Download Longitudinal Voltages, Induced by Parallel Overhead Transmission

Progress In Electromagnetics Research Symposium Proceedings, Stockholm, Sweden, Aug. 12-15, 2013 305
Longitudinal Voltages, Induced by Parallel Overhead Transmission
Lines Magnetic Field
N. B. Rubtsova2 , M. Sh. Misrikhanov1 , S. G. Murzin1 , and A. Yu. Tokarskij3
1
2
JSC Federal Network Company, Moscow, Russian Federation
Federal State Budgetary Institution ‘Research Institute of Occupational Health’ under the Russian
Academy of Medical Sciences, Moscow, Russian Federation
3
JSC Federal Network Company Branch “Main Power Networks of the Center”
Moscow, Russian Federation
Abstract— Longitudinal voltage values, induced by magnetic fields currents in parallel overhead transmission line are determined usually by derivative of Carson’ integral equations. These
equations application have broad “dead space” conditionally the spacing of transmission lines
and ground conductivity that lead to great calculative error. The method of longitudinal voltage
values induced by magnetic field in parallel overhead transmission lines calculation for any distance between lines is suggested with full coincidence with Carson’ equation calculation (in work
space) without “dead space” (with high calculation error).
Under parallel overhead transmission line (TL) repair work linemen safety requires to know
the voltage that equal voltage (electromotive force) induced in repaired TL by working parallel
TL currents magnetic field (MF), since linemen may fall this longitudinal voltage under conductor
disconnection possibility.
Two parallel single-wire TL treat: line 1 and line 2 disposed at “a12 ” distance apart (see Fig. 1).
Line 1 is live at operative mode, and current I˙1 flows in its wire, line 2 is disconnected, and its
section l length is grounded on ends. Accept equation µA = µE = µ0 for air and earth magnetic
conductivity. Reverse line, similar to line 2 wire, will arrange under line 2 wire at a depth hw . This
line 2 wire will metal connect with grounding conductor’ slopes closed circuit “wire 2 — grounding
conductor-reverse wire” forming. Magnetic field created by I˙1 direct current induces Ė2 voltage in
derived circuit that calculates excluding earth conductivity (earth is ideal dielectric) by equation:
Ė2 = −j
ωµ0 lI˙1 a212 + (h1 + hw )2
ωµ0 l I˙1 r1R.w.
ln 2
=
−j
ln
,
4π
2π
r12
a12 + (h1 − h2 )2
(1)
where: ω = 2πf — angular frequency, f — AC current I˙1 frequency, µ0 — permeability of vacuum.
Figure 1: Two parallel single-wire TL.
PIERS Proceedings, Stockholm, Sweden, Aug. 12–15, 2013
306
In case of length l reverse wire section of line 2 is absent, and earth has finite resistivity ρE ,
voltage Ė2 may be determined by equation F12 = −2jJ(r, θ), where J(r, θ) = P + jQ — is Carson’
integral [1, 2], r and θ — integral characteristics:
/
r = r12
p
ωµ0 /ρE ,
θ = arctga12 /(h 1 + h2 ),
/
r12 = r1R.w.
at hw = h2 .
(2)
Ė2 voltage is determined by equation [1, 2]:
ωµ0 lI˙1
Ė2 = −j
2π
Ã
/
r
ln 12 + F12
r12
!
Z∞
,
F12 = 2
0
1
−ν(h1 +h2 )
q
cos νa12 dν,
¢e
¡ 2
2
2
ν + ν − kE − k0
(3)
2 −k 2 = −jωµ /ρ .
where kE — earth characteristic parameter, k0 — air characteristic parameter, kE
0 E
0
F12 is series expansion [1, 2], and for r ≤ 0.25 parameter is calculate by expansion:
µ
¶
r 2 cos ψ
2 sin ψ
F12 = −0.0772 − ln +
r cos θ − j ψ −
r cos θ ,
(4)
2
3
3
and for r ≥ 5 — by expansion:
µ
¶
2
cos 2θ −j2ψ cos 3θ −j3ψ 3 cos 5θ −j5ψ
−jψ
F12 =
cos θ e
−
e
+
e
−
e
.
r
r
r2
r4
(5)
√
/
pTaking into account that ψp= π/4, sin ψ = cos ψ = 2/2, from the first Equation (2) r12 =
r ρE /(ωµ0 ), intake [2] δE = 2ρE /(ωµ0 ) — as the earth penetration depth (the depth of elec√ h +h
/
2
1
2
√E , cos θ = h 1 +h
=
tromagnetic wave attenuation in e = 2.72 times), will result r12 = rδ
2 rδ
,
/
E
2
r12
from first Equation (2) consideration (3) will receive for r ≤ 0.25 parameter:
" √
#
˙
ωµ
l
I
2
δ
π
2
h
+
h
(6)
0 1
1
2
E
Ė2 = −j
ln
−j +
(1 + j) − 0.0772 .
