Bus Impedance Matrix (Z-bus)

• \[\begin{aligned} \mathrm{Z_{bus}} & = \mathrm{Y_{bus}^{-1}} \\ \mathrm{V} & = \mathrm{Z_{bus}} \cdot \mathrm{I} \end{aligned}\]
  Bus impedance matrix and its relationship with bus voltages & Currents

• \(\mathrm{Z_{bus}}\) is also symmetrical similar to \(\mathrm{Y_{bus}}\) around the principal diagonal

• Elements on the main diagonal are called driving point impedances of the buses

• Off diagonal elements are called transfer impedances of the buses

• \(\mathrm{Z_{bus}}\) is very useful in fault analysis

• \(\mathrm{Z_{bus}}\) can be determined by two methods:

  • Determine \(\mathrm{Y_{bus}}\) and take its inverse

  • Directly formed from reactance diagram which requires knowledge of modification of existing \(\mathrm{Z_{bus}}\) matrix due to addition of new buses or addition of new line (or impedances) between existing buses

• Step-by-step technique which proceeds branch by branch.

• \[\mathrm{Z_{BUS}\left(old\right)\ \underrightarrow{Z_{b}=\text{branch impedance}}}\ \mathrm{Z_{Bus}\left(new\right)}\]
  formulation Advantage that any modification of the network does not require complete rebuilding of

• Upon adding a new branch, one of the following situations is presented

\(\checkmark\) Adding \(Z_b\) from new bus-p \(\rightarrow\) reference bus

\[Z_{\text{bus, new}}=\left[\begin{array}{cccc|c} & & & & 0\\ & Z_{\text{orig}} & & & 0\\ & & & & \vdots\\ & & & & 0\\ \hline 0 & 0 & \cdots & 0 & Z_{b} \end{array}\right]\]

• dimension of \(\mathrm{Z_{BUS}}\) goes up by one

• \((n+1)^{th}\) column and row are all zeros except the diagonal element

\(\checkmark\) Adding \(Z_b\) from new bus-p \(\rightarrow\) existing bus-q

\[ Z_{\text{bus, new}}=\left[\begin{array}{cccc|c} & & & & Z_{1q}\\ & Z_{\text{orig}} & & & Z_{2q}\\ & & & & \vdots\\ & & & & Z_{nq}\\ \hline Z_{q1} & Z_{q2} & \cdots & Z_{qn} & Z_{qq}+Z_{b} \end{array}\right] \]

• dimension of \(\mathrm{Z_{BUS}}\) goes up by one

• elements of \((n+1)^{th}\) column are the elements of \(q^{th}\) column and elements of \((n+1)^{th}\) row are the elements \(q^{th}\) row

\(\checkmark\) Adding \(Z_b\) from existing bus-q \(\rightarrow\) reference bus

• a new loop is formed but the dimension of \(\mathrm{Z_{BUS}}\) does not change

\[Z_{jk,act}=Z_{jk}-\dfrac{Z_{j\left(n+1\right)}Z_{\left(n+1\right)k}}{Z_{\left(n+1\right)\left(n+1\right)}}\]

• \(Z_{jk,act}\) corresponding to row-j and column-k of actual new \(Z_{bus}\)

• \(Z_{jk}\), \(Z_{(n+1)k}\), \(Z_{j(n+1)}\), \(Z_{(n+1)(n+1)}\) impedances of new \(Z_{bus}\) or order \((n+1)\)

• \(Z_{jk,act} = Z_{kj,act}\)

\(\checkmark\) Adding \(Z_b\) between two existing buses h and q

• new loop is formed but the dimension of \(\mathrm{Z_{BUS}}\) does not change

\[ Z_{\text{bus, new}}=\left[\begin{array}{cccc|c} & & & & Z_{1h}-Z_{1q}\\ & Z_{\text{orig}} & & & Z_{2h}-Z_{2q}\\ & & & & \vdots\\ & & & & Z_{nh}-Z_{nq}\\ \hline Z_{h1}-Z_{q1} & Z_{h2}-Z_{q2} & \cdots & Z_{hn}-Z_{qn} & Z_{\left(n+1\right)\left(n+1\right)} \end{array}\right] \]

• elements of \((n+1)^{th}\) column is the difference between the elements of column-h and column-q

• elements of \((n+1)^{th}\) row is the difference between the elements of row-h and row-q

• What is the size of \(Z_{Bus}\) ?

• Can we directly find \(Z_{Bus}\) from \(Y_{Bus}\)?