Application of One-Cycle Control to Stator Field-Oriented Control

Alberto S. Lock

1

, Edison R. C. da Silva

1

, Malik E. Elbuluk

2

, Cursino B. Jacobina

1 1

Departamento de Engenharia Elétrica/Universidade Federal de Campina Grande C.P. 10.105, Campina Grande, PB, 58109-970 Brasil. Tel.: +55(83) 3310-1407, Fax: +55(83) 33101418 e-mail: alberto.lock@ee.ufcg.edu.br, [edison. jacobina]@dee.ufcg.edu.br

2

University of Akron, Ohio, USA e-mail:melbuluk@uakron.edu

Abstract.-

Usually, Rotor Field Oriented Control (RFOC) controls the speed of the motor, while the Stator Field Oriented Control (RFOC) and the Direct Torque Control (DTC) control the torque. This paper presents a new approach for SFOC drive, using One Cycle Control (OCC). In this new approach the motor speed is controlled by SFOC, which is simplified and has its dynamic response improved. Theoretical foundation for both SFOC and OCC approaches is provided. Simulation and DSP-based experimental results confirm the feasibility of the proposed technique.

I.

INTRODUCTION

DC motors were largely used in the past although they had the drawbacks of being bulky, heavy, spenders, needing a periodical mechanical maintenance. However, their control simplicity was one of a major advantages i.e. separately dc motor control owns a separately armature and a field current controllers. FOC [1-3][5-7] and DTC [4] were invented to emulate dc motor control. As there is no a field coil in ac motor, an artificial one was invented for control, utilizing a

dq

decomposing of the spatial vector, conformed by its three phase components. Herein, these are named vector controllers. In general, FOC controllers can be split into two large groups: Rotor FOC (RFOC) [1-3][8], and Stator FOC (SFOC) [5-7], according with which magnetic field is oriented to. RFOC greatly depends on rotor constant and leakage factor and therefore on temperature [82]. SFOC depends also on these factors, but its temperature dependence is minor [4],[5]. RFOC is implemented using an inner current loop, i.e. by hysteresis control, while SFOC is implemented directly by voltage, i.e. using space vector modulation (SVM) [6] exhibiting some control simplicity. Both control methods are based on slip frequency control, but as SFOC utilizes stator frequency, slip control can also be used to estimate rotor speed. However, in spite of its little advantages, according to [2] there is always a coupling effect between the

dq

control components, but in [9] it is pointed out that the effect disappears on steady state. Above all, in contrast to RFOC, SFOC is not used for speed control, but torque control. One Cycle Control (OCC) is a technique commonly utilized to control power factor in controlled rectifiers [10], [11], active power filters [12], and single phase inverters [13]. Similar to PWM, OCC compares the modulating waves to a triangular carrier to generate drive pulses for (AC/DC or DC/AC) converter controlling. OCC integrates the input control variable until this integrating value matches the output control variable in one control cycle, generating a variable amplitude sawtooth carrier. In this sense, OCC is considered a generalized (fixed frequency) PWM modulator [14]. This paper presents a new approach of flux and rotor speed controllers for SFOC drives, using One Cycle Control (OCC). New approach utilizes SFOC to control speed motor, improving SFOC dynamic response, as well as simplifying even more, SFOC control strategy, avoiding complicate SVM control operations. Theoretical basis is provided for both SFOC and OCC approaches. Simulation and DSP based experimental results show the good performance of proposed technique and confirm the theoretical assumptions.

II. INDUCTION MOTOR MODELING

Equations (1)-(6) constitute the space vector model of an induction motor with single rotor cage and core loss neglected [15].

S xS S S S

jdt d i RV

Ψ+Ψ+=

ω

(1)

Rm x R R R

jdt d i R

Ψ−+Ψ+=

)(0

ω ω

(2)

r mS S S

i Li L

+=Ψ

(3)

S mr R
R

i Li L

+=Ψ

(4)

]*)(Im[.

