Common-Mode Voltage Suppression of a Five-Level Converter Based on Multimode Characteristics of Selective Harmonic Elimination PWM

Abstract

The combination of five-level converters with selective harmonic elimination pulse-width modulation (SHEPWM) is a practical need in medium-voltage, high-power applications. However, how to suppress the common-mode voltage (CMV) in this case becomes a difficult problem. Although CMV suppression under high switching frequency (SF) modulations and three-level SHEPWM has been discussed in many studies, these methods are not applicable to five-level SHEPWM. This is partly because the zero-sequence voltage under SHEPWM is difficult to adjust and partly because the solution spaces of three- and five-level SHEPWM are completely different. Moreover, conventional CMV suppression in three-level SHEPWM must sacrifice the switching angles to control the zero-sequence voltage, which makes the equivalent SF increase. Therefore, in this article, we propose a novel CMV suppression method that effectively utilizes the multimode characteristics of five-level SHEPWM. Multimode characteristics refers to the output waveform containing different levels of jump patterns. Therefore, there are a large number of switching angle trajectories in five-level SHEPWM, which outputs the same fundamental voltage with different CMVs. The proposed method uses the special multimode characteristics to reduce the CMV without sacrificing the switching angles. Its effectiveness and feasibility are verified by experiments.

1. Introduction

With the penetration of electric power in various industries, medium-voltage, high-power application scenarios become increasingly extensive [1], such as renewable energy generation, AC–DC hybrid microgrids, electrified rail transportation, electric vehicles and charging stations, etc. However, conventional two-level converters are not adaptable to medium-voltage, high-power applications due to the rated voltage of the switching devices. Even three-level converters can only be used in low-to-medium-voltage applications (up to 3 kV) [2]. When an engineering application has a rated voltage between 3 kV and 10 kV, the most-suitable topology is a five-level converter represented by an active neutral-point-clamped structure, as shown in Figure 1. Without the need for an additional transformer, this kind of converter has the significant advantage of being directly connected to the distribution network (10 kV). Five-level converters based on this topology are also used commercially by international companies such as ABB Group [3]. For these reasons, it was chosen as the study object in this article.

Five-level converters only satisfy the requirement of medium voltage at the topology level, while the requirement of high power needs to be satisfied by implementing low-switching-frequency (SF) modulations. Commonly used low-SF modulations include discontinuous modulation [4], synchronized space vector modulation (SSVPWM) [5], and selective harmonic elimination pulse-width modulation (SHEPWM) [6]. These modulation strategies can effectively reduce the SF while ensuring relatively good output performance, thereby reducing switching losses, improving system efficiency, and simplifying the cooling system [7,8]. In particular, SHEPWM is able to completely eliminate the specified low-order harmonics and has received much attention for industrial applications and in academic research [9]. Therefore, we propose the application of SHEPWM to five-level converters to meet the practical needs of medium-voltage, high-power applications.

However, depending on the application scenario, five-level converters may be connected to photovoltaic or wind power as grid-connected converters or linked to medium-voltage electrical motors as power drivers, requiring that the common-mode voltage (CMV) of the converter be reduced as much as possible. CMV can also cause common-mode current (CMC), adversely affecting the insulation of the loads [10,11]. Thus, CMV suppression has always been a hot topic of research for scholars at home and abroad.

CMV suppression is a common problem for three-phase converters; therefore, valuable solutions under high SF have been proposed by a large number of scholars. Although the specific steps to achieve CMV suppression differ depending on the converter, they can be broadly classified into two categories. The first category is based on the direct analysis of space vectors. CMV under the action of different space vectors was discussed in [12], leading to the proposal of a new switching sequence. Since the new switching sequence directly discards the space vector that causes a high CMV, the output waveform at this point can significantly suppress the CMV [13]. An improved fixed-switching-frequency (FSF)-based model predictive control (MPC) method was proposed to select the control voltage vectors to reduce CMV [14]. These voltage vectors with high CMV are abandoned to limit the CMV amplitude to one-sixth of the DC-link input voltage. The second category is based on the analysis of the zero-sequence voltage in carrier modulation mode. The authors of [15,16] provided an in-depth analysis of the relationship between the zero-sequence voltage and CMV, leading to the proposal of a method for adjusting the zero-sequence voltage to suppress the CMV. Unfortunately, both method types are aimed at high SF modulations. The concept of the space vector does not exist under SHEPWM, and the zero-sequence voltage is difficult to freely regulate under SHEPWM. For a low switching frequency, the nearest zero CMV vector (NZCMVV) was proposed to suppress the CMV [17]. Unlike NLM, which uses the nearest level, NZCMVV selects two candidate levels for each phase to provide eight degrees of freedom to select the zero-CMV voltage vector. It is simple to implement and does not increase the complexity due to an increasing number of levels. However, it cannot be applied to SHEPWM and significantly reduces harmonic performance.

