## 2 Dual-Loop Decoupled Control Strategy of PWM Converter  
The control of a single converter was the basis for studying the ZSCC problem. The three-phase full-bridge voltage-source PWM converter was used in this study. Its input is three-phase balanced AC current, and the output is DC current. Figure 1 illustrates the main circuit structure of this kind of converter [25], where \(u_{i}(i=\mathrm{a},\mathrm{b},\mathrm{c})\) is the voltage of the AC side, \(L\) is the filter inductance at the AC side, \(R\) is the equivalent resistance of the converter, \(i_{k}(k=\mathrm{a},\mathrm{b},\mathrm{c})\) is the three-phase current at the AC side, \(S_{i}(i=\mathrm{a},\mathrm{b},\mathrm{c})\) is the three bridge arms, \(C_{\mathrm{dc}}\) is the capacitor at the DC side, \(u_{\mathrm{dc}}\) is the DC bus voltage, \(i_{\mathrm{o}}\) is the load current at the DC side, and \(R_{\mathrm{L}}\) is the DC-side resistance. Assuming the negative pole of the DC-side capacitor as the reference point, the mathematical model of a single PWM converter shown in Figure 1 can be expressed as follows:  
\[\left\{ \begin{array}{l}L\frac{\mathrm{d}i_{\mathrm{a}}}{\mathrm{d}t}+Ri_{ \mathrm{a}}=u_{\mathrm{a}}-d_{\mathrm{a}}u_{\mathrm{dc}}-u_{\mathrm{NO}}\\ L\frac{\mathrm{d}i_{\mathrm{b}}}{\mathrm{d}t}+Ri_{\mathrm{b}}=u_{\mathrm{b}}-d_ {\mathrm{b}}u_{\mathrm{dc}}-u_{\mathrm{NO}}\\ L\frac{\mathrm{d}i_{\mathrm{c}}}{\mathrm{d}t}+Ri_{\mathrm{c}}=u_{\mathrm{c}}-d_ {\mathrm{c}}u_{\mathrm{dc}}-u_{\mathrm{NO}}\\ C_{\mathrm{dc}}\frac{\mathrm{d}u_{\mathrm{dc}}}{\mathrm{d}t}=d_{\mathrm{a}}i_{ \mathrm{a}}+d_{\mathrm{b}}i_{\mathrm{b}}+d_{\mathrm{c}}i_{\mathrm{c}}-i_{ \mathrm{o}}\end{array} \right. \tag{1}\]  
where \(u_{\mathrm{NO}}\) is the neutral voltage at the AC side, and \(d_{i}(i=\mathrm{a},\mathrm{b},\mathrm{c})\) represents the output duty ratio of the three-phase bridge arm. The above mathematical model of a converter in static three-phase coordinates is very intuitive. However, because the voltage and current on the AC side are time-varying AC variables, in order to realize the zero steady-state error control, it is necessary to change the three-phase sinusoidal quantities into DC quantities. At present, the transformation from three-phase static coordinate system \(ABC\) to two-phase synchronous rotating coordinate system \(d\)-\(q\) is mostly realized by three-dimensional coordinate transformation [26]. The \(d\)-\(q\) type mathematical model of the converter can be derived as follows:  
\[\left\{ \begin{array}{l}L\frac{\mathrm{d}i_{\mathrm{d}}}{\mathrm{d}t}+Ri_{ \mathrm{d}}=u_{\mathrm{d}}-d_{\mathrm{d}}u_{\mathrm{dc}}+\omega Li_{\mathrm{q}} \\ L\frac{\mathrm{d}i_{\mathrm{q}}}{\mathrm{d}t}+Ri_{\mathrm{q}}=u_{\mathrm{q}}-d_ {\mathrm{q}}u_{\mathrm{dc}}-\omega Li_{\mathrm{d}}\\ C_{\mathrm{dc}}\frac{\mathrm{d}u_{\mathrm{dc}}}{\mathrm{d}t}=\frac{3}{2}\!\left( d_{\mathrm{d}}i_{\mathrm{d}}+d_{\mathrm{q}}i_{\mathrm{q}}\right)-i_{\mathrm{o}} \end{array} \right. \tag{2}\]  
where \(\omega\) is the fundamental frequency of three-phase alternating current. Figure 1: Structure of three-phase voltage-source pulse width modulation (PWM) converter. According to Equation (2), in the mathematical model of \(d\)-\(q\) reference frame, there is a coupling phenomenon between the \(d\)-axis component and the \(q\)-axis component, which leads to the inability to adjust the active and reactive components, respectively, of the system. Therefore, it is necessary to decouple these coupling terms. In addition, PI control was utilized in this study to achieve the goal of zero steady-state error tracking [27]. Based on these theories, the inner current control loop is as follows:  
\[\left\{ \begin{array}{l}u_{\mathrm{d}}^{*}=-\Big{(}k_{\mathrm{ip}}+\frac{k_{ \mathrm{i}}}{s}\Big{)}\Big{(}i_{\mathrm{d}}^{*}-i_{\mathrm{d}}\Big{)}+u_{ \mathrm{d}}+\omega Li_{\mathrm{q}}\\ u_{\mathrm{q}}^{*}=-\Big{(}k_{\mathrm{ip}}+\frac{k_{\mathrm{i}}}{s}\Big{)} \Big{(}i_{\mathrm{q}}^{*}-i_{\mathrm{q}}\Big{)}+u_{\mathrm{q}}-\omega Li_{ \mathrm{d}}\end{array} \right.