Insulation Coordination

Deterministic Line Design and Switching-Impulse Gap Factors

The deterministic method for designing transmission-line air clearances from a switching-surge study (\(V_3 = E_m\)), a worked 500 kV example, the switching-impulse strength of station post insulators, and the general gap-factor approach — the Paris–Cortina and Gallet equations and the gap factors for every common gap configuration.

Reading time ≈ 30 min

Section 1

The Deterministic Design Method

This page explains how a switching-surge study result is converted into a deterministic line-clearance design using \(V_3 = E_m\), CFO equations and gap factors.

Using the statistical insulation strength of tower insulation, a simple deterministic method can be built. This method was used to design the first 500 kV and 765 kV lines; only later was the improved probabilistic method adopted. The deterministic method is the natural starting point — it turns a switching-surge study directly into a strike distance.

This note covers the deterministic design rule and a worked 500 kV example, then the switching-impulse strength of post insulators and the general gap-factor approach that lets the same idea be applied to any gap geometry.

What this page teaches

how deterministic line design differs from probabilistic flashover assessment;
why the maximum switching overvoltage \(E_m\) is compared with \(V_3\);
how a 500 kV example is worked through;
why strike distance and insulator length are different design quantities;
how gap factors modify the reference CFO equation;
when Paris–Cortina or Gallet-type equations are used;
why post-insulator switching strength must be checked separately.

Key message

Deterministic design sets the required withstand strength equal to the maximum study overvoltage:

\(V_3 = E_m\)

Section 2

The Design Rule

Assume an EMTP® or TNA study has determined the maximum switching surge \(E_m\). The deterministic rule is to set the statistical withstand voltage equal to that surge:

\[ V_3 = E_m \]

Since \(V_3 = \text{CFO}\,(1 - 3\,\sigma/\text{CFO})\), the required CFO follows by rearranging:

\[ \text{CFO}_A = \frac{E_m}{1 - 3\,(\sigma/\text{CFO})} = \frac{E_m}{0.85} \qquad (\sigma/\text{CFO}=5\%) \]

\(E_m\): maximum switching surge from the EMTP® study
\(V_3\): statistical withstand voltage \((=\text{CFO}-3\sigma)\)
\(\text{CFO}_A\): CFO required at the installation (altitude/atmosphere)

\(E_m\) is obtained from the switching-surge study. \(V_3\) is the statistical withstand voltage selected for deterministic insulation design. The clearance is then chosen so that the insulation withstand is not lower than the maximum study overvoltage.

From \(\text{CFO}_A\), the strike distance is found. Because the gap factor \(k_g\) and the correction exponent both depend on the strike distance, this cannot be done in one step — an iteration is needed (usually two or three passes suffice).

E_mstudy surge → V₃withstand → CFOrequired → k_ggap factor → Sstrike dist. → Clearanceline design

Table 1 — Design quantities — what each one actually means.

Quantity	Meaning
Strike distance	Electrical air-gap distance controlling flashover
Insulator length	Physical insulation / string length
Clearance	Practical minimum distance in the tower geometry
Gap factor	Geometry correction applied to the reference CFO equation
\(\text{CFO}\)	50% flashover voltage
\(V_3\)	Statistical withstand voltage

Section 3

Worked Example: A 500 kV Line at 1000 m

Design the centre-phase strike distance and the number of standard insulators for a 500 kV (550 kV max) line at altitude 1000 m.

Inputs

System: 500 kV line, 550 kV maximum; altitude 1000 m.
Maximum switching surge: 2.0 per unit, with 1 pu = 450 kV (peak).
Tower geometry: \(W = 1.5\) m, \(h = 15\) m.
Wave front: equal to the critical wave front; design for wet conditions.
Statistical spread: \(\sigma/\text{CFO} = 5\%\).

\[ E_m = 2.0 \times 450 = 900\ \text{kV} \qquad \text{CFO}_A = \frac{900}{0.85} = 1059\ \text{kV} \]

The relative air density at 1000 m is \(\delta = e^{-1/8.6} = 0.890\). Because the gap factor and the correction exponent both depend on \(S\), iterate: guess \(S\) → get \(k_g\) → CFO from the Gallet equation → the altitude correction → updated \(S\). Starting from \(k_g=1.2\), the iteration converges in two or three steps.

Table 2 — Worked-example result (550 kV max, 1000 m, wet).

Phase	Strike distance \(S\)	Min. insulator length \((1.05\,S)\)	Standard Insulators
Centre phase	3.18 m (10.4 ft)	3.34 m	23 units
Outside phase	2.91 m	—	20 units

A subtlety on the outside phase

The outside phase is about 8% stronger, so it is tempting to take its strike distance as \(3.18/1.08 = 2.94\) m. But that assumes a linear relationship, which is untrue. The correct way is to repeat the iteration with \(k_g\) increased by a factor of 1.08, which gives 2.91 m (20 insulators) — close to, but not the same as, the linear estimate.

Conclusion

From a study surge of \(E_m = 900\) kV the design needs \(\text{CFO}_A \approx 1059\) kV, giving a centre-phase strike distance of 3.18 m with 23 standard insulators, and a 2.91 m outside-phase gap with 20 units. The clearance is then set so the insulation withstand is never below the maximum study overvoltage.

Caution — do not mix peak and rms

Do not mix peak and rms quantities. Switching impulse withstand and CFO values are peak impulse quantities. They must not be combined with rms power-frequency quantities unless the conversion is made explicitly. In this example the 450 kV per-unit base and all CFO/\(V_3\) values are peak.

Section 4

Switching-Impulse Strength of Post Insulators

Station post insulators behave differently from line insulation. As the steel pedestal height increases, the positive-polarity strength increases but the negative-polarity strength decreases — so for some pedestal height the two could be equal (though not at practical heights; for a 1000 µs front and a 20 ft pedestal, the negative CFO is only ~3% above the positive).

For the critical wave front (~120 µs) and an 8 ft (2.4 m) pedestal, the CFO is approximated by the Gallet form:

\[ \text{CFO} = k_g \cdot \frac{3400}{1 + 8/S} \quad\begin{cases} k_g = 1.4 & \text{positive polarity} \\ k_g = 1.7 & \text{negative polarity} \end{cases} \qquad \frac{\sigma}{\text{CFO}} \approx 7\% \]

Wet tests on these vertical columns are erratic; the strength depends on the number of post units — an all-porcelain column (no intervening metal caps) shows a higher CFO. From the IEC 273-1990 standard BIL/BSL values (interpreted as wet-condition BSLs), a regression gives a wet gap factor of 1.18, versus 1.40 dry/positive — so wet conditions reduce the CFO by about 16%. The standard BIL scales roughly with insulator length:

\[ \text{BIL}_S \approx 450 \, S \ \ \text{kV} \qquad (S \text{ in metres}) \]

Section 5

The General CFO Equations

Paris and Cortina (1968) noted that CFO-versus-spacing curves all have essentially the same shape, with the rod-plane gap the lowest. This led to a general equation for positive polarity, dry conditions:

\[ \text{CFO} = 500 \, k_g \, S^{0.6} \qquad \text{(Paris–Cortina, } 250\ \mu\text{s front)} \]

\(S\): gap spacing / strike distance (metres)
\(k_g\): gap factor (1.00 for rod-plane, up to ~1.9 for conductor-rod)

Crucially, Paris–Cortina used a fixed 250 µs front, so it is not the critical wave front and does not give the minimum strength. Gallet et al. then proposed an equation for the critical wave front (minimum CFO), now used exclusively:

\[ \text{CFO} = k_g \cdot \frac{3400}{1 + 8/S} \qquad \text{(Gallet — positive polarity, dry)} \]

For very long gaps the Gallet equation loses validity (see Section 8). For a rod-plane gap with \(S\) in the 13–30 m range, Pigini, Rizzi and Bramilla proposed:

\[ \text{CFO} = 1400 + 55\,S \qquad (13 \le S \le 30\ \text{m, rod-plane}) \]

IEC Publication 71 also gives positive- and negative-polarity rod-plane forms:

\[ \text{CFO} = 1080 \, k_g \, \ln(0.46\,S + 1) \quad (\text{positive, } S \le 25\ \text{m}) \] \[ \text{CFO} = 1180 \, k_g \, S^{0.45} \quad (\text{negative, } 2 \le S \le 14\ \text{m}) \]

The positive IEC form gives essentially the same result as the Gallet equation (within ~1.8% at \(S=6\) m), so there is little reason to change the basic rod-plane equation. The standard deviation is about 5–6% (positive) and about 8% (negative).

Section 6

Gap Factors for Common Configurations

The gap factor \(k_g\) carries all the geometry: same distance, different shape, different strength. Typical positive-polarity values (rod-plane = 1.0 reference) are:

Table 3 — Typical gap factors (positive polarity).

Gap Configuration	Gap factor \(k_g\)
Rod-plane (reference)	1.00
Rod-structure (under)	~1.05–1.10
Conductor-plane	~1.15
Conductor-window (centre phase, lattice / steel pole)	1.20 / 1.25
Conductor-crossarm (outside phase)	~1.28–1.30
Conductor-structure (tower leg / lower structure)	~1.30
Conductor-lateral structure (tower leg to the side)	~1.41–1.43
Conductor-rod / conductor-rope (most divergent field)	up to ~1.65–1.90

IEC 71-2 selected \(k_g = 1.30\) for a conductor-to-structure gap (e.g. a tower leg) and \(k_g = 1.10\) as a conservative value for a rod-structure gap (e.g. an apparatus bushing top with little or no grading ring to a tower leg). With further study it became clear the gap factor is not a single number but varies with the detailed parameters of each configuration; CIGRE later published general gap-factor equations.

Section 7

Configurations in Detail

The CIGRE gap-factor equations cover the main practical geometries. The headline results, with \(W\) = structure width, \(h\) = conductor height, \(S\) = the controlling strike distance:

Conductor-window (centre phase)

Applicable for \(S = 2\) to 10 m, \(W/S = 0.1\) to 1.0, \(h/S = 2\) to 10. For the usual conditions (\(h/S\) of 4–5), a normal lattice tower has \(W/S = 0.5\) to 0.6 giving \(k_g \approx 1.20\); a steel pole has \(W/S \approx 0.2\) giving \(k_g \approx 1.25\). Not much variation. The controlling \(S\) is usually to the lower tower where the conductor exits the window (or from the vibration dampener if fitted).

