- Email: [email protected]
Saturation model for squirrel-cage induction motors
Electric Power Systems Research 79 (2009) 1054–1061
Contents lists available at ScienceDirect
Electric Power Systems Research journal homepage: www.elsevier.com/locate/epsr
Saturation model for squirrel-cage induction motors J. Pedra a,∗ , I. Candela b , A. Barrera c a
Department of Electrical Engineering, ETSEIB-UPC, Av. Diagonal 647, 08028 Barcelona, Spain Department of Electrical Engineering, ETSEIT-UPC, Colom 1, 08222 Terrassa, Spain c Asea Brown Boveri, S.A. Fabrica de Motores, Poligono Industrial S.O., 08192 Sant Quirze del Valles, Barcelona, Spain b
a r t i c l e
i n f o
Article history: Received 27 March 2008 Received in revised form 14 October 2008 Accepted 7 January 2009 Available online 10 February 2009 Keywords: Induction motor modeling Parameter estimation Saturation
a b s t r a c t An induction motor model which includes stator leakage reactance saturation, rotor leakage reactance saturation and magnetizing reactance saturation is presented. This improved model is based on experimental data from 96 motors. The power range of the motors is between 11 and 90 kW. The effects on the torque–speed and current–speed curves of each kind of saturation have been studied. In addition, the parameters of magnetizing reactance saturation and stator leakage reactance saturation have been studied for each motor, and an average value and its dispersion for each parameter are given. This model is considerably more accurate than other models. In particular, it explains the significant differences between theoretical and experimental torque–speed curves in the braking regime (s > 1). © 2009 Elsevier B.V. All rights reserved.
1. Introduction The incorporation of magnetic saturation into induction machine models is a complex topic that has received considerable attention in the past few years. For simulation study, different nonlinear models for saturated induction motors have been elaborated based on the equivalent circuit approach [1–14]. Ref. [1] presents a comparative table with the main features of different models in the bibliography. Most of the papers in the bibliography present a dynamic d–q axis model with saturation. In this paper a steady-state model with saturation has been chosen because all the experimental data have been obtained in steady-state test. Refs. [2,3] also present dynamic double-cage saturated models, but the former has saturation in the magnetizing reactance only whereas the latter includes the mutual rotor reactance saturation. In the authors’ opinion, the fitting of the model parameters to experimental data becomes more complicated if dynamical models are used. This paper experimentally justifies the need for a model with three different saturations: magnetizing reactance saturation, stator leakage reactance saturation and rotor leakage reactance saturation. The new saturation model for the induction motor has been developed from the experimental data of the locked-rotor test, the no-load test and the torque–speed curve. As is pointed out in Ref. [2], representation of the rotor with two rotor windings rather than with a single winding is known to lead to significant improvement in the accuracy of simulation results for both deep-
∗ Corresponding author. Tel.: +34 934016728; fax: +34 934017433. E-mail addresses: [email protected] (J. Pedra), [email protected] (I. Candela), [email protected] (A. Barrera). 0378-7796/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.epsr.2009.01.004
bar and double-cage induction machines. Ref. [4] gives a detailed justification of the use of a double-cage model. The influence of saturation on the skin effect has been represented by the saturation of a fictitious outer-cage. This saturation explains the big difference between theory and test in the braking regime (s > 1), noted in Ref. [5]. In the paper a study of the parameters’ influence on the torque–speed curve has been made to justify the proposed model.
2. Induction motor model discussion The aim of the paper is to propose a new model of induction motor saturation and to check that a set of parameters in good agreement with experimental data can be found. Fig. 1 shows significant differences between the theoretical and experimental torque–speed curves described in Ref. [5]. This book points out that the industry has accumulated large amounts of data on the torque–speed curves of induction motors of all power ranges and that, in general, it has been noticed that there are significant differences between theory and test, especially in the braking regime (s > 1), where a substantial rise of torque at the braking regime is observed. A torque–speed curve similar to that in Fig. 1, where the maximum torque has a value of 2 (relative value), the starting torque has a value of 1 and the torque in s = 2 is 3.5, is shown in this book. Fig. 2 shows the torque–speed measured values of an induction motor of 18.5 kW, where an anomalous torque increase near zero speed can be observed. The torque–speed curves of 96 motors, whose powers range between 90 and 11 kW, have been studied to find out how many motors exhibit this special effect. The result is that the torque–speed curve of about 40% of the motors have this
J. Pedra et al. / Electric Power Systems Research 79 (2009) 1054–1061
1055
To reproduce the special behavior of the torque–speed curve near zero speed (present in 40% of the studied motors), a new saturation is introduced in the model. This behavior is characterized by:
Fig. 1. Torque–speed curve similar to that plotted in Ref. [8], showing the anomalous torque in the braking regime (s > 1).
Fig. 2. Torque–speed measured points where the anomalous increase in torque near zero speed can be clearly observed.
shape. This effect can only be justified by outer-cage rotor leakage reactance saturation. Four motors of 18.5, 30, 45 and 75 kW, which clearly show this effect, have been chosen for detailed study. The motors of 75 and 45 kW have two pole pairs and the motors of 30 and 18.5 kW have three pole pairs. Fig. 3 shows the proposed non-linear model, where a new saturation in the fictitious outer-cage of the rotor (saturation of X2d in Fig. 3) is proposed. In the authors’ opinion, this saturation is necessary to represent the anomalous torque values near zero speed. If the induction motor model is tested with only magnetizing and stator leakage reactance saturation, the following can be stated: - Magnetizing reactance saturation has little influence on torque–speed and current–speed curves for nominal voltages. - Stator leakage reactance saturation has a great influence on the starting torque and current, and on the maximum torque. - The torque–speed curve in the braking regime is always flat; thus, this model (with only two saturations) cannot explain the change in the curvature of the torque–speed curve near zero speed. - Inner-cage rotor leakage saturation has little influence on the torque–speed curve near zero speed, because the torque produced by this cage is low at this speed. This is the main reason why all the leakage saturation is concentrated in the stator reactance. Moreover, if this saturation is considered in two reactances, the authors do not know any experimental test to measure their value. The locked-rotor test only allows the saturation of one element to be determined.
- Outer-cage rotor leakage reactance saturation changes very abruptly with the outer-cage rotor current. This is essential to obtain the correct curvature of the torque–speed curve near zero speed. - Outer-cage rotor leakage reactance saturation works at greater currents than stator leakage reactance saturation. - Outer-cage rotor leakage reactance saturation has a great influence on the torque–speed curve, but its influence on the current–speed curve is very small. This behavior is confirmed by the manufacturer’s experience. Mutual leakage inductance saturation affects both the outercage and inner-cage leakage inductances. Moreover, this saturation cannot be separated from the stator leakage inductance saturation. Therefore, it has been considered that the saturation effects of the mutual leakage inductance are included in the stator leakage inductance saturation. Finally, for simplicity, the mutual rotor inductance has not been considered in the equivalent circuit of Fig. 3, since, in the case of a linear model, the behavior of the motor model with or without mutual rotor inductance can be equivalent, as justified in Ref. [15]. 3. Magnetic saturation function In this paper the function used to represent non-linear reactance behavior is Xk (ik ) =
Xak − Xbk p
(1 + (|ik |/i0k ) k )
qk /pk
+ Xbk ,
(1)
where for Xak , Xbk , pk , qk and i0k , k = m corresponds to the magnetizing reactance saturation, k = s to the stator leakage reactance saturation and k = r to the rotor leakage reactance saturation. These five parameters have a clear physical interpretation: -
Xak reactance value in the linear zone Xbk reactance value in the saturated zone pk , qk influence the shape of the curve. i0k is the current where saturation begins.
