In order to explore how to avoid Braess paradox when adding transmission lines in interconnected power grid, this study uses the second order Kuramoto-like model to model the dynamics of the power grid, and connects the subnets through the largest nodes to construct interconnected power grid. When there is power transmission between the two subnets, consider adding transmission lines inside and between subnets to explore the probability of Braess paradox in interconnected power grid and analyze its reasons. The results show that the probability of Braess paradox is related to the transmission power between the subnets. When the power between subnets reaches a critical value, adding transmission lines in the power receiving subnet or between two subnets, the probability of Braess paradox in interconnected power grid will be reduced to nearly zero. Therefore, the Braess paradox phenomenon should be avoided. The reasons for this phenomenon are analyzed in this paper.

The optimal site selection of distributed power stations on a weighted network is studied by combining the topology and electrical characteristics of the network. Firstly, the second-order Kuramoto-like phase oscillator model is used to model the power grid, and the network power flow value under equal coupling strength is used to weight the power grid lines to construct a weighted network model of the power grid; by calculating the transmission efficiency from each load node to each generator node and the power absorbed by each load node from each generator node, which defines the power supply efficiency index of source-load node pair. Then, according to the power supply efficiency value of the source-load nodes of each load node, three grid access methods are defined. The influence of three different grid access methods of distributed power stations on the synchronizability of the grid is studied on the coupling weighted network. Study shows that the access mode of distributed power stations is the best in order to arrange the power supply efficiency value from small to large, followed by the way of randomly selects load nodes, and the way in which the power supply efficiency value is arranged in descending order is the worst. Therefore, the power grid can achieve better synchronizability when the distributed power stations connect to the power grid on the load node with the lower power supply efficiency index of the source-load node pair.

To explore a better power grid structure and reduce the probability of Braess paradox. In this paper, the second order Kuramoto-like phase oscillator model is used to reasonably model the power grid, and several networks are generated by ER random model in which the number of network nodes and links is the same as IEEE14, IEEE30, and IEEE39, respectively. By increasing the number of generator nodes and changing the position of generators in the random network, the occurrence probability of Braess paradox caused by a new adding transmission lines in the different power grid is studied. Study shows that appropriately increasing the number of generator nodes in the power grid can reduce the probability of the Braess paradox in the power grid, and taking large-degree nodes as generator nodes is beneficial to improve the power grid synchronizability and reduce the probability of Braess paradox. This study has certain guiding significance for the topology design and optimization of the power grid.

By dividing the community of the power grid, the information network community is divided according to the current situation and the actual coupling relationship of the corresponding hierarchical partition construction of the power grid and the information network, and the node with the largest betweenness and the node with the largest degree of each community in the information network is selected for full connection. The power information interdependence network model based on the hybrid point-to-point structure of the blockchain, combined with the consensus mechanism of the blockchain, analyzes the advantages of the model, and uses the two attack strategies of high number attack and high betweenness attack to study the proposed interdependence The robustness of the network is compared with the traditional centralized control power information network and the fully decentralized point-to-point power information network. The research results show that the hybrid point-to-point power information interdependence network model can effectively reduce the communication overhead and consensus delay of the practical Byzantine fault-tolerant algorithm on the basis of improving the robustness of the network. In the hybrid peer-to-peer power information network, the system shows stronger vulnerability under the high-number attack mode.

To explore allocation strategy of network coupling strength for improving performance of power grids, we analyze synchronizability of power grids by using critical synchronization coupling strength, and compare robustness of power grids by imposing disturbance power on network nodes. It is found that local topological structure and power of nodes in power grids restrict the local synchronizability of nodes. In general, the greater the power and the smaller the degree value of the node, the weaker its local synchronizability is. And it is more difficult for the node to reach a local synchronization state with its neighbors. Based on this, a non-uniform distribution strategy of network coupling strength is proposed. Keep the total coupling strength the same, we increase the coupling strength between nodes with weak local synchronizability, and reduce the coupling strength between nodes with strong local synchronizability. It shows that this method optimizes synchronizability of the network and enhance robust performance of power grids to a certain extent.

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.