Transient Stability-Consideration of Time
The Swing Curve
In the previous section, attention has been mainly directed towards the determination of the angular position of the rotor; in practice, the corresponding times are more important. The protection engineer requires allowable times rather than angles when specifying relay settings. The solution of equation (8.1) with respect to time is performed by means of numerical methods and the resulting time-angle curve is known as the swing curve. A simple step-by-step method will be given in detail and references will be made to more sophisticated methods used for digital computation.
In this method the change in the angular position of the rotor over a short time interval is determined. In performing the calculations the following assumptions are made:
- 1. The accelerating power DP at the commencement of a time interval is considered to be constant from the middle of the previous interval to the middle of the interval considered.
- 2. The angular velocity is constant over a complete interval and is computed for the middle of this interval.
These assumptions are probably better understood by reference to Figure 8.6. From Figure 8.6,
The change in d over the (n — 1)th interval, that is from times (n — 2) to (n — 1)
as v _{3} is assumed to be constant
^{n—} 2
Figure 8.6 (a), (b), and (c) Variation of DP, v and DS with time. Illustration of step-bystep method to obtain S-time curve
Figure 8.7 Discontinuity of Д? in middle of a period of time
Over the nth interval,
From the above,
It should be noted that Ad_{n} and A<5_{n}_j are the changes in angle.
Equation (8.7) is the basis of the numerical method. The time interval At used should be as small as possible (the smaller At, however, the larger the amount of labour involved), and a value of 0.05 s is frequently used. Any change in the operational condition causes an abrupt change in the value of AP. For example, at the commencement of a fault (t = 0), the value of AP is initially zero and then immediately after the occurrence it takes a definite value. When two values of AP apply, the mean is used. The procedure is best illustrated by an example.
Example 8.3
In the system described in Example 8.2 the inertia constant of the generator plus turbine is 2.7 p.u. Obtain the swing curve for a fault clearance time of 125 ms.
Solution
A time interval Dt — 0.05 s will be used. Hence The initial operating angle
Just before the fault the accelerating power DP — 0. Immediately after the fault,
The first value is that for the middle of the preceding period and the second is for the middle of the period under consideration. The value to be taken for DP at the commencement of this period is (0.78/2), that is 0.39 p.u. At t — 0, S — 33.8°.
Dt2
Dt3
The fault is cleared after a period of 0.125 s. As this discontinuity occurs in the middle of a period (0.1-0.15 s), no special averaging is required (Figure 8.7).
If, on the other hand, the fault is cleared in 0.15 s, an averaging of two values would be required.
Figure 8.8 Typical swing curve for generator
From t — 0.15 s onwards,
P — 1 —1.3 sin S (note change to P-S curve of Figure 8.5)
Dt_{4}
Dt5
Dt6
If this process is continued, S commences to decrease and the generator remains stable.
If computed by hand, a tabular calculation is recommended, as shown in Table 8.1
This calculation should be continued for at least the peak of the first swing, but if switching or auto-reclosing is likely to occur somewhere in the system, the calculation of S needs to be continued until oscillations are seen to be dying away. A typical swing curve shown in Figure 8.8 illustrates this situation.
Different curves will be obtained for other values of clearing time. It is evident from the way the calculation proceeds that for a sustained fault, S will continuously increase and stability will be lost. The critical clearing time should be calculated for conditions which allow the least transfer of power from the generator. Circuit breakers and the associated protection operate in times dependent upon their design; these times can be in the order of a few cycles of alternating voltage. For a given fault position a faster
Table 8.1 Tabular calculation of dn
t(s) |
DP |
M |
Adn |
dn |
0- |
0.00 |
— |
_{—} |
33.8 |
0+ |
0.78 |
— |
— |
33.8 |
0.05 |
0.39 |
3.25 |
3.25 |
37.05 |
0.1 |
0.76 |
6.33 |
9.58 |
46.63 |
0.15 |
0.71 |
5.91 |
15.49 |
62.12 |
0.2 |
-0.149 |
-1.24 |
14.25 |
76.37 |
0.25 |
-0.26 |
-2.17 |
12.08 |
88.39 |
Figure 8.9 Typical stability boundary
clearing time implies a greater permissible value of input power P_{0}. A typical relationship between the critical clearing time and input power is shown in Figure 8.9 - this is often referred to as the stability boundary. The critical clearing time increases with increase in the inertia constant (H) of turbine generators. Often, the first swing of the machine is sufficient to indicate stability.