Field-weakening control schemes for high-speed drives based on induction motors

Field-Weakening Control Schemes for High-Speed Drives Based on Induction Motors: a Comparison

D. Casadei, M. Mengoni, G. Serra, A. Tani, L. Zarri

University of Bologna, Department of Electrical Engineering, Bologna, Italy

Abstract—Three sensorless control schemes for the operation of induction motors in the field-weakening region are compared and assessed in terms of performance and complexity. These three control schemes fully utilize the maximum available voltage and current for steady-state torque production at any speed, and thus provide the maximum possible torque in the entire field weakening region.

For the comparison, the three control schemes are implemented on the same experimental platform, i.e. the same Digital Signal Processor (DSP) board, power inverter and motor drive. In this way, it is possible to judge not only the performance of each solution, but also its requirements in terms of computational time, tuning complexity, parameter knowledge and stability of operation.

I.I NTRODUCTION

Power electronics has deeply changed the use of induction motors in automotive or automation applica

tions, giving them the capability of fast torque response, and consequently a full control of the drive speed.

路灯远程控制系统When the induction motors are used for applications at high speed, it is desirable to retain the maximum torque capability in the field weakening region. Several papers about this issue were presented [1]-[4]. According to these field weakening algorithms, the optimal flux value of the motor should be updated by means of look-up tables or explicit expressions containing the motor parameters and quantities such as the motor speed, the motor currents, the dc-link voltage and the requested torque. However, the performance of these algorithms is strictly related to the accuracy by which the parameters are known. A further problem is represented by the variable value of the leakage and magnetizing inductances, to which the rotor-flux-oriented scheme is particularly sensitive [5]. In addition, the drive performance in the high speed range may depend on the correct determination of the base speed, which is function of the actual dc-link voltage and the overload capability. As a consequence, new methods for compensating the parameter variations and the uncertainties of the models have been investigated. Among these, some adaptive schemes have been proposed in order to provide a suitable estimation of the varying parameters [6]-[8]. These methods provide good drive performance to the detriment of the complexity of the control scheme and the regulator tuning.

For the reasons stated above, the stator-flux-oriented drive, more insensitive to parameter variations than the rotor-flux-oriented one, has received an increasing attention for field weakening applications [9]-[11].

A suitable method for robust field weakening is to determine the optimal flux level using closed-loop schemes that analyze the motor behavior, rather than look-up tables or explicit expressions containing the motor parameters.

In the last ten years several important contributions towards robust field weakening strategies were proposed in [12], [13] for stator-flux-oriented induction motor drives and in [14]-[18] for rotor-flux-oriented induction motor drives. According to these papers, the flux level is adjusted on the basis of the supply voltage requested by the regulators and the maximum torque capability is exploited by means of a suitable control strategy.

In this paper three sensorless control schemes for the operation of induction motors in the field-weakening region are compared and assessed in terms of performance and complexity.

These three control schemes fully utilize the available inverter voltage and the maximum inverter current for steady-state torque production at any speed, and thus provide the maximum possible torq

ue in the entire field weakening region. In addition, all these control algorithms are insensitive to changes of the machine parameters and to variations of the dc link voltage.

The performance of the three schemes seems quite alike but the operating principles of the three control schemes are indeed different in terms of number and type of regulators, complexity of implementation and transient behavior. It is rather difficult to assess the performance of different control schemes, since they are often presented with reference to different experimental set-up. For the comparison, the three control schemes have been implemented on the same experimental platform, i.e. the same DSP, power inverter and induction motor, and use the same basic functions, such as the voltage modulator. In this way, it is possible to judge not only the performance of each solution, but also its requirements in terms of computational burden, calibration complexity, parameter requirements and operating stability.

II.O PERATING P RINCIPLES OF THE R OBUST F IELD

W EAKENING C ONTROL S CHEMES

In the high-speed range the motor performance is limited by the maximum inverter voltage V s,max, related to the dc-link voltage, and the inverter/machine current rating, represented by the maximum s

tator current I s,max. These limits sensibly influence the motor behavior, especially at high speed. It is known that the operation of

an induction motor can be divided into three speed ranges, namely the low speed range (region I), the constant-power speed range (region II) and the decreasing-power speed range (region III).

The basic principles for obtaining the maximum torque in region I, II and III are explained in the following paragraphs.

A.Effect of the Current Limit

The current limit determines the maximum torque that can be generated in region I and in region II. In particular, in the low speed range, the maximum torque corresponds

to the current limit and to the rated flux level, whereas in region II, it is necessary to reduce the stator flux

magnitude to keep the back emf approximately constant.

