Estimation For The Weibull Power Law Parameters In The Step Up ...

Engineering Letter, 17:2, EL_17_2_10
____________________________________________________________________________________
Estimation for the Weibull Power Law
Parameters in the Step-up Voltage Test
Kotaro Tsuru
Hideo Hirose ∗
Abstract—In assessing the insulation withstand
application test is crucial for insulation design. Although
level of the electric apparatus,
the step-up test
the test method is rather simple, we have still many un-
method is used.
However, we have still many un-
known matters; e.g., how we deal with the test results.
known matters regarding the treatment of the re-
Assuming, e.g., that the voltage level imposed to the ap-
sults. In this paper, we assume that the underlying
paratus is continuously increased. Then, we can obtain
probability distribution of failure time with a con-
the breakdown voltage results; using the basic statistics
stant voltage level follows a Weibull distribution and
we may estimate the voltage blow which the breakdown
that an inverse power law relationship between the
probability is speciﬁed to be low, e.g., µ − 3σ is a simple
mean lifetime and the imposed voltage holds; that
is, the Weibull power law is assumed.
Under such
guess to that (see [4]) where µ and σ are the sample mean
a condition, we ﬁrst investigate whether we can es-
and the sample standard deviation. However, we know
timate the unknown Weibull power law parameters
that values µ and σ are aﬀected by the voltage increasing
using the breakdown voltage results obtained from
rate per second; the faster the rate, the higher the value
the step-up test. We assume two models: one is the
of µ. Using the step-up test method (Figure 1), a similar
independence model, and the other is the cumulative
tendency will be observed. Unlike the breakdown time
exposure model.
When we use the maximum like-
observation with same constant voltage stress imposing,
lihood estimation (MLE) method, the estimation is
we aware how diﬃcult to assess the insulation withstand
well performed in both the models. On the contrary,
level by using the step-up voltage test [1, 6, 7, 8, 9]. This
the method of least squares (LS), commonly used for
is because the step-up voltage test includes the random
electric engineers in obtaining the Weibull parameters
variable of time T and the explanation variable of voltage
for the breakdown voltage, performs badly. We com-
pare the estimation results between those using the
stress v together.
MLE and those using the LS both the models. The LS
has a tendency to yield a bias for the Weibull shape
parameter, and it generates a larger standard devia-
tion. Consequently, the RMSE using the LS becomes
larger than that using the MLE. We conclude that
the MLE is superior to the LS. Regarding the model
selection of which model between the independence
model and the cumulative exposure model should be
used, we recommend the cumulative exposure model
from both viewpoints of the model derivation and the
RMSE.
Keywords:
Weibull distribution, power law, step-up
voltage test, maximum likelihood estimation, method of
least squares
Figure 1: Step-up voltage test. The test is performed as
1
Introduction
follows: Stress vi is imposed to the insulation one minute;
if the insulation is not broken, then the stress level is
To assess the insulation withstand level of the electric ap-
raised higher to vi+1 and vi+1 is imposed one minute;
paratus in electric power substations by using the voltage
this continues until the insulation is broken, and the ﬁnal
∗Manuscript received May 2, 2009. The authors would like
stress vf is used for the estimation.
to thank Dr. Okabe and Mr. Tsuboi for their cooperation and
valuable comments.
This research was supported by the Min-
istry of Education, Science, Sports and Culture, Grant-in-Aid for
Here, we assume that the underlying probability distribu-
Scientiﬁc Research (20510159), and by Tokyo Electric Power Co.
tion of failure time with a constant voltage level follows
Correspondence: Department of Systems Design and Informatics,
Kyushu Institute of Technology, Fukuoka 820-8502, Japan Tel/Fax:
a Weibull distribution with shape parameter a and that
+81(948)29-7711/7709, Email: hirose@ces.kyutech.ac.jp
there is a inverse power law between the mean lifetime
(Advance online publication: 22 May 2009)

Engineering Letter, 17:2, EL_17_2_10
____________________________________________________________________________________
tmeanlife and the imposed voltage v, that is
voltages) independently aﬀects the insulation failure re-
gardless of its history such as aging or deterioration of
tmeanlife = kv−n,
(1)
the insulation. Then, the probability distribution model,
when we observe the ﬁnal breakdown voltage of vf , is
where n is the power law constant, and k is a constant.
expressed as,
Therefore, we assume the Weibull power law model [2, 3],
f
∑
F (t; v) = 1 − exp[−K(vnt)a],
(2)
F (v) = 1 − exp[−A
(vna · ta)],
(4)
i
i
i=1
where, t expresses the random variable and v expresses
the explanation variable. In this paper, we ﬁrst investi-
where A is a constant.
