Application of the PSS Tuning Concepts to a Multi-Machine Power System
10.4 The P-Vr characteristics of the generators and the associated synthesized characteristics
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
10.4 The P-Vr characteristics of the generators and the associated
To avoid unnecessary complexity it should be noted in this analysis that the limited number of encompassing operating conditions on which the power flows - and thus the P-Vr char- acteristics - are based are normal operating conditions. In practice, the P-Vr characteristics for a relevant encompassing set of contingency conditions must be included in the deter- mining the synthesized characteristic.
Examination of Figures 10.8 to 10.21 reveals that, over the modal frequency range of 1 to 15 rad/s, the bands of P-Vr characteristics1 for any generator under normal operating con- ditions may possess the following features:
• Magnitude plots: The width of the bands is typically less than 6 dB; the variation about a characteristic lying in the centre of the band is therefore dB or less.
• Phase plots: The maximum width of the bands at the relevant frequency is typically less than ; the variation about a central characteristic is thus or less.
1. The word “characteristics” is shortened to “Xstics” in the following figure captions.
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15 7.5
Figure 10.8 P-Vr Xstics, HPS_1
Case 1 Case 2 Case 3 Case 5 Synthesized PVr
10−1 100 101 102
−200
−150
−100
−50 0
Frequency (rad/s)
Phase (deg)
10−1 100 101 102
−60
−40
−20 0
Magnitude (dB)
Figure 10.9 P-Vr Xstics, BPS_2
Case 1 Case 2 Case 3 Case 4 Case 5 Case 6 Synthesized PVr
10−1 100 101 102
−200
−150
−100
−50 0
Frequency (rad/s)
Phase (deg)
10−1 100 101 102
−30
−20
−10 0 10 20
Magnitude (dB)
Sec. 10.4 P-Vr characteristics of the generators 489
Figure 10.10 P-Vr Xtics, EPS_2
Case 1 Case 2 Case 3 Case 4 Case 5 Case 6 Synthesized PVr
10−1 100 101 102
−250
−200
−150
−100
−50 0
Frequency (rad/s)
Phase (deg)
10−1 100 101 102
−40
−20 0 20
Magnitude (dB)
Figure 10.11 P-Vr Xtics, MPS_2
Case 1 Case 2 Case 3 Case 4 Case 5 Case 6 Synthesized PVr
10−1 100 101 102
−200
−150
−100
−50 0
Frequency (rad/s)
Phase (deg)
10−1 100 101 102
−30
−20
−10 0 10 20
Magnitude (dB)
Figure 10.12 P-Vr Xtics,VPS_2
Case 1 Case 2 Case 3 Case 4 Case 5 Case 6 Synthesized PVr
10−1 100 101 102
−150
−100
−50 0
Frequency (rad/s)
Phase (deg)
10−1 100 101 102
−20
−10 0 10 20
Magnitude (dB)
Case 1 Case 2 Case 3 Case 4 Case 5 Case 6 Synthesized PVr
10−1 100 101 102
−200
−150
−100
−50 0
Frequency (rad/s)
Phase (deg)
10−1 100 101 102
−40
−30
−20
−10 0 10
Magnitude (dB)
Figure 10.13 P-Vr Xtics, LPS_3
Figure 10.14 P-Vr Xtics, YPS_3
Case 1 Case 2 Case 3 Case 4 Case 5 Case 6 Synthesized PVr
10−1 100 101 102
−300
−250
−200
−150
−100
−50 0
Frequency (rad/s)
Phase (deg)
10−1 100 101 102
−60
−40
−20 0 20
Magnitude (dB)
Figure 10.15 P-Vr Xtics, CPS_4
Case 1 Case 2 Case 3 Case 4 Case 5 Case 6 Synthesized PVr
10−1 100 101 102
−200
−150
−100
−50 0
Frequency (rad/s)
Phase (deg)
10−1 100 101 102
−30
−20
−10 0 10 20
Magnitude (dB)
Figure 10.16 P-Vr Xtics, GPS_4
Case 1 Case 2 Case 3 Case 4 Case 5 Case 6 Synthesized PVr
10−1 100 101 102
−200
−150
−100
−50 0
Frequency (rad/s)
Phase (deg)
10−1 100 101 102
−30
−20
−10 0 10 20
Magnitude (dB)
Figure 10.17 P-Vr Xtics, SPS_4
Case 1 Case 2 Case 3 Case 4 Case 5 Case 6 Synthesized PVr
10−1 100 101 102
−150
−100
−50 0
Frequency (rad/s)
Phase (deg)
10−1 100 101 102
−20
−10 0 10 20
Magnitude (dB)
Sec. 10.4 P-Vr characteristics of the generators 491
For each set of generator P-Vr characteristics a synthesized P-Vr characteristic is derived based on the following:
Case 1 Case 2 Case 3 Case 4 Case 5 Case 6 Synthesized PVr
10−1 100 101 102
−200
−150
−100
−50 0
Frequency (rad/s)
Phase (deg)
10−1 100 101 102
−50
−40
−30
−20
−10 0 10 20
Magnitude (dB)
Figure 10.18 P-Vr Xtics, TPS_4 Figure 10.19 P-Vr Xtics, NPS_5
Case 1 Case 2 Case 3 Case 4 Case 5 Case 6 Synthesized PVr
10−1 100 101 102
−300
−250
−200
−150
−100
−50 0
Frequency (rad/s)
Phase (deg)
10−1 100 101 102
−40
−30
−20
−10 0 10 20
Magnitude (dB)
Case 1 Case 2 Case 3 Case 4 Case 5 Case 6 Synthesized PVr
10−1 100 101 102
−250
−200
−150
−100
−50 0
Frequency (rad/s)
Phase (deg)
10−1 100 101 102
−50
−40
−30
−20
−10 0 10 20
Magnitude (dB)
Figure 10.20 P-Vr Xtics,TPS_5 Figure 10.21 P-Vr Xtics, PPS_5
Case 1 Case 2 Case 3 Case 4 Case 5 Case 6 Synthesized PVr
10−1 100 101 102
−200
−150
−100
−50 0
Frequency (rad/s)
Phase (deg)
10−1 100 101 102
−30
−20
−10 0 10 20
Magnitude (dB)
• The synthesized characteristic is a best fit of a generator’s P-Vr characteristics for the range of cases examined over the modal frequency range of interest, 1.5 to 15 rad/s.
As outlined in Section 5.10.6.1 the ‘best fit’ characteristic for these studies is consid- ered to be that lying in the centre of the magnitude and phase bands formed by the P- Vr characteristics1.
• If particular P-Vr characteristics tend to lie outside the bands formed by the majority of the characteristics, the synthesized P-Vr may be offset towards the band formed by the majority (e.g. see Figures 10.17 and 10.19). However, weighting of P-Vrs depends on knowledge of the system, the contingencies and engineering judgement.
The transfer function of the synthesized P-Vr characteristic, PVR(s), for each of the 14 gen- erators is given in Table 10.5.
In several figures, e.g. Figures 10.12 and 10.21 for generators VPS_2 and PPS_5 respective- ly, the bands of the low-frequency responses for the magnitude plots are much wider than those in other figures, e.g. Figure 10.20 for TPS_5. An examination of the generation con- ditions for the six power flow cases in Table 10.2 reveals that the generator real power out- puts vary from 45% to 98% of rated real power for VPS_2, and 60% to 100% for PPS_5;
on the other hand, the variation for TPS_5 is much smaller, 90-100%. These observations are consistent with those in Section 5.11 and Figure 5.16, namely, that the low-frequency magnitude response (the gain) of the P-Vr characteristic decreases as the real power output of the generator is reduced. This phenomenon is explained in Section 9.4.1. It is shown that the gain of the P-Vr characteristic varies only with the scalar gain vdo, the steady-state d-axis component of the terminal voltage, but retains its shape over the range of operating condi- tions. At rated power output vdo is relatively large, but tends to zero as the real power output is reduced. However, from 70% to 100% of real power output the magnitude characteristic is, for practical purposes, lie within a band of less than dB from the Design Case2. (Sim- ilarly, at constant real power output the magnitude of the gain decreases as the reactive pow- er output is varied from maximum leading to maximum lagging power factor. See Tables 9.7 and 9.8).
The more-or-less invariant nature of the phase responses of the P-Vr characteristics is also explained in Section 9.4.1.
1. A least squares estimation procedure, or the MATLAB® Signal Processing Toolbox rou- tine ‘invfreqs.m’, can be employed to determine.the parameters for the synthesized trans- fer function.
2. ‘Design Case’ is defined in Section 5.10.6.1.
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Sec. 10.5 Synthesized P-Vr and PSS transfer functions 493 Table 10.5 Transfer functions of synthesized P-Vr characteristics, PVR(s)