Computer simulation and mathematical modelling of ...

Procedia Computer Science 3 (2011) 1009–1017 Procedia Computer Science 00 (2009) 000–000

Procedia Computer www.elsevier.com/locate/procedia Science www.elsevier.com/locate/procedia

WCIT 2010

Computer Simulation and Mathematical Modelling of Static Rotor Resistance Chopper Control of WRIM by Reference Frame Theory Hilmi F. Ameen Salahaddin University, College of Engineering, Electrical Department, Erbil, Iraq, [email protected]

Abstract A detailed analysis and software engineering applied to the static rotor resistance-capacitor drive system under steady state and dynamic conditions are presented. The nonlinear differential equations that describe the system in synchronously rotating reference frame theory are solved by using MATLAB software package. The mathematical state space model utilizing the representation of the machine in synchronously rotating reference frame, where by the direct and quadrature axes are rotating at synchronous speed are presented. The digital simulation results obtained for the torque-speed characteristic of this speed control system is essentially linear for a particular duty cycle. The solution sequence is controlled by a series of conditional statement direct the solution to take into account the type of load characteristics chosen and also the various operational changes, with the effect of duty cycle of the performance of the motor. A WRIM with static rotor-capacitor drive system with constant air gap flux, the electromagnetic torque is directly proportional to rotor rectified current. This control scheme has many advantages, such as smooth and step less control, fast response, less maintenance, longer life and compact size of overall system. c 2010 Published by Elsevier Ltd. Open access under CC BY-NC-ND license. ⃝ Selection and/or peer-review under responsibility of the Guest Editor. Keywords: Software MATLAB Program, Reference Frame Theory; Computer Simulation, Static Rotor-Capacitor Resistance Control, Wound Rotor Induction Motor(WRIM)

1.

Introduction

The WRIM offers a lot of flexibility for wide range of speed compared to squirrel cage motor. The disadvantage is low efficiency due to additional loses in resistance connected in the rotor circuit. As the losses mainly take place in the external resistance they do no-heat the motor itself. The basic analysis of the drive system using the equivalent circuits representation are valid only in steady state operation. In an adjustable speed drive system, the analysis can be conveniently achieved in terms of a mathematical model utilizing the representation of the machine in synchronously rotating reference frame, where by the direct and quadrature axes are rotating at synchronous speed [1]. The simplest speed control scheme for wound- rotor induction motors is achieved by changing the rotor resistance. For the control of induction motors, it is desirable to keep the air gap flux constant. In the rotor control the stator voltage and frequency are kept constant. Therefore, the control of the rotor currents to obtain variable speed is c 2010 Published by Elsevier Ltd. Open access under CC BY-NC-ND license. 1877-0509 ⃝ doi:10.1016/j.procs.2010.12.166

1010

H.F. Ameen / Procedia Computer Science 3 (2011) 1009–1017 Hilmi F. Ameen/ Procedia Computer Science 00 (2010) 000–000

preferred. The rotor resistance control method can provide high starting torque with low starting current and variation of speed over a wide rang below the synchronous speed of the motor. Moreover, the power factor is generally improved. Thus, it is extensively used where a starting current may cause serious line disturbances when the simplicity of operation is desired. However, there are certain applications that require enormous variation of the motor speed. With the increasing in availability of high current power electronic devices, smooth and quick variation of external resistance introduce in the rotor circuit of WRIM to control speed can be accomplished electronically [2]. This circuit is widely used in industrial applications when the drive operation is intermittent such as hoists, cranes, conveyer, lifts and high starting torque are more important with low starting current to avoid voltage drop [3]. In the basic chopper circuit, the rotor power is rectified by a full three-phase diode bridge. A filter inductor (Ld) and external resistor Rex with capacitor in series are connected in series across the diode bridge as shown in figure (1). A chopper is a power switch electronically monitored by a control circuit [3,4].

