Adaptive Model Predictive Control for DAB Converter Switching Losses Reduction

Abstract

The solid-state transformer is the enabling technology for the future of electric power systems. The increasing relevance of this equipment demands higher standards for efficiency and losses reduction. The dual active bridge (DAB) topology is the most usual DC-DC converter used in the solid-state transformer, and is responsible for part of its switching losses. The traditional phase-shift modulation used on DAB converters presents significant switching losses during the operation with reduced loads. The alternative Triangular and Trapezoidal Modulations have been proposed in recent literature; however, there are limitations on the maximum power these techniques can deal with. This paper presents an adaptive model predictive control to select among these three techniques, according to the converter model, the one that minimizes the switching losses and allows the current demanded by the load. Moreover, an alternative cost function is proposed, including the output voltage and current. Through real-time simulation, using a 1000 V/600 V 12 kW DAB converter, it is shown that the proposed control is able to reduce the losses on the converter. Furthermore, the proposed control presents fast and accurate response, and precise transition between the modulation techniques.

1. Introduction

The future of electric power systems demands an efficient distribution network. This network shall be able to manage the power flow, to respond quickly to transients, to easily integrate power sources, and to incorporate energy storage systems [1].

The solid-state transformer (SST) is being presented as the technology that enables the new power system construction [2]. Among the advantages of the SST, one can cite: reduced weight and volume in comparison with traditional transformers; power factor compensation; accurate output voltage regulation; harmonic mitigation; short-circuit current limitation; and voltage dip immunity under certain limitation, for instance. Its main drawbacks are related to cost, reliability, and efficiency [3].

Besides the application on distribution grid, the SST has been used in offshore energy generation, traction systems, DC grids, microgrids, and so on. In many of these applications, the SST topology consists of an AC-DC conversion stage, a DC-DC conversion through a resonant magnetic link to connect high-voltage DC to low-voltage DC, and an output DC-AC conversion stage [4].

This paper deals with the SST DC-DC conversion stage, as shown in Figure 1. The DAB is an isolated converter formed by the union of two active bridges through an inductor and a high-frequency transformer (HFT). The DAB topology stands out for the application in SSST due to the insulation between the HCDC and LVDC sides, the bidirectional power flow, the precision for control, and the reduced losses [5]. In this work, a novel control scheme is proposed to the DAC converter to enhance its efficiency.

1.1. Literature Review

As a new and promising structure, several studies have been conducted to select the best topology for the solid-state transformer. A comprehensive review on these topologies was made by [6]. This review compared several representative topologies for the implementation of SST and explored how it is beneficial to the distribution systems in regards to the reduction in size, controllability, reliability, resiliency, and end-use applications. This paper identifies the most suitable topology capable of supporting additional functionalities such as on-demand reactive power support to grid, voltage regulation, and current limiting. The comparison considered switch losses, switch count, supported functionalities, and control characteristics. The conclusion is that a three-stage configuration comprising distinct AC-DC, DC-DC, and DC-AC stages results in the most suitable implementation.

The increasing relevance of the SST converters to the electric systems is pushing the development of more efficient equipment. In this matter, the switching losses are a crucial aspect, and the DC-DC conversion stage represents a great part of these losses. In [7,8] converter design proposals are based on control dynamics and thermal models for converters to increase the efficiency of converters.

Regarding the modulation techniques employed on the DAB converter, the single phase shift (SPS) modulation is the traditional choice. This modulation uses 2-level square waveforms at $v_{AC1}\left(t\right)$ and $v_{AC2}\left(t\right)$ with 50% duty cycle. The main advantage of this strategy is the zero-voltage switching (ZVS) at a wide operation range [9]. However, when the relation between the input and output voltage is far from one, especially with small loads, soft switching is not possible. Moreover, the converter has relevant backflow power in this situation, which leads to high currents circulation. These currents increase the switches’ stress and the conduction losses. By this, one can conclude that the SPS modulation does not ensure high efficiency of the DAB converter whole operation range.

Many works have proposed solutions for the DAB soft-switching range expansion. The concept of an extended-phase-shift (EPS) is presented in [10,11]. This technique employs 3-level modulation at one side of the converter while the other side uses 50% duty cycle. It was shown that the EPS can reduce the backflow power [10]. In this work the Karush–Kuhn–Tucker function is used in many scenarios to determine the optimum pairs of phase shift and PWM. The drawback of this strategy is that the RMS value of the DAB current is still high. Moreover, the EPS is usually implemented with lookup tables, making it hard to be applied in real equipment [11].

Alternatively, the dual-phase-shift was presented [12]. The DPS uses 3-level modulation on both sides of the converter with the same duty cycle. It demonstrated an efficiency enhance in comparison with the SPS. Nevertheless, as in EPS, this modulation requires an optimization method to the variable choice.

In addition to the modulations presented before, the triple-phase-shift (TPS) was proposed [13,14]. Similar to DPS it uses 3-level modulation on both sides of the converter; however, with different duty cycles on each bridge. This modulation has enough degrees of freedom to ensure high performance on the DAB converter. The possible operation modes with its AC currents waveforms are presented in [15]. Despite the good results, this method has several operation mode possibilities that complicate the DAB parameter choice and complex optimization methods must be used.

The methods presented in the literature have in common the use of complex optimization methodologies, demanding high computational performance, which may compromise the project.

Several control strategies have been used widely in power electronics. Feedback control with proportional-integral compensator (PI) is the most simple method to adjust the output voltage. The phase shift is modified dependent on the error in the output voltage [16]. The linearized control method eliminates the nonlinear terms in the DAB converter. The control can reduce the sensitivity of the system stability to the load condition and reference voltage and help to enlarge the stable margin [17]. The Virtual Direct Power Control (VDPC) is a feedforward control that that has the advantage of eliminating the need to use information about the power transfer inductance. The disadvantage is that it cannot operate efficiently under low load conditions [18]. The Sliding Mode Control (SMC) presents stability and robustness against parameter uncertainties. The disadvantage of this method is from the small-signal analysis, the system can be unstable [19,20].

