Comparative Stability Study of a Dual Active Bridge-Based
Battery Charger With Advanced Modulation Schemes

Martin Scohier, Olivier Deblecker, Bashir Bakhshideh Zad
Electrical Engineering Division, Engineering Faculty, University of Mons
31, Bd. Dolez
Mons, Belgium
Tel.: +3265374154
E-Mail: martin.scohier@umons.ac.be
URL: www.epeu-umons.be

*Index Terms*—Small-signal stability, Battery charger, Dual Active Bridge (DAB), Modulation scheme

***Abstract***—Thanks to their galvanic isolation and bidirectional power transfer features, the Dual Active Bridge (DAB) converters can offer a promising solution for interconnecting battery chargers to emerging DC grid applications. When cascaded with LC filters, the charger stability can be affected by the impedance interactions caused by the closed-loop charging process. This is exacerbated by the wide range of operating points in terms of DC voltages as well as the numerous Modulation Schemes (MS). To address this, the current paper extends the averaged small-signal model to the case of a Constant Current Constant Voltage (CC-CV) charging process. A stability analysis method is proposed to compare Single, Extended, Dual and Triple Phase Shift MS during a charging cycle. Simulation results validate the proposed models and highlight the flaws of advanced MS under varying operating conditions.

I. INTRODUCTION

Nowadays, the emerging of energy storage systems
and the growing demand for Electric Vehicles (EV)
are key drivers promoting research on battery system
enhancement. Beside the rapid innovation in the field
of battery technology and monitoring, advanced power
electronics and control methods are required to safely
manage the battery charge and discharge. Hence, battery
chargers are power converters, often cascaded with fil-
ters and front-end converters that ultimately control the
power at the battery ports, ensuring efficient and life-
lasting operation. Conventional chargers use front-end
AC-DC converters in AC grids but the growing adoption
of DC grids allows to consider single stage DC-DC
converter topologies [1]. With their medium frequency

transformer, Isolated Bidirectional DC-DC (IBDC) con-
verters offer seamless voltage matching at high power
density.

A versatile bidirectional battery charger configuration
is presented in Figure 1 where LC filters are connected in
cascade to an IBDC converter for switching harmonics
rejection on both sides. However, such a power con-
version chain is challenged by the inherent impedance
interactions of each individual system. Indeed, The sta-
bility challenge of LC cascaded systems in DC grids is
well-known and passive and active damping approaches
are discussed in [2]. Those approaches however require
the evaluation of the cascaded elements input and output
impedances, that significantly depend on the operating
point, the control and modulation schemes. In addition,
the inherent variations in battery cell voltage, along
with voltage disturbances on the DC grid side, call for
consideration of significant DC voltage variations around
the rated operating point.

The Dual Active Bridge (DAB) is an IBDC con-
verter widely studied for its applications in solid state
transformers, railway and aerospace onboard equipment.
It allows for the conversion at high power density of a

<figure><img src="image_1.png" /><figcaption>Fig. 1: Power conversion chain of a DC grid-tied battery charger.</figcaption></figure>
wide range of input and output voltages thanks to its medium frequency transformer [3], making it a popular topology in battery systems especially in EV onboard and off-board chargers. Despite the ease of implementing Single Phase Shift (SPS) modulation for the power transfer control, higher losses occur as the input and output DC voltages depart from their rated values. To address this, advanced modulation techniques such as Extended Phase Shift (EPS), Dual Phase Shift (DPS) and Triple Phase Shift (TPS) [4] have been proposed in literature. These modulations utilize the inner duty cycles of the primary- and/or secondary-side AC voltages to shape the inductor current depending on the desired objective. Each of these modulation comes with operational boundaries. Therefore, in order to extend the operating range of the DAB converter, multiple modes are combined to form a holistic Modulation Scheme (MS) [5]–[7]. The stability challenge of LC-DAB systems is most often analyzed with regard to SPS modulation only [8], [9]. A two-stage Constant Current Constant Voltage (CC-CV) battery charger is considered in [10] but it emphasizes on the impedance shaping in the AC-DC converter. Concerning the DAB converter with advanced MS, modulator comparison and active damping strategy are proposed in [11], [12] but are limited to a restricted operating range. Moreover, the study in [13] develops comprehensive small-signal stability analysis able to decouple the modulator from the impedance of the converter. This however complicates the practical modulator implementation as it introduces an intermediate control variable.

