Insulation Coordination

Switching-Overvoltage Distributions: Brute Force, Case Peak and E₂

How the switching-overvoltage stress is collected and represented before computing the flashover rate — the brute-force, case-peak and phase-peak methods, and fitting a Gaussian or extreme-value positive-skew distribution using the 2% statistical switching overvoltage E₂.

Reading time ≈ 35 min

Section 1

Where This Fits: Collecting the SOV Distribution

In probabilistic switching-surge design there are two random quantities — the stress (switching overvoltage, SOV) and the strength (insulation withstand) — and flashover happens when stress exceeds strength. The previous note combined the two to compute the SSFOR (Switching Surge Flashover Rate). This note answers one upstream question:

The question here

How should the switching-overvoltage distribution be collected and represented?

Three collection methods are covered — the brute-force, case-peak and phase-peak methods — followed by two continuous distributions used to represent the data: the Gaussian (normal) and the extreme-value positive-skew distribution.

Section 2

How SOVs Are Obtained

SOVs are obtained from a transient simulation — today EMTP®, ATP, PSCAD or EMTP® (historically a TNA). The breaker is switched many times with random pole-closing instants, and each case records the maximum overvoltage on each phase: \(V_A\), \(V_B\), \(V_C\) (the phase-to-ground SOVs).

All three phases matter, because a three-phase operation produces three different overvoltages. For example, one random case might give \(V_A = 1.55\), \(V_B = 1.92\), \(V_C = -1.70\) pu; another \(V_A = -1.85\), \(V_B = 1.40\), \(V_C = 1.65\) pu. The question is how to use these three values in the statistical SOV distribution — which is exactly what the case-peak and phase-peak methods address.

Section 3

The Brute-Force Method

The most exact method keeps the three phase voltages exactly as they occur. With \(q_A, q_B, q_C\) the no-flashover probabilities of each phase, the product \(q_A q_B q_C\) is the probability that no phase flashes over, so the per-case flashover probability is:

\[ P(F) = 1 - q_A\,q_B\,q_C \]

\(q_A, q_B, q_C\): probability of no flashover on phases A, B, C
\(P(F)\): probability that at least one phase flashes over (this case)

Averaging \(P(F)\) over all simulated cases gives the SSFOR. Typically only one phase controls: a small voltage gives \(q \approx 1\); a high positive voltage gives \(q < 1\); and a negative voltage gives \(q \approx 1\) (negative-polarity strength is much higher).

Brute force is the most faithful method — it keeps the actual voltages, polarities, phase correlation and system-specific behaviour. Its drawback is that it is specific to one system and one simulation set: it gives accurate numbers but little general insight into how parameters (E₂, \(\sigma\), closing resistors, CFO) affect the result.

Section 4

Why Negative SOVs Are Usually Ignored

The polarity factor of ½

Because negative-polarity switching-impulse strength is significantly higher than positive, a negative SOV of the same magnitude is less dangerous — so \(q \approx 1\) for negative surges and negative polarity is neglected. With roughly half the surges positive, only that half is critical, which is why the equations carry a factor of \(\tfrac{1}{2}\).

Section 5

The Case-Peak Method

For each switching operation, keep only the largest crest among the three phases (positive or negative), and store it as positive. For example, with \(V_A = 1.5\), \(V_B = -1.9\), \(V_C = 1.7\) pu, the largest crest is \(-1.9\) pu, stored as \(+1.9\) pu — because the opposite breaker switching sequence could produce the same magnitude with positive polarity. So one value is kept per case.

The method assumes one SOV dominates each operation: usually only the highest positive-equivalent phase matters, with \(q \approx 1\) for the other two, so \(P(F) = 1 - q_A q_B q_C \approx 1 - q_{max}\). Keeping one value per three-phase operation means the result is directly the line SSFOR per breaker operation — the practical question: “if the three-phase breaker operates once, what is the probability the line flashes over?”

\[ P(F) \approx 1 - q_{max} \qquad P(F)_{\text{effective}} \approx \tfrac{1}{2}\,(1 - q_{max}) \]

The \(\tfrac{1}{2}\) is applied because positive and negative polarities are equally likely but only the positive half is treated as critical.

Section 6

The Phase-Peak Method

Instead of one value per case, the phase-peak method keeps all three phase peaks as separate data points, each treated as positive. So \(N\) switching cases give \(3N\) values, producing a distribution of phase overvoltages and a per-phase flashover rate \(\text{SSFOR}_p\).

