## 4 Control Strategy for Suppressing ZSCC
Moreover, because only one converter is needed to control the entire ZSCC, \(\chi_{1}\) is set to zero. What is more, the sum of the duty ratio of the two nonzero vectors and the zero vector is 1. Therefore, Equation (13) can be simplified as follows:  
\[(L_{1}+L_{2})\frac{\text{d}i_{x}}{\text{d}t}=\frac{1}{2}(\Delta D_{12}-12\chi )u_{\text{dc}} \tag{14}\]  
where \(\Delta D_{12}=d_{11}-d_{12}-d_{21}+d_{22}\). In order to analyze the system function generating the ZSCC, the Laplace transform of Equation (14) is obtained as follows:  
\[I_{\text{z}}(s)=\frac{-6u_{\text{dc}}/(L_{1}+L_{2})}{s}\bigg{(}\chi(s)-\frac{ 1}{12}\Delta D_{12}(s)\bigg{)}. \tag{15}\]  
Therefore, Equation (15) shows that the generating system of ZSCC can be regarded as a first-order system with disturbance. In this system, the input is the adjusting factor, and the output is the ZSCC. For the first-order system, PI control is generally used. The input of the PI controller here is the difference between the actual value and the reference value of ZSCC, and the output is the adjusting factor. However, as shown in [32], the main low-frequency harmonics in ZSCC are the 3 rd, 9 th, and 15 th. Hence, to suppress these harmonics, the resonant controller needs to be added to the PI control. As the gain of an ideal resonant controller at the resonant frequency is infinite, which cannot be achieved in practice, the quasi-resonant controller is currently used the most [33]. This kind of controller can increase system bandwidth, suppress higher harmonics, and is easy to implement. In this study, a PI controller with quasi-resonant control (PIQR) was used. Its system function can be expressed as follows:  
\[G_{PIQR}(s)=k_{\text{pz}}+\frac{k_{\text{iz}}}{s}+\sum_{n=3,9,15}\frac{2k_{ \text{rzzn}}\omega_{\text{c}}s}{s^{2}+2\omega_{\text{c}}s+(n\omega)^{2}} \tag{16}\]  
where \(k_{\text{pz}}\) and \(k_{\text{iz}}\) are the proportional and integral coefficients of the ZSCC control loop, respectively, \(k_{\text{rzzn}}\) is the resonance coefficient for suppressing the \(n\)th harmonic, \(\omega_{\text{c}}\) is the cut-off angular frequency, and \(\omega\) is the fundamental angular frequency of the AC-side voltage. According to Equation (16), the generation of zero-vector duty ratio adjusting factor can be derived as follows:  
\[\chi=\left(k_{\text{pz}}+\frac{k_{\text{iz}}}{s}+\sum_{n=3,9,15}\frac{2k_{ \text{rzzn}}\omega_{\text{c}}s}{s^{2}+2\omega_{\text{c}}s+(n\omega)^{2}} \right)(i_{x}^{*}-i_{x}) \tag{17}\]where the ZSCC reference value \(i_{\text{z}}^{*}\) is set to zero. Due to the differences in line parameters, the currents on the branches of different converters are unequal, so the reference voltages generated by the current regulators are also different, which means nonzero vector duty ratio differences are bound to occur. It can be seen from Equation (15) that this difference will have a certain effect on the adjusting factor. However, the PIQR controller does not eliminate this disturbance, so this paper proposes a control strategy that combines a feedforward compensation scheme with PIQR. The improved adjusting factor generation can be expressed as follows:  
\[\chi^{\prime}=\left(k_{\text{Pz}}+\frac{k_{\text{iz}}}{s}+\sum_{n=3,9,15}\frac {2k_{\text{rzrz}}\omega_{\text{c}}s}{s^{2}+2\omega_{\text{c}}s+(n\omega)^{2}} \right)\left(i_{\text{z}}^{*}-i_{\text{z}}\right)+\frac{\Delta D_{12}(s)}{12}. \tag{18}\]  
The ZSCC control loop proposed in this paper and obtained according to Equation (18) is shown in Figure 7.  
Based on the above analysis, the control strategy of the two parallel converter system is depicted in Figure 8. In this control scheme, the two converters are controlled by an independent voltage-current loop. The difference is that when the switching signal is generated through space vector modulation, the first converter uses the modulation method, as shown in Figure 6a, while the second converter needs to use the modulation method after adding the zero-vector adjusting factor, as shown in Figure 6b.