Emergent Failures and Cascades in Power Grids: A Statistical Physics Perspective
Tommaso Nesti,
1
Alessandro Zocca,
2
and Bert Zwart
1
1
CWI, Amsterdam 1098 XG, Netherlands
2
California Institute of Technology, Pasadena, California 91125, USA
(Received 25 July 2017; revised manuscript received 4 April 2018; published 21 June 2018)
We model power grids transporting electricity generated by intermittent renewable sources as complex
networks, where line failures can emerge indirectly by noisy power input at the nodes. By combining
concepts from statistical physics and the physics of power flows and taking weather correlations into
account, we rank line failures according to their likelihood and establish the most likely way such failures
occur and propagate. Our insights are mathematically rigorous in a small-noise limit and are validated with
data from the German transmission grid.
DOI:
10.1103/PhysRevLett.120.258301
Understanding cascading failures in complex networks is
of great importance and has received a lot of attention in
recent years
[1
–
17]
. Despite proposing different mecha-
nisms for their evolution, a common feature is that cascades
are triggered by some
external
event. This initial attack is
chosen either (i) deliberately, to target the most vulnerable
or crucial network component, or (ii) uniformly at random,
to understand the average network reliability. This dis-
tinction led to the insight that complex networks are
resilient to random attacks but vulnerable to targeted
attacks
[7,18,19]
. However, both lead to the
direct
failure
of the attacked network component.
In this Letter, we focus on networks in which edge
failures occur in a fundamentally different manner.
Specifically, we consider networks where fluctuations of
the node inputs can trigger edge failures. The realization
(which we call configuration) of the noise at the nodes not
only is the cause of edge failures but can also impact the
way they propagate in the network.
We present our results in the context of power grids that
transport electricity generated by solar and wind parks. In
power grids, line failures can arise when the network is
driven from a stable state to a critically loaded state by
external factors; intermittent power generation at the nodes
causes random fluctuations in the line power flows,
possibly triggering outages and cascading failures. Thus,
line failures can emerge
indirectly
due to the interplay
between noisy correlated (due to weather) power input at
the nodes, the network structure, and power flow physics.
This interplay is challenging to analyze, yet this problem is
urgent as the penetration of renewable energy sources is
increasing
[20,21]
.
We analyze this interplay using statistical physics and
large-deviations theory. We consider a parsimonious static
stochastic model similar to Ref.
[22]
, introduce a scaling
parameter
ε
describing the magnitude of the noise, and
consider the regime
ε
→
0
. In the limit, we can identify the
most vulnerable lines and explicitly determine the most
likely configuration of power inputs leading to failures and
subsequent propagating failures. These results are validated
using real data for the German transmission network.
Previous works applying large-deviations techniques to
problems in complex networks dynamics, such as epidemic
extinction and biophysical networks, include Refs.
[23,24]
.
We model a transmission network by a connected graph
G
with
n
nodes representing the
buses
and
m
directed edges
modeling
transmission lines
. The nominal values of net
power injections at the nodes are given by
μ
¼f
μ
i
g
i
¼
1
;
...
;n
.
We model the stochastic fluctuation of the power injections
around
μ
, due to variability in renewable generation, by
means of the random vector
p
¼f
p
i
g
i
¼
1
;
...
;n
, which is
assumed to follow a multivariate Gaussian distribution with
density
φ
ð
x
Þ¼
exp
½
−
1
2
ð
x
−
μ
Þ
T
ð
ε
Σ
p
Þ
−
1
ð
x
−
μ
Þ
ð
2
π
Þ
n=
2
det
ð
ε
Σ
p
Þ
1
=
2
;
ð
1
Þ
with
ε
Σ
p
∈
R
n
×
n
being the covariance matrix of
p
. In our
theoretical analysis, we assume that
Σ
p
is known and
let
ε
→
0
.
The Gaussian assumption is debatable, for both solar and
wind. While consistent with atmospheric physics
[25]
and
recent wind park statistics
[26,27]
, different models are
preferred for different timescales
[28
–
31]
. An extension of
our framework to the dynamic model in Ref.
