Recent Researches in Automatic Control, Systems Science and Communications
The Application of Genetic Algorithms in Electrical Drives to Optimize the PWM Modulation ANDRÉS FERNANDO LIZCANO VILLAMIZAR, JORGE LUIS DÍAZ RODRÍGUEZ, ALDO PARDO GARCÍA. Universidad de Pamplona, Pamplona, Colombia
[email protected],
[email protected],
[email protected].
Abstract: - This paper describes a new strategy for pulsewidth modulation (PWM) optimized by a direct method and the application of genetic algorithms (GA) to minimize the harmonic content specifically the fifth and seventh harmonic of the total content, based on the minimum total harmonic distortion (THD). Described the development of the method, the guidelines to take into account including the algorithm development, the strategy within the Digital Signal Processor (DSP), and the obtained results can be visualized.
Key-Words: - Pulse Width Modulation, Selective Harmonic Elimination, Genetic Algorithms, Harmonic Distortion, Digital Signal Processing, Fast Fourier Transform.
I.
harmonics are to be the most representative of lower order produced by the inverter which generates no even harmonics and the 3rd order harmonic are suppressed proper connection with a motorcycle
Introduction.
The great development of industrial processes supported by AC power has increased the demand for highly reliable strategies for optimal control and operation thereof, that is why, it is necessary to study and investigate various processes to meet those needs, which give rise to the development of applications such as the one presented in this work [1-4]. One of the major reasons is the reduction of energy consumption of the devices, in addition to this reduction in switching losses and reducing harmonic content. The use of artificial intelligence techniques such as genetic algorithms are, establish a new optimization tool, despite its relative current implementation is not complex, so it is an efficient and attractive tool in solving a optimization problem [5]. Reviewing the literature on the subject can verify that the application of genetic algorithms in electrical drives to optimize the PWM modulation frequency inverters, data of the last decade, in the early work on the subject include that of the Authors K. L. Shi and H. Li [6], this paper applies a genetic algorithm to optimize a PWM inverter, with the results obtained over sinusoidal PWM modulation (SPWM) standard triangular and random PWM. However, both modulations are not a good starting point, since you can add other parameters such as modulation SPWM triangular asymmetry [7, 8] or use other PWM modulation to ensure lower content of harmonic distortion [9]. Others can also be used as optimization criteria harmonic is the harmonic content of the 5th and 7th harmonics, which
ISBN: 978-1-61804-103-6
II. PWM modulation with the Direct method The rise of programmable digital devices, such as Digital Signal Processors or DSP's, have enabled these techniques PWM can be implemented in a comfortable and efficient way, allowing the great evolution of the modulation strategies including decreased costs for its development [4, 9]. To generate a PWM requires a certain set of switching angles (positions) are determined using numerical and computational methods. Figure 1 shows a direct method PWM of 3 pulses [4, 9]. The direct method PWM has the advantage of not using a carrier signal, as the case of the Sine PWM modulation (SPWM), since the algorithm generates the switching angles by means of calculation (simulation) and then is recorded to the digital electronic device.
Fig 1. Direct method PWM modulation.
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Recent Researches in Automatic Control, Systems Science and Communications
We discuss the procedure used to generate the Direct Method PWM from the following expressions [4, 9]. From the sine reference expression:
2 sin
V
Sign
2
fase 4
1
(1)
u Vm sen t
Mod
Xi i
fix Xi
Where: u Desired output voltage [V]. Vm Maximum value (amplitude) [V]. Angular frequency ( = 2f) [rad/sec]. f Frequency [Hz]. The area of a half period (0 to / ).
Aseno Vm sen t dt 0
Vm
cos t
0
2Vm
Scope D
pi
3
cos
Q
C !Q
Np
cos
Tpi
PWM 1
5 Pp
(2)
1
Fig. 2. Simulink® Model of Direct Method PWM modulation With this predecessor block is generated which has the following inputs: the amplitude (V), frequency (), the number of pulses (Np) and the asymmetry of the triangular carrier signal (Pulse position or Pp), as shown in figure 3, it varies in a range of 0 to 1 and determines the position of the pulses.
This can be divided into Np (number of pulses) regular time intervals as shown below [4, 9]: t1 np 2 i t2 ti np n p n tn p np
Xi_a Xi_a
Tiempo
t0 0
(3) i 0, 1, 2,, n p
The area of each of the intervals is: ti
Ai Vm sen t dt ti 1
Ai
Vm
Vm
cos t
ti ti 1
(4)
cos ti 1 cos ti
Knowing the area of each interval and setting the maximum value of the PWM waveform equal to the amplitude of the sinusoidal signal, one can determine the width of the pulses. t pi
Ai Vm
Fig. 3. Pulse position values. Figure 4 shows the result of modulation to an inverter stage and the next shows the spectrum of harmonics to a fundamental frequency of 60 Hz, a pulse number Np = 33 and pulse position Pp = 0.45 asymmetrical with respect to central axis reference. In figure 5 shows the simulation program for the description of PWM developed and the respective calculation of harmonic content (THD) total and 5th and 7th harmonics both graphically and numerically.
