IEEE Power & Energy Magazine - July/August 2020 - 68

in both obvious and complicated ways, symbolized by the
white soap film on the rings in Figure 1.
Borromean rings are either whole or broken, but we do
not know how to measure how stressed the generation and
transmission equipment is. Further, there is no way to know
if everything in the two ancillary elements (the control and
protection devices as well as the practices and procedures) is working properly, as they are largely unobservable.
But the power system has built-in redundancies. Unlike the
Borromean rings, the power system usually holds together
when none of its three elements functions perfectly, yet all
three are necessary to its operation. So, to conclude this illuminating, but imperfect, analogy, it is hard to know when
the power system is relatively secure or at immediate risk of
a cascading failure.
Based on past events, Table 1 lists three conditions that
seem necessary for a cascading blackout to occur. Every
cascading blackout we know of met all three conditions.
More robust systems withstand greater stress or more
extreme failures.

A Cascading Outage Network
Graph (network) theory arguably began with Euler's 1736
proof that it was impossible to stroll across all seven bridges
in the city of Konigsberg without crossing any of them twice.
Most of network theory's development occurred in the last
century. Figure 2 displays a generic model of a network. Networks are composed of vertices (nodes) and edges connecting the nodes. For our purposes, network theory is motivated
by the three following observations:
1) Large networks are common.
2) Large networks are too big to study element by element.
3) "Big is different." Large networks behave differently
than small networks.
What constitutes a large network is not defined in the literature, as systems grow in different ways. In Six Degrees,

Vertex
Edge

figure 2. Network nomenclature. (Inspired by M.E.J.
Newman.)
ieee power & energy magazine

Power-Flow Network
On a daily basis, power system operators and transmission
planners use power-flow software of various kinds. These
programs simulate how power produced by generators flows
through the network to the customers, in accordance with the
laws of electric circuits. In these simulations, the generators
and customers are connected to the network at its vertices,
called buses. The edges are power-transmitting branches,
mainly lines and transformers. The math that drives this
software operates on a large matrix of numbers, called Ybus
(bold type denotes matrices). Ybus represents the electrical
properties of each branch and identifies the nodes to which
each branch is connected.

Cascading Failure Network

Network Theory

68

Duncan J. Watts observed that the problem with systems
like the power grid is that they are built of many components whose individual behavior is well understood but
whose collective behavior can sometimes be orderly and
sometimes chaotic, confusing, and even destructive. For
large networks, one must therefore study their structural
properties and metrics.

The study of cascading outages needs a different network
model because the objective is to study how branch failures propagate throughout the system, rather than how
power flows through the various branches. It turns out that
the failure of essentially any nonradial branch is linked to
essentially every other such branch, not by individual lines
but by the network as a whole. In the cascading blackout
network, the vertices are branches (lines, transformers, and
interfaces), and the edges are line-outage distribution factors (LODFs) [or, more pronounceably, distribution factors
(DFAXes)]. This is radically different from the vertices
and nodes of the power-flow network. But like the powerflow models, the cascading failure model is built around a
large matrix, this one consisting of all the DFAXes. Each
DFAX specifies how much the outage of a particular branch
changes the flow in another (monitored) branch, with the
generation and demand at the various buses not changing.
The higher the postoutage flow on a monitored branch, the
greater the risk of cascading.
The cascading failure network calculations operate on a
matrix consisting of all the DFAXes. This is similar in concept to the power-flow software operating on Ybus, although
the math is very different.
DFAXes usually have values between −1 and +1. A DFAX
of zero value means that the outage of a particular branch will
not change the flow on the monitored branch. A value of 0.5
(or −0.5) means that half of the flow carried by the outaged
branch will be added to (or subtracted from) the monitored
branch's precontingency flow. The DFAX matrix is much
larger than the Ybus matrix and fundamentally different.
Commercial power-flow programs compute the DFAXes
from Ybus using conventional circuit analysis. DFAXes are
july/august 2020



IEEE Power & Energy Magazine - July/August 2020

Table of Contents for the Digital Edition of IEEE Power & Energy Magazine - July/August 2020

Contents
IEEE Power & Energy Magazine - July/August 2020 - Cover1
IEEE Power & Energy Magazine - July/August 2020 - Cover2
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