Time-Varying Reliability Analysis of Integrated Power System Based on Dynamic Bayesian Network

Abstract

In response to the limitations of traditional static reliability analysis methods in characterizing the reliability changes of the Integrated Power System, this paper proposes a time-varying reliability analysis framework based on a Dynamic Bayesian Network. By embedding a multi-physics coupled degradation model into the conditional probability nodes of the Dynamic Bayesian Network, a joint stochastic differential equation for the degradation process was constructed, and the dynamic correlation between continuous degradation and discrete fault events throughout the entire life cycle was achieved. A modified method for modeling continuous degradation systems was proposed, which effectively solves the numerical stability problem of modeling complex degradation systems. Finally, the applicability and correctness of the model were verified through numerical examples, and the results showed that the analysis framework can be effectively applied to time-varying reliability assessment and dynamic health management of complex equipment systems such as the Integrated Power System.

1. Introduction

The Integrated Power System (IPS) for warships combines a traditional warship power system with an electric system, resulting in increased systemic interdependencies and complexity. It is a typical complex equipment system, and its reliability will directly affect the stable operation of the system and the smooth completion of tasks [1]. During service, the system is affected by factors such as dynamic degradation, time-varying loads, and multi-state coupling. Consequently, the focus of its reliability analysis has been shifting from static scenarios to high-dynamic time-varying scenarios. For dynamic reliability, its reliability index can be considered as the probability of equipment operating without faults during a certain period of time. Traditional static reliability analysis methods have difficulty effectively describing its time-varying characteristics. Therefore, accurately analyzing the reliability variation law of the IPS, obtaining more accurate life assessment parameters, is conducive to mastering the health of the system, guiding maintenance work, and improving economic benefits [2].

The operation of the IPS involves multiple dynamic variables and is influenced by various factors. Firstly, during its service life, the system is affected by factors such as temperature changes and vibrations. The coupled stress of temperature vibration triggers a chain reaction of multi-scale multi-physics field failure. At the micro level, it manifests as material performance degradation and structural dynamic characteristic shift, forming nonlinear coupling effects. At the macro level, it manifests as cascading failures of functional chains under dynamic load interactions, leading to reliability degradation. The degradation has non-stationary and polymorphic correlation characteristics [3]. Second, the complex and ever-changing marine environment, coupled with frequent switching of system operating condition (SOC), often has an impact on key nodes in the system, thereby affecting system reliability [4]. In addition, the IPS has numerous subsystems, a large number of key nodes, and coupling relationships between nodes, which cannot be ignored in terms of system reliability. Therefore, system failures often exhibit significant dynamic accumulation and uncertainty, with distinct time-varying characteristics [5].

Traditional reliability analysis methods, such as Markov Chain (MC) [6,7,8] and Monte Carlo Simulation (MCS) [9,10], typically assume that system state transitions satisfy static probability or independent identically distributed conditions, are typically based on the assumption that system state transitions satisfy static probability or independent identically distributed conditions. This makes dynamic coupling failure mechanisms difficult to be characterized, leading to prediction results deviating from engineering reality. Therefore, building a time-varying reliability analysis method that combines dynamic modeling capabilities and computational feasibility has become a research hotspot in this field. Among them, the Dynamic Bayesian Network (DBN) has become an effective tool for dynamic reliability analysis due to its modeling ability for time-varying failure logic and bidirectional reasoning advantages, demonstrating potential in fault diagnosis and risk prediction [11].

The DBN algorithm is mainly divided into two categories: the Discrete Time Bayesian Network (DTBN) algorithm and the Continuous Time Bayesian Network (CTBN) algorithm. Posterior probability of the system at predetermined discrete time points is solved by DTBN through setting a Conditional Probability Table (CPT) and a Marginal Probability Table (MPT). However, with the continuous increase in system complexity, the computational burden of DTBN has sharply increased. In contrast, CTBN significantly reduces computation time in the continuous time domain by constructing analytical solutions of Probability Density Functions (PDFs) for nodes [12].

With the flourishing development of machine learning [13] and data mining [14] technologies, the application scope of DBN continues to expand. Guo et al. [15] proposed a dynamic system reliability modeling method based on DTBN to address the issue of common cause failures in the nuclear power plant control system, achieving precise control of reliability. Huang et al. [16] addressed the issue of dynamic reliability assessment of hierarchical multi-state systems when observation data are inaccurate and improved analysis accuracy by fusing multi-level evidence using Dempster’s rule within the DTBN framework. Wang et al. [17] established a dynamic reliability model for remote-controlled unmanned underwater vehicles under the combined effects of human and equipment factors, combining GO method, fault tree, and human factor analysis, further expanding the application scenarios of DTBN. Cheng et al. [18] proposed a dynamic reliability analysis method for the vehicle transmission system based on the direct conversion between DTBN and reliability block diagrams. Through sensitivity analysis, key subsystems were identified, providing a quantitative basis for system optimization. Wei et al. [19] converted the dynamic fault tree into a DTBN model and applied it to the dynamic time series reliability analysis of the main feedwater system of pressurized water reactors. The results showed that it is closer to the actual operating state than traditional methods and improves decision accuracy. However, as the number of nodes and time slices increases, the dimensionality of CPT&MPT increases exponentially [20], and the computational complexity also increases exponentially. CTBN provides a solution to this problem.

