Question

(a) Consider complex power transmission via the three-phase short line for which the per-phase circuit is shown in Figure 5.19. Express ${\mathit{S}}_{\mathbf{12}}$, the complex power sent by bus $\mathbf{1}$(or${\mathit{V}}_{\mathbf{1}}$), and ($\mathbf{-}{\mathit{S}}_{\mathbf{21}}$), the complex power received by bus $\mathbf{2}$(or${\mathit{V}}_{\mathbf{2}}$), in terms of ${\mathit{V}}_{\mathbf{1}}\mathbf{,}{\mathit{V}}_{\mathbf{2}}\mathbf{,}\mathit{Z}\mathbf{\angle }\mathit{Z}\mathbf{,}$and${\mathit{\theta }}_{\mathbf{12}}\mathbf{=}{\mathit{\theta }}_{\mathbf{1}}\mathbf{-}{\mathit{\theta }}_{\mathbf{2}}$, which is the power angle.

(b) For a balanced three-phase transmission line in per-unit notation with$\mathbf{1}\mathbf{\angle }\mathbf{85}\mathbf{°}\mathbf{,}$,${\mathit{\theta }}_{\mathbf{12}}\mathbf{=}\mathbf{10}\mathbf{°}$determine ${\mathit{S}}_{\mathbf{12}}$and ($\mathbf{-}{\mathit{S}}_{\mathbf{21}}$) for

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

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

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

Short answer

(a) The complex power $S_{12}$is$\frac{V_1^2}{Z}\angle Z-\frac{V_1{V}_2}{Z}\angle \left({\theta }_{12}+z\right)°$ and$s_{21}$ is$\frac{V_2^2}{Z}\angle Z-\frac{V_1{V}_2}{Z}\angle \left(-{\theta }_{12}+Z\right)$

(b) (i)The complex power$S_{12}$ is$0.174+0j\text{\hspace{0.17em}VA}$and$S_{21}$ is$-0.172+0.03\text{\hspace{0.17em}VA}$ .

(ii) The complex power$S_{12}$ is $0.192+0.22j\text{\hspace{0.17em}VA}$and $S_{21}$is $-0.186+0.15\text{\hspace{0.17em}VA}$.

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 $S_{12}$is given as follows.

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

Here,${\mathit{S}}_{\mathbf{12}}$ is the complex power from sending end to receiving end, ${\mathit{V}}_{\mathbf{1}}$ is the sending end voltage and ${\mathit{I}}_{\mathbf{12}}$ is the current from sending end to receiving end.

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

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

Here, ${\mathit{S}}_{\mathbf{21}}$is the complex power from sending end to receiving end,role="math" localid="1655956588854" ${\mathit{V}}_{\mathbf{1}}$ is the receiving end voltage and ${\mathit{I}}_{\mathbf{21}}$is the current from receiving end to sending end.

Determine the complex power S12and -S21.

(a)

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

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

Calculate the complex power$S_{12}$

Substitute$\frac{V_1}{Z}\angle \left({\theta }_1-Z\right)°-\frac{V_2}{Z}\angle \left({\theta }_2-Z\right)°$for $I_{12}$and $v_1<0$for$V_1$into equation (1).

$\begin{array}{l}{S}_{12}=V_1\angle \theta {\left(\frac{V_1}{Z}\angle \left({\theta }_1-Z\right)°-\frac{V_2}{Z}\angle \left({\theta }_2-Z\right)°\right)}^{*}\\ S_{12}=\frac{V_1^2}{Z}\angle Z-\frac{V_1{V}_2}{Z}\angle \left({\theta }_1-{\theta }_2+z\right)°\\ S_{12}=\frac{V_1^2}{Z}\angle Z-\frac{V_1{V}_2}{Z}\angle \left({\theta }_{12}+z\right)°\end{array}$

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

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

Determine the complex power$S_{21}$.

Substitute $\frac{V_2}{Z}\angle \left({\theta }_2-Z\right)°-\frac{V_1}{Z}\angle \left({\theta }_1-Z\right)°$for $I_{21}$and $V_2\angle {\theta }_2$for $V_2$into equation (2)

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

Hence the complex power$S_{12}$ is $\frac{V_1^2}{Z}\angle Z-\frac{V_1{V}_2}{Z}\angle \left({\theta }_{12}+z\right)°$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 powerS12 and -S21.

(b)

The impedance,$Z=1\angle 85°$

The angle,${\theta }_{12}=10°$

(i)

Determine the complex power $S_{12}$.

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

Substitute$1\angle 85°$for$Z$,${\theta }_{12}$for and$1$for$V_1,V_2$into above equation.

$\begin{array}{l}{S}_{12}=\frac{1}{1}\angle 85-\frac{\left(1\right)\left(1\right)}{1}\angle \left(10+85\right)°\\ S_{12}=1\angle 85°-1\angle 95°\\ S_{12}=0.174+0j\text{\hspace{0.17em}VA}\end{array}$

Determine the complex power$S_{21}$.

$\begin{array}{l}{S}_{21}=\frac{1}{1}\angle 85-\frac{\left(1\right)\left(1\right)}{1}\angle \left(-10+85\right)°\\ S_{21}=1\angle 85°-1\angle 75°\\ S_{21}=-0.172+0.03\text{\hspace{0.17em}VA}\end{array}$

Hence the complex power$S_{12}$is $0.174+0j\text{\hspace{0.17em}VA}$and $S_{21}$is $-0.172+0.03\text{\hspace{0.17em}VA}$.

(ii)

Determine the complex power $S_{12}$.

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

Substitute$1\angle 85°$for $Z$, for${\theta }_{12}$and$1.1$for $V_1,$and $0.9$for$V_2$into above equation.

$\begin{array}{l}{S}_{12}=\frac{{1.1}^2}{1}\angle 85-\frac{\left(1.1\right)\left(0.9\right)}{1}\angle \left(10+85\right)°\\ S_{12}=1.21\angle 85°-0.99\angle 95°\\ S_{12}=0.192+0.22j\text{\hspace{0.17em}VA}\end{array}$

Determine the complex power .$S_{21}$

$\begin{array}{l}{S}_{21}=\frac{{0.9}^2}{1}\angle 85-\frac{\left(1.1\right)\left(0.9\right)}{1}\angle \left(-10+85\right)°\\ S_{21}=0.81\angle 85°-0.99\angle 75°\\ S_{21}=-0.186+0.15\text{\hspace{0.17em}VA}\end{array}$

Hence the complex power $S_{12}$is$0.192+0.22j\text{\hspace{0.17em}VA}$ and $S_{21}$is $-0.186+0.15\text{\hspace{0.17em}VA}$.

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.