A Backflow Power Suppression Strategy for Dual Active Bridge Converter Based on Improved Lagrange Method

Abstract

Dual-active bridge (DAB) converters are receiving increasing attention from researchers as a critical part of the power transmission of energy routers. However, the DAB converter generates a large backflow power in conventional control mode, and when the load is mutated, its output voltage takes longer to return to the reference value accompanied by large fluctuations. To solve the above problems, a hybrid strategy is proposed in this paper to optimize the converter. The mathematical models of the transmitted power and the backflow power were firstly derived through in-depth analysis of the DAB converter under extended-phase-shift (EPS) modulation, and the suppression of the backflow power was performed according to the improved Lagrange method utilized in the obtained results. Moreover, considering the poor dynamic characteristics of DAB converters under PI control, according to the state space average model of output voltage in the paper, a model prediction control equation is established to improve the dynamic response of the converter by predicting the output voltage value at the next moment. The simulation results verify the effectiveness of the optimization strategy presented in the text.

1. Introduction

There has been a rapid development of alternating DC hybrid power networks in recent years. Many countries and regions are accelerating the construction of future energy networks based on conventional AC grids [1,2]. The energy router is considered the key electricity-changing equipment for the future energy network [3,4], as shown in Figure 1. The energy router is designed for energy to flow in both directions, and the intermediate DC link utilizes a bi-directional DC-DC converter. In practical applications, the dual-active bridge (DAB) converter is often used [5,6]. The DAB converter has the advantage of higher power density and easy implementation of soft switches, in addition to meeting the bi-directional flow of energy [7,8]. However, the traditional control methods can result in backflow power, which reduces the efficiency of the DAB converter’s transmission [9].

The DAB converter is commonly operated using phase-shift modulation [10], which can be further categorized into single-phase-shift (SPS) modulation [11], extended-phase-shift (EPS) modulation [12], dual-phase-shift (DPS) modulation [13], and triple-phase-shift (TPS) modulation [14]. SPS modulation is the easiest to implement, as it only requires a single control variable between bridges. Nowadays, SPS modulation is widely utilized in cascade power electronic transformer intermediate DC link power equilibrium control [15] and to eliminate transient DC bias of the DAB converter [16].

SPS modulation only has one control variable. It must ensure the power transmission, which cannot easily inhibit the backflow power at the same time [17]. Although there is a study that redefined and analyzed SPS modulation to reshape DAB working modes from the perspective of underlying circuit elucidation and elaborating the method of the suppression of backflow power, both simulation and experimental results indicate that the backflow power still exists [18].

The KKT condition can be utilized when EPS modulation is used. It can optimize both the current stress and backflow power simultaneously, although the inhibition of backflow power is less effective [19]. Additionally, the gradient descent algorithm can also be utilized to achieve the suppression of the backflow power, but its effectiveness remains uncertain for more step-down applications [20]. Further, DPS can be achieved by increasing the same amount of phase shift in the back bridge as in the front bridge, based on EPS. The optimization of dynamic characteristics can be accomplished by combining dynamic matrix control with conventional DPS modulation [21]. On this basis, the phase-shift mode of the DAB converter’s back bridge under DPS modulation can be improved to achieve simultaneous optimization of current stress and backflow power [22]. Nevertheless, both methods under DPS modulation have limited effectiveness in suppressing backflow power.

Compared to the SPS, EPS, and DPS modulation modes, the TPS modulation mode presents a more complex situation for converter modal analysis due to the numerous combinations among the three control variables. Although it is relatively easy to suppress backflow power, it can be challenging to apply in practical engineering applications [23,24].

The DAB converter is required to ensure that the transmission power is unchanged while suppressing the backflow power. In order to achieve such a constrained minima problem, a widely used approach is the Lagrange method [25,26]. This method has also been successful in reducing current stress within the DAB converter [27,28]. However, traditional Lagrange methods may prove challenging in attaining desired transmission power levels, which often results in the need for introducing a PI controller for stabilizing transmission power. Unfortunately, this leads to a reduction in the dynamic characteristics of the DAB converter [29].

