# Vector Control Algorithm for Electric Vehicle AC Induction Motor Based on Improved Variable Gain PID Controller

Authors: Gang Qin; Mushuang Liu; Jianxiao Zou; Xiaoshuai Xin
Journal: Mathematical Problems in Engineering
 (2015)

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## Abstract

The acceleration performance of EV, which affects a lot of performances of EV such as start-up, overtaking, driving safety, and ride comfort, has become increasingly popular in recent researches. An improved variable gain PID control algorithm to improve the acceleration performance is proposed in this paper. The results of simulation with Matlab/Simulink demonstrate the effectiveness of the proposed algorithm through the control performance of motor velocity, motor torque, and three-phase current of motor. Moreover, it is investigated that the proposed controller is valid by comparison with the other PID controllers. Furthermore, the AC induction motor experiment set is constructed to verify the effect of proposed controller.


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## Body

## 1. Introduction

With the increased emphasis on saving energy and reducing emission, electric vehicles (EVs) have emerged as very strong candidates to achieve these goals [1–5]. Moreover, the acceleration performance of EV, which affects a lot of performances of EV such as start ability, passing ability, driving safety, and ride comfort, is the key point of EV researches.Vector control algorithm, which can accurately control the torque and has a wide control range of motor velocity and also has a current loop which can be used for current limiting protection, is widely used in EV driving control. However, the velocity loop controller of vector control algorithm, which uses traditional PID control algorithm generally, limits the dynamic performance of driving system and the acceleration performance of EV. During the last few years, the velocity loop controller of EV AC induction motor (ACIM) controller system is researched and improved unceasingly and many methods are presented. One method is using the fuzzy controller to replace velocity loop traditional PID controller and current loop traditional PID controller of vector control algorithm [6–8], which can make the control system track the different given velocity rapidly and without overshoot in different load and has strong ability against load disturbance, but its steady-state accuracy is not high because of no existing integration element. Another method is using the neutral network PID controller to replace velocity loop traditional PID controller of vector control algorithm [9–11], which has the advantages of adjusting velocity rapidly, zero overshoot, smooth and small-fluctuation control signals, and good system tracking, but it reduces the EV control performance due to learning slowly in learning process and long response time. Literature [12] also presented a method using the fuzzy-PI controller which executes fuzzy control algorithm when velocity deviation is greater than given threshold and executes traditional PID control algorithm when velocity deviation is less instead of velocity loop traditional PID controller [13]. The method can make velocity response rapidly with small overshoot [14], but it is difficult to achieve completely smooth switching and may cause velocity hop when control algorithm switches, thereby affecting the driving safety and ride comfort when EV accelerates.In this paper, we design a vector control algorithm for vehicle asynchronous motor based on improved variable gain PID controller which can make motor velocity rise rapidly and no overshoot. Moreover, it can satisfy the demands of EV driving system dynamic performance and acceleration performance no matter whether the EV runs in low velocity, normal velocity, high velocity, or variable velocity.The sections are organized as follows. In Section2, asynchronous motor model is studied. In Section 3, improved variable gain PID control algorithm is designed to improve the acceleration performance. In Section 4, vector control algorithm for EV asynchronous motor based on improved variable gain PID controller is proposed. Furthermore the stability condition is given. In Section 5, the effectiveness of controller is demonstrated by simulation with Matlab/Simulink (Figure 4). Section 6 presents some concluding remarks.


