# Investigation of Recent Metaheuristics Based Selective Harmonic Elimination Problem for Different Levels of Multilevel Inverters

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## Abstract

**:**

## 1. Introduction

- The switching moments obtained by the SHE method are applied to an MLI structure that exploits fewer elements than the classical cascade MLI structures, in which a large number of switching elements are used because the cost will increase rapidly with the level of a classical MLI.
- In order to solve the non-linear transcendental equation, set, new generation metaheuristic algorithms that have not been applied before in the literature are examined.
- By examining MLIs with different levels, the effect of increasing the number of levels on the cost, THD minimization, the quality of the desired output voltage, the computing time (related to computational complexity), and the convergence rate of metaheuristic algorithms are revealed in detail.
- Each method cannot produce an optimal solution over the entire modulation index range. Therefore, comparing the performances of algorithms within the entire modulation index range, such as 0.1 ≤ M
_{i}≤ 1.1, and a certain modulation index range, such as 0.4 ≤ M_{i}≤ 0.9, will allow the determination of the most efficient method. - In this study, the most comprehensive evaluation of classical and current metaheuristic methods, which are run with initial parameters such as the maximum number of iterations, search range limits, and the number of search agents, is applied to solve the SHE problem. The results obtained contribute to the field of metaheuristics in terms of better analysis of the performances of the algorithms.

## 2. Multilevel Inverter

#### 2.1. Output Voltage Waveform of Multilevel Inverter

_{1}, α

_{2}, …, and α

_{S}in MLI at a certain level and the stepping output voltage is obtained using the Fourier series expansion, f(t), given in (2).

_{n}and b

_{n}are the even and odd component amplitudes of the nth harmonic, respectively. a

_{0}is the DC coefficient. In this equation, considering the quarter-wave symmetry, the coefficients a

_{0}, and even harmonics will be zero and the equation will turn into Equation (3).

_{0}) is formed as in Equation (5).

_{dc}is the nominal DC voltage, n is the number of switching angles, and α

_{k}is the firing angles calculated in the order (0 < α

_{1}< … < α

_{S}≤ π/2).

#### 2.2. Selective Harmonic Elimination PWM Technique

_{i}) is defined by (7).

_{des}is the desired or calculated value of the output voltage and S is the switching number. For example, in a 7-level MLI, two unwanted harmonics can be eliminated while keeping the output voltage constant. Thus, Equation (6) becomes the set of equations in (9).

_{des}is the desired fundamental voltage and V

_{1}represents the fundamental output voltage of MLI. h

_{s}and V

_{hs}are the order and amplitude of the sth harmonic, respectively, e.g., h

_{2}= 3, h

_{3}= 5, and h

_{5}= 11. The variables (α) are limited by Equation (7).

## 3. Implementation of the Harmonic Problem

_{i}. $Ith{d}_{Mk}$ is the THD value of each algorithm.

_{1}and V

_{des}are the values of output voltage and desired output voltage, respectively.

_{i}), V

_{pu}is subtracted from 1. Then, the sum of pu errors in all M

_{i}is obtained using Equation (14).

_{pu}error.

_{i}< 0.9, the performance analysis was also carried out within this range because, in case M

_{i}> 0.9, the output voltage approaches the square waveform. The control parameters of the algorithms are tabulated in Table 3. The parameters in the table are set to the values used in the literature and recommended in the main article of each algorithm. The size of the search agents, i.e., the population in the search space, is tuned to 100. The maximum number of iterations of 500 is chosen to plot the convergence curves of the methods. The best cost value for each algorithm is obtained by running all methods independently 500 times. The algorithm that provides the most statistically optimal cost is selected.

_{i}range in terms of THD, SPBO is the best method, while MGA shows the worst performance. The BMO, GWO, MFO, GTO, TLBO, and GBO methods indicate better performance in terms of THD than others. Although the results are close in terms of V

_{pu}, MGA, SPBO, WHO, SCA, TLBO, and JS are the prominent methods.

_{i}< 0.9, SPBO, GA, GWO, MFO, SPSA, and BMO are spectacular methods in terms of THD minimization. JS, MGA, SPBO, WHO, SCA, and TLBO methods outperform in terms of output voltage quality.

