Dual-Active Bridge Series Resonant Electric Vehicle Charger: A Self-Tuning Method

electronics Article Dual-Active Bridge Series Resonant Electric Vehicle Charger: A Self-Tuning Method Alireza Namadmalan 1, Kumars Rouzbehi 2,* , Juan Manuel Escaño 2and Carlos Bordons 2 1 Department of Electrical and Computer Engineering, Jundi-Shapur University of Technology, 84154 Dezful, Iran; [email protected] 2Departamento de Ingeniería de Sistemas y Automática, Universidad de Sevilla, 41092 Sevilla, Spain; [email protected] (J.M.E.); [email protected] (C.B.) *Correspondence: kr[email protected] Received: 25 November 2019; Accepted: 27 January 2020; Published: 3 February 2020   Abstract: This paper presents a new self-tuning loop for a bidirectional dual-active bridge (DAB) series resonant converter (SRC). For different loading conditions, the two active bridges can be controlled with a minimum time displacement between them to assure zero voltage switching (ZVS) and minimum circulation current conditions. The tuning loop can instantly reverse the power direction with a fast dynamics. Moreover, the tuning loop is not sensitive to series resonant tank tolerances and deviations, which makes it a robust solution for power tuning of the SRCs. For simplicity, the power is controlled based on the power-frequency control method with a fixed time displacement between the active bridges. The main design criteria of the bidirectional SRC are the time displacement, operating frequency bandwidth, and the minimum and maximum power, which are simply derived and formulated based on the self-tuning loop’s parameters. Based on the parameters of the tuning loop, a simplified power equation and power control method is proposed for DAB-SRCs. The proposed control method is simulated in static and dynamic conditions for different loadings. The analysis and simulation results show the effectiveness of the new tuning method. Keywords: battery charger; dual active bridge; zero voltage switching; self-oscillating methods 1. Introduction Bidirectional direct current-direct current (DC-DC) converters are widely employed in various applications such as renewable energy, battery chargers of electric vehicles (EVs), DC micro-grids and other battery energy storage system (BESS) applications. Typically, to achieve soft switching with superior electro-magnetic interference (EMI) considerations, these converters are implemented based on resonant converters [ 1 – 4 ]. This kind of converters have different topologies with respect to their power ratings, voltage levels and input or output required voltage (current) source models. Among them, series resonant converters (SRCs) are the most resilient and practical topologies, which have various control methods, reduced numbers of passive elements and improved efficiency. Moreover, SRCs are more compatible with BESS due to their intrinsic voltage source behavior [3–7]. For high power applications such as fast charging of EVs, a dual-active bridge SRC (DAB-SRC) is the well-known structure for the bidirectional power flow control at EV charging infrastructures [ 4 – 7 ]. Figure 1shows a charging stage of EVs based on DAB-SRCs to achieve bidirectional power flow capability between the grid and the electric vehicle, i.e., V2G capability [ 8 – 10 ]. Another advantage of DAB-SRC is that it can be implemented in a modular structure without the need for extra input or output filters, as shown in Figure 1. Electronics 2020,9, 253; doi:10.3390/electronics9020253 www.mdpi.com/journal/electronics Electronics 2020,9, 253 2 of 15 Electronics 2020, 9, 253 2 of 15 Current researches are mainly based on algorithms which utilize phase-locked loop (PLL) techniques. PLLs are well-known techniques to tune the switching frequency of resonant converters; however, they are sensitive to uncertainty and tolerances in resonant tank circuits [3]. Due to high power rating of fast DC chargers, about 100–200 kW, robustness and ability to cope with deviations or tolerances in a resonant tank are essential [13]. AC Network AC-DC DC-DC BESS DC-DC EV charger-1 DC-DC EV charger-n r C Active Bridge r L DAB-SRC-1 LC Filter DAB-SRC-m Active Bridge LC Filter Figure 1. An electric vehicle (EV) charging station with n-number of charging stages, while each stage comprises of m-number of paralleled dual-active bridge series resonant converters (DAB-SRCs). To address this problem, a self-tuning method is proposed which has fast dynamics and is not sensitive to the tolerances. Recently, this method has been utilized for power and frequency tuning of wireless charging of EVs using inductive power transfer (IPT) technology, DC-DC converters and various battery chargers [3,14–17]. Moreover, in self-tuning methods, all the switching methods such as pulse width modulation (PWM), pulse density modulation (PDM) and the phase-shift control method can be implemented. However, in the available researches, the bidirectional capability of the SRCs is not considered in the self-tuning method. Moreover, the formulations and analysis of the selftuning are derived for an ohmic load [14–17]. In this paper, design considerations of the DAB-SRC such as DC-link voltages, bandwidth of the switching frequency, the minimum and maximum of the transferred power and the time displacement between the two active bridges are derived based on the self-tuning loop’s parameters. In addition, a simple control method is proposed which can change the direction of the output power with less transients and power fluctuations. Another advantage is direct control of the phase displacement between the dual active bridges hence; the minimum phase displacement can be achieved simply, which is essential for a DAB-SRC [11,12]. As presented in [16], the proposed method can be implemented based on phase shift and PWM modulations which is essential for DAB-SRCs with different DC-link voltages. However, for the sake of simplicity, the power and frequency tuning is devised based on phase shift controlling between the two active bridges which is a usual solution for approximately equal DC-link voltages [7]. The rest of this paper is organized as follows: a description of DAB-SRC with and its mathematical model are presented in section 2. Section 3 presents modeling and formulation of the proposed tuning loop for the DAB-SRC. Section 4 presents simulation results and modification of the analysis to verify the proposed control method in the transient and steady-state conditions and the main conclusions of the paper are summarized in section 5. Figure 1. An electric vehicle (EV) charging station with n-number of charging stages, while each stage comprises of m-number of paralleled dual-active bridge series resonant converters (DAB-SRCs). The most important problems associated with the state-of-the-art are the power and frequency tuning of these power converters with minimum phase displacement or circulating current [7,10–12]. Current researches are mainly based on algorithms which utilize phase-locked loop (PLL) techniques. PLLs are well-known techniques to tune the switching frequency of resonant converters; however, they are sensitive to uncertainty and tolerances in resonant tank circuits [ 3 ]. Due to high power rating of fast DC chargers, about 100–200 kW, robustness and ability to cope with deviations or tolerances in a resonant tank are essential [13]. To address this problem, a self-tuning method is proposed which has fast dynamics and is not sensitive to the tolerances. Recently, this method has been utilized for power and frequency tuning of wireless charging of EVs using inductive power transfer (IPT) technology, DC-DC converters and various battery chargers [ 3 , 14 – 17 ]. Moreover, in self-tuning methods, all the switching methods such as pulse width modulation (PWM), pulse density modulation (PDM) and the phase-shift control method can be implemented. However, in the available researches, the bidirectional capability of the SRCs is not considered in the self-tuning method. Moreover, the formulations and analysis of the self-tuning are derived for an ohmic load [14–17]. In this paper, design considerations of the DAB-SRC such as DC-link voltages, bandwidth of the switching frequency, the minimum and maximum of the transferred power and the time displacement between the two active bridges are derived based on the self-tuning loop’s parameters. In addition, a simple control method is proposed which can change the direction of the output power with less transients and power fluctuations. Another advantage is direct control of the phase displacement between the dual active bridges hence; the minimum phase displacement can be achieved simply, which is essential for a DAB-SRC [11,12]. As presented in [ 16 ], the proposed method can be implemented based on phase shift and PWM modulations which is essential for DAB-SRCs with different DC-link voltages. However, for the sake of simplicity, the power and frequency tuning is devised based on phase shift controlling between the two active bridges which is a usual solution for approximately equal DC-link voltages [7]. The rest of this paper is organized as follows: a description of DAB-SRC with and its mathematical model are presented in Section 2. Section 3presents modeling and formulation of the proposed tuning loop for the DAB-SRC. Section 4presents simulation results and modification of the analysis to verify Electronics 2020,9, 253 3 of 15 the proposed control method in the transient and steady-state conditions and the main conclusions of the paper are summarized in Section 5. 