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Selective Harmonic Elimination PWM Control for Cascaded Multilevel Voltage Source Converters: A Generalized Formula Mohamed S. A. Dahidah, Member, IEEE, and Vassilios G. Agelidis, Senior Member, IEEE

Abstract—This paper proposes a generalized formulation for selective harmonic elimination pulse-width modulation (SHE-PWM) control suitable for high-voltage high-power cascaded multilevel voltage source converters (VSC) with both equal and nonequal dc sources used in constant frequency utility applications. This formulation offers more degrees of freedom for specifying the cost function without any physical changes to the converter circuit, as compared to conventional stepped waveform technique, and hence the performance of the converter is greatly enhanced. The paper utilizes the merits of the hybrid real coded genetic algorithm (HRCGA) in finding the optimal solution to the nonlinear equation system with fast and guaranteed convergence. It is confirmed that multiple independent sets of solutions exist. Different operating points for both five- and seven-level converters including singleand three-phase patterns are documented. Selected experimental results are reported to verify and validate the theoretical and simulation findings. Index Terms—Cascaded multilevel converter, hybrid genetic algorithm, pulse-width modulation (PWM), selective harmonic elimination.

I. INTRODUCTION

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ULTILEVEL converters have drawn tremendous interest in recent years and have been studied for several high-voltage and high-power applications [1]. Switching losses in these high-power high-voltage converters represent an issue and any switching transitions that can be eliminated without compromising the harmonic content of the final waveform is considered advantageous. The term multilevel starts with the introduction of the three-level converter [1]. By increasing the number of levels in a given topology, the output voltages have more steps generating a staircase waveform, which approaches closely the desired sinusoidal waveform and also offers reduced harmonic distortion [1]–[6]. One promising technology to interface battery packs in electric and hybrid electric vehicles Manuscript received July 15, 2007; revised January 3, 2008. Published July 7, 2008 (projected). This paper is a revised and extended version of an earlier paper that was presented at the IEEE PESC 2006, Jeju, South Korea, June 18–22, 2006. Recommended for publication by Associate Editor J. Rodriguez. M. S. A. Dahidah is with the School of Electrical and Electronic Engineering, University of Nottingham, Jalan Broga, 43500 Semenyih, Selangor, Malaysia (e-mail: [email protected]). V. G. Agelidis is with the School of Electrical and Information Engineering, University of Sydney, Sydney, NSW 2006 Australia (e-mail: v.agelidis@ee. usyd.edu.au). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TPEL.2008.925179

are multilevel converters because of the possibility of high VA rating and low harmonic distortion without the use of a transformer [3]. Various multilevel converters structures are reported in the literature, and the cascaded multicell converter appears to be superior to other multilevel converters in application at high power rating due to its modular nature [3]. Selective harmonic elimination pulse-width modulation (SHE-PWM) has been mainly developed for two- and three-level converters in order to achieve lower total harmonic distortion (THD) in the voltage output waveform [2], [7]–[11]. The main challenge associated with SHE-PWM techniques is to obtain the analytical solution of the system of nonlinear transcendental equations that contain trigonometric terms which in turn provide multiple sets of solutions [8]. This has been reported in numerous technical articles [3]–[5], [7]–[9]. Several algorithms have been reported in the technical literature concerning methods of solving the resultant nonlinear transcendental equations, which describe the SHE-PWM problem. These algorithms include the well-known iterative approach, Newton–Raphson method [2], [10], [11]. This method is derivative-dependent and may end in local optima; further, a judicious choice of the initial values alone will guarantee convergence [6]–[9]. Another approach uses Walsh functions [12] where solving linear equations, instead of solving nonlinear transcendental equations, optimizes the switching angles. Recently, a technique based on a combination of an interval-search procedure and Newton’s method is proposed to find all solutions to the nonlinear transcendental equations [9]. However, the method was only applied to the two-level converters. The work presented in [13] and [14] aimed to develop a more complete relationship between conventional sine-triangle PWM and the SHE techniques by introducing a variable-carrier approach that produces harmonic elimination waveforms with high accuracy. This paper shows that the real-time implementation of the selected harmonic elimination could be easily achieved if the method of the modulation technique is followed. However, solutions to switching angles over the entire range of the modulation index were not provided, and moreover, the results were not experimentally validated. Furthermore, the technique has not been extended to multilevel converters where it apparently becomes more complicated. Genetic algorithm has been recently introduced to optimize the sequence of the carrier waveform of the PWM as to minimize the THD and the distortion factor (DF) of output line voltage [15]. However, the paper has not shown the possibility of

