Svpwm Variable Structure Control Of Im Drives

SVPWM Variable Structure Control of Induction Motor Drives P. Alkortaȥ, O. Barambones, A. J. Garrido and I. Garrido E.U.I.T.I. de Eibarȥ / E.U.I.T.I. de Bilbao University of the Basque Country Eibar, Spain Email: [email protected], [email protected]

Abstract— This paper presents a new proposal of speed vector control of induction motors based on robust adaptive VSC (Variable Structure Control) law and its experimental validation. The presented control scheme incorporates the SVPWM (Space Vector Pulse Width Modulation) instead of the traditional current hysteresis comparator. The SVPWM improves the quality of the stator currents, reducing the harmonics, while maintains the main characteristics that is usual in this kind of algorithm, like the fast response and good rejection to uncertainties and measurement noises. This regulator is also compared with the PI (Proportional Integral) controller designed in the frequency domain, in order to prove the good performance of the proposed controller. The two controllers have been tested using various simulation and real experiments, taking into account the parameter uncertainties and measurement noise in the loop signal, in the rotor speed and in the stator current. This work shows that the VSC regulator is more efficient than the traditional PI controller in both adverse conditions and suitable conditions. Finally, some practical recommendations for real experiment implementations are also given.

I.

INTRODUCTION

AC induction motors have been widely used in industrial applications such machine tools, steel mills and paper machines owing to their good performance provided by their solid architecture, low moment of inertia, low ripple of torque and high initiated torque. Some control techniques have been developed to regulate these induction motors servo drives in high-performance applications. One of the most popular technique is the indirect field oriented control method [1], [2], [3]. The field-oriented technique guarantees the decoupling of torque and flux control commands of the induction motor, so that the induction motor can be controlled linearly as a separated excited D.C. motor. However, the control performance of the resulting linear system is still influenced by uncertainties, which are usually composed of unpredictable parameter variations, external load disturbances, and unmodelled and nonlinear dynamics. Therefore, many studies have been made on the motor drives in order to preserve the performance under these parameter variations and external load disturbances, such as nonlinear control, optimal control, variable structure system control, adaptive control and neural control [4], [5], [6].

1-4244-0755-9/07/$20.00 '2007 IEEE

In the past decade, the variable structure control strategy using the sliding-mode has been focussed on many studies and research for the control of the AC servo drive systems [7], [8], [9], [10].

Fig. 1. Diagram of PI speed control of induction motor.

The sliding-mode control offers many good properties, such as good performance against unmodelled dynamics, insensitivity to parameter variations, external disturbance rejection and fast dynamic response [11]. These advantages of the sliding-mode control may be employed in the position and speed control of an AC servo system. Since V. Utkin proposes in 1993 [11] the sliding mode design principles and applications to electric drives, a lot of authors have used this advanced technique in induction motors speed control. The most important characteristics that may offer this control algorithm are the good rejection to parameter uncertainties and to measurement noises, good behavior with non-modeled dynamics and fast response. Recently, it has been proposed [12] an induction motor speed control based on VSC algorithm that may eliminate the speed tracking error in spite of the presence of important uncertainties and measure noises. However, the control scheme proposed in [12] uses a current control based on a hysteresis-band which may cause an undesirable harmonics generation. In the present paper the author proposes a SVPWM control that improves the quality of the stator currents and reduces the harmonics generation that is usual in the traditional current hysteresis comparator. This report is organized as follows. The PI speed controller design is introduced in Section II. Then, the proposed variable structure robust speed control is presented in Section III. In the

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Section IV, the variable structure speed control with current PI controller is introduced. The induction motor speed control simulation and experimental results are presented in Section V. Finally some concluding remarks are stated in the last Section. II. PROPORTIONAL INTEGRAL SPEED CONTROLLER DESIGN The PI control algorithm is well known classic control algorithm. This control strategy has a great popularity, and a extended use in the industry because its simple design and good performance. Moreover, many times the PI control is used as a reference in order to compare other controllers with it. As it is well know, the closed loop stability of the motor with the PI controller is guaranteed if the PMȦ margin phase has a positive and suficiently high value. Therefore, the gains of the PI controller may be calculated ussing the frequency domain [13].

