Question

As shown in Figure 9.21 (a), two three-phase buses $\mathbf{abc}$and ${\mathbf{a}}^{\mathbf{\text{'}}}{\mathbf{b}}^{\mathbf{\text{'}}}{\mathbf{c}}^{\mathbf{\text{'}}}$are interconnected by short circuits between phases $\mathbf{b}$and ${\mathbf{b}}^{\mathbf{\text{'}}}$ and between c and c^', with an open circuit between phases a and a^' . The fault conditions in the phase domain are I_a=I_a^'=0 and V_bb^'=V_cc^'=0. Determine the fault conditions in the sequence domain and verify the interconnection of the sequence networks as shown in Figure 9.15 for this one conductor-open fault.

Short answer

The value of fault condition in sequence domain where phases$\mathrm{b}$ and${\mathrm{b}}^{\text{'}}$ and between$\mathrm{c}$ and$\mathrm{c}\text{'}$ are short circuit and phases$\mathrm{a}$ and${\mathrm{a}}^{\text{'}}$ are open circuit is $\left[\begin{array}{c}{\mathrm{V}}_{00}\\ {\mathrm{V}}_{11}\\ {\mathrm{V}}_{22}\end{array}\right]=\left[\begin{array}{c}0\\ 0\\ 0\end{array}\right]$.

Step-by-step solution

Write the given data from the question.

The${\mathrm{I}}_0$ Is the zero sequence current.

The${\mathrm{I}}_1$ Is the positive sequence current.

The ${\mathrm{I}}_2$Is the negative sequence current.

The ${\mathrm{I}}_{\mathrm{a}}$,${\mathrm{I}}_{\mathrm{b}}$ and${\mathrm{I}}_{\mathrm{c}}$ are phase currents.

The${\mathrm{V}}_{{\mathrm{bb}}^{\text{'}}}$ is voltage between$\mathrm{b}-{\mathrm{b}}^{\text{'}}$ terminals and${\mathrm{V}}_{{\mathrm{cc}}^{\text{'}}}$ is voltage between$\mathrm{c}-{\mathrm{c}}^{\text{'}}$ terminals.

Determine the formula of fault condition in sequence domain.

Write the formula of fault condition in sequence domain.

$\left[\begin{array}{c}{\mathbf{V}}_{{\mathbf{00}}^{\mathbf{\text{'}}}}\\ {\mathbf{V}}_{{\mathbf{11}}^{\mathbf{\text{'}}}}\\ {\mathbf{V}}_{{\mathbf{22}}^{\mathbf{\text{'}}}}\end{array}\right]=\frac{\mathbf{1}}{\mathbf{3}}\left[\begin{array}{ccc}\mathbf{1}& \mathbf{1}& \mathbf{1}\\ \mathbf{1}& {\mathbf{a}}^{\mathbf{2}}& \mathbf{a}\\ \mathbf{1}& \mathbf{a}& {\mathbf{a}}^{\mathbf{2}}\end{array}\right]\left[\begin{array}{c}{\mathbf{V}}_{{\mathbf{aa}}^{\mathbf{\text{'}}}}\\ {\mathbf{V}}_{{\mathbf{bb}}^{\mathbf{\text{'}}}}\\ {\mathbf{V}}_{{\mathbf{cc}}^{\mathbf{\text{'}}}}\end{array}\right]$ …… (1)

Here, ${\mathbf{V}}_{{\mathbf{aa}}^{\mathbf{\text{'}}}}$is voltage between $\mathbf{a}\mathbf{-}\mathbf{a}\mathbf{\text{'}}$terminals, ${\mathbf{V}}_{{\mathbf{bb}}^{\mathbf{\text{'}}}}$is voltage between $\mathbf{b}\mathbf{-}{\mathbf{b}}^{\mathbf{\text{'}}}$terminals and ${\mathbf{V}}_{{\mathbf{cc}}^{\mathbf{\text{'}}}}$is voltage between $\mathbf{c}\mathbf{-}{\mathbf{c}}^{\mathbf{\text{'}}}$terminals.

