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Can complex power injections be estimated from voltage magnitudes?

In a recent paper we discussed when net complex power injections pi+jqi∈Cp_i + j q_i \in \mathbb{C} for PQ buses i=1,…,ni=1,\dots,n can be estimated from measurements of the magnitudes of the voltage phasors vi≜∣v‾i∣∈Rv_i \triangleq |\overline{v}_i| \in \mathbb{R}.

We called this idea "phaseless observability". This idea is useful because synchrophasor measurements (i.e., measurements of the voltage phase angles θi\theta_i, i=1,…,ni=1,\dots,n) are often unavailable, especially in distribution systems (see my recent talk for some additional exposition on this).

We came up with two conditions that use:

  1. The bus power factors αi, i=1,…,n\alpha_i, \ i =1,\dots,n, or ratios of active and reactive power.

  2. The voltage magnitude-power sensitivity matrices ∂v∂p\frac{\partial v}{\partial p} and ∂v∂q\frac{\partial v}{\partial q},

  3. and the implicit reactive-active power sensitivities ∂q∂p\frac{\partial q}{\partial p}, where

∂qi∂pi=±1αi1−αi2. \frac{\partial q_i}{\partial p_i} = \pm \frac{1}{\alpha_i}\sqrt{1-\alpha_i^2}.

You can check if the network is "phaselessly observable" at its current operating point by checking if the square matrix

S~=∂v∂p+∂v∂q∂q∂p \tilde{S} = \frac{\partial v}{\partial p} + \frac{\partial v}{\partial q} \frac{\partial q}{\partial p}

is invertible.

Check out all of the code in the full repository available here.

The broader idea of phase retrieval in power systems

In this paper, we discuss how to estimate the voltage phase angles θ\theta of a power system from measurements of the voltage magnitudes vv. 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

∂p∂θ(v,q)=diag⁡(v)∂q∂v−2diag⁡(q), \frac{\partial p}{\partial \theta}(v,q) = \operatorname{diag}(v)\frac{\partial q }{\partial v} - 2 \operatorname{diag}(q), ∂q∂θ(v,p)=−diag⁡v∂p∂v+2diag⁡(p), \frac{\partial q}{\partial \theta}(v,p) = -\operatorname{diag}{v} \frac{\partial p}{\partial v} + 2 \operatorname{diag}(p),

which says that the change in power injections with respect to changes in voltage phase angles depend only on the voltage magnitudes and power injections, given knowledge of the power-voltage magnitude sensitivities ∂p∂v\frac{\partial p}{\partial v} and ∂q∂v\frac{\partial q}{\partial v}.