7.11 Tuning of AVRs with Type 2B PID compensation in a three- generator system
7.11.2 The frequency response characteristics of the brushless exciter and genera- tor
The closed-loop terminal voltage control system of each generator is shown in the block di- agram in Figure 7.41; note that the generator model accounts for the effects of the external system when the unit is on-line. Because the models of both the generator and the exciter are non-linear, the parameters of the linearized model will change with conditions at the gen- erator terminals. In order to establish suitable parameters for the excitation control system it is necessary to determine the variation of the generator-exciter characteristics with termi- nal conditions.
Figure 7.41 Terminal voltage control system. The gain KAE accounts for the per unit sys- tem of the excitation control system which includes a brushless exciter.
Over a range of terminal conditions such a characteristic is the frequency response of the generator-exciter transfer function, as measured between the exciter field voltage as input
1. See definition in Section 10.2.2.
10
KAE Gex Ggen
Vref
+
VtExciter
Vexf Ef
Verr
KG Gpid
Generator/
Vpid
Gtrn Vtrn
Voltage transducer
external system AVR
and the transducer voltage output, . The use of the latter transfer func- tion is particularly pertinent to brushless excitation systems in which the exciter output volt- age is not accessible for measurement.
A 5th order model of the generator and a non-linear exciter model are available in a small- signal power system dynamic performance package. Such models are automatically line- arized at each operating condition by the software. The non-linear and linearized exciter model are shown in Figure 7.49 and 7.50 of Appendix 7–I.2.
Based on (i) the power system of Figure 7.40, (ii) the system and device parameters given in Section 7.11.1, the frequency responses of relevant blocks in the voltage control loop are calculated for selected operating conditions. The set of frequency responses for the genera- tor and exciter, Vtrn/Vexf , are shown in Table 7.42 when either one, two or three generators are on-line; the output of a generator is 25 or 50 MW at 1 pu terminal voltage.
For normal and N-1 operation of this system, operation at lagging power factors is more likely to occur. The selection of the PID parameters may be influenced accordingly.
It is noted from Figure 7.42 that, for feasible cases C01 to C95 the gain in the generator/
exciter frequency responses in the region of 1.0 rad/s varies within dB, and the phase varies by about . The variations in the responses over the frequency range are due not only to the range of steady-state conditions at the generator terminals but also to the asso- ciated parameter values in the small-signal model of the exciter. An example of the exciter parameters and the steady-state field voltage is illustrated in Table 7.10 of Appendix 7–I.2 for operating conditions C16 to C20 in which a single machine is on-line.
The significance of the phase variation is the following. Let us assume that when the PID is added to the forward loop the gain-cross-over frequency of the Bode plot of occurs at 1 rad/s. The gain variation in the generator/exciter frequency responses at 1 rad/s is small but the phase variation remains at about . This will result in a similar variation in the phase margin over the range of operating conditions with impli- cations for both stability and transient response to a step change in reference voltage. The Bode plots suggest that, when the units are under closed-loop voltage control, the greater phase lags in the leading power factor cases (i) are not conducive to stability, and (ii) result in the terminal voltage response to step changes in reference voltage being less-well or poor- ly damped.
To determine an appropriate set of PID parameters for the range of operating conditions let us base the analysis on a condition in the middle of the band of phase variations, say Case C17, in which the output of a single generator is 50MW, 12.5 Mvar lagging; the line ‘a’ is out of service and the load is disconnected.
VtrnjfVexfjf
6 25
VtrnjfVrefjf
25
Sec. 7.11 Tuning AVRs with type 2B PID compensation 375
Figure 7.42 Envelopes of frequency responses between the generator terminal voltage transducer and the exciter field voltage, Vtrn/Vexf , for the feasible range of operating
conditions shown in Table 7.9 7.11.2.1 Calculation of the PID Type 2B parameters
In order to establish a basis for the compensation to be provided by the PID, let us consider for Case C17 the frequency responses of the generator and exciter, , and
(a) (b)
(a) One unit, (b) Two units, (c) Three units.
Solid lines: Maximum lagging reactive power Dashed lines: Maximum leading reactive power Real power output is 50 MW in all cases except for C21-25, C61-65 and C92-95 when it is 25 MW.
