A Multi-state Degradation Model for Reliability Assessment of Multi-component Nuclear Safety Systems Considering Degradation Dependency and Random Shocks

Abstract

The degradation (e.g., wear, stress corrosion cracking, and fatigue) of nuclear safety systems is an inherently irreversible process, which will lead to system failure when the accumulated damage reaches a threshold level, resulting in catastrophic consequences. Therefore, it is essential to understand and model the degradation behavior of nuclear safety systems to predict and prevent potential failures and thus effectively avoid subsequent losses. This paper proposes a multi-state degradation model for multi-component nuclear safety systems, considering the dependency among the degradation processes and the effect of random shocks. The degradation processes of the system were modeled by the Semi-Markov process. The arrival of random shocks obeys a Poisson process. The transfer kernel function of the holistic model was derived, based on which the Monte Carlo algorithm for estimation of the system reliability was developed. Based on a simple case, the correctness of the proposed model is verified. The model is applied to the reliability analysis of one sub-system of the residual heat removal system of a nuclear power plant.

1 Introduction

For nuclear safety systems that are typically designed for high reliability, traditional approaches based on time-to-failure data may be inapplicable, and degradation models that can take advantage of the wealth of the infrastructure health information have received widespread attention and impetus in several decades [1].

The multi-state models (MSMs) have been widely used in degradation modeling, including the Markov process and its extended version Semi-Markov process. Fleming et al. [2] developed a generalized 4-state Markov model for nuclear power plant (NPP) piping systems applicable to different degradation mechanisms such as wall thickness thinning and weld defect growth. Unwin et al. [3] proposed a multi-state physic model (MSPM) for a dissimilar metal weld in a primary coolant system of an NPP. Giorgio et al. [4] presented an age- and state-dependent Markov model to describe the cylinder liner wear process in a marine heavy-duty diesel engine. Moghaddass and Zuo [5] studied a nonhomogeneous continuous-time hidden Semi-Markov process for holistic modeling of the degradation and observation processes.

Random shocks may influence degradation evolution and require to be accounted for. For example, the impacts of thermal and mechanical shocks (e.g., internal thermal shock and water hammer) [6,7,8] on power plant components can result in increases in temperature and stress, which could accelerate component degradation. In the literature, random shocks are generally modeled by Poisson processes [8,9,10,11], mainly divided into two categories, extreme shocks that directly lead the system to fail and cumulative shocks that cause an additional instantaneous increase in the degradation degree of the system cumulatively. Lin et al. [8, 9] analyzed the effect of the random shocks on the transition rates of MSMs and the possibility that the component degrades instantaneously from the current state to a more “deteriorated” state. Similar considerations can be found in the performance analysis by Srivastav et al. [10] for subsea safety valves, where the device degradation was modeled as a Markov chain, and the device demand was considered as random shocks. Yang et al. [11] assumed that the shock arrival rate is dependent on the current state of the system, and a shock would result in a sudden change in the system state. Eryilmaz [12] suggested that a shock is extreme when its load exceeds the threshold level and considered the randomness of the number of system states induced by random shocks.

Most of these models mentioned above, however, are based on a single degradation process. Nuclear safety systems are usually complex collections of different kinds of constituent units, exhibiting the phenomenon of multiple degradation processes that may be correlated and have competitive/non-competitive relationships. In addition, few studies have considered both dependency among multiple degradation processes and dependency between degradation processes and random shocks. In this paper, we develop a holistic multi-state degradation model for nuclear safety systems with considering degradation dependency and random shocks. The Monte Carlo (MC) algorithm for estimating the system reliability indexes was presented. The validity and applicability of the models were demonstrated by a numerical example. The remainder of the paper is structured as follows, Sect. 2 presents the framework of the model, Sect. 3 gives the model validation, Sect. 4 provides the application of the model, and Sect. 5 gives some conclusions.

2 Model Framwork

2.1 Assumptions

• The system has L components, each of which may subject multiple degradation processes/mechanisms. Denote by ${\varvec{M}} = \left\{ {1, \cdots ,M} \right\}$ the degradation processes of the system.

• The degradation process $n \in {\varvec{M}}$ can be modeled by a time-homogeneous Semi-Markov process (SMP) $\left\{ {X_{n} \left( t \right);t \ge 0} \right\}$ on a discrete state space $S_{{X_{n} }} = \left\{ {0,1, \cdots ,s_{n} } \right\}$, where $s_{n}$ is the perfect functioning state and 0 is the failure state. The transition rate $\lambda_{{i_{n} ,j_{n} }} \left( {\tau_{{i_{n} }} } \right)$ of $X_{n} \left( t \right)$ from state $i_{n}$ to state $j_{n}$ is a function of the sojourn time in the current state $i_{n}$ since the last transition of the process, i.e.,

