energies
Article
A Failure Probability Calculation Method for Power
Equipment Based on Multi-Characteristic Parameters
Hang Liu 1, *, Youyuan Wang 1, *, Yi Yang 2 , Ruijin Liao 1 , Yujie Geng 2 and Liwei Zhou 1
1
2
*
State Key Laboratory of Power Transmission Equipment & System Security and New Technology,
School of Electrical Engineering, Chongqing University, Chongqing 400044, China; [email protected] (R.L.);
[email protected] (L.Z.)
Shandong Electric Power Research Institute, Shandong Electric Power Company, Jinan 250002, China;
[email protected] (Y.Y.); [email protected] (Y.G.)
Correspondence: [email protected] (H.L.); [email protected] (Y.W.); Tel.: +86-23-6511-1795 (Y.W.)
Academic Editor: Issouf Fofana
Received: 30 March 2017; Accepted: 11 May 2017; Published: 17 May 2017
Abstract: Although traditional fault diagnosis methods can qualitatively identify the failure modes
for power equipment, it is difficult to evaluate the failure probability quantitatively. In this paper, a
failure probability calculation method for power equipment based on multi-characteristic parameters
is proposed. After collecting the historical data of different fault characteristic parameters, the
distribution functions and the cumulative distribution functions of each parameter, which are applied
to dispersing the parameters and calculating the differential warning values, are calculated by using
the two-parameter Weibull model. To calculate the membership functions of parameters for each
failure mode, the Apriori algorithm is chosen to mine the association rules between parameters and
failure modes. After that, the failure probability of each failure mode is obtained by integrating the
membership functions of different parameters by a weighted method, and the important weight of
each parameter is calculated by the differential warning values. According to the failure probability
calculation result, the series model is established to estimate the failure probability of the equipment.
Finally, an application example for two 220 kV transformers is presented to show the detailed process
of the method. Compared with traditional fault diagnosis methods, the calculation results not only
identify the failure modes correctly, but also reflect the failure probability changing trend of the
equipment accurately.
Keywords: failure probability; multi-characteristic parameters; the Weibull model; differential
warning value; association rule; failure modes
1. Introduction
With the development of smart grid, the Chinese power industry is entering the era of Big
Data [1–3]. Establishing the fault diagnosis models based on Big Data technology is of great importance
since it can realize the comprehensive utilization of multi-source data and improve the accuracy of
the fault diagnosis results, which is beneficial for ensuring reliability and safe operation, optimizing
the condition-based maintenance (CBM) strategies, and increasing the remaining useful life (RUL) of
power equipment.
With the development of online monitoring technology and information technology, the volume
and variety of the condition monitoring data of the fault characteristic parameters have increased
dramatically in recent years [4]. The fault diagnosis technology has become more automatic and
intelligent than before by using multi-source data. For example, evidence fusion theory [5,6], support
vector machine (SVM) [7], artificial neural network (ANN) [8–10], and other methods have been widely
Energies 2017, 10, 704; doi:10.3390/en10050704
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Energies 2017, 10, 704
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applied to fault diagnosis for power equipment. Though many research achievements have been
reached, there are still some shortcomings, which are summarized as follows:
(1)
(2)
(3)
Although the volume and variety of the monitoring data are large enough, most of the data
changes in the normal range. That is to say, there are little fault data or abnormal data. However,
many existing diagnosis methods often require a large amount of fault data to build and optimize
the models. The low value density of the condition monitoring data limits the application of
those methods [11–13].
Most existing methods have failed to reveal the association between the failure modes and various
characteristic parameters. The equipment failure modes are difficult to identify correctly when
some kinds of parameters are changed. Additionally, the values of some key parameters and
factors in the models are difficult to estimate. Thus, they are always determined by empirical
values or expert experience, which may cause large errors.
The insulation performance of power equipment is degraded gradually by the long-term impact
of mechanical and electrical stress, which causes an increasing trend in the failure probability of
the equipment. Many existing fault diagnosis methods can only qualitatively identify the failure
modes of the equipment under abnormal conditions, but not quantitatively evaluate the failure
probability and the changing trends of normal equipment. In fact, the equipment under normal
conditions still has a risk of failure.
With the increasing reliability requirements of the power supply, the traditional fault diagnosis
methods cannot meet the requirements of the fault diagnosis of the power system. How to make
full use of the multi-characteristic parameters to improve the accuracy and the effectiveness when
establishing the model becomes a significant challenge. This paper presents a failure probability
calculation method for power equipment based on multi-characteristic parameters, which achieves the
accurate identification of fault modes and quantifies the change trend of the failure probability of the
power equipment.
This manuscript is organized as follows: In Section 2, the probability density functions and the
cumulative distribution functions of different fault characteristic parameters are obtained, and the steps
to disperse the parameters and calculate the differential warning values are presented. In Section 3,
the method for association rules mining and the membership function calculations for parameters are
given. In Section 4, the model to estimate the failure probability of the failure modes, as well as the
equipment’s failure probability, are introduced. In Section 5, the procedure of the method is listed in
detail. An application example for two 220 kV power transformers is presented in Section 6, and the
conclusions are drawn in Section 7.
2. The Discrete Intervals and the Differential Warning Values Calculation
2.1. The Weibull Model
The probability density function and the cumulative probability distribution function of the
two-parameter Weibull distribution are shown in Equations (1) and (2), respectively [14].