[31]
looks
promising (using the Freidlin-Wentzell theory as in
Ref.
[32]
). For a static non-Gaussian extension, see
[33]
.
Assuming the vector
μ
has zero sum and using the dc
approximation
[20]
, the line power flows
f
¼f
f
i
g
i
¼
1
;
...
;m
are given by
f
¼
Vp
;
ð
2
Þ
where
V
is an
m
×
n
matrix encoding the grid topology and
parameters (i.e., line susceptances). The dc approximation
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=
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=
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=
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is commonly used in transmission system analysis
[52
–
55]
.
More realistic nonlinear models based on ac power flows
[56]
may be analyzed leveraging the contraction princi-
ple
[57]
.
The total net power injected in the network
P
n
i
¼
1
p
i
is
nonzero, as
p
is random. Automated affine response and
redispatch mechanisms take care of this issue in power
grids. Mathematically, this corresponds to a
“
distributed
slack
”
in our model: The total power injection mismatch is
distributed uniformly among all nodes (the matrix
V
accounts for this; see
[33]
).
In view of Eqs.
(1)
and
(2)
, the line power flows
f
also
follow a multivariate Gaussian distribution with mean
ν
and
covariance matrix
ε
Σ
f
. The vector
ν
¼
V
μ
∈
R
m
describes
the nominal line flows, while the covariance matrix
ε
Σ
f
¼
ε
V
Σ
p
V
T
describes the correlations between line
flow fluctuations, taking into account both the correlations
of the power injections (encoded by
Σ
p
) and correlations
created by the network topology due to power flow physics
(Kirchhoff
’
s laws) via
V
.
A line
overloads
if the absolute amount of power flowing
in it exceeds a given
line threshold
. We assume that such
overloads immediately lead to the outage of the corre-
sponding line, to which we will henceforth refer simply as a
line failure
. The rationale behind this assumption is that
there are security relays on high-voltage transmission lines
performing an emergency shutdown as soon as the current
exceeds a dangerous level. Without such mechanisms, lines
may overheat, sag, and eventually trip.
We can express the line flows in units of the line
threshold by incorporating the latter in the definition of
V
[33]
, so that
f
is the vector of
normalized
line power
flows and the failure of line
l
corresponds to
j
f
l
j
≥
1
.We
let the power grid operate on average safely by assuming
that max
l
¼
1
;
...
;m
j
ν
l
j
<
1
, so that only large fluctuations of
line flows lead to failures.
We are most interested in scenarios where power grids
are highly stressed, meaning that the nominal power
injections
f
μ
i
g
i
¼
1
;
...
;n
are such that the corresponding
nominal line power flows
f
ν
l
g
l
¼
1
;
...
;m
are close to their
thresholds. Such a stress could be caused by very high wind
generation
[58]
.
An illustrative scenario is reported in Fig.
1(a)
, which
depicts a snapshot of nominal line flows on the SciGRID
German network
[59]
. SciGRID is a detailed model of the
actual German transmission network with
n
¼
585
buses
and
m
¼
852
lines that we use as a main illustration. The
data set includes load and generation time series, line limits,
grid topology, and generation costs. In our case study, we
obtain
μ
by solving an optimal power flow (OPF) problem
[60]
based on realistic data for wind and solar generation,
and we estimate
ε
Σ
p
using autoregressive moving average
models; for details, see Supplemental Material
[33]
, which
also describes a setting covering conventional controllable
power plants.
We now turn to the analysis of emergent failures and
their propagation using large-deviations theory
[61]
.We
begin by deriving the exponential decay of probabilities of
single line failure events
j
f
l
j
≥
1
for
l
¼
1
;
...
;m
. As line
power flows are Gaussian, we obtain (see Example 3.1 in
Ref.
[61]
) that
I
l
¼
−
lim
ε
→
0
ε
log
P
ε
ðj
f
l
j
≥
1
Þ¼
ð
1
−
j
ν
l
jÞ
2
2
σ
2
l
;
ð
3
Þ
where
σ
2
l
¼ð
Σ
f
Þ
ll
. We call
I
l
the
decay rate
of the failure
probability of line
l
. Thus, for small
ε
, we approximate the
probability of the emergent failure of line
l
as
P
ðj
f
l
j
≥
1
Þ
≈
exp
ð
−
I
l
=
ε
Þ¼
exp
−
ð
1
−
j
ν
l
jÞ
2
2
εσ
2
l
ð
4
Þ
and that of the first emergent failure as
P
ð
max
l
j
f
l
j
≥
1
Þ
≈
exp
ð
−
min
l
I
l
=
ε
Þ
:
ð
5
Þ
These approximations for failure probabilities may not be
sharp, in general, even when
ε
is small, since all terms that
FIG. 1. (a) Nominal line flows
j
ν
l
j
at 11 am. (b) True overload probabilities log
10
P
ðj
f
l
j
≥
1
Þ
at 11 am. (c) Top 5% of most likely lines
to fail (red) at 11 am, according to
(3)
, and nominal injections from renewable sources.
PHYSICAL REVIEW LETTERS
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are decaying subexponentially in
1
=
ε
are ignored.
Nevertheless, Eq.
(4)
is quite useful for ranking purposes,
allowing us to explicitly identify the lines that are most
likely to fail. To verify this empirically, we note that the
expression in Eq.
(4)
depends only on the product
εσ
2
l
¼
ε
ð
V
Σ
p
V
T
Þ
ll
and thus, ultimately, only on the
product
ε
Σ
p
, which in our case study we estimate directly
from the SciGRID data; see
[33]
.
Figure
1(b)
shows the heat map for the exact line failure
probabilities
P
ðj
f
l
j
≥
1
Þ
, for the same day and hour as in
Fig.
1(a)
: It is clear that a larger
j
ν
l
j
does not necessarily
imply a higher chance of failure. Figure
1(c)
depicts the 5%
most likely lines to fail, ranked according to
I
l
. The
ranking based on the large-deviations approximation suc-
cessfully recovers the most likely lines to fail and, in fact,
yields the same ordering as the one based on exact
probabilities
[33]
, thus providing an accurate indicator of
system vulnerabilities.
Figure
1(c)
also illustrates the nominal renewable gen-
eration mix: The buses housing stochastic power injections
have different colors [blue (light blue) for wind offshore
(onshore), yellow for solar] and sizes proportional to the
absolute values of the corresponding nominal injections.
Many vulnerable lines are located where the most renew-
able energy production occurs. However, the interplay
between network topology, power flows physics, and
correlation in power injections caused by weather fluctua-
tions results in a spread-out arrangement of vulnerable
lines, which is hard to infer by looking at nominal
values only.
We proceed with an analysis of how emergent failures
occur, using again large-deviations theory. In particular, we
provide an explicit estimate of the most likely power
injection that caused a specific emergent failure. To this
end, we fix a line
l
and consider the conditional distribu-
tion of
p
, given
j
f
l
j
≥
1
. The mean of this distribution
greatly simplifies as
ε
→
0
to
p
ð
l
Þ
¼
arginf
p
∈
R
n
∶
j
ˆ
e
T
l
Vp
j
≥
1
1
2
ð
p
−
μ
Þ
T
Σ
−
1
p
ð
p
−
μ
Þ
:
ð
6
Þ
If
ν
l
≠
0
, the solution is unique and reads
p
ð
l
Þ
¼
μ
þ
½
sgn
ð
ν
l
Þ
−
ν
l
σ
2
l
Σ
p
V
T
ˆ
e
l
;
ð
7
Þ
where sgn
ð
a
Þ¼
1
if
a
≥
0
and
−
1
otherwise and
ˆ
e
l
∈
R
m
is the
l
th unit vector. As
ε
→
0
, the conditional variance of
p
given
j
f
l
j
≥
1
decreases to 0 exponentially fast in
1
=
ε
,
yielding that the conditional distribution of
p
given
j
f
l
j
≥
1
gets sharply concentrated around
p
ð
l
Þ
[33]
.
We interpret
p
ð
l
Þ
as the
most likely
power injection
profile, conditional on the failure of line
l
. The corre-
sponding line power flow profile
f
ð
l
Þ
¼
Vp
ð
l
Þ
is
f
ð
l
Þ
k
¼
ν
k
þ
½
sgn
ð
ν
l
Þ
−
ν
l
σ
2
l
Cov
ð
f
l
;f
k
Þ
;
∀
k
≠
l
:
ð
8
Þ
As such, our framework provides more explicit information
than the approach in Ref.
[62]
, which approximates the
most likely way events happen using the mode, without
leveraging large deviations. In our validation experiments,
we found that the error between
p
ð
l
Þ
and
E
½
p
jj
f
l
j
≥
1
is
typically less than 1% of the nominal values
[33]
.
A numerical illustration is given in Fig.
2(b)
.
A key finding is that an emergent line failure does not
occur due to large fluctuations only in neighboring nodes
but as a cumulative effect of small unusual fluctuations in
the entire network
“
summed up
”
by power flow physics
and correlations in renewable energy. Such an emergent
failure requires every line flow to be driven to an unusual
state
f
ð
l
Þ
k
, which deviates from the nominal value
ν
k
by an
amount proportional to the covariance Cov
ð
f
l
;f
k
Þ
, in view
of Eq.
(7)
.
FIG. 2. (a) After the emergent failure of line 27 (red), six additional lines (orange) fail, 4 pm. (b) Most likely power injection
p
ð
l
Þ
causing the isolated failure of line 720 (red) and subsequent failures (orange). The bus sizes reflect how much
p
ð
l
Þ
deviates from
μ
at 11
am (red, positive deviations; blue, negative). Left, with correlation in noise; right, without correlation in noise (setting to 0 all the off-
diagonals of
Σ
p
).
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We continue by investigating the propagation of failures,
combining our results describing the most likely power
injections configuration leading to the first failure and the
power flow redistribution in the network afterwards. To this
end, we first differentiate between different types of line
failures, by assessing whether the most likely way for
failure of line
l
to occur is as (i) an
isolated failure
,if
j
f
ð
l
Þ
k
j
<
1
for all line
k
≠
l
, or (ii) a
joint failure
, if there
exists some other line
k
≠
l
such that
j
f
ð
l
Þ
k
j
≥
1
.
Any type of line failure(s) cause(s) a global redistribution
of the line power flows according to Kirchhoff
’
slaws,
which could trigger further outages and cascades. In our
setting, the power injections configuration
p
ð
l
Þ
redistributes
across an altered network
̃
G
ð
l
Þ
(a subgraph of the original
graph
G
) in which line
l
(and possible other lines, in case
of a joint failure) has been removed, increasing stress on the
remaining lines. The way this redistribution happens on
̃
G
ð
l
Þ
is governed by power flow physics, and we assume
that it occurs instantaneously. Extending this to dynamic
models
[63,64]
is a natural future topic, as transient
oscillatory effects may aggravate the impact of line failures.
The power flow redistribution amounts to computing a
new matrix
̃
V
linking the power injections and the new
power flows, which can be constructed analogously to
V
[33]
. The most likely power flow configuration on
̃
G
ð
l
Þ
after redistribution is
̃
f
ð
l
Þ
¼
̃
Vp
ð
l
Þ
.
In the special case of an isolated failure (say, of line
l
),
it is enough to calculate the vector
φ
ð
l
Þ
∈
R
m
−
1
of
(normalized) redistribution coefficients, known as
line
outage distribution factors
(LODF)
[65]
. The quantity
φ
ð
l
Þ
j
takes values in
½
−
1
;
1
, and
j
φ
ð
l
Þ
j
j
represents the
percentage of power flowing in line
l
that is redirected
to line
j
after the failure of the former. The most likely
power flow configuration on
̃
G
ð
l
Þ
after redistribution then
equals
̃
f
ð
l
Þ
¼f
f
ð
l
Þ
k
g
k
≠
l
þ
f
ð
l
Þ
l
φ
ð
l
Þ
, where
f
ð
l
Þ
l
¼
1
depending on the way the power flow is most likely to
exceed the threshold 1. The power flow configuration
̃
f
ð
l
Þ
can be efficiently used to determine which lines sub-
sequently fail, by checking for which
k
we have
j
̃
f
ð
l
Þ
k
j
≥
1
; see
[33]
.
