# Fast Torque Control System of PMSM.pdf

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Fast Torque Control System of P…

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**简介：**本文档为《Fast Torque Control System of PMSMpdf》，可适用于电子通讯领域，主题内容包含FastTorqueControlSystemofPMSMbasedonModelPredictiveControlYuyaHozumi,Shinj符等。

Fast Torque Control System of PMSM
based on Model Predictive Control
Yuya Hozumi, Shinji Doki and Shigeru Okuma
Department of Electrical Engineering and Computer Science
Nagoya University, Aichi, Japan 464-8603
Telephone: +81-52-789-2777
Fax: +81-52-789-3140
E-mail: hozumi@nagoya-u.jp, doki@nagoya-u.jp, okuma@nagoya-u.jp
Abstract—This paper describes a fast torque control system
of permanent magnet synchronous motor (PMSM) based on
model predictive control (MPC). This torque controller selects
directly one of the voltage vector of voltage source PWM inverter
considering voltage saturation explicitly. To obtain the fast torque
response at the transient state and the stable current response
at the steady state, the problem with selecting the voltage vector
to output is formulated based on MPC. In this paper, real-time
implementation method using a look-up table, which is designed
beforehand, is discussed and the experimental results are shown.
I. INTRODUCTION
Permanent Magnet Synchronous Motor (PMSM) has been
used in a wide field because of high efficiency and high power
per volume and weight. Recently, PMSM is applied to traction
drive for electric vehicles and railway vehicles [1] [2].
In these applications, a fast torque response at high speed
region is important. However, large back EMF is generated by
permanent magnet at high speed region. And also, controller
makes more higher voltage reference to obtain a fast torque
response. On the other hand, inverter has a voltage limiter
depending on a DC-link voltage. Even if controller generates a
large voltage reference, inverter can’t output the same voltage
as the reference. In a word, voltage saturation occurs. As a
result, current and torque response gets worse [3].
When the voltage saturation occurs, voltage limiter methods
dominate a current response, which also does a torque re-
sponse. In the case of interior permanent magnet synchronous
motor (IPMSM), however, improvement of current response
does not guarantee one of torque response. Therefore, pro-
posed voltage limiters have restrictive performance [4], or need
controller switching for compatibility.
To overcome these problems, we propose a new torque
control system being able to obtain the fast torque response
at the transient state and the stable current response at the
steady state considering the various constraint on the inverter
explicitly. Proposed controller directly selects a switching
mode of the inverter by using Model Predictive Control (MPC)
[5]. It may be similar to Direct Torque Control (DTC). To be
precise, our proposed method can be a superset of DTC, which
means that it is able to control not only torque but also currents
and switching freqency explicitly, with prediction of future
current and torque behavior by using the inverter switching
mode and mathematical model of PMSM.
In this paper, the proposed MPC torque control system and
its real-time implementation methods using a look-up table,
which designed beforehand, are discussed and the experimen-
tal results are shown.
II. TORQUE CONTROL SYSTEM BASED ON MPC
Our proposed torque control system based on MPC is shown
in Fig.1. Current reference generator consists of Maximum
Torque per Ampere (MTPA) Control [6], Flux Weakening
Control [7], and so on.
In this following section, the MPC controller in Fig.1 is
described.
Input
Voltage
MPC
Controller
PMSM
vV
Switching
Mode
Mathematical Model of PMSM
Current
Inverter
Current
Reference
uvw/dq
Rotor Position θ, Speed ω
Position
Sensor
*
dq
i
DC
V
dq
i
DC-link Voltage
uvw
i
Current
Reference
Generator
Torque
Reference
*
T
Fig. 1. Torque contorl system based on MPC
A. Voltage limit and MPC controller
Initially, voltage limit should be discussed. Voltage refer-
ence vector for inverter generated by the upper controller
(e.g. PI current controller) must be limited in the voltage that
inverter is able to output.
In this limitation of the voltage vector, there is a redundancy
of not only the limitation of norm but also voltage phase
control. In a general vector control system, some limitation
methods, such as fixed voltage phase method and fixed d-axis
voltage method, are proposed [4] [8] [9] [10].
However, it is difficult to design a voltage limiter as the
limiter influences control performance designed by the upper
controller.
On the other hand, MPC controller directly generates one
of eight voltage vectors decide by limited DC-link voltage
and combinations of switch of each phase (Fig.2, Fig.3). Then
MPC controller searchs and selects a suitable voltage vector
from among eight voltage vectors based on a reference of
upper controller or control error. As a result, the voltage limiter
is naturally achieved [11].
