Harmonic analysis and FPGA implementation of SHE controlled three phase CHB 11-level inverter in MV drives using deterministic and stochastic optimization techniques
© Vesapogu et al.; licensee Springer. 2013
Received: 17 April 2013
Accepted: 29 July 2013
Published: 5 August 2013
With the advancements in semiconductor technology, high power medium voltage (MV) Drives are extensively used in numerous industrial applications. Challenging technical requirements of MV Drives is to control multilevel inverter (MLI) with less Total harmonic distortion (%THD) which satisfies IEEE standard 519-1992 harmonic guidelines and less switching losses. Among all modulation control strategies for MLI, Selective harmonic elimination (SHE) technique is one of the traditionally preferred modulation control technique at fundamental switching frequency with better harmonic profile. On the other hand, the equations which are formed by SHE technique are highly non-linear in nature, may exist multiple, single or even no solution at particular modulation index (MI). However, in some MV Drive applications, it is required to operate over a range of MI. Providing analytical solutions for SHE equations during the whole range of MI from 0 to 1, has been a challenging task for researchers. In this paper, an attempt is made to solve SHE equations by using deterministic and stochastic optimization methods and comparative harmonic analysis has been carried out. An effective algorithm which minimizes %THD with less computational effort among all optimization algorithms has been presented. To validate the effectiveness of proposed MPSO technique, an experiment is carried out on a low power proto type of three phase CHB 11- level Inverter using FPGA based Xilinx’s Spartan -3A DSP Controller. The experimental results proved that MPSO technique has successfully solved SHE equations over all range of MI from 0 to 1, the %THD obtained over major range of MI also satisfies IEEE 519-1992 harmonic guidelines too.
KeywordsC-GA Technique CHB inverter FPGA based Xilinx’s Spartan-3A DSP controller MPSO technique NR technique Selective harmonic elimination technique
Advancements of semiconductor switching devices like GTOs, IGBTs and IGCTs lead the pace of high power converters and MV Drives since 1980s. The MV drives cover the voltage ratings from 2.3 kV to 13.8 kV and power ratings from 0.4 MW to 40 MW. Medium voltage drives are extensively used in industrial applications such as steel rolling mills in the metals industry (Okayama et al. 1996), petrochemical industry (Rossmann & Ellis 1998), traction applications in the transportation industry (Bernert 2000), cement industry (Menz & Opprecht 2002) because of advantages such as: high efficiency, high reliability, effective fault protection, high dynamic performance, regenerative braking capacity, four quadrant operation and significant savings on energy cost (Bin Wu 2006).
However, a single power semiconductor switch alone cannot be connected to medium voltage grids (2.3kV, 3.3kV, 4.16kV or 6.9 kV). For this reason, different Multilevel Inverter topologies have emerged as a solution for working with higher voltage levels (Lai & Peng 1996). At present, there are three commercial topologies of Multilevel Inverters: Neutral point clamped (NPC) topology (Nabae et al. 1981), Cascaded H-bridge (CHB) topology (Meynard & Foch 1992) and Flying capacitor (FC) topology (Rodriguez et al. 2002; (Siva Kumar et al. 2010).
Among all the topologies, CHB topology has drawn significant importance because of the following advantages: (i) achievement of higher power levels by using lesser rating of semiconductor devices, (ii) no clamping diodes are required as in NPC topology, (iii) voltage balancing capacitors are not needed as in FC topology, (iv) Modularity in construction, (v) less dv/dt stresses which results in less Electro Magnetic Interference (EMI), (vi) lesser common node voltages (Renge et al. 2006) and producing output voltage with less Harmonic distortion (Malinowski et al. 2010). With an increasing of number of DC voltage sources, number of levels of output voltages increases which reduces the %THD.
However, as the number of levels increases, the control complexity increases. Since three phase CHB 11-level inverter is chosen for analysis, sixty semi conductor switches have to be controlled which further increases the control complexity. However, this problem is effectively addressed in this paper by successfully implementing MPSO technique in real time too.
In high power MV Drive applications, high converter efficiency with least %THD (which obeys IEEE 519-1992 harmonic guidelines) is of prime importance. Since, the switching losses are in direct relation with the modulation strategy used for controlling of MLI, they have been an active subject of research. High switching modulation control methods like Carrier based PWM techniques (Joshi et al. 2013) or Space vector modulation techniques leads to high switching losses and side bands around carrier frequency appears as lower order harmonics producing high %THD in output voltage and current (Samir Kouro et al. 2009).