The model predictive control (MPC) stands out for use in power converters. The dynamic response of MPC was evaluated and compared with traditional PI control, the improved response was demonstrated through simulation [21] and experimental results [22]. The MPC presents fast transient response and high precision in steady state. Its main drawback was the necessity of accurate modelling and high sensibility to parameter variation.

1.2. Contributions

In contrast with the complex and sophisticated modulations presented before, this paper investigates the trapezoidal and triangular modulation. These techniques were presented by [23], and it was demonstrated that they can naturally extend DAB soft-switching operation, both in the buck and boost operation [11]. They consist in the utilization of pulse widths different by 50%. Their proper application can extend the number of soft commutations to four (trapezoidal) or six (triangular) of a total of eight in a cycle even during the operation with reduced loads.

In this work, the main goal is to enhance the efficiency of the DAB converter operating far from its rated values, keeping a fast and accurate response. To achieve these goals, a novel technique is proposed, that is, the adaptative model predictive control (AMPC). The proposed technique is based on Moving-Discretized-Control-Set Model-Predictive-Control (MDSC MPC) [23,24,25,26], which was originally conceived for the SPS, and here is expanded to triangular and trapezoidal modulations. The main contribution is the performance enhancement without complex optimization methods, as the simple trapezoidal continuous conduction mode and triangular discontinuous conduction mode can naturally and sequentially extend the soft-switching boundary of the conventional SPS modulation to the full range [11].

1.3. Paper Organization

The paper is organized as follows: Section 2 presents the mathematical modelling of the SPS, Triangular and Trapezoidal modulations. The proposed control scheme is detailed in Section 3. Section 4 discusses the considerations for the power converter design and equipment choice. The obtained results are shown in Section 5 and discussed in Section 6. Finally, Section 7 shows the conclusions of this work.

2. Mathematical Modelling

The DAB converter is formed by two active bridges connected through a high frequency transformer and a series inductor. The classical topology for this converter is shown in Figure 2a. Its working principle is based on the phase shift between the voltages applied on the HFT terminals, $v_{AC1}\left(t\right)$ e $v_{AC2}\left(t\right)$. The phase shift signal ($\delta$) defines the power flow direction and its module is proportional to the active power flow until ∓90° [9,27].

Many modelling methods were presented in the literature [28,29,30]. For simplicity, in this work the HFT equivalent circuit is represented by the series inductor referred to the primary side, as shown in Equation (1).

$$L=L_{tr1}+{\left(\frac{N_1}{N_2}\right)}^2·L_{tr2}$$

(1)

where $L$ is the equivalent inductance, $L_{tr1}$ and $L_{tr2}$ are the leakage inductances on primary and secondary sides of the HFT, and $N_1$ and $N_2$ are the number of turns on primary and secondary sides, respectively.

The voltages produced by the active bridges may present variable pulse widths in order to allow different types of modulation. In general, the pulses produced by bridges may present phase shift ($\delta$) and inner phase shift, represented by ${\tau }_1$ and ${\tau }_2$, as shown in Figure 2c. In this situation the power flow between the bridges may be controlled by the phase shift or by the variation in the inner phase shift. The dashed line represents the SPS modulation (${\tau }_1+2·{\mathsf{\Omega }}_1={\tau }_2+2·{\mathsf{\Omega }}_2$ = 180°) and the continuous line represents the modulation using the phase shift and the inner phase shift.

The average value of DAB output current $I_{DC2}$ is expressed as Equation (2). This equation is obtained by the equivalent circuit, presented in Figure 2b using the phasor notation.

$$I_{DC2}=\frac{8·n·V_{DC1}·\sin \frac{{\tau }_1}{2}·\sin \frac{{\tau }_2}{2}·\sin \delta }{{\pi }^2·\omega ·L}$$

(2)

where $V_{DC1}$ is the mean value of the input voltage; ${\tau }_1$ and ${\tau }_2$ are inner phase shift angles; $\delta$ is the phase shift between the bridges; $\omega$ is the angular frequency of converter switching and $n$ is the transformer ratio ($n=N_1/N_2$).

The buck and boost modes are defined by the DAB transformation ratio ($d$) shown in Equation (3). When $d<1$ the converter works at buck mode, when $d>1$ it works at boost mode, and when $d=1$ the DAB operates at ZVS switching in the whole power range.

$$d=\frac{N_1·V_{DC2}}{N_2·V_{DC1}}$$

(3)

The following sections present the SPS, triangular and trapezoidal modulations for both buck and boost modes.

2.1. SPS Modulation

The SPS is the traditional modulation for DAB converters (${\tau }_1={\tau }_2$ = 180°). It allows the maximum power transferring and, when working with rated load, this modulation allows ZVS. Moreover, it has natural short-circuit protection, as $I_{DC2}$ is independent from $V_{DC2}$. The transferred power is provided by Equation (4).

$$P_{SPS}=\frac{8·n·V_{DC1}·V_{DC2}·\sin \delta }{{\pi }^2·\omega ·L}$$

(4)

The soft switching on this converter happens when the antiparallel diode is polarized before the transistor activation. Equation (5) presents the limits for the ZVS actuation of the primary and secondary active bridges, respectively [31].

$$\{\begin{array}{l}{P}_{ZVS1}=\left(\frac{V_{DC1}{}^2·\delta }{\omega ·L·\mathsf{\Phi }}\right)·\left(\frac{\pi -\left|\delta \right|}{\pi -2·\left|\delta \right|}\right)\\ P_{ZVS2}=\left(\frac{V_{DC1}{}^2·\delta }{\omega ·L·\mathsf{\Phi }}\right)·\left(\frac{\pi -\left|\delta \right|}{\pi }\right)·\left(\frac{\pi -2·\left|\delta \right|}{\pi }\right)\end{array}$$

(5)

where $\mathsf{\Phi }$ is rated phase shift. If the DAB power is above $P_{ZVS2}$ or below $P_{ZVS1}$, it is not possible to operate with ZVS. To explicit the ZVS region, Figure 3 presents the DAB power curve as a function of $\delta$ for three different values of $d$.

