Keywords

1 Introduction

The multilevel converter consists of a large number of low-voltage power switching devices. Compared with the two-level converter, the multilevel converter reduces the total harmonic distortion of the output voltage, and has the characteristics of high power factor, strong fault tolerance, etc. In recent years, the multilevel converter has been widely used in industry, such as flexible dc transmission [1], photovoltaic power generation systems [2], power electronic transformers [3, 4], etc. Multilevel converter topologies mainly include cascaded H-bridge multilevel converter (CHBMC), modular multilevel converter (MMC), and neutral point clamped multilevel converter (NPCMC) [5].

CHBMC and MMC are connected by star or delta, and are applied to three-phase systems. Under three-phase unbalanced conditions, the stability of the system will be greatly affected [6]. To solve this problem, a cascaded three-phase bridge converter (C3PBC) topology is proposed in [7]. It uses the three-phase voltage source inverter (VSI) as sub-module. The line voltages in ac side of the converter are produced through cascaded way.

Modulation strategy directly affects the performance of the multilevel converters, such as harmonic characteristics, control stability [8]. Therefore, it is necessary to study the modulation strategy of the multilevel converter. The modulation strategies of the multilevel converters mainly include nearest level modulation (NLM), carrier phase-shifted pulse width modulation (CPS-PWM), and space vector pulse width modulation (SVPWM). CPS-PWM is simple and easy to be implemented. It is the main modulation method of C3PBC at present. However, when the level number is high, the phase shift angle decreases, the phase shift control accuracy and synchronization accuracy are improved, which increases the difficulty of CPS-PWM implementation. NLM is simple to calculate and easy to be implemented. However, when the level number is low, the harmonic characteristics of the output voltage of the converter are poor. So NLM is more suitable for the converter with high level number. SVPWM has the advantages of low THD, multiple control degrees of freedom, and high dc voltage utilization [9]. However, the traditional SVPWM strategy involves a large number of irrational calculations. Especially when the level number of converter increases, the space vectors and sector triangles increase dramatically, which makes it difficult to locate the reference vector and calculate the switching state. SVPWM strategy is difficult to be applied to converters above seven levels.

In order to extend SVPWM to higher level converters, many simplified SVPWM strategies have been proposed. The reference vector is expressed as a sum of a contender vector and an error vector in [10]. The error vector is mapped to a two-level hexagon centered at the origin. The two-level SVPWM algorithm is used to locate and synthesize the error vector. Finally, the multilevel SVPWM strategy is completed by inverse mapping. This method needs to complete two mapping at each reference vector sampling point. When the level number is high, the switching states corresponding to the competition vector increases, and it is difficult to select the appropriate switching state. Therefore, this method is not suitable for high level converters. Reference [11] proposes a simplified SVPWM strategy based on non-orthogonal coordinates g-h. In the g-h coordinates, the coordinate components of the space vector are all integers, which simplifies the reference vector’s location and speeds up the calculation of the switching state’s duty cycle. However, due to the limitation of the C3PBC topology, its space vector distribution is different from that of the traditional converter, so this method is not suitable for the C3PBC topology.

This paper aims to explore an SVPWM strategy for the C3PBC. The space vector is located on the unit integer grid through coordinate rotation transformation, and it is convenient to locate the reference vector, simplifying the modulation process. The proposed SVPWM strategy is suitable for C3PBC of any level.

2 Cascaded Three-Phase Bridge Converters

The topology of C3PBC is shown in Fig. 1(a), and its sub-module is a three-phase inverter bridge with input dc voltage E. C3PBC cascades the line voltage of sub-modules to achieve multilevel output. According to the definition of the cascaded cell number n of C3PBC in [6], it can be obtained

$$n={u}_{max}/E$$
(1)

where \({u}_{max}\) is the amplitude of the output line voltage at the ac side of C3PBC.

Fig. 1.
figure 1

C3PBC topology (a) 2-cell C3PB converter topology, (b) 2-cell C3PB converter simplified model

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Figure 1 shows the topology and simplified topology of the 2-cell C3PBC. The 2-cell C3PBC consists of three sub-modules. Its three line voltages are \({u}_{AB}\), \({u}_{BC}\), and \({u}_{CA}\).