Cases C76, C80, C81, and others are omitted because operating constraints are infringed and are infeasible.
For some other cases the reactive power output per generator is reduced, e.g. from to Mvar for Case 79.
Case C17 is adopted as the Base Case and is shown in all three sets of plots.
10
– –5
C01 C05 C06 C10
C11 C15 C16 C20
C21 C25 C17
10−1 100 101
−200
−150
−100
−50 0
Frequency (rad/s)
Phase (deg)
10−1 100 101
−60
−50
−40
−30
−20
−10 0
Magnitude (dB)
C42 C44 C46 C50
C51 C55 C56 C60
C61 C65 C17
10−1 100 101
−200
−150
−100
−50 0
Frequency (rad/s)
Phase (deg)
10−1 100 101
−60
−50
−40
−30
−20
−10 0
Magnitude (dB)
C72 C75 C77 C79
C82 C84 C86 C88
C92 C95 C17
10−1 100 101
−250
−200
−150
−100
−50 0
Frequency (rad/s)
Phase (deg)
10−1 100 101
−60
−50
−40
−30
−20
−10 0
Magnitude (dB)
(c)
VtrnjfVexfjf
the AVR, . The PID parameters selected for trial are those for set No. 2 in Table 7.6 on page 350. These two responses are shown in Figure 7.43 together with the phase response of the open-loop transfer function .
Figure 7.43 Case C17 (one unit): Frequency responses of the component transfer functions in the open-loop system including the PID parameter Set No. 2
(see Table 7.6 on page 350),
i.e. KP= 14 pu, KI= 7.0 pu/s, KD= 8.0 pu-s, TD= 0.143 s, KG= 1.0.
Gain cross-over frequency of the open-loop transfer function is 1.51 rad/s.
From the open-loop transfer function in Figure 7.43 it is noted that (i) the gain-cross-over frequency occurs in the range 0.7 - 2.5 rad/s for which the possible variations in the loop gain lie in the range dB; (ii) the phase margin varies from to over the same frequency range.
In determining appropriate PID parameters the phase margin should be more-or-less con- stant about the gain-cross-over frequency for robustness to gain variations. Selecting a phase margin of , say, ensures the closed-loop response of terminal voltage to a step change in reference voltage is not significantly over-damped (for large values of the phase margin) or under-damped (for small values of the phase margin). In the case of higher values of loop gain associated with the gain-cross-over frequency exceeding 3 rad/s we note that the
VexfVerr = PID j f
VtrnVref
Exciter−Gen: V
trn/V
exf
PID #2: V
exf/V
err
Open−loop TF: V
trn/V
ref, PM = 91 deg
10−2 10−1 100 101
−20
−10 0 10 20 30 40
Magnitude (dB)
10−2 10−1 100 101
−200
−150
−100
−50 0 50
Frequency (rad/s)
Phase (deg)
4.5 103 75
65
Sec. 7.11 Tuning AVRs with type 2B PID compensation 377 closed-loop step response of terminal voltage is likely to contain a damped oscillatory com- ponent due to the electro-mechanical modal resonance at 5 to 6 rad/s.
We will therefore examine an approach to derive a more of less constant phase margin of over an appropriate frequency range for Case 17. The ‘phase matching’ method which achieves this objective is described in Appendix 7–I.5 in which it is shown in Figure 7.55 that the parameter set No. 4 for PID Type 2B in Table 7.6 provides the required phase mar- gin.
The significance of the analysis of the phase margin for the Base Case 17 is revealed in its effect on the terminal voltage response of the closed-loop system due to a step change in reference voltage for the full set of operating conditions. As shown in Figure 7.44 the re- sponse of the system incorporating PID Set No. 2 (phase margin ) is well damped. How- ever, with PID set 4 (phase margin ) a satisfactory, suitably-damped response results.
Moreover, the settling-time requirement that the response lies within 10% of its final value within 5 s is satisfied with both PID Sets 2 and 4.
Figure 7.44 Case C17. Single unit only on-line. Perturbations in terminal voltage (Vt) due to a step change in reference voltage from a steady-state value of 1.0 pu to 1.01 pu (1%).
