Storage reliability assessment for the stored equipment under periodical inspection

Abstract

Some equipment of special types can be stored for a very long time. During storage, the equipment can be affected by temperature, humidity, mechanical stress, and so on, and the study on their storage reliability is very significant. Many theories and methods are proposed for storage reliability assessment in the literature, but few are applied into practice or verified by the actual conditions of the engineering project. Besides, the performance degradation of the equipment with a long storage time is not evident, the performance degradation model is hard to build. Thus, in this article, a storage reliability assessment method under periodical inspection for the equipment with a long storage time is studied. In the method, the storage reliability curves of “repaired as good as new” and “degenerated after repaired” are constructed to describe the different states of the stored equipment after periodical inspection, the environmental factor is used to deal with the data coming from accelerated life tests, and the study is based on an engineering project, thus the effectiveness can be verified.

Introduction

There is some equipment of special types and they can be stored for a very long time. During storage, the equipment can be affected by many kinds of environmental factors such as temperature, humidity, and mechanical stress. The main failure mechanism during storage includes oxidation, aging, mildew, sealing failures, and other slow chemical or physical processes.^1,2 Storage reliability assessment can help find out the root causes of storage failure, eliminate the hidden faults, and provide a technical support for storage management, and thus it deserves much attention.^3–5

The reliability of the stored equipment gradually declines under the influence of various environmental stresses. For example, solder joints of electronic components are oxidized, mechanical components may be corroded or rusted, rubber parts will be aging, and fatigue cracks will grow.^6–8 These are the main reasons for performance degradation and reliability reduction of the stored equipment. ⁵ Most storage reliability assessment methods construct storage reliability models by analyzing the failure mechanism and performance parameter degradation based on the probability theory,^2,9 but these models may fail when the degradation is not evident. Also, some storage reliability models are studied on basis of conducting storage reliability tests which need rich sample and sufficient data.^10,11 As the research continues, new methods and theories are introduced into storage reliability assessment, such as the neural network, ¹² Bayesian statistics,^13–15 fuzzy theory, ¹⁶ interval analysis, ¹⁷ and multi-information fusion analysis theory ¹⁸ are introduced, and these methods are verified much more suitable for the storage and small-sample systems under multi-source stresses. However, the aforementioned methods are a little more complex and their models contain many unknown parameters which are difficult to be accepted by engineering applications. Thus, their applications are restricted. By considering the actual conditions of an engineering project, this article aims to propose a storage reliability assessment method under periodical inspection for the equipment with a long storage time.

The remainder of this article is organized as follows. The section “Storage reliability assessment under periodical inspection” gives key storage reliability indexes and storage reliability models under inspection period. In the section “Storage reliability assessment of an electronic equipment,” after a brief introduction to the project, the storage reliability assessment framework is constructed. Following the framework, the storage reliability model as well as its parameter estimation are studied under a data collection of 12 samples. Moreover, storage reliability indexes in life cycle are assessed at the end of this section. Finally, the conclusion and future studies are summarized in the “Conclusion” section.

Storage reliability assessment under periodical inspection

Several key storage reliability indexes

Storage reliability is to study the ability of the stored equipment to keep its required functions under stated storage conditions for a specified period of time or a time interval. According to the definition of storage reliability, three primary storage reliability indexes, namely, storage life $T$ , storage reliability $R_s(t)$ , and R-percentile storage life $t(R)$ are defined as follows.

• Definition 1: Storage life $T$ is the length of time that the stored equipment can be kept in storage under specified conditions without failing to meet its operation or performance specifications. ¹⁹

• Definition 2: Storage reliability $R_s(t)$ represents the failure probability of the stored equipment under stated storage conditions for a specified period of time or a specified time interval. Assume that if the storage life $T$ obeys exponential distribution, then storage reliability $R_s(t)$ can be computed by

$$R_s\left(t\right)=\exp (-\lambda t)$$

(1)

where $\lambda$ is the failure rate.

• Definition 3: Q-percentile storage life is the length of time that the stored equipment can be kept under specified conditions when satisfying the specified degree of storage reliability. Assume that if the storage life $T$ obeys exponential distribution, then R-percentile storage life $t(R)$ can be computed by

$$t\left(R\right)=-\frac{1}{\lambda }\ln R$$

(2)

where $\lambda$ is the failure rate and $R$ is the specified degree of storage reliability.

