Implementation of Genetic Algorithm in Control

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Advances in Electrical and Computer Engineering

Volume 14, Number 4, 2014

Implementation of Genetic Algorithm in Control Structure of Induction Motor A.C. Drive Pavel BRANDSTETTER, Marek DOBROVSKY, Martin KUCHAR VSB - Technical University of Ostrava, 70833, Czech Republic [email protected] 1

Abstract—Modern concepts of control systems with digital signal processors allow the implementation of time-consuming control algorithms in real-time, for example soft computing methods. The paper deals with the design and technical implementation of a genetic algorithm for setting proportional and integral gain of the speed controller of the A.C. drive with the vector-controlled induction motor. Important simulations and experimental measurements have been realized that confirm the correctness of the proposed speed controller tuned by the genetic algorithm and the quality speed response of the A.C. drive with changing parameters and disturbance variables, such as changes in load torque. Index Terms—Artificial intelligence, genetic algorithms, induction motor, variable speed drive, vector control.

I. INTRODUCTION The electrical drive is a basic element of electrical energy conversion that significantly influences the efficiency of the electrical energy conversion into mechanical energy. It also depends on control quality, connected with the dynamics of the controlled system that is ensured by modern control algorithms realized by high performance digital signal processors [1-3]. Induction motor A.C. drives are finding an increased number of industrial applications. The induction motors are often the preferred choice in variable speed drive applications. Nowadays, modern digital signal processors enable the development of electrical drives with high dynamic performance using new control methods that include soft computing methods [4-6]. Genetic algorithms (GA) belonging to the area of evolutionary algorithms also represent a very interesting application possibility in the field of controlled electrical drives [7-10]. In principle, GA is a stochastic optimization method to look for unknown parameter values pertaining to a known mathematical model within a predefined range of values. In its operations, the algorithm retains the best proposed solutions established so far and, on the basis thereof, determines the further direction in searching the space of possible solutions, in parts either in a random or controlled manner. The algorithm inputs are generally represented by the mathematical model of the analyzed problem along with the fitness function and the range of values the desired parameters may acquire. The outputs of algorithms are the The article has been elaborated in the framework of the IT4Innovations Centre of Excellence project, reg. no. CZ.1.05/1.1.00/02.0070 funded by Structural Funds of the European Union and state budget of the Czech Republic and in the framework of the project SP2014/119 which was supported by Student Grant Competition of VSB-TU of Ostrava.

best values found for the desired parameters and the fitness function value achieved. In certain cases, the parameters of specific, i.e. actual systems may be directly entered as inputs [11]. II. MATHEMATICAL MODEL OF THE INDUCTION MOTOR SUPPLIED BY FREQUENCY CONVERTER A mathematical model of the induction motor supplied by a frequency converter is the multi-parameter non-linear system. The mathematical model of the induction motor supplied by a frequency converter can be described in different coordinate systems: stator coordinate system (α, β), rotor coordinate system (d, q) and orientated coordinate system (x, y). In reference to simple decoupling between individual magnetizing and torque current components iSx, iSy, it is useful to select a relative coordinate system (x, y) that rotates with angular speed ωim and is oriented to the rotor flux vector ψR. The rotor flux vector ψR is expressed by the magnetizing current im and the main induction of the induction motor Lh. The squirrel-cage induction motor can be described by a system of equations in the reference frame system (x, y):  TS  TS

diSx u  iSx  Sx  d y dt RS diSy dt

 iSy 

uSy RS

 dx

dim dt d x  im TS iSx  1    imTS im d y  im TS iSy  1    TS

dim  im  iSx dt d m 3 Lh J  im iSy  TL  KT im iSy  TL dt 2 1  R

TR

TS 

LS L , TR  R RS RR

(1) (2) (3) (4) (5) (6) (7)

where iSx, iSy – stator current components of the stator current vector in [x, y] rotating coordinate system; uSx, uSy – components of the stator voltage vector; im – magnetizing current; J – total moment of inertia; KT – torque constant; Lh – magnetizing inductance; LR – rotor inductance; LS – stator inductance; RR – rotor phase resistance; RS – stator phase resistance;  – total leakage constant; R – rotor leakage constant; TM – induction motor torque; TL – load torque; TR – rotor time constant; TS – stator time constant; ωim – angular speed of the magnetizing current im; ωm – real rotor angular speed.

