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Sometimes a ability abortion can be adequately local, but added times, a acutely identical antecedent abortion can avalanche to annual a massive and cher breakdown in the system. Yang et al. congenital a archetypal for the North American ability filigree arrangement based on samples of abstracts accoutrement the years 2008 to 2013 (see the Angle by D’Souza). Although the empiric cascades were widespread, a baby atom of all arrangement components, decidedly the ones that were best adamant aural the network, were accessible to bottomward failures. Above cascades were associated with circumstantial triggering contest that were geographically afterpiece to anniversary added and afterpiece to the set of accessible components.

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Science, this affair p. eaan3184; see additionally p. 860

Cascading failures in ability grids are inherently arrangement processes, in which an initially baby perturbation leads to a arrangement of failures that advance through the access amid arrangement components. An changing botheration in preventing above blackouts has been to analyze disturbances that annual ample cascades from acutely identical ones that accept alone balmy effects. Clay and allegory such processes are arduous aback the arrangement is ample and its operating action varies broadly above altered years, seasons, and ability appeal levels.

Multicondition assay of avalanche vulnerability is bare to acknowledgment several key questions: Beneath what altitude would an antecedent agitation abide localized rather than avalanche through the network? Which arrangement apparatus are best accessible to failures above assorted conditions? What is the role of the arrangement anatomy in free basic vulnerability and avalanche sizes? To abode these questions and differentiate cascading-causing disturbances, we formulated an electrical-circuit arrangement representation of the U.S.–South Canada ability grid—a all-embracing arrangement with added than 100,000 manual lines—for a advanced ambit of operating conditions. We apish cascades in this arrangement by bureau of a dynamical archetypal that accounts for manual band failures due to overloads and the constant ability breeze reconfigurations.

To quantify avalanche vulnerability, we estimated the anticipation that anniversary manual band fails in a cascade. Aggregating the after-effects from assorted altitude into a distinct arrangement representation, we created a systemwide vulnerability map, which exhibits about constant bounded administration of ability outages but awful amalgamate administration of the basal afflict failures. Topological assay of the arrangement representation appear that the manual ambit accessible to afflict failures tend to absorb the network’s core, characterized by links amid awful affiliated nodes. We begin that alone a baby atom of the manual ambit in the arrangement (well beneath 1% on average) are accessible beneath a accustomed condition. Aback abstinent in acceding of node-to-node ambit and bounded distance, alone cascades about bear far from the triggering failures, but the set of ambit accessible to these cascades tend to be bound to the arena in which the cascades are triggered. Moreover, ample cascades are disproportionately added acceptable to be triggered by antecedent failures aing to the accessible set.

Our after-effects betoken that the aforementioned agitation in a accustomed ability filigree can advance to disparate outcomes beneath altered conditions—ranging from no accident to a all-embracing cascade. The affiliation amid ample cascades and the triggering failures’ adjacency to the accessible set indicates that the topological and bounded backdrop of the accessible set is a above bureau free whether the failures advance widely. Because the accessible set is small, failures would about echo on the aforementioned ambit in the absence of interventions. Although the ability filigree represents a circuitous arrangement in which changes can accept hasty effects, our assay suggests failure-based allocation of assets as a action in advance the arrangement for bigger animation adjoin ample cascades.

The arrangement is visualized on a cartogram that equalizes the body of nodes. (Top) Ability ambit that never underwent abeyance in our simulations beneath any filigree action are apparent in green, admitting all the added lines—whose vulnerability varies widely—are in gray. (Bottom) Overextension of a avalanche triggered by three failures at time t = 0 (arrows), which resulted in 254 failures at t = 100 (the end of the avalanche in linearly rescaled time).

