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Short Circuit Current Calculations Using Symmetrical Components

Ralph Fehr, Ph.D., P.E.

Course Outline

This 8-hour online course presents the theory necessary for understanding and the procedures necessary for calculating short circuit currents.  The method of symmetrical components is used to analyze unbalanced fault conditions.  The concept of sequence networks is presented in a novel and intuitive way.

This course includes a multiple-choice quiz at the end, which is designed to enhance the understanding of the course materials.

Learning Objective

At the conclusion of this course, the student will:

• understand the theory and origin of symmetrical components;
• realize that symmetrical components are akin to force vector components;
• be able to convert phase quantities to symmetrical components;
• be able to convert symmetrical components to phase quantities;
• understand the electrical behavior of positive- and negative-sequence currents;
• understand the unique behavior of zero-sequence current;
• be able to construct a positive-sequence reactance diagram from a one-line diagram;
• be able to construct a negative-sequence reactance diagram from a positive-sequence reactance diagram;
• be able to construct a zero-sequence reactance diagram from a negative-sequence reactance diagram and machine phasing symbols from a one-line diagram;
• be able to construct the circuit model for a three-phase fault;
• be able to construct the circuit model for a line-to-ground fault;
• be able to construct the circuit model for a double line-to-ground fault;
• be able to construct the circuit model for a line-to-line fault;
• understand the advantages of using the per-unit system to do power system calculations;
• realize that the per-unit system is simply a transformation / inverse transformation process much like Laplace and Fourier transforms;
• know the four per-unit base quantities that must be defined;
• understand the relationships and dependencies between the per-unit base quantities;
• be able to convert electrical quantities to per-unit quantities;
• be able to convert per-unit quantities to electrical quantities;
• be able to convert a per-unit quantity defined on one set of bases to a different set of bases;
• be able to analyze a three-phase fault circuit model to determine the short-circuit current;
• be able to analyze a line-to-ground fault circuit model to determine the short-circuit current;
• be able to analyze a double line-to-ground fault circuit model to determine the short-circuit current; and
• be able to analyze a line-to-line fault circuit model to determine the short-circuit current.

Intended Audience

This course is targeted to utility, plant, or consulting engineers and technicians involved in transmission or distribution system protection, planning, operations, or engineering.  Participants should have a proficiency in basic circuit analysis, familiarity with basic three-phase power system calculations, vector analysis, and complex arithmetic.

Benefit to Attendees

This course will enable attendees to properly analyze an electric power system under both balanced and unbalanced fault conditions.  These skills are essential when specifying electrical equipment and designing protection systems.

Course Introduction

In 1918, Charles Fortescue presented a paper to the American Institute of Electrical Engineers in Atlantic City describing how a system of n unbalanced but related phasors can be represented by n systems of balanced phasors.  Using this principle, any unbalanced three-phase system can be represented by three balanced sequence networks.

The theory of symmetrical components and the synthesis of sequence networks for three-phase power systems are instrumental for solving most unbalanced problems such as asymmetrical short-circuit and open-circuit faults.  Symmetrical components and sequence networks are also vital for understanding the unbalanced operating conditions of an otherwise balanced system, and the behavior and influence of harmonic voltages and currents.
Unfortunately, the theory of symmetrical components is often learned as a set of abstract algebraic equations, into which known values are substituted and, hopefully, out of which the correct answer emerges.  Sequence networks are often synthesized using a building block approach, where documented sequence impedance models of various power system elements are connected together much like building blocks to form the sequence network.  This method often leads to errors, since topological errors in connecting the blocks are easy to make, and topological errors in the sequence network will often produce inaccurate results.  But even if the networks are properly constructed, the engineer often lacks the insight and level of understanding to thoroughly comprehend the behavior of the system.

The novel approach for understanding symmetrical components and synthesizing sequence networks presented in this course enlightens the engineer to the reasons behind the behaviors observed on an unbalanced three-phase power system.  It is this enlightenment that allows the engineer to fully understand the behavior of the three-phase system under unbalanced conditions.  As an additional benefit to applying these approaches, commonly-made errors in unbalanced system calculations will be significantly decreased if not totally eliminated.

Many power system calculations involve analysis of a balanced three-phase system.  When this is the case, only one phase needs to be analyzed.  The symmetry of the problem allows the behavior of the other two phases to be determined based on the calculated behavior of the first phase.  This single-phase equivalent approach is taken to simplify the calculation process.

But when the conditions to be analyzed result in an unbalanced system of voltage and current phasors, the single-phase equivalent approach cannot be directly applied.  Such an example is determining the system response to an unbalanced short-circuit fault, such as a line-to-ground fault.  The option of analyzing the unbalanced system as a three-phase problem is not an appealing one, since the resulting mathematics would be cumbersome and very difficult to solve.  Using a single-phase approach would be possible if the unbalanced phasors could be resolved into balanced components.  Charles Fortescue’s theory of symmetrical components shows us that resolving an unbalanced set of voltage or current phasors into a set of balanced components is always possible.  Before developing the symmetrical components of an unbalanced set of three-phase phasors, let us look at a more straightforward example of resolving a vector into components.

Course Content

The course content is in a PDF file:

Course Summary

Symmetrical components are used to resolve unbalanced electrical systems, typically systems subjected to unbalanced short circuit faults, into balanced systems.  The balanced systems are called the positive, negative, and zero sequence networks.  The voltage across and the current through the positive sequence network are called the positive sequence voltage and positive sequence current, respectively.  Negative and zero sequence quantities are named similarly.

The positive sequence quantities relate to the generated voltages and currents.  The negative sequence quantities relate to the voltages and currents resulting from load imbalance, and the zero sequence quantities show the effects of ground on the system.

Determining the positive sequence network is done directly from the system impedance or one-line diagram.  The negative sequence network has the same topology as the positive sequence network, without the EMF sources.  The zero sequence network can only allow current to flow where a ground source is present, so open circuits and short circuits to the reference bus exist in the zero sequence network due to machine connections (delta, wye, or grounded wye).

Because of the widespread use of transformers in power systems, power calculations are more readily done using the per-unit system, which transforms electrical quantities to dimensionless quantities.  Base quantities are assigned for power and voltage, then current and impedance bases are calculated.  After the calculation is performed in the per-unit system, an inverse transformation restores the electrical units.

Balanced (three-phase) faults are calculated by analyzing the positive-sequence network.  Unbalanced short circuit faults (line-to-ground, double line-to-ground, and line-to-line) are analyzed by appropriately connecting the sequence networks and simplifying the resulting circuit. 