Novel Droop-Based Techniques for Dynamic Performance Improvement in a Linear Active Disturbance Rejection Controlled-Dual Active Bridge for Fast Battery Charging of Electric Vehicles

Abstract

Electric vehicles (EVs) are rapidly replacing fossil-fuel-powered vehicles, creating a need for a fast-charging infrastructure that is crucial for their widespread adoption. This research addresses this challenge by improving the control of dual active bridge converters, a popular choice for high-power EV charging stations. A critical issue in EV battery charging is the smooth transition between charging stages (constant current and constant voltage) which can disrupt converter performance. This work proposes a novel feedforward control method using a combination of droop-based techniques combined with a sophisticated linear active disturbance rejection control system applied to a single-phase shift-modulated dual active bridge. This combination ensures a seamless transition between charging stages and enhances the robustness of the system against fluctuations in both input voltage and load. Numerical simulations using MATLAB/Simulink R2024a demonstrated that this approach not only enables smooth charging but also reduces the peak input converter current, allowing for the use of lower-rated components in the converter design. This translates to potentially lower costs for building these essential charging stations and faster adoption of EVs.

1. Introduction

The rapid growth of electric vehicles (EVs) presents a significant opportunity to combat climate change and promote sustainable transportation [1,2]. However, one of the main hurdles to widespread adoption of EV remains the time required to charge the battery. Conventional charging methods can be time-consuming, discouraging potential EV owners. To overcome this limitation, researchers are actively developing fast charging technologies that can significantly reduce charging times and improve user experience.

A promising approach for fast EV battery charging utilizes dual active bridge (DAB) converters [3,4] shown in Figure 1. DAB converters offer several advantages over conventional DC-DC converters, including high power density, inherent zero-voltage-switching (ZVS) capabilities, and the ability to isolate the input and output sides electrically. ZVS operation minimizes switching losses, leading to higher efficiency and reduced thermal stress on the converter components. Additionally, the galvanic isolation provided by the transformer in the DAB converter improves safety by preventing leakage currents between the grid and the battery [5,6,7,8,9].

Several control strategies have been explored in the existing research to control power flow in DAB converters. Proportional–integral (PI) control is a commonly used approach due to its simplicity. However, PI controllers struggle with parameter variations and external disturbances, leading to slow response times and potential instability in fast charging scenarios [10,11]. Fuzzy logic control offers improved robustness compared to PI control, but it requires complex design and tuning procedures, making it less practical for real-world applications [12,13]. Model predictive control (MPC) has also been investigated, demonstrating a fast dynamic response. However, MPC requires significant computational resources, which can be challenging for the resource-constrained embedded systems used in EV chargers [14,15]. Sliding mode control (SMC) has also been thoroughly employed in DAB converters due to its inherent ability to handle system uncertainties and disturbances and even though it is very robust and produces a fast dynamic response, a significant drawback of this technique is its inherent chattering effect which can lead to increased losses, noise, and wear on power electronic components [16,17,18]. Passivity-based control (PBC) of DAB has also been studied due to its ability to handle nonlinearities (robust to parameter variations and external disturbances) and maintain system stability. However, designing a PBC system often requires a deep understanding of the energy properties of the system and involves complex mathematical formulations. This complexity makes the implementation of PBC in DAB converters challenging [19,20,21,22].

A significant drawback of the above-mentioned controllers is their dependence on precise mathematical models of the system or a deep understanding of its input–output behavior. This requirement can be challenging to achieve in real-world applications [23]. To address this limitation, there has been a growing interest in techniques that focus on estimating and rejecting disturbances, with active disturbance rejection control (ADRC) being a prominent example. ADRC is a widely used control strategy for DC-DC converters in EV charging applications [24], however, there are not so many works in the literature that apply this technique to the DAB converter for battery charging. ADRC is essentially model-independent, only requires information regarding the order of the plant, and offers good regulation and disturbance rejection capabilities, ensuring stable operation of the converter. However, achieving a fast dynamic response for the output current/voltage, particularly under varying input/output voltage conditions as is the case during battery charging, remains a challenge with ADRC alone due to the non-linear nature of both the battery load and the DAB converter. A fast dynamic response is crucial for fast charging applications, as it allows the converter to adjust quickly the charging current to meet the battery’s needs and optimize charging efficiency [25,26,27,28].

Several strategies are found in the literature that can be applied to charge EV batteries [29]. Amongst all the existing methods for battery charging, the constant current (CC)/constant voltage (CV) technique is the most used as it ensures safe charging and good health of the battery. In this method, the battery is first charged at CC, during which its voltage and state of charge (SoC) rise and once the voltage has risen to a pre-defined value, the charging is then switched to CV. In this last stage, the SoC continues to increase while the battery current decreases [30,31]. Furthermore, a critical aspect of EV battery charging is ensuring a smooth transition between the two primary charging stages (CC and CV) [32]. However, this transition can introduce disturbances that disrupt the performance of the DAB converter, potentially leading to increased thermal stress on components, and voltage and current spikes that can compromise system stability and battery health [33].

