INTRODUCTION
Direct Torque Control (DTC) of induction motor drives offers high performance
in terms of simplicity in control and fast torque response. Principle of the
classical DTC is decoupled control of flux and torque using hysteresis control
of flux and torque error and flux position. In this method, switching lookup
table is included for selection of voltage vectors feeding the induction motor
(Takahshi and Noguchi, 1986). However, during steady
state operation, the DTC produces high level of torque ripple and variable switching
frequency of inverter.
Development of DTC for overcoming the drawbacks of classical DTC is voltage
modulation application replacing lookup table of the voltage vector selection.
The voltage modulation is based on space vector modulation (SVM) with constant
switching frequency. The SVM strategy is based on space vector representation
of the converter AC side voltage and has become very popular because of its
simplicity. Contrary to conventional Pulse Width Modulation (PWM) method, in
the SVM method there is no separate modulators for each phase (Hebetler
et al., 1992).
Alternatively, SVM method is incorporated with direct torque control (socalled
DTCSVM) for induction motor drives to provide a constant inverter switching
frequency. Several methods with DTCSVM are presented in the literature. The
first method is based on deadbeat control derived from the torque and stator
flux errors (Hebetler et al., 1992). It offers
good steady state and dynamic performance. However, this technique has a limitation
since it is computationally intensive. Another method is based on fuzzy logic
and artificial neural network for decoupled stator flux and torque control (Grabowski
et al., 2000). Good steady state and dynamic response are achieved.
Another method uses two PI controllers instead of hysteresis controllers for
generating direct and quadrature components of voltage from stator flux and
torque errors, respectively (Lascu et al., 2000;
Lai and Chan, 2001). This method provides good transient
performance, robustness and reduced torque ripple. However, in any of above
methods, an analytical approach is not presented to direct reducing torque ripple.
Another objective in induction motor control is to achieve maximum efficiency.
Choosing the level of flux in the induction motor remains an open problem for
the perspective of maximizing motor efficiency and many researchers continue
to work on this problem. Numerous operation schemes have been proposed by many
researchers concerning the optimal choice of flux level for a given operating
point. In lowfrequency operation, core loss (hysteresis and eddy current losses)
is rather low compared with copper loss. As the speed goes up, the contribution
of the eddy current loss increases and finally becomes dominant (Lorenz
and Yang, 1992). The technique allowing efficiency improvement can be divided
into two categories. The first category is the lossmodelbased approach (Lorenz
and Yang, 1992), which consists of computing losses by using the machine
model and selecting a flux level that minimizes the losses. The second category
is the search controller (Kirschen et al., 1985),
in which, the flux is decreased until the input power settles down to the lowest
value for a given torque and speed. Important drawbacks of the search controller
are the slow convergence and high torque ripples. The lossmodelbased approach
is fast and does not produce torque ripple. However, the accuracy depends on
the accurate modeling of the motor and the losses. Different loss models for
loss minimization in induction motor can be found in the literature. In modelbased
loss minimization algorithms, the leakage inductance of stator and rotor are
usually neglected to simplify the loss model and minimization algorithm. However,
with these simplified models, the exact loss minimization cannot be achieved,
especially in highspeed operation of the motor, since a large voltage drop
across the leakage inductance is neglected (Lim and Nam,
2004). In this study, a simplified loss model without neglecting the inductance,
which presented by Lim and Nam (2004), is used.
The main contribution of this study is torque ripple minimization along with efficiency optimization of induction motor which is based on the analytical determination of torque ripple in DTCSVM. Torque ripple minimization is achieved by optimum selection of switching pattern in SVM and efficiency optimization is achieved by optimum selection of stator flux level in DTC.
INDUCTION MACHINE MODEL
The dynamic behavior of induction machine is described using the following equations in terms of space vectors in rotor flux oriented frame.
where, R_{s}, R_{r} represent the stator and rotor resistance. L_{s}, L_{r} and M are the self and mutual inductances. ω_{m} is the rotor angular speed expressed in electrical radians and ω_{e} is the speed of rotor flux oriented frame.
The electromagnetic torque is expressed as:
where, p is the pole pair number and (Sbita et al.,
2007).
In Eq. 5 the symbol "." represents scalar vector product.
Eliminating
from Eq. 14, leads to the state variable
form of the induction machine equations with stator and rotor fluxes as state
variables as:
ANALYSIS OF TORQUE VARIATION
Based on the principle of DTC method, at each sampling period Δt the proper voltage vector is selected. For small values of Δt the stator and rotor flux at time t_{k+1} can be calculated as:
Substituting Eq. 7 in Eq. 8 and 9
gives:
From Eq. 5, the electromagnetic torque at (k+1)th sampling instant can be written as:
Substituting Eq. 10 and 11 in Eq.
12 and neglecting terms proportional to the square of Δt, the torque
at t_{k+1} is given:
Where:
From Eq. 1315, torque variation can
be written as:
Where:
Equation 16 clearly shows the effects of the applied voltage
vector and stator and rotor flux on the torque variation.
PROPOSED SWITCHING PATTERN OF SVM FOR TORQUE RIPPLE MITIGATION
As shown in Fig. 1, the switching pattern of inverter control includes eight switching states, made up of six active and two zero switching states. Active vectors divide the plane into six sectors, where the reference voltage vector (V_{S}^{*} with γ angle) is obtained by switching on (for the proper time) two adjacent vectors.
The vector patterns associated with the switching states are expressed by Lai
and Chan (2001):

