Lightning Example
Consider a transmission line with towers that are 40m tall and spaced 280m apart. Assume that there is a single shield wire with a characteristic impedance of 520 ohm and assume that the tower ground strap has a characteristic impedance of 135 ohm and the tower has a footing resistance of 30 ohm. Assume a propagation velocity of the speed of light for the ground wires and 0.85 times the speed of light for the tower ground strap. Assume that the phase conductors are 75% of the way up the tower and that the ground wire is segmented and open at the top of each of the adjacent towers.
Consider the case of a lightning strike where the current rises to 40kA in 2 s and falls to 20kA after 40 s.
Define parameters: Rfoot 30ohm Zc_gw 520ohm
Htower 40m Zc_tower 135ohm
Hconductor 0.75 H tower Hconductor 30 m Dspan 280m
Define speed of light: μo 4 π 10 7H
m εo 8.854 10 12 F
m
c 1
μoεo
c 3105 km
sec
νgw c
νtower 0.85 c νtower 2.5483105 km
sec
μsec 106sec
(a) Find the peak voltage at the tower top, the bottom of the tower and at the conductor height if the lightning strike the top of the tower.
30 ohm TOWER CROSSA
RFOOT
TOWER TTOP
CROSSA I STROKE
Grnd Wire
VFARL Grnd Wire
VFARR
ATPDraw Schematic
ATP Data File
BEGIN NEW DATA CASE
C --- C Generated by ATPDRAW April, Wednesday 25, 2012
C A Bonneville Power Administration program
C by H. K. Høidalen at SEfAS/NTNU - NORWAY 1994-2009 C --- C dT >< Tmax >< Xopt >< Copt ><Epsiln>
2.5E-9 5.E-5
500 1 0 0 0 0 0 1 0
C 1 2 3 4 5 6 7 8
C 345678901234567890123456789012345678901234567890123456789012345678901234567890 /BRANCH C < n1 >< n2 ><ref1><ref2>< R >< L >< C > C < n1 >< n2 ><ref1><ref2>< R >< A >< B ><Leng><><>0 RFOOT 30. 2
-1CROSSARFOOT 135.2.55E8 30. 1 0 0
-1TTOP CROSSA 135.2.55E8 10. 1 0 0
-1VFARL TTOP 520. 3.E8 280. 1 0 0
-1TTOP VFARR 520. 3.E8 280. 1 0 0
/SWITCH C < n 1>< n 2>< Tclose ><Top/Tde >< Ie ><Vf/CLOP >< type > STROKETTOP MEASURING 1
/SOURCE C < n 1><>< Ampl. >< Freq. ><Phase/T0>< A1 >< T1 >< TSTART >< TSTOP > 13STROKE-1 4.E4 2.E-6 2.E4 4.2E-5 1.
/OUTPUT
TTOP CROSSA BLANK BRANCH
Plot of lightning current:
(f ile prob2a.pl4; x-v ar t) c:STROKE-TTOP
0 10 20 30 40 *10-6 50
0 5 10 15 20 25 30 35 40
*103
Voltages at tower top, cross arm and footing resistance.
(f ile prob2a.pl4; x-v ar t) v :TTOP v :CROSSA v :RFOOT -
0 10 20 30 40 *10-6 50
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6
*106
Zoom in on peak of voltage waveform:
(f ile prob2a.pl4; x-v ar t) v :TTOP v :CROSSA v :RFOOT -
0 2 4 6 8 10 12 14 *10-6 16
0.6 0.8 1.0 1.2 1.4 1.6
*106
As one would expect, the top shows the biggest peaks in voltage.
Note also the time delay in the voltage at the footing resistor.
Vpeak_top 1.5609 10 3kV at ttop 2.5075μsec Vpeak_crossarm 1.4876 10 3kV at tcross 2.5400μsec
Note that this is actually the second relative
peak in this waveform Vpeak_foot 1.3294 10 3kV at tfoot 4.3275μsec
(b) Repeat of the lightning strike occurs 100m away from the tower top (in either direction).
Grnd Wire VFARR
30 ohm Grnd Wire
VFARL Gnd Wire
STRIKE I STROKE
TOWER CROSSA
RFOOT
TOWER TTOP
ATP Draw Circuit Diagram
Voltages at tower top, cross arm and footing resistance.
