Reflux Power Optimization of a Dual-Active Hybrid Full-Bridge Converter Based on Active Disturbance Rejection Control

Abstract

The dual-active hybrid full-bridge (H-FDAB) DC–DC converter has great potential in medium-voltage high-power photovoltaic power station applications by introducing a three-level bridge arm to increase the output voltage range. However, its mathematical model and optimum modulation schemes have not been fully explored. Under the traditional PI control, the H-FDAB DC–DC converter will produce significant reflux power, which will lead to a decrease in converter efficiency and output voltage fluctuation. On this basis, this paper proposes a reflux power optimization strategy for an H-FDAB DC-DC converter based on active disturbance rejection control (ADRC). Firstly, the structure and power characteristics of the H-FDAB DC–DC converter are analyzed, and the relationship among the reflux power, the transmission power, and the phase shift angle is derived. Secondly, to reduce the complexity of the control calculation, upon the foundation of dual phase-shifting modulation, the Karush–Kuhn–Tucker (KKT) condition is used to solve for the phase shift angle that corresponds to the minimum reflux power. Simultaneously, we develop an ADRC loop utilizing an extended state observer (ESO) for the real-time estimation of system states. We also consider the sudden changes in input voltage, load switching, and transmission power fluctuations caused by reflux power optimization strategies as system disturbances and compensate for them accordingly. Finally, the experiments conclusively validate the designed control strategy’s correctness and feasibility.

1. Introduction

With the rapid development of new power systems, the interconnection between energy systems and the concept of energy efficiency have attracted more and more attention. The photovoltaic DC microgrid system based on renewable energy has been rapidly developed [1]. The bidirectional converter is the core device for the photovoltaic DC microgrid to realize voltage conversion; power transmission occurs bidirectionally, as well as the improvement in the utilization rate of electrical energy by equipment and load. Among many converter topologies, the dual-active-bridge (DAB) converter has attracted wide attention due to its advantages of simple control, electrical isolation, bidirectional energy transmission, and soft switching characteristics [2]. The conversion efficiency of DAB is one of the key problems of the photovoltaic DC microgrid. The reflux power generated during the operation of the converter will seriously affect the working efficiency of the system. Therefore, suppressing the reflux power is of great significance for improving the conversion efficiency of the converter during the operation of the photovoltaic microgrid.

Many researchers have studied the operating state of two-level DAB under different control strategies. Among these, the single-phase shift (SPS) control method is a prevalent application in DAB because of its advantages of simple control and the easy realization of soft switching. However, under this control strategy, the converter has only one degree of freedom. When the transmission voltage does not match, SPS control has problems, such as difficulty in realizing soft switching of the converter, serious power loss, and low transmission efficiency [3]. To address these issues effectively, in references [4,5,6], various multi-phase shift (MPS) symmetric control strategies are proposed, like double-phase shift (DPS), extended-phase shift (EPS), triple-phase shift (TPS), and five-level control. Moreover, beyond the aforementioned symmetrical control methodologies, some asymmetric control schemes can be used, such as changing the voltage at both ends of the high-frequency transformer of the converter, by modifying the duty cycle. This scheme can improve the light load efficiency of DAB by realizing soft switching and optimizing the reflux power [7,8,9].

The abovementioned studies only focus on the steady-state performance of the converter. In reference [10], a multi-port railway power conditioner (MP-RPC) with a renewable energy system (RES) is proposed for three-port isolated DC–DC converters, and small signal modeling and analysis are carried out. A power decoupling control strategy with a feedforward response is proposed to improve the dynamic performance of the system and the adaptability of control parameters and promote the complementarity and mutual benefit of the railway system and RES. However, multi-port design and complex control strategies increase the complexity of the system, and high-performance isolated DC–DC converters and control devices may increase the overall cost of the system. Reference [11] proposed an optimized power flow control strategy to achieve real-time control of the sectioning post-energy storage system (SPESS). It realizes the optimal power flow management and control and has economic superiority. However, the coordinated control of traction substations (TSS) and SPESS in multiple traction substations is involved, and the system design and implementation are complex. A sliding mode control proposed in the literature [12] has better steady-state and dynamic performance than traditional PID control, but its switching frequency is unstable, which can easily lead to loss and electromagnetic interference problems. In reference [13], aiming at the nonlinear characteristics of the DAB converter, a fuzzy logic controller is proposed to control the output voltage of the DAB converter. Although it has certain advantages, its calculation demand is high, which not only affects the response speed of the converter in the sudden change in load but also increases the cost of the converter. Reference [14] proposed a model predictive control strategy based on triple-phase shift, which optimizes loss while solving the port power coupling problem, but there is also the problem of a large number of calculations. In the control strategy of a phase-shifted full-bridge converter, the backpropagation (BP) neural network and particle swarm optimization BP neural network are used to realize various double-closed-loop PI control effects, which have the advantages of fast response speed and small overshoot. However, BP itself has the disadvantages of slow convergence speed and the fact that it is easy to fall into the local minimum, which is not suitable for industrial applications. Although these methods are superior to traditional PI control, their application is still limited because they rely too much on the parameters of the control system model and the natural frequency.

