Piecewise deterministic Markov processes and maintenance modeling: application to maintenance of a train air-conditioning system

Abstract

This paper deals with the preventive maintenance (PM) optimization of airconditioning systems used aboard regional trains in France by the SNCF (French Railway Company). Two kinds of PM policies are envisioned: one with a single overhaul in the whole lifetime of the air-conditioning system, another with opportunistic replacements of components that are too old at each system failure. The air-conditioning system is formed of about 20 ageing and stochastically independent components. The envisioned PM policies make them functionally dependent, however. Both PM optimizations are performed with respect to the same cost function, involving the mean number of component replacements on some finite horizon. In view of its numerical assessment, a piecewise deterministic Markov processes (PDMP) model is used, both to model the maintained and the unmaintained system; a deterministic numerical scheme is next proposed, based on finite volume (FV) methods for PDMPs; owing to difficulties in its implementation, an approximation of this scheme is next used, which is much easier to implement than the initial FV scheme. As a result of using this method, it was finally possible to optimize both PM policies, which are both proved to lower the cost function of about 7 per cent.

Numerical methods for piecewise deterministic Markov processes with boundary

“…We now state a first convergence result Theorem 2 Under hypotheses (H) and (HD), let T > δt. Let (p (K) n ) i∈E,n∈N,K∈M be the solution of Scheme (13), (14), (15), (16), (17) which satisfies (23)-(24). Then, assuming δt → 0, τ → 0, and h/τ → 0, P D (dx, dt) (resp.…”

“…n ) n∈N,K∈M be the solution of Scheme (13), (14), (15), (16), (17) which satisfies (23)-(24). Then, assuming δt → 0, τ → 0, and h/τ → 0, P D (t)dx (defined by (22)) tends to ρ t , unique solution of (5) (or (41)), for the weak topology of the measures, for almost every t > 0.…”

“…In all application areas, most of interest quantities depend on the distribution of the process at each time, which means that the approximation of the marginal distributions is requested. For this purpose, Monte Carlo methods are widely used (see for instance [8,14,17,18]), but it has been shown in [2,4,6,9,10,11,15] that Finite volume schemes could also provide an efficient approximation of these marginal distributions. These methods consist in solving numerically equations which are fulfilled by the marginal distributions, namely generalized Kolmogorov equations.…”

“…The characterization of the marginal distributions by these equations is studied in [5] for a PDMP without boundary. Some of these schemes deal with PDMP with boundary from a practical point of view ( [9,15]) without a precise study of their properties. The aim of this paper is to fill up this gap and to propose a more efficient numerical scheme, handling as well transient and asymptotic steady situations.…”

Numerical methods for piecewise deterministic Markov processes with boundary

“…The characterization of the marginal distributions by these equations is studied in [5] for a PDMP without boundary. Finite volume schemes are proposed in [2,4,6,[10][11][12]14].…”