In this paper, in order to better suppress the backflow power, a novel method of backflow power suppression (model predictive control, Lagrange method and extended phase shift, MPCL-EPS) was proposed based on the method of modulation of EPS, which was improved using the traditional Lagrange method and combined with model prediction control. This method also improved the dynamic characteristics of the DAB converter while suppressing the backflow power. Finally, the proposed method was validated using MATLAB/Simulink.

2. Working Principle of the DAB Converter

2.1. Basic Principle

The DAB converter topology is shown in Figure 2. In this figure, HB₁ is the converter front bridge, HB₂ is the converter back bridge, Q₁~Q₈ is a fully controlled switching tube, D₁~D₈ is a diode, T is an isolation transformer, C_in and C_o are input and output capacitors, L is the sum of series inductor and transformer leakage sense, and R is equivalent load. n is the transformer variation ratio, U_in and U_o are the converter input and output voltages, u_ab is HB₁ output voltage, u_cd is transformer side voltage, i_L is inductive current, and I_o is the output load current.

There are two control variables, D₁ and D₂, in the method of modulation of EPS. D₁ is the control variable of phase shift between Q₁ and Q₄. In this paper, only Q₁ is considered to lag behind Q₄; D₂ is the control variable of phase shift between Q₁ and Q₅, and in this paper, only Q₅ is considered to lag behind Q₁. Only D₂ is a control variable under SPS modulation.

The DAB converter can both step-up and step-down. In this paper, only the step-down mode is analyzed, and the step-up mode analysis process is the same. The EPS modulation mode and SPS modulation mode operating waveforms in the step-down mode are shown in Figure 3. D₁ meets with D₂ for 0 ≤ D₁ ≤ 1, 0 ≤ D₂ ≤ 1 and 0 ≤ D₁ + D₂ ≤ 1. Q₁~Q₈ are switching tube driving signal, u_L is inductive voltage, P_SPS and P_EPS are transmitted power under SPS and EPS modulation, and T_hs is half switching cycle.

In order to obtain the mathematical model under SPS and EPS modulation, the working mode of the DAB converter needs to be analyzed. It is known from Figure 3 that SPS modulation lacks the control variable of phase shift D₁ within the HB₁ bridge compared to EPS modulation, so only the analysis of the method of modulation in EPS modulation needs to be performed. Then, let D₁ in the resulting mathematical model be 0; then, the mathematical model under SPS modulation can be obtained. According to Figure 3b, the DAB converter in the next cycle of EPS modulation has eight modes of operation and has symmetry, so the analysis is half a cycle, and the specific switching state is shown in Figure 4.

(1) phase 1: t = t₀~t₁

The switching state in this phase is shown in Figure 4a. Q₂ conducts with Q₄ inside the HB₁ bridge, and Q₆ conducts with Q₇ inside the HB₂ bridge. But, the current i_L flowing through the inductors is negative at this point, so the continuous flow from Q₂ to D₄ inside the HB₁ bridge and D₆ and D₇ inside the HB₂ bridge can be derived, and the expression of the inductive current is as follows:

$$i_{\mathrm{L}}\left(t\right)=i_{\mathrm{L}}\left(t_0\right)+\frac{n{U}_{\mathrm{o}}}{L}\left(t-t_0\right)$$

(1)

(2) phase 2: t = t₁~t₂

The switching state in this phase is shown in Figure 4b. Q₁ conducts with Q₄ inside the HB₁ bridge and Q₆ conducts with Q₇ inside the HB₂ bridge. But, the current i_L flowing through the inductors is still negative at this point, so the continuous flow from D₁ to D₄ inside the HB₁ bridge and D₆ and D₇ inside the HB₂ bridge can be derived, and the expression of the inductive current is as follows:

$$i_{\mathrm{L}}\left(t\right)=i_{\mathrm{L}}\left(t_1\right)+\frac{U_{\mathrm{in}}+n{U}_{\mathrm{o}}}{L}\left(t-t_1\right)$$

(2)

