MIAO Zhongcui,LI Dongliang,WANG Zhihao,YU Xianfei,ZHANG Wenbin

(1. School of Automation and Electrical Engineering,Lanzhou Jiaotong University,Lanzhou 730070，China； 2. Key Laboratory of Opto-Technology and Intelligent Control of Ministry of Education， Lanzhou Jiaotong University，Lanzhou 730070，China)

Abstract：Considering the actual demand for high-speed operation of induction motors in industrial occasions,the characteristics of induction motors in different regions are analyzed,especially the field weakening characteristics of induction motors in high-speed operation are studied.A field weakening control method of induction motor based on model predictive control (MPC)algorithm is proposed,which can predict the future state of the controlled object,and then obtain the optimal control variables by colling optimization.The simulation results show that the field-weakening control method based on MPC algorithm has faster response speed,stronger robustness and better control performance than the traditional control methods.

Key words：induction motor;model predictive control (MPC);field-weakening control;rolling optimization

0 Introduction

Induction motors have been widely used in transportation,industrial and mining transportation because of their simple structure,reliable operation,good traction performance,low price and easy maintenance.To improve the control performance and operating efficiency of induction motors[1],in the industrial situation,it is required that the induction motors should have a wide speed range,or it can operate stably above the rated speed (i.e.in the field weakening state).When the induction motor speeds up,the voltage demand at the motor end increases accordingly.When the speed is higher than the rated speed,the motor end voltage remains at the inverter DC-side voltage allowable output .At this time,by reducing the flux,the motor speed can continue to rise,as a result,the system obtains a wide speed range.When the induction motor is running in a field weakening state,the torque output and power output capability of the motor can be improved as much as possible under the conditions of voltage and current constraints,and the control system of the induction motor can maintain good dynamic and static performance[1-3].

The traditional field weakening control uses a method in which the flux linkage is inversely proportional to the speed.This method is the simplest,but it does not consider the influence of DC bus voltage fluctuations in the field weakening state,so it is difficult to achieve the maximum torque output[4].In practical applications,the flux linkage is controlled by a look-up table method.Through repeated testing of the induction motor,the optimal excitation current and torque current components are obtained at high speeds.This method is sensitive to motor parameters and it is complicated to obtain the table[5-6].The field weakening control method based on voltage trajectory proposed in Ref.[7] does not need to look up the table and know the parameters of the DC bus voltage,whereas it needs the rated excitation current parameters.In Ref.[8],the 6-step voltage overmodulation method is used for field weakening control,which can maximize the use of bus voltage,but will generate large current harmonics that cannot be suppressed.Ref.[9] presents a field weakening control strategy with iron loss compensation mechanism that can ensure maximum torque in the field weakening area,but has a large amount of calculation and a slow response speed.In recent years,more and more attention has been paid to the voltage closed-loop field weakening control method based on vector control.This method uses the deviation between the voltage reference value and its maximum limit value to determine thed-axis current,but its calculation process is complex and its response speed is slow.How to improve the control performance of the field weakening control still needs to be studied[10].

Model predictive control (MPC)uses the real-time state and dynamic model of the controlled system to predict the operating state of the system after the next cycle or cycles,and then calculates the current control amount of the system according to the existing performance index function in each cycle,which can meet the requirements of fast response and stable control of the control system[11].MPC has better robustness.It can solve the optimization problem by the deviation between the current predictive value and the given value in the next few cycles,so as to obtain the current optimal control quantity of the control system[12].

Based on the above analysis,this study takes the induction motor as the research object.Firstly,a mathematical model of the induction motor is established to analyze the characteristics of the motor in different areas in detail.Then MPC algorithm is adopted to achieve high-performance control of induction motors.In order to increase the speed range of the system,the system determines whether the system enters field weakening operation based on the speed,and then uses the relationship among speed,flux linkage and torque to design the field weakening controller.Finally,its effectiveness is verified through simulation.

1 Mathematical models

1.1 Dynamic mathematical model of induction motor

Under ideal conditions,the state equation of an induction motor in two-phase stationary state can be expressed as[13]

(1)

where

isandψsare the stator current and stator flux of the induction motor,respectively;RsandLsare the motor stator resistance and inductance,respectively;RrandLrare the motor rotor resistance and inductance,respectively;Lmis mutual inductance;ωris the motor speed;anduis the voltage vector matrix.

The torque equation of an induction motor is

(2)

wherenpis the number of pole pairs of the induction motor and ? is.