There are few works in the literature aimed at tackling this transition phenomenon, ultimately improving the dynamic response of DC-DC converters when used for battery charging. Ref. [34] proposes a fast current controller (FCC) based on direct power control for tracking the desired output current under various input/output parameter disturbances/variations; however, this work utilizes only the CC charging technique having only one current loop, hence it does not address the issues involved when two cascaded loops are utilized. In [35], a feedforward scheme is proposed to solve this charging transition problem and even though the dynamic response is greatly improved, the transition time is quite long. Furthermore, ref. [36] proposes a duty ratio feedforward control method to obtain rapid current dynamic response in a bidirectional H-bridge. Again, only a single loop CC stage is employed without consideration for the CV stage.

The authors in [37] propose a Kalman filter based SoC estimation technique for reducing switching transients during charging mode selection, demonstrating enhanced performance through simulation results. Another study [38] uses a combination of fuzzy-PI control and the Bayesian regularization (BR) algorithm trained artificial neural network (ANN)-based control to help solve the potential discontinuity in the charging current and voltage resulting from the switch from CC to CV. Additionally, ref. [39] proposes a probabilistic fuzzy neural network (PFNN) controller for a phase shift full bridge (PSFB) DC-DC converter to significantly reduce the discontinuous charging voltage and current during the transition between the CC and CV charging modes; experimental results showed that the proposed controller can achieve smooth voltage and current responses, faster response times, and improved efficiency during the CC-CV transition. Even though the works presented in [37,38,39,40] achieve good results in providing a continuous and quick transition from CC to CV, they require complex controllers and algorithms for their implementation. Finally, ref. [40] proposes a Schmitt trigger technique for reducing switching transients during the CC-CV mode transition with hardware in the loop simulations demonstrating its effectiveness. The use of this auxiliary Schmitt trigger circuit increases the system component and associated power losses.

Building on these previous works, this research proposes novel power-based and droop-based feedforward techniques that complement the linear active disturbance rejection control (LADRC) to improve significantly the dynamic performance of single-phase shift (SPS) modulated DAB converters for fast EV battery charging.

Droop control is a decentralized control method typically used in power systems that allows multiple converters to share power automatically without requiring complex communication protocols [41,42]. By incorporating a droop-based technique into the LADRC framework, this research aims to achieve faster response times for the output and input converter currents/voltages, enabling the converter to adapt swiftly to changing battery voltage and grid conditions during the fast-charging process. The droop-based method provides an adaptive control mechanism that adjusts the reference values based on load conditions, i.e., it dynamically adjusts the converter’s operating point, ensuring a seamless and smoother transition between the CC and CV charging stages. These droop-based techniques offer a new approach to address the limitations of conventional methods and mitigate the disturbances typically associated with these transitions, thus maintaining the converter performance and reliability. Moreover, by combining the droop-based feedforward control with LADRC, the proposed system not only provides a robust solution for enhancing dynamic performance but also reduces peak input currents. This reduction allows for the use of lower-rated components in the converter design, potentially lowering the overall cost and improving the feasibility of deploying more efficient and reliable fast-charging stations on a larger scale. Finally, this work is an extension of the work done in [43] where new droop-based feedforward expressions are proposed to control SPS modulated DAB converters, different from that suggested in the mentioned work which uses an alternative approach to derive the feedforward expressions to control the DAB converter using dual phase shift (DPS) modulation. Therefore, the LADRC modelling in this work is specifically focused on SPS modulation unlike the previous work whose LADRC modelling is focused on DPS modulation. Additionally, this work introduces an extra feedforward term in the internal current loop which is absent in the previous work. This significantly improves the dynamic response especially in the presence of load and input voltage disturbances and parameter fluctuations.

The rest of this paper is organized as follows: Section 2 provides the basic working principle of a DAB DC-DC converter based on single phase shift modulation while Section 3 presents the basic principles of LADRC. The proposed LADRC system modeling based on SPS modulation is explained in Section 4 together with their mathematical derivations, while Section 5 presents the novel performance enhancement feedforward techniques. In Section 6, some numerical simulations are provided to validate the control algorithms proposed in the previous section, and the work is concluded in Section 7.

2. Working Principle of DAB Converter Based on SPS Modulation

The simplicity of SPS modulation makes it the most widely used power flow control technique for the DAB converter. As shown in Figure 1, the converter’s two H-bridges (full bridges) transform the DC input and output voltages into high-frequency 2-level square waves (V₁ and V₂). We can control the power flow by adjusting the phase shift between these square waves using SPS. Figure 1 shows the possible directions of power flow (positive or negative). The direction depends on the sign of the phase shift angle. A positive angle means V₂ lags behind V₁, resulting in positive power flow. On the contrary, a negative angle means that V₁ lags behind V₂, leading to negative power flow. Figure 2 shows typical waveforms for a DAB converter operating in buck mode (V_in > V_o) with positive power flow using SPS modulation. In this mode, both bridges operate at the same switching frequency with a 50% duty cycle [44,45].

Ignoring the converter’s resistive losses, the average transmitted power can be expressed as follows [46]:

$$P_{av}={\displaystyle \frac{V_{in}{V}_o\phi (\pi -|\phi \left|\right)}{2{\pi }^{2}nL{f}_s}}.$$

(1)

From Equation (1), the maximum positive power transfer is achieved when the phase shift angle (φ) is π/2.

