## 1 Introduction  
Micro-grid technology is a key part of the future intelligent power system [1] due to its high receptiveness capacity of distributed generations in power systems and high utilization efficiency of intermittent distributed generations [2]. In DC micro-grids, the DC-DC converters connect the DC bus to the renewable power generation devices [3], the energy storage devices [4], electric vehicles [5], and other loads [6], which can achieve reliable interconnection, fewer conversion devices, etc. [7] Compared with the traditional DC-DC converters such as the buck converter [8], dual active bridge (DAB) DC-DC converters have been widely used in the high power conversion, because they can achieve high power density, bidirectional power flow, and electrical isolation [9, 10]. During the development of DAB converters, their topologies have been improved to meet various requirements. Resonant-type DAB converters account for a significant proportion of the improved converters. The LC resonant type DAB converter [11, 12] replaces the filter inductor of the traditional DAB converter with a series inductor-capacitor tank, which can achieve the maximal voltage gain at the resonant frequency, but its regulating range of output voltage is limited. The LLC resonant type DAB converter [13, 14] can achieve zero-voltage-switching (ZVS) in the whole operating range, but the wide range of frequency modulation control brings a burden to the controller. Different from the LLC resonant type DAB converter, the CLLC resonant-type DAB converter [15, 16] can obtain the consistent forward and reverse characteristics, but the difficulty of parameter design and control is increased. Moreover, the LCL resonant-type DAB (LCL-DAB) converter [17, 18]was proposed to significantly improve the voltage gain and reduce the bridge currents, and the adopted phase shift modulation scheme is simple and easy to implement. Thus far, some research has been conducted on the operation principle, characteristic analysis, and optimized control methods of them. In [19], the operating mode with extended-phase-shift (EPS) modulation scheme and the principles of parametric design for the LCL-DAB converter were analyzed. Over the same period, the characteristics of the LCL-DAB converter with a dual-phase-shift (DPS) modulation scheme were mathematically derived in [20; 21], but lacked a further control method, same as [19]. To ensure the ZVS of the LCL-DAB converter, the half-bridge ZVS boundaries were predicted using the frequency domain analysis method, which considers all relevant harmonics in the derivation. In terms of optimized control method, improved control strategies for the LCL-DAB converter with DPS [22], and triple-phase-shift (TPS) [23] modulation schemes were successively proposed, which can achieve ZVS operation in a wide range, but the complication of the algorithms made them difficult to apply. However, for the LCL-DAB converter, most of these articles focus on the analysis and implementation of ZVS operation [24]. Compared with ZVS implementation, backflow power elimination and transient performance optimization also deserve, but lack, attention, because backflow power can increase current stress and conduction losses, and outstanding transient response can improve dynamic performance against disturbances. An optimized design method for the LCL-DAB converter was proposed in [25] to discuss the relationship between the backflow power and the harmonic components, but it cannot detail the exact design boundaries. A TPS modulation strategy for the LCL-DAB converter was presented in [26; 27] to achieve the united power factor, i.e., eliminating the reactive power, but only the reactive power generated by the fundamental component can be eliminated due to the 1 st harmonic approximation. Based on the ZVS conditions derived in [22], a model predictive control method for the LCL-DAB converter was proposed in [28] to improve the dynamic response under ZVS, but the adopted 1 st harmonic approximation method is similarly less precise.