Question

(a) The bus impedance matrix for a three-bus power system is

${\mathbf{Z}}_{\mathbf{bus}}\mathbf{=}\mathbf{J}\left[\begin{array}{ccc}0.12& 0.08& 0.04\\ 0.08& 0.12& 0.06\\ 0.04& 0.06& 0.08\end{array}\right]\mathbf{per}\mathbf{}\mathbf{unit}$

Where subtransient reactance were used to compute${\mathbf{Z}}_{\mathbf{bus}}$. Prefault voltage is and prefault current is1.0 per unit and prefault current neglected. (a) Draw the bus impedance matrix equivalent circuit (rake equivalent). Identify the per-unit self- and mutual impedances as well as the prefault voltage in the circuit. (b) A three-phase short circuit occurs at. Determine the subtransient fault current and the voltages at buses 1, 2 and 3 during the fault.

(c) Repeat for the case of

${\mathbf{Z}}_{\mathbf{bus}}\mathbf{=}\mathbf{J}\mathbf{\left[}\begin{array}{ccc}\mathbf{0}\mathbf{.}\mathbf{4}& \mathbf{0}\mathbf{.}\mathbf{1}& \mathbf{0}\mathbf{.}\mathbf{3}\\ \mathbf{0}\mathbf{.}\mathbf{1}& \mathbf{0}\mathbf{.}\mathbf{8}& \mathbf{0}\mathbf{.}\mathbf{5}\\ \mathbf{0}\mathbf{.}\mathbf{3}& \mathbf{0}\mathbf{.}\mathbf{5}& \mathbf{0}\mathbf{.}\mathbf{2}\end{array}\mathbf{\right]}\mathbf{per}\mathbf{}\mathbf{unit}$

Short answer

$$\begin{gathered} \mathrm{a})\mathrm{The}\mathrm{per}-\mathrm{unit}\mathrm{self}-\mathrm{impedance}\mathrm{in}\mathrm{the}\mathrm{circuit}\mathrm{is}{\mathrm{Z}}_{11}=0.12\mathrm{per}\mathrm{unit},{\mathrm{Z}}_{22}=0.12\mathrm{per}\mathrm{unit}\mathrm{and} \\ {\mathrm{Z}}_{33}=0.08\mathrm{per}\mathrm{unit}. \end{gathered}$$

$$\begin{gathered} \mathrm{The}\mathrm{per}-\mathrm{unit}\mathrm{mutual}\mathrm{impedances}\mathrm{in}\mathrm{the}\mathrm{circuit}{\mathrm{areZ}}_{12}={\mathrm{Z}}_{21}=0.08\mathrm{per}\mathrm{unit} \\ {\mathrm{Z}}_{13}={\mathrm{Z}}_{31}=0.04\mathrm{per}\mathrm{unit}\mathrm{and}{\mathrm{Z}}_{33}={\mathrm{Z}}_{32}=0.08\mathrm{per}\mathrm{unit}. \end{gathered}$$

$$\begin{gathered} \left(\mathrm{b}\right)\mathrm{The}\mathrm{subtransient}\mathrm{fault}\mathrm{current}\mathrm{and}\mathrm{voltages}\mathrm{at}\mathrm{buses}1,2\mathrm{and}3\mathrm{are}8.33\mathrm{per}\mathrm{unit} \\ \mathrm{and}0.333\angle 0°\mathrm{per}\mathrm{unit},0\mathrm{and}0.5\angle 0°\mathrm{per}\mathrm{unit}. \\ \left(\mathrm{c}\right) \\ \mathrm{Repeat}\mathrm{the}\mathrm{same}\mathrm{case}\mathrm{of}\mathrm{part}\left(\mathrm{a}\right)\mathrm{and}\mathrm{as}: \\ \left(\mathrm{a}\right)\mathrm{The}\mathrm{per}-\mathrm{unit}\mathrm{self}-\mathrm{impedance}\mathrm{in}\mathrm{the}\mathrm{circuit}{\mathrm{isZ}}_{11}=0.4\mathrm{per}\mathrm{unit},{\mathrm{Z}}_{22}=0.8\mathrm{per}\mathrm{unit} \\ {\mathrm{andZ}}_{33}=1.2\mathrm{per}\mathrm{unit}. \\ \mathrm{The}\mathrm{per}-\mathrm{unit}\mathrm{mutual}\mathrm{impedances}\mathrm{in}\mathrm{the}\mathrm{circuit}\mathrm{are}{\mathrm{Z}}_{12}={\mathrm{Z}}_{21}=0.1\mathrm{per}\mathrm{unit} \\ {\mathrm{Z}}_{13}={\mathrm{Z}}_{31}=0.3\mathrm{per}\mathrm{unit}\mathrm{and}{\mathrm{Z}}_{23}={\mathrm{Z}}_{32}=0.5\mathrm{per}\mathrm{unit}. \\ \left(\mathrm{b}\right)\mathrm{The}\mathrm{subtransient}\mathrm{fault}\mathrm{current}\mathrm{and}\mathrm{voltages}\mathrm{at}\mathrm{buses}1,2\mathrm{and}3\mathrm{are}1.25\mathrm{per}\mathrm{unit} \\ \mathrm{and}0.875\angle 0°\mathrm{per}\mathrm{unit},0\mathrm{and}0.375\angle 0°\mathrm{per}\mathrm{unit}. \end{gathered}$$