One downside of the SPS modulation approach is the huge circulating currents created during power flow due to the mismatched nature of the voltages on both sides of the transformer. This impact is reflected by the shaded area in the inductor current waveform in Figure 2. The considerable power backflow consequently limits the soft switching range of the converter. As a result, current stress (RMS current), switching losses, and magnetic component (inductor and transformer) losses are all quite high, restricting system efficiency. More advanced modulation techniques like extended phase shift (EPS), dual phase shift (DPS), or triple phase shift (TPS) modulation have been proposed and used to reduce the RMS current, thus improving the converter efficiency, as they offer more degrees of freedom [47].

Although EPS, DPS, and TPS control offer more control options than SPS, their implementation complexity increases modulator design challenges. Additionally, their efficiency suffers with varying voltage ratios and output power. Then, to prioritize simplicity and robustness, this paper explores a DAB converter using SPS control.

3. Basic Principle of Linear Active Disturbance Rejection Control (LADRC)

LADRC operates on the principle of accurately estimating and compensating for internal and external disturbances [48]. It comprises a linear state error feedback (LSEF) control law and a linear extended state observer (LESO), as illustrated in Figure 3. The system features two primary control loops: the inner loop, known as the disturbance rejection loop, and the outer loop, known as the feedback control loop. The LESO anticipates the system’s total disturbances, which are then mitigated by the inner loop. This inner loop works in conjunction with the feedback control loop to enhance the system’s transient response and stability [49].

In Figure 3, r represents the reference input signal, u₀ and u are the controlled variables, while d_v and d_w denote external disturbances affecting the system. The controlled object is represented by G_p, with y as the system output signal. The variables $z_1$ and $z_2$ are the estimated values of the system output and total disturbance, respectively. k_p is the controller parameter and b is the system gain.

Any first-order single-input single-output system can be approximated as follows:

$$\dot{y}=bu+f\left(y,u,d_v,d_w,t\right),$$

(2)

where f represents the total internal and external disturbances. The system output signal y is assigned to the state x₁ (i.e., y = x₁), and an extended state x₂ is assigned to the lump disturbance f (i.e., f = x₂). Assuming f is differentiable, the system dynamics can be written in state–space form as below:

$$\left\{\begin{array}{c}\dot{x}=Ax+Bu+E\dot{f}\\ y=Cx\end{array},\right.$$

(3)

where $A=\left[\begin{array}{cc}0& 1\\ 0& 0\end{array}\right],B=\left[\begin{array}{c}b\\ 0\end{array}\right],E=\left[\begin{array}{c}0\\ 1\end{array}\right],x=\left[\begin{array}{c}{x}_1\\ x_2\end{array}\right],C=[\begin{array}{cc}1& 0\end{array}]$. The corresponding Luenberger-based LESO constructed from the system in (3) is given as follows:

$$\left\{\begin{array}{c}\dot{z}=Az+Bu+H(y-\widehat{y})\\ \widehat{y}=Cz\end{array}\right.,$$

(4)

where $z={\left[\begin{array}{cc}{z}_1& z_2\end{array}\right]}^T$ is the estimate of the state variable x₁ and x₂ and $H={\left[\begin{array}{cc}{h}_1& h_2\end{array}\right]}^T$ is the observer gain vector.

The characteristic equation $\lambda \left(s\right)$ of the LESO is obtained from (4) and given as follows:

$$\lambda \left(s\right)=\left|sI-(A-HC)\right|=\left[\begin{array}{cc}s+h_1& -1\\ h_2& s\end{array}\right]=s^2+h_{1}s+h_2.$$

(5)

According to the bandwidth parameterization method [50], all poles of the characteristic equation are assigned to the observer bandwidth (w_o), resulting in the following:

$$s^2+h_{1}s+h_2={(s+{\omega }_o)}^2=s^2+2{\omega }_{o}s+{{\omega }_o}^2.$$

(6)

From this, the observer gains are determined as: $h_1=2{\omega }_o$ and $h_2={{\omega }_o}^2$.

Moreover, the estimated variables are used to implement the control law as shown in Figure 3, which includes the disturbance rejection and error feedback control laws, given by the following [51]:

$$\left\{\begin{array}{c}{u}_o=k_p(r-z_1)\\ u={\displaystyle \frac{u_o-z_2}{b}}={\displaystyle \frac{k_p(r-z_1)-z_2}{b}}=K_p(R-\widehat{z})\end{array}\right.,$$

(7)

where $R=\left[\begin{array}{c}r\\ 0\end{array}\right]$ and $K_p=\left[\begin{array}{cc}{k}_p& 1\end{array}\right]/b$.

If the observer’s state variable z₂ accurately tracks the system’s total disturbance f, the system can be simplified as below:

$$\dot{y}=bu+x_2=\left(u_o-z_2\right)+x_2\approx u_o.$$

(8)

Using (7) and (8), and considering that $\widehat{y}\cong y$, when the disturbance is compensated the transfer function of the closed-loop control is the following:

$$G\left(s\right)={\displaystyle \frac{y\left(s\right)}{r\left(s\right)}}={\displaystyle \frac{k_p}{k_p+s}}.$$

(9)

Using the bandwidth parameterization approach again, the controller bandwidth (w_c) is assigned to the poles of the transfer function G(s) resulting in the following:

$$s+k_p=s+{\omega }_c⟹k_p={\omega }_c.$$

(10)

Thus, the design of LADRC is simplified to setting the observer and controller bandwidths, which greatly eases the parameter tuning and design process. The controller bandwidth should be less than the observer bandwidth; typically ${\omega }_o=3~10{\omega }_c$.

