Direct Power Based Control Strategy for DAB DC-DC Converter With Cooperative Triple Phase Shifted Modulation

The cooperative triple-phase-shifted modulation (CTPS) for the isolated dual-active bridge (DAB) DC-DC converter has the advantages of zero dual-DC side flow back currents and best current characteristics. However, the mathematical model and corresponding closed-loop control configuration of the DAB converter driven by CTPS are not presented so far. In order to solve this problem, in this paper, the average state space model of the DAB converter driven by CTPS is established for the first time. It is proven that the DAB converter driven by CTPS is a zero-order system and it is not necessary to construct an inner inductor current closed-loop control loop in the entire closed-loop control configuration of DAB converter. Furthermore, the average power model is derived. Two control strategies based on the resulted average power model when the DAB converter is connected to DC microgrid or a pure resistive load are proposed to realize the precise control of the transfer power or the output voltage, respectively. The detailed experimental results verify the correctness of the established mathematical model and the closed-loop control strategies.


I. INTRODUCTION
The isolated dual active bridge (DAB) DC-DC converter has the advantages of bidirectional power flow and bidirectional boost/buck voltage conversion, and fast dynamic response. Due to these advantages, the DAB DC-DC converter has been widely applied in the fields of the renewable energy generation systems and the energy storage systems, microgrids, solid state transformer, aerospace, fuel cells generation system, and so on [1]- [6].
Nowadays, towards increasing the comprehensive performance of DAB converter, the researchers focus on the full soft-switching realization, the optimization of current characteristics, including reducing the current stress and real mean square (RMS) value of the inductor current and the closed-loop control strategies. In [7]- [9], the current stress is discussed and corresponding improvement methods are proposed. In [10], the current characteristics is discussed and a current characteristics optimization is proposed to improve the efficiency and reduce the current stress. In [11], [12], The associate editor coordinating the review of this manuscript and approving it for publication was Yuh-Shyan Hwang . the problem of flow back current, which is also defined as the circulating current or nonactive current, is discussed and several solutions are proposed.
The existing literatures mainly concern the improvement of some certain key problems of DAB converter mentioned above, however, these strategies cannot solve all of the above problems. In [13], a cooperative triple-phase-shifted modulation (CTPS) is proposed to realize the basic voltage/power conversion and at the same time, to eliminate the dual-DC sides flow back currents and minimize the current stress of the power switches and RMS value of the inductor current. Eliminating the dual-side flow back currents is helpful to reduce the DC side current ripples and minimizing the RMS value of the inductor current is helpful to increase the efficiency. Furthermore, under the same maximum transfer power, the current stress of the CTPS is not greater than the SPS and as a result, the CTPS is much better than the SPS in terms of the current characteristics, efficiency and cost.
More importantly, in the actual industrial applications, the DAB DC-DC converter is usually controlled with the closedloop configuration to achieve a better dynamic response and precise steady-state control performance. VOLUME 9, 2021 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ In order to analysis the open-loop characteristics and design the closed-loop controller parameters, the small-signal models of DAB DC-DC converters have been built and some linear control methods have been adopted based on the smallsignal models. In [14], [15], a full-order model taking into account the leakage inductance current and the resonant transition intervals is built. In [16], a full-order continuous-time average model using the dc terms and first order terms of transformer current is built. In [17], the small-signal model of DAB converter based on the generalized state space average modeling method is built.
Some closed-loop control strategies are also proposed to improve the performance of the DAB DC-DC converter. In [18], [19], a model-based phase-shift control has been proposed to improve the dynamic characteristics in the load disturbance conditions. In [20], the model prediction control method is introduced into DAB converter to increase the dynamic performance. In [21], when the step change of the current reference occurs, the inductor current peak value corresponding to the new current reference is calculated based on the inductor current expressions and used to control the actual inductor current to obtain a good dynamic performance. In [22], the expression of average transfer power as a function of outer phase shift with the single-phaseshifted modulation (SPS) is derived to calculate the actual phase shift according to the power reference. In this way, the small-signal model is not used and in theory, the dynamic performance keeps constant in the entire power range because the actual performance is in independent of the pre-chosen operation point, which is necessary in the small-signal model based control system. In [23], [24], the interconnection and damping assignment passivity based control is applied to the port-controlled hamiltonian model of DAB converter in a DC Microgrid. In [25], an improved instantaneous current control for three-phase DAB converter is proposed.
Generally, the mathematical model of the DAB converter is related to the modulation strategies adopted. The previously presented models are all based on SPS. The resulted small-signal models are not able to directly apply in the DAB converter driven by CTPS because its characteristics are quite different from the SPS based ones. However, the linear model of the CTPS based DAB converter are not presented so far.
Similarly, all of the presented closed-loop control strategies are based on the SPS and they are not able to be applied in the DAB converter driven by CTPS.
In order to realize combining the CTPS with the closedloop control system, it is necessary to build the mathematical model of the DAB converter driven by the CTPS and to propose some more closed-loop control strategies.
In this paper, the mathematical model and corresponding closed-loop control strategy of the DAB converter driven by the CTPS are researched originally. The innovation and contribution of this paper are as follows. (1) The average state-space model of the DAB converter with the CTPS is originally derived. It is proven that it is a zero-order system if neglecting the effect of secondary DC capacitor. (2) Based on the zero-order system characteristics, the control strategies of the DAB converter when it is connected to a DC microgrid or serves alone as a DC source supply are proposed based on the obtained power model, on the one hand, to realize the precise control of the transfer power and output voltage, on the second hand, to simplify the closed-loop control configuration and reduce the design burden of the controller parameters.  and v h2 are the AC side output voltages of HB1 and HB2, respectively; i 1 and i 2 are the DC side currents of HB1 and HB2, respectively. n is the turns ratio of HFT and i L is the DC inductor current. Fig. 2 shows the operating waveforms of the general TPS modulation in the scenario that the power flows from V 1 to V 2 and V 1 ≥ nV 2 . In Fig. 2, the inner phase shift ratios of HB1 and HB2 are defined as D 1 and D 2 , respectively; and the outer phase shift ratio between v h1 and v h2 is defined as D ϕ .

