Performance Analysis of Adaptive Fuzzy Sliding Mode for Nonlinear Control of the Doubly Fed Induction Motor

Received Jul 30, 2018 Revised Des 4, 2018 Accepted Dec 18, 2018 In this article, we propose a contribution to the control of a doubly fed induction motor by sliding mode with adaptive fuzzy logic. The technique of vector-control by classical field oriented applied to the doubly fed induction motor (DFIM) with mechanical sensors made it possible to have performances comparable with that of the direct current motor. However, it very sensitive to the parametric variations of the machine. The regulation speed by a classical regulator (PI) presents disadvantages: Poor robustness against parametric uncertainties of modeling and no the considering of the disturbances and little degree of freedom for the regulation. Because this effect, several robust controls were proposed in the technical literature to ensure the decoupling of the currents of the DFIM in a reference (d, q) leading to calculate simplified correctors. Among them, the variable structure control by sliding mode. It uses algorithms of regulations which ensure the robustness of the behavior of the process compared to the parametric variations and disturbances. Also, the impact of regulators based on artificial intelligence techniques such as adaptive fuzzy sliding mode controller are studied. In terms of results obtained, good dynamic performance and robustness with respect to load disturbances and parametric variation has been observed. Keyword:


INTRODUCTION
In the field of high-power drives (such as the rolling mill for example), there is a new and original solution, using an alternative machine operating in a mode. This is the doubly fed induction motor DFIM, [1], [2] where the stator is fed by a fixed network and the rotor by variable power supply which can be a voltage source or a power source, [3].
DFIM finds its application in high power drives, it is characterized by its robustness, its longevity and a wider range of speed variation (hypo-synchronous, synchronous and hyper-synchronous regime) However, the asynchronous machine has a major disadvantage: the structure dynamic is strongly non-linear and the existence of a strong coupling between the torque and the flux, which complicates its control [4], [5].
In order to obtain a dual power asynchronous machine whose performances are like a DC machine, it is necessary to ensure the decoupling between the flux and the electromagnetic torque. This is the idea of the appearance of the vector control technique, or the control by flux orientation, [6]. This technique was proposed in 1973 by Blaschke and Hasse. The purpose of this technique is to manage the asynchronous machine as an independent excitation DC machine where there is a natural decoupling between the magnitude controlling the flux (the excitation current) and that related to the torque (the armature current) [7], [8]. Conventional PI or PID control techniques cover a wide range in industrial applications. These are linear control techniques and have the advantage of simplicity of implementation and ease of synthesis. Over time, its applications will be inefficient, especially if the processes to be controlled have complex and nonlinear structures. Finding the substitute for these classic techniques is the concern of many researchers. Because the latter involves a compromise between the robustness of one side and the cost of another side [9], [10].
Sliding mode control, which was developed in the Soviet Union there are more than 30 years old, is a nonlinear control technique. It uses setting algorithms which ensure the robustness of the behavior of the process in relation to the variations parametric and disturbances. It has several advantages such as: Robustness, speed of response without exceeding and good efficiency at parametric and modeling uncertainties. However residual vibrations at high frequencies inherent or chattering to order remain considerable inconvenience, [11]- [13].
In order to eliminate the chattering [14], as it can damage the actuators by oscillations too frequent and affect the operation and performance of the system, it is proposed the application of fuzzy logic in sliding mode, [15], [16] for adjusting the speed of the DFIM. This paper will be organized as follows. Section II dynamic model of DFIM and stator flux orientation will be reported; principle of Sliding mode control will be given in Section III, as well as the proposed solution will be presented. In Section IV, results of simulation tests are reported. Concluding remarks will be given in section V.

DOUBLY FED INDUCTION MACHINE MODEL
By referring to a rotating reference frame, denoted by the superscript (d, q), the dynamic model of a DFIM can be expressed by, [1]: where: To simplify the control, it is necessary to make a judicious choice of reference. For this, one places oneself in a reference frame (d-q) linked to the rotating field with an orientation of the stator flux (the axis d aligned with the direction of the stator flux), as the following Figure 1 shows, [17]. The electromagnetic torque is expressed by: Replacing (2) in (3) and (5) we find: after arrangement we will have the following system: The position stator flux is calculated by the following equations: . s  is the electrical stator position,  is the electrical rotor position.

PRINCIPLE OF SLIDING MODE CONTROL
A Sliding Mode Controller (SMC) is a Variable Structure Controller (VSC). Basically, a VSC includes several different continuous functions that can map plant state to a control surface. As a result, the sliding mode control is based on three steps [18].

Choice of the switching surface
J. Slotine proposes a form of general equation to determine the sliding surface ) (

The condition of convergence
The condition of convergence is defined by the equation of Lyapunov.

Control Calculation
The control algorithm includes two terms, the first for the exact linearization, the second discontinuous one for the system stability.
eq u : corresponds to the equivalent order suggested by Utkin (1992). It is calculated starting from the expression: n u is given to guarantee the attractivity of the variable to be controlled towards the commutation surface. The simplest equation is the form of relay.
k is the controller gain. The speed error is defined by: (18) for n =1, the speed control manifold equation can be obtained by:

Speed control
substituting the expression of   in (10) and in (20), we obtain: we take: during the sliding mode and in permanent regime, we have: where the correction factor is given by: In our proposal which allows the combination between the fuzzy logic and the control by sliding mode, we call the controller resulting from this combination: controller by fuzzy sliding mode (FSMC), it has the same control law as the SMC set to apart from the parameters k and  the component n u that will be adapted by a system to a fuzzy inference.
The key idea of this combination is inspired by the fact that in the ideal case and when ( ) is far from the sliding surface, parameters k and  must have a large enough value and when ( ) is near of the sliding surface, adjusted parameters k and  must have a small value. In this voice one can propose an adaptation of these parameters by means of a system with a fuzzy inference while keeping the same laws of control of the controller SMC.
Substituting the expression of rd i  in (10) and in (30), we obtain: therefore, the correction factor is given by: where rd V K :positive constant 3.6. Rotor quadrature current control and limitation In order to limit all possible overshoot of the current rd i , we add a limiter of current defined by: the direct current error is defined by: for n =1, the direct current control manifold equation can be obtained by: Substituting the expression of rq i  equation (10) in equation (39), we obtain: during the sliding mode and in permanent regime, we have: where the equivalent control is:

Stator flux control
In the proposed control, the manifold equation can be obtained by: Substituting the expression of sd  equation (10) in equation (45), we obtain: The equivalent control is: where the correction factor is given by: )) ( (

RESULTS AND ANALYSIS
To evaluate the performance of the proposed method, the model for the control law applied to the DFIM Figure 5 has been implemented in the Matlab / Simulink software.
The simulation results illustrate the dynamic responses of the rotor speed, the electromagnetic torque, the components of the stator fluxes and the rotor current.
The motor is operated at Ωref=150 rad/s under no load and a load disturbance torque (10 N.m) is suddenly applied at t=1s and eliminated at t=2s (-10 N.m), To show the effect of the parameters uncertainties, we have simulated the system with +50% of value of the rotor resistance and compared to nominal value. This Figure 6 shows the dynamic response of the speed. The latter reaches its reference in a response time of 0.13 s. The introduction of charge has no influence on the evolution (stability) of the speed, which shows the robustness of the proposed control against these disturbances. It is also noted that the speed follows its setpoint rapidly during the reverse direction of rotation (identical to the response time at startup).
It can also be seen that increasing the rotor resistance Rr by + 50% of its nominal value does not cause any undesirable effects. During the rotation inversion the change of the direction of the torque does not degrade the orientation of the flux so the introduction of charge has no influence, and it is clearly seen that the decoupling is not influenced by the variation of Rr.  Figure 8 illustrates the dynamic response of the electromagnetic torque for two rotor resistance values (Rrn and + 50% of Rr). Given this result, we note that this command gives good performance. It is noted that the regulation of the two rotor currents by sliding mode Figure 9 gives faster responses during a vacuum start and more robust during the variation of the load and the rotor resistance. From the results obtained, all the tests applied to this control plus the variation of the rotor resistance does not cause any undesirable effect on all the dynamic responses, and this shows the robustness of the nonlinear control applied to the DFIM.
Finally, and after this detailed study, we note a significant improvement in the static and dynamic performance of the DFIM, which is proving the efficiency and robustness of the developed Adaptive Fuzzy Sliding Mode controller and the nonlinear control chosen compared to the classic PI controller developed in [19], [20]

CONCLUSION
This paper was the subject of the application of the sliding mode controller in the following control loops: stator flux and rotor currents on the Doubly Fed Induction Motor (DFIM) and it is applied Adaptive Fuzzy Sliding Mode controller only in the speed loop. in this context, we first presented a theoretical reminder on the sliding mode control of variable structure systems, then we approached the adaptive fuzzy control algorithm with its different steps, then applied it on the regulation of DFIM, and finally simulation results are presented and commented.
It can be concluded from these results that, the field-oriented control using a non-linear Fuzzysliding mode is more robust than that of conventional regulator-controlled vector control (IP or PI). The comparative study in the test results by speed reference variation and rotor resistance variation shows that the robustness quality of the proposed controller.