*2.1. Overhead or Underground Distribution Lines*

With the Carson's equations presented in [23], the primitive impedance matrices for three-phase overhead and underground lines can be formed. For a grounded neural system, these matrices are reduced to phase impedance matrices of 3 × 3 size using Kron reduction. Figure 1 shows the three-phase distribution line model, and its shunt admittance is neglected due to its small effect. The phase impedance matrix for the line section 'jk' is given in Equation (1).

$$\mathbf{[Z\_{abc}]}\_{\mathbf{jk}} = \begin{bmatrix} \mathbf{Z\_{aa}} & \mathbf{Z\_{ab}} & \mathbf{Z\_{ac}} \\ \mathbf{Z\_{ba}} & \mathbf{Z\_{bb}} & \mathbf{Z\_{bc}} \\ \mathbf{Z\_{ca}} & \mathbf{Z\_{cb}} & \mathbf{Z\_{cc}} \end{bmatrix}\_{\mathbf{jk}} \tag{1}$$

**Figure 1.** A sample three-phase distribution line.

From Figure 1, the relationship between the phase voltage matrices of bus-j and bus-k is given in Equation (2):

$$
\begin{bmatrix} \mathbf{V\_{a}} \\ \mathbf{V\_{b}} \\ \mathbf{V\_{c}} \end{bmatrix}\_{\mathbf{k}} = \begin{bmatrix} \mathbf{V\_{a}} \\ \mathbf{V\_{b}} \\ \mathbf{V\_{c}} \end{bmatrix}\_{\mathbf{j}} - \begin{bmatrix} \mathbf{Z\_{a}} & \mathbf{Z\_{ab}} & \mathbf{Z\_{ac}} \\ \mathbf{Z\_{ba}} & \mathbf{Z\_{bb}} & \mathbf{Z\_{bc}} \\ \mathbf{Z\_{ca}} & \mathbf{Z\_{cb}} & \mathbf{Z\_{cc}} \end{bmatrix}\_{\mathbf{j}\mathbf{k}} \cdot \begin{bmatrix} \mathbf{I\_{a}} \\ \mathbf{I\_{b}} \\ \mathbf{I\_{c}} \end{bmatrix}\_{\mathbf{j}\mathbf{k}} \tag{2}
$$

The reactance of line is regarded as proportionate to the harmonic order for HPFA. For h-order harmonic frequency, the self-impedance of phase 'a' is given in Equation (3),

$$(\mathbf{Z\_{aa}})^\mathbf{h} = \mathbf{R\_{aa}} + \mathbf{j} \cdot \mathbf{h} \cdot \mathbf{X\_{aa}} \tag{3}$$
