Nonlinear speed controller supported by direct torque control algorithm and space vector modulation for induction motors in electrical vehicles.
Direct torque control algorithms for induction motors use reference torque values. Therefore, the transformation of reference speed values into reference torque values is needed. Classical feedback controllers use speed error values for the transformation. In addition, different system parameters are incorporated into the transformation for a better control operation , . As it is well-known, resolution is important for controller calculations and frequency is important at time sharing switching operations , .
In this study, starting from the design of Lyapunov function-based nonlinear controller, mechanical speed value and speed error value are used for the control transformation. To obtain the required torque value, reference speed value is transformed by either classical speed control or nonlinear speed control methods. In this study, PI controller is preferred for the classical speed control and reference controller is obtained by using well-known Ziegler-Nichols tuning rule. Unit step function is used for input reference at design operation . For the nonlinear speed control, Lyapunov function-based nonlinear controller is proposed. The design of the proposed controller is based on an energy function and opposite signed dynamics of this function. Zero is fixity point for the functions , . In addition, a parameter of the controlled system is used in the energy functions and the dynamic functions. Pause point is expected to be almost 1 for the control parameters. The torque controller uses a reference value provided to the controller. A gas pedal is used to set a reference speed value. The speed controller transforms the speed value into the reference torque value. The torque controller calculates required voltage vector for the motor and sends the vector to inverter controller. For the motor system, a power electronic inverter is used to invert direct currents to alternative currents. Time sharing-based vector controller sets the required voltage vector. Angular and amplitude values of the vector are obtained by the inverter control system. For experimental evaluations in this study, a rotational loaded system was used. In the design steps, the variation of electromagnetic torque parameter was taken into consideration. Required reference torque was created by using the reference speed value for direct torque controller (DTC).
II. DIRECT TORQUE CONTROL
Induction machine model is described in the general reference frame by using the following equations:
[[bar.V].sup.g.sub.s] = [R.sub.s] x [[bar.i].sup.g.sub.s] + d[[bar.[psi]].sup.g.sub.s]/dt + j x [[omega].sup.g] x [[bar.[psi]].sup.g.sub.s], (1)
0 = [R.sub.r] x [[bar.i].sup.g.sub.r] + d[[bar.[psi]].sup.g.sub.r]/dt + j([[omega].sub.g] - [[omega].sub.r])[[bar.[psi]].sup.g.sub.r], (2)
[[bar.[psi]].sup.g.sub.s] = [L.sub.s] x [[bar.i].sup.g.sub.s] + [L.sub.m] x [[bar.i].sup.g.sub.r], (3)
[[bar.[psi]].sup.g.sub.r] = [L.sub.r] x [[bar.i].sup.g.sub.r] + [L.sub.m] x [[bar.i].sup.g.sub.s], (4)
where [[bar.V].sub.s] is supply voltage signal of induction motor stator, [[bar.i].sub.s] and [[bar.i].sub.r] are stator and rotor currents, [[omega].sub.g] is general reference speed, [[omega].sub.r] is rotor angular velocity speed, [R.sub.s] and [R.sub.r] are stator and rotor resistances, [[bar.[psi]].sup.g.sub.s] and [[bar.[psi]].sup.g.sub.r] are stator and rotor field linkages. [L.sub.s] represents the parameter of stator leakage inductance, [L.sub.r] represents the parameter of rotor leakage inductance, and [L.sub.m] represents the parameter of mutual inductance. The superscript "g" refers to reference frame , .
Mechanical dynamics of the induction motor is described as follows:
[T.sub.e] = [3/2] x [p/2] x Im([[bar.[psi]].sup.g.sub.s] x [[bar.i].sup.g.sub.s]), (5)
J x [d[[omega].sub.n]/dt] = J x [2/p] x [d[[omega].sub.r]/dt] = [T.sub.e] - [T.sub.load], (6)
where [T.sub.e] is the motor torque, p is the number of motor poles, J is system inertia, [[omega].sub.m] is rotor mechanical angular velocity speed, and Thad is [T.sub.load] torque value . One of the six non-zero and two zero voltage vectors is selected by the torque controller and required switch positions are sent to the inverter , :
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (7)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (8)
where [DELTA][T.sub.e] and [DELTA][[bar.[psi]].sup.g.sub.s] are the symbols of torque and flux errors, respectively. [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] represent the reference values of [T.sub.e] and [[bar.[psi]].sup.g.sub.s], respectively. For selecting voltage vector, the errors must be taken into the hysteresis bands. The [[bar.V].sup.g.sub.s] vectors can be calculated using (9) 
[[bar.V].sup.g.sub.s]] = 2/3 x [V.sub.dc] x ([S.sub.a] + [e.sup.jx2.[pi]/e] x [S.sub.b] + [e.sup.jx4.[pi]/3] x [S.sub.c]), (9)
where [S.sub.a], [S.sub.b] and [S.sub.c] represent stator phase A, phase B, and phase C, respectively.
