## 4 Control Strategy for Suppressing ZSCC  
In this study, each converter in the parallel system adopted the dual-loop decoupled control strategy described in Section 2. Therefore, by synthesizing the key element of the ZSCC suppression shown in Figure 5, the zero-sequence voltage could be adjusted by controlling the zero-sequence duty ratio of the converter. When the zero-sequence duty ratios of the branches are equal, the zero-sequence voltages are equal, and the ZSCC is naturally suppressed. In addition, for two paralleled converters, only the zero-sequence duty ratio of one converter needs to be controlled to suppress the ZSCC of the entire system. Extending to the parallel system with \(N(N\geq 3)\) converters, the suppression of ZSCC needs to control the zero-sequence duty ratios of \(N-1\) converters. As noted in Section 2, in space vector modulation, the zero-sequence duty ratio is related to the action time of the state vectors. Among them, two nonzero state vectors can uniquely determine a reference voltage vector under constant action time, and changing the action time of the zero-state vector will not affect the output voltage and current. Therefore, the zero-sequence duty ratio is usually regulated by controlling the zero-vector action time. Taking sector I as an example, the switching signal before and after adjustment in SVPWM control are presented in Figure 6, where the duty ratios of two nonzero vectors are \(d_{1}\) and \(d_{2}\), respectively, and the duty ratio of zero vector is \(d_{0}\). As shown in Figure 6a, the zero-sequence duty ratio before adjustment can be expressed as follows:  
\[ \begin{array}{l}d_{\rm z}=d_{\rm a}+d_{\rm b}+d_{\rm c}=(d_{1}+d_{2}+0.5d_{0} )+(d_{2}+0.5d_{0})+0.5d_{0}\\ =d_{1}+2d_{2}+1.5d_{0}\end{array} . \tag{10}\]  
According to Figure 6b, an adjusting factor \(\chi\) is added to the duty ratio of zero vector in a PWM period. The action time of \(\mathbf{V}_{7}\) and \(\mathbf{V}_{0}\) are changed to \((d_{0}/2-\chi)T_{\rm s}\) and \((d_{0}/2+\chi)T_{\rm s}\), respectively. After adjusting, the zero-sequence duty ratio \(d_{\rm z}\)' can be described as follows:  
\[ \begin{array}{l}d_{\rm z}\,{}^{\prime}=d_{\rm a}\,{}^{\prime}+d_{\rm b}\,{}^ {\prime}+d_{\rm c}\,{}^{\prime}=(d_{1}+d_{2}+0.5d_{0}-2\chi)+(d_{2}+0.5d_{0}-2 \chi)+0.5d_{0}-2\chi\\ =d_{1}+2d_{2}+1.5d_{0}-6\chi\end{array} . \tag{11}\]  
Figure 6: Switching signals for space vector pulse width modulation (SVPWM) control in sector I: (**a**) before adjustment; (**b**) after adjustment. Moreover, Figure 6b shows that adjusting the factor directly changes the switching time \(T_{A\text{on}}\), \(T_{\text{Bon}}\), and \(T_{\text{Con}}\) of the state vector in SVPWM. The switching time after correcting can be derived as follows:  
\[\left\{ \begin{array}{l}T_{A\text{on}}{}^{\prime}=T_{A\text{on}}+\chi T_{ \text{s}}\\ T_{\text{Bon}}{}^{\prime}=T_{A\text{on}}{}^{\prime}+\frac{d_{1}}{2}T_{\text{s}} \\ T_{\text{Con}}{}^{\prime}=T_{\text{Bon}}{}^{\prime}+\frac{d_{2}}{2}T_{\text{s}} \end{array} \right. \tag{12}\]  
where \(\chi\) should be within the range of \([-d_{0}/4,d_{0}/4]\), which means that the action time of the zero vector is not less than zero without changing the action time of the nonzero vector. Substituting Equation (11) into Equation (9) can get the modified ZSCC model, which can be expressed as follows:  
\[ \begin{array}{l}(L_{1}+L_{2})\frac{\text{d}i_{x}}{\text{d}t}+(R_{1}+R_{2})i _{\text{z}}=(d_{2}{}^{\prime}-d_{x1}{}^{\prime})u_{\text{dc}}\\ =[(d_{12}+2d_{22}+1.5d_{02}-6\chi_{2})-(d_{11}+2d_{21}+1.5d_{01}-6\chi_{1})]u _{\text{dc}}\end{array}  \tag{13}\]  
where \(d_{ij}\) represents the duty ratio of the related vector in the \(j\)th module, and \(\chi_{j}\) denotes the zero-vector duty ratio adjusting factor of the \(j\)th module, \(j=1,2\). In practical engineering, the equivalent resistance value of the converter is much smaller than its reactance value, so the \(R_{1}\) and \(R_{2}\) can be ignored [31].