Computational complexity is greatly reduced by CTBN through establishing analytical solutions for each node’s PDF and obtaining posterior probabilities. Yang et al. [21] proposed a generalized CTBN model with dual state CPT for reliability analysis of mixed temporal and non-temporal events in wireless communication networks, effectively reducing the errors of discrete models. Yao et al. [22] proposed the CTBN method considering common cause failure, which uses a time series conditional probability table and impulse function sampling method to solve the overlapping problem of system dynamic failure and correlation. The model’s analysis accuracy under long task times is verified to be better than traditional methods through examples. Although DBN has demonstrated many advantages, there are still shortcomings in the dynamic description of continuous degradation processes in the field of time-varying reliability analysis of the complex equipment system [23,24], as well as the high dependence of parameter learning on prior assumptions [25,26].

In response to the above issues, an improved DBN time-varying reliability analysis framework is proposed in this paper, which embeds a multi-physics coupling degradation model into the DBN node conditional probability, eliminates the limitation of discrete time points, dynamically fuses multi-source heterogeneous data, and provides support for the dynamic reliability analysis and health management of warship integrated electric propulsion system.

This section introduces the purpose and significance of the study and reviews previous research related to this study. Section 2 introduces the construction process of a multi-physics coupled degradation model. Section 3 conducted a system-level decomposition and node definition for the IPS, demonstrating the logical relationships. In Section 4, a DBN model for the IPS was constructed by setting dynamic evolution rules and time slicing, combined with the generation of conditional probability tables. Section 5 uses the method proposed in this article to conduct experimental verification of a certain case, obtain the reliability variation law, and verify the effectiveness of the method through comparison. Finally, Section 6 presents the conclusion and future research directions.

2. Multi-Physics Coupling Degradation Model

2.1. Model Description and Basic Assumptions

Research on the time-varying reliability of the IPS requires an accurate understanding of the physical characteristics, environmental factors, system hierarchy, and interaction relationships of key components. But in order to apply the model to practical engineering problems with sufficient accuracy, parameter abstraction and model assumptions need to be made. The following assumptions can be made:

1.

Continuous degradation: The degradation of equipment performance $X\left(t\right)$ is a right-continuous random process that satisfies ${\lim }_{h\to 0^{+}}X\left(t+h\right)=X\left(t\right)$ and ${\lim }_{h\to 0^{+}}X\left(t-h\right)$ exists;

2.

Interference randomness: The random interference events caused by SOC switching follow a Poisson process, and the degradation amplitude $J_k$ follows a normal distribution;

3.

Stress independence: The continuous degradation process, random interference events $N\left(t\right)$, and random noise $Y\left(t\right)$ are independent of each other, and the coupled effects on degradation are linearly superimposed;

4.

Temperature vibration coupling: There is a coupling relationship between temperature $T\left(t\right)$ and vibration stress $S\left(t\right)$.

2.2. Model Construction

2.2.1. Continuous Degradation Term

According to reference [27], temperature changes accelerate the degradation of system components to a certain extent. Therefore, based on the Arrhenius equation, the time-varying degradation rate caused by temperature can be expressed as follows:

$${\mu }_T\left(t\right)=A_1{e}^{-{\displaystyle \frac{E_a}{K_{B}T\left(t\right)}}}$$

(1)

where $A_1$ is the temperature stress acceleration coefficient, which is usually fitted by constant temperature tests; $E_a$ is apparent activation energy; $K_B$ is the Boltzmann constant; $T\left(t\right)$ is the thermodynamic temperature of the environment at a given moment $t$.

Vibration stress induces material fatigue damage accumulation and structural dynamic response mismatch, leading to microcrack initiation and propagation, resulting in natural frequency shift and modal parameter degradation of key components, significantly affecting system reliability [28].

According to reference [29], based on Miner’s linear cumulative damage theory, the time-varying degradation rate caused by vibration stress can be expressed as follows:

$${\mu }_S\left(t\right)=A_2\sqrt{{\displaystyle \frac{1}{t}}{\displaystyle {\int }_0^t{S}^2\left(t\right)}dt}$$

(2)

where $A_2$ is the vibration stress sensitivity coefficient, usually calibrated by experiments; ${\displaystyle {\int }_0^t{S}^2\left(t\right)}dt$ is the accumulated vibration energy from time $0$ to $t$.

