Modified Multiverse Optimizer Selective Harmonic Elimination for Five-Level Modified Packed U-Cells Inverter and Analysis of Motor Torque Ripple Suppression

Li, Guohua; Wang, Yibo; Fu, Zhenfang

doi:https://doi.org/10.1155/2022/8397836

Mathematical Problems in Engineering

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Research Article | Open Access

Volume 2022 | Article ID 8397836 | https://doi.org/10.1155/2022/8397836

Modified Multiverse Optimizer Selective Harmonic Elimination for Five-Level Modified Packed U-Cells Inverter and Analysis of Motor Torque Ripple Suppression

Guohua Li,¹Yibo Wang,¹and Zhenfang Fu¹

Academic Editor: Francisco Beltran-Carbajal

Received15 May 2022

Revised26 Jul 2022

Accepted30 Aug 2022

Published30 Sept 2022

Abstract

A selective harmonic elimination method of modified packed u-cells (MPUC) inverter fed induction motor built on modified multiverse optimizer (MMVO) algorithm is put forward. The proposed method adopts the MMVO algorithm to search the optimal solution in the SHEPWM equations through the two processes of exploring the optimal solution distribution and exploiting the optimal solution within it. On this basis, the influence of the different orders of power supply harmonics eliminated selectively on the motor’s torque ripple is compared and analyzed by using the Simulink and MAXWELL. From the results of simulation and experiments, it can be proved that the method can achieve the required fundamental voltage and selectively suppress the harmonic with particular order. At the same time, it is verified by ANSYS MAXWELL that the working quality of the motor can be improved by selectively eliminating the 5th, 7th, 11th, and 13th harmonics in the power source to effectively suppress the 6th and 12th periodic torque ripples that arise from these harmonics.

1. Introduction

The high-order harmonics of the inverter power supply are one of the reasons which affect the smoothness of the motor output torque [1]. From the perspective of controlling the harmonics of the inverter output, the methods of reducing the electromagnetic noise and torque ripple of the motor include software filter algorithms and hardware filters [2], increasing the switching frequency [3], dynamically adjusting the switching frequency [4, 5], the random PWM (pulse width modulation) strategy [6–8], improving the topology of the inverter [9, 10], and eliminating specific harmonics [11]. Ref. [2] adds a bandpass digital filter in the control algorithm to realize the selective elimination of harmonics in a specific frequency band, but the calculation of the digital filter is relatively large. Ref. [3] proves that increasing the switching frequency of the converter can suppress the operating noise of the motor effectively, but an excessively high switching frequency will cause negative effects such as increased energy loss and stress of the power switching device. Ref. [4] reduces the vibration of the motor system in different operating states by dynamically changing the switching frequency. Ref. [5] effectively reduces the electromagnetic noise located near the integral multiple of the switching frequency in the AC (alternating current) motor through randomizing the switching frequency. However, in the course of the random spread spectrum, it is possible that the harmonic content near the peak of the original harmonic will increase instead, resulting in new electromagnetic vibration and electromagnetic noise. Ref. [6] provides a method to reduce the noise and electromagnetic vibration caused by induction motor through RPWM (random pulse width modulation) selective spectrum shaping. This method can reduce the noise and electromagnetic vibration around the integer multiples of inverter switching frequency, while also selectively suppressing the noise of other frequencies. The MPUC (modified packed u-cells) multilevel inverter is proposed in [9, 10], which effectively reduce the harmonic content by increasing inverter levels. Compared with the traditional multilevel inverters, this topology can output the same number of levels with fewer power switch devices. Another thing worth noting is that the MPUC inverter does not have the problem of voltage imbalance, which improves system reliability.

SHEPWM (selective harmonic elimination PWM) strategy is a feasible method to reduce the low-order harmonics of the inverter and adapt to low switching frequency [10]. Numerical method, algebraic method, and intelligent algorithm are commonly used to solve SHEPWM nonlinear transcendental equations. Numerical methods include Newton’s iterative method [7], homotopy algorithm [8], and Walsh function [11], which usually have problems such as the initial values that need to be given, low accuracy, and convergence to the local optimal solution. In order to avoid these deficiencies, algebraic methods such as Wu's method [12], Groebner Basis [13], Resultant theory [14], and algebraic polynomials [15] can be used to reduce the convergence probability of the local optimum and do not require a given initial value. But these usually increase the complexity of the algorithm. In recent years, some new intelligent algorithms have been applied in SHEPWM. Ref. [16, 17] uses the genetic algorithms, but the selection of population size, mutation probability, crossover probability, and evolutionary algebra may cause the population to lose the evolutionary ability, reduce diversity, premature population maturity, or nonconvergence. Ref. [18, 19] uses the particle swarm optimization algorithm, whose performance and convergence are directly affected by parameters, and the setting of parameters depends to a large extent on experience. Ref. [20] applies the bee colony algorithm to the calculation of SHEPWM equations, which has the advantage of high accuracy, but the disadvantage is that the convergence speed is slow, and it is easy to premature. Ref. [21] gives the ant colony optimization algorithm, but the shortcoming is the initial pheromone is lacking, the global convergence is poor, and the efficiency is low. Ref. [22] uses the neural network algorithm to predict the iterative initial values of switching angles, which not only reduced the scale of the neural network but also improved the accuracy of switching angle calculation. Ref. [23] uses the multipopulation genetic algorithm to solve the SHEPWM harmonic elimination equations quickly without the initial values of switching angles.