Instead, low switching frequency modulation technique or fundamental switching frequency technique like Selective Harmonic Elimination (SHE) has been traditionally one of the widely used modulation method for control of MLI (Patel & Hoft 1973). SHE technique has attractive features such as: maintaining the good power quality, less switching losses, direct control over output voltage harmonics and the ability to leave triplen harmonics uncontrolled to take the advantage of circuit topology in three phase system (Al-Othman et al. 2007).
The equations which are produced by SHE technique are highly non-linear transcendental in nature that contains trigonometric terms and exhibit multiple sets of solutions. Thus, providing analytical solution (i.e. finding optimum switching angles) for SHE equations during complete range of MI from 0 to 1, has been a main challenge for researchers over decades.
Several researchers have proposed many optimization techniques to solve non-linear transcendental SHE equations. These optimization techniques include deterministic methods like NR method (Patel & Hoft 1973; 1974), Predicted initial values (Sun & Grotstollen 1992), WALSH functions (Liang et al. 1997), Elimination theory (Chiasson et al. 2004), Resultant theory (Chiasson et al. 2005), Function minimisation (Vassilios et al. 2008) and stochastic optimization like modified species based particle swarm optimization(MPSO) (Tarafdar et al. 2009) or continuous genetic algorithm(C-GA) (Reza et al. 2011).
The main objective of this paper is to present comparative harmonic analysis for deterministic optimization technique like NR method and stochastic method like continuous C-GA and MPSO which are applied to solve SHE equations of three phase CHB 11-level inverter during whole range of MI from 0 to 1 and to validate the simulated results with experimental approach.
This paper has been organized as follows: SHE technique and problem formation section briefly explains the SHE technique and formulation of objective function and cost function. Optimization techniques such as deterministic method like NR method and stochastic methods like C-GA and MPSO are briefly discussed in optimization techniques for Solving SHE Equations section. Results obtained from deterministic and stochastic optimization techniques are presented in simulation and analysis section. Real time implementation and comparative analysis are presented in hardware implementation section. Finally, conclusion section summarizes the merits and limitations of various algorithms and future scope of the work.
SHE technique and problem formation
The selective harmonic elimination method is also called fundamental switching frequency method based on the harmonic elimination theory proposed by Patel et al. (Patel & Hoft, 1973; 1974). Selective harmonic elimination (SHE) technique is one of the preferred modulation methods at fundamental switching frequency for control of high power inverters since early 1960s and developed into matured form in 1970s. Theoretically it provides highest power quality at fundamental switching frequency in comparison to other existing PWM methods like space vector modulation and carrier based PWM techniques (Holmes & Lipo 2003).
In SHE technique, all the switching angles are limited in between 0° and 90° (0 ≤ α ≤ 90°). Because of odd quarter-wave symmetry of output voltage waveform, all even order harmonics are zero. Here, it is not necessary to eliminate triplen harmonics because they will be eliminated in the line to line output voltage.
MI is defined as the ratio of fundamental output voltage (V1) to maximum obtainable fundamental voltage V1max.
The main objective is to find a set of switching angles such that the magnitude of fundamental harmonic reaches a desired value, i.e V1 * by minimizing lower order of harmonics (i.e 5th, 7th, 11th and 13th). The fitness value is a measure of the appropriateness of a solution with respect to the original objective and the amount of infeasibility. The fitness function is formed by adding original objective function or fundamental component function to penalty function (Homaifar, Gi & Lai 1994; Michalewicz & Schoenauer, 1996).
Where, V1 * is the desired fundamental harmonic, ‘s’ is the number of switching angles and hn is the order of nth viable harmonic at the output of a three phase multi-level inverter e.g h2 = 5 and h5; = 13. Detailed explanation about functioning of fitness function is presented in Reza et al. 2011.
Optimization techniques for solving SHE equations
In many industrial applications, it is required to operate the inverter over a range of MI from 0 to 1. As it is mentioned, finding analytical (feasible) solution for SHE equations which gives least %THD has been a challenging task for researchers. Among the existing optimization techniques to solve non linear transcendental equations, deterministic methods like NR, stochastic methods like C-GA and MPSO are considered and comparative analysis have been carried out. This section briefly explains different optimization algorithms to solve SHE equations.
