Formula Sheet: Fault Analysis

Symmetrical Fault Analysis

Three-Phase Fault Current (Balanced Faults)

Symmetrical Fault Current (Bolted Fault): \[I_{fault} = \frac{V_{pre-fault}}{Z_{eq}}\]

• I_fault: Symmetrical fault current (A)
• V_pre-fault: Pre-fault voltage at fault location (V)
• Z_eq: Equivalent impedance from source to fault point (Ω)

Three-Phase Fault Current at Generator Terminals: \[I_{f} = \frac{V_t}{Z_d}\]

• V_t: Terminal voltage (V)
• Z_d: Generator reactance (use X_d", X_d', or X_d depending on time frame) (Ω)

Subtransient Fault Current: \[I_f'' = \frac{E_g}{X_d''}\]

• I_f": Subtransient fault current (first cycle, 0-0.1 s) (A or pu)
• E_g: Internal generator voltage (V or pu)
• X_d": Direct-axis subtransient reactance (Ω or pu)

Transient Fault Current: \[I_f' = \frac{E_g}{X_d'}\]

• I_f': Transient fault current (0.1-1.0 s) (A or pu)
• X_d': Direct-axis transient reactance (Ω or pu)

Steady-State Fault Current: \[I_f = \frac{E_g}{X_d}\]

• I_f: Steady-state fault current (after 1-3 s) (A or pu)
• X_d: Direct-axis synchronous reactance (Ω or pu)

Short Circuit MVA Method

Three-Phase Fault MVA: \[MVA_{fault} = \sqrt{3} \times V_{L-L} \times I_{fault} \times 10^{-6}\]

• MVA_fault: Fault MVA (MVA)
• V_L-L: Line-to-line voltage (V)
• I_fault: Fault current (A)

Fault Current from Fault MVA: \[I_{fault} = \frac{MVA_{fault} \times 10^6}{\sqrt{3} \times V_{L-L}}\] System Equivalent Impedance from Fault MVA: \[Z_{eq} = \frac{(kV)^2}{MVA_{fault}}\]

• Z_eq: Equivalent system impedance (Ω)
• kV: System line-to-line voltage (kV)
• MVA_fault: Available fault MVA (MVA)

Unsymmetrical Fault Analysis (Sequence Components)

Symmetrical Components Transformation

Phase to Sequence Transformation: \[\begin{bmatrix} V_0 \\ V_1 \\ V_2 \end{bmatrix} = \frac{1}{3} \begin{bmatrix} 1 & 1 & 1 \\ 1 & a & a^2 \\ 1 & a^2 & a \end{bmatrix} \begin{bmatrix} V_a \\ V_b \\ V_c \end{bmatrix}\]

• V₀: Zero-sequence voltage (V or pu)
• V₁: Positive-sequence voltage (V or pu)
• V₂: Negative-sequence voltage (V or pu)
• V_a, V_b, V_c: Phase voltages (V or pu)
• a: Complex operator = \(1\angle120°\) = \(e^{j2\pi/3}\)
• a²: = \(1\angle240°\) = \(e^{j4\pi/3}\)

Zero-Sequence Component: \[V_0 = \frac{1}{3}(V_a + V_b + V_c)\] \[I_0 = \frac{1}{3}(I_a + I_b + I_c)\] Positive-Sequence Component: \[V_1 = \frac{1}{3}(V_a + aV_b + a^2V_c)\] \[I_1 = \frac{1}{3}(I_a + aI_b + a^2I_c)\] Negative-Sequence Component: \[V_2 = \frac{1}{3}(V_a + a^2V_b + aV_c)\] \[I_2 = \frac{1}{3}(I_a + a^2I_b + aI_c)\] Sequence to Phase Transformation: \[\begin{bmatrix} V_a \\ V_b \\ V_c \end{bmatrix} = \begin{bmatrix} 1 & 1 & 1 \\ 1 & a^2 & a \\ 1 & a & a^2 \end{bmatrix} \begin{bmatrix} V_0 \\ V_1 \\ V_2 \end{bmatrix}\] Phase Voltage in Terms of Sequence Components: \[V_a = V_0 + V_1 + V_2\] \[V_b = V_0 + a^2V_1 + aV_2\] \[V_c = V_0 + aV_1 + a^2V_2\] Phase Current in Terms of Sequence Components: \[I_a = I_0 + I_1 + I_2\] \[I_b = I_0 + a^2I_1 + aI_2\] \[I_c = I_0 + aI_1 + a^2I_2\]

