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.
where
Z R L L
= + = R
( 2 ω2 2 1 2)/ θ tan−1 ω
It can be seen in Equation 4.5 that the first term is a sinusoidal term and its value changes with time and that the second term is a nonperiodic term and its value decreases exponentially with time.
The second term is a unidirectional offset and is also called the direct current (dc) component of the fault current. It will in general exist, and its initial magnitude (i.e., at t = 0) can be as large as the magnitude of the steady-state current term, as shown in Figure 4.5a. If the fault occurs at t = 0 when the angle α – θ = –90°, the value of the transient current becomes twice the steady-state maximum value and can be expressed as
i t Vm t e
Rt
( )= −cos + L
−
2 ω (4.6)
and is shown in Figure 4.5a. The associated value of α is obtained from tanα
= −ωR
L (4.7)
On the other hand, if α = 0, at t = 0, the dc offset does not exist, as shown in Figure 4.5b, and the value of the transient current can be expressed as
i t V Zm t
( )= sinω (4.8)
Obviously, if a α − θ = π at t = 0, the dc offset current again cannot exist. Thus, the value of the transient current depends on the angle α of the voltage wave. However, the time of the fault cannot be predicted in practice, and therefore the value of cannot be known ahead of time. However, the dc component diminishes very fast, usually in 8−10 cycles.
Furthermore, since the voltages generated in the phases of a three-phase synchronous generator are 120° apart from each other, each phase will have, in general, a different offset.
Note that, in the aforementioned discussions, the value of L for the generator is assumed to be constant. In reality, however, the reactance of a synchronous machine varies with time immediately
(a) (b)
i Imax
Imax = 2(Idc, max) Idc
0
i
t 0 t
FIGURE 4.5 Balanced fault current wave shapes: (a) a – θ = −90°; (b) α = θ.
after the occurrence of the fault. Thus, it is customary to represent a synchronous generator by a constant driving voltage in series with impedance that varies with time.
This varying impedance consists primarily of reactance since X ≫ Ra, that is, X is much larger than the armature resistance. Hence, the value of impedance is approximately equal to its reactance.
For the purpose of fault current calculations, the variable reactances of a synchronous machine can be represented, as shown in Figure 4.6b, by the following three reactance values:
′′
Xd = subtransient reactance: determines the fault current during the first cycle after the fault occurs. In about 0.05−0.1 s, this reactance increases to
i(t)
i(t)
(a)
(b)
Subtransient Transient Steady state
0 0
I˝max
I˝max
I´max
I (t)max
I´max
Imax
Imax
X(t)
x(t)
Xd xd
t1 t2 t
t1 t2 t
xd' xd
x˝d x˝d
´
FIGURE 4.6 Balanced fault current and reactance for one phase of synchronous machine: (a) balanced instantaneous fault current without dc offset; (b) reactance X(t) vs. time with stepped approximation.
′
Xd = transient reactance: determines the fault current after several cycles at 60 Hz. In about 0.2−2 s, it reactance increases to
Xd = Xs = synchronous reactance: determines the fault current after a steady-state condition is reached
This representation of the machine reactance by three different reactances is due to the fact that the flux across the air gap of the machine is much greater at the instant the fault occurs than it is a few cycles later. Thus, when a fault occurs at the terminals of a synchronous machine, time is nec- essary for the decrease in flux across the air gap. As the flux lessens, the armature current lessens since the voltage produced by the air gap flux regulates the current.
Thus, the subtransient reactance Xd′′ includes the leakage reactance of the stator and rotor wind- ings of the generator, the influences of damper windings and of the solid parts of the rotor body being included in the rotor leakage. The subtransient reactance is also called the initial reactance.
The transient reactance Xd′′ includes the leakage reactance of the stator and excitation windings of the generator. It is usually larger than the subtransient reactance.
If, however, the rotor has laminated poles and yokes and no damper windings, the transient reactance is the same as the subtransient reactance. If, however, the rotor has laminated poles and yokes and no damper windings,* the transient reactance is the same as the subtransient reactance.
The synchronous reactance Xd is the total reactance of the armature winding, which includes the stator leakage reactance and the armature reaction reactance of the generator. It is much larger than the transient reactance Xd′′. Note that, all three reactances are considered to be the positive- sequence reactance of the synchronous machine.
In a salient-pole machine, the index d means that the reactances refer to a position of the rotor such that the axis of the rotor winding coincides with the axis of the stator winding so that the flux flows directly into the pole face. Therefore, it is called the direct axis, and thus the three reactances are also known as the direct-axis reactances.
In addition to these reactances, the generator also has reactances in the corresponding quadra- ture axis, that is, X′′ ′q, Xq and Xq, due to the flux path between poles, that is, midway between the field poles. The quadrature axis is 90 electrical degrees apart from the adjacent direct axes. However, the quadrature axis reactances are not relevant to the fault calculations.
Note that in a nonsalient-pole machine (i.e., cylindrical-rotor machine), values of Xd and Xq are basically equal. Therefore, there is no need to differentiate Xd from Xq but only call the synchronous reactance Xs. For the sake of simplification, in this book, all synchronous machines are assumed to have cylindrical rotors.
If the generator is operating at no load before the occurrence of a three-phase fault at its termi- nals, then the continuously varying symmetrical maximum current, Imax(t), and reactance can be approximated with the discrete current levels, as shown in Figure 4.6, so that
′′ = ′′
X E
d Imax max
(4.9)
′ = ′
X E
d Imax max
(4.10)
X E
d= Imax
max
(4.11)
*They are located in the pole faces of a generator and are used to reduce the effects of hunting.
where Emax is the no-load line-to-neutral maximum voltage of the generator. Alternatively, using the root mean square (rms) values,
′′ = ′′
I E X
g d
(4.12)
′ = ′ I E
X
g d
(4.13)
I E X
g d
= (4.14)
where
Eg = no-load line-to-neutral rms voltage
Iʺ = subtransient current,* rms value without dc offset Iʹ = transient current, rms value without dc offset
I = steady-state current, rms value
Note that, the importance of the reactances given by Equations 4.9 through 4.11 depends on what percentage they represent of the short-circuit impedance. For example, if the fault occurs right at the terminals of the generator, they are very important; however, if the fault is remote from the genera- tor, their importance is smaller.
The fault current will be lagging in power in a system where X ≫ R. Table 4.1 gives the typical values of the reactances for synchronous machines.
It is interesting to observe in Figure 4.6a that the total alternating component of armature current consists of the steady-state value and the two components that decay with time constants Td′ and Td′′. It can be expressed as
I I I t
T I I t
d T
ac= ′′ − ′ exp − exp
′′
+ ′ − −
′
( ) ( ) ++I (4.15)
where all quantities are in rms values and are equal but displaced 120 electrical degrees in the three phases.