Fig. 1

The chopper control resistance with capacitor in series of a three phase WRIM

When the chopper is in the ON mode all the time, the equivalent external resistance Reff with series capacitor connected to terminal of bridge rectifier. When the chopper is in the OFF mode all the time, Reff will be equal to infinity. If the chopper is periodically regulated so that, in each chopper period, it is ON for some time and OFF for the rest, it is possible to obtain variation of Reff between zero and infinity . Thus the chopper electronically alters the Rex in a continuos and contactless manner. In the time ratio control (TRC) strategy, the period of chopper T ch is kept constant and the duty cycle (G), which is the ratio of ON time to chopper period is controlled by a pulse width control. Attempts have been made by several authors to study the performance characteristics of the motor chopper drive system, such as: R.M.Crowder and G.A.Smith[5], presented a system combined of rotor chopper resistance control and statorvoltage control regulator for induction motor. Analyzed and experimental results show that stable four-quadrant operation suitable for hoist applications. Y.Jean, P.Viarouge, H.Lehuyand E.J.Dckinson[6], presented a new optimized design of a force commutated thyristor circuit using inductance in parallel with external resistance, then the combination of them in series with thyristor. It used in a wind tunnel up to 200kw, but the commutation circuit is still expensive. N.S.Wani and Ramamoorty [7], have used a thyristor controlled chopper circuit for speed control of slip ring induction motor. A small signal dynamic model is developed for drive system. The dynamic performance is very much improved with speed and current feedback loops. S.D.Joshi, G.K Dubey and S.K Pilliai [8], presented both theoretical and experimental investigation of the method by increasing the control range of pulse resistance, introduced a capacitor in series with external resistance, forced commutated thyristor used as a chopper switch. But the problem at low on periods, because the drive sensed to small changes in duty cycle and hence the commutation failure occur at high frequency due to transient region of low on periods. M.Y.Abdelfattah[9] have used the closed loop control system with two types of controller P and PI controller for WRIM chopper resistance control. Hilmi F. Ameen [10], presented and investigated the harmonic analysis and their effects on the speed control of WRIM by chopper resistance control at different duty cycle and chopper frequency. The aim of the present work of static rotor resistance control of slip ring induction motor to predict the motor drive performance characteristics under steady state and small signal, dynamic model is developed for systems. A novel form of rotor resistance control systems applicable for speed control of a 1.2 kW, three-phase wound rotor induction motor is designed and constructed in the laboratory. A combination of resistor-capacitor scheme is used to increase

1011


the speed range. The static rotor resistance control is analyzed based on a synchronously rotating reference frame theory. The system is simulated on a digital computer using a set of nonlinear simultaneous differential equation, describing the steady state and transient performance of the system. The equations are rearranged and solved by MATLAB package. 1. System Analysis The dynamic model cosiders the instantinous effects of varying voltages/currents, stator frequency, and torque disturbance. The dynamic model of the WRIM is derived by using a two phase motor in direct(d) and quadrature(q) axes. This approach is desirable because of the conceptual simplicity obtained with two sets of windings, one on the stator and the other on the rotor. The concept of power invariance is introduced the power must be equal in three phase machine and its equivalent two phase model. The differntial equations describing the induction motor are non linear. For simplicity and controller design applications, it is important to linearize the machine equations around a steady state operating point. The dynamic performance of an ac machine is quite complex because of the coupling effect between stator and rotor phases where the coupling coefficients vary with the rotor position [11]. Therefore, machine model can be described by differential equation with time varying coefficients. In the reference frame theory, the time varying parameters are eliminated and expressed in mutually decoupled direct (d) and qudrature (q) axis. The d-q transient equation of an induction motor can be expressed either in stationary or in rotating reference frame. From the generalized machine theory, the induction motor can be modeled using two-axis representation with the following assumptions [12,13]; 1-The mmf distribution in the rotor and stator winding is sinusoidal, 2- The uniform airgap, 3-Inductance aginst rotor position is sinsoidal, 4- The saturation is neglected,and 5-The voltage, current and impedance are balanced, the stator and rotor windings are considered identical, sinusoidally distributed, displaced by 120° as shown in figure 2. The transformed (d-q) axis stator voltage for a three-phase machine may be expressed[14];

ªV ds « «V qs «V ¬ os

º » » » ¼

ªcos T «sin T « «¬0.5

cos(T 2S / 3) sin(T 2S / 3) 0.5

º » sin(T 4S / 3) Vs max » »¼ 0.5

cos(T 4S / 3)

