0 Introduction

In a complex system, there are three common types of dependencies: economic dependence, stochastic dependence and structural dependence^[1]. Initially, economic dependence was widely considered in maintenance strategies, while the other two dependencies were rarely mentioned. In these years, Birnbaum importance measure is introduced to effectively consider the system structure, but it does not take the influence of maintenance group into consideration^[2-3]. To solve it, a group improvement factor was proposed to formulate a multi-level predictive maintenance^[4]. But it did not provide a specific corrective maintenance (CM) rule for component failure. So Ref. [5] proposed an SMRCR rule, but it does not consider component failure accelerates the degradation of other components, i.e., stochastic dependence^[2]. Gradually, different maintenance strategies were formulated for the degradation processes and degradation state-rate interactions of components^[6-7]. Further, Refs.[8-9] proposed degradation-threshold-shock (DTS) models and stress-strength models (SSM) based on degradation and shock. For the DTS models, the effects of degradation and shocks have been analyzed and different maintenance strategies have been formulated^[10-12]. For the SSM, a large number of research results on reliability have appeared based on two different situations. First, when stress and strength are not related, Ref. [13] estimates stress-strength reliability (SSR) and both stress and strength obey gamma distribution; Ref. [14] estimates SSR when both stress and strength follow exponential Weibull distribution. Second, when stress and strength are related, Ref. [15] analyzes a component SSR based on a general two-dimensional distribution; Ref. [16] uses the copula function to describe the dependence between strength and stress and establishes a dependent SSM. However, the above research based on SSM is limited to single-component systems and rarely used for reliability modeling of complex systems.

In response to the above problems, based on Ref.[4], an improved maintenance strategy is proposed. Firstly, system structure is described by component grades and is updated by survival signature. Secondly, considering degradation and shock, SSM and survival signature are introduced to establish a dynamic SSR. Moreover, using group importance measure designed by system dynamic SSR and PM cost, a two-level preventive maintenance (PM) rule is formulated. Finally, a corresponding maintenance strategy model is established, which is optimized by using simulation and genetic algorithm (GA). Further, using a numerical example, this study demonstrates that the developed strategy is useful and effective.

1 System Reliability Modeling 1.1 System Description

K(K≥2) types of n components make up a monotonous and coherent system. At initial moment, the system works normally and its components are all new. Let all components status vector be x =(x₁, x₂, L, x_n)∈{0, 1}ⁿ. x_i=1 indicates the i-th component is in working status. Define ϕ=ϕ(x): {0, 1}ⁿ→{0, 1} as system structure function. ϕ(x)=1 indicates the system works normally in view of x. Survival signature is introduced to defined system survival probability, which is rewritten as^[17]

$ \varPhi\left(l_1, l_2, L, l_K\right)=\left[\prod\limits_{k=1}^K\left(\begin{array}{c} m_k \\ l_k \end{array}\right)^{-1}\right] \times \sum\limits_{x \in S_{l_1, l_2, L, l_K}} \phi(x) $

(1)

where m_k is type k component number, l_k is the number of type k components that work normally with k=1, 2, L, K, and S_{l1, l2, L, lK} is the set of the status vector of components. A 10-component system is used to explain how the survival signature is used. Refer to Appendix for detailed explanation.

As the change of the component status affects the grades of the system component and ultimately affects the system structure, the four component grades defined in Ref. [5] is used to represent the system structure. The set of the four component grades are critical component set (CCS), temporary critical component set (TCS), failed component set (FLS) and non-critical idle component set (NCIS), respectively. Among the four component grades, the component in CCS will be repaired immediately after failure and the maintenance time is negligible, so it is considered that the components in the grade will not be transferred to other grades. To better exhibit the mutual transformation relationship of the other three component grades, the paper discusses the failure of any component i in the three component grades at time t (Fig. 1). t⁺ is the time just after time t. Refer to Section 2.1 for detailed description of Fig. 1.

