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Accelerated Monte-Carlo methods for piecewise deterministic Markov processes for a faster reliability assessment of power generation systems within the PyCATSHOO toolbox

Abstract : This thesis deals with the reliability assessment of nuclear or hydraulic power plants, which are built and exploited by the company EDF (Électricité de France). As the failures of such systems are associated to major human and environmental consequences, for both safety and regulatory reasons, EDF must ensure that the probability of failure of its power generation systems is low enough. The failure of a system occurs when the physical variables characterizing the system (temperature, pression, water level) enter a critical region. Typically, these physical variables can enter a critical region only when a sufficient number of the basic components within the system are damaged. So, in order to assess the probability of having a system failure, we have to jointly model the evolution of the physical variables and of the statuses of the components. To do so we use a model based on piecewise deterministic Markovian processes (PDMPs).This model allows to estimate the probability of failure of the system by simulation. Unfortunately the model is computationally intensive to run, and the classic Monte-carlo method, which needs a lot of simulations to estimate the probability of a rare event, is then too computationally intensive in our context. Methods requiering less simulations are needed, like for instance variance reduction methods.Among variance reduction methods, we distinguish importance sampling methods and splitting methods. The difficulty is that we need to use these methods on PDMPs, which raises a few issues.The theoretical foundations for the importance sampling methods with PDMPs are yet to be defined. Indeed these methods require to weight the simulations with likelihood ratios, and these likelihood ratios have not been properly defined so far for PDMP trajectories, which are degenerate processes. Also efficient biasing strategies (i.e. altered simulation processes yielding a small variance estimator) have not been proposed for PDMPs. This thesis presents how to build the likelihood ratios, it investigates the characteristics of the ideal optimal biasing strategy, and it presents a convenient and efficient way to specify practical biasing strategies for systems of reasonable size.Concerning particular filters methods, they tend to perform poorly on PDMPs with low jump rates and therefore they need to be adapted in order to be successfully applied to reliable power generation systems. Indeed in this context, splitting methods are sometimes less efficient than the naive Monte-Carlo method. This thesis investigates how it is possible to efficiently use these methods with PDMPs. Namely we propose an adaptation of the interacting particles system method (IPS) for PDMPs with low jump rates, and we investigate the convergence properties of the estimators of our methods. The efficiency of the method is tested on a reasonable size system showing a perfomance slightly better than or equivalent to the Monte-Carlo method.An additionnal result on the IPS method is also proposed in a general Markovian framework (beyond PDMPs). The IPS method takes as input certain potential functions that directly impact the variance of the estimator. In this PhD, we show that there are optimal potential functions for which the variance is minimized and we give their closed-form expressions.
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Thomas Galtier. Accelerated Monte-Carlo methods for piecewise deterministic Markov processes for a faster reliability assessment of power generation systems within the PyCATSHOO toolbox. General Mathematics [math.GM]. Université Paris Cité, 2019. English. ⟨NNT : 2019UNIP7176⟩. ⟨tel-03154815⟩

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