  $$ \begin{gathered} \lambda _{{i_{n} ,j_{n} }} \left( {\tau _{{i_{n} }} } \right) \hfill \\ = \mathop {\lim }\limits_{{\Delta t \to 0}} \rm{\mathbb{P}}\left( {X_{{n,k + 1}} = j_{n} ,T_{{n,k + 1}} \in \left[ {T_{{n,k}} ,T_{{n,k}} + \tau _{{i_{n} }} + \Delta t} \right]} \right. \hfill \\ \left. {\rm{\mid }\left\{ {X_{{n,l}} ,T_{{n,l}} } \right\}_{{l = 0}}^{{k - 1}} ,\left( {X_{{n,k}} = i_{n} ,T_{{n,k}} } \right)} \right)/\Delta t \hfill \\ = \rm{\mathbb{P}}\left( {X_{{n,k + 1}} = j_{n} ,T_{{n,k + 1}} \in \left[ {T_{{n,k}} ,T_{{n,k}} + \tau _{{i_{n} }} } \right]\rm{\mid }\left( {X_{{n,k}} = i_{n} ,T_{{n,k}} } \right)} \right)/\Delta t \hfill \\ \end{gathered}$$

  (1)

where $T_{n,k}$ and $X_{n,k}$ are the time and the arrival state of the kth jump of $X_{n} \left( t \right)$.

• The arrival of random shocks is governed by a homogeneous Poisson process $\left\{ {N\left( t \right);t \ge 0} \right\}$ with parameter $\mu$.

• A shock could cause the degradation process $X_{n} \left( t \right)$ to transfer instantaneously with probability $p_{{i_{n} ,j_{n} }}$ from the current state $i_{n}$ to the state $j_{n} \le i_{n}$. Note $p_{{i_{n} ,0}}$ that denotes the probability of the extreme shock.

• Considering repair, the system can recover from a “worse” to a “better” state except from the failed state.

2.2 Multi-state Model Considering Multiple Dependent Degradation Processes and Random shocks

Let ${\varvec{X}}\left( t \right) = \left( {X_{1} \left( t \right), \cdots ,X_{M} \left( t \right)} \right)$, then ${\varvec{X}}\left( t \right)$ is a stochastic process on an n-dimensional discrete state space $S_{{\varvec{X}}} = S_{{X_{1} }} \times \cdots \times S_{{X_{M} }}$, whose failure space ${\mathcal{F}}_{{\varvec{X}}}$ is dependent on the structure of the system and can be determined with the help of fault tree analysis or reliability block diagram. The correlation among degradation processes will have an impact on the evolution of $X_{n} \left( t \right),n \in {\varvec{M}}$. Assume that the transition rate of $X_{n} \left( t \right)$ from the current state $i_{n}$ is correlated with the state of ${\varvec{X}}\left( t \right)$ during the sojourn time of $X_{n} \left( t \right)$ in the current state $i_{n}$. It may be hypothesized that $X_{n} \left( t \right)$ jumps to state $j_{n}$ at time $t$ after sojourning $\tau_{{i_{n} }}$ in the current state $i_{n}$ since the last transition, then the transition rate can be expressed as

$$ \lambda_{{i_{n} ,j_{n} }} \left( {\tau_{{i_{n} }} |\left\{ {{\varvec{X}}\left( \tau \right)} \right\}_{{\tau = t - \tau_{{i_{n} }} }}^{t} } \right) $$

(2)

The state transition diagram of the system is shown in Fig. 1. We assume no simultaneous state mutation of any two degradation processes will occur. Given that the kth transfer of ${\varvec{X}}\left( t \right)$ occurs at time t to reach state ${\varvec{x}}_{i} = \left( {i_{1} , \cdots ,i_{n} , \cdots ,i_{M} } \right)$ and the sojourn times of $X_{1} \left( t \right), \cdots ,X_{M} \left( t \right)$ in states $i_{1} , \cdots ,i_{M}$ at time t are $\tau_{{i_{1} }}^{t} , \cdots ,\tau_{{i_{M} }}^{t}$ and the states of ${\varvec{X}}\left( t \right)$ on the time interval $\left[ {t - \tau_{max}^{t} ,t} \right]$ ($\tau_{max}^{t} = max\left\{ {\tau_{{i_{1} }}^{t} , \cdots ,\tau_{{i_{M} }}^{t} } \right\}$) are $\left\{ {{\varvec{X}}\left( \tau \right)} \right\}_{{\tau = t - \tau_{max}^{t} }}^{t}$, and let, the transition rate of the process ${\varvec{X}}\left( t \right)$ from the current state ${\varvec{x}}_{i}$ to ${\varvec{x}}_{j} = \left( {i_{1} , \cdots ,j_{n} , \cdots ,i_{M} } \right)$ is

$$ \begin{array}{*{20}c} {\lambda_{{{\varvec{x}}_{i} ,{\varvec{x}}_{j} }} \left( {\tau_{{{\varvec{x}}_{i} }} |{\varvec{\theta}}\left( t \right)} \right) = } \\ { = \lambda_{{i_{n} ,j_{n} }} \left( {\tau_{{n_{i} }}^{t} + \tau_{{{\varvec{x}}_{i} }} |\left\{ {{\varvec{X}}\left( \tau \right)} \right\}_{{\tau = t - \tau_{{i_{n} }} }}^{t} } \right)} \\ {{\varvec{\theta}}\left( t \right) = \left( {\tau_{{i_{1} }}^{t} , \cdots ,\tau_{{i_{M} }}^{t} ,\left\{ {{\varvec{X}}\left( \tau \right)} \right\}_{{\tau = t - \tau_{max}^{t} }}^{t} } \right)} \\ \end{array} $$