From the Fig. 2 diagram the open loop transfer function is obtained as: Ki · 1 § (3) TFOL ( s ) Zm ¨ Kp  ¸ s ¹ sJ © Considering that in the cross frequency the system has a unit gain (4) TFOL ( s ) Z 1 m

s jZc

and that the system phase is



arg TFOL ( s ) Zm



180º  PM Z

s jZc

It is obtained the following expressions for the PIȦ controller parameter’s tg( PM Z ) Ki (6) Kp

Zc

JZ c

Ki

2

1  tg( PM Z )

Fig. 2. Diagram for speed loop PI controller design

Figure 1, shows the PI clasic control for a induction motor and the function of the blocks that appear in this diagram are: The ABCÆdq block get the is space vector from the iA, iB e iC motor stator currents, using the Park’s transformation [3], while the dqÆABC block makes the reverse Park’s transformation. It should be noted that this transformations make use of the rotor flux angular position, șe and therefore this angle should be calculated using an indirect method. The design of the PI controller is quite easy but in this control scheme the induction motor parameters should be precisely know. The design consists on the one hand, in calculating the PI controller parameters, Kp and Ki, for the Zm mechanical rotor speed loop, and on other hand, in calculating the PI controllers parameters, Kpi and Kii. for the two current loops, isd and isq . The calculation of the PI controller parameters for the speed loop is based on the diagram shows in Fig. 2. We may choose a band width of Ȧc rad/s and a margin phase of PMȦ dB in open loop. The motor electromagnetic torque [3] has the following expression (1) in stationary state, taking into account that the torque and rotor flux components are decoupled in the rotating reference frame,

Te

3 p Lm \ rd isq 4 Lr

K T isq

(1)

where KT is the torque constant (2) and <*rd is the rotor flux command 3 p Lm * (2) KT \ rd 4 Lr

(5)

(7) 2

On the other hand, the function of the current loops PI controllers is to generate the voltage commands for the SVPWM modulator block, so that then this voltage signals are used as voltage commands for the VSI (Voltage Source Inverter) block. Taking equations (8) and (9), and considering that its components are decoupling, it is obtained the diagram of Fig. 3

v sd

v sq

disd Lm d\ rd   ZeVLs isq dt Lr dt disq L Rs isq  VLs  Zd m \ rd  ZeVLs isd dt Lr

Rs isd  VLs

(8) (9)

when 2

V

1

Lm Ls Lr

(10)

Fig. 3. Diagram for current loop PI controllers design

The calculation procedure for the Kpi and Kii parameters of the current loop is based on the diagram of figure 3 and is similar to the previously explained parameters calculation for the speed loop. However, in this case, the bandwidth is ten times higher than the speed loop bandwidth, Ȧci=10*Ȧc rad/s, even if the phase margin is the same PMi=PMȦ. Now, from diagram of Fig. 4, it is obtained the open loop transfer function of the system and considering that in the cross frequency, the system has a unit gain and an argument of -180º +PMi, the current PI controllers parameters may be obtained,

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Z LV· S § tg¨ PM i   arctg ci s ¸ 2 V ¹ ©

G

Z ci Rs 2  (Z ci LsV ) 2

Kii

d (t )

(11)

(12)

S (t )

Kii G

(20)

Next it is defined the sliding variable S(t) with an integral component as

1 G 2

Kpi

 'aZm (t )  'f (t )  'bi sq (t )

t

e(t )  ³ ( k  a )e(W )dW

(21)

0

(13)

Zci

III. VARIABLE STRUCTURE SPEED CONTROLLER DESIGN The VSC using the sliding mode offers some good properties just as a good behavior in the presence of non modelated dynamics, load disturbances, parameter variations and measurement noises, when also offers a fast dynamic response [11]. These advantages are used in the vector speed control of induction motor drives. The stability of the proposed motor controller is demosntrated using the Lyapunov stability theory. This powerful algorithm applied to speed control of induction motors, was proposed initially in [12].

Then the sliding surface is defined as: S(t) = 0. Now is designed a variable structure speed controller, that incorporates an adaptive sliding gain, in order to control the induction motor drive

u( t )

ke(t )  E (t ) sgn(S )

(22)

when sgn( ) is the sign function and ȕ is the switching gain.

From (1) and the mechanical equation (14)

JZ m  BZ m  TL

Te (14)

is obtained

Z m  aZm  f

bi sq

(15)

where the parameters are defined as

KT TL B ,b ,f J J J Now, is considered the previous mechanical equation mechanical (15), with a, f y b terms uncertainties ('a, 'f y 'c) a

 Z m

( a  'a )Zm  ( f  'f )  (b  'b)isq 

Fig. 4. Diagram of VSC speed control of induction motor with the current control based on hysteresis comparator.

In order to obtain the speed trajectory tracking, the following assumptions should be formulated: (A1) The gain k must be chosen so that the term (k-a) is strictly negative. Therefore the constant k should be k<0.

