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. [ 1 − d 2 − ( 1 − d 1 − d φ ) ] T s 2 v c L s = d φ T s 2 v d c n L s + ( 1 − d 1 − d φ ) T s 2 v d c − n v c n L s (1) d 2 = 1 + k ( d 1 − 1 ) (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 d φ = d 2 (3)

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 two-stage 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 closed-loop 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: T m = Z o _ r e c Z i n _ D A B (4) 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 (Tian et al., 2016), 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.

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: Z o _ r e c = − 2 ( s L g − V d c G c i ) 2 s 2 C d c L g − 2 s C d c V d c G c i + 3 ( D d 2 + D q 2 ) − 3 G c i D d I L d (5) 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.

The impedance model of the CTPS-DAB converter is derived first based on the simplified circuit shown in Figure 5. The voltage-controlled 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: L t d i L d t = v h 1 n − v h 2 − R t i L = g 1 v d c n − g 2 v c − R t i L (6) C b d v c d t = i 2 − i b = g 2 i L − v c R b (7) 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. g 1 ( t ) = { 1 ,          [ t 1 , t 3 ) 0 ,         [ t 0 , t 1 ) ∪ [ t 3 , t 4 ) − 1 ,       [ t 4 , t 6 ) (8) g 2 ( t ) = { 1 ,         [ t 2 , t 4 )   0 ,        [ t 1 , t 2 ) ∪ [ t 4 , t 5 ) − 1 ,      [ t 0 , t 1 ) ∪ [ t 5 , t 6 ) (9)

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, x = [ 〈 v c 〉 0 〈 v c 〉 1 R 〈 v c 〉 1 I 〈 i L 〉 0 〈 i L 〉 1 R 〈 i L 〉 1 I ] T (10) 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 〉 1 R , 〈 v c 〉 1 I and 〈 i L 〉 0 are decoupled from the rest of the system. Hence, the state-variable vector is reduced to x = [ 〈 v c 〉 0 〈 i L 〉 1 R 〈 i L 〉 1 I ] T .

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, 〈 g 1 〉 1 R = − sin ( d 1 π ) π (11) 〈 g 1 〉 1 I = − 1 + cos ( d 1 π ) π (12) 〈 g 2 〉 1 R = − sin ( d 1 π ) + sin ( d 1 + d 2 ) π π (13) 〈 g 2 〉 1 I = − cos ( d 1 π ) + cos ( d 1 + d 2 ) π π (14)

The resulting GSSA state equations can be obtained as Eq. 15. The output vector y = [ i 1 i b ] T can be expressed as Eq. 16. x ˙ = [ − 1 C b R b 2 〈 g 2 〉 1 R C b 2 〈 g 2 〉 1 I C b − 〈 g 2 〉 1 R L t − R t L t ω s − 〈 g 2 〉 1 I L t − ω s − R t L t ] x + [ 0 〈 g 1 〉 1 R n L t 〈 g 1 〉 1 I n L t ] 〈 v d c 〉 0 (15) where x ˙ is the derivative of the state vector. y = [ 0 2 n 〈 g 1 〉 1 R 2 n 〈 g 1 〉 1 I 1 R b 0 0 ] x + [ 0 0 ] 〈 v d c 〉 0 (16)

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 is x ^ 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, u ^ = [ v ^ d c d ^ 1 ] (17)

The open-loop small signal model of the CTPS-DAB converter is expressed in Eq. 18. x ^ ˙ = A x ^ + B u ^ y ^ = C x ^ + D u ^ (18) where A, B, C, D are system matrix: A = [ A 11 2 〈 g 2 〉 1 R C b 2 〈 g 2 〉 1 I C b A 21 − R t L t ω s A 31 − ω s − R t L t ] (18a) B = [ B 11 B 12 B 21 B 22 B 31 B 32 ] (18b) C = [ 0 2 n 〈 g 1 〉 1 R 2 n 〈 g 1 〉 1 I 1 R b 0 0 ] (18c) D = [ 0 2 ( I L R ∂ 〈 g 1 〉 1 R ∂ d 1 + I L I ∂ 〈 g 1 〉 1 I ∂ d 1 ) 0 0 ] (18d) A 11 = − 1 C b R b + 2 C b ( I L R ∂ 〈 g 2 〉 1 R ∂ v c + I L I ∂ 〈 g 2 〉 1 I ∂ v c ) (18e) A 21 = − 〈 g 2 〉 1 R L s − V c L s ∂ 〈 g 2 〉 1 R ∂ v c (18f) A 31 = − 〈 g 2 〉 1 I L t − V c L t ∂ 〈 g 2 〉 1 I ∂ v c (18g) B 11 = 2 C b ( I L R ∂ 〈 g 2 〉 1 R ∂ v d c + I L I ∂ 〈 g 2 〉 1 I ∂ v d c ) (18h) B 12 = 2 C b ( I L R ∂ 〈 g 2 〉 1 R ∂ d 1 + I L I ∂ 〈 g 2 〉 1 I ∂ d 1 ) (18i) B 21 = 〈 g 1 〉 1 R L t − V c L t ∂ 〈 g 2 〉 1 R ∂ v d c (18j) B 22 = − V c L t ∂ 〈 g 2 〉 1 R ∂ d 1 + V d c n L t ∂ 〈 g 1 〉 1 R ∂ d 1 (18k) B 31 = 〈 g 1 〉 1 I L t − V c L t ∂ 〈 g 2 〉 1 I ∂ v d c (18l) B 32 = − V c L t ∂ 〈 g 2 〉 1 I ∂ d 1 + V d c n L t ∂ 〈 g 1 〉 1 I ∂ d 1 (18m)

The partial derivatives are given by, ∂ 〈 g 1 〉 1 R ∂ d 1 = − cos ( D 1 π ) (19a) ∂ 〈 g 1 〉 1 I ∂ d 1 = sin ( D 1 π ) (19b) ∂ 〈 g 2 〉 1 R ∂ d 1 = − cos ( D 1 π ) − ( 1 + k ) cos ( ( D 1 + D 2 ) π ) (19c) ∂ 〈 g 2 〉 1 I ∂ d 1 = sin ( D 1 π ) + ( 1 + k ) sin ( ( D 1 + D 2 ) π ) (19d) ∂ 〈 g 2 〉 1 R ∂ v d c = − D 1 − 1 n V c cos ( ( D 1 + D 2 ) π ) (19e) ∂ 〈 g 2 〉 1 R ∂ v c = k D 1 − 1 V c cos ( ( D 1 + D 2 ) π ) (19f) ∂ 〈 g 2 〉 1 I ∂ v d c = D 1 − 1 n V c sin ( ( D 1 + D 2 ) π ) (19g) ∂ 〈 g 2 〉 1 I ∂ v c = − k D 1 − 1 V c sin ( ( D 1 + D 2 ) π ) (19h)

