Types of Power System Stabilizers
8.6 Conceptual explanation of the action of the pre-filter in the IAP PSS
8.6.2 Effect of the washout filters and integrators on the performance of the pre- filter
Sec. 8.6 Action of the pre-filter in the IAP PSS 419
Figure 8.12 Block diagram of the prefilter for the IAP PSS. The gain is set to unity in the following analysis.
8.6.2 Effect of the washout filters and integrators on the performance of the pre-
Figure 8.13 Path between the electric power input and the RTF output for a test input which replaces the electric power signal.
Note that, because the blocks in the path of Figure 8.13 represent linear elements, the prin- ciple of superposition permits the performance of this sub-system to be analysed inde- pendently of the rest of the pre-filter. The behaviour of this sub-system also reflects its behaviour when it is incorporated in the complete prefilter.
8.6.2.1 Dynamic response of the isolated path of Figure 8.13 to a ramp input
Useful insight is provided by examining both the dynamic and steady-state responses at the input and output of the RTF as well the tracking errors for a ramp in mechanical power. We are concerned only with the path of Figure 8.13.
Let us now demonstrate the nature of the response of the RTF for a ramp of rate pu/s in mechanical power output. For the current and later applications the parameters of the complete pre-filter of Figure 8.12 are given below:
• Washout filters: s, assuming the lowest (inter-area)
modal frequency is 1.5 to 2 rad/s (only Tw3 and Tw4 in Figure 8.12 are relevant to the signal path under study);
• Integrator: H=3 MWs/MVA; Pseudo-integrator (as derived in Section 8.3.1):
TH= 7.5 s;
• RTF: N= 1, M= 5, s, s. (The selection of s is
mentioned in Section 8.5.4).
It is shown in Figure 8.14(a) it is noted that, for the ideal integrator, the output of the RTF does not track the ramp in mechanical power but tends to a constant value
in the steady state. Furthermore, the output of the RTF tracks its input with negligible error which, as shown in Figure 8.14(b), tends to zero in the steady state. For the pseudo-integrator, however, it is observed in Figure 8.14(a) that the output of the RTF follows the ramp in mechanical power with zero following error in the steady-state (i.e. after some 50 s). This is because the pseudo-integrator ceases to act as an integrator and becomes a low pass filter at low frequencies.
sTw 1+sTw
--- sTw 1+sTw
--- 1+sT8
1+sT9
M
--- N Ramp-tracking filter
U s Ideal / Urtf Vrtf
Pseudo Integrator
U s
R0 = 0.0075
Tw1 = Tw2 = Tw3 = Tw4 = 7.5
T9 = 0.1 T8 = MT9 = 0.5 T9 = 0.1
TW2R02H = 0.0703 Vrtf t
Urtf t
Sec. 8.6 Action of the pre-filter in the IAP PSS 421 For a ramp in mechanical power it is also noted in Figure 8.14(a) that zero tracking error between the input and output of the RTF is achieved for both the ideal and the pseudo- in- tegrator.
Figure 8.14 Responses to a ramp in mechanical power for ideal and pseudo-integrators with two washout filters in the isolated path of Figure 8.13. The plots show (a) the input and
output responses of the RTF , , and (b) that the error across the RTF, , in the responses is very small and tends to zero in the steady state.
(To avoid a discontinuity at time zero in Figure 8.14(a) and (b), the initial slope of the me- chanical power output is varied in parabolic fashion from zero to the ramp rate of 0.0075 pu/s at 1 s.)
8.6.2.2 The steady-state tracking - and tracking errors - of the RTF
For the RTF with the parameters given in Figure 8.9 it is known that its output tracks a ramp change at its input with zero steady state error. However, for a parabolic input to the RTF its output tracks the input with a constant following error after any initial tran- sients have decayed away.
Let us examine the behaviour of the isolated path of Figure 8.13 in more detail. Firstly, for the sake of completeness, it is of interest to ascertain the performance of the RTF not only for the four types of mechanical power change , but also the effects of none, one and two washout filters on the tracking errors. Secondly, consideration is given to the effects of the ideal and pseudo-integrators, the transfer function of the latter being , (8.5). Of interest are not only the steady-state values of the input to the
Ideal Int. Pseudo−Int.
