Scenario generation methods that replicate crossing times in spatially distributed stochastic systems

The purpose of this paper is to bring to light the importance of time series simulation methods which accurately replicate the crossing times of stochastic processes. A crossing time is a contiguous block of time for which a stochastic process is above or below some benchmark such as a forecast. In addition to bringing attention to the issue, we present a family of models, which we call crossing state models (both univariate and multivariate models are introduced), that outperform standard time series modeling techniques in their ability to replicate these crossing times. This is verified using a weighted quadratic empirical distribution function statistic. In addition, in multivariate processes (which may be spatially distributed) we address the problem of replicating crossing times at both the disaggregate and aggregate levels. Proper modeling of crossing times is especially significant in applications in the realm of energy systems. For example, a robust control policy for an energy system with high penetrations of renewables must account for the possibility that energy from wind falls below forecasts for a long period of time. A policy that performs well on a set of renewable power scenarios in which the crossing times are accurately modeled will likely be robust in practice as well. Modeling crossing time behavior is pertinent in other problems involving stochastic optimization as well, such as portfolio management and inventory management problems.