# Optimal uncertainty quantification for renewable energy utilization

**Abstract**

**Main research theme **How to properly evaluate the performance of a stochastic system, when we do not have exact probability distribution of the underlying random variables, but only have statistics (mean, covariance, etc) from observed data? This problem arises in many modern large- scale and/or networked systems with uncertainties, and can be a major obstacle in designing of systems of this kind. Recent advances in mathematics and optimization theory have shown that such analysis is possible if one is interested in the worst case. Our research has pushed existing results from those areas towards more practical and efficient methods that can potentially scale to larger systems.

**Theoretical contribution **Our theory is based on recent results from the area of optimal un- certainty quantification (UQ). Optimal UQ is a powerful tool that performs data-driven worst-case evaluation of system performance. Specifically, it formulates the evaluation problem as an optimiza- tion problem, which searches over the space of probability distributions consistent with observed data. One difficulty in applying optimal UQ directly is that the optimization problem is generally non-convex and can be expensive to solve in a number of cases. To overcome this drawback, our research focused on canonical forms where the corresponding optimization can be reformulated as a convex problem, for which efficient solvers can be used. In particular, we have found that some problems in network analysis can be converted into these canonical forms using Lagrange duality from optimization theory.

**Application area **For demonstration, we have studied the problem of storage placement in power grids with renewables. Introducing storage is considered as one of the major candidate solutions to reduce the effect of uncertainties caused by integration of renewables (wind, solar, etc) in future power grids. One reason that has prevented adding a large amount of storage into the grids, aside from battery technology, is the lack of methods to rigorously evaluate the potential benefit of any storage placement strategy. Our method is suitable for such analysis for two reasons. First, it does not require building a detailed stochastic model of the system, which is intractable for the size of any practical power grids. Second, it is a worst-case analysis, and such conservatism may be preferred for design of systems that prioritizes over stability. Specifically, we performed analysis based on a standard network model (IEEE 14-bus) and statistics from realistic wind power generation time series. Our analysis has shown that time correlation of wind power generation may play an important role in designing such systems. Such factors has not been studied carefully in previous literatures, partly due to the lack of tools.

**Award ID:**** **0931746