B.Effect of the Voltage Limit When the stator voltage magnitude equals V s,max , it is possible to express the motor torque as follows [18],

D ¸¸¹

·¨¨©§Z V #2sin 432max ,22s r s V L L M p T (1)where p is the pole pairs number, L s ,L r and M are the motor self and mutual inductances, Z is the angular speed

of the rotor flux vector with respect to a stationary

reference frame, D is the angle between the stator and the rotor flux vectors, and V is defined as follows: r

s L L M

21 V . (2) From (1) it is clear that for any value of Z , the maximum torque is produced when the angle between the stator flux and the rotor flux vectors is 45°, i.e., D =r 45°.This fundamental relationship is used by the three control schemes to achieve the maximum torque operation in region III. It is worth noting that there are different ways to express the condition D =r 45°. An equivalent formulation considers the input voltage vector instead of the stator flux vector. Since the input voltage vector leads the stator flux vector by nearly 90 degrees, the condition of maximum torque happens when the angle between the input voltage vector and the rotor flux vector is 90r 45 degrees,

namely 135° for motor operation or 45° for generator operation. III.D ESCRIPTION OF THE C ONTROL S CHEMES

In this paper three robust field weakening control schemes for induction motors are compared.

The first one (scheme A) is suitable for a stator-flux oriented drive and its basic principle was presented in [13], the second one (scheme B) and the third one (scheme C) are rotor-flux oriented drives and their basic principles were presented in [14] and [18] respectively.

These control scheme were selected because they are relatively recent and are based on the common principle of analyzing the motor voltage to adjust the flux level.

A.Control Scheme (A) The block diagram of the control scheme (A) is shown in Fig. 1. The control scheme is implemented in a

reference frame synchronous with the stator flux vector. The main control variables are the stator flux magnitude

M s and the q component of the stator current i sq .The speed is adjusted by the PI regulator (a), that

generates the request of torque-producing current, i sq,req .The current reference is tracked in its turn by the PI regulator (d). Due to the action of the saturation block (g),

i sq,ref is limited so that the stator current magnitude cannot

exceed I s,max in region I and II. In this case, the maximum

value for i sq,ref depends on the current i sd used for the

generation of the flux. The greater is i sd , the lower is i sq,max .In region III the PI regulator (e) further decreases i sq,max

until the angle between the stator and rotor flux vectors is 45°, i.e., the maximum torque condition is reached.

　测量空间The stator flux command is generated by the PI

regulator (b) on the basis of the voltage request. If this request is greater than the available voltage, the field weakening algorithm reduces the flux, otherwise the flux is increased, but not beyond its rated value. Finally, the switch (s) can create a temporary voltage margin to enable a fast reaction of

the current controller, in order to improve the transient behavior. If the requested voltage is greater than the available voltage, i.e., the flux is being decreased, the switch (s) is closed and the angle T s of the reference frame is modified by adding a small quantity 'T s proportional to the speed error. As a consequence, this small rotation of the reference frame is applied to the stator voltage and has the effect to improve the torque production to the detriment of the flux,

especially in the beginning of the speed transient [13]. Although this last algorithm has the aim of imp

roving the behavior of the motor during the speed transients in the field weakening speed range, actually it is not essential for the field weakening operation. Hence for the sake of simplicity the effects related to the switch (s) have not been considered in this paper.

B.Control Scheme (B)

The block diagram of the control scheme (B) is shown in Fig. 2. The control scheme is implemented in a reference frame synchronous with the rotor flux vector. The motor currents, which are the main control variables, are adjusted by the PI regulators (c) and (d). The d component of the stator current is used to regulate the rotor flux, whereas the q component is used to vary the motor torque.

To adjust the field level, the scheme uses the same method proposed in Scheme (A), namely the reference value for i sd is set by the PI regulator (b) on the basis of the voltage request. If the voltage request is greater than the available voltage, the flux level is reduced, otherwise it is increased to the rated value.

The speed is controlled by the regulator (a), that generates the reference value for i sq . The limitation block (g) ensures that the constraint on the stator current is satisfied in region I and II, and also the exploitation of the maximum torque capability in region III. In fact the output signal i sq,max

of the limitation block (g) is equal

to 2

2sd s,max

i I in region I and II, whereas in region III it is decreased until the absolute value of the v sd is equal to 2

s,max

V . As explained in Section II, this condition means that, under the assumption that the maximum voltage is applied to the motor, the phase angle of the voltage vector in the rotor-flux oriented reference frame is 90°r 45°.