Newton-Raphson method is deterministic optimization technique which is extremely powerful and fast convergence iterative method to solve non linear transcendental equations (Patel & Hoft 1973). Newton's method was first described by Isaac Newton in 1969 and Joseph Raphson after twenty years later got close to Newtons approach but only for polynomials of degree 3, 4, 5 ……10. Finally in 1740, Thompson Simpson explained NR method as an iterative method to solve optimization problems by setting the gradient to zero (Peng et al. 1997).
Main steps for NR method are
Step 1: Assume any random initial guess for switching angles (say α0).
Step 2: Set MI =0.
Step 3: Calculation of F(α0),B(MI) and jacobian matrix J(α0).
Step 4: Computation of error Δα.
Step 5: Updating of the switching angles and Perform the transformation to bring switching angles in feasible range i.e. between zero and π/2.
Step 6: Repeat the steps (3) to (6) for sufficient number of iterations to attain error goal.
Step 7: Substitute α0 = α (k + 1).
Step 8: Repeat steps (2) to (8) for whole range of MI.
Step 9: Increment by a fixed step.
Step 10: Repeat steps (2) to (10) for complete range of MI.
The proposed algorithm is implemented in MATLAB programming environment and complete analysis of functioning of algorithm is present in simulation and results section.
Continuous genetic algorithms
In order to ease the complexity of controlling modern industries with multiple objectives and constraints, many researchers have developed biologically inspired algorithms and proved their effectiveness of control compared to derivative approaches. Genetic algorithm is one of the Stochastic optimization (SO) methods to solve non linear transcendental equations effectively. Stochastic optimization (SO) methods are optimization methods that generate and use random variables. GAs are a subclass of Evolutionary computing and are random search algorithms. Though, all minimum seeking algorithms uses the same basic approach of heading downhill from an arbitrary starting point but they differ in deciding in which direction to move and how far to move (Davis 1991).
After Successive improvements like increasing the speed of search process with good intelligence, without trapping at local minimum, powerful and widely accepted biologically inspired algorithm of Genetic Algorithm has been proposed by John Hallond in 1975 and finally it has been popularized by one of his student Goldberg who has solved the complex problem of control of gas-pipe line transmission for his dissertation work (Chamber 1995). Later it has been successfully implemented for solving number of engineering optimization problems due to the advantages such as Optimization with continuous or discrete variables, no need of calculus information, capability of dealing with a large number of variables, well suited for parallel computers, ability to find optimum global minimum instead of local minimum even in most complex objective functions (Sivanandam & Deepa 2011); (Deb 2001).
In this work, Continuous Genetic Algorithm has been used to solve non linear transcendental SHE equations during whole range of MI from 0 to 1.
Steps involved in Continuous GA are as follows
Step 1: Defining of optimization variables, cost function, cost.(cost minimization).
Step 2: Generation of initial population.
Step 3: Fitness/Cost evaluation.
Step 4: Selection of Mates.
Step 5: Mating.
Step 6: Mutation.
Step 7: Convergence check.
Step 8: Repeat step (2) to step (7) until requirements met.
Detailed explanation of algorithm is presented in (Ozpineci et al. 2005).
Modified particle swarm optimization
Swarm intelligence is defined as “. . . any attempt to design algorithms or distributed problem-solving techniques inspired by the collective behavior of social insect colonies and other animal societies.” (Karaboga & Akay 2009). One of the popular swarm-intelligence-based algorithm is the Particle Swarm Optimization (PSO) algorithm which was introduced by Eberhart and Kennedy in (1995 2002) (Kennedy & Eberhart, 1995); (2002). PSO is also a population-based stochastic optimization technique and is well adapted to solve complex optimization problems because of the advantages such as: with less computational effort, simplicity in computer coding, search techniques do not use gradient information but the values of objective function, initial guess is not needed like traditional iterative methods. Successful applications of PSO technique to engineering optimization problems are found in (Chan & Tiwari, 2007; Mohamed (Azab, 2009;); Hereford, & Siebold,); Hereford, & Siebold, ); Hereford, & Siebold,2008). Recently, Modified species –based PSO (MPSO) has proposed in (Tarafdar et al. 2009), which has better performance like fast convergence, less computational effort and calculation time than basic PSO. Hence, this algorithm has been considered for solving SHE equations set (6).