For a long time, SHEPWM has been regarded as an independent modulation strategy, with numerous works dedicated to exploring related techniques. A segment of research, as evidenced by studies such as [18,19], is centered on numerical algorithms designed to optimize the calculation process of the switching angles. Attempting to relax the symmetry constraints, half-wave symmetry SHEPWM was also discussed in-depth in [20,21]. Moreover, a large number of scholars aim to construct a unified mathematical model and implementation method for multilevel SHEPWM, as in [22,23]. As for the current state of research, only a limited number of studies have delved into control problems under SHEPWM. The study presented in [24] proposed an optimal control method for torque pulsation under SHEPWM, directly considering torque as a constraint. The implementation of low-frequency neutral point voltage suppression for three-level converters was detailed in [25], based on the optimal third harmonic. Since the switching angle of SHEPWM is determined by the harmonics eliminated, it is inherently weakly controlled. So, CMV suppression is difficult under SHEPWM, and only a few studies have tackled this aspect.

Drawing inspiration from the concept of zero-sequence voltage injection, a CMV suppression method was introduced for three-level converters in [26]. This approach actively incorporates the third harmonic component as a control degree of freedom into the constraint equation of SHEPWM. By combining SHEPWM and selective harmonic mitigation PWM (SHMPWM) with each other, harmonic mitigation and elimination were utilized to simultaneously control output voltage harmonic distortion and suppress CMV in [27,28]. While these methods currently represent the most-effective means of CMV suppression under SHEPWM, it is crucial to note that neither approach is deemed mature nor perfect. Firstly, the CMV suppression discussions in [26,27,28] specifically pertain to three-level SHEPWM, and transitioning from three-level to five-level SHEPWM is far from a straightforward generalization. These two modulation strategies are fundamentally different, and this distinction will be thoroughly analyzed in a subsequent section. If a similar approach is to be employed under five-level SHEPWM, extensive research is needed. Secondly, and significantly, both methods make trade-offs between harmonic elimination performance and the degrees of freedom to suppress CMV. In [26], the switching angles were sacrificed to control the third harmonic component, diminishing the low-order harmonics that SHEPWM can eliminate and resulting in poorer output performance. From another perspective, this implies an increase in the equivalent SF, contrary to the original intention of using SHEPWM. In [27,28], the limitations on the quality of harmonic elimination were relaxed, leading to inferior harmonic performance. A comparative summary of the aforementioned CMV suppression methods is provided in

Table 1. Comparison of different CMV suppression methods.

Methods	Levels Studied	Modulation	Switching Frequency	CMV	THD
voltage vector with low CMV [12,13,14]	three and multi-level	SVPWM and MPC	high	low	medium
zero-sequence voltage injection [15,16]	three level	SPWM	high	low	medium
voltage vector with zero CMV [17]	multi-level	NZCMVV	low	zero	high
$3n_{\mathrm{th}}$ harmonic elimination or mitigation [26,27,28]	three-level	SHEPWM and SHMPWM	low	low	low

.

Based on an extensive review of the existing literature and a rigorous analytical process, we propose a pioneering method for CMV suppression in five-level SHEPWM that avoids sacrificing the switching angles and decreasing quality of the elimination of the harmonics. This distinctive methodology arises from a nuanced exploration of the substantial distinctions between three- and five-level SHEPWM architectures. In the conventional three-level SHEPWM, a solitary level jump pattern prevails during the positive half-cycle, exclusively between 0 and $v_{dc}/2$. In stark contrast, the five-level SHEPWM introduces a multitude of level jump patterns, encompassing transitions such as 0 to $v_{dc}/4$, $v_{dc}/4$ to $v_{dc}/2$, $-v_{dc}/2$ to $-v_{dc}/4$, and $-v_{dc}/4$ to 0. The varied frequencies and quantities of each jump pattern give rise to multiple mode combinations, illustrating the concept of “multimode” articulated in this study. Our inquiry focuses on the meticulous analysis of the distinct multimode characteristics within five-level SHEPWM and their nuanced influence on CMV. Despite yielding an identical fundamental voltage, disparate modes exhibit markedly different switching angle trajectories, resulting in diverse CMV profiles. This scholarly exploration culminates in the identification of modes that minimize CMV under diverse operational conditions, thereby engendering a groundbreaking CMV suppression technique unique to five-level SHEPWM.

2. Illustration of Selective Harmonic Elimination Pulse-Width Modulation Technique and Common-Mode Voltage Problem for Five-Level Converter

In order to facilitate the discussion in subsequent sections, the basic principles of five-level SHEPWM and the quantitative relationship between the CMV and the $3n_{\mathrm{th}}$ harmonic are first illustrated.

2.1. Principle of Five Level Selective Harmonic Elimination Pulse-Width Modulation

Taking $v_{dc}/4$ as the base value for normalizing the level, five-level converters can output five types of levels, namely 2, 1, 0, −1, and −2. Therefore, a typical five-level SHEPWM waveform can be depicted as in Figure 2. The waveform consists of a set of rising and falling edges, which are shown by the red and blue arrows. Defining the number of switching angles in the first quarter-cycle as N, the switching angles can be expressed as ${\alpha }_{i(=1,2,3,\dots ,N)}$. Considering that the SHEPWM waveform has quarter- and half-wave symmetry, its Fourier series can be stated as in (1).

