Direct torque control of dual star induction motor using a fuzzy-PSO hybrid approach

Thispaperpresentstheparticleswarmoptimization(PSO)algorithminconjuctionwiththefuzzylogicmethod inordertoachieveanoptimizedtuningofaproportionalintegralderivativecontroller(PID)intheDTCcontrolloopsofdualstarinductionmotor(DSIM).Thefuzzycontrollerisinsensitivetoparametricvariations,however, withthePSO-basedoptimizationapproachweobtainajudiciouschoiceofthegainstomakethesystemmorerobust.AccordingtoMatlabsimulation,theresultsdemonstratethatthehybridDTCofDSIMimprovesthe speedloopresponse,ensuresthesystemstability,reducesthesteadystateerrorandenhancestherisingtime.Moreover,withthiscontroller,thedisturbancesdonotaffectthemotorperformances.


Number of pole pairs J
The moment of inertia f r The friction coefficient T em The electromagnetic torque T r The load torque Ω r is the mechanical rotation speed of the rotor R s1 , R s2 denote variables and parameters of stator 1 and 2 respectively V ds1 , V qs1 , V ds2 , V qs2 respectively stator voltages in the d-q axis I ds1 , I qs1 , I ds2 , I qs2 components of the stator currents in the d -q axis I dr , I qr the rotor currents ψ ds1 , ψ qs1 , ψ ds2, ψ qs2 components of the stator flux linkage vectors in d-q axis ψ dr , ψ qr respectively rotor fluxes

Introduction
In high-power applications, the dual star induction motor has largely replaced the induction machines whose roles were predominant in the industry [1][2][3][4][5].The dual star induction motor is constituted of two windings with phases shifted from one another by an angle of 30 electrical degrees.These windings are usually powered by a six-phase inverter fed by variable speed drives.The main advantages of the DSIM are: [6] their higher torque density, reduced harmonic content of the DC link current and the reliability of this machine which allows a functioning with one or several phases of defective motor.However the control of DSIM is considered to be complicated because the difficulty of obtaining the decoupling of the torque and the flux.To overcome these difficulties, high-performance algorithms have been developed [7][8][9][10][11][12][13][14].
To satisfy the performance of an electromechanical system drive, the generally used strategy consists in controlling the speed by a PID controller to cancel the static error and reduce the response time.This speed is often characterized by an overshoot at startup and depends on the parameters of the machine.In order to overcome these complications, several methods have been developed to adjust the PID regulator.
Auto-tuning is one of these methods, which is used in PID controllers [15].The performance of the control loops is improved by automatically adjusting the PID gain parameters of the conventional controllers.
The self-tuning method has been suggested by many researchers [16,17].A selfadjusting mechanism has been set up to adapt the PID regulator in case of any disturbances.
PSO was first used by Eberhart and Kennedy in 1995 [32].This approach is inspired by the social behavior shown by the natural species.In recent years, particle swarm optimization has appeared as a new and popular optimization algorithm due to its simplicity and efficiency.The role of the PSO in this study is to suggest an adequate adjustment of the parameters (kp, Ki, Kd) to satisfy some drive system requirements.
In last years, the FLC has improved results of nonlinear and complex processes [33].The main idea of this approach is that it does not need a precise mathematical model of the electric machine, FLCs are robust and their performance is insensitive to parameter variations.With the increasing evolution of approximation theory, the adaptive control methods have been presented to cope with the nonlinear systems with parametric uncertainty based on fuzzy logic system (FLS) [34], neural networks (NNs) [35], adaptive fuzzy and NN control approaches via backstepping methods [36,37].
There are two disadvantages in the conception of a FLC.The first one is the obtaining of a suitable rule-base for the application, while the second is the selection of scaling factors prior to fuzzification and after defuzzification, in order to overwhelm these drawbacks and expedite the determination of the design parameters and to reduce the time consumption.Several solutions are adapted to remedy these problems.In [38,39] the authors presents an on line method for adapting the scaling factors of the FLC, the authors suggest a solution to design an adaptive fuzzy controller.The objective of the proposed form is to adapt online scaling factors according to a performance measure in order to refine the controller and increase the performance of the drive system.
In this paper, we investigate the performance of PSO for optimizing the gains of the fuzzy-PID speed controller of the DSIM.

Modeling of the dual star induction motor
The DSIM dynamic equations in the reference d-q can be reported as follow [4]: where the fluxes equations are: For studying the dynamic behavior, the following equation of motion was added:

Direct torque control of dual star induction motor
The model of the DSIM has been completed by the expression of the electromagnetic torque T em given below: A schematic representation of the stator and rotor windings of dual star induction motor is given in Figure 1.

