Multi-Grid Schemes for Multi-Scale Coordination of Energy Systems

Abstract

We discuss how multi-grid computing schemes can be used to design hierarchical coordination architectures for energy systems. These hierarchical architectures can be used to manage multiple temporal and spatial scales and mitigate fundamental limitations of centralized and decentralized architectures. We present the basic elements of a multi-grid scheme, which includes a smoothing operator (a high-resolution decentralized coordination layer that targets phenomena at high frequencies) and a coarsening operator (a low-resolution centralized coordination layer that targets phenomena at low frequencies). For smoothing, we extend existing convergence results for Gauss-Seidel schemes by applying them to systems that cover unstructured domains. This allows us to target problems with multiple timescales and arbitrary networks. The proposed coordination schemes can be used to guide transactions in decentralized electricity markets. We present a storage control example and a power flow diffusion example to illustrate the developments.

Overlapping Schwarz Decomposition for Constrained Quadratic Programs

“…Structured quadratic programs (QPs) arise in a number of applications such as optimal power flow (OPF), optimization with embedded partial differential equations (PDEs), model predictive control, and multistage stochastic programming. A wide range of decomposition schemes have been proposed to tackle such problems; these include Lagrangian dual decomposition [1], the alternating direction method of multipliers (ADMM) [2], and Jacobi/Gauss-Seidel methods [3]. The basic tenet behind such algorithms is to decompose the original problem into subproblems and to coordinate subproblem solutions by using primal-dual information at the boundary of the subdomains.…”

A Parallel Decomposition Scheme for Solving Long-Horizon Optimal Control Problems

“…They also appear in other application domains such as chemical production planning [4] and electricity production planning [5]. Different decomposition techniques have been reported in the literature to improve computational tractability of these problems including dual decomposition [6], alternating direction method of multipliers [7], dual dynamic programming [8], Gauss-Seidel schemes [9], [10], and parallel Newton schemes [11]. Such decomposition techniques allow scalable solutions of longhorizon OCPs by the use of parallel computers.…”

Diffusing-Horizon Model Predictive Control

“…In the proposed approach, we aggregate the variables and constraints (equivalently, aggregate primal and dual variables) to reduce their numbers. This approach is known as algebraic coarsening; algebraic coarsening strategies were introduced to optimization problems in [23], where the authors used the strategy to design a hierarchical architecture to solve large optimization problems. Here we present a generalized version of the algebraic coarsening scheme that incorporates prior information of the primal-dual solution.…”

“…In the context of multigrid, it has been observed that grid coarsening can be seen as a variable aggregation strategy and this observation has been used to derive more general algebraic coarsening schemes [21], [22]. In these schemes, one can aggregate variables and constraints in a more flexible manner by exploiting underlying algebraic structures (e.g., networks) [23], [24]. The move-blocking MPC strategy reported in [25], [26] can be regarded as a special case of algebraic coarsening strategy (this aggregates controls but not states), although the authors did not explicitly mention such a connection.…”