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DAHIDAH AND AGELIDIS: Multilevel SHE SELECTIVE PWM 02 HARMONIC ELIMINATION PWM CONTROL

finding multiple sets of solutions; moreover, it has not been applied to multilevel converters where the problem becomes much more complicated. An optimization technique assisted with a hybrid genetic algorithm was successfully applied to find the switching transitions (i.e., switching angles) of the SHE-PWM ac/ac converter [16]. References [17] and [18] generalized the problem formulation where the constraint of quarter-wave symmetry is relaxed to half-wave symmetry where all the even harmonics are zero but the harmonic phasing is free to vary. However, only the twoand three-level waveforms were reported, and the technique has not been extended to the multilevel case where the problem becomes more challenging. On the other hand, SHE-PWM methods have also been introduced to multilevel converters in several technical articles. Initially, the switching frequency was restricted to line frequency, and therefore, the staircase multilevel waveform was arranged in such a way as to control the fundamental and eliminate the low-order harmonic from the waveform [19]. Specifically, the theory of resultants and its performance for a multilevel staircase waveform was reported in [20]. A unified approach was presented in [21]. More recently, the use of symmetric polynomials is combined with the resultant theory for a multilevel converter [22]. Previous work [20] has shown that the transcendental equations characterizing the harmonic content can be converted to polynomial equations which are then solved using the method of resultants from elimination theory. A difficulty with this approach [20] as suggested by [22], is that when there are several dc sources, the degrees of the polynomials are quite large, thus making the computational burden of their resultant polynomials (as required by elimination theory) quite high. An interesting method to overcome the previously mentioned drawbacks was reported in [20], where the theory of symmetric polynomials is exploited to reduce the degree of the polynomial equations that must be solved which in turn greatly reduces the computational burden. The method is also extended to find the solutions to the switching angles when the dc sources are unequal [23]. In recent work [24], the theory of the power sums was introduced to reduce the degree of the polynomial equations that must be solved so that they are well within the capability of existing computer algebra software tools. However, a major limitation of the resultant theory appears as reported by [25] if one wanted to apply the method to multilevel converters with several changing (unequal) dc sources (which is always the case in actual practical applications), where the set of the transcendental equations to be solved is no longer symmetrical and requires the solution of a set of high-degree equations, which is beyond the capability of contemporary computer algebra. More recently, a general genetic algorithm using Matlab GA Optimization Toolbox was applied to solve the same problem of PWM-SHE [6]. However, the paper shows only the solutions to the equal dc sources and with the fundamental frequency switching method which has lower degrees of freedom for specifying the cost function compared with the proposed PWM method for the same physical structure. Furthermore, hybrid genetic algorithm is applied to stepped multilevel converter with nonequal dc sources in [26]. As the primary limitation (disadvantage) of the stepped modulation techniques lies in its very

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Fig. 1. Generalized multilevel PWM waveform.

Fig. 2. Switching angles versus modulation index (PWM multilevel with equal dc sources, = 3 + 3).

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narrow modulation index range, several works have been attempting to overcome this shortcoming. Specifically, [2] proposed a modulation technique that widens the modulation index by swapping the polarity of some levels so that a low modulation index can be obtained and maintaining the fundamental switching technique. A new active harmonic elimination technique was recently introduced to the line frequency method aiming to eliminate higher order of harmonics by simply generating the opposite of the harmonics to cancel them [27], [25]. However, the disadvantage in that is that it uses a high switching frequency to eliminate higher order harmonics. Other approaches have also been reported, including one where the harmonic elimination is combined with a programmed method [5], and another where multilevel SHE-PWM defined by the well-known multicarrier phase-shifted PWM (MPS-PWM) was proposed in [28] and [29], where the modulation index defines the distribution of the switching angles, and then the problem of SHE-PWM is applied to a particular operating point aiming to obtain the optimum position of these switching transitions that offer elimination to a

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Fig. 3. Implementation of five-level PWM waveform with equal dc sources (single-phase pattern, simulation). (a) Output voltage waveform. (b) Spectrum of the output voltage waveform.

Fig. 4. Implementation of the five-level PWM waveform with equal dc sources (three-phase pattern, simulation). (a) Line-to-neutral output voltage waveform. (b) Spectrum of the line-to-neutral output voltage waveform. (c) Line-to-line output voltage waveform. (d) Spectrum of the line-to-line output voltage waveform.

selected order of harmonics. A generalized formulation for multilevel SHE-PWM converters with nonequal dc source was also reported in [30]. A practical harmonics elimination method for multilevel converters was recently reported by [31]. The method reduces the number of the equations defining the harmonic elimination into four simple equations with minimum calculation time. The equal area criterion is then employed, and the solution to

the angles is obtained through a simple iteration procedure. However, and as stated by the authors, the performance of this method is related with the number of switching angles and the number of selected harmonics. Therefore, directly applying this method would not guarantee finding best switching angles for all the modulation index and voltage step combinations. More generalized formulation when the symmetry requirement are completely abolished in the formulation of the

DAHIDAH AND AGELIDIS: Multilevel SHE SELECTIVE PWM 02 HARMONIC ELIMINATION PWM CONTROL

five-level SHE-PWM was recently reported in [32], where the switching angles are relaxed between 0 and 2 . In this paper, cascade multilevel PWM converters with both equal and nonequal dc sources are investigated. The work is aimed at high-power voltage source converter (VSC) multilevel systems in utility applications and the converter output frequency is fixed to the utility’s grid frequency. Moreover, the modulation index range does not change significantly and remains within a region between 0.7 p.u. to 1 p.u. The main objectives of this paper are first to reformulate the problem of SHE for multilevel converters based on the multilevel PWM waveform. Where individual cells are operated with a frequency that is close to the fundamental frequency so that the switching losses are relatively low and higher degrees of freedom for specifying the cost function and therefore higher bandwidth when compared to existing family of techniques such as conventional stepped (fundamental switching) waveform is gained. Second, to utilize the merits of the minimization technique assisted with a hybrid genetic algorithm in order to greatly reduce the computational burden associated with the nonlinear transcendental equations of the selective harmonic elimination method which was proposed in [16], [26], and [30]. This paper is organized as follows. Section II presents the formulation along with analysis for the generalized multilevel PWM voltage waveform. Section III discusses the implementation of the proposed formulation to both the equal and nonequal cases and also presents the solution and the simulation results for a number of selected cases. Additionally, Section III also briefly presents the implementation of the proposed minimization approach. Selected results are experimentally verified and reported in IV. Finally, conclusions are summarized in Section V. II. PROBLEM FORMULATION AND CIRCUIT ANALYSIS Fourier series expansion of the generalized multilevel PWM output waveform of the single-phase multilevel converter shown in Fig. 1 can be expressed as follows: (1) Owing to the PWM waveform characteristics of odd function symmetry and half-wave symmetry, the output voltage can be reduced to

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Fig. 5. Switching angles versus modulation index (PWM multilevel with equal dc sources, = 3 + 3 + 3).

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where for single-phase system; • • for three-phase system and is odd; for three-phase system and is • even; the number of dc sources (i.e., converter cells), and • is the value of the th dc source the product



are the number of pulses per-quarter , respectively; • (total number of pulses per-quarter cycle of the multilevel PWM output voltage waveform); is the th switching angle; • • in (3), the polarity is positive if is an odd number (otherwise it is negative). An objective function describing a measure of effectiveness of eliminating selected order of harmonics while controlling the fundamental must be solved to obtain the optimal switching angles and that is defined as cycle at converter

(4) where

(2) where is the Fourier coefficient. for any number of switching A generalized expression of angles and any number of voltage levels (even or odd, provided that the waveform is physically correct and can be implemented) is given by

(3)

(5) and

is the modulation index which is defined as (6)

It should be noted that is the normalized fundamental component, and is the input dc voltage source of each converter cell. The optimal switching angles are obtained by minimizing (4) when it is subject to the constraint of (7). Consequently, selected harmonics are eliminated. These switching angles are generated for different operating points and then stored into lookup tables

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Fig. 6. Implementation of seven-level PWM waveform with equal dc sources (single-phase pattern, simulation). (a) Output voltage waveform. (b) Spectrum of the output voltage waveform.

Fig. 7. Switching angles versus modulation index (PWM multilevel with = 3 + 3). nonequal dc sources,

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to be used to control the converter for a certain operating point or to be interpolated if the online control of the converter is desired (7) III. RESULTS AND DISCUSSION The mathematical model (cost function) of the proposed multilevel SHE-PWM technique with both equal and nonequal dc sources is first developed, and then the optimization technique assisted with a real-coded hybrid genetic algorithm proposed in [16], [26], and [30] is applied. Various examples including fiveand seven-level converters are studied and analyzed to show the ruggedness the proposed technique. Details on the proposed optimization technique can be found in [16]. However, for the reader’s convenience, the implementation of the proposed optimization method is summarized hereafter. The simulation results are ultimately obtained using the PSIM software package [33]. A. Hybrid Genetic Algorithm Implementation Genetic algorithms (GAs) are highly suited to search spaces which are not well defined or have a high number of local

