Dynamic compensation based feedforward control against output/input disturbance for LCL-DAB DC–DC converter operated under no backﬂow power

LCL-type resonant dual-active-bridge (LCL-DAB) DC–DC converters have more signif-icant advantages in eliminating backﬂow power and achieving higher voltage gain compared with traditional dual active bridge (DAB) DC–DC converters. Therefore, LCL-DAB DC–DC converters have a broad application prospect in DC micro-grids, electric vehicles, uninterruptible power supplies etc. Since at this stage only a few researches on LCL-DAB DC–DC converters have been done, here, the speciﬁc system operation region with triple-phase-shift (TPS) modulation scheme designed under no backﬂow power is presented. Furthermore, a dynamic compensation based feedforward control method under no backﬂow power is proposed, which enables the LCL-DAB DC–DC converter to operate under no backﬂow power in wide region and achieve outstanding dynamic performance against output/input disturbance. Finally, the experimental results verify the feasibility and effectiveness


INTRODUCTION
Bidirectional DC-DC converters are widely used in distributed generations [1], electric vehicles [2], uninterruptible power supplies [3] etc. Nowadays DC micro grid technology develops rapidly, thus bidirectional DC-DC converters will play more central role in power conversion system [4]. In order to achieve voltage matching and galvanic isolation, low-frequency or highfrequency transformer are often employed in bidirectional isolated DC-DC converters. Compared to low-frequency isolated DC-DC converter which is heavy and lossy, high-frequency isolated DC-DC converter will gradually become the developing trend for its high power density and low cost [5]. As a typical high-frequency isolated DC-DC converter, the dual-active-bridge (DAB) DC-DC converter has attracted increasing attention for their outstanding features: high power density, galvanic isolation, fast dynamic response etc. The symmetrical structure enables the converter to transmit power in both directions and the high-frequency transformer can achieve high power density [6]. For an DAB DC-DC converter, two This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited. © 2021 The Authors. IET Renewable Power Generation published by John Wiley & Sons Ltd on behalf of The Institution of Engineering and Technology sides of the inductor can be regarded as connected two PWM voltage sources, generated with different modulation schemes, such as single-phase-shift (SPS), extended-phase-shift (EPS), dual-phase-shift (DPS) and triple-phase-shift (TPS) modulation scheme. However, the direction of output voltage and current may be opposite at a specified instant, and the corresponding power flows back to the power supply u in . This part of power is called backflow power and it will have negative effects on the converter, for example, increasing conduction loss, copper loss, current stress and decreasing efficiency [7].
There are many methods proposed to reduce the backflow power. In [8][9][10][11][12], some researchers analysed the relationship between backflow power and phase shift ratio, and then calculated the optimal phase shift ratio at each state which can make the converter operate under the minimum backflow power. However, each transmitted power corresponds to only one set of phase shift ratios that can achieve no backflow operation and the model is assumed ideal, which may result in a certain deviation between the calculated optimal phase shift ratio and the real one in practical applications. In [13], reactive current, reactive power and power factor in time domain are uniformly defined and meanwhile the no backflow modulation for optimal power factor is derived. In addition, some other circuit topologies with the advantage of no backflow operation were proposed. A study in [14] connected the inductor between the primary bridge and the DC side, rather than between the primary bridge and the high frequency transformer. Previous work, such as [15,16], proposed an LCL-DAB DC-DC converter which can operate under no backflow power in a wide region because of the addition of an LCL-type resonant tank.
Although the LCL-DAB converters have excellent performance in eliminating backflow power, only a few researches have been done to study its specified system operation region under no backflow power. Researchers analysed the ZVS boundaries for LCL-DAB converters with TPS modulation scheme in [17]. The relationship between the harmonic components and the phase shift ratio in TPS modulation was discussed in [18], where however the but it did not specify how many the harmonic components correspond to no backflow power operation. Furthermore, an enhanced DPS (EDPS) modulation scheme was proposed in [19], which was able to make the converters operate not only under no backflow power but also under full-range soft-switching. This modulation scheme is interesting but lacks accuracy sometimes because the inductor current was approximated by the fundamental component. Besides the backflow power optimization, the optimization of dynamic performance of the converter is also a problem worthy to pay attention on due to usual disturbances. A feedforward compensation control method proposed in [20] was able to improve the dynamic response to a load change, but this control method contains a process of solving the trigonometric functions off-line, which lacks real-time control. A virtual direct power control scheme can also optimize the dynamic performance against an input voltage or a load change, and it is not sensitive to the circuit parameters. However, the sampling value of output voltage is in the denominator of the formula when calculating the control variable, which means that it is necessary to limit the range of the control variable [21]. This paper is organized as follows. Section 2 presents an LCL-DAB converter. Section 3 analyses the design region of the system under no backflow power. Section 4 gives the mathematic model of the LCL-DAB DC-DC converter to calculate the parameters of the controller. In order to eliminate the backflow power, Section 5 estimates an appropriate inner phase shift ratio. On basis of it, a dynamic compensation based feedforward control method under no backflow power is proposed, which can also optimize the transient response to an input voltage or a load change. Finally, with experimental results of a scale-down prototype in Section 6, the proposed dynamic compensation based feedforward control method against output/input disturbance under no backflow power is validated. Figure 1 depicts the typical configuration of a traditional DAB DC-DC converter, which mainly consists of two symmetrical H-bridges, a high frequency transformer, and an inductor. Main waveforms with SPS modulation scheme as an instance are given in Figure 2. Due to phase shift, there is a time difference between waveforms of u P and u S , which results in the voltage difference between the inductor. Therefore, the current of the inductor will change accordingly. It can be easily observed that u P > 0 and i L < 0 in time interval 0 < t < t 0 , which means power flows back to the power supply On the basis of the traditional DAB DC-DC converters, LCL-DAB DC-DC converters replace the single filter inductor with an LCL resonant tank. The topology diagram is shown in Figure 3, from which it can be seen that the resonant tank consists of two filter inductors L 1 and L 2 , and a filter capacitor C 1 . The leakage inductor of the transformer is converted to the primary side and becomes a part of filter inductor L 2 . For symmetric structure, inductances of L 1 and L 2 are considered equal in the following analysis and the turns ratio of the high frequency transformer is N:1. Figure 4 shows the equivalent model of LCL-DAB DC-DC converter. u P and u S are square waves, and both of them can be generated with different modulation scheme. Since this topology has two more components in the LCL resonant tank, the time domain analysis method described in the literature [6] is no longer applicable here. Therefore, the phasor method is adopted in this paper, which solves each sinusoidal variable at different frequencies and then  The equivalent model shown in Figure 4, from which an LCL-DAB DC-DC converter can be regarded as a two port circuit, the input and output characteristics are expressed in Equation (1). Here we define the resonant frequency as s = 1∕