Main steps in algorithm are as follows.
Step 1: Initialize population.
Step 2: Repeat.
Step 2: Calculate fitness values of particles.
Step 3: Sort particles from the best-fitness value to least- fitness value.
Step 4: Select species seed.
Step 5: Assign each species seed identified as the l best to all individuals identified in the same species.
Step 6: Replace redundant particles in species.
Step 7: Update particle positions according to (6) and (8).
Step 8: Until requirements met.
Vi k : velocity of ith particle at kth iteration,
χ : Constriction co-efficient
c1, c2: The cogitative and social parameters
rand1, rand2 : random numbers between 0 and 1
xi k : Current position of ith particle at kth iteration
pbesti : pbest of ith particle
gbest : gbest of the group
Clerc & Kennedy proposed constriction co-efficient (χ) to guarantee the convergence of PSO algorithm.
The complete explanation of MPSO algorithm is explained by Tarafdar et al. 2009.
Simulation and results
In MATLAB programming environment, codes are developed for the proposed algorithms like NR method, C-Genetic Algorithm and Modified Particle Swarm Optimization to find near global optimum solutions for SHE equations and SIMULINK model is also developed to observe %THD from FFT analysis and results are discussed.
Newton raphson method
In general, the main disadvantage of NR method is requirement of good initial guess which requires the past knowledge about convergence and good initial guess is not possible always. Hence, in this paper that drawback is eliminated by developing the algorithm which can work with any random initial guess. For an eleven level inverter, it is required to compute five switching angles at each MI from 0 to 1 with an increment of 0.001.
%THD & switching angles in degrees at various M I
By observing the computed results from Figure 3, the feasible solutions (α1 α2, α3 α4 and α5) at various modulation indices can be broadly classified into three regions: 1) No solution region 2) Single solution region 3) Multiple solution region. In Figure 3, switching angles α1 α2, α3 α4 , α5 and error are represented in blue, green, red, green, violet and black colors respectively and all the switching angles are measured in degrees and represents the possible solution. It can be seen from Figure 3 that solutions does not exist at lower ends of MI up to 0.460 and upper ends of the MI after 0.85, as the error is not zero. Multiple solution sets exists for MI = [0.510 0.569], [0.615 0.700]. Some solutions exist in very narrow range of MI i.e. at [0.365], [0.387], [0.900], [0.933] and [0.982].
From Table 1, it is observed that the developed algorithm successfully solved the SHE equations during major range of MI and interesting thing is that %THD produced during entire range of MI are less than 8.98% and minimum %THD of value 5.33% is observed at MI of value 0.755. From FFT analysis in Figure 4, it is clearly observed that all the desired lower order harmonics such as 5th, 7th, 11th and 13th are minimized to a greater extent and the %THD value also satisfies IEEE 519-1992 harmonic guidelines too.
Continuous genetic algorithm optimization technique
For all stochastic optimization technique computations in this paper, the values of the common parameters such as population size, maximum iteration number was considered same and the values are 300 and 1000 respectively.
%THD & firing angles in degrees at various M I
In order to overcome the problem of local trapping, the developed code is run for 10 times at each value of MI and near global optimum solution is considered. From Figure 5, it is seen that, the developed C-GA optimization technique has successfully solved the SHE equations set for entire range of MI from 0 to 1, which is unable to find in NR technique. The switching angles α1 α2, α3 α4 and α5 are in degrees and represented in red, blue, black, green and yellow colors respectively and represents possible solutions obtained from C-GA technique.
Table 2 presents feasible solutions (switching angles) at various MI from 0 to 1. As the MI increases %THD decreases and minimum %THD of value 6.19% has occurred at MI of 0.97. FFT analysis at MI of value 0.97 is presented in Figure 6. It reveals that the targeted lower order harmonics of 5th, 7th, 11th and 13th are significantly minimized and the value of %THD is 6.19%, which also complies with IEEE 519-1992 harmonic guidelines. MI at which maximum fundamental voltage is achieved, %THD is found to be significantly minimum of value 6.85%. Thus, the simulation results validate the effectiveness of C-GA optimization technique.
