Proceedings of the 9th WSEAS/IASME International Conference on ELECTRIC POWER SYSTEMS, HIGH VOLTAGES, ELECTRIC MACHINES
S. MAXIMOV, V. VENEGAS, J.L. GUARDADO, E. MELGOZA
Department of Electrical Engineering
Instituto Tecnológico de Morelia
Av. Tecnológico 1500, Col. Lomas de Santiaguito, C.P. 58120
MORELIA, MEXICO
sgmaximov@yahoo.com.mx
Abstract: In this work, a new method for calculating the parameters of electric arc model for SF6 power circuit
breaker is proposed. The methodology consists in the optimization of a theoretical function with respect to a
group of model parameters in order to achieve a better approximation to a set of experimental data. The
theoretical function is an asymptotic solution of the equation for the electric arc model obtained in a selected
period of time. The developed method is applied to calculate the parameters of an arc model previously
published in the literature. By using the variation of the cooling power, an improved electric arc model is also
obtained including its parameters. The new model has a very compact form and exhibit a good correlation
between voltage curves measured and calculated.
Key Words: Black box modeling, Mayr and Cassie models, Optimization, SF6 power circuit breaker.
Recently, new black box models for a SF6 circuit
breaker have been proposed (250 kV/ 50kA / 50Hz /
single:unit SF6 puffer circuit breaker) [1, 10]. In
reference [1], it is considered a linear relation
between the cooling power and the entry power, i.e.,
P0 + P1ui . The corresponding differential equation is of
the form:
The performance of high:voltage circuit breakers
during current interruption has been a topic of interest
for many years. A successful operation implies that
the circuit breaker is capable of interrupting currents
in a short period of time. In order to assess circuit
breaker capability during current interruption a test
program is required. Based on these tests, many
electric arc models have been proposed [1:10].
The developed models provide additional
information on the behavior of the electric arc under
different power system operating conditions. This
reduces test requirements in power breakers with
significant time and economic savings.
At present time, black box models are commonly
used to describe the interaction of the arc with the
electric network during a current interruption process.
Usually, black box models are modifications of the
models proposed by Cassie and Mayr [11,12], which
provide a qualitative description of the phenomena in
the low and high current regions respectively. These
models include a differential equation that depends
on a set of parameters x1 , x2 ,...xk which should be
obtained from experimental data. By selecting a
suitable set of arc parameters it is possible to obtain a
better approximation to the arc dynamics described
by experimental data. However, increasing the
number of parameters can lead to difficulties for
calculating their magnitudes. Therefore the number of
parameters should be reasonable and sufficient to
predict successfully the current interruption
capabilities of power circuit breaker.
ISSN: 1790-5117

d ln g 1 
ui
=
− 1

dt
τ  max(U arc i , P0 + P1ui ) 
(1)
where:
g
u
i
P0
P1
τ
U arc
arc conductance
arc voltage
arc current
cooling power constant in W
cooling constant
time constant
constant arc voltage in the high current area.
From Fig. 1, despite the fact that for electrical
currents above the critical current icr = 103 A , i. e. the
current for which the sudden voltage drop occurs, the
voltage curves calculated by means of equation (1)
show a significant difference with experimental
oscillograms, equation (1) has been considered in [1]
appropriate to describe the arc dynamics before
current zero. To obtain this conclusion, in the
computer simulations the authors took arbitrarily an
initial voltage almost 550V below its real value.
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ISBN: 978-960-474-130-4
Proceedings of the 9th WSEAS/IASME International Conference on ELECTRIC POWER SYSTEMS, HIGH VOLTAGES, ELECTRIC MACHINES
In [10], an improved arc model that successfully
describes the dynamic arc behavior in the high and
low current regions before current zero is proposed.
However, this model is difficult to implement
computationally.
In the first part of this work the aim is to develop
an improved arc model for SF6 power circuit breakers
capable of predicting the voltage behavior with a
great accuracy for a wide range of currents. This
model should be mathematically simple and contains
only the minimum necessary number of arc
parameters.
The second part addresses the development of a
methodology for the calculation of the parameters for
the model taking as a reference experimental data.
This is a very important stage in the modeling process
because the accuracy of the model is related with the
accuracy for determining these parameters.

d ln g 1 
ui
= 
− 1
dt
τ  P0 + P1ui 
In this paper we take equation (2) as the starting point
to develop an improved arc model.
The improved model should describe the irregular
behavior of the electric arc, which consists in the
sudden voltage drop as shown in Fig. 1. In reference
[1] it was mentioned that the “exact” cooling power
curve shows a small deviation from the linear
behavior around the electric power input 1.5 × 106W .
Such a non:linearity can be the cause of the irregular
behavior of the arc voltage, i. e., the cooling power
should be taken in the form P0 + (P1 + δP1 )ui , where
the small variation δP1 of the parameter P1 is
dependent on the arc voltage and current. As a result,
the arc model becomes:
#
d ln g
dt
=
δP1

1
ui

− 1
τ  P0 + (P1 + δP1 )ui 
(3)
! "
The variation of the left:hand side of equation (2),
caused by the variation of the parameter P1 , has the
form:
δ
d ln g d ln g
=
dt
dt
−
δP1
=−
d ln g
dt
=
reg
1 du
u dt
−
reg
1 du
u dt δP1
δP1
τ  P0
1

 P0
 + P1  + (P1 + δP1 )
ui
ui



Fig.1. Measured and calculated voltage and
current with model (1) (© [2000] IEEE).
(4)
where the derivative d ln g dt reg corresponds to the
The first problem is solved by varying one of the
parameters in model (1), which is taken as the
starting point for the analysis. The second one is
solved by obtaining an asymptotic solution for the
voltage dynamics in a certain segment of time [ta , tb ] ,
with the subsequent optimizing of this solution with
respect to the parameters of the model. As a result,
the developed model describes the electric arc
dynamics successfully, giving an excellent
correlation between theoretical and experimental
voltage curves for a wide range of currents.
!
(2)
regular electric arc behavior given by model (2). The
maximum contribution of the small variation δP1 to
the voltage dynamics is reached if the right:hand side
of equation (4) is the maximum, i. e.,
1
1
≈
 P0
 P
 P1 (P1 + δP1 )
 + P1  0 + (P1 + δP1 )
ui
ui



Substituting this result into equation (4) yields:
 1 du
1 du
−
 u dt δP u dt
1

τP12 
" #
δP1 ≈
Model (1) can be used to obtain an improved arc
model. Thus, for currents i < icr the cooling power is
a linear function of the electric power input and
equation (1) takes the form:
 1 du
1 + τP1 
 u dt

reg



reg 
1 du 
−
u dt δP1 
(5)
To obtain the form of the variation δP1 as a
ISSN: 1790-5117
134
ISBN: 978-960-474-130-4
Proceedings of the 9th WSEAS/IASME International Conference on ELECTRIC POWER SYSTEMS, HIGH VOLTAGES, ELECTRIC MACHINES
du di δP = u σ 2π according to expression (6). As
function of arc voltage and current we should model
the sudden voltage drop. In this case the voltage is
considered as a function of the current, i. e., u = u (i ) .
Then:
1
a result we find:
σ =−
du di du
=
dt dt di
1
where the sudden voltage drop occurs, behaves like
the Dirac delta:function [10,14,15]. Then, the
derivative du di δP can be approximated by the





