5 SIMULATION AND TEST RESULTS
5.2 Simulation results of excitation control
5.2.2 Excitation control in the field weakening range
In the field weakening range the flux must be reduced as a function of the rotating speed and a larger armature reaction, than in the nominal speed range, is obtained. Considering the excitation controller, this means, that it takes some more effort to respond to a certain level torque step in the field weakening range.
In the both simulation and laboratory tests a modified voltage reserve coefficient kres1 has been used. Unlike the coefficient kres defined in Eq. (4.21), the modified voltage reserve coefficient defines that particular voltage reserve, which can be used totally for the stator flux linkage acceleration by excluding the resistive voltage drop in the stator winding from the available stator voltage,
kres1= s,i ≈ n s sr
− e
u
e
u i , n ∈{1...6}. (5.7)
The approximation shown by Eq. (5.7) has bee used. It assumes, that the voltage vector and the current vector have approximately the same direction. The stator flux linkage reference in the field weakening range is calculated as
( )
ψs ref kres r
m n s s
= 1 −
ω u i (5.8)
In order to see the effect of the excitation control during a torque transient in the field weakening, three different excitation control methods has been simulated. Also the dynamic performance of the excitation unit has been varied by limiting the field current reference value with a ramp. In following examples rotating speed n=2.0 pu has been used, which requires a stator flux linkage modulus reduction larger than 50 % from the nominal value. The voltage reserve coefficient kres1=0.84 has been used.
In the first case an excitation control method with combined open loop and feedback control has been used. The field current dynamics has been adjusted by limiting the current slope with a ramp, where the slope has been defined by the ramp time Tex. Three different ramp times 0.2 pu/ms, 0.1 pu/ms and 0.05 pu/ms has been used. Fig.(5.11) shows the field current in different cases, Fig.
(5.12) shows the corresponding torque responses and Fig. (5.13) the corresponding load angle behaviour, when different field current slopes are used. The difference of the torque response between case A (Tex=0.2 pu/ms) and case B (Tex=0.1 pu/ms) is not significant. This is due to the fact, that the d-axis damper winding is able to keep the load angle below the limit value 85°deg, as illustrated in Fig. (5.13). If the excitation system dynamics is further reduced, case C (Tex=0.05 pu/ms), the load angle is getting too high and it must be limited in order to maintain the synchronism. A load angle limitation reduces significantly the torque response. In this case the direct load angle limitation based on the fast torque reference adjustment has been used (see also Fig. (5.30)).
0 2 4 6
580 620 660 700
[pu]
t[ ms]
if,A
if,B
if,C
Figure 5.11 A field current response in a 150 % torque step at the time instant t=600ms with three different slopes Tex= 0.2 pu/ms (A), Tex= 0.1 pu/ms (B) and Tex= 0.05 pu/ms (C). Speed n = 2.0 pu, kres1=0.84.
0 1 2
580 620 660 700
|ψs|A
[pu]
t[ ms]
|te|A
|te|B
|te|C
Figure 5.12 The torque response in a 150 % torque step at the time instant t=600ms with three different field current slopes corresponding to Fig. (5.11). Speed n = 2.0 pu, kres1=0.84. Also the stator flux linkage modulus in case A, |ψs|A has been shown. |ψs|A must be reduced due to the increasing voltage drop in the stator resistance. The fast field current response (A) gives the best torque response.
0
Figure 5.13 The load angle development in a 150 % torque step at the time instant t=600ms with three different field current slopes corresponding to Fig. (5.11). Speed n=2.0 pu, kres1=0.84. A fast field current response keeps the load angle smallest (A), in case of a the slowest field current response the load angle must be reduced actively (C).
A fast excitation control unit may not always be available. If a fast torque response in the field weakening is however required, the excitation slowness must be taken into account, when selecting the excitation method. One possibility is to use the partial load over excitation introduced before. If the maximal stator current value and the stator flux linkage modulus are known, the required excitation curve can be solved using Eq. (2.45). Fig. (5.14) shows the excitation curves for unity power factor, curve A, and for the constant d-axis air gap flux linkage corresponding to the maximal working point with |te|max=1.55 and |ψs|=0.33, curve B. The curve B has been solved from Eq. (2.45) with the given values and further approximated by using the polyfit function of MATLAB,
if,ref =0 705. ⋅te ref2 +0 497. ⋅ te ref +2 754. . (5.9)
One drawback of the excitation according to curve B occurs to be the high currents both in the stator and the rotor with small loads and thus increased losses. Another drawback of curve B is the numerical complexity involved with the iterative calculation. Note, that the approximation given in Eq. (5.9) is valid only for the given values |te|max=1.55 and |ψs|=0.33. Curve C represents a compromise between the unity power factor excitation and the constant d-axis air gap flux excitation, and is calculated by using the over excitation coefficient kex
i k
The excitation according to Eq. (5.10) does not significantly increase losses with small loads compared to unity power factor excitation, but improves stability with high loads. Also the numerical implementation is simple and it does not require lots of calculation power. In Fig. (5.14) over excitation coefficient kex=0.6 has been used. The relative over excitation below refers to this excitation method.
