Journal of Information & Computational Science 11:13 (2014) 4687–4696 Available at http://www.joics.com

September 1, 2014

Optimization of Permanent Magnet Synchronous Motor Vector Control System Based on Particle Swarm Optimization Weihua Li a,∗, a School b Qingyuan

Ziying Chen a , Wenping Cao b

of Electrical Engineering and Information, Jinan University, Zhuhai 519070, China Power Supply Bureau, Guangdong Power Grid Corporation, Qingyuan 511500, China

Abstract For permanent magnet synchronous motor control system at this stage has defects such as the poor robustness and control, this paper proposes an optimization scheme of permanent magnet synchronous motor vector control system based on particle swarm optimization. Establish the mathematical model of the permanent magnet synchronous motor, use the particle swarm optimization algorithm to optimize the PID controller parameters of the system speed loop on the basis of the theoretical analysis of vector control, PID controller and particle swarm optimization, build the optimization simulation model of the permanent magnet synchronous motor vector control system based on particle swarm optimization in MATLAB/Simulink and conduct the related research. The simulation results show that the optimization scheme is effective and it can significantly improve system control performances. Keywords: Permanent Magnet Synchronous Motor; Vector Control System; Particle Swarm Optimization Algorithm; PID Controller; System Optimization

1

Introduction

Permanent Magnet Synchronous Motor (PMSM) has advantages such as small size, light weight, fast response and high efficiency [1], and it has been widely used in aviation and aerospace, national defense, industrial and agricultural production and various fields of the daily life with the development of power electronics and control technology. PID controller has simple structure and its parameters can be adjusted easily, so it’s widely used in PMSM vector control field. But the outside interference and PMSM system characteristics like nonlinear, strong coupling and time-varying make it hard for the conventional PID vector control system to meet control requirements of the PMSM system with high performances and it usually has defects such as poor robustness and unsatisfactory control effect [2]. With the development of intelligent control technology, many scholars have used intelligent algorithms to optimize the PMSM vector control system and achieved the significant control effect. ∗

Corresponding author. Email address: [email protected] (Weihua Li).

1548–7741 / Copyright © 2014 Binary Information Press DOI: 10.12733/jics20104498

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Particle Swarm Optimization (PSO) is an algorithm which has fewer parameters and strong global search capability for both nonlinear and multimodal problems and it’s easy to be implemented, so it has been extensive attention in scientific research and engineering applications [3-4]. On the basis of establishing the mathematical model of PMSM, the PSO algorithm is applied to the PID parameter tuning of the speed loop in PMSM id = 0 vector control system to guarantee the global stability and the strong robustness while system parameters are changing, thus the control performances of the system can be significantly improved. Through the related simulation experiments, the optimization scheme of PMSM vector control system based on the PSO algorithm is verified to be effective.

2

Analysis of Theoretical Principles

2.1

Mathematical Model of PMSM

In order to establish the mathematical model of PMSM, make the following assumptions [5]. (1) PMSM magnetic circuit is linear, regardless of the effects of magnetic saturation, hysteresis and eddy current. (2) Three-phase windings are completely symmetrical, excluding edge effects. (3) The stator current in air gap produces only sinusoidal magnetomotive force, ignoring cogging and the higher harmonic. (4) Exclude the core loss. According to the assumptions above, establish the mathematical model of PMSM in the d, q-axis rotating coordinate system in a series of equations, including flux, voltage, mechanical motion and torque. Flux equation:

{

ψq = Lq iq ψd = Lq id + ψj

(1)

In formula (1), ψj is the flux generated by the magnetic pole and the stator. Voltage equation:

{

ud = Rs id + pψd − ωr ψq uq = Rs iq + pψq + ωr ψd

(2)

In formula (2), Rs is the phase resistance of the stator winding, ωr is the electrical angular velocity of the rotor. Mechanical motion equation: Tem − Tl = Jp Ωr

(3)

In formula (3), Ωr = ωr / pn is the mechanical angular velocity of PMSM, pn is the pole-pairs number of the rotor, Ti is the load torque. Torque equation: T = Pn (ψd id − ψq iq ) = Pn (ψj iq + (Ld − Lq )id iq )

(4)

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2.2

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PMSM Vector Control

Among PMSM vector control, id = 0 control is the simplest and the most widely used method. When the direct-axis component of the stator armature current of PMSM is always equal to 0 in the control process, the PMSM voltage equation can be simplified as: { ud = −ωr ψq (5) uq = Rs iq + pψq + ωr ψd Through id = 0 vector control, the speed can be estimated through PI regulator just using the error signal of q-axis between the estimated current and the actual current, which makes the whole system much simpler. Thus the research is conducted under the id = 0 vector control method.