^{−5}. While all methods arrive at the same best cost value at the end of the 500th iteration, SPSA and MFO converge to the value in the 130th iteration. In the case of 19 levels, all methods converge to the maximum value of 10 × 10

^{−4}at the 500th iteration. SPSA, GA, MFO, and SPBO offer the best cost faster than other methods. The method with the worst convergence performance at all levels emerges as GWO.

^{−5}. GA is followed by SPSA (8.0254 × 10

^{−5}), MFO (9.1466 × 10

^{−5}), GWO (1.1485 × 10

^{−4}), BMO (1.5272 × 10

^{−4}), and SPBO (0.0011), respectively. The SPBO is the method that undergoes the most changes in terms of a single iteration. It is obvious that the MFO method has the smallest iteration time at all levels in comparison with selected metaheuristics according to output voltage, THD minimization, and a single iteration time, respectively.

_{i}≤ 0.9 have been obtained at 11.8, 7.13, 1.54, and 2.29 for the 7, 11, 15, and 19 levels, respectively. The SPBO method reveals the best performance in 15-level MLI. In the BMO method, the THD value is seen as 4.96, 7.88, 4.16, and 8.36 depending on the level increase. The BMO method shows its best performance in the 15-level MLI. In the GA method, the THD value has been procured at 8.65, 3.63, 4.28, and 4.38, depending on the level increase. The GA method depicts its best performance in 11-level MLI. In the GWO method, the THD value has been calculated as 11.59, 6.26, 4.05, and 3 depending on the level increase. The GWO method exhibits better performance with each added harmonic component to the fitness function and reveals its best performance at the 19th level. On the other hand, the MFO method provides similar results to the GWO method. The best performance is exposed at level 15, with a very small margin compared to the MFO. Finally, although the SPSA method performs worse than the others in terms of THD at levels 11 and 15, it yields a better result at levels 15 and 19.

_{i}< 0.9). At this point, it should be taken into account that the output voltage is trying to be kept constant in the 7-level MLI optimization. In the absence of the first term corresponding to the optimization of the output voltage in Equation (10), it is expected that the harmonics will be lower because (10) becomes a THD minimization problem and three available arguments are used. Figure 5b shows the THD performances of the selected methods for the 11-level MLI. It is observed that the results are very close to each other in the range of 0.5 ≤ M

_{i}≤ 0.8, while the THD falls below 8%. If Figure 5b and Table 5 are investigated together, it reveals that the GA method is more successful in terms of THD in 11-level MLI. In Figure 5c, the THD performances of the selected methods for the 15-level MLI are given. The performances of all methods are similar and the THD (%) value provides the standard in a wide range, especially in the case of M

_{i}> 0.45. However, the performance of the SPBO at low M

_{i}values makes it stand out for 15-level optimization, as indicated in Table 5. Finally, THD performances for 19-level MLI are illustrated in Figure 5d. There is a significant decrease in THD values at low modulation indices due to the decrease in the value of the DC source voltages used and the presence of more switching angles. At this level, the IEEE 519—2014 standard has started to be met from M

_{i}= 0.35. The THD value for BMO, MFO, GWO, and GA methods with a modulation index in the range of 0.65–0.8 decreases below 2%. However, as can be seen in Table 5, the SPBO method provides successful results in 19-level MLI, as well as in the modulation index range of 0.4–0.9 at 15-level.

_{i}≥ 0.9, the output current also approximates the square waveform.

_{i}values. In Figure 12b, it is seen that the 3rd, 5th, 7th, and 9th harmonics are eliminated in a wider M

_{i}range compared to the 7-level for the GA method in the 11-level inverter. Since there are five degrees of freedom, four harmonic components can be eliminated. In addition, low-order harmonics were observed at lower values of the modulation index compared to 7-level. However, in case M

_{i}> 0.9, the values of the harmonic components increase again, as the operation of the inverter is similar to the square wave mode. Figure 12c describes that the 3rd, 5th, 7th, 9th, 11th, and 13th harmonics are eliminated in a wide range of modulation index values from 0.2 to 0.9 for the SPBO method applied to a 15-level inverter. In the 7-level inverter, the value of the 3rd harmonic, which is 0.7 PU at M

_{i}= 0.1, decreases to 0.2 PU. It is clearly seen from Figure 12d that the 3rd, 5th, 7th, 9th, 11th, 13th, 15th, and 17th harmonics are removed over a wide range of the modulation index value from 0.1 to 0.9 for the SPBO method in the 19-level inverter. Another result emerging from the figures is that as the inverter approaches the square wave mode, in the case of M

_{i}> 0.9 at all levels, its harmonic components increase and eventually settle at the same values.

