Ybus derivation for 3-bus π-model network

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.

                 (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.