Adaptive Maximum Torque Per Amp (Adaptive MTPA) control for induction motor drives seeks to achieve a desired torque with the minimum possible stator current regardless of operating points. This is favorable in terms of inverter operation and nearly optimal in terms of motor efficiency. However, the Adaptive MTPA control was validated only from the viewpoint of tracking a desired torque and was not shown that the desired torque is achieved with minimum possible stator current. This work experimentally demonstrates that optimal condition for Adaptive Maximum Torque Per Amp Control Strategy is achieved regardless of rotor resistance variation.
1. Introduction
Highly efficient operation of induction motor has been studied in the past years
[1

6]
. Among many efforts to obtain optimal performance, an Adaptive Maximum Torque Per Amp (Adaptive MTPA) control of induction motor drives was proposed
[7]
. One interesting approach to develop an Adaptive MTPA control is to employ an alternate
qd
induction machine model (AQDM) rather than the classical
qd
model (CQDM), such as in
[8

11]
. The Adaptive MTPA control strategy set forth in
[7]
has demonstrated the ability to achieve the commanded torque with good accuracy, regardless of temperature variation.
However, efforts have been made to develop many control algorithms for tracking a desired torque, particularly in the case of optimal controls
[1

6
,
8]
. Little study has been conducted on the optimality of the MTPA control that the desired torque was achieved with minimum possible stator current
[1

6
,
8]
.
Thus, this work demonstrates its optimality in a way that maximum torque was achieved for a certain stator current, which is also called maximum torque per amp condition. In addition, satisfaction of maximum torque per amp condition is also shown as rotor resistance is varied.
2. Alternate QD Induction Machine (AQDM)
Alternate QD Model (AQDM) included leakage saturation, magnetizing saturation, and distributed system effects in the rotor circuits which CQDM in
[8

13]
failed to represent over all possible operating conditions due to its constant parameters. Stator and rotor leakage inductance, and the absolute inverse magnetizing inductance are expressed as function of magnitude of magnetizing flux linkage,
λ_{m}
, which is equal to
. Stator and rotor leakage inductance, and the absolute inverse magnetizing inductance are denoted as
L_{ls}
(
λ_{m}
) ,
L_{lr}
(
λ_{m}
) , and Γ
_{m}
(
λ_{m}
), respectively. To consider distributed system effects in the rotor circuits, the rotor impedance,
Z_{r}
(
jω_{s}
) in Laplace form.
Z_{r}
(
jω_{s}
) is separated into a real and imaginary part, which are denoted
r_{r}
(
jω_{s}
) and
jω_{s}
L_{lrz}
(
jω_{s}
), respectively.
The steadystate equivalent circuit representing the AQDM in
[14]
is shown in
Fig. 1
. Additional details on the AQDM model and its nomenclature are found in
[14
,
15]
. In this work, the functional forms for AQDM parameters are specified as follows:
Steadystate equivalent circuit of AQDM model with rotor impedance represented as r_{r} ( jω_{s}) + jω_{s}L_{lrz} ( jω_{s})
The parameters in (1)(4) of AQDM were characterized by applying fitting process to the functional forms of (1)(4) on the laboratory experimental data taken for a 4pole, 460 V, 50 Hp, 60 Hz, deltaconnected squirrel cage induction motor. The resultant parameters in (1)(4) of AQDM for the test induction motor are listed in
Table 1
and are illustrated in
Fig. 2
. Detailed procedures of the parameter identification for AQDM is set forth in
[14
,
16]
but is omitted due to limited space.
Resultant parameters
AQDM parameters for the test induction motor
3. Adaptive Maximum Torque per Amp Control Strategy
 3.1 Objective and structure
The structure of Adaptive MTPA control strategy is to express rootmeansquare magnitude of the stator current
I_{s}
and slip frequency
ω_{s}
, as functions of the commanded torque and rotor resistance estimate
as shown in
Fig. 3.
Structure of adaptive maximum torque per amp control strategy
Its objective is to produce a desired torque with minimum stator current regardless of rotor resistance. To this end, two properties should be satisfied. One property is the tracking property and the other optimal condition in MTPA control strategy.
 3.1.1 Tracking property
The tracking property is to achieve a desired torque at steady states for the current of
, the slip frequency of
, and the estimated rotor resistance of
in the control strategy.
