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
\(\cscal{x}, \cscal{\uptheta}\)	complex scalars
\(\rvec{x}, \rvec{\theta}\)	real vectors
\(\cvec{x}, \cvecG{\uptheta}\)	complex vectors
\(\rmat{X}, \rmatG{\Theta}\)	real matrices
\(\cmat{X}, \cmatG{\Theta}\)	complex matrices
\(x, \cscal{x}, \rvec{x}, \cvec{x}, \rmat{X}, \cmat{X}\)	variables, functions
\(\param{x}, \param{\cscal{x}}, \param{\rvec{x}}, \param{\cvec{x}}, \param{\rmat{X}}, \param{\cmat{X}}\)	constants, parameters [1]
\(\hat{\rvec{x}}, \hat{\cvec{x}}, \hat{\rmat{X}}, \hat{\cmat{X}}\)	selected rows of interest of \(\rvec{x}, \cvec{x}, \rmat{X}, \cmat{X}\), respectively [2]

Operators

\(\diag{\cvec{a}}\)	diagonal matrix with vector \(\cvec{a}\) on the diagonal
\(\trans{\cmat{A}}\)	(non-conjugate) transpose of matrix \(\cmat{A}\)
\(\conj{\cscal{a}}\), \(\conj{\cvec{a}}\), \(\conj{\cmat{A}}\)	complex conjugate of \(\cscal{a}\), \(\cvec{a}\), and \(\cmat{A}\), respectively
\(\Re\{{\cvec{a}\}}\), \(\Im\{{\cvec{a}\}}\)	real and imaginary parts of \(\cvec{a}\), respectively
\(\cvec{a}^{n}\)	element-wise exponent [3] for vector \(\cvec{a}\)
\(\cmat{A}^{n}\)	matrix exponent [3] for matrix \(\cmat{A}\)
\(a^{\rvec{b}}\), \(a^\rmat{B}\)	element-wise exponent [3] for vector \(\rvec{b}\) and matrix \(\rmat{B}\), respectively
\(\f(\x), \F(\x)\)	scalar, vector functions of \(\x\), respectively
\(\f_\x, \F_\x\)	transpose of gradient of \(\f\), Jacobian of \(\F\), respectively, w.r.t. \(\x\)
\(\f_{\x\x}, \F_{\x\x}(\lam)\)	Hessian of \(\f\), Jacobian of \(\trans{\F_\x} \lam\), respectively, w.r.t. \(\x\)

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
\(n_\X, n_\V, n_\Z\)	dimension of vector \(\X\), \(\V\), \(\Z\), respectively.
\(\ones{n}, \Id{n}\)	\(n \times 1\) vector of all ones, \(n \times n\) identity matrix
\(\zeros\)	appropriately-sized vector or matrix of all zeros

Variables

\(\vvi{i}\)	complex voltage at node/port \(i\)
\(\vri{i}, \vii{i}\)	real and imaginary parts of voltage at node/port \(i\), \(\vvi{i} = \vri{i} + j \vii{i}\)
\(\vmi{i}, \vai{i}\)	voltage magnitude and angle at node/port \(i\), \(\vvi{i} = \vmi{i} e^{j \vai{i}}\)
\(\V\)	column vector of complex voltages \(\vvi{i}\)
\(\E\)	column vector \(\V\) with elements scaled to unit magnitude, \(\E = e^{j \Va}\)
\(\Vr, \Vi\)	column vectors of real (\(\vri{i}\)) and imaginary (\(\vii{i}\)) parts of voltage, respectively, \(\V = \Vr + j \Vi\)
\(\Vm, \Va\)	column vectors of voltage magnitudes \(\vmi{i}\) and angles \(\vai{i}\), respectively, \(\V = \dVm \E = \dVm e^{j \Va}\)
\(\inV\)	column vector of inverse of complex voltages \(\frac{1}{\vvi{i}}\), \(\inV = \V^{-1}\)
\(\z\)	column vector of real non-voltage state variables \(z_i\)
\(\Z\)	column vector of complex non-voltage state variables \(\cscal{z}_i\)
\(\Zr, \Zi\)	column vectors of real and imaginary parts of \(\Z = \Zr + j \Zi\)

Parameters

\(\J_\kk\)	matrix formed by taking selected rows, indexed by vector \(\kk\), from an identity matrix [4]
\(\YY\)	AC model admittance matrix
\(\LL\)	linear coefficient (of \(\Z\)) for affine complex current injections
\(\iv\)	vector of constant complex current injections
\(\MM\)	linear coefficient (of \(\V\)) for affine complex power injections
\(\NN\)	linear coefficient (of \(\Z\)) for affine complex power injections
\(\sv\)	vector of constant complex power injections
\(\BB\)	DC model susceptance matrix
\(\KK\)	linear coefficient (of \(\z\)) for affine active power injections
\(\pv\)	vector of constant active power injections
\(\CC\)	element-node incidence matrix for a given port
\(\DD\)	element-variable incidence matrix for a given state variable
\(\Aa\)	combined incidence matrix \(\Aa = \left[\begin{array}{ccc}\CC & \zeros \\ \zeros & \DD \end{array}\right]\)

[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 \(\J\) or \(\J_\kk\).

[3]

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

[4]

Often used simply as \(\J\) without the subscript.