A Cascading Failure Model by Quantifying Interactions

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by  Junjian Qi

A Cascading Failure Model by Quantifying Interactions
Junjian Qi and Shengwei Mei
Department of Electrical Engineering, Tsinghua University, Beijing, China 100084
Cascading failures triggered by trivial initial events are encountered in many complex systems.
Itistheinteractionandcouplingbetweencomponentsofthesystemthatcausescascadingfailures.
We propose a simple model to simulate cascading failure by using the matrix that determines
how components interact with each other. A careful comparison is made between the original
cascades and the simulated cascades by the proposed model. It is seen that the model can capture
general features of the original cascades, suggesting that the interaction matrix can well reflect
the relationship between components. An index is also defined to identify important links and the
distribution follows an obvious power law. By eliminating a small number of most important links
3 the risk of cascading failures can be significantly mitigated, which is dramatically different from
1 getting rid of the same number of links randomly.
0
2
Cascadingfailuresincomplexsystemsarecomplicated networks [19, 20]. In this paper we discuss the property
n
sequences of dependent outages. They can take place in of the directed weighted interaction network rather than
a
J electric power systems [1, 2], the Internet [3, 4], the road the network directly from the physical system.

4 system [5, 6], and the social and economic systems [7]. Assume there are m components in the system, we
1 For example, in power grid a line is tripped for some constructmatrixA∈Zm×m whoseentrya isthenum-
ij
reason, such as error of operators, bad weather, or tree ber of times that component i fails in one generation
]
h contact,andthepowertransmittedthroughthislinewill before the failure of component j among all original cas-
p beredistributedtootherlines,possiblycausingotherline cades. For each component failure in this generation we
- tripping or even cascading failures. find a component failure that most probably causes it.
c
o Several models have been proposed to study the gen- Specifically, the failure of component j is considered to
s eral mechanisms of cascading failures. The topological becausedbythecomponentfailurewiththegreatesta
. ij
s thresholdmodels[8–10]provideinsightintoseveraltypes amongallcomponentfailuresinthelastgeneration,thus
c ofcascadingfailures. Theinfluencemodel[11]comprises correcting A to be A(cid:48) ∈ Zm×m, whose entry a(cid:48) is the
i ij
s sites connected by a network structure and each site has number of times that the failure of component i causes
y
a status evolving according to Markov chain. The line thefailureofcomponentj. AnexampleisgiveninFig. 1.
h
p interaction graph [12] initiates a novel analysis method Thenwecangetthem×minteractionmatrixB∈Rm×m
[ for cascading failures in electric power systems by con- whose entry b is the empirical probability that the fail-
ij
sidering the interaction of the transmission lines. ure of component i causes the failure of component j.
3
v Thesystem-levelfailureoftightlycoupledcomplexsys- From the Bayes’ theorem we have bij = a(cid:48)ij/fi, where
5 tems is not caused by any specific reason but the prop- f is the number of times that component i fails. The
i
5 erty that components of the system are tightly coupled B matrix actually determines how components interacts
0 and dependent [13]. Based on this idea we quantify the with each other.
2
interaction of the components by an interaction matrix We propose a simple model to simulate cascading fail-
.
1 andthenproposeacascadingfailuremodelbyusingthis ureswithBmatrix. Initiallyallcomponentsareassumed
0
matrix. The interaction matrix can be obtained with to work well and the cascading failure is triggered by a
3
1 cascades from simulations or real statistical data. These smallfractionofcomponentfailures. Sincewewouldlike
: cascades can be grouped into different generations [14– tofocusontheinteractionofcomponentsratherthanthe
v
16] and are called original cascades to distinguish with triggering events, the component failures in generation 0
i
X the simulated cascades in the following discussion. A of an original cascade are directly considered as genera-
r typical cascade can be: tion 0 failures in the simulated cascade. The columns of
a
Bcorrespondingtotheinitialfailuresaresetzerosincein
generation0 generation1 ourmodelonceacomponentfailsitwillremainthatway
untiltheendofthesimulation. Thenthecomponentfail-
25, 40, 74, 102, 155 72, 73, 82
ures in generation 0 independently generate other com-
ponent failures. Specifically, if component i fails in gen-
Here the numbers are the serial number of the failed eration0itwillcausethefailureofanyothercomponent
components. Thiscascadehastwogenerationswhileoth- j with probability b . Once it causes the failure of a
ij
ers might contain one or several generations. component, the column of B corresponding to that com-
Topological properties such as small-world [17] and ponent will be set zero. All component failures caused
scale-free [18] behavior have been found in complex net- by generation 0 failures comprise generation 1. Genera-
works. Power-lawbehaviorisalsodiscoveredforweighted tion1failuresthengenerategeneration2. Thiscontinues
2
25 40 74 102 155 100
258229 592 1485 14134664 440168
258 5861487 1280 1270 359
502
72 73 82 10−1
y
(a) bilit
a
b
74 o
r
p 10−2
generation 0
original cascades
72 73 82 simulated cascades
10−3
(b) 100 101
number of components failed
FIG. 1. Illustration of correcting A to A(cid:48). Two consecu-
tive generations of a cascade are given. The last generation FIG. 2. CCD of the total number of line outages for orig-
comprises the failure of 25, 40, 74, 102, and 155 and in this inal cascades (circles) and simulated cascades (rectangles).
generation there are three component failures, which are 72, Downward-pointing triangles denote CCD of the generation
73, and 82. (a) The numbers on the edges are a , where i is 0 failures.
ij
the source vertex and j is the destination vertex. (b) When
constructing A(cid:48), for the two consecutive generations 72, 73,
and82areconsideredtobecausedby74sincethea starting
ij
from 74 are the greatest.
until no failure is caused.
An example for electric power systems is presented
to illustrate the proposed model. The original cascades
are generated by the Alternating Current ORNL-PSerc-
Alaska (AC OPA) simulation [21–23], which is the AC
counterpart of the OPA model [24, 25]. We use the form
of the AC OPA simulation in which the power system is
fixed and does not evolve or upgrade. As other variants
of the OPA model, AC OPA can also naturally produce
line outages in generations; each iteration of the “main
loop”ofthesimulationproducesanothergeneration. We
FIG. 3. Links for the original and simulated cascades. Blue
choose transmission lines as components and simulate a
linksaresharedbybothoriginalandsimulatedcascades;red
total of 5000 cascades on the IEEE 118 bus system [26], links are only for original cascades; green links are only for
which represents a portion of a past American Electric simulated cascades; dots denote component failures.
Power Company transmission system.
The complementary cumulative distributions (CCD)
of the total number of line outages for original and sim- starts from the failure of component i and ends with the
ulated cascades are shown in Fig. 2. It is seen that the failure of component j. These links form a directed net-
two distributions match well. In our case B is a rather work, for which the vertices are component failures and
sparse matrix, a 186×186 matrix with only 202 nonzero thedirectedlinksrepresentthatthesourcevertexcauses
elements. Just because of the interaction between com- the destination vertex with probability greater than 0.
ponentsdenotedbythissparsematrix,thecascadingfail- Fig. 3showsthenetworkforbothoriginalandsimulated
ure is able to propagate a lot, which is suggested by the cascades. ThesharedlinksbelongtosetL andthelinks
1
dramaticdifferencebetweenthegeneration0distribution onlyowedbytheoriginalandsimulatedcascadesrespec-
and the total line outage distribution. If the elements of tively belong to set L and L . In our case 133 links are
2 3
B are all zeros and the components do not interact, all shared by the original and simulated cascades. 69 links
cascadeswillstopimmediatelyaftergeneration0failures are owned only by the original cascades and 45 only by
and the distribution of the total line outages will be the the simulated cascades.
same as generation 0 failures. Itseemsthatthesimulatedcascadesarequitedifferent
The nonzero elements of B determine how one com- from the original cascades since there are many different
ponent affects another. They are called links. A link links between them. However, it is not the truth if we
l : i → j corresponds to the nonzero element b and take into account the intensities of the links. The links
ij
3
12 y 100
bilit original cascades
18 proba 10−2 simulated cascades (a)
14 34 88 89 155 100 101 102 103
link weights
25 71 74 148 178 y 100
abilit 10−1
79 72 73 82 83 90 62 128 149 154 169 26 ob (b)
pr 10−2
77 42 43 70 20 68 144 145 146 147 160 100 101 102 103
out−strength
22 59 60 61 ability 1100−01
ob (c)
FIG. 4. Diagram showing how I is calculated. pr 10−2
l 100 101 102 103
in−strength
are not equally important and can cause dramatically
different consequences. We quantify the importance of a FIG. 5. (a) CCD of the link weights. (b) CCD of the vertex
linkl: i→jbyanindexIl,whichmeasurestheexpected out-strength. (c) CCD of the vertex in-strength.
componentfailuresthataspecificlinkl cancauseonthe
conditionthatthenumberoftimesthatitssourcevertex
fails is known. When the important links of the original are close to
Specifically, assume a component i fails for Ni times, their counterparts for simulated cascades, S will be near
theexpectedfailuresofcomponentj willbeNj =Nibij. 1.0. InourcaseS =1.0522,whichmeansthatmostlinks,
Theexpectedfailurescausedbythefailureofcomponent at least the most important links, have similar propaga-
(cid:80)
j is Nj k∈jbjk, where k ∈ j denotes the destination tion capacity for the original and simulated cascades.
vertices starting from j. We continue to calculate the
TheCCDofthelinkweightsfortheoriginalandsimu-
expected failures until reaching the leaf vertices. All the
lated cascades are shown in Fig. 5(a). The two distribu-
expected failures are summated to be I . In fact, I =
l l tionsmatchverywell. Bothofthemfollowobviouspower
(cid:80)
N , where V is the set of vertices for which there
v∈V v law and can range from 1 to more than 1000, suggesting
exists a path starting from link l and N is the expected
v thatasmallnumberoflinkscancausemuchgreatercon-
failures of vertex v. Fig. 4 lists all vertices that can be
sequences than most of the others.
influenced by link 12→18.
The CCD of the vertex out-strength and in-strength
By using I as weights of the links we can get a di-
l for original and simulated cascades are shown in Figs.
rected weighted network corresponding to the nonzero
5(b)–5(c). Here the out-strength is defined as the sum-
elements of B matrix. Denote the index of link l for
mationofweightsoftheoutgoinglinksfromavertexand
the original and simulated cascades respectively as Iori
l thein-strengthisthesummationofweightsoftheincom-
and Isim. (cid:80) Isim/(cid:80) Iori is 0.9917,
l l∈(L1∪L3) l l∈(L1∪L2) l inglinkstoavertex. Anobviouspowerlawbehaviorcan
which means that the links of the original and simu-
beseen,whichmeansthatmostvertices(componentfail-
lated cascades have almost the same propagation capac-
ures) have small consequences while a small number of
ity on the whole. Besides, (cid:80) Iori/(cid:80) Iori
and (cid:80)l∈L1Ilsim/(cid:80)l∈(L1∪L3)Ilsiml∈La1rel separla∈t(eLl1y∪L02.)96l90 tthheemstrheanvgethmudcishtrgibreuattioenrsimofpathcte.oAringointahleranpdoinsitmiuslathteadt
and0.9620,indicatingthatthesharedlinksplaythema- cascadesmatchverywell,indicatingthattheysharesim-
jor role among all links. ilar features from an overall point of view.
For the shared links (cid:80) Isim/(cid:80) Iori is 0.9845.
l∈L1 l l∈L1 l We eliminate 5% of the links (10 links) by setting 10
This suggests that the overall effects of the shared links
nonzeroelementsinBmatrixtobezero. WegetB by
int
of the original and simulated cascades are close to each
eliminating 10 links with the greatest weights and B
rand
other. However,itisstillpossiblethattheweightsofthe
by randomly removing 10 links. In Fig. 6 we show the
same link for the original and simulated cascades can be
position of the removed links in the interaction matrix.
quite different. To show if the same link is close to each
Then we separately simulate cascading failures with the
other we define a similarity index S for the shared links.
proposed model by using B and B and the two
int rand
mitigation strategies are respectively called intentional
(cid:88)(cid:18) Isim+Iori Isim(cid:19) mitigation and random mitigation. Fig. 7 shows the
S = l l l (1)
(cid:80) (Isim+Iori) Iori effectsofthetwomitigationstrategies. Itisseenthatthe
l∈L1 l l l risk of large-scale cascading failures can be significantly
l∈L1
4
dramatically mitigated.
0
We are grateful for the financial support of NSFC
20 Grant No. 50525721, China’s 863 Program Grant No.
40 2011AA05A118, and Henan cascading failure project.
60
80
100
[1] NERC (North America Electric Reliability Council),
120
1996 System Disturbances (Princeton, 2002).
140 [2] U.S.-CanadaPowerSystemOutageTaskForce,FinalRe-
160 portontheAugust14thBlackoutintheUnitedStatesand
Canada (2004).
180
[3] R. Pastor-Satorras, A. Va´zquez, and A. Vespignani,
0 50 100 150
Phys. Rev. Lett. 87, 258701 (2001).
[4] K. I. Goh, B. Kahng, and D. Kim, Phys. Rev. Lett. 88,
108701 (2002).
FIG. 6. Position of the removed links. Black dots denote
[5] S. N. Dorogovtesev and J. F. Mendes, Evolution of Net-
nonzero elements of B, among which the nonzero elements
works (Oxford University Press, Oxford, 2003).
removedbyintentionalmitigationandrandommitigationare
[6] S. H. Strogatz, Nature (London) 410, 268 (2001).
denoted by red squares and green triangles.
[7] R. N. Mantegna and H. E. Stanley, An Introduction to
Econophysics: Correlations and Complexity in Finance
(Cambridge University Press, Cambridge, 2000).
100 [8] M.Granovetter,AmericanJ.Sociology,83,1420(1978).
[9] S. Morris, Review of Economic Studies, 67, 57 (2000).
[10] D. J. Watts, Proc. Natl. Acad. Sci. U.S.A., 99, 5766
(2002).
10−1 [11] C. Asavathiratham, Ph.D. thesis, EECS Dept., MIT
y (2000).
bilit [12] P.D.Hines,I.Dobson,E.Cotilla-Sanchez,andM.Epp-
ba stein, Hawaii International Conference on System Sci-
o
r ences (Maui, HI, 2013).
p 10−2
[13] C. Perrow, Normal accident: living with high-risk tech-
generation 0
nologies (Princeton University Press, Princeton, 1999).
original cascades
random mitigation [14] I. Dobson, J. Kim, and K. R. Wierzbicki, Risk Analysis,
intentional mitigation 30, 650 (2010).
10−3 [15] J. Kim, K. R. Wierzbicki, I. Dobson, and R. C. Hardi-
100 101
man, IEEE Systems Journal, 6, 548 (2012).
number of components failed
[16] I. Dobson, IEEE Trans. Power Syst., 27, 2146 (2012).
[17] D. J. Watts and S. H. Strongatz, Nature (London) 393,
FIG. 7. CCD under two mitigation strategies.
440 (1998).
[18] R. Albert and A.-L. Baraba´si, Rew. Mod. Phys. 74, 47
(2000).
[19] A. Barrat, M. Barth´elemy, R. Pastor-Satorras, and A.
mitigatedbyeliminatingonlyafewmostimportantlinks.
Vespignani, Proc. Natl. Acad. Sci. U.S.A., 101, 5766
By contrast, the mitigation effect is minor if we get rid (2004).
of the same number of links randomly. [20] A.Barrat,M.Barth´elemy,andA.Vespignani,Phys.Rev.
Lett. 92, 228701 (2004).
To conclude, we quantitatively determine the inter-
[21] S. Mei, Yadana, X. Weng, and A. Xue, in Proceedings
action of components in the system by calculating the
of the 25th Chinese Control Conference, 2006 (Harbin,
probability that one component failure causes another.
China, 2006).
By using this information we propose a simple model to [22] S.Mei, Y,Ni.Weng, G.Wang, andS.Wu, IEEETrans.
simulate cascading failures. The model is validated to Power Syst., 23, 1719 (2008).
be able to capture the general properties of the original [23] S. Mei, X. Zhang, and M. Cao, Power Grid Complexity
cascades through comparison between the original and (TsinghuaUniversityPress,BeijingandSpringer,2011).
[24] H. Ren, I. Dobson, and B. A. Carreras, IEEE Trans.
simulated cascades. An obvious power law is found in
Power Syst., 23, 1217 (2008).
the distributions of the link weights and the vertex out-
[25] B.A.Carreras,D.E.Newman,I.Dobson,andN.S.De-
strength and in-strength. The links have significantly
gala, 46th Hawaii Intl. Conference on System Sciences,
different propagation capacity. A small number of links 2013 (Maui, HI, 2013).
are much more crucial than most of the others and by [26] The IEEE 118 Bus Test System. [Online]. Available:
eliminating them the risk of cascading failures can be http://www.ee. washington.edu/research/pstca/.