Given the continued integration of intermittent renewable generators in electrical power grids, connection overloads are of increasing concern for grid operators. The risk of an overload due to injection variability can be described mathematically as a barrier crossing probability of a function of a multidimensional stochastic process. Crude Monte Carlo is a wellknown technique to estimate probabilities, but it may be computationally too intensive in this case as typical modern power grids rarely exhibit connection overloads. In this paper we derive an approximate rate function for the overload probability using results from large deviations theory. Based on this large deviations approximation, we apply a rare event simulation technique called splitting to estimate overload probabilities more efficiently than Crude Monte Carlo simulation.We show on example power grids with up to eleven stochastic power injections that for a fixed accuracy Crude Monte Carlo would require tens to millions as many samples than the proposed splitting technique required. We investigate the balance between accuracy and workload of three splitting schemes, each based on a different approximation of the rate function. We justify the workload increase of large deviations based splitting compared to naive splitting -that is, splitting based on merely the Euclidean distance to the rare event set. For a fixed accuracy naive splitting requires over 60 times as much CPU time as large deviation based splitting, illustrating its computational advantage. In these examples naive splitting -unlike large deviations based splitting -requires even more CPU time than CMC simulation, illustrating its pitfall.
As intermittent renewable energy penetrates electrical power grids more and more, assessing grid reliability is of increasing concern for grid operators. Monte Carlo simulation is a robust and popular technique to estimate indices for grid reliability, but the involved computational intensity may be too high for typical reliability analyses. We show that various reliability indices can be expressed as expectations depending on the rare event probability of a so-called power curtailment, and explain how to extend a Crude Monte Carlo grid reliability analysis with an existing rare event splitting technique. The squared relative error of index estimators can be controlled, whereas orders of magnitude less workload is required than when using an equivalent Crude Monte Carlo method. We show further that a bad choice for the time step size or for the importance function may endanger this squared relative error.
A key challenge for an efficient splitting technique is defining the importance function. If the rare event set consists of multiple separated subsets this challenge becomes bigger since the most likely path to the rare event set may be very different from the most likely path to an intermediate level. We propose to mitigate this problem of path deviation by estimating the subset probabilities separately using a modified splitting technique. We compare the proposed separated splitting technique with a standard splitting technique by estimating the probability of entering either of two separated intervals on the real line. The squared relative error of the estimator is shown to be significantly higher when using standard splitting than when using separated splitting. We show that this difference increases if the rare event probability becomes smaller, illustrating the advantage of the separated splitting technique.
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