β x β −1
x β
f (x) =
exp −
(1)
α α
α
x β
(2)
F = 1 − exp −
α
where x is the value of a kind of fault characteristic parameter; α and β are the scale and shape
parameters. When these two parameters are determined, the Weibull model is uniquely determined. α
and β can be estimated by using the maximum likelihood estimation method [15], and the steps are
as follows:
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(1) The
parameter
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2017,
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time series X = [x1, x2, …, xn] are Substituted into the logarithmic likelihood
3 of 15
function equation shown in Equation (3). The likelihood functions are established in Equation (4):
(1)
β −1
  x  β into

The parameter time series X = [x1 , x2 , . .m. ,βxn ]xiare
the logarithmic likelihood
 Substituted
ln L ( X ) = ln ∏   exp  −  i  
(3)
function equation shown in Equation (3). iThe
likelihood
functions
are
established in Equation (4):
=1 α  α 
  α  
m
xi) β−1
x β
β
∂
X
ln

(
exp − i
(3)
ln L( X ) = ln ∏
=
0

α α
α
i =1
 ∂α
(
( (XX))
∂ ln
 ∂ ln
=
=00
 ∂∂lnβ∂α(X )
=0
∂β
(4)
(4)
(2) Substitute Equation (3) into Equation (4), then α and β are calculated.
(2) Substitute Equation (3) into Equation (4), then α and β are calculated.
Taking the dissolved gas in transformer oil as an example, the dissolved gas content data in
Taking
dissolved
gas in220
transformer
oil as an
dissolved
gas
content
data in
recent three the
years
of different
kV transformers
in example,
a region the
were
collected.
The
gas includes
recent
three
years
of
different
220
kV
transformers
in
a
region
were
collected.
The
gas
includes
hydrogen, methane, ethane, ethylene, and acetylene, whose chemical formulae are H2, CH4, C2H4,
hydrogen,
ethane, ethylene,
and to
acetylene,
whose
chemical formulae
are H2 , CH
4 , C2 H
4,
C2H6, and methane,
C2H2, respectively.
According
the steps,
the parameters
of the Weibull
model
are
C
H
,
and
C
H
,
respectively.
According
to
the
steps,
the
parameters
of
the
Weibull
model
are
2
6
2
2
estimated and shown in Table 1, and the probability density function curves and frequency
estimated
shown
Tableof1,gas
andare
theshown
probability
density
function curves and frequency histograms
histogramsand
of the
fiveintypes
in Figure
1, respectively.
of the five types of gas are shown in Figure 1, respectively.
Table 1. The parameters calculation results of the Weibull model.
Table 1. The parameters calculation results of the Weibull model.
Parameters
CH4 C2H6 C2H4 C2H2
H2
Parametersα
H2 17.03 CH
4
10.26
C2.94
2 H6
C2 H41.46 C2 H2
1.68
β
17.030.81 10.26
1.06
0.81
1.06
2.94
0.79
0.79
0.811.68 0.89 1.46
0.81
0.89
α
β
(a) The content of H2 (uL/L)
(b) The content of CH4 (uL/L)
0.3
0.35
0.3
0.25
0.25
0.2
0.2
0.15
0.15
0.1
0.1
0.05
0.05
0
0
5
10
15
20
25
30
0
(c) The content of C2H6 (uL/L)
0
5
10
15
20
25
(d) The content of C2H4 (uL/L)
Figure 1.
1. Cont.
Figure
Cont.
30
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10, 704
704
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44 of
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15
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0.3
0.3
0.25
0.25
0.2
0.2
0.15
0.15
0.1
0.1
0.05
0.05
0
0
0
5
10
15
20
25
30
0
5
10
15
20
25
30
(e) The content of C2H2 (uL/L)
(e) The content of C2H2 (uL/L)
Figure 1. The probability density function curves and frequency histograms of the dissolved gas.
Figure 1.
1. The probability density function curves and frequency histograms of the dissolved gas.
Figure
From Figure 1, the probability density functions are consistent with the actual distribution of
From
Figureindicates
theprobability
probability
density
consistent
the actual
distribution
of
Figure
1,1,the
density
functions
areare
consistent
withwith
the actual
distribution
of each
each From
gas, which
the effectiveness
offunctions
the Weibull
model.
eachwhich
gas, which
indicates
the effectiveness
the Weibull
model.
gas,
indicates
the effectiveness
of theofWeibull
model.
2.2. The Discrete Intervals Calculation
2.2. The Discrete Intervals Calculation
The fault characteristic parameters value is continuous, but the association rule mining
fault
characteristic
parameters
value
is continuous,
the association
rule
mining
The
fault
characteristic
value
isparameters
continuous,
but the but
association
ruletomining
algorithms
algorithms
require
discreteparameters
input.
Thus,
the
should
be
discretized
different
classes
algorithms
require
discrete
input.
Thus,
the
parameters
should
be
discretized
to
different
classes
require
discrete
input.[16].
Thus,
thepaper
parameters
should
be discretized
classes
before
data
before data
mining
This
presents
a discrete
method to
fordifferent
parameters
based
on
the
before
data
mining
[16].
This
paper
presents
a
discrete
method
for
parameters
based
on
the
mining
[16].
This
paper
presents
a
discrete
method
for
parameters
based
on
the
distribution
probability
distribution probability of the equipment condition. The steps are as follows:
distribution
probability
of the
equipment
of
the equipment
condition.
The
steps are condition.
as follows:The steps are as follows:
(1) According to [17], the equipment condition is divided into normal condition, attentive condition,
(1) According
condition
isisdivided
into
attentive
(1)
Accordingto
to[17],
[17],the
theequipment
equipment
condition
divided
intonormal
normalcondition,
condition,
attentive
condition,
abnormal
condition,
and
serious
condition.