2.2.2. Random Interference Term

According to the basic assumption, the interference events caused by the switching of SOC are random. Therefore, the degradation caused by the switching of SOC can be described by a composite Poisson process:

$$J\left(t\right)={\displaystyle {\sum }_{k=1}^{N\left(t\right)}{J}_k}$$

(3)

where $N\left(t\right)$ represents the number of SOC switching occurrences within time $\left[0,t\right]$ and obeys the Poisson distribution $N\left(t\right)\sim P\left(\lambda t\right)$; $J_k$ represents the degradation amplitude of the $k-\mathrm{th}$ time SOC switch and obeys the normal distribution $J_k\sim N\left({\mu }_j,{\sigma }_j^2\right)$.

2.2.3. Random Noise Term

To describe small disturbances that have not been explicitly modeled, such as measurement errors and manufacturing defects, a random noise term is introduced, and the resulting degradation can be expressed as follows:

$$Y\left(t\right)=\sigma W\left(t\right)$$

(4)

where $\sigma$ is the diffusion coefficient; $W\left(t\right)$ is the standard Wiener process; $dW\left(t\right)$ is the increment of the standard Wiener process, following a normal distribution with a mean of $0$ and a variance of $dt$, namely, $dW\left(t\right)\sim N\left(0,dt\right)$.

2.2.4. Coupled Degradation Model

In order to consider the coupling effect of temperature vibration, quantify the synergistic amplification effect of vibration stress on thermal damage rate, and describe the nonlinear modulation effect of temperature on the cumulative rate of vibration damage, it is necessary to modify Equations (1) and (2).

Therefore, Equation (1) can be expressed as follows:

$${\mu }_T^{\prime }\left(t\right)=A_1{e}^{-{\displaystyle \frac{E_a}{K_{B}T\left(t\right)}}}·\left[1+\alpha \mathrm{rms}\left(S\left(t\right)\right)\right]$$

(5)

where $\alpha$ is the vibration enhancement coefficient, which is usually obtained by measuring the growth rate of $\mu$ at different $S\left(t\right)$ values through vibration temperature dual-stress acceleration tests.

Equation (2) can be expressed as follows:

$${\mu }_S^{\prime }\left(t\right)=A_2\sqrt{{\displaystyle \frac{1}{t}}{\displaystyle {\int }_0^t{S}^2\left(t\right)}{e}^{\beta T\left(t\right)}dt}$$

(6)

where $\beta$ is the temperature modulation coefficient, which is obtained by changing the temperature under constant amplitude vibration and fitting the slope of the $\ln \left({\mu }_S\right)\mathrm{vs}.T$ curve.

By linearly superimposing the continuous degradation term, random interference term, and random noise term and expressing them in the form of stochastic differential equations, the multi-physics coupled degradation model can be obtained as follows:

$$dX\left(t\right)={\mu }^{\prime }\left(t\right)dt+\sigma dW\left(t\right)+dJ\left(t\right)$$

(7)

3. System-Level Decomposition and Node Definition

The first step in DBN modeling is to clarify the hierarchical structure, node definitions, and dynamic coupling relationships of the IPS [21]. This section proposes a four-layer decomposition integration framework based on a multi-physics coupled degradation model to support the requirements of time-varying reliability analysis.

A typical complex equipment system can be decomposed into the following four levels:

1.

Environment Layer (EL)

External dynamic input variables $E\left(t\right)$, including temperature stress sequence $T\left(t\right)$ and vibration stress sequence $S\left(t\right)$.

2.

Component Layer (CL)

Contains a set of basic functional units $\mathrm{C}=\left\{C_1,C_2,\dots ,C_m\right\},i\in \left[1,m\right]$, and the degradation process of each component $C_i$ is driven by a multi-physics coupled degradation model.

3.

Subsystem Layer (SL)

The set ${\mathrm{S}}_k=\left\{S_{k1},S_{k2},\dots ,S_{kn}\right\},j\in \left[1,n\right]$ containing $n$ components and functionality associated, whose reliability logic is determined by the architecture:

$$R_{{\mathrm{S}}_k}\left(t\right)={\mathsf{\Psi }}_k\left(R_{k_1}\left(t\right),R_{k_2}\left(t\right),\dots \right)$$

(8)

where series subsystem ${\mathsf{\Psi }}_k={\displaystyle \prod R_{ki}\left(t\right)}$; parallel subsystem ${\mathsf{\Psi }}_k=1-{\displaystyle \prod \left(1-R_{ki}\left(t\right)\right)}$.

4.

System Layer (SysL)

Global reliability is defined jointly by the subsystem:

$$R_{\mathrm{Sys}}\left(t\right)=\mathsf{\Phi }\left(R_{S_1}\left(t\right),R_{S_2}\left(t\right),\dots \right)$$

(9)

where $\mathsf{\Phi }\left(·\right)$ is the mapping function, used to describe the significant nonlinear relationship between degradation $X\left(t\right)$ and reliability $R\left(t\right)$.

