1. Introduction

2. MMC Technology

3. Proposed Non-Isolated DC-DC MMC Model

3.2. The Proposed Non-isolated DC-DC MMC Design's Mathematical Model

The use of mathematical modeling plays an extremely critical role in adequately evaluating the recommended architectures in MMC technology. Several presumptions concerning the suggested MMC topologies must be established in order to carry out a more comprehensive mathematical analysis of the mathematical modeling under examination.

The provided non-isolated DC-DC MMC architecture in this study includes several hypothetical principles that validate the previously mentioned circumstance. Equivalent circuit models of MMC designs have been established based on these hypothetical assumptions. Figure 6 demonstrates the equivalent circuit of the novel DC-DC MMC design that has been suggested. The 2-arm and 1-arm equivalent circuit designs of the suggested DC-DC MMC design are illustrated in Figure 7a and Figure 7b, correspondingly. Each branch's SM groups are represented by a voltage source with a variable characteristic in this equivalent circuit structure. Voltage sources that alter over time consist of both DC and AC components. These DC and AC components' mathematical counterparts are presented following.

According to the 1-arm equivalent circuit model (arm-a) of the suggested DC-DC MMC architecture, the mathematical computations provided in the following section have been carried out. Since similar mathematical analyses are valid for the other arms of the design (arms b and c), mathematical evaluations for the two arms under consideration have not been included in this study. The following mathematical parameters can be determined by applying the fundamental Kirchhoff Voltage Law and Kirchhoff Current Laws to the architecture of the 1-arm circuit illustrated in Figure 7b. First and foremost, the following are the voltage equations for SM clusters.

$${\mathrm{v}}_{\mathrm{s}1\mathrm{a}}={\mathrm{v}}_{\mathrm{s}1\mathrm{a},\mathrm{d}\mathrm{c}}+{\mathrm{v}}_{\mathrm{s}1\mathrm{a},\mathrm{a}\mathrm{c}}=\frac{{\mathrm{V}}_{\mathrm{p}\mathrm{H}}}{2}-\frac{{\mathrm{V}}_{\mathrm{s}\mathrm{L}}}{2}+2\mathrm{L}\frac{\mathrm{d}}{\mathrm{d}\mathrm{t}}{{\mathrm{i}}_{\mathrm{s}1\mathrm{a}}+2\mathrm{r}{\mathrm{i}}_{\mathrm{s}1\mathrm{a}}+\mathrm{V}}_{\mathrm{s}1\mathrm{a},\mathrm{m}\mathrm{a}\mathrm{x}}\sin \mathsf{\omega }\mathrm{t}$$

(1)

$${\mathrm{v}}_{\mathrm{s}2\mathrm{a}}={\mathrm{v}}_{\mathrm{s}2\mathrm{a},\mathrm{d}\mathrm{c}}+{\mathrm{v}}_{\mathrm{s}2\mathrm{a},\mathrm{a}\mathrm{c}}=\frac{{\mathrm{V}}_{\mathrm{s}\mathrm{L}}}{2}+2\mathrm{L}\frac{\mathrm{d}}{\mathrm{d}\mathrm{t}}{{\mathrm{i}}_{\mathrm{s}2\mathrm{a}}+2\mathrm{r}{\mathrm{i}}_{\mathrm{s}2\mathrm{a}}+\mathrm{V}}_{\mathrm{s}2\mathrm{a},\mathrm{m}\mathrm{a}\mathrm{x}}\sin (\mathsf{\omega }\mathrm{t}+{\mathsf{\theta }}_1)$$

(2)

$${\mathrm{v}}_{\mathrm{s}3\mathrm{a}}={\mathrm{v}}_{\mathrm{s}3\mathrm{a},\mathrm{d}\mathrm{c}}+{\mathrm{v}}_{\mathrm{s}3\mathrm{a},\mathrm{a}\mathrm{c}}=\frac{{\mathrm{V}}_{\mathrm{s}\mathrm{L}}}{2}+2\mathrm{L}\frac{\mathrm{d}}{\mathrm{d}\mathrm{t}}{{\mathrm{i}}_{\mathrm{s}3\mathrm{a}}+2\mathrm{r}{\mathrm{i}}_{\mathrm{s}3\mathrm{a}}-\mathrm{V}}_{\mathrm{s}3\mathrm{a},\mathrm{m}\mathrm{a}\mathrm{x}}\sin (\mathsf{\omega }\mathrm{t}+{\mathsf{\theta }}_2)$$