To tackle this issue, a study highlighted the role of MS design in the stability of LC-DAB systems [14]. To that end, an averaged small-signal model, valid for advanced MS, was proposed. Nevertheless, the stability analysis was conducted with only one TPS MS on a resistive load. To address this limitation, the main contribution on this paper is to extend the models to a DAB-based battery charger. The closed-loop model of the input impedance is therefore developed, considering CC-CV charging in a wide range of voltage variations. Secondly, a stability analysis of a battery charging cycle is performed to compare the use of three advanced MS.

The remainder of the paper is organized as follows. Section II details the operation and the control of the DAB-based battery charger. Section III then extends the averaged small-signal models to a CC-CV charging process. Finally in Section IV, the stability analysis is performed over an entire charging cycle. It is shown that the models accurately expose the stability margin enhancement or degradation caused by an advanced MS.

## II. DESCRIPTION OF THE DAB-BASED BATTERY CHARGER

The DAB-based battery charger circuit is shown in Figure 2. The battery, DAB converter and filters are represented along with the modulator and controller. For the DAB converter, $v_d$ ($i_d$) and $v_o$ ($i_o$) are the input and output voltages (currents), respectively. Their average values over one switching period $T_s$ ($= 1/f_s$) are denoted as $V_d$, $V_o$, $I_d$, and $I_o$.

1) DC grid side: To represent a scenario that is both challenging and realistic, we consider a 1.5 kV railway DC grid, characterized by significant variations in average voltage, ±33% of the rated value, as well as frequent voltage spikes. The filter inductor $L_f$ and its equivalent series resistor (ESR) $R_f$ form with the converter input capacitor $C_d$ the LC filter required for harmonic rejection and voltage spike protection. The resonant frequency is typically set to a few tens of Hertz.

2) Battery side: A Li-Ion EV battery pack with rated voltage and charging current of 700 V and 150 A is considered here. The battery is modeled with a constant open circuit (OC) voltage $V_b$ in series with an impedance $Z_b$. The latter is the combination of the battery internal resistance $R_b$ and the filtering inductor $L_b$ that smooths out the battery current ripple. The OC voltage $V_b$ is related to the State of Charge (SoC) of the battery following the model in [15]. Figure 3 shows the evolution of the electrical characteristics of the battery during a common CC-CV charging process.

First, $V_b$ rises from 500 to 770 V while charged at a constant current of 150 A. Also, the output voltage of the converter $V_o$ rises along $V_b$ to compensate the voltage drop on $R_b$. When $V_b = V_b^{cv}$ (770 V), $V_o$ reaches the maximum voltage and is thus fixed as a reference. The

<figure><img src="image_2.png" /><figcaption>Fig. 2: Power converter and control diagram of the DAB-based battery charger.</figcaption></figure>
<figure><img src="image_3.png" /><figcaption>Fig. 3: Electrical characteristics of the battery pack with a CC-CV charging process.</figcaption></figure>

charging current and power then slowly decrease until the battery is fully charged (i.e. when $V_b = V_b^{max}$). The CC-CV charging process is implemented as two controllers with fixed parameters throughout the entire operating range for current and voltage regulation.

3) DAB converter with advanced MS: As indicated above, voltage variations from the DC grid and battery sides will significantly alter the operation of the DAB converter. For convenience, we define the voltage ratio as

$$d = \frac{V_o}{nV_d} \qquad (1)$$

The current in the AC power inductor $i_L$ is controlled by the two-level square wave AC voltages $v_1$ and $v_2$. The resulting average power $P$ is considered positive during the battery charging. Hence, the converter operates in *boost* mode when $d$ is greater than one (i.e. $V_d < V_o/n$). Inversely, *buck* mode stands for values of $d < 1$. In conventional designs, the transformer turns ratio $n$ is chosen to ensure $d \approx 1$ at rated conditions. Thereafter, the normalized power is given by