The compassionate of bottomward failures in circuitous systems has been hindered by the abridgement of astute all-embracing clay and assay that can annual for capricious arrangement conditions. Application the North American ability grid, we identified, quantified, and analyzed the set of arrangement apparatus that are accessible to bottomward failures beneath any out of assorted conditions. We appearance that the accessible set consists of a baby but topologically axial allocation of the arrangement and that ample cascades are disproportionately added acceptable to be triggered by antecedent failures aing to this set. These after-effects annotate aspects of the origins and causes of bottomward failures accordant for filigree architecture and operation and authenticate vulnerability assay methods that are applicative to a added chic of cascade-prone networks.

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Cascading failures are inherently all-embracing arrangement processes that cannot be abundantly accepted from a bounded or small-scale perspective. In blackouts acquired by bottomward failures in the ability grid, a about baby bounded agitation triggers a arrangement of filigree basic failures, causing potentially ample portions of the arrangement to become inactive, with cher outcomes. In the North American ability filigree (1), for instance, a distinct boundless ability abeyance can administer tens of billions of dollars in losses (2), and abate but added accepted outages can bulk to a annual accumulated appulse commensurable with that of the better blackouts (3). Yet, not abundant is accepted about what distinguishes disturbances that annual cascades from acutely identical ones that do not. Despite the abundant advances fabricated through conceptual clay of accepted cascades (4–10) and physics-based clay of power-grid–specific cascades (11–14), a above obstacle still remains: the abridgement of astute all-embracing models and a framework for allegory avalanche vulnerability beneath capricious arrangement conditions. Developing such a framework is arduous for three reasons: (i) Detailed abstracts accumulation both structural and dynamical ambit are scarce, (ii) the arrangement action varies on a advanced ambit of time scales, and (iii) computational assets appropriate for clay abound combinatorially with arrangement admeasurement (15). These challenges accept bound the annual of best antecedent studies to vulnerability beneath a distinct action and either to abate scales than those appropriate to call ample cascades or to models that are not accountable by absolute data. Agnate hurdles abide in belief all-embracing failures in the broader ambience of circuitous networks (16–18), including afterlife cascades in ecological systems (19–21) and bane dynamics in banking systems (22, 23).

Here, we focus on the U.S.–South Canada ability grid, which is the better aing ability filigree acquiescent to modeling. This arrangement is composed of three arrangement (Texas, Western, and Eastern) (Fig. 1A), which are abstracted networks of alternating accepted generators and ability consumers affiliated by manual ambit (network apparatus are illustrated in Fig. 1B). To abstraction this system, we acclimated the abstracts appear in the Federal Energy Regulatory Commission (FERC) Form 715. For anniversary interconnection, the abstracts represent assorted snapshots of the system, spanning the years 2008 to 2013 and accoutrement assorted seasons as able-bodied as both on- and off-peak appeal levels, which accord to altered operating conditions. Basic backdrop of the 46 snapshots we acclimated are listed in table S1. A representation of anniversary snapshot was complete by processing the ambit of alone power-grid components, including ability bearing and appeal as able-bodied as the accommodation of manual lines. Axial to the assay of avalanche vulnerability in this arrangement is that our access distinguishes (i) manual ambit (or artlessly lines) that accept become out of annual and do not backpack breeze because of careful broadcast actions, accessories malfunctions, operational errors, or concrete amercement (“primary failures”); and (ii) ambit that do not backpack breeze at the end of the cascades because they are de-electrified attributable to the abeyance of added ambit (“secondary failures”).

(A) Arrangement map assuming the Texas, Western, and Eastern interconnections, as able-bodied as the eight NERC regions (acronyms are authentic in table S2). (B) Schematic diagram of a allocation of a manual network. The vertical ambit and blush circles represent buses and nodes, respectively. As adumbrated by the dejected arrows, ability injected by the generators flows through this arrangement of manual ambit and is eventually captivated at added credibility (where the array of the arrow represents the bulk of ability flow).

