E3S Web of Conferences 124, 01036 (2019) https://doi.org/10.1051/e3sconf/201912401036 SES-2019

Minimization of the current in with defined load on the shaft by maintaining optimum absolute slip

V. N. Meshcheryakov1,*, V. V. Danilov1, Sh. R. Khasanov2, and S. Valtchev3

1 Lipetsk State Technical University, Lipetsk, Russia 2 Kazan State Power Engineering University, Kazan, Russia 3 Universidade Nova de Lisboa, Caparica, Portugal

Abstract. This paper presents the study of induction motor operation with regulated frequency and voltage while maintaining the absolute slip of the two optimal modes - with a minimum of the “stator current / ” ratio and a minimum of the “winding loss / torque” ratio; the results of experimental studies confirm the validity of the analytical expressions.

1 Introduction where α – relative frequency; β – absolute slip; Sα – In standard open-loop electric drive systems with relative slip; f – frequency of the stator supply voltage; frequency regulated induction motor and scalar control, 1 magnitude of the motor absolute slip in steady state f1n – rated frequency of the stator supply voltage; f 2 –

depends on the rigidity of mechanical characteristic and frequency of the current; ω1 el – circular frequency on torque on the motor shaft. The absolute slip is usually of the stator supply voltage from the frequency not controlled and not regulated. Since the rigidity of the converter; ω – angular frequency of the rated voltage; mechanical characteristic depends on the supply voltage 0 el of the stator winding, it is possible to change the rigidity ω0 – rotation frequency of the stator field at the nominal

of static mechanical characteristic with the given load on frequency f1n ; ω1 – frequency of the stator field rotation the motor shaft by adjusting the voltage reference signal. at a frequency ; Δω – absolute deviation of the rotor Frequency regulated electric drive systems are angular velocity; Δω = Δωp = ω – reduced to an widely used in industry, therefore, their research and el n 2 improvement continues [1–3]. An important area of electric circuit, the absolute deviation of the rotor speed, research is energy efficiency [4, 5]. It is possible to equal to the circular electric frequency of the emf and the increase the energy performance of a frequency rotor current; p n – number of the motor poles pairs. regulated electric drive by using the optimal control From formula (2) it follows that the absolute slip system settings [6–10]. The optimal control of the linearly depends on the absolute deviation of the rotor inverter supplying the induction motor is studied in [11– angular velocity, which determines the slope of the static 13]. The energy performance of an induction motor can mechanical characteristic (Figure 1), expressed by be improved by regulating the voltage when the load of absolute value of rigidity [17]: the motor changes [14–15]. The impact of the absolute slip on the stator current ΔМ β = − (4) of induction motor with a given load on the motor shaft Δω is discussed below.

where ΔМ = МС – deviation of the torque that is equal 2 Theoretical part to the static torque of the motor.

Parameters of frequency-regulated induction motor are described by relative values as [16]: α = f /f = ω /ω 1 1n 1el 0el (1)

β = f 2 /f1H = Δω ω0 = α Sα ; (2)

Sα = (ω1 − ω)/ω1 = Δω/ω1 = Δωel/ω1el (3)

* Corresponding author: [email protected]

© The Authors, published by EDP Sciences. This is an open access article distributed under the terms of the Creative Commons Attribution License 4.0 (http://creativecommons.org/licenses/by/4.0/). E3S Web of Conferences 124, 01036 (2019) https://doi.org/10.1051/e3sconf/201912401036 SES-2019

The variable losses on active resistance of stator winding are determined by:

2 2 ΔP = 3 (I1 R1 + I2 R2 ) (11)

Solving equations (11), (6) and (7), taking into account expressions (2) and (3), we obtain the following expression:

M Δω ΔP3*= el 3Rp2n (12) (LL)R/++222 Δ m2σ2el  *RR2 12+ Fig. 1. Variation in mechanical characteristics of rigidity in Lm induction motor with different control system settings. Table 1 shows the results of finding the minimum The effective values of the rotor and stator currents ratio “stator current/torque” based on study of expression are connected by the following formula: (7), and “loss in windings/torque” based on study of expression (12). x m It is possible to adjust the rigidity of the mechanical I2 = I1  (5) 2 characteristics in induction motor at a fixed frequency of (x + x / ) 2 + R / /β 2 m 2 2 the supply voltage by changing the effective value of the voltage. In the practical implementation of the correction The equivalent circuit shows that the parameter α systems [19–22], it is more convenient to tune the “stator affecting absolute slip β and varying with frequency current/torque” ratio to a minimum, since the static control, has a direct impact on the resistance of the characteristics have a lower required rigidity in this case, circuit branches and the distribution of the currents. therefore, less voltage correction required than linear The torque of induction motor with frequency control frequency control u/f = const. The optimal operation in static mode is [16]: mode of the induction motor in the frequency electric drive system can be implemented using the unsaturated 2 2 2 3 R'  I' 3 R'  I' 3 p  R'  I' magnetic circuit, when the static torque does not exceed M = 2 2 = 2 2 = n 2 2 (6) ω β Δω Δω the nominal value of MC  MN. In this case, the drive 0 el control system, configured to a minimum of the “stator Solving equations (5) and (6) with regard to current / torque” ratio, should provide the optimum angle expressions (2) and (3), we obtain the relation of the value = 45º. stator current and the torque 3 Description of the experimental (L + L )Δω /R + R /ω m 2σ el 2 2 el (7) procedure I1 = М  2 3pn Lm To confirm the analytical results, we carry out The following relation connects the rotor flux linkage experimental studies of variable-frequency induction and the stator current of the induction motor in the drive with scalar control. The frequency converter was operator form [18]: controlled according to the law U/f = const using additional corrections of this control law. 1 The purpose of the experimental studies was to find Ψ (p) = I (p)  L  (8) 2 1 m T p +1 the dependence of the stator current on the difference 2  between the speed of the stator rotating field and the

where T2 is the rotor time constant: rotor speed, and the search for the optimal value opt in which the stator current is minimal. L2 L2σ + L m T2 = = (9) The experiment consists in changing the voltage and R2 R2 frequency supplied to the stator winding while ensuring that the drive operates with fixed static torque on the The phase frequency characteristic of the first order motor shaft and fixed rotation speed. In this case, the

aperiodic link at the frequency ω = Δωel is search for the optimal difference between the speed of the stator rotating field and the rotor speed is carried out by introducing an additional set point in increments of / L2Δωel  (Δω ) = arctg(T  Δω ) = arctg (10) 1 Hz into the frequency reference channel, and 0 el 2 el  R 2 stabilizing the motor shaft rotation speed by changing the voltage amplitude.

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Table 1. The relation between variables and criteria of optimal systems.

Criterion Minimal ratio “stator current/motor torque” Minimal energy loss in the motor windings ( I ) ( ) 1 min Pmin M   P = 3 el  2 2 2  (L + L ) + R /  3R2  pn Initial equations I2 = M el  m 2 2 el with effective 1 3R  p L2  (L + L )2 + R2 / 2  variables values 2 n m  m 2 2 el R + R   2 1 2   Lm 

dP/del = 0 , d(I1/ М) dΔel = 0 Equation for finding 2 2 2 2 extremum 2 2 2 (Lm + L2 ) − R2 / el R1 + R2 Lm = 0 (Lm + L2 ) − R2 / el = 0

R Δω = 2 = el opt (ΔP) 2 2 R (Lm + L2) + Lm R2/R1 Optimal frequency Δω = 2 deviation el opt(I) L R 1 2 = 2 L 2 2 2 (1+ Lm R2 )/L2 R1 R 1 R / β = 2 Optimal absolute 2 opt(ΔР) 2 2 βopt(I) = L2 0el (1+ L  R )/L  R slip L/ ω m 2 2 1 2 0el L tg =   2 = LΔω 0 opt(Р) el opt(Р) R Optimal value 2 el opt(I) 2 tg = = 1  0 opt(I) tg0 R 1 2 = 2 2 (1+ Lm  R2 )/L2  R1 L  = arctg(  2 ) ; Optimal value 0 opt(Р) el opt(Р)  = 45 R  0 opt(I) 2 0 0   39...40 0 opt(Р)