$$P_n = P \frac{8n f_s L}{V_d V_o} \qquad (2)$$

Assuming a fixed switching frequency, the adequate switching of the transistors $Q_{1-8}$ adjusts the amplitude and phase of $v_1$ and $v_2$. The inner duty cycles $D_1$ and $D_2$, and the phase-shift ratio $D_3$ are the three control variables used to generate the gate pulse signals (see Figure 4). An infinite number of control variable combinations exist for the same operating point (OP). A modulation scheme is therefore defined here as set of functions that maps an operating point ($d, P_n$) to a control variable combination ($D_1, D_2, D_3$). Numerous MS are derived in literature and differ from their optimization objectives (e.g. conduction, switching or total losses minimization). While TPS modulation generally defines the use of the three control variables, the SPS, EPS

<table><caption>TABLE I: Selected Modulation Schemes.</caption><thead><tr><th>Advanced MS</th><th>Minimization objective</th><th>Number of Modes</th></tr></thead><tbody><tr><td>EPS [6]</td><td>Conduction losses</td><td>5 { Low P (buck, boost)<br>High P (buck, boost)<br>SPS</td></tr><tr><td>DPS [5]</td><td>Peak inductor current</td><td>4 { Low P (buck, boost)<br>High P (buck, boost)</td></tr><tr><td>TPS [7]</td><td>Backflow power</td><td>4 { Low P (buck, boost)<br>High P (buck, boost)</td></tr></tbody></table>

and DPS appellations refer to sub-categories respectively defined as ($D_1 \wedge D_2 = 1$), ($D_1 \vee D_2 = 1$), and ($D_1 = D_2$). In addition to SPS, three advanced MS are selected in literature based on their high performances with low complexity. As shown in Table I, a small amount of modes is preferred where the control variables calculation is simple and unified between each mode. To that end, we use the fundamental phase shift angle between $v_1$ and $v_2$ as a unified variable to control the transferred power. It is expressed as

$$D_f = D_3 + \frac{D_2}{2} - \frac{D_1}{2} \qquad (3)$$

For each selected mode, the operating range as well as the unified control variable to transferred power function are found in Appendix, Table III. Moreover, the tables mapping $D_f$ into ($D_1, D_2, D_3$) can be found in the respective papers. Figure 4 displays an example of AC voltages and inductor current waveforms using the selected MS at identical operating conditions (Low power in *boost* mode).

Then, for illustration purposes, Figure 5 depicts the range of normalized operating points of the modes that constitute the four selected MS. While SPS operates uniformly across the entire range (see Figure 5a), different modes are employed depending on the power level with the advanced MS. On top of that, dashed arrows represent the trajectory of operating points during the charging process, distinguishing between low and high supply voltage conditions. This highlights the multi-modal operation of the DAB-based battery charger with advanced MS.

Finally, the specifications of the studied system are found in Table II. The converter parameters are taken from the previous design of a DAB converter for a DC railway auxiliary power supply that operates in similar input and output ranges (see [16], [17]).
<figure><img src="image_4.png" /><figcaption>(a) SPS
$D_1=D_2=1$, $D_3 = 0.09$</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>(c) DPS [5]
$D_1=D_2=0.68, D_3=0.14$</figcaption></figure>

<caption>(d) TPS [7]
$D_1=0.86, D_2=0.57, D_3=0.3$</caption>

<figure><img src="image_6.png" /><figcaption>(a) SPS</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>(c) DPS [5]</figcaption></figure>