(3)

where $\tau_{{{\varvec{x}}_{{\varvec{i}}} }}$ is the sojourn time of ${\varvec{X}}\left( t \right)$ in current state ${\varvec{x}}_{i}$.

Fig. 1.

The state transition diagram of the system

Fig. 2.

The multi-state model of the system considering random shocks

Construct the stochastic process.${\varvec{X}}^{S} \left( t \right) = \left( {{\varvec{X}}\left( t \right),N\left( t \right)} \right)$., the state space and failure space of ${\varvec{X}}^{S} \left( t \right)$. Are $S_{{{\varvec{X}}^{S} }} = S_{{\varvec{X}}} \times {\mathbb{N}}$ and, respectively. The multi-state model considering multiple dependent degradation processes and random shocks is shown in Fig. 2.

Given that the kth transfer of ${\varvec{XS}} \left( t \right)$. Occurs at time t to reach state $\left( {{\varvec{x}}_{i} = \left( {i_{1} , \cdots ,i_{n} , \cdots ,i_{M} } \right),s} \right)$ and the sojourn times of $X_{1} \left( t \right), \cdots ,X_{M} \left( t \right)$ in states $i_{1} , \cdots ,i_{M}$ at time t are $\tau_{{1_{i} }}^{t} , \cdots ,\tau_{{M_{i} }}^{t}$ and the states of ${\varvec{X}}\left( t \right)$ on the time interval $\left[ {t - \tau_{max}^{t} ,t} \right]$ are $\left\{ {{\varvec{X}}\left( \tau \right)} \right\}_{{\tau = t - \tau_{max}^{t} }}^{t}$, the possible transition rates of the process from the current state $\left( {{\varvec{x}}_{i} ,s} \right)$ include

$$ \lambda_{{\left( {{\varvec{x}}_{i} ,s} \right),\left( {{\varvec{x}}_{j} ,s} \right)}} \left( {\tau_{{\left( {{\varvec{x}}_{i} ,s} \right)}} |{\varvec{\theta}}\left( t \right)} \right) = \lambda_{{i_{n} ,j_{n} }} \left( {\tau_{{n_{i} }}^{t} + \tau_{{\left( {{\varvec{x}}_{i} ,s} \right)}} |{\varvec{\theta}}\left( t \right)} \right) $$

(4)

the rate of occurrence of ${\varvec{X}}\left( t \right)$ jumping to state ${\varvec{x}}_{j} = \left( {i_{1} , \cdots ,j_{n} , \cdots ,i_{M} } \right)$;

$$ \lambda_{{\left( {{\varvec{x}}_{i} ,s} \right),\left( {{\varvec{x}}_{i} ,s + 1} \right)}} \left( {\tau_{{\left( {{\varvec{x}}_{i} ,s} \right)}} |{\varvec{\theta}}\left( t \right)} \right) = \mu \mathop \prod \limits_{n = 1}^{M} p_{{i_{n} ,i_{n} }} $$

(5)

the rate of occurrence of a shock which doesn’t cause the state change of ${\varvec{X}}\left( t \right)$;

$$ \lambda_{{\left( {{\varvec{x}}_{i} ,s} \right),\left( {{\varvec{x}}_{j} ,s + 1} \right)}} \left( {\tau_{{\left( {{\varvec{x}}_{i} ,m} \right)}} |{\varvec{\theta}}\left( t \right)} \right) = \mu \mathop \prod \limits_{n = 1}^{M} p_{{i_{n} ,j_{n} }} $$

(6)

the rate of occurrence of a shock which causes ${\varvec{X}}\left( t \right)$ to go to state ${\varvec{x}}_{j} = \left( {i_{1} , \cdots ,j_{n} , \cdots ,i_{M} } \right) \ne {\varvec{x}}_{i}$.

Obtaining analytical solutions of the Semi-Markov process ${\varvec{X}}^{S} \left( t \right)$ with complex state transition rates related to the state of the process at some time in the past and the sojourn time in the current state is a complicated task. The integral/differential equations describing the evolution of the system states over time for multi-component systems with dependent degradation processes may be challenging to develop. The number of the equations is usually extensive, which demands high computational resource requirements. Monte Carlo (MC) simulation is an effective way to overcome the above challenges and has been widely used in system reliability analysis studies [13,14,15].

