Electr Eng (2007) 89: 285–291 DOI 10.1007/s00202-006-0003-9
O R I G I NA L PA P E R
Mohamed S. A. Dahidah · M. V. C. Rao
A hybrid genetic algorithm for selective harmonic elimination PWM AC/AC converter control
Received: 10 May 2005 / Accepted: 19 November 2005 / Published online: 14 March 2006 © Springer-Verlag 2006
Abstract This paper presents an optimal solution for eliminating pre-specified orders of harmonics from the output waveform of AC/AC converter. The main challenge of solving the associated nonlinear equations, which are transcendental in nature and therefore have multiple solutions, is the convergence and therefore an initial point selected considerably close to the exact solution is required. The paper discusses an efficient hybrid real coded genetic algorithm (HRCGA) that reduces significantly the computational burden resulting in a fast convergence. An objective function describing a measure of effectiveness of eliminating selected orders of harmonics, while controlling the fundamental is derived and a comparison of different operating points is reported. It is observed that the modulation index can reach unity value. The theoretical findings are verified through simulation results using the PSIM software package. Keywords Genetic algorithms · Selective harmonic elimination (SHE) · Pulse width modulation (PWM) · AC/AC converter
1 Introduction The developments recently achieved in the field of power electronics made it possible to improve the performance of electrical system utilities. Some of the advantages offered by using such devices are fast response, compactness and low power demands of control circuitry. On the other hand, control by switching is often accompanied by extra losses due to time harmonics presented in output voltage waveform, which leads to poor power factor. An AC voltage regulator is used as one of the power electronics system to control an output AC M. S. A. Dahidah (B) · M. V. C. Rao Faculty of Engineering and Technology, Multimedia University, 75450, Ayer Keroh Lama-Melaka, Malaysia E-mails:
[email protected];
[email protected] Tel.: +606–2523176 Fax: +606–2316552
voltage due to its simplicity and the ability of controlling large amount of power economically [1]. The pulse-width modulation (PWM) AC chopper has been suggested as an alternative to conventional AC regulator due to improved power factor and the possibility of eliminating low -order harmonics with a wide voltage control range. The harmonic elimination methods which introduced to AC converters are similar to those employed in PWM inverters or in AC to DC converters. The common characteristic of the selective harmonic elimination–pulse–width modulation (SHE–PWM) method is that the waveform analysis is performed using Fourier theory. Sets of nonlinear transcendental equations are then derived, and the solution is obtained using an iterative procedure, mostly by Newton–Raphson method [2]. This method is derivative-dependent and may end in local optima; further, a judicious choice of the initial values alone will guarantee convergence [3]. Another approach uses Walsh functions [4] where solving linear equations, instead of solving nonlinear transcendental equations, optimizes the switching angles. A number of cost functions have been studied in the literature such as eliminating selected harmonics [2–6], minimizing THD [6] or minimizing motor losses [7]. All abovementioned methods are off-line techniques that suffer from the requirement of the large memory to accompany the different operating points. The on-line real-time elimination techniques, which are outside the scope of this paper, are presented in several publications where nonlinear curves of the nonlinear transcendental equations are replaced with piecewise representations, with each curve made up of one or more straight line segments [7]. It also has been reported in [8] that by using accurate modeling techniques it is possible to define an analytic expression describing the law governing the switching angles of PWM that is implemented using microprocessors or digital signal processing (DSP) [8]. In contrast, microprocessor-based techniques require a large memory, whereas most of the computational effort of the microprocessor is spent on tedious timing tasks associated with the generation of the switching signals for individual
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Fig. 1 a Circuit of Pulse-width modulation (PWM) AC converter. b Output voltage
phases of the inverter. The application of neural-network technology has been adapted to allow construction of a simple microprocessorless PWM that realizes optimal switching angles for the inverters, for any value of the modulation index [9]. The objective of this paper is to introduce a minimization technique assisted with a hybrid genetic algorithm in order to reduce the computational burden associated with the nonlinear transcendental equations of the SHE–PWM method. An accurate solution is guaranteed even for a number of switching angles that is higher than other techniques would be able to calculate for a given computational effort. The proposed algorithm could calculate angles even beyond the point where other methods fail to converge and hence it seems to be a promising method for applications when overmodulation is required. It is observed that the modulation index can reach unity without resulting in failure of convergence.