1.2 Inverter model

In this study,a three-phase two-level inverter is used to drive the induction motor.The three-phase two-level inverter has eight working states,corresponding to eight basic space voltage vectors,as shown in Fig.1,whereu0andu7are two zero-voltage vectors,andu1tou6are six effective working vectors[14].

Fig.1 Vector diagram of basic voltage space

The corresponding relationship between the switching state of the inverter and the phase voltage is shown in Table 1.

Table 1 Corresponding relationship between inverter switch state and phase voltage

According to Table 1 and Eq.(3),the voltage vector valueusunder different switch states can be obtained by

(3)

2 Field weakening control theory of induction motor

The operation section of induction motor is divided into three parts:constant torque area,constant power area and constant voltage area,as shown in Fig.2[15].

Fig.2 Operation areas of induction motor

The maximum voltage and current that the induction motor bears when operating in the weak magnetic state are

(4)

Combining Eqs.(1)and (4),we can get

(5)

Then it can be obtained from Eq.(5)as

(6)

Eq.(6)gives the current limit circle and voltage limit ellipse when the motor is running.Taking the stator current componentsisαandisβas the coordinate axes,the voltage and current limit curves are shown in Fig.3,where pointain the figure is the maximum torque point of the motor.

Fig.3 Voltage and current limit curve

At present,the most widely used flux weakening control method is voltage closed-loop flux weakening control based on vector control,and its structure is shown in Fig.4.

Fig.4 Block diagram of voltage closed-loop flux weakening control based on vector control

The basic principle of this algorithm is to compare the stator voltage of induction motor with the maximum limit voltage.If the stator voltage is higher than the maximum limit voltage,the induction motor enters the weak magnetic operation state.At this time,the excitation component of the stator current should be reduced so as the stator voltage of induction motor can be reduced until the maximum limit voltage is met.

3 Weak magnetic control based on MPC

3.1 MPC

MPC system is mainly composed of three parts:predictive model,rolling optimization and feedback correction.Its structure is shown in Fig.5[16].

Fig.5 Structure diagram of MPC

The basic principle of model prediction is to predict the follow-up state quantity based the state quantity and control quantity of the system,and then carry out repeated online optimization and feedback correction according to the given constraints and rolling optimization strategy,so as to obtain the optimal control quantity that can achieve the performance index of the control system and implement this control quantity[17].

Rolling optimization is a major feature of MPC.It means that the control quantity is obtained by repeated online optimization from the perspective of a certain performance index optimization.The rolling optimization is different from the global optimization in traditional control,Instead,there is an optimized performance index corresponding to that moment at different times.The relative forms of the performance indexes at different times are the same,but the corresponding specific time intervals are different.The time interval is scrolled forward as the time advances.The feedback correction link forms a closed-loop mechanism for MPC.At each sampling moment,the real-time state of the object is first detected.Before optimization of the optimal control quantity,the feedback information is used to build the next prediction and optimization on a more realistic basis by refreshing or correcting[18].

3.2 Weak magnetic control based on MPC

The voltage closed-loop flux weakening control system based on vector control shown in Fig.4 is to reduce the flux of the induction motor by controlling the current after judging that the induction motor has entered the state of flux weakening operation,so that the speed of the induction motor can continue to rise in case of the stable operation of the motor.The steps of the flux weakening control system of induction motor based on the MPC is as follows:Taking the rated speed as the reference value,if the motor speed is less than the rated speed,judging that the motor does not enter the flux weakening operation,at this time,the given flux of the system is a constant value;If the motor speed is greater than the rated speed,judging that the motor enters the flux weakening operation,at this time,the flux linkage and torque of the induction motor are controlled by the flux weakening control method proposed,so that the speed of the induction motor can continue to rise to meet the expected requirements.The structural block diagram is shown in Fig.6.

Fig.6 Block diagram of weak magnetic control

Based on the above analysis,comprehensive considering the requirements of control accuracy and calculation,the first-order Euler discrete method is adopted to discretize the flux and torque equations of induction motor.

