Reliability growth assessment under several operating stress conditions
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Key Words: Reliability growth, Accelerated testing, Reliability, Operating stresses, Cox’s model, AMSAA, Rocket engine SUMMARY & CONCLUSIONS
Developed by the common research program of CNES and SAFRAN-SNECMA, this paper presents a new methodology for reliability growth model during development tests plan in which the reliability is estimated on the rocket engine by integrating the operating conditions. The proposed method evaluates the reliability of mechanical systems under variable constraints which contain the information of failure modes observed on the materials, the condition of repairs, the operating data linked to each development test.
Applying to rocket engines data provided by SAFRAN- SNECMA society, the new reliability growth model gives better performance than the case of classical reliability growth model. The obtained result by a case study simulation illustrates the impact of operational data to estimate the reliability growth of system. The estimations obtained on real data on rocket engines allowed to emphasize the results of reliability estimated during operational flights. This reliability growth estimation method is thus valued in term of industrial application.
1 INTRODUCTION
In the domain of aeronautics, aerospace or military, reliability is an important indicator for development stage of systems. To achieve requirements formulated by the Specifications of Management and standards ECSS, SAFRAN-SNECMA company has developed reliability plans in order to build, manage and estimate the reliability of products [1]. The rules of the followed RAMS approach remain traditionally the identification and the elimination of the known or potential critical points. This approach leads to an improvement of the intrinsic reliability of the rocket engine during development tests plan.
The reliability estimation is made from the knowledge accumulated by development (cycles, duration) and failures observed during tests. Due to the technical improvements brought to the design throughout a development andthe implementation in practice of a test-fix-test approach, the reliability growth assessment is of the most relevant problematic. For such a complex mechanical system, as rocket engine, the most popular reliability growth model, AMSAA
[2], has been developed by US Department of Defense.
However, it's not taken into account the realistic assumptions of different failure modes, which occur from different materials during a development tests plan, and especially the effects of operational and environmental stresses.
Practically, the rocket engine development test divides to different stages such as tuning, maturation, qualification. Each stage may be developed with different mission profiles with particular operational setup in order to validate several requirements on development test plan. The mission profiles of each test are generally divided in three domains: operational flight, qualification and flight in extreme conditions.
Moreover, a mission profile contains several individual phases of functional point characterized by sixty main parameters.
These parameters are strongly correlated due to less degree of freedom associated to a system design (control valves, IHM conditions...). In fact, a functioning point of rocket engine can be resumed by two typical parameters: the global flow and a ratio of mixture between Oxygen and Hydrogen.
Therefore, the functioning data can be used to accelerate the reliability improvement during test plans and to reduce the cost of test plans in order to obtain the requirements of reliability improvement. Recently, it can be cited the work of Krasich [3] developing the reliability growth model which has applied operational stresses and environmental stresses under different measurements (vibration, thermal cycling, thermal dwell, humidity). However, for such complex system as rocket engine, the operational conditions, characterized by several variables, cannot be directly correlated to physical characteristics by usingan accelerated law associated with failure modes.
In order to develop a reliability growth model which can take into account the operating stresses conditions to reliability estimation on the industrial demand of SAFRAN- SNECMA society, this paper addresses to proportional hazards model (PHM). This model has developed by Cox which derives robust, consistent, and efficient estimates of co- variate effects. Based on reliability studies during design phases and on the real data base of rocket engine, the influence of different operating stresses to reliability calculation is consequently considered. Afterward a corrective factor is modeled by Cox's proportional hazards model.
Technically, in the framework of industrial application the operational variables can be preliminary analyzed and classified by statistical approaches. Principal component analysis (PCA) is used to insure effectively the number of principal components before using Cox's model and an associate statistic test (Wald test, Likelihood Ratio test, etc.).
A graphical approach is performed on real data to check proportional hazard assumption.
On repairable systems, after a failure observed under hypothesis that materials are replaced as new or upgraded, the system reliability is improved during development tests by a growth factor corresponding to the system global evolution dependent of each redesign on the fault equipment configuration [4]. In order to asset the productive quality after the development tests plan, the mission reliability of a reference flight is evaluated. On the order hand, the industrial requirement considers the impact of functioning points to reliability growth assessment. Therefore, the Cox's proportional hazards model is integrated to reliability parameters estimation and to mission reliability calculation.
The development for integrating these assumptions on the classic AMSAA model is also carried out in this work.
In the remaining part of the paper, the theory development of Cox's model is firstly presented with parameters estimation and interval confidence. Thereafter, the reliability growth model is considered with AMSAA Crow model and with taking into account functioning data to estimate the reliability parameters in the next section. In order to point out the interest of proposed approach, a case study is next shown, with the simulation results and remarks, to evaluate the proposed model. Finally, in the purpose of validation of our methodology, an industrial application of rocket engine data is detailed.
2 PROPORTIONAL HAZARDS MODEL
The influence of operational conditions is an important factor to estimate more accuracy the system reliability. In the literature, this problematic has been investigated by many authors. For example, Singpurwalla in [5] has considered the items time-to-failure modeling under dynamic environmental conditions. Guérineau in [6] developed the proportional hazard model for piecewise failure rate under environment conditions (temperature and humidity) stresses. However, the influence information of physical parameters is not practically available for the complex system materials. In the purpose of taking into account the operational stresses to reliability growth estimation, the Cox model of proportional hazard risks is ideally considered. The base of Cox's model can be cited in several reference, such as [7] [8] [9] and [10]. In this case, the failure intensity ( , )λ t Z depends on the operating measurement co-variables Z.
Let λ0( )t be the baseline intensity which does not depend on the co-variables, the failure rate is defined as:
( , )t 0( ) exp(t h )
λ Z =λ Z (1)
where exp(hZ is the Cox's risks function as exponential ) distribution, h is the weight vector ( ,h h1 2,...,hk) associated to
each of co-variables vector Z.
2.1 Cox's parameters estimation
In order to estimate the risks function, we consider the partial likelihood developed in [11] to model the failure times.
The tests on rocket engine correspond to reparable system and provide data with ties between failure times.
Let the during ( , ,..., )t t1 2 tn which contains r failures and censured times (n-r), by assembling the failure times into the same vector ( , ,..., )t t1 2 tr , an approximation of the partial likelihood function following Breslow [12] is:
1
exp( )
( ) [ exp( )]
i i
i T r l
l T d
i j
j
h Z
L h h Z
∈
=
∈
=
∏ ∑ ∑
D
R
(2)
where d is the failure number at i t , and i R is risk set at i time t i.e.i Ri ={ :j Xj ≥ti}.
The log-likelihood function is:
1
log ( ) { log[ exp( )]}
i i
r
T T
l i j
i l j
L h h Z d h Z
= ∈ ∈
=
∑ ∑
−∑
D R
(3) Applying the likelihood maximization (ML), we can obtain the estimated regression parameters of Cox. The confidence interval of Cox's parameters is also deduced from variance-covariance matrix by inverting the Fisher information matrix. Considering the parameters vector
( ,...,1 p)
h= h h , the Fisher information matrix F can be obtained as:
11 12 1
21 22 2
1
...
...
... ... ... ...
... ...
p p
p pp
f f f
f f f
F
f f
=
(4)
where
2
, , 1,..., .
ij
i j
f L i j p
h h δ
= −δ δ = (5)
where L is likelihood function.
2.2 Hypothesis testing
Supposes that we consider q components of regression parameters vector h, h=(h h1T, 2T)Twhere h is of dimension 1 q and h is of dimension2 p q− . In order to test the estimation by partial likelihood, the likelihood ratio (LR) criterion is studied by testing the null hypothesis H0:h1=h0 against
1: 1 0 0
H h ≠ ∀h h fixed.
2[log ( ) log ( )]ˆ
TSLR= L h − L h% (6)
where h%=( ,h h0 %2) and h%2 maximizeL under condition h is 1 fixed by h . Under null hypothesis, the asymptotic distribution 0 of TSLR is a Chi-squared law of q degree.
By using the presented Cox's model, the observed operating measurements are studied and taken into account Cox's factor exp(hZ . Estimating independently, these Cox's ) parameters h, which represent the performance of operational
data. These parameters are integrated to reliability growth estimation in order to evaluate the impact of operational data on reliability improvement. In the next section, the reliability growth model is presented by considering the assumptions above.
3 RELIABILITY GROWTH ASSESSMENT 3.1 Development test plan
For cost reasons, the development tests take place with a limited number of engines following several operating cycles [13].
During fault detection, repair is insured by replacing the failing materials by new sub-systems as before. Then an improvement of the sub-system is scheduled to correct the observed defects. In order to improve the system reliability during development test plan, some failure correction policies are proposed in the framework of industrial requirements [14]:
• Immediate correction: applied in the case of blocked failure, the test plan is stopped until repairing the failed materials.
• Correction to the next configuration: in the case of uncritical failure, the correction will be applied to the next configuration (new testing system) during development test plan.
• Correction following a delay period: the observed failure is not critical to system operation; the correction is indeed to delay due to strong constraints of development test plan.
This approach leads to an improvement of the intrinsic reliability of rocket engines during development test plan. In the general framework of the development test plan, the various growth factors of reliability are bound to:
• technical improvements on design,
• test-fix-test process on observed failures,
• acquired experience on control valves during tuning phase.
Because of the expected and effective rarefaction of incidents, the implementation of a reliability growth model is then of the most relevant [1].