in the PWM form at the load terminals. By the proper choice of PWM switching angles, the fundamental component can be controlled and a selected low order of harmonics can be eliminated. Setting up N switching angles per-quarter cycle allow the elimination of (N − 1) low-order harmonics and the remaining angle is used to control the fundamental component. Consider for example, an output voltage of AC converter with N = 5 pulses perquarter cycle as in the case of Fig. 1b. Fourier series can easily express the general output voltage of the inverter as follows:
2 Circuit analysis and problem formulation
vo (ωt) =
vo (ωt) =
∞
An cos nωt + Bn sin nωt
(1)
n=1
Owing to the PWM waveform characteristics of odd function symmetry and halfwave symmetry, the output voltage can be reduced to: ∞
Bn sin nωt
(2)
n=1,3,5,...
The basic power circuit of a PWM AC converter that consists of two bi-directional switches (SW1 and SW2 ) is illustrated in Fig. 1a. The series connected switch (SW1 ) regulates the power delivered to the load, and the parallel switch (SW2 ) provides the freewheeling path to discharge the stored energy when the series one is turned off [1]. When a switching function shown in Fig. 1b, is applied, the output voltage appears
where, Bn is the Fourier coefficient Hence, one can write the magnitude of the harmonic components including the fundamental as: 5 2 sin 2αi i B1 = 1 − (−1) αi − π 2 i=1
(3)
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5 2 sin(n − 1)αi sin(n + 1)αi Bn = − π n−1 n+1
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(4)
i=1
where n = 3, 5, . . . , 2N − 1, N is the number of switching angles per-quarter cycle (i.e. N = 5 in this case), and αi is the ith switching angle. An objective function describing a measure of effectiveness of eliminating selected order of harmonics by controlling the fundamental, i.e. eliminating low order of harmonics is specified as F(α1 , α2 , . . . , α N ) (5) = (B1 − m i )2 + B32 + B52 + · · · + Bn2 where m i is the modulation index which is defined as m i = (V1 /Vm ), noting that V1 and Vm are the values of the fundamental component and the peak of the input voltage, respectively Minimizing (5), that is subject to the constraint of (6), the optimal switching angles are generated and consequently selected harmonics are eliminated. These switching angles are generated for different values of modulation index and then stored in look-up tables to be used to control the converter for certain operating point. π 0 ≤ α1 ≤ α2 ≤ · · · ≤ α N ≤ (6) 2 3 Genetic algorithm The genetic algorithm (GA) is a search mechanism based on the principle of natural selection and population genetics, which is transformed by three genetic operators: selection, crossover and mutation. Each string (chromosome) has a possible solution to the problem being optimized and each bit (or group of bits) represent a value or some variable of the problem (gene). These solutions are classified by an evaluation function, giving better values, or fitness, to better solutions. Each solution must be evaluated by the fitness function to produce a value. In selection, a number of selected exact copies of chromosomes in the current population become a part of the offspring. In crossover, randomly selected subsections of two individual chromosomes are swapped to produce the offspring. In mutation, randomly selected genes in chromosomes are altered by a probability equal to the specified mutation rate. 3.1 Simple crossover This kind of crossover operation is analogous to that of the binary implementation. The basic one is one-cut point crossover. Let two parents be x = [x1 , x2 , . . . , xn ] and y = [y1 , y2 , . . . , yn ]. They are crossed after the kth position; the resulting offsprings are given by; (7) x = [x1 , x2 , . . . , xk , yk+1 , yk+2 , . . . , yn ] (8) y = [y1 , y2 , . . . , yk , xk+1 , xk+2 , . . . , xn ] The further extensions of one-point crossover are multi-cut-point, and uniform crossover.
