2.2.4. Converter Gain

The generalized relationship between the output voltage and the input voltage of the hybrid ISIP-OSOP DC-DC converter can be found by performing two steps. The first step is adding the *L* number of KVL equations presented in (11) for the primary multimodule DC-DC converters, assuming ˆ *dLj* = 0, *j* = 1, 2, ... , *L*, and substituting (2), (4), (14), (17) and (18) in the added equation. However, the second step is adding the *M* number of KVL equations presented in (11) for the secondary multimodule DC-DC converters, assuming ˆ *dMj*=0, *j* = 1, 2, ... , *M*, and substituting (7), (9), (16), (19) and (20) in the added equation.

Adding the *L* number of KVL equations in (11), assuming ˆ *dLj* = 0, *j* = 1, 2, ... , *L*, and substituting (2), (4) and (14) would result in:

$$\begin{split} \frac{D\_{eff1}}{K\_{1}} \sum\_{j=1}^{L} \mathfrak{d}\_{\mathrm{cdl},j} + \frac{\beta\_{l\perp}}{\overline{\beta\_{l\perp}}} \frac{V\_{in}}{K\_{1}} \Bigg( \begin{split} &-\frac{\beta\_{l\perp} K\_{1} R\_{\mathrm{d\overline{l}}}}{\overline{\beta\_{l\perp}}} \mathfrak{d}\_{\mathrm{out},l} \bigg( \frac{s R \mathcal{L}\_{\mathrm{c}} + s \psi\_{\parallel 1} R\_{\mathrm{d\overline{l}}} \mathcal{L}\_{\mathrm{c}} + b\_{l\perp}}{R (1 + s R\_{\mathrm{d\overline{l}}} \mathcal{L}\_{\mathrm{c}})} \bigg) + \\ &\sum\_{j=1}^{L} \frac{a\_{l\perp} \psi\_{\parallel 1} \mathfrak{d}\_{\mathrm{l\overline{l}}} \mathfrak{d}\_{\mathrm{c}} \varPi\_{\mathrm{c\overline{l}}} \mathfrak{d}\_{\mathrm{c\overline{l}}}}{a\_{l\perp} \psi\_{\parallel 1} \mathfrak{d}\_{\mathrm{c\overline{l}}} \varPi\_{\mathrm{c\overline{l}}} + b\_{l\perp}} \bigg) \\ &= s L\_{\mathrm{d}} \Big( \frac{s R \mathcal{C}\_{\mathrm{c}} + s \psi\_{\parallel 1} R\_{\mathrm{c\overline{l}}} \mathcal{C}\_{\mathrm{c\overline{l}}} + \psi\_{\parallel 1}}{R (1 + s R\_{\mathrm{c\overline{l}}} \mathcal{C}\_{\mathrm{c\overline{l}}})} \bigg) \mathfrak{d}\_{\mathrm{out},\mathrm{L}} + \sum\_{j=1}^{L} \mathfrak{d}\_{\mathrm{out},l,j} \end{split} \tag{44}$$

Substituting (17) and (18) in (44) would give:

$$\begin{split} \frac{D\_{eff1}}{K\_{1}} \mathcal{V}\_{\mathcal{L}} \Big( \mathbf{1} + \frac{a\_{\perp \mathcal{L}} \mathbb{H}\_{\perp} \mathcal{R}\_{\text{dL}}}{a\_{\perp \mathcal{L}} \mathbb{H}\_{\perp \mathcal{L}} \mathcal{R}} \Big) \mathfrak{d}\_{\text{in}L} \\ = \left( \mathrm{s} \mathcal{L}\_{\mathcal{L}} + \mathcal{R}\_{\text{dL}} \right) \Big( \Big( \frac{\mathrm{s} \mathcal{R} \mathcal{L}\_{\mathcal{L}} + \mathrm{s} \flat\_{\perp \mathcal{L}} \mathcal{R}\_{\text{dL}} \mathcal{L}\_{\text{L}} + b\_{\perp \mathcal{L}}}{\mathcal{R} (1 + \mathrm{s} \mathcal{R}\_{\text{dL}} \mathcal{L}\_{\text{L}})} \Big) \mathfrak{d}\_{\text{out}L} \Big) + c\_{\perp} \mathfrak{d}\_{\text{out}L} \end{split} \tag{45}$$

Simplifying (45) would result in (46):

$$\begin{split} G\_{\text{tr}\mathcal{G}L} &= \frac{\phi\_{\text{val}L}}{\psi\_{\text{in}L}} \\ &= \frac{\frac{D\_{\text{eff}}f1}{R\_{\text{L}}}\chi\_{\text{L}}\left(1+\frac{\nu\_{\text{L}}\gamma\_{\text{L}}R\_{\text{L}}}{\nu\_{\text{L}}+\Gamma\_{\text{L}}}\right)(1+sR\_{\text{cL}}\mathcal{C}\_{L})}{s^{2}L\_{\text{L}}\mathcal{C}L\left(1+\frac{\nu\_{\text{L}}R\_{\text{cL}}}{R}\right)+s\left(\frac{\nu\_{\text{L}}L\_{\text{L}}}{R}+R\_{\text{cL}}\mathcal{C}L\left(1+\frac{\nu\_{\text{L}}R\_{\text{cL}}}{R}\right)+c\_{\text{L}}\mathcal{C}L\right)+\frac{\nu\_{\text{L}}R\_{\text{cL}}}{R}+c\_{L}} \end{split} \tag{46}$$

ˆ

Similarly, Adding the *M* number of KVL equations in (11), assuming *dMj* = 0, *j* = 1, 2, ... , *M*, and substituting (7), (9) and (16) would result in:

$$\begin{aligned} \frac{D\_{eff1}}{K\_{2}} \sum\_{f=1}^{M} \boldsymbol{\vartheta}\_{c0Mf} + \frac{\boldsymbol{\theta}\_{M2}}{\boldsymbol{\theta}\_{M1}} \frac{V\_{in}}{K\_{2}} \left( \frac{-\frac{\boldsymbol{\mathcal{B}}\_{M1} K\_{2} R\_{dM}}{\boldsymbol{\mathcal{B}}\_{M2} V\_{in}} \boldsymbol{\vartheta}\_{outM} \left( \frac{s R \boldsymbol{C}\_{M} + s \boldsymbol{\mathsf{b}}\_{M1} R\_{cM} \boldsymbol{C}\_{M} + \boldsymbol{\mathsf{b}}\_{M1}}{R (1 + s \boldsymbol{\mathsf{R}}\_{cM} \boldsymbol{C}\_{M})} \right)}{\sum\limits\_{f=1}^{M} \frac{a\_{M2} \boldsymbol{\mathsf{b}}\_{M1} \boldsymbol{\mathsf{b}}\_{M1} \boldsymbol{\mathsf{R}}\_{M1} R\_{cM} \boldsymbol{D}\_{eff1}}{a\_{M1} \boldsymbol{\mathsf{b}}\_{M2} \boldsymbol{\mathsf{b}}\_{R2} \boldsymbol{R} V\_{in}} \boldsymbol{\vartheta}\_{cMf} + \boldsymbol{\mathcal{d}}\_{M1}} \right) \\ = \boldsymbol{\mathcal{S}} \boldsymbol{L}\_{M} \left( \frac{s \boldsymbol{\mathcal{R}} \boldsymbol{C}\_{M} + s \boldsymbol{\mathsf{b}}\_{M1} \boldsymbol{\mathsf{R}}\_{cM} \boldsymbol{C}\_{M} + \boldsymbol{\mathsf{b}}\_{M1}}{R (1 + s \boldsymbol{\mathsf{R}}\_{cM} \boldsymbol{C}\_{M})} \right) \boldsymbol{\vartheta}\_{outM} + \sum\_{f=1}^{M} \boldsymbol{\mathcal{C}}\_{outMf} \end{aligned} \tag{47}$$

Substituting (19) and (20) in (47) would give:

$$\begin{aligned} \frac{\partial \boldsymbol{J}\_{off2}}{\partial \boldsymbol{K}\_{2}} \boldsymbol{\Psi\_{M}} \left( 1 + \frac{\boldsymbol{a}\_{M2} \boldsymbol{b}\_{M1} \boldsymbol{R}\_{dM}}{\boldsymbol{a}\_{M1} \boldsymbol{b}\_{M2} \boldsymbol{R}} \right) \boldsymbol{\boldsymbol{\upphi}}\_{\text{in}M} \\ &+ \boldsymbol{c}\_{M} \boldsymbol{\upphi}\_{\text{out}M} \left( \begin{aligned} &-\frac{\boldsymbol{\uptheta}\_{M1} \boldsymbol{K}\_{2} \boldsymbol{R}\_{d}}{\boldsymbol{\upbeta}\_{M2} \boldsymbol{V}\_{\text{in}}} \boldsymbol{\upphi}\_{\text{out}M} \left( \frac{\boldsymbol{s} \boldsymbol{R} \boldsymbol{C}\_{M} + \boldsymbol{s} \boldsymbol{b}\_{M1} \boldsymbol{R}\_{dM} \boldsymbol{C}\_{M} + \boldsymbol{b}\_{M1}}{R \left(1 + \boldsymbol{s} \boldsymbol{R}\_{dM} \boldsymbol{C}\_{M}\right)} \right) + \boldsymbol{1} \\ &\sum\_{j=1}^{M} \frac{\boldsymbol{a}\_{M2} \boldsymbol{R}\_{j1} \boldsymbol{R}\_{M1} \boldsymbol{R}\_{dM} \boldsymbol{R}\_{dj2} \boldsymbol{D}\_{off2}}{\boldsymbol{a}\_{M1} \boldsymbol{R}\_{dM} \boldsymbol{R}\_{dM} \boldsymbol{R}\_{d}} \boldsymbol{\upphi}\_{\text{cMM}} + \boldsymbol{\upbeta}\_{\text{cM1}} \end{aligned} \right) \begin{aligned} \boldsymbol{\upphi}\_{\text{M}} \\ &+ \boldsymbol{\upphi}\_{\text{out}M} \end{aligned} \tag{48}$$
 
$$\begin{aligned} \boldsymbol{\upphi}\_{M} &= \operatorname{s\small{L}}\_{M} \left( \frac{\boldsymbol{s} \boldsymbol{R} \boldsymbol{C}\_{M} + \boldsymbol{s} \boldsymbol{b}\_{M1} \boldsymbol{R}\_{dM} \boldsymbol{C}\_{M} + \boldsymbol{b}\_{M1}}{$$

Simplifying (48) would result in (49):

$$\begin{split} G\_{vylM} &= \frac{\hat{\upsilon}\_{outM}}{\hat{\upsilon}\_{inM}}\\ &= \frac{\frac{D\_{eff1}}{K\_{2}}\gamma\_{M}\left(1+\frac{a\_{M2}b\_{M1}R\_{dM}}{a\_{M1}b\_{M2}R}\right)\left(1+sR\_{cM}C\_{M}\right)}{s^{2}L\_{M}C\_{M}\left(1+\frac{b\_{M1}R\_{dM}}{R}\right)+s\left(\frac{b\_{M1}L\_{M}}{R}+R\_{dM}C\_{M}\left(1+\frac{b\_{M1}R\_{CM}}{R}\right)+c\_{M}C\_{M}\right)+\frac{b\_{M1}R\_{dM}}{R}+c\_{M}}\end{split} \tag{49}$$

By adding *GvgL* and *GvgM*, the output impedance transfer function can be found.

#### **3. Hybrid Input-Series Output-Parallel (ISOP) Multimodule DC-DC Converter**

In this section, the hybrid ISOP multimodule power converter circuit diagram, as well as the hybrid ISOP multimodule converter small-signal analysis, are discussed. The analysis carried out in this section is not limited to unidirectional power flow and can be applied for bidirectional power flow. The generalized small-signal modeling presented in Section 2 is used to derive the small-signal model for the eight-module hybrid ISOP power converter.

## *3.1. ISOP Circuit Diagram*

The conventional ISOP converter shown in Figure 5a consists of multiple DAB units that are connected in series and in parallel at the input and the output sides, respectively, where all the modules are assumed identical. However, the concept of the hybrid ISOP power converter shown in Figure 5b is dividing conventional ISOP multimodule DC-DC converters into two groups. The primary group consists of identical ISOP DC-DC converters and is responsible for delivering a large portion of the total required power with lower switching frequency. This is shown in Figure 5c. However, the secondary group consists of another identical ISOP multimodule DC-DC converters. It is responsible for delivering the remaining power with higher switching frequency. This is shown in Figure 5d. This would result in two groups of multimodule converters operating at a di fferent switching frequency, and utilizing di fferent leakage inductance, transformers, filter inductors, and capacitors.

In this paper, the EV UFC specifications are assumed to deliver a total power of 200 kW using a battery voltage of 400 V, and assuming a grid voltage of 10 kV. It is assumed that the primary group handles 80% of the total battery charging power, while the secondary group handles 20% of the total battery charging power. Therefore, the primary group is responsible for delivering 160 kW, which 4 5 of the total required power. The primary group is assumed to operate at switching frequency *fsL* of 10 kHz. However, the secondary group is responsible for delivering the remaining 40 kW, which is 1 5 of the total required power. The secondary group is assumed to operate at switching frequency of *fsM* 100 kHz. Accordingly, the input voltage of the primary group *VinL* is 8 kV, which is 4 5 of the total input voltage *Vin*, while the input voltage of the secondary group *VinM* is 2 kV, which is 1 5 of the total input voltage *Vin*. Similarly, the output current of the primary group *IoL* is 400 A, which is 4 5 of the total output current *Iout*, while the output current of the secondary group *IoM* is 100 A, which is 1 5 of the total output current *Iout*. It is essential to mention that the portions 4 5 and 1 5 are denoted as *KL* and *KM*, respectively.

By ensuring equal IVS for the primary group and secondary group, the input voltage per module in the primary group is reduced to *VinL* 4 , while the input voltage per module in the secondary group is *VinM* 4 . Besides, by ensuring equal OCS for the primary and secondary group, the output current of each module in the primary group is reduced to *IoL* 4 , while the output current of each module in the secondary group is reduced to *IoM* 4 . In which, *VinL*, *VinM*, *IoL*, and *IoM* are the input voltages and output currents of the primary group and secondary group, respectively.

**Figure 5.** Eight-module ISOP multimodule DC-DC converter circuit diagram; (**a**) Conventional ISOP DC-DC Converter, (**b**) Hybrid ISOP DC-DC Converter, (**c**) DAB Converter based on IGBTs, (**d**) DAB Converter based on MOSFETs.

#### *3.2. Hybrid ISOP Small Signal Analysis*

The eight-module hybrid ISOP converter small-signal model shown in Figure 6 is derived using [36]. The generalized model derived in the previous section is used to derive the small-signal functions for the presented converter in Figure 6, as shown in Table 2. The presented transfer functions are used in the control strategy scheme presented in the power balancing section.

**Figure 6.** Hybrid ISOP DC-DC converter small-signal model for eight-modules.

**Table 2.** Generalized model verification with the eight-module hybrid ISOP DC-DC converter.


#### **4. E**ffi**ciency and Power Density Assessment of the Hybrid Multimodule DC-DC Converter**
