Abstract
This research work proposes unique MLSTM-ZOA to quickly measure SAs for an MLI. Within a certain MI range, the suggested method may calculate a greater count of SAs with various solutions. Here, the main objective lies in minimizing the THD with consideration of three parameters such as MI, number of pulses per quarter cycle, and duty cycle. Furthermore, by achieving a lower fitness value in less iteration, the suggested method has definitely outperformed other methods with respect to convergence behavior, according to the data. Lastly, analysis and performance associated with the experimental validation of SHE in multi-level inverter are also carried out. The proposed MLSTM-ZOA model in terms of THD is 56.25%, 81.58%, 53.33%, and 41.67% better than MPA, HHO, MGMPA, and SF-BOA respectively. Similarly, the proposed MLSTM-ZOA model with respect to HDP is 64.03%, 47.16%, 84.01%, and 27.62% advanced than MPA, HHO, MGMPA, and SF-BOA respectively.
Keywords: Multi Level Inverter, Selective Harmonics Elimination, Pulse width modulation, Modified long short-term memory, Zebra optimization algorithm
Subject terms: Engineering, Electrical and electronic engineering
Introduction
In the past, PWM methods have been thoroughly studied and employed to power electronics converters to ensure their effective functioning1–4. With the fewest undesirable harmonic components, it generates the required basic component at the output5–8. Nevertheless, high switching frequency-oriented PWM methods, like Space Vector oriented modulation (SVPWM) and carrier-oriented modulation (SPWM), were the major emphasis9. The primary benefit of employing high switching frequency (measured in kHz) approaches is that they yield the intended output components while simultaneously moving the harmonic component at the switching frequency as a sideband10. Consequently, a higher switching frequency will need less filtering11. Higher processing power requirements, yet open up the possibility of creating novel converter configurations to satisfy growing power requirements with constrained current and voltage ratings for semiconductor devices12.
In recent years, a number of lowered devices count emerging configurations as well as the main multilevel layouts like Cascaded H-Bridge (CHB)13, Neutral Point Clamped (NPC)14, Flying Capacitor (FC)15, Active Neutral Point Clamped (ANPC)16, Modular Multilevel Converter (MMC)17, and others have been studied and introduced18. These topologies’ key components involve smaller dv/dt at switches, reduced device switching frequencies, higher quality output waveforms (Stepped), current rating and low voltage semiconductor switches, which reduce THD19. The non-elimination harmonics associated with the output waveform as well as varying THD are produced by the various solutions in a certain MI range20–25. Finding every solution from the resultant set of non-linear non-transcendental equations is therefore crucial26.
Over the years, many techniques have been presented in the literature for handling the framework of simultaneous SHE equations27. These may be roughly categorized as algebraic approaches, metaheuristic-oriented optimization techniques, and iterative techniques on the basis of numerical approaches28. Iterative methods on the basis of numbers are rapidly convergent and yield solutions with the required level of precision. The primary difficulty in using these methods, nevertheless, is choosing the right starting points and computing derivatives for each iteration29–35. This leads to divergence in the solution as well as singularity issues. A single fitness function is addressed using various harmonic level restrictions in metaheuristic approaches. The primary drawbacks of applying a metaheuristic-oriented optimization approach involve an initial guess, appropriate algorithm parameter selection, a sluggish rate of convergence, and increased processing time throughout the whole MI range36. The SHE equation can have all of its solutions provided with precise values via algebraic techniques. These techniques are only utilized for the computation of a small count of SAs since, as the count of SAs rises, the polynomial degree grows several times37. After the SAs are initially calculated utilizing any of the previously stated methods, real-time implementation is used for a variety of applications38.
The paper contribution is as below.
To propose a unique MLSTM-ZOA to quickly measure SAs for an MLI.
To calculate a greater count of SAs with various solutions among a certain MI range.
To consider the main objective as THD minimization with consideration of three parameters such as MI, number of pulses per quarter cycle, and duty cycle.
The paper organization is as follows. Section 1 is the introduction of CHBMLI. Section 2 is a literature survey. Section 3 is proposed methodology with problem formulation, CHBMLI, PWM approaches, SHEPWM, objective function, MLSTM, and ZOA algorithm. Section 4 is the results. Section 5 is the conclusion.
Motivation of the work
SHE-PWM describes a critical technology used in MLI operation with the goal of reducing harmonic distortion and optimizing power quality. Enhanced modulation schemes are becoming increasingly important as the requirement for renewable energy sources rises and efficient power conversion systems39–42. The computational effectiveness as well as flexibility of conventional approaches are sometimes limited, particularly when system complexity rise. Deep learning provides a revolutionary method for addressing SHE-PWM problems43. With the use of NNs, it is capable of forecasting the best SAs for eliminating particular harmonics and model complex patterns found in the PWM waveform. These speeds up the design procedure and improves performance simultaneously, enabling real-time modifications under dynamic operating situations44. Additionally, by overcoming the limitations of traditional methodologies, DL’s capacity to learn from enormous datasets facilitates the creation of more reliable as well as comprehensive solutions. The integration of DL with SHE-PWM in MLIs presents a potential frontier in the pursuit of more environmentally friendly and efficiently operating energy systems45. This integration can stimulate innovation and pave the path for sustainable technological developments. Taking advantage of this synergy can result in notable enhancements to system dependability as well as power quality, which will ultimately help the environment and consumers.
Related works of PWM methodologies
This article46 proposed an IPD-TPWM approach on the basis of power balancing. The modulation scheme could achieve power-balancing of H-bridge units within a full modulation ratio range by adjusting the configuration of triangular carriers in the vertical plane while utilizing the carrier segment in the half carrier period associated with the IPD-TPWM scheme as the fundamental unit47. This could be achieved by choosing an accurate trapezoidal wave triangulation rate δ, which will also ensure the waveform nature related to the output phase voltage. As a result, the issues of uneven output power from every unit and poor DC-side voltage utilization rate have been resolved. It was suggested to use a hybrid multi-carrier PWM method on the basis of carrier reconstruction. This method optimized the modulation effectiveness by regularly altering its layout in the vertical plane. It did this by using the carrier segment in the half carrier period associated with the IPD-PWM approach as the fundamental unit. The output power balance among the cascaded H-bridge cells was naturally attained when the CHB MLI used the Hybrid Multi-carrier PWM approach, similar to the CPS-PWM approach48. At the exact time, the output voltage’s harmonic spectrum was similar to that of the IPD-PWM approach, meaning that the inverters’ output line voltage’s harmonic features were efficiently enhanced.