$$\begin{array}{cc}\hfill & f\left(\theta \right)=\sum _{n=1}^{+\infty }[a_{n}sin\left(n\theta \right)+b_{n}cos\left(n\theta \right)]\hfill \\ \hfill & \left\{\begin{array}{c}{a}_n=\frac{v_{dc}}{n\pi }{\displaystyle {\sum }_{i=1}^N}{(-1)}^{k_i}cos\left(n{\alpha }_i\right),(n=1,3,5,\dots )\hfill \\ a_n=0,(n=2,4,6,\dots ),b_n=0,(n=1,2,3,\dots )\hfill \end{array}\right.\hfill \end{array}$$

(1)

$\theta$ represents the phase of the fundamental voltage. There is a very special parameter $k_i$ in (1) whose value depends on whether the level jumps according to the rising or falling edge at ${\alpha }_i$. $k_i$ follow the constraint of (2).

$$k_i=\left\{\begin{array}{c}0,forrisingedge\hfill \\ 1,forfallingedge\hfill \end{array}\right.$$

(2)

In general, when the SHEPWM technique is used, we are looking for an accurate output of the fundamental voltage while eliminating the low-order harmonic components. Especially in three-phase systems, where the $3n_{\mathrm{th}}$ harmonics are automatically canceled, they are not specifically considered. Thus, the constraint equation of five-level SHEPWM can be written as in (3).

$$\left\{\begin{array}{c}{\displaystyle {\sum }_{i=1}^N}{(-1)}^{k_i}cos\left({\alpha }_i\right)=\frac{\pi }{2}m\hfill \\ {\displaystyle {\sum }_{i=1}^N}{(-1)}^{k_i}cos\left(h{\alpha }_i\right)=0\hfill \\ h=6l\pm 1,l=1,2,3,\dots ,\frac{N-1}{2}\hfill \end{array}\right.$$

(3)

where m indicates the modulation ratio that is defined as $\frac{v_{ref}}{v_{dc}/2}$. $v_{ref}$ is exactly the desired fundamental voltage. The following steps are the same as the conventional two- and three-level SHEPWM. A suitable numerical algorithm is used to solve the constraint equation of (3) and, finally, obtain the switching angles. In practical applications, these switching angles can be read using a look-up table (LUT).

2.2. Mechanism of Common-Mode Voltage Problem under Selective Harmonic Elimination Pulse-Width Modulation

Subsequently, we briefly describe the relationships among the switching states and output voltage and the corresponding CMV problem for five-level converters.

Firstly, the simplified five-level converter shown in Figure 1 contains eight fully controlled switching devices in each phase. Taking phase u as an example, the PWM signals of switching devices $S_{u1}\sim S_{u4}$ are complementary to those of switching devices $\overline{S_{u1}}\sim \overline{S_{u4}}$. Therefore, there exists a relationship among the output voltage and switching states, as shown in

Table 2. Relationship among switching states and output voltage (taking phase u as an example).

Level	2	1	1	0	0	−1	−1	−2
$S_{u1}$	on	on	on	on	off	off	off	off
$S_{u2}$	on	on	on	on	off	off	off	off
$S_{u3}$	on	on	off	off	on	off	on	off
$S_{u4}$	on	off	on	off	on	on	off	off

. For output levels of 1, 0, and −1, there are pairs of redundant switching states, respectively. This kind of hardware redundancy can be used for capacitor voltage balance. Since this content is not the area of concern in this article, it will not be expanded further.

In Figure 1, the connection point of two DC-link capacitors is denoted by the symbol o, and the neutral point of the loads is denoted by the symbol n. Therefore, the CMV can be written as $v_{on}$, which can be calculated by (4) in most topologies. Here, $v_u$, $v_v$, and $v_w$ are the three phase voltages, which are the potential difference between the output of the bridge arm to the point o.

$$v_{on}=\frac{v_u+v_v+v_w}{3}$$

(4)

According to (3), it is known that the low-order harmonics are eliminated except for the fundamental voltage and the $3n_{\mathrm{th}}$ harmonic components. Therefore, $v_u$, $v_v$, and $v_w$ can be expressed as in (5).

$$\left\{\begin{array}{c}{v}_u=a_{1}sin\left(\theta \right)+a_{3n}sin\left(3n\theta \right)\hfill \\ v_v=a_{1}sin(\theta -\frac{2\pi }{3})+a_{3n}sin\left(3n\theta \right)\hfill \\ v_w=a_{1}sin(\theta +\frac{2\pi }{3})+a_{3n}sin\left(3n\theta \right)\hfill \end{array}\right.$$

(5)

Substituting (5) into (4), $v_{on}$ can be rewritten as (6).

$$v_{on}=a_{3n}sin\left(3n\theta \right)$$

(6)

Thus, we can easily obtain a very straightforward idea of adding (6) directly to the constraint equation of SHEPWM. In this way, the calculated switching angles are naturally able to suppress the CMV. However, upon deeper consideration, we found that the problem is far more complex than we thought. First, solving for the switching angles of five-level SHEPWM is inherently difficult. There is no study that gives the switching angle trajectories for the full range of the modulation ratio. If (6) is to be added to the constraint equation, a very large number of topics still need to be investigated. Also, and more importantly, if the $3n_{\mathrm{th}}$ harmonics are to be suppressed, some switching angles need to be sacrificed. This leads to poor output performance and higher equivalent SF, defeating the original purpose of using SHEPWM. Therefore, the CMV problem under five-level SHEPWM needs further in-depth study.

