# Writings

This is a personal blog, where I post comments, errata, and code snippets. Nothing here is peer-reviewed, and should be treated as such.

## 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

$\frac{\partial p}{\partial \theta}(v,q) = \operatorname{diag}(v)\frac{\partial q }{\partial v} - 2 \operatorname{diag}(q),$ $\frac{\partial q}{\partial \theta}(v,p) = -\operatorname{diag}{v} \frac{\partial p}{\partial v} + 2 \operatorname{diag}(p).$

## Can complex power injections be estimated from voltage magnitudes?

In a recent paper we discussed when net complex power injections $$p_i + j q_i \in \mathbb{C}$$ for PQ buses $$i=1,\dots,n$$ can be estimated from measurements of the magnitudes of the voltage phasors $$v_i \triangleq |\overline{v}_i| \in \mathbb{R}$$.

We called this idea "phaseless observability". This idea is useful because synchrophasor measurements (i.e., measurements of the voltage phase angles $$\theta_i$$, $$i=1,\dots,n$$) 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 $$\alpha_i, \ i =1,\dots,n$$, or ratios of active and reactive power.

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

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

$\frac{\partial q_i}{\partial p_i} = \pm \frac{1}{\alpha_i}\sqrt{1-\alpha_i^2}.$

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

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

is invertible.

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

## Checking if an electric power system is radial with the admittance matrix

A power system is "radial" if there are no loops in the circuit.

If we call $$L$$ and $$U$$ the lower and upper off-diagonal components the network admittance matrix $$Y \in \mathbb{C}^{n \times n}$$, then the network is not radial if

$\sum_{i} \sum_j L_{ij} \geq n,\\ \text{where} \ (i,j) \in \{(i,j): L_{ij} \neq 0 \},$

or alternatively,

$\sum_i \sum_j U_{ij} \geq n,\\ \text{where} \ (i,j) \in \{(i,j): U_{ij} \neq 0 \}.$

This function tells you if a PowerModels.jl network model is radial or not.

using LinearAlgebra

"""
Given a network's admittance matrix Y, determine if that network is radial.
In other words, determine if there are no electrical loops.
Params:
"""

n = size(Y)[1]

#Upper and lower off-diagonal elements
U(A::AbstractMatrix) = [A[i] for i in CartesianIndices(A) if i[1]>i[2]]
L(A::AbstractMatrix) = [A[i] for i in CartesianIndices(A) if i[1]<i[2]]

#Get the nonzero upper and lower off diagonal elements
nz_upper = [1 for y_ij in U(Y) if y_ij != 0]
nz_lower = [1 for y_ij in L(Y) if y_ij != 0]

return !(sum(nz_upper)>n-1 || sum(nz_lower) >n-1)
end

println(is_radial(Y))
true