Types of Power System Stabilizers
8.2 Dynamic characteristics of washout filters
8.2.1 Time-domain responses
In Section 5.8.6.1 the washout filter is introduced with the purpose of eliminating any steady-state offset, or DC level, in the input signal to a PSS. In this section, in addition to those of the single washout filter, the dynamic characteristics of two identical washout filters in series are examined and a comparison made with the dynamic performance of a single washout.
The transfer functions of one and two washout filters having a washout time constant of (seconds) are, respectively:
, and (8.1)
. (8.2)
In analog terms, the analysis assumes a low impedance source drives the filters which then feed into a high impedance sink. Expressions for the time-domain responses of each of the filters to a step input of units and a ramp input of units/s are shown in Table 8.1.
Based on the definition of settling times in Section 2.8, the time-domain response of the sin- gle washout filter to a step input decays to zero with a 2% settling time of s. However, for a ramp input the response of the single filter tends to a finite value - also with a settling time s. Consequently, for a PSS having electrical power as the stabilizing signal, and with the input being a slow ramp in electrical power, the single washout filter produces a potentially undesirable offset in the terminal voltage of the generator. For the single wash- out filter the forms of the step and ramp responses are illustrated in Figure 8.1 for washout time constants of 4 and 8 s.
TW
G1W s sTW 1+sTW ---
=
G2W s s2TW2 1+sTW
2
---
=
A0 R0
4TW R0TW 4TW
Table 8.1 Analytical expressions for responses of washout filters to step and ramp inputs.
Figure 8.1 Responses of a single washout filter to step and ramp inputs of 1 unit and 1 unit/s, respectively, for washout time constants of 4 and 8 s.
For the case of two washout filters in series the following time-domain characteristics are of interest.
1. The responses of two identical washout filters in series to a step input of 1 unit are shown in Figure 8.2 for values of the washout time constant of 4 and 8 s. For a pos- itive step input the response decays from the initial value , passes through zero at time s and under-shoots by a value at s. It then decays to within of zero after approximately s.
Input signal at time
Output responses
One washout filter Two washout filters in series
Step, units
Ramp, R0 units/s t0 +
A0 Y1WS A0e
t – TW
=
Y1WS0 as t
Y2WS= A0e–tTW1–t T W Y2WS0 as t
Y1WR=R0TW 1 e–tTW
–
Y1WRR0TW as t
Y2WR R0te–tTW
=
Y2WR0 as t
0 10 20 30 40 50
0 0.2 0.4 0.6 0.8 1
Washout Responses
Time (s)
Step in Tw=4 Tw=8
0 10 20 30 40 50
0 2 4 6 8 10
Washout Responses
Time (s)
Ramp in Tw=4 Tw=8
A0
TW –A0e–2= –0.135A0 2TW
0.02
– A0 5.5TW
Sec. 8.2 Dynamic characteristics of washout filters 401
Figure 8.2 Responses of two identical washout filters in series to a step input of 1 unit for washout time constant values of 4 and 8 s.
2. For a positive ramp input the time domain response reaches a maximum value of at time s. It then decays to zero. Notice that the maxi- mum value of the response depends on the ramp rate and the value of the washout time constant.
The responses to a ramp input of two identical washout filters in series is of particular inter- est in the discussion of the ‘integral-of-accelerating-power’ PSS considered in Section 8.5.
Accordingly, the responses of two such filters to a ramp of 1 unit/s are shown in Figure 8.3 for a range of values of the washout time constant from 1 to 10 s.
In Figures 8.1 to 8.3 the time-domain characteristics listed in Table 8.1 are clearly illustrated.
8.2.2 Frequency-domain responses
The nature of the frequency response of a single washout filter, and its role in the dynamic performance of speed-PSSs, are discussed in Section 5.8.6.1. Because the application of two washouts is of interest in this chapter the frequency response of two identical washouts in series, time constant TW, is shown in Figure 8.4. The response is normalised to a corner fre- quency of 1 rad/s (i.e. TW= 1 s). For example, if TW= 5 s the associated corner frequency is 0.2 rad/s, the magnitude and phase of the response at say 0.02 rad/s (as read off Figure 8.4 at rad/s) are then dB and , respectively.
