Algebraic multigrid for the nonlinear powerflow equations
Algebraic multigrid for the nonlinear powerflow equations
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DOI：
10.1002/nla.2347
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发表时间：
2020-11
影响因子：
4.3
通讯作者：
Barry Lee;Enrique Pereira Batista
中科院分区：
文献类型：
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作者：
Barry Lee;Enrique Pereira Batista
在最近的一篇文章中，一位作者开发了一种用于粗粒度动态电网模型的多重网格技术。该技术的一个关键组成部分是对由电网网络及其加权图给出的图拉普拉斯进行基于松弛的粗化，该图由导纳矩阵表示。在本文中，我们使用这种粗化策略开发了一种多重网格方法，用于求解通过欧姆定律产生的静态非线性方程组，即所谓的潮流方程。这些静态方程与动态模型紧密结合，因为完整的电网模型是一个代数微分系统，其潮流方程描述了代数约束。我们假设动态模型对应于稳定运行的电网，因此，功率流方程与物理稳定的系统相关联。这种稳定性允许功率流方程的粗化基于嵌入功率流系统中的近似图拉普拉斯算子。通过代数构建近似加权图拉普拉斯算子的层次结构，非线性功率流方程的层次结构立即变得显而易见。然后，后一个层次结构可以在完全逼近方案 (FAS) 框架中使用，该框架导致非线性求解器通常具有比牛顿方法更大的吸引盆。考虑到近似拉普拉斯算子的代数多重网格 (AMG) 粗化，求解器是 AMG-FAS 方案。或者，使用为近似图拉普拉斯层次结构生成的粗网格节点和插值运算符，可以导出乘法校正方案。将介绍和分析这两种方案的推导，并给出数值示例来证明这些方案的性能。
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.