  Voltage phase angle measurements are often unavailable from sensors in
distribution networks and transmission network boundaries. Therefore, this
paper addresses the conditions for estimating sensitivities of voltage
magnitudes with respect to complex (active and reactive) electric power
injections based on sensor measurements. These sensitivities represent
submatrices of the inverse power flow Jacobian. We extend previous results to
show that the sensitivities of a bus voltage magnitude with respect to active
power injections are unique and different from those with respect to reactive
power. The classical Newton-Raphson power flow model is used to derive a novel
representation of bus voltage magnitudes as an underdetermined linear operator
of the active and reactive power injections; parameterized by the bus power
factors. Two conditions that ensure the existence of unique complex power
injections given voltage magnitudes are established for this underdetermined
linear system, thereby compressing the solution space. The first is a
sufficient condition based on the bus power factors. The second is a necessary
and sufficient condition based on the system eigenvalues. We use matrix
completion theory to develop estimation methods for recovering sensitivity
matrices with varying levels of sensor availability. Simulations verify the
results and demonstrate engineering use of the proposed methods.
