Analysis Of Prevention Of Induction Motors Stalling By Capacitor Switching

16th NATIONAL POWER SYSTEMS CONFERENCE, 15th-17th DECEMBER, 2010 capacitor is chosen so that the switching point comes inside the region, it will ultimately accelerate and settle at its stable equilibrium point. If the switching choice is such that it falls either in the region DEF or BAG at the instant of switching, the machine will be in the stable region with Peout Pm . Hence, it will accelerate for a short time and settle at the stable equilibrium point. If the switching choice is such that it falls within the regions DEH or BAI, then at the instant of switching, the motor torque output will be less than the load and the machine being in the stable region continues to decelerate and ultimately settle at the stable operating point(at a higher slip). Note that these two regions are outside the region determined by the relation (7). Hence the condition given by (7)(motor power load power at the instant of switching) is only a sufficient condition and not a necessary condition. This condition becomes a necessary condition if the slip at the instant of switching is greater than smaxpm corresponding to the post switched condition. Hence, we see that the condition for determining the magnitude of the capacitance turns out to be different depending on whether the slip at the instant of capacitor switching is lesser than smaxpm or otherwise. If the switching choice is such that the corresponding point lies either on segment DE or AB, then the switching slip will be the final slip at which the rotor will settle and therefore there will be no additional dynamics due to the capacitor switching. For a particular post disturbance Pmf inal the following can be noted. 262 B ω C2 (smax,ω C2max ) C (scritical,ω Ccritical ) A ωC sinitial E (smin,ω C1min ) D 0 ω C1 1 sswitch Fig. 2. Variation of ωC1 and ωC2 with sswitch for a particular post disturbance Pm B I C ωC A G F 1) If sinitial sswitch smin , the value of ωCswitch has to be within the boundary ωC1min ωCswitch ωC2max . Note that this value is different from that obtained from (9). 2) If smin sswitch smax , then ωCswitch should be strictly greater than ωC1 at sswitch and lesser than or equal to ωC2max (not the ωC2 computed from (9)). 3) If smax sswitch scritical , then ωCswitch should be strictly greater than ωC1 and strictly lesser than ωC2 at sswitch . 4) For sswitch scritical the stalling of induction motor cannot be prevented by insertion of any value of capacitance. For different values of Pmf inal , contours similar to that shown in the Fig 2/Fig 3 can be obtained. A family of ωC Vs s characteristics has been obtained and shown in Fig 4. It can be observed from the Fig 4 that for higher values of Pmf inal , the region within the contour shrinks. The slip scritical occurs at a lower value for higher values of Pmf inal . This study of switching capacitances at various slips, based on the results in Figs 2, 3 and 4 is extremely useful in getting an insight into the process of stabilization. This also provides us the theoretical limits for the possible switching options, the basis for such limits, as well as the nature of the post switching stabilization process. E H D slip Fig. 3. Fig. 4. ωC Vs s contour divided into different switching regions ωCswitch Vs sswitch contour for various values of Pmf inal Department of Electrical Engineering, Univ. College of Engg., Osmania University, Hyderabad, A.P, INDIA.