2. Modeling of a Dual-Active Bridge Series Resonant Converter (DAB-SRC) In this section, the principle of operation for a DAB-SRC is set forth as follows. Figure 2shows the two active bridges with DC-link voltage of V in for the charger side and V Batt for the EV side as the battery voltage level. In this figure, for simplicity, the input/output filters are not considered and a high-frequency transformer (HFT) is utilized for isolation and better impedance matching between these two sides. In the proposed tuning loop, a current transformer (CT) is used for the resonant tank current sensing, io, while Iorepresents phasor state of io. Electronics 2020, 9, 253 3 of 15 2. Modeling of a Dual-Active Bridge Series Resonant Converter (DAB-SRC) In this section, the principle of operation for a DAB-SRC is set forth as follows. Figure 2 shows the two active bridges with DC-link voltage of Vin for the charger side and VBatt for the EV side as the battery voltage level. In this figure, for simplicity, the input/output filters are not considered and a high-frequency transformer (HFT) is utilized for isolation and better impedance matching between these two sides. In the proposed tuning loop, a current transformer (CT) is used for the resonant tank current sensing, io, while Io represents phasor state of io. Two simple first order phase shifters are considered for phase displacement between the two bridges, while the Micro-Controller Unit (MCU) can digitally implement these two phase shifters in the discrete time mode. The active bridge that is supposed to send the power is responsible to regulate the power, hence only the leading phase shifter is considered variable. By digital implementation, the system will be more compact and less sensitive to tolerances occurring in the phase shifter circuits. For analog implementation, the variable leading phase shifter resistor, RT1, can be implemented using a digital-potentiometer or combination of a light-emitting diode and light-dependent resistor (LEDLDR) as discussed in [17]. In this implementation, the digital-potentiometer or light of LED can be controlled directly from the MCU [17]. In this paper, the two bridges are switching with duty cycle of about 50%, i.e., neglecting the dead time, td, between the low side and high side switches. Hence, the two bridges can be considered as two square wave voltage sources connected together by the series resonant tank, i.e., resonant inductor, Lr and resonant capacitor, Cr. Using fundamental components, each square voltage source can be modeled as a sinusoidal voltage source as shown in Figure 3. In this Figure, Rs is the winding resistance of the HFT and Io is considered as the reference phasor. E1 and E2 represent fundamental harmonics of the output voltage of the charger side and EV side referred to the primary side of the HFT, while e1 and e2 represent their instantaneous state, respectively. Moreover, δ1 and δ2 are phase angles of the aforementioned sinusoidal voltages with respect to the reference phasor, i.e., Io. Hence, for the forward power transfer, battery charging, and considering zero voltage switching (ZVS) for the two active bridges, the phase angles must be derived as δ1 > 0 and δ2 < 0. This rule is reversed when the battery sends its energy back to the station side while ZVS is achieved [6–10]. In the following equations, each phasor is considered based on its maximum value. E1, E2 and the natural angular frequency of the power converter are derived by the following equations according to the system parameters: 1 1 2 2 4 4 in Batt V E, nV E. n π π = = (1a) (1b) Sb3 Sb2 Sb1 Sb4 Batt V GateDrivers&dead-time circuits Sa3 Sa2 Sa1 Sa4 in V o i 1 e r C CT ZeroDetector Lead Phase Shifter GateDrivers&dead-time circuits 1T R 1T C r L 12 :nn Constant Lag Phase Shifter 2T R 2T C ZeroDetector MCU ADC 1 To T R 2 e Trigger1 Trigger2 Sb1 Sb4 Sb2 Sb3 Sa1 Sa4 Sa2 Sa3 S C S C S C S C S C S C S C S C Figure 2. Schematic of the DAB-SRC for a bidirectional battery charging system using the proposed self-tuning method. Figure 2. Schematic of the DAB-SRC for a bidirectional battery charging system using the proposed self-tuning method. Two simple first order phase shifters are considered for phase displacement between the two bridges, while the Micro-Controller Unit (MCU) can digitally implement these two phase shifters in the discrete time mode. The active bridge that is supposed to send the power is responsible to regulate the power, hence only the leading phase shifter is considered variable. By digital implementation, the system will be more compact and less sensitive to tolerances occurring in the phase shifter circuits. For analog implementation, the variable leading phase shifter resistor, R T1 , can be implemented using a digital-potentiometer or combination of a light-emitting diode and light-dependent resistor (LED-LDR) as discussed in [ 17 ]. In this implementation, the digital-potentiometer or light of LED can be controlled directly from the MCU [17]. In this paper, the two bridges are switching with duty cycle of about 50%, i.e., neglecting the dead time, t d , between the low side and high side switches. Hence, the two bridges can be considered as two square wave voltage sources connected together by the series resonant tank, i.e., resonant inductor, L r and resonant capacitor, C r . Using fundamental components, each square voltage source can be modeled as a sinusoidal voltage source as shown in Figure 3. In this Figure, R s is the winding resistance of the HFT and Iois considered as the reference phasor. Electronics 2020,9, 253 4 of 15 Electronics 2020, 9, 253 4 of 15 1 n rr . LC ω = (2) The impedances presented in Figure 3 and the equivalent