Step-by-step solution

Write the given data from the question.

The bus impedance matrix for a three-bus power system is

$$\begin{gathered} {\mathrm{Z}}_{\mathrm{bus}}=j\left[\begin{array}{ccc}0.12& 0.08& 0.04\\ 0.08& 0.12& 0.06\\ 0.04& 0.06& 0.08\end{array}\right]\mathrm{per}\mathrm{unit}. \\ \mathrm{The}\mathrm{prefault}\mathrm{voltage}\mathrm{is}\mathrm{given}1.0\mathrm{per}\mathrm{unit}. \\ \mathrm{The}\mathrm{prefault}\mathrm{current}\mathrm{is}\mathrm{neglected}. \\ \mathrm{Again}\mathrm{the}\mathrm{bus}\mathrm{impedance}\mathrm{matrix}\mathrm{for}\mathrm{a}\mathrm{three}-\mathrm{bus}\mathrm{power}\mathrm{system}\mathrm{is} \\ {\mathrm{Z}}_{\mathrm{bus}}=\mathrm{j}\left[\begin{array}{ccc}0.4& 0.1& 0.3\\ 0.1& 0.8& 0.5\\ 0.3& 0.5& 1.2\end{array}\right]\mathrm{per}\mathrm{unit}. \end{gathered}$$

Determine the required formula.

Write the formula of sub-transient fault current.

$$\begin{gathered} I_N=I_F \\ =\frac{V_F}{Z_{22}}......\left(1\right) \end{gathered}$$

$$\begin{gathered} \mathrm{Here},{\mathrm{V}}_{\mathrm{F}}\mathrm{is}\mathrm{fault}\mathrm{voltage}\mathrm{and}{\mathrm{Z}}_{22}\mathrm{is}\mathrm{per}\mathrm{unit}\mathrm{self}-\mathrm{impedances}. \\ \mathrm{Write}\mathrm{the}\mathrm{formula}\mathrm{of}\mathrm{voltage}\mathrm{at}\mathrm{bus}1\mathrm{during}\mathrm{fault}. \\ {\mathbf{E}}_{\mathbf{1}}\mathbf{=}\left(1-\frac{Z_{12}}{Z_{22}}\right){\mathbf{V}}_{\mathbf{F}}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}......\left(2\right) \\ \mathrm{Here},{\mathrm{Z}}_{12}\mathrm{are}\mathrm{per}-\mathrm{unit}\mathrm{mutual}-\mathrm{impedances},{\mathrm{Z}}_{22}\mathrm{is}\mathrm{per}\mathrm{unit}\mathrm{self}-\mathrm{impedances}\mathrm{and}{\mathrm{V}}_{\mathrm{F}}\mathrm{is} \\ \mathrm{fault}\mathrm{voltage}. \\ {\mathbf{E}}_{\mathbf{2}}\mathbf{=}\mathbf{(}\mathbf{1}\mathbf{-}\frac{{\mathbf{Z}}_{\mathbf{22}}}{{\mathbf{Z}}_{\mathbf{22}}}\mathbf{)}{\mathbf{V}}_{\mathbf{F}}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}......\left(3\right) \\ \mathrm{Here},{\mathrm{Z}}_{22}\mathrm{is}\mathrm{per}\mathrm{unit}\mathrm{self}-\mathrm{impedances}\mathrm{and}{\mathrm{V}}_{\mathrm{F}}\mathrm{is}\mathrm{fault}\mathrm{voltage}. \\ \mathrm{Write}\mathrm{the}\mathrm{formula}\mathrm{of}\mathrm{voltage}\mathrm{at}\mathrm{bus}3\mathrm{during}\mathrm{fault}. \\ {\mathbf{E}}_{\mathbf{3}}\mathbf{=}\mathbf{(}\mathbf{1}\mathbf{-}\frac{{\mathbf{Z}}_{\mathbf{32}}}{{\mathbf{Z}}_{\mathbf{22}}}\mathbf{)}{\mathbf{V}}_{\mathbf{F}}\mathbf{}\dots \dots \left(4\right) \\ \mathrm{Here},{\mathrm{Z}}_{32}\mathrm{are}\mathrm{per}-\mathrm{unit}\mathrm{mutual}-\mathrm{impedances},{\mathrm{Z}}_{22}\mathrm{is}\mathrm{per}\mathrm{unit}\mathrm{self}-\mathrm{impedances}\mathrm{and}{\mathrm{V}}_{\mathrm{F}}\mathrm{is} \\ \mathrm{fault}\mathrm{voltage}. \end{gathered}$$

