Section 1
Phase-Phase Coordination Inside the Station
The companion note covered phase-ground switching-overvoltage coordination in substations. This note takes the same probabilistic method to the phase-phase insulation of a station — estimating the required phase-phase clearance and BSL, and the effect of gap geometry, the \(V_+\)/\(V_-\) split and arresters.
Phase-phase insulation matters a great deal in a station because there are many phase-to-phase air gaps: between busbars, breaker and disconnector terminals, grading rings, bushing terminals, supported bus fittings and conductor-to-conductor arrangements.
Section 2
Two Ways to Describe Phase-Phase Strength
As on lines, phase-phase strength is described two ways — the same insulation, two forms. The alpha method fixes the voltage split and is the basis of the IEC standard test; the \(V_+\)–\(V_-\) method separates the polarities and is the convenient form for probabilistic calculation.
Section 3
The Alpha Method
Opposite-polarity voltages are applied to the two phases. A fraction of the total \(V_p\) goes positive on one phase, the rest negative on the other; \(\alpha\) is held constant while \(V_p\) is raised to flashover:
\[ V_+ = \alpha\,V_p, \quad V_- = (1-\alpha)\,V_p \qquad \text{CFO}_p = k_{gp}\,\frac{3400}{1 + 8/S} \]
- \(S\)
- phase-phase spacing (m)
- \(k_{gp}\)
- phase-phase gap factor (alpha method)
- \(\text{CFO}_p\)
- phase-phase critical flashover voltage
Section 4
The IEC Phase-Phase BSL Test
IEC defines the phase-phase BSL with the alpha method at \(\alpha = 0.5\) — a symmetrical test, equal and opposite on the two terminals:
\[ \alpha = 0.5 \;\Rightarrow\; V_+ = V_- = \frac{\text{BSL}_p}{2} \]
What the standard test applies
A phase-phase BSL of \(\text{BSL}_p = 1267\) kV is verified by applying +633.5 kV to one terminal and −633.5 kV to the other at the same instant.
Section 5
The V⁺/V⁻ Method
Hold a constant \(V_-\) on one phase and raise \(V_+\) on the other until flashover. The positive CFO falls linearly with the negative voltage:
\[ \text{CFO}_+ = \text{CFO}_0 - K_L\,V_- \qquad \text{CFO}_0 = k_g\,\frac{3400}{1 + 8/S} \]
- \(\text{CFO}_0\)
- phase-phase CFO with one electrode grounded (\(V_- = 0\))
- \(K_L\)
- slope — contribution of the negative phase
- \(k_g\)
- gap factor for the one-electrode-grounded test
When \(V_- = 0\) the second phase is effectively at ground, so \(\text{CFO}_+ = \text{CFO}_0\) — a one-electrode-grounded gap, which is why the gap factor here is simply written \(k_g\).
Section 6
What Kₗ Means — and When It Is 1
\(K_L\) is how strongly the negative phase contributes. \(K_L = 1\) means the total phase-phase voltage controls the gap; \(K_L < 1\) means the split between positive and negative components matters. The dividing line:
- \(K_L \approx 1\) for small air gaps below ~2–3 m, or nominal voltages below ~500 kV — no need to separate \(V_+\) and \(V_-\), the total \(V_p\) is enough.
- \(K_L < 1\) for larger external EHV air clearances — the split matters.
\(K_L = 1\) as a conservative assumption
Using \(K_L = 1\) as a simplifying assumption is conservative where the actual gap has \(K_L < 1\), because it assumes the negative component contributes fully to the equivalent stress — inflating it and producing larger clearances. It is therefore a safe choice below EHV, but it is not a physical property of the gap: a real EHV gap has \(K_L < 1\) and a smaller true stress.
Section 7
Station Gap Factors
Station gaps are short — the long conductor-to-conductor span used for transmission lines is not appropriate here. For station bus clearances the 10 m conductor–conductor case is the relevant one; for breakers, the ring–ring grading-ring spacing applies. The two configurations used in the examples:
Table 1 — Phase-phase gap parameters for the two key station configurations.
| Configuration | \(K_L\) | \(k_g\) | \(\sigma_f/\text{CFO}_0\) | Represents |
|---|
| Conductor–conductor, 10 m | 0.67 | 1.35 | 0.035 | Short bus clearances |
| Ring–ring | 0.70 | 1.53 | 0.05 | Circuit-breaker grading rings |
Other tabulated geometries include rod–rod (smaller grading rings, lower-voltage fittings), supported busbar fittings, asymmetrical rod–conductor gaps, and jumper–shield-ring gaps for EHV.