Fig. 4 shows the influence of the saturation parameters on the shape of the non-linear reactance X(i). This function is similar to that used in Ref. [16]. The difference is the inclusion of parameter q, which is necessary to obtain an abrupt variation of X(i), as can be observed in Fig. 4. Fig. 12 shows the influence of different parameter q values on the torque–speed curves of four motors. The current on the axis has been normalized by the nominal current IN . The nominal values will be indicated by subscript N, the values of the locked-rotor test by subscript LR and the values of the no-load test by subscript NL. 4. Experimental data All the experimental data used in this paper have been measured in the ABB laboratory, equipped with five test beds for testing motors of 1–90 kW. The test bed used for testing the induction motor consisted of the following main parts:
Fig. 3. Steady-state equivalent circuit for the double-cage model of the three-phase induction motor.
- loading machine and speed controller (DC machine and DC adjustable speed drive),
1056
J. Pedra et al. / Electric Power Systems Research 79 (2009) 1054–1061
Fig. 4. Relation of impedance–current, X(i), of the proposed saturation curve.
- torque transducer mounted on the motor axis and speed and current sensors, - variable three-phase sinusoidal voltage source. The experimental data on the non-linear behavior of the squirrel-cage induction motor have been obtained from: - Locked-rotor test. - No-load test. - Torque–speed curve. The locked-rotor test plays a major role in stator leakage reactance saturation. The no-load test determines the magnetizing reactance saturation. The shape of the torque–speed curve near zero speed is greatly influenced by the rotor leakage reactance saturation.
4.2. No-load test: magnetizing reactance saturation
4.1. Linear parameters Induction motors are usually Delta or isolated Wye connected. The parameters in the steady-state equivalent circuit have been calculated considering that the motor is Wye connected. Table 1 shows the linear parameters of the four induction motors expressed in p.u. The real values of the motor parameters are Rs = rs Zb ; R1 = r1 Zb ; R2 = r2 Zb ; Xm = xm Zb Xsd = xsd Zb ; X1d = x1d Zb ; X2d = x2d Zb ,
(2)
where the impedance base is Zb =
Fig. 5. Relation of impedance–current in the no-load test.
U2 , P
(3)
where U is the rated line voltage and P the rated mechanical power.
Table 1 Linear parameters of double-cage model. P (kW)
rs
xsd
xm
r1
x1d
r2
x2d
75 45 30 18.5
0.0039 0.0045 0.0076 0.0075
0.1106 0.0641 0.1124 0.1270
3.4424 3.0145 3.7235 3.2218
0.0176 0.0181 0.0355 0.0208
0.1261 0.1644 0.1588 0.1472
0.1358 0.1416 0.1466 0.0662
0.0829 0.1479 0.1111 0.0397
In the no-load test (s = 0), the stator leakage inductance has been considered linear because its influence is negligible in this test. The no-load test has been made at voltages: 0.25, 0.35, 0.5, 0.75, 0.95, 1, 1.0375, 1.05, 1.1 and 1.25. The impedance that is measured in this case is ZNL (im ) = |Rs + jXsd + jXm | Xm (im ), Xsd Xm .
(4)
The dots in Fig. 5 represent the experimental measurements of the impedance in the no-load test for four motors of 18.5, 30, 45 and 75 kW, respectively. The saturation parameters of xm (im ) for the four motors, xam , xbm , pm , qm and i0m , are detailed in Table 2, with the value k = m. Fig. 5 shows a good agreement between the experimental data and the values predicted by the non-linear model, which are represented by a continuous line. In this type of saturation, the non-linear parameter qm always has value 1, the reason being that the rate of change with current in this type of saturation is slow. Magnetizing reactance saturation has been studied in 96 induction motors. The range of power of the motors varies from 11 to 90 kW at a line voltage of 400 V. The non-linear parameters, xam , xbm , pm , qm and i0m , have been calculated for each motor using a least-square algorithm to fit test data to the curve (1). Table 3 shows the average value and its dispersion (standard deviation) for the saturation parameters of the 96 induction motors studied. The parameters which best characterize non-linear behavior are pm , u0 and xbm /xam . The saturation voltage u0 is a better parameter than
J. Pedra et al. / Electric Power Systems Research 79 (2009) 1054–1061
1057
Table 2 Saturation parameters of double-cage model. P (kW)
k
pk
qk
i0k
xbk /xak
75
m s r
4.49 10.0 10.0
1.0 2.0 3.0
0.3468 0.90 1.716
0.03 0.65 0.2
45
m s r
4.74 10.0 10.0
1.0 1.5 4.5
0.357 0.90 2.05
0.03 0.50 0.38
30
m s r
4.3 10.0 9.0
1.0 2.0 7.0
0.27 0.90 1.80
0.035 0.70 0.1
18.5
m s r
4.9 10.0 10.0
1.0 1.8 10.0
0.32 0.9 2.98
0.04 0.75 0.1
Table 3 Manufacturer data of no-load test (s = 0). pm = 4.096 ± 1.204 i0m = 0.363 ± 0.118
xam = 3.396 ± 0.718 xam i0m = u0 = 1.164 ± 0.128
xbm = 0.245 ± 0.137 xbm /xam = 0.0705 ± 0.0352
Table 4 Manufacturer data of locked-rotor test (s = 1). i = I/IN
u = U/UN
z(i) = Z/ZN
zR (i) = z(i)/z(1)
(zR )
0.75 1 1.25 iST ≈ 7
0.1447 0.1822 0.2166 1
0.1929 0.1822 0.1733 0.1428
1.059
±0.043
0.951 0.784
±0.034 ±0.070
the saturation current i0 because it has a lower dispersion. With the help of Table 3, and given the magnetizing reactance xam = xm , a typical set of non-linear parameters can be determined as pm = 4.096;
i0m =
1.164 ; xam
xbm = 0.0705xam .
(5) Fig. 6. Relation of impedance–current in the locked-rotor test.
4.3. Locked-rotor test: stator leakage reactance saturation It should be noted that the locked-rotor test only allows the variation of the sum of the stator and rotor reactances to be determined. Thus, it is impossible to determine the degree of saturation affecting each reactance. This is the main reason why, in this paper, it has been assumed that saturation in the locked-rotor test is entirely assigned to the stator leakage reactance. For simplicity the mutual rotor reactance has not been considered in Fig. 3. Three measurements of the locked-rotor current and voltage at 0.75IN , IN and 1.25IN have been made by the usual testing procedure in the ABB factory. These measurements, together with that of the starting current (measured at nominal voltage, u = 1), allow statorleakage inductance saturation to be estimated. The expression of the starting impedance is Z ST = Rs + jXsd +
1 , (1/jXm ) + 1/(R1 + jX1d ) + 1/(R2 + jX2d )
ZST (is ) = |Rs + jXsd (is ) + Z p |.
