## 1 Introduction
### CSI Theory Review  
Voltage-source inverter space vector modulation strategies have been extensively studied and applied in power electronics. The fundamental advantage of VSI SVM is a two-pole inverter state that is, in essence, less complicated than that of the CSI, which has three states for its inverter pole. As shown in [29,30], this inherent simplicity of VSI SVM, combined with its well-established characteristics, makes it possible to use and adapt it to drive the CSI efficiently, which may boost its performance and advantages.  
This method is feasible and straightforward to implement. The basic topology under consideration is illustrated in Figure 1[31].  
The VSI operates with three switches active at any given time. In contrast, the CSI operates with only two switches active simultaneously. The fundamental vectors of the CSI are orthogonal to the missing phase vector; for instance, vector **bc** is orthogonal to vector **a**, **ca** is orthogonal to **b**, and **ab** is orthogonal to **c**[32]. Consequently, the base vectors of the CSI are phase-shifted by \(\pi/6\) (or \(30^{\circ}\)) when compared to a VSI.  
The CSI space vectors, corresponding to active switching states, align with the vertices of a regular hexagon and have the magnitude specified in Equation (1):  
\[i_{j}=(\frac{2}{\sqrt{3}})\times I_{DC} \tag{1}\]  
Similar to the VSI operation [29], the reference space vector in the CSI [32] can be represented as a complex summation of two adjacent base vectors. For Sector 1, the calculation proceeds as follows:  
\[\overrightarrow{i}_{1}=\frac{1}{T_{p}}\int\limits_{0}^{T_{p}}\overrightarrow{i }_{1,k}\ dt_{p}=d_{(ac)}\cdot\overrightarrow{i}_{1,(ac)}+d_{(ab)}\cdot \overrightarrow{i}_{1,(ab)}=\overrightarrow{i}_{1}^{*} \tag{2}\]  
Figure 1: Grid-connected CSI topology. The circuit includes a DC link inductor at the input and an output low-pass filter connected to the grid at the output.  
\[\begin{split}& d_{(ac)}=m\sin(\frac{\pi}{b}+\varphi_{\stackrel{{ \rightarrow}}{{i}}_{1}})\\ & d_{(ab)}=m\sin(\frac{\pi}{b}-\varphi_{\stackrel{{ \rightarrow}}{{i}}_{1}})\\ & d_{(aa)}=1-(d_{(ac)}+d_{(ab)})\end{split} \tag{3}\]  
where \(\stackrel{{\rightarrow}}{{i}}_{1}^{*}\) denotes the reference of the input phase's current fundamental component; and \(d_{(ac)},d_{(ab)}\), and \(d_{(aa)}\) represent the duty cycle of base vectors \(\stackrel{{\rightarrow}}{{i}}_{2}\), \(\stackrel{{\rightarrow}}{{i}}_{1}^{*}\), and zero state (\(aa\) indicates shorted phase leg), respectively. The notation in Figure 2 is such that the first letter denotes the positive rail and the second letter, the negative rail. For instance, \(\stackrel{{\rightarrow}}{{i}}_{1,(ab)}\) means that \(a\) is positive and \(b\) is negative (S156). The modulation index \(m\) is defined as the ratio of the peak phase current to the DC link current.  
\[m=\frac{\stackrel{{\rightarrow}}{{i}}_{k}}{I_{DC}}\,\quad k=\textbf{a,b,c} \tag{4}\]  
By neglecting converter losses and applying the principle of input/output power balance, the modulation index can also be expressed as [31]:  
\[m=\frac{\sqrt{2}}{3}\frac{V_{DC}}{V_{AC}} \tag{5}\]  
This leads to an important constraint between the input source voltage and load voltage:  
\[V_{DC}\leq\frac{3}{\sqrt{2}}V_{AC}\cdot\cos\phi \tag{6}\]  
where \(\phi\) is the phase shift between the fundamental component of the pole output current (square wave current) and the load (AC source) voltage.