(d) Type E(e) Type F(f) Type G

Figure 3. Flux trajectory in rotor coordination for unsymmetrical voltage sag (types B, C, D, E, F, and G), for sag duration of 2, 2.25, and 2.5 cycles.

sag be a quarter of a period plus any number of half cycles of the supply, the flux has maximum distance to the initial value, therefore torque transients when the voltage is restored will be maximum.

Flux trajectory for sag types C, E, and G detours from the circle, caused by terms with twice the supply frequency. This distortion appears in torque oscillations too. For these types, the effect of sag duration on peak torque at the voltage restoration moment is as same as type A.

For sag type F, flux trajectory distortion is the severest and a small circle appears in large circle turns in opposite direction. Nevertheless, the influence of sag duration on peak torque when the voltage restores is similar to sag types B and D.

4. Simulation

In order to observe the responses, a salient-pole 4150 kVA synchronous machine is modeled using generalized Park model [10] in MATLAB/SIMULINK based on [11], and different types of voltage sags with s = 0.5 are applied to it. The machine model details can be found in [10]. The motor parameters and rated quantities are given in Appendix B.

Before voltage sag, the machine consumes 1 pu real power in nominal terminal voltage at unity power factor. In t = 0.5 s, sag begins by changing motor terminal voltages. During voltage sag, load torque and excitation voltage are assumed to be constant. Figure 4 shows electrical torque, for type A voltage sag with duration of 10, 10.25, and 10.5 cycles.

As we expect, there is damped torque pulsation with supply frequency during and after voltage sag and the worst case for peak torque when voltage amplitude restores occurs when sag duration equals (0.5 + n)T, that T

(a) Sag duration = 10 cycles

(b) Sag duration = 10.25 cycles

(c) Sag duration = 10.5 cycles

Figure 4. Electrical torque for type A voltage sags with duration of (a) 10; (b) 10.25; and (c) 10.5 cycles.

is supply period-time.

Since the results for types B and D are identical, we only represented the curves for type B in Figure 5. The curves show that there are undamped torque oscillations during voltage sag. In this case, the worst peak torque occurs for a voltage sag with a duration of (2n + 1)T∕4.

Figure 6 shows the results for voltage sag of type C. It is observed that the torque curve has been deformed based on flux deformation. Like type A, the torque transient when the voltage is restored will be maximum, when the duration of voltage sag is a multiple of the period-time of the supply plus half a cycle.

For type F, the flux trajectory in complex coordination was different from other types. As we see in Figure 7, torque oscillations have been more deformed similar to its flux trace. Whereas in this case, the effect of the sag duration on the torque peak at the voltage recovery moment is almost as same as type B.

The difference will be more obvious when we change the sag duration and plot peak torque after restoring voltage magnitude as in Figure 8. In the case of type B voltage sag, peak torque changes sinusoidal respect to sag duration with a frequency twice the supply frequency that certificates last results. But for type F, it has a periodic form that has two local minimum per cycle, originnate from inner circle and outer circle in the flux trajectory. Whereas, its maximums occur at the durations similar to type B.

5. Conclusions

The effects of different sags types on the synchronous machine torque transient have been studied by the flux

(a) Sag duration = 10 cycles

(b) Sag duration = 10.25 cycles

(c) Sag duration = 10.5 cycles

Figure 5. Electrical torque for type B voltage sags with duration of (a) 10; (b) 10.25; and (c) 10.5 cycles.

trajectory analysis. In the analysis, the sag magnitude and the point of voltage wave that sag begins are not included.

Varying flux trajectory as sag type varies, the torque pulsation during and after voltage sag will be different. For sags types A, C, E, and G, if the duration of sag be a multiple of the period-time of the supply plus a half cycle, the peak torque at the voltage restoring moment will be maximum. For types B and D, if the duration of sag be a quarter of a period plus any number of half cycles of the supply, peak torque when the voltage is restored will be

(a) Sag duration = 10 cycles

(b) Sag duration = 10.25 cycles

(c) Sag duration = 10.5 cycles

Figure 6. Electrical torque for type C voltage sags with duration of (a) 10; (b) 10.25; and (c) 10.5 cycles.

(a) Sag duration = 10 cycles

(b) Sag duration = 10.25 cycles

(c) Sag duration = 10.5 cycles

Figure 7. Electrical torque for type F voltage sags with duration of (a) 10; (b) 10.25; and (c) 10.5 cycles.

(a) For sag of type B (b) For sag of type F

Figure 8. Peak torque when voltage magnitude restores vs. sag duration for voltage sags of type B and F.

maximum. For sag type F, the relation between peak torque and sag duration is different, but its worst case for peak torque is as same as types B and D.

With obtained results we can identify the fault types that have the more intensive effect on the machine. After considering motor behavior against torque pulsations, the preventive decision can be made, Such as changing the transformer or load connections to change the voltage sag type or changing protective relays settings to prevent unessential tripping or damage to the shaft.

REFERENCES

- B. W. Kennedy, “Power Quality Primer,” McGraw-Hill, New York, 2000.
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- C. Sankaran, “Power Quality,” CRC Press LLC, New York, 2002.
- IEEE Standard, “IEEE Recommended Practice for Monitoring Electric Power Quality1159-1995,” Institute of Electrical and Electronics Engineers, New York, 1995.
- M. H. J. Bollen, “IEEE tutorial on Voltage Sag Analysis,” IEEE Press TP139-0, New York, 1999.
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- J. C. Das, “Effects of Momentary Voltage Dips on the operation of induction and Synchronous Motors,” IEEE Transaction on Industry Applications, vol. 26, No. 4, 1990, pp. 711-718. doi:10.1109/28.55998
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Appendix A

For symmetrical sags, the magnitude of sag is the remained RMS voltage in per unit or percent of rated voltage that expressed by “s”. For unsymmetrical sags, “s” is a coefficient that appears in voltage equations and causes difference in their magnitude and/or phase angles. Voltage equation for phase “a” at sag occurrence moment is as. Voltage equations for different sag types in synchronous frame are expressed as:

Type A:

(A.1)

Type B:

(A.2)

Type C:

(A.3)

Type D:

(A.4)

Type E:

(A.5)

Type F:

(A.6)

Type G:

(A.7)

Appendix B

Ratings and parameters of 4150 kVA synchronous motor are stated in Table B1 [12].

Table B1. Rated data and parameters of synchronous machine.