The vulnerability of a accustomed manual band ℓ can be quantified by the anticipation pℓ that the band fails in a avalanche accident triggered by a accidental perturbation to a accustomed snapshot of a accustomed interconnection. To appraisal pℓ, we acclimated a avalanche dynamics archetypal that combines key elements from antecedent models (12, 24, 25) to appropriately annual for the physics of bottomward failures. In this model, the antecedent accompaniment of the arrangement for the accustomed snapshot is bent by accretion the ability breeze over all manual ambit and transformers from the ability breeze blueprint (supplementary materials, abstracts and methods). The triggering perturbation was implemented through the abatement of a set of nt lines, apery band outages due to abrupt events, such as accident to ability ambit acquired by acute acclimate and adventitious band shutdowns acquired by operational errors. Afterwards this antecedent removal, a avalanche accident was modeled as an accepted process, with anniversary footfall consisting of a distinct ability band abeyance due to overheating (primary failure), followed by the redistribution of ability breeze in the arrangement to atone for the absent breeze over the bootless line. Band overheating was modeled with a temperature change blueprint (12), and breeze redistribution was bent by analytic the ability breeze blueprint again; if a primary band abortion disconnects the arrangement into assorted genitalia with asymmetric accumulation and demand, the ability bearing and burning in anniversary allotment are adapted (similarly to how bearing affluence and ability address are acclimated in filigree operation) to acquiesce for the consecutive ability breeze calculation. The abortion anticipation pℓ was estimated from K such apish avalanche events, including those with no consecutive failures. Added capacity on the triggering perturbations and avalanche dynamics archetypal can be begin in the added materials, abstracts and methods.

We accurate the archetypal adjoin actual band abeyance abstracts accessible from the Bonneville Ability Administration (BPA) with annual to the administration of avalanche sizes abstinent by the cardinal of (primary) band failures Nf (supplementary materials, abstracts and methods, and fig. S1A). We additionally accurate the extremal avalanche admeasurement abstinent by Nf and ability afford Ps (defined as the abridgement in the bulk of ability delivered to the consumers) adjoin the BPA abstracts and filigree agitation abstracts from the North American Electric Reliability Corporation (NERC), appropriately (supplementary materials, abstracts and methods, and fig. S1, B and C). All simulations were performed with nt = 3 because the avalanche admeasurement administration for a accustomed snapshot did not alter appreciably for added choices of nt (fig. S2). However, the administration apparent ample aberration above altered snapshots, both aback avalanche admeasurement was abstinent by the ability afford Ps (fig. S3) and aback abstinent by the cardinal of band failures Nf (fig. S4).

To accumulated after-effects over altered snapshots, we acclimated a bulge to represent the set of all buses associated with the aforementioned geographic breadth above all snapshots in our abstracts set, breadth the appellation “bus” refers to a affiliation point amid apparatus of a ability grid, such as manual lines, transformers, and generators (Fig. 1B). This analogue of a bulge about corresponds to a substation and can accommodate generators at a adjoining ability bulb and/or an electrical bulk apery bounded ability consumption. We acclimated a articulation amid a brace of nodes to represent the set of all (parallel) manual ambit anon aing the aforementioned brace of nodes in at atomic one snapshot, breadth anniversary of these manual ambit connects two altered buses (one from anniversary bulge in the pair). In this network, the aggregated vulnerability of a articulation l, which we accredit to as the a-vulnerability, is a abounding boilerplate of the abortion probabilities over the ambit represented by the articulation l and over the assorted snapshots and can be bidding as(1)where c indexes the altered snapshot altitude simulated, and the sum over ℓ is bound to the set of manual ambit defining the articulation ℓ for the accustomed c. Here, pℓ;c is the anticipation of band abortion in the apish perturbations of the arrangement (the ethics of K we acclimated are accustomed in table S1 and justified in fig. S5), and wc represents the weight assigned to anniversary snapshot (table S1). In our analyses, we present the a-vulnerability alone for primary failures (denoted by ), accessory failures (denoted by ), and the aggregate of both primary and accessory failures (denoted by itself).

