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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. Keywords Dual stator induction motor (DSIM), Direct torque control (DTC), Speed control, particle swarm optimization (PSO), Fuzzy logic control (FLC) Paper type Review Article Nomenclature P Number of pole pairs R Rotor resistance J The moment of inertia L ,L Stators inductances s1 s2 f The friction coefficient L Rotor Inductance r r T The electromagnetic L Mutual inductance em m torque n number of particles in T The load torque the group Ω is the mechanical d dimension index rotation speed of t pointer of iterations the rotor (generations) R ,R Stators resistances s1 s2 © Ghoulemallah Boukhalfa, Sebti Belkacem, Abdesselem Chikhi and Said Benaggoune. Published in Applied Computing and Informatics. Published by Emerald Publishing Limited. This article is published under the Creative Commons Attribution (CC BY 4.0) license. Anyone may reproduce, distribute, translate and create derivative works of this article (for both commercial and non-commercial purposes), subject to full attribution to the original publication and authors. The full terms of this license may be seen at http://creativecommons.org/licences/by/4.0/legalcode Publishers note: The publisher wishes to inform readers that the article “Direct torque control of dual star induction motor using a fuzzy-PSO hybrid approach” was originally published by the previous publisher of Applied Computing and Informatics and the pagination of this article has been subsequently changed. There has been no change to the content of the article. This change was Applied Computing and necessary for the journal to transition from the previous publisher to the new one. The publisher Informatics Vol. 18 No. 1/2, 2022 sincerely apologises for any inconvenience caused. To access and cite this article, please use pp. 74-89 Boukhalfa, G., Belkacem, S., Chikhi, A., Benaggoune, S. (2020), “Direct torque control of dual star Emerald Publishing Limited e-ISSN: 2210-8327 induction motor using a fuzzy-PSO hybrid approach”, New England Journal of Entrepreneurship. p-ISSN: 2634-1964 DOI 10.1016/j.aci.2018.09.001 Vol. 18 No. 1/2, pp. 76-91. The original publication date for this paper was 06/09/2018. t v velocity of particle at stators and rotor, i,m Direct torque iteration I respectively control of dual w inertia weight factor subscripts 1, 2 denote variables and star induction rand ( ) random number between parameters of stator 1 motor 0 and 1 and 2 respectively pbest best previous position of V ,V ,V ,V respectively stator i ds1 qs1 ds2 qs2 the ith particle voltages in the d-q axis gbest best particle among all I ,I ,I ,I components of the stator ds1 qs1 ds2 qs2 the particles in currents in the d - q axis the population I ,I the rotor currents dr qr C ,C acceleration constant ψ , ψ , ψ ψ components of the stator 1 2 ds1 qs1 ds2, qs2 subscripts s, r indicate the variables flux linkage vectors in and parameters of d-q axis ψ , ψ respectively rotor fluxes dr qr 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 ACI main idea of this approach is that it does not need a precise mathematical model of the electric 18,1/2 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]: d ψ ds1 V ¼ R i þ ω ψ ds1 s1 ds1 s qs1 dt d ψ > qs1 > V ¼ R i þ þ ω ψ qs1 s1 qs1 s ds1 > dt > d ψ > ds2 V ¼ R i þ ω ψ < ds2 s2 ds2 s qs2 dt (1) d ψ qs2 > V ¼ R i þ þ ω ψ qs2 s2 qs2 s > ds2 dt d ψ dr V ¼ 0 ¼ R i þ ðω ω Þψ > dr r dr s r qr > dt > d w qr V ¼ 0 ¼ R i þ þðω ω Þw qr r qr s r dr dt where the fluxes equations are: ψ ¼ L i þ L ði þ i þ i Þ > s1 ds1 m ds1 ds2 dr ds1 ψ ¼ L i þ L ði þ i þ i Þ > s1 qs1 m qs1 qs2 qr qs1 ψ ¼ L i þ L ði þ i þ i Þ s2 ds2 m ds1 ds2 dr ds2 (2) ψ ¼ L i þ L ði þ i þ i Þ > s1 qs1 m qs1 qs2 qr qs1 ψ ¼ L i þ L ði þ i þ i Þ dr r dr m ds1 ds2 dr ψ ¼ L i þ L ði þ i þ i Þ r qr m qs1 qs2 qr qr For studying the dynamic behavior, the following equation of motion was added: dΩ J ¼ T T f Ω (3) em r r r dt The model of the DSIM has been completed by the expression of the electromagnetic torque Direct torque T given below: em control of dual star induction T ¼ p ðψ ði þ i Þ ψ ði þ i ÞÞ (4) em qs1 qs2 ds1 ds2 dr qr L þ L m r motor 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: – 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. Sb1 Vsb1 isb1 Vsb2 V isa2 sa2 b2 sb2 Ra Sa2 ira sa1 α=30° isa1 Rb Sa1 rc irb Figure 1. Schematic representation of dual Vsc1 sc2 star induction isc1 motor (DSIM). Sc1 sc2 Sc2 ACI 3 2 Vdc 18,1/2 4 5 6 (, ,1,2)s Inverter1 1 s sref (cflx) (a ,b,c,1)s Switching DSIM Table Τ (ccpl) (a ,b,c,2)s