---
title: "Ybus with nominal π model, line lengths"
subject: "Power System Analysis II"
chapter: 1
key: "semester_5/power_system_analysis_ii:ybus-km-lines-pi-model"
source: https://www.bctengineer.com/q/semester_5/power_system_analysis_ii/ybus-km-lines-pi-model
site: BCT Engineering
---

# Ybus with nominal π model, line lengths

Power System Analysis II · Chapter 1: Interconnected Power System

- Asked in: 2070 Asar, 6 marks; 2069 Chaitra, 6 marks (asked 2 times)
- Answer checked by Claude Sonnet 5.5 on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_5/power_system_analysis_ii/ybus-km-lines-pi-model
- Whole chapter: https://www.bctengineer.com/md/semester_5/power_system_analysis_ii/chapter-1

## Question

The single line diagram of a power system is shown in figure below. If the per km line series reactance is 0.001 pu and shunt susceptance is 0.0016 pu, find bus admittance matrix using nominal pi-model of lines. [Figure: three buses; line 1-2 is 100 km, line 1-3 is 200 km, line 2-3 is 400 km]

## Short answer

Calculate each line's total series reactance (0.001 pu/km × length) and shunt susceptance (0.0016 pu/km × length), split half at each end. Series admittance is 1/(jX); diagonal elements sum series admittances plus half susceptances; off-diagonals are negative of series admittances. Ybus = [[-ij14.76, j10, j5], [j10, -ij12.10, j2.5], [j5, j2.5, -ij7.02]] pu.

## Answer

With the nominal π model, each line's total series reactance is $x \times \text{length}$ and its total shunt susceptance $B = b \times \text{length}$ is split as $B/2$ at each end. Resistance is not given, so it is taken as zero.

Given: $x = 0.001$ pu/km, $b = 0.0016$ pu/km.

#### Step 1: Line parameters
| Line | Length (km) | $X$ (pu) | $y = 1/jX$ (pu) | Total $B$ (pu) | $B/2$ (pu) |
|---|---|---|---|---|---|
| 1-2 | 100 | 0.1 | $-j10$ | 0.16 | 0.08 |
| 1-3 | 200 | 0.2 | $-j5$ | 0.32 | 0.16 |
| 2-3 | 400 | 0.4 | $-j2.5$ | 0.64 | 0.32 |

#### Step 2: Diagonal elements
$Y_{ii}$ = sum of series admittances at bus $i$ + sum of half-line-charging susceptances at bus $i$:

$$
\begin{aligned}
Y_{11} &= (-j10 - j5) + j(0.08 + 0.16) = -j14.76 \\
Y_{22} &= (-j10 - j2.5) + j(0.08 + 0.32) = -j12.10 \\
Y_{33} &= (-j5 - j2.5) + j(0.16 + 0.32) = -j7.02
\end{aligned}
$$

#### Step 3: Off-diagonal elements
$$
\begin{aligned}
Y_{12} &= Y_{21} = -y_{12} = j10 \\
Y_{13} &= Y_{31} = -y_{13} = j5 \\
Y_{23} &= Y_{32} = -y_{23} = j2.5
\end{aligned}
$$

#### Result
$$
Y_{bus} =
\begin{bmatrix}
-j14.76 & j10 & j5 \\
j10 & -j12.10 & j2.5 \\
j5 & j2.5 & -j7.02
\end{bmatrix}\ \text{pu}
$$

**Answer:** $Y_{11} = -j14.76$, $Y_{22} = -j12.10$, $Y_{33} = -j7.02$, $Y_{12} = j10$, $Y_{13} = j5$, $Y_{23} = j2.5$ pu. The shunt charging only reduces the magnitude of the diagonal terms; off-diagonal terms are unaffected.

## Check yourself

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

1. For the 200 km line 1-3 with x = 0.001 pu/km, the series admittance is:
   A. -j10 pu
   B. -j2.5 pu
   C. -j5 pu
   D. j5 pu
   Answer: C. X = 0.2 pu, y = 1/j0.2 = -j5. (Key point: Line reactance)

2. In the nominal pi model the total line charging B is:
   A. placed fully at one end
   B. ignored
   C. added to off-diagonal elements
   D. split as B/2 at each end
   Answer: D. B/2 appears at each bus and adds to the diagonal. (Key point: Half charging)

3. Y22 equals:
   A. -j12.10 pu
   B. -j12.90 pu
   C. -j14.76 pu
   D. j12.10 pu
   Answer: A. Y22 = -j10 - j2.5 + j(0.08 + 0.32) = -j12.10. (Key point: Y22)

## 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
- [G-S one iteration with π-model lines](https://www.bctengineer.com/question.md?key=semester_5%2Fpower_system_analysis_ii%3Ags-one-iteration-pi-lines-2069) — Power System Analysis II, chapter 2
- [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
- [Zbus vs Ybus; off-nominal transformer model](https://www.bctengineer.com/question.md?key=semester_5%2Fpower_system_analysis_ii%3Aybus-off-nominal-transformer) — 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
- [Ybus derivation for 3-bus π-model network](https://www.bctengineer.com/question.md?key=semester_5%2Fpower_system_analysis_ii%3Aybus-three-bus-derivation) — Power System Analysis II, chapter 1, asked 2×

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