ªsin Z s t º «sin(Z t 2S / 3)» s « » «¬sin(Z s t 4S / 3)»¼

(1)

For synchronously rotating reference frame theory (ș= Ȧst). Since at zero, the direct axis of the synchronously rotating reference frame coincides magnetic axis of rotor frame, the direct component of the supply voltage follows the direct axis components of the rotor voltage by angle 90º. . Figure (2) represents the schematic diagram of magnetic axis of the machine stator and rotor, for balanced condition of the three phase supply, it is useful to model the machine in synchronous reference frame [13,14], where related equations are, 2

2

( 2)

2

2

(3)

Vs max

V ds V qs

I sma x

I ds I qs

Where Vos is zero sequence voltage component. It is convenient to set ș=0, so that the equation is simplified to,

Vds

V s max sin(T Z st ) & Vqs

V s max cos(T Z st )

( 4)

Because of the sinusoidal variation of mutual inductances with respect to the displacement angle (șr), time varying coefficients will appear in the voltage equations, by transforms the voltage and current of both stator and rotor to a common frame of reference. The complete voltage equations of the induction motor in the synchronously rotating

1012


reference frame are below. This matrix describes the relation between stator and rotor voltages and currents in term of motor parameter.

Fig. 2 Transformation from 3-phase(a-b-c) to 2-phase( d-q) modelling of three phase induction motor. The stator voltage equation of d and q frame.

dM qs dM ds (5) Z sM qs & Vqs Rs I ds Z sM ds dt dt and Z sM ds are the speed of electromagnetic field due to rotation of the axis. Also the rotor voltage

Vds

Rs I ds

Where

Z sM qs

equation when rotor actually moves at speed Zr;

Vdr

Rr I dr

dM dr Z s Z r M qr & Vqr dt

Rr I qr

dM qr dt

(Z s Z r )M dr

(6)

The flux linkage expressions in terms of the currents can be written as follows,

M ds ° ° M qs ° °M dr ® °M qr °M ° dm °M ¯ qm

Lls I ds Lm ( I ds I dr ) ½ ° Lls I qs Lm ( I qs I qr ) ° ° Llr I dr Lm ( I ds I dr ) ° ¾ Llr I dr Lm ( I qs I qr ) ° ° Lm ( I ds I dr ) ° ° Lm ( I qs I qr ) ¿

(7 )

Combining the equation 7 with equations 5&6, the electrical transient model in terms of voltages and currents for stator and rotor can be given in matrix form as,

ªVds º « » «Vqs » «V ' » « dr » «V ' » ¬ qr ¼

ª R s SL s « «Z s L s « « SL m « ¬ (Z s Z r ) L m

Z s Ls R s SL s (Z s Z r ) L m SL m

SL m

Z s Lm '

Z s Lm SL m

' (Z s Z r ) L r ' ' (Z s Z r ) L r R r SL r

R r SL r

º » » » » » ¼

ªids º « » «iqs » «i ' » « dr » «iqr' » ¬ ¼

(8)

1013


Where S is the Laplace operator and the voltage equation for filter circuit and chopper, with neglecting effect of stator and rotor leakage inductance compared to them . may be expressed as, Vdc =SLdidc +Rd idc+Rex (1-į)/ į idc

(9)

When Vdc is the rectified mean voltage and idc is the instantaneous dc link current. And if the dc link current idc is assumed to be perfectly filtered, then didc/dt o 0, and if Rd is very small hence the ripple voltage will be reduced. Depending on the basis of the models of the induction motor, the bridge rectifier, smoothing inductor and chopper scheme, the static rotor resistance control of wound rotor induction motor equation are rearranged and formulated for the digital simulation of the system using computer simulation MATLAB software. Since the rectifier voltage is transformed to the synchronously rotating reference frame, the instantaneous value of the q-axis always coincides with the maximum value of the rotor phase (a) voltage and (E1max= Vsmax), but the daxis is shifted by then; & Vqr´ =E1max (10) Vdr´ = 0 , Where E1max is the peak phase voltage, the relation between rotor rectified current and peak phase voltage, 3 3 3 3 ' V dc = V qr E 1 max =

S

S

(11)

