1 1.2. Main concepts 11 N F ωN,F (a) With 2 states N C F ωN,C ωC,F (b) With 3 states Figure 1.8: Markov chains, where N: nominal behavior; C: non-nominal behavior e.g., due to a crack; F: failure; ω is the hazard rate. modelled using a three-state Markov chain, as shown in Figure 1.8(b). Note that the y-axis in Figure 1.7 represents state probability, indicating the probability of being in a state given the bearing’s age. For completeness, the Markov chain in Figure 1.7 operates in continuous-time, which di!ers fromdiscrete-time Markov chains, which operate at specific equally spaced intervals. Further details are provided in Section II.4.1 (page 95). Additionally, in Appendix A.1, this example is further elaborated with mathematical details. Regarding the application of Markov processes for prognostics applications, we encounter: X. Zhang, Xu, Kwan, et al., 2005; Tobon-Mejia, Medjaher, Zerhouni, et al., 2012 apply hidden Markov chains for degradation assessment and RUL estimation in bearings; Ranjith, Setunge, Gravina, et al., 2013 models five degradation condition states (good, condition, minor decay, decay, crushing) in timber bridge elements; D. Zhou, Yu, H. Zhang, et al., 2016 uses Markov chains for degradation assessment in gas turbines; J. Chiachío, Jalón, M. Chiachío, et al., 2020 proposes a Markov chains prognostics framework and uses a case study on fatigue crack propagation to perform stochastic damage modelling, estimating time-dependent reliability; Tanwar, H. Park, and Raghavan, 2021 uses hidden Markov chains to model the degradation in lubricating oil. In this dissertation, we use Markov chains for Multi-State Deterioration (MSD) modelling, where the states in the model explicitly represent a well-defined state associated with deterioration. Our case study focuses on sewer networks, where the inspection data contains damage severities. More on this case study is presented in the next section. Modelling deterioration in sewer mains Sewer mains are vital components of urban infrastructure, necessary for maintaining sanitary conditions and public health (M.A. Cardoso and Silva, 2016). Figure 1.9(a) illustrates the sewer network system of Breda, The Netherlands. This network comprises approximately 25,000 pipes of various materials and functionalities, covering an area of 10 by 12 kilometres, highlighting the system’s large scale. Figure 1.9(b) displays the data type collected via Closed Circuit Television camera inspections, with severities classified according to the EN 13508:1 norm. Here, a higher severity level (k) indicates larger and more pronounced cracks in the sewer main. Degradation modelling in sewer mains is crucial for prioritising them in annual programming and renovation activities; however, developing accurate deterioration
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