Section 8
From CFO₀ to CFOₚ
The probabilistic calculation produces \(\text{CFO}_0\) (one-electrode-grounded), but the standard BSL is based on \(\text{CFO}_p\) (the symmetric \(\alpha = 0.5\) test). For \(\alpha = 0.5\) they are linked by:
\[ \text{CFO}_p = \frac{2\,\text{CFO}_0}{1 + K_L} \]
The design chain
Every phase-phase station calculation follows \(\text{CFO}_0 \rightarrow \text{CFO}_p \rightarrow \text{BSL}_p\) — solve for \(\text{CFO}_0\) from the stress, convert to \(\text{CFO}_p\), then to the standard BSL.
Section 9
The Equivalent Stress Vz
The same key simplification as on lines. Flashover occurs when \(V_+ > \text{CFO}_0 - K_L V_-\), i.e. \(V_+ + K_L V_- > \text{CFO}_0\). Define:
\[ V_z = V_+ + K_L\,V_- = (1 - K_L)\,V_+ + K_L\,V_p \;\Rightarrow\; \text{flashover when } V_z > \text{CFO}_0 \]
The second form — in terms of \(V_+\) and \(V_p\) — is the practical one for EMTP®/PSCAD post-processing, because both are read directly from the transient study. For small gaps \(K_L = 1\) and it collapses to \(V_z = V_p\).
Section 10
The Phase-Phase SOV Distribution
Three voltages relate by \(V_p = V_+ + V_-\); collect \(V_p\) and \(V_+\) (the latter is already needed for phase-ground), then \(V_- = V_p - V_+\). If both are Gaussian, \(V_z\) is Gaussian with:
\[ \mu_z = (1-K_L)\mu_+ + K_L\mu_p \]
\[ \sigma_z = \sqrt{(1-K_L)^2\sigma_+^2 + K_L^2\sigma_p^2 + 2K_L(1-K_L)\rho_{p+}\sigma_p\sigma_+} \]
- \(\rho_{p+}\)
- correlation of \(V_p\) and \(V_+\), normally 0.8–1.0
Taking the conservative \(\rho_{p+} = 1.0\) simplifies the spread to \(\sigma_z = (1-K_L)\sigma_+ + K_L\sigma_p\). Where studies are thin, the 2% phase-phase value is estimated from its phase-ground counterpart, \(E_{2p} \approx 1.55\,E_2\) (e.g. \(E_2 = 1.8\) pu → \(E_{2p} \approx 2.8\) pu).
Time-instant warning
The maximum phase-phase and maximum phase-ground SOVs do not necessarily occur at the same instant. Ideally collect at both relevant times; in practice engineers take each maximum at its own time and treat them as coincident. This is not exact but is conservative — and for the highest SOVs that drive the risk, the two maxima often coincide anyway.
Section 11
The Station Phase-Phase SSFOR
The substation estimating method assumes all correlations are 1.0, reversed parameters are not used, and the number of parallel insulations is taken as \(n = 1\). Because \(n = 1\), the method keeps both the strength scatter \(\sigma_f\) and the stress scatter \(\sigma_0\) — it cannot assume a near-vertical strength curve as the many-tower line case does. For Gaussian stress and strength:
\[ Z_e = \frac{\text{CFO}_0 - \mu_z}{\sqrt{\sigma_z^2 + \sigma_f^2}} \qquad \text{SSFOR} = \tfrac{1}{2}\bigl[\,1 - F(Z_e)\,\bigr] \]
The \(\tfrac{1}{2}\) appears because only one polarity orientation is considered, the reverse being neglected.
