---
title: "Ybus with shunt admittances"
subject: "Power System Analysis II"
chapter: 1
key: "semester_5/power_system_analysis_ii:ybus-shunt-admittances-2073"
source: https://www.bctengineer.com/q/semester_5/power_system_analysis_ii/ybus-shunt-admittances-2073
site: BCT Engineering
---

# Ybus with shunt admittances

Power System Analysis II · Chapter 1: Interconnected Power System

- Asked in: 2073 Chaitra, 8 marks
- Video: [The Ybus Admittance Matrix for Solving Power Flow Equations Part 1](https://www.youtube.com/watch?v=8jNuSwnL7HE) — GeneralPAC: Power System Tutorials
- Answer checked by Claude Sonnet 5.5 on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_5/power_system_analysis_ii/ybus-shunt-admittances-2073
- Whole chapter: https://www.bctengineer.com/md/semester_5/power_system_analysis_ii/chapter-1

## Question

Compute Y-bus matrix for the following power system network shown in figure below. [Figure: G1 at bus 1 and G2 at bus 3; line 1-2 x1 = j0.1 pu; two parallel lines between buses 2 and 3, x2 = j0.2 pu and x3 = j0.2 pu; shunt admittances y1 = j5 pu at bus 1 and y2 = j5 pu at bus 2 to ground]

## Short answer

Parallel lines 2-3 combine to –j10. Shunt admittances j5 at buses 1 and 2 add to diagonal. Y11 = j5–j10 = –j5, Y22 = j5–j10–j10 = –j15, Y33 = –j10; off-diagonals Y12=Y21=j10, Y23=Y32=j10, Y13=Y31=0. Ybus = [[–j5, j10, 0]; [j10, –j15, j10]; [0, j10, –j10]] pu.

## Answer

#### Step 1: Branch admittances
- Line 1-2: $y_{12} = 1/j0.1 = -j10$
- Two parallel lines 2-3: $j0.2 \parallel j0.2 = j0.1$, so $y_{23} = 1/j0.1 = -j10$ (or $-j5 - j5$)
- No line 1-3: $y_{13} = 0$
- Shunt admittances (given as admittances): $y_{10} = j5$, $y_{20} = j5$, $y_{30} = 0$

Generators are sources at buses 1 and 3; no internal reactance is given, so they do not add to $Y_{bus}$.

#### Step 2: Diagonal elements
$$
\begin{aligned}
Y_{11} &= y_{10} + y_{12} = j5 - j10 = -j5 \\
Y_{22} &= y_{20} + y_{12} + y_{23} = j5 - j10 - j10 = -j15 \\
Y_{33} &= y_{23} = -j10
\end{aligned}
$$

#### Step 3: Off-diagonal elements
$$
Y_{12} = Y_{21} = j10,\qquad Y_{23} = Y_{32} = j10,\qquad Y_{13} = Y_{31} = 0
$$

**Answer:**

$$
Y_{bus} =
\begin{bmatrix}
-j5 & j10 & 0 \\
j10 & -j15 & j10 \\
0 & j10 & -j10
\end{bmatrix}\ \text{pu}
$$

Note the zero at $Y_{13}$ (no direct line), which shows the sparsity of $Y_{bus}$, and that the capacitive shunts reduce $|Y_{11}|$ and $|Y_{22}|$.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. Two parallel lines 2-3 with x = j0.2 pu each give y23 of:
   A. -j5
   B. -j10
   C. -j20
   D. -j2.5
   Answer: B. j0.2 || j0.2 = j0.1, so y23 = -j10. (Key point: Parallel lines)

2. Y11 = y10 + y12 with y10 = j5 and y12 = -j10 is:
   A. -j15
   B. -j5
   C. j15
   D. -j10
   Answer: B. j5 - j10 = -j5. (Key point: Y11)

3. Why is Y13 zero?
   A. Shunts cancel
   B. No direct line joins buses 1 and 3
   C. G1 and G2 are equal
   D. Buses 1 and 3 are generator buses
   Answer: B. Only existing branches contribute to off-diagonals. (Key point: Zero element)

4. Compared with no shunt, the capacitive shunts at buses 1 and 2:
   A. Increase |Y11| and |Y22|
   B. Reduce |Y11| and |Y22|
   C. Change Y12
   D. Make Ybus unsymmetrical
   Answer: B. Positive j shunt cancels part of the negative imaginary series admittance. (Key point: Effect of capacitive shunt)

## Related in Power System Analysis II

- [Ybus and bus types (parallel lines)](https://www.bctengineer.com/question.md?key=semester_5%2Fpower_system_analysis_ii%3Aybus-bus-types-2074) — Power System Analysis II, chapter 1
- [Ybus for 3-bus lines with shunt susceptance](https://www.bctengineer.com/question.md?key=semester_5%2Fpower_system_analysis_ii%3Aybus-three-bus-shunt-2071) — Power System Analysis II, chapter 1
- [Node equations and Ybus, six-bus network](https://www.bctengineer.com/question.md?key=semester_5%2Fpower_system_analysis_ii%3Aybus-six-bus-node-equations) — Power System Analysis II, chapter 1
- [Ybus of five-bus system](https://www.bctengineer.com/question.md?key=semester_5%2Fpower_system_analysis_ii%3Aybus-five-bus) — Power System Analysis II, chapter 1
- [L-G fault at bus 3, parallel lines](https://www.bctengineer.com/question.md?key=semester_5%2Fpower_system_analysis_ii%3Alg-fault-bus3-two-generators) — Power System Analysis II, chapter 5
- [Ybus for 3-bus system (2078)](https://www.bctengineer.com/question.md?key=semester_5%2Fpower_system_analysis_ii%3Aybus-three-bus-2078) — Power System Analysis II, chapter 1

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