Calculation of Lightning-Induced Overvoltages using MODELS
H. K. Høidalen
SINTEF Energy Rearch
7034 Trondheim, NORWAY
hans.hoidalen@
Abstract - The paper prents a method for calculation of lightning-induced overvoltages (LIO). The modeling of the lightning channel as well as the coupling to overhead lines is formulated analytically and implemented in MODELS in ATP. The analytical approach is possible as long as the overhead line and the ground is assumed lossless and the transmission line model is ud for the lightning channel with constant current velocity. A large number of overhead lines can be included in an ATP data ca, and each line can consist of veral conductors. The paper prents calculation results of LIOs in a larger low voltage system with and without a neutral wire. The dependency of stroke location, overhead line configuration and the low voltage in the system is studied as well as the effect of arresters.
Keywords: Lightning, induced-volages, low-voltage, protection, MODELS, ATP-EMTP
I. INTRODUCTION
The calculation of lightning-induced overvoltages (LIO) in overhead lines was analytically formulated by Rusck [1]. Assuming the transmission line (TL) model [2] for the lightning channel and an infinite long overhead line and ignoring loss effects he established well-known engineering equations for LIOs. Today, more sophisticated calculation models exist [3], but the approach propod by Rusck is still believed to give acceptable results [4, 5].
An advantage with Rusck’s formulation is that it is analytical. This results in fast computation and makes it simple to study the effect of specific parameters and as well as induced voltages in larger systems. This paper extends the results from Rusck, using the Agrawal coupling model [6], which is bad on measurable electromagnetic field quantities. A finite length of the overhead line is taken into account as well as the option of multiple phas. The lossy ground effect, handled in [7, 8], is ignored. This is reasonable for high conducting ground and line lengths shorter than about 1 km. The equations for induced voltage are completely analytical.
A model of a lossless overhead line excited by electromagnetic fields from a lightning channel has been established in the ATP [9], using the MODELS language [10]. In this model the lightning current
parameters, the overhead line length geometry and orientation is ur lectable. The graphical tool ATPDraw [11] is ud to visuali the electrical network.
II. MODELLING
In this ction models for induced voltage calculations are
prented. 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 ud.
· The transmission line (TL) model is ud for the
lightning channel, assuming a pure step current at
ground.
· The electrical field from the lightning leader is
constant, and the resulting induced voltage is zero.
The assumptions enable a completely analytical
solution. The actual lightning current shape is taken into
account by a convolution integral of the inducing voltages.
If g0(t) is the step respon of a current I0, then the
respon of a time varying current i(t) is:
t
t
t
d
t
知音难觅的经典诗句
i
I
t
g
t
g
t
×
¶
¶
×
-
=ò)(
)五带划分
(
)(
00
0 (1)
The configuration of the system is shown in Fig. 1. The
end A of the overhead line is oriented so that x A > x B.
A B
obrvation
point
(x, y, z)
x
A x B
overhead line
Fig. 1. Co-ordinate system and configuration.
The overhead line can further be modelled electrically as
shown in Fig. 2.
Fig. 2. EMTP model of overhead line.
where
)
-
(t
i
Z
+)
-
(t
U
+
(t)
U
=
(t)
U B
B
indA
rA
t
t×¢
)
-
(t
i
Z
+)
-
(t
U
+
(t)
U
=
(t)
U A
A
indB
rB
t
t×¢ (2)
Z’ is the overhead line’s characteristic impedance and t is
the travelling time (t=L/c where L is the line length and c
is the speed of light).
Several phas are handled straightforward by using matrix expressions in eq. (2).
The impact of the incident field from a return stroke is embedded in the terms U indA and U indB , called the inducing voltages . The two terms become equivalent source terms. Loss effects on the electrical field due to propagation above a lossy ground can be taken into account by modifying the terms [8]. The model in Fig. 2 is implemented in ATP-EMTP without modifying the source code as in [12].