3.2 AMSAA models
Reliability growth models are developed to extract the growth process observed and afterward to define the applicable performance indicators to reliability. Based on the observed incidents, the system reliability is improved by redesigning the failure equipment configuration. In the literature, the most of growth model is based on the working of Duane [15], in which the failure intensity gives back the system performance by exponential evolution. One of usual reliability growth models, AMSAA-Crow, developed by Crow [2], considers the failure number during the development times as a non-homogenous Poisson process [4].
( ( ))
E N t =λtβ (7)
In equivalent of Weibull distribution by considering 1/ β
λ= η . Hence, β and η are respectively the shape
parameter and scale parameter. The shape parameter characterizes the form trend of reliability evolution, and1−β, the growth rate indicates the improvement degree or system degradation during development time.
Following the development test, the AMSAA model allows to estimate a mission reliability of d duration at each development time t by the relation:
{ }
( ) exp [ ( ) ]
R t = −λ t+d β −λtβ (8) In the case of development test plan for the system that contains multi-components in parallel, AMSAA multi- components is considered that allows to evaluate the reliability growth in function of cumulative test times [4] [2]. In fact, the failure correction has no effect to intensity discontinuity that corresponds to minimal repair. Hence, this model is used to unique repairable system. In the next section, AMSAA multi- components model is improved by taking into account the functioning data in failure rate to reliability estimation in order to asset the impact of operational conditions to mission reliability.
4 METHODOLOGY
The failure intensity of reliability growth model (ex.
AMSAA) is not constant and depends on development times.
In order to estimate the reliability parameters by taking the Cox's coefficients estimated in advance, the following hypothesis is proposed.
4.1 Hypothesis
The Cox’s factor estimated by the partial likelihood in the equation (1) does not depend on the times on a cycle of operating data. Following [16], the failure intensity can be estimated by different cycle of operating data and λ0( )t is supposed to be constant between the operating data cycles. It can be deduced that the regression coefficients are estimated independently in term of reference failure rate.
4.2 Reliability growth parameters estimation by taking into account Cox's coefficients
In order to estimate reliability growth parameters by taking into account the Cox's coefficients associated to operating data, the failure intensity is decomposed by the different operating data stages. Supposes that a development test contains p stages corresponding to q operating data, Zj
r is the vector of operating data at timet , the change time at stage j j, 0≤ ≤j p, with the failure intensity at t is defined as:
( )
( )t 0( ) expt h Z. j
λ =λ (9)
where t∈[tj−1,tj]∀ ≤ ≤1 j p and λ0( )t =( /β ηβ).tβ−1for the case of Weibull distribution. This model corresponds to a stratified Cox model in [7] (Chapter 21.4).
Let ,t ii =1,...,n be the failure times, and θr is parameters vector that needs be estimated, the likelihood function is:
1
1 1 1 1
( ,..., ) 1
( , ,..., , ) ( | ,..., )
( ,..., , )
n
n n n n
T T n
L t t t P t T T t T t
f t t
θ
θ
= < + = =
× r
r (10)
By taking account the operating data, the density of failure times is the derivation of failure probability function.
{
1}
, ,
, 1
1
,0 1
( | ) 1 exp ( )
1 exp ( )
i
i i
p
i p i p
i p t
T i i t i
N t hZ
t i p
F t t t dt
t e dt λ
λ
−
−
−
=
= − −
= − −
∫
∑∫
(11)where
• h is Cox coefficients vector
• N is the number of operating data betweenp (ti−1, )ti ,
• Zi p, ,p=1,...,Np are the operating data vector at t , i p,
• t are the change times of operating data stages between i j,
1
ti− and , 0ti ≤ ≤j p, with ,0 1, ,
i i i Np i
t =t− t =t.
The conditional density is calculated by deriving failure probability function:
, , ,
, 1
1 1
,0 ,0
1
( | ) ( | )
( ) exp ( )
i i
p
i N p i p i p
i p
T i i
T i i
i
N t
hZ hZ
i i t i
p
F t t f t t
t
t e t e dt
δ δ
λ λ
−
−
−
=
=
= −
∑∫
(12)
Therefore, the likelihood can be rewritten as:
,
, ,
, 1
. ,0 1
1 .
,0
1 1
( )
exp ( )
i N p
p
i p i p
i p
n h Z
i i
i
n N t h Z
t i
i p
L t e
t e dt λ
λ
−
= +
= =
=
× −
∏
∑∑∫
(13)
4.3 Application to AMSAA model
In the case of AMSAA multi-components model, the reference failure rate isλ0( )t =λβtβ−1, equivalent to Weibull distributionλ0( )t =( /β ηβ)×tβ−1.
Hence, the log-likelihood function for AMSAA model is:
( )
,
, 1
1 , , 1 .