Where V´dc is the rectifier mean voltage referred to stator windings, the fundamental currents in the rectifier are in phase with input voltage. Since the q-axis is always positioned at maximum value of the rotor voltage, and Vdr´ is always zero. If the losses in the rectifier are neglected, then the instantaneous power on the ac side of the rectifier must be equal to the power on the dc side for the balance condition. Hence, (12) 3/2 Vqr´ iqr´ = - V´dc idc´ Substituting equation (11) into equation (12) , gives that, idc´ = - 0.906 iqr´ , Where iqr´ is quadrature component of the rotor current fed to the rectifier. The negative sign in return to the input power to the rotor. Substituting equation (11&12) into (9) yields that ,

3 3 ' V qr = Ld d/dt(-0.906 iqr´ ) +(-0.906 iqr´) Rd + Rex (1-į) (-0.906 iqr´ ) , S

Vqr´ = -0.55 iqr´ [ Rd + Rex (1-į)/į ] - 0.55 SLd iqr´

(13)

By substituting equation (13) in equation (8) gives;

ªVds º «V » « qs » «0 » « » «¬Vqr/ »¼

ª R s SL s «Z L « s s « SL « m « (Z Z ) L r m ¬ s

Z s Ls R s SL s (Z s Z r ) L m SL m

Z s Lm

SL m

' R r SL r ' (Z s Z r ) L r

º » » '» (Z s Z r ) L r » ' ' R tr SL tr »¼ Z s Lm

SL m

'

ª i ds º «i » « qs » «i ' » « dr » « i qr ' » ¼ ¬

(14 )

The electromagnetic torque developed may be expressed for a (P) pole machine, three-phase induction motor in quadrate form; (15) Te = (3/2)(P/2) Lm (ids iqr´ - iqs idr´) In addition the equations in the mechanical axes, comprising the motion of the motor and driven load are given by Te = J (2/P) dȦr/dt + (2/p) B Ȧr +TL Where Rrt' =Rr' + 0.55(Rd + Rex(1- į)/ į), Lrt' = Lr' + 0.55Ld , J is the moment of inertia in (Kg.m2), TL is the mechanical load torque , B is damping coefficient in N.m/(rad/sec), and į is duty cycle

(16)

1014


Computer Simulation For computer digital simulation of static rotor resistance drive control of WRIM, it is convenient to rearrange the equations in a form of the state space variables, which is suitable for the numerical solution. In case of balanced three phase supply of induction motor the synchronously reference frame is more convenient, and the equations of WRIM controlled by static chopper resistance enhanced with dc capacitor in series (Rex + C) in state space [14] are below, ªV ds º «V » « qs » «0 » » « ' «¬V qr »¼

ªLs «0 « «L « m «0 ¬

0 ª0 «0 0 « Zr « 0 Lm « «¬ L m 0

Lm

0 Ls

0

0

Lr

Lm

0

0

0

0 0 Lr

'

'

º 0 » » ' Lr » » 0 »¼

0

º L m »» » 0 » ' L rt »¼

ª i ds «i « qs «i « dr « i qr ¬

d dt

ª « « « « « ¬

º » ª« R s » + «0 ' » « » «0 ' » «0 ¼ ¬

i ds º i qs »» Z ' i dr » » ' i qr »¼

s

ª0 «L « s «0 « «¬ L m

0

0

Rs

0

0

'

0

Rr

0

Ls

Lm

0 Lm

0 Lm

0 0

Lr

º » 0 » » 0 » ' R rt »¼


0

c

º » » '» » '» ¼

º 0 »» L rc » » 0 »¼


º » » ' » » ' » ¼

( 17 )

The mathematical arrange and differentiation of the different currents are grouped as a separate column vector on the left hand side of the equations, consequently the voltage equation (17) rearranged accordingly into four simultaneous equations which can be solved by using MATLAB software toolbox. This equations rearranged for synchronously rotating reference frame static rotor resistance control of wound rotor induction motor drive system as given in the following forms,

d [i]/dt =[L]-1{[V]-[R]-Ȧs[M]-Ȧr[N]}[i] Where according to the equation 20, [i] = [ids iqs idr´ iqr´], [R] = Diag[Rs Rs Rr´ Rrt´], [V] = [Vds Vqs 0 Vqr’ ], [N] = matrix of inductance with Ȧr.