em T Inverter2 Ω em PID ref Controller + - -1 Speed r ˆ em sensor V(, ,1,2)s Figure 2. Flux and Torque Block diagram of the Estimation a,b,c DTC of DSIM. Τ (, ,1,2)s em 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: ψ ¼ ðV R i Þdt sα1;2 s sα1;2 sα1:2 (5) ψ ¼ ðV R i Þdt sβ1;2 s sβ1;2 sβ1:2 where V and V are the estimated components of the stator vector voltage. They are sα1;2 sβ1;2 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 xi5(x ,x ,....... ,x ). The finest previous positions of the ith particle are i1 i2 id considering by [32]: cflx ccpl 123456 Corrector 01 V V V V V V 2 levels 2 3 4 5 6 1 0V V V V V V 7 0 7 0 7 0 1V V V V V V 3 levels 6 1 2 3 4 5 11 V V V V V V 2 levels Table 1. 3 4 5 6 1 2 0V V V V V V Control strategy with 0 7 0 7 0 7 hysteresis comparator. 1V V V V V V 3 levels 5 6 1 2 3 4 Direct torque pbest ¼ðpbest ; pbest ; :::; pbest Þ (6) i i1 i2 id control of dual The index of the best particle amongst the group is gbest . The velocity of particle ith is star induction represented as: motor v ¼ðv ; v ; ::::; v Þ (7) i i1 i2 id The modified speed and position of each particle can be calculated using the current and the distance from pbest to gbest as expressed in the following equations: i,d d tþ1 t t v ¼ w:v þ C * randðÞ* pbest X þ C * randðÞ* gbest x (8) 1 2 i:m i:m i;m i;m i;m i;m |ﬄﬄ{zﬄﬄ} |ﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ} |ﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ} inertia personelinfluence socialinfluence tþ1 t tþ1 x ¼ x þ v i ¼ 1; 2; :::; n ; m ¼ 1; 2; ... ; d (9) i;m i;m i;m where: n is the number of particles in the group, d is the dimension index, t is the pointer of iterations (generations), v is the velocity of particle at iteration i, w is the inertia weight i,m factor, rand ( ) is the random number between 0 and 1, pbest is the best previous position of the ith particle, gbest is the best particle among all the particles in the population, C ,C are 1 2 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 W5 0.8, with acceleration constants C15 C25 2. The PSO main program has to optimize in this case three parameters, K ,K and β to the fuzzy controller, and search optimal value of the e d three-dimensional search space. Step 2: So we arbitrarily configured a swarm of “50” population in three-dimensional search space using (x ,x ,x ), and (v ,v ,v ) as preliminary situation along with velocity. i1 i2 i3 i1 i2 i3 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. Figure 3. Description of a searching point by PSO. Step 3: Run the program by means of PSO algorithm through n numbers of iterations, as well, ACI the program finds the final optimal value of the fitness function as “best fitness” with the last 18,1/2 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]: Start Generate initial populations Run the Fuzzy-DTC control model for each set of parameters Calculate parameters (k ,k , β ) e d of fuzzy-PID controller Calculate the fitness function of fuzzy-PID controller Calculate the pbest of each particle and gbest of population Update the velocity, position, gbest and pbest of particles No Maximum iteration number reached? Yes Output the parameters (k ,k , β ) e d of fuzzy-PID controller Figure 4. Flowchart for fuzzy PSO-DTC-DSIM. Stop Z Direct torque IAE ¼ jeðtÞjdt (10) 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 } as follows: c 81 eðtÞ¼ Ω Ω ðtÞ (11) ref r deðtÞ e ðtÞ¼ (12) 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. Descriptions Parameters Number of particles in the swarm 50 Number of Iterations 10 Number of components or dimension 3 Table 2. Inertia weight factor w 0.8 Parameters of PSO C15C2 2 algorithms. Controllers Parameters K 37.5 p Table 3. K 0.35 Performance of PID K 0 controller. ref e E - U Fuzzy Controller em DTC-DSIM k r α + dt Figure 5. The proposed control IAE PSO structure for PID-PSO Feedback type FLC. The output u of the fuzzy PID is presented by: ACI 18,1/2 Z u ¼ αU þ βUdt (13) where: U is the output of the FLC. The relationship between the input and output variables is given by [42]: U ¼ A þ PE þDdE=dt (14) where:E 5 K .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: u ¼ αA þ βA:t þ αK :Pe þ βK D:e þ βK P edt þ αK D:de=dt (15) e d: e: d: Finally, the components of PID-FLC can be deducted as follows: The proportional gain: αK . PþβK .D; The integral gain: β K .P; The derivative e d e gain: αK .