## 4. Discussion

_{i}< 0.85. According to the IEEE 519—2014 standard, a maximum THD value of 8% is allowed for 1 kW inverters. In terms of THD, since the harmonic elimination in 15-level MLI falls to the desired values, exceeding this level will only increase the cost significantly. By using topologies that employ fewer switching elements, the cost increase can be somewhat reduced.

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 3.**Convergence curves of the optimization algorithms for MLI with (

**a**) 7-level; (

**b**) 11-level; (

**c**) 15-level; and (

**d**) 19-level.

**Figure 5.**THD variations versus modulation indexes for MLI with (

**a**) 7-level, (

**b**) 11-level, (

**c**) 15-level, and (

**d**) 19-level applied to suggested metaheuristics.

**Figure 6.**FFT analysis of the output current of the 19-level MLI at a modulation index of 0.8 for (

**a**) BMO; (

**b**) GA; (

**c**) GWO; (

**d**) MFO; (

**e**) SPBO; and (

**f**) SPSA.

**Figure 7.**The output voltage (per unit) variations versus modulation indexes for MLI with (

**a**) 7-level, (

**b**) 11-level, (

**c**) 15-level, and (

**d**) 19-level applied to suggested metaheuristics.

**Figure 8.**The output currents and voltages depend on the modulation index for a 7-level MLI inverter using a BMO.

**Figure 9.**The effect of modulation index variations on the output currents and voltages of 11-level MLI employing the GA method.

**Figure 10.**The effect of modulation index variations on the output currents and voltages of 15-level MLI employing the SPBO method.

**Figure 11.**The effect of modulation index variations on the output currents and voltages of 19-level MLI employing the SPBO method.

**Figure 12.**Variation of harmonic components depending on modulation index: (

**a**) 7-level MLI (BMO); (

**b**) 11-level MLI (GA); (

**c**) 15-level MLI (SPBO); and (

**d**) 19-level MLI (SPBO).

Optimization Method | Inspirer |
---|---|

Particle Swarm Optimization (PSO) | Inspired by a school of fish or a flock of birds moving in a group [14] |

Genetic Algorithm (GA) | Inspired by the natural evolutionary process [12] |

Whale Optimization Algorithm (WOA) | Inspired by the air bubble behavior of humpback whales use while hunting [25] |

Grey Wolf Optimization (GWO) | Inspired by the hunting behavior and social leadership of grey wolves [19] |

Moth Flame Optimization (MFO) | Inspired by the transverse orientation behavior of moths [23] |

Sine Cosine Algorithm (SCA) | Inspired by the concept of trigonometric sine and cosine functions [42] |

Teaching Learning based Optimization (TLBO) | Inspired by the teaching and learning behavior in a classroom [33] |

Sparrow Search Algorithm (SPSA) | Inspired by sparrows’ group erudition, foraging, and anti-predation behaviors [78] |

Student Psychology based Optimization (SPBO) | Inspired by the psychology of students striving to be the best student in the class [79] |

Barnacles Mating Optimizer (BMO) | Inspired by the mating process of barnacles [80] |

Dingo Optimization Algorithm (DOA) | Inspired by the social, cooperative, and hunting action of dingoes [31] |

Gradient-Based Optimizer (GBO) | Inspired by Newton’s method that integrates both the gradient search rule and local escaping operator [81] |

Gorilla Troops Optimizer (GTO) | Inspired by gorilla troops’ social intelligence in nature [82] |

Jellyfish Search Optimizer (JS) | Inspired by the act and foraging behavior of jellyfish in the ocean [83] |

Lichtenberg Algorithm (LA) | Inspired by the Lichtenberg figure patterns [84] |

Material Generation Algorithm (MGA) | Inspired by the configuration of chemical compounds and reactions in the production of new materials [85] |

Political Optimizer (PO) | Inspired by the mathematical mapping of the multistage process of politics [86] |

Peafowl Optimization Algorithm (POA) | Inspired by the group foraging behavior of the peafowl swarm [87] |

Parasitism—Predation algorithm (PPA) | Inspired by the multi-interactions between cuckoos, crows, and cats [88] |

Smell Agent Optimization (SAO) | Inspired by the relationship between a smell agent and an object that vaporizes a small molecule [89] |

Sperm Swarm Optimization (SSO) | Inspired by sperm-ovum interactions in the fertilization procedure [90] |

Wild Horse Optimizer (WHO) | Inspired by the social life behavior of wild horses [91] |

S.No | Parameters | Specification |
---|---|---|

1 | Number of Levels | 7, 11, 15, and 19 |

2 | Voltage Source (DC) | 100, 60, 42.85, and 33.33 V |

3 | Fundamental frequency | 50 Hz |

4 | Load | R = 50 Ω, L = 20 mH |

Method | Parameter | Value |
---|---|---|

GWO | Convergence parameter a r _{1}, r_{2} | linearly decreased from 2 to 0 [0, 1] |

SSA | c_{2}, c_{3} | [0, 1] |

WOA | Convergence parameter a | decreases linearly from 2 to 0 |

SCA | r_{1}r _{2}, r_{3}, r_{4} | decreases linearly from 2 to 0 [0, 2π], [0, 2], [0, 1] |

PSO | cognitive coefficient social coefficient inertia constant | 2 2 decrease from 0.9 to 0.2 |

MFO | convergence constant spiral constant b | [−1, −2] 1 |

GA | type, selection crossover mutation | real coded, roulette wheel probability = 0.8 Gaussian (probability = 0.05) |

TLBO | Teaching factor T | [1, 2] |

DOA | MOP, sensitivity parameter α control parameter μ | [0.2, 1], 9 0.1 |

BMO | penis length of the barnacle pl | 7 |

GBO | probability parameter pr | 0.5 |

GTO | p, β, w | 0.03, 3, 0.8 |

JS | distribution coefficient β motion coefficient γ | 3 0.1 |

LA | addition of a refinement ref stick probability S creation radius R _{c} | 0.4 1 150 |

MGA | the probabilistic component e^{−} | Gauss Distribution |

PO | party switching rate λ number of parties, constituencies, candidates in each party n | linearly decreased from 1 to 0 8 |

POA | number of peacocks rotation radius R _{s} θ _{0}, θ_{1} coefficient γ | 5 0.5 0.1, 1 1.5 |

PPA | the intrinsic growth rate of crows, r_{1}the death rate of cuckoos, r _{2}the death rate of cats, r _{3} α _{1}, α_{2} β _{1}, β_{2} c _{1}, c_{2} d _{1}, d_{2} | 1 0.1 0.1 0.2, 0.25 0.1 0.1 0.01 |

SPBO | Not Available | |

SPSA | threshold value ST number of the producers PD number of the danger-perceivers SD | 0.8 20% 10% |

SSO | pH value a factor of velocity damping D temperature T | [7, 14] [0, 1] [35.1, 38.5] |

WHO | crossover percentage, PC stallions’ percentage, PS | 0.13 0.2 |

**Table 4.**Comparison of statistical values of the fitness function’s cost, current total harmonic distortions, and output voltages in algorithms applied to the 11-level SHEPWM problem (the prominent values are highlighted in bold).

Algorithms | A Single Iteration Time (s) | Fitness Function Cost | Total Current THD Error 0.1 < M _{i} < 1.1 | Total Output Voltage PU Error 0.1 < M _{i} < 1.1 | Total Current THD Error 0.4 < M _{i} < 0.9 | Total Output Voltage PU Error 0.4 < M _{i} < 0.9 | |||
---|---|---|---|---|---|---|---|---|---|

Best | Mean | Worst | Std | ||||||

BMO | 1.3202 × 10^{−3} | 2.8741 × 10^{−10} | 7.3633 × 10^{−7} | 3.0686 × 10^{−5} | 3.4150 × 10^{−6} | 68.96 | 2.08 | 6.12 | 0.55 |

GBO | 1.6676 × 10^{−3} | 7.7066 × 10^{−20} | 8.3334 × 10^{−11} | 5.4120 × 10^{−10} | 1.4612 × 10^{−10} | 70.76 | 2.06 | 8.09 | 0.56 |

GTO | 1.0636 × 10^{−3} | 9.2000 × 10^{−23} | 1.7044 × 10^{−10} | 5.4099 × 10^{−10} | 1.9544 × 10^{−10} | 70.20 | 2.10 | 7.30 | 0.55 |

JS | 6.3162 × 10^{−4} | 8.4924 × 10^{−11} | 1.2737 × 10^{−5} | 2.5813 × 10^{−4} | 3.2158 × 10^{−5} | 71.69 | 2.05 | 8.70 | 0.52 |

LA | 2.9378 × 10^{−3} | 1.0620 × 10^{−7} | 7.9679 × 10^{−6} | 7.7236 × 10^{−5} | 1.3306 × 10^{−5} | 82.00 | 2.13 | 12.86 | 0.55 |

MGA | 6.0866 × 10^{−5} | 2.7317 × 10^{−2} | 1.8867 × 10^{−1} | 3.9264 × 10^{−1} | 8.0063 × 10^{−2} | 131.04 | 1.76 | 18.89 | 0.51 |

PO | 1.4118 × 10^{−3} | 1.6449 × 10^{−17} | 6.9237 × 10^{−11} | 5.4082 × 10^{−10} | 1.1998 × 10^{−10} | 73.94 | 2.12 | 8.54 | 0.57 |

POA | 2.8153 × 10^{−3} | 2.9787 × 10^{−17} | 2.1614 × 10^{−10} | 5.4112 × 10^{−10} | 2.1120 × 10^{−10} | 71.57 | 2.05 | 8.59 | 0.53 |

PPA | 8.8035 × 10^{−4} | 9.6400 × 10^{−22} | 2.2238 × 10^{−10} | 7.9866 × 10^{−10} | 2.0503 × 10^{−10} | 70.92 | 2.06 | 8.03 | 0.54 |

SPBO | 1.3096 × 10^{−3} | 3.7682 × 10^{−9} | 1.6368 × 10^{−6} | 3.2202 × 10^{−5} | 3.9299 × 10^{−6} | 8.03 | 1.79 | 4.82 | 0.52 |

SPSA | 8.6769 × 10^{−4} | 1.3969 × 10^{−16} | 1.3471 × 10^{−10} | 5.3942 × 10^{−10} | 1.8617 × 10^{−10} | 75.70 | 2.14 | 6.43 | 0.58 |

SSO | 1.4437 × 10^{−3} | 5.4589 × 10^{−3} | 7.5589 × 10^{−2} | 1.8221 × 10^{−1} | 3.7680 × 10^{−2} | 80.52 | 2.07 | 10.51 | 0.61 |

WHO | 2.1154 × 10^{−3} | 1.6449 × 10^{−17} | 1.1348 × 10^{−10} | 1.0401 × 10^{−9} | 1.6571 × 10^{−10} | 72.01 | 2.04 | 8.96 | 0.52 |

SSA | 8.5082 × 10^{−4} | 1.1341 × 10^{−13} | 1.5287 × 10^{−10} | 5.4140 × 10^{−10} | 1.9452 × 10^{−10} | 70.16 | 2.09 | 7.40 | 0.57 |

WOA | 1.0745 × 10^{−3} | 9.1607 × 10^{−7} | 1.6122 × 10^{−4} | 2.5524 × 10^{−3} | 3.4030 × 10^{−4} | 71.71 | 2.42 | 7.71 | 0.58 |

GWO | 6.5974 × 10^{−4} | 2.6246 × 10^{−8} | 2.1951 × 10^{−4} | 6.5700 × 10^{−3} | 8.4363 × 10^{−4} | 69.09 | 2.13 | 5.67 | 0.60 |

PSO | 4.6609 × 10^{−3} | 5.3836 × 10^{−18} | 9.6393 × 10^{−11} | 5.3938 × 10^{−10} | 1.4142 × 10^{−10} | 77.39 | 2.16 | 7.70 | 0.61 |

MFO | 6.5258 × 10^{−4} | 6.4324 × 10^{−14} | 8.3350 × 10^{−11} | 5.4035 × 10^{−10} | 1.3430 × 10^{−10} | 70.00 | 2.14 | 5.69 | 0.56 |

SCA | 6.2341 × 10^{−4} | 3.0773 × 10^{−3} | 2.9946 × 10^{−2} | 7.9005 × 10^{−2} | 1.4711 × 10^{−2} | 70.73 | 2.03 | 7.35 | 0.50 |

TLBO | 1.1286 × 10^{−2} | 9.7908 × 10^{−15} | 3.5904 × 10^{−7} | 1.4051 × 10^{−5} | 1.6061 × 10^{−6} | 70.25 | 2.04 | 7.44 | 0.52 |

GA | 2.3372 × 10^{−3} | 1.5440 × 10^{−13} | 1.2567 × 10^{−5} | 1.3202 × 10^{−4} | 2.3307 × 10^{−5} | 76.74 | 2.17 | 5.48 | 0.60 |

DOA | 1.5837 × 10^{−3} | 1.7300 × 10^{−17} | 2.6695 × 10^{−8} | 2.6615 × 10^{−6} | 2.6614 × 10^{−7} | 72.46 | 2.13 | 9.20 | 0.59 |

**Table 5.**Comparison of the error values of the output voltages (V

_{pu}), the current total harmonic distortions (Ithd), and an iteration time in selected algorithms applied to the 7, 11, 15, and 19 levels of MLI.

7 Level MLI | 11 Level MLI | |||||

V_{pu} Error0.4 ≤ M_{i ≤} 0.9 | Ithd Error0.4 ≤ M_{i} ≤ 0.9 | A Single Iteration Time (s) | V_{pu} Error0.4 ≤ M_{i} ≤ 0.9 | Ithd Error0.4 ≤ M_{i} ≤ 0.9 | A Single Iteration Time (s) | |

SPBO | 0.0606 | 11.8 | 6.347161 × 10^{−4} | 0.0634 | 7.128 | 1.223483 × 10^{−3} |

BMO | 0.0580 | 4.96 | 1.125343 × 10^{−3} | 0.0538 | 7.878 | 1.235453 × 10^{−3} |

GA | 0.0495 | 8.65 | 1.592762 × 10^{−3} | 0.0625 | 3.63 | 1.606038 × 10^{−3} |

GWO | 0.0623 | 11.59 | 2.764809 × 10^{−4} | 0.0624 | 6.258 | 3.541279 × 10^{−4} |

MFO | 0.0554 | 11.75 | 2.724207 × 10^{−4} | 0.0571 | 6.618 | 3.213458 × 10^{−4} |

SPSA | 0.0536 | 15.16 | 6.837816 × 10^{−4} | 0.0737 | 11.158 | 7.411829 × 10^{−4} |

15 Level MLI | 19 Level MLI | |||||

V_{pu} Error0.4 ≤ M_{i} ≤ 0.9 | Ithd Error0.4 ≤ M_{i} ≤ 0.9 | A Single Iteration Time (s) | V_{pu} Error0.4 ≤ M_{i} ≤ 0.9 | Ithd Error0.4 ≤ M_{i} ≤ 0.9 | A Single Iteration Time (s) | |

SPBO | 0.0661 | 1.54 | 2.120586 × 10^{−3} | 0.0651 | 2.29 | 3.247140 × 10^{−3} |

BMO | 0.0660 | 4.16 | 1.317429 × 10^{−3} | 0.0673 | 8.36 | 1.487826 × 10^{−3} |

GA | 0.0643 | 4.28 | 1.651683 × 10^{−3} | 0.0647 | 4.38 | 1.675692 × 10^{−3} |

GWO | 0.0651 | 4.05 | 4.464614 × 10^{−4} | 0.0654 | 3 | 5.419105 × 10^{−4} |

MFO | 0.0681 | 3.34 | 3.915422 × 10^{−4} | 0.0664 | 3.45 | 4.830447 × 10^{−4} |

SPSA | 0.0643 | 3.77 | 8.338084 × 10^{−4} | 0.0642 | 5.34 | 8.553041 × 10^{−4} |

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**MDPI and ACS Style**

Ürgün, S.; Yiğit, H.; Mirjalili, S.
Investigation of Recent Metaheuristics Based Selective Harmonic Elimination Problem for Different Levels of Multilevel Inverters. *Electronics* **2023**, *12*, 1058.
https://doi.org/10.3390/electronics12041058

**AMA Style**

Ürgün S, Yiğit H, Mirjalili S.
Investigation of Recent Metaheuristics Based Selective Harmonic Elimination Problem for Different Levels of Multilevel Inverters. *Electronics*. 2023; 12(4):1058.
https://doi.org/10.3390/electronics12041058

**Chicago/Turabian Style**

Ürgün, Satılmış, Halil Yiğit, and Seyedali Mirjalili.
2023. "Investigation of Recent Metaheuristics Based Selective Harmonic Elimination Problem for Different Levels of Multilevel Inverters" *Electronics* 12, no. 4: 1058.
https://doi.org/10.3390/electronics12041058