Efﬁciency Optimization Design of L-LLC Resonant Bidirectional DC-DC Converter

: The new type of L-LLC resonant bidirectional DC-DC converter (L-LLC-BDC) has merits of high efﬁciency, high-power density and wide gain and power ranges, and it is suitable for energy interface between energy storage systems and DC micro grid. However, the resonances are sensitive to the parasitic parameters, which will deteriorate the efﬁciency. This paper investigates the intrinsic mechanism of parasitic parameters on the L-LLC-BDC operating principle and working characteristics based on the analysis of working modes and resonance tank. By taking the oscillation of parasitic parameters produced in the stage for the freewheeling stage into consideration, a parameter optimization method is proposed to reduce the resonant current oscillation while maintaining the characteristic of the natural soft switching. The experiment results not only validated the proposed parameter optimization design method, but also testiﬁed to the improvement of the efﬁciency through the minimization of the conduction and switching loss.


Introduction
The energy storage systems (ESSs) in the DC micro grid have a wide range of output voltages and currents, which requires a bi-directional DC-DC converter with a wide range of voltage gain and power receptivity. L-LLC resonant bidirectional DC-DC converter (L-LLC-BDC) can be used as the key equipment for interaction between ESSs and the DC micro-grid for stabilization of the DC bus voltage.
In the literature [1], the additional inductance is added to the traditional LLC resonant converter to form L-LLC-BDC, enabling a symmetrical topology that can work in both the forward and reverse directions. As this converter's operating characteristics not only depend on its working point, but also its working frequency and load condition as well, it is difficult to analyze. Ref. [2] redesigned the converter with the fundamental harmonic approximation, and the resonant voltage and current are proximately equivalent to sinusoidal waveform at resonant frequency. However, the low approximation accuracy at those points outside the resonance point cannot meet the requirements, and the gain formula obtained is deviated from the actual values to some extent. Although the parameters of the converter are designed, the key parameters are empirical values, and the design method is difficult to popularize and apply. In practical application, due to the influence of parasitic parameters, the real resonant frequency is deviated from the theoretical value, and the operating characteristics of the converter will be different from the theoretical analysis as well. Methods such as the third harmonic approximation and extended describing function are used to improve the accuracy of analysis [3,4]. However, the analytic solutions are in nature obtained through approximation.
In order to improve the efficiency of the traditional LLC resonant converter, other methods are used, e.g., optimal control strategy [5][6][7], optimization of the dead time and magnetic inductance [8], and peak gain method [9,10]. Recently, the new design method considering peak gain is proposed [11,12], the LLC resonant converter will get the peak gain point when the input square-wave voltage is synchronous with the resonant current. A modified gain model and its corresponding design method for LLC resonant converter are proposed [13], in which both resonant and load factor are considered and discussed in detail by combining time and frequency domain together. As complete charging and discharging within the dead time, it is necessary for the converter to achieve soft switching, then influence of the output capacitance of the switch should be taken into consideration [14][15][16][17]. The existing literature usually considers the output capacitance of the primary side switch, but rarely the secondary side capacitance. Ref. [18] proposes a primary side switch in parallel with the capacitance to mitigate the problem of the high output voltage under light load condition. Ref. [19] analyzes the output voltage oscillation caused by the parasitic capacitance of the rectifier side diode under different switching frequencies for the centertapped LLC converter, and they proposed a new parameter design idea, which has certain limitations. Ref. [20] proposed a new topology of LLC to improve the efficiency with the increase of components; however, the power flow was unidirectional. The more accurate time domain analysis was adopted in [21][22][23][24][25].
Based on an accurate time domain analysis of the L-LLC-BDC operation process, this paper proposes a parameter optimization design method, which takes into account the effect of the output capacitance of the switch. The proposed method effectively eliminated the current oscillation when switches turn on by configuring switching frequency, the conduction and switching loss of the converter are reduced by larger magnetizing inductance of the converter on the basis of satisfying the voltage gain of the converter. The efficiency is increased. This paper is organized as follows: Section 2 introduces the working principle of the Bi-directional L-LLC resonant converter; Section 3 describes the time-domain analysis of the converter; Section 4 proposed a parameter optimization design; Section 5 provides the experimental results, and Section 6 concludes this paper. Figure 1 shows the topology of L-LLC-BDC, whose primary and secondary sides are connected to a full-bridge converter, respectively. Where V in and V O represent the voltage of ESSs and the DC bus, respectively, L m1 represents magnetizing inductance of the transformer, L r and C r are resonant inductor and capacitor, and L m2 is the additional inductor. S 1 -S 4 and S 5 -S 8 are switches on the primary and secondary converter, respectively. D 1 -D 8 and C 1 -C 8 are body diodes and output capacitors corresponding to S 1 -S 8 . As indicated in Figure 1, the forward power direction is defined as from the primary side to the secondary side. To simplify the analysis as well as control, L m2 is set the same as L m1 so that the working principles in forward and reverse power direction are exactly the same. Therefore, only the working principle analysis in the forward mode is provided. In the forward mode, the output voltage is regulated with the frequency modulation method. When the converter works in the continuous conduction mode, L r and C r form a resonant tank, while in the discontinuous conduction mode, L m1 and L m2 participates the resonance in the forward and reverse mode, respectively. For example, in the forward mode, the L m2 does not participate in the resonance, but it helps the switches on the primary-side switch to realize ZVS. Therefore, so does L m1 in the reverse mode.

