Question

Consider complex power transmission via the three-phase short

line for which the per-phase circuit is shown in Figure 5.19. Express ${\mathbf{S}}_{\mathbf{12}}$, the

complex power sent by bus 1$\mathbf{(}\mathbf{or}\mathbf{}{\mathbf{V}}_{\mathbf{2}}\mathbf{)}$,and$\left(\mathbf{-}{\mathbf{S}}_{\mathbf{21}}\right)$, the complex power received

by bus 2$\mathbf{(}\mathbf{or}\mathbf{}{\mathbf{V}}_{\mathbf{2}}\mathbf{)}$,in terms of$\mathbf{1}\mathbf{\angle }{\mathbf{85}}^{\mathbf{\circ }}\mathbf{,}\mathbf{}{\mathbf{\theta }}_{\mathbf{12}}\mathbf{=}{\mathbf{10}}^{\mathbf{\circ }}$and${\mathbf{\theta }}_{\mathbf{12}}\mathbf{=}{\mathbf{\theta }}_{\mathbf{1}}\mathbf{-}{\mathbf{\theta }}_{\mathbf{2}}$,which is the power

angle.

(b) For a balanced three-phase transmission line in per-unit notation with

$\mathbf{1}\mathbf{\angle }{\mathbf{85}}^{\mathbf{\circ }}\mathbf{,}\mathbf{}{\mathbf{\theta }}_{\mathbf{12}}\mathbf{=}{\mathbf{10}}^{\mathbf{\circ }}$,determine${\mathbf{S}}_{\mathbf{12}}$and$\left(\mathbf{-}{\mathbf{S}}_{\mathbf{21}}\right)$for

(i)${\mathbf{V}}_{\mathbf{1}}\mathbf{=}{\mathbf{V}}_{\mathbf{2}}\mathbf{=}\mathbf{1}\mathbf{.}\mathbf{0}$

(ii)${\mathbf{V}}_{\mathbf{1}}\mathbf{=}\mathbf{1}\mathbf{.}\mathbf{0}\mathbf{}\mathbf{and}\mathbf{}{\mathbf{V}}_{\mathbf{2}}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{9}$

Comment on the changes of real and reactive powers from parts (i) to (ii)

Short answer

(a) The complex power$S_{12}$is $\frac{{\mathrm{V}}_1^2}{\mathrm{Z}}\angle \mathrm{Z}-\frac{{\mathrm{V}}_1{\mathrm{V}}_2}{\mathrm{Z}}\angle {\left(\angle {\mathrm{\theta }}_{12}+\mathrm{z}\right)}^{\circ }$and$s_{21}$is

$\frac{{\mathrm{V}}_2^2}{\mathrm{Z}}\angle \mathrm{Z}-\frac{{\mathrm{V}}_1{\mathrm{V}}_2}{\mathrm{Z}}\angle \left(\angle -{\mathrm{\theta }}_{12}+\mathrm{z}\right)$

(b) (i) The complex power$S_{12}$is 0.174 + 0jVA and$S_{21}$is -0.172+0.03 VA .

(ii) The complex power$S_{12}$is 0.192+0.22jVA and$S_{21}$is -0.186+0.15VA .

When the voltage of bus 1 is increased, the real and the reactive power by bus 1

increased and when voltage of bus 2 decreased, the real and reactive by bus 2 are

increased.

Step-by-step solution

determine the formulas to calculate the complex power.

The equation to calculate the complex power is given as follows.

${\mathrm{S}}_{12}={\mathrm{V}}_1{\mathrm{I}}_{12}^{*}$ …… (1)

Here,${\mathbf{S}}_{\mathbf{12}}$is the complex power from sending end to receiving end,${\mathbf{V}}_{\mathbf{1}}$is the

sending end voltage and${\mathbf{I}}_{\mathbf{12}}$is the current from sending end to receiving end.

The equation to calculate the complex power$S_{21}$is given as follows.

${\mathbf{S}}_{\mathbf{21}}\mathbf{=}{\mathbf{V}}_{\mathbf{2}}{\mathbf{I}}_{\mathbf{21}}^{\mathbf{*}}$ …… (2)

Here,${\mathbf{S}}_{\mathbf{12}}$is the complex power from sending end to receiving end,${\mathbf{V}}_{\mathbf{1}}$is the

receiving end voltage and${\mathbf{I}}_{\mathbf{21}}$is the current from receiving end to sending end.

Determine the complex powerS12and-S21.