0 0 .2 0 .4 0 .6 0 .8 1 1 .2 1 .4 1 .6
0 1 2 3 4 5 6
te[p u]
A
if [p u]
B C
Figure 5.14 Different excitation curves with |te|max=1.55 and |ψs|=0.33. A nominal excitation curve (A), constant d-axis air gap flux linkage (B), Eq. (5.9) and relative over excitation (C), Eq. (5.10).
All curves minimise the stator current in the maximal working point, where |is|=4.65.
Figs. (5.15) and (5.16) show a torque step when the field current reference according to Eq. (5.9) has been used. The torque and the excitation curve slopes have been synchronised. The d-axis damper winding peak value during the transient is small and the magnetic energy oscillates only with a very small amplitude. Since the d-axis air gap flux component is kept approximately constant, the rate of saturation is also kept constant, which partially reduces oscillations and improves dynamic current model accuracy on the d-axis. The torque response is very good despite of a small voltage reserve on the stator side, and the load angle reacts well during the transient.
-6 -4 -2 0 2 4 6
580 620 660 700
if
iD
id
[pu]
t[ m s]
Figure 5.15 A torque step 150% at the time instant t=600ms with approximately constant d-axis air gap flux linkage, speed n=2.0 pu, kres1=0.84. The field current reference has been calculated using Eq. (5.9). The field current reference and the torque reference have been synchronised. Field current if, d-axis damper winding current iD and stator current id.
0 0.5 1 1.5 2
580 620 660 700
|te| [pu]
[rad]
δ
ψmd
|ψs|
t[ ms]
uDC
Figure 5.16 The torque step corresponding to Fig. (5.15). Stator flux linkage modulus |ψs|, d-axis air gap flux linkage ψmd, load angle δ, torque response |te| and DC link voltage uDC. The torque response is fast and the load angle achieves a new steady state value immediately after the torque step.
When using the relative over excitation according to Eq. (5.10), the torque dynamics is reduced and a larger d-axis damper winding current is generated. However, the same well behaving load angle is obtained. The torque step with this excitation method has been shown in Figs. (5.17) and (5.18).
-6 -4 -2 0 2 4 6
580 620 660 700
[pu]
t[ ms]
if
iD
id
Figure 5.17 A torque step 150% at the time instant t=600ms with the relative over excitation according to Eq. (5.10), kex=0.6, speed n=2.0 pu, kres1=0.84. Field current reference and the torque reference have been synchronised. Field current if, d-axis damper winding current iD and stator current id.
0 0.5 1 1.5 2
580 620 660 700
[pu]
[rad]
t[ ms]
uDC
|te|
δ ψmd
|ψs|
Figure 5.18 The torque step corresponding to Fig. (5.17). Stator flux linkage modulus |ψs|, d-axis air gap flux linkage ψmd, load angle δ, torque response |te| and DC link voltage uDC. Torque response is reduced compared to the previous case, Fig. (5.16), where the d-axis air gap flux linkage has been kept constant. Also a larger DC link voltage oscillation can be observed. The load angle is kept small due to the relative over excitation.
In Figs. (5.16) and (5.18) also the DC link voltage has been shown. In both cases significant oscillation are seen, which is one reason also for the magnetic energy oscillation. Sudhoff et al (1998) has shown a method for damping DC link oscillations. DC link control was not a subject in this work and damping methods were not tested. However, this is a typical real world limitation, which must be taken into consideration in the control design. The electrical drive should be considered as a system, where single controllers should not be sensitive to each others unideal behaviour.
Fig. (5.19) and (5.20) still compares the torque response and load angle behaviour during the transient with the described three different excitation methods. The large initial air gap flux linkage gives a fast torque response and keeps the drive stable. If the relative over excitation is used according to Eq. (5.10), the load angle is kept smaller than when the unity power factor excitation is used, but the torque response remains almost equal.
0 0.5 1 1.5
580 620 660 700
[pu]
t[ ms]
|te|A
|te|B
|te|C
Figure 5.19 A comparison of torque responses with different excitation methods. Speed n=2.0 pu, kres1=0.84. Constant d-axis air gap flux linkage (A), relative over excitation according to Eq. (5.10) with excitation coefficient kex=0.6 (B) and the unity power factor excitation with Tex=0.2 pu/ms (C).
0 0.5 1 1.5
580 620 660 700
[rad]
t[ ms]
δΑ δΒ
δC
Figure 5.20 A comparison of load angle behaviour by different excitation methods during a torque step corresponding Fig. (5.19). Speed n=2.0 pu, kres=0.84. Constant d-axis air gap flux linkage (A), relative over excitation according to Eq. (5.10) with excitation coefficient kex=0.6 (B) and unity power factor excitation (C).