2π
r12
4 3 δE
(6)
Similarly for r ≥ 5 parameter:
"
µ
¶#
/
ωµ0 lI˙1
r12 2
cos
2θ
cos
3θ
3
cos
5θ
(7)
Ė2 = −j
ln
+
cos θ e−jψ −
e−j2ψ +
e−j3ψ −
e−j5ψ
, (7)
2π
r12 r
r
r2
r4
q
where
=
+ (h 1 + h2 ) and r12 = a212 + (h 1 − h2 )2 .
There is break space or “dead space” on r parameter, from 0.25 to 5, in which Equations (6)
and (7) have very high error, and as result — miscount.
From the Equation (2) receive:
q
¯
¯p
¯
.¯p
¯
¯
¯
¯
/
r12 = a212 + (h 1 + h2 )2 = r ¯ jωµ0 /ρE ¯ = r · ¯ ρE /(jωµ0 )¯ .
/
r12
q
a212
2
If consider that a11/ À h1 + h1/ , then h1 + h1/ ≈ 0 may be accepted, in that case:
¯p
¯
.√
p
¯
¯
a12 = r ¯ ρE /(jωµ0 )¯ = r ρE /(ωµ0 ) = rδE
2.
(8)
“Dead space” borders for earth resistivity ρ3 different values are shown in Table 1: aδ12E max —
maximal distances between lines under which expression (6) use is possible, and minimal aδ12E min
distance, from which expression (7) use is admissible.
Consequently, voltages induced by parallel lines magnetic field calculation by equations derived
by Carson’s integral with δE electromagnetic wave earth penetration depth use, in the r parameter
disruption space gives incorrect result.
Progress In Electromagnetics Research Symposium Proceedings, Stockholm, Sweden, Aug. 12-15, 2013 307
Table 1: aδ12E max and aδ12E min values for ρE earth resistivity.
ρE
aδ12E max
aδ12E min
Oh · m
m
m
1
13
252
5
28
563
10
40
796
50
89
1800
100
126
2516
500
281
5627
1000
398
7958
For this defect elimination, we introduce into voltage induced by line 2 in “l length wire —
grounding conductor — earth” circuit calculation reverse current [2] in this circuit and earth equivalent depth:
r
r
r
r
r
r
r
2
2
eρE
e
ρE
2
2, 718
ρE
ρE
ρE
= /
hEQ = /
=
= 658, 898
≈ 660
,
−7
ωµ
2
π
µ
f
1,
781
2
π
4
π
·
10
f
f
f
γ
γ
0
0
where: e — natural logarithmic base, γ / = 1, 781 from Euler’ constant ln γ / = 0.5772, f — line
current frequency.
We assume that magnetic field created by I˙1 current, penetrate into the earth not more than
hEQ m and reverse wire of concerned line 2 circuit is placed at hEQ depth. In that case substitution
hEQ instead of hw into Equation (1), we derive the expression for calculation of voltage induced by
created I˙1 current in line 1 wire MF in line 2 grounded circuit:
Ė2w1 = −j
ωµ0 l I˙1 a212 + (h 1 + hEQ )2
ln 2
.
4π
a12 + (h 1 − h 2 )2
(9)
Current I˙1 magnetic field induces in the earth electric field (EF), its strength Ė1 is determined
by expression:
ωµ0 I˙1 y 2 + (h 1 + hEQ )2
Ė1 (x, y) = −j
ln
.
4π
y 2 + (h 1 + x)2
Considering the earth as isotropic medium we can determine current density η̇1 (x, y) in it created
by EF strength Ė1 (x, y) induced by I˙1 current MF by equation:
η̇1 (x, y) =
ωµ0 I˙1 y 2 + (h 1 + hEQ )2
Ė1 (x, y)
= −j
ln
.
ρE
4πρE
y 2 + (h 1 + x)2
The currents induced in the earth by magnetic field are vortex.
Generated by η̇1 (x, y) current density MF induction Ḃηy (X, Y ) component, is determined by
expression:
+
Zy 2h
Z EQ
µ0
η̇1 (x, y) (X − x)
Ḃηy (X, Y ) =
dxdy.
2π
(X − x)2 + (Y − y)2
y−
0
Ė2η voltage, generated in grounding part of line 2 wire circuit by magnetic flux of Ḃηy (X, Y )
induction, is determined as:
Ė2η =
ω 2 µ20 I˙1 l
−
8π 2 ρE
+
h
ZEQZy 2h
Z EQ
ln
−h2 y
−
0
y 2 + (h1 + hEQ )2
y2
2
+ (h1 + x)
(X − x)
dxdydX.