The distribution
probability
of thecondition,
different
abnormal
condition,
and
serious
condition.
The
distribution
probability
of
the
different
abnormal
condition,
and
serious
condition.
The
distribution
probability
of
the
different
conditions
P
=
p
p
p
p
[ norm atten abni ser ] are calculated after collecting the actual condition
conditions
h
P =p[ atten
pnorm ppabn
p
p calculated
] are calculated
conditions
after collecting
actual data
condition
P
=
after collecting
the actualthe
condition
of all
atten
p
norm
serabn areser
data of all of the same
type
ofpequipment.
data
of same
all of type
the same
type of equipment. F = [ F F F
of
the
of
equipment.
(2) The cumulative distribution probability
hI
II
III 1] isiobtained by accumulating the
F
=
F
F
F
[
by accumulating
the
(2) The
cumulative
distribution
probability
The cumulative
distribution
probability F = I FI IIFII IIIFIII 1] 1is obtained
is obtained
by accumulating
distribution
probability
P.
P. the inverse cumulative probability distribution function based on
probability
P.into
(3) distribution
Fthe
I, Fdistribution
II, and FIII
areprobability
brought
IIII
, ,and
into the
cumulative
probability
distribution
function
based
(3) F
FII,,FFtwo-parameter
andFFIIIIIIare
arebrought
brought
theinverse
inverse
cumulative
probability
distribution
xII xIIIoni]
the
Weibullinto
model
expressed
in Equation
(5). Three
values Xfunction
ph= [ xI
X
=
x
x
xIII ]
[
the
Weibullmodel
modelexpressed
expressedininEquation
Equation
(5).
Three
values
the two-parameter
Weibull
(5).
Three
values
X p =p 2.xI I xII II xIII
are
calculated and four
discrete intervals
s1–s4 are obtained,
as
shown
in Figure
are calculated
calculated and
and four
four discrete
discrete intervals s11–s
–s44are
areobtained,
obtained,as
asshown
shownin
inFigure
Figure2.
2.
are
1/ β
x = α − ln (1 − F ) 1/ β
(5)
xx ==αα[−
− lnln(1(1−−F F) )]1/β
(5)
Figure 2. The discrete intervals of parameters.
Figure 2. The discrete intervals of parameters.
The distribution
distribution probability
probability and
the cumulative
cumulative probability
probability of
of the
the 220
220 kV
kV transformers
transformers are
are
The
and the
The
distribution
probability
and
the
cumulative
probability
of
the
220
kV
transformers
are
collected and
andlisted
listedin
inTable
Table2,2,and
andthe
thecalculation
calculation
result
discrete
intervals
of the
types
of
collected
result
of of
thethe
discrete
intervals
of the
fivefive
types
of the
collected
and
listedininTable
Table3.2, and the calculation result of the discrete intervals of the five types of
the
gas
are
shown
gas are shown in Table 3.
the gas are shown in Table 3.
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Table 2. The data of 220kV transformers.
Equipment Condition
Normal
Condition
Attentive
Condition
Abnormal
Condition
Serious
Condition
Distribution probability
Cumulative probability
95.46%
95.46%
3.53%
98.99%
0.94%
99.93%
0.07%
1
Table 3. The discrete intervals of the gas (uL/L).
Gas
xI
xII
xIII
H2
CH4
C2 H6
C2 H4
C2 H2
82.44
29.75
6.79
5.19
12.21
139.89
43.22
11.08
8.10
20.12
257.84
66.57
19.54
13.55
35.84
2.3. The Differential Warning Values Calculation
The abnormal increasing of fault characteristic parameters, such as the dissolved gas in oil, will
increase the failure probability of the equipment. To ensure the safe and reliable operation of the power
equipment, the warning value of each characteristic parameter is given by standard or by guide. If the
parameter value exceeds the warning value, it indicates that the equipment is under poor condition
or has a fault. For example, the warning value of hydrogen content is 150 uL/L, and the acetylene
content is 5 uL/L for the 220 kV power transformers [17].
However, due to the operating environment, the service age and health status of each equipment
are different, and the actual warning value of the equipment may deviate from the warning value
in [17]. According to the method in [18], this paper presents a differential warning values calculation
method for the equipment. The steps are as follows:
(1)
(2)
Count the average failure probability Pave of the same types of equipment in a region.
The probability Pave is substituted into the inverse cumulative probability distribution functions
with different characteristic parameters, as shown in Equation (5), and the differential warning
value Vdi f of each parameter is obtained.
Table 4 is the differential warning value of the dissolved gas when the average failure probability
of the 220 kV transformers is 0.18.
Table 4. The differential warning value of the gas (uL/L).
Gas
Differential Warning Value
H2
CH4
C2 H6
C2 H4
C2 H2
37.52
17.06
3.27
2.67
5.80
3. Membership Function of the Fault Characteristic Parameter
3.1. The Association Rules Mining for Failure Modes and the Fault Characteristic Parameters
Different failure modes will have different effects on the change trend of the fault characteristic
parameters, which are quantified by association rule mining algorithms, such as Apriori and FP-Growth
algorithms [19,20]. The Apriori algorithm scans the whole database in each loop to calculate the
support and confidence coefficient of the candidate item sets. The association rule mining by the
Apriori algorithm includes the following steps.
