Arrester separation for a 500 kV/μs surge

A surge reaching the arrester rises at $S = 500$ kV/μs. It travels on to the transformer, $d$ metres beyond, and arrives after $T = d/v$. The transformer (open end) doubles it, and the reflected wave gets back to the arrester at $2T$. From then on, the arrester voltage rises at $2S$, so the arrester sparks over at $V_a = 420$ kV at time $t_s = V_a/(2S) + T$. The clamping effect takes a further time $T$ to reach the transformer, and until then the transformer voltage keeps rising. The standard result is

$$V_t = V_a + 2S\frac{d}{v}, \quad v = 300\ \text{m}/\mu\text{s}$$

This holds when the reflection returns before the arrester would spark over on the incident wave alone, i.e. $2T < V_a/S = 420/500 = 0.84\ \mu$s ($d < 126$ m), which is true here.

(i) Coordination check (arrester right at the transformer)

$$PR = \frac{\text{BIL}}{V_a} = \frac{500}{420} = 1.19$$

The required PR is 1.25, which means the protective level must not exceed $500/1.25 = 400$ kV. Since $420 > 400$ kV, i.e. $1.19 < 1.25$:

Not properly coordinated, even with zero separation.

(ii) Transformer voltage with the arrester 40 m away

$$\begin{aligned} T &= \frac{40}{300} = 0.1333\ \mu\text{s} \\ t_s &= \frac{420}{2 \times 500} + 0.1333 = 0.553\ \mu\text{s} \\ V_t &= 420 + 2 \times 500 \times 0.1333 \\ &= 420 + 133.3 = 553.3\ \text{kV} \end{aligned}$$

This exceeds the BIL of 500 kV, so the transformer insulation would fail. The achieved PR is $500/553.3 = 0.90$.

(iii) Maximum allowed distance

With PR = 1.25: the allowed transformer voltage is 400 kV, which is already below the arrester level of 420 kV. So no distance works ($d_{max}$ is negative). The arrester rating must be lowered or the BIL raised.

Without margin, $V_t \le$ BIL = 500 kV:

$$\begin{aligned} 420 + 2 \times 500 \times \frac{d}{300} &\le 500 \\ d &\le \frac{(500 - 420) \times 300}{2 \times 500} = 24\ \text{m} \end{aligned}$$

  500 kV/us                  d
 ---------->  Arrester  ----------- Transformer
             (420 kV)               (BIL 500 kV)

Answer: (i) not coordinated (PR = 1.19); (ii) 553.3 kV; (iii) the arrester can be at most 24 m away to keep the voltage within the BIL. With the 1.25 margin, no distance satisfies the requirement.