An Improved Maximum Torque Per Amp
Control Strategy for Induction Machine Drives
C. Kwon, student member, IEEE, and Scott D. Sudhoff, Senior Member, IEEE
School of Electrical and Computer Engineering
Purdue University
West Lafayette, IN, 47907, USA
Abstract- In this paper, an improved maximum torque per
amp control strategy for induction machine drives is proposed.
The proposed control strategy is simple in structure and includes
the effects of magnetizing and leakage saturation. It is
experimentally shown that the proposed control strategy
produces the desired torque with the minimum stator current for
the entire torque range and that this control strategy yields
superior performance over a control strategy based on the
classical qd induction machine model.
strategy is experimentally validated. It is shown that the open
loop MTPA control strategy achieves commanded torque with
good accuracy and that the maximum torque per amp condition
is, in fact, achieved. The control is aIso experimentaIly
compared to the one set forth in 171and shown to yield superior
performance.
1I.NOTATION
I. INTRODUC~ION
With improper control, induction machines operate with
poor efficiency at light or moderate loads, where some
machines are driven for a significant portion of their service life.
To improve induction machine efficiency in those operating
conditions, there have been a variety of techniques proposed
[ 11-[7]. These methods aim to find an optimum slip frequency,
at which the maximum efficiency of the induction motor i s
achieved. These methods possess a variety of disadvantages.
The loss model based approaches [1142) require substantial
look-up tables. On-line search techniques [3]-[5] have the
potential to exhibit hunting. Other approaches include [6],
wherein Pontriagin's maximum principle is used to improve
efficiency and [7], wherein a particularly elegant approach to
minimize the stator current amplitude for a given load torque is
set forth. However, since these methods [6]-[7] are based on
classical qd model whose deficiencies such as failure to
represent leakage saturation, magnetizing saturation, and
distributed system effects in the rotor circuits have long been,
noted in the literature [8]-[13], degradation of their ability to
find an optimum slip frequency is inevitable.
In this paper, a new control strategy is proposed based on an
altemate qd induction machine model (AQDM) [14], which
addresses the deficiencies in classical qd model (CQDM).This
AQDM has been experimentally shown to be a good
mathematical representation of the machine, and one that is
sufficiently accurate for control design [ 141-[16].
The proposed maximum torque per amp (MTPA) control
In this work, electrical rotor position is designated 8, and
electrical rotor speed U,. These quantities are related to the
mechanical rotor position 19, and mechanical rotor speed
w, by a factor of P / 2 where P is the number of poles.
This work will make use of a transformation of variables to
both the rotor and synchronous reference frames. The
transformation of stator or inverter quantities may be expressed
where ' y ' may be ' s for stator, ' i ' for inverter, ' r ' for rotor,
or ' m ' for magnetizing, ' f ' may be a ' v ' for voltage, ' i ' for
This work was supported in part by ONR Grant N00014-02-1-0990
"
Polytopic Modeling Based Stability Analysis and Genetic Optimization of
Elecmc Warship Power Systems " and by ONR Grant N00014-02-1-0623"
National Naval Responsibility for Naval Engineering Education and Research
for the Electric Naval Engineer "
0-7803-8975-1/05r$20.0002W5 IEEE.
current, or ' h ' for flux linkage, and where
K:(8)
cOs(I9-2x13)
sin(@) s i n ( 8 - 2 x / 3 )
:[
=-
1
c o s ( ~ + 2 ~ ~ q
s i n ( 8 + 2 a / 3 ) (2)
112
112
112
The superscript ' x ' will denote frame of reference. In a break
with the usual notation, omission of a superscript will designate
the rotor reference kame in which 0 = 8,. This is done for
notational simplicity since most quantities used herein are
expressed in the rotor reference frame. Setting ' x ' to ' e will
denote the synchronous reference frame.
A similar
transformation is used for rotor variables with the exception
that B is replaced by B - 0, in (1H2).
A related transformations used herein is that the q- and daxis inverter currents may be expressed as
where
cos(B-xi6)
K;,;(8)= - sin(B-x/6)
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sin(@)
-cos(Q)
This variation arises from the assumption that the sum of the
inverter currents is zero. Additional discussion of reference
frame theory and notation is set forth in [SI.
111. REVIEW OF INDUCTION MACHINE DRIVE
The test system for this work utilizes a 4-pole, 460 V, 50 Hp,
60 Hz, delta-connected induction machine whose
specifications are listed at Table I. The associated inverter is
illustrated in Fig. 1.
