2. Notation

This section introduces and summarizes the mathematical notation used throughout this manual.

This notation is consistent with what was used in the MP-Element technical note, MATPOWER Technical Note 5 [TN5] where you can find more detail.

Styles

$x,\theta$	real scalars
$\mathrm{x},\mathrm{\theta }$	complex scalars
$\mathit{x},\mathit{\theta }$	real vectors
$\mathbf{x},\mathbf{\theta }$	complex vectors
$\mathit{X},\mathit{\Theta }$	real matrices
$\mathbf{X},\mathbf{\Theta }$	complex matrices
$x,\mathrm{x},\mathit{x},\mathbf{x},\mathit{X},\mathbf{X}$	variables, functions
$\underset{―}{x},\underset{―}{\mathrm{x}},\underset{―}{\mathit{x}},\underset{―}{\mathbf{x}},\underset{―}{\mathit{X}},\underset{―}{\mathbf{X}}$	constants, parameters [1]
$\hat{\mathit{x}},\hat{\mathbf{x}},\hat{\mathit{X}},\hat{\mathbf{X}}$	selected rows of interest of $\mathit{x},\mathbf{x},\mathit{X},\mathbf{X}$, respectively [2]

Operators

$\left[{}^{\setminus }{\mathbf{a}}_{\setminus }\right]$	diagonal matrix with vector $\mathbf{a}$ on the diagonal
${\mathbf{A}}^{\mathsf{T}}$	(non-conjugate) transpose of matrix $\mathbf{A}$
${\mathrm{a}}^{\ast }$, ${\mathbf{a}}^{\ast }$, ${\mathbf{A}}^{\ast }$	complex conjugate of $\mathrm{a}$, $\mathbf{a}$, and $\mathbf{A}$, respectively
$\mathrm{\Re }\{\mathbf{a}\}$, $\mathrm{\Im }\{\mathbf{a}\}$	real and imaginary parts of $\mathbf{a}$, respectively
${\mathbf{a}}^n$	element-wise exponent [3] for vector $\mathbf{a}$
${\mathbf{A}}^n$	matrix exponent [3] for matrix $\mathbf{A}$
$a^{\mathit{b}}$, $a^{\mathit{B}}$	element-wise exponent [3] for vector $\mathit{b}$ and matrix $\mathit{B}$, respectively
$\mathrm{f}(\mathit{x}),\mathbf{f}(\mathit{x})$	scalar, vector functions of $\mathit{x}$, respectively
${\mathrm{f}}_{\mathit{x}},{\mathbf{f}}_{\mathit{x}}$	transpose of gradient of $\mathrm{f}$, Jacobian of $\mathbf{f}$, respectively, w.r.t. $\mathit{x}$
${\mathrm{f}}_{\mathit{x}\mathit{x}},{\mathbf{f}}_{\mathit{x}\mathit{x}}(\mathit{\lambda })$	Hessian of $\mathrm{f}$, Jacobian of ${{\mathbf{f}}_{\mathit{x}}}^{\mathsf{T}}\mathit{\lambda }$, respectively, w.r.t. $\mathit{x}$

Constants and Dimensions

$e,j$	constants, $e$ is base of natural log ($\approx 2.71828$), $j$ is $\sqrt{-1}$
$n_k,n_n,n_p,n_p^k$	number of elements, nodes, ports, ports for element $k$, respectively
$n_{\mathbf{x}},n_{\mathbf{v}},n_{\mathbf{z}}$	dimension of vector $\mathbf{x}$, $\mathbf{v}$, $\mathbf{z}$, respectively.
${\mathbf{1}}_n,\left[{}^{\setminus }{{\mathbf{1}}_n}_{\setminus }\right]$	$n\times 1$ vector of all ones, $n\times n$ identity matrix
$\mathbf{0}$	appropriately-sized vector or matrix of all zeros

Variables

${\mathrm{v}}_i$	complex voltage at node/port $i$
$u_i,w_i$	real and imaginary parts of voltage at node/port $i$, ${\mathrm{v}}_i=u_i+j{w}_i$
${\nu }_i,{\theta }_i$	voltage magnitude and angle at node/port $i$, ${\mathrm{v}}_i={\nu }_i{e}^{j{\theta }_i}$
$\mathbf{v}$	column vector of complex voltages ${\mathrm{v}}_i$
$\mathbf{e}$	column vector $\mathbf{v}$ with elements scaled to unit magnitude, $\mathbf{e}=e^{j\mathit{\theta }}$
$\mathit{u},\mathit{w}$	column vectors of real ($u_i$) and imaginary ($w_i$) parts of voltage, respectively, $\mathbf{v}=\mathit{u}+j\mathit{w}$
$\mathit{\nu },\mathit{\theta }$	column vectors of voltage magnitudes ${\nu }_i$ and angles ${\theta }_i$, respectively, $\mathbf{v}=\left[{}^{\setminus }{\mathit{\nu }}_{\setminus }\right]\mathbf{e}=\left[{}^{\setminus }{\mathit{\nu }}_{\setminus }\right]e^{j\mathit{\theta }}$
$\mathbf{\Lambda }$	column vector of inverse of complex voltages $\frac{1}{{\mathrm{v}}_i}$, $\mathbf{\Lambda }={\mathbf{v}}^{-1}$
$\mathit{z}$	column vector of real non-voltage state variables $z_i$
$\mathbf{z}$	column vector of complex non-voltage state variables ${\mathrm{z}}_i$
${\mathit{z}}_r,{\mathit{z}}_i$	column vectors of real and imaginary parts of $\mathbf{z}={\mathit{z}}_r+j{\mathit{z}}_i$

Parameters

${\underset{―}{\mathit{J}}}_{\mathit{k}}$	matrix formed by taking selected rows, indexed by vector $\mathit{k}$, from an identity matrix [4]
$\underset{―}{\mathbf{Y}}$	AC model admittance matrix
$\underset{―}{\mathbf{L}}$	linear coefficient (of $\mathbf{z}$) for affine complex current injections
$\underset{―}{\mathbf{i}}$	vector of constant complex current injections
$\underset{―}{\mathbf{M}}$	linear coefficient (of $\mathbf{v}$) for affine complex power injections
$\underset{―}{\mathbf{N}}$	linear coefficient (of $\mathbf{z}$) for affine complex power injections
$\underset{―}{\mathbf{s}}$	vector of constant complex power injections
$\underset{―}{\mathit{B}}$	DC model susceptance matrix
$\underset{―}{\mathit{K}}$	linear coefficient (of $\mathit{z}$) for affine active power injections
$\underset{―}{\mathit{p}}$	vector of constant active power injections
$\underset{―}{\mathit{C}}$	element-node incidence matrix for a given port
$\underset{―}{\mathit{D}}$	element-variable incidence matrix for a given state variable
$\underset{―}{\mathit{A}}$	combined incidence matrix $\underset{―}{\mathit{A}}=\left[\begin{array}{cc}\underset{―}{\mathit{C}}& \mathbf{0}\\ \mathbf{0}& \underset{―}{\mathit{D}}\end{array}\right]$

[1]

Constants and parameters are underlined, with the following exceptions: constants $e$ and $j$, $p$, $q$, $m$ and $n$ when used as dimensions, and $i$, $j$, and $k$ as indices.

[2]

Obtained by multiplying by matrix $\underset{―}{\mathit{J}}$ or ${\underset{―}{\mathit{J}}}_{\mathit{k}}$.

[3]

Superscripts may also be used as indices, indicated by context.

[4]

Often used simply as $\underset{―}{\mathit{J}}$ without the subscript.