(18) [L] = is the inductance element matrix, [M] = matrix of inductance with Ȧs, and

The above state space equation of WRIM motor as obtained by considering mechanical system dynamics is added, when speed changes, the instantaneous electromagnetic torque (Te) and the instantaneous load torque (T L) are related as given in equation (16). Now the five nonlinear simultaneous differential equations can set to be solved by using the MATLAB software package of integration. The solution sequence is controlled by a series of conditional statement direct the solution to take into account the type of load characteristics chosen and also the various operational changes, with the effect of duty cycle(G), load torque (TL), external resistance (Rex), chopper frequency, and the rotor speed the motor. Simulation Results The parameters of drive system and rated variables of WRIM under the computer simulation test are ( 1.5 kW, 380V, 3.5A, 50Hz, 4-pole, Y-connected, 1440rpm, Rs = Rr´= 2.4ȍ, Ls = Lr´ =332mH, J( motor + load ) (kg.m2)= 0.06, B( motor + load ) (kg.m2 /sec)= 0.05), Rex = 50:, C = 600PF, chopper frequency = 1kHz, smoothing inductor = 0.15H, smoothing inductor resistance= 0.5:. Performance of static rotor chopper resistance-capacitor control drive system are simulated by Matlab software package using the set of equations presented in synchronously rotating reference frame. Figure 3 shows the variation of the Ids and Iqs against speed in(rpm) at different duty cycles where q-axis is assumed to be positioned so that the

1015


Fig. 3 Ids ,Iqs against speed at different duty cycles. 1400

70

G(Duty cycle) =0.85 TL(load Torque)= 10N.m

50

(a)

1200

1000

T e (N .m )

40

30

600

20

400

10

200

0 0

0.5

1


800

S p e ed (r p m )

60

0 0

1.5

(b)

0.5

1

1.5

Time(sec)

Time (sec) 25

30

20


20

10

Id s,Iq s,Id r',Iq r'(A )

R o to r R e c tif ie d C u r r e n t I d c (A )

Ids Iqs

(d)

15

0

10

-10

Idr'

(c)

Iqr'

-20

-30 0


0.5

Time (sec) Time(sec)

1

5

1.5

0 0

0.5

TimeTime(Sec) (sec)

1

Fig. 4 a- The electromagnetic torque, b-The rotor speed, c- The Ids ,Iqs, Idr’ , Iqr’ and d- The rotor rectified current responses at full load torque and 0.85 duty cycle depending on synchronously rotating reference frame theory.

1.5

1016


30

1400


25

1200

1000

(a)


20

T e(N .m )

S p e ed (rp m )

800 15

(b) 600 400

10

200 5

0 0 0

0.5

1

1.5

2

2.5

3

3.5

4

Time(sec)

15

-200 0

0.5

1

1.5

2 Time(sec)

2.5

3.5

4

9



8

(c)

10

(d)

7 R o t o r R e c t i f ie d C u r r e n t I d c ( A )

Ids I d s ,I q s ,I d r ', I q r ' ( A )

3

Iqs

6

5

5 4

0

3

Idr'

-5

2

Iqr'

1 -10 0

0.5

1

1.5

2 Time(sec)

2.5

3

3.5

4

0 0

0.5

1

1.5

2 Time(sec)

2.5

3

3.5

4

Fig. 5 a- The electromagnetic torque, b-The rotor speed, c- The Ids ,Iqs, Idr’ , Iqr’ and d- The rotor rectified current responses at full load torque and 0.85 duty cycle depending on synchronously rotating reference frame theory. quadrature axis stator voltage equal to peak value of stator phase voltage. The values of I ds for (G=0.15, G=0.45 and G=0.85) are constant with varying speed from standstill to full load speed, but the values of Iqs for (G=0.15, G=0.45 and G=0.85) are varied from full load current to zero at no load speed approximately. From the values of the figure 3 shown that the root mean square summation of direct and quadrature axis current are equal to the peak value of the stator input current. Figure 4 shows the electromagnetic torque response, the rotor speed response, the (direct axis stator current Ids , quadrature axis stator current Iqs, direct Idr' and quadrature axis rotor current Iqr' referred to stator side Ids, Iqs, , Idr' , and Iqr' ) response and the rotor rectified current at starting for G=0.85 and full load torque of TL=10 N.m. Figure 5 shows the electromagnetic torque response, the rotor speed response, the (direct axis stator current I ds , quadrature axis stator current Iqs, direct Idr' and quadrature axis rotor current Iqr' referred to stator side) response and the rotor rectified current at starting for G=0.5 and full load torque of TL=5 N.m.

