A Robust Fuzzy Speed Control Applied to a Three-Phase Inverter Feeding a Three-Phase Induction Motor. A.T. Leão (MSc) E.P. Teixeira (Dr) J.R. Camacho (PhD) H.R de Azevedo (Dr) Universidade Federal de Uberlândia School of Electrical Engineering P.O.Box: 593 - Campus Santa Mônica CEP: 38400-90 - Uberlândia MG Brazil e.mail: rcamacho@ufu.br ABSTRACT In agreement with extensive development in power electronics, the induction motor became widely used in industrial, commercial and residential applications, being responsible for a huge consumption of energy. Basic inverter switching, normally results in non-sinusoidal output voltages and currents, which may affect the performance of a non-linear load. The generation by the inverter of an "a.c." waveform with low harmonic content is extremely important, harmonic filters are not an option, when controlling speed, due to the large range in the frequency spectrum at the inverter output. With the inverter feeding an "a.c." motor, the frequency variation at the inverter output terminals must be followed by a change in the applied voltage, in order to keep unchanged the magnetic properties at the machine's air-gap. A double control is necessary. The first through an external robust fuzzy speed control loop and the second through an internal "PID" current control loop. This work presents a driving scheme in which the machine's three windings are fed through a static converter. Such tool is responsible for the machine's speed and current control. The aim is to improve the speed response and to eliminate operational undesirable torque oscillations. and rotor. Therefore it is possible to obtain the motor performance neglecting the saturation effects and considering the inverter coupling effect. A comparative study is made between an open loop inverter - induction motor driving system and the proposed double loop (speed and current) control, being respectively robust fuzzy logic speed control and conventional "PID" current control. MATHEMATICAL MODEL FOR THE THREE-PHASE INDUCTION MOTOR Among the electrical machines, the induction motor is one of the most applied machines and, detailed studies in order to improve the performance of its speed control is a very ustifiable matter. Following this path many researches[4] have been propelled by new discoveries in the electronics and in the semi-conductor components field. In order to simulate an induction motor driven by an inverter, it is mandatory for its mathematical model to represent the machine in steady-state and transient states, for any kind of source sinusoidal or not, balanced or unbalanced. That is the reason why the induction motor is represented here in phase coordinates. Keywords: Induction motor, inverter, fuzzy control. 1 INTRODUCTION Electronic driving systems made possible the speed and/or torque variation for induction motors through the use of static "ac-ac" converters, responsible for the conversion of fixed to variable voltage and frequency. The inverter, last stage of a electronic converter, presents a speed control with good efficiency. The speed control is possible through electronic switches allowing such variation through the very known pulse wih modulation (PWM), where the pulses are the result of a comparison of two input signals. The present work is a theoretical evaluation of the behavior of a three-phase induction motor fed by a PWM inverter with feedback control loops of speed and current.. Therefore, was implemented a computational program which is able to simulate the symmetrical three-phase induction motor speed control through the use of a fuzzy logical controller assembled in Matlab /Simulink TM. A model for the three-phase induction motor with squirrel cage rotor having as a reference the "ABC" stator phase axis, is also presented. The model uses the concept of phase windings distribution and windings space harmonics. Then, it is obtained the model for the magnetic field density distribution in the stator (a) (b) Figure 1 (a) Winding generic coil and (b) its mmf (FMM b(α)) distribution..1 Time Domain Harmonic Analysis The influence of the windings over the voltage and/or current waveforms and over the induction machine torque is the first step to obtain the mathematical modeling for the induction motor
taking in consideration the space harmonics for the stator magnetomotrice forces mmf distribution[3]. Considering a coil belonging to the mentioned winding, with N turns, polar step given by βπ and centered in a position α b. Figure 1 shows this coil in its schematic shape and the ideal mmf spatial distribution for an instantaneous value for the current i in the coil. The analytical expression who defines the mmf distribution can be seen in Figure 1(b) it is called FMM b (α). Applying Fourier series to FMM b (α), and considering the induction motor winding, one of its phases will be nominated as. This phase is composed of q coils distributed with a central reference position α, where is circulating the phase current i. Every coil has N turns and a coil step equals to β π. Each of the q coils originates a mmf order h harmonic component. Superimposing the harmonic components for the q coils it obtained the following harmonic equation: 1 FMM h ( α ) =. N. q. k ph. kdh. i..cos[ h. ( α α )] (1) π. Electrical Equations For an induction machine generic phase i, the voltage equation can be written as: dλi Vi = Ri. ii + (4) where: R i is the phase i winding resistance, i i is the phase current, λ i is the mutual total flux, including the leakage flux. Using the total mutual flux of Equation (4) the following expression can be developed for the induction motor stator phase a : d( L) d ( ) ( i ) Va = Ra. ia + K. ω r.. i + K. L. el dθ (5) where: θ el is the rotor electrical angular position related to a stator reference, is equivalent to a, b, and c for the stator and A, B and C for the rotor phases, R a is the winding resistance at phase a stator phase, L is the motor inductance X matrices as a θ el function, K = m 3π.. f, X is the motor m magnetization reactance, f is the motor frequency. el.3 Induction Motor Swing Equation Figure - Phase "" winding coil distribution. Where k ph and k dh are respectively nominated as step and distribution factors. Clarifying the fact that the magnetic circuit reluctance of iron parts are neglected, applying Ampere s law at the air-gap, the magnetic field density distribution B h can be obtained, given by: µ 0 1 B α =.. N. q. k. k. i..cos h. α α () h ( ) [ ( )] ph π δ h where µ 0 is the air magnetic permeability and δ is the air-gap length. Considering now any another phase i of the motor, which has all the already previously defined parameters indicated by the i index. With the help of Equation () the magnetic flux produced by phase and embracing phase i for a given harmonic h, can be described as: dh [ h ( α α )] kwih. k ' wh λih = Ki.. i.cos. i (3) h p. L. R. µ 0. q... ' i q Ni N where: K i = 4., k wih = k pih.k dih is the π. δ phase i winding factor, k wh = k ph.k dh is the phase winding factor, and p = number of pole pairs. Starting from a system formed by generically represented windings by phases i and and, having the structural conditions to the electromechanical energy conversion, the following expression can be written for the electromagnetic torque[3]: p diih Tel =. ii. i. (6) dθ h i el where I ih are the motor harmonic order inductances between two generic phases, defined as previously. Defining T c as the motor load torque the swing equation in this case can be written as: dw J r = Tel T (7) c where: J is the moment of inertia, and w r is the rotor mechanical angular speed. Equation (6) is the complement for the induction motor mathematical model by conecting electrical and mechanical characteristics. 3 THREE-PHASE INDUCTION MOTOR SIMULATION FED THROUGH A PWM INVERTER WITHOUT CONTROL This item shows the model of a three-phase induction motor fed by a balanced sinusoidal waveform. The validation of the developed model has been done in a previous publication through the comparison of simulation results and laboratory tests.[6] Such mathematical model is linear and the differential equations representing the motor are using current as state variables. Simulation can represent the model fed directly from the mains (sinusoidal waveform) or from an electronic driver. This model takes in account time and space harmonics in the electromechanical conversion process. It is considered a load that offers a torque that changes proportionally to the rotor speed squared. Figure 3 - Mutual flux for phase "i" winding.
Where at the block "Motor Constants" are the input data and the induction motor parameters. Those parameters are obtained at no-load and short-circuit tests, presented in the following table. Figure 4 Three-phase induction motor fed by an inverter, the main block diagram. It is possible then, to verify the variation of parameters that pictures the induction motor behavior such as: - stator and rotor currents; - developed torque; - load torque; - angular speed from starting to steady-state condition. 3.1 Induction motor main block diagram. Table 1. Parameters for the induction motor. Machine Parameters Obtained Values Stator resistance 4.913 Ω Rotor resistance 4.913 Ω Stator leakage inductance 9.3e-3 H Rotor leakage inductance 9.3e-3 H Magnetization inductance 0.196 H Pole number 4 Inertia moment 0.0045 Kg.m Power.0 HP (1.5 KW) Nominal speed 170 rpm Power factor 0.78 Nominal current 6.9/3.99 A Nominal voltage 0/380 V The Fharm block is responsible for the calculation of harmonic parameters necessary to the complete model development, while the Fmit block is in charge of the state variables calculation for the model using a matrix formulation, and the last stage the Ftmotor block which calculates the electromagnetic torque. A detailed vision of the block diagram "Switch 1-4" is given at Figure 7. Figure 4 shows the induction motor main block diagram, extracted from the matrix formulation and representing the machine when submitted to voltages created by a pulse wih modulation source, without feedback control. Figure 5 shows in detail the induction motor block diagram from Figure 4. The induction motor is modeled using the ABC reference frame and can calculate up to the 55 th harmonic. Figure 7 Block diagram for Switch 1-4. Block diagrams for "switch 3-6 and switch 5- at Figure 6 are identical to this one shown in Figure 7. Figure 5 Three-phase symmetrical induction motor block diagram. The "PWM Inverter" block diagram at Figure 4 can be seen in detail at Figure 6. Simulation results for the motor fed by a PWM inverter at no load condition, with constant voltage and frequency, can be seen in the following figures. Figure 6 "PWM Inverter" block diagram. Figure 8 Phase to phase voltage.
Figure 9 - Phase stator current. Figure 1 - Main block for the current and speed controller. Figure 13 - Fuzzy speed controller block. Figure 10 Machine's angular speed in rpm. The behavior of angular speed and the electromagnetic torque for the machine at no-load can be seen respectively at Figures 10 and 11. The output variable is the increment to be applied in the action variable at the controller, in this case this variable is the output frequency for the controller. Figure 13 shows the three variables at the fuzzy speed controller block. The current controller block at Figure 1 is shown in detail in Figure 14. Figure 14 - Current controller internal structure. Figure 11- Machine's electromagnetic torque in N.m. A detailed block in Figure 15 shows the reference block A. 4 THREE-PHASE INDUCTION MOTOR SIMULATION WITH SPEED AND CURRENT CONTROLERS In this section are presented results of digital simulation through a fuzzy logic speed controller and a fixed frequency PWM controller for the phase currents. Figure 1 shows a block diagram of all the system including machine, speed controller and current controller. Figure 13 shows the configuration of a block called "Fuzzy Speed Controller" at Figure 1. The fuzzy logic controller input variables are defined by the error, the difference between reference speed and mechanical speed (W ref W mec ) and through error variation which is how much the error changes in time. Figure 15 - Reference block A. The reference blocks B and C are identical to the one presented in Figure 15, except that the "Fcn" block, which deals with each output signal sinusoidal waveform, should be set to operate with a phase angle difference of 10 degrees.
5 SIMULATION RESULTS A number of simulations has been done in order to follow the behavior of the speed and current controllers, some of the results obtained are shown in the figures below: fed by pulse-wih modulation (PWM) inverter. In order to get speed control it was used a fuzzy logic controller. Phase currents were adusted and controlled, current high frequency oscillations were observed due inverter switching. Figure 16 Stator current - Isa, Isb e Isc Figure 16 above shows the stator current for phases A, B and C, under no-load operation. The mechanical speed pick-up with the motor operating at 60 rad/s can be seen below in Figure 17. Figure 19 Mechanical speed (rotor) at 170 rad./s. Through the obtained results, can be observed that both controllers (fuzzy logic and current controller) are able to generate together a good result. This can be easily verified through the speed and current figures along this text with machine being operated with and without controller. As a contribution for further future work concerning this matter, should be indicated the electromagnetic torque behavior with the induction motor operating at nominal load. This should be done in order to properly tune both controls. 7 REFERENCES Figure 17 Mechanical speed (rotor) at 60 rad./s. Mechanical speed pick-up with the motor operating at 10 rad/s can be seen in the following figure. Figure 18 Mechanical speed (rotor) at 10 rad./s. [1] B.P. ALVARENGA, Induction Machine Torque Calculation Model Including Winding and Magnetic Saturation Effects, MSc Thesis (In Portuguese), Electrical Engineering, Universidade Federal de Uberlândia, August 1993. [] A.E. FITZGERALD, C. KINGSLEY, S.D. UMANS, Electric Machinery. 5 th edition. New York: McGraw-Hill, 1990. [3] G.B. KLIMAN, A.B. PLUNKETT, Development of a Modulation Strategy for a PWM Inverter Drive. IEEE Transactions on Industry Applications, v.ia-15, n.1, p.7-79, January/February. 1979. [4] I. BARBI, Basic Theory on Induction Motors, (In Portuguese), Editor: Universidade Federal de Santa Catarina, Florianópolis, Santa Catarina, 1985. [5] A.T. LEÃO, Computational Model of the Driving of a Three- Phase Induction Motor in A, B, C Coordinates Through a Sinusoidal Modulation PWM Inverter, MSc Dissertation (in Portuguese), School of Electrical Engineering, Universidade Federal de Uberlândia, December 1998. [6] L. M. NETO, J.R. CAMACHO, C.H. SALERNO E B.P. ALVARENGA, Analysis of a Three-Phase Induction Machine Including Time and Space Harmonic Effects: The A, B, C Reference Frame, PES/IEEE Transactions on Energy Conversion, Vol. 14, Number 1, pp. 80-85, March 1999, Piscataway, NJ, USA. Mechanical speed pick-up with the motor operating at 170 rad/s can be seen in Figure 19. 6 CONCLUSION As a contribution to this work it was developed a computer program which could verify the behavior of a symmetrical threephase induction motor under controlled speed and current, and