Section 12
Solving for Clearance and BSL
Given a target SSFOR, first find \(Z_e\). For \(\text{SSFOR} = 1/100\), \(F(Z_e) = 1 - 2(0.01) = 0.98\), so \(Z_e \approx 2.054\). Solve the \(Z_e\) equation for \(\text{CFO}_0\) (a short iteration, since \(\sigma_f\) depends on \(\text{CFO}_0\)), then convert:
\[ S_p = \frac{8}{\dfrac{3400\,k_g}{\text{CFO}_0} - 1} \qquad \text{CFO}_p = \frac{2\,\text{CFO}_0}{1+K_L}, \quad \text{BSL}_p = \text{CFO}_p\Bigl(1 - 1.28\,\tfrac{\sigma_f}{\text{CFO}_p}\Bigr) \]
For \(\sigma_f/\text{CFO}_p = 0.05\), \(\text{BSL}_p = 0.936\,\text{CFO}_p\). Note the basis: the tabulated scatter \(\sigma_f/\text{CFO}_0\) (Table 1) is referenced to \(\text{CFO}_0\), while the BSL step uses \(\sigma_f/\text{CFO}_p\); keep the basis consistent throughout and convert if a precise value is needed. The phase-phase station design SSFOR usually ranges from 0.10/100 to 1.0/100, with \(1.0/100\) the de-facto standard unless a more conservative requirement applies.
Section 13
Example 4 — Bus and Breaker Gaps
A 500/500 kV system, \(1\,\text{pu} = 449\) kV (the phase-to-ground crest of the 550 kV highest system voltage, \(1\,\text{pu} = \sqrt{2/3}\,U_m\) — not the 500 kV nominal line-to-line RMS), phase-phase SOV \(E_{2p} = 2.8\) pu with \(\sigma_p/E_{2p} = 0.10\), positive SOV \(E_2^+ = 1.8\) pu with \(\sigma_+/E_2^+ = 0.10\), \(\rho_{p+} = 1.0\), \(\text{SSFOR} = 1/100\). Using the 10 m conductor–conductor gap (\(K_L = 0.67\), \(k_g = 1.35\), \(\sigma_f/\text{CFO}_0 = 0.035\)):
\[ \text{CFO}_0 = 2.50\ \text{pu} = 1122.5\ \text{kV} \;\Rightarrow\; S_p = 2.59\ \text{m} \quad(\text{computer: } 2.59\ \text{m}) \]
Now the ring–ring breaker gap (\(K_L = 0.70\), \(k_g = 1.53\), \(\sigma_f/\text{CFO}_0 = 0.05\)):
\[ S_p = 2.27\ \text{m}, \quad \text{CFO}_p = 1354\ \text{kV}, \quad \text{BSL}_p = 0.936\times 1354 = 1267\ \text{kV} \;(\text{computer: } 1268) \]
So the required phase-phase BSL is \(\text{BSL}_p \approx 1267\) kV — verified by the IEC \(\alpha = 0.5\) test as \(+633.5\) / \(-633.5\) kV.
Section 14
Altitude
The examples are at sea level. Above sea level the required phase-phase spacing increases, corrected by the same atmospheric methods used for phase-ground: the required actual-site \(\text{CFO}_A\) is fixed by the design, the standard \(\text{CFO}_S\) must rise, and the clearance \(S_p\) grows accordingly.
Section 15
Arresters and Phase-Phase SOVs
If the SOVs are high enough for at least one arrester to operate, the phase-phase SOVs fall — reducing both clearance and BSL. The phase-phase case is more involved than phase-ground because the stress carries both polarities. The procedure:
- Determine whether one or more arresters operate.
- Modify the SOV parameters for the affected phase(s), keeping the positive and negative distributions.
- For conservatism set \(\rho_{+-} = 1.0\), build the \(V_z\) distribution, and solve for clearance / BSL.
If arresters operate on both phases, the original and reversed cases become equal, so a desired total \(1/100\) is entered as \(0.5/100\) (and \(0.1/100\) as \(0.05/100\)).
