Application of the PSS Tuning Concepts to a Multi-Machine Power System
10.9 Correlation between small-signal dynamic performance and that following a major disturbance
10.9.1 A transient stability study based on the fourteen-generator system
We shall examine the dynamic behaviour of the simplified fourteen-generator system of Figure 10.1 to a three-phase fault at a major busbar on the high-voltage side of a large power station, i.e. busbar #206 at BPS_2. Because there is no line switching or other system chang- es associated with this busbar fault, which is cleared in 0.120 s, the system configuration and
Generator Generator output Intra-station modes
MW Mvar PSSs off PSSs on Mode shift
SPS_4 no. 1 330 4.5
SPS_4 no. 2 320 3.4
SPS_4 no. 3 310 2.4
All PSS damping gains set to 20 pu on machine MVA rating
†
0.20j12.0 0.24j12.0
2.79 – j13.3
2.80 – j13.3
2.99j1.36 –
3.04 – j1.35
†
steady-state operating conditions in the post- and pre-fault periods are the same. The system modes are therefore unchanged.
In order to reveal features of the dynamic responses following the clearance of the fault, the low value of the damping gain of 5 pu on machine MVA rating is adopted for all the PSSs.
As is seen in Table 10.14 or Figure 10.26(a) for Case 1, a heavy load condition, the system is stable and the real parts of the rotor modes lie between -0.05 and -1.00. The mode behav- iour shown in the table does not differ significantly from that of Table 10.3 when all PSSs are out of service.
Table 10.14 Behaviour and type of the rotor modes, Case 1;
damping gain of PSSs is 5 pu.
The responses of speed perturbations about synchronous speed following the incidence of the three-phase fault are shown in Figure 10.32 for selected generators. As stated, the fault occurs at the 330 kV bus at BPS_2 (bus 206) and is cleared in 0.120 s. The responses are divided into three time intervals so that the various features of the modal behaviour in each interval can be examined; the time intervals are (a) 0 to 7 s, (b) 7 to 16 s, (c) 16 to 30 s. (Note the changes of scales on both axes.)
Mode
Mode Behaviour Mode Type
No. Real Imag
A B C D E F G H I J K
L M
-0.68 -0.39 -0.42 -1.00 -0.68 -0.88 -0.81 -0.40 -0.64 -0.92 -0.18 -0.05 -0.14
10.47 9.65 9.06 8.73 8.38 8.27 7.80 7.82 7.83 7.48 3.93 2.57 1.98
0.065 0.041 0.046 0.114 0.081 0.106 0.103 0.052 0.082 0.123 0.046 0.021 0.073
VPS_2<-->EPS_2 SPS_4<-->CPS_4, GPS_4 BPS_2<-->EPS_2, VPS_2 NPS_5<-->TPS_5 CPS_4, SPS_4<-->TPS_4, GPS_4, HPS_1, EPS_2<-->MPS_2, LPS_3 HPS_1, MPS_2<-->EPS_2, BPS_2
TPS_4<-->GPS_4, SPS_4, MPS_2 YPS_3, MPS_2, HPS_1<-->LPS_3, EPS_2
PPS_5<-->TPS_5, NPS_5 Area 3 <--> Area 5, Area 2 Area 4, Area 5 <--> Area 2 Area 5, Area 3 <--> Area 4
Local Area
“
“
“
“
“
“
“
“ Local Area
Inter-area
“ Inter-area
<--> means ‘... swings against ...’. Generators or areas are listed under ‘Mode Behaviour’ are in descending order of their participation factors.
Sec. 10.9 Small- and large-signal dynamic performance 511
Figure 10.32 Rotor speed perturbations of selected generators following a 3-phase fault
Rotor speed perturbations (pu)
0 1 2 3 4 5 6 7
-200 -150 -100 -50 0 50 100
150 x10-4
BPS_2, EPS_2 & HPS_1 respond to the local mode . Although remote from the fault, PPS_5 is excited by inter-area mode .
0.42
– j9.06
0.05
– j2.57
(a)
16 18 20 22 24 26 28 30
-1 0 1 2 3 4 5 6 7
Time (s)
BPS_2 EPS_2 GPS_4 LPS_3 HPS_1 PPS_5 Except for LPS_3,the responses are dominated by the inter-area mode . LPS_3 responds to the inter-area mode .
0.05
– j2.57
0.18
– j3.93
(c) x10 -4
7 8 9 10 11 12 13 14 15 16
-10 -5 0 5 10 15
Mode decays and BPS_2, EPS_2 & HPS_1 re- sponses merge into the inter-area mode . LPS_3 response is that of inter-area mode .
0.42 – j9.06
0.05 – j2.57
0.18 – j3.93
(b) x10-4
During the interval 0 to 4 s the responses shown in Figure 10.32(a) for selected rotor speed perturbations is dominated by the mode C, , subject to the caveat discussed later. The phase relationship between the principal participants in the response appears close to that predicted by the mode shape in Figure 10.33. Although remote from the faulted bus, PPS_5 is excited by the inter-area mode L, , in which machines in Area #2 also participate, as revealed in Figure 10.33. The same comment applies to LPS_3 with respect to the inter-area mode K, .
During the interval 7 to 14 s shown in Figure 10.32(b) the responses principally associated with mode C, , decay away and merge into the modal behaviour revealed in the mode shape in Figure 10.33 for the inter-area mode L, . After 16 s, except for LPS_3, the machines participate in the slowly decaying mode L, with a 5% settling time of ~56 s. LPS_3 continues to participate in the more rapidly-decaying mode K,
, (see Figure 10.33(c)).
Figure 10.33 Participation factors and mode shapes for the principal modes in the re- sponse, the local-area mode C ( ) and the inter-area modes K ( ) and L ( ); all PSS damping gains in Case 1 are all set to 5 pu on machine MVA
rating. (<*PS_area> refers to all generators in the numbered area.) 10.9.1.1 Benefits of small-signal analysis of large power systems
This example demonstrates how small-signal analysis complements that based on transient stability studies.
0.42 – j9.06
0.05j
– 2.57
0.18 – j3.93
0.42 – j9.06
0.05j
– 2.57
0.18
– j3.93
* *
* *
*
*
(a) (b) (c)
0.42
– +j9.06 –0.18+j3.93
0.05 – +j2.57
Sec. 10.9 Small- and large-signal dynamic performance 513 The above example reveals the important features of small-signal analysis, that is, it furnish- es not only an understanding of the underlying modal structure of the power system and but also provides insights into a system's dynamic characteristics that cannot easily be derived from time-domain simulations for large magnitude disturbances. It is the case in Figure 10.33 that only a few of the thirteen modes appear to be excited; the nature and lo- cation of the fault does not significantly excite the local-area modes outside the faulted area at all. Understanding the nature of the small-signal modal behaviour therefore yields a syn- optic view of the system characteristics which would require many large-signal studies of faults in different locations to gain similar, but not exact, information [9].
Knowledge of the behaviour of certain local and inter-area modes has revealed the nature of the responses of the speed states following a major disturbance on the system. However, as stated earlier, the behaviour of the system is highly non-linear during the initial phase of the response. During the first 0.6 s certain exciters reach their ceiling voltages and some PSSs, together with most SVCs, hit limits on their outputs. In the context of the magnitude of rotor speed oscillations, the question is asked in Section 1.10, “how small is small?”. The peak amplitudes of the speed perturbations in Figure 10.32(a) are 1.5 to 2% which are not small. The functional non-linearities come into play and therefore the small-signal analysis is based is not strictly accurate. In the following section the applicability and validity of the small-signal analysis that has been conducted in this section is reviewed.