Distance Relaying Fundamentals | What is Apparent Impedance?

In our introductory article on distance relaying, we presented a simple apparent impedance calculation for a phase-to-phase fault on a radial system without load and with fault impedance assumed to be zero. The calculation used the faulted phase voltages and currents based on the impedance fault loop from a phase-to-phase fault.

In this article, we will derive the impedance equations using symmetrical fault analysis. These impedance equations are widely recognized in distance relaying.

Derivation of Impedance Equations

Analyses in protective relaying are heavily based on the theory of symmetrical components and distance relaying is no exception. The derivation of the impedance equations for all fault types assumes a radial system with fault impedance assumed to be zero. The following derivations will be based on the network shown in figure 2.

The sample network shows the location of the distance relay. The relay continuously measures voltages and currents which are used in the calculation of the ‘apparent impedance’. The impedance ‘Z’ is designated as the impedance of the line with ‘mZ‘ representing a section of the line and where ‘m‘ is the per-unit distance to the fault, e.g. the distance to the fault in kilometers divided by the total line length in kilometers.

The calculation of impedance in terms of measured voltages and currents is referred to as ‘apparent impedance’ when discussing distance relaying.

F. Calero in “Distance Elements: Linking Theory and Testing”

Phase-to-Phase Faults

The symmetrical component network for a phase-to-phase fault is shown below.

Let us start by recognizing the following:

Using the above relationships, the apparent impedance can be calculated as a function of the positive and negative sequence component values of the voltages and currents.

The apparent impedance equation can be expressed in terms of the measured voltages, VR_B and VR_C, and currents, IR_B and IR_C, by using the following relationships.

The above equation applies to a B-C fault. For A-B, and C-A faults the following impedance equation applies.

Phase-to-Phase-to-Ground Faults

The foregoing analysis can be extended to a phase-to-phase-to-ground fault by recognizing that VF₁=VF₂=VF₀ from its symmetrical component network as shown in figure 4.

Therefore, the apparent impedance equation for a phase-to-phase-ground fault is the same as the phase-to-phase fault.

Three Phase Faults

The apparent impedance equation for phase-to-phase faults also applies to three phase faults by recognizing that only positive sequence voltages and currents are present.

Phase-to-Ground Faults

The analysis of phase-to-ground faults starts by recognizing from its symmetrical component network that VF₁+VF₂+VF₀=0,

and using the following relationships,

the apparent impedance equation can be derived.

where

The above equation is used by numerical relays that can obtain symmetrical components for the current during ground faults.

Other numerical relays calculate only the zero-sequence I₀ from the measured current. For this type of relay, the apparent impedance equation is derived as follows.

where

For electromechanical and static relays, and other numerical relays which uses the neutral or residual current 3I₀, the apparent impedance equation is derived as follows.

where

K₀, K’₀, K”₀ are referred to as the earth fault compensation factor or zero-sequence compensation factor.

The apparent impedance equation for B-Ground and C-Ground faults are derived similarly.

Summary

The impedance equations for different fault types is summarized below.

Take note that the above equations were derived by using a radial system with fault impedance assumed to be zero.

References

[1] G. Ziegler, “Numerical Distance Protection. Principles and Applications,” Publicis Erlangen, Zweigniederlassung der PWW GmbH. Germany, 2011.

[2] F. Calero, “Distance Elements: Linking Theory with Testing,” Schweitzer Engineering Laboratories, Inc., August 2009.

[3] K. Zimmerman, and D. Costello, “Impedance-Based Fault Location Experience,” Schweitzer Engineering Laboratories, Inc., July 2010.

[4] M. Ibrahim, “Case Studies Related to Overhead Transmission-Line System Disturbances,” in Disturbance Analysis for Power Systems, IEEE, 2012, pp.461-570, DOI: 10.1002/9781118172094.ch6.

[5] G. Pradeep Kumar, “Transmission Line Protection,” notes on Power System Protection Training, Visayan Electric Company, Cebu City, Philippines, 2016.

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