In the following we derive the transfer kernel function of ${\varvec{X}}^{S} \left( t \right)$ for MC simulation, which is denoted by $Q_{{\left( {{\varvec{x}}_{i} ,s} \right),\left( {{\varvec{x}}_{j} ,m} \right)}}^{S} \left( {\tau_{{\left( {{\varvec{x}}_{i} ,s} \right)}} |{\varvec{\theta}}\left( t \right)} \right)$ and represents the probability of that ${\varvec{X}}^{{\varvec{S}}} \left( t \right)$ go to state $\left( {{\varvec{x}}_{j} ,m} \right)$ during infinitesimal time interval $\left[ {t + \tau_{{\left( {{\varvec{x}}_{i} ,s} \right)}} ,t + \tau_{{\left( {{\varvec{x}}_{i} ,s} \right)}} + d\tau_{{{\varvec{x}}_{i} }} } \right]$, given that the kth transfer of ${\varvec{X}}^{{\varvec{S}}} \left( t \right)$ occurs at time t to reach state $\left( {{\varvec{x}}_{i} = \left( {i_{1} , \cdots ,i_{n} , \cdots ,i_{M} } \right),s} \right)$ and the condition ${\varvec{\theta}}\left( t \right)$. Satisfies:

$$ Q_{{\left( {{\varvec{x}}_{i} ,s} \right),\left( {{\varvec{x}}_{j} ,m} \right)}}^{S} \left( {\tau_{{\left( {{\varvec{x}}_{i} ,s} \right)}} |{\varvec{\theta}}\left( t \right)} \right)d\tau_{{\left( {{\varvec{x}}_{i} ,s} \right)}} = P_{{\left( {{\varvec{x}}_{i} ,s} \right)}} \left( {\tau_{{\left( {{\varvec{x}}_{i} ,s} \right)}} |{\varvec{\theta}}\left( t \right)} \right) \cdot \lambda_{{\left( {{\varvec{x}}_{i} ,s} \right),\left( {{\varvec{x}}_{j} ,m} \right)}} \left( {\tau_{{\left( {{\varvec{x}}_{i} ,s} \right)}} |{\varvec{\theta}}\left( t \right)} \right)d\tau_{{\left( {{\varvec{x}}_{i} ,s} \right)}} $$

(7)

where ${P}_{\left({{\varvec{x}}}_{i},s\right)}\left({\tau }_{\left({{\varvec{x}}}_{i},s\right)}|{\varvec{\theta}}\left(t\right)\right)$ is the probability of that there is no transition of ${{\varvec{X}}}^{{\varvec{S}}}\left(t\right)$ during time interval $\left[t+{\tau }_{\left({{\varvec{x}}}_{i},s\right)},t+{\tau }_{\left({{\varvec{x}}}_{i},s\right)}\right]$, given that the kth transfer of ${{\varvec{X}}}^{{\varvec{S}}}\left(t\right)$ occurs at time t to reach state $\left({{\varvec{x}}}_{i}=\left({i}_{1},\cdots ,{i}_{n},\cdots ,{i}_{M}\right),s\right)$ and the condition ${\varvec{\theta}}\left(t\right)$. ${P}_{\left({{\varvec{x}}}_{i},s\right)}\left({\tau }_{\left({{\varvec{x}}}_{i},s\right)}|{\varvec{\theta}}\left(t\right)\right)$ satisfies

$$ P_{{\left( {\varvec{x}_{i} ,s} \right)}} \left( {\tau _{{\left( {\varvec{x}_{i} ,s} \right)}} + d\tau _{{\left( {\varvec{x}_{i} ,s} \right)}} |\varvec{\theta }\left( t \right)} \right) = P_{{\left( {\varvec{x}_{i} ,s} \right)}} \left( {\tau _{{\left( {\varvec{x}_{i} ,s} \right)}} |\varvec{\theta }\left( t \right)} \right) \cdot \left( {1 - \lambda _{{\left( {\varvec{x}_{i} ,s} \right)}} \left( {\tau _{{\left( {\varvec{x}_{i} ,s} \right)}} |\varvec{\theta }\left( t \right)} \right)} \right)d\tau _{{\left( {\varvec{x}_{i} ,m} \right)}} $$

(8)

$$ \lambda _{{\left( {\varvec{x}_{i} ,s} \right)}} \left( {\tau _{{\left( {\varvec{x}_{i} ,s} \right)}} |\varvec{\theta }\left( t \right)} \right) = \mathop \sum \limits_{{\varvec{x}_{j} }} \lambda _{{\left( {\varvec{x}_{i} ,s} \right),\left( {\varvec{x}_{j} ,m} \right)}} \left( {\tau _{{\left( {\varvec{x}_{i} ,s} \right)}} |\varvec{\theta }\left( t \right)} \right) $$

(9)