1.2 Degradation and Shock Modeling of System

Ref.[8] proposes an SSM for components due to degradation and shock. In the SSM, component strength and component stress are two random variables. Component strength refers to the inherent strength for component itself and component stress refers to the external shock, which are denoted by X and Y, respectively. At this point, the dynamic SSR of system or component is defined as the strength greater than the stress, and its mathematical expression is as follows:

$ R=P(X>Y) $

(2)

Eq. (2) reflects the possibility (probability) that the system or component can fully perform their intended function when they work under specified environmental conditions. However, in the field of engineering systems, the strength of some components and systems may deteriorate with time under random stress (such as corrosion, aging, etc.), and the effect of random stress on components and systems may be a cumulative process. So, it is very important to study the multi-system dynamic SSR. The following are the basic assumptions:

1) The initial strength X and random stress Y of the component obey the exponential distribution with parameters α and β, respectively;

2) System components only bear one kind of random stress;

3) Each time a component is subjected to stresses, its strength will degrade by b units and b=0.4. So, when the component is subjected to j cycles of stress Y_j=Y, its corresponding strength is X_j=X－jb, and j=1, 2, L.

4) In interval (0, t], the random stress arrival number N(t) acting on the system obeys Poisson distribution with the parameter t/u, u=0.2 is a parameter that number of random stress cycles obey the Poisson distribution.

Based on above assumptions, the cumulative distribution functions (CDF) of related variables are obtained:

$ F_X\left(x^{\prime}\right)=1-\exp \left(-\alpha x^{\prime}\right), x^{\prime}>0 $

(3)

$ F_Y(y)=1-\exp (-\beta y), y>0 $

(4)

$ \begin{gathered} P(N(t)=j)=\frac{\exp \left(-\frac{t}{u}\right)\left(\frac{t}{u}\right)^j}{j !} \\ t \geqslant 0, j=0, 1, 2, L \end{gathered} $

(5)

where x' is an arbitrary real number, y is also an arbitrary real number.

Theorem 1    Let T_sys be the life of the system, then the system dynamic SSR expression at time T_sys=t is

$ \begin{aligned} & R_{T_{\mathrm{sys}}}(t)=\sum\limits_{l_1=0}^{n_1} L \sum\limits_{l_K=0}^{n_K}\left[\prod\limits_{k=1}^K\left(\begin{array}{c} m_k \\ l_k \end{array}\right)^{-1}\right] \times \\ & \quad \sum\limits_{x \in S_{l_1, L, l_K}} \phi(x) \prod\limits_{k=1}^K\left(\begin{array}{c} m_k \\ l_k \end{array}\right)\left[\frac{\beta_k \gamma}{\alpha_k+\beta_k}\right]^{m_k-l_k}\left[1-\frac{\beta_k \gamma}{\alpha_k+\beta_k}\right]^{l_k} \end{aligned} $

(6)

where $\gamma=\exp \left(-\frac{\left(1-\exp \left(-\alpha_k b\right)\right) t}{u}\right)$.

Proof    Let T' be the life of the component, σ_j(y) indicates an event that the component has not failed after j cycles of fixed stress Y=y. So the reliability of the component after bearing fixed stress of jth cycle is

$ \begin{aligned} r_i= & P\left(\sigma_1(y) \cap \sigma_2(y) \cap L \cap \sigma_j(y)\right)= \\ & P(X-j b>Y)=P(X>j b+y)= \\ & \exp (-\alpha(y+j b)) \end{aligned} $

(7)

When Y is unknown, the reliability of the component after bearing random stress of jth cycle is

$ \begin{aligned} R_j= & \int_0^{\infty} \exp (-\alpha(y+j b)) f_Y(y) \mathrm{d} y=\frac{\beta}{\alpha+\beta} \exp (- \\ & j \alpha b) \end{aligned} $

(8)

Combining Eq. (5), component SSR at time T'=t can be obtained, namely,

$ \begin{aligned} & R_{T^{\prime}}(t)=\sum\limits_{j=0}^{\infty} R_j P(N(t)=j)= \\ & \sum\limits_{j=0}^{\infty} \frac{\beta}{\alpha+\beta} \exp (-j \alpha b) \frac{\exp \left(-\frac{t}{u}\right)\left(\frac{t}{u}\right)^j}{j !}= \\ & \frac{\beta}{\alpha+\beta} \exp \left(-\frac{(1-\exp (-\alpha b)) t}{u}\right), t \geqslant 0 \end{aligned} $

