*3.1. Modeling of Three-Phase Interleaved Parallel DC/DC Converter Circuit*

Based on the circuit structure and working principle of a three-phase interleaved parallel bidirectional DC-DC converter, this article divides it into three identical Buck–Boost circuits, without considering the parasitic components of capacitors and inductors. The control flowchart of the composite controller is shown in Figure 8.

**Figure 8.** Control flowchart of composite controller.

The equivalent circuit diagram is shown in Figure 9a for when the converter is used as a Boost converter. The equivalent circuit diagram is shown in Figure 9b for when the converter is used as a Buck converter. The topology parameters are shown in Table 2.

**Figure 9.** Equivalent variable structure model of Boost converter and Buck converter: (**a**) Boost mode and (**b**) Buck mode.

**Table 2.** Topology diagram parameters of Buck mode and Boost mode.


The results of variable structure theory analysis can be used to obtain the state equation of the bidirectional DC-DC converter in Buck mode with continuous inductance current as follows:

Firstly, the Buck circuit is modeled and studied, and its equivalent circuit topology is shown in Figure 9. Write the state equation in stages and calculate the average variable.

(1) In 0 ≤ t ≤ dTS, switch the tube S conduction and diode VD cutoff, and, at this time, there is the following equation of state.

$$\begin{cases} \mathcal{L}\frac{\text{di}\_{\text{L}}(\mathbf{t})}{\text{d}\mathbf{t}} = \mathbf{u}\_{\text{in}}(\mathbf{t}) - \mathbf{u}\_{\text{o}}(\mathbf{t})\\ \mathcal{C}\frac{\text{du}\_{\text{o}}(\mathbf{t})}{\text{d}\mathbf{t}} = -\frac{\mathbf{u}\_{\text{o}}(\mathbf{t})}{\text{R}\_{\text{CPL}}} + \dot{\mathbf{u}}\_{\text{L}}(\mathbf{t}) \text{ }^{\prime} \end{cases} \tag{2}$$

(2) IndTS ≤ t ≤ TS, switch S off, diode VD conduction, and the inductor L release magnetic field can supply constant power load at the same time to charge the capacitor. The equation of state is as follows.

$$\begin{cases} \mathbf{L} \frac{\text{d}\mathbf{i}\_{\text{L}}(\mathbf{t})}{\text{d}\mathbf{t}} = -\mathbf{u}\_{\text{0}}(\mathbf{t})\\ \mathbf{C} \frac{\text{d}\mathbf{u}\_{\text{0}}(\mathbf{t})}{\text{d}\mathbf{t}} = -\frac{\mathbf{u}\_{\text{0}}(\mathbf{t})}{\text{R}\mathbf{c}\mathbf{P}} + \mathbf{i}\_{\text{L}}(\mathbf{t}) \text{ \textdegree } \end{cases} \tag{3}$$

By averaging (2) and (3), the following matrix equation can be obtained.

$$
\begin{pmatrix}
\frac{\text{d}\mathbf{i}\_{\text{L}}}{\text{d}\mathbf{t}}\\\frac{\text{d}\mathbf{u}\_{\text{0}}}{\text{d}\mathbf{t}}
\end{pmatrix} = \begin{pmatrix}
0 & -\frac{1}{\mathsf{L}}\\\frac{1}{\mathsf{C}\_{1}} & -\frac{1}{\mathsf{C}\_{1}\mathsf{R}}
\end{pmatrix} \begin{pmatrix}
\mathbf{i}\_{\text{L}}\\\mathbf{u}\_{\text{0}}
\end{pmatrix} + \begin{pmatrix}
\frac{\mathbf{u}\_{\text{in}}}{\mathsf{L}}\\\mathbf{0}
\end{pmatrix} \mathbf{u}\_{\text{\prime}}\tag{4}
$$

The state space equation in Boost mode with a continuous inductance current is as follows:

$$
\begin{pmatrix}
\frac{d\mathbf{i}\_{\rm l}}{dt} \\
\frac{d\mathbf{u}\_{\rm o}}{dt}
\end{pmatrix} = \begin{pmatrix}
0 & -\frac{1}{\mathsf{L}} \\
\frac{1}{\mathsf{C}\_{2}} & -\frac{1}{\mathsf{C}\_{2}\mathsf{R}}
\end{pmatrix} \begin{pmatrix}
\mathbf{i}\_{\rm l} \\
\mathbf{u}\_{\rm o}
\end{pmatrix} + \begin{pmatrix}
\frac{\mathsf{u}\_{\rm o}}{\mathsf{L}} \\
\end{pmatrix} \mathbf{u} + \begin{pmatrix}
\frac{\mathsf{u}\_{\rm l\rm in}}{\mathsf{L}} \\
0
\end{pmatrix},\tag{5}
$$

The transfer function can be derived through Laplace transform, using the average state space equation:

$$\begin{cases} \mathbf{G}\_{\rm id}(\mathbf{s}) = \frac{\mathbf{v}\_{\rm ln}(1 + \mathbf{R}\_{\rm L}\mathbf{C}\mathbf{s})}{\mathbf{R}\_{\rm L} + \mathbf{L}\mathbf{s} + \mathbf{R}\_{\rm L}\mathbf{C}\mathbf{L}\mathbf{s}^{2}}\\ \quad \mathbf{G}\_{\rm vd}(\mathbf{s}) = \frac{\mathbf{R}}{1 + \mathbf{R}\_{\rm L}\mathbf{C}\mathbf{s}} \end{cases} \tag{6}$$

This article first analyzes the Buck pattern.

$$\mathbf{P} = \frac{\mathbf{u}\_0 \mathbf{^2}}{\mathbf{R}\_{\mathbf{L}}} \mathbf{'} \tag{7}$$

where d is the duty cycle of the converter, and T is the switching cycle. A dynamic model of the Buck converter was established using the state-space averaging method. By substituting Equation (7) into (4) and linearizing it, we obtain the following:

$$\begin{cases} \frac{\text{di}\_{\text{L}}}{\text{dt}} = \frac{\text{v}\_{\text{ln}}}{\text{L}} \mathbf{u} - \frac{\text{v}\_{\text{O}}}{\text{L}}\\ \frac{\text{du}\_{\text{o}}}{\text{dt}} = \frac{\text{i}\_{\text{L}}}{\text{C}} - \frac{\text{P}}{\text{C} \text{v}\_{\text{o}}} - \frac{\text{v}\_{\text{o}}}{\text{R} \text{C}} \end{cases} \tag{8}$$

The voltage tracking error is defined as x1 = e = vo − vref, where vref is the reference voltage. The dynamic model in Equation (8) can be rewritten as follows:

$$
\dot{\mathbf{x}}\_1 = \frac{\dot{\mathbf{i}}\_\mathcal{L}}{\mathcal{C}} - \frac{\mathbf{v}\_\mathcal{O}}{\mathcal{R}\_\mathcal{L}\mathcal{C}} - \dot{\mathbf{v}}\_{\text{ref}} + \mathbf{d}\_1(\mathbf{t}) \tag{9}
$$

where d1(t) <sup>=</sup> <sup>−</sup> <sup>P</sup> Cvo , and another state variable is defined as x2 <sup>=</sup> iL <sup>C</sup> <sup>−</sup> vo RLC , so take the derivative of that and obtain the following:

$$\dot{\mathbf{x}}\_{2} = \frac{\mathbf{u}}{\mathbf{L}\mathbf{C}} \mathbf{v}\_{\text{in}0} - \frac{\mathbf{x}\_{1} + \mathbf{v}\_{\text{ref}}}{\mathbf{L}\mathbf{C}} - \frac{\mathbf{x}\_{2}}{\mathbf{R}\_{\text{L}}\mathbf{C}} + \mathbf{d}\_{2}(\mathbf{t}) \tag{10}$$

where d2(t) = <sup>1</sup> RC d1(t) <sup>+</sup> vin−vin0 LC , d2(t) is a more complex form of time varying, consisting of a constant power load and fluctuations in input voltage. The following equation can be obtained by sorting out Equations (9) and (10):

$$\begin{cases} \dot{\mathbf{x}}\_1 = \mathbf{x}\_2 + \mathbf{d}\_1\\ \dot{\mathbf{x}}\_2 = \frac{\mathbf{u}}{\mathbf{L}\mathbf{C}} \mathbf{v}\_{\text{in}0} - \frac{\mathbf{x}\_1 + \mathbf{v}\_{\text{ref}}}{\mathbf{L}\mathbf{C}} - \frac{\mathbf{x}\_2}{\mathbf{R}\mathbf{L}} + \mathbf{d}\_2 \end{cases} \tag{11}$$
