With power grid topology and power flow tracing technology, a method for identifying key nodes of a power grid based on subnet division is proposed. Firstly, generator nodes are divided into subsets according to their neighborhood information and power. Then, with power flow tracking technology, a coefficient distribution matrix of the power grid is obtained. Next, a load node is divided into a generator node subset which offer the maximum power according to the coefficient distribution matrix. A multi-attribute decision-making method is used to sort the nodes in each subnet. The structure coefficient of subnet is further improved and calculated an index for measuring importance of the subnet. According to the importance of subnets, a specific proportion of candidate key nodes are extracted from each subnet. These candidate nodes are reordered with multi-attribute decision-making method to obtain the final ranking of the key nodes. Taking IEEE14, IEEE57 and IEEE118-node systems as examples, subnet division results and ranking results of important nodes of standard networks are obtained. Our method, PageRank method and multi-attribute decision-making method are used to sort key nodes, respectively. Cascade fault performance experiment and network efficiency performance are carried out on the key nodes with top ranking. It shows that the key nodes selected by the proposed algorithm have the greatest impact on propagation performance of the entire network.

To explore Braess paradox phenomenon in an interconnected power grid, a second order Kuramoto-like phase oscillator model is applied to model dynamics of the power grid. The two subnets are connected by the largest degree nodes to build an interconnected power grid. As power transmission between two subnets occurs, new transmission lines are added in the two subnets repectively to explore the probability of Braess paradox phenomenon and analyze the reasons. It is found that as the power transmission between the two subnets reaches a critical value, synchronizability of the power receiving subnet is much better than that of the power supply subnet. The probability of Braess paradox caused by adding a new transmission line in the power supply subnet is much higher than that caused by adding a new transmission line in the power receiving subnet. The phenomena are analyzed in depth by defining subnet order parameter. This study is important for the topology optimization of a interconnected power grid.

Based on complex network theory, a modified admittance model of a power system is constructed. Cascading failures of power grids are studied with topological and electrical characteristics of power grids. Cascading failures are made by removing transmission lines randomly. Effects of the number of nodes, average degree, number of power stations and distribution of power stations on system robustness are studied. Braess paradox phenomenon in the cascading failures of small world power grids is studied. It shows that robustness of a power grid is closely related to its topological structure. As the average degree is great, there exist several bifurcation points in the robustness curve of the nearest-neighbor coupled network and the small world network. In small world structure power grids, generally, the greater the average degree and the number of nodes, the more power stations, the better robustness of the power grid. Robustness of a power grid with distributed power stations is better than that with centralized distribution. In addition, the Braess phenomenon, which leads to the decrease of robustness due to the increase of network capacity, is explained.