When the vector is selected, stator flux is rotated to desired frequency, which is inside a specified band. If the motor stator ohmic drops are neglected, stator flux dynamic is calculated using (10)
d[[bar.[psi]].sup.g.sub.s] = [[bar.V].sup.g.sub.s] x [DELTA]T. (10)
If (11) below and (5) are used together, a sinusoidal function of [gamma] is obtained on motor torque. [gamma] is the angle value between stator and rotor fields of induction motor. The angle is adapted and required stator field is obtained using (13) and (14), respectively :
[[bar.i].sub.s] = [[bar.[psi]].sup.g.sub.s]/[sigma][L.sub.s] - [[L.sub.m]/[sigma][L.sub.s][L.sub.r]]/[[bar.[psi]].sup.g'.sub.r] and [sigma] = 1 - [L.sup.2.sub.m]/[L.sub.s][L.sub.r], (11)
[T.sub.e] = [3/2][p/2][[L.sub.m]/[sigma] x [L.sub.s] x [L.sub.r]] x [absolute value of [[bar.[psi]].sup.g.sub.s]] x [absolute value of [[bar.[psi]].sup.g'.sub.s]] x sin [gamma], (12)
[[bar.[psi]].sup.s.sub.s] = [integral]([[bar.V].sup.s.sub.s] - [[bar.i].sup.s.sub.s] x [R.sub.s])dt, (13)
[T.sub.e] = 3/2 x p x ([[phi].sup.s.sub.ds] x [i.sup.s.sub.qs] - [[phi].sup.s.sub.qs] x [i.sup.s.sub.ds]), (14)
where the superscript "s" refers to stator reference frame. [[phi].sup.s.sub.ds] is the real axis component of stator magnetic field space vector fixed coordinate system, [[phi].sup.s.sub.qs] is the imaginary axis component of stator magnetic field space vector fixed coordinate system, [i.sub.ds] is the real axis of stator flux in the fixed coordinate system, and [i.sub.qs] is the imaginary axis of stator flux in the fixed coordinate system.
Measured stator voltage and stator current signals are represented with [[bar.V].sup.s.sub.s] and [[bar.i].sup.s.sub.s]. [[bar.[psi]].sup.s.sub.s] represents calculated stator field vector. [DELTA][[bar.[psi]].sup.g.sub.s] and [DELTA][T.sub.e] states are enough for voltage vector selection . The details of the proposed control system and data flow in the proposed control system are shown in Fig. 1.
III. TIME SHARING-BASED SPACE VECTOR MODULATION
The disadvantage of classical direct torque control systems is the use of inverter control modulator. This disadvantage is reinforced with time sharing-based space vector modulator which supplies required voltage vector at angle position. The hysteresis band values and flux vector components are placed into the time sharing-based space vector modulator. Measured voltage and current parameters are used for the calculation of stator flux vector. These parameters are also used for the calculation of electromagnetic torque. Torque error value and flux vector positions on their hysteresis bands are enough for selecting required supply voltage vector , .
The position, stator flux band position and torque error band position are used in three different cases. In the first case, flux angular position [[epsilon].sub.R]: 0[degrees] ... 60[degrees] is placed in the band area and both of the band values are positive. First active vector is selected as V1 and second active vector is selected as [[bar.V].sub.2]. [t.sub.vectornumber] is relevant vector switching time and [t.sub.total]. is total switching period. Then, [[epsilon].sub.R] is determined in 0[degrees] ... 360[degrees] circular area. When these parameters are used, following time parts are obtained:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (15)
In the second case, [[epsilon].sub.R]: 120[degrees] ... 180[degrees] is placed in the band area, the flux band value is positive and torque error band value is equal to zero. This means [t.sub.1] = [t.sub.total] and [t.sub.2] = [t.sub.0] = [t.sub.7] = 0 time values must be chosen in time domain. In the third case, torque error band value and stator flux band value have negative signs and [[epsilon].sub.R]: 0[degrees] ... 60[degrees] angular position. Then, [[bar.V].sub.5] and [[bar.V].sub.6] active vectors are selected and switching time parts are calculated as follows
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (16)
IV. PERFORMANCE EVALUATIONS
In the proposed system, nonlinear speed controller and time sharing-based space vector modulator are integrated into the induction motor control system. For experimental evaluations in this study, induction motor parameters, [R.sub.1] = 1.027 [OMEGA], [R.sub.2] = 1.475 [OMEGA], [L.sub.0] =112.7 mH, [L.sub.[sigma]1] = [L.sub.[sigma]2] = 8.07 mH and J = 0.089 kg.[m.sup.2]. DC supply voltage is 336 V.