Sequence Impedances

Positive-Sequence Impedance: \[Z_1 = \frac{V_1}{I_1}\]

• Z₁: Positive-sequence impedance (Ω or pu)
• Represents normal balanced operation impedance

Negative-Sequence Impedance: \[Z_2 = \frac{V_2}{I_2}\]

• Z₂: Negative-sequence impedance (Ω or pu)
• For static equipment: Z₂ = Z₁
• For rotating machines: Z₂ ≠ Z₁

Zero-Sequence Impedance: \[Z_0 = \frac{V_0}{I_0}\]

• Z₀: Zero-sequence impedance (Ω or pu)
• Depends heavily on grounding and return path
• Typically: Z₀ > Z₁ for overhead lines

Sequence Network Connections for Common Faults

Single Line-to-Ground (SLG) Fault:

• Fault Type: Phase A to ground
• Network Connection: Series connection of Z₁, Z₂, and Z₀

Fault Current (Phase A-to-Ground): \[I_a = I_0 + I_1 + I_2 = 3I_0 = 3I_1 = 3I_2\] \[I_1 = I_2 = I_0 = \frac{V_f}{Z_1 + Z_2 + Z_0 + 3Z_f}\] \[I_a = \frac{3V_f}{Z_1 + Z_2 + Z_0 + 3Z_f}\]

• V_f: Pre-fault voltage at fault point (V or pu)
• Z_f: Fault impedance (Ω or pu)
• I_b = I_c = 0 for SLG fault

Line-to-Line (L-L) Fault:

• Fault Type: Phase B to Phase C (no ground)
• Network Connection: Parallel connection of Z₁ and Z₂; Z₀ not involved

Fault Current (Line-to-Line): \[I_1 = -I_2 = \frac{V_f}{Z_1 + Z_2 + Z_f}\] \[I_0 = 0\] \[I_b = -I_c = \sqrt{3} \times I_1 \angle-90°\] \[I_a = 0\]

• I_b: Current in phase B (A or pu)
• I_c: Current in phase C (A or pu)

Magnitude of Line-to-Line Fault Current: \[|I_{L-L}| = \sqrt{3} \times |I_1| = \frac{\sqrt{3} \times V_f}{Z_1 + Z_2 + Z_f}\] Double Line-to-Ground (2LG) Fault:

• Fault Type: Phase B and C to ground
• Network Connection: Z₁ in series with parallel combination of Z₂ and Z₀

Sequence Currents (Double Line-to-Ground): \[I_1 = \frac{V_f}{Z_1 + \frac{Z_2(Z_0 + 3Z_f)}{Z_2 + Z_0 + 3Z_f}}\] \[I_2 = -I_1 \times \frac{Z_0 + 3Z_f}{Z_2 + Z_0 + 3Z_f}\] \[I_0 = -I_1 \times \frac{Z_2}{Z_2 + Z_0 + 3Z_f}\] \[I_a = 0\] Phase Currents (Double Line-to-Ground): \[I_b = I_0 + a^2I_1 + aI_2\] \[I_c = I_0 + aI_1 + a^2I_2\]

Three-Phase Fault (Balanced)

Sequence Currents: \[I_1 = \frac{V_f}{Z_1 + Z_f}\] \[I_2 = 0\] \[I_0 = 0\]

• Only positive-sequence network carries current
• Most severe fault in terms of magnitude (typically)