(f ile prob2b.pl4; x-v ar t) v :TTOP v :CROSSA v :RFOOT -
0 10 20 30 40 *10-6 50
0.0 0.5 1.0 1.5 2.0 2.5
*106
Note that the peak voltages are higher in this case.
Zoom in on peak of voltage waveform:
(f ile prob2b.pl4; x-v ar t) v :TTOP v :CROSSA v :RFOOT -
0 3 6 9 12 *10-6 15
0.2 0.6 1.0 1.4 1.8 2.2
*106
Again,the top shows the biggest peaks in voltage. Note also the time delay in the voltage at the footing resistor.
Vpeak_top 2.0707 10 3kV at ttop 2.8275μsec
Note that the time to the peak is delayed, due to travel time along the ground wire.
Vpeak_crossarm 1.9251 10 3kV at tcross 2.8650μsec
Vpeak_foot 1.6244 10 3kV at tfoot 4.0525μsec
Note that this is actually the second relative peak in this waveform, but it is earlier than the last case.
(c) Repeat part (a) if the adjacent towers are each grounded, with the same ground strap characteristics and the same footing resistances.
For this case, we will just model the adjactent towers in detail, since we aren't concerned
with waiting for further reflections to come back.
We do model both ground wires at the top of each though.
TOWER CROSSA
RFOOT 30 ohm Grnd Wire Grnd Wire VLFT
VFARL
TOWER TTOP I STROKE
TOWER
30 ohm
TOWER
TOWER
30 ohm
TOWER Grnd Wire
VRGHT
Grnd Wire VFARR
(f ile prob2c.pl4; x-v ar t) v :TTOP v :CROSSA v :RFOOT -
0 10 20 30 40 *10-6 50
0.0 0.3 0.6 0.9 1.2 1.5
*106
(f ile prob2c.pl4; x-v ar t) v :TTOP v :CROSSA v :RFOOT -
0 2 4 6 8 10 *10-6 12
0.0 0.3 0.6 0.9 1.2 1.5
*106
(f ile prob2c.pl4; x-v ar t) v :VFARR v :VFARL
0 10 20 30 40 *10-6 50
0 50 100 150 200 250 300 350
*103
(a) Find the peak voltage at the tower top, the bottom of the tower and at the conductor height if the lightning strike the top of the tower.
PSCAD Schematic from file Lightning1.psc
Surge
Simple Lightning Surge:
I = 40kA/T1 * t - C1(t-T1) TIME
Tower1
Tow er1 T
Tower1Tower2
Tow er2 T
Tower2 CrossA
30[ohm]
VFOOTVtop GWL
GWL T
GWL
10e10 [ohm]
GWR
GWR T
GWR
10e10 [ohm]
Vtop
CrossA
Vfoot
Ia
Notes:
Note the transmission line models used for the ground
straps and static wires
PSCAD requires a terminating impedance at the end
of a line, it can't be open circuited. A very large resistance has been placed at the end of the line
Current surge source dialog box:
Plot of lightning current:
Main : Graphs
0.000 0.010m 0.020m 0.030m 0.040m 0.050m
0.0 5.0k 10.0k 15.0k 20.0k 25.0k 30.0k 35.0k 40.0k
y (kA)
Surge
Voltages at tower top, cross arm and footing resistance.
Main : Graphs
0.000 0.010m 0.020m 0.030m 0.040m 0.050m
0.0 0.2M 0.4M 0.6M 0.8M 1.0M 1.2M 1.4M 1.6M
y (kV)
Vtop CrossA Vfoot
Zoom in on peak of voltage waveform:
Main : Graphs
0.000 0.002m 0.004m 0.006m 0.008m 0.010m 0.012m 0.014m 0.016m 0.018m 0.0
0.2M 0.4M 0.6M 0.8M 1.0M 1.2M 1.4M 1.6M
y (kV)
Vtop CrossA Vfoot
As one would expect, the top shows the biggest peaks in voltage.
Note also the time delay in the voltage at the footing resistor.
Vpeak_top 1.5609 10 3kV at ttop 2.5075μsec Vpeak_crossarm 1.4876 10 3kV at tcross 2.5400μsec
Note that this is actually the second relative
peak in this waveform Vpeak_foot 1.3294 10 3kV at tfoot 4.3275μsec
(b) Repeat of the lightning strike occurs 100m away from the tower top (in either direction).