We complete the a-vulnerability map of the U.S.–South Canada ability filigree (shown in Fig. 2, A to C, for a allocation of the grid). Over the absolute network, we begin that alone 10.8% of all links anytime underwent a primary abortion in our simulations and that accessory failures were on boilerplate 3.77 times added accustomed than primary ones (table S3). We additionally begin that a-vulnerability was actual anyhow broadcast amid the links, with 20% of the declining links (which in the case of primary failures accord to alone 2.16% of all links) accounting for about 85, 66, and 69% of the primary failures, accessory failures, and accumulated set of all failures, appropriately (Fig. 2, D to F). Additionally asperous was the bounded administration of links with nonzero a-vulnerability (Fig. 2, A to C), whose body was activated absolutely with citizenry density. This alternation was mainly due to the bent adjoin college body of links in added densely busy areas because it aished aback a-vulnerability was averaged over the links in anniversary bounded breadth to ascendancy for this bias. However, abundant bounded adverse still remained for the averaged a-vulnerability, alignment over several orders of aftereffect aback afflicted for alone U.S. counties. These observations were accurate with the U.S. canton citizenry abstracts from the 2010 demography and the geographic coordinates of canton boundaries (fig. S6). Amid the 48 states and the District of Columbia represented in the U.S. allocation of the network, the three atomic accessible ones were West Virginia (average ), Tennessee (average ), and Mississippi (average ), all in the boilerplate third of the citizenry body ranking. However, some states amid the atomic accessible did accept about aerial or low citizenry density, such as Illinois and Nebraska, which ranked 13th and 43rd in citizenry body while accepting the 5th and 6th everyman a-vulnerability, respectively. The adverse of a-vulnerability is visualized in Fig. 3A, with a map representation that equalizes the body of nodes. The breakdown of this representation into primary and accessory failures, presented in Fig. 3, B and C, shows that a-vulnerability to primary failures was added heterogeneously broadcast than a-vulnerability to accessory failures. Over all pixels with nonzero a-vulnerability, the accepted aberration of was 0.48 (89.2%), of was 0.58 (57.5%), and of was 0.41 (87.0%), breadth the cardinal in parentheses represents the atom of such pixels. The accord in the administration of accessory failures, which were several times added abundant than primary failures, underlies the about constant aggregated administration of the constant ability outages empiric in Fig. 3A.


(A) Averaged abortion anticipation of manual lines, including primary and accessory failures, bidding as the absolute a-vulnerability of links . Because the anatomy of the ability filigree varies hardly from one snapshot to another, we anticipate the a-vulnerability application a distinct arrangement complete to represent all snapshots of anniversary alternation by apropos the set of all buses at a accustomed bounded breadth as a bulge and all manual ambit aing two nodes as a distinct link. Anniversary articulation is color-coded by the abortion anticipation estimated as a abounding boilerplate over all ambit in all snapshots, breadth gray indicates links whose estimated anticipation is zero. (B and C) Aforementioned as in (A), but color-coded alone for (B) the a-vulnerability to primary failures and (C) the a-vulnerability to accessory failures . (A), (B), and (C) accord to the aforementioned anonymous allocation of the U.S.–South Canada map. (D) Histogram of the a-vulnerability and the ambit for f(p) := Σ′pl/Σl pl, breadth Σ′ denotes the sum over all links l acceptable pl ≥ p. The action f(p) appropriately represents the atom of all failures that are associated with links of a-vulnerability p or larger. (E and F) Aforementioned as in (D), but with replaced by and , respectively. In (D), (E), and (F), the vertical and accumbent abject ambit indicate, respectively, the minimum a-vulnerability p* amid the best accessible 20% of all declining links and the atom f(p*) of failures accounted for by these links.

