[PSA] Chapter 9 - Network Matrices

9.0 Network Matrices

• Representing network with two matrix relationships

\[\bf I = Y_{bus}V, V=Z_{bus}I\]

Where Admittance and impedance matrices are in inverse relationship

9.1 Bus admittance Matrix

\[I_i=\sum_{j=1}^{n}Y_{ij}V_j\]

• Diagonal term : Sum of admittance connected to $i$th bus
• Off-diagonal term : $-1 $ times admittance between two busses

Transmission-line $\pi$ model

\[\begin{bmatrix} I_p\\ I_q \end{bmatrix} = \begin{bmatrix} y+\dfrac{y_{\mathrm{shunt}}}{2}&-y\\[2mm] -y&y+\dfrac{y_{\mathrm{shunt}}}{2} \end{bmatrix} \begin{bmatrix} V_p\\ V_q \end{bmatrix}\]

• $y$ : admittance between two busses

Transformer

• If transformer has $jx$ leakage inductance,

\[\begin{bmatrix} I_p\\ I_q \end{bmatrix} = \begin{bmatrix} y&-y\\ -y&y \end{bmatrix} \begin{bmatrix} V_p\\ V_q \end{bmatrix}\]

• Where $y = {1\over jx}$

Off-nominal transformer

Transformer with tap ratio $a:1$

\[\begin{bmatrix} I_p\\ I_q \end{bmatrix} = \begin{bmatrix} \dfrac{y}{a^2}&-\dfrac{y}{a}\\[2mm] -\dfrac{y}{a}&y \end{bmatrix} \begin{bmatrix} V_p\\ V_q \end{bmatrix} \tag{9.2b}\]

9.3 Kron Reduction

• Unconnected (generator nor load) bus $k$ can be eliminated

\[\mathbf Y_{\mathrm{red}} = \mathbf Y_{aa} - \mathbf Y_{ab} \mathbf Y_{bb}^{-1} \mathbf Y_{ba}\]

• $a$ : remaining bus, $b$ : bus to be eliminated by zero-injection

9.4 $Y_{bus}$ structure

• Three key properties
  1. ${\bf Y}_{bus}$ is normally sparse matrix
  2. Symmetric matrix
  3. $Y_{ij}=0\Leftrightarrow$ no direct connection between $i, j$ bus

9.5 Bus impedance matrix

• Connect new bus to the branch

Connect new bus to reference

\[\mathbf Z_{\mathrm{bus}}^{\,n} = \begin{bmatrix} \mathbf Z_{\mathrm{bus}}&\mathbf0\\ \mathbf0^T&z_b \end{bmatrix} \tag{9.22}\]

Connect new bus to bus $i$

\[\mathbf Z_{\mathrm{bus}}^{\,n} = \begin{bmatrix} \mathbf Z_{\mathrm{bus}}&\mathbf Z_i\\ \mathbf Z_i^T&Z_{ii}+z_b \end{bmatrix} \tag{9.23}\]

Add new branch between bus $i, j$

\[\mathbf Z_{\mathrm{bus}}^{\,n} = \mathbf Z_{\mathrm{bus}} - \gamma\mathbf b\mathbf b^T \tag{9.25}\]

Where $b = {\bf Z}_i - {\bf Z}_j$, $\gamma = (z_b+Z_{ii}+Z_{jj}-2Z_{ij})^{-1}$

Can figure certain column of $Z_{bus}$ without calculating entire matrix using LU decomposition