IEEE Power & Energy Magazine - May/June 2022 - 40

converters and renewables makes this computational effort
significantly larger for the following two reasons:
1) Power electronic converters (e.g., from wind turbines,
solar photovoltaics, batteries, high-voltage dc lines,
electric vehicles, and so on) have much more complex
models based on differential equations, with time
constants that interfere with the electromagnetic transients
of transmission lines.
2) Renewables add significant degrees of uncertainty
with their fluctuating production, meaning that we
need to assess a much larger set of possible operating
points for each disturbance.
Neural networks could assist here by substituting traditional
numerical solvers and providing a fast estimate of the actual
solution (up to 1,000 times faster). This creates the following
two major benefits:
1) A much faster estimate (within a few milliseconds) of
what the frequency will be 2 or 5s after the disturbance,
enabling the activation of necessary counteractions
earlier.
2) The ability to assess up to 1,000 potentially critical
disturbances in the same time frame that conventional
software will calculate only a single disturbance. This
allows us to screen a very large set of disturbances
for many different operating points and select only the
most critical ones, which we can then assess in more
detail and with higher accuracy than with conventional
software.
To train such a neural network with conventional methods,
we need to provide data points (the black dots in
Figure 6) and fit network parameters so they provide a
prediction that matches the data points. As the training
process progresses, the prediction function (orange line)
becomes more sophisticated and fits the data points better.
However, when comparing the prediction to the function
that produced the data points (i.e., ground truth), which
is governed by the model's differential equations, we see
clear mismatches for all three examples. The prediction in
Figure 6(c), in particular, highlights what is referred to as
overfitting. The prediction matches all data points nearly
perfectly but shows large approximation errors between
the data points.
Now imagine that we want our neural network to predict
a series of different trajectories that correspond to different
disturbances and operating points. Poor fitting or overfitting
becomes a major issue when checking whether the underfrequency
or overfrequency thresholds have been violated.
One way to achieve a better fit is to use more data points
from more trajectories. However, this can be difficult if, for
example, the data are obtained from measurement devices
that do not sample the frequency to an adequate degree of
granularity. It also can be computationally expensive if we
need to generate more trajectories with simulation software
to obtain more points.
Recent advances in neural network training, however,
offer us a new way. We now can numerically compute the
derivatives of neural network outputs with respect to inputs
during the neural network process. If the frequency is a
neural network output, we can calculate, for example, the
derivative of the frequency over time inside the network
training. This enables us to include differential equations,
such as the swing equation, inside the neural network training
to drive the training procedure to a neural network
where the frequency (which is a neural network output) and
its derivative validate the swing equation. This additional
requirement leads to a much better approximation of the
true functions.
Figure 7 (dashed black lines) shows the derivatives at
selected points. Intuitively speaking, by adding physical
equations inside the neural network training, the network
prediction needs to cross a data point (black dot) and create
a trajectory that fits the " shape " of its neighborhood.
Increases During Training
" Underfitting "
Frequency
Frequency
Frequency
" Overfitting "
Time
Time
(a)
(b)
(c)
figure 6. The training process of a simple regression neural network, showing the evolution of network output during
training (orange) versus ground truth (blue). (a) At the start of the training, the neural network output is " underfitting. "
(b) As the training progresses, the neural network output fits the data points better. (c) If the training continues,
the neural network output is " overfitting. "
40
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