Herein,
ε
is a very small number. As can be seen in
Fig. 4
, torques at point B and C does not achieve torque command
, resulting in failure of the tracking property. However, torques at points A and A’ satisfies this property.
Optimal condition of (Adaptive) Maximum Torque Per Amp controlled induction machine drive
 3.1.2 Optimal condition: MTPA condition
The other is optimal condition in MTPA control strategy at which condition the generated torque
T_{ℯ}
(
,
,
) is not only the desired torque but also maximum torque for a given stator current of
. This is also called Maximum Torque Per Amp (MTPA) condition which the control strategy was named after. In other words, the desired torque was achieved with minimum stator current by minimizing conduction loss
In
Fig. 4
, the desired torque was achieved at point A’ with
I
_{s2}
and at point A with
. Since
I
_{s2}
is larger than
, operation with
has higher efficiency than the one with
I
_{s2}
due to higher conduction loss. Thus, MTPA condition is satisfied ONLY at A point.
 3.2 Derivation of torque equation
To derive Adaptive MTPA control strategy, we can start by expressing the general electromagnetic torque Eq. (6) in the synchronous reference frame in terms of rms magnitude of the applied stator current
I_{s}
, slip frequency
ω_{s}
, and rotor resistance estimate
as the generated torque in
Fig. 3.
The relationship between the
q
 axis stator current in the phasor representation, and the
q
 and
d
 axis currents in the synchronous reference frame
[17]
is
By letting all the stator current in the qaxis through appropriate selection of the phase reference, (7) reduces to
where
I_{s}
is the magnitude of
ĩ_{qs}
. Like relations of currents in (7), the magnetizing flux linkage may be expressed as
After the algebraic manipulation of (6)(9), the electromagnetic torque may be rewritten in terms of stator current and magnetizing flux linkage phasor as
where the overbar ‘ − ’ indicates complex conjugate.
From the AQDM steadystate equivalent circuit in
Fig. 1
,
is expressed as
where
Z_{ag}
is the impedance in parallel with two branches,
jω_{ℯ}
/ Γ
_{m}
(
λ_{m}
) and
jω_{ℯ }
L_{lr}
(
λ_{m}
) +
Z_{r}
(
jω_{s}
) /
S
, resulting in
Thus, substitution of (11) into (10) yields the electromagnetic torque in terms of
ω_{s}
,
I_{s}
, and
r_{r}
.
where, since
λ_{m}
=
,
λ_{m}
(
ω_{s}
,
I_{s}
,
r_{r}
) can be computed as
With these two nonlinear simultaneous Eqs. (13) and (14), torque can be found when
ω_{s}
,
I_{s}
, and
r_{r}
are given. In this work, NewtonRaphson method was utilized to calculate
λ_{m}
(
ω_{s}
,
I_{s}
,
r_{r}
) in (14). But any useful nonlinear algebraic equation solver can be used.
 3.3 Derivation of AQDM based MTPA Control
 3.3.1 Setting up optimization problem
Thus, maximum torque can be obtained by finding optimal slip frequency,
for the given stator current,
, and rotor resistance estimate,
, by applying any optimization technique to (15)
Therein,
ω_{s}
can be any value in real numbers but confined from 0
rad/s
to 4
rad/s
by our experience in order to avoid finding a local maximizer. Note that maximization of the torque in (13) for a given stator current means the minimization of the current for a given torque. The processing to solve the optimal slip frequency,
, will be repeated for all combinations of
I_{s}
and
r_{r}
. It is assumed that
I_{s}
ranges from nearly 0 A to a somewhat over rated current and
r_{r}
is selected to vary from 0.01 Ω to 0.21Ω. The
j
−th point of
I_{s}
and k−th point of
r_{r}
will be denoted
and
r_{r,k}
, respectively.
The optimal slip frequency,
, for a given pair of (
,
r_{r,k}
) can be obtained by solving (15) with
I_{s}
and
r_{r}
replaced by
and
r_{r,k}
. The resulting optimum slip frequency and the corresponding maximum value of torque for
and
r_{r,k}
will be denoted
and
. These resulting data points are recorded for future data processing and are illustrated in
Fig. 5.