Energies 2017, 10, 704
(1)
(2)
(3)
(4)
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The fault data, which have m kinds of fault characteristic parameters and g kinds of failure modes,
are collected. All of the parameters are discretized in four classes.
The dataset of fault data is established and denoted as Df = [G, df1 , df2 , ···, df m ], where G is the
transaction of the failure modes, G = [G(1), . . . , G(g)]. The dfi = [dfi (1) dfi (2), dfi (3), dfi (4)] is the
transaction of the fault characteristic parameter i (i = 1, . . . , m).
The Df is divided into m datasets according to each parameter, namely Df1 = [G, df1 ], Df2 = [G, df2 ],
. . . , Dfm = [G, dfm ].
Set the minimum support (Supmin) the minimum confidence coefficient (Confmin), the Apriori
algorithm is applied to finding out all of the frequent two-item sets G → d f , the support
coefficient Sup G → d f , and the confidence coefficient Con f G → d f based on Equations (6)
and (7):
count G → d f
Sup G → d f =
(6)
count D f
count G → d f
Con f G → d f =
(7)
count D f
where Count(·) is the counting function of the item set.
Support is used to measure the statistical importance of association rules in the
entire dataset.
The greater the support, the more frequent the item sets or rules appear in Df . If Sup G → d f is no
less than the Supmin, G → d f is regarded as a frequent item set. Otherwise, G → d f is regarded as an
infrequent item set.
Confidence is used tomeasurethe trustworthiness ofassociation
rules. If an association rule
G → d f satisfies both Sup G → d f ≥ Supmin and Con f G → d f ≥ Con f min, G → d f will be
regarded as a strong association rule. Otherwise, it is a weak association rule.
3.2. Membership Function Calculation Method
According to association rules between the failure modes and the parameters, the distribution
probability, that is, the confidence coefficient between each item set is obtained. For any kind of failure
mode j and characteristic parameter i, Equation (8) is satisfied:
4
∑
Con f G ( j) → d f i (k) = 1
(8)
k =1
The membership function of the characteristic parameter i for the failure mode j is expressed in
Equation (9):

Cij (1)

x ∈ s1
 xI · x



C
2
(
)
C (1) + ij

x ∈ s2

xII − xI · ( x − xI )
 ij
Cij (3)
Fij = Cij (1) + Cij (2) +
(9)
x ∈ s3
xIII − xII · ( x − xII )



C
4
(
)
ij


Cij (1) + Cij (2) + Cij (3) + xM − xIII · ( x − xIII ) x ∈ s4



1
x > xM
where Cij (sk ) = con f G ( j) → d f i (sk ) , x is the value of characteristic parameters. xM is the value
under the condition that the cumulative distribution probability is (FIII + 1)/2.
Energies 2017, 10, 704
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4. The Failure Probability of Failure Modes and the Equipment
4.1. The Failure Probability of the Failure Mode
According to the method referred to in Section 3, the membership function of different failure
modes resulting from different fault characteristic parameters can be figured out. Those different
membership functions have different contribution to the failure probability of the failure modes.
Therefore, the failure probability of each failure mode is calculated by a weighted method. For example,
the failure probability of the jth failure mode is calculated by Equation (10):
m
PG( j) = F · W =
∑ Fij · ωij
(10)
i =1
m
∑ ωij = 1
(11)
i =1
where Fij and ωij are the membership function and important weight from the ith characteristic
parameter to the jth failure mode, respectively. The important weight ωij of each fault characteristic
parameter for different failure modes is calculated in Section 4.2.
4.2. The Calculation for Important Weight
The existing weight calculation methods, such as analytic hierarchy process (AHP) [21], may lack
accuracy because of the dependence of subjective judgments from experts. To ensure the rationality of
the result, a method for weight calculation is presented according to the differential warning values of
the characteristic parameters, which is shown as follows:
(1)
(2)
Obtain the differential warning values of the different characteristic parameters.
Count the number (N) of samples in which the characteristic parameters are greater than, or equal
to, their differential warning values shown in Equation (12):



N=

N11
N21
···
Nm1
N12
N22
···
Nm2
···
···
···
···
N1m
N2m
···
Nmg





(12)
For every Nij , it is the number of samples of the ith characteristic parameter to the jth failure mode.
The important weights of all of the fault characteristic parameters for the jth failure mode are obtained
by the normalized calculation expressed in Equation (13):
ωij =
Nij
m
(13)
∑ Nij
i =1
According to Equations (12) and (13), the result of the importance weight W is obtained based
on the calculation of the fault data and the differential warning value rather than the experience of
experts so that the accuracy and rationality are ensured.
4.3. The Failure Probability of the Equipment
Every type of the failure mode will cause equipment failure, and the series network is
established between equipment failure probability and the failure probability of the failure modes [14].
The calculation method of the equipment failure probability is given by Equation (14):
Energies 2017, 10, 704
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g
Pequ = 1 −
∑
1 − Pj
(14)
j =1
where Pequ is the equipment failure probability, and Pj is the failure probability of the jth failure mode.
Energies 2017, 10, 704
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5. The Procedure of the Method
5. The
Procedure
of method
the Method
The
flow of the
is shown in Figure 3. The key steps are as follows:
The flow of the method is shown in Figure 3. The key steps are as follows:
(1) According to the method in Section 2, the probability density functions and cumulative
distributiontofunctions
of different
fault2,characteristic
parameters
on aand
two-parameter
(1) According
the method
in Section
the probability
density based
functions
cumulative
Weibull distribution
model
are calculated.