0 10 20 30 40 50
−4
−2 0 2 4 6 8 10
Units of speed
Time (s) x 10−5
0 10 20 30 40 50
0 1 2 3 4 5 6 7
Units of speed (%)
Time (s)
RTF In RTF Out Ideal Int.
RTF In RTF Out Pseudo−Int.
Mechanical Power Ramp (a) (b)
Urtf Vrtf Urtf t –Vrtf t
Vrtf Urtf
U t
TH2H1+sTH
RTF but also how closely the output of the RTF tracks the input to the RTF. Consequently, in Appendix 8–I.2 expressions are derived which analyse the nature of the tracking error for power changes of a general form .1. The results are summarised in Table 8.6.
The upper value in each row of the table is the steady-state input to the RTF, (not the power changes at the input, ). The steady-state input is
.2 .
The lower value is the steady-state tracking error of the RTF, i.e. the difference between the steady-state input to the RTF and its output, i.e. . Note that:
• means the quantity increases indefinitely with time.
• When the both the mechanical power and the input to the RTF are increasing indefi- nitely with time the tracking error may be zero or finite (e.g. columns 5 to 8, parabolic input).
Although the tracking error is zero for a ramp applied directly to the RTF (column 1), when a ramp is applied to an ideal integrator in the path the tracking error is non-zero (Table 8.6, ramp, col. 4). The conceptual discussion in Section 8.6 surrounding Figure 8.11, in which there is an ideal integrator in the path, is based on the assumption that the tracking error is zero. However, it can be shown that this error is small even for fast ramps. In practice of course, there are one or more washout filters in the power-signal path in which case the tracking error of the RTF is zero.
In summary, the practical case is the replacement of the ideal integrator by the pseudo-inte- grator of (8.5) with one or two washout filters in the electric power input path. As noted in Table 8.6 - and analysed in Appendix 8–I.2 - the steady-state tracking errors of the RTF are zero if a pseudo-integrator is employed when the mechanical power input is a step, ramp, or parabola.
1. The expressions are for the input to the RTF and the tracking error between RTF input and output. For the ideal integrator these are (8.29) and (8.32), respectively; for the pseudo-integrator they are (8.33) and(8.34).
2. Final Value Theorem. See Section 2.10.
U t = antn
U s
Urtf ss Urtf
tlim t sUrtf
slim0 s
= =
efss = Urtf ss–Vrtf ss x
Sec. 8.6Action of the pre-filter in the IAP PSS423 Table 8.6 Steady-state input to the RTF and the tracking errors following changes in mechanical power
Type of mechanical power change
,
Input to RTF, , output and tracking error of the RTF, as
, for mechanical power changes applied to the path in Figure 8.13: Notes
Input to RTF ---
Tracking error
RTF only.(
Ideal integrator
& RTF (no washouts)
Pseudo- integrator
& RTF (no washouts)
One washout,
ideal integrator
& RTF
One washout,
pseudo- integrator
& RTF
Two washouts,
ideal integrator
& RTF
Two washouts,
pseudo- integrator
& RTF
1 2 3 4 5 6 7 8 9 10
Step,
n= 0 efss 0 0 0 0
0 0
0 0
0
0 ,
, , Ramp,
n= 1 efss 0 0 0 0 0
0 0 Parabola,
n= 2 efss 0 0 0
Cubic,
n= 3 efss 0
RTF: N=1, M=5; Inertia constant H (MWs/MVA); Integrators: Ideal (I) , Pseudo (P) Urtf ss
U t t0
Urtf ss V
rtf ss efss U
rtf ss V
rtf ss
–
=
t U t
R0 Urtf ss R0 K0PR0 K1IR0 KG = 10T92
K0I = 1 2H K0P = TH2H K1I = TW2H
K1P = TWTH2H K2I = TW2 2H K2P = TW2TH2H
R0t Urtf ss
K0IKGR0
K1PR0 K2IR0
R0t22 Urtf ss R0KG
K0PKGR0
K1IKGR0
K2PR0
R0t36 Urtf ss
K1PKGR0
K2IKGR0
1 2Hs TH2H1+sTH