The speed tracking error is defined as  e(t ) Z m (t )  Zm* (t )  

(A2) The gain ȕ must be chosen so that ȕ • |d(t)| for all time.

when Ȧm* is the rotor speed command. Taking the derivative of the previous equation respect to time yield

Theorem 1. Consider the induction motor given by equation (16), and if assumptions (A1) and (A2) are verified, the control law (22) leads the rotor mechanical speed Zm(t) so that speed tracking error (17) tends to zero as the time tends to infinity. This is demonstrated in [12], using for it the function candidate (23) and the Lyapunov stability theory,

e(t ) Z m (t )  Z m* (t ) ae(t )  u (t )  d (t )

(18)

where

u (t )

bi sq (t )  aZm* (t )  f (t )  Z m* (t )

(19) V (t )

and the uncertainty terms have been collected in the signal d(t)

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1 S (t ) S (t ) 2

(23)

Finally is concluded that S(t) tends to zero as the time t tends to infinity. Moreover, all trajectories starting off the sliding surface S(t)=0, must reach it in finite time and then will remain on this surface. This system’s behavior once this sliding surface is called sliding mode [11]. When the sliding mode occurs on the sliding surface, then S (t ) S (t ) 0, and therefore the dynamic behavior of the tracking problem is equivalently governed by the following equation:

S (t ) 0 Ÿ e(t )

(k  a )e(t )

V. SIMULATION AND EXPERIMENTAL RESULTS The Fig. 6 shows the blocks diagram of the control platform for the induction motors designed in the EUITI of Eibar [14] that may be used to real time implementation of the advanced induction motor control algorithms.

(24)

Then, under assumption (A1), the tracking error e(t) converges to zero exponentially. Finally, the torque current * command, isq (t ) , can be obtained directly substituting (22) in (19):

i sq* (t )

1 ke  E sgn( S )  aZm*  Z m*  f b

>

@

(25)

Fig. 6. The detailed blocks diagram of experiments system

IV. VARIABLE STRUCTURE WITH CURRENT PI CONTROLLER The current control based in the hysteresis-band is easy to understand and to implement, but it is not adequate for real implementations because this method produces a lot of harmonics in the stator currents of the induction motor. It is well know that the highest efficiently of the induction machines is obtained when the stator currents are a pure sinusoidal ones. That is, when this currents do not have any harmonics. In this way, in this paper it is proposed the replacement of hysteresis-band module for the SVPWM modulator in order to eliminate the harmonics generated for the hysteresis-band current control. However, this modulator needs the stator voltage three-phase command but the VSC controller gives the current command. To overcome the previous problem in two PI controllers are used to convert the two current commands in two voltage commands and subsequently they can be convert through the dqÆABC block in the required three-phase voltage command, Fig. 5.

Fig. 5. Diagram of VSC speed control of induction motor with the PI current control and SVPWM.

This control platform allows to verify the real time performance of the induction motor controls in a real induction motors. The designed platform can be divided in three main blocks. The first one is the design block that consists of a PC with Windows 2000 in which it is installed MatLab7/Simulink and dsControl 2.5.5 of dSpace and the DS1103 Controller Board real time interface. The second block is the power block that is formed of a three-phase rectifier connected to 380 V/50 Hz AC electrical net and a capacitor bank of 27.200 µF for to get a DC bus of 540 V, two three-phase IGBT/Diode bridges of 100A, a M2AA 132M4 ABB induction motor of 7.5KW of squirrel-cage type, with the following parameters: Rs, stator resistance, 0.57 : Rr, rotor resistance, 0.81 : Lm, magnetizing inductance, 0.117774 mH Ls, stator inductance, 0.120416 mH Lr, rotor inductance, 0.121498 mH p, number of poles, 4 J, moment of inertia, 0.043 kg m2 B, viscous friction coefficient, 0.001 Nm/(rad/s) Finally, the third block is the measures-securitycommunications block, which makes the rotor speed and stator currents measurements, a PLC that watches the DC bus charge, and all the communications needed between the different modules. The simulation tests has been realized in Simulink with a simulation period of 2 µs and ode1(Euler) Fixed-step type numeric algorithm. The power electronics modules used are the Universal Bridge with 3 arms of IGBT/Diode and Asynchronous Machine of Squirrel-cage type. The sampling time used in real tests for controller modules is of 100 µs, also used for the same controller modules in the simulation tests.

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Fig. 7. Graphs of the simulation test of the original VSC model: it arrives speed reference and response, in means iA stator current, and down, sliding variable.

Fig. 9. Graphs of the simulation and real tests of the proposed VSC-SVPSW model: it arrives speed reference and response, in means iA stator current, and down, sliding variable.

Fig. 8. Graphs of speeds and electromagnetic torque of simulation and real test of PI control with Ȧc =25 rad/s y Ȧci =250 rad/s

Fig. 10. Graphs of speed and electromagnetic torque of simulation and real test of VSC-SVPWM control, with Ȧci =250 rad/s for PI current controllers

The rotor flux current command, isd*, has been fixed to the 8.61 A constant value to keep the rotor flux in its nominal value of 1.01 Wb. On the other hand, the electromagnetic torque current command (1), isq*, also has been fixed to 20 A, to limit and to protect the over currents in the induction motor’s stator fed.

The graphs of Fig. 8 and 10 show to the speed response and the electromagnetic torque obtained in the simulation and in the experimental tests, using the PI control and the new proposed VSC control respectively. In this test it is used a 1000 rpm and 1 Hertz saw-toothed input reference. As it may be observed, VSC-SVPWM control offers an answer with better dynamics than PI control. In addition, it may be observed that, after a transitory time, the speed tracking error is eliminated. Making several tests with different designs from PI controller and using a greater bandwidth every time, it is observed that the answer of the system is faster indeed, but from approximately 650 rad/s of cross frequency in current control loop appear interferences and/or noises in the laboratory where the platform is, and they are introduced in the system, producing instability in form of strong mechanical vibrations in the motor. Then, we must stop the application that supports the experiment before the equipment could be damaged. Fig 11 shows the effect of this noise in the answer of the motor when the reference input is a 100 rpm and 1 Hertz saw-toothed signal, with control PI Ȧc = 70 rad/s and Ȧci = 700 rad/s. Therefore the PI controller can not obtain so good performance as it is obtained with VSC controller.

A. Speed Tracking The smaller is the hysteresis band, the better is the answer of the motor, Ȧm , but at the cost of the frequency increment of the IGBT. Since the frequency of commutation of VCI module of the platform is limited to 8 kHz, this band is fixed to 13A, and the simulation results may be seen in Fig 7. The following values have been chosen for the VSCSVPWM controller parameters: Ȧci = 600 rad/s, and k = -50 and ȕ = 30, these last ones are equal that the value used in [12]. The uncertainties in the parameters have been fixed to + 20 % for all tests in VSC controllers. Fig 9 shows to the graphs of the simulation test and real implementation. Comparing the graphs of Fig. 7 and 9, it is possible to be observed that the new proposed design offers the same effectiveness of control that the original one, and that in addition the stator currents are practically smaller and without ripple, as much in simulation as in real implementation.

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Fig. 11. Graphs of reference and response speed of PI control real test with Ȧc =70 rad/s and Ȧci =700 rad/s

B. Uncertainties and measurement noise reject Simulation and real test have been made using different noise levels in the two feedback signals: Ȧm and iA. In addition, the tests have been also made introducing the uncertainty in the moment of inertia: the controllers are designed supposing that J’ = 0.03 kg m2. The graphs of Fig. 12 and 13 show the response for 200 rpm and 1 Hz square reference signal, and iA stator current, using the PI control and the VSC control, respectively.

motors. The main contribution of this work consists of the elimination of the stator current harmonics. In order to eliminate the current harmonics, the current control based on the hysteresis comparator is replaced by the SVPVM modulation. This new control, VSC-SVPWM control scheme, has been also compared with the PI control, by means of some simulation and some real tests. The controllers have been compared both in favorable conditions and in very adverse situations of measurement noise and uncertainties, and it is concluded that VSC-SVPWM control offers a better answer than the one of PI control in all cases. The real experiments allow us to conclude that the control platform is a very efficient tool due to the great similarity existing between the results of the simulation text and the experimental ones. Finally, several practical recommendations like the limitation of the currents, frequency of commutation of the IGBT of VCI inverter and the bandwidth in the design of PI control have been also presented. ACKNOWLEDGMENTS The author are grateful to the Basque Country University for partial support of this work through the research Projects 1/UPV 00146.363-E-16001/2004 and EHU06/88, and also to the Ministry of Education and Science of Spain for the Project MEC DPI2006-01677.

Fig. 12. Rotor speed and stator current graphs for PI control, with high level in Ȧm and iA measure, and uncertainty in J.

REFERENCES [1] [2] [3] [4] [5] [6]

Fig. 13. Rotor speed and stator current graphs for VSC-SVPWM control, with high level in Ȧm and iA measure, and uncertainty in J.

The simulation tests have been done supposing that the moment of inertia is J’ and w without noise measurement, while the real tests have been done with J = 0.043 kg m2 (+ 43,33 % of J’) and a with the measurement noise inherent to the real systems. In this sense there is a high level of noise in the measurement of the rotor speed Ȧm and in the iA signals. However, it may be observed that the speed tracking is good in both controllers but the VSC controller has a better performance than the PI controller both with and without system uncertainties and measurement noises.

[7] [8] [9] [10] [11] [12]

VI. CONCLUSIONS

[13]

In this work it is presented and experimentally validated in real time a new VSC controller for speed control of induction

[14]

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