(9)

Using Eq. (9) can calculate the SSR of the type k components at time T'=t, namely,

$ R_{T^{\prime}}^k(t)=\frac{\beta_k}{\alpha_k+\beta_k} \exp \left(-\frac{\left(1-\exp \left(-\alpha_k b\right)\right) t}{u}\right) $

(10)

In a monotonous and coherent system, survival signature can be used to describe the working status of the system, and the system structure can be separated from the probability model that describes component random failure. So, survival signature is widely used to calculate the system reliability, i.e.,

$ \begin{aligned} & R_{T_{\mathrm{sys}}}(t)=\sum\limits_{l_1=0}^{n_1} L \sum\limits_{l_K=0}^{n_K} \varPhi\left(l_1, L, l_K\right) P\left(\bigcap\limits_{k=1}^K\left(C_t^k=l_k\right)\right)= \\ & \quad \sum\limits_{l_1=0}^{n_1} L \sum\limits_{l_K=0}^{n_K}\left[\prod\limits_{k=1}^K\left(\begin{array}{c} m_k \\ l_k \end{array}\right)^{-1}\right] \times \sum\limits_{x \in S_{l_1, L, l_K}} \phi(x) \prod\limits_{k=1}^K\left(\begin{array}{c} m_k \\ l_k \end{array}\right) . \\ & \quad\left[1-R_{T^{\prime}}^k(t)\right]^{m_k-l_k}\left[R_{T^{\prime}}^k(t)\right]^{l_k} \end{aligned} $

(11)

Based on Eq. (1) and Eqs. (10)-(11), the system dynamics SSR at time T_sys=t can be obtained, namely,

$ \begin{aligned} & R_{T_{\mathrm{sys}}}(t)=\sum\limits_{l_1=0}^{n_1} \cdots \sum\limits_{l_K=0}^{n_K}\left[\prod\limits_{k=1}^K\left(\begin{array}{c} m_k \\ l_k \end{array}\right)^{-1}\right] \times \\ & \sum\limits_{x \in S_{l_1, L, l_K}} \phi(x) \prod\limits_{k=1}^K\left(\begin{array}{c} m_k \\ l_k \end{array}\right) \cdot \\ & {\left[\frac{\beta_k}{\alpha_k+\beta_k} \exp \left(-\frac{\left(1-\exp \left(-\alpha_k b\right)\right) t}{u}\right)\right]^{m_k-l_k} \cdot }\\ & {\left[1-\frac{\beta_k}{\alpha_k+\beta_k} \exp \left(-\frac{\left(1-\exp \left(-\alpha_k b\right)\right) t}{u}\right)\right]^{l_k}} \end{aligned} $

(12)

The proof is completed.

The following takes the substation system^[18] as an example to analyze the system dynamic SSR by comparing it with the system reliability that only considers the internal degradation of the components. Their topology and configuration are given, as shown in Fig. 2 and Fig. 3. The substation system relevant data are listed in Table 1.

The calculating result of the two system reliabilities are shown in Fig. 4. The results indicate both system reliabilities decrease with time, but the system dynamic SSR decreases faster, because for the system dynamic SSR, it takes into account internal degradation and external shock. Obviously, the results are realistic.

Fig. 5 and Fig. 6 show sensitivity analysis result with parameters b and u, respectively. Fig. 5 shows when b increases, the system dynamic SSR also increases. Because the larger the value of b, the larger the degree of strength degradation becomes, and ultimately the smaller the system survival probability becomes. Fig. 6 shows when u increases, the system dynamic SSR decreases. Because when u increases, the arrival frequency of external shock is reduced, which makes the number of the component subjected to external shock reduce.

The analysis indicates that it is meaningful to establish the system dynamic SSR. The following applies the dynamic SSR to system maintenance strategy. To simplify the subsequent related model, the following assumptions are made:

1) At a certain moment, only one component fails;

2) Preventive maintenance and corrective replacement are the replacement of an original component with a brand new one, while minimal repair is the repair of the original faulty component;

3) Maintenance durations of system components are 0;

4) Necessary maintenance resources are always available.