<caption>(d) TPS [7]</caption>

<caption>Fig. 5: Normalized operating range of the selected MS.</caption>

<table><caption>TABLE II: Specifications of the considered system.</caption><thead><tr><th></th><th>Symbol</th><th>Value</th></tr></thead><tbody><tr><td>Rated supply voltage</td><td>V<sub>s</sub></td><td>1 ~ 2 kV</td></tr><tr><td>Filter inductance</td><td>L<sub>f</sub></td><td>4 mH</td></tr><tr><td>Filter ESR</td><td>R<sub>f</sub></td><td>300 mΩ</td></tr><tr><td>Input Capacitor</td><td>C<sub>d</sub></td><td>1 mF</td></tr><tr><td>Output capacitor</td><td>C<sub>o</sub></td><td>1 mF</td></tr><tr><td>Output filter inductance</td><td>L<sub>b</sub></td><td>5 μH</td></tr><tr><td>Battery internal resistance</td><td>R<sub>b</sub></td><td>200 mΩ</td></tr><tr><td>Battery OC voltage range</td><td>V<sub>b</sub></td><td>500 ~ 800 V</td></tr><tr><td>Battery rated charging current</td><td>I<sub>b</sub><sup>max</sup></td><td>150 A</td></tr><tr><td>Rated Power</td><td>P<sup>max</sup></td><td>120 kW</td></tr><tr><td>Switching frequency</td><td>f<sub>s</sub></td><td>5 kHz</td></tr><tr><td>Transformer turns ratio</td><td>n</td><td>21/41</td></tr><tr><td>AC inductance</td><td>L</td><td>187.5 μH</td></tr><tr><td>Current controller parameters</td><td>k<sub>v</sub>, T<sub>v</sub></td><td>0.001, 5 ms</td></tr><tr><td>Voltage controller parameters</td><td>k<sub>i</sub>, T<sub>i</sub></td><td>0.005, 5 ms</td></tr></tbody></table>

### III. SMALL-SIGNAL MODEL

Because the multi-mode modulator serves as the intermediate between the controller and the DAB converter, different dynamics are to be observed between the different MS. A linear approach for controller design is preferred as it allows for well established stability analysis. Therefore, small-signal linearization and PI controllers are used. The averaged small-signal model of the DAB converter, presented in [14], is valid for the studied MS given in Table I. It assumes that the DAB converter is made out of two current sources by neglecting the high-order dynamics in the AC link. Next, the input and output currents are averaged over one switching period and are equal to $I_d = P/V_d$ and $I_o = P/V_o$, by assuming no losses in the converter. The large-signal current sources are therefore expressed as

$$ \begin{cases} I_d = \frac{1}{8n f_s L} \cdot V_o \cdot P_n(D_f, V_d, V_o) \\ I_o = \frac{1}{8n f_s L} \cdot V_d \cdot P_n(D_f, V_d, V_o) \end{cases} \qquad (4) $$

where $P_n(D_f, V_d, V_o)$ for each mode is obtained in Table III. By applying small-signal perturbations $\tilde{d}_f, \tilde{v}_d, \tilde{v}_o$ to (4) around an equilibrium point $(D_f, V_d, V_o)$, the small-signal current sources can be expressed as a product of matrix

$$ \begin{pmatrix} \tilde{i}_d \\ \tilde{i}_o \end{pmatrix} = \begin{pmatrix} G_{id}d_f & G_{id}v_d & G_{io}v_o \\ G_{io}d_f & G_{io}v_d & G_{io}v_o \end{pmatrix} \begin{pmatrix} \tilde{d}_f \\ \tilde{v}_d \\ \tilde{v}_o \end{pmatrix} = G_{ij} \begin{pmatrix} \tilde{d}_f \\ \tilde{v}_d \\ \tilde{v}_o \end{pmatrix} \qquad (5) $$
where $G_{ij}$, the matrix of small-signal current gains, is the Jacobian matrix of the function in (4) expressed as

$$G_{ij} = \frac{1}{8nf_sL} \cdot \left( \begin{array}{ccc} \frac{\partial(V_o \cdot P_n)}{\partial D_f} & \frac{\partial(V_o \cdot P_n)}{\partial V_d} & \frac{\partial(V_o \cdot P_n)}{\partial V_o} \\ \frac{\partial(V_d \cdot P_n)}{\partial D_f} & \frac{\partial(V_d \cdot P_n)}{\partial V_d} & \frac{\partial(V_d \cdot P_n)}{\partial V_o} \end{array} \right) \quad (6)$$