Based on the inverse
cumulative
function,
distribution
functions
of different
fault characteristic
parameters
based distribution
on a two-parameter
the condition
distribution
and the
average
probability
of thedistribution
equipment
Weibull
distribution
modelprobability
are calculated.
Based
on failure
the inverse
cumulative
are applied
calculating
the discrete
intervals and the differential
warning
value ofofeach
function,
thetocondition
distribution
probability
average failure
probability
the
characteristic
equipment
areparameter.
applied to calculating the discrete intervals and the differential warning value of
characteristic
parameter.
(2) each
According
to the method
in Section 3, the Apriori algorithm is applied to finding the association
(2) According
the method
in Section
3, the
Apriori
is applied
to findingand
the calculating
association
rules in thetofault
data between
failure
modes,
asalgorithm
well as different
parameters
rules
the fault data
betweenoffailure
modes, item
as well
as different
parameters
and calculating
of
of theinconfidence
coefficient
the frequent
sets.
After that,
the cumulative
probability
the
confidence
coefficient
ofcharacteristic
the frequentparameter
item sets.is After
that, the cumulative probability
distribution
function
of each
obtained.
function
of each
characteristic
parameter
obtained.
(3) distribution
The membership
function
and
the important
weight ofiseach
parameter are obtained in Section 4.
(3) The
function
weight
ofcalculated
each parameter
are obtained
in Sectionand
4.
(4)
Themembership
failure probability
of and
all ofthe
theimportant
failure modes
are
by a weighted
calculation,
(4) The
failure
probability
of
all
of
the
failure
modes
are
calculated
by
a
weighted
calculation,
and
the series model is proposed to calculate the failure probability of the equipment.
the series model is proposed to calculate the failure probability of the equipment.
1 Historical data of the fault characteristic parameters
Two-parameter Weibull function
Discrete intervals and the differential warning value
2 Fault data
The important weight of parameters;
Association rule mining by Apriori Algorism;
The membership function Fij
3 The monitoring data of the characteristic parameters
The failure probability of the failure modes ;
The failure probability of the equipment
Figure 3. The flow of the failure probability calculation.
Figure 3. The flow of the failure probability calculation.
6. Application Example
6. Application Example
6.1. Basic
6.1.
Basic Information
Information
According to
to the
thefault
faultdata
datasamples
samplesofofthe
the220
220
transformers,
four
types
of the
failure
modes
According
kVkV
transformers,
four
types
of the
failure
modes
are
are
defined,
namely,
low-temperature
overheating
(G
1), high-temperature overheating (G2), lowdefined, namely, low-temperature overheating (G1 ), high-temperature overheating (G2 ), low-energy
energy discharging (G3), and high-energy discharging (G4). The quantitative distribution of each
failure mode is shown in Figure 4.
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discharging (G3 ), and high-energy discharging (G4 ). The quantitative distribution of each failure mode
is shown
in10,
Figure
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0.2
0.2
0.11
0.11
0.21
0.21
0.48
0.48
G1
G1
G2
G2
Figure4.
4.The
Thequantitative
quantitativedistribution
distributionof
ofeach
eachfailure
failure mode.
mode.
Figure
Figure 4. The quantitative distribution of each failure mode.
6.2.
6.2. The
TheFailure
Failure Probability
Probability and
and Important
Important Weight
Weight
6.2. The Failure Probability and Important Weight
After
After setting
setting the
the minimum
minimum support
support Supmin
Supmin ==0.02
0.02and
andthe
theminimum
minimumconfidence
confidenceConfmin
Confmin==0.02,
0.02,
After setting the minimum support Supmin = 0.02 and the minimum confidence Confmin = 0.02,
the
association
rules
among
all
of
the
failure
modes
and
the
different
ways
by
which
the
gas
dissolves
the association rules among all of the failure modes and the different ways by which the gas dissolves
the association rules among all of the failure modes and the different ways by which the gas dissolves
are
are calculated
calculated by
by Apriori
Apriori algorithm.
algorithm. Taking
Taking methane
methane and
and ethane
ethane as
as examples,
examples, the
the frequent
frequent item
item sets
sets
are calculated by Apriori algorithm. Taking methane and ethane as examples, the frequent item sets
and
the
confidence
results
are
shown
in
Figure
5a,b.
and the confidence results are
and the confidence results are shown in Figure 5a,b.
Confidence
Confidence
0.8
0.8
0.7
0.7
0.6
0.6
0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0
0
G1 →s1
G1→s1
→s1
G3
G3 →s1
0.54
0.54
0.41
0.41
0.08
0.08
G1 →s2
G1→s2
→s2
G3
G3 →s2
0.69
0.69
0.44
0.44
0.25
0.25 0.21
0.21
0.17
0.08 0.09 0.17 0.09
0.09
0.08 0.09
G1 →s3
G1→s3
→s3
G3
G3 →s3
Frequent item sets
Frequent item sets
G1 →s4
G1→s4
→s4
G3
G3 →s4
G2 →s1
G2→s1
→s1
G4
G4 →s1
0.31
0.31 0.25
0.25 0.17
0.14
0.17 0.14
0.08
0.08
G2 →s2
G2→s2
→s2
G4
G4 →s2
G2 →s3
G2→s3
→s3
G4
G4 →s3
G2 →s4
G2→s4
→s4
G4
G4 →s4
(a) The association rule between failure modes and methane.
(a) The association rule between failure modes and methane.