$$\left\{\begin{array}{c}{u}_{AB}={u}_{a1b1}+{u}_{a2b2}\\ {u}_{BC}={u}_{b2c{2}}+{u}_{b3c3}\\ {u}_{CA}={u}_{c3a3}+{u}_{c1a1}\end{array}\right.$$
(2)

where \({u}_{aibi}\), \({u}_{bici}\), and \({u}_{ciai}\) are the line voltages output by the i-th sub-module respectively. According to (2), the output line voltage of 2-cell C3PBC contains five voltages of \(\pm 2E\), \(\pm E\), and 0, corresponding to five levels of  ±2, ±1, and 0, respectively. Similarly, n-cell C3PBC consists of 3(n-1) sub-modules, and its output line voltage includes 2n + 1 levels of ± n, ± (n−1), …, 0. The output line voltage is

$$\left\{\begin{array}{c}{u}_{AB}={u}_{a1b1}+{u}_{a2b2}+...+{u}_{anbn}\\ {u}_{BC}={u}_{bncn}+{u}_{b(n+1)c(n+1)}+...+{u}_{b(2n-1)c(2n-1)}\\ {u}_{CA}={u}_{c(2n-1)a(2n-1)}+{u}_{c(2n)a(2n)}+...+{u}_{c(3n-3)a(3n-3)}+{u}_{c1a1}\end{array}\right.$$
(3)

The line voltages \({u}_{AB}\), \({u}_{BC}\), and \({u}_{CA}\) are normalized with the reference value E to obtain the line voltage levels a, b, and c as

$$\left\{\begin{array}{c}a={u}_{AB}/E\\ b={u}_{BC}/E\\ c={u}_{CA}/E\end{array}\right.$$
(4)

According to the connection mode of C3PBC and the symmetry of the three-phase system,

$$a+b+c=0$$
(5)

3 Coordinate Transformation and Space Vector Distribution

Three line voltage reference signals of C3PBC are

$$\left\{\begin{array}{c}{u}_{rab}=mncos\omega t\\ {u}_{rbc}=mncos(\omega t-\frac{2\pi }{3})\\ {u}_{rca}=mncos(\omega t+\frac{2\pi }{3})\end{array}\right.$$
(6)

where m is the modulation index, \(0<m\le 1\), and m reflects the dc voltage utilization rate of C3PBC, \(\omega \) represents the angular frequency of the reference signals.

According to the principle of SVPWM and the definition of traditional coordinates \(\alpha \beta \), transform the line voltage reference signals into the reference vector \({V}_{r}\) in the \(\alpha \beta \mathrm{coordinates}\). The coordinate components of \({V}_{r}\) can be expressed as:

$$\left[\begin{array}{c}{\alpha }_{r}\\ {\beta }_{r}\end{array}\right]=\frac{1}{3}\left[\begin{array}{c}2{u}_{rab}-{u}_{rbc}-{u}_{rca}\\ \sqrt{3}\left({u}_{rbc}-{u}_{rca}\right)\end{array}\right]$$
(7)

The reference vector can also be expressed as \({V}_{r}({\alpha }_{r},{\beta }_{r})\). The trajectory of \({V}_{r}\left({\alpha }_{r},{\beta }_{r}\right)\) is expressed as

$$\frac{{{\alpha }_{r}}^{2}+{{\beta }_{r}}^{2}}{{\left(mn\right)}^{2}}=1$$
(8)

Equation (8) shows that the trajectory of the reference vector in the \(\alpha \beta \) coordinates is a circle with a radius of mn. Same as (7), the space vector \(V\) in the \(\alpha \beta \) coordinates can be expressed as

$$\left[\begin{array}{c}\alpha \\ \beta \end{array}\right]=\frac{1}{3}\left[\begin{array}{c}2a-b-c\\ \sqrt{3}\left(b-c\right)\end{array}\right]$$
(9)

where \(\alpha \) and \(\beta \) are the coordinate components of the space vector V, the space vector can also be expressed as \(V(\alpha ,\beta )\). \((a,b,c)\) satisfying (9) is called the switching state corresponding to the \(V(\alpha ,\beta )\), denoted as \(S(a,b,c)\). The trajectory of the reference vector and space vector diagram of 2-cell C3PBC in \(\alpha \beta \) coordinates are shown in Fig. 2(a).

Equation (7) indicates that the space vectors in the \(\alpha \beta \) coordinates are located in non-integer grid, which makes it difficult to locate the reference vector. In the modulation process, there are a large number of irrational calculations, which makes the implementation of SVPWM strategy complicated. To solve this problem, the coordinates \(\alpha \beta \) is transformed into a new coordinates \({\alpha }^{^{\prime}}{\beta }^{^{\prime}}\). The space vector \(V(\alpha ,\beta )\) in the \(\alpha \beta \) coordinates is mapped to the space vector \({V}^{^{\prime}}({\alpha }^{^{\prime}},{\beta }^{^{\prime}})\) in the \({\alpha }^{^{\prime}}{\beta }^{^{\prime}}\) coordinates.

$$\left[\begin{array}{c}\alpha {^{\prime}}\\ \beta {^{\prime}}\end{array}\right]={C}_{r}{C}_{c}\left[\begin{array}{c}\alpha \\ \beta \end{array}\right]=\frac{1}{3}\left[\begin{array}{ccc}1& -2& 1\\ 1& 1& -2\end{array}\right]\left[\begin{array}{c}a\\ b\\ c\end{array}\right]$$
(10)

where \({C}_{c}\) is the stretching matrix, which stretches the \(\beta \)-axis of the \(\alpha \beta \) coordinates by a factor of \(\sqrt{3}\), \({C}_{c}=\left[\begin{array}{cc}1& 0\\ 0& \sqrt{3}\end{array}\right]\), and \({C}_{r}\) is the rotation matrix, which rotates the \(\alpha \beta \) coordinates clockwise by 45°, \({C}_{r}=\left[\begin{array}{cc}1/2& -1/2\\ 1/2& 1/2\end{array}\right]\), and The principle of the coordinate transformation is shown in Fig. 2(b). Substitute (5) into (10) to get

$$\left[\begin{array}{c}{\alpha }^{^{\prime}}\\ {\beta }^{^{\prime}}\end{array}\right]=\frac{1}{3}\left[\begin{array}{ccc}1& -2& 1\\ 1& 1& -2\end{array}\right]\left[\begin{array}{c}-b-c\\ b\\ c\end{array}\right]=\left[\begin{array}{c}-b\\ -c\end{array}\right]$$
(11)

Equation (11) shows that the space vector in the \({\alpha }^{^{\prime}}{\beta }^{^{\prime}}\) coordinates is located on the unit integer grid. Same as (11), the reference vector \(V_{r}^{^{\prime}} (\alpha_{r}^{^{\prime}} ,\beta_{r}^{^{\prime}} )\) in the \({\alpha }^{^{\prime}}{\beta }^{^{\prime}}\) coordinates is

$$\left[\begin{array}{c}{\alpha }_{r}^{^{\prime}}\\ {\beta }_{r}^{^{\prime}}\end{array}\right]=\frac{1}{3}\left[\begin{array}{ccc}1& -2& 1\\ 1& 1& -2\end{array}\right]\left[\begin{array}{c}-{u}_{rbc}-{u}_{rca}\\ {u}_{rbc}\\ {u}_{rca}\end{array}\right]=\left[\begin{array}{c}-{u}_{rbc}\\ -{u}_{rca}\end{array}\right]$$
(12)
Fig. 2.
figure 2

Trajectory of the reference vector and space vector diagram (a) \(\alpha \beta \) coordinates, (b) \({\alpha }^{^{\prime}}{\beta }^{^{\prime}}\) coordinates

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$${\left(\frac{{\alpha }_{r}^{^{\prime}}+{\beta }_{r}^{^{\prime}}}{mn}\right)}^{2}+{\left(\frac{{\alpha }_{r}^{^{\prime}}-{\beta }_{r}^{^{\prime}}}{\sqrt{3}mn}\right)}^{2}=1$$
(13)

In the \({\alpha }^{^{\prime}}{\beta }^{^{\prime}}\) coordinates, Eq. (13) shows the trajectory of the reference vector \({V}_{r}^{^{\prime}}\) is an ellipse, as shown in Fig. 2(b). 2-cell C3PBC contains 19 space vectors that form two nested hexagons. n-cell CHBMC has \((1+6\sum_{i=1}^{n}i)\) space vectors that form n nested hexagons.

4 Locating and Synthesizing Reference Vector

The reference vector \({V}_{r}^{^{\prime}}({\alpha }_{r}^{^{\prime}},{\beta }_{r}^{^{\prime}})\) locates in a unit square formed by \({V}_{0}^{^{\prime}}({\alpha }_{0}^{^{\prime}},{\beta }_{0}^{^{\prime}})\), \({V}_{1}^{^{\prime}}({\alpha }_{0}^{^{\prime}}+1,{\beta }_{0}^{^{\prime}})\), \({V}_{2}^{^{\prime}}({\alpha }_{0}^{^{\prime}},{\beta }_{0}^{^{\prime}}+1)\), and \({V}_{3}^{^{\prime}}({\alpha }_{0}^{^{\prime}}+1,{\beta }_{0}^{^{\prime}}+1)\), as shown in Fig. 3.

Fig. 3.
figure 3

Location of reference vector

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$$\left\{\begin{array}{c}{\alpha }_{0}^{^{\prime}}=floor({\alpha }_{r}^{^{\prime}})\\ {\beta }_{0}^{^{\prime}}=floor({\beta }_{r}^{^{\prime}})\end{array}\right.$$
(14)

where floor(*) represents the largest integer not greater than the number.

When \(\left[\left({\alpha }_{r}^{^{\prime}}-{\alpha }_{0}^{^{\prime}}\right)+\left({\beta }_{r}^{^{\prime}}-{\beta }_{0}^{^{\prime}}\right)\right]\le 1\), the reference vector is located in sector I composed of the space vectors \({V}_{0}^{^{\prime}}\), \({V}_{1}^{^{\prime}}\), and \({V}_{2}^{^{\prime}}\). \({V}_{0}^{^{\prime}}\), \({V}_{1}^{^{\prime}}\), and \({V}_{2}^{^{\prime}}\) synthesize the reference vector. According to the volt-second balance principle, the duty cycles \({t}_{0}\), \({t}_{1}\), and \({t}_{2}\), of the space vectors \({V}_{0}^{^{\prime}}\), \({V}_{1}^{^{\prime}}\), and \({V}_{2}^{^{\prime}}\) are calculated.

$$\left\{\begin{array}{c}{t}_{0}={T}_{s}\cdot \left[\left({\alpha }_{0}^{^{\prime}}+{\beta }_{0}^{^{\prime}}+1\right)-\left({\alpha }_{r}^{^{\prime}}+{\beta }_{r}^{^{\prime}}\right)\right]\\ {t}_{1}={T}_{s}\cdot \left({\alpha }_{r}^{^{\prime}}-{\alpha }_{0}^{^{\prime}}\right)\\ {t}_{2}={T}_{s}\cdot \left({\beta }_{r}^{^{\prime}}-{\beta }_{0}^{^{\prime}}\right)\end{array}\right.$$
(15)

When \(\left[\left({\alpha }_{r}^{^{\prime}}-{\alpha }_{0}^{^{\prime}}\right)+\left({\beta }_{r}^{^{\prime}}-{\beta }_{0}^{^{\prime}}\right)\right]>1\), the reference vector is located in sector II composed of the space vectors \({V}_{1}^{^{\prime}}\), \({V}_{2}^{^{\prime}}\), and \({V}_{3}^{^{\prime}}\). \({V}_{1}^{^{\prime}}\), \({V}_{2}^{^{\prime}}\), and \({V}_{3}^{^{\prime}}\) synthesize the reference vector. According to the volt-second balance principle, the duty cycles \({t}_{1}\), \({t}_{2}\), and \({t}_{3}\) of the space vectors \({V}_{1}^{^{\prime}}\), \({V}_{2}^{^{\prime}}\), and \({V}_{3}^{^{\prime}}\) are calculated.

$$\left\{\begin{array}{c}{t}_{1}=\left({\alpha }_{0}^{^{\prime}}+1-{\alpha }_{r}^{^{\prime}}\right)\cdot {T}_{s}\\ {t}_{2}=\left({\beta }_{0}^{^{\prime}}+1-{\beta }_{r}^{^{\prime}}\right)\cdot {T}_{s}\\ {t}_{3}={T}_{s}\cdot \left[\left({\alpha }_{r}^{^{\prime}}+{\beta }_{r}^{^{\prime}}\right)-\left({\alpha }_{0}^{^{\prime}}+{\beta }_{0}^{^{\prime}}+1\right)\right]\end{array}\right.$$
(16)

According to (8), the switching state corresponding to the space vector \(V{^{\prime}}(\alpha {^{\prime}},\beta {^{\prime}})\) is \(S(\alpha {^{\prime}}+\beta {^{\prime}},-\alpha {^{\prime}},-\beta {^{\prime}})\). So, \({S}_{0}({\alpha }_{0}^{^{\prime}}+{\beta }_{0}^{^{\prime}},-{\alpha }_{0}^{^{\prime}},-{\beta }_{0}^{^{\prime}})\), \({S}_{1}({\alpha }_{0}^{^{\prime}}+{\beta }_{0}^{^{\prime}}+1,-{\alpha }_{0}^{^{\prime}}-1,-{\beta }_{0}^{^{\prime}})\), \({S}_{2}({\alpha }_{0}^{^{\prime}}+{\beta }_{0}^{^{\prime}}+1,-{\alpha }_{0}^{^{\prime}},-{\beta }_{0}^{^{\prime}}-1)\), and \({S}_{3}({\alpha }_{0}^{^{\prime}}+{\beta }_{0}^{^{\prime}}+2,-{\alpha }_{0}^{^{\prime}}-1,-{\beta }_{0}^{^{\prime}}-1)\) are the switching states corresponding to the space vectors \({V}_{0}^{^{\prime}}, {V}_{1}^{^{\prime}}, {V}_{2}^{^{\prime}}\) and \({V}_{3}^{^{\prime}}\), respectively.

The five-segment switching method is used to synthesize the reference vector. Taking the reference vector \({V}_{r}^{^{\prime}}({\alpha }_{r}^{^{\prime}},{\beta }_{r}^{^{\prime}})\) in sector I as an example, the switching sequence composed of the switching states \({S}_{0}\), \({S}_{1}\) and \({S}_{2}\) has six cases, namely, \({S}_{0}\)→\({S}_{1}\)→\({S}_{2}\)→\({S}_{1}\)→\({S}_{0}\), \({S}_{0}\)→\({S}_{2}\)→\({S}_{1}\)→\({S}_{2}\)→\({S}_{0}\), \({S}_{1}\)→\({S}_{2}\)→\({S}_{0}\)→\({S}_{2}\)→\({S}_{1}\), \({S}_{1}\)→\({S}_{0}\)→\({S}_{2}\)→\({S}_{0}\)→\({S}_{1}\), \({S}_{2}\)→\({S}_{1}\)→\({S}_{0}\)→\({S}_{1}\)→\({S}_{2}\), \({S}_{2}\)→\({S}_{0}\)→\({S}_{1}\)→\({S}_{0}\)→\({S}_{2}\). Taking the switching sequence \({S}_{0}\)→\({S}_{1}\)→\({S}_{2}\)→\({S}_{1}\)→\({S}_{0}\) as an example. Figure 4 shows the switching sequence and switching time distributions.

Fig. 4.
figure 4

Switching sequence and switching time distribution

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5 Simulation Verification

In order to verify the proposed SVPWM strategy, the 4-cell C3PBC system is simulated in MATLAB/Simulink. The 100V voltage source is applied as the sub-module DC input, and the reference signal frequency f is 50 Hz, the sampling period is 0.1 ms, and the modulation index m is 0.96.

Figure 5 shows the line voltage waveforms and the spectra of the line voltage. The output line voltage has 9 levels, which verifies the previous analysis. The THD of the line voltage is 15.83%, the amplitude of its fundamental component is 382.2 V, and its rms value is 270.3 V. The rms value of the line voltage calculated with (6) is 271.5 V, with a small difference of 1.2 V from the simulation result, which is caused by the internal resistances of the switching devices in simulation. Figure 6 shows the output line voltage waveform and its corresponding input reference voltage waveform, which indicates that the line voltage tracks the reference voltage well.

Fig. 5.
figure 5

Simulation results (m  =  0.96) (a) Line voltage, (b) Spectra of the line voltage

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Table 1. Comparison of Modulation Strategy
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Fig. 6.
figure 6

Reference voltage \({u}_{ref}\) (white) and output line voltage \({u}_{out}\) (red)

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Table 1 shows the comparison results between the proposed SVPWM strategy and the CPS-PWM strategy in [6]. As can be seen from the table, under the same modulation index, the THD of the proposed SVPWM strategy is significantly better than that of the CPS-PWM strategy.

6 Conclusion

An SVPWM strategy suitable for C3PBC is proposed in this paper. The traditional coordinates \(\alpha \beta \) is transformed into a new coordinates \({\alpha }^{^{\prime}}{\beta }^{^{\prime}}\). The trajectory of reference vector and space vector distributions in the new coordinates \(\alpha {^{\prime}}\beta {^{\prime}}\) are calculated. The space vector is located on unit integer grid in the \(\alpha {^{\prime}}\beta {^{\prime}}\) coordinates, which simplifies the reference vector’s location and speeds up the calculation of the switching state’s duty cycle. The special relationship between the coordinate components of the space vector and the line voltage levels in the \(\alpha {^{\prime}}\beta {^{\prime}}\) coordinates simplifies the calculation of the switching state, and the modulation process is simple without calculating many redundant switching states. The proposed SVPWM strategy can be easily extended to any level C3PBC without increasing the computational burden. Finally, the proposed strategy is verified by MATLAB/Simulink simulation.