(a)

Calculate the current$I_{12}$in the circuit.

$$\begin{gathered} I_{12}=\frac{V_1\angle {\theta }_1-V_2\angle {\theta }_2}{Z\angle Z} \\ {I}_{12}=\frac{V_1}{Z}\angle {\left({\theta }_2-Z\right)}^{\circ }-\frac{V_2}{Z}\angle {\left({\theta }_2-Z\right)}^{\circ } \end{gathered}$$

Calculate the complex power$S_{12}$

Substitute$\frac{V_1}{Z}\angle {\left({\theta }_1-Z\right)}^{\circ }-\frac{V_2}{Z}\angle {\left({\theta }_2-Z\right)}^{\circ }$for$I_{12}$and$V_1\angle \theta$for$V_1$into equation (1).

$$\begin{gathered} S_{21}=V_1\angle \theta \left(\frac{V_1}{Z}\angle {\left({\theta }_1-Z\right)}^{\circ }-\frac{V_2}{Z}\angle {\left({\theta }_2-Z\right)}^{\circ }\right) \\ {S}_{21}=\frac{V_1^2}{Z}\angle Z-\frac{V_1{V}_2}{Z}\angle {\left({\theta }_1+{\theta }_2+Z\right)}^{\circ } \\ {S}_{21}=\frac{V_1^2}{Z}\angle Z-\frac{V_1{V}_2}{Z}\angle {\left({\theta }_{12}+Z\right)}^{\circ } \end{gathered}$$

Calculate the current$I_{21}$in the circuit.

$$\begin{gathered} I_{21}=\frac{V_2\angle {\theta }_2-V_1\angle {\theta }_1}{Z\angle Z} \\ {I}_{21}=\frac{V_2}{Z}\angle {\left({\theta }_2-Z\right)}^{\circ }-\frac{V_1}{Z}\angle {\left({\theta }_1-Z\right)}^{\circ } \end{gathered}$$

Determine the complex power .

Substitute for and for into equation (2)

$$\begin{gathered} S_{21}=V_2\angle \theta \left(\frac{V_2}{Z}\angle {\left({\theta }_2-Z\right)}^{\circ }-\frac{V_1}{Z}\angle {\left({\theta }_1-Z\right)}^{\circ }\right) \\ {S}_{21}=\frac{V_2^2}{Z}\angle Z-\frac{V_1{V}_2}{Z}\angle \left(-{\theta }_1+{\theta }_2+Z\right) \\ {S}_{21}=\frac{V_2^2}{Z}\angle Z-\frac{V_1{V}_2}{Z}\angle \left(-{\theta }_{12}+Z\right) \end{gathered}$$

Hence the complex power$S_{12}$is $\frac{V_1^2}{Z}\angle Z-\frac{V_1{V}_2}{Z}\angle {\left({\theta }_{12}+Z\right)}^{\circ }$and$S_{21}$is.

$\frac{V_2^2}{Z}\angle Z-\frac{V_1{V}_2}{Z}\angle \left(-{\theta }_{12}+Z\right)$

Determine the complex powerS12and-S21.

(b)

The impedance,$Z=1\angle {85}^{\circ }$

The angle,${\theta }_{12}={10}^{\circ }$

(i)

Determine the complex power$S_{12}$.

$S_{12}=\frac{V_1^2}{Z}\angle Z-\frac{V_1{V}_2}{Z}\angle {\left({\theta }_{12}+Z\right)}^{\circ }$

Substitute$1\angle {85}^{\circ }$, for Z , ${10}^{\circ }$and 1 for $V_1,V_2$into above equation.

$$\begin{gathered} S_{12}=\frac{1}{1}\angle 85-\frac{\left(1\right)\left(1\right)}{1}\angle {\left(10+85\right)}^{\circ } \\ {S}_{12}=1\angle {85}^{\circ }-1\angle {95}^{\circ } \\ {S}_{12}=0.174+0jVA \end{gathered}$$

Determine the complex power$S_{21}$.

$$\begin{gathered} S_{21}=\frac{1}{1}\angle 85-\frac{\left(1\right)\left(1\right)}{1}\angle {\left(10+85\right)}^{\circ } \\ {S}_{21}=1\angle {85}^{\circ }-1\angle {85}^{\circ } \\ {S}_{21}=0.172+0.03jVA \end{gathered}$$

Hence the complex power$S_{12}$is 0.174+0jVA and$S_{21}$is$-0.172+0.03\mathrm{VA}$.

(ii)

Determine the complex power$S_{12}$.

$S_{12}=\frac{V_1^2}{Z}\angle Z-\frac{V_1{V}_2}{Z}\angle {\left({\theta }_{12}+Z\right)}^{\circ }$

Substitute for$1\angle {85}^{\circ }$for Z,${10}^{\circ }$for${\theta }_{12}$and 1.1 for$V_1$,and 0.9 for$V_2$into above

equation.

$$\begin{gathered} S_{12}=\frac{1.1^2}{1}\angle 85-\frac{\left(1.1\right)\left(0.9\right)}{1}\angle {\left(10+85\right)}^{\circ } \\ {S}_{12}=1.21\angle {85}^{\circ }-0.99\angle {95}^{\circ } \\ {S}_{12}=0.192+0.22jVA \end{gathered}$$

Determine the complex power$S_{21}$.

$$\begin{gathered} S_{21}=\frac{0.9^2}{1}\angle 85-\frac{\left(1.1\right)\left(0.9\right)}{1}\angle {\left(-10+85\right)}^{\circ } \\ {S}_{21}=0.81\angle {85}^{\circ }-0.99\angle {75}^{\circ } \\ {S}_{21}=-0.186+0.15\mathrm{VA} \end{gathered}$$

Hence the complex power$S_{12}$is 0.192+0.22jVA and$S_{21}$is -0.186+0.15VA.

When the voltage of bus 1 is increased, the real and the reactive power by bus 1

increased and when voltage of bus 2 decreased, the real and reactive by bus are

increased.