## 3 Analysis of ZSCC in Paralleled PWM Converters  
The parallel system studied in this work consisted of two three-phase voltage-source PWM converters, as shown in Figure 1, connected to the common DC link, which is depicted in Figure 4.  
Figure 4: Topology of two paralleled three-phase voltage-source PWM converters.  
Figure 3: Control block of three-phase PWM converter.  
From Figure 4 and Equation (1), the average model of the parallel system in static three-phase coordinates can be expressed as follows:  
\[\left\{\begin{array}{l}L_{i}\frac{\mathrm{d}i_{\mathrm{d}i}}{\mathrm{d}t}+R_{i }i_{\mathrm{a}i}=u_{\mathrm{a}}-d_{\mathrm{a}i}u_{\mathrm{dc}}-u_{\mathrm{NO}} \\ L_{i}\frac{\mathrm{d}i_{\mathrm{d}i}}{\mathrm{d}t}+R_{i}i_{\mathrm{b}i}=u_{ \mathrm{b}}-d_{\mathrm{b}i}u_{\mathrm{dc}}-u_{\mathrm{NO}}\\ L_{i}\frac{\mathrm{d}i_{\mathrm{d}i}}{\mathrm{d}t}+R_{i}i_{\mathrm{c}i}=u_{ \mathrm{c}}-d_{\mathrm{c}i}u_{\mathrm{dc}}-u_{\mathrm{NO}}\\ 2C_{\mathrm{dc}}\frac{\mathrm{d}u_{\mathrm{dc}}}{\mathrm{d}t}=\sum\limits_{i= 1}^{2}(d_{\mathrm{a}i}a_{\mathrm{i}i}+d_{\mathrm{b}i}i_{\mathrm{b}i}+d_{ \mathrm{c}i}i_{\mathrm{c}i})-i_{\mathrm{o}}\end{array}\right. \tag{5}\]  
where expressions with subscript \"\(i\)\" refer to the related variables of the \(i\)th converter, \(i=1,2\).  
For a single PWM converter, no ZSCC is generated because there is no circulating current path. However, when two converters are connected in parallel, a path will be formed. If a zero-sequence voltage exists, ZSCC will also be generated on the loop impedance. The ZSCC \(i_{\mathrm{z}}\) is defined in this study as follows:  
\[\left\{\begin{array}{l}i_{\mathrm{z}1}=i_{\mathrm{a}1}+i_{\mathrm{b}1}+i_{ \mathrm{c}1}\\ i_{\mathrm{z}}=i_{\mathrm{z}1}=-i_{\mathrm{z}2}\end{array}\right. \tag{6}\]  
where \(i_{\mathrm{z}x}\) denotes the ZSCC of the \(x\)th module, \(x=1,2\). The ZSCC of Module 1 and the circulating current of Module 2 are equal in size and in opposite directions.  
In the SVPWM control, because the third harmonic is injected, the sum of the output duty ratios of the three-phase bridge arms of the PWM converter will not be equal to zero. The zero-sequence duty ratio of the \(i\)th\((i=1,2)\) converter can be defined as follows:  
\[d_{\mathrm{z}i}=d_{\mathrm{a}i}+d_{\mathrm{b}i}+d_{\mathrm{c}i}. \tag{7}\]  
The voltage of the three-phase alternating current was balanced in this study, which means that \(u_{\mathrm{a}}+u_{\mathrm{b}}+u_{\mathrm{c}}=0\). Therefore, combining Equations (5)-(7) results in the following:  
\[\left\{\begin{array}{l}L_{1}\frac{\mathrm{d}i_{\mathrm{z}1}}{\mathrm{d}t}+R _{1}i_{\mathrm{z}1}+u_{\mathrm{dc}}d_{\mathrm{z}1}+3u_{\mathrm{NO}}=0\\ L_{2}\frac{\mathrm{d}i_{\mathrm{z}2}}{\mathrm{d}t}+R_{2}i_{\mathrm{z}2}+u_{ \mathrm{dc}}d_{\mathrm{z}2}+3u_{\mathrm{NO}}=0\end{array}\right.. \tag{8}\]  
The equivalent mathematical model of ZSCC can be derived from Equation (8) as follows:  
\[(L_{1}+L_{2})\frac{\mathrm{d}i_{\mathrm{z}}}{\mathrm{d}t}+(R_{1}+R_{2})i_{ \mathrm{z}}=(d_{\mathrm{z}2}-d_{\mathrm{z}1})u_{\mathrm{dc}}. \tag{9}\]  
The physical circuital model of Equation (9) is shown in Figure 5, where \(u_{\mathrm{z}i}=d_{\mathrm{z}i}u_{\mathrm{dc}}\) denotes the zero-sequence voltage of the \(i\)th converter, \(i=1,2\).  
Figure 5: Equivalent physical model of zero-sequence circulating current (ZSCC) in paralleled converters.  
Figure 5 demonstrates that, when analyzing the ZSCC problem, each converter module can be equivalent to a branch in the physical circuit, and two branches can be combined into a circulating current path. It can be seen that the magnitude of ZSCC is related to the AC-side inductance, equivalent resistance, and zero-sequence voltage of the two branches, and controlling the zero-sequence voltage difference to zero is the key to eliminating ZSCC.