The equivalent small-signal circuit of the battery charger is shown in Figure 6 where the grid and battery sides are represented as their Thevenin equivalent voltage sources $\tilde{v}_{td}$, $\tilde{v}_{to}$ and impedances $Z_f$, $Z_o$ defined as

$$Z_f(s) = \frac{sL_f + R_f}{s^2 L_f C_d + sR_f C_d + 1} \quad (7)$$

$$Z_o(s) = \frac{Z_b}{sC_oZ_b + 1} \quad (8)$$

with the battery impedance equal to $Z_b(s) = R_b + sL_b$. Finally, the open-loop model is completed by the following equations

$$\tilde{v}_d = \tilde{v}_{td} - Z_f \tilde{i}_d \quad (9)$$

$$\tilde{v}_o = \tilde{v}_{to} + Z_o \tilde{i}_o \quad (10)$$

$$\tilde{i}_b = \tilde{v}_o / Z_b \quad (11)$$

The diagram of the control structure is shown in Figure 7. It includes the open-loop model and the control structure of the CC-CV charging.

Depending on the charging state, the controller outputs the unified control variable, defined as

$$\tilde{d}_f = \begin{cases} C_i(\tilde{i}_b^* - \tilde{i}_b) & , CC \\ C_v(\tilde{v}_o^* - \tilde{v}_o) & , CV \end{cases} \quad (12)$$

with $C_i(s) = k_i(1 + 1/sT_i)$ and $C_v(s) = k_v(1 + 1/sT_v)$, two PI controllers responsible of the constant current

<figure><img src="image_8.png" /><figcaption>Fig. 6: Averaged small-signal model of the battery charger.</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>Fig. 7: Block diagram of the CC-CV charger control.</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>Fig. 8: Simplified diagram for the calculation of $Z_{in}^d$.</figcaption></figure>

and constant voltage regulation, respectively. Here, the reference terms $\tilde{v}_o^*$, $\tilde{i}_b^*$ and the voltage perturbations $\tilde{v}_{td}$, $\tilde{v}_{to}$ are set to zero. By rearranging the controllers as in eq. (13), a simplified diagram can be drawn in Figure 8 for the calculation of the input impedance.

$$C_{iv} = \begin{cases} C_i / Z_b & , CC \\ C_v & , CV \end{cases} \quad (13)$$

Hence, by combining eqs. (5), (9), (10) and (12), the input impedance is calculated as follows :

$$\begin{aligned} Z_{in}^d &= \left( \frac{\tilde{i}_d}{\tilde{v}_d} \right)^{-1} \\ &= \left( \frac{Z_o G_{i_o v_d} (G_{i_d v_o} - G_{i_d d_f} C_{io})}{1 - Z_o (G_{i_o v_o} - G_{i_o d_f} C_{iv})} + G_{i_d v_d} \right)^{-1} \end{aligned} \quad (14)$$

Now, the interacting impedances can be fully derived for any operating point, modulation scheme, filter and controller parameters.

### A. Stability criterion

At the DC grid side of the converter (see Figure 6), the input voltage resulting from the impedance interaction is expressed as

$$\tilde{v}_d = \frac{1}{1 + Z_f / Z_{in}^d} \tilde{v}_{td} \quad (15)$$

From Eq. (15), the Nyquist stability criterion states that the impedance ratio $Z_f / Z_{in}^d$ should not encircle the (0 dB, 180°) critical point. Both the Phase Margin (PM)
and Gain margin (GM), defined in Eq. (16), must be positive for small-signal stability [2]. We define each margin as follows

$$ \begin{cases} GM(\text{dB}) = |Z_{in}^d(j2\pi f_p)| - |Z_f(j2\pi f_p)| > 0 \\ PM(^\circ) = 180^\circ + \angle Z_{in}^d(j2\pi f_g) - \angle Z_f(j2\pi f_g) > 0 \end{cases} \quad (16) $$

where $f_g$ and $f_p$, are the crossing frequencies.

Regarding the battery side, the stability criterion is analogous to that described but with the ratio $Z_o/Z_{in}^o$. This is however not addressed in this paper.

## B. Model validation on studied system

For validation purposes, the DAB-based battery charger and control structure are implemented on a switching model in PLECS. An operating point at maximum charging power is simulated at low and high $V_s$. Once steady-state is reached, a small multitone signal is added to $v_d$ while the harmonic response of $i_d$ is observed. The Bode diagram of the interacting impedances and the Black diagram of the impedance ratio are displayed in Figure 9. The linear modeling of $Z_{in}^d$ is depicted using continuous lines, while the harmonic response of the switched model is represented by discrete markers. One can note that the developed models match the switched model with high accuracy within this frequency range.