These curves show that for small angles, which means reduced power, the SPS modulation cannot operate with ZVS; therefore, the converter efficiency is compromised in these regions.

2.2. Triangular Modulation

An alternative to reducing the switching losses of SPS modulation is the triangular modulation. The name comes from the inductor current waveform, as can be seen in Figure 4 for both buck and boost modes.

With this modulation the soft switching occurs six times in a total of eight switching for each period. However, it can only be used when voltages $V_{DC1}$ and $n·V_{DC2}$ are different. In addition, the maximum power transferred is limited in this modulation, restraining its application to low power [23].

The conditions for the triangular modulation on buck mode are reached, as shown in Equation (6).

$$\{\begin{array}{l}{\tau }_1=2·\delta ·\left(\frac{n·V_{DC2}}{V_{DC1}-n·V_{DC2}}\right)\\ {\tau }_2=2·\delta ·\left(\frac{V_{DC1}}{V_{DC1}-n·V_{DC2}}\right)\end{array}$$

(6)

The conditions for the triangular modulation on boost mode are reached, as shown in Equation (7).

$$\{\begin{array}{l}{\tau }_1=2·\delta ·\left(\frac{n·V_{DC2}}{n·V_{DC2}-V_{DC1}}\right)\\ {\tau }_2=2·\delta ·\left(\frac{V_{DC1}}{n·V_{DC2}-V_{DC1}}\right)\end{array}$$

(7)

As seen in Figure 4a, in the buck mode the voltages must start at the same time. In contrast, for the boost mode the voltages must end at the same time, as shown in Figure 4b.

2.3. Trapezoidal Modulation

The trapezoidal modulation enables superior power flow in comparison with triangular modulation. This modulation is picked if the input and output voltages are both equal or different. For this modulation the soft switching occurs four times in a total of eight switchings per period. The voltage and current waveforms at AC link are shown in Figure 5.

This modulation starts when the maximum power of triangular modulation is reached. The waveforms are shown in Figure 5a for buck mode and Figure 5b for boost mode. The conditions for the trapezoidal modulation on buck mode are reached, as shown in Equation (8).

$$\{\begin{array}{l}{\tau }_1=\left(2\pi -2·{\tau }_{blank}-2·\delta \right)·\left(\frac{n·V_{DC2}}{n·V_{DC2}+V_{DC1}}\right)\\ {\tau }_2=\left(2\pi -2·{\tau }_{blank}-2·\delta \right)·\left(\frac{V_{DC1}}{n·V_{DC2}+V_{DC1}}\right)\end{array}$$

(8)

where ${\tau }_{blank}$ is the angle referred to the IGBT or MOSFET dead time, typically around 1 μs and 7 μs. This work used the dead time of 1 μs.

3. Proposed Adaptive Model Predictive Control (AMPC)

Modern control strategies have been used widely in power electronics. Among these techniques the model predictive control (MPC) stands out.

For application with power converters, the MPC can be defined in two modes: Finite Control Set (FCS-MPC) and Continuous Control Set (CCS-MPC) [32,33]. The FCS-MPC considers a discrete set of possibilities to reach the desired result. The CCS-MPC, on the other hand, analyses the variables continuously. This method usually needs an optimization solver to determine a control signal to the modulation, which increases the computational effort.

This paper presents a new strategy for DAB control that uses the MDCS-MPC [22], originally proposed for the SPS modulation, and expands it to the triangular and trapezoidal modulation.

3.1. Discrete Model

The control scheme, in this work, is based on the MDCS-MPC. The MPC uses a discrete model to predict the voltages and currents on the converter.

For the determination of the discrete model it is necessary to predict the DAB output current. By the discretization of Equation (2), Equation (9) is obtained.

$$I_{DC2}\left(k+1\right)=\frac{8·n·V_{DC1}\left(k\right)·\sin \left(\delta \left(k+1\right)\right)}{{\pi }^2·\omega ·L}·\sin \left(\frac{{\tau }_1\left(k+1\right)}{2}\right)·\sin \left(\frac{{\tau }_2\left(k+1\right)}{2}\right)$$

(9)

where $I_{DC2}\left(k+1\right)$, $\delta \left(k+1\right)$, ${\tau }_1\left(k+1\right)$, and ${\tau }_2\left(k+1\right)$ are the predicted current, phase shift angle, and inner phase shift at $t=\left(k+1\right)$ and $V_{DC1}\left(k\right)$ is the input voltage at $t=k$.

The output voltage mean value ($V_{DC2}$) is obtained from the relation of the output currents, described in Equation (10).

$$i_C\left(t\right)=C_{out}\frac{d v_{DC2}\left(t\right)}{dt}=i_{DC2}\left(t\right)-i_{LOAD}\left(t\right)$$

(10)

where $i_C\left(t\right)$ is the output capacitor current, $i_{LOAD}\left(t\right)$ is the load current, and $C_{out}$ is the output capacitance.