II. BRIEF INTRODUCTION TO CTPS
The flow back current is briefly discussed below. Assuming that the power transfers from V 1 to V 2 , the ideal situation is that both the primary DC side current flowing through V 1 to HB1 and the secondary DC side current flowing through HB2 to V 2 should be not less than zero. Otherwise, if i 1 or i 2 is less than zero in some regions, the power will be transferred back from HB1 to V 1 or from V 2 to HB2 for a short time during the switching period. This part current is defined as the flow back current and it is not expected. The negative effect of the flow back currents have been described in detail in [13] and it is not mentioned here. CTPS in [13] is proposed to not only eliminate the dual-DC sides flow back currents, but also optimize the current characteristics and achieve the full soft switching in the entire power range.
The principle of the CTPS is briefly explained here. The details please refer to [13]. Fig. 2(a) shows the operating waveforms of the CTPS. i 1 and i 2 are never less than zero in the entire switching period and the flow back currents are eliminated. From [13], the relationship between the three phase shifts in order to eliminate the dual-side flow back currents can be expressed as where k = V 1 /nV 2 is the voltage conversion ratio.
If it is assumed k is greater than 1, when D 2 is equal to 1, D 1 is less than 1 and the corresponding operating waveforms of the CTPS are shown in Fig. 2(b). This point is called the critical point. In [13], the corresponding D 1 and the average  power at the critical point (P av_cri ) have been solved and are expressed as where p av_cri is the per-unit value of P av_cri . When P av < P av_cri , D ϕ should keep equal to zero and the relationship between D 1 and D 2 is the same as that in (1). The corresponding operating waveforms are shown in Fig. 2 The relationship between D ϕ and D 1 in the entire power range can be expressed as When the relationship between the three phase shifts meet (3), P av can be expressed as (5) (shown at the bottom of the page).
From [13], the maximum average power and D 1 at the maximum average power point can be expressed as It has been indicated in [13] that, when P av ≥ P av_cri , D 1 should be tuned in the region of D 1_cri , D 1_ max to obtain a good current characteristics. Similarly, when P av < P av_cri , D 1 should be tuned in the region of D 1_cri , 1 .
The distribution curve of the average power as a function of D 1 is drawn and shown in Fig. 3. When D 1 is tuned in the region of D 1_ max , 1 , the relationship between the average power and D 1 is not of one-to-one correspondence. In addition, the exact value of D ϕ is also determined by the actual average power. Therefore, the desired average power instead of D 1 is chosen as the input of CTPS. The actual expression of D 1 as a function of P av is derived from (4) and it is shown below.
When p av ≥ p av_cri , the actual expression of D 1 as a function of p av is When p av < p av_cri , the expression of D 1 is

