IEEE Power & Energy Magazine - July/August 2020 - 69

well known and often used in planning studies. What is new
here is using them to define a failure network and derive
metrics for network stress. These metrics, in turn, are used
to assess the risk of a cascading blackout. Most nonradial
branches are both monitored and outaged. The computations
are practical even for the largest systems.

Vulnerability Metrics
The vulnerability of a branch is a measure of how its flow
is affected by outages of the other branches in the system.
This measure of stress is reasonable, as cascading always
begins with one outage causing other branches to become
highly loaded and fail. Two vulnerability metrics have
been defined: vulnerability rank (RankV) and vulnerability
degree (DegreeV).
For illustration, the postoutage flows for a small network
were computed from DFAXes and preoutage flows. We skip
the computations as they are not new or interesting, but the
results are (see Table 2).
Two branches are outaged, one at a time. The right-most
column in the gold field states that after an outage of branch 3,
the three monitored branches will carry 115, 443, and 0 MW,
respectively. The column just to the left in the gold field
shows what happens after the outage of branch 1: the three
monitored branches will carry 0, 481, and 125 MW. Most
branches are both monitored and outaged, but there may be
exceptions. In Table 2, branch 2 is monitored but not outaged.
Are the postoutage flows in Table 2 a problem? That
depends on how they compare to each monitored branch's
rating, given in the first column of the table. (Parenthetically,
line ratings depend on such parameters as the thickness of
the conductors and their height above ground or obstacles.
Transformer ratings are provided by the manufacturer but
can be increased by improving the transformer cooling or by
limiting the hours they are heavily loaded, as transformers
can get hot enough to be damaged when they are heavily
loaded for long periods.)
For instance, in Table 2, the outage of branch 1 will cause
125 MW to flow in branch 3. This is greater than branch 3's
102-MW rating. Expressed differently, the postoutage flow
will be 125/102 = 1.23 times branch 3's rating.
The postoutage flows in megawatts (the gold field in Table 2)
are converted to fractions of the branch ratings in the gold
field of Tables 3 and 4. (See the 1.23 calculated above in the
lower left of the gold field in each table.) The rest of the gold
fields of Tables 3 and 4 is calculated from the gold field of
Table 2 exactly the same way.
The simple gold fields in Tables 3 and 4 are the heart
and soul of this article. For each of the branches we monitor
through calculations, the gold field in Table 3 tells how it
will be loaded after every outage, one outage at a time. And
Table 4 tells how seriously each possible outage will affect
the monitored branches.
The gold field can be huge for a real system. We found two
simple but powerful apps to mine very useful information
july/august 2020

from all of this data on how stressed the branches and the
network are as a whole. Tables 3 and 4 illustrate these two
apps. They can be defined mathematically, but TablesĀ 3
and 4 summarize the branch calculations. Later in this article, we show how to extract the metrics of system stress from
the branch metrics.
Our first stress metrics measure vulnerability-how
much a particular branch is at risk of overloading and

table 2. Postoutage flows and ratings.
Monitored Branches

Outaged Branches

Ratings (MW)

Branches

1

3

235

1

0

115

512

2

481

443

102

3

125

0

Postoutage flows (MW)

table 3. Postoutage flows and vulnerability metrics.
Vulnerability (Monitored Branches)

Outaged Branches

RankV

DegreeV

Branches

1

3

0.49

0

1

0

0.49

0.94

2

2

0.94

0.87

1.23

1

3

1.23

0

Postoutage flows
(fractions of monitored
branch ratings)
RankV = the maximum postoutage flow in each row.
DegreeV = the number of postoutage flows greater than the
threshold in each row.
Threshold = 0.75 or 75% of the monitored branch ratings.

table 4. Postoutage flows and criticality metrics.
Criticality (Outaged Branches)

Monitored
branches

RankC

1.23

0.87

DegreeC

2

1

Branches

1

3

1

0

0.49

2

0.94

0.87

3

1.23

0

Postoutage flows
(fractions of monitored
branch ratings)
RankC = the maximum postoutage flow in each column.
DegreeC = the number of postoutage flows greater than the
threshold in each column.
Threshold = 0.75 or 75% of the monitored branch ratings.
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IEEE Power & Energy Magazine - July/August 2020

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