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Other example(s)
1. Lightning induced voltage calculation
In this example a model for induced voltage calculations in overhead lines is presented. The example is taken from [1, 2]. The basic assumptions are:
The overhead line can be treated as loss-less.
The electrical field is assumed to propagate unaffected by the ground.
The Agrawal coupling model is used.
The transmission line (TL) model is used for the lightning channel, assuming a pure step current at ground. The actual current shape at ground is taken into account by a final convolution integral of the induced voltages. This assumption is believed to be valid the first few microseconds where oftens the maximum induced voltage occurs.
The electrical field from the lightning leader is constant, and the resulting induced voltage is zero.
The same inducing voltage is assumed in all phases, but this could easily be extended since the inducing voltage is proportional to the line height, z.
The circuit shown in Fig. 1 shows the equivalent circuit of and overhead line excited inducing fields from a lightning channel. It can be connected to any component in the ATP, and several line segments are allowed.
Fig. 1. Equivalent circuit of 2-phase overhead line, excited by nearby lightning. Modeled in ATPDraw.
Below, an example of the electric circuit part of the two-phase model in Fig. 1 is listed in ATP file format.
/BRANCH
51X1 P1 300.
52X2 N1 200. 500.
51P2 X3 300.
52N2 X4 200. 500.
/SOURCE
60X1
60X2
60X3
60X4
A further simplification of this model is to rewrite it into a type94 Norton-transmission component.
References
[1] H.K. Høidalen, Lightning induced voltages in low-voltage systems, PhD thesis, ISBN 82-471-0177-1, University of Trondheim 1997.
[2] H.K. Høidalen, "Calculation of Lightning-induced Overvoltages using MODELS ", Proc. int. conf on Power Syst. Transients IPST'99, June 20-24, Budapest.
MODEL INDUS2
CONST
Tmax {VAL:500} --number of time steps
Im {VAL:30.e3} --current amplitude
T1 {VAL:2.e-6} --front time constant
T2 {L:50.e-6} --decay time constant
m {VAL:5} --slope factor
c {VAL:3.e8} --speed of light
v {VAL:1.1e8} --lightning velocity
Io {VAL:1} --step current ampl.
z {VAL:6} --line height
INPUT UAP,UBP,UAN,UBN --terminal voltages
DATA Y,XA,XB --orientation parameters
OUTPUT
UrAP,UrBP,UrAN,UrBN --to type 60 sources
VAR
UindA[0..1000],UindB[0..1000],dI[0..1000],
Tr,Ti,I,e,dt,UrAP,UrBP,UrAN,UrBN,
ta,tb,b,n,L,x,Ko,Ui,Tj
FUNCTION
SQR(x):=x*x
FUNCTION
F(x,tr):= (x+b*b*(c*tr-x))/sqrt(sqr(r(v*tr)+(1-b*b)*(x*x+y*y))
FUNCTION
U0(x,tr):= 60*Io*z*b*(c*tr-x)/(y*y+sqr(b*(c*tr-x)))
HISTORY
UrAP {t:0}, UrBP {dflt:0}
UrAN {dflt:0}, UrBN {dflt:0}
UAP {dflt:0}, UBP {dflt:0}
UAN {flt:0}, UBN {dflt:0}
INIT
dt:= timestep
b:=v/c
L:=XA-XB
FOR Tj:=1 TO 2 DO
if Tj=1 then
x:=XA else
x:=-XB
endif
ta:=sqrt(x*x+y*y)/c
tb:=sqrt(sqr(x-L)+y*y)/c
FOR Ti:=0 TO Tmax DO
Tr:=Ti*dt
if Tr>ta
then
if Tr>tb+L/c
then
Ui:=U0(x,Tr)*(f(x,Tr)-f(x-L,Tr-L/c))
elselse
Ui:=U0(x,Tr)*(f(x,Tr)+1)
endif
else
Ui:=0
endif
if Tj=1 then
UindA[Ti]:=Ui else
UindB[Ti]:=Ui
endif
ENDFOR
ENDFOR
FOR Ti:=0 TO Tmax DO --Heidler current:
Tr:=Ti*dt
IF (Ti=0) THEN dI[0]:=0
ELSE
e:=exp(-(T1/T2)*exp(ln(m*T2/T1)/m))
I:=Im/e*exp(m*ln(Tr/T1))/
(exp(m*ln(Tr/T1))+1)*exp(-Tr/T2)
dI[Ti]:=I*((m/Tr)/
(exp(m*ln(Tr/T1))+1)-1/T2)
ENDIF
ENDFOR--Convolution integral. Small Io required!!
Ti:=Tmax
WHILE Ti>1 DO
FOR Tr:=1 TO Ti-1 DO
UindA[Ti]:=UindA[Ti] + UindA[Tr]*dI[Ti-Tr]*dt
UindB[Ti]:=UindB[Ti] + UindB[Tr]*dI[Ti-Tr]*dt
ENDFOR
Ti:=Ti-1
ENDWHILE--Possible to scale Uind
Tr:=L/c
ENDINIT
EXEC
UrAP:=UindA[t/dt]+2*delay(UBP,Tr-dt,1)-delay(UrBP,Tr,1)
UrAN:=UindA[t/dt]+2*delay(UBN,Tr-dt,1)-delay(UrBN,Tr,1)
UrBP:=UindB[t/dt]+2*delay(UAP,Tr-dt,1)-delay(UrAP,Tr,1)
UrBN:=UindB[t/dt]+ 2*delay(UAN,Tr-dt,1)-delay(UrAN,Tr,1)
ENDEXEC
ENDMODEL