III. AVERAGE STATE-SPACE MODEL OF DAB CONVERTER WITH CTPS
In this section, the average state-space model of DAB converter with CTPS is derived first. The reason why uses this kind of modelling method is explained below. It is because that during each positive-half switching period, the inductor current always starts at zero and ends at zero. It means the operation process in the positive-half switching period is decoupled from that in the negative-half switching period. In addition, since the starting and ending values of all the variables, including the inductor currents and voltages across it are known, only the average model of the converter in the positive-half period needs to build. Thus the problem existed in the conventional SPS or DPS does not exist. As a result, the average state-space modelling method can be applied in the CTPS based DAB converter. The resulted model by using this modelling method is precise because there is no any approximation is taken during the modelling process, which is usually necessary in the popular fundamental analysis method.
The average state-space equation can be derived from (10)- (12) and it is expressed as where i 2_av is the average value of secondary DC side current, which is proportional to the actual load. By substituting k = V 1 /nV 2 into (13), it yields During the region of [t 3 , t 4 ], the state space equation is The average state-space equation in a switching period is From (13), (14) and (17), the unified average state-space model of the DAB converter with CTPS in the entire power range can be described in the following matrix form.
From the built average state-space model of DAB converter with CTPS, the derivative of the average inductor current in a switching period is always equal to zero. It is also independent of the input and output voltages and the variation of three phase shifts.
It means there is a certain relationship between the three phase shifts and the exact average inductor current in a switching period. As a result, the inductor current is directly determined by the phase shifts. In other words, when it is desired to realize the step change of the transfer power, it can be realized by updating the corresponding phase shifts directly and the inductor current becomes the corresponding value immediately without any transient process. This unique feature leads to the following two merits of the DAB converter. The first one is that, there is no transient transition between the variation of phase shifts and the actual inductor current. The transient transition process mentioned here is the transient process of the actual inductor current from the initial steady state to the new steady state caused by the step change of the phase shifts or the input/output voltages. The second one is that the inductor current is always under control because it is always determined by the present phase shifts and input/output voltages. Therefore, even the inductor current is not specially considered in the entire closed-loop control system, it can still realize a good dynamic performance and the over-current problem existed in the common power electronics devices does not occur.
From this unique characteristics, it does not need to set a inductor current control loop in the entire control configuration of the DAB converter and the inductor current does not varies uncontrollably. This characteristic simplifies the closed-loop control configuration of the DAB converter.

IV. CLOSED-LOOP CONTROL STRATEGIES OF DAB CONVERTER WITH CTPS
Since it is not necessary to set an inductor current control loop in the DAB converter with CTPS. The closed-loop control of the DAB converter can be implemented in a simply way. From the actual application fields of the DAB converter, the DAB converter is possible to connect to the DC microgrid or works alone as a DC source supply. For the scenario of using in DC microgrid, V 2 is regarded as constant and only the transfer power should be controlled. If the DAB converter is used as a DC source supply, both the transfer power and the terminal DC voltage should be controlled. The control strategies of the two application scenarios are discussed separately.

A. DAB CONVERTER CONNECTED TO DC MICROGRID
In this scenario, V 2 is regarded as constant. Considered that the DAB DC-DC converter is usually used to connect the PV or battery to the DC microgrid. The single output/input power control approach is adopted. Known from the average state-space model, the DAB converter controlled by CTPS is a zero-order system and there is no transient process between the average transfer power or the inductor current and the phase shifts. It means that, when a certain phase shifts are applied to the DC-DC converter, the corresponding power can be achieved immediately. In this scenario, a simple power control approach is used and the schematic is shown in Fig. 4. The desired power is compared with the critical power first, if the desired power is larger than the critical power, the actual phase shift D 1 is determined by (8), otherwise, D 1 is determined by (9). Then D 2 and D ϕ are calculated by (4). It should be indicated that, this system is essentially an openloop system because the expressions of (8) or (9) used to calculate D 1 do not consider the power loss of the DAB converter. This method is still effective because even though there is the transfer power error, the controlled variables of the DC microgrid are still able to be controlled at their desired values because of the closed-loop control algorithm performed in the main control unit of DC microgrid.