(5)
Finally, for the formation of the PMW signal under the criterion above, are located each of the pulses obtained at the center of each of the intervals, mathematically expressing this set of pulses in an x,y coordinate plane, where the axis x corresponds to the time axis y to the signal amplitude [4], as shown: xi 1 xi t pi x xi t pi x i 1 , Pulsoi x, y 2 2 2 2 0 y Vm
(6)
where i 0, 1, 2,, n p
A. System Simulation To validate the above methodology and obtain the PWM modulation optimized by the direct method and a model is developed using the Simulink® for generating the PWM (Figure 2).
ISBN: 978-1-61804-103-6
Fig. 4. PWM modulation for np = 33 and pp = 0.45.
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Recent Researches in Automatic Control, Systems Science and Communications
position) representing possible solutions to the problem in particular you want to optimize and find the lowest harmonic distortion of the system, which makes the objective function (fitness) to optimize the harmonic distortion is equivalent to the 5th and 7th harmonics of the PWM output signal. Each of the values THD is subtracted from the constant α which corresponds to the maximum permitted standard distortion value within the genetic algorithm. The objective function is: F
(7)
Where: F Objective function (fitness). Maximum fitness value. v5 Magnitude of 5th harmonic voltage. v7 Magnitude of 7th harmonic voltage. v1 Magnitude of main harmonic voltage. In the above equation are taken only the magnitudes of the 5th and 7th harmonic, since the shape of the signal must present only odd harmonics and the third harmonic is then removed from the delta connection of three phase asynchronous motor. The representative harmonics by low order would be the 5th and 7th harmonic, the other harmonics for being away from the fundamental frequency much less influence on the functioning of the machine. In figure 7 shows the block diagram of the structure of the genetic algorithm implemented [7]. As seen from this figure the first step performed is to create an initial population that refers to possible solutions to the problem, then makes the evaluation of the objective function and only the fittest individuals will advance to the next stage in the which is applied to the mutation and crossover to generate new individuals with better characteristics to be evaluated again. This procedure is repeated successively until you find the best possible outcome in 100 generations. The genetic algorithm can be stopped in two ways, first is whether it meets the desired optimization and the second if you end up running the stipulated number of generations. One of the great advantages of using a genetic algorithm is that it ensures that this does not lead to an answer of zero error achieved by the end of the process the individual best suited to achieve the lowest possible error. As mentioned above genetic algorithms based on the fact of random solutions, in our case study we decided to take a range of individuals from 100 according to the available hardware to apply the tool which is restricted to the number of possible solutions, which prevents us from establishing a wider range of individuals.
Fig. 5. Simulink® model for harmonic distortion. Figure 6 shows the application of the Fast Fourier Transform (FFT) to the PWM modulation.
Fig 6. FFT analysis for PWM signal. Input parameters are positioned randomly, which does not ensure that the PWM modulation obtained is the best, as can be seen the frequency spectrum, including harmonic content has direct component, which is undesirable. But he notes that the fifth and seventh harmonics are quite low in amplitude that makes it not all bad the response. Numerically obtain the result of total harmonic distortion for the fifth and seventh harmonic equal to 23%, which is adequate but can be reduced further with an optimization process.
III. Optimization of the PWM Modulation with Genetic Algorithms The purpose of the application of genetic algorithms is the optimization of the PWM modulation, this is achieved by creating a set of individuals (different values of frequency, pulse number and pulse
ISBN: 978-1-61804-103-6
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Recent Researches in Automatic Control, Systems Science and Communications
The algorithm is developed in Simulink tool of Matlab, which works interactively with the data generated by the model of three-phase inverter, so the algorithm is applied immediately and are generated optimal solutions for reducing of harmonic distortion.
To show the algorithm by which to obtain the optimum individual and get the least distortion is shown in Figure 8 the Fast Fourier Transform (FFT) of the PWM signal optimized for 60 Hz.
Fig. 8. PWM Signal Spectrum for 60 Hz. The figure above shows the decrease of 5th and 7th harmonics in a considerable way, the absence of even harmonics and the increase of the 3rd harmonic is not important because the motor delta connection of phase does not allow these to circulate the power supply. When applying the PWM modulation can be optimized to get the motor current waveform and its frequency spectrum (Figure 9) which yields a harmonic distortion percentage less than 1%.
Fig 7. Genetic algorithm diagram.
Fig. 9. Spectrum of power to 60 Hz.
Genetic algorithms, have a common feature with the various artificial intelligence techniques, it is significant or considerable time it takes to run and find possible solutions to our case and with the number of individuals is taken from 4 to 6 hours to run as a whole, the idea of taking a genetic algorithm as a tool for finding solutions is to fully optimize the result. For each of the frequencies algorithm 100 evaluates different possible solutions (individuals), statistics are generated when the algorithm classifies each of the solutions and placed in the first position the response clearer and better results (optimal) to obtain a significant reduction in harmonic distortion. A. Results Table I shows the results for some of the frequencies near 60 Hz, which presents the best-fit parameters for the implementation of modulation. Table I: Optimization for frequency near (60 Hz). Frequency (Hz)
Pulses (Np)
Position (0
