ece 476 power system analysis lecture 8: transmission line parameters, transformers prof. tom...

ECE 476 Power System Analysis

Lecture 8: Transmission Line Parameters,

Transformers

Prof. Tom Overbye

Dept. of Electrical and Computer Engineering

University of Illinois at Urbana-Champaign

overbye@illinois.edu

Special Guest Lecturer: TA Won Jang

Announcements

• Please read Chapters 5 and then 3• Quiz today on HW 3• H4 is 4.34, 4.41, 5.2, 5.7, 5.16

• It should be turned in on Sept 24 (hence no quiz next week)

2

Leidos Engineering

Transmission Line Equivalent Circuit

•Our current model of a transmission line is shown below

For operation at frequency , let z = r + j L

and y = g +j C (with g usually equal 0)

Units on

z and y are

per unit

length!

4

Derivation of V, I Relationships

We can then derive the following relationships:

( )

( ) ( )

dV I z dx

dI V dV y dx V y dx

dV x dI xz I yV

dx dx

5

Setting up a Second Order Equation

2

2

2

2

( ) ( )

We can rewrite these two, first order differential

equations as a single second order equation

( ) ( )

( )0

dV x dI xz I yV

dx dx

d V x dI xz zyVdxdx

d V xzyV

dx

6

V, I Relationships, cont’d

2 2

Define the propagation constant as

where

the attenuation constant

the phase constant

Use the Laplace Transform to solve. System

has a characteristic equation

( ) ( )( ) 0

yz j

s s s

7

Equation for Voltage

1 2

1 2 1 2

1 1 2 2 1 2

1 2

1 2

The general equation for V is

( )

Which can be rewritten as

( ) ( )( ) ( )( )2 2

Let K and K . Then

( ) ( ) ( )2 2

cosh( ) sinh( )

x x

x x x x

x x x x

V x k e k e

e e e eV x k k k k

k k k k

e e e eV x K K

K x K x

8

Real Hyperbolic Functions

•For real x the cosh and sinh functions have the following form:

cosh( ) sinh( )sinh( ) cosh( )

d x d xx x

dx dx

9

Complex Hyperbolic Functions

•For x = + j the cosh and sinh functions have the following form

cosh cosh cos sinh sin

sinh sinh cos cosh sin

x j

x j

10

Determining Line Voltage

R R

The voltage along the line is determined based upon

the current/voltage relationships at the terminals.

Assuming we know V and I at one end (say the

"receiving end" with V and I where x 0) we can

1 2determine the constants K and K , and hence the

voltage at any point on the line.

11

Determining Line Voltage, cont’d

1 2

1 2

1

1 2

2

c

( ) cosh( ) sinh( )

(0) cosh(0) sinh(0)

Since cosh(0) 1 & sinh(0) 0

( )sinh( ) cosh( )

( ) cosh( ) sinh( )

where Z characteristic

R

R

R RR

R R c

V x K x K x

V V K K

K V

dV xzI K x K x

dx

zI I z zK I

yyz

V x V x I Z x

zy

impedance

12

Determining Line Current

By similar reasoning we can determine I(x)

( ) cosh( ) sinh( )

where x is the distance along the line from the

receiving end.

Define transmission efficiency as

RR

c

out

in

VI x I x x

Z

PP

13

Transmission Line Example

R

6 6

Assume we have a 765 kV transmission line with

a receiving end voltage of 765 kV(line to line),

a receiving end power S 2000 1000 MVA and

z = 0.0201 + j0.535 = 0.535 87.8 mile

y = 7.75 10 = 7.75 10 90

j

j

.0

Then

zy 2.036 88.9 / mile

262.7 -1.1 c

mile

zy

14

Transmission Line Example, cont’d

*6

3

Do per phase analysis, using single phase power

and line to neutral voltages. Then

765 441.7 0 kV3

(2000 1000) 101688 26.6 A

3 441.7 0 10

( ) cosh( ) sinh( )

441,700 0 cosh(

R

R

R R c

V

jI

V x V x I Z x

2.036 88.9 )

443,440 27.7 sinh( 2.036 88.9 )

x

x

15

Transmission Line Example, cont’d

16

Lossless Transmission Lines

c

c

c

For a lossless line the characteristic impedance, Z ,

is known as the surge impedance.

Z (a real value)

If a lossless line is terminated in impedance

Z

Then so we get...

R

R

R c R

jwl ljwc c

VI

I Z V

17

Lossless Transmission Lines

2

( ) cosh sinh

( ) cosh sinh

( )( )

V(x)Define as the surge impedance load (SIL).

Since the line is lossless this implies

( )

( )

R R

R R

c

c

R

R

V x V x V x

I x I x I x

V xZ

I x

Z

V x V

I x I

If P > SIL then line consumes

vars; otherwise line generates vars.

18

Transmission Matrix Model

•Oftentimes we’re only interested in the terminal characteristics of the transmission line. Therefore we can model it as a “black box”.

VS VR

+ +

- -

IS IR

Transmission

Line

S

S

VWith

IR

R

VA B

IC D

19

Transmission Matrix Model, cont’d

S

S

VWith

I

Use voltage/current relationships to solve for A,B,C,D

cosh sinh

cosh sinh

cosh sinh

1sinh cosh

R

R

S R c R

RS R

c

c

c

VA B

IC D

V V l Z I l

VI I l l

Z

l Z lA B

l lC DZ

T

20

Equivalent Circuit Model

The common representation is the equivalent circuit

Next we’ll use the T matrix values to derive the

parameters Z' and Y'.

21

Equivalent Circuit Parameters

'' 2

' '1 '

2

' '2 2

' ' ' '' 1 1

4 2

' '1 '

2' ' ' '

' 1 14 2

S RR R

S R R

S S R R

S R R

S R

S R

V V YV I

ZZ Y

V V Z I

Y YI V V I

Z Y Z YI Y V I

Z YZ

V V

Z Y Z YI IY

22

Equivalent circuit parameters

We now need to solve for Z' and Y'. Using the B

element solving for Z' is straightforward

sinh '

Then using A we can solve for Y'

' 'A = cosh 1

2' cosh 1 1

tanh2 sinh 2

C

c c

B Z l Z

Z Yl

Y l lZ l Z

23

Simplified Parameters

These values can be simplified as follows:

' sinh sinh

sinhwith Z zl (recalling )

' 1tanh tanh

2 2 2

tanh 2 with Y2

2

C

c

z l zZ Z l l

y l z

lZ zy

l

Y l y l y lZ z l y

lY

yll

24

Medium Length Line Approximations

For shorter lines we make the following approximations:

sinh' (assumes 1)

' tanh( / 2)(assumes 1)

2 2 / 2

50 miles 0.998 0.02 1.001 0.01

100 miles 0.993 0.09 1.004 0.0

lZ Z

l

Y Y ll

sinhγl tanh(γl/2)Length

γl γl/2

4

200 miles 0.972 0.35 1.014 0.18

25

Three Line Models

(longer than 200 miles)

tanhsinh ' 2use ' ,2 2

2 (between 50 and 200 miles)

use and 2

(less than 50 miles)

use (i.e., assume Y is zero)

ll Y Y

Z Zll

YZ

Z

Long Line Model

Medium Line Model

Short Line Model

26

Power Transfer in Short Lines

•Often we'd like to know the maximum power that could be transferred through a short transmission line

V1 V2

+ +

- -

I1 I1Transmission

Line with Impedance Z

S12 S21

1

** 1 2

12 1 1 1

1 1 2 2 2

21 1 2

12 12

with , Z

Z Z

V VS V I V

Z

V V V V Z Z

V V VS

Z Z

27

Power Transfer in Lossless Lines

21 1 2

12 12 12

12 12

1 212 12

If we assume a line is lossless with impedance jX and

are just interested in real power transfer then:

90 90

Since - cos(90 ) sin , we get

sin

Hence the maximu

V V VP jQ

Z Z

V VP

X

1 212

m power transfer is

Max V VP

X

28

Limits Affecting Max. Power Transfer

• Thermal limits– limit is due to heating of conductor and hence depends

heavily on ambient conditions.– For many lines, sagging is the limiting constraint.– Newer conductors limit can limit sag. For example, in

2004 ORNL working with 3M announced lines with a core consisting of ceramic Nextel fibers. These lines can operate at 200 degrees C.

– Trees grow, and will eventually hit lines if they are planted under the line.

29

Other Limits Affecting Power Transfer

• 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. As reactive power increases the voltage falls, resulting in a potentially cascading voltage collapse.

30

Transformers Overview

• Power systems are characterized by many different voltage levels, ranging from 765 kV down to 240/120 volts.

• Transformers are used to transfer power between different voltage levels.

• The ability to inexpensively change voltage levels is a key advantage of ac systems over dc systems.

• In this section we’ll development models for the transformer and discuss various ways of connecting three phase transformers.

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Transmission to Distribution Transfomer

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Transmission Level Transformer

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