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

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Addressing the progressive failure risks inherent in electric ship battery management systems (BMS), this study introduces a novel, quantitative reliability evaluation framework. It simultaneously characterizes multi-state degradation, sequential failure propagation, and common cause failure (CCF) coupling across cell, module, pack, and system levels. Employing a modular dynamic fault tree and continuous-time Markov chain model, validated through closed-form solutions and Monte Carlo simulation, the framework offers bias-free reliability metrics for informed operation and maintenance decisions, ultimately bolstering ocean intelligence applications.
A multi-state dynamic framework for electric ship BMS reliability evaluation: time-interval sequential rules, multi-domain CCF, and closed-form CTMC solutions

The maritime sector's transition to electric propulsion presents a compelling, yet complex engineering challenge. While electric ships promise reduced emissions and improved efficiency, ensuring the reliability of their core systems, particularly battery management systems (BMS), is paramount. This recent study, detailing a multi-state dynamic framework for evaluating BMS reliability, directly addresses a critical gap in current methodologies. Existing approaches often fall short in accounting for the progressive degradation of battery components, the cascading effects of sequential failures, and the pervasive influence of common cause failures (CCF) – events that compromise multiple components simultaneously. This research builds upon foundational work in reliability engineering, such as that explored in Fault Tree Analysis: A Reliable Approach to System Safety and extends it significantly to the unique demands of a marine environment. The development of a quantitative framework, as described here, is a necessary step towards bolstering the safety and operational longevity of electric vessels.

The methodology outlined in the paper is particularly noteworthy for its sophistication. The integration of a four-level multi-state fault tree model, encompassing cell, module, pack, and system levels, allows for a nuanced understanding of failure progression. The adoption of an extended multi-group β-factor model to quantify CCF across power supply, communication bus, thermal environment, and software domains represents a significant advancement. The modular dynamic fault tree (DFT) architecture, with its sequential discrimination capabilities, and the use of closed-form analytical solutions to avoid truncation errors, further solidify the rigor of the approach. Moreover, the validation through Monte Carlo simulation and the cross-validation of importance metrics provides strong evidence of the framework’s accuracy and robustness. As evidenced by the stark difference in failure probability between the CCF and independent failure assumptions, the consideration of these shared vulnerabilities is not merely academic; it’s a critical determinant of system reliability. A complementary perspective on the importance of data-driven reliability assessment can be found in Data-Driven Reliability Assessment of Maritime Systems, highlighting the potential for real-time monitoring and predictive maintenance.

The findings regarding the time heterogeneity of CCF and the reshaped criticality rankings of components are particularly valuable for practical engineering applications. The observation that bus-related basic events (total voltage and current detection) become significantly more critical under CCF conditions underscores the need for targeted redundancy and fault decoupling strategies. Similarly, the increased importance of relay control faults highlights the necessity for robust and independent redundancy design. The study's conclusion advocating for a phased approach to reliability management – global CCF monitoring in the early stages, redundancy optimization in the medium stage, and independent redundancy for relay units in the long term – offers a pragmatic roadmap for ship operators and designers. The acknowledgement of parameter uncertainty and the associated confidence intervals further enhances the framework’s utility for risk-informed decision-making. Understanding the interplay between component degradation and shared vulnerabilities is increasingly vital, as demonstrated by research on predictive maintenance strategies in Predictive Maintenance in Maritime Industry: A Review.

Looking ahead, the authors rightly point to the potential for incorporating component repair behaviors and cross-domain secondary CCF propagation mechanisms to further refine the model. A particularly intriguing area for future investigation would be the integration of real-time sensor data and machine learning techniques to dynamically update the CCF risk assessment based on operational conditions and historical performance. As electric ships become increasingly prevalent, the ability to proactively identify and mitigate potential failure modes will be paramount to ensuring their safe and reliable operation. The question remains: how can this framework be scaled and integrated into existing ship design and maintenance workflows to maximize its impact and drive a new era of maritime safety and efficiency?

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.

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