Modified particle swarm optimization technique
MATLAB code is developed using MPSO technique by considering algorithmic specific parameters as follows: cognitive parameter (c1), social parameter (c2) are considered as 1 and 3.5 respectively. r 1 and r 2 are random values uniformly distributed within [0, 1].
%THD & switching angles in degrees at various M I
From Figure 7, it is clearly observed that, the developed MPSO algorithm has successfully solved SHE equations set (6) during entire range of MI. The switching angles α1 α2, α3 α4 and α5 are in degrees and represented in red, blue, black, green and yellow colors respectively and represents possible solutions obtained from MPSO technique. It is seen that feasible solutions obtained are multiple at some values of MI and single at some values of MI. In the case of multiple solutions, the feasible solutions which produce least %THD are considered.
Figures 8 & 9 represents simulated output phase voltage which contains eleven steps and line to line voltage wave forms of three phase CHB 11-level inverter. Table 3 shows, feasible switching angles and %THD at each value of MI. As the value of MI has increased linearly from 0 to 1 in steps of 0.1 the %THD decreases. Figure 10 represents the graph of %THD at various MI. It can be seen that the value of %THD over a major range of MI (greater than 0.5) found to be nearer to 6.0% and also satisfies IEEE 519-1992 harmonic guidelines.
From Table 3, the minimum value of %THD of 4.10% is obtained at MI of value 1.0 and FFT analysis at that MI is also shown in Figure 11. It can be seen that all the lower order of harmonics such as 5th, 7th, 11th and 13th are significantly reduced to a greater extent and the value of %THD is 4.10%. Thus, it proves the effectiveness of proposed MPSO algorithm for solving this particular optimization problem.
%THD values of simulation and experiment approach at various M I
From experimental results, it is clearly observed that %THDs obtained at various MI validates the simulation results and are very close with each other. It is also observed for the MI of value greater than 0.5, %THD values are less and significant less values of %THDs are observed at 0.7, 0.75, 0.8, 0.9 and 1.0. As the value of MI increases linearly from 0 to 1.0, the %THD value decreases and minimum of value 5.145% has occurred at MI of 1.0
Figure 15 represents experimental FFT analysis representing order of harmonics at MI of 1.0. It can be seen that, desired lower order harmonics such as 5th, 7th, 11th and 13th are significantly minimized and the value of %THD at that MI is found to be 5.145%, which can be seen clearly from experimental FFT analysis list as shown in Figure 16. This value of %THD is significantly minimum and complies with IEEE 519-1992 harmonic guidelines. In addition to that the values of %THD which are obtained above MI of value 0.7 comply with IEEE 519-1992 harmonic guidelines too.
This paper mainly focused on developing efficient, rugged algorithm with less computational effort to solve SHE equations set during entire range of MI from 0 to 1.0 to get feasible switching angles, in order to obtain less %THD which satisfies IEEE 519-1992 harmonic guidelines. A three phase CHB 11-level inverter has been chosen for analysis. In order to solve SHE optimization problem, deterministic method like NR method, stochastic optimization methods like C-GA and MPSO techniques have been applied.
By observing the computational results from NR method, though the drawback of good initial guess has been overcome by any random initial guess here, NR algorithm could not solve SHE equations set during lower ends of MI up to 0.460 and upper ends of MI greater than 0.850. But the algorithm is rugged and has effectively solved SHE equations from MI of value 0.460 to 0.850 with significantly less %THD. The maximum value of %THD obtained is 8.9% and minimum is 5.33%. Also, in NR method by comparing FFT analysis obtained from all the techniques, all the lower order harmonics like 5th, 7th, 11th and 13th are completely minimized which could not be observed in optimization techniques mentioned here.
C-GA has effectively solved SHE equations set during complete range of MI from 0 to 1, which could not be solved by NR technique. The values of %THDs are significantly minimum has observed at above 0.7 MI of value and least value of 6.10% has occurred at of value 0.97 MI. The magnitude of fundamental voltage at which minimum %THD has occurred is much greater than that obtained in NR technique. Though, it could overcome the drawbacks of NR methods, the complexity of writing long coding is more because of processes such as cross over, mutation etc. It is observed for this problem, in order to eliminate the problem of local trapping, the algorithm needs several runs, say 10 and minimum %THD among them has to be considered.