When the ground is assumed lossless the inducing voltage at terminal A can be written:
()()()ò
ò×---×÷ø
öçèæ--=
×z
B
z A z A
x B
x A x i B i A xA indA dz
t z y x E t z y x E dx c x x t z y x E )-(t U - (t)U + (t)U 2 = (t)U 0
,,,,,,,,,t t (3)
where E x (x,y,z,t) and E z (x,y,z,t) are the incident horizontal and vertical field, respectively and U x and U i are the induced voltage terms t up by the fields. The height of the overhead line, z, is further assumed to be much less than the distance y and the length of the overhead line L, and the vertical field is thus assumed to be constant between the ground and the line. The horizontal electrical field in the integrand in (3) is zero when x x x y c t A -++<
×22 which is the distance from the
lightning stroke to the point of obrvation, (x,y,0) and
further along the line to the endpoint A.清炖黑鱼
Using the TL model for the lightning channel, the inducing voltage at terminal A is:
()()()()
从理论到实践[]
(]
()
ïïî
ïïíì¥ÎÎÎ---+×=,for ,for ,0for ,,1
,0,)(0
B B A A A
A
A A
indA t t t t t t t c
L t L f t f t f t U t U x
x x x (4)
where
()()()[]V x t c y x t c z I t x u 2
220060,-××+-×××××=
b b (5)
()()
()()()
222221,x y t v x t c x t x f +×-+×-××+=
b b (6)
c v an
d ,22
22=++=+=b c
L c y x t y x t B B A A (7) v and I 0 are the lightning current wave velocity and step amplitude, respectively.
For times less than t B the expression in (4) gives the same result as Rusck’s model [1, eq. (105)] (called U 1) even though a different coupling model is ud and only the field from the return stroke is considered in (4). Rusck’s
expression is equivalent since he assumes an infinitely long line, which caus the static contribution from the charged leader to vanish.
The inducing voltage at terminal B is found by substituting -x B for x A in (4).
A model called INDUS2 using the expressions in eq. (2) to calculate the LIOs in a 2-pha overhead line, is written in the MODELS language and shown in appendix 1. Combining this model with Fig. 2, assuming two overhead line conductors and using the graphical pre-processor ATPDraw, gives an equivalent circuit shown in Fig. 3. The overhead line is modelled with a lumped resistive matrix, reprenting the lossless characteristic impedances. The
model INDUS2 measures the terminal voltage of two phas at each end of the line and calculates the inducing voltage terms and reflections at the line terminals. The current into the line terminal is calculated internally, and (2) is efficiently reformulated into:
)(t U )-(t U + (t)U = (t)U j rB j B indA j rA t t --×2 )(t U )-(t U + (t)U = (t)U j rA j A indB j
rB t t --×2
(8)
where j is the pha number (1..2).
The timestep between measured and calculated quantities must be taken into account, as shown in the appendix.
In this paper the same inducing voltage is assumed in all line conductors, but this could easily be extended since the inducing voltage is proportional to the line height, z.
The circuit shown in Fig. 3 can be connected to any component in the ATP, and veral line gments are allowed.
Fig. 3. Equivalent circuit of 2-pha overhead line, excited by
nearby lightning. Modelled in ATPDraw.
Below, an example of the electric circuit part of the two-pha model in Fig. 3 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 [10].
III. CALCULATIONS
关于雪Fig. 4 shows the configuration of the low-voltage network ud as an example in this paper. The network consists 5 overhead line gment, in an H-shape. Four lines are 250 m long and the one line in the middle is 500 m. The height of the line is in all cas 6 m. The same voltage is assumed to be induced in all three pha-conductors and they are
reprented by one conductor with characteristic impedance 300 W. The mutual impedance between the equivalent pha conductor and the neutral conductor is 200 W, and the characteristic impedance of the neutral is 500 W. Loads are attached at each point (1-6). Measurements on distrib
ution transformers and low voltage power installations (LVPI) networks and developed models in the frequency range 10 kHz to 1 MHz are prented in [13, 14]. The simplified results for LVPI networks are summarid in tab. 1. In point 1 a 3-leg distribution transformer is modelled with a zero quence inductance of Z T = 10 m H. In point 2-6 low-voltage power installations (LVPI) are modelled with a simple reprentation according to tab. 1 and fig. 4. The transformer has grounding impedance of 5 W, while the LVPI networks have 50 W each.