C.Control Scheme (C)

The block diagram of the control scheme (C) is shown in Fig. 3. In this rotor-flux-oriented control scheme the

main control variables are the stator flux components instead of the stator current components.

To understand the control principle, it is useful to recall the main motor equations written in terms of stator flux components in a reference frame synchronous with the rotor flux vector [18]:

sd s

r r

r r L M dt d R L M M M V (3)

sq r r

s L L M

p T M M V

23 (4) As can be seen, (3)-(4) are quite similar to the corresponding equations of the traditional field oriented control based on d-q stator current components. In fact the rotor flux depends only on M sd , whereas the motor torque is proportional to M sq .

According to (4), the torque demand is transformed by the speed regulator (a) in the request of q-component of the stator flux.

The limitation block (b) ensures the respect of the current limit in region II and the maximum torque capability in region III. Unlike the control schemes (A) and (B), these conditions are obtained without using additional regulators, but only with algebraic relationships. To satisfy the condition D =45°,M sq has to be equal to M sd , whereas the overcoming of the maximum current is prevented by guaranteeing that the absolute value of M sq,ref is lower than the quantity M sq,available .

The stator flux regulator behaves as a proportional controller, with some additional terms compensating the stator back-EMF and the voltage drop caused by the stator resistance. The equations of the stator flux regulator can be expressed as follows:

ref sd sq r sd s req sd i R v W M M

M Z ,, (5) q

ref sq sd r sq s req sq i R v W M M

M Z ,,

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(6)

where 1/Ĳd and 1/W q are the gains of the controller, and Z r

worth noting that it is possible to select W d equal to W q , but it could be convenient to adopt two different time constants to the advantage of flexibility in the tuning of the regulators.旅游电商平台

The rotor flux is controlled by adjusting the d-component of the stator flux. However, the basic principle that regulates the flux weakening request is quite different from that of schemes (A) and (B). It is widely known that, if the motor operates at constant speed, fast torque responses can be ac

hieved only if the control scheme keeps the flux level constant during the torque transients. In particular, the flux level should be always set to the value required to generate the maximum achievable torque at any operating speed. In this way any demand of torque variations within the admissible values is achieved without changing ĳsd but only ĳsq .

For a given value of the d-component of the stator flux, and consequently of the rotor flux, the maximum torque is achieved when ĳsq,ref = ±ĳsq,max .

Taking this equation into account, the voltage required to generate the maximum torque can be determined from (5) and (6) as follows:

ref sd max sq sq max r sd s req max sd i R v W M M M M Z ,,,,sign (7) q

ref sq sd max r max sq s req max sq i R v W M M M Z ,,,, (8) where i sq,max is defined as follows s max

sq max sq L i V M

,, (9) and Ȧr,max is the corresponding angular frequency of the rotor flux, expressed by

sd sq max sq r r sq

r max r L R M M M V M Z Z |

|sign ,, . (10) It is worth noting that in practical applications it is possible to approximate Ȧr,max with Ȧ and therefore the knowledge of the rotor parameters is not necessary.

In scheme (C) the flux request M sd,req is reduced only if

speed requires a voltage greater than V s,max . In other words, the flux level is always set to the value required to generate the maximum achievable torque at any speed. IV.F LUX O BSERVERS

The aim of the flux observer is the determination of stator flux and phase angle of the rotor flux, which are necessary for the field oriented control of the induction machine. The flux observer operates in the stator reference frame for all the control schemes. Hereafter the superscript “s” will be used to identify quantities expressed in the stator reference frame.

The rotor flux can be estimated as follows

热交换设备

s s s s

s s r i L M

L V M

M . (11)

The stator flux could be determined integrating the stator voltage:

s s s s s

s i R v dt

d M . (12) It is evident from (12) that th

e estimation o

f the stator flux vector can be affected by stator resistance mismatch, sensor offsets and the inverter non-linearity (inverter dead-times, voltage drop on the conductin

g switches, etc.). However, at hig

h speed, and hence in the field weakening region, the estimation error is lower than that at low speed, because the input voltage becomes the most prominent

term in the right-hand side of (12).

In order to obtain good performance at low speed, it is

preferable to adopt a closed-loop flux estimator, that could reduce the effect of parameter mismatch and sensor offsets.

A closed-loop estimator is based on the principle that feeding back the difference between the measured output

of the observed system and the estimated output, and

continuously correcting the model by the error signal, the error should be minimized. In the case of a flux estimator,

the motor flux cannot be directly measured, but the idea of realizing a closed-loop system is still applicable if the

difference between a signal representing the steady-state value of the reference flux and the signal of the estimated

flux vector is used as feedback signal.