Confidence
Confidence
0.9
0.9
0.8
0.8
0.7
0.7
0.6
0.6
0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0
0
G1 →s1
G1→s1
→s1
G3
G3 →s1
0.82
0.82
0.61
0.61
0.37
0.37
0.48
0.48
0.33 0.31
0.33 0.31
0.15 0.18
0.18 0.1 0.12
0.07 0.15
0.1 0.12
0.07
G1 →s2
G1→s2
→s2
G3
G3 →s2
G1 →s3
G1→s3
→s3
G3
G3 →s3
0.17
0.17
Frequent item sets
Frequent item sets
G1 →s4
G1→s4
→s4
G3
G3 →s4
G2 →s1
G2→s1
→s1
G4
G4 →s1
0.16
0.16
0.05 0.03
0.05 0.03
G2 →s2
G2→s2
→s2
G4
G4 →s2
G2 →s3
G2→s3
→s3
G4
G4 →s3
0.05
0.05
G2 →s4
G2→s4
→s4
G4
G4 →s4
(b) The association rule between failure modes and ethane.
(b) The association rule between failure modes and ethane.
Figure 5. The calculation result.
Figure 5. The calculation result.
Figure 5. The calculation result.
The membership functions of different kinds of gas are calculated, and the curves of the
The membership functions of different kinds of gas are calculated, and the curves of the
functions
are shown infunctions
Figure 6.
The
functionsmembership
are shown in Figure 6.of different kinds of gas are calculated, and the curves of the functions
are shown in Figure 6.
10 of 15
10 of 15
1.0
1.0
0.8
0.8
Failure Probability
Failure probability
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0.6
0.4
0.2
0.0
0
50
100
150
H2(uL/L)
200
G1
G2
G3
G4
250
G1
G2
G3
G4
0.6
0.4
0.2
0.00
300
20
40
1.0
1.0
0.8
0.8
0.6
0.4
0.2
G1
2
4
6
8
10
80
(b) CH4
Failure Probability
Failure Probability
(a) H2
0.00
60
CH4(uL/L)
12
G2
G3
G4
16
18
20
14
G1
G2
G3
G4
0.6
0.4
0.2
0.00
10
20
30
40
50
C2H4(uL/L)
C2H6(uL/L)
(c) C2H6
(d) C2H4
Failure Probability
1.0
0.8
0.6
0.4
0.2
0.00
5
10
15
20
G1
G2
G3
25
G4
30
C2H2(uL/L)
(e) C2H2
Figure 6.
6. The
The membership
membership functions
functions of
of the
the dissolved
dissolved gas.
gas.
Figure
The important weights of the five types of gas to the four failure modes are calculated and shown
The important weights of the five types of gas to the four failure modes are calculated and shown
in Figure 7.
in Figure 7.
It is obvious that the weight of acetylene is small in overheating failure modes while the alkanes
It is obvious that the weight of acetylene is small in overheating failure modes while the alkanes
is larger. Additionally, the weight of hydrogen is much larger than the weight of other gases in
is larger. Additionally, the weight of hydrogen is much larger than the weight of other gases in
discharging failure modes.
discharging failure modes.
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H2
H2
0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0.0
0.0
C2H2
C2H2
G1
G1
G3
G3
G2
G2
G4
G4
CH4
CH4
C2H4
C2H4
C2H6
C2H6
Figure7.7.The
Theimportant
importantweight
weightofofeach
eachgas.
gas.
Figure
Figure 7. The important weight of each gas.
6.3.Case
CaseStudy
Study11
6.3.
6.3. Case Study 1
Shownin
inFigure
Figure88 are
are the
the time
time series
toto
30
Shown
series data,
data, which
whichinclude
include236
236samples
samplesfrom
from1 1January
January2016
2016
Shown in Figure 8 are the time series data, which include 236 samples from 1 January 2016 to 30
June
2016,
transformer (named
(named T1)
T1) where
where the
the value
valueofof
30
June
2016,ofofthe
thedissolved
dissolvedgas
gascontent
contentofofaa 220
220 kV
kV transformer
June 2016, of the dissolved gas content of a 220 kV transformer (named T1) where the value of
ethyleneisiszero.
zero.
ethylene
ethylene is zero.
120
120
90
90
60
60
30
300
0
10
108
86
64
42
20
0
H2
H2
50
50
100
100
150
150
200
200
C2 H6
C2 H6
50
50
100
100
150
150
200
200
7
7
6
6
5
5
4
4
3
3
0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0.0
0.0
CH4
CH4
50
50
100
100
150
150
200
200
C2 H4
C2 H4
50
50
100
100
150
150
200
200
Time(Day)
Time(Day)
Figure 8. Thecontent
content of gas(uL/L).
(uL/L).
Figure
Figure8.8.The
The contentofofgas
gas (uL/L).
The data of the five kinds of dissolved gas are brought to each kind of membership function,
Thedata
dataofofthe
thefive
five
kinds
dissolved
brought
to each
of membership
function,
The
kinds
of of
dissolved
gasgas
areare
brought
to each
kindkind
of membership
function,
and
and the failure probability of the four failure modes and the failure probability of T1 are calculated
andfailure
the failure
probability
the four
failure
modes
andfailure
the failure
probability
T1 calculated
are calculated
the
probability
of theoffour
failure
modes
and the
probability
of T1ofare
by
by Equations (10) and (14). The calculation result is shown in Figure 9.
by Equations
calculation
result
is shown
in Figure
Equations
(10) (10)
and and
(14).(14).
The The
calculation
result
is shown
in Figure
9. 9.
