The purpose of the fault analysis (also called short-circuit study or analysis) is to calculate the maximum and minimum fault currents and voltages at different locations of the power system for various types of faults so that the appropriate protective schemes, relays, and circuit breakers can be selected in order to rescue the system from the abnormal condition within minimum time.
In practice, to perform the fault analysis, the following simplifying assumptions are usually made:
1. The normal loads, line-charging (i.e., shunt) capacitances, shunt elements in transformer equivalent circuits for representing magnetizing reactances or core loss, and other shunt connections to the ground are neglected.
2. All generated (i.e., internal) system voltages are equal (in magnitude) and are in phase.
3. Normally, the series resistances of lines and transformers are neglected if considered small in comparison with their reactances.
4. All the transformers are considered to be at their nominal taps.
5. The generator is represented by a constant-voltage source behind (i.e., in series with) a proper reactance that may be subtransient (Xd′′), transient (Xd′), or synchronous, at steady state (Xd) reactance. Usually, the subtransient reactance (Xd′′) is selected for the positive- sequence reactance. Therefore, such a representation is sufficient to calculate the magni- tudes of the fault currents in the first three to four cycles after the fault takes place.
The first assumption is based on the fact that the fault circuit has predominantly lower impedance than the shunt impedances. Therefore, the saving in computational effort due to this assumption usually justifies the slight loss in accuracy.
The second assumption results from the first assumption. With the first assumption, the power system network becomes open-circuited, and therefore the normal load currents (i.e., prefault cur- rents) are consequently neglected, and thus all the prefault bus voltages will have the same magni- tude and phase angle.
Thus, in per-unit analysis, the prefault bus voltages are set equal to 1.0∠0° pu. In the rare event that taking into account for load current is desirable, this can be done by applying the superposition theorem. The third assumption is usually done for hand calculations and educational purposes. With this assumption, the power system network will contain only reactances, and therefore the system can be represented by its most simplified reactance diagram. However, this assumption is not neces- sary if the computation will be done using a digital computer.
The fourth assumption neglects the transformer tapings so that the fault analysis can be carried out in a per-unit system. Thus, with this representation, transformers will be out of circuit. As men- tioned before, the subtransient reactance, Xd′′, is usually selected as the positive-sequence reactance.
The value of the negative-sequence reactance is slightly different than the positive-sequence reac- tance for the salient-type machines.
In the event that the generator is a nonsalient type (i.e., a cylindrical rotor machine) and if the subtransient reactance (X′′d) is selected for the positive-sequence reactance, the negative-sequence reactance becomes identical to the positive-sequence reactance, as shown in Table 4.1. In practice, to calculate the maximum fault currents, it is common to make an assumption that the fault is
“bolted,” that is, one having no fault impedance (Zf = 0) resulting from fault arc. (This assumption not only simplifies the fault calculations but also provides a safety factor since the calculated values become larger than the ones calculated using a fault impedance value.)
In the event that the fault has a short-circuit path that is not a metallic path but an arc or a path through the ground, nonlinear impedances are included that tend to inject harmonics into the cur- rent and/or voltage. Fault resistance has two components: the resistance of the arc [3] and the resis- tance of the ground [4]. If the fault is between phases such as line to line, the fault includes only the arc. Thus, the fault resistance includes only the arc resistance. Fault arc resistance is given by Warrington [5] as
Rarc=8750I ×
1 4
. Ω (4.2)
where
ℓ = length of arc in still air in feet I = fault current in amperes
If time is involved, the arc resistance is calculated from
R d vt
arc=8750I ×3
1 4
( )
. Ω (4.3)
where
d = conductor spacing in feet (from Figure 4.3) v = wind velocity in miles per hour
t = duration in seconds
The relation between the fault current and the arc voltage is given as
R d vt
arc=8750I ×3
0 4
( )
. Ω (4.4)
If a high-resistance line-to-ground fault occurs, the important impedances in the fault circuit are the contact to ground and the path through it. Warrington [5] has shown that the resistance of the ground fault is somewhat nonlinear partly because there are small arcs between conducting particles and partly due to the compounds of silicon, carbon, etc., which have nonlinear resistance (see Figure 4.3). It is interesting to note that, in practice, such fault resistance is erroneously called the fault impedance, and it is assumed to include a fictitious reactive component.
In general, the fault types that may occur in a three-phase power system can be categorized as follows.
25
20
15
10
5
Feet
100 200 300
kV Conductor sp
acing
Numb er of insu
lator s (suspen
sion )
Insulator string length
Arcinghorngap length
FIGURE 4.3 Minimum arcing distances on overhead lines (based on average tower dimensions in the United Kingdom and the United States). (From Warrington, A. R. van C., Protective Relays: Their Theory and Practice, vol. 2. Chapman & Hall, London, 1969.)
A. Shunt faults:
1. Balanced (also called symmetrical) three-phase faults a. Three-phase direct (L–L–L) faults
b. Three-phase faults through a fault impedance to ground (L–L–L–G) 2. Unbalanced (also called unsymmetrical) faults
a. Single line-to-ground (SLG) faults b. Line-to-line (L–L) faults
c. Double line-to-ground (DLG) faults B. Series faults:
1. One line open (OLO) 2. Two lines open (TLO)
3. Unbalanced series impedance condition C. Simultaneous faults:
1. A shunt fault at one fault point and a shunt fault at the other 2. A shunt fault at one fault point and a series fault at the other 3. A series fault at one fault point and a series fault at the other 4. A series fault at one fault point and a shunt fault at the other
Shunt faults are more severe than series faults. Balanced faults are simpler to calculate than unbalanced faults. Simultaneous faults, involving two or more faults that occur simultaneously, are usually considered to be the most difficult fault analysis problem. In this chapter, only balanced faults and series faults are reviewed; unbalanced faults will be reviewed in Chapter 6. The prob- ability of having a simultaneous fault is much less than the shunt fault. Therefore, the discussion of the simultaneous faults is kept beyond the scope of this book. However, for those readers interested in the subject matter, the book by Anderson [6] is highly recommended.