Working Principle of the L-LLC Resonant Bi-Directional DC-DC Converter
Considering the influence of the MOSFET output capacitor on the working process of L-LLC-BDC, Figure 2 is intended to show the main steady-state waveforms of the converter, according to the status of the switches, and there are a total 6 modes. Due to the symmetric waveforms in the positive and negative period, the working principles in these periods are similar, only the analysis in the first half of the cycle is provided in this paper. The primary side currents are i p = i Lr + i Lm2 and i s = n (i Lr − i Lm1 ).

Mode I
Mode I: t∈ [t 0 , t 1 ] (see Figure 2), during which switches S 1 and S 4 are turned on. Before t 0 , the output capacitors of the switches S 1 and S 4 discharge to zero voltage through i p , thus the switches S 1 and S 4 can be turned on under the ZVS condition, v AB = V in . The body diodes of the secondary side switches S 5 and S 8 are conducted, which clamping v CD = V O energy is delivered to the load.

Mode II
Mode II: t∈ [t 1 , t 2 ] (see Figure 2), during which switches S 1 and S 4 are still turned on, L r and L m1 resonate with C r , the resonant current is equal to the magnetizing current and the secondary side current i s is zero. With the current being zero, the body diodes D 5 and D 8 are shut off naturally without reverse recovery loss, realizing ZCS of the secondary switch. The output voltage no longer clamps the two points of C-D, and the output capacitor C 5 -C 8 participate in the resonance. This mode is actually a continuous current stage.

Mode III
Mode III: t∈ [t 2 , t 3 ] (see Figure 2), during which switches S 1 and S 4 are off, entering into the dead time. The i p charges the C 1 and C 4 until their voltage reach to V in meanwhile. C 2 and C 3 discharges and their voltage drops to zero, which enables S 2 and S 3 to be turned on under the ZVS condition. The secondary current i s charges C 5 and C 8 to a voltage of V O , and the voltage S 6 and S 7 decreases to zero. At the end of this stage, after the charging and discharging, the voltage at A-B is −V in , and the voltage at C-D is clamped at −V O . Figures 3-5 show the equivalent circuits for Modes I, II and III and the differential equations are established, respectively, and we can get the expression of resonant inductor current and resonant capacitor voltage.

The Time-Domain Analysis of L-LLC Resonant Bi-Directional DC-DC Converter
The first harmonic approximation (FHA) only considers the component at the fundamental frequency, while the high order harmonics are neglected, which makes it lack the capability to identify different operating modes. Therefore, it is necessary to build an accurate steady-state model to describe the converter's operating process and characteristics. In the period of Mode II, the output capacitor of the secondary side switch participates in resonance, and the four output capacitors are connected in series in pairs and then in parallel, which can be then equivalent to the capacitance C o . In the Mode III, which corresponds to the dead time, C 1~C4 and C 5~C8 are connected in series in pairs and then in parallel, equivalent to the capacitor C in and C o , respectively. Due to the short time of Mode III, both i Lm1 and i Lm2 can be taken as unchanged. With the simplified circuit and initial conditions of each resonant operating mode in Figures 3-5, the voltage of the resonant capacitor, i.e., u Cr , i p as well as i s can be obtained in each mode.

Mode I
The time-domain expression of Mode I: where, I Lr0 and U cr0 are the current of resonant inductor L r and voltage of the capacitor C r at time t 0 , respectively; I m represents the peak current of magnetizing inductance L m1 and additional inductance L m2 ; Z 0 the characteristic impedance, the initial value of the voltage of resonant capacitor at time t 0 , and ω 0 is the resonant frequency, which can be calculated as follows:

Mode II
The time-domain expression of Mode II: where, A1, A2, A3 and A4 are where t 1 < t < t 2 , I Lr1 is the initial value of the current of resonant inductor at time t 1 , U Cr1 the initial value of the voltage of resonant capacitor at time t 1 , and U Co1 the initial value of the voltage of output capacitor at time t 1 .