The discretization of Equation (10) using the Euler method, yields Equation (11).

$$V_{DC2}\left(k+1\right)=\frac{I_{DC2}\left(k+1\right)-I_{LOAD}\left(k\right)}{C_{out}·f_s}+V_{DC2}\left(k\right)$$

(11)

where $V_{DC2}\left(k+1\right)$ and $V_{DC2}\left(k\right)$ are the predicted and measured output voltages, respectively; $I_{LOAD}\left(k\right)$ is the measured load current; and $f_s$ is the DAB switching frequency.

3.2. Traditional MPC

The MDCS-MPC consists in the discretization of the phase shift and the utilization of a small number of angles [22], as shown in Equation (12). The choice of these angles is a trade of between the system transient response and the controller precision.

$${\delta }_{set}=\left[\left({\delta }_{old}-{\delta }_{step}\right);\hspace{1em}{\delta }_{old};\hspace{1em}\left({\delta }_{old}+{\delta }_{step}\right)\right]$$

(12)

where ${\delta }_{old}$ is the phase shift applied at the before time $\left(k-1\right)$ and ${\delta }_{step}$ is the angle increment used to increase or decrease the transferred power.

Aiming to obtain the best possible results, the adopted strategy uses an adaptive angle choice [22]. When the error of the controlled variable is high, larger angles are used to accelerate the response. By the other side, if the error is small, small angles are used to increase the precision.

The benefits of using this adaptive strategy also includes a small computational cost, the utilization of a fixed switching frequency, a fast transient response, and the suppression of steady state error.

The ${\delta }_{step}$ is dynamic evaluated according to Equation (13).

$${\delta }_{step}={\delta }_{min}·\left(1+\alpha ·V_{adp}\right)$$

(13)

where

$$V_{adp}=\{\begin{array}{lll}\left|V_{ref}-V_{DC2}\right|;& & \mathrm{if} \left|V_{ref}-V_{DC2}\right|\le Vm\\ Vm;& & \mathrm{if} \left|V_{ref}-V_{DC2}\right|>Vm\end{array}$$

(14)

where the lowest phase shift angle is ${\delta }_{min}$ and the maximum value for $V_{adp}$ is $Vm$, $\alpha$ is a gain, and $V_{ref}$ is the desired output voltage of the DAB converter.

3.3. Adaptive MPC

The proposed AMPC aims to enhance the DAB converter efficiency during the steady state operation. This method applies different modulations with traditional MPC, in contrast with the utilization of only SPS on the previous proposals. Different from existing modulation methods that are usually dependent on complex mathematical optimization processes, the AMPC does not use complex mathematical methods once the trapezoidal continuous conduction mode and triangular discontinuous conduction mode can naturally and sequentially extend the soft-switching boundary of the conventional single phase shift (SPS) modulation to the full range [11]. Their proper application can extend the number of soft commutations to four (trapezoidal) or six (triangular) on a total of eight in a cycle even during the operation with reduced loads.

The working principle of the AMPC is shown in Figure 6. Once the control defines the ${\delta }_{set}$, the modulation strategy can be selected. According to the load, these choices consider the reduction in the switching losses.

The first block of the algorithm checks the relation between $V_{DC1}$ and $V_{DC2}$, to determine the operation mode, buck or boost. Next, the algorithm calculates the inner phase shift (Equation (6) for buck mode or Equation (7) for boost mode) to verify if the triangular modulation can be used. If the inner phase shift is smaller than π the converter can operate with the triangular modulation. Otherwise, the algorithm checks if it can operate with trapezoidal modulation or SPS. By matching Equations (4) and (5) it is possible to determine the maximum values for the angles (${\delta }_{lim}$) with soft switching with the SPS modulation. According to the operation mode (buck or boost) for angles greater than ${\delta }_{lim}$ the ZVS is achieved. Beyond these limits, the trapezoidal modulation is used.

3.4. Cost Function and Loop Compensation

The last step of AMPC control consist in the compensation of the error presented by mathematical model and the use of the cost function to choose the phase shift ($\delta$) and inner phase shift (${\tau }_1$ e ${\tau }_2$) to reach the smallest error between the output voltage and setpoint.

The compensation of errors associated with the mathematical modelling is based on [11] and [34]. The predicted voltage is shown in Equation (15).

$$V_{DC{2}_c}={\lambda }_1·Erro{r}_1+{\lambda }_2·Erro{r}_2+V_{DC{2}_P}\left(k+1\right)$$

(15)

where

$$\{\begin{array}{l}Erro{r}_1={\lambda }_1·\left(V_{DC{2}_S}\left(k\right)-V_{DC{2}_P}\left(k\right)\right)\\ Erro{r}_2={\lambda }_2·\left(V_{DC{2}_S}\left(k-1\right)-V_{DC{2}_P}\left(k-1\right)\right)\end{array}$$

(16)

where $V_{DC{2}_c}$ is the voltage corrected at $\left(k+1\right)$, $V_{DC{2}_P}$ is the predicted voltage (Equation (11)), $V_{DC{2}_S}$ is the measured output voltage, ${\lambda }_1$ and ${\lambda }_2$ are correction factors. The values for the correction factors were obtained experimentally as ${\lambda }_1$= 0.5 and ${\lambda }_2$= 0.25.

The traditional cost function for the predictive control is provided by Equation (17). The output voltage is evaluated using the three possible values of ${\delta }_{set}$. The one that presents the smallest error is defined according to the cost function.

$$G_1={\left(V_{ref}-V_{DC2}\right)}^2$$

(17)

In order to compensate variation on $V_{DC2}$ caused by transients, the cost function used is according to Equation (18).

$$G={\alpha }_1·G_1+{\alpha }_2·G_2$$

(18)

where

$$G_2={\left(I_{DC2}\left(k+1\right)-I_{LOAD}\right)}^2$$

(19)

Therefore, the cost function is formed by two terms. The first term, $G_1$, is related to the voltage regulation. This term is predominant when the output voltage is far from the reference value. As the voltage nears the reference value, $G_2$ becomes dominant, reducing the oscilations arround the reference. The adjustment of ${\alpha }_1$ and ${\alpha }_2$ is very important to the proposed control scheme performance. According to the method presented in [35] their values are ${\alpha }_1$ = 1 and ${\alpha }_2$ = 1.

The proposed control strategy is shown in Figure 7. The input variables are the input voltage ($V_{DC1}$), the output voltage ($V_{DC2}$), the output current ($I_{LOAD}$), the desired DAB output voltage (setpoint); and the output variables are the phase shift ($\delta$) and the inner phase shift angles (${\tau }_1$ and ${\tau }_2$).

4. DAB Converter Design Procedure

The design procedure for the DAB converter passive and active devices is discussed in this section. The initial considerations are: The SST that contains the DAB has 12 kW, the HVDC bus operates with 1 kV, and the LVDC bus operates with 0.6 kV.

4.1. Frequency Selection

The phase shift is naturally continuous. However, in digital control, the phase shift needs to be discretized. The discretization depends on the control’s platform. The smaller phase shift ${\delta }_{min}$ is the value that can be achieved in a digital control platform. For unidirectional power flow, the DAB converter works in the range shown in Equation (20).

$$\delta \in \left(0,{\delta }_{min},2·{\delta }_{min},\dots ,90°\right)$$

(20)

The discretization of waveforms of the DAB converter is shown in Figure 8.

The value of ${\delta }_{min}$ is calculed in Equation (21).

$${\delta }_{min}=360·f_s·t_{sample}$$

(21)

where $t_{sample}$ is the sample time of digital platform control.

The validation of the proposed control is made with Real-Time Digital Simulation (RTDS) using Typhoon-HIL HIL402. The sample time of HIL402 is 0.5 μs. In

Table 1. Smaller phase shift ${\delta }_{min}$ for frequency of the DAB converter for $t_{sample}$ = 0.5 μs.

Frequency	${\delta }_{min}$	Range of δ
50 kHz	9°	(0, 9°, 18°, …, 90°)
10 kHz	1.8°	(0, 1.8°, 3.6°, …, 90°)
1 kHz	0.18°	(0, 0.18°, 0.36°, …, 90°)
500 Hz	0.09°	(0, 0.09°, 0.18°, …, 90°)

is shown the ${\delta }_{min}$ value for some operation frequency of the DAB converter for $t_{sample}$ = 0.5 μs.

Among the values shown in Table 1, the highest frequencies work with large ${\delta }_{min}$, which hampers a soft control action. Thus, the chosen frequency value for the DAB operation is 1 kHz, guaranteeing a good discretization and allowing smaller oscillations in the steady-state output voltage. Despite being a low frequency for applications in real situations, this frequency is acceptable to demonstrate the operation of AMPC control because the objective of this control is to choose which modulation allows the greater efficiency, being easily extended to other operating frequencies.

4.2. Waveforms Construction

A fundamental part of the proposed system is the modulator, as it allows for making the waveforms shown in Figure 4 and Figure 5. In order to make the centralized PWM possible, it was necessary to generate a triangular waveform at the desired switching frequency with a unit peak value that can be shifted in time.

Figure 9 shows the step-by-step process to generate the time-phased triangular waveform. The phase shift $\delta$ is the input to this block. The Step 1 uses phase shift to calculate the shift value at time $t^{\prime }$. Since the time is shifted, Step 2 uses the remainder of the division between $t^{\prime }$ and the switching period $T$ to generate a sawtooth waveform. Step 3 makes the sawtooth waveform assume a unitary peak value. The Step 4 makes the interpolation of the sawtooth waveform in order to generate the triangular waveform with a unitary peak value $TR$.

The centered PWM is generated by comparing the $TR$ triangular waveform with the desired reference value. Figure 10 shows the algorithm to generate this PWM. The inputs of this block are the inner phase shift $\tau$ and the triangle wave $TR$. The outputs are the triggers for the full bridge. Where, $S_{1UP}$ and $S_{2UP}$ refer to the upper switches of full bridge and $S_{1DOWN}$ and $S_{2DOWN}$ refer to the lower switches of the full bridge.

Bridge 1 was used as a reference, that is, the phase shift for this bridge is zero. While Bridge 2 received the phase shift value desired by the controller. Thus, the modulator has as inputs the inner phase shift values (${\tau }_1$ and ${\tau }_2$) and the phase shift $\delta$. The output values are the triggers of the active bridge switches of the DAB converter. In Figure 11 is shown the modulator used in the present work.

4.3. Passive Elements Calculations

The sizing expressions for inductance on the primary side of the transformer and the capacitor on the DC output side to the DAB converter was derived from [7,27]. Equation (22) represents the inductance on the primary side of the transformer, where $P$ is the rated output power for rated angle $\mathsf{\Phi }=45°$. Equation (23) represents the output capacitance of the DAB converter, where $V_{max}$ and $V_{min}$ are maximum and minimum value of output voltage, respectively. These values were chosen considering an output voltage ripple of 10% ($V_{max}=1.05V_{DC2}$ and $V_{min}=0.95V_{DC2}$).

$$L_{min}\ge \frac{3·n·V_{DC1}·V_{DC2}}{32·f_s·P}$$

(22)

$$C_{out}\ge \frac{P}{V_{max}{}^2-V_{min}{}^2}$$

(23)

According to the above mentioned specifications, the parameters used for the experiments are shown in

Table 2. DAB converter parameters.

Variable	Symbol	Value
Switch Frequency	$f_s$	1 kHz
Rated Power	$P$	12 kW
Voltage Input DAB	$V_{DC1}$	1000 V
Voltage Output DAB	$V_{DC2}$	600 V
Rated phase shift	$\mathsf{\Phi }$	45°
HFT Transformer Ratio	$a=N_1/N_2$	1515
Transfer Power Inductor	$L$	7.8 mH
Input Capacitor	$C_{in}$	330 µF
Output Capacitor	$C_{out}$	670 µF
Lowest phase-shift angle	${\delta }_{min}$	0.18°
Gain	$\alpha$	1 rad/V
Maximum value for $Vadp$	$Vm$	10 V

.

5. Results

The AMPC controller and the DAB converter simulation were made with Real-Time Digital Simulation (RTDS) using Typhoon-HIL HIL402. The measure was carried out using the Tektronix TBS1064 Digital Oscilloscope with four channels and using the computer, as shown in Figure 12. A dead time of 1 μs was set on the switches drivers in order to simulate the delay on real devices.

The experiments represent a microgrid formation by the DAB converter. The output voltage was adjusted to be stable at 600 V. The proposed AMPC is able to choose among the triangular, trapezoidal, and SPS modulation. Therefore, the control strategy may assume five operation modes:

• Mode 1: Triangular Modulation operating as Buck ($V_{DC1}=1 \mathrm{kV}, 1.28 \mathrm{kW}$);
• Mode 2: Trapezoidal Modulation operating as Buck ($V_{DC1}=1 \mathrm{kV}, 4.28 \mathrm{kW}$);
• Mode 3: Triangular Modulation operating as Boost ($V_{DC1}=850 \mathrm{V},0.69 \mathrm{kW}$);
• Mode 4: Trapezoidal Modulation operating as Boost ($V_{DC1}=850 \mathrm{V}, 3.69 \mathrm{kW}$);
• Mode 5: SPS Modulation.

Source voltage disturbance rejection, steady state, transient behavior, $G_2$ component of cost function (Equation (18)) and parameters variation analysis were carried out. Aspects such as output voltage, AC voltages, inductor current and switching losses were observed and discussed. The switching losses were calculated using Typhoon-HIL toolbox, considering the actual current and voltage, and the ambient temperature of 25 °C. The switch used at simulation was 5SNG 0150Q170300. The transient behavior for buck and boost modes using triangular and trapezoidal modulations was analyzed observing the converter output voltage. In addition, the contribution of the $G_2$ term on the cost function was evaluated.

5.1. Source Voltage Disturbance Rejection

In real-world situations the DAB converter is often connected in cascade with an active rectifier (see Figure 1) causing voltage fluctuations at the converter input. In order to evaluate the performance of the proposed method in these situations, two experiments were carried out.

The first experiment was based on [7]. Figure 13 shows the variation in the response of the DAB converter for a step change, from 550 V to 600 V, in the output voltage setpoint. Several input voltages were considered in a range from $-30\%$ to $+30\%$ of the rated value. Figure 13 shows the results.

It can be observed that value above $30\%$ of the nominal value have a greatest impact on the step response, causing an overshot of $8\%$ and a response time of approximately $100 \mathrm{ms}$.

The second experiment was conducted to evaluate the source voltage disturbance rejection [36], as shown in Figure 14. The source voltage can be decomposed by DC ($V_{DC}$) and AC ($v_{ripple}$) components, where $v_{ripple}$ is the injected voltage perturbation.

The source voltage disturbance rejection was analyzed through the ratio between peak of $v_{ripple}$ and peak of output voltage ripple $V_{DC2\text{\_}ripple\text{\_}peak}$, as shown in Equation (24):

$$G_{vdr}=20·\log \frac{V_{DC2\text{\_}ripple\text{\_}peak}\left(f\right)}{v_{rippl{e}_{peak}}\left(f\right)}$$

(24)

The $V_{peak}$ of the DAB converter was measured for each injected frequency. The source voltage disturbance was adjusted for $v_{rippl{e}_{peak}}$ = 50 V. $G_{vdr}$ was plotted as shown in Figure 15a. An example of the measurement is shown in Figure 15b, where the AC components of the input and output voltage of the DAB converter for the frequency of 200 Hz are shown.

The frequency response shown that $G_{vdr}$ rises from −31 dB to −20 dB in the range from 10 Hz to 30 Hz and decreases from −20 dB to −28 dB in the range from 30 Hz to 100 Hz. The AMPC control presents a good performance for the analyzed frequency range.

5.2. Variations of Circuit Parameters

In real situations it is common for the parameters of the converter devices to vary. This analysis is important in order to verify that the control is able to stabilize the system even when it does not coincide with the nominal model.

In the following, a robustness analysis is proposed with regard to the variation in circuit parameters. This analysis is important in order to verify that the control is able to stabilize the system even when it does not coincide perfectly with the nominal model [7].

Figure 16 shows the variation in the response of the DAB converter for parameter changes, from 550 V to 600 V, in the output voltage setpoint. Several capacitance and inductance were considered in a range from $-30\%$ to $+30\%$ of the rated value. Figure 16a,b shows the variation in the response of the DAB converter when output capacitance and inductance changed, respectively.

It can be clearly seen that inductance causes greatest changes in the behavior of the converter. However, in steady state the voltage presented errors less than 4%. Furthermore, the dynamic response was similar for different parameters.

Therefore, the AMPC controller was able to keep the output voltage stable, being robust to parameter variations.

5.3. Steady State Analysis

During the experiments, the waveforms for the DAB $v_{AC1}\left(t\right)$, $v_{AC2}\left(t\right)$, and the AC link current—$i_L\left(t\right)$ behavior are observed.

The converter waveforms for the buck operation are presented in Figure 17a (triangular modulation) and Figure 17b (trapezoidal modulation). The waveforms with the boost operation are presented in Figure 17c (triangular modulation) and Figure 17d (trapezoidal modulation).