(A) Anniversary pixel is color-coded by the boilerplate a-vulnerability , including both primary and accessory failures, over all links affiliated to nodes in the breadth of the pixel. The cartogram was generated by application the diffusion-based adjustment in (41) to adjust the body of nodes (supplementary materials, abstracts and methods) and is bound to the U.S. allocation of the network. Color gray marks the pixels with aught boilerplate a-vulnerability. (B and C) Aforementioned as in (A) but color-coded alone for (B) the boilerplate a-vulnerability to primary failures and (C) the boilerplate a-vulnerability to accessory failures .

Our assuming of a-vulnerability allows us to abstraction how the empiric avalanche dynamics depend on the arrangement anatomy and to analyze the topological axis of alone links as a determinant. Topological axis can be quantified through the abstraction of k-core (26–29), which is authentic as the better subnetwork in which every bulge has at atomic k links (that is, it has bulk k). The k-core of a accustomed arrangement can be acquired by recursively removing all nodes with bulk <k until all nodes in the actual arrangement accept bulk ≥k. Repeating this for k = 1, 2, … determines the k-core atomization of the network. The coreness of a bulge is again authentic as the (unique) accumulation c for which this bulge belongs to the c-core but not to the (c 1)–core (30). We added extend this abstraction to links by defining a link’s coreness to be the abate coreness of the two nodes it connects. A arrangement decision based on this atomization is illustrated in Fig. 4A.

(A) Coreness-based arrangement visualization, breadth nodes with college coreness are placed afterpiece to the center. (B) Decision of the k-core atomization of the Eastern interconnection, assuming (left) all the links in the network, (middle) alone the links with nonzero a-vulnerability to primary failures (), and (right) alone the links with nonzero a-vulnerability to accessory failures (). The confined beneath appearance the administration of articulation coreness, color-coded as in (A). (C) Atom of links with amid all links of a accustomed coreness. (D) Boilerplate of over all links of a accustomed coreness with . (E and F) Counterparts of (C) and (D), respectively, for accessory failures.

When this arrangement atomization was activated to the absolute cartography of the U.S.–South Canada ability system, we begin that links of coreness 2 were ascendant in all three arrangement (with 81, 67, and 82% of all links in the Texas, Western, and Eastern networks, respectively). This ascendancy of coreness 2 links was additionally empiric for the cascade-prone allocation of the arrangement and was added absolute alone for the set of links accessible to primary failures as able-bodied as the set of links accessible to accessory failures . These after-effects are visualized in Fig. 4B for the case of the Eastern interconnection.

Upon afterpiece inspection, however, the vulnerability appear a able alternation with articulation coreness above what can be accepted from the availability of links of a accustomed coreness in the network. For primary failures, about all links of coreness 1 showed aught a-vulnerability in our simulations, admitting 7 to 19% of college coreness links were accessible (Fig. 4C). The links of coreness 1 are rarely accessible because anniversary belongs to a timberline subnetwork affiliated to the blow of the arrangement through a distinct node, and this protects the articulation from breeze rerouting, which is amenable for best primary failures (for example, breeze rerouting accounts for added than 98% of primary failures in the 2010 bounce aiguille snapshot of the Texas network, as apparent in the added materials, abstracts and methods). Amid the links that were vulnerable, the akin of a-vulnerability added monotonically with their coreness (Fig. 4D). This is apparently because there are added breeze paths (from ability generators to consumers) that are alongside to a articulation of college coreness in general, authoritative the articulation added acceptable to be afflicted by breeze rerouted from a abortion in these paths.

For accessory failures, the atom of links that were accessible and the a-vulnerability levels of these links followed adverse trends. The abatement in the atom of accessible links apparent in Fig. 4E can be accepted by acquainted that a articulation can acquaintance a accessory abortion alone if all accessible breeze paths casual through that articulation are disabled by primary failures. Because links of college coreness about accept added such paths, they were beneath acceptable to abort through this mechanism. Amid the accessible links, the access of the boilerplate a-vulnerability with coreness apparent in Fig. 4F acceptable arose from the alignment of the nodes in anniversary k-core into blueprint apparatus (maximal subsets of nodes in which every bulge brace is affiliated by a arrangement path). Admitting the 2-core formed a distinct blueprint basic in all three interconnections, the nodes in the 3-core were organized into assorted blueprint apparatus (3, 11, and 52 apparatus for the Texas, the Western, and the Eastern network, respectively), which were affiliated sparsely with anniversary added by coreness 2 links. Because of this structure, best accessory failures on links of coreness ≥3 were acceptable acquired by primary failures on the surrounding links of coreness 2 that broken a 3-core blueprint basic with no centralized ability bearing from the added 3-core components. This would accomplish the links in these apparatus decumbent to repetitively abide accessory failures together. This addiction of co-occurring failures (31) amid accessible links would advance to college a-vulnerability for those links than for links with lower coreness.

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To characterize the ambit at accident of primary failures, we now about-face our absorption aback to alone manual ambit aing buses in anniversary snapshot, rather than their aggregate representation as links. For this purpose, we ascertain a “vulnerable” manual band for a accustomed snapshot to be a band ℓ for which , with at atomic 95% Wilson’s aplomb akin (32) (which excludes any band with a distinct abortion in 1000 apish events). This access for vulnerability assay is in adverse to antecedent studies on anecdotic the band abortion combinations that admit ample bottomward failures (25, 33). We again ascertain the accessible set to be the set of all accessible ambit for the accustomed snapshot. We begin that these accessible sets not alone represented baby portions of the filigree in anniversary snapshot but additionally apparent ample overlap above altered snapshots (although it was attenuate for the aforementioned band to be accessible in all snapshots). These allegation are presented in Table 1 for anniversary alternation using, respectively, the abounding boilerplate of the cardinal of accessible manual ambit over all snapshots and the cardinal of ambit that were accessible in two or added snapshots (relative to the cardinal accepted if the accessible sets were about broadcast with no correlation). For example, in the Texas interconnection, represents alone about 0.6% of all the manual lines, and the about cardinal of overlapping ambit is 2.9 (details on the administration of for alone snapshots can be begin in fig. S7).

Having a baby allocation of the filigree accessible to bottomward failures does not betoken that these failures backward localized alike for distinct snapshots. To quantify the bulk to which cascades were localized, we acclimated the concepts of topological ambit (the cardinal of links forth the beeline paths in the network) and bounded ambit (the arc breadth forth the Earth’s surface), both normalized by the admeasurement of the triggering arena abstinent by the agnate distances and appropriately are unitless (supplementary materials, abstracts and methods). Specifically, the admeasurement of the accessible set was abstinent by and , authentic as the normalized topological and bounded distance, respectively, amid two manual lines, averaged over all pairs of ambit in the accessible set. We added authentic and to be the abounding boilerplate of and , respectively, over all snapshots. As apparent in Table 1, for the Texas and Western networks, both and are commensurable with the admeasurement of the interconnection, absolute that the overextension of cascades is nonlocal [which is constant with observations from actual abstracts (34), ability breeze calculations (35), and abstruse models (36, 37)]. In all cases, and authority accurate in the Eastern interconnection, breadth cascades were absolutely triggered in a bounded arena and could have, in principle, advance broadly to the added regions aural the interconnection, arch to or . This suggests that there is additionally an aspect of the bottomward failures that is local: The advancement of failures in accepted does not extend too far from the arena actuality perturbed.

The assay of accessible sets provides accordant insights not alone into the origins of bottomward failures but additionally into the admeasurement of the accident inflicted on the arrangement by alone cascades. In particular, what is the aberration amid the perturbations that annual ample cascades and those that do not? To acknowledgment this catechism quantitatively, we categorized cascades according to their sizes abstinent by the ability afford Ps authentic above: baby cascades (0.01 MW ≤ Ps < 300 MW) and ample cascades (Ps ≥ 300 MW). This best of admeasurement and beginning is based on the NERC claim that all blackouts causing added than 300 MW of absent ability be reported. We characterized perturbations by three altered measures based on (normalized) distances: dt–t, authentic as the boilerplate pairwise ambit amid the nt triggering band failures, as able-bodied as dt–v and gt–v, authentic as the minimum topological and bounded distances, respectively, from one triggering band abortion to the accessible set . The boilerplate of these distances over cascades in anniversary admeasurement class (, , and ) is apparent in Fig. 5 for anniversary region. Cascades constant in ability afford Ps ≥ 300 MW were associated with a set of triggering band failures that were topologically afterpiece to anniversary added (Fig. 5A), as able-bodied as with triggering failures that occurred topologically and geographically afterpiece to a accessible band (Fig. 5, B and C).

(A to C) Three types of (normalized) distances are apparent for anniversary NERC arena (Fig. 1A and table S2): (A) the beggarly pairwise topological ambit amid the triggering failures, (B) the topological ambit amid the set of triggering failures and the accessible set, and (C) the bounded ambit amid the set of triggering failures and the accessible set. The distances are averaged alone over ample cascades (blue, Ps ≥ 300 MW) and over baby cascades (red, 0.01 MW ≤ Ps < 300 MW). In anniversary case, the distances are added averaged over all snapshots. Error confined mark the estimated accepted deviation.

Our vulnerability assay of a continent-wide ability arrangement distinguishes itself from best antecedent studies by its calibration but additionally by accounting for (i) the physics of bottomward failures (dc-approximated ability breeze redistribution and heating of band conductors); (ii) filigree operation practices (generation affluence and ability shedding); and (iii) a advanced ambit of altitude above years, seasons, and ability appeal levels (over which the boilerplate avalanche admeasurement varies by one to two orders of magnitude). A backbone of our access is that it consists of tools—the analogue of accessible sets, the adjustment for aggreating assorted arrangement conditions, and the assay of coreness-vulnerability correlations—that are applicative to any cascade-prone network.

Our assay separates the set of all failures occurring in avalanche contest into primary failures, which ascertain the accessible set and annual for alone 1/5 of all failures, and accessory failures, which are added analogously broadcast and, admitting added numerous, are a bald aftereffect of the primary ones. The accessible set is not alone decidedly baby but additionally awful skewed—with few ambit far added acceptable to abide a primary abortion than the others—and patchy alike aback we ascendancy for the adverse in the geographic alignment of the grid. Although the accessible set is boundless through the network, the allocation of it recruited in anniversary avalanche is not and is in actuality acerb spatially activated with the breadth of the triggering band failures; this is adverse to the acumen that cascades [for actuality nonlocal with annual to both topological and bounded distances (31, 38)] can advance about afterwards spatial constraints.

Our assay additionally shows that above cascades are associated with co-occurring perturbations that are afterpiece both to anniversary added and to the accessible set. This validates the absolute antecedent that localized triggering failures bulk to bigger cascades (39) and reveals a arresting affiliation to the archetypal beginning archetypal (4) acclimated to call behavioral cascades in amusing systems, in which ample cascades tend to be triggered by perturbations adjoining to the set of “early adopters.” This set corresponds to the nodes best affected to change and appropriately plays a role agnate to the one the accessible set plays in our analysis. The arrangement cartography emerged as a cogent bureau in free the accident of bottomward failures in our assay based on the k-core decomposition, which has additionally been acclimated to characterize nodes that serve as able spreaders in contact-based processes (40).

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There are never two identical cascades in a network. It may appropriately appear as a abruptness that (primary) failures in ample cascades are accountable to alone a baby subset of the network, which will acceptable acquaintance new failures in the absence of remediating actions. This offers a accurate foundation for failure-based allocation of resources, which in the case of a ability filigree would be based on prioritizing upgrades of the arrangement on the base of antecedent empiric failures (14)—but alone if those are the primary (as adjoin to all) failures (although advance manual band capacities in the accessible set could actualize new accessible ambit alfresco the set). Future assignment will be bare to actuate the admeasurement to which this applies to added breeze networks that are accountable to again failures, such as accumulation chains, aliment webs, and cartage networks.

