Speed Sensorless Control of a Doubly Fed Induction Motor Drives using MRAS Estimator

In this paper, direct vector control by rotor flux orientation for doubly fed induction motor without mechanical sensor based on the MRAS estimator is discussed, this method consists in developing two models one of reference and the other adjustable for the estimation of the two components of the rotor flux from the measurement of currents, statoric and rotor voltages respectively, the speed estimated is obtained by canceling the difference between the rotor flux of the reference model and the adjustable one, while using the theory of hyperstability to obtain the adaptive mechanism the simulation results are presented to validate the proposed method.


Introduction
The doubly fed induction machine (DFIM) is a very attractive solution for variable-speed applications such as electric vehicles and electrical energy production. Obviously, the required variable-speed domain and the desired performance depend on the application, [1], [2].
The DFIM has some distinct advantages compared to the conventional squirrel-cage machine. The DFIM can be fed and controlled stator or rotor by various possible combinations. Indeed, the input-commands are done by means of four precise degrees of control freedom relatively to the squirrel cage induction machine where its control appears quite simpler, [3], [4].
However, these advantages have long been inhibited by the complexity of the control, [5]. In order to obtain an DFIM having similar performance to a DC machine where there is a natural decoupling between the magnitude controlling the flux (the excitation current) and the magnitude related to the torque (the Armature current) [6], [7]. Several methods are used to control the induction motor among which the vector control (field oriented control) that also the decoupling between the torque and the flux, in order to obtain an independent control of torque and flux like DC motor, [8].
But, the knowledge of the rotor speed is necessary, this necessity requires additional speed sensor which adds to the cost and the complexity of the drive system. Over the past few years, ongoing research has concentrated on the elimination of the speed sensor at the machine shaft without deteriorating the dynamic performance of the drive control system. The advantages of speed sensorless induction motor drives are reduced hardware complexity and lower cost, reduces size of the drive machine, elimination of the sensor cable, better noise immunity, increased reliability and less maintenance requirements, [9].
In order to achieve good performance of sensorless vector control, different speed estimation schemes have been proposed, and a variety of speed estimators exist nowdays. Such as direct calculation method, model reference adaptive system (MRAS), Extended Kalman Filters (EKF), Extended Luenberger observer (ELO), sliding mode observer ect... Among different rotor speed estimation techniques, model reference adaptive systems schemes are the most common strategies employed due to their relative simplicity and low computational effort, [5], [9]. This paper is organized as follows. Section 2 dynamic model of a DFIM is reported; principle of field-oriented controller is given in Section 3. The proposed solution is presented in Section 4. In Section 5, results of simulation tests are reported. Finally, Section 6 draws conclusions.

Dynamic Model of Dfim
By referring to a rotating reference frame, denoted by the superscript (d, q), the dynamic model of a DFIM can be expressed by, [  The mathematical model for the mechanical parts is written as the following state equations: Where p ω Ω = j is the moment of inertia of the revolving parts, f is the coefficient of viscous friction, arising from the bearings and the air flowing over the motor, and T r is the load torque, p is the number of pole pairs.
The electromagnetic torque is expressed by:

Principle of Field Oriented Controller
The main objective of the vector control DFIM is, as in DC machines, to independently control the torque and the flux; this is done by using a d-q rotating reference frame synchronously with the rotor flux space vector. The d-axis is then aligned with the rotor flux space vector [5]. Under this condition we get: rd r φ φ = and rq 0 φ = (4) Figure 1 shows the structure for the rotor field orientation on the d-q axis

The Equations linking the flux Are
He comes then: A relation between the quadrature component of the stator flux and the current i rq can be formulated, and a relation between the electromagnetic couple and the same current is written as below

Current Control and Compensation Term
The following method was introduced by D. LECOQ. It requires the use of four current correctors, [6], [10].
In order to obtain a good decoupling between the magnitudes according to the axes d and q, it defines new transformed voltages as follows Then, the model (1) becomes: From equations of the model (12), we obtain the following control equations:

Decoupling by Compensation
The static compensation method consists in introducing terms, called equal compensation Fem, but with opposite signs so as to eliminate their influence and thus to separate the mutual actions on the two axes d and q.
The model (15) can be re-expressed in the following form , , We can define two new command variables such as: We can then represent the model (19) of the MADA by the following figure:   These transfer functions are first-order and almost identical; each is a function of the machine parameters (rotor and stator respectively) To obtain the different references of the currents to be regulated, we use the equations linking the following flows and currents: By imposing an orientation of the rotor flux and an operation at a unit power factor at the rotor we will have We obtain the following system:

Stator Current Regulation
For currents, direct and quadrature, highlight two symmetric control loops equipped with regulators defined by the parameters ( , ) The block diagram of the direct current regulation is shown in the following figure Let's impose two complex and conjugated poles with negative real parts for which the denominator of the corresponding transfer functions is of the form

Rotor Current Regulation
Recall the transfer function connecting the rotor components of each axis of the DFIM .
The same as for the stator part, the components of the rotor current (i rd , i rq ), have the same control loop given in Figure 4.
The same procedure performed for the correctors of the stator currents is applied to the correctors of the rotor currents. The parameters of the correctors are therefore the same. They are given by.

Rotor Speed Regulation
The external speed control loop will be defined by the parameters ( , ) From the equation of the mechanics governing the dynamics of rotating bodies The relationship linking the speed to the electromagnetic torque is given by The block diagram of the speed loop regulation is in Figure 5: By imposing the closed loop poles, obtain the parameters of the PI corrector:

Estimation of the Speed by the Technique of MRAS
The adaptive reference model system is based on the comparison of the outputs of the two estimators. The first, which does not introduce the magnitude to be estimated (the speed in our case), is called reference model and the second is the adjustable model. The error between these two models drives an adaptation mechanism that generates the speed by applying Popov's criterion of hyper-stability. This speed is used in the adjustable model, [11].
To ensure the convergence of the system, Schauder proposed an adaptation law that satisfies Popov's stability criterion given by the relation The structure of the mechanical sensorless control of the DFIM by rotor flux orientation according to the rotor fluxbased MRAS principle is shown in Figure 8 and the one used for the simulation.

Simulation Results and Discussion
Simulated the system for a speed reference of 150 rad / s, under the application of a load torque equal to 10 Nm between instants t1 = 1s and t2 = 2s respectively and reversal of direction of rotation to instant t = 2.5s to -150 rad / s. Got the results above ( Figure 8) With regard to the simulation results, the actual speed perfectly follows the estimated speed with a low tracking error during the transient phases and canceling out in steady state but during the transition from 150 ras / s to -150 rad / s that there is an estimation error. An excellent orientation of the rotor flux on the direct axis is also observed. This affects the electromagnetic torque. During the changes of the setpoints, and in particular during the inversion of rotation, the change of direction of the torque does not degrade the orientation of the flow. Likewise for the estimated components of the rotor flux that are little influenced by this inversion of speed.
Good sensitivity to load disturbances is observed, with a relatively low rejection time.

Influence of parametric variations
In order to study the influence of parametric variations on the behavior of vector control without speed sensor based on the MRAS technique, we introduced a variation of + 50% of Rr in the first test, then a variation of + 50% of Rs. That obtained the results as shown in Figure 9 and 10, respectively: It should be noted that during the variation of the rotor resistance Rr all the magnitudes of the machine present a small disturbance (Figure 9).
For a nominal value of Rr, the stator resistance is increased by + 50% of its nominal value, the results of the figure 10 are obtained. These results prove that the variation of Rs further deteriorates the accuracy of the estimate, especially in the low-speed zone "-150 rad / s", will lead to a significant estimation error plus the instability of the estimator. the high speed zone the estimation error remains low. It conclude that this technique is very sensitive to variations in machine parameters.

Conclusion
In this article, we have exploited MRAS technique for DFIM speed estimation.
A numerical simulation in the Matlab-Simulink environment was performed to validate these performances.
From the simulation results obtained, it can be concluded that this proposed estimation technique is valid for the nominal conditions. On the other hand, the studied estimator has a good robustness with respect to the variation of the load and the pursuit, makes it possible to reach good functional performances with a low cost and low volume installation.
By contrast, the MRAS technique is not robust to