Furthermore, it can be observed that the input impedance is independent of the MS at low frequencies while it greatly differs above 10 Hz. In this case where the resonant frequency is of the same order of magnitude than the controller bandwidth, the stability will be significantly impacted by the MS. Concerning the Black diagram, because the critical point is never encircled, the charger is stable for all presented cases. Regarding stability margins, the PM is always infinite as the 0 dB axis is never crossed. Still, the GM, measured by the distance between the critical point and the crossing of the 180° axis, depends on the supply voltage as well as the selected MS. Under low supply voltage condition (i.e. in *boost* mode), the advanced modulation schemes result in a higher GM than for SPS. However, in *buck* mode, the opposite is observed where DPS and TPS modulation almost lead to instability.

# IV. STABILITY ANALYSIS OF A CHARGING CYCLE

This section provides a stability analysis during one complete charging cycle. The stability at the input side is analyzed for each operating point of the CC-CV charging under high and low supply voltage conditions following the flowchart shown in Figure 10. First, the equilibrium state values are calculated according to the SoC (see

<figure><img src="image_11.png" /><figcaption>Fig. 9: Bode and Black diagrams of $Z_f$, $Z_{in}^d$ and their ratio at $P^{max}$.</figcaption></figure>

Figure 3). Then, the small-signal model is obtained for each MS. Finally, the input impedance is calculated by mean of (14), enabling the stability margin evaluation.

Figure 11 shows the modeled GM during the charging process for each MS under low and high supply voltage conditions. We first notice that the CV mode shows no concern due to the reduced transferred power. In Figure 11a, the use of an advanced MS increases the GM in comparison with SPS. However, Figure 11b shows a significant stability margin degradation when using EPS, DPS and TPS MS. This indicates that in *buck* mode, the advanced modulations have a negative impact on the system stability.

In order to compare with those observations, a 1ms, 100 V pulse is generated as a disturbance on the supply voltage of the PLECS model. The responses of $v_d$, $i_o$ and $i_L$ are observed in Figure 12. It can be seen from the oscillations of $v_d$ that SPS exhibits a quicker recovery after the disturbance. Inversely, as predicted by the linear models, DPS and TPS show a slow recovery which translates into a reduced stability margin. Finally, it must be reminded that advanced MS are mainly used to minimize the power losses. Despite the fluctuating results
<figure><img src="image_12.png" /><figcaption>Fig. 10: Flowchart of the stability analysis over the charging cycle.</figcaption></figure>

<figure><img src="image_13.png" /><figcaption>(a) $V_s = 1$ kV (boost mode)</figcaption></figure>

<figure><img src="image_14.png" /><figcaption>(b) $V_s = 2$ kV (buck mode)</figcaption></figure>

<caption>Fig. 11: Modeled GM as a function of the SoC and MS.</caption>

with regard to stability, EPS and TPS are expected to reach higher efficiency as indicated by their lower RMS inductor current. This confirms that a comprehensive stability analysis is required for MS selection.

<figure><img src="image_15.png" /><figcaption>(a) SPS</figcaption></figure>

<figure><img src="image_16.png" /><figcaption>(c) DPS</figcaption></figure>

<figure><img src="image_17.png" /><figcaption>(b) EPS</figcaption></figure>

<figure><img src="image_18.png" /><figcaption>(d) TPS</figcaption></figure>

<caption>Fig. 12: Supply voltage pulse response at $P^{max}$, $V_s = 2$ kV (buck mode).</caption>

V. CONCLUSION

In this work, the averaged small-signal model of a DAB-based charger with a CC-CV charging process is proposed. It is shown that despite their optimized performances in steady-state, the advanced MS have a large impact on the converter input impedance. The models are validated by simulation and a method for stability margin evaluation is proposed for each operating point of a charging cycle, highlighting the poor dynamic response of advanced MS in *buck* mode. Finally this work enables the comprehensive design of the modulator,
controllers and filters in order to achieve safe and efficient battery charging. Further research will explore the non-linearity of the advanced MS for increased accuracy. Nonetheless, the impedance interactions in the case of multiple parallel chargers will be considered.