Digital Object Identifier 10.4316/AECE.2014.04003

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Advances in Electrical and Computer Engineering

Volume 14, Number 4, 2014

It is possible to separately control the flux producing component iSx and torque producing component iSy of the stator current vector iS that is defined by stator phase currents. The frequency converter can be described by transfer constant KFC, time constant of the frequency converter TFC and control voltages uCx, uCy. After neglecting the weak mutual connection between the x and y components, we can write the following equations for the frequency converter: uSx  K FC

1 uCx 1  sTFC

(8)

uSy  K FC

1 uCy 1  sTFC

(9)

K FC  kC

Ud T , TFC  sw uC max 2

(10)

where uCx, uCy – components of the control voltage vector; KFC – transfer constant of the frequency converter; kC – control constant of the frequency converter; Ud – D.C. link voltage; uCmax – maximal value of the control voltage; TFC – time constant of the frequency converter; Tsw – switching period of the frequency converter. The control constant of the frequency converter kC respects the used method of pulse-width modulation (PWM), for example comparative and vector PWM. III. CONTROL STRUCTURE OF THE INDUCTION MOTOR DRIVE WITH VECTOR CONTROL For the variable speed electric drive with the induction motor, a cascade structure is often used because of its flexibility (see Fig.1). The cascade structure consists of several control loops, whereas the current control loops for the magnetizing and torque current components iSx, iSy are subordinate and the flux and speed control loops are superior loops. The cascade structure requires a frequency separation of all control loops in the controlled system.

dy from the output signal of the current controllers in each axis to cancel this mutual connection. It is so-called decoupling. The current control loops play a critical role in the overall structure of the vector-controlled induction motor. The reason is that the design of superior flux and speed control loop assumes their ideal behavior; this means precise control without delay. Using a subordinate current control loop, we ensure a quick response to its reference value that comes from the superior flux and speed control loop, thus we ensure maximum machine dynamics during a sudden change of the reference magnetizing current imref and reference speed ωmref. The next task of the current control loop is to keep the current within safe limits and a quality decoupling current loops for magnetizing and torque current component iSx, iSy. The open current control loop is described using transfer function: Foi ( s )  FRi ( s )

K FC K I K AD 1 1 RS 1  sTi  1  sTNCi 

LS , TNCi  1.5 TSi  TFC  RS

Ti   TS  

(11) (12)

where Foi – transfer function of the current control loop; FRi – transfer function of the current controller; KFC – transfer constant of the frequency converter; KI – transfer constant of the current sensor; KAD – transfer constant of the A/D converter; LS – stator inductance; RS – stator resistance; Ti – time constant of the current control loop (CL); TNCi – uncompensated time constant of the current CL; TSi – sampling period of the current CL; TS – stator time constant; TFC – time constant of the frequency converter;  – total leakage constant. For the control of current components iSx and iSy we can use PI controllers with the parameters KRi and TRi: FRi ( s )  K Ri

1  sTRi 

(13)

sTRi

The closed current control loop with the PI controller designed using the optimal module method is then described using transfer function: FCi ( s ) 

1

(14)

1  s 2TNCi 

The speed controller determines the reference quantity iSyref for the current controller. Its design is based on the motion equation (4). The open speed control loop is described using transfer function: Fo ( s )  FR ( s ) FCi ( s )

Figure 1. Control structure of induction motor drive with the vector control

The design of the controllers in individual control loops is based on the equations (1) to (10). The equations (1) and (2) show that there is a mutual connection between the magnetizing and torque current components iSx, iSy that is expressed by terms dx and dy. We then have to subtract the calculated quantities dx and

16

K IS KT im sK I K AD J

1 T 1  s S 2

(15)

where Foω – transfer function of the speed control loop; FRω – transfer function of the speed controller; FCi – transfer function of the closed current control loop; im – magnetizing current; J – total moment of inertia; KAD – transfer constant of the A/D converter; KI – transfer constant of the current sensor; KIS – transfer constant of the incremental sensor IS; KT – torque constant; TSω – sampling period of the speed CL. For the speed control we can design a PI controller with the parameters KRω and TRω: FR ( s )  K R

1  sTR  sTR

(16)

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Advances in Electrical and Computer Engineering

Volume 14, Number 4, 2014

We can also design a flux controller in similar way using the equations (5) and (14). IV. CLASSICAL PI SPEED CONTROLLER The quality of the control is assessed according to the response of the control loop to step change of input variable (see Fig. 2). From a practical point of view, four factors are most important for the assessment of control quality: rise time tr, settling time ts, overshoot ym and steady state error.

