The dual active bridge (DAB) converter has been widely used because of its various advantages such as high power density, bidirectional power flow, galvanic isolation, and so on [

25,

26]. Most DAB converters consist of half-bridge or full-bridge circuits, which produce 2-level square waves across the HFIT. These 2-level DAB converters are usually cascaded for medium and high voltage applications [

27,

28]. In this structure, the individual voltage of DC-links cannot be higher than the voltage rating of the switching devices used in the 2-level DAB converters. Therefore, there is a disadvantage in that the number of required subunits increases due to the voltage rating of the switching devices when 2-level DAB converters are cascaded in an SST system.

The 3-level DAB converters can overcome this voltage constraint of semiconductor switches to a great extent. Power converters synthesizing more than two voltage levels, such as NPC, and flying capacitor converter (FCC), are traditionally known as 3-level converters. The FCC has a disadvantage in that a complicated switching scheme is required to keep the flying capacitor voltage charged at the appropriate constant level [

29]. On the other hand, NPC has an advantage in that all of the switching devices in a leg withstand only half of the individual DC-link voltage using simple switching scheme. Therefore, in this paper, a 3-level NPC circuit is employed on the primary side of the DC/DC converter.

The loads of this DC/DC converter should work under a bipolar DC distribution system. To simply meet this requirement, two 2-port DC/DC converters can be connected in parallel to the individual DC-link. However, this structure has the disadvantage of increasing the number of HFITs and converters. In addition, when a power transfer occurs between the two bipolar DC outputs, the losses are increased because the circulating power is transmitted through two primary circuits. Therefore, the TAB converter using a three windings coupled transformer is advantageous for bipolar DC distribution. Studies have been conducted to expand to a TAB converter using 2-level DAB converters [

30,

31].

However, unlike the conventional 2-level TAB, the TAB converter used in this paper is configured as follows: a 3-level NPC circuit on the primary side, a three windings coupled transformer, and a secondary side is configured as a multi-terminal in which two half-bridge circuits are connected in series (as shown in

Figure 1b). In addition, the outputs of the each TAB converter are connected in parallel, and each subunit has to be rated for only a fraction of the full power.

This section discusses the modeling and controller design procedure of the TAB for this particular structure.

#### 4.1. Modeling of the Three Windings Coupled Transformer

Conceptually, the 2-port DAB converter can be viewed as an interfacing inductor driven at either end by a controlled square-wave voltage source [

32]. In the 2-port DAB converter, the interfacing inductance is the sum of the transformer leakage inductance and the external inductance that is needed in order to adjust the output power as well as to extend the zero-voltage switching operation [

33]. The interfacing inductor between each port is a dominant parameter as the energy transfer component in the DAB converter [

32,

33]. Therefore, interfacing inductances between each port should be obtained for the power transfer modeling of the TAB converter.

Theoretically, the extended cantilever model (ECM) for the transformer is convenient in terms of the parameter extraction, since each model parameter can be extracted from a single measurement of an open-circuit voltage or a short-circuit current [

34]. For this reason, the ECM is applied for the three windings coupled transformer to extract interfacing inductances between each port, as shown in

Figure 3, where

${L}_{\mathrm{W}1}$ is the magnetizing inductance seen from the primary winding,

${W}_{1}$ (

${L}_{ij}$,

i,

j = 0∼2,

i≠

j) is the leakage inductance between each pair of windings, and

${n}_{2}$ and

${n}_{3}$ are turns ratios and have the same value in this application.

The equivalent circuit of the TAB converter using an extended cantilever is shown in

Figure 4, where (

${L}_{ij}^{{}^{\prime}}$,

i,

j = 0∼2,

i≠

j) is the leakage inductance between each pair of windings referred to secondary and tertiary sides, the 3-level voltage at its primary NPC circuit is referred to secondary and tertiary sides and it is denoted as

${V}_{0}$.

${V}_{1}$ and

${V}_{2}$ are the square-wave voltage at its secondary and tertiary bridges, respectively, and

${P}_{0}$ is the amount of power transferred from the DC-link to the input port.

${P}_{1}$ and

${P}_{2}$ are the amount of power supplied to each load from the bipolar output ports.

#### 4.2. Power Transfer Modeling of the TAB Converter

According to the equivalent circuit shown in

Figure 4, it can be seen that the power flow paths between any two of the three ports

${P}_{ij}$ (the power transferred from port

i to port

j;

i,

j = 0∼2,

i≠

j).

${P}_{01}$ and

${P}_{02}$ are the power flow from the primary side to each bipolar output, respectively. They can be represented by the power flowing through the 2-port DAB converter, consisting of a primary NPC leg and a secondary half-bridge leg.

Figure 5a,b show the theoretical voltage and current waveforms of this 2-port DAB converter, where

${V}_{\mathrm{ac}.\mathrm{P}}$ is the 3-level voltage produced by the primary NPC leg,

$\alpha $ is half of the angular distance for zero state,

${V}_{\mathrm{ac}.\mathrm{S}j}$ is the 2-level voltage produced by the secondary or tertiary half-bridge leg,

${\varphi}_{0j}$ is the phase-shift angle between the active bridges, and

${V}_{\mathrm{L}0j}$ and

${I}_{\mathrm{L}0j}$ are the voltage and current of the interfacing inductor between port 0 and port

j, respectively.

${I}_{\mathrm{o}j}$ is the current of the each bipolar output.

The switching scheme for the 2-port DAB converter with a 3-level NPC topology was presented in Refs. [

35,

36]. Based on the value of

${\varphi}_{0j}$, there may be two cases such as Case I: 0 ≤

${\varphi}_{0j}$ ≤

$\alpha $, Case II:

$\alpha $ <

${\varphi}_{0j}$ ≤

$\pi $/2. Referring to

Figure 5, the power equations for two cases can be obtained as follows:

where

$\omega $ is the switching angular frequency.

According to Equations (

3) and (

4), the smaller the value of

$\alpha $, the more power can be transferred to the load. Therefore,

$\alpha $ should theoretically have the minimum value for freewheeling. This means that the 2-port DAB converter with a 3-level NPC topology works in Case I over a very short interval in practice. In addition, the amount of power calculated by Equations (

3) and (

4) are almost the same under light load conditions. Considering the above statements, the power equation of the 2-port DAB converter with a 3-level NPC topology may be considered as the generalized equation and can be rewritten as follows:

Equation (

5) is equivalent to the equation of a 2-level half-bridge DAB converter because the

$\alpha $ term that generates the zero-state in the NPC leg is removed. Therefore,

${P}_{12}$, which is another power flow between two bipolar DC outputs, can also be obtained from Equation (

5). As a result, in the TAB converter, the amount of power between any two of the three ports can be obtained from Equation (

5). To reduce computing burden, (

5) can be simplified as follows through the fundamental approximation in Ref. [

32]:

According to (

6), the amount of power flowing through port 1 and port 2 are calculated as follows, respectively:

Equations (

7) and (

8) can be rewritten in matrix form as follows:

where

$\overrightarrow{G}$ is the transfer function matrix that describes the system parameters.

As shown in (

9), the TAB converter is a two-input and two-output (TITO) system. The TITO system is characterized by significant interactions between their inputs and outputs. Because of these interactions, a control action in one loop gets transmitted to the other loop as a disturbance. Therefore, decoupling techniques are required to eliminate or minimize the interaction. The simplified decoupling technique is most widely used in practice because of its robustness and simple decoupling network for TITO systems [

37].

Figure 6 shows the control block diagram with simplified decoupling. The decoupler transfer function matrix

$\overrightarrow{D}$ and the decoupled transfer function matrix

$\overrightarrow{T}$ can be given as in (

10) and (

11), respectively:

#### 4.3. Controller Design

After applying the simplified decoupling, the decoupled controller is represented by two single-input and single-output (SISO) controllers as shown in

Figure 7. Two SISO controllers hold each bipolar DC output voltage at a set point by automatically adjusting control signals as follows. The reference voltage and the measured voltage are given as inputs to the comparator, from which an error signal is passed as input to the integral proportional (IP) controller. The use of the IP controller enhances the system dynamic response and reduces undesirable peak overshoot compared to the proportional integral (PI) controller [

38]. The IP controller generates the control signal based on the error signal. The output of the IP controller sets the output current of the converter required to keep the output capacitor voltage at the appropriate constant level. In addition, a load current feedforward is added into the controller to improve the dynamic response against the load changes [

39].

Using the formula of the IP controller and the mathematical relationship between the voltage and current of the capacitor, the closed-loop transfer function is given by:

where

${C}_{\mathrm{o}j}$ is the output capacitance,

${K}_{\mathrm{p}}$ and

${K}_{\mathrm{i}}$, all non-negative, denote the coefficients for the proportional and integral terms, respectively. To order the closed-loop system, it is necessary to choose the coefficients

${K}_{\mathrm{p}}$ and

${K}_{\mathrm{i}}$.

The transfer function of a second-order system in standard form is given by:

By comparing (

12) and (

13), the resulting IP controller parameters are obtained by:

where

$\zeta $ is the damping factor and

${\omega}_{\mathrm{n}}$ is the natural frequency.

The outputs of the DC/DC stage are connected in parallel, and each subunit has to be rated for only a fraction of the full power. The primary goal of these parallel-connected converters is to share the load current among the constituent converters. Improper load sharing can lead to the converter overloading and overheating, which decreases the system reliability and can eventually lead to the failure of the overall system [

40]. Therefore, a robust control technique that ensures synchronizing and proper load sharing is essential to ensure the stability and reliability of the system.

Figure 8 shows the control block diagram of the current-sharing controller based on the outer loop regulation (OLR) mode studied in Ref. [

41]. The OLR mode uses the current-sharing error signal to adjust the voltage reference of the outer voltage loop until equal load current distribution is achieved. The current data of each module is transmitted via the CAN protocol, and the average current value can be calculated based on this information. The current-sharing error signal is processed through the average current method (ACM), which compares the difference between the average current and the individual current. The key features of the scheme, generally, are that the reference voltage, output voltage feedback, and the voltage compensator of each converter module are independent.