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

Abstract

This paper presents the particle swarm optimization (PSO) algorithm in conjuction with the fuzzy logic method in order to achieve an optimized tuning of a proportional integral derivative controller (PID) in the DTC control loops of dual star induction motor (DSIM). The fuzzy controller is insensitive to parametric variations, however, with the PSO-based optimization approach we obtain a judicious choice of the gains to make the system more robust. According to Matlab simulation, the results demonstrate that the hybrid DTC of DSIM improves the speed loop response, ensures the system stability, reduces the steady state error and enhances the rising time. Moreover, with this controller, the disturbances do not affect the motor performances.

Citation

Boukhalfa, G., Belkacem, S., Chikhi, A. and Benaggoune, S. (2022), "Direct torque control of dual star induction motor using a fuzzy-PSO hybrid approach", Applied Computing and Informatics, Vol. 18 No. 1/2, pp. 74-89. https://doi.org/10.1016/j.aci.2018.09.001

Nomenclature

P

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

Stators resistances

R_r

Rotor resistance

L_s1, L_s2

Stators inductances

L_r

Rotor Inductance

L_m

Mutual inductance

n

number of particles in the group

d

dimension index

t

pointer of iterations (generations)

v^t_i,m

velocity of particle at iteration I

w

inertia weight factor

rand ( )

random number between 0 and 1

pbest_i

best previous position of the ith particle

gbest

best particle among all the particles in the population

C₁,C₂

acceleration constant

subscripts s, r

indicate the variables and parameters of stators and rotor, respectively

subscripts 1, 2

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

1. Introduction

In high-power applications, the dual star induction motor has largely replaced the induction machines whose roles were predominant in the industry [1–5]. 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 [7–14].

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 self-adjusting mechanism has been set up to adapt the PID regulator in case of any disturbances.

The use of optimization algorithms as alternative methods for tuning PID controllers has been a recent topic of research in electric machines control. New optimization techniques are proposed, for instance, the Imperialist Competitive Algorithm (ICA) [18], evolutionary algorithm [19], Genetic Algorithm (GA) [20–21], BAT algorithm [22] ,Particle Swarm Optimization (PSO) [23–26], and Ant Colony Optimization (ACO) algorithm [27], Harmony Search (HS) [28], hybrid GA [29–30], adaptive Cuckoo Search algorithm (CS) [31].

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.

2. Modeling of the dual star induction motor

The DSIM dynamic equations in the reference d-q can be reported as follow [4]:

(1){Vds1=Rs1ids1+dψds1dt−ωsψqs1Vqs1=Rs1iqs1+dψqs1dt+ωsψds1Vds2=Rs2ids2+dψds2dt−ωsψqs2Vqs2=Rs2iqs2+dψqs2dt+ωsψds2Vdr=0=Rridr+dψdrdt−(ωs−ωr)ψqrVqr=0=Rriqr+dϕqrdt+(ωs−ωr)ϕdr

where the fluxes equations are:

(2){ψds1=Ls1ids1+Lm(ids1+ids2+idr)ψqs1=Ls1iqs1+Lm(iqs1+iqs2+iqr)ψds2=Ls2ids2+Lm(ids1+ids2+idr)ψqs1=Ls1iqs1+Lm(iqs1+iqs2+iqr)ψdr=Lridr+Lm(ids1+ids2+idr)ψqr=Lriqr+Lm(iqs1+iqs2+iqr)

For studying the dynamic behavior, the following equation of motion was added:

(3)JdΩrdt=Tem−Tr−frΩr

The model of the DSIM has been completed by the expression of the electromagnetic torque T_em given below:

(4)Tem=pLmLm+Lr(ψdr(iqs1+iqs2)−ψqr(ids1+ids2))

A schematic representation of the stator and rotor windings of dual star induction motor is given in Figure 1.

3. Direct torque control (DTC) of the DSIM

The classical DTC, proposed by [40], is based on the following algorithm:

1. –

   Divide the time domain into periods of reduced duration Ts;

2. –

   For each clock struck, measure the line currents and phase voltages of the DSIM;

3. –

   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;

4. –

   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 +1 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;

5. –

   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 +1 to increase the flux and 0 to reduce it;

6. –

   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.

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:

(5)ψsα1.2=∫0t(Vsα1,2−Rsisα1,2)dtψsβ1.2=∫0t(Vsβ1,2−Rsisβ1,2)dt

where Vsα1,2 and Vsβ1,2 are the estimated components of the stator vector voltage. They are expressed from the model of the inverter.

4. 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 xi=(x_i1, x_i2,….… ,x_id). The finest previous positions of the ith particle are considering by [32]:

(6)pbesti=(pbesti1,pbesti2,...,pbestid)

The index of the best particle amongst the group is gbest_d. The velocity of particle ith is represented as:

(7)vi=(vi1,vi2,....,vid)

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:

(8)vi.mt+1=w.vi.mt︸inertia+C1∗rand()∗(pbesti,m−Xi,mt)︸personelinfluence+C2∗rand()∗(gbesti,m−xi,mt)︸socialinfluence

(9)xi,mt+1=xi,mt+vi,mt+1 i=1,2,...,n; m=1,2,…,d

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, C₁,C₂ are acceleration constant. Velocity vector for position update is depicted in Figure 3.