(X − x)2 + (a12 − y)2
(10)
Since line 1 is single-wire, the I˙1 current of Ė1 source (Fig. 2) passes through the wire 1 to ZL
load’ resistance the another end of its is grounded, and passing through the earth in the form of
reverse current I˙RC1 returns to the grounding end of Ė1 source.
Considering all earth volume from the surface to 2hEQ depth as isotropic medium, elementary
reverse current dI˙RC1 , passing through elementary channel with length
p
lRC + 2r = lRC + 2 x2 + y 2
PIERS Proceedings, Stockholm, Sweden, Aug. 12–15, 2013
308
Figure 2: To the definition of η̇OT line 1 reverse current to the earth density
and cross-sectional area dSE , may be evaluated as:
dI˙RC1 =
³
ρE lRC
U̇E
´ dSE ,
p
+ 2 x2 + y 2
where U̇E — is voltage between source and load groundings.
In ABCD plane, transversely to line phase wires, disposed at ζ distance from Ė1 source, reverse
current density η̇RC is determined by equation:
η̇RC (x, y) = η̇RC =
³
ρE lRC
U̇E
´.
p
+ 2 x2 + y 2
Since reverse current is line 1 current I˙1 but passes in reverse direction, the voltage U̇E value is
determined by expression:
U̇E = −
I˙1
yR 2hREQ
+
0
y−
.
(11)
1√
dxdy
ρE (lRC +2 x2 +y 2 )
MF induction ḂηRC y (X, Y ) component OY -axis, created by reverse current I˙RC1 , is determined
by expression:
Zy 2h
Z EQ
.h
i
µ0
ḂηRC y (X, Y ) =
[η̇RC (x, y) (X − x)] (X − x)2 + (Y − y)2 dxdy.
2π
+
y−
0
Voltage Ė2RC , created in line 2 wire grounded part circuit by magnetic flux of ḂηRC y (X, Y )
induction, can be determined by equation:
Ė2RC
ωµ0 l
= −j
2π
+
h
ZEQ Zy 2h
Z EQ
³
−h1/ y −
0
ρE lRC
U̇E (X − x)
´h
i dx dy dX.
p
+ 2 x2 + y 2 (X − x)2 + (a12 − y)2
Voltage full value is calculated as:
(12)
Ė2
= Ė2w1 + Ė2η + Ė2RC .
(12)
Progress In Electromagnetics Research Symposium Proceedings, Stockholm, Sweden, Aug. 12-15, 2013 309
(6)
(7)
We can examine Ė2 value changes, calculated by expressions (6) Ė2 and (7) Ė2 that were
(12)
received by Carson’ integral, and by Equation (12) Ė2 under distance between lines a12 increase
from 10 to 50000 m, take the earth resistivity value ρE = 50 Ohm · m. Line 1 current I˙1 = 4000 A,
its frequency f = 50 Hz. Then reverse current equivalent depth hEQ = 660 m. Taking y + = −y − =
100000 m, by expression (11) we can calculate voltage value U̇E = −50.13 V.
(12)
(12)
There are shown in Fig. 3 the changes of voltage modules Ė2w1 , Ė 2η , Ė2RC , Ė2 and arg(Ė2 )
under a12 distance increase from 100 to 2000 m.
(12)
Figure 3: Voltage module Ė2w1 , Ė2η , Ė2RC , Ė2
(12)
and arg(Ė2
(6)
) values for 100 ≤ a12 ≤ 2000 m.
(7)
(12)
Figure 4 shows superposed curves of voltage modules Ė2 , Ė2 , Ė2w1 and arguments Ė2
under a12 distance increase from 100 m to 2000 m (in “dead space” by r parameter).
(6)
(7)
(12)
Figure 4: Voltage module and argument changes Ė2 , Ė2 , Ė2w1 and Ė2
from 100 m to 2000 m.
(6)
under a12 distance increase
(12)
There is good match of voltage module and argument curves Ė2 and Ė2 up to “dead space”.
(6)
(7)
Voltage curves Ė2 and Ė2 by r parameter in “dead space” (see Fig. 4) are in the area of large
(12)
errors while voltage module Ė2
curve changes smoothly without going into the zone of large
(7)
errors. After “dead space” by r parameter (a12 > 1800 m) voltage module Ė2 curve continuous
to approach and coincide with voltage module Ė2w1 curve practically, but remain in lower position
(12)
than voltage module Ė2 and Ė 2η curves.
Discussed mathematical model allows to define voltage induced in parallel lines its current
magnetic field more exactly, especially in Carson’ integral “dead space” by r parameter.
REFERENCES
1. Carson, J. R., “Wave propagation in overhead wires with ground return,” Bell Syst. Techn.
J., Vol. 5, No. 4, 1926.
2. Kostenko, M. V., L. S. Perelman, and Y. P. Shkarin, “Wave processes and electrical disturbances in multi-wire high voltage lines,” M.: Energy, 272, 1973 (in Russian).