(3) phase 3: t = t₂~t₃

The switching state in this phase is shown in Figure 4c. Q₁ is conducted with Q₄ inside the HB₁ bridge, and Q₆ is conducted with Q₇ inside the HB₂ bridge. At t₂, the flow through the inductive current i_L is negatively changed. So, the transport current inside the HB₁ bridge is conducted with Q₁ and Q₄, and the transport current inside the HB₂ bridge is conducted with Q₆ and Q₇, and the expression of the inductive current is as follows:

$$i_{\mathrm{L}}\left(t\right)=i_{\mathrm{L}}\left(t_2\right)+\frac{U_{\mathrm{in}}+n{U}_{\mathrm{o}}}{L}\left(t-t_2\right)$$

(3)

(4) phase 4: t = t₃~t₄

The switching state in this phase is shown in Figure 4d. Q₁ is conducted with Q₄ inside the HB₁ bridge, and Q₅ is conducted with Q₈ inside the HB₂ bridge. But, the flow through the inductive current i_L is positive, so the transport current is conducted through Q₁ and Q₄ within the HB₁ bridge, and the continuous flow is conducted through D₅ and D₈ within the HB₂ bridge, and the expression of the inductive current is as follows:

$$i_{\mathrm{L}}\left(t\right)=i_{\mathrm{L}}\left(t_3\right)+\frac{U_{\mathrm{in}}-n{U}_{\mathrm{o}}}{L}\left(t-t_3\right)$$

(4)

2.2. Transmitted Power

It is known from the previous analysis that the inductive current i_L has symmetry when the DAB converter is stably operated, so the values of the inductive current at each moment can be given, as shown in (5), where the ratio of the DAB converter is k = U_in/nU_o, and f_s is the switching period.

$$\left\{\begin{array}{l}{i}_{\mathrm{L}}\left(t_0\right)=-\frac{n{U}_{\mathrm{o}}}{4f_{\mathrm{s}}L}\left[k\left(1-D_1\right)+\left(2D_1+2D_2-1\right)\right]\\ i_{\mathrm{L}}\left(t_1\right)=-\frac{n{U}_{\mathrm{o}}}{4f_{\mathrm{s}}L}\left[k\left(1-D_1\right)+\left(2D_2-1\right)\right]\\ i_{\mathrm{L}}\left(t_3\right)=\frac{n{U}_{\mathrm{o}}}{4f_{\mathrm{s}}L}\left[k\left(2D_2+D_1-1\right)+1\right]\\ i_{\mathrm{L}}\left(t_4\right)=\frac{n{U}_{\mathrm{o}}}{4f_{\mathrm{s}}L}\left[k\left(1-D_1\right)+\left(2D_1+2D_2-1\right)\right]\end{array}\right.$$

(5)

Combining (5) with the previous analysis, the expression for the transmitted power P_EPS of the DAB converter from the HB₁ bridge to the HB₂ bridge can be expressed as:

$$P_{\mathrm{EPS}}=\frac{1}{T_{\mathrm{hs}}}{\displaystyle {\int }_0^{T_{\mathrm{hs}}}{u}_{\mathrm{ab}}\cdot i_{\mathrm{L}}\left(t\right)}\mathrm{d}t=\frac{n{U}_{\mathrm{in}}{U}_{\mathrm{o}}\left[2D_2-2D_2^2+D_1-D_1{}^2-2D_1{D}_2\right]}{4f_{\mathrm{s}}L}$$

(6)

Let D₁ be 0 in (6); the expression of the transmitted power P_SPS is expressed by:

$$P_{\mathrm{SPS}}=\frac{n{U}_{\mathrm{in}}{U}_{\mathrm{o}}\left(D_2-D_2^2\right)}{2f_{\mathrm{s}}L}$$

(7)

To facilitate the later analysis and calculation, Equation (6) needs to be normalized, and the maximum transmission power P_MAX under SPS modulation is set as the baseline value. It can be expressed as:

$$P_{\mathrm{MAX}}=\frac{n{U}_{\mathrm{in}}{U}_{\mathrm{o}}}{8f_{\mathrm{s}}L}$$

(8)

Next, substituting (6) and (7) into (8), the transmitted power expressions of p_EPS and p_SPS after normalization are as follows:

$$\left\{\begin{array}{l}{p}_{\mathrm{EPS}}=\frac{P_{\mathrm{EPS}}}{P_{\mathrm{MAX}}}=4D_2-4D_2^2+2D_1-2D_1{}^2-4D_1{D}_2\\ p_{\mathrm{SPS}}=\frac{P_{\mathrm{SPS}}}{P_{\mathrm{MAX}}}=4D_2-4D_2^2\end{array}\right.$$