None of the SHEPWM algorithms mentioned above have been analyzed and designed uniformly according to the torque ripple characteristics of motor loads. In this paper, a five-level MPUC inverter SHEPWM method build on the MMVO (modified multiverse optimizer) algorithm is proposed, which does not need the initial value and can accurately calculate the switching angles required for selective harmonic elimination of the five-level MPUC inverter. Using it as a base, the relationship between the harmonics produced by the inverter and the torque pulsation of the induction motor is analyzed, and the 6th and 12th periodic torque pulsation of the induction motor is effectively suppressed by selectively eliminating the 5th, 7th, 11th, and 13th harmonics.

The specific content of this article is as follows. The article mainly expounds the topology and working mode of MPUC inverter in Section 2. Then, Section 3 analyzes the torque ripple of the induction motor fed by the inverter. Section 4 proposes a MMVO algorithm for the five-level MPUC inverter SHEPWM method. The practicability and validity of the proposed method are proved through the simulation and experiment in Section 5. Also, finally, Section 6 summarizes the theory proposed in this paper.

2. Theory of MPUC Inverter

Figure 1 shows the topology of a three-phase five-level MPUC inverter, which is composed of three groups of the identical MPUC inverter. Taking a single-phase MPUC five-level inverter as an example, the structure is composed of 2 separate voltage sources and 6 groups of power switching elements [24].

Table 1 lists all the switching states of the MPUC inverter, among which 1 is the on-state of the switching device, 0 is the off-state of the switching device, and u_AB represents the DC (direct current) voltage output by the inverter. It can be seen from Table 1 that there are 8 switching states of the MPUC inverter, and the switching devices T₁T₄, T₂T₅, and T₃T₆ are complementary in the on-off process. Different power supply voltage amplitude and modulation control method will produce different waveform of output voltage.

Figure 2 shows the voltage and current flow direction of the MPUC inverter under the operating conditions in Table 1.

(a)

(b)

(c)

(d)

(e)

3. Analysis of Torque Ripple of Induction Motor Fed by Inverter

The harmonic components in the input current of the induction motor will result in the harmonic electromagnetic torques. The motor stator current fed from the PWM converter is nonsinusoidal, which results in the change of magnetic flux density and the generation of harmonics in time and space. Harmonic torques can be divided into asynchronous torques and pulsating torques. When the air gap harmonic flux and the rotor harmonic current are the same number of times, the asynchronous torques will be produced by the interaction of electromagnetic effects [25].where C_T = m₁p N₂k_w2/ is a torque constant, m₁ is the number of phases, p is the number of pole pairs, N₂ is the number of rotor turns each phase, k_w2 is the rotor winding coefficient for fundamental, is the stator magnetic flux generated by the nth-degree harmonics, I_2n is the rotor harmonic current, and Ψ_2n is the phase shift between I_2n and electromotive force. The direction of the torques depends on the order of high-frequency harmonics. When the order of the harmonics is equal to 1, 7, 13,…, the direction of torques is the same as the speed. On the contrary, when harmonics order n = 5,11,17,…, the direction of torques is opposite to the speed. The value of asynchronous torques produced by the harmonic current is small and can be offset by harmonic torques in the opposite direction, so they have small impact on the vibration and noise of the motor, which can be neglected in practical operation.

The principal pulsating torques are generated by the interaction of magnetic flux and current at different frequencies in the air gap. Due to the uncertainty of harmonic frequency, there are many reasons for the torque pulsation generated by each group of current and magnetic flux. The most important ones are caused by the fundamental flux of the stator and the harmonic current of the rotor. This type of torque can be expressed aswhere is the fundamental flux, and ѱ_1,2n is the phase shift between the rotor current and the electromotive force for fundamental. For a three-phase motor, the symbol “−” represents the current harmonics n = 6k+1, while the symbol “+” corresponds to the current harmonics n = 6k−1. Due to the 5th, 7th, 11th, and 13th harmonic currents have a relatively large amplitude, moreover, the amplitudes of harmonics will decrease with the increase of the harmonic order. Therefore, the pulsating harmonic torques caused by the mutual effect between the fundamental magnetic field and the harmonic currents are mainly considered.