H.F. Ameen / Procedia Computer Science 3 (2011) 1009–1017

1017

Hilmi F. Ameen/ Procedia Computer Science 00 (2010) 000–000

Conclusion A WRIM with static rotor-capacitor drive system can match the dc motor performance with constant airgap flux, the electromagnetic torque is directly proportional to rotor rectified current. This control scheme has many advantages, such as smooth and stepless control, fast response, less maintenance, longer life and compact size of overall system. The model that describes the open loop static rotor resistance-capacitor drive system based on the synchronously rotor rotating reference frame theory. Variables like the rotor rectified current, the electromagnetic torque produced, the rotor current, the stator current and the rotor speed were predicted from solving nonlinear differential equations using MATLAB software package. This model used to predict the operation of static rotor resistance-capacitor drive system both in dynamic and steady state to find the speed and rotor rectified current response for step change in į and load torque disturbance. Also, the system simulated by using simulation program SIMULINK of MATLAB package realized all components of the system with its values, three phase induction motor, bridge rectifier, smoothing inductor and power semiconductor devises with its drive circuit, to show the waveforms and responses of all variables.

References 1. Zuhair D. Shebeeb,Slip Power Recovery Induction Motor Drive System , Ph.D. Thesisin the Electrical and Electronics Engineering Department, University of Technology, 1998. 2. Hilmi F. Ameen, “ High Chopper Frequency for WRIM with Resistively Load Rotor Chopper”, ASTF-SRO4 Conference , Damascus, Syria, 2005. 3. P. C. Sen and K. H. Ma, " rotor chopper control for induction drive: TRC strategy " , IEEE Trans. on Ind. Appl. ,vol. IA-11 , No.1, January/February 1975. 4. N. S. Wani and M. Ramamoorty ," chopper controlled slip ring induction motor" , IEEE Trans. on IECI, vol.24, no.2,May 1977. 5 R. M. Crowder and G. A Smith," induction motors for crane applications", IEE Jour. of Electric power Applications. Vol.2, No.6, December 1979. 6. Y. Jean , P.Viarouge, H.lehuy and E. J Dikinson, " auto-adaptive chopper for speed regulation of wound rotor induction machine", IEEE Trans. on Ind.Appl., vol.19,No.6,November/December 1983. 7. N .S. Wani and M. Ramamoorty," dynamic model for a chopper- controlled slip-ring induction motor", IEEE Trans. on IECI, vol. 25, No.3, August 1978. 8. S. D Joshi, G.K Dubey and S.K Pilli, " Extension of control Range of pulse resistance controlled wound rotor induction motor " I.E.(I) Journal-EL, EE Division, Vol.61,December,1980. 9. M.Y.Abdelfattah,” SPeed Control of WRIM using Chopper Controlled External Resistance Enhanced with a dc capacitor”, Alexandria Engineering Journal, Vol. 42, No.1, 2003, pp 25-34. 10.Hilmi F. Ameen,, “ Stator Current Harmonic Analysis and Torque pulsation of WRIM speed control by Chopper Resistance in Rotor Circuit”, Zanko Journal of pure and Applied Science, Vol 19, No.1, 2007, pp 123-134. 11.Bimal.K. Bose, " Adjustable Speed AC Drives- A Technology Status Review" , proceeding of the IEEE, vol 70, No.2, February 1982. 12. P. C. Krause and C. H. Thomas, " Simulation of Symmetrical Induction Machinery ", IEEE Trans. PAS-84, Nov.1965. 13. P. C. Krause, " Method of Multiple Reference Frames Applied to the Analysis of Symmetrical Induction Machinery ", IEEE Trans. PAS -87, No.1, January 1968. 14.John R. Smith, " Response Analysis of A.C. Electrical machines Computer Models and Simulation " , John Wiley & Sons Inc. 1990.