## 2. AC Induction Motor Model

ACIM is widely applied to EV driving system, which has many good characteristics such as robustness, durability, simple structure, reliable operation, low cost, low torque ripple, low noise, no position sensor, and high velocity limit. The design of ACIM for EV which is different from normal ACIM and must satisfy the power performance of EV must have the following characteristics: (1) constant power output and big velocity adjustable range for satisfying the demands for flat road, overtaking, and so on when run in high/low velocity,(2) smaller mass and volume in the condition of certain power level, and (3) robust structure and resistance to vibrations [15].This paper which takes ACIM for example researches motor mathematical model of EV driving system. The mathematical model of ACIM is a nonlinear, high order, close coupling multivariable system. Ignore these factors such as core loss, space harmonics, the change of frequency, the change of temperature, and the saturation of magnetic circuit on the impact of winding resistances when establishing motor model [16].A physical model of ACIM is shown in Figure1. The three-phase winding resistances which are 120° phase different in the space are symmetrical, and the mutual inductance and self-inductance of every winding resistance are constant. The mathematical models of ACIM, which consist of voltage matrix equation, magnetic linkage matrix equation, and torque equation, can be obtained according to the physical model of ACIM.Figure 1
The physical model of ACIM.Based on the voltage balance principle of three-phase stator winding resistances, the voltage matrix equation [17] can be written as(1)uAuBuCuaubuc=Rs000000Rs000000Rs000000Rr000000Rr000000RriAiBiCiaibic+pψAψBψCψaψbψc,where uA, uB, and uC are stator phase voltage, ua, ub, and uc are rotor phase voltage, iA, iB, and iC are stator phase current, ia, ib, and ic are rotor phase current, ψA, ψB, and ψC are magnetic linkage of stator winding resistance, ψa, ψb, and ψc are magnetic linkage of rotor winding resistance, Rs is stator resistance, Rr is rotor resistance, and p is differential operator.Based on the principle that the magnetic linkage of every winding resistance is equal to its self-inductance magnetic linkage plus mutual inductance magnetic linkage with other winding resistances, the magnetic linkage matrix equation can be written as(2)ψAψBψCψaψbψc=LAALABLACLAaLAbLAcLBALBBLBCLBaLBbLBcLCALCBLCCLCaLCbLCcLaALaBLaCLaaLabLacLbALbBLbCLbaLbbLbcLcALcBLcCLcaLcbLcciAiBiCiaibic,where LAA is self-inductance and LAB is mutual inductance.The torque equation can be expressed as(3)Te=npLmiAia+iBib+iCicsinθiiiiiiiii+iAib+iBic+iCiasinθ+120°iiiiiiiii+iAic+iBia+iCibsinθ-120°,where np is pole pairs, Lm is mutual inductance, and θ is electrical degree difference between a axis and A axis.The ACIM, whose mathematical model is very complex, is very difficult to be controlled in practical application. The vector control algorithm controls ACIM as DC motor through coordinate transformation for the problem that the mathematical model ACIM is very complex so that the governing performance of ACIM can be comparable with DC motor.


## 3. Improved Variable Gain PID Control Algorithm Design

The fundamental thought of traditional variable gain PID control algorithm is matching the cumulative velocity of integral value with the magnitude of deviation. The integral action reduces to nothing for preventing integral saturation when system deviation is large and is reinforced for improving the stability of velocity when system deviation is small. The more desirable situation is matching the magnitude of proportional coefficient with deviation. The action of proportional part is reinforced for improving the dynamic performance of system when system deviation is large and reduces for preventing overshoot when system deviation is small. This paper designs an improved variable gain PID control algorithm based on improving the variable gain PID control algorithm.The proportional and integral term of improved variable gain PID control algorithm can be expressed as(4)uk=kp+xekek+ki∑i=0k-1ei+yekekT,where x[e(k)] and y[e(k)] are the functions of deviation e(k). As e(k) increases, x[e(k)] increases and y[e(k)] reduces. As e(k) reduces, x[e(k)] reduces and y[e(k)] increases.The expression ofx[e(k)] can be described as(5)xek=kp1′,ek≤e2,kp2′-kp1′e1ek-e2+kp1′,e2<ek≤e1+e2,kp2′,ek>e1+e2,where parameters e1, e2, kp1′, and kp2′ are necessary to be ensured, 0≤kp1′<kp2′. On one hand, the chosen values of these four parameters must satisfy the condition of system stability. On the other hand, the chosen values of e1 and kp2′ must meet the condition of velocity, and the chosen values of e2 and kp1′ must meet the condition of no velocity overshoot.The value ofx[e(k)] varies in the range of interval [kp1′,kp2′].When|e(k)|>e1+e2, x[e(k)] is equal to kp2′ and the proportional coefficient of control algorithm is equal to kp+kp2′, for improving the dynamic performance of system.When|e(k)|≤e2, x[e(k)] is equal to kp1′ and the proportional coefficient of control algorithm is equal to the minimum value kp+kp1′, for preventing overshoot.Whene2<|e(k)|≤e1+e2, the value of x[e(k)] which is in the range of interval [kp1′,kp2′] and the proportional coefficient of control algorithm which is in the range of interval [kp+kp1′,kp+kp2′] vary with the magnitude of |e(k)|.The expression ofx[e(k)] can be described as (6)yek=1,ek≤e4,ki′-1e3ek-e4+1,e4<ek≤e3+e4,ki′,ek>e3+e4,where parameters e3, e4, and ki′ are necessary to be ensured, 0≤ki′<1. On one hand, the chosen values of these three parameters must satisfy the condition of system stability. On the other hand, the chosen values of e3 and ki′ must meet the condition of no integral saturation and velocity overshoot, and the chosen value of e4 must make velocity stability rapidly.The value ofy[e(k)] varies in the range of interval [ki′,1].When|e(k)|>e3+e4, the value of y[e(k)] is equal to ki′, for reducing the integral action to the lowest or not accumulating the current value of e(k).When|e(k)|≤e4, the integral term is the same as the general, for increasing the integral action to the highest and accumulating the current value of e(k).Whene4<|e(k)|≤e3+e4, the value of y[e(k)], which is in the range of interval [ki′,1], varies with the magnitude of |e(k)| and the integral term accumulates part current value of e(k). Thus, the integral velocity is in the range of ki{∑i=0k-1e(i)+ki′e(k)}T to ki∑i=0ke(i)T.In order to increase the regulating range of improved variable gain PID control algorithm, the values of parameterse1, e2, e3, and e4 must be decided by the maximum value of deviation after desired value varies which are not fixed values. Thus, we can get(7)e1=emaxm1,e2=emaxm2,e3=emaxm3,e4=emaxm4,where emax is the maximum value of deviation between desired value and feedback value after the desired value of controller input changes and parameters m1, m2, m3, and m4 are necessary to be ensured, which must satisfy 0<mi<1, i=1,2,3,4, 0<m1+m2<1, and 0<m3+m4<1. First of all, the chosen values of these four parameters must satisfy the condition of system stability. Secondly, the chosen value of m1 must make velocity stability rapidly, and the chosen value of m2 must meet the condition of no velocity overshoot, and the chosen value of m3 must meet the condition of no integral saturation and velocity overshoot, and the chosen value of m4 must make velocity stability rapidly.Finally, the improved variable gain PID control algorithm is obtained as(8)uk=kp+xekek+ki∑i=0k-1ei+yekekT+kdek-ek-1T.Because requirement of improved variable gain PID control algorithm to the values of parametersm1, m2, m3, m4, kp1′, kp2′, and ki′ is not accurate, the values are easy to be ensured.


