Modeling power transmission networks is an important area of research with applications such as vulnerability analysis, study of cascading failures, and location of measurement devices. Graph-theoretic approaches have been widely used to solve these problems, but are subject to several limitations. One of the limitations is the ability to model a heterogeneous system in a consistent manner using the standard graph-theoretic formulation. In this paper, we propose a network-of-networks approach for modeling power transmission networks in order to explicitly incorporate heterogeneity in the model. This model distinguishes between different components of the network that operate at different voltage ratings, and also captures the intra and inter-network connectivity patterns. By building the graph in this fashion we present a novel, and fundamentally different, perspective of power transmission networks. Consequently, this novel approach will have a significant impact on the graph-theoretic modeling of power grids that we believe will lead to a better understanding of transmission networks.
Many researchers have explored the use of random graphs to model power grid @cite_11 @cite_13 @cite_2 @cite_6 @cite_7 @cite_10 , but in each case the resulting random graph does not fully model the structure of interest. In most cases, authors attempt to match graph characteristics from power grid networks, such as degree distribution, average path length, and clustering coefficient, in their random models.
@cite_7 @cite_10 @cite_6 @cite_2 @cite_13 @cite_11
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Considerable attention has been focused on the properties of graphs derived from Internet measurements. Router-level topologies collected via traceroute-like methods have led some to conclude that the router graph of the Internet is well modeled as a power-law random graph. In such a graph, the degree distribution of nodes follows a distribution with a power-law tail. We argue that the evidence to date for this conclusion is at best insufficient We show that when graphs are sampled using traceroute-like methods, the resulting degree distribution can differ sharply from that of the underlying graph. For example, given a sparse Erdos-Renyi random graph, the subgraph formed by a collection of shortest paths from a small set of random sources to a larger set of random destinations can exhibit a degree distribution remarkably like a power-law. We explore the reasons for how this effect arises, and show that in such a setting, edges are sampled in a highly biased manner. This insight allows us to formulate tests for determining when sampling bias is present. When we apply these tests to a number of well-known datasets, we find strong evidence for sampling bias. In order to design an efficient communication scheme and examine the efficiency of any networked control architecture in smart grid applications, we need to characterize statistically its information source, namely the power grid itself. Investigating the statistical properties of power grids has the immediate benefit of providing a natural simulation platform, producing a large number of power grid test cases with realistic topologies, with scalable network size, and with realistic electrical parameter settings. The second benefit is that one can start analyzing the performance of decentralized control algorithms over information networks whose topology matches that of the underlying power network and use network scientific approaches to determine analytically if these architectures would scale well. With these motivations, in this paper we study both the topological and electrical characteristics of power grid networks based on a number of synthetic and real-world power systems. The most interesting discoveries include: the power grid is sparsely connected with obvious small-world properties; its nodal degree distribution can be well fitted by a mixture distribution coming from the sum of a truncated geometric random variable and an irregular discrete random variable; the power grid has very distinctive graph spectral density and its algebraic connectivity scales as a power function of the network size; the line impedance has a heavy-tailed distribution, which can be captured quite accurately by a clipped double Pareto lognormal distribution. Based on the discoveries mentioned above, we propose an algorithm that generates random topology power grids featuring the same topology and electrical characteristics found from the real data. Numerous recent papers have found important relationships between network structure and risks within networks. These results indicate that network structure can dramatically affect the relative effectiveness of risk identification and mitigation methods. With this in mind this paper provides a comparative analysis of the topological and electrical structure of the IEEE 300 bus and the Eastern United States power grids. Specifically we compare the topology of these grids with that of random [1], preferential-attachment [2] and small-world [3] networks of equivalent sizes and find that power grids differ substantially from these abstract models in degree distribution, clustering, diameter and assortativity, and thus conclude that these abstract models do not provide substantial utility for modeling power grids. To better represent the topological properties of power grids we introduce a new graph generating algorithm, the minimum distance graph, that produces networks with properties that more nearly match those of known power grids. While these topological comparisons are useful, they do not account for the physical laws that govern flows in electricity networks. To elucidate the electrical structure of power grids, we propose a new method for representing electrical structure as a weighted graph. This analogous representation is based on electrical distance rather than topological connections. A comparison of these two representations of the test power grids reveals dramatic differences between the electrical and topological structure of electrical power systems. Random graph theory is used to examine the "small-world phenomenon"– any two strangers are connected through a short chain of mutual acquaintances. We will show that for certain families of random graphs with given expected degrees, the average distance is almost surely of order log n logd where d is the weighted average of the sum of squares of the expected degrees. Of particular interest are power law random graphs in which the number of vertices of degree k is proportional to 1 k β for some fixed exponent β. For the case of β > 3, we prove that the average distance of the power law graphs is almost surely of order log n log d. However, many Internet, social, and citation networks are power law graphs with exponents in the range 2 < β < 3 for which the power law random graphs have average distance almost surely of order log log n, but have diameter of order log n (provided having some mild constraints for the average distance and maximum degree). In particular, these graphs contain a dense subgraph...