The Power Grid as a Complex Network: a Survey
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paper_title: Emergence of scaling in random networks

paper_content:

Systems as diverse as genetic networks or the World Wide Web are best described as networks with complex topology. A common property of many large networks is that the vertex connectivities follow a scale-free power-law distribution. This feature was found to be a consequence of two generic mechanisms: (i) networks expand continuously by the addition of new vertices, and (ii) new vertices attach preferentially to sites that are already well connected. A model based on these two ingredients reproduces the observed stationary scale-free distributions, which indicates that the development of large networks is governed by robust self-organizing phenomena that go beyond the particulars of the individual systems.
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paper_title: Predictability and epidemic pathways in global outbreaks of infectious diseases: the SARS case study

paper_content:

BackgroundThe global spread of the severe acute respiratory syndrome (SARS) epidemic has clearly shown the importance of considering the long-range transportation networks in the understanding of emerging diseases outbreaks. The introduction of extensive transportation data sets is therefore an important step in order to develop epidemic models endowed with realism.MethodsWe develop a general stochastic meta-population model that incorporates actual travel and census data among 3 100 urban areas in 220 countries. The model allows probabilistic predictions on the likelihood of country outbreaks and their magnitude. The level of predictability offered by the model can be quantitatively analyzed and related to the appearance of robust epidemic pathways that represent the most probable routes for the spread of the disease.ResultsIn order to assess the predictive power of the model, the case study of the global spread of SARS is considered. The disease parameter values and initial conditions used in the model are evaluated from empirical data for Hong Kong. The outbreak likelihood for specific countries is evaluated along with the emerging epidemic pathways. Simulation results are in agreement with the empirical data of the SARS worldwide epidemic.ConclusionThe presented computational approach shows that the integration of long-range mobility and demographic data provides epidemic models with a predictive power that can be consistently tested and theoretically motivated. This computational strategy can be therefore considered as a general tool in the analysis and forecast of the global spreading of emerging diseases and in the definition of containment policies aimed at reducing the effects of potentially catastrophic outbreaks.
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paper_title: Analysis of the Nordel power grid disturbance of January 1, 1997 using trajectory sensitivities

paper_content:

This paper uses trajectory sensitivity analysis to investigate a major disturbance of the Nordel power system which occurred on January 1, 1997. The Nordel system is described, and the details of the disturbance are presented. Background to trajectory sensitivity analysis is also provided. Results of the investigation indicate the usefulness of trajectory sensitivities for exploring the influence of various system parameters on the large disturbance behaviour of the system. The trajectory sensitivities provide a way of judging the relative importance of various factors which affected behaviour.
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paper_title: Network Topology of a Potential Energy Landscape: A Static Scale-Free Network

paper_content:

Here we analyze the topology of the network formed by the minima and transition states on the potential energy landscape of small clusters. We find that this network has both a small-world and scale-free character. In contrast to other scale-free networks, where the topology results from the dynamics of the network growth, the potential energy landscape is a static entity. Therefore, a fundamentally different organizing principle underlies this behavior: The potential energy landscape is highly heterogeneous with the low-energy minima having large basins of attraction and acting as the highly connected hubs in the network.
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paper_title: Prediction and predictability of global epidemics: the role of the airline transportation network

paper_content:

The systematic study of large-scale networks has unveiled the ubiquitous presence of connectivity patterns characterized by large scale heterogeneities and unbounded statistical fluctuations. These features affect dramatically the behavior of the diffusion processes occurring on networks, determining the ensuing statistical properties of their evolution pattern and dynamics. In this paper, we investigate the role of the large scale properties of the airline transportation network in determining the global evolution of emerging disease. We present a stochastic computational framework for the forecast of global epidemics that considers the complete world-wide air travel infrastructure complemented with census population data. We address two basic issues in global epidemic modeling: i) We study the role of the large scale properties of the airline transportation network in determining the global diffusion pattern of emerging diseases; ii) We evaluate the reliability of forecasts and outbreak scenarios with respect to the intrinsic stochasticity of disease transmission and traffic flows. In order to address these issues we define a set of novel quantitative measures able to characterize the level of heterogeneity and predictability of the epidemic pattern. These measures may be used for the analysis of containment policies and epidemic risk assessment.
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paper_title: An Experimental Study of the Small World Problem

paper_content:

Arbitrarily selected individuals (N=296) in Nebraska and Boston are asked to generate acquaintance chains to a target person in Massachusetts, employing “the small world method” (Milgram, 1967). Sixty-four chains reach the target person. Within this group the mean number of intermediaries between starters and targets is 5.2. Boston starting chains reach the target person with fewer intermediaries than those starting in Nebraska; subpopulations in the Nebraska group do not differ among themselves. The funneling of chains through sociometric “stars” is noted, with 48 per cent of the chains passing through three persons before reaching the target. Applications of the method to studies of large scale social structure are discussed.
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paper_title: Directed-graph epidemiological models of computer viruses

paper_content:

The strong analogy between biological viruses and their computational counterparts has motivated the authors to adapt the techniques of mathematical epidemiology to the study of computer virus propagation. In order to allow for the most general patterns of program sharing, a standard epidemiological model is extended by placing it on a directed graph and a combination of analysis and simulation is used to study its behavior. The conditions under which epidemics are likely to occur are determined, and, in cases where they do, the dynamics of the expected number of infected individuals are examined as a function of time. It is concluded that an imperfect defense against computer viruses can still be highly effective in preventing their widespread proliferation, provided that the infection rate does not exceed a well-defined critical epidemic threshold. >
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paper_title: On power-law relationships of the Internet topology

paper_content:

Despite the apparent randomness of the Internet, we discover some surprisingly simple power-laws of the Internet topology. These power-laws hold for three snapshots of the Internet, between November 1997 and December 1998, despite a 45% growth of its size during that period. We show that our power-laws fit the real data very well resulting in correlation coefficients of 96% or higher.Our observations provide a novel perspective of the structure of the Internet. The power-laws describe concisely skewed distributions of graph properties such as the node outdegree. In addition, these power-laws can be used to estimate important parameters such as the average neighborhood size, and facilitate the design and the performance analysis of protocols. Furthermore, we can use them to generate and select realistic topologies for simulation purposes.
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paper_title: Error and Attack Tolerance of Complex Networks

paper_content:

Communication/transportation systems are often subjected to failures and attacks. Here we represent such systems as networks and we study their ability to resist failures (attacks) simulated as the breakdown of a group of nodes of the network chosen at random (chosen accordingly to degree or load). We consider and compare the results for two different network topologies: the Erdos–Renyi random graph and the Barabasi–Albert scale-free network. We also discuss briefly a dynamical model recently proposed to take into account the dynamical redistribution of loads after the initial damage of a single node of the network.
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paper_title: Collective dynamics of ‘small-world’ networks

paper_content:

Networks of coupled dynamical systems have been used to model biological oscillators1,2,3,4, Josephson junction arrays5,6, excitable media7, neural networks8,9,10, spatial games11, genetic control networks12 and many other self-organizing systems. Ordinarily, the connection topology is assumed to be either completely regular or completely random. But many biological, technological and social networks lie somewhere between these two extremes. Here we explore simple models of networks that can be tuned through this middle ground: regular networks ‘rewired’ to introduce increasing amounts of disorder. We find that these systems can be highly clustered, like regular lattices, yet have small characteristic path lengths, like random graphs. We call them ‘small-world’ networks, by analogy with the small-world phenomenon13,14 (popularly known as six degrees of separation15). The neural network of the worm Caenorhabditis elegans, the power grid of the western United States, and the collaboration graph of film actors are shown to be small-world networks. Models of dynamical systems with small-world coupling display enhanced signal-propagation speed, computational power, and synchronizability. In particular, infectious diseases spread more easily in small-world networks than in regular lattices.
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paper_title: Modeling blackout dynamics in power transmission networks with simple structure

paper_content:

A model for blackouts in electric power transmission systems is implemented and studied in simple networks with a regular structure. The model describes load demand and network improvements evolving on a slow timescale as well as the fast dynamics of cascading overloads and outages. The model dynamics are demonstrated on the simple power system networks. The dynamics depend weakly on the network topologies tested. The probability distribution functions of measures of the cascading events show the existence of power-dependent tails.
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paper_title: Exploring complex networks

paper_content:

The study of networks pervades all of science, from neurobiology to statistical physics. The most basic issues are structural: how does one characterize the wiring diagram of a food web or the Internet or the metabolic network of the bacterium Escherichia coli? Are there any unifying principles underlying their topology? From the perspective of nonlinear dynamics, we would also like to understand how an enormous network of interacting dynamical systems — be they neurons, power stations or lasers — will behave collectively, given their individual dynamics and coupling architecture. Researchers are only now beginning to unravel the structure and dynamics of complex networks.
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paper_title: Network topology of the interbank market

paper_content:

We provide an empirical analysis of the network structure of the Austrian interbank market based on Austrian Central Bank (OeNB) data. The interbank market is interpreted as a network where banks are nodes and the claims and liabilities between banks define the links. This allows us to apply methods from general network theory. We find that the degree distributions of the interbank network follow power laws. Given this result we discuss how the network structure affects the stability of the banking system with respect to the elimination of a node in the network, i.e. the default of a single bank. Further, the interbank liability network shows a community structure that exactly mirrors the regional and sectoral organization of the current Austrian banking system. The banking network has the typical structural features found in numerous other complex real-world networks: a low clustering coefficient and a short average path length. These empirical findings are in marked contrast to the network structures that have been assumed thus far in the theoretical economic and econo-physics literature.
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paper_title: The large-scale organization of metabolic networks

paper_content:

In a cell or microorganism, the processes that generate mass, energy, information transfer and cell-fate specification are seamlessly integrated through a complex network of cellular constituents and reactions1. However, despite the key role of these networks in sustaining cellular functions, their large-scale structure is essentially unknown. Here we present a systematic comparative mathematical analysis of the metabolic networks of 43 organisms representing all three domains of life. We show that, despite significant variation in their individual constituents and pathways, these metabolic networks have the same topological scaling properties and show striking similarities to the inherent organization of complex non-biological systems2. This may indicate that metabolic organization is not only identical for all living organisms, but also complies with the design principles of robust and error-tolerant scale-free networks2,3,4,5, and may represent a common blueprint for the large-scale organization of interactions among all cellular constituents.
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paper_title: Is the Boston Subway a Small-World Network?

paper_content:

The mathematical study of the small-world concept has fostered quite some interest, showing that small-world features can be identified for some abstract classes of networks. However, passing to real complex systems, as for instance transportation networks, shows a number of new problems that make current analysis impossible. In this paper we show how a more refined kind of analysis, relying on transportation efficiency, can in fact be used to overcome such problems, and to give precious insights on the general characteristics of real transportation networks, eventually providing a picture where the small-world comes back as underlying construction principle.
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paper_title: Global disease spread: statistics and estimation of arrival times

paper_content:

We study metapopulation models for the spread of epidemics in which different subpopulations (cities) are connected by fluxes of individuals (travelers). This framework allows one to describe the spread of a disease on a large scale and we focus here on the computation of the arrival time of a disease as a function of the properties of the seed of the epidemics and of the characteristics of the network connecting the various subpopulations. Using analytical and numerical arguments, we introduce an easily computable quantity which approximates this average arrival time. We show on the example of a disease spread on the world-wide airport network that this quantity predicts with a good accuracy the order of arrival of the disease in the various subpopulations in each realization of epidemic scenario, and not only for an average over realizations. Finally, this quantity might be useful in the identification of the dominant paths of the disease spread.
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paper_title: Stochastic Model for Power Grid Dynamics

paper_content:

We introduce a stochastic model that describes the quasi-static dynamics of an electric transmission network under perturbations introduced by random load fluctuations, random removing of system components from service, random repair times for the failed components, and random response times to implement optimal system corrections for removing line overloads in a damaged or stressed transmission network. We use a linear approximation to the network flow equations and apply linear programming techniques that optimize the dispatching of generators and loads in order to eliminate the network overloads associated with a damaged system. We also provide a simple model for the operator's response to various contingency events that is not always optimal due to either failure of the state estimation system or due to the incorrect subjective assessment of the severity associated with these events. This further allows us to use a game theoretic framework for casting the optimization of the operator's response into the choice of the optimal strategy which minimizes the operating cost. We use a simple strategy space which is the degree of tolerance to line overloads and which is an automatic control (optimization) parameter that can be adjusted to trade off automatic load shed without propagating cascades versus reduced load shed and an increased risk of propagating cascades. The tolerance parameter is chosen to describes a smooth transition from a risk averse to a risk taken strategy. We present numerical results comparing the responses of two power grid systems to optimization approaches with different factors of risk and select the best blackout controlling parameter
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paper_title: Small Worlds: The Dynamics of Networks between Order and Randomness.

paper_content:

Transport brackets are affixed to the container and include provisions for achieving transportability with either a crane or a fork lift. Variable storage capacity is provided by an extension bin which is detachably affixed as part of the container. The top and bottom of the container are compatibly configured to provide an interlock between a plurality of stacked containers and a dolly is configured for use with these interlocking provisions. The container is configured to achieve stacking interchangeability when any vertically peripheral sides are aligned and brackets are disposed around the vertical periphery of the container to achieve fork lift accessibility from any direction perpendicular to the vertically peripheral sides.
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paper_title: An Experimental Study of the Small World Problem

paper_content:

Arbitrarily selected individuals (N=296) in Nebraska and Boston are asked to generate acquaintance chains to a target person in Massachusetts, employing “the small world method” (Milgram, 1967). Sixty-four chains reach the target person. Within this group the mean number of intermediaries between starters and targets is 5.2. Boston starting chains reach the target person with fewer intermediaries than those starting in Nebraska; subpopulations in the Nebraska group do not differ among themselves. The funneling of chains through sociometric “stars” is noted, with 48 per cent of the chains passing through three persons before reaching the target. Applications of the method to studies of large scale social structure are discussed.
---
paper_title: Error and Attack Tolerance of Complex Networks

paper_content:

Communication/transportation systems are often subjected to failures and attacks. Here we represent such systems as networks and we study their ability to resist failures (attacks) simulated as the breakdown of a group of nodes of the network chosen at random (chosen accordingly to degree or load). We consider and compare the results for two different network topologies: the Erdos–Renyi random graph and the Barabasi–Albert scale-free network. We also discuss briefly a dynamical model recently proposed to take into account the dynamical redistribution of loads after the initial damage of a single node of the network.
---
paper_title: Collective dynamics of ‘small-world’ networks

paper_content:

Networks of coupled dynamical systems have been used to model biological oscillators1,2,3,4, Josephson junction arrays5,6, excitable media7, neural networks8,9,10, spatial games11, genetic control networks12 and many other self-organizing systems. Ordinarily, the connection topology is assumed to be either completely regular or completely random. But many biological, technological and social networks lie somewhere between these two extremes. Here we explore simple models of networks that can be tuned through this middle ground: regular networks ‘rewired’ to introduce increasing amounts of disorder. We find that these systems can be highly clustered, like regular lattices, yet have small characteristic path lengths, like random graphs. We call them ‘small-world’ networks, by analogy with the small-world phenomenon13,14 (popularly known as six degrees of separation15). The neural network of the worm Caenorhabditis elegans, the power grid of the western United States, and the collaboration graph of film actors are shown to be small-world networks. Models of dynamical systems with small-world coupling display enhanced signal-propagation speed, computational power, and synchronizability. In particular, infectious diseases spread more easily in small-world networks than in regular lattices.
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paper_title: The very small world of the well-connected

paper_content:

Online networks occupy an increasingly larger position in how we acquire information, how we communicate with one another, and how we disseminate information. Frequently, small sets of vertices dominate various graph and statistical properties of these networks and, because of this, they are relevant for structural analysis and efficient algorithms and engineering. For the web overall, and specifically for social linking in blogs and instant messaging, we provide a principled, rigorous study of the properties, the construction, and the utilization of subsets of special vertices in large online networks. We show that graph synopses defined by the importance of vertices provide small, relatively accurate portraits, independent of the importance measure, of the larger underlying graphs and of the important vertices. Furthermore, they can be computed relatively efficiently.
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paper_title: Topology of technology graphs: small world patterns in electronic circuits.

paper_content:

Recent theoretical studies and extensive data analyses have revealed a common feature displayed by biological, social, and technological networks: the presence of small world patterns. Here we analyze this problem by using several graphs obtained from one of the most common technological systems: electronic circuits. It is shown that both analogic and digital circuits exhibit small world behavior. We conjecture that the small world pattern arises from the compact design in which many elements share a small, close physical neighborhood plus the fact that the system must define a single connected component (which requires shortcuts connecting different integrated clusters). The degree distributions displayed are consistent with a conjecture concerning the sharp cutoffs associated to the presence of costly connections [Amaral et al., Proc. Natl. Acad. Sci. USA 97, 11 149 (2000)], thus providing a limit case for the classes of universality of small world patterns from real, artificial networks. The consequences for circuit design are outlined.
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paper_title: Do topological models provide good information about electricity infrastructure vulnerability

paper_content:

In order to identify the extent to which results from topological graph models are useful for modeling vulnerability in electricity infrastructure, we measure the susceptibility of power networks to random failures and directed attacks using three measures of vulnerability: characteristic path lengths, connectivity loss, and blackout sizes. The first two are purely topological metrics. The blackout size calculation results from a model of cascading failure in power networks. Testing the response of 40 areas within the Eastern U.S. power grid and a standard IEEE test case to a variety of attack/failure vectors indicates that directed attacks result in larger failures using all three vulnerability measures, but the attack-vectors that appear to cause the most damage depend on the measure chosen. While the topological metrics and the power grid model show some similar trends, the vulnerability metrics for individual simulations show only a mild correlation. We conclude that evaluating vulnerability in power ...
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paper_title: A topological analysis of the Italian electric power grid

paper_content:

Large-scale blackouts are an intrinsic drawback of electric power transmission grids. Here we analyze the structural vulnerability of the Italian GRTN power grid by using a model for cascading failures recently proposed in Crucitti et al. (Phys. Rev. E 69 (2004)).
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paper_title: Complex Networks Theory: A New Method of Research in Power Grid

paper_content:

The rapid development of complex network theory provide a new insight into the research of interconnected complex power grid. On the basis of mapping the power grid into a network graph, the topological parameters, such as degree distribution, characteristic path length, clustering coefficient etc, are introduced to describe the network structure characters. And the statistical results of these parameters in power grids of China and America are shown and analyzed. Then, small-world model are explained and simulated to the power grid. In order to measure the performance of the power grid globally and locally, the concept efficiency and its application is also introduced. The mechanism and the model of cascading failure in power grid are successively discussed. Finally, some possible directions of future research are proposed at the end of the paper
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paper_title: Cascade-based attack vulnerability on the US power grid

paper_content:

The vulnerability of real-life networks subject to intentional attacks has been one of the outstanding challenges in the study of the network safety. Applying the real data of the US power grid, we compare the effects of two different attacks for the network robustness against cascading failures, i.e., removal by either the descending or ascending orders of the loads. Adopting the initial load of a node j to be Lj !" kj# Rm2Cj km$% a with kj and Cj being the degree of the node j and the set of its neighboring nodes, respectively, where a is a tunable parameter and governs the strength of the initial load of a node, we investigate the response of the US power grid under two attacks during the cascading propagation. In the case of a < 0:7, our investigation by the numerical simulations leads to a counterintuitive finding on the US power grid that the attack on the nodes with the lowest loads is more harmful than the attack on the ones with the highest loads. In addition, the almost same effect of two attacks in the case ofa ! 0:7 may be useful in furthering studies on the control and defense of cascading failures in the US power grid.
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paper_title: Evaluating North American Electric Grid Reliability Using the Barabasi-Albert Network Model

paper_content:

The reliability of electric transmission systems is examined using a scale-free model of network topology and failure propagation. The topologies of the North American eastern and western electric grids are analyzed to estimate their reliability based on the Barabasi–Albert network model. A commonly used power system reliability index is computed using a simple failure propagation model. The results are compared to the values of power system reliability indices previously obtained using standard power engineering methods, and they suggest that scale-free network models are usable to estimate aggregate electric grid reliability.
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paper_title: Vulnerability assessment of bulk power grid based on complex network theory

paper_content:

The vulnerability assessment algorithm of the bulk power grid based on complex network theory is proposed in this paper. The traditional research model of the power grid based on complex network theory is a graph with no direction and no weight at present. Because this model is far from the real power system, sometimes the wrong results may be gotten. In this paper, the models of components in the power grid are constructed detailedly, furthermore the weighted and directional graph is provided. First, the connecting formings among buses (instead in the traditional methods only substations are considered), lines, transformers and generators are simulated detailedly. Second, the power flow direction in the power grid is considered, and the components' tolerance to disturbances is also considered. Based on the proposed model, the power grid's tolerance to errors and attacks is analyzed. The key components and weak areas are indicated. At last, the robustness and fault dissemination mechanism of the power grid under cascading failures are analyzed based on the proposed method. The North China power grid is used as an example to validate the proposed method.
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paper_title: Critical node identification of smart power system using complex network framework based centrality approach

paper_content:

A method of critical node identification in a power system is presented in this paper. Maximum flow in a network is used to measure centrality of various nodes in the power system. The proposed approach is an improvement of previous methodology in the sense that it does not take into consideration shortest electrical distance as path to flow power. Instead of using normal steady-state condition maximum possible flow is considered. Simulation of various standard test systems are carried out to identify critical nodes.
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paper_title: Suppressing cascades of load in interdependent networks

paper_content:

Understanding how interdependence among systems affects cascading behaviors is increasingly important across many fields of science and engineering. Inspired by cascades of load shedding in coupled electric grids and other infrastructure, we study the Bak–Tang–Wiesenfeld sandpile model on modular random graphs and on graphs based on actual, interdependent power grids. Starting from two isolated networks, adding some connectivity between them is beneficial, for it suppresses the largest cascades in each system. Too much interconnectivity, however, becomes detrimental for two reasons. First, interconnections open pathways for neighboring networks to inflict large cascades. Second, as in real infrastructure, new interconnections increase capacity and total possible load, which fuels even larger cascades. Using a multitype branching process and simulations we show these effects and estimate the optimal level of interconnectivity that balances their trade-offs. Such equilibria could allow, for example, power grid owners to minimize the largest cascades in their grid. We also show that asymmetric capacity among interdependent networks affects the optimal connectivity that each prefers and may lead to an arms race for greater capacity. Our multitype branching process framework provides building blocks for better prediction of cascading processes on modular random graphs and on multitype networks in general.
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paper_title: Small Worlds: The Dynamics of Networks between Order and Randomness.

paper_content:

Transport brackets are affixed to the container and include provisions for achieving transportability with either a crane or a fork lift. Variable storage capacity is provided by an extension bin which is detachably affixed as part of the container. The top and bottom of the container are compatibly configured to provide an interlock between a plurality of stacked containers and a dolly is configured for use with these interlocking provisions. The container is configured to achieve stacking interchangeability when any vertically peripheral sides are aligned and brackets are disposed around the vertical periphery of the container to achieve fork lift accessibility from any direction perpendicular to the vertically peripheral sides.
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paper_title: Modeling Cascading Failures in the North American Power Grid

paper_content:

The North American power grid is one of the most complex technological networks, and its interconnectivity allows both for long-distance power transmission and for the propagation of disturbances. We model the power grid using its actual topology and plausible assumptions about the load and overload of transmission substations. Our results indicate that the loss of a single substation can result in up to \(25\%\) loss of transmission efficiency by triggering an overload cascade in the network. The actual transmission loss depends on the overload tolerance of the network and the connectivity of the failed substation. We systematically study the damage inflicted by the loss of single nodes, and find three universal behaviors, suggesting that \(40\%\) of the transmission substations lead to cascading failures when disrupted. While the loss of a single node can inflict substantial damage, subsequent removals have only incremental effects, in agreement with the topological resilience to less than \(1\%\) node loss.
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paper_title: Robustness of the European power grids under intentional attack

paper_content:

The power grid defines one of the most important technological networks of our times and sustains our complex society. It has evolved for more than a century into an extremely huge and seemingly robust and well understood system. But it becomes extremely fragile as well, when unexpected, usually minimal, failures turn into unknown dynamical behaviours leading, for example, to sudden and massive blackouts. Here we explore the fragility of the European power grid under the effect of selective node removal. A mean field analysis of fragility against attacks is presented together with the observed patterns. Deviations from the theoretical conditions for network percolation (and fragmentation) under attacks are analysed and correlated with non topological reliability measures.
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paper_title: Structural vulnerability of the North American power grid

paper_content:

The magnitude of the August 2003 blackout affecting the United States has put the challenges of energy transmission and distribution into limelight. Despite all the interest and concerted effort, the complexity and interconnectivity of the electric infrastructure precluded us for a long time from understanding why certain events happened. In this paper we study the power grid from a network perspective and determine its ability to transfer power between generators and consumers when certain nodes are disrupted. We find that the power grid is robust to most perturbations, yet disturbances affecting key transmision substations greatly reduce its ability to function. We emphasize that the global properties of the underlying network must be understood as they greatly affect local behavior.
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paper_title: Electrical centrality measures for electric power grid vulnerability analysis

paper_content:

Centrality measures are used in network science to rank the relative importance of nodes and edges of a graph. Here we define new measures of centrality for power grid structure that are based on its functionality. We show that the relative importance analysis based on centrality in graph theory can be generalized to power grid network with its electrical parameters taken into account. In the paper we experiment with the proposed electrical centrality measures on the NYISO-2935 system and the IEEE 300-bus system. We analyze the centrality distribution in order to identify important nodes or branches in the system which are of essential importance in terms of system vulnerability. We also present and discuss a number of interesting discoveries regarding the importance rank of power grid nodes and branches.
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paper_title: Assessing European power grid reliability by means of topological measures

paper_content:

Publicat originalment a: WIT transactions on ecology and the environment, 2009, vol. 121, p. 527-537
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paper_title: Towards Decentralization: A Topological Investigation of the Medium and Low Voltage Grids

paper_content:

The traditional power grid has been designed in a hierarchical fashion, with energy pushed from the large scale production factories towards the end users. 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. Of course, end users need incentives to do so and want 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 influence the attractiveness of participating in such decentralized energy market, thus identifying the important topological parameters to work on to facilitate such open decentralized markets.
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paper_title: The Node Degree Distribution in Power Grid and Its Topology Robustness under Random and Selective Node Removals

paper_content:

In this paper we numerically study the topology robustness of power grids under random and selective node breakdowns, and analytically estimate the critical node-removal thresholds to disintegrate a system, based on the available US power grid data. We also present an analysis on the node degree distribution in power grids because it closely relates with the topology robustness. It is found that the node degree in a power grid can be well fitted by a mixture distribution coming from the sum of a truncated Geometric random variable and an irregular Discrete random variable. With the findings we obtain better estimates of the threshold under selective node breakdowns which predict the numerical thresholds more correctly.
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paper_title: Power Grids as Complex Networks: Topology and Fragility

paper_content:

We almost certainly agree that the harnessing of electricity can be considered the most important technological advance of the twentieth century. The development that allows this control is the power grid, an intricate system composed of generators, substations and transformers connected by cable lines hundreds of kilometers long. Its presence is nowadays so intertwined with ours, and so much taken for granted, that we are only capable of sensing its absence, disguised as a cascading failure or a blackout in its extreme form. Since these extreme phenomena seem to have increased in recent years and new actors, like highly liberalized markets or environmental and social constraints, are taking a leading role, different paths other than traditional engineering ones have been explored. In the last ten years, and mainly due to increasing computational capability and accessibility to data, the conceptual frame of complex networks has allowed different approaches in order to understand the usual (and not-so-usual) outcomes of this system. This paper considers power grids as complex networks. It presents some recent results that correlate their topology with their fragility, together with major malfunctions analysis, and for the European transmission power grid in particular.
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paper_title: A Centrality Measure for Electrical Networks

paper_content:

We derive a measure of "electrical centrality" for AC power networks, which describes the structure of the network as a function of its electrical topology rather than its physical topology. We compare our centrality measure to conventional measures of network structure using the IEEE 300-bus network. We find that when measured electrically, power networks appear to have a scale-free network structure. Thus, unlike previous studies of the structure of power grids, we find that power networks have a number of highly-connected "hub" buses. This result, and the structure of power networks in general, is likely to have important implications for the reliability and security of power networks.
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paper_title: Identifying vulnerable lines in a power network using complex network theory

paper_content:

The latest developments in complex network theory have provided a new direction to power system research. Based on this theory a power system can be modelled as a graph with nodes and vertices and further analysis can help in identifying the important lines. This paper proposes a new betweenness index using the reactance of the transmission lines as the weight and criteria to measure of vulnerability of a power network. The reactance is a simplified measure of power flow in a lossless transmission line based on the power flow analysis equations. The weighted line index is defined as the reactance of the electric path taken to reach from one node to another node. More power is transmitted along the lines with less reactance, to reach from the source node to destination node, which gives the edges with less reactance a higher weight in the analysis. The analyzes have been carried out on the IEEE 39 bus system and IEEE 118 Bus System. The new betweenness index can identify the critical lines of the system, either due to their position in the network or by the power they transmit along the network.
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paper_title: Topological properties of high-voltage electrical transmission networks

paper_content:

The topological properties of high-voltage electrical power transmission networks in several UE countries (the Italian 380 kV, the French 400 kV and the Spanish 400 kV networks) have been studied from available data. An assessment of the vulnerability of the networks has been given by measuring the level of damage introduced by a controlled removal of links. Topological studies could be useful to make vulnerability assessment and to design specific action to reduce topological weaknesses.
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paper_title: The Improvement of the Small-world Network Model and Its Application Research in Bulk Power System

paper_content:

Due to the complexity of cascading failures and blackouts, the small world theory for complex network research is employed in probing the mechanism of them. In the paper, the original small world network model has been modified in two ways. Firstly, the magnitude of the line impedance regarded as the weight of edges is taken into consideration which is not considered in the original model. The formulae of corresponding parameters of the model are also modified. The calculation results of some practical grids validate the correctness and effectiveness of the new model. Secondly, the complex number format of power flow is introduced as the network flow in the small world model. With the power flow and the topology of grids, the two main factors to the system vulnerability, considered, the improved small world model can be used as a tool for vulnerability analysis of grids.
---
paper_title: Topological analysis of the power grid and mitigation strategies against cascading failures

paper_content:

This paper presents a complex systems overview of a power grid network. In recent years, concerns about the robustness of the power grid have grown because of several cascading outages in different parts of the world. In this paper, cascading effect has been simulated on three different networks, the IEEE 300 bus test system, the IEEE 118 bus test system, and the WSCC 179 bus equivalent model. Power Degradation has been discussed as a measure to estimate the damage to the network, in terms of load loss and node loss. A network generator has been developed to generate graphs with characteristics similar to the IEEE standard networks and the generated graphs are then compared with the standard networks to show the effect of topology in determining the robustness of a power grid. Three mitigation strategies, Homogeneous Load Reduction, Targeted Range-Based Load Reduction, and Use of Distributed Renewable Sources in combination with Islanding, have been suggested. The Homogeneous Load Reduction is the simplest to implement but the Targeted Range-Based Load Reduction is the most effective strategy.
---
paper_title: Analysis of structural vulnerabilities in power transmission grids

paper_content:

Abstract Power transmission grids play a crucial role as a critical infrastructure by assuring the proper functioning of power systems. In particular, they secure the loads supplied by power generation plants and help avoid blackouts that can have a disastrous impact on society. The power grid structure (number of nodes and lines, their connections, and their physical properties and operational constraints) is one of the key issues (along with generation availability) to assure power system security; consequently, it deserves special attention. A promising approach for the structural analysis of transmission grids with respect to their vulnerabilities is to use metrics and approaches derived from complex network (CN) theory that are shared with other infrastructures such as the World-Wide Web, telecommunication networks, and oil and gas pipelines. These approaches, based on metrics such as global efficiency, degree and betweenness, are purely topological because they study structural vulnerabilities based on the graphical representation of a network as a set of vertices connected by a set of edges. Unfortunately, these approaches fail to capture the physical properties and operational constraints of power systems and, therefore, cannot provide meaningful analyses. This paper proposes an extended topological approach that includes the definitions of traditional topological metrics (e.g., degrees of nodes and global efficiency) as well as the physical/operational behavior of power grids in terms of real power-flow allocation over lines and line flow limits. This approach provides two new metrics, entropic degree and net-ability, that can be used to assess structural vulnerabilities in power systems. The new metrics are applied to test systems as well as real power grids to demonstrate their performance and to contrast them with traditional, purely topological metrics.
---
paper_title: TOPOLOGICAL VULNERABILITY OF THE EUROPEAN POWER GRID UNDER ERRORS AND ATTACKS

paper_content:

We present an analysis of the topological structure and static tolerance to errors and attacks of the September 2003 actualization of the Union for the Coordination of Transport of Electricity (UCTE) power grid, involving thirty-three different networks. Though every power grid studied has exponential degree distribution and most of them lack typical small-world topology, they display patterns of reaction to node loss similar to those observed in scale-free networks. We have found that the node removal behavior can be logarithmically related to the power grid size. This logarithmic behavior would suggest that, though size favors fragility, growth can reduce it. We conclude that, with the ever-growing demand for power and reliability, actual planning strategies to increase transmission systems would have to take into account this relative increase in vulnerability with size, in order to facilitate and improve the power grid design and functioning.
---
paper_title: The Concept of Betweenness in the Analysis of Power Grid Vulnerability

paper_content:

Vulnerability analysis in power systems is a key issue in modern society and many efforts have contributed to the analysis. Recently, complex network metrics applied to assess the topological vulnerability of networked systems have been used in power systems, such as betweenness metric, since transmission of power systems is in basis of a network structure. However, a pure topological approach fails to capture the specificity of power systems. This paper redefines, starting from the concept of complex networks, an electrical betweenness metric which considers several of specific features of power systems such as power transfer distribution and line flow limits. The electrical betweenness is compared with the conventional betweenness in IEEE-300 bus network according to the un-served energy after network is attacked. The results show that the tested network is more vulnerable when the components of the network are attacked according to their criticalities ranked by electrical betweenness.
---
paper_title: LOCATING CRITICAL LINES IN HIGH-VOLTAGE ELECTRICAL POWER GRIDS

paper_content:

Electrical power grids are among the infrastructures that are attracting a great deal of attention because of their intrinsic criticality. Here we analyze the topological vulnerability and improvability of the spanish 400 kV, the french 400 kV and the italian 380 kV power transmission grids. For each network we detect the most critical lines and suggest how to improve the connectivity.
---
paper_title: Analyzing power network vulnerability with maximum flow based centrality approach

paper_content:

Complex network theory has been studied extensively in solving large scale practical problems and the recent developments have given a new direction to power system research. This theory allows modeling a power system as a network and the latest developments incorporate more and more electrical properties as opposed to the topological analysis in the past. In the past, such networks have been analyzed based on shortest path travel and betweenness index. In a power system, the power might not necessarily flow only through the shortest path so this paper proposes a centrality index based on the maximum power flow through the edges. The links which carry more portion of power from the source (generator) to sink (load) are given a higher weight in this analysis. Few simple cases have been explained and then the algorithm has been demonstrated on the IEEE 39 bus system.
---
paper_title: TOPOLOGY AND CASCADING LINE OUTAGES IN POWER GRIDS

paper_content:

Motivated by the small world network research of Watts & Strogatz, this paper studies relationships between topology and cascading line outages in electric power grids. Cascading line outages are a type of cascading collapse that can occur in power grids when the transmission network is congested. It is characterized by a self-sustaining sequence of line outages followed by grid breakup, which generally leads to widespread blackout. The main findings of this work are twofold: On one hand, the work suggests that topologies with more disorder in their interconnection topology tend to be robust with respect to cascading line outages in the sense of being able to support greater generation and demand levels than more regularly interconnected topologies. On the other hand, the work suggests that topologies with more disorder tend to be more fragile in that should a cascade get started, they tend to break apart after fewer outages than more regularly interconnected topologies. Thus, as has been observed in other complex networks, there appears to be a tradeoff between robustness and fragility. These results were established using synthetically generated power grid topologies and verified using the IEEE 57 bus and 188 bus power grid test cases.
---
paper_title: Vulnerability Assessment of Power Grid Using Graph Topological Indices

paper_content:

This paper presents an assessment of the vulnerability of the power grid to blackout using graph topological indexes. Based on a FERC 715 report and the outage reports, the cascading faults of summer WSCC 1996 are reconstructed, and the graphical property of the grid is compared between two cases: when the blackout triggering lines are removed simulating the actual sequence of cascading outages and when the same number of randomly selected lines are removed. The investigation finds that the critical path lengths of the triggering events of the July and August outages of 1996 WSCC blackout are higher than those of no-outage and arbitrary events. In addition, the small world-ness index for each of the outage triggering events is much smaller than that of normal or any no-outage scenario, indicating that events of shifting a network from small world to a random network would be more likely cascaded to wide area outage.
---
paper_title: Using Graph Models to Analyze the Vulnerability of Electric Power Networks

paper_content:

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.
---
paper_title: Generating Statistically Correct Random Topologies for Testing Smart Grid Communication and Control Networks

paper_content:

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.
---
paper_title: Do topological models provide good information about electricity infrastructure vulnerability

paper_content:

In order to identify the extent to which results from topological graph models are useful for modeling vulnerability in electricity infrastructure, we measure the susceptibility of power networks to random failures and directed attacks using three measures of vulnerability: characteristic path lengths, connectivity loss, and blackout sizes. The first two are purely topological metrics. The blackout size calculation results from a model of cascading failure in power networks. Testing the response of 40 areas within the Eastern U.S. power grid and a standard IEEE test case to a variety of attack/failure vectors indicates that directed attacks result in larger failures using all three vulnerability measures, but the attack-vectors that appear to cause the most damage depend on the measure chosen. While the topological metrics and the power grid model show some similar trends, the vulnerability metrics for individual simulations show only a mild correlation. We conclude that evaluating vulnerability in power ...
---
paper_title: Complex Networks Theory: A New Method of Research in Power Grid

paper_content:

The rapid development of complex network theory provide a new insight into the research of interconnected complex power grid. On the basis of mapping the power grid into a network graph, the topological parameters, such as degree distribution, characteristic path length, clustering coefficient etc, are introduced to describe the network structure characters. And the statistical results of these parameters in power grids of China and America are shown and analyzed. Then, small-world model are explained and simulated to the power grid. In order to measure the performance of the power grid globally and locally, the concept efficiency and its application is also introduced. The mechanism and the model of cascading failure in power grid are successively discussed. Finally, some possible directions of future research are proposed at the end of the paper
---
paper_title: Cascade-based attack vulnerability on the US power grid

paper_content:

The vulnerability of real-life networks subject to intentional attacks has been one of the outstanding challenges in the study of the network safety. Applying the real data of the US power grid, we compare the effects of two different attacks for the network robustness against cascading failures, i.e., removal by either the descending or ascending orders of the loads. Adopting the initial load of a node j to be Lj !" kj# Rm2Cj km$% a with kj and Cj being the degree of the node j and the set of its neighboring nodes, respectively, where a is a tunable parameter and governs the strength of the initial load of a node, we investigate the response of the US power grid under two attacks during the cascading propagation. In the case of a < 0:7, our investigation by the numerical simulations leads to a counterintuitive finding on the US power grid that the attack on the nodes with the lowest loads is more harmful than the attack on the ones with the highest loads. In addition, the almost same effect of two attacks in the case ofa ! 0:7 may be useful in furthering studies on the control and defense of cascading failures in the US power grid.
---
paper_title: Evaluating North American Electric Grid Reliability Using the Barabasi-Albert Network Model

paper_content:

The reliability of electric transmission systems is examined using a scale-free model of network topology and failure propagation. The topologies of the North American eastern and western electric grids are analyzed to estimate their reliability based on the Barabasi–Albert network model. A commonly used power system reliability index is computed using a simple failure propagation model. The results are compared to the values of power system reliability indices previously obtained using standard power engineering methods, and they suggest that scale-free network models are usable to estimate aggregate electric grid reliability.
---
paper_title: Vulnerability assessment of bulk power grid based on complex network theory

paper_content:

The vulnerability assessment algorithm of the bulk power grid based on complex network theory is proposed in this paper. The traditional research model of the power grid based on complex network theory is a graph with no direction and no weight at present. Because this model is far from the real power system, sometimes the wrong results may be gotten. In this paper, the models of components in the power grid are constructed detailedly, furthermore the weighted and directional graph is provided. First, the connecting formings among buses (instead in the traditional methods only substations are considered), lines, transformers and generators are simulated detailedly. Second, the power flow direction in the power grid is considered, and the components' tolerance to disturbances is also considered. Based on the proposed model, the power grid's tolerance to errors and attacks is analyzed. The key components and weak areas are indicated. At last, the robustness and fault dissemination mechanism of the power grid under cascading failures are analyzed based on the proposed method. The North China power grid is used as an example to validate the proposed method.
---
paper_title: Suppressing cascades of load in interdependent networks

paper_content:

Understanding how interdependence among systems affects cascading behaviors is increasingly important across many fields of science and engineering. Inspired by cascades of load shedding in coupled electric grids and other infrastructure, we study the Bak–Tang–Wiesenfeld sandpile model on modular random graphs and on graphs based on actual, interdependent power grids. Starting from two isolated networks, adding some connectivity between them is beneficial, for it suppresses the largest cascades in each system. Too much interconnectivity, however, becomes detrimental for two reasons. First, interconnections open pathways for neighboring networks to inflict large cascades. Second, as in real infrastructure, new interconnections increase capacity and total possible load, which fuels even larger cascades. Using a multitype branching process and simulations we show these effects and estimate the optimal level of interconnectivity that balances their trade-offs. Such equilibria could allow, for example, power grid owners to minimize the largest cascades in their grid. We also show that asymmetric capacity among interdependent networks affects the optimal connectivity that each prefers and may lead to an arms race for greater capacity. Our multitype branching process framework provides building blocks for better prediction of cascading processes on modular random graphs and on multitype networks in general.
---
paper_title: Small Worlds: The Dynamics of Networks between Order and Randomness.

paper_content:

Transport brackets are affixed to the container and include provisions for achieving transportability with either a crane or a fork lift. Variable storage capacity is provided by an extension bin which is detachably affixed as part of the container. The top and bottom of the container are compatibly configured to provide an interlock between a plurality of stacked containers and a dolly is configured for use with these interlocking provisions. The container is configured to achieve stacking interchangeability when any vertically peripheral sides are aligned and brackets are disposed around the vertical periphery of the container to achieve fork lift accessibility from any direction perpendicular to the vertically peripheral sides.
---
paper_title: Modeling Cascading Failures in the North American Power Grid

paper_content:

The North American power grid is one of the most complex technological networks, and its interconnectivity allows both for long-distance power transmission and for the propagation of disturbances. We model the power grid using its actual topology and plausible assumptions about the load and overload of transmission substations. Our results indicate that the loss of a single substation can result in up to \(25\%\) loss of transmission efficiency by triggering an overload cascade in the network. The actual transmission loss depends on the overload tolerance of the network and the connectivity of the failed substation. We systematically study the damage inflicted by the loss of single nodes, and find three universal behaviors, suggesting that \(40\%\) of the transmission substations lead to cascading failures when disrupted. While the loss of a single node can inflict substantial damage, subsequent removals have only incremental effects, in agreement with the topological resilience to less than \(1\%\) node loss.
---
paper_title: Robustness of the European power grids under intentional attack

paper_content:

The power grid defines one of the most important technological networks of our times and sustains our complex society. It has evolved for more than a century into an extremely huge and seemingly robust and well understood system. But it becomes extremely fragile as well, when unexpected, usually minimal, failures turn into unknown dynamical behaviours leading, for example, to sudden and massive blackouts. Here we explore the fragility of the European power grid under the effect of selective node removal. A mean field analysis of fragility against attacks is presented together with the observed patterns. Deviations from the theoretical conditions for network percolation (and fragmentation) under attacks are analysed and correlated with non topological reliability measures.
---
paper_title: Structural vulnerability of the North American power grid

paper_content:

The magnitude of the August 2003 blackout affecting the United States has put the challenges of energy transmission and distribution into limelight. Despite all the interest and concerted effort, the complexity and interconnectivity of the electric infrastructure precluded us for a long time from understanding why certain events happened. In this paper we study the power grid from a network perspective and determine its ability to transfer power between generators and consumers when certain nodes are disrupted. We find that the power grid is robust to most perturbations, yet disturbances affecting key transmision substations greatly reduce its ability to function. We emphasize that the global properties of the underlying network must be understood as they greatly affect local behavior.
---
paper_title: Electrical centrality measures for electric power grid vulnerability analysis

paper_content:

Centrality measures are used in network science to rank the relative importance of nodes and edges of a graph. Here we define new measures of centrality for power grid structure that are based on its functionality. We show that the relative importance analysis based on centrality in graph theory can be generalized to power grid network with its electrical parameters taken into account. In the paper we experiment with the proposed electrical centrality measures on the NYISO-2935 system and the IEEE 300-bus system. We analyze the centrality distribution in order to identify important nodes or branches in the system which are of essential importance in terms of system vulnerability. We also present and discuss a number of interesting discoveries regarding the importance rank of power grid nodes and branches.
---
paper_title: Assessing European power grid reliability by means of topological measures

paper_content:

Publicat originalment a: WIT transactions on ecology and the environment, 2009, vol. 121, p. 527-537
---
paper_title: The Node Degree Distribution in Power Grid and Its Topology Robustness under Random and Selective Node Removals

paper_content:

In this paper we numerically study the topology robustness of power grids under random and selective node breakdowns, and analytically estimate the critical node-removal thresholds to disintegrate a system, based on the available US power grid data. We also present an analysis on the node degree distribution in power grids because it closely relates with the topology robustness. It is found that the node degree in a power grid can be well fitted by a mixture distribution coming from the sum of a truncated Geometric random variable and an irregular Discrete random variable. With the findings we obtain better estimates of the threshold under selective node breakdowns which predict the numerical thresholds more correctly.
---
paper_title: Power Grids as Complex Networks: Topology and Fragility

paper_content:

We almost certainly agree that the harnessing of electricity can be considered the most important technological advance of the twentieth century. The development that allows this control is the power grid, an intricate system composed of generators, substations and transformers connected by cable lines hundreds of kilometers long. Its presence is nowadays so intertwined with ours, and so much taken for granted, that we are only capable of sensing its absence, disguised as a cascading failure or a blackout in its extreme form. Since these extreme phenomena seem to have increased in recent years and new actors, like highly liberalized markets or environmental and social constraints, are taking a leading role, different paths other than traditional engineering ones have been explored. In the last ten years, and mainly due to increasing computational capability and accessibility to data, the conceptual frame of complex networks has allowed different approaches in order to understand the usual (and not-so-usual) outcomes of this system. This paper considers power grids as complex networks. It presents some recent results that correlate their topology with their fragility, together with major malfunctions analysis, and for the European transmission power grid in particular.
---
paper_title: Identifying vulnerable lines in a power network using complex network theory

paper_content:

The latest developments in complex network theory have provided a new direction to power system research. Based on this theory a power system can be modelled as a graph with nodes and vertices and further analysis can help in identifying the important lines. This paper proposes a new betweenness index using the reactance of the transmission lines as the weight and criteria to measure of vulnerability of a power network. The reactance is a simplified measure of power flow in a lossless transmission line based on the power flow analysis equations. The weighted line index is defined as the reactance of the electric path taken to reach from one node to another node. More power is transmitted along the lines with less reactance, to reach from the source node to destination node, which gives the edges with less reactance a higher weight in the analysis. The analyzes have been carried out on the IEEE 39 bus system and IEEE 118 Bus System. The new betweenness index can identify the critical lines of the system, either due to their position in the network or by the power they transmit along the network.
---
paper_title: Topological properties of high-voltage electrical transmission networks

paper_content:

The topological properties of high-voltage electrical power transmission networks in several UE countries (the Italian 380 kV, the French 400 kV and the Spanish 400 kV networks) have been studied from available data. An assessment of the vulnerability of the networks has been given by measuring the level of damage introduced by a controlled removal of links. Topological studies could be useful to make vulnerability assessment and to design specific action to reduce topological weaknesses.
---
paper_title: Collective dynamics of ‘small-world’ networks

paper_content:

Networks of coupled dynamical systems have been used to model biological oscillators1,2,3,4, Josephson junction arrays5,6, excitable media7, neural networks8,9,10, spatial games11, genetic control networks12 and many other self-organizing systems. Ordinarily, the connection topology is assumed to be either completely regular or completely random. But many biological, technological and social networks lie somewhere between these two extremes. Here we explore simple models of networks that can be tuned through this middle ground: regular networks ‘rewired’ to introduce increasing amounts of disorder. We find that these systems can be highly clustered, like regular lattices, yet have small characteristic path lengths, like random graphs. We call them ‘small-world’ networks, by analogy with the small-world phenomenon13,14 (popularly known as six degrees of separation15). The neural network of the worm Caenorhabditis elegans, the power grid of the western United States, and the collaboration graph of film actors are shown to be small-world networks. Models of dynamical systems with small-world coupling display enhanced signal-propagation speed, computational power, and synchronizability. In particular, infectious diseases spread more easily in small-world networks than in regular lattices.
---
paper_title: Topological analysis of the power grid and mitigation strategies against cascading failures

paper_content:

This paper presents a complex systems overview of a power grid network. In recent years, concerns about the robustness of the power grid have grown because of several cascading outages in different parts of the world. In this paper, cascading effect has been simulated on three different networks, the IEEE 300 bus test system, the IEEE 118 bus test system, and the WSCC 179 bus equivalent model. Power Degradation has been discussed as a measure to estimate the damage to the network, in terms of load loss and node loss. A network generator has been developed to generate graphs with characteristics similar to the IEEE standard networks and the generated graphs are then compared with the standard networks to show the effect of topology in determining the robustness of a power grid. Three mitigation strategies, Homogeneous Load Reduction, Targeted Range-Based Load Reduction, and Use of Distributed Renewable Sources in combination with Islanding, have been suggested. The Homogeneous Load Reduction is the simplest to implement but the Targeted Range-Based Load Reduction is the most effective strategy.
---
paper_title: Topological analysis of eastern region of Indian power grid

paper_content:

The modern power grid is tending towards more complexity as power has become an integral part of life even in remote location. Thus the vulnerability of such grids has become more prominent; necessitating some simulation based modeling to extract some important features for such sprawling network. The spatial structure of the power grid has been used for topological analysis of the power grid. The case studies pertaining to eastern region power grid of India validate such approach.
---
paper_title: Analysis of structural vulnerabilities in power transmission grids

paper_content:

Abstract Power transmission grids play a crucial role as a critical infrastructure by assuring the proper functioning of power systems. In particular, they secure the loads supplied by power generation plants and help avoid blackouts that can have a disastrous impact on society. The power grid structure (number of nodes and lines, their connections, and their physical properties and operational constraints) is one of the key issues (along with generation availability) to assure power system security; consequently, it deserves special attention. A promising approach for the structural analysis of transmission grids with respect to their vulnerabilities is to use metrics and approaches derived from complex network (CN) theory that are shared with other infrastructures such as the World-Wide Web, telecommunication networks, and oil and gas pipelines. These approaches, based on metrics such as global efficiency, degree and betweenness, are purely topological because they study structural vulnerabilities based on the graphical representation of a network as a set of vertices connected by a set of edges. Unfortunately, these approaches fail to capture the physical properties and operational constraints of power systems and, therefore, cannot provide meaningful analyses. This paper proposes an extended topological approach that includes the definitions of traditional topological metrics (e.g., degrees of nodes and global efficiency) as well as the physical/operational behavior of power grids in terms of real power-flow allocation over lines and line flow limits. This approach provides two new metrics, entropic degree and net-ability, that can be used to assess structural vulnerabilities in power systems. The new metrics are applied to test systems as well as real power grids to demonstrate their performance and to contrast them with traditional, purely topological metrics.
---
paper_title: TOPOLOGICAL VULNERABILITY OF THE EUROPEAN POWER GRID UNDER ERRORS AND ATTACKS

paper_content:

We present an analysis of the topological structure and static tolerance to errors and attacks of the September 2003 actualization of the Union for the Coordination of Transport of Electricity (UCTE) power grid, involving thirty-three different networks. Though every power grid studied has exponential degree distribution and most of them lack typical small-world topology, they display patterns of reaction to node loss similar to those observed in scale-free networks. We have found that the node removal behavior can be logarithmically related to the power grid size. This logarithmic behavior would suggest that, though size favors fragility, growth can reduce it. We conclude that, with the ever-growing demand for power and reliability, actual planning strategies to increase transmission systems would have to take into account this relative increase in vulnerability with size, in order to facilitate and improve the power grid design and functioning.
---
paper_title: The Concept of Betweenness in the Analysis of Power Grid Vulnerability

paper_content:

Vulnerability analysis in power systems is a key issue in modern society and many efforts have contributed to the analysis. Recently, complex network metrics applied to assess the topological vulnerability of networked systems have been used in power systems, such as betweenness metric, since transmission of power systems is in basis of a network structure. However, a pure topological approach fails to capture the specificity of power systems. This paper redefines, starting from the concept of complex networks, an electrical betweenness metric which considers several of specific features of power systems such as power transfer distribution and line flow limits. The electrical betweenness is compared with the conventional betweenness in IEEE-300 bus network according to the un-served energy after network is attacked. The results show that the tested network is more vulnerable when the components of the network are attacked according to their criticalities ranked by electrical betweenness.
---
paper_title: Analyzing power network vulnerability with maximum flow based centrality approach

paper_content:

Complex network theory has been studied extensively in solving large scale practical problems and the recent developments have given a new direction to power system research. This theory allows modeling a power system as a network and the latest developments incorporate more and more electrical properties as opposed to the topological analysis in the past. In the past, such networks have been analyzed based on shortest path travel and betweenness index. In a power system, the power might not necessarily flow only through the shortest path so this paper proposes a centrality index based on the maximum power flow through the edges. The links which carry more portion of power from the source (generator) to sink (load) are given a higher weight in this analysis. Few simple cases have been explained and then the algorithm has been demonstrated on the IEEE 39 bus system.
---
paper_title: TOPOLOGY AND CASCADING LINE OUTAGES IN POWER GRIDS

paper_content:

Motivated by the small world network research of Watts & Strogatz, this paper studies relationships between topology and cascading line outages in electric power grids. Cascading line outages are a type of cascading collapse that can occur in power grids when the transmission network is congested. It is characterized by a self-sustaining sequence of line outages followed by grid breakup, which generally leads to widespread blackout. The main findings of this work are twofold: On one hand, the work suggests that topologies with more disorder in their interconnection topology tend to be robust with respect to cascading line outages in the sense of being able to support greater generation and demand levels than more regularly interconnected topologies. On the other hand, the work suggests that topologies with more disorder tend to be more fragile in that should a cascade get started, they tend to break apart after fewer outages than more regularly interconnected topologies. Thus, as has been observed in other complex networks, there appears to be a tradeoff between robustness and fragility. These results were established using synthetically generated power grid topologies and verified using the IEEE 57 bus and 188 bus power grid test cases.
---
paper_title: Using Graph Models to Analyze the Vulnerability of Electric Power Networks

paper_content:

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.
---
paper_title: Do topological models provide good information about electricity infrastructure vulnerability

paper_content:

In order to identify the extent to which results from topological graph models are useful for modeling vulnerability in electricity infrastructure, we measure the susceptibility of power networks to random failures and directed attacks using three measures of vulnerability: characteristic path lengths, connectivity loss, and blackout sizes. The first two are purely topological metrics. The blackout size calculation results from a model of cascading failure in power networks. Testing the response of 40 areas within the Eastern U.S. power grid and a standard IEEE test case to a variety of attack/failure vectors indicates that directed attacks result in larger failures using all three vulnerability measures, but the attack-vectors that appear to cause the most damage depend on the measure chosen. While the topological metrics and the power grid model show some similar trends, the vulnerability metrics for individual simulations show only a mild correlation. We conclude that evaluating vulnerability in power ...
---
paper_title: A topological analysis of the Italian electric power grid

paper_content:

Large-scale blackouts are an intrinsic drawback of electric power transmission grids. Here we analyze the structural vulnerability of the Italian GRTN power grid by using a model for cascading failures recently proposed in Crucitti et al. (Phys. Rev. E 69 (2004)).
---
paper_title: Evaluating North American Electric Grid Reliability Using the Barabasi-Albert Network Model

paper_content:

The reliability of electric transmission systems is examined using a scale-free model of network topology and failure propagation. The topologies of the North American eastern and western electric grids are analyzed to estimate their reliability based on the Barabasi–Albert network model. A commonly used power system reliability index is computed using a simple failure propagation model. The results are compared to the values of power system reliability indices previously obtained using standard power engineering methods, and they suggest that scale-free network models are usable to estimate aggregate electric grid reliability.
---
paper_title: Analysis of weighted networks

paper_content:

The connections in many networks are not merely binary entities, either present or not, but have associated weights that record their strengths relative to one another. Recent studies of networks have, by and large, steered clear of such weighted networks, which are often perceived as being harder to analyze than their unweighted counterparts. Here we point out that weighted networks can in many cases be analyzed using a simple mapping from a weighted network to an unweighted multigraph, allowing us to apply standard techniques for unweighted graphs to weighted ones as well. We give a number of examples of the method, including an algorithm for detecting community structure in weighted networks and a simple proof of the maximum-flow--minimum-cut theorem.
---
paper_title: Model for cascading failures in complex networks.