Per-Unit System for Fault Analysis

Per-Unit Base Quantities

Base Power (Single-Phase or Three-Phase): \[S_{base} = \text{Selected base MVA (typically 100 MVA)}\] Base Voltage (Line-to-Line): \[V_{base(L-L)} = \text{Nominal system voltage (kV)}\] Base Voltage (Line-to-Neutral): \[V_{base(L-N)} = \frac{V_{base(L-L)}}{\sqrt{3}}\] Base Current (Three-Phase): \[I_{base} = \frac{S_{base} \times 10^6}{\sqrt{3} \times V_{base(L-L)} \times 10^3} = \frac{S_{base(MVA)} \times 10^3}{\sqrt{3} \times V_{base(kV)}}\]

• I_base: Base current (A)
• S_base(MVA): Base power (MVA)
• V_base(kV): Base voltage line-to-line (kV)

Base Impedance: \[Z_{base} = \frac{V_{base(L-L)}^2}{S_{base}} = \frac{(kV_{base})^2}{MVA_{base}}\]

• Z_base: Base impedance (Ω)
• Commonly expressed in ohms when kV and MVA are used

Alternative Base Impedance: \[Z_{base} = \frac{V_{base(L-L)}}{I_{base}} = \frac{(V_{base})^2}{S_{base}}\]

Per-Unit Conversions

Impedance in Per-Unit: \[Z_{pu} = \frac{Z_{actual}}{Z_{base}}\] Change of Base for Impedance: \[Z_{pu(new)} = Z_{pu(old)} \times \frac{MVA_{base(new)}}{MVA_{base(old)}} \times \left(\frac{kV_{base(old)}}{kV_{base(new)}}\right)^2\] Reactance from Percent to Per-Unit: \[X_{pu} = \frac{X_{\%}}{100}\] Generator/Transformer Reactance Conversion: \[X_{pu(system)} = X_{pu(nameplate)} \times \frac{MVA_{base(system)}}{MVA_{nameplate}} \times \left(\frac{kV_{nameplate}}{kV_{base(system)}}\right)^2\] Current in Per-Unit: \[I_{pu} = \frac{I_{actual}}{I_{base}}\] Voltage in Per-Unit: \[V_{pu} = \frac{V_{actual}}{V_{base}}\] Power in Per-Unit: \[S_{pu} = \frac{S_{actual}}{S_{base}}\]

Per-Unit Fault Calculations

Per-Unit Fault Current: \[I_{fault(pu)} = \frac{V_{pre-fault(pu)}}{Z_{eq(pu)}}\]

• Typically, V_{pre-fault(pu)} = 1.0 pu

Actual Fault Current from Per-Unit: \[I_{fault(actual)} = I_{fault(pu)} \times I_{base}\]

Sequence Impedance of Power System Components

Generators

Positive-Sequence Impedance:

• X_d": Subtransient reactance (pu) - used for first cycle fault calculations
• X_d': Transient reactance (pu) - used for transient stability and breaker duty
• X_d: Synchronous reactance (pu) - used for steady-state analysis
• Typical values: X_d" ≈ 0.12-0.20 pu, X_d' ≈ 0.15-0.30 pu, X_d ≈ 1.0-2.0 pu

Negative-Sequence Impedance: \[Z_2 \approx X_2 \approx X_d''\]

• Typically: X₂ ≈ 0.10-0.20 pu
• Close to but slightly less than X_d"

Zero-Sequence Impedance: \[Z_0 \approx X_0\]

• Typically: X₀ ≈ 0.02-0.10 pu
• Depends on generator grounding method
• For ungrounded generators: X₀ → ∞

Transformers

Positive and Negative-Sequence Impedance: \[Z_1 = Z_2 = Z_{transformer}\]

• Static device: positive and negative sequence impedances are equal
• Use nameplate impedance (typically given as % or pu on nameplate rating)

Zero-Sequence Impedance:

• Depends on winding configuration and grounding
• Wye-Grounded/Wye-Grounded: Z₀ ≈ Z₁
• Wye-Grounded/Delta: Z₀ present on grounded side only
• Delta/Delta: Zero-sequence current cannot flow; infinite Z₀
• Wye-Ungrounded: Zero-sequence current blocked; infinite Z₀

Transformer Impedance (Leakage Reactance): \[Z_{pu} = \frac{Z_{\%}}{100} = \frac{V_{sc}}{V_{rated}}\]

• Z_%: Percent impedance from nameplate
• V_sc: Short-circuit test voltage

Transmission Lines

Positive-Sequence Impedance: \[Z_1 = R_1 + jX_1\]

• Calculated from line geometry and conductor properties
• Balanced three-phase impedance

Negative-Sequence Impedance: \[Z_2 = Z_1\]

• Equal to positive-sequence for static transmission lines

Zero-Sequence Impedance: \[Z_0 = R_0 + jX_0\]

• Overhead lines: Z₀ ≈ 2-3.5 × Z₁ (due to ground return path)
• Cable: Z₀ varies based on sheath grounding and arrangement
• Depends on ground resistivity and geometric mean distance to ground return

Zero-Sequence Resistance: \[R_0 \approx R_1 + 3R_g\]

• R_g: Ground resistance per unit length

Motors (Induction and Synchronous)

Positive-Sequence Impedance:

• Synchronous motors: Use X_d" (similar to generators)
• Induction motors: Z₁ ≈ 0.15-0.25 pu (locked rotor impedance)

Negative-Sequence Impedance: \[Z_2 \approx Z_1\]

• For motors, Z₂ ≈ Z₁ (locked rotor impedance)

Zero-Sequence Impedance:

• Typically very high or infinite (motors usually not grounded through neutral)
• Z₀ → ∞ for most motor connections

Fault Current Calculation Methods

Thevenin Equivalent Method

Thevenin Equivalent Impedance at Fault Point: \[Z_{th} = Z_{eq} = \text{Equivalent impedance looking into network from fault point}\] Fault Current: \[I_f = \frac{V_{th}}{Z_{th} + Z_f}\]

• V_th: Thevenin voltage (pre-fault voltage at fault location)
• Z_th: Thevenin equivalent impedance
• Z_f: Fault impedance

Bus Impedance Matrix Method

Fault Current Using Z_bus: \[I_f = \frac{V_f}{Z_{ii} + Z_f}\]

• Z_ii: Diagonal element of bus impedance matrix at fault bus i
• V_f: Pre-fault voltage at bus i

Voltage at Bus k During Fault at Bus i: \[V_k = V_{k(pre-fault)} - Z_{ki} \times I_f\]

• Z_ki: Off-diagonal element of Z_bus between bus k and bus i

Change in Bus Voltage: \[\Delta V_k = -Z_{ki} \times I_f\]

Sequence Network Analysis

General Procedure:

1. Construct positive-, negative-, and zero-sequence networks
2. Connect sequence networks based on fault type
3. Calculate sequence currents
4. Transform sequence currents to phase currents

Circuit Breaker Duties and Ratings

Interrupting Current and Duty

Symmetrical Interrupting Current: \[I_{sym} = I_{rms(ac)}\]

• RMS value of AC component of fault current
• Used for breaker selection

Asymmetrical Fault Current: \[I_{asym} = I_{sym} \times \sqrt{1 + 2e^{-4\pi t/(X/R)}}\]

• I_asym: Asymmetrical RMS current (A)
• I_sym: Symmetrical RMS current (A)
• t: Time from fault initiation (s)
• X/R: System reactance to resistance ratio at fault point

DC Component of Fault Current: \[i_{dc}(t) = \sqrt{2} \times I_{sym} \times e^{-t/(L/R)} = \sqrt{2} \times I_{sym} \times e^{-2\pi ft/(X/R)}\]

• L/R: Circuit time constant (s)
• f: System frequency (Hz)
• DC component decays exponentially

