Selective Harmonic Elimination Equations Research Articles

A hybrid search space reduction algorithm and Newton–Raphson based selective harmonic elimination for an asymmetric cascade H-bridge multi-level inverter

Abstract In this paper, a selective harmonic elimination (SHE) method for an asymmetric cascade H-bridge multi-level inverter (ACHB-MLI) has been performed by using hybrid search space reduction and Newton–Raphson (SSR-NR) algorithm. Three H-bridges are cascaded in a single phase configuration, while each of the bridge is fed with V dc , 3V dc and 6V dc respectively. This particular asymmetric combination of dc sources will produce 10 possible switching combinations and 21-levels in the output voltage ranging from −10V dc to +10V dc . Hence, there will be 10 SHE equations corresponding to fundamental and harmonics, and the solution will be the 10 switching angles which satisfy those SHE equations. While traditional gradient-based techniques like the Newton–Raphson method can yield the global optimal solution for the SHE equations, they necessitate an initial guess. Conversely, meta-heuristic methods often face challenges related to sub-optimal convergence. To address these issues, a novel the hybrid SSR-NR algorithm has been introduced in this paper. This method combines stochastic and deterministic elements, offering a promising solution for solving the SHE equations. In the first step, the evolutionary SSR algorithm will be used and the output of the SSR algorithm is given to the NR method as an initial guess to converge to the final solution, this way the proposed hybrid SSR-NR algorithm is able to address the drawbacks of both the algorithms such as initial guess requirement for NR method and sub-optimal convergence characteristics of evolutionary algorithms. The simulations were carried out in Matlab/simulink environment for a 21-level ACHB-MLI. The results obtained showcase the efficacy of the proposed SSR-NR algorithm in solving the SHE equations. For demonstrating the effectiveness of SSR algorithm the results were also compared with other algorithms such as particle swarm optimization and grey wolf optimizer as well. Further, the experimental results on a laboratory prototype using dSPACE DS1104 R&D controller board were also presented in the paper, to validate the proposed methodology.

Dandelion Optimizer and Gold Rush Optimizer Algorithm-Based Optimization of Multilevel Inverters

With the increasing integration of renewable energy sources into distribution and transmission networks, the efficiency of cascade H-bridge multilevel inverters (MLIs) in power control applications has become increasingly significant for sustainable electricity generation. Traditionally, obtaining optimal switching angles of MLIs to minimize total harmonic distortion (THD) requires solving the selective harmonic elimination equations. To this end, this research aims to use two recently developed intelligent optimization algorithms, dandelion optimizer and gold rush optimizer, to solve this problem. To evaluate the effectiveness of the proposed algorithms, an eleven-level cascaded H-bridge MLI (CHB-MLI) was considered in the study. Simulation results for different modulation indices were obtained, and the accuracy and solution quality were compared with genetic algorithm and particle swarm optimization algorithms. MATLAB/Simulink-based models were used to verify numerical computations, ensuring the reliability of the findings. This research contributes to the field by providing insights into obtaining optimal switching angles and minimizing THD in MLIs by applying intelligent optimization algorithms.

Application of Newton Identities in Solving Selective Harmonic Elimination Problem With Algebraic Algorithms

Algebraic algorithms are powerful methods in solving the selective harmonic elimination (SHE) problem, which can find all exact solutions without the requirements of choosing initial values. However, the huge computational burden and long solving time limit the solving capability of algebraic algorithms. This article presents a novel Newton’s identifies-based method to simplify the SHE equations including the order reduction and the variable elimination, thereby reducing the computational burden and the solving time of algebraic algorithms or in other words improving the solving capability of the algebraic algorithms. Compared with existing simplification methods, the proposed method significantly improves the efficiency of solving SHE equations. With the proposed method, the degree of reduction is no longer the bottleneck of solving the SHE equations by using algebraic algorithms. By using the proposed method, the SHE equations with ten switching angles are completely solved with the algebraic algorithm for the first time. The simulation and experimental results indicate that the proposed method is effective and correct.

Real-Time Harmonic Optimization in Multilevel Inverter Using Artificial Neural Network (ANN)

Multilevel inverters (MLIs) are increasingly used in real time applications. Among several pulse width modulation (PMW) techniques currently deployed for the control of MLIs, selective harmonic elimination PWM (SHEPWM) technique arguably gives the best performance due to its direct harmonic mitigation capability. However, real time application of SHEPWM technique is presently infeasible due to the heavy computational cost involved in solving the transcendental nonlinear equations known as selective harmonic elimination (SHE) equations, which characterize the harmonics that are selected for elimination or mitigation. This paper presents a twostage approach to the online generation of switching angles that mitigate selected lower-order harmonics in multilevel inverters. The first stage involves an offline solution of SHE equations using ant colony optimisation (ACO). In the second stage, ACO computed results are used to train an artificial neural network (ANN) predictive model. The results obtained from the simulation of the proposed method in MATLAB/SIMULINK environment show that the method is highly efficient and accurate.

