Finite element modelling for electric field distribution around positive streamers in oil

equation shown in (3) can be 
established from equation (1) and equation (2) 
s
(a)
z
r

Main channel
Surrounding 
channels
 
(b)
Plane electrode
Point electrode
Hollow cones
Main channel
m= 0.1mm
Channel tip field Et
Rotational
symmetry
s
t = 0.05 mm
Nguyen Van Dung, Le Vinh Truong 
62 
 r
V

0
2 
 (3) 
As the charge ρ = 0, the Poisson’s equation can be converted into Laplace’s equation as follows 
 0
2  V (4) 
For 2D axial symmetry problem, the distribution of potential is dependent upon the 
coordinate. Thus, the equation (4) is rewritten as follows 
 0
12 








 
z
V
zr
V
r
rr
V (5) 
The FEM method reported in references 8-9 is used to solve equation (5) as all boundary 
conditions are known. An open domain is applied to the point-plane gap problem, i.e. the electric 
field is zero at infinity. For simplification of simulation, the outermost boundaries are at infinity. 
Due to these assumptions, the conditions of boundary are set as bellows. 
V = Vapplied on the point electrode (high voltage); V = 0 on the plane electrode (ground); nD = 0 
on outermost boundaries 
Figure 10 shows the typical mesh of one case of simulation with elements of triangles. The 
density of elements is higher while its size is smaller for regions around electrodes and streamer 
branches. Similar results were observed for other cases. 
 Figure 10. The mesh of the model for simulation of shielding effect ( = 75 mm, = 5o). 
4. RESULTS AND DISCUSSION 
4.1. The electric field distribution in the electrode gap 
Figure 11 shows the distribution of electric field around streamers at 210 kV derived from 
simulation results with mesh parameters shown in Table 1. The electric field reaches the 
maximum value (Et) at the surface of streamer envelope in the direction of gap axis, and 
decreases with an increase in distance away from a streamer region. From Fig. 11, Et is 
Finite Element Modelling for Electric Field Distribution around Positive Streamers in Oil 
63 
determined and plotted with increasing streamer extension as presented in Fig. 12. It is observed 
that Et reduces gradually with increasing streamer length due to an increase in diameter of 
streamer envelope until it reaches the minimum value at about 60% of gap crossing. Then, Et 
increases again because of the approaching of streamers to the plane electrode. Similar results 
were reported in the 2
nd
 mode and 3
rd
 mode streamers by other researchers 1, 2, 4. Apparently, 
Et obtains the value of about 8.3 MV/cm, which is higher than the electric field of about 7 
MV/cm at the tip of the point electrode, for the 4
th
 mode streamers (Fig. 4a) and drops to about 
0.16 MV/cm for the 3
rd
 mode streamers (Fig. 4b) and reduces to the minimum value of 0.1 
MV/cm for the 2
nd
 mode streamers (Fig. 4c). Fig. 12 also shows the propagation velocity of 
streamers exhibited in Fig. 4. It seems that there is a correlation between the velocity and the 
electric field Et during streamer propagation. Streamers with high electric field at their tips 
propagate with high velocity, and vice versa. With the electric field of about 8.3 MV/cm at their 
tips, streamer velocity reaches the value of about 45 km/s. However, when the electric field at 
streamer tips drops to approximately 0.1-0.2 MV/cm, the streamer velocity reduces to 2-4 km/s. 
Therefore, it is inferred that if the electric field at streamer tips exceeds a value of about 8.3 
MV/cm, streamers will travel at high speed of approximately 45 km/s over the entire electrode 
gap. 
 Figure 11. Plots of electric field. The letter symbols referred to streamer images shown in Fig. 4. 
Table 1. Mesh statistics (210 kV). 
No Parameters 
Value 
(Fig. 11a) 
Value 
(Fig. 11b) 
Value 
(Fig. 11c) 
Value 
(Fig 11d) 
1 Number of elements 5509 5036 4970 4903 
2 Minimum element quality 0.07562 0.07562 0.07562 0.07562 
3 Average element quality 0.821 0.8183 0.8159 0.8165 
4 Element area ratio 4.325 10-6 4.325 10-6 4.325 10-6 4.325 10-6 
Figure 13 shows the surface plots of the electric field for the 4
th
 mode streamers at 270 kV. 
Again, the electric field gets the maximum value at the streamer tips. From these plots, the 
