New Unbalance Factor for Estimating Performance of a ... - IEEE Xplore

IEEE TRANSACTIONS ON ENERGY CONVERSION, VOL. 25, NO. 3, SEPTEMBER 2010

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New Unbalance Factor for Estimating Performance of a Three-Phase Induction Motor With Under- and Overvoltage Unbalance Makbul Anwari, Member, IEEE, and Ayong Hiendro

Abstract—This paper evaluates the influence of the unbalance factors in calculating total copper losses, efficiency, power factor, input power, output torque, peak currents, and derating factor of the motor operating under unbalanced voltage system. The complete definition for the voltage unbalance is using complex voltageunbalance factor that consists of its magnitude and angle. A coefficient to determine either under- or overunbalance conditions is inserted. The analysis of the motor is performed using the method of symmetrical component, and MATLAB software is used to investigate the performance of the induction motor. The simulation results show that the International Electrotechnical Commission definition of the voltage unbalance combined with the coefficient of unbalance condition can be applied to evaluate total copper losses, input power, power factor, and total output torque precisely. However, the phase angle of the unbalance factor must be included for accurate prediction of peak current and peak copper losses of the phase windings and derating factor of the motor. Index Terms—Complex voltage-unbalance factor (CVUF), overvoltage, three-phase induction motor, undervoltage, voltage unbalance.

I. INTRODUCTION OLTAGE unbalance combined with under- or overvoltage is a power-quality problem. It is a common phenomenon found in a three-phase power system. Although three-phase voltage supply is balanced in both magnitude and phase angle at generation and transmission levels, the voltages at distribution end and utilization side can become unbalance. It is practically impossible to be obviated due to the uneven distribution of single-phase loads in the three-phase supply system and asymmetry of transmission line and transformer winding impedances. Such condition has severe impacts on the performance of an induction motor. This motor is designed and built so that it can be able to tolerate only a low degree of voltage unbalance and has to be derated if the unbalance is excessive. As per the National Electrical Manufactures Association (NEMA) guidelines [1],

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Manuscript received February 28, 2010; accepted May 23, 2010. Date of publication June 28, 2010; date of current version August 20, 2010. Paper no. TEC-00094-2010. M. Anwari is with the Department of Energy Conversion, Faculty of Electrical Engineering, Universiti Teknologi Malaysia, Skudai 81310 UTM, Malaysia (e-mail: [email protected]). A. Hiendro is with the Department of Electrical Engineering, Faculty of Engineering, Tanjungpura University, Pontianak 78124, Indonesia (e-mail: [email protected]). Digital Object Identifier 10.1109/TEC.2010.2051548

operation of the motor for any length of time above 5% of voltage-unbalance condition “is not recommended.” In most studies [2]–[6], the method of evaluating the degree of unbalanced voltage is based on either the NEMA standard [voltage-unbalance percentage (VUP)] or the International Electrotechnical Commission (IEC) definition [voltage-unbalance factor (VUF)]. Both the NEMA and the IEC definitions are only considering “magnitude” of voltage unbalance to indicate the degree of unbalanced voltage. The three-phase voltage supply has not only magnitude, but also has the phase angle that contribute to the voltage unbalance. Hence, both definitions lead to a comparatively large error in predicting the performance of the induction motor [7]. A more precise approach in predicting the performance of an induction motor operation with unbalanced three-phase voltage is using complex quantity to specify the degree of voltage unbalance. This definition is also known as the complex voltageunbalance factor (CVUF) in some literatures. The CVUF is an extension of the IEC definition, first time evaluated by Wang [8] that consists of magnitude and angle of the voltage unbalance. However, there are many voltage-unbalance conditions that have the same magnitudes and angles of voltage unbalance. The effects of unbalanced voltage cannot be evaluated precisely without indicating the exact voltage-unbalance conditions. Lee [9] compared the balanced voltage with the positivesequence voltage component of the unbalanced voltage to judge the voltage-unbalance conditions. Faiz and Ebrahimpour plotted the variation of the unbalanced voltage in a 3-D locus to reduce the range of variation of the terminal voltages [10]. They suggested that the mean terminal voltage must be known in addition to the CVUF so that a value of the derating factor would be obtained as a unique numerical value. Indulkar [11] applied Monte Carlo analysis to handle this large range of unbalanced voltage magnitudes and angles on the performance of the motor. Gnacinski [12] used the normalized negative sequence as a substitution of the VUF to analyze the derating factor of the motor. He believed that the VUF was not a perfect parameter describing voltage unbalance. In this paper, the performance of an induction motor is studied precisely using both magnitude and angle of the voltage unbalance. A coefficient of unbalance condition is defined to distinguish between under- and overvoltage unbalance conditions. The magnitude and angle of unbalanced factor along with the coefficient of unbalance are taken into consideration to specify an exact value of the unbalanced-voltage condition. The symmetrical component theory approach is used to analyze the