(6) (7)
The dots in Fig. 6 represent the experimental measurements of the impedance in the locked-rotor test for four motors of 18.5, 30, 45 and 75 kW, respectively. The saturation parameters of xs (is ) for the four motors, xas , xbs , ps , qs and i0s , are detailed with the value k = s in Table 2. Fig. 6 shows a good agreement between the experimental data and the values predicted by the non-linear model, which are represented by a continuous line. Stator leakage reactance saturation has been studied in 96 motors. The relative impedance zR , defined in Table 4, has been cal-
culated for each motor with the three measurements of impedance at 0.75IN , IN , 1.25IN and the starting impedance at IST . The average value and dispersion (standard deviation) of zR (0.75IN ), zR (1.25IN ) and zR (IST ) are shown in Table 4. Assuming that the influence of magnetizing reactance saturation and rotor leakage reactance saturation is negligible in the locked-rotor test, the average values of the impedances in Table 4 indicate that |Rs + jXsd (0.75) + Z p | ZST (0.75) = = 1.059 ± 0.043, ZST (1) |Rs + jXsd (1) + Z p |
(8)
|Rs + jXsd (1.25) + Z- p | ZST (1.25) = − 0.951 ± 0.034, ZST (1) |Rs + jXsd (1) + Z- p |
(9)
|Rs + jXsd (7) + Z p | ZST (7) = 0.784 ± 0.070. = ZST (1) |Rs + jXsd (1) + Z p |
(10)
The above values indicate that there is a significant change in the motor impedance in the locked-rotor test when the current has a value near the nominal value (i ≈ 1). 4.4. Torque–speed curve: rotor leakage reactance saturation The dots in Fig. 7 represent the torque–speed measurements of four induction motors. The measurements have been made at √ reduced voltage (u = 1/ 3 = 0.577) to avoid overheating of the motor during the test. The solid line in Fig. 7 represents the torque–speed curve calculated with the linear parameters of Table 1 and the
1058
J. Pedra et al. / Electric Power Systems Research 79 (2009) 1054–1061
Fig. 7. Torque–speed and current–speed curves and measured torque and current data of four squirrel-cage induction motors at three different voltages.
non-linear parameters of Table 2 at reduced voltage (u = 0.577). A good agreement between the experimental data and the values predicted by the non-linear model can be observed. In the same figure a dashed line represents the torque–speed curve calculated with the nominal voltage, u = 1, and a dotted line represents the curve calculated at reduced voltage (u = 1/3 = 0.333). All the curves have been normalized with their own nominal torque and nominal current to make different torque–speed and current–speed curves comparable. The relation between the different nominal torque and current is TN(u=1) = 3TN(u=0.577) = 9TN(u=0.333) IN(u=1) = 1.732IN(u=0.577) = 3IN(u=0.333)
(11)
The particular behavior of the torque–speed curve, where the torque increases near zero speed can be noted. The four motors in Fig. 7 show this effect. This behavior is perfectly explained by the model proposed in Fig. 3, as can be verified in Fig. 7. In the author’s opinion, this effect can only be justified by the outer-cage rotor leakage reactance saturation. The effects of outer-cage saturation agree with the significant differences between theoretical and experimental torque–speed curves explained in Ref. [5]. The saturation parameters of the inner-cage rotor leakage reactance, X2d (i2 ), for the four motors, xar , xbr , pr , qr and i0r , are detailed with the value k = r in Table 2. A more detailed justification of the shape of the torque–speed curve is provided in the Section 5.3.
the measured data in the locked-rotor test. This saturation substantially affects the torque–speed curve and some iterations (by means of which, first the torque–speed measurements and then the locked-rotor test measurements are adjusted) must be made to obtain good parameters. Next, the non-linear parameters of the magnetizing reactance are calculated by trial and error to obtain good agreement with the measured data in the no-load test. This saturation does not significantly affect the torque–speed curve and affects only slightly the current–speed curve near zero slip. Finally, the non-linear parameters of the rotor leakage reactance are calculated by trial and error to obtain good agreement with the measured data in the torque–speed curve and current–speed curve near zero speed. The influence of this saturation on the current–speed curve near zero speed is very little. This fact agrees with the manufacturer experience. As an example, the parameters of the 75 kW motor in Tables 1 and 2 are rs = 0.0039; r1 = 0.0176; r2 = 0.1358; xm = xm (im ) xsd = xsd (is ); x1d = 0.1261; x2d = x2d (i2 ), where xm (im ) =
xs (is ) = 4.5. Parameter determination The process to obtain linear and non-linear parameters is mainly a trial and error method. In the first step, a set of linear parameters is obtained from the manufacturer data using the algorithm of Ref. [17]. The manufacturer data are conveniently modified to obtain good agreement with the torque–speed and current–speed measured points (the starting torque must normally be undervalued). Then, the non-linear parameters of the stator leakage reactance are calculated by trial and error to obtain good agreement with
(12)
x2 (i2 ) =
3.4424 − 0.03 · 3.4424 4.49 1/4.49
(1 + ((|im |/0.3468))
0.1106 − 0.65 · 0.1106 10 2/10
+ 0.03 · 3.4424
)
+ 0.65 · 0.1106
(1 + (|is |/0.9) )
0.0829 − 0.2 · 0.0829 10 3/10
(13)
+ 0.2 · 0.0829,
(1 + (|i2 |/1.716) )
5. Study of the saturation model effects In this section, the effects of each type of saturation on the torque–speed curve are studied. In each case, the linear model (parameters of Table 1) is compared to the saturated model with
J. Pedra et al. / Electric Power Systems Research 79 (2009) 1054–1061
1059
Fig. 8. Influence of magnetizing reactance saturation on the torque–speed and current–speed curves.
only one type of saturation, i.e., the other two saturations have been eliminated. 5.1. Magnetizing reactance saturation Fig. 8 shows the influence of magnetizing reactance saturation, xm (im ), on the torque–speed and current–speed curves for two motors. The range of speed is between s = 0.5 and s = 0, since the influence of saturation for higher slips is negligible. The torque–speed and current–speed curves for three different voltages, u = 1.732, 1 and 0.577, are also plotted in Fig. 8. The change in current with voltage at s = 0 shown in Fig. 8 is in agreement with the experimental data of the no-load test in Fig. 5.
When the voltage is lower than u = 1, the influence on the torque and current is negligible, and when the voltage is higher, then the current increases but the torque decreases. It must be remembered that the current values are normalized with different values of nominal current for each voltage (11). 5.2. Stator leakage reactance saturation Fig. 9 shows the influence of stator leakage reactance saturation, xsd , on the torque–speed and current–speed curves for two motors. The torque–speed and current–speed curves for two different voltages, u = 1 and 0.333, and the case when the motor is linear are also represented.
Fig. 9. Influence of stator leakage reactance saturation on the torque–speed and current–speed curves.
Fig. 10. Influence of rotor leakage reactance saturation on the torque–speed and current–speed curves.
1060
J. Pedra et al. / Electric Power Systems Research 79 (2009) 1054–1061
Fig. 11. Influence of the outer- and inner-cage on torque–speed and current–speed curves. Variation of the outer-cage impedance with the slip.