3.2 Arithmetic crossover The basic concept of this kind of operator is borrowed from the convex set theory [10]. Simple arithmetic operators are defined as the combination of two vectors (chromosomes) given as: x = λx + (1 − λ)y x = (1 − λ)x + λy
(9) (10)
where λ is a uniformly distributed random variable between zero and one. 3.3 Dynamic mutation Michaelewicz [10] proposed the mutation operator also called nonuniform mutation. It is designed for fine-tuning capabilities aimed at achieving high precision. For a given parent x, if the element xk of it is selected for mutation, the resulting offspring is x = [x1 . . . xk . . . xn ], where xk is randomly selected from the two possible choices; xk = xk + (t, xkU − xk ) or xk = xk − (t, xk − xkL ) (11) The function (t, d x)returns a value in the range [0, d x] such that the value of (t, d x) approaches zero as t increases. This property causes the operator to search the space uniform initially (when t is small) and very locally at later stages. The function (t, d x) is given by; d (t, d x) = d x · r · 1 − t T (12) where r is a random number from [0,1]. T is the maximal generation number and d is a parameter determining the degree of nonuniformity (usually assumed as 2). 3.4 Solution acceleration technique The convergence of the real coded GA can be accelerated greatly by assuming that the best solution in the population is closest to the global optimum. If it is true, then searching the solution spacing this neighborhood will produce solutions to the global optimum. This can be accomplished by remapping the population after each competition stage in the GA algorithm so that all solutions are moved a random distance towards the best solution at that generation. This results in more solutions in the neighborhood of the minimum and hence allows a more thorough search of its surrounds [4]. Mathematically, the solution-acceleration technique can be described as: x = xb + r (xb − x)
(13)
where xb = best solution vector (best individual) in the population, x = solution vector (individual) to be remapped, x = remapped solution vector, and r = uniformly distributed random number between zero and one.
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4 Hybrid genetic algorithm implementation
4.2 Phase-2 (direct search optimization method) algorithm
In general, local search techniques have the advantage of solving the problem quickly, though their results are very much dependent on the initial starting point; therefore they can be easily trapped in a local optimum. On the other hand, genetic algorithm samples a large search space, climbs many peaks in parallel, and is likely to lead search towards the most promising area. However, GA difficulties in a fining– tuning of local search; it spends most of the time competing between different hills, rather than improving the solution along a single hill that the optimal point locates. Hence using the advantage of both the local and GA techniques, the search algorithm can be improved both effectively and efficiently [4,10]. The proposed hybrid genetic algorithm combines a standard real coded GA and the second phase of conventional search technique. Real coded GA takes the place of the first phase of the search providing the potential near optimum solution, and phase-2 of a search technique using optimization by direct search and systematic reduction of the size of search region. Phase-2 algorithm is applied to rapidly generate a precise solution under the assumption that the evolutionary search has generated a solution near the global optimum.
After the phase-1 is halted, satisfying the halting condition described in the previous section, optimization by direct search and systematic reduction of the size of search region method is employed in the phase-2. In the light of the solution accuracy, the success rate, and the computation time, the best vector obtained from the phase-1 is used as an initial point for the phase-2. The optimization procedure based on direct search and systematic reduction in search region is found effective in solving various problems in the field of nonlinear programming [10,11]. This direct search optimization procedure is implemented as follows:
Michaelewicz [10] indicates that for real valued numerical optimization problems floating-point representation outperform binary representations because they are more consistent, more precise and lead to faster execution. For most application of genetic algorithms to constrained optimization problems, the real coding technique is used to represent a solution to a given problem. Hence, real coded algorithm is considered as phase-1 [4] and it is implemented as follows: 1. A population of N P trail solution is initialized. Each solution is taken as a real valued vector with their dimensions corresponding to the number of variables (α1 , α2 , . . . , α N ). The initial components of αi are selected in accordance with a uniform distribution ranging between 0 and 1. 2. The fitness score for each solution vector αi is evaluated, after converting each solution into corresponding switching instants αa using upper and lower bounds. 3. Roulette wheel based selection method is used to produce N P offspring from parents. 4. Arithmetic crossover and non-uniform mutation operators are applied to offspring to generate next generation parents. 5. The algorithm proceeds to step 2, unless the best solution does not change for a pre-specified interval of generations.