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 and, ΔT ) are shown in Figures 6 and 7. Linguistic variables are (NB, NM, NS, ZE, PS, c em 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 m(ΔT ) associated em with every rule, for example, m(ΔT )5Min[m(e),m(e )]. em c In the defuzzification stage, a crisp value of the electromagnetic torque is obtained by the normalized output function as [33]: μðΔT ÞΔT emj emj j¼1 du ¼ (16) μðΔT Þ emj j¼1 where: m is the total number of rules (7*7), m(ΔT ) is the membership grade for the n rule, em ΔT is the position of membership functions in rule n in U (15,10,-5,0,10,15). em Controller Fuzzy-PSO Input scaling factor k optimized 3.1604 Input scaling factor k optimized 3.6741 Table 4. d β is the gain of the integral component 0.8081 Performance of fuzzy- PSO controller. α scaling factor for the output u 1 Direct torque NB NM NS ZE PS PM PB control of dual star induction motor 0.8 0.6 83 0.4 0.2 Figure 6. Membership functions 0 for e and e c. -1.5 -1 -0.5 0 0.5 1 1.5 NB ZE PS PM PB NM NS 0.8 0.6 0.4 0.2 Figure 7. Membership functions 0 for ΔT em. -15 -10 -5 0 5 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 is PB) is required to bring the speed to its reference and if the speed reaches its em reference and is established (e is ZE and e is ZE) so impose a small gain ΔT is ZE. c em Table 5 represents the inference rules. 6. Simulation results and discussion ACI The results were obtained using a PSO algorithm programmed and implemented in 18,1/2 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 t5 0.6 s. In fact, this behaviour represents a clear improvement in dynamic response with a hybrid controller ΔT NB NM NS ZE PS PM PB em 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 Table 5. PB ZE PS PM PB PB PB PB Inference rules. Rated Power 4.5KW Stator Resistance R 5 R 3.72U s1 s2 Rotor Resistance R 2.12U Stator Inductance L 0.022H Rotor Inductance L 0.006H Mutual Inductance L 0.3672H Pole Pairs P 1 Machine Inertia J 0.0662 kg.m2 Table 6. Viscous Friction Coefficient f 0.001 kg.m2/s DSIM parameters [12]. r 3 2 Vdc 4 1 (, ,1,2)s Inverter1 1 s sref (cflx) (a ,b,c,1)s s 3 IAE PSO Switching DSIM Table T (ccpl) T (a ,b,c,2)s Ω em ref Inverter2 em Fuzzy controller - ˆ -1 3 Speed Ωr em Figure 8. V(, ,1,2)s sensor Block diagram of the ψ Flux and proposed DTC-fuzzy- Torque PSO tuning speed a,b,c Estimation controller. Τ I(, ,1,2)s em that is adjusting strictly the values of the parameters by increasing the constant of Direct torque integration without an overshoot at the level of the dynamic response of the speed, contrary to control of dual a drive with a standard DTC-PID where the speed has underwent slightly rejected. star induction Performance with each controller is also analyzed through these of Integral Squared Error motor (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. PID Fuzzy-PSO 98 100 Figure 9. 0.2 0.22 0.24 0.6 0.605 0.61 Comparison of the rotor speed regulation of the standard DTC 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 and hybrid DTC. Time (s) Controllers IAE ISE ITSE Table 7. PID 0.5473 0.1498 0.1348 Comparison of Fuzzy-PSO 0.2072 0.0215 0.0193 performance index. PID Fuzzy-PSO 30 8 0.58 0.6 0.62 0.64 0 Figure 10. Electromagnetic torque comparison of -10 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 the two strategies. Time (s) Speed (rad/s) Torque (N.m) In Figure 11, it can be observed that the currents are sinusoidal and current ripples have ACI also a notable reduction in fuzzy-PSO controller compared to the standard controller. 18,1/2 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 PID fuzzy-PSO -20 -40 Figure 11. Phase current in the stator 1for both hybrid -60 DTC and standad DTC. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Time(s) 1.5 PID Fuzzy-PSO 0.5 -0.5 -1 Figure 12. -1.5 Stator flux trajectory in -1.5 -1 -0.5 0 0.5 1 1.5 the stator1. phisd1 (Wb) Stator current isa1(A) phisq1(Wb) 120 Direct torque control of dual star induction motor Figure 13. Evolution of the fitness function relative to 1 2 3 4 5 6 7 8 9 10 Fuzzy-PSO algorithm. Number of Iterations 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 References [1] Y. Zhao, T.A. 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Corresponding author Sebti Belkacem can be contacted at: belkacem_sebti@yahoo.fr For instructions on how to order reprints of this article, please visit our website: www.emeraldgrouppublishing.com/licensing/reprints.htm Or contact us for further details: permissions@emeraldinsight.com
Applied Computing and Informatics – Emerald Publishing
Published: Mar 1, 2022
Keywords: Dual stator induction motor (DSIM); Direct torque control (DTC); Speed control; particle swarm optimization (PSO); Fuzzy logic control (FLC)
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