Abstract
Emerging technologies such as electric vehicles, solid-state transformers, and DC transformers are implemented using the closed-loop bi-directional dual-active-bridge (DAB) converter. In this context, it is necessary to have average models that provide an efficient way of tuning the proportional integral (PI) compensator parameters for large- and small-signal applications. We present a novel small-signal model (SSM) for DAB converter with a single closed-loop PI controller and the total elimination of reactive current (). The method applies a modulation technique for and introduces a composite function in the controller while reducing the original non-linear switching model, which allows to decrease the order of the transfer function and analyze the closed-loop operation. The proposed SSM is analyzed using different response time, load, and DC voltage changes. The simulation and experimental results demonstrate the ease of implementation and effectiveness of the proposed model with respect to other SSM techniques.
DUAL-ACTIVE-BRIDGE (DAB) converter is an attractive, modern, and opportune technology for several applications that involve DC-DC conversion stages [

Fig. 1 Topology and applications of DAB.
The voltage difference between the HFT windings produces a current flow IL dependent upon: the leakage inductance Lh, the parasitic resistance , the phase shift carrier , and the duty ratio between the modulation signals and m2. and represent the DC currents of the H-bridge converters. and represent the DC power. More specifically, PDC2 represents the dissipated power in the resistive load RL.
The reactive current appears as negative peak currents in both the port converters and . The elimination of the reactive current, i.e., a reactive current () in DAB converters presents multiple advantages [
In [
The PVM technique demonstrates advantages over the rest of the modulation methods, as it preserves the bi-directional active power transfer without generating reactive currents for all operation modes. However, small-signal model (SSM) can guarantee the operation of an optimal DAB converter in modern applications, as it proposes a proportional integral (PI) criterion to adjust the parameters of feedback circuit. The following section discusses some of the literature on SSM for DAB converters.
In [
In [
In [
The SSM for two DAB converters operating in parallel is presented in [
In [
Applying SPSC modulation again in [
In [
A three-phase DAB configuration is discussed in [
The above studies do not consider the total reactive power elimination attached to a simple feedback. We propose an SSM based on the PVM technique to eliminate reactive power in the DAB converter [
This rest of the paper is organized as follows. In Section II, the non-linear switching model is introduced. Average models as well as the process for obtaining the closed-loop SSM based on the PVM, are given in Section III. In Section IV, the simulations and comparative models are introduced. In Section V, the experimental results are presented. And Section VI draws the conclusion.
The switching states are considered to perform a closed-loop dynamic analysis of DAB, which helps to obtain an operation model based on independent voltage and current sources. The PVM technique analyzes the DAB as a constant DC at both and ports with the transformation ratio coefficient . is defined as the ratio of the turn number at the primary to the turn number at the secondary NP and NS of HFT, that is:
(1) |
According to (1), the performance of the PVM technique depends upon the topological states of .
For , the PVM must follow the basic rule, i.e., to equalize the charged voltage per second of charged flowing through to the discharged voltage per second.
Using
(2) |
(3) |
in the inductance L of HFT is given by:
(4) |
where is the initial condition for ; and is the switching frequency period. Based on (1), is defined as:
(5) |
The currents flowing through the DC converter terminals are:
(6) |
(7) |
(8) |
For the DAB converter,

Fig. 2 Equivalent circuit of DAB converter.
Based on the equivalent circuit of DAB converter, the time domain voltage and current equations are:
(9) |
(10) |
The differential equation in (9) describes on the high-frequency AC side of the DAB converter, while (10) describes model of the DC port. Using (9) and (10), we obtain a transfer function in the frequency domain of each DC and AC circuit as a precursive step towards developing a full model of the converter plant.
By solving AC part in (9) for the DAB converter model and then applying Laplace transform, can be written as:
(11) |
By substituting (2) and (3) into (11), can be re-written as:
(12) |
Similarly, by solving the DC part of the DAB model, the stored energy in the capacitor is given by:
(13) |
where depends on the difference between the injected power at the capacitor input and the power at its output . The power difference is given by:
(14) |
(15) |
(16) |
By substituting (15) and (16) into (14), and then solving in Laplace domain, we have:
(17) |