2.3

PID Controller

The error e(t), which exists between the given value R(t) and the actual output value Y(t), is used by PID controller to constitute the control through the linear combination of proportion (P), integral (I) and differential (D) so as to achieve the effective control of the controlled object. The principle of a conventional PID controller is shown in Fig. 1. Proportion (P) R(t)+ _

+

e(t)

Integral (I)

+ +

Controlled object

Y(t)

Differential (D)

Fig. 1: Block diagram of PID controller The conventional PID control process is written in the form of transfer function as: G(s) =

1 1 U (s) = Kp (1 + + Td s) = Kp + Ki + Kd s R(s) Ti s s

(6)

In formula (6), Kp is the proportional action coefficient, Ki is the integral action coefficient and Kd is the differential action coefficient. Transfer function can be easily changed by adjusting the parameters Kp , Ki and Kd , for which the control effect of the PMSM vector control system can be greatly improved.

2.4

Basic Principle of PSO

Particle swarm optimization, a swarm intelligence-based global random search algorithm, is proposed by Kennedy and Eberhart inspired from artificial life research results. It regards all individuals in the population as particles without mass and volume in the D-dimensional search

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space, and each particle moves at a certain speed to the best position of its own history pbest and the best position of its neighborhood history gbest in the solution space, in order to achieve the evolution of candidate solutions [6]-[8]. Assume that there is a community composed of N particles in a D-dimensional search space, the best position that the i − th particle has been searched so far is called the individual extremum, and it is denoted as: pbest = (pi1 , pi2 , · · · , piD ) i = 1, 2, · · · , n (7) The best position that the whole particle swarm has been searched so far is called the global extremum, and it is denoted as: gbest = (pg1 , pg2 , · · · , pgD )

(8)

The particle updates its own velocity and position according to formula (9) and (10) [9]: vid (k + 1) = vid (k) + c1 r1 (pid (k) − xid (k)) + c2 r2 (pgd (k) − xid (k))

(9)

xid (k + 1) = xid (k) + vid (k + 1) , 1 ≤ i ≤ N, 1 ≤ d ≤ D

(10)

In formula (9) and (10), vi d is the particle velocity, which is between −vm ax and vm ax, and vmax is a constant set by the user to limit the particle velocity; xi d the particle position; c1 and c2 are learning factors which are also known as the acceleration constant, and c1 adjusts the particle step in the direction of moving to the best position of its own history while c2 adjusts the particle step in the direction of moving to the best position of the global history, and they are usually set as c1 = c2 = 2, i = 1, 2, 3, ..., N ; r1 and r2 are uniform random numbers within the range of [0, 1]; k is the evolutionary iterative algebra. The PSO algorithm achieves the parameter optimization through the following steps [10, 11]. Step 1 Initialize the population size of the particle swarm, and initialize the position and the velocity of each particle. Step 2 Calculate the fitness value of each particle Fit [i]. Step 3 Compare each particle’s fitness value Fit [i] and individual extremum pbest (i), if Fit [i] > pbest (i), then replace pbest (i) with Fit [i]. Step 4 Compare each particle’s fitness value Fit [i] and global extremum gbest , if Fit [i] > gbest , then replace gbest with Fit [i]. Step 5 Update the velocity vi and the position xi of the particle according to formula (9) and formula (10). Step 6 Determine whether the algorithm termination condition is met, which means whether the maximum number of iterations is reached. If the termination condition is met, then the algorithm stops and outputs the optimal value; otherwise the algorithm returns to Step 2 to continue the search. In summary, the optimization process of the PSO algorithm is shown in Fig. 2.

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Population and parameter initialization

Calculate the fitness value of each particle

Find the individual optimum

Find the global optimum of population Generate a new solution

Evolution of the particle velocity according to formula (9) Evolution of the particle position according to formula (10)

Whether the termination condition is met

N

Y Output the optimal value

Fig. 2: Parameter optimization process of the PSO algorithm

3

PID Optimization Strategy Based on PSO

The optimization of PID controller parameters is necessary for a better control effect of the PMSM vector control system, that is determine the optimal values of three PID controller parameters Kp , Ki , and Kd according to the control performances of the PMSM vector control system. Combining the basic principle of PMSM vector control system, the control characteristics of PID controller and the optimization capability of PSO, this paper proposes an optimization strategy called PMSM vector control system based on the PSO algorithm, which regards a set of parameters (Kp , Ki , Kd ) of PID controller as a food source of the PSO algorithm in order to convert the PID controller parameter tuning problem into the PSO optimization process containing the threedimensional vector. The optimization of PID controller parameters can be quickly and easily completed using the PSO algorithm, and the excellent optimization capability of the algorithm guarantees a better control effect of the PMSM vector control system.

3.1

Fitness Function Selection and Parameter Settings

PID controller parameter tuning is essentially to find a set of parameters (Kp , Ki , Kd ) to make the value of fitness function the minimum, therefore, before using the PSO algorithm it need to be

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determined how to take the PID controller performance index as the fitness function. ITAE, IAT, ISTE, ISE, etc. are commonly used fitness functions reflecting the PID controller performance index of PMSM vector control system, and this paper takes ITAE criterion as the fitness function for the optimization of PMSM id = 0 vector control system, considering the fact that the system which takes ITAE criterion as the fitness function usually has good characteristics like smooth, fast, small overshoot [12]. The formula (11) below is the expression of ITAE. ∫ +∞ IT AE = t |e(t)| dt + c |min(y(t))| (11) 0

In formula (11), e(t) is the system error, c is the weighting factor whose value for the nonminimum phase systems is 20 while for the minimum phase systems is 0. In this paper, PID controller parameter optimization of PMSM vector control system based on the PSO algorithm is completed through the interaction between M-files and Simulink. Firstly build the simulation model of the PMSM vector control system in Simulink, then take the PID parameters of the speed regulator as the variables wihch need to be optimized and set them as global variables, and the ITAE values of the PI regulator are the fitness values. The calculation of ITAE index can be completed by building the related module in Simulink which is shown in Fig. 3. 1 s

Time Product

1 Error

Integrator

1 Out 1

|u| Abs

Display

Fig. 3: Calculation module of ITAE in Simulink In Fig. 3, time is the clock module which provides the simulation time information. ITAE module makes the integral after that the absolute value of error is multiplied by the time, then obtains the ITAE index value and outputs it through the Out1. The PSO algorithm is achieved by the M-files and the model simulation is implemented through the sim () function. After the simulation, output the data which need to be processed to the workspace, so that the M-files can process them according to the predetermined performance evaluation function and obtain the ITAE value by the final value access from Out1. Finally three PID parameters can be adjusted by the PSO algorithm in order to achieve the PID parameter controller optimization.

3.2

Implementation Method of the PID Controller Optimization Based on PSO

The structure of PID controller optimization based on the PSO algorithm is shown in Fig. 4. Compared to the general control process of PMSM vector control system, the difference lies in the selection of control system speed loop PID parameters which is achieved by the PSO algorithm instead of the manual test. This paper converts PID controller parameter tuning into a three-dimensional PSO optimization problem, searches for three parameter values meeting the

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best fitness value according to the setting fitness function and finds the optimal PID control parameters. PSO algorithm kp r(t)

+

e(t)

ki PID

kd SVPWM

PMSM

−

Fig. 4: Block diagram of PID optimization based on PSO The process of the PMSM control system speed loop PID controller optimization based on the PSO algorithm is shown as follows. Step 1 Initialize solution space and parameters, such as particle population, inertia weight and particle velocity. Step 2 Start the simulation model by sim () function and output the results to the workspace. When the simulation is completed, the value output from Out1 in workspace is the value of fitness function ITAE index. Step 3 Update the individual extremum and the global extremum. Step 4 Divide the particles except the global optimal particle to particles with better fitness value, particles with centered fitness value and particles with poor fitness value by the size of the ITAE index value, and update the particle velocity vi and position xi according to formula (9) and formula (10). Step 5 Determine whether the algorithm termination condition is met, which means whether the maximum number of iterations is reached. If the termination condition is met, then the algorithm stops and returns the optimal PID parameters; otherwise the algorithm returns to Step 2 to continue the search.

4

Simulation and Results Analysis

The parameters of PMSM are as in Table 1. According to PMSM id = 0 vector control system mathematical model and its parameters, build the PMSM vector control system simulation model based on the PSO algorithm in MATLAB/Simulink software to optimize the speed loop PID controller and the simulation model is shown in Fig. 5. The global fitness value curve of speed loop PID controller optimized by the PSO algorithm is shown in Fig. 6. It can be seen from Fig. 6 that the global fitness value of PID controller decreases with the number of iterations increasing and that the global fitness value is minimum and then keeps constant when the number of iterations is 40. Therefore, the PID controller minimum fitness value can be quickly and accurately obtained by the application of the PSO algorithm

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Table 1: Parameters list of PMSM