According to the above formula, the frequency of pulsating torques generated by the 5th and 7th harmonics is 6f, and their directions are opposite in the three-phase motor. That is, f-(-5f) = 6f and f-7f = -6f. Similarly, the frequency of pulsating torques generated by the 11th and 13th harmonics is 12f. That is, f−(−11f) = 12f and f −13f = −12f. Then, it can be extended to the pulsating torques generated by any order of the harmonic currents in the stator and fundamental flux; namely, the main source of pulsating torques is 6 times the fundamental frequency. Thus, the instantaneous torques can also be expressed aswhere T_(em) (t) is the instantaneous pulsating torque of the motor, T₀ is the instantaneous pulsating torque of fundamental, and T_6n is the pulsating torque generated by 6n−th harmonics. Harmonic torques are a nonnegligible excitation source during the operation of the motor, which leads to the periodic change of motor torques and the oscillation of speed and affects the vibration of the mechanical device through the magnetism and solid coupling. Therefore, the pulsating harmonic torques must be suppressed to prevent the influence of resonance on the system when the frequency of harmonic torques is consistent with the resonance frequency of the motor mechanical device.

4. Algorithm of MMVO-Based SHEPWM

4.1. Fundamentals of MVO Algorithm

Figure 3 shows the phase voltage waveform output by a five-level MPUC inverter. In half a period, the output voltage waveform is 1/4 couple symmetry, and there are m + k switching points. In the figure, δ₁, δ_m, and δ_m ₊ _k are switching angles. In the three-phase system, due to the symmetry of the load, the line voltage does not contain the harmonics of phase voltage nor the harmonics of the third order and its multiple after selectively eliminating the specific order harmonics in phase voltage.

According to the symmetry characteristics of the amplitude and waveform of output voltage, the nonlinear SHEPWM equations of five-level inverter output can be expressed as follows:

The M is the amplitude modulation ratio, n = 5, 7, 11, 13, … is the harmonics times to be eliminated, p_k is the position coefficient, the rising edge is 1, the falling edge is -1. δ_k is the value of the kth switching angle.

MVO is a process algorithm based on simulating the exchange of substances in the universe [26]. Different universes have different normalized inflation rate (NI). Depending on the NI, the substances can be transferred from a white hole with high NI to a black hole with low NI. This process can be realized by the method of the roulette wheel.

The and are the qth variables of the pth universe and the variate selected through the above-mentioned mechanism, NI(U_p) represents the normalized inflation rate of the pth universe, and r₁ represents a random number ranging from 0 to 1.

Substances can be transferred not only through white holes and black holes but also through wormholes. Suppose the wormhole tunnels are invariably set up between the universe and the optimal universe. The mechanism of the wormhole establishment can be expressed as

The x_q is the qth variable of the optimal universe at present; ub_q and lb_q are the upper limit and lower limits of this variate; and r₂, r₃, and r₄ represent random numbers ranging from 0 to 1.

This mechanism mainly relies on the two coefficients of wormhole existence probability (WEP) and travelling distance rate (TDR).

The i and I in the formula are the current iterations, and the maximum iterations, man, mix, and k are constants.

As can be seen from the formula, TDR is exponentially decreased and WEP is linearly increased. Before TDR and WEP intersect, TDR greater than WEP can prevent the occurrence of the local optimal solution. After the intersection, the iterative process can improve the evaluation of global optimization result with the decrease of TDR and the increase of WEP. This is a mechanism to improve search accuracy during iteration. By emphasizing the development and exploration proportional to the number of iterations, the convergence of the algorithm can be guaranteed.

4.2. Principle of MMVO Algorithm

When the universe cannot show a better solution (falling into local optimum), the search mechanism can be adjusted to prevent the universe from falling into local optimum by modifying the universe. Therefore, a method based on the universe mutation and the position update is proposed, and the SCA (sine cosine algorithm) is introduced to improve the MVO algorithm.

SCA is an intelligent algorithm based on the population optimization, which was proposed by Mirjalili [27]. Compared with other algorithms, this method has a stronger exploration ability rather than development ability.