paper_content:

Large but rare cascades triggered by small initial shocks are present in most of the infrastructure networks. Here we present a simple model for cascading failures based on the dynamical redistribution of the flow on the network. We show that the breakdown of a single node is sufficient to collapse the efficiency of the entire system if the node is among the ones with largest load. This is particularly important for real-world networks with a highly hetereogeneous distribution of loads as the Internet and electrical power grids.
---
paper_title: Structural vulnerability of the North American power grid

paper_content:

The magnitude of the August 2003 blackout affecting the United States has put the challenges of energy transmission and distribution into limelight. Despite all the interest and concerted effort, the complexity and interconnectivity of the electric infrastructure precluded us for a long time from understanding why certain events happened. In this paper we study the power grid from a network perspective and determine its ability to transfer power between generators and consumers when certain nodes are disrupted. We find that the power grid is robust to most perturbations, yet disturbances affecting key transmision substations greatly reduce its ability to function. We emphasize that the global properties of the underlying network must be understood as they greatly affect local behavior.
---
paper_title: Electrical centrality measures for electric power grid vulnerability analysis

paper_content:

Centrality measures are used in network science to rank the relative importance of nodes and edges of a graph. Here we define new measures of centrality for power grid structure that are based on its functionality. We show that the relative importance analysis based on centrality in graph theory can be generalized to power grid network with its electrical parameters taken into account. In the paper we experiment with the proposed electrical centrality measures on the NYISO-2935 system and the IEEE 300-bus system. We analyze the centrality distribution in order to identify important nodes or branches in the system which are of essential importance in terms of system vulnerability. We also present and discuss a number of interesting discoveries regarding the importance rank of power grid nodes and branches.
---
paper_title: Impact of Smart Grid on distribution system design

paper_content:

There has been much recent discussion on what distribution systems can and should look like in the future. Terms related to this discussion include smart grid, distribution system of the future, and others. Functionally, a smart grid should be able to provide new abilities such as self-healing, high reliability, energy management, and real-time pricing. From a design perspective, a smart grid will likely incorporate new technologies such as advanced metering, automation, communication, distributed generation, and distributed storage. This paper discussed the potential impact that issues related to smart grid will have on distribution system design.
---
paper_title: Towards Decentralization: A Topological Investigation of the Medium and Low Voltage Grids

paper_content:

The traditional power grid has been designed in a hierarchical fashion, with energy pushed from the large scale production factories towards the end users. 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. Of course, end users need incentives to do so and want 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 influence the attractiveness of participating in such decentralized energy market, thus identifying the important topological parameters to work on to facilitate such open decentralized markets.
---
paper_title: A Centrality Measure for Electrical Networks

paper_content:

We derive a measure of "electrical centrality" for AC power networks, which describes the structure of the network as a function of its electrical topology rather than its physical topology. We compare our centrality measure to conventional measures of network structure using the IEEE 300-bus network. We find that when measured electrically, power networks appear to have a scale-free network structure. Thus, unlike previous studies of the structure of power grids, we find that power networks have a number of highly-connected "hub" buses. This result, and the structure of power networks in general, is likely to have important implications for the reliability and security of power networks.
---
paper_title: Analysis of structural vulnerabilities in power transmission grids

paper_content:

Abstract Power transmission grids play a crucial role as a critical infrastructure by assuring the proper functioning of power systems. In particular, they secure the loads supplied by power generation plants and help avoid blackouts that can have a disastrous impact on society. The power grid structure (number of nodes and lines, their connections, and their physical properties and operational constraints) is one of the key issues (along with generation availability) to assure power system security; consequently, it deserves special attention. A promising approach for the structural analysis of transmission grids with respect to their vulnerabilities is to use metrics and approaches derived from complex network (CN) theory that are shared with other infrastructures such as the World-Wide Web, telecommunication networks, and oil and gas pipelines. These approaches, based on metrics such as global efficiency, degree and betweenness, are purely topological because they study structural vulnerabilities based on the graphical representation of a network as a set of vertices connected by a set of edges. Unfortunately, these approaches fail to capture the physical properties and operational constraints of power systems and, therefore, cannot provide meaningful analyses. This paper proposes an extended topological approach that includes the definitions of traditional topological metrics (e.g., degrees of nodes and global efficiency) as well as the physical/operational behavior of power grids in terms of real power-flow allocation over lines and line flow limits. This approach provides two new metrics, entropic degree and net-ability, that can be used to assess structural vulnerabilities in power systems. The new metrics are applied to test systems as well as real power grids to demonstrate their performance and to contrast them with traditional, purely topological metrics.
---
paper_title: The Concept of Betweenness in the Analysis of Power Grid Vulnerability

paper_content:

Vulnerability analysis in power systems is a key issue in modern society and many efforts have contributed to the analysis. Recently, complex network metrics applied to assess the topological vulnerability of networked systems have been used in power systems, such as betweenness metric, since transmission of power systems is in basis of a network structure. However, a pure topological approach fails to capture the specificity of power systems. This paper redefines, starting from the concept of complex networks, an electrical betweenness metric which considers several of specific features of power systems such as power transfer distribution and line flow limits. The electrical betweenness is compared with the conventional betweenness in IEEE-300 bus network according to the un-served energy after network is attacked. The results show that the tested network is more vulnerable when the components of the network are attacked according to their criticalities ranked by electrical betweenness.
---
paper_title: Analyzing power network vulnerability with maximum flow based centrality approach

paper_content:

Complex network theory has been studied extensively in solving large scale practical problems and the recent developments have given a new direction to power system research. This theory allows modeling a power system as a network and the latest developments incorporate more and more electrical properties as opposed to the topological analysis in the past. In the past, such networks have been analyzed based on shortest path travel and betweenness index. In a power system, the power might not necessarily flow only through the shortest path so this paper proposes a centrality index based on the maximum power flow through the edges. The links which carry more portion of power from the source (generator) to sink (load) are given a higher weight in this analysis. Few simple cases have been explained and then the algorithm has been demonstrated on the IEEE 39 bus system.
---
paper_title: Generating Statistically Correct Random Topologies for Testing Smart Grid Communication and Control Networks

paper_content:

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.
---
paper_title: Do topological models provide good information about electricity infrastructure vulnerability

paper_content:

In order to identify the extent to which results from topological graph models are useful for modeling vulnerability in electricity infrastructure, we measure the susceptibility of power networks to random failures and directed attacks using three measures of vulnerability: characteristic path lengths, connectivity loss, and blackout sizes. The first two are purely topological metrics. The blackout size calculation results from a model of cascading failure in power networks. Testing the response of 40 areas within the Eastern U.S. power grid and a standard IEEE test case to a variety of attack/failure vectors indicates that directed attacks result in larger failures using all three vulnerability measures, but the attack-vectors that appear to cause the most damage depend on the measure chosen. While the topological metrics and the power grid model show some similar trends, the vulnerability metrics for individual simulations show only a mild correlation. We conclude that evaluating vulnerability in power ...
---
paper_title: A topological analysis of the Italian electric power grid

paper_content:

Large-scale blackouts are an intrinsic drawback of electric power transmission grids. Here we analyze the structural vulnerability of the Italian GRTN power grid by using a model for cascading failures recently proposed in Crucitti et al. (Phys. Rev. E 69 (2004)).
---
paper_title: Complex Networks Theory: A New Method of Research in Power Grid

paper_content:

The rapid development of complex network theory provide a new insight into the research of interconnected complex power grid. On the basis of mapping the power grid into a network graph, the topological parameters, such as degree distribution, characteristic path length, clustering coefficient etc, are introduced to describe the network structure characters. And the statistical results of these parameters in power grids of China and America are shown and analyzed. Then, small-world model are explained and simulated to the power grid. In order to measure the performance of the power grid globally and locally, the concept efficiency and its application is also introduced. The mechanism and the model of cascading failure in power grid are successively discussed. Finally, some possible directions of future research are proposed at the end of the paper
---
paper_title: Vulnerability assessment of bulk power grid based on complex network theory

paper_content:

The vulnerability assessment algorithm of the bulk power grid based on complex network theory is proposed in this paper. The traditional research model of the power grid based on complex network theory is a graph with no direction and no weight at present. Because this model is far from the real power system, sometimes the wrong results may be gotten. In this paper, the models of components in the power grid are constructed detailedly, furthermore the weighted and directional graph is provided. First, the connecting formings among buses (instead in the traditional methods only substations are considered), lines, transformers and generators are simulated detailedly. Second, the power flow direction in the power grid is considered, and the components' tolerance to disturbances is also considered. Based on the proposed model, the power grid's tolerance to errors and attacks is analyzed. The key components and weak areas are indicated. At last, the robustness and fault dissemination mechanism of the power grid under cascading failures are analyzed based on the proposed method. The North China power grid is used as an example to validate the proposed method.
---
paper_title: Suppressing cascades of load in interdependent networks

paper_content:

Understanding how interdependence among systems affects cascading behaviors is increasingly important across many fields of science and engineering. Inspired by cascades of load shedding in coupled electric grids and other infrastructure, we study the Bak–Tang–Wiesenfeld sandpile model on modular random graphs and on graphs based on actual, interdependent power grids. Starting from two isolated networks, adding some connectivity between them is beneficial, for it suppresses the largest cascades in each system. Too much interconnectivity, however, becomes detrimental for two reasons. First, interconnections open pathways for neighboring networks to inflict large cascades. Second, as in real infrastructure, new interconnections increase capacity and total possible load, which fuels even larger cascades. Using a multitype branching process and simulations we show these effects and estimate the optimal level of interconnectivity that balances their trade-offs. Such equilibria could allow, for example, power grid owners to minimize the largest cascades in their grid. We also show that asymmetric capacity among interdependent networks affects the optimal connectivity that each prefers and may lead to an arms race for greater capacity. Our multitype branching process framework provides building blocks for better prediction of cascading processes on modular random graphs and on multitype networks in general.
---
paper_title: Small Worlds: The Dynamics of Networks between Order and Randomness.

paper_content:

Transport brackets are affixed to the container and include provisions for achieving transportability with either a crane or a fork lift. Variable storage capacity is provided by an extension bin which is detachably affixed as part of the container. The top and bottom of the container are compatibly configured to provide an interlock between a plurality of stacked containers and a dolly is configured for use with these interlocking provisions. The container is configured to achieve stacking interchangeability when any vertically peripheral sides are aligned and brackets are disposed around the vertical periphery of the container to achieve fork lift accessibility from any direction perpendicular to the vertically peripheral sides.
---
paper_title: Robustness of the European power grids under intentional attack

paper_content:

The power grid defines one of the most important technological networks of our times and sustains our complex society. It has evolved for more than a century into an extremely huge and seemingly robust and well understood system. But it becomes extremely fragile as well, when unexpected, usually minimal, failures turn into unknown dynamical behaviours leading, for example, to sudden and massive blackouts. Here we explore the fragility of the European power grid under the effect of selective node removal. A mean field analysis of fragility against attacks is presented together with the observed patterns. Deviations from the theoretical conditions for network percolation (and fragmentation) under attacks are analysed and correlated with non topological reliability measures.
---
paper_title: Structural vulnerability of the North American power grid

paper_content:

The magnitude of the August 2003 blackout affecting the United States has put the challenges of energy transmission and distribution into limelight. Despite all the interest and concerted effort, the complexity and interconnectivity of the electric infrastructure precluded us for a long time from understanding why certain events happened. In this paper we study the power grid from a network perspective and determine its ability to transfer power between generators and consumers when certain nodes are disrupted. We find that the power grid is robust to most perturbations, yet disturbances affecting key transmision substations greatly reduce its ability to function. We emphasize that the global properties of the underlying network must be understood as they greatly affect local behavior.
---
paper_title: Electrical centrality measures for electric power grid vulnerability analysis

paper_content:

Centrality measures are used in network science to rank the relative importance of nodes and edges of a graph. Here we define new measures of centrality for power grid structure that are based on its functionality. We show that the relative importance analysis based on centrality in graph theory can be generalized to power grid network with its electrical parameters taken into account. In the paper we experiment with the proposed electrical centrality measures on the NYISO-2935 system and the IEEE 300-bus system. We analyze the centrality distribution in order to identify important nodes or branches in the system which are of essential importance in terms of system vulnerability. We also present and discuss a number of interesting discoveries regarding the importance rank of power grid nodes and branches.
---
paper_title: Towards Decentralization: A Topological Investigation of the Medium and Low Voltage Grids

paper_content:

The traditional power grid has been designed in a hierarchical fashion, with energy pushed from the large scale production factories towards the end users. 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. Of course, end users need incentives to do so and want 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 influence the attractiveness of participating in such decentralized energy market, thus identifying the important topological parameters to work on to facilitate such open decentralized markets.
---
paper_title: Power Grids as Complex Networks: Topology and Fragility

paper_content:

We almost certainly agree that the harnessing of electricity can be considered the most important technological advance of the twentieth century. The development that allows this control is the power grid, an intricate system composed of generators, substations and transformers connected by cable lines hundreds of kilometers long. Its presence is nowadays so intertwined with ours, and so much taken for granted, that we are only capable of sensing its absence, disguised as a cascading failure or a blackout in its extreme form. Since these extreme phenomena seem to have increased in recent years and new actors, like highly liberalized markets or environmental and social constraints, are taking a leading role, different paths other than traditional engineering ones have been explored. In the last ten years, and mainly due to increasing computational capability and accessibility to data, the conceptual frame of complex networks has allowed different approaches in order to understand the usual (and not-so-usual) outcomes of this system. This paper considers power grids as complex networks. It presents some recent results that correlate their topology with their fragility, together with major malfunctions analysis, and for the European transmission power grid in particular.
---
paper_title: A Centrality Measure for Electrical Networks

paper_content:

We derive a measure of "electrical centrality" for AC power networks, which describes the structure of the network as a function of its electrical topology rather than its physical topology. We compare our centrality measure to conventional measures of network structure using the IEEE 300-bus network. We find that when measured electrically, power networks appear to have a scale-free network structure. Thus, unlike previous studies of the structure of power grids, we find that power networks have a number of highly-connected "hub" buses. This result, and the structure of power networks in general, is likely to have important implications for the reliability and security of power networks.
---
paper_title: Identifying vulnerable lines in a power network using complex network theory

paper_content:

The latest developments in complex network theory have provided a new direction to power system research. Based on this theory a power system can be modelled as a graph with nodes and vertices and further analysis can help in identifying the important lines. This paper proposes a new betweenness index using the reactance of the transmission lines as the weight and criteria to measure of vulnerability of a power network. The reactance is a simplified measure of power flow in a lossless transmission line based on the power flow analysis equations. The weighted line index is defined as the reactance of the electric path taken to reach from one node to another node. More power is transmitted along the lines with less reactance, to reach from the source node to destination node, which gives the edges with less reactance a higher weight in the analysis. The analyzes have been carried out on the IEEE 39 bus system and IEEE 118 Bus System. The new betweenness index can identify the critical lines of the system, either due to their position in the network or by the power they transmit along the network.
---
paper_title: The Improvement of the Small-world Network Model and Its Application Research in Bulk Power System

paper_content:

Due to the complexity of cascading failures and blackouts, the small world theory for complex network research is employed in probing the mechanism of them. In the paper, the original small world network model has been modified in two ways. Firstly, the magnitude of the line impedance regarded as the weight of edges is taken into consideration which is not considered in the original model. The formulae of corresponding parameters of the model are also modified. The calculation results of some practical grids validate the correctness and effectiveness of the new model. Secondly, the complex number format of power flow is introduced as the network flow in the small world model. With the power flow and the topology of grids, the two main factors to the system vulnerability, considered, the improved small world model can be used as a tool for vulnerability analysis of grids.
---
paper_title: Topological analysis of eastern region of Indian power grid

paper_content:

The modern power grid is tending towards more complexity as power has become an integral part of life even in remote location. Thus the vulnerability of such grids has become more prominent; necessitating some simulation based modeling to extract some important features for such sprawling network. The spatial structure of the power grid has been used for topological analysis of the power grid. The case studies pertaining to eastern region power grid of India validate such approach.
---
paper_title: Analysis of structural vulnerabilities in power transmission grids

paper_content:

Abstract Power transmission grids play a crucial role as a critical infrastructure by assuring the proper functioning of power systems. In particular, they secure the loads supplied by power generation plants and help avoid blackouts that can have a disastrous impact on society. The power grid structure (number of nodes and lines, their connections, and their physical properties and operational constraints) is one of the key issues (along with generation availability) to assure power system security; consequently, it deserves special attention. A promising approach for the structural analysis of transmission grids with respect to their vulnerabilities is to use metrics and approaches derived from complex network (CN) theory that are shared with other infrastructures such as the World-Wide Web, telecommunication networks, and oil and gas pipelines. These approaches, based on metrics such as global efficiency, degree and betweenness, are purely topological because they study structural vulnerabilities based on the graphical representation of a network as a set of vertices connected by a set of edges. Unfortunately, these approaches fail to capture the physical properties and operational constraints of power systems and, therefore, cannot provide meaningful analyses. This paper proposes an extended topological approach that includes the definitions of traditional topological metrics (e.g., degrees of nodes and global efficiency) as well as the physical/operational behavior of power grids in terms of real power-flow allocation over lines and line flow limits. This approach provides two new metrics, entropic degree and net-ability, that can be used to assess structural vulnerabilities in power systems. The new metrics are applied to test systems as well as real power grids to demonstrate their performance and to contrast them with traditional, purely topological metrics.
---
paper_title: The Concept of Betweenness in the Analysis of Power Grid Vulnerability

paper_content:

Vulnerability analysis in power systems is a key issue in modern society and many efforts have contributed to the analysis. Recently, complex network metrics applied to assess the topological vulnerability of networked systems have been used in power systems, such as betweenness metric, since transmission of power systems is in basis of a network structure. However, a pure topological approach fails to capture the specificity of power systems. This paper redefines, starting from the concept of complex networks, an electrical betweenness metric which considers several of specific features of power systems such as power transfer distribution and line flow limits. The electrical betweenness is compared with the conventional betweenness in IEEE-300 bus network according to the un-served energy after network is attacked. The results show that the tested network is more vulnerable when the components of the network are attacked according to their criticalities ranked by electrical betweenness.
---
paper_title: Analyzing power network vulnerability with maximum flow based centrality approach

paper_content:

Complex network theory has been studied extensively in solving large scale practical problems and the recent developments have given a new direction to power system research. This theory allows modeling a power system as a network and the latest developments incorporate more and more electrical properties as opposed to the topological analysis in the past. In the past, such networks have been analyzed based on shortest path travel and betweenness index. In a power system, the power might not necessarily flow only through the shortest path so this paper proposes a centrality index based on the maximum power flow through the edges. The links which carry more portion of power from the source (generator) to sink (load) are given a higher weight in this analysis. Few simple cases have been explained and then the algorithm has been demonstrated on the IEEE 39 bus system.
---
paper_title: TOPOLOGY AND CASCADING LINE OUTAGES IN POWER GRIDS

paper_content:

Motivated by the small world network research of Watts & Strogatz, this paper studies relationships between topology and cascading line outages in electric power grids. Cascading line outages are a type of cascading collapse that can occur in power grids when the transmission network is congested. It is characterized by a self-sustaining sequence of line outages followed by grid breakup, which generally leads to widespread blackout. The main findings of this work are twofold: On one hand, the work suggests that topologies with more disorder in their interconnection topology tend to be robust with respect to cascading line outages in the sense of being able to support greater generation and demand levels than more regularly interconnected topologies. On the other hand, the work suggests that topologies with more disorder tend to be more fragile in that should a cascade get started, they tend to break apart after fewer outages than more regularly interconnected topologies. Thus, as has been observed in other complex networks, there appears to be a tradeoff between robustness and fragility. These results were established using synthetically generated power grid topologies and verified using the IEEE 57 bus and 188 bus power grid test cases.
---
paper_title: Generating Statistically Correct Random Topologies for Testing Smart Grid Communication and Control Networks

paper_content:

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.
---
paper_title: Complex Networks Theory: A New Method of Research in Power Grid

paper_content:

The rapid development of complex network theory provide a new insight into the research of interconnected complex power grid. On the basis of mapping the power grid into a network graph, the topological parameters, such as degree distribution, characteristic path length, clustering coefficient etc, are introduced to describe the network structure characters. And the statistical results of these parameters in power grids of China and America are shown and analyzed. Then, small-world model are explained and simulated to the power grid. In order to measure the performance of the power grid globally and locally, the concept efficiency and its application is also introduced. The mechanism and the model of cascading failure in power grid are successively discussed. Finally, some possible directions of future research are proposed at the end of the paper
---
paper_title: Small Worlds: The Dynamics of Networks between Order and Randomness.

paper_content:

Transport brackets are affixed to the container and include provisions for achieving transportability with either a crane or a fork lift. Variable storage capacity is provided by an extension bin which is detachably affixed as part of the container. The top and bottom of the container are compatibly configured to provide an interlock between a plurality of stacked containers and a dolly is configured for use with these interlocking provisions. The container is configured to achieve stacking interchangeability when any vertically peripheral sides are aligned and brackets are disposed around the vertical periphery of the container to achieve fork lift accessibility from any direction perpendicular to the vertically peripheral sides.
---
paper_title: An Experimental Study of the Small World Problem

paper_content:

Arbitrarily selected individuals (N=296) in Nebraska and Boston are asked to generate acquaintance chains to a target person in Massachusetts, employing “the small world method” (Milgram, 1967). Sixty-four chains reach the target person. Within this group the mean number of intermediaries between starters and targets is 5.2. Boston starting chains reach the target person with fewer intermediaries than those starting in Nebraska; subpopulations in the Nebraska group do not differ among themselves. The funneling of chains through sociometric “stars” is noted, with 48 per cent of the chains passing through three persons before reaching the target. Applications of the method to studies of large scale social structure are discussed.
---
paper_title: Towards Decentralization: A Topological Investigation of the Medium and Low Voltage Grids

paper_content:

The traditional power grid has been designed in a hierarchical fashion, with energy pushed from the large scale production factories towards the end users. 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. Of course, end users need incentives to do so and want 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 influence the attractiveness of participating in such decentralized energy market, thus identifying the important topological parameters to work on to facilitate such open decentralized markets.
---
paper_title: The Improvement of the Small-world Network Model and Its Application Research in Bulk Power System

paper_content:

Due to the complexity of cascading failures and blackouts, the small world theory for complex network research is employed in probing the mechanism of them. In the paper, the original small world network model has been modified in two ways. Firstly, the magnitude of the line impedance regarded as the weight of edges is taken into consideration which is not considered in the original model. The formulae of corresponding parameters of the model are also modified. The calculation results of some practical grids validate the correctness and effectiveness of the new model. Secondly, the complex number format of power flow is introduced as the network flow in the small world model. With the power flow and the topology of grids, the two main factors to the system vulnerability, considered, the improved small world model can be used as a tool for vulnerability analysis of grids.
---
paper_title: Collective dynamics of ‘small-world’ networks

paper_content:

Networks of coupled dynamical systems have been used to model biological oscillators1,2,3,4, Josephson junction arrays5,6, excitable media7, neural networks8,9,10, spatial games11, genetic control networks12 and many other self-organizing systems. Ordinarily, the connection topology is assumed to be either completely regular or completely random. But many biological, technological and social networks lie somewhere between these two extremes. Here we explore simple models of networks that can be tuned through this middle ground: regular networks ‘rewired’ to introduce increasing amounts of disorder. We find that these systems can be highly clustered, like regular lattices, yet have small characteristic path lengths, like random graphs. We call them ‘small-world’ networks, by analogy with the small-world phenomenon13,14 (popularly known as six degrees of separation15). The neural network of the worm Caenorhabditis elegans, the power grid of the western United States, and the collaboration graph of film actors are shown to be small-world networks. Models of dynamical systems with small-world coupling display enhanced signal-propagation speed, computational power, and synchronizability. In particular, infectious diseases spread more easily in small-world networks than in regular lattices.
---
paper_title: Exploring complex networks

paper_content:

The study of networks pervades all of science, from neurobiology to statistical physics. The most basic issues are structural: how does one characterize the wiring diagram of a food web or the Internet or the metabolic network of the bacterium Escherichia coli? Are there any unifying principles underlying their topology? From the perspective of nonlinear dynamics, we would also like to understand how an enormous network of interacting dynamical systems — be they neurons, power stations or lasers — will behave collectively, given their individual dynamics and coupling architecture. Researchers are only now beginning to unravel the structure and dynamics of complex networks.
---
paper_title: TOPOLOGICAL VULNERABILITY OF THE EUROPEAN POWER GRID UNDER ERRORS AND ATTACKS

paper_content:

We present an analysis of the topological structure and static tolerance to errors and attacks of the September 2003 actualization of the Union for the Coordination of Transport of Electricity (UCTE) power grid, involving thirty-three different networks. Though every power grid studied has exponential degree distribution and most of them lack typical small-world topology, they display patterns of reaction to node loss similar to those observed in scale-free networks. We have found that the node removal behavior can be logarithmically related to the power grid size. This logarithmic behavior would suggest that, though size favors fragility, growth can reduce it. We conclude that, with the ever-growing demand for power and reliability, actual planning strategies to increase transmission systems would have to take into account this relative increase in vulnerability with size, in order to facilitate and improve the power grid design and functioning.
---
paper_title: TOPOLOGY AND CASCADING LINE OUTAGES IN POWER GRIDS

paper_content:

Motivated by the small world network research of Watts & Strogatz, this paper studies relationships between topology and cascading line outages in electric power grids. Cascading line outages are a type of cascading collapse that can occur in power grids when the transmission network is congested. It is characterized by a self-sustaining sequence of line outages followed by grid breakup, which generally leads to widespread blackout. The main findings of this work are twofold: On one hand, the work suggests that topologies with more disorder in their interconnection topology tend to be robust with respect to cascading line outages in the sense of being able to support greater generation and demand levels than more regularly interconnected topologies. On the other hand, the work suggests that topologies with more disorder tend to be more fragile in that should a cascade get started, they tend to break apart after fewer outages than more regularly interconnected topologies. Thus, as has been observed in other complex networks, there appears to be a tradeoff between robustness and fragility. These results were established using synthetically generated power grid topologies and verified using the IEEE 57 bus and 188 bus power grid test cases.
---
paper_title: Vulnerability Assessment of Power Grid Using Graph Topological Indices

paper_content:

This paper presents an assessment of the vulnerability of the power grid to blackout using graph topological indexes. Based on a FERC 715 report and the outage reports, the cascading faults of summer WSCC 1996 are reconstructed, and the graphical property of the grid is compared between two cases: when the blackout triggering lines are removed simulating the actual sequence of cascading outages and when the same number of randomly selected lines are removed. The investigation finds that the critical path lengths of the triggering events of the July and August outages of 1996 WSCC blackout are higher than those of no-outage and arbitrary events. In addition, the small world-ness index for each of the outage triggering events is much smaller than that of normal or any no-outage scenario, indicating that events of shifting a network from small world to a random network would be more likely cascaded to wide area outage.
---
paper_title: Using Graph Models to Analyze the Vulnerability of Electric Power Networks

paper_content:

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.
---
paper_title: Generating Statistically Correct Random Topologies for Testing Smart Grid Communication and Control Networks

paper_content:

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.
---
paper_title: A topological analysis of the Italian electric power grid

paper_content:

Large-scale blackouts are an intrinsic drawback of electric power transmission grids. Here we analyze the structural vulnerability of the Italian GRTN power grid by using a model for cascading failures recently proposed in Crucitti et al. (Phys. Rev. E 69 (2004)).
---
paper_title: Complex Networks Theory: A New Method of Research in Power Grid

paper_content:

The rapid development of complex network theory provide a new insight into the research of interconnected complex power grid. On the basis of mapping the power grid into a network graph, the topological parameters, such as degree distribution, characteristic path length, clustering coefficient etc, are introduced to describe the network structure characters. And the statistical results of these parameters in power grids of China and America are shown and analyzed. Then, small-world model are explained and simulated to the power grid. In order to measure the performance of the power grid globally and locally, the concept efficiency and its application is also introduced. The mechanism and the model of cascading failure in power grid are successively discussed. Finally, some possible directions of future research are proposed at the end of the paper
---
paper_title: Evaluating North American Electric Grid Reliability Using the Barabasi-Albert Network Model

paper_content:

The reliability of electric transmission systems is examined using a scale-free model of network topology and failure propagation. The topologies of the North American eastern and western electric grids are analyzed to estimate their reliability based on the Barabasi–Albert network model. A commonly used power system reliability index is computed using a simple failure propagation model. The results are compared to the values of power system reliability indices previously obtained using standard power engineering methods, and they suggest that scale-free network models are usable to estimate aggregate electric grid reliability.
---
paper_title: Structural vulnerability of the North American power grid

paper_content:

The magnitude of the August 2003 blackout affecting the United States has put the challenges of energy transmission and distribution into limelight. Despite all the interest and concerted effort, the complexity and interconnectivity of the electric infrastructure precluded us for a long time from understanding why certain events happened. In this paper we study the power grid from a network perspective and determine its ability to transfer power between generators and consumers when certain nodes are disrupted. We find that the power grid is robust to most perturbations, yet disturbances affecting key transmision substations greatly reduce its ability to function. We emphasize that the global properties of the underlying network must be understood as they greatly affect local behavior.
---
paper_title: Electrical centrality measures for electric power grid vulnerability analysis

paper_content:

Centrality measures are used in network science to rank the relative importance of nodes and edges of a graph. Here we define new measures of centrality for power grid structure that are based on its functionality. We show that the relative importance analysis based on centrality in graph theory can be generalized to power grid network with its electrical parameters taken into account. In the paper we experiment with the proposed electrical centrality measures on the NYISO-2935 system and the IEEE 300-bus system. We analyze the centrality distribution in order to identify important nodes or branches in the system which are of essential importance in terms of system vulnerability. We also present and discuss a number of interesting discoveries regarding the importance rank of power grid nodes and branches.
---
paper_title: Assessing European power grid reliability by means of topological measures

paper_content:

Publicat originalment a: WIT transactions on ecology and the environment, 2009, vol. 121, p. 527-537
---
paper_title: Towards Decentralization: A Topological Investigation of the Medium and Low Voltage Grids

paper_content:

The traditional power grid has been designed in a hierarchical fashion, with energy pushed from the large scale production factories towards the end users. 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. Of course, end users need incentives to do so and want 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 influence the attractiveness of participating in such decentralized energy market, thus identifying the important topological parameters to work on to facilitate such open decentralized markets.
---
paper_title: The Node Degree Distribution in Power Grid and Its Topology Robustness under Random and Selective Node Removals

paper_content:

In this paper we numerically study the topology robustness of power grids under random and selective node breakdowns, and analytically estimate the critical node-removal thresholds to disintegrate a system, based on the available US power grid data. We also present an analysis on the node degree distribution in power grids because it closely relates with the topology robustness. It is found that the node degree in a power grid can be well fitted by a mixture distribution coming from the sum of a truncated Geometric random variable and an irregular Discrete random variable. With the findings we obtain better estimates of the threshold under selective node breakdowns which predict the numerical thresholds more correctly.
---
paper_title: A Centrality Measure for Electrical Networks

paper_content:

We derive a measure of "electrical centrality" for AC power networks, which describes the structure of the network as a function of its electrical topology rather than its physical topology. We compare our centrality measure to conventional measures of network structure using the IEEE 300-bus network. We find that when measured electrically, power networks appear to have a scale-free network structure. Thus, unlike previous studies of the structure of power grids, we find that power networks have a number of highly-connected "hub" buses. This result, and the structure of power networks in general, is likely to have important implications for the reliability and security of power networks.
---
paper_title: Topological properties of high-voltage electrical transmission networks

paper_content:

The topological properties of high-voltage electrical power transmission networks in several UE countries (the Italian 380 kV, the French 400 kV and the Spanish 400 kV networks) have been studied from available data. An assessment of the vulnerability of the networks has been given by measuring the level of damage introduced by a controlled removal of links. Topological studies could be useful to make vulnerability assessment and to design specific action to reduce topological weaknesses.
---
paper_title: Collective dynamics of ‘small-world’ networks

paper_content:

Networks of coupled dynamical systems have been used to model biological oscillators1,2,3,4, Josephson junction arrays5,6, excitable media7, neural networks8,9,10, spatial games11, genetic control networks12 and many other self-organizing systems. Ordinarily, the connection topology is assumed to be either completely regular or completely random. But many biological, technological and social networks lie somewhere between these two extremes. Here we explore simple models of networks that can be tuned through this middle ground: regular networks ‘rewired’ to introduce increasing amounts of disorder. We find that these systems can be highly clustered, like regular lattices, yet have small characteristic path lengths, like random graphs. We call them ‘small-world’ networks, by analogy with the small-world phenomenon13,14 (popularly known as six degrees of separation15). The neural network of the worm Caenorhabditis elegans, the power grid of the western United States, and the collaboration graph of film actors are shown to be small-world networks. Models of dynamical systems with small-world coupling display enhanced signal-propagation speed, computational power, and synchronizability. In particular, infectious diseases spread more easily in small-world networks than in regular lattices.
---
paper_title: TOPOLOGICAL VULNERABILITY OF THE EUROPEAN POWER GRID UNDER ERRORS AND ATTACKS

paper_content:

We present an analysis of the topological structure and static tolerance to errors and attacks of the September 2003 actualization of the Union for the Coordination of Transport of Electricity (UCTE) power grid, involving thirty-three different networks. Though every power grid studied has exponential degree distribution and most of them lack typical small-world topology, they display patterns of reaction to node loss similar to those observed in scale-free networks. We have found that the node removal behavior can be logarithmically related to the power grid size. This logarithmic behavior would suggest that, though size favors fragility, growth can reduce it. We conclude that, with the ever-growing demand for power and reliability, actual planning strategies to increase transmission systems would have to take into account this relative increase in vulnerability with size, in order to facilitate and improve the power grid design and functioning.
---
paper_title: Using Graph Models to Analyze the Vulnerability of Electric Power Networks

paper_content:

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.
---
paper_title: A topological analysis of the Italian electric power grid

paper_content:

Large-scale blackouts are an intrinsic drawback of electric power transmission grids. Here we analyze the structural vulnerability of the Italian GRTN power grid by using a model for cascading failures recently proposed in Crucitti et al. (Phys. Rev. E 69 (2004)).
---
paper_title: Structural vulnerability of the North American power grid

paper_content:

The magnitude of the August 2003 blackout affecting the United States has put the challenges of energy transmission and distribution into limelight. Despite all the interest and concerted effort, the complexity and interconnectivity of the electric infrastructure precluded us for a long time from understanding why certain events happened. In this paper we study the power grid from a network perspective and determine its ability to transfer power between generators and consumers when certain nodes are disrupted. We find that the power grid is robust to most perturbations, yet disturbances affecting key transmision substations greatly reduce its ability to function. We emphasize that the global properties of the underlying network must be understood as they greatly affect local behavior.
---
paper_title: Electrical centrality measures for electric power grid vulnerability analysis

paper_content:

Centrality measures are used in network science to rank the relative importance of nodes and edges of a graph. Here we define new measures of centrality for power grid structure that are based on its functionality. We show that the relative importance analysis based on centrality in graph theory can be generalized to power grid network with its electrical parameters taken into account. In the paper we experiment with the proposed electrical centrality measures on the NYISO-2935 system and the IEEE 300-bus system. We analyze the centrality distribution in order to identify important nodes or branches in the system which are of essential importance in terms of system vulnerability. We also present and discuss a number of interesting discoveries regarding the importance rank of power grid nodes and branches.
---
paper_title: Towards Decentralization: A Topological Investigation of the Medium and Low Voltage Grids

paper_content:

The traditional power grid has been designed in a hierarchical fashion, with energy pushed from the large scale production factories towards the end users. 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. Of course, end users need incentives to do so and want 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 influence the attractiveness of participating in such decentralized energy market, thus identifying the important topological parameters to work on to facilitate such open decentralized markets.
---
paper_title: Do topological models provide good information about electricity infrastructure vulnerability

paper_content:

In order to identify the extent to which results from topological graph models are useful for modeling vulnerability in electricity infrastructure, we measure the susceptibility of power networks to random failures and directed attacks using three measures of vulnerability: characteristic path lengths, connectivity loss, and blackout sizes. The first two are purely topological metrics. The blackout size calculation results from a model of cascading failure in power networks. Testing the response of 40 areas within the Eastern U.S. power grid and a standard IEEE test case to a variety of attack/failure vectors indicates that directed attacks result in larger failures using all three vulnerability measures, but the attack-vectors that appear to cause the most damage depend on the measure chosen. While the topological metrics and the power grid model show some similar trends, the vulnerability metrics for individual simulations show only a mild correlation. We conclude that evaluating vulnerability in power ...
---
paper_title: A topological analysis of the Italian electric power grid

paper_content:

Large-scale blackouts are an intrinsic drawback of electric power transmission grids. Here we analyze the structural vulnerability of the Italian GRTN power grid by using a model for cascading failures recently proposed in Crucitti et al. (Phys. Rev. E 69 (2004)).
---
paper_title: Cascade-based attack vulnerability on the US power grid

paper_content:

The vulnerability of real-life networks subject to intentional attacks has been one of the outstanding challenges in the study of the network safety. Applying the real data of the US power grid, we compare the effects of two different attacks for the network robustness against cascading failures, i.e., removal by either the descending or ascending orders of the loads. Adopting the initial load of a node j to be Lj !" kj# Rm2Cj km$% a with kj and Cj being the degree of the node j and the set of its neighboring nodes, respectively, where a is a tunable parameter and governs the strength of the initial load of a node, we investigate the response of the US power grid under two attacks during the cascading propagation. In the case of a < 0:7, our investigation by the numerical simulations leads to a counterintuitive finding on the US power grid that the attack on the nodes with the lowest loads is more harmful than the attack on the ones with the highest loads. In addition, the almost same effect of two attacks in the case ofa ! 0:7 may be useful in furthering studies on the control and defense of cascading failures in the US power grid.
---
paper_title: Evaluating North American Electric Grid Reliability Using the Barabasi-Albert Network Model

paper_content:

The reliability of electric transmission systems is examined using a scale-free model of network topology and failure propagation. The topologies of the North American eastern and western electric grids are analyzed to estimate their reliability based on the Barabasi–Albert network model. A commonly used power system reliability index is computed using a simple failure propagation model. The results are compared to the values of power system reliability indices previously obtained using standard power engineering methods, and they suggest that scale-free network models are usable to estimate aggregate electric grid reliability.
---
paper_title: Vulnerability assessment of bulk power grid based on complex network theory

paper_content:

The vulnerability assessment algorithm of the bulk power grid based on complex network theory is proposed in this paper. The traditional research model of the power grid based on complex network theory is a graph with no direction and no weight at present. Because this model is far from the real power system, sometimes the wrong results may be gotten. In this paper, the models of components in the power grid are constructed detailedly, furthermore the weighted and directional graph is provided. First, the connecting formings among buses (instead in the traditional methods only substations are considered), lines, transformers and generators are simulated detailedly. Second, the power flow direction in the power grid is considered, and the components' tolerance to disturbances is also considered. Based on the proposed model, the power grid's tolerance to errors and attacks is analyzed. The key components and weak areas are indicated. At last, the robustness and fault dissemination mechanism of the power grid under cascading failures are analyzed based on the proposed method. The North China power grid is used as an example to validate the proposed method.
---
paper_title: Suppressing cascades of load in interdependent networks

paper_content:

Understanding how interdependence among systems affects cascading behaviors is increasingly important across many fields of science and engineering. Inspired by cascades of load shedding in coupled electric grids and other infrastructure, we study the Bak–Tang–Wiesenfeld sandpile model on modular random graphs and on graphs based on actual, interdependent power grids. Starting from two isolated networks, adding some connectivity between them is beneficial, for it suppresses the largest cascades in each system. Too much interconnectivity, however, becomes detrimental for two reasons. First, interconnections open pathways for neighboring networks to inflict large cascades. Second, as in real infrastructure, new interconnections increase capacity and total possible load, which fuels even larger cascades. Using a multitype branching process and simulations we show these effects and estimate the optimal level of interconnectivity that balances their trade-offs. Such equilibria could allow, for example, power grid owners to minimize the largest cascades in their grid. We also show that asymmetric capacity among interdependent networks affects the optimal connectivity that each prefers and may lead to an arms race for greater capacity. Our multitype branching process framework provides building blocks for better prediction of cascading processes on modular random graphs and on multitype networks in general.
---
paper_title: Modeling Cascading Failures in the North American Power Grid

paper_content:

The North American power grid is one of the most complex technological networks, and its interconnectivity allows both for long-distance power transmission and for the propagation of disturbances. We model the power grid using its actual topology and plausible assumptions about the load and overload of transmission substations. Our results indicate that the loss of a single substation can result in up to \(25\%\) loss of transmission efficiency by triggering an overload cascade in the network. The actual transmission loss depends on the overload tolerance of the network and the connectivity of the failed substation. We systematically study the damage inflicted by the loss of single nodes, and find three universal behaviors, suggesting that \(40\%\) of the transmission substations lead to cascading failures when disrupted. While the loss of a single node can inflict substantial damage, subsequent removals have only incremental effects, in agreement with the topological resilience to less than \(1\%\) node loss.
---
paper_title: Robustness of the European power grids under intentional attack

paper_content:

The power grid defines one of the most important technological networks of our times and sustains our complex society. It has evolved for more than a century into an extremely huge and seemingly robust and well understood system. But it becomes extremely fragile as well, when unexpected, usually minimal, failures turn into unknown dynamical behaviours leading, for example, to sudden and massive blackouts. Here we explore the fragility of the European power grid under the effect of selective node removal. A mean field analysis of fragility against attacks is presented together with the observed patterns. Deviations from the theoretical conditions for network percolation (and fragmentation) under attacks are analysed and correlated with non topological reliability measures.
---
paper_title: Structural vulnerability of the North American power grid

paper_content:

The magnitude of the August 2003 blackout affecting the United States has put the challenges of energy transmission and distribution into limelight. Despite all the interest and concerted effort, the complexity and interconnectivity of the electric infrastructure precluded us for a long time from understanding why certain events happened. In this paper we study the power grid from a network perspective and determine its ability to transfer power between generators and consumers when certain nodes are disrupted. We find that the power grid is robust to most perturbations, yet disturbances affecting key transmision substations greatly reduce its ability to function. We emphasize that the global properties of the underlying network must be understood as they greatly affect local behavior.
---
paper_title: Assessing European power grid reliability by means of topological measures

paper_content:

Publicat originalment a: WIT transactions on ecology and the environment, 2009, vol. 121, p. 527-537
---
paper_title: Towards Decentralization: A Topological Investigation of the Medium and Low Voltage Grids

paper_content:

The traditional power grid has been designed in a hierarchical fashion, with energy pushed from the large scale production factories towards the end users. 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. Of course, end users need incentives to do so and want 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 influence the attractiveness of participating in such decentralized energy market, thus identifying the important topological parameters to work on to facilitate such open decentralized markets.
---
paper_title: The Node Degree Distribution in Power Grid and Its Topology Robustness under Random and Selective Node Removals

paper_content:

In this paper we numerically study the topology robustness of power grids under random and selective node breakdowns, and analytically estimate the critical node-removal thresholds to disintegrate a system, based on the available US power grid data. We also present an analysis on the node degree distribution in power grids because it closely relates with the topology robustness. It is found that the node degree in a power grid can be well fitted by a mixture distribution coming from the sum of a truncated Geometric random variable and an irregular Discrete random variable. With the findings we obtain better estimates of the threshold under selective node breakdowns which predict the numerical thresholds more correctly.
---
paper_title: Power Grids as Complex Networks: Topology and Fragility

paper_content:

We almost certainly agree that the harnessing of electricity can be considered the most important technological advance of the twentieth century. The development that allows this control is the power grid, an intricate system composed of generators, substations and transformers connected by cable lines hundreds of kilometers long. Its presence is nowadays so intertwined with ours, and so much taken for granted, that we are only capable of sensing its absence, disguised as a cascading failure or a blackout in its extreme form. Since these extreme phenomena seem to have increased in recent years and new actors, like highly liberalized markets or environmental and social constraints, are taking a leading role, different paths other than traditional engineering ones have been explored. In the last ten years, and mainly due to increasing computational capability and accessibility to data, the conceptual frame of complex networks has allowed different approaches in order to understand the usual (and not-so-usual) outcomes of this system. This paper considers power grids as complex networks. It presents some recent results that correlate their topology with their fragility, together with major malfunctions analysis, and for the European transmission power grid in particular.
---
paper_title: A Centrality Measure for Electrical Networks

paper_content:

We derive a measure of "electrical centrality" for AC power networks, which describes the structure of the network as a function of its electrical topology rather than its physical topology. We compare our centrality measure to conventional measures of network structure using the IEEE 300-bus network. We find that when measured electrically, power networks appear to have a scale-free network structure. Thus, unlike previous studies of the structure of power grids, we find that power networks have a number of highly-connected "hub" buses. This result, and the structure of power networks in general, is likely to have important implications for the reliability and security of power networks.
---
paper_title: Identifying vulnerable lines in a power network using complex network theory

paper_content:

The latest developments in complex network theory have provided a new direction to power system research. Based on this theory a power system can be modelled as a graph with nodes and vertices and further analysis can help in identifying the important lines. This paper proposes a new betweenness index using the reactance of the transmission lines as the weight and criteria to measure of vulnerability of a power network. The reactance is a simplified measure of power flow in a lossless transmission line based on the power flow analysis equations. The weighted line index is defined as the reactance of the electric path taken to reach from one node to another node. More power is transmitted along the lines with less reactance, to reach from the source node to destination node, which gives the edges with less reactance a higher weight in the analysis. The analyzes have been carried out on the IEEE 39 bus system and IEEE 118 Bus System. The new betweenness index can identify the critical lines of the system, either due to their position in the network or by the power they transmit along the network.
---
paper_title: Topological properties of high-voltage electrical transmission networks

paper_content:

The topological properties of high-voltage electrical power transmission networks in several UE countries (the Italian 380 kV, the French 400 kV and the Spanish 400 kV networks) have been studied from available data. An assessment of the vulnerability of the networks has been given by measuring the level of damage introduced by a controlled removal of links. Topological studies could be useful to make vulnerability assessment and to design specific action to reduce topological weaknesses.
---
paper_title: Topological analysis of the power grid and mitigation strategies against cascading failures

paper_content:

This paper presents a complex systems overview of a power grid network. In recent years, concerns about the robustness of the power grid have grown because of several cascading outages in different parts of the world. In this paper, cascading effect has been simulated on three different networks, the IEEE 300 bus test system, the IEEE 118 bus test system, and the WSCC 179 bus equivalent model. Power Degradation has been discussed as a measure to estimate the damage to the network, in terms of load loss and node loss. A network generator has been developed to generate graphs with characteristics similar to the IEEE standard networks and the generated graphs are then compared with the standard networks to show the effect of topology in determining the robustness of a power grid. Three mitigation strategies, Homogeneous Load Reduction, Targeted Range-Based Load Reduction, and Use of Distributed Renewable Sources in combination with Islanding, have been suggested. The Homogeneous Load Reduction is the simplest to implement but the Targeted Range-Based Load Reduction is the most effective strategy.
---
paper_title: Analysis of structural vulnerabilities in power transmission grids

paper_content:

Abstract Power transmission grids play a crucial role as a critical infrastructure by assuring the proper functioning of power systems. In particular, they secure the loads supplied by power generation plants and help avoid blackouts that can have a disastrous impact on society. The power grid structure (number of nodes and lines, their connections, and their physical properties and operational constraints) is one of the key issues (along with generation availability) to assure power system security; consequently, it deserves special attention. A promising approach for the structural analysis of transmission grids with respect to their vulnerabilities is to use metrics and approaches derived from complex network (CN) theory that are shared with other infrastructures such as the World-Wide Web, telecommunication networks, and oil and gas pipelines. These approaches, based on metrics such as global efficiency, degree and betweenness, are purely topological because they study structural vulnerabilities based on the graphical representation of a network as a set of vertices connected by a set of edges. Unfortunately, these approaches fail to capture the physical properties and operational constraints of power systems and, therefore, cannot provide meaningful analyses. This paper proposes an extended topological approach that includes the definitions of traditional topological metrics (e.g., degrees of nodes and global efficiency) as well as the physical/operational behavior of power grids in terms of real power-flow allocation over lines and line flow limits. This approach provides two new metrics, entropic degree and net-ability, that can be used to assess structural vulnerabilities in power systems. The new metrics are applied to test systems as well as real power grids to demonstrate their performance and to contrast them with traditional, purely topological metrics.
---
paper_title: TOPOLOGICAL VULNERABILITY OF THE EUROPEAN POWER GRID UNDER ERRORS AND ATTACKS

paper_content:

We present an analysis of the topological structure and static tolerance to errors and attacks of the September 2003 actualization of the Union for the Coordination of Transport of Electricity (UCTE) power grid, involving thirty-three different networks. Though every power grid studied has exponential degree distribution and most of them lack typical small-world topology, they display patterns of reaction to node loss similar to those observed in scale-free networks. We have found that the node removal behavior can be logarithmically related to the power grid size. This logarithmic behavior would suggest that, though size favors fragility, growth can reduce it. We conclude that, with the ever-growing demand for power and reliability, actual planning strategies to increase transmission systems would have to take into account this relative increase in vulnerability with size, in order to facilitate and improve the power grid design and functioning.
---
paper_title: The Concept of Betweenness in the Analysis of Power Grid Vulnerability

paper_content:

Vulnerability analysis in power systems is a key issue in modern society and many efforts have contributed to the analysis. Recently, complex network metrics applied to assess the topological vulnerability of networked systems have been used in power systems, such as betweenness metric, since transmission of power systems is in basis of a network structure. However, a pure topological approach fails to capture the specificity of power systems. This paper redefines, starting from the concept of complex networks, an electrical betweenness metric which considers several of specific features of power systems such as power transfer distribution and line flow limits. The electrical betweenness is compared with the conventional betweenness in IEEE-300 bus network according to the un-served energy after network is attacked. The results show that the tested network is more vulnerable when the components of the network are attacked according to their criticalities ranked by electrical betweenness.
---
paper_title: LOCATING CRITICAL LINES IN HIGH-VOLTAGE ELECTRICAL POWER GRIDS

paper_content:

Electrical power grids are among the infrastructures that are attracting a great deal of attention because of their intrinsic criticality. Here we analyze the topological vulnerability and improvability of the spanish 400 kV, the french 400 kV and the italian 380 kV power transmission grids. For each network we detect the most critical lines and suggest how to improve the connectivity.
---
paper_title: Analyzing power network vulnerability with maximum flow based centrality approach

paper_content:

Complex network theory has been studied extensively in solving large scale practical problems and the recent developments have given a new direction to power system research. This theory allows modeling a power system as a network and the latest developments incorporate more and more electrical properties as opposed to the topological analysis in the past. In the past, such networks have been analyzed based on shortest path travel and betweenness index. In a power system, the power might not necessarily flow only through the shortest path so this paper proposes a centrality index based on the maximum power flow through the edges. The links which carry more portion of power from the source (generator) to sink (load) are given a higher weight in this analysis. Few simple cases have been explained and then the algorithm has been demonstrated on the IEEE 39 bus system.
---
paper_title: TOPOLOGY AND CASCADING LINE OUTAGES IN POWER GRIDS

paper_content:

Motivated by the small world network research of Watts & Strogatz, this paper studies relationships between topology and cascading line outages in electric power grids. Cascading line outages are a type of cascading collapse that can occur in power grids when the transmission network is congested. It is characterized by a self-sustaining sequence of line outages followed by grid breakup, which generally leads to widespread blackout. The main findings of this work are twofold: On one hand, the work suggests that topologies with more disorder in their interconnection topology tend to be robust with respect to cascading line outages in the sense of being able to support greater generation and demand levels than more regularly interconnected topologies. On the other hand, the work suggests that topologies with more disorder tend to be more fragile in that should a cascade get started, they tend to break apart after fewer outages than more regularly interconnected topologies. Thus, as has been observed in other complex networks, there appears to be a tradeoff between robustness and fragility. These results were established using synthetically generated power grid topologies and verified using the IEEE 57 bus and 188 bus power grid test cases.
---
paper_title: Vulnerability Assessment of Power Grid Using Graph Topological Indices

paper_content:

This paper presents an assessment of the vulnerability of the power grid to blackout using graph topological indexes. Based on a FERC 715 report and the outage reports, the cascading faults of summer WSCC 1996 are reconstructed, and the graphical property of the grid is compared between two cases: when the blackout triggering lines are removed simulating the actual sequence of cascading outages and when the same number of randomly selected lines are removed. The investigation finds that the critical path lengths of the triggering events of the July and August outages of 1996 WSCC blackout are higher than those of no-outage and arbitrary events. In addition, the small world-ness index for each of the outage triggering events is much smaller than that of normal or any no-outage scenario, indicating that events of shifting a network from small world to a random network would be more likely cascaded to wide area outage.
---
paper_title: Using Graph Models to Analyze the Vulnerability of Electric Power Networks

paper_content:

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.
---
paper_title: Generating Statistically Correct Random Topologies for Testing Smart Grid Communication and Control Networks

paper_content:

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.
---
paper_title: Do topological models provide good information about electricity infrastructure vulnerability

paper_content:

In order to identify the extent to which results from topological graph models are useful for modeling vulnerability in electricity infrastructure, we measure the susceptibility of power networks to random failures and directed attacks using three measures of vulnerability: characteristic path lengths, connectivity loss, and blackout sizes. The first two are purely topological metrics. The blackout size calculation results from a model of cascading failure in power networks. Testing the response of 40 areas within the Eastern U.S. power grid and a standard IEEE test case to a variety of attack/failure vectors indicates that directed attacks result in larger failures using all three vulnerability measures, but the attack-vectors that appear to cause the most damage depend on the measure chosen. While the topological metrics and the power grid model show some similar trends, the vulnerability metrics for individual simulations show only a mild correlation. We conclude that evaluating vulnerability in power ...
---
paper_title: A topological analysis of the Italian electric power grid

paper_content:

Large-scale blackouts are an intrinsic drawback of electric power transmission grids. Here we analyze the structural vulnerability of the Italian GRTN power grid by using a model for cascading failures recently proposed in Crucitti et al. (Phys. Rev. E 69 (2004)).
---
paper_title: Cascade-based attack vulnerability on the US power grid

paper_content:

The vulnerability of real-life networks subject to intentional attacks has been one of the outstanding challenges in the study of the network safety. Applying the real data of the US power grid, we compare the effects of two different attacks for the network robustness against cascading failures, i.e., removal by either the descending or ascending orders of the loads. Adopting the initial load of a node j to be Lj !" kj# Rm2Cj km$% a with kj and Cj being the degree of the node j and the set of its neighboring nodes, respectively, where a is a tunable parameter and governs the strength of the initial load of a node, we investigate the response of the US power grid under two attacks during the cascading propagation. In the case of a < 0:7, our investigation by the numerical simulations leads to a counterintuitive finding on the US power grid that the attack on the nodes with the lowest loads is more harmful than the attack on the ones with the highest loads. In addition, the almost same effect of two attacks in the case ofa ! 0:7 may be useful in furthering studies on the control and defense of cascading failures in the US power grid.
---
paper_title: Modeling Cascading Failures in the North American Power Grid

paper_content:

The North American power grid is one of the most complex technological networks, and its interconnectivity allows both for long-distance power transmission and for the propagation of disturbances. We model the power grid using its actual topology and plausible assumptions about the load and overload of transmission substations. Our results indicate that the loss of a single substation can result in up to \(25\%\) loss of transmission efficiency by triggering an overload cascade in the network. The actual transmission loss depends on the overload tolerance of the network and the connectivity of the failed substation. We systematically study the damage inflicted by the loss of single nodes, and find three universal behaviors, suggesting that \(40\%\) of the transmission substations lead to cascading failures when disrupted. While the loss of a single node can inflict substantial damage, subsequent removals have only incremental effects, in agreement with the topological resilience to less than \(1\%\) node loss.
---
paper_title: Robustness of the European power grids under intentional attack

paper_content:

The power grid defines one of the most important technological networks of our times and sustains our complex society. It has evolved for more than a century into an extremely huge and seemingly robust and well understood system. But it becomes extremely fragile as well, when unexpected, usually minimal, failures turn into unknown dynamical behaviours leading, for example, to sudden and massive blackouts. Here we explore the fragility of the European power grid under the effect of selective node removal. A mean field analysis of fragility against attacks is presented together with the observed patterns. Deviations from the theoretical conditions for network percolation (and fragmentation) under attacks are analysed and correlated with non topological reliability measures.
---
paper_title: Structural vulnerability of the North American power grid

paper_content:

The magnitude of the August 2003 blackout affecting the United States has put the challenges of energy transmission and distribution into limelight. Despite all the interest and concerted effort, the complexity and interconnectivity of the electric infrastructure precluded us for a long time from understanding why certain events happened. In this paper we study the power grid from a network perspective and determine its ability to transfer power between generators and consumers when certain nodes are disrupted. We find that the power grid is robust to most perturbations, yet disturbances affecting key transmision substations greatly reduce its ability to function. We emphasize that the global properties of the underlying network must be understood as they greatly affect local behavior.
---
paper_title: The Node Degree Distribution in Power Grid and Its Topology Robustness under Random and Selective Node Removals

paper_content:

In this paper we numerically study the topology robustness of power grids under random and selective node breakdowns, and analytically estimate the critical node-removal thresholds to disintegrate a system, based on the available US power grid data. We also present an analysis on the node degree distribution in power grids because it closely relates with the topology robustness. It is found that the node degree in a power grid can be well fitted by a mixture distribution coming from the sum of a truncated Geometric random variable and an irregular Discrete random variable. With the findings we obtain better estimates of the threshold under selective node breakdowns which predict the numerical thresholds more correctly.
---
paper_title: TOPOLOGICAL VULNERABILITY OF THE EUROPEAN POWER GRID UNDER ERRORS AND ATTACKS

paper_content:

We present an analysis of the topological structure and static tolerance to errors and attacks of the September 2003 actualization of the Union for the Coordination of Transport of Electricity (UCTE) power grid, involving thirty-three different networks. Though every power grid studied has exponential degree distribution and most of them lack typical small-world topology, they display patterns of reaction to node loss similar to those observed in scale-free networks. We have found that the node removal behavior can be logarithmically related to the power grid size. This logarithmic behavior would suggest that, though size favors fragility, growth can reduce it. We conclude that, with the ever-growing demand for power and reliability, actual planning strategies to increase transmission systems would have to take into account this relative increase in vulnerability with size, in order to facilitate and improve the power grid design and functioning.
---
paper_title: Modeling Cascading Failures in the North American Power Grid

paper_content:

The North American power grid is one of the most complex technological networks, and its interconnectivity allows both for long-distance power transmission and for the propagation of disturbances. We model the power grid using its actual topology and plausible assumptions about the load and overload of transmission substations. Our results indicate that the loss of a single substation can result in up to \(25\%\) loss of transmission efficiency by triggering an overload cascade in the network. The actual transmission loss depends on the overload tolerance of the network and the connectivity of the failed substation. We systematically study the damage inflicted by the loss of single nodes, and find three universal behaviors, suggesting that \(40\%\) of the transmission substations lead to cascading failures when disrupted. While the loss of a single node can inflict substantial damage, subsequent removals have only incremental effects, in agreement with the topological resilience to less than \(1\%\) node loss.
---
paper_title: Identifying vulnerable lines in a power network using complex network theory

paper_content:

The latest developments in complex network theory have provided a new direction to power system research. Based on this theory a power system can be modelled as a graph with nodes and vertices and further analysis can help in identifying the important lines. This paper proposes a new betweenness index using the reactance of the transmission lines as the weight and criteria to measure of vulnerability of a power network. The reactance is a simplified measure of power flow in a lossless transmission line based on the power flow analysis equations. The weighted line index is defined as the reactance of the electric path taken to reach from one node to another node. More power is transmitted along the lines with less reactance, to reach from the source node to destination node, which gives the edges with less reactance a higher weight in the analysis. The analyzes have been carried out on the IEEE 39 bus system and IEEE 118 Bus System. The new betweenness index can identify the critical lines of the system, either due to their position in the network or by the power they transmit along the network.
---
paper_title: Topological analysis of the power grid and mitigation strategies against cascading failures

paper_content:

This paper presents a complex systems overview of a power grid network. In recent years, concerns about the robustness of the power grid have grown because of several cascading outages in different parts of the world. In this paper, cascading effect has been simulated on three different networks, the IEEE 300 bus test system, the IEEE 118 bus test system, and the WSCC 179 bus equivalent model. Power Degradation has been discussed as a measure to estimate the damage to the network, in terms of load loss and node loss. A network generator has been developed to generate graphs with characteristics similar to the IEEE standard networks and the generated graphs are then compared with the standard networks to show the effect of topology in determining the robustness of a power grid. Three mitigation strategies, Homogeneous Load Reduction, Targeted Range-Based Load Reduction, and Use of Distributed Renewable Sources in combination with Islanding, have been suggested. The Homogeneous Load Reduction is the simplest to implement but the Targeted Range-Based Load Reduction is the most effective strategy.
---
paper_title: Analysis of structural vulnerabilities in power transmission grids

paper_content:

Abstract Power transmission grids play a crucial role as a critical infrastructure by assuring the proper functioning of power systems. In particular, they secure the loads supplied by power generation plants and help avoid blackouts that can have a disastrous impact on society. The power grid structure (number of nodes and lines, their connections, and their physical properties and operational constraints) is one of the key issues (along with generation availability) to assure power system security; consequently, it deserves special attention. A promising approach for the structural analysis of transmission grids with respect to their vulnerabilities is to use metrics and approaches derived from complex network (CN) theory that are shared with other infrastructures such as the World-Wide Web, telecommunication networks, and oil and gas pipelines. These approaches, based on metrics such as global efficiency, degree and betweenness, are purely topological because they study structural vulnerabilities based on the graphical representation of a network as a set of vertices connected by a set of edges. Unfortunately, these approaches fail to capture the physical properties and operational constraints of power systems and, therefore, cannot provide meaningful analyses. This paper proposes an extended topological approach that includes the definitions of traditional topological metrics (e.g., degrees of nodes and global efficiency) as well as the physical/operational behavior of power grids in terms of real power-flow allocation over lines and line flow limits. This approach provides two new metrics, entropic degree and net-ability, that can be used to assess structural vulnerabilities in power systems. The new metrics are applied to test systems as well as real power grids to demonstrate their performance and to contrast them with traditional, purely topological metrics.
---
paper_title: LOCATING CRITICAL LINES IN HIGH-VOLTAGE ELECTRICAL POWER GRIDS

paper_content:

Electrical power grids are among the infrastructures that are attracting a great deal of attention because of their intrinsic criticality. Here we analyze the topological vulnerability and improvability of the spanish 400 kV, the french 400 kV and the italian 380 kV power transmission grids. For each network we detect the most critical lines and suggest how to improve the connectivity.
---
paper_title: Analyzing power network vulnerability with maximum flow based centrality approach

paper_content:

Complex network theory has been studied extensively in solving large scale practical problems and the recent developments have given a new direction to power system research. This theory allows modeling a power system as a network and the latest developments incorporate more and more electrical properties as opposed to the topological analysis in the past. In the past, such networks have been analyzed based on shortest path travel and betweenness index. In a power system, the power might not necessarily flow only through the shortest path so this paper proposes a centrality index based on the maximum power flow through the edges. The links which carry more portion of power from the source (generator) to sink (load) are given a higher weight in this analysis. Few simple cases have been explained and then the algorithm has been demonstrated on the IEEE 39 bus system.
---
paper_title: TOPOLOGY AND CASCADING LINE OUTAGES IN POWER GRIDS

paper_content:

Motivated by the small world network research of Watts & Strogatz, this paper studies relationships between topology and cascading line outages in electric power grids. Cascading line outages are a type of cascading collapse that can occur in power grids when the transmission network is congested. It is characterized by a self-sustaining sequence of line outages followed by grid breakup, which generally leads to widespread blackout. The main findings of this work are twofold: On one hand, the work suggests that topologies with more disorder in their interconnection topology tend to be robust with respect to cascading line outages in the sense of being able to support greater generation and demand levels than more regularly interconnected topologies. On the other hand, the work suggests that topologies with more disorder tend to be more fragile in that should a cascade get started, they tend to break apart after fewer outages than more regularly interconnected topologies. Thus, as has been observed in other complex networks, there appears to be a tradeoff between robustness and fragility. These results were established using synthetically generated power grid topologies and verified using the IEEE 57 bus and 188 bus power grid test cases.
---
paper_title: Vulnerability Assessment of Power Grid Using Graph Topological Indices

paper_content:

This paper presents an assessment of the vulnerability of the power grid to blackout using graph topological indexes. Based on a FERC 715 report and the outage reports, the cascading faults of summer WSCC 1996 are reconstructed, and the graphical property of the grid is compared between two cases: when the blackout triggering lines are removed simulating the actual sequence of cascading outages and when the same number of randomly selected lines are removed. The investigation finds that the critical path lengths of the triggering events of the July and August outages of 1996 WSCC blackout are higher than those of no-outage and arbitrary events. In addition, the small world-ness index for each of the outage triggering events is much smaller than that of normal or any no-outage scenario, indicating that events of shifting a network from small world to a random network would be more likely cascaded to wide area outage.
---
paper_title: A topological analysis of the Italian electric power grid

paper_content:

Large-scale blackouts are an intrinsic drawback of electric power transmission grids. Here we analyze the structural vulnerability of the Italian GRTN power grid by using a model for cascading failures recently proposed in Crucitti et al. (Phys. Rev. E 69 (2004)).
---
paper_title: Evaluating North American Electric Grid Reliability Using the Barabasi-Albert Network Model

paper_content:

The reliability of electric transmission systems is examined using a scale-free model of network topology and failure propagation. The topologies of the North American eastern and western electric grids are analyzed to estimate their reliability based on the Barabasi–Albert network model. A commonly used power system reliability index is computed using a simple failure propagation model. The results are compared to the values of power system reliability indices previously obtained using standard power engineering methods, and they suggest that scale-free network models are usable to estimate aggregate electric grid reliability.
---
paper_title: Vulnerability assessment of bulk power grid based on complex network theory

paper_content:

The vulnerability assessment algorithm of the bulk power grid based on complex network theory is proposed in this paper. The traditional research model of the power grid based on complex network theory is a graph with no direction and no weight at present. Because this model is far from the real power system, sometimes the wrong results may be gotten. In this paper, the models of components in the power grid are constructed detailedly, furthermore the weighted and directional graph is provided. First, the connecting formings among buses (instead in the traditional methods only substations are considered), lines, transformers and generators are simulated detailedly. Second, the power flow direction in the power grid is considered, and the components' tolerance to disturbances is also considered. Based on the proposed model, the power grid's tolerance to errors and attacks is analyzed. The key components and weak areas are indicated. At last, the robustness and fault dissemination mechanism of the power grid under cascading failures are analyzed based on the proposed method. The North China power grid is used as an example to validate the proposed method.
---
paper_title: Structural vulnerability of the North American power grid

paper_content:

The magnitude of the August 2003 blackout affecting the United States has put the challenges of energy transmission and distribution into limelight. Despite all the interest and concerted effort, the complexity and interconnectivity of the electric infrastructure precluded us for a long time from understanding why certain events happened. In this paper we study the power grid from a network perspective and determine its ability to transfer power between generators and consumers when certain nodes are disrupted. We find that the power grid is robust to most perturbations, yet disturbances affecting key transmision substations greatly reduce its ability to function. We emphasize that the global properties of the underlying network must be understood as they greatly affect local behavior.
---
paper_title: Towards Decentralization: A Topological Investigation of the Medium and Low Voltage Grids

paper_content:

The traditional power grid has been designed in a hierarchical fashion, with energy pushed from the large scale production factories towards the end users. 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. Of course, end users need incentives to do so and want 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 influence the attractiveness of participating in such decentralized energy market, thus identifying the important topological parameters to work on to facilitate such open decentralized markets.
---
paper_title: The Concept of Betweenness in the Analysis of Power Grid Vulnerability

paper_content:

Vulnerability analysis in power systems is a key issue in modern society and many efforts have contributed to the analysis. Recently, complex network metrics applied to assess the topological vulnerability of networked systems have been used in power systems, such as betweenness metric, since transmission of power systems is in basis of a network structure. However, a pure topological approach fails to capture the specificity of power systems. This paper redefines, starting from the concept of complex networks, an electrical betweenness metric which considers several of specific features of power systems such as power transfer distribution and line flow limits. The electrical betweenness is compared with the conventional betweenness in IEEE-300 bus network according to the un-served energy after network is attacked. The results show that the tested network is more vulnerable when the components of the network are attacked according to their criticalities ranked by electrical betweenness.
---
paper_title: Complex Networks Theory: A New Method of Research in Power Grid

paper_content:

The rapid development of complex network theory provide a new insight into the research of interconnected complex power grid. On the basis of mapping the power grid into a network graph, the topological parameters, such as degree distribution, characteristic path length, clustering coefficient etc, are introduced to describe the network structure characters. And the statistical results of these parameters in power grids of China and America are shown and analyzed. Then, small-world model are explained and simulated to the power grid. In order to measure the performance of the power grid globally and locally, the concept efficiency and its application is also introduced. The mechanism and the model of cascading failure in power grid are successively discussed. Finally, some possible directions of future research are proposed at the end of the paper
---
paper_title: Cascade-based attacks on complex networks.

paper_content:

We live in a modern world supported by large, complex networks. Examples range from financial markets to communication and transportation systems. In many realistic situations the flow of physical quantities in the network, as characterized by the loads on nodes, is important. We show that for such networks where loads can redistribute among the nodes, intentional attacks can lead to a cascade of overload failures, which can in turn cause the entire or a substantial part of the network to collapse. This is relevant for real-world networks that possess a highly heterogeneous distribution of loads, such as the Internet and power grids. We demonstrate that the heterogeneity of these networks makes them particularly vulnerable to attacks in that a large-scale cascade may be triggered by disabling a single key node. This brings obvious concerns on the security of such systems.
---
paper_title: Modeling Cascading Failures in the North American Power Grid

paper_content:

The North American power grid is one of the most complex technological networks, and its interconnectivity allows both for long-distance power transmission and for the propagation of disturbances. We model the power grid using its actual topology and plausible assumptions about the load and overload of transmission substations. Our results indicate that the loss of a single substation can result in up to \(25\%\) loss of transmission efficiency by triggering an overload cascade in the network. The actual transmission loss depends on the overload tolerance of the network and the connectivity of the failed substation. We systematically study the damage inflicted by the loss of single nodes, and find three universal behaviors, suggesting that \(40\%\) of the transmission substations lead to cascading failures when disrupted. While the loss of a single node can inflict substantial damage, subsequent removals have only incremental effects, in agreement with the topological resilience to less than \(1\%\) node loss.
---
paper_title: Model for cascading failures in complex networks.

paper_content:

Large but rare cascades triggered by small initial shocks are present in most of the infrastructure networks. Here we present a simple model for cascading failures based on the dynamical redistribution of the flow on the network. We show that the breakdown of a single node is sufficient to collapse the efficiency of the entire system if the node is among the ones with largest load. This is particularly important for real-world networks with a highly hetereogeneous distribution of loads as the Internet and electrical power grids.
---
paper_title: Assessing European power grid reliability by means of topological measures

paper_content:

Publicat originalment a: WIT transactions on ecology and the environment, 2009, vol. 121, p. 527-537
---
paper_title: Power Grids as Complex Networks: Topology and Fragility

paper_content:

We almost certainly agree that the harnessing of electricity can be considered the most important technological advance of the twentieth century. The development that allows this control is the power grid, an intricate system composed of generators, substations and transformers connected by cable lines hundreds of kilometers long. Its presence is nowadays so intertwined with ours, and so much taken for granted, that we are only capable of sensing its absence, disguised as a cascading failure or a blackout in its extreme form. Since these extreme phenomena seem to have increased in recent years and new actors, like highly liberalized markets or environmental and social constraints, are taking a leading role, different paths other than traditional engineering ones have been explored. In the last ten years, and mainly due to increasing computational capability and accessibility to data, the conceptual frame of complex networks has allowed different approaches in order to understand the usual (and not-so-usual) outcomes of this system. This paper considers power grids as complex networks. It presents some recent results that correlate their topology with their fragility, together with major malfunctions analysis, and for the European transmission power grid in particular.
---
paper_title: A Centrality Measure for Electrical Networks

paper_content:

We derive a measure of "electrical centrality" for AC power networks, which describes the structure of the network as a function of its electrical topology rather than its physical topology. We compare our centrality measure to conventional measures of network structure using the IEEE 300-bus network. We find that when measured electrically, power networks appear to have a scale-free network structure. Thus, unlike previous studies of the structure of power grids, we find that power networks have a number of highly-connected "hub" buses. This result, and the structure of power networks in general, is likely to have important implications for the reliability and security of power networks.
---
paper_title: Attacks and Cascades in Complex Networks

paper_content:

This paper reviews two problems in the security of complex networks: cas- cades of overload failures on nodes and range-based attacks on links. Cascading failures have been reported for numerous networks and refer to the subsequent failure of other parts of the network induced by the failure of or attacks on only a few nodes. We investigate a mechanism leading to cascades of overload failures in complex networks by constructing a simple model incorporating the flow of physical quantities in the network. The second problem is motivated by the fact that most existing works on se- curity of complex networks consider attacks on nodes rather than on links. We address attacks on links. Our investigation leads to the finding that many scale-free networks are more sensitive to attacks on short-range than on long-range links. Besides its im- portance concerning network security, our result has the unexpected implication that the small-world phenomenon in these scale-free networks is mainly due to short-range links.
---
paper_title: Suppressing cascades of load in interdependent networks

paper_content:

Understanding how interdependence among systems affects cascading behaviors is increasingly important across many fields of science and engineering. Inspired by cascades of load shedding in coupled electric grids and other infrastructure, we study the Bak–Tang–Wiesenfeld sandpile model on modular random graphs and on graphs based on actual, interdependent power grids. Starting from two isolated networks, adding some connectivity between them is beneficial, for it suppresses the largest cascades in each system. Too much interconnectivity, however, becomes detrimental for two reasons. First, interconnections open pathways for neighboring networks to inflict large cascades. Second, as in real infrastructure, new interconnections increase capacity and total possible load, which fuels even larger cascades. Using a multitype branching process and simulations we show these effects and estimate the optimal level of interconnectivity that balances their trade-offs. Such equilibria could allow, for example, power grid owners to minimize the largest cascades in their grid. We also show that asymmetric capacity among interdependent networks affects the optimal connectivity that each prefers and may lead to an arms race for greater capacity. Our multitype branching process framework provides building blocks for better prediction of cascading processes on modular random graphs and on multitype networks in general.
---
paper_title: Robustness of the European power grids under intentional attack

paper_content:

The power grid defines one of the most important technological networks of our times and sustains our complex society. It has evolved for more than a century into an extremely huge and seemingly robust and well understood system. But it becomes extremely fragile as well, when unexpected, usually minimal, failures turn into unknown dynamical behaviours leading, for example, to sudden and massive blackouts. Here we explore the fragility of the European power grid under the effect of selective node removal. A mean field analysis of fragility against attacks is presented together with the observed patterns. Deviations from the theoretical conditions for network percolation (and fragmentation) under attacks are analysed and correlated with non topological reliability measures.
---
paper_title: Topological properties of high-voltage electrical transmission networks

paper_content:

The topological properties of high-voltage electrical power transmission networks in several UE countries (the Italian 380 kV, the French 400 kV and the Spanish 400 kV networks) have been studied from available data. An assessment of the vulnerability of the networks has been given by measuring the level of damage introduced by a controlled removal of links. Topological studies could be useful to make vulnerability assessment and to design specific action to reduce topological weaknesses.
---
paper_title: LOCATING CRITICAL LINES IN HIGH-VOLTAGE ELECTRICAL POWER GRIDS

paper_content:

Electrical power grids are among the infrastructures that are attracting a great deal of attention because of their intrinsic criticality. Here we analyze the topological vulnerability and improvability of the spanish 400 kV, the french 400 kV and the italian 380 kV power transmission grids. For each network we detect the most critical lines and suggest how to improve the connectivity.
---
paper_title: Using Graph Models to Analyze the Vulnerability of Electric Power Networks

paper_content:

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.
---
paper_title: Self-organized criticality: An explanation of the 1/f noise.

paper_content:

We show that dynamical systems with spatial degrees of freedom naturally evolve into a self-organized critical point. Flicker noise, or 1/f noise, can be identified with the dynamics of the critical state. This picture also yields insight into the origin of fractal objects.
---
paper_title: Generating Statistically Correct Random Topologies for Testing Smart Grid Communication and Control Networks

paper_content:

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.
---
paper_title: Complex Networks Theory: A New Method of Research in Power Grid

paper_content:

The rapid development of complex network theory provide a new insight into the research of interconnected complex power grid. On the basis of mapping the power grid into a network graph, the topological parameters, such as degree distribution, characteristic path length, clustering coefficient etc, are introduced to describe the network structure characters. And the statistical results of these parameters in power grids of China and America are shown and analyzed. Then, small-world model are explained and simulated to the power grid. In order to measure the performance of the power grid globally and locally, the concept efficiency and its application is also introduced. The mechanism and the model of cascading failure in power grid are successively discussed. Finally, some possible directions of future research are proposed at the end of the paper
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paper_title: Cascading Link Failure in the Power Grid: A Percolation-Based Analysis

paper_content:

Large-scale power blackouts caused by cascading failure are inflicting enormous socioeconomic costs. We study the problem of cascading link failures in power networks modelled by random geometric graphs from a percolation-based viewpoint. To reflect the fact that links fail according to the amount of power flow going through them, we introduce a model where links fail according to a probability which depends on the number of neighboring links. We devise a mapping which maps links in a random geometric graph to nodes in a corresponding dual covering graph. This mapping enables us to obtain the first-known analytical conditions on the existence and non-existence of a large component of operational links after degree-dependent link failures. Next, we present a simple but descriptive model for cascading link failure, and use the degree-dependent link failure results to obtain the first-known analytical conditions on the existence and non-existence of cascading link failures.
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paper_title: Classes of small-world networks

paper_content:

We study the statistical properties of a variety of diverse real-world networks. We present evidence of the occurrence of three classes of small-world networks: (a) scale-free networks, characterized by a vertex connectivity distribution that decays as a power law; (b) broad-scale networks, characterized by a connectivity distribution that has a power law regime followed by a sharp cutoff; and (c) single-scale networks, characterized by a connectivity distribution with a fast decaying tail. Moreover, we note for the classes of broad-scale and single-scale networks that there are constraints limiting the addition of new links. Our results suggest that the nature of such constraints may be the controlling factor for the emergence of different classes of networks.
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paper_title: Towards Decentralization: A Topological Investigation of the Medium and Low Voltage Grids

paper_content:

The traditional power grid has been designed in a hierarchical fashion, with energy pushed from the large scale production factories towards the end users. 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. Of course, end users need incentives to do so and want 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 influence the attractiveness of participating in such decentralized energy market, thus identifying the important topological parameters to work on to facilitate such open decentralized markets.
---
paper_title: Power Grids as Complex Networks: Topology and Fragility

paper_content:

We almost certainly agree that the harnessing of electricity can be considered the most important technological advance of the twentieth century. The development that allows this control is the power grid, an intricate system composed of generators, substations and transformers connected by cable lines hundreds of kilometers long. Its presence is nowadays so intertwined with ours, and so much taken for granted, that we are only capable of sensing its absence, disguised as a cascading failure or a blackout in its extreme form. Since these extreme phenomena seem to have increased in recent years and new actors, like highly liberalized markets or environmental and social constraints, are taking a leading role, different paths other than traditional engineering ones have been explored. In the last ten years, and mainly due to increasing computational capability and accessibility to data, the conceptual frame of complex networks has allowed different approaches in order to understand the usual (and not-so-usual) outcomes of this system. This paper considers power grids as complex networks. It presents some recent results that correlate their topology with their fragility, together with major malfunctions analysis, and for the European transmission power grid in particular.
---
paper_title: Collective dynamics of ‘small-world’ networks

paper_content:

Networks of coupled dynamical systems have been used to model biological oscillators1,2,3,4, Josephson junction arrays5,6, excitable media7, neural networks8,9,10, spatial games11, genetic control networks12 and many other self-organizing systems. Ordinarily, the connection topology is assumed to be either completely regular or completely random. But many biological, technological and social networks lie somewhere between these two extremes. Here we explore simple models of networks that can be tuned through this middle ground: regular networks ‘rewired’ to introduce increasing amounts of disorder. We find that these systems can be highly clustered, like regular lattices, yet have small characteristic path lengths, like random graphs. We call them ‘small-world’ networks, by analogy with the small-world phenomenon13,14 (popularly known as six degrees of separation15). The neural network of the worm Caenorhabditis elegans, the power grid of the western United States, and the collaboration graph of film actors are shown to be small-world networks. Models of dynamical systems with small-world coupling display enhanced signal-propagation speed, computational power, and synchronizability. In particular, infectious diseases spread more easily in small-world networks than in regular lattices.
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paper_title: Topological analysis of the power grid and mitigation strategies against cascading failures

paper_content:

This paper presents a complex systems overview of a power grid network. In recent years, concerns about the robustness of the power grid have grown because of several cascading outages in different parts of the world. In this paper, cascading effect has been simulated on three different networks, the IEEE 300 bus test system, the IEEE 118 bus test system, and the WSCC 179 bus equivalent model. Power Degradation has been discussed as a measure to estimate the damage to the network, in terms of load loss and node loss. A network generator has been developed to generate graphs with characteristics similar to the IEEE standard networks and the generated graphs are then compared with the standard networks to show the effect of topology in determining the robustness of a power grid. Three mitigation strategies, Homogeneous Load Reduction, Targeted Range-Based Load Reduction, and Use of Distributed Renewable Sources in combination with Islanding, have been suggested. The Homogeneous Load Reduction is the simplest to implement but the Targeted Range-Based Load Reduction is the most effective strategy.
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paper_title: Mean-field theory for scale-free random networks

paper_content:

Random networks with complex topology are common in Nature, describing systems as diverse as the world wide web or social and business networks. Recently, it has been demonstrated that most large networks for which topological information is available display scale-free features. Here we study the scaling properties of the recently introduced scale-free model, that can account for the observed power-law distribution of the connectivities. We develop a mean-field method to predict the growth dynamics of the individual vertices, and use this to calculate analytically the connectivity distribution and the scaling exponents. The mean-field method can be used to address the properties of two variants of the scale-free model, that do not display power-law scaling.
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paper_title: Generating Statistically Correct Random Topologies for Testing Smart Grid Communication and Control Networks

paper_content:

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.
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paper_title: Power Grid Network Evolutions for Local Energy Trading

paper_content:

The shift towards an energy Grid dominated by prosumers (consumers and producers of energy) will inevitably have repercussions on the distribution infrastructure. Today it is a hierarchical one designed to deliver energy from large scale facilities to end-users. Tomorrow it will be a capillary infrastructure at the Medium and Low Voltage levels that will support local energy trading among prosumers. In [74], we analyzed the Dutch Power Grid and made an initial analysis of the economic impact topological properties have on decentralized energy trading. In this paper, we go one step further and investigate how dierent networks topologies and growth models facilitate the emergence of a decentralized market. In particular, we show how the connectivity plays an important role in improving the properties of reliability and pathcost reduction. From the economic point of view, we estimate how the topological evolutions facilitate local electricity distribution, taking into account the main cost ingredient required for increasing network connectivity, i.e., the price of cabling.
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paper_title: Do topological models provide good information about electricity infrastructure vulnerability

paper_content:

In order to identify the extent to which results from topological graph models are useful for modeling vulnerability in electricity infrastructure, we measure the susceptibility of power networks to random failures and directed attacks using three measures of vulnerability: characteristic path lengths, connectivity loss, and blackout sizes. The first two are purely topological metrics. The blackout size calculation results from a model of cascading failure in power networks. Testing the response of 40 areas within the Eastern U.S. power grid and a standard IEEE test case to a variety of attack/failure vectors indicates that directed attacks result in larger failures using all three vulnerability measures, but the attack-vectors that appear to cause the most damage depend on the measure chosen. While the topological metrics and the power grid model show some similar trends, the vulnerability metrics for individual simulations show only a mild correlation. We conclude that evaluating vulnerability in power ...
---
paper_title: Complex Networks Theory: A New Method of Research in Power Grid

paper_content:

The rapid development of complex network theory provide a new insight into the research of interconnected complex power grid. On the basis of mapping the power grid into a network graph, the topological parameters, such as degree distribution, characteristic path length, clustering coefficient etc, are introduced to describe the network structure characters. And the statistical results of these parameters in power grids of China and America are shown and analyzed. Then, small-world model are explained and simulated to the power grid. In order to measure the performance of the power grid globally and locally, the concept efficiency and its application is also introduced. The mechanism and the model of cascading failure in power grid are successively discussed. Finally, some possible directions of future research are proposed at the end of the paper
---
paper_title: Evaluating North American Electric Grid Reliability Using the Barabasi-Albert Network Model

paper_content:

The reliability of electric transmission systems is examined using a scale-free model of network topology and failure propagation. The topologies of the North American eastern and western electric grids are analyzed to estimate their reliability based on the Barabasi–Albert network model. A commonly used power system reliability index is computed using a simple failure propagation model. The results are compared to the values of power system reliability indices previously obtained using standard power engineering methods, and they suggest that scale-free network models are usable to estimate aggregate electric grid reliability.
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paper_title: Vulnerability assessment of bulk power grid based on complex network theory

paper_content:

The vulnerability assessment algorithm of the bulk power grid based on complex network theory is proposed in this paper. The traditional research model of the power grid based on complex network theory is a graph with no direction and no weight at present. Because this model is far from the real power system, sometimes the wrong results may be gotten. In this paper, the models of components in the power grid are constructed detailedly, furthermore the weighted and directional graph is provided. First, the connecting formings among buses (instead in the traditional methods only substations are considered), lines, transformers and generators are simulated detailedly. Second, the power flow direction in the power grid is considered, and the components' tolerance to disturbances is also considered. Based on the proposed model, the power grid's tolerance to errors and attacks is analyzed. The key components and weak areas are indicated. At last, the robustness and fault dissemination mechanism of the power grid under cascading failures are analyzed based on the proposed method. The North China power grid is used as an example to validate the proposed method.
---
paper_title: Small Worlds: The Dynamics of Networks between Order and Randomness.

paper_content:

Transport brackets are affixed to the container and include provisions for achieving transportability with either a crane or a fork lift. Variable storage capacity is provided by an extension bin which is detachably affixed as part of the container. The top and bottom of the container are compatibly configured to provide an interlock between a plurality of stacked containers and a dolly is configured for use with these interlocking provisions. The container is configured to achieve stacking interchangeability when any vertically peripheral sides are aligned and brackets are disposed around the vertical periphery of the container to achieve fork lift accessibility from any direction perpendicular to the vertically peripheral sides.
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paper_title: Robustness of the European power grids under intentional attack

paper_content:

The power grid defines one of the most important technological networks of our times and sustains our complex society. It has evolved for more than a century into an extremely huge and seemingly robust and well understood system. But it becomes extremely fragile as well, when unexpected, usually minimal, failures turn into unknown dynamical behaviours leading, for example, to sudden and massive blackouts. Here we explore the fragility of the European power grid under the effect of selective node removal. A mean field analysis of fragility against attacks is presented together with the observed patterns. Deviations from the theoretical conditions for network percolation (and fragmentation) under attacks are analysed and correlated with non topological reliability measures.
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paper_title: Structural vulnerability of the North American power grid

paper_content:

The magnitude of the August 2003 blackout affecting the United States has put the challenges of energy transmission and distribution into limelight. Despite all the interest and concerted effort, the complexity and interconnectivity of the electric infrastructure precluded us for a long time from understanding why certain events happened. In this paper we study the power grid from a network perspective and determine its ability to transfer power between generators and consumers when certain nodes are disrupted. We find that the power grid is robust to most perturbations, yet disturbances affecting key transmision substations greatly reduce its ability to function. We emphasize that the global properties of the underlying network must be understood as they greatly affect local behavior.
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paper_title: Electrical centrality measures for electric power grid vulnerability analysis

paper_content:

Centrality measures are used in network science to rank the relative importance of nodes and edges of a graph. Here we define new measures of centrality for power grid structure that are based on its functionality. We show that the relative importance analysis based on centrality in graph theory can be generalized to power grid network with its electrical parameters taken into account. In the paper we experiment with the proposed electrical centrality measures on the NYISO-2935 system and the IEEE 300-bus system. We analyze the centrality distribution in order to identify important nodes or branches in the system which are of essential importance in terms of system vulnerability. We also present and discuss a number of interesting discoveries regarding the importance rank of power grid nodes and branches.
---
paper_title: Towards Decentralization: A Topological Investigation of the Medium and Low Voltage Grids

paper_content:

The traditional power grid has been designed in a hierarchical fashion, with energy pushed from the large scale production factories towards the end users. 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. Of course, end users need incentives to do so and want 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 influence the attractiveness of participating in such decentralized energy market, thus identifying the important topological parameters to work on to facilitate such open decentralized markets.
---
paper_title: Power Grids as Complex Networks: Topology and Fragility

paper_content:

We almost certainly agree that the harnessing of electricity can be considered the most important technological advance of the twentieth century. The development that allows this control is the power grid, an intricate system composed of generators, substations and transformers connected by cable lines hundreds of kilometers long. Its presence is nowadays so intertwined with ours, and so much taken for granted, that we are only capable of sensing its absence, disguised as a cascading failure or a blackout in its extreme form. Since these extreme phenomena seem to have increased in recent years and new actors, like highly liberalized markets or environmental and social constraints, are taking a leading role, different paths other than traditional engineering ones have been explored. In the last ten years, and mainly due to increasing computational capability and accessibility to data, the conceptual frame of complex networks has allowed different approaches in order to understand the usual (and not-so-usual) outcomes of this system. This paper considers power grids as complex networks. It presents some recent results that correlate their topology with their fragility, together with major malfunctions analysis, and for the European transmission power grid in particular.
---
paper_title: Topological properties of high-voltage electrical transmission networks

paper_content:

The topological properties of high-voltage electrical power transmission networks in several UE countries (the Italian 380 kV, the French 400 kV and the Spanish 400 kV networks) have been studied from available data. An assessment of the vulnerability of the networks has been given by measuring the level of damage introduced by a controlled removal of links. Topological studies could be useful to make vulnerability assessment and to design specific action to reduce topological weaknesses.
---
paper_title: Collective dynamics of ‘small-world’ networks

paper_content:

Networks of coupled dynamical systems have been used to model biological oscillators1,2,3,4, Josephson junction arrays5,6, excitable media7, neural networks8,9,10, spatial games11, genetic control networks12 and many other self-organizing systems. Ordinarily, the connection topology is assumed to be either completely regular or completely random. But many biological, technological and social networks lie somewhere between these two extremes. Here we explore simple models of networks that can be tuned through this middle ground: regular networks ‘rewired’ to introduce increasing amounts of disorder. We find that these systems can be highly clustered, like regular lattices, yet have small characteristic path lengths, like random graphs. We call them ‘small-world’ networks, by analogy with the small-world phenomenon13,14 (popularly known as six degrees of separation15). The neural network of the worm Caenorhabditis elegans, the power grid of the western United States, and the collaboration graph of film actors are shown to be small-world networks. Models of dynamical systems with small-world coupling display enhanced signal-propagation speed, computational power, and synchronizability. In particular, infectious diseases spread more easily in small-world networks than in regular lattices.
---
paper_title: Analysis of structural vulnerabilities in power transmission grids

paper_content:

Abstract Power transmission grids play a crucial role as a critical infrastructure by assuring the proper functioning of power systems. In particular, they secure the loads supplied by power generation plants and help avoid blackouts that can have a disastrous impact on society. The power grid structure (number of nodes and lines, their connections, and their physical properties and operational constraints) is one of the key issues (along with generation availability) to assure power system security; consequently, it deserves special attention. A promising approach for the structural analysis of transmission grids with respect to their vulnerabilities is to use metrics and approaches derived from complex network (CN) theory that are shared with other infrastructures such as the World-Wide Web, telecommunication networks, and oil and gas pipelines. These approaches, based on metrics such as global efficiency, degree and betweenness, are purely topological because they study structural vulnerabilities based on the graphical representation of a network as a set of vertices connected by a set of edges. Unfortunately, these approaches fail to capture the physical properties and operational constraints of power systems and, therefore, cannot provide meaningful analyses. This paper proposes an extended topological approach that includes the definitions of traditional topological metrics (e.g., degrees of nodes and global efficiency) as well as the physical/operational behavior of power grids in terms of real power-flow allocation over lines and line flow limits. This approach provides two new metrics, entropic degree and net-ability, that can be used to assess structural vulnerabilities in power systems. The new metrics are applied to test systems as well as real power grids to demonstrate their performance and to contrast them with traditional, purely topological metrics.
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paper_title: Using Graph Models to Analyze the Vulnerability of Electric Power Networks

paper_content:

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.
---

Title: The Power Grid as a Complex Network: a Survey
Section 1: Introduction
Description 1: Outline the purpose and scope of the survey, provide background on Complex Network Analysis (CNA), and explain its relevance to the study of power grids.

Section 2: Background and Survey Methodology
Description 2: Define common terms and concepts used in CNA, provide essential graph theory definitions, and describe the methodology used to analyze and compare studies in the survey.

Section 3: The Power Grid as a Complex Network
Description 3: Discuss the application of complex network analysis to power grids and describe the specific studies reviewed in this survey.

Section 4: Basic Power Grid Characteristics
Description 4: Identify and compare basic non-technical characteristics of the power grids analyzed in various studies, such as geography, type of Grid (high, medium, low voltage), and sample size.

Section 5: Statistical Global Graph Properties
Description 5: Summarize the main statistical properties of the power grids under analysis, including order, size, average degree, node degree distribution, and betweenness distribution.

Section 6: The Small-World Property
Description 6: Examine whether the power grids analyzed exhibit the small-world property and compare findings from various studies.

Section 7: Node Degree Distribution
Description 7: Summarize findings on the degree of nodes within the power grids and fit these findings into distribution models, such as exponential or power-law distributions.

Section 8: Betweenness Distribution
Description 8: Describe the distribution of betweenness centrality in the power grids and its implications on the infrastructure's robustness.

Section 9: Resilience Analysis
Description 9: Compare the resilience of different power grids to random failures and targeted attacks on nodes and edges, and describe the metrics used to assess this resilience.

Section 10: Studies with Infrastructure Improvement Analysis
Description 10: Review studies that propose and assess strategies for improving the reliability and robustness of the power grids.

Section 11: Further Studies
Description 11: Highlight additional analyses and investigations from the surveyed studies that provide unique insights into the power grid as a complex network.

Section 12: Discussion and Conclusion
Description 12: Summarize the overall findings of the survey, discuss common patterns and notable differences, and provide recommendations for future research directions.