Analyzing Faults with Symmetrical Components

## Basic Concepts

The basic reason for sequence components is to shift the analysis of unbalanced faults to a separate domain to make it easier to study the effects of unbalanced faults. The new domain, sequence space, was chosen due to each of the components having some meaning. Positive sequence represents the phase rotation that is associated with the system during normal operation. The choice of it being A-B-C or C-B-A (phase rotation can be changed by just swapping two phases) is an arbitrary decision. Negative sequence represents the reverse rotation of positive sequence. Zero sequence represents a set of phasors that are all equal in magnitude and angle and can be looked at as the separation of the neutral point of positive and negative sequence set of phasors from ground. Zero sequence can't change the line-to-line phasors because it shifts each line-to-neutral equally.

All faults that produce ground current create a zero sequence potential difference that causes zero sequence or ground currents to flow. During faults, the zero sequence current can represent the tightness of the neutral to ground. In a solidly grounded system, there usually is a lot of available fault current for ground faults due to the thevenin impedance of the zero component being small and the positive and negative sequence thevenin impedences being small by design to prevent large voltage drops. In an ungrounded system, the thevenin zero sequence impedance is infinite (not exactly due to capacitive grounding in the system) and the neutral point will separate from the ground easily. High resistance grounded systems provide the best of both worlds by providing enough ground fault current to make it easy to locate the fault while at the same time reducing the magnitude of the ground fault currents to protect equipment and personal and preventing over voltages due to line-to-neutral voltages increasing to line-line voltages.

The interconnections of the sequence networks are based on mathematical properties of each fault and are not intuitive. For example, for a single line to ground fault there will only be fault current in one phase, the faulted phase. That is a property of that type of a SLG fault. If you transfer the phasors for a SLG fault to the sequence domain, you will find I1=I2=I0. The sequence network connection that represent represent all three components having the same sequence currents (I1=I2=I0) is for all three networks to be connected in series. This then is the sequence network connection for a SLG fault because it represents how I1,I2, and I0 are related in the sequence domain. The connection of the sequence networks ,basically, represents the transformation of the characteristics of the unbalanced fault in the phase domain to the sequence domain. The sequence network connections for the other types of unbalanced faults can be derived similarly. For multiple contingency faults, the networks get pretty complex and using something like the Clarke's transformation might allow for simpler network connections.

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For a phase to phase fault, all the current the goes out on one phase returns the other phase, Ia = -Ib. This in the sequence domain is represented by I1 = -I2. This network involves only the positive and negative sequence portions and will be the positive sequence thevein network connected in series with the negative sequence network with reversed polarity. This fault is independent of the grounding of the system (ungrounded, high impedance grounding, or ungrounded) since it doesn't involve the zero sequence network.

Ground faults with a few exceptions need a faulted ground connection to produce ground current. One exception will occur if the system has untransposed lines. Untransposed lines have the effect of making the transmission line impedances not the same for all three lines. This will create imbalance and produce zero sequence flows from on grounded connection to another. The magnitude of these normally flowing zero sequence currents will be small in comparison to the available ground fault current but maybe should be taken into consideration when setting the pickup for the ground overcurrent relaying. Imbalance on the phases due to unbalanced load can also produce zero sequence current flows. This though at least at the transmission level usually is blocked by a transformer connection that prevents zero sequence currents from passing through. The effect that unbalanced loads have in this case is in introducing negative sequence currents on the system.

An open phase is another exception that will produce ground current. This will create an imbalance and cause zero sequence current to flow in a grounded system. The magnitude of these ground currents will be dependent on the loading or power flows in the system. It might be possible to only detect the open phase after the system loading has increased enough to produce sufficient ground current. For that reason, it may be preferable to detect an open phase with negative sequence overcurrent ,which also has load depedancy issues, or with using a ratio of positive sequence current to negative sequence currents, which has load independence.

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Here is an example of an ungrounded system (neutral lacking a hold on ground) separating for a SLG fault. During a SLG fault on an ungrounded system, the ground point shifts from the neutral to the the point of the faulted phase's phasor. If you look at the diagram below, no current (I1,I2,I0) can flow for the fault since the zero sequence network provides no return path.  The voltages on the ungrounded system ,however, will change as the result of the fault and this can be seen by transferring the sequence network voltages at H ,in the diagram below, from the sequence domain to the phasor domain.  The voltages below at point H are V(positive) = E1G, V(negative) = 0, and V(zero) = -E1G.  If these voltages are transferred from the sequence domain to the phase domain, the ground has shifted to the faulted phase and the line-to-neutral voltage on the other phases has increased to the line-to-line voltages and the angle between the two line-to-ground phasors has reduced from 60 to 30 degrees . No negative sequence voltage developed to change the line-to-line voltages ,which are unaffected by zero sequence, and the change was due entirely to the neutral shifting from ground due to the introduction of zero sequence voltage.

Another important item to keep in mind is the sequence networks for the different types of transformers. During unbalanced faults, a transformer can source, pass, or block zero sequence currents depending on how the transformers are grounded and connected. A delta on a transformer will never be able to pass zero sequence current due to it being connected line to line but is necessary on a wye-grounded-delta transformer to make the wye a ground source. The ground currents that flow up the grounded wye circulate on the delta side bookending the zero sequence current and making the wye side into a ground source. From a protection point of view,  the delta on a  delta-wye-grounded transformer also provides the benefit of isolating each side's ground currents.  Neither side can pass zero sequence currents through the transformer so the ground protection on each side can be coordinated independently. A wye-ground-wye-ground transformer will only pass zero sequence currents but can't act as a ground source (yeah, it looks kind of weird to have ground-wye's on both sides but no ground source). A delta-delta transformer cannot source or pass zero sequence currents. Here is a table for the sequence networks for all the two and three winding transformers.

Another reason for having a delta is that odd triplen harmonics on the system will behave like zero sequence currents and having something like a delta-wye-ground transformer nearby can provide a path for them to flow so that they don't cause problems on the system.  Odd triplen harmonics can be generated by non-linear loads like  VFDs, lighting ballast, and transformer excitation current. The delta also can be used in line-to-line connected loads to trap the odd triplen harmonics in the delta.

**Good References:**

* FREE Power Quality Teaching Toy by Alex McEachern of Power Standards Lab- I couldn't recommend this more. It beats playing with your calculator to try to get a rough idea of how the sequence and phase domains relate to each other.
* Symmetrical Components for Power Systems Engineering, J.L. Blackburn
* Protective Relaying: Principles and Applications J.L. Blackburn
* *Power Systems Analysis*, J.J. Grainger, W.D. Stevenson, McGraw-Hill, © 1994
* ABB's Electrical T&D Ref - Red Book - this is the old Westinghouse T&D book that is being reprinted by ABB. This is a neat book if you are interested in how things used to be done.
* Sequence Network Connections for Faults
* Presentation to IAS "Power System Calculations - Part 2" by Kurt Ederhoff
* SEL "Tutorial on Symmetrical Components Part 1" by Ariana Amberg and Alex Rangel
* SEL "Tutorial on Symmetrical Components Part 2" by Ariana Amberg and Alex Rangel