Therein,, for a given commanded torque T," , an MTPA
control strategy determines a corresponding inverter current
command in the synchronous reference frame as well as a slip
[
frequency command, which are denoted i$ = i$
i$T and
SuatCgy
U,' ,,respectively. Later in this work, an MTPA control strategy
to determine these quantities will be set forth. Other variables
which appear in Fig. 1 but which are not defined by the figure
include the measured a- and 6- phase inverter currents,
and
w;
Fig. 1. The current controlled inverter-fed drive
ti
-ibi , :the
electrical frequency we (which is integrated to
determine the position of the synchronous reference frame Se),
the vector of inverter phase current commands
izbci
iii
and finally the vector of switch
=[&
ci],
commands sa,, =[s:
s; s:] . Setting :s high indicates the
upper transistor of x- phase should be turned on (and the lower
transistor off), where ' x ' may be ' U ' b ', and ' c '.
The SCR block designates a synchronous current regulator,
whose implementation is depicted in Fig. 2 [I 71. Therein, it can
:be seen that the
Gi are transformed to the synchronous
n- e
I
Fig. 2. Schematic diagram of SCR
I,
ti,
by KF'I ,which denotes the left two columns of K g ' .
lic
A delta modulator is used to generate the switching
commands,
, for switching devices of the inverter,
...,
T6
.
In
this modulator, every T,, seconds, the phase
T,,
reference frame by K:,i and used to generate the error between
the actual and commanded q- and d- axis currents. This error is
multiplied by the integral gain (1 / r,,, ) , integrated, limited to
sibc
current error between i:i from SCR and
rt i k I , and added back to isi.The modified q- and d- axis
.*
eXi= lxi where ' x ' m a y be 'a', ' b ', and ' c '
current commands are transferred back to abc variables using
TABLE I
error in each phase in
determined by
SPEClFtCATlON OF BALDOR ZDM41 IST-AM1 INDUCTION MACHME
HorsepowerKilowatt
Voltage
Phase
Full Load Amps
WM.
Frame Size
Ratin
Full Load Efficienc
Power Factor
Enclosure
tiare calculated as
7
ixi
(5)
.
Based on the sign of
(9, switching
command :s
is
50137.3
2301460
The switching of the three phases is evenly staggered in time.
For the study here, T,, is 5et to 10 kHz.
3
114/57
1775
326TC
IV. MTPA CONTROL STRATEGY BASED ALTERNATE QD
40C AMB-CONT
INDUCTION MACHINE MODEL
87.0
In order to design an MTPA control strategy, an accurate
mathematical induction machine model is essential. Therefore,
prior to setting forth the design of an MTPA control strategy,
TEBC
741
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the AQDM is briefly reviewed.
The steady-state equivalent circuit corresponding to the
AQDM is depicted in Fig. 3 [15]. Therein, W, is defined as
me - w, , A,,,is equal to
fi.l&m
I, and slip S is defined as
0 --o
s=e--L
-
me
Fig. 3. Steady-state equivalent circuit of AQDM
The functional forms of machine parameters in this
equivalent circuit are taken to be
= I,, (const)
(7)
2
' m ( ~ ) = m l -
.am + , ~ ( L - m 4 ) + e m d k - m s l
-,timi;)
Te ="(ljmi:s
2 2
(1 1)
It is convenient to express (1 1) in terms of slip frequency and
rms magnitude of the applied stator current. The relationship
between the q- axis stator current in phasor representation and
the q- and d- axis currents in the synchronous reference frame is
(9)
y r ( $ ) = . 2 L + A L + - Ya3
(10)
YrlS + 1 Y r 2 J + 1 Yr3S + I
For this work, the AQDM parameters of the test induction
machine were characterized using the methods set forth in [ 161.
The resultant machine parameters of AQDM for this machine
are listed in Table 11.
The stage is now set to set forth a control strategy based on
the AQDM model and its parameters listed in Table 11. The
objective of this control strategy is to produce a desired torque
with the minimum current, which is favorable in terms of
inverter losses, and nearly optimal in terms of efficiency [8].
The structure of this control strategy is such that
root-mean-square magnitude of the stator current I, and slip
frequency w, are expressed as functions of the commanded
Without loss of generality, selecting the phase reference such
that all the current is in the q- axis, (12) reduces to
&I,
(13)
= iGS
where 1, is the magnitude of
kS which
has only a real
component. Likewise in (12), the relationship of magnetizing
flux linkages may be expressed
(14)
fi,iqm
=A J ~ j.a:m
After algebraic manipulation of (1 tH14), the
electromagnetic torque may be rewritten in terms of stator
current and magnetizing flux linkage phasors as
torque, T=*.
The derivation begins by noting that the electromagnetic
torque in the synchronous reference frame may be written as
-
where the overbar '
indicates complex conjugate. From the
AQDM steady-state equivalent circuit in Fig. 3,
is
&,
expressed as
TABLE 11
RESULTANTPARAMETERS
where
ml
1.12e-1
8SleO
m2
3.53e-1
m3
Yc2
8.76eO
7.89e0
3.18e0
1S2e0
1.07eO
7.71e0
6.07e-2
2.26e-5
Yo3
5.61e0
fr4
r m (.I
m+
m5
m6
Y.1
Yd
K
Yal
I
i
The subscript ' u g ' in Zag indicates the impedance looking
into the air-gap of the induction machine.