Proposed stochastic technique like MPSO algorithm has successfully solved SHE equations set during complete range of MI from 0 to 1. Implementation of MPSO algorithm is easy as the algorithm is robust and it effectively provides near global optimum solutions. The values of %THD above 0.4 Modulation indices are minimum and significantly less %THDs are observed above 0.7 MI. The least %THD of value 4.10% is obtained at MI of 1.0. Thus, the magnitude of fundamental voltage at which least %THD has occurred is more compared to above mentioned techniques. Hence, this proposed MPSO technique has been considered for experimental validation.
FPGA Xilinx’s SPARTAN 3-A DSP controller is used to generate the required sixty gate pulses for chosen three phase CHB 11-level inverter. From experimental results listed in Table 4, it is clearly observed that %THDs obtained at different MI from 0 to 1.0 are very close with each other and the %THDs obtained above MI of value 0.7, comply with IEEE 519-1992 harmonic guidelines too. Least %THD of value 5.145% is obtained at MI of value 1.0 from experimental analysis and from simulation approach is 4.10%. Since the value of %THD obtained from experimental and simulation analyses are nearly equal, thus it proves the effectiveness of proposed algorithm in real time too.
From above analysis, it is understood that, for this optimization problem among all the mentioned optimization techniques, MPSO has successfully solved SHE equations and the experimental results also validates the effectiveness of proposed MPSO algorithm and %THDs obtained over a range of MI also comply with IEEE 519-1992 harmonic guidelines too. Hence, this can be successfully used in applications such as MV Drives, renewable sources, hybrid electrical vehicles and power quality improvement devices.
On the other hand, in all stochastic optimization techniques, algorithm specific parameters such as mutation rate, social parameter, cognitive parameter and constriction factor are needed. However, during recent advancements in optimization techniques, teaching learning based optimization (TLBO) algorithm has newly introduced (Rao, & Patel 2012). TLBO algorithm does not use such specific algorithm parameters. Thus, it may be implemented for solving SHE optimization problem which may reduce the burden further in developing code.
JM received B.Tech degree in Electrical & Electronics Engineering from Nagarjuna University, Guntur, AP, India in 2000 and M.Tech degree in Power Electronics from Visveswaraiah Technological University, Belgaum, KA, India in 2004. Presently he is pursuing Ph.D degree from J.N.T.U. College of Engineering, Anantapur AP, India. Currently he is with the Dept of Electrical and Electronics Engineering, Guntur Engineering College, Guntur,AP,India. His area of research includes multilevel inverter control, Industrial Drives and Optimisation Techniques. He is Life member of ISTE.
SP is presently working as Professor & HOD of Electrical and Electronics Engineering department, JNTUCEA, Anantapur, AP, India. She completed her B.Tech, M.Tech and Ph.D. degrees in Electrical Engineering from J.N.T.U.Anantapur, Anantapur, AP, India, in 1993, 2003 and 2012 respectively. Her area of interest includes Reliability Engineering with emphasis to Power Systems and Real time Energy Management.
KSR has completed his B.Tech, M.Tech and Ph.D. degrees in Electrical Engineering from JNTUH, Hyderabad, AP, India in 1982, 1985 and 1999 respectively. Currently he is Principal JNTU College of Engg, Anantapur, India. His research interest includes Power Systems and Intelligent Techniques. He is Fellow of Institution of Engineers (I), Life Member of ISTE and Indian Society of Power Engineers (ISPE).
