Ybus elements and line shunt effect

The bus admittance matrix relates bus injected currents to bus voltages, $\mathbf{I} = \mathbf{Y}_{bus}\mathbf{V}$. Its elements are found from KCL at each bus.

3-bus network

Buses 1, 2 and 3 are connected by lines 1-2, 2-3 and 1-3. Each line is represented by a π model: series admittance $y_{ik}$ and half of the line-charging admittance $y'_{ik}/2 = jB_{ik}/2$ at each end. $I_1, I_2, I_3$ are injected currents.

                I2
                |
               (2)--- y20 --- ground
              /     \
        y12  /       \  y23
            /         \
 I1 --(1)--------------(3)-- I3
    |       y13         |
   y10                 y30
    |                   |
  ground              ground

 y10 = y'12/2 + y'13/2,  y20 = y'12/2 + y'23/2,  y30 = y'13/2 + y'23/2

KCL at each bus

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

Rearranging,

$$\begin{bmatrix} I_1 \\ I_2 \\ I_3 \end{bmatrix} = \begin{bmatrix} y_{10} + y_{12} + y_{13} & -y_{12} & -y_{13} \\ -y_{12} & y_{20} + y_{12} + y_{23} & -y_{23} \\ -y_{13} & -y_{23} & y_{30} + y_{13} + y_{23} \end{bmatrix} \begin{bmatrix} V_1 \\ V_2 \\ V_3 \end{bmatrix}$$

General expressions

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

so that for an $n$-bus system $I_i = \sum_{k=1}^{n} Y_{ik}V_k$.

• $Y_{ii}$ (self/driving-point admittance) = sum of all admittances terminating at bus $i$.
• $Y_{ik}$ (mutual/transfer admittance) = negative of the series admittance between $i$ and $k$; zero if not connected.

Effect of line shunt parameters

With line charging, $y_{i0} = \sum_k \dfrac{jB_{ik}}{2}$ (plus any shunt capacitor/reactor at bus $i$).

1. Only diagonal elements change. Each $Y_{ii}$ gains the term $+j\sum B_{ik}/2$. Off-diagonal elements $Y_{ik} = -y_{ik}$ are unchanged, because shunt branches connect a bus to ground, not to another bus.
2. Since series admittance is inductive (negative imaginary part) and charging is capacitive (positive), the shunt term reduces the magnitude of the imaginary part of $Y_{ii}$.
3. Without shunts, each row of $Y_{bus}$ sums to zero (matrix is singular, no reference). Shunt elements give a path to ground, making $Y_{bus}$ non-singular so $Z_{bus} = Y_{bus}^{-1}$ exists.
4. In load flow, the shunt terms add reactive generation, raising voltages at light load; neglecting them is acceptable only for short lines.