PID parameter Sets 2 and 4, Table 7.6. The % band about the final value is also shown.
The dynamic performance of the single generator with PID Set No. 4, parameter values KP= 14 pu, KI= 14 pu/s, KD= 5.89 pu-s and TD= 0.1053 s, appears satisfactory. The ap- plication of this PID set to all the feasible operating cases and conditions for a generator off- line and one, two and three units on-line is now examined.
7.11.2.2 Dynamic performance of a unit off-line under closed-loop terminal voltage control When the generator is operating off-line at rated speed and under closed-loop voltage con- trol it is required to satisfy the relevant performance specifications. For example, such spec- ifications may require that the measured terminal voltage settles within 10% of the final value in less than 5 s for a step change of 1% in the terminal voltage (see Section 7.4). in the 65
96
66
PID #2 PID #4
0 2 4 6 8 10
0 0.2 0.4 0.6 0.8 1
ΔV t (%)
Time (s)
10
analysis.When the generator is off-line in the following analysis the 10% settling time in ter- minal voltage for a 1% change in reference voltage is 2.5 s.
In previous examples simple models of the exciter and the generator have been employed, i.e. and . However, (i) at 1 pu voltage the small-signal gain of the generator is determined by the slope of the saturation curve and is less than unity; (ii) the perturbations in generator field current modulates the generator field voltage by two mechanisms represented in the model of the exciter in Appendix 7–I.2, Figure 7.50. The mechanisms are (i) the effect of the demagnetization term, KDE, and (ii) the non-linear re- duction in rectifier average output voltage with increase in the rectifier load, i.e the generator field current. The latter mechanism is represented by the value of the gain KCE and the as- sociated mode of operation of the rectifier.
The linearized model of the off-line, fifth-order salient-pole generator and the exciter are formed automatically. The off-line unit operates a rated voltage and speed. The other ele- ments in the voltage control loop are PID Set No. 4 (see Appendix 7–I.5), the per unitizing gain KAE and the terminal voltage transducer, time constant Ttrn. The generator and exciter parameters are given in Appendix 7–I.1.2. The Bode Plot of the open voltage-control loop, , and the associated closed-loop response in generator terminal voltage due to a +1% step in the reference voltage, are shown respectively in (a) and (b) of Figure 7.45.
The closed-loop step response is adequately damped, as predicted by the Bode plot, and sat- isfies the performance specification.
7.11.2.3 Dynamic performance over a range of operating conditions; one, two and three units on-line based on PID parameter Set No. 4.
The open-loop frequency responses for one, two and three units on-line are examined to derive information on both the damping of the voltage control loop and the stability of the power system under closed-loop conditions. The margins of rotor angle stability under closed-loop conditions are also examined, assuming for planning purposes a 5 s halving time for the dominant mode. Finally, the closed-loop responses of the generator terminal voltage to step changes in its reference voltage are assessed to determine if the requirement that the response lies within 10% of its final value within 5 s is satisfied over the range of operating conditions.
1KE+sTE 1 1 +sTd0
VtrnjfVrefjf
Sec. 7.11 Tuning AVRs with type 2B PID compensation 379
Figure 7.45 Generator off-line, operating at rated speed under terminal voltage control with PID Set No. 4. (a) Open-loop Bode Plot. (b) Perturbation in closed-loop terminal volt-
age step-response.
7.11.2.3.1 Open-loop frequency responses; one, two and three units on-line.
In the following Bode plots for generator #1 the terminal voltage feedback path is open on that generator, but is closed on the other generators when more than one unit is on-line. The Bode plots are shown in Figure 7.46 for the cases when one, two or three generators are on- line. When all machines are under closed-loop voltage control these open-loop plots should reveal the nature of (i) the damping in the voltage control loop on generator #1, (ii) the sta- bility of the system, and (iii) the terminal voltage response of generator #1 to a step in its reference voltage. One can equally well apply the above analysis to unit #2 or #3 instead.