Figure 2. Typical step response for an under damped second order system

The classical PI speed controller for the induction motor drive can be expressed using proportional and integral gain Kp, Ki as follows: (17) iSyref  K p (mref  m )  K i  (mref  m )dt where iSyref – reference value of the torque current component; Kp, Ki – proportional and integral gain of the PI controller; ωmref, ωm – reference and real rotor angular speed; eω, Δeω – speed error and speed error change. The change of the torque current component in discrete form using the sampling period TSω is described as follows: (18) e ( k )  mref ( k )  m ( k ) iSy ( k )  iSyref ( k )  iSyref ( k 1)   K p [e ( k )  e ( k 1) ]  K i TS e ( k ) iSyref ( k )  iSyref ( k 1)  iSy ( k )

(19) (20)

The speed difference eω is processed by the PI speed controller whose parameters determine the behavior of the electrical drive. The parameters of the PI controller can be set by various mathematical and experimental methods [1214]. V. GA SPEED CONTROLLER The genetic algorithms (GA) are global optimization procedures inspired by the laws of natural selection and genetics [15-18]. The GA speed controller differs from the classical PI controller. The PI controller parameters are not constant, but their values are adapted to new conditions in the controlled system [19]. The block diagram of the GA speed controller is shown in Fig. 3.

Figure 3. Block diagram of the GA speed controller

The GA part of the controller performs parameter setting that the genetic algorithm optimizes using fitness evaluation of different controller parameters. The first simulations of GA speed controller were performed at our department in 2011. The simulations results are listed in [20]. The GA algorithm has been improved and implemented in a real control system of the induction motor A.C. drive (see Chapters VI and VII). The Genetic Algorithm block ensures modifications and tuning of the controller constants according to a specified algorithm the best suited values of the proportional gain Kp and the integral gain Ki that then sends it to the conventional part of the speed controller. The activity of the improved speed controller with the genetic algorithm includes the following steps: 1. Determination of an initial population of Kp and Ki gains with the chromosomes cj (j=1, 2 … n) using random selection; n is population size. 2. Coding chromosomes. Decimal number was used because controller parameters Kp and Ki can be coded into a chromosome. The corresponding chromosome coded by Kp and Ki has six digits in decimal number. For example, the gains Kp = 18.6 A/rad.s-1 and Ki = 1120 A/rad are coded as {186112}. 3. Measuring and sampling the speed signal of the speed incremental sensor. Calculation of the speed error eω, speed error change Δeω and change of torque current component ΔiSyref. 4. Calculation of the parameter pij for each chromosome cj and calculated speed error eω, speed error change Δeω and current change ΔiSyref (see Table I). (21) pij   K p e ( k )  K i T e ( k )   iSyref ( k )  iSyref ( k 1)  TABLE I. GA PARAMETERS PROCESSING Chromosome (j) 1 2 j Calculation pij Generation (i) 1 Kp11Ki11 Kp12Ki12 Kp1jKi1j p11 p12 p1j . A new generation (i +1) is . created by crossover and . mutation of generation i. . i Kpi1 Kii1 Kpi2 Kii2 Kpij Kiij pi1 pi2 pij

5. Test if i = N, the desired number of generations is achieved (the maximal number of generations is N). If yes, proceed to step 7. 6. Performing GA operators (crossover, mutation), production of the next generation and continuing to step 3. An example of the crossover and mutation is shown in Fig.4.

Figure 4. Example of the crossover and mutation

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7. Computation of criterion function Ji: N

J j   wi pij2  w1 p12j  w2 p22 j  ...  wN pN2 j  i 1

(22)

2  1.05 p12j  1,1 p22 j  ...  2.0 pNj

wi  1 

i 20

having to use a transistor switch and resistor in the voltage intermediate circuit. The induction machines are mechanically connected to the frame and together they form a set. Both machines are connected mechanically by a coupling.