(9)

2.3. Analysis of Backflow Power

Defining the HB₁ output voltage u_ab opposite to the direction of the inductive current i_L, the power transmitted to the input side supply U_in is the backflow power, as shown in the black shaded section in Figure 3. Comparing the magnitudes of the shadow areas of the transmission power curves under the two modulation modes of EPS and SPS, it is clear that EPS has a more obvious effect on the inhibition of the backflow power for two control modes, because under EPS modulation mode, switch tubes Q₁ and Q₄ can no longer act simultaneously, which will have a positive effect on the inhibition of the backflow power.

Based on the results of the analysis of the four working modes of the converter, it can be concluded that the Q_EPS and Q_SPS expressions of the backflow power are as follows:

$$\left\{\begin{array}{l}{Q}_{\mathrm{EPS}}=\frac{n{U}_{\mathrm{in}}{U}_{\mathrm{o}}{\left[k\left(1-D_1\right)+2D_2-1\right]}^2}{16f_{\mathrm{s}}L\left(k+1\right)}\\ Q_{\mathrm{SPS}}=\frac{n{U}_{\mathrm{in}}{U}_{\mathrm{o}}{\left[k+2D_2-1\right]}^2}{16f_{\mathrm{s}}L\left(k+1\right)}\end{array}\right.$$

(10)

Equation (10) was subjected to the unitization using (8), from which the backflow power expressions q_EPS and q_SPS after unitization can be obtained as follows:

$$\left\{\begin{array}{l}{q}_{\mathrm{EPS}}=\frac{{\left[k\left(1-D_1\right)+2D_2-1\right]}^2}{2\left(k+1\right)}\\ q_{\mathrm{SPS}}=\frac{{\left[k+2D_2-1\right]}^2}{2\left(k+1\right)}\end{array}\right.$$

(11)

The q_EPS and q_SPS three-dimensional plots plotted according to (11) are shown in Figure 5, where k was taken to be 1, 1.5, and 2, respectively. It is known from Figure 5a that there is little inhibitory effect on the backflow power under SPS modulation, and the backflow power increases as k increases. Figure 5b shows that, compared with SPS, only one control variable, EPS, has an obvious inhibitory effect on backflow power after adding a control variable D₁ inside the HB₁ bridge. When D₂ takes a smaller value and D₁ also takes a smaller value or D₂ takes a larger value and D₁ also a larger value, it has the most desirable inhibitory effect on backflow power, and even zero backflow power can be achieved. When D₂ takes a smaller value but D₁ takes a larger value, the inhibition effect becomes worse for the backflow power. However, given that the 0 ≤ D₁ + D₂ ≤ 1 constraint is to be satisfied, it should be satisfied that D₂ less D₁ is also smaller.

3. Backflow Power Inhibition

The traditional Lagrange method does not achieve the best result when suppressing the backflow power, so it needs to be improved. The specific method is to re-represent the mathematical model after taking the deviance of the Lagrange original equation in order to obtain the expression of the phase-shift amount that inhibits the best effect of the backflow power.

The equation relationship of the Lagrange method for the backflow power under transmission power constraints is first defined by:

$$L\left(q,p,\lambda \right)=q_{\mathrm{EPS}}+\lambda \left(p_{\mathrm{EPS}}-p^{\ast }\right)$$

(12)

In this equation, λ is the Lagrange multiplier and p* is the given transmission power.

To obtain the optimized expression for the amount of phase shift, Equation (12) needs to be biased, as shown in (13).

$$\left\{\begin{array}{l}\frac{\partial L}{\partial D_1}=0\\ \frac{\partial L}{\partial D_2}=0\end{array}\right.$$

(13)

Adding the p_EPS in (9) and the q_EPS in (11) into (13), it is possible that Equation (12) will bias D₁ and D₂ as follows:

$$\left\{\begin{array}{l}\frac{\partial L}{\partial D_1}=\frac{-k\left(2D_2-1+k\left(1-D_1\right)\right)}{2k+2}+\lambda \left(1-2D_1-2D_2\right)=0\\ \frac{\partial L}{\partial D_2}=\frac{\left(2D_2-1+k\left(1-D_1\right)\right)}{2k+2}+\lambda \left(1-D_1-2D_2\right)=0\\ \eta =\frac{\left(2D_2-1+k\left(1-D_1\right)\right)}{2k+2}\end{array}\right.$$

(14)

In this equation, η is the common part of L after the derivation of D₁ and D₂.