1 1
ln ln ( 1) ln .
( )
p
p
i p n
i i N
i n N
h Z
i p i p
i p
logL t h Z
t t
e
β β
β
β β η β
η
=
+ −
= =
= − + − +
− −
∑
∑∑
(14)
By maximizing of log-likelihood function, the optimal parameters of reliability growth model are obtained, noted
{ , }β ηˆ ˆ =Argmax{ ( , )}L β η . 4.4 Mission reliability calculation
From the estimated parameters, the mission reliability of a reference flight at time t, the end of development test plan, is calculated with (8) as:
0 0
0
( )exp(ˆ )
( ) exp(ˆ )
( )
t Tmission t
mission
s h
mission
t T t
h
R t e ds
e
β β β
λ
η
+
+ −
− ×
= ∫
=
Z
Z (15)
where λ0( )t is the mission failure rate, Z is operating data of 0 a mission flight, Tmission is the mission duration. The mission reliability indicator in (15) is used to validate the reliability growth model after finishing the development tests plan.
Hence, the reference functioning data Z is chosen and taken 0 into account the reliability calculation to illustrate for field conditions of a mission flight.
5 CASE STUDY : SIMULATION
In the case study, a simulation example is illustrated in order to validate the proposed methodology in section 3. The Cox's parameters are fixed for simulation. The operating stresses are generated for each cycle of the plan following a standard normal distribution. Considering a typical development test plan of rocket engine system with the parameters of plan as follows:
• 4 development test phases: tuning, maturation, qualification, reference.
• 15 engines following the phases respectively [4 4 2 5].
• 240 development tests for the plan.
As discussed in section 1, a functioning point of rocket engine can be described by two typical parameters: the global flow and a ratio of mixture between Oxygen and Hydrogen.
So the simulation are made with these two parameters, named respectively Z and 1 Z . 2
On the other hand, the reliability growth parameters of AMSAA model are chosen as:
• Shape parameterβs =0.7;
• Scale parameterηs =200;
• Cox’s parameters , 2.59, 2.59 to generate failure times.
For each replication of test plan, the development phases are simulated by generating random failure times with respect to censoring time of each engine cycles. On this simulation, first the Cox's parameters are estimated by MLE and integrated to estimation of reliability growth. The Nelder- Mead algorithm is used to find the optimum variables with fminsearch function in MATLAB. By 500 replications on the fixed development plan, the reliability parameters are estimated in two cases: taking into account the operating stresses level and without operating stresses. The evolution of the mean of the estimated reliability growth parameters are shown in Figure1 and Figure 2.
Figure 1 - Evolution of estimated parameters Beta
Figure 2 - Evolution of estimated parameters Eta As well-known, the reliability growth estimation of the scale parameter by AMSAA model is biased due to multiple censoring times. Nevertheless, the estimations of the proposed model including operating stresses are more closed to the theoretical value.
In additive information, it shows in Figure 3 the relative errors of each reliability growth parameters.
Figure 3 - Relative errors of estimated parameters Finally, the impact of operating stresses to reliability growth assessment is studied in Figure 4 by the comparison of mission reliability at the end of development test plan in two cases of taking into account the operating stresses and without operating stresses.
Figure 4 - Mission reliability estimation
The obtained results show a gain of mission reliability in comparison with the case without taking into account the operational stresses.
6 INDUSTRIAL APPLICATION
Based on the requirement of industrial application, we propose a methodology to apply the reliability growth estimation on real data of rocket engines provided by SAFRAN-SNECMA society. However, the methodology cannot be applied directly due to several co-variables that do not allow the Cox parameters estimation.
For a complex system such as rocket engine, the operational conditions are defined by a functional point characterized by several variables which cannot be directly correlated to physical characteristics and failure modes. This fact allows us to use the simplifying assumption enounced in section 4.1. As the parameters are strongly correlated, Principal Components Analysis (PCA) allows to the number of co-variables to characterize operating stress.
The algorithm of application is divided as following steps:
1. Operating measurements analysis: PCA to subtract the principal operating variables.
2. Classification of the Cox model co-variables.
3. Cox's parameters estimation by partial log-likelihood, on the selected principal operating variables (see section 2) 4. Hypothesis testing, (see section 2.2) and test of
proportionality
5. Reliability growth parameters estimation with Cox's parameters by MLE
The process is also illustrated on the diagram in Figure 5.
The obtained results on real data show a gain of mission reliability in comparison with the case without taking into account the operational stresses and emphasize the results of reliability estimated during operational flights of rocket engines.
Figure 5 - Taking into account the operating data for reliability growth estimation
For the further works, the methodology can be improved by considering reliability growth model more complexity including the different failure modes on different materials of system and risk concurrent on the materials as in [14].
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