JOAN PRATS I CATALÀ
III. Reforma institucional y reforma organizativa en la reforma del Estado
As usual in electrical machine control, two regions of speeds are considered. In the first region, the speed of the machine is less than or equal to the rated (base) speed. In this region, the applied voltage (Vb)
changes proportionally with the frequency (fb) so that Vb/fb is constant.
In the second region, the speed of the motor is higher than the rated value and Vb is kept constant at the rated value [31].
In this section, we show the influence of including and neglecting the magnetic saturation in the inductances of the SynRM model at steady state operation for several speeds. In this analysis, the SynRM performance is investigated in open loop control. Model 3, with unsaturated Ld and Lq is compared with model 2, where the magnetic
saturation is included. Two cases of the unsaturated model 3 are investigated.
The first case considers three different q-axis inductance (Lq) values
while the d-axis inductance (Ld) value is fixed. The values of Lq are
0.0051 H, 0.0037 H and 0.0032 H and Ld=0.0203 H. The selection of
0 0.5 1 1.5 2 2.5 3 -70 -50 -30 -100 10 30 50 70 t[s] iabc [A ] 0.5 0.51 0.52 0.53 0.54 0.55 -30 30 t[s] iabc [A ] (b) (a)
Ld=0.0203 H is to represent approximately the average value of Ld in
the linear region (neglecting saturation and cross-saturation effects) of the d-axis flux linkage, see Fig 2.8. The selection of the three q-axis inductance values is to represent approximately the average value of Lq
in the linear, knee and saturated regions of the q-axis flux linkage respectively, see Fig. 2.8. The second case considers three different d- axis inductance (Ld) values while the q-axis inductance (Lq) value is
fixed. The values of Ld are 0.0110 H, 0.0152 H and 0.0203 H and the
value of Lq is 0.0051 H. The following paragraphs give a brief summary
of the results. A detailed analysis of these two cases is provided in the
Appendix A.
Figure 2.22 and Fig. 2.23 show the results of the first and second case respectively.
Figure 2.22: Variation of SynRM maximum torque Tm (a) and power
factor PFm at Tm (b) with different speeds ωr for
unsaturated (different Lq and Ld=0.0203 H) and saturated
(blue solid-line) machines. (a) and (b) have the same legend. 0 200 400 600 800 1000 1200 0 10 20 30
!
r[rad/s]
T
m[N
.m
]
L q=0.0051 H L q=0.0037 H L q=0.0032 H L q Saturated 0 200 400 600 800 1000 1200 0.5 0.6 0.7 0.8!
r[rad/s]
P
F
m (a)base speed line
41 2.8 Performance of the SynRM at different speeds including flux weakening
Fig. 2.22 shows the variation of the maximum torque Tm of the
SynRM at different speeds from 10% up to 200% of the rated value for the saturated (Model 2) and unsaturated (Lq= 0.0051 H, 0.0037 H and
0.0032 H at Ld=0.0203 H) machines. The region below the curves in
Fig. 2.22-(a) as well as in Fig. 2.23-(a) represents the region where the machine can work stably and synchronize with the supply frequency, while the region above the curves shows the instability region (in the direction of the plotted arrow in the figures). The stability region of the unsaturated machine increases with decreasing Lq because of increasing
the saliency ratio (Ld/Lq). Moreover, the machine considering the
magnetic saturation has the larger stability region (the blue solid-line) for all the considered speeds.
The machine power factor at the maximum torque Tm for different
speeds is shown in Fig. 2.22-(b). The machine considering the magnetic saturation (blue solid-line) has a better power factor compared to the unsaturated cases for all speeds less or equal than the rated value. However, the machine with Lq=0.0032 H (magenta dashed-line) has the
best power factor for speeds higher than the rated value.
Figure 2.23-(a) illustrates the variation of the maximum torque Tm of
the SynRM as a function of the speed, ranging from 10% to 200% of the rated value. Curves are shown for saturated (Model 2) and unsaturated (Ld=0.0110 H, 0.0152 H and 0.0203 H at Lq=0.0051 H)
machines. It is evident that the machine including the magnetic saturation (blue solid line) has a higher stability region. On the other hand, the variation of the Ld at constant Lq has a lower influence on the
stability region compared to Fig. 2.22 where the Lq varies at constant Ld. Figure 2.23-(b) shows the variation of the power factor of the
SynRM for different speeds at the maximum torque Tm. The saturated
machine has the best power factor for all the considered speeds.
Notice that the maximum torque (Tm) in Figs. 2.22 and 2.23 should
be constant for all speeds up to the rated value. However, it has a slightly lower values for low speeds compared to the value at rated speed. This influence may be due to the inaccurate representation of the stator resistance and the error in the V/f control for low speeds.
Figure 2.23: Variation of SynRM maximum torque Tm (a) and power
factor PFm at Tm (b) with different speed ωr for unsaturated
(different Ld and Lq=0.0051 H) and saturated (blue solid-
line) machines. (a) and (b) have the same legend.
2.9 Conclusions
This chapter has investigated deeply the modelling of SynRMs, taking into account the magnetic saturation and rotor position effects in open loop and closed loop controlled methods. Moreover, the stability limits of operation for the SynRM have been indicated. A simple and very fast efficient model for the SynRM has been proposed based on an accurate representation of the dq-axis flux linkages. The dq-axis flux linkages are computed from FEM, considering the magnetic saturation and rotor position effects. Look-up tables (LUTs) are generated for the dq-axis flux linkages and can be used in the simulations of the SynRM, obtaining an accurate prediction for its performance and control.
Three models of the dq-axis flux linkages are investigated based on an open loop controlled method:
0 200 400 600 800 1000 1200 0 10 20 30
!
r[rad/s]
T
m[N
.m
]
Ld=0.0203 H Ld=0.0152 H Ld=0.0110 H Ld Saturated 0 200 400 600 800 1000 1200 0.4 0.6 0.8!
r[rad/s]
P
F
mbase speed line
(a)
43 Biography
Model 1: Considering the magnetic saturation and rotor position effects.
Model 2: Considering only the magnetic saturation, without the rotor position effect.
Model 3: Considering constant values for both Ld and Lq.
It is found that the SynRM torque capability and stability operation region depend mainly on the dq-axis flux linkages characteristics. Including magnetic saturation in the model of a SynRM is mandatory to have an accurate prediction of its performance (output torque, power factor and stable region of operation). This means choosing constant inductances (Ld and Lq) to represent the SynRM in a very simple way,
is not good enough and can lead to a large deviation in the prediction of the torque capability compared with the real motor. However, the rotor position has almost no influence on the SynRM torque capability or stability region.
In the closed-loop controlled method, it is noticed that considering the magnetic saturation effect on the control of the SynRM results in an 8% increase in the output torque compared to neglecting the saturation effect for the same conditions.
Finally, the SynRM torque capability and power factor have been indicated at several speeds from 10% up to 200% of the rated value; showing the necessity of including the magnetic saturation in the SynRM modelling for accurate performance prediction.
Biography
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