Maintenance policy for two-stage deteriorating mode system based on cumulative damage model

2.1. System description

The considered system is assumed to be an observable system which degrading stochastically. The degradation level at time $t$ is supposed to be presented by a random variable $Y\left(t\right)$ [19, 20]. The system degradation process is an increasing stochastic process with initial state $Y\left(t\right)=0$. System will be declared as failed when deterioration level $Y\left(t\right)$ exceeds a failure threshold $Y_f$ (namely $Y\left(t\right)\ge Y_f$). $T_f$ is defined as system failure time. System failed does not mean that the system cannot work, but implies that it’s economic and safety impacts will be unacceptable if it still in operation.

Fig. 1Two-stage degradation process

System degradation rate changes at time $t_c$ during a life cycle (as shown in Fig. 1). It means that the parameters of system degradation process undergo transitions at change-point $t_c$. The system is supposed to be in a nominal degradation mode and denoted by $M_1$ before $t_c$. After $t_c$, the system degradation mode evolves to an accelerated mode $M_2$. Degradation rate is usually small in the first stage $M_1$ and large in the second stage $M_2$. Therefore, system degradation process can be modeled by two stochastic processes under the same law but with different parameters [6].

Most physical degradation observation and the property of the Levy processes have shown that system deterioration can be thought as the accumulation of large numbers of small shocks [21], and deterioration level can be defined as the sum of damage values due to shock. When the system is in mode $M_k$, damage values $x_{ki}$ ($k=$ 1, 2; $i=$ 0, 1, 2, …) are assumed to be normally distribution, $x_{ki}~N\left(u_k,{\sigma }_k^2\right)$. $x_{ki}$ is a constant in mode $M_k$ (see Fig. 1). Random variable $N_S$ represents the number of shock times during time [0, $t_s$] and it obeys to homogeneous Poisson distribution with strength ${\lambda }_k$ in mode $M_k$. Therefore, the probability of shock times $N_S$ just as $n$ in mode $M_k$ is:

For the convenience of expression, in this paper $x_{ki}$ is denoted as shock strength and $1/{\lambda }_k$ is denoted as shock frequency. It is not difficult to find that shock strength and shock frequency decide the size of degradation rate.

2.2. Degradation level modeling

In the two-stage degradation process, the value of damage time $t_s$ may be in the first stage ($0\le t_s\le t_c$) or in the second stage ($t_s>t_c$). The calculation methods of degradation level are not alike in different degradation stage. According to the above notation, degradation level at time $t_s$ can be written as:

where $Y_{t_s}^k$stands for degradation level in mode $M_k$, $I_{\left\{E\right\}}=1$if $E$ is true and $I_{\left\{E\right\}}=0$ otherwise. When $0\le t_s\le t_c$, the degradation level is:

In this case, the probability of shock times equal to $n$ within time [0, $t_s$] is the same to Eq. (1). Due to every shock damage is independently and unrelated, it can be known from the characters of normal distribution (the sum of normal distribution parameters still in line with normal distribution) that $Y$ is obey to normal distribution, namely:

When $t_s>t_c$, degradation level consists of the damage in first stage $M_1$ and the damage in second stage $M_2$. In this case, degradation time length in the first stage is $t_c$, $t_s–t_c$ to the second stage. Hence, the system degradation level is:

Similar to the Eq. (4), system degradation level is in line with normal distribution, there is:

Because shocks between the two stages are independently, the probability of the number of shock times just as $m$ in mode $M_1$ and equal to $n$ in mode $M_2$ is:

2.3. Reliability modeling

In engineering practice, the change-point $t_c$ for degradation rate is not fixed, but is distributed in a certain time interval. Shown as Fig. 1, the time distribution interval of change-point $t_c$ for degradation rate is [$t_A$, $t_B$], in other words, the deteriorating mode may change at any time from $M_1$ to $M_2$ when system works during time [$t_A$, $t_B$].

System reliability is the probability for degradation level $Y$ less than failure threshold $Y_f$ when damage time is $t_s$. Shock strength, shock frequency and change-point should be considered in reliability modeling, which are main factors in cumulative damage model. Reliability modeling for two-stage degraded system is specific expressed as following.

When $0\le t_s\le t_c$, system reliability is:

When $t_s>t_c$, system reliability is:

where $g\left(t\right)$ is the probability density distribution function for change-point during time [$t_A$, $t_B$].

3.1. Global maintenance policy

In order to show the importance of considering the changes of system degradation rate, traditional maintenance decision-making method is presented in the first place, which called global maintenance policy. The method just defines a single alarm threshold $Y_A$ and a single inter-inspection time $∆T$, as done in Dieulle et al. [22]. It is not difficult to find that the method only pay attention to system degradation level and ignore the degradation rate.

