A Cascade PI-SMC Method for Brushless Doubly-Fed Induction Machine with Matrix Converter

This paper proposes a cascade proportional-integral continuous second-order sliding mode control method for brushless doubly-fed induction machine. This method consists of the speed out-loop proportional-integral controller and the current inner-loop sliding mode control. Proportional-integral controller provides references to the inner-loop sliding mode control with constraints according to the system requirements in terms of maximum current and speed limits. Sliding mode control method is designed to achieve excellent robustness and anti-interference ability, which is suitable for machines with nonlinear structures. Experimental results demonstrate the fast-dynamic performance and excellent robust tracking of the proposed method.


I. INTRODUCTION
To obtain high dynamic performance, many available studies regarding speed and torque have been proposed for the brushless doubly-fed induction machine (BDFIM) configuration [1][2][3]. In most of the literature, linear controllers are used, such as proportional integral (PI) [1,2] or model predictive control (MPC) [3]. In these methods, the speed controller typically covers the second-order dynamics of speed and current. However, the current inner loop of the BDFIM has a complex nonlinear structure and is easily interfered by various disturbances. Since SMC has the characteristics of easy decoupling, disturbance rejection, insensitive to parameter variations, this method is often applied to control AC drives. However, due to the inherent characteristics of SMC, it does not include solutions with constraints. It is an urgent problem to impose restrictions on SMC and use it in BDFIM [4,5]. This paper presents the application of the design of cascade PIsecond-order SMC for the BDFIM. In this method, the innerloop SMC method controls CW current and the out-loop PI controller track the reference rotor speed. Since the matrix converter possesses the advantages of bi-directional power flow with full four-quadrant operation, sinusoidal input and output currents, controllable input power factor, high power density and no DC-link energy storage elements [6,7], it is selected as the AC driver of BDFIM. The main advantages of this method are that the outer loop control provides constraints on the SMC through maximum current and speed limits and the super-twisting-based second-order SMC achieves fast and fixed convergence times.

II. MATHEMATICAL MODEL OF MATRIX CONVERTER AND BDFIM
In this paper, the modulation method based on mathematical construction in [8] is applied. This method uses a mathematical construction method to directly obtain the switching duty cycle of matrix converter, which can make the voltage utilization ratio of the matrix converter reach 0.866.
The mathematical model of the BDFIM is shown in (1):

III. SLIDING MODE CONTROL OF BDFIM WITH MATRIX CONVERTER
The sliding mode control diagram of BDFIM with matrix converter is shown in Fig.1. The control method adopts a double closed-loop control structure. Among them, the outer loops adopt the traditional PI control strategy to realize the tracking of the reactive power of the power winding and the reference value of the rotor speed. The output of the outer loop controller is the reference of the stator current of the inner loop control side. In order to improve dynamic response of the stator current on the control side and the immunity of the system, a robust second-order sliding mode control is introduced as the current loop and obtain the desired reference voltage and duty cycle. ( ) (0) t X t X e λ = (4) The control law of the sliding mode controller is selected as shown in (5)  the system can be set as a first-order system only when the proportional term exists, but this will reduce the system's immunity. Therefore, in order to solve this problem, this paper introduces the active damping link in the traditional PI controller, as shown in (8) IV. EXPERIMENTAL VERIFICATION In order to verify the correctness of the cascade PI-SMC method for BDFIM with MC, the experimental prototype is set up. Fig. 2(a) shows the waveforms of the stator voltage, stator current and grid voltage on the control side of the BDFIM at a given speed of 400r/min. According to Fig. 2(a), the sliding mode control is also used when the error is bounded. The matrix converter output current can be guaranteed to be sinusoidal, achieving similar control effects as conventional vector control.   Fig. 2(b) shows the dynamic performance comparison between the proposed sliding mode control and the traditional vector control when the given speed is given by the ramp and the speed is changed from 200r/min to 400r/min. As shown in Fig. 2(b), when the speed changes suddenly, sliding mode control and vector control can track the given speed very well and it also has good dynamic performance.

V. CONCLUSION
In this paper, the cascade PI-SMC method for BDFIM with MC is proposed. The outer loop adopts the traditional PI controller to follow the reference value of the rotational speed and reactive power. The inner loop uses a second-order sliding mode controller to control the CW current of the brushless doubly-fed motor and obtain the desired output voltage. Finally, the desired output voltage is modulated by a mathematical construction modulation method to obtain the switching duty cycle of the matrix converter. The experiment verifies the correctness and effectiveness of the proposed method.