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The purpose of this paper is to improve the performance of the conventional direct torque control (DTC) method of a dual star induction motor (DSIM) by enhancing speed control and reducing the ripples of electromagnetic torque and stator current. To achieve this, we propose a new optimal tuned controller based on the combination of a fractional order proportional integral controller (FOPI) and particle swarm optimization (PSO) algorithm. The aim of this highperformance controller is to reduce the rise time, settling time, steadystate error, and effects of load disturbances in the speed response of the DSIM, as well as minimize oscillations in the torque and stator currents, particularly at low speeds. This strategy named DTCFOPIPSO will be investigated, and its performances will be compared with the traditional DTC strategy based on the classical PI controller. Simulation tests using MATLAB/Simulink software are conducted under different operating conditions to demonstrate that the proposed DTCFOPIPSO strategy has a direct impact on improving speed dynamic, reducing torque fluctuations, minimizing steadystate error and provides excellent performance for load variation and reference speed inversion.
DTC, DSIM, Control, FOPI, PSO
In highpower applications, AC machines powered by static converters occupy increasingly industrial areas. However, the constraints on the power components limit the switching frequency and, thus, the performance. Therefore, the power must be segmented to enable the use of components with higher switching frequencies [13]. One way to ensure this is to use large phase (or multiphase) machines, and as an example of this type is the dual star induction motor.
Dual star induction motor has noticeable potential because of their reliability and ability to operate in degraded conditions [4, 5]. However, despite these advantages, their control remains rather complicated compared to the DC machines because the mathematical model is nonlinear and intrinsically coupled [6, 7].
To guaranty good performance and decoupling control, one can use direct torque control. This is a vector control method introduced by TAKAHASHI and DEPENBROCK. It’s considered as an alternative approach because of its efficiency and simplicity of implementation [8].
This technique is based on the calculation of the stator flux and the electromagnetic torque from stator currents and voltages measurements without the need of a speed sensor [9]. It has been proven that the DTC has remarkable dynamic performance and is recognized as a reliable and robust solution to meet high control requirements [1012].
Improving system performances is the main objective of the control techniques. For this purpose, different techniques are used, including linear and nonlinear approaches. The simpler and effective solutions that exist nowadays in the majority of engineering control applications employ proportional integral derivative (PID) in the regulation loop [13]. On the other hand, lots of nonlinear control laws have been proposed including sliding mode control (SMC) [14, 15], synergetic control [1618], predictive control [19], backstepping control [20, 21], and highorder SMC techniques [5]. In addition, the response and the speed dynamic of a system can be improved using intelligent techniques, such as fuzzy logic and neural networks [2224].
However, PID controllers are limited and cannot meet all the needs of a variable speed drives.
There is strong evidence that PI and PI tuned by PSO controllers continue to be poorly understood and, in particular, inadequately tuned in many applications during the last few years [25].
Numerous strategies have been suggested to deal with these problems. One recommendation is to reinforce the proportionalintegralderivative controller's flexibility, robustness, and stability by introducing two additional fractional noninteger differentiation and integration control parameters and then get a fractionalorder controller. Some recent researches recommended fractional order (FOPID) control for robust control [26, 27].
In this context, with the aim of improving the features of the traditional DTC, we propose an approach based on FOPI controller and the particle swarm optimization technique, named FOPIPSO.
The fact that the FOPI controller contains an additional parameter, will makes the FOPIPSO more flexible than the classic PI controller. Moreover, the performance of the DSIM system will be enhanced as a result of the designed FOPIPSO method.
The strategy of this work is to tune the parameters of the FOPI controller using the PSO algorithm in order to achieve accurate tuning that corresponds to the high performance required in DTC control of the DSIM. This includes minimizing the rise time, settling time, steadystate error, impact of disturbances on speed control, as well as reducing torque and stator current ripples.
The study is organized as follows: Section 1 presents the DSIM mathematical model; Section 2 presents the conventional DTC control of the dual star induction motor. Sections 3 and 4 introduce the concept of a fractional order PID controller and the PSO optimization method, respectively. Section 5 includes a detailed description of the proposed strategy with the DTC using the PSO algorithm. The results, discussions, and conclusion complete the study.
The stator of the DSIM consists of two threephase systems (a_{s}_{1}, b_{s}_{1}, c_{s}_{1}) and (a_{s}_{2}, b_{s}_{2}, c_{s}_{2}), separated by an electrical angle α of 30 degrees, as shown in Figure 1. The rotor consists of a classic threephase system (a_{r}, b_{r}, c_{r}), which is identical to that of the induction motor.