Solved by the traditional Lagrange method, the process is described in (14), where η and λ simultaneous depletion yields the shortest expression for D₁ versus D₂ as follows:

$$D_1=\frac{\left(1+k\right)\left(1-2D_2\right)}{k+2}$$

(15)

The plot of D₁ versus D₂ plotted by using (15) is shown in Figure 6 when k is taken as 1.4. As can be seen from the figure, when D₁ is larger and D₂ is small, taken together with the conclusion from the previous analysis, it is known that under this condition, it is difficult to effectively suppress the backflow power. So, a reframing of (14) is required.

The improved Lagrange method requires that when solving (10), not for the public part η, proceed to elimination and solve for the resulting new D₁ expression as follows:

$$D_1=\frac{k^2+k-1+\left(2-k^2\right)D_2-\sqrt{\left(k^4+4k^3+8k^2+8k+4\right)D_2^2-\left(2k^2+4k+4\right)D_2+1}}{k^2+2k}$$

(16)

The plot of D₁ versus D₂ plotted by using (16) is shown in Figure 7, when k is similarly taken as 1.4. As can be seen from the figure, the improved results satisfy the expectation that D₁ is also smaller when D₂ is smaller compared to when the resulting D₁ versus D₂ relationship is solved using the traditional Lagrange method. Combined with Figure 5b, it is known that the D₁ versus D₂ combined relationship at this time has a positive effect on inhibiting the backflow power.

Further, combining (16) with the p_EPS from (9) to eliminate the intermediate D₂ yields the final expression D_1.end, and the specific solution procedure and results are shown in Appendix A.

With traditional control methods, D₁ was ablated to obtain D₂ by bringing the end back to the p_EPS in (9) at this time. However, in this way, the resulting D₂ acts on the DAB converter at the same time as D_1.end, making it difficult to guarantee that the converter can achieve the rated transmission power. In practical applications, D₂ is mostly taken through the PI controller. But, when the resulting D₂ of the PI controller is acting simultaneously with D_1.end on the DAB converter, it, in turn, causes the converter’s dynamic characteristics to become poor. When the load changes, the converter output voltage cannot return to a given value in a shorter time [29].

4. Model Predictive Control

In order to be able to improve the dynamic characteristics of the DAB converter while suppressing the backflow power, the model prediction control is adopted in this paper to replace the traditional PI control. Model prediction control is needed to establish the state space average equation. This paper chooses output capacitor voltage as the state variable by establishing the output voltage state space average model of the DAB converter to obtain the outer phase shift D₂. The specific steps are as follows.

An equivalent schematic diagram for the output side circuit is shown in Figure 8, where I_av is the mean current normalized to the side edge and I_C is the current flowing through the capacitance C_o.

Then, using Kirchhoff’s law, the node current equation for C_o is expressed as:

$$I_{\mathrm{C}}=C_{\mathrm{o}}\frac{\mathrm{d}{U}_{\mathrm{o}}}{\mathrm{d}t}=I_{\mathrm{av}}-I_{\mathrm{o}}$$

(17)

I_av in (17) can be obtained using P_EPS. Hence,

$$I_{\mathrm{av}}=\frac{P_{\mathrm{EPS}}}{U_{\mathrm{o}}}$$

(18)

Finally, collating (6) with (17) and (18), we obtain the state space average equation for the output voltage under EPS modulation as follows:

$$C_{\mathrm{o}}\frac{\mathrm{d}{U}_{\mathrm{o}}}{\mathrm{d}t}=\frac{n\left[2D_2-2D_2^2-2D_1{D}_2+D_1-D_1{}^2\right]}{4f_{\mathrm{s}}L}{U}_{\mathrm{in}}-\frac{U_o}{R}$$

