Phase retrieval in power systems

In this paper we discuss how to estimate the voltage phase angles \(\theta_i\) of a power system from measurements of the voltage magnitudes \(v_i\). This is called "phase retrieval" in the signal processing literature. We derived a circuit law that describes the power-voltage phase angle submatrices of the power flow Jacobian as a function of the voltage magnitudes, active power injections, and reactive power injections. This law is given as

\[\begin{aligned} \frac{\partial p}{\partial \theta}(v,q) = \operatorname{diag}(v)\frac{\partial q }{\partial v} - 2 \operatorname{diag}(q),\\ \frac{\partial q}{\partial \theta}(v,p) = -\operatorname{diag}{v} \frac{\partial p}{\partial v} + 2 \operatorname{diag}(p). \end{aligned}\]