Determine the per-unit self- and mutual impedances.

a)

Construct a bus impedance matrix equivalent circuit (rake equivalent).

The per-unit self-impedances are,

$$\begin{gathered} Z_{11}=0.12\mathrm{per}\mathrm{unit} \\ {\mathrm{Z}}_{22}=0.12\mathrm{per}\mathrm{unit} \\ {\mathrm{Z}}_{33}=0.08\mathrm{per}\mathrm{unit} \\ \mathrm{The}\mathrm{per}-\mathrm{unit}\mathrm{mutual}-\mathrm{impedances}\mathrm{are}, \\ \mathrm{Write}\mathrm{the}\mathrm{impedance}{\mathrm{Z}}_{12}. \\ {\mathrm{Z}}_{12}={\mathrm{Z}}_{21} \\ =0.08\mathrm{per}\mathrm{unit} \\ \mathrm{Write}\mathrm{the}\mathrm{impedance}{\mathrm{Z}}_{13}. \\ {\mathrm{Z}}_{13}={\mathrm{Z}}_{31} \\ =0.04\mathrm{per}\mathrm{unit} \\ \mathrm{Write}\mathrm{the}\mathrm{impedance}{\mathrm{Z}}_{23}. \\ {\mathrm{Z}}_{23}={\mathrm{Z}}_{32} \\ =0.06\mathrm{per}\mathrm{unit} \end{gathered}$$

Determine the sub-transient fault current and voltage at bus 1, bus 2 and bus 3.

(b)

For a three-phase short circuit at bus 2, the fault is generated at the bus.

Consider that the neutral current is $I_N$and fault current be $I_F$.

Construct a bus impedance matrix equivalent circuit (rake equivalent).

Determine the sub-transient fault current.

$$\begin{gathered} \mathrm{Substitute}1\mathrm{for}{\mathrm{V}}_{\mathrm{F}}\mathrm{and}0.12\mathrm{for}{\mathrm{Z}}_{22}\mathrm{into}\mathrm{equation}\left(1\right). \\ {I}_{\mathit{N}}\mathit{=}\frac{1}{0.12} \\ \mathit{}\mathit{}\mathit{}=8.33\mathrm{per}\mathrm{unit} \\ \mathrm{Determine}\mathrm{the}\mathrm{voltage}\mathrm{at}\mathrm{bus}1\mathrm{during}\mathrm{fault}. \\ \mathrm{Substitute}0.08\mathrm{for}{\mathrm{Z}}_{12},0.12\mathrm{for}{\mathrm{Z}}_{22}\mathrm{and}1\mathrm{for}{\mathrm{V}}_{\mathrm{F}}\mathrm{into}\mathrm{equation}\left(2\right). \\ {\mathrm{E}}_1=\left(1\frac{0.08}{0.12}\right)\left(1\right) \\ =\left(1-0.666\right)\left(1\right) \\ =0.333\angle 0°\mathrm{per}\mathrm{unit} \\ \mathrm{Determine}\mathrm{the}\mathrm{voltage}\mathrm{at}\mathrm{bus}2\mathrm{during}\mathrm{fault}. \\ \mathrm{Substitute}1\mathrm{for}{\mathrm{Z}}_{22},1\mathrm{for}{\mathrm{Z}}_{22}\mathrm{and}1\mathrm{for}{\mathrm{V}}_{\mathrm{F}}\mathrm{into}\mathrm{equation}\left(3\right) \\ {\mathrm{E}}_2=\left(1\frac{1}{1}\right)\left(1\right) \\ =0 \\ \mathrm{Determine}\mathrm{the}\mathrm{voltage}\mathrm{at}\mathrm{bus}3\mathrm{during}\mathrm{fault}. \\ \mathrm{Substitute}0.06\mathrm{for}{\mathrm{Z}}_{32},0.12\mathrm{for}{\mathrm{Z}}_{22}\mathrm{and}1\mathrm{for}{\mathrm{V}}_{\mathrm{F}}\mathrm{into}\mathrm{equation}\left(4\right). \\ {\mathrm{E}}_3=\left(1\frac{0.06}{10.12}\right)\left(1\right) \\ =0.5\angle 0°\mathrm{per}\mathrm{unit} \end{gathered}$$

