---
title: "Ybus derivation for 3-bus π-model network"
subject: "Power System Analysis II"
chapter: 1
key: "semester_5/power_system_analysis_ii:ybus-three-bus-derivation"
source: https://www.bctengineer.com/q/semester_5/power_system_analysis_ii/ybus-three-bus-derivation
site: BCT Engineering
---

# Ybus derivation for 3-bus π-model network

Power System Analysis II · Chapter 1: Interconnected Power System

- Asked in: 2078 Kartik, 8 marks; 2075 Asoj, 8 marks (asked 2 times)
- Video: [Bus Admittance Matrix | Ybus Matrix | Derivation of Ybus Matrix | Load Flow](https://www.youtube.com/watch?v=KXYiU_3QcAY) — Dr. Hemant Mahala Academy 
- Answer checked by Claude Sonnet 5.5 on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_5/power_system_analysis_ii/ybus-three-bus-derivation
- Whole chapter: https://www.bctengineer.com/md/semester_5/power_system_analysis_ii/chapter-1

## Question

With an example of a 3-Bus system and π-model of inter-connecting transmission lines derive (explain the steps of forming) the bus admittance matrix (YBUS). Also define the diagonal and off-diagonal elements of the matrix.

## Short answer

Apply KCL at each bus with lines in π-model. Diagonal Y_ii is sum of all admittances (series and shunt) connected to bus i. Off-diagonal Y_ik is negative of series admittance between buses i and k. For a 3-bus system, write I = YV equations, yielding symmetric Ybus.

## Answer

The **bus admittance matrix** $Y_{bus}$ relates the bus current injections to the bus voltages, $\mathbf{I}_{bus} = \mathbf{Y}_{bus}\mathbf{V}_{bus}$. It is formed by applying KCL at each bus of the network, with lines represented by their π-model.

#### Network and π-model
Consider three buses connected by lines 1-2, 1-3 and 2-3. Each line $i$-$k$ has series admittance $y_{ik} = 1/(R_{ik} + jX_{ik})$ and total shunt charging admittance $y'_{ik}$, half of which ($y'_{ik}/2$) is placed at each end.

```text
                 (2)
                /   \
          y12  /     \  y23
              /       \
   I1 --- (1)---------(3) --- I3
              y13

  I1, I2, I3 are the currents injected at buses 1, 2, 3.
  Shunt admittance to ground at each bus (half-line charging):
    bus 1: y'12/2 + y'13/2
    bus 2: y'12/2 + y'23/2
    bus 3: y'13/2 + y'23/2
```

#### Steps of forming Ybus
**Step 1: KCL at bus 1.** Current injected = current in series branches + current in shunt branches:

$$
\begin{aligned}
I_1 &= y_{12}(V_1 - V_2) + y_{13}(V_1 - V_3) + \frac{y'_{12}}{2}V_1 + \frac{y'_{13}}{2}V_1 \\
&= \left(y_{12} + y_{13} + \frac{y'_{12}}{2} + \frac{y'_{13}}{2}\right)V_1 - y_{12}V_2 - y_{13}V_3
\end{aligned}
$$

**Step 2: KCL at bus 2 and bus 3** in the same way:

$$
\begin{aligned}
I_2 &= -y_{12}V_1 + \left(y_{12} + y_{23} + \frac{y'_{12}}{2} + \frac{y'_{23}}{2}\right)V_2 - y_{23}V_3 \\
I_3 &= -y_{13}V_1 - y_{23}V_2 + \left(y_{13} + y_{23} + \frac{y'_{13}}{2} + \frac{y'_{23}}{2}\right)V_3
\end{aligned}
$$

**Step 3: Write in matrix form.**

$$
\begin{bmatrix} I_1 \\ I_2 \\ I_3 \end{bmatrix}
=
\begin{bmatrix} Y_{11} & Y_{12} & Y_{13} \\ Y_{21} & Y_{22} & Y_{23} \\ Y_{31} & Y_{32} & Y_{33} \end{bmatrix}
\begin{bmatrix} V_1 \\ V_2 \\ V_3 \end{bmatrix}
$$

**Step 4: Read off the elements**, e.g. $Y_{11} = y_{12} + y_{13} + \frac{y'_{12}}{2} + \frac{y'_{13}}{2}$, $Y_{12} = Y_{21} = -y_{12}$.

#### Diagonal elements (self or driving-point admittance)
$$
Y_{ii} = \sum_{k \ne i} y_{ik} + \sum_{k \ne i} \frac{y'_{ik}}{2}
$$

$Y_{ii}$ is the sum of all admittances connected to bus $i$ (series and shunt). Physically, $Y_{ii} = I_i/V_i$ with all other buses short-circuited ($V_k = 0$).

#### Off-diagonal elements (mutual or transfer admittance)
$$
Y_{ik} = Y_{ki} = -y_{ik}
$$

It is the negative of the series admittance between buses $i$ and $k$; it is zero if no line joins them. Physically, $Y_{ik} = I_i/V_k$ with all buses except $k$ shorted.

#### Properties
- Symmetric ($Y_{ik} = Y_{ki}$) for networks without phase shifters.
- Sparse: in large systems most off-diagonal terms are zero.
- Easy to form directly by inspection and to modify when a line is added or removed.

## Check yourself

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

1. Ybus is formed by:
   A. applying KVL around each mesh only
   B. using the Jacobian matrix
   C. applying KCL at each bus with lines in pi-model
   D. inverting the impedance matrix only
   Answer: C. Current injection at each bus equals the sum of series and shunt branch currents. (Key point: Basis of formation)

2. The diagonal element Yii equals:
   A. negative of the series admittance to bus k
   B. only the shunt admittance
   C. the sum of off-diagonal terms of a different row
   D. the sum of all admittances connected to bus i (series plus half-line charging)
   Answer: D. Yii = sum yik + sum y'ik/2. (Key point: Diagonal element)

3. The off-diagonal element Yik is:
   A. -yik, and zero if no line joins buses i and k
   B. +yik for all buses
   C. yik plus the shunt terms
   D. always non-zero
   Answer: A. Mutual admittance is the negative of the series admittance between the buses. (Key point: Off-diagonal element)

4. Which is a property of Ybus (no phase shifters)?
   A. Always full and non-symmetric
   B. Symmetric and sparse in large systems
   C. Diagonal only
   D. Singular for every network with shunts
   Answer: B. Yik = Yki and most off-diagonal terms are zero in large networks. (Key point: Properties)

## Related in Power System Analysis II

- [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
- [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
- [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 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
- [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

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