The possible maintenance actions which can put into practice after inspection time $T_i$ are defined as follows:

• If $Y\left(T_i\right)<Y_A$, do nothing and the system is left as it is until next inspection time $T_{i+1}=T_i+∆T$.

• If $Y_A\le Y\left(T_i\right)<Y_f$, the system is too badly deteriorated so it is necessary to perform preventive maintenance.

• If $Y\left(T_i\right)\ge Y_f$, the system is considered as failed and it has to be performed corrective maintenance.

The rule of global maintenance policy is illustrated in Fig. 2.

Fig. 2Global maintenance policy

3.2. Time-depended maintenance policy

As the degradation rate in mode $M_2$ larger than in mode $M_1$, the inter-inspection time should be shorter and shorter in term of work time. This kind of maintenance decision-making method called time-depended maintenance policy. For example, the inter-inspection time of $i$th monitor is $∆T_i$, the next inter-inspection time is $∆T_{i+1}$, $∆T_{i+1}=q·∆T_i$ and $q<1$.

The possible maintenance actions which can put into practice after inspection time $T_i$ are defined as follows:

• If $Y\left(T_i\right)<Y_A$, do nothing and the system is left as it is until next inspection time $T_{i+1}=T_i+∆T_{i+1}$.

• If $Y_A\le Y\left(T_i\right)<Y_f$, the system is too badly deteriorated so it is necessary to perform preventive maintenance.

• If $Y\left(T_i\right)\ge Y_f$, the system is considered as failed and it has to be performed corrective maintenance.

The rule of time-depended maintenance policy is illustrated in Fig. 3.

Fig. 3Time-depended maintenance policy

3.3. Adaptive maintenance policy

According to the characteristics of system degradation rate suddenly changes from nominal mode $M_1$ to accelerated mode $M_2$, Saassouh et al. [6, 12] put forward adaptive maintenance policy. The method is different with global maintenance policy, it considers system degradation level and degradation rate. As a result, this maintenance decision-making method is more responsive to systems with two-stage deteriorating mode.

The alarm threshold and inter-inspection time for adaptive maintenance policy are defined as follows:

Set $Y_{nom}$ as the alarm threshold and $∆T_{nom}$ is the inter-inspection time for nominal degradation mode $M_1$. When the inspection time $T_i$ is less than change-point $t_c$ ($T_i\le t_c$), the possible maintenance actions which can put into practice are defined as follows:

• If $Y\left(T_i\right)<Y_{nom}$, do nothing and the system is left as it is until next inspection time $T_{i+1}=T_i+∆T_{nom}$.

• If $Y_{nom}\le Y\left(T_i\right)<Y_f$, the system is too badly deteriorated so it is necessary to perform preventive maintenance.

• If $Y\left(T_i\right)\ge Y_f$, the system is considered as failed and it has to be performed corrective maintenance.

Set $Y_{acc}$ as the alarm threshold and $∆T_{acc}$ is the inter-inspection time for accelerated degradation mode $M_2$. When the inspection time $T_j$ is greater than change-point $t_c$ ($T_j>t_c$), the possible maintenance actions which can put into practice are defined as follows:

• If $Y\left(T_j\right)<Y_{acc}$, do nothing and the system is left as it is until next inspection time $T_{j+1}=T_j+∆T_{acc}$.

• If $Y_{acc}\le Y\left(T_j\right)<Y_f$, the system is too badly deteriorated so it is necessary to perform preventive maintenance.

• If $Y\left(T_j\right)\ge Y_f$, the system is considered as failed and it has to be performed corrective maintenance.

As the degradation rate for mode $M_2$ is greater than mode $M_1$, so the maintenance policy parameters $Y_{acc}<Y_{nom}$ and $∆T_{acc}<∆T_{nom}$. The rule of adaptive maintenance policy is illustrated in Fig. 4.

Fig. 4Adaptive maintenance policy

3.4. Simplified adaptive maintenance policy

Adaptive maintenance policy so complex that not suitable for engineering application. Therefore, adaptive maintenance policy is simplified in application by some researchers [12]. Inter-inspection time $∆T$ is a constant value and never changes in simplified adaptive maintenance policy.

The maintenance policy rule (alarm threshold, possible maintenance action) is similar to adaptive maintenance policy, the only difference is that: no matter $Y\left(T_i\right)<Y_{nom}$ or $Y\left(T_i\right)<Y_{acc}$, the next inspection time always $T_{i+1}=T_i+∆T$ (namely $∆T_{nom}=∆T_{acc}=∆T$).

