Nathan Gray | Searcharxiv

arXiv2022

Simulation-Integrated Distributed Optimal Power Flow for Unbalanced Power Distribution Systems

Distributed optimization methods have been extensively applied for the optimization of electric power distribution systems, especially for grid-edge coordination. Existing distributed optimization algorithms applied to power distribution systems require many communication rounds among the distributed agents and may pose convergence challenges in difficult nonlinear settings. The communication network parameters also significantly impact the algorithm's performance. In this paper, we propose a scalable, equivalent network approximation-based, distributed optimization algorithm that employs simulation within optimization using the system's digital twin (DT) to solve the optimal power problems (OPF) for a three-phase unbalanced distribution system. The proposed approach is implemented using a cyber-physical co-simulation platform to validate the robustness of the proposed distributed algorithm under stressed communication. The proposed approach is thoroughly validated using the IEEE 123-bus test system.

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arXiv2017

Duality for metaplectic ice

We interpret values of spherical Whittaker functions on metaplectic covers of the general linear group over a nonarchimedean local field as partition functions of two different solvable lattice models. We prove the equality of these two partition functions by showing the commutativity of transfer matrices associated to different models via the Yang-Baxter equation.

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arXiv2017

Metaplectic Ice for Cartan Type C

We use techniques from statistical mechanics to provide new formulas for Whittaker coefficients of metaplectic Eisenstein series on odd orthogonal groups, matching Friedberg and Zhang. We study a particular variation/generalization of the six-vertex model of Cartan type C having "domain-wall boundary conditions" dependent on a given integer partition $\lambda$ of length at most $r$, where $r$ is a fixed positive integer. More precisely, we examine a planar, non-nested, U-turn model whose partition functions $Z_{\lambda}$ are a generalization of a deformation of characters of the symplectic group $\operatorname{Sp}(2r, \mathbb{C})$. Special cases appeared in: Kuperberg; Hamel and King; Brubaker, Bump, Chinta, and Gunnells; Ivanov. Our main result is that these new families of "metaplectic" models are solvable---i.e., they possess Yang--Baxter equations. We use this to derive two types of functional equations involving $Z_{\lambda}$ corresponding to the two root lengths for simple reflections of the symplectic Weyl group. It is widely believed that the local component of metaplectic Eisenstein series is a metaplectic Whittaker function, though this is subtle owing to the lack of uniqueness of Whittaker models and only verified in type A by McNamara. Thus, we also give evidence for the conjecture that $Z_{\lambda}$ is a spherical Whittaker function by showing that $Z_{\lambda}$ satisfies the same identities under our solution to the Yang--Baxter equation as the metaplectic Whittaker function under intertwining operators on the unramified principal series of an $n$-fold metaplectic cover of $\operatorname{SO}(2r + 1)$, for $n$ odd.

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