4. Control System Design

The control system proposed for battery charging is illustrated in Figure 4. It comprises an external LADRC voltage loop that generates the reference battery current for the inner LADRC battery current loop. This loop in turn generates the required phase shift to feed to the SPS modulator to control power flow. There is also the presence of a novel feedforward loop (FF1) which is added to the output of the external voltage loop. Moreover, an internal feedforward expression (FF2) is also added to improve the dynamic response of the DAB converter. The FF2 expression is obtained by readjusting (1) with the result given in (11):

$${\phi }_{ff}=\pi \left({\displaystyle \frac{1}{2}}-\sqrt{{\displaystyle \frac{1}{4}}-{\displaystyle \frac{2n{f}_{s}L{i_o}^{*}}{V_{in}}}}\right).$$

(11)

Regarding the feedforward block FF1, the equations used to calculate the output current set point are provided in Section 5.

The design specifications of the DAB converter for the control system design are shown in

Table 1. DAB converter specifications.

Parameter	Value
Input voltage (Vin)	756 V
Battery nominal voltage (Vo)	900 V
Switching frequency (fs)	100 kHz
Rated power (Po)	250 kW
External inductance plus transformer leakage inductance (L)	1.8 µH
Parasitic resistance of inductor and transformer	50 mΩ
Drain-source on resistance of MOSFET (Rds-on)	35 mΩ
Transformer turns ratio	5:6
Output filter capacitor (Cf)	200 µF
Filter capacitor ESR	50 mΩ

.

The LADR modeling of the various subsystems is provided next.

4.1. LADRC-Based Output Current Loop Control

This loop controls the output current, hence the battery average current, and generates the required phase shift angle to control the power flow. The output current control using LADR begins with the dynamic equation of the inductor current derivative using the DAB equivalent circuit shown in Figure 5 and is given as follows:

$${\displaystyle \frac{d{i}_L}{dt}}={\displaystyle \frac{v_1}{L}}-{\displaystyle \frac{i_L{r}_L}{L}}-{\displaystyle \frac{v_2}{L}}.$$

(12)

In Figure 5, S₁ and S₂, respectively, represent the switching functions of the primary and secondary H-bridges based on SPS modulation and are defined as follows:

$$S_1\left(\theta \right)=\left\{\begin{array}{c}1,0\le \theta <\pi \\ -1,\pi \le \theta <2\pi \end{array}\right.,$$

(13)

$$S_2\left(\theta \right)=\left\{\begin{array}{c}1,\phi \le \theta <\pi +\phi \\ -1,\pi +\phi \le \theta <2\pi +\phi \end{array}\right.,$$

(14)

where $\phi$ represents the phase shift between S₁ and S₂.

The exponential Fourier series representation of S₁ and S₂ are given, respectively, as below:

$$\left\{\begin{array}{c}{S}_1\left(\theta \right)={\displaystyle \sum _{m=odd}^{\infty }}\left({\displaystyle \frac{2}{j\pi m}}\right)e^{jm{\theta }_s}\\ S_2\left(\theta \right)=S_1\left(\theta -\phi \right)={\displaystyle \sum _{m=odd}^{\infty }}{\displaystyle \frac{2}{j\pi m}}{e}^{jm({\theta }_s-\phi )}={\displaystyle \sum _{m=odd}^{\infty }}\left({\displaystyle \frac{2}{j\pi m}}{e}^{-jm\phi }\right)e^{jm{\theta }_s}\end{array}.\right.$$

(15)

From Figure 5, the DAB converter output and inductive currents are related by the following relationship:

$$i_o={\displaystyle \frac{S_2{i}_L}{n}},$$

(16)

where n is the transformer turns ratio.

To simplify the analysis, the output capacitor ESR is considered small enough to be neglected. In this case $v_o\approx v_c$. Moreover, (11) can be reformulated to include the switching functions and this is given by the following:

$${\displaystyle \frac{d{i}_L}{dt}}={\displaystyle \frac{v_{in}{S}_1}{L}}-{\displaystyle \frac{i_L{r}_L}{L}}-{\displaystyle \frac{v_c{S}_2}{nL}}.$$

(17)

By substituting (17) into the time derivative of (16), the following equation is obtained:

$${\displaystyle \frac{d{i}_o}{dt}}={\displaystyle \frac{v_{in}{S}_1{S}_2}{nL}}-{\displaystyle \frac{S_2{i}_L{r}_L}{nL}}-{\displaystyle \frac{{\left(S_2\right)}^2{v}_c}{n^{2}L}}.$$

(18)

Letting $u_1=S_1{S}_2$ and $u_2={\left(S_2\right)}^2$ simplifies (18) to

$${\displaystyle \frac{d{i}_o}{dt}}={\displaystyle \frac{v_{in}{u}_1}{nL}}-{\displaystyle \frac{S_2{i}_L{r}_L}{nL}}-{\displaystyle \frac{u_2{v}_c}{n^{2}L}}.$$

(19)

To make (19) in a form that makes $\phi$ a control variable and hence easy to apply the concept of LADRC, an intuitive technique is to utilize the concept of generalized average modeling of a converter which is based on the complex exponential Fourier series expansion of periodic time-dependent variables. According to [52], the derivative of the k^th coefficient for a state variable x(t) is given by the following:

$${\displaystyle \frac{d{〈x〉}_k(t)}{dt}}=-jk\omega {〈x〉}_k(t)+{\langle {\displaystyle \frac{dx}{dt}}\rangle }_k(t).$$

(20)

Furthermore, the k^th coefficient of the product of two variables x and y is given as follows:

$${〈xy〉}_k(t)=\sum _i{〈x〉}_{k-i}\left(t\right){〈y〉}_i\left(t\right).$$

(21)

(20) and (21) are based on the differentiation and convolution properties of complex exponential Fourier series. For this work, k = 0 for DC signals while k = ±1 for the transformer AC signals and switching function components at the switching frequency ω_s.

Applying (20) to (19) yields

$${\displaystyle \frac{d{〈i_o〉}_o}{dt}}={\displaystyle \frac{{〈v_{in}{u}_1〉}_o}{nL}}-{\displaystyle \frac{r_L{〈S_2{i}_L〉}_o}{nL}}-{\displaystyle \frac{{〈u_2{v}_c〉}_o}{n^{2}L}}.$$

(22)

Now, the following expression can be derived by making use of (21):

$$\begin{gathered} {〈i_L{S}_2〉}_o=\underset{=0}{\underbrace{{〈i_L〉}_o{〈S_2〉}_o}}+{〈i_L〉}_1{〈S_2〉}_{-1}+{〈i_L〉}_{-1}{〈S_2〉}_1 \\ =2\left[Re\left({〈i_L〉}_1\right)Re\left({〈S_2〉}_1\right)+Im\left({〈i_L〉}_1\right)Im\left({〈S_2〉}_1\right)\right]. \end{gathered}$$

(23)

Letting ${〈i_L〉}_1=i_{Ld}+j{i}_{Lq}$ simplifies (23) to the following:

$${〈i_L{S}_2〉}_o=-{\displaystyle \frac{4i_{Ld}}{\pi }}\sin \left(\phi \right)-{\displaystyle \frac{4i_{Lq}}{\pi }}\cos \left(\phi \right).$$

(24)

For a very small $\phi$, $\sin \left(\phi \right)\approx \phi$ and $\cos \left(\phi \right)\approx 1$, then (24) becomes as follows:

$${〈i_L{S}_2〉}_o=-{\displaystyle \frac{4i_{Ld}}{\pi }}\phi -{\displaystyle \frac{4i_{Lq}}{\pi }}.$$

(25)

Using (21), the harmonic terms for $u_1$ and $u_2$ can be obtained as below:

$$\begin{array}{c}{〈u_1〉}_o={〈S_1{S}_2〉}_o={〈S_1〉}_1{〈S_2〉}_{-1}+{〈S_1〉}_{-1}{〈S_2〉}_1+{〈S_1〉}_3{〈S_2〉}_{-3}+{〈S_1〉}_{-3}{〈S_2〉}_3+\dots \hfill \\ ={\displaystyle \frac{8}{{\pi }^2}}{\displaystyle \sum _{p=0}^{\infty }}{\displaystyle \frac{\cos \left[\left(2p+1\right)\phi \right]}{{\left(2p+1\right)}^2}},\hfill \end{array}$$

(26)

$$\begin{array}{c}{〈u_2〉}_o={〈S_2{S}_2〉}_o={〈S_2〉}_1{〈S_2〉}_{-1}+{〈S_2〉}_{-1}{〈S_2〉}_1+{〈S_2〉}_3{〈S_2〉}_{-3}+{〈S_2〉}_{-3}{〈S_2〉}_3+\dots \hfill \\ ={\displaystyle \frac{8}{{\pi }^2}}{\displaystyle \sum _{p=0}^{\infty }}{\displaystyle \frac{1}{{\left(2p+1\right)}^2}}.\hfill \end{array}$$

(27)

Evaluating the expressions in (26) and (27) yields

$$\left\{\begin{array}{c}{〈u_1〉}_o={\displaystyle \frac{8}{{\pi }^2}}\left({\displaystyle \frac{\pi \phi -{\phi }^2}{4}}\right)={\displaystyle \frac{2}{{\pi }^2}}\left(\pi \phi -{\phi }^2\right)\\ {〈u_2〉}_o={\displaystyle \frac{8}{{\pi }^2}}\left({\displaystyle \frac{{\pi }^2}{8}}\right)=1\end{array}.\right.$$

(28)

In a similar way, ${〈u_1〉}_1$ and ${〈u_2〉}_1$ can be obtained as

$$\left\{\begin{array}{c}{〈u_1〉}_1={〈S_1{S}_2〉}_1={〈S_1〉}_1{〈S_2〉}_0+{〈S_1〉}_0{〈S_2〉}_1+{〈S_1〉}_{-1}{〈S_2〉}_2+{〈S_1〉}_2{〈S_2〉}_{-1}+\dots =0\\ {〈u_2〉}_1={〈S_2{S}_2〉}_1={〈S_2〉}_1{〈S_2〉}_0+{〈S_2〉}_0{〈S_2〉}_1+{〈S_2〉}_{-1}{〈S_2〉}_2+{〈S_2〉}_2{〈S_2〉}_{-1}+\dots =0\end{array}\right..$$

