Failure Localization in Power Systems via Tree Partitions
Linqi Guo, Chen Liang, Alessandro Zocca, Steven H. Low, and Adam Wierman
⇤
Computing and Mathematical Sciences, California Institute of Technology
1200 E. California Blvd, Pasadena, CA, 91125, USA
Email:
{
lguo, cliang2, azocca, slow, adamw
}
@caltech.edu
.
ABSTRACT
Cascading failures in power systems propagate non-locally,
making the control and mitigation of outages extremely hard.
In this work, we use the emerging concept of the
tree parti-
tion
of transmission networks to provide an analytical char-
acterization of line failure localizability in transmission sys-
tems. Our results rigorously establish the well perceived in-
tuition in power community that failures cannot cross bridges,
and reveal a finer-grained concept that encodes more precise
information on failure propagations within tree-partition re-
gions. Specifically, when a non-bridge line is tripped, the
impact of this failure only propagates within well-defined
components, which we refer to as
cells
, of the tree par-
tition defined by the bridges. In contrast, when a bridge
line is tripped, the impact of this failure propagates
globally
across the network, a
↵
ecting the power flow on all remain-
ing transmission lines. This characterization suggests that it
is possible to improve the system robustness by
temporarily
switching o
↵
certain transmission lines, so as to create more,
smaller components in the tree partition; thus spatially lo-
calizing line failures and making the grid less vulnerable to
large-scale outages. We illustrate this approach using the
IEEE 118-bus test system and demonstrate that switching
o
↵
a negligible portion of transmission lines allows the im-
pact of line failures to be significantly more localized without
substantial changes in line congestion.
1. INTRODUCTION
Power system reliability is a crucial component in the
development of sustainable modern power infrastructure.
Recent blackouts, especially the 2003 and 2012 blackouts
in Northwestern U.S. [1] and India [2], demonstrated the
devastating economic impact a grid failure can cause. In
even worse cases, where facilities like hospitals are involved,
blackouts pose direct threat to people’s health and lives.
Because of the intricate interactions among power sys-
tem components, outages may cascade and propagate in a
very complicated, non-local manner [3–5], exhibiting very
⇤
This work has been supported by Resnick Fellowship,
Linde Institute Research Award, NWO Rubicon grant
680.50.1529., NSF grants through PFI:AIR-TT award
1602119, EPCN 1619352, CNS 1545096, CCF 1637598,
ECCS 1619352, CNS 1518941, CPS 154471, AitF 1637598,
ARPA-E grant through award DE-AR0000699 (NODES)
and GRID DATA, DTRA through grant HDTRA 1-15-1-
0003 and Skoltech through collaboration agreement 1075-
MRA.
Copyright is held by author/owner(s).
di
↵
erent patterns for di
↵
erent networks [6]. Such complex-
ity originates from the interplay between network topology
and power flow physics, and is aggravated by possible hid-
den failures [7] and human errors [8]. This complexity is the
key challenge for research into the modeling, control, and
mitigation of cascading failures in power systems.
There are three traditional approaches for characterizing
the behavior of cascades in the literature: (i) using sim-
ulation models [9] that rely on Monte-Carlo approaches to
account for the steady state power flow redistribution on DC
[5,8,10,11] or AC [12,13] models; (b) studying purely topo-
logical models that impose certain assumptions on the cas-
cading dynamics (e.g., failures propagate to adjacent lines
with high probability) and infer component failure propa-
gation patterns from graph-theoretic properties [14–16]; (c)
investigating simplified or statistical cascading failure dy-
namics [3,17–19]. In each of these approaches, it is typically
challenging to make general inferences across di
↵
erent sce-
narios due to the lack of structural understanding of power
redistribution after line failures.
A new approach has emerged in recent years, which seeks
to use spectral properties of the network graph in order to
derive precise structural properties of the power system dy-
namics, e.g., [20–22]. The spectral view is powerful as it of-
ten reveals surprisingly simple characterizations of the com-
plicated system behaviors. In the cascading failure context,
a key result from this approach is about the
line outage
distribution factor
[6, 23]. Specifically, it is shown in [21]
that the line outage distribution factor is closely related to
transmission graph spanning forests.
While this literature has yet to yield a precise characteri-
zation of cascades, it has suggested a new structural repre-
sentation of the transmission graph called the
tree partition
,
which is particularly promising. For example, [21] shows
that line failures in a transmission system cannot propagate
across di
↵
erent regions of the tree partition (for more back-
ground on the tree partition, see Section 2).
Contributions of this paper:
We prove that the tree
partition proposed in [21] can be used to provide an analyti-
cal characterization of line failure localizability, under a DC
power flow model, and we show how to use this characteriza-
tion to mitigate failure cascades by temporarily switching o
↵
asmallnumberoftransmissionlines.
Our results rigorously
establish the well perceived intuition in power community
that failures cannot cross bridges, and reveal a finer-grained
concept that encodes more precise information on failure
propagations within tree-partition regions. This work builds
on the recent work focused on the line outage distribution
factor, e.g., [6, 21, 24], and shows that the tree partition is
a particularly useful representation of this factor, one that
Performance
Evaluation
Review,
Vol.
46,
No.
2, September
2018
57
captures many aspects of how line failures can cascade.
Our formal characterization of localizability is given in
Theorem 3. It shows that the impact of tripping a non-
bridge line only propagates within well-defined components,
which we refer to as cells, inside the tree partition regions.
In constrast, the failure of bridge lines, in normal operating
conditions, propagate globally across the network and im-
pact the power flow on all transmission lines. In order to
prove these results, we depend on properties of the tree par-
tition proved in [21] as well as some novel properties derived
in [25]. Further, we make use of the block decomposition
of tree partition regions to completely eliminate the graph
spanning forests among distinct cells, which in the spectral
view means failure localization [21]. Lastly, we apply classi-
cal techniques from algebraic geometry to address potential
pathological system specifications and establish our results.
The characterization we provide in Theorem 3 yields many
interesting insights for the planning and management of
power systems and, further, suggests a new approach for
mitigating the impact of cascading failures. Specifically, our
characterization highlights that switching o
↵
certain trans-
mission lines
temporarily
in responds to the real-time injec-
tion profile can lead to more, smaller regions/cells, which
localize line failures, thus making the grid less vulnerable
against line outages. In Section 4, we illustrate this approach
using the IEEE 118-bus test system. We demonstrate that
switching o
↵
only a negligible portion of transmission lines
can lead to significantly better control of cascading failures.
Further, we highlight that this happens without significant
increases in line congestion across the network.
2. PRELIMINARIES
We use the graph
G
=(
N
,
E
) to describe a power trans-
mission network, where
N
=
{
1
,...,n
}
is the set of buses
and
E
⇢
N
⇥
N
is the set of transmission lines. The terms
bus/vertex and line/edge are used interchangeably. An edge
in
E
between vertices
i
and
j
is denoted either as
e
or (
i, j
).
We assume
G
is connected and simple, and assign an arbi-
trary orientation over
E
so that if (
i, j
)
2
E
then (
j, i
)
/
2
E
.
The line susceptance of
e
is denoted as
B
e
and the branch
flow on
e
is denoted as
P
e
. The susceptance matrix is defined
to be the diagonal matrix
B
=diag(
B
e
:
e
2
E
).
We denote the power injection and phase angle at bus
i
as
p
i
and
✓
i
,anduse
n
and
m
to denote the number of
buses and transmission lines in
G
. The vertex-edge incidence
matrix of
G
is the
n
⇥
m
matrix
C
defined as
C
ie
=
8
>
<
>
:
1 if vertex
i
is the source of
e