Question

The bus admittance ($Y_{\text{bus}}$) matrix of a 3-bus power system is given below. \[ Y_{\text{bus}} = \begin{bmatrix} -j15 & j10 & j5 \\ j10 & -j13.5 & j4 \\ j5 & j4 & -j8 \end{bmatrix} \] Considering that there is no shunt inductor connected to any of the buses, which of the following can NOT be true?

• A. Line charging capacitor of finite value is present in all three lines
• B. Line charging capacitor of finite value is present in line 2–3 only
• C. Line charging capacitor of finite value is present in line 2–3 only and shunt capacitor of finite value is present in bus 1 only
• D. Line charging capacitor of finite value is present in line 2–3 only and shunt capacitor of finite value is present in bus 3 only

Hint:

In $Y_{\text{bus}}$ analysis, compare the diagonal elements with the negative sum of corresponding off-diagonal terms. Any difference reveals the presence of shunt capacitances or shunt elements.

Answer 1

The Correct Option is B

Step 1: Recall $Y_{\text{bus}$ properties.}
- Off-diagonal entry $Y_{ij}$ corresponds to $-Y_{line}(i,j)$. - Diagonal entry $Y_{ii}$ is the sum of admittances connected to bus $i$, i.e. \[ Y_{ii} = \sum_{j\neq i} Y_{line}(i,j) + Y_{\text{shunt},i}. \]

Step 2: Check line admittances from off-diagonal terms.
From the matrix: - Between bus 1 and bus 2: $Y_{12} = j10 \;\Rightarrow\; Y_{line}(1,2)= -j10$. - Between bus 1 and bus 3: $Y_{13} = j5 \;\Rightarrow\; Y_{line}(1,3)=-j5$. - Between bus 2 and bus 3: $Y_{23} = j4 \;\Rightarrow\; Y_{line}(2,3)=-j4$. Thus, three lines exist: 1–2, 1–3, 2–3.

Step 3: Check diagonal entries.
- For bus 1: \[ Y_{11} = -j15, \text{expected from lines } = -(j10+j5) = -j15. \] So no shunt element at bus 1. - For bus 2: \[ Y_{22} = -j13.5, \text{expected from lines } = -(j10+j4) = -j14. \] Difference: $0.5j$, indicating a shunt capacitor at bus 2. - For bus 3: \[ Y_{33} = -j8, \text{expected from lines } = -(j5+j4) = -j9. \] Difference: $j1$, indicating a shunt capacitor at bus 3.

Step 4: Interpret results.
- All three lines (1–2, 1–3, 2–3) have line charging capacitances. - Shunt capacitances are present at bus 2 and bus 3.

Step 5: Evaluate options.
- (A) True: All three lines have finite capacitances. - (B) False: Not only line 2–3, all lines exist. This can NOT be true. - (C) False: Shunt capacitor is not at bus 1, but at buses 2 and 3. - (D) False: Shunt capacitor is not only at bus 3, but also at bus 2. % Final Answer \[ \boxed{\text{Option (B)}} \]