(29)

From the results in (28) and (29), the following can then be obtained:

$$\left\{\begin{array}{c}{〈v_{in}{u}_1〉}_o={〈v_{in}〉}_o{〈u_1〉}_o+\underset{=0}{\underbrace{{〈v_{in}〉}_1{〈u_1〉}_{-1}}}+\underset{=0}{\underbrace{{〈v_{in}〉}_{-1}{〈u_1〉}_1}}={\displaystyle \frac{2v_{in}}{{\pi }^2}}\left(\pi \phi -{\phi }^2\right)\\ {〈v_c{u}_2〉}_o={〈v_c〉}_o{〈u_2〉}_o+\underset{=0}{\underbrace{{〈v_c〉}_1{〈u_2〉}_{-1}}}+\underset{=0}{\underbrace{{〈v_c〉}_{-1}{〈u_2〉}_1}}=v_c\end{array}\right..$$

(30)

Inserting (25) and (30) into (22), (31) given below is obtained:

$${\displaystyle \frac{d{〈i_o〉}_o}{dt}}={\displaystyle \frac{2v_{in}}{{\pi }^{2}nL}}\left(\pi \phi -{\phi }^2\right)-{\displaystyle \frac{r_L}{nL}}\left(-{\displaystyle \frac{4i_{Ld}}{\pi }}\phi -{\displaystyle \frac{4i_{Lq}}{\pi }}\right)-{\displaystyle \frac{v_c}{n^{2}L}}.$$

(31)

(31) can be rewritten in a form suitable for LADRC:

$${\displaystyle \frac{d{i}_o}{dt}}={\displaystyle \frac{2}{\pi nL}}\left(v_{in}+i_{Ld}{r}_L\right)\phi +f\left(v_c,i_{Lq},\phi \right).$$

(32)

(32) is in the form $\dot{y}=bu+f$ which then makes it easy to apply LADRC hence their tuning follows the principles explained in Section 3.

4.2. Battery Voltage Loop Control Based on LADRC

This controller regulates the battery voltage during the CV mode of battery charging. For LADRC implementation, the dynamic equation of the output capacitor voltage derivative is required and given as follows:

$${\displaystyle \frac{d{v}_c}{dt}}={\displaystyle \frac{i_o}{C}}-{\displaystyle \frac{i_b}{C}}={\displaystyle \frac{i_o}{C}}+f\left(i_b\right).$$

(33)

Since (33) is already in the form $\dot{y}=bu+f$, $v_c$ can then be directly controlled to produce the reference output current ($i_o$) based on the LADR principle of Section 3.

5. Proposed Feedforward for Dynamic Performance Enhancement

The transition from CC to constant voltage CV charging is a major challenge for the system, often causing voltage and current spikes that disrupt performance. To address this, feedforward compensation (FF1 block) that generates a feedforward current (I_of) is typically introduced at the output of the external voltage loop controller and helps the system to transition smoothly from CC to CV charging, thus improving the dynamic response and efficiency of the system. The most common way to compute the feedforward expression [53] is based on the balance of the total equivalent load conductance at each time step:

$${G_{Load}}^{*}=G_{Load}⟹\left\{\begin{array}{c}{G_{Load}}^{*}={\displaystyle \frac{{i_{of}}^{*}}{{V_o}^{*}}}\\ G_{Load}={\displaystyle \frac{i_o}{V_o}}\end{array}\right.⟹{i_{of}}^{*}=\left({\displaystyle \frac{{V_o}^{*}}{V_o}}\right)i_o.$$

(34)

While LADRC with its ESO is powerful at handling disturbances, the non-linear nature of DAB converters and battery systems creates problems for the LADRC controller. This non-linearity can lead to system instability and difficulty reaching a stable operating state. As a result, the existing feedforward expression in (34) cannot be directly applied here. To address this challenge, three new methods are proposed for computing the feedforward terms, which will be described next.

5.1. Approach 1

This approach is derived from the load balance equation given in (34). Expression (34) is modified by introducing a convergence factor, $\alpha$, which determines the system’s ability to eliminate disturbances. The modified expression is given in (35):

$${i_{of1}}^{*}={\left({\displaystyle \frac{{V_o}^{*}}{V_o}}\right)}^{\alpha }{i}_o.$$

(35)

Even though (35) helps the converter transition smoothly from CC to CV and eliminates disturbances almost instantly, it requires very large values of the convergence factor to perform this function which could be a problem for the controller.

5.2. Approach 2

This approach is a modification of (35) brought about to overcome the drawback of (34), the need for very large convergence factors. The modified expression introduces a new fractional term that computes the ratio between a given input current reference and its instantaneous value. This is shown in (36):

$${i_{of2}}^{*}={\left[\left({\displaystyle \frac{{V_o}^{*}}{V_o}}\right)\left({\displaystyle \frac{{i_{in}}^{*}}{i_{in}}}\right)\right]}^{\beta }{i}_o.$$

(36)

$\beta$ is the convergence factor, just like $\alpha$, but with values that could be very small in relation to $\alpha$. This is proven in the curve shown in Figure 6 which shows how the normalized output current changes as α and β change for different values of normalized output voltage. If a normalized output voltage of 1.1 is considered, then for a per unit output current of 1.05, the values of α and β obtained are 0.5 and 0.25, respectively.