The solution of Eq. (9) is

$$ P_{{\left( {{\varvec{x}}_{i} ,s} \right)}} \left( {\tau_{{\left( {{\varvec{x}}_{i} ,s} \right)}} |{\varvec{\theta}}\left( t \right)} \right) = exp\left( { - \mathop \smallint \limits_{0}^{{\tau_{{\left( {{\varvec{x}}_{i} ,s} \right)}} }} \lambda_{{\left( {{\varvec{x}}_{i} ,s} \right)}} \left( {\tau_{{\left( {{\varvec{x}}_{i} ,m} \right)}} |{\varvec{\theta}}\left( t \right)} \right)du} \right) $$

(10)

Then, we have

$$ Q_{{\left( {{\varvec{x}}_{i} ,s} \right),\left( {{\varvec{x}}_{j} ,m} \right)}}^{S} \left( {\tau_{{\left( {{\varvec{x}}_{i} ,s} \right)}} |{\varvec{\theta}}\left( t \right)} \right) = \pi_{{\left( {{\varvec{x}}_{i} ,s} \right),\left( {{\varvec{x}}_{j} ,m} \right)}} \left( {\tau_{{\left( {{\varvec{x}}_{i} ,s} \right)}} |{\varvec{\theta}}\left( t \right)} \right) \cdot \psi_{{\tau_{{\left( {{\varvec{x}}_{i} ,s} \right)}} }} \left( {\tau_{{\left( {{\varvec{x}}_{i} ,s} \right)}} |{\varvec{\theta}}\left( t \right)} \right) $$

(11)

$$ \pi_{{\left( {{\varvec{x}}_{i} ,s} \right),\left( {{\varvec{x}}_{j} ,m} \right)}} \left( {\tau_{{\left( {{\varvec{x}}_{i} ,s} \right)}} |{\varvec{\theta}}\left( t \right)} \right) = \frac{{\lambda_{{\left( {{\varvec{x}}_{i} ,s} \right),\left( {{\varvec{x}}_{j} ,m} \right)}} \left( {\tau_{{\left( {{\varvec{x}}_{i} ,s} \right)}} |{\varvec{\theta}}\left( t \right)} \right)}}{{\lambda_{{\left( {{\varvec{x}}_{i} ,s} \right)}} \left( {\tau_{{\left( {{\varvec{x}}_{i} ,s} \right)}} |{\varvec{\theta}}\left( t \right)} \right)}} $$

(12)

$$ \psi_{{\tau_{{\left( {{\varvec{x}}_{i} ,s} \right)}} }} \left( {\tau_{{\left( {{\varvec{x}}_{i} ,s} \right)}} |{\varvec{\theta}}\left( t \right)} \right) = \lambda_{{\left( {{\varvec{x}}_{i} ,s} \right)}} \left( {\tau_{{\left( {{\varvec{x}}_{i} ,s} \right)}} |{\varvec{\theta}}\left( t \right)} \right) \cdot exp\left( { - \mathop \smallint \limits_{0}^{{\tau_{{\left( {{\varvec{x}}_{i} ,s} \right)}} }} \lambda_{{\left( {{\varvec{x}}_{i} ,s} \right)}} \left( {u|{\varvec{\theta}}\left( t \right)} \right)du} \right) $$

(13)

According to Eq. (11) to Eq. (13), we can implement the MC simulation procedure for the system as follows: repeatedly using ${\psi }_{{\tau }_{\left({{\varvec{x}}}_{i},s\right)}}\left({\tau }_{\left({{\varvec{x}}}_{i},s\right)}|{\varvec{\theta}}\left(t\right)\right)$ and ${\pi }_{\left({{\varvec{x}}}_{i},s\right),\left({{\varvec{x}}}_{j},m\right)}\left({\tau }_{\left({{\varvec{x}}}_{i},s\right)}|{\varvec{\theta}}\left(t\right)\right)$ to sample the sojourn time ${\tau }_{\left({{\varvec{x}}}_{i},s\right)}$ and the arrival state $\left({{\varvec{x}}}_{j},m\right)$, until the cumulative sojourn time reaches the preset task time or the system enters the failure space. Algorithm 2.1 provides the procedure to sample ${\tau }_{\left({{\varvec{x}}}_{i},s\right)}$ and $\left({{\varvec{x}}}_{j},m\right)$. Algorithm 2.2 presents the MC simulation procedure for the system.