In the exploration stage, SCA searches the feasible region of the solution in the search space through the randomly generated random solution. In the development stage, the random solution gradually changes smoothly. This method makes use of the cyclic nature of sine and cosine functions to reciprocate from exploration to development, and the specific position update is shown in the formula. The algorithm flow is shown in Figure 4.where is the i-th position of the t-th iteration, and is the optimal position. A represents a constant, t is the current iterations, and T is the maximum iterations. , , and represent random numbers between 0 and 1.

In order to prevent the MVO algorithm from falling into the local optimal situation, this paper proposes an improvement in the way of selecting universe. The current position is randomly selected by using the position in the equilibrium pool x_eq, which is made up of the 4 best position vectors so far, and combined with their average positions for position replacement. This method enables better exploration to obtain the optimal solution.where x_best1∼x_best4 are the current optimal universal positions, and x_ave=(x_best1+x_best2+x_best3+x_best4)/4.

Sine and cosine functions in the SCA algorithm are used to reconstruct the universe. The global optimization ability of the MVO algorithm is improved through the above mechanism, and formula (10) gives the improved position formula of the MVO algorithm based on the SCA algorithm.where , the r₉, r₁₀, r₁₁, and r₁₂ represent random numbers ranging from 0 to 1.

Taking the harmonic elimination of 5th, 7th, 11th, 13th, and 17th as an example, the (11) are established according to Formula (4).

The (11) are constrained by the formula.

The equations consist of 6 subequation and belong to a multigoal optimization problem, among which ξ is the value of the subequation. For the convenience of solving, the problem can be transformed into a single-objective optimization problem by using the formula (12). The smaller the value of f(ξ), the more accurate the equation to be solved.

Through the MMVO algorithm, the position of the universe is updated according to formula (10), the new fitness value is calculated, and the corresponding switching angle is finally obtained.

Under the same conditions, MVO and MMVO are used to search for optimization. The calculation times of both are 10, and the optimal value is recorded. Figures 5 and 6 show the switching angle trajectory obtained by the two algorithms when the modulation degree is between 0.51–1. Figure 7 depicts the fitness values for different modulation depths under different algorithms. Compared with the more times of angle fluctuations in Figure 5, the number of switching angle fluctuations calculated by the MMVO algorithm is less, which shows the advantages of MMVO. Meanwhile, it can be seen from Figure 7 that the fitness of the MMVO algorithm is lower, which means that the accuracy of the solution is higher.

The SHEPWM nonlinear equations of the five-level MPUC inverter are solved by the above method, and the specific order harmonics which have a significant effect on the torque pulsation of the induction motor in the inverter are selectively eliminated, which can availably suppress the torque ripple of the motor and improve the working quality.

5. Simulation and Experiment

5.1. Simulation of MMVO-Based SHEPWM

The effectiveness of selective harmonic elimination based on the MMVO algorithm is validated with the example of the three-phase MPUC inverter with LR load. The followings are the simulation parameters of the system: the load side parameters of the inverter are R = 5 Ω and L = 5mH, and the DC side reference voltage of the MPUC inverter is 24 V. The harmonics to be eliminated are 5th, 7th, 9th, 11th, and 17th, and M are between 0.7 and 1. Taking M = 0.8 as an example, the switching angles are solved by the MMVO algorithm. Figure 8 shows the relationship between the objective function and the iterations. It is obvious that the MMVO algorithm has a fast convergence speed and strong searching ability. The optimal solution can be calculated in a very short time and can avoid falling into local optimal.

Relevant simulations are implemented according to the above principle. The waveform of phase voltage and line voltage output by MPUC inverter and their corresponding FFT (fast Fourier transformation) analysis waveform are shown in Figures 9 and 10.

(a)

(b)

As shown in Figure 9(a) and Figure 10(a), the phase voltage levels output by the inverter are five, and line voltage levels are seven. Figures 9(b) and 10(b) show that the phase voltage does not contain the 5th, 7th, 11th, 13th, and 17th harmonics. Simultaneously, due to the symmetry of the three-phase circuit, the line voltage does not contain the harmonics of phase voltage nor the harmonics of the third order and its multiple.

(a)

(b)

5.2. Experiment of MMVO-Based SHEPWM

Consistent with the simulation parameters, the experimental prototype and the test system are shown in Figure 11. The system is mainly controlled by 32-bit DSP TMS320F28379. IGBT BSM50GB120DN2 is selected as the power switching device for the main circuit of the MPUC inverter, and the oscilloscope of DS1052 E is used to observe the output waveform of the inverter in the experiment. The experimental waveforms of phase voltage and line voltage output by the MPUC inverter and the corresponding FFT waveforms are shown in Figures 12 and 13, among which PSD is the abbreviation of power spectral density.