2 Maintenance Strategy

Internal degradation and external shock will cause system component failure, which affects the system structure. So, an SMRCR is formulated. In addition, to reduce maintenance cost, a preventive maintenance rule is formulated considering the group importance measure.

2.1 System Structure Updating

Let Stru (t₀) be system structure at time t₀. The dependency relationship of system structure and component grade sets is given at time t₀, namely,

$ \begin{aligned} {\bf{Stru}}(t) & =[\operatorname{CCS}(t), \operatorname{TCS}(t), \operatorname{NCIS}(t), \\ & \operatorname{FLS}(t)] \end{aligned} $

(13)

Based on Eq. (13), a component failure leads to component grade updating, which ultimately leads to system structure updating. Next, the system structure updating process with component failure is given based on Fig. 1.

Firstly, a status-survival matrix is defined based on the survival signature at time t₀, i.e.,

$ \operatorname{Mat}\left(i, t_0\right)=\left(S_{l_1, l_2, L, l_K} \varPhi\left(l_1, L, l_K\right)\right)_{x=\left(x_1, L, 0_i, L, x_n\right)} $

(14)

Mat(i, t₀) reflects the status vector when component i fails at time t₀. Then, a function is defined as follows:

$ \operatorname{summ}\left(i, t_0\right)=\sum\limits_{x=\left(x_1, L, 0_i, L, x_n\right)} \varPhi\left(l_1, l_2, L, l_K\right) $

(15)

Eq. (15) is calculated by the last column of the row corresponding to x_i=0 in the status-survival matrix at time t₀, and it reflects and quantifies the impact degree of component i failure on the system at time t₀. If summ(i, t₀)=0, it means component i failure causes system failure. When component w fails at time tf_w, let tf_w^- and tf_w⁺ be the time before and after tf_w, respectively.

(ⅰ) If the component w is in CCS(tf_w^-), Stru(tf_w^-)=Stru(tf_w⁺) is obtained.

(ⅱ) If the component w is not in CCS(tf_w^-) Mat(w, tf_w) and summ(w, tf_w) are obtained based on Eq. (1) and Eqs. (14)-(15). The following step is to renew component grades after component w fails based on Fig. 1 as follows:

Case A    If summ(w, tf_w)≠0, component w failure does not cause system failure, so the component w is removed from the original component grade set and stored in FLS(tf_w⁺). At this time, if the working component q is in CCS(tf_w^-), the component grade of q remains unchanged, and summ(q, tf_w) is obtained. If summ(q, tf_w)=0, q is removed from the original component grade set and stored in TCS(tf_w⁺). Otherwise, q is stored in NCIS(tf_w⁺).

Case B    If summ(w, tf_w)=0, the component w is temporary critical component at time tf_w^-. Let C_i^tf_w be CM cost of component i at time tf_wand be defined as follows:

$ C_i^{t f_w}=c_{m, i} I_{\left\{n_0[i]<l\right\}}+c_{c, i} I_{\left\{n_0[i]=l\right\}} $

(16)

where n₀[i] is the number of consecutive minimal repair of component i. Let SE(tf_w) be a set of failed components, and when any component in SE(tf_w) is restored to working status, the failed system can be restored to working status at time tf_w.

For any component g in SE(tf_w) and g≠w, C_h^tf_w=min(C_w^tf_w, C_g^tf_w) can be obtained based on Eq. (16) and component h needs to be executed by CM immediately.

1) If h=w, system component grades are constant, i.e., Stru(tf_w^－)=Stru(tf_w⁺).

2) If h=g, w is removed from the original component grade set and stored in FLS(tf_w⁺), and g is removed from FLS(tf_w^-) and stored in TCS(tf_w⁺). At this time, Mat(w, tf_w⁺) is obtained. For the working component d, if d is in CCS(tf_w^-), component d grade remains unchanged. Otherwise, summ(d, tf_w⁺) is calculated based on Mat(w, tf_w⁺). If summ(d, tf_w⁺)=0, component d is stored in TCS(tf_w⁺). Otherwise, it is stored in NCIS(tf_w⁺).

