DLG fault sequence network connection

In a double line to ground (L-L-G) fault on phases b and c, all three sequence voltages at the fault are equal and the sequence currents add to zero, so the positive sequence network is in series with the parallel combination of the negative and zero sequence networks.

Generator sequence equations (unloaded)

$$V_{a1} = E_a - Z_1I_{a1}, \qquad V_{a2} = -Z_2I_{a2}, \qquad V_{a0} = -Z_0I_{a0}$$

where $Z_0 = Z_{g0} + 3Z_n$.

Boundary conditions (bolted fault, b and c to ground)

  a o-------------  Ia = 0
  b o-----+         Ib -->
  c o-----+         Ic -->
          |  If = Ib + Ic
        ground

$$I_a = 0, \qquad V_b = 0, \qquad V_c = 0$$

Sequence voltages

$$\begin{bmatrix} V_{a0}\\V_{a1}\\V_{a2} \end{bmatrix} = \frac13\begin{bmatrix} 1&1&1\\1&a&a^2\\1&a^2&a \end{bmatrix}\begin{bmatrix} V_a\\0\\0 \end{bmatrix} \quad\Rightarrow\quad V_{a0} = V_{a1} = V_{a2} = \frac{V_a}{3}$$

Equal voltages across the three networks at the fault point means they are connected in parallel at that point.

Sequence currents

$$I_a = I_{a0} + I_{a1} + I_{a2} = 0$$

So the positive sequence current $I_{a1}$ divides into $-I_{a2}$ and $-I_{a0}$, i.e. it returns through the negative and zero sequence networks in parallel.

Solving

From $V_{a1} = V_{a2} = V_{a0}$:

$$I_{a2} = -\frac{V_{a1}}{Z_2}, \qquad I_{a0} = -\frac{V_{a1}}{Z_0}$$

Substituting in $I_{a0} + I_{a1} + I_{a2} = 0$ with $V_{a1} = E_a - Z_1I_{a1}$:

$$I_{a1} = (E_a - Z_1I_{a1})\left(\frac{1}{Z_2} + \frac{1}{Z_0}\right) \;\Rightarrow\; I_{a1} = \frac{E_a}{Z_1 + \dfrac{Z_2Z_0}{Z_2 + Z_0}}$$

This is exactly the current in a circuit with $E_a$ and $Z_1$ in series with ($Z_2 \parallel Z_0$), which proves the connection.

$$I_{a2} = -I_{a1}\frac{Z_0}{Z_2 + Z_0}, \qquad I_{a0} = -I_{a1}\frac{Z_2}{Z_2 + Z_0}$$

Fault (ground) current: $I_f = I_b + I_c = 3I_{a0}$.

Interconnection

        Ia1 -->        F
  +--(Ea)--Z1----------o------+---------+
  |                           |         |
  |                          Z2        Z0
  |                           | Ia2     | Ia0
  N1 -------------------------+---------+
                              N2        N0

Positive sequence network in series with the parallel combination of negative and zero sequence networks. With fault impedance $Z_f$ to ground, $3Z_f$ is added in series with $Z_0$.