1. Introduction
Research interest in DC/DC converters has surged recently, largely driven by their application in electric vehicle technologies [
1]. The dual active bridge (DAB) converter is particularly popular among different DC/DC converter topologies because it can manage power flow in both directions [
2]. It also offers several benefits, such as high power density, galvanic isolation, and high efficiency, making it an excellent option for various power conversion scenarios [
3,
4,
5]. The leakage inductance of the high-frequency transformer is the key to power transfer between the DC-link source and the load in a DAB converter. Choosing the right leakage inductance is critical. An incorrect choice can lead to large reactive currents with small changes in the DC-link voltages [
6,
7]. In electric vehicle applications, variations from the nominal DC-link voltage can be as high as 34% [
8]. A study [
9] of a DAB converter that operates over a broad voltage span from 100–700 V, representing a 75% variation, was presented for microgrid applications. Choosing the right leakage inductance to reduce RMS currents and thereby minimize copper losses across the entire range of operations is crucial for achieving peak performance [
10]. Choosing a lower value than the required leakage inductance can boost power transfer capacity but comes at the cost of increased reactive power, especially at low loads [
11]. Conversely, a higher leakage inductance could limit the power transfer capacity.
In [
12], a detailed investigation was conducted on optimizing the leakage inductance in a single-phase DAB converter, specifically for vehicular applications. This involved methodically adjusting the phase shift between the two bridges, the leakage inductance, and the DC conversion ratio to pinpoint an optimal value of leakage inductance that effectively minimizes the RMS current. A similar strategy was employed in the study [
13], where the focus was on selecting the best leakage inductance for a three-phase multilevel DAB converter used in wind turbine installations. Both studies [
12,
13] utilized a thorough brute-force optimization technique, which, while robust, was noted for being computationally intensive and requiring considerable time to execute. In the study [
14], a sophisticated methodology for optimizing leakage inductance was detailed, which was specifically designed for a three-phase DAB converter used in utility applications. This method uniquely blended the analytical expressions for power and current specific to the three-phase DAB converter with the capabilities of numerical solvers to accurately identify the most effective leakage inductance value. While this technique achieved results more expediently than the exhaustive brute-force method utilized in the studies [
12,
13], it fell short of providing a thorough analysis of variations in the leakage inductance relative to changes in the design parameters. This highlights a need for flexibility in the optimization process to ensure its applicability across various system designs. Therefore, selecting an appropriate leakage inductance is essential for optimal DAB converter performance across various modulation schemes. The strategic selection of the leakage inductance not only improves the efficiency but also enhances the reliability and durability of the converter system. Furthermore, optimizing the design of DAB converters can lead to better adaptability in power systems, ensuring more stable and robust power supply solutions in advanced technological applications.
This study explores the optimization of leakage inductance in single-phase DAB converters to enhance their performance. An analytical formula was developed to identify the ideal leakage inductance. Additionally, a power controller was designed to independently manage the active and reactive powers of the first harmonic components of the voltages and currents in the converter. Implementing the analytical formula to determine the optimal leakage inductance ensured that the power controller effectively eliminated reactive power caused by the first harmonic components under various operating conditions. By adopting this approach, this study demonstrated a significant reduction in inductor current compared to conventional modulation methods. Furthermore, this optimized strategy allowed for a substantial decrease in the size of the leakage inductance when compared with that of the conventional method, thus not only enhancing the converter’s efficiency, but also its compactness and applicability in space-constrained applications such as electric vehicles and portable devices.
2. Steady State Analysis
A single-phase DAB DC/DC converter incorporates a high-frequency transformer, which is essential for power flow from source to load. It employs two H-bridges to manage the conversion from the input DC-link voltage (
Vin) to the output voltage (
Vo).
Figure 1 illustrates the schematic layout of the DAB converter, illustrating its key components and the interconnections between the H-bridges and the transformer. The leakage inductance of the transformer winding is denoted by
Le and the combined parasitic resistance of the transformer winding and the on-state resistance of the semiconductor switches is represented by
Re, while
vpr(
t) and
vsr(
t) represent the input and output voltages of the high-frequency transformer. The variables used in this section are listed in
Table 1.
