Section 1
The Four Parameters That Control SSFOR
The full SSFOR equation — \(\text{SSFOR} = \tfrac{1}{2}\int f_s(V)\,[1 - \prod q_i(V)]\,dV\) (see the SSFOR note) — is accurate but needs numerical integration. This note asks: which parameters really matter, and can SSFOR be estimated quickly? The answer to both is yes.
Table 1 — Parameters affecting SSFOR, in order of importance.
| Rank | Parameter | Sensitivity |
|---|
| 1 | Strength-to-stress ratio \(V_3/E_2\) | Dominant |
| 2 | SOV profile along the line \((E_S/E_R)\) | Next most important |
| 3 | SOV spread \(\sigma_0/E_2\) or \(\beta/E_2\) | Matters when \(V_3/E_2 \ne 1\) |
| 4 | Number of towers \(n\) | Weak over normal ranges |
The key message: the most important factor is not the number of towers — it is the ratio between insulation strength and statistical switching stress, \(V_3/E_2\).
Section 2
The Strength-Stress Ratio V₃/E₂
Both quantities are tail values, which is why comparing them is meaningful. The statistical withstand voltage is three standard deviations below CFO:
\[ V_3 = \text{CFO}\Bigl(1 - 3\,\tfrac{\sigma_f}{\text{CFO}}\Bigr) = 0.85\,\text{CFO} \quad (\sigma_f/\text{CFO} = 0.05) \]
- \(V_3\)
- statistical withstand voltage (lower tail of strength)
- \(\sigma_f\)
- standard deviation of insulation strength
\(E_2\), the statistical switching overvoltage, is the upper tail of the stress — the voltage exceeded by 2% of operations \((P(V \ge E_2) = 0.02)\). So if \(V_3 > E_2\) the withstand exceeds the 2% overvoltage and the SSFOR is low; if \(V_3 < E_2\) the SSFOR is high. The mean SOV and the CFO themselves are not the best indicators — the tail ratio \(V_3/E_2\) is.
Section 3
The SOV Profile Along the Line
The maximum SOV is normally at the opened/receiving end; it is lower at the switched end. With \(E_S\) at the switched end and \(E_R\) at the receiving end, the profile ratio is \(\gamma = E_S/E_R\), and the profile is assumed linear:
\[ E(x) = E_S + (E_R - E_S)\,\frac{x}{L} \qquad \gamma = \frac{E_S}{E_R} \]
- \(x,\ L\)
- distance from the switched end, and total line length
- \(\gamma\)
- 1.0 = all towers see the same SOV; <1.0 = switched end lower
A key definition
The SOV distribution \(f_s(V)\) always refers to the voltage at the opened/receiving end. The ratio \(\gamma\) is then used to scale the voltage at every other tower. The stress is not defined separately per tower — it is defined at the receiving end and scaled along the line.
Section 4
Only the Tails — and the Design Rule
Plotting the integrand of the SSFOR equation shows it is non-zero only in a narrow band — about 1.9 to 2.3 pu in the standard example. The flashover rate is produced only where high SOV meets low strength, confirming that \(V_3/E_2\) is the best compact indicator. For \(n = 500\) towers, \(\sigma_f/\text{CFO} = 5\%\), \(E_S/E_R = 1.0\), and either a Gaussian or extreme-value SOV distribution, the curves cross at:
\[ \frac{V_3}{E_2} \approx 1.0 \quad\Longrightarrow\quad \text{SSFOR} \approx \frac{1}{100} \]
The probabilistic design rule
For a target of about 1 flashover per 100 switching operations, set \(V_3 \approx E_2\) — independent of the SOV spread \(\sigma_0/E_2\). This is the central practical result of the probabilistic method.
Section 5
Probabilistic vs Deterministic
Table 2 — The two design rules compared.
| Method | Rule | Compares withstand to… |
|---|
| Deterministic | \(V_3 = E_m\) | the maximum switching overvoltage |
| Probabilistic | \(V_3 = E_2\) | the 2% statistical switching overvoltage |
Because \(E_m > E_2\), the deterministic method is usually more conservative.
Section 6
Sensitivity to the SOV Spread
A wider SOV distribution puts more values in the high-voltage tail, raising SSFOR — but only when the insulation margin is thin. For a Gaussian distribution:
- At \(V_3/E_2 = 0.95\): SSFOR rises from 3.0/100 to 8.7/100 as \(\sigma_0/E_2\) goes 0.05 → 0.11.
- At \(V_3/E_2 = 1.05\): SSFOR rises only from 0.19/100 to 0.38/100.
So when the strength is below the stress, the spread matters a lot; once the margin is good \((V_3/E_2 > 1)\), it matters little.
Section 7
Sensitivity to Towers and Strength Scatter
Both are surprisingly weak over normal ranges, for a target SSFOR of 1/100:
- Number of towers: for \(n = 300\) to 1000 (lines ~60 to 300 km), the required \(V_3/E_2\) changes only by about \(\pm 1\%\). Once the line is long enough, more towers barely move the required ratio.
