*2.4. Dynamic Model*

A state-space averaged model of converters is a key aspect in the control of power electronic systems. In this study, averaged models were developed based on common techniques to complement the study of the proposed converter under previously mentioned applications. The resulting model, when a voltage source is used in the input port, is given by Equations (33).

$$\begin{aligned} L\_1 \frac{di\_{L1}}{dt} &= E - (1 - d)v\_{C1} - R\_{p1}i\_{L1} \\ L\_2 \frac{di\_{L2}}{dt} &= d(v\_{C1} - E) - (1 - d)v\_{C2} - R\_{p2}i\_{L2} \\ C\_1 \frac{dv\_{C1}}{dt} &= (1 - d)i\_{L1} - di\_{L2} \\ C\_2 \frac{dv\_{C2}}{dt} &= (1 - d)i\_{L2} - \frac{v\_{C2}}{R} .\end{aligned} \tag{33}$$

The averaged model, when a PV module with a coupling capacitor is used in the input port and a resistive load in the output port, results in

$$\begin{aligned} L\_1 \frac{di\_{L1}}{dt} &= v\_{Ci} - (1 - d)v\_{C1} - R\_{p1}i\_{L1} \\ L\_2 \frac{di\_{L2}}{dt} &= d(v\_{C1} - v\_{Ci}) - (1 - d)v\_{C2} - R\_{p2}i\_{L2} \\ C\_1 \frac{dv\_{C1}}{dt} &= (1 - d)i\_{L1} - di\_{L2} \\ C\_2 \frac{dv\_{C2}}{dt} &= (1 - d)i\_{L2} - \frac{v\_{C2}}{R} \\ C\_i \frac{dv\_{Ci}}{dt} &= i\_{pv} - i\_{L1} + di\_{L2} \end{aligned} \tag{34}$$

Finally, the averaged model of the converter in a PV application connected to a DC voltage system corresponds to

$$\begin{aligned} \mathbf{L}\_{1}\frac{d\dot{l}\_{L1}}{dt} &= v\_{\text{Ci}} - (1 - d)v\_{\text{C1}} - R\_{p1}\dot{i}\_{L1} \\ \mathbf{L}\_{2}\frac{d\dot{l}\_{L2}}{dt} &= d(v\_{\text{C1}} - v\_{\text{Ci}}) - (1 - d)v\_{\text{C2}} - R\_{p2}\dot{i}\_{L2} \\ \mathbf{C}\_{1}\frac{d\mathbf{v}\_{\text{C1}}}{dt} &= (1 - d)\dot{i}\_{L1} - d\dot{i}\_{L2} \\ \mathbf{C}\_{2}\frac{d\mathbf{v}\_{\text{C2}}}{dt} &= (1 - d)\dot{i}\_{L2} - \dot{i}\_{o\prime} \\ \mathbf{C}\_{i}\frac{d\mathbf{v}\_{\text{Ci}}}{dt} &= \dot{i}\_{pv} - \dot{i}\_{L1} + \dot{d}\dot{i}\_{L2} \end{aligned} \tag{35}$$

In the above models, *d*(*t*) represents the averaged duty ratio; *Rpn*, with *n* = 1, 2, represents the parasitic resistances in the converter in Equation (36).

$$R\_{pn} = R\_{Ln} + R\_{Mn}d + R\_{Dn}(1 - d),\tag{36}$$

where *RLn* is the resistance in the inductor, and *RMn* and *RDn* are the on-resistances of the active and passive switches, respectively.