In order to solve the abovementioned problems and improve the response speed of the system, this paper proposes an ADRC method. ADRC does not depend on the accurate internal information of the object and has the advantages of a simple structure and strong adaptability [15]. It can solve the contradiction between the fast response and overshoot of the PI controller well and has the characteristics of good dynamic response and strong anti-interference ability. Parameter tuning is an important issue in ADRC design. Reference [16] proposed to linearize the traditional ADRC to reduce the number of adjustable parameters and give a parameter tuning method, which inherits the advantages of the original controller. In reference [17], ADRC is applied to the control of a distribution network inverter, which verifies that the stability of the ADRC controller is good and that the controller has dynamic tracking response speed and anti-interference ability. In references [18,19], ADRC was applied to the servo system, and the speed controller and current controller of the permanent magnet synchronous motor in the servo system were designed to achieve a high dynamic response, strong anti-interference, and noise suppression of the controller. In the abovementioned study, the advantages of ADRC technology have been fully verified.

If DAB is better applied to the medium-voltage DC (MVDC) system, as shown in Figure 1, a three-level bridge arm can be introduced into DAB [20] for mitigating the voltage stress on the switch tube of the three-level converter. At the same time, the switch tube with a lower voltage withstand level can be selected to reduce the cost when selecting the device. The H-FDAB DC–DC converter is composed of a primary two-level H-bridge and a secondary three-level NPC bridge. Compared to the conventional two-level DAB (2L-DAB), the H-FDAB DC–DC converter exhibits superior voltage-blocking capabilities and achieves a higher boost ratio while offering more degrees of freedom to refine conversion efficiency and improve the dynamic performance of the DAB system [21]. In addition, compared with the cascaded converter applied to the MVDC system, the H-FDAB DC–DC converter efficiently minimizes the requirement for components such as semiconductors and transformers, thereby streamlining the overall system architecture, and avoids problems such as uneven power distribution and power circulation between different modules, thereby simplifying control.

In summary, when designing the control system of the H-FDAB DC-DC converter, it is not only required that the output side voltage tracks the given value quickly but also required that the system has a certain anti-disturbance ability when disturbances occur on the load side and the power side. The ADRC strategy has the advantage of not relying on accurate mathematical models, coupled with robust immunity, which enhances its applicability and reliability and provides a good solution for the control strategy of these strongly nonlinear components of the converter itself. Based on this, on the basis of suppressing the reflux power of the H-FDAB DC–DC converter, this paper overcomes the limitations of traditional control methods in dealing with complex nonlinear dynamics and disturbances and improves the dynamic response speed and work efficiency of the H-FDAB DC–DC converter.

2. Modeling of the Dual-Active Hybrid Full-Bridge Converter

Figure 2 depicts the topological structure of the H-FDAB DC–DC converter. It consists of a high-frequency transformer, an inductance L_s (external inductance plus transformer leakage inductance), and an H-bridge circuit on the primary and secondary sides. In the diagram, C₁ is the primary side DC-blocking capacitor and C₄ is the secondary side DC-blocking voltage stabilizing capacitor. V₁ is the DC bus voltage, and V₂ is the output voltage of the converter. Among these, V_H1 and V_H2 are the midpoint voltages of the two respective bridge arms, I_L is the inductor current, R is the side load, and n is the ratio of the high-frequency transformer.

Figure 3 shows the working waveform of the converter under DPS control. D₁ is the in-bridge phase shift angle of the switch tube S₁ leading S₄ and Q₁Q₂ leading Q₇Q₈ (D₁ = φ₁/π), and D₂ is the inter-bridge phase shift angle of the switch tube S₁ leading Q₁Q₃ (D₂ = φ₂/π). According to reference [22], the phase shift angle D₁ and D₂ need to satisfy D₁ + D₂ < 1, otherwise, the transmission power of the converter cannot be controlled by D₁ and D₂. According to the relationship between D₁ and D₂, DPS control is divided into the following two working modes: mode 1: 1 > D₂ > D₁ > 0 and mode 2: 1 > D₁ > D₂ > 0.

