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Importance sampling the union of rare events with an application to power systems analysis. (English) Zbl 1416.62088
The authors consider importance sampling for estimating the probability \(\mu\) of a union of \(J\) rare events defined by a random variable \(x\), denoted by \(\mu=\mathrm{Prob.} (\bigcup_{j=1}^J H_j)\). The method described as ‘at least one rare event’ (ALOE) consists of choosing a rare event at random and then taking samples from the system given that the random event occurs. For each such sample, the total number of events is recorded. Denote by \(\bar\mu\) the upper bound for \(\sum_{j=1}^J \mathrm{Prob.}(H_j)\geq\mu\). The ALOE estimate \(\hat\mu\) is derived and it is shown that \(\operatorname{variance}(\hat\mu)\leq(J+(1/J)-2)\mu^2/4n\) when the system is sampled \(n\) times and the coefficient of variation \[\operatorname{c.v.}(\hat\mu)\leq \{(J+(1/J)-2)/4n\}^{1/2}.\] The motivation for this work is drawn from power system reliability, when the phase differences between connected nodes have a joint normal distribution and the rare events occur whenever there are large phase differences. The random inputs here are fluctuating demands by users and irregular production as in wind farms. The methodology uses a simpler linear Direct Current (DC) model for an electric grid rather than a complicated Alternating Current (AC) model. Further as stated, normality is assumed to model randomness. Numerical examples are provided to compare ALOE to ‘pmvnorm’ described in R-package ‘mvtnorm’.

MSC:
62D05 Sampling theory, sample surveys
62P30 Applications of statistics in engineering and industry; control charts
Software:
AS 24; MATPOWER; mvtnorm; R
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