The usual value of i0s for stator leakage reactance saturation is near the nominal current, i0s ≈ 1. Therefore, the influence of saturation is very important for slip values greater than the nominal slip, where the current is high. The change in current with voltage at s = 1 shown in Fig. 9 is in agreement with the experimental data of the locked-rotor test in Fig. 6. There is a great distance between the linear curve, which represents the case without saturation, and the curve with voltage u = 0.333. The main change caused by saturation is produced near u ≈ 1/7 = 0.14, which corresponds to a current close to the nominal current. It must be remembered that the current values are normalized with different values of nominal current for each voltage (11). 5.3. Outer-cage rotor leakage reactance saturation Fig. 10 shows the influence of outer-cage rotor leakage reactance saturation on the torque–speed and current–speed curves for two different motors and three different voltages. The other saturations, magnetizing and stator leakage, have been eliminated. It can be observed that outer-cage rotor leakage reactance saturation may explain the significant differences between the theoretical and experimental torque–speed curves mentioned in Ref. [5]. Fig. 11 shows the torque–speed and current–speed curves of the outer-cage, the inner-cage and their sum (the modulus of their fasorial sum), for two motors at voltage u = 0.577. In this case, the three saturations are active. The upper plots of Fig. 11 show the great influence of the outercage on the braking regime (s > 1) and the low influence of the innercage. The middle plots of Fig. 11 show the currents of the outerand inner-cage. The outer-cage current has a clear dependence of the slip, which is “quasi” linear and enables the saturation model to be developed. The lower plots of Fig. 11 show the variation of the outer-cage rotor leakage reactance with the slip. The plot of the impedance variation with the slip for the 18.5 kW motor shows how the rotor leakage reactance change suddenly and this justifies the change in the torque–speed curve near s = 1, as observed in Fig. 11.
Fig. 12. Influence of parameter qr of the rotor leakage reactance saturation on the torque–speed and current–speed curves.
J. Pedra et al. / Electric Power Systems Research 79 (2009) 1054–1061
Fig. 12 shows the influence of parameter qr on the shape of the torque–speed curve for the four studied motors at the voltage u = 0.577. In this case the three saturations are active. The importance of parameter qr to achieve the shape of the torque–speed curve which matches the experimental data of Fig. 7 can also be observed. 6. Conclusions This paper contributes to a better understanding of induction motor saturation with experimental data from 96 motors of 11–90 kW. A new saturation model of the induction motor correctly explaining the experimental data is proposed. The model has three different saturation effects, which have been characterized in four motors in great detail. The paper calculates typical data on stator leakage reactance saturation and magnetizing reactance saturation and gives the average value and the dispersion for the main saturation parameters. The most important conclusion of this study is that outer-cage rotor leakage saturation can satisfactorily explain the high values of the torque–speed curve in the braking regime (s > 1). Acknowledgments The authors acknowledge the financial support of the Comisión Interministerial de Ciencia y Tecnología (CICYT) under the project (DPI2004-00544). The authors would like to thank Francesc Quintana from Asea Brown Boveri, S.A., Fabrica de Motores for providing the experimental data. References [1] S.D. Sudhoff, D.C. Aliprantis, B.T. Kuhn, P.L. Chapman, An induction machine model for predicting inverter-machine interaction, IEEE Trans. Energy Convers. 17 (2) (2002) 203–210. [2] E. Levi, General method of magnetising flux saturation modelling in d–q axis models of double-cage induction machines, IEE Proc. Electr. Power Appl. 144 (2) (1997) 101–109. [3] A.C. Smith, R.C. Healey, S. Williamson, A transient induction motor model including saturation and deep bar effect, IEEE Trans. Energy Convers. 11 (1) (1996) 8–15.
1061
[4] P. Vas, Electric Machines and Drives. A Space-vector Theory Approach, Clarendon Press, 1992, pp. 279–280. [5] I. Boldea, S.A. Nasar, The Induction Machine Handbook, CRC Press, 2002, pp. 319–320. [6] T.A. Lipo, A. Consoli, Modeling and simulation of induction motors with saturable leakage reactances, IEEE Trans. Industry Appl. 20 (1) (1984) 180–189. [7] R.J. Kerkman, Steady-state and transient analyses of an induction machine with saturation of the magnetizing branch, IEEE Trans. Industry Appl. 21 (1) (1985) 226–234. [8] J.C. Moreira, T.A. Lipo, Modeling of saturated AC machines including air gap flux harmonic components, IEEE Trans. Industry Appl. 28 (2) (1992) 343–349. [9] R.C. Healey, S. Williamson, A.C. Smith, Improved cage rotor models for vector controlled induction motors, IEEE Trans. Industry Appl. 31 (4) (1995) 812–822. [10] S. Williamson, R.C. Healey, Space vector representation of advanced motor models for vector controlled induction motors, IEE Proc. Electric Power Appl. 143 (1) (1996) 69–77. [11] V. Donescu, A. Charette, Z. Yao, V. Rajagopalan, Modeling and simulation of saturated induction motors in phase quantities, IEEE Trans. Energy Convers. 14 (3) (1999) 386–393. [12] S.D. Sudhoff, D.C. Aliprantis, B.T. Kuhn, P.L. Chapman, Experimental characterization procedure for use with an advanced induction machine model, IEEE Trans. Energy Convers. 18 (1) (2003) 48–56. [13] M. Ikeda, T. Hiyama, Simulation studies of the transients of squirrel-cage induction motors, IEEE Trans. Energy Convers. 22 (2) (2007) 233–239. [14] P. Kundur, Power System Stability and Control, McGraw-Hill, 1994. [15] F. Corcoles, J. Pedra, M. Salichs, L. Sainz, Analysis of the induction machine parameter identification, IEEE Trans. Energy Convers. 17 (2) (2002) 183–190. [16] J. Pedra, L. Sáinz, F. Córcoles, R. López, M. Salichs, PSpice computer model of a non-linear three-phase three-legged transformer, IEEE Trans. Power Deliv. 19 (1) (2004) 200–207. [17] J. Pedra, F. Corcoles, Double-cage induction motor parameters estimation from manufacturers data, IEEE Trans. Energy Convers. 19 (2) (2004) 310–317. Joaquín Pedra was born in Barcelona (Spain) in 1957. He received his B.S. degree in Industrial Engineering and his Ph.D. degree in Engineering from the Universitat Politècnica de Catalunya, Barcelona, Spain, in 1979 and 1986, respectively. Since 1985 he has been professor in the Electrical Engineering Department of the Universitat Politècnica de Catalunya. His research interest lies in the areas of power system quality and electrical machines. J. Ignacio Candela was born in Bilbao (Spain) in 1962. He received his B.S. degree in Industrial Engineering from the Universitat Politècnica de Catalunya, Barcelona, Spain, in 2000. Since 1991 he has been professor in the Electrical Engineering Department of the Universitat Politècnica de Catalunya. His main field of research is power system quality and electrical machines. Amalia Barrera was born in Barcelona (Spain) in 1967. She received her B.S. degree in Industrial Engineering from the Universitat Politècnica de Catalunya, Tarrasa, Spain, in 1994. Since 1994 she is with the Asea Brown Boveri S.A. Motors Factory technical department and since 2000 she is the Electrical Design Engineer responsible for I + D and applications.