Several key storage reliability indexes

The storage reliability of the stored equipment gradually declines under the influence of various environmental stresses. Oxidation, corrosion, aging, and so on are the main reasons for performance degradation and reliability deterioration of the stored equipment. ⁵ In practical application, the stored equipment mostly adopts regular inspection approach to control their reliability and safety. Thus, studies on storage reliability change rules and the construction of storage reliability models under the inspection period for the stored equipment are the essential issues. ¹⁹

In general, the storage life of the stored equipment obeys exponential distributions.^3–5 After the inspection, some equipment can be restored as good as new while some degenerate to a certain extent. Obviously, if the states of the stored equipment are different after being repaired, then the characters of their storage reliability curve are various. Based on different states, storage reliability curves can be described as the state of “repaired as good as new” and “degenerated after repaired,” and characteristic curves under two states are shown in Figures 1 and 2, where $R_s^0$ is initial reliability and $\tau$ refers to the testing periodic time.

Figure 1.

The characteristic curve of storage reliability under the state of “repaired as good as new.”

Figure 2.

The characteristic curve of storage reliability under the state of “degenerated after repaired.”

For these two states, storage reliability models can be constructed as follows.

• Model I (“Repaired as good as new”): For the stored equipment to be inspected every $\tau$ years, the failed components will be repaired or replaced. Assuming the stored equipment will be as good as new after inspected, the storage reliability will return to its initial degree. However, their failure rates will increase for the remaining components degenerate with different levels of degradation. If ${\lambda }_i$ is the mean failure rate of the $i\mathrm{th}$ inspection period, then

$${\lambda }_1\le {\lambda }_2\le \cdots \le {\lambda }_k$$

$R_s^0$ is the initial storage reliability, $k$ refers to the number of times that the stored equipment has been inspected, and $\tau$ is the time length of the inspection period. For the stored equipment under the state of “repaired as good as new,” the storage reliability can be computed by

$$R_s\left(t\right)=R_s^0\exp (-{\lambda }_i(t-k\tau ))$$

(3)

where ${\lambda }_i$ is monotone and never decreases with the increase of $k$ and sequence ${\lambda }_1,\dots ,{\lambda }_k$ reflects the degradation mechanism of storage reliability. The degradation of storage reliability can be approximately represented by a Weibull process,^2,9 that is

$${\lambda }_k={\lambda }_0{(k+1)}^{\beta }$$

(4)

where ${\lambda }_0$ is the mean failure rate without any repair and $\beta$ is the degradation parameter irrelevant to environmental stresses.

• Model II (“Degenerated after repaired”): For the stored equipment to be stored for a significantly long time, its components experience different levels of degradation. However, just several faulty parts are repaired or replaced after the inspection and the remaining parts still degenerate, that is to say, the stored equipment cannot be repaired as good as new and the storage reliability cannot return to its previous reliability level. Here, the faulty parts are defined as a whole as an exchange part and the rest is defined as a non-renewal part. Then, the storage reliability of the stored equipment can be computed by

$$R_s\left(t\right)=R_{s1}\left(t\right)·R_{s2}\left(t\right)$$

(5)

where $R_{s1}(t)$ is the storage reliability of the renewed part, while $R_{s2}(t)$ is the storage reliability of the non-renewal part. After inspection, the problematic part is repaired or replaced and the reliability of this part returns to $R_{s1}^0$ . The storage reliability of the exchange part can be modeled by Model I, and then the storage reliability of the exchange part after $k$ times of inspection can be computed by

$$R_{s1}\left(t\right)=R_{s1}^0\exp (-{\lambda }_{k+1}(t-k\tau ))$$

(6)

For the non-renewal part, its components have degraded but have not reached such an extent as to be repaired or replaced. If $\delta$ denotes the mean failure rate of the non-renewal part, $R_{s2}^0$ is the initial reliability, then the storage reliability of the non-renewal part can be calculated by

$$R_{s2}\left(t\right)=R_{s2}^0\exp (-\delta t)$$

(7)

Thus, the storage reliability of the whole system can be computed by

$$R_s\left(t\right)=R_{s1}^0·R_{s2}^0·\exp (-{\lambda }_{k+1}(t-k\tau ))·\exp (-\delta t)=R_{s1}^0·R_{s2}^0\exp (-({\lambda }_{k+1}+\delta )t+k\tau {\lambda }_{k+1})$$

(8)

where $R_{s1}^0$ and $R_{s2}^0$ are the initial storage reliabilities of the exchange part and the non-renewal part, respectively; $\delta$ is the mean failure rate of the non-renewal components; $k$ is the number of times that the stored equipment has been inspected; $\tau$ is the time length of the inspection period; and $t$ is the storage time when $k\tau <t\le (k+1)\tau$ .