The configuration investigated in this paper is a simplification with the purpo to illustrate the LIO level in low-voltage systems, and how the lightning parameters and loads or terminations influence it.
Fig. 4. Low-voltage system configuration. Overhead line system, and electrical load at the bottom (left: loads, right: transformer). Tab. 1. Models of LVPI-networks ud in the calculations, Z LVPI. Type
small medium large
TN
10 uH
5 uH 2 uH
IT
10 uH50 nF 5 uH100 nF 2 uH 200 nF Fig. 5 shows how the LVPI network loads are connected to the overhead line in the IT- and TN-system. In Figs. 8-11 the voltage across the LVPI loads, Uj, are denoted Pj-Nj with j=1..6. In Fig. 12-13 the maximum of this voltage is shown.
+
Uj
-
Node j
+
Uj
-
Node j
(left) and the TN-system (right).
The metal oxide arrester ud in the investigation has a rated voltage of 440 V and an energy capacity of 650 J. The current-voltage characteristic ud by ATP [9] is shown in Fig. 6, with the data for a standard 8/20 µs impul shown as circles.
江西茶
0.20.40.60.81 1.2 1.4 1.6 1.8
0.000001
0.00001
0.0001
0.001
0.01
0.1
1
10
100
u [kV]
i [kA]
Calculated
Data
Fig. 6. Current-voltage characteristic for MOV.
The Heidler model is ud to reprent the lightning current in all the calculations with front time constant t1=2 m s, decay constant t2=50 m s and slope parameter m=5. The amplitude is 30 kA and the time to crest is just above 4 m s. Eq. (1) is ud to take this current shape into account. The velocity of the lightning current in the lightning channel according to the TL model is v=1.1e8 m/s.
Fig. 7 shows how this system is modelled in ATPDraw, using the developed model. Tab. 2 shows the geometry of the system and the ttings of the three co-ordinates (Y, XA, XB) for each of the five line gments.
Tab. 2. G eometry of the five line gments. Ref. Fig. 7.
Line length L=XA-XB, XA>XB.
Lightning location: Angle a=90º and distance r0=100 m. Line Y XA XB
1-2 100 250 -250 1-3 250 100 -150 1-4 250 350 100 2-5 250 150 -100 2-6 250 -100 -350
Fig. 7. System model in ATPDraw, using MODELS.
Fig. 8 shows the lightning induced voltages across loads at
point 1, 2, and 5, with no arresters installed in the system. The loads (LVPI networks) are of TN type small (inductance of 10 m H). Fig. 9 shows the same simulation, but with arresters installed at all loads 2-6. The voltages at the other points 3, 4 and 6 are always lower then at point 5.
2
4
6810
Time [us]
-2000
-1500-1000-500
05001000
150020002500V o l t a g e [V ]
Fig. 8. LIO across loads. TN-system. No arresters. Lightning
location: Angle a =90º and distance r 0=100m.
The voltage is highest across the transformer equivalent, reaching just above 2 kV, and this voltage is unaffected by arresters in the system. At load position 2, the voltage is limited by the surge arreste
r, and at the other positions 3-6 the voltage is below the protective level of the arrester. The voltage across the transformer in Fig. 8 is higher than across load in position 2 due to the lower grounding impedance of the transformer.
Figs. 10 and 11 shows the same calculation as Figs. 8-9, but now for an IT system (without a neutral conductor). A
capacitance of 50 nF is added in ries with all loading inductances, reprenting a typical small IT system installation. The connection to the neutral conductor is removed, and the coupling between the neutral conductor and the pha conductors is t to zero. The transformer is still modelled as an inductance of 10 m H, since its low-voltage neutral is assumed grounded. An ungrounded neutral will give even larger voltages.
2
4
6810
Time [us]
-2000
-1500-1000-500
05001000
150020002500V o l t a g e [V ]
Fig. 9. LIO across loads. TN-system. With arresters. Lightning
location: Angle a =90º and distance r 0=100m.