Contents lists available at ScienceDirect
Electric Power Systems Research journal homepage: www.elsevier.com/locate/epsr
Saturation model for squirrel-cage induction motors J. Pedra a,∗ , I. Candela b , A. Barrera c a
Department of Electrical Engineering, ETSEIB-UPC, Av. Diagonal 647, 08028 Barcelona, Spain Department of Electrical Engineering, ETSEIT-UPC, Colom 1, 08222 Terrassa, Spain c Asea Brown Boveri, S.A. Fabrica de Motores, Poligono Industrial S.O., 08192 Sant Quirze del Valles, Barcelona, Spain b
a r t i c l e
i n f o
Article history: Received 27 March 2008 Received in revised form 14 October 2008 Accepted 7 January 2009 Available online 10 February 2009 Keywords: Induction motor modeling Parameter estimation Saturation
a b s t r a c t An induction motor model which includes stator leakage reactance saturation, rotor leakage reactance saturation and magnetizing reactance saturation is presented. This improved model is based on experimental data from 96 motors. The power range of the motors is between 11 and 90 kW. The effects on the torque–speed and current–speed curves of each kind of saturation have been studied. In addition, the parameters of magnetizing reactance saturation and stator leakage reactance saturation have been studied for each motor, and an average value and its dispersion for each parameter are given. This model is considerably more accurate than other models. In particular, it explains the significant differences between theoretical and experimental torque–speed curves in the braking regime (s > 1). © 2009 Elsevier B.V. All rights reserved.
1. Introduction The incorporation of magnetic saturation into induction machine models is a complex topic that has received considerable attention in the past few years. For simulation study, different nonlinear models for saturated induction motors have been elaborated based on the equivalent circuit approach [1–14]. Ref. [1] presents a comparative table with the main features of different models in the bibliography. Most of the papers in the bibliography present a dynamic d–q axis model with saturation. In this paper a steady-state model with saturation has been chosen because all the experimental data have been obtained in steady-state test. Refs. [2,3] also present dynamic double-cage saturated models, but the former has saturation in the magnetizing reactance only whereas the latter includes the mutual rotor reactance saturation. In the authors’ opinion, the fitting of the model parameters to experimental data becomes more complicated if dynamical models are used. This paper experimentally justifies the need for a model with three different saturations: magnetizing reactance saturation, stator leakage reactance saturation and rotor leakage reactance saturation. The new saturation model for the induction motor has been developed from the experimental data of the locked-rotor test, the no-load test and the torque–speed curve. As is pointed out in Ref. [2], representation of the rotor with two rotor windings rather than with a single winding is known to lead to significant improvement in the accuracy of simulation results for both deep-
∗ Corresponding author. Tel.: +34 934016728; fax: +34 934017433. E-mail addresses: [email protected] (J. Pedra), [email protected] (I. Candela), [email protected] (A. Barrera). 0378-7796/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.epsr.2009.01.004
bar and double-cage induction machines. Ref. [4] gives a detailed justification of the use of a double-cage model. The influence of saturation on the skin effect has been represented by the saturation of a fictitious outer-cage. This saturation explains the big difference between theory and test in the braking regime (s > 1), noted in Ref. [5]. In the paper a study of the parameters’ influence on the torque–speed curve has been made to justify the proposed model.
2. Induction motor model discussion The aim of the paper is to propose a new model of induction motor saturation and to check that a set of parameters in good agreement with experimental data can be found. Fig. 1 shows significant differences between the theoretical and experimental torque–speed curves described in Ref. [5]. This book points out that the industry has accumulated large amounts of data on the torque–speed curves of induction motors of all power ranges and that, in general, it has been noticed that there are significant differences between theory and test, especially in the braking regime (s > 1), where a substantial rise of torque at the braking regime is observed. A torque–speed curve similar to that in Fig. 1, where the maximum torque has a value of 2 (relative value), the starting torque has a value of 1 and the torque in s = 2 is 3.5, is shown in this book. Fig. 2 shows the torque–speed measured values of an induction motor of 18.5 kW, where an anomalous torque increase near zero speed can be observed. The torque–speed curves of 96 motors, whose powers range between 90 and 11 kW, have been studied to find out how many motors exhibit this special effect. The result is that the torque–speed curve of about 40% of the motors have this
J. Pedra et al. / Electric Power Systems Research 79 (2009) 1054–1061
1055
To reproduce the special behavior of the torque–speed curve near zero speed (present in 40% of the studied motors), a new saturation is introduced in the model. This behavior is characterized by:
Fig. 1. Torque–speed curve similar to that plotted in Ref. [8], showing the anomalous torque in the braking regime (s > 1).
Fig. 2. Torque–speed measured points where the anomalous increase in torque near zero speed can be clearly observed.
shape. This effect can only be justified by outer-cage rotor leakage reactance saturation. Four motors of 18.5, 30, 45 and 75 kW, which clearly show this effect, have been chosen for detailed study. The motors of 75 and 45 kW have two pole pairs and the motors of 30 and 18.5 kW have three pole pairs. Fig. 3 shows the proposed non-linear model, where a new saturation in the fictitious outer-cage of the rotor (saturation of X2d in Fig. 3) is proposed. In the authors’ opinion, this saturation is necessary to represent the anomalous torque values near zero speed. If the induction motor model is tested with only magnetizing and stator leakage reactance saturation, the following can be stated: - Magnetizing reactance saturation has little influence on torque–speed and current–speed curves for nominal voltages. - Stator leakage reactance saturation has a great influence on the starting torque and current, and on the maximum torque. - The torque–speed curve in the braking regime is always flat; thus, this model (with only two saturations) cannot explain the change in the curvature of the torque–speed curve near zero speed. - Inner-cage rotor leakage saturation has little influence on the torque–speed curve near zero speed, because the torque produced by this cage is low at this speed. This is the main reason why all the leakage saturation is concentrated in the stator reactance. Moreover, if this saturation is considered in two reactances, the authors do not know any experimental test to measure their value. The locked-rotor test only allows the saturation of one element to be determined.
- Outer-cage rotor leakage reactance saturation changes very abruptly with the outer-cage rotor current. This is essential to obtain the correct curvature of the torque–speed curve near zero speed. - Outer-cage rotor leakage reactance saturation works at greater currents than stator leakage reactance saturation. - Outer-cage rotor leakage reactance saturation has a great influence on the torque–speed curve, but its influence on the current–speed curve is very small. This behavior is confirmed by the manufacturer’s experience. Mutual leakage inductance saturation affects both the outercage and inner-cage leakage inductances. Moreover, this saturation cannot be separated from the stator leakage inductance saturation. Therefore, it has been considered that the saturation effects of the mutual leakage inductance are included in the stator leakage inductance saturation. Finally, for simplicity, the mutual rotor inductance has not been considered in the equivalent circuit of Fig. 3, since, in the case of a linear model, the behavior of the motor model with or without mutual rotor inductance can be equivalent, as justified in Ref. [15]. 3. Magnetic saturation function In this paper the function used to represent non-linear reactance behavior is Xk (ik ) =
Xak − Xbk p
(1 + (|ik |/i0k ) k )
qk /pk
+ Xbk ,
(1)
where for Xak , Xbk , pk , qk and i0k , k = m corresponds to the magnetizing reactance saturation, k = s to the stator leakage reactance saturation and k = r to the rotor leakage reactance saturation. These five parameters have a clear physical interpretation: -
Xak reactance value in the linear zone Xbk reactance value in the saturated zone pk , qk influence the shape of the curve. i0k is the current where saturation begins.
