# Scalable Computation of 2D-Minkowski Sum of Arbitrary Non-Convex Domains: Modeling Flexibility in Energy Resources

- Soumya Kundu, Optimization and Control Group, Pacific Northwest National Laboratory, Richland, Washington, United States
- Vikas Chandan, Optimization and Control Group, Pacific Northwest National Laboratory, Richland, Washington, United States
- Karan Kalsi, Optimization and Control Group, Pacific Northwest National Laboratory, Richland, Washington, United States

**Abstract**The flexibility of active ($p$) and reactive power ($q$) consumption in distributed energy resources (DERs) can be represented as a (potentially non-convex) set of points in the $p$-$q$ plane. Modeling of the aggregated flexibility in a heterogeneous ensemble of DERs as a Minkowski sum (M-sum) is computationally intractable even for moderately sized populations. In this article, we propose a scalable method of computing the M-sum of the flexibility domains of a heterogeneous ensemble of DERs, which are allowed to be non-convex, non-compact. In particular, the proposed algorithm computes a guaranteed superset of the true M-sum, with desired accuracy. The worst-case complexity of the algorithm is computed. Special cases are considered, and it is shown that under certain scenarios, it is possible to achieve a complexity that is linear with the size of the ensemble. Numerical examples are provided by computing the aggregated flexibility of different mix of DERs under varying scenarios.