Fig. 3 Main value of , , and for PVM.
The topological control states can be used to exhibit the advantages of PVM over the methods of SPSC and DPSC. These advantages are the IQ elimination through the DAB power range and the current stress reduction on semiconductor devices. According to [
and are also depicted in
The average is given as:
(18) |
where is the switching frequency. The process necessary to obtain (18) is described in [
The external feedback loop is implemented using as a constant reference to control the output of DC-bus voltage. This scheme maintains voltage regulation of for a wide range of load values using a single PI, giving as the output of the PI compensator of the form:
(19) |
where is the error voltage; is the proportional gain of ; and is the integral gain of .
Finally, between the output variable of the PI compensator and required in the modulation system, it is possible to propose a new intermediate function , which is useful in the linearization process of the complete converter scheme defined in Section III.
The control structure is implemented using (12), (17) and (19), generating a non-linear system. This scheme is built considering the dynamic characteristics of the instantaneous voltages and currents within the DAB converter.
The control structure shown in

Fig. 4 Controller and plant model for DAB converter using PVM.
This study proposes a replacement of the current generation process with the approximation of (18), i.e., to eliminate AC switching frequency in all converter variables. In the steady state, . Therefore, the plant model depicted in

Fig. 5 Controller and plant for average DAB converter model using PVM.
(20) |

Fig. 6 Controller and plant for reduction of average DAB converter.
From (18), it is considered that operates in the range of , therefore:
(21) |
Under the above assumption, a new equation is generated between the compensator output value and , where the objective is to create a directly proportional relationship between and , which is given by:
(22) |
The compound equation is achieved by substituting (22) in (21):
(23) |
(24) |
is given as:
(25) |
The equation can be further reduced if the following equality is achieved:
(26) |
Canceling the zero of the compensator with the plant pole , (25) is reduced as:
(27) |
The closed-loop equation of the converter is obtained as:
(28) |
where is represented as a first-order equation, which is one of the main contributions in this paper. The response time is considered as:
(29) |
remains as:
(30) |
In (30), can be used to control the voltage at the inductor terminals . This control action also helps set the flow direction and magnitude of , which in turn helps manage the charge and discharge of . Finally, the switched model of

Fig. 7 Controller and plant for reduction of average DAB converter.
The analysis of the transfer function is performed on the optimal values for the compensators and . It is proposed to use the desired response times, where is given as:
(31) |
Based on the equalization proposed in (26), is given as:
(32) |
To build a system in (28), the numerical values of the system elements are replaced and corroborated by the simulation results, and discussed in the following section.
After modeling a transfer function for the dynamics of closed-loop DAB converter and eliminating the switching components and the non-linearities shown in
Assume that the phase shift at full load is the intermediate point of the operation range, i.e., . Using a switching frequency of 5 kHz, and considering that , the inductance value considering is given by:
(33) |
In this case, the obtained value is and a parasite resistance is assigned. Finally, the used capacitance value is , for a voltage ripple with the value of 5%. The desired time response at is calculated using (31) and (32), and denotes that the elapsed time reaches 63% of the final output value. The values of and are obtained as and respectively. The closed-loop function given in (23) is a function of which generates . In the same way, the PI values are calculated for different response time. The simulation parameters are shown in
Once the parameters are identified, the proposed PI tuning criteria for a DAB converter are applied. This is achieved by considering the operation with the PVM technique and the obtained results with the proposed average model in
These results are analyzed for a wide range of different steps such as start-up operation, load step changes, and voltage variations at , which are attached to the waveforms of the operation of DAB converter in the scheme of PVM.

