Question

Determine the equivalent $\mathbf{\pi }$circuit for the line in Problem 5.16. Compare the equivalent $\mathbf{\pi }$circuit with the nominal $\mathbf{\pi }$circuit.

Short answer

The diagram for the equivalent pi circuit is shown in Figure below.

The value of resistance R' is 0.33% smaller than the nominal resistance R.

The value of equivalent reactance X' is 2.415% smaller than the reactance X.

The value of Susceptance $\frac{B\text{'}}{2}$is 2.24% larger than value of admittance role="math" localid="1655903554728" $\frac{y}{2}$.

The value of Shunt conductance is introduced in the circuit as $\frac{G\text{'}}{2}$is $-0.3925\times {10}^{-6}S$.

Step-by-step solution

Write the given data from the question.

The given supply voltage at the sending end is V_s = 500 kV.

The given frequency is 60-Hz.

The given length of the line, I = 300 km.

Consider a conductor temperature is${50}^{\circ }C$.

The three phase totally transposed line with three ASCR 1113 kcmil conductors per bundle.

Consider the distance between conductors in the bundle d = 0.5 m.

Consider the horizontal phase gaping between bundle centres is 10 m, 10 m and 20 m.

The series impedance per km $z=0.336\angle 86.{56}^{\circ }\Omega /km$.

The shunt admittance per km $y=j4.801\times {10}^{-6}S/km$.

Determine the required formula.

Write the formula ofvalue of characteristic impedance.

$\mathit{Z}\mathbf{\text{'}}\mathbf{=}\mathit{z}\mathbf{I}$ …… (1)

Here, z is impedance and I is length of line.

Write the formula of $\mathbf{\gamma }\mathbf{I}$.

$\mathit{\gamma }\mathit{I}\mathbf{=}\left(\sqrt{zy}\right)\mathit{I}$ …… (2)

Here, $\mathit{\gamma }$ is the propagation constant, z is impedance, yis admittance and I is the length of line.

Write the formula of Aparameter.

$\mathit{A}\mathbf{=}\mathbf{1}\mathbf{+}\frac{\mathbf{Y}\mathbf{\text{'}}\mathbf{Z}\mathbf{\text{'}}}{\mathbf{2}}$ …… (3)

Here, Y' is admittance and Z' is impedance.

Write the formula of Bparameter.

$\mathit{B}\mathbf{=}\mathit{Z}\mathbf{\text{'}}$ …… (4)

Here, Z' is impedance.

Write the formula of Cparameter.

$\mathit{C}\mathbf{=}\mathit{Y}\mathbf{\text{'}}\left[1+\frac{Y\text{'}Z\text{'}}{4}\right]$ …… (5)

Here, Y' is admittance and Z' is impedance.

Write the formula of Dparameter.

$\mathit{D}\mathbf{=}\mathbf{1}\mathbf{+}\frac{\mathbf{Y}\mathbf{\text{'}}\mathbf{Z}\mathbf{\text{'}}}{\mathbf{2}}$ …… (6)

Here, Y' is admittance and Z' is impedance.

Determine the value of characteristics impedance Z, γI and ABCD parameters.

Determine the expression of characteristics impedance Z.

Substitute $0.336\angle 86.{56}^{\circ }$ for z and 300 for I into equation (1).

$\begin{array}{rcl}Z& =& (0.336\angle 86.{56}^{\circ })\left(300\right)\\ & =& 100.8\angle 86.{56}^{\circ }\\ & =& 6+j100.62\Omega \\ & =& R+jX\end{array}$

Since, the characteristics impedance Z is $6+j100.62\Omega$.

Determine the value of $\gamma I$.

Substitute $0.336\angle 86.{56}^{\circ }$for z, $4.801\times {10}^{-6}\angle {90}^{\circ }$ for y and 300 for I into equation (2).

$\begin{array}{rcl}\gamma I& =& \left[\sqrt{\left(0.336\angle 86.{56}^{\circ }\right)\left(4.801\times {10}^{-6}\angle {90}^{\circ }\right)}\right]\left(300\right)\\ & =& \left[\sqrt{\left(1.613\times {10}^{-6}\angle 176.{28}^{\circ }\right)}\right]\left(300\right)\\ & =& (1.27\times {10}^{-6}\angle 88.{28}^{\circ })\left(300\right)\\ & =& 0.381\angle 88.{28}^{\circ }\end{array}$

Solve further as,

$\gamma I=0.0114+j0.381pu$

Determine the value of $\left(\gamma I\right)$.