Max: 9.2 MV/cm Max: 4.4 10-1 MV/cm Max: 3.3 10-1 MV/cm Max: 2.6 10-1 MV/cm
a b c d
c = 23 mm
s = 50 mm
s = 65 mm
m = 0.2 mm
Nguyen Van Dung, Le Vinh Truong 
64 
maximum electric field Et was obtained, and Et versus streamer growth is shown in Fig. 14. It is 
observed that Et gradually increases from 8 MV/cm to 25.5 MV/cm when streamers propagate 
across the electrode gap distance with the speed of about 100 km/s at 270 kV. The growth of 
streamer channels leads to a phenomenon that resembles the extension of the point electrode 
resulting in electrode gap reduction and thus an increase in Et. The high magnitude of Et (8 - 
25.5 MV/cm) could be used to explain why the 4
th
 mode streamers (Fig. 5) propagate with very 
high velocity ( 100 km/s). Compared to the value of Et in Fig.12, it was observed that if Et 
increase to the value of about 10 MV/cm after initiation, streamers will keep travel with high 
velocity. Otherwise, streamers will propagate with a decrease in velocity. Thus, the critical value 
of approximately 10 MV/cm can be considered as a threshold value to convert low mode 
streamers into fast mode streamers. However, it is aware that this threshold value is estimated 
without regard to the existence of charges surrounding the tips of branches. Thus, the real value 
of the threshold electric field at the channel tips could be lower close to the tips and could 
increase further away from the tips. The mesh parameters for FEM simulation of this case is 
presented in Table 2. 
 Figure 12. Velocity versus electric field at the channel tips of streamers at 210 kV. 
Figure 13. Distribution of the electric field around streamer channel tip at 270 kV. 
0,01
0,1
1
10
100
0,01
0,1
1
10
100
0 20 40 60 80 100
A
v
er
ag
e 
v
el
o
ci
ty
 (
k
m
/s
)
E
le
ct
ri
c 
fi
el
d
 (
M
V
/c
m
)
Gap crossing of streamers (%)
Electric field
Streamer velocity
a
b
c
d
Max: 2.6 10 MV/cm
l = 0 l = 30 mm l = 60 mm l = 78 mm
2.5
2
1.5
1
0.5
Finite Element Modelling for Electric Field Distribution around Positive Streamers in Oil 
65 
Figure 14. Channel tip field Et versus streamer growth at 270 kV. 
Table 2. Mesh statistics (270 kV). 
No Parameters 
Value 
(l = 0 mm) 
Value 
(l = 30 mm) 
Value 
(l = 60 mm) 
Value 
(l = 78 mm) 
1 Number of elements 5341 5478 2343 2126 
2 Minimum element quality 0.07562 0.07562 0.5061 0.561 
3 Average element quality 0.8220 0.8219 0.8323 0.8323 
4 Element area ratio 4.325 10-6 4.325 10-6 4.325 10-6 4.325 10-6 
4.2. The influence of the shielding effect on electric field at channel tips of streamers 
Figure 15 shows some typical simulation results for the spherical model (s = 75 mm). The 
modelling results shows that the maximum electric field (Et) was found at the main channel tip 
and edges of hollow cones. However, Et at the tip of main channel is much higher than that of 
edges of hollow cones. Outside the channel tip and cone edges, the electric field significantly 
reduces. From simulation results with varying the value of angle , Et is determined and plotted 
as shown in Fig. 16. It is found that Et significantly increases with less branching, i.e. higher 
angle between channels, and become saturated with of about 30o and 60o for streamer 
envelope diameter of 75 mm and 40 mm, respectively. This means that an increase in streamer 
branching raises the shielding effect resulting in lower Et and vice versa. The similar results are 
obtained between two cases. However, Et of the bigger diameter of streamer envelope (s = 75 
mm) with higher branching degree still higher than that of the smaller diameter (s = 40 mm) 
with lower degree of branching. This indicates that the influence of the shielding effect on Et 
possibly reduces as streamers approach the plane electrode. The mesh parameters for FEM 
simulation of this case is presented in Table 3. 
0.1
1
10
100
0 20 40 60 80 100
E
le
ct
ri
c 
fi
el
d
 E
t
(M
V
/c
m
)
Gap crossing of streamers (%)
Exxsol oil - 270 kV
Nguyen Van Dung, Le Vinh Truong 
66 
Figure 15. The distribution of electric field for s = 75 mm. 
Figure 16. The channel tip field versus angle between branches. 
Main channel
Hollow cones
Max: 3 10 MV/cm
 = 5
o
 = 60
o
Main channel
