1. Introduction
In recent years, the proportion of renewable energy in the power grid has rapidly increased. However, as a significant source of renewable energy, solar power could not be connected to the AC grid directly. As a result, the DC microgrid was proposed and has developed greatly [
1]. The DC microgrid could not only realize local consumption but also relatively reduce the times of electric energy conversion, which significantly improved efficiency [
2]. In a DC microgrid, solar power could not work stably. It would lead to damage to the converter, and even DC-link voltage fluctuation, causing grid fault [
3]. Bidirectional isolated DC-DC converters can not only transmit power bidirectionally between DC buses of different voltage levels, but also have the advantages of higher power density, simple structure, and isolation, which have vital research value.
R.W. De Doncker proposed DAB converters for the first time in the late 20th century [
4]. Their excellent properties soon led to a series of studies by different researchers. They are now widely used in renewable energy, energy storage, electric vehicle charging, energy routers, and other fields [
5,
6,
7].
The most common modulation of DAB converters is phase shift modulation (PSM), mainly including single phase shift (SPS) modulation [
4], extended phase shift (EPS) modulation [
8,
9], dual phase shift (DPS) modulation [
10,
11], and triple phase shift (TPS) modulation [
12]. To avoid magnetic bias, PSM can only operate at a fixed 50% duty cycle. For the same reason, pulse width modulation (PWM) also cannot be used directly. In [
13], a three-level DAB is proposed. Its modulation is a combination of PSM and PWM. Although the complexity of control is reduced, the cost and the complexity of hardware increase. Furthermore, [
14] proposed a modulation that applies PWM on both sides of the DAB converter.
For the optimization of current stress, the most direct method is derivation [
15]. However, it is challenging to derive or obtain the analytical conditional extreme points when the equation of current stress is complex. On this basis, the Lagrange multiplier method is applied [
16]. However, this method could only solve the current stress optimization problem with equality conditions, namely transmission power. When it needs to meet both the ZVS conditions and mode boundary, inequality conditions should be added. To solve this, the Karush–Kuhn–Tucker method (KKT) is applied [
17]. However, with the increase in constraint conditions, the difficulty of the analytical solution increases exponentially. So, in [
18], a graphic method was proposed. The characteristics of the optimal operating point can be determined by analyzing the variation in current stress and constraint conditions. This simplifies the analytical process and helps achieve optimization. However, when the ZVS conditions, boundary conditions, and current stress equations are complex and their trends are difficult to analyze, the above methods may not be effective. To address this issue, this paper presents an optimization method that involves discretizing and numerical solutions.
The research on the topology, modulation, and optimization of DAB converters has been relatively detailed, while the research based on DC blocking capacitors is still insufficient. The research on DC blocking capacitors is currently still focused on SPS under the hybrid bridge modulation [
19,
20].
Due to the insufficient analysis of PWM and current stress optimization, a novel ADM based on DC blocking capacitors is proposed. By using PWM to generate a DC component that is then converted into DC bias, the proposed modulation enables DAB to adapt to a wide range of voltage ratios. By adjusting the duty cycle of the full bridge output voltage on the input side and the phase between the full bridge output voltages on both sides, the power and transmission direction can be controlled without changing its fundamental topology. The different operating modes of the converter are classified based on the output voltage waveforms of the full bridge on both sides of the power transmission process. The transmission power and ZVS boundary of all modes are mathematically deduced. Furthermore, the OADM is proposed, which can be used to optimize the current stress. By this method, minimum current stress and full ZVS for voltage ratios of
m ≤ 0.5 is achieved. Finally, the proposed OADM is validated through comparative experiments. It also makes it possible to achieve a smoother transition of mode switching in the method proposed in [
20].
The structure of this article is as follows: In
Section 2, the basic definition of the proposed ADM is introduced. In
Section 3, the working principles of the proposed ADM are analyzed in detail. In
Section 4, the optimization of the ADM with ZVS and inductor current, namely OADM, is given. In
Section 5, the experimental results are given to verify the analysis and compare with those of the conventional modulation. Finally, the conclusions are presented in
Section 6.
2. ADM with the Aid of DC Blocking Capacitors
The topology of a DAB converter with DC blocking capacitors is shown in
Figure 1.
V1 and
V2 are input and output DC-link voltage, respectively. The turn ratio of the transformer is
n.
VAB is the primary side bridge output voltage, and
VCD is the secondary side bridge output voltage.
Vp is the primary side voltage of the transformer, and
Vs is the secondary side voltage of the transformer.
Cbp is the DC blocking capacitor of the primary side, and
Cbs is the DC blocking capacitor of the secondary side.
L is the sum of external series inductance and transformer leakage inductance.
iL is the inductor current.
io is the output current.
C1 and
C2 are the DC bus capacitors for input and output. With the aid of DC blocking capacitors, ADM can adjust the DC bias of DC blocking capacitors to control the output voltage.
The typical ADM waveforms based on DC blocking capacitors are shown in
Figure 2.
Ths represents half of a switching cycle.
Vcbp and
Vcbs represent DC bias voltage on the primary and secondary side DC blocking capacitors, respectively.
D is the duty cycle of primary bridge output in one switching cycle. The phase shift angle
φ, which is calculated in radians as the difference between the rising edges of the primary and secondary side voltages of the transformer, varies between −π and π. A positive phase shift (
φ > 0) is shown in
Figure 2, corresponding to the phase shift ratio
Dφ in half a switching period.
Dφ is the ratio of the difference between the rising edges of the primary and secondary side voltages of the transformer to half a switching period.
According to the definitions of
D and
Dφ, the ranges are as follows:
3. Principles of the Proposed ADM
The principle of ADM control is relatively easy to understand. Its main idea is to apply different DC biases to the DC blocking capacitors by changing the duty cycle to match the voltage of the primary side and the secondary side of the transformer, thus reducing the current stress and increasing the soft switching range. There are eight operating modes. This section analyzes each operating mode in detail. The transmission power and the conditions of ZVS for each mode are discussed with a specific analysis of mode A as an example.
3.1. Analysis of Operation Mode
D and Dφ are controlled variables in the ADM based on DC blocking capacitors. Distinctions between different control combinations are mainly reflected in the waveforms and voltages across the DC blocking capacitors on the changed duty cycle side. Whether D ≥ 1/2 determines the polarity of the voltage on the DC blocking capacitors. So, the relationship between the waveforms of Vp and Vs and the voltage across DC blocking capacitors can be used to classify the operation modes of the converter.
Firstly, when
Dφ ≥ 0 and
D ≤ 1/2, the rising edge of
Vp is ahead of V
s. In this case, when
D >
Dφ/2, the operation state is mode A, and when
D ≤
Dφ/2, the operation state is mode B. Secondly, when
Dφ ≥ 0 and
D ≥ 1/2, the rising edge of
Vp is ahead of V
s. Therefore, when
D >
Dφ/2 + 1/2, the operation state is mode C, and when
D ≤
Dφ/2 + 1/2, the operation state is mode D. Thirdly, when
Dφ ≤ 0 and
D ≤ 1/2, the rising edge of
Vs is ahead of
Vp. In this situation, when
D >
Dφ/2 + 1/2, the operation state is mode E, and when
D ≤
Dφ/2 + 1/2, the operation state is mode F. Lastly, when
Dφ ≤ 0 and
D ≥ 1/2, the rising edge of
Vs is ahead of
Vp. Thus, when
D >
Dφ/2 + 1, the operation state is mode G, and when
D ≤
Dφ/2 + 1, the operation state is mode H. The waveforms of
Vp and
Vs in different modes are shown in
Figure 3, and the ranges of corresponding variables are presented in
Table 1.
The range of modes is shown in
Figure 4.
There is no overlap between the modes. The range of modes exactly covers (1), ensuring the wholeness and uniqueness of classification.
3.2. Transmission Power Analysis
Transmission power varies across different modes. The operation mode shown in
Figure 2 is used as an example to illustrate the deduction.
The equivalent circuit of each stage in mode A from
t0 to
t4 can be drawn according to
Figure 2, and the results are shown in
Figure 5. The red lines in
Figure 5 indicate the current flow of each stage in mode A.
Stage 1 (
t0 −
t1′): At
t0, S
1 and S
4 turn on, while S
2 and S
3 turn off. As the current on
L is negative, diodes SD
1 and SD
4 conduct, allowing the power devices to realize ZVS. The current through diodes QD
2 and QD
3 on the secondary side of the transformer remains continuous. The voltage across
L can be derived as [
V1 −
V1(2
D − 1)] +
nV2. So,
iL at this stage can be derived as:
Stage 2 (
t1′ −
t1): At
t1′, the current through
L begins to increase from zero. The current flows through S
1 and S
4 on the primary side of the transformer and through Q
2 and Q
3 on the secondary side. The voltage on
L can be deduced as [
V1 −
V1(2
D − 1)] +
nV2. So,
iL at this stage can be derived as:
Stage 3 (
t1 −
t2): At
t1, Q
1 and Q
4 turn on, while Q
2 and Q
3 turn off. The current on
L increases; the current on the primary side of the transformer flows through S
1 and S
4, and the current on the secondary flows through QD
1 and QD
4. The voltage across the inductor is [
V1 −
V1(2
D − 1)] −
nV2, so the power devices achieve ZVS. The current through
L reaches its maximum absolute value at this stage and can be expressed as:
Stage 4 (
t2 −
t3′): At
t2, S
2 and S
3 realize ZVS while S
1 and S
4 turn off. The current flows through QD
1 and QD
4 on the secondary side of the transformer. The voltage on
L is [−
V1 −
V1(2
D − 1)] −
nV2. At this stage,
iL falls to zero, and the expression is:
Stage 5 (
t3′ −
t3): At
t3′, the current on
L begins to increase negatively from zero. The current on the primary side flows through S
2 and S
3, and the current on the secondary side flows through Q
2 and Q
3. The voltage across
L at this stage is [−
V1 −
V1(2
D − 1)] −
nV2, and
iL increases in the reverse direction, expressed as:
Stage 6 (
t3 −
t4): At
t3, Q
2 and Q
3 realize ZVS, while Q
1 and Q
4 turn off. The current on
L increases in the reverse direction. The primary side current of the transformer flows through S
2 and S
3, while the secondary side current flows through QD
2 and QD
3. The voltage across
L is [−
V1 −
V1(2
D − 1)] +
nV2, and the expression of
iL is:
According to the condition of zero integral of current on the DC blocking capacitors in a steady state:
Solving Formulas (2) to (8), the inductor current at different times is shown below:
m represents the voltage ratio and m = nV2/V1.
The transmission power of mode A is derived as follows:
The maximum power that a DAB converter with DC blocking capacitors can transmit under SPS control is as follows:
For the simplification of the analysis, the transmission power expression of Formula (10) is normalized by
PN expressed as a function of
D and
Dφ as follows:
Similar to the derivation process of transmission power for mode A, the expression for each mode is as follows:
Normalized transmission power is shown in
Figure 6 based on Formula (13).
Under the ADM based on DC blocking capacitors, the transmission power characteristics are as follows: (1) the forward and reverse transmission powers are symmetrical and have a maximum value that is equal to 1; (2) there exist equal power points, which provide a basis for optimizing current stress; (3) the transmission power in modes B, C, F, and G can range from −0.5 to 0.5.
3.3. ZVS Analysis
ZVS has different boundary conditions in different modes. The software Mathematica is used to simplify boundary conditions and the figure of the ZVS range is verified and plotted using MATLAB. To make a switch achieve ZVS, the current has to flow reversely when an on signal is set. Take mode A in
Figure 2 as an example. At
t0, S
1 and S
4 are turned on, and S
2 and S
3 are turned off. At this time, the current on the inductor is less than zero to achieve ZVS. At
t1, Q
1 and Q
4 are turned on, and Q
2 and Q
3 are turned off. At this moment, the current on the inductor is more than zero to realize ZVS. At
t2, S
2 and S
3 are turned on, and S
1 and S
4 are turned off. At this moment, the current on the inductor is greater than or equal to zero to realize ZVS. At
t3, Q
2 and Q
3 are turned on, and Q
1 and Q
4 are turned off. At this moment, the current on the inductor is less than or equal to zero to realize ZVS. The same methods can be used to analyze the boundary conditions of every mode. The boundary conditions are shown in
Table 2.
Take the boundary conditions of ZVS under mode A as an example:
To simplify the calculation, the maximum transmission power current is taken as the reference current, and the current is normalized.
The normalized current at each moment under mode A is:
By combining Formulas (16) and (14), Formula (17) can be derived, which represents the ZVS region under mode A. The boundary of this region varies with
m.
The same method can be used to solve the other seven modes. Thus, the range of ZVS in the full operating area is shown in
Figure 7.
Figure 7 shows the range of ZVS in the full operating area when
m = 0.1,
m = 0.1,
m = 0.3, and
m = 0.4, respectively.
6. Conclusions
By applying DC bias to the DC blocking capacitors, the proposed ADM provides a new method of optimization. With a detailed derivation of the operation of a DAB converter with DC blocking capacitors under ADM, the eight operating modes of the converter are analyzed based on theoretical considerations, and the relationship between the control variables and the transmission power is investigated for each mode. By studying the boundary conditions of ZVS for each power device of the converter, an OADM is proposed that ensures full ZVS in the range of voltage ratio m ≤ 0.5 and effectively reduces the current stress of the system. The proposed OADM is optimized using numerical methods, and experimental results confirm its effectiveness. Specifically, when the voltage ratio is low, the OADM can modify the bias to compensate for it, allowing the DAB to maintain consistent performance. It can increase efficiency by 3.58%, 6.57%, 8.81%, and 10.33% compared with DPS when P is equal to 0.36 and m is equal to 0.4, 0.3, 0.2, and 0.1, respectively.
Overall, this approach provides greater flexibility and robustness for DAB, which may be especially valuable in applications where voltage levels can vary widely. With the increasing demand for high-efficiency photovoltaic converters and on-board chargers (OBCs), the OADM proposed in this paper offers more opportunities for pursuing higher efficiency.