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Linearizing the Power Flow Equations

Over the past few years, I've given several tutorial presentations on linearizing the AC power flow equations. I've gotten some positive feedback on this, so I wanted to share some of the key ideas here on my blog.

The "Flat Start" Linearization

The "flat start" linearization approximates the power flow equations around a nominal operating point where all voltage magnitudes are 1.0 per unit and all voltage angles are 0 radians. This linearization is useful for gaining insights into power system behavior, and also has very nice graph-theoretic properties that make it useful to work with.

You may have seen this approximation before–it looks something like this:

$$\begin{bmatrix} \boldsymbol{p}\\ \boldsymbol{q} \end{bmatrix} \approx \begin{bmatrix} \boldsymbol{G} & -\boldsymbol{B}\\ -\boldsymbol{B} & -\boldsymbol{G} \end{bmatrix} \begin{bmatrix} \boldsymbol{v} - \mathbf{ 1}\\ \boldsymbol{\theta} \end{bmatrix};$$

where $\boldsymbol{G}$ is the network conductance matrix and $\boldsymbol{B}$ is the network susceptance matrix; $\boldsymbol{p}$ and $\boldsymbol{q}$ are the active and reactive power injection vectors; $\boldsymbol{v}$ is the voltage magnitude vector; $\boldsymbol{\theta}$ is the voltage phase angle vector; and $\mathbf{ 1}$ is the vector of all ones.

In particular, $\boldsymbol{G}$ and $\boldsymbol{B}$ are the real and imaginary parts of the admittance matrix $\boldsymbol{Y} = \boldsymbol{G} + j\boldsymbol{B} \in \mathbb{C}^{n \times n}$; moreover, $\boldsymbol{G}$ is a positive-semidefinite graph Laplacian matrix and $\boldsymbol{B}$ is a negative-semidefinite graph Laplacian matrix for typical power systems.

Deriving the Flat Start Linearization from First Principles

What you may not have seen before is:

1. how to derive (1) from the full AC power flow equations and

2. how (1) is equivalent to the LinDistFlow approximation and the DC power flow approximation.

You can access my tutorial slides for a detailed derivation of the flat start linearization here:

Tutorial: Linearizing the Power Flow Equations (PDF)

For further reading, check out the references to the relevant literature at the end of the slides.

Other resources

• Chris Yeh's blog post on DistFlow and LinDistFlow

• Vassilis Kekatos' lecture notes on DistFlow and LinDistFlow

• Steven Low's lecture notes on power system analysis

• Dan Molzahn's monograph

• Collection of my past talk slides

• The appendix of my paper on admittance matrix concentration, which contains a derivation of the flat start linearization.

• Self-contained tutorial note on the flat start linearization (coming soon!)