Simulation and Hardware Verification of a PD Fuzzy Speed Controller for a Three Phase Induction Motor Chamorro Vera Harold ReneҰ, Toro BillyҰ, Trujillo CesarҰ, Guarnizo Marin José GuillermoҰ Ұ Alternative Energy Sources Research Laboratory (LIFAE) Faculty of Engineering, Universidad Distrital Bogotá, Colombia
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Abstract – This paper presents a PD fuzzy controller to regulate the nominal speed for a three phase induction motor. Some non linear characteristic of this electrical machine are presented to show the importance of a nonlinear controller. The application of fuzzy controllers in difficult mathematical models is what gives its real importance, it is only necessary some technical parameters of the motor to check the speed response in simulation. Based on some linguistic variables, the fuzzification and the rule base are made, and a tuning of the antecedent membership functions are presented. Finally some simulation regulation tests are used to demonstrate the performance of the fuzzy controller in simulation and hardware. Keywords – Fuzzy Controller, Non Linear Systems, Three Phase Induction Motor, Error, Change in Error.
I.
INTRODUCTION
The recent trend to save energy and efficiency, the development in alternative energy sources, and the daily technological advances in the power field, have created a necessity of a global conscience about the consumption of this. The electronic advances in recent decades in control algorithms to generation, transmission and electric machines [5], have achieved a diversification in power electronics. The electric machines have a great role in human life and energy conversion, the classification of electric machines is based on generators and motors. The three phase induction motors (or asynchronous motors) are widely used in the industry due to its characteristics like resistance, easy maintenance, low cost and durability. The speed variation of these motors is made from zero to nominal speed. The speed depends on frequency and voltage so is a requirement an electronic control. The typical topologies of activation to control are the scalar control [11], vector control or DTC (direct torque control), however these methods require a difficult mathematical model, or not sense the speed directly like current or the slip. The main purpose of fuzzy logic control [1, 3, 7, 13] is use linguistic labels to obtain the rules to control only with the behaviour of the system, so the model is no necessary, just a few parameters are introduced in Matlab® to simulate the control and the system, and the FIS® tool is a good form to obtain to design the controller [2], nevertheless a tuning over the membership functions should be done [8].
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II. MECHANICAL CHARACTERISTICS Clearly the induction motor has nonlinearities like other electrical machines; here two mechanical nonlinear characteristics are exposed briefly. A.
Load Mechanical Characteristics
These characteristics are determined from the dependence between the rotor speed and torque achieved by the mechanism, and this is one of the more important topics in the industry as to classify the kinds of loads as to classify the electric machines, this fact allow to know the operation points of each designed systems [10]. In order to classify the load kinds it is necessary to resort to an empirical equation to graphic a trajectory defining the relation between the speed rotation and torque [10, 5], the equation is:
⎛ ω T = T0 + (TN − T0 )⎜⎜ ⎝ ωN
m
⎞ (1) ⎟⎟ ⎠
where: T = mechanism production torque, T0 = inital torque, TN = torque nominal speed, ω = speed, m = characteristic factor resistance torque changing the speed, Using (1) it is possible to find four basic characteristics depending of the m factor, the induction motor operates with a quadratic load factor m, in other words when m = 2, it is a parabolic characteristic, and this is typical in fans, centrifugal pipes and propellers too. When, m=0, the developed torque is independent of the speed and is itself of systems like cranes or plunger pipes. When, m=1, its a lineal characteristic, there a lineal relation between the speed and torque, and T0 is cero. There is another characteristic when m=-1, is known as decreasing non linear, its varying is inversely proportional between torque and speed, is presented in milling machine, lathes and reels [10] [5]. The figure 1 shows the behaviour of torque achieved and speed, for the different m factor, for all characteristics shown.
The main specifications of induction motor used are presented in Table I. TABLE I. INDUCTION MOTOR SPECIFICATIONS CHARACTERISTIC NOMINAL FREQUENCY LINE VOLTAGE POLES POWER FACTOR NOMINAL CURRENT
Figure 1. Load Factor
B. Mechanical Characteristics of Electric Machinery In rotary electric machines have been defined the mechanical characteristics and its performance according to the next behaviours: - Rigid characteristics: the torque do not affect the speed, this is presented in synchronous motor. - Semi Rigid characteristics: appear when the variation torque produces a speed variation, this is presented in asynchronous motors. - Soft characteristics: meanwhile the torque change the speed have a drastic change, this is presented in motor excitement series. In agreement with the mentioned characteristics of the main electric motors, the figure 2 shows the behaviour of these characteristics.