impedance, Xt, between these two bridges neglecting the series resistor, Rs ≈ 0, are defined as follows: () 1 Crs Lrs X C, X L. ω ω − = = (3a) (3b) () 22 1− =−= − tLC L ns X XX X . ωω (4) where, ωs is the angular switching frequency. Regarding Figure 3 and Rs ≈ 0, the power-angle equation is derived as follow in the forward power direction, i.e., the battery is charging [7]. L jX s R C jX− 11 E δ ∠ 22 E δ ∠ 0 0 o I∠ Figure 3. Phasor model of the DAB-SRC at the steady-state condition. () 12 12 2t EE P sin . X δδ =− (5) Regarding (5), there are different strategies for power flow regulation which are the frequency control, phase control or combination of them [4–10]. Using frequency control, Xt changes to regulate the power while phase control has direct effect on the two phase angles, δ1 and δ2. The relation between E1, E2 and Io, is derived by (6), considering Rs ≈ 0. Hence, the relation between angles and amplitude of the input and output voltages are derived by (7b). 11 2 2 to so E EjXIRI. δδ ∠= ∠+ + (6) 112 2 12 21 in Batt E cos E cos , Vncos . Vncos δδ δ δ = ≈ (7a) (7b) Moreover, the relation between amplitude of resonant tank current, angles and voltage amplitudes are derived as: 112 2 112 2 to o t Esin Esin XI, Esin Esin I. X δδ δδ =+ − = (8a) (8b) To achieve ZVS, the snubber capacitors, Cs, of the power switches should be considered. Hence, a minimum required phase displacement for each active bridge is defined for the worst case, the minimum current or light load conditions. For the charger and the battery side bridges, the minimum phase displacements for the active bridges with respect to the minimum resonant tank current Iomin Figure 3. Phasor model of the DAB-SRC at the steady-state condition. E 1 and E 2 represent fundamental harmonics of the output voltage of the charger side and EV side referred to the primary side of the HFT, while e 1 and e 2 represent their instantaneous state, respectively. Moreover, δ1 and δ2 are phase angles of the aforementioned sinusoidal voltages with respect to the reference phasor, i.e., I o . Hence, for the forward power transfer, battery charging, and considering zero voltage switching (ZVS) for the two active bridges, the phase angles must be derived as δ1 >0 and δ2 <0. This rule is reversed when the battery sends its energy back to the station side while ZVS is achieved [ 6 – 10 ]. In the following equations, each phasor is considered based on its maximum value. E 1 ,E 2 and the natural angular frequency of the power converter are derived by the following equations according to the system parameters: E1=4Vin π, (1a) E2=4n1VBatt n2π. (1b) ωn=1 √LrCr . (2) The impedances presented in Figure 3and the equivalent impedance, X t , between these two bridges neglecting the series resistor, Rs≈0, are defined as follows: XC=(Crωs)−1, (3a) XL=Lrωs. (3b) Xt=XL−XC=XL1−ω2 nω−2 s. (4) where, ωs is the angular switching frequency. Regarding Figure 3and R s≈ 0, the power-angle equation is derived as follow in the forward power direction, i.e., the battery is charging [7]. P=|E1||E2| 2Xt sin(δ1−δ2). (5) Regarding (5), there are different strategies for power flow regulation which are the frequency control, phase control or combination of them [ 4 – 10 ]. Using frequency control, X t changes to regulate the power while phase control has direct effect on the two phase angles, δ1 and δ2 . The relation between E 1 ,E 2 and I o , is derived by (6), considering R s≈ 0. Hence, the relation between angles and amplitude of the input and output voltages are derived by (7b). |E1|∠δ1=|E2|∠δ2+jXtIo+RsIo. (6) |E1|cosδ1=|E2|cosδ2, (7a) Vin VBatt ≈n1cosδ2 n2cosδ1 . (7b) Electronics 2020,9, 253 5 of 15 Moreover, the relation between amplitude of resonant tank current, angles and voltage amplitudes are derived as: |E1|sinδ1=|E2|sinδ2+XtIo, (8a) Io=|E1|sinδ1−|E2|sinδ2 Xt. (8b) To achieve ZVS, the snubber capacitors, C s , of the power switches should be considered. Hence, a minimum required phase displacement for each active bridge is defined for the worst case, the minimum current or light load conditions. For the charger and the battery side bridges, the minimum phase displacements for the active bridges with respect to the minimum resonant tank current I omin are derived by (9) and (10), respectively, to assure ZVS for the two active bridges. Equations (9) and (10) are derived assuming that the resonant inductor is large enough to keep the current constant at the dead time, td, of switching. δmin 1=sin−1 2CsVin Imin otd!. (9) δmin 2=sin−1 2Csn2VBatt Imin otdn1!. (10) For DAB-SRC there are six possible conditions regarding voltage ratio of the two DC-link sides and three possible conditions are depicted in Figure 4in the forward power flow, the battery is charging. Regarding Figure 4, the equivalent impedance is considered as a pure inductive, meaning that R s≈ 0 and ωs > ωn . For Figure 4a,c, the corresponding phase angle of the lower voltage is significantly derived from the lower rather than the other side phase angle. Hence, in these conditions the lower angle should be considered at the minimum required phase angle for ZVS even for full load condition. This fact is essential for minimum phase displacement achieving and reducing the circulation current. This approach requires precise controlling methods which are depend on system parameters identification. Electronics 2020, 9, 253 5 of 15 are derived by (9) and (10), respectively, to assure ZVS for the two active bridges. Equations (9) and (10) are derived assuming that the resonant inductor is large enough to keep the current constant at the dead time, td, of switching. 