  Phase retrieval is a prevalent problem in digital signal processing and
experimental physics that consists of estimating a complex signal from
magnitude measurements. This paper expands the classical phase retrieval
framework to electric power systems with unknown network models and limited
access to observations of voltage magnitudes, active power injections, and
reactive power injections. The proposed method recovers the phase angles and
the power-phase angle submatrices of the AC power flow Jacobian matrix. This is
made possible by deriving topology and parameter-free expressions for the
structural symmetries of the power flow Jacobian that do not depend on the
phase angles. These physical laws provide structural constraints for the
proposed phase retrieval method. The paper then presents sufficient conditions
for guaranteed recovery of the voltage phase angles, which also depend solely
on voltage magnitudes, active power injections, and reactive power injections.
The method offers two significant benefits: both estimating the voltage phase
angles and recovering the power flow Jacobian matrix, a basis for approximating
the power flow equations. Simulations on widely studied open-source test
networks validate the findings.