Peak Asymmetrical Current (First Cycle): \[I_{peak} = \sqrt{2} \times I_{sym} \times K\] \[K = 1 + e^{-\pi/(X/R)}\]

• K: Asymmetry factor (typically 1.4-2.0)
• I_peak: Maximum instantaneous current (A)

Asymmetry Factor (Simplified): \[K \approx 1.0 + e^{-\pi/(X/R)}\]

• For X/R = 0: K = 2.0 (maximum asymmetry)
• For X/R → ∞: K = 1.0 (no DC offset)

RMS Asymmetrical Current (at time t): \[I_{asym(rms)} = \sqrt{I_{sym}^2 + I_{dc(rms)}^2} = I_{sym}\sqrt{1 + 2e^{-4\pi t/(X/R)}}\]

Interrupting Duty Calculation

Interrupting MVA: \[MVA_{int} = \sqrt{3} \times V_{L-L} \times I_{int} \times 10^{-6}\]

• MVA_int: Interrupting duty (MVA)
• I_int: Interrupting current (symmetrical) (A)
• V_L-L: System line-to-line voltage (V)

Breaker Interrupting Time:

• Typical range: 3-8 cycles (0.05-0.13 s for 60 Hz)
• Use transient or subtransient current depending on interrupting time

Momentary Rating and Closing/Latching Current

Momentary Current Rating: \[I_{momentary} = 1.6 \times I_{sym}\]

• Used for mechanical stress calculation
• Based on asymmetrical current in first cycle

Closing and Latching Current:

• Typically equal to asymmetrical peak current
• Breaker must withstand mechanical forces when closing into a fault

Grounding and Ground Fault Protection

Ground Fault Current

Ground Fault Current in Neutral: \[I_n = 3I_0\]

• I_n: Neutral current (A)
• I₀: Zero-sequence current (A)
• Valid for grounded wye systems

Solidly Grounded System (SLG Fault): \[I_{fault} = \frac{3V_{L-N}}{Z_1 + Z_2 + Z_0}\]

• Maximum ground fault current for solidly grounded system

Resistance Grounded System: \[I_0 = \frac{V_f}{Z_1 + Z_2 + Z_0 + 3R_n}\]

• R_n: Neutral grounding resistance (Ω)
• Limits ground fault current

Reactance Grounded System: \[I_0 = \frac{V_f}{Z_1 + Z_2 + Z_0 + 3X_n}\]

• X_n: Neutral grounding reactance (Ω)

Ground Fault Overvoltage

Voltage Rise in Unfaulted Phases (SLG Fault): \[V_{b(fault)} = V_0 + a^2V_1 + aV_2\] \[V_{c(fault)} = V_0 + aV_1 + a^2V_2\]

• Unfaulted phase voltages can exceed line-to-neutral voltage
• For ungrounded or high-impedance grounded systems, can approach line-to-line voltage

Coefficient of Grounding: \[COG = \frac{V_{L-G(max)}}{V_{L-L}/\sqrt{3}} \times 100\%\]

• COG: Coefficient of grounding (%)
• V_L-G(max): Maximum line-to-ground voltage during fault
• Solidly grounded: COG ≤ 80%
• Impedance grounded: 80% < cog="" ≤="">
• Ungrounded or resonant grounded: COG > 100%

Fault Current Decay and Time Constants

Generator Fault Current Decay

Total Fault Current (Generator): \[i(t) = i_{ac}(t) + i_{dc}(t)\] AC Component Decay: \[i_{ac}(t) = \frac{E_g}{X_d''} e^{-t/\tau''} + \frac{E_g}{X_d'}\left(1 - e^{-t/\tau'}\right) + \frac{E_g}{X_d}\]

• τ": Subtransient time constant (typically 0.01-0.05 s)
• τ': Transient time constant (typically 0.5-2.0 s)

Simplified AC Decay (Three Regions):

