Electrical Fault Calculation | Positive Negative Zero Sequence Impedance

Designing an electrical protection system starts with calculating how the electrical power system behaves during a fault. The results help engineers select relay settings and check whether equipment can withstand the expected fault duty.

The study determines maximum and minimum fault currents and the corresponding voltages at selected buses and branches. Their magnitudes and phase relationships support the correct application of each protection relay across the electrical power system. This process is known as electrical fault calculation.

A fault study calculates the current produced by a specified fault. The usual workflow has three main steps.

1. Choice of impedance rotations.
2. Reduction of complicated electrical power system network to single equivalent impedance.
3. Electrical fault currents and voltages calculation by using symmetrical component theory.

Impedance Notation of Electrical Power System

An electrical power system often operates at several voltage levels. For example, a generator may produce power at 6.6 kV, a step-up transformer may raise it to 132 kV for transmission and substations may reduce it to 33 kV, 11 kV and then 0.4 kV for distribution.

Impedances stated at different voltage levels cannot be combined directly. A fault calculation would become difficult and error-prone if every component value had to be converted separately whenever the calculation crossed a transformer.

The calculation becomes simpler when every component impedance is referred to a consistent set of base values. This technique is called impedance notation. Before an electrical fault calculation, the engineer selects the base quantities and expresses the model in ohms, percentage values or per-unit values on those bases.

Apparent power and voltage are commonly selected as base quantities. For a balanced three phase system, the base for three phase power is normally stated in MVA or kVA, while base voltage is the line-to-line value in kV. Base impedance follows from the selected power and voltage bases.

Per unit impedance is the ratio of actual impedance to base impedance.

A Percentage impedance is 100 times the corresponding per unit value.

A value must be converted when it is given on a different base from the one used in the study.

The chosen impedance notation must be applied consistently. A base voltage is selected for one part of the network, and the corresponding bases at other voltage levels follow transformer ratios.
A 132 kV base may be convenient for a network dominated by 132 kV lines, but 33 kV or 11 kV can also be used if all values are converted correctly for the fault calculation.

Network Reduction

After selecting the bases, convert the impedances of generators, overhead lines, cables and each transformer to the common base. The resulting impedance diagram shows which components and sources contribute to the fault point.

The network can then be reduced to an equivalent impedance by applying series, parallel and star-delta relationships. Unbalanced-fault studies require separate positive-, negative- and zero-sequence networks because each sequence can follow a different path.

Three phase faults are symmetrical when the three phases have equal fault impedances. Under that assumption, the fault is represented by the positive-sequence network alone. The three phase fault current is then obtained from the relation below.

Here, I_f is the fault current, v is the prefault phase-to-neutral voltage and Z₁ is the equivalent positive-sequence impedance to the fault point. The voltage and impedance units must be consistent.

Symmetrical Component Analysis

The preceding calculation assumes a balanced three-phase system and a symmetrical fault. One phase can represent the system because the voltage and current magnitudes are equal in all three phases, with the expected phase displacement.

Many faults in an electrical power system are unbalanced. Examples include phase-to-earth, phase-to-phase and double-phase-to-earth faults. Their phase voltages and currents are not symmetrical, so engineers solve them with symmetrical component analysis.

A set of unbalanced phase quantities in a three phase vector diagram can be resolved into three balanced component sets. The positive sequence has the normal phase rotation, the negative sequence has the opposite rotation and the zero-sequence quantities are in phase.

The phase and sequence quantities are related by the following equation.

The inverse transformation is shown below.

All quantities are referred to reference phase r.
The same transformations apply to sequence currents. The transformed voltage and current quantities are then used with the sequence networks.

In a balanced linear network, each sequence current produces symmetrical component analysis results and voltage drops of the same sequence. The positive-, negative- and zero-sequence networks can therefore be reduced separately before their fault-point connections are solved.

Let Z₁, Z₂ and Z₀ represent the equivalent positive-, negative- and zero-sequence impedances at the fault point. The following ideal-fault relations depend on the stated network and fault assumptions.
For earth fault

Phase to phase faults


Double phase to earth faults

Three phase faults

To find the current in a branch, solve the connected sequence networks, distribute each sequence current according to the branch impedances and transform the result back to phase quantities. Voltages at any bus can be found from the sequence currents and the impedance of each sequence network.

Sequence Impedance

Positive Sequence Impedance

A network’s positive sequence impedance is the impedance presented to positive-sequence current. This sequence normally represents a balanced three-phase fault.

Negative Sequence Impedance

A network’s negative sequence impedance is the impedance presented to negative-sequence current. This value is needed when the phase conditions are unbalanced.

Zero Sequence Impedance

The impedance that a network presents to zero-sequence current is its zero sequence impedance.
In the preceding equations, Z₁, Z₂ and Z₀ represent positive-, negative- and zero-sequence impedance. A usable zero-sequence model must include the available neutral or earth return path. Each sequence impedance also depends on the component being modelled.

1. In static and balanced power system components like transformer and lines, the sequence impedance offered by the system are the same for positive and negative sequence currents. In other words, the positive sequence impedance and negative sequence impedance are same for transformers and power lines.
2. But in case of rotating machines the positive and negative sequence impedance are different.
3. The assignment of zero sequence impedance values is a more complex one. This is because the three zero sequence current at any point in a electrical power system, being in phase, do not sum to zero but must return through the neutral and /or earth. In three phase transformer and machine fluxes due to zero sequence components do not sum to zero in the yoke or field system. The impedance very widely depending upon the physical arrangement of the magnetic circuits and winding.
4. The reactance of transmission lines of zero sequence currents can be about 3 to 5 times the positive sequence current, the lighter value being for lines without earth wires. This is because the spacing between the go and return(i.e. neutral and/or earth) is so much greater than for positive and negative sequence currents which return (balance) within the three phase conductor groups.
5. The zero sequence reactance of a machine is compounded of leakage and winding reactance, and a small component due to winding balance (depends on winding tritch).
6. The zero sequence reactance of transformers depends both on winding connections and upon the construction of the core.