Selective harmonic elimination in T-type multilevel inverter with reduced switch count using Sea-Horse Algorithm

Abstract

Multi-level inverters (MLIs) are also used in relatively high-power systems; but the goals of low overall harmonic distortion (THD) and fewer power semiconductor devices remain a design issue of serious difficulty. Despite using different metaheuristic optimization methods to selective harmonic elimination (SHE), there are still drawbacks in convergence stability, finding accurate solutions, and the ability of the method to suppress harmonic. The proposed study will have a three-stage, 9-level, and reduced-switch-count T-type multilevel inverter design with pulse width modulation (PWM) and selective harmonic elimination (SHE). The equations of the nonlinear SHE switching angle are optimized with an objective of reducing harmonic distortion within basic voltage limitations over a broad modulation index depth (0.1–1.0). A fairly recent algorithm is the Sea-Horse Optimization (SHO) used to calculate the optimal switching angles of the SHE-PWM setup. The application of SHO is relatively compared with Particle Swarm Optimization (PSO) and Genetic Algorithm (GA). The simulation outcomes show that SHO is better in harmonic suppression and convergence is more stable. At M 1, SHO has a THDe of 0.03%, which is dramatically better than GA (0.19%) and PSO. The findings prove that the balance mechanism of exploration-exploitation and the search strategy of SHO which uses the Levy-flight mechanism promotes the improvement of optimization robustness. The suggested SHO-based SHE-PWM method enhances the quality of power and the number of switches is kept less providing a relevant and effective solution to a high-level multilevel inverter.

Introduction

Due to the usefulness of the applications semiconductor devices offer, the latest electronic power converter technologies are enlarging their sphere of application rather quickly. These devices are favored due to their high benefit, which is reduced energy use, enhanced performance, and generated efficiency together with reduced maintenance^1–3. It is believed that direct medium-voltage grid interfaces to power semiconductor switches perform optimally in this area. Multilevel inverter structures offer many solutions and are a strong alternative, especially at high and medium voltages^4,5. These systems provide an extremely cost-efficient solution to applications with low power needs, such as renewable power sources such as solar⁵ and wind⁶, as well as to high power needs.

The multilevel inverters have numerous applications, which include locomotives, static converters, rail-side systems, etc.^7–9. There are numerous advantages to power systems using these inverters, and they have lower effects on the environment. These inverters have the capability to meet high- and medium-level energy requirements, and thus they can be utilized in a wide variety of ways, including in the integration of renewable energy systems, electric transportation systems, and industrial power systems^10–13.

There are several key benefits of using multilevel inverters instead of 2-level inverters. First, multilevel inverters make the AC output voltage smoother by making it practically sinusoidal. This is a significant advantage for improving power quality. Moreover, the multilevel inverters produce a smaller electromagnetic interference (EMI)¹⁴ since the decreased harmonic content causes a smaller voltage distortion. This feature is what makes them be specifically applicable in cases of sensitive electronic equipment. Another advantage is that it doesn’t put as much strain on the switches. Multilevel inverters have a lower dv/dt (voltage change rate), which means the switches don’t have to work as hard. This makes the devices last longer and work better. Furthermore, level inverters transform energy more effectively, which helps the overall system perform better and consume less energy^5,15. These inverters are also superior at handling enormous power demands, which makes them perfect for big energy systems and industrial usage. Systems perform better when multiple inverters flip less frequently because they stay cooler. 2-level inverters have to flip more frequently, which might use up energy¹⁶. Additionally, multilevel inverters allow for the use of smaller filters because the harmonic content in the output voltage is lower¹⁷. This makes it simpler and less expensive to plan systems. Because they have less harmonic distortion, multilevel inverters provide higher power quality¹⁸.

Multi-level inverters are effective, and they also possess certain issues. First, the multilevel inverters are not as easy to design and control as well since they possess a more complex structure. That is, the control algorithms will be more complicated and will include more parts. The system is further complicated and costly in that every switch requires its own driver circuit. This problem is simplified by 2-level inverters, which are simpler and require fewer components¹⁹. Another disadvantage is that multilevel inverters are larger. Such devices may not perform well when the power required is low²¹. The number of switches and driver circuits is such that the system requires more physical space, and this increases the costs. Moreover, uniform distribution of voltages among the switches in multilevel inverters can be challenging and, hence, makes designing and controlling them more difficult²⁰. Another problem is that there is a higher chance of multiple inverters overheating. Since driving each switch could make it hot, better cooling systems are needed. 2-level inverters usually don’t have as many problems with overheating because they need fewer switches. Multilevel inverters need more complicated PWM management systems, which means they have to run at higher frequencies and lose more energy^21,24. Last but not least, multilevel inverters can’t be made better for low-voltage and low-power uses because they are too complicated, too expensive, and not very efficient in those situations. In this investigation, a three-phase, T-type, reduced-switch-count²², 9-level multilevel inverter structure is presented as a means of resolving the problems. The proposed topology deals with certain shortcomings of standard T-type inverters, such as large number of switches, complexity of switching and large overall harmonic distortion (THD) and has a better harmonic performance and is easier to implement in comparison with traditional T-type inverters because it has fewer power semiconductor devices and uses optimized SHE-PWM control. For the purpose of inverter control, the Selective Harmonic Elimination (SHE) pulse width modulation (PWM) approach is the recommended choice since it reduces harmonic distortion. A new algorithm called the Sea-Horse Algorithm (SHO) is employed to calculate the desired switching angles to SHE-PWM²³. Being a fairly recent technique of optimization, the Sea-Horse method is reputed to have strong local search capabilities and find solutions in a comparatively limited amount of time. This method has more expedient and effective results when compared with more traditional methods of optimization.

Additional recent studies further demonstrate ongoing efforts to improve harmonic suppression in inverter systems.²⁴ proposed a symmetrical super-lift DC–AC inverter structure optimized using the Firefly algorithm for photovoltaic applications.²⁵ investigated advanced optimization techniques for inverter control strategies aimed at reducing harmonic distortion and improving system efficiency. Other recent works have explored improved harmonic elimination strategies and inverter structures to enhance power quality and converter efficiency in energy systems^26–29.

Along with the harmonic performance, the number of required power semiconductor devices to produce the required voltage levels is also one of the most significant design criteria in multilevel inverter structures. The more switches, the more switching losses, control complexity and cost of the system. Thus, a lot of recent research is devoted to the inverter design of reduced switch count inverter structures with acceptable harmonic performance.

Table 1 is a comparison of various common multilevel inverter schemes with respect to voltage level capability, number of switches required, DC source requirement and auxiliary components. Traditional topologies like the Neutral Point Clamped (NPC) and Flying Capacitor (FC) need more clamping diodes/capacitors, which increase the overall number of components and circuitry considerably^30,31. CHB inverters offer modular expansion, but each H-bridge modules need several isolated DC sources that raise the cost and complexity of the system³².

Table 1.

Comparison of MLI topologies complexity.

