IEEE Power & Energy Magazine - July/August 2020 - 70

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.
cascading due to an outage. One measure of vulnerability,
RankV, is the highest postoutage flow on the branch, considering all possible outages, one at a time. This can be picked
right out of the gold field. The highest number in the bottom
row of the gold field, 1.23, is the maximum postoutage flow
(RankV) of branch 3, considering all (both) outages. For
the next row up, the highest number is 0.94, the RankV of
branch 2. And from the top gold row, the RankV of branch 1
is 0.49. Each of these numbers, 0.49, 0.94, and 1.23, is listed
in the leftmost column of Table 3. These are the maximum
postoutage flows on each monitored branch in terms of the
branch's rating, a measure of its stress. So, our first app finds
the maximum loading in each row.
Another measure of vulnerability, DegreeV, is the number of outages that will load each branch above some worrisome threshold. We will use 0.75 as the threshold here. The
DegreeV can be found easily from the gold field of Table 3 using
a simple counting app. Just one outage will load branch 3
over the threshold, so the DegreeV is one for branch 3. Similarly, the DegreeV is two for branch 2 and zero for branch 1.
The DegreeV metric values for the three monitored lines are
in the second column in the tan field.
Why would it be reasonable to use a threshold other than
1? In the 1965 and 2003 U.S./Canada cascading blackouts
described previously, the cascading began below the official
ratings of the lines due to hidden failures. As discussed later
in this article, the precise value of the threshold is not critical.
We have presented two apps, maximizing and counting,
to define the reasonable vulnerability stress metrics that
measure the risk of particular branches overloading. We
will next use these two apps to define the metrics that measure how critical particular outages are, which is quite a
different question.

Criticality Metrics
In the rightmost column of the gold field in Table 4, 0.87 is
the maximum load on any of the three monitored branches
after the outage of branch 3. So the criticality rank (RankC),
of the outage of branch 3 is 0.87. In the left gold column,
the maximum load (RankC) on any monitored branch after
the outage of branch 1 is 1.23. These two RankC values are
recorded in the blue area of Table 4 in the RankC row.
We next use the counting app to compute the degree of
criticality (DegreeC) of the outages. After the outage of
branch 3, one monitored branch will be loaded above 0.75.
After the outage of branch 1, two monitored branches will be
loaded over 0.75. These two DegreeC values, one and two,
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are recorded in the bottom row (DegreeC) of the blue area
of Table 4.
In summary, two simple apps, maximizing and counting, are applied to the rows and columns of the gold field
of Table 3. The resulting metrics measure the vulnerability
of the monitored lines to cascading and the criticality of the
various outages. A mnemonic may be helpful: a vulnerable
branch is a victim, and a critical branch is a culprit. Vulnerability metrics are computed by rows of the gold-field matrix
of postoutage flows and the criticality metrics by columns.
The four rank and degree metrics are complementary measures of the cascading risk of a network in terms of individual branches and outages. The evaluation of the metrics
across the network gives a multidimensional measurement
of the overall cascading risk.
We have gone to great lengths to show how Tables 3
and 4 were developed. Having computed the stress metrics,
planners and operators may make special efforts to protect
the most vulnerable and critical branches. Possible actions
include verifying the branch ratings, scheduling maintenance activities for off-peak periods, and reducing the flows
on these branches. We will return to this topic in discussing
our WI study.

Former Studies and Disclaimer
The analyses of stress for networks in the eastern United
States and Peru were presented previously (see Merrill and
Feltes in the "For Further Reading" section). Here we discuss in-depth analyses of the WI of North America, roughly
the contiguous United States and Canada west of the Rocky
Mountains (omitting the northern-most portions) and part of
northern Mexico. Preliminary results were presented previously and can be found in Hossain et al. in the "For Further
Reading" section.
Although we discuss weaknesses of the network in certain conditions, we are not critical of the operation of the WI.
The problems described are typical in the industry. Also,
the WI system evolves constantly as facilities are added and
upgraded. The results we present do not necessarily reflect the
state of the system at any other time or for other conditions.

Western Interconnection Study
The WI study conducted by the WECC and the University
of Utah used high-quality power-flow base cases, which are
listed in Table 5. The 2012 and 2016 cases were built by WI
utilities and the WECC for operating studies and are not
snapshots of moments in time. They are highly realistic and
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IEEE Power & Energy Magazine - July/August 2020

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