PSCAD Circuit Diagram from file Lightning2.psc
Surge
Simple Lightning Surge:
I = 40kA/T1 * t - C1(t-T1) TIME
Tower1
Tow er1 T
Tower1Tower2
Tow er2 T
Tower2 CrossA
30[ohm]
VFOOT
Vtop
GWL
GWL T
GWL
10e10 [ohm]
GWR
GWR T
GWR
10e10 [ohm]
Vtop CrossA
Vfoot
Ia
GWL2
GWL2 T
GWL2
Voltages at tower top, cross arm and footing resistance.
Main : Graphs
0.000 0.010m 0.020m 0.030m 0.040m 0.050m
0.0 0.3M 0.5M 0.8M 1.0M 1.3M 1.5M 1.8M 2.0M 2.3M
y (kV)
Vtop CrossA Vfoot
Note that the peak voltages are higher in this case.
Main : Graphs
0.000 0.002m 0.004m 0.006m 0.008m 0.010m 0.012m 0.014m 0.016m 0.0
0.3M 0.5M 0.8M 1.0M 1.3M 1.5M 1.8M 2.0M 2.3M
y (kV)
Vtop CrossA Vfoot
Zoom in on peak of voltage waveform:
Again,the top shows the biggest peaks in voltage. Note also the time delay in the voltage at the footing resistor.
Vpeak_top 2.0707 10 3kV at ttop 2.8275μsec
Note that the time to the peak is delayed, due to travel time along the ground wire.
Vpeak_crossarm 1.9251 10 3kV at tcross 2.8650μsec
Vpeak_foot 1.6244 10 3kV at tfoot 4.0525μsec
Note that this is actually the second relative
peak in this waveform, but it is earlier than the last case.
(c) Repeat part (a) if the adjacent towers are each grounded, with the same ground strap characteristics and the same footing resistances.
For this case, we will just model the adjacent towers in detail, since we aren't concerned
with waiting for further reflections to come back.
We do model both ground wires at the top of each though.
The model includes the static wires beyond the next tower as well, terminated with a very
large resistance
From file Lightning3.psc
Surge
Simple Lightning Surge:
I = 40kA/T1 * t - C1(t-T1) TIME
Tower1
Tow er1 T
Tower1Tower2
Tow er2 T
Tower2 CrossA
30[ohm]
VFOOT Vtop GWL
GWL T
GWL
10e10 [ohm]
GWR GWR
T
GWR
10e10 [ohm]
Vtop CrossA
Vfoot
Ia
GWR2 GWR2
T
GWR2
Tow er3 T
Tower3
Tower3 30[ohm]
Tower4
Tow er4 T
Tower4 30[ohm]
GWL2 GWL2
T
GWL2
Main : Graphs
0.000 0.010m 0.020m 0.030m 0.040m 0.050m
0.0 0.2M 0.4M 0.6M 0.8M 1.0M 1.2M 1.4M
y (kV)
Vtop CrossA Vfoot
Voltages at tower top, cross arm and footing resistance.
Main : Graphs
0.000 0.002m 0.004m 0.006m 0.008m 0.010m 0.012m 0.014m 0.016m 0.0
0.2M 0.4M 0.6M 0.8M 1.0M 1.2M 1.4M
y (kV)
Vtop CrossA Vfoot
Zoomed voltages at
tower top, cross arm and footing resistance.