(3)

$${\mathrm{v}}_{\mathrm{s}4\mathrm{a}}={\mathrm{v}}_{\mathrm{s}4\mathrm{a},\mathrm{d}\mathrm{c}}+{\mathrm{v}}_{\mathrm{s}4\mathrm{a},\mathrm{a}\mathrm{c}}=\frac{{\mathrm{V}}_{\mathrm{p}\mathrm{H}}}{2}-\frac{{\mathrm{V}}_{\mathrm{s}\mathrm{L}}}{2}+2\mathrm{L}\frac{\mathrm{d}}{\mathrm{d}\mathrm{t}}{{\mathrm{i}}_{\mathrm{s}4\mathrm{a}}+2\mathrm{r}{\mathrm{i}}_{\mathrm{s}4\mathrm{a}}-\mathrm{V}}_{\mathrm{s}4\mathrm{a},\mathrm{m}\mathrm{a}\mathrm{x}}\sin (\mathsf{\omega }\mathrm{t}+{\mathsf{\theta }}_3)$$

(4)

$${\mathrm{i}}_{\mathrm{s}1\mathrm{a}}={\mathrm{I}}_{\mathrm{P}\mathrm{H}}={\mathrm{i}}_{\mathrm{s}2\mathrm{a}}+{\mathrm{i}}_{\mathrm{s}1\mathrm{a},\mathrm{u}\mathrm{o}}$$

(5)

$${\mathrm{i}}_{\mathrm{s}2\mathrm{a}}={\mathrm{i}}_{\mathrm{s}3\mathrm{a}}+{\mathrm{i}}_{\mathrm{s}1\mathrm{a},\mathrm{c}\mathrm{o}}$$

(6)

$${\mathrm{i}}_{\mathrm{s}4\mathrm{a}}={\mathrm{i}}_{\mathrm{s}3\mathrm{a}}+{\mathrm{i}}_{\mathrm{s}1\mathrm{a},\mathrm{l}\mathrm{o}}$$

(7)

where ${\mathrm{v}}_{\mathrm{s}1\mathrm{a}},{\mathrm{v}}_{\mathrm{s}2\mathrm{a}},{\mathrm{v}}_{\mathrm{s}3\mathrm{a}},\mathrm{a}\mathrm{n}\mathrm{d} {\mathrm{v}}_{\mathrm{s}4\mathrm{a}}$ denote the voltage expressions for the SM sets which areset1a, set2a, set3a and set4a in arm a, respectively. In keeping with the same logic ${\mathrm{i}}_{\mathrm{s}1\mathrm{a}},{\mathrm{i}}_{\mathrm{s}2\mathrm{a}},{\mathrm{i}}_{\mathrm{s}3\mathrm{a}},\mathrm{a}\mathrm{n}\mathrm{d} {\mathrm{i}}_{\mathrm{s}4\mathrm{a}}$ represent the currents flowing through the SM sets which are set1a, set2a, set3a and set4a in arm a, correspondingly. The AC magnitude of the AC voltage of sets which are set1a, set2a, set3a, and set4a is represented by ${\mathrm{V}}_{\mathrm{s}1\mathrm{a},\mathrm{m}\mathrm{a}\mathrm{x}},{\mathrm{V}}_{\mathrm{s}2\mathrm{a},\mathrm{m}\mathrm{a}\mathrm{x}},{\mathrm{V}}_{\mathrm{s}3\mathrm{a},\mathrm{m}\mathrm{a}\mathrm{x}},\mathrm{a}\mathrm{n}\mathrm{d} {\mathrm{V}}_{\mathrm{s}4\mathrm{a},\mathrm{m}\mathrm{a}\mathrm{x}}$ respectively. ${\mathsf{\theta }}_1,{\mathsf{\theta }}_2,\mathrm{a}\mathrm{n}\mathrm{d} {\mathsf{\theta }}_3$ are corresponding initial phase angles of SM sets voltage expression for arm-a. The angular frequency is represented by $\mathsf{\omega }$. The primary and secondary DC voltage levels of the suggested DC-DC MMC design are denoted by the ${\mathrm{V}}_{\mathrm{p}\mathrm{H}}$ and ${\mathrm{V}}_{\mathrm{s}\mathrm{L}}$, correspondingly. Secondly, the current expressions of SM clusters are as follows.

$${\mathrm{i}}_{\mathrm{s}1\mathrm{a}}={\mathrm{i}}_{\mathrm{s}1\mathrm{a},\mathrm{d}\mathrm{c}}+{\mathrm{i}}_{\mathrm{s}1\mathrm{a},\mathrm{a}\mathrm{c}}={{\mathrm{I}}_{\mathrm{p}\mathrm{H}}+\mathrm{I}}_{\mathrm{s}1\mathrm{a},\mathrm{m}\mathrm{a}\mathrm{x}}\sin (\mathsf{\omega }\mathrm{t}+{\mathsf{\beta }}_1)$$

(8)

$${\mathrm{i}}_{\mathrm{s}2\mathrm{a}}={\mathrm{i}}_{\mathrm{s}2\mathrm{a},\mathrm{d}\mathrm{c}}+{\mathrm{i}}_{\mathrm{s}2\mathrm{a},\mathrm{a}\mathrm{c}}={{\mathrm{I}}_{\mathrm{p}\mathrm{H}}-\frac{{\mathrm{V}}_{\mathrm{p}\mathrm{H}}}{\mathrm{V}{\mathrm{s}}_{\mathrm{L}}}{\mathrm{I}}_{\mathrm{p}\mathrm{H}}+\mathrm{I}}_{\mathrm{s}2\mathrm{a},\mathrm{m}\mathrm{a}\mathrm{x}}\sin (\mathsf{\omega }\mathrm{t}+{\mathsf{\beta }}_2)$$

(9)

$${\mathrm{i}}_{\mathrm{s}3\mathrm{a}}={\mathrm{i}}_{\mathrm{s}3\mathrm{a},\mathrm{d}\mathrm{c}}+{\mathrm{i}}_{\mathrm{s}3\mathrm{a},\mathrm{a}\mathrm{c}}={{\mathrm{I}}_{\mathrm{p}\mathrm{H}}-\frac{{\mathrm{V}}_{\mathrm{p}\mathrm{H}}}{\mathrm{V}{\mathrm{s}}_{\mathrm{L}}}{\mathrm{I}}_{\mathrm{p}\mathrm{H}}-\mathrm{I}}_{\mathrm{s}3\mathrm{a},\mathrm{m}\mathrm{a}\mathrm{x}}\sin (\mathsf{\omega }\mathrm{t}+{\mathsf{\beta }}_3)$$

(10)

$${\mathrm{i}}_{\mathrm{s}4\mathrm{a}}={\mathrm{i}}_{\mathrm{s}4\mathrm{a},\mathrm{d}\mathrm{c}}+{\mathrm{i}}_{\mathrm{s}4\mathrm{a},\mathrm{a}\mathrm{c}}={{\mathrm{I}}_{\mathrm{p}\mathrm{H}}-\mathrm{I}}_{\mathrm{s}4\mathrm{a},\mathrm{m}\mathrm{a}\mathrm{x}}\sin (\mathsf{\omega }\mathrm{t}+{\mathsf{\beta }}_4)$$

(11)

Examining the voltage and current expressions of the SM sets above reveals that each voltage and current parameter is made up of two parts which are DC and AC parts. The DC voltage components of the SM sets are represented by ${\mathrm{v}}_{\mathrm{s}1\mathrm{a},\mathrm{d}\mathrm{c}},{\mathrm{v}}_{\mathrm{s}2\mathrm{a},\mathrm{d}\mathrm{c}},{\mathrm{v}}_{\mathrm{s}3\mathrm{a},\mathrm{d}\mathrm{c}},\mathrm{a}\mathrm{n}\mathrm{d} {\mathrm{v}}_{\mathrm{s}4\mathrm{a},\mathrm{d}\mathrm{c}}$. ${\mathrm{v}}_{\mathrm{s}1\mathrm{a},\mathrm{a}\mathrm{c}},{\mathrm{v}}_{\mathrm{s}2\mathrm{a},\mathrm{a}\mathrm{c}},{\mathrm{v}}_{\mathrm{s}3\mathrm{a},\mathrm{a}\mathrm{c}},\mathrm{a}\mathrm{n}\mathrm{d} {\mathrm{v}}_{\mathrm{s}4\mathrm{a},\mathrm{a}\mathrm{c}}$ are the AC voltage components of sets of SM. Considering a comparable methodology, the AC components of the currents flowing through the SM sets are expressed by ${\mathrm{i}}_{\mathrm{s}1\mathrm{a},\mathrm{a}\mathrm{c}},{\mathrm{i}}_{\mathrm{s}2\mathrm{a},\mathrm{a}\mathrm{c}},{\mathrm{i}}_{\mathrm{s}3\mathrm{a},\mathrm{a}\mathrm{c}},\mathrm{a}\mathrm{n}\mathrm{d} {\mathrm{i}}_{\mathrm{s}4\mathrm{a},\mathrm{a}\mathrm{c}}$, while the DC components are expressed by ${\mathrm{i}}_{\mathrm{s}1\mathrm{a},\mathrm{d}\mathrm{c}},{\mathrm{i}}_{\mathrm{s}2\mathrm{a},\mathrm{d}\mathrm{c}},{\mathrm{i}}_{\mathrm{s}3\mathrm{a},\mathrm{d}\mathrm{c}},\mathrm{a}\mathrm{n}\mathrm{d} {\mathrm{i}}_{\mathrm{s}4\mathrm{a},\mathrm{d}\mathrm{c}}$. The AC magnitude of the AC voltage of sets which are set1a, set2a, set3a, and set4a is represented by ${\mathrm{I}}_{\mathrm{s}1\mathrm{a},\mathrm{m}\mathrm{a}\mathrm{x}},{\mathrm{I}}_{\mathrm{s}2\mathrm{a},\mathrm{m}\mathrm{a}\mathrm{x}},{\mathrm{I}}_{\mathrm{s}3\mathrm{a},\mathrm{m}\mathrm{a}\mathrm{x}},\mathrm{a}\mathrm{n}\mathrm{d} {\mathrm{I}}_{\mathrm{s}4\mathrm{a},\mathrm{m}\mathrm{a}\mathrm{x}}$ respectively. The phase angle of the currents flowing through the SMets is represented by the letters ${\mathsf{\beta }}_1$, ${\mathsf{\beta }}_2$, ${\mathsf{\beta }}_3$, and ${\mathsf{\beta }}_4$.The equation that follows is the current flowing from point a2 through the grounding branch line. It is evident from equation (12) that the DC component is zero.

$${\mathrm{i}}_{\mathrm{s}1\mathrm{a},\mathrm{c}\mathrm{o}}={\mathrm{i}}_{\mathrm{s}1\mathrm{a},\mathrm{c}\mathrm{o},\mathrm{d}\mathrm{c}}+{\mathrm{i}}_{\mathrm{s}1\mathrm{a},\mathrm{c}\mathrm{o},\mathrm{a}\mathrm{c}}={2\mathrm{I}}_{\mathrm{s}1\mathrm{a},\mathrm{c}\mathrm{o},\mathrm{m}\mathrm{a}\mathrm{x}}\sin (\mathsf{\omega }\mathrm{t}+\gamma )$$

(12)

The power expressions have been presented as follows, following the voltage and current expressions of the SM sets. Branch inductances and resistors are disregarded in this case though, to prevent an excessively involved mathematical examination and to provide a more efficient power assessment.

$${\mathrm{p}}_{\mathrm{s}1\mathrm{a}}={\mathrm{v}}_{\mathrm{s}1\mathrm{a}}{\mathrm{i}}_{\mathrm{s}1\mathrm{a}}=\left[\frac{{\mathrm{V}}_{\mathrm{p}\mathrm{H}}}{2}-\frac{{\mathrm{V}}_{\mathrm{s}\mathrm{L}}}{2}+2\mathrm{L}\frac{\mathrm{d}}{\mathrm{d}\mathrm{t}}{{\mathrm{i}}_{\mathrm{s}1\mathrm{a}}+2\mathrm{r}{\mathrm{i}}_{\mathrm{s}1\mathrm{a}}+\mathrm{V}}_{\mathrm{s}1\mathrm{a},\mathrm{m}\mathrm{a}\mathrm{x}}(\sin \mathsf{\omega }\mathrm{t})\right]\left[{{\mathrm{I}}_{\mathrm{p}\mathrm{H}}+\mathrm{I}}_{\mathrm{s}1\mathrm{a},\mathrm{m}\mathrm{a}\mathrm{x}}\sin (\mathsf{\omega }\mathrm{t}+{\mathsf{\beta }}_1)\right]$$

(13)

$$\begin{gathered} {\mathrm{p}}_{\mathrm{s}1\mathrm{a}}=\frac{{\mathrm{V}}_{\mathrm{p}\mathrm{H}}}{2}{\mathrm{I}}_{\mathrm{p}\mathrm{H}}+\frac{{\mathrm{V}}_{\mathrm{p}\mathrm{H}}}{2}{\mathrm{I}}_{\mathrm{s}1\mathrm{a},\mathrm{m}\mathrm{a}\mathrm{x}}\sin (\mathsf{\omega }\mathrm{t}+{\mathsf{\beta }}_1)-\frac{{\mathrm{V}}_{\mathrm{s}\mathrm{L}}}{2}{\mathrm{I}}_{\mathrm{p}\mathrm{H}}-\frac{{\mathrm{V}}_{\mathrm{s}\mathrm{L}}}{2}{\mathrm{I}}_{\mathrm{s}1\mathrm{a},\mathrm{m}\mathrm{a}\mathrm{x}}\sin (\mathsf{\omega }\mathrm{t}+{\mathsf{\beta }}_1)+{\mathrm{V}}_{\mathrm{s}1\mathrm{a},\mathrm{m}\mathrm{a}\mathrm{x}}(\sin \mathsf{\omega }\mathrm{t}){\mathrm{I}}_{\mathrm{p}\mathrm{H}} \\ +{\mathrm{V}}_{\mathrm{s}1\mathrm{a},\mathrm{m}\mathrm{a}\mathrm{x}}\sin (\mathsf{\omega }\mathrm{t}){\mathrm{I}}_{\mathrm{s}1\mathrm{a},\mathrm{m}\mathrm{a}\mathrm{x}}\sin (\mathsf{\omega }\mathrm{t}+{\mathsf{\beta }}_1) \end{gathered}$$

(14)

$$\begin{gathered} {\mathrm{p}}_{\mathrm{s}1\mathrm{a}}=\frac{{\mathrm{V}}_{\mathrm{p}\mathrm{H}}}{2}{\mathrm{I}}_{\mathrm{p}\mathrm{H}}-\frac{{\mathrm{V}}_{\mathrm{s}\mathrm{L}}}{2}{\mathrm{I}}_{\mathrm{p}\mathrm{H}}+\frac{{\mathrm{V}}_{\mathrm{p}\mathrm{H}}}{2}{\mathrm{I}}_{\mathrm{s}1\mathrm{a},\mathrm{m}\mathrm{a}\mathrm{x}}\sin (\mathsf{\omega }\mathrm{t}+{\mathsf{\beta }}_1)-\frac{{\mathrm{V}}_{\mathrm{s}\mathrm{L}}}{2}{\mathrm{I}}_{\mathrm{s}1\mathrm{a},\mathrm{m}\mathrm{a}\mathrm{x}}\sin (\mathsf{\omega }\mathrm{t}+{\mathsf{\beta }}_1)+\frac{{\mathrm{V}}_{\mathrm{s}1\mathrm{a},\mathrm{m}\mathrm{a}\mathrm{x}}{\mathrm{I}}_{\mathrm{s}1\mathrm{a},\mathrm{m}\mathrm{a}\mathrm{x}}}{2}\cos \left({\mathsf{\beta }}_1\right) \\ +{\mathrm{V}}_{\mathrm{s}1\mathrm{a},\mathrm{m}\mathrm{a}\mathrm{x}}{\mathrm{I}}_{\mathrm{p}\mathrm{H}}\sin (\mathsf{\omega }\mathrm{t})-\frac{{\mathrm{V}}_{\mathrm{s}1\mathrm{a},\mathrm{m}\mathrm{a}\mathrm{x}}{\mathrm{I}}_{\mathrm{s}1\mathrm{a},\mathrm{m}\mathrm{a}\mathrm{x}}}{2}\left[\cos \left(2\mathsf{\omega }\mathrm{t}\right)\cos \left({\mathsf{\beta }}_1\right)-\sin \left(2\mathsf{\omega }\mathrm{t}\right)\sin \left({\mathsf{\beta }}_1\right)\right] \end{gathered}$$

(15)

Where, ${\mathrm{p}}_{\mathrm{s}1\mathrm{a}}$ represents the power of the SM set st1a of arm-a. As seen in Equation (13), the power expression consists of both DC and AC components, as in the case of voltage and current parameters, since it is inherently the product of the voltage falling on the relevant SM set and the current flowing through it. The power expressions of the set 2a set 3a, and set 4a SM sets of arm-a have been gathered with a similar approach and provided as ${\mathrm{p}}_{\mathrm{s}2\mathrm{a}}$, ${\mathrm{p}}_{\mathrm{s}3\mathrm{a}}$, and ${\mathrm{p}}_{\mathrm{s}4\mathrm{a}}$, correspondingly.

$${\mathrm{p}}_{\mathrm{s}2\mathrm{a}}={\mathrm{v}}_{\mathrm{s}2\mathrm{a}}{\mathrm{i}}_{\mathrm{s}2\mathrm{a}}=\left[\frac{{\mathrm{V}}_{\mathrm{s}\mathrm{L}}}{2}{-\mathrm{V}}_{\mathrm{s}3\mathrm{a},\mathrm{m}\mathrm{a}\mathrm{x}}\sin (\mathsf{\omega }\mathrm{t}+{\mathsf{\theta }}_2)\right]\left[{{\mathrm{I}}_{\mathrm{p}\mathrm{H}}-\frac{{\mathrm{V}}_{\mathrm{p}\mathrm{H}}}{\mathrm{V}{\mathrm{s}}_{\mathrm{L}}}{\mathrm{I}}_{\mathrm{p}\mathrm{H}}+\mathrm{I}}_{\mathrm{s}2\mathrm{a},\mathrm{m}\mathrm{a}\mathrm{x}}\sin (\mathsf{\omega }\mathrm{t}+{\mathsf{\beta }}_2)\right]$$

(16)

$$\begin{gathered} {\mathrm{p}}_{\mathrm{s}2\mathrm{a}}=\frac{{\mathrm{V}}_{\mathrm{s}\mathrm{L}}}{2}{\mathrm{I}}_{\mathrm{p}\mathrm{H}}-\frac{{\mathrm{V}}_{\mathrm{s}\mathrm{L}}}{2}\frac{{\mathrm{V}}_{\mathrm{p}\mathrm{H}}}{\mathrm{V}{\mathrm{s}}_{\mathrm{L}}}{\mathrm{I}}_{\mathrm{p}\mathrm{H}}+\frac{{\mathrm{V}}_{\mathrm{s}\mathrm{L}}}{2}{\mathrm{I}}_{\mathrm{s}2\mathrm{a},\mathrm{m}\mathrm{a}\mathrm{x}}\sin (\mathsf{\omega }\mathrm{t}+{\mathsf{\beta }}_2)+{\mathrm{V}}_{\mathrm{s}2\mathrm{a},\mathrm{m}\mathrm{a}\mathrm{x}}\sin (\mathsf{\omega }\mathrm{t}+{\mathsf{\theta }}_1){\mathrm{I}}_{\mathrm{p}\mathrm{H}} \\ -{\mathrm{V}}_{\mathrm{s}2\mathrm{a},\mathrm{m}\mathrm{a}\mathrm{x}}\sin (\mathsf{\omega }\mathrm{t}+{\mathsf{\theta }}_1)\frac{{\mathrm{V}}_{\mathrm{p}\mathrm{H}}}{\mathrm{V}{\mathrm{s}}_{\mathrm{L}}}{\mathrm{I}}_{\mathrm{p}\mathrm{H}}+{\mathrm{V}}_{\mathrm{s}2\mathrm{a},\mathrm{m}\mathrm{a}\mathrm{x}}\sin (\mathsf{\omega }\mathrm{t}+{\mathsf{\theta }}_1){\mathrm{I}}_{\mathrm{s}2\mathrm{a},\mathrm{m}\mathrm{a}\mathrm{x}}\sin (\mathsf{\omega }\mathrm{t}+{\mathsf{\beta }}_2) \end{gathered}$$

(17)

$$\begin{gathered} {\mathrm{p}}_{\mathrm{s}2\mathrm{a}}=\frac{{\mathrm{V}}_{\mathrm{s}\mathrm{L}}}{2}{\mathrm{I}}_{\mathrm{p}\mathrm{H}}-\frac{{\mathrm{V}}_{\mathrm{p}\mathrm{H}}}{2}{\mathrm{I}}_{\mathrm{p}\mathrm{H}}+\frac{{\mathrm{V}}_{\mathrm{s}\mathrm{L}}}{2}{\mathrm{I}}_{\mathrm{s}2\mathrm{a},\mathrm{m}\mathrm{a}\mathrm{x}}\sin (\mathsf{\omega }\mathrm{t}+{\mathsf{\beta }}_2)+\left[{[\mathrm{V}}_{\mathrm{s}2\mathrm{a},\mathrm{m}\mathrm{a}\mathrm{x}}{\mathrm{I}}_{\mathrm{p}\mathrm{H}}-{\mathrm{V}}_{\mathrm{s}2\mathrm{a},\mathrm{m}\mathrm{a}\mathrm{x}}{\mathrm{I}}_{\mathrm{p}\mathrm{H}}\frac{{\mathrm{V}}_{\mathrm{p}\mathrm{H}}}{{\mathrm{V}}_{\mathrm{s}\mathrm{L}}}\right]\sin (\mathsf{\omega }\mathrm{t}+{\mathsf{\theta }}_1) \\ +\frac{{\mathrm{V}}_{\mathrm{s}2\mathrm{a},\mathrm{m}\mathrm{a}\mathrm{x}}{\mathrm{I}}_{\mathrm{s}2\mathrm{a},\mathrm{m}\mathrm{a}\mathrm{x}}}{2}\cos ({\mathsf{\theta }}_1-{\mathsf{\beta }}_2)-\frac{{\mathrm{V}}_{\mathrm{s}2\mathrm{a},\mathrm{m}\mathrm{a}\mathrm{x}}{\mathrm{I}}_{\mathrm{s}2\mathrm{a},\mathrm{m}\mathrm{a}\mathrm{x}}}{2}\left[\cos \left(2\mathsf{\omega }\mathrm{t}+2{\mathsf{\theta }}_1\right)\cos \left({\mathsf{\theta }}_1-{\mathsf{\beta }}_2\right)+\sin \left(2\mathsf{\omega }\mathrm{t}+2{\mathsf{\theta }}_1\right)\sin \left({\mathsf{\theta }}_1-{\mathsf{\beta }}_2\right)\right] \end{gathered}$$

(18)

$${\mathrm{p}}_{\mathrm{s}3\mathrm{a}}={\mathrm{v}}_{\mathrm{s}3\mathrm{a}}{\mathrm{i}}_{\mathrm{s}3\mathrm{a}}=\left[\frac{{\mathrm{V}}_{\mathrm{s}\mathrm{L}}}{2}+{\mathrm{V}}_{\mathrm{s}3\mathrm{a},\mathrm{m}\mathrm{a}\mathrm{x}}\sin (\mathsf{\omega }\mathrm{t}+{\mathsf{\theta }}_2)\right]\left[{{\mathrm{I}}_{\mathrm{p}\mathrm{H}}-\frac{{\mathrm{V}}_{\mathrm{p}\mathrm{H}}}{\mathrm{V}{\mathrm{s}}_{\mathrm{L}}}{\mathrm{I}}_{\mathrm{p}\mathrm{H}}-\mathrm{I}}_{\mathrm{s}3\mathrm{a},\mathrm{m}\mathrm{a}\mathrm{x}}\sin (\mathsf{\omega }\mathrm{t}+{\mathsf{\beta }}_3)\right]$$

(19)

$$\begin{gathered} {\mathrm{p}}_{\mathrm{s}3\mathrm{a}}=\frac{{\mathrm{V}}_{\mathrm{s}\mathrm{L}}}{2}{\mathrm{I}}_{\mathrm{p}\mathrm{H}}-\frac{{\mathrm{V}}_{\mathrm{s}\mathrm{L}}}{2}\frac{{\mathrm{V}}_{\mathrm{p}\mathrm{H}}}{\mathrm{V}{\mathrm{s}}_{\mathrm{L}}}{\mathrm{I}}_{\mathrm{p}\mathrm{H}}-\frac{{\mathrm{V}}_{\mathrm{s}\mathrm{L}}}{2}{\mathrm{I}}_{\mathrm{s}3\mathrm{a},\mathrm{m}\mathrm{a}\mathrm{x}}\sin (\mathsf{\omega }\mathrm{t}+{\mathsf{\beta }}_3)+{\mathrm{V}}_{\mathrm{s}3\mathrm{a},\mathrm{m}\mathrm{a}\mathrm{x}}\sin (\mathsf{\omega }\mathrm{t}+{\mathsf{\theta }}_2){\mathrm{I}}_{\mathrm{p}\mathrm{H}} \\ -{\mathrm{V}}_{\mathrm{s}3\mathrm{a},\mathrm{m}\mathrm{a}\mathrm{x}}\sin (\mathsf{\omega }\mathrm{t}+{\mathsf{\theta }}_2)\frac{{\mathrm{V}}_{\mathrm{p}\mathrm{H}}}{\mathrm{V}{\mathrm{s}}_{\mathrm{L}}}{\mathrm{I}}_{\mathrm{p}\mathrm{H}}-{\mathrm{V}}_{\mathrm{s}3\mathrm{a},\mathrm{m}\mathrm{a}\mathrm{x}}\sin (\mathsf{\omega }\mathrm{t}+{\mathsf{\theta }}_2){\mathrm{I}}_{\mathrm{s}3\mathrm{a},\mathrm{m}\mathrm{a}\mathrm{x}}\sin (\mathsf{\omega }\mathrm{t}+{\mathsf{\beta }}_3) \end{gathered}$$

(20)