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The sequence line voltages can be transformed directly to the sequence phase voltages by      1 0 0 Usab Vs0  Vs1  =  0 t 0   Usbc  (4) 0 0 t∗ Vs2 Usca √ where t = (1 3) exp(−jπ/6) and its conjugate is t∗ = √ (1 3) exp(jπ/6). B. Stator and Rotor Currents If the sequence phase voltages are applied to the sequence equivalent circuits (see Fig. 1), then the sequence currents can be computed. Positive-sequence stator and rotor currents are Fig. 1.

Is1 =

Sequence equivalent circuit.

Vs1 Z1

(5)

and operation of induction motor under such conditions. MATLAB program is also used for the simulation. Finally, recommendations are suggested for accurately calculating performance values of the induction motor operation under unbalanced-voltage conditions.

Ir 1 = CD1 Is1 CD1 =

Is2 =

(7)

Vs2 Z2

(8)

and

Analysis of an induction motor operating with unbalanced voltage supply using symmetrical component approach requires positive- and negative-sequence equivalent circuit of a threephase induction motor, as represented in Fig. 1. Each circuit performs to both positive- and negative-sequence circuits. The only difference between the circuits is the load resistance defined by positive- and negative-sequence slips. Positive-sequence slip is s1 = s, negative-sequence slip is s2 = 2 − s, and slip s is s=

ns − nr ns

(1)

where ns is synchronous speed and nr is rotor speed. Input sequence impedances for the positive- and negativesequence circuit are as follows: Zi = Rs + jXs +

(jXm ) . (Rr /s1 ) + j(Xm + Xr )

Negative-sequence stator and rotor currents are

II. SYMMETRICAL COMPONENT THEORY A. Induction Motor Model

(6)

(jXm ) ((Rr /si ) + jXr ) (Rr /si ) + j (Xm + Xr )

(2)

where i = 1 for positive sequence and i = 2 for negative sequence. Let Usab , Usbc , and Usca be the line voltages of the stator. The corresponding zero-, positive-, and negative-sequence voltages (Us0 , Us1 , and Us2 ) are given by      1 1 1 Us0 Usab 1  Us1  =  1 a a2   Usbc  (3) 3 Us2 1 a2 a Usca where a = 1 ∗ exp(j2π/3) is the Fortescue operator and a2 = 1 ∗ exp(−j2π/3).

Ir 2 = CD2 Is2

CD2 =

(jXm ) (Rr /s2 ) + j(Xm + Xr )

(9)

(10)

where CD1 and CD2 are positive- and negative-sequence current dividers, respectively. Calculating stator and rotor losses requires phase stator and rotor currents. The phase currents are determined by performing the transformation back. Transforming the stator currents using the Fortescue matrix      1 1 1 Is0 Isa  Isb  =  1 a2 a   Is1  . (11) Isc Is2 1 a a2 Similarly, the rotor currents of calculated by transformation    Ir a 1 1  Ir b  =  1 a2 Ir c 1 a

the induction motor can be   1 Ir 0 a   Ir 1  . Ir 2 a2

(12)