Hardware-in-the-Loop Implementation of Projectile Target Search Algorithm for Selective Harmonic Elimination in a 3-Phase Multilevel Converter

Multilevel converters (MLCs) are extensively used in medium- and high-power applications. Currently, MLCs are also used in grid integration of renewable sources of energy. In this paper a three-phase 11-level MLC topology is investigated which requires a smaller number of active switches compared to similar existing topologies for the same number of levels. In order to eliminate lower order harmonics, selective harmonic elimination (SHE) pulse width modulation technique is used. A projectile target search (PTS) algorithm is used to solve nonlinear SHE equations to determine the optimum values of switching angles to eliminate 5 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">th</sup> , 7 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">th</sup> , 11 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">th</sup> and 13 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">th</sup> harmonics from the converter output voltage. The PTS algorithm finds the best value of switching angles for different modulation indices (M) and these values are used in simulating the converter for eliminating lower order harmonics. Total harmonic distortion (THD) analysis is conducted for PTS optimization techniques at different modulation index. Simulation results are verified by hardware-in-the-loop test of the proposed three-phase converter using a Typhoon HIL-402 emulator.

Multi-verse optimization algorithm- and salp swarm optimization algorithm-based optimization of multilevel inverters

Renewable energy sources are installed into both distribution and transmission grids more and more with the introduction of smart grid concept. Hence, efficient usage of cascaded H-bridge multilevel inverters (MLIs) for power control applications becomes vital for sustainable electricity. Conventionally, selective harmonic elimination equations need to be solved for obtaining optimum switching angles of MLIs. The objective of this study is to obtain switching angles for MLIs to minimize total harmonic distortion. This study contributes to the solution of this problem by utilizing two recently developed intelligent optimization algorithms: multi-verse optimization algorithm and salp swarm algorithm. Moreover, well-known particle swarm optimization is utilized for MLI optimization problem. Seven-level, 11-level and 15-level MLIs are used to minimize total harmonic distortions. Simulation results with different modulation indexes for seven-, 11- and 15-level MLIs are calculated and compared in terms of the accuracy and solution quality. Numerical calculations are verified by using MATLAB/Simulink-based models.

Unified Selective Harmonic Elimination Control for Four-Level Hybrid-Clamped Inverters

A unified selective harmonic elimination (SHE) control for four-level hybrid-clamped (4L-HC) inverters is presented in this article. With this unified strategy, all four-level switching patterns and the corresponding switching angles can be obtained simultaneously by solving one group of unified four-level SHE equations. Therefore, the optimal switching pattern of each modulation index with the designed optimization goal can be evaluated, and the best overall output performance is achieved. In order to ensure the proper operation of the 4L-HC inverter, the voltages across the three-phase flying capacitors and three dc-link capacitors must be controlled and balanced at one-third of the dc-link voltage. The proposed voltage control method is based on redundant switching states and introducing a slight variation to the precalculated switching angles, which extends or limits the conduction time of capacitors depending on voltage deviation and current direction. The effect of switching angle variations on the harmonic performance is also studied. Simulation and experimental results are presented to confirm the validity of the unified model and the proposed voltage-balancing strategy. Comparisons with phase-shifted pulsewidth modulation (PWM) are also presented to demonstrate that the implemented SHE-PWM could significantly reduce the switching frequency while keeping the capacitor voltages well balanced within the expected band limits.

Generalized Formulations and Solving Techniques for Selective Harmonic Elimination PWM Strategy: A Review

Multilevel inverters (MLIs) are widely recommended for DC to AC power conversion as per the load requirements. The MLIs are preferred over other types of multilevel and conventional converters due to their ability to produce staircase output voltage waveforms with superior harmonics profile. However, the performance of MLIs largely depends on the modulation strategy employed. Among the various modulation techniques, the selective harmonic elimination (SHE) method is more efficient in eliminating low-order harmonics from the voltage waveform produced by MLIs. Proper mitigation of the dominant low-order harmonics results in reduction in switching losses in the semiconductor devices and improves the efficiency of the inverter. However, the analytical solution of the nonlinear SHE equations constitutes the main challenge. Over last few decades, various solving algorithms and techniques such as algebraic methods such as resultant theory, numerical techniques and optimization algorithms have been developed and proposed for selective elimination of unwanted harmonics. This review paper presents the various aspects of the problem formulations for SHE technique. Moreover, the detailed principle operation for five different SHE problem-solving techniques is also discussed in this paper. Solution trajectories corresponding to multiple solutions, PWM waveforms with single and multiple transitions at each voltage level and also for pulse-width-amplitude modulation are evaluated and presented to help the researchers gain better insight into the SHE problem and its solution.