The control law and data points for Adaptive MTPA control strategy based on AQDM with different r_{r}
 3.3.2 Curve fitting
The data points {
,
,
r_{r,k}
} in
Fig. 5 (a)
are used to construct a stator current control law. Note that
I_{s}
is not a function of
r_{r}
and so the form of stator current control law can be formulated as
where 𝑎
_{1}
, 𝑎
_{2}
, 𝑎
_{3}
, 𝑏
_{1}
, and 𝑏
_{2}
are selected by maximizing the objective fitness function
f
_{AMTPA}
defined as
where
ε
is a small number (10
^{−3}
) added to the denominator in order to prevent singularities in the unlikely event of a perfect fit,
N_{J}
is the number of a set of the stator current command selected, and
is given by (16) with
.
As for the slip frequency
ω_{s}
, it is a function of rotor resistance as well as the electromagnetic torque as shown in
Fig. 5 (b)
. The data points {
,
,
r_{r,k}
} are fit to the functional form
where
d
_{0}
,
d
_{1}
,
n
_{1}
,
n
_{2}
, and
n
_{3}
are parameters to be identified. These parameters are identified by maximizing the objective function defined by (19)
where
N_{K}
is the number for a set of rotor resistances selected, and
is given by (18) with
=
and
. Both (16) and (18) are composed of the AQDM based Adaptive MTPA control law. To obtain coefficients in (16) and (18), any fitting technique could be used. In this work, a genetic algorithm is employed, which was part of the Genetic Optimization System Engineering Tool (GOSET 1.02), a Matlab based toolbox. Details are set forth in
[18]
.
The resulting control laws for
and
for the test machine may be expressed as
and are also depicted, along with the individual data points, in
Fig. 5.
It can be seen that (20) and (21) fit the calculated data points {
,
,
} in a good accuracy.
4. Test Setup and Configuration for Study on Optimal Condition of Adaptive MTPA Control Strategy
The optimal condition of the proposed Adaptive MTPA control strategy was experimentally investigated. To this end, a current controlled inverterfed drive was used to operate the test induction motor. The configuration of the motor drive used in this study is depicted in
Fig. 6
. Therein, the upper part in
Fig. 6
is the power converter topology. The lower part is composed of a speed control block with an antiwindup integrator, synchronous current regulator (SCR), and delta modulator, to determine the switching signal for switching devices, T
_{1}
~T
_{6}
.
The configuration of the NonAdaptive and Adaptive MTPA control based induction machine drive
For this study, the optimal condition of the Adaptive MTPA control strategy was compared with that of the MTPA control strategy whose slip frequency,
, control be obtained by substituting the value of 0.176 Ω for rotor resistance,
r_{r}
, in (21), as in
Fig. 4
. Thus, it results in
in
[7]
. Herein, 0.176 Ω is the value of the rotor resistance at the temperature of 43 ℃ at which the test induction motor was characterized. For convenience purposes, MTPA control strategy is referred to as NonAdaptive in that the commanded current and slip frequency are not adaptive to rotor resistance variation.
Additional details on the configuration for the drive and nomenclature are set forth in
[16]
.
Fig. 7
shows the experimental setup for this work. The induction motor being tested is enclosed in the small chamber and is exactly same as the motor used for the dynamometer.
Experimental setup snapshot
The experiment focuses on a single operating condition. As shown in
Fig. 8
, the test induction machine was driven at a speed of 900 rpm and the torque command was set to 150 Nm in both cases. The torque estimator used in this work was shown to be highly accurate when the induction motor is rotating at moderate to high speeds
[19

20]
.
Experimental flow chart
With the experimental setup mentioned above, due to the difficulty in directly measuring the actual rotor resistance of the test motor, the rotor resistance estimator proposed in
[21]
was utilized and incorporated into the proposed control strategy.
5. Review of Tracking Performance of Adaptive MTPA Control Strategy
To help readers understand the performance of the Adaptive MTPA control strategy, the results demonstrated in
[7]
are rewritten in this section. In
[7]
, the Adaptive MTPA control strategy was validated by comparing its performance to that of the MTPA control strategy with slip frequency control law of (22).