The electromagnetic torque now can be expressed in terms of
onlyw, and I,. Substitution of (16) into (15) yields
It should be observed that the we from the definition of Zag in
(1 8) cancels the me in the denominator at (1 8),
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SO
(1 8) is not
actuafly a function of .q. It remains to show how A,, (q,
I, )
is computed in (1 8). In (1X), Am , which is equal to
fi.I&,,,/,
can be catculated by solving the nonlinear aIgebraic equation
I
we
I f i ~ l ~ ~)I
.Am =
119)
.zag (Am 9 ~ s
by the Newton-Raphson method for a given usand 1,.
The procedure to derive an MTPA control strategy using
(18) will proceed as follows. First, a set of stator current
commands for I, will be assumed (from nearly 0 A to
somewhat over rated current). The j-th point will be denoted
The optimum slip frequency for this current is then
identified by numerically maximizing (IS) with I , = 13,j. The
20
40
60
80 100 120 140
TMque m m a n d (Nm)
160
Is0 200 220
and the
resulting value o f slip frequency will be denoted
corresponding value of torque T&.
‘0
[a) Stator current command I: versus torque command T,‘
These data points are
illustrated in Fig. 4.
Next, these data points are used to construct a stator current
and slip frequency control law. As for a stator current control
Te,j)are used to formulate a
law, these data points {
.
.
.
20
40
60
stator current control law of the form
I: = oI .T‘,
where
ulr a 2 , a,,
4,
+
(20)
and b2 are selected by maximizing the
T ~ +a3
* .~T ~~* ~ *
objective fitness function fmpA defined as
’0
80
t20
100
140
160
180
200 220
Torque command (Nm)
where
E
is a small number (IO”)
and added to the
(b) Slip frequency command
denominator in order to prevent singularities in the unlikely
event of a perfect fit, NJ is the size of a set of stator current
commands, and
{ q j ,Te,j) are
V. VALIDATION
used to
The performance of the AQDM based MTPA control
strategy derived in the previous section is now investigated.
For comparison purposes, the performance of a CQDM based
MTPA control strategy is also presented. In [7], a derivation of
the MTPA control strategy based on CQDM is set forth.
However, while [7] demonstrated the ability to achieve a
desired torque, it was not demonstrated to be optimal. In fact,
the performance of the MTPA control strategy set forth in [7] is
sub-optimal.
To show this, consider the 50 Hp machine characterized
herein. The parameters of the CQDM were selected to
maximize the function, fceoM
formulate a slip frequency control law of the form
0,’ = CO + cI T,’ + cz
+ c3 .Te*3+ c4 . Te*4
(22)
where c,, C ] , c,, c3, and c4 are also chosen by maximizing
cb2
(21) with I, replaced with
U,.
m:,j
is given by (22) with
T‘, = T,,j,Together, (20) and (22) form the MTPA control
strategy. The resultant MTPA control law based on AQDM
whose parameters are listed in Table 11, by the aforementioned
procedure, is given by
+ 18.5, ~ e * 0 ~ 0 7 9 p (23)
I; (c*)
= 0.109.T,’ 17.0, T:,
~
w:
(c’) 1.4-2.4 .lo”<’
=
+ 176.1O”T;*
versus torque command T,‘
Fig. 4. The MTPA control law based on AQDM
is given by (20) with Te*= T e , j .
Similarly, the data points
03
+ I .74. 10-9q*4
- 933.10-9Te*3
(24)
The final analytical forms of this resulting MTPA control law
based on the AQDM are also depicted in Fig. 4.
743
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where
E
is a small number ( lo6 in (25)) and Ncq is total
zl,
is the measured
number of measured test points,
per-phase fundamental frequency impedance at the l-th test
point, and
i;, is given by
2iSf= r- + jueL,s +
me
(26)
L+ w,I
jLm
r- + j O L r
T h e experimentally measured input data set ( w e , q }and
were obtained for the following conditions: the tested
induction machine was driven with the rated line-to-line
voltage, q+,,,,s,of 480 V, rated electrical frequency we of
, ,Z
Commandedtorque (Nm)
377 radk, and the rotor speed was selected to vary linearly from
1782 to 1800 rpm in 10 steps. The optimization of (25) yields
the parameters: LlS = 4.16 mH,
= 4.16 mH, L, = 91.5 mH,
rr = 0.159 Tz . Using these parameters and the control strategy
set forth in [73yields a CQDM based MTPA control law, which
consists of the stator current command,
and slip
(a) Stator current command I:,CQDM versus torque command T,'
frequency command, o~~cqDM.
Since the machine is
delta-connected, the inverter current command in the
synchronous frame shown in Fig. 1 is obtained by scaling the
stator current command by & . Fig. 5 depicts the MTPA
control law developed using the control strategy set forth in [7].
To investigate whether the CQDM based MTPA control
strategy achieves the maximum torque per ampere condition,
the measured torque was recorded as the commanded torque
was varied from 10 to 200 Nm. In addition, two additional sets
of torque measurements were taken at 110 YOand 90
the estimated optimal slip frequency command,
YOtimes
The
Cnmmanded torque (Nm)
~ z , ~ ~ ~ ~ ,
(b) Slip frequency command O:,,-QDM
torque was measured by a torque estimator which was shown to
be highly accurate when the induction machine is rotating at
moderate to high speeds [18], [19]. Fig. 6 illustrates T,/T,*
versus torque command
Ti
Fig. 5 . CQDM based MTPA control strategy in [7]
versus Te* corresponding to T,' for three different slip
frequency commands. For a given
re*,the solid line, dotted
line, and dashed line represents the measured torque when
estimated optimal slip frequency command u&,M, 90% of
..
w~,,-qoM, and 110% of w&M
are used, respectively. While
overall the MTPA control strategy set forth in [7] performs well,
the error in achieving the commanded torque is up to 25 %.
Further, it is apparent that by varying the slip frequency for
some torque commands approximately 3% mare torque could
have been obtained. This is a modest potential improvement,
but can nevertheless be significant given the maturity of the
field.
To show that maximum torque per ampere condition is
achieved by the AQDM based MTPA control strategy, the
electromagnetic torque is also measured at three different slip
..
..
..
..
..
..
..
..
..
..
..
..
0'9
20
40
60
Bo
100
120 140
Commanded torque (Nm)
._
..
..
160
180
x)(1
Fig. 6. Measured torque / Commanded torque versus Commanded torque for
CQDM based MTPA control strategy in 171
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..
..
..
.
.
.
.
.
..
..
., 1.. ..,...;.. .....i , , ......i...,.....i .... ....~.........
I..
. . .......i-....... .,
..
.
_
_
.
.
.
t
40
60
BO 103 120 140
C0“anded bXqUE (Nm)
if30
180
200
Fig. 7. Measured torque I Commanded torque versw Commanded toque for
AQDM based MTPA control strategy
[lo]
frequency commands, U:, 90% o f U : , and 110% of U : ,
respectively, as it was in CQDM based MTPA control strategy
in [7]. Fig. 7 depicts measured torque divided by commanded
torque, T, IT,*,versus commanded torque T,‘ . The solid line,
1111
[I21
dotted line, and dashed line represents T,/ Te* at m: , T, / T,’ at
[ 131
90 % of U:, and T, / T,‘ at 110 YOof a:, respectively.
As can be seen, T, /
c’ is maximized with a slip frequency
[I41
command of U: over the vast majority of the commanded
torque range, indicating that the maximum torque per amp
condition is achieved. It is also observed that T, /T‘* is close to
unity, indicating that the torque produced closely tracks the
commanded torque. In fact, the maximum open loop torque
error i s less than 3% (as opposed to 25% for the CQDM based
MTPA control strategy).
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Vi. CONCLUSION
In this work, an improved maximum torque per amp control
strategy for induction machine drives was proposed. The
proposed control strategy is simple in structure and produces
the desired torque with the minimum stator currents and shown
to yield superior performance when compared to a CQDM
based MTPA control.
Chunki Kwon (S’OI) received the B.S. and M.S. degrees in electrical
engineering from Korea University in 1992 and 1994, respectively. He is in the
pursuit of Ph.D. degree at Purdue University, West Lafayem, IN. His research
interests include control and modeling of electric machines.
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Drives,” IEEE trunraclions on Indusftial Applications, Vol. 20, NO. 5,
September/October 1984, pp. 1244-1251,
S.K.Sul and M.H.Park, “A Novel Technique for Optimal Efficiency
Control of a Current-Source inverter-Fed Induction Motor,” IEEE
trnnraclionr on Power Electronics, Vol. 3, No. 2, April 1988, pp.
192-198.
M.S.,and
Ph.D. degrees in electrical engineering from Purdue University in 1986, 1989,
and 1991, respectively. From 1991-1993, he served as visiting faculty with
Purdue University. From 1993 to 1997, he served as a faculty member at the
University of Missouri - Rolla, and in 1997 he joined the faculty of Purdue
University, where he currently hoIds the rank of Full Professor. His interests
include electric machines, power electronics, finite-inertia power system,
applied conkol, and genetic algorithms. He has published over fifty journal
papers, including four prize papers, in these areas,
Scott D. Sudhoff (SM’Ol) received the B.S. (highest distinction),
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