3. Analysis of Multimode Characteristics in Five-Level Selective Harmonic Elimination Pulse-Width Modulation and Discussion on Their Corresponding Common-Mode Voltage

As we introduced at the end of Section 1, the essential difference between the conventional three-level SHEPWM and five-level SHEPWM is whether a large number of modes in the SHEPWM waveform can be achieved. There is only one mode for three-level SHEPWM, while multiple modes can be obtained for five-level SHEPWM. Different modes correspond to different switching angle trajectories, which, in turn, will result in a different CMV. For this reason, it is necessary to analyze the multimode characteristics of five-level SHEPWM in depth and, then, discuss the corresponding CMV. This is an important entry point for the proposed method in this article.

3.1. Multimode in Five-Level Selective Harmonic Elimination Pulse-Width Modulation

It is clear from (3) that the constraint equations are directly determined by $k_i$. However, $k_i$ is completely uncertain, which creates a significant impact on the calculation of the switching angles. In order to make a uniform process for solving the switching angles uniformly, some necessary processing needs to be conducted.

The first step is to define a new set of switching angles ${\overline{\alpha }}_{i(i=1,2,3,\dots ,N)}$, which are equal to $\pi -{\alpha }_{i(i=1,2,3,\cdots ,N)}$. Then, we can find the relationship of (7).

$$\left\{\begin{array}{c}cos\left(h{\overline{\alpha }}_i\right)=cos\left(h(\pi -{\alpha }_i)\right)=\hfill \\ cos\left(h\pi \right)cos\left(h{\alpha }_i\right)=-cos\left(h{\alpha }_i\right)\hfill \\ h=6l\pm 1,l=1,2,3,\cdots ,\frac{N-1}{2}\hfill \end{array}\right.$$

(7)

In conjunction with (3), the constraint equation can be rewritten as (8).

$$\left\{\begin{array}{c}{\displaystyle {\sum }_{i=1}^N}cos\left({\overline{\alpha }}_i\right)=\frac{\pi }{2}m\hfill \\ {\displaystyle {\sum }_{i=1}^N}cos\left(h{\overline{\alpha }}_i\right)=0\hfill \\ h=6l\pm 1,l=1,2,3,\cdots ,\frac{N-1}{2}\hfill \end{array}\right.$$

(8)

where ${\overline{\alpha }}_i$ takes values in the range of 0 to $\pi$. The observation of (8) reveals that the constraint equation no longer contains any unknown quantity except ${\overline{\alpha }}_i$ with the solution. Therefore, the second step is to solve the switching angles directly using the commonly used numerical algorithm.

The third step is to calculate the true ${\alpha }_{i(i=1,2,3,\cdots ,N)}$ and $k_i$ based on the obtained ${\overline{\alpha }}_i$ through (9).

$$[{\alpha }_i,k_i]=\left\{\begin{array}{c}[{\overline{\alpha }}_i,0],if0\le {\overline{\alpha }}_i<\pi /2\hfill \\ \pi -{\overline{\alpha }}_i,1],if\pi /2\le {\overline{\alpha }}_i<\pi \hfill \end{array}\right.$$

(9)

It is important to note that, although (9) contains all the modes of five-level SHEPWM, some of them are physically unrealizable. Therefore, a new constraint needs to be added for (8). We introduce a new state quantity to represent the rising and falling edges of the level jumps, and its relationship with $k_i$ is shown in (10).

$$s_i=\left\{\begin{array}{c}1,if{k}_i=0\hfill \\ -1,if{k}_i=1\hfill \end{array}\right.$$

(10)

The last step is to ensure that the new constraint of (11) is satisfied.

$$L_i=|\sum _{i=1}^j{s}_i|,j=1,2,3,\cdots ,N$$

(11)

Based on the above processing, a unified five-level SHEPWM formula is established. However, how to solve it to obtain the appropriate switching angle trajectory in five-level multimode is the biggest problem in current research. Numerous studies have been published on SHEPWM calculation methods, including (1) algebraic methods (AMs), (2) biological algorithms (BIAs), and (3) numerical methods (NMs). Among them, AMs have a complicated calculation, so that they are not suited to multilevel SHEPWM, and the precision of BIAs is low, while NMs have a very high convergence speed and convergence accuracy, which is conducive to the process of searching for multiple solutions at one modulation ratio. But, an appropriate initial value of the algorithm is the key to convergence performance. Both multiple random initial value searches and small range search supplements at the endpoints were used in this investigation. The specific flow of the algorithm is shown in Figure 3:

• Random search: Although the initial value calculation method of the equal area method under five-level SHEPWM has been studied, it is not suitable for the five-level multimode case. At multiple levels, there are multiple switching angle solutions from multiple modes or multiple switching angles within one mode under one modulation ratio. Fortunately, under the uniform constraint formula (8), the probability of obtaining the switching angle solution is very high with a set of random initial values from 0 to $\pi$. So, for one modulation ratio, it is necessary to carry out several random value calculation processes to find a complete switching angle solution under the modulation ratio as much as possible. In the random search phase, most angle solutions can be obtained efficiently for the full modulation ratio of each mode. At the same time, the greater the number of search (NS) times under one modulation ratio, the higher the possibility of obtaining a complete solution is, but the lower the search efficiency. A mere random search is a purposeless search, which is suitable for the first stage of the full solution space search.
• Endpoints’ replenishment: According to the continuity feature of the SHEPWM switching angle solution trajectory, there is a high probability that a solution exists near the endpoint to form a continuous solution trajectory. The endpoint is defined as the switching angle solution ($m_{ept},angl{e}_{ept}$), where the nearby angle solution, ($m_{ept}-0.01,$$angl{e}_{addL}$) or ($m_{ept}+0.01,angl{e}_{addR}$), is not found. The value of 0.01 refers to the search step size of the modulation ratio. So, a certain random disturbance in the endpoint superposition is used as the initial value for the NMs, which is as in (12).

  $$x_0=c·angle+(1-c)·sort(\pi ·rand(1,N))$$

  (12)

  where c is the parameter for controlling the magnitude of the random disturbance that takes a value between 0.95 and 1. $rand$ takes a value between 0 and 1 and sorts them in ascending order. After solving by the fsolve function, the switching angle solution that satisfies the condition is added to the angle solution set. This process is repeated until the endpoints no longer add new angular solutions. The endpoints’ replenishment phase is the process of extending and supplementing the switching angles at the endpoint until the solution trajectory is continuous.

Considering the span of space, the results are shown and illustrated here with N equal to seven as an example. Since there are multiple modes in five-level SHEPWM, a function $P_n$ is defined here to describe them, shown in (13).

$$P_n=\sum _{i=1}^N{2}^{i-1}{k}_i$$

(13)

Finally, there is a total of 17 modes that are effective for $N=7$ in the five-level inverter under $2^7$ switching edge combinations. However, the modulation ratio range ($m_{range}$) with an effective switching angle solution for partial mode is small, while the specific $m_{range}$ performance is 0.01–0.04 for mode 30, 0.58–0.61 for mode 58, 0.58 for mode 78, 0.31–0.32 for mode 88, and 0.33–0.36 for mode 99. Limited by the number of mode switches and the article length, Figure 4 gives the switching angle solution trajectories of the other 12 modes with a wide $m_{range}$. The solid trajectory represents the rising edge, while the dashed trajectory represents the falling edge. It is worth explaining that these trajectories are the most-continuous trajectories in each mode within a wide range of the effective modulation ratio, which helps with the control.

To better understand the multimode situation and the necessity of optimization based on multiple modes, two important pieces of information can be discovered and discussed: (1) No mode can have full modulation ratio switching angles, and each mode has solutions within only a portion of the modulation ratio interval, so that a combination of switching angles from multiple modes is necessary for a full or broad modulation ratio application of five-level SHEPWM. (2) More than one mode is available at one modulation ratio, which means these modes cannot simply be merged together. Meanwhile, each switching angle from different modes has different characteristics, so that choosing the applicable switching angles’ solution during these modes is the key to achieving our optimization goals.

In addition, the other number of switches has the same situation and even more-efficient modes, 5 modes ($N=5$) and 50 modes ($N=9$).

Next, the CMV characteristics of the multiple modes need to be studied.

3.2. Common-Mode Voltage and Characteristics under Multiple Modes

The previous analysis in Section 3.1 fully demonstrates that multimodality does exist for five-level SHEPWM, and the switching angle solution trajectory of each mode is also solved. In the next step, the CMV results with different switching angle trajectories need to be discussed. As shown in (6), CMV is determined by the zero-sequence harmonics (ZSH). Hence, we can use the total zero-sequence harmonics ($TZSH$) to measure the magnitude of the CMV as in (14).

$$TZSH=\sqrt{\frac{{\sum }_{n=1,2,\cdots }{a}_{3n}^2}{{(V_{dc}/4)}^2}}$$

(14)

Figure 5 shows the $TZSH$ of the angle trajectories from different modes ($N=7$). It is obvious that the $TZSH$, i.e., the CMV in different modes, has a huge difference. The maximum difference is 1.06 between mode 104 ($TZSH=1.49$) and mode 86 ($TZSH=0.43$) when m = 0.45, while the average difference is also up to 0.57. Combined with the effective $m_{range}$ from the different modes shown in Figure 5, for a particular modulation ratio (0.1–0.97), the cases including modes and the corresponding $TZSH$ ($TZSH\left(mode\right)$) can be seen in

Table 3. $TZSH$ for multiple modes under partial modulation ratio.

m	0.1	0.2	0.3	0.4	0.5	0.6	0.7	0.8	0.97
case 1	1.06 (54)	1.09 (98)	1.01 (98)	0.42 (86)	0.43 (86)	0.36 (86)	0.44 (90)	0.51 (90)	0.54 (90)
case 2	1.10 (100)	1.15 (54)	1.22 (46)	0.66 (54)	0.77 (54)	0.52 (90)	0.73 (54)	0.55 (106)	0.73 (106)
case 3	1.15 (98)	1.20 (100)	1.24 (54)	0.88 (89)	0.80 (101)	0.75 (89)	0.79 (105)	0.57 (54)
case 4	1.24 (46)	1.24 (46)	1.25 (100)	0.89 (101)	1.08 (105)	0.88 (54)	0.93 (104)	1.08 (105)
case 5	1.39 (104)	1.54 (104)	1.63 (104)	1.25 (104)	1.20 (100)	1.26 (104)
case 6	1.87 (97)	1.79 (97)	1.71 (97)	1.26 (46)	1.29 (105)
case 7				1.31 (100)

Data format: $TZSH$ (mode).

. There are nearly five cases on average of freedom per modulation ratio to optimize the control. The CMV is significantly higher at a low modulation ratio ($m\le 0.3$). As for the higher modulation ratio ($m>0.97$), optimization based on multiple modes is not possible since there are only angle solutions from one mode (mode 106). As a whole, according to the difference in the CMV performance of different switching angles, the optimal design of the CMV can be realized by choosing the appropriate mode and the corresponding switching angle under each modulation ratio.

Based on the five-level multiple modes and CMV characteristics, the formation process of the full modulation switching angle trajectory to suppress the CMV will be introduced next.

4. Optimized-Mode-Trajectory-Based Common-Mode Voltage Suppression Method

4.1. Optimized Mode Trajectory

The multimode angle trajectories and corresponding CMV for $N=7$ was calculated in Section 3. In order to suppress the CMV, we can simply choose those switching angle solutions with the minimum $TZSH$. Figure 6 shows the $TZS{H}_{min}$ of the full modulation ratio, the bottom solid line, while the $TZS{H}_{max}$ is also provided, the top dotted line, for comparison. The maximum proportion of the slump is up to $71.1\%$ when m = 0.45, which corresponds to the aforementioned maximum difference. Since there is only one mode (mode 106), there is no decrease when m > 0.97. As for the full modulation ratio, the values of $TZS{H}_{min}$ decrease by an average of $44.7\%$ compared with the $TZS{H}_{max}$, so that a substantial reduction in the CMV can be achieved in this way. But, this also brings the problem of a large amount of mode switchovers, a total of 10 switchovers between six modes at the full modulation ratio, which is represented by a series of points in Figure 6. Mode switchover brings the current impulse problem and also greatly increases the control complexity. Therefore, it is necessary to minimize the number of mode switchovers while suppressing the CMV.

Based on these modes in $TZS{H}_{min}$, partial modes can be replaced by other modes to reduce the number of mode switchovers, the premise being that the other modes also have solutions for the partial modulation ratio. The modulation ratio ranges for these modes in $TZS{H}_{min}$ are shown in Figure 7. By looking at each modal modulation ratio interval at the minimum CMV, we can identify two sections that have the following two characteristics: the first and last belong to the same mode and the intermediate modulation ratio can also be replaced by that mode for successive operations, so that mode 98 (0.16–0.32) and mode 89 (0.33) can be replaced by mode 54, corresponding to $TZS{H}_{opt1}$, and mode 90 (0.79–0.86 and 0.91–0.97) can be replaced by mode 106, corresponding to $TZS{H}_{opt2}$. The final post-replacement curve of the optimal $TZSH$ ($TZS{H}_{opt}$) consists of the rest of $TZS{H}_{min}$, $TZS{H}_{opt1}$ and $TZS{H}_{opt2}$. The value of $TZS{H}_{opt}$ still decreased by an average of $41.2\%$ compared with the $TZS{H}_{max}$, which is only a $5.8\%$ increase over the $TZS{H}_{min}$. But, finally only, three mode switchovers between four modes are required. The $TZSH$ trajectories from mode 54, mode 86, mode 90, and mode 106 are shown in Figure 6, which can constitute the full modulation ratio. It is worth stating that the replaced modulation ratio proposed is not exclusive, and there is a balance between the number of mode switchovers and CMV suppression for this mode-optimal model. However, we can be sure that, as the number of modal modulation ratio interval substitutions increases, this will inevitably lead to an increase in the level of the $TZSH$, thus affecting the suppression of the common-mode voltage.

Through the above discussion, we were able to determine the optimized mode trajectory that can effectively suppress the CMV under five-level SHEPWM ($N=7$). The trajectories of the switching angles for N equal to 5 and 9 have the same characteristics.

Table 4. Optimal mode selection corresponding to modulation ratio range.

N-Feature	Range 1	Range 2	Range 3	Range 4	Range 5
5-$[m_L,m_R]$	$[0.01,0.43]$	$[0.44,0.65]$	$[0.66,0.69]$	$[0.7,1.16]$
5-mode	mode 14	mode 22	mode 25	mode 26
7-$[m_L,m_R]$	$[0.01,0.39]$	$[0.4,0.61]$	$[0.62,0.7]$	$[0.71,1.14]$
7-mode	mode 54	mode 86	mode 90	mode 106
9-$[m_L,m_R]$	$[0.01,0.32]$	$[0.33,0.38]$	$[0.39,0.6]$	$[0.61,0.66]$	$[0.67,1.15]$
9-mode	mode 402	mode 214	mode 342	mode 346	mode 362

gives the distribution of the switching modes for N equal to 5, 7, and 9. As there are up to 50 modes for $N=9$, the optimized mode trajectory for the full modulation ratio only includes four mode switches from five modes.

Once the switching angles have been obtained, the subsequent implementation steps will be exactly the same as those for the conventional SHEPWM. In practice, these switching angles are read by looking up the table and the corresponding PWM pulses are generated with the help of a digital signal processor (DSP) or field programmable gate array (FPGA).

4.2. Discussion on Strengths and Weaknesses

Finally, we provide a discussion on the advantages and disadvantages of the proposed method, which will help to clarify the main contributions of this article. The advantage is mainly reflected in the fact that the proposed method does not use any switching angle to suppress the $3n_{\mathrm{th}}$ harmonic components, which makes the order of the lowest harmonic that can be eliminated by five-level SHEPWM constant. In other words, the proposed method does not significantly affect the elimination of the low-order harmonics and does not increase the equivalent SF. In addition, this article provides a new idea or perspective that actively utilizes the multimode characteristics that only exist in multilevel SHEPWM to suppress the CMV. The disadvantage is that the CMV cannot be completely eliminated. It can only be suppressed to a certain extent with the proposed method. If complete elimination of the CMV is pursued, the proposed method will not be applicable.

5. Experimental Verification

In order to fully verify the correctness and effectiveness of the proposed method, a small-scale prototype was built, as shown in Figure 8. The five-level converter was constructed on the basis of discrete IGBTs (IKW50N60T, Infineon, Munich, Germany). These IGBTs are driven by driver chips (2ED020I12-F2, Infineon, Munich, Germany). The input DC power is obtained by diode rectification, for which the amplitude is regulated via an s voltage slider (220 V input and 0–250 V output). The concerned algorithms were implemented in real-time on a DSP board (TMS320F28335, TI, Dallas, TX, USA). The voltages of the DC-link capacitor and flying capacitor are read into the DSP via sensors (LV25, LEM, Geneva, Switzerland). The currents of the three-phase output and DC-link midpoint are read into the DSP via sensors (LA50, LEM, Geneva, Switzerland). The experimental results were measured directly using the hardware voltage and current probes and observed by a hardware oscilloscope.

The experimental parameters are presented in

Table 5. Parameters of the 5L-ANPC experiment models.

Parameters	Experiment Value
DC-link voltage ($v_{dc}$)	180 V
Upper and bottom capacitor ($c_1,c_2$)	$2.2$ mF
Flying capacitor ($c_{uf}$)	$4.7$ mF
Base frequency (BF)	50 Hz
R-load	20 $\mathsf{\Omega }$
-load	30 mH

. The operation conditions in experiment verification are organized as follows.

In order to verify the superiority of the selected modes in CMV suppression, the switching angle solutions from $TZS{H}_{opt}$ and $TZS{H}_{max}$ ($N=7$) were used for the experimental comparison. The specific angles and corresponding $TZSH$ under the selected modulation ratio are shown in

Table 6. $TZSH$ and switching angles for the experiments ($N=7$).

${\mathit{m}}_{\mathit{a}}$	Mode	TZSH	${\mathit{\theta }}_1$	${\mathit{\theta }}_2$	${\mathit{\theta }}_3$	${\mathit{\theta }}_4$	${\mathit{\theta }}_5$	${\mathit{\theta }}_6$	${\mathit{\theta }}_7$
0.2	97	1.790	6.0907	18.1079	43.8777	57.8349	71.7076	84.0559	87.7112
	54	1.153	14.7970	42.8632	55.7531	60.2542	68.6148	81.0791	87.8051
0.5	100	1.204	13.2686	22.2327	40.482	53.1922	56.2091	75.1309	86.9406
	86	0.426	27.8713	34.7755	44.3154	50.6552	54.6971	76.2578	79.9691
0.66	104	1.141	12.6403	21.3068	43.236	64.3369	67.6133	78.8194	89.9732
	90	0.316	12.5836	16.6783	21.263	64.2222	67.4298	76.4245	78.5191
0.85	105	1.058	3.4746	18.1354	24.6972	31.4997	59.4013	76.1961	79.0356
	106	0.588	18.4544	27.864	35.218	58.2564	63.5534	66.6092	80.8947

, while the direction of level jumping can be deduced from the backwardsmode according to (13). The sets of switching angles were stored in the LUT together with the corresponding level jumping states.

Using angles from m = 0.66 in Table 6 as an example firstly, during this experiment, a change of mode from mode 104 $\left(TZS{H}_{max}\right)$ to mode 90 $\left(TZS{H}_{opt}\right)$ was forced at the instant of 0.06 s. The experimental result of the phase voltage, line current, CMV, and FFT analysis is shown in Figure 9. Both before and after the switching modes, the level jump patterns from the phase voltage behaved as the mode set, and the unwanted lower harmonics were effectively eliminated. It can be observed that, however, the CMV of $\left(TZS{H}_{opt}\right)$ was much less than that of $\left(TZS{H}_{max}\right)$, which was as high as 73.655V for $(TZS{H}_{max}=116.4\%)$, while only 21.663 V for $(TZS{H}_{opt}=29.7\%)$. In order to make a comprehensive analysis of the relationship between the CMV and $3n_{\mathrm{th}}$ harmonics, Figure 9 also shows the detailed comparison of the $3n_{\mathrm{th}}$ harmonics. It can be observed that the lower $3n_{\mathrm{th}}$ harmonic amplitudes in $\left(TZS{H}_{opt}\right)$ showed a significant decrease from those in $\left(TZS{H}_{max}\right)$. When looking at the higher $3n_{\mathrm{th}}$ harmonics, there was a slight increase of the harmonic amplitudes in $\left(TZS{H}_{opt}\right)$ compared with $\left(TZS{H}_{max}\right)$, which will not affect the reduction of the overall $3n_{\mathrm{th}}$ harmonics. Besides, the current THD in $\left(TZS{H}_{opt}\right)$ was also lower, which means that a suppression of the CMV can be achieved by $\left(TZS{H}_{opt}\right)$ with a good output performance at the same time.

Figure 10 shows the experimental results and FFT of the other angles in Table 6, and the summary statistics are shown in

Table 7. The summary statistics of the experiment.

$[{\mathit{m}}_{\mathit{a}},{\mathit{v}}_{\mathit{ref}}]$	Mode	$[{\mathit{v}}_{\mathit{act}},{\mathit{v}}_{\mathit{dev}}]$(V)	TZSH	CMV(V)	Current THD (%)
[0.2, 17.2 V]	97	[16.58, 0.62]	1.740	89.3	16.78
	54	[15.62, 1.58]	1.146 (⇓34.1%)	63.2 (⇓29.2%)	20.78 (⇑23.8%)
[0.5, 43 V]	100	[42.3, 0.7]	1.247	73.7	7.83
	86	[41.8, 1.2]	0.418 (⇓66.5%)	34.7 (⇓53.0%)	7.26 (⇓7.3%)
[0.66, 56.7 V]	104	[55.3, 1.4]	1.164	73.655	5.66
	90	[56.2, 0.5]	0.297 (⇓74.5%)	21.663 (⇓70.6%)	4.45 (⇓21.4%)
[0.85, 73.1 V]	105	[72.7, 0.4]	1.013	58.5	4.79
	106	[70.6, 2.5]	0.617 (⇓39.1%)	43.3 (⇓26.0%)	4.26(⇓11.1%)

Rising (⇑) or falling (⇓) of the statistical value from $TZS{H}_{opt}$ compared to $TZS{H}_{max}$.

. The same as discussed in Figure 9, it can be seen that all modes could achieve the set jumping levels and the desired fundamental amplitude. It is worth explaining that there was some deviation ($v_{dev}$) between the actual fundamental amplitude ($v_{act}$) and the reference value ($v_{ref}$), but this deviation is acceptable and understandable. From the perspective of software control, a possible explanation for this might be the fluctuation of the midpoint potential and suspension capacitance voltage. In order to realize the control of the voltage, the actual angle issued will deviate from the theoretical value, resulting in the difference in the amplitude of the fundamental wave. This is not the focus of the research in this paper. From the experimental platform, sensors and non-ideal switching devices can also have an effect on the output waveforms. But, as is clear, the CMV in the optimized mode is effectively suppressed because of the lower $3n_{\mathrm{th}}$ harmonic amplitude. The specific decline is shown in percentage terms in Table 7. The decline ratio for the CMV completely conforms to the transformation law of the $TZSH$ with little differences in the numerical value. For the output performance, there is no doubt that the unwanted specific harmonics are eliminated for all modes realizing the basic goal of SHEPWM. Interestingly, $TZS{H}_{opt}$ also had a good performance in terms of the output current harmonics compared with $TZS{H}_{max}$.

What emerges from the results reported here is that the harmonic characteristics of multiple modes with the same modulation ratio are quite different, including the $3n_{\mathrm{th}}$ harmonics, which are closely related to the CMV. According to the optimization method proposed in this paper, minimizing the $TZSH$ and considering the mode switchover frequency simultaneously, the resulting mode can ensure good output performance while effectively realizing the suppression effect of the CMV.

6. Conclusions

In medium-voltage, high-power applications, SHEPWM usually needs to be applied to five-level converters to suppress the switching losses, improve system efficiency, and reduce heat. In this case, the control bandwidth of the system is very low because the switching angles of SHEPWM are difficult to adjust significantly in real-time. This causes control problems such as CMV suppression becoming very difficult. This article discarded the idea of using the switching angles to specify the zero-sequence voltage, because this approach will lead to an increase in the equivalent SF. Instead, a new solution idea was developed in this article, which utilizes the unique multimode characteristics of five-level SHEPWM that three-level SHEPWM does not have for a new control design. By analyzing the relationship between the switching angle trajectories and CMV in different modes of five-level SHEPWM, we successfully found the optimal mode and its corresponding switching angles that are favorable to suppress the CMV in various operation conditions. Thus, the CMV suppression can be achieved without sacrificing the switching angles under five-level SHEPWM. The experimental results were also able to prove that the proposed method is effective.