Fig. 4 shows the influence of the saturation parameters on the shape of the non-linear reactance X(i). This function is similar to that used in Ref. [16]. The difference is the inclusion of parameter q, which is necessary to obtain an abrupt variation of X(i), as can be observed in Fig. 4. Fig. 12 shows the influence of different parameter q values on the torque–speed curves of four motors. The current on the axis has been normalized by the nominal current IN . The nominal values will be indicated by subscript N, the values of the locked-rotor test by subscript LR and the values of the no-load test by subscript NL. 4. Experimental data All the experimental data used in this paper have been measured in the ABB laboratory, equipped with five test beds for testing motors of 1–90 kW. The test bed used for testing the induction motor consisted of the following main parts:
Fig. 3. Steady-state equivalent circuit for the double-cage model of the three-phase induction motor.
- loading machine and speed controller (DC machine and DC adjustable speed drive),
1056
J. Pedra et al. / Electric Power Systems Research 79 (2009) 1054–1061
Fig. 4. Relation of impedance–current, X(i), of the proposed saturation curve.
- torque transducer mounted on the motor axis and speed and current sensors, - variable three-phase sinusoidal voltage source. The experimental data on the non-linear behavior of the squirrel-cage induction motor have been obtained from: - Locked-rotor test. - No-load test. - Torque–speed curve. The locked-rotor test plays a major role in stator leakage reactance saturation. The no-load test determines the magnetizing reactance saturation. The shape of the torque–speed curve near zero speed is greatly influenced by the rotor leakage reactance saturation.
4.2. No-load test: magnetizing reactance saturation
4.1. Linear parameters Induction motors are usually Delta or isolated Wye connected. The parameters in the steady-state equivalent circuit have been calculated considering that the motor is Wye connected. Table 1 shows the linear parameters of the four induction motors expressed in p.u. The real values of the motor parameters are Rs = rs Zb ; R1 = r1 Zb ; R2 = r2 Zb ; Xm = xm Zb Xsd = xsd Zb ; X1d = x1d Zb ; X2d = x2d Zb ,
(2)
where the impedance base is Zb =
Fig. 5. Relation of impedance–current in the no-load test.
U2 , P
(3)
where U is the rated line voltage and P the rated mechanical power.
Table 1 Linear parameters of double-cage model. P (kW)
rs
xsd
xm
r1
x1d
r2
x2d
75 45 30 18.5
0.0039 0.0045 0.0076 0.0075
0.1106 0.0641 0.1124 0.1270
3.4424 3.0145 3.7235 3.2218
0.0176 0.0181 0.0355 0.0208
0.1261 0.1644 0.1588 0.1472
0.1358 0.1416 0.1466 0.0662
0.0829 0.1479 0.1111 0.0397
In the no-load test (s = 0), the stator leakage inductance has been considered linear because its influence is negligible in this test. The no-load test has been made at voltages: 0.25, 0.35, 0.5, 0.75, 0.95, 1, 1.0375, 1.05, 1.1 and 1.25. The impedance that is measured in this case is ZNL (im ) = |Rs + jXsd + jXm | Xm (im ), Xsd Xm .
(4)
The dots in Fig. 5 represent the experimental measurements of the impedance in the no-load test for four motors of 18.5, 30, 45 and 75 kW, respectively. The saturation parameters of xm (im ) for the four motors, xam , xbm , pm , qm and i0m , are detailed in Table 2, with the value k = m. Fig. 5 shows a good agreement between the experimental data and the values predicted by the non-linear model, which are represented by a continuous line. In this type of saturation, the non-linear parameter qm always has value 1, the reason being that the rate of change with current in this type of saturation is slow. Magnetizing reactance saturation has been studied in 96 induction motors. The range of power of the motors varies from 11 to 90 kW at a line voltage of 400 V. The non-linear parameters, xam , xbm , pm , qm and i0m , have been calculated for each motor using a least-square algorithm to fit test data to the curve (1). Table 3 shows the average value and its dispersion (standard deviation) for the saturation parameters of the 96 induction motors studied. The parameters which best characterize non-linear behavior are pm , u0 and xbm /xam . The saturation voltage u0 is a better parameter than
J. Pedra et al. / Electric Power Systems Research 79 (2009) 1054–1061
1057
Table 2 Saturation parameters of double-cage model. P (kW)
k
pk
qk
i0k
xbk /xak
75
m s r
4.49 10.0 10.0
1.0 2.0 3.0
0.3468 0.90 1.716
0.03 0.65 0.2
45
m s r
4.74 10.0 10.0
1.0 1.5 4.5
0.357 0.90 2.05
0.03 0.50 0.38
30
m s r
4.3 10.0 9.0
1.0 2.0 7.0
0.27 0.90 1.80
0.035 0.70 0.1
18.5
m s r
4.9 10.0 10.0
1.0 1.8 10.0
0.32 0.9 2.98
0.04 0.75 0.1
Table 3 Manufacturer data of no-load test (s = 0). pm = 4.096 ± 1.204 i0m = 0.363 ± 0.118
xam = 3.396 ± 0.718 xam i0m = u0 = 1.164 ± 0.128
xbm = 0.245 ± 0.137 xbm /xam = 0.0705 ± 0.0352
Table 4 Manufacturer data of locked-rotor test (s = 1). i = I/IN
u = U/UN
z(i) = Z/ZN
zR (i) = z(i)/z(1)
(zR )
0.75 1 1.25 iST ≈ 7
0.1447 0.1822 0.2166 1
0.1929 0.1822 0.1733 0.1428
1.059
±0.043
0.951 0.784
±0.034 ±0.070
the saturation current i0 because it has a lower dispersion. With the help of Table 3, and given the magnetizing reactance xam = xm , a typical set of non-linear parameters can be determined as pm = 4.096;
i0m =
1.164 ; xam
xbm = 0.0705xam .
(5) Fig. 6. Relation of impedance–current in the locked-rotor test.
4.3. Locked-rotor test: stator leakage reactance saturation It should be noted that the locked-rotor test only allows the variation of the sum of the stator and rotor reactances to be determined. Thus, it is impossible to determine the degree of saturation affecting each reactance. This is the main reason why, in this paper, it has been assumed that saturation in the locked-rotor test is entirely assigned to the stator leakage reactance. For simplicity the mutual rotor reactance has not been considered in Fig. 3. Three measurements of the locked-rotor current and voltage at 0.75IN , IN and 1.25IN have been made by the usual testing procedure in the ABB factory. These measurements, together with that of the starting current (measured at nominal voltage, u = 1), allow statorleakage inductance saturation to be estimated. The expression of the starting impedance is Z ST = Rs + jXsd +
1 , (1/jXm ) + 1/(R1 + jX1d ) + 1/(R2 + jX2d )
ZST (is ) = |Rs + jXsd (is ) + Z p |.
