Cuts using for modeling the propagation of cascading failures in electrical power grids

Analysis of network reliability is extremely important for their design and operation. For various types of networks, various models have been proposed that take into account network particular features, within which different indicators of network reliability arc considered. As a rule, random graphs in various modifications arc taken as a basis. Usually, the probability of connectivity of the corresponding random graph in the ease of unreliable edges that fail independently and absolutely reliable nodes is used as an indicator of network reliability. The problems of exact calculating of various reliability indicators arc NP-hard. When network elements arc subject to dependent failure, reliability analysis becomes a much more time consuming task. A typical example of dependent network failures is cascading failures in power networks. The initiating event of the failure propagation process is caused by external circumstances: it can be a fallen tree, a strong gust of wind, a line break due to overload, etc. If its failure caused overloading of other lines or equipment, then this, in turn, can generate new outages, etc. Thus, a sequence of dependent failures occurs. An important property of cascade outages in power grids is both their locality and their nonlocality, as practice shows. The examples of real cascading outages show that there is a failure of lines along the sections that cut off certain subnets. Such scenarios for the propagation of cascading failures arc explained by the fact that the failure of a power transmission line leads to an almost instantaneous redistribution of electricity to other power transmission lines, primarily to those lines that arc included in the cut with the failed one. This paper proposes a model for the propagation of dependent failures in a network along its structural cuts. As a structural model of a power grid, we consider an undirected graph G {V, E), where V is the set of vertices, and E is the set of edges of the graph G. Let the presence probability be given for each edge. We will interpret this value as the probability of failure-free operation of the corresponding transmission line within a given time interval. If a failure occurs, then a cascade failure begins along the network cuts, the development of which is described by influence graphs. In this ease, it is assumed that the vertices arc absolutely reliable, i.e. arc present with probability equal to 1. In such conditions, several characteristics arc considered as indicators of power network reliability: the probability of network connectivity, the probability of each consumer can connect to any power center, the probability of the availability of any power source for a given proportion of consumers. The last indicator can be more informative than the previous ones when considering a power grid of large dimension, for example, on a national scale, or several countries, if the corresponding networks arc interconnected. The article proposes an algorithm for the accurate calculation of reliability indices, based on the use of the total probability formula, and an estimation algorithm, based on the Monte Carlo approach. In addition, a method of cumulative updating of the bounds of reliability indicators is proposed, which makes it possible to make a conclusion about the sufficient reliability (or unreliability) of a network in relation to a given threshold. The pseudocodes of the proposed algorithms and the results of numerical experiments are given.