Fig. 8 Simulation results for dynamics of closed-loop DAB converter. (a) Step response for s. (b) Step response for s. (c) Wave forms of , , , and IL in steady state. (d) Wave forms of , , , and in steady state. (e) VDC2 and PDC2 for step response, step load, and VDC2,ref change.
Subsequently,
In
In the experimental bed, a DC source is included as the first converter port. A half-bridge converter–SiC-Mosfet (KIT8020-CRD-8FF1217P-1) is included, which is controlled by the generated pulses in the DSP (TMF320F28335) device. The pulses are programmed using the MATLAB-Simulin
The performance of control scheme is experimentally tested using a scaled-down 10:1 laboratory prototype, with a transformation ratio . To obtain the same results in the prototype, a similar design criterion is applied. The values used for solving (34) are mH and . The parameter values for the prototype are presented in

Fig. 9 Experimental results of closed-loop DAB prototype for s. (a) Step response of and . (b) Step response of and . (c) Step response of . (d). Step response of (e) Step response of . (f) Step response of
It is also observed that during the transient, tends to increase the compensation of the new energy demand, while preserving the DC output voltage.
The minor differences are attributed to the small variances between the ideal constant values of the elements in the simulations and the real ones in the experimental set-up.
In the state of the art, the numbers of SSMs for DAB converters is no more than ten. These have been shown with mathematical techniques for different operation types and study cases. The proposed model contemplates an SSM with an elimination of complete reactive current. As future work, it is interesting to generate new SSMs using other modulation techniques with the ability to eliminate reactive power.
The DC-DC DAB converter is considered as a key element for emerging applications in the field of DC power management due to its advantages of energy conversion such as electrical isolation, high input/output voltage ratios, and bi-directional power flow. Eliminating the reactive current in the DAB converters is also achieved to offer multiple advantages such as less current stress.
In this paper, we demonstrate a closed-loop linear model of the DAB converter obtained from the non-linear plant model and a PI compensator. This has been achieved through a new insertion of transfer function integrated between the compensator and plant, which generates a SSM and obtains the operation gains in a closed loop.
The paper provides a solid foundation for future researchers towards the developments of the DAB converter for specific applications.
REFERENCES
A. K. Jain and R. Ayyanar, “PWM control of dual active bridge: comprehensive analysis and experimental verification,” IEEE Transactions on Power Electronics, vol. 26, no. 4, pp. 1215-1227, Apr. 2011. [百度学术]
L. Jingxin, W. Dongzhi, W. Wang et al., “Minimize current stress of dual-active-bridge DC-DC converters for electric vehicles based on lagrange multipliers method,” Elsevier Energy Procedia, vol. 105, pp. 2733-2738, May 2017. [百度学术]
J. A. Martinez-Velasco, S. Alepuz, F. González-Molina et al., “Dynamic average modeling of a bidirectional solid-state transformer for feasibility studies and real-time implementation,” Elsevier Electric Power Systems Research, vol. 117, pp. 143-153, Dec. 2014. [百度学术]
Q. Ye, R. Mo, and H. Li, “Low-frequency resonance suppression of a dual-active-bridge DC/DC converter enabled DC microgrid,” IEEE Journal of Emerging and Selected Topics in Power Electronics, vol. 5, no. 3, pp. 982-994, Sept. 2017. [百度学术]
Y. Shi, R. Li, Y. Xue et al., “Optimized operation of current-fed dual active bridge DC-DC converter for PV applications,” IEEE Transactions on Industrial Electronics, vol. 62, no. 11, pp. 6986-6995, Nov. 2015. [百度学术]