(23)

where i is generation number, N is maximal number of generations, wi is generation weight, j is chromosome number. The criterion function represents the sum of all the parameters pij of each chromosome. This means that this function captures the whole of its history in all generations. The parameter weight grows with the generation of the chromosome. The last generation should have the highest quality. 8. Fitness evaluation using the fitness function: Fj 

1 1 J j

(24)

9. Selection of the combination of the controller parameters Kp and Ki according to the selection probability that is defined as follows: Pj 

Fj

(25)

n

F j 1

j

where n is a number of the chromosomes, j is chromosome number. New parameters for the speed controller are chosen from the last N-th generation according to the selection probability Pj. The correct values of the controller parameters Kp and Ki are set in the speed controller and are used for speed control. 10. Resetting the counter of generations. The last N-th generation is now used as the initial generation for the next cycle (creation of N new generations of the chromosomes). It continues with step 3. If random selection is used for the initial generation again, so it continues to step 1. VI. EXPERIMENTAL LABORATORY WORKPLACE An experimental laboratory workplace with two induction machines supplied by frequency converters was realized (see Fig. 5) to verify the simulation models and the speed control of the induction motor using the GA speed controller. This laboratory stand allows research of new control methods and solution of problems of A.C. variable speed drives. The active load unit is realized as a vector-controlled induction motor drive that allows choosing different load characteristics, for example a load with constant torque, fan and lift characteristics. The basic parts are: two induction machines, frequency converters, DSP control systems, personal computers and the necessary measuring instruments. The induction machines and the incremental sensor (IS) are located on a common shaft. Each machine is connected to a separate voltage inverter that is supplied from a voltage D.C. link. To increase the efficiency of the drive at loading, a concept with a common D.C. link was chosen, which allows the use of regenerative energy for the A.C. drive, without

18

Figure 5. Laboratory stand with A.C. drive and the load unit

Induction motor parameters are presented in Table II. A rotor position is estimated by an incremental sensor that generates 2048 pulses per revolution forms. TABLE II. PARAMETERS OF THE INDUCTION MACHINE Parameter Value Rated power 2.2 kW Rated speed 1420 rpm Rated voltage 230 V/400 V Rated current 8.4 A/4.7 A Rated torque 14.8 Nm Number of pole-pairs 2 Stator resistance 3.71 Ω Rotor resistance 3.06 Ω Stator inductance 330 mH Rotor inductance 336 mH Rotor time constant 0.1098 s Moment of inertia 0.005 kgm2

The frequency converters were made at the Department of Electronics. For practical realization, switching elements IGBTs SKM 75GB123D (UCE = 1200 V; IC = 75 A) and drivers Concept 6SD106EI were used. In the control system, a Freescale 56F8037 digital signal processor is used. The program is loaded and tuned via the USB interface. VII. IMPLEMENTATION OF CONTROL ALGORITHMS Control algorithms were developed according to the control structure of the induction motor drive with the vector control. All interrupt routines are listed in Table III. TABLE III. INTERRUPT ROUTINES OF CONTROL ALGORITHMS Interrupt routine Priority Function Parameter setting, PC Operation of buttons 1 communication PI current controllers, Current control loop 2 transformation, vector rotation, IM current model, decoupling PI speed controller or GA speed Speed control loop 3 controller, speed estimation Flux control loop 3 PI flux controller, field weakening Error routine 4 PWM modulator turn off

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Advances in Electrical and Computer Engineering

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Sampling frequencies of individual loops are the following: PWM - 10 kHz, current control loop - 5 kHz, flux and speed control loop - 0.2 kHz. The DSP works in an endless loop (back program) most of the time. If the interrupt occurs, the program counter can leave the back program. The interrupts are caused by the timers with the sampling period 5 kHz (current control) and the sampling period 0.2 kHz (flux and speed control). The interrupt that switches off the PWM modulator has the highest priority 4 (error routine). The DC link voltage or stator currents are higher than the limit values. The program routine with pressing buttons has the lowest priority 1. The computing time of selected control algorithms are listed in Table IV. TABLE IV. COMPUTING TIME OF CONTROL ALGORITHMS Control algorithm Time [μs] Transformation 3/2 1.5 Vector rotation 2.5 Current model of IM 70 Speed estimation 10 Current controller, flux controller 4 Classical speed controller 5.5 GA speed controller 125 PWM 15

The developed GA speed controller has implemented in real-time. Computing time is 125 μs.