Then, the estimation of the state probability vector $\hat{\varvec{P}}\left( t \right) = \left( {\hat{p}_{1} \left( t \right), \ldots ,\hat{p}_{N} \left( t \right)} \right)$ at time t is

$$ \hat{\varvec{P}}\left( t \right) = \frac{1}{{N_{max} }}\left( {n_{1,t} , \ldots ,n_{{N_{{\varvec{X}}} ,t}} } \right) $$

(14)

where $\left\{ {n_{i,t} |i = 1, \ldots ,N_{{\varvec{X}}} ,t \le T} \right\}$ is the total number of visits to state ${\varvec{x}}_{j} \in {\varvec{S}}_{{\varvec{X}}}$ by time t, $N_{{\varvec{X}}}$ is the total number of states in ${\varvec{S}}_{{\varvec{X}}}$. The sample variance of $\hat{p}_{i} \left( t \right)$ is defined as

$$ var_{{\hat{p}_{i} \left( t \right)}} = \frac{1}{{N_{max} - 1}}\hat{p}_{i} \left( t \right)\left( {1 - \hat{p}_{i} \left( t \right)} \right) $$

(15)

Then, the reliability function of the system is

$$ R\left( t \right) = 1 - \mathop \sum \limits_{{i \in {\mathbf{\mathcal{F}}}_{{\varvec{X}}} }} \hat{p}_{i} \left( t \right) $$

(16)

3 Model Validation Based on a Simple Case

Fig. 3.

Degradation models for validation calculation

We selected a simple system with 2 degradation processes (as shown in Fig. 3) for model validation. The correctness of the model is verified by comparing the computational results of the model with the results estimated using the analytic formulations of the functioning states of the system derived subsequently.

We assume that sojourn times in the functioning state of ${X}_{1}(t)$ and ${X}_{2}(t)$ obey exponential and Weibull distributions, respectively. The transition rate ${\lambda }_{ij}$ of ${X}_{1}(t)$ is constant. The degenerate evolution of ${X}_{2}(t)$ is influenced by the state of ${X}_{1}(t)$. Suppose ${X}_{2}(t)$ jumps to state j at time t after a sojourn of τ in state i, and the parameters of the distribution of the state sojourn time of ${X}_{2}(t)$ under the state k of ${X}_{1}(t)$ are $\left({\alpha }^{k},{\beta }^{k}\right)$. Given the state set of ${X}_{1}(t)$ on time interval $\left(t-\tau ,t\right)$ is $\left\{k;k\in {S}_{{X}_{1}}^{^{\prime}}\subset {S}_{{X}_{1}}\right\}$, the transition rate of ${X}_{2}(t)$ is assumed as

$$ \overline{\lambda }_{ij}^{2} \left( \tau \right) = \overline{\alpha }^{2} \left( {\overline{\beta }_{ij}^{2} } \right)^{{\overline{\alpha }^{2} }} \tau^{{\overline{\alpha }^{2} - 1}} $$

(17)

$$ \overline{\alpha }^{2} = \mathop \sum \limits_{{k \in S_{{X_{1} }}^{^{\prime}} }} \frac{{\alpha^{k} }}{{N_{k} }},\overline{\beta }_{ij}^{2} = \mathop \sum \limits_{{k \in S_{{X_{1} }}^{^{\prime}} }} \frac{{\beta^{k} }}{{N_{k} }} $$

(18)

where $N_{k}$ is the state number of $S_{{X_{1} }}^{^{\prime}}$.

The probability model [9, 10] for a shock leading to a sudden transition in the state of $X_{1} \left( t \right)$ or $X_{2} \left( t \right)$ is

$$ p_{ij}^{x} = \frac{{9 \times 0.1^{{\left( {i - j + 1} \right)}} }}{{1 - 0.1^{{\left( {i + 1} \right)}} }},i \ge j,x = 1,2 $$

(19)

The model satisfies $\mathop \sum \limits_{j = i}^{0} p_{ij}^{x} = 1$.

Given that $X_{1} \left( t \right)$ state $i_{1}$ and $X_{2} \left( t \right)$ state $i_{2}$ at time s, the probability that $X_{1} \left( t \right)$ and $X_{1} \left( t \right)$ do not experience a sudden state change caused by random shocks on the time interval [s, s + u] is

$$ \varphi_{{\left( {i_{1} ,i_{2} } \right)}} \left( u \right) = \mathop \sum \limits_{n = 0}^{\infty } \left( {p_{{i_{1} ,i_{1} }}^{1} } \right)^{n} \left( {p_{{i_{2} ,i_{2} }}^{2} } \right)^{n} \frac{{\left( {\mu u} \right)^{n} }}{n!}e^{ - \mu u} = e^{{ - \mu u\left( {1 - p_{{i_{1} ,i_{1} }}^{1} p_{{i_{2} ,i_{2} }}^{2} } \right)}} $$

(20)

Suppose ${X}_{1}\left(0\right)=2$ and ${X}_{2}\left(0\right)=1$, and use ${P}_{\left({i}_{1},{i}_{2}\right)}\left(t\right)$ to denote the probability that the system is in state $\left({i}_{1},{i}_{2}\right)$ at time t. For ${X}_{1}(t)$ and ${X}_{2}(t)$ to be in the initial state at time t, the 2 degradation processes should be free of state transfers due to degradation or random shocks, then we have

$$ P_{{\left( {2,1} \right)}} \left( t \right) = e^{{ - \lambda_{21} t}} e^{{ - \left( {\beta^{2} t} \right)^{{\alpha^{2} }} }} \varphi_{{\left( {2,1} \right)}} \left( t \right) $$