Continuous

Parameter

Mark

Value

Unit

Stator resistance

Rs

0.62

Omega

D-axis inductance

Ld

2.1

mH

Q-axis inductance

Lq

2.1

mH

Rotational inertia

J

0.0003617

Kgm2

Damping coefficient

B

0

Damping coefficient

P

2

1 s

time

powergui

1

|u| 〈signal 1〉 Display PID

1500 speed ref

PI

PSO-PID

Uq Ud

〈signal 2〉 Ualfa fcn Ubeta

theta Unpark 0

〈signal 3〉

Ualfa pulse Ubeta

〈signal 5〉

SVPWM

〈signal 6〉

〈signal 4〉

PI

Isabc 1

Constant 2

Vd Vq

theta

Step

Universal bridge +

id iq DC

−

g

Tm

A

A

B

B

C

C

PMSM

m

+ v _

〈Stator voltage Vs_d(V)〉 〈Stator voltage Vs_q(V)〉 〈Stator current is_d(A)〉 〈Stator current is_q(A)〉 Gain 1 〈Rotor speed wm (rad/s)〉 4 〈Rotor angle thetam (rad)〉 〈Electromagnetic torque Te(N*m)〉 〈Stator current is_a(A)〉 〈Stator current is_b(A)〉 〈Stator current is_c(A)〉

Gain

id iq1

−K− speed

Vab 0.5*pi Ialfa Ibeta theam

Id Iq Park

Ialfa Ibetaa Clarke

Te

iabc

Iabc

Fig. 5: PMSM vector control system simulation model based on the PSO algorithm

The fitness value

1.25

The optimal global fitness value

1.20 1.15 1.10 1.05 0

20 40 60 80 The number of iterations

100

Fig. 6: Global fitness value curve of speed loop PID controller optimized by the PSO algorithm

in PMSM vector control system, which shows that the PSO algorithm has a good optimization capability for PMSM vector control system.

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The curve of PID controller parameters in the PSO algorithm optimization process is shown in Fig. 7. As Fig. 6 and Fig. 7 show, three PID controller parameters constantly change to find the smallest fitness value in the optimization process of PSO algorithm. When the number of iterations is 40, the PSO algorithm has found the global minimum fitness value, and the parameter values (Kp , Ki , Kd ) are the optimal PID controller parameters which remain unchanged after that. It can be seen that the PSO algorithm can efficiently achieve PID parameter tuning in PMSM vector control system to obtain a set of optimal PID parameters, for which the PID controller has the best control effect. In the parameter tuning process based on the PSO algorithm, the speed waveform, the torque waveform and the current waveform of the PMSM servo system can be obtained, which are shown in Fig. 8, Fig. 9 and Fig. 10. Kp，　Ki，　Kd optimization curve

300 Kd 200

n(r/s)

Parameter values

400

Kp Ki

100 0 0

20 40 60 80 The number of iterations

100

1600 1400 1200 1000 800 600 400 200 0 −200 0

Fig. 7: Curve of three PID parameters in the optimization

0.1

0.2

0.3 t(s)

0.4

0.5

0.6

Fig. 8: Curve of speed

25

Te(N·m)

20 15 10 5 0 0

0.1

0.2

0.3 t(s)

0.4

0.5

0.6

I(A)

Fig. 9: Waveform of torque

10 0 −10 −20

0

0.1

0.2

0.3 t(s)

0.4

0.5

0.6

Fig. 10: Waveform of electric current As Fig. 8, Fig. 9 and Fig. 10 show, the speed loop responses fast with short rising time and adjusting time and no speed overshoot phenomena at the starting moment of the PMSM, then the electromagnetic torque waveform is stable and smooth and the current waveform is a sine wave after the speed is stable. Given a sudden load torque, the speed landing is small and the

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speed can recover in a short time, then the current waveform and the torque waveform are quickly stable after a slight jitter, showing that the system torque can quickly return to a stable state in the load disturbance with small torque pulsation, fast response and good anti-interference. When the speed is adjusted, the system adjustment time is short and the speed overshoot is small. Once the system is stable, there is no static speed error, and the waveform that the electromagnetic torque outputs is smooth and stable. Therefore, the optimization scheme of the PMSM vector control system based on the PSO algorithm is effective and the system optimized by the PSO algorithm has better control performances and achieves significant control effect.

5

Conclusion

The PMSM vector control system optimized by the PSO algorithm has shown satisfactory results in terms of accuracy, robustness, reliability, speed response and torque control, confirming that the optimization of PMSM vector control system speed loop PID controller based on the PSO algorithm is effective. It’s of great significance to apply the PSO algorithm to design and develop the PMSM vector control system with high performances to meet the rapid development of modern industry.

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