minima, which plague more traditional calculus-based search methods. By removing the need for auxiliary information regarding the optimization surface, the computational requirements are greatly reduced. The GA necessitates the need for the optimization variables to be coded as population of strings transformed by three Genetic operators: selection, crossover, and mutation. The proposed hybrid genetic algorithm combines a standard real coded GA and the phase-2 of conventional search technique which are briefly described in the following subsections. 1) Phase-1 (Real Coded) Algorithm: Real coded genetic algorithm is implemented as follows. trial solution is initialized. Each 1) A population of solution is taken as a real-valued vector with their dimensions corresponding to the number of variables . The initial components of are selected in accordance with a uniform distribution ranging between 0 and 1. is evalu2) The fitness score for each solution vector ated, after converting each solution into corresponding using upper and lower bounds. switching instants 3) A roulette wheel-based selection method is used to produce offspring from parents. 4) Arithmetic crossover and nonuniform mutation operators are applied to offspring to generate next-generation parents. The algorithm proceeds to step 2, unless the best solution does not change for a prespecified interval of generations. 2) Phase-2 (Direct Search Optimization Method) Algorithm: 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 phase-2. In 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. This direct search optimization procedure is implemented as follows.

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Fig. 8. Implementation of the five-level PWM waveform with nonequal dc sources (single-phase pattern, simulation). (a) Output voltage waveform. (b) Spectrum of the output voltage waveform.

Fig. 9. Implementation of the five-level PWM waveform with nonequal dc sources (three-phase pattern, simulation). (a) Line-to-neutral output voltage waveform. (b) Spectrum of the line-to-neutral output voltage waveform. (c) Line-to-line output voltage waveform. (d) Spectrum of the line-to-line output voltage waveform.

1) The best solution vector obtained from the first phase of the for phase-2 hybrid algorithm is used as an initial point and an initial range vector is defined as (8)

and 0, respectively), and switching angle here are RMF is a range multiplication factor. The value of RMF varies between 0.0 and 1.0. trial solution vectors around are generated using 2) following relationship:

where Range is defined as the difference between the upper and lower bound (the upper and lower bound for each

(9)

RMF

Range

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Fig. 10. Switching angles versus modulation index (seven-level PWM con= 3 + 3 + 3). verter with nonequal dc sources,

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where is the th trial solution vector, represents element-by-element multiplication operation, and is a random vector, whose element value varies from 0.5 to 0.5. 3) For each feasible trial solution vector, find objective function value and find the trial solution set, which minimizes and equate it to as follows: (10) is the trial solution set with minimum . where 4) Reduce the range by an amount given by , where is the range reduction factor, whose typical value is 0.05. The algorithm proceeds to step 2, unless the best solution does not change for a prespecified interval of generations. B. Case I: Multilevel Converters With Equal DC Sources In this section, it is assumed that the levels of the dc sources of the cascaded converter cells are equal and constant, i.e., p.u. Therefore, one can rewrite (3) as follows:

fundamental. It is interesting to note that even though the solution does not exist for the whole region of modulation index, multiple sets of solution have been found as plotted in Fig. 2. The investigation to define the optimal set that might have lower harmonic distortion is beyond the scope of this paper as its application dependant and will be addressed in future work. Howfrom set 3 is chosen, and the ever, one particular point associated waveform and its spectrum are illustrated in Fig. 3. A closer look at Fig. 3(b) reveals that selected harmonics are eliminated for the given fundamental component value. For completeness, another combination of the switching angles is also considered for a three-phase system. In this case, and , allowing control/elimination of seven low-order of harmonics (i.e., 5th, 7th, 11th, 13th, 17th, 19th, and 23rd) while controlling the fundamental. It is worth noting that all triplen harmonics need not to be controlled as they are absent in the line-to-line voltage which in turn increases the bandwidth of the converter. Fig. 4 shows the implementation of the aforementioned case. Looking at the spectrum of the line-to-neutral output voltage [Fig. 4(b)], one can see that selected harmonics being eliminated, and only triplen harmonics appear which are absent from the line-to-line voltage [Fig. 4(d)]. Furthermore, the next significant harmonic appearing in the line-to-line output voltage is the 25th. 2) Seven-Level Converter: In this case, a seven-level converter with equal dc sources is considered, and each converter cell is operated with a frequency of three times the fundamental. As a result, there are nine switching angles per-quarter cycle. The variation of the optimal switching angles against a certain range of the modulation index is plotted in Fig. 5. Multiple sets of solution have been also found for this case (i.e., five sets). As an example, one point is chosen from set 3 to show the elimination of the selected harmonics. Fig. 6 depicts the implementation of the seven-level waveform (single-phase pattern) for a given value of the fundamental component (i.e., 2.4 p.u.). Closer inspection at the spectrum shown in Fig. 6(b) reveals that the fundamental is maintained at 2.4 p.u. while all selected harmonics are eliminated which strongly confirms the validity of the proposed technique. C. Case II: Multilevel Converters With Nonequal DC Sources