Since a three-phase operation has three phases, convert per-phase to line SSFOR by treating the three as independent: \((1-\text{SSFOR}_p)^3\) is the probability of no flashover on all three, so:

\[ \text{SSFOR} = 1 - (1 - \text{SSFOR}_p)^3 \;\approx\; 3\,\text{SSFOR}_p \quad (\text{small } \text{SSFOR}_p) \]

The approximation follows because \(1-(1-x)^3 \approx 3x\) for small \(x\).

Section 7

Which Method to Use

Table 1 — The three SOV-collection methods.

Method	Data Kept per Case	Result	Accuracy / Use
Brute force	All three phase SOVs (\(V_A, V_B, V_C\))	Direct line SSFOR	Highest accuracy — detailed EMT study
Case peak	One largest crest (stored positive)	Line SSFOR per operation	Good approximation — general design method
Phase peak	All three phase peaks (\(3N\) values)	Per-phase \(\text{SSFOR}_p\)	Approximate — phase-based statistical analysis

The chapter uses the case-peak method because it appears to be the superior approximation — while noting that brute force is the only method that considers all variations of the SOV distribution. The hierarchy: brute force (most exact) → case peak (preferred simplified) → phase peak (alternative approximate).

Section 8

Practical EMTP® Workflow

Best — brute force

For each simulation, record each phase peak and polarity, compute \(q_A, q_B, q_C\) directly, then \(P(F) = 1 - q_A q_B q_C\), and average over all cases. Most accurate; uses the real phase voltages and polarities.

Simplified — case peak

For each simulation, take the largest crest as the case peak, build the case-peak distribution, fit a Gaussian or extreme-value curve, compute SSFOR, and apply the \(\tfrac{1}{2}\) polarity factor with positive-only strength. Best for general analytical work.

Section 9

The Gaussian Distribution

Rather than raw data or a histogram, the SOV is often fitted to a continuous density. The Gaussian has the familiar bell shape, described by mean \(\mu_0\) and standard deviation \(\sigma_0\):

\[ f(V) = \frac{1}{\sigma_0\sqrt{2\pi}}\, e^{-\frac{1}{2}\left(\frac{V-\mu_0}{\sigma_0}\right)^2} \qquad Z = \frac{V - \mu_0}{\sigma_0} \]

\(\mu_0\): mean SOV
\(\sigma_0\): standard deviation of the SOV
\(Z\): reduced (standard) normal variate

The cumulative \(F(Z) = \int_{-\infty}^{Z} f(z)\,dz\) gives the probability that the SOV is at or below a value. In SOV work, the distribution is usually defined not by \(\mu_0, \sigma_0\) directly, but by E₂ and the ratio \(\sigma_0/E_2\).

Section 10

E₂ — The Statistical Switching Overvoltage

\(E_2\), the statistical switching overvoltage, is the voltage exceeded by 2% of the SOVs — the 98th percentile:

\[ P(V \ge E_2) = 0.02 \qquad E_2 = \mu_0 + 2.05\,\sigma_0 \]

So 98% of switching overvoltages are below \(E_2\) and 2% are at or above it — a high-percentile measure used instead of the maximum. Given \(E_2\) and \(\sigma_0/E_2\), the whole Gaussian can be reconstructed. Worked example: with \(E_2 = 900\) kV and \(\sigma_0/E_2 = 0.10\):

\[ \sigma_0 = 0.10\times 900 = 90\ \text{kV} \qquad \mu_0 = 900 - 2.05\times 90 = 715.5\ \text{kV} \]

The probability of exceeding 800 kV is then \(Z = (800-715.5)/90 = 0.939\), giving:

\[ P(V \ge 800) = 1 - F(0.939) \approx 0.173 \quad (17.3\%) \]

A caveat on the Gaussian

The Gaussian extends from \(-\infty\) to \(+\infty\), but SOVs are physically bounded (between ~1.0 pu and a maximum \(E_m\)). This is not a severe limitation: taking \(E_m \to \infty\) is conservative enough for practical integration.

Section 11

Typical Values and the Closing-Resistor Benefit

CIGRE-based estimates show the strong effect of closing resistors:

Table 2 — Typical statistical switching overvoltage E₂.

Condition	E₂	\(\sigma_0/E_2\)
High-speed reclosing, no closing resistors	~2.8 pu	~0.08 to 0.11
With closing resistors	~1.8 pu	~0.08 to 0.11

The maximum SOV is approximated as about one standard deviation above \(E_2\):

\[ E_m \approx E_2 + \sigma_0 = E_2\bigl(1 + \sigma_0/E_2\bigr) \]

For example, \(E_2 = 1.8\) pu with \(\sigma_0/E_2 = 0.08\) gives \(E_m \approx 1.94\) pu; \(E_2 = 2.8\) pu with \(0.11\) gives \(E_m \approx 3.11\) pu.

Why closing resistors matter

Closing resistors cut \(E_2\) from ~2.8 to ~1.8 pu — about a 36% reduction in the statistical switching overvoltage. That means lower SOV stress, lower SSFOR, smaller required clearances, and easier EHV line insulation design.

Section 12

The Extreme-Value Positive-Skew Distribution

A Gaussian is symmetric, but real SOV distributions may be skewed — many values near the mode with a long tail toward high overvoltages. The extreme-value positive-skew distribution captures this: most SOVs are moderate, but a small number of cases produce high overvoltages, which is physically reasonable since the rare high-voltage tail is what matters.

It uses a modal value \(u\) and a slope parameter \(\beta\) (loosely like a standard deviation); for SOV work the practical parameters are \(E_2\) and \(\beta/E_2\). Mean and standard deviation can be derived from \(u\) and \(\beta\) (the relationship involves Euler’s constant, \(\gamma \approx 0.5772\)). Typical spread:

\[ \frac{\beta}{E_2} \approx 0.05 \text{ to } 0.09 \quad\text{vs Gaussian}\quad \frac{\sigma_0}{E_2} \approx 0.08 \text{ to } 0.11 \]

Gaussian

Simple, familiar, described by mean and standard deviation, widely used — but symmetric and unbounded, so the tail may not represent extreme events perfectly.

Extreme value (positive skew)

Better represents the high-voltage tail and rare high-SOV events — but less familiar, still mathematically unbounded, and its parameters are less intuitive than mean/standard deviation.

Like the Gaussian, the extreme-value distribution is mathematically unbounded; its upper truncation \(E_m\) is estimated from \(E_2\) and \(\beta\), and for conservative calculations \(E_m\) may again be taken as infinity.

Section 13

Summary

The complete logic: a random breaker closing produces \(V_A, V_B, V_C\); choose a collection method (brute force / case peak / phase peak) to build the SOV stress distribution \(f_s(V)\); then combine with the insulation strength \(F_S(V)\) to compute the SSFOR. The key formulas:

\[ \text{brute force: } P(F) = 1 - q_A q_B q_C \qquad \text{case peak: } P(F) \approx \tfrac{1}{2}(1 - q_{max}) \] \[ \text{phase peak: } \text{SSFOR} = 1 - (1-\text{SSFOR}_p)^3 \approx 3\,\text{SSFOR}_p \qquad E_2 = \mu_0 + 2.05\,\sigma_0 \]

Key messages

Brute force is the most exact — it uses the actual three-phase SOVs, polarities and correlation from every case, but is system-specific.
Case peak (preferred here) keeps one dominant crest per operation and gives the line SSFOR directly; phase peak gives a per-phase rate, converted by \(\text{SSFOR}=1-(1-\text{SSFOR}_p)^3\).
Negative-polarity surges are usually neglected (higher strength) — hence the \(\tfrac{1}{2}\) factor.
The SOV is represented by a Gaussian or an extreme-value positive-skew distribution, parameterised by E₂ (the 2% statistical overvoltage) and \(\sigma_0/E_2\) or \(\beta/E_2\).
Closing resistors cut E₂ from ~2.8 to ~1.8 pu (~36%) — a major reduction in switching-surge stress.

Part Two Reading now

Switching-Overvoltage Distributions

How the switching-overvoltage stress is collected and represented — the brute-force, case-peak and phase-peak methods, and fitting a Gaussian or extreme-value distribution via the 2% statistical overvoltage E₂.

Series progress 2 of 3

Part Three

SSFOR Sensitivity and Brown’s Method

Which parameters control SSFOR (the strength-stress ratio V₃/E₂ dominates), and Brown’s quick method for estimating SSFOR or the required V₃/E₂ using an equivalent multi-tower CFO_n.

Previous Probabilistic Line Design and SSFOR

The stress-strength concept, the histogram and continuous (integral) methods, many-tower and voltage-profile effects, and the design quantity SSFOR — flashovers per 100 breaker closings.