The converter utilizes three control variables to manage the active and circulating powers: the inner phase shift of the primary H-bridge (
d1), the inner phase shift of the secondary H-bridge (
d2), and the phase shift between the voltages of the two H-bridges (
d3).
Figure 2 illustrates the primary and secondary voltage waveforms along with the control variables
d1,
d2, and
d3. The range of these control parameters is limited to 0 ≤
d1 ≤ 1, 0 ≤
d2 ≤ 1, and −1 ≤
d3 ≤ 1 ensuring precise control over power flow in the system. By choosing appropriate values for these control variables, the power flow within the converter can be effectively controlled.
The transformer windings voltage can be expressed as [
15]
where
φ represents the total phase shift between the two voltages and
ωs denotes the angular frequency.
The current flowing through the inductor can be expressed as
where
.
It can be observed from Equations (1), (2), and (4) that the voltage on the primary and secondary sides of the transformer, as well as the inductor current, can be represented as infinite summations of odd sinusoids. The first, third, and fifth harmonics of the primary side voltage (
vpr1(
t),
vpr3(
t), and
vpr5(
t)) alongside corresponding harmonics of the inductor current (
ile1(
t),
ile3(
t), and
ile5(
t)) are graphically depicted in
Figure 3. It is evident from
Figure 3 that the first harmonics of both voltage and current contribute most significantly among all the frequency components. This indicates that the first harmonics carry the majority of the energy in these waveforms.
Focusing primarily only on the first harmonics in a DAB DC/DC converter significantly simplifies the power flow analysis in the converter. In the modeling and analysis of the DAB converter, only the first harmonic components of both the voltages (vpr1(t), vsr1(t)), and current (ile1(t)) are considered. This reduces the complexity of the mathematical model and enhances the understanding of the converter’s fundamental operation without the complications introduced by higher-order harmonics, which typically have less impact on the overall system performance.
To analyze the system in the
dq rotating reference frame, the voltage and current phasors are projected on the
d and
q axes. Thus, the voltage vectors are expressed as
The projection of the first harmonics of the inductor current vector
ile1(
t) on the
d and
q axis of the
dq rotating reference frame can be obtained via
The active and reactive powers seen at the transformer’s primary side can be expressed as
The aforementioned power equation can be represented in discrete differential form as
The rate of change of the
dq voltages and current in discrete time can be expressed as
where the
Ts denotes the sampling period and the superscript “*” indicates the corresponding command variables.
Using Equations (9)–(11), the rate of change in discrete time of active and reactive power can be illustrated as
The discrete-time large signal model of the converter in the
dq rotating reference frame can be expressed as
By substituting Equation (9) into Equation (8), we obtain
where the mathematical expressions of the coefficients
M1,
M2,
M3,
M4,
N1,
N2,
N3, and
N4 are provided in
Appendix A.
A unique voltage command is derived from Equation (10) as a function of
dq voltages and active/reactive powers for the subsequent sampling time, expressed as
The unique voltage command
v*pdq in Equations (17) and (18) is transformed into a stationary reference frame, with an amplitude of
Vm and a phase shift
θ to achieve the requisite power commands. Then, the ensuing phase shifts
d1 and
d3 for the subsequent sample are determined via
Here, d1 and d3 serve as the control variables enabling independent control over active and reactive powers. This independence is crucial for efficient energy management. The control of active power not only regulates the output voltage but also influences the overall power efficiency of the system. Meanwhile, controlling reactive power minimizes the current flowing through the inductor, which helps in reducing losses and improves the power factor. Together, these controls contribute to optimizing the performance of the converter.
The control block diagram of the power level controller is depicted in
Figure 4. An active power command is generated by a Proportional-Integral voltage controller to adjust the output voltage, ensuring that it tracks the reference voltage (
Vref). Simultaneously, a reactive power generator produces a reactive power command. This command is specifically aimed at reducing the current that flows through the inductor. By minimizing the current, the converter can achieve lower losses and enhanced efficiency.
4. Design of High-Frequency Transformer
This section details a systematic approach to the design of high-frequency transformers for DAB DC/DC converters. It covers the selection process of core materials, determination of the number of turns for transformer windings, and calculation of losses.
4.1. Selection of the Core Material
The selection of the core material is a vital aspect in designing a transformer as it directly influences factors such as cost, losses, and size. The core materials [
19] with high flux density, high Curie temperatures, and low losses are most preferred. These characteristics ensure efficient performance and reliability of the transformer. Popular core materials for high-frequency transformers in DAB converters include ferrite, nanocrystalline, and amorphous metal. Ferrite offers a good balance of cost and performance at lower frequencies, while nanocrystalline provides superior performance at higher frequencies but comes at a higher cost. Amorphous metal exhibits the best high-frequency performance but is the most expensive option. Additionally, the core’s ability to handle power is assessed by its area product (
Ap). For a given power (
Prated), the area product is calculated from [
20] as part of the design process to ensure optimal performance and efficiency of the transformer using the following equation:
where the parameters
J,
Kf, and
Ku denote the current density, waveform coefficient, and utilization factor, respectively. To meet the specific requirements of the transformer, a ferrite core 9478116002 with a cross-sectional area (
Ac) of 284.289 mm
2 [
21] is selected.
4.2. Transformer Winding Turns
The formula to calculate the number of turns for transformer windings is derived from [
22] as follows:
Taking into consideration the frequency (f), voltage (V), and flux density (Bac), the optimal number of turns for both the primary (Np) and secondary (Ns) windings of the transformer is determined to be 22 each. The Litz wire is used as transformer windings to reduce the high-frequency current and conduction losses.
4.3. Inductance Calculation
The leakage inductance of a transformer is influenced by several design aspects, such as core shapes and winding styles. An estimate of the leakage inductance can be calculated via [
23]
where the permeability of the vacuum is denoted by
μ0, while
NT represents the total number of turns, and
D0,
Dw, and
Hw stand for the core’s airgap, width, and height, respectively. The resulting leakage inductance for the presented method equals 130.61 µH, satisfying the condition outlined in Equation (30).
4.4. Core Losses
For a single-phase transformer design, core loss calculation typically involves the application of the Steinmetz equation. However, given the non-sinusoidal nature of the input voltage, the Improved Generalized Steinmetz Equation [
24] is preferred for a more accurate estimation of the core losses. This equation yields the core losses (
Pc) per volume formulated as follows:
where
α,
β, and
Ki represent the Steinmetz coefficients.
The required leakage inductance for the proposed approach is significantly less than that of the SPS-based design approach. As a result, the size of the transformer needed for the converter is considerably reduced. This size reduction is further highlighted through a normalized comparison, as depicted in
Figure 9.
The designed transformer to validate the proposed methodology has its leakage inductance determined using Equation (30) for the laboratory prototype of the converter. The specifications for this prototype are provided in
Section 5. The cross-sectional area (
Ac) and core volume (
Vc) are measured at 356.76 mm
2 and 94,310.6 mm
3, respectively. For identical converter specifications, the SPS-based method necessitates a transformer with 321.43 μH of leakage inductance, which is more than twice the size compared to the proposed method. Opting for the same material, the most suitable core for this inductor is the 9478117002 E-type core. A comprehensive comparative analysis between the two transformers, including considerations of volume, weight, and core losses, is detailed in
Table 3.
One significant aspect to consider is the heat generation in the single-phase transformer. Heat generation primarily arises from core losses and copper losses. In the proposed method, core losses are reduced due to the smaller size of the required core. Copper losses occur in the transformer windings due to the resistance of the transformer windings and are proportional to the square of the current flowing through the windings.
When comparing the heat generated in transformers for different approaches, the conventional SPS-based approach typically results in higher core and copper losses due to the need for larger transformers and higher current flows. This leads to more heat generation, affecting the transformer’s efficiency and thermal management. Conversely, the proposed method requires a smaller transformer because the required leakage inductance is smaller, which is made possible by the reduced leakage inductance required with the developed power controller-based modulation method for the same active power requirement. Additionally, the current flowing through the transformer is up to 51% less with the same load requirement using the proposed modulation method. This reduction in current substantially decreases copper losses, and the smaller core size reduces core losses. By minimizing both core and copper losses and reducing the current flow, the proposed method generates less heat, thus enhancing the overall efficiency and reliability of the transformer. This approach is crucial for better thermal performance and a longer operational lifespan of the transformer.