d ln g 1 
iu
= 
− 1
2
τ
dt
P1iu


 P0 +
− P3 (i − i cr )2
P2 e
+u


1
normal distribution:
du
u
e
=
di δP1 σ 2π
2σ 2
(6)
where u is the magnitude of the voltage drop which
can be obtained directly from the experimental data
as shown in Fig. 2.
The derivative
1 du
u dt
(8)
Substituting (8) into (7), we finally find the form
of the variation δP1 as a function of the arc voltage
and current. As a result the model equation for this
SF6 power circuit breaker takes the form:
The derivative du di δP around the current icr ,
(i −icr )2
−
u di
tan ψ
2π dt
(9)
This mathematical model has three additional
parameters: P2 , P3 and icr . The current icr can be
obtained directly from Fig. 1. The parameters P2 and
P3 can be calculated according to the formulations:
is much smaller then
reg
P2 = τP1 cotψ
1 du
and therefore can be neglected. Substituting
u dt δP1
(6) into (5), it can be obtained:
$
P3 =
π
 di

u tanψ 

dt


2
(10).
% "
The proposed methodology for parameters
calculation consists in the optimization of a
theoretical function u (t , P0 , P1 , τ ) with respect to the
set of parameters of the model P0 , P1 and τ , with the
purpose of obtaining the best approach to a set of
experimental values u1 , u 2 ,..., u n measured in the
corresponding moments of time
t1 , t 2 ,..., t n .
Nevertheless, the theoretical voltage, i. e., an analytic
solution of equation (9) with respect to the arc
voltage, is required.
Due to the complexity of equation (9), the
determination of analytic results has to be carried out
for certain ranges of time in which some terms of the
equation can be neglected. Firstly, the variation of the
cooling power obtained above is significant only for
the small region near the current icr . Outside this
region, equation (9) approximately coincides with
equation (2). Therefore, equation (9) is solved in a
time segment [t a , tb ] which does not contain the
voltage drop.
Equation (2) can be represented as a nonlinear
differential equation with respect to the electric
Fig. 2. Parameters Hu and Ψ in the measured voltage.
τP12
δP1 ≈
di
u
dt
(i − icr )2
uσ 2π e
2σ 2
− τP1
(7)
di
u
dt
To find the parameter σ it can be notice that in the
critical point i = icr , du di δP di dt = − cotψ , where
1
the angle ψ can be obtained from the voltage drop as
shown in Fig. 2 and in the critical point
ISSN: 1790-5117
135
ISBN: 978-960-474-130-4
Proceedings of the 9th WSEAS/IASME International Conference on ELECTRIC POWER SYSTEMS, HIGH VOLTAGES, ELECTRIC MACHINES
for an arbitrary current i(t). For the current shown in
Fig. 1 it is necessary to substitute ia − α (t − t a ) into
(14). Finally we obtain the following approximate
solution:
power input w :
dw
+ q(t )w = Q (w(t ) )
dt
(11)
where:

 2 di
1 1
q(t ) ≡  − 1 −
τ  P1
 i dt
Q(w) =
u (t , P0 , P1 ,τ ) = (ia − α (t − t a )) ⋅ e
P0 
P Pw 
1 − 0 1

2 
P1 τ  1 + P0 P1w 
ia  1 
 ξ  1 −1
 −1 
 e τ  P1  e ατ  P1 

×
+
ατ
 ia − αξ

and the current i (t ) is a linear function of time, as
shown in Fig. 1, i. e., di dt = −α . By selecting the
circuit impedance higher than the arc resistance, it is
always possible to reach the arc current behavior
being linear or almost linear, Fig. 1. Equation (11)
can be transformed into the following integral
equation:
w(t ) =
ua 2
i (t )
ia
+ i 2 (t )e
∫
ta
e
t1 −ta  1 
 −1 
τ  P1 
i 2 (t1 )
P0
P1w(t )
P12τ
t −t a
0







∞
∫
e− x
dx
x
where the principal value of the integral is taken.
Q(w(t1 ) )dt1
&
"
'
The least squares method can be used for the
optimization of function (15). The method consists in
the minimization of the function:
L (P0 , P1 ,τ ) =
(13)
n
∑ [u (t
k , P0 , P1 ,τ
) − uk ] 2
(16)
k =1
The minimization of the function (16) can be
significantly simplified if one of the unknown
parameters is expressed through other parameters.
The necessary information to eliminate one of the
parameters can be obtained from the experimental
results shown in Fig. 1. As a reference point it is
possible to take the local maximum of the voltage
curve (t m , um ) , where t m is the time moment in which
the voltage arrives to its maximum um . Substituting
du dt t = 0 into equation (2) we can express the
is satisfied, i. e., the electrical power input w(t) is
high enough. This assumption should be verified
later. Then we can write:
P0
P0
P12ατ