The article49 proposes a modified MLI with step-up operation utilizing minimal components. With the help of three capacitors and eight switches, the suggested circuit may provide an output with seven levels. The charge pump technique, which could increase the output voltage by 1.5 times the input voltage, was the foundation of boost functioning. To drive switches, a SPWM was utilized50. The primary goal of the switching pattern was to handle capacitor discharge and charging. Reducing overall harmonic distortion was the primary goal of this architecture. This letter uses a fuzzy logic method to develop an interesting switching strategy51. By doing away with the standard logic-gate architecture, the suggested solution streamlines the conventional approach. In addition to producing pulses, the fuzzy logic pulse generator also functions as a lookup table. Controlled MFs and FLC rules provide several ways to directly produce pulses based on the MI as input52. SHEPWM was used to assess the suggested method on a cascaded MLI with asymmetric and symmetric operations. The pre-computed firing constraints for various MI values serve as the foundation for the modelling of MFs53. Hardware verification was done in order to validate the suggested switching method.
In this paper, a 5-level CHB MLI powered by a 36-pulse ac-dc converter-oriented IMD was suggested to use a novel level-shifted carrier-oriented PWM approach54. A 36-pulse ac-dc converter was created by coupling a modular transformer to fulfil the grid side power quality requirement of IEEE-519. On the motor side, a 5-level CHB MLI was used, and it switches actions using the suggested carrier-oriented PWM approach to guarantee the IMD performs well. The THI 2 SCPWM was the name of the suggested PWM technology55. It forms the recommended carrier signal by combining the inverted sinusoidal signal with the THI approach. This study introduces the evaluation of logic gates for the reduced switch MLI switching sequence operation56. Two MLI variations having fewer switches were taken into consideration while analyzing the logic gates to determine the appropriate output voltage level. A symmetrical voltage was applied to one MLI, while an asymmetrical voltage was applied to the other. The single-phase, seven-level output voltage for asymmetrical as well as symmetrical inverters was used to analyze the suggested logical operation. PD, POD, and APOD were the multi carrier PWM approaches from which the input pulse pattern for the logic gates was selected.
A PWM template for CHB MLI was presented in this paper. The suggested control approach modifies a sinusoidal modulating waveform to fit inside a single triangle carrier signal range, generating suitable modulation templates for the CHB inverter. These templates don’t require any further control adjustment to be utilized on CHB inverters of any level. The suggested modulation produced an equal distribution of the total real power within the power switches that make up the power switches, a nearly uniform distribution of switching pulses, and improved output voltage quality. Findings from simulations as well as experiments were reported for an R-L load on a 3-phase, 7-level CHB. This research has conducted a thorough analysis of the link between the multilevel SVPWM technique and the CBPWM technique. Multi-modulation waves CBPWM may be used to create SVPWM by breaking down the modulation wave. Additionally, the association among SVPWM and CBPWM at the N level having any count of sequence segments is found. Therefore, switching sequences associated with MLIs with greater DC voltage utilization and lower THD in the spectrum may be achieved by using a basic multi-modulation waves CBPWM method. The findings of the experiment and simulation show that the link between SVPWM and CBPMM was accurate.
To calculate SAs, a novel IWO approach was presented in this study. The suggested method is capable of calculating SAs for MLIs that are asymmetrical or symmetrical. As a result, it outperforms popular optimization strategies like PSO, GA, ACO, DE, etc. Furthermore, as the study demonstrates, it offers more appropriate findings and faster convergence in specific MI ranges. The practical findings on the laboratory-developed prototype have validated the computational outcomes. The suggested method was implemented using a controller that was dependent on FPGA. The computational findings and hardware outcomes were found to be nearly in accord. Compared to common topologies like MLDCL and DHB, this work offers a unique dual bridge asymmetric cascaded MLI to provide 21 level output having just twelve switches and three sources. Using MATLAB/Simulink to simulate the predicted architecture validates its successful operation. THD was decreased when a MGWO-PI-PWM approach was used to control the switches. The count of switches as well as DC sources needed to provide the twenty-one-level output for the suggested topology was thoroughly analyzed, compared, and contrasted with the traditional MLI configurations.
This study looked at the optimization-oriented CHB 27-level inverters for hybrid renewable energy sources and their selective harmonic removal. This technique made advantage of single MPPT hybrid renewable energy sources, including tidal, wind, and solar photovoltaics. The Dwarf Mungo optimization (DMO) technique was used with a single MPPT to get excellent efficiency from hybrid sources concurrently. Ten semiconductor switches that were linked to three DC sources make up the suggested inverter. The 27-level inverter used the Osprey optimization algorithm (OOA) to determine the SAs for SHE-PWM. Efficiency was evaluated using the suggested method on a Simulink platform, and its effectiveness was shown by comparing it to a traditional method. The simulation’s OOA with SHE-PWM yielded 1.98% THD.
A three-phase half-bridge cascaded MLI-powered photovoltaic source control technique using modified Sinusoidal Pulse Width Modulation (SPWM) was presented in this study. The MLI configuration was chosen because it has fewer switching components, improving system dependability and making experimental implementation easier. The suggested control technique only needed three signals: a triangle waveform, a carrier signal, and a modulating signal. In contrast to SPWM strategies, which needed (m − 1) carriers and resulted in an extremely complicated pulse circuit generation, this approach only needed three signals. This method made control simpler to understand and made it easier to put into practice. With the aid of MATLAB/SIMULINK software, the suggested control system simulation was confirmed. The best SAs for the suggested control system were found by applying the Grey Wolf Optimization (GWO) method.