Direct torque control (DTC) of the DSIM
The classical DTC, proposed by [40], is based on the following algorithm: -Divide the time domain into periods of reduced duration Ts; -For each clock struck, measure the line currents and phase voltages of the DSIM; -Reconstitute the components of the stator flux vector and estimate the electromagnetic torque, through the estimation of the stator flux vector and the measurement of the line current; -The error between the estimated torque and the reference one is the input of a three level hysteresis comparator when this latter generates at its output the value of þ1 to increase the flux and 0 to reduce it and thus increasing the torque -1 it reduce this flux and 0 to keep it constant in a band; -The error between the estimated stator flux magnitudes is the input of a two levels of the hysteresis comparator, which generates at its output the value þ1 to increase the flux and 0 to reduce it; -Select the state of the switches to determine the operating sequences of the inverter using the switching table.The input quantities are the stator flux sector and the outputs of the two hysteresis comparators.
The block diagram of the DTC of DSIM is shown in Figure 2.

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Moreover, Table 1 presents the sequences corresponding to the position of the stator flux vector to the different sectors.The flux and the torque are controlled by two hysteresis comparators at 2 and 3 levels, respectively, in the case of a two-level voltage inverter Table 5.
The expression of the stator flux is described by: where V sα1;2 and V sβ1;2 are the estimated components of the stator vector voltage.They are expressed from the model of the inverter.

Particle swarm optimization algorithm
PSO uses a population of individuals to discovery the high solution in a search area among the neighboring solutions.The individual is defined by a particle, which displaces stochastically in the guidance of its preceding finest position and the best past location of the swarm.Presume that the size of the swarm is n and the search area is m , next the position of the ith particle is given as xi5(x i1 , x i2 ,. .... . .,x id ).The finest previous positions of the ith particle are considering by [32]: The index of the best particle amongst the group is gbest d .The velocity of particle ith is represented as: The modified speed and position of each particle can be calculated using the current and the distance from pbest i,d to gbest d as expressed in the following equations: where: n is the number of particles in the group, d is the dimension index, t is the pointer of iterations (generations), v t i,m is the velocity of particle at iteration i, w is the inertia weight factor, rand ( ) is the random number between 0 and 1, pbest i is the best previous position of the ith particle, gbest is the best particle among all the particles in the population, C 1 ,C 2 are acceleration constant.Velocity vector for position update is depicted in Figure 3.

Algorithm steps for PSO
The working of PSO algorithm is interpreted in the consecutive steps.
Step 1: We establish the values of PSO algorithm constants as an inertia weight factor W 5 0.8, with acceleration constants C1 5 C2 5 2. The PSO main program has to optimize in this case three parameters, K e , K d and β to the fuzzy controller, and search optimal value of the three-dimensional search space.
Step 2: So we arbitrarily configured a swarm of "50" population in three-dimensional search space using (x i1 , x i2 , x i3 ), and (v i1 , v i2 , v i3 ) as preliminary situation along with velocity.Considered the primary fitness function of any also to the position with minimum fitness function is displayed as best, so the optimal fitness function as pbest1.Step 3: Run the program by means of PSO algorithm through n numbers of iterations, as well, the program finds the final optimal value of the fitness function as "best fitness" with the last overall optimal point as "gbest".The PSO parameters are described in Table 2 in the Appendix.
The flowchart for fuzzy PSO-DTC-DSIM is shown in Figure 4.

Fitness function 4.2.1
The conception of fitness function.To evaluate the static and dynamic conduct of the control system, it is found that IAE offers good system stability with reduced oscillations the IAE criterion is widely adopted [41]:

Direct torque control of dual star induction motor
5. Design of PID-PSO controller type FLC for the DSIM The optimization of the FLC gains using PSO can be given by the input variable {e}, and the error change {e c } as follows: e c ðtÞ ¼ deðtÞ dt (12) Table 3 illustrated the performance of PID controller in the Appendix The fuzzy PI controller is the commonly used because the PD one encounters difficulties in deleting the steady state error.However, the fuzzy-PI gives a poor performance in the transient response in higher order systems because of its inherent internal integration operation.It is therefore more convenient to combine PI and PD actions to design a fuzzy PID controller (FLC-PID) to achieve proportional, integral, and derivative control action.It is imperative to obtain an FLC-PID controlled by adding the fuzzy-PD controller output and its embedded part.The fuzzy-PID controller is depicted in Figure 5. Table 4

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The output u of the fuzzy PID is presented by: where: U is the output of the FLC.The relationship between the input and output variables is given by [42]: where: E 5 K e .e and dE/dt 5 Kd.de/dt according to Figure 5. Therefore, from Eqs. ( 13) and ( 14) the controller output is expressed by the following equation: Finally, the components of PID-FLC can be deducted as follows: The proportional gain: αK e .PþβK d .D; The integral gain: β K e .P; The derivative gain: αK d .D.

Fuzzification
The inputs to the Fuzzy-PSO have to be fuzzified before being fed into the control rule and gain rule determinations.The triangular membership functions (MFs) used for the input (e, e c and, ΔT em ) are shown in Figures 6 and 7. Linguistic variables are (NB, NM, NS, ZE, PS, PM, PB).
Where: NB is Negative Big, NM is Negative Medium, NS is Negative Small, EZ is Equal Zero, PS is Positive Small, PM is Positive Medium, PB is Positive Big.