RS Oe

ii L K m

=

(5)

Lem

mmdt d p J

−=

ω

(6) where the symbol “ “indicates space vector;

R

S

,

R

R

are the stator and rotor resistances;

L

S

,

L

R

,

L

m

are the stator, rotor and magnetizing inductances;

ω

x

is an arbitrary speed,

ω

Ψ

S

is the magnetic flux speed;

ω

m

is the mechanical speed;

p

is the pole pair number,

J

is the shaft inertia;

m

e

and

m

L

are the electrical and load torque, respectively. In addition

+=+=

O R ROS S

L L L L

)1()1(

σ σ

(7)

)1)(1(
11

RS

σ σ σ

++−=

(8)

RS S R R

i L

=+Ψ−Ψ

])1([1

σ σ

(9)

S R RS S

i L

=+Ψ−Ψ

])1([1

σ σ

(10) where

σ

,

σ

S

, and

σ

R

are general, stator and rotor leakage factors. Space vector

D

is defined as

978-1-4577-0541-0/11/$26.00 ©2011 IEEE 3595

)3/2exp(
][3/2

2

π

ja Da Da D D

cba

=++=

(11) where

D

represents stator voltage (

V

S

), stator current (

i

S

), rotor current (

i

R

), stator (

Ψ

S

) or rotor flux (

Ψ

R

). The quantities

D

k

(

k

=

a

,

b

,

c

) represents

k

component of

D

in relation to a neutral point.

III. VECTOR CONTROL SYSTEM

Analysis of the conventional SFOC principles is given, followed with the present proposal. Note that RFOC [2] as well as SFOC theory [4, 5] are based on (2).

A. Conventional SFOC approach

As in the RFOC, the projection of magnetic flux

Ψ

R

on

d

current axis gives a means to visualize electrical variable relations [2], then

mRO R

i L

=Ψ

(12)

i

mR

is rotor magnetizing current. Similarly, for SFOC, assume that the stator flux is proportional to the magnetizing current

i

mS

, see Fig.1, that is,

Fig.1. References frames and space vectors for SFOC control

mS OS

i L

=Ψ

(13) Substituting former equation into stator equation (1) and taking a gyrating synchronous

dq

axis,

ω

x

=

ω

Ψ

S

, it is obtained

Sd S Sd mS O

i RV dt di L

−=

(14)

SqS SqO

i RV dt d L

−=

µ

(15) where

µ=

ω

b

dt

,

µ

is angle corresponding to slip frequency

ω

b

, since

mS b

ω ω ω

−=

Ψ

(16) On the other hand, from (12), (13) and the rotor equation (2)

S S mS R

iii

)1(

σ

+−=

(17)

Substituting former equation into (5) and simplifying

SqmS e

ii K m

..

=

(18)

Equation above emulates the torque of a dc motor, as a product of field (

i

mS

) and armature current (

i

Sq

). This confirms the goal of FOC controllers. On the other hand,

from (10)

,

)1]([

RS S S R

i L

σ σ

+−Ψ=Ψ

(19) Substituting (17) and equation above into (2)

])[1(
][)1(])1([0

S S S Rb
S S S RS S mS R

i L ji Ldt d ii R

σ σ ω σ σ σ

−Ψ++
−Ψ+++−=

(20) Combining (7), and equations (12)-(14), (

dt d

/

=∆

) it is obtained, for

d

axis,

Sq RbS mS RSd R

iii

τ σ ω σ τ τ σ

++∆+=∆+

)1()1()1(

(21)

τ

R

=

L

R

/

R

R

is constant time rotor. Analogously, for

q

axis

Sd RbS mS RbSq R

iii

τ σ ω σ τ ω στ

−+=∆+

)1()1(

(22) Equations (21) and (22) depict SFOC control system [5-7]. To decouple the system, it is assumed that current

i

Sd

can be an output flux controller plus a decoupling term

i

Sdq

,

see Fig.2, that is,

SdqS S d

iGi

+Ψ−Ψ=

)ˆ(

**

(23) where

∆+=

i p

K K G

(24) and

K

p

,

K

i

are proportional and integral constants controller,

respectively. Then, substituting (21) into (22)

Fig.2. SFOC control [104]

Sq RbS mS RSdq RS S R

iiiG

τ σ ω σ τ τ σ τ σ

++∆+=∆++Ψ−Ψ∆+

)1()1()1()
ˆ()1(

*

(25)

As the first terms of both sides of former equation correspond to stator flux, the second terms must be equal, which then gives:

)1(

∆+=

RSq RbSdq

T ii

σ τ σ ω

(26) slip frequency

ω

b

is found from (22):