$$\begin{gathered} {\mathrm{p}}_{\mathrm{s}3\mathrm{a}}=\frac{{\mathrm{V}}_{\mathrm{s}\mathrm{L}}}{2}{\mathrm{I}}_{\mathrm{p}\mathrm{H}}-\frac{{\mathrm{V}}_{\mathrm{p}\mathrm{H}}}{2}{\mathrm{I}}_{\mathrm{p}\mathrm{H}}-\frac{{\mathrm{V}}_{\mathrm{s}\mathrm{L}}}{2}{\mathrm{I}}_{\mathrm{s}3\mathrm{a},\mathrm{m}\mathrm{a}\mathrm{x}}\sin (\mathsf{\omega }\mathrm{t}+{\mathsf{\beta }}_3)-{\mathrm{V}}_{\mathrm{s}3\mathrm{a},\mathrm{m}\mathrm{a}\mathrm{x}}{\mathrm{I}}_{\mathrm{p}\mathrm{H}}+{\mathrm{V}}_{\mathrm{s}3\mathrm{a},\mathrm{m}\mathrm{a}\mathrm{x}}{\mathrm{I}}_{\mathrm{p}\mathrm{H}}\frac{{\mathrm{V}}_{\mathrm{p}\mathrm{H}}}{\mathrm{V}{\mathrm{s}}_{\mathrm{L}}}\sin (\mathsf{\omega }\mathrm{t}+{\mathsf{\theta }}_2) \\ +\frac{{\mathrm{V}}_{\mathrm{s}3\mathrm{a},\mathrm{m}\mathrm{a}\mathrm{x}}{\mathrm{I}}_{\mathrm{s}3\mathrm{a},\mathrm{m}\mathrm{a}\mathrm{x}}}{2}\cos ({\mathsf{\theta }}_2-{\mathsf{\beta }}_3)-\frac{{\mathrm{V}}_{\mathrm{s}3\mathrm{a},\mathrm{m}\mathrm{a}\mathrm{x}}{\mathrm{I}}_{\mathrm{s}3\mathrm{a},\mathrm{m}\mathrm{a}\mathrm{x}}}{2}\left[\cos \left(2\mathsf{\omega }\mathrm{t}+2{\mathsf{\theta }}_2\right)\cos ({\mathsf{\theta }}_2-{\mathsf{\beta }}_3)+\sin \left(2\mathsf{\omega }\mathrm{t}+2{\mathsf{\theta }}_2\right)\sin \left({\mathsf{\theta }}_2-{\mathsf{\beta }}_3\right)\right] \end{gathered}$$

(21)

$${\mathrm{p}}_{\mathrm{s}4\mathrm{a}}={\mathrm{v}}_{\mathrm{s}4\mathrm{a}}{\mathrm{i}}_{\mathrm{s}4\mathrm{a}}=\left[\frac{{\mathrm{V}}_{\mathrm{p}\mathrm{H}}}{2}-\frac{{\mathrm{V}}_{\mathrm{s}\mathrm{L}}}{2}{-\mathrm{V}}_{\mathrm{s}4\mathrm{a},\mathrm{m}\mathrm{a}\mathrm{x}}\sin (\mathsf{\omega }\mathrm{t}+{\mathsf{\theta }}_3)\right]\left[{{\mathrm{I}}_{\mathrm{p}\mathrm{H}}-\mathrm{I}}_{\mathrm{s}4\mathrm{a},\mathrm{m}\mathrm{a}\mathrm{x}}\sin (\mathsf{\omega }\mathrm{t}+{\mathsf{\beta }}_4)\right]$$

(22)

$$\begin{gathered} {\mathrm{p}}_{\mathrm{s}4\mathrm{a}}=\frac{{\mathrm{V}}_{\mathrm{p}\mathrm{H}}}{2}{\mathrm{I}}_{\mathrm{p}\mathrm{H}}-\frac{{\mathrm{V}}_{\mathrm{p}\mathrm{H}}}{2}{\mathrm{I}}_{\mathrm{s}4\mathrm{a},\mathrm{m}\mathrm{a}\mathrm{x}}\sin (\mathsf{\omega }\mathrm{t}+{\mathsf{\beta }}_4)-\frac{{\mathrm{V}}_{\mathrm{s}\mathrm{L}}}{2}{\mathrm{I}}_{\mathrm{p}\mathrm{H}}+\frac{{\mathrm{V}}_{\mathrm{s}\mathrm{L}}}{2}{\mathrm{I}}_{\mathrm{s}4\mathrm{a},\mathrm{m}\mathrm{a}\mathrm{x}}\sin (\mathsf{\omega }\mathrm{t}+{\mathsf{\beta }}_4){-\mathrm{V}}_{\mathrm{s}4\mathrm{a},\mathrm{m}\mathrm{a}\mathrm{x}}\sin (\mathsf{\omega }\mathrm{t}+{\mathsf{\theta }}_3){\mathrm{I}}_{\mathrm{p}\mathrm{H}} \\ +{\mathrm{V}}_{\mathrm{s}4\mathrm{a},\mathrm{m}\mathrm{a}\mathrm{x}}\sin (\mathsf{\omega }\mathrm{t}+{\mathsf{\theta }}_3){\mathrm{I}}_{\mathrm{s}4\mathrm{a},\mathrm{m}\mathrm{a}\mathrm{x}}\sin (\mathsf{\omega }\mathrm{t}+{\mathsf{\beta }}_4) \end{gathered}$$

(23)

(24)

4. Operational Principle and Simulation Analysis of the DC-DC MMC Topology

5. Conclusions

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