The traditional Power Grid has been designed in a hierarchical fashion, with Energy pushed from the large scale production facilities towards the end users. But with the increasing availability of micro and medium scale generating facilities, the situation is changing. Many end users can now produce energy and share it over the Power Grid. Naturally, end users need to have incentives to do so and might want to be able to act in an open decentralized energy market. In the present work, we offer a novel analysis of the Medium and Low Voltage Power Grids of the North Netherlands using statistical tools from the Complex Network Analysis field. We use a weighted model based on actual Grid data and propose a set of statistical measures to evaluate the adequacy of the current infrastructure for a decentralized energy market. Further, we use the insight gained by the analysis to propose parameters that tie the statistical topological measures to economic factors that might influence the attractiveness to the end users in participating in such a decentralized energy market, thus identifying what are the important topological parameters to work on to facilitate such open decentralized markets.
There are also some brief studies related to the Power Grid that appear as examples in more general discussions about Complex Networks. In particular, Amaral al in @cite_29 show a study of the Southern California and the model following an exponential decay for node degree distribution. Watts and Strogatz in @cite_27 show the Small-world phenomenon applied to the Western States Power Grid while Newman, within a more general work @cite_45 , shows the exponential node degree distribution for the same Grid, while Barabasi @cite_46 model the as a Scale-free network characterized by a power-law node degree distribution.
@cite_46 @cite_27 @cite_29 @cite_45
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Long a matter of folklore, the small-world phenomenon'''' --the principle that we are all linked by short chains of acquaintances --was inaugurated as an area of experimental study in the social sciences through the pioneering work of Stanley Milgram in the 1960''s. This work was among the first to make the phenomenon quantitative, allowing people to speak of the six degrees of separation'''' between any two people in the United States. Since then, a number of network models have been proposed as frameworks in which to study the problem analytically. One of the most refined of these models was formulated in recent work of Watts and Strogatz; their framework provided compelling evidence that the small-world phenomenon is pervasive in a range of networks arising in nature and technology, and a fundamental ingredient in the evolution of the World Wide Web. But existing models are insufficient to explain the striking algorithmic component of Milgram''s original findings: that individuals using local information are collectively very effective at actually constructing short paths between two points in a social network. Although recently proposed network models are rich in short paths, we prove that no decentralized algorithm, operating with local information only, can construct short paths in these networks with non-negligible probability. We then define an infinite family of network models that naturally generalizes the Watts-Strogatz model, and show that for one of these models, there is a decentralized algorithm capable of finding short paths with high probability. More generally, we provide a strong characterization of this family of network models, showing that there is in fact a unique model within the family for which decentralized algorithms are effective. The emergence of order in natural systems is a constant source of inspiration for both physical and biological sciences. While the spatial order characterizing for example the crystals has been the basis of many advances in contemporary physics, most complex systems in nature do not offer such high degree of order. Many of these systems form complex networks whose nodes are the elements of the system and edges represent the interactions between them. Traditionally complex networks have been described by the random graph theory founded in 1959 by Paul Erdohs and Alfred Renyi. One of the defining features of random graphs is that they are statistically homogeneous, and their degree distribution (characterizing the spread in the number of edges starting from a node) is a Poisson distribution. In contrast, recent empirical studies, including the work of our group, indicate that the topology of real networks is much richer than that of random graphs. In particular, the degree distribution of real networks is a power-law, indicating a heterogeneous topology in which the majority of the nodes have a small degree, but there is a significant fraction of highly connected nodes that play an important role in the connectivity of the network. The scale-free topology of real networks has very important consequences on their functioning. For example, we have discovered that scale-free networks are extremely resilient to the random disruption of their nodes. On the other hand, the selective removal of the nodes with highest degree induces a rapid breakdown of the network to isolated subparts that cannot communicate with each other. The non-trivial scaling of the degree distribution of real networks is also an indication of their assembly and evolution. Indeed, our modeling studies have shown us that there are general principles governing the evolution of networks. Most networks start from a small seed and grow by the addition of new nodes which attach to the nodes already in the system. This process obeys preferential attachment: the new nodes are more likely to connect to nodes with already high degree. We have proposed a simple model based on these two principles wich was able to reproduce the power-law degree distribution of real networks. Perhaps even more importantly, this model paved the way to a new paradigm of network modeling, trying to capture the evolution of networks, not just their static topology. In this article, we model electric power delivery networks as graphs, and conduct studies of two power transmission grids, i.e., the Nordic and the western states (U.S.) transmission grid. We calculate values of topological (structural) characteristics of the networks and compare their error and attack tolerance (structural vulnerability), i.e., their performance when vertices are removed, with two frequently used theoretical reference networks (the Erdos-Renyi random graph and the Barabasi-Albert scale-free network). Further, we perform a structural vulnerability analysis of a fictitious electric power network with simple structure. In this analysis, different strategies to decrease the vulnerability of the system are evaluated. Finally, we present a discussion on the practical applicability of graph modeling. We show that nodes of high degree tend to occur infrequently in random graphs but frequently in a wide variety of graphs associated with real world search problems. We then study some alternative models for randomly generating graphs which have been proposed to give more realistic topologies. For example, we show that Watts and Strogatz's small world model has a narrow distribution of node degree. On the other hand, Barabasi and Albert's power law model, gives graphs with both nodes of high degree and a small world topology. These graphs may therefore be useful for benchmarking. We then measure the impact of nodes of high degree and a small world topology on the cost of coloring graphs. The long tail in search costs observed with small world graphs disappears when these graphs are also constructed to contain nodes of high degree. We conjecture that this is a result of the small size of their "backbone", pairs of edges that are frozen to be the same color.