Using an improved logistic function as the mapping function:

$$\mathsf{\Phi }\left(X\left(t\right)\right)={\displaystyle \frac{1}{1+\exp \left(\gamma \left(X\left(t\right)-X_{th}+\frac{\delta }{\gamma }\right)\right)}}$$

(10)

where $X_{th}$ is the failure threshold, defined as the critical value of degradation, calibrated through historical data. $\gamma$ is the curvature parameter that controls the rate of reliability decay. $\delta$ is the offset parameter, which modifies the symmetry of the function to adapt to the asymmetric degradation process. $\gamma$ and $\delta$ can be calibrated based on historical failure data using maximum likelihood estimation.

Therefore, the hierarchical structure of the system is shown in Figure 1.

Based on system-level decomposition, DBN nodes can be divided into three categories, as shown in

Table 1. Node classification.

Node Type	Symbolic Representation	Variable Definition
Degenerate nodes	$X_i^t$	Degradation of components $C_i$
Environmental nodes	$E_j^t$	Environment variable
Reliability nodes	$R_k^t$	Reliability of the system

.

4. Construction of DBN

4.1. Time Slice Design and Dynamic Evolution Rules

To accurately characterize the time-varying reliability of a warship’s integrated electric propulsion system using dynamic Bayesian networks, it is necessary to systematically construct time-slice design criteria and dynamic evolution rules from the time dimension.

To ensure that the increment of degradation $X_t$ within $\mathsf{\Delta }t$ does not exceed the failure threshold, thereby maintaining the smoothness of the degradation process. Therefore, based on Equation (7), the upper bound of time slicing $\mathsf{\Delta }t$ is determined and expressed as follows:

$$\mathsf{\Delta }t\le {\displaystyle \frac{X_{th}-X_0}{\max \left({{\mu }^{\prime }}_T\left(t\right),{{\mu }^{\prime }}_S\left(t\right)\right)}}$$

(11)

At the same time, in order to achieve a computable expression of the continuous degradation process, $X_t$ need to be discretized into $m$ state intervals (such as normal, slight degradation, failure), whose boundaries are calibrated by historical data quantiles. The discretization process needs to balance the physical meaning and computational efficiency of state partitioning, ensuring a smooth transition of probability distributions between adjacent states.

The dynamic evolution rules unify the representation of multi-source uncertainty through the modeling of composite degradation processes. In DBN, the state evolution of degraded nodes satisfies Markov properties, namely:

$$P\left(X^{t+1}|X^t,X^{t-1},\dots ,X^0\right)=P\left(X^{t+1}|X^t\right)$$

(12)

So, the evolution of degradation includes continuous degradation terms, random interference terms, and noise terms. The evolution process can be represented by the joint distribution of independent random processes, that is:

$$P\left(X_i^{t+1}|X_i^t,E_j^t\right)=N\left(X_i^t+{\mu }^{\prime }\left(t\right)\mathsf{\Delta }t,{\sigma }^2\mathsf{\Delta }t\right)\ast CP\left(\lambda \mathsf{\Delta }t,N\left({\mu }_J,{\sigma }_J^2\right)\right)$$

(13)

where $N$ represents the joint normal distribution of the continuous degradation term and random noise term, $CP$ term represents the composite Poisson process of the random interference term, and $\ast$ represents the convolution operation of the two.

4.2. Generation and Update of Conditional Probability Table

The multi-physics coupled degradation model is a complex nonlinear model, and it is difficult to obtain conditional probabilities through analytical solutions. Therefore, the MCS method is introduced to achieve refined modeling.

First, discretize Equation (7) into a recursive form with a time step of $\mathsf{\Delta }t$, expressed as follows:

$$X_i^{l+1}=X_i^l+{\mu }^{\prime }\left(t_l\right)\mathsf{\Delta }t+\sigma \sqrt{\mathsf{\Delta }t}·\epsilon +{\displaystyle {\sum }_{k=1}^K{J}_k}$$

(14)

where ${\mu }^{\prime }\left(t_l\right)={{\mu }^{\prime }}_T\left(t_l\right)+{{\mu }^{\prime }}_S\left(t_l\right)$ is the degradation rate at the current moment. $\epsilon \sim N\left(0,1\right)$ is the standard normal distribution noise. $K\sim P\left(\lambda \mathsf{\Delta }t\right)$, is the number of random interferences within $\mathsf{\Delta }t$.

Then, generate a degradation trajectory, as shown in Figure 2.

