Application of the PSS Tuning Concepts to a Multi-Machine Power System
10.3 Eigen-analysis, mode shapes and participation factors of the 14- generator system, no PSSs in service
10.3.2 Application of Participation Factor and Mode Shape Analyses to Case 1 Consider in Figure 10.2 the unstable, oscillatory mode (designated ‘Mode L’)
Let us view the plots not only of the magnitudes of its participation factors (PFs) but also of its mode shape (MS); the plots are shown in Figure 10.3.
Figure 10.3 Magnitude of the participation factors (left) and the mode shape (right) for the unstable mode L, . No PSSs are in service.
(In the plot of the participation factors ‘W’ and ‘DEL’ are the rotor speed and angle pertur- bations, respectively.)
Recall that the concepts of participation factors (PFs) and mode shapes (MSs) were dis- cussed in Chapters 3 and 9. In this case the participation factor is the participation of the states in the selected mode arranged in decreasing values of the magnitude of the PFs. For the selected mode the mode shape is the plot of the normalised magnitude and phase of the
0.088+j2.60
0.088+j2.60
Sec. 10.3 Eigen-analysis, mode shapes and participation factors 483 right speed eigenvectors and reveals, for example, that a group of generators swing with - or against - another group of machines.
According to the plot of the PFs in Figure 10.3 the two states, rotor speed and angle, of a number of generators dominate the involvement of the states in mode L; this mode is there- fore an electro-mechanical mode. There are a total of 125 states in this system model. The MS reveals that the generators in Areas 5 and 4 swing against those in Areas 1 and 2; the participation factors of those in Area 3 are small. Mode L is therefore classified as an ‘inter- area’ mode. However, note that:
• although the magnitude of the MS phasor of generator NPS_5 is the second largest, the PF of its speed state is the twelfth largest;
• as highlighted in Section 9.2.2, some care should therefore be attached to interpreting the lengths of the MS phasors. The length of the MS phasors for some generators is shorter than for others because their inertias on system MVA base may be signifi- cantly greater. In Figure 10.3, for example, the relative lengths of the MS phasors for BPS_2 and PPS_5 are in the ratio 0.274:1, the ratio of their inertias is 2.56:1. The most useful feature of the MS plot is therefore the relative phase information that it pro- vides.
• the PFs for mode L are nearly real (e.g. PF is for both the speed and rotor angle states of PPS_5). When in a later Chapter 13 we analyse the mode shift contributed by the PSS of a given generator we shall find that the complex value of its PF plays a major role [4].
For a second unstable mode, , the PF plot in Figure 10.4 reveals that this mode is also an electro-mechanical mode; the MS shows that generators in Areas 3 and 1 swing against machines in Areas 2 and 5. This mode, called ‘K’, is also an inter-area mode.
For reference in later studies the PFs and MS for the third inter-area mode (‘M’) are shown in Figure 10.5. In this case Areas 5, 3 and 2 swing against Area 4.
0.101+j0.013
0.115+j3.97
Figure 10.4 Participation factors and mode shape for the unstable mode K ( ), no PSSs are in service.
Figure 10.5 Case 1. Participation factors and mode shape for the inter-area mode M ( ), no PSSs are in service.
0.115+j3.97
(a) (b)
0.016 – +j2.03
Sec. 10.3 Eigen-analysis, mode shapes and participation factors 485 To determine the nature of other lightly-damped or unstable oscillatory modes with fre- quencies between 7 and 11 rad/s, shown in Figure 10.2, the plots of their MSs and PFs are examined. Such plots are displayed in Figure 10.6; each plot reveals a rotor mode of oscilla- tion. All three are found to be local-area modes:
• in mode VPS_2 swings against EPS_2;
• in the unstable mode SPS_4 swings mainly against CPS_4 and GPS_4;
• in the unstable mode BPS_2 swings mainly against EPS_2, VPS_2 and TPS_4.
Figure 10.6 Case 1. Participation factors and mode shapes for three lightly-damped
modes, A, B & C, respectively ; the latter
two modes are unstable.
The behaviour and type of the thirteen electro-mechanical modes in the fourteen machine system are summarised in Table 10.3.
Though not shown in the eigen-plot of Figure 10.2 there is an oscillatory mode at about which could be of interest since it lies in the frequency range of the rotor modes.
In the PF plot, shown in Figure 10.7, it is observed that the states mainly participating in the mode are associated with the direct axis of the generators at NPS_5, i.e. the field voltage and the AVR. Thus, this mode is likely to be a controller mode associated with the AVR and
0.17 – +j10.4
0.11+j9.58 0.04+j8.96
0.17
– +j10.4 0.11 +j9.58 0.04 +j8.96
1.4
– j2.8
generator dynamics of NPS_5. Because such an examination of the PFs of a selected mode quickly establishes the nature of the mode, participation factor analysis proves to be a very useful tool.
Table 10.3 Behaviour and type of the electromechanical modes, Case 1;
no PSSs in service
Figure 10.7 Participation factor plot for an oscillatory mode that participates mainly in states associated with the direct axis of NPS_5.
The analysis of the behaviour of the electro-mechanical modes demonstrated above for Case 1, Table 10.3, is repeated for the other cases 2 to 6 for all PSSs out of service. In Tables 10.4, 10.15 and 10.16 the modes for each case are sorted such that each row contains the
Mode
Mode Behaviour Mode Type
No. Real Imag
A B C D E F G H I J K L M
-0.17 0.11 0.04 -0.56 -0.26 -0.61 -0.44 0.01 -0.19 -0.62 0.12 0.09 -0.02
10.44 9.58 8.96 8.63 8.37 8.05 7.96 7.81 7.72 7.43 3.97 2.60 2.03
0.02 -0.01 -0.01 0.06 0.03 0.08 0.06 -0.00 0.02 0.08 -0.03 -0.03 0.01
VPS_2<-->EPS_2, BPS_2 SPS_4<-->CPS_4, GPS_4 EPS_2, VPS_2<-->BPS_2
NPS_5<-->TPS_5 CPS_4, SPS_4<-->GPS_4, TPS_4, HPS_1, MPS_2<-->EPS_2, VPS_2, LPS_3 MPS_2, HPS_1<-->EPS_2, BPS_2, VPS_2 TPS_4<-->GPS_4, SPS_4, MPS_2 YPS_3, MPS_2<-->LPS_3, EPS_2
PPS_5<-->TPS_5, NPS_5 Area 3 <--> Area 5, Area 2 Area 5, Area 4 <--> Area 2 Area 5, Area 3 <--> Area 4
Local Area
“
“
“
“
“
“
“
“ Local Area
Inter-area
“ Inter-area
<--> means ‘... swings against ...’. - damping ratio.
In ‘Mode Behaviour’, generators or areas are listed in descending order of their participation factors.
Sec. 10.4 P-Vr characteristics of the generators 487 modes of the same behaviour and type. For example,. in Table 10.4 the modes ‘J’ in row 10 for Cases 3 and 4, and , respectively, are modes in which the same generators are the main participants and both are local-area modes. This type of infor- mation will prove useful in a later chapter.
Table 10.4 Rotor modes of oscillation and damping ratios, Cases 3 and 4, peak and light loads. No PSSs in service