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# Polynomial multi-variable control strategy for flux balancing in Dual Active Bridge Converter

Pierre-Baptiste Steckler, Jean-Yves Gauthier, Xuefang Lin-Shi

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Pierre-Baptiste Steckler, Jean-Yves Gauthier, Xuefang Lin-Shi. Polynomial multi-variable control strategy for flux balancing in Dual Active Bridge Converter. EPE'20 ECCE Europe, Sep 2020, Lyon, France. 10.23919/EPE20ECCEEurope43536.2020.9215740. hal-03160043

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Submitted on 16 Jun 2025

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# Polynomial multi-variable control strategy for Flux Balancing in Dual Active Bridge Converter

Pierre-Baptiste STECKLER†§, Jean-Yves GAUTHIER†§,
Xuefang LIN-SHI†§, François WALLART†
†SUPERGRID INSTITUTE

23 rue Cyprian, Villeurbanne 69628, France

E-Mail: pierre-baptiste.steckler@supergrid-institute.com

URL: https://www.supergrid-institute.com

§UNIVERSITY OF LYON, INSA LYON, CNRS, AMPERE
Villeurbanne 69621, France

E-Mail: jean-yves.gauthier@insa-lyon.fr

URL: http://www.ampere-lab.fr

## Acknowledgements

This work was supported by a grant overseen by the French National Research Agency (ANR) as part of the Investissements d'Avenir Program (ANE-ITE-002-01).

## Keywords

«Converter control», «Modelling», «Microgrid», «Digital control», «Transformer»

## Abstract

An attractive option for the Dual Active Bridge converter is to integrate the AC coil inside the transformer, using its leakage inductance. However, this complicates the flux balancing process. To address that, a novel model and a multi-variable linear control for power and magnetizing average currents are proposed.

## Introduction

During the last decade, many attentions have been paid to Dual Active Bridge (DAB) converter (Fig. 1) [1] [2] for high power, reversible DC-DC conversion. Its control and modulation scheme have been widely investigated, and additional results have been provided concerning flux balancing of its transformer. This process is mandatory to avoid flux walking and core saturation. When a discrete AC power inductor $L_{ext}$ is used in series with the transformer [3] [4], flux balancing is easier. However, in order to improve both power density and efficiency, it is more interesting to remove this coil and to use the leakage inductance of the transformer itself, sized accordingly. Unfortunately, this increases the flux balancing process complexity as the leakage inductance is split into two parts, one at each winding, with a distribution linked to the transformer physical layout. It may induce important changes in the dynamic response of the converter. The objective of this paper is to propose a general flux balancing control, applicable for any leakage distribution. The control method is based on a polynomial multi-variable control law, allowing a much faster dynamic behavior compared to classical cascaded loop controls.
<figure><img src="image_3.png" /><figcaption>Fig. 1: Dual Active Bridge converter schematic (with an external coil)</figcaption></figure>

The paper will be organized as follows: first, average model of the converter with any leakage distribution will be established. Then, a polynomial multi-variable control strategy will be proposed. It will be compared with the conventional method, using an average model of the converter. Finally, simulation results will be given, using a detailed model of a DAB converter with its modulation scheme, with an analysis regarding quantization noise.

## System modelling

Only the flux-balancing control of the transformer in a DAB converter will be studied. The transformer can be modelled by the general low frequency circuit diagram (Fig. 2) where the primary and secondary average voltages $V_p$ and $V_s$ can be converted to duty cycles of the DAB converter, thanks to modulation stage (which will be considered later in the paper). The model (1) can be obtained, where $I_p$, $I_s$ and $I_m$ are respectively the primary, secondary and magnetizing average currents. $R_p$, $L_{fp}$, $R_s$ and $L_{fs}$ are respectively the primary and secondary resistances and inductances. $R_m$ corresponds to core losses and $L_m$ is the magnetizing inductance. $m$ is the transformer turns ratio.

<figure><img src="image_4.png" /><figcaption>Fig. 2: Non-ideal low-frequency linear transformer model of DAB</figcaption></figure>

$$ \frac{d}{dt} \begin{bmatrix} I_p \\ I_s \\ I_m \end{bmatrix} = \begin{bmatrix} -\frac{R_p + R_m}{L_{fp}} & \frac{R_m \cdot m}{L_{fp}} & \frac{R_m}{L_{fp}} \\ \frac{R_m \cdot m}{L_{fs}} & -\frac{R_s + m^2 \cdot R_m}{L_{fs}} & -\frac{R_m \cdot m}{L_{fs}} \\ \frac{R_m}{L_m} & -\frac{R_m \cdot m}{L_m} & -\frac{R_m}{L_m} \end{bmatrix} \cdot \begin{bmatrix} I_p \\ I_s \\ I_m \end{bmatrix} + \begin{bmatrix} \frac{1}{L_{fp}} & 0 \\ 0 & -\frac{1}{L_{fs}} \\ 0 & 0 \end{bmatrix} \cdot \begin{bmatrix} V_p \\ V_s \end{bmatrix} \quad (1) $$

For control issues, $R_m$ can be neglected. However, the omission of $R_m$ induces a dependency between $I_p$, $I_s$ and $I_m$, since $I_m = I_p - m \cdot I_s$. In this case, the rank of the state matrix downs to 2. To recover full rank, a model reduction is required. For the sake of simplicity, the secondary resistance $R_s$ and the inductance $L_{fs}$ are reflected on the primary side as shown in Figure 3.
<figure><img src="image_5.png" /><figcaption>Fig. 3: Simplified and reflected transformer model of DAB</figcaption></figure>

Defining $\Delta = L_m \cdot L_{fs}/m + L_{fp} \cdot L_m \cdot m + L_{fp} \cdot L_{fs}/m$, the resulting system is presented in (2). This model, shown under the form $\dot{x} = Ax + Bu$, $y = Cx$, is general and applies for any leakage distribution and transformer layout. When all the leakage flux is totalized at one side, for instance at primary winding, it corresponds to a specific case of (2) where $L_{fp} = L_{ft}$ and $L_{fs} = 0$.

$$ \left\{ \begin{aligned} \frac{d}{dt} \begin{bmatrix} I_p \\ I_m \end{bmatrix} &= \begin{bmatrix} -\frac{L_m \cdot m \cdot R_p}{\Delta} - \frac{L_m \cdot R_s + L_{fs} \cdot R_p}{m \cdot \Delta} & \frac{L_m \cdot R_s}{m \cdot \Delta} \\ -\frac{L_{fs} \cdot R_p}{m \cdot \Delta} + \frac{L_{fp} \cdot R_s}{m \cdot \Delta} & -\frac{L_{fp} \cdot R_s}{m \cdot \Delta} \end{bmatrix} \cdot \begin{bmatrix} I_p \\ I_m \end{bmatrix} + \begin{bmatrix} \frac{m \cdot L_m + \frac{L_{fs}}{m}}{\Delta} & -\frac{L_m}{\Delta} \\ \frac{L_{fs}}{m \cdot \Delta} & \frac{L_{fp}}{\Delta} \end{bmatrix} \cdot \begin{bmatrix} V_p \\ V_s \end{bmatrix} \\ \begin{bmatrix} I_p \\ I_s \end{bmatrix} &= \begin{bmatrix} \frac{1}{m} & 0 \\ -\frac{1}{m} & 0 \end{bmatrix} \cdot \begin{bmatrix} I_p \\ I_m \end{bmatrix} \end{aligned} \right. \quad (2) $$

## Polynomial multi-variable control strategy

There are many control strategies for MIMO linear systems, such as multi-variable decoupling state feedback control [5], but it could require a more complex algorithm to ensure no steady-state error. We propose here another solution with multi-variable pole-zero compensation. It consists first to determine the Transfer Function matrix (TFM) from (2) by calculating $M(s) = C(sI_n - A)^{-1}B$ with $s$ the Laplace variable. The control structure is given in Figure 4, where $R(s)$ is the reference vector for $I_p$ and $I_m$. Every connection is a two-element vector. The controller $K(s)$ is a $(2 \times 2)$ TFM, as is $M(s)$. For a desired closed-loop behavior represented by its $(2 \times 2)$ TFM $\Gamma(s)$, the controller is given by $K(s) = M^{-1}(s) \cdot \Gamma(s) \cdot [I_2 - \Gamma(s)]^{-1}$.

<figure><img src="image_6.png" /><figcaption>Fig. 4: Proposed control structure</figcaption></figure>

The choice of $\Gamma(s)$ has a strong physical sense. First, to obtain a decoupled response between power and magnetizing currents, $\Gamma(s)$ must have a diagonal structure. Secondly, care must be taken with the possible Right Half Plane Zeros (RHPZ), as their compensation would lead to an unstable closed loop system. No RHPZ are encountered in this case. The diagonal terms describe the individual dynamics of both currents. For example, a first-order dynamics can be chosen and leads to (3). Here, $\tau_p$ and $\tau_m$ represent the individual time constants chosen for both first-order sub-systems in closed loop.

$$ \Gamma(s) = \begin{bmatrix} \frac{1}{1+\tau_p \cdot s} & 0 \\ 0 & \frac{1}{1+\tau_m \cdot s} \end{bmatrix} \quad (3) $$
An important remark is that, unlike cascaded loops as proposed in [4], there is no “hierarchical” relationship between $τ_p$ and $τ_m$. Their values are chosen according to the open-loop dynamics and allow a much faster primary current response than expectable with cascaded loops.

## Comparative analysis of the control for the transformer part

A physical 100kW DAB converter [6] is studied and an external tapped inductor $L_{ext}$ was used. For this converter, the leakage distribution is almost (1 − 0) with approximately 50µH of total leakage inductance seen from primary side. The measured parameters of the converter and its transformer are given in Table I. For future designs, the AC inductor will be included inside the transformer. The leakage distribution should be about (1/3 − 2/3). To test the new design with the proposed control scheme, the equivalent values shown in Table II are used for simulations. Only the transformer part of the DAB is simulated, based on Matlab/Simulink and Simpower Systems toolbox, but the resistances $R_p^*$ and $R_s^*$ take into account the on-state resistance of corresponding H-bridge MOSFETs. More details on the converter are given in [6]. The simulation scheme is shown in Figure 5.

<table><caption>**Table I: Measured converter parameters**</caption><thead><tr><th>L<sub>f<sub>t</sub></sub></th><th>R<sup>*</sup><sub>p</sub></th><th>R<sup>*</sup><sub>s</sub></th><th>m</th><th>L<sub>m</sub></th><th>L<sub>ext</sub></th></tr></thead><tbody><tr><td>6 µH</td><td>22.4 mΩ</td><td>11.0 mΩ</td><td>0.5</td><td>8 mH</td><td>44.8 µH</td></tr></tbody></table>

<table><caption>**Table II: Simulated future converter parameters**</caption><thead><tr><th>L<sub>f<sub>p</sub></sub></th><th>L<sub>f<sub>s</sub></sub></th><th>R<sup>*</sup><sub>p</sub></th><th>R<sup>*</sup><sub>s</sub></th><th>m</th><th>L<sub>m</sub></th></tr></thead><tbody><tr><td>33.3 µH</td><td>4.17 µH</td><td>22.4 mΩ</td><td>11.0 mΩ</td><td>0.5</td><td>8 mH</td></tr></tbody></table>

<figure><img src="image_7.png" /><figcaption>Fig. 5: Simulation scheme for the proposed controller</figcaption></figure>

With parameters of Table II, the system (2) has two poles at −1331 rad/s and −1.853 rad/s which correspond respectively to the current (power) and flux dynamics. After choosing closed-loop dynamics four times faster than the corresponding open-loop ones, Figure 6 shows the simulation results of the proposed control with $I_p = 1$ A, $I_m = 1$ A and consequently $I_s = 0$ as initial conditions. The desired currents have been set to zero. The control behavior is compared with the one proposed in [4] where two PI controllers are used to generate respectively $V_s$ and $V_p$: one for the magnetizing current loop which has the same bandwidth than the proposed control, and another for the primary current loop. The latter is set 10 times slower than the former, as recommended in [4]. It can be seen that the proposed control results in a faster primary current dynamics. For the flux dynamics, the difference is smaller but a better damping is obtained with the proposed control.

To test the robustness of the proposed control, the same control parameters are simulated with a “bad model” where $L_{f_p} = 50$ µH and $L_{f_s} = 0$. As shown in Figure 7, a similar damping is achieved with both nominal and uncertain model. There is no significant difference in response time.
<figure><img src="image_8.png" /><figcaption>Fig. 6: Primary and magnetizing currents: comparison between the proposed controller and the double (cascaded) PI controller</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>Fig. 7: Primary and magnetizing currents: robustness of the proposed controller with an uncertain model</figcaption></figure>

Simulation results with DAB

To assess the performance of the proposed control scheme in a more realistic way, the parameters described in Table II are used to simulate with a detailed model of the Dual Active Bridge converter. The Single Phase Shift (SPS) [7] modulation scheme is used, with three degrees-of-freedom: the phase shifting angle (Φ) controls the power flow, whereas the duty cycles (D₁, D₂) of both voltages are the output variables of the proposed average current controller. The simulations scheme is presented in Fig. 8. The PWM waveforms and their associated degrees of freedom are presented Fig. 9
<figure><img src="image_10.png" /><figcaption>Fig. 8: Simulation scheme using a detailed DAB model.</figcaption></figure>

<figure><img src="image_11.png" /><figcaption>Fig. 9: PWM waveforms of the DAB converter (SPS control)</figcaption></figure>

For this simulation, the power controller is not studied, and the phase shift is controlled in open loop. Its value corresponds to the nominal output power of the converter (100 kW). As shown in Fig. 8, both input and output voltages (V<sub>IN</sub> and V<sub>OUT</sub>) are constant, and equal to 1200 V and 600 V respectively. The switching frequency is equal to 20 kHz. The average value of the primary and secondary currents is obtained using a second-order Butterworth filter, with a corner frequency of 2 kHz. The closed-loop dynamics is defined by τ<sub>m</sub> = 10.8 ms (50 times faster than open-loop) and τ<sub>p</sub> = 376 µs (2 times faster). As the primary current is already fast enough, it is not accelerated to a large extent, in order to mitigate the noise/performance tradeoff.

The state variables of the model are initialized such that the average values of I<sub>p</sub> and I<sub>m</sub>, which are controlled, equal respectively 100 A and 10 A, and as both references are zero, the controller needs to bring both to zero. The simulation results are presented in Fig. 10. The filtered currents, which are used by the controller, are shown too.
<figure><img src="image_12.png" /><figcaption>Figure 10: General view of evolution of the state and control variables</figcaption></figure>

First of all, it can be seen that the proposed control scheme effectively forces $I_p$ and $I_m$, and according to the chosen dynamics, the regulation of $I_p$ is much faster than its magnetization counterpart. These two independent dynamics also appear in the waveforms of $D_1$ and $D_2$: the control of $I_p$ seems to be mostly related to the difference between $D_1$ and $D_2$, whereas $I_m$ is correlated to their average value. Indeed, both duty cycles are separated in the first part of the transient and approximately equal as soon as $I_p$ reaches zero, then they slowly converge together to their final value (0.5). Note that this approximation would be exact if the leakage distribution of the transformer was 50%-50%. A zoom on the primary current transient is presented Fig. 11.

<figure><img src="image_13.png" /><figcaption>Fig. 11: Zoom on the primary current dynamics and evolution of the state and control variables</figcaption></figure>

Even though the proposed control scheme shows its effectiveness, it can be observed that the variations in $D_1$ and $D_2$ are extremely small. It is not a consequence of the control scheme, but only concerns the parameters of the converter, which has low winding and inverter resistances due to its high efficiency (98%).

It can be expected that, for real-time implementation, quantization phenomenon on the control variables could disturb the controller and limits its performance. To evaluate the effects of quantization, the finite resolution of the PWM is taken into account. On the converter demonstrator, it is implemented using a FPGA with a 100 MHz clock. As the PWM frequency is 20 kHz, only 5000 values are realizable for the duty cycle, from 0 to 1. This constraint is added to the simulation, and the results are shown in Fig. 12 (overview) and Fig. 13 (zoom).
<figure><img src="image_14.png" /><figcaption>Fig. 12: General view of evolution of the state and control variables, with quantization</figcaption></figure>

<figure><img src="image_15.png" /><figcaption>Fig. 13: Zoom on the primary current dynamics and evolution of the state and control variables, with quantization</figcaption></figure>

It can be seen that even though the control variables are affected by the quantization noise, the effect on $I_p$ and $I_m$ is mitigated, and the overall performance is maintained.

## Conclusion

In this paper, a general flux-balancing process based on a multi-variable polynomial control is proposed for a DAB converter without AC inductor in order to reduce the converter size. Two benefits can be noted compared to existing works: on the one hand the proposed flux-balancing control is valid for any leakage distribution, on the other hand the multi-variable control approach allows independent closed-loop dynamics for primary and magnetizing currents. The effectiveness of the proposed control scheme is assessed in simulation, using a detailed model of the converter and its modulation scheme, even with quantization noise on the control variables. As a perspective, the proposed method can be extended to multi-phase and multi-winding transformers, making it a powerful tool for many high power converter topologies. Moreover, experimental validation will be performed in a future work.

## References

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[2] N. Soltau, Z. Shen, and R. W. D. Doncker, "Design of series inductances for high-power dc-dc converters," in 2015 International Conference on Renewable Energy Research and Applications (ICRERA), Nov 2015, pp. 890-895.

[3] Z. Wang; J. Chai; X. Sun, "A Method to Control Flux Balancing of High-frequency Transformers in Dual Active Bridge DC-DC Converters", IET-The Journal of Engineering, vol. 2018, Issue: 17, pp. 1835–1843, 11 2018.
[4] Y. Panov, M. M. Jovanovi, and B. T. Irving, "Novel transformer-flux-balancing control of dual-active-bridge bidirectional converters," in 2015 IEEE Applied Power Electronics Conference and Exposition (APEC), March 2015, pp. 42-49.
[5] S. Skogestad, I. Postlethwaite, "Multivariable feedback control: Analysis and Design" Wiley-Blackwell, 303 page, mai 1996.
[6] T. Lagier, P. Ladoux, and P. Dworakowski, "On the potential of silicon carbide mosfets in the dc/dc converters for future hvdc offshorewind-farms," IET High Voltage, 07 2017.
[7] O. Yade, J.-Y. Gauthier, X. Lin-Shi, M. Gendrin and A. Zaoui, "Modulation strategy for a Dual Active Bridge converter using Model Predictive Control", 2015 IEEE International Symposium on Predictive Control of Electrical Drives and Power Electronics (PRECEDE), Valparaiso, 2015, pp. 15-20.