The reliable functioning of the electric power grid forms a critical part of a modern industrial country. The failure to maintain the reliability of the grid leads to significant societal, national security problems and environmental costs. With the ever-increasing use of intermittent power sources such as wind, solar, battery, along with new types of disruptions that are arising, the overall reliability and stability of the electric grid is threatened. The proposed research will have an impact on understanding the stability of the electric grid given these intermittent power sources. This can lead to significant cost savings based on modeling, in addition to a direct positive impact on the environment, and a more stable civil society as well as economy.
With the ever-increasing presence of intermittent (stochastic) generators (wind, solar, battery, etc.) and loads, it is difficult to integrate these power fluctuations into an electric grid while maintaining reliability and safety. This is a hard problem - in particular it is high dimensional, non-Gaussian, and it is not feasible with current computational resources to obtain sufficient predictive accuracy. The aim of this project is the development of mathematically rigorous numerical methods for computing the statistics of a high dimensional Quantity of Interest (QoI) of the power flow. In particular, the investigators propose: i) To derive stochastic analytic regularity of the power flow with respect to the topology of the electric grid, power generation and load uncertainties to help estimate convergence rates of sparse grid type methods. ii) To develop a multi-level sparse grid reproducing kernel method that is adapted to the stochastic regularity of the electric grid topology. This method will be used to compute the statistics of the stochastic (transient) power flow. iii) If resources permit, as an alternative mitigation strategy for very large dimensional problems, the investigators propose a splitting method of the QoI into non-linear (large deviation) and linear (small deviation) components. The high dimensional small deviations are approximated with a perturbation method with at most quadratic computational cost in dimension. The large deviations are approximated with a multi-level sparse grid reproducing kernel method. This work proposes to significantly improve the state of the art in high dimensional UQ methods that are adapted to topologies such as those of the electric grid. This is a very relevant topic since the the application and theory of UQ methods to electric grids are in their infancy. A National Academy of Sciences report from last year underscores the current importance of this issue.