−+∆+=

Sd S mS RSq Rb

iii

σ σ τ στ ω

)1()1(

(27) Thus, decoupling factors are given by (26) and (27). Above all, decoupling effect seems a little more serious when consider steady state (

∆

=0). For instance, from (21)

Sq RbSd
S mS

iii

τ σ ω σ

−=+

)1(

(28) Substituting (28) into (22) and simplifying gives:

Rb RbSqSd

ii

τ ω σ τ ω σ

)1()(1

2

−+=

(29)

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In accordance, from (21) and (22)

22

)(1)(1)1(

Rb RbS mS Sd

ii

τ σω τ ω σ σ

+++=

(30)

2

)(1)1()1(

Rb RbS mS Sq

ii

τ σω τ ω σ σ

+−+=

(31) Equations (30) and (31) show a coupling effect between

d

and

q

control components [2]. In fact, dividing (30) and (31) leads to expression (29). Notice that (29) depends on the square of the slip frequency and rotor constant. This coupling effect does not disappear in steady state.

B.

Proposed approach

Note that (13) can be also expressed as, see Fig. 1

Sd S

Ψ=Ψ

(32) Substituting former equation into stator equation (1)

Sd Sd S Sd

i Rv

∆Ψ+=

(33)

Sd S SqS Sq

i Rv

Ψ+=

Ψ

ω

(34) Then, from (13), (18) and (26)

SqSd t e

i K m

Ψ=

(35) Replacing equation above into (28)

Sd S Sd et Sq

m K v

Ψ+Ψ=

Ψ

ω

(36) From (21), (22) in steady state (

∆

=0) and from (26)

Sd RbSd S
R RSd Sd

ii L

τ σ ω σ
τ τ

−∆++
Ψ−=∆Ψ

)1(

(37) On the other hand, from (32) and Fig.1

Sd Sd

i K

=Ψ

(38) From (33) and (37)

SqS b RSd Sd RS S Sd

i Li L Rv

σω τ σ τ

−Ψ+∆++=

)(

(39) Former equation can be expressed as a function of a PI flux controller, since from (33)

S Sd Sd Sd

Rvi

/)(

∆Ψ−=

(40) Then, replacing former equation into (39)

SqS bS S I pd

i L K K v

σω

−Ψ−Ψ∆+=

))(/(

*11*

(41)

K

p1

,

K

i1

are proportional and integral constants of PI controller,

Ψ

*S

is stator flux reference. On the other hand, from (6) and (36), and considering constant flux

Sd S Lumt Sq

m K K v

Ψ++∆=

Ψ

ω ω

(42) If load torque

m

L

varies slowly with respect to mechanical speed variations, then

Sd S mt Sq

K v

Ψ+∆≈

Ψ

ω ω

(43) The above equation can be expressed as a PI speed controller. Then

Sd S mm I pq

K K v

Ψ+−∆+=

Ψ

ω ω ω

))(/(

*22*

(44)

K

p2

,

K

i2

are the proportional and integral constants of PI controller, respectively,

ω

*m

is the mechanical speed reference.

IV. CONTROL SYSTEM DESCRIPTION

The full control algorithm of the DTC-OCC proposed in this work is presented in Fig.3. Limiters at output flux

(PI

Ψ

)

and speed

(PI

ω

)

controllers protect motor against overcurrent. The main difference with conventional DTC-SVM (PWM) drives, where flux and speed controllers are independently used to form the three phase modulating waves, is the fact that in proposed system the output speed controller is managing the sinusoidal modulating waves while it is utilized to manage the OCC variable amplitude carrier. Moreover, an angle

φ

depending on the coefficient of speed and flux controllers is added to the stator flux angle to form the sinusoidal modulating waves.

A.