(11) The cost function of (4) is recalled here and optimized (solved) using the proposed minimization technique. It is interesting to note that multiple solutions were found and reported which confirms the theory. Selected results are then chosen to demonstrate the validity of the proposed formulation. The results are discussed in the following subsections. 1) Five-Level Converter: In this case, the five-level output waveform is produced by connecting two H-bridge cells in series. First, it is assumed that each cell operates at a frequency of three times the fundamental; therefore, there are three switching angles per quarter-cycle at the output voltage of each cell (i.e., producing an overall six angles per quartercycle of the five-level waveform (i.e., . The optimization technique is applied to this case to find the optimal switching angles that eliminate five lower harmonics and controlling the

We now recall the problem presented in case I and with assumption that the levels of the dc sources are nonequal and can be measured; furthermore, each dc source has a nominal value of 1 p.u. (i.e., ; one can easily modify the generalized expression of and for any number of switching angles and voltage levels (even or odd, provided that the waveform is physically correct and can be implemented) as follows:

(12)

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Fig. 11. Implementation of the seven-level PWM waveform with nonequal dc sources (single-phase pattern, simulation). (a) Output voltage waveform. (b) Spectrum of the output voltage waveform.

Fig. 12. Block diagram of the prototype-based experimental verification.

where the product is the value of the th dc source. Similarly, the five- and seven-level converters are once again analyzed and discussed in this section. It is interesting to note that multiple solutions have been also found and reported which confirms the theory. Selected results are chosen to demonstrate the validity of the proposed generalized formulation. p.u.: In this 1) Five-Level Converter With V example, a five-level converter with different dc source levels p.u.) was considered. It is assumed (i.e., once again that each individual converter cell is operating at

frequency of three times of the fundamental one. As a result, there are three switching angles per-quarter cycle at each level of the output waveform. Hence, there are six switching angles per-quarter cycle of the output waveform of the multilevel inverter, which offer elimination of five low-order harmonics and controlling the fundamental at a certain value. The switching anfor the said case is graphed in Fig. 7. gles variation against It is confirmed that more than one set of solutions exist and for (i.e., set 1, , set 2, a certain range of , and set 3, . As an example, an operating point when was chosen to illustrate the effectiveness of the proposed method and the simulation results are shown in Fig. 8. It is evident that the targeted harmonics (3rd, 5th, 7th, 9th, and 11th are eliminated, and the next significant harmonic appearing in the output voltage is the 13th. A three-phase system with different combination of the dc p.u.) is also considered here for sources (i.e., completeness. Although the variation of the switching angles with the entire range of modulation index is not reported, the was obtained solution of an operating point when and the implementation of the resultant waveform is presented in Fig. 9. ,V , and 2) Seven-Level Converter With V p.u.: Three converter cells with unequal dc sources V are cascaded in series for this case. Once again, each converter cell is operated at three times of the fundamental frequency. Therefore, there are three switching angles per-quarter cycle at the output waveform of each converter cell. As a result, there are nine degrees of freedom offering the elimination of eight low-order harmonics while controlling the fundamental component. The solution sets for the switching angles for a certain (set 1, , set 2, range of , and set 3, are illustrated in Fig. 10. Fig. 11(a) depicts the output voltage waveform, and its spectrum is shown in Fig. 11(b), where the absence of selected harmonics (3rd, 5th, , 17th) is clearly evident for the given fundamental value which is 2.4 p.u. It is worth noting that the most significant order of harmonic in the output waveform is the 19th, and

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Fig. 13. Implementation of five-level PWM waveform with equal dc sources (single-phase pattern, experimental). (a) Output voltage waveform. (b) Spectrum of the output voltage waveform.

Fig. 14. Implementation of the five-level PWM waveform with equal dc sources (three-phase pattern, experimental). (a) Output voltage waveform. (b) Spectrum of the output voltage waveform.