IEEE 2009 1147 Preprint of IECON 2009 Proceedings
2
DC
V
2
DC
V
phase phase phase
u
w
u
v
v
v
w
v
v
Neutral
point
＋
－
Fig. 2. Voltage source PWM inverter
phase
u
w
v
V
1
V
5
V
4
V
0
,V
7
V
2
V
3
V
6
phase
phase
Fig. 3. Output voltage vector of the inverter
B. Selection of voltage vector based on MPC
MPC controller predicts current behavior and torque be-
havior by using a mathematical model of PMSM. And then,
it selects a voltage vector that the inverter outputs based on
prediction results.
The state equation of PMSM is given as follows:
i
A
i
B
v
e
(1)
where i
and v
are the - axis
vectors which denote the stator currents and the stator voltages.
is the electric rotor speed. is the resistance of a stator
winding,
,
are the -axis and -axis inductances of a
stator winding, and
is the permanent magnet flux.
The discrete-time state equation of (1) (with sampling time
) is given as follows:
i
A
A
i
A B
B
v
A e
e
(2)
The variable
is the discrete-time instant and
points
the present time. MPC controller predicts the current response
and the torque response by using the equation (2).
A concrete process of selection is shown as follows.
First, the finite-time period from
to
(
is finite positive integer.) is defined as the prediction period
as shown in Fig.4. Next, the finite sequence of the inverter
voltage vectors
V
to V
in the prediction period V
is
shown as follows:
V
V
V
V
(3)
V
V
V
Current
(Torque)
Reference
V3 V3 V3 V3 V3 V3 V4
Selected voltage
vector sequence
Voltage vector
sequence for prediction
Prediction period
Sampling time
Predicted behavior
Time
Present
time
(n = 0 )
Behavior
of the past
Fig. 4. Prediction of the current(torque) behavior
The superscript
denotes that V
is -th sequence of all
possible sequences. The coordinate transformation using the
rotor position eventually yields the voltage vector sequence in
- axis
v
v
v
.
Then, assuming that the rotor speed is constant in the
prediction period, which indicates A
, B
and e
in (2) are
constants, the equation (2) and the voltage sequence yield the
sequence of the predicted current behavior in the prediction
period I
as follows:
I
i
i
i
(4)
For the current limiter calculation, if the calculated i
is larger than the maximum current, then v
in that case
will be rejected.
If the predicted current behavior can be obtained like (4), the
torque behavior in the future can be predicted. In this paper,
torque equation is as follows:
(5)
where
is number of pole pairs.
Finally, the evaluation function (for example, the equation
(6)) is calculated from the predicted current sequence i
and torque sequence
and the current reference i
and
torque reference . In addition,
,
and
indicate
the weights of each term.
(6)
IEEE 2009 1148 Preprint of IECON 2009 Proceedings
The prediction and the evaluation mentioned above are
done to all possible voltage sequences. If
-th sequences,
V
and I
, minimize the value of the objective function,
MPC controller selects the switching mode corresponding to
V(0) ( V
to V
) as optimal output. This process is
repeated at every sampling time.
C. Design of weight in the evaluation function
An example of evaluation function which was proposed
in the equation (6) includes three design parameters (
,
,
). These design parameters decide a characteristics
of torque and current control.
For instance, setting weight for , axis current error
,
to zero gives priority to torque control and abandons
current control. Therefore fast torque resoponse at the transient
state is achieved and steady state errors of , axis current get
worse because they are not controlled. It makes copper loss
increase at the steady state greatly. On the other hand, setting
weight for torque error
to zero gives priority to , axis
current control. As a result, torque response depends on ,
axis current reference. It is, however, difficult to calculate cur-
rent reference to attain a fastest torque response analytically,
because it is necessary to consider various nonlinear factors
caused by currents and voltages limitation and overmodulated
operation of inverter [8].
In this paper, to obtain a fast torque response at the transient
state and stable current response at steady state for the salient
pole permanent magnet synchronous motor, the weights are
proposed as follows:
(7)
First, the weight of -axis current
is discussed. The
reference literature describes that the fast torque response is
obtained when a negative -axis current is applied greatly to
use the reluctance torque effectively [4]. In fact, at the torque
transient state, the -axis current should be quite different from
reference current in steady state, such as the MTPA condition,
for fast torque response. And, in the steady state, if the torque
and the -axis current follow in the reference, the torque
equation (5) makes the -axis current follow the reference
automatically. According to the above-mentioned two facts, it
is not necessary to control explicitly -axis current whether at
the transient state or at the steady state. As a result, the weight
of -axis current
is set as zero.