In (11) and (12), there are no zero-sequence currents (Is0 = Ir 0 = 0) for the motor connecting in delta or ungrounded wye. C. Copper Losses, Input Power, Power Factor, Output Power, Output Torque, and Efficiency Stator copper losses of the motor is Pslosses = |Isa |2 + |Isb |2 + |Isc |2 Rs

(13)

ANWARI AND HIENDRO: NEW UNBALANCE FACTOR FOR ESTIMATING PERFORMANCE OF A THREE-PHASE INDUCTION MOTOR

and its rotor copper losses is Pr losses = |Ir a |2 + |Ir b |2 + |Ir c |2 Rr .

621

(14)

Input active power of the motor is Pin = Re3 (Vs1 (Is1 )∗ + Vs2 (Is2 )∗ )

(15)

and its input reactive power is Qin = Im3 (Vs1 (Is1 )∗ + Vs2 (Is2 )∗ ). The power factor can be calculated with

Qin −1 pf = cos tan . Pin

(16)

(17)

As the motor has no core and mechanical losses, output power is

Fig. 2.

Pout = P1 + P2 where

P1 = 3 |Ir 1 |2 P2 = 3 |Ir 2 |2

1 − s1 s1

1 − s2 s2

(18)

Rr

(19)

Rr .

(20)

P1 3I 2 Rr = r1 T1 = ωm s1 ωs P2 3I 2 Rr = − r2 ωm s2 ωs

(22)

(23)

where angular speed is ωm = ωs (1 − s) and ωs is synchronous speed (in radians per second). Efficiency of the motor is defined as follows: Pout × 100%. Pin

(24)

D. Complex Voltage-Unbalance Factor The CVUF, defined in [8], is as follows: Vs2 = kv θv Vs1

(25)

where kv is the magnitude and θv is the angle of the CVUF. If the angle of the CVUF is neglected, it becomes the VUF, which is defined by the IEC [13]: kv = VUF =

Usca = aUs1 + a2 Us2 .

(21)

Each positive and negative sequences of the output torques are

kv =

Usbc = a2 Us1 + aUs2 (27)

By the CVUF definition applied to the line voltages, we have

Tout = T1 + T2 .

η=

In contrary, the line voltages can be obtained from their positive and negative sequences, but minus the zero-sequence component (Us0 = 0) Usab = Us1 + Us2

The motor output torque is

T2 =

Terminal voltage of a three-phase induction motor with k u = 5%.

Vs2 × 100%. Vs1

(26)

Let Usab , Usbc , and Usca be the line voltages of the stator. Their symmetrical components are given by (3), as mentioned earlier.

ku =

Us2 = ku θu . Us1

(28)

Using (27) and (28), the definition of the CVUF, combined with a coefficient of unbalance, can be viewed more closely as a cylinder in 3-D. As an example, the variation of terminal voltages from 0.8 to 1.2 (in per-unit value) for ku = 5% with phase angle in 0◦ –360◦ range is shown in Fig. 2. There are infinite combinations of terminal voltages of the induction motor for ku = 5%. Every point of the outer surface of the cylinder will give the same level of the voltage unbalance. The large range of variation of the terminal voltages for a given value of the voltage unbalance can be reduced by considering the phase angle θu and the coefficient of under- or overvoltage unbalance conditions. For instant, the variation of the terminal voltages under a 0% voltage unbalance and the coefficient of under- or overvoltage unbalance condition varied by 0.8–1.2 is shown in Fig. 3. By considering the unbalanced conditions, there are two sections of the graphic. The first, ku = 0% is for undervoltage if Usab , Usbc , and Usca < 1, and the other, ku = 0% is for overvoltage if Usab , Usbc , and Usca > 1. III. RESULTS AND DISCUSSION The motor parameters used in simulation are [8]: three-phase, Y-Connected, 220 V (line-to-line), 7.5 kW, 60 Hz, and six poles. The impedances of its equivalent circuit in Ohm/phase referred to the stator are: Rs = 0.294, Rr = 0.144, Xs = 0.503, Xr = 0.209, and Xm = 13.25. In this paper, the rated motor current is assumed to be 30 A.

