Electrical grids are large-sized complex systems that require strong computing power for monitoring and analysis. Kron reduction is a general reduction method in graph theory and is often used for electrical circuit simplification. In this paper, we propose a novel formulation of the weighted Laplacian matrix for directed graphs. The proposed matrix is proved to be strictly equivalent to the conventionally formulated Laplacian matrix and is verified to well model a lossless DC power flow network in directed graphs. We as well present significant properties of the proposed weighted Laplacian and conditions of Kron reduction in directed graphs and in lossless DC power flow networks. The reduction method is verified via simulation models of IEEE-3, IEEE-5, IEEE-9, IEEE-14, and IEEE RTS-96 test systems.

翻译：电网是规模庞大的复杂系统，需要强大的计算能力进行监测与分析。Kron约简是图论中一种通用的约简方法，常用于电路简化。本文提出了一种针对有向图的加权拉普拉斯矩阵新公式。该矩阵被证明在严格意义上等价于传统定义的拉普拉斯矩阵，并验证了其能良好地建模有向图中的无损直流潮流网络。我们还展示了所提出的加权拉普拉斯矩阵的重要性质，以及有向图和无损直流潮流网络中Kron约简的条件。该约简方法通过IEEE-3、IEEE-5、IEEE-9、IEEE-14和IEEE RTS-96测试系统的仿真模型进行了验证。