## 1 Introduction
Further disconnections of 110-220 kV overhead PTLs triggered a voltage avalanche with the widespread disconnection of consumers [60]. The successive disconnection of 220 kV overhead PTLs until the emergency disconnection of the 220 kV Kedrovo-Ochakovo overhead PTL was not a cascading process. However, it did result in the transition of the EPS to the region of inadmissible power flows. In [61], the non-homogeneity of an electrical network is understood as the constancy of the value of the ratio of active resistance to inductive resistance (\(R_{\mathrm{ij}}/X_{\mathrm{ij}}\)) for any section of the network (i, j,..., \(n\), i \(\
eq\) j). However, this is not enough, since there are other factors that influence the occurrence and development of cascading failure processes. The study [62] examined the non-homogeneity of EPS parameters by assessing the coherence of generator rotation and identifying the groups of generators that had electrical connections with different inductive resistances. To this end, a mathematical tool, including spectral, singular, and cluster analysis, was utilized. Such calculations require large computational resources and, therefore, are not used in the real-time assessment of EPS reliability. In [63], the influence of different types of emergency disturbances in a test network circuit, as well as a German 380 kV transmission network, was investigated. The authors took into account the distributed placement of consumers and power plants across the network. The work calculated the probability of the disconnection of a certain number of consumers and its impact on the development of accidents. The authors drew conclusions about the impact of the magnitude of load disconnection on the speed of emergency processes and also found that when one power transmission line failed, a greater number of power receivers were disconnected in large load clusters than in small clusters. The authors noted that with an increase in the EPS size, when critical connections were disconnected (causing cascading shutdowns), the number of critical connections decreased. The study [64] noted that cascading failures occurred when the load of nodes affected by electrical equipment failures switched part of the load to adjacent nodes. In this case, it was unimportant for what reason the emergency shutdown of the network element occurred. The authors performed modeling in a small-sized test circuit (21 nodes). The article concluded that there was a high probability of node shutdowns due to trigger disturbances at adjacent nodes. In [65], the authors present an overview of various studies on cascading shutdowns, and in the conclusion, they use the term heterogeneity from the position of the interdependence between electrical networks and cyber networks. The article does not provide a definition of the term \"heterogeneity\" or a reference to another source. The work [66] presents a spectral clustering algorithm for developing EPS shutdown decisions, which allows minimizing deviations from planned values of power flows between islands. The developed method allows preventing power supply disruptions by dividing an EPS into stable islands. However, it is not always possible to divide an EPS into balanced islands, especially in the process of developing an emergency cascade process, since individual parts of an EPS, as a rule, contain more power plants, while others contain electricity consumers. In [67], calculations of modes for assessing the impact of the failure of one power transmission line on flows in other lines were performed using the Moore-Penrose pseudoinverse matrix. The authors used the pseudoinverse matrix of admittances to develop an algorithm for identifying the process of cascade failure development. The article noted that the use of the algorithm was superior to the random enumeration of options and allowed obtaining optimal solutions in simple cases. Evidence from analyzing cascade accidentsshowed that they could not be classified as simple cases.