Proposed control

Consider the block diagram shown in Fig.2, utilized for SFOC-SVM (PWM) controller. It can be observed that the voltage references for PWM modulator depend on flux, speed controllers, and also the estimated flux angle. This relation can be written as

−−−+−+−=

*****

)3/2
ˆ()3/2
ˆ()3/2
ˆ()3/2
ˆ(ˆˆ

qd cba

vvSinCosSinCosSinCos Avvv

π θ π θ
π θ π θ
θ θ

(45) where

S S

t

ΨΨ

==

θ ω θ

ˆ

is the estimated flux angle and

A

is an arbitrary sinusoidal amplitude wave. (45) can be written as

)./(

**

d q

vvarctg

=

φ

(46.a)

−+++++=

)3/2
ˆ()3/2
ˆ()ˆ()()(

2*2*
***

π φ θ π φ θ φ θ

CosCosCosvv Avvv

qd cba

(46.b) Stator flux must be kept constant during control operation, even when rotor speed varies roughly due to a speed or torque command change i.e. motor stop or reverse speed. So it is reasonable to assume that flux output controller is much less than speed output controller. This can also be noticeable from second term of expressions (41) and (44), where in general

σω

b

L

S

i

Sq

<<

ω

Ψ

S

Ψ

S

and can be forced choosing

PI constants of flux and speed controllers

K

P1

,

K

I

1

<<

K

P

2

,

K

I

2

.

Then, if

v

d *

<<

v

q*

−++++=

)3/2
ˆ()3/2
ˆ()ˆ(

****

π φ θ π φ θ φ θ

CosCosCosv Avvv

qcba

(47) As it was pointed out above, amplitude

A

is an arbitrary value in conventional PWM modulator. However, OCC can vary its carrier amplitude for hardware proposes [10,11]. Therefore, the amplitude

A

could vary with inverse of output flux controller to improve PWM performance. Thus, if

A

=1/

v

d *

then

3597

Fig.3. Proposed control system for DTC-OCC

−++++=

)3/2
ˆ()3/2
ˆ()ˆ(

*****

π φ θ π φ θ φ θ

CosCosCosvvvvv

d qcba

(48) Since in conventional PWM, gate power switches turn on (off) when modulating waves equal the carrier amplitude. Then, gate signal can be defined as

=>=≤

;0,1,

**

k k k k

q K vq K v

(49) where

k

denotes each one of phase voltages,

k

=

a

,

b

,

c

;

K

is triangular carrier amplitude. Then substituting (48) into former relation

=>+−
=≤++

0,)3/2
ˆ(1,)3/2
ˆ(

****

k d qk d q

qv K nCosv
qv K nCosv

φ π θ φ π θ

(50) Since

======

ck for n
bk for n
ak for n

,2,1,0

(51) As it can be observed from Eq. (51), speed output controller

v

q*

, controls instantaneous voltage amplitude and the flux output controller

v

d *

, controls variable amplitude triangular carrier. Note that (47) can be inferred directly from PWM modulator depicted in Fig.2, while (48) is an external condition to satisfy OCC constrains. Note also from Equation (51) that modulating waves, controlled by

v

q*

, are now bipolar (ac) waves instead of unipolar (dc) modulating waves obtained for OCC controllers [10-14].

V. OCC FUNDAMENTALS

Conventional PWM modulators are implemented by comparing a modulation reference to a constant amplitude and frequency triangular carrier. OCC varies PWM carrier amplitude to realize modulation of the output, since OCC integrates the output signal until the integration value reaches the reference signal [10]. Fig.4 (a) shows a block diagram of one cycle controller. On this way, a fixed frequency, variable duty-cycle pulse is generated comparing a reference

V

ref

to a fixed-frequency variable amplitude sawtooth wave. The comparison process is realized by hardware using an integrator, a few Flip Flops and a comparator. Thus, a resettable integrator is implemented [11] (see the Fig.4(b) for generated waveforms). This modulation technique is referred to, as a general constant-frequency PWM [12]. Hence, a sawtooth or triangular carrier is generated, but always tied to a hardware scheme. As an example, let us analyze briefly a three phase rectifier as shown in Fig.4, where X(t) and Y(t) are input and output variables, respectively, so as it was reported previously reported in [11]

−−−===

)1()1()1(

cObOaOC B A

d V d V d V V V V

(52) Using matrix or vector notation, the former equation can be expressed as

)()()(

t Dt Y t X

=

(53) where

V

A

,

V

B

,

V

C

are input voltages;

d

a

,

d

b

,

d

c

are duty cycles and

V

O

is dc link voltage. Then

O

V t Y

=

)( (54)

−−−=

)1()1()1()(

cba

d d d t D

(55)

S S cba

I RV V V t X

=

=

000

)(

(56) where

S

I

is the input current vector given by

[ ]

