Impedance Modeling and Stability-Oriented Parameter Optimization of Isolated Dual Active Bridge-Based Two-Stage AC-DC-DC Converter

The isolated dual active bridge (DAB)-based two-stage AC-DC-DC converters have been widely applied in grid-connected power electronics systems. However, the impedance interactions between the DAB converter and the AC-DC recti ﬁ er can cause instability problems to the two-stage converter systems. The accurate terminal impedance characteristics of the DAB converters are still not clear due to the high frequency ac conversion stage. This makes it dif ﬁ cult to assess the impacts of the DAB circuit parameters on the stability of the two-stage converters. To address these issues, the impedance models of the DAB converters are originally derived in this paper. Based on the developed impedance models, the stability of the DAB-based two-stage AC-DC-DC converters is analyzed. The impacts of the DAB converter circuit parameters on the stability of the two-stage AC-DC-DC converters are comprehensively revealed by the Bode and Nyquist plots. The analysis results offer instructive implications to ﬁ netune the design rules of the DAB converters. The conclusions are validated by comprehensive simulation and experimental results.


INTRODUCTION
In order to connect various DC energy storage sources, such as the batteries and supercapacitors, to the AC utility grids, one of the most common topologies is the two-stage AC-DC-DC converter which consists of an isolated DC-DC conversion stage and an AC-DC stage (Blaabjerg et al., 2004;Wu et al., 2015;Feng et al., 2020). Among the bidirectional DC-DC converter topologies, the dual active bridge (DAB) converter is becoming more and more popular because of its advantages on the high-power density, galvanic isolation, and soft switching (Krismer and Kolar 2009;Roggia et al., 2013;Ye et al., 2017). Figure 1 shows the DAB-based two-stage AC-DC-DC converter. Literature (Wu et al., 2018) proposed the cooperative triple phase shift (CTPS) control is proposed to eliminate the dual-side flow back currents and reduce the current stress of the DAB converter. The CTPS-based DAB converter has a great potential for being used as the interface converter for the energy storage. Therefore, it is also applied in the twostage AC-DC-DC converter in this paper.
However, it is found that this two-stage converter has instability issues. Although the DAB converter and the AC-DC rectifier are stable individually, the instable problem may occur when they are cascaded together. Similar problems have also been reported in other twostage systems (Radwan and Mohamed 2012;Wu et al., 2012;Liu et al., 2014;Wen et al., 2015). Driven by this issue, a lot of efforts have been made to investigate the stability criterion of the two-stage systems. The concept of the impedance-based stability criterion is first proposed in Literature Middlebrook and Cuk (1976), and later becomes the base for the stability analysis of cascaded systems (Sun 2011;Zhang et al., 2015). A general representation of a two-stage cascaded converter configuration is shown in Figure 2. The stability of the two-stage converter depends on the minor loop gain T m , expressed as the ratio of the source converter output impedance Z out_S to the load converter input impedance Z in_L . If the impedance ratio T m satisfies the Nyquist criterion, the two-stage converter will be stable.
Hence, to analyze the stability of the two-stage converter, it is necessary to derive the impedance models of the AC-DC rectifier and the DAB converter. The impedance model of AC-DC rectifier has been well established. However, the high frequency ac conversion stage brings difficulties to the model generation of the DAB converter. The reduced-order impedance models are derived in Literature , based on the "smallripple" assumption. Since the dc average of the transformer current over a switching period is 0, the dynamics of the transformer current are neglected, thus leading to the sacrifice of accuracy. A full-order discrete-time model, which can obtain higher accuracy compared to the reduced-order model, is developed in Literature Zhao et al. (2010). However, for control design, the continuous model is more desirable. Based on the generalized averaging method, a full-order continuous-time model of the single-phase shift (SPS) modulated DAB converter is proposed and verified to be more accurate than the reduced order models at dc and low frequency (Qin and Kimball 2012). But the model is not extended to the derivation of the DAB converter input or output impedances. In Literature (Mueller and Kimball 2017), the closed-loop impedance of the SPS-DAB converter is calculated based on the Extra Element Theorem with a dummy controller. However, the derivation process is cumbersome and time-consuming for practical use. So far, the existing impedance models are derived from SPS-DAB converters and are not suitable for CTPS-DAB converters. Since the CTPS modulation is drawing increasing attention among the DAB converter applications, the impedance model of the CTPS-based DAB converters should be developed. Moreover, in previous applications, the design of the DAB converter circuit parameters is focused on the steady and dynamic performance (Tan et al., 2012). The influences of the circuit parameters on the stability of the twostage converter have not been studied yet.
To fill up the gap, the full-order small signal impedance model of the CTPS-based DAB converter is originally derived in this paper. The developed impedance model is used to analyze the stability of the two-stage converter and the influences of the circuit parameters on the stability. Furthermore, the design of the DAB circuit parameters is revisited. The optimization guideline of the DAB circuit parameters is proposed to improve the stability of the two-stage converter.
The rest of this paper is organized as follows. The operation mechanism of the CTPS based DAB converter is analyzed in Section 2. And the small signal impedance models of the two-stage converter are derived in Section 3. The stability of the system and the effects of the filter capacitances are analyzed in Section 4. The simulation and hardware-in-the-loop (HIL) experimental results are presented in Section 5 to verify the theoretical analysis. And Section 6 gives the conclusions of this work.

Operation Principle of CTPS-DAB Converters
The phase-shift modulation methods are the most widely used to regulate the power transferred in the DAB converters. It has been verified in Literature Wu et al. (2018) that the CTPS modulation  can eliminate the dual side flow back currents in the DAB converter and achieve better current characteristics, including the inductor current stress and RMS value. The CTPS-DAB converter is more efficient and reliable compared to other traditional phase-shift control methods. Therefore, the CTPS modulation is applied to the DAB-based two-stage converter in this paper. For the purpose of impedance modeling derivation, the relationship of the three control degrees, i.e., the phase-shift ratios, is given here. The details of the CTPS operation principle is omitted due to space limit but can be found in Literature Wu et al. (2018). Figure 3 depicts the typical operation waveforms of CTPS-DAB converter. S 1 -S 8 are the driving signals for the switches of the DAB converter. i L is the inductor current. I 1 and I 2 are the input and output current of the DAB converter, respectively. v h1 and v h2 are the output voltages of the two H-bridges on the primary and secondary side of the transformer, respectively. T s is the switching cycle of the DAB converter. d φ is the phase-shift ratio between v h1 and v h2 . d 1 and d 2 are the phase-shift ratios of both primary and secondary H-bridges. All the time points t 1 -t 6 can be found in Table 1. The driving signals S 4 or S 8 is reversed only when the I 1 or I 2 is zero. Thus, the polarities of v h1 and v h2 can be consistent with that of i L . To meet this requirement, the total increment of i L should be zero during half of a switching cycle, which can be expressed as in Eq. 1. And this constraint can be further simplified as in Eq. 2.
where k = nv dc /v c ; v dc is the DC bus voltage; v c is the voltage across the capacitance C b ; n is the turns ratio of the transformer; L s is the inductance.
The outer phase shift ratio d φ should be equal to d 2 to keep the same polarity of v h1 and v h2 , namely

Stability Analysis of Two-Stage Converters
According to Literature Zhang et al. (2015), the stability of the two-stage converter relies on the minor loop gain T m , which is organized as the ratio of the voltage-controlled converter's output impedance to the current-controlled converter's input impedance. Figure 4 shows the schematic and the control system block diagram of the two-stage AC-DC-DC converter, which is used to connect a battery to the utility grid. Between the DAB converter and the AC-DC rectifier, a DC-bus capacitor C dc is used to smooth the DC-bus voltage ripple. The control of the twostage AC-DC-DC converter can be split into two parts: the DC-DC conversion stage and the AC-DC conversion stage. The AC-DC rectifier is responsible for maintaining the DC bus voltage V dc constant and realizing the grid current closedloop control. The DAB converter closes the control loop regulating the battery charging current I b . It should be noted that the control strategy of the two-stage converter keeps the same in bidirectional operation. From Figure 4, T m can be calculated as: where Z o_rec is the output impedance of the DC-AC rectifier and Z in_DAB is the input impedance of the DAB converter. The stability analysis of the two-stage converter therefore relies on the model-based determinations of the required impedances in Eq. 4. According to Literature , two-stage converters have instable problems only when the power is transferred from the voltage-controlled sub-   converter to current-controlled sub-converter. Considering the control arrangement in Figure 4, the stability of the two-stage converter in battery charging mode, i.e., the power is transferred from the rectifier to the DAB converter, should be analyzed. Hence, the input impedance of the DAB converter in the battery charging mode are derived in the following section.

Output Impedance of the AC-DC Rectifiers
The performance and control of the three-phase AC-DC rectifier have been well studied in the existing literature. The derivation process of the small signal model of the rectifier which can be found in Literature Twining and Holmes (2003) is omitted here for conciseness. The output impedance Z o_rec of the rectifier can be expressed as: where C dc is the dc-link capacitance; L g is the input filter inductance of the rectifier; G ci is the controller of the rectifier; I Ld is the d-axis component of the inductor current; D d and D q denote the d-and q-axes components of the duty ratio of the rectifier, respectively.

Small Signal Impedance Modeling
The impedance model of the CTPS-DAB converter is derived first based on the simplified circuit shown in Figure 5. The voltagecontrolled AC-DC rectifier can be regarded as a DC voltage source. R b is the equivalent load of the battery in the charging operation mode. R t is the winding resistance of the transformer. The capacitor voltage v c and the inductor current i L are taken as the state variables. The state equations of the CTPS-DAB converter can be expressed as: where g 1 and g 2 are the switching functions of the two H-bridges at the primary and secondary sides of the transformer. They can be expressed as follows according to Figure 3.
Due to the high-frequency ac link in the DAB converter, the generalized state-space averaging (GSSA) method is applied (Caliskan et al., 1999;Qin and Kimball 2012). The averages of the state variables and the switching functions are approximated  by the sum of the index-0 and index-1 coefficient of their Fourier series. This leads to the following state-variable vector, where the subscripts denote the number of the coefficients of the Fourier series; the superscripts "R" and "I" mean the real and imaginary parts of the first coefficients, respectively. These rules apply in the whole paper.
In Literature Qin and Kimball (2012), the authors show that the dynamics of 〈v c 〉 R 1 , 〈v c 〉 I 1 and 〈i L 〉 0 are decoupled from the rest of the system. Hence, the state-variable vector is reduced to Taking the Fourier series of the switching functions, the zeroth coefficients 〈g 1 〉 0 〈g 2 〉 0 0 due to the symmetry of the operation waveforms. The complex number index-1 coefficients of g 1 (t) and g 2 (t) are given by, The resulting GSSA state equations can be obtained as Eq. 15.
where _ x is the derivative of the state vector.
The equilibrium point can be derived by solving _ x 0. The small signal model can be obtained by perturbating the input variables around the equilibrium point. Note that in this paper, for the variable x, its small signal notation isx and its steady state value is denoted by the uppercase letter X. d 2 is calculated from Eq. 2. Substituting Eq. 2 into Eqs 15, 16, d 2 is replaced by d 1 , v c and v dc . The input vector is thus given by, The open-loop small signal model of the CTPS-DAB converter is expressed in Eq. 18.
where A, B, C, D are system matrix: The partial derivatives are given by, z〈g 2 〉 I 1 zd 1 sin(D 1 π) + (1 + k) sin((D 1 + D 2 )π) (19d) The open-loop transfer functions can be obtained as Eqs 20a-d by solving the open-loop small signal model Eq. 18 using MATLAB.
The battery current is measured in the control loop of the DAB converter. The control block diagram is shown in Figure 6. G c is the controller of the DAB converter based on PI structure. From Figure 6, the closed-loop input impedance Z in_DAB can be obtained in Eq. 21 based on Mason's gain formula (Nise 2007).
where T is the loop gain of the DAB converter and T G c G OL i b d 1 . And Z in_DAB can be further simplified as Eq. 22.
Eq. 22 can be used to determine the closed-loop input impedance of DAB converters around a certain operating point when the battery voltage changes V c due to the state-ofcharge variations.
It should be mentioned that the battery voltage varies between 270 and 300 V due to the SOC change. Therefore, it is meaningful to investigate the influences of the battery voltage on the DAB input impedance models. Figure 7 shows the bode plots of a set of linearized input impedance models around four different operating points, when V c is incrementally changed from 270 to 300 V. The magnitude of Z in_DAB is slightly decreased with the increase of V c , although the phase angles of Z in_DAB are almost the same. According to the Middlebrook criterion, the stability of the two-stage converter requires Z in_DAB larger than Z o_rec . The unstable problem of the two-stage converter most likely happens when the input impedance of the DAB converter presents the lowest magnitude. Therefore, to ensure the stability within the whole operation range, the stability of the   two-stage converter should be assessed when the battery voltage is 300 V.

Validation of Input Impedance Models
The validity of the developed input impedance model is verified using fully detailed simulation in MATLAB/Simulink environment. The measurement of Z in_DAB is shown in Figure 8. Similar impedance measurement methods have been successfully applied in (Huang et al., 2009;Rygg and Molinas 2017). The parameters of the two-stage converter are shown in Table 2. The deadtime of the switches and the sampling period are set as 0.2 and 250 μs, respectively. The simulation time step is fixed to 0.1 μs. The AC-DC rectifier is replaced by a controllable ripple voltage source connected in series with a 660 V DC voltage source. A sinusoidal perturbation is generated at the DC bus, which is superimposed on steady-state DC voltage as the ripple voltage, v dc_r . The magnitude of v dc_r is set to 1 and the frequency of v dc_r varies logarithmically from 2 Hz to 10 kHz. The corresponding DC bus currents are also measured. The magnitude and phase of the current ripple, i dc_r , are extracted by FFT analysis at each frequency. The impedance of the DAB converter at different frequencies can be calculated by Eq. 23.
The bode plots of Z in_DAB with measured values and model predictive values are shown in Figure 9. The impedance model predicted results match well with the measured results from the low frequencies up to the half of the switching frequency f s . Therefore, the developed impedance model can provide fairly accurate prediction for the input impedance of the CTPS-DAB converter. Furthermore, Z in_DAB shows negative resistance characteristics at the low frequencies. Moreover, the magnitude of Z in_DAB is reduced at about 200 Hz. This is the potential threat for stability consideration of the two-stage converter. If the magnitude of Z in_DAB is lower than that of Z o_rec within the low frequency range, the two-stage converter will be unstable. Hence, it is necessary to increase the minimum magnitude value of Z in_DAB . Since the DAB converter circuit parameters are included in the impedance model, the most straightforward method to modify the DAB input impedance is to optimize the DAB circuit parameters. Thus, the effects of the DAB converter circuit parameters on the stability of the two-stage converter should be analyzed.

STABILITY-ORIENTED OPTIMIZATION OF CIRCUIT PARAMETERS
For the stability consideration, the magnitude intersection between Z in_DAB and Z o_rec should be avoided. The minimum value of Z in_DAB should be designed in the range above the peak value of Z o_rec . From Eq. 18 to Eq. 22, the inductor L s and DC side output filter capacitor C b are included in the impedance model Z in_DAB . Therefore, to improve the stability of the two-stage converter, the most effective and easiest method is to optimize the value of L s and C b .

Stability Improvement by Optimization of L s
To analyze the effects of L s on the minimum of Z in_DAB , Figure 10A shows the bode plots of Z in_DAB with different values of L s . The rest of the circuit parameters are listed in Table 2. The minimum of Z in_DAB increases with the reduction of L s . When L s is 200 µH or 100 μH, Z in_DAB intersects with Z o_rec , indicating that the two-stage converter is unstable. However, for L s smaller than 50 μH, the minimum value of Z in_DAB is larger than the magnitude of Z o_rec within all operation frequencies. According to the Middlebrook criterion, the two-stage converter is stable under such conditions. The influences of L s on the stability of two-stage converter can be further illustrated by the Nyquist plots of T m shown in Figure 10B.
The trajectory of T m encircles the critical point (−1, 0) when L s is 200 µH or 100 μH, suggesting that there is a right-hand plane pole in the s-plane. The two-stage converter is hence unstable with these two inductors. And the stability of the two-stage converter can be improved by reducing the value of L s . As seen from Figure 10B, with the reduction of L s , the gain margin and phase margin are both   increased. When L s is smaller than 50 μH, the trajectory of T m no longer encircles (−1, 0) anymore. This can also be deduced from Figure 10A, where the minimum value of Z in_DAB is gradually increased with the reduction of L s . Therefore, for improving the stability of the two-stage converter, a relatively small value of L s is preferred, although the other factors such as power transfer ability should also be considered.

Stability Improvement by Optimization of C b
Similarly, to analyze the influences of C b on the stability of the twostage converter, a set of C b are selected. The relationship between the value of C b and the minimum magnitude of Z in_DAB is shown in Figure 11A. The minimum magnitude of Z in_DAB keeps decreasing with the reduction of C b . From Figure 11A, when C b is larger than 200 μF, Z in_DAB and Z o_rec intersect with each other, which can cause the unstable problem to the two-stage converter.
The effects of C b on system stability are further illustrated by the Nyquist plots shown in Figure 11B. From Figure 11B, When C b is larger than 200 μF, the trajectory of T m encircles the point (−1, 0). This indicates that the two-stage converter cannot maintain its stability. However, with C b decreasing from 1,000 µF to 50 μF, the gain margin and the phase margin of the two-stage converter increase gradually. And the two-stage converter is stable when C b is smaller than 100 µF. The conclusion that can be drawn from the Nyquist plot together with the bode plots is that a relatively small value of C b is good for the stability consideration.

EXPERIMENTAL RESULTS AND DISCUSSION
The hardware-in-the-loop (HIL) experimental system has been proven to be an effective method for stability analysis of power electronic converters (Zhang et al., 2018). To verify the theoretical analysis, the stability of the two-stage converter is tested based on the OPAL-RT based HIL experimental system. The parameters of the experimental system are shown in Table 2. It should be noted that the waveform of V dc in Figures 12-15 are only its AC components to show the stability. Figure 12A shows the waveforms of the two-stage converter with original circuit parameters (L s = 200 μH, C b = 1,000 μF). As seen, the oscillation at V dc can be found. Moreover, the oscillating frequency is around 180 Hz, which is consistent with the prediction of the bode plot in Figure 10A. The instable problem of the two-stage converter can be more visualized from the FFT spectra of V dc shown in Figure 12B. The amplitude of the 180 Hz harmonic component is significant. This is the root cause of the waveform oscillation.
Considering the effects of L s on the minimum magnitude of Z in_DAB , L s should be reduced to improve the stability. Figure 13A shows the waveforms of the two-stage converter with reduced L s = 50 µH and other parameters being unchanged. The oscillation at V dc is mitigated as expected. This is because the minimum magnitude of Z in_DAB is increased and there is no intersection between Z in_DAB and Z o_rec as shown in Figure 10A. It can be also observed from the FFT spectra of V dc in Figure 13B that the quantities of the 180 Hz harmonic response are suppressed.
Similarly, according to the above-mentioned analysis, C b should be reduced to for stability consideration. Figure 14A shows the waveforms of the two-stage converter with C b = 100 µF and other parameters being constant. The waveforms show that the two-stage converter is stable with reduced C b . According to FFT spectra of V dc in Figure 14B, the 180 Hz harmonic response is also greatly suppressed.
The dynamic performances of the two-stage converter with reduced L s and C b are also tested and shown in Figure 15. When I b steps up from 6 to 12 A, V dc can be stabilized with reduced L s or C b after a short transient period. However, when reduced L s is applied, the overshoot of V dc is about 14 V. When reduced C b is applied, the overshoot of V dc is only about 5 V.
Hence, the stability of the two-stage converter can be improved by optimizing L s or C b . Furthermore, the dynamic performance of the two-stage converter with reduced C b is better than that of the two-stage converter with reduced L s according to the experimental waveforms.

CONCLUSION
The full-order impedance model of the CTPS based DAB converter has been derived in this paper. The developed impedance model can fully represent the dynamics of the ac conversion stage of the DAB converters. Based on the developed impedance model, the stability of two-stage converter has been analyzed. Furthermore, the effects of the DAB circuit parameters on the stability of the two-stage converter have been presented and analyzed. The design of the DAB circuit parameters is revisited. The stability-oriented optimization guideline of circuit parameters is provided for the DAB converter. The stability of the two-stage converter can be improved by reducing the leakage inductance or the DC side output filter capacitance of the DAB converter.

DATA AVAILABILITY STATEMENT
The original contributions presented in the study are included in the article/Supplementary Material, further inquiries can be directed to the corresponding authors.

AUTHOR CONTRIBUTIONS
FF contributed to the conception of the study and wrote the manuscript; JF performed the data analyses; HG contributed to analysis and manuscript preparation. UM and LW performed the experiment.