## 4. Vector Control Algorithm for Vehicle Asynchronous Motor Based on Improved Variable Gain PID Controller

The block diagram of vector control algorithm for EV ACIM based on improved variable gain PID controller is obtained in Figure2. This algorithm, which uses rotator flux oriented, uses velocity and current double closed-loop control algorithm in control structure.Figure 2
The block diagram of vector control algorithm based on improved variable gain PID controller motor.In outer loop control, collect motor rotor velocityωr via revolution velocity transducer from the ACIM side. Then, set the deviation between expected rotor velocity ωr∗ and feedback rotor velocity ωr as the input of automatic voltage regulator (AVR), and the output is expected electromagnetic torque Te∗. The expected electromagnetic torque via torque-current transformation and slicing obtains the inner loop expected torque current isq∗. The requisite parameter θ of Park transformation and Park inverse transformation is provided by flux linkage observer. The inputs of weak magnetic block are usα, usβ, the maximum output voltage value of inverter is usm, and the maximum motor current value in safe running is ism, and the outputs are the slicing values of expected excitation current isd∗ and expected torque current isq∗.The transfer function block diagram of vector control algorithm for EV asynchronous motor based on improved variable gain PID controller can be obtained in Figure2.In Figure3, 1/Tcqs+1 is closed-loop transfer function of torque control system. So the transfer function of controlled object can be expressed as (9)Gs=1Tcqs+1∗1Js=1σLs/kiqs+1∗1Js,where kiq is q loop integral coefficient of vector control algorithm current loop, σ is leakage inductance coefficient of ACIM, Ls is stator inductance of ACIM, and J is moment of inertia of ACIM.Figure 3
The transfer function block diagram of vector control algorithm based on improved variable gain PID controller.Figure 4
Simulation diagram constructed by Matlab/Simulink.The differentiation element of PID control algorithm, which is sensitive to the noise of input signal, is not used in the system which has bigger noise in general. Thus, only PI control in the velocity loop controller of vector control algorithm for EV ACIM in general is used.The transfer function of velocity loop controller based on improved variable gain PID control algorithm is given as(10)Cs=kp+xeks+kiyeks,where kp is proportional coefficient of traditional PID control algorithm, ki is integral coefficient of PID control algorithm, kp1′≤x[e(k)]≤kp2′, and ki′≤y[e(k)]≤1. Here kp1′, kp2′, and ki′ are parameters of improved variable gain PID control algorithm to be determined.Therefore, the closed-loop transfer function of system is obtained as(11)φs=CsGs1+CsGs=kp+xeks+kiyekσLsJ/kiqs3+Js2+kp+xeks+kiyek.Using Routh stability criterion [18] to judge the stability of system can obtain the stability condition of system as (12)kp1′>σLskikiq-kp,ki′>0.The system can be stable if the values of parameterskp1′ and ki′ satisfy the condition of (9) when designing the vector control algorithm for EV ACIM based on improved variable gain PID controller.


