A multi-state dynamic framework for electric ship BMS reliability evaluation: time-interval sequential rules, multi-domain CCF, and closed-form CTMC solutions

IntroductionAiming at the progressive failure and multi-domain coupling risks of battery management systems (BMS) for electric ships operating in marine environments, existing reliability evaluation methods fail to simultaneously characterize multi-state degradation evolution, sequential failure propagation, and common cause failure (CCF) coupling effects. This study constructs a quantitative reliability evaluation framework for ship BMS to support failure risk control and full-lifecycle operation and maintenance decision-making.MethodsA four-level multi-state fault tree model covering the cell, module, pack, and system layers is established, defining three operating states—intact, performance-degraded, and complete failure—to match the system failure evolution logic. An extended multi-group β-factor model is adopted to quantify CCF risks across four coupling domains: power supply, communication bus, thermal environment, and software. A modular dynamic fault tree (DFT) framework is proposed, with dedicated two-event and three-event DFT modules featuring sequential discrimination capability; an interval partitioning strategy is implemented to eliminate double-counting errors in sequential analysis. Closed-form analytical solutions are derived based on the inclusion–exclusion principle to avoid truncation errors inherent in numerical integration. A Multi-State continuous-time Markov chain (CTMC) model is built for analytical dynamic reliability assessment, with Monte Carlo simulation adopted for comparative verification of calculation results. Cross-validation of Birnbaum importance and Spearman rank correlation coefficients is conducted to distinguish inherent parameter risks from CCF-amplified secondary risks.ResultsValidation results demonstrate that the optimized sequential interval rule table eliminates double-counting defects of traditional algorithms, and all analytical expressions for system state probabilities satisfy the probability normalization axiom. The absolute error between the Multi-State CTMC analytical solutions and the closed-form solutions derived via the inclusion–exclusion principle remains negligible across the full-time horizon, with the relative error consistently approaching zero, verifying the mathematical consistency and accuracy of both solution frameworks. Compared with the independent failure assumption, CCF exhibits marked time heterogeneity: the probability of complete system failure at 1000 h is 10.1 times higher than that under the independent condition, while it decreases by 10.0% at 100000 h, an effect originating from the component screening mechanism. CCF reshapes component criticality rankings: the Birnbaum importance of bus-related basic events Total voltage detection and Total current detection increase by 159.9% and 168.8%, respectively, under CCF conditions, whereas that of the relay control fault rises by only 2.8%, establishing it as the core failure trigger throughout the entire service cycle. Parameter uncertainty analysis reveals that the 95% confidence interval of the system complete failure probability at 100000 h is [0.3737, 0.3971], which is 13.0% higher than the deterministic point estimate.DiscussionThe proposed framework addresses the limitation of traditional static fault trees in handling sequential and multi-state failures, and achieves compatibility between CCF modeling and hierarchical degradation logic. The modular DFT architecture balances computational accuracy and efficiency, making it applicable to reliability evaluation of complex marine electronic equipment. Comparative results between the Multi-State CTMC and closed-form approaches confirm that the proposed model is free of truncation errors, yielding bias-free reliability metrics over the entire mission duration. For engineering applications, global CCF monitoring is recommended in the early service stage; redundancy optimization and fault decoupling for bus-associated basic events should be prioritized in the medium stage; and independent redundancy design of relay units must be strengthened in the long-term service stage. Future research should incorporate component repair behaviors and cross-domain secondary CCF propagation mechanisms to improve model applicability further. This study provides a quantitative tool for the reliability design, risk assessment, and operation and maintenance strategy formulation of electric ship BMS.