Multigrid for model reduction of power grid networks
Multigrid for model reduction of power grid networks
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DOI：
10.1002/nla.2201
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发表时间：
2018-07
影响因子：
4.3
通讯作者：
Barry Lee
中科院分区：
文献类型：
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作者：
Barry Lee
本文提出了一种用于确定由微分代数方程组描述的电网网络的简化模型的相关节点以及用于构造粗粒度动态电网系统的方法。为了确定这些总线，不需要微分方程的时间积分，而是分析一个静止的系统。然而，与通过近似发电机的相干性仅确定粗略发电机总线的固定系统方法不同，所提出的方法分析与导纳矩阵相关联的图形拉普拉斯算子。选择用于简化模型的总线以确保简化模型的图拉普拉斯算子是完整系统的图拉普拉斯算子的精确近似。由于在所有母线上定义了拉普拉斯算子，因此可以通过该程序选择负荷母线和发电机母线。该方法的基础在于系统的同步性和该拉普拉斯算子的谱性质之间的密切关系，即该拉普拉斯算子的谱上的条件几乎肯定保证系统的同步性。因此，假设整个系统是同步的，我们的策略是粗化整个系统的拉普拉斯算子，使得粗拉普拉斯算子具有对这些谱条件的良好近似。对这些条件的精确近似可以更好地导致同步简化模型。粗拉普拉斯算子定义在粗自由度（DOF）上，这些自由度与要包括在简化模型中的相关总线相关联。为了实现这种粗自由度选择，我们使用基于兼容松弛的多网格粗化技术。多重网格是自然的选择，因为它已被广泛用于粗化所产生的椭圆型偏微分方程的离散拉普拉斯算子，并正在积极扩展到图形拉普拉斯算子。通过为简化模型选择总线，通过使用标准电网技术或通过使用在粗自由度选择过程中构造的网间算子来构造粗导纳矩阵值和其他物理参数，来完成简化模型。不幸的是，粗节点的选择和导纳矩阵和物理参数的粗化本身不足以产生稳定的简化系统。为了实现稳定的系统，细粒度模型的系统结构必须保留在简化模型中。我们分析这一点，开发一个多网格的方法来构建稳定的电网系统的简化模型。数值例子验证了这种方法。
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.