Question

Calculate the inductive reactance in $\frac{\mathbf{\Omega }}{\mathbf{km}}$ of a bundled$\mathbf{500}\mathbf{kV}\mathbf{,}\mathbf{60}\mathbf{Hz}$ , three-phase completely transposed overhead line having three ACSR$\mathbf{1113}\mathbf{kcmil}$ conductors per bundle, with$\mathbf{0}\mathbf{.}\mathbf{5}\mathbf{m}$ between conductors in the bundle. The horizontal phase spacings between bundle centers are $\mathbf{10}\mathbf{,}\mathbf{10}$and$\mathbf{20}\mathbf{m}$ .

Short answer

The inductive reactance of completely transposed overhead line is$0.335\frac{\mathrm{\Omega }}{\text{km}}$ respectively.

Step-by-step solution

Write the given data from the question.

The voltage of the line $V=500\text{\hspace{0.17em}kV}$.

The frequency of the system $f=60\text{\hspace{0.17em}Hz}$.

The bundle spacing $d=0.5\text{\hspace{0.17em}m}$.

The horizontal spacing between the conductor are $10,10$and $20$.

Refer to table A4, the geometric mean radius of the overhead conductor for $1113\text{\hspace{0.17em}kcmil}$,

$\begin{array}{l}{D}_s=0.0435\text{\hspace{0.17em}ft}\\ {\text{D}}_s=0.0435\times 0.305\text{\hspace{0.17em}m}\\ D_s=0.0132\text{\hspace{0.17em}m}\end{array}$

Determine the equation to calculate the inductance and inductive reactance.

The equation to calculate the geometric mean distance of completely transposed overhead line is given as follows.

${\mathbf{D}}_{\mathbf{eq}}\mathbf{=}\sqrt[\mathbf{3}]{{\mathbf{D}}_{\mathbf{12}}{\mathbf{D}}_{\mathbf{23}}{\mathbf{D}}_{\mathbf{31}}}$ …… (1)

Here, ${\mathbf{D}}_{\mathbf{12}}$is the distance between conductor role="math" localid="1656484507965" $\mathbf{1}$and role="math" localid="1656484523235" $\mathbf{2}$, role="math" localid="1656484456552" ${\mathbf{D}}_{\mathbf{23}}$is the distance between conductor role="math" localid="1656484517198" $\mathbf{2}$and role="math" localid="1656484533928" $\mathbf{3}$, ${\mathbf{D}}_{\mathbf{31}}$is distance between $\mathbf{3}$and $\mathbf{1}$.

The equation to calculate the bundle geometric mean radius of completely transposed overhead line is given as follows.

${\mathbf{D}}_{\mathbf{SL}}\mathbf{=}\sqrt[\mathbf{3}]{{\mathbf{D}}_{\mathbf{s}}{\mathbf{d}}^{\mathbf{2}}}$ …… (2)

The equation to calculate the inductance transposed overhead line inductance is given as follows.

$\mathbf{L}\mathbf{=}\mathbf{2}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{7}}\mathbf{ln}\left(\frac{D_{\mathrm{eq}}}{D_{\mathrm{SL}}}\right)\frac{\mathbf{H}}{\mathbf{m}}$ …… (3)

The equation to calculate inductive reactanceof transposed overhead lineis given as follows,

$\mathbf{X}\mathbf{=}\mathbf{2}\mathbf{\pi fL}$ …… (4)

Conversion from foot to metres.

$\mathbf{1}\mathbf{ft}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{305}\mathbf{m}$

Calculate the inductive reactance of the transposed overhead line.

Calculate the geometric mean distance.

Substitute$10m$for$D_{12}$,$10m$for$D_{23}$and$20m$for$D_{31}$into equation (1).

$\begin{array}{c}{D}_{eq}=\sqrt[3]{10\times 10\times 20}\\ D_{eq}={\left(2000\right)}^{\frac{1}{3}}\\ D_{eq}=12.59\text{\hspace{0.17em}m}\end{array}$

Calculate the bundle GMR.

Substitute$0.0132\text{\hspace{0.17em}m}$ for$D_s$and$0.5m$for $d$into equation (2).

$\begin{array}{l}{D}_{SL}=\sqrt[3]{0.0132\times {0.5}^2}\\ D_{SL}={\left(0.0033\right)}^{\frac{1}{3}}\\ D_{SL}=0.148\text{\hspace{0.17em}m}\end{array}$

Calculate the inductance of the transposed overhead conductors.

Substitute $12.59\text{\hspace{0.17em}m}$for $D_{eq}$, $0.148\text{\hspace{0.17em}m}$for $D_{SL}$into equation (3).

$\begin{array}{c}L=2\times {10}^{-7}\ln \left(\frac{12.59}{0.148}\right)\\ L=2\times {10}^{-7}\ln \left(85.067\right)\\ L=2\times {10}^{-7}\times 4.443\\ L=8.886\times {10}^{-7}\frac{\text{H}}{\text{m}}\end{array}$

Calculate the inductive reactance of the transposed overhead conductors.

Substitute $8.886\times {10}^{-7}\frac{\text{H}}{\text{m}}$for $L$and $60\text{\hspace{0.17em}Hz}$for $f$into equation (4).

$\begin{array}{l}X=2\pi \times 60\times 8.886\times {10}^{-7}\\ X=3349.94\times {10}^{-6}\\ X=0.335\frac{\Omega }{\text{km}}\end{array}$

Hence the inductive reactance of completely transposed overhead line is $0.335\frac{\Omega }{\text{km}}$respectively.