SYSTEM DESCRIPTION
√ LC 1 and the resonant impedance as Z 0 = s L= √ L∕C 1 , where L is the inductance of filter inductors L 1 and L 2 . When the switching frequency equals the resonant frequency, Equation (1) can be converted to Equation (2).
From Equation (2), it can be seen thatU p is 90 • ahead oḟ I S anḋI p is 90 • ahead ofU S . IfU p is 90 • ahead ofU S , the voltage will be in phase with the current at both sides, which means there is no reactive power. According to the relationship between backflow power and instantaneous power described in [22,23], backflow power can be reduced by eliminating reactive power. Due to paper length limitation, the detailed analysis process is not described here. In conclusion, three preconditions of no backflow power operation for LCL-DAB converter can be summarised as follows.
1. The switching frequency equals the resonant frequency of the LCL resonant tank. 2. The phase shift angle between the fundamental waves of u P and u S is 90 • . 3. Harmonic components in converter should be as few as possible.
The first two conditions are easy to meet, so the focus of the literature [18] is on what relationship between the backflow power and harmonic components in LCL-DAB DC-DC converters. However, it only gives the qualitative relationship, but does not give the specific modulation region for the converter operated under no backflow power, which is incomplete. Therefore, the specific operation region under no backflow power is shown below, and the first two conditions are satisfied by default.

SYSTEM OPERATION REGION DESIGN UNDER NO BACKFLOW
In this paper, TPS modulation scheme is adopted for LCL-DAB DC-DC converters because it is a unified form of all phaseshifting modulation methods, where d 1 and d 2 denote the inner phase shift ratio respectively, and d denotes the phase shift ratio between the primary and secondary bridge. According to the second condition in Section 2, ω = ω s and d = 0.25 are defaulted in the following analysis.
To facilitate the analysis, u P can be seen as an odd function, from Figure 5, then the Fourier decompositions of u P and u S are derived as: where U in and U o are the effective value of u in and u o respectively. Equations (3) and (4) can be used to derive the phasor expressions of u P and u S at each frequency, and then the phasor expressions of i P and i S can be solved using phasor method, as shown in Equations (5) and (6).
. (6) If loss-less power transmission is assumed, the active power transmitted by LCL-DAB DC-DC converter can be expressed  The relationship between the two inner phase shift ratio and the ratio of the power generated by the fundamental wave to the actual power by Equation (7). The equation shows that the active power is the superposition of the active power generated by each harmonic component.
Due to the complexity of the above expression, it is essential to simplify the expression. What is shown in the Figure 6 is the ratio of the power generated by the fundamental component to the actual power, where it can be seen that the error is just under 4%. Hence, it is feasible to approximate the power transmitted by LCL-DAB DC-DC converter as the power generated by the fundamental component, and thus the calculation of power in Equation (7) can be simplified as Equation (8).
As the operation mode is symmetric every half cycle and the objective of this paper is to make LCL-DAB DC-DC converters operate under no backflow power, in addition to the first two conditions in Section 2 (ω = ω s and d = 0.25), it is necessary to make the voltage and the current at both sides of the LCL resonant tank have the same polarity. From the analysis above, waveforms of i P and i S are approximately sinusoidal, so it can be simplified to the following four inequalities. Among them Equations (9) and (10) are sufficient conditions for no backflow power at 'P' side, and Equations (11) and (12) are sufficient conditions for no backflow power at 'S' side.
According to Equation (6) it can be found that as long as Equations (9) and (12) are satisfied, there is no backflow power at both 'P' and 'S' side, because the solutions of the first inequality must be contained in the second one, and the third and the forth one are the same. In a word, the goal of no backflow power control at 'P' side can be achieved as long as Equation (9) is satisfied. Similarly, the goal of no backflow power control at 'S' side can be achieved as long as Equation (12) is satisfied. The detailed expressions of Equations (9) and (12) can be respectively written as:  with the analysis of the relationship between harmonic contents and modulation region in the literature [18]. Compared with literature [22], the design region gets wider due to TPS modulation scheme adopted. Figure 8 shows that each design region under no backflow power at 'S' side shrinks with the increase of k. Similarly, due to the symmetry of the converter, the design region of d 1 is maximized when d 2 = 0.167. In this paper, the load is a resistor so the focus of this paper is on the backflow power at 'P' side. Therefore, from Figure 7 it can be seen that choosing an appropriate value of d 1 is able to ensure that the converter operate under no backflow power in most cases.