References	Topology	Number of power switches	Number of DC sources	Auxiliary components
30	Neutral Point Clamped (NPC)	High	1	Clamping diodes
31	Flying Capacitor (FC)	High	1	Large number of capacitors
32	Cascaded H-Bridge (CHB)	High	1 per H-bridge modules	-
33	Reduced Switch Count MLI	Medium	Reduced	-
This work	T-Type RCS	Reduced	Reduced	-

Minimal-switch-count multilevel inverter designs are intended to overcome these issues through the reduction of the switching devices, but retain the capability to produce a variety of voltage levels. This minimization of control structure reduces the control complexity and enhances reliability by reducing switching losses and the complexity of hardware³³. The 9-level T-type multilevel inverter with reduced number of switches chosen in this work based on these considerations offers a good trade-off between the capability to adjust voltages and the simplicity of hardware. The lower number of components also allows the optimization of nonlinear switching angles under the Sea-Horse Optimization (SHO) algorithm to allow an efficient harmonic elimination without a large inverter structure. Moreover, enhanced harmonic elimination method minimizes the need to have large output filters as well as minimizing the switching losses. This helps in having a cheaper implementation of the inverter since it reduces the complexity of the hardware as well as the power losses incurred and therefore SHO is an appropriate optimization method to use in high performance applications of multilevel inverters.

Performance of the Sea-Horse Algorithm is compared to that of two other optimization methods that are commonly used. These include the Genetic Algorithm (GA)³⁷ and the PSO³⁵. GA and PSO are optimization strategies that are usually chosen due to the fact that they can conduct global searches. Even though many metaheuristic algorithms like GA and PSO have been utilized to solve the nonlinear switching angle problem of SHE, currently, literature is mainly concerned with finding a solution assuming feasibility, but not high-precision convergence, stability at a large range of modulation indices, and predictable harmonic suppression at a small hardware setup. Further, a vast majority of works consider the inverter topology and optimization strategy separately, and do not consider the hardware complexity and algorithm robustness together. Thus, there is a definite gap in the research concerning the combined optimization of low switching-count multilevel inverter configurations with an algorithm that offers both effective globally-searching properties and consistent convergence properties.

This study assesses the Sea-Horse Algorithm’s effectiveness in solving large, complex problems. The performance of these methods on the inverter is evaluated and compared at modulation index values of between 0.1 and 1.0. The proposed work extends beyond the incremental optimization, by incorporating a new algorithm, the Sea-Horse Optimization (SHO), with their balanced exploration-exploitation scheme and Levy-flights-based search-strategy into the SHE-PWM algorithm of a 9-level reduced-switch-count T-type inverter. In comparison to traditional methods, this research has been shown to achieve a better convergence stability, high precision in harmonic suppression, and stable performance. Drawing upon the obtained findings, it may be concluded that the Sea-Horse Algorithm is more efficient than GA and PSO regarding its capabilities to minimize harmonic distortion and its use at lower switching angles. This paper highlights the advantages of the Sea-Horse Algorithm, especially in applications involving SHE-PWM. In addition, the results help in the development of solutions for multilevel inverter designs that are more efficient and cost-effective.

This study presents a model developed to optimize reduced-switch-count multilevel inverters:

• The main objective of the study is to minimize the Total Harmonic Distortion (THD) by eliminating selected harmonics.

• For this purpose, the switching angles used for inverter control are calculated using the newly developed Sea-Horse Algorithm (SHO). SHO is an algorithm that increases efficiency in the optimization process with its strong local search capabilities and quick solution times. Its performance is compared with commonly used methods like Genetic Algorithm (GA) and Particle Swarm Optimization (PSO).

• Additionally, the performance of each algorithm on the inverter is evaluated and compared for different modulation index values ranging from 0.1 to 1.0.

The organization of the paper is structured as follows: In section “T-type 9-level 3 phase MLI”, we will explain the basic operating principles of the three-phase 9-level T-type reduced-switch-count multilevel inverter and the aim of this study. Section “Optimization based selective harmonic elimination (SHE)” will introduce the Sea-Horse Algorithm (SHO) and SHE-PWM methods, detailing how the switching angles are calculated. In section “Seahorse optimizer (SHO)”, Seahorse algorithm basis and advantages are presented. Section “Test functions and performance comparison” presents test function comparisons in detail. In section “Results and discussion”, the simulation results will be presented, and in section “Conclusion”, the results of the study will be evaluated, with suggestions for future research.

T-type 9-level 3 phase MLI

Neutral-point-clamped (NPC), flying capacitor (FC), and cascaded H-bridge (CHB) topologies are more common conventional multilevel inverter (MLI) topologies that are utilized in high-power applications. Yet such constructions often demand a great number of power semiconductor devices, isolated DC sources or complicated capacitor voltage balancing circuits. The more levels of voltage, the more switches, the more the gate driver circuitry, and the system complexity which contribute to higher cost, larger size and less reliability^{30–32,39,40}.

A reduced-switch-count (RSC) T-type multilevel inverter topology is chosen to be used to handle these engineering constraints, in this study. The 9-level T-type RSC MLI offers a good trade-off between the quality of the voltage and the complexity of the hardware³³. It allows more output voltage resolution than less semiconductor devices than traditional MLIs, and eliminates the requirement of multiple switched DC sources or complicated balancing schemes^34,35.

T-type reduced-switch-count multilevel inverters are used primarily in systems that require high-voltage applications and energy efficiency. These inverters produce a high-level output voltage waveform to achieve lower harmonic distortion and higher efficiency^5,13. In T-type multilevel inverters with a lower number of switch, the number of active switching devices is reduced when compared to the traditional T-type topologies. Peak inverse voltage (PIV) are also reduced with individual switches and brings about inverter’s better thermal distribution. Consequently, the electrical and thermal stresses are reduced and this enhances the overall reliability of the inverter³³. Thus, reduced-switch T-type MLIs offer benefits in the safety of the devices and their working life with similar performance^22,23. T-type inverters are considered a simplified version of NPC MLI with less component count and less complexity in implementation; but the simplicity is introduced at the cost of higher voltage stress of main switches^30,31. Thus, the targeted engineering problem that will be dealt with in this work is accomplishment of: lower THD, decrease in switch number and improvement of power quality with uncomplex circuit.

The generalized circuit of a single-phase 9-level T-type inverter with a four-input DC source and ten power switches is shown in Fig. 1a, while the output waveform is presented in Fig. 1b). Figure 2 shows the output voltage values based on switch states, and Table 2 summarizes the general conditions.

Fig. 1.

Three-phase 9-level T-type RSC-MLI (a) circuit structure, (b) output voltage waveform.

Fig. 2.

Different switching states of 9-level T-Type MLI.

Table 2.

Switching states of Single-Phase 9-Level T-Type Reduced-Switch-Count Multilevel Inverter (RSC-MLI).