(21)

Define $\eta_{2} = min\left\{ {\delta_{2} ,\tau_{1} } \right\}$, where $\delta_{2}$ is the sojourn time of $X_{1} \left( t \right)$ in state 2 considering the effect of random shocks, and $\tau_{1}$ is the time interval from t = 0 when no transfer of $X_{2} \left( t \right)$ from the initial state 1 caused by random shocks occurs. The probability density function (PDF) of $\eta_{2}$ is

$$ \begin{array}{*{20}c} g(u)=\lambda_{21} e^{-\lambda_{21} u} \varphi_{(2,1)}(u) \\+e^{-\lambda_{21} u} \sum_{n=1}^{\infty}\left(p_{22}^1\right)^{n-1} p_{21}^1\left(p_{11}^2\right)^n \frac{(\mu u)^{n-1}}{(n-1) !} \mu e^{-\mu u} =\left(\lambda_{21}+\mu p_{21}^1 p_{11}^2\right) \\\cdot e^{-\left(\mu\left(1-p_{22}^1 p_{11}^2\right)+\lambda_{21}\right) u} \\ \end{array} $$

(22)

Assume that $X_{1} \left( t \right)$ transfers from state 2 to state 1 at time u and $X_{2} \left( t \right)$ is in initial state 1 at this time, let $\rho^{u} = min\left\{ {\delta_{u,1} ,\delta_{u,2} } \right\}$, where $\delta_{u,1}$ and $\delta_{u,2}$ denote the sojourn times in the current states since time u for $X_{1} \left( t \right)$ and $X_{2} \left( t \right)$, respectively. Without considering the effect of random shocks, $\rho^{u}$ has the cumulative distribution function (CDF)

$$ F_{{\rho^{u} }} \left( x \right) = 1 - exp\left( { - \mathop \smallint \limits_{0}^{x} \left( {\lambda_{10} + \overline{\alpha }\left( {\overline{\beta }} \right)^{{\overline{\alpha }}} \left( {v + u} \right)^{{\overline{\alpha } - 1}} } \right)dv} \right) $$

(23)

where $\overline{\alpha } = \left( {\alpha^{2} + \alpha^{1} } \right)/2,\overline{\beta } = \left( {\beta^{2} + \beta^{1} } \right)/2$.

Then we have

$$ P_{{\left( {1,1} \right)}} \left( t \right) = \mathop \smallint \limits_{0}^{t} g\left( u \right)e^{{ - \left( {\beta^{2} u} \right)^{{\alpha^{2} }} }} \cdot \left( {1 - F_{{\rho^{u} }} \left( {t - u} \right)} \right)\varphi_{{\left( {1,1} \right)}} \left( {t - u} \right)du $$

(24)

Table 1. Parameters for model validation

Based on the parameters in Table 1, the functioning state probabilities of the system calculated using the model are shown in Fig. 4, where the results calculated using Eqs. (21) and (24) are also presented. A quantitative comparison of the results shows a high degree of agreement between the results using the model and the analytical formulas, with relative errors below 6%, indicating the correctness of the model.

Fig. 4.

Probabilities of functioning states estimated with different methods

4 Numerical Example

The model is applied to one sub-system of the residual heat removal system (RHRS) of a nuclear power plant [9]. The system consists of a centrifugal pump and a pneumatic valve in series, as shown in Fig. 5.

Fig. 5.

A Subsystem of RHRS, consisting of a centrifugal pump and a pneumatic valve

4.1 Degradation Modeling

Table 2. Model parameters

The pump's performance will deteriorate if wear, corrosion, or erosion occurs to the parts that comprise the pump. The primary degradation mechanism of the valve is the external leakage due to corrosion at the actuator and bottom pneumatic port connection. We assumed that the pump's performance could be divided into four mutually exclusive levels, and there are three states for the valve. Let ${X}_{1}(t)$ denote the degradation level of the pump by time t, with a state space ${S}_{{X}_{1}}=\left\{\mathrm{0,1},\mathrm{2,3}\right\}$ where 3 is the perfect functioning state and 0 is the failure state. Let ${X}_{2}(t)$ denote the degradation level of the valve by time t, with a state space ${S}_{{X}_{2}}=\left\{\mathrm{0,1},2\right\}$ where 2 is the perfect functioning state and 0 is the failure state. The multi-state models describing the degradation process of the pump and valve are shown in Fig. 6, where ${\lambda }_{ij}^{1}\left(\tau \right)$ and ${\lambda }_{ij}^{1}\left(\tau \right)$ denote the derogation transition rates of the pump and valve associated with the sojourn time τ in the current state, respectively.