Table 1 Hybrid genetic algorithm parameters Phase-1
Phase-2
Population size (N P ) = 80 Max. no. of generations = 300 Crossover probability = 0.2 Mutation probability = 0.06
No. of trails (N S ) = 80 Max. no. of generations = 300 Range reduction factor = 0.01 Range = 0.5 β = 0.3
100
Switching angles (degrees)
4.1 Phase-1 (real coded) algorithm
1. The best solution vector obtained from the first phase of the hybrid algorithm is used as an initial point α(0) for phase-2 and an initial range vector is defined as R(0) = RMF × Range (14) where Range is defined as the difference between the upper and lower bounds (the upper and lower bounds for each switching angle here are π/2 and 0, respectively) and RMF is a range multiplication factor. The value of RMF varies between 0.0 and 1.0. 2. N S trail solution vectors around α(0) are generated using following relationship, αi = α(0) + α(0) ·∗ rand(1, n) (15) ∗ where, αi is the ith trail solution vector, (· ) represents element-by-element multiplication operation, and rand
80
α3
60
α2
40
α1 20
0
0.1
0.2
0.3
0.4 0.5 0.6 0.7 Modulation index ( mi )
0.8
Fig. 2 Switching angles versus modulation index (N = 3)
0.9
1
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Fig. 3 The spectrum of the output waveform (case I)
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Switching angles ( degrees)
α5 80
α4 60
α3 α2
40
α1
20
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Modulation index (m i ) Fig. 4 Switching angles versus modulation index (N = 5)
(1, n) is a random vector, whose element value varies from – 0.5 to 0.5. 3. For each feasible trail solution vector find objective function value and find the trail solution set, which minimizes F(α) and equate it to α(0) as follows: α(0) = αbest
(16)
where αbest is the trail solution set with minimum F(α). 4. Reduce the range by an amount given by R(0) = R(0) ∗ (1 − β). Where β is the range reduction factor, whose typical value is 0.05. 5. The algorithm proceeds to step 2, unless the best solution does not change for a pre-specified interval of generations.
5 Results and discussion To verify the validity of the proposed algorithm, a program was developed using the software package MATLAB 6.0 [12]. The program is run for a number of independent trials on Pentium-IV computer operating at 1.4 GHz clock speed. The parameters of GA such as crossover and mutation probability, population size and number of generations are usually selected as common values given in the literature or by means of a trial and error process to achieve the best solution set [3]. The parameters thus selected for the implementation of HRCGA are tabulated in Table 1.
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Fig. 5 The spectrum of the output waveform (case II)
0.25
Case I (N = 3)
HLF (p.u.)
0.2
0.15
0.1
0.05
0
Case II (N = 5)
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Modulation index (mi ) Fig. 6 Harmonic loss factor (HLF) variations versus modulation index
The program is run for different values of modulation index as well as different number of pulses per quarter cycle of the output voltage including odd and even numbers. The calculated switching angles are then simulated using the software package PSIM [13]. The results are presented in the following sections.
5.1 Case I: eliminating the 3rd and 5th harmonic In this case, three switching angles per quarter cycle (N = 3) are considered. The fundamental component was maintained at 0.75 p.u. Figure 2 shows the calculated switching angles for the entire range of modulation indices (0.02 ≤ m i ≤ 0.99).
Fig. 3 illustrates the spectrum of the output voltage waveform where the absence of 3rd and 5th harmonics is clearly evident. 5.2 Case II: eliminating the 3rd, 5th, 7th, and 9th harmonics Five switching angles per quarter cycle (N = 5) are chosen in this case that is aimed to eliminate the 3rd, 5th, 7th, and 9th order of harmonics while maintaining the fundamental component at 0.60 p.u. The solution sets for the switching angles for the entire range of modulation indices are illustrated in Fig. 4 & 5. With reference to Fig. 5, it can be clearly seen that the selected harmonics are totally eliminated. It is worth noting that the first significant harmonic in this case
A hybrid genetic algorithm
(i.e. 11th harmonic) is relatively high, as the fundamental is pulled down. 5.3 Performance index In order to indicate the usefulness and effectiveness of the proposed technique, a quality factor has to be chosen. Several quality factors have been reported in [2] such as Harmonic loss factor (HLF), second order distortion factor (DF2 ) and total harmonic distortion (THD). With the fact that most of the loads under consideration here are induction motors; The harmonic equivalent circuit of an induction machine can be assumed to be its total leakage reactance at the harmonic frequency, hence HLF is considered in this paper. The HLF, which is proportional to total rms harmonic current, can be defined by (17). For practical reasons, the HLF is calculated up to 31st order of harmonics. 31 2 Vn 1
HLF = p.u. (17) V1 n n=2
where V1 and Vn are the amplitudes of the fundamental and the nth harmonics, respectively. Furthermore, this factor is calculated for both case I and case II and for the whole range of the modulation index as depicted in Fig. 6. As it can be obviously seen from Fig. 6, a poor HLF is obtained when the converter is operated at low modulation index. However, at higher modulation index, the HLF improve tremendously. On the other hand, the higher the number of pulses per-quarter cycle the lower the HLF is. However, the switching losses would be increased if higher number of pulses are considered. 6 Conclusion A method to generate optimal switching angles in order to eliminate a certain order of harmonics is introduced in this paper. A two phase genetic algorithm, namely the real-coded and direct search is proposed to overcome the computational burden and to ensure the accuracy of the calculated angles. The algorithm was developed using MATLAB software and is run for a number of times independently to ensure the feasibility and the quality of the solution. This method was found to be superior to conventional techniques that may fail
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to converge if higher pulses per quarter-wave are sought. It is also proved to be an appropriate technique when an overmodulation operation is demanded since it still finds the solution for modulation index as high as unity. In order to prove the feasibility and effectiveness of the proposed method, the algorithm is tested with different operating points/cases including three- and fine-pulse per quarter waveforms. The harmonic loss factor (HLF) was chosen as performance index for optimality and is calculated for different operating points. Good agreement between the theoretical findings and the simulation results is achieved. Acknowledgements The authors gratefully acknowledge the support of Multimedia University (MMU) for this work.
References 1. Addoweesh KE, Mohamadein AL (1990) Microprocessor based harmonic elimination in chopper type AC voltage regulators. IEEE Trans Power Electron 5(2):191–200 2. Enjeti PN, Ziogas PD, Lindsay JF (1990) Programmed PWM techniques to eliminate harmonics: a critical evaluation. IEEE Trans Ind Appl 26(2):302–316 3. Liang TJ, O’Connnell RM, Hoft RG (1997) Inverter harmonic reduction using walsh function harmonic elimination method. IEEE Trans Ind Electron 12(6):971–982 4. Baskar S, Subbaraj P, Rao MVC (2001) Performance of hybrid real coded genetic algorithms. Int J Comput Eng Sci 2(4):583–601 5. Deib DA, Hill HW (1993) Optimal harmonic reduction in ac/ac chopper converters In Proceedings of 24th annual IEEE power electronics specialiste conference, pp. 1055–1060 (1993) 6. Holmes DG, Lipo TA (2003) Pulse-width modulation for power converters principles and practice. IEEE press series on power engineering, New York (2003) 7. Maswood AI, Rashid MH, Jian L. (1998) Optimal PWM–SHE switching on NPC inverter: a winning match for high power conversion. Electr Power Syst Res 48:19–24 8. Sun J, Beineke S., Grotstollen H. (1996) Optimal PWM based on real-time solution of harmonic elimination equations. IEEE Trans Ind Electron 11:612–621 9. Trzynadlowski AM, Legowski S (1994) Application of neural networks to the optimal control of three-phase voltage-controlled inverters. IEEE Trans Power Electron 9:397–404 10. Goldberg DE (1989) Genetic algorithms in search, optimization, and machine learning, Addison-Wesley, Reading 11. Michaelwicz Z (1996) Genetic algorithms + data structures = evolutionary programs. Springer, Berlin Heidelberg New York 12. MATLAB 6.0 software package. http://www.Mathworks.com 13. PSIM software package version 6.0, Powersim Inc. http://www. powersimtech.com