The reference input current introduced in (36) is computed based on a linear droop control approach explained next.

Suppose that $M({V_o}^{*},{i_{in}}^{*})$ is a fixed point on a non-vertical line, $L_1$, whose slope is $k_d$ (droop constant). Let $N(V_o,i_{in})$ be an arbitrary point on $L_1$ as shown in Figure 7a. Then from linear algebra, the slope of $L_1$ is given by (37):

$$k_d={\displaystyle \frac{{i_{in}}^{*}-i_{in}}{{V_o}^{*}-V_o}}⟹{i_{in}}^{*}=k_d\left({V_o}^{*}-V_o\right)+i_{in}.$$

(37)

The slope $k_d$ is a conductance which can be considered equal to the converter’s input conductance $\left(k_{d1}\right)$ or output load conductance $\left(k_{d2}\right)$. Hence, ${i_{in}}^{*}$ can have two values depending on $k_d$ and given by the following:

$${i_{in}}^{*}=\left\{\begin{array}{c}{k}_{d1}\left({V_o}^{*}-V_o\right)+i_{in}for{k}_{d1}={\displaystyle \frac{i_{in}}{V_{in}}}\\ k_{d2}\left({V_o}^{*}-V_o\right)+i_{in}for{k}_{d2}={\displaystyle \frac{i_o}{V_o}}\end{array}\right..$$

(38)

(38) can then be used in conjunction with (36) to improve the dynamic response of the converter during its transition from CC to CV and also in the presence of other disturbances of the system.

5.3. Approach 3

This approach uses a similar linear droop-based approach as before. Here, a feedforward term is derived from the linear relationship between the input and output currents, as shown in Figure 7b. If the fixed point $R({i_o}^{*},{i_{in}}^{*})$ is considered and an arbitrary point $S(i_o,i_{in})$ on the line $L_2$, then the droop slope ($m$) of $L_2$ is given as follows:

$$m={\displaystyle \frac{{i_{in}}^{*}-i_{in}}{{i_o}^{*}-i_o}}={\displaystyle \frac{V_o}{{V_o}^{*}}}.$$

(39)

After rearranging (39), the following feedforward expression is obtained:

$${i_{of3}}^{*}=\left({\displaystyle \frac{{V_o}^{*}}{V_o}}\right)\left({i_{in}}^{*}-i_{in}\right)+i_o.$$

(40)

To deduce ${i_{in}}^{*}$, the following expression is used:

$${i_{in}}^{*}=\left({\displaystyle \frac{i_o}{V_{in}}}\right){V_o}^{*}.$$

(41)

Furthermore, according to the power balance theory, the following can be deduced:

$$i_o={\displaystyle \frac{V_{in}{i}_{in}}{V_o}}.$$

(42)

Now, in the presence of any disturbance, the ratio of the desired average battery voltage to its instantaneous voltage is expected to be equivalent to the ratio of the desired average input voltage to its instantaneous voltage, as given in (43):

$${\displaystyle \frac{{V_o}^{*}}{V_o}}={\displaystyle \frac{{V_{in}}^{*}}{V_{in}}}⟹V_{in}={\displaystyle \frac{{V_{in}}^{*}{V}_o}{{V_o}^{*}}}.$$

(43)

Inserting (43) into (42) leads to

$$i_o={\displaystyle \frac{{V_{in}}^{*}{i}_{in}}{{V_o}^{*}}}.$$

(44)

(41) and (44) can then be inserted into (40) to obtain the complete robust feedforward expression in (45), which effectively eliminates system disturbances and enhances system dynamics:

$${i_{of3}}^{*}=\left({\displaystyle \frac{{V_o}^{*}}{V_o}}\right)\left[\left({\displaystyle \frac{i_o}{V_{in}}}\right){V_o}^{*}-i_{in}\right]+{\displaystyle \frac{{V_{in}}^{*}{i}_{in}}{{V_o}^{*}}}.$$

(45)

6. Numerical Simulations

To validate the different SPS-based DAB LADR control techniques proposed in this paper, MATLAB Simulink was used. The parameters of the two loop controllers together with the feedforward constants are given in

Table 2. Controller parameters.

Description	Parameter	Value
LADRC for output current	${\omega }_{c1},{\omega }_{o1}$	2000 rad/s, 6000 rad/s
LADRC for output voltage	${\omega }_{c2},{\omega }_{o2}$	500 rad/s, 3000 rad/s
Feedforward parameters	α, β	15, 6

.

The battery charging is performed by first charging the battery at a constant current of 270 A, and then a constant voltage of 904 V. Figure 8 shows the impact of the proposed feedforward techniques on the battery’s voltage and current response during the transition between CC and constant CV charging modes. Both scenarios, with and without the techniques, follow the same charging profile. Initially, the battery undergoes a constant current charging phase (0–7.5 ms) where the current reaches its desired value of 270 A while the voltage remains unregulated. At t = 7.5 milliseconds, the charging mode switches to CV. Here, the control system begins regulating the battery voltage to a target value of 904 V.