1 1 2 min sin min od CV s in . It δ − =  (9) 12 2 1 2 min sBatt min od CnV s in . Itn δ − =  (10) For DAB-SRC there are six possible conditions regarding voltage ratio of the two DC-link sides and three possible conditions are depicted in Figure 4 in the forward power flow, the battery is charging. Regarding Figure 4, the equivalent impedance is considered as a pure inductive, meaning that Rs ≈ 0 and ωs > ωn. For Figure 4a and Figure 4c, the corresponding phase angle of the lower voltage is significantly derived from the lower rather than the other side phase angle. Hence, in these conditions the lower angle should be considered at the minimum required phase angle for ZVS even for full load condition. This fact is essential for minimum phase displacement achieving and reducing the circulation current. This approach requires precise controlling methods which are depend on system parameters identification. Another fact is that for unequal voltages, the system requires higher operating frequencies to assure ZVS condition due to large phase difference between the two bridges. In this condition, the wave forms are far from a sinusoidal equation, also achieving ZVS is more difficult for the lower voltage active bridge. Figure 4b is an idea condition for DAB-SRCs by achieving an equal phase displacement for the two bridges and intrinsic ZVS for simply operating frequencies higher than the natural frequency of the system. This condition E1 ≈ E2 is proposed for high power application and power and frequency control methods can be applied simply. Moreover, in this condition, the operating frequency can be selected near the natural frequency to achieve almost pure sinusoidal waveforms to reduce power losses in the HFT. 2 E 1 E o I to j XI 2 δ 1 δ 2 E o I to j XI 2 δ 1 δ 1 E 2 E 1 E o I to j XI 2 δ 1 δ (a) (b) (c) Figure 4. Phasor diagram of E1, E2 and Io for different voltage ratio conditions in forward power transfer. Regardless of the three possible conditions and their corresponding applications, previous research required complex control methods for power and frequency-tuning methods which mainly depend on parameters of the DAB-SRC [7]. These control methods are sensitive to possible system tolerances. Moreover, systems that are based on PLLs have transients to achieve ZVS which imposes extra EMI at transient conditions and is not recommended for high power applications. In the next section, the proposed tuning loop is devised based on self-oscillating methods presented in [17] and the main formulation of the system is derived based on DAB-SRC. The new tuning method can simply assure ZVS condition, regardless of system tolerances. Moreover, the tuning loop has fast dynamics in start-up and power direction changing. Figure 4. Phasor diagram of E 1 ,E 2 and I o for different voltage ratio conditions in forward power transfer. Another fact is that for unequal voltages, the system requires higher operating frequencies to assure ZVS condition due to large phase difference between the two bridges. In this condition, the wave forms are far from a sinusoidal equation, also achieving ZVS is more difficult for the lower voltage active bridge. Figure 4b is an idea condition for DAB-SRCs by achieving an equal phase displacement for the two bridges and intrinsic ZVS for simply operating frequencies higher than the natural frequency of the system. This condition E 1≈ E 2 is proposed for high power application and power and frequency control methods can be applied simply. Moreover, in this condition, the operating frequency can be selected near the natural frequency to achieve almost pure sinusoidal waveforms to reduce power losses in the HFT. Regardless of the three possible conditions and their corresponding applications, previous research required complex control methods for power and frequency-tuning methods which mainly depend on parameters of the DAB-SRC [ 7 ]. These control methods are sensitive to possible system tolerances. Electronics 2020,9, 253 6 of 15 Moreover, systems that are based on PLLs have transients to achieve ZVS which imposes extra EMI at transient conditions and is not recommended for high power applications. In the next section, the proposed tuning loop is devised based on self-oscillating methods presented in [ 17 ] and the main formulation of the system is derived based on DAB-SRC. The new tuning method can simply assure ZVS condition, regardless of system tolerances. Moreover, the tuning loop has fast dynamics in start-up and power direction changing. 3. Modeling of the Proposed Tuning Loop Similar to the other self-tuning methods for voltage source inverters [ 3 ], the switching signals are generated by the series resonant load current, i o , which is received by the MCU using CT, as shown in Figure 2. In this figure, output of the CT is passed through the two phase shifters. For forward power flow, the variable leading phase shifter is considered for the left active bridge and the constant lagging phase shifter is considered for the battery side active bridge. For changing the power flow direction, the two phase shifters are replaced by each other while the CT signal must be considered with 180 0 phase shift, which can be done simply by using NOT gates at the outputs of the two zero detector units. Without loss of generality, the system formulation and analysis are considered in the forward direction. For the DAB-SRC start-up, the series resonant tank should be charged up by turning Sa1, Sa4, Sb1 and Sb3, as shown in Figure 2, i.e., Trigger1 is applied. Hence considered zero initial condition for the tank circuit, the instantaneous resonant current, i o , is derived by the following for the first half cycle and considering that the leading phase shifter makes no phase displacement at start-up. io=qCr Lrsin(ωnt), 0< ωnt< π.(11) After the first half cycle, the EV side bridge is activated for switching by applying the Trigger2 signal, while the leading phase shifter makes the minimum time constant, τ1min , defined by the MCU, which corresponds to the minimum output power, P min . The lagging phase shifter has a fixed time constant, τ2 . In the self-tuning method, the state of the switches reversed at the zero crossing of the phase shifters’ output signals. In the positive half cycles of leading phase shifter output, Sa2 and Sa3 are turned on and in the negative half-cycle Sa1 and Sa4 are turned on. For the EV side, Sb1 and Sb4 are turned on for the negative half cycles and Sb2 and Sb3 are turned on for the positive cycles of the output signal of the lagging phase shifter. Hence, considering the time constants for the two phase shifters, the phase displacements for the two active bridges are derived as follows: δ1=tan−1ω−1 sτ−1 1, (12a) τ1=RT1CT1. (12b) δ2=−tan−1(ωsτ2), (13a) τ2=RT2CT2. (13b) where C T1 ,R T1 are the tuning capacitor and resistor of the leading phase shifter while R T1 is a variable resistor which can be implemented with a digital potentiometer connected to the MCU. R T2 and C T2 are the tuning resistor and capacitor of the lagging phase shifter which are constant and should be designed to make the minimum phase displacement that assures ZVS at the predefined light load condition, I omin or P min . Regarding (10) and (13b), τ2 is derived by (14b), where ωmax is the predefined maximum switching frequency of the DAB-SRC at the light load condition: ωmaxτ2=tan sin−1 2Csn2VBatt Imin otdn1!!, (14a) Electronics 2020,9, 253 7 of 15 τ2=1 ωmax v u u t1 1−2Csn2VBatt Imin otdn12−1. (14b) Despite previous research on the self-tuning methods that are based on one active bridge, here the two phase displacements of the active bridges must be satisfied. Hence, to derive a relationship between the time constants of the proposed tuning loop and the angular switching frequency of the converter Equation (7) should be considered. Equation (15b) derives the relationship between the switching frequency and the time constants: E1 E2≈cosδ2 cosδ1 =costan−1(ωsτ2) costan−1ω−1 sτ−1 1. (15a) E1 E2 =v t1+(ωsτ1)−2 1+(ωsτ2)2. (15b) For conditions where E 1≈ E 2 , as described in Figure 4b, Equation (15b) is simplified in (16), which shows that the angular switching frequency is equal to geometric mean of the two time constants.        ωs≈1 √τ1τ2for 1 √τ1τ2≤ωn. ωs=ωnfor 1 √τ1τ2 > ωn.(16) Hence, the minimum and maximum values for τ1 whit respect to the predefined minimum, ωmin , and maximum, ωmax, operating frequencies of the converter can be derived by the followings: τmax 1=Rmax T1CT1=1 τ2ω2 min , (17a) τmin 1=Rmin T1CT1=1 τ2ω2 max . (17b) The phase displacements can be derived by (18) with respect to the time constants. It is worth noting that regarding (18), the converter intrinsically reduces the required phase displacement for high power conditions. Equation (19) derives the time displacement, t δ , between these two bridges and shows that for ωmin < ωs < ωmax , the time displacement is approximately constant which is essential for high power loads, where a better power factor is needed. δ1=−δ2=tan−1 rτ2 τ1!. (18) tδ=δ1−δ2 ωs =2√τ1τ2tan−1 rτ2 τ1!≈2√τ1τ2rτ2 τ1 =2τ2. (19) Regarding (5) and (17), the transferred power between the two bridges is derived according to the time constants of τ1and τ2, as follows: P=8n1VBattVin π2n2Lrωs1−ω2 nω−2 ssin(ωstδ). (20) Regarding (2), (16) and (18), the resonant inductor and capacitor can be designed according to the maximum power and ωmin . Comparing Equations (5) and (20), it is concluded that the new tuning method proposes a simplified equation for power regulation, which not only depends on ωs or variable Electronics 2020,9, 253 8 of 15 τ1 but also intrinsically applies the required phase displacement at light load and gradually decreases the phase displacement for high power conditions, as described. 