4.1 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 = 0.8, with acceleration constants C1 = C2 = 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.

4.2 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]:

(10)IAE=∫0∞|e(t)|dt

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:

(11)e(t)=Ωref−Ωr(t)

(12)ec(t)=de(t)dt

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 represented the performance of fuzzy controller is in the Appendix.

The output u of the fuzzy PID is presented by:

(13)u=αU+∫βU dt

where: U is the output of the FLC.

The relationship between the input and output variables is given by [42]:

(14)U=A+PE+D dE/dt

where: E = K_e.e and dE/dt = Kd.de/dt according to Figure 5.

Therefore, from Eqs. (13) and (14) the controller output is expressed by the following equation:

(15)u=αA+βA.t+αKe.Pe+βKd.D.e+βKe.P∫edt+αKd.D.de/dt

Finally, the components of PID-FLC can be deducted as follows:

The proportional gain: αK_e. P+βK_d.D; The integral gain: β K_e.P; The derivative gain: αK_d.D.

5.1 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.

5.2 Inference and defuzzification

The present paper uses MIN operation for the calculation of the degree µ(ΔT_em) associated with every rule, for example, µ(ΔT_em)=Min[µ(e),µ(e_c)].

In the defuzzification stage, a crisp value of the electromagnetic torque is obtained by the normalized output function as [33]:

(16)du=∑j=1mμ(ΔTemj)ΔTemj∑j=1mμ(ΔTemj)

where: m is the total number of rules (7*7), µ(ΔT_em) is the membership grade for the n rule, ΔT_em is the position of membership functions in rule n in U (−15,−10,-5,0,10,15).

5.3 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.

6. 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 = 0.6 s. In fact, this behaviour represents a clear improvement in dynamic response with a hybrid controller 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.

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.

7. 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 tracking accuracy in presence of external disturbances. In future, the work can be extended by the applications of the intelligent hybrid techniques like neuro-fuzzy-GA, neuro-fuzzy-PSO, neuro-fuzzy-ACO.

Abbreviations

DSIM

dual star induction motor

FLC

fuzzy logic controller

PSO

particle swarm optimization

DTC

direct torque control

PID

proportional integral derevative

IAE

the integral of absolute value of the error

ISE

the integral of square error

ITSE

the integral of time multiply square error

Figures

Figure 1

Schematic representation of dual star induction motor (DSIM).

Figure 2

Block diagram of the DTC of DSIM.

Figure 3

Description of a searching point by PSO.

Figure 4

Flowchart for fuzzy PSO-DTC-DSIM.

Figure 5

The proposed control structure for PID-PSO type FLC.

Figure 6

Membership functions for e and e_c.

Figure 7

Membership functions for ΔT_em.

Figure 8

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

Figure 9

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

Figure 10

Electromagnetic torque comparison of the two strategies.

Figure 11

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

Figure 12

Stator flux trajectory in the stator1.

Figure 13

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

Table 1

Control strategy with hysteresis comparator.

cflx	ccpl	1	2	3	4	5	6	Corrector
0	1	V2	V3	V4	V5	V6	V1	2 levels
	0	V7	V0	V7	V0	V7	V0
	−1	V6	V1	V2	V3	V4	V5	3 levels
1	1	V3	V4	V5	V6	V1	V2	2 levels
	0	V0	V7	V0	V7	V0	V7
	−1	V5	V6	V1	V2	V3	V4	3 levels

Table 2

Parameters of PSO algorithms.

Descriptions	Parameters
Number of particles in the swarm	50
Number of Iterations	10
Number of components or dimension	3
Inertia weight factor w	0.8
C1=C2	2

Table 3

Performance of PID controller.

Controllers	Parameters
Kp	37.5
Ki	0.35
Kd	0

Table 4

Performance of fuzzy-PSO controller.

Controller	Fuzzy-PSO
Input scaling factor ke optimized	3.1604
Input scaling factor kd optimized	3.6741
β is the gain of the integral component	0.8081
α scaling factor for the output u	1

Table 5

Inference rules.

					ec
ΔTem		NB	NM	NS	ZE	PS	PM	PB
E	NB	NB	NB	NB	NB	NM	NS	ZE
	NM	NB	NB	NB	NM	NS	ZE	PS
	NS	NB	NB	NM	NS	ZE	PS	PM
	ZE	NB	NM	NS	ZE	PS	PM	PB
	PS	NM	NS	ZE	PS	PM	PB	PB
	PM	NS	ZE	PS	PM	PB	PB	PB
	PB	ZE	PS	PM	PB	PB	PB	PB

Table 6

DSIM parameters [12].

Rated Power	4.5KW
Stator Resistance Rs1= Rs2	3.72Ω
Rotor Resistance Rr	2.12Ω
Stator Inductance Ls	0.022H
Rotor Inductance Lr	0.006H
Mutual Inductance Lm	0.3672H
Pole Pairs P	1
Machine Inertia J	0.0662 kg.m2
Viscous Friction Coefficient fr	0.001 kg.m2/s

Table 7

Comparison of performance index.

Controllers	IAE	ISE	ITSE
PID	0.5473	0.1498	0.1348
Fuzzy-PSO	0.2072	0.0215	0.0193

Appendix

See.