Comparing dynamics of cascading failures betweennetwork-centric and power flow models

Valentina Cupac,1 Joseph T. Lizier,2, 1, ∗ and Mikhail Prokopenko1

1CSIRO Information and Communication Technologies Centre,PO Box 76, Epping, NSW 1710, Australia

2Max Planck Institute for Mathematics in the Sciences,Inselstraße 22, 04103 Leipzig, Germany

(Dated: November 27, 2012)

Small initial failures in power grids can lead to large cascading failures, which have significanteconomic and social costs, and it is imperative to understand the dynamics of these systems and tomitigate risks of such failures. We directly compare two prominent models for cascading failures inpower grids: a power flow model — ORNL-PSerc-Alaska (OPA) and a complex networks model —Crucitti-Latora-Marchiori (CLM). Quantitative comparison of these two models is not trivial andhas not been previously performed, so we present a method to quantitatively compare the two usingtransmission capacity as a common variable. Primarily, we find that the two models exhibit similarphase transitions in average network damage (load shed/demand in OPA, path damage in CLM)with respect to transmission capacity, sharing a common critical region and similar transitions inprobability distributions of network damage size. Furthermore, we find that both OPA and CLMreveal similar impacts of network topology, size and heterogeneity of transmission capacity, withrespect to vulnerability to large cascades. Thus, our analysis indicates that the CLM model, despiteneglecting realistic power flow assumptions and exhibiting differences in behaviour at the local scale,nonetheless exhibits ensemble properties which are consistent with the more realistic OPA fast-scalemodel. Given the advantages of simplicity and scalability of CLM, these results provide impetusfor the use of complex networks models to study ensemble properties of cascading failures in largerpower grid networks.

Keywords: cascading failures, vulnerability of infrastructure, power grids, OPA model, complex networks

I. INTRODUCTION

Complex networks are susceptible to cascading fail-ures, whereby small initial disturbances may propagatethrough the system causing significant damage [1–4].Cascading failures pose a significant risk to power trans-mission systems. When a line outages, power flow be-comes shifted onto the remaining lines. Consequently,the lines carrying extra flow may become overloadedthemselves, forcing further redistribution to other linesin turn. The spread of the cascading failure may causelarge blackouts across the power grid [2]. Since blackoutshave significant social and economic costs, it is impera-tive to model the complex dynamics of cascading failuresin power grids, and to undertake preventive or correctivemeasures in mitigating the risk of large blackouts [5–7].

A major challenge lies in developing models that are(i) realistic, adhering to basic power flow and networkconstraints, (ii) scalable, mapping to large topologies ofreal power grids, and (iii) computationally efficient, per-mitting real-time applications of the model. These trade-offs are apparent when considering two prominent classesof cascading failure models: power flow models, such asORNL-PSerc-Alaska (OPA) [8–10], and complex networktheory models, such as Crucitti-Latora-Marchiori (CLM)model [1, 11]. These models are described in detail inSection II.

∗ joseph.lizier@csiro.au; phone: +61 (0) 2 9372 4711

OPA offers a more faithful representation of powersystems as it explicitly incorporates the DC power flowequations and minimisation of generation cost and loadshed [9]. However, this also results in notable compu-tational limitations, whereby use of OPA is limited tonetworks with relatively smaller number of nodes andrequires considerable time for solving constrained linearoptimisation functions [2]. Furthermore, OPA requiresa more involved process in setting parameters and hasmore parameters compared to CLM.

CLM is a simpler and more abstract model, designed tostudy common features of flows in many systems [1], e.g.transportation flows, internet traffic flows, and powerflows. It offers advantages in modelling cascading failureswith fewer parameters (simplicity) and can be more read-ily applied to larger networks (scalability). On the otherhand, CLM neglects basic power flow constraints and isunable to directly represent generator capacity and loaddemand. Furthermore, it does not provide direct physicalmeasures of blackout size (load shed), but rather resortsto abstract measures (path damage).

Thus, OPA is able to offer a more realistic model atthe expense of scalability, whilst CLM is able to offer amore scalable and simpler abstract model at the expenseof neglecting basic power flow network constraints.

Whilst CLM provides theoretically interesting appli-cations of graph theory in modelling cascading fail-ures, it is not immediately clear whether or not it hasadequate practical significance in specifically modellingpower transmission systems due to its abstract nature.

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A critical issue lies in model validation, the extent towhich abstract variables within CLM are meaningful andtranslatable to more realistic measures used within OPA.While CLM is certainly a scalable and computationallyefficient model, it is necessary to ascertain whether it pro-vides valid measures of damage resulting from cascadingfailures.

Most previous research in this area typically focused onsingle models [1, 10–12], with a lack of cross-validationand in-depth comparative analysis. Some studies [2, 5]have provided qualitative comparison of these models —identifying similarities and differences, and evaluatingadvantages and disadvantages. We provide a detailedqualitative comparison in Section III. Crucially though,there has been no rigorous attempt to directly quanti-tatively compare the behaviour of CLM with respect towell-established power flow models such as OPA, using aset of common network topologies and comparable trans-mission capacities.

Consequently, our primary motivation is to directlycompare the critical behaviour and resultant distribu-tions of power grid damage in CLM versus OPA, andto evaluate the extent to which abstract variables andmeasures within CLM are meaningful in the context ofmore well-understood measures within OPA.

Such a comparison is not straightforward due to dif-ferences in representing system capacity, differences initerative algorithms they rely on, and differences in mea-sures of system damage. In Section IV, we describe howwe will set parameters and measure responses to damagein each model to ensure they are comparable.

The scope of this paper is restricted to studying theOPA fast dynamics model which is analogous in tempo-ral horizons to the CLM model, and ignore the OPA slowdynamics model with does not have a CLM counterpart.Furthermore, we study transmission capacity which is de-fined for both CLM and OPA, and ignore generation ca-pacity which is defined within OPA but not within CLM.

Certainly, OPA and CLM are inconsistent at the lo-cal scale, and we show that load shed/damage in OPAis poorly correlated with path damage in CLM whentriggered by the same simulated individual failure (seeAppendix A). Nonetheless, we seek to examine whetherthere are similarities in global or ensemble behaviour atthe network level (e.g. susceptibility to large cascades asa function of transmission capacity). Specifically, we aimto evaluate the extent to which CLM is consistent withOPA regarding whether CLM and OPA exhibit similar:

1. phase transitions in average network damage andprobability distributions of network damage, withrespect to transmission capacity;

2. ranking of grid topologies and grid sizes, based ontheir susceptibility to large cascading failures; and

3. shifts in probability distributions of failure, withrespect to heterogeneity in transmission capacityacross the network.

The analysis of probability distributions of resultantfailures and damage is crucial to our paper. In the caseof OPA, we examine the probability distribution of loadshed/demand. In the case of CLM, we examine the prob-ability distribution of efficiency loss (i.e. damage). Wethen analyse in Section V the extent to which the twomodels exhibit similarities in their distributions, and theextent to which they exhibit similarities in changes withindistributions arising from changes in transmission capac-ity. This is motivated by the hypothesis that even thoughCLM and OPA may model power flow dynamics very dif-ferently, they may agree in terms of the resultant high-level ensemble properties and probability distributions ofcascading failures.