NOMINAL SPEED NOMINAL POWER
VALUE 50 HZ 208 VRMS 4 0.85 3A 1400 RPM 746W
Table II an III show the specific values of the different tests to obtain the equivalent circuit of the induction motor as specialised literature recommends [5]. TABLE II. INDUCTION MOTOR TESTS NO LOAD TEST BLOCKED ROTOR TEST DC TEST VDC=15V VLINE=VNL=38.7 V VLINE=VBR=38.7V IDC=3A IAV=2.7A IARB=2.9A IBV=2.7A IBRB=2.9A ICV=2.7A ICRB=2.9A FV=60HZ FBR=60HZ PINV=34.66W PINBR=100W
The parameters to introduce in the Simulink® block properties of the three phase induction motor are the nominal power, voltage line, frequency, stator resistance and inductance, rotor resistance and inductance and mutual inductance. TABLE III. INDUCTION MOTOR PARAMETERS
Figure 2. Mechanical Characteristic
CHARACTERISTIC VALUE (Ω) R1 (STATOR RESISTANCE) 5 R2 (ROTOR RESISTANCE) 1.46 X1 (STATOR INDUCTANCE) 3.96 X2 (ROTOR INDUCTANCE) 1.58 43 XM (MUTUAL INDUCTANCE)
III. INDUCTION MOTOR TESTS
IV. SPEED RESPONSE INDUCTION MOTOR
In order to obtain the induction motor parameters to its electrical equivalent model, as the figure 3 depicts, is necessary make the three tests of real motor, such as the DC test, blocked rotor test and no load test [5].
To simulate the system is used Matlab® Simulink®, is generating the AC signals from VSC (Voltage Source Converter) [6] built with IGBT`s [12] from SPWM (Sinusoidal Pulse Width Modulation) trough modulation index and frequency.
Figure 3. Electrical Equivalent Motor
The VSC is the basic part in the electric drive, this synthesize with SPWM signals switching the full bridge to generate an AC signal with three phases, with these signals is not complicated make the variations in frequency and voltage to establish the constant v/f relation to obtain an torque operation in the induction motor [12]. When the motor start from zero speed the carrier signal could achieve big values, staying constant until a frequency range (synchronous operation), in this way the wave generated is almost sinusoidal achieving a great soft rotation.
Figure 5 illustrates the complete system with SPWM, VSC and the induction motor blocks, to measure the speed which is the feedback variable, is necessary an extra block in SimPower® library, to change the speed in rad/s to rpm.
V. FUZZY CONTROLLER DESIGN The overall system with controller is represented in the next figure, in this picture are two subsystems, the first one have the controller with a normalization scale with the two inputs, the speed reference and the speed, those generate the error and change in error, for this reason is called PD fuzzy controller. The control outputs are the increment or decrement in the modulation index and frequency.
Figure 9. Complete System Figure 5. Simulink diagram of the system
Figure 6 show the results of simulation system with an index modulation near to 1, with those parameters the induction motor reach its nominal speed, this response is typical in theoretical models [2] and real probes [5]. A. Input Signals and Fuzzy Sets Configuration The maximum limits to error and change in error are between [0.18, -0.05] with the scale factor, as it can be seen in figures 10 and 11.
Figure 6. Speed Response
Some torque loads are applied to the induction motor, in different changes as it is shown in figure 8, and the speed decline in steps as it should happen.
Figure 10. Change in Error
Figure 8. Speed Variation due to change in Load
Figure 11. Error
The proposed linguistic variables according to the Error (E) and Change in Error (CiE) are condensed in the Table IV, where (N) is Negative, (P) Positive, (G) Large, (M) Medium, (I) Increasing, (D) Decrement, (C) Change and another P is for Small, for example PP (Positive Small). TABLE IV. INPUT AND OUTPUT LINGUISTIC VARIABLES E
CIE
NG
CNG
DG
NP
CNP
DM
CE
ΔM, ΔF
DP
PP
CPG
PG
CPP
SC IP IM
The inference engine that relates inputs with output is performed using Mamdani’s implication [9] [14], which could be defined with a minimum (2) or a product (3), the latter one is also denoted as Larsen implication.
μ AMM ( x, y ) = min[μ p ( x), μ q ( y )]
(2)
μ AMP ( x, y ) = μ p ( x) ⋅ μ q ( y )
(3)
With the design choices the control surfaces degenerates to a diagonal plane for the two membership outputs deltam and deltaf, these surfaces (figure 15) are similar due to as is mentioned the relation v/f should be constant. The Matlab RuleViewer is shown too, to see a specific rule (figure 14).
IG
The concept of tuning fuzzy controls explains how construct a rule base with a linear input – output mapping that acts like a summation an is exposed in [8], there are a list to achieve this, such as the use of triangular input sets that cross near to μ = 0.5, use the connective and the rule base must be the outer and product of all inputs, and is made a recommendation of the use of CoG (centre of gravity) defuzzification. Following those rules and changing the membership functions to the limits in error and change in error is obtained the figures 12 and 13.