- Strength scatter: changing \(\sigma_f/\text{CFO}\) from 3% to 7% shifts the required \(V_3/E_2\) by only about \(\pm 1\%\). The conservative practical value \(\sigma_f/\text{CFO} = 5\%\) is fine.
Section 8
Sensitivity to the SOV Profile
After \(V_3/E_2\), the SOV profile is the most sensitive parameter. A flat profile \((E_S/E_R = 1.0)\) is worst because every tower sees the maximum; a falling profile reduces risk. For \(\sigma_f/\text{CFO} = 5\%\) and \(\sigma_0/E_2 = 11\%\):
- Changing \(E_S/E_R\) from 1.0 to 0.9 decreases the required \(V_3/E_2\) by about 4%.
- A further change to \(E_S/E_R = 0.6\) decreases it by only another 2%.
So most of the benefit comes from moving from a flat profile to a realistic 0.9 — and the approximate rule \(V_3 = E_2\) becomes conservative. Directly on SSFOR:
\[ E_S/E_R: \;\; 1.0 \rightarrow 0.9 \quad\Longrightarrow\quad \text{SSFOR}: \;\; \frac{1.19}{100} \rightarrow \frac{0.55}{100} \]
A realistic voltage profile roughly halves the estimated flashover rate versus assuming every tower sees the maximum.
Section 9
Brown’s Method — The Idea
The full equation needs numerical integration. Brown’s method is an excellent approximation that gives both the SSFOR and the required \(V_3/E_2\) directly. Its core idea: replace the gradual multi-tower strength curve with a single-valued strength at an equivalent voltage \(\text{CFO}_n\).
For many towers, the probability of at least one flashover rises sharply, so the line’s effective strength curve becomes steep — like a vertical step at \(\text{CFO}_n\), the voltage at which the line has a 50% chance of at least one flashover. Because there are many towers, this happens at a lower voltage than a single tower’s CFO:
\[ \text{CFO}_n < \text{CFO} \qquad (\text{the more towers, the lower } \text{CFO}_n) \]
Section 10
Finding the Equivalent CFOn
With single-tower flashover probability \(p\) and \(q = 1-p\), the line reaches 50% at \(\text{CFO}_n\) when \(1 - q^n = 0.5\):
\[ q^n = 0.5 \;\Rightarrow\; p = 1 - 0.5^{1/n} \qquad \text{CFO}_n = \text{CFO}\Bigl(1 + Z_f\,\tfrac{\sigma_f}{\text{CFO}}\Bigr) \]
- \(p\)
- single-tower flashover probability at the line’s 50% point
- \(Z_f\)
- reduced normal variate for that \(p\) (usually negative)
Example, \(n = 200\): \(p = 1 - 0.5^{1/200} = 0.003460\), giving \(Z_f \approx -2.70\). With \(\sigma_f/\text{CFO} = 0.05\):
\[ \text{CFO}_n = \text{CFO}\,(1 - 2.70\times 0.05) = 0.865\,\text{CFO} \]
So a 200-tower line has an equivalent strength of about 86.5% of the single-tower CFO.
Section 11
Brown’s SSFOR, and the Voltage Profile
With strength as a step at \(\text{CFO}_n\), the SSFOR is just half the probability that the SOV exceeds \(\text{CFO}_n\). For a Gaussian SOV:
\[ \text{SSFOR} \approx \tfrac{1}{2}\Bigl[\,1 - F\bigl(\tfrac{\text{CFO}_n - \mu_0}{\sigma_0}\bigr)\Bigr] \]
The upper limit \(E_m\) is usually high enough that \(F(E_m)\approx 1\), so \(E_m\) seldom needs evaluating. For an extreme-value SOV the same idea uses the Gumbel CDF; with \(\text{CFO}=1000\) kV, \(n=200\) \((\text{CFO}_n=865)\), \(E_2=900\) kV and \(\beta/E_2=0.11\) \((u=513.7)\):
\[ \text{SSFOR} = \tfrac{1}{2}\bigl[\,1 - e^{-e^{-y}}\,\bigr], \;\; y = \tfrac{865 - 513.7}{99} = 3.548 \;\Rightarrow\; \frac{1.42}{100} \]
Equivalent number of towers (for the profile)
Brown initially assumes \(E_S/E_R = 1.0\). For a falling profile, fewer towers are effectively at the maximum, captured by an equivalent number of towers \(n_e\) (with \(k_n \approx 0.4\) over the practical 30–500 range):
\[ n_e = \frac{0.4}{1-\gamma}\,\Bigl(\tfrac{\sigma_f}{\text{CFO}}\Bigr)\,n \qquad (n_e \le n) \]
Example: \(n=200\), \(\gamma = E_S/E_R = 0.9\), \(\sigma_f/\text{CFO}=0.05\) gives \(n_e = \frac{0.4}{0.1}(0.05)(200) = 40\). A 200-tower line with a 0.9 profile behaves like just 40 towers all at the maximum. Then \(\text{CFO}_n = 895\) kV and the SSFOR falls to 1.15/100.