B. Zhao, Q. Song, W. Liu et al., “Overview of dual-active-bridge isolated bidirectional DC-DC converter for high-frequency-link power-conversion system,” IEEE Transactions on Power Electronics, vol. 29, no. 8, pp. 4091-4106, Aug. 2014. [百度学术]
H. Wen, W. Xiao, and B. Su, “Nonactive power loss minimization in a bidirectional isolated DC-DC converter for distributed power systems,” IEEE Transactions on Industrial Electronics, vol. 61, no. 12, pp. 6822-6831, Dec. 2014. [百度学术]
H. Bai and C. Mi, “Eliminate reactive power and increase system efficiency of isolated bidirectional dual-active-bridge DC-DC converters using novel dual-phase-shift control,” IEEE Transactions on Power Electronics, vol. 23, no. 6, pp. 2905-2914, Nov. 2008. [百度学术]
N. Hou, W. Song, Y. Zhu et al., “Dynamic and static performance optimization of dual active bridge DC-DC converters,” Journal of Modern Power Systems and Clean Energy, vol. 6, no. 3, pp. 607-618, May 2018. [百度学术]
J. R. Rodriguez-Rodrıguez, E. L. Moreno-Goytia, V. Venegas-Rebollar et al., “The proportional-values modulation (PVM), a technique for improving efficiency and power density of bidirectional DAB converters,” Electric Power Systems Research, vol. 144, pp. 280-289, Mar. 2017. [百度学术]
C. Zhao, S. D. Round, and J. W. Kolar, “Full-order averaging modelling of zero-voltage-switching phase-shift bidirectional DC-DC converters,” IET Power Electronics, vol. 3, no. 3, pp. 400-410, May 2010. [百度学术]
F. Krismer and J. W. Kolar, “Accurate small-signal model for the digital control of an automotive bidirectional dual active bridge,” IEEE Transactions on Power Electronics, vol. 24, no. 12, pp. 2756-2768, Dec. 2009. [百度学术]
H. Qin and J. W. Kimball, “Generalized average modeling of dual active bridge DC-DC converter,” IEEE Transactions on Power Electronics, vol. 27, no. 4, pp. 2078-2084, Apr. 2012. [百度学术]
A. Rodríguez, A. Vázquez, D. G. Lamar et al., “Different purpose design strategies and techniques to improve the performance of a dual active bridge with phase-shift control,” IEEE Transactions on Power Electronics, vol. 30, no. 2, pp. 790-804, Feb. 2015. [百度学术]
X. Pan and A. K. Rathore, “Small-signal analysis of naturally commutated current-fed dual active bridge converter and control implementation using cypress PSoC,” IEEE Transactions on Vehicular Technology, vol. 64, no. 11, pp. 4996-5005, Nov. 2015. [百度学术]
K. Zhang, Z. Shan, and J. Jatskevich, “Large- and small-signal average-value modeling of dual-active-bridge DC-DC converter considering power losses,” IEEE Transactions on Power Electronics, vol. 32, no. 3, pp. 1964-1974, Mar. 2017. [百度学术]
C. Liu, H. Liu, G. Cai et al., “Novel hybrid LLC resonant and DAB linear DC-DC converter: average model and experimental verification,” IEEE Transactions on Industrial Electronics, vol. 64, no. 9, pp. 6970-6978, Sept. 2017. [百度学术]
S. S. Shah and S. Bhattacharya, “Large & small signal modeling of dual active bridge converter using improved first harmonic approximation,” in Proceedings of 2017 IEEE Applied Power Electronics Conference and Exposition (APEC), Tampa, USA, May 2017, pp. 1175-1182. [百度学术]
L. Shi, W. Lei, Z. Li et al., “Stability analysis of digitally controlled dual active bridge converters,” Journal of Modern Power Systems and Clean Energy, vol. 6, no. 2, pp. 375-383, Mar. 2018. [百度学术]
M. Faïçal, “New tuning rules of PI-like controllers with transient performances for monotonic time-delay systems,” ISA Transactions, vol. 47, no. 4, pp. 401-406, Oct. 2008. [百度学术]
R. O. Núñez, G. G. Oggier, F. Botterón et al., “A comparative study of three-phase dual active bridge converters for renewable energy applications,” Sustainable Energy Technologies and Assessments, vol. 23, pp.1-10, Oct. 2017. [百度学术]
M. Berger, I. Kocar, H. Fortin-Blanchette et al., “Hybrid average modeling of three-phase dual active bridge converters for stability analysis,” IEEE Transactions on Power Delivery, vol. 33, no. 4, pp. 2020-2029, Aug. 2018. [百度学术]