  Fast and accurate knowledge of power flows and power injections is needed for
a variety of applications in the electric grid. Phasor measurement units (PMUs)
can be used to directly compute them at high speeds; however, a large number of
PMUs will be needed for computing all the flows and injections. Similarly, if
they are calculated from the outputs of a linear state estimator, then their
accuracy will deteriorate due to the quadratic relationship between voltage and
power. This paper employs machine learning to perform fast and accurate flow
and injection estimation in power systems that are sparsely observed by PMUs.
We train a deep neural network (DNN) to learn the mapping function between PMU
measurements and power flows/injections. The relation between power flows and
injections is incorporated into the DNN by adding a linear constraint to its
loss function. The results obtained using the IEEE 118-bus system indicate that
the proposed approach performs more accurate flow/injection estimation in
severely unobservable power systems compared to other data-driven methods.