There is much evidence that failures propagate non-
locally in power grids
[66
–
70]
. To analyze this in our
framework, we first consider a ring network with
μ
¼
0
and
Σ
p
¼
I
. In this network, there are two paths along which
power can flow between any two nodes, using the con-
vention that a positive flow corresponds to a counterclock-
wise direction. If line
l
fails, the power originally flowing
on line
l
must now flow on the remaining path in the
opposite direction. To make this rigorous, we show in
Ref.
[33]
that
φ
ð
l
Þ
k
¼
−
1
for every
k
≠
l
. As power flows
must sum to zero by Kirchhoff
’
s law, neighboring lines
tend to have positively correlated power flows, while flows
on distant lines exhibit negative correlations. Hence, the
power injections that make the power flows in line
l
exceed
the line threshold (say, by becoming larger than 1) also
make the power flows in the antipodal half of the network
negative. These will go beyond the line threshold
−
1
after
the power flow redistributes; cf. Fig.
3
.
In the SciGRID example, Fig.
2(a)
shows how the
emergent isolated failure of line
l
¼
27
causes the failure
of six more lines
k
1
;
...
;k
6
, two of which are far way from
the original failure. For validation purposes, we found
numerically that
P
ð
line
k
j
fails
∀
j
¼
1
;
...
;
6
jj
f
27
j
≥
1
Þ
≥
0
.
9987
. Conversely, the failure of line 27 under the nominal
power injection profile leads to only two subsequent
failures. The nontypical input caused other lines to be
more loaded than expected, and these lines get more
vulnerable as the cascades progresses, resulting in more
subsequent failures.
To validate this insight, we have looked at the first two
stages of emergent cascading failures for several IEEE test
networks and compare them with those of classical cascad-
ing failures, obtained using nominal power injection values
rather than the most likely ones and deterministic removal
of the initial failing line; see
[33]
for a precise description of
the experiment. As before, emergent cascades tend to lead
to a higher number of subsequent failures in each stage.
A nondiagonal noise matrix
Σ
p
exacerbates these effects.
Experiments [see Fig.
2(b)
] with our SciGRID case study
suggest that, if there is a correlation in noise, for example,
due to fluctuations in weather patterns, the number of
subsequent failures can become higher. Furthermore, it is
easier for a failure to be triggered by many small disturb-
ances across the network, compared to the case where these
correlations are not taken into account. In the latter case, we
see a more local effect with relatively larger disturbances.
In conclusion, we illustrated the potential of concepts
from statistical physics and large-deviations theory to
analyze emergent failures and their propagation in complex
networks. Exogenous noise disturbances at the nodes,
potentially amplified by correlations, push a complex
network into a critical state in which edge failure may
emerge. Large-deviations theory provides a tool to rank
such failures according to their likelihood and predicts how
such failures most likely occur and propagate. When an
FIG. 3. Left: Most likely power injections
p
ð
l
Þ
leading to the
failure of line
l
(orange), visualized using the color and size of
the nodes (red, positive deviations; blue, negative), together with
power flows
f
ð
l
Þ
k
. Right: Situation after the power flow redis-
tribution with three subsequent failures and the values
̃
f
ð
l
Þ
k
¼
f
ð
l
Þ
k
−
1
,
k
≠
l
.
PHYSICAL REVIEW LETTERS
120,
258301 (2018)
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emergent edge failure occurs, its impact on the network can
be more significant than a purely exogenous failure,
possibly resulting in cascades that propagate quicker than
in a classical vulnerability analysis.
The accuracy of the small noise limit has been validated in
our case study, making the case for applying large-deviations
techniques to more realistic models. In Ref.
[33]
,wepropose
a promising economic application of our approach, showing
how our framework can shed light on the trade-off between
network reliability and societal costs.
We thank the referees for many useful comments, in
particular, for suggesting SciGRID. NWO Vici
639.033.413 and NWO Rubicon 680.50.1529 grants pro-
vided financial support. A. Z. acknowledges the support of
Resnick Sustainability Institute at Caltech.
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