Finite Element Modelling for Electric Field Distribution around Positive Streamers in Oil 
67 
Table 3. Mesh statistics (the shielding effect). 
No Parameters 
 = 40 mm  = 75 mm 
 = 0o = 5o = 60o = 0o = 5o = 60o 
1 Number of elements 4955 13876 5791 4890 14679 5796 
2 Minimum element quality 0.07562 0.07562 0.07562 0.07562 0.07562 0.07562 
3 Average element quality 0.8174 0.803 0.8209 0.8161 0.8035 0.8161 
4 Element area ratio 4.33 10-6 2.86 10-11 1.91 10-6 4.33 10-6 2.93 10-11 4.33 10-6 
4.3. The relationship among electric field, branching and velocity 
From Fig. 4, Fig. 5, Fig. 12, Fig. 14 and Fig. 16, it is observed that the 2
nd
 mode streamers 
consisting of numerous filamentary branches propagate with the velocity of about 1-2 km/s, and 
the electric field at the channel tips of streamers is estimated to be about 0.13-0.19 MV/cm. It is 
also obtained that the 3
rd
 mode streamers propagating with velocity of about 4-10 km/s has the 
electric field at their tips of about 0.2 MV/cm. The 4
th
 mode streamers with few branches travel 
with velocity of 50 km/s-100 km/s and reach the estimated field at the streamer tips of about 8-
25.5 MV/cm. This indicates that more branching, which is manifested with high number of 
branches, is associated with low velocity (1-2 km/s) and low electric field ( 0.2 MV/cm) at the 
streamer channel tips and vice versa. Therefore, the relationship between branching and velocity 
of positive streamers is suggested as follows. Streamers initiating with the speed of about 1-2 
km/s allow the development of branches, i.e. more branching. Due to branching, the 
macroscopic field of streamers becomes lower leading to a reduction in streamer velocity. By 
contrast, when streamers start with the high velocity ( 50 km/s), the chance for streamer 
branches to develop is low. Therefore, the electric field in front of the dominating branches 
raises greatly, which further increases the speed of streamers. The relationship among velocity, 
branching and electric field is summarized as shown in Fig. 17. This suggested hypothesis is also 
supported by experimental results that streamer propagation across the electrode gap was 
observed to be controlled by the macroscopic electric field of streamers 5, and guiding tubes 
that suppressed branching accelerated streamers 4, 10. 
Figure 17. The diagram describing the correlation between branching and velocity of streamers. 
Low 
velocity
More 
branching
High 
velocity
Less 
branching
Reduced 
macroscopic/
microscopic 
field
Increased 
macroscopic/
microscopic 
field
High 
electric 
field
(10 
MV/cm)
Streamers start 
at the point 
electrode
Terminating 
at the plane 
electrode 
Yes
No
Nguyen Van Dung, Le Vinh Truong 
68 
4. CONCLUSIONS 
Simulation of electric field distribution for positive streamers during propagation and the 
influence of the shielding effect on the electric field were performed. The simulation results 
show that the electric field reduces with streamer extension and reaches the minimum value at 
the position of about 60 % of the electrode gap before increases again due to streamer proximity 
to the plane electrode. The channel tip field of streamers at the 2
nd
 mode is determined to be 
about 0.1 MV/cm while 8.3 MV/cm was received for the 4
th
 mode streamers. It was also 
observed that the shielding effect formed by streamer branching greatly reduces the electric field 
at streamer channel tips. The shielding effect reduces with increasing the angle between 
channels and has a tendency of saturation at the angle of about 60o and 30o for 40 mm and 75 
mm of diameters of streamer envelope, respectively. The hypothesis on the relationship among 
electric field, velocity and branching of streamers is proposed as follows. If starting with high 
electric field at the tips (10 MV/cm), streamers will propagate with very high velocity which 
results in less branching and thus high electric field (10 MV/cm) which further raises the 
velocity, and vice versa. However, the accuracy of the hypothesis should be further checked with 
simulation results from higher applied voltage, e.g. 540 kV, in next study. 
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