# Reliability Modelling and Evaluation for LTD System Based on Load-Sharing Model
## 1 Introduction  
The linear transformer driver (LTD) is a rapidly developing technology used to generate pulsed outputs with the requirement of high voltage, high current, and high power. By contrast with the conventional pulsed-power-accelerator scheme, such as a Marx generator, the LTD system adopts power-adding technology rather than pulse compression and, correspondingly, has several noticeable advantages [1; 2; 3]. As a modified induction voltage adder (IVA) [4; 5], the LTD also consists of several induction cavities called \"LTD stages\". A hierarchical and modular structure is adopted in the general design process of the LTD systems [6; 7], which decreases the inherent complexity of the system and makes it possible to conduct module maintenance, recycling, and reconfiguration. Aimed to improve the output performance, many research projects focusing on LTD stages have been conducted. Thus, there have been many improved designs, such as Fast LTDs [8; 9] and solid-state LTD [10]. All phenomena show that the LTD technology is a hot issue. With the increasing tendency of the system scale and number of components, the reliability and stability of the LTD system become important problems. Serval fault modes that could affect the pulsed outputs of the LTD system have been discussed in recent research, such as switch self-breakdown and capacitor failure. Robust design and fault analysis for basic components have been conducted recently [11; 12], but there is still a lack of studies considering whole system reliability. However, we always need to evaluate whole system reliability to balance the benefits of successful experiments with the serious economic losses and even catastrophic accidents caused by failures. Another challenge is that it is dangerous and costly to directly conduct real-world experiments of the LTD system because the experiment is a large-scale cycle of charge and discharge involving numerous capacitors. Therefore, lack of enough experimental data on the whole LTD system, there is an urgent need to make attempts to evaluate LTD system reliability with analytical and/or simulation methods. As mentioned above, all the LTD-based pulsed-power accelerators adopt a hierarchical and modular structure. Considering the redundancy design, a small number of failure modules are allowed during the experiments. Thus, the LTD system is a typical k-out-of-n: F system [13]. In most reliability analysis for a k-out-of-n: F system, independence is always assumed across the components within the system [14; 15]. Nevertheless, it is more realistic to consider the feature of fault dependence among the modules during reliability analysis. Load-sharing mechanism can be used to describe the fault dependence, which assumes that the total load will be redistributed into the remaining modules with some rules as there is a module failure. Therefore, the load-sharing k-out-of-n: F system is a more appropriate description of the LTD system. Since proposed by [PERSON] [16] to describe how the strain on yarn fibers increases as individual fibers within a bundle break, the load-sharing mechanism has been wildly considered in the system analysis process at various stages of the life cycle, such as load allocation strategy [17], maintenance analysis [18], and system architecture design [19]. When it comes to reliability analysis, there are also many research works based on load-sharing mechanism. [PERSON] et al. [20] and [PERSON] [21] respectively presented reliability analysis of the load-sharing k-out-of-n system with iid components and non-iid components. A tampered failure rate model and a cumulative exposure model were used to calculate the system lifetime with the assumption that the fixed baseline fault distribution of the components is known in advance [22; 23]. Moreover, new failure mechanisms, such as components degradation, were gradually considered [24; 25; 26]. Meanwhile, due to the difficulty of acquiring the baseline distribution, [PERSON] et al.