(19)

Model prediction control is used to improve the system dynamic characteristics by making predictions on the output voltage at the next moment, so the output voltage needs to be discretized. The discretization method used in this paper is forward Euler, and its original expression is as follows:

$$y^{\prime }=\frac{y\left(k+1\right)-y\left(k\right)}{T}$$

(20)

By using (20), the discretized output voltage equation is as follows:

$$\frac{\mathrm{d}{U}_{\mathrm{o}}}{\mathrm{d}t}=\frac{U_{\mathrm{o}}\left(t_{i+1}\right)-U_{\mathrm{o}}\left(t_i\right)}{t_{i+1}-t_i}=\frac{U_{\mathrm{o}}\left(t_{i+1}\right)-U_{\mathrm{o}}\left(t_i\right)}{2T_{\mathrm{hs}}}$$

(21)

In this equation, U_o (t_i+1) is the predicted value of the output voltage at t_i+1, and U_o (t_i) is the output voltage at t_i.

By carrying (21) to (19), the predicted equation for the output voltage is expressed as:

$$U_{\mathrm{o}}\left(t_{i+1}\right)=U_{\mathrm{o}}\left(t_i\right)-\frac{I_{\mathrm{o}}\left(t_i\right)}{f_{\mathrm{s}}{C}_{\mathrm{o}}}+\frac{n\left[2D_2-2D_2^2-2D_1{D}_2+D_1-D_1{}^2\right]}{4f_{\mathrm{s}}^{2}L{C}_{\mathrm{o}}}{U}_{\mathrm{in}}\left(t_i\right)$$

(22)

In this equation, I_o(t_i) is the current flowing over the load at time t_i, and U_in(t_i) is the input voltage at time t_i.

Therefore, the output voltage can follow the reference U_o.ref. Let U_o (t_i+1) be equal to the output voltage reference, and the expression is as follows:

$$U_{\mathrm{o}}\left(t_{i+1}\right)=U_{\mathrm{o}.\mathrm{ref}}$$

(23)

By substituting (23) to (22), the external phase shift D₂ can be expressed as:

$$D_2=\frac{1}{2}-\frac{D_1}{2}-\sqrt{\frac{1}{4}-\frac{D_1{}^2}{4}-2\frac{f_{s}L{I}_{\mathrm{o}}\left(t_i\right)+f_{\mathrm{s}}^{2}L{C}_{\mathrm{o}}\left(1-U_{\mathrm{o}.\mathrm{b}}\right)}{n{U}_{\mathrm{in}}\left(t_i\right)}}$$

(24)

In this equation, U_o.b is the normalized output voltage, and its expression is U_o.b = U_o(t_i)/U_o.ref.

To more intuitively demonstrate the model prediction control lifting over the dynamic characteristics, it is now given the various element parameters in the main circuit from Figure 2, and the specific numerical values are shown in

Table 1. Main circuit parameters of the DAB converter.

Parameter	Value
Uin	140 V
Uo	100 V
Cin	2000 μF
Co	2000 μF
L	150 μH
n	1:1
fs	10 kHz

.

Let the load R = 15 Ω and D₁ = 0.2. While bringing the parameters in Table 1 to (24), the plot of U_o.b versus D₂ is shown in Figure 9. It is known from the figure that when the output voltage differs more from the reference, D₂ outputs at its upper or lower limit; when the output voltage is near the reference, the converter adjusts relatively smoothly, ensuring that the output voltage can be quickly followed to the reference, as a way to improve the dynamic properties of the system.

Equation (24) is a mathematical model obtained under ideal conditions, which, in practical operation, is affected by such factors as dead time, pressure drop of the switch tube, and output time delay, and the mathematical model of power of (6) will have some deviation from the actual situation [29]. In order to keep the output power at steady-state operation as close as possible to the expected value, the output voltage is now compensated to allow it to stabilize at the reference value at steady state. The difference between the converter’s reference and output voltages is fed to the PI controller, and the resulting value is denoted as ΔU, and its diagram is shown in Figure 10.