Finally, the system structure is updated based on (ⅰ) and (ⅱ) at time tf_w⁺ as follows:

$ \begin{aligned} \operatorname{Stru}\left(t f_w^{+}\right)= & {\left[\operatorname{CCS}\left(t f_w^{+}\right), \operatorname{TCS}\left(t f_w^{+}, \right) \operatorname{NCIS}\left(t f_w^{+}\right)\right.} \\ & \left.\operatorname{FLS}\left(t f_w^{+}\right)\right] \end{aligned} $

(17)

2.2 SMRCR rule

Minimal repair (MR) reduces cost and corrective replacement (CR) improves system reliability. Considering their advantages, this subsection proposes an SMRCR rule by updating system structure. When component w fails at time tf_w, three cases are discussed:

1) If w is in CCS(tf_w^-), it needs to be repaired immediately. If n₀[w] < l, MR is selected. Otherwise, CR is selected.

2) If wis in TCS(tf_w^-), for q in SE(tf_w) and q≠w, C_h^tf_w=min(C_w^tf_w, C_q^tf_w) is calculated by using Eq. (16). Now, component h is repaired immediately. If n₀[h] < l, h is executed MR. If n₀[h]=l, h is executed CR.

3) If w is in NCIS(tf_w^-), w keeps fail.

2.3 Group Importance Measure Based on System Dynamics SSR

In the existing PM rules, it is rare to take into account the importance measure of several components together. So group importance measure is designed as follows:

$ I M^{G_z}\left(T_r\right)=\frac{R_{T_{\mathrm{sys}}}^{G_z}\left(T_r\right)}{C^{G_z}} $

(18)

where

1) R_Tsys^G_z(T_r) is system dynamics SSR after all components in G_z are preventively replaced at time T_r, G_z is the z-th group of components and T_sys is system life. It should be noted that when the type k components in group G_z are preventively replaced, SSR of all corresponding components at this time is 1, i.e.,

$ R_{T^{\prime}}^k\left(T_r\right)=1 $

(19)

For the type k components outside group G_z, their SSR are obtained based on Eq. (10), i.e.,

$ R_{T^{\prime}}^k\left(T_r\right)=\frac{\beta_k}{\alpha_k+\beta_k} \exp \left(-\frac{\left(1-\exp \left(-\alpha_k b\right)\right) T_r}{\lambda}\right) $

(20)

where T' is component life. Finally, substituting Eq. (19) and Eq. (20) into Eq. (11), R_Tsys^G_z(T_r) is obtained.

2) C^G_z is PM cost for G_z.

$ \begin{aligned} & C^{G_z}=\sum\limits_{i \in G_z}\left(c_{p, i}+c_s\right)-c_s\left(\left|G_z\right|-1\right)- \\ & c_{\mathrm{psd}}\left(\sum\limits_{i \in G_z} I_{\left\{i \in \operatorname{CCS}\left(T_r\right) \cup \operatorname{TCS}\left(T_r\right)\right\}}-I_{\left\{G_z \text { is a critical group }\right\}}\right) \end{aligned} $

(21)

where I_{{ }} is an indicator function. I_{{ }}=1 if { } is true and I_{{ }}=0 if not. G_z is defined as a critical group when the failures of one or more components in G_z cause system failure. c_s(|G_z|-1) is set-up cost saving because of positive economic dependence (PED) between components.

$ c_{\mathrm{psd}}\left(\sum\limits_{i \in G_z} I_{\left\{i \in \operatorname{CCS}\left(T_r\right) \cup \operatorname{TCS}\left(T_r\right)\right\}}-I_{\left\{G_z \text { is a critical group }\right\}}\right) $

is planned shutdown cost saving owing to G_z.

2.4 Two-Level PM rule

A two-level PM rule is executed at time T_r=rT, r=1, 2, L, which considers group importance measure. The two-level includes system level and component level.

2.4.1 System level

System dynamics SSR, i.e., R_Tsys(T_r^-) is firstly calculated by Eq. (6), then compared with reliability threshold R₀(0 < R₀≤1). If R_Tsys(T_r^-) < R₀, the next step is to seek an optimal component group to be replaced, which occurs at the component level.

