# Research on Composite Control Strategy of Quasi-Z-Source DC-DC Converter for Fuel Cell Vehicles
## 2 Topology Operation Principle Analysis and Dynamic Modeling
### Dynamic Modeling  
In this paper, the converter operates in continuous conduction mode (CCM). The state space averaging method is used to model the converter. It is assumed that the inductors, capacitors and power semiconductors are ideal components. When the power switch \(Q\) is turned on, capacitor \(C_{3}\) and capacitor \(C_{5}\) are connected in parallel as shown in Figure 2. Equation (11) can be obtained in the loop circuit of \(C_{3}\) and \(C_{5}\) in Figure 2.  
\[C_{3}\frac{du_{C3}}{dt}=-C_{5}\frac{du_{C5}}{dt} \tag{11}\]  
Therefore, there is an invalid state space variable between \(C_{3}\) and \(C_{5}\) in the state space equation. The coupling relationship between \(C_{3}\) and \(C_{5}\) can be eliminated by introducing series equivalent resistance \(r_{1}\) into the loop circuit of \(C_{3}\) and \(C_{5}\). Equation (11) can be written as Equation (12). \[C_{3}\frac{du_{C3}}{dt}=-\frac{u_{C5}-u_{C3}}{r_{1}} \tag{12}\]  
where, \(r_{1}\) is the series equivalent resistance in the loop circuit of \(C_{3}\) and \(C_{5}\). \(r_{1}=0.1\)\(\Omega\). When the power switch \(Q\) is turned off, the series equivalent resistance \(r_{2}\) (\(r_{2}=0.1\)\(\Omega\)) can be adopted to eliminate the coupling relationship between \(C_{1}\), \(C_{2}\), \(C_{3}\), \(C_{4}\) and \(C_{5}\). The state variables are \(x=[i_{L1},i_{L2},u_{C1},u_{C2},u_{C3},u_{C4},u_{C5}]\). The input variable is \(u=u_{\text{in}}\) and the output variable is \(y=u_{\text{o}}\). The open-time of the power switch is \(dT\) and the close-time is (1-\(d\))\(T\) in a switching period. Therefore, when power switch \(Q\) is turned on, the state space equation can be obtained as Equation (13). When power switch \(Q\) is turned off, the state space equation can be obtained as Equation (14). \[\left\{ \begin{array}{l}\dot{x}=A_{1x}+B_{1u}\\ y=C_{1x}+D_{1u}\\ \end{array} \right. \tag{13}\]  
\[\left\{ \begin{array}{l}\dot{x}=A_{2x}+B_{2u}\\ y=C_{2x}+D_{2u}\\ \end{array} \right. \tag{14}\]where the coefficient matrixes \(A_{1}\) and \(A_{2}\) can be obtained as in Equation (15). The coefficient matrixes \(B_{1}\), \(C_{1}\), \(D_{1}\), \(B_{2}\), \(C_{2}\) and \(D_{2}\) can be obtained as in Equation (16). \[A_{1}=\left[ \begin{array}{cccccc}0&0&0&\frac{1}{L_{1}}&0&0&0\\ 0&0&\frac{1}{L_{2}}&0&0&0&0\\ 0&\frac{-1}{L_{1}}&0&0&0&0&0\\ \frac{-1}{C_{2}}&0&0&0&0&0&0\\ 0&0&0&0&\frac{-1}{C_{3}r_{1}}&0&\frac{1}{C_{3}r_{1}}\\ 0&0&0&0&\frac{-1}{C_{4}R}&\frac{-1}{C_{4}R}\\ 0&0&0&0&\frac{1}{C_{5}r_{1}}&-\frac{1}{C_{5}r_{1}}&\frac{-R-r_{1}}{C_{5}R_{1}} \\ \end{array} \right] \tag{15}\] \[A_{2}=\left[ \begin{array}{cccccc}0&0&\overline{L_{1}}&0&0&0&0\\ 0&0&0&\overline{-1}&0&0&0\\ \frac{1}{C_{1}}&0&\frac{-1-r_{2}}{C_{1}r_{1}}&\frac{-r_{1}r_{2}}{C_{1}r_{2}}& \frac{-1}{C_{1}r_{2}}&\frac{1}{C_{1}r_{2}}&\frac{r_{1}+r_{2}}{C_{1}r_{2}}\\ 0&\frac{1}{\overline{u}_{0}}&\frac{-r_{1}-r_{2}}{C_{1}r_{2}}&\frac{-1}{C_{1}r_ {2}}&\frac{1}{C_{1}r_{2}}&\frac{r_{1}+r_{2}}{C_{1}r_{2}}\\ 0&0&\frac{1}{\overline{u}_{0}r_{1}}&\frac{-1}{\overline{u}_{0}r_{1}}&\frac{-1 }{\overline{u}_{0}r_{1}}&\frac{-1}{\overline{u}_{0}r_{1}}&\frac{1}{C_{3}r_{2}} &\frac{r_{1}+r_{2}}{C_{1}r_{2}}\\ 0&0&\frac{1}{\overline{C}_{3}r_{2}}&\frac{1}{\overline{r}_{2}r_{2}}&\frac{-1} {\overline{r}_{2}r_{2}}&\frac{1}{\overline{r}_{2}r_{2}}&\frac{1}{\overline{r} _{2}}\\ 0&0&\frac{1}{\overline{C}_{4}r_{2}}&\frac{1}{\overline{C}_{4}r_{2}}&\frac{1}{ \overline{C}_{4}r_{2}}&\frac{-R-r_{2}}{C_{4}R_{2}}&\frac{-R_{1}-r_{2}}{C_{4}R_ {2}}\\ 0&0&\frac{\overline{r}_{1}+r_{2}}{C_{5}r_{1}r_{2}}&\frac{\overline{r}_{1}+r_{2} }{C_{5}r_{1}r_{2}}&\frac{-R-r_{2}}{C_{5}R_{2}}&\frac{-R_{1}-r_{1}r_{2}-R_{2}} {C_{5}R_{1}r_{2}}\\ \end{array} \right] \tag{16}\]  
According to Equations (13) and (14), the state space average equation is obtained as in Equation (17). \[\left\{ \begin{array}{l}\overline{x}=(A_{1}\overline{d}+A_{2}(1-\overline{d} ))\overline{x}+(B_{1}\overline{d}+B_{2}(1-\overline{d}))\overline{u}\\ \overline{y}=(C_{1}\overline{d}+C_{2}(1-\overline{d}))\overline{x}+(D_{1} \overline{d}+D_{2}(1-\overline{d}))\overline{u}\end{array} \right.