3.5. Maintenance policy evaluation

3.5.5. Expected time length of renewal cycle

As Eq. (14) shown, expected time length of renewal cycle $E\left[T\right]$ is affected by system life $T_f$ and average work time length $T_P$ when system ends with preventive replacement. If system faults, it is considered that system will not work any time. Therefore, the expected time length $T_f$ when system ends with corrective maintenance is the time interval for degradation level from initial value 0 to failure threshold $Y_f$. However, the expected time length $T_P$ when system ends with preventive maintenance is different. System will no longer work if monitoring shows preventive replacement should be performed, so system life when system ends with preventive maintenance is times of inter-inspection time $∆T$.

System fault occurs in the second stage when system degradation with two-stage mode. System life $T_f$ is affected by shock strength $x_{1i}$, $x_{2j}$ and change-point $t_c$. The system mean time to failure is:

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$E\left[T_f\right]={\int }_{t_c}^{\infty }R\left(t\right)dt$
$=\sum _{m=0}^{\infty }\sum _{n=0}^{\infty }\mathrm{\Phi }\left(\frac{Y_f-\left(m{\mu }_1+n{\mu }_2\right)}{\sqrt{m{\sigma }_1^2+n{\sigma }_2^2}}\right)\cdot \frac{{\lambda }_1^m\cdot {\lambda }_2^n}{m!\cdot n!}\cdot {\int }_{t_A}^{t_B}{\int }_{\tau }^{\infty }\left(e^{-{\lambda }_1\tau -{\lambda }_2\left(t-\tau \right)}\cdot {\tau }^m\cdot {\left(t-\tau \right)}^n\cdot g\left(\tau \right)\right)dtd\tau .$

The average system life when system ends with preventive maintenance is:

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$E\left[P_P{T}_P\right]=P\left(A_{P1}\right)z_{P1}\Delta T_{nom}+P\left(A_{P2}\right)\left(t_c+{\int }_0^{\Delta T_{nom}}\omega f_{\omega }d\omega +z_{P2}\Delta T_{acc}\right)$
$+P\left(A_{P3}\right)\left(t_c+{\int }_0^{\Delta T_{nom}}\omega f_{\omega }d\omega \right).$

Fig. 5The definition of ω

Where $z_{p1}$ is the number of monitoring in mode $M_1$ for event $A_{p1}$, $z_{p2}$ is the number of monitoring in mode $M_2$ for event $A_{p2}$, the time length from $t_c$ to next inspection time is $w$ (as shown in Fig. 5), $f_w$ is the distribution density function of w during [$T_{z-1}$, $T_z$].

4.1. Choice of parameters values

In this study, the time distribution of change-point $t_c$ is assumed to follow uniform distribution. In order to emphasize the influence of distribution of change-point $t_c$, different uniform distribution parameters are considered as follows:

• Change-point of two-stage degradation mode: $t_c~U\left(\text{1,}\text{200}\right)$.

• Early change-point of two-stage degradation mode: $t_c~U\left(\text{1,}\text{100}\right)$.

• Middle change-point of two-stage degradation mode: $t_c~U\left(\text{50,}\text{150}\right)$.

• Late change-point of two-stage degradation mode: $t_c~U\left(\text{100,}\text{200}\right)$.

The upper limit of change-point distribution 200 is considered that a majority of system failures occur in degradation mode $M_2$ and seldom in degradation mode $M_1$. Early and late time distributions present the first and second half of full change-point distribution, respectively.

In order to make the influence analysis of different parameters more close to the actual situation of gearbox degradation process, the selection of failure threshold $Y_f$ is based on the actual value of gearbox life-cycle experiment in Section 5. Therefore, the failure threshold is evaluated as $Y_f=$ 10000 g² in this study. Meanwhile, for the purpose of ensuring the credibility of optimization results for global maintenance policy (the optimization results will be regarded as a basis of comparison), the unit maintenance costs are evaluated as other literatures [1, 12, 24], so $C_I$ = 5, $C_P$ = 50, $C_C$ = 100.

