GATE EE 2023 Set 1 — Question 28

In a bus admittance matrix, the off-diagonal elements are $Y_{ij} = -y_{ij}$ and the diagonal elements are $Y_{ii} = \sum_{j \neq i} y_{ij} + y_{i0}$, where $y_{ij}$ is the series admittance of the line between buses $i$ and $j$, and $y_{i0}$ is the total shunt admittance at bus $i$.From the given matrix:
$y_{12} = -Y_{12} = -j10$
$y_{13} = -Y_{13} = -j5$
$y_{23} = -Y_{23} = -j4$Now check the diagonal elements for shunt admittances:
For bus 1: $Y_{11} = y_{12} + y_{13} + y_{10} \Rightarrow -j15 = (-j10) + (-j5) + y_{10} \Rightarrow y_{10} = 0$. Thus, there is no shunt at bus 1.
For bus 2: $Y_{22} = y_{21} + y_{23} + y_{20} \Rightarrow -j13.5 = (-j10) + (-j4) + y_{20} \Rightarrow y_{20} = +j0.5$ (Capacitive shunt).
For bus 3: $Y_{33} = y_{31} + y_{32} + y_{30} \Rightarrow -j8 = (-j5) + (-j4) + y_{30} \Rightarrow y_{30} = +j1.0$ (Capacitive shunt).Analysis of options:
(A) If line charging capacitors were present in all three lines, they would contribute to shunts at all connected buses. Since bus 1 has zero shunt, line charging cannot be present in all three lines. Thus, (A) is NOT true.
(B) If line charging is only in line 2-3, it contributes shunts to buses 2 and 3, which matches our findings. This can be true.
(C) We found $y_{10} = 0$, so a shunt capacitor at bus 1 is impossible. Thus, (C) is NOT true.
(D) This matches our findings that shunts exist at buses 2 and 3. This can be true.Therefore, the statements that can NOT be true are A and C.