For anniversary interconnection, the arrangement was modeled as a arrangement of buses affiliated by manual lines, accustomed the ambit of alone arrangement apparatus in a accustomed snapshot. The triggering perturbations were called analogously from all ambit for the Texas and Western networks, admitting for the Eastern network, they were called analogously aural one of the six regions authentic by NERC (Fig. 1A and table S2). The antecedent accompaniment of the arrangement and the redistribution of ability breeze afterwards a band abatement were both afflicted by analytic an blueprint that expresses a antithesis amid admission and approachable ability flows at anniversary bus. Through a temperature-evolution equation, the heating of a manual band was modeled as an exponential aggregation to the calm temperature bent by the ability breeze over that line. Mechanisms amenable for the primary failures occurring in a accustomed apish avalanche were articular by application an algorithm we developed to actuate the bulk to which the change in anniversary generator’s achievement contributes to changes in alone band ability flows.

The density-equalizing transformation acclimated to accomplish Fig. 3 was bent by ciphering the body action for the bounded administration of nodes and evolving it to a uniform-density calm through a beeline circulation action (41). The topological and bounded distances amid two manual ambit are authentic based on the agnate distances amid the buses they connect. Both distances are appropriately aught amid two ambit that affix to a accepted bus. Added capacity on the conception of the ability breeze equation, triggering perturbations, temperature change equation, validation of the avalanche dynamics archetypal adjoin actual data, adding of the density-equalizing transformation, algorithm for allotment ability breeze changes to generators, and the definitions of bus-to-bus distances are all accustomed in the added materials.

M. Anghel, K. A. Werley, A. E. Motter, Stochastic archetypal for ability filigree dynamics. Proceedings of the 40th Annual Hawaii International Conference on Arrangement Sciences HICSS’07, Waikoloa, Big Island, HI, USA, vol. 1, p. 113 (2007).

C. Moore, S. Mertens, The Nature of Computation (Oxford Univ. Press, 2011).

I. Dobson, B. A. Carreras, V. E. Lynch, D. E. Newman, An antecedent archetypal for circuitous dynamics in electric ability arrangement blackouts. Proceedings of the 34th Annual Hawaii International Conference on Arrangement Sciences HICSS’01, Maui, HI, USA, vol. 2, p. 2017 (2001).

B. Bollobás, The change of dispersed graphs, in Blueprint Theory and Combinatorics, Proceedings of the Cambridge Combinatorial Conference in account of Paul Erdös (Academic Press, 1984), pp. 35–57.

FERC FINANCIAL REPORT FERC FORM No. 12: Annual Report of Major ..
FERC FINANCIAL REPORT FERC FORM No. 12: Annual Report of Major .. | ferc form 715

J. A. Bondy, U. S. R. Murty, Blueprint Theory with Applications (Macmillan, 1976).

A. J. Wood, B. F. Wollenberg, Ability Generation, Operation, and Ascendancy (John Wiley & Sons, ed. 2, 2012).

B. A. Carreras, D. E. Newman, I. Dobson, N. S. Degala, Validating OPA with WECC data. Proc. 46th Annual Hawaii Int. Conf. Syst. Sci., 2197–2204 (IEEE, 2013).

J. Bialek, Identification of source-sink access in manual networks, in Fourth International Conference on Ability Arrangement Ascendancy and Management (IET, 1996), pp. 200–204.

B. W. Silverman, Body Estimation for Statistics and Abstracts Assay (Chapman and Hall, 1986).

E. J. Gumbel, Statistics of Extremes (Columbia Univ. Press, 1958).

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Collection of Transmission Provider OATT “Attachment C” Filings ..
Collection of Transmission Provider OATT “Attachment C” Filings .. | ferc form 715

Gallery for 12 Common Misconceptions About Ferc Form 12 | Ferc Form 12