2.4.2 Component level

Under certain conditions, comparison of their group importance measures can find the optimal component group to be replaced. At time T_r, if a components are working, possible group number is 2^a-1. Group G_z is acceptable if R_Tsys^G_z(T_r)>R₀. By comparing all acceptable groups, an optimal group G_z^* is found as the one with the highest value of group importance measure. The specific expression is as follows:

$ I M^{G_z^*}\left(T_r\right)=\max\limits_{z=1, 2, L, 2^a-1}\left\{I M^{G_z}\left(T_r\right) \mid R_{T_{\mathrm{sys}}}^{G_z}\left(T_r\right)>R_0\right\} $

(22)

2.5 Maintenance Strategy Model

Based on the proposed maintenance strategy, cost-rate is defined as follows^[3]:

$ \begin{aligned} & \operatorname{CR}\left(T, R_0\right)=\lim _{M \rightarrow \infty} \operatorname{CR}\left(M \mid T, R_0\right)= \\ & \lim _{M \rightarrow \infty} \frac{\sum\limits_{r=1}^M\left(C_{T_r}^{\mathrm{insp}}+C_{T_r}^{\mathrm{prev}}+C_{T_r}^{\text {corr }}+C_{\left(T_{r-1}, T_r\right)}^{\mathrm{corr}}\right)}{M T} \simeq \\ & \frac{\sum\limits_{r=1}^{\tilde{M}}\left(C_{T_r}^{\mathrm{insp}}+C_{T_r}^{\text {prev }}+C_{T_r}^{\text {corr }}+C_{\left(T_{r-1}, T_r\right)}^{\mathrm{corr}}\right)}{\tilde{M} T} \end{aligned} $

(23)

where CR(M|T, R₀|) is cost-rate within M first planned maintenance periods. $\tilde{M}$ is planned maintenance period number, which is large enough.

C_Tr^insp is only the inspection operations cost of surviving components at time T_r. It is calculated as follows:

$ C_{T_r}^{\mathrm{insp}}=c_{\text {insp }}\left(n-\sum\limits_{i \in \varOmega \backslash G_z^*} I_{\left\{i \in \operatorname{FLS}\left(T_r\right)\right\}}\right) $

(24)

where Ω\G_z^* is the set of components outside the group G_z^*.

C_Tr^prev is total PM cost at time T_r, which contains preventive replacement cost for every component in G_z^*, set-up cost and the planned shutdown cost. Owing to the PED of components in group G_z^*, set-up cost is distributed only once when maintenance is executed at time T_r. C_Tr^prev is calculated by

$ \begin{aligned} C_{T_r}^{\text {prev }} & =\sum\limits_{i \in G_z^*}\left(c_{p, i}+c_s\right)-c_s\left(\left|G_z^*\right|-1\right)- \\ & c_{\mathrm{psd}}\left(\sum\limits_{i \in G_z^*} I_{\left\{i \in \operatorname{CCS}\left(T_r^{-}\right) \cup \operatorname{TCS}\left(T_r^{-}\right)\right\}}-I_{\left\{G_z^* \text { is a critical group }\right\}}\right) \end{aligned} $

(25)

C_(Tr－1^corr, T_r) is total CM cost in the r-th maintenance cycle, which contains CR cost, MR cost, set-up cost and unplanned shutdown cost. If component w fails at time tf_w with T_r－1 < tf_w < T_r, C_(Tr-1^corr, T_r) is calculated as follows:

$ \begin{array}{r} C_{\left(T_{r-1}, T_r\right)}^{\mathrm{corr}}=\sum\limits_{w \in \mathrm{FLS}\left(tf_w\right)}\left(c_{m, w} I_{\left\{n_0[w]<l\right\}}+c_{c, w} I_{\left\{n_0[w]=l\right\}}+\right. \\ \left.c_s\right)+\sum\limits_{w \in \operatorname{FLS}\left(t f_w\right)} c_{\text {usd }} I_{\left\{w \in \operatorname{CCS}\left(t f_w^{-}\right) \cup \operatorname{TCS}\left(t f_w^{-}\right)\right\}} \end{array} $