Let us denote with T s and T r the phase angle of s

s M and s

r M in the stator reference frame. Hence, (12) is replaced for scheme (A) by the following equation:

s s j ref s s s s s s s s

s s e G i R v dt

d M M M T , (13)

and for schemes (B) and (C) by the following equation:

s r s ref r r s s s s s s s G i R v dt

d M M M ,. (14)

The quantities G s and G r in (13) and (14) are the gains of the flux observers. The reference flux vector can be

calculated for scheme (B) as

r j ref sd s

ref r e Mi T M ,, (15) and for scheme (C) as

r j ref sd s

s ref r e L M

T M M ,,. (16) The performance of these observers at low speed is not equal. Furthermore they require the knowledge of different motor parameters. Therefore it could seem unfair the use of different observers for the comparison of three different control schemes.

These solutions have been adopted mainly for the sake of simplicity. Nevertheless, it is opportune to recall

that the purpose of this paper is to compare the field-weakening algorithms of the three control schemes and it is sufficient that the observers present approximately the same behavior before entering the field weakening speed range. In fact, at high speed, the integration of the voltage back-emf provides an estimation of the stator flux vector that is sufficiently reliable for all the three observers, and

the effect of the feedback signal is less important.

V.T UNING OF THE C ONTROL S CHEMES The three schemes presents a different complexity in terms of tuning of the regulators.

In total, scheme (A) requires 5 PI regulators (two PI regulators are used for the flux and current control, one for the speed control and the other two for the robust field-weakening algorithm), and if a fast torque response is needed, it is necessary to tune also the two constant gains shown in block (m).

Scheme (B) requires 5 PI regulators (two PI regulators

are used for the current control, one for the speed control and the other two for the robust field-weakening

algorithm). Finally, scheme (C) requires two PI regulators (the first one for the speed control and the second one for the robust field-weakening control), and two gain constants for the flux regulator (7) and (8).

The criteria used for the tuning during the experimental test are beyond the scope of this paper. For the regulators of the inner loops, i.e., regulators (d) and (e) in scheme (A), regulators (c) and (d) in scheme (B) and the stator flux regulators in scheme (C), some

simple design rules can be derived, generally based on zero-pole cancellations. The tuning of the other regulators, instead, is more

difficult, because the drive dynamics depends on the motor inertia and on the field-weakening algorithm itself. So the tuning of these regulators has been initially faced by means of numerical simulations, and then it has been

refined during the experimental tests by using a trial-and-error procedure.

VI.E XPERIMENTAL R ESULTS A complete drive system has been realized to verify the performances of the control schemes. The experimental

set-up consists of an IGBT inverter and a 4 kW, 4-pole

squirrel cage induction motor. The control algorithms, written in C code, are implemented on a DSP

TMS320C28. The sampling period (coinciding with the

switching period) is 100 P s.The parameters of the electric drive are shown in Table

I and the rated speed is about 700 rpm.

It is important to note that the performance of each control scheme depends on many factors that are not directly related to the field-weakening control scheme, such as the use of fixed-point or floating-point math, the compensation for the inverter dead-time, or just the skill

of the programmer. Therefore, the results stated in this Section should be considered with care, as a particular case, which depends on the adopted experimental set-up. A.Comparison of the Steady-State Behavior

From the analysis of the experimental tests, it is possible to note that the three control schemes have practically the same performance in terms of speed粉底原料

response and field-weakening speed range. Each of them have reached a maximum speed that is about seven times the base speed (the maximum speed is practically imposed by the friction torque of the inverter bench).

However, each control scheme has shown its own advantages and disadvantages, that are presented hereafter.

Figs. 4, 5 and 6 show the behavior of the three control schemes during a speed step command up to 700% of the

base speed. Each figure shows the speed response (at the top) and the corresponding phase current waveform (at the bottom). The two intermediate traces of each figure show

the waveforms of the main control variables of each control scheme, i.e., the stator flux and the current i sq for

scheme (A), the stator current components for scheme (B)

and the stator flux components for scheme (C). In Figs. 4, 5, 6 the extension of region II and region III is also represented.

The main comments that can be done are as follows: i) The speed responses of schemes (B) and (C) are quite similar, whereas the one of scheme (A) shows some small oscillations in region III. ii) The best quality of the motor current is obtained by T ABLE I –M OTOR PARAMETERS

P rated = 4 kW R s = 0.45 : I s,rated

=16A rms R r = 0.44 :V s,rated = 110 V rms L s = 56 mH Z s = 2S 50rad/s L r 56mH J = 0.03 Kg m 2

M = 53 mH J tot =0.22 Kg m 2

p = 2

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