Finally, calculate the probability of state transition. After generating $N$ trajectories, calculate the frequency of the degradation amount $X_i^{l+1}$ falling into each state $Z$ interval at the next moment, and normalize it to obtain the conditional probability:

$$P\left(X_i^{l+1}=Z_n|X_i^l=Z_m,T^l,S^l\right)={\displaystyle \frac{\mathrm{count}\left(X_i^{l+1}\in Z_n\right)}{N}}$$

(15)

To achieve online parameter updates and dynamic adjustment of CPT, the Particle Filter (PL) is introduced. The core process can be decomposed into the following steps [30]:

• Particle Initialization

Extract $M$ particles ${\left\{{\theta }_0^{\left(m\right)}\right\}}_{m=1}^M$ from the prior distribution $P\left({\theta }_0\right)$, each particle represents the possible values of the degradation model parameter $\theta$, and these are ${{\mu }^{\prime }}_T,{{\mu }^{\prime }}_S,\sigma ,\lambda ,{\mu }_j,{\sigma }_j$. The initial weights of all particles are set to $w_0^{\left(m\right)}=1/M$, indicating that the initial credibility of each parameter is consistent.

2.

State Prediction

Generate degradation trajectories for each particle ${\theta }^{\left(m\right)}$ through a Monte Carlo simulation.

3.

Weight Update

After obtaining real-time observation data $Y^{\left(l+1\right)}$, it is necessary to adjust the weights of each particle to reflect its degree of matching with the real state. The weight update equation is as follows:

$$w_{l+1}^{\left(m\right)}\propto w_l^{\left(m\right)}·\exp \left(-{\displaystyle \frac{{\left(Y^{\left(l+1\right)}-X_i^{\left(l+1\right),\left(m\right)}\right)}^2}{2{\sigma }_Y^2}}\right)·{\mu }_d\left({\lambda }^{\left(m\right)}\right)$$

(16)

where $\propto$ represents that the new weight is proportional to the right-hand equation. $\exp \left(·\right)$ is the Gaussian likelihood function of the observed data, used to quantify the deviation between the predicted degradation $X$ and the measured value $Y$. ${\sigma }_Y^2$ is the variance of observation data errors. ${\mu }_d\left({\lambda }^{\left(m\right)}\right)$ is a fuzzy membership function introduced for maintaining records and modify the parameter distribution through implicit knowledge, to achieve the fusion of observation data and maintenance data. The fuzzy membership function can be expressed as follows:

$${\mu }_d\left({\lambda }^{\left(m\right)}\right)=\exp \left(-{\displaystyle \frac{{\left({\lambda }^{\left(m\right)}-k/\tau \right)}^2}{2{\epsilon }^2}}\right)$$

(17)

where $k$ is the maintenance frequency within the time window $\tau$; $\epsilon$ is the tolerance for ambiguity, and the larger the value of $\epsilon$, the more deviation of ${\lambda }^{\left(m\right)}$ from the expected value is allowed.

Realize the fusion weight adjustment of observation data and maintenance knowledge through Equations (16) and (17). As shown in Figure 3, this process can effectively track the SOC switching frequency of PMSM.

4.

Resampling

To avoid the degradation of particle weights (i.e., most particle weights tend to zero), resampling operations need to be performed. To ensure that the starting point of each sampling is different and evenly covers the entire weight distribution, it is necessary to first generate uniformly distributed random numbers $u_1=U\left[0,1/M\right]$, then construct a sequence $u_k=u_1+\left(k-1\right)/M$, $\left(k=1,2,\dots ,M\right)$, and then select high-weight particles for replication based on the cumulative weight distribution, eliminating low-weight particles, thereby ensuring that the total number of particles remains unchanged while improving the accuracy of parameter estimation.

5.

Parameter Estimation

By fusing all particle information through a weighted average, calculate the posterior estimation parameters as follows:

$${\widehat{\theta }}_{l+1}={\displaystyle {\sum }_{m=1}^M{w}_{l+1}^{\left(m\right)}}{\theta }_{l+1}^{\left(m\right)}$$

(18)

Based on the updated particle set, the state transition frequency of the degraded trajectory is recalculated, normalized to generate a new CPT, and fed back to the DBN to complete the model iteration. Adopting an adaptive dual-mode update strategy to balance computational efficiency and tracking accuracy, with a time step of 0.1 h.

• Regular update mechanism

The structural parameters that characterize the long-term degradation trend are updated regularly every 50 steps.

• Event triggering mechanism

Real-time response updates are performed on transient parameters that are sensitive to operating conditions, and parameter re-estimation is completed within 2 steps to capture dynamic changes.

This dual-mode update mechanism, based on parameter characteristics and system state, not only ensures the dynamic tracking ability of key degradation processes but also optimizes resource allocation.

