Comparison of numerical methods for the assessment of production availability of a hybrid system

doi:10.1016/j.ress.2006.12.001

Reliability Engineering & System Safety

Volume 93, Issue 1, January 2008, Pages 168-177

https://doi.org/10.1016/j.ress.2006.12.001 Get rights and content

Abstract

A finite volume (FV) scheme is proposed in order to compute different probabilistic measures for systems from dynamic reliability field. The FV scheme is tested on a small but realistic benchmark case stemmed from gas industry [Labeau PE, Dutuit Y. Fiabilité dynamique et disponibilité de production: un cas illustratif. Proceedings of $λ μ$ 14, Bourges, France, vol. 2. 2004. p. 431–6 [in French]]. The point is to compute the production availability and the annual frequency of loss of nominal production (among other quantities) for a system of gas production. The results of the FV method are compared to those obtained by Monte Carlo simulation, showing the accuracy of the method.

Introduction

The assessment of production availability, namely the probabilistic measure of the production regularity of a system, is a major concern in reliability studies. A recent benchmark [1] has been proposed, in order to compare different numerical methods allowing for such an evaluation in case of a small but realistic industrial situation. Such a benchmark has already been studied in [2], [3]. The problem is to assess the availability, for a system of gas production, to deliver the nominal production it is meant to do, as well as other quantities, such as the annual mean number of loss of nominal production. The gas production device is what is called a hybrid system, in the sense that its time evolution is governed by two different types of dynamics: a discrete dynamic, which is related to the existence of discrete events such as failures of components and a continuous dynamic, linked to the evolution of a continuous characteristic, here the liquid level in a tank. Such hybrid systems are typical of dynamic reliability or dynamic PSA, see [4] for instance. They may be modelled with stochastic Petri nets (see [5] with references therein) or piecewise deterministic Markov processes (PDMP), see [6] or [2]. Their numerical assessment is frequently done through Monte Carlo (MC) simulations, see [7] or [8] with references therein. It may also be done through cell to cell mapping techniques (CCMT) where the evolution of the system is approximated by a Markov chain with finite state space, see [9], [10] e.g. Recently, it has been shown that finite volume (FV) methods could also be used within the same context [11]. Both methods (CCMT and FV) have already proved their capacity to be competitive with MC simulations from a computing time point of view, at least in case of small systems, see [6], [10], [12]. Though emerging from different scientific communities, FV methods and continuous-time CCMT (as presented in [10]) actually appear as very near: indeed, both methods start from a system of integro-differential equations fulfilled by the marginal distributions of the involved process; dividing the state space into cells, such equations are then discretized both in time and in space; such discretized equations provide what is called a FV scheme in [11] and an approximating Markov chain derived from what is called CCMT in [10]. Note, however, that the integro-differential equation used in [10] is the classical Chapman–Kolmogorov equation (as in [11]) whereas we use here another system of equations.

The aim of this paper is twofold. On one hand, we present an extension of the FV method from [11] which allows for handling such problems as the benchmark. On the other hand, in order to validate the results of the FV method, we provide confidence bands for MC simulation using a regenerative property of the system.

The paper is therefore organized as follows. We first recall the problem to be studied in Section 2 and we present the numerical results provided by regenerative Monte Carlo (RMC) simulation and the FV method. A mathematical formulation of the benchmark problem and of the different probabilistic quantities to assess is then given in Section 3 in term of a PDMP. We next recall in Section 4 the mathematical background which is needed to estimate the accuracy of RMC simulations. The FV scheme, and some of its basic properties, are presented in Section 5. Note that an important mathematical work remains to be done in order to show the convergence of the scheme towards the marginal distribution of the PDMP. Finally, we draw some future research lines in Section 6.

Section snippets

Description of the benchmark

For the sake of completeness, we recall here the benchmark proposed in [1] and studied in [2], [3]. A gas production device is composed of two parallel production units, denoted by $U_{1}$ and $U_{2}$ , which can be up or down. Maximal production rates for the two units, respectively, are $φ_{1, \max} = 3200 m^{3} / h$ and $φ_{2, \max} = 5500 m^{3} / h$ . The device is required to produce gas at the nominal rate $φ_{nom} = 7500 m^{3} / h$ which cannot be produced by a single unit. When both units are up, the production is then dispatched between

A mathematical formulation of the problem

Before developing the numerical methods used to get the previous computational results, we first specify the mathematical model for the benchmark, using notations from dynamic reliability. The time-evolution of the system is described by a PDMP $(I_{t}, X_{t})_{t ⩾ 0}$ (see [13] or [14]). The first part $I_{t}$ is discrete with values in $E = {(1, 1), (1, 0), (0, 1), (0, 0)},$ and describes the state of both units: for $η = (e_{1}, e_{2}) \in E$ and $i = 1, 2$ , $U_{i}$ is working if $e_{i} = 1$ and in repair if $e_{i} = 0$ . The second part $X_{t}$ stands for

RMC simulation

The problem here is to provide confidence bands for asymptotic quantities by MC simulation. A first method might be to compute the requested quantities at some large time T, namely to simulate n independent samples up to time T and then compute the associated confidence bands. We have better use here a second method, based on the regenerative property of the system. Indeed, as already noted in the previous section, $(I_{t}, X_{t})_{t ⩾ 0}$ is a regenerative process with successive entrances in state $((1, 1), R)$

The FV scheme

We finally come to the main part of this paper with the presentation of the FV scheme. Such schemes are not classically used in the framework of probabilistic studies, since they have mainly been developed by engineers, in order to approximate the solutions of balance equations of the type $\frac{\partial A}{\partial t} + \sum_{i = 1}^{d} \frac{\partial F_{i}}{\partial x_{i}} = 0 .$ In this equation, d is the dimension of the space variable $(x_{1}, \dots, x_{d})$ , (frequently, $d = 2$ or 3), A is a given expression of some unknown functions of the space and time variables, and the

Conclusion

This paper shows that the FV method can be successfully used to approximate the marginal distributions of stochastic processes involved in the model of hybrid systems, as an alternative to MC simulations. It leads to accurate results for an admissible computing time. Nevertheless, some mathematical work remains to be performed as the proof of the convergence of the numerical scheme. Also, some computational work must be done in order to make possible the handling of high-dimensional problems.

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Comparison of numerical methods for the assessment of production availability of a hybrid system

Abstract

Introduction

Section snippets

Description of the benchmark

A mathematical formulation of the problem

RMC simulation

The FV scheme

Conclusion

Reliab Eng Syst Safety

Reliab Eng Syst Safety

Reliab Eng Syst Safety