Ybus, Zbus and 4-bus Ybus formulation

Bus admittance and bus impedance matrices

• Bus admittance matrix $Y_{bus}$: relates injected bus currents to bus voltages, $\mathbf{I} = \mathbf{Y}_{bus}\mathbf{V}$. It is sparse and formed by inspection.
• Bus impedance matrix $Z_{bus}$: the inverse, $\mathbf{V} = \mathbf{Z}_{bus}\mathbf{I}$, with $Z_{bus} = Y_{bus}^{-1}$. It is a full matrix, formed by the building algorithm or inversion, and used mainly in fault analysis.

Formulation for a 4-bus system

Assume lines 1-2, 1-3, 2-3, 2-4 and 3-4 with series admittances $y_{ik}$, and shunt admittances $y_{i0}$ at each bus (half line charging).

   (1)---------y12---------(2)
    |                    / |
    |                 /    |
   y13            y23     y24
    |          /           |
    |       /              |
   (3)---------y34---------(4)

   No line 1-4. Each bus i also has y_i0 to ground.

KCL at each bus:

$$\begin{aligned} I_1 &= y_{10}V_1 + y_{12}(V_1 - V_2) + y_{13}(V_1 - V_3) \\ I_2 &= y_{20}V_2 + y_{12}(V_2 - V_1) + y_{23}(V_2 - V_3) + y_{24}(V_2 - V_4) \\ I_3 &= y_{30}V_3 + y_{13}(V_3 - V_1) + y_{23}(V_3 - V_2) + y_{34}(V_3 - V_4) \\ I_4 &= y_{40}V_4 + y_{24}(V_4 - V_2) + y_{34}(V_4 - V_3) \end{aligned}$$

In matrix form:

$$\begin{bmatrix} I_1 \\ I_2 \\ I_3 \\ I_4 \end{bmatrix} = \begin{bmatrix} Y_{11} & -y_{12} & -y_{13} & 0 \\ -y_{12} & Y_{22} & -y_{23} & -y_{24} \\ -y_{13} & -y_{23} & Y_{33} & -y_{34} \\ 0 & -y_{24} & -y_{34} & Y_{44} \end{bmatrix} \begin{bmatrix} V_1 \\ V_2 \\ V_3 \\ V_4 \end{bmatrix}$$

with $Y_{11} = y_{10} + y_{12} + y_{13}$, $Y_{22} = y_{20} + y_{12} + y_{23} + y_{24}$, $Y_{33} = y_{30} + y_{13} + y_{23} + y_{34}$, $Y_{44} = y_{40} + y_{24} + y_{34}$.

General equations of elements

$$Y_{ii} = y_{i0} + \sum_{k \ne i} y_{ik},\qquad Y_{ik} = Y_{ki} = -y_{ik}$$

Properties of Ybus

1. Symmetric about the leading diagonal (no phase shifters).
2. Sparse: $Y_{ik} = 0$ when no line joins $i$ and $k$ (here $Y_{14} = 0$).
3. Generally diagonally dominant: $|Y_{ii}| \ge \sum_{k\ne i} |Y_{ik}|$ when the shunt admittances are small or absent.
4. Non-singular only when shunt elements to ground exist.
5. Easy to form by inspection and to modify for adding or removing a line.