Fig. 1: 
Switching states in SVM 
where, t_{s} is sampling time and
In this study, for minimizing torque ripple, time t_{0} is divided
into two parts: t_{01} and t_{02}. In proposed strategy for
space vector modulation, switching pattern is shown in Fig. 2.
From Eq. 16, torque slope during t_{A} and t_{B} can be written as:
where, i is A and B and torque slope during t_{01} , t_{02} can be written as:
Figure 3 shows the torque ripple during sampling time t_{s}.
According to Fig. 3, the square of the RMS value of torque ripple during t_{s} can be expressed as:
The optimum value of t_{01} to minimize the torque ripple can be obtained by the partial derivatives of Eq. 23 with respect to t_{01}. By solving Eq. 24, proper time t_{01} can be obtained from Eq. 25.

Fig. 2: 
Proposed switching pattern 

Fig. 3: 
Torque variation during t_{s} 
LOSS MINIMIZATION SOLUTION
In this study, a simplified induction motor model with iron loss is used. Equivalent circuit of this model in the stationary reference frame is shown in Fig. 4. This model in synchronous frame is described by following equations:
where, ω_{sl }= ω_{e}ω_{m }and S is slip of motor and R_{m} is core loss equivalent resistance.
It is assumed that the rotor flux oriented scheme is utilized. The rotor flux oriented scheme is realized by aligning the reference frame on the rotor flux axis in which, Φ_{qr }= 0. In the steady state condition, i_{dr} = 0 and:

Fig. 4: 
Equivalent circuit of induction motor. In stationary reference
frame 
The total motor losses can be expressed as:
Where:
The electromagnetic torque in the rotor flux oriented scheme at steady state is given by:
Where:
From Eq. 34 and 37, the cost function
is defined by:
The optimal solution for Eq. 38 is (Lim
and Nam, 2004):
From Eq. 39, 40, the optimum flux level
is obtained from:
PROPOSED STRATEGY AND SIMULATION RESULTS
Figure 5 shows the block diagram of the proposed strategy
for DTCSVM of induction motor. In this method,the reference torque (T_{ref})
is obtained from speed PI controller.

Fig. 5:  Block
diagram of proposed method 
Table 1: 
Parameters of induction motor 

The optimum components of stator flux in rotor flux reference frame are calculated
from reference speed and T_{ref}. These components are transformed in
stationary reference frame. The reference voltage components are calculated
from:
These components in stationary frame are applied to space vector modulator. The values of t_{A}, t_{B}, t_{01}, t_{02} are calculated based on the reference flux and applied to space vector modulator.
The rating and parameters of the standard induction motor are given in Table 1. The rated power of motor is 2 KW and the rated speed is 3500 rpm.
The validity of the proposed method is shown by comparing the simulation results of proposed DTCSVM and conventional DTCSVM.
As shown in Fig. 6a and b the torque ripple
in proposed method is reduced comparing with conventional method. Total Harmonic
Distortion (THD) of torque in proposed method is 2.89% and in conventional method
is 4.12%. Both methods are simulated in constant sampling frequency (f_{s}
= 10 KHz).

Fig. 6: 
Simulation results in torque ripple: (a) conventional DTCSVM
(b) proposed DTCSVM 
Figure 7a shows the optimal stator flux that is calculated
for step load change from 10 to 5 Nm and Fig. 7b shows the
speed of motor. The estimated stator flux is calculated by:

Fig. 7: 
Simulation results in: (a) Optimal stator flux and (b) speed
of motor 

Fig. 8:  Simulation
results in the input power of motor at: (a) Rated stator flux and (b)
optimal stator flux 
For validation of lossminimization in proposed method, Fig. 8 shows the input power of motor when the rated flux is applied to DTC and when the optimal flux is applied to DTC (for constant load of P_{out} = 800 w).
As shown in Fig. 8, when the optimal flux is applied to DTC,
the input power of motor reduced about 13.6% and efficiency of motor improved
about 15%.
Table 2: 
Efficiency improvement of motor 

Efficiency improvement of the induction motor for different rotor speeds and
torques has been illustrated in Table 2.
This research project was conducted from August 2009 to March 2010 in Isfahan University of Technology, Isfahan, Iran.
CONCLUSION
In this study, a new strategy for direct torque control of induction motor based on space vector modulation is presented. The features of the proposed method include, first, minimizing the torque ripple of motor by optimal selection of switching pattern in SVM and second, minimizing the loss of motor by optimal selection of stator flux level. The proposed method is evaluated using simulation result which shows very good performance of the method.