This paper presents a method for determining the relevant buses for reduced models of power grid networks described by systems of differential‐algebraic equations and for constructing the coarse‐grain dynamical power grid systems. To determine these buses, time integration of differential equations is not needed, but rather, a stationary system is analyzed. However, unlike stationary‐system approaches that determine only coarse generator buses by approximating the coherency of the generators, the proposed method analyzes the graph Laplacian associated with the admittance matrix. The buses for the reduced model are chosen to ensure that the graph Laplacian of the reduced model is an accurate approximation to the graph Laplacian of the full system. Both load and generator buses can be selected by this procedure since the Laplacian is defined on all the buses. The basis of this proposed approach lies in the close relationship between the synchrony of the system and the spectral properties of this Laplacian, that is, conditions on the spectrum of this Laplacian that almost surely guarantee the synchrony of the system. Thus, assuming that the full system is in synchrony, our strategy is to coarsen the full‐system Laplacian such that the coarse Laplacian possesses good approximation to these spectral conditions. Accurate approximation to these conditions then can better lead to synchronous reduced models. The coarsened Laplacian is defined on coarse degrees of freedom (DOFs), which are associated with the relevant buses to include in the reduced model. To realize this coarse DOF selection, we use multigrid coarsening techniques based on compatible relaxation. Multigrid is the natural choice since it has been extensively used to coarsen Laplacians arising from discretizations of elliptic partial differential equations and is actively being extended to graph Laplacians. With the selection of the buses for the reduced model, the reduced model is completed by constructing the coarse admittance matrix values and other physical parameters using standard power grid techniques or by using the intergrid operators constructed in the coarse DOFs selection process. Unfortunately, the selection of the coarse buses and the coarsening of the admittance matrix and physical parameters are not sufficient by themselves to produce a stable reduced system. To achieve a stable system, system structures of the fine‐grain model must be preserved in the reduced model. We analyze this to develop a multigrid methodology for constructing stable reduced models of power grid systems. Numerical examples are presented to validate this methodology.
In a recent article, one of the authors developed a multigrid technique for coarse‐graining dynamic powergrid models. A key component in this technique is a relaxation‐based coarsening of the graph Laplacian given by the powergrid network and its weighted graph, which is represented by the admittance matrix. In this article, we use this coarsening strategy to develop a multigrid method for solving a static system of nonlinear equations that arises through Ohm's law, the so‐called powerflow equations. These static equations are tightly knitted to the dynamic model in that the full powergrid model is an algebraic‐differential system with the powerflow equations describing the algebraic constraints. We assume that the dynamic model corresponds to a stable operating powergrid, and thus, the powerflow equations are associated with a physically stable system. This stability permits the coarsening of the powerflow equations to be based on an approximate graph Laplacian, which is embedded in the powerflow system. By algebraically constructing a hierarchy of approximate weighted graph Laplacians, a hierarchy of nonlinear powerflow equations immediately becomes apparent. This latter hierarchy can then be used in a full approximation scheme (FAS) framework that leads to a nonlinear solver with generally a larger basin of attraction than Newton's method. Given the algebraic multigrid (AMG) coarsening of the approximate Laplacians, the solver is an AMG‐FAS scheme. Alternatively, using the coarse‐grid nodes and interpolation operators generated for the hierarchy of approximate graph Laplacians, a multiplicative‐correction scheme can be derived. The derivation of both schemes will be presented and analyzed, and numerical examples to demonstrate the performance of these schemes will be given.
more » « less Award ID(s):
 1734727
 NSFPAR ID:
 10453938
 Publisher / Repository:
 Wiley Blackwell (John Wiley & Sons)
 Date Published:
 Journal Name:
 Numerical Linear Algebra with Applications
 Volume:
 28
 Issue:
 2
 ISSN:
 10705325
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
More Like this

Summary 
Vassilevski, Panayot (Ed.)
Abstract In a recent paper, the author examined a correlation affinity measure for selecting the coarse degrees of freedom (CDOFs) or coarse nodes (C nodes) in systems of elliptic partial differential equations (PDEs). This measure was applied to a set of relaxed vectors, which exposed the near‐nullspace components of the PDE operator. Selecting the CDOFs using this affinity measure and constructing the interpolation operators using a least‐squares procedure, an algebraic multigrid (AMG) method was developed. However, there are several noted issues with this AMG solver. First, to capture strong anisotropies, a large number of test vectors may be needed; and second, the solver's performance can be sensitive to the initial set of random test vectors. Both issues reflect the sensitive statistical nature of the measure. In this article, we derive several other statistical measures that ameliorate these issues and lead to better AMG performance. These measures are related to a Markov process, which the PDE itself may model. Specifically, the measures are based on the diffusion distance/effective resistance for such process, and hence, these measures incorporate physics into the CDOF selection. Moreover, because the diffusion distance/effective resistance can be used to analyze graph networks, these measures also provide a very economical scheme for analyzing large‐scale networks. In this article, the derivations of these measures are given, and numerical experiments for analyzing networks and for AMG performance on weighted‐graph Laplacians and systems of elliptic boundary‐value problems are presented.

Algebraic Multigrid (AMG) is an extremely popular linear system solver and/or preconditioner approach for matrices obtained from the discretization of elliptic operators. However, its performance and scalability for large systems obtained from unstructured discretizations seem less consistent than for geometric multigrid (GMG). To a large extent, this is due to loss of sparsity at the coarser grids and the resulting increased cost and poor scalability of the matrixvector multiplication. While there have been attempts to address this concern by designing sparsification algorithms, these affect the overall convergence. In this work, we focus on designing a specialized matrixvector multiplication (matvec) that achieves high performance and scalability for a large variation in the levels of sparsity. We evaluate distributed and shared memory implementations of our matvec operator and demonstrate the improvements to its scalability and performance in AMG hierarchy and finally, we compare it with PETSc.more » « less

Summary This paper proposes improving the solve time of a bootstrap algebraic multigrid (AMG) designed previously by the authors. This is achieved by incorporating the information, a set of
algebraically smooth vectors, generated by the bootstrap algorithm, in a single hierarchy by using sufficiently large aggregates, and these aggregates are compositions of aggregates already built throughout the bootstrap algorithm. The modified AMG method has good convergence properties and shows significant reduction in both memory and solve time. These savings with respect to the original bootstrap AMG are illustrated on some difficult (for standard AMG) linear systems arising from discretization of scalar and vector function elliptic partial differential equations in both 2D and 3D. 
Multigrid is one of the most effective methods for solving elliptic PDEs. It is algorithmically optimal and is robust when combined with Krylov methods. Algebraic multigrid is especially attractive due to its blackbox nature. This however comes at the cost of increased setup costs that can be significant in case of systems where the system matrix changes frequently making it difficult to amortize the setup cost. In this work, we investigate several strategies for performing lazy updates to the multigrid hierarchy corresponding to changes in the system matrix. These include delayed updates, value updates without changing structure, process local changes, and full updates. We demonstrate that in many cases, the overhead of building the AMG hierarchy can be mitigated for rapidly changing system matrices.more » « less