Analyzing Impacts of Amplitude Modulation Index on Performance Characteristics of Three-Phase Nine-Level Modular Multilevel Converters

Abstract

Modular multilevel converters (MMCs) represent the forefront of power converter technology, with wide-ranging applications across diverse industries. Over recent decades, substantial research and development efforts have been dedicated to enhancing MMCs’ performance. A significant challenge in power conversion processes is the presence of total harmonic distortion (THD) in output waveforms, which can have adverse effects on electrical equipment. In response, extensive studies have been conducted to address THD-related challenges by refining the control and operation of MMCs. This study investigates the effect of the amplitude modulation index (Ma) on the total harmonic distortion (THD) in nine-level MMC output waves. For this, a standard three-phase and nine-level MMC model was built and simulated in MATLAB/ Simulink environment, and the Ma value was shifted between 0.1 and 1.5. The output current and voltage waves were analyzed, and the optimal limits for the Ma values yielding the lowest THD values were determined. The simulation outcomes reveal a crucial Ma range between 0.6 and 1.2, where THD is significantly minimized. Ma values below 0.6 introduce significant harmonic distortion in the voltage waves, while values surpassing 1.2 lead to appreciable harmonic distortion in the current wave. This study contributes valuable insights for engineers and researchers and aids in the refinement of MMC control strategies and the mitigation of THD-related challenges in power systems.

1. Introduction

Multilevel converters have gained significant attention in industry and power systems as preferred solutions for energy conversion applications [1,2,3,4,5]. Their widespread use is attributed to their ability to power various applications, including milling machines, gas turbine starters, wind energy conversion, HVDC transmission, conveyors, fans, pumps, and compressors, among others [3,6,7,8,9]. High-power multilevel converters offer numerous advantages over conventional counterparts, such as reduced harmonics, lower voltage fluctuations, integration of durable electronic circuit breakers, more compact and cost-effective I/O filters, and higher efficiency [2,6,10,11,12]. These advantages have spurred industrial and academic research centers to develop commercial modular multilevel converter (MMC) products for applications like motor drives, HVDC transmission, and static synchronous compensators (STATCOM). Several companies are actively engaged in the commercialization of MMCs; compact modular versions are now available, featuring operational voltages ranging from 2.3 kV to 4.16 kV [8].

Research literature emphasizes the growing significance of MMCs in high-energy applications [2,6,8,13,14]. The inception of MMCs dates back to R. Marquardt’s proposal in 2001, where a DC-to-three-phase converter structure was introduced that sequentially connected sub-modules (SMs), arm inductors, and half/full bridge SMs [15]. Recent publications showcase diverse MMC configurations, spanning from single-phase to single-phase [16] and single-phase to three-phase [17], enabling the transformation of low-frequency voltage into the medium-frequency voltage needed for machine motor operation. Fundamental structures of single-phase MMCs with basic element control methods are presented, adaptable for use in three-phase MMCs [18]. Subsequent publications verify the accuracy of the proposed MMC structure and control methods [19]. Substantial efforts have since been dedicated to developing and enhancing MMC performance, covering theoretical models for steady-state analysis, simulation techniques to streamline computations, modulation methods for improving output voltage and current quality, DC and AC element control approaches, and efficiency and reliability studies. A particularly active area of research centers on methods that can enhance MMC output quality and performance [15].