4.2. Influence of maintenance policy

Fig. 6EC∞ is affected by adaptive maintenance policy parameters

a)$E\left(C_{\infty }\right)$ is affected by $Y_{nom}$ and $T_{acc}$

b)$E\left(C_{\infty }\right)$ is affected by $Y_{acc}$ and $T_{acc}$

c)$E\left(C_{\infty }\right)$ is affected by $T_{nom}$ and $T_{acc}$

For the purpose of studying on the influence of different monitoring methods, the four kinds of condition-based maintenance policies (global, time-depended, simplified adaptive and adaptive) presented in Section 3 are assessed. Because adaptive maintenance policy is the most complex in the four methods, an example focusing on adaptive policy analyzing is presented to show the approach of obtaining minimal average long-run maintenance cost rate $E\left(C_{\infty }\right)$. When $x_{1i}~N$(10, 20²), $x_{2j}~N$(40, 80²) and ${\lambda }_1={\lambda }_2=1$, $E\left(C_{\infty }\right)$ is affected by adaptive maintenance policy parameters $Y_{nom}$, $Y_{acc}$, $T_{nom}$, $T_{acc}$, as shown in Fig. 6. It illustrates that $Y_{nom}$, $Y_{acc}$, $T_{nom}$, $T_{acc}$ should be considered at the same time when optimizing $E\left(C_{\infty }\right)$. Contour map (Fig. 7) shows $E\left(C_{\infty }\right)$ for different values of $Y_{acc}$ and $T_{acc}$ under adaptive maintenance decision when $Y_{nom}$ = 8000, $∆T_{nom}$ = 70. $E\left(C_{\infty }\right)$ in the same contour are equal. It can be seen that optimal parameter values which minimize the cost rate ($E_4\left(C_{\infty }\right)$ = 0.3547) are $Y_{acc}$ = 7000 and $∆T_{acc}$ = 37 when $Y_{nom}$ = 8000, $∆T_{nom}$ = 70.

Fig. 7EC∞ under adaptive maintenance policy parameters (Ynom = 8000, ∆Tnom = 70)

As shown in Table 1, for the global maintenance policy, the optimal inter-inspection time $\Delta T^1$ falls in between $\Delta T_{nom}^4$ and $\Delta T_{acc}^4$ for adaptive maintenance policy. The situation of $\Delta T^3$ for simplified adaptive maintenance policy is in between $\Delta T_{nom}^4$ and $\Delta T_{acc}^4$, too. $\Delta T^2$, the first inter-inspection time for time-depended maintenance policy, is the largest optimal inter-inspection time for the four kinds of condition-based maintenance policies.

The expected costs for the four kinds of maintenance policies are different because they are impacted by alarm thresholds and inter-inspection times. As the monitoring method for global maintenance policy does not consider the influence of degradation rate changes, taking the expected cost $E_{1\left(C_{\infty }\right)}$ of global maintenance policy as a basis of comparison, the expected cost $E_{2\left(C_{\infty }\right)}$ for time-depended maintenance policy have a decrease of 0.0286, the optimal value is 7.69 % of $E_{1\left(C_{\infty }\right)}$. As shown in Table 1, it is obviously that the expected costs for other three maintenance policies are optimized compare with the average cost for global maintenance policy. Adaptive maintenance policy is better than simplified adaptive maintenance policy. Time-depended maintenance policy is the best replacement strategy for the system with two-stage degradation process.

4.3. Influence of change-point distribution

$E\left(C_{\infty }\right)$ for different change-point distributions are computed, see results in Table 1. It can be noticed that the decreases of expected costs for different maintenance policies (time-depended, simplified adaptive, adaptive) are 3.94 %, 1.59 %, 3.14 %, respectively, when the change-point is in early distribution $U$(1, 100). The impact of average long-run maintenance costs rate when the change-point is in middle distribution $U$(50, 150) and late distribution $U$(100, 200) are also given in Table 1. These results show that more the time of change-point occurs late, more the maintenance policies have a decrease on $E\left(C_{\infty }\right)$.

As seen previously, decreases of 7.69 %, 2.71 %, 4.68 % can be obtained respectively for different maintenance policies when the change-point distribution is $U$(1, 200), and they are 6.53 %, 2.61 %, 4.34 %, respectively for the case that the change-point distribution is $U$(50, 150). It is not difficult to find that the decreases for former are larger than latter. Meanwhile, in the two situations, the mean value of change-point distribution is the same, both equal to 100. Therefore, it is shown that more profits can be obtained in using different maintenance policies for a larger interval of time distribution.

The results show that the distribution of change-point impact on expected costs for different maintenance policies. More late for change-point occurs and more a large time interval of change-point distribution, more benefits can be obtained.

4.4. Influence of shock strength

In degradation modeling based on cumulative damage model, degradation rate is decided by shock strength and shock frequency. The degradation rate is in direct proportion to shock strength and shock frequency. Hence, the degradation rate can be expressed by shock strengths if the shock frequencies are the same.

The influence of different maintenance policies and change-point distribution have been analyzed for a two-stage degraded system in Table 1 when $x_{1i}~N$(10, 20²), $x_{2j}~N$(40, 80²). The same computed results with $x_{1i}~N$(10, 20²) and $x_{2j}~N$(20, 40²) are shown in Table 2. The degradation rate size of mode $M_2$ is four times superior than the mode $M_1$ in Table 1, while it is twice in Table 2. From Table 1 and Table 2, it can be known that the decreases are more considerable in Table 1. That is say, the profits is more considerable when degradation rate changes more significantly between mode $M_2$ and mode $M_1$.