As in
Fig. 9
, the performance for the two controls was recorded as the stator surface temperature varies. Therein, the red solid line with dot marks indicates the resultant torque of the Adaptive MTPA control strategy whose slip frequency command is
in (21) and the dashed line indicates the resultant torque of the NonAdaptive MTPA control strategy whose slip frequency command is
in (22). As can be seen, Adaptive MTPA control strategy produced the resultant torque closer to the torque command of 150 Nm and larger than resultant torque which the NonAdaptive MTPA control strategy
Performances by the Adaptive MTPA control strategy and NonAdaptive MTPA control strategy at the torque command of 150 Nm
6. Study on Optimal Condition of MTPA Control Strategy due to Rotor Temperature Variation
It would be of interest to investigate how MTPA condition is affected by rotor temperature variation.
 6.1 MTPA condition of NonAdaptive MTPA control strategy
Fig. 10
shows that the resultant electromagnetic torques at estimated optimal slip frequency command,
, in (22) as well as two additional sets of resultant torques taken at 0.9 and 1.1 times of
. In the initial part of each study as in
Fig. 10 (a)
, when the temperature is low and where rotor resistance is smaller than the rotor resistance used to design Nonadaptive MTPA control strategy, the largest torque and closest to the commanded torque was not produced at
but at 0.9 times
, resulting in failure of MTPA condition. Likewise, at the last part of the studies, the largest torque was again obtained at 1.1 times
, which also means failure of MTPA condition. However, for some temperature region, maximum torque per ampere condition is in fact achieved at the estimate optimal slip frequency command
. In the middle part of the study as in
Fig. 10 (b)
, when rotor resistance was close to the design value, it can be seen that a slip value of
ω_{s}
=
yields the most torque, thus satisfying MTPA condition.
MTPA conditions along temperature variation with = 150 (Nm) when controlled by NonAdaptive
 6.2 MTPA condition of adaptive MTPA control strategy
The study on optimal condition, which is Maximum Torque Per Amp condition, of the proposed Adaptive MTPA control strategy was made by comparing its MTPA condition to that of the NonAdaptive MTPA control strategy to show optimal control of the Adaptive MTPA control strategy.
Fig. 11
illustrates the MTPA condition of the two controls as the stator surface temperature increased during the duration of the study. The red solid line with dot marks indicates the estimated torque of the Adaptive MTPA control strategy whose slip frequency command is
given by (21). For NonAdaptive MTPA control strategy, three sets of estimates over neighborhoods of
in (22) have been included to investigate the satisfaction of the Maximum Torque Amp Condition. Therein, the dashed line indicates the estimated torque of the nonadaptive MTPA control strategy whose slip frequency command is
given by (22). In the initial part of study, when the temperature is low and where rotor resistance is smaller than the rotor resistance used to design NonAdaptive MTPA control strategy, the torque estimated at 0.9 times of
is largest (of the nonadaptive controls). In the middle part (in time) of the study, when the rotor resistance was close to the design value, it can be seen that a slip value of
ω_{s}
=
in (22) yields the most torque (again, at the NonAdaptive controls). Finally, as the studies proceed in time, eventually the largest torque estimate was obtained using 1.1 times of
. On the other hand, Adaptive MTPA control strategy adjusted the slip frequency command,
, given by (21) such that the largest torque are always achieved at
, thus satisfying MTPA condition.
MTPA conditions along temperature variation when controlled by Adaptive MTPA control strategy with = 150Nm
The observations from
Fig. 11
indicate that the Adaptive MTPA control strategy satisfies MTPA condition regardless of rotor temperature variation as well as achieve the desired torque accurately as validated in
[7]
.
7. Conclusion
It was experimentally shown that the Adaptive MTPA control strategy performs optimally regardless of rotor resistance variation by showing that the torque produced at the optimal slip frequency is the largest and desirable at the same time, indicating that optimal condition (MTPA condition) of Adaptive MTPA control strategy is satisfied.
As can be seen from experimental results in the previous section, the Adaptive MTPA control strategy makes true optimal performance regardless of rotor resistance variation by reflecting rotor resistance variation in the design of optimal slip frequency control law.
Acknowledgements
This research was supported in part by Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Education, Science and Technology (Grant 20130235 ) and was supported in part by the Soonchunhyang University Research Fund
BIO
ChunKi Kwon received B.S. and M.S. degrees in electrical engineering from Korea University in 1992 and 1994, respectively. He is with Soonchunhyang University, Asan, Chungnam, Korea. His research interests include control and modeling of electric machines, and expands this area to medical engineering such as rehabilitation devices.
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