Total harmonic distortion (THD) is a critical factor affecting the quality of current and voltage waveforms in modular multilevel converters (MMCs). THD is precisely defined as the ratio of the sum of the powers of all harmonic components to the power of the fundamental frequency [20,21]. It serves as a vital indicator widely employed to evaluate the output power quality of MMCs [21,22]. The application of MMCs in any context necessitates the implementation of a control methodology to effectively address THD concerns and attain high-quality output signals [23]. Various control techniques are available for MMCs, with pulse width modulation (PWM) technology being one of the most prevalent choices. PWM technology is favored for its ability to regulate the voltages of sub-module (SM) capacitors, thereby ensuring the generation of high-quality output signals [24]. Among PWM-based control techniques, the amplitude modulation index emerges as a pivotal parameter. This index quantifies the ratio between the amplitudes of the reference signal (typically a 50 Hz sinusoidal signal) and the carrier signal (usually a triangular signal). It plays a fundamental role in governing the PWM signal’s output and serves as the basis for generating triggering pulses for the converter switches. The amplitude modulation index directly determines the pulse width of the average voltage over one period, with larger values resulting in wider pulse widths [21]. Studies conducted by N. Ismail et al. [21], as well as Kwang and Masri [25], have explored the impact of the amplitude modulation index on the current and voltage waveforms of single-phase MMCs. Their findings indicate a clear trend: an increase in the amplitude modulation index corresponds to a reduction in THD. For instance, with an amplitude modulation index of 1.4, the THD values were observed to be 27.06% in the current waveform and 27.08% in the voltage waveform [21]. Similarly, investigations involving three modulation index values (0.6, 0.9, and 1) revealed that the minimum THD value for both current and voltage waves aligns with a modulation index value of 1 [24]. These results underscore the importance of the amplitude modulation index in controlling THD in MMCs and provide valuable insights for optimizing their performance.

However, a critical research gap still exists within this domain. The literature, while comprehensive in covering MMCs’ advantages and applications, lacks a comprehensive investigation into Ma’s impact on THD in nine-level MMCs. The Ma parameter is pivotal in determining the quality of output voltage and current waves, yet this crucial aspect remains unexplored. This research gap is of paramount significance in the field of power generation and conversion as understanding and optimizing Ma’s role can lead to enhanced performance, improved efficiency, and higher-quality output signals.

This study seeks to address this significant research gap by making valuable contributions to the field of power generation and conversion. Our research aims to thoroughly investigate the impact of the amplitude modulation index (Ma) on the performance of three-phase nine-level modular multilevel converters (MMCs). Our contributions can be summarized as follows:

Clarifying the Impact: We elucidate the direct influence of the amplitude modulation index (Ma) on the total harmonic distortion (THD) in the output waves of nine-level MMCs, providing a comprehensive understanding of this critical parameter.

Optimization: By conducting extensive simulations and analysis, we identify the optimal Ma values that minimize THD, thus enabling engineers and researchers to fine-tune MMC performance for various applications.

Enhancing Quality: Our findings contribute to enhancing the quality of output voltage and current waves in MMCs, which is crucial for improving the overall performance of electrical systems.

Overall, our research sheds light on the practical implications of Ma in the context of MMCs, offering valuable guidance for engineers and researchers working in the field of power generation and conversion.

To fulfill these research objectives, a model of the nine-level MMC was built and simulated using Simulink/MATLAB, with the Ma value ranging from 0.1 to 1.5. The analysis focused on the output current and voltage waves, assessing them based on the total harmonic distortion (THD) value. The study subsequently determined the optimal amplitude modulation index values that yield the lowest THD values.

2. Typical Three-Phase MMC Topology

A three-phase MMC can be achieved by connecting multiple modules in a cascading arrangement, forming a robust and efficient topology for energy conversion. The typical circuit of the three-phase MMC is shown in Figure 1. The input power is DC, while the output power is AC. Each branch in the MMC consists of an upper arm (positive arm) and a lower arm (negative arm). Each arm has a group of SMs that are connected through arm inductors. The arm inductors help limit the surge current resulting from the instantaneous voltage difference between the arms and mitigate high-frequency components [26]. Their selection is contingent upon factors such as the voltage of the SMs, the switching frequency, and the modulation technique employed [27]. Generally, IGBT transistors are used as control switches in the SMs due to their ability to withstand high voltages, pass high currents, and operate at relatively high frequencies. The upper IGBT (S1) and the lower IGBT (S2) in the SM are designed to complement each other.

The three-phase MMC configuration offers distinct advantages over single-phase alternatives, making it an ideal choice for various industrial and power system applications. Notably, its ability to provide precise and reliable AC output power positions it as a preferred choice. The three-phase MMC excels in terms of lower harmonics, improved voltage stability, and enhanced efficiency. These inherent advantages make it suitable for powering a wide range of applications [3]. The integration of durable and cost-effective electronic circuit breakers, the ability to use less bulky and less expensive I/O filters, and the achievement of higher efficiency further enhance its appeal [2,6,10]. The versatility of MMCs is highlighted by their adaptability to diverse applications and working voltage requirements. MMCs exhibit excellent scalability, allowing them to adapt to a wide range of voltage and power levels. The design and performance of MMCs change significantly when scaling up or down, affecting component requirements, topology, control complexity, losses, and suitability for different applications. The choice of scaling should align with the unique demands of the specific MMC application. For instance, in applications involving motor drives with voltage ratings ranging from 3.3 to 13.8 kV, the choice of 5 to 20 SMs per arm is made based on specific voltage requirements, with lower ratings typically necessitating fewer SMs and higher ratings requiring a greater number. This decision is influenced by factors such as the desired voltage quality, motor power, and system constraints. In contrast, for high-voltage DC (HVDC) power transmission applications, the MMC configuration may comprise between 200 and 400 SMs per arm, enabling the achievement of operating voltages as high as 320 kV [28]. In such high-voltage applications, a significantly larger number of SMs is needed to accommodate the elevated voltage levels and ensure reliable power transmission.

Furthermore, MMCs can be tailored to specific application demands by adopting various submodule configurations. Among these, the half-bridge (HB) configuration is one of the most prevalent choices, where each submodule consists of two IGBT transistors and a capacitor [2,29,30]. The function of the MMC is related to the charging and discharging of the capacitor voltage, which in turn depends on the direction of current flow into or out of the SM, as depicted in the simplified single-phase equivalent circuit illustrated in Figure 1b. Here, the submodule is represented as a voltage source, and each arm encompasses a cascade of N voltage sources. Here, Vdc and 𝒾dc represent the dc-link voltage and dc-link current, respectively. Symbol ‘L’ indicates the inductor arm, while ‘Vx’ and ‘𝒾x’ symbolize the converter’s output voltage and output current. 𝒾1 and 𝒾2 are the upper arm current and the lower arm current, respectively [27,31].

Table 1. Switching states of the HB-SM.

Mode	S1	S2	D1	D2	𝒾x	State	Vc	VX
1	1	0	1	0	>0	On	Vc↑ (charging)	Vc
2	1	0	0	0	<0	On	Vc↓ (discharging)	Vc
3	0	1	0	0	>0	Off	Vc ≈ (uncharged)	0
4	0	1	0	1	<0	Off	Vc ≈ (uncharged)	0

presents the fluctuations in output voltage and capacitor voltage across different switching states using the half-bridge (HB) submodule (SM) configuration. Each SM has two operating states, namely “ON” and “OFF.” In the “ON” state, the SM produces an output voltage of Vc, whereas in the “OFF” state, the output voltage is 0. During the “ON” state, the capacitor either charges or discharges, contingent upon the direction of 𝒾x (current through the SM). For instance, if 𝒾x > 0, the capacitor will charge, whereas if 𝒾x < 0, it will discharge. The capacitor voltage remains constant during the “OFF” state. The MMCs based on HB circuits are simple, easily controlled, and have low losses compared to other types [29,32]. Accordingly, this configuration will be used to model the MMC in this study.

Notably, the three-phase MMC’s superiority lies in its ability to provide precise and reliable AC output power. The inherent advantages of the three-phase configuration, including lower harmonics, improved voltage stability, and enhanced efficiency, position it as a preferred choice for numerous energy conversion applications. In this study, we focus on elucidating the intricate interplay of the amplitude modulation index (Ma) on the performance of three-phase nine-level modular multilevel converters (MMCs) to further enhance the understanding and application of this robust technology.

3. Technical Challenges in MMC Applications

Using MMCs in various industrial applications has brought about numerous advantages, but it has also introduced several technical challenges that must be addressed for successful implementation. These challenges are closely related to the nature of the application and the control techniques employed [33,34]. Some of the primary technical challenges associated with the use of MMCs include:

• Design Limitations: The design of MMCs is constrained by factors such as the size of arm inductors, which filter harmonics in arm currents and limit the continuous component of short-circuit currents. Similarly, the selection of SM capacitors is constrained by allowable voltage ripple. Overcoming these design limitations while maintaining optimal converter performance is a challenge.
• Pre-Charging Mechanisms: Before operation, the capacitors of SMs in MMCs need to be pre-charged to their nominal voltage. This pre-charging process introduces high transient currents that can potentially damage the converter components. Developing effective pre-charging mechanisms to mitigate these transient effects is essential.
• Capacitor Voltage Balancing: Ensuring that the voltage across each SM capacitor remains balanced is a challenge. Imbalances can lead to uneven power distribution, stress on components, and decreased converter performance.
• Total Harmonic Distortion (THD): THD is a critical factor that affects the quality of both output voltage and output current waves. Minimizing THD requires advanced control strategies and modulation techniques to mitigate harmonic components.
• Circulating Currents: Voltage differences between the upper and lower arms of MMC branches can lead to circulating currents. Although these currents do not appear in the output, they cause losses and negatively impact SM capacitor voltages.
• Fault Handling: MMCs often include standby SMs that come into play in the event of a failure in the main SMs. Integrating these standby elements smoothly into the circuit without causing high transient currents is a challenge.
• Modeling and Simulation; Accurate modeling and simulation tools are necessary for designing and analyzing MMC systems. Developing accurate models that capture the complexities of MMC behavior can be challenging.
• Component Reliability; The reliability of individual components like IGBTs and capacitors is crucial for the overall reliability of the MMC. Ensuring the longevity and durability of these components under high-stress conditions is a challenge. Component reliability and durability are paramount in MMCs, especially under high-stress conditions. Strategies such as advanced cooling, overvoltage protection, current limiting, careful component selection, condition monitoring, predictive maintenance, and quality assurance contribute to enhancing the reliability and longevity of critical components like IGBTs and capacitors [35]. These measures collectively ensure the sustained and dependable performance of MMC-based power conversion systems, even in demanding operational environments.
• High-Current Operation; In applications involving high-power MMCs, managing high currents is a challenge. Current sharing between SMs, ensuring uniform current distribution, and minimizing losses are important considerations.
• Scalability: Designing MMCs for different voltage and power levels requires scalability. Ensuring that the converter’s performance remains consistent while scaling up or down is a challenge.

While these challenges may appear as a list, they are interconnected and collectively represent the multifaceted nature of issues that arise when working with MMCs in different applications. For example, challenges such as “Design Limitations”, “Pre-Charging Mechanisms”, and “Capacitor Voltage Balancing” are all interrelated, as addressing design limitations may have implications for pre-charging mechanisms and capacitor voltage balancing. Similarly, “Total Harmonic Distortion (THD)” is affected by design choices and control techniques and, in turn, can impact “Component Reliability” and “High-Current Operation”.

The mentioned challenges constitute many research areas, as each issue can be divided into several issues to be solved. In the context of these challenges, this paper focuses on investigating the impact of the amplitude modulation index on reducing THD in a three-phase MMC. By determining optimal Ma values, high-quality output waves can be produced. This contributes to enhancing the overall performance of MMCs in various applications.

4. Modulation and Control Strategy of MMC

The voltage difference across the arm inductors arises due to the disparity between the total output voltages of the SMs in the branches and the DC link voltage (Vdc). This voltage difference results in the passing of circulating current through the arms, leading to ripples in capacitor voltages and power losses. It should be mentioned that circulating currents in MMCs can impact efficiency, stability, and voltage quality. Implementing appropriate control strategies, such as a phase disposition modulation (PD-PWM) strategy, virtual impedance control, and advanced algorithms, can effectively manage circulating currents, reduce power losses, improve system stability, and enhance the overall performance of MMC-based power conversion systems. These strategies should be tailored to the specific requirements of the MMC application to ensure optimal results. The arm voltage is equalized by a single voltage source, where its magnitude equals the total voltages of SMs in the arm [23,27,36]. According to the equivalent circuit shown in Figure 1b, the upper arm current and the lower arm current are given as:

$${𝒾}_1=\frac{{𝒾}_T}{2}+{𝒾}_C+\frac{{𝒾}_{dc}}{3} ,$$

(1)

$${𝒾}_2=\frac{{𝒾}_T}{2}-{𝒾}_C+\frac{{𝒾}_{dc}}{3} ,$$

(2)

From this, the circulating current and the output current can be calculated as follows:

$${𝒾}_C=\frac{{𝒾}_1-{𝒾}_2}{2}-\frac{{𝒾}_{dc}}{3} ,$$

(3)

$${𝒾}_T={𝒾}_1+{𝒾}_2 ,$$

(4)

According to [31], the upper and lower arm reference voltages are given as:

$$V_1^{ref}=\frac{V_{dc}}{2}-V_{inductor}-V_{xO} ,$$

(5)

$$V_2^{ref}=\frac{V_{dc}}{2}-V_{inductor}+V_{xO} ,$$

(6)

where $V_{inductor}$ and $V_{xO}$ are the voltage across the arm inductor and the output phase voltage (phase modulating signal), respectively. They are given as follows:

$$V_{inductor}=\frac{(V_{dc}-V_1-V_{2)}}{2} ,$$

(7)

$$V_{xo}=M\times \frac{V_{dc}}{2}\times \sin \left(\omega t+\phi \right) ,$$

(8)

where $M$ is the modulation index ranges $0 \le M \le 1$, $\omega$ is the supply frequency in rad/s, and $\phi$ is the initial phase angle.

According to references [27,31], the normalized reference modulating signals for both the upper and lower arm can be outlined as follows:

$$V_1^{\prime ref}=\frac{1}{2}[1-M\times \sin \left(\omega t+\phi \right)] ,$$

(9)

$$V_2^{\prime ref}=\frac{1}{2}[1+M\times \sin \left(\omega t+\phi \right)] ,$$

(10)

In this study, we employ an amplitude modulation index (Ma) adjustment for THD control. However, it’s worth noting that several other techniques exist for THD management in modular multilevel converters (MMCs), such as selective harmonic elimination (SHE), which focuses on eliminating specific harmonics in the output voltage waveform by carefully selecting the modulation indices for the MMC. SHE is renowned for its capability to precisely target and eliminate particular harmonics while preserving the desired fundamental voltage levels.

Additionally, advanced control strategies like model predictive control (MPC) and artificial intelligence (AI)-based methods are gaining prominence. These strategies leverage predictive algorithms to dynamically optimize the MMC’s switching patterns in real-time, effectively reducing THD. While these methods offer exceptional THD reduction, they may introduce increased complexity in terms of implementation compared to the more straightforward Ma adjustments.

The choice of THD control method should align with the specific requirements of the MMC application in question. While Ma adjustment provides flexibility and ease of implementation, it may not deliver the same precision as other methods, such as SHE or advanced control strategies. Each approach comes with its own set of strengths and weaknesses, and the selection should be based on the desired level of precision, complexity tolerance, and adaptability required for the particular MMC system under consideration.

The PWM technique employed to control the conventional converter can be adapted for application in MMCs [37]. Different modulation techniques, such as phase disposition modulation, sinusoidal pulse width modulation, carrier phase shift modulation, and nearest level modulation, are commonly employed to control the active devices in MMCs. Among these techniques, phase disposition modulation (PD-PWM) stands out due to its superior harmonic characteristics and easy implementation [34]. Consequently, this paper opts for the PD-PWM as the fundamental modulation approach for subsequent analysis. In the PD-PWM technique, all of the High-frequency carrier waves are synchronized in the same phase, as shown in Figure 2. The control process involves comparing n number of carrier triangular waves with a low-frequency sinusoidal wave to generate control pulses, which in turn produce an n-level staircase output voltage waveform. The carrier triangular signals are organized into “upper triangles” for segments above the zero reference point and “lower triangles” for segments below the zero reference point. The PWM strategy and control process are depicted in Figure 3. At first, the controller tracks the targeted output signal and then computes the reference signal accordingly. Subsequently, the pulse width modulator obtains the reference input to produce the digital switching signal. This signal then switches the IGBTs of the SM via the gate driver. The AC output signal of the MMC is generated by summing all the SMs output signals. The modulator boosts the digital signal into an analog voltage output, thus making the AC output approximate the reference waveform. When the modulator’s digital signal is at a high level of 1, the output from the SM becomes Vdc, as shown in Figure 1b. Consequently, the digital signals computed by the modulator hold significant importance, as they have a direct impact on the quality of the MMC output signals. The resulting signal has the same number of steps as the summed signal. The IGBT switching pulses are obtained through the decoding process of the aggregated signal in accordance with the switching sequence.

5. Modeling and Simulation Outcomes

In this study, a model of a three-phase, nine-level MMC was built, and simulations were performed using MATLAB/Simulink. Each branch of the MMC has four SMs in the upper arm and four SMs in the lower arm. Each SM contains two IGBT breakers and a capacitor. Simulations were performed at carrier triangular signal frequency $F_C=2 \mathrm{kHz}$, reference signal frequency $F_{ref}=50 \mathrm{Hz}$, and DC input voltage $V_{DC}=600 \mathrm{V}$. The choice of a carrier triangular signal frequency of 2 kHz and a reference signal frequency of 50 Hz in our MMC simulation aligns with industry standards and practical considerations for power electronics applications. These frequencies enable precise control, low harmonic distortion, and compatibility with both power grid operation and electrical loads. The amplitude modulation index varied within a range from 0.1 to 1.5, but in general, its value can be found through the following equation:

$$M_a=\frac{A_{ref}}{A_C} ,$$

(11)

$A_{ref}$: the amplitude of the reference signal.

$A_C$: the amplitude of the carrier signal.

Ma values exceeding 1 in an MMC simulation represent a state of over modulation where nonlinear behavior and increased harmonic content occur in the output voltage waveform. While this may not be the typical operating condition, understanding these implications is essential for assessing the MMC’s response to transient events, specialized applications requiring controlled harmonic injection, and rare fault scenarios. It ensures that the MMC system is capable of handling a range of conditions and remains reliable in practical situations.

Key parameters in our model include capacitance (C) values for the capacitors within each SM, selected to match the desired voltage level and power rating. Voltage ratings (V_rating) are chosen to align with the specified DC input voltage (V_DC) of 600 V. IGBT breakers, tailored to match the MMC’s operational voltage and current levels. We assume ideal components without losses in our simulation, allowing focused analysis of the amplitude modulation index (Ma) variation’s effects on MMC behavior. The simulation employs a carrier triangular signal frequency (F_C) of 2 kHz to determine IGBT switching frequencies and a reference signal frequency (F_ref) of 50 Hz to ensure compatibility with power grid operation and electrical loads. These parameters and assumptions underpin our simulation model, enabling a comprehensive analysis of Ma’s impact on MMC performance, particularly in relation to voltage and current waveforms and Total Harmonic Distortion (THD).

Figure 4 shows the output current of a single phase of the MMC at Ma = 0.1, accompanied by a Fourier analysis that reveals the harmonic content of the output current wave. Figure 5 shows the output phase voltage, while Figure 6 shows the output line voltage of the MMC, both accompanied by Fourier analyses illustrating their respective harmonic characteristics at Ma = 0.1.

As shown in Figure 4, the output current wave is an imperfect sine waveform with a total harmonic distortion of THD = 1.27%. Fourier analysis of this waveform reveals that the amplitude of the fundamental component is 2.42. Figure 5 illustrates that the phase voltage wave at the MMC’s output of THD = 225.06% exhibits substantial harmonic distortion. Additionally, it is observed that the wave lacks discernible levels, as if the MMC behaves like a conventional two-level converter. The phase voltage wave is distinctly characterized by both odd and even-order harmonics, with the odd harmonics having the most pronounced effect. It is noted from Figure 6 that the line voltage wave at the output of the MMC is also characterized by levels less than it should be, as it is assumed to be nine levels. In addition, this wave exhibits significant harmonic distortion THD = 217.01%.

From the analysis of the collected simulation results, it was observed that, when increasing the value of the Ma, the absence of levels in the output voltage persists until Ma reaches 0.5. Alongside this, the amplitude of the fundamental component of the voltage and current waves also increased. Furthermore, a decrease in the THD value was observed. The summarized outcomes of these findings are presented in

Table 2. Total harmonic distortion values, current and voltage amplitudes at the MMC output when changing Ma between 0.1 and 0.5.

Amplitude Modulation Index (Ma)		0.1	0.2	0.3	0.4	0.5
Output Phase voltage	THD%	225.06	166.12	105.1	76.4	52.21
	Amplitude (V)	31.15	58.91	88.95	118.94	149.4
Output Line voltage	THD%	217.01	145.71	94.92	66.01	38.95
	Amplitude (V)	52.11	98.8	155.11	205.9	258.1
Output Current	THD%	1.27	0.92	0.78	0.57	0.34
	Amplitude (A)	2.42	5.21	7.51	8.86	12.14

.

Starting from the threshold value of Ma = 0.6, the voltage levels in the output wave begin to manifest following the principles of the MMCs. Figure 7 illustrates the output current wave, while Figure 8 and Figure 9 illustrate the phase and line voltage waves of the investigated MMC, with Fourier analysis applied to each respective wave.

Figure 7 clearly illustrates that the current wave is a semi-pure sine wave, characterized by relatively low total harmonic distortion THD = 0.31%. Concurrently, Figure 8 demonstrates that the total harmonic distortion within the phase voltage wave is notably higher at 44.04%. The line voltage wave shown in Figure 9 has nine distinct levels and is characterized by a total harmonic distortion of THD = 24.92%. Therefore, we can infer that Ma = 0.6 represents the optimal value for minimizing the total harmonic distortion in current wave.

Continued increase of the amplitude modulation index to values beyond 0.6 revealed a gradual rise in THD values within the current wave. At Ma = 1.35, the total harmonic distortion in the current reached THD = 4.15%. This value is close to the allowable limit value according to standard values approved by IEEE. When Ma is increased to higher values, the harmonic distortion in the current wave exceeds acceptable limits. On the other hand, increasing Ma amplified the amplitudes of both current and voltage waves, which contributed to a gradual improvement in the voltage waveform. This progression is illustrated in Figure 10, Figure 11 and Figure 12, which show the output current and voltage waves of the MMC investigated at Ma = 1.35.

Although the output current waveform is periodic, it lacks a regular sinusoidal pattern (shown in Figure 10). This accounts for the notably elevated total harmonic distortion, measuring THD = 4.15%. Furthermore, it can be deduced that the dominant influential harmonic is the third harmonic, followed by the fifth harmonic exerts a significantly lesser influence. Figure 11 and Figure 12 demonstrate that increasing the modulation index decreased THD values, leading to enhanced output voltage quality of the MMC.

Table 3. Total harmonic distortion values and current and voltage amplitudes at the MMC output when changing Ma between 0.6 and 1.5.

Amplitude Modulation Index (Ma)		0.6	0.7	0.8	0.9	1	1.2	1.35	1.5
Output Phase voltage	THD%	44.04	40.9	38.19	30	24.9	21.84	20.75	21.5
	Amplitude (V)	177	208.1	237.2	281	306	325	335.9	344
Output Line voltage	THD%	24.92	26.97	29.85	27	23	18	14.61	13.01
	Amplitude (V)	309.1	360	411	486	530	563.1	583	596
Output Current	THD%	0.31	0.33	0.34	0.36	0.58	2.85	4.15	6.71
	Amplitude (A)	14.04	16.36	18.67	22.11	24.09	25.6	26.51	27.1

displays the changes in THD values and the amplitudes of fundamental components for both current and voltage at the output of the examined MMC. These changes occur as the modulation index is varied within a range from 0.6 to 1.5. It is evident that increasing in the amplitude modulation index leads to an enhancement in the amplitudes of the fundamental components and a reduction in the total harmonic distortion of the voltage waves.

6. Discussion of Results

The simulation results have highlighted the pivotal role of the amplitude modulation index in influencing the quality of the MMC output waveforms, particularly concerning curbing the total harmonic distortion value. It also plays a significant role in controlling the amplitudes of the output waves. To better explain this effect, we plotted the changes in the total harmonic distortion and the amplitudes of both current and voltage waves in the MMC output. These plots will reflect changes corresponding to the adjustments in the amplitude modulation index.

The changes in both the total harmonic distortion and the phase voltage amplitude of the analyzed MMC, according to the adjustments in the amplitude modulation index, are depicted in Figure 13. It is evident that the phase voltage amplitude changes linearly in response to variations in the amplitude modulation index in the range from 0.1 to 1. Subsequently, these changes transition to a non-linear pattern as Ma increases. The figure clearly illustrates that the total harmonic distortion of the phase voltage waves experiences a rapid decrease as Ma changes from 0.1 to 0.6. Subsequently, this reduction continues, albeit at a slower pace. Figure 14 illustrates the variations in the amplitude and the total harmonic distortion value of the line voltage wave. Similar to previous observations, a linear correlation between voltage amplitude and Ma is evident until Ma = 1. Beyond this point, the relationship becomes non-linear. Notably, the THD value of the line voltage wave demonstrates a substantial reduction as Ma transitions from 0.1 to 0.6, followed by a less pronounced change after that. Changes in both amplitude and total harmonic distortion for the current wave are depicted in Figure 15. In line with previous observations, there is a discernible linear correlation in the amplitude of the current wave as Ma varies within the range from 0.1 to 1. Regarding the THD alteration, a decline in its values is noticeable as Ma ranges from 0.1 to 0.6. However, after Ma surpasses 0.6, the THD values begin to increase once more. At Ma = 1.35, the THD value approaches the acceptable threshold as defined by international standards, set at 5%. This necessitates attention to maintain adherence to desired quality levels. It should be noted that defining THD threshold values in MMC applications is context-dependent and governed by industry standards, grid codes, load sensitivity, and the specific goals of the application. Typically, THD levels below 5% are considered acceptable in many industrial applications, while optimal THD levels vary based on the priorities of the system, such as energy efficiency, grid compatibility, or load reliability.

Building upon the preceding findings, it becomes feasible to establish optimal modulation index values for nine-level MMCs. These values can be selected within the range from 0.6 to 1.2. Notably, values below 0.6 result in the absence of voltage levels in the output voltage wave, leading to elevated total harmonic distortion. Conversely, values exceeding 1.2 yield increased total harmonic distortion in the output current wave. Worth noting is that the recorded THD value at Ma = 1.2 was 2.85%. Furthermore, these outcomes offer insights into calibrating the amplitude of both current and voltage outputs of the nine-level MMCs. This calibration is viable within the range from Ma = 0.6 to Ma = 1. The wave amplitude, and consequently its effective value, maintains a direct proportionality with Ma. Therefore, achieving the desired effective value can be achieved through a direct calibration of the modulation index.

7. Conclusions

A three-phase and nine-level MMC has been effectively modelled and simulated using MATLAB/Simulink for different amplitude modulation index values. The primary objective of conducting this analysis has been successfully met, resulting in the determination of an optimal modulation index value that minimizes the total harmonic distortion (THD). The study’s outcomes revealed that the modulation index significantly influences both the output current and output voltage amplitudes in the MMC. Additionally, it plays a crucial role in regulating the total harmonic distortion value, consequently impacting the quality of the output signals. The study demonstrated that the optimal values for the modulation index fall within the range from 0.6 to 1.2. In this range, the effective current and voltage values can be controlled by directly adjusting the modulation index, ensuring high-quality current and voltage waveforms. In addition, Ma values lower than 0.6 increase THD in the output voltage wave, while values exceeding 1.2 increase THD in the output current wave.

Our research on Ma optimization in MMCs holds significant practical implications across various real-world applications. It enhances harmonic control, improves energy efficiency, and reduces component stress, making MMC-based systems more reliable and efficient. These implications extend to renewable energy integration, HVDC transmission, industrial processes, and electric vehicle charging, among others. Our research holds the potential to make a significant impact by improving efficiency, power quality, and reliability in these sectors, ultimately contributing to the advancement of cleaner and more sustainable energy systems and industrial processes.

In our future endeavors, we are dedicated to enhancing the credibility and practical applicability of our research findings. This commitment includes the implementation of a pilot version of the system, which will validate the accuracy of our simulation results and enable real-world testing. Our plans involve constructing a physical prototype that replicates the simulation model, comprehensive data collection, performance evaluation, and an iterative improvement process to validate and enhance the accuracy of our simulation findings. This approach underscores our commitment to bridging the gap between theoretical simulations and practical MMC deployment, ultimately making a tangible and positive impact in various industrial and power system contexts.