line to a load

Lossless Line Example

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For a 765kV lossless transmission line with receiving end line-

to-line voltage of 765kV and surge impedance loading .

z=j0.535 Ω/mile, y=j 7.75* 10-6 mho/mile. Calculate the ABCD

parameter, the equivalent Π circuit, and the theoretical

maximum real power that that a 200-mile line can deliver.

( ) ( )' 104.06

2 1 2 0.9182 1'0.0016

2 104.06

Z B j

AYj

B j

= =− −

= = =

max

765*7655623.92

' 104.06s RV V

P MWX

= = =

Page 12: Lecture EE333 - lecture 13people.clarkson.edu/~lwu/ee333/Lectures/Lecture EE333 - lecture 13.pdf · Surge impedance loading (SIL) The power delivered by a lossless line to a load

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Thermal limits

Thermal limit is due to heating of conductor and hence depends

heavily on ambient conditions.

These lines can operate at 200 degrees C.

For many lines, sagging is the limiting constraint. Trees grow,

and will eventually hit lines if they are planted under the line.

Angle limits

While the maximum power transfer occurs when line angle

difference is 90 degrees, actual limit is substantially less due to

multiple lines in the system.

Voltage stability limits

As power transfers increases, reactive losses increase as I2X.

This would cause the voltage drops, resulting in a potentially

cascading voltage collapse.

Limits Affecting Power Transfer

Page 13: Lecture EE333 - lecture 13people.clarkson.edu/~lwu/ee333/Lectures/Lecture EE333 - lecture 13.pdf · Surge impedance loading (SIL) The power delivered by a lossless line to a load

Midterm Exam

Is scheduled on March 8th during the class.

Covers all materials we have discussed in chapters 2-5.

Close book, close note, you can bring one letter-size double-

sided information sheet.

Calculator.

Problems

4 true/false

3 multiple choices

3 calculations

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Page 14: Lecture EE333 - lecture 13people.clarkson.edu/~lwu/ee333/Lectures/Lecture EE333 - lecture 13.pdf · Surge impedance loading (SIL) The power delivered by a lossless line to a load

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