B. DAB CONVERTER CONNECTED TO RESISTIVE LOAD
In this scenario, V 2 should be constant and the closed-loop control of V 2 is adopted. In addition, considering that when the power varies in the entire range, the variation of D 1 is not monotonous. D 1 cannot be determined directly by the controller output of V 2 . The direct power control is still used in this scenario. The control loop of V 2 is set as the outer loop and the output of the V 2 control loop is set as the transfer power reference. Then the actual D 1 can be calculated from (8) or (9) and D 2 and D ϕ are calculated from (4).
The transfer function of the transfer power to V 2 is derived as follows. The average transfer power in a switching period can be derived as Substituting (19) into (17), the differential equation of V 2 can be derived as It can be further derived as Setting a new variable V 22 = V 2 2 and the s-domain transfer function of the average power to V 22 can be derived as Then the DAB DC-DC converter controlled by CTPS is equivalent to a first-order inertial element. In order to simplify the system, the controller of V 2 is set as a proportionalintegral (PI) controller. The entire control approach in this scenario is drawn in Fig. 4(b).
The design of the PI controller parameters is descripted as follows. The closed-loop transfer function is By using the principle of zero-pole points cancellation, the design rule of the coefficients of the PI controller is obtained. The proportional coefficient of V 2 controller is designed as C 2 /2 and the integral coefficient is designed as 1/R rated , where R rated is the rated load of the DAB converter. Then the final closed-loop transfer function yields It is obvious that the resulted transfer function is the firstorder inertial element and it is unconditionally stable.
Since C 2 and R rated are predesigned and are easy to obtain, the two coefficients are also easy to calculate.

V. EXPERIMENTAL RESULTS AND DISCUSSION
The experimental prototype of DAB DC-DC converter based on DSP chip TMS320F28335 is built as shown in Fig. 5 to validate the proposed control strategies. The experimental parameters are listed in Table 1. A DC source supply is connected to HB1 as the main power supply. A resistor and another DC source supply are connected to the DC side of HB2. The second DC source supply is used to emulate the DC microgrid. If it is assumed that the DAB converter serves as a DC source supply, the second DC source supply should be removed to ensure that only the DAB converter supplies the power to the load to evaluate its dynamic performance.
Considering that in the actual system, the phase shifts should vary in real time to tune the transfer power, the implementation of CTPS using the DSP chip is briefly described below. Fig. 6 illustrates the operating waveforms of the CTPS implemented by the DSP considering the update process of the phase shifts. The four timers T1-T4 in the DSP chip are synchronous and the phase shifts are updated by updating the various compare registers. At the starting moment of various timers, these registers are calculated and updated based on the new phase shifts.
In addition, the rising edges of the control signals of S 4 and S 8 always align at the starting moment of the timers. It ensures that the phase shifts are always updated at the zerocrossing point of the inductor current. From Fig. 6, If it is assumed that the phase shifts should be updated at t k , the compare registers CR1A, B and CR4A, B are directly updated by replacing D 1 and D ϕ by D 1 and D ϕ and anything else does not need to do. From Fig. 6, the actual phase shifts are equal to the desired ones immediately after the update moment without any delay or error. More importantly, there is no DC bias current caused during the transient process.
The experimental verification is performed in the following steps. First, the built average state-space model is verified via suddenly varying the unified average power reference from 0.2 to 0.55. In this case, the two DC side voltages   Fig. 7. From Fig. 7(a), after the unified average power reference jumps from 0.2 to 0.55, the three phase shifts are calculated immediately based on the new power reference. The inductor current increases immediately and reaches its new steady state in the same switching period. There is no transient transition process in the inductor current. It can be easily verified by the profile of the inductor waveforms. Similarly, after the unified average power reference jumps from 0.55 to 0.2, as shown in Fig. 7(b), the inductor current decreases immediately and reaches to the new steady state without the transient process. It means the actual inductor current, but not its average derivate is directly determined by the phase shifts. For any phase shifts, there is a certain inductor current corresponding to them. These experimental results shown in Fig. 7 verify that the unique characteristics of DAB converter controlled by the CTPS, namely it is essentially a zero-order system. It means the built model is correct.
This unique feature significantly simplifies the closed-loop control configuration of the DAB converter. Furthermore, the corresponding waveforms of the averaged i 2 are also shown in Fig. 7. Due to the constant V 2 , the averaged i 2 is proportional to the actual average transfer power. From Fig. 7(a) and (b), the averaged i 2 reaches its new steady state value very fast after the step change of the power reference occur. The ratios between the initial and the final values of the averaged i 2 in these two scenarios are the same as those of the power reference. It means the desired transfer power are achieved. It should be indicated that the transfer power error is not seriously evaluated in this paper. The reason is power error can be compensated by higher-level closed-loop controller, i.e., the DC bus voltage closed-loop controller operated in the central control system of the DC microgrid.
Secondly, the dynamic performance of the DAB converter with secondary voltage closed-loop control approach is evaluated. In this situation, the second DC source supply is removed and only the resistor is connected to the DC side of HB2. Considering that the voltage closed-loop control approach should achieve the zero-steady state even under the conditions of the sudden changes of the voltage reference or the load resistance, it is verified in two separate ways.
The first way is that the voltage reference jumps from 18V to 25V and jumps back after a certain period. The corresponding experimental waveforms are shown in Fig. 8. In order to further verify the built average state model, the waveforms of the inductor current along with the controlled variable, namely V 2 and the averaged i 2 (denoting the transfer power) are all shown. From Fig. 8, in these two cases, after the voltage reference jumps, the actual inductor current and the averaged i 2 vary immediately and the actual secondary DC side voltage increases to its reference within 30ms. There is no overshoot in the actual V 2 because the parameters of the PI controller are designed only to ensure the stability, but not to   pursue an extreme dynamic performance. More importantly, there is no transient overshoot or delay in the inductor current. The correctness of the built model of the DAB converter is verified once again and it means even though the actual inductor current is not controlled in a closed-loop way, it is still controllable and the safety and reliability of the DAB converter are ensured.
The second way to evaluate the dynamic performance of the voltage closed-loop control approach is to jump the resistance of the load from 100% to 50% and from 50% to 100%, respectively. The corresponding experimental waveforms are shown in Fig. 9(a) and (b), respectively. From Fig. 9, the actual inductor current varies immediately and the actual averaged i 2 varies very fast as well. After about 50ms, the actual secondary DC side voltage reaches to the new reference and there is no overshoot in the both cases. It achieves a good dynamic performance. During the dynamic process, the actual inductor current is always under control, there is no overshoot or additional transient process in it.
Moreover, the experimental waveforms of the DAB dc-dc converter driven by the CPTS control under reverse mode are tested and shown in Fig. 10. Fig. 10 (a), (b) show the openloop experimental waveforms with different power reference. When the transfer power vary in a wide range, v h2 is always ahead of v h1 and both i 1 and i 2 are less than zero or equal to zero, it means the actual power flows from V 2 to V 1 and the dual side flow back currents are completely eliminated. The waveforms of all of the variables under reverse mode are similar with the ones under forward mode shown in [13]. Fig. 10 (c) and (d) show the closed-loop experimental waveforms under reverse mode. The dynamic performance under reverse mode is quite similar with that under forward mode.
From the both dynamic experimental results, the following two aspects are verified. (1) The unique feature of CTPS, that is, the relationship between the inductor current and the three phase shifts is zero-order, is correct. (2) It is not necessary to control the inductor current in a closed-loop way and the inductor current is always under control even it is not directly controlled. (3) The direct power based control approach achieves a good control performance and it significantly simplify the closed-loop control configuration.
Furthermore, the efficiency of the DAB converter with the CTPS is test with V 1 =100V and V 2 =25V. The corresponding efficiency curve is shown in Fig.11. The efficiency increases gradually with the increase of the transfer power first and when the power is greater than 0.7, the efficiency decreases gradually with the increase of the transfer power. The maximum efficiency is about 96%.
Finally, the dynamic performance of the proposed approach is compared with several of several closed-loop control strategies of the DAB converter with SPS. In [22], the performance of the traditional voltage closed-loop control (TVL), the load current feed-forward control (LCFF), the model-based phase-shifted control (MPC) and the virtual direct power control (VDPC) are tested and compared with each other. We read the dynamic response data of the output voltage step change and load step change of these strategies from [22] and listed them together with those of the proposed one in Table 2. From Table 2, the dynamic response time and overshoot voltage of the proposed strategies is better than those of CVL, LCFF, MPC and is similar with that of VDPC. It means the proposed approach realizes the closed-loop control of the output voltage with a good steady-state and dynamic performance.
In addition, the reason that the dynamic performance of the CVL, LCFF, MPC is not so good is analysed below. From the built linear model of the DAB converter based on SPS modulation using the fundamental analysis method, the transfer function of phase shift to inductor current is at  least three-order. However, all of these control strategies are based on the average model and do not consider the transient relationship between the phase shift and the inductor current. It slows the dynamic performance.

VI. CONCLUSION
The established average state-space model of DAB DC-DC converter with the CTPS proves that the DAB converter becomes a zero-order system and it makes the closed-loop control of the DAB converter easier. The direct power control based on the expression of the average transfer power as a function of the three phase shifts achieves the good dynamic performance in the two application scenarios of connecting the DAB converter to the DC microgrid or severing as a DC source supply. The research work is helpful to increase the practicability of the CTPS and ensures that the CTPS is able to apply in the actual systems to best develop its advantages of the good current characteristics and full soft-switching operation.