## 1 Introduction
Petri net models were developed by [4] to assess the resilience of nuclear power plants, and they assess the ability of a system to respond to disruptive events and focus on both the speed and efficiency used to restore the system to its pre-event operating state. The method involves three steps: firstly, the vulnerability of subsystems to external disruptive events is defined using probabilities; secondly, the responses of the reactor system are simulated to reduce the impact of events on the system, and the physical state of the reactor is assessed; lastly, the process of maintenance and restoration of the system is modeled. Reliability studies of photovoltaic systems have been carried out by [5] using Petri nets and considering several failure modes for each repairable component. In addition, a sensitivity study of the various factors has been conducted to highlight the influence of input data on the lifetime and reliability of a photovoltaic module. Given that a key element for a system to achieve its design reliability is adequate logistical support, the author proposes in [6] a methodology for coupling a system-oriented approach to spare management, defining analytically which spare parts combination is required to keep the system at a given operational availability level for a particular scenario, with reliability and maintainability simulation using Monte Carlo and stochastic Petri nets. The author in [7] presents a method for modeling a reactor protection system during maintenance by employing Petri nets to provide a reliability calculation model. The same Petri net model is simulated to study the influence of maintenance status and maintenance compliance distribution probability on the system's availability. The author in [8] suggests an approach to evaluate the reliability and availability of some typical subsea safety systems. This approach works by first identifying the stakeholders' requirements, and then generating various design alternatives. The method Design Failure Mode, Effects, and Criticality Analysis DFMECA was used for dysfunctional analysis. Toestimate the failure rates, the author took the original data from an existing recognized database for subsea applications together with the judgment from designers. To model and estimate reliability and availability, the stochastic Petri Net was chosen. The author in [9] proposes a methodology, based on stochastic and deterministic Petri nets, to evaluate the reliability, availability, and global production time of an industrial renewable energy system. It is primarily based on a qualitative analysis, including both functional and dysfunctional analysis, allowing for the identification of failure modes, this is followed by a quantitative analysis to evaluate the performances. Ref. [10] suggests a hybrid graphical formalism that enhances the capabilities of stochastic Petri nets with those of Bayesian networks. It enables the analysis of continuous input data without the necessity of a time-slicing discretization process. The model is used to assess reliability and dynamic safety. Ref. [11] provides a generalized approach to dynamic and dependent tree theory that is intended to overcome the restrictions of an ordinary fault tree and simplify the modeling of subsystem dependencies avoiding the representation of their individual components by leveraging the features of stochastic Petri net, Markov chains, and binary decision diagrams. The suggested approach comprises seven steps and is essentially founded on the identification of train components covered by the dependency relationship and the extraction of relative sub-trees in the fault tree. The work of [12] is built around the application of a systematic approach to evaluate the reliability of mechatronic systems in the design phase. It commences with a qualitative functional and dysfunctional analysis, then proceeds to a dynamic modeling of the mechatronic system and its various functional and dysfunctional transitions using stochastic Petri nets and different probability distributions. Ref.