16th NATIONAL POWER SYSTEMS CONFERENCE, 15th-17th DECEMBER, 2010 III. N UMERICAL R ESULTS The actual choice of the switching strategy in a particular situation has to be made based on the theoretical consideration discussed earlier; in addition, issues such as the practical availability of the designed capacitor sizes, acceptable over voltages, limits on rates of acceleration/deceleration must also be taken into account. In this section, we illustrate the concepts presented above considering the system shown in the Fig 1. We choose a particular value of Pmf inal and obtain the ωC Vs s contour similar to that given in the Fig 3. Then we choose various switching options so as to have points from all the regions of the C s plane and obtain the dynamic behavior of the system and show how the general nature of this dynamic behavior can be predicted based on the location of the switching point in Fig 3. The initial operating conditions of the system in the Fig 1 are given as follows. Pe 0.2967, Qe 0.0045, V 0.9946, θ 0.0896, s 0.003. The system parameters are given as follows [1]. E 1.0, Xe 0.3, Rr 0.01 and Xr 0.05. Pm is suddenly increased to 1.47 p.u. The ωC Vs s contour ABCDE for Pmf inal 1.47 is obtained solving (9) for various values of s and is shown in the Fig 5. The values of scritical , smin , smax , ωC1min and ωC2max are identified for this Pmf inal and marked in the diagram. The lines DH and IB are drawn parallel to the s axis. The lines BG and F D are determined by finding the slip at which the maximum power occurs considering various values of the switching capacitance between ωC1min and ωC2max . These lines/contours divide the s C plane to a number of regions. One value of capacitance and slip from each of these regions is chosen to understand the dynamics due to these capacitance-slip switching options. The capacitance values and the slips at which the capacitances are switched(sswitch ) are given in the Table I. Consider the switching capacitance and the switching slip corresponding to the point 38 given in the Table I. The induction motor Pm is increased to 1.47 p.u at t 0.001s. The switching capacitance corresponding to the point 38 is 7.2 and the slip is 0.075. The capacitor is inserted when the slip of the motor reaches 0.075 at t 0.3228s. The pre switching and post switching Pm Vs s characteristics are given in the Fig 6. It can be seen from the Fig 5 that the point 38 lies in the region BCDF G of the contour. It can be seen from the Fig 6 that at the instant of switching, Pm 1.4952 which is greater than 1.47 at sswitch 0.075, and it lies in the unstable region of the motor Pm Vs s characteristics. Since the Pm at sswitch is greater than 1.47, the motor will accelerate. The final settling point is at a slip of 0.0266 which is the intersecting point of the motor Pm Vs s characteristics and the load Pm Vs s characteristics corresponding to Pm 1.47 in the stable region of the motor characteristics as evident from the Fig 6. All these features can be qualitatively predicted by observing that the point 38 lies inside the contour. To study the dynamics of the rotor due to capacitor switching, the variation of the slip of the induction motor with time for this switching option has been obtained and is given in 263 Fig 7. It can be observed from the Fig 7 that as soon as the initial load Pm is increased, the slip starts increasing. When the capacitor is switched, at t 0.3228s, the slip starts decreasing at a slower rate. However, after t 0.45s , the slip starts decreasing at a faster rate, crosses s 0.0457, which is the slip at the maximum Pm for this value of switched capacitance and settles down at a slip of 0.0266 at t 0.82s. We have studied the impact of switching the capacitances for each of the 25 points marked in the C s plane in Fig 5 in the similar manner. The summary of the impact of switching each of the capacitance values at their respective switching slips is given in the Table I. For each case, most of the qualitative and quantitative aspects of the post switching behavior are tabulated in Table I. From the results in Table I, it is easy to see that the predicted post switching behavior based on Fig 3 in the section II is actually realized. 9 18 22 8 I 28 B 21 12 26 7 11 6 ωC 32 36 smax 0.0567 31 ω C2max 7.9638 38 scritical 0.0808 ω C1 ω C2 6.1396 17 C 35 5 A 15 33 25 4 G 3F 34 2E smin 0.0266 ω C1min 0.1597 41 16 1 14 13 D H 0 0 23 24 0.01 0.02 0.03 0.04 37 27 0.05 0.06 0.07 0.08 0.09 slip Fig. 5. ωC Vs s characteristics for Pmf inal 1.47p.u at t 0.001s 1.8 (0.0457,1.6978) 1.6 (0.075,1.4952) 1.4 1.2 Pm Pm 1.47 (0.0278,1.3883) (0.0266,1.47) 1 (0.075,0.8792) 0.8 0.6 pre switching characteristics 0.4 s 0.075 0.2 0 Fig. 6. 0 0.05 slip 0.1 0.15 Induction motor Pm Vs s characteristics for ωC 7.2 IV. E XTENSION TO P RACTICAL S ITUATIONS The procedure presented in the last section to determine the capacitances to be inserted considering a simple equivalent Department of Electrical Engineering, Univ. College of Engg., Osmania University, Hyderabad, A.P, INDIA.