The functional block diagram of control system used in the experiment is shown in Figure 2. Blocks 1–12 are standard for the frequency control system with the 4 Discussion of the results control law U/f = const. The block 13 was added to the Figure 3 shows the dependences I1 = f (Δω) for different standard control system in order to introduce an rotation frequencies and load of induction motor. * additional set point fa dd into the frequency setting Therefore, there is a value opt at which the stator channel. Output of block 13 goes to the adder that results current is minimal at the fixed load torque and rotor * reference signal to the frequency f . The correction speed. Figure 4 illustrates the average the optimal values device in the voltage reference channel is aimed to obtained for different load torques and rotor stabilize the motor shaft rotation speed by changing the voltage amplitude. speeds. The average optimal difference can be The speed reference signal generated in block 16 is established independent of the rotor speed. subtracted from the actual motor speed. The speed error signal goes to the proportional controller 15 and further The average optimal differences between the to the limiting unit 14. The resulting correction signal speed of the stator rotating field and the rotor speed goes to the adder that results the reference amplitude of calculated analytically is: the supply voltage. R 7,73  =(  −  p ) =   =2 = = 16,38 rad / s el opt el1 n opt opt L2 0,472

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2 1 U 3 4 5 6 U* U * ref a + - f cont- * U voltage rol b PWM 13 7 inverter unit * Uc + 11 12 I 8 9 10 max I |I| current a align- Ic ment

14 15 16

Fig. 2. Functional diagram of the control system of induction motor drive used for the experiment.

Fig. 3. Stator current I1 dependence on the rotor rotational frequency difference Δω: 1- ω=0.42 о.е., М=0.28 о.е.; 2- ω=0.19 о.е., М=0.28 о.е.; 3- ω=0.86 о.е., М=0.28 о.е.; 4- ω=0.17 о.е., М=0.5 о.е.; 5- ω=0.39 о.е., М=0.5 о.е.; 6- ω=0.84 о.е., М=0.5 о.е.; 7- ω=0.29 о.е., М=1 о.е.; 8- ω=0.65 о.е., М=0.19 о.е.; 9- ω=0.63 о.е., М=0.38 о.е.; 10-ω=0.26 о.е., М=0.57 о.е.; 11- ω=0.72 о.е., М=0.57 о.е.; 12- ω=0.81 о.е., М=0.7 о.е.

Further, we implemented experiment on squirrel cage induction motor and obtained optimal values . opt 5 Conclusions Comparison of the analytical and experimental results performed good convergence. The average relative 1. Experimentally proven the possibility of obtaining difference between the values obtained does not exceed near-optimal modes of frequency regulated induction 8%. electric drive. The minimal ratio “stator current/motor

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Fig. 4. Dependence of the optimal rotor speed difference Δω on the rotor speed ω.