Table 2 — How the arrester-operation assumption changes the result.
| Assumption | Effect on the Calculation | Result Trend |
|---|
| Case A — one phase arrester operates | Only one polarity distribution is modified | Clearance / BSL usually reduced |
| Case B — both phase arresters operate | Original and reversed cases become equal; target risk halved (input \(0.5/100\)) | Clearance / BSL may increase significantly |
Example 5. The same 318 kV MCOV arrester (switching discharge 823 kV at 2 kA) modifies the positive phase but not the negative phase. With the ring–ring gap:
\[ S_p = 2.14\ \text{m}, \quad \text{BSL}_p = 1210\ \text{kV} \qquad(\text{vs } 1267\ \text{kV without arresters}) \]
The assumption dominates the answer
If arresters are conservatively assumed to operate on both phases (identical arrester-modified values, input \(0.5/100\)), the computer gives \(S_p = 2.83\) m and \(\text{BSL}_p = 1495\) kV — far more onerous. How arrester operation is assumed has a major effect on the phase-phase requirement.
Section 16
Workflow, Differences and Lessons
Table 3 — Phase-ground vs phase-phase station coordination.
| Item | Phase-Ground Station | Phase-Phase Station |
|---|
| Stress variable | \(V_+\) | \(V_z = (1-K_L)V_+ + K_L V_p\) |
| Strength | CFO, \(V_3\), BSL | \(\text{CFO}_0\), \(\text{CFO}_p\), \(\text{BSL}_p\) |
| Gap factor | usually \(k_g = 1.3\) | strongly geometry-dependent |
| Standard test | one phase to ground | opposite-polarity, two terminals |
| Split matters? | No | Yes, if \(K_L < 1\) |
| Arrester effect | modifies phase-ground SOV | may modify one or both phases |
EMTP® post-processing workflow
- Collect \(V_+\) and \(V_p\) from the EMT study, preferably at consistent instants.
- Select the gap geometry and read off \(K_L,\ k_g,\ \sigma_f/\text{CFO}\).
- Build \(V_z = (1-K_L)V_+ + K_L V_p\) and fit a Gaussian (\(\mu_z,\ \sigma_z\)) or read \(E_{2z}\).
- Choose the design SSFOR (typically 1/100, or 0.1/100), find \(Z_e\), and solve for \(\text{CFO}_0\).
- Get the spacing \(S_p\), then \(\text{CFO}_p = 2\text{CFO}_0/(1+K_L)\) and \(\text{BSL}_p\).
- Apply altitude correction; recalculate if arresters operate.
Document the time instants. When extracting \(V_+\), \(V_-\) and \(V_p\) from EMT simulations, record the instants used for the maxima — the maximum phase-ground and maximum phase-phase stresses may not occur at exactly the same instant.
Key lessons
- For large station air gaps the total phase-phase voltage alone is not enough — \(V_+\) and \(V_-\) matter separately.
- The equivalent stress \(V_z = V_+ + K_L V_-\) reduces it to one variable; for small gaps \(K_L \approx 1\) and \(V_z = V_p\).
- IEC phase-phase BSL is the symmetric \(\alpha = 0.5\) test (\(V_+ = V_-\)).
- 10 m conductor–conductor: \(K_L = 0.67,\ k_g = 1.35,\ \sigma_f/\text{CFO} = 0.035\). Ring–ring: \(K_L = 0.70,\ k_g = 1.53,\ \sigma_f/\text{CFO} = 0.05\).
- Arresters cut phase-phase clearance and BSL — but only if they actually operate on the relevant phase(s), and the assumption made dominates the result.
Equation Summary
Equivalent stress
\(\displaystyle V_z = V_+ + K_L V_- = (1-K_L)V_+ + K_L V_p\)
Flashover criterion
\(\displaystyle V_z > \text{CFO}_0\)
One-electrode-grounded CFO
\(\displaystyle \text{CFO}_0 = k_g\,\frac{3400}{1+8/S_p}\)
Phase-phase clearance
\(\displaystyle S_p = \frac{8}{\dfrac{3400\,k_g}{\text{CFO}_0} - 1}\)
Symmetric phase-phase CFO
\(\displaystyle \text{CFO}_p = \frac{2\,\text{CFO}_0}{1+K_L}\)
Phase-phase BSL
\(\displaystyle \text{BSL}_p = \text{CFO}_p\Bigl(1 - 1.28\,\frac{\sigma_f}{\text{CFO}_p}\Bigr)\)
