3
magnitude,” and “frequency.” The meaning of the latter two
axes will differ by context. However, the products of these
tensors will be written with the standard matrix multiplication
notation. If Aand Bare both tensors with three axes, then
AB is a new three-axis tensor given by contracting the third
axis of Awith the second axis of B. Put another way, AB
is the tensor obtained by looping through and holding each
index of the first axis of Aand B(resulting in 4standard
matrices each), performing matrix multiplication on each of
the holds, and then re-stacking the resulting matrices back
into a 3-tensor. For further reference, this is the convention
used in the “matmul” function in both NumPy and PyTorch.
IV. THE EVENT-PARTICIPATION DECOMPOSITION
A. Decomposition Properties and Transformation Tricks
In this subsection, we give a set of desired properties
that the event-participation decomposition ought to have, and
present some general tips to obtain such properties. A full
description of the decomposition steps will be provided in the
next subsection.
1) Algorithmic Requirements: There are a myriad of tensor
decomposition methods available for use. Selecting which to
use depended on two key design constraints. First, the method
must have guaranteed convergence over a widely varying
set of tensors. Second, the method should yield independent
participation factors.
The reason for this first constraint is that the dataset used
for training the model involves almost 1000 power system
event tensors, with event causes ranging from downed lines,
generator tripping, lightning strikes, and more. A wide variety
of dynamic phenomena can be observed in this dataset. Thus,
uniform convergence of the chosen tensor decomposition
technique is critical.
The second constraint is not as straightforward as the first.
Recall that each participation factor is a set of samples from
an event-dependent distribution. Orthogonality means that the
distributions corresponding to each co-occurring event signa-
ture are statistically uncorrelated. In the “PMUs-as-samples”
viewpoint, this is the only variable observed in multiple
instances. Unlike participation factors, event signatures are
viewed holistically, not as samples of a random variable. Thus,
the independence of event-participation pairs from one another
can only be tested by the independence of participation factors.
Independence of event-participation pairs has three impor-
tant outcomes:
1) The distributions are less taxed by the curse of dimen-
sionality, and are therefore easier to learn.
2) An outlying or otherwise unrealistic sample of one
participation factor has no effect on the realism of the
other participation factors.
3) A given power system event signature can be perturbed
independently without sacrificing the realism of the
others.
The first two of these outcomes depend only on the in-
dependence of participation factors. The last outcome relies
on assuming that this independence also carries over to the
independence of event signatures.
These latter outcomes are expanded upon next. For the sec-
ond outcome, suppose there are several coinciding event sig-
natures e1, e2,· · · , er. These event signatures are sent through
the learned functions µθto obtain µθ(e1, e2,· · · , er) =
µθ1(e1)µθ2(e2)· · · µθr(er). Then Nvalues are sampled from
this joint distribution and stacked in vectors to obtain partici-
pation factors p1, p2,· · · , pr. If there is an immense outlying
value in p1, then only the outlier in p1needs to be resampled.
If they are not independent, then all rparticipation factors
need to be resampled. As the number of co-occurring event
signatures increases, the likelihood of one of these participa-
tion factors being an outlier increases as well.
For the third outcome, suppose there are rcoinciding
event signatures {ei}i=1,···r, which can be used to obtain
the distributions {µθ(ei)}i=1,··· ,r, and participation factors
{pi}i=1,··· ,r. If the first event signature is perturbed by a small
amount ∆e1, then the log-likelihood of the full event tensor
changes by the following expression:
∆P(x) = log P(e1+ ∆e1)−log P(e1)
+
N
X
i=1
log P(pi
1(e1+ ∆e1)|e1+ ∆e1)−log P(pi(e1)|e1),
where the index iruns over the sampled participation factors.
The first two terms only depends on the perturbed event
signature and the latter two depend only on the adaptability of
the first participation factor map (the terms in the sum, which
should be near zero for a well trained participation factor map).
Independent Component Analysis (ICA) [25] seems to fit
the tensor decomposition method best given this latter algorith-
mic requirement. In ICA, the independent vector components
can be interpreted as participation factors and view the mixing
matrix as event signatures. Unfortunately, ICA fails to con-
verge on about 10% of the data tensors. As such, uncorrelated
(orthogonal) participation factors instead of truly independent
ones will be used in this study. A variant of Singular Value
Decomposition (SVD) was applicable. While SVD does yield
orthogonal event signatures, this property is not required since
distinct event signatures may overlap in time while still being
statistically independent in the PMU space. Removing this
requirement means that it is appropriate to either perturb event
signatures or replace them entirely - so long as any changes
made preserve the orthogonality of the participation factors.
2) The QR Re-Orthogonalization Trick: The participation
factors can always be re-orthogonalized with a QR factor-
ization. To be specific, suppose there is an existing Event-
Participation decomposition X=P ET. If Eis changed to
˜
E, then a change in participation factor is induced resulting
in ˜
P(to maintain the equality). If a QR decomposition is
performed on ˜
P, then X=QR ˜
ET. Denoting Qas the new
set of participation factors and R˜
ETas the new set of event
signatures, then we have a new orthogonal-satisfying event
participation decomposition. This does mix the event signa-
tures. However, since Ris upper-triangular, some subspace
uniqueness can be kept by placing signatures that should be
preserved at the bottom of the event-signature matrix.