## 4 Direct Output Current Control Based on Virtual Voltage Compensation under No Backflow Power
### Power Characteristic Analysis  
For a two-port network with sinusoidal voltages and currents, the instantaneous power can be expressed as (26), where \(P_{*}\) and \(Q_{*}\) are the active and reactive power, respectively. As shown in Figure 4, the first part of (26) is always positive while the second part is sometimes negative. Therefore, \(p_{*}(t)\) will be always positive if \(Q_{*}\) is eliminated, i.e., the backflow will be eliminated.  
\[p_{*}(t)=P_{*}[1-\cos(2\omega t)]-Q_{*}\sin(2\omega t) \tag{26}\]  
If the parasitic parameters and losses of the converter are ignored, the output active power can be seen as the sum of the active power transferred by each harmonic, i.e.,  
\[P=\sum P_{n}=\frac{8 NU_{\text{in}}U_{0}}{\pi^{2}Z_{0}}\sum_{n=3,5,7\dots}\frac {\cos^{2}(n\pi d_{1})\sin(2n\pi d)}{n^{3}(2-n^{2})} \tag{27}\]  
Similarly, the reactive power can be derived as  
\[Q=\sum Q_{n}=\frac{8 NU_{\text{in}}U_{0}}{\pi^{2}Z_{0}}\sum_{n=3,5,7\dots}\frac {\cos^{2}(n\pi d_{1})\left[\cos(2n\pi d)+k(n^{2}-1)\right]}{n^{3}(2-n^{2})} \tag{28}\]  
According to (28), it can be seen that \(Q_{1}\) = 0 when \(d\) = 0.25, which means that the backflow power corresponding to the fundamental component is eliminated. However, the reactive power of other harmonic components is not zero, meaning that the backflow power actually exists. To analyze the proportion of power generated by the harmonic components, the ratios of active and reactive generated by the harmonic components to the active power generated by the fundamental component are defined as the active power ratio \(Y_{P}\) and the reactive power ratio \(Y_{Q}\), expressed as  
\[Y_{P}=\sum_{n=1,3,5\dots}\left(\frac{P_{n}}{P_{1}}\right)\times 100\% \tag{29}\]  
\[Y_{Q}=\sum_{n=1,3,5\dots}\left(\frac{Q_{n}}{P_{1}}\right)\times 100\% \tag{30}\]  
Figure 5 shows the relationship between the active power ratio and phase shift ratios, indicating the active power ratio is closed 0 in most regions. Taking \(k\) = 1 as an example, Figure 6 shows the relationship between the reactive power ratio and phase shift ratios. Different from the active power ratio, the reactive power ratio changes significantly for different phased shift ratios. As a result, the transmission power expression can be expressed using the 1 st harmonic approximation method, reduced from (27) to (31). However, 1 st harmonic approximation method should not be utilized in the backflow elimination analysis, so it is necessary to adopt another method in the following calculation of design regions of no backflow power.  
\[P=\frac{8 NU_{\text{in}}U_{0}\cos^{2}(\pi d_{1})\sin(2\pi d)}{\pi^{2}Z_{0}} \tag{31}\]  
Figure 4: The relationship between backflow power and reactive power.  
The DC voltage gain of LCL-DAB converters is defined as \(G_{\text{DC}}=U_{\text{o}}/U_{\text{in}}\). With the resistive load R, the DC voltage gain can be calculated as  
\[G_{\text{DC}}=\frac{8 NR\cos^{2}(\pi d_{1})\sin(2\pi d)}{\pi^{2}Z_{0}} \tag{32}\]  
After removing the filter capacitor, the LCL-DAB converter can be regarded as a traditional DAB converter. Figure 7 shows the voltage gain curves of the LCL-DAB and the traditional DAB converter varying with phase shift ratios \(d_{1}\) and \(d\). For both types of converters, the voltage gain \(G_{\text{DC}}\) increases and then decreases with the increase in \(d\) and the curves are symmetric about the plane \(d=0.25\). Furthermore, \(G_{\text{DC}}\) decreases with the increase in \(d_{1}\) and reaches zero when \(d_{1}=0.5\). It can be seen that the LCL-DAB converter can achieve higher voltage gain than the traditional DAB converter under the same inductance.  
The above analysis indicates that it is inaccurate and infeasible to eliminate backflow power by eliminating reactive power using 1 st harmonic approximation. Therefore, the exact design regions of no backflow power are presented and analyzed in the following section.