Question

Compute the $\mathbf{4}\mathbf{\times }\mathbf{4}$per-unit zero-, positive-, and negative-sequence bus impedance matrices for the power system given in Problem 9.5. Use a base of and $\mathbf{20}\mathbf{}\mathit{k}\mathit{V}$in the zone of generator$\mathit{G}\mathbf{3}$.

Short answer

Answer

The value of positive-sequence bus impedance matrix is, $Z_{bus1}=1\left[\begin{array}{cccc}0.194& 0.1152& 0.0845& 0.0672\\ 0.1152& 0.1873& 0.1373& 0.1093\\ 0.0854& 0.1373& 0.2473& 0.0801\\ 0.0672& 0.1093& 0.0801& 0.1804\end{array}\right]$.

The value of negative-sequence bus impedance matrix is, role="math" localid="1656141698713" $Z_{bus2}=1\left[\begin{array}{cccc}0.1947& 0.1164& 0.0866& 0.0679\\ 0.1164& 0.1892& 0.1408& 0.1104\\ 0.0866& 0.1408& 0.2535& 0.0821\\ 0.0679& 0.1104& 0.0821& 0.1811\end{array}\right]$.

The value of zero-sequence bus impedance matrix is, $Z_{bus0}=j\left[\begin{array}{cccc}0.22& 0& 0& 0\\ 0& 0.2& 0& 0\\ 0& 0& 0.47& 0\\ 0& 0& 0& 0.3\end{array}\right]$

Step-by-step solution

Write the given data from the question.

As reference of problem 9.5.

The value of base MVA is$1000MVA$.

The value of voltage base zone of generator $G3$is$20kV$.

Determine the formula of sequence impedance bus matrix.

Write the formula of positive-sequence bus impedance matrix

${\mathit{Z}}_{\mathbf{b}\mathbf{u}\mathbf{s}\mathbf{1}}\mathbf{=}\frac{\mathbf{1}}{{\mathbf{Y}}_{\mathbf{b}\mathbf{u}\mathbf{s}\mathbf{1}}}$ …… (1)

Here, ${\mathit{Z}}_{\mathbf{bus}\mathbf{1}}$is per-unit positive-sequence bus admittance matrix.

Write the formula of negative-sequence bus impedance matrix

${\mathit{Z}}_{\mathbf{bus}\mathbf{2}}\mathbf{=}\frac{\mathbf{1}}{{\mathbf{Y}}_{\mathbf{b}\mathbf{u}\mathbf{s}\mathbf{2}}}$ …… (2)

Here, ${\mathit{Z}}_{\mathbf{bus}\mathbf{2}}$is per-unit negative-sequence bus admittance matrix.

Write the formula of zero-sequence bus impedance matrix

${\mathit{Z}}_{\mathbf{bus}\mathbf{0}}\mathbf{=}\frac{\mathbf{1}}{{\mathbf{Y}}_{\mathbf{bus}\mathbf{0}}}$ …… (3)

Here, ${\mathit{Z}}_{\mathbf{bus}\mathbf{0}}$is per-unit zero-sequence bus admittance matrix.

(a) Determine the value of positive sequence, negative sequence and zero sequence bus impedance matrix of the system.

Refer the figure 7.14 from the textbook.

Determine the base impedance of transmission line.

$$\begin{gathered} Z_{base}=\frac{{\left(500\times {10}^3\right)}^2}{1000\times {10}^6} \\ =\frac{25\times {10}^{10}}{1000\times {10}^6} \\ =250\Omega \end{gathered}$$

Determine the per unit quantities for lines.

Determine the per unit zero positive and negative quantities for lines.

role="math" localid="1656142251704" $$\begin{gathered} X_1=X_2 \\ =\left(\frac{50}{250}\right) \\ =0.2p.u. \end{gathered}$$

Determine the per unit zero sequence reactance for lines.

$$\begin{gathered} X_0=\left(\frac{150}{250}\right) \\ =0.6p.u. \end{gathered}$$

Draw the circuit diagram of positive sequence network.

Figure 1

The circuit's positive-sequence bus admittance matrix is shown below.

$$\begin{gathered} Y_{bus2}=j\left[\begin{array}{cccc}\frac{1}{0.32}+\frac{1}{0.2}& -\frac{1}{0.2}& 0& 0\\ -\frac{1}{0.2}& -\frac{1}{0.2}+\frac{1}{0.2}+\frac{1}{0.2}& -\frac{1}{0.2}& -\frac{1}{0.2}\\ 0& -\frac{1}{0.2}& \frac{1}{0.2}+\frac{1}{0.58}& 0\\ 0& -\frac{1}{0.2}& 0& \frac{1}{0.2}+\frac{1}{0.28}\end{array}\right] \\ =\frac{1}{j}\left[\begin{array}{cccc}8.125& -5& 0& 0\\ -5& 15& -5& -5\\ 0& -5& 6.72& 0\\ 0& -5& 0& 8.57\end{array}\right] \end{gathered}$$

Determine the per unit positive sequence bus impedance matrix of the system.

Here,is positive-sequence admittance matrix.

Substituteforinto above equation.

Therefore, the value of per unit positive-sequence bus impedance matrix is,.

Figure 2

The circuit's negative-sequence bus admittance matrix is shown below.

Determine the per unit negative sequence bus impedance matrix of the system.

Here,is negative-sequence bus admittance matrix.

Substituteforinto above equation.

Therefore, the value of per unit negative-sequence bus impedance matrix is,.

Figure 3

The circuit's zero-sequence bus admittance matrix is shown below.

Determine the per unit zero sequence bus impedance matrix of the system.

Here,is zero-sequence bus admittance matrix.

Substituteforinto above equation.

Therefore, the value of per unit zero-sequence bus impedance matrix is,.