It is clear from Figure 9 that the hydrogen content in T1 abnormally increased from the 45th day
clearfrom
fromFigure
Figure99that
thatthe
thehydrogen
hydrogencontent
contentin
inT1
T1abnormally
abnormallyincreased
increasedfrom
fromthe
the45th
45thday
day
ItItisisclear
to the 63rd day, and surpassed the warning value (37.52 uL/L) so that the failure probability of the
tothe
the 63rd
63rd day,
day, and
and surpassed the warning
soso
that
thethe
failure
probability
of the
to
warning value
value(37.52
(37.52uL/L)
uL/L)
that
failure
probability
of
low-energy discharging (G3) and low-temperature overheating (G1) increased, resulting in the
3) 3and
in the
the
low-energy
discharging
(G(G
the
low-energy
discharging
) andlow-temperature
low-temperatureoverheating
overheating(G
(G1)1 ) increased,
increased, resulting in
increment of the equipment failure probability. With the hydrogen content decreasing after the 63rd
increment of
of the equipment
thethe
hydrogen
content
decreasing
afterafter
the 63rd
increment
equipment failure
failureprobability.
probability.With
With
hydrogen
content
decreasing
the
day, the equipment failure probability decreased to a lower level, and remained lower than 0.3. The
day,day,
the equipment
failure
probability
decreased
to a to
lower
level,
and and
remained
lower
thanthan
0.3. 0.3.
The
63rd
the equipment
failure
probability
decreased
a lower
level,
remained
lower
calculation results are consistent with the actual situation of T1.
calculation
results
are are
consistent
with
thethe
actual
situation
of T1.
The
calculation
results
consistent
with
actual
situation
of T1.
6.4. Case Study 2
The monitoring data of a 220 kV transformer (named T2) in 2015 are listed in Table 5. The overheat
defect was detected in the oil pump on the 96th day, and it became serious after the 99th day.
The diagnosis resulting from three-ratio method is 022 which means the failure mode of T2 is
Energies 2017, 10, 704
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1.0
0.9
Faiulre Probability
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G1
G2
G3
G4
T1
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0.8
0.7
Faiulre Probability
0.6
high-temperature overheating
(G2 ). The content of total hydrocarbon is the sum of CH4 , C2 H6 ,
0.5
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C2 H4 and
H
.
2 2
6.4. Case Study 2
0.4
0.3
1.0
0.2
0.9
0.1
0.8
0.0
0.7
G1
0
G2
G3
50
G4
T1
100
150
200
Time(Day)
0.6
0.5
0.4
Figure 9. The failure probability calculation result of T1.
0.3
0.2
0.1
The monitoring
0.0 data of a 220 kV transformer (named T2) in 2015 are listed in Table 5. The
0
50
100
150
200
overheat defect was detected in the oil pump on the 96th day, and it became serious after the 99th
Time(Day)
day. The diagnosis resulting from three-ratio method is 022 which means the failure mode of T2 is
high-temperature overheating
(GThe
2). The content of total hydrocarbon is the sum of CH4, C2H6, C2H4
Figure9.
9.
failureprobability
probability calculation
calculation result
result of
of T1.
T1.
Figure
The failure
and C2H2.
Table 5. The content of the dissolved gas in T2 (uL/L).
Table 5. The content of the dissolved gas in T2 (uL/L).
6.4. Case Study 2
The monitoring
220 kV transformer
(named
T2) in 2015
are listed in Table 5. The
Day data
H2 of aH
CH
C2CH2H
C H
Total Hydrocarbon
66
2H
2
Day
2 4 CH4
CC
2H
4 4 C2H2 2 Total
Hydrocarbon
overheat defect was detected in the oil pump on the 96th day, and it became serious after the 99th
1st
0 1.6 1.6
00
0.60.6
0
2.2
1st
0
0
2.2
day. The diagnosis
which means
the
31st
146.6 three-ratio
6.6
26 method
1.41.4 is 0022 0
34
14 from
26
31st resulting
34 failure mode of T2 is
high-temperature
(G
2). 110
The content
of190
total 0hydrocarbon
sum of CH4, C2H6, C2H4
73
110
41
0
96thoverheating
96th
73
41
190
341is the341
77
99th
374.8
99th
77120 120 44
44
210210 0.8 0.8
374.8
and C2H2.
95
406.2
105th
95130 130 55
55
220220 1.2 1.2
406.2
76
140
64
230
1.1
435.1
107th
140
64 of the
230
1.1 gas in T2 435.1
Table765. The
content
dissolved
(uL/L).
86
416.1
110th
86137 137 48
48
230230 1.1 1.1
416.1
135 H
320
634.4
Day
2230 CH
2H6
C320
2H
4
C21.4
H2 1.4
Total Hydrocarbon
146th
135
2304 C83
83
634.4
140 0140
240 1.6
736.1
1st
0.6
2.2
196st
240 085
85
410410 01.1 1.1
736.1
31st
14
6.6
26
1.4
0
34
96th
73
110
41
190
0
341
Thefailure
failureprobability
probabilitycalculation
calculationresult
resultof
ofT2
T2isisshown
shownin
inFigure
Figure10,
10,which
whichindicates
indicatesthat
thatfailure
failure
The
99th
77
120
44
210
0.8
374.8
probability
was
increasing
rapidly
from
the
31th
day
to
the
96th
day,
and
the
failure
mode
of
T2
was
probability was increasing
from the
day to1.2
the 96th day,
and the failure mode of T2 was
105th rapidly
95
130
55 31th
220
406.2
G
1
or
G
2
.