T cbaS

I I I I

=

(57) Resettable integrator srcinates a variable amplitude ramp signal, DC output voltage controlled (by

Y or a PI controller (

Y

(

K

p

+

K

i

/

s

)), that when compared to an average reference signal given by input vector (

τ

f

corresponds to cut-off frequency of a low pass filter)

S S f f ref

I R st X sV

)1(1)()1(1

τ τ

+=+=

(58) provokes the duty cycle vector

)(

t D

and so on. As the three phase rectifier should keep its power factor nearly “1”, and as it is suggested by former equation, current vector must be in phase and varies proportionally with the input sinusoidal voltage vector [11]. Note, however, that constant filter

τ

f

must be

large enough to satisfy the former constraint, and

3598

Fig.4. (a) OCC block diagram (b) Generated waveforms

then srcinates a unipolar (modulating) voltage reference

ref

V

[10-24] and some current delay in the control system.

A.

OCC control system approach

Fig.5(a) shows a block diagram for the proposed DTC-OCC control system. It consists of a general integrator, a magnitude-controlled triangular carrier and a comparator. Fig.5(b) shows the generated waveforms for one of the phase voltages: as one of the

ref

V

components reaches the magnitude of the triangular carrier controlled V

+

(by a PI flux controller) a pulse Q is generated. To model the proposed control from an OCC approach, the idea of an artificial filtrated delayed pole voltage is used, that is,

P g S

V sV

)1(1

τ

+=

(59.a)

{ }

S S

V t v

1

)(

−

Λ=

(59.b) where

P

V

is pole voltage (

P

=

A

,

B

,

C

), see Fig.3,

S

V

is the filtered pole voltage vector,

1

−

Λ

is Laplace inverse transform,

)(

t v

S

is equivalent to

S

V

in time domain,

τ

g

corresponds to cut off frequency of the low pass filter. If time constant

τ

g

is chosen

in order to provoke that stator current

S

i

and voltage

S

v

be in phase, then there exists an input resistor

0

e

R

such that

)(

00

t D RV Rvi

dqeOeSdqSdq

==

(60) where

==

qd dqdq

d d t D D

)(

(61) Accordingly,

=

=

qd eOSqSd Sdq

d d RV iii

0

(62)

==

SqqSd d

i K d i K d

21

(63) On the other hand, using Fig.5(a)

))(),(()(

t Dt X Y t Y

dq

=

(64)

==Ψ=

))(()()(

t X f V
t X t Y

ref mS

ω

(65) As from Eq. (16), assuming a small slip frequency

ω

b

0, then

ω

m

ω

Ψ

S

and from Eq. (34), if for control purposes, it is assumed

v

Sq

=k i

Sq

, where

Sd mSq

k i

Ψ≈

ω

(66) Substituting (32) into (66) gives

mSqS

i K

ω

2

=Ψ

(67)

mqS

d K

ω

2

=Ψ

(68) in vector or matrix notation

)(.)(.)(

2

t X ut D K t Y

S

=

(69) where

[ ]

10

=

S

u

is a unity vector. Note that former equation is equivalent to (27), see (13), with constant

K

2

absorbing dimensional quantities. On the other hand, the voltage reference, see (47) and Fig.5 (a), is given by

[ ]

T cbaref

vvvV

***

=

(70)

ref

V

could be interpreted as an instantaneous vector depending on mechanical speed X(t), see (42). Consequently, to control the power converter, modulating waves (depending on speed controller

v

*q

) are compared with triangular carrier (depending on flux controller

v

*d

). This is sort of similar to OCC strategy presented for rectifiers [10-14]. Notice, on the other hand, that the proposed control system is deduced from conventional SVM (PWM) modulator shown in Fig.2. The stator flux must be seen as the output. Otherwise, the control system can be viewed as an oscillating one. This idea is verified in Section VII.

VI. FLUX AND SPEED ESTIMATORS

Stator flux can be estimated from the stator voltages and currents on

αβ

frame according to [16]

−−=

ΨΨ=Ψ

dt i RV
dt i RV

S S S S S S S S S

)ˆ()ˆ(ˆˆˆ

β β α α β α

(71) In order to avoid accumulated offset effect on flux estimator, a lead lag compensators array is used instead of low pass filters array. This could be seen basically as a generalization of [17]. Then

3599