Fig. 15. Implementation of the five-level PWM waveform with nonequal dc sources (single-phase pattern, experimental). (a) Output voltage waveform. (b) Spectrum of the output voltage waveform.

it could be further shifted to the 31st if the three-phase system was considered. IV. EXPERIMENTAL VERIFICATION (SELECTED RESULTS) Fig. 12 shows the block diagram of the laboratory-based prototype of a single-phase five-level converter that is implemented with eight insulated gate bipolar transistors (IGBT) switches with internal anti-parallel diodes (IRG4BC20FD). Four high-

voltage high-speed IGBT drivers (IR2112) were used to provide proper and conditioned gate signals to the power switches. The precalculated PWM signals are implemented using lowcost high-speed Texas Instruments TMS320F2812 digital signal processor (DSP) board with an accuracy of 20 s. The optically coupled isolators SFH610 are used to provide an electrical isolation between the DSP board and the power circuit. A digital real-time oscilloscope (Tektronix TDS210) was used to display and capture the output waveforms and with the feature of the fast

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Fig. 16. Implementation of the five-level PWM waveform with nonequal dc sources (three-phase pattern, experimental). (a) Output voltage waveform. (b) Spectrum of the output voltage waveform.

Fourier transformer (FFT), the spectrum of the output voltage is obtained for different operating points as discussed hereafter. Selected experimental results based on a low-power five-level converter were obtained and validated the simulation results. Specifically, the simulation results of Fig. 3 are experimentally verified and shown in Fig. 13, where it is clear that the next significant harmonic is the 13th and that confirms the simulation results presented in Fig. 3. The three-phase pattern was also experimentally validated, and the results are shown in Fig. 14, which are very much in good agreement with the simulation ones presented in Fig. 4. It can be seen from Fig. 14(b) that only triplen harmonics are present, and the next significant harmonic in the output voltage is the 25th (i.e., cursor 2 at 1250 Hz). Experimental results have been obtained for the nonequal case as well confirming the simulation results. Specifically, Fig. 15 shows the output voltage waveform and its spectrum in Fig. 15(b), where a very good agreement with the simulation results of Fig. 8 is evident. The three-phase pattern was also experimentally verified and the results are illustrated in Fig. 16 which are very much identical to the simulation results presented in Fig. 9 where one can easily see from Fig. 16(b) that only triplen harmonics are present within the selected bandwidth and the next significant harmonic in the output voltage is the 23rd. V. CONCLUSION A generalized formula of SHE-PWM suitable for high-power high-voltage cascaded multilevel converters with both equal and nonequal dc voltage sources was proposed and demonstrated in this paper. It has been shown that the proposed formulation offers higher converter bandwidth and thus better harmonic performance compared to the conventional fundamental switching multilevel converters for the same physical structure (i.e., number of cells). An efficient optimization technique of hybrid genetic algorithm was applied to the problem of SHE-PWM, and the optimal solutions of the switching angles are obtained. Although multiple sets of solutions have been reported, the evaluation to identify the optimal set that offers a better performance was beyond the scope of this paper as it is application dependant and will be addressed in future

work. The proposed formulation has been applied to various operating points including five- and seven-level converters and for both single- and three-phase systems. Selected simulation and experimentally verified results are presented to confirm the validity of the theoretical analysis. REFERENCES [1] J. Rodriguez, J. S. Lai, and F. Z. Peng, “Multilevel inverters: A survey of topologies, controls, and applications,” IEEE Trans. Ind. Electron., vol. 49, no. 4, pp. 724–738, Aug. 2002. [2] S. Siriroj, J. S. Lai, and T. H. Liu, “Optimum harmonic reduction with a wide range of modulation indexes for multilevel inverters,” in Proc. IEEE-IAS Annu. Meeting, Rome, Italy, Oct. 2000, pp. 2094–2099. [3] L. M. Tolbert, J. N. Chiasson, D. Zhong, and K. J. McKenzie, “Elimination of harmonics in a multilevel converter with nonequal dc sources,” IEEE Trans. Ind. Applicat., vol. 4, no. 1, pp. 75–82, Jan./Feb. 2005. [4] J. N. Chiasson, L. M. Tolbert, K. J. McKenzie, and Z. Du, “Control of a multilevel converter using resultant theory,” IEEE Trans. Control Syst. Technol., vol. 11, no. 3, pp. 345–354, May 2000. [5] Z. Du, L. M. Tolbert, and J. N. Chiasson, “Harmonic elimination in multilevel converter with programmed PWM method,” in Proc. IEEE Ind. Applicat. Soc. Annu. Meeting, Seattle, WA, Oct. 2004, pp. 2210–2215. [6] B. Ozpineci, L. M. Tolbert, and J. N. Chiasson, “Harmonic optimization of multilevel converters using genetic algorithms,” IEEE Power Electron. Lett., vol. 3, no. 3, pp. 92–95, Sep. 2005. [7] V. G. Agelidis, A. Balouktsis, and I. Balouktsis, “On applying a minimization technique to the harmonic elimination PWM control: The bipolar waveform,” IEEE Power Electron. Lett., vol. 2, no. 2, pp. 41–44, Jun. 2004. [8] V. G. Agelidis, A. Balouktsis, I. Balouktsis, and C. Cossar, “Multiple sets of solutions for harmonic elimination PWM bipolar waveforms: Analysis and experimental verification,” IEEE Trans. Power Electron., vol. 21, no. 2, pp. 415–421, Mar. 2006. [9] R. A. Jabr, “Solution trajectories of the harmonic-elimination problem,” IEE Proc. Electr. Power Applicat., vol. 153, no. 1, pp. 97–104, Jan. 2006. [10] H. S. Patel and R. G. Hoft, “Generalized harmonic elimination and voltage control in thyristor inverters: Part I—Harmonic elimination,” IEEE Trans. Ind. Applicat., vol. 9, no. 3, pp. 310–317, May/Jun. 1973. [11] H. S. Patel and R. G. Hoft, “Generalized harmonic elimination and voltage control in thyristor inverters: Part II—Voltage control technique,” IEEE Trans. Ind. Applicat., vol. 10, no. 5, pp. 666–673, Sep./ Oct. 1974. [12] T. J. Liang, R. M. O’Connnell, and R. G. Hoft, “Inverter harmonic reduction using walsh function harmonic elimination method,” IEEE Trans. Ind. Electron., vol. 12, no. 6, pp. 971–982, Nov. 1997. [13] P. T. Krein, B. M. Nee, and J. R. Wells, “Harmonic elimination switching through modulation,” in Proc. IEEE Workshop Comput. Power Electron., 2004, pp. 123–126. [14] J. R. Wells, X. Geng, P. L. Chapman, P. T. Krein, and B. M. Nee, “Modulation-based harmonic elimination,” IEEE Trans. Power Electron., vol. 22, no. 1, pp. 336–340, Jan. 2007.

