Nuclear Engineering and Design 77 (1984) 49-62 49 North-Holland ... - Ju Li

Nuclear Engineering and Design 77 (1984) 49-62

49

North-Holland, Amsterdam

MONTE CARLO SIMULATION OF MARKOV UNRELIABILITY MODELS

E.E. LEWIS and Franz BOHM * Department of Mechanical and Nuclear Engineering, The Technological Institute, Northwestern University, Evanston, Illinois 60201, USA

Received 9 June 1983

A Monte Carlo method is formulated for the evaluation of the unrealibility of complex systems with known component failure and repair rates. The formulation is in terms of a Markov process allowing dependencies between components to be modeled and computational efficiencies to be achieved in the Monte Carlo simulation. Two variance reduction techniques, forced transition and failure biasing, are employed to increase computational efficiency of the random walk procedure. For an example problem these result in improved computational efficiency by more than three orders of magnitudes over analog Monte Carlo. The method is generalized to treat problems with distributed failure and repair rate data, and a batching technique is introduced and shown to result in substantial increases in computational efficiency for an example problem. A method for separating the variance due to the data uncertainty from that due to the finite number of random walks is presented.

1. Introduction

Fault tree methodologies [1-3] are widely employed in probabilistic risk assessment of nuclear reactors. Reactor shutdown, emergency core cooling and other safety systems require such low failure probabilities that sufficient reliability estimates often cannot be made from operating experience or system test data. Fault trees provide a method for estimating system reliability parameters in terms of more easily obtainable data for component failure and repair rates, For large fault trees computer analysis is needed both to express the logical structure of the tree in terms of minimal cut sets, and for the quantitative evaluation of the system unrealibility or unavailability. In what follows we shall consider Markov Monte Carlo methods for the evaluation of system unreliability.

Early use of Monte Carlo techniques was made for the quantitative evaluation of fault trees [4,5]. While some effort has continued in the use of purely Monte Carlo methods, they have largely been supplanted deterministic techniques often referred to as Kinetic Tree methods [5-9]. Two limitations, however, present themselves in the use of Kinetic Tree methods.

* Current address: Institut fur Kerntechnik und Energiesysteme, Universit~t Stuttgart, Pfaffenwaldring 32, 7000 Stuttgart 80, Fed. Rep. Germany.

First, in Kinetic Tree methods the reliability characteristics of each component are modeled separately. To evaluate the fault tree by combining component failure probabilities, the components are assumed to behave independently of one another. In fact, dependencies often arise from common mode failures, from the increased stress in partially disabled systems, and from a variety of errors in testing, maintenance and repair.

Due to this limitation of the Kinetic Tree formulation there is increasing use of Markov models for reliability analysis [2,3,10-12], for with such models quite general dependencies between components may be treated. For systems with more than a few components, however, Markov analysis by deterministic means becomes a prodigious task. For even while innovative methods have been employed to reduce the complexity of the computations [10-12], the fact remains that one must solve a set of 21 coupled first-order differential equations, where I is the number of components, Thus even a system with only ten components will result in a system of over one thousand coupled equations with a transition matrix with over a million elements. Moreover, if some of the components are repairable, the equations are likely to be quite stiff, requiring that very small time steps be used in the numerical integration.

A second limitation on Kinetic Tree methods is a result of the lack of precision to which the component failure and repair rates are normally known. A means is

0029-5493/84/$03.00 ? Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

50

E.E. Lewis+, F. B6hm / Monte Carlo simulation

required to determine the variance in the result of the fault tree analysis in terms of the variance of the failure and repair rate data from which the component characteristics are calculated. Invariably this is accomplished by Monte Carlo sampling of the failure rate data using log-normal or other distributions [13 18]. The fault tree is evaluated deterministically with data from each data sampling, and the mean, variance and other characteristics of the system are estimated. A similar procedure is also applied to Markov models [10], requiring that the solution of the coupled set of differential equations be repeated thousands of times.

What follows is the formulation of a class of Monte Carlo methods which provides a natural framework for the treatment of both component dependencies and data uncertainties. In section 2 we formulate Monte Carlo simulation of the unreliability of systems with repairable components within the framework of a Markov process. This approach retains the power of deterministic Markov methods in modeling component dependencies that would not be possible if direct Monte Carlo simulation were to be carried out. At the same time the Monte Carlo simulation requires very little computer memory. In section 3, variance reduction techniques, similar to those that have been highly developed for neutral particle transport calculations [19 22], are applied to greatly increase the computational efficiency of Monte Carlo reliability calculations. In section 4 the Monte Carlo formulation is generalized to include probability distributions that represent the uncertainty in the component failure and repair rate data. The variance in the result is then due to two causes: the finite number of random walk simulations, and the uncertainty in the data. A batching technique is introduced and is shown to further reduce that part of the variance due to the finite number of random walks without a commensurate increase in computing effort.

2. Analog Markov Monte Carlo

2.1. Markov formulation

To formulate a Markov process [23,24] for system failure in a form suitable for Monte Carlo simulation of the random walks, we assume a system consisting of I components, each of which may be either operating or failed. There are then 2/ system states arising from all possible combinations of operating and failed components; let 12 represent the set of all possible system states. Certain combinations of component failures correspond to system failure; let/" be the set of all system failure states.

The equations governing system failure are constructed from two probability density functions. Let

Probability density that the system

f ( t l t ' , k') -~ i will make a state transition at t given (1) 1 that it is state k ' at time t ' ( t ' ~ ................
................

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