0 10 20 30 40 50
−0.2 0 0.2 0.4 0.6 0.8 1
Washout Responses
Time (s)
Step in Tw=4 Tw=8
R0TWe–1 = 0.368R0TW TW
0.02 0.2 = 0.1 –40 169
Figure 8.3 Responses of two identical washout filters in series to a ramp input of R0= 1 unit/s as the washout time constant TW is varied from 1 to 10 s.
Time-frames: (i) 0-15 s, (ii) 0-80 s. Solid lines TW 1-5 s; dashed lines TW 6-10 s.
Peak occurs at TW s.
Figure 8.4 Frequency response for two identical washouts filters in series normalised to a corner frequency of 1 rad/s. (For a single washout filter, halve all vertical-axis quantities.)
Input ramp
Tw=1 Tw=2 Tw=3 Tw=4 Tw=5
0 5 10 15
0 0.5 1 1.5 2 2.5 3 3.5 4
Washout Responses
Time (s)
0 20 40 60 80
0 0.5 1 1.5 2 2.5 3 3.5 4
Washout Responses
Time (s) Input ramp
Tw=6 Tw=7 Tw=8 Tw=9 Tw=10
(i) (ii)
10−2 10−1 100 101 102
−80
−60
−40
−20 0
Magnitude (dB)
10−2 10−1 100 101 102 0
30 60 90 120 150 180
Frequency (rad/s)
Phase (deg)
Sec. 8.2 Dynamic characteristics of washout filters 403 8.2.3 Comparison of dynamic performance between a single and two washout fil- ters.
Let us compare the features of a single washout filter with two identical washouts in series.
Consider a power system in which the lowest inter-area modal frequency is 2 rad/s. For the purpose of the design of the associated PSS, let us assume that if a single washout filter is employed the corner frequency of the washout would be 0.2 rad/s, say, a decade below the modal frequency. The time constant of the single filter is T1W= 5 s; the phase lead intro- duced by the filter at the modal frequency is . To introduce the same phase lead at the modal frequency for two identical washouts the corner frequency of each should be 0.1 rad/
s, i.e. T2W= 10 s. Based on these assumptions a comparison of dynamic performance is summarized in Table 8.2.
Table 8.2 Characteristics of a single washout and two identical washout filters in series
Some observations on the characteristics of the washout filters of Table 8.2 are listed below.
1. A reduction in time constants for both a single washout and two washouts in series improves their dynamic performance through lower settling times.
2. The performance of the single washout filter in Table 8.2 is superior to that of two washouts, except that the ramp response of the single washout filter tends to a finite value. As mentioned in Section 8.2.1, in the case of an electrical power PSS this char- acteristic can produce an offset in generator terminal voltage and reactive power output when a ramp in electrical power output occurs.
3. As noted, the time-domain performance of two identical washouts in series can be improved by reducing the time constant. However, such a reduction is a compro-
One washout filter Two washout filters in series Corner frequency,
TW 0.2 rad/s,
5 s
0.1 rad/s, 10 s Phase lead introduced at 2
rad/s
Step response of 1 unit:
Settling time Under-shoot Final value
4TW= 20 s - 0
~5.4TW= 54 s -0.135A0= -0.135 at 2TW s
0 Ramp response of 1 unit/s:
Settling time Peak value Final value
4TW= 20 s Peak is the final value
TWR0= 5
By calculation 0 5.7
5.7 5.7
0.368R0TW = 3.68
mise with the increase in phase lead at the lower modal frequencies. In the case of a reduction in the time constant, say from 10 s to 5 s, the phase lead at the modal fre- quency of 2 rad/s is increased from to . If desired, the increased phase lead so introduced by the two washouts can be compensated for in the tuning of the PSS main compensation blocks.
4. If the inter-area modes are not of concern, the washout filter time constants can likewise be determined based on the relatively higher frequency of the local-area mode(s).