The pseudocode of the particle filter algorithm is as follows (Algorithm 1):

Algorithm 1 Particle Filter for CPT Adjustment [30]

Input: Prior distribution $P\left({\theta }_0\right)$, Observations $Y$, Maintenance parameters $\left(k,\tau ,\epsilon \right)$, Particle count $M$ Output: Posterior estimate $\widehat{\theta }$, Updated CPT 1: Initialize particles:   for $m$ = 1 to $M$:     ${\theta }_0^{\left(m\right)}~P\left({\theta }_0\right)$    //  ${\mu }_T,{\mu }_S,\sigma ,\lambda ,{\mu }_j,{\sigma }_j$     $w_0^{\left(m\right)}=1/M$ 2: for each time step $t$ = 1 to $T$: 3:  // State prediction 4:  for each particle $m$: 5:    $X_t^{\left(m\right)}$ ← MCS $\left({\theta }_{t-1}^{\left(m\right)}\right)$    //  Degradation trajectory 6:  // Weight update 7:  for each particle $m$: 8:    $\mathcal{L}=\exp \left(-{\left(Y_t-X_t^{\left(m\right)}\right)}^2/\left(2{\sigma }_i^2\right)\right)$    //  Gaussian likelihood (Equation (16)) 9:    ${\mu }_d=\exp \left(-{\left({\lambda }^{\left(m\right)}-k/\tau \right)}^2/\left(2{\epsilon }^2\right)\right)$    //  Fuzzy fusion (Equation (17)) 10:   $w_t^{\left(m\right)}\propto w_{t-1}^{\left(m\right)}·\mathcal{L}·{\mu }_d$        // Weight update (Equation (16)) 11:  Normalize weights: ${\displaystyle \sum w_t^{\left(m\right)}}=1$ 12:  // Resampling 13:  $u_1$ ∼ Uniform [0, 1/$M$] 14:  for $k$ = 1 to $M$: 15:     $u_k=u_1+\left(k-1\right)/M$ 16:     Select particle with cumulative weight > $u_k$ 17:     Reset $w_t^{\left(m\right)}=1/M$ 18:  // Parameter estimation 19:  ${\widehat{\theta }}_{l+1}={\displaystyle {\sum }_{m=1}^M{w}_{l+1}^{\left(m\right)}}{\theta }_{l+1}^{\left(m\right)}$   // Posterior estimate (Equation (18)) 20:  // CPT update 21:  Calculate state transition frequency 22:  Normalize → New CPT 23:  Update DBN model 24: end for

5. Case Study

5.1. Case Background and Experimental Design

In order to evaluate the time-varying reliability analysis method proposed in this study, the IPS used in a certain type of warship is taken as an example. The system includes key subsystems such as power generation, transmission and distribution, and propulsion system. It is greatly affected by temperature and vibration and has the characteristics of multi-physical field coupling degradation. There is also a switching of SOC under different power requirements, which meets the complexity requirements of time-varying reliability analysis. The composition structure is shown in Figure 4.

The IPS comprises four mission-critical subsystems working in concert to generate, distribute, manage, and utilize electrical energy. The power generation system converts mechanical energy from prime movers into electrical power via generators, with governors ensuring stable frequency output. This electricity is routed through the power transmission and distribution system—the central power backbone operating at medium-to-high-voltage levels (typically 4 kV–15 kV DC/AC). This subsystem employs distribution panelboards for load allocation, circuit-breakers for millisecond-level fault protection, and transformers for voltage conversion, collectively minimizing ohmic losses during ship-wide power transfer. Simultaneously, the energy management system monitors operational parameters via data acquisition units and dynamically optimizes energy flow through predictive control modules. Finally, the propulsion system transforms this regulated electrical energy into thrust via permanent magnet synchronous motors, which are driven by IGBT-based frequency converters and transfer torque through transmission shafting to high-efficiency screw propellers [4,5].

According to the structural diagram of the IPS, its DBN can be constructed as shown in Figure 5, and the node numbers are shown in

Table 2. Node number.

Node Number	Name	Node Number	Name
SF	IPS	B2	Circuit-breaker
AF	Power generation system	B3	Transformer
BF	Power transmission and distribution system	C1	Data acquisition unit
CF	Energy management system	C2	Control module
DF	Propulsion system	D1	Permanent magnet synchronous motor
A1	Prime mover	D2	Frequency converter
A2	Generator	D3	Transmission shafting
A3	Governor	D4	Screw propeller
B1	Distribution panelboard

.

For the convenience of discussion, taking the permanent magnet synchronous motor (PMSM) component in the system as an example, the degradation model parameters are set as shown in

Table 3. Multi-physics coupled degradation model parameters for PMSM.

