App. 7–I.1 Generator and exciter parameters
App. 7–I.1.1 Parameters for the 6th order generator and a simple exciter
Note. These models are only used in the application of all types of compensation except that in Section 7.8 and 7.11 for Type 2B PIDs.
Generator model: 6th order, classical. Saturation ignored. Values in pu on machine MVA base (the MVA base is stated in the application). (These parameters are the same as those used for generator TPS_4 in Table 10.23 on page 527). System frequency is 50 Hz.
D= 0, H= 2.6 s, ra= 0, = 2.3, = 1.7, = 0.30,
= 5.0 s, xl= 0.2, = 0.40, = 2.0 s, = 0.25, = 0.25 s,
= 0.25, = 0.03 s.
Exciter: Simple linear first-order lag model:KE = 1.0 pu, TE = 0.1 s.
App. 7–I.1.2 Parameters for the 5th order salient-pole generator and a brushless AC exciter
Note. These models are only used in Section 7.8 and 7.11 for the application of the analysis of Type 2B PIDs.
Generator model: 5th order, salient-pole generator; saturation is included. All values are in per unit on machine rating (58.8 MVA) unless otherwise stated. System frequency is 50 Hz.
D= 0, H= 5.5 s, ra= 0, = 1.5, = 0.7, = 0.22,
= 8.0 s, = 0.10, = 0.16, = 0.12 s, = 0.16, = 0.04 s,
= 0.15, = 0.45.
The exciter is an AC generator with a rotating rectifier and is represented by an AC8B Ex- citation System Model [12] and is shown in Figure 7.49. Its parameters are:
KE= 1.0 TE= 0.7 s, KCE= 0.1, KDE= 1.25, KAE= 1.75, TA= 0.
The terminal voltage transducer is represented by a first-order lag block, Ttrn= 20 ms.
App. 7–I.2 Models of the brushless AC exciter
The model of the brushless AC exciter is shown in Figure 7.49.
xd xq xd
Td0 xq Tq0 xq Tq0
xd Td0
xd xq xd
Td0 xl xq Tq0 xd Td0
Sd1.0 Sd1.2
App. 7–I.2 Models of the brushless AC exciter 387
Figure 7.49 The brushless AC exciter is based on the AC8B Excitation System Model [12]
The rectifier regulation modes are expressed by the three equations:
.
The AVR comprises the Type 2B PID (TG= 0) in which the gain KG may represent power amplification. The gain KAE, included in the exciter model, is a factor which accounts for the per unitization of exciter and generator quantities.
Figure 7.50 Small-signal model of the AC8B Excitation System Model
KP
KD 1+sTD
KI s
KG 1+sTG
1 sTE
KDE VESE(VE)
KE FEX=fn(IN) KCE IFD VE
VE
IN FEX
EFD
IFD Vexf
+
+ + +
+
+ + + + + VC
VS Vref
Exciter AVR
KAE Vr
FEX f I N
1.0 0.577I– N
IN0.433 0.75–IN2 0.433IN0.75 1.732 1.0 –IN 0.75IN1.0
= =
1 sTE
KDE KS KE
FEX0
VE0 +
+ +
+ +
+
+ KEX
VE EFD
IFD
IN
Vexf
FEX
KAE Vr +
KIV
KIF
The parameters in the model are defined as follows:
, .
The gain KS is related to the saturation function of the exciter and is dependent on the initial steady-state value of the field voltage . It is given by the following expression:
.
Saturation in the exciter is assumed to be negligible under steady-state operating conditions and thus KS= 0.
As an example, the values of the parameters of the linearized exciter model for cases C16 - C20 are provided in Table 7.10. The parameters of the exciter model are listed in Appendix 7–I.1.2.
Table 7.10 Parameters of the small-signal model of the exciter
App. 7–I.3 PI Compensation using positive feedback
A simple positive feedback implementation can be employed for PI Compensation. The di- agram of the associated control blocks is shown in Figure 7.51.