Determine the per-unit self- and mutual impedances.

(c)

(a)

Construct a bus impedance matrix equivalent circuit (rake equivalent).

The per-unit self-impedances are,

$$\begin{gathered} Z_{11}=0.4\mathrm{per}\mathrm{unit} \\ {\mathrm{Z}}_{22}=0.8\mathrm{per}\mathrm{unit} \\ {\mathrm{Z}}_{33}=1.2\mathrm{per}\mathrm{unit} \\ \mathrm{The}\mathrm{per}-\mathrm{unit}\mathrm{mutual}-\mathrm{impedances}\mathrm{are}, \\ {\mathrm{Z}}_{12}={\mathrm{Z}}_{21}=0.1\mathrm{per}\mathrm{unit} \\ {\mathrm{Z}}_{13}={\mathrm{Z}}_{31}=0.3\mathrm{per}\mathrm{unit} \\ {\mathrm{Z}}_{23}={\mathrm{Z}}_{32}=0.5\mathrm{per}\mathrm{unit} \end{gathered}$$

Determine the sub-transient fault current and voltage at bus 1, bus 2 and bus 3.

(c)

(b)

For a three-phase short circuit occurs at bus 2, the fault is generated at the bus.

Consider that the neutral current is $I_{\mathrm{N}}$and fault current belocalid="1655449025830" $I_{\mathrm{F}}$.

Construct a bus impedance matrix equivalent circuit.

Determine the sub-transient fault current.

Substitute 1 for$V_{\mathrm{F}}$and 0.8 for$Z_{22}$into equation (1).

localid="1655448988470" $$\begin{gathered} I_N\mathit{=}\frac{1}{0.8} \\ \mathit{}\mathit{}\mathit{}=1.25\mathrm{per}\mathrm{unit} \\ \mathrm{Determine}\mathrm{the}\mathrm{voltage}\mathrm{at}\mathrm{bus}1\mathrm{during}\mathrm{fault}. \\ \mathrm{Substitute}0.1\mathrm{for}{\mathrm{Z}}_{12},0.8\mathrm{for}{\mathrm{Z}}_{22}\mathrm{and}1\mathrm{for}{\mathrm{V}}_{\mathrm{F}}\mathrm{into}\mathrm{equation}\left(2\right). \\ {E}_1=\left(1-\frac{0.1}{0.8}\right)\left(1\right) \\ =\left(1-0.125\right)\left(1\right) \\ =0.875\angle 0°\mathrm{per}\mathrm{unit} \\ \mathrm{Determine}\mathrm{the}\mathrm{voltage}\mathrm{at}\mathrm{bus}2\mathrm{during}\mathrm{fault}. \\ \mathrm{Substitute}1\mathrm{for}{\mathrm{Z}}_{22},1\mathrm{for}{\mathrm{Z}}_{22}\mathrm{and}1\mathrm{for}{\mathrm{V}}_{\mathrm{F}}\mathrm{into}\mathrm{equation}\left(3\right). \\ {\mathrm{E}}_2=\left(1-\frac{1}{1}\right)\left(1\right) \\ =0 \\ \mathrm{Determine}\mathrm{the}\mathrm{voltage}\mathrm{at}\mathrm{bus}3\mathrm{during}\mathrm{fault}. \\ \mathrm{Substitute}0.5\mathrm{for}{\mathrm{Z}}_{32},0,8\mathrm{for}{\mathrm{Z}}_{22}\mathrm{and}1\mathrm{for}{\mathrm{V}}_{\mathrm{F}}\mathrm{into}\mathrm{equation}\left(3\right). \\ \mathrm{E}3=\left(1-\frac{0.5}{0.8}\right)\left(1\right) \\ =0.375\angle 0°\mathrm{per}\mathrm{unit} \end{gathered}$$