Bus admittance matrix and its elements

The bus admittance matrix $Y_{bus}$ is an $n \times n$ matrix that relates the currents injected at the $n$ buses of a power network to the bus voltages measured from the reference (ground):

$$\mathbf{I}_{bus} = \mathbf{Y}_{bus}\,\mathbf{V}_{bus},\qquad I_i = \sum_{k=1}^{n} Y_{ik}V_k$$

For a 3-bus system:

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

Diagonal elements: self (driving-point) admittance

$$Y_{ii} = y_{i0} + \sum_{k \ne i} y_{ik} = \left.\frac{I_i}{V_i}\right|_{V_k = 0,\ k \ne i}$$

$Y_{ii}$ is the sum of all admittances connected to bus $i$, including series line admittances and shunt elements (half line charging, capacitors, reactors). It equals the current injected at bus $i$ per unit voltage at bus $i$ when all other buses are shorted to ground.

Off-diagonal elements: mutual (transfer) admittance

$$Y_{ik} = -y_{ik} = \left.\frac{I_i}{V_k}\right|_{V_j = 0,\ j \ne k}$$

$Y_{ik}$ is the negative of the series admittance between buses $i$ and $k$. It is the current injected at bus $i$ per unit voltage applied at bus $k$ with all other buses shorted. If buses $i$ and $k$ are not directly connected, $Y_{ik} = 0$.

Properties

• Symmetric: $Y_{ik} = Y_{ki}$ (no phase-shifting transformers).
• Sparse: most off-diagonal elements are zero in large networks.
• Usually diagonally dominant, since $Y_{ii}$ is the sum of the admittances that appear (with opposite sign) in the off-diagonal terms of its row; it is formed easily by inspection.
• Used directly in Gauss–Seidel and Newton–Raphson load flow.