Question

A30 Km, 34.5 k V, 60 Hz ,three-phase line has a positive-sequence series impedance$\mathbf{z}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{19}\mathbf{+}\mathbf{j}\mathbf{}\mathbf{0}\mathbf{.}\mathbf{34}\mathbf{}\frac{\mathbf{\Omega }}{\mathbf{km}}$The load at the receiving end absorbs10 M V Aat 33 k V. Assuming a short line, calculate: (a) the ABCDparameters, (b) the sending-end voltage for a load power factor of 0.9lagging, and (c) the sending-end voltage for a load power factor of0.9leading.

Short answer

(a)The A , B , C , Dparameters are $1\angle 0^{°}\mathrm{pu},(5.7+\mathrm{j}10.2)\mathrm{\Omega },\mathrm{o}\mathrm{s}$and $1\angle 0^{°}\mathrm{pu}.$.

(b) The sending end voltage is $20.76\angle 3.{23}^{°}\mathrm{k}\mathrm{V}$.

(c) The sending end voltage is $19.28\angle 6.{077}^{°}\mathrm{k}\mathrm{V}$.

Step-by-step solution

Given data.

The load absorbs S = 10 M V A at receiving end voltage ${\mathrm{V}}_{\mathrm{R}}=33\mathrm{k}\hspace{0.17em}\mathrm{V}$.

The line impedance is $\mathrm{z}=0.19+\mathrm{j}0.34\frac{\mathrm{\Omega }}{\mathrm{km}}$.

The length of the line is l = 30 k m.

The power factor is p.f = 0.9.

Determine the formulas of ABCD parameters, line current, power angle and sending end voltage.

Write the formulas of ABCD parameters for a short line.

$$\begin{gathered} \mathbf{A}\mathbf{=}\mathbf{1}\mathbf{\angle }{\mathbf{0}}^{\mathbf{°}}\mathit{p}\mathit{u}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{\left(}\mathbf{1}\mathbf{\right)} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\mathbf{D} \\ \mathbf{C}\mathbf{=}\mathbf{}\mathbf{0}\mathbf{}\mathbf{S}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{\left(}\mathbf{2}\mathbf{\right)} \\ \mathbf{B}\mathbf{=}\mathit{z}\mathit{l}\mathbf{}\mathbf{\Omega }\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{=}\mathbf{Z}\mathbf{}\mathbf{\Omega }\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{\left(}\mathbf{3}\mathbf{\right)} \end{gathered}$$

Here, A , B , C , Dare the transmission line parameters.

Write the formula of sending end voltage in terms of transmission line parameters.

${\mathbf{V}}_{\mathbf{S}}\mathbf{=}{\mathbf{AV}}_{\mathbf{R}}\mathbf{+}{\mathbf{BI}}_{\mathbf{R}}$

Here, $V_S$is the sending end voltage, $V_R$is the receiving end voltage, and $l_R$is the transmission line current.

Write the formula of line current.

${\mathbf{l}}_{\mathbf{R}}\mathbf{=}\frac{\mathbf{S}}{\sqrt{\mathbf{3}}{\mathbf{V}}_{\mathbf{R}}}\mathbf{}\mathbf{}\mathbf{}..........\left(5\right)$

Here, Sis apparent power consumed by load.

Write the formula of power angle.

$\mathbf{\theta }\mathbf{=}{\mathbf{cos}}^{\mathbf{-}\mathbf{1}}\mathbf{(}\mathbf{p}\mathbf{.}\mathbf{f}\mathbf{)}\mathbf{}..............\left(6\right)$

Here,p.f is a power factor.

Determine the ABCD parameters.

$$\begin{gathered} \left(\mathrm{a}\right) \\ \mathrm{From}\mathrm{equation}\left(1\right),\mathrm{the}\mathrm{A}\mathrm{and}\mathrm{D}\mathrm{parameters}\mathrm{for}\mathrm{short}\mathrm{line}\mathrm{are}, \\ \mathrm{A}=1\angle 0^{°}\mathrm{pu}\mathrm{and}\mathrm{D}=1\angle 0^{°}\mathrm{pu} \\ \mathrm{From}\mathrm{equation}\left(2\right),\mathrm{the}\mathrm{c}\mathrm{parameter}\mathrm{is},\mathrm{c}=0\mathrm{S} \\ \mathrm{Determine}\mathrm{the}\mathrm{B}\mathrm{parameter}. \\ \mathrm{Substitute}0.19+\mathrm{j}0.34\frac{\mathrm{\Omega }}{\mathrm{km}}\mathrm{for}\mathrm{z},\mathrm{and}30\mathrm{km}\mathrm{for}\mathrm{l}\mathrm{in}\mathrm{equation}\left(3\right). \\ \mathrm{Z}=\left(0.19+\mathrm{j}0.34\frac{\mathrm{\Omega }}{\mathrm{km}}\right)\left(30\mathrm{km}\right) \\ =\left(5.7+\mathrm{j}10.2\right)\mathrm{\Omega } \\ \mathrm{Therefore},\mathrm{the}\mathrm{A},\mathrm{B},\mathrm{C},\mathrm{D}\mathrm{parameters}\mathrm{are}1\angle 0^{°}\mathrm{pu},(5.7+\mathrm{j}10.2)\mathrm{\Omega }0\mathrm{S}\mathrm{and} \\ 1\angle 0^{°}\mathrm{pu}. \end{gathered}$$

Determine the sending end voltage for lagging power factor.

(b)

Determine the line current.

Substitute 10 MVA for S and 33kV for$V_R$in equation (5).

Determine the sending end voltage for leading power factor.

(c)

Determine the sending end voltage for leading power factor.