If we use unit time for each
gate whether we can estimate the unknown values of a
ti, then, t is hidden. Note that in this model the ran-
and n using the results obtained by the step-up test. We
dom variable is superﬁcially expressed by v; then, the
use the maximum likelihood estimation method (MLE).
unknown shape parameter in this Weibull model is na,
resulting that the parameters n and a are not obtained
Superﬁcially, one sometimes uses the relationship of
simultaneously. Therefore, the use of the parameter m is
common.
m = na,
(3)
By a simple calculation when time duration to each
to evaluate the voltage stress endurance parameter, as if
stress is small enough, the estimated shape parameter
v works as the random variable and m behaves as the
in Weibull analysis using this independence model has a
Weibull shape parameter. As seen in the literature, e.g.,
bias for m even if the sample size is large enough; this
[7], it is common for electric engineers to use the method
is because, by approximating the summation part by in-
of least squares (LS) in obtaining the Weibull parameters
tegration, (4) tends to the ordinary Weibull distribution
for the breakdown voltage, i.e., the Weibull plot method
with shape parameter m + 1; thus, the bias of quantity
is used. In such a case, the shape parameter m is often
1 is always observed if the Weibull plot is used. See ap-
addressed.
pendix.
We next compare the results between those using the
2.2
Cumulative exposure model
MLE and those using the LS. We ﬁnally show the su-
periority of the MLE over the LS.
The cumulative exposure model [3], also called the ac-
cumulation model by [7, 8], is a probability model that
2
Step-up Test Method
every step vi, where stress vi is imposed to the insulation,
aﬀects the next step vi+1 failure probability. This may
The step-up test method is performed as follows: 1) stress
be interpreted that we assume that the quantity vnt is
v1 is imposed to the insulation one minute, 2) if the insu-
accumulated to the insulation for future failure probabil-
lation is not broken, then the stress level is raised higher
ity. Then, the probability distribution model, when we
to v2 and v2 is imposed one minute, 3) this continues
observe the ﬁnal breakdown voltage vf , is expressed as,
until the insulation is broken, and the ﬁnal stress vf is
used for the estimation. Here, the initial stress level is
f
∑
set such that the breakdown would not occur, and the
F (v) = 1 − exp[−B{
(vn · t
i
i)}a],
(5)
step-up distance d is set such that too many stress levels
i=1
are not imposed; in actual case, v1 is 70% to 90% of the
mean value of v
where B is a constant.
f , i.e., the sample mean µ, and d is 4%
to 8% of the mean value of vf . In the simulation study
By a similar approximation to the independence model,
in this paper, we set v1 = 0.1 and d = 0.1; the number of
we mention that we observe a bias of quantity a to the
samples is 200; the number of trials is 100 each.
shape parameter m when we admit (3).
We have two cases of random number generation method
according to the two proposed methods for insulation
2.3
Bias for the Weibull shape parameter
evaluation. One is the independence model and the other
is the cumulative exposure model. In reliability ﬁelds, the
As mentioned above, the bias will always be observed
latter is common.
even if the sample size is large; an exception is seen when
a = 1, the case of the exponential model. Otherwise,
2.1
Independence model
by looking at the functions in the exponential function
in (4) and (5), we can see that the convexity varies by
The independence method, called by [7, 8] and adopted
the value of a; see appendix. This is explained in [7] in
by JEC-012 [4], is a probability model that each insu-
a more concrete case when the actual step-up setting is
lation event phenomenon such as TOV (temporary over
assumed.
