The Study of Large Equipments Condition Maintenance Policy

based on Kalman Filtering

Shuai Zhang 1, a, Jian Zhang 2, b, Shuiguang Tong 3, c, Chaowei Wu 4, d

1~4Institute of Thermal Science and Power Engineering, Zhejiang University, Hangzhou;310027,

China

aqingdaozhang2011@126.com, bjian_zhang_zju@zju.edu.cn,

ccetongsg@zju.edu.cn, d21127085@zju.edu.cn

Key words: PHM; Kalman Filtering; Reliability Estimation; Condition Based Maintenance

Abstract. Aimed at the problem that because common proportional hazards model (PHM) cannot

fuse new failure data of long-life complex equipment, which features a small-sample, the reliability

estimation accuracy will decline, a new condition-based maintenance strategy based on dynamic

PHM was proposed. Kalman filtering theory was adopted to fuse in-time new failure data and

expand sample size. Extended Kalman filtering method was used to solve the nonlinearity of the

observation equation of PHM and then its regression coefficient was online updated, according to

which the residual life was estimated and the optimal maintenance decision was made. Finally, the

condition monitoring data and historic operation data of a certain kind of wind power gearbox were

used to validate this method. The result indicates that this method has good dynamic estimation

ability under the condition of small sample with a 20.6% increase in the accuracy of regression

coefficient estimation and a 8.7% decrease in optimal preventive maintenance interval estimation

error.

0 Introduction

The reliability estimation accuracy during the operation period of mechanical equipment

determines the downtime and maintenance cost. It is difficult to obtain a lot of failure data in a short

term, so adopting classical probability theory cannot obtain satisfied reliability estimation accuracy.

Due to comprehensive consideration of performance parameters, failure types and maintenance

history data, CBM is becoming the main method in maintenance decision. The existing CBM

method can fuse condition monitoring data into reliability estimation, but reliability estimation can

not be able to take advantage of these new failure data. The small sample problem of historical

failure data can’t be resolved radically [1,2]. Therefore, reliability estimation accuracy during

operation period is becoming a bottleneck of improving the equipment safety and economy.

Regression coefficient in PHM based on weibull distribution is generally estimated by

maximum likelihood method and least square method. These methods based on a great sample data

are static parameter estimation methods. For reliability estimation of complex equipment, which has

a long design life and small sample characteristic, these methods have defects in parameter

estimation accuracy, information fusion ability and self-learning ability. In this paper based on

CBM method, kalman filter was adopted to fuse new failure data into PHM, expanding the sample

size of failure data, and eventually reducing the influence of small sample on reliability estimation

accuracy. In order to eliminate the nonlinearity of the measurement equation, EKF iterative

Advanced Materials Research Vols. 706-708 (2013) pp 2128-2132Online available since 2013/Jun/13 at www.scientific.net© (2013) Trans Tech Publications, Switzerlanddoi:10.4028/www.scientific.net/AMR.706-708.2128

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algorithm was used to dynamic parameter estimation. Based on results of reliability estimation, the

optimal preventive maintenance interval of equipment was estimated and optimal maintenance

strategy was made. Finally, condition monitoring data and historical operating data of a certain

wind turbine gearbox were used to validate this method.

1. Kalman filter model based on PHM

1.1 Proportional Hazards Model.

PHM, which was used for joint covariant risk analysis, was proposed by Cox in 1972. Recently,

because it can consider the system internal state, the operating environment, the size of the load,

and the influence of historical running data on system life, PHM has become a research hotspot in

reliability field[3].

The basic idea of PHM considers that failure rate function of different individuals is

proportional to each other. A mathematical mapping between operating state parameters and failure

rate is estimated by PHM. The mathematical expression of PHM:

0( / ) ( ) exp( )h t h t= γγγγZ Z (1)

In the model, ( / )h t Z ——failure rate function

0 ( )h t ——basic failure rate function, only related to time

Z——covariate, reflecting the running state of the equipment

γγγγ ——regression coefficient, indicating the influence of covariate on failure rate

In the model, 1 1 2 2 n nZ Z Zγ γ γ= + + +�γγγγ Z .The weibull distribution function, which has strong

adaptability to three failure period of the product life "bathtub curve" and can fully reflect the

influence of material defects and stress concentration source on fatigue life, was used as basic

failure rate function. It has widely used for reliability analysis of mechanical equipment in

engineering practice. Generally, the weibull distribution function can be obtained in delivery test.

PHM based on weibull distribution [4]:

1( / ) ( ) exp( )t

h tββ

η η−= γγγγZ Z (2)

In the model, β —shape parameter of weibull distribution

η—scale parameter of weibull distribution

1.2 Parameter Estimation based on EKF.

The covariate Z, which is obtained by condition monitoring, contains the latest equipment

information. Regression coefficients can be estimated by using this information. With the constant

enlargement of sample data, γγγγ can be online updated. Considering γγγγ as state variable, Z as

observation variable, Kalman filtering model was established. In order to eliminate the influence of

different orders of magnitude between covariant data, "maximization processing" was used for

dimensionless condition monitoring data.

It is assumed that the estimated value of γγγγ is [ ]^

1 2( )n

k γ γ γ= �γγγγ at time k . A group of

covariates monitoring data [ ]1 2( )n

k Z Z Z= �Z was got, according to kalman filtering theory:

Advanced Materials Research Vols. 706-708 2129

^ ^

( ) ( 1) ( 1)k k k= − + −γ γγ γγ γγ γ w (3)

( / ) 1

( ) ln ( )( ) ( )

h tk k

h t k= +

γγγγZ

Z v (4)

Let ( )kg be( / ) 1

ln( ) ( )

h t

h t k

Z

γγγγ.

In the equation, ( 1)k −w and ( )kv represent Gaussian white noise added to the system. Their

mathematical expectations are both 0 and their covariance matrix are ( 1)k −Q and ( )kR respectively.

Obviously, γγγγ is nonlinear with Z in observation equation. In order to meet the linear

requirement of kalman filtering, EKF was used for first order Taylor expansion in the optimal state.

( ) ( ( )) ( ) ( ) ( ) ( )k k k k k k

∧ ∧ = + − + Z g g vγγγγγ γ γγ γ γγ γ γγ γ γ (5)

Where ( )kgγγγγ is the Jacobian matrix of ( )kg

Gain matrix of Kalman filtering is:

T( 1) ( 1| ) ( 1)k k k k+ + + ×γγγγK = P g1

T( 1) ( 1| ) ( 1) ( 1)k k k k k−

+ + + + + γ γγ γγ γγ γg P g R (6)

Where ( 1| ) ( | ) ( )k k k k k+ = +P P Q , ( | )k kP is the predicted covariance matrix of influence

coefficient at time k . The filtering estimation equation:

( 1| +1) ( +1| ) ( +1)k k k k k

∧ ∧

+ = + ×γ γγ γγ γγ γ K ( 1) ( 1) ( 1| )k k k k

∧ + − + + γγγγ γγγγZ g (7)

The renewal equation of filtering covariance:

( 1| 1) ( 1| )k k k k+ + = + ×P P ( 1) ( 1)k k − + + I K gγγγγ (8)

2. CBM Policy based on Maximum Availability.

If we define the preventive maintenance cost asPMC , the corrective maintenance cost as

CMC ,

and the total cost of production asC , we can determine the optimal preventive maintenance interval

PMT so as to minimize the total cost of production per unit time.

The expectation of the total cost of production per unit time is represented by:

( )

( ) PM CMC C n tC

Et t

+= (9)

and ( )n t is the mean value of failure number in t.

1

0( ) ( ) exp( )

t un t du

ββη η

−= ∫ Zγγγγ (10)

According to equation (9), (10), the optimal preventive maintenance interval PMT can be got:

1

( 1) exp( )

PM

PM

CM

CT

C

β

ηβ

=

− Zγγγγ (11)

3. Experimental Verification

As typical large equipment, wind turbine gearbox generally has a design life of 20 years. It is

difficult to get a lot of failure data in a short term, so studying dynamic parameter estimation in the

case of small sample data is of great significance to reliability and life prediction.

In this paper, the W2000DF 50 Hz TCIII wind turbine gearbox in a certain wind farm is taken

as the research object and lubricating oil temperature (T) is selected as the condition monitoring

variable [5]. The weibull distribution function of this gearbox can be obtained in its delivery test

( 3.5301β = , 523.2544η = ). According to the 180 groups of historical operating data from April to

September in 2011, we got that the actual value of γγγγ was 0.30 and the actual gearbox optimal

preventive maintenance interval was 71.6d by using classical probability estimation method.

2130 Mechatronics and Intelligent Materials III

Tab. 1 Dimensionless condition monitoring data state variable ten groups of new condition monitoring data

T 1.2901 1.1668 1.3459 1.3621 1.4187 1.3524 1.3522 1.3396 1.3661 1.3508

Tab. 2 Dimensionless historical operating data state variable ten groups of historical operating data

T 1.4035 1.2270 1.4235 1.1905 1.6529 1.2821 1.4286 1.5152 1.2821 1.4706

In order to verify the dynamic estimation ability of this method under the condition of small

sample, two methods were adopted to estimateγγγγ .Then we can calculate the optimal preventive

maintenance interval, according to the estimation results. Method 1: Ten groups of new condition

monitoring data (shown in table 1) were selected and thenγγγγ was online updated by adopting EKF.

Method 2: Ten groups of data (shown in table 2) were selected randomly from the 180 groups of

historical operating data and then γγγγ was estimated by classical probability method. In method 1,

measurement error existing in the condition monitoring and quantization error of non-linear

processing were respectively integrated into system noise Q and measurement noise R. In this

article 0.1Q = and 0.1R = . Initial value of state variables and prediction covariance have little effect

on the estimation results, because kalman filter always reach convergence. Assume

that 0.37=γγγγ , 1P = [6,7]. The estimation results of γγγγ and root mean square error obtained by two

methods are presented in Figs.1 and 2.

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

0.50

0 1 2 3 4 5 6 7 8 9 10

the sample number

the

esti

mat

ed v

alu

e o

f

reg

ress

ion

co

effi

cien

t

method 1

method 2

Fig.1 Dynamic estimation result of regression coefficient

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

1 2 3 4 5 6 7 8 9 10

the sample number

the r

oot

mean s

quare

err

or

method 1

method 2

Fig.2 Root mean square error between estimated value and actual value

As can be seen from fig.1, the initial value of γγγγ has little effect on estimation results. Even if

there are larger errors between initial value and actual value, estimation results can quickly

converge to actual value nearby after 10 EKF iterations. Although adopting method 2 can also

achieve parameter estimation, the estimation results are disordered divergence. In fig.2, by adopting

method 1, the root mean square errors between initial value and actual value decrease gradually

Advanced Materials Research Vols. 706-708 2131

with the increase of sample number. However, root mean square errors have no obvious decreased

trend by adopting method 2. When sample number is 10, the root mean square errors are 2.47% and

3.11% respectively and the accuracy of regression coefficient estimation is increased by 20.6%.

For this gearbox, the ratio of preventive maintenance cost to corrective maintenance cost is

1/300. According to the equation (11) and the estimation results ofγγγγ , the optimal preventive

maintenance interval can be got. The calculation results are 69.5d and 73.9d respectively and the

estimation error was reduced by 8.7%.

4. Conclusion

This paper presents a new condition-based maintenance strategy based on dynamic PHM. EKF

theory was adopted to update the regression coefficient in PHM. According to updated regression

coefficient, the optimal preventive maintenance interval can be calculated. The state monitoring

data and historic operation data of a certain kind of wind power gearbox were used to validate this

method. The result indicates that the accuracy of regression coefficient estimation increase by

20.6% and the optimal preventive maintenance interval estimation error decrease by 8.7%. So, the

suggested method has certain significance for large equipment reliability research.

Acknowledgments

This study was partially supported by science and technology Major projects of Zhejiang

province. We are grateful to the assistance of Jian Zhang, corresponding author of this paper, for his

assistance in the preparation and review of the manuscript.

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2132 Mechatronics and Intelligent Materials III

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