1
Transmission line parameters
• Aim
– Learn how to use ATP to obtain series
impedance parameters;
• Contents
– Introducing the ground reference
– Self and mutual impedances
– Matrix description
p
– Look into ATPDraw LCC module
– Examples
MTU-Houghton, 2010
Internett: www.elkraft.ntnu.no/
2
Briefly about the speaker
• Professor at Norwegian Univ
Univ. Science and
Technology – Dept. Electrical Engineering
– Power system
y
transients and p
protection
– High voltage engineering, stress calculations
– Recent focus on Power Transformers
• Honorary member of European EMTP user’s group
– User of ATP for 20 years
• Developer of ATPDraw
• Sabbatical at MTU
– Room 628, phone 487-2910
– hhoidale@mtu.edu
3
Relevance of series impedance
parameters
• Why do we have to understand the details?
– The manufacturer provides only positive
sequence 50/60 Hz data!
– Zero sequence data important for ground fault
situations!
it ti
!
– Mutual coupling between parallel transmission
lines is important for protection settings!
– What is the influence of
•
•
•
•
Transmission line height, h
Phase separation, D
Bundling, duplex/triplex
Ground resistivity,
resistivity 
MTU, Houghton, 2010
www.elkraft.ntnu.no/
4
Ground plane
• The text book chapt. 4 handles only
conductors in free space.
p
Let us introduce
a ground plane:
Ia
D
Ib
Air
Earth
Ia
Air
Air
h1
h1
-Ia
Field lines
perpendicular
to earth surface
D
Ib
Ia
Ib
h2
h2
-Ib
Ideal case:
Imaging concept
Air
h1
h2
‘Air’ h1
-IIa
h2

-IIb
Real case:
Penetration
depth of earth
MTU, Houghton, 2010
www.elkraft.ntnu.no/
5
Internal self impedance
• Self impedance is split in internal and
external part
p
Z s  Zi  Ze
• Internal
I t
l impedance
i
d
((round,
d solid
lid cond.):
d)

Z i  Ri  j
8
Eq. 4.2 & 4.13 in text book.
Depends on skin effect and
geometry. GMR available.
• The last part is often written on the form
 4r 
0 r
0

Eq. 4.23
4 23 in text book.
book
j  j   j  ln  e  Eq
8
2 4
2  
MTU, Houghton, 2010
www.elkraft.ntnu.no/
6
External self impedance
• A conductor over an ideal, lossless ground
2r
((Eq.
q 4.22 in text book):
)
h
– Imaging:
Z e
j

0
 2h 

 ln
l  
2
 r 
=0
h
/ ]
/m]
image
• A conductor over a real earth surface
– Penetration depth (or Carson’s formula)


[m]
– For low frequencies (>>h):
( h):
j
0
 Dj 
j 0
 2h     0 j 0
 ln 

 ln 
Ze 
 /m], with


r
r
2
8
2




MTU, Houghton, 2010
D j  660 
 [Ωm]
[m]
f [Hz]
www.elkraft.ntnu.no/
7
Generalized self impedance
• The inductive part of the internal and
external impedances
p
can be merged
g
 Dj 
  0 j 0

 ln 
Z s  Z i  Z e  Ri  j 

8
8
2
r


 Dj 
 0 j 0
 [Ωm]
 Ri 

 ln 
[m]
/m], with D j  660 

f [Hz]
8
2
 r' 
• Geometric mean radius:
– General
G
l r '  GMR . Tables
T bl exist,
i ref.
f A
A.3
3
– For solid, circular, non-magnetic material
r '  e1/4  r  0.7788  r
MTU, Houghton, 2010
www.elkraft.ntnu.no/
8
Mutual impedance
• The conductor will link with both the other
conductor and its image:
D’
I

D’’
D
-II
• According to Eq. 4.36 this gives
 D 2  (h  h  ) 2
j0
 D ''  j0
1
2
 ln 
 ln 
Zm 

2
2

2
2
 D' 


D
(
h
h
)
1
2





• Which for low frequencies becomes:
 Dj 
0
0
Zm 
 j  ln 

8
2  D ' 
MTU, Houghton, 2010
www.elkraft.ntnu.no/
9
Multiple conductors - Matrix
• Th
The conceptt off selflf and
d mutual
t l impedances
i
d
is easily expandable to multiple conductors
– Conductors on the same potential can be handled
with equivalent conductors, ref Chapt. 4.8 in text
book, or by reduction of the full matrix
– Conductors on ground potential has to be
eliminated
• Th
The series
i iimpedance
d
matrix
t i iis symmetrical
ti l
on the form  Z sa Zmab Zmac Zmag 


Z 



Z sb
Z mbc
Z sc

Z mbg 
Z mcg 

Z sg 
MTU, Houghton, 2010
www.elkraft.ntnu.no/
10
Positive and zero sequence
• Let the series impedance matrix now be
reduced to a 3x3 matrix on the form
Zs
Z  