(a)

(b)

(a)

(b)

As shown in Figure 12, the phase voltage varies by five levels, and the harmonics to be eliminated in the FFT waveform are effectively suppressed. Compared with the line voltage waveform shown in Figure 13, in addition to the difference in the number of levels, the line voltage does not contain the harmonics of the third order and its multiple except for the eliminated harmonics. Experiments verify the effectiveness of the MMVO algorithm based on the MPUC five-level inverter.

5.3. Torque Ripple Suppression of Induction Motor

In this section, the harmonic torques of the motor are researched by ANSOFT MAXWELL to prove the accuracy of the theoretical analysis presented in this paper. The parameters of the motor fed by the inverter are shown in Table 2.

The rated voltage signal produced by the MPUC inverter is input into the finite element model of the 3-phase induction motor. Under the same condition that the modulation depth is 0.8 and the simulation time is 0.5 s, the first group of SHEPWM control method is aimed at eliminating the 5th, 7th, 11th, 13th, and 17th harmonics; the second group of SHEPWM control method is aimed at eliminating the 17th, 19th, 23th, 25th, and 29th harmonics. The stabilized torque waveform is selected as the analysis object. After the intercepted torques are processed by FFT, the torque spectrum obtained by normalization is shown as follows.

As can be seen from Figures 14 and 15, the influence of harmonics on motor torque ripple is gradually reduced with the increase of harmonic orders; that is, the motor torque is dominated by low-order ripple. According to the results of the torque ripple simulation in Figure 15, it can be seen that the first group of SHEPWM eliminates the 5th, 7th, 11th, 13th, and 17th harmonics caused by the inverter. In addition, the percentages of the 6th and 12th torque harmonics of the motor are reduced to 0.17% and 0.05%. The 18th torque harmonics still have obvious content, because only the 17th harmonics are eliminated and the 19th harmonics are not eliminated in the phase voltage. In contrast, the second group of SHEPWM eliminates the 17th, 19th, 23th, 25th, and 29th harmonics in the phase voltage. Affected by it, the percentages of 18th and 24th torque harmonics of the corresponding motor are reduced to 0.03%. Since the 5th, 7th, 11th, and 13th harmonics of the second group SHEPWM are not eliminated, the percentages of 6th and 12th torque harmonics of the motor are as high as 16.33% and 1.27%, and the low-order torque ripple is strong.

6. Conclusions

Aiming at the research blank that the SHEPWM algorithm proposed at present has not been analyzed and designed uniformly according to the torque ripple characteristics of motor loads, a selective harmonic elimination method based on the modified multiverse optimizer algorithm for the motor fed by the five-level MPUC inverter is proposed. The contribution of this paper can be summarized as follows:(1)The influence of harmonic torques generated by the inverter on the motor load is analyzed, and it is proved that the pulsating torques in harmonic torques have a greater impact on the operating performance of the motor. Therefore, the pulsating harmonic torques caused by the mutual effect between the fundamental magnetic field and the harmonic currents are mainly considered when the motor fed by the inverter. Moreover, the ANSOFT MAXWELL software platform was used to verify that low-order harmonics have a greater impact on the motor torque ripple; that is, the motor torque is dominated by low-order ripple.(2)Based on the MVO algorithm, the five-level SHEPWM equations of the MPUC inverter are established to achieve selective harmonic elimination without the initial conditions. In addition, in view of the situation that the traditional MVO algorithm is easy to fall into the local optimum, this paper introduces the equilibrium pool to increase the diversity of the universe on the basis of improving the position of the universe by using the sine and cosine factors. Modified multiverse optimizer algorithm improves the global optimization ability of the MVO algorithm. Compared with the traditional MVO algorithm, the MMVO algorithm has the advantages of fewer switching angle fluctuations, lower fitness, higher accuracy of the solution, and not easy to fall into local optimum.(3)The method proposed in this paper is verified by simulation software, and the experimental platform with DSP TMS320F28379 as the control core is established according to the simulation parameters. It can be seen from the results of simulation and experiments that this method can achieve the desired fundamental voltage, and the content of the specific harmonic to be eliminated is basically zero. At the same time, the working quality of the motor is improved by selectively eliminating the 5th, 7th, 11th, 13th, and other harmonics to suppress the 6th and 12th periodic torque pulsation caused by the harmonics, which has important reference significance in practical engineering applications.

Data Availability

The inverter parameters data used to support the findings of this study are included within the article.

Conflicts of Interest

The authors declare no conflicts of interest.

Acknowledgments

This work was supported by the National Natural Science Foundation of China under Grant 51307076, Natural Science Foundation of Liaoning Province, China under Grant 20180550268, and The Innovation Team Project of Liaoning Technical University under Grant LNTU20TD-32.

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