After
the
96th
day,
the
failure
probability
of
T2
approached
1,
which
means
that
T2
had
G1 or G2 . After the 96th
day, the
107th
76 failure
140 probability
64
230 of T2
1.1 approached
435.11, which means that T2 had aa
seriousoverheating
overheatingfault.
fault.
The86result
result
isconsistent
consistent
withthe
theactual
actualsituation.
situation.
110thThe
137is
48
230with
1.1
416.1
serious
146th 135 230
83
320
1.4
634.4
196st 140 240
85
410
1.1
736.1
105th
107th
110th
146th
196st
G1
G2
G3
G4
Equipment
Failure Probability
Failure Probability
1.0
The failure probability
calculation result of T2 is shown in Figure 10, which indicates that failure
0.9
probability was increasing rapidly from the 31th day to the 96th day, and the failure mode of T2 was
0.8
G1 or G2. After the 96th day, the failure probability of T2 approached 1, which means that T2 had a
0.7
serious overheating
0.6fault. The result is consistent with the actual situation.
0.5
0.4
0.3
1.0
0.2
0.9
0.1
0.8
0.0
0
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
0
G1
50
G2
G3
100
G4
Equipment
150
200
Time(Day)
Figure
Figure10.
10. The
The result
result of
offailure
failureprobability
probabilitycalculation.
calculation.
50
100
150
Time(Day)
Figure 10. The result of failure probability calculation.
200
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6.5. Comparative Analysis
To make a comparison between the method in this paper and the traditional methods, this paper
presents two types of common fault diagnosis methods based on classifier and the threshold values.
6.5.1. The Fault Diagnosis Method Based on a Classifier
Three kinds of common classifiers are presented to establish the fault diagnosis models,
respectively, namely, decision tree classifier (J48), random forest classifier (RF), and SVM. After
the training process of each model, the accuracy results of the three models are obtained and listed
in Table 6. According to the precision results, the models based on RF and SVM are selected for
fault diagnosis.
Table 6. Precision results of each model.
Day
Precision of the Model
J48
RF
SVM
0.873
0.997
0.952
6.5.2. The Fault Diagnosis Method Based on the Threshold Value
The content and the relative increase rate of the dissolved gas in oil are important criteria for fault
diagnosis. The threshold values of different kinds of gas are listed in Table 7 according to [17].
Table 7. The threshold values of the dissolved gas in oil.
Fault Severity
The Fault Criterion
Slight fault
The content of total hydrocarbon or H2 is greater than 150 µL/L.
The content of C2 H2 is greater than 5 µL/L;
Serious fault
The relative increase rate of total hydrocarbon is greater than 10% per month;
The increase rate of total hydrocarbon is greater than 10 mL per day.
The monitoring data of Case 1, shown in Figure 8, is applied to the fault diagnosis model based
on RF, SVM, and the threshold value. The results indicate the equipment is in a normal state without
any fault, which is consistent with the actual situation. Similarly, the fault diagnosis result of Case 2 is
also obtained and listed in Table 8. It indicates that the failure mode of the T2 is the high-temperature
overheating fault (G2 ), and it becomes serious after the 96th day.
Table 8. The fault diagnosis results of three models.
Day
RF
SVM
Guide
1st
31st
96th~196st
Normal
Normal
G2
Normal
Normal
G2
Normal
Normal
Serious
According to the results, three traditional fault diagnosis methods have high accuracy in
determining whether the equipment fails. However, the model of the threshold value fails to identify
the failure modes for the equipment while the method based on the classifier is unable to determine the
severity of the fault. Furthermore, any traditional methods can not quantify the failure probability of
an equipment failure and reflect the changing trend of the failure probability. Taking T2 as an example,
the fault cannot be found before the 96th day by traditional methods, so that maintenance strategy for
T2 is not be made or arranged in time. The equipment failure may cause serious power interruption,
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and significant losses of equipment and to the power system. These shortcomings of the traditional
models make it impossible to meet the effectiveness requirements for the power system.
7. Conclusions
In this paper, a failure probability calculation method for power equipment based on
multi-characteristic parameters is proposed. The historical data of different fault characteristic
parameters are collected to obtain the cumulative distribution function and the differential warning
value of each parameter. Then the association relations between different failure modes and the
characteristic parameter and the important weight of each parameter are obtained. Finally, the
weighted calculation and the series model are applied to calculating the failure probability of all of the
failure modes and the equipment.
The failure probability calculation method realizes the comprehensive utilization of
multi-characteristic parameters. To avoid the influence of subjective factors on the calculation results,
the process of the fault probability function calculation is driven by data, but not judgment from
experts, so that the results are ensured.
An application example for two 220 kV transformers is presented, and the procedure of the
method is described in detail. The results indicate that this novel method has obvious advantages in
accurately quantifying the failure probability, reflecting the failure changing trend of the equipment
and helping to create a timely maintenance strategy so that the remaining useful life and the reliability
of the power equipment can be markedly increased.
Acknowledgments: The work is supported by the National High-tech R and D Program of China (863 Program,
2015AA050204) and the China State Grid Corporation of Science and Technology Project (520626150032).
Author Contributions: The research presented in this paper was a collaborative effort among all authors.
Hang Liu, Youyuan Wang, and Liwei Zhou proposed the methodology of the model and contributed to the
manuscript at all stages. Ruijin Liao discussed the results and revised the manuscript critically. Yi Yang and
Yujie Geng prepared the data for the paper.
Conflicts of Interest: The authors declare no conflict of interest.
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