Unified Selective Harmonic Elimination for Cascaded H-Bridge Asymmetric Multilevel Inverter

The unequal dc-link voltages in the cascaded H-bridge (CHB) asymmetric multilevel inverters (AMLIs) lead to asymmetry of the equations for selective harmonic elimination (SHE), which increases the difficulty in solving the switching angles. On the other hand, the unequal dc-link voltages provide a large amount of synthesized waveforms which can be used to optimize the performance of inverters. However, as most of the traditional SHE approaches are derived according to a given output waveform, it cannot handle multiple synthesized waveforms. In this paper, the unified SHE equations are derived for all the possible synthesized waveforms under a specific switching angles distribution among the cells. By using the polynomial homotopy continuation (PHC) algorithm, all the possible solutions of the unified SHE equations can be found without the selection of initial values. As the waveforms generated by H-bridge are limited to three levels, some of the solutions are not physically realizable and should be discarded, hence, the criterion is proposed to distinguish these specious solutions. The case of a CHB converter with two cells and six switching angles is studied; in total, there exist 86 groups of candidate solutions and 14 of them are physically realizable. Experiments are also shown to validate the correctness of this unified SHE approach for CHB AMLIs.

Simplified SHE pulse‐amplitude modulation for multilevel inverters

In this study, a selective harmonic elimination (SHE)-pulse-amplitude modulation method is suggested for the cascaded multilevel inverters. This method is based on the balancing of the voltage-second area under the output voltage waveform and reference sinusoidal signal. The proposed method is free from the conventional transcendental SHE equations as it is based on a composite midpoint rule of numerical integration which works on linear equations. Reduced computational burden enables real-time implementation of the proposed method with great ease even for the inverter with a huge number of levels. The switching angles remain constant for an inverter while the output voltage is controlled by varying voltage of dc sources linearly. The proposed method eliminates more harmonics and offers reduced total harmonic distortion (THD) in comparison to the conventional SHE methods without amplifying the amplitude of higher order harmonics. THD of the output voltage remains constant with variation in modulation index. Performance analysis of the suggested method is validated with simulation and experimental studies on cascaded H-bridge inverter.

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 $m=\text{0.72}$ 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.

GA Based Selective Harmonic Elimination for Five-Level Inverter Using Cascaded H-bridge Modules

Multilevel inverters (MLI) have been commonly used in industry especially to get quality output voltage in terms of total harmonic distortion (THD). In addition, development in semiconductor technology and advanced modulation techniques make MLI implementation more attractive. Selective Harmonic Elimination (SHE) that can be applied MLI at desired switching frequency offers elimination of harmonics in the output voltage. Also, by using SHE technique with cascaded multilevel inverters, the necessity of using filter in the output can be minimized. In this paper, SHE equations have been solved by using of Genetic Algorithm (GA) Toobox&Matlab and it has been aimed to eliminate desired harmonic orders at fundamental output voltage. Simulation results have clearly demonstrated that GA based SHE techniques can eliminate the demanded harmonic orders.

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.

Parallel resultant elimination algorithm to solve the selective harmonic elimination problem

Although the resultant elimination method can get all the possible solutions for the selective harmonic elimination (SHE) problem without the selection of initial values, it still has some fatal shortcomings, such as the high computation burden and the huge memory consumption caused by the intermediate expression swell in the procedure of computing the symbolic determinant of the Sylvester matrix. On the basis of the principle of polynomial interpolation, an algorithm framework is proposed to compute the resultant polynomials, which contains the following two major steps: the evaluation of numerical interpolation points and the solution of linear equations. This approach avoids symbolic computing whose computation complexity is usually very high, furthermore, both of these two steps are suitable for parallel implementing which can speed up the computing tremendously. By using the extended n-dimensional Bjorck–Pereyra's algorithm, this algorithm framework is implemented on a parallel computing system, and it has been used to solve the SHE equations for two-level, three-level, and multilevel inverters. As all the possible solutions can be found by this algorithm, the optimal solutions which have the lowest total harmonic distortion can be identified. Experiment results verify the correctness and effectiveness of the proposed method.

Selective Harmonic Elimination for a Single-Phase 13-level TCHB Based Cascaded Multilevel Inverter Using FPGA

This paper presents an implementation of selective harmonic elimination (SHE) modulation for a single-phase 13-level transistor-clamped H-bridge (TCHB) based cascaded multilevel inverter. To determine the optimum switching angle of the SHE equations, the Newton-Raphson method is used in solving the transcendental equation describing the fundamental and harmonic components. The proposed SHE scheme used the relationship between the angles and a sinusoidal reference waveform based on voltage-angle equal criteria. The proposed SHE scheme is evaluated through simulation and experimental results. The digital modulator based-SHE scheme using a field-programmable gate array (FPGA) is described and has been implemented on an Altera DE2 board. The proposed SHE is efficient in eliminating the 3 rd , 5 th , 7 th , 9 th and 11 th order harmonics, which validates the analytical results. From the results, it can be seen that the adopted 13-level inverter produces a higher quality with a better harmonic profile and sinusoidal shape of the stepped output waveform.

Harmonic analysis and FPGA implementation of SHE controlled three phase CHB 11-level inverter in MV drives using deterministic and stochastic optimization techniques.

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.