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Fig. 3.


Terminal voltage of a three-phase induction motor with k u = 5%.

Fig. 5.

Variations of phases a, b, and c rotor losses with θv .

TABLE I COMPARISON OF PEAK LOSSES AND PEAK CURRENT UNDER k v = 6% AND T o u t = 40.3578 N/M2

Fig. 4.

Variations of phases a, b, and c stator losses with θv .

To facilitate the analysis, the positive-sequence voltage is assumed to be constant as

Vsa + Vsb + Vsc 0◦ . (29) Vs1 = f 3 Hence, the negative-sequence voltage can be obtained by Vs2 = kv Vs1

(30)

where f is the coefficient of unbalance and is defined as f < 1, for undervoltage unbalance condition, and f > 1, for overvoltage unbalance condition. A. Influence of θv and Unbalance Conditions Upon the Induction Motor Losses Figs. 4 and 5 describe the effects of phase angle of the unbalance factor on stator and rotor copper losses. The phases a, b, and c of the stator losses (Pisa , Pisb , and Pisc ) and the rotor losses (Pir a , Pir b , and Pir c ) vary from θv = 0◦ to 360◦ for a constant kv = 6% and output torque = 40.3578 N/m2 at slip s = 0.0224. The losses in the three phases are extremely nonuniform distributed under the unbalance voltage condition

and vary significantly with θv , but total of the stator and rotor losses remain constant and are not dependent on the phase angle of the unbalance factor. The peak losses of each phase refer to the peak currents. The peak current of the stator occurs when the phase angle of the unbalance factor is as follows: 2nπ , n = 0, 1, 2 (31) θv = angle(Z2 ) − angle(Z1 ) + 3 and the phase angle of the unbalance factor for the rotor peak current is as follows: θv = (angle(Z2 ) − angle(Z1 )) − (angle(CD2 ) 2nπ , n = 0, 1, 2. (32) 3 If at an angle of θv , the current in phase a has a peak value, then, in the next 120◦ interval (=θv + 120◦ ), phase b would have a peak and phase c would have its peak current in the subsequent 120◦ interval (=θv + 240◦ ). Once a phase has the value of peak current above the rate current, it will lead to local excessive heating in the motor’s windings. The motor is not allowed to operate at above the rated current or it will harm to the stator and rotor windings. Table I shows variations of peak losses and peak currents occurred in the rotor and stator phases of the induction motor operation under kv = 6% and output torque = 40.3578 N/m2 by considering over- or undervoltage unbalance conditions. By comparing Tables I and II, it can be found that peak losses and peak current in one of the stator and the rotor phase windings − angle(CD1 )) +


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TABLE II PEAK LOSSES AND PEAK CURRENT UNDER BALANCED VOLTAGE AND T o u t = 40.3578 N/M2

Fig. 7. N/m2 .

Variations of efficiency with unbalance conditions at torque = 33.8046

Fig. 6. Variations of total copper losses with unbalance conditions at torque = 33.8046 N/m2 .

become very high, if the induction motor is operated with either overvoltage unbalance or undervoltage unbalance conditions. Under the same value of kv , the undervoltage unbalance gives higher peak losses and peak currents than the three-phase overvoltage unbalance does. The stator current is always larger than the rotor current, because it is a vector summation of the magnetizing current and the rotor current. The total copper losses are summation of the stator and rotor copper losses and do not depend on the phase angle of the voltage unbalance. At the same degree of voltage unbalance, the total losses depend upon the unbalance conditions. As can be seen in Fig. 6, the motor that operates in undervoltage unbalance condition gives greater losses, compared to when it is in overvoltage unbalance condition at kv = 0%, 1%, 2%, 3%, 4%, 5%, and 6%. B. Efficiency, Input Power, and Power Factor Fig. 7 shows that the larger degree of voltage unbalance will decrease the efficiency of the induction motor. The motor operating with undervoltage unbalance will suffer lower efficiency than if it operates with overvoltage unbalance conditions. The efficiency of the motor decreases steeply if the undervoltage unbalance increases. The lower induction motor efficiency leads to the higher electric power consumption (see Fig. 8), although it has better power factor than it was in overvoltage unbalance conditions (see Fig. 9). From Figs. 6 to 9, it is obvious that only specifying the degree of voltage unbalance is not sufficient to calculate total copper losses, efficiency, input power, and power factor of the motor operating with unbalanced supply. It is also needed to specify