## 5. Simulation and Interpretation of Results

To study the improvements of the improved variable gain PID control algorithm, it is imperative to compare it to classical PID control algorithm through simulation. The motor parameters of 20 kW ACIM which is used in simulation are given in Table1.Table 1
The parameters of ACIM.
Rated line voltageUN
180 V
Rated torqueTeN
53 Nm
Rated velocitynN
3600 rpm
Stator resistanceR1
0.0205 ohm
Rotor resistanceR2
0.0097 ohm
Stator leakage inductanceLls
9.2668e − 05 H
Rotor leakage inductanceLlr
10.9033e − 07 H
Mutual inductanceLm
0.0055887 H
Pole pairs
2Through debugging the simulation model of specific ACIM described in Table1, proportional and integral gains as 0.63797 and 30.158, respectively, can be obtained, and the values of parameters of improved variable gain PID control algorithm in the velocity loop controller are kp=2.26, ki=3.58, m1=0.06, m2=m3=0.04, m4=0.16, kp1′=0, kp2′=4, and ki′=0.01. kp1′>σLski/kiq-kp=-2.2538 and ki′>0; these satisfy the stability condition of system.As shown in Figure3, take a PID controller for example for comparison. Sample time Ts is 500 μs which is the sample period of the closed-loop system.


### 5.1. Low-Velocity Range

EV low running velocity is about 10 km/h, and the corresponding motor velocity is about 50 rad/s. If the expected velocity is motor nominal velocity of 50 rad/s in simulation, the control result is illustrated in Figure5.Figure 5
The simulation result of motor in low-velocity range.In Figure5, the response time of the PID controller is about 3 s, and the overshoot is about 14%. The response time of variable gain PID controller is about 2 s, and the overshoot is 0. However, the proposed controller can still track the desired velocity less than 1 s without 0 overshoots.In Figure6, compared with PID controller and variable gain PID controller, the peak of three-phase current is less than 100 A. In addition, it is investigated that the proposed controller is valid by comparison with the other PID controllers.Figure 6
The simulation result of three-phase current.
(a)
Three-phase current with PID controller
(b)
Three-phase current with variable gain PID controller
(c)
Three-phase current with improved variable gain PID controller


### 5.2. Moderate-Velocity Range

EV normal running velocity is 80 km/h to 100 km/h, and the corresponding motor velocity is about 370 rad/s. If the expected velocity is motor nominal velocity of 370 rad/s in simulation, the control result is illustrated in Figure7.Figure 7
The simulation result of motor in moderate-velocity range.In Figure7, the response time of the PID controller is about 3.5 s, and the overshoot is about 10.8%. The response time of variable gain PID controller is about 3 s, and the overshoot is 0. However, the proposed controller can still track the desired velocity less than 2 s without 0 overshoots.In Figure8, compared with PID controller and variable gain PID controller, three-phase current changes slowly and the response time is smaller. In addition, it is investigated that the proposed controller is valid by comparison with the other PID controllers.Figure 8
The simulation result of three-phase current.
(a)
Three-phase current with PID controller
(b)
Three-phase current with variable gain PID controller
(c)
Three-phase current with improved variable gain PID controller


### 5.3. High-Velocity Range

EV high running velocity is about 120 km/h, and the corresponding motor velocity is about 630 rad/s. If the given expected velocity is motor nominal velocity of 630 rad/s in simulation, the control result is illustrated in Figure9.Figure 9
The simulation result of motor in high-velocity range.In Figure9, the response time of the PID controller is about 5 s, and the overshoot is about 11.1%. The response time of variable gain PID controller is about 7 s, and the overshoot is 0. However, the proposed controller can still track the desired velocity less than 4 s without 0 overshoots.In Figure10, compared with PID controller and variable gain PID controller, three-phase current changes slowly and the response time is smaller. In addition, it is investigated that the proposed controller is valid by comparison with the other PID controllers.Figure 10
The simulation result of three-phase current.
(a)
Three-phase current with PID controller
(b)
Three-phase current with variable gain PID controller
(c)
Three-phase current with improved variable gain PID controller


### 5.4. Variable Velocity Range

In order to test wheter the method designed in this paper can satisfy the application needs in velocity, we make simulation of motor in variable velocity range with the initial expected velocity as 630 rad/s and turn to 300 rad/s at the 4th second. The control result is illustrated in Figure11.Figure 11
The simulation result of motor in variable velocity range.As shown in Figure11, the response time of the PID controller is about 2.5 s, and the overshoot is about 7.9%. The response time of variable gain PID controller is about 3 s, and the overshoot is 0. However, the proposed controller can still track the desired velocity less than 4 s without 0 overshoots.