When the converter operates at a steady state over a single switching cycle, the inductor current exhibits symmetry, and its average value equals zero, as follows:

$$\left\{\begin{array}{l}{I}_L\left(t_0\right)=-I_L\left(t_4\right)\\ I_L\left(t_1\right)=-I_L\left(t_5\right)\\ I_L\left(t_2\right)=-I_L\left(t_6\right)\\ I_L\left(t_3\right)=-I_L\left(t_7\right)\end{array}\right.$$

(1)

In the H-FDAB DC–DC converter under DPS control, the expression of the half-switching cycle inductor current in mode 1 is as follows:

$$\left\{\begin{array}{l}{i}_{\mathrm{L}}\left(t_0\right)=\left[n{V}_1\left(D_1-1\right)-V_2\left(D_1+2D_2-1\right)\right]/\left(4n{L}_{\mathrm{r}}{f}_{\mathrm{s}}\right)\\ i_{\mathrm{L}}\left(t_1\right)=\left[n{V}_1(D_1-1)-V_2(2D_2-1-D_1)\right]/\left(4n{L}_{\mathrm{r}}{f}_{\mathrm{s}}\right)\\ i_{\mathrm{L}}\left(t_2\right)=\left[n{V}_1(2D_2-1-D_1)-V_2(D_1-1)\right]/\left(4n{L}_{\mathrm{r}}{f}_{\mathrm{s}}\right)\\ i_{\mathrm{L}}\left(t_3\right)=\left[n{V}_1(D_1+2D_2-1)-V_2(D_1-1)\right]/\left(4n{L}_{\mathrm{r}}{f}_{\mathrm{s}}\right)\end{array}\right.$$

(2)

The term f_s represents the switching frequency employed within the H-FDAB DC–DC converter. Similarly, under DPS control, the current expression of the H-FDAB DC–DC converter operating in mode 2 is as follows:

$$\left\{\begin{array}{l}{i}_{\mathrm{L}}\left(t_0\right)=\left[\begin{array}{c}n{V}_1(D_1-1)-V_2(D_1+2D_2-1)\end{array}\right]/(4n{L}_{\mathrm{r}}{f}_{\mathrm{s}})\\ i_{\mathrm{L}}\left(t_1\right)=\left[\begin{array}{c}n{V}_1(D_1-1)+V_2(1-D_1)\end{array}\right]/(4n{L}_{\mathrm{r}}{f}_{\mathrm{s}})\\ i_{\mathrm{L}}\left(t_2\right)=\left[\begin{array}{c}n{V}_1(D_1-1)+V_2(D_2+1-D_1)\end{array}\right]/(4n{L}_{\mathrm{r}}{f}_{\mathrm{s}})\\ i_{\mathrm{L}}\left(t_3\right)=\left[\begin{array}{c}n{V}_1(D_1+2D_2-1)+V_2(1-D_1)\end{array}\right]/(4n{L}_{\mathrm{r}}{f}_{\mathrm{s}})\end{array}\right.$$

(3)

The average transmission power P of the H-FDAB DC–DC converter is as follows:

$$P={\displaystyle \frac{1}{T_h}}{\displaystyle {\int }_{t_0}^{t_0+T_h}{V}_{H1}}{I}_L\mathrm{d}t$$

(4)

For the sake of simplicity and clarity in analysis, according to the maximum transmission power $Q_N=P_N=(n{V}_1{V}_2/8L_r{f}_{\mathrm{s}})$ of the DAB under SPS control fiducial value, the transmission power per unit value P₀ of the H-FDAB DC–DC converter in a switching cycle is as follows:

$$P_0={\displaystyle \frac{P}{P_N}}=\left\{\begin{array}{ll}2(2D_1-2{D_1}^2-{D_2}^2)\hfill & 1>D_2>D_1>0\hfill \\ 2(2D_1-{D_1}^2-2D_1{D}_2)\hfill & 1>D_1>D_2>0\hfill \end{array}\right.$$

(5)

Figure 4 presents the variation in the transmission power of the H-FDAB DC-DC converter as a function of the phase shift angle.

Similarly, the reflux power per unit value Q of the H-FDAB DC-DC converter is as follows:

$$Q={\displaystyle \frac{{\left[K(1-D_1)+(2D_2-2D_1-1)\right]}^2}{2(K+1)}}$$

(6)

The voltage regulation ratio is defined as K = nV₁/(V₂). According to Formula (5), there is a coupling relationship between transmission power P₀ and D₁, D₂. Therefore, optimizing the reflux power necessitates an adjustment in the internal phase shift angle, which in turn, impacts the stability of the output voltage, ultimately affecting the overall system stability. Addressing these issues, this paper proposes an optimization design for the reflux power in the H-FDAB converter based on ADRC. The objective is to solve the output voltage fluctuations and improve the slow dynamic response caused by the reflux power optimization in the converter.

2.1. Soft Switching Characteristics of H-FDAB DC–DC Converter

Since the topology proposed in this paper is suitable for the boost scenario, only mode 1 of the H-FDAB DC–DC converter working in the BOOST mode is discussed, that is, when K < 1. Its mode 2 analysis method is the same, and this article does not elaborate.