II. MODELS OF CASCADING FAILURES

Modelling cascading failures poses a number of chal-lenges due to the diversity of mechanisms which are in-volved in initial failures and in subsequent interactionswithin the power system [5]. These challenges have moti-vated the development of various cascading failure mod-els. These models can be classified into several broadcategories [2, 13]: artificial power flow, conventional reli-ability, and complex network theory approaches.

Artificial power flow approaches seek to provide realis-tic power flow models, and examine the changes in flowsas a result of initial failures. OPA, which as describedin Section II A is the most prominent of these, considersthe dynamics of the flows in time on a given underly-ing network structure. Another power flow model is theCASCADE model [12], though it is considered “too sim-ple” in that it “disregards the system structure, neglectsthe times between adjacent failures and generation adap-tation during failure” [13].

Conventional reliability approaches, including the FTAmodel [14], provide simple insights to end-users, but typ-ically require models specialized to individual networksand events, and complex analysis which is difficult toverify and construct [13].

Complex network theory approaches including theCLM model (as described in Section II B) formulate moreabstract models, using the tools of graph theory to ob-tain general insights incorporating network structure inan efficient manner. 1

In seeking to evaluate how accurately the CLM modelperforms on a given network structure, we compare itto the OPA model, since this is the most prominent ofthe power flow models, and is specifically designed toconsider the dynamics of flows in time and the underlying

1 Separate to considerations of modelling cascading failures, othercomplex systems approaches to studying cascading failures in-clude evaluation of the goodness of fit of failure event sizes topower-laws [15], and analysing the dynamics of cascading failureevents using state transition graphs [16].

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network structure. We describe these models in detail inthe following section.

A. OPA model

ORNL-PSerc-Alaska (OPA) is a model proposed byresearchers at Oak Ridge National Laboratory (ORNL),Power System Engineering Research Center of Wiscon-sin University (PSerc), and Alaska University (Alaska)[8–10]. The power grid is represented as a set of nodeswith power injections (Pi) which are positive for genera-tors (G) and negative for loads (L). Generators have lim-ited power supply (Pmax

i ) and loads have fixed demand(P dem

i ). A transmission line connecting nodes i and jhas a power flow Fij , which is limited by the maximumpower flow Fmax

ij that it can carry. A transmission lineconnecting nodes i and j has susceptance bij , impedancezij , and resistance is neglected.

The OPA model considers dynamics in terms of two in-terdependent time scales. The slow time scale describeshow the system evolves as demand grows over longertimeframes, and the subsequent network upgrades whichoccur in response to demand growth and blackouts. Thefast time scale, on the other hand, describes cascadingfailures of transmission lines at a fixed point during theslow dynamics. Our comparative study focuses on thefast dynamics, which are described below.

Power dispatch is formulated as a linear optimisationproblem, where the goal is to minimise a cost function2

which gives preference to generation shift whilst assigninga high cost to (W = 100) to load shedding (reductionof power supply to consumers). It is assumed that allgenerators operate at the same cost and that all loadsare served with equal priority:

Cost =∑i∈G

Pi(t) +W∑j∈L

Pj(t). (1)

The optimisation is subject to the following con-straints. Generators always have positive power injec-tions and cannot exceed generator capacity limits:

0 ≤ Pi ≤ Pmaxi , i ∈ G. (2)

Loads always have negative power injections (cannotgenerate power)3:

P demj ≤ Pj ≤ 0, j ∈ L. (3)

The absolute power flow along edges is subject topower flow limits:

− Fmaxij ≤ Fij ≤ Fmax

ij , i, j ∈ N. (4)

2 There was a typographical error in the optimisation functionwritten in [9, 10], where there was an incorrect minus sign inthe optimisation function, which should have been a plus signinstead [17].

3 Papers including [9, 10] omitted the lower bound for Eq. 3 [17].

Total power generation and consumption remain bal-anced: ∑

i∈G∪LPi = 0. (5)

Based on standard DC power flow analysis, power flowsare expressed as:

F = AP, (6)

where F is a vector of power flows along the edges, P isa vector of power flows injected at each node Pi, and Ais a matrix that depends on the network structure andimpedances (see [8] for details about the computation ofA).

After solving the linear optimisation problem, we ex-amine which lines are overloaded. A line is considered tobe overloaded if Fij is within 1% of Fmax

ij . Each over-loaded line may outage with probability p1. When lineoutage occurs, its power flow limit Fmax

ij is divided bya very large number (κ2) to ensure that practically nopower may flow through the line. Furthermore, to avoida matrix singularity resulting from the line outage, lineimpedance is multiplied by a large number (κ1), result-ing in changes to the network matrix A. The updatedset of line flow limits and the updated network matrixlead to the recomputation of the optimisated state and,once again, some lines may outage, leading to furtheriterations. A cascade is represented as a series of prob-abilistic line failures and the subsequent power flow re-distribution. The process terminates when a solution isfound with no line outages.

During the cascade process, load shed occurs whenlines are outaged due to an imbalance between supplyand demand. Load shed (Si) for an individual node i isdefined as the difference between its power injection anddemand, and is always non-negative:

Si = |P demi − Pi|. (7)

Consequently, total load shed (S) for the system is:

S =∑i∈L

Si. (8)

Finally, total load shed is normalised with respect tototal demand D, and used as a measure of damage to thesystem resulting from the cascade:

S/D =

∑i∈L Si∑

i∈L Pdemi

. (9)

A crucial result arising from OPA simulations is thepresence of scale-free (power-law) regions in the distribu-tion of blackout size, consistent with historical blackoutdata for real power systems [10]. In addition, the ob-served power law distribution of blackout size providedevidence for self-organised criticality in power systems.

OPA is able to incorporate basic elements of power sys-tem components, power flows, and the relations between

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supply and demand. However, the applications of OPAhave generally been limited to networks with a relativelysmall number of nodes (typically hundreds), compared toreal power grids [2].

B. CLM model

Complex network theory has been used to formulatemore abstract models, where the power grid is repre-sented as an undirected graph with nodes (substations)and edges (transmission lines). A prominent approachis the CLM model [1], including extensions to differenti-ate generators and loads [11]. The power grid is repre-sented as a set of N nodes representing NG generatorsand NL loads representing distribution substations, in-terconnected by a set of edges representing transmissionlines.

An edge connecting nodes i and j is characterised byedge efficiency eij(t), representing relative link capac-ity. Initially we assume that edges are perfectly efficienteij(0) = 1, although the efficiency is subject to changeover consequent iterations. At any time step, edge weightis computed as the inverse of edge efficiency. It is as-sumed that electricity flowing between any two nodes iand j takes the most efficient path connecting those twonodes. The most efficient path is the shortest path be-tween i and j, where the length of a given path is definedas the sum of edge weights on that path in the graph be-tween i and j. The efficiency εij(t) of the most efficientpath from i to j is the inverse of the shortest path length[18].

Each node is characterised by a loading Li(t)4, which

varies over time and a fixed limited capacity Ci. Theloading of each node with transmitting capabilities is de-fined as the number of most efficient paths from gener-ators to distribution substations that pass through thatnode [19]. (In the original model [1] which consideredpaths between all node pairs, the loading was equivalentto betweenness centrality [19].) Consequently, the moreshortest paths that pass through a node, the higher theloading it has. The loading at a given node amounts tothe total flow passing through that node when all gen-erator/distribution substation pairs transmit energy be-tween them [19].

Node capacity Ci represents the maximum loading thatit can handle without performance degradation. The ca-pacity is assumed to be directly proportional to the initialloading of the node [20]. The fixed network tolerance pa-rameter α ≥ 1 represents the ability of nodes to handle

4 CLM papers typically use the term load to qualitatively describethe amount of power flowing through a node. At the same time, itcan be used to signify node type, whereby a load node representsa distribution substation. Thus, to avoid potential ambiguity, weuse the term loading for the former case.

increased loading thereby resisting perturbations:

Ci = αLi(0). (10)

An initial breakdown of edges surrounding a nodecauses power to be redistributed in the network, reflectedby changes in the most efficient paths and, consequently,changes in the loading at each node. Some nodes arethen forced to operate above capacity (being overloaded),represented by decreases in efficiency of the edges of thatnode (slightly altering the definition of [1], as per [21]):

eij(t+ 1) =

eij(0) min( Ci

Li(t),

Cj

Lj(t)) if Li(t) > Ci

or Lj(t) > Cj

eij(0) otherwise.

(11)The changes in the efficiency of these edges may change

the most efficient paths between node pairs, and indeedchanges in most efficient paths alter the loading acrossthe network. Convergence in this iterative process occurswhen the state dynamics become stable or periodic.

Network efficiency is computed as the average pathefficiency of the most efficient path from each generatorto each load [11]5:

E(t) =1

NGNL

∑i∈G

∑j∈L

εij(t). (12)

Network damage D caused by the cascade is definedas the normalised loss of average path efficiency [11]:

D(t) =E(0)− E(t)

E(0). (13)

CLM can specifically examine the effect of networkstructure on cascades and their consequences (for exam-ple, random graphs appear to be more resistant to cas-cading failures than scale-free graphs [1]). Results basedon the North American power network indicate that thecritical tolerance levels for loading-based node removalwere substantially higher than the critical tolerance lev-els for random node removals [11]. Furthermore, it wasshown that nodes can be divided into different classes:the removal of some nodes causes very little damage forany tolerance levels, whilst other nodes’ removal leads toan efficiency damage which is a function of their tolerancelevel.

A key advantage of complex network models, such asCLM, is that they enable us to utilise techniques fromgraph theory and perform simulations on large networksdue to their scalability; for example, Kinney et al. [11]have demonstrated the applicability of CLM to at leasttens of thousands of nodes (an increase of two orders of

5 Note that the original formulation in [1] considered most efficientpaths between all node pairs, failing to differentiate between gen-erators and loads.

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magnitude over the capability of OPA [2]), though thereare challenges in reflecting particular features of powergrid operations. Nonetheless, such an approach providesimportant insights and has gained increasing applicationsin modelling cascading failures in power grids.

III. QUALITATIVE COMPARISON

Since OPA and CLM rely on different variables andalgorithms outlined in the previous section, we presenthere a general qualitative comparison between these twomodels, followed by a quantitative comparative analysisin the next section.

A. Demand and generation

The two models differ in the extent to which they areable to represent demand and generator capacity. OPAprovides direct representation of load demand and gener-ator capacity. For example, when mapping any network,it enables us to specify the load demand of individualdistribution substations, and to specify the capacity ofindividual generation substations. On the other hand,CLM does not quantitatively represent demand and gen-eration capacity (although it does quantitatively repre-sent transmission capacity). OPA allows variations inthe total demand / generator capacity ratios. CLM doesnot allow such variation, total demand and total genera-tor capacity are implicitly equal since every generator isconnected to every load and vice versa.

Furthermore, CLM assumes that every distributionsubstation is connected to every generator, whereby thereis only one shortest path from any distribution substa-tion and every generator. This implies that every distri-bution substation attempts to extract an equal amountof power from every generator. OPA does not have sucha simplistic restriction, a distribution substation may bereceiving power from just some generators (rather thanfrom all generators, as is the case in CLM) and further-more, it may disproportionately receive more power fromsome generators than others (rather than attempting toconnect equally to all generators, as is the case in CLM).In CLM, every generator is used at all times, whilst inOPA, some generators may be fully utilised, some par-tially utilised, and some might not provide any power atall (see Fig. 4. in [9]).

Therefore, OPA is able to represent transitions in rel-ative generation capacity of the power grid, whilst CLMis unable to do so.

B. Transmission

The input variable that is directly represented in bothmodels is transmission capacity, the capacity to carrypower. OPA represents transmission capacity as the

maximum line flow, which is associated with edges in themodel. On the other hand, CLM represents transmis-sion capacity as the maximum number of shortest paths,which is associated with nodes in the model. In bothcases, transmission capacity varies across the system, i.e.in OPA, different lines may have different limits, and inCLM, different nodes may have different limits.

In both models, transmission capacity is the crucialfactor in the cascade process, described below. In OPA,edges fail when transmission capacity is reached — anedge is overloaded when its power flow reaches its powerflow limit. In CLM, nodes fail when transmission capac-ity is exceeded — a node is overloaded when the numberof shortest paths exceeds the maximum number of short-est paths that can pass through the node.

Therefore, both OPA and CLM are able to representtransitions in relative transmission capacity of the powergrid.

C. Initial failure

We mentioned that edges fail in OPA, whilst nodes failin CLM, in alignment with the terminology used in thepaper which describe the CLM model. It should be notedthat in CLM, when we speak of nodes failing, it does notimply that actual substations fail, but rather that thetransmission lines immediately attached to those nodeswould fail. Consequently, in the implementation of CLM,node failure is actually represented as the failure of itsadjacent edges.

D. Transmission paths

A major difference is that in OPA, power flows throughmultiple paths from one node to another, whilst inCLM, power is considered to flow only through a sin-gle (shortest) path. That said, the two models still ex-hibit some similarities in determining the paths throughwhich power will flow. Edge impedance in OPA is relatedto edge efficiency in CLM, and edge susceptance in OPAis related to edge weight in CLM. When an initial failureoccurs, the susceptance and efficiency of affected edges isdecreased in OPA and CLM, respectively. This leads toa corresponding increase of those edges’ impedances andweights in OPA and CLM, respectively.

Both algorithms iteratively consider updates to thepower flows based on changes in impedance and weights,though they do this in different ways. We have the fol-lowing important differences regarding how the modelsrespond to overloading: (i) magnitude of the change and(ii) permanence of the change. Firstly, in OPA, thechanges are more extreme: edge failure leads to settingthe susceptance to almost zero and the impedance to avery large number, and the result is that there is theoret-ically no flow through that edge. On the other hand, in

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CLM, edge efficiency is decreased depending on the ex-tent of overloading, hence there may be smaller or largerdecreases (unlike OPA, where the susceptance is guar-anteed to go to almost zero). Similar principles applyto weights in CLM: they are not necessarily increased toa very large number. Secondly, in OPA, the changes arepermanent for the duration of the cascade, whilst in CLMthe changes are temporary (efficiency may recover afterit is decreased, during a single cascade). It should alsobe noted that whilst failure is deterministic in CLM (anoverloaded node fails with certainty), it is probabilisticin OPA (an overloaded edge fails with probability p1).

E. Measurement of cascade damage

At the completion of the cascade, both models mea-sure the extent of damage caused by the cascade. OPAprimarily measures blackout size using load shed, rep-resenting the difference between total power demandedand total power supplied to the distribution substations.Thus, it measures the extent to which consumers failed toreceive the power they demanded which occurred eitherdue to line outages or inadequate generator capacity.

CLM is unable to represent load shed because, as men-tioned before, it quantifies neither power demand norpower injections which are necessary for the computationof load shed. What it does do is to represent damage bycomputing the average efficiency reduction. These arethe only two measures that we can use to evaluate theextent of damage in both models.

F. Power-law distribution of cascade damage

When analysing cascading failures, an important con-sideration is the probability distribution of extent of dam-age, and in particular the tail behaviour. In a normaldistribution, failures are symmetrically distributed abouta mean, and extreme failures are relatively rare. In apower law distribution, the occurrence of extreme failuresis much more probable than in a normal distribution.

Analysis of North American blackouts, based on 15years of normalised NERC data, indicates the presenceof power law tails in the distribution of blackout sizes [9],demonstrating that the probability of large blackouts ishigher than what would be expected if blackouts werenormally distributed. It is important for models to beable to capture this tail behaviour to ensure that realpower systems are adequately modelled.

Studies performed with OPA have shown that theprobability distribution of normalised load shed haspower law tails — in a remarkable agreement with theNorth American historical blackout data, indicating thatthe North American power systems have operated closeto a critical point [9]. Furthermore, it was shown withan analytically tractable loading-dependent model of cas-cading failure that, at a critical loading, a power-law re-

gion emerges in the distribution of number of componentsfailed [12].

A study performed using the CLM model applied tothe North American power grid found that the cumula-tive damage distributions with α = 1.1 and α = 1.2 werein approximate agreement with the power-law distribu-tions found previously using OPA [11].

IV. METHODOLOGY

We seek to compare the OPA and CLM models bysimulating cascading failure experiments, defined by thefollowing parameters:

• A model for cascading failures (OPA, CLM). En-suring comprability between the two models neces-sitated some adjustments as described in the re-mainder of this section;

• A network specifying the grid topology (several al-ternatives are described in subsection IV A);

• A transmission capacity factor (α), representingthe average or global transmission capacity for thesystem; and

• A heterogeneity factor (r), representing local fluc-tuations in α across the network.

For each set of parameters considered, we ran experi-ments as follows:

1. Initialise network capacity, which incorporates αand r, as described in Section IV B.

2. Initiate random failure and model the iterative cas-cade process, as described in Section IV C.

3. Measure the resulting damage caused by the cas-cading failure, as described in Section IV D.

We studied network damage for 21 levels α (α = 0.90 toα = 1.2 with increments of 0.02, and α = 1.2 to α = 1.40with increments of 0.04). For each of these levels, weran 1,000 experiments — each with a different randomly-selected point of the initial failure. Furthermore, for anygiven level, even though the global transmission capac-ity would be fixed for all experiments performed at thatlevel, local variations would result from the renewed ap-plication of the heterogeneity factor r for each experi-ment, as explained in Section IV B.

We also examined probability distributions for a sub-set of levels α (α = 0.80, 1.00, 1.10, 1.20, 1.60, 2.00). Foreach of these subset levels, we ran 10,000 experiments togather enough results to show the probability distribu-tion of network damage.6

6 The number of experiments was selected by qualitatively ana-lyzing a. stability to increase in number of experiments, and b.

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A. Network topology

The graph is specified by a set of nodes (a node is ei-ther a load or generator) connected by edges (represent-ing transmission lines). In both OPA and CLM case, weconstructed a network with nodes which were either gen-erator nodes (representing net generation substations) orload nodes (representing net distribution substations).In our OPA implementation, to facilitate comparabilitywith CLM, all the generators were assumed to have equalcapacity, and all the loads were assumed to have equaldemand. Furthermore, all edge impedances were set to 1in OPA, to align with the uniform edge weight initialisa-tion in CLM.

We considered different types of networks, compris-ing of the standard 300-node IEEE300 bus test case [22]as well as a randomly-generated 300-node scale free net-work (SCALE300) and a 300-node Erdos-Renyi network(ER300). In constructing IEEE300, nodes with genera-tor buses were classed as generator nodes and all othernodes were classed as load nodes. When constructingSCALE300 and ER300 we ensured that they had thesame percentage of generator nodes as IEEE300 (approx-imately 23%) — for each node, the probability of it beingselected as a generator was 0.23 and the probability ofbeing selected as a load was 0.77. The average degree dwas relatively similar between these networks (d = 2.73for IEEE300, d = 2.94 for SCALE, and d = 3.23 forER300).

We also considered two larger networks, IEEE418 andSCALE400. We constructed “IEEE418” [23] by combin-ing the standard IEEE300 and IEEE118 bus test cases[22]. These two networks were combined by inserting30 additional edges to connect the two networks, re-sulting in a combined network with average path effi-ciency which approximated the weighted average pathefficiency of the networks considered separately. We con-structed SCALE400 using the same method as generatingSCALE300, and ensured that the proportion of genera-tors in SCALE400 was same as in IEEE418.

B. Initialisation of transmission capacity

We used α as a common measure of relative trans-mission capacity for both networks. The CLM modelassumes that α ≥ 1, representing excess transmission ca-pacity. In our study, we use a more relaxed approachα ≥ 0 which enables us to consider cases with deficienttransmission capacity. This ensured comparability with

smoothness, of the probability distribution function. In particu-lar, we examined these qualities in the tail behaviour. We foundthat 5,000 samples was suitable, but doubled this to 10,000 asa buffer. For the investigations of average behaviour, we foundthat 500 samples provided stable results, but again double thisto 1,000 as a buffer.

studies performed in OPA which considered both under-and over-capacity system.

For both models, we computed initial transmission(initial loading in CLM, initial flows in OPA), and ap-plied the α multiplier to compute transmission capacity,which remained fixed for the duration of the experiment.

In CLM, for each node i, initial loading Li(0) was com-puted as described in Section II B. To incorporate hetero-geneity r, the capacity for node i is initialised as follows,where q is a uniformly distributed random variable:

Ci = αqL(0), q ∈ [1− r, 1 + r]. (14)

Regarding OPA, prior studies set the initial limits (de-mand, generator capacity, line flow limits) by evolvingthe network using combined fast-slow dynamics until thenetwork reaches a steady state (stabilisation of averagefractional overload) [17]. In order to ensure compara-bility with CLM, and taking into consideration that welimited the scope of our comparsion to fast dynamics,we used a simpler initialisation strategy which did notrequire the use of the slow model.

To do this, we make certain assumptions in order to es-timate the initial flows, and then set the flow limits basedon these estimates. Demand for all load nodes was fixedto a constant amount (we used 100), and total generationcapacity was set to be equal to total demand, and equallydivided among the generators. Then, we estimated powerflows along the lines, on the assumption that every loadnode would obtain an equal amount of power from everygenerator. These are aligned with assumptions in theCLM model regarding generation and demand (see Sec-tion III A). The initial flow estimations were implementedby selecting a generator (one at a time), setting all othergenerator capacities to zero (to prevent them from beingused by the optimiser) and then computing power flowsto each load node. When the power flow magnitudes weresummed (resulting in F ∗ij(0)), they would reflect equalcontribution from every generator, analogous to the ini-tialisation process in CLM. The power flow along eachline would be multiplied by α and, incorporating hetero-geneity r, the maximum capacity for an edge connectingnodes i and j is initialised as follows, where q is a uni-formly distributed random variable:

Fmaxij = αq|F ∗ij(0)|, q ∈ [1− r, 1 + r]. (15)

The F ∗ij(0) were used only for the purposes of settingthe flow limits, and it is important to note that they areneither computed using the full OPA algorithm nor dothey feed into the dynamics of the algorithm after theinitialisation. They are only used to select the initialflow limits in a manner that is comparable to the limitsused by CLM.

We attempt to experimentally verify whether thetransmission capacity initialisation schemes applied toCLM and OPA were indeed comparable. The transmis-sion capacity for a node i was computed directly in CLM(Ci). In OPA, there is no direct transmission capacity

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Transmission capacity (CLM)

Tra

nsm

issi

on c

apacit

y (

OP

A)

FIG. 1. Scatterplot of normalised node transmission capaci-ties in OPA versus CLM, both using the IEEE300 network,with homogeneous transmission capacity (r = 0). Node trans-mission capacity initialisation in CLM is highly correlatedwith transmission capacity in OPA (rOPA,CLM = 0.88). Theline of best fit was added to the plot.

measure for each node i, so we modelled it based on thetransmission capacity Fmax

ij of lines connected to it asfollows: the transmission capacity for node i was mod-elled as the sum of maximum power flows of adajcentedges

∑j∈J F

maxij where J is the set of nodes directly

connected to i via an edge.In Fig. 1, we plot the transmission capacities deter-

mined for each node under both models (with r = 0).This verified that transmission capacity in CLM washighly correlated with transmission capacity in OPA(with correlation coefficient 0.88). That is, nodes withlow transmission capacity in CLM also tended to havelow transmission capacity in OPA, and nodes with hightransmission capacity in CLM tended to have high trans-mission capacity in OPA. This indicates that our initial-isation scheme was consistent for CLM and OPA.

For both models, we ran the bulk of our experimentswith relatively low heterogeneity factor (r = 0.1) whichenabled us to ensure that the capacity of individual com-ponents did not substantially deviate from the global ca-pacity factor α, whilst being able to represent diversityin cascade events. However, in experiments where wespecifically studied the impact of heterogeneity, we stud-ied the effect the behaviour of the network using a higherheterogeneity factor (r = 0.5).

C. Cascade algorithm

To ensure comparability with CLM, we ignored theeffects of network upgrades over time, and thus appliedfast dynamics only.

The cascade was initiated by the failure of a singlenode in each model. The single node is randomly selectedfrom the set of nodes in the network, where each node is

equally likely to be selected. In CLM, the initial single-node failure is represented by the removal of the edgesconnected to it. In OPA, the initial single-node failure isrepresented by outaging the edges connected to it.

Then, we applied the algorithms which were defined inSection II. The algorithms ran over several iterations un-til they either converged or exceeded the maximum num-ber of steps (we used maximum 15 iterations for CLMand OPA). Normal convergence in CLM would be whenthe state dynamics become stable or periodic, whilst nor-mal convergence in OPA would be when no edge fails inthe prior iteration, or if there is no solution to the linearoptimisation (which is treated as a blackout [8]).

In OPA, we set the probability of an overloaded edgeto outage to p1 = 1, to ensure comparability with CLM,where an overloaded node failed with certainty.

D. Cascade damage

We examined the extent of blackout, or damage to thenetwork that were caused by the different types of noderemovals. In CLM, we computed path damage, i.e. nor-malised efficiency loss (previously defined by [11]), whichrepresents damage to the network. In OPA, we com-puted the load shed/demand ratio, representing extentof blackout. The measurements used in our results aremade at the end of the cascade (i.e. at the completion ofthe simulation algorithm, as described above).

In an abstract sense, we will use the term network dam-age to refer to both cases.

V. RESULTS

It is evident that OPA and CLM provide different re-sults at the local scale (see Appendix A), however weevaluated the extent to which the two models are consis-tent at the network level. Firstly, we examined whetherCLM and OPA exhibit similar characteristics in phasetransitions of network damage with respect to transmis-sion capacity (see Section V A). Secondly, we investigatedwhether they provide a consistent ranking of grid topolo-gies, based on their tolerance to large cascading failures(see Section V B). Thirdly, we verified whether CLM andOPA show consistent behaviour as network size increases(see Section V C). Fourthly, we examined whether net-work susceptibility to large failure changes consistentlywith variations in heterogeneity of transmission capacityacross the system (see Section V D).

We note again that it is the better computational ef-ficiency and scalability of CLM which motivates us toexamine its consistency with OPA at the network level.While we do not specifically investigate scalability here(see [2, 11]), we report that our CLM simulations in thissection were generally 10-20 times faster than OPA foranalysis of corresponding networks here. If CLM is foundto produce consistent results, this will prompt further use

9

0.001

0.01

0.1

1

0.9 1.0 1.1 1.2 1.3 1.4

Trasmission capacity (α)

Avera

ge n

etw

ork

dam

age

CLM

OPA

FIG. 2. Phase transition in average network damage (pathdamage for CLM, load shed/demand for OPA) for a range ofα values. Error bars (standard error of the mean) were toosmall to be visible on this plot.

of CLM to study ensemble properties of cascading fail-ures in larger networks.

A. Phase transitions

The network damage resulting from cascading fail-ures is dependent on the relative capacity of the sys-tem. When relative capacity is quite high, a system hashigh tolerance to failure, whereby a small disturbancewould tend to cause relatively low damage on average,with low probability of very high damage. As relativecapacity decreases and passes a critical capacity level, itincreases the probability of large failure. We focus on howthe probability distribution of average network damagefor IEEE300 changes over different transmission capacitylevels α.

We first investigated the change in average networkdamage as a function of relative capacity α. This is thetypical analysis used in CLM (e.g. see Fig. 4 [1]) andis akin to the identification of critical demand levels inOPA (e.g. See Fig. 5 in [9]). This comparison betweenthe two models is only possible with the adjustments weintroduced. Figure 2 shows that both models exhibita strongly stable regime for large α ≥ 1.14 where net-work damage is small and remains relatively constant asα changes. Additionally, both models exhibit an unsta-ble regime for small α < 1.10 where small disturbancescan cause large amounts of network damage. Most im-portantly, this figure shows that both models exhibit asharp increase in network damage at the same critial αregion (1.10 ≤ α ≤ 1.12). That is to say, we have identi-fied very similar phase transitions in the space of α (con-trol parameter), with respect to average path damage oraverage load shed/demand (order parameter).

The presence of the critical α region is characterisedby a rapid increase in load shed/demand as transmis-sion capacity is reduced. This is consistent with prior

OPA studies which have identified a second order transi-tion point occurring at a particular power demand, whenpower flows in transmission lines reach limits set by thetransmission capacity of the grid (e.g. see Fig. 5 [9]).The critical phase transition shown for CLM is also con-sistent with prior studies (e.g. see Fig. 2(b) in [1]).

We then examined the probability distributions of mag-nitude of network damage for particular values of α.7

This is the typical analysis conducted in OPA studies(e.g. see Fig. 8 in [9]), and while a similar examinationhas been presented for CLM (e.g. see Fig. 3 in [11])there has been no quantitative comparison on the samenetwork with a common measure of transmission capac-ity. Figures 3(a) and 3(b) plot these probability distri-butions for several α (of course, the expectation valuesof each probability distribution correspond to the singleaverages plotted for each α on Fig. 2). The most impor-tant result is that both OPA and CLM show the sametrends in these probability distributions. These figuresindicate that when the relative capacity is in the stableregime (α ≥ 1.2 for this figure), probability distributionsdo not change with α and are skewed towards low net-work damage. This indicates that both models show sim-ilar susceptibility to failure when transmission capacityis sufficiently above the critical value and that, in thosecircumstances, increasing transmission capacity has noimpact on overall failure reduction. However, as relativecapacity enters the critical region (see α = 1.1 here), theprobability distribution shifts to the right and the likeli-hood of large failures increases significantly in both mod-els. The distribution continues shifting to the right asrelative capacity is reduced even further (see α = 0.8, 1.0here).

Certainly, the exact averages and distributions of net-work damage may not be identical between the models:CLM shows larger network damage than OPA in the sta-ble regime and lower network damage in the unstableregion, and appears slightly less sensitive to early criticalloading effects. Indeed, the two models may not providethe same results on a local level (see Appendix A), so onecannot use the resulting measure of path damage (CLM)from a specific individual failure to infer the measure ofload shed/demand (OPA) for the same specific failure.However, what is important is that both models exhibitthe same network-wide or ensemble behaviour with re-spect to transmission capacity α. Crucially, they identifya common critical region, and probability distributionsundergo similar shifts as relative transmission capacity αchanges.

7 Note that the magnitude or “scale” of network damage here is avery different concept to our use of the term “scale” to refer to mi-cro or local scale features of the cascading failures (i.e. specificsof power flows on specific edges due to specific failures) versusmacro or network-wide scale features (i.e. probability distribu-tions of failure event sizes with respect to capacity or structuretype).

10

0.001

0.01

0.1

1

0.01 0.1 1

Load shed / Demand

Pro

bab

ilit

y

0.8

1.0

1.1

1.2

1.6

2.00.001

0.01

0.1

1

0.01 0.1 1

Path damage

Pro

bab

ilit

y

0.8

1.0

1.1

1.2

1.6

2.0

FIG. 3. Probability distribution functions (PDFs) for network damage for IEEE300: (a) PDFs for load shed/demand in OPA;(b) PDFs for path damage in CLM. Different curves correspond to different levels α.

B. Impact of network topology

We examined how network topology affects suscepti-bility to large cascading failures in CLM and OPA. Westudied three networks with identical node count, com-prising the standard IEEE300 network, a Erdos-Renyi300-node network (ER300) and a 300-node scale-free net-work (SCALE300).

We found similar results from OPA and CLM by com-puting the maximum damage caused by the cascade overa range of relative capacity (α) levels. Figures 4(a) and4(b) illustrate that rankings of topology-dependent casesof high damage were generally preserved over a range oftransmission capacity levels, whereby the topology-baseddifferences in ranking were most evident in high capacity(α ≥ 2) cases. At critical and lower capacities (α ≤ 1.1),topology appeared to have a less significant impact: CLMshowed convergence in maximum path damage across allthree networks (see Fig. 4(b)) and OPA showed overlap-ping of two networks (see Fig. 4(a)).

Figures 5(a) and 5(b) show that network structure doesimpact tail behaviour, at relatively high capacity levels(α = 2.0). Both models produced consistent rankingsof high-cascade risk associated with network topology,evident in tail behaviour for load shed/demand in OPA,and path damage in CLM.

Thus, our results indicate that network structure hasan impact on susceptibility to large cascading failures inboth models, being of particular importance for highertransmission capacity levels. Analysis of tail behaviourfor OPA and CLM showed that, for an identical nodecount, Erdos-Renyi graphs are the most tolerant to largefailure, and scale-free graphs were most susceptible tolarge failure. This is because scale-free graphs have largehubs and we know that the network is prone to large cas-cades when these hubs are affected [24]. Importantly, thestandard IEEE300 network showed intermediate toler-ance. This is consistent with prior research which shown

that the power grid topology is an intermediate betweenErdos-Reyni graphs and scale-free graphs [19, 25] (ascited in [11]).

We can confirm that OPA and CLM reveal similarphase transitions in average damage with respect to ca-pacity for scale-free graphs and Erdos-Renyi graphs (aswe observed for the IEEE300 in Fig. 3), though the tran-sition revealed by CLM for Erdos-Renyi was significantlymore broad. Importantly, the critical regions identifiedby the two models overlapped in each case. We shouldalso note that the correspondence of ranking of differenttopologies with respect to damage size by OPA and CLMonly holds for maximal damage events, not the averagedamage events.

C. Impact of network size

We investigated the impact of increased network sizeon susceptibility to large cascades. Firstly, we studied theprobability distributions for IEEE300 versus IEEE418.Figures 6(a) and 6(b) show that OPA and CLM gen-erate similar distributions as network size changes fromIEEE300 to IEEE418. The larger network exhibited ear-lier and sharper tail cut-offs, indicating that increasingnetwork size increases network tolerance to large failure.

Similar results were found when comparing the pro-files of SCALE300 and SCALE400 (not shown) wherethe larger network again exhibited sharper tail cut-offs.

Thus, our results indicate that increasing network size(but keeping size of initial failure the same) increases tol-erance to large cascades, and importantly, this is detectedby both OPA and CLM models. This may be because thelarger networks offer more redundant transmission pathswhich can mitigate failures.

11

0.001

0.01

0.1

1

0.9 1.0 1.1 1.2 1.3

Trasmission capacity (α)

Max

imu

m l

oad

sh

ed /

dem

and

.

SCALE300

IEEE300

ER300

0.001

0.01

0.1

1

0.9 1.0 1.1 1.2 1.3

Trasmission capacity (α)

Max

imu

m p

ath

dam

age

SCALE300

IEEE300

ER300

FIG. 4. Maximum observed network damage for SCALE300, IEEE300, ER300: (a) maximum observed load shed/demand inOPA; (b) maximum observed path amage in CLM. Results are shown for a range of levels α.

0.001

0.01

0.1

1

0.01 0.1 1

Load shed / Demand

Pro

bab

ilit

y

SCALE300

IEEE300

ER300

0.001

0.01

0.1

1

0.01 0.1 1

Path damage

Pro

bab

ilit

y

SCALE300

IEEE300

ER300

FIG. 5. Probability distribution of network damage for SCALE300, IEEE300, ER300 for α = 2.0: (a) probability distributionof load shed/demand in OPA; (b) probability distribution of path damage in CLM.

0.001

0.01

0.1

1

0.01 0.1 1

Load shed / Demand

Pro

babil

ity

IEEE300

IEEE418

0.001

0.01

0.1

1

0.01 0.1 1

Path damage

Pro

bab

ilit

y

IEEE300

IEEE418

FIG. 6. Probability distribution of network damage for IEEE300 and IEEE418: (a) probability distribution of load shed/demandin OPA; (b) probability distribution of path damage in CLM

12

D. Impact of heterogeneity in transmissioncapacity

We examined the extent to which OPA and CLM ex-hibit consistent behaviour as heterogeneity in transmis-sion capacity increases across the system. Our previ-ous results were computed using r = 0.1 as describedin Section IV B to represent random local fluctuations inrelative transmission capacity. Here we compare thoseresults to an experiment using a larger level of hetero-geneity (r = 0.5), for a sample transmission capacityα = 1.2.

Figures 7(a) and 7(b) indicate that increasing the het-erogeneity factor r shifts the probability distribution tolarger network damage values, i.e. increasing the prob-ability of larger cascades. We find that certain α values(e.g. α = 1.2 here) which are stable for low heteregeneitylevels (r = 0.1 here, see Section V A) become unstable forhigher heteregeneity levels (r = 0.5 here). Both OPA andCLM reveal this behaviour with increasing heteregeneity.

The tendency for heterogeneity to increase the proba-bility of larger cascades, thereby shifting the critical re-gion to a higher transmission capacity α level, is consis-tent with prior OPA studies which examined the effectof random demand fluctuations and found that the levelof fluctuations causes the network exhibit criticality evenwhen total demand is below total generator capacity lim-its (see Section V in [9]).

VI. DISCUSSION AND CONCLUSION

We investigated the behaviour of two approaches tomodelling cascading failures: a power flow model —ORNL-PSerc-Alaska (OPA), and a complex networksmodel — Crucitti-Latora-Marchiori (CLM) model. Ouranalysis indicates that the CLM model, despite neglect-ing realistic network constraints and being inconsistentwith OPA at the local scale, nonetheless exhibits ensem-ble or network-level properties which are consistent withthe more realistic OPA fast-scale model with respect totransmission capacity α.

Primarly, we found that the OPA and CLM exhib-ited similar phase transitions in network damage withrespect to transmission capacity α: both models exhib-ited a strongly stable regime for larger α and an unstableregime for smaller α. Crucially, they shared a commoncritical α region. Furthermore, we found both models ex-hibited similar probability distribution of network dam-age, in response to changes in α.

Additionally, we found that OPA and CLM providedconsistent rankings of network topologies with respectto susceptibility to large failures (i.e. tail behaviour):both models showed that scale-free networks were mostsusceptible to large cascades, whilst Erdos-Renyi net-works were most resilient to large cascades. Importantly,IEEE300 (a more realistic network) showed intermedi-ate susceptibility to large cascades relative to these ref-

erence structures. We also found both models showedthat larger networks were more resilient than smallernetworks. Furthermore, OPA and CLM exhibited sim-ilar transitions in the probability distribution of networkdamage with respect to changes in heterogeneity in trans-mission capacity α across the network: increasing hetero-geneity increased susceptibility to large cascades, and ledto instability even for higher α.

The results from this comparative paper provide animportant contribution regarding the usefulness of anabstract model (CLM). Even though CLM is inconsis-tent with OPA at the local scale, it is nonetheless use-ful in studying properties of a power system at a globalscale and profiling the risk of cascading failures in vari-ous network topologies, with the advantages of simplicityand scalability compared to a detailed and more realis-tic power flow model (OPA). That is to say, despite themodelling power of OPA, it is much less computationallyefficient and there are limits to the size of networks it canstudy [2] in comparison to CLM; as such, our finding thatCLM exhibits ensemble properties consistent with OPAis particularly important for the study of larger networks.

Our study provides a starting point for more extensivequantitative validation of abstract models against morerealistic ones, and to identify conditions under which theyexhibit comparable behaviour. It is important to notethat the current comparative study was performed un-der a specific set of restrictions, constraints and simplifi-cations. Firstly, we introduced α to OPA to enable us todirectly compare the behaviour of the two models withrespect to a common measure of transmission capacityα. Secondly, we outlined an ad hoc method of using αto initialise transmission capacity of edges in OPA, withthe assumption that demand and generation are equaland (only for the purposes of capacity initialisation) thateach distribution substation attempts to equally extractpower from each generator. We showed that this ini-tialisation scheme ensured that the transmission capac-ity was comparable between OPA and CLM. Nonethe-less, it should be noted that typically OPA would beinitialised by evolving the network with both slow- andfast-scale dynamics, and only the latter were consideredin our study. Thirdly, the following simplifications wereapplied to OPA to ensure comparability with CLM forthe purposes of this study: nodes were treated as eitherpurely generators or purely loads, load nodes had uni-form demand, generator nodes had uniform generationcapacity, edges had uniform impedance, overloaded edgesfailed with certainty.

Future studies may consider extensions to CLM whichcould enable more comprehensive comparative studiesand broaden the scope of applications of abstract modelsof power systems. Firstly, it may be useful to model notonly variations in transmission capacity, but also varia-tions in generation capacity, and to consider situationswhere generation capacity and demand are not equallydistributed, which is aligned with more realistic casesof power grids. One could for example consider supply

13

0.001

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0.1

1

0.01 0.1 1

Load shed / Demand

Pro

bab

ilit

y

0.1

0.5

0.001

0.01

0.1

1

0.01 0.1 1

Path damage

Pro

bab

ilit

y

0.1

0.5

FIG. 7. Probability distribution of network damage for IEEE300 for α = 1.2 with r = 0.1 and r = 0.5: (a) probabilitydistribution of load shed/demand in OPA; (b) probability distribution of path damage in CLM.

from a given generator to only a portion of, rather thanall, distributed substations. Secondly, it may be usefulto consider cases with non-uniform initial edge weights,corresponding to non-uniform edge impedances in realitywhich are an important element in dictating power flows.Thirdly, slight modifications to betweenness centralitymay be applied so that loading and transmission capacityare computed for edges rather than nodes. While we haveconsidered heterogeneous transmission capacities acrossthe network here, using random uniformly distributedfluctuations around α, it may be worthwhile to considerdirect individual settings for α (with fluctuations as here)in different parts of the network. These extensions wouldenable us to map real power grid data to CLM, and en-able more comprehensive studies and evaluations of the

extent to which CLM remains useful in these more gen-eral cases.

ACKNOWLEDGMENTS

We thank B.A. Carreras [17] for providing valu-able clarification regarding the implementation of theOPA model and providing the IEEE118 data usedin prior OPA papers. We also thank the HighPerformance Computing and Communications Centre(http://www.hpccc.gov.au/) for the use of their super-computer clusters in performing the experiments for thispaper.

[1] Paolo Crucitti, Vito Latora, and Massimo Marchiori,“Model for cascading failures in complex networks,”Physical Review E 69, 045104 (2004).

[2] Ke Sun and Zhen-Xiang Han, “Analysis and compari-son on several kinds of models of cascading failure inpower system,” in Transmission and Distribution Confer-ence and Exhibition: Asia and Pacific, 2005 IEEE/PES(2005) pp. 1 – 7.

[3] Adilson E. Motter and Ying-Cheng Lai, “Cascade-basedattacks on complex networks,” Physical Review E 66,065102 (2002).

[4] Charles D. Brummitt, Raissa M. D’Souza, and E. A.Leicht, “Suppressing cascades of load in interdependentnetworks,” Proceedings of the National Academy of Sci-ences 109, E680–E689 (2012).

[5] R. Baldick, B. Chowdhury, I. Dobson, Zhaoyang Dong,Bei Gou, D. Hawkins, H. Huang, M. Joung, D. Kirschen,Fangxing Li, Juan Li, Zuyi Li, Chen-Ching Liu, L. Mili,S. Miller, R. Podmore, K. Schneider, Kai Sun, D. Wang,Zhigang Wu, Pei Zhang, Wenjie Zhang, and Xiaop-ing Zhang, “Initial review of methods for cascading fail-

ure analysis in electric power transmission systems IEEEPES CAMS task force on understanding, prediction, mit-igation and restoration of cascading failures,” in IEEEPower and Energy Society General Meeting – Conver-sion and Delivery of Electrical Energy in the 21st Cen-tury, 2008 IEEE (2008) pp. 1–8.