Note that the peak
voltage is significantly lower than the other cases
Main : Graphs
0.000 0.005m 0.010m 0.015m 0.020m 0.025m 0.030m 0.035m 0.00
50.00k 100.00k 150.00k 200.00k 250.00k 300.00k 350.00k
y (kV)
VTopL VTopR
Voltages at tops of the next towers to the left and right
Now add the phase conductors to the model. Closest to case 3 above
Note that this is simplified to illustrate how to model
30 ohm TOWER CRSA3
RFOOT3
TOWER TTOP3 I
STROKE
V
ZSRC V
SWS BUS1
LCC F1
40. km
V
F2 VS
F3 BUS2
LCC
10. km
TTOP2
LCC
0.28 km
F1 TTOP1
LCC FS2
0.28 km
ZSRC VR
I
V
30 ohm 30 ohm
30 ohm
V
30 ohm TOWER CRSA2
RFOOT2
TOWER TTOP2
CRSA2
30 ohm TOWER CRSA1
RFOOT1
TOWER TTOP1
CRSA1 ABC
(file ltngcase4.pl4; x-var t) v:TTOP2 v:CRSA2 v:RFOOT2-
0 10 20 30 40 [us] 50
0.0 0.3 0.6 0.9 1.2 1.5 [MV]
Compare cross arm voltage to case 3:
ltngc as e4.pl4: v:C R S A2 ltngc as e3.pl4: v:C R O S S A
0 2 4 6 8 1 0 [us ] 1 2
0 .0 0 .3 0 .6 0 .9 1 .2 1 .5 [M V ]
Voltages on the footing resistors of the other uprights:
(file ltngcase4.pl4; x-var t) v:G1R - v:G2R - v:G3R -
0 10 20 30 40 [us] 50
-30 -20 -10 0 10 20 30 40 50 [kV]
Phase conductor voltages (with cross arm level on ground strap)
(file ltn g c a s e 4 .p l4 ; x-va r t) v:F 2 A v:F 2 B v:F 2 C v:C R S A2
0 1 0 2 0 3 0 4 0 [us ] 5 0
-0 .4 -0 .1 0 .2 0 .5 0 .8 1 .1 1 .4 [M V ]
Same voltages without the lightning strike:
( file ltn g c a s e 4 .p l4 ; x-v a r t) v:F 2 A v:F 2 B v:F 2 C v:C R S A 2
0 1 0 2 0 3 0 4 0 [us ] 5 0
-3 0 0 -2 0 0 -1 0 0 0 1 0 0 2 0 0 3 0 0 4 0 0 5 0 0 [k V ]
Phase Currents Don't show much effect
( f il e lt n g c a s e 4 . p l 4 ; x - v a r t ) c : F S 2 A - F 2 A c : F S 2 B - F 2 B c : F S 2 C - F 2 C
0 . 0 0 . 1 0 . 2 0 . 3 0 . 4 [ m s ] 0 . 5
- 2 0 0 0 - 1 0 0 0 0 1 0 0 0 2 0 0 0 3 0 0 0 [ A ]
Zoom on one phase:
(file ltngc as e4.pl4; x-var t) c :FS2A -F2A
0.0 0.1 0.2 0.3 0.4 [ms] 0.5
2500 2530 2560 2590 2620 2650
[A ]
V
FS2 F2
TTOP2
LCC
0.28 km
+ Vf - UI
F1 TTOP1
LCC FS2
0.28 km
30 ohm 30 ohm
V
30 ohm CRSA2
RFOOT2
TTOP2
+ Vf - UI
+ Vf - UI
Now add voltage dependent flashover
switches:
Switch voltages:
(file ltn g c a s e 4 .p l4 ; x -va r t) v :C R S A 2 -F 2 A v :C R S A2 -F 2 B v :C R S A 2 -F 2 C
0 .0 0 0 .0 2 0 .0 4 0 .0 6 0 .0 8 [m s ] 0 .1 0
-1 .0 -0 .5 0 .0 0 .5 1 .0 1 .5 [M V ]
Phase currents (between towers):
( file ltn g c a s e 4 .p l4 ; x -v a r t) c :F S 2 A -F 2 A c :F S 2 B -F 2 B c :F S 2 C - F 2 C
0 .0 0 0 .0 2 0 .0 4 0 .0 6 0 .0 8 [m s ] 0 .1 0
-5 0 0 0 -3 5 0 0 -2 0 0 0 -5 0 0 1 0 0 0 2 5 0 0 4 0 0 0 [A ]
Phase voltages at Bus 2 (closer bus)
(file ltn g c a s e 4 .p l4 ; x-va r t) v:B U S 2 A v:B U S 2 B v:B U S 2 C
0 .0 0 0 .0 2 0 .0 4 0 .0 6 0 .0 8 [m s ] 0 .1 0
-1 .0 -0 .5 0 .0 0 .5 1 .0 1 .5 [M V ]
Phase Currents at Bus 2:
(file ltngcase4.pl4; x-var t) c:SWRA -BUS2A c:SWRB -BUS2B c:SWRC -BUS2C
0.00 0.02 0.04 0.06 0.08 [ms] 0.10
-3000 -2000 -1000 0 1000 2000 [A]