Parameter	Physical Meaning	Unit	Value Range	Calibration and Verification
$A_1$	Temperature stress acceleration coefficient	h−1	2.5×10−4 ± 0.3×10−4	80 °C constant-temperature accel-erated aging tests [27]
$A_2$	Vibration stress sensitivity coefficient	(m/s2)−1·h−1/2	1.2×10−3 ± 0.2×10−3	Vibration table acceleration test (100 Hz sine vibration)
$E_a$	Apparent activation energy	eV	0.45 ± 0.05	Material properties of NdFeB magnets [27]
$\alpha$	Vibration enhancement coefficient	(m/s2)−1	0.15 ± 0.03	Dual-stress accelerated tests (100 Hz vibration platform)
$\beta$	Temperature modulation coefficient	K−1	0.08 ± 0.01	Variable-temperature vibration tests (20–120 °C)
$\sigma$	Diffusion coefficient	h−1/2	0.015 ± 0.002	Factory test data (100-unit sample)
$\lambda$	SOC switching frequency	h−1	0.06 ± 0.01	Warship power system operation logs
${\mu }_j$	Mean degradation amplitude from transient load shocks	Dimensionless	0.02 ± 0.005	Sudden load test data [4]
${\sigma }_j$	Standard deviation of transient load degradation	Dimensionless	0.003 ± 0.001	Failure event statistics
$X_{th}$	Failure threshold (15% efficiency drop)	Dimensionless	0.25 ± 0.03	Historical maintenance records (200 failure cases)

. Environmental loads, random interference, and noise caused by SOC switching are monitored. After inputting DBN-related parameters, the model is solved to obtain the time-varying reliability curve of PMSM.

The following four steps are followed to solve and then referred to other components. The time-varying reliability curve of the IPS can ultimately be obtained.

• Data preprocessing

Integrating real-time monitoring data (temperature, vibration spectrum) with maintenance records, calibrating the initial degradation state $X_0$, and removing abnormal data points.

Monitor temperature data through K-type thermocouples (accuracy ± 0.5 °C) on the stator winding of permanent magnet synchronous motors, and monitor vibration data through ICP accelerometers (0.5–5 kHz frequency response) at a sampling rate of 10 kHz. Select temperature and vibration stress data of the permanent magnet synchronous motor within a certain task cycle, as shown in Figure 6.

From Figure 6, it can be seen that within the 10 h task cycle, the PMSM power is low within 0–1 h, normal power within 1–6 h, and system failure due to external impact at 6 h, resulting in a sudden increase in temperature and vibration stress. To clearly indicate the impact stage, highlight it in pink line in the figure. After troubleshooting, the vibration stress immediately returns to the normal range, and the temperature gradually returns to the normal range. The magnified image of the impact phase is shown in Figure 7.

In the actual operation of IPS, sensors will monitor a large amount of data, but these data are multi-source and heterogeneous, and there is a large amount of invalid data. After data cleaning, it is difficult to support subsequent reliability analysis. Therefore, it is necessary to combine historical operational data of the system. The Bayesian estimation method can obtain more accurate estimates through maximum a posteriori estimation by integrating population information, prior information, and sample information, significantly improving the accuracy of reliability analysis. In practical operation, acceleration test data, expert experience, and historical databases can be integrated as prior information, and real-time monitoring data can be used as sample information.

Due to the coupling effect of temperature and vibration stress in characterizing degradation, it is necessary to consider the synergistic amplification effect of vibration stress on thermal damage rate and the nonlinear modulation effect of temperature on vibration damage rate. Based on the temperature vibration data during the PMSM task period, the degradation rate variation law is obtained by linearly superimposing random interference terms and random noise terms, as shown in Figure 8.

The degradation amount is the accumulation of the degradation rate over time, obtained by integrating the degradation rate. Therefore, the degradation variation curve can be obtained, as shown in Figure 9.

From Figure 9, it can be seen that the overall degradation amount shows a slightly fluctuating upward trend, and the incremental degradation amount is also clearly segmented at different stages, such as low power, normal power, and under impact in PMSM. At low power, the temperature and vibration stress are relatively low, and the degradation caused is not significant. After being impacted, there is a significant increase in temperature, vibration stress, and degradation.

2.

Dynamic evolutionary simulation

Generate degradation trajectories based on Equation (14) and update environmental nodes $E\left(t\right)$ synchronously; dynamically adjust model parameters $\theta$ and update CPT through particle filtering.

3.

Reliability calculation

According to the logistic mapping function of Equation (10), convert the degradation $X\left(t\right)$ into subsystem and global reliability $R\left(t\right)$.

4.

Validation design

Compare the prediction error of traditional Static Bayesian Networks (SBNs) with the method proposed in this paper and validate the advantages of the proposed model.

5.2. Result Analysis

By running custom code in MATLAB R2023a software and following the above steps to calculate the reliability of components and subsystems, the variation pattern of the IPS reliability is finally obtained, as shown in Figure 10.

As shown in Figure 10, the time-varying reliability evolution process of the IPS can be divided into three stages:

• Initial Stable Stage (0–1000 h)

The system maintains an ideal reliability ($R_{\mathrm{Sys}}=1$), indicating that the IPS has complete reliability characteristics in the initial stage of operation, which conforms to the typical “bathtub curve” initial stable stage of engineering systems.

2.

Initial degradation stage (1000–1200 h)

As can be seen from the magnified local image, the system has entered the initial degradation stage, and the reliability shows a linear decay trend, with a cumulative reliability loss of $\mathsf{\Delta }{R}_{\mathrm{Sys}}=1.3\%$ This stage conforms to the early failure characteristics described by the Weibull distribution.