16th NATIONAL POWER SYSTEMS CONFERENCE, 15th-17th DECEMBER, 2010 264 TABLE I S UMMARY OF CAPACITOR SWITCHING Point 11 12 13 14 15 16 17 18 21 22 23 24 25 26 27 28 31 32 33 34 35 36 37 38 41 ωCswitch 6.8096 7.1 0.2531 0.125 4 0.7 6 8.2 7.7241 8.1 0.481 0.24 4 7.2 0.075 7.9 7.4537 8 4.1445 3.8 5 7.8 0.075 7.2 0.5 -:For FOR THE POINTS ON THE CONTOUR IN THE F IG 5 Contour region sswitch Pm at sswitch Pmmax smaxpm sswitch pos sf inal On the segment AB 0.02 1.47 1.8547 0.0403 Lesser 0.02 BAI 0.02 1.2945 1.7348 0.0443 Lesser 0.0247 On the segment DE 0.02 1.47 1.5217 0.026 Lesser 0.02 outside the contour 0.02 1.3889 1.4515 0.0269 Lesser unstable BCDF G 0.02 5.2072 8.5612 0.0068 Greater 0.0006 DEF 0.02 1.8062 1.8214 0.0227 Lesser 0.0116 BAG 0.02 2.1529 2.331 0.0298 Lesser 0.0106 outside the contour 0.02 0.8384 1.4147 0.0603 Lesser unstable On the segment AB 0.04 1.47 1.5326 0.0532 Lesser 0.04 outside the contour 0.04 1.3332 1.4374 0.0588 Lesser unstable On the segment BCD 0.04 1.47 1.6626 0.0243 Greater 0.04/0.0147 outside the contour 0.04 1.3808 1.5143 0.0261 Greater unstable BCDF G 0.04 2.7714 8.5612 0.0068 Greater 0.0006 BAG 0.04 1.6823 1.6978 0.0457 Lesser 0.0266 outside the contour 0.04 1.3228 1.4257 0.0272 Greater unstable BAI 0.04 1.4043 1.486 0.0558 Lesser 0.0483 On the segment BCD 0.075 1.47 1.6123 0.0493 Greater 0.075/0.0321 outside the contour 0.075 1.4056 1.4612 0.0573 Greater unstable On the segment BCD 0.075 1.47 7.0783 0.0084 Greater 0.075/0.0009 outside the contour 0.075 1.4308 12.1305 0.0048 Greater unstable BCDF G 0.075 1.5403 3.5708 0.0179 Greater 0.0039 outside the contour 0.075 1.4306 1.512 0.0543 Greater unstable outside the contour 0.075 0.8892 1.4257 0.0272 Greater unstable BCDF G 0.075 1.4952 1.6978 0.0457 Greater 0.0266 DEH 0.01 1.1788 1.6754 0.0242 Lesser 0.0144 23, 31 and 33 ideally sf inal sswitch . But the motor will accelerate and settle in the stable region due to small disturbance sswitch pos-: sswitch position, whether sswitch is greater than or equal to or lesser than smaxpm sf inal -: final settling slip(if it exists) Rotor dyn-: rotor dynamics, Acc-:Accelerate, Dec-:Decelerate, NC-:No change Rotor dyn NC Dec NC Acc Acc Acc NC Acc Acc Acc Dec Acc Acc Acc Acc Dec Zth -:equivalent impedance of the system seen from the induction motor bus including all the loads(treated as impedances) ZIM -:induction motor equivalent impedance. ZIM Rs Re (s) j(Xs Xe (s)). It is easy to see that at any slip s1 0.08 0.07 0.06 slip 0.05 Pm (Re ) V 2 2 2 1 s1 (Rs Re ) (Xs Xe ) 0.04 0.03 0.02 0.01 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 time(s) Fig. 7. Variation of slip with time for switching ωC 7.2 at t 0.3228s circuit of the induction motor can be extended to a more general setting. Consider the Fig 8(A), which is a schematic of a large system. An induction motor connected to one of the buses as shown in the Fig 8(A) is considered here. The induction motor in the Fig 8(A) is represented by its complete equivalent circuit shown in Fig 8(C) which can be replaced by an exact equivalent as shown in the Fig 8(D) which is 0 determined as the impedance of the motor seen from AA . In Fig 8(D), jRr Xm sXm Xr Rr jXr ) s Rr jsXm jsXr (10) The rest of the system in Fig 8(A) is replaced by its Thevenin equivalent circuit as shown in the Fig 8(B) where Re jXe jXm lel( Eth -:represents the Thevenin voltage at the induction motor bus