With buck (modes 1 and 2) and boost (modes 3 and 4) operation the control demonstrates accurate responses, showing $\left(601-600\right)/600=0.17\%$ error on the voltage mean value for the trapezoidal modulation for buck. The error in the other modes was negligible.

The switching losses on DAB Bridge 1 and Bridge 2 were evaluated for both traditional and proposed control methods. The results are shown in

Table 3. Losses in primary and secondary bridges of DAB for AMPC and traditional MPC in buck mode.

Operation Point $V_{DC1}$ = 1000 V	AMPC (Triangular and Trapezoidal)	Traditional MPC (SPS Modulation)
$Triangular$ $V_{OUT}=600 \mathrm{V}-1.28 \mathrm{kW}$	$Bridge 1: 45.9 \mathrm{W}$ $Bridge 2: 21.9 \mathrm{W}$ $Total: 67.8 \mathrm{W}$	$Bridge 1: 70.4 \mathrm{W}$ $Bridge 2: 38.1 \mathrm{W}$ $Total: 108.5 \mathrm{W}$
$Trapezoidal$ $V_{OUT}=600 \mathrm{V}-4.28 \mathrm{kW}$	$Bridge 1: 43 \mathrm{W}$ $Bridge 2: 41.4 \mathrm{W}$ $Total: 84.4 \mathrm{W}$	$Bridge 1: 69 \mathrm{W}$ $Bridge 2: 40.9 \mathrm{W}$ $Total: 109.9 \mathrm{W}$

for buck mode and in

Table 4. Losses in primary and secondary bridges of DAB for AMPC and Traditional MPC in boost mode.

Operation Point $V_{DC1}$ = 850 V	AMPC (Triangular and Trapezoidal)	Traditional MPC (SPS Modulation)
$Triangular$ $V_{OUT}=600 \mathrm{V}-0.69 \mathrm{kW}$	$Bridge 1: 2.4 \mathrm{W}$ $Bridge 2: 6.6 \mathrm{W}$ $Total: 9 \mathrm{W}$	$Bridge 1: 5.6 \mathrm{W}$ $Bridge 2: 12.9 \mathrm{W}$ $Total: 18.5 \mathrm{W}$
$Trapezoidal$ $V_{OUT}=600 \mathrm{V}-3.69 \mathrm{kW}$	$Bridge 1: 15.6 \mathrm{W}$ $Bridge 2: 34 \mathrm{W}$ $Total: 49.6 \mathrm{W}$	$Bridge 1: 15 \mathrm{W}$ $Bridge 2: 46.5 \mathrm{W}$ $Total: 61.5 \mathrm{W}$

for boost mode.

The results show that the converter performance is enhanced by the utilization of the AMPC. For buck mode, the triangular modulation reduced the losses by $\left(\left|67.8-108.5\right|\right)/108.5=37.5\%$ and the trapezoidal modulation reduced the losses in $\left(\left|84.4-109.9\right|\right)/109.9=23.2\%$ in comparison with the SPS. For boost mode, with the triangular modulation the reduction is $\left(\left|9-18.5\right|\right)/18.5=51.3\%$ and with the trapezoidal modulation the reduction is $\left(\left|49.6-61.5\right|\right)/61.5=19.3\%$ in comparison with the SPS.

5.4. Transient Analysis

For these RTDS the DAB setpoint is changed and the control system response is evaluated by the observation of the output voltage $V_{DC2}$, output current $I_{LOAD}$, and the operation mode. The setpoint change is caused by a load change, and the voltage stabilization response times and the operation mode transitions are observed.

In the first simulation, for buck mode, the transition from mode 1 (triangular) to mode 2 (trapezoidal) and the transition from mode 2 (trapezoidal) to mode 5 (SPS modulation) are observed in Figure 18a,b, respectively. To implement this transition a load change from 1.28 kW to 4.28 kW in Figure 18a and from 6.6 kW to 10.6 kW in Figure 18b were carried out. In both cases the reference output voltage is 600 V and the input voltage is 1000 V.

The maximum overshoot for this simulation was $\left(\left|628-600\right|\right)/600=4.7\%$ and the maximum settling time was 120 ms for this experiment.

In the second simulation, for boost mode, the transition from mode 3 (triangular) to mode 4 (trapezoidal) and the transition from mode 4 (trapezoidal) to mode 5 (SPS Modulation) are observed, in Figure 19a,b, respectively. To implement this transition a load change from 0.69 kW to 3.69 kW in Figure 19a and from 5.49 kW to 9.09 kW in Figure 19b were carried out. In both cases the reference output voltage is 600 V and the input voltage is 850 V.

The maximum overshoot for this simulation was $\left(\left|628-600\right|\right)/600=4.7\%$ and the maximum settling time was 170 ms on the Trapezoidal to SPS transition.

The effectiveness of $G_2$ is shown by the repetitive transition among triangular and trapezoidal modulations. These tests were made for buck and boost modes, on the experiments early implemented with ${\alpha }_2=0$. Figure 20a shows the transitions from triangular (mode 1) to trapezoidal (mode 2) and Figure 20b shows the transitions from trapezoidal to SPS (mode 5) during the buck operation. Figure 20c shows the transition from triangular (mode 3) to trapezoidal modulation (mode 4) and Figure 20d shows the transition from trapezoidal to SPS during the boost operation.

The results show that the transient behavior become oscillatory when ${\alpha }_2=0$. The response time is greater than 600 ms. Thus, the importance of the proposed $G_2$ compontent can be clearly seen.

Table 5. Results compiled.