4. Simulation Results and Improved Feedback Circuit In this section, the DAB-SRC is tuned based on the proposed self-oscillating power and frequency control loop. In the following simulations, the main parameters of the power converter are L r =100 µ H, C r =100 nF, V in =200 V, V Batt =200 V, τ2 =1 µ s and τ1 varies between 2 µ s up to 10 µ s depending on the desired output power. The DC-links of the two active bridges are compensated by LC filters with filter’s inductor of 100 µ H and capacitor of 200 µ F. For simplicity, the power switches and HFT are considered as ideal elements and the maximum and minimum operating frequencies are considered 85.5 kHz and 53.5 kHz, respectively. The maximum and minimum power for equal 200 V level for the two active bridges are designed as 4.5 kW and 700 W, respectively. Figure 5shows the output voltages of the active bridges at start-up, while the tuning loop is set at the lowest output power, the worst case for start-up and achieving ZVS condition. Electronics 2020, 9, 253 8 of 15 Lr = 100 µH, Cr = 100 nF, Vin = 200 V, VBatt = 200 V, τ2 = 1 µs and τ1 varies between 2 µs up to 10 µs depending on the desired output power. The DC-links of the two active bridges are compensated by LC filters with filter’s inductor of 100 µH and capacitor of 200 µF. For simplicity, the power switches and HFT are considered as ideal elements and the maximum and minimum operating frequencies are considered 85.5 kHz and 53.5 kHz, respectively. The maximum and minimum power for equal 200 V level for the two active bridges are designed as 4.5 kW and 700 W, respectively. Figure 5 shows the output voltages of the active bridges at start-up, while the tuning loop is set at the lowest output power, the worst case for start-up and achieving ZVS condition. 1.10 1.151.05 1.00 Time(ms) Resonant Tank Current[A] Voltageof Charger Bridge[V] Voltageof EV Bridge[V] 0 20− 0 200− 200 0 200− 200 Start-up Sb2&Sb3 are ON 20 Figure 5. Output voltage of charger bridge, EV bridge and io under equal voltages at start-up condition with τ1 = 2 µs and τ2 = 1 µs. In this simulation, Sb2 and Sb3 are switched on for about one cycle of the resonant current, about 20 µs, for charging-up of the series resonant tank, as can be seen from Figure 5. The time constants of the tuning loop are considered as τ2 = 1 µs and τ1 = 2 µs and the converter is considered in forward power flow, meaning that the switching states of the charger bridge is determined by zero crossings of the lead phase shifter. As can be seen from Figure 5, the converter properly starts without current stresses according to its nominal peak current rating for io, which is about 40 A, moreover, ZVS is achieved at the beginning of the cycles. Figure 6 shows the output voltages of the active bridges and the resonant tank current at steadystate condition and forward power direction for τ2 = 1 µs and τ1 = 2 µs. It can be seen that ZVS is achieved for the two bridges with minimum phase displacements or circulating currents. In condition of Figure 6, the transferred power from the right side bridge to the EV side is almost 700 W with operating frequency of about 87.5 kHz. Regarding Equation (16) and τ2 = 1 µs and τ1 = 2 µs, the operating frequency is derived about 112.5 kHz. The difference between analysis and simulation result is caused by the non-sinusoidal behavior of io at light load conditions. To improve this, a new feedback circuit is devised at the end of this section. Figure 5. Output voltage of charger bridge, EV bridge and i o under equal voltages at start-up condition with τ1=2µs and τ2=1µs. In this simulation, Sb2 and Sb3 are switched on for about one cycle of the resonant current, about 20 µ s, for charging-up of the series resonant tank, as can be seen from Figure 5. The time constants of the tuning loop are considered as τ2 =1 µ s and τ1 =2 µ s and the converter is considered in forward power flow, meaning that the switching states of the charger bridge is determined by zero crossings of the lead phase shifter. As can be seen from Figure 5, the converter properly starts without current stresses according to its nominal peak current rating for i o , which is about 40 A, moreover, ZVS is achieved at the beginning of the cycles. Figure 6shows the output voltages of the active bridges and the resonant tank current at steady-state condition and forward power direction for τ2 =1 µ s and τ1 =2 µ s. It can be seen that ZVS is achieved for the two bridges with minimum phase displacements or circulating currents. In condition of Figure 6, the transferred power from the right side bridge to the EV side is almost 700 W with operating frequency of about 87.5 kHz. Regarding Equation (16) and τ2 =1 µ s and τ1 =2 µ s, the operating frequency is derived about 112.5 kHz. The difference between analysis and simulation result Electronics 2020,9, 253 9 of 15 is caused by the non-sinusoidal behavior of i o at light load conditions. To improve this, a new feedback circuit