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

: 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


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 bidirectional 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-phaseshift (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  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. 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.

Basic Principle
The DAB converter topology is shown in Figure 2. In this figure, HB 1 is the converter front bridge, HB 2 is the converter back bridge, Q 1~Q8 is a fully controlled switching tube, D 1~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 HB 1 output voltage, u cd is transformer side voltage, i L is inductive current, and I o is the output load current.

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, Cin and Co 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, Uin and Uo are the converter input and output voltages, uab is HB1 output voltage, ucd is transformer side voltage, iL is inductive current, and Io 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 stepdown 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, uL is inductive voltage, PSPS and PEPS are transmitted power under SPS and EPS modulation, and Ths is half switching cycle. There are two control variables, D 1 and D 2 , in the method of modulation of EPS. D 1 is the control variable of phase shift between Q 1 and Q 4 . In this paper, only Q 1 is considered to lag behind Q 4 ; D 2 is the control variable of phase shift between Q 1 and Q 5 , and in this paper, only Q 5 is considered to lag behind Q 1 . Only D 2 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. D 1 meets with D 2 for 0 ≤ D 1 ≤ 1, 0 ≤ D 2 ≤ 1 and 0 ≤ D 1 + D 2 ≤ 1. Q 1~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  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 D 1 within the HB 1 bridge compared to EPS modulation, so only the analysis of the method of modulation in EPS modulation needs to be performed. Then, let D 1 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.
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: 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 iL 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:

Transmitted Power
It is known from the previous analysis that the inductive current iL 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 = Uin/nU o, and fs is the switching period. (1) phase 1: The switching state in this phase is shown in Figure 4a. Q 2 conducts with Q 4 inside the HB 1 bridge, and Q 6 conducts with Q 7 inside the HB 2 bridge. But, the current i L flowing through the inductors is negative at this point, so the continuous flow from Q 2 to D 4 inside the HB 1 bridge and D 6 and D 7 inside the HB 2 bridge can be derived, and the expression of the inductive current is as follows: The switching state in this phase is shown in Figure 4b. Q 1 conducts with Q 4 inside the HB 1 bridge and Q 6 conducts with Q 7 inside the HB 2 bridge. But, the current i L flowing through the inductors is still negative at this point, so the continuous flow from D 1 to D 4 inside the HB 1 bridge and D 6 and D 7 inside the HB 2 bridge can be derived, and the expression of the inductive current is as follows: The switching state in this phase is shown in Figure 4c. Q 1 is conducted with Q 4 inside the HB 1 bridge, and Q 6 is conducted with Q 7 inside the HB 2 bridge. At t 2 , the flow through the inductive current i L is negatively changed. So, the transport current inside the HB 1 bridge is conducted with Q 1 and Q 4 , and the transport current inside the HB 2 bridge is conducted with Q 6 and Q 7 , and the expression of the inductive current is as follows: The switching state in this phase is shown in Figure 4d. Q 1 is conducted with Q 4 inside the HB 1 bridge, and Q 5 is conducted with Q 8 inside the HB 2 bridge. But, the flow through the inductive current i L is positive, so the transport current is conducted through Q 1 and Q 4 within the HB 1 bridge, and the continuous flow is conducted through D 5 and D 8 within the HB 2 bridge, and the expression of the inductive current is as follows:

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.
Combining (5) with the previous analysis, the expression for the transmitted power P EPS of the DAB converter from the HB 1 bridge to the HB 2 bridge can be expressed as: Let D 1 be 0 in (6); the expression of the transmitted power P SPS is expressed by: 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: Next, substituting (6) and (7) into (8), the transmitted power expressions of p EPS and p SPS after normalization are as follows:

Analysis of Backflow Power
Defining the HB 1 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 Q 1 and Q 4 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: 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: 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 D 1 inside the HB 1 bridge. When D 2 takes a smaller value and D 1 also takes a smaller value or D 2 takes a larger value and D 1 also a larger value, it has the most desirable inhibitory effect on backflow power, and even zero backflow power can be achieved. When D 2 takes a smaller value but D 1 takes a larger value, the inhibition effect becomes worse for the backflow power. However, given that the 0 ≤ D 1 + D 2 ≤ 1 constraint is to be satisfied, it should be satisfied that D 2 less D 1 is also smaller.

