Relay Protection Academy Module 04 of 25
04
Module 04 Intermediate

Fault Calculations &
Symmetrical Components

⌛ ~2.5 hours 📚 IEC 60909 family 📑 12 slides

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4.1

Why Fault Calculations?

Before any relay can be set, the engineer must know three boundary values at each relaying point:

Maximum Fault Current
Sets the upper limit for relay pick-up. Must not exceed CT saturation or CB interrupting rating.
Minimum Fault Current
Sets the lower limit; relay must still detect the weakest credible fault with minimum connected sources.
Maximum Through-Fault
Current in a healthy branch during an external fault. Used for differential relay stability margins.
Single driving voltage substitution To simplify multi-source network analysis, all normal driving voltages are short-circuited and a single voltage source equal to the pre-fault open-circuit voltage is injected at the fault point. This is valid by the principle of superposition and is the standard method used in fault calculation software and analytical standards.
Scope of analysis Three-phase faults are balanced (symmetrical) and can be solved with positive-sequence networks alone. All other fault types create unbalanced conditions and require symmetrical component analysis to decompose into manageable sequence networks.
4.2

Symmetrical Components

Any unbalanced set of three-phase quantities can be resolved into three balanced sets:

Positive Sequence (1)
Equal magnitudes, normal phase rotation A-B-C, 120° apart. Represents the healthy-system component.
Negative Sequence (2)
Equal magnitudes, reversed rotation A-C-B. Only exists during unbalanced conditions (faults, asymmetric loading).
Zero Sequence (0)
Equal magnitudes, all co-phasal (in phase). Requires an earth return path; absent in isolated or delta-wound circuits.
Transformation
where is the rotation operator from Module 03.
Key simplification For large transmission networks, . Only rotating machines have significantly different and (sub-transient vs. negative-sequence reactance). The zero-sequence network topology differs from positive and negative because it depends on transformer winding connections and neutral earthing.
4.3

The Three Sequence Networks

Each sequence has its own Thevenin equivalent network. Only the positive-sequence network contains driving voltages (the pre-fault source EMFs).

Z1 Network
Contains all source EMFs. Impedances are line and transformer positive-sequence values. Load is normally ignored. Source impedance includes generator sub-transient reactance.
Z2 Network
Passive (no voltage sources). Topology mirrors Z1 but uses negative-sequence impedances. For non-rotating plant .
Z0 Network
Passive. Topology differs from Z1: delta windings block zero-sequence current; solid-earthed neutrals provide a low-impedance return path; isolated neutrals present infinite Z0.
Z0 boundary conditions at transformers A delta winding interrupts zero-sequence current flow. A star winding with an earthed neutral allows zero-sequence current to flow. A star winding with an isolated (unearthed) neutral blocks zero-sequence current just as a delta does. These boundary conditions must be traced carefully when building the Z0 network for any power system.
4.4

Shunt Fault Types and Network Connections

The sequence networks are interconnected differently for each fault type. The boundary conditions at the fault determine the connection.

Single Phase-to-Earth (A-E)
Most common fault type. Sequence networks connected in series: Z1 + Z2 + Z0.
Fault current:
Phase-to-Phase (B-C)
No earth involvement. Z1 and Z2 in parallel. Zero-sequence not connected ().
Fault current:
Phase-Phase-Earth (B-C-E)
All three networks in parallel. Both phase and zero-sequence currents flow. Fault current depends on Z0/Z1 ratio.
Three-Phase (3-ph)
Balanced fault. Only Z1 network active. .
Typically the highest fault current; most destabilising.
4.5

Residual Current and Voltage

Residual quantities are the vector sum of the three phase quantities. They are zero in a balanced system and non-zero only when an earth-return current path exists.

Residual Current

Measured by a residual (Holmgreen) CT connection or by summing three CT secondary currents. The relay that measures this is an earth fault relay operating on zero-sequence current.

Residual Voltage

Measured from the open delta (broken delta) secondary of a VT. Used for earth fault detection on isolated neutral systems and for directional earth fault polarisation.
Common error: neutral current vs residual current The neutral current flowing in the star point of an earthed transformer is (the residual). This is sometimes confused with the zero-sequence current itself. The relay measures ; the sequence network equation uses . Always be clear which quantity a relay input is wired to before entering settings.
Decoupled from load In a balanced three-phase load, exactly. Earth fault relays connected to residual CTs are therefore inherently immune to load current and can be set with very sensitive pickups, down to a few percent of rated current.
4.6

Sequence Voltage Gradients

Sequence voltages distribute differently across the network during a fault. This is critical for directional and distance relay polarisation.