is devised at the end of this section. Electronics 2020, 9, 253 9 of 15 3010 3015 3005 Time(μs) 3020 3025 3030 Voltageof Charger Bridge Voltageof EV Bridge 200V 158t.s δ μ ≈ 50A. o i Figure 6. Output voltage of charger bridge, EV bridge and io under equal voltages condition with τ1 = 2 µs and τ2 = 1 µs. For the maximum power, the converter is simulated for τ2 = 1 µs and τ1 = 10 µs and the power direction is considered forward. The output voltages and io are shown in Figure 7 which shows that the converter properly operates with the proposed minimum phase displacement at the high power condition. In Figure 7, the transferred power is about 4.5 kW with operating frequency of 53.4 kHz, which is approximately in fair agreement with Equations (16) and (20). To show the effectiveness of the proposed tuning loop, the unequal DC-link voltages condition is simulated for Vin = 200 V and VBatt = 180 V. In the following simulation, the maximum output power is achieved for τ1 ≈ 4.7 µs while the operating frequency is about 54.5 kHz, as shown in Figure 8. As can be seen, in this simulation, minimum phase displacement required for ZVS is achieved which shows the proper performance of the proposed tuning loop under unequal DC-link voltages. Regarding simulations, for VBatt < Vin, the required time constant to achieve maximum output power occurs between τ1max and τ1min derived by (17). Regarding Figure 6 and Figure 7, tδ is approximately constant from low power up to high power conditions. Moreover, regarding simulation results, Equation (19) should be modified to Equation (21). 9020 9030 9010 Time(μs) Voltageof Charger Bridge Voltageof EV Bridge o i 200V 36 5A. 160t.s δ μ ≈ Figure 7. Output voltage of charger bridge, EV bridge and io under equal voltages condition with τ1 = 10 µs and τ2 = 1 µs. Figure 6. Output voltage of charger bridge, EV bridge and i o under equal voltages condition with τ1 = 2µs and τ2=1µs. For the maximum power, the converter is simulated for τ2 =1 µ s and τ1 =10 µ s and the power direction is considered forward. The output voltages and i o are shown in Figure 7which shows that the converter properly operates with the proposed minimum phase displacement at the high power condition. In Figure 7, the transferred power is about 4.5 kW with operating frequency of 53.4 kHz, which is approximately in fair agreement with Equations (16) and (20). Electronics 2020, 9, 253 9 of 15 3010 3015 3005 Time(μs) 3020 3025 3030 Voltageof Charger Bridge Voltageof EV Bridge 200V 158t.s δ μ ≈ 50A. o i Figure 6. Output voltage of charger bridge, EV bridge and io under equal voltages condition with τ1 = 2 µs and τ2 = 1 µs. For the maximum power, the converter is simulated for τ2 = 1 µs and τ1 = 10 µs and the power direction is considered forward. The output voltages and io are shown in Figure 7 which shows that the converter properly operates with the proposed minimum phase displacement at the high power condition. In Figure 7, the transferred power is about 4.5 kW with operating frequency of 53.4 kHz, which is approximately in fair agreement with Equations (16) and (20). To show the effectiveness of the proposed tuning loop, the unequal DC-link voltages condition is simulated for Vin = 200 V and VBatt = 180 V. In the following simulation, the maximum output power is achieved for τ1 ≈ 4.7 µs while the operating frequency is about 54.5 kHz, as shown in Figure 8. As can be seen, in this simulation, minimum phase displacement required for ZVS is achieved which shows the proper performance of the proposed tuning loop under unequal DC-link voltages. Regarding simulations, for VBatt < Vin, the required time constant to achieve maximum output power occurs between τ1max and τ1min derived by (17). Regarding Figure 6 and Figure 7, tδ is approximately constant from low power up to high power conditions. Moreover, regarding simulation results, Equation (19) should be modified to Equation (21). 9020 9030 9010 Time(μs) Voltageof Charger Bridge Voltageof EV Bridge o i 200V 36 5A. 160t.s δ μ ≈ Figure 7. Output voltage of charger bridge, EV bridge and io under equal voltages condition with τ1 = 10 µs and τ2 = 1 µs. Figure 7. Output voltage of charger bridge, EV bridge and i o under equal voltages condition with τ1 = 10 µs and τ2=1µs. To show the effectiveness of the proposed tuning loop, the unequal DC-link voltages condition is simulated for V in =200 V and V Batt =180 V. In the following simulation, the maximum output power is achieved for τ1≈ 4.7 µ s while the operating frequency is about 54.5 kHz, as shown in Figure 8. As can be seen, in this simulation, minimum phase displacement required for ZVS is achieved which shows the proper performance of the proposed tuning loop under unequal DC-link voltages. Regarding simulations, for V Batt <V in , the required time constant to achieve maximum output power occurs between τ1max and τ1min derived by (17). Regarding Figures 6and 7,t δ is approximately constant from low power up to high power conditions. Moreover, regarding simulation results, Equation (19) should be modified to Equation (21). tδ≈1.59τ2. (21)