Reliability analysis of a two-dimensional voting system equipped with protective devices considering triggering failures

•A two-dimensional voting system with multiple protective devices is first built.•Components operate under a new proposed mixed δ-shock model.•New protective effect of devices is investigated, i.e. isolating failed components.•Triggering failures of protective devices and multiple attempts are considered.•FMCIA and UGF technique are combined to obtain a series of reliability indexes. Some engineering systems are supported by protective devices to mitigate system failure risks and extend system lifetime. Nevertheless, existing research on the reliability of systems with protective devices has some limitations regarding the system composition, shock impact mechanisms, protection mechanisms and triggering mechanisms of the protective device. To fulfill these research gaps, this paper proposes a reliability model for a two-dimensional voting system consisting of n subsystems with multi-state protective devices in a shock environment, driven by practical engineering applications. The components in such a system degrade gradually under a novel mixed δ-shock model. Each subsystem is supported by a protective device responsible for isolating failed components in the subsystem to ensure the stable operation of the entire system. The triggering failure of the protective device is taken into account and a maximum number of triggering attempts is preset for the protective device in each working state. Probabilistic indices of system reliability and the protective device's performance are obtained by using the finite Markov chain imbedding approach and universal generating function technique. Ultimately, a case study based on the power distribution system with protection relays is presented to demonstrate the applicability of the proposed model.