Next, the weight of torque
and the weights of -axis
current
are discussed. Weights should be set as equation
(7) by considering the steady state. where
is a coefficient
that converts the -axis current into the torque. As a result,
the torque and -axis current can be evenly controlled in the
dimension of torque.
In addition, the weight of the torque
is made to increase
when the difference between the predicted torque and the
torque reference is large, to obtain a fast torque response.
Finally, the weights are proposed as equation (8).
(8)
The reference literature deals with another approach of
the MPC [12]. This method uses the approximation and
parameters obtained empirically, therefore, it can’t apply to
other PMSMs because it is designed for a particular PMSM.
On the other hand, our proposed method can apply to all of
salient pole permanent magnet synchronous motors.
III. IMPLEMENTATION OF MPC CONTROLLER
MPC controller selects voltage vector as the result of
predicting current and torque behavior and evaluating by
evaluation function at every sampling time. However, it is hard
to predict current and torque behavior and calculate evaluation
function in such a short time with a current embedded pro-
cessor performance.
In order to solve the problem, the voltage vectors are stored
in the look-up table beforehand, and selected by referring it in
every control period. The look-up table is designed beforehand
by using simulation results at all the assumed operating points.
This table is configured on the multidimensional state space
that consists of rotor speed, rotor position, - axis currents, -
axis current references, and so on. The process of designing
this table is shown as follows [13].
First, operating range of PMSM is decided from the spec-
ifications, for example maximum rotor speed and maximum
current. Next, rotor speed and - axis current of operating
range are quantized from continuous quantities to discrete
quantities. The domain of definition is given by the specifica-
tions mentioned above, and the width of division are decided
by the pilot simulation beforehand. Finally, by predicting the
states (current and torque) and evaluating the function at each
operating point, the inverter output voltage vector is selected
and stored in the multidimensional look-up table. For example,
two demensional subspace of the look-up table is shown in
Fig.5. Where
and
are -axis and -axis current errors.
Then, the voltage vector is selected by referring to the table
with sensed states in every control period. As a result, the
cost of the calculation can be greatly decreased, and real-time
implementation of MPC controller becomes possible.
IEEE 2009 1149 Preprint of IECON 2009 Proceedings
V4 V4 V5 V6 V1 V1 V2 V3
V4 V5 V5 V6 V1 V2 V3 V3
V4 V5 V6 V1 V2 V2 V4 V4
V4 V5 V6 V6 V1 V2 V3 V4
V4 V6 V1 V1 V2 V3 V4 V4
V6 V6 V1 V2 V3 V3 V4 V5
V1 V1 V2 V3 V4 V4 V5 V6
V6 V1 V1 V2 V3 V4 V5 V6
Δiq
Δid
max(Δiq)
max(Δid)
min(Δiq)
min(Δid)
Fig. 5. Two dimensional subspace of a look-up table
(One example of a look-up table)
IV. EXPERIMENTAL VERIFICATION
In this section, we show the performance of proposed
torque control system in experiment. Experiment system is
shown in Fig.6. DSP is TMS320C6713-225 made by the Texas
Instruments, Inc.
A. Experimental condition
This experiment verifies torque response and - axis cur-
rent response under the condition that a demanded operating
point of PMSM transfers from 0[Nm] to 1[Nm]. Rotor speed
is set to be 1000[rpm] constant, because we want to compare
the torque response. The block diagram of the experiment is
shown in Fig.1. The MTPA control is employed in the current
reference generator. To prevent the degauss, the -axis current
limiter is set to -6[A].
The parameters of PMSM and the inverter are shown in
TABLE I. The configuration parameters of proposed controller
are shown in TABLE II. And parameters of look-up table for
the MPC controller are shown in TABLE III.
To evaluate our MPC controller, the performance is com-
pared with one of conventional torque control system which
consists of MTPA and PI current controller with fixed -axis
voltage method as voltage limiter and integration stop method
as antiwindup [14]. The configuration parameters of PI current
controller are shown in TABLE IV.