Without the proposed feedforward techniques, a significant challenge arises during the CC to CV transition. The plot reveals a noticeable dip in both the battery current and voltage. This dip, lasting roughly 4 ms, can negatively impact battery health and system stability by introducing sudden changes during a critical phase. In stark contrast, the plot with the droop-based feedforward technique implemented (using approach 3) showcases a remarkably smooth and seamless process, taking only about 1.5 ms, significantly faster than the scenario without the technique. This smoother transition not only minimizes stress on the system components but also potentially improves battery charging efficiency by avoiding unnecessary fluctuations.

In Figure 8, there is also an input voltage variation that takes place at t = 15 ms. This constitutes a disturbance to the system. It can be observed that without the feedforward technique, there is a small undershoot in both the battery voltage and current and it takes about 3 ms for the controller to regain control. However, in the presence of the proposed feedforward technique, this input disturbance is almost instantly eliminated and observed. This shows the robustness of the LARDC in combination with the proposed droop-based feedforward techniques.

The overall effect of the switch from CC to CV on the converter is shown in the input current and inductor current waveforms of Figure 9. A large dip in both the inductor and input currents is noticed in the absence of the proposed technique as well as the gradual progression to steady state when an input voltage disturbance takes place at t = 15 ms. In contrast, a very smooth transition in the waveforms is observed when the proposed feedforward technique is applied in the control loop. It is also noticed how the controller reacts almost instantly when there is an input voltage change at t = 15 ms shown by the instantaneous rise in the current amplitudes at this time.

Next, the three proposed feedforward techniques are compared as shown in Figure 10. All three proposed feedforward approaches demonstrate the ability to switch quickly and seamlessly from CC to CV modes, hence avoiding the current and voltage dips as well as long transients typically seen in this kind of system. It should also be noted that approaches 1 and 2 have better dynamic stability in comparison to approach 3 when used in conjunction with LADRC. An optimum stability point for all three approaches will require proper tuning of the LADRC bandwidth parameters, which is not the focus of this paper. Moreover, the oscillatory effect, that may potentially appear due to improper tuning of the bandwidth parameters when approach 3 is used, can be eliminated with the use of a non-linear active disturbance rejection controller.

The robustness of the proposed feedforward methods given by (35), (36), and (45) is compared with the conventional approach based on the input–output port power balance approach given by (34). The results obtained for charging the battery both with approach 3 denoted by (45) and the conventional approach are shown in Figure 11. In both cases, the battery is first charged at CC of 270 A after which the control switches to CV of 904 V. Moreover, during the CV stage, an input voltage disturbance occurs at t = 15 ms. The proposed approach smoothly transitions from CC to CV and the voltage immediately becomes controlled and continues being controlled even in the presence of the input voltage disturbance. On the contrary, during the switch from CC to CV, the conventional approach cannot effectively control the voltage to its desired value even though the voltage dips found in Figure 8 due to the absence of a feedforward term are absent. However, after the input voltage changes at t = 15 ms, the conventional approach eventually starts controlling the voltage. This shows that the conventional approach leads to the DAB converter stability at specific operating points whereas with the proposed approach, the converter stability is guaranteed at all operating points within the battery limits; thus, it is able to deal with the highly non-linear nature of the converter. The same behavior is noticed with the battery current.

7. Conclusions

This work methodologically presented the design and dynamic control of a dual active bridge (DAB)-based battery charging system using linear active disturbance rejection control (LADRC). Starting from the fundamental operating principle of a DAB converter under single phase shift (SPS) modulation, the core principles of LADRC were elucidated. Building on these foundations, a detailed modeling approach was developed, enabling effective control of various parameters within the proposed cascaded two-loop control system. Furthermore, three distinct feedforward approaches were proposed to smooth the transition process from constant current (CC) to constant voltage (CV) during the battery charging process. Two of these approaches leverage the system’s droop-like properties. Numerical simulations demonstrated the effectiveness of these three approaches, highlighting their potential to enhance significantly battery charging performance.

In the introduction, several existing control strategies, including PI, fuzzy logic, MPC, SMC, and PBC, were explored in the context of DAB converters. However, these approaches suffer from limitations such as poor disturbance rejection, complex design processes, or undesirable side effects like chattering. The proposed droop-based LADRC method addresses these limitations by offering a simpler, more robust solution that enhances system stability and dynamic performance, particularly during the critical CC to CV charging transition. Simulation results presented in Figure 8, Figure 9, Figure 10 and Figure 11 confirmed that this method not only ensures smooth transitions but also maintains system robustness at all operating points even under input voltage disturbances. This is in contrast to conventional methods that either focus on single loop battery charging or introduce significant complexity and can moreover lose effectiveness during specific load conditions.

Future work will investigate the integration of non-linear active disturbance rejection control to enhance system performance further. Despite the introduction of non-linear control, the proposed smoothing loops will continue to be essential for ensuring optimal battery charging dynamics. This research paves the way for more efficient and reliable battery charging systems, which are crucial for the widespread adoption of electric vehicles.