---
title: "Ybus of five-bus system"
subject: "Power System Analysis II"
chapter: 1
key: "semester_5/power_system_analysis_ii:ybus-five-bus"
source: https://www.bctengineer.com/q/semester_5/power_system_analysis_ii/ybus-five-bus
site: BCT Engineering
---

# Ybus of five-bus system

Power System Analysis II · Chapter 1: Interconnected Power System

- Asked in: 2082 Baishakh, 6+2 marks
- Video: [Formation of Y-BUS using inspection method](https://www.youtube.com/watch?v=jaS0DWPHeFE) — Fun N Facts
- Answer checked by Claude Sonnet 5.5 on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_5/power_system_analysis_ii/ybus-five-bus
- Whole chapter: https://www.bctengineer.com/md/semester_5/power_system_analysis_ii/chapter-1

## Question

The figure below shows a five-bus power system. Each line has a series impedance of 0.05 + j0.15 pu. The line shunt admittance may be neglected. Compute the Bus-admittance matrix. What are the driving point and transfer admittances? [Figure: five buses with lines 1-2, 1-5, 2-3, 2-5, 3-4 and 4-5]

## Short answer

Each line series admittance y = 2 - j6 pu. Driving-point admittances: Y11=Y33=Y44=4-j12 pu, Y22=Y55=6-j18 pu. Transfer admittances between directly connected buses are -2+j6 pu; zero otherwise.

## Answer

#### Series admittance of each line
$$
y = \frac{1}{0.05 + j0.15} = \frac{0.05 - j0.15}{0.025} = 2 - j6\ \text{pu}
$$

Shunt admittances are neglected, so $Y_{ii}$ = (number of lines at bus $i$) × $y$ and $Y_{ik} = -y$ for each connected pair.

#### Lines at each bus
| Bus | Connected to | No. of lines | $Y_{ii}$ |
|---|---|---|---|
| 1 | 2, 5 | 2 | $4 - j12$ |
| 2 | 1, 3, 5 | 3 | $6 - j18$ |
| 3 | 2, 4 | 2 | $4 - j12$ |
| 4 | 3, 5 | 2 | $4 - j12$ |
| 5 | 1, 2, 4 | 3 | $6 - j18$ |

Off-diagonal: $Y_{12} = Y_{15} = Y_{23} = Y_{25} = Y_{34} = Y_{45} = -2 + j6$; $Y_{13} = Y_{14} = Y_{24} = Y_{35} = 0$.

**Answer:**

$$
Y_{bus} =
\begin{bmatrix}
4 - j12 & -2 + j6 & 0 & 0 & -2 + j6 \\
-2 + j6 & 6 - j18 & -2 + j6 & 0 & -2 + j6 \\
0 & -2 + j6 & 4 - j12 & -2 + j6 & 0 \\
0 & 0 & -2 + j6 & 4 - j12 & -2 + j6 \\
-2 + j6 & -2 + j6 & 0 & -2 + j6 & 6 - j18
\end{bmatrix}\ \text{pu}
$$

#### Driving-point and transfer admittances
- **Driving-point (self) admittances** are the diagonal elements $Y_{ii}$: $Y_{11} = Y_{33} = Y_{44} = 4 - j12$ pu and $Y_{22} = Y_{55} = 6 - j18$ pu. $Y_{ii}$ is the current injected at bus $i$ per unit voltage at bus $i$ with all other buses shorted to ground.
- **Transfer (mutual) admittances** are the off-diagonal elements $Y_{ik}$: $-2 + j6$ pu for directly connected buses and 0 otherwise. $Y_{ik}$ is the current at bus $i$ per unit voltage at bus $k$ with all other buses shorted.

## Check yourself

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

1. With identical lines and no shunts, Y_ii equals:
   A. y
   B. Number of lines at bus i times y
   C. Sum over all buses of y
   D. Zero
   Answer: B. Y_ii is the sum of the admittances of lines at bus i. (Key point: Diagonal with no shunts)

2. Bus 2 has three lines (to 1, 3, 5). Y22 is:
   A. 4 - j12
   B. 6 - j18
   C. 2 - j6
   D. -6 + j18
   Answer: B. 3(2 - j6) = 6 - j18. (Key point: Y22)

3. Y13 is zero because:
   A. Bus 3 is a PV bus
   B. No line joins buses 1 and 3
   C. y=0
   D. Shunt neglect
   Answer: B. No direct connection gives a zero transfer admittance. (Key point: Zero entries)

4. Driving-point admittance refers to:
   A. Off-diagonal elements
   B. Diagonal elements Y_ii
   C. The inverse of Ybus
   D. Shunt susceptance only
   Answer: B. Y_ii is the self (driving-point) admittance; Y_ik are the transfer admittances. (Key point: Driving point)

## Related in Power System Analysis II

- [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 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
- [Ybus modified by mid-line tap changer](https://www.bctengineer.com/question.md?key=semester_5%2Fpower_system_analysis_ii%3Aybus-tap-changer-midpoint) — Power System Analysis II, chapter 1
- [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
- [Bus admittance matrix and its elements](https://www.bctengineer.com/question.md?key=semester_5%2Fpower_system_analysis_ii%3Aybus-definition-elements) — Power System Analysis II, chapter 1

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