torque” is achieved when the absolute slip has the [5] J. Rodriguez, R.M. Kennel, J.R. Espinoza, M. / Trincado, C.A. Silva, C.A. Rojas, High performance R 2 optimal value opt(I) = / and tg0 = 1. control strategies for electrical drives: an experimental L2  0el assessment. IEEE Trans. Ind. Electron. 59, 812–820 2. To implement the optimal mode with a minimum (2012) of the “stator current / motor torque” ratio, it is necessary [6] H. Rehman, Detuning minimization for to adjust the voltage applied to the stator winding with alternative energy vehicular drive system. IEEE Vehicle respect to the linear frequency regulation law U/f = Power and Propulsion Conference, Seoul. 42-47 (2012) const. [7] R. Issa, Three-Phase Induction Motor Stator Current Optimization. IJCA Special Issue on This research was supported by grant RFBR 19-48-480001 Evolutionary Computation 2, 41–50 (2010) “Development, investigation and optimization of energy-saving [8] A. Boukhelifa, M. Kherbouch, A. Cheriti, R. electrical and electrically driven automated systems for plasma, Ibtiouen, O. Touhami, R. Tahmi, Stator current electrometal slag and induction technologies and units”. minimization by field optimization in induction machine. International Conference on Electrical, Electronic and References Computer Engineering, ICEEC’04 10.1109/ICEEC.2004.1374611 (2004) [1] V.Ya. Ushakov, Isolyatsiya ustanovok [9] S. Lim, K. Nam, Loss-Minimizing Control Blaschke F.A., 1971 New Method for Structural Scheme for Induction Motors. IEEE Proc. - Electr. Decoupling of A.C. Induction Machines. Proceedings of Power Appl. 151(4), (2004) the Conference Rec. IFAC (Duesseldorf, Germany) 1-15 [10] T. Chelliah, S. Srivastava, P. Agarwal, (1994) Energy Efficient Control of Three-Phase Induction [2] I. Boldea, A. Moldovan, L. Tutelea, Scalar Motor - A Review. International Journal of Computer V/f and I-f control of AC motor drives: An overview. and Electrical Engineering. 1(1), 61-70 (2009) 2015 Intl Aegean Conference on Electrical Machines & [11] R. Issa, H. Sarhan, Improving Mechanical (ACEMP), 2015 Intl Conference on Characteristics of Inverter- Induction Motor Drive Optimization of Electrical & Electronic Equipment System. American Journal of Applied Sciences. 3(8), (OPTIM) & 2015 Intl Symposium on Advanced (2006) Electromechanical Motion Systems [12] B.K. Bose, Modern Power Electronics and (ELECTROMOTION) (Side) 8-17 (2015) AC Drives, Prentice Hall (2007) [3] J.M. Peña, E.V. Díaz, Implementation of V/f [13] B. Singh, B.N. Singh, Experience in the scalar control for speed regulation of a three-phase design optimization of a voltage source inverter fed induction motor. ANDESCON. Arequipa. 1-4 (2016) squirrel cage induction motor. Electric Power Systems [4] I. Takahashi, T. Noguchi, A New Quick Research. 26, 155-161 (1993) Response and High Efficiency Control Strategy for an [14] R. Issa, The Effect of Non- Nominal Voltage Induction Motor. IEEE Trans. Ind. Appl. IA-22, 820– on Indices of AC Drive Systems. Journal of Engineering 827 (1985) Studies & Research, Baghdad, 2(1), (1997)

5 E3S Web of Conferences 124, 01036 (2019) https://doi.org/10.1051/e3sconf/201912401036 SES-2019

[15] B.B. Palit, Energy saving operation at induction motors by voltage reduction at no-and low partial load. Proc. IEEE Ind. Appl. Annual Meeting pp 147-151 (1989) [16] S.A. Kovchin, Theory of electric drive [Text] Saint-Petersburg: Energoatomizdat 496 (2000) [17] V.I. Klyuchev, Theory of electric drive: the Textbook for high schools. - 2 nd ed. revised. and add. [Text]. Moscow: Energoatomizdat 704 (2001) [18] G.G. Sokolovsky, 2006 AC electric drives with frequency control [Text] Мoscow: «Akademia» 272 [19] V.N. Meshcheryakov, V.S. Cherkasova, and Meshcheryakova O.V., Correction of system of of the asynchronous electric drive [Text] Control system and information technologies. 3(61), 36–38 (2015) [20] V.N. Meshcheryakov, P.N. Levin, Optimization of mutual position of stator current vectors and magnetic flux of induction motor under vector control [Text] Proceedings of Universities. Electromechanics 1, 25–27 (2006) [21] R.T. Schreiner, Y. Dmytrenko, Optimal frequency control of asynchronous electric drive [Text] Kishinev: Shtiintsa. 223 (1982) [22] V.N. Meshcheryakov, V.G. Karantayev, Application of the search-free adaptive system for control of the electric drive with the valve motor [Text] Electrotechnical complexes and systems 2, 38–40 (2006)

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