Unbalanced fault analysis — sequence networks (LG / LL / LLG / 3-φ)

Reference notes

Most transmission-system faults are unbalanced. Balanced 3-phase faults occur in only 5-10% of cases; single line-to-ground (LG) faults dominate at 70-80%. Unbalanced fault analysis uses symmetrical components — the Fortescue decomposition into positive, negative, and zero sequence networks — with sequence networks connected differently per fault type.

Fault-type frequency distribution

Symmetrical components — Fortescue (1918)

• Any unbalanced 3-phase phasor set decomposes UNIQUELY into three balanced sets:
  • Positive sequence — balanced rotation A-B-C (same as normal operation).
  • Negative sequence — balanced rotation A-C-B.
  • Zero sequence — all three phases EQUAL in magnitude and phase.
• Inverse transform: I_a = I_a1 + I_a2 + I_a0; I_b = a²·I_a1 + a·I_a2 + I_a0; I_c = a·I_a1 + a²·I_a2 + I_a0, where a = 1∠120°.
• Each element (generator, transformer, line) has its own Z1 (positive), Z2 (negative), Z0 (zero-sequence) impedance.

LG fault — single line-to-ground (70-80% of faults)

• Phase A faults to ground; phases B and C are healthy.
• Boundary conditions: I_a = I_fault, I_b = 0, I_c = 0, V_a = 0 (or I × Z_fault for impedance fault).
• Sequence currents: I_a1 = I_a2 = I_a0 = I_fault / 3 (equal across all three networks).
• Sequence-network connection: POSITIVE, NEGATIVE, and ZERO networks all in SERIES.
• Fault current: I_fault = 3·V_pre / (Z1 + Z2 + Z0). Multiply by 3 because I_a = 3·I_a1 when sequence currents are equal.
• On solidly-grounded systems where Z0 ≈ Z1, LG fault current ≈ 3-φ fault. If Z0 < Z1 (near a grounded transformer), LG can EXCEED 3-φ.
• Causes: lightning, vegetation contact, animal contact, insulator flashover, equipment failure.

LL fault — line-to-line (10-15%)

• Phases B and C are shorted to each other; phase A is healthy. No path to ground.
• Boundary conditions: I_a = 0, I_b = -I_c = I_fault, V_b = V_c.
• Sequence currents: I_a0 = 0 (no zero-sequence — no ground path); I_a1 = -I_a2.
• Sequence-network connection: POSITIVE in PARALLEL with NEGATIVE. Zero sequence does NOT participate.
• Fault current: I_LL = (√3/2) · I_3φ ≈ 0.866 · I_3φ — always SMALLER than balanced 3-φ fault.
• Causes: conductor swing in wind, jumper failures, animal contact between phases.

LLG fault — double line-to-ground (5-10%)

• Phases B and C both faulted to ground simultaneously.
• Boundary conditions: I_a = 0, V_b = V_c = 0.
• Sequence relations: I_a0 + I_a1 + I_a2 = 0; V_a1 = V_a2 = V_a0.
• Sequence-network connection: POSITIVE in SERIES with the PARALLEL combination of NEGATIVE and ZERO networks.
• Fault current depends heavily on Z0/Z1 ratio:
  • Z0 ≫ Z1 (high-Z grounded) → LLG fault much smaller than 3-φ.
  • Z0 ≈ Z1 (solidly grounded) → LLG fault 50-90% of 3-φ.
• Detection: distance relays AND ground overcurrent both pick up — significant positive AND zero-sequence content.

3-φ balanced fault (5-10%)

• All three phases simultaneously short to each other and optionally to ground.
• Boundary conditions: I_a + I_b + I_c = 0; V_a = V_b = V_c = 0.
• Sequence content: POSITIVE sequence ONLY. I_a2 = I_a0 = 0.
• Fault current: I_3φ = V_pre / Z1. Basic single-line short-circuit calculation.
• Causes: failed switchgear during repair, inrush switching with prefault charging, falling tower bringing all three conductors down.
• Standard sizing case for switchgear interrupting ratings — though LG can exceed 3-φ on solidly-grounded systems with low Z0.

Sequence impedances by component

• Generator: Z1 = Z2 ≈ Xd'' (subtransient reactance, 15-25%); Z0 typically 5-15% (grounded) or ∞ (ungrounded).
• Transformer: Z1 = Z2 = nameplate impedance (5-15%) regardless of connection.
  • Y-grounded both sides: Z0 ≈ Z1, zero-sequence passes normally.
  • Δ (delta) on either side: Z0 looking from the Δ side = ∞. Delta blocks zero-seq from passing into the external network (circulates internally only).
  • Y-grounded / Δ: Z0 from the Y side is finite; from the Δ side = ∞. Wye-delta transformer ISOLATES zero-sequence between primary and secondary.
• Transmission line: Z1 = Z2 (same line geometry, balanced currents); Z0 ≈ 2-4 × Z1 due to earth-return path with ground resistance plus mutual coupling.
• Load: typically modeled as Z1 only at full system V_pre.

LG vs 3-φ fault current — depends on Z0/Z1 ratio

• Z0 = Z1 → ratio = 1.0 (LG ≈ 3-φ).
• Z0 = 0.5 Z1 → ratio = 1.2 (LG > 3-φ — LG is the worst-case interrupting duty).
• Z0 = 3 Z1 → ratio = 0.6 (LG < 3-φ).
• Practical: on solidly-grounded systems near transformers, LG can EXCEED 3-φ. Modern fault studies always compute ALL four fault types at EVERY bus.

Modern fault-study workflow

• Engineer enters network model: generators (with Xd''), transformers (nameplate Z + connection), lines (Z1, Z0), loads (Z1).
• Software builds sequence Y-bus matrices: Y_bus_1, Y_bus_2, Y_bus_0.
• Inverts to get sequence Z-bus matrices: Z_bus_1 = inv(Y_bus_1), etc.
• For a fault at bus k: Thevenin impedance from sequence Z-bus diagonal: Z_th_1 = Z_bus_1[k,k], Z_th_2 = Z_bus_2[k,k], Z_th_0 = Z_bus_0[k,k].
• Connect networks per fault-type rule, solve sequence currents at fault, use Z-bus sensitivity factors to compute currents and voltages everywhere.
• Output used for: switchgear interrupting rating, protection coordination, ground-grid design (using max LG fault current per IEEE 80), arc-flash analysis (IEEE 1584).
• Industry-standard tools: ETAP, SKM PowerTools, EasyPower, PowerWorld, DIgSILENT PowerFactory.