Mode III
Since the dead time is very short, it can be considered that in this period both current of magnetizing inductance L m1 and additional inductance L m2 are constant at I m . The voltage of the equivalent capacitor C in and C o are constant at U 1 and U 2 , respectively.
The time-domain expression of Mode III: In the expression, I Lr2 is the initial value of the resonant inductor current at time t 2 , and U Cr2 the initial value of the resonant capacitor voltage at time t 2 . Z 2 represents the characteristic impedance, and ω 2 the resonant frequency.
During this period, the switches S 1 -S 4 are all turned off, i p and i s are charged and discharged C 1 -C 4 and C 5 -C 8 , respectively. The operating principle is shown in Figure 6 below. The voltage expressions of u C2 (t) and u C6 (t) in this period can be obtained.

The Parameter Optimization Design of L-LLC Resonant Bi-Directional DC-DC Converter
It can be seen from Figure 2 and the mathematical analysis for each working mode that the output capacitor of the switch participates in the operation of the circuit only in Mode II and III. When t 1 < t < t 2 , v CD is no longer clamped to V O , the output capacitors of the switches and L m1 form a resonant tank, which results in the high frequency oscillation at both primary and secondary side switches. Such, oscillation not only increases the conduction loss of the switch, but also affects the charging and discharging of the output capacitor during the dead time., i.e., t 2 < t < t 3 . At the dead time, both side switches are disconnected, C 1 -C 4 and C 5 -C 8 are charged by i p and discharged by i s , respectively. To maintain the same topology in forward and reverse modes, parameters such as the output capacitance of MOSFET are kept to be symmetric, so that capacitance referred from the secondary side is equal to that on the primary side. The decrease of the primary side current i p will affect the soft switch of the converter. If the primary side current flows in reverse and drops the zero before the end of this period, the capacitors cannot be fully discharged, which results in the failure of the primary side switch to realize ZVS. Moreover, the change of current in dead-time makes the converter continue to a continuous current stage even when the switching frequency equals the resonant frequency.
The effect of the aforementioned output capacitor of MOSFET toward the operation of the converter can be eliminated by parameter optimization to improve the performance of the converter. The first consideration in the design of the resonant component is to reduce the loss of the converter to improve its efficiency. As L-LLC-BDC has the characteristic of the natural ZVS soft switching, the conduction loss accounts for a major part. It was testified that the current circulation can be reduced by increasing the magnetic inductance. However, there are trade-offs, with the increasing of magnetizing inductor, its current will be decreased, which prevents switches from achieving ZVS soft-switching.
As can be seen from Figure 2, the voltage variation range of the output capacitance is the largest in the dead time, large i p and i s are required for charging and discharging the capacitance. In practice, due to the presence of the stray inductance in the circuit, the voltage and current oscillation in the stage of Mode II are both attenuated. Therefore, the design should be based on the first zero crossing of i s to guarantee the ZVS soft-switching. Then, there is no operating Mode II.
According to the optimization goal of L-LLC-BDC, the starting point of the proposed improvement design method is to ensure that the magnetizing current fully charges and discharges the output capacitor of the switch within the dead time T d , so as to achieve ZVS.
From Equation (5), the peak magnetizing current can be obtained: It can be seen from Equation (6) that with the selected device and given dead time, the peak magnetizing inductor current I m can be obtained. When i p and i s fully charges and discharges the primary side output capacitor and secondary side capacitor of the switch within the dead time, the waveform of the converter is the same as Figure 7. This is the resonant mode of the converter, and the resonant frequency is slightly higher than the resonant frequency. Under this condition, the charge and discharge of the capacitors are completed in the dead time, but due to the current on the additional inductance, the primary switch can realize ZVS completely. The resonant current has little distortion when the switch is turned on, and the expressions of the primary and secondary side current in the period T B (from t 0 to t 12 ) can be obtained. Since the instantaneous value of i Lr at time t 0 is 0, and the instantaneous value of i Lr at time t 12 is I m , then i s can be obtained: The field current i Lm1 at t 12 in the period of Mode 1 can be linearly approximated as: When the converter works in the resonant mode, energy from i s to the load within period as shown in the following expressions: Given the resonant frequency f 0 , rated power P O , output voltage V O and dead time T d , the value of the period T B from t 0 to t 12 can be obtained. At this time, the ideal resonant To analyze the effect of output capacitance toward the voltage gain M, simulation studies are carried under the different switching frequency with and without output capacitance. Figure 8 shows that voltage gain vs. switching frequency under the different load with the parasitic capacitance. Its characteristic curve is basically the same as that of the converter that does not consider the output capacitance. However, the gain has a sudden increase with the increase of switching frequency when there is no load or the load is very light. According to reference [9], the required maximum gain expressed by the minimum and maximum input voltages U inmax and U inmin is the maximum gain at full load: where, M max is the maximum gain which is achieved at full load when f s = f min . More details of M max can be found in Equation (A1). U inmax and U inmin are the maximum and minimum input voltage, f max is the maximum switching frequency. As the L m1 is known, the resonant inductance ratio L r can be calculated from Equation (11), and the resonant capacitor can be derived by the resonant frequency f 0 . In practical design, the current and voltage oscillation in the continuous current stage can also be reduced by increasing the resonant capacitance value properly.
The optimum design flow is a simple mathematical expression, the design flow of L-LLC-BDC resonance parameters is shown in Figure 9. According to the initial conditions, the RMS current expressions of the magnetizing inductance I p and the additional inductance I Lm1 and I Lm2 at the quasi-resonant frequency are as follows: As the ideal resonant period is less than the resonance period, i.e., T B < T 0 , the optimal L m1 (meaning the magnetic inductance is large enough) is calculated according to charging and discharging of the output capacitance of the switch in the dead time. When the output voltage and the load is constant, the effective value of the primary side current I p and the effective values of the magnetic inductance current as shown in expressions Equation (12) will decrease, leading to improved efficiency of the converter. Even working at the other frequency, the reduced magnetizing current can also improve the efficiency of the converter.

Experimental Verification
To verify the optimization design method proposed in this paper, the prototype of L-LLC resonant bidirectional DC-DC converter is built. The overall experiment set-up is shown in Figure 10. The input voltage varies from 630 V to 890 V, while the output voltage is 380 V. The operating frequency range is 80-120 kHz, the actual working resonant frequency f is 104 kHz, dead time is 100 ns, and output capacitor is 92 pF. A 220 pF capacitor is connected in parallel to the secondary side switch to ensure the forward and reverse topology of the converter are exactly the same. The all SiC MOSFET are mounted on the heat sink, and adopts air cooling at the same time. The comparison between parameter II before [1] and after optimization is listed in Table 1. To regulate the output voltage, a feedback controller is designed. Figure 11 shows the simplified control block diagram. A conventional linear regulator, such as PI compensator, controls the switching frequency to regulate the output voltage V O to deal with the load change. The pulse frequency modulation (PFM) is applied to the converter, and the control system is built in TMS320F28335.

Steady State Waveforms
The waveforms of the voltage U ds1 at both ends of the primary side switch which work with the forward direction and the primary side current i Lr and their detailed waveforms are shown in Figure 12. As shown in Figure 12a, when the switch is off, there is high frequency oscillation before the parameter's optimization because the capacitor of the switch is not fully charged or discharged during the dead time. It can be seen from Figure 12b that in the continuous flow stage, the output capacitance of S 1 is fully discharged in the dead time, the oscillation has been significantly reduced compare what is shown in Figure 12a, which validates the effectiveness of the parameter optimization. The waveforms of the voltage U ds1 at both ends of the primary side switch that work with the forward direction and the primary side current i Lr and their detailed waveforms are shown in Figure 12. As shown in Figure 12a, when the switch is off, there is a high frequency oscillation before the parameter's optimization because the capacitor of the switch is not fully charged or discharged during the dead time. It can be seen from Figure 12b that in the continuous flow stage, the output capacitance of S 1 is fully discharged in the dead time, and the oscillation has been significantly reduced compare the shown in Figure 12a, which validates the effectiveness of the parameter optimization.   Figures 13-15 show the experiment waveforms with the reverse transmission when the converter is half loaded, lightly loaded, and fully loaded, respectively. Figure 13 shows the driving waveforms of switch, the waveforms of the voltage U ds5 at both ends of the primary side switch that works in the reverse direction at light load, the primary side current i s , the resonant capacitor voltage U cr , and their detailed waveforms. As can be seen from Figures 13-15, the ZVS is achieved in a wide range of loads, e.g., the ZVS is achieved when the output current is 0.4 A, 4 A and 8 A.   Similarly, Figure 16 compares the waveforms of the secondary side with and without optimization. U ds5 has oscillation in the forward direction, which is caused by the parasitic oscillation between the resonant inductor and the MOSFET parasitic capacitor when all switches are turned off after half a resonant period. Figure 17 shows the waveforms of the voltage U ds5 and the secondary side current i s when the load current is 0.4 A in forward transmission power. The waveforms of the voltage U ds1 of the secondary side switch S 1 , and the secondary side current i P when the load current is 0.4 A in reverse transmission power is shown in Figure 18. It can be seen that both voltage and current oscillations are decreased with forward transmission and reverse transmission significantly after the optimization.      In order to carry out a bi-directional power transmission experiment, the control block diagram of charging and discharging battery experiment system is designed, as shown in Figure 21. When the reference current is −10 A, the DC power supply charges the battery, the converter operates in forward power transmission. When the reference current is 10 A, the battery is discharged to the DC power supply, and the converter runs in reverse. The dynamic waveforms of the charging and discharging battery are shown in Figure 22. The figure shows that the charge and discharge current can follow the reference, and forward and reverse power flow can be achieved.  The dynamic waveforms of charging and discharging battery using L-LLC resonant converter.

Efficiency
The operation efficiency is measured by a power analyzer under different input voltage levels and load conditions, and electronic load is used for load regulation. Figures 23 and 24 show the efficiency curves of the design method under different loading conditions with forward and inverse transmission power. As can be seen from Figure 23 especially, when the output power is 3 kW, the maximum efficiency of the experimental device can reach up to 94.9% with forward transmission power. The design method that has been optimized delivers improved loading efficiency compared with that in the literature [4]. This is due to uncontrolled rectifier is used in the secondary side to reduce the driving loss and maximum magnetic inductance also reduces the turn-off current of the switch. In particular, light or moderate loading efficiency is 1% higher than that in the literature [4], because the time domain analysis method is taken to obtain the maximum value of L m1 , which is independent of load change. The loss ratio i Lm1 increases under light load, so the efficiency increases greatly under light load. The oscillation at full load is significantly reduced, and the efficiency is increased.     The figures show that the optimized parameters in this paper can realize the closedloop control of the converter by adjusting the frequency of the switch under different input voltages. Figure 28 shows the operation efficiency of L-LLC-BDC resonant converter in forward mode under the different input voltage. The efficiency can reach up to 96% at rated power with an input voltage of 760 V. From reference [26], for the Sic device, the ratio of hysteresis energy loss E diss and Coss stored energy E oss is less than 0.1, even to 0.01. The device selected in this paper is C2M0080120D, E OSS is 27 µJ when the input voltage is 760 V from the datasheet, so the maximum hysteresis energy loss E diss is less than 2.7 µJ, even to 0.27 µJ, and maximum hysteresis loss of a single device is 0.27 W when the switching frequency is 100 kHz, even to 0.027 W. The proportion of the hysteresis loss is very small for the 6 kW converter.
Switching loss includes turn-on loss and turn-off loss.
(1) Turn-on loss: The resonant converter achieves soft switching, which ideally yields zero losses, but various studies show that even if the soft switching is realized, there is hysteresis loss in the switch. The loss is very small by analysis quantitatively in the second question. (2) Turn-off loss: The turn-off loss of the switch in the primary side is about 2 W, turn-off loss of the switch in the secondary side ideally yields zero because the ZCS is realized when the converter operates at the main mode. In the actual state, C oss transfers most loss to the conduction loss, However, the overall loss of the switch remains unchanged.
The main loss of the resonant converter is the conduction loss of the switch. The efficiency is improved because the current value passing through the switch of primary side is reduced in the design. Due to the large magnetic inductance of the transformer, the core loss of the transformer is slightly larger than that before optimization.
The power loss distribution and comparison of the before optimization and after optimization have been done and presented in Figure 29, and the loss breakdown has been down at full load and 760 V input.

Conclusions
For L-LLC resonant bidirectional DC-DC converter, with the effect of the output capacitance of the power device taken into account, a parameter optimization design method is proposed by configuring switching frequency and satisfying the gain range of the converter to achieve high efficiency.
Based on the complete charging and discharging of the output capacitance of the switch in the dead time, the magnetizing inductance is calculated. The configured switching frequency is calculated according to energy conservation law, and the inductance ratio satisfying the gain condition can be calculated according to the maximum voltage gain, and finally all the resonant parameters can be calculated. The experimental results show that the voltage and current oscillation is obviously eliminated, and the conduction and switching loss of the converter are reduced. The high efficiency can be ensured.  where, 0 ≤ ωt ≤ ωt 3 , I Lr3N , I LM13N , ωt, ωt 30 and M are unknown quantities, u Lm13N (ωt) is the voltage of magnetizing inductance. Stage indirect continuity condition: Symmetry condition of switching moment: ω(t + t 10 ) + ω(t + t 20 ) + ω(t + t 30 ) = f s π f (A7) The zero current peak gain is: