2.2.2. Proposed Virtual Space Vector Modulation

(a) Sector division and duty cycles.

In the proposed VSVM, each virtual vector is synthesized by the two nearest active vectors, and each virtual sector is denoted as the area between two virtual vectors. There are totally six virtual vectors → *I a* ∼ → *I f* and six virtual sectors. The concept of virtual vector in this paper is similar to the virtual vector concept in [26]. Figure 3 presents the virtual sector divisions according to the virtual vectors and the synthesis of input current reference vector in the virtual sector I of the proposed method. As it can be seen from Figure 3b, when the input current reference vector → *I re f* is in virtual sector I, two virtual vectors → *I a*, → *I b* and one zero vector are used to synthesize the reference vector depending on the magnitude of modulation index *mi*. The selection of zero vector in the proposed VSVM is similar to zero vector selection of the C-SVM method.

**Figure 3.** Space vector diagram of proposed virtual space vector modulation (VSVM) for AC–DC matrix converter. (**a**) Sector division; (**b**) Input current reference vector synthesis.

The duty cycles *da*, *db*, *d*0 of virtual vectors → *I a*, → *I b* and zero vector → *I* 0, when the input current reference vector is in the virtual sector I, are expressed as:

$$d\_d = m\_l \sin\left(\frac{\pi}{3} - \theta\right) \tag{7}$$

$$d\_b = m\_i \sin(\theta) \tag{8}$$

$$d\_0 = 1 - d\_a - d\_b.\tag{9}$$

where *mi*: modulation index; *mi* = *ii*1/*idc*; *mi* ∈ [0, 1]; *ii*1: the peak value of the fundamental-frequency component in *ii*; θ: input current reference vector angle θ ∈ 0, π3 . →

The durations *Ta*, *Tb*, *T*0 of virtual vectors → *I a*, *I b* and zero vector are determined as:

$$T\_a = d\_a T\_s \tag{10}$$

$$T\_b = d\_b T\_s \tag{11}$$

$$T\_0 = T\_s - T\_a - T\_b.\tag{12}$$

The virtual vectors → *I a*, → *I b* are synthesized by three original active vectors → *I* 1, → *I* 2, and → *I* 3. The dwell times *T*1, *T*2, *T*3 of three original active vectors can be derived by:

$$T\_1 = \frac{T\_a}{2} \tag{13}$$

$$T\_2 = \frac{T\_a}{2} + \frac{T\_b}{2} \tag{14}$$

$$T\_3 = \frac{T\_b}{2}.\tag{15}$$

The dwell times of other virtual and original active vectors in remaining sectors are determined in the similar manner of the sector I.

(b) Switching patterns.

The effectiveness of the proposed VSVM not only depends on the modulation of virtual vectors, but also on the switching patterns.

Figure 4 presents the switching patterns of conventional VSVM (C-VSVM) in [26]. This switching pattern, which is a seven-segmen<sup>t</sup> pattern, is not optimized for reducing the DC current ripple. In this paper, the optimized switching patterns are proposed to further reduce the DC current ripple at both high- and low-modulation operation. The proposed switching patterns of the VSVM strategy for AC–DC MC under different modulation index ranges in sector I are illustrated in Figure 5. At low modulation index operation, the switching period of the zero vector is the longest switching period compared with other switching periods of the remaining active vectors. The longer the switching period of the zero vector, the higher the current ripple due to the longer period of decreasing output DC current. Most of the approaches for DC ripple reduction are not optimized for the zero vector at low-modulation operation. Thus, the optimized switching patterns for the zero vector are proposed in this paper to further reduce the DC current ripple of AC–DC MC.

(c) Controller system.

The control block diagram of the proposed control strategy is illustrated in Figure 6. The battery voltage is sensed and compared with the reference voltage, then passed through the proportional-integral (PI) controller and compared with the DC current or using direct DC current reference depending on the selection of constant voltage (CV) control mode or constant current (CC) control mode. After that, the signal is passed through the PI controller to obtain the modulation index. At the input terminal, three-phase source voltages, three-phase source currents are sensed and passed through the αβ-transformation, then combined with the PLL block for the calculation of input current. Finally, the VSVM algorithm is applied to compute the duty cycles for AC–DC MC.

 **Figure 4.** Switching patterns of C-VSVM (conventional VSVM) for AC–DC matrix converter.

**Figure 5.** Switching patterns of proposed VSVM for AC–DC matrix converter under different modulation index and input current reference angle. (**a**) Low modulation index; (**b**) High modulation index.

**Figure 6.** Control block diagram of proposed VSVM for AC–DC matrix converter.