The open-loop transfer functions can be obtained as Eqs 20a–d by solving the open-loop small signal model Eq. 18 using MATLAB. Z i n _ D A B O L = ( s − A 11 ) [ ( s + R t L t ) 2 + ω s 2 ] − 2 〈 g 2 〉 1 R C b [ A 21 ( s + R t L t ) + A 31 ω s ] − 2 〈 g 2 〉 1 I C b [ A 31 ( s + R t L t ) − A 21 ω s ] { [ ( 2 n 〈 g 1 〉 1 R A 21 + 2 n 〈 g 1 〉 1 I A 31 ) s + 2 n 〈 g 1 〉 1 R ( A 21 R t L t + A 31 ω s ) + 2 n 〈 g 1 〉 1 I ( A 31 R t L t − A 21 ω s ) ] B 11 + [ 2 n 〈 g 1 〉 1 R ( s 2 − ( A 11 − R t L t ) s − A 11 R t L t − 2 〈 g 2 〉 1 I C b A 31 ) + 2 n 〈 g 1 〉 1 I ( − ω s s + A 11 ω s + 2 〈 g 2 〉 1 I C b A 31 ) ] B 21 + [ 2 n 〈 g 1 〉 1 R ( ω s s − A 11 ω s + 2 〈 g 2 〉 1 I C b A 21 ) + 2 n 〈 g 1 〉 1 I ( s 2 − ( A 11 − R t L t ) s − A 11 R t L t − 2 〈 g 2 〉 1 R C b A 21 ) ] B 31 } (20a) G i 1 d 1 = { [ ( 2 n 〈 g 1 〉 1 R A 21 + 2 n 〈 g 1 〉 1 I A 31 ) s + 2 n 〈 g 1 〉 1 R ( A 21 R t L t + A 31 ω s ) + 2 n 〈 g 1 〉 1 I ( A 31 R t L t − A 21 ω s ) ] B 12 + [ 2 n 〈 g 1 〉 1 R ( s 2 − ( A 11 − R t L t ) s − A 11 R t L t − 2 〈 g 2 〉 1 I C b A 31 ) + 2 n 〈 g 1 〉 1 I ( − ω s s + A 11 ω s + 2 〈 g 2 〉 1 I C b A 31 ) ] B 22 + [ 2 n 〈 g 1 〉 1 R ( ω s s − A 11 ω s + 2 〈 g 2 〉 1 I C b A 21 ) + 2 n 〈 g 1 〉 1 I ( s 2 − ( A 11 − R t L t ) s − A 11 R t L t − 2 〈 g 2 〉 1 R C b A 21 ) ] B 32 } ( s − A 11 ) [ ( s + R t L t ) 2 + ω s 2 ] − 2 〈 g 2 〉 1 R C b [ A 21 ( s + R t L t ) + A 31 ω s ] − 2 〈 g 2 〉 1 I C b [ A 31 ( s + R t L t ) − A 21 ω s ] (20b) G i b v d c = C b [ ( s + R t L t ) 2 + ω s 2 ] B 11 + 2 〈 g 2 〉 1 R ( s + R t L t ) B 21 − 2 〈 g 2 〉 1 I ω s B 21 + 2 〈 g 2 〉 1 R ω s B 31 + 2 〈 g 2 〉 1 I ( s + R t L t ) B 31 R b C b ( s − A 11 ) [ ( s + R t L t ) 2 + ω s 2 ] − 2 R b 〈 g 2 〉 1 R [ A 21 ( s + R t L t ) + A 31 ω s ] − 2 R b 〈 g 2 〉 1 I [ A 31 ( s + R t L t ) − A 21 ω s ] (20c) G i b d 1 = C b [ ( s + R t L t ) 2 + ω s 2 ] B 12 + 2 〈 g 2 〉 1 R ( s + R t L t ) B 22 − 2 〈 g 2 〉 1 I ω s B 22 + 2 〈 g 2 〉 1 R ω s B 32 + 2 〈 g 2 〉 1 I ( s + R t L t ) B 32 R b C b ( s − A 11 ) [ ( s + R t L t ) 2 + ω s 2 ] − 2 R b 〈 g 2 〉 1 R [ A 21 ( s + R t L t ) + A 31 ω s ] − 2 R b 〈 g 2 〉 1 I [ A 31 ( s + R t L t ) − A 21 ω s ] (20d)

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). 1 Z i n _ D A B = i ^ 1 v ^ d c = 1 Z i n _ D A B O L − G i b v d c O L G c G i 1 d 1 O L 1 + T = ( 1 + T ) − G i b v d c O L G c G i 1 d 1 O L Z i n _ D A B O L ( 1 + T ) Z i n _ D A B O L (21) where T is the loop gain of the DAB converter and T = G c G i b d 1 O L .

And Z _{in_DAB} can be further simplified as Eq. 22. Z i n _ D A B = ( 1 + T ) Z i n _ D A B O L ( 1 + T ) − G i b v d c O L G c G i 1 d 1 O L Z i n _ D A B O L (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-of-charge 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.

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. Z i n _ D A B = v d c _ r i d c _ r (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.

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.

Similarly, to analyze the influences of C _b on the stability of the two-stage 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.