Zm
Zs
Zm 
Z m 
Z s 
For simplicity a perfectly
transposed system is assumed
• Then
e tthe
e pos
positive
t e and
a d zero
e o seq
seq. imps.
ps a
are
e
 D j   0
 Dj 
 0

0
ln
 j   0  ln 





j



8
2
'
8
2
'

r
D





 D '
 R i  j   0  ln 

Influence of ground disappears!
2
 r' 
Z   Z s  Z m  Ri 
Z 0  Z s  2 Z m  Ri 

Dj
3   0

 3 j   0  ln
l 
 3 2010
MTU,
2
8
2  Houghton,
 D ' r

Strong ground
www.elkraft.ntnu.no/
'
influence!
11
Coupling between transmission lines
• Consider two transmission lines:
S
• This gives a 6x6 series impedance matrix:
Zs



Z 




Zm
Zs
Zm
Zm
Z m11
Z m12
Z m 22
Zs
Zm
Zs
Zs
Z m12 
Z m 23 
Z m 22 

Zm 
Zm 

Z s 
As th
A
the di
distance
t
b
between
t
the lines increases, the
mutual impedances Zmij
tends to become equal
Z mij
 Dj 


0
0

 j  ln 

8
2  S 
MTU, Houghton, 2010
www.elkraft.ntnu.no/
12
Coupling between transmission lines
• Now consider a zero-sequence
q
component (I02) in one line, what is the
consequence on the other?
 Va   Z s
V  
 b 
 Vc  
 
V02  
V02  
  
V02  
Zm
Zs
Zm
Zm
Z m11
Z m12
Z m 22
Zs
Zm
Zs
Zs
Z m12   I a 
Z m 23   Ib 
Z m 22   I c 
 