5. Results
The parameters for the dual active bridge (DAB) converter were determined to fulfill the specific needs of the Level-1 electric vehicle (EV) charging station. It is important to note that the charging voltage varies depending on the level of the EV charging system [
25,
26]. For example, a Level 1 charging station typically has a charging voltage ranging from 200 to 400 V. Higher levels of charging stations, such as Level 2 and above, operate at significantly higher voltages, with Level 3 and DC fast charging stations delivering voltages from 400 V up to 800 V or more. Additionally, the average rectified voltage derived from a 120 V AC input is around 108 V.
The efficacy of the presented method was evaluated through a comparative analysis using a Matlab Simulink 2022b software with the conventional SPS and dual-phase shift (DPS) [
27] control methods. This assessment involved employing converters with identical parameters, facilitating a thorough examination of their performances under identical conditions. The leakage inductance was determined through the proposed analytical formula outlined in Equation (30) for a 1.5 kW converter for the developed power controller-based modulation method. The key parameters of the converter included input and output voltages of 108/250 V, a leakage inductance of 33.3 μH, and a switching frequency of 30 kHz, as illustrated in
Table 4.
The power controller underwent comprehensive testing across various load conditions to evaluate its performance.
Figure 10 illustrates the steady-state primary side voltage (green), secondary side voltage (blue), and current (red) waveform at 20%, 60%, and full load, providing insights into its performance across different load levels.
The proposed method was compared to the SPS and DPS techniques at 0.6 pu of load, as shown in
Figure 11. The analysis of reactive power and inductor current waveforms revealed that the proposed method effectively eliminates circulating power and significantly reduces inductor current under the same load conditions.
A scale-down laboratory prototype with a power rating of 350 W was constructed, as shown in
Figure 12. The voltages and switching frequency were kept constant to facilitate accurate comparison with theoretical predictions. The TMS320F28335 microcontroller (Texas Instruments, Dallas, TX, USA) was employed for control and operation. Furthermore, the leakage inductance was calculated from (30) to be 130.61 µH to ensure optimal performance. To evaluate the effectiveness of the developed power controller, a comprehensive test was carried out. Initially, the connected load was set to 20% of the converter’s full load capacity. Subsequently, the load was abruptly increased to reach the full capacity of the designed converter.
Figure 13 illustrates the transition in active power from 0.2 per unit to full load. Remarkably, even as the load conditions changed, the reactive power of the first harmonic components remained at zero during steady-state conditions.
A comparative experiment was carried out to assess the efficacy of the developed method against the SPS and DPS modulation techniques. The findings demonstrated that, across all load conditions, the presented method consistently minimizes the current through the inductor compared to both SPS and DPS modulation methods.
Figure 14 illustrates the inductor current waveform during steady-state conditions, comparing the SPS, DPS, and proposed approaches at a 0.6 pu load. The experimental results closely match the simulation outcomes, validating the effectiveness of the power controller-based method. By leveraging the proposed approach, the reactive power due to the first harmonics of voltage and current is effectively eliminated. For instance, at a 0.6 pu load, the reactive power was observed to be −482.9 Vars and −325.4 Vars with the SPS and DPS techniques, respectively, while the presented method eliminates reactive power, indicative of its superior performance. Furthermore, the inductor rms current was significantly reduced with the proposed approach. While the SPS and DPS methods result in an inductor rms current of 5.6 A and 5 A, respectively, the proposed power controller-based approach achieves a mere 4.1 A.
Figure 15 provides a detailed comparison of the root mean square (rms) currents of the inductor under various load conditions for the SPS, DPS, and the proposed method. This bar graph not only showcases the performance variations between the conventional SPS and DPS methods but also emphasizes the efficacy of the proposed method. The considerable reduction in the inductor current achieved by the proposed method not only enhanced efficiency but also played a crucial role in minimizing energy losses and boosting the performance of the converter. This reduction is particularly relevant in applications where power efficiency and minimal energy wastage are critical. Furthermore, the enhanced control over the inductor current under varying loads led to more stable operations and the extended lifespan of the hardware components, including the inductor itself. These advancements are pivotal for optimizing the design and function of power converters in modern electrical engineering applications, contributing to more sustainable and cost-effective energy solutions.