- Agelidis VG, Balouktsis AI: On Attaining the Multiple Solutions of Selective Harmonic Elimination PWM Three-Level Waveforms Through Function Minimization. IEEE Trans on Ind Electron 2008, 55(3):996-1004. 10.1109/TIE.2007.909728View ArticleGoogle Scholar
- AKAl O, Nabil A, Ahmed AM, Kandari A, Ebraheem HK: Selective Harmonic Elimination of PWM AC/AC Voltage Controller using Hybrid RGA-PS Approach. World Acad Sci Eng Technol 2007, 29: 142-146. 10.1.1.193.242Google Scholar
- Azab M: Global maximum power point tracking for partially shaded PV arrays using particle swarm optimization. Int J Renew Energy Technol 2009, 1(2):211-235. 10.1504/IJRET.2009.027991View ArticleGoogle Scholar
- Bernert S: Recent developments of high power converters for industry and traction applications. IEEE Trans Power Electron 2000, 15(6):1102-1117. 10.1109/63.892825View ArticleGoogle Scholar
- Bin W: High –Power Converters and AC Drives. Piscataway, NJ: IEEE Press; 2006.Google Scholar
- Cetin A, Ermis M: VSC-Based D-STATCOM with Selective Harmonic Elimination. IEEE Trans Ind Electron 2009, 45(3):1000-1015. 10.1109/TIA.2009.2018926Google Scholar
- Chambers L: Practical handbook of Genetic Algorithms. Boca Raton: CRC Press; 1995.View ArticleGoogle Scholar
- Chiasson JN, Tolbert LM, Mckenzie KJ, Du Z: A complete solution to the harmonic elimination problem. IEEE Trans Power Electron 2004, 19: 491-499. 10.1109/TPEL.2003.823207View ArticleGoogle Scholar
- Chiasson J, Tolbert L, McKenzie K, Du Z: Eliminating harmonics in a multilevel converter using the resultant theory of symmetric polynomials and resultants. IEEE Trans Control Syst Technol 2005, 13(2):216-223. 10.1109/TCST.2004.839556View ArticleGoogle Scholar
- Clerc M, Kennedy J: The particle swarm: Explosion, stability and convergence in a multi-dimensional complex space. IEEE Trans Evol Comput 2002, 6(1):58-73. 10.1109/4235.985692View ArticleGoogle Scholar
- Davis L: The Handbook of Genetic Algorithms. New York: Van Nostrand reinhold; 1991.Google Scholar
- Deb K: Multi-Objective Optimisation using Evalutionary Algorithms. New York: John Wiley & sons; 2001.Google Scholar
- Dervis K, Bahriye A: A comparative study of Artificial Bee Colony algorithm. Appl Math Comput 2009, 214(1):108-132. org/10.1016/j.amc.2009.03.090View ArticleGoogle Scholar
- Felix TS, Chan M, Kumar T: Swarm intelligence. Vienna, Austria: I-Tech Education and Publishing; 2007. 10.5772/48Google Scholar
- Hammond PW: A new approach to enhance power quality for medium voltage AC drives. IEEE Trans Ind Appl 1997, 33(1):202-208. 10.1109/28.567113View ArticleGoogle Scholar
- Hereford J, Siebold M: Multi robot search using a physically embedded Particle Swarm optimization. Int J Comput Intell Res 2008, 4(2):197-209.View ArticleGoogle Scholar
- Holmes DG, Lipo TA: Pulse Width Modulation for Power Converters-Principle and Practice. New York: IEEE Press/Wiley & son-Interscience; 2003.View ArticleGoogle Scholar
- Homaifar CXG, Lai SH: Constrained optimization via Genetic Algorithms. Simul 1994, 62(4):242-254. 10.1177/003754979406200405View ArticleGoogle Scholar
- Rodriguez J, Lai JS, Peng FZ: Multilevel Inverters: A Survey of Topologies, Controls, and Applications. IEEE Trans Ind Electron 2002, 49(4):724-738. 10.1109/TIE.2002.801052View ArticleGoogle Scholar
- Joshi Manohar V, Sujatha P, Anjaneyulu KSR: A Comparative Analysis of Modulation Control Strategies for Cascade H-Bridge 11-Level Inverter. World Acad Sci Eng Technol 2013, 77: 791-795.Google Scholar
- Kennedy J, Eberhart R: Particle swarm optimization. in Proc IEEE Int Conf Neural Netw 1995, 4: 1942-1948. 10.1109/ICNN.1995.488968View ArticleGoogle Scholar
- Lai JS, Peng FZ: Multilevel converters – A new breed of converters. IEEE Trans Ind Appli 1996, 32(3):509-517. 10.1109/28.502161View ArticleGoogle Scholar
- Liang TJ, Connnell RMO, Hoft HRG: Inverter harmonic reduction using Walsh function harmonic elimination method. IEEE Trans Ind Electron 1997, 12: 971-982.Google Scholar
