Healthy-phase overvoltage in SLG fault; effective grounding

During a single line-to-ground (SLG) fault on phase $a$, the voltages of the healthy phases $b$ and $c$ rise above normal. How much they rise depends on how the system neutral is grounded, that is, on the ratio $X_0/X_1$.

Derivation of the healthy-phase voltage

Fault on phase $a$ (zero fault impedance): $V_a = 0$, $I_b = I_c = 0$. From symmetrical components:

$$I_{a1} = I_{a2} = I_{a0} = \frac{E_a}{Z_1 + Z_2 + Z_0}$$

The sequence voltages at the fault are

$$V_{a1} = E_a - I_{a1}Z_1, \quad V_{a2} = -I_{a1}Z_2, \quad V_{a0} = -I_{a1}Z_0$$

Phase $b$ voltage, using $a^2 + a + 1 = 0$:

$$\begin{aligned} V_b &= V_{a0} + a^2V_{a1} + aV_{a2} \\ &= a^2E_a - I_{a1}(Z_0 + a^2Z_1 + aZ_2) \end{aligned}$$

Neglect resistance and take $Z_1 = Z_2 = jX_1$, $Z_0 = jX_0$, $k = X_0/X_1$. Then $Z_0 + a^2Z_1 + aZ_2 = jX_1(k - 1)$ and $I_{a1} = E_a/[jX_1(k+2)]$, so

$$V_b = E_a\left[a^2 - \frac{k-1}{k+2}\right]$$

With $a^2 = -\tfrac{1}{2} - j\tfrac{\sqrt3}{2}$:

$$V_b = E_a\left[-\frac{3k}{2(k+2)} - j\frac{\sqrt3}{2}\right]$$ $$\boxed{\frac{|V_b|}{E_a} = \frac{\sqrt3\sqrt{k^2 + k + 1}}{k + 2}}$$

$|V_c|$ is the same by symmetry.

Condition of effective grounding

The coefficient of earthing (COE) is the highest healthy-phase voltage during an SLG fault, as a fraction of the line-to-line voltage:

$$\text{COE} = \frac{V_{b,max}}{\sqrt3E_a}$$

A system is effectively grounded when COE ≤ 80%, that is, $V_b \le 0.8\sqrt3 E_a = 1.386E_a$.

$$\begin{aligned} \frac{\sqrt3\sqrt{k^2+k+1}}{k+2} &\le 0.8\sqrt3 \\ k^2 + k + 1 &\le 0.64(k+2)^2 \\ 0.36k^2 - 1.56k - 1.56 &\le 0 \\ k &\le 5.17 \end{aligned}$$

So, with resistance neglected, the limiting case is $X_0 \approx 5.16X_1$. Check: $k = 5.16$ gives $V_b = \sqrt3\sqrt{32.79}/7.16 = 1.385$ pu, which is 80% of line voltage. This is exactly the limit of effective grounding.

In practice resistance also raises the overvoltage. So standards (IEEE/IEC) define effective grounding as

$$\frac{X_0}{X_1} \le 3 \quad \text{and} \quad \frac{R_0}{X_1} \le 1$$

which keeps COE ≤ 80% with a margin. This allows 80% arresters (rated at 80% of line voltage) and reduced insulation levels.