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power system pdf book free 11
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Americans' safety, productivity, comfort, and convenience depend on the reliable supply of electric power. The electric power system is a complex "cyber-physical" system composed of a network of millions of components spread out across the continent. These components are owned, operated, and regulated by thousands of different entities. Power system operators work hard to assure safe and reliable service, but large outages occasionally happen. Given the nature of the system, there is simply no way that outages can be completely avoided, no matter how much time and money is devoted to such an effort. The system's reliability and resilience can be improved but never made perfect. Thus, system owners, operators, and regulators must prioritize their investments based on potential benefits.
Enhancing the Resilience of the Nation's Electricity System focuses on identifying, developing, and implementing strategies to increase the power system's resilience in the face of events that can cause large-area, long-duration outages: blackouts that extend over multiple service areas and last several days or longer. Resilience is not just about lessening the likelihood that these outages will occur. It is also about limiting the scope and impact of outages when they do occur, restoring power rapidly afterwards, and learning from these experiences to better deal with events in the future.
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It was a little more than 20 years ago when two papers gave impetus to the field of network sciences. In their paper1 Watts and Strogatz presented the concept of small-world networks, describing systems that are highly clustered but have small characteristic path lengths, thus showing similarity in certain aspects to lattices and random graphs as well. A year later Barabási and Albert reported2 the discovery of a high degree of self-organization in large complex networks based on the nature of the interaction between vertices (nodes), an attribute to become known as scale-free behaviour. Both papers demonstrated their concepts on real-world networks, among which a common choice was the electrical power grid of the Western United States (modelled as nodes being generators, transformers and substations and the edges being the power lines between them).
Despite these early critiques, a large number of papers were published on the topic in the first decade of the new millennium. Authors have examined the small-world behaviour for various power grids, including the Western US6,7,8, North America9,10, China9,11,12,13, Scandinavia7, Europe14 and The Netherlands15. Different distributions were fitted to the cumulative probability distributions of node degrees, including exponential4,9,13,14,16,17,18, power-law19 and mixed models8,15,20,21, leading to somewhat controversial results.
While it seems that the focus of publications has shifted to new topics, the debate is far from over, as highlighted by the comment of Holme22. Influential papers include the ones published by Clauset, Shalizi and Newman23 and Broido and Clauset24 claiming that technological networks exhibit very weak or no evidence of scale-free structure. This work was recently re-evaluated by Artico et al.25, leading to opposite results, classifying almost two-thirds of the examined networks as scale-free. While being very comprehensive, the latter two studies only marginally touched upon power grids (as of today, the Colorado Index of Complex Networks database includes only 4 power grid topologies of the total 5410 entries), thus leave the questions of small-world and scale-free behaviour open in that field.
The number of papers specialized on this topic is also very low. Buzna et al.26 analysed the 40-year-long evolution of the French 400 kV transmission network, which was characterized by a slow phase, followed by an intensive growth and a saturation. Their most important findings were that a small-world property was only seen in the saturation phase of the process, and that the clustering coefficient of the network has started to decrease after 1996. Buzna et al. did not consider the importance of multiple voltage levels in small-world behaviour, which was addressed by Espejo et al.27 by examining 400 kV and 220 kV networks of 15 countries. While concluding that all selected networks can be considered as small-world, this is only true if voltage levels are considered jointly. Their work did not consider the evolutionary aspect of power grid development either.
Analytical models of power grid evolution were presented by Deka et al.28, showing that the node degree distribution of the generated synthetic networks was a weighted sum of shifted exponentials, similar to what is observed in the case of many European and American power grids.
A small-world model was used to simulate the 50-year-long growth and evolution of a power grid consisting of multiple voltage levels by Mei et al.12. It was shown that the used evolution model did not lead to consistent characteristics of node degree distribution. While certain snapshots showed scale-free behaviour, no generalization could be made. It is also worth noting that exponential distributions were not a good fit for the node degree distribution either.
It is evident that these periods also influenced the small-world coefficient \(\left( \sigma \right)\) of the network. The value of \(\sigma\) becomes bigger than unity for the first time in 1968 and reaches its final range (above 7) in 1990. In the literature, networks with \(\sigma > 1\) are usually considered small-world, which would imply that the Hungarian power grid shows such properties after the introduction of multiple voltage levels into the transmission network. The technical description of such network development (connecting distant points of the network with higher voltage levels to decrease transmission losses) also resembles the method of creating small-world networks. This implication is is similar to the conclusions drawn in Ref.27 as presented in the introduction, emphasizing the importance of multiple voltage levels.
Power-law and exponential fits to the cumulative node degree distribution for 10-year snapshots are shown in Figs. 5 and 6, respectively. Both figures show that parameters of the fitted distribution show little variance from 1979, despite that the size of the network has increased from 220 nodes and 281 edges to 385 nodes and 504 edges. While both fits reach adequate accuracy (R square values for the power-law and the exponential fit are 0.881 and 0.957, respectively), it can be seen that performance of both fits decreases with increasing node number. Considering performance on the long-term data, the exponential fits are better between 1949 and 1969, while from 1979, no significant difference is seen between the two types. Mixed distributions were also fitted for the node degree distributions, but no generalisations could be made based on the results, except that power-laws fits perform poorly for high-degree nodes, while in the case of the exponential fit, such a major issue is not seen. Numerical values of the fits were compared to the ones reported in the literature and are in the same range for exponential fits but show a large difference for power-law fits. Based on these results, the authors conclude that the node distribution of the examined long-term model does not show scale-free behaviour and that the scaling of the network varies in a relatively small range, which may characterise the specific network evolution process examined.
It is also worth comparing the presented results to the only paper discussing a long-term grid evolution12, since the parameters of the grid simulated by Mei et al. shows remarkable similarities to the Hungarian power grid. Selected values after 50 years of development are compared in Table 1. The number of nodes and edges, the installed capacity of power plants, the clustering coefficient and the average node degree are very close to each other. A difference is seen in the distribution of lines with various voltage levels; this is mainly due to the slightly different physical area of the grids and the waiting time before the introduction of new voltage levels. Mei et al. allow 220 and 500 kV lines after 19 and 39 years, respectively; while in the Hungarian grid it actually took 13 and 29 years, respectively. As for the small-world properties of the networks, the simulated grid was considered to show such properties after approximately 20 years, which was the same in the case of the historical data of the Hungarian grid. 2ff7e9595c
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