Case EF0 KIF KIV VE0
C16 C17 C18 C19 C20
2.16 1.92 1.61 1.37 1.20
0.044 0.049 0.059 0.069 0.079
0.041 0.046 0.056 0.065 0.075
2.26 2.03 1.70 1.45 1.27 KEX= -0.577, FEX0= 0.945, KS= 0 KEX
0.577
– IN00.433 IN0
0.75–IN02 ---
– 0.433IN00.75
1.732
– 0.75IN01.0
= KIV KCEIFD0
VE02 ---
=
, and KIF KCE VE0 ---
=
VE0
KS SEVE0 VE0 SEVE VE ---
VE=VE0
+
=
App. 7–I.3 PI Compensation using positive feedback 389
Figure 7.51 PI Compensator
The transfer function of the system from x to y, i.e. excluding the gain , is:
(7.50)
where, for the approximation , the low and high frequency corners are and rad/s, respectively. These corners should be respectively about a decade or more be- low and above the extremes of the range of frequencies of rotor oscillations.
Equation (7.50) can be rearranged into the following form representative of the PI structure:
For the compensator the effective integrator gain is and the proportional gain is over the range of rotor frequencies. The respective gains of the compensator
become and if .
A plot of frequency responses of is shown in Figure 7.52 for a range of values of . As explained below, the value of is such that the phase angle approaches zero degrees in the mid-range of rotor frequencies, e.g. 4 rad/s.
1 1+sT2 ---
1 1+sT1 --- +
+
KG
G1(s)
x y
KG G1 s 1+sT2
s sT 1T2+T1+T2 ---
=
1+sT2
T2s1+sT1
---, for T2»T1
T2»T1 1T2
1T1
G1 s 1 T1+T2 --- 1
---s T2 T1+T2 ---
2 1
1+s T 1T2T1+T2 ---
+
= 1 T2 --- 1
s--- 1 1+sT1
---, for T2»T1
+
G1 s 1T2
1
KGG1 s KGT2 KG KG1
G1 s T2
T1
Figure 7.52 Frequency responses of the PI compensation using positive feedback for val- ues of T2 from 1 to 5 s; associated values of T1 are such that the maximum phase angle is at
4 rad/s.
For design of the parameters of the PI compensator it may be desirable to place the phase angle characteristic such that phase is close to zero degrees over the range of rotor modal frequencies. From the figure we note that we can choose a frequency at which the phase an- gle is a maximum. This frequency, , occurs at the geometric mean of the corner fre- quencies of the exact transfer function (7.50), i.e.
. (7.51)
Thus, given T2and , the value of T1 can be derived from (7.51) to yield:
. (7.52)
Based on (7.50) and (7.51), the value of the phase characteristic when is
. (7.53)
10−1 100 101 102
−20
−10 0 10 20
Magnitude (dB)
10−1 100 101 102
−80
−60
−40
−20 0
Frequency (rad/s)
Phase (deg)
T2=1 T2=2 T2=3 T2=4 T2=5
mx
mx 1 T2
--- T1+T2 T1 ---
=
mx
T1= T2T22mx2 –11T2mx2 for T2»T1
G1jf f = mx G1
mx = atanmxT2–90–atanmxT1T2T1+T2 2atan T1T1+T2
–
=
App. 7–I.4 Integrator Wind-up Limiting 391 Figure 7.52 is based on the selection rad/s with T2 being varied from 1 to 5 s; the corresponding values of T1and at are calculated from (7.52) and (7.53), re- spectively.
App. 7–I.4 Integrator Wind-up Limiting
Two types of limiter, anti-windup1 and windup, are encountered in excitation system mod- els. Examples of these types are shown in Figure 7.53 in the case of a simple integrator. The upper and lower limits are UL and LL, respectively.
Figure 7.53 Integrator with (a) anti-windup limiting, and (b) windup limiting.