3.

Accelerated Failure Stage (>1200 h)

At this stage, the system exhibits nonlinear accelerated degradation characteristics, with a significant increase in the rate of reliability degradation and a clear change in curvature indicating the existence of a multi-factor coupled failure mechanism. After 1800 h, due to the saturation of microcrack propagation inside the material, further degradation requires higher energy input, resulting in a gradual trend at the end of the accelerated failure stage.

To verify the adequacy of the multi-physics coupled degradation model, this study used the Kolmogorov–Smirnov (K-S) test method for statistical verification: Empirical distribution analysis was conducted on 10⁵ degradation trajectories generated by Monte Carlo simulation and compared with the theoretical distribution based on Equation (7). The test results at the 2000 h task endpoint (t = 2000 h) showed that the K-S statistic calculated p = 0.18 (greater than the significance level of 0.05), indicating that there is no significant deviation between the simulated distribution and the theoretical distribution (the null hypothesis H₀ cannot be rejected at the α = 0.05 level). This result statistically confirms that the model can accurately capture the intrinsic coupling mechanism of multi-physics field degradation.

In order to evaluate the performance differences between the proposed DBN method and SBN, MCS, and MC in long-term system operation, Figure 11 shows the comparison results of the system reliability of the four methods within 1400–2000 h. The stability and robustness characteristics of the four methods can be intuitively compared through the curve variation pattern.

Table 4. Performance comparison of four methods.

Method	Run Time (s)	RMSE vs. MCS
Proposed DBN	15.20	0.0164
MCS	85.72	0.0000
SBN	9.54	0.0392
MC	11.37	0.0517

compares the performance and root mean square error (RMSE) of four methods in actual operation.

As shown in Figure 11, the overall decreasing trend of $R_{\mathrm{Sys}}$ for the four methods is similar, showing a trend of slow first, then fast, and then slow again. The curve solved by the Proposed DBN is located in the middle of the other three methods, which preliminarily indicates the feasibility of the method. To further quantify the comparison, as shown in Table 4, RMSE vs. MCS was used to represent the root mean square error of the other three methods relative to MCS. The results further showed that the average error between Proposed DBN and MCS was 1.64%, while SBN’s was 3.92% and MC’s was 5.17%. This ratio further validates the feasibility and correctness of the Proposed DBN method in this paper. Meanwhile, although the computation time of the Proposed DBN is higher than SBN and MC, it is significantly lower than MCS and within an acceptable range.

To evaluate the impact of each subsystem on the global reliability of the IPS, Figure 12 visually presents the sensitivity levels of the AF, BF, CF, and DF subsystems through sensitivity analysis.

Analysis shows that the DF subsystem has a decisive impact on the global reliability of the IPS, with a sensitivity value 11.8% higher than the second-ranked AF, highlighting its critical role in system reliability. The main reasons for the highest sensitivity of DF are as follows: The terminal nature of the load makes it a stress convergence point, the multi-field coupling catalyzes an increase in damage index, and the topological criticality amplifies the consequences of failure. BF and CF exhibit moderate and low sensitivity characteristics, respectively, forming significant gradient differences, which is consistent with the redundancy setting in engineering practice. This further validates the superiority of the multi-factor coupling analysis method proposed in this paper.

6. Conclusions

A time-varying reliability analysis framework based on dynamic Bayesian networks is proposed in this study. By embedding a multi-physics coupled degradation model into the conditional probability nodes of DBN, a joint stochastic differential equation for the degradation process is constructed, achieving dynamic correlation between continuous degradation and discrete fault events. The improved modeling method for continuous degradation systems effectively solves the numerical stability problem of complex degradation systems, and the effectiveness of the model is verified through numerical simulations and case studies. Taking a certain type of ship’s IPS as an example, the influence of temperature vibration coupling stress and system operating condition switching on reliability was quantitatively analyzed. The results showed that this method can accurately capture the three-stage evolution law of system reliability (initial stability, linear regression, accelerated failure), which is highly consistent with the characteristics of the “bathtub curve” in engineering practice. Compared with SBN and MC, its root mean square error relative to MCS is the smallest, and the sensitivity of the propulsion subsystem (DF) is 11.8% higher than other subsystems, which is consistent with engineering redundancy design practices. The dynamic fusion of multi-source heterogeneous data has been achieved through a particle filtering algorithm, which supports online updating of model parameters and significantly improves the prediction robustness under extreme working conditions. The research results provide theoretical support for the dynamic health management of complex equipment systems, and their universal design can be extended to fields such as energy and transportation.

In future research, deep learning-driven degradation failure coupling modeling and cross-scale correlation mechanisms can be further explored to enhance predictive adaptability under non-stationary conditions. Meanwhile, the universality of the verification method can be considered for applications in different fields, such as hydropower plant power systems and mine power supply systems.