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 q p q p p (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).
Adding the pEPS in (9) and the qEPS in (11) into (13), it is possible that Equation (12) will bias D1 and D2 as follows:

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: 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).
Adding the p EPS in (9) and the q EPS in (11) into (13), it is possible that Equation (12) will bias D 1 and D 2 as follows: (14) In this equation, η is the common part of L after the derivation of D 1 and D 2 . Solved by the traditional Lagrange method, the process is described in (14), where η and λ simultaneous depletion yields the shortest expression for D 1 versus D 2 as follows: The plot of D 1 versus D 2 plotted by using (15) is shown in Figure 6 when k is taken as 1.4. As can be seen from the figure, when D 1 is larger and D 2 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.
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: 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: 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. The improved Lagrange method requires that when solving (10), not for the public part η, proceed to elimination and solve for the resulting new D 1 expression as follows: The plot of D 1 versus D 2 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 D 1 is also smaller when D 2 is smaller compared to when the resulting D 1 versus D 2 relationship is solved using the traditional Lagrange method. Combined with Figure 5b, it is known that the D 1 versus D 2 combined relationship at this time has a positive effect on inhibiting the backflow power. With traditional control methods, D1 was ablated to obtain D2 by bringing the end back to the pEPS in (9) at this time. However, in this way, the resulting D2 acts on the DAB converter at the same time as D1. 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 D1. 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 With traditional control methods, D 1 was ablated to obtain D 2 by bringing the end back to the p EPS in (9) at this time. However, in this way, the resulting D 2 acts on the DAB converter at the same time as D 1.end , making it difficult to guarantee that the converter can Energies 2023, 16, 5683 9 of 17 achieve the rated transmission power. In practical applications, D 2 is mostly taken through the PI controller. But, when the resulting D 2 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].

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 D 2 . 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 .
Further, combining (16) with the pEPS from (9) to eliminate the intermediate D2 yields the final expression D1. 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 pEPS in (9) at this time. However, in this way, the resulting D2 acts on the DAB converter at the same time as D1. 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 D1. 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].

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 Iav is the mean current normalized to the side edge and IC is the current flowing through the capacitance Co. Then, using Kirchhoff's law, the node current equation for Co is expressed as: Iav in (17) can be obtained using PEPS. Hence, Then, using Kirchhoff's law, the node current equation for C o is expressed as: I av in (17) can be obtained using P EPS . Hence, Finally, collating (6) with (17) and (18), we obtain the state space average equation for the output voltage under EPS modulation as follows: 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: By using (20), the discretized output voltage equation is as follows: (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: 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: (23) By substituting (23) to (22), the external phase shift D 2 can be expressed as: In this equation, U o.b is the normalized output voltage, and its expression is 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. Let the load R = 15 Ω and D 1 = 0.2. While bringing the parameters in Table 1 to (24), the plot of U o.b versus D 2 is shown in Figure 9. It is known from the figure that when the output voltage differs more from the reference, D 2 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 ∆U, and its diagram is shown in Figure 10.
Let the load R = 15 Ω and D1 = 0.2. While bringing the parameters in Table 1 to (24), the plot of Uo.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 D2. end can be obtained with a final expression as follows: 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 pEPS are calculated; (2) D2. end was taken according to Equation (25), and D1. 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. 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 D2. end can be obtained with a final expression as follows: 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 pEPS are calculated; (2) D2. end was taken according to Equation (25), and D1. 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. Upon re-entry into (24), the D 2.end can be obtained with a final expression as follows: 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.  Figure 11. Optimization algorithm control block diagram.

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.

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.

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.

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 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.

Dynamic Characteristics
The output voltage waveforms and output current waveforms under SPS, EPS, and MPCL-EPS modulation are shown in Figures 14 and 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.

Dynamic Characteristics
The output voltage waveforms and output current waveforms under SPS, EPS, and MPCL-EPS modulation are shown in Figures 14 and 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.

Dynamic Characteristics
The output voltage waveforms and output current waveforms under SPS, EPS, and MPCL-EPS modulation are shown in Figures 14 and 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 Figures 16 and 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.  The output voltage waveforms and output current waveforms under SPS, EPS, and MPCL-EPS modulation are shown in Figures 16 and 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.
The output voltage waveforms and output current waveforms under SPS, EPS, and MPCL-EPS modulation are shown in Figures 16 and 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.

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.

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.

Conflicts of Interest:
The authors declare no conflict of interest.

Appendix A
The expression of D 1 after realignment is as follows: In this equation, the expression of each parameter is as follows: Equation (A1) is the standard form of the quaternion equation and, theoretically, has four different solutions, respectively: