
Research Article


Search Method for Wind Turbine Gearbox Failures Based on Grey and Fuzzy Fault Tree Analysis 

Wang Shoubin,
Li Chengwei
and
Sun Xiaogang



ABSTRACT

According to the characteristics of complicated structure
as well as failure probability for shortage, failure mechanism is not clear,
the fault degree are grey and fuzzy properties in Wind turbine gearbox transmission
system. The grey and fuzzy theory were introduced into fault tree analysis method
and the fault tree analysis method of Wind turbine gearbox transmission system
was proposed based on grey and TS fuzzy fault tree analysis. The method used
fuzzy Numbers to describe system failure probability to solve the failure probability
of uncertainty. Using TS logic fuzzy gate to solve the connections and using
grey correlation degree to get fault search order. Example of fault tree analysis
of the Wind turbine gearbox system was given for illustration. Results show
that the method is feasible.





Received: February 19, 2013;
Accepted: May 04, 2013;
Published: July 19, 2013


INTRODUCTION
With the increasing shortage of energy and more serious environmental issues,
wind power as a clean and renewable energy has developed on a large scale (Hameeda
et al., 2009). Fault rate of wind turbines is much higher as the
installed capacity and scale increase much more. Therefore, monitoring and fault
diagnosis of wind turbines have become more important. The inherent uncertainty
and changeability of wind power bring enormous challenge to the operation and
control of transmission system. As a result, fault diagnosis of transmission
system which core is the gearbox has been an important problem to be not neglected
on that of mechanical system in wind turbines. And it has become a burning issue
of wind power (Long et al., 2012).
For the complex structure and the incomplete fault information, gearbox transmission
system of wind turbines can be regarded as a grey system. What’s more,
it is fuzzy because of the indeterminate failure mechanism and probability.
The gearbox fault diagnosis can provide technical support for the equipment
and provide convenience for the maintenance. Vibration signal can mostly reflect
the running state of the equipment of gearbox’s various monitoring signals
(Gao et al., 2011), set vibration signal as the
research object and draw whether the gearbox operation is normal after the time
and frequency domain analysis (Gao et al., 2011;
HongShan et al., 2012; Yang
et al., 2011).
Considering the gray characteristic and fuzziness, this study applies the theory
of grey system and fuzzy mathematics and combines that with the method of fault
tree analysis to find the failure probability and grey correlation of minimal
segmental sets. Then the correlation sequence and the relative approach degree
of ideal solution of fault search are obtained, so as to determine the order
of fault search.
BASIC TS FUZZY THEORY Trapezoidal subordinate function: In the analysis of system, fuzzy numbers
and linguistic values are always used to represent the state of the system or
components. The trapezoidal subordinate function of fuzzy number is shown in
Fig. 1 and its expression is:
where, F_{0} the center of fuzzy number supporting set is, a is the
supporting radius and b is fuzzy region. As a is equal to 0, trapezoidal subordinate
function becomes into a triangle. While b equals zero, the fuzzy number is some
definite value.

Fig. 1: 
Fuzzy subordinate function 
TS fuzzy gate algorithm: TS fuzzy gate is composed of IFTHEN rules.
Assume that the failure degree of bottom events x_{1}, x_{2},…,x_{k}
and top event y are represented in fuzzy numbers as, (x^{1}_{1},
x^{1}_{1},…,x^{n1}_{1}), (x^{1}_{k},
x^{1}_{k},…,x^{nk}_{k}) and (y^{1},
y^{2},…y^{ny}) where:
If the rule l(l = 1, 2,…, m) is known and the fault fuzzy probability
of the bottom events are p(x_{1}^{il}), p(x_{2}^{i2}),…,
p(x_{k}^{ik}), the execution probability of the rule l is:
And fuzzy probabilities of superior events are:
If the bottom events are known as x’ = (x’_{1}, x’_{2},…,
x’_{k}), the fault fuzzy probabilities of superior events can be
estimated by TS model as followed:
where,
is the fault degree of the component j in the rule l which also referred to
as the membership of fuzzy sets x’_{j}.
GREY SYSTEM THEORY
Grey system theory is a new method to study the issues of uncertainty and lack
of data or information. Through the main relationship in various elements (or
subsystems) of systems, the important factors affecting the target value is
found out which promotes and guides the coordinated development of system rapidly,
efficiently and healthy.
Correlation degree is referred to correlation changing with time between two
systems or two factors of the same system. Grey correlation grade is the index
of the similarity between two grey systems.
In a case with two series {X_{i}(t), X_{j}(t)}, as t equals
k, correlation degree is:
Where:
where, ε_{ij}(k) is the correlation coefficient and Δ_{ij}(k)
is absolute difference of the two series at the moment k. Δ_{max}
is the maximum absolute difference Δ_{min} and is the minimum one.
ρ is the distinguishing coefficient.
FAULT SEARCH ALGORITHM BASED ON GREY THEORY AND FUZZY FAULT TREE
Based on grey theory and fuzzy fault tree, fault search steps are illustrated
as following:
• 
Minimal segmental sets are found out according to the system
fuzzy fault tree 
• 
Based on the TS fuzzy algorithm, the critical importance of every bottom
event is calculated and the pattern vector to be tested is determined 
• 
Fault feature matrix is established 
• 
Dimension of the original data is eliminated and the data is converted
into comparable data columns 
• 
Grey difference, Δ_{oi}(k), is calculated. Δ_{oi}(k)
is the absolute difference of two comparing series at the moment k, i.e.,
Δ_{oi}(k) = x_{O}(k)x_{i}(k) 
• 
Correlation coefficient, ε_{ij}(k), is calculated: 

where, ρ is the distinguishing coefficient, generally
taking 0.5 
• 
Correlation degree, γ_{i}, is calculated: 

where, γ_{i} is correlation degree of series
x_{i} and x_{0} series. Its value is greater and the relationship
between x_{i} and x_{0} is more closely 
• 
Correlation order is arranged. The fault search order is obtained by comparing
the failure occurring probability of bottom events as the failure occurs
in the top events 
EXAMPLE ANALYSIS
Establishment of TS fuzzy fault tree of gearbox transmission system in
wind turbines: Data shows that the failure ratio of gear to the gearbox
reaches up to 60% (Tavner et al., 2007; Ribrant
and Bertling, 2007). Consider the wind turbines of a wind farm in a Zhangbei
County, Zhangjiakou City. Without regard to work environment and human factors,
the TS fuzzy fault tree is established in which the top event is that the gear
transmission system can not operate normally, as shown in Fig.
2.
In Fig. 2, the top event, y4, is the output of TS gate 4
which represents that the gear transmission system can not run. Intermediate
event, y1, is the output of TS gate 1 representing the gear teeth failure.
The event y2 as the output of TS gate 2 is represented the gear wheel failure.
The output of TS gate 3 is the intermediate event y3 and it is the bearing
failure. X1, X2, X3, x4, x5, x6, x7 and x8 are the components’ partial
failures of gear transmission system while y1, y2, y3 and y4 represent the system
partial failures of gear transmission system. Consider that the common fault
degree is (0, 0.5, 1). In this case, 0 represents no failure which means that
the system can work normally. 0.5 means half fault or mild fault which means
that the system can work partially.

Fig. 2: 
Tree of TS fuzzy fault 
Table 1: 
Names of the events 

Table 2: 
TS gate 1 of the rules 

1 which represents the complete failure or serious fault, means that the system
does not work. Names of the events are shown in Table 1.
Considering the trapezoidal subordinate function shown in Fig.
1, let the parameter a equal 0.1 and the parameter b equal 0.3. According
to the relevant data of the Dali wind field in Inner Mongolia and the Zhangbei
wind field in Zhangjiakou (MingMing, 2009), the TS
gate rules are established with the combination of expert experience and human
estimation.
Each row in Table 25 represents a fuzzy
rule. As shown in Table 2, the first row represents that if
x2, x3 and x4 are all 0, the possibility that y5 equals 0 is 1 while the possibility
y5 being 0.5 or 0 is 0 and so on.
Table 3: 
TS gate 2 of the rules 

Table 4: 
TS gate 3 of the rules 

Table 5: 
TS gate 4 of the rules 

Table 6: 
Importance degree of each bottom event 

Analysis of fuzzy importance degree: An important index of fault tree
analysis is importance degree, in which critical importance can reflect not
only the status of bottom events in the fault tree of the system, but also the
uncertainty of the event itself. What’s more, it can reflect more objectively
the impact of components on the system fault tree (Yao
et al., 2011; Yao and Zhao, 2009) and calculating
method of critical importance is illustrated in the reference (Song
et al., 2005; Ni et al., 2008; YongJian
et al., 2010). The importance degree of each bottom event is given
in Table 6.
Failure searching based on the grey theory and fuzzy fault tree:
• 
Finding out the minimum cut set: According to the fault
tree shown in Fig. 2 and the ascending method, the minimum
cut set is found out as following: 
T_{k1} = {1}, T_{k2} = {2}, T_{k3}
= {3}, T_{k4} = {4}, T_{k5} = {5},
T_{k6} = {6}, T_{k7} = {7}, T_{k8} = {8} 
• 
Determination of the pattern vector to be tested: On
the basis of the critical importance of each bottom event, the test pattern
vector is: 
X_{n} = {0.0744567, 0.2233701, 0.0372284,
0.0956897,
0.1913793, 0.1381771, 0.1381771, 0.1015014} 
• 
Establishment of fault feature matrix: The number of
bottom events is 8, i.e., n = 8. In the feature matrix, let the bottom events
comprising the minimal cut set equal one and the others equal zero. Then: 
• 
Initialization of original data: The original data
is initialized as following, considering the test pattern vector, X_{n},
as the maternal factor and the minimum cut set, T_{ki}, as the subfactor: 
T_{ki} = {1, 3, 0.5, 1.2851725, 2.5703436,
1.8558048,
1.8558048, 1.3632272} 
• 
Calculation of the difference sequence Δ_{zki}(k):
Difference sequence is defined as the absolute difference between the two
comparison series 
Δ_{zki}(k) = X_{n}(k)T_{ki}(k) 

where, i = 1, 2…8 and k = 1, 2…8. Then the biggest
difference is 3 and the smallest difference is 0 
• 
Calculation of the correlation coefficient: 

where, ρ = 0.5, the grey correlation coefficient is shown
in Table 7 
Table 7: 
Grey correlation coefficient 

• 
Calculation of the correlation degree: 
Then:
γ_{k1} = 0.546214, γ_{k2}
= 0.512939, γ_{k3} = 0.501035,
γ_{k4} = 0.538746, γ_{k5} = 0.516038, γ_{k6}
= 0.548469,
γ_{k7} = 0.501035, γ_{k8} = 0.536181 
• 
Arrangement of correlation order and obtaining of fault
search order: The correlation order of X_{ki} to X_{n}
is: 
γ_{k6}>γ_{k1}>γ_{k4}>γ_{k8}>γ_{k5}>γ_{k2}>γ_{k3}
= γ_{k7} 
As a result, eight failure modes leading to the abnormal operation of gear
transmission system in wind turbines have the sequence of occurring possibility
as (from high to low, no comma meaning same) {X_{6}}, {X_{1}},
{X_{4}}, {X_{8}}, {X_{5}}, {X_{2}}, {X_{3}}
{X_{7}}. That is the fault search order.
CONCLUSION
According to the fuzzy and gray characteristics of the gear system fault in
wind power gearbox, the uncertainty of fault probabilistic is determined effectively
by the description of system failure probability with fuzzy number; and the
fuzzy characteristics caused by the ambiguous relationship between events and
failure mechanism are solved instead of the traditional fault tree with TS
fuzzy logic gates. Using grey system and fuzzy theory and combined with the
fault tree analysis method, grey correlation degree of the minimum cut sets
is calculated and correlation sequence is obtained on the basis of the failure
analysis of the abnormal operation of gear transmission system in wind turbine
gearbox. Thus the fault search order is determined. And the research provides
a theoretical basis for the order dealing with accidents, control of accidents
occurrence and improvement of system reliability.
ACKNOWLEDGMENT
This study was supported by the National Natural Science Foundation of China
(No. 61071036) and the Tianjin Science and Technology Development Foundation
for Colleges and Universities (No. 20110713).

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