(6) (7)
The dots in Fig. 6 represent the experimental measurements of the impedance in the locked-rotor test for four motors of 18.5, 30, 45 and 75 kW, respectively. The saturation parameters of xs (is ) for the four motors, xas , xbs , ps , qs and i0s , are detailed with the value k = s in Table 2. Fig. 6 shows a good agreement between the experimental data and the values predicted by the non-linear model, which are represented by a continuous line. Stator leakage reactance saturation has been studied in 96 motors. The relative impedance zR , defined in Table 4, has been cal-
culated for each motor with the three measurements of impedance at 0.75IN , IN , 1.25IN and the starting impedance at IST . The average value and dispersion (standard deviation) of zR (0.75IN ), zR (1.25IN ) and zR (IST ) are shown in Table 4. Assuming that the influence of magnetizing reactance saturation and rotor leakage reactance saturation is negligible in the locked-rotor test, the average values of the impedances in Table 4 indicate that |Rs + jXsd (0.75) + Z p | ZST (0.75) = = 1.059 ± 0.043, ZST (1) |Rs + jXsd (1) + Z p |
(8)
|Rs + jXsd (1.25) + Z- p | ZST (1.25) = − 0.951 ± 0.034, ZST (1) |Rs + jXsd (1) + Z- p |
(9)
|Rs + jXsd (7) + Z p | ZST (7) = 0.784 ± 0.070. = ZST (1) |Rs + jXsd (1) + Z p |
(10)
The above values indicate that there is a significant change in the motor impedance in the locked-rotor test when the current has a value near the nominal value (i ≈ 1). 4.4. Torque–speed curve: rotor leakage reactance saturation The dots in Fig. 7 represent the torque–speed measurements of four induction motors. The measurements have been made at √ reduced voltage (u = 1/ 3 = 0.577) to avoid overheating of the motor during the test. The solid line in Fig. 7 represents the torque–speed curve calculated with the linear parameters of Table 1 and the
1058
J. Pedra et al. / Electric Power Systems Research 79 (2009) 1054–1061
Fig. 7. Torque–speed and current–speed curves and measured torque and current data of four squirrel-cage induction motors at three different voltages.
non-linear parameters of Table 2 at reduced voltage (u = 0.577). A good agreement between the experimental data and the values predicted by the non-linear model can be observed. In the same figure a dashed line represents the torque–speed curve calculated with the nominal voltage, u = 1, and a dotted line represents the curve calculated at reduced voltage (u = 1/3 = 0.333). All the curves have been normalized with their own nominal torque and nominal current to make different torque–speed and current–speed curves comparable. The relation between the different nominal torque and current is TN(u=1) = 3TN(u=0.577) = 9TN(u=0.333) IN(u=1) = 1.732IN(u=0.577) = 3IN(u=0.333)
(11)
The particular behavior of the torque–speed curve, where the torque increases near zero speed can be noted. The four motors in Fig. 7 show this effect. This behavior is perfectly explained by the model proposed in Fig. 3, as can be verified in Fig. 7. In the author’s opinion, this effect can only be justified by the outer-cage rotor leakage reactance saturation. The effects of outer-cage saturation agree with the significant differences between theoretical and experimental torque–speed curves explained in Ref. [5]. The saturation parameters of the inner-cage rotor leakage reactance, X2d (i2 ), for the four motors, xar , xbr , pr , qr and i0r , are detailed with the value k = r in Table 2. A more detailed justification of the shape of the torque–speed curve is provided in the Section 5.3.
the measured data in the locked-rotor test. This saturation substantially affects the torque–speed curve and some iterations (by means of which, first the torque–speed measurements and then the locked-rotor test measurements are adjusted) must be made to obtain good parameters. Next, the non-linear parameters of the magnetizing reactance are calculated by trial and error to obtain good agreement with the measured data in the no-load test. This saturation does not significantly affect the torque–speed curve and affects only slightly the current–speed curve near zero slip. Finally, the non-linear parameters of the rotor leakage reactance are calculated by trial and error to obtain good agreement with the measured data in the torque–speed curve and current–speed curve near zero speed. The influence of this saturation on the current–speed curve near zero speed is very little. This fact agrees with the manufacturer experience. As an example, the parameters of the 75 kW motor in Tables 1 and 2 are rs = 0.0039; r1 = 0.0176; r2 = 0.1358; xm = xm (im ) xsd = xsd (is ); x1d = 0.1261; x2d = x2d (i2 ), where xm (im ) =
xs (is ) = 4.5. Parameter determination The process to obtain linear and non-linear parameters is mainly a trial and error method. In the first step, a set of linear parameters is obtained from the manufacturer data using the algorithm of Ref. [17]. The manufacturer data are conveniently modified to obtain good agreement with the torque–speed and current–speed measured points (the starting torque must normally be undervalued). Then, the non-linear parameters of the stator leakage reactance are calculated by trial and error to obtain good agreement with
(12)
x2 (i2 ) =
3.4424 − 0.03 · 3.4424 4.49 1/4.49
(1 + ((|im |/0.3468))
0.1106 − 0.65 · 0.1106 10 2/10
+ 0.03 · 3.4424
)
+ 0.65 · 0.1106
(1 + (|is |/0.9) )
0.0829 − 0.2 · 0.0829 10 3/10
(13)
+ 0.2 · 0.0829,
(1 + (|i2 |/1.716) )
5. Study of the saturation model effects In this section, the effects of each type of saturation on the torque–speed curve are studied. In each case, the linear model (parameters of Table 1) is compared to the saturated model with
J. Pedra et al. / Electric Power Systems Research 79 (2009) 1054–1061
1059
Fig. 8. Influence of magnetizing reactance saturation on the torque–speed and current–speed curves.
only one type of saturation, i.e., the other two saturations have been eliminated. 5.1. Magnetizing reactance saturation Fig. 8 shows the influence of magnetizing reactance saturation, xm (im ), on the torque–speed and current–speed curves for two motors. The range of speed is between s = 0.5 and s = 0, since the influence of saturation for higher slips is negligible. The torque–speed and current–speed curves for three different voltages, u = 1.732, 1 and 0.577, are also plotted in Fig. 8. The change in current with voltage at s = 0 shown in Fig. 8 is in agreement with the experimental data of the no-load test in Fig. 5.
When the voltage is lower than u = 1, the influence on the torque and current is negligible, and when the voltage is higher, then the current increases but the torque decreases. It must be remembered that the current values are normalized with different values of nominal current for each voltage (11). 5.2. Stator leakage reactance saturation Fig. 9 shows the influence of stator leakage reactance saturation, xsd , on the torque–speed and current–speed curves for two motors. The torque–speed and current–speed curves for two different voltages, u = 1 and 0.333, and the case when the motor is linear are also represented.
Fig. 9. Influence of stator leakage reactance saturation on the torque–speed and current–speed curves.
Fig. 10. Influence of rotor leakage reactance saturation on the torque–speed and current–speed curves.
1060
J. Pedra et al. / Electric Power Systems Research 79 (2009) 1054–1061
Fig. 11. Influence of the outer- and inner-cage on torque–speed and current–speed curves. Variation of the outer-cage impedance with the slip.