(Advance online publication: 22 May 2009)

Engineering Letter, 17:2, EL_17_2_10
____________________________________________________________________________________
3
Parameter Estimation using the MLE
5.2
Simulation results
In both the models of 2.1 and 2.2, we use the likelihood
The simulation results for the independence model are
function such that
shown in Table 2. In the table, the RMSE is computed
as
√
N
∏
RMSE =
bias2 + variance,
L =
{F (vf
)},
(6)
k +1 ) − F (vfk
k=1
where variance is s.d.2.
where N is the sample size. The maximum likelihood es-
timates are obtained by searching for the parameters such
Table 2: Simulation results for m in the independence
that the log-likelihood function becomes the maximum.
model
We, here, use the simplex method [5] in optimization.
true
estimation
estimate
m
method
bias
s.d.
RMSE
When we adopt the independence model, the unknown
6.67
MLE
−0.003
0.427
0.427
parameters are m and A. When we adopt the cumulative
LS
0.754
1.109
1.341
MLEc
0.951
0.448
1.052
exposure model, the unknown parameters are a, n, and
10
MLE
0.028
0.612
0.613
B.
LS
0.339
2.322
2.346
MLEc
0.900
0.706
1.144
15
MLE
−0.065
0.896
0.898
4
Parameter Estimation using the LS
LS
0.418
2.452
2.487
MLEc
0.764
1.028
1.280
33.3
MLE
−0.005
1.913
1.913
When we use the LS, the Weibull plot is required. To ﬁt
LS
−0.079
5.075
5.076
the straight line on the Weibull plot to the observed data,
MLEc
0.975
1.676
1.939
we transform the voltage values vf (k = 1, 2, . . . , N ) to
k
its logarithmic values, and the probability of kth order
From the table, we can see that the large bias is not
statistics is set usually to
k
, where v
N +1
(f1) ≤ . . . ≤ v(fN ).
observed in the MLE, but is in the LS as indicated in
The superﬁcial estimate for the shape parameter in the
2.1. The more important matter is that the standard
Weibull model is then obtained.
deviation by the LS is markedly larger than that by the
MLE. Consequently, the estimates by using the MLE are
When we adopt the independence model, the unknown
always far superior to those using the LS. In the table, we
parameters are m and A. When we adopt the cumulative
can see MLEc for reference; this value is the maximum
exposure model, the unknown parameters are m and B.
likelihood estimate for m when we regard the data as the
continuous data. We discuss this in the next section.
5
Simulation
The simulation results for the cumulative exposure model
are shown in Tables 3 and 4. In Table 3, the results by
5.1
Simulation cases
the MLE are shown, and in Table 4, the value for m using
the MLE by (3) and using the LS are shown.
For the independence model we consider the cases where
m = 6.67, 10, 15, 33.3. The simulation cases for the cumu-
From the tables, we do not see the large bias in the MLE,
lative exposure model are shown in Table 1. The value of
but we do in the LS as indicated in 2.2. Similar to the
m is set according to (3). The value of A is set to 0.053m.
results in the independence model, the standard deviation
by the LS is larger than that by the MLE. Consequently,
the estimates by using the MLE are always superior to
Table 1: Simulation cases for the cumulative exposure
those using the LS.
model
a
n
m
6.67
2
6
Discussion
0.3
10
3
33.3
10
As stated, using the independence model, we can estimate
6.67
6.67
the superﬁcial Weibull shape parameter induced from (4)
1
10
10
33.3
33.3
by the LS, which is no more the same as m. On the con-
6.67
10
trary, the MLE will not yield the large bias. Similarly
1.5
10
15
in the cumulative model, the MLE will not produce the
33.3
49.95
large bias, while the LS will. In addition, the standard
deviation is larger in the LS than in the MLE. As a con-
In all the cases, we set v1 = 0.1 and d = 0.1; the number
sequence, the RMSE tends to larger in the LS than in the
of samples is 200; the number of trials is 100 to each case.
MLE. This is a ﬁrst merit in using the MLE.
(Advance online publication: 22 May 2009)

Engineering Letter, 17:2, EL_17_2_10
____________________________________________________________________________________
Table 3: Simulation results in the cumulative exposure
Table 4: Simulation results in the cumulative exposure
model (1): a and n by the MLE
model (2): m by the MLE and the LS
true
estimate
a
true
estimation
estimate
a
n
bias
s.d.