Research gaps
Even with the DL-based improvements in SHE-PWM for MLIs, there are still a number of unanswered questions that need to be investigated. Firstly, there exists a lack of development in the DL method’s ability to generalize across different inverter topologies as well as operating situations. The methods that are in use today frequently work better on certain datasets but have trouble adjusting to changes in voltage, load, and switching frequency. Secondly, a major obstacle in this situation describes the interpretability of DL algorithms. Since numerous current methods function as “black boxes,” it might be challenging for engineers to comprehend the thought procedures that went into choosing the particular harmonics reduction procedures. Improving the clarity of the method might boost confidence and make it easier for it to be integrated into real-world uses. Furthermore, the majority of research has concentrated on offline training methods, ignoring the possible advantages of real-time adaptability and online learning. This might greatly improve SHE-PWM techniques’ ability to adapt to dynamic variations in system circumstances. Finally, there is still little integration of DL with other optimization methods as Particle Swarm Optimization (PSO) or Genetic Algorithms (GAs). Investigating hybrid strategies may produce more reliable as well as effective results. Filling up these gaps will progress the industry and help create MLI systems that are more intelligent and effective.
Proposed methodology
Problem formulation of the proposed work
Obtaining switching angles that result in the appropriate fundamental frequency component as well as the removal of particular low order harmonic components from the output waveform are the primary goals of, SHE PWM. Following the use of the Fourier series decomposition related to the output waveform, a group of simultaneous mathematical equations is formed. The switching angles are next calculated by the use of homotropy perturbation. Figure 1 depicts the generalized cascaded H-bridge multilevel inverter circuit. When the semiconductor switches from each individual H-cell are properly switched, three voltage levels, like 
, 0 or 
, are produced. The total count of harmonics that can be eliminated depends on how many positive as well as negative pulses there are in a cycle. The resulting waveform is analyzed using the Fourier series expansion. The stepped waveform’s general Fourier series expansion may be found in (1).
 |
1 |
Fig. 1.
Generalized cascaded H-bridge multilevel inverter circuit.
The waveforms are regarded as having strange quarter-wave symmetry for the sake of simplicity. In this case, the even harmonic components will also be zero, as well as the Fourier coefficients 
and 
. After simplification, the Fourier coefficient 
is obtained as (2), in which 
denotes the harmonic order and 
the total count of switching transitions in quarter periods.
 |
2 |
Additionally, the waveform for the staircase will be provided by (3) from the cascaded H bridge inverter.
 |
3 |
In a balanced three-phase framework, the triplen harmonics will automatically cancel. Thus, the output waveform, which may be described as (4) and (5) for three-level as well as multilevel waveforms, may be regarded as containing the required basic structure plus multiple nontriplen odd-order harmonics components.
 |
4 |
 |
5 |
If 
count of switching transitions are examined quarterly, the 
harmonics (typically the lowest order) may be removed from the output waveform. To get the appropriate basic component magnitude, one degree of freedom is set aside. In three-level as well as multilevel stepped waveforms, the generalized formulation for having the required fundamental component as well as the exclusion of particular low order harmonics components from the output waveform are provided in (6) and (7), correspondingly.
 |
6 |
 |
7 |
shows the modulation index, while 
shows the count of steps. Equation (8) gives the ratio of the magnitude of the fundamental component to the highest possible magnitude.
 |
8 |
The restrictions of minimum, maximum, as well as non-equality, as stated in (9), should be satisfied by the aforementioned Eq.
 |
9 |
The simultaneous, very non-linear, transcendental equations in Eq. (6) as well as (7) are regarded as such. Their answer is not simple since it might provide several solutions, a single solution, or no solution at various modulation index values.
Cascaded H-bridge multilevel inverter
Two or more H-bridge cells linked in series make up a CHBMLI. An 
level inverter requires 
DC sources or firing angles, in which 
; they might be unequal or equal. 
, 
, and 
can all be produced by one H-bridge cell. H-Bridges are linked in series, and each has a distinct DC source attached to it.
A power electronic circuit known as an inverter has the ability to convert AC to DC on a base of two-level voltages. There are several problems with the switching losses, distortion factors, and reduced efficiency of the basic inverter. Numerous scholars create and analyze multilayer inverters to give improved characteristics in order to address these problems. Due to aberrations in the converted power, it was challenging to integrate inverted power from several power electronic applications in the previous decades. Among the power electronic circuit family, multilevel inverters are far superior because they reduce issues caused by distortions in current and voltage. There are three voltage values at the beginning level of the multilayer inverter. It has been chosen to use multilevel inverters with levels ranging from 3, 5, 7, 9, 11, 27, and so on. On the basis of the choice of voltage as well as connection, multilevel inverters are typically needed for commercial, industrial, and residential applications. They come with a lot of functions.
Numerous research studies have proposed three topologies that are dependent on how multilevel inverters operate. A single DC voltage source is used to run diode clamped and flying capacitors, whereas 
DC voltage sources are used to control CMLI, in which 
shows the CMLI’s level. There exist additional nonlinear and linear components, power semiconductor switches, and DC sources to use. Numerous modulation techniques have been examined in order to enable the chosen MLI to operate as intended. A large number of researchers are studying reduced switch MLI. The major compact switch MLI represents the chosen option within the work of several researchers.
PWM approaches
Low as well as high switching frequency modulation approaches have been used in the research to run the MLI. Many modulation approaches, including trapezoidal modulation, staircase modulation, SVPWM, harmonic injection modulation, and SPWM, are employed on the basis of the appropriateness of function as well as the output needs. Multiple carriers PWM is superior to single carrier PWM in a number of ways within these entire PWM techniques. In order to operate the PWM method, the count of carrier signals is selected as
 |
10 |
Level shifting and phase shifting represents two categories of multiple carrier modulation methods. The two types of level shifting approaches are constant switching frequency multi carrier signal and variable switching frequency multi carrier signal. The divisions associated with the switching frequency approaches are POD, PD, and APOD.
The level shifting PWM approach is applied with both constant and variable switching frequencies, as well as their subdivisions. Using 
triangular carrier signals, 
amplitude and 
frequency are utilized. Using the signal comparison related to the carrier and reference of the PWM technique, a modulating signal associated with the sinusoidal waveform with the amplitude of 
and frequency of 
is generated to create the pulse for the ON/OFF states of the switch. The modulation index 
as well as frequency ratio 
are computed as shown in (11) and (12) on the basis of the frequency and amplitude of the carrier and the modulating signal.
 |
11 |
 |
12 |
The carrier signal frequency is the same as the switching frequency. To create pulses, the carrier signal and modulating signal are compared using the relational operator. The count of carrier signals matches the count of pulses produced. Using the available pulses to activate the switches in a way that produces the desired output is difficult.
SHEPWM
Fourier series analysis associated with the phase voltage of a three-phase CHBMLI may be used to derive the SHE equations for the system. Because of the odd nature related to the function, considering a quarter wave symmetry, the even harmonics as well as the cosine terms both become zero. The 
harmonics can be reduced for an 
-level three phase CHBMLI with 
count of h-bridge cells. The fundamental as well as certain low order non-triplen harmonics that make up the SHE equations may be expressed as follows:
 |
13 |
 |
14 |
The basic component may be expressed as below with respect to MI:
 |
15 |
 |
16 |
 |
17 |
Here, 
shows the phase voltage fundamental value that the H-bridge produces. 
shows the highest phase voltage that CHBMLI could measure, and 
shows the number of H-Bridge cells. When all of the switching angles are zero, or when 
is reached,
 |
18 |
SHE equations associated with a 9-Level CHBMLI may be expressed as follows:
 |
19 |
 |
20 |
 |
21 |
 |
22 |
With respect to MI, Eq. (19) describes the basic component, and Eqs. (20–22) provide the formulas for removing harmonics.
Objective function
Equation (23) has been used to calculate the (%THD) in line voltage for an 11-level cascaded H-bridge inverter. The results show that alternative solutions for a given MI have varying values. As a result, the least percentage THD value associated with the solution can be used to determine which solution is best.
 |
23 |
In the above equation, the term 
describes harmonic order and 
describes the time steps respectively.
MLSTM
The MLSTM is mainly selected for performing the THD minimization that is considered as the major objective or the fitness function with respect to three parameters such as modulation index, number of pulses per quarter cycle, and duty cycle. Here, the considered parameters are tuned by ZOA with the intention of deriving the fitness function, thus referred as MLSTM. In Natural Language Processing, large temporal aspects such as sequences are thought to be well captured by LSTM recall. Every LSTM layer in the LSTM module is made up of LSTM cells. The LSTM module can have one or more LSTM layers. An essential component of the LSTM module that creates the temporal connections between layers in an LSTM is the LSTM cell, also known as the LSTM internal memory cell.
A LSTM cell’s architecture is made up of two distinct recurring features: cell state 
and hidden state 
. To update the current time step 
, which may be written as follows, an LSTM unit employs a prior state, which includes a previous hidden state and previous cell state
, and a current input 
.
 |
24 |
Here, 
shows a function that can transfer an input sequence of arbitrary length to an output sequence. Three gates—an input gate, a forget gate, as well as an output gate—are present in a fundamental LSTM unit to regulate the cell state. Specifically, at every time step 
, an LSTM cell receives the input sequence 
. The three gates (functions) in every cell receive the input 
as well as the hidden state associated with the current time step 
before passing via a sigmoid activation function 
. At time step 
, an LSTM cell’s update with respect to 
is stated as below:
 |
25 |
Here, the weight parameters in an LSTM cell are 
, 
, 
, 
, 
, 
, 
; the bias parameters are 
, 
, 
, and 
. The training procedure updates both weights as well as biases. The symbols 
, 
, and 
describe three gates: input, forget, as well as output. The forget gate 
determines the amount of data that must be kept in the cell state 
from the earlier state. The amount of newly acquired data that is considered through cell update 
is determined by input gate 
. Identical to the three gates before, the calculation for 
employs the Tanh activation function, whose value range is (− 1, 1) as
 |
26 |
The amount of content in the cell state 
that must leave the cell in order to reach the hidden state 
is controlled by the output gate 
. At time 
, the cell state is shown as
 |
27 |
The hidden state associated with an LSTM cell’s current step, 
, when applying the Tanh activation function, is always in the interval (− 1, 1) and is represented as
 |
28 |
The input, output, as well as forget gates from Eq. (25), together with the cell update from Eq. (26), are used to calculate the cell state in Eq. (27) as well as the hidden state in Eq. (28). Since it may receive an input time-series sequence of any length, such as 
, and generate an output sequence of any length, such as 
, an LSTM cell is versatile.
LSTM offers various advantages such as better in processing data sequences having long range dependencies, effective with time series analysis, etc. But it is limited from the fact that it needs vast training information for learning in an efficient manner, slower in training vast datasets, etc. Thus, to limit its shortcomings, the parameters of LSTM are tuned by ZOA, thereby referred to as MLSTM. This novel MLSTM overcome the limitations of conventional LSTM and also handles small and large datasets in a very efficient manner. The diagrammatic model of the proposed MLSTM-ZOA for the SHEPWM in MLI is shown in Fig. 2.
Fig. 2.
Proposed MLSTM-ZOA for the SHEPWM in MLI.
ZOA algorithm
The ZOA is mainly selected here for tuning the parameters of the LSTM model with the consideration of attaining the fitness function. Here, the main objective lies in minimizing the THD with consideration of three parameters such as MI, number of pulses per quarter cycle, and duty cycle. ZOA mimics zebras’ feeding habits as well as their defence mechanisms against predators. Zebras are part of the population of ZOA, a population-oriented optimizer. From a mathematical perspective, every zebra represents a potential solution to the issue, and the plain where the zebras are located represents the problem’s search space.
The values associated with the decision variables are determined by every zebra’s location inside the search space. The values related to the problem variables can therefore be represented by the components of a vector that represents every zebra as an individual of the ZOA. A matrix may be used to numerically design the zebra population. The zebras are first positioned in the search space at random. Equation (29) contains information about the ZOA population matrix.
 |
29 |
Here, 
shows the count of population individuals (zebras), 
shows the count of choice variables, 
shows the zebra population, 
shows the 
zebra, and 
shows the solution related to the 
problem variable provided by the 
zebra. Every zebra symbolizes a potential resolution to the optimization issue. As a result, the suggested values of every zebra for the issue variables may be used to assess the fitness function. By utilizing (30), the values acquired for the fitness function are provided as a vector.
 |
30 |
Here, 
shows the fitness function value acquired for the 
zebra and 
shows the vector of fitness function values. The optimal candidate solution for the provided issue is found by comparing the values attained for the fitness function, which efficiently analyses the quality of their associated candidate solutions.
The optimal zebra in the group is known as the pioneer zebra in ZOA, and it guides remaining zebras in the population towards its location in the search space. Hence, (31) and (32) may be used to mathematically simulate how zebras’ positions are updated throughout the foraging stage.
 |
31 |
 |
32 |
Here, 
shows its fitness function value, 
shows the novel status associated with the 
zebra on the basis of first stage, 
shows its 
dimension value, 
shows its 
dimension, 
shows a random count in the interval [0, 1]. Furthermore, 
shows the pioneer zebra, which describes the optimal individual. 
, in which 
represents a random count within the range [0, 1]. As a result, 
, and if parameter 
, then the movement of the population varies considerably more.
It is considered in the ZOA model that there exists an equal possibility of one among the below two situations occurring: In the first method, zebras are attacked by lions and flee from the onslaught by gathering around the region where they are. Hence, the mode 
in (33) may be used to quantitatively represent this method. When a zebra is attacked by another predator, the remaining zebras in the herd move towards it in an attempt to create a protective format that would terrify as well as confuse the predator. This is the second method. The mode 
in (33) is used to mathematically represent this zebra technique. When zebra positions are updated, a zebra’s novel location is allowed if it involves a superior value for the fitness function there. (34) is used to simulate this updating circumstance.
 |
33 |
 |
34 |
Here, 
shows the novel status associated with the 
zebra on the basis of second stage, 
shows its 
dimension value, 
shows its fitness function value, 
shows the iteration contour, 
shows the maximum count of iterations, 
shows the constant number equal to 0.01; 
shows the probability of selecting one among two strategies that are randomly produced in the interval [0 1]; 
shows the status associated with the attacked zebra, and 
shows its 
dimension value.
Updating the population individuals on the basis of the first as well as second stages concludes every ZOA iteration. Up to the algorithm’s complete implementation, the algorithm population is updated depending on steps (31) through (34). Through several cycles, the optimal candidate solution is updated and stored. ZOA presents the top contender as the optimal solution to the provided problem when it has been completely developed. Algorithm 1 presents the ZOA stages as pseudocode.
Algorithm 1.
ZOA.
Results and discussion
Experimental setup
The optimized switching angles for SHEPWM of CHBMLI are obtained by implementing the suggested MLSTM-ZOA method in MATLAB. Table 1 lists the essential parameters needed for optimization. This approach is strongly advised for MLIs with both reduced and conventional component formats. Using deep learning techniques, optimization algorithms of any kind may readily remove the harmonics. The Simulink model related to the proposed system is shown in Fig. 3.
Table 1.
Parameter specification.
| Parameters | Specification |
|---|---|
| Maximum attainable voltage | 5 V |
| Count of harmonics to be eliminated | 3 |
| Population size | 20 |
| Voltage levels | 11 |
| Count of switching angles | 15 |
| Iterations count | 50 |
| DC input | 100 V |
| MI | 0.4, 0.6, 0.8, 1.0 |
| Load inductor | 30 mH |
| Chromosome length | 1 |
| Load resistor | 90 ohms |
| Fundamental frequency | 45 Hz |
Fig. 3.
Proposed system’s Simulink model.
Line voltage analysis
Figure 4 displays the simulated output line voltages for the 11-level CHBMLI line voltages under three distinct MIs. As seen in Fig. 4, it is evident that the SHEPWM successfully removed the 5th, 7th, and 11th order harmonics in each of the three cases. Because of this, it seems that determining its rms value is more complex than determining the phase voltage. Additionally, it is impossible to create a single advanced technique that can calculate the rms value for the whole range of SAs.
Fig. 4.
Line voltage analysis at MI = 0.6, 0.8, and 1.0.
THD analysis
The suggested MLSTM-ZOA method yields the lowest current-THD among the optimization techniques when the load’s neutral is independent to the MLI’s points. Figure 5; Table 2 present the findings as MI vs. current THD. The MI of the MLSTM-ZOA is 1.0, meaning that it outperforms MPA, HHO, MGMPA, and SF-BOA by 56.25, 81.58, 53.33, 41.67%, respectively. Thus, for the proposed MLI model, the MLSTM-ZOA strategy is better than other methods with consideration of THD analysis.
Fig. 5.
THD analysis.
Table 2.
THD analysis.
HDP analysis
The voltage generated by a PV-fed CHB 11 level employing the MLSTM-ZOA and other low THD methods is displayed in this section. At a frequency of 45 Hz, the matching basic line voltage is produced. Figure 6 displays the HDP analysis at various MIs, and Table 3 displays the values of the analysis. It can be demonstrated that MLSTM-ZOA attains minimal parameter distortion than the prior methods in most of the MIs. The proposed MLSTM-ZOA model with respect to HDP is 64.03%, 47.16%, 84.01%, and 27.62% advanced than MPA, HHO, MGMPA, and SF-BOA respectively. Consequently, it is evident that in terms of HDP analysis, the suggested MLSTM-ZOA is better than the alternative methods.
Fig. 6.
HDP analysis.
Table 3.
HDP analysis.
| Methods | MI | |||
|---|---|---|---|---|
| 0.4 | 0.6 | 0.8 | 1.0 | |
| MPA [26] | 35.01 | 20.22 | 12.11 | 4.05 |
| HHO [27] | 36.20 | 23.32 | 14.53 | 5.95 |
| MGMPA [28] | 30.30 | 17.33 | 10.12 | 1.80 |
| SF-BOA [29] | 39.23 | 25.88 | 19.10 | 8.15 |
| Proposed method | 41.34 | 28.99 | 22.21 | 11.26 |
Convergence analysis
Table 4; Fig. 7, respectively, illustrate the connection between the MIs and the cost function. As can be seen, the MLSTM-ZOA cost function is superior at MI = 1 with a cost function of 2.76%. Compared to existing methods, the suggested MLSTM-ZOA connection maintains lower output distortion (0.5-1.0) across the modulation duration (0.4-1.0). In this method, the recommended strategy reduces convergence, hence improving the output waveform. The proposed MLSTM-ZOA model in terms of convergence is 78.46%, 84.34%, 61.07%, and 49.54% higher than MPA, HHO, MGMPA, and SF-BOA respectively. As the primary fundamental concept, the suggested MLSTM-ZOA technique has been examined from a variety of angles and has been shown to be accurate when compared to a number of antiquated practices.
Table 4.
Convergence analysis.
| Methods | MI | |||
|---|---|---|---|---|
| 0.4 | 0.6 | 0.8 | 1.0 | |
| MPA [26] | 36.01 | 21.23 | 13.12 | 5.06 |
| HHO [27] | 37.21 | 24.33 | 15.54 | 6.96 |
| MGMPA [28] | 30.31 | 18.34 | 10.14 | 2.80 |
| SF-BOA [29] | 29.84 | 17.89 | 10.11 | 2.16 |
| Proposed method | 26.95 | 14.96 | 7.22 | 1.09 |
Fig. 7.
Convergence analysis.
SAs analysis
The proposed MLSTM-ZOA algorithm for the cascaded H-bridge 11-level inverter were developed in MATLAB. After gathering offline data on ideal SAs, they are put into practice in a SIMULINK setting. Table 5 contrasts suggested and current methods. The proposed MLSTM-ZOA model with respect to SAs is 58.38%, 59.39%, 54.31%, and 55.33% superior to MPA, HHO, MGMPA, and SF-BOA respectively. The MI vs. SAs (degree) plot of a cascaded 11-level H-bridge inverter utilizing the MLSTM-ZOA algorithm is shown in Fig. 8.
Table 5.
SAs analysis.
| Methods | MI | |||
|---|---|---|---|---|
| 0.4 | 0.6 | 0.8 | 1.0 | |
| MPA [26] | 1.30 | 1.19 | 1.08 | 0.82 |
| HHO [27] | 1.28 | 1.17 | 1.06 | 0.80 |
| MGMPA [28] | 1.38 | 1.27 | 1.16 | 0.90 |
| SF-BOA [29] | 1.36 | 1.25 | 1.14 | 0.88 |
| Proposed method | 2.47 | 2.26 | 2.25 | 1.97 |
Fig. 8.
SAs analysis.
Conclusion
A novel MLSTM-ZOA was proposed in this study to swiftly compute SAs for an MLI. The proposed approach might compute a higher count of SAs with different solutions within a given MI range. Here, the primary goal was to minimize the THD while taking into account three different factors: duty cycle, MI, and the number of pulses every quarter cycle. The results also show that the recommended strategy has done better than other ways in terms of convergence behavior, obtaining a lower fitness value in fewer rounds. Finally, performance and analysis related to SHE’s experimental validation in multi-level inverters were also done. The proposed MLSTM-ZOA model in terms of THD was 56.25%, 81.58%, 53.33%, and 41.67% better than MPA, HHO, MGMPA, and SF-BOA respectively. Similarly, the proposed MLSTM-ZOA model with respect to HDP was 64.03%, 47.16%, 84.01%, and 27.62% advanced than MPA, HHO, MGMPA, and SF-BOA respectively. Future work may concentrate on strengthening interpretability, including real-time learning capabilities, and expanding the method’s generalization under other circumstances. Further investigation into hybrid optimization approaches may result in resilient, flexible, as well as effective PWM schemes, thereby enhancing power quality.
Acknowledgements
This work was supported by the Researchers Supporting Project number (RSP2024R467), King Saud University, Riyadh, Saudi Arabia.
Abbreviations
- PWM
Pulse width modulation
- CHB
Cascaded H-bridge
- SHE
Selective harmonics elimination
- NPC
Neutral point clamped
- MLSTM-ZOA
Modified long short-term memory-based zebra optimization algorithm
- FC
Flying capacitor
- SAs
Switching angles
- ANPC
Active neutral point clamped
- MLI
Multi level inverter
- MMC
Modular multilevel converter
- THD
Total harmonic distortion
- IPD-TPWM
In-phase disposition trapezoidal pulse width modulation
- MI
Modulation index
- SPWM
Sinusoidal pulse width modulation
- MFs
Membership functions
- THI 2 SCPWM
Third-harmonic injected inverted sinusoidal carrier pulse width modulation
- FLC
Fuzzy logic controller
- PD
Phase disposition
- IWO
Invasive weed optimization
- POD
Phase opposition disposition
- FPGA
Field programming gate arrays
- APOD
Alternate phase opposition disposition
- MLDCL
Multi-level DC link
- CMLI
Cascaded multi level inverters
- DHB
Dual H-bridge
- SVPWM
Space vector pulse width modulation
- MGWO-PI-PWM
Modified GWO-based PI controller with pulse width modulation
- HDP
Harmonic distortion parameter
Author contributions
All authors contributed to the study, conception, and design. all authors commented on the manuscript. All authors read and approved the final manuscript.
Funding
The authors did not receive support from any organization for the submitted work.
Data availability
The datasets used and/or analyzed during the current study are available from the corresponding author upon reasonable request.
Declarations
Competing interests
The authors declare no competing interests.
Ethical approval
This paper does not contain any studies with human participants or animals performed by any of the authors.
Footnotes
Publisher’s note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
References
- 1.Zhang, H. et al. High-dynamic and low-cost sensorless control method of high-speed brushless DC motor. IEEE Trans. Ind. Inform.19(4), 5576–5584 (2023). [Google Scholar]
- 2.Ye, M., Kang, L., Xiao, Y., Song, P. & Li, S. Modified hybrid modulation strategy with power balance control for H-bridge hybrid cascaded seven-level inverter. IET Power Electron.11(6), 1046–1054 (2018). [Google Scholar]
- 3.Sivamani, S., Mangaiyarkarasi, S. P., Gandhi Raj, R. & Senthilkumar, S. A quad DC source switched three-phase multilevel DC-link inverter topology. Sci. Rep.14, 2065 (2024). [DOI] [PMC free article] [PubMed] [Google Scholar]
- 4.Jammala, V., Yellasiri, S. & Panda, A. K. Development of a new hybrid multilevel inverter using modified carrier SPWM switching strategy. IEEE Trans. Power Electron.33(10), 8192–8197 (2018). [Google Scholar]
- 5.Lin, S., Zhang, J. & Qiu, C. Asymptotic analysis for one-stage stochastic linear complementarity problems and applications. Mathematics11(2), 482 (2023). [Google Scholar]
- 6.Liang, X. & He, J. Load model for medium voltage cascaded H-bridge multi-level inverter drive systems. IEEE Power Energy Technol. Syst. J.3(1), 13–23 (2016). [Google Scholar]
- 7.Liu, J., Sun, Y., Li, Y. & Fu, C. Theoretical harmonic analysis of cascaded H-bridge inverter under hybrid pulse width multilevel modulation. IET Power Electron.9(14), 2714–2722 (2016). [Google Scholar]
- 8.Chitra, A. & Himavathi, A. Reduced switch multilevel inverter for performance enhancement of induction motor drive with intelligent rotor resistance estimator. IET Power Electron.8(12), 2444–2453 (2015). [Google Scholar]
- 9.Gao, J., et al. Design and optimization of a novel double-layer Helmholtz coil for wirelessly powering a capsule robot. IEEE Trans. Power Electron. (2023).
- 10.Ma, K., Yang, J. & Liu, P. Relaying-assisted communications for demand response in smart grid: Cost modeling, game strategies, and algorithms. IEEE J. Sel. Areas Commun.38(1), 48–60 (2019). [Google Scholar]
- 11.Chen, C., et al. Prediction of magnetic field for PM machines with irregular rotor cores based on an enhanced conformal mapping model considering magnetic-saturation effect. IEEE Trans. Transp. Electrif. (2024).
- 12.Jiao, N. et al. Sideband harmonic suppression analysis based on vector diagrams for CHB inverters under unbalanced operation. IEEE Trans. Ind. Electron.71(1), 427–437. 10.1109/TIE.2023.3247797 (2024). [Google Scholar]
- 13.Zhang, J., et al. A novel multiple-medium-AC-port power electronic transformer. IEEE Trans. Ind. Electron. (2023).
- 14.Sahoo, S. K. & Bhattacharya, T. Phase-shifted carrier-based synchronized sinusoidal PWM techniques for a cascaded H-bridge multilevel inverter. IEEE Trans. Power Electron.33(1), 513–524 (2018). [Google Scholar]
- 15.Choi, J. S. & Kang, F. S. Seven-level PWM inverter employing series connected capacitors paralleled to a single DC voltage source. IEEE Trans. Ind. Electron.62(6), 3448–3459 (2015). [Google Scholar]
- 16.Salam, Z., Majed, A. & Amjad, A. M. Design and implementation of 15-level cascaded multi-level voltage source inverter with harmonics elimination pulse-width modulation using differential evolution method. IET Power Electron.8(9), 1740–1748 (2015). [Google Scholar]
- 17.Venkataramanaiah, J., Suresh, Y. & Panda, A. K. A review on symmetric, asymmetric, hybrid and single dc sources based multilevel inverter topologies. Renew. Sustain. Energy Rev.76, 788–812 (2017). [Google Scholar]
- 18.Zhang, J. et al. An embedded DC power flow controller based on full-bridge modular multilevel converter. IEEE Trans. Ind. Electron.71(3), 2556–2566 (2024). [Google Scholar]
- 19.Xiang, X., et al. Identifying the generator matrix of a stationary Markov chain using partially observable data. Chaos: Interdiscip. J. Nonlinear Sci. 34 (2) (2024). [DOI] [PubMed]
- 20.Wang, Z., et al. Underwater moving object detection using superficial electromagnetic flow velometer array based artificial lateral line system. IEEE Sens. J. (2024).
- 21.Venkataramanaiah, J., Suresh, Y. & Panda, A. K. Design and development of a novel 19-level inverter using an effective fundamental switching strategy. IEEE J. Emerg. Sel. Top. Power Electron.6(4), 1903–1911 (2018). [Google Scholar]
- 22.Ahmed, I. et al. Simplified space vector modulation techniques for multilevel inverters. IEEE Trans. Power Electron.31(12), 8483–8499 (2016). [Google Scholar]
- 23.Karthikeyan, D., Krishnaswamy, V. & Sathik, M. A. J. Development of a switched diode asymmetric multilevel inverter topology. J. Power Electron.18(2), 418–431 (2018). [Google Scholar]
- 24.Kathiresan, N. & Samuel, M. J. A comparative analysis of fusion-based techniques based on multi resolution transforms. Natl. Acad. Sci. Lett.38(1), 61–65 (2015). [Google Scholar]
- 25.Krithiga, G., Mohan, V. & Senthilkumar, S. A brief review of the development path of artificial intelligence and its subfields. Int. J. Eng. Technol. Manag. Res.10(6), 1–12. 10.29121/ijetmr.v10.i6.2023.1331 (2023). [Google Scholar]
- 26.Wang, Z., et al. Permanent magnet-based superficial flow velometer with ultra low output drift. IEEE Trans. Instrum. Meas. (2023).
- 27.Yuntao, J. et al. Distributed three-phase power flow for AC/DC hybrid networked microgrids considering converter limiting constraints. IEEE Trans. Smart Grid13(3), 1691–1708 (2022). [Google Scholar]
- 28.Li, M. et al. Scaling-basis Chirplet transform. IEEE Transactions on Industrial Electronics68(9), 8777–8788 (2021). [Google Scholar]
- 29.Wang, H., et al. A MTPA and flux-weakening curve identification method based on physics-informed network without calibration. IEEE Trans. Power Electron. (2023).
- 30.Ye, M. et al. Research on power-balance control strategy of CHB multilevel inverter based on TPWM. IEEE Access7, 157226–157240 (2019). [Google Scholar]
- 31.Ye, M. et al. Hybrid multi-carrier PWM technique based on carrier reconstruction for cascaded H-bridge inverter. IEEE Access7, 53152–53162 (2019). [Google Scholar]
- 32.Suresh, K. & Parimalasundar, E. A modified multi level inverter with inverted SPWM control. IEEE Can. J. Electr. Comput. Eng.45(2), 99–104 (2022). [Google Scholar]
- 33.Azeem, H., Yellasiri, S., Jammala, V., Naik, B. S. & Panda, A. K. A fuzzy logic based switching methodology for a cascaded H-bridge multi-level inverter. IEEE Trans. Power Electron.34(10), 9360–9364 (2019). [Google Scholar]
- 34.Biswas, S. P. et al. A new level shifted carrier based PWM technique for a cascaded multilevel inverter based induction motor drive. IEEE Trans. Ind. Appl.59(5), 5659–5671 (2023). [Google Scholar]
- 35.Kumar, K. V. & Kumar, R. S. Analysis of logic gates for generation of switching sequence in symmetric and asymmetric reduced switch multilevel inverter. IEEE Access7, 97719–97731 (2019). [Google Scholar]
- 36.Sun, Q., et al. Virtual current compensation-based quasi-sinusoidal-wave excitation scheme for switched reluctance motor drives. IEEE Trans. Ind. Electron. (2023).
- 37.Yang, Y. et al. Design of a simultaneous information and power transfer system based on a modulating feature of magnetron. IEEE Trans. Microw. Theory Tech.71(2), 907–915 (2022). [Google Scholar]
- 38.Shirkhani, M. et al. A review on microgrid decentralized energy/voltage control structures and methods. Energy Rep.10, 368–380 (2023). [Google Scholar]
- 39.Liang, J. et al. A direct yaw moment control framework through robust TS fuzzy approach considering vehicle stability margin. IEEE/ASME Trans. Mechatron.29(1), 166–178 (2023). [Google Scholar]
- 40.Odeh, S. I., Lewicki, A. & Morawiec, M. A single-carrier-based pulse-width modulation template for cascaded H-bridge multilevel inverters. IEEE Access9, 42182–42191 (2021). [Google Scholar]
- 41.He, Y., Liu, Y., Lei, C. & Liu, J. Equivalent space vector output of diode clamped multilevel inverters through modulation wave decomposition under carrier-based PWM strategy. IEEE Access8, 104918–104932 (2020). [Google Scholar]
- 42.Ahmad, S., Iqbal, A., Ashraf, I. & Meraj, M. Improved power quality operation of symmetrical and asymmetrical multilevel inverter using invasive weed optimization technique. Energy Rep.8, 3323–3336 (2022). [Google Scholar]
- 43.Niraimathi, R. & Seyezhai, R. Analysis, simulation and implementation of a novel dual bridge asymmetric cascaded multi-level inverter using MGWO-PI-PWM controller. Microprocess. Microsyst.77. (2020).
- 44.Fang, Li., Li, D. & Ronghai, Qu. Torque improvement of vernier permanent magnet machine with larger rotor pole pairs than stator teeth number. IEEE Trans. Ind. Electron.70(12), 12648–12659 (2023). [Google Scholar]
- 45.Liu, Z. et al. A review of common-mode voltage suppression methods in wind power generation. Renew. Sustain. Energy Rev.203, 114773 (2024). [Google Scholar]
- 46.Yaghoubi, E. et al. A systematic review and meta-analysis of machine learning, deep learning, and ensemble learning approaches in predicting EV charging behavior. Eng. Appl. Artif. Intell.135, 108789 (2024). [Google Scholar]
- 47.Yang, Y., et al. Real time aggregation control of P2H loads in a virtual power plant based on a multi-period Stackelberg game. Energy 131484 (2024).
- 48.Zhang, J. et al. Cascade ADRC speed control base on FCS-MPC for permanent magnet synchronous motor. J. Circuits Syst. Comput.30(11), 2150202 (2021). [Google Scholar]
- 49.Yang, Y. et al. Broadband electrical impedance matching of sandwiched piezoelectric ultrasonic transducers for structural health monitoring of the rail in-service. Sens. Actuators A Phys.364, 114819 (2023). [Google Scholar]
- 50.Duan, Y., Zhao, Y. & Jiangping, Hu. An initialization-free distributed algorithm for dynamic economic dispatch problems in microgrid: Modeling, optimization and analysis. Sustain. Energy Grids Netw.34, 101004 (2023). [Google Scholar]
- 51.Yang, M., et al. Symmetric differential demodulation-based heterodyne laser interferometry used for wide frequency-band vibration calibration. IEEE Trans. Ind. Electron. (2023).
- 52.Krithiga, G. & Mohan, V. Elimination of harmonics in multilevel inverter using multi-group marine predator algorithm-based enhanced RNN. Int. Trans. Electr. Energy Syst.2022, 8004425 (2022). [Google Scholar]
- 53.Ali, M. et al. Robust ANN-based control of modified PUC-5 inverter for solar PV applications. IEEE Trans. Ind. Appl.57(7), 3863–3876 (2021). [Google Scholar]
- 54.Ahmad, S., Meraj, M., Iqbal, A. & Ashraf, I. Selective harmonics elimination in multilevel inverter by a derivative-free iterative method under varying voltage condition. ISA Trans.92, 241–256 (2019). [DOI] [PubMed] [Google Scholar]
- 55.Krishnaram, K., Padmanabhan, T. S., Alsaif, F. & Senthilkumar, S. Performance optimization of interleaved boost converter with ANN supported adaptable stepped-scaled P&O based MPPT for solar powered applications. Sci. Rep.14, 8115 (2024). [DOI] [PMC free article] [PubMed] [Google Scholar]
- 56.Nasser, A. M., Refky, A., Shatla, H. & Abdel-hamed, A. M. A grey wolf optimization-based modified SPWM control scheme for a three-phase half bridge cascaded multilevel inverter. Sci. Rep.14, 7016 (2024). [DOI] [PMC free article] [PubMed] [Google Scholar]
Associated Data
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Data Availability Statement
The datasets used and/or analyzed during the current study are available from the corresponding author upon reasonable request.