Inference and defuzzification
The present paper uses MIN operation for the calculation of the degree m(ΔT em ) associated with every rule, for example, m(ΔT em )5Min[m(e),m(e c )].
In the defuzzification stage, a crisp value of the electromagnetic torque is obtained by the normalized output function as [33]: where: m is the total number of rules (7*7), m(ΔT em ) is the membership grade for the n rule, ΔT em is the position of membership functions in rule n in U (À15,À10,-5,0,10,15).  Direct torque control of dual star induction motor

Control rule determination
The logic of determining this rule matrix is based on a global knowledge of the system operation.As an example, we consider the following two rules: if e is PB and ec is PB then ΔTem is PB if e is ZE and ec is ZE then ΔTem is ZE They indicate that if the speed is too small compared to its reference (e is PB), so a big gain (ΔT em is PB) is required to bring the speed to its reference and if the speed reaches its reference and is established (e is ZE and e c is ZE) so impose a small gain ΔT em is ZE.Table 5 represents the inference rules.

Simulation results and discussion
The results were obtained using a PSO algorithm programmed and implemented in MATLAB.The parameters of the DSIM are presented in Table 6 (Appendix).To illustrate the performances of the DTC of the DSIM we replaced the classical PID controller by a fuzzy-PSO technique in Figure 8.The simulation is carried out under the following conditions: the hysteresis band of the torque comparator is set to ±0.25 Nm and that of the flux comparator to ± 0.5 Wb.
Figure 9 depicts the waveforms of the improved performances of speed control.It can be noticed that the use of the fuzzy-PSO controller allows the speed to judiciously follow its reference value of 100 rad/s despite the presence of a load torque of 14Nm at t 5 0.6 s.In fact, this behaviour represents a clear improvement in dynamic response with a hybrid controller

Direct torque control of dual star induction motor
that is adjusting strictly the values of the parameters by increasing the constant of integration without an overshoot at the level of the dynamic response of the speed, contrary to a drive with a standard DTC-PID where the speed has underwent slightly rejected.
Performance with each controller is also analyzed through these of Integral Squared Error (ISE), Integral Absolute Error (IAE) and Integral Time Squared Error (ITSE), and the results described in Table 7 confirm the improved performance with the fuzzy-PSO algorithm.
In Figure 10 the electromagnetic torque produced by the DSIM controlled by DTC-PID and DTC-fuzzy-PSO is presented.In this figure, it can be noticed that the ripple is not the same for the two techniques.It is clear that the classical DTC-PID present two problems, steady state error and high torque ripples.On the other hand, the DTC-fuzzy-PSO corrects the steady state error and reduces the torque ripples.

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In Figure 11, it can be observed that the currents are sinusoidal and current ripples have also a notable reduction in fuzzy-PSO controller compared to the standard controller.
Figure 12 shows the trajectory of stator flux for the standard DTC and the hybrid DTC.It can be seen that this hybrid strategy has less ripple.
Figure 13 summarizes the evolution of the fitness function with respect to the number of iterations.

Conclusion
In this paper, a comparative study between the conventional DTC of the DSIM with PID controller and DTC-fuzzy-PSO has been presented for a speed controller.From the simulation studies, hybrid controller produced better performances in terms of a fast rise time, a small overshoot, reduced torque and flux ripples.Therefore very satisfactory performances have been achieved.Furthermore, the effectiveness of the proposed algorithms is evaluated and justified from performance indices IAE, ISE and ITSE.According to the yielded simulation results one can conclude that this algorithm is suitable for applications requiring a high

Figure 3 .
Figure 3. Description of a searching point by PSO.

Figure 5 .
Figure 5.The proposed control structure for PID-PSO type FLC.
Input scaling factor k e optimized 3.1604 Input scaling factor k d optimized 3.6741 β is the gain of the integral component 0.8081 α scaling factor for the output u 1 Figure 7. Membership functions for ΔT em.

Figure 6 .
Figure 6.Membership functions for e and e c.

Figure 8 .
Figure 8. Block diagram of the proposed DTC-fuzzy-PSO tuning speed controller.

Figure 10 .
Figure 10.Electromagnetic torque comparison of the two strategies.

Figure 9 .
Figure 9.Comparison of the rotor speed regulation of the standard DTC and hybrid DTC.

Figure 11 .
Figure 11.Phase current in the stator 1for both hybrid DTC and standad DTC.

Figure 13 .
Figure 13.Evolution of the fitness function relative to Fuzzy-PSO algorithm.

Table 1 .
Control strategy with hysteresis comparator.

Table 3 .
represented the performance of fuzzy controller is in the Appendix.Performance of PID controller.

Table 2 .
Parameters of PSO algorithms.

Table 7 .
Comparison of performance index.