localid="1655907523885" $\begin{array}{rcl}\cos h\left(\gamma i\right)& =& \cos h(0.0114+j0.381)\\ & =& \frac{e^{0.0114+j0.381}+e^{-0.0114+j0.381}}{2}\\ & =& \frac{e^{0.0114}{e}^{j0.381}+e^{0.0114-}{e}^{-j0.381}}{2}\\ & =& \frac{1.0114\angle 0.381+0.988\angle -0.381}{2}radians\end{array}$

Solve further as,

$\begin{array}{rcl}\sin h\left(\gamma i\right)& =& \frac{0.744\angle 1.54}{2}radians\\ & =& 0.372\angle 88.{24}^{\circ }\end{array}$

Determine the value of equivalent impedance

$Z\text{'}=\frac{Z\sin h\left(\gamma I\right)}{\left(\gamma I\right)}$

Here, Z is characteristics impedance and $\gamma I$is dimensionless quantity.

Substitute $100.8\angle 86.{56}^{\circ }$for Z and $0.372\angle 88.{24}^{\circ }$for $\left(\gamma l\right)$into above equation Z'.

$\begin{array}{rcl}Z\text{'}& =& \frac{(100.8\angle 86.{56}^{\circ })(0.372\angle 88.{24}^{\circ })}{0.0114+j0.381}\\ & =& \frac{37.49\angle 174.8^{\circ }}{0.381\angle 188.{28}^{\circ }}\\ & =& 5.98+j98.17\Omega \end{array}$

Determine the value of characteristics admittance.

Y = yl

Here, y is admittance and l is length of line.

Substitute $j4.801\times {10}^{-6}$ for y and 300 for l into above equation .

localid="1655964183132" $\begin{array}{rcl}Y& =& (j4.801\times {10}^{-6})\left(300\right)\\ & =& j0.00144S\end{array}$

Determine the value of equivalent admittance

$\frac{Y\text{'}}{2}=\frac{Y\tan h{(\gamma l/2)}_{}}{2(\gamma l/2)}$

Here, Y is characteristics admittance and $\gamma l$is dimensionless quantity.

Substitute j0.00144 for Y and $0.381\angle 88.{28}^{\circ }$for $\gamma l$into above equation $\frac{Y\text{'}}{2}$.

$\begin{array}{rcl}\frac{Y\text{'}}{2}& =& \left(\frac{2Y}{\gamma l}\right)\left(\frac{\cos h\left(\gamma l\right)-1}{\sin h\left(\gamma l\right)}\right)\\ & =& \left(\frac{2(j0.00144)}{0.381\angle 88.{28}^{\circ }}\right)\left(\frac{(0.928\angle 0.{268}^{\circ })-1}{0.372\angle 88.{24}^{\circ }}\right)\\ & =& \left(\frac{j0.00288}{0.381\angle 88.{28}^{\circ }}\right)\left[\frac{0.0721\angle 176.{55}^{\circ }}{0.381\angle 88.{28}^{\circ }}\right]\\ & =& (0.00756\angle 1.{72}^{\circ })(0.194\angle 88.{31}^{\circ })\end{array}$

Solve further as,

$Y\text{'}=-0.785\times {10}^{-6}+j1.473\times {10}^{-3}S$

As we know that

$\frac{Y\text{'}}{2}=\frac{G\text{'}+jN\text{'}}{2}$

Here, G' is shunt conductance and B' is shunt Susceptance.

Draw the circuit diagram of nominal $\pi$circuit.

Draw a circuit diagram of equivalent $\pi$circuit.

Figure 2

Differentiate the nominal $\pi$circuit and equivalent $\pi$circuit parameters.

The value of resistance R is $6\Omega$ and value of equivalent resistance is changed to R' as $5.98\Omega$.

Hence, the value of resistance R' is 0.33% smaller than the nominal resistance R.

The value of reactance X is $100.6\Omega$ and value of equivalent reactance X' is $98.17\Omega$.

Hence, the value of equivalent reactance X' is 2.415% smaller than the reactance X.

The value of Susceptance $\frac{B\text{'}}{2}$is $0.7365\times {10}^{-3}S$and value of admittance $\frac{y}{2}$is $7.2\times {10}^{-4}S$.

Hence, the value of Susceptance $\frac{B\text{'}}{2}$is 2.24% larger than value of admittance $\frac{y}{2}$.

The value of Shunt conductance $\frac{G\text{'}}{2}$is $-0.3925\times {10}^{-6}S$.

In the corresponding $\pi$circuit, the shunt conductance is introduced.