- Malinowski M, Gopakumar K, Rodriquez J, Perez MA: A survey on cascaded multilevel inverters. IEEE Trans Ind Electron 2010, 57(7):2197-2206. 10.1109/TIE.2009.2030767View ArticleGoogle Scholar
- Menz R, Opprecht F: Replacement of a wound rotor motor with an adjustable speed drive for a 1400 kW kiln exhaust gas fan. Proc 44th IEEE IAS Cement Ind Tech Conf 2002, 85-93. 10.1109/.2002.1006497Google Scholar
- Meynard T, Foch H: Multi-level choppers for high voltage applications. Eur Power Electron J 1992, 2(1):45-50.Google Scholar
- Michalewicz Z, Schoenauer M: Evolutionary algorithms for constrained parameter optimization problems. Evol Comput J 1996, 4(1):1-32. 10.1.13.9279View ArticleGoogle Scholar
- Nabae I, Takahashi Akagi H: A new neutral-point-clamped PWM inverter. IEEE Trans Ind Appl 1981, IA-17(5):518-523. 10.1109/TIA.1981.4503992View ArticleGoogle Scholar
- Okayama H, Koyama M, et al.: Large capacity high performance 3-level GTO inverter system for steel main rolling mill drives. in Conf Rec IAS Annu Meet 1996, 174-179. 10.1109/IAS.1996.557012Google Scholar
- Ozpineci B, Tolbert LM, Chiasson JN: Harmonic Optimization of MultilevelConverters Using Genetic Algorithms. IEEE Power Electron Lett 2005, 3(3):92-95. 10.1109/PESC.2004.1355167View ArticleGoogle Scholar
- Patel HS, Hoft RG: Generalized harmonic elimination and voltage control in thyristor inverters: Part I—Harmonic elimination. IEEE Trans Ind Appl 1973, IA-9(3):310-317.View ArticleGoogle Scholar
- Patel HS, Hoft RG: Generalized harmonic elimination and voltage control in thyristor inverters: Part II—Voltage control technique. IEEETrans Ind App 1974, IA-10(5):666-673. 10.1109/TIA.1974.349239View ArticleGoogle Scholar
- Peng FZ, McKeever JW, Adams DJ: Cascade Multilevel Inverters for Utility Applications. IECON Proc (Ind Electron Conf) 1997, 2: 437-442.Google Scholar
- Renge MM, Suryawanshi H, Borghate VB, Ramteke MR: Common mode voltage in induction motor drives. IEEE int conf on ind technol 2006. Mumbai 10.1109/ICIT.2006.372665Google Scholar
- Reza S, Naeem F, Mehrdad A, Seyed Hamid F: Elimination of Low order harmonics in Multilevel Inverter using Genetic Algorithm. J Power Electron 2011, 11(2):132-139. doi:3_JPE_10230.pdf 10.6113/JPE.2011.11.2.132View ArticleGoogle Scholar
- Rossmann WC, Ellis RG: Retrofit of 22 pipeline pumping stations with 3000-hp motors and variable-frequency drives. IEEE Trans Ind Appl 1998, 34(1):178-186. 10.1109/28.658744View ArticleGoogle Scholar
- Samir Kouro S, La Rocca B, Cortes P, Alepuz S, Bin W, Rodriguez J: Predictive control based selective harmonic elimination with low switching frequency for multilevel converters. Energy Convers Congr Exposition 2009, 3130-3136. 10.1109/ECCE.2009.5316041Google Scholar
- Sivanandam SN, Deepa SN: Principles of Soft Computing, Second edition page(s). 2011, 385-401.Google Scholar
- Siva Kumar K, Das A, Ramchand R, Patel C, Gopa kumar K: A hybrid multilevel topology for an open-end winding induction motor drive using two-level inverters in series with a capacitor fed H-bridge cell. IEEE Trans Ind Electron 2010, 57(11):3707-3714. 10.1109/TIE.2010.2040565View ArticleGoogle Scholar
- Sun J, Grotstollen H: Solving nonlinear equations for selective harmonic eliminated PWM using predicted initial values. Proc Int Conf Ind Electron Control Instrum Automation 1992, 1: 259-264. 10.1109/IECON.1992.254623Google Scholar
- Tarafdar M, Taghizadeh H, Razi K: Harmonic Minimization in Multilevel Inverters Using Modified Species- Based Particle Swarm Optimization. IEEE Trans Power Electron 2009, 24(10):2259-2266. 10.1109/TPEL.2009.2022166View ArticleGoogle Scholar
- Venkata Rao R, Vivek P: An elitist teaching-learning based optimization algorithm for solving complex constrained optimization problems. Int J Ind Eng Comput 2012, 3: 535-560.Google Scholar
This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.