without the motor. (11) The slip at the instant of capacitor switching is assumed to be known and hence V can be calculated from (11). The circuit in the Fig 8(B) is similar to that in the Fig 1. So an equation similar to (9) can be used to determine the range of capacitances by making the appropriate changes. This approach has an implicit assumption that the power system loads are linear. The accuracy of the computed range of capacitances depends on the validity of this assumption. If the system loads are highly nonlinear, the computed range could be considered as only a reasonable approximation. V. C ONCLUSIONS Switching the capacitors at the induction motor terminals is a known method of preventing the stalling of induction motors following a sudden load increase. Results in this paper help us to understand all aspects of this phenomenon. In addition to providing insight into the basis of capacitor selection, it also can be used as a practical method to choose the value of the capacitor as well as its instant of switching. Even though most of the concepts are developed considering a small sample system, it is demonstrated in this paper that these can be extended to induction motors working as a part of a large system. R EFERENCES [1] Yasuji Sekine and Hiroshi Ohtsuki, Cascaded Voltage Collapse, IEEE transactions on Power Systems, Vol 5, No 1, February 1990, pp 250-256. Department of Electrical Engineering, Univ. College of Engg., Osmania University, Hyderabad, A.P, INDIA.

16th NATIONAL POWER SYSTEMS CONFERENCE, 15th-17th DECEMBER, 2010 B A Rs I jXs V A jXr X th Rr s jXm Ir A' Zim Eth (B) B' (C) A' A Rs Re (s) j ( X s X e ( s ) ) SYSTEM IM A (A) V I A' (D) Fig. 8. (A)A general power system containing an induction motor load (B)Thevenin equivalent of the circuit in the Fig (A) seen from A into the system (C)The actual equivalent circuit of the induction motor(D)The 0 equivalent circuit of the induction motor seen from AA . [2] D. H. Popovic, I. A. Hiskens and D. J. Hill, Stability Analysis of Induction Motor Networks, Electric Power and Energy Systems, Vol 20, No 7, 1998, pp 475-487. [3] A. E. Hammad and M. Z. El-Sadek, Prevention of Transient Voltage Instabilities due to Induction Motor Loads by Static VAR Compensators, IEEE Transactions on Power Systems, Vol 4, No 3, August 1989, pp 11821190. [4] C. W. Taylor, Power System Voltage Stability, McGraw-Hill, 1994. [5] A. Kurita and T. Sakurai, The Power System Failure on July 23, 1987 in Tokyo, Proceedings of the 27th IEEE conference on Decision and Control, Austin, Texas, December 1988, pp 2093-2097. [6] M. K. Pal, Assessment of Corrective Measures for Voltage Stability Considering Load Dynamics, Electric Power and Energy Systems, Vol 17, No 5, 1995, pp 325-334. [7] W. Xu, Y. Mansour and P. G. Harrington. Planning Methodologies for Voltage Stability Limited Power Systems, International Journal of Electrical Power and Energy Systems, Vol 15(4), 1993, pp 475-487. Department of Electrical Engineering, Univ. College of Engg., Osmania University, Hyderabad, A.P, INDIA. 265

prevention of voltage instability due to induction motors is an important concern. The prevention of induction motor stalling by switching a capacitor at the induction motor bus is studied in [1], [2], [4], [6] and [7]. It has been mentioned in [2] that if a capacitance of a particular value has to be switched, it has to be done