By using Park’s method, the threephase scheme of the DSIM (specified in Figure 1) is transformed into a twophase model (dq). Hence, the stator and rotor voltages elements are expressed in dq frame as follows [28]:
$\left\{\begin{array}{l}V_{s 1 d}=R_{s 1} I_{s 1 d}+\frac{d \Phi_{s 1 d}}{d t}\omega_s \Phi_{s 1 q} \\ V_{s 1 q}=R_{s 1} I_{s 1 q}+\frac{d \Phi_{s 1 q}}{d t}+\omega_s \Phi_{s 1 d} \\ V_{s 2 d}=R_{s 2} I_{s 2 d}+\frac{d \Phi_{s 2 d}}{d t}\omega_s \Phi_{s 2 q} \\ V_{s 2 q}=R_{s 2} I_{s 2 q}+\frac{d \Phi_{s 2 q}}{d t}+\omega_s \Phi_{s 2 d} \\ V_{r d}=0=R_r I_{r d}+\frac{d \Phi_{r d}}{d t}\omega_{s r} \Phi_{r q} \\ V_{r q}=0=R_r I_{r q}+\frac{d \Phi_{r q}}{d t}+\omega_{s r} \Phi_{r d}\end{array}\right.$ (1)
where,
V_{s1d}, V_{s1q}, V_{s2d }and V_{s2q} are the dq voltage components of the two stator threephase systems.
I_{s1d}, I_{s1q}, I_{s2d} and I_{s2q} are the dq stator current components of the two stator systems.
I_{rd} and I_{rq} are the dq current components of the rotor.
R_{s1} and R_{s2} are the stator resistances.
Rr is the rotor resistance.
Figure 1. Representation of the DSIM winding
The stator and rotor flux components Φ_{s1d}, Φ_{s1q}, Φ_{s2d}, Φ_{s2q}, Φ_{rd} and Φ_{rq} are given by:
$\left\{\begin{array}{l}\Phi_{s 1 d}=L_{s 1} I_{s 1 d}+L_m\left(I_{s 1 d}+I_{s 2 d}+I_{r d}\right) \\ \Phi_{s 1 q}=L_{s 1} I_{s 1 q}+L_m\left(I_{s 1 q}+I_{s 2 q}+I_{r q}\right) \\ \Phi_{s 2 d}=L_{s 2} I_{s 2 d}+L_m\left(I_{s 2 d}+I_{s 2 d}+I_{r d}\right) \\ \Phi_{s 2 q}=L_{s 2} I_{s 2 q}+L_m\left(I_{s 1 q}+I_{s 2 q}+I_{r q}\right) \\ \Phi_{r d}=L_r I_{r d}+L_m\left(I_{s 1 d}+I_{s 2 d}+I_{r d}\right) \\ \Phi_{r q}=L_r I_{r q}+L_m\left(I_{s 1 q}+I_{s 2 q}+I_{r q}\right)\end{array}\right.$ (2)
where,
L_{s1} and L_{s2} are the stator inductances.
L_{r} is the rotor inductance.
Lm is the mutual inductance.
The mechanical speed formulation is given by:
$J \frac{d \Omega}{d t}=T_{e m}T_LF_r \Omega$ (3)
where, the relation of the electromagnetic torque T_{em} is as below:
$T_{e m}=p \frac{L_m}{L_r+L_m}\left[\Phi_{r d}\left(I_{s 1 q}+I_{s 2 q}\right)\Phi_{r q}\left(I_{s 1 d}+I_{s 2 d}\right)\right]$ (4)
where,
p denotes the number of pole pairs.
J represents the total moment of inertia.
F_{r} is the viscous friction coefficient.
T_{L} denotes the load torque.
