Kehu Yang

Selective Harmonic Elimination With Groebner Bases and Symmetric Polynomials

Selective harmonic elimination (SHE) technology has been widely used in many medium- and high-power converters which operates at very low switching frequency; however, it is still a challenging work to solve the switching angles from a group of nonlinear transcendental equations, especially for the multilevel converters. Based on the Groebner bases and symmetric polynomial theory, an algebraic method is proposed for SHE. The SHE equations are transformed to an equivalent canonical system which consists of a univariate high-order equations and a group of univariate linear equations, thus the solving procedure is simplified dramatically. In order to solve the final solutions from the definition of the elementary symmetric polynomials, a univariate polynomial equation is constructed according to the intermediate solutions and two criteria are given to check whether the results are true or not. Unlike the commonly used numerical and random searching methods, this method has no requirement on choosing initial values and can find all the solutions. Compared with the existing algebraic methods, such as the resultant elimination method, the calculation efficiency is improved, and the maximum solvable switching angles is nine. Experiments on three-phase two-level and 13-level inverters verify the correctness of the switching angles solved by the proposed method.

A Groebner Bases Theory-Based Method for Selective Harmonic Elimination

An algebraic method is proposed for selective harmonic elimination PWM (SHEPWM). By computing its Groebner bases under the pure lexicographic monomial order, the nonlinear high-order SHE equations are converted to an equivalent triangular form, and then a recursive algorithm is used to solve the triangular equations one by one. Based on the proposed method, a user-friendly software package has been developed and some computation results are given. Unlike the commonly used numerical and intelligent methods, this method does not need to choose the initial values and can find all the solutions. Also, this method can give a definite answer to the question of whether the SHE equations have solutions or not, and the accuracy of the solved switching angles are much higher than that of the reference method. Compared with the existing algebraic methods, such as the resultant elimination method, the calculation efficiency is improved. Experimental verification is also shown in this paper.

Unified Selective Harmonic Elimination for Multilevel Converters

For multilevel converters, as their selective harmonic elimination (SHE) equations are highly related to the switching patterns, it is almost impossible to give a complete study for all the switching patterns under the traditional SHE framework. In this paper, a unified SHE approach is proposed, in which the various SHE equations for different switching patterns are merged into one group of unified SHE equations and the inequality constraints on the switching angles are eliminated at all. This unified SHE equations can be solved by any methods, which have been used in solving the traditional SHE problems. By using the theory of Groebner bases and symmetric polynomials, all the possible switching patterns and the corresponding switching angles can be solved simultaneously. The study on the case for four switching angles discovers some unusual switching patterns that have the lowest total harmonic distortion for low-modulation indices. Also, the case for nine switching angles with modulation index m=0.72 is studied; in total, there exist 43 groups of switching angles, which belong to 33 different switching patterns. Experiments on a 13-level cascaded-H bridge converter verify the correctness of this unified multilevel SHE approach.

Real solution number of the nonlinear equations in the SHEPWM technology

This paper studied the real solution number of the switch angles for the inverters which are based on the Selective Harmonic Eliminated PWM (SHEPWM) technology. By the method of variable substitution, the nonlinear transcendental equations can be transformed to a Semi-Algebraic system. Then, with the help of the latest progress in the mechanical proving software for the Semi-Algebraic systems, an analytical method to classify the real solution number of the switching angles is proposed. In order to verify the effectiveness of this method, the real solution classifications for the three phase bipolar and unipolar inverters with N = 3 are given. Compared with the exist numerical results, this method can find out the exact boundary points and the final results are analytical.

Harmonic elimination for multilevel converter with Groebner bases and symmetric polynomials

Selective harmonic elimination technology has been widely used in many medium and high power converters which operating at very low switching frequency, however, it is still a challenging work to solve the switching angles from a group of nonlinear transcendental equations, especially for the multilevel converters. In this paper, an algebraic method is proposed for selective harmonic elimination. First, the order of the SHE equations is reduced by using the symmetric polynomials theory, and then, its Groebner bases is computed. After this two conversions, the solving of multivariable high order SHE equations is converted to solve two univariate equations and a set of univariate linear equations which simplifies the solving procedure dramatically and abandon the requirement on initial values and can find all the solutions. Compared with the existing algebraic method, such as the resultant elimination method, the calculation efficiency is improved. Experimental verification is also shown in this paper.

Switching angles generation for selective harmonic elimination by using artificial neural networks and quasi-newton algorithm

A hybrid method based on artificial neural networks (ANNs) and Quasi-Newton algorithm is proposed to generate the switching angles for selective harmonic elimination (SHE), which makes a compromise among the memory consumption, executing efficiency and the solution precision. Unlike the other ANNs based methods which use ANNs to directly give the final switching angles, this hybrid method just uses ANNs to give the initial values, which lowers the precision requirement on training the ANNs, so, the number of the neurons can be reduced and less on-chip memories are required. Then, the Quasi-Newton algorithm is used to solve the exact switching angles from the initial values given by the ANNs, which guarantees the solving efficiency and the solution precision. The case of 11-level staircase modulated converter is studied by using the single-layer back-propagation (BP) neural networks. The trained neural networks have only 9 neurons in the hidden layer and the output initial values can meet the convergent requirement of the Quasi-Newton algorithm in the full range of modulation index. The total executing time is about 70ms on a STM32F407 microcontroller, as the executed code is automatically generated by MATLAB, the executing time could be further reduced if the code is manual optimized. Experiment results are also shown to verify the correctness of the switching angles generated by the proposed hybrid method.

Unified selective harmonic elimination for fundamental frequency modulated multilevel converter with unequal DC levels

In this paper, a unified SHE approach is proposed for the fundamental frequency modulated multilevel converters with unequal dc levels. Unlike the traditional SHE methods whose equations depend on the switching patterns, this approach unifies all the switching patterns into one group of equations and the inequality constraints on the switching angles are eliminated. In order to obtain all the possible switching angles and their corresponding switching patterns, a polynomial homotopy continuation (PHC) algorithm based method is proposed to solve this unified SHE equations. Experiments on an 11-level cascaded H-bridge (CHB) multilevel converter verify the correctness of this method.