Figure 14. Matlab RuleViewer Picture Figure 12. Change in Error Membership Function
Figure 15. Control Surfaces Figure 13. Error Membership Function
B. Rules Basis and Inference Engine The fuzzy rules basis is tabulated in the table shown in V. This table is used to determine the fuzzy controller output for a given deltam and deltaf. It is seen that there are (3x4) 12 combinations of the mentioned variables. A typical rule could be: Rule 1: IF (E is PG) AND (CiE is CN) THEN (deltam is SC) (deltaf is SC). TABLA V. FUZZY CONTROL RULE BASE DECISION TABLE ∆E/E N NP CE PP CN DG DM DP DP CPG DP IP IP IM CPP SC SC SC IP
C. Defuzzification Once the values from the inference engine are obtained, is necessary to find a crisp value at the output system that represents in a good way the fuzzy set. From the obtained values at the inference, it requires now a specific value at the output system that represents the best mode to diffuse the whole crisp obtained [14]. The defuzzifier elected is the average of centres, which is an approximation of the centre of gravity and as defined in (4):
∑ y w y* = ∑ w M
−n
n
n=1 M
n=1
(4)
n
Where M is the number of fuzzy sets, w are the weights of set defined for its height y y-1 is the centre of n-esimal fuzzy set.
The output system ‘fuzzy of inference engine’ is defined according to basis rule, so in this case match with the sets of increment or decrement labels. These outputs sets of membership functions represent the increment or decrement in the modulation index and frequency that is why is called deltam and deltaf (figure 16). The label SC means No change and it happens when is no necessary make an increment or decrement.
The second probe is similar to the first, however in a moment the load is changed to zero, and then the speed falls for a time and the controller seek to bring it to nominal speed again, it has a time with some oscillations with a new change of load but the nominal speed is regulated. The load reference changes and speed picture are in the figure 18.
Figure 16. Output Membership Fuctions
VI. REGULATION PROBES There are many regulation tests to demonstrate the control speed regulation action, so in this document it has been made two probes, always with the same objective: regulate the nominal speed regardless the load of the induction motor, of course the load in a suitable work range. The first one is changing the loads torque, but the first load to probe start after the motor reach the rated speed, otherwise could be enter in a total unstable zone. The next load enter a few seconds after the regulation speed, first increasing the load and then decreasing, this is showed in the figure 17.
Figure 18. Speed Response with suddenly change load to cero and return
These probes confirm the good results of fuzzy logic control tuned in a specific speed reference with load variation. VII. PROBES IN HARDWARE IMPLEMENTATION Due to the control variable is the speed, is necessary take a direct measure of this, is implemented a electronic tachometer with two outputs to achieve it, a 8 bits microcontroller (in this application is used the MC68HC908AP16® of FreeScale®), the first output shows the speed in rpm in a LCD and the other a dc value according with the speed, where the saturation value is 4V. The dc value is the input for a DSC (Digital Signal Controller), this is the dsPIC30F3011™ from Microchip®, which has the characteristics to process the feedback signal and the fuzzy engine inference, also it has DSP functions to make simpler the programming the fuzzy logic controller in C language. The algorithm program is not developed in this document because is not the study object in this document. It is possible observe the speed response from the dc value, to verify the open loop and control action of the three phase motor speed using the digital multimeter UNI-T 60A, which has a PC graphical interface as the figure shown.
Figure 17. Test changing torque loads
Figure 19. Tool Graphical Interface
Next figure shows the speed response zoom, in open loop where the nominal speed 1400 rpm fits with 3.9V dc.
Figure 22. Board Hardware Implementation
VIII.
Figure 20. Open loop speed response
As it can be seen in figure 20 the simulated speed response corresponds with the data acquired by the tool. Now a mechanical load is tested, the figure 21 shows the good response of fuzzy PD control regulating the nominal speed response in spite of different mechanical loads.
CONCLUSIONS
It has been demonstrated that the fuzzy logic control is a good solution in nonlinear systems and complicated mathematical models like this application, and the versatility and capability of Matlab® software to simulate electric machines with a desired performance. Notice a non symmetrical membership are proposed, this is a not recommendation however works for this application and has been tuned trying to minimize the rule basis; maybe there is another possibility to use less rules with a better response. The tuning control surfaces demonstrate to be near to a diagonal plane which is the theoretical control surface. The typical tests in three phase induction motor demonstrate that a fuzzy control is effective to the regulation action having account is a speed control with a feedback of the error and change in error speed. The hardware development verified the good control action of simulation results and motor test with the real induction motor. IX. REFERENCES
Figure 21. Control action with load test
Finally the prototype test implemented is presented. The VSC converter designed use MOSFETS (IRFP450), it can be seen down the motor and the boards with the DSC and tachometer are in the right side from it (figure 22).
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