2
4
6810
Time [us]
-3000
-2000-100001000
20003000400050006000V o l t a g e [V ]
Fig. 10. LIO across loads. IT-system. No arresters. Lightning
location: Angle a =90º and distance r 0=100m.
2
4
6
8
10
Time [us]
-2500
-2000-1500-1000-500
050010001500200025003000
V o l t a g e [V ]
Fig. 11. LIO across loads. IT-system. With arresters. Lightning
location: Angle a =90º and distance r 0=100m.
A
Figs. 12 and 13 show the maximum induced voltage across the load (Z T and Z LVPI ) in the network in Fig. 4 (with a lightning distance r 0=500) as a function of angle a . All the
load types shown in tab. 1 are investigated. The transformer in point 1 is in all cas modelled with Z T = 10 m H. No arresters are installed in the system. The markers
on the curves indicate where in the network (point 1-6) the maximum voltage occurs.
U [kV]MAX
a [deg] Fig. 12. Maximum LIO in TN system, dependency on LVPI
model. Distance r 0=500 m.
U [kV]
MAX
a [deg] Fig. 13. Maximum LIO in IT system, dependency on LVPI如何提高服务质量
model. Distance r 0=500 m.
Figures 12 and 13 show that the amplitude of induced
voltages in a TN and IT system strongly dependent of the
nature and size of the LVPI. In general larger installations
result in lower overvoltages. The voltages induced in an
IT-system are substantially higher than in a TN-system.
IV. DISCUSSION
The MODELS language makes it possible to interface
induced voltages calculations with the ATP, and thus to
investigate induced overvoltages in a larger and practical
system. The following main simplifications are applied:
· The transmission line model is ud for the lightning
channel. This is reasonable for the first few
microconds.
·
The same voltage is assumed induced in all pha
conductors and only the common mode system is
studied. This is also reasonable for the first few
microconds, but is doubtful for strong unsymmetry in load configurations.
· Lossy ground effects are ignored. It is reasonable to assume the overhead line to be lossless for lengths shorter than 1 km, but ignoring the propagation effect
on the electromagnetic fields is more doubtful. Ignoring the lossy ground effects for nearby lightning (<1 km) could be reasonable.
· The lightning-induced voltage has a frequency content below 500 kHz. This justifies the simple load models, and the assumption is reasonable.
· The static voltage from the cloud and lightning leader is ignored, and this is reasonable if the power system has a connection to ground, and the distance from the stroke location to the line is above ca. 100 m.
Completely analytical expressions are established, and the calculation speed is limited by the convolution integrals
performed in MODELS, taking the lightning current shape
into account. The development of a type94 component
would simplify the equivalent circuit further.
A main motivation factor initiating this work was the
discussion regarding IT- versus TN- systems and the assumed vulnerability of IT-system to lightning-induced
effects. The IT-system is the common standard in Norway
and overhead lines, without a ground or neutral wire, are in frequent u, particularly in rural areas. The calculations show that the IT-system in general results in much higher voltage across loads (and in general pha-to-ground) than
the TN-system. This is mainly caud by:
· Lower terminating impedances of overhead lines in the TN-system, due to electrical loads. · The prence of a ground-wire in the overhead line in TN-systems. · The grounding of the distribution transformer’s LV-neutral in TN-systems.
The capacitive behaviour of IT-system power installations
up to at least 100 kHz is the most important factor. In
general larger installations result in lower overvoltages.
V. CONCLUSION
An analytical model for calculation of lightning-induced
voltages in overhead lines has been developed bad on a
few specific assumptions. In order to be able to calculate
the voltages in a complex power system the model is
implemented in the MODELS language of ATP-EMTP. In
this way no external calculation program is needed, and人事主管
multiple overhead lines can be included with no additional
limits than given by the ATP program. The calculation
speed is mostly limited by the convolution integral
required to take a specific lightning current shape at
ground into account. Development of a type 94 component
would further simplify the connection between the induced
voltage calculation and the electric circuit.
Bad on the listed assumptions, the developed model is
believed to be accurate for the first few microconds, or
up to the first peak in the lightning-induced voltage. Most
likely, the model gives the best results for a distance between lightning and an overhead lines in the medium-clo range of 100-1000 m.