Z m   I 02 
Z m   I 02 
  
Z s   I 02 
V a  Z s  I a  Z m  I b  Z m  I c  ( Z m11  Z m 12  Z m13 )  I 02
 ( Z s  Z m )  I a  Z 012  I 02
A zero sequence component is
coupled to the other line
MTU, Houghton, 2010
www.elkraft.ntnu.no/
13
Using Line Constants in ATP
• LCC interface in ATPDraw
– Get g
geometrical data
– Start ATPDraw, File New
– Start LCC (right click in empty space)
MTU, Houghton, 2010
www.elkraft.ntnu.no/
14
LCC model input
• Choose PI model and Standard data
• On Data page type in conductor data
MTU, Houghton, 2010
www.elkraft.ntnu.no/
15
Creating an LCC model
• Click on View to inspect
• Click on Run ATP to create model (Cancel
the plotting window that pops up)
• Where
Wh
iis th
the resultlt ((note
t th
the name off liline)?
)?
– Check Tools|Options/Files&Folders
| p
(ATP)
(
)
– Lib file is final model, lis contains sub-results
1IN___AOUT__A
1IN
AOUT A
2IN___BOUT__B
3IN___COUT__C
6.64863719E-01
6 64863719E 01
5.08928089E-01
6.66163048E-01
4.86898502E-01
5 08928089E 01
5.08928089E-01
6.64863719E-01
4.79819218E+00
4 79819218E+00 1.20191093E-01
1 20191093E 01
1.57302035E+00 -1.58574976E-02
4.72564369E+00 1.22277240E-01
1.12911067E+00 -3.57568321E-03
1.57302035E+00
1 57302035E+00 -1.58574976E-02
1 58574976E 02
4.79819218E+00 1.20191093E-01
MTU, Houghton, 2010
www.elkraft.ntnu.no/
Impedance matrix, in units of [ohms/kmeter ] for the system of physical conductors.
16
Rows and columns proceed in the same order as the sorted input.
1
1.163069E-01
8.404390E-01
2
5.667074E-02
2.955284E-01
1.163069E-01
8.404390E-01
3
5.657221E-02
2.432963E-01
5.667074E-02
2.955284E-01
1.163069E-01
8.404390E-01
4
5.670445E-02
5.498391E-01
5.666840E-02
2.929874E-01
5.656775E-02
2.420161E-01
1.163069E-01
8.404390E-01
5
5.666462E-02
5.237530E-01
5.662880E-02
2.929909E-01
5.652871E-02
2.420503E-01
5.666466E-02
5.498841E-01
1.162273E-01
8.405289E-01
6
5
5.666466E-02
666466E 02
5.498841E-01
5
5.663112E-02
663112E 02
2.955291E-01
5
5.653314E-02
653314E 02
2.433302E-01
5
5.666462E-02
666462E 02
5.237530E-01
5
5.662488E-02
662488E 02
5.499291E-01
1
1.162273E-01
162273E 01
8.405289E-01
7
5.667300E-02
2.981582E-01
5.670445E-02
5.498391E-01
5.666840E-02
2.929874E-01
5.667074E-02
2.955284E-01
5.663112E-02
2.955291E-01
5.663336E-02
2.981557E-01
1.163069E-01
8.404390E-01
8
5.663336E-02
5.663336E 02
2.981557E-01
5.666462E-02
5.666462E 02
5.237530E-01
5.662880E-02
5.662880E 02
2.929909E-01
5.663112E-02
5.663112E 02
2.955291E-01
5.659158E-02
5.659158E 02
2.956183E-01
5.659381E-02
5.659381E 02
2.982481E-01
5.666466E-02
5.666466E 02
5.498841E-01
1.162273E-01
1.162273E 01
8.405289E-01
9
5.663112E-02
2.955291E-01
5.666466E-02
5.498841E-01
5.663112E-02
2.955291E-01
5.662880E-02
2.929909E-01
5.658927E-02
2.930773E-01
5.659158E-02
2.956183E-01
5.666462E-02
5.237530E-01
5.662488E-02
5.499291E-01
1.162273E-01
8.405289E-01
10
5.657660E-02
2.445987E-01
5.667300E-02
2.981582E-01
5.670445E-02
5.498391E-01
5.657221E-02
2.432963E-01
5.653314E-02
2.433302E-01
5.653751E-02
2.446322E-01
5.667074E-02
2.955284E-01
5.663112E-02
2.955291E-01
5.663336E-02
2.981557E-01
5.653751E-02
2.446322E-01
5.663336E-02
2.981557E-01
5.666462E-02
5.237530E-01
5.653314E-02
2.433302E-01
5.649414E-02
2.433861E-01
5.649849E-02
2.446885E-01
5.663112E-02
2.955291E-01
5.659158E-02
2.956183E-01
5.659381E-02
2.982481E-01
5.666466E-02
5.498841E-01
1.162273E-01
8.405289E-01
5.653314E-02
2 433302E 01
2.433302E-01
5.663112E-02
2
2.955291E-01
955291E 01
5.666466E-02
5
5.498841E-01
498841E 01
5.652871E-02
2
2.420503E-01
420503E 01
5.648973E-02
2
2.421059E-01
421059E 01
5.649414E-02
2
2.433861E-01
433861E 01
5.662880E-02
2
2.929909E-01
929909E 01
5.658927E-02
2
2.930773E-01
930773E 01
5.659158E-02
2
2.956183E-01
956183E 01
5.666462E-02
5.237530E-01
5.662488E-02
5.499291E-01
1.162273E-01
8.405289E-01
5.582845E-02
3.131729E-01
5.581227E-02
2.901020E-01
5.573753E-02
2.469376E-01
5.582795E-02
3.121978E-01
MTU, Houghton,
2010
www.elkraft.ntnu.no/
5.578980E-02
5.579030E-02
5.581381E-02
5.577574E-02 5.577421E-02
3.153077E-01 3.163665E-01 2.917828E-01 2.935802E-01 2.918212E-01
Inspecting the lis file
Full system (14x14)
1.163069E-01
8.404390E-01
11
12
13
17
Reduced system (3x3)
Impedance matrix, in units of [ohms/kmeter ]
for the system of equivalent phase conductors.
Rows and columns proceed in the same order as the sorted input
input.
1
6.648637E-02
4.798192E-01
2
5.089281E-02
1.573020E-01
6.661630E-02
4.725644E-01
3
4.868985E-02 5.089281E-02 6.648637E-02
1.129111E-01 1.573020E-01 4.798192E-01
Both "R" and "X" are in [ohms];
MTU, Houghton, 2010
www.elkraft.ntnu.no/
18
Check the result I
• User Verify in LCC module
MTU, Houghton, 2010
www.elkraft.ntnu.no/
19
Check the result II
• Line Check module
– Select a line sections in the circuit
– Click ATP|Line Check
MTU, Houghton, 2010
www.elkraft.ntnu.no/
20
Line Check results
• Results differ somewhat from Verify
because an improved method is used
MTU, Houghton, 2010
www.elkraft.ntnu.no/
21
Double circuit line
• Example
17.5 m
18 0 m
18.0
100 m
h=(2Vmid+Vtow)/3 m
=100 m
• Verify (1 km line):
• Homework:
– Reproduce
– Check with hand
calculations
MTU, Houghton, 2010
www.elkraft.ntnu.no/
22
Summary
• The concept of Self and Mutual
impedances of a transmission line over
lossy ground introduced
• Hand-calculation formulas presented and
linked to text book chapt.
chapt 4
• Multi-conductor matrix systems introduced
• Line Constants of ATP introduced via the
LCC module of ATPDraw
– Verify
– Inspection of lis
lis-file
file
MTU, Houghton, 2010
www.elkraft.ntnu.no/