The DTC was proposed by Takahashi and Noguchi [8] in the mid1980s. The principle of this control is to directly control the torque and the stator flux of the machine. In this context, the hysteresis comparators are used which allow comparing the estimated values with those of references, then the inverter states are directly controlled through a predefined selection table. Compared to fieldoriented control, DTC is less sensitive to parametric variations of the machine and allows obtaining precise and fast torque dynamics. The DTC control strategy of the DSIM is illustrated in Figure 2. The main components of its structure with a speed loop regulator are defined as follows:
Figure 2. The classical DTC control structure of DSIM using a PI controller for the speed loop
Table 1. Switching table of the classical DTC control strategy
Sector 

1 
2 
3 
4 
5 
6 
Comparator 
C_{flx}=1 
C_{cpl}=1 
V_{2} 
V_{3} 
V_{4} 
V_{5} 
V_{6} 
V_{1} 
2 levels 
C_{cpl}=0 
V_{7} 
V_{0} 
V_{7} 
V_{0} 
V_{7} 
V_{0} 

C_{cpl}=1 
V_{6} 
V_{1} 
V_{2} 
V_{3} 
V_{4} 
V_{5} 
3 levels 

C_{flx}=0 
C_{cpl}=1 
V_{3} 
V_{4} 
V_{5} 
V_{6} 
V_{1} 
V_{2} 
2 levels 
C_{cpl}=0 
V_{0} 
V_{7} 
V_{0} 
V_{7} 
V_{0} 
V_{7} 

C_{cpl}=1 
V_{5} 
V_{6} 
V_{1} 
V_{2} 
V_{3} 
V_{4} 
3 levels 
The following equations give the expression for the stator flux estimation:
$\left\{\begin{array}{l}\Phi_{s \alpha 1,2}=\int_0^t\left(V_{s \alpha 1,2}R_s i_{s \alpha 1,2}\right) d t \\ \Phi_{s \beta 1,2}=\int_0^t\left(V_{s \beta 1,2}R_s i_{s \beta 1,2}\right) d t\end{array}\right.$ (5)
where, $\mathrm{V}_{s \alpha 1,2}$ and $\mathrm{V}_{s \beta 1,2}$ are the stator vector voltage components in the αβ fixed frame. They are measured or obtained using the inverter model.
Fractional order PID controller indicated by FOPID was suggested by Podlubny et al. [29] in 1997.
The FOPID is an extension of the PID controller, vastly used in the industrial control system. The FOPID is based on fractional calculus. FOPID has five parameters which are accountable for providing good behavior of dynamical systems and low sensitivity to parameters changing in a controlled system [30].
The integerdifferential Equation (6) gives the control law of the fractional order PID controller [31].
$u(t)=K_p e(t)+K_i D^{\lambda} e(t)+K_d D^\mu e(t)$ (6)
The input signal e(t), the command signal u(t), the proportional gain K_{p} , the integration gain K_{i} , and the derivative gain K_{d} are all represented in Equation 6. The integral and derivative terms that are not integer orders are named $\lambda$ and $\mu$ , respectively.
An operator for fractional derivatives and integrals is called D^{a} . The most commonly used definition of the fractional operator, known as the RiemannLiouville description, is as follows [32, 33]:
${ }_a D_t^\alpha f(t)=\frac{1}{\Gamma(n\alpha)} \frac{d^n}{d t^n}\left[\int_a^t \frac{f(\tau)}{(t\tau)^{\alphan+1}} d \tau\right]$ (7)
The gamma expression of Euler is represented by Г(.), the integration limits are a and t, and n is an integer that fits the condition n1< α < n.
The following equation provides the Laplace transform of Equation (6) for zero initial conditions:
$L\left\{{ }_a D_t^\alpha f(t)\right\}=\int_0^{\infty} e^{s t}{ }_a D_t^\alpha f(t) d t=s^\alpha F(s)$ (8)
Then, the transfer function of the FOPID controller is given by Equation (9) as follows:
$G(s)=K_p+K_i s^{\lambda}+K_d s^\mu$ (9)
Obviously, taking $\lambda=1$ and $\mu=1$ , one can obtain a classic PID.
Figure 3. Classical PID controller versus FOPID controller
The FOPID controller expands the traditional PID controller from a point to a plane, as depicted in Figure 3. The flexibility and robustness of this generalized controller allows us to more precisely manage the dynamics of the control system, and as a result, the system becomes robust and more stable [31].
The structure of a FOPI ( $K_d=0$ ) is selected, which is the PI controller's general form.
The particle swarm optimization algorithm is one of the most widely used methods today for solving optimization problems. Due to its low computational cost, high performance, and simplicity, it has been hailed as a potential and successful optimization technique. PSO algorithms are used in many scientific fields, which successfully solve most optimization problems [34, 35].
Midway through the 1990s, Kennedy and Eberhart [36] created the PSO algorithm as a heuristic search technique. The social behavior and movement of a collection of birds or a group of fish looking for food is the basis for PSO. Particles (possible solutions) fly across the search space, looking for attractive locations based on previous experiences and those of their neighbors, according to how the PSO algorithm operates [37, 38]. The PSO technique is recommended in this study to tune the controller gains to achieve the optimal controller parameters and, as a result, the best system output.