Storage reliability assessment of an electronic equipment

The purpose of the project is to assess electronic equipment which will be periodically inspected every 2 years. The main design requirements of this electronic equipment are that the initial reliability is not less than $R_s^0=0.99$ and they should be stored for 10.5 years under the testing period ( $\tau =2\mathrm{years}$ ) with storage reliability not less than 0.96.

The process of storage reliability assessment is shown in Figure 3.

Figure 3.

Storage reliability assessment framework.

Storage lifetime data for 12 samples

In this project, 12 test samples produced on 30 January 2013 are chosen for tests, and storage reliability tests are subsequently conducted. In general, the electronic equipment is featured with extremely high reliability and it will be stored for a significantly long time. The existing approaches to obtain storage lifetime data can be divided into field storage tests and accelerated storage tests. ²⁰ Field storage tests are time-consuming and they are difficult to be managed. Moreover, the storage reliability of the electronic equipment is very few in the storage period of more than a dozen years. Thus, field storage tests are inapplicable for the electronic equipment. Since the sample quantity is extremely small (only 12 samples can be used) and the cost of production is very expensive, it is impossible to conduct a large amount of experiments. According to the actual situation, here we take the idea of accelerated life tests and increase the environmental stress for samples. Besides, some reliability lifetime data are also obtained in a short period.

The detailed situations of storage lifetime tests and key time nodes are introduced as follows.

1. Twelve samples chosen for tests are produced on 30 January 2013 and their initial reliability $R_s^0=0.99$ is adopted.

2. After being stored for 1 year, the first storage lifetime test is conducted from 1 February to 10 February 2014. In this test, two environmental stresses, namely temperature and humidity, are considered, and thus the humid heat test is conducted. Eight samples are taken to do the humid heat test and the remaining four samples are stored as usual. In the humid heat test, all eight samples experienced 10 cycles in total with every 24 h as a cycle. The samples are detected at the end of every cycle. The test condition is 120°C, 95% RH, where RH means the relative humidity. The result of this test is that one of the eight samples failed after the fifth cycle and the failed one does not take part in the remaining tests.

3. After conducting the humid heat test, the failed sample will be repaired and the failed components will be replaced. This sample will not be used in the remaining tests and the remaining seven samples are stored in the warehouse again.

4. The second storage reliability test is conducted from 10 April to 20 April 2015. The test process is the same as the first one but there is an adjustment of environmental stresses. Then, eight samples are chosen to undergo the humid heat test and the rest four samples are stored as usual. The test condition is 150°C, 105% RH. All eight samples experienced 10 cycles in total with every 24 h as a cycle. The samples are inspected at the end of every cycle. The result of this test shows that one of eight samples failed after the fifth cycle.

Storage lifetime data related to these 12 samples can be summarized in Table 1, where $n_i$ is the total number of test samples and $r_i$ refers to the number of failed samples under the $i\mathrm{th}$ environment.

Table 1.

Storage lifetime data for 12 samples.

Number	Time	$n_i$	$r_i$	States	Conditions
I	30 June 2013	12
II	1–10 February 2014	8	1	Environment A	120°C, 95% RH
		4	0	Environment B	5°C–30°C, 50% RH
III	10–20 April 2015	8	1	Environment C	150°C, 105% RH
		3	0	Environment B	5°C–30°C, 50% RH

RH: relative humidity.

Storage reliability modeling for electronic equipment

In each test, only the failed part is replaced and the reliability of this part cannot return to its initial reliability degree. Thus, the storage reliability assessment model for this type of electronic equipment should be constructed by Model II. In this article, parameter estimation of the storage reliability assessment model is based on processed data of humid heat tests. Furthermore, lifetime data coming from the humid heat tests will be conversed to lifetime data under the normal storage state via an environmental factor, which will be introduced in the next subsection.

