Question

Statement:
In the following network, the voltage magnitudes at a buses are equal to 1 p.u., the voltage phase angles are very small, and the line resistances are negligible. All the line reactances are equal to $j1\Omega$

Q: The voltage phase angles in rad at buses 2 and 3 are

• A. ${\theta }_2=-0.1,{\theta }_3=-0.2$
  
• B. ${\theta }_2=0,{\theta }_3=-0.1$
• C. ${\theta }_2=0.1,{\theta }_3=0.1$
• D. ${\theta }_2=0.1,{\theta }_3=0.2$

Answer 1

The correct option is B ${\theta }_2=0,{\theta }_3=-0.1$

Lossless system
$P_1+P_2+P_3=0$
$P_1+j0.1-j0.2=0$
$P_1=j0.1p.u.$
No power flow in between bus 1 and 2
$P_{12}\propto sin({\theta }_1-{\theta }_2)=0$
${\theta }_1-{\theta }_2=0$
As bus 1 is a slack bus
${\theta }_1=0$
$\Rightarrow {\theta }_2=0$
Now, $P_{23}=\frac{V_2{V}_3}{X}sin({\theta }_2-{\theta }_3)$
or $0.1=\frac{1\times 1}{1}sin({\theta }_2-{\theta }_3)$
$=0.10016\simeq 0.1$
${\theta }_3=-0.1$