At each cycle, the following calculations are made to determine each population particle's new velocity and position:
$\left\{\begin{array}{l}v_i^{k+1}=w v_i^k+c_1 r_1\left(\text { Pbest }_i^kX_i^k\right)+c_2 r_2\left(\text { Gbest }_i^kX_i^k\right) \\ r_1, r_2 \in[0,1] \text { are random numbers }\end{array}\right.$ (10)
$X_i^{k+1}=X_i^k+v_i^{k+1}$ (11)
where, Gbest ${ }_i^k$ is the global best solution and Pbest ${ }_i^k$ is the personal best solution.
Equation (12) represents the integral of absolute error (IAE) between the speed $\Omega$ and its reference $\Omega_{\text {ref }}$ . The IAE is used to build the objective function that will be optimized,
$I A E=\int_0^{\infty}(\Delta \Omega) d t$ (12)
where,
$\Delta \Omega=\Omega_{r e f}\Omega$ (13)
To optimize the parameters of the FOPI controller offline, it is recommended to vary the speed and load in different scenarios during the execution of the PSO algorithm. This optimization approach can improve speed dynamics, reduce torque oscillations, and minimize steadystate errors.
Figure 4. Proposed FOPIPSO controller concept
The proposed FOPIPSO controller is illustrated in Figure 4, it contains three parameters (K_{p}, K_{i} and λ) to be optimized offline by the PSO method. This optimized controller will provide at its output the reference torque necessary to control the speed with precision and more robustness than the conventional PI controller.
The PSO method is used to implement the optimization steps of the FOPI controller for controlling the DSIM speed. Figure 5 depicts these actions in the order they should be taken.
Figure 5. Flowchart of FOPIPSODTCDSIM
Table 2 provides a list of the control parameters for the PSO method used to optimize the FOPI controller.
Table 2. Parameters of the PSO algorithm
Parameter 
Value 
Swarm size 
50 
Maximum iteration 
100 
C_{1} 
0.1 
C_{2} 
1.2 
W 
0.8 
The proposed DTC method with the DSIM's FOPIPSO algorithm is shown in Figure 6.
Figure 6. Schematic of DTC of DSIM with FOPIPSO
To validate the performance of the proposed approach and compare it with to the classical technique, a simulation was carried out in MATLAB/Simulink software, the control diagram depicted in Figure 6 is implemented using the parameters that are presented in Table 3.
Two control techniques which are: classical PI and the proposed one (FOPIPSO) are tested in two scenarios.
Table 3. Specifications parameters of DSIM [39]
Parameter 
Value 
Rated power 
4.5 kW 
Stator resistance (R_{s1}=R_{s2} ) 
3.72 Ω 
Rotor resistance (Rr ) 
2.12 Ω 
Stator selfinductance (L_{s1}=L_{s2} ) 
0.022 H 
Rotor selfinductance (Lr ) 
0.006 H 
Cyclic mutual inductance (Lm ) 
0.3672H 
Pole pair number (P ) 
1 
Moment of inertia (J ) 
0.0662 kg.m^{2} 
Friction coefficient (Fr ) 
0.001N.m.s/rad 
7.1 Tracking test
The first scenario is when the DSIM is started with a reference command until the speed reaches the set point (100 rad/s) under no load. The second scenario aims to test the dynamic response after the application of a load step.
Figure 7 shows that the speed follows its references perfectly for the two controllers, PI and FOPIPSO. However, better speed control precision is obtained with the proposed FOPIPSO strategy. It is observed that the FOPIPSO has superior performances in terms of the rise time and settling time, where the rise time is 0.1751 s and 0.1908 s for FOPIPSO and PI respectively, and the settling times are 0.1941s and 0.2123s.
After introducing the load (14 N.m) at t=0.7s, the proposed FOPIPSO method shows a fast disturbance rejection in a very short time with no steadystate error. On the other hand, we can observe that the classical PI controller is suffering from a speed drop and a permanent steadystate error.
Figure 8 illustrates the torque dynamic performance. It is visible that the torque ripples obtained by the proposed FOPIPSO controller are lower than those obtained by the traditional techniques based on the PI controller. The use of FOPIPSO controller contributes in mitigating fluctuations and refining the stator currents wave form.
Figure 7. Speed response of the DSIM
Figure 8. The electromagnetic torque response
Figure 9. Stator current in winding (1)
Figure 9 shows the phase stator current (I_{sa1}). It has a sinusoidal shape and less current ripples in the case of the proposed FOPIPSO controller compared to the PI controller which has a large current ripple. To obtain a good system output response, the FOPIPSO controller has good results and welltuned values. As a result of this action, the performance constraints, including rise time, settling time, steadystate error and disturbance rejection, are reduced considerably. Table 4 shows the results obtained for the different controllers. The high reduction ratios, with percentages of 8.22% and 8.57% for each of the rise time and the settling time respectively, indicates that the proposed FOPIPSO controller is more effective in improving the rise time, settling time and steady state error than the classical PI controller.
Table 4. Comparison between PI and FOPIPSO
Criteria 
PI 
FOPIPSO 
Rise time (sec) 
0.1908 
0.1751 
Settling time (sec) 
0.2123 
0.1941 
Dynamic response(sec) 
Medium 
Very fast 
Speed tracking 
Good 
Excellent 
Steady state error (%) 
>3 
Negligible 
Torque ripples reduction 
Medium 
Very well 
Minimizing the stator current ripples 
Medium 
Very well 
Stator current quality 
Acceptable 
Excellent 
Table 4 lists a comparison between performances of the proposed method and the classical PI technique. The proposed FOPIPSO improved all the performance criteria, compared to the PI controller.
7.2 Robustness test
Concerning the speed changing and load variation, this test examines the effectiveness and validity of the proposed method. The following conditions are applied to the system during simulation:
Figure 10. Speed tracking of the DSIM
Figure 11. Electromagnetic torque
Figures 1012 exhibit the results of the test, where we can observe the curves of the stator current (I_{sa1}), the speed, and the electromagnetic torque.
Figure 10 illustrates the variation of the rotor speed. For both methods PI and FOPIPSO, it can be seen that the speed absolutely complies with their references when we apply a various amount of the load.
Figure 12. Stator current in winding (1)
However, the proposed FOPIPSO technique yields greater speed control accuracy, and performs better than classical method in terms of rise time, settling time, steadystate error and disturbance rejection.
The dynamic response of the torque is shown in Figure 11. It is clear that the suggested FOPIPSO controller produces lower torque ripples than those produced by conventional PI method under speed and load variation.
The phase stator current (I_{sa1}) is depicted in Figure 12. Using FOPIPSO controller leads to a sinusoidal current shape with fewer ripples compared to the conventional PI controller, which exhibits significant current ripple when the speed and load torque are varied.
In this research paper, we have presented a DTCFOPIPSO method to enhance the system performance of the control of a DSIM.
Because it includes more parameters compared to the classic PI regulator, the proposed FOPIPSO controller has been proved more adaptable and efficient, despite that, tuning several parameters was a challenging task. For that, the PSO optimization method was employed to determine the optimal parameters of the FOPI controllers.
According to simulation results, the proposed DTCFOPIPSO method showed superior behavior in different tests when compared to classical methods and other existing works.
The benefits of the proposed DTCFOPIPSO can be summarized in the following points:
For future research, we recommend exploring the following suggestions:
s 
Stator index 
r 
Rotor index 
$\begin{aligned} & V_{d s 1}, V_{q s 1} \\ & V_{d s 2}, V_{q s 2}\end{aligned}$ 
Voltages in the dq axis for stator 1 and 2, respectively 
$\begin{aligned} & I_{d s 1}, I_{q s 1}, \\ & I_{d s 2}, I_{q s 2}\end{aligned}$ 
Currents in the dq axis for stator 1 and 2, respectively 
$I_{d r}, I_{q r}$ 
Rotor currents dq axis components 
$\Phi_{d s 1}, \Phi_{q s 1}$ 
Stator flux vectorsdq axis components 
$\Phi_{d r}, \Phi_{q r}$ 
Rotor flux vectors dq axis components 
$T_{e m}$ 
Electromagnetic torque 
$T_L$ 
Load torque 
$\Omega_r$ 
Mechanical speed 
$\Omega_{\text {ref }}$ 
Mechanical speed reference 
$\omega_r$ 
Angular speed for rotor 
$\omega_S$ 
Angular speed for stator 
P 
Number of pole pairs 
J 
Moment of Inertia 
$F_r$ 
Friction coefficient 
$R_{s 1}, R_{s 2}$ 
Stator resistances 
$R_r$ 
Rotor resistance 
$L_{s 1}, L_{s 2}$ 
Stator selfinductances 
$L_m$ 
Cyclic mutual inductance 
$L_r$ 
Rotor selfinductances 
$W$ 
Inertia weight factor 
$\mathrm{C}_1, \mathrm{C}_2$ 
The acceleration constant 
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