Because the Phase Margins derived from the Bode plots in Figure 7.46 are all positive the system is stable over the range of operating conditions. However, the Phase Margins are much less than the desired value of at higher values of leading reactive power output, e.g. for C20 the PM is at 1.26 rad/s. Thus under leading power factor operation and closed-loop voltage control the system damping is degraded. However, in the cases of one, two or three generators on-line at rated real power output it should be noted that the higher values of leading reactive power output are unlikely to arise in practice. In such cases the re- active power import to the system at the infinite bus is somewhat greater than that absorbed by the generator. For example, in case C55 the output of two units is 100 MW -40 Mvar and the reactive import from the infinite bus is 64 Mvar. Similarly in case C15 for one generator, output 50 MW -20 Mvar, 15 Mvar is imported from the system. Such reactive flows from the real power sink to the real power source are unwarranted and uneconomic - especially
10−1 100 101 102
−270
−225
−180
−135
−90
Frequency (rad/s)
Phase (deg)
10−1 100 101 102
−80
−60
−40
−20 0 20 40
Magnitude (dB)
0 2 4 6 8 10
0 0.2 0.4 0.6 0.8 1 1.2
Terminal Voltage (%)
Time (s)
(a) Bode Plot: Phase Margin 60 deg at 2.2 rad/s.
(b) Perturbation in generator terminal voltage due to a step change in reference voltage from a steady-state value of 1.0 pu to 1.01 pu (1%).
10% Settling time < 2.5 s.
65
44
for the outage of line ‘a’. Thermal limits of transmission lines and transformers, which may be relevant under outage conditions, have been ignored. Clearly, under leading power factor operation the reactive power absorbed by the generators must be limited. Such limits would need to be determined by further studies.
Figure 7.46 All feasible Cases, C01 to C95: Bode plots of the terminal voltage when the feedback path is open on generator #1 with PID Set No. 4.
C42 C44 C46 C50
C51 C55 C56 C60
C61 C65 C17
10−2 10−1 100 101 102
−200
−180
−160
−140
−120
−100
−80
Frequency (rad/s)
Phase (deg)
10−2 10−1 100 101 102
−80
−60
−40
−20 0 20 40 60
Magnitude (dB)
C01 C05 C06 C10
C11 C15 C16 C20
C21 C25 C17
10−2 10−1 100 101 102
−200
−180
−160
−140
−120
−100
−80
Frequency (rad/s)
Phase (deg)
10−2 10−1 100 101 102
−80
−60
−40
−20 0 20 40 60
Magnitude (dB)
(a) One unit, (b) Two units, (c) Three units.
PID Set No.4 installed on all generators Solid lines: Maximum lagging reactive power output.
Dashed lines: Maximum leading reactive power output.
Unit real power output is 50 MW in all cases ex- cept for C21-25, C61-65 and C92-95 when it is 25 MW.
Cases C76, C80, C81, and others are omitted be- cause operating constraints are infringed and are infeasible. For some other cases the reactive pow- er output per generator is reduced, e.g. from to Mvar for Case 79.
Case C17 is adopted as the Base Case and is shown in all three sets of plots.
10 – 5
–
C72 C75 C77 C79
C82 C84 C86 C88
C92 C95 C17
10−2 10−1 100 101 102
−200
−160
−120
−80
Frequency (rad/s)
Phase (deg)
10−2 10−1 100 101 102
−80
−40 0 40
Magnitude (dB)
(c)
(a) (b)
VtrnjfVrefjf
Sec. 7.11 Tuning AVRs with type 2B PID compensation 381 7.11.2.3.2 System eigenvalues when generators are under closed-loop voltage control While the Phase Margins derived from the Bode plots show that the system is stable, an ex- amination of the eigenvalues for the rotor modes reveals the degree of stability of these modes. The most onerous conditions most likely to yield rotor angle instability are the cases for which line ‘a’ in Figure 7.40 is out of service and the load at bus 5 is off, i.e. the rated output of the station is carried over line ‘b’. The local and inter-machine modes for one or more units on-line are seen in Figure 7.47.
Figure 7.47 Eigenvalues for one, two and three units on-line. Line ‘a’ is out-of-service and the load at bus 5 is off. Conditions are shown for feasible maximum lagging and leading
reactive power outputs at rated real power. PID Set No. 4 installed on all generators.