(26)

C_Tr^corr is CM cost at time T_r, and contains CR or MR cost. C_Tr^corr is calculated as follows:

$ C_{T_r}^{\text {corr }}=\sum\limits_{w \in \varOmega \backslash G_z^*}\left(c_{m, w} I_{\left\{n_0[w]<l\right\}}+c_{c, w} I_{\left\{n_0[w]=l\right\}}\right) I_{\left\{w \in \operatorname{FLS}\left(T_r\right)\right\}} $

(27)

Consequently, from Eqs. (23)-(27), CR(T, R₀) is calculated. Finally, by minimising cost-rate, the optimal values (T, R₀) is found, i.e.,

$ \operatorname{CR}\left(T^*, R_0^*\right)=\min\limits_{T>0, R_0 \in(0, 1]}\left\{\operatorname{CR}\left(T, R_0\right)\right\} $

(28)

Based on Eqs.(23)-(27), simple analytical method cannot evaluate the cost-rate. So, GA is considered in Ref. [3], which is done only with $\tilde{M}$ first planned maintenance periods. The condition to ensure the cost-rate convergence is

$ \begin{aligned} \Delta \mathrm{CR}=&\max _{j=\tilde{M}-\varepsilon, L, \tilde{M}} \mathrm{CR}\left(j \mid T, R_0\right)- \\ & \min _{j=\tilde{M}-\varepsilon, L, \tilde{M}} \operatorname{CR}\left(j \mid T, R_0\right) \leqslant \triangle \mathrm{CR}^0 \end{aligned} $

(29)

where △CR⁰=0.01 is convergence tolerance. The integer $\varepsilon(\varepsilon<\tilde{M})$ is consecutive planned maintenance period number when cost-rate change is always smaller than △CR⁰.

3 Numerical Example

This section takes the substation system of Fig. 2 as an example to optimize and evaluate the proposed strategy. Finally, the paper analyses parameter sensitivity. The relevant cost parameters are listed in Table 2.

3.1 Optimization of Maintenance Strategy Model

The maintenance strategy model in Section 2.5 is very complex, so the strategy is optimized by simulation analysis and GA^[3].

Firstly, to prove the convergence of the cost-rate, a simulation-analysis method is used by supposing T=40 and R₀=0.6. To eliminate the chance of the simulation results and make it more convincing, the simulation is repeated five times. The result is shown in Fig. 7. It shows when the planned maintenance period number is big enough, five curve difference is small enough. So, it can infer the cost-rate is convergent when planned maintenance period number is big enough.

After cost-rate is proved to be convergent, the following is to obtain the optimal solutions of the cost-rate by using the GA solver in the global optimization package of software R. The GA's parameters are set to 50 for the initial possible solutions, 0.05 for the uniform mutation probability, 1 for the crossover probability, and 100 for the number of iterations. After 100 iterations, the optimal values are received, i.e., (T^*, R₀^*)=(41.30, 0.60), and the minimal cost-rate is CR(41.30, 0.60)=8.99. Finally, the optimal strategy is obtained.

3.2 Optimal Strategy Evaluation

Optimal strategy is evaluated based on system reliability and maintenance cost.

Firstly, to evaluate the impact of proposed strategy on system reliability, system reliability before and after maintenance is analyzed in the optimal case T^*=41.30 and R₀^*=0.60, as shown in Fig. 8. The result shows that system reliability is significantly improved after system maintenance. This is because PM is triggered when system reliability is less than R₀^* at time T_r^-. Afterwards, group importance measure will select G_z^* for preventive replacement, which improves system reliability.