Fig. 8. Variations of input power with unbalance conditions at torque = 33.8046 N/m2 .

either the coefficient of over or undervoltage unbalance in which the motor operates. In these cases, the phase angle of the voltage unbalance can be omitted. Therefore, it will be satisfied enough to apply the IEC definition and specify the coefficient of over or undervoltage unbalance condition for calculating total copper losses, efficiency, input power, and input power of the induction motor operating with unbalance voltage supply. C. Derating Factor It has been discussed earlier that a large unbalanced voltage may generate the current flowing in one of the phase windings to exceed the rated value. To protect the induction motor against such harm situation, a derating factor must be inserted to reduce the peak current that may occur in its phase winding when operating with unbalance voltage supply. The derating factor is the ratio of the motor output power under unbalanced voltage supply to that under balanced ratedvoltage condition or it can be written as follows: df =

the motor output power under unbalanced voltage . the motor output power under balanced rated voltage (33)

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Fig. 9. Variations of power factor with unbalance conditions at torque = 33.8046 N/m2 .

Fig. 10.

Variations of derating factor at f = 1 and different θv .

Effect of the phase angle of unbalance factor to derating factor is shown in Figs. 10 and 11. Fig. 10 shows the variations of the derating factor for f = 1 versus θv . There are different curves for each θv at the specified value of f . The relation of the derating factor and θv is not linear at a specified value of kv and it can be seen in Fig. 11. In this case (f = 1 and kv = 1%), the derating factor has peak values at θv = 100◦ , 220◦ , 240◦ . The same values occurred periodically with an interval of 120◦ . Further more, the derating factor is involved by voltageunbalance conditions, as shown in Figs. 12 and 13. The derating factor greater than one (df > 1) is obtained in overvoltage unbalance condition. It means that the terminal voltage of the motor is greater than its rated voltage, and it can offer power above the rated power without exceeding the rated current of the motor. The curves in Figs. 10–13 show that the derating factor in any degree of voltage unbalances depend not only on kv , but also on θv and f factor. Therefore, it is not sufficient to determine the derating factor if one of the parameters is ignored in the calculation. The derating factor will vary over a wide range, and it will not be evaluated precisely if one of the parameters is unknown

Fig. 11.

Variations of derating factor with θv at f = 1 and k v = 1%.

Fig. 12.

Variations of derating factor at θv = 0 ◦ and different f .

Fig. 13.

Variations of derating factor with f at θv = 0 ◦ and k v = 1%.

IV. CONCLUSION The degree of voltage unbalance is a quantity required in studying the performance of a three-phase induction motor operating with an unbalanced voltage supply. It has been asserted that the prevailing operating conditions of the motor are not


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accurately assessed without knowing the phase angle of unbalance factor and the coefficient of unbalance condition. The IEC definition of the VUF is reliable enough, which is applied to evaluate total copper losses, input power, power factor, and total output torque, because they are not dependent on the phase angle of the unbalance factor. However, the CVUF that consists of both magnitude and phase angle of the voltage unbalance should be used to compute peak current and peak copper losses of the phase windings and derating factor of the motor. All the performance quantities cannot be calculated precisely without knowing the condition of voltage deviation. As a result, a coefficient of unbalance may be inserted to define the unbalance condition, which is either under- or overvoltage unbalance.