RMSE
m (a, n)
method
bias
s.d.
RMSE
6.67
0.001
0.018
0.018
2 (0.3, 6.67)
MLE
−0.027
0.125
0.128
0.3
10
0.005
0.023
0.023
LS
0.224
0.180
0.288
33.3
0.003
0.019
0.019
MLEc
0.251
0.136
0.286
6.67
−0.003
0.046
0.046
3 (0.3, 1.0)
MLE
0.004
0.176
0.176
1.0
10
0.021
0.048
0.053
LS
0.259
0.226
0.344
33.3
0.000
0.038
0.038
MLEc
0.308
0.200
0.368
6.67
−0.001
0.025
0.025
10 (0.3, 33.3)
MLE
-0.015
0.630
0.630
1.5
10
−0.003
0.045
0.045
LS
0.108
0.693
0.702
33.3
−0.008
0.031
0.032
MLEc
0.331
0.655
0.733
true
estimate
n
6.67 (1, 6.67)
MLE
−0.123
0.296
0.321
a
n
bias
s.d.
RMSE
LS
0.799
0.469
0.927
6.67
0.112
0.313
0.332
MLEc
0.902
0.435
1.002
0.3
10
0.157
0.480
0.505
10 (1, 10)
MLE
−0.135
0.434
0.454
33.3
0.415
0.289
0.506
LS
0.900
0.787
1.196
6.67
0.100
0.118
0.155
MLEc
0.881
0.591
1.060
1.0
10
0.332
0.203
0.389
33.3 (1, 33.3)
MLE
−0.164
1.206
1.217
33.3
0.170
0.142
0.222
LS
0.468
2.248
2.296
6.67
0.171
0.083
0.190
MLEc
1.160
1.822
2.160
1.5
10
0.146
0.120
0.189
10 (1.5, 6.67)
MLE
−0.262
0.222
0.343
33.3
0.070
0.078
0.105
LS
1.277
0.754
1.483
MLEc
1.376
0.623
1.510
15 (1.5, 10)
MLE
−0.250
0.432
0.499
LS
1.166
0.963
1.512
Which model between the independence model and the
MLEc
1.298
0.844
1.548
cumulative exposure model should be used? This is a
49.95 (1.5, 33.3)
MLE
−0.358
0.982
1.045
next question. We can adopt the cumulative exposure
LS
0.795
3.604
3.691
model because of the natural derivation of the model.
MLEc
1.168
2.679
2.923
The independence model has long been used because the
estimation method is simple and is easy to use. There
than the LS can.
seems no diﬀerence between the two models as long as
we use the LS. However, if we use the MLE, a big diﬀer-
ence between the two models is seen apparently. In the
7
Conclusion
independence model (4), the parameter m can be deﬁned
by m = na. However, we cannot estimate the param-
To assess the insulation withstand level of the electric ap-
eters a and n simultaneously. On the contrary, we can
paratus, the step-up test method is often used. However,
estimate both the parameters a and n simultaneously, if
we have still many unknown matters regarding the treat-
we use the MLE. Moreover, from Tables (2) and (4), the
ment of the results. Assuming that the underlying proba-
RMSE for m is smaller in the cumulative exposure model
bility distribution of failure time with a constant voltage
than in the independence model. We can regard that the
level follows a Weibull distribution with shape parame-
cumulative exposure model can be recommended from
ter a and that an inverse power law tmeanlife = kv−n be-
both viewpoints of the model derivation and the RMSE.
tween the mean lifetime tmeanlife and the imposed voltage
v holds, then, we ﬁrst investigate whether we can estimate
We have shown the MLEc values for reference in Tables 2
the unknown values of n and a using the breakdown volt-
and 4; these value are the maximum likelihood estimates
age results obtained by the step-up test. We have used
for m when we regard the data as the continuous data.
the maximum likelihood estimation (MLE) method for
That is, we ﬁt the continuous Weibull model,
this purpose. We have assumed two proposed models:
m v
v
1) the independence model, 2) the cumulative exposure
f (v) =
( )m−1 exp[−( )m].