Why Brown’s method works — and a caution
For many towers the line-strength standard deviation \(\sigma_{fn}\) becomes much smaller than the SOV spread \(\sigma_0\), so the strength can be treated as a vertical step at \(\text{CFO}_n\). The caution: this requires \(\sigma_{fn} \ll \sigma_0\), which is not always true — for \(n = 1\), substations, or a small number of insulation points, the strength scatter can be important.
Section 12
Estimating the Required V₃/E₂
The usual design problem is the reverse: given an acceptable SSFOR, what strength is required? Brown’s method gives \(V_3/E_2\) directly as a product of a strength-side factor and a stress-side factor:
\[ \frac{V_3}{E_2} = K_f\,K_G \quad(\text{Gaussian}) \qquad \frac{V_3}{E_2} = K_f\,K_E \quad(\text{extreme value}) \]
- \(K_f = V_3/\text{CFO}_n\)
- strength side — depends on \(n\) (or \(n_e\)) and \(\sigma_f/\text{CFO}\)
- \(K_G,\ K_E\)
- stress side — depend on the target SSFOR and the SOV spread
The stress factor uses the reduced variate found from \(F = 1 - 2\,(\text{SSFOR})\) (the “twice the SSFOR” rule, from the polarity \(\tfrac{1}{2}\)). For \(\text{SSFOR} = 1/100\), \(F = 0.98\), giving \(Z_e = 2.054\) (Gaussian) or \(y_e = 3.902\) (extreme value).
Fix — the “twice the SSFOR” figure
The extreme-value reduced variate \(y_e = 3.902\) corresponds to a target SSFOR = 1/100, entering with \(F = 1 - 2\times 0.01 = 0.98\). (A source mislabels this as “SSFOR = 2/100” — that 2/100 is actually the intermediate \(2\times\text{SSFOR}\) value, not the target rate.)
Worked example. For \(\text{SSFOR}=1.5/100\), \(n=200\), \(\sigma_0/E_2=0.08\), \(\sigma_f/\text{CFO}=0.05\): \(p=0.003460\), \(Z_f=-2.70\), so \(K_f = 0.9827\). With \(F(Z_e)=1-2(0.015)=0.97\), \(Z_e=1.88\), \(K_G=0.9862\). Therefore:
\[ \frac{V_3}{E_2} = K_f\,K_G = 0.9827 \times 0.9862 = 0.9691 \]
With a realistic profile \((E_S/E_R = 0.9 \Rightarrow n_e = 40)\), \(K_f = 0.9507\) and \(V_3/E_2 = 0.9376\) — a lower required ratio, because fewer towers see the maximum. And for the headline case \(n = 500\), \(\text{SSFOR}=1/100\): \(K_f \approx K_G \approx 1\), confirming \(V_3/E_2 \approx 1\).
Section 13
Design Workflow and Summary
Putting it together for a transmission-line switching-surge design:
Design workflow
- Run EMT studies for random switching; determine \(E_2\) \((P(V\ge E_2)=0.02)\).
- Decide the acceptable SSFOR (e.g. 1/100).
- Estimate \(V_3/E_2\): for many EHV lines \(\approx 1\); for detail use \(V_3/E_2 = K_f K_G\) (or \(K_f K_E\)).
- Compute \(V_3 = (V_3/E_2)\,E_2\), then \(\text{CFO} = V_3/0.85\).
- Convert CFO to a strike distance via the Gallet equation \(\text{CFO}=k_g\cdot\frac{3400}{1+8/S}\), with wet, altitude, geometry, phase and wavefront corrections.
Key messages
- SSFOR is controlled mainly by the strength-stress ratio \(V_3/E_2\); the SOV profile is next; \(n\) and \(\sigma_f/\text{CFO}\) are weak over normal ranges.
- The probabilistic design rule is \(V_3 \approx E_2\) for SSFOR ≈ 1/100 — versus the deterministic \(V_3 = E_m\) (more conservative).
- A realistic voltage profile roughly halves the SSFOR (e.g. 1.19 → 0.55/100 for \(E_S/E_R = 0.9\)).
- Brown’s method replaces many towers by an equivalent \(\text{CFO}_n\) (e.g. \(0.865\,\text{CFO}\) for 200 towers), giving SSFOR or \(V_3/E_2\) without integration — valid while \(\sigma_{fn} \ll \sigma_0\).
- The voltage profile enters through the equivalent number of towers \(n_e = \frac{0.4}{1-\gamma}(\sigma_f/\text{CFO})\,n\).