The operation of the two types of limiters are illustrated in principle in Figure 7.53, (a) and (b). In illustration (b), with windup limiting, the output of the integrator y(t) continues to increase once the limit UL is reached but starts to decrease only when the input u(t) changes sign. Limiting ceases only when the output y falls below UL. With anti-windup limiting, however, it ceases limiting as soon as the input changes sign. The advantage of anti-windup limiting is that it eliminates the time delay between sign reversal and wind-down to UL that occurs in windup limiting.
Anti-windup and windup limiting occur in other types of transfer function blocks incorpo- rating lead-lag and PI compensation for example (see [7], [12]).
1. Anti-windup limiting is also known as “non-windup” limiting ([7], [12]).
mx = 4 G1jf
mx
1 s 1
s UL
LL
UL
LL
UL y
UL u u
u u y
y
w
w y
If UL y LL then w= y If y UL then w= UL
If y LL then w= LL (a) Integrator with anti-windup limiting (b) Integrator with windup limiting
time time
If If If
UL y LL then dy dt = u y UL and u 0 then set dy dt = 0 y LL and u 0 then set dy dt = 0
App. 7–I.5 A ‘phase-matching’ method for constant phase margin over an appropriate frequency range
Consider the phase responses of for Case C17 and the parameter Set No. 2 for PID Type 2B in Table 7.6; the responses are shown in Figure 7.54.
Figure 7.54 Case C17: Frequency responses of the phase of the component transfer func- tions of the open-loop system. The phase response of the open-loop transfer function with
PID Set No. 2 is shown by x-x-x.
Let and (deg.) be the phase responses of the generator-exciter and the PID with parameter Set No. 2, respectively, as shown in Figure 7.54. The phase of the open-loop transfer function is shown by x-x-x in the figure. Depending on the location of the gain-crossover in the range 0.7 - 2.5 rad/s, the phase margin is the difference between the open-loop phase response ( ) and . At the gain-crossover-frequency, , the phase margin is
. (7.54)
At low frequencies and, for the PID, . Let
(7.55) so that at low frequencies both and .
Assume the desired phase margin is PMdes, e.g. . The required values of , based on (7.54) and (7.55), are
(7.56) (If the actual value of is greater than the desired value of then the phase margin is greater (i.e more stable) than the desired phase margin ; and vice-a-versa.)
VtrnjfVexfjf
α, Exciter−Gen: V
trn/V β, AVR: V exf
exf/V
ref = PID #2
10−2 10−1 100 101
−180
−120
−60 0 60
Frequency (rad/s)
Phase (deg)
x x x xx x x xx x x x α+β: Phase OLTF, V
trn/V
ref
α
β
jf jf
+
+ –180 f = c
PM = c+c––180 = c+c+180
= 0 = –90
= +90
= 0 = 0
65
des = PMdes––90
des
PMdes
App. 7–I.5 A ‘phase-matching’ method 393 Thus, in order to match the phase margin with the desired phase margin it is necessary to find the PID frequency response, , that closely matches the line over the potential range of gain-crossover-frequencies. Let
, thus (7.57)
. (7.58)
In order to illustrate a design procedure based on (7.58) let us consider the following steps.
1. Given a selected system operating condition, choose (i) a set of parameters for a Type 2B PID as in Table 7.6, (ii) the design case C17 for the generator-exciter trans- fer function (see Figure 7.42), and (iii) set PMdesto , say.
2. Plot (i) , the negated phase angle of the transfer function for the selected operating condition, (ii) , the phase angle of the Type 2B PID advanced by , and (iii) the line showing where the response of must lie with respect to the plot of to satisfy the Phase Margin requirement,
. The plots of and is
shown by ‘x x x’ in Figure 7.55
3. Based on the plot in Step 1 adjust the PID parameters systematically so that the desired phase margin is satisfied, i.e. plots of and match closely - or overlap -over the desired frequency range.
4. Check that the resulting PID satisfies the system performance criteria over the range of operating conditions in which one or more units are on-line.
Let us consider the determination of the PID parameters based on the above steps.
For Step 1 the system operating condition Case C17 and a set of parameters have already been selected for the analysis associated with Figure 7.54. The parameters are those in Set No. 2, Table 7.6, KP= 14 pu, KI= 7.0 pu/s, KD= 8.0 pu-s, TD= 0.143 s, KG= 1.0. Let us base our analysis in this step on this set of PID parameters and Case C17.
The plots of and associated with the transfer function for Case C17 and the PID parameter set, respectively, are shown in Figure 7.55. Also shown is a plot (x x x) along which the angle of the desired PID must lie in order for the open-loop trans- fer function to have the desired phase margin (assuming for this study that the gain-cross-over frequency for the resulting open-loop transfer function (OLTF) lies in the range 0.7 to 2.5 rad/s).
des
= –
des = PMdes+–90
VtrnjfVexfjf 65
VtrnjfVexfjf
90
des = PMdes+–90 = –25 des = –25
des
VtrnjfVexfjf
VtrnVref
Figure 7.55 Plots of for exciter-generator transfer function (TF) for Case C17, and of the phase-advanced PID TF together with the plot of (x x x) which rep-
resents the desired location of the phase plot of the PID TF.
It is clear from Figure 7.55 that PID parameter Set No. 2 produces excessive phase lead and therefore the phase margin of the OLTF is greater than the desired value PMdes = . Re- ferring to Table 7.6 or Figure 7.25 it is seen that, by increasing the values of the corner fre- quencies , and for the PID sets, the plot of in Figure 7.55 approaches the desired phase margin plot. For PID Set No. 4 with parameters KP= 14 pu, KI= 14 pu/s, KD= 5.89 pu-s, TD= 0.105 s, KG= 1.0, the plot of coincides with desired phase margin plot for gain-cross-over frequencies in the range 0.9 to 2.5 rad/s.
Based on the PID parameter Sets 2 and 4, the composite OLTF for Case C17 is plotted in Figure 7.56. From this Bode plot it is observed that (i) the gain cross-over frequencies for the two sets are 1.5 and 1.3 rad/s, respectively, (ii) with parameter Set 4 the phase margin is close to the desired value of . The phase margin variations for PID Sets 2 and 4 for a loop-gain variation of dB are shown in Table 7.11.
α′, Exciter−Gen TF V
trn/V
exf, Phase Negated β′, PID #2 with phase advanced by 90 deg.
β′, PID #4 with phase advanced by 90 deg.
10−1 100 101
0 20 40 60 80 100 120 140 160 180 200
Frequency (rad/s)
Phase Response (deg)
x x
x x
x x
x x
x x
x x x x Desired PM=65 deg
α′
β′
β′
des = –25
65
1 2 D
65
6
App. 7–I.5 A ‘phase-matching’ method 395
Figure 7.56 Case C17. Bode Plots of the OLTF comprising and PID Sets 2 or 4.
Table 7.11 Case C17. Robustness: Phase Margin variation for loop-gain change of dB
From Figure 7.56 and Table 7.11, it is evident that the phase margin variation of about associated with PID Set No. 4 implies it is robust to variation in the loop gain; this is revealed by the relatively small changes in its phase in the figure. In Set 2, however, not only is the phase margin variation considerably more but the phase margin exceeds the desired value of
over the gain variation of dB.
Set Gain
change (dB)
Phase Margin at
frequency of ... Gain change (dB)
Phase Margin at frequency of ...
PM (deg) Frequency
(rad/s) PM (deg) Frequency
(rad/s)
2 102 0.54 +6 69 3.0
4 70 0.80 +6 64 2.4
Desired phase margin is .
Open−loop TF with PID #2, PM=91 deg.
Open−loop TF with PID #4, PM=66 deg.
10−1 100 101
−180
−160
−140
−120
−100
−80
−60
Frequency (rad/s)
Phase (deg)
10−1 100
−20
−10 0 10 20 30 40
Magnitude (dB)
VtrnjfVrefjf
6
6 –
6 –
65
6
65 6
397