Conductor-crossarm (outside phase)

For \(h/S_1 = 4\) to 5 and \(W/S_1 = 0.5\): \(k_g = 1.35 + 0.135\,(S_2 - 1.5)\); if \(S_1 = S_2\) then \(k_g = 1.28\). This confirms the practical rule from line testing — the outside-phase gap factor is about 1.08 times the centre-phase value \((1.08\times 1.20 = 1.30 \approx 1.28)\).

Conductor-lower structure (e.g. a vehicle under the line)

With \(h' = 0\) and \(W = 0\) the gap reverts to conductor-plane \((k_g = 1.15)\). A worked case: a truck under the line with \(W = 8\) m, \(h' = 3\) m, \(h = 10\) m, \(S = 7\) m gives \(k_g = 1.181\) and \(\text{CFO} = 1875\) kV. Without the truck, \(k_g = 1.15\) applied to \(S = 10\) m gives \(\text{CFO} = 2172\) kV. The truck cuts the CFO by only 14% even though the strike distance falls 30% — because the higher gap factor partly compensates.

Conductor-lateral structure

For a tower leg to the side with \(W/S = 0.5\) to 0.6 and \(h/S = 4\) to 5, \(k_g \approx 1.41\) to 1.43 — higher than the crossarm (1.28–1.30), which makes sense since the crossarm adds an extra “arm” to the conductor-lateral-structure case.

Rod-rod with lower structures

The most complex arrangement, with two gap factors \(k_{g1}\) and \(k_{g2}\) for spacings \(S_1\) and \(S_2\). The limiting cases check out: \(h'=0,\ W=0\) gives \(k_{g2}=1\) (rod-plane); a vertical rod-rod recovers the Table 3 form; and a horizontal rod-rod tends toward \(k_g \approx 1.4\) for small \(S_1/h\).

Section 8

Validity Limits — Mind the Asymptote

A common trap

As \(S \to \infty\), the Gallet equation makes the rod-plane CFO approach 3400 kV, which seems to imply a maximum CFO exists for any gap. This is untrue — it is a limitation of the equation, not real physics. The Gallet equation is valid only up to about 15 m.

Beyond ~15 m, use the Pigini rod-plane equation \((\text{CFO}=1400+55\,S)\). The two agree at the boundary and then diverge as the gap grows:

Table 4 — Rod-plane CFO: Gallet vs Pigini for long gaps.

Strike distance \(S\)	Gallet \(\dfrac{3400}{1+8/S}\)	Pigini \(1400+55\,S\)
15 m	2217 kV	2225 kV
20 m	2429 kV	2500 kV
25 m	2579 kV	2775 kV

The Gallet equation flattens toward its artificial ceiling, while the physical strength (Pigini) keeps climbing. For practical line and substation clearances this rarely matters — gaps are well under 15 m — but it is essential to know the limit before extrapolating.

Section 9

Summary

The key points

Deterministic design sets \(V_3 = E_m\); the required CFO is \(\text{CFO}_A = E_m/0.85\), and the strike distance is found by iteration (\(k_g\) and the altitude correction both depend on \(S\)).
For a 500 kV line at 1000 m with a 2.0 pu surge, the centre phase needs \(S = 3.18\) m (23 insulators), the outside phase \(S = 2.91\) m (20) — redone with \(k_g\times 1.08\), not by dividing the distance.
Post insulators: Gallet form with \(k_g = 1.4\) (positive) / 1.7 (negative) dry; wet reduces CFO ~16% \((k_g \approx 1.18)\); \(\text{BIL}_S \approx 450\,S\) kV.
Use the Gallet equation (critical wave front, minimum CFO) for design — not Paris–Cortina \((500\,k_g\,S^{0.6}\), 250 µs front).
The gap factor encodes geometry (1.0 rod-plane → ~1.9 conductor-rod); outside phase ≈ 1.08 × centre phase.
The Gallet 3400 kV asymptote is an equation limit, not physics — valid to ~15 m; beyond that use \(\text{CFO}=1400+55\,S\).

Part Two Reading now

Deterministic Line Design and Gap Factors

The deterministic method (V₃ = E_m), a worked 500 kV example, post-insulator strength, and the general gap-factor approach — Paris–Cortina and Gallet equations for common gap configurations.

Series progress 2 of 3

Part Three

Lightning Impulse Strength of Line Insulation

The practical lightning-impulse CFO gradients (560/605 kV/m), time-lag curves, wood and fibreglass in series, the impulse-tail effect, the flashover mechanism, and Gaussian vs Weibull statistics.

Previous Switching Impulse Strength of Transmission-Line Insulation

Why switching impulses can exceed lightning for long air gaps, the statistical CFO and withstand voltage V₃, the critical wave front, the Gallet equation, and corrections for polarity, rain, phase and altitude.