Then, to verify the impact of the optimal maintenance strategy on maintenance cost, it is compared with another strategy^[4]. Finally, the cost-rate analysis result is shown in Table 3. It shows the proposed strategy cost-rate is smaller than comparative strategy cost-rate. Because proposed strategy optimal PM maintenance period is less than that of the comparison strategy and the proposed strategy optimal reliability threshold is larger than that of the comparison strategy, this makes PM activities in the comparison strategy frequently executed. Finally, comparison result shows the proposed strategy can reduce maintenance costs by

$ \frac{34.41-8.99}{34.41} \times 100 \%=73.87 \% $

3.3 Sensitivity Analysis

Sensitivity analysis of cost parameters can summarize the impact of changes in cost parameters on cost-rate. To be precise, sensitivity analysis is to change the value of one cost parameter, while other cost parameters remain unchanged. In this section, let CR₁(41.30, 0.60) be the proposed strategy cost-rate under T^*=41.30 and R₀^*=0.60, and CR₂(14.28, 0.56) be the comparative strategy cost-rate under T^* = 14.28 and R₀^*=0.56, (as shown in Table 4). The sensitivity results of these four cost parameters are analyzed separately as follows:

1) c_insp ranges from 0 to 20 with 5 cost unit increments while other parameters remain unchanged. When c_insp is higher, cost-rate is higher. However, the proposed strategy cost-rate is always lower than comparative strategy cost-rate. So, c_insp is positively related with two maintenance strategies, respectively.

2) c_s ranges from 0 to 20 with 5 cost unit increments while other parameters remain unchanged. When c_s increases, cost-rate from two strategies also increases, but comparative strategy increases relatively faster. This is because in the optimal PM period, the comparison strategy is relatively small, so that the comparative strategy must frequently execute PM, and ultimately the cost-rate increases relatively quickly.

3) c_psd ranges from 0 to 200 with 50 cost unit increments while other parameters remain unchanged. Based on Eq. (21), c_psd is saved maintenance cost owing to group G_z, which is positive or negative value. Obviously, the cost-rate saved by c_psd is a positive value for the proposed strategy, but the cost-rate saved by c_psd is a negative value for the comparative strategy. In other words, c_psd is positively related to the comparative strategy and is negatively related to the proposed strategy.

4) c_usd ranges from 100 to 300 with 50 cost unit increments while other parameters remain unchanged. When c_usd is higher, maintenance cost-rate is higher.So, c_usd is positively related to two maintenance strategies, respectively.

4 Conclusions

Firstly, the proposed maintenance strategy introduces a stress-strength model to depict component failure process because of internal degradation and external shock, and the system dynamic SSR is obtained in combination with the survival signature. Compared with the system reliability based on internal degradation, the system dynamic SSR is more realistic. Then, applying system dynamic SSR to system maintenance, an SMRCR rule considering the system structure updating and a two-level PM rule considering the group importance are formulated. Finally, using numerical example shows that the proposed maintenance strategy can effectively increase system reliability and reduces maintenance cost by 73.87%.

Appendix

To better illustrate survival signature definition, this section takes the 10-component system shown in Fig. A1 as an example for detailed explanation, in which components "1" and "2" belong to the same type 1, and components "3", "4" and "9" belong to type 2, components "5" and "10" belong to type 3, and components "6", "7" and "8" belong to type 4. The results of the survival probability of the system are shown in Table A1, in which all rows with survival probability 0 and 1 in the table are deleted.

Take l₁=2, l₂=1, l₃=0, l₄=2 as examples to explain the results in Table A1. This situation indicates that two components of type 1 are all working properly, a component of three components of type 2 is working normally, two components of type 3 are faulty, two components of three components of type 4 are in normal operation. At this time, all possible results of the component state vector are

$ \left(\begin{array}{l} 2 \\ 2 \end{array}\right)\left(\begin{array}{l} 3 \\ 1 \end{array}\right)\left(\begin{array}{l} 2 \\ 0 \end{array}\right)\left(\begin{array}{l} 3 \\ 2 \end{array}\right)=9 $

so

$ \prod\limits_{k=1}^4\left(\begin{array}{c} m_k \\ l_k \end{array}\right)^{-1}=\frac{1}{9} $

In the 9 possible results, there are only two combinations that components "1", "2", "3", "6", "8" and components "1", "2", "3", "7", "8" are all working properly to make the system work, i.e.,

$ \sum\limits_{x \in S_{l_1, \cdots, l_K}} \phi(x)=2 $

Finally, the result is

$ \varPhi(2, 1, 0, 2)=\frac{1}{9} \times 2=\frac{2}{9} $