The state space of the process ${\varvec{X}}\left(t\right)=\left({X}_{1}\left(t\right),{X}_{2}(t)\right)$ is ${S}_{{\varvec{X}}}=\left\{\left(\mathrm{0,0}\right),\left(\mathrm{1,0}\right),\left(\mathrm{2,0}\right),\left(\mathrm{3,0}\right),\left(\mathrm{0,1}\right),\left(\mathrm{1,1}\right),\left(\mathrm{2,1}\right),\left(\mathrm{3,1}\right),\right.$$\left.\left(\mathrm{0,2}\right),\left(\mathrm{1,2}\right),\left(\mathrm{2,2}\right),\left(\mathrm{3,2}\right)\right\}$. The failure state space is ${\mathcal{F}}_{{\varvec{X}}}=\left\{\left(\mathrm{0,0}\right),\left(\mathrm{1,0}\right),\left(\mathrm{2,0}\right),\left(\mathrm{3,0}\right),\left(\mathrm{0,1}\right),\left(\mathrm{0,2}\right)\right\}$.

Fig. 6.

Degradation models for the pump and valve

We assumed that the sojourn time τ in the current state i of ${X}_{n}\left(t\right),n=\mathrm{1,2}$ before the process jumps to state i-1 follows the Weibull distribution. We take the shape parameter $\alpha >1$ to consider the increase of the component degradation transition rate with the growth of the sojourn time, namely, the aging effect. Then the transition rates of ${X}_{n}\left(t\right),n=\mathrm{1,2}$ are

$$ \lambda_{i}^{x} \left( \tau \right) = \alpha^{x} \left( {\beta_{ii - 1}^{x} } \right)^{{\alpha^{x} }} \tau^{{\alpha^{x} - 1}} ,x = 1,2 $$

(25)

The dependency of the two degradation processes is that the pump degradation will vibrate, causing the valve to vibrate, which accelerates the external leakage process of the valve. The assumption on the transition rate of (Eq. (17)) is used again. Furthermore, the shock impact model (Eq. (19)) in Sect. 3 is applied.

The degradation transition rates could be evaluated from degradation/failure data from historical field collection. The transition rates associated with maintenance tasks could be estimated from the data on maintenance activities. For example, for the pump, the parameters of the postulated distribution of state transition times can be estimated by statistics of the state transition times of numerous identical or similar pumps using the method of maximum likelihood or Bayesian estimation. The optimal distribution could be selected by the goodness-of-fit hypothesis test. Correspondingly, the state transition rates of the pump are obtained. Due to the lack of data, we assumed the model parameters as presented in Table 2 for the illustrative calculations.

4.2 Results and Analysis

Let the number of simulations be taken as 1E4, 1E5, and 1E6, respectively, and we estimate the reliability of the system over [0, 13176 h] (18 months, a typical nuclear plant refueling cycle), as shown in Fig. 7, which tends to stabilize when reaches 1E5.

The time-dependent probabilities of functioning states of the system are shown in Fig. 8. In many trials, the system jumps out of the initial state (3, 2) in a very short time, so the probability of this state drops quickly (at about t = 52 h) to 0. Due to the degradation of the pump and valve, the probabilities of the other functioning states go through an increasing phase and then a decreasing phase.

The reliability of the system, the pump, and the valve with/without random shocks are estimated, as shown in Fig. 9. Furthermore, the numerical comparisons on the reliabilities at t = 13176 h are provided in Table 3. The results reveal that ignoring random shocks will lead to an overestimation of the reliability of the system and the components.

Fig. 7.

System reliability estimated with different Nmax

Fig. 8.

Probabilities of functioning states of the system

Fig. 9.

The reliability of the system, the pump, and the valve without/with random shocks

Table 3. Reliability with/without random shocks at t = 13176 h

Fig. 10.

The reliability of the system, the pump, and the valve without/with dependency

Figure 10 presents the reliability of the system, the pump, and the valve with/without degradation dependency. Table 4 gives the numerical comparisons on the reliabilities at t = 13176 h. Neglecting degradation dependency will result in an overestimation of the reliability of the system and the valve. Since the degradation process of the pump is independent of the degradation level of the valve, the reliabilities of the pump with/without degradation dependency are equal.

Table 4. Reliability with/without dependency at t = 13176h

5 Conclusions

In this paper, a multi-state degradation model for reliability assessment of multi-component nuclear safety systems is proposed, accounting for the dependency between degradation processes and random shocks. The degradation process of the system is modeled by the Semi-Markov process, and the random shocks are modeled by the Poisson process. Degradation dependency is modeled explicitly in the transition rates of the integrated degradation process of the system. The transfer kernel function of the holistic model for Monte Carlo simulation is derived. Based on a simple case, the correctness of the proposed model is verified by comparing the system state probabilities estimated using the proposed model and the analytical solution, respectively. The model is applied to the reliability analysis of one sub-system of the RHRS of a nuclear power plant.

The present model uses relatively simple models for the effects of random shocks and maintenance. In future work, more realistic and complex models for the effects of random shocks and maintenance will be investigated.