[9] C. Y. Lee, “Effects of unbalanced voltages on the operation performance of a three-phase induction motor,” IEEE Trans. Energy Convers., vol. 14, no. 2, pp. 202–208, Jun. 1999. [10] J. Faiz and H. Ebrahimpour, “Precise derating of three-phase induction motors with unbalanced voltages,” IEEE Trans. Ind. Appl., vol. 6, no. 5, pp. 485–491, Oct. 2005. [11] C. S. Indulkar, “Monte-Carlo analysis of a three-phase induction motor for unbalanced supply voltage,” J. Inst. Eng. (India), vol. 87, pp. 37–41, Mar. 2007. [12] P. Gnacinski, “Derating of an induction machine under voltage unbalance combined with over or undervoltage,” Energy Convers. Manage., vol. 50, pp. 1101–1107, 2009. [13] Rotating Electrical Machines, Part 26: Effects of Unbalanced Voltages on the Performance of Induction Motors, IEC Standard 60034-26, 2002.

REFERENCES

Makbul Anwari (S’04–M’06) was born in Pontianak, Indonesia. He received the B.Eng. degree in electrical engineering from Tanjungpura University, Pontianak, Indonesia, in 1995, the M.Eng. degree in electrical engineering from Bandung Institute of Technology, Bandung, Indonesia, in 2000, and the Dr. Eng. degree from Nagaoka University of Technology, Nagaoka, Japan, in 2005. From 1995 to 2006, he was a Lecturer in the Department of Electrical Engineering, Tanjungpura University. He is currently an Associate Professor in the Department of Energy Conversion, Faculty of Electrical Engineering, Universiti Teknologi Malaysia, Skudai, Malaysia. Dr. Anwari is a member of the IEEE Power and Energy Society and the Industry Application Society.

[1] Motor and Generators, Part 14.36: Effects of Unbalanced Voltages on the Performance of Polyphase Induction Motors, NEMA Standard MG1-1998 (Revision 3, 2002), 2002. [2] R. F. Woll, “Effect of unbalanced voltage on the operation of poly-phase induction motor,” IEEE Trans. Ind. Appl., vol. IA-11, no. 1, pp. 38–42, Jan./Feb. 1975. [3] P. B. Cummings, J. R. Dunki-Jacobs, and R. H. Kerr, “Protection of induction motors against unbalanced voltage operation,” IEEE Trans. Ind. Appl., vol. IA-21, no. 4, pp. 778–792, May/Jun. 1985. [4] W. H. Kersting and W. H. Philips, “Phase frame analysis of the effects of voltage unbalance on induction machine,” IEEE Trans. Ind. Appl., vol. 33, no. 2, pp. 415–420, Mar./Apr. 1997. [5] C. Y. Lee, B. W. Chin, W. J. Lee, and Y. F. Hsu, “Effects of various unbalanced voltages on the operation performance of an induction motor under the same voltage unbalance factor condition,” in Proc. IEEE Ind. Commercial Power Syst. Conf., 1997, pp. 51–59. [6] P. Pillay, P. Hoftman, and M. Manyage, “Derating of induction motors operating with the combination of unbalanced voltages and over or under voltages,” IEEE Trans. Energy Convers., vol. 17, no. 4, pp. 485–491, Dec. 2002. [7] J. Faiz, H. Ebrahimpour, and P. Pillay, “Influence of unbalanced voltage on the steady-state performance of a three-phase squirrel-cage induction motor,” IEEE Trans. Energy Convers., vol. 19, no. 4, pp. 657–662, Dec. 2004. [8] Y. J. Wang, “Analysis of effects of three-phase voltage unbalance on induction motors with emphasis on the angle of the complex voltage unbalance factor,” IEEE Trans. Energy Convers., vol. 16, no. 3, pp. 270– 275, Sep. 2001.

Ayong Hiendro was born in Ketapang, Indonesia. He received the B.Eng. degree in electrical engineering from Tanjungpura University, Pontianak, Indonesia, and the M.Eng. degree in electrical engineering from Bandung Institute of Technology, Bandung, Indonesia, in 2000. He is currently a Senior Lecturer in the Department of Electrical Engineering, Faculty of Engineering, Tanjungpura University.