1
 
  ia − αξ  1
 − 1  
 − 1 Ei −
ατ  P1  
 P1  

Ei(z ) = −
(12)
< ε << 1
+
Ei( z ) is the exponential integral function:
Let also the time segment [t a , tb ] be such that the
inequality
Q (w ) =
 ia

−z
t −t a  1  t
 −1
τ  P1 
0<
t −t a  1 
 −1  
τ  P1  u a

(15)
t −t  1 
− a  −1 
τ  P1 
e
−
−
(1 + O(ε ) )
As a result the following asymptotic solution of
equation (12) can be obtained:
m
u (t ) =
ua
i (t )
ia
parameter P0 through the parameters P1 and τ :
t −t  1 
− a  −1
τ  P1 
e
+ i (t )
t −t  1 
− a  −1
τ  P1 
e
P0
P12τ
t
(1 + O(ε ))∫ e
ta
i 2 (t1 )

d ln i
P0 ( P1 ,τ ) = imu m 1 − P1 + τ

dt t = t m

(14)
t1 −ta  1 
 −1 
τ  P1 
dt1

 i0 − αt m
= um (i0 − αt m ) ⋅ 
− P1 

 i0 − α (τ + t m )
where ia and ua are the current and voltage at the
moment of time t a . Equation (13) has been obtained
ISSN: 1790-5117




(17)
By substitution of equation (17) into (15) we obtain
136
ISBN: 978-960-474-130-4
Proceedings of the 9th WSEAS/IASME International Conference on ELECTRIC POWER SYSTEMS, HIGH VOLTAGES, ELECTRIC MACHINES
the analytic solution u (t , P1 , τ ) as function of two
parameters, P1 and τ . The solution u (t , P1 , τ ) should
be optimized, i. e., the following function should be
minimized:
L(P1 ,τ ) =
n
∑ [u(t , P ,τ ) − u ]
k
1
2
k
(18)
k =1
The minimization of the function (18) is based on
solving the following system of equations:
 ∂L(P1 ,τ )
=0
 ∂P
1

 ∂L(P1 ,τ ) = 0
 ∂τ
Fig.3. Minimum of the function L(P1 ,τ ) at the point τ = 0.06936 and
P1 = 0.999275 .
with respect to unknown variables P1 and τ . In Fig.
3 the minimum of the function L(P1 ,τ ) is shown. The
parameters involved in the function (15) are shown in
Table I.
The optimization is fulfilled with the use of the
Mathematica 5 software. The results of the
optimization is presented in Table II.
* (
The model (9) can be validated by means of two
ways. The first one considers the asymptotic solution
(15). This solution has been obtained under the
assumption of (13) in the segment [ta , tb ] where
t a = 42.7 -s and tb = 85-s . Fig. 4 shows two voltage
curves, the asymptotic solution (continuous line) and
the voltage calculated by means of model (9) under
the assumption of a linear behavior of the electric
current (dashed line). The parameters used in the
model are given in Table II. The comparison of both
curves shows clearly that the asymptotic solution
(15), the calculated voltage (9) and the measured
voltage (Fig. 1) are very well correlated in the time
range of interest [ta , tb ] . This is clear evidence that
the asymptotic solution (15) is obtained with a very
good accuracy, i. e., the assumption (3) is satisfied in
the time segment [ta , tb ] .
From Fig. 5, the second validation is made by
comparing the results of model (9) with the irregular
behavior of the measured voltage shown in Fig. 1 for
i > icr . It can be noticed that model (9) successfully
describes the sudden voltage drop at t = 40 -s .
Moreover, the calculated voltage in Fig. 5 has an
excellent correlation with the measured voltage
shown in Fig. 1 for all time ranges.
Fig. 6 shows the behavior of the cooling power
versus the electrical power input in the region of the
electric arc voltage drop. It can be noticed a small
deviation of the cooling power from the linear
behavior. Such a deviation from the linear behavior
can be observed in the corresponding experimental
Table I. Parameters obtained from measurements, Fig 1.
ta
tb
ua
ia
um
tm
α = − di dt
u
ψ
(
42.7
85.0
1171.87
946.0
2328.12
88.0
20
500
0.00259
)
-s
-s
V
A
V
-s
A -s
V
:
Table II. Parameters for the model (9).
(
icr
P2
P3
P0
P1
τ
ISSN: 1790-5117
103
26.788
4.6923 × 10:3
3413.35
0.999275
0.069365
)
A
V
A −2
W
:
-s
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ISBN: 978-960-474-130-4
Proceedings of the 9th WSEAS/IASME International Conference on ELECTRIC POWER SYSTEMS, HIGH VOLTAGES, ELECTRIC MACHINES
cooling power curve published in [1]. In this sense,
the obtained shape of the cooling power curve is in a
good agreement with experimental data.
experimental data. The above results give confidence
in the models developed and the approach used for
calculating arc parameters.
Fig. 4. Arc voltage calculated with the model (9), and the asymptotic
solution (13) in the time range [t a , tb ] .
Fig. 7. d ln g dt :curve calculated with the model (9)
In this paper a new method for obtaining the
parameters of electric arc for SF6 power circuit
breakers is presented. The methodology consists in
solving analytically the model equation and the
optimization of the solution with respect to a group of
parameters in order to achieve a better approximation
to a set of experimental data. Nevertheless, the
solution of the model equation is complicated,
especially because the arc behavior is closely related
with the electrical circuit. Therefore the solution can
be obtained for the regions where one or more terms
(or parameters) can be neglected and the equation can
be solved asymptotically. Such an asymptotic
solution is optimized.
The developed methodology is applied to a SF6
power circuit breaker. The asymptotic solution (15)
of the model equation (9) for the arc voltage is
obtained for a selected period of time [ta , tb ] . Then,
the asymptotic solution (15) is optimized and the arc
parameters calculated.
The model (9) is obtained by means of variation of
the constant P1 of the linear cooling power P0 + P1iu
introduced in reference [1]. In the proposed model
(9), the cooling power depends on the arc current and
voltage, as shown in the following equation:
Fig. 5. Arc voltage and current calculated with the model (9).
Fig. 6. Calculated cooling power versus electrical power input. A
small deviation from the linear behavior.
Fig. 7 shows the d ln g dt curve obtained with the
model (9). It can be noticed a sharp peak in the region
of the electric arc voltage drop. Such a curve shape
can be observed in the corresponding experimental
curve published in [1]. Unlike the model (1) (see
reference [1]), the d ln g dt curve calculated with
model (9) has a very good correlation with the
ISSN: 1790-5117
P(i, u ) = P0 +
P1iu 2
P2 e − P3 (i − icr ) + u
2
It has been shown in Fig. 5 that this approach to the
cooling power is in a good agreement with the
experimental data published in [1].
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ISBN: 978-960-474-130-4
Proceedings of the 9th WSEAS/IASME International Conference on ELECTRIC POWER SYSTEMS, HIGH VOLTAGES, ELECTRIC MACHINES
The new model has a very compact form and the
results show a good correlation between the
theoretical voltage curves and the experimental
oscillograms. The model given by (9) describes in a
very efficient way the irregular behavior of the
voltage when the current reaches the value icr = 1 kA .
In comparison with the model obtained in [10] the
model (9) has only two additional parameters, P2 and
P3 , equation (9) is quite suitable for simulation.
References:
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
[10]
[11]
[12]
[13]
[14]
[15]
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ISSN: 1790-5117
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ISBN: 978-960-474-130-4