Figure 6. Speed control using GA speed controller, reference speed (blue) and real speed (black)

Figure 7. Speed control using PI speed controller, reference speed (blue) and real speed (black)

been

VIII. EXPERIMENTAL RESULTS The speed control of the vector controlled induction motor was tested in different conditions of rotor angular speed and load torque. The calculations of the genetic algorithm are realized on-line during the work of the A.C. drive. The maximum generation and population size influence the computing time of the GA speed controller. Larger values of these parameters cause longer computing time. Parameters of the speed controllers are shown in Table V. Initial values of the proportional gain Kp and the integral gain Ki were selected experimentally (see Fig. 11 and Fig. 12; Kp_in = 20 A/rad.s-1, Ki_in = 100 A/rad). The output signal of the GA speed controller has a limitation that determines the maximum torque component of the stator current vector.

Figure 8. Speed control using GA speed controller, reference speed (blue) and real speed (black), details about speed of 200 rpm

Figure 9. Speed control using PI speed controller, reference speed (blue) and real speed (black), details about speed of 200 rpm

TABLE V. PARAMETERS OF SPEED CONTROLLERS Classical PI GA speed Parameter controller controller -1 Proportional gain Kp 10 A/rad.s [0 - 20] A/rad.s-1 Integral gain Ki 1000 A/rad [0 - 2000] A/rad Number of generations (i) 20 Number of pairs Kp, Ki 20 in population (j)

The experimental results of the speed control were recorded by the program for setting control variables in the LabVIEW environment. The reference speed is set, first at 0 rpm, and then it is changed to 200 rpm, resp. -200 rpm. At the time 0.6 s, the A.C. drive was loaded by torque TL = 13 Nm, resp. and at the time 2.4s by negative torque s TL = -13 Nm. Figures 6-10 show the time responses of important quantities of the A.C. drive that were obtained by measurement on the laboratory stand.

Figure 10. Induction motor torque (black) TM = f(t), reference load torque (red) TL = f(t)

Figures 11, 12 show changes of the GA speed controller gains Kp, Ki during the speed control. From the design of the classical PI controller, we obtain controller parameters that are constants (Kp = 10 A/rad.s-1, Ki = 1000 A/rad). The GA speed controller provides variable gains Kp, Ki according to the actual state of the control process and optimizes its

19

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Advances in Electrical and Computer Engineering transients. This means, proportional gain Kp is changing between the values Kpmin, Kpmax (Kp = 1 - 20 A/rad.s-1) and integral gain Ki is changing between the values Kimin, Kimax (Ki = 100 - 2000 A/rad; see Fig.11 and Fig. 12).

Volume 14, Number 4, 2014 controller is suitable for the speed control of the vector controlled induction motor. It is necessary to underline the clear conclusion that all the benefits of the control using different soft computing methods are not free of cost and add to the computing performance expenses of the digital signal processor. REFERENCES [1] [2]

[3] Figure 11. Variation of the proportional gain Kp during speed control [4]

[5]

[6]

Figure 12. Variation of the integral gain Ki during speed control

For the assessment of control quality, we can compare the speed waveforms during acceleration and load jump (see Fig. 8 and Fig. 9). There are results for both types of controllers in the following Table VI. TABLE VI. CONTROL QUALITY PARAMETERS GA speed PI speed Quality parameter controller controller Start Load Start Load Rise time [ms] 115 80 110 250 Settling time [ms] 330 110 450 570 Overshoot [%] 5 5 15 40 Steady state error [rpm] 2 2 5 5

The classical controller design is based on the mathematical model of the controlled system that is often multivariable and nonlinear with parameter variation. The classical PI controller has fixed gains. The main advantage is relatively simple structure which can be easily implemented in industrial applications. Therefore, the use of the classical PI controller does not meet the requirements of robust performance. The control quality parameters listed in Table VI confirms this conclusion.

[7]

[8] [9]

[10]

[11] [12] [13] [14]

[15] [16]

IX. CONCLUSION The paper shows a possible application of genetic algorithms in the control of induction motor A.C. drives. Properties of the GA speed controller have been tested in various operating modes with a very low and normal drive speed. The proof of stability for the used GA method is given by the measured experimental results that were obtained in different testing regimes. The experimental results confirm the expected application possibilities. It can be said that the parameters designed by genetic algorithm achieved higher control quality compared to the conventional designing method. The presented GA speed

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