When I_L is negative at time t₁ and the inductor possesses sufficient energy to effectively charge and subsequently discharge the parasitic capacitance, switches S₁, S₄, Q₅Q₆, and Q₃Q₄ can achieve zero voltage switching (ZVS). When the inductor current is positive at t₅, switches S₂, S₃, Q₁Q₂, and Q₇Q₈ can achieve ZVS characteristics. The ZVS current boundary constraints on both sides of the H-FDAB DC–DC converter are as follows:

$$\left\{\begin{array}{l}{i}_{\mathrm{L}}\left(t_1\right)\le 0\\ i_{\mathrm{L}}\left(t_5\right)\ge 0\end{array}\right.$$

(7)

Substituting Formula (7) into Formula (2), we can obtain the following:

$$\left\{\begin{array}{l}\left[n{V}_1(D_1-1)-V_2(-D_1+2D_2-1)\right]/(4n{L}_{\mathrm{r}}{f}_s)\le 0\\ \left[n{V}_1(-D_1+2D_2-1)-V_2(D_1-1)\right]/(4n{L}_{\mathrm{r}}{f}_s)\ge 0\end{array}\right.$$

(8)

When Formula (8) is normalized, we can obtain the following:

$$\left\{\begin{array}{l}{\displaystyle \frac{(K+2)D_1-K+1}{2}}⩽D_2\\ {\displaystyle \frac{K{D}_1+K-1}{2K}}⩽D_2\end{array}\right.$$

(9)

2.2. Optimization of Reflux Power Based on DPS

Under DPS control, when the product of I_L and V_H1 is less than zero, it indicates that the transmission power is transmitted from the secondary side to the primary side, which is defined as the reflux power. The objective of this paper is to find the best shift ratio combination to minimize the reflux power. When the H-FDAB DC–DC converter has inequality constraints, to find the optimal solution of the global shift ratio combination under DPS control, the traditional extreme value method becomes complicated. Therefore, the KKT condition is adopted in this paper. By taking Formula (6) as the objective function and adding soft switching constraints, the optimization problem for minimizing the reflux power can be described as follows:

$$\left\{\begin{array}{l}min\hspace{1em}Q(X)\hspace{1em}\\ P(X)-P_0*=0\\ f_i(X)\le 0,\hspace{1em}i=1,2,\dots ,m\end{array}\right.$$

(10)

The control variables are represented by the tuple X = (D₁, D₂), with P₀* denoting the reference transmission power. The set of constraints is defined as f_i(X), ensuring adherence to operational specifications. The numerical solution is usually applied to the transmission power or output voltage changes in a very narrow range. Otherwise, additional memory will be required for the heavy computational burden of the microcontroller used to pre-store the numerical solution. On the other hand, the analytical solution represented by transmission power P and K can enable the automatic online optimization of control in response to varying operating conditions, thereby ensuring optimal performance. Therefore, the analytical solution is more suitable for practical applications. The KKT condition is usually used to obtain the analytical solution of the optimization problem, which can be succinctly characterized as follows:

$$\left\{\begin{array}{l}E(X,\lambda ,{\mu }_i)=Q(X)+\lambda (P(X)-P_0*)+{\displaystyle \sum _{i=1}^m{\mu }_i}{f}_i(X)\hfill \\ {\displaystyle \frac{\partial E}{\partial X}}{|}_{X=X^{*}}=0, {\mu }_i{f}_i(X^{*})=0, f_i(X^{*})\le 0, \lambda \ne 0, {\mu }_i\ge 0\hfill \end{array}\right.$$

(11)

In the Lagrange quantity, denoted as E(X, λ, μ_i), λ is the constraint coefficient of transmission power in brackets; μ is the constraint coefficient corresponding to f; P₀* is the per-unit value of transmission power; and Q(X) is the set of equality constraints composed of the normalized reflux power model. P(X) is the equality constraint condition of the mathematical model of the normalized power, and f_i(X) is an inequality constraint condition composed of the internal and external phase shift angle relations. The local optimum of each working condition is pre-screened by numerical solution, and the global optimum candidates in different power domains are located, so as to simplify the analytical solution process, as shown in Figure 5.

