  High impedance faults (HIFs) in distribution grids may cause wildfires and
threaten human lives. Conventional protection relays at substations fail to
detect more than 10\% HIFs since over-currents are low and the signatures of
HIFs are local. With more $\mu$PMU being installed in the distribution system,
high-resolution $\mu$PMU datasets provide the opportunity of detecting HIFs
from multiple points. Still, the main obstacle in applying the $\mu$PMU
datasets is the lack of labels. To address this issue, we construct a
physics-informed convolutional auto-encoder (PICAE) to detect HIFs without
labeled HIFs for training. The significance of our PICAE is a physical
regularization, derived from the elliptical trajectory of voltages-current
characteristics, to distinguish HIFs from other abnormal events even in highly
noisy situations. We formulate a system-wide detection framework that merges
multiple nodes' local detection results to improve the detection accuracy and
reliability. The proposed approaches are validated in the IEEE 34-node test
feeder simulated through PSCAD/EMTDC. Our PICAE outperforms the existing works
in various scenarios and is robust to different observability and noise.