[6] R. Hooshmand and M. Moazzami, “Optimal design ofadaptive under frequency load shedding using artificialneural networks in isolated power system,” InternationalJournal of Electrical Power & Energy Systems 42, 220–228 (2012).

[7] P. A. Trodden, W. A. Bukhsh, A. Grothey, and K. I. M.McKinnon, “MILP formulation for controlled islandingof power networks,” International Journal of ElectricalPower & Energy Systems 45, 501–508 (2013).

[8] I. Dobson, B.A. Carreras, V.E. Lynch, and D.E. New-man, “An initial model for complex dynamics in elec-tric power system blackouts,” in Proceedings of the 34thAnnual Hawaii International Conference on System Sci-ences, 2001 (2001) pp. 710 – 718.

[9] B.A. Carreras, V.E. Lynch, I. Dobson, and D.E. New-

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man, “Critical points and transitions in an electricpower transmission model for cascading failure black-outs,” Chaos 12, 985 – 994 (2002).

[10] B.A. Carreras, D.E. Newman, I. Dobson, and A.B.Poole, “Evidence for self-organized criticality in a timeseries of electric power system blackouts,” IEEE Trans-actions on Circuits and Systems I: Regular Papers 51,1733 – 1740 (2004).

[11] R. Kinney, P. Crucitti, R. Albert, and V. Latora, “Mod-eling cascading failures in the North American powergrid,” European Physical Journal B 46, 101 – 107 (2005).

[12] I. Dobson, B.A. Carreras, and D.E. Newman, “Aloading-dependent model of probabilistic cascading fail-ure,” Probability in the Engineering and InformationalSciences 19, 15 – 32 (2005).

[13] R. Leelaruji and V. Knazkins, “Modeling adequacy forcascading failure analysis,” in Australasian UniversitiesPower Engineering Conference (AUPEC ’08) (IEEE,2008) pp. 1–6.

[14] M. Rausand and A. Hoyland, System Reliability TheoryModels, Statis- tical Methods, and Application (John Wi-ley and Sons, Hoboken, NJ, 2004).

[15] Martı Rosas-Casals and Ricard Sole, “Analysis of majorfailures in europe’s power grid,” International Journal ofElectrical Power & Energy Systems 33, 805–808 (2011).

[16] Liang Chang and Zhigang Wu, “Performance and relia-bility of electrical power grids under cascading failures,”International Journal of Electrical Power & Energy Sys-tems 33, 1410–1419 (2011).

[17] B.A. Carreras, personal communication (2012).[18] Vito Latora and Massimo Marchiori, “Efficient behavior

of small-world networks,” Physical Review Letters 87,198701 (2001).

[19] K. I. Goh, B. Kahng, and D. Kim, “Universal behavior ofload distribution in scale-free networks,” Physical ReviewLetters 87, 278701 (2001).

[20] Adilson E. Motter and Ying-Cheng Lai, “Cascade-basedattacks on complex networks,” Physical Review E 66,065102 (2002).

[21] Joseph T. Lizier, David J. Cornforth, and MikhailProkopenko, “The information dynamics of cascadingfailures in energy networks,” in Proceedings of EuropeanConference on Complex Systems, Coventry, UK, Sept2009 (2009) p. 54.

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www.ee.washington.edu/research/pstca/.[23] Valentina Cupac, Joseph T. Lizier, and Mikhail

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Appendix A: Local behaviour in response to specificfailures

It can be illustrated that CLM and OPA models ex-hibit differences in local behaviour in response to specificfailures. We construct a simple network (see Fig. 8) with

TABLE I. Network damage for each possible line failure

Edge Load shed Path Absoluteremoved / demand damage difference

ID (OPA) (CLM)

E12 0.17 0.10 0.07E23 0.33 0.24 0.09E34 0.50 0.50 0.00E45 0.00 0.37 0.37E56 0.50 0.50 0.00E67 0.33 0.24 0.09E78 0.17 0.10 0.07

nodes Ni, i ∈ 1...8 and edges Eij , i, j ∈ 1...8, and com-pare the two models’ response to line failure, regarding:(i) the changes in state of network components, and (ii)the subsequent measure of network damage.

Figure 8(a) shows the impact of outaging edge E67 us-ing OPA. After the outage, no power can flow throughedges E67, E78. Thus load nodes N7, N8 receive no power,generator node N5 supplies less power, and edge E56

transmits less power. We note that in the final state,the power output of generators is assymetrical, despitethe symmetry of the network — this assymetry in gener-ator output, due to the behaviour of the linear optimiser,has been previously documented (see Fig. 4 in [9]). It itimportant to note that, in theory, there may be severalpossible optimal solutions which satisfy power flow con-straints in OPA, however the linear optimiser producesonly a single optimal solution.

Figure 8(b) shows the impact of outaging edge E67 us-ing CLM. The efficiency of edge E67 becomes zero, reduc-ing the efficiency of any shortest paths passing throughthat edge to zero. Consequently, less paths are offered,reducing the transmission loading at nodes N5 −N7.

Therefore, OPA and CLM exhibit some differencestheir network components responded to the outage ofedge E67. In the case of OPA, edges E56, E67, E78 andnodes N5, N7, N8 were affected. In the case of CLM,edges E67 and nodes N5 − N7 were affected by the out-age.

Table I shows the load shed/demand (S/D) and pathdamage (D) resulting from all possible edge failures (onefailure at a time). In general, the measures of loadshed/demand and path damage differed, with a promi-nent difference observed for edge E45. The ratios in net-work damage also differed, e.g. the impact of outagingedge E23 relative to E12 was 0.33/0.17 = 1.94 in OPAand 0.24/0.10 = 2.4 in CLM.

Based on individual failures shown in Table I, we com-puted the linear correlation between the measures of loadshed/demand in OPA and path damage in CLM, findingsome weak correlation (R2 = 0.35). Thus, there was nostrong correlation between these two measures even on avery simple network.

Furthermore, we examined the correlation of loadshed/demand and path damage for individual node re-movals for each specific α value, using IEEE300 —

15

1 2 3 6 7 8

-100 -100 -100 +300 +300 -100 -100 -100

-100 -200 -300 0 +300 +200 +100

1 2 3 6 7 8

-100 -100 -100 +300 +100 -100 0 0

-100 -200 -300 0 +100 0 0

Initial state:

After line outage:

4 5

4 5

1 2 3 6 7 8

1

0 2 4 3 3

1 1 1 1 1 1

4 2 0

1 2 3 6 7 8

1

0 2 4 3 1

1 1 1 1 0 1

0 0 0

Initial state:

After line outage:

4 5

4 5

FIG. 8. Simple network with load nodes (blue circles) and generator nodes (orange squares), numbered to indicate position:(a) using OPA, indicating power flows Fij shown above edges, power injections Pi shown below nodes (positive for generatornodes and negative for load nodes); (b) using CLM, indicating edge efficiencies eij shown above edges, and transmission loadingLi shown below nodes.

analagous to the experiments on the network level inSection V A, however with homogenous transmission ca-pacity (i.e. r = 0). We found low levels of correlation:R2 = 0.27 when α = 1.50, R2 = 0.04 when α = 1.10,R2 = 0.03 for α = 1.05, R2 = 0.002 for α = 1.01. This in-

dicates that, when we simulate the failure of one specificnode, the resulting measure of load shed/demand (OPA)and path damage (CLM) are poorly correlated. Thus,OPA and CLM are not consistent at the local scale.