Substituting Equations (5), (6), (9), and (10) into Equation (11), the shift ratio calculation formula for minimizing the reflux power can be obtained as follows:

$$\left\{\begin{array}{l}L={\displaystyle \frac{2{\left[K\left(1-D_1\right)+\left(2D_2-2D_1-1\right)\right]}^2}{2(K+1)}}+{\mu }_1\left(D_1-D_2\right)\\ +{\mu }_2\left[\left(D_1-1\right)-K\left(D_1+2D_2-1\right)\right]+{\mu }_3(D_2-1)\\ +{\mu }_4(-D_1)+{\mu }_5\left[K\left(D_1-1\right)-\left(D_1+2D_2-1\right)\right]\\ +\lambda \left(4D_2-4D_2^2-2D_1^2-P_0\right)\\ \lambda \ne 0,{\mu }_1,{\mu }_2,{\mu }_3,{\mu }_4,{\mu }_5\ge 0\\ {\displaystyle \frac{\partial L}{\partial D_1}}=0,{\displaystyle \frac{\partial L}{\partial D_2}}=0\\ D_1-D_2\le 0,D_2-1\le 0,-D_1\le 0\\ K\left(D_1-1\right)-\left(D_1+2D_2-1\right)\le 0\\ \left(D_1-1\right)-k\left(D_1+2D_2-1\right)\le 0\\ 4D_2-4D_2^2-2D_1^2-P_0=0\\ {\mu }_1\left(D_1-D_2\right)=0,{\mu }_2\left[\left(D_1-1\right)-K\left(D_1+2D_2-1\right)\right]=0\\ {\mu }_3(D_2-1)=0,{\mu }_4\left(-D_1\right)=0\\ {\mu }_5\left[K\left(D_1-1\right)-\left(D_1+2D_2-1\right)\right]=0\end{array}\right.$$

(12)

According to Formula (12) (The detailed derivation process is shown in Appendix A.1), the phase shift angle expression for minimizing the reflux power can be derived as follows:

$$\left\{\begin{array}{l}{D}_1={\displaystyle \frac{3+K-\sqrt{21K^2+20P_0+26K+4}}{10}}\\ D_2={\displaystyle \frac{1-K+(K+2)D_1}{2}}\end{array}\right.$$

(13)

Substituting D₁ and D₂ into Formula (6), the minimum reflux power Q_min under the DPS control is as follows:

$$Q_{min}=0$$

(14)

Since reflux power is a useless power, the actual power transmitted by the H-FDAB DC–DC converter exceeds the power required by the load, which will generate unnecessary energy loss and reduce the transmission efficiency of the converter. The control strategy in this paper can suppress the reflux power, reduce the switching loss, and improve the conversion efficiency, and the control method is simple.

3. Control Algorithm

3.1. Traditional PI Control Algorithm

In order to maintain the stability of the output voltage. The PI control of the H-FDAB DC–DC converter obtains the error signal e(t) according to the comparison between the reference voltage U_ref and the actual output voltage signal V₂, which is obtained by the linear superposition of the proportional link K_p and the integral link K_i/s.

$$u(t)={K_p}^{*}e(t)+{K_i}^{*}{\displaystyle {\int }_0^{t}e}(t)dt$$

(15)

The transfer function of Equation (13) after the Laplace transform can be expressed as follows:

$$G_c(s)={\displaystyle \frac{u(s)}{e(s)}}=K_p+{\displaystyle \frac{K_i}{s}}$$

(16)

The initial parameters of K_p and K_i in Equation (16) need to be adjusted. In this paper, the Ziegler–Nichols (ZN) parameter setting method [23] in the critical proportional setting method is used to obtain K_p = 6.01 and K_i = 10.5.

The reflux power optimization control block diagram under the traditional DPS control is shown in Figure 6. The reflux power optimization control adopts the pre-established reflux power model, combined with the phase shift angle D₁ in the output bridge and the phase shift angle D₂ between the output bridges adjusted by the PI controller, to achieve the optimization of the reflux power while maintaining the stability of the output voltage. However, when the system is subject to large disturbances or the reference voltage changes abruptly, the pi control integral term may accumulate rapidly and reach saturation, resulting in the controller output exceeding the limit range of the actuator. This not only cannot effectively suppress the disturbance but may aggravate the instability of the system and affect the stability of the output voltage. When the operation has significantly deviated from the original design point, the transmission power will change, and the interaction between the proportional term and the integral term may lead to overshooting or the oscillation of the system. In the H-FDAB DC–DC converter, this may affect the optimization effect of the reflux power, reduce the overall efficiency of the system, and increase the fluctuation of the output voltage.

3.2. Active Disturbance Rejection Control Algorithm

To reduce the reflux power of the H-FDAB DC–DC converter and optimize its dynamic performance, this paper introduces the ADRC under the DPS control. The extended state observer facilitates the real-time estimation of the output voltage. The change in the internal shift angle D₁ caused by the reflux power optimization is regarded as the system disturbance, and the feedforward compensation of the ADRC is used to obtain an ADRC method for the dual-active hybrid full-bridge converter combined with the reflux power optimization. According to reference [24], The small-signal model of the H-FDAB DC-DC converter is shown in Figure 7:

The differential equation of the output voltage is as follows:

$$C_4{\displaystyle \frac{\mathrm{d}{\tilde{u}}_{\mathrm{o}}}{\mathrm{d}t}}={\displaystyle \frac{n{U}_{\mathrm{in}}\left(1-2D_2\right)}{2L{f}_{\mathrm{s}}}}{\tilde{d}}_2-{\displaystyle \frac{{\tilde{u}}_{\mathrm{o}}}{R}}$$

(17)

Among these, ${\tilde{u}}_0$ represents the small signal model of the output voltage and ${\tilde{d}}_2$ is the small signal value of the inter-bridge phase shift angle D₂, and Formula (17) is written as follows:

$$\dot{y}=-a_{1}y+w+bu$$

(18)

The primary control objective of this paper is to guarantee stability and consistency in the output voltage. The key control variable revolves around the external duty cycle shift relative to D₂, with y designated as the output voltage of the converter. The output u of the controller, on the other hand, represents the internal duty cycle shift in relation to D₁. Within this system, w encompasses both internal and external disturbances, while a₁ represents the internal, yet unknown, parameters of the H-FDAB DC–DC converter. Notably, b is the input control gain and partially known (b₀ = nV₁ (1 − 2D₁)/2fsLs), then the above formula can be simplified to the following:

$$\dot{y}=-a_{1}y+w+(b-b_0)u+b_{0}u=f+b_{0}u$$

(19)

The abovementioned equation includes the internal disturbance caused by the uncertainty of the internal parameters of the system and the external disturbance caused by the environmental factors of the system, where $f=-a_{1}y+w+(b-b_0)u$. Although b₀ contains the inductance parameter Ls, the influence of inductance can be ignored due to the compensation effect of the total disturbance. The H-FDAB DC–DC converter is taken as the control object. The control block diagram of the ADRC algorithm under DPS control is shown in Figure 8. N₁ and N₂ are the external and internal disturbances of the H-FDAB DC–DC converter.

The expression of the first-order tracking differentiator (TD) is as follows:

$$\left\{\begin{array}{l}{e}_1=v_1-U_{\mathrm{ref}}\hfill \\ v=-r{e}_1\hfill \end{array}\right.$$

(20)

The ESO expression is as follows:

$$\left\{\begin{array}{l}{\dot{z}}_1=z_2-{\beta }_1(z_1-y)+b_{0}u\\ {\dot{z}}_2=z_3-{\beta }_2(z_1-y)\\ {\dot{z}}_3=-{\beta }_3(z_1-y)\end{array}\right.$$

(21)

The expression of the linear state error feedback (LSEF) controller is as follows:

$$\left\{\begin{array}{l}{e}_3={\nu }_1-z_1\\ u={\displaystyle \frac{u_0-z_2}{b}}\\ u_0=k{e}_3\end{array}\right.$$

(22)

$$u_0=k({\nu }_1-z_1)$$

(23)

In Formula (23), u₀ is expressed as the output term of LSEF, and k is expressed as the gain of the proportional controller. If the estimation error of z₂ for the total disturbance is neglected, the substitution of Equation (23) into Equation (19) can be reduced to an integral part, as follows:

$$\dot{y}=(f(y,w)-z_2)+u_0\approx u_0$$

(24)

According to Formulas (21) and (24), the closed-loop transfer function of the H-FDAB DC–DC converter can be sorted out as follows:

$$\phi (s)={\displaystyle \frac{k}{s+k}}={\displaystyle \frac{1}{1/k\cdot s+1}}$$

(25)

It can be seen from Formula (25) that the bandwidth of LSEF is ω_c = k, so in order to finally make the system stable, a better proportional gain value should be selected.

Among these, β₁, β₂, and β₃ are used to improve the observation gain of ESO, and e₁ and e₃ are the observation errors of ESO. z₁ is the observation output y, z₂ is the observation total disturbance f, and z₃ is the observation total disturbance differential.

From Formula (20) (The detailed derivation process is shown in Appendix A.2), the output z₁ and z₂ of the observer can be obtained using Laplace transform, as follows:

$$\left\{\begin{array}{l}{z}_1={\displaystyle \frac{({\beta }_2+{\beta }_{1}s)s+{\beta }_3}{s^3+{\beta }_1{s}^2+{\beta }_{2}s+{\beta }_3}}y(s)-{\displaystyle \frac{b_0{s}^2}{s^3+{\beta }_1{s}^2+{\beta }_{2}s+{\beta }_3}}u(s)\\ z_2={\displaystyle \frac{({\beta }_3+{\beta }_{2}s)s}{s^3+{\beta }_1{s}^2+{\beta }_{2}s+{\beta }_3}}y(s)-{\displaystyle \frac{({\beta }_3+{\beta }_{2}s)b_0}{s^3+{\beta }_1{s}^2+{\beta }_{2}s+{\beta }_3}}u(s)\end{array}\right.$$

(26)

According to Formulas (19) and (21), the transfer function of the observation z₂ relative to the disturbance f can be obtained as follows:

$${\varphi }^{\prime }(s)={\displaystyle \frac{z_2}{f}}={\displaystyle \frac{({\beta }_3+{\beta }_{2}s)}{s^3+{\beta }_1{s}^2+{\beta }_{2}s+{\beta }_3}}$$

(27)

The pole assignment method is to configure the pole at ω₀. Compared with the traditional ESO, it does not increase the adjustable parameters, and the difficulty of the engineering setting is similar to that of the traditional ESO. In this way, the observation ability of ESO is improved without increasing the difficulty of the engineering setting. According to the pole assignment method, the observer bandwidth configuration is as follows:

$$s^3+{\beta }_1{s}^2+{\beta }_{2}s+{\beta }_3={(s+{\omega }_0)}^3$$

(28)

$$\left\{\begin{array}{l}{\beta }_1=3{\omega }_0\hfill \\ {\beta }_2=3{\omega }_0^2\hfill \\ {\beta }_3={\omega }_0^3\hfill \end{array}\right.$$

(29)

Substituting Equation (29) into Equation (26), the expressions of z₁ and z₂ with respect to ω₀ can be obtained as follows:

$$\left\{\begin{array}{l}{z}_1={\displaystyle \frac{3{\omega }_0{s}^2+3{\omega }_0^{2}s+{\omega }_0^3}{{\left(s+{\omega }_0\right)}^3}}y(s)+{\displaystyle \frac{b_0{s}^2}{{\left(s+{\omega }_0\right)}^3}}u(s)\\ z_2={\displaystyle \frac{(3{\omega }_0^{2}s+{\omega }_0^3)s}{{\left(s+{\omega }_0\right)}^3}}y(s)-{\displaystyle \frac{({\omega }_0^3+3{\omega }_0^{2}s)b_0}{{\left(s+{\omega }_0\right)}^3}}u(s)\end{array}\right.$$

(30)

To verify the control performance of ADRC, the open loop transfer function Bode diagram of the system under ADRC and PI control is drawn as shown in Figure 9. By comparing the frequency domain characteristics of the ADRC and PI control, it can be seen from Figure 9 that in the low-frequency band, the amplitude of the ADRC curve is larger, the command tracking effect is stronger, and the low-frequency disturbance suppression ability is better; at a high frequency, the amplitude of the ADRC curve is low, and the suppression ability of high-frequency noise is better than that of the PI control.

Figure 10 is the Bode diagram of the transfer function of z₂ relative to the perturbation f under different bandwidths, as shown in Figure 10.

Here, the influence of the noise δ_y observer of observation y is considered. The transfer function of the observation noise δ_y obtained from Equation (30) is as follows:

$${\displaystyle \frac{z_1}{{\delta }_y}}={\displaystyle \frac{3{\omega }_0{s}^2+3{\omega }_0^{2}s+{\omega }_0^3}{{\left(s+{\omega }_0\right)}^3}}$$

(31)

From Formula (29), the transfer function Bode diagram can be obtained, as shown in Figure 11.

Due to the negative correlation between the noise immunity and the immunity of the converter, the larger the ω₀ is, the weaker the noise immunity of the ESO is, and the smaller the ω₀ is, the weaker the immunity of the ESO is. Therefore, the selection of ω₀ should be balanced between immunity and noise immunity. From Equations (18) and (23), it can be seen that adjusting controller bandwidth ω_c and observer bandwidth ω₀ can realize the controller parameter design. Observer bandwidth ω₀ is generally taken as 4~10 times of ω_c [25]. Combined with the parameter tuning method in the literature [16], the control effect is optimal when ω_c = 100 and ω₀ = 400.

It can be obtained that the phase shift angle D₂ between the output bridges controlled by ADRC is as follows:

$$D_2={\displaystyle \frac{K_{\mathfrak{p}}(U_{\mathrm{ref}}-z_1)-z_2}{b_0}}$$

(32)

where K_p (U_ref − z₁)/b₀ is the feedback control component, and −z₂/b₀ is the disturbance compensation component. When the ESO observation is accurate, z₁ = V₂, z₂ = f, and z₃ = $\stackrel{·}{f}$. Substituting them into Formula (30), we can obtain the following:

$$D_2={\displaystyle \frac{K_{\mathfrak{p}}\left(U_{\mathrm{ref}}-V_2\right)-f}{b}}$$

(33)

The comprehensive system block diagram depicted in Figure 12 outlines the control scheme adopted in this paper.

4. Experiment Verification Results

To validate the efficacy of the reflux power optimization algorithm presented herein, a low-power experimental prototype with TMS320F28335 produced by Texas Instruments (TI, Dallas, TX, USA) as the control core was built, and four groups of experiments are carried out using the control variable method. For a comparative analysis, this paper juxtaposes the reflux power optimization strategy against the conventional DPS PI control method, examining their performance under varying load sizes and input voltage values. An experimental setup is showcased in Figure 13 and accompanied by its parameter specifications listed in

Table 1. Experimental platform parameters.

System Parameters	Numerical Value
Input voltage	40 V
Output voltage	150 V
Auxiliary inductance	100 μH
Transformer ratio	1:3
Input side capacitance	470 μF
Output side capacitance	300 μF
Switching frequency	10 kHz
Rated load	30 Ω

.

The output voltage comparison of two different control schemes is as follows: when the input voltage V₁ = 40 V, the output voltage V₂ = 150 V, the load R = 30 Ω and K = 0.8, and the output voltage is shown in Figure 14. Figure 14a shows the output voltage curve of the converter under ADRC control, and Figure 14b depicts the performance curve of the converter when the output voltage is operated under PI control. In the figure, the converter startup time under ADRC control is 30 ms, and the overshoot is 0 V. The converter startup time under PI control is 42 ms, and the overshoot is 7 V. Therefore, the dynamic performance of the ADRC control scheme is better than that of the PI control.

The comparison of reflux power under two different control schemes is as follows: when the input voltage V₁ = 40 V, the output voltage V₂ = 150 V, the load R = 30 Ω and K = 0.8, and the reflux power is shown in Figure 15. Figure 15a shows the reflux power curve of the converter under ADRC control, and Figure 15b shows the reflux power curve of the converter under PI control. Under the PI control scheme, the instantaneous reflux power of the converter is 1100 W. Under the ADRC control scheme, the instantaneous reflux power of the converter is 0 W. ADRC control can better suppress the reflux power generation.

Two different control schemes are compared when the load changes, as follows: when the input voltage V₁ = 40 V, the output voltage V₂ = 150 V, and K = 0.8, the experimental waveforms of the output voltage and output current of the converter when the load changes from 30 Ω to 15 Ω are shown in Figure 16. Under ADRC control, the voltage fluctuation is about 2 V, and the adjustment duration for the output voltage is 5 ms. Under PI control, the voltage fluctuation is about 6 V, and the adjustment duration for the output voltage is 30 ms. ADRC control has better anti-interference ability than PI control.

Illustrating the response of the output voltage to a sudden input voltage variation, Figure 17 compares the behavior under two distinct control schemes, as follows: when the output voltage V₂ = 150 V, the load R = 30 Ω, the input voltage V₁ changes from 40 V to 50 V, the output voltage regulation time of the PI control scheme is about 20 ms, and the output voltage fluctuation is about 6.5 V; the adjustment time of the ADRC control scheme is about 15 ms and the voltage fluctuation is about 2 V. Compared with the PI control, the ADRC control has better dynamic response when the input voltage changes abruptly.

In this paper, the driving signal of the switch tube S₁, the voltage at both ends of the switch tube, and the current flowing through the switch tube are shown in Figure 18. The drain source voltage of S₁ has dropped to zero before the corresponding driving voltage rises, achieving ZVS. The analysis of other switches is similar, and this article does not elaborate.

When the input voltage of V₁ = 100 V and R = 30, Ω gradually increases from 40 V to 100 V, and the voltage ratio K gradually increases from 0.8 to 1.6. Figure 19 shows the relationship between the input voltage and the transmission efficiency of the converter. The transmission efficiency of the scheme in this paper is always the highest, and the efficiency of the converter can reach 96.2%.

5. Conclusions

This paper delves into the operational features and principles of the dual-active bridge converter under DPS control, formulating models for both transmission and reflux power. Leveraging the KKT equation, we derive the optimal phase shift angle that minimizes reflux power within DPS operational parameters, enhancing converter efficiency. In addition, to solve the dynamic characteristics and overshoot problems of the converter, this paper adopts the ADRC strategy and compensates for the input voltage, load mutation, and internal phase shift angle change caused by reflux power optimization through ESO to maintain the output voltage stability. The ADRC constructed in this paper only samples the output voltage for the calculation, which has the advantages of strong anti-interference ability and simple control.

(1)

Dual-phase shift control can effectively increase the control freedom of the H-FDAB DC–DC converter, which creates conditions for the optimization of H-FDAB DC–DC converter performance;

(2)

The ADRC model introduced herein efficiently optimizes reflux power during H-FDAB DC–DC converter operation, thereby notably enhancing power transmission efficiency.