MODELLING OF LCL-DAB CONVERTER
In order to design the parameters of the controller, an accurate model of LCL-DAB DC-DC converter needs to be built. Generalized average modelling method is utilised whereby the variables are approximated to the zero and first order, which not only avoids the cost of complexity but also has relatively high accuracy. In this method, the dc terms and the first order terms of the current of inductors and the voltage of capacitors are used as state variables and then a full-order continuous-time average model can be built.
According to Figure 5, the time-domain expressions of u P and u S can be expressed as: where s 1 (t) is the switching function at 'P' side and s 2 (t) is the switching function at 'S' side. These two functions can be estab-lished as: Due to the time-varying and nonlinear nature of the converter, it is necessary to represent an average state variable to obtain the time-invariant model, so the Fourier series can be solved by selecting the waveforms in time interval (t-T < τ < T) (19) where ⟨x⟩ nR is the real part of the coefficients of the nth Fourier series, and ⟨x⟩ nI is the imaginary part of the coefficient of the nth Fourier series. As switching signal s 1 (t) and s 2 (t) are all symmetrical in each half cycle, the dc terms of them are both zero. The first coefficient of s 1 (t) and s 2 (t) are: Then the real and the imaginary part of them can be expressed as: If the current of inductors and the voltage of capacitor are determined to be the state variables, the large-signal differential equations of an LCL-DAB converter are Substituting Equations (25) and (26) in the literature [24] into the above equations, the DC and first coefficient terms of Fourier series for each variable above are naturally derived. It is obvious that ⟨i P ⟩ 0 = 0, ⟨i S ⟩ 0 = 0, ⟨u C1 ⟩ 0 = 0 and ⟨u C2 ⟩ 1 = 0, then we can get the new equations, expressed as matrix Equation (27) (shown in the bottom).
d dt According to (27) it can be recognised that it is a seventhorder system. The reason of the increase of order is additional passive filter elements are inserted (an inductor and a capacitor) compared with the traditional DAB converter. For the controller design and the analysis for this converter, it is necessary to build an input-output small-signal model. Assume that each variable is the sum of the large-signal state and the small-signal state, that is: Substituting Equations (28)-(34) into Equation (27) and removing the large-signal variables, the small-signal model of an LCL-DAB converter can be derived in Equation (35) (shown in the bottom), which shows that the system is seventh order. In order to avoid unnecessary calculation and reasonably simplify the mathematic model in practical application, only the relative terms of the output voltage Δ⟨u o ⟩ 0 and the inner phase shift ratio Δd 2 are retained. As a result, the control-to-output transfer function is:

PROPOSED DYNAMIC COMPENSATION BASED FEEDFORWARD CONTROL UNDER NO BACKFLOW POWER
In order to enable the LCL-DAB DC-DC converter to operate under no backflow power and obtain a satisfactory dynamic response against an input voltage change or a load change, this paper proposes a dynamic compensation based feedforward control method under no backflow power. First and foremost, an appropriate value of d 1 is estimated to ensure that the LCL-DAB DC-DC converter operates under no backflow power. Second, the steady-state value of d 2 should be determined to ensure enough margin. Third, a feedforward method is introduced to improve the dynamic performance of the converter. Finally, the parameters of PI regulator are designed according to the established system model.

5.1
The estimation of d 1 According to Section 3, the operation of an LCL-DAB DC-DC converter under no backflow power should be restricted by the design region in Figure 7. The closer d 1 is to 0.167, the larger the design regions of d 2 under no backflow power is. However, other control objectives such as output voltage and transmission power should also be taken in to consideration in actual applications. For example, for a specific d 1 , D 1 denotes the steady-state of it, so the maximum power that an LCL-DAB DC-DC converter can transmit is: Since the power transmitted by the converter is consumed in the load (a resistor), the value of the resistance will constrain the output voltage. Then, the maximum output voltage can be Compared with the traditional DAB converter, an additional filter capacitor is added in an LCL-DAB converter. Therefore, the discrepancy in the ability to regulate the output voltage exists for both types of converters. On the premise of the same inductance, that is, L 1 + L 2 = L, the maximum output voltage can be derived as: where inductor L e is the sum of L 1 and L 2 , as shown in the Figure 9, the solid line represents the maximum voltage gain for an LCL-DAB DC-DC converter for 0 < d 1 < 0.5 and the dotted line represents the maximum voltage gain for a traditional DAB converter with arbitrary modulation method. It is obvious that the voltage gain in an LCL-DAB converter is maximized when d 1 = 0 and decreases with the increase of d 1 , which becomes 0 when d 1 = 0.5. Moreover, when 0 < d 1 < 0.34, the voltage gain for an LCL-DAB converter is higher than that for a traditional DAB converter and lower when 0.34 < d 1 < 0.5. Therefore, it can be seen that introducing an LCL resonant tank do benefit to improve the voltage gain in many cases. However, an excessively small value of d 1 will result in a small or none operation region under no backflow power according to Section 3, which is contradiction with the analysis of the maximum voltage gain mentioned above. Furthermore, large inner phase shift ratio of primary bridge or secondary bridge will lead to greater current stress under the same transmission power conditions, which can cause higher power loss. Trading a significant reduction in one performance for a modest improvement in another thing is not what we want, so the value of d 1 should be weighed and chosen according to the actual situation and requirement.

5.2
The determination of steady-state value D 2 In this part, d 2 denotes the instantaneous value and D 2 denotes the steady-state value of d 2 , and the relationship between them can be seen in Figure 10. According to Figure 7, an appropriate value of d 1 just ensures that an LCL-DAB DC-DC converter can operate under no backflow power in most cases, but not in all cases. Since d 2 is the control variable for this proposed control method, the value of d 2 and D 2 are not exactly the same-the value of d 2 is constantly adjusted around the value of D 2 , even if the error between them is tiny. When d 2 exceeds the corresponding design boundary, there will be backflow power. Therefore, the steady-state operation point of the converter should be determined thoughtfully in order to ensure enough margin when d 2 is adjusted in the design region under no backflow power, so as to avoid d 2 beyond the design region in adjustment.

5.3
Proposed dynamic feedforward compensation control Figure 10 shows output voltage control block diagram with the traditional constant feedforward compensation, where a constant D 2 is feedforward to improve system dynamic performance. It is worth noting that a delay term e −T d s is included after the PI controller, due to the calculation delay of digital micro-controllers. This control method is simple and feasible whereas the transient response to disturbance is unsatisfactory, because the control variable always takes a constant as a steadystate point and adjusts around it. When the input voltage or the load changes suddenly, the burden of adjusting the control variable will be all on the PI regulator. Hence, an optimized control method is proposed here, as shown in Figure 11, which is based dynamic feedforward compensation.
For any LCL-DAB converter with TPS modulation scheme, Equation (40) is always satisfied, and thus the feedforward expression can be redesigned. Nevertheless, the derived expression contains resonant impedance Z 0 , which means a dependence on main circuit parameters of the LCL-DAB DC-DC converter. In practical applications, it is difficult to obtain accurate inductance, and the inductance also varies with the current. Therefore, two approaches can be chosen according to actual requirement to calculate the feedforward compensation. One is calculating the feedforward compensation directly by inverting Equation (40), that is an approximation approach; the other is an accurate approach. In order to avoid the inverse influence of inaccurate inductance on control performance, a set of steadystate operation points is utilized to cancel out Z 0 term in the formula, as shown in Equation (41) (42), where k ff = cos(πD 2 )U in /I o , U in , I o and d 2 are a set of steadystate operation points used to replace the existence of Z 0 , and D ′ 2 represents the calculated feedforward compensation according to u in and i o . It can be seen that the compensation is able to optimize the dynamic response against output/input disturbance according to the current state of the converter, which has better flexibility compared with the traditional output voltage control method with constant feedforward compensation mentioned before.

The parameters of PI regulator
As shown in Figure 11, the PI regulator is also chosen for the closed-loop control of output voltage. The parameters of the PI regulator can be calculated according to the method presented in [20]. From Figure 11, the open-loop transfer function of the control system is derived as: where m is a negative constant according to Section 4. The crossover frequency of this control system is expressed as ω c , and the phase margin is expressed as φ m . Therefore, the phase of open-loop transfer function at ω c is given by: Define integral time as: Equations (47) and (48) present the design rules of PI regulator parameters with a specified phase margin when the closeloop output voltage control method is adopted. In the following experiments, a PI regulator with the same parameters will be used to compare the responses using the two different control strategies.

EXPERIMENTAL RESULTS
In order to verify the dynamic compensation based feedforward control against output/input disturbance for LCL-DAB DC-DC converter operated under no backflow power, the prototype of an LCL-DAB DC-DC converter has been built. The photo of this prototype is shown in Figure 12 and main system parameters are listed in Table 1. For experimental comparison, the converter was operated with the traditional constant First of all, the effectiveness of the proposed control method to eliminate backflow power was verified. Three groups of experimental parameters were set, that is, (a) U in = 100 V, Figures 13 and 14 are the steady-state waveforms respectively with two control methods. To highlight the experimental phenomenon, the value of d 1 was estimated to 0 in Figure 13, and the value of d 1 was estimated to 0.1 in Figure 14 (according to Section 5.1). Although two control methods in Figures 13 and 14 were based on feedforward compensation in order to make a unified comparison, there is no significant difference in the steady-state results with the two feedforward compensation methods. Actually, only the value of d 1 had effect on the steady-state results. After the output voltage stable around the reference value, it can be easily seen the difference on backflow power with two control method. Figure 13(a-c) show that the waveforms of u P and i P have different polarity in certain time interval because of the inappropriate value of d 1 , which means that the backflow power is not eliminated in the input voltage source. On the contrary, since d 1 is determined to be 0.1 in Figure 14(a-c), i P always passed the zero scale in the period u P = 0. As a result, the backflow power was basically eliminated in the input voltage source with the proposed control method for three experimental parameters. Generally, Figures 13 and 14 show that the value of d 1 significantly affect the amount of backflow power in the operation of an LCL-DAB DC-DC converter, and this also proves the analysis in Section 3.
In order to better verify the relationship between control variables (d, d 1   results confirmed the analysis in Figure 9. In addition, these five groups of u P and i P waveforms indicated that the backflow power was eliminated in the input voltage source, throughout the whole experiments. In general, the LCL-DAB DC-DC converter was able to operate under no backflow power in wide region and achieve high voltage gain. From Figure 16(a,b) it can be seen that the settling time with the traditional control method is 265 ms, slightly shorter than that with the proposed control method 319 ms. Moreover, the overshoot with traditional output voltage control method is 32.6% and the overshoot with dynamic compensation based feedforward control method is 49.7%. The reason of this phenomenon is that the steady-state point of the d 2 in Figure 16(b) was dynamic compensated. The proposed control method in this paper made the converter operated at a control variable adjusted on the base of the sampled data. In contrary, the traditional control method made the LCL-DAB DC-DC converter operated at an expectant steady-state value of d 2 . Although this  Figure 17(a,b) compared the transient response with two control methods when input voltage U in changed from 100 to 140 V. Output voltage was first increased due to the input voltage step, and then was adjusted back around the reference value by the controller. It is easily to observe the process of output voltage adjustment in Figure 17(a) with a long settling time 231 ms and a voltage overshoot 4.4 V. Due to dynamic feedforward compensation, the voltage overshoot in Figure 17(b) was tiny 2.3 V and the settling time was about 188 ms.
As shown in Figure 18(a,b), when the load changed from 100 to 50 Ω, since output voltage was supporting by capacitor C 2 , the change of output voltage lag slightly behind load current i o . The output voltage dropped immediately and then was adjusted to the original reference value. However, the output voltage fluctuated violently and the undershoot reached 21.9 V, and the settling was u P to 328 ms. Figure 18(b) shows the transient response with dynamic compensation based feedforward control method, from which it can be seen that output voltage undershoot reached just 4.5 V at the most serious situation. Furthermore, the settling time was about 206 ms, obviously less than the one above. According to the above analyses of experimental results shown in Figures 17 and 18, the proposed control method in this paper has a superior dynamic performance against both input voltage and load disturbance.

CONCLUSION
This paper proposes a dynamic compensation based feedforward control method against output/input disturbance for LCL-DAB DC-DC converters operated under no backflow power. By analysing the relationship between the inner phase shift ratio in primary bridge and the operation region under no backflow power, a specified value of d 1 can be estimated, enabling the LCL-DAB DC-DC converter operate under no backflow power. The inner phase shift ratio d 2 in secondary bridge is set to be the control variable, making output voltage follow the reference voltage stably. In order to optimize the transient response to an input voltage change or a load change, a dynamic feed-forward compensation loop is added. Finally, experimental results obtained from a scale-down prototype are given to verify the feasibility and effectiveness of the proposed control method.