Upon re-entry into (24), the D_2.end can be obtained with a final expression as follows:

$$D_{2.\mathrm{end}}=\frac{1}{2}-\frac{D_1}{2}-\sqrt{\frac{1}{4}-\frac{D_1{}^2}{4}-2\frac{f_{s}L{I}_{\mathrm{o}}\left(t_i\right)+f_{\mathrm{s}}^{2}L{C}_{\mathrm{o}}\left(1-U_{\mathrm{o}.\mathrm{b}}+\mathsf{\Delta }U\right)}{n{U}_{\mathrm{in}}\left(t_i\right)}}$$

(25)

From the above analysis, the specific steps of the MPCL-EPS method are: (1) from voltage and current values of the sampling converter, k and p_EPS are calculated; (2) D_2.end was taken according to Equation (25), and D_1.end was taken according to Equations (A1)–(A6) in Appendix A; (3) the desired result is acted on the transformer by a phase-shifting modulation module, and the specific control block diagram is shown in Figure 11.

5. Simulation Verification

To verify the proposed optimization control strategy, MPCL-EPS, in this paper, the DAB converter circuit simulation model is built using Matlab/Simulink, and the circuit parameters are shown in Table 1.

5.1. Backflow Power

The transmission power waveforms under SPS, EPS, and MPCL-EPS modulation are shown in Figure 12 at an output power unit value of p = 0.41 (R = 21 Ω). It can be seen that the peak of backflow power under SPS modulation is the largest and can reach 1395 W, and the peak of backflow power under EPS modulation is significantly reduced compared with SPS, but the peak of backflow power still reaches 409 W. MPCL-EPS modulation has the most desirable effect on the inhibition of backflow power and can achieve zero backflow power.

The transmission power waveforms under SPS, EPS, and MPCL-EPS modulation are shown in Figure 13 at an output power unit value of p = 0.68 (R = 12.5 Ω). It can be seen from the figure that the peak of backflow power under SPS modulation is still larger, reaching 1903 W, while the peak of backflow power under EPS modulation is 896 W, and the inhibition effect of backflow power under MPCL-EPS modulation is still more desirable, achieving zero backflow power, which is consistent with the results of analysis in Figure 7.

5.2. Dynamic Characteristics

The output voltage waveforms and output current waveforms under SPS, EPS, and MPCL-EPS modulation are shown in Figure 14 and Figure 15, where the output power unit p was mutated from 0.41 (R = 21 Ω) to 0.68 (R = 12.5 Ω). It can be seen from the figure that the dynamic properties under traditional SPS modulation and EPS modulation are both poor, and they take 347 ms and 309 ms, respectively, from the voltage under mutation to following the reference value. The dynamic characteristics under MPCL-EPS modulation are desirable, and they can follow the reference voltage with little reaction time. Moreover, compared with the dramatic voltage fluctuations that accompany the mutation process under SPS with EPS modulation, the voltage under MPCL-EPS modulation shows little fluctuation.

The output voltage waveforms and output current waveforms under SPS, EPS, and MPCL-EPS modulation are shown in Figure 16 and Figure 17, where the output power unit p was mutated from 0.68 (R = 12.5 Ω) to 0.41 (R = 21 Ω). It can be seen from the figure that the dynamic characteristics under SPS modulation and EPS modulation are basically consistent, and it takes 519 ms and 513 ms to recover from the mutation of voltage to the nominal value, respectively, while there are obvious voltage fluctuations in the dynamic process. The output voltage under MPCL-EPS modulation, which can quickly follow the rated voltage after a slight drop, has an extremely short reaction time, which is consistent with the results of analysis from Figure 9.

6. Conclusions

In order to improve the problem of the high backflow power in the DAB converter under the traditional method of control and simultaneously raise the dynamic response capability of the DAB converter, in this paper, on the basis of the in-depth analysis of the operational characteristics of the DAB converter, an optimized control strategy based on the modulation of EPS by MPCL-EPS was proposed. This strategy suppresses the backflow power by modifying the Lagrange method and using model predictive control to improve the dynamic characteristics of the system. The theoretical analysis and simulation results confirm that the MPCL-EPS modulation strategy can significantly suppress the backflow power of the DAB converter and achieve a significant improvement in the dynamic properties of the DAB converter.