 Determine the fault condition in sequence domain where phases and and between and are short circuit and phases and is open circuit.

Draw the interconnection network between two three phase buses where phases $\mathrm{b}$and role="math" localid="1656481023409" ${\mathrm{b}}^{\text{'}}$and between c and role="math" localid="1656480966137" $\mathrm{c}\text{'}$are short circuit and phases $\mathrm{a}$and ${\mathrm{a}}^{\text{'}}$are open circuit are shown in figure 1.

Figure 1

Determine the fault condition in the given phase domain is,

$\begin{array}{rcl}{\mathrm{I}}_{\mathrm{a}}& =& \left({\mathrm{I}}_0+{\mathrm{I}}_1+{\mathrm{I}}_2\right)\\ & =& 0\end{array}$

$\begin{array}{rcl}{\mathrm{I}}_{{\mathrm{a}}^{\text{'}}}& =& \left({\mathrm{I}}_0+{\mathrm{I}}_1+{\mathrm{I}}_2\right)\\ & =& 0\end{array}$

Determine the fault condition in the given phase voltage.

${\mathrm{V}}_{{\mathrm{bb}}^{\text{'}}}={\mathrm{V}}_{{\mathrm{cc}}^{\text{'}}}=0$ …… (2)

Here,${\mathbf{V}}_{{\mathbf{bb}}^{\mathbf{\text{'}}}}$ is voltage between$\mathbf{b}\mathbf{-}{\mathbf{b}}^{\mathbf{\text{'}}}$ terminals and${\mathbf{V}}_{{\mathbf{cc}}^{\mathbf{\text{'}}}}$ is voltage between$\mathbf{c}\mathbf{-}{\mathbf{c}}^{\mathbf{\text{'}}}$ terminals.

Determine the fault condition in the given sequence domain by using equation (2).

Substitute$0$ for${\mathrm{V}}_{{\mathrm{bb}}^{\text{'}}}$ and$0$ for${\mathrm{V}}_{{\mathrm{cc}}^{\text{'}}}$ into equation (1).

$\begin{array}{rcl}\left[\begin{array}{c}{\mathrm{V}}_{{00}^{\text{'}}}\\ {\mathrm{V}}_{{11}^{\text{'}}}\\ {\mathrm{V}}_{{22}^{\text{'}}}\end{array}\right]& =& \frac{1}{3}\left[\begin{array}{ccc}1& 1& 1\\ 1& {\mathrm{a}}^2& \mathrm{a}\\ 1& \mathrm{a}& {\mathrm{a}}^2\end{array}\right]\left[\begin{array}{c}{\mathrm{V}}_{{\mathrm{aa}}^{\text{'}}}\\ 0\\ 0\end{array}\right]\\ & =& \left[\begin{array}{c}\frac{{\mathrm{V}}_{{\mathrm{aa}}^{\text{'}}}}{3}\\ \frac{{\mathrm{V}}_{{\mathrm{aa}}^{\text{'}}}}{3}\\ \frac{{{\mathrm{V}}_{\mathrm{aa}}}^{\text{'}}}{3}\end{array}\right]\end{array}$

We know that for open circuit voltage at ${\mathrm{V}}_{{\mathrm{aa}}^{\text{'}}}=0$.

$\left[\begin{array}{c}{\mathrm{V}}_{00}\\ {\mathrm{V}}_{11}\\ {\mathrm{V}}_{22}\end{array}\right]=\left[\begin{array}{c}0\\ 0\\ 0\end{array}\right]$

Hence, the required fault condition is $\begin{array}{c}{\mathrm{V}}_{00}=0\\ {\mathrm{V}}_{11}=0\\ {\mathrm{V}}_{22}=0\end{array}$.

Draw the diagram of sequence network is shown in figure 2.

Figure 2