As shown in Figure 7.47, when three machines are on-line at rated real power output the 5 s halving time is breached, or nearly breached (cases C86-C88). To provide an adequate mar- gin of stability for the most onerous condition it is therefore necessary to install power sys- tem stabilizers on the generators. (This is not considered here.)
7.11.2.3.3 Step responses for the range of feasible operating conditions; one, two or three units on-line.
Based on the PID parameter Set 4 in Table 7.6, let us determine the terminal voltage re- sponse of the closed-loop system to a +1% step change in reference voltage of generator
#1 over the range of operating conditions C01 to C95 considered in Figure 7.42.
The perturbations in the generator # 1 terminal voltage from its initial steady-state value are shown in Figure 7.48. For each of the Case sets in Table 7.9, e.g. C01 - C05, C50 - C60, only the maximum feasible lagging and the maximum leading reactive power cases are plotted.
We observe the following.
1. All Cases C01 to C95 satisfy the terminal voltage settling-time criterion. As intended, the choice of Case C17 as the base case results in a satisfactory set of responses. The
−1.5 −1 −0.5 0
0 2 4 6 8
Imaginary Part (rad/s)
Real Part (Np/s) BA
BA C
D
BA DC
Local Mode
A - Maximum lagging B - Maximum leading Inter-machine Mode
C - Maximum lagging D - Maximum leading
Halving time
=5 s
One unit Two units Three units
overshoot of the terminal voltage response at leading power factors may be of con- cern when one unit is on-line.
2. In Figure 7.46(a) the phase of the extreme Case C21 (25 MW 25 Mvar lag) is some less than that of Case C17 at a gain cross-over frequency of ~1 rad/s. Conse- quently the phase margin for Case C21 is likely to be 65+20 = , a value which results in an over-damped response - as is evident in Figure 7.48(a). The converse argument applies to Case C20 (50 MW 20 Mvar lead), i.e. the resulting step response is lightly damped.
Figure 7.48 Closed-loop operation. The perturbations are shown in the terminal voltage of generator #1 due to a step change in reference voltage from a steady-state value of 1.0 pu to 1.01 pu (1%) for the feasible operating conditions. Responses lie within the 10% of the
final value of the step amplitude in less than 5 s.
The lack of damping at leading power factors, highlighted in Section 7.11.2.3.1, results in the excessive over-shoot of the terminal voltage responses when one or two generators are on- line. The otherwise satisfactory small-signal performance of the three-generator power sys- tem based on parameter set No. 4 for PID Type 2 compensation (Table 7.6) is demonstrated in Figure 7.48 for N and N-1 conditions. Studies examining the provision of PSSs for the generators, the limiting of reactive power absorption by the generators, and the performance
20
85
0 2 4 6 8 10
0 0.2 0.4 0.6 0.8 1 1.2
ΔVt (%)
Time (s)
C01 C05 C06 C10
C11 C15 C16 C20
C21 C25 C17
0 2 4 6 8 10
0 0.2 0.4 0.6 0.8 1 1.2
ΔVt (%)
Time (s)
C42 C44 C46 C50
C51 C55 C56 C60
C61 C65 C17
0 2 4 6 8 10
0 0.2 0.4 0.6 0.8 1 1.2
ΔVt (%)
Time (s)
C72 C75 C77 C79
C82 C84 C86 C88
C92 C95 C17
(b) (a) One unit, (b) Two units, (c) Three units.
Solid lines: Maximum lagging reactive power output.
Dashed lines: Maximum leading reactive power output.
Unit real power output is 50 MW in all cases except for C21- 25, C61-65 and C92-95 when it is 25 MW.
Cases C76, C80, C81, and others are omitted because oper- ating constraints are infringed and are infeasible. For some other cases the reactive power output per generator is re- duced, e.g. from -10 to -5 Mvar for Case 79.
Case C17 is adopted as the Base Case and is shown in all three sets of plots.
(a)
(c)
Sec. 7.12 Summary, Chapter 7 383 of the system for major disturbances would be undertaken in practice, but are beyond the scope of this chapter.