Transition	Settling Time ${\alpha }_2=1$	Settling Time ${\alpha }_2=0$	Overshoot ${\alpha }_2=1$	Overshoot ${\alpha }_2=0$
$Mode 1 \to Mode 2$	120 ms	680 ms	$\frac{\left(\left|596-600\right|\right)}{600}=0.7\%$	$\frac{\left(\left|592-600\right|\right)}{600}=1.3\%$
$Mode 2\to Mode 1$	120 ms	1.24 s	$\frac{\left(\left|624-600\right|\right)}{600}=4.0\%$	$\frac{\left(\left|620-600\right|\right)}{600}=3.3\%$
$Mode 2 \to Mode 5$	120 ms	720 ms	$\frac{\left(\left|588-600\right|\right)}{600}=2.0\%$	$\frac{\left(\left|588-600\right|\right)}{600}=2.0\%$
$Mode 5\to Mode 2$	120 ms	1.28 s	$\frac{\left(\left|628-600\right|\right)}{600}=4.6\%$	$\frac{\left(\left|628-600\right|\right)}{600}=4.6\%$
$Mode 3 \to Mode 4$	130 ms	600 ms	$\frac{\left(\left|592-600\right|\right)}{600}=1.3\%$	$\frac{\left(\left|592-600\right|\right)}{600}=1.3\%$
$Mode 4 \to Mode 3$	160 ms	880 ms	$\frac{\left(\left|624-600\right|\right)}{600}=4.0\%$	$\frac{\left(\left|624-600\right|\right)}{600}=4.0\%$
$Mode 4 \to Mode 5$	170 ms	960 ms	$\frac{\left(\left|584-600\right|\right)}{600}=2.7\%$	$\frac{\left(\left|584-600\right|\right)}{600}=2.7\%$
$Mode 5\to Mode 4$	170 ms	1.12 s	$\frac{\left(\left|628-600\right|\right)}{600}=4.7\%$	$\frac{\left(\left|628-600\right|\right)}{600}=4.7\%$

shows the settling time and overshoot for scenarios with ${\alpha }_2=1$ (with $G_2$ terms) and ${\alpha }_2=0$ (without $G_2$ terms) in the trasient analysis.

The proposed control was compared with the controls presented in the literature in terms of dynamic performance, robustness against parameter variation, and implementation cost. The implementation cost includes voltage/current sensors and computational burden. The dynamic performance and robustness to parameter variation was based on simulations performed in [21].

Table 6. Comparison of DAB control methods.

Control Method	Dynamic Performance	Robustness to Parameter Variation	Implementation Cost	Switch Losses with (Low Load d ≠ 1)
Feedback [16]	Medium	Good	Low (1 sensor)	Medium
VDPC [18]	Good	Poor	High (3 sensors)	Medium
SMS [19]	Medium	Medium	Low (1 sensor)	Medium
MDCS [22]	Good	Medium	High (3 sensor)	Medium
AMPC	Good	Medium	High (3 sensor)	Good

shows this comparison.

6. Discussion

The DAB converter has many attractive features such as bidirectional power flow, inherent soft-switching capability, buck and boost operation, and high power density. These characteristics enable the utilization of SST in grid applications, energy storage, and electric transportation systems.

In order to expand the DAB converter’s soft-switching range, several operating points can be used. The problem with this solution is the increase in computational burden since complex optimization algorithms must choose between ${\tau }_1$, ${\tau }_2$, and $\delta$. However, in the present work a solution is presented that does not demand high computational costs since the controlled values are related to each other, it is only necessary to choose which modulation to use.

The proposed control combines the benefits of both the SPS, triangular, and trapezoidal modulations. The SPS modulation is used for loads near the DAB rated values, and the triangular and trapezoidal modulations are used for reduced loads in order to reduce the number of hard switching and, consequently, reducing losses.

The insertion of the $G_2$ term in the cost functions demonstrates its importance to the oscilatory transient behavior. A significant reduction on the time response is observed.

The proposal validation was carried out using real-time digital simulator Typhoon HIL. The results show the output voltage transient behavior, the steady state values and waveforms for AC link voltages and current, and the power losses in each modulation.

The results corroborate the performance of the proposed control scheme. The transient behavior presents fast and accurate response. The steady state waveforms shows the possibility of more soft switching per cycle and the losses reduction that it implies.

Future works can use the methodology presented in this paper in other DAB converter topologies as three-phase DAB and ANPC DAB, for instance. There are several microcontrollers that can be used for controllers for this application, such as DSpace and TMS320F2837xD from Texas Instruments. For higher switching frequencies, such as $f_s$ = 20 kHz, the time available to perform the calculations is 1/20 kHz = 50 μs. The MDCS-MPC used approximately 7.8 μs [22] to perform the necessary calculations for the control, so the AMPC has approximately 42.2 μs available to perform its modulations choice calculations.

7. Conclusions

This paper proposed a novel control strategy for the DAB converter, which enhances the efficiency in its whole operation range. The strategy uses the MDCS-MPC, originally proposed for the SPS modulation, and expands it to the triangular and trapezoidal modulation. The expansion consists in adjusting the phase shift and inner phase shift angles, without requiring complex optimization methods.

There is a maximum error of $0.17\%$ on the DAB output voltage during the steady state operation. In this condition, for both boost and buck modes the switching losses were reduced in comparison with the traditional MPC. Using the triangular modulation, a $37.5\%$ reduction on the buck mode and $51.3\%$ on the boost mode is observed. With the trapezoidal modulation the reduction is $23.2\%$ on the buck mode and $19.3\%$ on the boost mode.

In addition to the proposed control scheme, this work presents a modification on the cost function. The presence of the $G_2$ term includes the contribution of the load current on this function and maintains the predicted current near the measured current. During the transients the maximum overshoot observed was around 4.7%. The settling time for the converter operation on buck mode was 120 ms, and on boost mode was 170 ms.