C. Sombattheera et al. (Eds.): MIWAI 2012, LNCS 7694, pp. 81–89, 2012. © Springer-Verlag Berlin Heidelberg 2012

Evolutionary Multi-objective Optimization Based Proportional Integral Controller Design

for Induction Motor Drive

Moulay Rachid Douiri and Mohamed Cherkaoui

Mohammadia Engineering School, Department of Electrical Engineering, Avenue Ibn Sina, 765, Agdal-Rabat, Morocco

douirirachid@hotmail.com

Abstract. A new proportional-integral controller optimization methodology based on a multi-objective genetic algorithm for indirect field oriented controlled induction motor drive was proposed in this paper. GA-PI offers pos-sibility of using the mathematical precision of PI algorithm with adaptability, and flexibility of genetic algorithm. This approach is independent of the system parameters, independent of the mathematical model, and can handle the system nonlinearity, allowing eliminates and reduces the overshoot, rise-time, settling time, load disturbance, and near-zero steady state error. The validity of proposed methods is confirmed by simulation results.

Keywords: genetic algorithm, PI controller, field oriented control, induction motor.

1 Introduction

The concept of field orientation (also called vector control), proposed by Hasse [1] in 1969 using an indirect method, and by Blaschke [2] in 1971, using a direct method, constitutes one of the most important paradigms in the theory and practice of induc-tion motor control. The objective of field orientation is to make the induction motor emulate the separately excited dc motor as a source of adjustable torque, in which the field flux and the armature current are orthogonal [3]. The vector control has the dis-advantage of requiring the use of a mechanical sensor (speed or position), this impos-es an over cost and increases the complexity of the arrangements [3], [4], [5]. Also, it is necessary to control the speed of the induction motor to meet desired specifica-tions in various industrial applications. Classical control techniques such as propor-tional-integral-derivative (PID), proportional-integral (PI), and various adaptive controllers have been used for a long time to control the speed of the induction motor. The disadvantages of using conventional controllers are that they are sensitive to variation in the motor parameters and load disturbance. In addition, it is difficult to tune PI or PID gains to eliminate and reduce the overshoot and load disturbance. In order to avoid the shortcomings of conventional controllers, researchers applied adaptive control techniques for induction motor speed control to achieve parameter insensitivity and fast speed response [6].

82 M.R. Douiri and M. Cherkaoui

To overcome the shortcoming of these controllers, the proportional and integral gain of PI controller is optimized by genetic algorithm. Based on population genetics and Darwin's theory of natural selection, genetic algorithms are a type of evolutionary computing that solves problems by probabilistically searching the solu-tion space [7], [8]. In contrast to most algorithms which work by successively improv-ing a single estimate of the desired optimum via iterations, GA's work with several estimates at once, which together form a population. Given an initial population of individuals representing possible solutions to the problem, genetic algorithms simu-late evolution by allowing the most fit individuals to reproduce to form subsequent generations [7], [9]. After several generations, convergence to an optimal solution is often accomplished. Determining the fitness of an individual is problem dependent and the fitness function usually incorporates a priori knowledge of the desired opti-mum. The basic genetic algorithm is improved by using problem specific knowledge in specifying the various operations required to direct the evolution [9], [10], [11].

This paper is organized as follows: The principle of field oriented control is pre-sented in the second part, the optimization PI speed controller based on genetic algorithm in section three, the fourth part is devoted to illustrate the simulation performance of this control strategy, a conclusion and reference list at the end.

2 Field Oriented Control

Field Oriented Control (FOC) is very popular in IM control for its simplicity. Both the torque and flux are decoupled and controlled independently. The theory of FOC was first introduced by Blaschke [2], who suggested that the three phase IM can be set to imitate the dynamics of separately excited DC machines in a way similar to (1). The generated torque in separately excited DC machines is given by:

em t f ak iψΓ = (1)

To make IM torque in (2) similar to that of a DC machine in (1), ψrq must be set to 0.

( )23

mem p sq rd sd rq

r

Lp i i

Lψ ψΓ = − (2)

In order to eliminate ψrq component, the rotor flux is aligned to the d-axis of the re-volving reference frame of the stator current, so that ψr = ψrd. When ψr deviates away from the d-axis, ψr can be realigned by adjusting the quadratic component of the sta-tor current isq.

23

mem p sq rd

r

Lp i

LψΓ = (3)

The slip speed estimation is necessary to determine the synchronous angle. It can be calculated as follows:

Evolutionary Multi-objective Optimization Based Proportional Integral Controller 83

mrq sq

r

Li i

L= − (4)

rsl sq

rd

Riω

ψ= − (5)

sqm

slr rd

iLωτ ψ

= (6)

Hence, the synchronous angle can be obtained by integrating the mechanical and slip speeds.

( )s p m slp dtθ ω ω= + (7)

The speed controller processes the error between command and actual speeds and generates the command torque. The small change in speed Δωr, produces a corres-ponding change in torque ΔΓem, and taking the load torque Γl as a constant, we find (1):

( )r

em r

dJ F

dt

ω ωΔΔΓ = + Δ (8)

Integrating (8) gives us the total change of torque as:

0

( )t

em r rJ F t dtω ωΓ = Δ + Δ (9)

Equation (9) represents the PI algorithm for the speed controller that may be rewritten as:

*

0

( )t

em p r i rk k t dtω ωΓ = Δ + Δ (10)

where kp is the proportional constant, ki is the integral constant and Δωr=ωr*-ωr is the

speed error between the command speed ωr* and the actual motor speed ωr. In Lap-

lace domain (10) can be written as:

* ( ) ( )iem p r

ks k s

sω Γ = + Δ

(11)

84 M.R. Douiri and M. Cherkaoui

The block diagram of the drive with the PI speed controller is shown in Fig. 1.

ip

kk

s+ tk

1

Js F+

*sqi

emΓlΓ

rω*rω

Fig. 1. Block diagram of speed control loop

with

2

*

32

em mt

sq r

pLk

i L

Γ= =

(12)

Therefore, the closed loop transfer function, G(s) can be described by:

( )2

* 2 22

( )( )

( ) 2r i t n

r n np t i t

s k kG s

s s sJs F k k s k k

ω ωω ξω ω

= = =+ ++ + +

(13)

where 2

p t

i t

F k k

Jk kξ

+= and i t

n

k k

Jω =

At first, the PI controller gains are selected (ki = 0.06 and kp = 0.3085) for critically damping to avoid any overshoot but the controller becomes too sluggish. In order to get a reasonable settling time, the PI controller gains (kp = 0.06 and ki = 0.628) are designed for a slightly underdamped condition (ζ = 0.7). The gains of the transfer functions are normally tuned around the calculated values for best performance such as minimum overshoot, minimum settling time and zero steady state error.

3 PI Controller Tuning Using Genetic Algorithms

Genetic Algorithms (GA’s) were developed as non-traditional optimization methods that simulate the natural evolutionary process of survival of the fittest. A GA is an optimization procedure that operates on a population of parents (possible solutions). Each population member (solution) is called a chromosome that consists of series of genes (binary or digital numbers) representing the values of the variables associated with the solution [9], [11]. The fitness of each chromosome is a measurement of its performance, as defined by the objective function and the constraints. In its simplest form, GA’s evolve the population until an optimum is reached. The evolutionary process involves cycles of generating and testing new offspring solution, one of three operations: reproduction; crossover; and mutation [9], [10], [11].

Genetic algorithms are used to tune the proportional (kp) and integral (ki) gains of PI controller to ensure optimum performance using the principles of evolution and

Evolutionary Multi-objective Optimization Based Proportional Integral Controller 85

genetics to select and adapt the controller parameters [12], [13], [14]. ki and kp are coded by decimal numbers in chromosome. The members of the population are represented by the candidate controller's genetic PI controller. The algorithm tries to eliminate the bad traits of individuals and develop them on how well it minimizes the time absolute speed error (Fig. 3). The equations of input/output controller written at time t are:

*( ) ( ) ( ) ( )r r re t t t tω ω ω= Δ = − (14)

* * *( ) ( 1) ( )em em emt t tΓ = Γ − + ΔΓ (15)

For GA-PI controller gains, consider a step speed response where the goal is a short rise time, small overshoot, and near-zero steady state error. Since most performance indices in classical control do not necessarily fulfill these requirements simultaneous-ly, a multiple objective function is required. For this we define the objective function as follows:

*

0 0 0

(1) (2) (3)

4 ( ). ( )t t t

obj

dzf e dt z z t dt e tdt

dtδ= + − +

(16)

with: (1) Measure of a fast dynamic response; (2) The penalty on the multiple overshoot of the response, where δ(dz/dt) detects

the instances that overshoots (or undershoots) occur:

0

0

1 0( )

0 0

dzif

dz dtdzdt

ifdt

δ+

−

== ≠

(17)

and |z*-z(t)| determines the response deviation from the desired value; (3) Measure the steady state error. The equations of the new populations generated from crossover are:

01 1 2

02 2 1

(1 )

(1 )p p

p p

c c c

c c c

α αα α

= − + = − +

(18)

where cp1 and cp2 are the old chromosomes, α is the random from 0 to 1, c01 and c02 are the new chromosomes.

The genetic algorithm with the free parameters shown in Table 1 was able to find the near-optimum solution with a population of 12 individuals, in almost 169 genera-tions Fig. 2. This is due to the large number of design parameters involved in concur-rent optimization.

86 M.R. Douiri and M. Cherkaoui

Table 1. Genetic algorithm parameters

GA Property Value GA Property Value/ Method

Number of generations 169 Selection method Roulette wheel

No of chromosomes in each generation 12 Crossover method Double-point

No of genes in each chromosome 2 Crossover probability 0.8

Chromosome length 40 bit Mutation rate 0.05

Fig. 2. Speed of convergence

)()1( ** tt emem ΔΓ+−Γ( )e t

)(* trω

)1(ˆ −trω

)(* temΓ)(temΔΓ

pk ik

Fig. 3. Genetic algorithms PI controller structure

The optimization algorithm and the motor drive response are then verified under loading and unloading conditions. The unloaded drive is started at 0.15 sec and full load is applied at 0.5 sec; then load is completely removed at 1 sec. Later, after speed reversal, full load is applied at 2 sec and the load is fully removed at 2.5 sec. The machine is brought to rest at 2.9 sec. Fig. 4 shows the speed optimization result and response of the drive system.

The starting transient performance and disturbance rejection property of the induction motor under the different control strategies is shown in Fig. 5. Table 2 show the performance of genetic PI controller compared to PI conventional controller (Fig. 5).

Evolutionary Multi-objective Optimization Based Proportional Integral Controller 87

Fig. 4. Application and removal of load; PI controller and GA-PI controller

(a)

(b)

Fig. 5. (a) Starting transient performance and overshoot, (b) Disturbance rejection property for GA-PI and PI conventional

88 M.R. Douiri and M. Cherkaoui

Table 2. Summary of results

Rise time (sec) Overshoot (%) Settling time (%) Steady state error (%)

PI 0.026 2.78 0.06 0.5 GA-PI 0.024 2.5 0.0012 0.9

4 Conclusions

In this paper, it was demonstrated that genetic algorithms can be employed effectively as a search tool to optimize the parameters of PI in the speed control loop of induction motor, allowing eliminates and reduces the overshoot, rise-time, settling time, load disturbance, and near-zero steady state error (see Table 2). The paper shows the pos-sibility to use genetic algorithms to design complicated controller with adaptation and learning capabilities with reduced time and minimum effort, it also guarantees the best performance for the system using the optimized controller.

References

1. Hasse, K.: On the Dynamics of Speed Control of a Static ac Drive with a Squirrel-Cage Induction Machine. PhD Dissertation, Tech. Hochsch. Darmstadt (1969)

2. Blaschke, F.: The Principle of Field Orientation as Applied to the New Transvector Closed-Loop Control System for Rotating Field Machines. Siemens Review 34(5), 217–219 (1972)

3. Vas, P.: Sensorless Vector and Direct Torque Control. University Press, London (1998) 4. Abbondante, A., Brennen, M.B.: Variable Speed Induction Motor Drives use Electronic

Slip Calculator Based on Motor Voltages and Currents. IEEE Trans. Ind. Appl. 1A 11(5), 483–488 (1975)

5. Takahashi, I., Noguchi, T.: A New Quick-Response and High-Efficiency Control Strategy of an Induction Mîtor. IEEE Transactions on Industry Applications IA 22(5) (1986)

6. Shin, E.C., Park, T.S., Oh, W.H., Yoo, J.Y.: A Design Method of PI Controller for an In-duction Motor with Parameter Variation. In: The 29th Annual Conference of the IEEE Industrial Electronics Society IECON 2003, Roanoke, VA, USA, pp. 408–413 (2003)

7. Beasley, D., Bull, D.R., Martin, R.R.: An Overview of Genetic Algorithms. Part 1, Hin-damentals, Technical report, Inter-University Commitee on Computing (1993)

8. Back, T.: Evolutronary Algorithm in Tleory and Practice. Oxford University Press, New York (1996)

9. Davis, L.: Handbook of Genetic Algorithms. Van Nostrand Reinhold, New York, NY (1991)

10. Holland, J.: Adaptation in Natural and Artificial Systems: An Introductory Analysis With Applications to Biology, Control, and Artificial Intelligence. University of Michigan Press, Ann Arbor (1975)

11. Goldberg, D.E.: Genetic Algorithms in Search, Optimization and Motor Learning. Addi-son-Wesley Publishing Company, Reissue (1989)

12. Wang, J., Zhao, Y.: The Optimization of PI Controller Parameters using Genetic Algorithm in the DC Speed Control System. In: Proceedings of the 3rd World Congress on Intelligent Control and Automation, pp. 545–548 (2000)

Evolutionary Multi-objective Optimization Based Proportional Integral Controller 89

13. Vlachos, C., Williams, D., Gomm, J.B.: Genetic Approach to Decentralized PI Controller Tuning for Multivariable Processes. IEE Proceedings Control Theory and Applica-tions 146(1), 58–64 (1999)

14. Silva, W.G., Acamley, P.P., Finch, J.W.: Application of Genetic Algorithm to the Online Tuning of Electric Drive Speed Controllers. Trans. Ind. Electron of IEEE 47(1), 217–219 (2000)

Appendix

Induction motor parameters: Pn=3Kw, Vn=230V, Rs=2.89Ω, Rr=2.39Ω, Ls=0.225H, Lr=0.220H, Lm=0.214H,

J=0.2Kg.m², pp=2. Nomenclature

d,q Direct and quadrature components Rs , Rr Stator and rotor resistance [Ω] isd , isq Stator current dq –axis [A] ird , irq Rotor current dq –axis [A] Ls , Lr, Lm Stator, rotor and mutual inductance [H] ψrd, ψrq dq rotor fluxes [Wb] Γem Electromagnetic torque [N.m] Γl Load torque [N.m] kt Torque constant [N.m] ψf Excitation flux [Wb] ia Armature current [A] ωr, ωm Rotor and mechanic speed [rad/sec] ωe, ωsl Synchronous and slip frequency [rad/sec] θs Slip angle [rad] τr Rotor time constant [sec]

J Inertia moment [Kg.m2] pp Motor pole number F Viscous friction coefficient [N.m.s/rad] fobj Objective function s Laplace operator * Reference symbol Estimated quantity