## 5.1. Low-Velocity Range

EV low running velocity is about 10 km/h, and the corresponding motor velocity is about 50 rad/s. If the expected velocity is motor nominal velocity of 50 rad/s in simulation, the control result is illustrated in Figure5.Figure 5
The simulation result of motor in low-velocity range.In Figure5, the response time of the PID controller is about 3 s, and the overshoot is about 14%. The response time of variable gain PID controller is about 2 s, and the overshoot is 0. However, the proposed controller can still track the desired velocity less than 1 s without 0 overshoots.In Figure6, compared with PID controller and variable gain PID controller, the peak of three-phase current is less than 100 A. In addition, it is investigated that the proposed controller is valid by comparison with the other PID controllers.Figure 6
The simulation result of three-phase current.
(a)
Three-phase current with PID controller
(b)
Three-phase current with variable gain PID controller
(c)
Three-phase current with improved variable gain PID controller


## 5.2. Moderate-Velocity Range

EV normal running velocity is 80 km/h to 100 km/h, and the corresponding motor velocity is about 370 rad/s. If the expected velocity is motor nominal velocity of 370 rad/s in simulation, the control result is illustrated in Figure7.Figure 7
The simulation result of motor in moderate-velocity range.In Figure7, the response time of the PID controller is about 3.5 s, and the overshoot is about 10.8%. The response time of variable gain PID controller is about 3 s, and the overshoot is 0. However, the proposed controller can still track the desired velocity less than 2 s without 0 overshoots.In Figure8, compared with PID controller and variable gain PID controller, three-phase current changes slowly and the response time is smaller. In addition, it is investigated that the proposed controller is valid by comparison with the other PID controllers.Figure 8
The simulation result of three-phase current.
(a)
Three-phase current with PID controller
(b)
Three-phase current with variable gain PID controller
(c)
Three-phase current with improved variable gain PID controller


## 5.3. High-Velocity Range

EV high running velocity is about 120 km/h, and the corresponding motor velocity is about 630 rad/s. If the given expected velocity is motor nominal velocity of 630 rad/s in simulation, the control result is illustrated in Figure9.Figure 9
The simulation result of motor in high-velocity range.In Figure9, the response time of the PID controller is about 5 s, and the overshoot is about 11.1%. The response time of variable gain PID controller is about 7 s, and the overshoot is 0. However, the proposed controller can still track the desired velocity less than 4 s without 0 overshoots.In Figure10, compared with PID controller and variable gain PID controller, three-phase current changes slowly and the response time is smaller. In addition, it is investigated that the proposed controller is valid by comparison with the other PID controllers.Figure 10
The simulation result of three-phase current.
(a)
Three-phase current with PID controller
(b)
Three-phase current with variable gain PID controller
(c)
Three-phase current with improved variable gain PID controller


## 5.4. Variable Velocity Range

In order to test wheter the method designed in this paper can satisfy the application needs in velocity, we make simulation of motor in variable velocity range with the initial expected velocity as 630 rad/s and turn to 300 rad/s at the 4th second. The control result is illustrated in Figure11.Figure 11
The simulation result of motor in variable velocity range.As shown in Figure11, the response time of the PID controller is about 2.5 s, and the overshoot is about 7.9%. The response time of variable gain PID controller is about 3 s, and the overshoot is 0. However, the proposed controller can still track the desired velocity less than 4 s without 0 overshoots.


## 6. Conclusion

This paper has presented a novel approach in the automotive field to implement the improved variable gain PID controller to control EV ACIM. In this paper, the design of variable gain PID controller and stability analysis have been presented along with simulation. The simulation results qualitatively demonstrate that the improved variable gain PID controller could improve on the control of motor velocity in the EV versus using the classical PID control method. In addition, this control method can satisfy the demands of EV driving system dynamic performance and acceleration performance when EV runs in low velocity, moderate velocity, high velocity, and variable velocity.



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