Level	S1	S2	S3	S4	S5	S6	S11	S12	S13	S14	Output voltage (Van)
4	0	0	0	0	0	0	1	1	0	0	4 V
3	1	0	0	0	0	0	0	1	0	0	3 V
2	1	0	1	1	0	0	0	1	0	0	2 V
1	0	0	0	0	1	0	0	1	0	0	1 V
0	0	0	0	0	0	0	0	1	0	1	0 V
– 1	0	1	0	0	0	0	0	0	1	0	– 1 V
– 2	0	0	0	1	0	0	0	0	1	0	– 2 V
– 3	0	0	0	0	0	1	0	0	1	0	– 3 V
– 4	0	0	0	0	0	0	0	0	1	1	– 4 V

It is to be noted that, as illustrated in Table 2, each output voltage level of the suggested 9-level RSC-MLI is produced by a distinct switching condition. Because of the reduced-switch configuration and the lack of floating capacitors, there are no redundant switching states to be used in balancing capacitor voltage. Consequently, the proposed topology does not need any further balancing strategy. In the following cases, the switching states of the switches and the output voltage change at specific levels: When switches S11 and S13 are conducting, the output voltage is zero (Fig. 2a). When switches S5 and S12 are conducting, the output voltage becomes +Vdc (Fig. 2b). When switches S3 and S12 are conducting, the output voltage becomes +2Vdc (Fig. 2c). When switches S1 and S12 are conducting, the output voltage becomes +3Vdc (Fig. 2d). When switches S11 and S12 are conducting, the output voltage becomes +4Vdc (Fig. 2e). When switches S13 and S2 are conducting, the output voltage becomes – Vdc (Fig. 2f). When switches S13 and S4 are conducting, the output voltage becomes – 2Vdc (Fig. 2g). When switches S13 and S6 are conducting, the output voltage becomes – 3Vdc (Fig. 2h). Finally, when switches S13 and S14 are conducting, the output voltage becomes -4Vdc (Fig. 2i). When switches S11 and S12 are conducting, positive (+) half-cycle levels can be produced, while when switches S13 and S14 are conducting, negative (–) half-cycle levels can be produced.

Optimization based selective harmonic elimination (SHE)

By solving the SHE equations at the fundamental frequency, we may get the fundamental voltage we want and get rid of undesirable or chosen low-order harmonics. The switching angles are then calculated. Simplifying electromagnetic interference (EMI) and reducing switching losses are two benefits of using a single switch for each module. That is why the relationship between the number of DC sources (k) and the number of switching angles is linear. The inverter’s output voltage may have low-order harmonics removed, where k is the number of DC sources or the number of positive/negative steps in quarter wave at the same time.

In a 9-level RCS-MLI, the Fourier series expansion of the output voltage may be written as nonlinear harmonic equations. These equations are required to find the best switching angles. Fig. 2b shows the Fourier transform of a stepped waveform for the 9-level system, which is given in Eq. (1).

1

Here, represents the angular speed, denotes the voltage magnitude of the DC source, and refers to the conducting angle of the i-th step. The coefficients in Eq. (1) can be determined as follows:

2

, , , and represent the switching angles. According to the quarter-wave symmetry, the switching angles must satisfy the condition given by Eq. (3). Also, in a three-phase system, the triplen harmonics (harmonics of third order or multiples of 3rd order) are cancelled naturally in the line-to-line voltages. As such, the SHE formulation in this paper only focuses on odd, non-triplen harmonics (including the 5th, 7th, 11th, and 13th harmonics) to enhance quality output voltage.

3

Equation (4) provides the formula for calculating Total Harmonic Distortion (THD), which is the ratio of the sum of the squares of the harmonics to the square of the fundamental harmonic. This ratio may be used to compute THD.

4

The %THD number encompasses lower-order harmonics but fails to convey their influence. Consequently, the THDe value, defined as the ratio of the square root of the sum of the squares of the chosen harmonic amplitudes to the amplitude of the fundamental harmonic, is used. The threshold value represents the chosen maximum harmonic value. Given the absence of the third and its multiple harmonics in the system, the THDe value for the three-phase RSC-MLI may be computed as shown in Eq. (5), omitting the third and its multiple harmonics³⁶.

5

where the threshold value is equal to 11 for the 9-level inverter.

Traditionally, SHE-PWM optimization models aim to minimize only lower-order harmonics. The number of harmonics that can be eliminated is one less than the number of DC sources or the number of positive/negative steps in quarter wave, k. SHE-PWM optimization focuses on eliminating specific harmonics, whereas higher-order harmonics are generally not considered.

Considering the above, the optimization equivalent of the SHE-PWM equations for the 9-level CHB-MLI inverter is given by Eq. (6). Optimization problems are adjusted according to their function. The fitness function for 9-levels is given in Eqs. (7) and (8).

The system of non-linear equations for the objective functions is defined as:

6

7

8

where M denotes the modulation index. M is determined by the ratio of , the peak value of the required fundamental voltage, to the overall DC source value, as specified in Eq. (9).

The modulation index M is defined as the ratio of the peak fundamental output voltage to the total DC source voltage:

9

The optimization variables are the switching angles , which are determined by using the Sea-Horse Optimization (SHO) algorithm given in section “Seahorse optimizer (SHO)”.

Seahorse optimizer (SHO)

The proposed SHO algorithm consists of three main components: hunting, reproduction, and mobility. To balance exploration and exploitation, various search strategies have been developed for mobility and hunting behaviors. The reproduction behavior is executed following the completion of these two behaviors. The mathematical models for these behaviors will be demonstrated and thoroughly explained below.

As depicted in Fig. 3, seahorses are unique in the animal kingdom due to their male counterparts giving birth. Male seahorses possess a brood pouch located on the front or side of their abdomen. During mating, the female deposits her eggs in this pouch, and the male is responsible for fertilizing them. After fertilization, the male seahorse carries the eggs in the pouch until they mature and are ready to be released into the ocean.

Fig. 3.

The behaviors of the sea horse²³.

Hunting, reproduction, and mobility are the three primary parts of the suggested SHO algorithm. Different search strategies for mobility and hunting behaviors have been developed to strike a balance between exploration and exploitation. After these two behaviors are finished, reproduction is carried out. In the following section, a detailed explanation and demonstration of the mathematical models for these behaviors will be provided.

Initialization

Initializing the population is the first step in SHO, as it is in other current metaheuristic algorithms. If each seahorse as a potential solution is considered in the problem’s search space, the whole population can be expressed as follows (Eq. 10):

10

where pop is the population size and Dim is the variable’s dimension. The boundaries of a given issue, LB and UB, are used to create solutions at random within those constraints. The i-th person in the search space could be represented as (Eq. 11):

11

assuming that rand is a random number between 0 and 1, and that is the value of the j-th dimension in the i-th person. A positive integer between 1 and the population size is the index i, and a positive integer between 1 and Dim, where Dim is the problem’s number of dimensions, is j. In an optimization problem, the lower and upper limits of the j-th dimension are denoted by and , respectively.

The elite person, represented by , is the one with the lowest fitness value. Using Eq. (12), the value of can be obtained as follows:

12

where represents the objective function value of a given problem.

The movement behavior of sea horses

Sea horses’ initial behavior shows movement patterns that roughly correspond to a normal distribution, which is shown as We set as the threshold to balance exploration and exploitation, allocating 50% of the movements to local exploration and 50% to global search. In the conventional SHO formulation, the exploration and exploitation phases are given by the random parameter . When , the algorithm carries out an exploitation whereas when it is less, exploration is carried out. Because is randomly chosen, and it follows a symmetric distribution, the likelihood of choosing exploration or exploitation is more or less the same in each round of optimization. Thus, exploration to exploitation ratio is not dynamically varying with the number of iterations, but that is automatically kept in check due to the stochasticity of the algorithm. As seen below, this process results in two distinct movement examples.