# APPENDIX

<table><caption>TABLE III: Operating range and transferred power function of the selected modes.</caption><thead><tr><th>Mode</th><th>Operating Range</th><th>|P<sub>n</sub>| = f(δ<sup>1</sup>, D<sub>f</sub>)</th></tr></thead><tbody><tr><td>SPS</td><td>∀(δ, P<sub>n</sub>)</td><td>4|D<sub>f</sub>|(1 - |D<sub>f</sub>|)</td></tr><tr><td>EPS<br>Low P</td><td>|P<sub>n</sub>| ∈ [0, 2δ(1 - δ)]</td><td>(4δ/(2-δ))|D<sub>f</sub>|(1 + 2|D<sub>f</sub>|)</td></tr><tr><td>EPS<br>High P</td><td>|P<sub>n</sub>| ∈ [2δ(1 - δ),<br>1 - (√(1-δ<sup>2</sup>-1)/δ)]<sup>2</sup></td><td>4|D<sub>f</sub>|(1 - |D<sub>f</sub>|) - (1 - D<sub>α</sub>)<sup>2</sup></td></tr><tr><td>DPS<br>Low P</td><td>|P<sub>n</sub>| ∈ [0, (1+2δ-3δ<sup>2</sup>)/2]</td><td>(1+3δ/2)2|D<sub>f</sub>|<sup>2</sup></td></tr><tr><td>DPS<br>High P</td><td>|P<sub>n</sub>| ∈ [(1+2δ-3δ<sup>2</sup>)/2, 1]</td><td>1 - (3δ<sup>2</sup>-2δ+1)/(2δ<sup>2</sup>) (1 - 2|D<sub>f</sub>|)<sup>2</sup></td></tr><tr><td>TPS<br>Low P</td><td>|P<sub>n</sub>| ∈ [0, 2δ(1 - δ)]</td><td>(8δ/(1-δ)|D<sub>f</sub>|)<sup>2</sup></td></tr><tr><td>TPS<br>High P</td><td>|P<sub>n</sub>| ∈ [2δ(1 - δ), 1]</td><td>1 - (2δ<sup>2</sup>-2δ+1)/(δ<sup>2</sup>) (1 - 2|D<sub>f</sub>|)<sup>2</sup></td></tr></tbody></table>

# REFERENCES

[1] M. Berrehil El Kattel, R. Mayer, F. Ely, and B. de Jesus Cardoso Filho, “Comprehensive review of battery charger structures of EVs and HEVs for levels 1–3,” *International Journal of Circuit Theory and Applications*, vol. 51, DOI 10.1002/cta.3579, no. 7, pp. 3514–3542, 7 2023.

[2] R. Kumar and C. N. Bhende, “Active Damping Stabilization Techniques for Cascaded Systems in DC Microgrids: A Comprehensive Review,” 2 2023.

[3] F. Krismer, “Modeling and optimization of bidirectional dual active bridge DC-DC converter topologies,” Ph.D. dissertation, ETH Zurich, Zürich, 2010.

[4] N. Hou and Y. W. Li, “Overview and Comparison of Modulation and Control Strategies for a Nonresonant Single-Phase Dual-Active-Bridge DC-DC Converter,” *IEEE Transactions on Power Electronics*, vol. 35, DOI 10.1109/TPEL.2019.2927930, no. 3, pp. 3148–3172, 2020.

[5] M. Kim, M. Rosekeit, S. K. Sul, and R. W. De Doncker, “A dual-phase-shift control strategy for dual-active-bridge DC-DC converter in wide voltage range,” *8th International Conference on Power Electronics - ECCE Asia: "Green World with Power Electronics"*, ICPE 2011-ECCE Asia, DOI 10.1109/ICPE.2011.5944548, pp. 364–371, 2011.

[6] B. Liu, P. Davari, and F. Blaabjerg, “An Optimized Hybrid Modulation Scheme for Reducing Conduction Losses in Dual Active Bridge Converters,” *IEEE Journal of Emerging and Selected Topics in Power Electronics*, vol. 9, DOI 10.1109/JESTPE.2019.2956323, no. 1, pp. 921–936, 2 2021.

[7] S. Shao, M. Jiang, W. Ye, Y. Li, J. Zhang, and K. Sheng, “Optimal Phase-Shift Control to Minimize Reactive Power for a Dual Active Bridge DC-DC Converter,” *IEEE Transactions on Power Electronics*, vol. 34, DOI 10.1109/TPEL.2018.2890292, no. 10, pp. 10 193–10 205, 2019.

[8] F. Lin, X. Zhang, and F. Feng, “Modeling and Stability Analysis of the Cascaded Dual Active Bridge Converter System with Considerations of the Controller Parameters,” in *2019 10th International Conference on Power Electronics and ECCE Asia (ICPE 2019 - ECCE Asia)*, DOI 10.23919/ICPE2019-ECCEAsia42246.2019.8796930, pp. 1817–1820, 2019.

[9] Y. Eto, Y. Noge, M. Shoyama, and T. Babasaki, “Stability Analysis of Bidirectional Dual Active Bridge Converter With Input and Output LC Filters Applying Power-Feedback Control,” *IEEE Transactions on Power Electronics*, vol. 38, DOI 10.1109/TPEL.2022.3225113, no. 3, pp. 3127–3139, 2023.

[10] V. M. Iyer, S. Gulur, and S. Bhattacharya, “Small-signal stability assessment and active stabilization of a bidirectional battery charger,” *IEEE Transactions on Industry Applications*, vol. 55, DOI 10.1109/TIA.2018.2871101, no. 1, pp. 563–574, 2019.

[11] F. Feng, H. B. Gooi, and U. Manandhar, “Comparison of SPS, DPS and CTPS Modulations with Full Consideration of Stability of DAB Converters with Input LC Filter,” in *Proceedings of the Energy Conversion Congress and Exposition - Asia, ECCE Asia 2021*, DOI 10.1109/ECCE-Asia49820.2021.9478977, pp. 98–103. Institute of Electrical and Electronics Engineers Inc., 5 2021.

[12] F. Feng, X. Zhang, F. Lin, and H. B. Gooi, “Stability Analysis and Optimization of Dual Active Bridge Converter with LC Input Filter,” in *2019 IEEE International Conference on Industrial Technology (ICIT)*, DOI 10.1109/ICIT.2019.8754992, pp. 544–549, 2019.

[13] Y. Chen, Y. Zhang, E. Qu, and M. He, “Active Damping Method for LC-DAB System Based on a Power-Based Impedance Model,” *IEEE Transactions on Power Electronics*, vol. 38, DOI 10.1109/TPEL.2022.3230218, no. 4, pp. 4405–4418, 4 2023.

[14] M. Scohier, M. L. Gajardo, O. Deblecker, and T. Dragičević, “An Informed Modulation Strategy for Stability Enhancement of Dual Active Bridge Converters under Varying Supply Conditions,” DOI 10.36227/techrxiv.173747676.69845353/v1, 1 2025.

[15] C. Zhang, J. Jiang, L. Zhang, S. Liu, L. Wang, and P. C. Loh, “A generalized SOC-OCV model for lithium-ion batteries and the SOC estimation for LNMCO battery,” *Energies*, vol. 9, DOI 10.3390/en9110900, no. 11, 11 2016.

[16] M. Scohier, O. Deblecker, and C. Valderrama, “Triple-Phase-Shift Controlled Dual Active Bridge Converter with Variable Input Voltage in Auxiliary Railway Supply,” *24th European Conference on Power Electronics and Applications (EPE'22 ECCE Europe)*, Hanover, Germany, pp. 1–10, 2022.

[17] M. Scohier, O. Deblecker, and B. Bakhshideh Zad, “A Simple and Effective Active Damping Method for Triple-Phase-Shift DAB Converter in DC Railway,” in *IECON 2023- 49th Annual Conference of the IEEE Industrial Electronics Society*, DOI 10.1109/IECON51785.2023.10312325, pp. 1–6, 2023.

<footnote n="1">¹ δ = d<sup>sgn</sup>(1-d) is used to handle *buck* and *boost* modes in a unified manner.</footnote>

<footnote>II D<sub>α</sub> is a control variable used in the EPS MS (see [6]).</footnote>