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[15] K. L. Shi and H. Li, “Optimized PWM strategy based on genetic algorithms,” IEEE Trans. Ind. Electron., vol. 52, no. 5, pp. 1458–1461, Oct. 2005. [16] M. S. A. Dahidah and M. V. C. Rao, “A hybrid genetic algorithm for selective harmonic elimination PWM ac/ac converter control,” Elect. Eng., vol. 89, no. 4, pp. 285–291, Mar. 2007, Springer. [17] J. R. Wells, B. M. Nee, P. L. Chapman, and P. T. Krein, “Selective harmonic control: A general problem formulation and selected solutions,” IEEE Trans. Power Electron., vol. 20, no. 6, pp. 1337–1345, Nov. 2005. [18] J. R. Wells and P. L. Chapman, “Generalization of selective harmonic control/elimination,” in Proc. IEEE Power Electron. Specialists Conf., 2004, pp. 1358–1362. [19] P. M. Bhagwat and V. R. Stefanovic, “Generalized structure of a multilevel PWM inverter,” IEEE Trans. Ind. Applicat., vol. IA-19, no. 6, pp. 1057–1069, Nov. 1983. [20] J. N. Chiasson, L. M. Tolbert, K. J. McKenzie, and Z. Du, “Control of a multilevel converter using resultant theory,” IEEE Trans. Control Syst. Technol., vol. 11, no. 3, pp. 345–354, May 2003. [21] J. N. Chiasson, L. M. Tolbert, K. J. McKenzie, and Z. Du, “A unified approach to solving the harmonic elimination equations in multilevel converters,” IEEE Trans. Power Electron., vol. 19, no. 2, pp. 478–490, Mar. 2004. [22] J. N. Chiasson, L. M. Tolbert, K. J. McKenzie, and D. Zhong, “Elimination of harmonics in a multilevel converter using the theory of symmetric polynomials and resultants,” IEEE Trans. Control Syst. Technol., vol. 13, no. 2, pp. 216–223, Mar. 2005. [23] L. M. Tolbert, J. N. Chiasson, Z. Du, and K. J. McKenzie, “Elimination of harmonics in a multilevel converter with non equal dc sources,” IEEE Trans. Ind. Applicat., vol. 41, no. 1, pp. 75–82, Jan./Feb. 2005. [24] J. N. Chiasson, L. M. Tolbert, Z. Du, and K. J. McKenzie, “The use of power sums to solve the harmonic elimination equations for multilevel converters,” Eur. Power Electron. Drives J., vol. 15, no. 1, pp. 19–27, Feb. 2005. [25] Z. Du, L. M. Tolbert, and J. N. Chiasson, “Active harmonic elimination in multilevel converters using fpga control,” in Proc. IEEE Workshop Comput. Power Electron., Urbana-Champaign, IL, Aug. 2004, pp. 127–132. [26] M. S. A. Dahidah and V. G. Agelidis, “A hybrid genetic algorithm for selective harmonic elimination control of a multilevel inverter with non-equal dc sources,” in Proc. 6th IEEE Power Electron. Drives Syst. Conf., Kuala Lumpur, Malaysia, Nov./Dec. 2005, pp. 1205–1210. [27] Z. Du, L. M. Tolbert, and J. N. Chiasson, “Active harmonic elimination for multilevel converters,” IEEE Trans. Power Electron., vol. 21, no. 2, pp. 459–469, Mar. 2006. [28] V. G. Agelidis, A. Balouktsis, I. Balouktsis, and C. Cossar, “Five-level selective harmonic elimination PWM strategies and multicarrier phaseshifted sinusoidal PWM: A comparison,” in Proc. IEEE Power Electron. Specialists Conf., Recife, Brazil, Jun. 2005, pp. 1685–1691. [29] V. G. Agelidis, A. Balouktsis, and M. S.A. Dahidah, “A five-level symmetrically defined selective harmonic elimination PWM strategy: Analysis and experimental validation,” IEEE Trans. Power Electron., vol. 23, no. 1, pp. 19–26, Jan. 2008. [30] M. S. A. Dahidah and V. G. Agelidis, “Generalized formulation of multilevel selective harmonic elimination PWM: Case I—Non-equal dc sources,” in Proc. IEEE Power Electron. Specialists Conf., Jeju, Korea, Jun. 2006, pp. 472–1477. [31] J. Wang, Y. Huang, and F. Z. Peng, “A practical harmonics elimination method for multilevel inverters,” in Proc. Conf. Rec. IEEE-IAS Annu. Meeting, Hong Kong, Oct. 2005, pp. 1665–1670.

IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 23, NO. 4, JULY 2008

[32] M. S. A. Dahidah, V. G. Agelidis, and M. V. C. Rao, “On abolishing symmetry requirements in the formulation of a five-level selective harmonic elimination pulse width modulation technique,” IEEE Trans. Power Electron., vol. 21, no. 6, pp. 1833–1837, Nov. 2006. [33] [Online]. Available: http://www.powersimtech.com PSIM software package, version 6.0, Powersim, Inc.

Mohamed S. A. Dahidah (M’02) was born in Tripoli, Libya. He received the B.S. degree in electrical and electronic engineering from Bright Star University of Technology, Briga, Libya, in 1998, the M.A.S. degree from the Universiti Putra Malaysia, Malaysia, in 2002, and the Ph.D. degree in electrical engineering from Multimedia University, Malaysia in 2007. From May 2002 to October 2007, he was a full-time Lecture with the Faculty of Engineering and Technology, Multimedia University. In November 2007, he was appointed Assistant Professor in the School of Electrical and Electronic Engineering, University of Nottingham, Malaysia Campus. He has authored or coauthored a number of refereed journal and conference papers. His research interests include multilevel converters, selective harmonic elimination (SHE) modulation techniques, PWM converter controls, matrix converters, fuel cells, and renewable energy. Dr. Dahidah has been a reviewer for the IEEE TRANSACTIONS ON POWER ELECTRONICS.

Vassilios G. Agelidis (SM’00) was born in Serres, Greece. He received the B.S. degree in electrical engineering from Democritus University of Thrace, Thrace, Greece, in 1988, the M.S. degree in applied science from Concordia University, Montreal, QC, Canada, in 1992, and the Ph.D. degree in electrical engineering from the Curtin University of Technology, Perth, Australia, in 1997. From 1993 to 1999, was with the School of Electrical and Computer Engineering, Curtin University of Technology. In 2000, he joined the University of Glasgow, Glasgow, U.K., as a Research Manager for the Center for Economic Renewable Power Delivery. In addition, he has authored/coauthored several journal and conference papers as well as Power Electronic Control in Electrical Systems (Newnes, 2002). From January 2005 to December 2007, he was the inaugural Chair in Power Engineering in the School of Electrical, Energy, and Process Engineering, Murdoch University, Perth. Since January 2007, he has held the Energy Australia Chair of Power Engineering at the University of Sydney, Sydney, Australia. Dr. Agelidis received the Advanced Research Fellowship from the U.K.’s Engineering and Physical Sciences Research Council (EPSRC-UK) in 2004. He was the Vice President Operations within the IEEE Power Electronics Society for 2006-2007. He was an Associate Editor of the IEEE POWER ELECTRONICS LETTERS from 2003 to 2005, and served as the PELS Chapter Development Committee Chair from 2003 to 2005. He is currently an AdCom member of IEEE PELS for 2007-2009. He has been the Technical Chair of the 39th IEEE PESC’08, Rhodes, Greece.

Selective Harmonic Elimination PWM Control for ... -

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