Case 1

When the standard random variable is positioned to the right of the threshold, the seahorse primarily focuses on the localized use of the SHO while navigating a spiral trajectory inside the sea vortex. As a result, the seahorse travels in a spiral pattern in the direction of the ideal solution, . The step size of the seahorse’s movement is represented by the Lévy flight³⁷, which improves its ability to identify new areas in the early stages and keeps the SHO from focusing too much on local exploitation. Additionally, the spiral motion constantly adjusts its rotational angle, which broadens the area around the current local solutions. In this context, the seahorse’s new location can be expressed numerically as follows (Eq. 13):

13

In the context of spiral movement, the three-dimensional coordinates (x, y, z) are expressed as

which facilitate the updating of search agents’ locations. is the length of the stems determined by the logarithmic spiral constants u and v, where is a random variable within the interval . denotes the Lévy flight distribution function and is determined by Eq. (14). Lévy flight implementation of the Sea-Horse Optimization (SHO) algorithm improves it by allowing it to escape local minima since it may sometimes make long-distance moves, thus expanding the search power of the search process. This process also helps in the strength and convergence behavior of the algorithm, particularly in multimodal optimization problems.

14

where represents a random number within the interval [0, 2] (designated as in this study). s is a constant value of 0.01. w and k are random variables within the interval [0, 1]. is determined using Eq. (15):

15

Case 2

In this case, the seahorse’s exploration is affected by drifting, especially when is to the left of the cut-off point. The main focus is on the search operation that helps avoid local extrema of the SHO. Brownian motion is used to facilitate exploring the search space better by simulating the seahorse’s movement over a long distance. This section explains how to formulate the mathematics for this situation generally with Eq. (16):

16

Here, L is a constant coefficient, defined as . The coefficient denotes the stochastic component of Brownian motion. It is a stochastic variable that adheres to a typical normal distribution. The value of may be calculated using Eq. (17):

17

Both situations can be combined to compute the updated location of the seahorse at iteration t as follows (Eq. 18):

18

The random integer that follows a normal distribution is . Fig. 4 shows how the seahorse updates its location using two different types of movement: spiral and Brownian motion. Both methods show how the seahorse’s movement is unpredictable due to changing underwater conditions.

Fig. 4.

Different motion patterns of the sea horse in sea²³.

Case 3

The predation behavior of sea horses: When a seahorse is looking for zooplankton and tiny crustaceans, it can either succeed or fail. The seahorse has a success rate of over 90% when it comes to catching prey. The random number in the SHO is used to tell the difference between these two outcomes, which is set at a critical threshold of 0.1. The elite position supplies a rough idea of where the prey is, and the predation success shows how effective the SHO is at taking advantage of the situation. If , it indicates that predation is successful, meaning the seahorse outperforms the prey (elite), accelerates past it, and ultimately captures it. If predation fails, the reaction speed of both the seahorse and the prey reverses, showing that the seahorse is now exploring the search space. This is how math shows this predation behavior (Eq. 19).

19

Here, denotes the updated location of the seahorse after the movement at iteration t. The value of is a stochastic variable inside the interval [0,1]. The parameter , which diminishes linearly over repetitions, is used to modulate the step size of the seahorse during prey hunting, and it is computed using Eq. (20).

20

Figure 5, where the blue star shows the new location of the seahorse, while the red dot shows where the prey is thought to be, shows the two possible outcomes of the seahorse’s predatory behavior. In Fig. 5a, the seahorse moves to the best position after it has successfully caught its prey. As iterations go on, the seahorse gets closer and closer to the global best individual thanks to parameter . A thorough search is done in Fig. 5(b) because the prey is not caught. The parameter is used for the vector that connects the current person to the elite, and is used to change where the seahorse is right now. This method aims to do a more thorough search at first and stop over-exploitation later.

Fig. 5.

Predation process of the sea horse²³.

Based on their fitness levels, the population is split into two groups: men and women. Male seahorses are the ones that breed in the SHO algorithm. As a result, half of the people with the highest fitness values are called ’fathers’, and the other half are called ’mothers’. This method helps pass on beneficial traits from parents to their children, which helps make the next generation stronger. This separation also stops the algorithm from being too localized, which encourages different approaches to finding the best answers. The mathematical models for assigning tasks to the seahorses are defined as follows (Eqs. 21 and  22), During the reproduction phase of the Sea Horse Optimizer, the “fathers” are selected from the top half of the sorted population :

21

22

In the SHO algorithm, denotes the whole set of values, arranged in ascending order based on their fitness levels. The term “fathers” denotes the male population, whereas “mothers” refers to the female population. The populations are randomly coupled to produce new progeny. To simplify the SHO method, it is assumed that each couple of seahorses generates just a single child. The formula for the offspring (Eq. 23) is then determined from the genetic information of the progenitors:

23

Where i is a positive integer that falls between the range [1, pop/2], and is a random number that falls within the range. When considering this particular context, the terms and are used to denote people who have been randomly picked from the male and female populations, respectively.

Figure 6 illustrates the reproductive process of seahorses. Figure 6a demonstrates that people are sorted in increasing order according to their fitness levels. Figure 6b illustrates the approximate location of a freshly produced offspring. This progeny is randomly generated along a continuum between the two progenitors, enabling the exchange of genetic material between two subpopulations.

Fig. 6.

Predation process of the sea horse and breeding process of sea horses²³.

The pseudocode for the SHO algorithm is shown in Table 3. Regarding space complexity, it is considered that maximal space utilization occurs when offspring are generated throughout iterations. Consequently, the space complexity of the SHO method is denoted as .

Table 3.

Pseudocode of Sea-Horse Optimizer (SHO)³⁸.

Step	Description
1	Initialize sea horses
2	Calculate the fitness value of each sea horse
3	Determine the best sea horse
4	while do
	/* Movement behavior */
5	if do
6	Set , constant parameters
7	Rotation angle
8	Generate Levy coefficient by using Eq. (5)
9	Update positions of the sea horses by using Eq. (4)
10	else if do
11	Set constant parameter
12	Update positions of the sea horses by using Eq. (7)
13	else if
	/* Predation behavior */
14	Update positions of the sea horses by using Eq. (10)
15	Handle variables out of bounds
16	Calculate the fitness value of each sea horse
	/* Breeding behavior */
17	Select mothers and fathers by using Eq. (12)
18	Breed offspring by using Eq. (13)
19	Handle variables out of bounds
20	Calculate the fitness value of each offspring
21	Select the next iteration population from the offspring and parents ranked top pop in fitness values
22	Update elite () position
23
24	end while

Test functions and performance comparison

When assessing whether or not optimization strategies are effectively applied, it is common practice to make use of benchmark test functions that are based on standards. These functions provide an objective technique of evaluating the search capabilities, convergence speed, and ability to avoid local optima of any algorithm. This evaluation may be done in order to determine whether or not the algorithm.

A comparison of the performance of three distinct algorithms is shown in this part. These algorithms are the GA, the PSO, and the SHO. The benchmark functions that are used in this section are diverse. Within the context of this comparison, the focus is placed on significant aspects such as the quality of the solution, the rate of convergence, and the stability. The purpose of this investigation is to shed light on the benefits and drawbacks that each algorithm has when it comes to the resolution of a wide variety of optimization difficulties. Table 4 contains the parameter values associated with each procedure.

Table 4.

Algorithm parameters.

Algorithm	Parameter	Value
GA	Population size	50
	Selection mechanism	Roulette Wheel
	Mutation rate	0.2
	Crossover rate	0.8
	Number of iterations	500
PSO	Population size	50
	Inertia weight (Wmin, Wmax)	0.4, 0.9
	Acceleration coefficients (C1, C2)	1.2, 1.2
	Number of iterations	500
SHO		0
	Probability of success	0.1
	(Lévy distribution parameter)	1.5
	s (step size constant)	0.01
	w	[0, 1]
	k	[0, 1]

To ascertain the best possible value, each approach is applied to the resolution of a total of twenty-three different benchmark issues. These problems are composed of both unimodal and multimodal functions in their overall structure. Multimodal functions are shown in Table 6, whereas unimodal functions are presented in Table 5.

Table 6.

Multimodal benchmark functions.

Functions	Dimension	Domain	Global minimum
	50		0
	50		0
	50		0
	50		0
	50		0
	2		1
	4		0.00030
	2		-1.0316
	2		0.398
	2		3
	3	[1, 3]	– 3.86
	6	[0, 1]	– 3.32
	4	[0, 10]	– 10.1532
	4	[0, 10]	– 10.4028
	4	[0, 10]	– 10.5363

Table 5.

Unimodal benchmark functions.

Functions	Dimension	Domain	Global minimum
	50		0
	50		0
	50		0
	50		0
	50		0
	50		0
	50		0
	50		-418.98

The performance of each method (GA, PSO, and SHO) is shown in Table 7 which includes the results of several test functions and Figs. 7, 8 that includes convergence curves. When it comes to discovering the best answers, GA is often effective; nonetheless, it demonstrates significant variances in the worst outcomes and the variety of options.

Table 7.

Benchmark functions performance comparison.

Function	Algorithm	Best	Worst	Median	StdDev
F1	GA
	PSO
	SHO
F2	GA
	PSO
	SHO
F4	GA				4.77
	PSO				3.49
	SHO				5.30
F5	GA
	PSO
	SHO
F6	GA
	PSO
	SHO	2.69		4.31
F7	GA	5.55		5.83	7.93
	PSO				2.53
	SHO				2.28
F8	GA
	PSO
	SHO
F9	GA
	PSO
	SHO	0.00		0.00
F10	GA
	PSO	9.59		9.59
	SHO				1.37
F11	GA
	PSO	9.11		9.11
	SHO	0.00
F12	GA
	PSO
	SHO

Function	Algorithm	Best	Worst	Median	StdDev
F13	GA
	PSO
	SHO
F14	GA	0.998003878	16.93873492	0.998004061	1.261889752
	PSO		13.11075583	5.032382488	0.492311499
	SHO		0.181	6.67	0.733
F15	GA	0.005694638	0.038997929	0.006046746	0.002177705
	PSO	0.00269904	0.049777657	0.00269904	0.003108569
	SHO		0.0531		0.00371
F16	GA				0.054594538
	PSO				0.028682334
	SHO				0.0145
F17	GA	0.398146645	0.865441819	0.398929087	0.036417685
	PSO	0.397887358	0.819225978	0.397887358	0.020241569
	SHO	0.398	0.652	0.402	0.0211
F18	GA	3.470247856	25.87043642	3.473872535	2.861960681
	PSO	3.00	14.86687399	3.00	0.565411847
	SHO	3.00	8.63	3.00	0.26
F19	GA				0.014829231
	PSO				0.006989047
	SHO				0.0443
F20	GA				0.114958353
	PSO				0.086734598
	SHO				0.15
F21	GA				0.961371036
	PSO				0.622550865
	SHO				0.767
F22	GA				1.028203626
	PSO				0.578228017
	SHO				0.185
F23	GA				0.792078998
	PSO				0.442726216
	SHO				0.722

Fig. 7.

Behavior of algorithms in each benchmark function from 1 to 15.

Fig. 8.

Behavior of algorithms in each benchmark function from 16 to 23.

In order to assure the stability of the presented findings such as the very small value of of , each algorithm was repeated several times (30 times) with random seeds that were different. In these runs, consistent convergence patterns were found, and it cannot be concluded that the high precision is due to overfitting to the benchmark functions but rather a property of the algorithm.

In some test functions, including F1, F2 and F5, the best values that GA can provide may be fairly low, while the worst values that it can produce can be extremely high. This experiment shows that although GA is capable of finding outstanding answers in some circumstances, it also has a high probability of reaching a point where it becomes trapped in a local minimum or delivers unsuccessful results. By way of illustration, the GA algorithm is capable of achieving a very low maximum value of in the F1 function, but its worst value may go up to . This demonstrates that GA often provides a diverse selection of answers; nonetheless, there are moments when it may have the most unfavorable effects. The large values of the standard deviation also indicate that the answers are very much varied and uncertain. Due to this, GA would perform well in many different fields, but with its reliability there are also certain risks (see Table 7).

The results that PSO provides are more reliable and stable compared to GA. The difference between the worst and best values of PSO is usually less, which implies that PSO has a disposition to be less volatile and deliver results that are more predictable. As an example, the PSO algorithm receives a comparatively small maximum value of in the F9 function, although its minimum value remains all the way through the algorithm. Due to this trend, PSO is the more appropriate option to consider. The values of PSO standard deviation are also less than those of GA, meaning that the solutions will be unlikely to vary. PSO has been discovered to be better in the F2 function with a smaller standard deviation and a limited solution range. Due to this fact, PSO is preferable in problem-solving situations that require steadiness and uniformity. PSO performs very well with medium-sized solution areas (see Table 7).

Despite the fact that SHO has very low “Best” scores in general, it manages to attract attention with high “Worst” levels. This suggests that SHO may generate findings that are very successful in some tests while producing results that are not very effective in other tests. Within the context of functions like F1, F2, and F3, SHO is capable of achieving very low best values (for example, ), but its worst values may go as high as . These extreme levels of SHO imply that the algorithm may demonstrate poor performance in certain situations, despite the fact that it may be extremely successful in other situations to a certain extent. A tighter and more stable route towards the solution is followed by the algorithm, despite the fact that it sometimes faces extremely large fluctuations. However, the standard deviation values of SHO typically remain low, suggesting that the method adheres to this path. Although SHO has the potential to be effective, especially in difficult optimization situations, it does not always deliver results that are dependable and consistent.

In conclusion, as compared to GA and SHO, PSO often yields outcomes that are superior in terms of consistency and stability. Although SHO has the ability to deliver the highest results, it may sometimes create high variances and results that are not very good. There is a possibility that GA will have a lower level of dependability owing to being trapped in local minima, but it may provide excellent results in huge solution spaces. Because PSO is the algorithm that yields the most consistent and balanced results, it is the algorithm of choice for applications that need dependability and stability. On the other hand, SHO could be more suited for situations that call for a greater amount of exploration and variety in broad solution areas; yet, it might also provide outcomes that are vulnerable to instability and danger.

The SHO algorithm has computational complexity, like any other population-based metaheuristics (PSO and GA). The SHO algorithm is linear in the number of inverter levels since every switching angle is a decision variable and thus the algorithm could solve higher-dimensional problems effectively. Stochastic search schemes like Levy flight and Brownian motion increase the quantitative stability and sturdiness of the algorithm by guaranteeing variety in population and avoiding untimely convergence. Just like any other metaheuristic, SHO algorithm can take longer to compute very high dimensional problems or very large inverter levels. The selection of algorithm parameters and population size has an effect on the convergence performance (see Table 7 and Figs. 7, 8).

The Sea-Horse algorithm has been chosen in this work as it is the most reliable, easy to implement, and perfectly compatible with the MATLAB/Simulink environment. All the parameters of the algorithm such as population size, the number of iterations, as well as control settings, are clearly described so that an independent replication of the results can be carried out. Table 4 specifications are clarified with the convergence performance; also, simulation details are given in section “Results and discussion” for transparency and reproducibility.

Results and discussion

Simulations are carried out on a personal notebook equipped with an AMD Ryzen 9 7845HX (3.00 GHz) processor, an NVIDIA GeForce RTX 4060 8GB graphics card, 32 GB of RAM, and a 64-bit Windows 11 operating system¹. MATLAB 2020b is used for the simulations. The performance of the SHO algorithm is assessed and compared with that of PSO and GA, using a 9-level T-type Reduced-Switch-Count Multilevel Inverter (RSC-MLI). For a thorough comparison, the modulation indices are systematically tested across the range from 0.1 to 1.0, with a step size of 0.1.

The model of the 9-level T-type RSC-MLI system, modelled using MATLAB/Simulink, is shown in Fig. 9. The subsystem model for phase A is presented in Fig. 10. The switching angles calculated using GA, PSO, and SHO algorithms, along with simulation results, are presented in Tables 8, 9 and 10. The performance of GA, PSO, and SHO optimization algorithms is evaluated considering the THD and THDe values in the output voltage, along with the voltage control error.

Fig. 9.

Simulink model for 9-level T-type RSC-MLI.

Fig. 10.

Subsystem for phase A.

Table 8.

Simulation results of GA for 9-level T-Type RSC-MLI.

M					Vref(max)	V1p(max)	Error (%)	THD%	THDe%	5th	7th	11th
0.1	79.711	83.984	88.372	89.873	20	19.96	0.20	153.56	139.09	91.58	83.76	62.8
0.2	59.266	86.443	87.261	89.572	40	39.88	0.30	58.17	35.27	32.68	5.13	12.23
0.3	32.997	84.807	89.579	89.66	60	59.82	0.30	27.01	22.29	9.88	19.97	0.2
0.4	53.798	59.46	82.201	88.734	80	79.8	0.25	29.11	22.48	18.84	6.23	10.57
0.5	42.763	53.843	81.213	84.615	100	99.67	0.33	21.83	19.78	3.86	0.34	16.33
0.6	41.024	42.463	73.007	84.265	120	119.6	0.33	16.57	8.39	2.94	5.86	5.23
0.7	37.199	55.483	56.113	73.832	140	139.5	0.36	14.11	4.98	2.8	4.11	0.22
0.8	28.259	47.357	53.918	68.523	160	159.5	0.31	8.08	3.71	2.99	2.2	0.21
0.9	7.659	29.522	41.634	77.361	180	179.4	0.33	7.16	4.76	0.36	4.73	0.39
1	10.054	21.973	40.624	61.93	200	199.3	0.35	6.41	0.19	0.11	0.15	0.04

Table 9.

Simulation results of PSO for 9-level T-Type RSC-MLI.

M					Vref(max)	V1p(max)	Error (%)	THD%	THDe%	5th	7th	11th
0.1	71.690	90.000	90.000	90.000	20	19.950	0.25	81.51	73.97	63.88	35.74	10.65
0.2	54.711	87.099	90.000	90.000	40	39.970	0.08	35.56	21.96	9.99	12.98	14.62
0.3	43.080	77.756	90.000	90.000	60	59.870	0.22	16.57	13.01	1.27	7.25	10.72
0.4	41.523	63.147	86.776	90.000	80	79.740	0.33	17.30	2.59	1.69	1.26	1.51
0.5	38.627	56.196	76.513	90.000	100	99.680	0.32	12.89	2.91	1.79	1.43	1.80
0.6	37.634	50.374	67.421	85.911	120	119.600	0.33	9.41	0.89	0.31	0.19	0.81
0.7	37.306	46.854	61.611	75.855	140	139.600	0.29	8.32	0.35	0.19	0.16	0.25
0.8	24.700	45.531	57.040	68.889	160	159.500	0.31	7.29	0.04	0.02	0.02	0.03
0.9	12.550	34.239	48.879	68.471	180	179.400	0.33	8.28	0.19	0.07	0.04	0.17
1	10.009	22.126	40.731	61.792	200	199.300	0.35	6.27	0.04	0.02	0.02	0.01

Table 10.

Simulation results of SHO for 9-level T-Type RSC-MLI.

M					Vref(max)	V1p(max)	Error (%)	THD%	THDe%	5th	7th	11th
0.1	71.643	90.000	90.000	90.000	20	19.98	0.10	81.34	73.78	63.65	35.76	10.66
0.2	54.680	87.079	90.000	90.000	40	39.94	0.15	35.40	21.99	9.93	13.01	14.69
0.3	43.138	77.660	90.000	90.000	60	59.93	0.12	16.49	12.94	1.47	7.11	10.70
0.4	41.482	63.119	87.993	88.782	80	79.87	0.16	19.61	2.43	1.69	1.11	1.35
0.5	38.792	56.164	76.342	89.999	100	99.81	0.19	12.86	2.93	1.93	1.21	1.85
0.6	36.877	51.089	67.026	86.091	120	119.8	0.17	8.92	0.07	0.01	0.02	0.01
0.7	35.706	48.138	60.772	76.246	140	140	0.00	8.03	0.07	0.02	0.02	0.00
0.8	24.480	45.321	57.045	68.757	160	160	0.00	7.29	0.05	0.02	0.01	0.03
0.9	12.963	33.712	49.369	67.708	180	180	0.00	7.75	0.35	0.16	0.08	0.30
1	0.175	0.386	0.711	1.078	200	199.40	0.30	6.21	0.03	0.02	0.02	0.02

In the proposed 9-level T-type RSC-MLI system, a DC source balancing strategy is implemented to maintain equal voltage across all DC links, ensuring stable operation and minimizing circulating currents. The ideal DC sources of each DC link are represented in Simulink as DC voltage sources, and capacitors (4700 F) are added to the DC-link input to represent realistically the voltage smoothing and energy storage at the DC input to the inverter. Smooth balancing helps to avoid the overvoltage stress on individual devices and helps to increase the reliability of inverters.

The full specifications of the simulation are as follows: The ODE45 solver with a fixed-step sampling time of 1 s is used. Such parameters provide real and reproducible simulation outcomes, enabling the reader to gain a comprehensive understanding of the behavior of the system, control strategy, and design assumptions. DC link voltages are 100 V. The load is a three-phase R-L load (,  mH) and an output LC filter ( mH, F) is also added to minimize voltage ripple.

During the polarity generation phase (H-bridge structure), opposing gating signals are used and applied to the upper and bottom switches of each leg to avoid simultaneous conduction. The explicit dead-time physical modeling is not present since there is no explicit dead-time in the MATLAB/Simulink environment because ideal semiconductor switches are utilized. The applied control logic, however, makes sure that opposite switches in the same leg never get triggered at the same moment and prevents shoot-through conditions. In real-world applications of hardware, some kind of dead-time is required to avoid short-circuit faults across the DC link.

As the modulation index increases, the findings indicate that the GA algorithm causes the THD and THDe values to improve. The obtained results also indicate that the error in the output voltage varies between 0.20% and 0.36%. The SHO algorithm, however, reduces the values of the true harmonic distortion (THD) and total harmonic distortion (THDe) to a minimum. This keeps the voltage control error very near zero. PSO algorithm also performs well with increased modulation index, which is indicated by a large reduction in the values of THD and THDe.

Overall, the SHO algorithm helps to simplify everything, reduces the values of THD and THDe, and allows adjusting the output voltage in a more predictable manner. The findings of the simulation prove the effectiveness of the SHO algorithm in harmonic removal. At , SHO has a THD of 6.21% and THDe of 0.03, which is much better than GA (0.19%), indicating that SHO has much better performance in reducing harmonic distortion. Another advantage is that it is also more functional. All three algorithms ensure the output voltage remains constant, and the error in voltage control is minimal. Nevertheless, SHO has the lowest voltage control errors. SHO algorithm is effective at high levels of modulation in reducing the values of THD and THDe. Algorithms PSO and GA, however, perform quite well at mid-levels of modulation. Such algorithms can be optimal in programs that require high-resolution voltage control and contain significant harmonic distortion.

Table 11 demonstrates the performance of GA, PSO and SHO algorithms with varying modulation indices. Table 11 indicates the relationship between the best, the worst, and the standard deviation (Std) values of each of the three methods to each other. There is a varying performance of the respective algorithms that increases with the modulation index. Of particular interest is that the SHO algorithm is highly efficient and the variance is very low. GA is very variable at low modulation levels, but with medium modulation levels PSO can obtain good results. It is generally believed that the SHO algorithm is superior to the other algorithms due to low values of THD and THDe or its use is consistent. The SHO algorithm ensures that the difference between the optimal and the worst readings at the beginning of and at the end of the process is not large even with a high modulation intensity. Because of this, the standard deviation statistics are very low. SHO’s results are much more stable and have fewer errors than GA and PSO’s results, which are more variable, especially when modulation levels are high.

Table 11.

Statistical analysis of all algorithms.

M	GA			PSO			SHO
	Best	Worst	Std	Best	Worst	Std	Best	Worst	Std
0.1	6.49E+02	6.50E+04	1.17E+02	2.18E+02	8.39E+04	0.00E+00	2.18E+02	1.22E+05	2.14E–12
0.2	2.81E+02	7.23E+04	4.60E+02	7.66E+01	1.14E+05	4.47E+04	7.66E+01	1.28E+05	2.13E–01
0.3	2.76E+02	4.43E+04	3.16E+02	6.09E+01	5.80E+04	1.28E–14	6.09E+01	4.70E+04	4.57E+00
0.4	3.92E+01	4.05E+04	1.18E+02	4.26E+00	1.03E+05	4.47E+04	3.95E+00	1.01E+05	3.77E+01
0.5	7.41E+01	4.11E+04	1.12E+02	8.30E+00	1.01E+05	1.92E–06	8.32E+00	1.05E+05	5.50E+01
0.6	1.15E+02	2.41E+04	1.07E+02	4.31E–29	2.45E+04	3.88E–06	4.31E–29	2.45E+04	3.88E–06
0.7	4.21E+01	1.41E+04	1.29E+02	3.66E–29	3.00E+04	5.09E–01	3.66E–29	3.00E+04	5.09E–01
0.8	7.55E+01	1.00E+05	8.43E+01	6.51E–30	3.62E+04	6.07E+00	6.51E–30	3.62E+04	6.07E+00
0.9	3.57E+01	8.09E+03	2.99E+01	1.19E–01	1.61E+04	1.07E+01	1.19E–01	1.61E+04	1.07E+01
1.0	8.02E+01	1.00E+05	1.73E+02	4.26E–29	1.01E+05	0.00E+00	4.26E–29	1.01E+05	0.00E+00

The SHO method works better for optimization and gives more consistent results as the modulation index goes up. This is due to the aforementioned reason. Both PSO and GA can give good results at medium levels, but they can’t compete with SHO when it comes to errors and stability. When harmonic distortion is very important and it is needed to change the output voltage very precisely, SHO is the best algorithm to use. In conclusion, SHO is the best algorithm to use. Table 12 indicates the number of function calls and the time required to search the optimum value by GA, PSO and SHO algorithms. The GA algorithm makes 2.25 run-time calls, calculated on average as 2.27381306 calls. This implies that the GA algorithm must execute more times, which implies that this algorithm is more expensive to execute. PSO algorithm is more effective since it invokes an average of 1.18 function calls per run, and the average function calls are 1.18089564. This demonstrates that PSO achieves the best value faster and has lesser time periods.

Table 12.

The number of function calls to reach the optimum value and computation time.

Algorithm	Run 1	Run 2	Run 3	Run 4	Run 5	AVG
GA	2.2854976	2.2711851	2.2786764	2.273858	2.2598482	2.27381306
PSO	1.1854933	1.1856388	1.1806828	1.1814692	1.1712137	1.18089956
SHO	0.4488576	0.3242404	0.3091586	0.3263483	0.325701	0.34686118

Favorable algorithm is SHO. The mean number of calls is 0.34686118 for SHO, comparatively very low compared to the other two algorithms. SHO also has the shortest computation time, averaging at 0.34686118. This demonstrates that SHO is the quickest and the most effective algorithm in terms of invoking functions as well as the computation time. Due to that reason, the results produced by the SHO algorithm are quicker and more practical than the results of GA and PSO. Optimization performance of the 9-level T-type RSC-MLI is indicated over the range of modulation indexes 0.1 to 1.0. Figure 11 depicts the unit modulation output voltage waveform. Figures 12 and 13 indicate the values of THD (Total Harmonic Distortion) and THDe (Total Harmonic Elimination), respectively.

Fig. 11.

Output waveforms of the 9-level T-Type RSC-MLI at : (a) GA, (b) PSO, (c) SHO.

Fig. 12.

THD analysis of the 9-level T-Type RSC-MLI at : (a) GA, (b) PSO, (c) SHO.

Fig. 13.

THDe analysis of the 9-level T-Type RSC-MLI at : (a) GA, (b) PSO, (c) SHO.

Figures 11, 12, and 13 clearly indicate that the SHO algorithm has the lowest peak-to-peak voltage error and harmonic distortion, but at medium and low modulation indices, GA has greater variations. PSO offers compromise performance, implying a compromise between the speed of convergence and harmonic elimination. For the modulation index (M) of 1.0, the algorithms could be compared as follows, and other modulation indices can be evaluated correlated with algorithm’s performance results at 1.0 M index:

GA (Genetic Algorithm): The system’s voltage error is 0.35%, with a peak voltage of 199.30 V and an RMS voltage of 140.9 V. The Total Harmonic Distortion (THD) is 6.41%, and the Total Harmonic Elimination (THDe) is 0.19%. The 5th, 7th, and 11th harmonics had distortions of 0.11%, 0.15%, and 0.04%, respectively.

PSO (Particle Swarm Optimization): The system had a voltage error of 0.35%, with a peak voltage of 199.30 V and an RMS voltage of 141 V. The THD is 6.27%, and the THDe is 0.04%. The 5th, 7th, and 11th harmonics had distortions of 0.02%, 0.02%, and 0.01%, respectively.

SHO (Sea-Horse Optimization): The system had a voltage error of 0.35%, with a peak voltage of 199.40 V and an RMS voltage of 141 V. The THD is 6.21%, and the THDe is 0.03%. For the 5th, 7th, and 11th harmonics, the distortions are 0.02%, 0.02%, and 0.02%, respectively.

Increasing the modulation index lowers the best and worst values for GA, which makes the results more stable. The worst and standard deviation values of PSO are lower than those of GA, and PSO produces better results, especially when the modulation levels are higher. High values of THD are recorded at low modulation indices (M 0.2) because of the lack of sufficient voltage levels to provide an accurate approximation of the sinusoidal waveform. For experimental implementation, this could lead to more harmonics on loads because of low modulation index, which could be avoided with extra filtering.

The SHO algorithm has much lower worst and standard values than either GA or PSO, which means that it works much more consistently. The SHO algorithm works very well, with almost no convergence failures or fluctuations, especially when modulation levels are low. SHO also works better at higher modulation levels, showing the lowest THD and THDe values. Both PSO and GA work well at medium levels, but the SHO method always provides results that are more accurate and useful. It can be demonstrated that the SHO algorithm gives the best results because it is very stable and has very little volatility. The PSO and GA algorithms, on the other hand, are less predictable and take longer to solve SHE equations.

All presented results indicate that SHO provides the lowest THD and THDe values while maintaining stable voltage control, especially at high modulation indices, compared to GA and PSO. Table 13 shows that the GA, PSO, and SHO algorithms work very well with a wide range of modulation indices. The GA method has worse values than other algorithms, especially when modulation levels are low. Because of this, more calculations and iterations are needed.

Table 13.

Statistical analysis of algorithms.

m	GA			PSO			SHO
	Best	Worst	Std	Best	Worst	Std	Best	Worst	Std
0.1	6.49E+02	6.50E+04	1.17E+02	2.18E+02	8.39E+04	0.00E+00	2.18E+02	1.22E+05	2.14E–12
0.2	2.81E+02	7.23E+04	4.60E+02	7.66E+01	1.14E+05	4.47E+04	7.66E+01	1.28E+05	2.13E–01
0.3	2.76E+02	4.43E+04	3.16E+02	6.09E+01	5.80E+04	1.28E–14	6.09E+01	4.70E+04	4.57E+00
0.4	3.92E+01	4.05E+04	1.18E+02	4.26E+00	1.03E+05	4.47E+04	3.95E+00	1.01E+05	3.77E+01
0.5	7.41E+01	4.11E+04	1.12E+02	8.30E+00	1.01E+05	1.92E–06	8.32E+00	1.05E+05	5.50E+01
0.6	1.15E+02	2.41E+04	1.07E+02	4.31E–29	2.45E+04	3.88E–06	4.31E–29	2.45E+04	3.88E–06
0.7	4.21E+01	1.41E+04	1.29E+02	3.66E–29	3.00E+04	5.09E–01	3.66E–29	3.00E+04	5.09E–01
0.8	7.55E+01	1.00E+05	8.43E+01	6.51E–30	3.62E+04	6.07E+00	6.51E–30	3.62E+04	6.07E+00
0.9	3.57E+01	8.09E+03	2.99E+01	1.19E–01	1.61E+04	1.07E+01	1.19E–01	1.61E+04	1.07E+01
1.0	8.02E+01	1.00E+05	1.73E+02	4.26E–29	1.01E+05	0.00E+00	4.26E–29	1.01E+05	0.00E+00

In order to place the contribution of the proposed method in the existing literature, a comparison was drawn with recent results of 9-level and other related multilevel cascaded H-bridge inverters based on multicarrier PWM techniques; the researchers reported rather high values of the THD³⁹. The literature also included optimization-based methods⁴⁰, testing GA, PSO, GOA, and GWO algorithms on a 9-level cascaded H-bridge inverter, with the GWO algorithm performing best. Findings of a 9-level inverter with multicarrier PWM techniques are compared by³⁹ and reported minimum THD of around 11.43% with average THD scores of compared studies around 14.5%. Reference⁴⁰ tested GA, PSO, GOA, and GWO on a 9-level CHB inverter, with THD about 6.69% (GWO) at 0.8 modulation index.

Comparatively, the proposed SHO-based SHE PWM approach for a reduced switch-count T-type 9-level inverter delivers a THD of 6.21% featuring a THDe of 0.03%, which is superior in harmonic suppression, better voltage control, and far more consistent even at high modulation index. Convergence rate is platform-dependent; hence, no direct quantitative comparisons of convergence among studies are described. Even at low modulation indices, quantitative validation versus recent literature standards shows that SHO exhibits lower THD than similar GA or PSO implementations, though absolute harmonic levels are high and must be mitigated in practical design.

Conclusion

This paper has optimized reduced-switch-count 9-level T-type multilevel inverters with the use of the Sea-Horse Optimization (SHO) algorithm and compared its performance with that of the Genetic Algorithm (GA) and Particle Swarm Optimization (PSO). The findings show that SHO is superior, especially at high modulation indices, in reducing harmonic distortion.

SHO offers a better convergence speed than GA and PSO, with more stable output voltage control, with lower THD and THDe values. Moreover, SHO works more efficiently and requires fewer function calls, which dramatically lessens the time of computation. These characteristics suit SHO to serve as an ideal solution with high efficiency and low harmonic distortion in both industrial and renewable energy.

This paper demonstrates that SHO is a high-performance algorithm to optimize T-type multilevel inverters and provides more cost-effective solutions in comparison with other optimization strategies. Future work will be directed at the further development of SHO for new inverter topologies and areas of use. Also, considerations of hybrid methodologies between SHO and other optimization strategies, e.g., Genetic Algorithm (GA) or Particle Swarm Optimization (PSO), may be discussed to achieve better performance by providing the advantages of both strategies. Hybrid algorithms can be more adaptable to complex problems, converge faster, and be more accurate in optimization, and thus are good candidates to take forward in future developments in multilevel inverter optimization.

The proposed SHO-based SHE PWM method reduces THD to 6.21% and THDe to 0.03% at , outperforming GA and PSO in both stability and computational efficiency. Limitations include higher harmonic levels at very low modulation indices and the idealized modeling of semiconductor switches without explicit dead-time. The result of the study, the effectiveness of the Sea-Horse Optimization (SHO) algorithm, may be also confirmed with the help of the hardware-in-the-loop (HIL) testing in order to address the gap between the simulation and the reality. Moreover, the exploration-exploitation ratio adaptive-controlling could be considered as a way to improve the performance of the convergence. The research of hybrid optimization methods and their implementation to larger, multi-objective systems is also an active area of research of the extension of applicability and resilience of the algorithm.

Despite the fact that the current study is not designed to be experimentally validated, the MATLAB/Simulink model offers an elaborate and dependable framework to analyze the performance of the suggested optimization-based selective harmonic elimination strategy. The next step in the research will be the real-time application of the proposed approach with the help of DSP or FPGA-based inverter control platforms.

Author contributions

Abadal-Salam T. Hussain: The author conceived and planned the study. Taha Almulaisi (Corresponding Author): Conceived the research study and prepared and finalised the manuscript. Hazry Desa: Contributed analysis tools. Enes Bektaş: Contributed analysis tools. Ravi Sekhar Provided analytical tools and reviewed the manuscript. Pritesh Shah: Contributed analysis tools and reviewed/corrected the manuscript. Ahmed Dheyaa Radhi: Provided analytical tools. Noor N. Abed: Conceived part of the research study. S. B. Yaakob: Contributed analysis tools.

Funding

Open access funding provided by Symbiosis International (Deemed University).

Data availability

This published article includes all the data generated or analysed during this study. Additional data can be made available from the corresponding author upon reasonable request.

Competing interests

The authors declare no competing interests.