We introduce the Skipping Sampler, a novel algorithm to efficiently sample from the restriction of an arbitrary probability density to an arbitrary measurable set. Such conditional densities can arise in the study of risk and reliability and are often of complex nature, for example having multiple isolated modes and non-convex or disconnected support. The sampler can be seen as an instance of the Metropolis-Hastings algorithm with a particular proposal structure, and we establish sufficient conditions under which the Strong Law of Large Numbers and the Central Limit Theorem hold. We give theoretical and numerical evidence of improved performance relative to the Random Walk Metropolis algorithm.
The Skipping Sampler is an adaptation of the MH algorithm designed to sample from targets which have areas of zero density. It skips' across such areas, much as a flat stone can skip or skim repeatedly across the surface of water. Our original motivation is the study of rare events in stochastic systems, such as that in @cite_36 and @cite_27 . Suppose the system has been designed to satisfy some constraint with high probability, under a random exogenous input vector @math which is known only at the level of its distribution. For example, a power system should remain secure despite the randomness in renewable generation. Although the subset @math where the constraint is violated has a small probability mass @math (hence corresponding to a rare event), nevertheless violations may be extremely costly, so that simulating their occurrence is important to improve system design.
@cite_36 @cite_27
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The frequency stability of power systems is increasingly challenged by various types of disturbance. In particular, the increasing penetration of renewable energy sources is increasing the variability of power generation while reducing system inertia against disturbances. In this paper we explore how this could give rise to rate of change of frequency (RoCoF) violations. Correlated and non -Gaussian power disturbances, such as may arise from renewable generation, have been shown to be significant in power system security analysis. We therefore introduce ghost sampling which, given any unconditional distribution of disturbances, efficiently produces samples conditional on a violation occurring. Our goal is to address questions such as “which generator is most likely to be disconnected due to a RoCoF violation?” or “what is the probability of having simultaneous RoCoF violations, given that a violation occurs?” We model power grids transporting electricity generated by intermittent renewable sources as complex networks, where line failures can emerge indirectly by noisy power input at the nodes. By combining concepts from statistical physics and the physics of power flows and taking weather correlations into account, we rank line failures according to their likelihood and establish the most likely way such failures occur and propagate. Our insights are mathematically rigorous in a small-noise limit and are validated with data from the German transmission grid.