Load
Position
Sensor
PMSM
DSP Inverter
DC Power
Supply
Rotor Position, Rotor Speed
Gate
Signal
Stator Current
Fig. 6. Block diagram of experimental instrument
TABLE I
PARAMETERS OF PMSM AND INVERTER
Rated power 0.5 [kW]
Rated current 5.0 [A]
Rated speed 2500 [rpm]
Resistance (R) 0.45 []
Inductance (-axis)
4.15 [mH]
Inductance (-axis)
16.74 [mH]
EMF constant
0.104 [V/(rad/s)]
Rotor inertia 1.5 [kgm]
Number of pole pairs (p) 2
DC-link voltage
50 [V]
TABLE II
PARAMETERS OF PROPOSED CONTROLLER
Control period 20 [s]
Prediction time 100 [s]
B. Experimental result
The experimental results of torque response and - axis
current response in proposed torque control system based
on MPC are shown in Fig.7, Fig.8. And the results in the
conventional torque control system with PI current controller
are shown in Fig.9, Fig.10.
First, comparison of Fig.7 and Fig.9 shows that both tran-
sient response and steady state error of torque are improved
by using MPC controller.
0 5 10 15 20 25
-0.2
0
0.2
0.4
0.6
0.8
1
T
T
*
Time [ms]
T
o
r
q
u
e
[
N
m
]
Fig. 7. Experimental Result :
Torque response by proposed MPC controller
0 5 10 15 20 25
-8
-6
-4
-2
0
id
id
*
0 5 10 15 20 25
0
2
4
iq
iq
*
C
u
r
r
e
n
t
(
d
)
[
A
]
C
u
r
r
e
n
t
(
q
)
[
A
]
Time [ms]
Time [ms]
Fig. 8. Experimental Result :
- axis current response by MPC controller
IEEE 2009 1150 Preprint of IECON 2009 Proceedings
TABLE III
PARAMETERS OF LOOK-UP TABLE
state min max number ofdivisions
-6.0[A] 6.0[A] 50
-3.0[A] 3.0[A] 50
0[deg] 360[deg] 18
TABLE IV
PARAMETERS OF PI CURRENT CONTROLLER
Control period 100 [s]
PI-Gain of current controller 2000 [rad/s]
Inverter carrier frequency 5 [kHz]
Next, Fig.8 and Fig.10 show -axis and -axis current re-
sponse of each controller. Proposed MPC controller generates
negative large -axis current at transient of torque. It makes
proposed MPC torque control system possible to increase
reluctance torque to get fast torque response. In addition, it
is understood that the current limiter functions properly from
-axis current response. And steady state current shown in
Fig.8 is kept stable, though -axis current is not controlled
directly in our proposed system.
These results show that proposed MPC torque controller
has faster torque response and more stable steady current
response compossible without any switch of control strategy
than conventional torque controller.
0 5 10 15 20 25
-0.2
0
0.2
0.4
0.6
0.8
1
T
T
*
Time [ms]
T
o
r
q
u
e
[
N
m
]
Fig. 9. Experimental Result :
Torque response by PI current controller
0 5 10 15 20 25
-8
-6
-4
-2
0
id
id
*
0 5 10 15 20 25
0
2
4
iq
iq
*
C
u
r
r
e
n
t
(
d
)
[
A
]
C
u
r
r
e
n
t
(
q
)
[
A
]
Time [ms]
Time [ms]
Fig. 10. Experimental Result :
- axis current response by PI current controller
V. CONCLUSION
This paper proposed the fast torque control system based
on Model Predictive Control. And the evaluation function for
MPC controller, which can realize a fast torque response at
the transient state and a stable current response at the steady
state without any switch of control strategy was discussed.
Moreover, its real-time implementation methods using a look-
up table, which designed beforehand, was shown.
Good performance of proposed MPC torque control system
was shown by experimental results compared with conven-
tional torque control system.
REFERENCES
[1] H.Nakai, H.Ohtani, E.Satoh, Y.Inaguma, ”Development and Testing of
the Torque Control for the Permanent-Magnet Synchronous Motor”,
IEEE Trans. Industrial Electronics, Vol.52, No.3, June, 2005, pp. 800-
806.
[2] G.Pugsley, ”Electric Motor Specifications and Sizing for Hybrid Electric
Vehicles”, Automotive Power Electronics, 2006.
[3] S.Morimoto, M.Sanada, Y.Takeda, ”Wide Speed Operation of Interior
Permanent Magnet Synchronous Motors with High Performance Current
Regulator”, IEEE Trans. Industry Applications, Vol.30, No.4, pp. 920-
926.
[4] S.Lerdudomsak, M.Kadota, S.Doki, S.Okuma, ”Novel Techniques for
Fast Torque Response of IPMSM Based on Spac