To compare the proposed system with pulse width controlled system, time constant, 20 [micro]s, and carrier frequency, 20 kHz, are used for discrete time applications with pulse width modulator controlled inverter in MATLAB. Sampling time, 50 [micro]s, and [t.sub.total] = 800 [micro]s are used for [down arrow]discrete time application system with time sharing-based space vector modulator controlled inverter in MATLAB. Fig. 2 shows the values of the block parameters of insulated gate bipolar transistor (IGBT) inverter used in this study.
Due to the disadvantage that the PCI card added to the system together with the optoelectronic switches has only one analog to digital converter and one digital to analog converter, stator currents-voltages and encoder signals slide on time phase. Hence, in this system, desired performance characteristics may not be obtained by control algorithms.
Upper valued time parameters are used for pulse width modulator-based system. DC supply voltage is selected as 141 V for both systems. If voltage signals are taken into consideration, the result of the pulse modulator is 110-155 V and the result of the space modulator is 140 [+ or -] 8 V. Desired sinusoidal current signal, [i.sub.continuous] = 2-sin(2[pi] x 50 x t)A, is generated in the stator windings by the space vector modulator controlled inverter. As it is well-known, circular shaped flux is desired for the best performance of induction motors , . Fig. 3 (a) and (b) show the results of stator phase-A voltage signals obtained from the pulse width modulator and the time shared space vector modulator, respectively.
Fig. 4 (a) and (b) show the results of stator phase-A current signals obtained from the pulse width modulator and the time shared space vector modulator, respectively. Fig. 5 (a) and (b) show the results of stator flux positions in D-Q axis obtained from the pulse width modulator and the time shared space vector modulator, respectively. It is seen that desired flux structure is formed by the space vector modulator. The reason of axis slidings is the PCI card used for analog to digital and digital to analog conversions.
In Fig. 6 (a) and (b), the positions of rotor mechanical speeds in time domain obtained from the pulse width modulator and the time shared space vector modulator are shown. After 0.1 sec, pulse width controlled system takes its continuous time position. On the other hand, the proposed space vector controller is slower. It takes its continuous time position after 0.4 sec. But, ~[+ or -] 80rpm fluctuation of the pulse width controlled motor shows that its performance is worse in continuous time.
To compare the proposed system with classical feedback control and Lyapunov function, 10 (is time constant is used for all discrete time simulations in MATLAB. The same system parameters used in the previous experiments are used. As shown in Fig. 7, stator phase-A currents have nearly same characteristics for both the systems. Also, as shown in Fig. 8, stator flux positions in D-Q axis have nearly
As shown in Fig. 9 (a) and (b), the PI controller is 25% faster than the proposed nonlinear controller. However, different from the PI controller, overshoot value of the proposed controller is zero. In Fig. 10 (a) and (b), stator electromagnetic torque positions in time domain obtained from the PI controller and the proposed nonlinear controller are shown.
In this paper, the details of nonlinear speed controller supported by direct torque control algorithm and space vector modulation are explained. With the results of comparative experimental studies, the performance of the proposed system and its advantages over classical controllers are shown.
Proposed nonlinear speed controller has several advantages over classical controllers such as the use of different Lyapunov functions and easy integration of different system parameters.
Manuscript received June 24, 2012; accepted April 19, 2013.
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I. Aliskan (1), K. Gulez (2), G. Tuna (3), T. V. Mumcu (2), Y. Altun (2)
(1) Department of Electrical and Electronics Engineering, Bulent Ecevit University, Zonguldak, Turkey
(2) Department of Control and Automation Engineering, Yildiz Technical University, 34220, Istanbul, Turkey
(3) Department of Computer Programming, Edirne Vocational School of Technical Sciences, Trakya University, Edirne, Turkey
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|Author:||Aliskan, I.; Gulez, K.; Tuna, G.; Mumcu, T.V.; Altun, Y.|
|Publication:||Elektronika ir Elektrotechnika|
|Date:||Jun 1, 2013|
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