## 2 Single-Phase Grid-Connected Power Conversion System  
Consider a typical dual-stage off-grid operating power conversion system depicted in Figure 1a, formed by a mains-interfacing AC/DC converter and a downstream DC-DC converter, interconnected via a DC link capacitance \(C_{DC}\). The grid is assumed to act as energy source in the subsequent discussion, yet the methodology remains valid in case the energy flows from the DC side to the AC side. Moreover, all the variables below are switching-cycle-average, i.e., free from switching components. Considering harmonic-free mains and unity power factor operation for brevity (yet without loss of generality), the grid-side voltage and current are given by [52]  
\[v_{G}(t)=v_{M}(t)\sin(\int\omega(t)dt),\ \ \ i_{G}(t)=i_{M}(t)\sin(\int\omega(t)dt) \tag{1}\]  
with \(\omega\) denoting the instantaneous base frequency of the grid and \(v_{M}\), \(i_{M}\) representing the corresponding magnitudes. In steady state, the above quantities are constant so that  
\[v_{M}(t)=V_{M},\ \ \ i_{M}(t)=I_{M},\ \ \ \omega(t)=\omega_{G}. \tag{2}\]  
In steady state, the base frequency of the mains may attain the following values,  
\[\omega_{G}=\alpha\cdot \begin{cases}100\pi,&50\ \mathrm{Hz\ \textit{mains}}\\ 120\pi,&60\ \mathrm{Hz\ \textit{mains}}\end{cases}  \tag{3}\]  
with \(\alpha\in[\alpha_{\min}<1,\ \alpha_{\max}>1]\) representing the corresponding deviation from nominal values (i.e., \(1-\alpha\mid\) indicates the corresponding uncertainty). On the other hand, DC-side variables are also constant in steady state so that  
\[v_{L}(t)=V_{L},\ \ \ i_{L}(t)=I_{L}. \tag{4}\]  
Assuming lossless conversion, a functional diagram of the power conversion system is shown in Figure 1b with  
\[p_{L}(t)=v_{L}(t)i_{L}(t),\ \ \ p_{G}(t)=v_{G}(t)i_{G}(t),\ \ \ v_{DC}(t)i_{DC}(t)=p_{G}(t)-p_{L}(t), \tag{5}\]  
where \(v_{DC}(t)\) and \(i_{DC}(t)\) represent the voltage across the DC link capacitance \(C_{DC}\) and the corresponding current, respectively. Combining (1) with (5) yields [10]\[v_{DC}(t)i_{DC}(t)=v_{DC}(t)\underbrace{C_{DC}\frac{dv_{DC}(t)}{dt}}_{i_{DC}(t)}=0.5v _{M}(t)i_{M}(t)-0.5v_{M}(t)i_{M}(t)\cos\biggl{(}2\int\omega(t)dt\biggr{)}-v_{L}(t )i_{L}(t), \tag{6}\]  
Figure 1: Typical single-phase dual-stage grid-connected power conversion system. (**a**) Generalized topology; (**b**) low-frequency functional diagram; (**c**) bridge-rectifier-based AC/DC converter; (**d**) bridgeless rectifier/inverter. reducing in steady state to  
\[C_{DC}v_{DC}^{\rm ss}(t)\frac{d\
u_{DC}^{\rm ss}(t)}{dt}=0.5V_{M}I_{M}-0.5V_{M}I_{M }\cos(2\omega_{G}t)-V_{L}I_{L}, \tag{7}\]  
which may be split into  
\[0.5V_{M}I_{M}=V_{L}I_{L} \tag{8}\]  
and  
\[C_{DC}v_{DC}^{\rm ss}(t)\frac{d\
u_{DC}^{\rm ss}(t)}{dt}=-0.5V_{M}I_{M}\cos(2 \omega_{G}t), \tag{9}\]  
respectively. Consequently, the steady-state expressions of grid-side current and DC link voltage are given by  
\[I_{M}=\frac{2P_{L}}{V_{M}},\ \ P_{L}=V_{L}I_{L} \tag{10}\]  
and  
\[v_{DC}^{\rm ss}(t)=V_{DC}^{*}\sqrt{1-\frac{P_{L}}{\omega_{G}\big{(}V_{DC}^{*} \big{)}^{2}C_{DC}}\sin(2\omega_{G}t)}, \tag{11}\]  
respectively, with \(V_{DC}^{*}\) denoting the DC link voltage set point value. In practice,  
\[\frac{P_{L}}{\omega_{G}\big{(}V_{DC}^{*}\big{)}^{2}C_{DC}}<<1 \tag{12}\]  
typically holds; hence, (11) may be further reduced to [11]  
\[p_{L}(t)=v_{L}(t)i_{L}(t),\ \ \ p_{G}(t)=v_{G}(t)i_{G}(t),\ \ \ v_{DC}(t)i_{DC}(t)=p_{G}(t)-p_{L}(t), \tag{5}\]0  
Consequently, steady-state DC link voltage would always be formed by the dominating DC component and the significantly smaller double-base-frequency pulsating ripple component, proportional to the load power and inversely proportional to the DC link capacitance, DC link voltage set point, and base frequency of the mains. It must be emphasized that two typical realizations of the grid-interfacing AC/DC converter exist in practice.