1. Introduction

2. Fundamental topologies of SST

3. Continuous-time average state-space model

3.2. Isolation stage (DC-DC)

In this stage, the DAB converter is considered for DC-DC conversion. A high-frequency transformer is placed between DAB, as depicted in Figure 4. (a), and the equivalent diagram is depicted in Figure 4. (b). The DAB converter deals with three main parameters, namely, phase shift between two bridges (Ø), duty cycle ratios (D), and switching frequency (fs). The other parameters are as follows: ${\mathrm{R}}_{\mathrm{tf}}$ represents the equivalent of winding resistance and switching on resistance; ${\mathrm{L}}_{\mathrm{tf}}$ is the leakage inductance of the proposed transformer referred to as the secondary side; ${\mathrm{T}}_1$, ${\mathrm{T}}_2$, ${\mathrm{T}}_3$, ${\mathrm{T}}_4$, ${\mathrm{T}}_4$, ${\mathrm{T}}_5$, and ${\mathrm{T}}_6$ are the converter switches; and ${\mathrm{U}}_1$, and ${\mathrm{U}}_2$ are the primary and secondary winding voltages, respectively. The leakage inductance is calculated using Eq. (20). Refer to Eq. (15), the variation in transformer output with different frequencies and leakage inductance is shown in Figure 5.

Moreover, the duty cycle ratio of the proposed scheme is fixed to 0.5. Subsequently, the phase shift modulation technique is used here to control the operation of DAB. The transferred power from one stage of DAB to the other stage can be expressed in Eq. (16). Further, the state-space model of the DAB can be formulated as Eq. (17) by assuming the two-state variables of the proposed systems, namely, the current through the inductor ($i_L$) and the voltage across the capacitor (${\mathrm{U}}_0$). Eqs. (17) and (18) are time-variant and nonlinear. To ensure that the system is time-invariant and linear, the traditional state-space averaging procedure is applied to characterize a state variable x(t) by utilizing its Fourier series expansion as given in Eq. (19). When the Fourier analysis is applied in Eqs. (17) and (18), Eq. (20)-(25) are obtained. In this Fourier analysis, only the 0^th and 1^th order terms are considered, while other higher-order terms of state variables are neglected.

$$\mathrm{P}=\frac{{\mathrm{U}}^2}{2\times \mathsf{\pi }\times \mathrm{f}\times \mathrm{L}}\varnothing \left(1-\frac{\varnothing }{\mathsf{\pi }}\right)$$

(15)

$${\mathrm{P}}_{\mathrm{DAB}}=\frac{{\mathrm{U}}_{\mathrm{dc}}{\mathrm{U}}_0}{2{\mathrm{Nf}}_{\mathrm{s}}{\mathrm{L}}_{\mathrm{tf}}}\mathrm{D}\left(1-\mathrm{D}\right)$$

(16)

here, ${\mathrm{L}}_{\mathrm{tf}}=\frac{{\mathrm{L}}_1}{{\mathrm{N}}^2}+{\mathrm{L}}_2$

$$\frac{{\mathrm{di}}_{\mathrm{L}}\left(\mathrm{t}\right)}{\mathrm{dt}}=-\frac{{\mathrm{R}}_{\mathrm{tf}}}{{\mathrm{L}}_{\mathrm{tf}}}{\mathrm{i}}_{\mathrm{L}}\left(\mathrm{t}\right)+\frac{{\mathrm{T}}_1\left(\mathrm{t}\right)}{{\mathrm{L}}_{\mathrm{tf}}}{\mathrm{U}}_{\mathrm{dc}}\left(\mathrm{t}\right)-\frac{{\mathrm{T}}_3\left(\mathrm{t}\right)}{{\mathrm{L}}_{\mathrm{tf}}}{\mathrm{U}}_0\left(\mathrm{t}\right)$$

(17)

$$\frac{\mathrm{dU}\left(\mathrm{t}\right)}{\mathrm{dt}}=-\frac{1}{{\mathrm{RC}}_0}\mathrm{U}\left(\mathrm{t}\right)+\frac{{\mathrm{T}}_2\left(\mathrm{t}\right)}{{\mathrm{C}}_0}{\mathrm{i}}_{\mathrm{L}}\left(\mathrm{t}\right)-\frac{{\mathrm{I}}_0}{{\mathrm{C}}_0}$$

(18)

$$\mathrm{X}\left(\mathrm{t}\right)={\displaystyle \sum }_{\mathrm{k}=\infty }^{\infty }{\mathrm{X}}_{\mathrm{k}}\left(\mathrm{t}\right){\mathrm{e}}^{\mathrm{jk}\mathsf{\omega }\mathrm{t}}$$

(19)

where ω =2πf and X_k(t) is the k^th coefficient in the Fourier series analysis.

$$\frac{{\mathrm{d}\mathrm{U}}_{\mathrm{o}}^0}{\mathrm{dt}}=-\frac{{\mathrm{d}\mathrm{i}}_{\mathrm{o}}^0}{{\mathrm{C}}_0}-\frac{{\mathrm{U}}_{\mathrm{o}}^0}{{\mathrm{RC}}_0}+\frac{{\mathrm{T}}_2^0}{{\mathrm{C}}_0}{\mathrm{i}}_{\mathrm{L}}^{\mathrm{o}}+\frac{2{\mathrm{T}}_2^{1\mathrm{real}}}{{\mathrm{C}}_0}{\mathrm{i}}_{\mathrm{L}}^{1\mathrm{real}}+\frac{2{\mathrm{T}}_2^{1\mathrm{imag}}}{{\mathrm{C}}_0}{\mathrm{i}}_{\mathrm{L}}^{1\mathrm{imag}}$$

(20)

$$\frac{{\mathrm{d}\mathrm{U}}_{\mathrm{o}}^{1\mathrm{real}}}{\mathrm{dt}}=-\frac{{\mathrm{i}}_{\mathrm{o}}^{1\mathrm{real}}}{{\mathrm{C}}_0}-\frac{{\mathrm{U}}_{\mathrm{o}}^{1\mathrm{real}}}{{\mathrm{RC}}_0}-{\mathsf{\omega }\mathrm{U}}_{\mathrm{o}}^{1\mathrm{imag}}+\frac{{\mathrm{T}}_2^0}{{\mathrm{C}}_0}{\mathrm{i}}_{\mathrm{L}}^{1\mathrm{real}}+\frac{{\mathrm{T}}_2^{1\mathrm{real}}}{{\mathrm{C}}_0}{\mathrm{i}}_{\mathrm{L}}^0$$

(21)

$$\frac{{\mathrm{d}\mathrm{U}}_{\mathrm{o}}^{1\mathrm{imag}}}{\mathrm{dt}}=-\frac{{\mathrm{i}}_{\mathrm{o}}^{1\mathrm{imag}}}{{\mathrm{C}}_0}-\frac{{\mathrm{U}}_{\mathrm{o}}^{1\mathrm{imag}}}{{\mathrm{RC}}_0}-{\mathsf{\omega }\mathrm{U}}_{\mathrm{o}}^{1\mathrm{real}}+\frac{{\mathrm{T}}_2^0}{{\mathrm{C}}_0}{\mathrm{i}}_{\mathrm{L}}^{1\mathrm{imag}}+\frac{{\mathrm{T}}_2^{1\mathrm{imag}}}{{\mathrm{C}}_0}{\mathrm{i}}_{\mathrm{L}}^0$$

(22)

$$\frac{{\mathrm{d}\mathrm{i}}_{\mathrm{L}}^0}{\mathrm{dt}}=-\frac{{\mathrm{R}}_{\mathrm{tf}}}{{\mathrm{L}}_{\mathrm{tf}}}{\mathrm{i}}_{\mathrm{L}}^0+\frac{1}{{\mathrm{L}}_{\mathrm{tf}}}{\mathrm{U}}_{\mathrm{dc}}^0{\mathrm{T}}_1^0+\frac{2{\mathrm{T}}_1^{\mathrm{real}}}{{\mathrm{L}}_{\mathrm{tf}}}{\mathrm{U}}_{\mathrm{dc}}^{1\mathrm{real}}+\frac{2{\mathrm{T}}_1^{1\mathrm{imag}}}{{\mathrm{L}}_{\mathrm{tf}}}{\mathrm{U}}_{\mathrm{dc}}^{1\mathrm{imag}}-\frac{1}{{\mathrm{L}}_{\mathrm{tf}}}{\mathrm{U}}_{\mathrm{oc}}^0{\mathrm{T}}_2^0+\frac{2{\mathrm{T}}_2^{\mathrm{real}}}{{\mathrm{L}}_{\mathrm{tf}}}{\mathrm{U}}_{\mathrm{o}}^{1\mathrm{real}}+\frac{2{\mathrm{T}}_2^{1\mathrm{imag}}}{{\mathrm{L}}_{\mathrm{tf}}}{\mathrm{U}}_{\mathrm{o}}^{1\mathrm{imag}}$$

(23)

$$\frac{{\mathrm{d}\mathrm{i}}_{\mathrm{L}}^{1\mathrm{real}}}{\mathrm{dt}}=-\frac{{\mathrm{R}}_{\mathrm{tf}}}{{\mathrm{L}}_{\mathrm{tf}}}{\mathrm{i}}_{\mathrm{L}}^{\mathrm{real}}+{\mathsf{\omega }\mathrm{i}}_{\mathrm{L}}^{1\mathrm{imag}}+\frac{{\mathrm{U}}_{\mathrm{dc}}^{\mathrm{real}}}{{\mathrm{L}}_{\mathrm{tf}}}{\mathrm{T}}_1^0+{\mathrm{T}}_1^{1\mathrm{real}}{\mathrm{U}}_{\mathrm{dc}}^0-\frac{{\mathrm{T}}_2^0}{{\mathrm{L}}_{\mathrm{tf}}}{\mathrm{U}}_{\mathrm{o}}^{1\mathrm{real}}+{\mathrm{T}}_2^{1\mathrm{real}}{\mathrm{U}}_{\mathrm{o}}^0$$

(24)

$$\frac{{\mathrm{d}\mathrm{i}}_{\mathrm{L}}^{1\mathrm{imag}}}{\mathrm{dt}}=-\frac{{\mathrm{R}}_{\mathrm{tf}}}{{\mathrm{L}}_{\mathrm{tf}}}{\mathrm{i}}_{\mathrm{L}}^{\mathrm{imag}}+{\mathsf{\omega }\mathrm{i}}_{\mathrm{L}}^{1\mathrm{real}}+\frac{{\mathrm{U}}_{\mathrm{dc}}^{\mathrm{imag}}}{{\mathrm{L}}_{\mathrm{tf}}}{\mathrm{T}}_1^0+{\mathrm{T}}_1^{1\mathrm{imag}}{\mathrm{U}}_{\mathrm{dc}}^0-\frac{{\mathrm{T}}_2^0}{{\mathrm{L}}_{\mathrm{tf}}}{\mathrm{U}}_{\mathrm{o}}^{1\mathrm{imag}}+{\mathrm{T}}_2^{1\mathrm{imag}}{\mathrm{U}}_{\mathrm{o}}^0$$

(25)

The transient behavior of the DAB converter in terms of voltage variations is considerably slower in the input than in the output. Therefore, the 0^th term of input voltage is $U_{dc}^0$ =${\mathrm{U}}_{\mathrm{dc}}$ and $i_o^0$=${\mathrm{i}}_0$. The transient behavior of the first term of input voltage and load current is neglected, which are ${\mathrm{U}}_{1\mathrm{real}}$ and ${\mathrm{i}}_{1\mathrm{real}}$. In this design, the converters are operating in 50% of the duty cycle. Thus, the first-order terms of ${\mathrm{T}}_1$ (t) and $T_2$ (t) are similar to zero order and are presented in Eqs. (26)-(29). Moreover, in Eqs. (17) and (18), the independent state variables are the inductor current and capacitor voltage, which form the state-space model of the DAB converter with the state vector x(t) and the control vector u(t). Further, Eqs. (30) and (31) represent the state-space model with the input and output signals of the DAB converter. Therefore, the model parameters are input voltage source (${\mathrm{U}}_{\mathrm{dc}}^{}$), inductor current ( $i_L$ ), and duty cycle (D). When $T_1^{1imag}$ and $T_1^{1real}$are introduced into Eqs. (20), and (25), the state-space model becomes Eq. (31). Furthermore, the zero initial states are assumed in Eq. (32), which are, ${\mathrm{i}}_{\mathrm{L}}^0$, ${\mathrm{U}}_{\mathrm{o}}^{1\mathrm{real}}$, ${\mathrm{U}}_{\mathrm{o}}^{1\mathrm{imag}}$ are similar to zero order, and Eq. (32) becomes Eq. (33).

$$T_1^{1imag}=-\frac{2}{\mathsf{\pi }}$$

(26)

$$T_1^{1real}=0$$

(27)

$$T_2^{1imag}=-\frac{2\cos \mathrm{D}\mathsf{\pi }}{\mathsf{\pi }}$$

(28)

$$T_2^{1real}=-\frac{2\sin \mathrm{D}\mathsf{\pi }}{\mathsf{\pi }}$$

(29)

$$\frac{\mathrm{X}\left(\mathrm{t}\right)}{\mathrm{dt}}={\mathrm{A}}_1\mathrm{x}\left(\mathrm{t}\right)+{\mathrm{B}}_1\mathrm{u}\left(\mathrm{t}\right)$$

(30)

$$\mathrm{Y}\left(\mathrm{t}\right)={\mathrm{C}}_1\mathrm{x}\left(\mathrm{t}\right)+{\mathrm{D}}_1\mathrm{u}\left(\mathrm{t}\right)$$

(31)

(32)

(33)

A small signal control for the output transfer function is needed to measure the system stability of the DAB converter with the controller structure. Thus, a small deviation in D causes a distinction in the values of ${\mathrm{U}}_o^0$ from its steady-state value. Hence, the corresponding small signal model is defined in Eqs. (34)-(37). In addition, the nonlinearity of the system is introduced when the control and state variables are multiplied. The approximation solution of nonlinear terms is presented in Eq. (38). To this end, given the value of parameters defined in Eqs. (34)-(37) are placed into Eq. (33), then the state-space model can be expressed as in Eq. (39) while the corresponding values are given in A, B, C, and D matrix form. Further, the transfer function of the DAB converter can be expressed using a conversion formula, which is given in Eq. (40), and the values are assigned and given in Eq. (41), where ${\mathrm{G}}_{\mathrm{open}-\mathrm{loop}}$ is the open-loop transfer function of the DAB.

$$△\mathrm{D}=\mathrm{D}-\mathrm{d}$$

(34)

$$△{\mathrm{U}}_o^0={\mathrm{U}}_o^0-{\mathrm{U}}_o^0$$

(35)

$$△i_L^{1real}=i_L^{1real}-I_L^{1real}$$

(36)

$$△i_L^{1imag}=i_L^{1imag}-I_L^{1imag}$$

(37)

$$\sin \left(\mathsf{\pi }\mathrm{D}\right){\mathrm{U}}_o^0=\sin \left(\mathsf{\pi }\mathrm{D}\right)△{\mathrm{U}}_o^0+{\mathrm{U}}_o^0\sin \left(\mathsf{\pi }\mathrm{D}\right)\cos \left(\mathsf{\pi }△\mathrm{D}\right)+{\mathrm{U}}_o^0\cos \left(\mathsf{\pi }\mathrm{D}\right)\sin \left(\mathsf{\pi }△\mathrm{D}\right)=\sin \left(\mathsf{\pi }\mathrm{D}\right)△{\mathrm{U}}_o^0+{\mathrm{U}}_o^0\sin \left(\mathsf{\pi }\mathrm{D}\right)+{\mathrm{U}}_o^0\cos \left(\mathsf{\pi }\mathrm{D}\right)$$

(38)

(39)

$${\mathrm{G}}_{\mathrm{open}-\mathrm{loop}}=\left[\mathrm{C}{\left(\mathrm{sI}-\mathrm{A}\right)}^{-1}\mathrm{B}+\mathrm{D}\right]$$

(40)

$${\mathrm{G}}_{\mathrm{open}-\mathrm{loop}}=\frac{4.5\times {10}^4{\mathrm{s}}^2+4.15\times {10}^9\mathrm{s}+6.072\times {10}^{15}}{{\mathrm{s}}^3+92398{\mathrm{s}}^2+1.798\times {10}^{10}+6.264\times {10}^{12}}$$

(41)

4. Results and discussion

5. Conclusions

Bibilogpahy

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