The usual value of i0s for stator leakage reactance saturation is near the nominal current, i0s ≈ 1. Therefore, the influence of saturation is very important for slip values greater than the nominal slip, where the current is high. The change in current with voltage at s = 1 shown in Fig. 9 is in agreement with the experimental data of the locked-rotor test in Fig. 6. There is a great distance between the linear curve, which represents the case without saturation, and the curve with voltage u = 0.333. The main change caused by saturation is produced near u ≈ 1/7 = 0.14, which corresponds to a current close to the nominal current. It must be remembered that the current values are normalized with different values of nominal current for each voltage (11). 5.3. Outer-cage rotor leakage reactance saturation Fig. 10 shows the influence of outer-cage rotor leakage reactance saturation on the torque–speed and current–speed curves for two different motors and three different voltages. The other saturations, magnetizing and stator leakage, have been eliminated. It can be observed that outer-cage rotor leakage reactance saturation may explain the significant differences between the theoretical and experimental torque–speed curves mentioned in Ref. [5]. Fig. 11 shows the torque–speed and current–speed curves of the outer-cage, the inner-cage and their sum (the modulus of their fasorial sum), for two motors at voltage u = 0.577. In this case, the three saturations are active. The upper plots of Fig. 11 show the great influence of the outercage on the braking regime (s > 1) and the low influence of the innercage. The middle plots of Fig. 11 show the currents of the outerand inner-cage. The outer-cage current has a clear dependence of the slip, which is “quasi” linear and enables the saturation model to be developed. The lower plots of Fig. 11 show the variation of the outer-cage rotor leakage reactance with the slip. The plot of the impedance variation with the slip for the 18.5 kW motor shows how the rotor leakage reactance change suddenly and this justifies the change in the torque–speed curve near s = 1, as observed in Fig. 11.
Fig. 12. Influence of parameter qr of the rotor leakage reactance saturation on the torque–speed and current–speed curves.
J. Pedra et al. / Electric Power Systems Research 79 (2009) 1054–1061
Fig. 12 shows the influence of parameter qr on the shape of the torque–speed curve for the four studied motors at the voltage u = 0.577. In this case the three saturations are active. The importance of parameter qr to achieve the shape of the torque–speed curve which matches the experimental data of Fig. 7 can also be observed. 6. Conclusions This paper contributes to a better understanding of induction motor saturation with experimental data from 96 motors of 11–90 kW. A new saturation model of the induction motor correctly explaining the experimental data is proposed. The model has three different saturation effects, which have been characterized in four motors in great detail. The paper calculates typical data on stator leakage reactance saturation and magnetizing reactance saturation and gives the average value and the dispersion for the main saturation parameters. The most important conclusion of this study is that outer-cage rotor leakage saturation can satisfactorily explain the high values of the torque–speed curve in the braking regime (s > 1). Acknowledgments The authors acknowledge the financial support of the Comisión Interministerial de Ciencia y Tecnología (CICYT) under the project (DPI2004-00544). The authors would like to thank Francesc Quintana from Asea Brown Boveri, S.A., Fabrica de Motores for providing the experimental data. References [1] S.D. Sudhoff, D.C. Aliprantis, B.T. Kuhn, P.L. Chapman, An induction machine model for predicting inverter-machine interaction, IEEE Trans. Energy Convers. 17 (2) (2002) 203–210. [2] E. Levi, General method of magnetising flux saturation modelling in d–q axis models of double-cage induction machines, IEE Proc. Electr. Power Appl. 144 (2) (1997) 101–109. [3] A.C. Smith, R.C. Healey, S. Williamson, A transient induction motor model including saturation and deep bar effect, IEEE Trans. Energy Convers. 11 (1) (1996) 8–15.
1061
[4] P. Vas, Electric Machines and Drives. A Space-vector Theory Approach, Clarendon Press, 1992, pp. 279–280. [5] I. Boldea, S.A. Nasar, The Induction Machine Handbook, CRC Press, 2002, pp. 319–320. [6] T.A. Lipo, A. Consoli, Modeling and simulation of induction motors with saturable leakage reactances, IEEE Trans. Industry Appl. 20 (1) (1984) 180–189. [7] R.J. Kerkman, Steady-state and transient analyses of an induction machine with saturation of the magnetizing branch, IEEE Trans. Industry Appl. 21 (1) (1985) 226–234. [8] J.C. Moreira, T.A. Lipo, Modeling of saturated AC machines including air gap flux harmonic components, IEEE Trans. Industry Appl. 28 (2) (1992) 343–349. [9] R.C. Healey, S. Williamson, A.C. Smith, Improved cage rotor models for vector controlled induction motors, IEEE Trans. Industry Appl. 31 (4) (1995) 812–822. [10] S. Williamson, R.C. Healey, Space vector representation of advanced motor models for vector controlled induction motors, IEE Proc. Electric Power Appl. 143 (1) (1996) 69–77. [11] V. Donescu, A. Charette, Z. Yao, V. Rajagopalan, Modeling and simulation of saturated induction motors in phase quantities, IEEE Trans. Energy Convers. 14 (3) (1999) 386–393. [12] S.D. Sudhoff, D.C. Aliprantis, B.T. Kuhn, P.L. Chapman, Experimental characterization procedure for use with an advanced induction machine model, IEEE Trans. Energy Convers. 18 (1) (2003) 48–56. [13] M. Ikeda, T. Hiyama, Simulation studies of the transients of squirrel-cage induction motors, IEEE Trans. Energy Convers. 22 (2) (2007) 233–239. [14] P. Kundur, Power System Stability and Control, McGraw-Hill, 1994. [15] F. Corcoles, J. Pedra, M. Salichs, L. Sainz, Analysis of the induction machine parameter identification, IEEE Trans. Energy Convers. 17 (2) (2002) 183–190. [16] J. Pedra, L. Sáinz, F. Córcoles, R. López, M. Salichs, PSpice computer model of a non-linear three-phase three-legged transformer, IEEE Trans. Power Deliv. 19 (1) (2004) 200–207. [17] J. Pedra, F. Corcoles, Double-cage induction motor parameters estimation from manufacturers data, IEEE Trans. Energy Convers. 19 (2) (2004) 310–317. Joaquín Pedra was born in Barcelona (Spain) in 1957. He received his B.S. degree in Industrial Engineering and his Ph.D. degree in Engineering from the Universitat Politècnica de Catalunya, Barcelona, Spain, in 1979 and 1986, respectively. Since 1985 he has been professor in the Electrical Engineering Department of the Universitat Politècnica de Catalunya. His research interest lies in the areas of power system quality and electrical machines. J. Ignacio Candela was born in Bilbao (Spain) in 1962. He received his B.S. degree in Industrial Engineering from the Universitat Politècnica de Catalunya, Barcelona, Spain, in 2000. Since 1991 he has been professor in the Electrical Engineering Department of the Universitat Politècnica de Catalunya. His main field of research is power system quality and electrical machines. Amalia Barrera was born in Barcelona (Spain) in 1967. She received her B.S. degree in Industrial Engineering from the Universitat Politècnica de Catalunya, Tarrasa, Spain, in 1994. Since 1994 she is with the Asea Brown Boveri S.A. Motors Factory technical department and since 2000 she is the Electrical Design Engineer responsible for I + D and applications.