Impedance-based Sensitivity Analysis of Dual Active Bridge DC-DC Converter

The sensitivity analysis of a circuit reflects the robustness of this circuit to the relevant parameters. This paper investigates the sensitivity of input impedance of DAB converter, which is widely adopted in solid-state transformer (SST) and dc microgrid on more electric aircraft (MEA). In this paper, an improved impedance-based small-signal averaged model is developed to carry out the sensitivity analysis. Bode plots will be used as main analysis tool. The small signal model is compared to a complete switching model to confirm the validity of the proposed approach.


I. INTRODUCTION
Compared to conventional aircraft and all electric aircraft (AEA), the concept of more electric aircraft (MEA) is more preferred by aerospace industries as a step-wise approach, in order to limit risks and gaining experience [1]. During the last several decades, significant progress has been made to move toward MEA. Many onboard hydraulic, mechanical and pneumatic devices have been partially or fully replaced with electrical devices [2]. Also, the shortage of conventional energies accelerates the irresistible process of electrifying onboard devices. For example, in electrification of hydraulic systems, it is stated in [2]- [4] that the conventional hydraulic actuator is supposed to be replaced with electro-hydraulic actuator (EHA) or electromechanical actuator (EMA) because of its bulky size, easily jamming and leaking. Both EHA and EMA are controlled by solid state power electronics and variable speed motor drives. It is also reported in [3] that the EHA system has been adopted in the newest civilian aircrafts Boeing 787 and Airbus 380, to replace the conventional hydraulic actuator. This leads to a more flexible and leakless framework. However, to achieve a completely jamming free and leaking free actuator system with high reliability and safety, it is envisaged to use EMA system to get rid of hydraulic power entirely in future.
Consequently, by introducing more electrical devices into aircraft, the onboard electrical power generation increases while the challenges in designing an efficient and secure electrical power distribution system arise [3]. The onboard electrical power distribution system can be regarded as a microgrid since the aircraft is an isolated system with generators and loads [4]. Several different voltage standards applied for electrical system on large civilian aircraft are listed in [5], containing both ac and dc levels. Compared with ac microgrid, dc microgrid has the advantage of featuring fewer components due to reduced conversion stages, resulting in lower possibility to failures and lower weight [4]. A conventional dc microgrid contains multiple dc-dc converters to interface multiple sources and loads. Although this pattern allows power flow simultaneously, it is too complex and expensive because of more conversion stages and communication devices between converters [6]. To solve this problem, multiport dc-dc converter is proposed in which all ports are bidirectional, to reduce the number of conversion stages. Furthermore, voltage droop control as a kind of decentralized control methods, is introduced in [4] to achieve power sharing between converters with different sources/generators by establishing a virtual resistor, eliminating the communication devices.
When it comes to dc-dc converters applied to the microgrid on MEA, the converters must have the characteristic of galvanic isolation for safety purposes [7]. Hence, a converter containing a dc-ac inverter, a main transformer and a diode bridge rectifier can be the basic structure to achieve galvanic isolation. By applying the phase-shift modulation to inverter part, high efficiency unidirectional power flowing can be achieved. Furthermore, the diodes in rectifier also can be replaced with switches to realize bidirectional power flowing and enable both boost and buck modes. Among variety of bidirectional power flowing converters, dual active bridge (DAB) converter is the most classical topology which has drawn many researchers' attention in aerospace applications by its features in terms of high-power density, high efficiency and bidirectional power flow [8]. Although DAB converter is widely used in both researches and industrial applications, the stability of it needs to be thoroughly assessed when it is adopted in the microgrid for avionic use. In this paper, Section II gives the structure and operation principle of DAB converter. Section III presents an improved impedance-based small-signal model of DAB converter and also sensitivity analysis of the model by changing parameters of the circuit. Section IV provides specifications for DAB converter. Section V investigates the sensitivity of input impedance of DAB converter by changing specification parameters. Section VI presents the simulation results of impedance-based model and switching model. Section VII draws a conclusion and summarizes the contribution of this work.

II. DUAL ACTIVE BRIDGE CONVERTERS
The simplified structure of DAB converter is presented in Fig. 1. By employing phase-shift modulation to full bridges on both sides, the conversion between dc power and ac power is realized. In addition, the phase shift between switching signals of both sides determine the direction of power flowing. When the primary side bridge leads the secondary side bridge, the power will flow to load from supply. Conversely, when the primary side bridge lags, the power will flow back to supply from load. Fig. 2 shows the ideal operating waveforms of the DAB converter at the primary side bridge leading mode, in which power is flowing from source to load. The idealized output power equation of DAB converter at steady state is derived as where Vi is the input voltage, IH is the input current, Vo is the output voltage, Io is the output current. N is the turn ratio of the HF transformer, fs is the switching frequency, Lt is the leakage inductance of the HF transformer, d is phase shift ratio equal to . By controlling the phase shift, the transferred power can be controlled.
III. IMPEDANCE-BASED SMALL-SIGNAL MODEL OF DAB CONVERTER Generally, state-space equation averaging technique and circuit averaging technique are often used to obtain the smallsignal averaged model of a converter circuit [9]. However, it is less easy when they are applied to converters with more complex structure such as DAB converter. So far, several kinds of models of DAB converter has been proposed by researchers in [10]- [12]. In this section, an alternative method is proposed to obtain the small-signal averaged model of DAB converter, which is to model the input impedance by using the small-signal equations derived from the power equation of DAB converter. By using Equation (2.1), the input current IH and output current Io can be obtained as By observing these four gains, it is easy to find that 2 and 4 are same, meantime 1 is proportional to 3. A similar block scheme based on small-signal equations was obtained in [13]. However, it misses the consideration of time delay. In Fig. 3

the function '
' is added as a first-order time delay function. The symbol 'τ' is the time constant to represent the time delay from the moment when switches are operated to the moment the output is rectified. In this case the time delay is assumed to be one switching period. In addition, in Fig. 3 the input current IH and output current Io are also connected by the gain block N which is the turn ratio of transformer. According to this block scheme, the transfer function of input admittance can be figured out as (3.6) where 'Ctrl' is the transfer function of controller, TD represents the first-order time delay function , G1, G2, G3 and G4 are the gains. Based on that, the input impedance Zi including the input capacitor CIN can be calculated in a more detailed form as (3.7) IV. SPECIFICATIONS Before analyzing the sensitivity of input impedance of DAB converter, specifications should be given. Then the sensitivity can be analyzed by changing the parameters in a reasonable range. The specifications are shown in Table. 1. In this case, the controller is determined to be the PI (proportional + integral) type because PI control has not only fast dynamic response but also good immunization to noise.   V. SENSITIVITY ANALYSIS To analyze the sensitivity of input impedance, several important parameters are supposed to be changed: the phase shift ratio (the default value is 0.2), the load, the output capacitor, the parameters of controller, the type of controller and the switching frequency. To observe the change of input impedance in a more intuitional way, Matlab is used to plot the Bode diagram of input impedance by changing parameters. The sensitivity of input impedance of DAB converter to parameters plotted in Fig. 4. In Fig. 4, six bode diagrams (a), (b), (c), (d), (e) and (f) show the curves of input impedance of DAB converter according to different parameters of circuit. All parameters are changed in a proper range to keep the whole system still stable. In (a), the input impedance changes slightly in both magnitude and phase when the phase shift ratio d changes (load also changes because the power changes). In (b), (c), (d), (e) and (f) the input impedance almost keeps same when the output capacitor, the parameters of controller, the type of controller and the switching frequency changes. This means the sensitivity of input impedance to circuit parameters are very low if the transferred power is constant. In addition, it can be observed that the input impedance performs as a constant power load which has negative incremental impedance characteristic with phase close to -180° at low frequencies, while the input impedance starts to perform as a capacitor with phase close to -90° at higher frequencies.

VI. VALIDATION
To validate the small-signal model stated in Section III, the method is adding an external sinusoidal current into input current and then observe the changes in input voltage. Two simulation models are built by PLECS: 1) A switching model and 2) An impedance-based model. In the switching model, all components are ideal, and they are connected like the schematic shown Fig. 1, the value of them are same as the parameters shown in Table. 1. The purpose of impedancebased model is to obtain the same results by using transfer function. By comparing the results of these two models, the small-signal model can be verified.
In the step of validation, sinusoidal ac current with amplitude of 10A and various frequencies is injected at t=0.1s. The phase shift ratio d is adjusted to be 0.2. By comparing the input voltage of both two simulation models, the transfer function of input impedance can be verified. Moreover, the characteristic of input impedance can be figured out by observing the variations of both input current and input voltage, and then be verified with the bode diagrams in Fig. 4. The input voltage and input current waveforms of two simulation models injected by current with different frequencies are shown in Fig. 5. The high consistency of input voltage in both two simulation models proves that the transfer function of input impedance is correct. In addition, it is worth to mention that the input current of the impedance-based circuit model is negative, indicating that the input current flowing to source. It is reasonable because the transfer function of input impedance must have phase close to -180° at low frequencies to make the converter stable. Besides, the characteristic of input impedance can be clarified by calculating the ratio of the peak-to-peak value of input voltage to input current in fluctuating state and then verifying with the bode diagram in

VII. CONCLUSION
An improved impedance-based small-signal model of phase-shift controlled DAB converter is proposed in this paper. It uses small-signal equations derived from the power equation to obtain an appropriate block scheme including time delay, and then the transfer function of input impedance is worked out. The Bode diagram of input impedance is plotted by using Matlab. In addition, the sensitivity of input impedance to the phase shift ratio (load), the output capacitor, the parameters of controller, the type of controller and the switching frequency is analyzed. It shows that the sensitivity of input impedance to these important circuit parameters is very low if the transferred power is constant. Besides, to validate whether the transfer function is correct, PLECS is used to create a switching model and impedance-based model injected by current with different frequencies. The input voltage of these two models is nearly consistent, proving that the transfer function is correct. Moreover, the characteristic of input impedance is verified by calculating the ratio of the peak-to-peak value of input voltage to input current in fluctuating state. As a result, the input impedance behaves as a constant power load at low frequencies and behaves as a capacitor at high frequencies.