The first-order Euler discrete method is used to discretize Eqs.(1)and (2),and then we obtain the corresponding prediction model of the induction motor as

(7)

In the rolling optimization module,the control objective is to minimize the sum of the absolute errors of the flux and the torque.The objective function of rolling optimization is defined as

(8)

In order to meet the operating conditions of the motor during the high-speed operation,it is necessary to reduce the magnetic flux.After improvement on the traditional field weakening control method,both flux and torque have better real-time performance in the full speed range.Induction motor flux is given by

(9)

While adjusting the flux,the electromagnetic torque is also adjusted accordingly.The electromagnetic torque is obtained by

(10)

Because the method of setting the electromagnetic torque changes,the maximum value of the electromagnetic torque is also limited accordingly,namely

(11)

Combining Eqs.(9)and (10),the electromagnetic torque of induction motor is given by

(12)

4 Simulation

In order to verify the effectiveness and advancement of the proposed field-weakening control system,a simulation model is built on the Matlab/Simulink based on Fig.6.The relevant parameters of the induction motor used in the test are shown in Table 2.Udis 511 V;the sampling period of the system is 10 μs;kψis 95;and the parameters of the PI speed regulator in the system arekp=30 andki=0.025.

Table 2 Relevant parameters of induction motor

The rated speed of the induction motor is 1 000 r·min-1,and the speed of the motor reaches 2 000 r·min-1through the weak magnetic control.The field-weakening control system based on model prediction is system 1,and the control system based on vector control is system 2.

4.1 Transition from non weak magnetic region to weak magnetic region

When the motor runs at the rated speed and reaches 0.5 s,the speed is increased to 2 000 r·min-1,and the speed,flux,torque and current response are shown in Figs.7-10.

It can be seen from Fig.7 that the time for the system speed to reach 1 000 r·min-1is 0.24 s,and the time to reach 2 000 r·min-1is 0.98 s.The time for system 2 to reach 1 000 r·min-1is 0.37 s,and the time to reach 2 000 r·min-1is 1.03 s.It can be seen that system 1 has a faster response speed in the full speed range.

Fig.7 Speed response

Fig.8 is the stator flux response curves of system 1 and system 2.It can be seen that the flux response of system 1 is relatively stable in the whole process,and the transition from non-magnetic field weakening area to weak magnetic area is obviously more stable and smooth,while the flux response of system 2 fluctuates greatly.

(a)Stator flux response of system 1

Fig.9 shows the stator current response curves of system 1 and system 2.It can be seen that the stator current response of system 1 is more stable and smooth,while the stator current response of system 2 fluctuates greatly.Especially when the system transits from non-magnetic field weakening area to weak magnetic area,the current fluctuation of system 2 is obviously more severe.

(a)Current response of system 1

The performance of the stator flux and stator current directly determines the stability of output torque.Fig.10 shows the torque output response curves of system 1 and system 2.It can be seen that after the system reaches steady state,the torque ripple of System 1 is ±0.5 N·m,while that of system 2 is ±1.5 N·m.It is obvious that system 1 effectively reduces the torque ripple of the system,making the torque output more stable.

(a)Torque response of system 1

It can be concluded from experiment 1 that system 1 has better stability and faster response speed during the transition from the non-magnetic field weakening area to the magnetic field weakening area.

4.2 Speed change in field weakening area

After the motor is started,at the rated speed,it runs to 0.6 s,increasing the speed to 1 500 r·min-1;then runs stably to 1 s,increasing the speed to 2 000 r·min-1.The system speed and flux response are shown in Figs.11-12,respectively.

(a)Stator flux response of system 1

It can be seen from Fig.11 that in the two transition stages of speed from 1 000 r·min-1to 1 500 r·min-1and then from 1 500 r·min-1to 2 000 r·min-1in the magnetic field weakening area,the time of system 1 is 0.75 s and 1.3 s,respectively,while those of system 2 are 0.88 s and 1.32 s,respectively.It can be seen that system 1 has a faster response speed in the magnetic field weakening area.

Fig.11 Speed response

Fig.12 shows the stator flux response of system 1 and system 2 when the rotating speed is from 1 000-1 500 r·min-1and then from 1 500 r·min-1to 2 000 r·min-1.The results show that the flux of system 1 is smoother and more stable in the process of transition,and there is an obvious three-stage flux,while the flux response of system 2 fluctuates greatly but there is no obvious three-stage flux.

From experiment 2,it can be concluded that system 1 has better stability and control performance when the speed changes in the field weakening area.

5 Conclusions

As for the field weakening control strategy for induction motor,the disadvantages of the traditional field weakening control strategy are analyzed,response speed and large torque ripple.Based on this,such as slow a field weakening control system of induction motor based on model prediction is proposed.The corresponding simulation established by Matlab/Simulink and comparison with the traditional voltage closed-loop field weakening control system show that the control method proposed has the advantages of fast response speed,small torque ripple,etc.,therefore it has good control effect.