Positive sequence: drops toward the fault
Positive-sequence voltage is lowest at the fault point (approaches zero for a solid fault) and highest at remote sources. Distance relays measure impedance using V1 as the polarising quantity.
Negative and zero sequence: highest at the fault
No pre-fault driving voltage exists for negative or zero sequence. These voltages are generated at the fault and decay towards the network sources. Directional earth fault relays polarised by V0 or I0 rely on this gradient.
🖼
Illustration prompt

Technical line graph showing sequence voltage distribution along a transmission line during an A-E fault. X-axis: distance from source to fault point. Y-axis: per-unit voltage magnitude. Three curves: V1 (blue) starts at 1.0 pu at source, dips to near zero at the fault; V2 (orange) starts near zero at source, rises to maximum at fault; V0 (red) mirrors V2. Dark background, clean grid lines, labeled axes.

4.7

System Earthing and the Z0/Z1 Ratio

The ratio is the single most important parameter governing earth fault behaviour. It is set entirely by the neutral earthing method.

Solid Earthing (K < 3)
Low Z0. High earth fault currents. Residual voltage small. EHV and HV systems. Sensitive earth fault relays. Transient overvoltages on healthy phases are limited.
Resistance Earthing
Z0 increased by the earth resistance. Limits earth fault current to a defined level. Common on 11-33 kV industrial systems. Reduces equipment damage but increases healthy-phase overvoltage slightly.
Isolated / Petersen Coil
Very high Z0. Earth fault currents very small. Residual voltage rises to on healthy phases. Service can continue with single-phase fault present. Requires sensitive watt-metric earth fault relaying.
Isolated neutral: overvoltage hazard On an isolated neutral system with a solid earth fault, the capacitive charging voltage on the two healthy phases rises to times the normal phase-to-neutral voltage. This is equal to the full line voltage and determines the insulation class required for VTs and cables connected to those phases.
4.8

Current Distribution: C1 and C0 Factors

In a meshed network, fault current splits between parallel branches. The fraction reaching a relay through a specific branch is given by the sequence distribution factors.

Phase current in a branch during an A-E fault
where is the fraction of the positive-sequence fault current in the branch and is the fraction of the zero-sequence fault current.
C0 is not the same as C1 The zero-sequence network topology differs from the positive-sequence network due to transformer earthing methods and the absence of source EMFs. As a result, in almost all practical networks. Using in place of produces incorrect phase currents and invalidates distance relay settings. Always calculate both factors separately.
Unfaulted phase currents are non-zero Even though phases B and C carry no current at the fault point itself, branches parallel to the faulted path carry finite phase B and C currents equal to . Relay engineers call this "through fault current" and must verify it does not cause spurious pickup on phase overcurrent elements.
4.9

Fault Study Workflow

  1. Define operating conditions: establish maximum and minimum connected generation scenarios.
  2. Choose system base: select a system MVAb and nominal kVb for each voltage level.
  3. Convert all plant to per-unit: generators (), transformers, overhead lines, cables.
  4. Build sequence networks: construct Z1 (with sources), Z2 (passive, topology = Z1), Z0 (passive, different topology based on earthing).
  5. Inject fault at each relaying point: calculate using the appropriate sequence interconnection.
  6. Calculate distribution factors: find and for each relay branch; derive phase currents.
  7. Calculate voltage distribution: determine V1, V2, V0 at each relaying node.
  8. Select and set protection: use the current/voltage boundaries to choose relay type, set pickup, and verify grading margins.
Software vs. hand calculation Modern power system studies use simulation software (PSS/E, PSCAD, DigSILENT). However, a hand calculation of at least the maximum three-phase fault and the minimum A-E fault is essential to validate software output and to understand relay setting sensitivities. An engineer who cannot perform the hand check cannot confidently audit a computer result.
Worked Example

Phase Current Distribution in Branch OB

Given: A-E fault at Bus A. Pre-fault voltage V = 63.5 V (phase-neutral, secondary). System sequence impedances at the fault: . Branch OB distribution factors: , .

Step 1: Sequence currents at fault (networks in series for A-E)
and
Step 2: Phase A current in branch OB
Step 3: Phase B and C in branch OB
Engineering interpretation The 63.5 V reference is the secondary of a 110 V (L-L) VT: . Phase A in branch OB carries 26.7 A at -90 deg. Phases B and C carry 8.1 A at +90 deg. The opposite phase angles show B/C current flows against the fault current direction. An overcurrent relay on phases B or C in this branch will see 8.1 A of "through fault" current and must not be set below this value if security against false tripping is required.
Module 04

Knowledge Check