• Subtransient (0-0.1 s): \(I'' = E_g / X_d''\)
• Transient (0.1-1.0 s): \(I' = E_g / X_d'\)
• Steady-State (>1-3 s): \(I = E_g / X_d\)

DC Component Time Constant: \[\tau_a = \frac{L}{R} = \frac{X}{2\pi fR} = \frac{X/R}{2\pi f}\]

• τ_a: Armature time constant (s)
• f: System frequency (Hz)

Special Fault Conditions

Arc Resistance

Arc Resistance (Empirical): \[R_{arc} = \frac{28700 \times s}{I_{arc}^{1.4}}\]

• R_arc: Arc resistance (Ω)
• s: Arc length (ft)
• I_arc: Arc current (A)
• Arc resistance reduces fault current magnitude

Effect on Fault Current: \[I_f = \frac{V_f}{Z_{eq} + R_{arc}}\]

Motor Contribution to Fault

Motor Fault Contribution: \[I_{motor} = \frac{V_t}{Z_{motor}}\]

• Motors act as sources during fault (back-feed)
• Contribution decays rapidly (within 3-5 cycles)
• Use locked-rotor impedance for motor reactance

Total Fault Current with Motor Contribution: \[I_{total} = I_{utility} + I_{motor}\]

Fault Between Two Points

Fault Current Between Buses i and j: \[I_f = \frac{V_i - V_j}{Z_{ij} + Z_f}\]

• V_i, V_j: Pre-fault voltages at buses i and j
• Z_ij: Transfer impedance between buses

Voltage Sag and Swell During Faults

Voltage During Fault

Voltage at Non-Faulted Bus: \[V_k = V_{k(pre-fault)} - I_f \times Z_{kf}\]

• V_k: Voltage at bus k during fault (V or pu)
• Z_kf: Transfer impedance between bus k and fault location

Voltage Sag: \[V_{sag} = V_{pre-fault} - V_{during-fault}\] \[\%V_{sag} = \frac{V_{sag}}{V_{pre-fault}} \times 100\%\] Remaining Voltage (Retention): \[V_{retention} = \frac{V_{during-fault}}{V_{pre-fault}}\]

Protection Coordination Parameters

Fault Current for Relay Settings

Maximum Fault Current:

• Use minimum source impedance (maximum generation)
• Use subtransient reactance for generators
• Include all motor contributions

Minimum Fault Current:

• Use maximum source impedance (minimum generation)
• End-of-line or remote faults
• Include arc resistance effects
• Used for relay sensitivity check

Fault Current Multiplier for Relay Pickup: \[M = \frac{I_{fault(min)}}{I_{pickup}}\]

• M: Sensitivity margin (typically M > 2 for reliable detection)

Important Relationships and Ratios

Comparison of Fault Currents

Typical Fault Current Magnitudes (Relative):

• Three-phase fault: Highest in most systems (reference = 1.0)
• Single line-to-ground: 1.0-1.3 times three-phase (if Z₀ <>₁)
• Line-to-line: 0.866 times three-phase (if Z₁ = Z₂)
• Double line-to-ground: Between L-L and SLG

Ratio of L-L to 3-Phase Fault (Z₁ = Z₂): \[\frac{I_{L-L}}{I_{3\phi}} = \frac{\sqrt{3}/2}{1} = 0.866\] X/R Ratio Effects:

• Higher X/R → Greater DC offset → Higher asymmetrical current
• Typical X/R ratios:
  • Distribution systems: 2-5
  • Transmission systems: 10-30
  • Near generators: 30-100

Fault Analysis Assumptions

Standard Assumptions

• Pre-fault conditions: System operating at nominal voltage (1.0 pu), balanced, steady-state
• Fault impedance: Zero (bolted fault) unless otherwise specified
• Load currents: Neglected (small compared to fault current)
• Generator excitation: Constant (for transient and subtransient analysis)
• Transformer tap position: Nominal unless specified
• Transmission line capacitance: Neglected for short-circuit studies
• Resistance: Often neglected (X >> R) unless X/R ratio needed