Environmental factor estimation under exponential distributions

The environmental factor $K$ is a quantitative index commonly used to quantify different degradation degrees under different environmental stresses. When using the environmental factors, lifetime data under one environmental stress can be conversed to another.^21,22 Assume that if there are two groups of one certain equipment that have been stored for $n$ years, Group A is put into a much more adverse circumstance (Environment A) and Group is still stored as usual (Environment B). According to Liu and Zhang, ²³ for the system whose lifetime obeys the exponential distribution, the environmental factor can be computed by

$$K=\frac{{\lambda }_A}{{\lambda }_B}=\frac{\ln {\hat{R}}_A}{\ln {\hat{R}}_B}$$

(9)

where $K$ is the environmental factor, and ${\lambda }_A$ and ${\lambda }_B$ are failure rates under Environments A and B. Approximate equations for the environmental factor $K$ under confidence coefficient $\gamma$ are given in Table 2.

Table 2.

Interval evaluation of environmental factor $K$ under confidence coefficient $\gamma$ .

$(T_i,r_i)$	$K_L$	$K_U$
$\begin{array}{c}({\tau }_A,Z_A),({\tau }_B,Z_B),\hfill \\ Z_A\ne 0,Z_B\ne 0\hfill \end{array}$	$K_L=\frac{{\tau }_B}{{\tau }_A}·\frac{Z_A}{Z_B}·F_{2Z_A,2Z_B,1-\gamma }$	$K_U=\frac{{\tau }_B}{{\tau }_A}·\frac{Z_A}{Z_B}·F_{2Z_A,2Z_B,\gamma }$
$\begin{array}{c}({\tau }_A,0),({\tau }_B,Z_B),\hfill \\ Z_B\ne 0\hfill \end{array}$	$K_L=\frac{{\tau }_B}{3{\tau }_A}·\frac{1}{Z_B}·F_{\frac{2}{3},2Z_B,1-\gamma }$	$K_U=\frac{{\tau }_B}{3{\tau }_A}·\frac{1}{Z_B}·F_{\frac{2}{3},2Z_B,\gamma }$
$\begin{array}{c}({\tau }_A,Z_A),({\tau }_B,0),\hfill \\ Z_A\ne 0\hfill \end{array}$	$K_L=\frac{3{\tau }_B}{{\tau }_A}·Z_A·F_{2Z_A,\frac{2}{3},1-\gamma }$	$K_U=\frac{3{\tau }_B}{{\tau }_A}·Z_A·F_{2Z_A,\frac{2}{3},\gamma }$
$({\tau }_A,0),({\tau }_B,0)$	$K_L=\frac{{\tau }_B}{{\tau }_A}·F_{\frac{2}{3},\frac{2}{3},1-\gamma }$	$K_U=\frac{{\tau }_B}{{\tau }_A}·F_{\frac{2}{3},\frac{2}{3},\gamma }$

Here, $T_i$ and $r_i$ are the entire test time and the number of failures under the $i\mathrm{th}$ environment; accordingly, ${\tau }_A$ and ${\tau }_B$ are the entire test time under the Environments A and B; ${\tau }_A,{\tau }_B>0$ , $Z_A$ and $Z_B$ are the number of failures under the Environments A and B; $Z_A,Z_B>0$ , $K_U$ and $K_L$ are the upper and lower confidence limits of the environmental factor $K$ , respectively; $F_{a,b,\gamma }$ is a F-distribution with parameters $a,b$ under $\gamma$ -quantile.

Reliability lifetime data conversion based on environmental factors

For the electronic equipment, the confidence coefficient is expressed as $\gamma =0.6$ . According to data II, the upper and lower confidence limits of the environmental factor $K$ can be obtained as

$$K_L=\frac{3{\tau }_B}{{\tau }_A}·Z_A·F_{2Z_A,\frac{2}{3},1-\gamma }=6.8057$$

(10)

$$K_U=\frac{{\tau }_B}{{\tau }_A}·Z_A·F_{2Z_A,\frac{2}{3},\gamma }=27.4219$$

(11)

Similarly, the interval evaluation of the environmental factor $K$ based on data III can be obtained as

$$K_L=\frac{3{\tau }_B}{{\tau }_A}·Z_A·F_{2Z_A,\frac{2}{3},1-\gamma }=9.07425$$

(12)

$$K_U=\frac{{\tau }_B}{{\tau }_A}·Z_A·F_{2Z_A,\frac{2}{3},\gamma }=36.5625$$

(13)

The storage reliability model for the electronic equipment

If the reliability lifetime data are transferred from a much more adverse circumstance to a normal environment, the lower confidence limits of environmental factors should be used.^24–26 Thus, the electronic equipment storage reliability under the normal storage conditions can be computed by

$$\hat{R}=\exp \left(\frac{\ln {\hat{R}}_A}{K_L}\right)$$

(14)

where ${\hat{R}}_A$ is the reliability point estimation under the Environment A, $K_L$ is the lower confidence limit of the environmental factor $K$ and $\hat{R}$ is the storage reliability under the normal storage conditions.

The storage reliability assessment model for these 12 samples can be constructed by Model II, that is

$$R_s\left(t\right)=0.99\exp (-(\lambda +\delta )t+2k\lambda )$$

(15)

According to life test data II, the reliability of these samples under the condition 120°C, 95% RH is

$$R_A=\frac{n_A-r_A}{n_A}=0.875$$

(16)

Thus, the reliability of this type of electronic equipment after stored for 1 year is

$${\hat{R}}_{\mathrm{II}}=\exp \left(\frac{\ln 0.875}{6.8057}\right)=0.9806$$

(17)

According to equations (15)–(17), we have

$$\lambda +\delta =0.0095$$

(18)

According to data III, on 30 January 2015, these 11 samples are inspected and failed components are replaced. On 10 April 2015, the humidity test concerning 8 samples is conducted again and 1 of 8 samples fails after the fifth cycle. Thus, the reliability point estimation for these samples can be obtained by

$${\hat{R}}_{\mathrm{III}}=\exp \left(\frac{\ln 0.875}{9.07425}\right)=0.9854$$

(19)

According to equation (15), we can obtain that

$$R(2.25)=0.99\exp (-0.25\lambda -2.25\delta )=0.9854$$

(20)

Taking equation (18) and equation (20) into equation (15), we can obtain that

$$R\left(t\right)=0.99\exp (-0.0095t+0.016675k)$$

(21)

and the reliability lifetime can be calculated by

$$t\left(R\right)=-\frac{1}{0.0095}(\ln \left(\frac{R}{0.99}\right)-0.016675k)$$

(22)

Storage reliability assessment for the electronic equipment

According to the requirements made by a customer, this type of the electronic equipment should be stored for 10.5 years with the storage reliability of more than 0.96 under $\tau =2\mathrm{years}$ . Here, the storage reliability model constructed in the “Storage reliability modeling for electronic equipment” section will be adopted to assess whether these samples satisfy the requirements.

Storage reliability assessment

As analyzed above, the storage reliability of this electronic equipment can be computed by

$$R\left(t\right)=0.99\exp (-0.0095t+0.016675k)$$

From Figure 2, it can be concluded that the minimum reliability appears at the end of every inspection period and thus, the storage reliability can be estimated by computing the storage reliability at the end of each inspection period. The results are shown in Table 3.

Table 3.

Storage reliability before the inspection.

$t$	2	4	6	8	10	10.5
$R$	0.9714	0.9691	0.9669	0.9646	0.9624	0.9739

It can be seen that the storage reliability at the end of every inspection period as well as 10.5 years is more than 0.96, that is to say, the storage reliability of the electronic equipment satisfies the requirement.

Storage reliability lifetime assessment

Assume that ${\hat{R}}_0$ is the infimum allowed value of storage reliability, and according to equation (22), the maximum storage lifetime can be computed as

$$t\left({\hat{R}}_0\right)=-\frac{1}{0.0095}(\ln \left(\frac{{\hat{R}}_0}{0.99}\right)-0.016675k)$$

(23)

For the electronic equipment, the storage reliability should not be less than 0.96 in 10.5 years, that is $t(0.96)\ge 10.5$ . If ${\hat{R}}_0=0.96$ , the maximum storage lifetime is

$$t(0.96)=-\frac{1}{0.0095}(\ln \left(\frac{0.96}{0.99}\right)-0.016675k)=13.77\ge 10.5$$

(24)

Thus, the storage lifetime of the electronic equipment satisfies the requirement.

Conclusion

Many theories and methods have been developed for storage reliability assessment in the literature, but few are put into practice or verified by engineering examples. Thus, this article proposed a storage reliability assessment method under periodical inspection for the equipment with a long storage time. Moreover, this article gave a detailed procedure description of storage reliability tests, test data processing, model selecting, model construction, parameter estimation, and so on. In the future, the optimization model for the best periodic time of inspection will be studied, because excessive inspection will increase life cycle costs while overlong periodic time of inspection violates the original intention of periodical inspection.