text: The development of algorithms for enhancing the resilience and efficiency of the power grid requires performance evaluation with real topologies of power transmission networks. However, due to security reasons, such topologies and particularly the locations of the substations and the lines are usually not publicly available. Therefore, we study the structural properties of the North American grids and present an algorithm for generating synthetic spatially embedded networks with similar properties to a given grid. The algorithm uses the Gaussian Mixture Model (GMM) for density estimation of the node positions and generates a set of nodes with similar spatial distribution to the nodes in a given network. Then, it uses two procedures, which are inspired by the historical evolution of the grids, to connect the nodes. The algorithm has several tunable parameters that allow generating grids similar to any given grid. Particularly, we apply it to the Western Interconnection (WI) and to grids that operate under the SERC Reliability Corporation (SERC) and the Florida Reliability Coordinating Council (FRCC), and show that it generates grids with similar structural and spatial properties to these grids. To the best of our knowledge, this is the first attempt to consider the spatial distribution of the nodes and lines and its importance in generating synthetic power grids.
  @cite_9: 
  @cite_29: We propose a model to create synthetic networks that may also serve as a narrative of a certain kind of infrastructure network evolution. It consists of an initialization phase with the network extending tree-like for minimum cost and a growth phase with an attachment rule giving a trade-off between cost-optimization and redundancy. Furthermore, we implement the feature of some lines being split during the grid's evolution. We show that the resulting degree distribution has an exponential tail and may show a maximum at degree two, suitable to observations of real-world power grid networks. In particular, the mean degree and the slope of the exponential decay can be controlled in partial independence. To verify to which extent the degree distribution is described by our analytic form, we conduct statistical tests, showing that the hypothesis of an exponential tail is well-accepted for our model data.
  @cite_0: A random network is grown by introducing at unit rate randomly selected nodes on the Euclidean space. A node is randomly connected to its ith predecessor of degree k i with a directed link of length l using a probability proportional to k i l α . Our numerical study indicates that the network is scale free for all values of α>α c and the degree distribution decays stretched exponentially for the other values of a. The link length distribution follows a power law: D(l)∼l δ , where δ is calculated exactly for the whole range of values of a.
  @cite_27: 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.
  @cite_13: We consider a growing network, whose growth algorithm is based on the preferential attachment typical for scale-free constructions, but where the long-range bonds are disadvantaged. Thus, the probability of getting connected to a site at distance d is proportional to @math where is a tunable parameter of the model. We show that the properties of the networks grown with @math are close to those of the genuine scale-free construction, while for @math the structure of the network is quite different. Thus, in this regime, the node degree distribution is no longer a power law, and it is well represented by a stretched exponential. On the other hand, the small-world property of the growing networks is preserved at all values of .
summary: More detailed models that are specifically tailored to the power grid characteristics were proposed in @cite_27 @cite_29 but they did not consider the nodes' and the length distribution of the lines. The spatial distribution of the nodes is correlated with the length of the lines, and as mentioned above, it is important to consider line lengths when designing a method for synthetic power grid generation. While there are several models for generating spatial networks @cite_9 @cite_0 @cite_13 , most of them were not designed to generate networks with properties similar to power grid networks. To the best of our knowledge, this paper is the first to consider the spatial distribution of the nodes in power grids and its importance in generating synthetic networks with similar structural properties.