(7)
u u
u
model. The maximum likelihood estimation is well per-
formed in both the models.
The estimated values by the LS and by the M LEc are,
in principle, dealt with the data as continuous. On the
It is common for electric engineers to use the method of
contrary, the estimates by the MLE are dealt with the
least squares (LS) in obtaining the Weibull parameters
data as grouped. It would be recommended that we use
for the breakdown voltage. In such a case, the shape
the maximum likelihood method in the continuos model
parameter m is often addressed. We next compare the
too. However, the diﬀerence of the RMSE value between
results between those using the MLE and those using the
the LS and the M LEc is smaller than that between the
LS in both the models. The LS is inclined to yield a bias
MLE and the LS. The important point is that the MLE
for parameter m, and it generates a larger standard de-
can deal with the data more accurately to each model
viation. Consequently, the RMSE using the LS becomes
(Advance online publication: 22 May 2009)

Engineering Letter, 17:2, EL_17_2_10
____________________________________________________________________________________
larger than that using the MLE. That is, the MLE is
[4] JEC-0102-1994,“ Standard for Test Voltages ”Stan-
superior to the LS.
dard of the Japanese Electrotechnical Committee,
(1994) (in Japanese)
Regarding the model selection of which model between
the independence model and the cumulative exposure
[5] J.A. Nelder and R. Mead, Computer Journal, 7, pp.
model should be used, we recommend the cumulative ex-
308-313, (1965)
posure model from both viewpoints of the model deriva-
[6] S. Okabe, T. Tuboi, J. Takami: Reliability Evalua-
tion and the smaller RMSE.
tion Method with Weibull Distribution for Tempo-
Appendix
rary Overvoltages of Substation Equipment Trans.
IEEJ, 127, pp.994-1001, (2007)
We consider the case that time duration t is 1 for all the
stress levels. We use a simple approximation such that
[7] S. Okabe, T. Tuboi, J. Takami: Inﬂuence of Voltage
Application History on Insulation Test with One-
f
∑
Minute Step-up Method IEEE Transactions on Di-
vna = (dv)na + (2dv)na + · · · + (v
i
f )na
electric and Electrical Insulation, 15, pp.1261-1270,
i=1
∫
(2008)
vf
≈ C
unadu = C (vf )na+1.
(8)
[8] S. Okabe, T. Tuboi, J. Takami: Design of Insula-
0
tion Test with One-minute Step-up Method for Sub-
station Equipment IEEE Transactions on Dielectric
In the independence model, from (4),
and Electrical Insulation, 15, pp.1271-1280, (2008)
f
∑
F (v) = 1 − exp[−D
vna]
[9] T.
Takuma:
Consideration
on
AC
(Power-
i
Frequency) Insulation Tests for High Voltage Power
i=1
≈
Equipment Trans. IEEJ, 124-B, pp.977-983, (2004)
1 − exp[−D vna+1].
(9)
f
Transforming this into the Weibull plot structure,
1
log log
= (na + 1) log vf + const.
(10)
1 − F (v)
We can obtain the estimated mean of na+1 for true value
of na; i.e., the bias is 1.
In the cumulative exposure model, transforming (5) into
the Weibull plot structure,
f
1
∑
log log
= a log[
(vn)].
(11)
1 − F (v)
i
i=1
By the approximation described above,
1
log log
≈ a(n + 1) log vf + const,
(12)
1 − F (v)
which deduces the bias of a.
References
[1] H. Goshima, M. Yashima, T. Shindo, T. Takuma:
Feasibility of Replacement Postponement Based
on Insulation Test for Aged Gas-Insulated Power
Equipment T. IEE Japan, 126-B, pp.694-700, (2006)
[2] H. Hirose: Estimation of The Threshold Stress in
Accelerated Life Testing, IEEE Transactions on Re-
liability, 42 , pp.650-657, (1993)
[3] H. Hirose: Theoretical Foundation for Residual Life-
time Estimation, T.IEE Japan , 116-B, pp.168-173,
(1996)
(Advance online publication: 22 May 2009)